# 5645.[Berichte Aus Der Luft- Und Raumfahrttechnik] Alan Celic - Performance of modern Eddy-Viscosity turbulence models (2004 Shaker Verlag GmbH Germany).pdf

код для вставкиСкачатьPerformance of Modern Eddy-Viscosity Turbulence Models Von der Fakultät für Luft- und Raumfahrttechnik und Geodäsie der Universität Stuttgart zur Erlangung der Würde eines Doktor-Ingenieurs (Dr.-Ing.) genehmigte Abhandlung Vorgelegt von Alan Celić geboren in Tübingen Hauptberichter: Prof. Dr.-Ing. habil. Ernst H. Hirschel 1. Mitberichter: Prof. Dr.-Ing. Siegfried Wagner 2. Mitberichter: Prof. Peter Bradshaw Tag der mündlichen Prüfung: 23.07.2004 Institut für Aerodynamik und Gasdynamik Universität Stuttgart 2004 Berichte aus der Luft- und Raumfahrttechnik Alan Celi ´c Performance of Modern Eddy-Viscosity Turbulence Models . D 93 (Diss. Universität Stuttgart) Shaker Verlag Aachen 2004 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the internet at http://dnb.ddb.de. Zugl.: Stuttgart, Univ., Diss., 2004 . Copyright Shaker Verlag 2004 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publishers. Printed in Germany. ISBN 3-8322-3517-5 ISSN 0945-2214 Shaker Verlag GmbH • P.O. BOX 101818 • D-52018 Aachen Phone: 0049/2407/9596-0 • Telefax: 0049/2407/9596-9 Internet: www.shaker.de • eMail: [email protected] Acknowledgments This work was conducted during my time as a research engineer at the Institut für Aerodynamik und Gasdynamik (IAG) of the University of Stuttgart, Germany, and was funded by the Deutsche Forschungsgemeinschaft (Grants Hi 342/4-1 to 342/4-4). I am deeply grateful to my thesis adviser Professor Dr.-Ing. habil. Ernst H. Hirschel for his great personal support and help, and technical advice. Professor Hirschel always believed in my work, which gave me conﬁdence especially during diﬃcult periods when things did not go as smoothly as hoped. I highly appreciate that I could be one of his doctoral students. I am also deeply grateful to Professor Peter Bradshaw from Stanford University who was a distant adviser and a “Mitberichter” (co-referee) for this Ph.D. thesis. Professor Bradshaw’s invaluable professional and linguistic advice as well as his personal support made this work a great experience and joy for me. I have learned so much from him. I also wish to thank Professor Dr.-Ing. Siegfried Wagner for his commitment as a Mitberichter and for oﬀering me the opportunity to perform this study at his institute. The IAG is a great place to work at and I also thank all my former colleagues for creating such a great atmosphere. Very special thanks go to Dr.-Ing. Werner Haase from EADS Munich. Dr. Haase put me on track at the beginning of this work by supplying his CFD code and his great experience in turbulence modeling. He always took the time to teach me about CFD and turbulence modeling which I sincerely appreciate. I am likewise grateful to Dr.-Ing. Markus Kloker from the IAG for providing his invaluable advice when I had questions concerning numerics. I am indebted to Professor Stefan Staudacher who, on short notice, attended my Ph.D. exam in place of Professor Bradshaw who unfortunately could not take the burden of the long travel to Stuttgart because at that time he had not completely recovered from an accident. Last but not least, I wish to thank my family and my close friends who have supported me in many ways and without whom I would have not been able to master this work. Alan Celić Toulouse, October 24th, 2004 Contents Notation 9 Abstract 15 Zusammenfassung 17 1 Introduction 1.1 The Present Study . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Contents and Organization of the Thesis . . . . . . . . . . . . 25 33 34 I 37 Topological Approach to Turbulence Modeling 2 Basic Considerations 3 Governing Equations and Numerical Method 3.1 Governing Equations of the Mean Flow . . . . 3.2 The Baldwin-Lomax Model . . . . . . . . . . . 3.3 The Johnson-King Model . . . . . . . . . . . . 3.4 Numerical Method (I) . . . . . . . . . . . . . . 38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 42 44 46 48 4 Demonstration 4.1 Description of Flow Case . . . . . . . . . . . . . . . . . . . . 4.2 Computational Grid . . . . . . . . . . . . . . . . . . . . . . . 4.3 Computational Results and Discussion . . . . . . . . . . . . . 4.3.1 Topology of the Velocity Field . . . . . . . . . . . . . 4.3.2 Pressure and Skin-Friction Distributions . . . . . . . . 4.3.3 Boundary-Layer Proﬁles . . . . . . . . . . . . . . . . . 4.3.4 Numerical Experiment in the Recirculation Zone . . . 4.3.5 Comments Regarding Hidden Three-Dimensional Effects in Nominally Two-Dimensional Flows . . . . . . 50 50 51 53 53 55 60 67 II 73 Analysis of Modern Turbulence Models 5 Numerical Method (II) 70 74 6 Models Investigated 6.1 The k, ω Models of Wilcox . . . . . . . . . . . . . . . . . . . 6.2 The k, ω Shear-Stress Transport (SST) Model of Menter . . . 6.3 The Turbulent/Non-turbulent (TNT) k, ω Model of Kok . . . 6.4 The Local Linear Realizable (LLR) k, ω Model of Rung . . . 6.5 The Explicit Algebraic Reynolds-Stress Model (EARSM) of Wallin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Boundary Conditions for the k, ω Models . . . . . . . . . . . 6.6.1 Free-Stream Boundary Conditions . . . . . . . . . . . 6.6.2 Wall Boundary Conditions . . . . . . . . . . . . . . . 6.7 The One-Equation Model of Spalart & Allmaras . . . . . . . 6.8 The One-Equation Model of Edwards & Chandra . . . . . . . 6.9 The Strain-Adaptive Linear Spalart-Allmaras (SALSA) Model 6.10 Boundary Conditions for the One-Equation Models . . . . . . 75 76 79 81 82 7 Test Cases Selected 7.1 Flat-Plate Boundary Layer (Case FPBL) . . . . . . . . . . . 7.1.1 Computational Setup . . . . . . . . . . . . . . . . . . 7.1.2 Computational Results and Discussion . . . . . . . . . 7.1.3 Some Modiﬁcations of the k, ω SST Model . . . . . . 7.1.4 Eﬀects of Low-Reynolds-Number Modiﬁcations . . . . 7.2 Boundary Layer with Adverse Pressure Gradient (Case BS0) 7.2.1 Computational Setup . . . . . . . . . . . . . . . . . . 7.2.2 Computational Results and Discussion . . . . . . . . . 7.3 Boundary Layer with Pressure-Induced Separation (Case CS0) 7.3.1 Computational Results and Discussion . . . . . . . . . 7.4 Separated Airfoil Flow (Case AAA) . . . . . . . . . . . . . . 7.4.1 Computational Results and Discussion . . . . . . . . . 94 95 95 96 101 105 107 108 111 118 119 127 127 8 Numerical Issues 8.1 Grid Convergence . . . . . . . . . . . . . . . . . 8.2 Local Preconditioning for Low Mach Numbers . 8.3 Transition . . . . . . . . . . . . . . . . . . . . . 8.4 Artiﬁcial Damping in Boundary Layers . . . . . 8.5 Boundary-Value Dependences . . . . . . . . . . 8.5.1 Dependences on Wall Value of ω . . . . 8.5.2 Dependence on Free-Stream Value of ω 137 137 141 146 149 153 153 154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 87 87 88 89 91 91 93 9 Summary and General Conclusions 156 10 Outlook 160 III 161 Appendices A RANSLESS – A New Approach to RANS/LES Coupling A.1 Brief Review of Turbulence Physics at Turbulent Separation A.2 RANS/LES Coupling for Separated Flows . . . . . . . . . . A.2.1 Inﬂow Conditions for LES . . . . . . . . . . . . . . . A.2.2 Outﬂow Conditions for LES . . . . . . . . . . . . . . A.2.3 Inﬂow Conditions for RANS . . . . . . . . . . . . . . A.2.4 Outﬂow Conditions for RANS . . . . . . . . . . . . . A.3 Closing Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 162 162 164 167 167 167 167 B Details of the Johnson-King Model 169 C Graphs of Computational Results C.1 Boundary Layer with Adverse Pressure Gradient (Case BS0) C.2 Boundary Layer with Pressure-Induced Separation (Case CS0) C.3 Separated Airfoil Flow (Case AAA) . . . . . . . . . . . . . . 172 172 185 207 D Overview of Algorithmic Accomplishments 232 E Typical FLOWer Input Decks 234 E.1 Typical FLOWer Input Deck for Case FPBL . . . . . . . . . 234 E.2 Typical FLOWer Input Deck for Case BS0 and CS0 . . . . . 238 E.3 Typical FLOWer Input Deck for Case AAA . . . . . . . . . . 242 Bibliography 254 Notation General Conventions • Scalar variables are italicized; vector, matrix and tensor variables are bold and italicized. • To avoid departing too much from conventions usually used in literature on turbulence modeling and general ﬂuid mechanics, some symbols denote more than one quantity. • Following the usual Einstein summation convention, repetition of an index in a single term denotes summation with respect to that index: ui ui ≡ 3 ui ui = u1 u1 + u2 u2 + u3 u3 i=1 Latin Symbols a a1 aex , aex ij A A+ b c cf cv cp cD cL cM D e E FKleb (y; ymax /CKleb ) √ speed of sound, γRT Townsend’s constant; Bradshaw’s constant extra-anisotropy tensor wing area Van Driest damping coeﬃcient wingspan chord length skin-friction coeﬃcient, τw /(0.5ρ∞ |v∞ |2 ) speciﬁc heat at constant volume speciﬁc heat at constant pressure; pressure coeﬃcient, p − p∞ /(0.5ρ∞ |v∞ |2 ) global drag coeﬃcient global lift coeﬃcient global moment coeﬃcient cylinder diameter; Van Driest damping function volume-speciﬁc inner energy volume-speciﬁc total energy Klebanoﬀ intermittency function H I i, j, k k ks lmix M ṁ p P Pr P rt q R Re Ret S, Sij t T Tη u, v, w ui ui uj u+ uτ v vmix w x, y, z y+ volume-speciﬁc total enthalpy identity matrix grid-point index in the ξ, η, ζ direction kinetic energy of turbulent ﬂuctuations per unit mass surface roughness height mixing length Mach number mass per unit time mean static pressure preconditioning matrix Prandtl number turbulent Prandtl number heat-ﬂux vector perfect-gas constant Reynolds number turbulent Reynolds number mean strain-rate tensor time static temperature Kolmogorov time scale of dissipation Cartesian components of v Cartesian components of v in index notation tensor of one-point central moments of ﬂuctuating velocity components, speciﬁc Reynolds-stress tensor dimensionless, sublayer-scaled velocity, u/uτ friction velocity, τw /ρ mean velocity in vector notation mixing velocity vector of dependent ﬂow variables Cartesian coordinates dimensionless, sublayer-scaled wall distance, yuτ /ν Greek Symbols α γ δ δv∗ ∆ λ Λ ν µ µt µ ti µ to ψ ξ, η, ζ ζ θ κ ρ σ(x) σ, σij τ , τij τw ω |ω| Ω Ω, Ωij angle of attack speciﬁc-heat ratio, cp /cv boundary-layer thickness δ kinematic displacement thickness, 0 (1 − u/ue ) dy relative diﬀerence between computed and measured values dissipation rate per unit mass heat-conductivity coeﬃcient wing aspect ratio, b2 /A kinematic viscosity, µ/ρ dynamic viscosity eddy viscosity inner-layer eddy viscosity outer-layer eddy viscosity streamfunction curvilinear coordinates second viscosity coeﬃcient δ momentum thickness, 0 ρeρuue (1 − u/ue ) dy Kármán constant ﬂuid density non-equilibrium parameter viscous stress tensor Reynolds-stress tensor shear stress at the wall speciﬁc dissipation rate per unit mass magnitude of the vorticity vector absolute value of the vorticity mean rotation tensor Subscripts e exp value at boundary-layer edge measured value, value from experiment local m max min neq r, ref s w ∞ local value value at position of maximum Reynolds shear stress maximum value minimum value non-equilibrium reference quantity streamline wall (surface) value free-stream value Superscripts + − ﬂuctuating part of a ﬂow variable sublayer-scaled value Reynolds-averaged value, time-averaged value Acronyms AAA CFD DNS DES EADS EARSM FPBL GCI IAG LES LLR MBC ONERA RANS RANSLESS Aerospatiale-A airfoil computational ﬂuid dynamics direct numerical simulation detached-eddy simulation European Aeronautic Defence and Space Company explicit algebraic Reynolds-stress model ﬂat-plate boundary layer grid-convergence index Institut für Aerodynamik und Gasdynamik large-eddy simulation local linear realizable Menter’s boundary condition (for ωw ) Oﬃce National d’Etudes et de Recherches Aerospatiales Reynolds-averaged Navier-Stokes (computation) RANS surrounded LES scenario RBC RSTM SALSA SST TNT WBC Rudnik’s boundary condition (for ωw ) Reynolds-stress transport model strain-adaptive linear Spalart Allmaras shear-stress transport turbulent/non-turbulent Wilcox’s boundary condition (for ωw ) Abstract Turbulent ﬂows of engineering interest are frequently computed by solving the Reynolds-averaged Navier-Stokes equations in combination with an eddyviscosity turbulence model. In situations where the turbulent boundary layer separates from the surface of the body under consideration poor predictive accuracy of the computational results is encountered more often than not. In this work, the topology of velocity ﬁelds serves as a basis to generally classify separated ﬂows. It is speculated whether the computation of a single ﬂow class can be improved if turbulence modeling is adjusted to topological structures. A separated airfoil ﬂow is investigated in detail using the ﬂow topology as a guideline for identifying ﬂow regions of possible importance for turbulence modeling. It is found that modiﬁcations of the turbulence model downstream of separation do not improve or inﬂuence computational results. It appears that boundary-layer development upstream of separation is the key issue for accurately predicting the primary separation. Therefore, diﬀerent boundary-layer ﬂows with increasing physical complexity are selected to study the eﬀect of adverse pressure gradient on model predictions. A broad set of modern turbulence models including some recently developed models is employed for the computation of the test cases selected. Computational results are compared to experimental data. None of the turbulence models investigated is able to predict important ﬂow quantities for all test cases in good agreement with experimental data. The “best” turbulence model changes from ﬂow case to ﬂow case. The predictions of models based on one transport equation for the eddy viscosity are much closer to each other than predictions of the k, ω models employed in this work. Compared to experiment, transport-equation models are found to respond qualitatively better to pressure gradient than the algebraic model investigated. No general conclusion can be drawn on what model to use for computing pressure-induced separated ﬂows. Besides turbulence model performance, several additional important issues for the computation of separated ﬂows are investigated. These range from purely numerical issues, like local preconditioning for low Mach numbers and artiﬁcial damping of the numerical scheme, to issues regarding boundary conditions for the turbulence variables and speciﬁcation of transition locations. It is found that local preconditioning is essential for obtaining accurate ﬂow solutions for small Mach numbers with a compressible ﬂow solver. Moreover, it proves to be necessary to reduce standard artiﬁcial damping terms in the direction normal to the wall in boundary layers in order to prevent spurious momentum loss. The separated airfoil ﬂow is seen to be very sensitive to the prescribed location of transition and to the rate at which the turbulence model reaches the fully turbulent state. A partly novel method for coupling RANS and LES for the computation of turbulent ﬂows is proposed. It is intended to form the basis for future work. Zusammenfassung Einleitung Abgelöste turbulente Strömungen spielen eine zentrale Rolle in vielen ingenieurtechnischen Anwendungen. Die zuverlässige und exakte Berechnung solcher Strömungen stellt trotz jahrzehntelanger intensiver Forschungsbemühungen auf diesem Gebiet eine große Herausforderung dar. Viele verschiedene statistische Turbulenzmodelle wurden zu diesem Zweck entwickelt und vorgeschlagen. In dieser Arbeit wurde zum einen der topologieorientierte Ansatz nach Hirschel (1999) für die Berechnung von abgelösten turbulenten Strömungen diskutiert und anhand eines Beispieles demonstriert. Zum anderen wurden elf moderne Wirbelviskositätsmodelle, wie sie in der europäischen Luft- und Raumfahrtindustrie heutzutage zum Einsatz kommen, an verschiedenen Strömungsfällen unterschiedlicher physikalischer Komplexität untersucht. Außerdem wurden wichtige numerische Aspekte untersucht und dargestellt. Ein topologieorientierter Ansatz zur Turbulenzmodellierung Grundlegende Ideen Nach Hirschel (1999) wurde die Topologie von Geschwindigkeitsfeldern als Ordnungsprinzip benutzt, um eine Einteilung abgelöster aerodynamischer Strömungen in verschiedene Klassen vorzunehmen. Es wurden zwei Hauptklassen deﬁniert, die sich beide auf starre, unbewegte Körper bei stationärer Anströmung beziehen. Zur Klasse 1 gehören statistisch stationäre, abgelöste turbulente Strömungen, während in Klasse 2 statistisch instationäre Strömungen zusammengefasst sind. Diese Hauptklassen unterteilen sich in weitere Unterklassen, welche sich aus den topologischen Strukturen des gemittelten Geschwindigkeitsfeldes ergeben. Beispielsweise besteht Unterklasse 1.1 aus statistisch zweidimensionalen abgelösten Strömungen mit einem oder zwei Rezirkulationsgebieten. Wie von Hirschel (1999) vorgeschlagen, wurde untersucht, ob die Berechenbarkeit von Strömungen einer Klasse verbessert werden kann, wenn das verwendete Turbulenzmodell innerhalb der ausgezeichneten topologischen Strukturen modiﬁziert wird. Dies wird als topologieorientierte Turbulenzmodellierung bezeichnet. Des Weiteren regte Hirschel (1999) an, die Strömungstopologie als einen Leitfaden zur Analyse von abgelösten Strömungen heranzuziehen. Beide Aspekte wurden im ersten Teil dieser Arbeit behandelt. 18 Zusammenfassung Erhaltungsgleichungen und numerisches Verfahren Für die Berechnung der Strömung wurden die kompressiblen Reynolds-gemittelten NavierStokes Gleichungen in konservativer und integraler Schreibweise numerisch gelöst. Das numerische Verfahren ist ein so genanntes Jameson-Verfahren, das eine zentrale Raumdiskretisierung zweiter Ordnung verwendet. Die Zeitintegration erfolgt durch ein fünfstuﬁges Runge-Kutta-Zeitschrittverfahren, wobei Konvergenzbeschleunigung mit Hilfe lokaler Zeitschritte, Mehrgittertechnik und implizitem Residuum-Glätten erzielt wird. Künstliche viskose Terme zweiter und vierter Ordnung erhöhen die Stabilität des Verfahrens, indem sie die Wiggle-Moden dämpfen. Um eine Verbesserung der Genauigkeit und der Konvergenz bei niedrigen Machzahlen zu erzielen, wird ein so genanntes Local Preconditioning“ verwendet. Der Reynods-Spannungstensor, der ” den Einﬂuss der Turbulenz auf die gemittelte Strömung beschreibt, wurde im ersten Teil der Arbeit mit Hilfe des Johnson-King Turbulenzmodells berechnet. Johnson & King (1985) entwickelten dieses Modell gezielt für die Berechnung von abgelösten Proﬁlumströmungen. Es basiert auf dem algebraischen Modell von Baldwin & Lomax (1978), wobei zusätzlich eine gewöhnliche Differentialgleichung für den Transport der Reynods-Schubspannung gelöst wird, um Konvektionseﬀekte in der Turbulenz zu berücksichtigen. Im zweiten Teil der Arbeit wurden mehrere, verschiedene Turbulenzmodelle untersucht, die weiter unten aufgeführt sind. Demonstration an einer abgelösten Proﬁlumströmung Betrachtet wurde die zweidimensionale, abgelöste Umströmung des Aerospatiale-AProﬁls bei M = 0.15, Re = 2.0 · 106 und α = 13.3◦ . Dabei bildete sich ein einzelnes Rezirkulationsgebiet in der Nähe der Hinterkante aus. Somit gehört diese Strömung zu der oben diskutierten Unterklasse 1.1. Ausgehend von der Proﬁlnase wurden auf der Oberseite folgende topologische Strukturen identiﬁziert: Staupunkt (der auch ein Anlegepunkt ist), laminare Grenzschicht, transitionale Ablöseblase, turbulente Grenzschicht, Ablösepunkt, freie Scherschicht, Wiederanlegepunkt, Wirbelfokus und ,,rücklaufende” Grenzschicht. Die berechneten Ergebnisse wurden mit experimentellen Daten qualitativ und quantitativ verglichen. Während die Simulation die Strömungstopologie qualitativ richtig berechnete, stellten sich deutliche quantitative Unterschiede zwischen experimentellen und berechneten Ergebnissen heraus. Insbesondere stimmte die vorhergesagte Lage des Ablösepunktes nicht mit der gemessenen Lage überein. Bei den berechneten Geschwindigkeitsproﬁlen in der turbulenten Grenzschicht und im Rezirkulationsgebiet gab es zunehmen- Zusammenfassung 19 de Abweichungen der Berechnungsergebnisse von den gemessenen Daten bei Annäherung an die Proﬁlhinterkante. Des Weiteren berechnete das Turbulenzmodell von Null verschiedene Werte für die Reynods-Spannungen im Rezirkulationsgebiet, obwohl das Experiment dort nur sehr geringe Werte zeigte. Ausgehend von diesem Ergebnis wurde das Johnson-King-Modell derart modiﬁziert, dass in Rezirkulationsgebieten der Unterklasse 1.1 keine ReynodsSpannungen mehr vorhergesagt werden. Es zeigte sich, dass der Einﬂuss dieses numerischen Experimentes auf die Umströmung des Proﬁls nahezu nicht existent war. Folglich haben Modiﬁkationen an der Turbulenzmodellierung stromab der primären Ablösung im Rezirkulationsgebiet keinen Einﬂuss auf den Ablösepunkt. Vielmehr ist die Entwicklung und der Zustand der Grenzschicht, und deshalb auch deren korrekte Vorhersage, stromauf der Ablösestelle von zentraler Bedeutung. Dieses Ergebnis war nicht unmittelbar zu erwarten, denn bei einer Proﬁlumströmung können kleine Änderungen im Hinterkantenbereich aufgrund der Zirkulation um das Proﬁl einen Einﬂuss auf Bereiche, die weit stromauf liegen, ausüben. Die gegenseitige Abhängigkeit zwischen Grenzschicht, Zirkulation um das Proﬁl und Druckgradient entlang der Grenzschicht erschwert bei Proﬁlumströmungen allerdings die systematische Untersuchung von Ursachen für das Versagen eines Turbulenzmodells. Deshalb wurde die anschließende vergleichende Untersuchung moderner Turbulenzmodelle für abgelöste Strömungen anhand von Strömungsfällen durchgeführt, bei denen keine enge Kopplung des gesamten Strömungsfeldes vorliegt und bei welchen der Druckgradient direkt oder indirekt vorgegeben werden kann. Die einzige Ausnahme bildete dabei die Umströmung des Aerospatiale-A-Proﬁls, da diese bereits mit dem Johnson-King-Modell untersucht wurde. Auch war diese Strömung bereits Gegenstand in vielen anderen vergleichenden Untersuchungen. Die genaue Vorhersage abgelöster Proﬁlumströmungen ist außerdem ein zentraler Punkt bei der Entwicklung von Hochauftriebsystemen im Flugzeugbau. Vergleichende Untersuchung moderner Turbulenzmodelle Eingesetzte Turbulenzmodelle Bei den untersuchten Turbulenzmodellen handelte es sich um ein algebraisches Turbulenzmodell sowie um mehrere Ein- und Zweigleichungsmodelle. Ein nichtlineares, explizites, algebraisches Reynods-Spannungsmodell wurde ebenfalls untersucht. Im Einzelnen kamen folgende Turbulenzmodelle für die vergleichende Untersuchung zum Einsatz: • das algebraische Modell von Baldwin & Lomax (1978) 20 Zusammenfassung • die beiden k-ω-Transportgleichungsmodelle nach Wilcox (1988, 1998) • das k-ω-SST-Transportgleichungsmodell von Menter (1993) • das k-ω-TNT-Transportgleichungsmodell von Kok (2000) • das k-ω-LLR-Transportgleichungsmodell von Rung & Thiele (1996) • das explizite, algebraische Reynods-Spannungsmodell (EARSM) von Wallin & Johansson (2000) • das Eingleichungsmodell von Spalart & Allmaras (1992) • das Eingleichungsmodell von Edwards & Chandra (1996) • das Eingleichungsmodell SALSA von Rung et al. (2003) Zusätzlich zu den genannten Modellen wurde das k-ω-SST-Modell von Menter im Rahmen dieser Arbeit modiﬁziert. Ergebnisse mit diesem Modell wurden ebenfalls für die vergleichende Untersuchung verwendet. Untersuchte Strömungsfälle Die Strömungsfälle für die vergleichende Untersuchung der Turbulenzmodelle wurden so gewählt, dass mit jedem Strömungsfall eine Zunahme an physikalischer Komplexität erzielt wurde. Dadurch sollte geprüft werden, bei welchem ,,Komplexitätsgrad” ein Turbulenzmodell im Vergleich zum Experiment eine unzureichende VorhersageGenauigkeit liefert. Außerdem wurden nur solche Strömungsfälle ausgewählt, für welche umfangreiche und zuverlässige experimentelle Daten zur Verfügung standen. Als erstes wurde eine turbulente Grenzschicht an der ebenen Platte, das heißt mit verschwindend geringem Druckgradienten, berechnet. Hier zeigte sich, dass insbesondere bei der Vorhersage des Reibungsbeiwertes die einzelnen Modelle sehr unterschiedliche Ergebnisse lieferten. Das Gleiche galt auch für die Vorhersage des Geschwindigkeitsproﬁls in der Grenzschicht. Dabei hatten die Ergebnisse des k-ω-TNT-Modells von Kok und des modiﬁzierten k-ω-SST-Modells die besten Übereinstimmungen mit dem Experiment von DeGraaﬀ & Eaton (2000) und dem logarithmischen Wandgesetz von Coles. Als nächster Strömungsfall wurde eine stark verzögerte, anliegende Grenzschicht untersucht, bei welcher der Druckgradient ein starkes Nichtgleichgewicht zwischen mittlerer Strömung und Turbulenz verursachte. Die Vorhersage solcher Strömungen stellt eine schwierige Aufgabe für die Turbulenzmodelle dar. (Das zugehörige Experiment wurde von Driver & Johnston Zusammenfassung 21 (1990); Driver (1991) durchgeführt und ausführlich dokumentiert.) Auch für diesen Strömungsfall lieferten die Turbulenzmodelle deutlich unterschiedliche Vorhersagen. Im Vergleich zum Experiment zeigte das k-ω-Modell nach Wilcox von 1998 die besten Ergebnisse, insbesondere für den Verlauf der Wandschubspannung. Das Baldwin-Lomax-Modell lieferte für alle untersuchten Strömungsgrößen die größten Abweichungen von den gemessenen Werten. Der nächste Strömungsfall war im Prinzip identisch mit dem zuletzt diskutierten mit der Ausnahme, dass der Druckgradient größer war und somit Ablösung eintrat. (Das Experiment ist ebenfalls von Driver & Johnston, 1990). Wieder zeigte sich ein große Streuung der Berechnungsergebnisse der verschiedenen Turbulenzmodelle. Dabei war die Streuung zwischen den Ergebnissen der k-ω-Modelle größer als zwischen den Resultaten der Eingleichungsmodelle. Insbesondere die Position des Ablösepunktes und des Wiederanlegepunktes wurde von allen Modellen unterschiedlich wiedergegeben. Keines der Modelle war in der Lage, die Position des Ablösepunktes in guter Übereinstimmung mit dem Experiment vorherzusagen. Die Stromabentwicklung der maximalen Reynods-Schubspannung in der Grenzschicht wurde ebenfalls von keinem der Turbulenzmodelle in Übereinstimmung mit dem Experiment wiedergegeben: Alle Modelle lieferten das Maximum zu weit stromauf und berechneten stromab davon einen Abfall der maximalen ReynodsSpannung. Im Gegensatz dazu zeigte das Experiment einen durchgehenden Anstieg der maximalen Reynods-Spannung. Die Ergebnisse der Transportgleichungsmodelle für den Verlauf der maximalen Reynods-Schubspannung lagen allerdings näher an den experimentellen Daten als die des BaldwinLomax-Modells. Dies wird auf die ansatzweise Berücksichtigung von Konvektionseﬀekten bei den Transportgleichungsmodellen zurückgeführt. Insgesamt lagen die Ergebnisse des modiﬁzierten k-ω-SST-Modells am nächsten bei den experimentellen Daten für diesen Strömungsfall. Als letzter Fall wurde wieder die abgelöste Strömung um das AerospatialeA-Proﬁl betrachtet, die jetzt mit allen untersuchten Modellen berechnet wurde. Bei der Auswertung der Strömungstopologie des Ablösegebietes zeigte sich, dass alle Modelle ein sehr viel kleineres Rezirkulationsgebiet voraussagten, als gemessen worden ist. Im Vergleich zum Experiment wurde der Ablösepunkt von allen Modellen zu weit stromab berechnet. Das BaldwinLomax-Modell und das k-ω-Modell von Wilcox von 1988 berechneten fast keine Ablösung, während das Johnson-King-Modell, das k-ω-SST-Modell und das SALSA-Modell die größten Ablösegebiete voraussagten. Aufgrund der zu klein berechneten Ablösegebiete lieferten die Turbulenzmodelle im Vergleich 22 Zusammenfassung zu den gemessenen Daten zu große Auftriebs- und zu geringe Widerstandsbeiwerte. Auch wurde wieder eine große Streuung zwischen den Ergebnissen der einzelnen Modelle beobachtet. Das Johnson-King-Modell lieferte Ergebnisse mit den geringsten Abweichungen zum Experiment. Bei allen untersuchten Strömungsfällen war festzustellen, dass die Vorhersage-Genauigkeit eines Turbulenzmodells für verschiedene Strömungsgrößen unterschiedlich gut war. Außerdem lieferte bei jedem Strömungsfall ein anderes Turbulenzmodell die beste Übereinstimmung mit den jeweiligen experimentellen Daten. Somit konnte keines der untersuchten Turbulenzmodelle als zuverlässig bezüglich der erzielten Vorhersage-Genauigkeit bewertet werden. Numerische Aspekte Die Empﬁndlichkeit der berechneten Lösungen gegenüber Netzverfeinerung beziehungsweise -vergröberung wurde anhand ausgiebiger und systematischer Gitterkonvergenzstudien überprüft. Für alle Strömungsfälle konnte eine ausreichende Netzfeinheit mit Hilfe des so genannten Grid-Convergence Index“, der auf der Richardson-Extrapolations” Methode basiert, nachgewiesen werden. Der Einﬂuss des eingesetzten Local Preconditioning für kleine Machzahlen wurde anhand einer Proﬁlumströmung untersucht. Es zeigte sich, dass der Einsatz von Local Preconditioning eine deutliche Erhöhung der Berechungsgenauigkeit erzielte. Beispielsweise wurden unerwünschte Druckoszillationen in der Nähe der Hinterkante von Proﬁlen eliminiert. (Diese Druckoszillationen bei geringen Machzahlen sind typisch für numerische Verfahren zur Berechnung kompressibler Strömungen.) Auch wurde korrektes asymptotisches Verhalten des Reibungsbeiwertes für verschwindend geringe Machzahlen festgestellt. Folglich ist bei Verwendung eines Verfahrens zur Berechnung von kompressiblen Strömungen, in denen Ablösegebiete mit geringen lokalen Machzahlen existieren, der Einsatz von Local Preconditioning notwendig. Der Einﬂuss der expliziten künstlichen Dämpfung wurde ebenfalls untersucht. Es stellte sich heraus, dass die künstliche Dämpfung in Grenzschichten normal zur Wand reduziert werden musste. Dort waren die physikalischen viskosen Flüsse ausreichend groß, um das numerische Verfahren zu stabilisieren. Ein zusätzliches Dämpfen erzeugte insbesondere im laminaren Teil der Grenzschicht einen zu hohen Impulsverlust. Beim Aerospatiale-A-Proﬁl wurde der Einﬂuss der Transitionsvorgabe in der Simulation untersucht. Es zeigte sich eine hohe Empﬁndlichkeit der Strömungslösung gegenüber den vorgegebenen Transitionslagen am Proﬁl. Zusammenfassung 23 Wurde zum Beispiel das Einschalten des Turbulenzmodells um ein Prozent der Proﬁltiefe stromab verschoben, stellte sich ein stark abgelöstes Strömungsgebiet mit zwei gegensinnig rotierenden Rezirkulationsgebieten ein. Schlussfolgerungen Modiﬁkationen des Turbulenzmodells, die stromab und innerhalb eines Rezirkulationsgebietes greifen, haben keinen Einﬂuss auf die Lage der primären Ablösung und die Größe des berechneten Rezirkulationsgebietes. Die untersuchten Turbulenzmodelle lieferten unterschiedliche Ergebnisse für gleiche Strömungsfälle. Dies traf selbst für Modelle zu, die der gleichen Modellklasse angehören, wie zum Beispiel die k-ω-Modelle. Bei jedem Strömungsfall lieferte ein anderes Modell die beste Übereinstimmung mit dem Experiment. Transportgleichungsmodelle hatten die grundsätzliche Tendenz, bei abgelösten Strömungen eine höhere Vorhersage-Genauigkeit für die untersuchten Strömungsgrößen zu erzielen als das algebraische Baldwin-Lomax-Modell. Es konnte keine allgemeine Aussage darüber gemacht werden, welches der untersuchten Turbulenzmodelle sich für die Berechnung von abgelösten turbulenten Strömungen am besten eignet. Jedoch wurde nach Hirschel (1999) der Versuch unternommen, eine solche Aussage für einzelne Klassen abgelöster turbulenter Strömungen zu erhalten. Die Lage der für die Berechnung üblicherweise vorzugebenden Transitionsorte kann große Auswirkungen auf die erzielten Rechenergebnisse haben. Eine sorgfältige Modellierung der Transition im Rahmen eines Turbulenzmodells, dass heißt die Art, wie ein Turbulenzmodell den vollturbulenten Zustand erreicht, erscheint insbesondere für abgelöste Proﬁlumströmungen von großer Bedeutung zu sein. Nur mit Hilfe eines geeigneten Local Preconditioning kann mit einem kompressiblen Verfahren eine befriedigende Genauigkeit der Strömungslösung bei kleinen Machzahlen (z.B. M ≤ 0.15) erzielt werden. Des Weiteren müssen die üblichen künstlichen Dämpfungsglieder in Grenzschichten reduziert werden, da sonst unphysikalisch hohe Impulsverluste in der Grenzschicht auftreten können. In Anbetracht der teilweise enttäuschenden Vorhersage-Genauigkeit, die mit den betrachteten Turbulenzmodellen erzielt wurde (auch in anderen Arbeiten), erscheint eine zuverlässige und genaue Vorhersage von abgelösten turbulenten Strömungen im Allgemeinen nicht möglich. Um die heutige Modellierung der Turbulenz und damit die Berechenbarkeit abgelöster turbulenter Strömungen zu verbessern, erscheint ein streng kombinierter und systema- 24 Zusammenfassung tisch koordinierter Einsatz von Experiment, direkter numerischer Simulation, Large-Eddy-Simulation und Reynolds-gemittelter Simulation einzig erfolgversprechend. Die einzelnen Disziplinen sollten dabei in direkter gegenseitiger Wechselwirkung stehen. Ausblick Eine mögliche Verbesserung der Vorhersage-Genauigkeit für turbulente Strömungen bei gleichzeitig vertretbarem Rechenaufwand kann eine Kopplung von Reynolds-gemittelter Simulation (RANS) und Large-EddySimulation (LES) ergeben. Die LES kommt dabei nur in Strömungsbereichen zum Einsatz, in denen eine physikalisch genauere Erfassung der Turbulenz erzielt werden soll, als dies mit RANS möglich ist. Ein kritischer Punkt bei dieser Methode ist die Erzeugung von physikalisch sinnvollen turbulenten Schwankungen beim Übergang vom RANS-Gebiet zum LES-Bereich. Zu diesem Zweck wurde ein teilweise neuer Ansatz vorgeschlagen, der als Basis für künftige Arbeiten dienen soll. 1 Introduction Turbulence plays a vital role in many ﬂows of engineering interest. Fast and accurate computations of turbulent ﬂow ﬁelds are therefore demanded by engineers in a wide variety of technical ﬁelds. It is commonly accepted that the Navier-Stokes equations permit accurate description of laminar, turbulent and transitional ﬂows of simple ﬂuids (Bradshaw, 1999; Krause, 1999). Resolving the entire spectrum of the ﬂuid’s turbulent motion in space and time by solving the Navier-Stokes equations, i.e. performing a direct numerical simulation (DNS), can yield a highly accurate solution of the ﬂow ﬁeld. Yet, the smallest turbulent length and time scales decrease with increasing Reynolds number (Re) and, as a consequence, computational power needed for the resolution of all turbulent scales is approximately proportional to Re3 (Rodi, 2000; Bradshaw, 1999). Despite ever-increasing computational power of modern supercomputers DNS of ﬂows at high Reynolds number is still not feasible. In fact, no matter how powerful a computer may be there will always be a limit to the achievable Reynolds number. In most practical cases computations of ﬂows at high Reynolds numbers are performed using some form of statistical turbulence model to account for the eﬀect of turbulence on the ﬂow. In the statistical approach to turbulence modeling, mean values are studied, which vary relatively smoothly with time and space. This idea dates back to 1895 when Reynolds introduced his concepts of averaging. In the averaging process instantaneous ﬂow variables are expressed as the sum of a mean and a ﬂuctuating part. This sum is inserted into the conservation equations of mass, momentum and energy of the ﬂow. Finally, Reynolds averaging of the equations is performed and the Reynoldsaveraged Navier-Stokes equations (RANS), which describe the mean motion of the ﬂuid, are obtained. Due to the non-linearity of the convection terms, so-called Reynolds stresses appear as additional terms in the mean momentum equations. Since smoothly-varying mean values are computed, resolution demands are relaxed by several orders of magnitude compared to DNS, and a solution of the mean motion of the ﬂow is obtained at much lower computational cost. Before Reynolds introduced his theory of averaging, it was recognized by some researchers that turbulence acts very much like additional stresses. In order to develop a mathematical description of turbulent stresses, Boussinesq suggested in 1877 the concept of eddy viscosity. In this framework, turbulent 26 1 Introduction stresses are treated analogously to molecular stresses, with eddy viscosity being the turbulent counterpart of molecular viscosity. Many diﬀerent turbulence models have been proposed for the computation of Reynolds stresses since the days of Reynolds and Boussinesq and a complete list of all published models cannot be given. Instead, only the most important or best known “landmarks” in statistical turbulence modeling shall be mentioned here. In 1925, Prandtl introduced his mixing length theory for the computation of eddy viscosity. He did not assume the eddy viscosity to be constant, as was usually done at that time. Instead, Prandtl made it a function of local ﬂow quantities, namely the gradient of mean ﬂow velocity, and a characteristic turbulent length scale, the mixing length. The rationale behind Prandtl’s ideas is that the eddy viscosity can, on dimensional grounds, be regarded as a product of a suitable length and velocity scale. Today, virtually all of the so-called algebraic turbulence models in use originate from Prandtl’s mixing length theory. The Baldwin-Lomax (Baldwin & Lomax, 1978) model is one of the most common models for computational ﬂuid dynamics belonging to this category. Its strengths are the capability to deliver reasonable results for many boundary-layer ﬂows while being numerically robust and computationally inexpensive and fast. More advanced eddy-viscosity models solve partial diﬀerential transport equations for either one or two turbulent quantities in order to incorporate non-local and ﬂow-history eﬀects into the eddy viscosity. Prandtl was the ﬁrst to suggest a one-equation model where the desired velocity scale is computed from the turbulent-kinetic-energy transport equation. In this model, an algebraic prescription for computing the length scale is still needed. In the one-equation model of Spalart & Allmaras (1992), a partial diﬀerential equation for the eddy viscosity itself is solved and the need to compute the length scale separately is circumvented. In two-equation models, the eddy viscosity is related solely to turbulent quantities, in contrast to algebraic models where it is related purely to mean ﬂow quantities. In almost all two-equation models the turbulent kinetic energy k serves as the velocity scale. In order to compute an appropriate length scale any expression of the form km n may be employed, with being the turbulent dissipation rate. (m and n don’t have to be integers.) Pioneering work on two-equation models was done as early as 1942 by Kolmogorov (see Wilcox, 1998) but due to the computational demands of such models computers had to come into general engineering use in order to be able to solve the models’ equations for ﬂows of interest. By 27 far the most popular two-equation model is the k, (m = 0, n = 1) model that was suggested by Jones & Launder (1972). It has often been modiﬁed and re-tuned and became a quasi-standard model in industrial use although its defects, like even poorer performance than algebraic models for boundary layers in adverse pressure gradients, cannot be dismissed. In addition, models based on k and cannot be integrated through the viscous sublayer without modiﬁcations. To permit integration through the sublayer, so-called low-Reynolds-number modiﬁcations, which usually consist of viscous damping functions, have to be incorporated into the model. Alternatively, wall functions are applied in order to entirely “bridge” the near-wall region in the solution of the model equations. Besides , the speciﬁc dissipation rate ω has gained increasing popularity as the second variable in two-equation models. This corresponds to setting m = −1, n = 1. As noted by Wilcox (1998), k, ω models oﬀer very appealing advantages over the k, models. First, k, ω models perform better in boundary-layer ﬂows, especially with adverse pressure gradients, and, secondly, they can be integrated through the viscous sublayer without any near-wall modiﬁcations. It was recognized very early that the eddy-viscosity assumption has major shortcomings and there is still very active research underway to entirely avoid the use of eddy viscosity. This can be done, in principle, by solving the exact Reynolds-stress transport equations, which can be deduced from the Navier-Stokes equations. However, additional unknown terms like pressurestrain correlation and turbulent diﬀusion, which require modeling, appear in the Reynolds-stress equations and, hence, introduce new uncertainties. Rotta (1951) was the ﬁrst to introduce a complete Reynolds-stress transport model (RSTM), but as with two-equation models, computers at that time did not oﬀer suﬃcient computational power to solve the model equations. One of the ﬁrst RSTMs that was computationally “aﬀordable” at the time of its development is the model by Bradshaw, Ferriss & Atwell (1967). In this model, a direct proportionality between Reynolds shear stress and turbulent kinetic energy k in a two-dimensional boundary layer is assumed. k is computed from a diﬀerential transport equation and, hence, the concept of eddy viscosity is avoided in the computation of the Reynolds stress. Another model that is also mandatory to mention in connexion with RSTMs is the model devised by Launder, Reece & Rodi (1975). It is the best known and most extensively tested model that computes the complete Reynolds-stress tensor from modeled Reynolds-stress equations. 28 1 Introduction A diﬀerent approach, which is expected to give a more accurate description of the Reynolds-stress tensor than linear eddy-viscosity models, is to assume a non-linear constitutive relation between the Reynolds-stress tensor and the strain-rate and rotation tensors. Several diﬀerent ways have been proposed to derive such relations (e.g. Lumley, 1972; Saﬀman, 1976; Speziale, 1987; Rodi, 1976; Gatski & Speziale, 1993) and, especially for ﬂows where system rotation and anisotropy of the Reynolds-stress tensor play an important role in computing the eﬀect of turbulence on the mean ﬂow, these kind of models improve ﬂow predictions, at least qualitatively. A recent survey and analysis of models belonging to this category can be found in Rung (2000). All models mentioned so far focus on the computation of Reynolds stresses. The latter are a result of averaging the entire turbulent wave-number spectrum. In large-eddy simulations (LES), averaging is performed only for high wave numbers belonging to small eddies while large and energy-bearing eddies are resolved. This is done by applying low-pass ﬁltering to the Navier-Stokes equations. The ﬁltering process introduces subgrid stresses, which account for the interaction between resolved turbulent structures and subgrid scales. Because most of turbulent energy is carried by larger eddies, modeling the high wave-number part of the spectrum seems to be an attractive alternative to full Reynolds-stress modeling on the one hand and DNS on the other. However, LES is only about one order of magnitude “cheaper” in computing costs than DNS for wall bounded ﬂows. Approaching the wall, large turbulent scales decrease in size and hence a “quasi” DNS must be utilized to resolve the energy bearing eddies. Therefore, qualitatively similar restrictions on achievable Reynolds numbers hold in both LES and DNS, although being less restrictive for LES. There are attempts to use “oﬀ-the-wall” boundary conditions so that the viscous wall region can be excluded from the main computation. These boundary conditions often rely in some form on the law of the wall and are not trustworthy for ﬂows where the law of the wall does not hold, as for separated boundary layers. Besides development of suitable numerical methods for LES, derivation and application of subgrid-scale models and appropriate near-wall treatment are subject to current research in LES. Recently, a lot of research on hybrid methods combining RANS simulation in one part of a ﬂow and LES in another has been performed. Detachededdy simulation (DES) proposed by Spalart, Jou, Strelets & Allmaras (1997), for example, uses a conventional eddy-viscosity model, namely the SpalartAllmaras model, for the near wall region. Away from the wall, the computa- 29 tion converts to LES and the Spalart-Allmaras model acts as a subgrid-scale model. This is ensured by the modiﬁed formulation of the underlying SpalartAllmaras model, which “detects” whether the grid is ﬁne enough or not to use LES, and not by explicit switching from RANS to LES. Other hybrid LES/RANS approaches are of explicit zonal character. There, LES is performed in regions of the ﬂow where a richer ﬂow description is necessary and classical RANS in other regions. This method has been applied in the computation of aero-engine gas-turbines where LES has been used in the combustion chamber while RANS models have been applied in the compressor and turbine sections (see Schlüter & Pitsch, 2001). Despite several decades of research on statistical turbulence models and the wide variety of models proposed the computed solution of a ﬂow ﬁeld frequently does not meet desired engineering accuracy. In most such situations, the failure can be attributed to poor performance of the employed turbulence model(s). This is primarily the case for separated ﬂows and for types of ﬂows where the applied model was not tested or calibrated before. An important fact in regard to occasionally disappointing performance of turbulence models is the general lack of useful possibilities to make estimates of accuracy of results a priori, i.e. without knowing the “real” solution. In light of the remarks in the above paragraph and considering documented experience of the research community, it may be deduced that, whatever model is considered, one can always draw the same conclusion about a model’s performance: There exists a variety of ﬂow situations where the model performs reasonably well. However, important ﬂow scenarios will be encountered where the model will unexpectedly produce results of insuﬃcient and unpredictable accuracy (compared to experiment or DNS). This indicates that, although a single universal turbulence model remains the ultimate goal in turbulence modeling, development of such an universal model is highly uncertain. From this premise, several researchers have abandoned development of universal models in favor of construction of models that are superior in a rather limited number of ﬂow classes. Johnson & King (1985), for example, developed a very successful model for the treatment of two-dimensional, subsonic, pressure-driven separated ﬂows and shock-induced separated ﬂows. Straightforward extension of this concept is the combination of “optimal” models in order to broaden the ﬁeld of application: In diﬀerent regions of a ﬂow diﬀerent models are applied using for each region an as optimal model as possible. The model-selection process is guided by the ﬂow type encountered in the considered region. This concept of zonal modeling was frequently dis- 30 1 Introduction cussed, see for example Kline et al. (1981), but Ferziger et al. (1988) were one of the ﬁrst to rigorously apply the zonal approach. They regard a complex ﬂow ﬁeld as a compound of several diﬀerent structural ﬂow zones where each ﬂow zone comprises a single part of the ﬂow with similar turbulent structures. Ferziger et al. (1988) argue that a turbulence model should be linked to local properties of the ﬂow; the turbulence model itself should vary as a function of relevant ﬂow parameters. This is based on the fact that modeling ﬂows with a single kind of turbulent structure can be successfully accomplished with existing turbulence models. To demonstrate their approach, they use the same baseline model throughout the entire domain and adapt the model’s closure constants and functions from zone to zone using so-called bridges. Because a structural ﬂow zone is supposed to contain similar turbulent structures, zoning of the ﬂow is guided by physical insight of and knowledge about turbulence structure. The topological approach to turbulence modeling for separated ﬂows according to Hirschel (1999) suggests a classiﬁcation of separated aerodynamic ﬂow ﬁelds guided by the observed or expected ﬂow topology. The term “ﬂow topology” denotes in this case the topology of streamlines of the Reynoldsaveraged velocity ﬁeld. (More frequently, in turbulence research, the term “topology” is used in the context of coherent turbulent structures.) An excerpt of a possible classiﬁcation of separated ﬂows according to Hirschel (1999) is shown in Table 1.1. Following this line of reasoning, separated turbulent ﬂows around rigid ﬁxed bodies in steady ﬂow at inﬁnity are sub-divided into ﬂow classes with and without vortex shedding denoted as class 1 and 2, respectively. Sub-classes with two- and three-dimensional ﬂows are deﬁned in each major class using topological and geometrical arguments. For example, class 1.1 in Table 1.1 comprises two-dimensional separated ﬂows with one or two recirculation zones featuring so-called critical or nodal points. These are points within the ﬂow domain where all velocity components are zero and the streamlines’ slopes are indeterminate (see Chong et al., 1990). The ﬂow topologies of class 1.1 are frequently encountered in RANS solutions of ﬂows past airfoils at high angle of attack near or at stall conditions. Indeed, many test cases for turbulence models are of this kind, (see Haase et al., 1993, 1997; Dervieux et al., 1998). However, it is not clear whether, or in what parameter range, ﬂows of class 1.1 exist in a statistically steady sense. Trailing-edge separation of an airfoil ﬂow may exist statistically only in an unsteady, vortex-shedding (sub-class 2.1) or vortex-ﬂapping motion. In this case, sub-class 1.1 comprises time-averaged topologies which are not coin- 31 cident with the time-varying statistical ensemble. One the one hand, it is generally questionable whether a RANS model is able to yield the statistical ensemble of a statistically unsteady ﬂow. On the other hand, for engineering purposes, it may be suﬃcient to deﬁne a time-average solution and postulate that solving the RANS equations together with a turbulence model should deliver this time average. In other situations, it may be necessary to predict the low-frequency part of a solution. Here, comparison of the computed solution with experimental data is especially diﬃcult since consistent distinctions between low- and high-frequency parts are needed for both the turbulence model and the measurement. The main distinctive factors between class 1.2 and 1.3 stem from diﬀerent geometrical properties of the bodies under consideration. Slender bodies lead to diﬀerent ﬂow topologies than bodies with high aspect ratios, and it has been observed that ﬂows of class 1.3 are free from large-scale unsteadiness for a wide range of parameters even if the vortices become turbulent. Based on this classiﬁcation of ﬂow ﬁelds, it is hypothesized that computation of a single class can be improved if turbulence modeling is adjusted to topological structures like recirculation zones, separating boundary layers, free shear layers etc. In this aspect, the topological approach is quite similar to the zonal idea discussed above, the main diﬀerence being the general classiﬁcation and partition of separated ﬂows by means of the topology of the velocity ﬁeld. In addition, the ﬂow topology serves as a guideline for discussion of results and model performance. 32 1 Introduction Table 1.1: Possible classes of separated aerodynamic ﬂows (schematic, averaged structures; excerpt from Hirschel, 1999) Rigid ﬁxed bodies in steady ﬂow at inﬁnity 1. Flows without vortex shedding 2. Flows with vortex shedding 1.1 Steady separated ﬂow past airfoils with recirculation area(s) – 2D 2.1 Unsteady separated ﬂow past airfoils – 2D a) α V∞ b) α α V∞ V∞ 1.2 Steady separated ﬂow past wings with large aspect ratio – 3D 2.2 Unsteady separated ﬂow past wings with large aspect ratio – 3D ? 1.3 Steady separated ﬂow past wings and slender bodies with longitudinal vortices – 3D 2.3 Unsteady separated ﬂow with longitudinal vorices past slender bodies – 3D 1.1 The Present Study 1.1 33 The Present Study The motivation for this work was twofold: The one aspect was testing and demonstration of the topological approach to turbulence modeling for separated ﬂows suggested by Hirschel (1999). The other aspect was to conduct a comparative study of various modern eddy-viscosity turbulence models in current production use in European aerospace industry with the objective to assess the models’ performance for separated turbulent ﬂows. Embarking on the topological concepts of Hirschel (1999) the starting point for this work was the group of ﬂows belonging to class 1.1 a) in Table 1.1. The computed ﬂow around an airfoil at high angle of attack with a single recirculation zone near the trailing edge was investigated in detail and compared with data from experiment. Subsequently, the applied turbulence model was modiﬁed in the recirculation zone according to arguments of Hirschel (1999) in order to study the feasibility of this approach. The major outcome of this feasibility test was that a modiﬁcation of a turbulence model does not improve computational results if it is restricted to the recirculation zone. In particular, the point of separation is not aﬀected by modiﬁcations inside of the recirculation zone and the size of the computed recirculation zone does not change. Consequently, following conclusions were drawn: 1. The boundary-layer development inﬂuenced by pressure gradient upstream of separation has to be considered in order to correctly capture the position of the primary separation point (or separation line in threedimensional ﬂows). 2. One major challenge in computing separated ﬂow ﬁelds is that the physical nature of turbulence structure changes from that of a boundary layer to that of a free shear layer. Turbulence models have to properly mimic the eﬀect of this change on the mean ﬂow. In order to investigate a model’s ability to perform this task, several ﬂow cases have to be studied which feature a successively increased strength of change in the ﬂow structure from case to case. Based on these conclusions the following strategy was chosen: First, the performance of several modern turbulence models was investigated in a classical ﬂat-plate boundary layer with zero pressure gradient. This was deemed to be necessary to basically assess the ability of turbulence models to correctly compute boundary layers in the frame of full Navier-Stokes solutions 34 1 Introduction as opposed to boundary-layer methods where these models typically have been tested. Secondly, an attached boundary layer under substantial adverse pressure gradient was computed using the same models; results obtained with the diﬀerent models were compared with each other and with experimental data. This ﬂow case is very well suited to test the models in a non-equilibrium boundary layer without separation. By increasing adverse pressure gradient a separating boundary layer was obtained. As before, computations with all models were performed; experimental data served for comparison of results. Finally, the separated airfoil ﬂow was again analyzed utilizing the available set of turbulence models. Questions regarding surface boundary conditions and dependence on freestream values for the turbulence equations were addressed as well as purely numerical issues like grid convergence and low-Mach-number preconditioning. The sensitivity of the computational solution to the choice of transition location was also investigated. The major contribution of this work is seen in thorough testing and performance assessment of various modern turbulence models for computing ﬂows with pressure-induced boundary-layer separation. The applied models range from the algebraic model of Baldwin & Lomax (1978) to the explicit algebraic Reynolds-stress model of Wallin & Johansson (2000), covering also a range of one- and two-equation eddy-viscosity models. Although the selected ﬂow cases may seem somewhat simple, they oﬀer a natural increase in complexity. It is evident from results obtained that the computation of the considered cases poses a challenging task. In particular, variances among results computed with diﬀerent models show that the task of successful turbulence modeling for such ﬂows is not completed, and further improvements are necessary. For this purpose, a partially new method of coupling RANS and LES was proposed. The method is presented in the appendix of this thesis. 1.2 Contents and Organization of the Thesis The present thesis is organized in three major parts. Part I is concerned with ideas proposed by Hirschel (1999). Accordingly, in Section 2, the ﬂow topology of velocity ﬁelds is used as a guideline for classifying separated turbulent ﬂows and for identifying ﬂow regions of possible importance to turbulence modeling. In Section 3, the governing equations of the ﬂow, the numerical method, and the turbulence model employed in Part I of this work are brieﬂy discussed. Subsequently, in Section 4, the ﬂow ﬁeld of a separated airfoil ﬂow 1.2 Contents and Organization of the Thesis 35 is investigated using the ﬂow topology as a guideline for the analysis of results. In addition, a numerical experiment is presented where the turbulence model was modiﬁed in the separated region of the airfoil ﬂow. Comments on hidden three-dimensional eﬀects in nominally two-dimensional ﬂows are placed in Section 4 at the end of Part I. Part II deals with a comparative study of various modern turbulence models for computing separated boundary-layer ﬂows. First, in Section 5, the numerical method employed in Part II is brieﬂy discussed. Secondly, in Section 6, the turbulence models investigated are presented. Thirdly, in Section 7, the test cases selected are introduced and computational results obtained with the models are compared with experimental data. Numerical issues, like grid convergence, local preconditioning and artiﬁcial damping are subject of Section 8. Sections 9 and 10 contain the conclusions and the future work, respectively. Part III consists of the appendices. In the ﬁrst section of the appendices, Section A, a partly novel approach to coupling RANS and LES for separated ﬂows is proposed. In Section B, details of the Johnson-King model are presented and Section C contains comparative graphs of computational and experimental results. A brief overview of the work performed concerning algorithmic issues is given in Section D. In the last section, Section E, typical input decks used for the computations with the FLOWer code are presented. Part I Topological Approach to Turbulence Modeling 38 2 2 Basic Considerations Basic Considerations It was already mentioned in the Introduction, that, more often than not, the computed solution of a turbulent ﬂow ﬁeld does not oﬀer the required accuracy. The reasons for unsatisfactory results are manifold; numerous possibilities have to be kept in mind when considering a particular ﬂow case. The lack of reliable models for determining locations of transition from laminar to turbulent in general ﬂow situations is one such possibility. Due to this lack, mostly, the locations have to be explicitly speciﬁed by the user before starting the computation. In some cases, transition locations on the surface of a wind-tunnel model are known from experiments, but in general they are unknown. However, even if this kind of transition is given from experimental data this may not comprise enough information about all the transition mechanisms potentially encountered in the ﬂow case considered. For instance, in complex, three-dimensional, separated ﬂow ﬁelds, like the ﬂow around a delta wing at high angle of attack, transition can occur in shear layers away from the surface before the turbulent ﬂuid “hits” the surface of the body. In this way, transition can signiﬁcantly inﬂuence the momentum and heat transfer at the surface of the body as well as the overall ﬂow ﬁeld. This type of transition and its location can hardly, if possible at all, be identiﬁed in experiments. One more example where the modeling of transition can have major impact on the ﬂow solution is dynamic stall of airfoil ﬂows. Ekaterinaris & Menter (1994) point out that correct modeling of leading edge transition is a key issue in the computation of such ﬂows. Clearly, in all turbulent ﬂow ﬁelds where transition plays a crucial role incorrect speciﬁcation of transition location(s) may become a major source of error. Another possible source of error in the computation of separated ﬂow ﬁelds is “hidden” three-dimensionality in nominally two-dimensional ﬂows. Frequently, in the computation and analysis of a particular ﬂow case it is implicitly assumed that the ﬂow ﬁeld is two-dimensional in the usual sense of RANS. This assumption may not be correct, especially when considering a massively-separated ﬂow. Similarly, it is often assumed that the ﬂow ﬁeld is statistically steady and that time-averaged mean values are a valid and accurate representation of probability mean values. Neither may be the case, and this can lead to discrepancies between measured and computed results. (For a more detailed discussion of statistical concepts for describing turbulent ﬂows and the problems arising in this context the reader is referred to Monin & Yaglom (1977); Rotta (1951); Celić & Hirschel (1999).) 39 Numerical issues, like accuracy of the numerical method, iterative convergence and grid convergence, have to be treated with appropriate thoroughness in order to eliminate numerical eﬀects on the results to greatest possible extent. These questions will be addressed in more detail in Section 8. The following list summarizes some of the most important sources of errors when computing turbulent aerodynamic ﬂows: • speciﬁcation of transition locations and transition modeling • hidden three-dimensional eﬀects in nominally two-dimensional ﬂows • the ﬂow case considered is statistically unsteady; probability mean values and time-averaged mean values diﬀer (it is questionable whether the concept of Reynolds averaging is applicable to unsteady ﬂows) • strong non-equilibrium between mean ﬂow and turbulence – the Boussinesq assumption becomes really wrong (this applies only to eddyviscosity turbulence models) In order to identify possible turbulence-model failures, only those ﬂow cases should be considered where all other possible sources of errors can be regarded as relatively small or non-existent. This is postulated of the group of ﬂow ﬁelds belonging to class 1.1 in Table 1.1, which were considered ﬁrst in this work. At this point, the validity of this assumption is not questioned but it will be discussed again later. The schematic ﬂow topology is repeated in Figure 2.1 and is discussed in the following; topological structures shown in the ﬁgure are listed in Table 2.1. Following the stagnation streamline, the ﬂow reaches the surface of the airfoil at the stagnation point A1, where it is split into two attached laminar boundary layers a1 and a4 on the upper and lower sides, respectively (see Figure 2.1). If the Reynolds number of the ﬂow is high enough the laminar boundary layers will become turbulent at some position downstream of the stagnation point. In Figure 2.1, the region of laminar-turbulent transition T on the upper side of the airfoil is denoted by a2; on the lower side it is denoted by a5. Generally, transition can be forced by some tripping device or it can occur “naturally”, i.e. free transition takes place. In the latter case, a small transitional bubble with laminar separation and turbulent reattachment may be found on the upper side of the airfoil, depending on the type of pressure distribution. Downstream of the transition region a2 the turbulent boundary layer a3 separates from the surface of the body at the separation 40 2 Basic Considerations transitional separation bubble (time averaged) a2 T a1 a3 A1 a4 b1 S1 T a5 a7 a6 F1 A2 a8 S2 b2 Figure 2.1: Schematic ﬂow topology of airfoil ﬂows with one recirculation zone at high angle of attack (class 1.1 a) of Table 2.1. Table 2.1: Two-dimensional topological structures shown in Figure 2.1 with a classiﬁcation according to Peake & Tobak (1980) Symbol Phenomena A1, A2 attachment point (stagnation point) Classiﬁcation half saddle S1 (primary) squeeze-oﬀ separation point half saddle S2 ﬂow-oﬀ separation point half saddle F1 recirculation zone around focus focus a1, a4 attached laminar boundary layer a2, a5 laminar-turbulent transition region (T) – attached turbulent boundary layer – attached boundary layer in recirculation zone – shear-layer skeleton streamline – a3, a6, a8 a7 b1, b2 – point S1. This type of separation is herein called “squeeze-oﬀ separation” following Hirschel (1986) since the two boundary layers a3 and a7 converge and “squeeze” each other oﬀ the surface at S1. Enclosed by the dividing streamline b1 a recirculation zone with focus F1 containing slowly moving ﬂuid is present at the trailing edge. The situation in the vicinity of the reattachment point A2 (and the boundary layer a8) is not as clear as for the other nodal points A1, S1, F1 and S2. Topologically speaking, A2 is simply a half-saddle. However, the ﬂuid is moving very slowly and the streamlines 41 found in the computations are highly curved in the vicinity of A2 in order to satisfy the Reynolds-averaged continuity equation after the turbulent shear layer b1 has deﬂected away from the wall (Collins & Simpson, 1978). Following Hirschel (1986), the turbulent boundary layer a6 on the lower side of the airfoil remains attached until it reaches the “ﬂow-oﬀ separation point” S2 at the trailing-edge . The formation of a transitional bubble at the upper surface depends on the shape of the airfoil and angle of attack, i.e. pressure distribution, and the Reynolds number. Detailed numerical and experimental investigations of a transitional bubble were performed for example by Lang et al. (2000). The time-averaged streamline topology of the transitional bubble, as shown in Figure 2.1, is very similar to the recirculation zone located further downstream at the trailing edge of the airfoil. However, underlying ﬂow physics diﬀers substantially. The separation point of a laminar boundary layer is welldeﬁned while the separation point S1 of the turbulent boundary layer exists only in an averaged sense. (The latter is “artiﬁcially” obtained through an averaging procedure.): In a time-accurate view, the bubble sheds spanwise vortices at its rear end and these are the beginning of the turbulent boundary layer. The time-accurate behavior of the recirculation zone at the trailing edge is not very well understood. It is assumed that some kind of non-periodic vortex-ﬂapping or vortex-shedding mechanisms combined with small scale turbulent motion take place. Experiments show strong interaction between wakes of the pressure and suction sides (Collins & Simpson, 1978). Since turbulence models for the Reynolds-averaged Navier-Stokes equations are the subject of this work, the subtle time-accurate features of the ﬂow are not considered and only time-averaged ﬂow topologies are discussed. Nevertheless, one should keep the complex time-accurate ﬂow structure in mind when considering computational solutions of such ﬂow ﬁelds. 42 3 3.1 3 Governing Equations and Numerical Method Governing Equations and Numerical Method Governing Equations of the Mean Flow For the mathematical description of turbulent ﬂows the Reynolds-averaged Navier-Stokes equations in integral and conservation form are considered. Strictly speaking, Favre- or mass-averaged equations are employed in order to eliminate density ﬂuctuations from the equations. However, for simplicity, and because all ﬂows computed in this work are at low Mach numbers where compressibility eﬀects are negligible, the term “Reynolds-averaged” is used. Consequently, ﬂuctuating quantities are denoted by a single prime as is common practice in Reynolds averaging as opposed to double primes in Favre averaging. In contrast to the notation frequently encountered in text books about turbulence, the overbar indicating an averaged quantity is omitted for all variables but the Reynolds stresses. This is in compliance with the usual notation of the Reynolds- or Favre-averaged equations used in most texts about computational ﬂuid dynamics. Results of the present work are not aﬀected by these conventions. Equations for mass, momentum and energy transport derived with the help of a ﬁnite control volume ﬁxed in space read, in symbolic notation, ∂ w dV + F · n dS = 0. ∂t V S In the above equation, n denotes the outward normal vector of the surface element dS of the control volume V . The vector w contains the conserved variables, w = (ρ, ρu, ρv, ρw, ρE)T , (3.1) and the generalized ﬂux vector is written as ⎡ ⎤ ρv ⎢ ⎥ F=⎣ ρv ⊗ v + pI − σ − τ ⎦ ρvH − σ · v − τ · v − λ∇T (3.2) with v = (u, v, w)T being the ﬂow-velocity vector and I the identity matrix. (⊗ denotes the dyadic product of two vectors.) The total energy E is the sum of internal and kinetic energy, E =e+ 1 2 |v| , 2 3.1 Governing Equations of the Mean Flow 43 where for a calorically perfect gas the internal energy becomes e = cv T . T denotes static temperature, p static pressure and cv the speciﬁc heat coefﬁcient at constant volume. The relation between total energy E and total enthalpy H is given by p H=E+ . ρ For a Newtonian ﬂuid and with the usual bulk-viscosity assumption ζ = − 23 µ the viscous stress tensor is deﬁned as 1 (3.3) σ = 2µ S − (∇ · v) · I , 3 with the strain-rate tensor 1 ∇ ⊗ v + (∇ ⊗ v)T . 2 Kinetic gas theory relates the coeﬃcient of dynamic viscosity µ to static temperature. This relation can be expressed through Sutherland’s formula in conjunction with a reference viscosity µref at a reference temperature Tref : 3 2 T T ref + 110K µ = µref . Tref T + 110K S= While the speciﬁc formulas by which the Reynolds-stress tensor τ in Equation 3.2 is computed depend on the turbulence model employed, it is generally deﬁned by one-point central moments of ﬂuctuating velocity components and reads in index notation τ ≡ τij = −ρui uj . (3.4) This is a result of averaging the Navier-Stokes equations. The computation of τ will be addressed when discussing the various turbulence models. All models investigated compute an eddy viscosity µt which, in turn, is used to derive an expression for the eﬀective heat-conductivity coeﬃcient λ: µ γR µt λ= + . γ − 1 Pr P rt R denotes the perfect-gas constant. While the molecular Prandtl number P r is a ﬂuid property and can be regarded as approximately constant for most gases, with P r = 0.72 for air at standard conditions, this is not the case for the turbulent Prandtl number P rt . However, in the framework of eddy-viscosity models it is assumed that P rt = 0.9 throughout the ﬂow ﬁeld. γ denotes the ratio of speciﬁc heat coeﬃcients with γ = 1.4 for air. 44 3.2 3 Governing Equations and Numerical Method The Baldwin-Lomax Model Although all computations for this chapter were performed using the model of Johnson and King, the Baldwin-Lomax Model is presented ﬁrst since it can be regarded as a “pre-requisite” for the former. The Reynolds-stress tensor in Equation 3.4 is modeled with the help of Boussinesq’s assumption, which states that the turbulent stresses are proportional to the strain-rate tensor, 1 τ = 2µt S − (∇ · v) · I . (3.5) 3 This is in analogy to the molecular stresses in Equation 3.3 but with the fundamental diﬀerence that the eddy viscosity µt is not a ﬂuid property. It is dependent on the particular ﬂow case and has to be computed by the turbulence model. Equation 3.5 is also referred to as a “constitutive relation”. Bearing in mind thin-shear-layer ﬂows, like boundary layers and wakes, Baldwin & Lomax (1978) suggested a two-layer eddy-viscosity model with separate algebraic expressions in each layer. Hence, the eddy viscosity is deﬁned by µti , y ≤ ymin µt = (3.6) µto , y > ymin with y being the coordinate in the wall-normal direction in case of boundarylayer ﬂows. In wake ﬂows, the wake center line is used as the origin for y. ymin denotes the smallest value of y for which µti = µto . The inner and outer viscosities are computed as follows: Inner Layer: 2 µti = ρ lmix |ω|, lmix = κyD, , κ = 0.40, −y + /A+ 0 D = 1−e (3.7) A+ 0 = 26. Expression 3.7 is based on Prandtl’s mixing length lmix = 0.40y multiplied by Van Driest’s damping function D. This damping function is only active in the close vicinity of solid walls in order to account for the damping eﬀect of the wall on the eddy viscosity. The dimensionless, sublayer-scaled wall-distance is deﬁned as τw yuτ y+ = with uτ = ν ρ 3.2 45 The Baldwin-Lomax Model where the friction velocity uτ serves as a velocity scale; it is related to the shear stress at the wall. ν is the kinematic viscosity of the ﬂuid and |ω| denotes the magnitude of the vorticity vector. Outer Layer: µto = αρCcp Fwake FKleb , 2 Cwake ymax Udif Fwake = min ymax Fmax ; , (3.8) Fmax Fmax = 1 max (lmix |ω|) , κ y FKleb (y; ymax /CKleb ) = 1 + 5.5 α = 0.0168, Ccp = 1.6, 6 −1 y ymax CKleb CKleb = 0.3, , (3.9) Cwake = 1. In the above relations, ymax is the value of y at which lmix |ω| achieves its maximum. Udif is the maximum velocity for boundary-layer ﬂows, and for free shear layers it is the diﬀerence between the maximum value of |v| in the layer and the value of |v| at ymax . Corrsin & Kistler (1954) as well as Klebanoﬀ (1954) experimentally showed that the ﬂow at the boundary-layer edge is intermittent. This means that at a ﬁxed location in space close to the boundary-layer edge, the ﬂow is sometimes non-turbulent and sometimes turbulent. The reason for this is that the shape of the sharp interface between laminar and turbulent regions is highly distorted and moving. In order to account for intermittency and its eﬀect on the outer eddy viscosity, µto is multiplied by the empirical function FKleb computed from Equation 3.9. Figure 3.1 shows a typical eddy-viscosity proﬁle computed with the Baldwin-Lomax model for a ﬂat-plate boundary layer. As a ﬁnal comment, the Baldwin-Lomax model can be viewed as a reformulated and extended model of Cebeci & Smith (1974). The main difference is that the latter requires the computation of boundary-layer properties which are diﬃcult to determine in general Navier-Stokes computations. Speciﬁcally, for the Cebeci-Smith model, the product Ccp Fmax in Equation 3.8 is replaced by ue δv∗ , where ue stands for the velocity at the boundary-layer edge and δv∗ denotes the kinematic displacement thickness. In boundarylayer methods, where these properties are readily available, the Cebeci-Smith model can be easily applied. 46 3 Governing Equations and Numerical Method 10-2 -3 y/L 10 -4 10 0 20 40 µturb/µ∞ 60 Figure 3.1: Typical Baldwin-Lomax eddy-viscosity proﬁle for a ﬂat-plate boundary layer. 3.3 The Johnson-King Model For the investigation of the topological concepts of Hirschel (1999) the turbulence model devised by Johnson & King (1985) was employed. This model has been extensively tested for separated airfoil ﬂows and has shown either equivalent or even superior performance for such ﬂows compared to many other models (see Haase, 1997; Haase & Fritz, 1993). Its development was guided by physical insight while avoiding signiﬁcant increase of mathematical complexity in comparison with the Cebeci-Smith or Baldwin-Lomax models. The main physical aspects that inspired the development of the JohnsonKing model are the following: 1. For rapidly-changing turbulent ﬂows, like separating boundary layers, convection of turbulence is an essential eﬀect and must be properly accounted for in the model (Bradshaw et al., 1967; Johnson & King, 1985). 2. Perry & Schoﬁeld (1973) carried out experiments of adverse pressure gradient boundary layers near separation. They found that descriptions for determining the mean velocity proﬁle based on velocity scales related to the maximum shearing stress and its distance from the wall gave 3.3 47 The Johnson-King Model the best results. Hence, the maximum of (−u v )1/2 is used as the controlling velocity scale in the Johnson-King model. In the style of the Cebeci-Smith and Baldwin-Lomax models, the JohnsonKing model uses a two-layer approach for the eddy viscosity and Boussinesq’s Equation 3.5 for the Reynolds-stress tensor. However, instead of the switch in Equation 3.6, a smooth exponential blending between the inner and outer eddy viscosities is employed: µt = µto 1 − e(−µti /µto ) . (3.10) Inner Layer: µti = ρD2 κy(−u v m )1/2 , −y(−u v m )1/2 D = 1 − exp , νA+ κ = 0.40, (3.11) (3.12) A+ = 15. It can be seen from the above equations that the inner eddy viscosity µti is strongly dependent on the velocity scale (−u v m )1/2 . Additionally, the resulting eddy viscosity µt is functionally dependent on µto for the greatest part of the boundary layer, see Equation 3.10. The re-formulation of Van Driest’s damping function D in terms of the velocity scale (−u v m )1/2 offers computational advantages compared to the original expression used in Equation 3.7 in cases where uτ = 0 or uτ < 0, as occurs for boundary-layer separation. Outer Layer: (3.13) µto = σ(x)0.0168ρue δv∗ FKleb Here, the intermittency function FKleb is basically the one given in Equation 3.9 with ymax /CKleb being replaced by the boundary-layer thickness δ. σ(x) is a modeling parameter that varies with the streamwise position x. It allows for adaption of the resulting eddy-viscosity distribution to non-equilibrium conditions. The heart of the model is an ordinary diﬀerential transport equation for (−u v m )1/2 which accounts for convection, diﬀusion, production and dissipation eﬀects. This equation is a simpliﬁed form of the shear-stress transportequation developed by Bradshaw et al. (1967). It assumes that the path of maximum turbulent kinetic energy km coincides with the downstream direction x and, additionally, the ratio of maximum shear stress to maximum 48 3 Governing Equations and Numerical Method turbulent kinetic energy is constant (−u v m /km = constant) in boundary layers, which is experimentally supported. The equation is given by Lm um d(−u v m ) Lm Dm , − (−u v m )1/2 = (−u v m, eq )1/2 − dx a1 (−u v m ) (−u v m ) dissipation production convection diﬀusion (3.14) ym /δ ≤ 0.225 , Lm = ym /δ > 0.225 Cdif (−u v m )1/2 Dm = 1 − σ(x) , a1 δ(0.7 − (y/δ)m ) a1 = 0.25, Cdif = 0.5. 0.4ym , 0.09δ, (3.15) The subscript m denotes that the quantity is evaluated where −u v assumes its maximum in the wall-normal direction. Lm is the dissipation length scale. It is constructed such that it resembles Prandtl’s expression for the mixing length for ym /δ ≤ 0.225, and Escudier’s expression for ym /δ > 0.225. Expression 3.15 for Dm is a diﬀusion model based on an “eﬀective” velocity at which turbulent energy is transported by turbulent diﬀusion eﬀects. This concept diﬀers from the gradient-diﬀusion model frequently employed. It is based on observations discussed by Townsend (1976) and supported by experiments of Bradshaw which suggest that most of the diﬀusion transport depends on the large eddies. Following this line of reasoning, turbulent diffusion can be viewed as either convection of smaller eddies by the motion of larger eddies or, simply, as a transfer of energy from one part of a large eddy to another. In this case, modeling diﬀusion by an “eﬀective” or “convective” velocity seems to be more appropriate than the usual gradient-diﬀusion model employing the eddy viscosity as a diﬀusion coeﬃcient. In the Johnson-King model, (−u v m )1/2 serves as the “eﬀective” velocity scale. Johnson & King (1985) point out that turbulent diﬀusion plays an important role especially in ﬂow recovery regions after reattachment of a separated boundary layer while its contributions to the rate equations of turbulence upstream of separation are secondary. Further details of the Johnson-King model will be given in Appendix B. 3.4 Numerical Method (I) The RANS equations were solved with a customized version of the Jamesontype ﬂow solver MUFLO, which was originally developed by Haase at the 3.4 Numerical Method (I) 49 European Aeronautic Defence and Space Company (EADS). MUFLO uses a two-dimensional cell centered ﬁnite-volume method for structured grids. Both convective and diﬀusive/dissipative ﬂuxes in the Navier-Stokes equations are approximated by a second-order central space discretization. To stabilize the central scheme and prevent spurious oscillations, a blend of second- and fourth-order artiﬁcial damping terms is utilized. An optimized ﬁve-stage Runge-Kutta method combined with a multigrid scheme is employed for the explicit time integration. Additional convergence acceleration to steady state is achieved through local time stepping and implicit residual smoothing. A detailed description of the numerical method is given in Haase & Fritz (1993); Jameson et al. (1981); Jameson & Baker (1983, 1984). In order to increase the physical accuracy of the computed solution for low-Mach-number ﬂows, local preconditioning of the Navier-Stokes equations was implemented in this work into MUFLO. Especially for two-dimensional separated ﬂows, where relatively large and slowly moving recirculation zones exist, local preconditioning oﬀers an improved accuracy of the ﬂow solution in these regions. The preconditioning method and the inﬂuence on the computational results will be presented in Section 8. Various algebraic and half-equation turbulence models, including the models of Baldwin-Lomax and Johnson-King, are implemented in MUFLO. However, the Johnson-King model produced computational results which were closest to the experimental data for the investigated ﬂow case compared to the results obtained with the other models. Therefore, it is the only model which was applied for the demonstration of the topological approach. 50 4 4.1 4 Demonstration Demonstration Description of Flow Case The ﬂow around the Aerospatiale-A airfoil at a Reynolds number of 2 · 106 , a Mach-number of 0.15 and an angle of attack of 13.3◦ was investigated. This ﬂow case exhibits the time-averaged ﬂow topology of class 1.1 a) shown in Figure 2.1. Experiments in two diﬀerent wind tunnels (F1 and F2) were conducted at ONERA (Oﬃce National d’Etudes et de Recherches Aerospatiales) and a complete database of the experimental results is available in Chaput (1997). In the experiments, transition at the lower surface was ﬁxed with a trip at 30 percent of chord length. On the upper side, free transition was allowed and a laminar separation bubble with turbulent re-attachment at 0.12 chord length was observed during the tests. The experimental values for the lift (cL ), drag (cD ), pressure (cp ) and skin friction (cf ) coeﬃcients obtained from measurements in ONERA’s wind tunnel F1 are judged to be more accurate than those obtained in wind tunnel F2 (Chaput, 1997). This is mainly attributed to the larger cross-sectional area of wind tunnel F1. However, measurements of boundary-layer proﬁles of the ﬂow variables including turbulence quantities require measurement techniques like Laser Doppler Velocimetry (LDV) which were not available in wind tunnel F1. These quantities were measured using LDV in experiments conducted in wind tunnel F2. The diﬀerences between the data measured in the two wind tunnels can lead to various discrepancies. For example, the cf distribution obtained from measurements in wind tunnel F1 is not “compatible” with the cf values that can be determined from the velocity proﬁles since the latter were measured in wind tunnel F2. In particular, the locations of the separation point (cf = 0) determined in the two wind tunnels diﬀer by almost 10 percent of the chord length. Consequently, all computational results, like proﬁles of ﬂow variables and turbulence statistics as well as coeﬃcients for pressure and skin friction, are compared to the values measured in wind tunnel F2. Only the lift and drag coeﬃcients are additionally compared to those obtained from experiments in wind tunnel F1 following the recommendations in Chaput (1997). Although complicating the validation process, this situation is unavoidable. However, while many other experimental data of separating airfoil ﬂows exist, frequently, these data do not comprise detailed measurements of the turbulent quantities. These 4.2 Computational Grid 51 quantities are available for the selected ﬂow case which makes it especially attractive for the validation of turbulence models for separating airfoil ﬂows. 4.2 Computational Grid The computation of the ﬂow over airfoil A was performed on a structured, body-aligned grid with a so-called C topology. Special care was taken in the generation of the computational grid. In particular, it was focused on resolving all topological structures discussed in Section 2. In order to reduce the required number of cells, grid points were clustered around the stagnation point A1 as well as in the region of the transitional bubble on the upper side and at the transition location at the lower surface. At the trailing edge, where the recirculation zone was encountered and where the converging boundary layers from the upper and lower sides experience a discontinuity in the boundary condition from no-slip to wake treatment, grid-point clustering was also applied. In regions and in the directions where the spatial variations of the ﬂow ﬁeld were not severe, that is, where second derivatives of the ﬂow variables with respect to the spatial coordinates were relatively small, coarser grid spacing was allowed. In addition to ensuring ﬁne grid spacing where needed, a smooth distribution of the metric terms was achieved by a combination of algebraic grid generation, using geometrical stretching, and grid smoothing. In boundary layers, the grid lines were forced to be normal and parallel to the surface, resulting in almost rectangular cells, which minimizes discretization errors. In the wall-normal direction, geometrical stretching of the grid point spacing was applied in order to further reduce the number of cells without sacriﬁcing the ﬁdelity of the solution. During the current study, computations were performed with many different grids in order to ﬁnd the most suitable conﬁguration. In order to investigate grid convergence of the computational solution, an extremely ﬁne grid was created, which was coarsened by successively skipping every other grid node. The “standard” grid level used in this section has the following dimensions: • a total of 512 cells in the wraparound direction ξ with 384 cells placed on the airfoil surface, and in each case 64 cells located in the upper and lower wake (ξ = (i − 1)/(imax − 1) with 1 ≤ i ≤ 513 being the grid line index and imax = 513 the total number of grid lines in that direction) 52 4 Demonstration • a total of 128 cells were used in the wall-normal direction η with 64 cells in the boundary layer (η = (j − 1)/(jmax − 1) with 1 ≤ j ≤ 129 being the grid line index and jmax = 129 the total number of grid lines in the considered direction; since a C-mesh is used there are, in fact, 256 cells across the wake region) • farﬁeld boundaries were located at least 18 chords away from the nearest surface in order to minimize errors resulting from the numerical farﬁeld boundary treatment. In addition, vortex correction was applied at farﬁeld boundaries to account for the eﬀect of circulation associated with the lift produced by the airfoil. The distribution of the metric term ∂ξ/∂x along the upper surface of the airfoil is shown in Figure 4.1. It can be seen that particularly in the convection direction and inside the boundary layer the distribution of the metric term is smooth and continuous. The same is true for the other metric terms ∂ξ/∂y, ∂η/∂x and ∂η/∂y (not shown). The grid point clustering in the above-mentioned areas is clearly recognizable in Figure 4.2 where the grid in the vicinity of the airfoil is shown. flow direction in the boundary layer 400 ∂ξ/∂x 300 200 100 0.85 0.8 0.75 0.7 ξ 0 0 0.65 0.6 0.55 0.5 0.5 1 η Figure 4.1: Distribution of ∂ξ/∂x metric along upper surface of airfoil A. 4.3 53 Computational Results and Discussion 0.4 x/c 0.2 0 -0.2 0 0.5 y/c 1 Figure 4.2: Grid with 512 × 128 cells for computation of ﬂow over airfoil A (only a cutout of the grid in the vicinity of the airfoil is shown). 4.3 Computational Results and Discussion To mimic the transition in the computation, the turbulence model was activated only downstream of x/c = 0.12 on the upper side and downstream of x/c = 0.3 on the lower side as well as in the wake. (This appeared to be common practice in all publications found where this ﬂow case was computed.) 4.3.1 Topology of the Velocity Field The topology of the computed ﬂow ﬁeld is shown in Figure 4.3. A very thin laminar separation bubble and a small, single recirculation zone close to the trailing edge are predicted by the code. It is noted that the streamlines shown in Figure 4.3 are iso-contour lines of the streamfunction ψ, which was evaluated from the total diﬀerential dψ = with ∂ψ ∂ψ dx + dy ∂x ∂y ∂ψ = −ρv, ∂x (4.1) ∂ψ = ρu. ∂y Setting ψ = 0 at the airfoil surface as the initial condition for the integration of Equation 4.1 leads to the straightforward deﬁnition of the dividing streamline enclosing the recirculation zone (denoted b1 in Figure 2.1): Since 54 y/c 4 Demonstration 0.087 y/c 0.02 0.086 0 -0.02 0.085 0.115 0.118 0.121 x/c 0.9 0.95 1 x/c y/c 0.1 0 -0.1 0 0.2 0.4 x/c 0.6 0.8 1 Figure 4.3: Computed topology of ﬂow around airfoil A (JohnsonKing model). a dividing streamline is identical to the separating streamline, it “carries” the same value of ψ as the surface, namely ψ = 0. An additional advantage of representing streamlines as iso-contour lines of ψ is the inherent avoidance of spiraling streamlines inside two-dimensional recirculation zones. This is called a repelling or attracting focus and is not possible in two dimensions without sources or sinks of ﬂuid. In contrast, the common method to compute streamlines, which is also used in three-dimensional ﬂows, is to integrate the path of imaginary particles “released” in the velocity vector ﬁeld. However, this method can lead to a summation of the numerical errors along the path of integration and, frequently, results in weakly-spiraling streamlines even in purely two-dimensional ﬂows. In Figure 4.4, dividing streamlines of the rear recirculation zone obtained from experimental data (experiment F2) and from the computation with the Johnson-King model are compared. It can be seen that the recirculation zone obtained from the Johnson-King computation is much smaller than the one evaluated from measured data. Although the “grid” of the experimental data is very coarse compared to the computational grid and, hence, the integration of Equation 4.1 based on the measured data is much more inaccurate, the 4.3 55 Computational Results and Discussion 0.04 S1exp experiment S1comp Johnson-King y/c 0.02 0 trailing edge -0.02 -0.04 A2exp 0.85 0.9 x/c 0.95 A2comp 1 Figure 4.4: Comparison of recirculation zones for airfoil A (dashed lines: experiment F2; solid lines: computed with JohnsonKing model). large diﬀerences in the size of the recirculation zone cannot be attributed to inaccuracies in the determination of the experimental streamlines alone. Besides possible errors produced solely by the turbulence model, weak threedimensional eﬀects are believed to be present in this region which add to discrepancies between the computed and measured solution. In order to get some basic information about possible three-dimensional eﬀects, the experimental streamlines computed from imaginary particle traces and the iso-contour line ψ = 0 are compared in Figure 4.5. The separating streamlines coincide well up to x/c ≈ 0.98 for the two methods. Downstream of x/c ≈ 0.98, however, large diﬀerences in the patterns of the streamlines are encountered. On the one hand, this is attributed to the discussed inaccuracies in the evaluation of streamlines. But on the other hand, the pronounced spiraling motion of the particle traces is taken as a strong indication of threedimensional eﬀects in the experiment which are not accounted for in the computation. 4.3.2 Pressure and Skin-Friction Distributions In Figure 4.6, the computed pressure distribution is compared with the measured values. The overall agreement is fair. Signiﬁcant diﬀerences are encountered only downstream of x/c ≈ 0.8 where the larger separation region found in the experiment leads to a more pronounced plateau in the experimental pressure distribution. The small transitional separation bubble, clearly recog- 56 4 Demonstration 0.125 0.1 S1 0.075 y/c 0.05 0.025 0 trailing edge -0.025 -0.05 ψ=0 A2 -0.075 0.8 0.85 0.9 x/c 0.95 1 1.05 Figure 4.5: Streamlines constructed from experimental data for ﬂow around airfoil A; solid, thin lines: particle traces; dashed, thick lines: ψ = 0. nizable in the computed pressure distribution at x/c ≈ 0.12, is not visible in the experimental data due to the inherently coarse positioning of the pressure oriﬁces at the airfoil surface. The corresponding distributions of the skin-friction coeﬃcients on the upper side of the airfoil are shown in Figure 4.7. Note that the oscillations in the computed skin friction at the trailing edge are mainly due to inaccuracies of the numerical scheme. This will be discussed in more detail in Section 8. A relatively large negative peak at x/c ≈ 0.12 and a subsequent steep rise of the skin friction indicate the transitional separation bubble and the onset of the turbulent boundary layer, respectively. Computed cf values compare well with measured data in the aft part of the airfoil (x/c > 0.8). This is somewhat surprising since the computed pressure distribution (Figure 4.6) and the size of the recirculation zone (Figure 4.4) found in the computation do not agree well with the corresponding experimental data in this region. In particular, the separation point S1 estimated from the iso-contour line ψ = 0 is located at x/c ≈ 0.825 in the experiment while in the computed solution it is found to be at x/c ≈ 0.90 (Figure 4.4). Note that there are also inconsistencies within the measured data itself: The position of the separation S1 obtained from the skin-friction distribution (x/c ≈ 0.87) and the one obtained from ψ = 0 (x/c ≈ 0.825) diﬀer by approximately 0.045 4.3 57 Computational Results and Discussion -4 -3 Experiment Johnson-King cp -2 -1 S1 0 1 0 0.25 0.5 x/c 0.75 1 Figure 4.6: Pressure distribution on surface of airfoil A; S1 relates to cf = 0 inferred from experiment F2, see Figure 4.7. chord lengths, although both are evaluated using data measured in the same wind tunnel (F2). These deviations are believed to result from measurement uncertainties. In particular, even small uncertainties in the cf distribution favor large diﬀerences in the location of the separation point cf = 0 due to the small slope of the cf distribution in this region. In fact, measurements of cf are subject to higher measurement uncertainties than measurements of the velocity. Therefore, the separation position x/c ≈ 0.825 is taken to be the more accurate one. In addition, looking at the velocity proﬁles in Figure 4.11, it is evident that the separation point must be located upstream of x/c ≈ 0.87 since x/c ≈ 0.87 is the ﬁrst downstream position where backﬂow in the boundary layer, i.e. negative tangential velocity u, is encountered. The global lift and drag coeﬃcients measured in wind tunnel F1 and F2, as well as the ones computed with the Johnson-King model, are listed in Table 4.1. Since no experimental data regarding the moment coeﬃcient cM are available, only the computed cM is reported in the table. The good agreement of the computed lift coeﬃcient with that measured in wind tunnel F2 is consistent with the close agreement of the corresponding pressure distributions (Figure 4.6). 58 4 Demonstration 0.015 Experiment Johnson-King cf 0.01 0.005 S1ψ S1cf 0 0 0.25 0.5 x/c 0.75 1 Figure 4.7: Skin-friction distribution on upper side of airfoil A; S1ψ denotes the position of separation evaluated from the streamfunction ψ, and S1cf denotes the position of separation evaluated from the skin-friction distribution (cf = 0). Table 4.1: Force and moment coeﬃcients for Airfoil A experiment F1 experiment F2 component total total Johnson-King computation total pressure friction cL 1.56 1.52 1.53 1.531 −0.001 cD 0.021 0.031 0.026 0.020 0.006 cM − − 0.0055 0.0050 0.0005 The value of cD measured in wind tunnel F2 is larger than the one obtained in wind tunnel F1 indicating that the recirculation zone at the trailing edge is also larger in the wind tunnel F2. This is also supported by comparing the cf distributions from the two experiments: The separation point S1 (cf = 0) in experiment F1 is located further downstream than in case F2 (Figure 4.8). Since the drag coeﬃcient for this separated ﬂow case is mainly due to pressure drag (see Table 4.1) the size of the recirculation zone at the 4.3 59 Computational Results and Discussion trailing edge greatly inﬂuences cD . Hence, the smaller recirculation zone in the experiment F1 leads to a signiﬁcantly lower drag coeﬃcient compared to the value from experiment F2. Signiﬁcant diﬀerences exist in the computed and measured drag coeﬃcients with the computed value being approximately 16 percent lower than the one obtained from the experiment F2 and approximately 24 percent larger than cD from F1. This can again be explained with the diﬀerent sizes of the trailing edge recirculation zone: The computed length of the separated region, and hence the size of the recirculation zone, is larger than the one found in experiment F1 but shorter than in experiment F2. The separation location in experiment F2 was discussed above by means of the streamline topology; the separation point was found to be at 0.825 ≤ x/c ≤ 0.87 (Figure 4.4). The separation location S1 (cf = 0) for case F1, on the other hand, can be inferred from Figure 4.8. Using a best-ﬁt polynomial representation of the measured values for this purpose yields cf = 0 at x/c ≈ 0.94. Hence the position of the separation point obtained from computational results is in between the positions of the two separation points evaluated from experimental data. 0.002 Experiment F1 Experiment F2 cf 0.001 S1F2 S1F1 0 -0.001 0.6 0.7 0.8 x/c 0.9 1 Figure 4.8: Skin-friction distributions from experiment F1 and F2 for airfoil A. 60 4 Demonstration The reattachment point A2 could not be evaluated from the experimental cf distribution because no change of sign from negative to positive values of cf is seen in the experimental data. The reason for this is that the spacing of the skin-friction measurements was too coarse to suﬃciently resolve the cf distribution in the immediate vicinity of the trailing edge in order to capture the change of sign. 4.3.3 Boundary-Layer Proﬁles Boundary-layer proﬁles at diﬀerent downstream positions are investigated in order to compare the computed and measured ﬂow data in more detail. The goal is to gain a deeper insight into the deviations already encountered between the computed and measured results. Since no measurements were taken on the lower side of the airfoil, only proﬁles on the upper side can be compared. The positions of the proﬁles on the airfoil are shown in Figure 4.9. The local coordinate systems are oriented such that the coordinate axes are parallel and normal, respectively, to the surface of the airfoil at the indicated downstream position x/c. For this purpose, a post-processing tool called newmono was developed that performs the appropriate coordinate transformations of the velocity vectors and the stress tensors of the computed ﬂow solution. See Section D in the appendices for a short description of newmono. 0.87 0.9 0.93 0.96 0.99 0.775 0.825 0.7 0.6 y/c 0.1 0. 5 0. 4 0.3 0.2 0 -0.1 0.2 0.4 0.6 x/c 0.8 1 Figure 4.9: Locations of measurement stations at the upper surface of airfoil A. The proﬁles of the normalized tangential velocity component u/U∞ are plotted in Figures 4.10, 4.11 and 4.12. In the ﬁgures, z/c denotes the wallnormal coordinate with z/c = 0 at the wall. Note, while in Figure 4.10 all 4.3 61 Computational Results and Discussion proﬁles have the same origin, in Figure 4.11 the proﬁles for x/c = 0.825 and x/c = 0.87 are shifted by constant values ∆u = 0.6 and ∆u = 0.3, respectively. Similarly, in Figure 4.12, the proﬁles for x/c = 0.93 and x/c = 0.96 are shiftedb y ∆u = 0.6 and ∆u = 0.3, respectively. This is done for better recognizability while still being able to show several proﬁles in the same diagram. 1.6 0.3 1.4 0.4 0.5 1.2 0.6 0.7 u/U∞ 1 0.775 0.8 0.6 0.4 Experiment Johnson-King 0.2 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 z/c Figure 4.10: Velocity proﬁles for airfoil A at x/c = 0.3, x/c = 0.4, x/c = 0.5, x/c = 0.6, x/c = 0.7, and x/c = 0.775. Up to the position x/c = 0.5 the computed proﬁles match the measured data very well. Downstream of x/c = 0.5 the computed proﬁles do not appropriately adjust to the pressure gradient; they are “fuller” in shape than the measured proﬁles. Due to the fuller velocity proﬁles of the computed ﬂow solution, more momentum is transported normal to the wall leading to a separation location that is located more downstream than in the experiment. As a consequence, the recirculation zone predicted is shorter and the backﬂow region (u < 0) does not extend as far away from the wall as in the experimental data. The discrepancies in the velocity proﬁles increase considerably after x/c = 0.6. This increase is far too large to be attributed only to inaccuracies of the 62 4 Demonstration 1.6 0.825 (shifted by ∆u=0.6) 1.4 1.2 0.87 (shifted by ∆u=0.3) u/U∞ 1 0.9 0.8 0.6 0.4 Experiment Johnson-King 0.2 0 0 0.01 0.02 0.03 0.04 z/c 0.05 0.06 0.07 Figure 4.11: Velocity proﬁles for airfoil A at x/c = 0.825, x/c = 0.87, and x/c = 0.9 (the graphs for x/c = 0.825 and x/c = 0.87 are shifted by ∆u = +0.6 and ∆u = +0.3, respectively). turbulence model. Hence, it corresponds to loss of momentum and mass ﬂux in the experiment evidently due to spanwise outﬂow. Regarding the proﬁles of Reynolds shear stresses (Figures 4.13 to 4.15), it is noted that Reynolds shear stresses are subject to much larger measurement uncertainties than mean velocities. With this in mind, one can say that the proﬁles of the computed Reynolds shear stresses match the measured ones rather well up to a downstream position of x/c = 0.6. At x/c = 0.7 large diﬀerences between the computed and measured values arise in the outer part of the boundary layer (Figure 4.13 d)). Especially the kink found in the measured proﬁle at a wall distance 0.025 ≤ z/c ≤ 0.035 is not reproduced at all by the computation. This kink is also visible in the experimental proﬁles further downstream, its position moving further away from the wall with increasing boundary-layer thickness. So, it is not believed to be an artifact due to measurement errors. Looking at the proﬁles further downstream reveals that the agreement between the measured and computed Reynolds shear stresses becomes successively poorer when approaching the trailing edge (Figures 4.14 and 4.15). 4.3 63 Computational Results and Discussion 1.6 0.93 (shifted by ∆u=0.6) 1.4 1.2 0.96 (shifted by ∆u=0.3) 1 0.8 u/U∞ 0.99 0.6 0.4 Experiment Johnson-King 0.2 0 0 0.02 0.04 0.06 z/c 0.08 0.1 Figure 4.12: Velocity proﬁles for airfoil A at x/c = 0.93, x/c = 0.96, and x/c = 0.99 (the graphs for x/c = 0.93 and x/c = 0.96 are shifted by ∆u = +0.6 and ∆u = +0.3, respectively). The maximum level of the Reynolds stress is underpredicted by the model for x/c ≥ 0.825 with an increasing deviation from the experimental values when approaching the trailing edge. In contrast, the Reynolds-stress level inside the recirculation zones is clearly overpredicted by the computation. In the experiment, the Reynolds shear stress is negligible up to a wall distance of z/c = 0.015, i.e. up to the center of the recirculation zone (Figure 4.15 d)). The turbulence model, however, predicts signiﬁcant Reynolds shear stress in this region. This ﬁnding led to a numerical experiment, i.e. a “topological modiﬁcation” of the turbulence model which is discussed in the next section. 0.004 2 2 0.003 0.004 0 0.001 0.002 0.003 -<u’v’>/U∞ z/c 0.015 0.02 0.025 Experiment Johnson-King 0.015 0.002 0.003 0.004 0 0 0 0.01 0.02 0.005 z/c 0.03 0.01 0.04 0.06 Experiment Johnson-King 0.015 Experiment Johnson-King 0.05 d) x/c=0.7 z/c b) x/c=0.5 Figure 4.13: Reynolds-shear-stress proﬁles for airfoil A at x/c = 0.4 (a), x/c = 0.5 (b), x/c = 0.6 (c), and x/c = 0.7 (d). 0 0.01 c) x/c=0.6 z/c 0.01 0.001 0.002 0.003 0.004 0.005 0 0.005 0.005 Experiment Johnson-King 0.001 0 0 a) x/c=0.4 0.001 0.002 -<u’v’>/U∞ 2 -<u’v’>/U∞ 2 -<u’v’>/U∞ 0.005 64 4 Demonstration 0.004 2 2 0.004 0.005 0 0.001 0.002 0.003 -<u’v’>/U∞ 0.04 z/c 0.04 0.05 0.06 0.08 Experiment Johnson-King 0.06 c) x/c=0.87 z/c 0.03 0.002 0.003 0.004 0.005 0.006 0 0 0 0.01 0.02 0.02 z/c 0.04 0.04 0.06 0.06 0.08 Experiment Johnson-King 0.05 d) x/c=0.9 z/c 0.03 Experiment Johnson-King b) x/c=0.825 Computational Results and Discussion Figure 4.14: Reynolds-shear-stress proﬁles for airfoil A at x/c = 0.775 (a), x/c = 0.825 (b), x/c = 0.87 (c), and x/c = 0.9 (d). 0 0.02 0.02 0 0.01 0.001 0.002 0.003 0.004 0.001 0 0 Experiment Johnson-King a) x/c=0.775 0.001 0.002 0.003 -<u’v’>/U∞ 2 -<u’v’>/U∞ 2 -<u’v’>/U∞ 4.3 65 0.005 2 2 0.005 0.006 0 0.001 0.002 0.003 0.004 -<u’v’>/U∞ 0 0.001 0.002 0.003 0.004 -<u’v’>/U∞ 0 0 0.04 0.04 0.06 z/c 0.06 0.08 0.1 Experiment Johnson-King 0.08 c) x/c=0.99 z/c Experiment Johnson-King 0 0.001 0.002 0.003 0.004 0.005 0 0.001 0.002 0.003 0.004 0.005 0.006 0 0 0.01 Experiment Johnson-King z/c 0.04 z/c 0.08 Experiment 0.06 Experiment Johnson-King position of separating streamline b1 0.02 0.03 Johnson-King d) x/c=0.99 0.02 b) x/c=0.96 Figure 4.15: Reynolds-shear-stress proﬁles for airfoil A at x/c = 0.93 (a), x/c = 0.96 (b), x/c = 0.99. (d) shows cut-out of (c) in recirculation zone. 0.02 area shown in figure d) 0.02 a) x/c=0.93 2 -<u’v’>/U∞ 2 -<u’v’>/U∞ 0.006 66 4 Demonstration 4.3 67 Computational Results and Discussion 4.3.4 Numerical Experiment in the Recirculation Zone The original Equation 3.10 for the eddy viscosity was modiﬁed to yield a damping of the eﬀective eddy viscosity in the recirculation zone. This was accomplished by scaling the eddy viscosity with an appropriate non-dimensional function which depends on the streamfunction. min(ψ, 0) ζ (−µti /µto ) , µ t = µ to 1 − e (4.2) 1− ψmin ζ = 0.1 − 0.3. Since the streamfunction is negative in the recirculation zone with its minimum value at the focus F1, Equation 4.2 leads to a modiﬁcation of the eddy viscosity only inside the recirculation zone. The modeling parameter ζ controls the functional behavior close to the edges of the recirculation zone. A low value for ζ, for example ζ = 0.1, gives an abrupt onset of the damping when ψ changes from positive to negative values, i.e. when “entering” the recirculation zone. Increasing ζ results in a less abrupt onset of the damping. Results Iso-contours of the Reynolds shear stress obtained with the original model cross the separating streamline without being aﬀected by the recirculation zone (Figure 4.16). In contrast, employing Equation 4.2 for the y/c 0.02 0 -0.02 0.9 x/c 0.95 1 Figure 4.16: Iso-contour lines of Reynolds shear stress for the standard Johnson-King model near the trailing edge of airfoil A on its upper side. computation of the eddy viscosity yields iso-contour lines that end abruptly at the separating streamline (Figure 4.17). The Reynolds shear stress obtained with the modiﬁed model is nearly zero inside the recirculation zone and it 68 4 Demonstration changes quickly to a ﬁnite value when crossing the separating streamline, see 2 Figure 4.18. Outside of the recirculation zone the value of (−u v )/U∞ computed with the modiﬁed model is very close to the value obtained with the standard model formulation (Figure 4.18). y/c 0.02 0 -0.02 0.9 x/c 0.95 1 Figure 4.17: Iso-contour lines of Reynolds shear stress for the topologically-modiﬁed Johnson-King model near the trailing edge of airfoil A on its upper side. 0.005 Experiment Johnson-King modified Johnson-King original 2 -<u’v’>/U∞ 0.004 0.003 Experiment Johnson-King modified 0.002 position of separating streamline b1 0.001 0 0 0.01 0.02 z/c 0.03 Figure 4.18: Reynolds-shear-stress proﬁles for airfoil A at x/c = 0.99 with and without modiﬁcation of the Johnson-King model. 4.3 69 Computational Results and Discussion However, the modiﬁcation of the Johnson-King model has a very small eﬀect on the velocity distribution (Figure 4.19). Similarly, the inﬂuence on the global force coeﬃcients and the position of separation S1 is virtually non-existent (Table 4.2). 1 0.8 u/U∞ 0.6 0.4 Experiment Johnson-King modified Johnson-King original 0.2 0 0 0.02 0.04 0.06 z/c 0.08 0.1 0.12 Figure 4.19: Velocity proﬁles for airfoil A at x/c = 0.99 with and without modiﬁcation of the Johnson-King model. Table 4.2: Eﬀects of the topological modiﬁcation of the model on the global force coeﬃcients for airfoil A Johnson-King model standard cL 1.5324 modiﬁed 1.5324 cD 0.025995 0.025996 cM 0.0055042 0.0055045 separation location (x/c) 0.898898 0.898898 The results obtained with the topologically modiﬁed Johnson-King model show that the deviations between the computational and experimental results encountered in the preceding sections are not mainly due to inaccurate 70 4 Demonstration turbulence modeling inside the recirculation zone. In fact, the correct response of the turbulence model to the adverse pressure gradient upstream of the separation point S1 is the key to accurately predicting the boundary-layer development and the separation point. On the one hand, this seems to be obvious since a boundary layer is governed by parabolic diﬀerential equations saying that the state of a boundary layer is inﬂuenced only by the upstream region and the local pressure gradient. On the other hand, the considered ﬂow case is an airfoil ﬂow where small changes at the trailing edge can have strong inﬂuence on the overall circulation. Hence, changes at the trailing edge can easily aﬀect the boundary-layer development further upstream by modifying the pressure distribution via the circulation. Bearing these considerations in mind, it was decided to perform the envisaged comparative study of turbulence models for separating ﬂows by means of ﬂow cases where the pressure distribution is held “ﬁxed”. This ensures that the pressure gradient is basically the same for each model and is not sensitive to the computed ﬂow solution which may change with the employed model. Thus, an objective study of the eﬀect of pressure gradient on the turbulence models is possible. 4.3.5 Comments Regarding Hidden Three-Dimensional Eﬀects in Nominally Two-Dimensional Flows Würz & Althaus (1995) investigated various cases of separated ﬂows around airfoil FX 63-137 close to, and at, stall conditions. In particular, they varied the aspect ratio of the wing section and the Reynolds number to study the eﬀect on the trailing-edge separation. Oil-ﬂow patterns on the model’s surface were evaluated to visualize wall streamlines and discuss three-dimensional eﬀects. A standard scenario in their investigations comprised a wing aspect ratio of Λ = 1.46 and a Reynolds number of Re = 1·106 . For these parameters, the separation line was a straight line in the spanwise direction indicating a basically two-dimensional separation zone. Increasing the aspect ratio of the wing section resulted in a wavy shape of the separation line and triggered the formation of spanwise cellular vortex patterns leading to a three-dimensional ﬂow topology. For example, Würz and Althaus found two spanwise vortex cells in a ﬂow past a wing section with Λ = 2.92 and Re = 1 · 106 (Figure 4.20). The formation of such cellular vortex patterns was also discussed by Weihs & Katz (1983). 4.3 Computational Results and Discussion 71 Figure 4.20: Two vortex cells at airfoil FX 63-137 for Λ = 2.92 and Re = 1 · 106 (Würz & Althaus, 1995, the picture is courtesy of Werner Würz, IAG, University of Stuttgart). For the Aerospatiale-A airfoil, the aspect ratio of the wing in wind tunnel F1 was Λ = 2.5; in wind tunnel F2 it was Λ = 2.33. Both values are just slightly lower than Λ = 2.92 in Würz & Althaus (1995). According to the results of Würz and Althaus this favors the formation of three-dimensional vortex cells or, at least, a wavy shape of the separation line in the spanwise direction. In light of this discussion, the spiraling streamlines found at the trailing edge of airfoil A (Figure 4.5) strongly suggest that three-dimensional eﬀects are, in fact, present in this ﬂow case. However, due to the much smaller separation region at airfoil A, compared to the separation regions found in the ﬂows past airfoil FX 63-137 investigated by Würz and Althaus, the three-dimensionality was probably less pronounced for our case. Note, that these conjectures are in contradiction to the statement “... the ﬂow was two-dimensional up to an incidence of 13◦ ...” which is made in the discussion of the experimental results for the ﬂow over airfoil A in Chaput (1997). Possibly, the measurements of the ﬂow over airfoil A were taken close to a center plane of a vortex cell which would reduce three-dimensional eﬀects in the measured data. Nevertheless, it remains an open question what solution can be expected of a purely two-dimensional computation and whether it is legit- 72 4 Demonstration imate to expect that such a computation can reproduce the ﬂow ﬁeld found in the symmetry plane of a three-dimensional vortex system. This question can only be answered by comparing two- and three-dimensional computations with each other and with experimental data. For this purpose, it is mandatory that the experimental data comprise accurate information about the three-dimensionality of the ﬂow ﬁeld. Part II Analysis of Modern Turbulence Models 74 5 5 Numerical Method (II) Numerical Method (II) The ﬂow computations for the comparative study of the turbulence models were performed using the FLOWer code of the MEGAFLOW software system. This software system has been, and is still, developed at the Deutsches Zentrum für Luft- und Raumfahrt (DLR) in Braunschweig, Germany, and is becoming a standard CFD tool in the European aircraft industry. FLOWer solves the three-dimensional unsteady and compressible Reynolds-averaged Navier-Stokes equations in integral form. For this purpose, a cell vertex or, optionally, a cell-centered ﬁnite-volume formulation on block-structured grids is utilized. On user input, either a second-order central scheme or one of the various available ﬂux-diﬀerence or ﬂux-vector upwind schemes is applied for the space discretization of the convective ﬂuxes. In either case, the diﬀusive ﬂuxes are centrally discretized. The central discretization of the convective ﬂuxes is augmented by a blend of second- and fourth-order artiﬁcial damping terms in order to prevent odd-even decoupling, damp spurious oscillations and allow for sharp shock resolution. For the time integration, an explicit ﬁve-stage Runge-Kutta scheme with optimized damping properties for multigrid is employed. Convergence to steady state is accelerated by means of local time stepping, implicit residual smoothing, simple or full multigrid and local preconditioning for low Mach numbers. For the computation of unsteady ﬂows a dual-time-stepping procedure is available. The eﬀect of turbulence on the mean ﬂow is modeled by algebraic or transport-equation turbulence models. In the transport-equation models the convective terms are discretized using either a ﬁrst-order upwind scheme or a second-order central scheme with artiﬁcial damping. The time integration of the turbulence equations is performed explicitly, according to the solution method of the mean ﬂow equations, or implicitly. In the latter case, two diﬀerent point-implicit schemes, where only the source terms are treated implicitly, or a line-implicit and a fully implicit treatment are available. All computations performed for this work were done using the central space discretization for the convective ﬂuxes of the mean ﬂow and the ﬁrst-order Roe type upwind scheme for the convective terms in the turbulence-transport equations. Additionally, a point-implicit treatment of the turbulence-transport equations was selected. A detailed presentation of the numerical algorithms utilized in FLOWer can be found in Kroll et al. (1995) and Aumann et al. (2000); a more general discussion of the MEGAFLOW project is available in Kroll et al. (2000). 75 6 Models Investigated Transport-equation models, in general, have been developed to account for non-local and non-equilibrium eﬀects, also called ﬂow-history eﬀects. There are two main types of transport-equation models, one in which diﬀerential equations are solved for the transport of the Reynolds stresses and one in which the transport equations yield the eddy viscosity. Stress-transport models are not in very common use. In the present thesis we therefore focus on eddy-viscosity-transport models. (For the sake of brevity, we will also denote “eddy-viscosity-transport models” simply by “transport-equation models”.) These models enjoy large popularity in today’s computational ﬂuid dynamics (CFD) applications. In this work, two diﬀerent categories of eddy-viscosity-transport models were investigated. The ﬁrst category comprises two-equation models which utilize the square root of the turbulent kinetic energy k as a velocity scale of turbulence while the speciﬁc dissipation rate ω is used as a reciprocal time scale. A partial diﬀerential transport equation must be solved for both k and ω. The eddy viscosity is a linear function of these two parameters. The other category consists of one-equation models that solve a partial diﬀerential transport equation for the eddy viscosity itself. For the comparative study, a total of eleven diﬀerent eddy-viscosity turbulence models from three diﬀerent model classes were considered. They range from the algebraic Baldwin-Lomax model over various one- and two-equation models to a non-linear, explicit algebraic Reynolds-stress model. The model of Baldwin & Lomax was already presented in detail in Section 3.2 and no description will be repeated in this section. The Johnson-King model in its original formulation is conﬁned to two-dimensional applications. Although extensions to three-dimensional ﬂows exist (see Abid et al., 1989) the resulting performance does not reach the level of the two-dimensional version (Haase, 1997). Consequently, an implementation of the Johnson-King model into the three-dimensional FLOWer code was not performed in the MEGAFLOW project nor in the present work. Since most of the turbulence models employed in this work are discussed in great detail in the referenced literature only short descriptions, basically consisting of the important equations, are presented in this section. The models investigated are listed in Table 6.1. 76 6 Models Investigated Table 6.1: Turbulence models investigated in this work 6.1 Model Developer(s) / Reference Baldwin-Lomax Baldwin & Lomax (1978) see Subsection Johnson-King Johnson & King (1985) k, ω 1988 Wilcox (1988) 6.1 k, ω 1998 Wilcox (1998) 6.1 k, ω SST Menter (1993) 6.2 k, ω TNT Kok (2000) 6.3 k, ω LLR Rung & Thiele (1996) 6.4 EARSM of Wallin Wallin & Johansson (2000) 6.5 Spalart-Allmaras Spalart & Allmaras (1992) 6.7 3.2 3.3, B Edwards-Chandra Edwards & Chandra (1996) 6.8 SALSA Rung et al. (2003) 6.9 k, ω SST modiﬁed Celić 7.1.3 The k, ω Models of Wilcox Wilcox developed two diﬀerent versions of a k, ω type of model. The ﬁrst model (Wilcox, 1988) proved to be superior to the k, models of Jones & Launder (1972) or Launder & Sharma (1974) for boundary-layer ﬂows under adverse pressure gradients. In these ﬂows, k, models typically produce too high levels of turbulence. Since the 1988 model of Wilcox does not yield the correct spreading rate of free shear layers in all cases Wilcox reﬁned his model in 1998 to further improve the predictive accuracy in such situations (Wilcox, 1998). Unlike turbulence models based on the transport equations for k and , the k, ω models of Wilcox do not require viscous damping, that is, low-Reynoldsnumber modiﬁcations, to predict a realistic value of the additive constant in the law of the wall. Nevertheless, Wilcox suggested low-Reynolds-number modiﬁcations for his models to enhance the models’ solutions for k in the viscous sublayer. An additional and intended outcome of the low-Reynoldsnumber modiﬁcations devised by Wilcox is the possibility to predict laminarturbulent transition. This desirable feature, however, is restricted to ﬂat-plate boundary layers since the modiﬁcations were validated based on the minimal critical Reynolds number obtained from linear stability theory in the Blasius boundary layer. The models are unable to predict transition in general ﬂows 6.1 77 The k, ω Models of Wilcox which is no surprise if one bears in mind that transition is a very complicated and still not fully-understood unsteady physical process. Since only the 1988 k, ω model of Wilcox was available in FLOWer, the 1998 model and the low-Reynolds-number extensions for both models were implemented into the FLOWer code during this work. Before presenting the models’ equations, it is noted that from the deﬁnition of the turbulent kinetic energy, k= 1 1 u u + v v + w w = ui ui , 2 2 it follows that the trace of the Reynolds-stress tensor is −2ρk. In order to insure this condition in the framework of the applied two-equation models, the constitutive relation, Equation 3.5, must be appropriately extended: 2 1 −ρui uj ≡ τ = 2µt S − (∇ · v) · I − ρk · I. 3 3 Although the model equations were discretized and solved in integral form they are presented for simplicity and space reasons in the usual diﬀerential form and in index notation. In some situations, the equivalent tensor notation is also shown in order to provide the link to equations which were above introduced in tensor notation. Both model versions of Wilcox are based on the same transport equations for k and ω. The models’ equations read as follows: Eddy Viscosity: k µt = ρ . (6.1) ω Turbulent Kinetic Energy: ∂ρk ∂ui ∂ ∂k ∂ρkuj = −ρui uj − β ∗ ρkω + + (µ + σ ∗ µt ) . (6.2) ∂t ∂xj ∂xj ∂xj ∂xj Speciﬁc Dissipation Rate: ω ∂ui ∂ ∂ω ∂ρωuj ∂ρω = α (−ρui uj ) − βρω 2 + + (µ + σµt ) . ∂t ∂xj k ∂xj ∂xj ∂xj However, the two models apply diﬀerent closure coeﬃcients and auxiliary functions. 1988 Closure Coeﬃcients: α= 5 , 9 β= 3 , 40 β∗ = 9 , 100 σ= 1 , 2 σ∗ = 1 . 2 78 6 Models Investigated 1998 Closure Coeﬃcients: α= 13 , 25 9 , 100 σ= 1 , 2 σ∗ = 1 . 2 Ωij Ωjk Ski 1 + 70χω , , χω = 1 + 80χω (β0∗ ω)3 1 ∇ ⊗ v − (∇ ⊗ v)T , Ωij ≡ Ω = 2 1, χk ≤ 0 1 ∂ω ∂k . , χk = 3 fβ∗ = 1+680χ2 k ω ∂xj ∂xj , χ > 0 k 1+400χ2 β0 = β0∗ = β ∗ = β0∗ fβ ∗ , β = β0 fβ , 9 , 125 fβ = (6.3) (6.4) k One can see from the above equations that the two models of Wilcox diﬀer in the destruction terms. The destruction term in the equation for ω is altered in the 1998 version only for three-dimensional ﬂows since the product Ωij Ωjk Ski in Equation 6.3 is zero in two dimensions. In addition, the crossdiﬀusion parameter χk is only active in the outer part of turbulent shear layers. Hence, for two-dimensional boundary-layer ﬂows the two models are expected to yield very similar results. For the low-Reynolds-number versions of the two models Wilcox suggested the following modiﬁcations of the eddy viscosity and the closure coeﬃcients: k µt = α ∗ . ω 1988 Low-Reynolds-Number Coeﬃcients: α∗ = α= β∗ = α0∗ + Ret /Rek , 1 + Ret /Rek 5 α0 + Ret /Reω 1 · , · 9 1 + Ret /Reω α∗ 5/18 + (Ret /Reβ )4 9 , · 100 1 + (Ret /Reβ )4 1 β 1 3 , σ ∗ = σ = , α0∗ = , α0 = , 40 2 3 10 k , Reβ = 8, Rek = 6, Reω = 2.7. Ret = ων 1998 Low-Reynolds-Number Coeﬃcients: β= α∗ = α0∗ + Ret /Rek , 1 + Ret /Rek 6.2 79 The k, ω Shear-Stress Transport (SST) Model of Menter α= β∗ = β= 9 fβ , 125 Ret = k , ων 13 α0 + Ret /Reω 1 · , · 25 1 + Ret /Reω α∗ 4/15 + (Ret /Reβ )4 9 · fβ ∗ , · 100 1 + (Ret /Reβ )4 σ∗ = σ = Reβ = 8, 1 , 2 α0∗ = Rek = 6, β0 , 3 α0 = 1 , 9 Reω = 2.95. Ret is frequently referred to as the turbulent Reynolds number. 6.2 The k, ω Shear-Stress Transport (SST) Model of Menter Menter (1993) modiﬁed the 1988 model of Wilcox in two ways. First, in order to remove the model’s sensitivity to the free-stream value of ω, Menter incorporated a blending function that converts the k, ω model to the k, model of Jones & Launder in the outer part of a boundary layer and in the free stream. This is plausible since the k, model shows almost no sensitivity to the free-stream value of . Since the k, ω model is employed for the largest part of the boundary layer the resulting model retains the favorable features of the k, ω model for boundary-layer ﬂows. The change between the two models is accomplished with the help of a blending function which turns on a so-called cross-diﬀusion term in the ω equation of the k, ω model in the outer part of the boundary layer. This cross-diﬀusion term renders the k, ω model a k, model. It is obtained when re-writing the equation in the k, model in terms of ω. Additionally, the closure coeﬃcients of the k, ω model are converted to the appropriate values of the k, model also by employing the blending function. The second modiﬁcation applied by Menter is based on Townsends’ and Bradshaw’s assumption that the Reynolds shear stress in a boundary layer is proportional to the turbulent kinetic energy, τxy = ρa1 k. This concept has already been introduced in the discussion of the Johnson-King model in Section 3.3. Menter uses this assumption to limit the eddy viscosity in regions where production of k exceeds dissipation of k which would normally lead to too high levels of µt . The model equations read in detail: Eddy Viscosity: a1 ρk µt = , max (a1 ω; ΩF2 ) 80 6 Models Investigated F2 = tanh (arg2 ) , arg2 = max √ 2 k 500ν ; . 0.09ωd d2 ω d is the wall-normal distance and Ω denotes the absolute value of the vorticity. Turbulent Kinetic Energy: ∂ρk ∂ui ∂ ∂k ∂ρkuj = −ρui uj − β ∗ ρkω + + (µ + σ ∗ µt ) . ∂t ∂xj ∂xj ∂xj ∂xj Speciﬁc Dissipation Rate: ρ ∂ui ∂ ∂ω ∂ρω ∂ρωuj = α (−ρui uj ) −βρω 2 + + (µ + σµt ) +(1−F1 )CD . ∂t ∂xj µt ∂xj ∂xj ∂xj CD is the cross-diﬀusion term. It is obtained from the k, model written in k, ω formulation and reads CD = 2ρσ2 1 ∂k ∂ω . ω ∂xk ∂xk Global Coeﬃcients: a1 = 0.31, κ = 0.41, β∗ = 9 . 100 σ1∗ = 0.85, α1 = κ2 β1 √ − σ . 1 β∗ β∗ k, ω Coeﬃcients (set 1): β1 = 3 , 40 σ1 = 1 , 2 k, Coeﬃcients (set 2): β2 = 0.0828, σ2 = 0.856, σ2∗ = 1.0, α1 = κ2 β2 − σ2 √ ∗ . ∗ β β The blending between set 1 and set 2 of the closure coeﬃcients is performed with the following function: φ = F1 φ1 + (1 − F1 ) φ2 . φ1 stands for any of the closure coeﬃcients from set 1 and φ2 , respectively, for closure coeﬃcients from set 2. The blending function F1 is deﬁned as follows: F1 = tanh arg14 , 6.3 81 The Turbulent/Non-turbulent (TNT) k, ω Model of Kok arg1 = min max √ k 500ν ; 0.09ωd d2 4ρσ2 k ; max(CD ; 10−20 ) d2 . Both blending functions F1 and F2 are unity inside the boundary layer and fall to zero approaching the boundary-layer edge. In order to insure the desired behavior of the model it is important that F1 goes to zero well inside the boundary layer, i.e. at about 50 percent of the boundary-layer thickness, while F2 is unity for most of the shear layer. Both functions are zero in free shear layers away from walls. The reader is referred to the original paper of Menter (1993) for a detailed discussion of the model and its blending functions. 6.3 The Turbulent/Non-turbulent (TNT) k, ω Model of Kok The k, ω TNT model was developed by Kok (2000) with a goal similar to that Menter had in mind for the development of the SST model, namely to reduce the dependence of Wilcox’s models on the free-stream value of ω. For this purpose, Kok added the cross-diﬀusion term obtained from the equation re-written in terms of ω to the ω equation in Wilcox’s model. But instead of using a blending function, which requires computation of the wall distance, the cross diﬀusion is taken into account only if it is positive. This makes the use of a blending function superﬂuous. In addition, Kok re-tuned the diﬀusion coeﬃcients of the model in order to correct the model behavior at turbulent/non-turbulent interfaces. The model uses Equation 6.1 for the eddy viscosity and Equation 6.2 for the turbulent kinetic energy. The modiﬁed transport equation for ω is given by: Speciﬁc Dissipation Rate: ∂ρω ω ∂ui ∂ ∂ω ∂ρωuj = α (−ρui uj ) − βρω 2 + + (µ + σµt ) + CD . ∂t ∂xj k ∂xj ∂xj ∂xj The cross-diﬀusion term CD is deﬁned as: ∂k ∂ω ρ CD = σD max ;0 . ω ∂xk ∂xk The following closure coeﬃcients were suggested by Kok: TNT Closure Coeﬃcients: α= 5 , 9 β= 3 , 40 β∗ = 9 , 100 σ= 1 , 2 σ∗ = 2 , 3 σD = 1 . 2 82 6 Models Investigated 6.4 The Local Linear Realizable (LLR) k, ω Model of Rung Another k, ω type of turbulence model which was tested in this work is the LLR model of Rung & Thiele (1996). It is closely related to the realizable k, model developed by Shih et al. (1995). The rationale behind these models is to explicitly secure realizability, i.e. 2 2 2 u2 i ≥ 0 and ui uj ≥ (ui uj ) . Conventional two-equation transport models, for example, can yield u2 i < 0 for large strain rates, which is not physical. In addition to incorporating the realizability constraints in their model, Shih et al. developed a novel transport equation for . Instead of directly deriving a transport equation for they chose to ﬁrst model the exact equation for the mean-square vorticity ﬂuctuation ωi ωi . By multiplying the model equation for ωi ωi by ν and considering that for large Reynolds numbers = ν ωi ωi Shih et al. then obtain a new transport equation for . Rung, in turn, reformulated the model of Shih et al. in terms of ω and applied further modiﬁcations. These mainly comprise low-Reynolds-number damping functions in order to achieve the correct asymptotic near-wall behavior of the turbulent kinetic energy, k ∝ y 2 . The equations of the resulting k, ω LLR model are given in the following: Eddy Viscosity: k µt = cµ ρ . 0.09 ω Turbulent Kinetic Energy: µt ∂k ∂ρk ∂ui ∂ ∂ρkuj µ+ = −ρui uj − βk ρkω + + . ∂t ∂xj ∂xj ∂xj 2 ∂xj Speciﬁc Dissipation Rate: µt ∂ω ω ∂ui ∂ ∂ρω ∂ρωuj µ+ = αω (−ρui uj ) − βω ρω 2 + + . ∂t ∂xj k ∂xj ∂xj 2 ∂xj Auxiliary Relations for the Eddy Viscosity: cµ = fµ c∗µ , c∗µ Rµ = Rt 70 α , fµ = 1/80 + Rµ , 1 + Rµ 1 , 0.12 , = max 0.04, min (b0 + As )Ũ 2 3 Rt Rt k 1 Rt = +2 , , α = + 1.6 3 0.09 ων 2 150 150 6.5 The Explicit Algebraic Reynolds-Stress Model (EARSM) of Wallin 83 √ III 1 181 a0 , b0 = max 48 arccos , S 2 , As = 3 cos 2 S Ũ 140 + 0.09 ω 1 (Ω2 + S 2 ) 2 S Ũ = , a0 = 8.0 − 4.1 tanh , 0.09 ω 1.8 · 0.09 ω S = 2Sij Sij , Ω = 2Ωij Ωij , III = Sij Sjk Ski . Auxiliary Relations for the k-Destruction Term: 0.83/3 + Rk , βk = 0.09 1 + Rk ∗ Rk = A A∗ = tanh 4 Rt 100 2.5 ∗ + (1 − A ) Rt , 100 Rt 100 . Auxiliary Relations for the ω-Production Term: αω = fβ = 5 5 0.09 , (1 − fβ ) + 9 9 c∗µ fβ 1/80 + Rtf , 1 + Rtf Rtf = Rt 10000 2 . Auxiliary Relation for the ω-Destruction Term: 1.83 βω = 0.09 . 1 + µcµ /(µ + µt ) It is noted that the above LLR model equations are not identical to the ones published in Rung & Thiele (1996). Rather, the present equations describe the model version that was implemented by Rung into the FLOWer code. This version of the model yields improved numerical robustness without aﬀecting the underlying physical assumptions that were employed for the model development outlined in Rung & Thiele (1996); Shih et al. (1995). 6.5 The Explicit Algebraic Reynolds-Stress Model (EARSM) of Wallin All models, except the EARSM model of Wallin & Johansson (2000), evaluated in this work belong to the category of linear eddy-viscosity models. 84 6 Models Investigated They rely on a constitutive relation, the well-known Boussinesq approximation, that is linear with respect to the mean strain-rate tensor S. There are many types of applications where this assumption is not a powerful enough model to yield even qualitatively correct results. In ﬂows with strong inﬂuence of streamline curvature, system rotation, pressure gradient or secondary ﬂows of Prandtl’s second kind the Boussinesq approximation can badly fail. A general Reynolds-averaged approach to computing these eﬀects would be based on modeling the full Reynolds-stress transport equations. These Reynoldsstress transport models promise a great potential for high predictive accuracy since they naturally incorporate such complex phenomena as inter-component transfer of Reynolds stresses, non-equilibrium between mean ﬂow and turbulence, anisotropy of the Reynolds-stress tensor and system rotation, to name only a few. Yet, besides questions pertaining to appropriate modeling of unknown terms in the Reynolds-stress transport equations, ﬂow computations with such models bear non-trivial numerical diﬃculties in complex three-dimensional ﬂows. The latter issue, especially, motivated development of turbulence models that do not introduce additional diﬀerential equations compared to two-equation models but still oﬀer the physical advantages of Reynolds-stress transport models yet at a lower computational eﬀort. One of the ﬁrst models for this purpose was suggested by Rodi (1976) who converted the diﬀerential transport equations of the Reynolds-stresses into implicit, algebraic expressions. Rodi assumed that the sum of all terms containing derivatives of the Reynolds stresses, that is, the convection and diﬀusion terms, can be written as the sum of the convection and diﬀusion of the turbulent kinetic energy k multiplied with the individual Reynolds stress and normalized with k. This algebraic Reynolds-stress model “inherits” most of the desirable features of the “parent” diﬀerential Reynolds-stress transport model. The main physical simpliﬁcation is that the advection and diﬀusion of the Reynolds-stress anisotropy tensor a ≡ aij = ui uj /k − 2δij /3 are neglected. This is frequently referred to as the algebraic Reynolds-stress model assumption. Due to the implicit character of the resulting equations the model has a very diﬃcult mathematical behavior like multiple solutions or singularities (see Wilcox, 1998). Therefore, explicit formulations are highly desirable and have been suggested for example by Gatski & Speziale (1993). In order to derive an explicit formulation of the algebraic Reynolds-stress model rewritten in terms of the anisotropy a, Gatski & Speziale embarked on ideas which were pioneered by Spencer (1971) and Pope (1975) and are based on integrity basis methods and the Cayley-Hamilton theorem. The 6.5 The Explicit Algebraic Reynolds-Stress Model (EARSM) of Wallin 85 resulting model yields non-linear, anisotropic constitutive relations that are closely related to the underlying Reynolds-stress transport model. This route is also taken in the explicit algebraic model of Wallin & Johansson (2000). Additionally, new developments like a near-wall treatment that ensures the realizability constraints and a formulation for compressible ﬂows are incorporated into this model. Other ingredients are the model of Rotta (1951) for the slow pressure strain, the rapid pressure-strain model of Launder et al. (1975), and the assumption of an isotropic dissipation rate tensor. The most important advantage of the explicit algebraic Reynolds-stress model over linear eddy-viscosity models is the ability to predict the anisotropy of normal Reynolds-stresses. The resulting model equations read as follows: Constitutive Relation: 2 ∗ −ρui uj = −ρk δij + 2µt Sij − ρkaex ij . 3 Eddy Viscosity: 1 µt = − f1 (β1 + IIΩ β6 )ρkτ, 2 1 µ τ = max , 20 . 0.09ω ρωk Extra-Anisotropy Tensor: aex ij = 3B2 − 4 1 (1 − f12 ) S − δ S II S ij ik kj max (IIS , IISeq ) 3 1 2 +f1 β3 Ωik Ωkj − IIΩ δij 3 B2 + f12 β4 − (1 − f12 ) (Sik Ωkj − Ωik Skj ) eq 2 max (IIS , IIS ) 2 +f1 β6 Sik Ωkl Ωlj + Ωik Ωkl Slj − IIΩ Sij − IV δij 3 +f12 β9 (Ωik Skl Ωlm Ωmj − Ωik Ωkl Slm Ωmj ) . Invariants: IIS = tr{Sik Skj }, IIΩ = tr{Ωik Ωkj }, IISeq = 405c21 , 216c1 − 160 IV = tr{Sik Ωkl Ωlj }, c1 = 1.8. 86 6 Models Investigated Normalized Mean Strain Rate and Rotation Tensors: τ ∂ui ∂uj 2 ∂uk τ ∂ui ∂uj + − δij , Ωij = − Sij = . 2 ∂xj ∂xi 3 ∂xk 2 ∂xj ∂xi β-Coeﬃcients: β1 = − β4 = − N (2N 2 − 7IIΩ ) , Q 2(N 2 − 2IIΩ ) , Q β3 = − β6 = − 12N −1 IV , Q 6N , Q β9 = 6 . Q Auxiliary Relations: f1 = 1 − exp −CyA Rey − CyB Re2y , √ ρ kd Rey = , µ CyA = 0.092, CyB = 0.00012, 5 2 N − 2IIΩ 2N 2 − IIΩ , 6 √ 1/3 √ 1/3 + P1 + P2 + P1 − P2 1/6 + 2 P12 − P2 cos 13 arccos √ P21 Q= ⎧ ⎨ N= P1 = ⎩ C1 3 C1 3 P1 −P2 C12 2 9 + IIS − IIΩ C1 , 27 20 3 C1 = P2 = P12 − ,P2 ≥ 0 , ,P2 < 0 C12 2 9 + IIS + IIΩ 9 10 3 3 9 (c1 − 1). 4 Turbulent Kinetic Energy: ∂ui ∂ρk ∂ ∂k ∂ρkuj = −ρui uj − β ∗ ρkω + + (µ + σ ∗ µt ) . ∂t ∂xj ∂xj ∂xj ∂xj Speciﬁc Dissipation Rate: ω ∂ui ∂ ∂ω ∂ρωuj ∂ρω = α (−ρui uj ) − βρω 2 + + (µ + σµt ) + CD , ∂t ∂xj k ∂xj ∂xj ∂xj CD = σD ρ max ω ∂k ∂ω ;0 . ∂xk ∂xk , 6.6 87 Boundary Conditions for the k, ω Models Low-Reynolds-Number Coeﬃcients: α∗ = 5 α0 + Ret /Reω 1 · , · 9 1 + Ret /Reω α∗ α= β∗ = β= 3 , 40 σ∗ = α0∗ + Ret /Rek , 1 + Ret /Rek 5/18 + (Ret /Reβ )4 9 , · 100 1 + (Ret /Reβ )4 2 , 3 σ = σD = 1 , 2 α0∗ = β , 3 α0 = 1 , 10 k , Reβ = 8, Rek = 6, Reω = 2.7. ων One can see from the above equations that the EARSM model investigated in this work relies on the transport equations for k and ω pertaining to the TNT model of Kok in combination with the low-Reynolds-number modiﬁcations suggested by Wilcox for the 1988 model. The application of the low-Reynolds-number coeﬃcients is essential in order to recover the correct asymptotic near-wall behavior of the Reynolds stresses and the turbulent kinetic energy. The low-Reynolds number extensions for the Wallin model were implemented into FLOWer during this work and all presented results were computed using the low-Reynolds number version. Ret = 6.6 6.6.1 Boundary Conditions for the k, ω Models Free-Stream Boundary Conditions The free-stream boundary conditions applied to the turbulence variables were the same for all k, ω models: k∞ = 3 (0.005 |v∞ |)2 , 2 µt∞ = 10−3 µ, ω∞ = ρ∞ k∞ . µt∞ The above conditions were speciﬁed at the farﬁeld inﬂow boundaries of the computational domain. Generally speaking, k∞ and ω∞ are transported streamwise through the irrotational outer ﬂow ﬁeld by the corresponding transport equations towards the body under consideration. Actual values of the turbulence variables at the boundary-layer edge can be quite diﬀerent from speciﬁed free-stream values since k and ω decay during the streamwise transport process. Hence, the values of k and ω at the boundary-layer edge 88 6 Models Investigated depend on the ﬂow and on the distance between the farﬁeld boundary and the body. In situations where the farﬁeld inﬂow boundary is located far away from the body a variation of the free-stream values has a weak eﬀect on the computed values at the boundary-layer edge. However, it is ω∞ on which results obtained with the k, ω models of Wilcox can depend on. It was encountered that the dependence was especially an issue if the farﬁeld inﬂow boundary could not be positioned far enough, that is 1.5 to 2.0 characteristic lengths, from the body. This will be discussed in detail in Section 8.5.2. On outﬂow boundaries standard convective outﬂow boundary conditions were applied. These are boundary conditions where zero streamwise gradients of the ﬂow variables are speciﬁed. (However, special outﬂow treatment was applied for the test cases BS0 and CS0, see Subsection 7.2.1.) 6.6.2 Wall Boundary Conditions The boundary condition for the turbulent kinetic energy at a solid surface is straightforward; k is set to zero at all no-slip walls. The speciﬁcation of ω, however, is not unique and several diﬀerent methods for setting the wall value of ω exist. Three of them were tested in this work. 1. In the method according to Wilcox a so-called slightly-rough-surface boundary condition is assumed. This bears the advantage that surface roughness is simply modeled by adjusting the surface value of ω accordingly. The resulting boundary condition for ω at the wall reads: 2 2500 uτ uτ ks ωw = , ks+ = . (6.5) + νw νw ks ks is the surface roughness height and for hydraulically smooth surfaces ks+ ≤ 5. In the present work ks+ = 5 was used. A disadvantage of this method is that the wall value of ω depends on uτ and hence on the skin friction. This means that in points where the skin friction is zero, as for separation and attachment points in a two-dimensional ﬂow, the wall value of ω is zero, too. However, there is no physical reason why the wall value of ω should vanish in such points. 2. Menter (1993) suggested a method in which the wall value of ω depends on the distance of the ﬁrst grid point from the wall, ∆y: ωw = 60νw . 0.075∆y 2 (6.6) 6.7 The One-Equation Model of Spalart & Allmaras 89 This expression mimics the fact that ω ∼ y −2 approaching the wall as pointed out by Wilcox (1998). 3. A very similar method to the one proposed by Menter is the procedure used by Rudnik (1997). However, in contrast to Menter’s approach Rudnik uses a ﬁxed reference length scale yr and the surface boundary condition for ω is then given by ωw = 60νw . 0.075yr2 (6.7) Rudnik recommends yref = 10−5 as an appropriate value for the reference length scale for a large variety of airfoil ﬂow cases (see Rudnik, 1997). This is also the value adopted in the present work if not otherwise noted. In FLOWer, one of the above boundary conditions is speciﬁed by user input. 6.7 The One-Equation Model of Spalart & Allmaras Early one-equation models were based on the transport equation of the turbulent kinetic energy which serves as the velocity scale for the computation of the eddy viscosity. These models are “incomplete” in a sense that they require the speciﬁcation of a turbulent length scale which varies from ﬂow case to ﬂow case and must be speciﬁed a priori, i.e. the length scale is not part of the obtained ﬂow solution. By contrast, one-equation models solving a transport equation for the eddy viscosity itself inherently provide the necessary velocity and time scale and are thus complete. One of the main motivations for the development of turbulence models employing one transport equation for the eddy viscosity is to reduce the computational eﬀort which is required for the solution of two-equation transport models. This comes in addition to the intention to utilize the principal advantages of transport-equation models over algebraic ones. As noted earlier, the latter do not account for ﬂow-history eﬀects and rely on physical assumptions that are in the spirit of equilibrium boundary layers and become incorrect when separated and multiple shear layers are present. Besides these physical arguments various issues at the implementation level favor the use of transport equation models. For example, algebraic models typically evaluate the velocity and/or vorticity proﬁle normal to the wall; additionally, 90 6 Models Investigated some of the models, like the Cebeci-Smith model, require the computation of the boundary-layer thickness. Reliable generalizations of these procedures for three-dimensional ﬂows around complex geometries and, possibly, on unstructured grids are extremely diﬃcult to develop and computationally expensive. Spalart & Allmaras (1992) proposed a very successful one-equation eddyviscosity-transport model. Unlike other attempts, for example the model of Baldwin & Barth (1991) or, more recently, Menter’s one-equation model (Menter, 1997), they did not derive their one-equation model by simplifying an existing k, two-equation model. Rather, they constructed a transport equation term by term using “empiricism, arguments of dimensional analysis, Galilean invariance and selective dependence on the molecular viscosity”. The model constants and closure functions were calibrated using building block ﬂow cases like diﬀerent kinds of free shear ﬂows and boundary layers. The interested reader is referred to the papers of Spalart & Allmaras (1992, 1994) for the details of the model derivation. The model uses the same constitutive relation as the Baldwin-Lomax model, Equation 3.5. The additional model equation read as follows: Eﬀective Eddy Viscosity: µt = ρ ν̃fv1 . Transport of Eddy Viscosity: ∂ρν̃ ∂ ∂ρν̃uj = Pν̃ − Dν̃ + + ∂t ∂xj ∂xj 2 µ + ρν̃ ∂ ν̃ cb2 ∂ ν̃ +ρ σ ∂xj σ ∂xj (6.8) Diﬀusion with the production and destruction terms Pν̃ = cb1 ρS̃ ν̃, Dν̃ = cw1 fw ρ 2 ν̃ . d Closure Coeﬃcients: cb1 = 0.1355, cb2 = 0.622, cb1 1 + cb2 + , κ2 σ Auxiliary Relations: cw1 = fv1 = χ3 , χ3 + c3v1 cv1 = 7.1, cw2 = 0.3, fv2 = 1 − χ , 1 + χfv1 σ= cw3 = 2, κ = 0.41. fw = g 2 , 3 1 + c6w3 g 6 + c6w3 1/6 , 6.8 χ= ν̃ , ν g = r + cw2 r6 − r , S̃ = S + 6.8 91 The One-Equation Model of Edwards & Chandra ν̃ fv2 , κ2 d2 S= r= ν̃ , S̃κ2 d2 2Ωij Ωij . (6.9) The One-Equation Model of Edwards & Chandra Edwards & Chandra (1996) performed a comparative study of several oneequation transport models including the Spalart-Allmaras model. They found that the original formulation of the strain-rate norm S̃ in Equation 6.9 can lead to a singular behavior of S̃ in the near-wall region. The achievable level of residual convergence of the numerical scheme is limited when this happens. In order to increase the numerical robustness of the Spalart-Allmaras model for such cases they suggested the following more stable way of computing S̃ as well as the near-parameter r which is used for the computation of the wall-blockage function fw : √ 1 ν̃ S̃ = S + fv1 , r = tanh / tanh(1.0), χ S̃κ2 d2 2 ∂ui ∂uj ∂ui 2 ∂uk + − . S= ∂xj ∂xi ∂xj 3 ∂xk All other model equations including the transport of the eddy viscosity, Equation 6.8, are identical to the original Spalart-Allmaras model. Since the modiﬁcations of Edwards and Chandra are purely numerically motivated the new model is expected to yield results very similar to the original SpalartAllmaras model. 6.9 The Strain-Adaptive Linear Spalart-Allmaras (SALSA) Model It is often reported in the literature that the Johnson-King model and the k, ω SST model of Menter show signiﬁcant improvements of predictive accuracy for separating boundary-layer ﬂows compared to their “parent” models, namely the Baldwin-Lomax model and the 1988 k, ω model of Wilcox, respectively. These improvements are mainly attributed to a forced limitation of the eddy viscosity in separated ﬂow regions. The limitation mechanisms used in the two models are based on Bradshaw’s assumption of constant ratio between the turbulent kinetic energy and the Reynolds shear stress in a boundary-layer ﬂow. Motivated by the success of limiting the eddy viscosity 92 6 Models Investigated in regions of excessive production, Rung et al. (2003) transfered the limitation concept to the framework of the one-equation model of Spalart & Allmaras. They developed an eddy-viscosity-transport model that is based on the transport equation of the original Spalart-Allmaras model, Equation 6.8, with a modiﬁed production term. In detail, following relations are employed in the SALSA model: Constitutive Relation: 2 ∗ −ρui uj = µt Sij − ρkδij 3 where S ∗ µt ρk = √ , cµ = 0.09 cµ and 1 ∂ui ∂uj 1 ∂uk ∗ ∗ ∗ = + δij , S ∗ = 2Sij Sij . Sij − 2 ∂xj ∂xi 3 ∂xk The same closure coeﬃcients and closure function are used as for the original Spalart-Allmaras model. However, the near-wall parameter r and the strain-rate norm S̃ are redeﬁned following a route similar to that of Edwards & Chandra: ν̃ ρ0 r = 1.6 tanh 0.7 , ρ S̃κ2 d2 1 S̃ = 1.04 S ∗ + fv1 . χ The key feature of the SALSA model is the reduction of the production term Pν̃ for excessive strains. For this purpose, the ratio between ν̃ obtained from the transport equation, Equation 6.8, and an eddy viscosity based on Prandtl’s mixing-length theory, νP randtl = lmix · vmix = κd · κd S ∗ , is used to yield the non-equilibrium factor ν̃ . κ2 d2 S ∗ This ratio is typically less than unity for separated boundary layers and is used to damp the production term. The employed expressions are: σneq = cb1 √ = 0.1355 Γ, Pν̃ = cb1 ρS̃ ν̃, Γ = min(1.25, max(γ, 0.75)), γ = max(α1 , α2 ). χ . α1 = (1.01 σneq )0.65 , α2 = max 0, 1 − tanh 68 6.10 6.10 Boundary Conditions for the One-Equation Models 93 Boundary Conditions for the One-Equation Models The speciﬁcation of boundary conditions for the models based on the transport equation for the eddy viscosity, Equation 6.8, is straightforward compared to procedures applied for the two-equation models. Two aspects must be considered: First, the free-stream value of µt must be suﬃciently small in order not to inﬂuence the irrotational part of the ﬂow solution. This is ensured by setting the free-stream value of µt to a small fraction of the ﬂuid’s viscosity. In this work the following relation was employed: ν̃∞ = 10−5 ν∞ . Secondly, since all ﬂuctuations are zero at the wall and, hence, all Reynolds stresses vanish, the eddy viscosity at the wall must be zero as well: ν̃w = 0. 94 7 7 Test Cases Selected Test Cases Selected The ﬂow cases investigated have, on the one hand, a simple geometry in order to simplify grid generation, to exclude numerical errors associated with extremely skewed grids, and to be easily able to study grid-convergence effects by simply doubling the number of grid points. On the other hand, they are well suited for validation purposes since for each ﬂow case accurate experimental data are available and each ﬂow case oﬀers the necessary physical complexity. Table 7.1 gives an overview of the considered ﬂow cases including the basic ﬂow parameters which were speciﬁed for the computations. A closer description of the investigated ﬂow will be given in the corresponding section. Table 7.1: Test cases selected including basic ﬂow parameters Test case Case ID Flow class Mref Reref Tref [K] Boundary layer with dp/dx = 0 FPBL BL 1.1 (Table 7.2) 0.02848 4.47 · 106 296.4 Non-equilibrium boundary layer with dp/dx > 0 BS0 BL 2.1 (Table 7.2) 0.08772 280000 291.1 Non-equilibrium boundary layer with dp/dx > 0 and separation CS0 BL 3.1 (Table 7.2) 0.08772 280000 291.1 Separated ﬂow around airfoil A AAA 1.1 (Table 1.1) 0.15 2.00 · 106 294.4 In the introduction, aerodynamic ﬂows are classiﬁed using the ﬂow topology as a guideline. Similarly, following Hirschel (2003), the boundary-layer ﬂows investigated in this section can also be classiﬁed using the topology of the velocity ﬁelds (see Table 7.2). Of course, the topologies shown in Table 7.2 are very simple and the classiﬁcation is straightforward. However, the table is intended for again demonstrating the concept of classifying ﬂows on the basis of ﬂow topology. The purpose of the classiﬁcation is to investigate the performance of the present turbulence models for each class of ﬂows. It is hoped that future investigations of this kind will show whether general conclusions can be drawn on what model to use for a given ﬂow class. 7.1 95 Flat-Plate Boundary Layer (Case FPBL) Table 7.2: Possible classes of two-dimensional turbulent boundary layers on ﬂat surfaces following Hirschel (2003) (ﬂows with favorable pressure gradient are not considered in the present work) Class BL 1.1 ∂p ∂x ∂p ∂x =0 Case FPBL 7.1 Class BL 2.1 >0 Case BS0 Class BL 3.1 ∂p ∂x > 0 and separation Case CS0 Flat-Plate Boundary Layer (Case FPBL) A ﬂat-plate boundary-layer ﬂow with zero pressure gradient was studied in order to investigate the models’ ability to predict the law of the wall. New and very accurate experiments were performed by DeGraaﬀ & Eaton (2000) and are available via the Internet. The advantage of DeGraaﬀ’s data over earlier measurements is that the ﬁrst measurement point was located very close to the wall, well within the viscous sublayer, so that model predictions of the sublayer can be checked in detail. In the experiments, the boundary-layer proﬁles were evaluated at a downstream position corresponding to Reθ = 2900, where Reθ denotes the Reynolds number based on the momentum thickness θ. The free-stream velocity in the experiments was u∞ = 9.83 m/s. In order to resemble this value for u∞ in the computation the following input parameters were speciﬁed: Mref = 0.02848 and Reref = 4471085 in combination with L = 7 m. The transition location was speciﬁed at x/L = 0.04. Reθ = 2900 was then obtained at approximately 40 percent of the plate. 7.1.1 Computational Setup The grid for the ﬂat-plate boundary-layer computations is shown in Figure 7.1. The plate is located at the lower boundary between x/L = 0 and x/L = 1 where no-slip and adiabatic-wall conditions were applied. In the front part of the lower boundary (−0.5 ≤ x/L < 0) symmetry was enforced by setting all derivatives in the y-direction to zero. The entire upper boundary was modeled by characteristic boundary conditions while free-stream conditions 96 7 Test Cases Selected were prescribed at the left inﬂow boundary. At the right boundary of the computational domain the pressure was ﬁxed while zero streamwise gradients of the velocity and the density were speciﬁed. y/L 0.3 0.2 0.1 0 -0.5 0 x/L 0.5 1 Figure 7.1: Grid for ﬂat-plate boundary-layer computations. The grid contained a total of 144 cells in the x-direction, where 96 cells were located along the no-slip boundary. 64 cells were used in the wall-normal direction. In order to resolve the boundary layer algebraic grid clustering normal to the wall was employed ensuring that the ﬁrst grid point above the surface was located below y + = 1. However, the grid spacing was uniform at the entry of the computational domain leading to the downwards slope of the “horizontal” grid lines in the region −0.5 ≤ x/L ≤ 0. Additionally, grid clustering in the x-direction was performed at the leading and trailing edges of the plate. These regions pose discontinuities in the boundary conditions resulting in large streamwise gradients. A high grid resolution is necessary in these regions in order to limit the inﬂuence of these discontinuities and to accurately resolve the large gradients. It is noted that the grid employed has highly smooth distributions of all metric terms which is deemed to be an essential prerequisite for obtaining accurate numerical solutions, especially in conjunction with numerical methods using a central space discretization. 7.1.2 Computational Results and Discussion Pressure Distribution One can see from the wall-pressure distribution shown in Figure 7.2 that, except at the leading edge and around the transition location (x/L = 0.04), a virtually zero pressure gradient with cp = 0 along the wall is predicted. The presented cp distribution is taken from the computation with the Baldwin-Lomax model. However, all other turbulence 7.1 97 Flat-Plate Boundary Layer (Case FPBL) models applied in this work show very similar results conﬁrming that no signiﬁcant pressure gradient acts on the boundary layer. 0.01 cp 0.005 0 -0.005 -0.01 0 0.25 0.5 x/L 0.75 1 Figure 7.2: Pressure coeﬃcient along ﬂat plate. Velocity Proﬁles and Skin Friction To compare the computed and measured velocity proﬁles a typical semi-logarithmic presentation was chosen in which the velocity is plotted over the wall distance in dimensionless wall coordinates. 1 Coles’ logarithmic law of the wall (u+ = 0.41 ln y + + 5.0) reproduces the experimental velocity proﬁle in the logarithmic region (30 ≤ y + ≤ 250) with high accuracy (Figure 7.3). As it is expected from theory, in the lower part of the viscous sublayer, that is for y + ≤ 5, the linear relation u+ = y + yields a good approximation to the experimental data. The velocity proﬁles obtained with the diﬀerent turbulence models are compared to the experimental data in Figures 7.4a) to 7.4d). WBC, MBC and RBC denote the speciﬁcation of ω at the wall according to Wilcox, Menter or Rudnik, respectively (see Subsection 6.6.2). In the case of the Wilcox and SST models the wall value of ω was determined following the procedure suggested by the respective model developer. For the TNT model, LLR model and the model of Wallin, Rudnik’s method for determining ωw was adopted. This is because the method of Rudnik is recommended by the FLOWer implementation team and the model developers do not recommend any speciﬁc surface boundary treatment for ω for the latter models. One can see from Figures 7.4b) and 7.4d) that the EARSM model of Wallin and the SST model of Menter overpredict the velocity in the outer region of the boundary layer. Both models of Wilcox, however, underpredict u+ compared to the experimental data for the largest part of the boundary 98 7 Test Cases Selected 25 Experiment (DeGraaff) u+=1/0.41 ln(y+) + 5.0 u+=y+ 15 u + 20 10 5 0 0 10 y+ 101 102 103 Figure 7.3: The log-law compared with the experiment of DeGraaﬀ at cf = 3.362 × 10−3 , that is at Reθ = 2900. layer (Figure 7.4a)); they slightly overpredict the slope of the velocity proﬁle in the logarithmic region. Note that the Wilcox 1998 model yields a velocity proﬁle that is closer to the measured data in the wake region of the boundary layer than the proﬁle obtained with the 1988 model. This diﬀerence is due to the function χk in Equation 6.4 of the 1998 model: In the wake region, ω is small compared to values encountered close to the wall, and gradients of both turbulence variables are non-zero. This combination leads to large values of χk which, in turn, increases the dissipation coeﬃcient β ∗ in the transport equation for k (Equation 6.2) and, hence, reduces k and the eddy viscosity. With the exception of the models of Menter, Wallin, and Wilcox, all of the models employed yield velocity proﬁles which are in very close agreement with the experiment. Both over- and underprediction of the sublayer-scaled velocity u+ are frequently rooted in the prediction of the shear stress at the wall. This is best seen when writing u+ and y + in terms of τw : u+ = u =u uτ ρw , τw y+ = yuτ y = ν ν τw . ρw 7.1 99 Flat-Plate Boundary Layer (Case FPBL) It follows from these relations that an overprediction of the wall shear stress τw leads to an underprediction of u+ in Figures 7.4a) to 7.4d) (and vice versa) even if the absolute velocity u was correctly predicted by the computation. Hence, the presentation of the velocity proﬁles in wall coordinates must always be discussed in conjunction with the corresponding skin-friction coeﬃcients. Table 7.3: Local cf at Reθ = 2900 (case FPBL) Source cf × 103 ∆ Experiment (DeGraaﬀ & Eaton, 2000) 3.362 – k, ω TNT (RBC) 3.2636 −2.9% Spalart-Allmaras 3.2502 −3.3% SALSA 3.2093 −4.6% k, ω LLR (RBC) 3.1788 −5.4% Baldwin-Lomax 3.1474 −6.4% Edwards-Chandra 3.1315 −6.8% k, ω 98 (WBC) 3.6128 +7.5% Wallin (RBC) 3.0739 −8.6% k, ω 88 (WBC) 3.7262 +10.8% k, ω SST (MBC) 2.9009 −13.7% Results obtained for cf (Table 7.3) are consistent with the discussion about over- and underprediction of u+ proﬁles in Figure 7.4: The model of Wallin and the k, ω SST model of Menter yield skin-friction coeﬃcients that are signiﬁcantly lower than the experimental value and they overpredict u+ . Both models of Wilcox compute cf that is signiﬁcantly larger than measured and they underpredict u+ . The k, ω TNT model gives the best prediction of the skin-friction coefﬁcient; it agrees within three percent with the experiment. Note that the Spalart-Allmaras model yields a very similar value for the skin-friction coeﬃcient. However, the velocity proﬁle in the transition region between the sublayer and the logarithmic part, that is for 15 ≤ y + ≤ 45, agrees even better with the experimental data than the velocity proﬁle obtained with the TNT model. All other model predictions of the skin friction range in between these extrema. + u + u 103 d) b) 101 101 102 y+ 102 Experiment (DeGraaff) kω SST MBC y+ Experiment (DeGraaff) kω TNT RBC Wallin RBC kω LLR RBC Figure 7.4: Velocity proﬁles for ﬂat plate at Reθ = 2900. 0 0 10 102 0 0 10 15 20 25 0 0 10 5 y+ 103 5 101 102 Experiment (DeGraaff) Spalart-Allmaras Edwards-Chandra SALSA y+ 5 10 15 20 25 10 c) 101 Experiment (DeGraaff) Baldwin-Lomax kω 88 WBC kω 98 WBC 10 15 20 25 0 0 10 5 10 15 20 a) + u + u 25 103 103 100 7 Test Cases Selected 7.1 101 Flat-Plate Boundary Layer (Case FPBL) 7.1.3 Some Modiﬁcations of the k, ω SST Model In order to improve the model predictions for the ﬂat-plate boundary layer two simple ad hoc modiﬁcations of the k, ω SST model were tested. These are discussed in the following. Eddy Viscosity in the Viscous Sublayer All linear k, ω models investigated in this work set the eddy viscosity proportional to k/ω. It can be shown that this gives the following asymptotic sublayer behavior: µt ∝ y n for y→0 (7.1) with n ≈ 5.28 for the Wilcox 1998 model and n ≈ 5.23 for all the other linear k, ω models. Expanding the ﬂuctuating velocities in Taylor series near the wall and employing the no-slip condition and the continuity equation yields u v ∝ y 3 for y → 0. Since the velocity gradient in the sublayer is constant (du+ /dy + = 1) and µt = −ρu v /(du/dy), the exact exponent for the eddy viscosity following from this asymptotic analysis is n = 3. Hence, a diﬀerent functional dependence of µt on k and ω that yields an exponent closer to the theoretical value of 3 in the viscous sublayer would be more appropriate. This is discussed below. In the viscous sublayer the turbulent motion is strongly aﬀected by the presence of the solid surface; the turbulent eddies decrease in size approaching the wall. Since small scale turbulent motions are diﬀused and dissipated by viscosity it is therefore appropriate to base the characteristic turbulence time scale in the viscous sublayer on the viscosity and the turbulence dissipation: ν ν Tη = ∝ . kω Tη is typically referred to as the Kolmogorov time scale. Since the eddy viscosity can be written as a product of k and a time scale, it seems to be reasonable to use Tη to compute the eddy viscosity in the viscous sublayer. For this purpose the following expressions were implemented into the SST model: 1−F3 a1 ρk µt = (SL ρν)F3 , max (a1 ω; ΩF2 ) 102 7 Test Cases Selected k/ω arg3 = max 1 − ,0 , SL νk/ω 1 F3 = tanh arg34 , 2 SL = 2. SL is a sublayer model constant and F3 is an additional blending function that ensures a smooth transition from the sublayer expression µt = SL kν/ω to the standard SST formulation for the eddy viscosity. This reformulation of the eddy viscosity in terms of Tη yields for the asymptotic behavior approaching the wall n ≈ 2.62 in Equation 7.1. This is much closer to the theoretical value n = 3 than in the original model. Computations show that the diﬀerences between the absolute values of µt obtained with the original expression and the sublayer formulation are small with respect to µt . Due to the large velocity gradient in the sublayer, however, the eﬀect on the Reynolds shear stress and, hence, on the momentum equation is signiﬁcant. Reducing ω in the Logarithmic Region Analysis of the eddy-viscosity proﬁle at Reθ = 2900 reveals that µt computed with the original k, ω SST model is lower than µt inferred from the experimental data in the region 40 ≤ y + ≤ 200 (Figure 7.5). 0.025 Experiment (DeGraaff) kω SST MBC νturb/(θUe) 0.02 0.015 0.01 0.005 0 0 10 101 y+ 102 103 Figure 7.5: Eddy viscosity at Reθ = 2900 (case FPBL). 7.1 103 Flat-Plate Boundary Layer (Case FPBL) To investigate to what extent this diﬀerence does have an inﬂuence on the computed velocity proﬁle the eddy viscosity was increased in the region of interest. For this purpose a local damping of ω around y + = 75 was introduced which was achieved by increasing the destruction term in the ω equation. In particular, the destruction coeﬃcient β1 of the SST model was modiﬁed as follows: 3 1 β1 = , (1 − 1.3F4 ) 40 2 1 75 − y + − 0.7, 0 . (7.2) F4 = max exp − 2 85 Figure 7.6 shows the resulting distribution of β1 across the boundary layer. At y + = 75 a peak value of β1 ≈ 0.123 is obtained leading to a strong ampliﬁcation of the destruction term in the ω transport equation in this region. The original value β1 = 3/40 is maintained below y + = 3.2 and above y + = 147. 0.14 β1 0.12 0.1 0.08 0.06 0 10 101 y+ 102 103 Figure 7.6: Proﬁle of modiﬁed destruction coeﬃcient β1 at Reθ = 2900 (case FPBL). Like all functions depending on y + , and hence on the skin friction, Equation 7.2 bears the disadvantage that it is not deﬁned for vanishing wall shear 104 7 Test Cases Selected stress. Theoretically, in such situations, y + = 75 is located inﬁnitely far away from the wall. Due to the construction of Equation 7.2, however, this does not pose any numerical diﬃculties since then the original value β1 = 3/40 is recovered. Results The modiﬁcations of the k, ω SST model lead to excellent agreement of computed and measured velocity proﬁles (Figure 7.7). Similarly, the skin-friction coeﬃcient obtained with the modiﬁed model agrees within roughly one percent with the experimental value (Table 7.4). This comes as no surprise since the modiﬁcations were tailored in order to yield best possible agreement of computational results with measurements for this ﬂow case. It will be shown, however, that these modiﬁcations improve model predictions for separated ﬂows as well. 25 20 Experiment (DeGraaff) kω SST original kω SST modified u+ 15 10 5 0 0 10 101 y+ 102 103 Figure 7.7: Comparison of velocity proﬁles computed with the original and modiﬁed k, ω SST model (case FPBL). It is noted that the focus of this work was not the development of new turbulence models. Therefore, a further improvement and a generalization of the simple modiﬁcations presented was not pursued. 7.1 105 Flat-Plate Boundary Layer (Case FPBL) Table 7.4: Local cf at Reθ = 2900 obtained with the modiﬁed k, ω SST model (case FPBL) Source 7.1.4 cf × 103 ∆ Experiment (DeGraaﬀ) 3.362 – k, ω SST original 2.9009 −13.7% k, ω SST modiﬁed 3.3253 −1.1% Eﬀects of Low-Reynolds-Number Modiﬁcations Wilcox (1998) developed the low-Reynolds-number modiﬁcations for his k, ω models with two aspects in mind. One was modeling of laminar-turbulent transition which will be discussed in Section 8.3. The other aspect was improvement of near-wall behavior of k and ω. Yet, model predictions for skin friction and velocity proﬁles obtained with and without application of lowReynolds-number terms were virtually identical (Table 7.5, Figure 7.8); differences were encountered only in the k and ω proﬁles. Figure 7.9 shows that in the computation including low-Reynolds-number modiﬁcations a near-wall peak in the k proﬁle is obtained. This is consistent with results from DNS (Wilcox, 1998). Table 7.5: Local cf at Reθ = 2900 obtained with Wilcox’s 1988 k, ω model in combination with Rudnik’s surface boundary condition for ω (k, ω 88 RBC) with and without low-Reynolds-number modiﬁcations (case FPBL) Source cf × 103 ∆ Experiment (DeGraaﬀ) 3.362 – k, ω 88 (RBC) standard 3.4383 +2.3% k, ω 88 (RBC) including viscous corrections 3.4327 +2.1% Except for the Aerospatiale-A airfoil, the test cases considered in this work were computed without low-Reynolds-number modiﬁcations. This was done for the following reasons: • Viscous correction suggested by Wilcox do not alter model predictions for skin friction and velocity proﬁles in fully-turbulent ﬂows. Yet, for 106 7 Test Cases Selected 30 25 kω 88 RBC kω 88 RBC low Reynolds number u + 20 15 10 5 0 0 10 10 1 y+ 2 10 10 3 Figure 7.8: Eﬀect of low-Reynolds-number modiﬁcations on u+ proﬁle of a ﬂat-plate boundary layer; computation performed with the 1988 k, ω model of Wilcox in combination with Rudnik’s surface condition for ω. the test cases considered, no experimental data are available for k (and ω) in the near-wall region. Hence, direct validation of the low-Reynoldsnumber modiﬁcations was not possible. • Typically, in production codes, the simpler high-Reynolds-number versions of the k, ω models are implemented. • Viscous correction do not contribute to the solution of the core problems of turbulence modeling since these are Reynolds averaging and the assumption of an eddy viscosity. However, in cases where viscous eﬀects like transition are signiﬁcant the viscous corrections developed by Wilcox can have a large inﬂuence on computed results even in the turbulent part of the ﬂow. As noted above, this will be discussed in Section 8.3. 7.2 107 Boundary Layer with Adverse Pressure Gradient (Case BS0) 1.2E-05 1E-05 kω 88 RBC standard kω 88 RBC low Reynolds number k/(p∞/ρ∞) 8E-06 6E-06 4E-06 2E-06 0 10-4 y/L 10-3 10-2 Figure 7.9: Eﬀect of low-Reynolds-number modiﬁcations on the k proﬁle of a ﬂat-plate boundary layer; computation were performed with the 1988 k, ω model of Wilcox in combination with Rudnik’s surface condition for ω. 7.2 Boundary Layer with Adverse Pressure Gradient (Case BS0) The next ﬂow case considered is a turbulent boundary layer with a strong adverse pressure gradient that was experimentally investigated by Driver (see Driver & Johnston, 1990; Driver, 1991). In this ﬂow, an axisymmetric boundary layer developed in the axial direction on a circular cylinder; the cylinder was lengthwise mounted along the center of the wind tunnel (Figure 7.10). An adverse pressure gradient was imposed at the cylinder by diverging all four wind tunnel walls. To reduce separation of the tunnel wall boundary layers, boundary-layer suction was applied at the tunnel side walls. The level of suction was used to control boundary-layer thickness at the tunnel walls and, thus, to adjust the pressure gradient at the boundary layer of the cylinder surface. No separation of the boundary layer at the cylinder occurred in case BS0, which is the case discussed in this section. The pressure gradient, how- 108 7 Test Cases Selected Figure 7.10: Driver’s cylinder ﬂow; case CS0 with separation (schematic ﬁgure is courtesy of David Driver, NASA-Ames Research Center, used with permission). ever, was strong enough to cause signiﬁcant non-equilibrium between mean ﬂow and turbulence. Case CS0 with separation will be presented in Section 7.3. Driver’s extensive experimental data for these ﬂow cases are highly accurate and self-consistent and form an excellent basis to assess performance of turbulence models for adverse pressure-gradient ﬂows without and with separation. 7.2.1 Computational Setup For the computation of the cylinder ﬂow three grid planes were located closely spaced in the circumferential direction ψ, as indicated in Figure 7.11. The spacing between the planes was ∆ψ = 0.5◦ . Each plane contained 256 cells in the axial and 64 cells in the radial direction. Additionally, algebraic grid clustering was employed normal to the wall ensuring that the position of the ﬁrst grid point above the surface was below y + = 1. To achieve axial symmetry a three-dimensional computation was performed and appropriate boundary conditions were set in the circumferential direction. For this purpose, the two outer grid planes served as ghost layers in which symmetry boundary 7.2 109 Boundary Layer with Adverse Pressure Gradient (Case BS0) conditions were applied. Hence, only the ﬂow solution obtained on the center grid plane is compared to experimental data. 1.4 axissymmetric boundary conditions in ϕ-direction y/D 1.2 inviscid streamline (Euler wall) 1 0.8 0.6 0.4 -6 outflow prescribed inflow no slip wall -4 -2 0 x/D 2 4 6 Figure 7.11: Computational grid and boundary conditions for Driver’s cylinder ﬂow (not to scale). The computational grid and the boundary conditions applied are shown in Figure 7.11. Note that, for better recognizability, the x to y ratio is not unity in the ﬁgure. In addition, only every other grid line in the x-direction (axial) and every fourth grid line in the y-direction (radial) is shown. The usual no-slip and adiabatic wall conditions were applied at the cylinder surface. At the inﬂow boundary all ﬂow variables were prescribed and held ﬁxed during the computations. In particular, the vector of the conserved variables was determined from the experimental velocity proﬁle assuming constant density across the boundary layer. Regarding the turbulence variables, k was inferred from the experimental Reynolds-stress proﬁle and ω was computed with the usual deﬁnition of eddy viscosity, e.g. Equation 6.1. (The eddy viscosity had been determined from the shear-stress and velocity proﬁles.) The Mach number speciﬁed for the computation was Mref = 0.08872, and the Reynolds number based on the diameter of the cylinder was Reref = 280000. To impose the experimental pressure distribution in the computation, a procedure was applied that is based on modeling the upper boundary opposite to the cylinder surface as an inviscid streamline: An inviscid streamline was used to determine the shape and position of the upper-boundary grid line. The streamline was inferred from experimental data by applying the 110 7 Test Cases Selected requirement of mass preservation: y1 +ys (x) u(x, y)y dy = constant. ṁ = 2πρ (7.3) y1 ṁ is mass per unit time, y1 denotes the cylinder radius, and ys (x) is the unknown coordinate of the inviscid streamline. At the inﬂow boundary ys (x) was set equal to the wall-normal position of the last measurement point on the experimental velocity proﬁle. The experimental velocity proﬁle was then integrated using Equation 7.3 to yield ṁ. Subsequently, at all other downstream stations, Equation 7.3 was employed to compute ys (y) which determined the shape and position of the upper-boundary grid line. Along the upper boundary, a perfect slip-wall condition was applied which ﬁnally rendered the grid line an inviscid streamline. (Note that the cylinder surface and the perfect slip-wall were much closer to each other than the cylinder surface and the wind tunnel walls.) The inviscid-streamline method was applied also by Menter (1993) who showed that this procedure yields very similar results to the direct speciﬁcation of pressure. Directly specifying surface pressure, however, is numerically less consistent in the framework of full Navier-Stokes computations. As a result of the inviscid-streamline procedure, an internal ﬂow is obtained. For such ﬂows special attention must be paid to speciﬁcation of pressure at the inﬂow and outﬂow boundaries. Since the pressure at the inﬂow boundary was set ﬁxed to the measured value, the outﬂow pressure could not be independently set. In such situations, the outﬂow pressure depends on the total-pressure loss and, hence, on the particular ﬂow solution. Consequently, one can not simply specify the measured value since the totalpressure losses can diﬀer between computation and experiment. Therefore, the outﬂow pressure must be adapted to the ﬂow solution such that a smooth pressure distribution is obtained throughout the ﬂow ﬁeld. In particular, ﬁxed inﬂow pressure in combination with inappropriate outﬂow pressure can yield a non-physical pressure “jump” between the inﬂow boundary and the ﬁrst interior grid line in the inﬂow region. In addition, in the compressible RANS equations pressure carries the thermodynamic information: Merely specifying a pressure diﬀerence between inﬂow and outﬂow is not applicable in this context, as can be done for incompressible Navier-Stokes computations. For these reasons, the pressure at the outﬂow was adapted during the computation such that the surface pressure obtained from the ﬂow computation resembled the experimental value at the ﬁrst grid point downstream of the 7.2 Boundary Layer with Adverse Pressure Gradient (Case BS0) 111 inﬂow boundary. For the other ﬂow variables, standard convective outﬂow conditions were speciﬁed. 7.2.2 Computational Results and Discussion Pressure Distribution Figure 7.12 shows computed and measured pressure distributions along the cylinder surface. The graphs of the pressure distributions obtained with diﬀerent models lie very close to each other up to x ≈ 0. Downstream of this point diﬀerences among the pressure distributions are evident; all investigated models tend to overpredict surface pressure, though to diﬀerent extent. The Baldwin-Lomax model yields the highest surface-pressure level while the 1998 k, ω model of Wilcox, the original SST model, the SALSA model, and the model of Wallin predict pressure values that are also somewhat too high but in fairly close agreement with the measurements. Note that due to the small distance between the lower and upper “walls” in the computation, even small changes in the momentum thickness predicted by the models signiﬁcantly aﬀect the surface pressure. Skin-Friction Distribution Corresponding skin-friction distributions are shown in Figure 7.13. Relatively larger variances are encountered among skin-friction distributions obtained with diﬀerent models compared to the variances in results for the pressure distributions. Diﬀerences in cf obtained with diﬀerent one-equation models are smaller than between computational results from k, ω models. Regarding the prediction of measured skin friction for this ﬂow case, the 1998 k, ω model of Wilcox and the modiﬁed k, ω SST model give the best results. Comparing the graphs in Figure 7.12 and Figure 7.13 one can see that if a turbulence model yields accurate predictions of surface pressure (with regard to experimental data) it does not necessarily yield skin-friction predictions of comparable accuracy. This is best recognized from results obtained with the Wallin model: The Wallin model yields cp that is very close to measurements while it predicts cf in modest agreement with the experiment. 0.4 0.4 0.6 0 0.2 cp 0 0.2 cp c) a) -1 -1 x/D x/D 1 1 2 Experiment Spalart-Allmaras Edwards-Chandra SALSA 2 3 3 0.4 0.6 0 0.2 0 0.2 d) b) -1 -1 0 0 x/D x/D 1 1 2 Experiment kω SST MBC kω SST modified 2 Experiment kω TNT RBC Wallin RBC kω LLR RBC Figure 7.12: Pressure distributions at cylinder surface for case BS0 (arrows show positions of measured and computed boundary-layer proﬁles). 0 0 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 0.4 0.6 cp cp 0.6 3 3 112 7 Test Cases Selected cf cf 2 x/D 1 2 3 3 0.002 0.003 0 -1 -1 d) b) 0 0 1 2 x/D 1 2 Experiment kω SST MBC kω SST modified x/D Experiment kω TNT RBC Wallin RBC kω LLR RBC Figure 7.13: Skin-friction distributions at cylinder surface for case BS0 (arrows show positions of measured and computed boundary-layer proﬁles). 0 0 1 Experiment Spalart-Allmaras Edwards-Chandra SALSA x/D 0.001 0.002 0.003 0 c) 0 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 0.001 -1 -1 a) 0.001 0.002 0.003 0 0.001 0.002 0.003 cf cf 3 3 7.2 Boundary Layer with Adverse Pressure Gradient (Case BS0) 113 114 7 Test Cases Selected Velocity Proﬁles Velocity proﬁles are compared at diﬀerent downstream positions. These positions are given by the experiment and are marked with arrows in Figures 7.12a) and 7.13a). So as not to overload this section with an excessively large number of graphs the corresponding ﬁgures are placed in Appendix C.1. All models but the Baldwin-Lomax model yield velocity proﬁles that are in fair agreement with experimental results (Figures C.1 to C.5). With increasing downstream position x/D, however, the agreement decreases and the diﬀerences among the results increase (Figure C.6). (Note that the measurement points are sparsely distributed in the direction normal to the wall at all downstream positions except x/D = 1.08857. This makes a thorough comparison of computed and measured results somewhat diﬃcult.) Results obtained with the Baldwin-Lomax model are in poor agreement with experimental data at x/D = 1.08857 and x/D = 1.63286. Inspection of the velocity proﬁles and the pressure distributions shows that the velocity at the boundary-layer edge and the surface pressure are obviously linked by Bernoulli’s equation (p + 0.5(u2 + v 2 ) = constant). Reynolds-Shear-Stress Proﬁles Computed and measured Reynolds shear stresses are compared in Figures C.7 to C.12; the Reynolds stresses were evaluated at the same downstream positions as the velocity proﬁles discussed above. It is noted that measurement uncertainties are much larger in case of Reynolds stresses than for mean velocities. This should be kept in mind when comparing computational and experimental results. The Baldwin-Lomax model and Wilcox’s 1988 k, ω model generally overpredict the level of Reynolds shear stress while other models yield Reynolds stresses that are either close to or below measured values. Best overall agreement of computed u v with measurements is obtained with the oneequation models. Note that the largest diﬀerences between computed and measured Reynolds-stress proﬁles are found at the last downstream station (x/D = 1.63286). The reason for this is that in the experiment the Reynolds-stress level signiﬁcantly increases between x/D = 1.08857 and x/D = 1.63286. In the computations, this increase is much less pronounced, if at all existent. This can be also seen in Figure 7.14 where the streamwise development of maximum Reynolds shear stress is shown. In the ﬁgure, the experimental data are only crude estimates inferred from the graphs of u v ; an accurate determination of measured u v m is not possible for most downstream positions 7.2 Boundary Layer with Adverse Pressure Gradient (Case BS0) 115 because of sparse data. However, the qualitative development of u v m is assumed to be correctly captured by the graph and it may serve for comparing the computational results with the experiment. The Baldwin-Lomax model computes streamwise development of u v m that is in very poor qualitative agreement with the experimental estimate (Figure 7.14). In particular, the total maximum of u v is obtained too far upstream at x/D = 0.0907143. Downstream of this point, u v decreases which indicates that the Baldwin-Lomax model instantaneously responds to the change of pressure gradient. This behavior is not seen in the results obtained with the other models; they yield total maxima of u v further downstream at x/D = 1.63286, as it is encountered in the measurements. These models are based on transport equations which do, to some extend, account for ﬂow-history eﬀects. Flow-history eﬀects are believed to be the reason why in this ﬂow case Reynolds stresses do not immediately respond to changes in the pressure gradient. Yet, even transport-equation models do not suﬃciently account for ﬂow-history eﬀects. This leads to the general underprediction of Reynolds shear stress at the last downstream position considered (Figure C.12). 2 <u’v’>m /0.001Ur 2 <u’v’>m /0.001Ur -1 -1.2 -1.4 -1.6 -1.8 -2 -2.2 -2.4 -1 -1.2 -1.4 -1.6 -1.8 -2 -2.2 -1 -1 x/D x/D 1 Experiment Spalart-Allmaras Edwards-Chandra SALSA 1 -1 -1.2 -1.4 -1.6 -1.8 -2 -2.2 -2.4 -1 -1.2 -1.4 -1.6 -1.8 -2 -2.2 -2.4 -1 -1 x/D 0 x/D Experiment kω SST MBC kω SST modified 0 Experiment kω TNT RBC Wallin RBC kω LLR RBC Figure 7.14: Streamwise development of u v m for case BS0. 0 0 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2 <u’v’>m /0.001Ur 2 <u’v’>m /0.001Ur -2.4 1 1 116 7 Test Cases Selected 7.2 117 Boundary Layer with Adverse Pressure Gradient (Case BS0) Summary The performance of each turbulence model for computing case BS0 is summarized in Table 7.7. In the Table, models’ performances are rated according to the predictive accuracy of computational results in comparison with corresponding experimental data. Each ﬂow variable discussed above is separately considered. Table 7.6: Symbols used to rate turbulence-model performance Symbol Meaning +++ very good agreement with measurements ++ + good agreement with measurements fair agreement with measurements modest agreement with measurements − weak agreement with measurements −− −−− poor agreement with measurements very poor agreement with measurements Table 7.7: Turbulence-model performance for case BS0 Model k, ω 98 WBC k, ω SST modiﬁed Edwards-Chandra k, ω SST MBC Spalart-Allmaras −u v m " cp cf u 2(+ + +) 2(+ + +) +++ + 16+ 2(++) 2(++) ++ + 11+ 2(++) 2(+) +++ ++ 11+ 2(+ + +) 2(+) + + 10+ 2(++) 2(+) ++ ++ 10+ SALSA 2(+ + +) 2(+) ++ 10+ Wallin RBC 2(+ + +) 2(−) ++ + 7+ k, ω TNT RBC 2(+) − + 2+ k, ω LLR RBC 2(++) 2(− − −) +++ + 2+ 2(+) 2(−−) −− −− 6− 2(−−) −−− −−− 10− k, ω 88 WBC Baldwin-Lomax From an engineering point of view, cp and cf are frequently more important than u or u v . Therefore, ratings for cp and cf are weighted by a factor of two, as indicated in the Table 7.7. The last column in this table contains 118 7 Test Cases Selected the sum of ratings given for each model while symbols used for the rating procedure are deﬁned in Table 7.6. Of course, this rating system is somewhat subjective rather than purely quantitive. However, it is meant to give qualitative conclusions about the models’ performances and to enable the reader to quickly obtain an overview about the various models. This approach indicates that Wilcox’s k, ω 1998 model gives the best overall results for the BS0 ﬂow case. Moreover, the model yields good or very good agreement with measurements for all ﬂow variables considered (Table 7.7). 7.3 Boundary Layer with Pressure-Induced Separation (Case CS0) To investigate the performance of turbulence models for predicting ﬂow separation, a separated boundary-layer ﬂow was computed. For this purpose, the cylinder ﬂow of Driver (1991) was again considered. In the experiment, Driver used the same experimental setup as for case BS0 without separation (Section 7.2). However, the suction level at the wind-tunnel walls was increased to increase the adverse pressure gradient at the cylinder surface and to force boundary-layer separation. Consequently, for computing this ﬂow case, the same computational setup was utilized as for case BS0; the upper boundary was again modeled by an inviscid streamline inferred from measured velocity proﬁles. Details about the computational setup and grid employed can be found in Section 7.2 and will not be repeated here. The only diﬀerence compared to the procedure presented in Section 7.2 is that the inviscid streamline was obtained from experimental data for case CS0. The resulting (simple) topology of the separated ﬂow ﬁeld is schematically shown in Figure 7.15. It belongs to sub-class 1.1 a) in Table 1.1, i.e. statistically steady, two-dimensional separated ﬂows with a single recirculation zone. b1 S1 F1 A1 cylinder surface Figure 7.15: Flow topology of separated ﬂow ﬁeld along Driver’s cylinder (case CS0; symbols as in Table 1.1). 7.3 Boundary Layer with Pressure-Induced Separation (Case CS0) 7.3.1 119 Computational Results and Discussion Flow Topology and Velocity Proﬁles The computed and measured topologies of the dividing streamlines are shown in Figure 7.16. S1exp and A1exp denote the separation and reattachment points, respectively, inferred from the measured streamfunction. Note that in the ﬁgures diﬀerent scales are used for the x and y axes for better recognizability; the recirculation zone is much thinner in reality than suggested by the ﬁgures. Large diﬀerences among the computational results for the ﬂow topology of the recirculation zone are encountered. The Baldwin-Lomax, the k, ω 1988 and the k, ω TNT model predict recirculation zones that are much thinner than the one inferred from experimental data. In fact, the k, ω 1988 model yields such a thin separation zone that the corresponding dividing streamline is not visible in Figure 7.16a). All models but the k, ω 1988 model have in common that they yield recirculation zones that extend further in the streamwise direction than in the experiment, i.e the distance between S1 and A1 is larger in the computational results than in the measurements. This is also seen from the skin-friction distributions discussed below (see also Figure 7.18). For the k, ω 1998 model, predictions are in fair overall agreement with the experimental ﬂow topology. The SALSA model predicts a much larger recirculation zone compared to the experiment and to results obtained with other one-equation models. Modiﬁcations applied to the k, ω SST model improve streamline predictions in the upstream half of the recirculation zone while at the downstream end the modiﬁcations yield only slight improvements. The velocity proﬁles of the separated boundary-layer ﬂow (Figures C.13 to C.22) are placed in the Appendix. Analogously to the procedure applied in Subsection 7.2 for the attached boundary layer, the proﬁles were evaluated at diﬀerent downstream stations. x/D positions of the stations are indicated by arrows in Figures 7.17a) and 7.18a). At the ﬁrst station, that is at x/D = −1.08857, all models yield velocity proﬁles that are in close agreement with measurements. Only the BaldwinLomax model shows a tendency to compute a velocity proﬁle that is fuller in shape than the measured one. This tendency is ampliﬁed with increasing downstream distance. To compensate high mass ﬂux due to fuller proﬁles, the Baldwin-Lomax model predicts a lower velocity level in the outer region of the boundary layer than in the experiment. All other models are able to predict velocity proﬁles in reasonable agreement with measurements. Diﬀerences between computed and measured data are seen mainly in the inner region of 120 7 Test Cases Selected 0.7 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC y/D 0.65 0.6 0.55 0.5 0.45 S1exp 0 0.7 y/D 1 2 Experiment kω TNT RBC Wallin RBC kω LLR RBC 0.6 b) 0.55 0.5 0.45 S1exp 0 0.7 A1exp x/D 1 2 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.65 y/D A1exp x/D 0.65 0.6 c) 0.55 0.5 0.45 S1exp 0 A1exp x/D 0.7 1 2 Experiment kω SST MBC kω SST modified 0.65 y/D a) 0.6 d) 0.55 0.5 0.45 S1exp 0 A1exp x/D 1 2 Figure 7.16: Topology of recirculation zones for case CS0; dividing streamlines are shown. the boundary layer. Variances among computed results are, however, larger than for the attached case (Subsection 7.2) and they are more pronounced at stations located further downstream. The modiﬁed SST model yields best overall agreement with experimental velocity proﬁles. 7.3 Boundary Layer with Pressure-Induced Separation (Case CS0) 121 Pressure Distribution Up to x/D ≈ −0.4, all turbulence models yield surface-pressure distributions which are in close agreement with measurements (Figure 7.17). Downstream of x ≈ 0, however, large variances are found between pressure distributions obtained with diﬀerent models: While most models overpredict surface pressure in the region −0.4 ≤ x/D ≤ 3.0, the model of Wallin virtually duplicates experimental results. The BaldwinLomax model yields worst agreement of predicted and measured surface pressure; all other model predictions are in between these extrema. Skin-Friction Distribution Corresponding skin-friction distributions are shown in Figure 7.18. Boundary-layer separation S1 and reattachment A1 are deﬁned at points where cf = 0. (In the experiment, due to the high accuracy of the experimental data of Driver, the positions of S1 and A1 inferred from the streamfunction and the ones obtained from the cf distribution were found to be almost identical.) From inspection of Figure 7.18, large variances among computational results are found. Note, however, that the predictions of the one-equation models are closer to each other than those of the k, ω type of models. Wilcox’s 1988 k, ω model is the only turbulence model that has cf = 0 in close agreement with the experimental separation point. Furthermore, the model yields a skin-friction distribution that is close to measurements up to separation. However, it computes a tiny separation zone which is barely existent. Both Wilcox models predict skin-friction distributions that show a small “kink” in the vicinity of cf = 0. This behavior is a result of the applied wall treatment for ω: Wilcox’s procedure (Equation 6.5) yields ωw = 0 for vanishing uτ and this leads to the small kink around cf = 0. While the SALSA model yields cp that is clearly below the predictions obtained with the other one-equation models, it predicts cf that is very close to the predictions obtained with the latter models (Figures 7.17 and 7.18). The modiﬁed k, ω SST model yields the best overall agreement of computational results with measured skin-friction data. From looking at the pressure and skin-friction results, the same conclusion is drawn as in Subsection 7.2: An accurate prediction of surface pressure does not necessarily mean that skin friction is predicted with comparable accuracy. 0.4 0.4 0.6 0 0.2 cp 0 0.2 cp c) a) -1 -1 x/D S1exp x/D 1 1 2 Experiment Spalart-Allmaras Edwards-Chandra SALSA A1exp 2 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 3 3 0.4 0.4 0.6 0 0.2 0 0.2 d) b) -1 -1 0 0 x/D S1exp x/D S1exp 1 1 Figure 7.17: Pressure distributions at cylinder surface for case CS0 (arrows show positions of measured and computed boundary-layer proﬁles). 0 0 S1exp A1exp 0.6 cp cp 0.6 2 Experiment kω SST MBC kω SST modified A1exp 2 Experiment kω TNT RBC Wallin RBC kω LLR RBC A1exp 3 3 122 7 Test Cases Selected cf cf 0 0.001 0.002 0.003 0 0.001 0.002 0.003 c) a) 0 0 x/D S1exp 2 1 A1exp 2 Experiment Spalart-Allmaras Edwards-Chandra SALSA x/D 1 A1exp 3 3 0 0.001 0.002 0.003 0 0.001 0.002 0.003 -1 d) -1 b) 0 0 x/D S1exp 1 A1exp 2 1 A1exp 2 Experiment kω SST MBC kω SST modified x/D S1exp Experiment kω TNT RBC Wallin RBC kω LLR RBC Figure 7.18: Skin-friction distributions at cylinder surface for case CS0 (arrows show positions of measured and computed boundary-layer proﬁles). -1 -1 S1exp Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC cf cf 3 3 7.3 Boundary Layer with Pressure-Induced Separation (Case CS0) 123 124 7 Test Cases Selected Reynolds-Shear-Stress Proﬁles As was done for case BS0, proﬁles of computed and measured Reynolds shear stresses are compared at diﬀerent downstream stations (Figures C.23 to C.32); the Reynolds stresses were evaluated at the same downstream positions as the velocity proﬁles discussed above. Upstream of reattachment A1 in the experiment (x/D < 1.63286) and as for case BS0, the Baldwin-Lomax model and the k, ω 1988 model of Wilcox yield Reynolds shear stress that is too high compared to measurements. However, in contrast to results obtained for case BS0, also Wilcox’s 1998 model, the k, ω TNT model, and all one-equation models compute too high levels of Reynolds stress. This overprediction is most pronounced just upstream of separation S1 in the experiment, that is around x/D = 0.181429. Diﬀerences in the results obtained with one-equation models are again generally less pronounced than diﬀerences among results obtained with k, ω models. The modiﬁed SST model and, to less extend, the k, ω LLR model yield especially good agreement between measured and computed Reynolds shear stress. For x/D ≥ 1.63286, i.e. downstream of reattachment A1 in the experiment, the models tend to underpredict the Reynolds stress level compared to measured values. Regarding streamwise development of maximum Reynolds shear stress, the same conclusions are drawn as for the attached boundary-layer ﬂow discussed in Subsection 7.2: The Baldwin-Lomax model predicts the maximum of u v m too far upstream, thus responding too quickly to the decreasing pressure gradient compared to experimental data; the other models yield results in better qualitative agreement with the experiment and they predict the maximum of u v m further downstream than the Baldwin-Lomax model (Figure 7.19). This is attributed to the fact that all models but the BaldwinLomax model are based on transport equations and are therefore able to qualitatively account for ﬂow history eﬀects. However, even the transportequation models predict the maximum of u v m too far upstream compared with experimental results. Downstream of its maximum, all models yield a decrease of u v m in downstream direction. This is in contrast to the experiment where an increase in u v m up to the last measurement station is found. Hence, downstream of reattachment A1 in the experiment (x/D ≥ 1.63286) all model predictions are in poor agreement with the measured u v m . Up to x/D = 1.08857, the modiﬁed k, ω SST model yields the best agreement of streamwise development of u v m with the experimental results. 7.3 125 Boundary Layer with Pressure-Induced Separation (Case CS0) The non-linear model of Wallin is able to correctly predict anisotropy of the Reynolds stress tensor in the separation region (Figures C.33). However, model predictions for u u and v v are below experimental results. Underprediction of u u is especially pronounced near the wall while underprediction of v v is larger in the outer region than near the wall; predictions for v v are in better overall agreement with the experiment than for u u . Note that in the ﬂow case considered, u u and v v contribute only marginally to the momentum equations and a correct prediction is not important to get this ﬂow “right”. This statement is true for all ﬂow cases considered in this work and, therefore, u u and v v are not included in the discussions of results. Summary To gain a qualitative overview of turbulence-model performance for case CS0, the computational results are rated against the experimental results. Table 7.8 reports ratings in the same manner as for case BS0 shown in Subsection 7.2. In addition, the last column contains the sum of the ratings for both cases BS0 and CS0. Note that for the ratings regarding predictions for u not only the velocity proﬁles but also the predictions of the ﬂow topology are considered. For case CS0, the best overall results yield the modiﬁed k, ω SST model and the model of Wallin. Table 7.8: Turbulence-model performance for case CS0 Model k, ω SST modiﬁed Wallin RBC k, ω SST MBC k, ω 98 WBC cp cf u −u v m " BS0 + CS0 +++ + 4+ 15+ 2(+ + +) 2(−−) + 3+ 10+ 2(++) 2(−−) + + 2+ 12+ 2(+) 2(−−) ++ − 1− 15+ k, ω LLR RBC 2(++) 2(− − −) 2− SALSA 2(++) 2(− − −) − 3− 7+ Spalart-Allmaras 2(−−) + − 4− 6+ Edwards-Chandra 2(−−) + − 4− 7+ 2(−−) −− −− 8− 14− k, ω 88 WBC k, ω TNT RBC 2(−−) 2(−) −− − 9− 7− Baldwin-Lomax 2(− − −) 2(− − −) −−− −−− 18− 28− 2 <u’v’>m /0.001Ur 2 <u’v’>m /0.001Ur -1.5 -2 -2.5 -3 -3.5 -4 -4.5 -5 -1.5 -2 -2.5 -3 -3.5 -4 -4.5 -1 c) -1 a) -0.5 -0.5 0.5 x/D x/D 0.5 1 1 1.5 A1exp 1.5 A1exp 2 2 -1.5 -2 -2.5 -3 -3.5 -4 -4.5 -5 -1.5 -2 -2.5 -3 -1 d) -1 -0.5 -0.5 0.5 x/D 0 0.5 x/D S1exp Experiment kω SST MBC kω SST modified 0 S1exp b) -3.5 -4 Experiment kω TNT RBC Wallin RBC kω LLR RBC -4.5 -5 Figure 7.19: Streamwise development of u v m for case CS0. 0 S1exp Experiment Spalart-Allmaras Edwards-Chandra SALSA 0 S1exp Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2 <u’v’>m /0.001Ur 2 <u’v’>m /0.001Ur -5 1 1 1.5 A1exp 1.5 A1exp 2 2 126 7 Test Cases Selected 7.4 Separated Airfoil Flow (Case AAA) 7.4 127 Separated Airfoil Flow (Case AAA) This subsection focuses again on the separated ﬂow around the AerospatialeA airfoil that was discussed in detail in Section 4. Results computed with diﬀerent turbulence models are now compared; the same computational grid and setup as the one presented in Subsection 4.2 were employed to perform the computations. Although results obtained with the Johnson-King model are presented in Section 4 they are included in this subsection for reference. A similar procedure for presenting computational results is pursued as in Subsection 4.3: Proﬁles of streamwise velocity and Reynolds shear stress were evaluated at twelve diﬀerent downstream stations from x/c = 0.3 to x/c = 0.99 (see Figure 4.9) to investigate streamwise development. The graphs are placed, however, in Appendix C.3 so as not to overload this section with a large number of ﬁgures. 7.4.1 Computational Results and Discussion Flow Topology and Velocity Proﬁles Evaluating streamlines at the trailing edge reveals that none of the turbulence models is able to predict a recirculation zone that is of comparable size with the one found in the experiment: While the Johnson-King model, the k, ω SST model and the SALSA model yield the largest separation zones, recirculation zones predicted by the models are generally smaller than the measured one (Figure 7.20). (Note that S1exp and A2exp shown in Figure 7.20 were evaluated from the measured streamfunction while in Figures 7.21, 7.22 and 7.23, S1exp relates to the position of zero skin friction. A2exp could not be evaluated with the latter method; see Subsection 4.3.2.) Regarding prediction of streamwise velocity, a consistent trend is encountered for all turbulence models: All models predict too high values of u in the boundary layer resulting in fuller velocity proﬁles compared with measurements (Figures C.34 – C.45). Overall predictive accuracy is poor and diﬀerences between measured and computed velocity proﬁles increase when approaching the trailing edge of the airfoil. The models do not suﬃciently respond to decreasing pressure gradient. This ﬁnding is in contrast to the results for the separated boundary layer (case CS0) discussed in Subsection 7.3. There, it was found that models responded too quickly to changes in pressure gradient compared to measurements. Regarding the airfoil ﬂow, the BaldwinLomax model yields velocity proﬁles in poorest agreement with experimental results while the Johnson-King results are closest to measurements. 128 7 Test Cases Selected y/c 0.02 0 S1exp Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC -0.02 -0.04 0.85 0.9 x/c A2exp 0.95 a) 1 y/c 0.02 0 S1exp Experiment kω TNT RBC Wallin RBC kω LLR RBC -0.02 -0.04 0.85 0.9 x/c A2exp 0.95 b) 1 y/c 0.02 0 S1exp Experiment Spalart-Allmaras Edwards-Chandra SALSA -0.02 -0.04 0.85 0.9 x/c A2exp 0.95 c) 1 y/c 0.02 0 S1exp Experiment kω SST MBC kω SST modified Johnson-King -0.02 -0.04 0.85 0.9 x/c 0.95 A2exp d) 1 Figure 7.20: Topology of separation zones for airfoil A; separating streamlines are shown. 7.4 Separated Airfoil Flow (Case AAA) 129 Pressure and Skin-Friction Distribution For all models, computed cp distributions are above measured values at the rear part of the airfoil, that is for x/c > 0.8 (Figure 7.21). (Note that in the ﬁgure the direction of the cp axis is reversed and graphs of larger cp values appear below graphs of lower cp values. This way of presenting cp distributions is common practice for airfoil ﬂows and is therefore applied in this work.) The plateau in the experimental pressure distribution encountered at the trailing edge on the upper surface of the airfoil is characteristic for separated ﬂow regions; none of the turbulence models is able to reproduce this pressure plateau. Diﬀerences between computed surface-pressure distributions are visible mainly in the region of the suction peak on the upper surface: While results obtained with the Johnson-King model are in close agreement with measurements, all other models overpredict the suction peak compared to experimental values. The Baldwin-Lomax model yields the strongest suction peak and is therefore in poorest agreement with experimental results. This overprediction of suction peak correlates with predicted values for lift coeﬃcient: The model of Baldwin & Lomax yields the highest value for cL while the Johnson-King model predicts the lowest lift (Table 7.9); the latter model yields also the lowest suction peak. All models have in common that they overpredict cL compared to F2 measurements. Figure 7.22 compares skin-friction distributions on the upper surface of the airfoil for the region where experimental values are available (0.3 ≤ x/c ≤ 0.99). One can see that all models tend to predict cf = 0 further downstream than found in the experiment. However, for the SALSA model, the oneequation model of Edwards & Chandra, and Menter’s SST model, the x/c position where cf = 0 is in very close agreement with measurements; the models of Wilcox and Wallin do not predict separation at all. While both Menter’s SST model and the Johnson-King model yield similar separation zones, they yield clearly diﬀerent skin-friction results. cp cp 0.5 x/c 0.75 S1exp Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.75 1 1 -1 -2 -3 -4 -5 1 0 0 d) b) 0.25 0.25 Figure 7.21: Pressure distributions for airfoil A. 1 0.25 x/c 0.5 1 c) 0.25 0 -1 -2 -3 -4 0 0 0 S1exp Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC -5 0 -1 -2 -3 -4 -5 1 0 -1 -2 -3 -4 a) cp cp -5 0.5 x/c 0.5 x/c S1exp 0.75 S1exp Experiment kω SST MBC kω SST modified Johnson-King 0.75 Experiment kω TNT RBC Wallin RBC kω LLR RBC 1 1 130 7 Test Cases Selected 0 0.002 0.004 0.006 0.5 x/c 0.6 x/c 0.6 0.7 0.7 0.9 0.8 0.002 0.004 0.006 -0.002 0.3 0.3 d) b) 0.4 0.4 0.5 0.5 Figure 7.22: Skin-friction distributions for airfoil A. 0.9 S1exp Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.8 -0.002 0.4 0.5 0 0.002 -0.002 c) 0.4 S1exp 0.004 0.006 0 0.3 0.3 a) Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 0 0.002 0.004 0.006 -0.002 cf cf cf cf x/c 0.6 x/c 0.6 0.7 0.7 0.9 0.8 0.9 S1exp Experiment kω SST MBC kω SST modified Johnson-King 0.8 S1exp Experiment kω TNT RBC Wallin RBC kω LLR RBC 7.4 Separated Airfoil Flow (Case AAA) 131 132 7 Test Cases Selected Global Aerodynamic Coeﬃcients Computed and measured aerodynamic coeﬃcients are compared in Table 7.9. Because of separation, total drag consists mainly of pressure drag. Hence, the size of the recirculation zone is qualitatively reﬂected in the drag coeﬃcient: Turbulence models that yield larger recirculation zones yield also larger values for cD (Figure 7.20 and Table 7.9). For example, the Johnson-King model computes the largest separation zone and the largest drag coeﬃcient among all model predictions. On the contrary, the Baldwin-Lomax yields a tiny recirculation zone that is not recognizable in Figure 7.20 and, at the same time, it predicts the lowest drag of all turbulence models (Table 7.9). All models predict signiﬁcantly lower drag coeﬃcients than found in the experiment F2. This is consistent with the fact that all models yield recirculation zones that are smaller in size than the measured one. Besides inﬂuencing drag, trailing-edge separation also inﬂuences the lift coeﬃcient: Large separation regions reduce the circulation around the airfoil, and therefore lift, compared to attached ﬂow. Consistently, turbulence models, which yield large separation zones, predict low values of lift, the Johnson-King model being the most prominent example (Figure 7.20 and Table 7.9). In addition, circulation around the airfoil inﬂuences the pressure distribution which, in turn, has a direct impact on the separation itself. This complex viscous/inviscid interaction between boundary-layer separation and inviscid ﬂow makes separated airfoil ﬂow a demanding test case for turbulence models. Yet, thorough identiﬁcation of reasons for model failure are diﬃcult to perform due to the strong coupling between ﬂow phenomena. The coupling between boundary layer separation and pressure distribution also eﬀects the computational results for the moment coeﬃcient cM : A positive value for cM is predicted by the Johnson-King model and the two k, ω SST models (Table 7.9). These models yield the best agreement of cp with the measurements at the suction peak and the trailing edge, see Figure 7.21. Additionally, they yield also the largest separation regions (Figure 7.20). In contrast, negative values for cM are predicted by all other models. They compute cp in relatively poorer agreement with the experiment and yield a smaller recirculation zone. Due to the lack of experimental data for cM , no general statement about the computational results can be made. Summarizing, one can say that the turbulence models investigated yield too high a value of cL , show no characteristic trailing-edge plateau in the cp 7.4 133 Separated Airfoil Flow (Case AAA) Table 7.9: Force and moment coeﬃcients for airfoil A cL cD cM total total pressure friction total Experiment (F1) 1.56 0.021 – – – Experiment (F2) 1.52 0.031 – – – Baldwin-Lomax 1.725 0.0158 0.0097 0.0061 −0.0165 Johnson-King 1.531 0.0260 0.0197 0.0065 +0.0055 k, ω 88 WBC 1.647 0.0175 0.0108 0.0067 −0.0044 k, ω 98 WBC 1.630 0.0176 0.0112 0.0065 −0.0022 k, ω SST MBC 1.568 0.0181 0.0127 0.0054 +0.0052 k, ω TNT RBC 1.627 0.0166 0.0108 0.0058 −0.0014 k, ω LLR RBC 1.650 0.0168 0.0102 0.0066 −0.0045 Wallin RBC 1.657 0.0153 0.0099 0.0054 −0.0044 Spalart-Allmaras 1.671 0.0166 0.0108 0.0058 −0.0088 Edwards-Chandra 1.643 0.0166 0.0114 0.0052 −0.0050 SALSA 1.624 0.0173 0.0119 0.0054 −0.0029 k, ω SST modiﬁed 1.604 0.0167 0.0103 0.0064 +0.0021 distribution, and predict cf = 0 too far downstream (compared to measurements). Reynolds-Shear-Stress Proﬁles For all turbulence models, agreement between predicted and measured Reynolds shear stresses is fair up to x/c = 0.6 (see Figures C.46 to C.49). However, clear diﬀerences among computed Reynolds-shear-stress proﬁles are seen in the outer part of the boundary layer. Compare, for example, results obtained with the Wallin and k, ω LLR model in Figure C.46. One can see, that the Wallin model yields signiﬁcantly lower Reynolds stresses than the LLR model in the outer part of the boundary layer. Downstream of x/c = 0.6 basically three things are encountered (see Figures C.50 to C.57): 1. All turbulence models predict less Reynolds stress than measured. This can also be seen in Figure 7.23 where streamwise development of maximum Reynolds shear stress is shown. The model of Wallin yields the lowest values of −u v m while the Baldwin-Lomax model and Menter’s SST Model predict the highest values. 134 7 Test Cases Selected 2. In the experimental results downstream of x/c = 0.7, a second local maximum in the −u v proﬁles is found in the outer part of the boundary layer. This kink is not reproduced by any turbulence model, not even qualitatively, and it may be an additional hint for three-dimensional eﬀects in the ﬂow. 3. In the experimental Reynolds-stress proﬁles, the global maximum of −u v is moving further away from the wall with increasing downstream position. The SST models and the Johnson-King model are the only turbulence models that are able to predict this eﬀect in fair agreement with measurements. All other models underpredict outward movement of −u v m with streamwise position. As a ﬁnal note, turbulence models that make use of Bradshaw’s assumption (−u v /k = constant) yield Reynolds-shear-stress proﬁles in better agreement with measured proﬁles than models which do not build on this assumption. However, this holds only for −u v ; for example, the one-equation models yield better agreement of cf with the measurements than the SST models or the Johnson-King model. 2 2 0.002 0.3 0.003 0.004 0.005 0.006 0.007 0.002 0.3 0.003 0.004 0.005 0.006 0.007 0.4 c) 0.4 a) x/c 0.7 0.8 0.9 0.6 x/c 0.7 0.8 0.9 Experiment S1exp Spalart-Allmaras Edwards-Chandra SALSA 0.6 S1exp 1 1 0.002 0.3 0.003 0.004 0.005 0.006 0.007 0.002 0.3 0.003 0.004 0.005 0.006 0.007 0.4 d) 0.4 b) 0.5 0.5 x/c 0.7 0.8 0.6 x/c 0.7 0.8 Experiment kω SST MBC kω SST modified Johnson-King 0.6 Experiment kω TNT RBC Wallin RBC kω LLR RBC Figure 7.23: Streamwise development of −u v m for airfoil A. 0.5 0.5 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2 2 -<u’v’>m /U∞ -<u’v’>m /U∞ -<u’v’>m /U∞ -<u’v’>m /U∞ 0.9 S1exp 0.9 S1exp 1 1 7.4 Separated Airfoil Flow (Case AAA) 135 136 7 Test Cases Selected Summary of Model Performance A summary of predictive accuracy of turbulence models is given in the same manner as introduced in Subsections 7.2 and 7.3. Corresponding ratings for the ﬂow over airfoil A are listed in Table 7.10. According to the table, the Johnson-King model oﬀers the highest predictive accuracy for this ﬂow case followed by Menter’s original SST model. Generally, the performance of the models for the present ﬂow case is poor compared with the performance achieved for the ﬂow cases discussed above (see Tables 7.7 and 7.8). Table 7.10: Turbulence-model performance for airfoil A Model Johnson-King k, ω SST MBC cp cf u −u v m " 2(++) − + 4+ 2(+) − ++ 3+ 2(−) 2(+ + +) −− −− Edwards-Chandra 2(−−) 2(++) −−− −− 5− Spalart-Allmaras 2(−−) 2(+) −−− −− 7− 2(−) 2(−) −− − 7− SALSA k, ω SST modiﬁed k, ω TNT RBC 2(−−) −− −− 8− Baldwin-Lomax 2(− − −) 2(−−) −−− + 12− k, ω 98 WBC 2(−−) 2(−−) −−− −−− 14− Wallin RBC 2(−−) 2(−−) −−− −−− 14− k, ω 88 WBC 2(−−) 2(− − −) −−− −−− 16− k, ω LLR RBC 2(−−) 2(− − −) −−− −−− 16− 8 Numerical Issues 8.1 Grid Convergence Most of this work was concerned with the predictive accuracy of the turbulence models employed. This section, however, deals with an equally important issue, namely computational accuracy. Assessment of model accuracy is subject to validation while determination of computational accuracy is known as veriﬁcation. (For an in-depth discussion about veriﬁcation and validation the reader is referred to Roache (1998).) Note that rigorous grid-dependence analysis is mandatory for any computational study; computational results obtained without a thorough investigation of grid dependence cannot be trusted. Grid-convergence studies presented in the following rely on generalized Richardson extrapolation. The latter was used to obtain a grid-convergence index (GCI) which relates results from any grid-convergence test to the expected results from grid doubling using a second-order accurate method. This yields an “objective asymptotic approach to quantiﬁcation of uncertainty of grid convergence” (Roache, 1998). In addition, by performing computations on three diﬀerent grids the observed order of accuracy was determined. The following relations were applied: Observed Order of Accuracy: f3 − f2 p = ln / ln(2), (8.1) f2 − f1 Grid-Convergence Index: GCI [grid level 1] = 1.25 GCI [grid level 2] = 1.25 2 | | f1f−f 1 2p − 1 3 | | f2f−f 2 2p − 1 , (8.2) . (8.3) p denotes the observed order of accuracy; f1 , f2 and f3 are the solutions on the ﬁne, intermediate and coarse grid, respectively. Any ﬂow variable can be utilized for this purpose but in the present work skin-friction coeﬃcients or related variables are utilized since these are a good indicator for accuracy. The grid reﬁnement ratio between grids employed for the convergence tests was always two. This means that grid level one had twice the resolution of grid level two. Hence, grid level two was obtained by omitting every other grid 138 8 Numerical Issues point from level one. Correspondingly, level three was obtained by omitting every other grid point from level two. Using GCI values obtained with Equations 8.2 and 8.3 asymptotic range of (grid) convergence is achieved if the following relation holds: GCI [grid level 2] ≈ 2p GCI [grid level 1]. (8.4) In addition, if Equation 8.4 holds, that means if all solutions obtained are in the asymptotic range, then the GCI of grid level 3 is approximated by GCI [grid level 3] ≈ 2p GCI [grid level 2]. (8.5) Grid-convergence studies were performed for every ﬂow case discussed in this work to assess computational accuracy of the ﬂow solutions obtained. However, due to the high computational eﬀort necessary for computing ﬂow solutions on ﬁne grids grid convergence was veriﬁed in each case with only one turbulence model; transport-equation models were exclusively utilized for this purpose. In some cases, grid-convergence tests with diﬀerent models were pursued, and qualitatively similar results were obtained when using diﬀerent models. This indicates that a grid-convergence index obtained from computations with a given turbulence model is also representative for computations on the same grids with other models. Flat Plate To conduct the analysis of predictive accuracy presented in Section 7.1 the computational grid contained 144 × 64 cells. In order to investigate grid convergence, additional solutions were computed on two ﬁner grids consisting of 288 × 128 and 576 × 256 cells. Skin-friction coeﬃcients evaluated at x/L = 0.361511 served to compute the grid-convergence indices. (x/L = 0.361511 is the grid point closest to the position where Reθ = 2900 was found on the standard grid, that is grid level 3.) The results are reported in Table 8.1. Two conclusions follow from the results shown in Table 8.1: a) Asymptotic convergence of results is achieved and b) the GCI for the standard grid is approximately 3.78 percent. This means that the estimated error compared with the exact solution is 3.78 percent including 1.25 as a factor of safety. (This is a rather sloppy deﬁnition of GCI and it is used here for simplicity – see Roache (1998) for a more precise description of GCI). While the numerical method applied is formally second-order accurate, the observed order is as low as 0.82 for this ﬂow case. This shows that in 8.1 139 Grid Convergence Table 8.1: Grid-convergence results for ﬂat plate determined from solutions obtained with Wilcox’s 1988 k, ω model in combination with Rudnik’s surface condition for ω (k, ω 88 RBC) Variable cf at x/L = 0.361511 p 0.82 GCI [576 × 256] 1.21% GCI [288 × 128] 2.14% 2p GCI [576 × 256] 2.13% 2p GCI [288 × 128] 3.78% actual applications of numerical schemes the formal order is not necessarily achieved. This is especially true on extremely stretched or skewed grids. Even larger values of p than the formal order can be observed if explicit artiﬁcial damping terms of high order are used in the numerical scheme. This will be shown below. Driver’s Cylinder Flow Separated ﬂows are much more sensitive to grid reﬁnement or coarsening than attached ﬂows. Therefore, and for reasons of brevity, only grid-convergence results for case CS0 (with separation) are reported for the cylinder ﬂow of Driver. (For case BS0, smaller values of GCIs were obtained and, therefore, better grid convergence was achieved compared to case CS0.) For the cylinder ﬂow, local skin-friction coeﬃcients were the basis for the GCI evaluation. The coeﬃcients were evaluated at x/D = −0.511. On the standard grid, this is the coordinate of the surface grid point located closest to the boundary-layer proﬁles at −0.544286 (see for example Figure 7.18). The standard grid contained 256 × 64 cells and it was used as level two for the grid-convergence studies. This is also the grid which was used for validation of turbulence models discussed in Subsection 7.3. A coarser grid with 128×32 cells and a ﬁner grid featuring 512 × 128 cells were utilized as level three and one, respectively. From looking at Table 8.2, one can see that the asymptotic range is achieved and that the GCI on the standard grid is 5.32 percent demonstrating that good computational accuracy is obtained on the standard grid. Like 140 8 Numerical Issues Table 8.2: Grid-convergence results for the separated cylinder ﬂow (case CS0) determined from solutions obtained with Wilcox’s 1998 k, ω model in combination with Wilcox’s surface condition for ω (k, ω 98 WBC) Variable cf at x/D = −0.54428 p 1.44 GCI [512 × 128] 2.01% GCI [256 × 64] 5.32% 2p GCI [512 × 128] 5.45% in the case of the ﬂat plate computations, the observed order of accuracy is lower than the formal order of the numerical scheme. To conﬁrm the consistency of the GCI results shown in Table 8.2, gridconvergence indices for the minimum value of cf and for cD due to skin-friction (both not shown) were evaluated. Very similar results to the ones in Table 8.2 were obtained. Airfoil A For airfoil A, grid convergence for two diﬀerent variables was investigated, viz. global drag coeﬃcient due to friction, and global lift coeﬃcient. The ﬁnest grid level employed consisted of 512 × 128 cells, being also the grid which was used for studying the performance of turbulence models for this ﬂow case. Two coarser grids were obtained by keeping every other and every fourth grid point resulting in 256 × 64 and 144 × 32 cells for level two and three, respectively. Observed orders of convergence and GCIs diﬀer signiﬁcantly for cD and cL (Table 8.3). At the ﬁrst glance these larger disparities seem surprising but the following discussion is believed to oﬀer a plausible explanation: Computed skin friction is much more sensitive to artiﬁcial damping of the numerical scheme than computed pressure. (The latter contributes by far to the largest part to the lift coeﬃcient.) In the current ﬂow case exclusively fourth-order damping terms are active since no shocks or discontinuities are present. The artiﬁcial damping terms are designed to damp so-called Wiggle-modes which are the smallest resolvable wave lengths; inﬂuence of damping is therefore bound to the grid spacing. While the eﬀect of damping does not change relative to the grid spacing when reﬁning the grid it is successively reduced 8.2 141 Local Preconditioning for Low Mach Numbers compared to gradients of ﬂow variables. Therefore, increased grid resolution and decay of damping terms both promote a “sharper” resolution of ﬂow gradients. Since gradients of velocity normal to the wall provide skin friction, decay of artiﬁcial damping terms consequently contributes to the accuracy of computed skin friction. This means that the eﬀect of reﬁning the grid from level two to level one is more pronounced for skin friction than for pressure; (f3 −f2 )/(f2 −f1 ) is much smaller for cD than for cL (Equation 8.1). However, good computational accuracy as well as asymptotic range of the solution is achieved on the grid with 512 × 128 cells for airfoil A for both quantities (Table 8.3). Table 8.3: Grid-convergence results for airfoil A determined from solutions obtained with the SALSA model 8.2 Variable cD due to friction cL p 0.367 4.29 GCI [512 × 128] 5.98% 0.016% GCI [256 × 64] 7.82% 0.309% 2p GCI [512 × 128] 7.71% 0.308% Local Preconditioning for Low Mach Numbers All computations for this work were performed with numerical methods designed for the solution of the compressible Navier-Stokes equations. In the low-subsonic Mach-number regime, however, such methods generally do not perform well. When the magnitude of ﬂow velocity becomes small compared to speed of sound, convective terms render the ﬂow equations stiﬀ. This leads to numerical diﬃculties. The reason is that the numerical scheme has to account for the large disparity between the ﬂuid velocity u and the acoustic wave speed u + a at which pressure signals are transported downstream. In addition, in explicit methods the time step is restricted and set proportional to 1/(u + a). Hence, in low-Mach-number regions of a ﬂow ﬁeld, the local time step is mainly controlled through speed of sound. This means that the numerical solution of the time marching scheme is advanced using time steps well-adapted to the propagation speed of acoustic waves. However, waves associated with u are excessively “over-resolved”. For the latter, a much larger time step would be possible. 142 8 Numerical Issues A solution to this problem is the application of preconditioning for the Navier-Stokes equations, which was implemented into the MUFLO code during this work. In the following, the Euler equations are discussed for simplicity but the same arguments apply to the full Navier-Stokes equations. According to the preconditioning procedure presented in Turkel, Radespiel & Kroll (1997) the Euler equations are considered in two dimensions and in diﬀerential form: ∂w ∂w ∂w P −1 +A +B =0. (8.6) ∂t ∂x ∂y Matrices A and B depend on the set of variables w. In the numerical methods considered, w usually contains the conserved variables, i.e. w = (ρ, ρu, ρv, ρE)T . In the framework of preconditioning, sets of variables that yield sparse and (nearly) diagonal forms of matrices A and B are preferred. Analysis of eigenvalues, that is propagation speeds, and construction of an appropriate preconditioning matrix P are signiﬁcantly eased in this case. The task is to ﬁnd a matrix P −1 that substantially reduces or removes the disparity in the eigenvalues of the system. However, the choice of P −1 is not unique. Note that P −1 does not alter the solution in the limit of steady state since the time derivative vanishes. It does alter the way in which this limit is reached, though, and this is exactly what is wanted. Clearly, preconditioning can be used only for the solution of steady ﬂow problems or within dual-time-stepping schemes for unsteady ﬂows. The preconditioning method adapted in this work is closely patterned after the work of Turkel et al. (1997). In particular, the inverse of the preconditioning matrix P −1 is chosen to be ⎞ ⎛ βMr2 βM 2 0 0 − a2r δ 2 a ⎟ ⎜ αu ⎜ − αu2 1 0 δ ⎟ ρa ρa2 ⎟ , (8.7) P=⎜ ⎜ − αv 0 1 αv δ ⎟ ⎠ ⎝ ρa2 ρa2 0 0 0 1 and w = (p, u, v, T )T is the set of variables. The parameters α and δ are used to generalize the preconditioning matrix. For δ = 0 the preconditioner suggested by Turkel (1987) is obtained, while for δ = 1 and α = 0 the preconditioner introduced by Choi & Merkle (1993) is recovered. In the present study, best results were obtained when setting δ = 1 and α = 0. Special attention must be paid to element (1,1) of P when the local Mach 2 number approaches zero. Normally, βMr2 = a2local Mlocal but when the local 8.2 Local Preconditioning for Low Mach Numbers 143 Mach number becomes very small, for example around stagnation points or critical points in general, it is necessary to bound βMr2 away from zero, since βMr2 = 0 leads to a singular preconditioning matrix. This is done by setting 2 βMr2 = max(a2local Mlocal , εa2local M 2 ). The value of ε has to be speciﬁed by the user and it was found to be dependent on Reynolds number and grid spacing. For example, for an airfoil ﬂow, a Reynolds number based on chord length of a few millions and a very ﬁne grid may require ε values of six and higher in order to weaken the eﬀect of preconditioning and to stabilize the computation. This reﬂects the fact that although local preconditioning provides advantages like accuracy improvements and very often substantial convergence speed-up, it leads to reduced robustness. The reader may be referred to Lee (1996) for a detailed theoretical discussion of preconditioning and its implications. To study the eﬀect of preconditioning on the accuracy of the numerical solution, laminar ﬂow around a NACA 0012 airfoil at zero incidence and Re = 5000 was used as a test case. The ﬂow over the airfoil was computed for two diﬀerent Mach numbers, M = 0.15 and M = 0.05. In both cases the numerical solutions were obtained with and without local preconditioning. Without preconditioning the surface-pressure distribution along the airfoil degenerates with decreasing Mach number. Following the Prandtl-Glauert √ rule (cp = cp,0 / 1 − M 2 ), the curve of the pressure distribution for M = 0.15 should be slightly above the one for M = 0.05; this is obviously not the case (Figure 8.1). Lowering the Mach number also increases pressure oscillations near the trailing edge, for which compressible ﬂow solvers are known in general. These spurious oscillations are due to artiﬁcial damping terms which are scaled with the “acoustic” eigenvalues u + a and v + a. 2 With preconditioning (and setting βMr2 = max(a2local Mlocal , 0.05a2local M 2 )) the curves for cp diﬀer only by the Prandtl-Glauert factor (Figure 8.2). In addition, pressure oscillations near the trailing edge are absent due to correct scaling of artiﬁcial damping terms. The skin-friction distributions along the airfoil computed without preconditioning are presented in Figure 8.3. Results discussed by Schlichting (1982) indicate that, in general, cf should decrease for increasing Mach numbers if an adiabatic wall is assumed. This trend is reproduced qualitatively correct by the numerical results (Figure 8.3). However, for the very low Mach numbers considered in this study, cf depends only marginally on M and the curves for cf should be indistinguishable, which is obviously not correctly reproduced. In contrast, and according to theory, the curves obtained with preconditioning lie on top of each other and the small unphysical increase in 144 8 Numerical Issues -0.5 M = 0.05 M = 0.15 -0.4 -0.3 cp -0.2 -0.1 0 0.1 0.2 0 0.25 0.5 0.75 x/c Figure 8.1: Pressure distributions for NACA 0012, Re = 5000, α = 0◦ , laminar ﬂow, without preconditioning. cf at the trailing edge (Figure 8.3) is not present in the solution obtained with preconditioning (Figure 8.4). -0.5 M = 0.05 M = 0.15 -0.4 -0.3 cp -0.2 -0.1 0 0.1 0.2 0.3 0 0.25 0.5 0.75 1 x/c Figure 8.2: Pressure distributions for NACA 0012, Re = 5000, α = 0◦ , laminar ﬂow, with preconditioning. 8.2 145 Local Preconditioning for Low Mach Numbers Note that results presented in this section are not conﬁned to the present numerical methods with central diﬀerencing in combination with artiﬁcial damping terms. Numerical methods based on upwind schemes show similar diﬃculties in obtaining accurate solutions when the Mach number becomes small. 0.15 0.125 M = 0.05 M = 0.15 0.1 cf 0.075 0.05 0.025 0 -0.025 -0.05 0 0.25 0.5 0.75 1 x/c Figure 8.3: Skin-friction distributions for NACA 0012, Re = 5000, α = 0◦ , laminar ﬂow, without preconditioning. For the ﬂow over airfoil A computed with the Johnson-King model, viz. with the MUFLO code, ε had to be set to four in order to stabilize the computation. This explains the oscillations in cf obtained with the JohnsonKing model at the trailing edge (see Figure 7.22 d)): In this region, due to the relatively high value of ε, the eﬀect of preconditioning was not strong enough to completely remove the numerical artifacts. For this reason, preconditioning in FLOWer was modiﬁed and locally diﬀerent values of ε were employed. This procedure yielded nearly optimal preconditioning at the trailing edge of the airfoil. In order to completely resolve numerical diﬃculties associated with low Mach numbers, application of an alternative preconditioner might be considered, but this was out of scope of the present work. The rise of the friction coeﬃcient at the trailing edge, like that shown in Figure 8.3, is ampliﬁed if metric terms of the computational grid are not smooth in the streamwise direction across the trailing edge. Therefore, it is mandatory to use computational grids with a smooth distribution of metric 146 8 Numerical Issues 0.15 0.125 M = 0.05 M = 0.15 0.1 cf 0.075 0.05 0.025 0 -0.025 -0.05 0 0.25 0.5 0.75 1 x/c Figure 8.4: Skin-friction distributions for NACA 0012, Re = 5000, α = 0◦ , laminar ﬂow, with preconditioning. terms everywhere in the domain. Many grid generators, however, produce grids with smooth metrics along the airfoil and along the wake but not also across the trailing edge. 8.3 Transition Transition in the RANS computations was achieved by “activating” the turbulence models downstream of a prescribed plane in the ﬂow ﬁeld. In the case of algebraic models, this was performed by simply setting µt = 0 upstream of the transition location. In the framework of transport-equation models, the production term was explicitly set to a very small value (virtually to zero) in the laminar region. This limiter was released downstream of transition letting production grow to “standard” levels. Figure 8.5 shows iso-contours of k in the transition region computed by Wilcox’s 1988 k, ω model in combination with Wilcox’s wall treatment for ω. One can see that signiﬁcant k values are obtained only downstream of the ﬁrst turbulent “i-plane” in the region where the limiter on the production term in the k equation is disabled. Computations of the ﬂow around the Aerospatiale-A airfoil on very ﬁne grids showed that it is diﬃcult to obtain steady-state solutions on grids with more than 512 × 128 cells. Analysis of computations revealed that unsteadiness is triggered from the transitional separation bubble encountered on the 8.3 147 Transition 0.087 0.0005 y/c 1E-05 1E-05 0.086 first turbulent i-plane 0.085 0.115 0.117 x/c 0.119 0.121 Figure 8.5: Modeling of transition for ﬂow over airfoil A; isocontours of k are shown in the transition region on the upper surface of the airfoil; Wilcox k, ω 1988 model, WBC. upper surface of the airfoil. On ﬁne grids, the negative surge in skin friction (see for example Figure 4.7) randomly and rapidly moved up- and downstream and also changed in magnitude. On the one hand, this behavior is rooted in the inherent unsteady physics of the bubble, which is resolved to some degree on the ﬁne grid. On the other hand, however, Reynolds averaging should be active from the beginning of the turbulent boundary layer and therefore suppress unsteadiness arising from transition processes. Spalart & Strelets (1997) performed DNS and RANS computations of a transitional bubble. They showed that in the RANS computation, transition occurred at a lower rate than in the DNS. The turbulence model yielded a skin-friction distribution whose recovery to positive values was too slow compared with DNS. Regarding the ﬂow over airfoil A, numerical experiments revealed that setting the prescribed transition location a little bit more upstream of the experimentally-observed transition location improved steadiness of the ﬂow solution. In particular, the location of transition was prescribed such that cf = 0 in the recovery region was obtained at the experimentallyobserved transition location (x/c = 0.12). Good results were obtained using x/c = 0.1167 for the transition location. Thus, delayed transition of the kind reported by Spalart & Strelets is believed to be the reason for the unsteadiness encountered for airfoil A. As mentioned in Subsection 6.1, Wilcox developed low-Reynolds number modiﬁcations for his models. Computations of the ﬂow around airfoil A using 148 8 Numerical Issues Wilcox’s 1988 k, ω model were performed with and without application of lowReynolds-number terms. The main eﬀect of the modiﬁcations is that the rate at which the model approaches the fully-turbulent state is slightly increased compared to computations performed without low-Reynolds-number terms. By including viscous corrections in the computation, skin-friction recovery from negative values in the transitional separation bubble to the positive peak just downstream of transition is more rapid (Figure 8.6). This means that delay of transition is somewhat reduced by applying low-Reynolds-number terms. Thus, if transition is important it is recommended to include the low-Reynolds-number modiﬁcations in computations performed with Wilcox’s k, ω models. However, Figure 8.6 shows that despite the use of viscous corrections the turbulence model is not able to correctly capture the trailingedge separation. 0.015 Experiment kω 88 RBC kω 88 RBC low Reynolds number cf 0.01 0.005 0 0.2 0.4 x/c 0.6 0.8 1 Figure 8.6: Inﬂuence of low-Reynolds-number modiﬁcations on skin-friction; ﬂow over airfoil A computed with Wilcox’s 1988 k, ω model and Rudnik’s surface boundary condition for ω (k, ω 88 RBC). Great sensitivity of the ﬂow solution to transition location in general was found. For example, prescribing transition at x/c = 0.13 instead of 8.4 Artiﬁcial Damping in Boundary Layers 149 x/c = 0.1167 yielded in one case of the computations for airfoil A a massively separated ﬂow ﬁeld with two counter-rotating recirculation zones (Figure 8.8). Therefore, in ﬂow situations where location of transition plays an important role, like separated airfoil ﬂow or dynamic stall to name only two, careful modeling of transition is mandatory. In particular, correctly predicting the increase of skin-friction as well as the change of momentum thickness between the laminar and the turbulent part of the boundary layer is essential. To perform this task a special “transition strip” procedure is necessary where the turbulence model is modiﬁed in order to reproduce the high levels of turbulence, and the associated rapid increase in cf and θ just downstream of transition, found in the experiment. For k, ω models, in addition to viscous corrections, this could be accomplished through special treatment of ω at the wall in the transition region. For example, substantially reducing ωw increases k throughout the boundary layer which, in turn, increases the rate at which transition occurs. Transition modeling is a broad ﬁeld and is out of the scope of this work. However, it is noted and emphasized that transition and how it is modeled can signiﬁcantly inﬂuence computational results even for RANS computations. This cannot be ignored as is often done in commercial CFD codes. (However, many of the latter are intended for internal ﬂows where transition is less important.) 8.4 Artiﬁcial Damping in Boundary Layers It was noted in Section 5 that the numerical method employed for this work uses second- and fourth-order artiﬁcial damping terms to prevent odd-even decoupling and to damp spurious oscillations. For subsonic ﬂows, only the damping terms based on fourth-order diﬀerences are active. Although these terms are necessary to stabilize the central scheme they can pollute the solution through introduction of too much diﬀusion. In boundary layers, for example, large gradients of tangential momentum ﬂux normal to the wall lead to high values of damping terms. However, viscous momentum ﬂux normal to the wall is suﬃciently high to provide the necessary damping of the numerical scheme in this direction. Therefore, artiﬁcial damping in boundary layers in the wall-normal direction can be reduced or even entirely omitted. This was accomplished by pre-multiplying the corresponding artiﬁcial damping term by (M/M∞ )2 in the FLOWer code. 150 8 Numerical Issues To study eﬀects of reducing artiﬁcial viscosity in boundary layers, solutions of the ﬂow over airfoil A computed with full and reduced damping were analyzed. In Figure 8.7, corresponding skin-friction distributions on the upper side of the airfoil are shown. 0.015 full artificial damping reduced artificial damping cf 0.01 0.005 kω 88 RBC model transition at x/c=0.1113 0 0 0.25 x/c 0.5 0.75 Figure 8.7: Inﬂuence of explicit artiﬁcial damping on skinfriction; ﬂow over airfoil A computed with Wilcox’s 1988 k, ω model and Rudnik’s surface boundary condition for ω (k, ω 88 RBC). In the computation with reduced artiﬁcial damping the laminar part of the boundary layer remains attached while in the case with full damping the laminar boundary separates and a transitional bubble is encountered. It is noted that all computations of the ﬂow over airfoil A presented in the sections above show laminar separation despite the fact that reduction of artiﬁcial damping was applied in each case. The reason is, that in the current case transition was prescribed further upstream, namely at x/c = 0.1113, than in the computations presented above. This was necessary to yield a steadystate solution for the computation with full damping. (Specifying transition at x/c = 0.1167 led to an unsteady transitional bubble as discussed in Section 8.3.) 8.4 Artiﬁcial Damping in Boundary Layers 151 Reducing artiﬁcial damping obviously leads to an increased lateral momentum ﬂux towards the wall in the laminar boundary layer. This yields a higher level of cf compared to the computation with full artiﬁcial damping (Figure 8.7). Regarding the turbulent boundary layer, eﬀects of reducing artiﬁcial damping enter mainly through transition. In the case with full damping a transitional bubble is computed: The bubble gives rise to increased production of turbulence in the model compared with the case where no transitional bubble is encountered. More production of turbulence, in turn, yields a larger cf immediately downstream of transition than without a transitional bubble (Figure 8.7). Hence, reducing the amount of artiﬁcial damping increases cf in the laminar part of the boundary layer but reduces cf in the turbulent boundary layer just downstream of transition. Further downstream in the turbulent boundary layer, the diﬀerences in cf computed with and without reduced artiﬁcial viscosity are small. This means that reducing artiﬁcial damping in boundary layers is especially important when laminar and transitional ﬂow is present. Combined eﬀects of artiﬁcial viscosity and location of transition were also investigated. In the computation of the ﬂow over airfoil A with full artiﬁcial damping, shifting transition from x/c = 0.1113 to x/c = 0.13 yields a massively separated ﬂow with two counter-rotating recirculation zones (Figure 8.8). With reduced artiﬁcial damping, however, shift of transition does not lead to such a fundamental change in ﬂow topology: Both transition locations x/c = 0.1167 and x/c = 0.13 yield a single recirculation zone at the trailing edge. Moreover, the rear part of the turbulent boundary layer is barely aﬀected by changing the transition location (Figure 8.9). The sensitivity to transition and artiﬁcial damping of ﬂow computations for airfoil A demonstrates that transition modeling and numerical dissipation must be carefully treated since both can have a major impact on computational results. Note, however, that in the fully turbulent ﬂows investigated (cases BS0 and CS0), reducing artiﬁcial viscosity in the boundary layer had very little eﬀect on the ﬂow solution. 152 8 Numerical Issues 0.015 transition at x/c=0.1113 transition at x/c=0.13 full artificial damping kω 88 RBC cf 0.01 0.005 0 0.2 0.4 x/c 0.6 0.8 1 Figure 8.8: Skin-friction distributions for airfoil A: full artiﬁcial damping in combination with shift of transition location from x/c = 0.1113 to x/c = 0.13. 0.015 transition at x/c=0.1167 transition at x/c=0.13 0.01 cf reduced artificial damping kω 88 RBC 0.005 0 0.2 0.4 x/c 0.6 0.8 1 Figure 8.9: Skin-friction distributions for airfoil A: reduced artiﬁcial damping in combination with shift of transition location from x/c = 0.1167 to x/c = 0.13. 8.5 153 Boundary-Value Dependences 8.5 8.5.1 Boundary-Value Dependences Dependences on Wall Value of ω Regarding the k, ω models, it was noted in Subsection 7.1.2 that the various model developers suggest diﬀerent procedures for specifying ω at the wall. In order to investigate the eﬀect of ωw on the ﬂow solution, computations were performed of the ﬂat-plate boundary-layer ﬂow with a given turbulence model but with diﬀerent speciﬁcation procedures for ωw . For this purpose, the wall treatment of ω according to Wilcox (Equation 6.5) and to Rudnik (Equation 6.5) were applied in combination with the 1998 k, ω model of Wilcox. From looking at Table 8.4 one can see that Rudnik’s method yields a much larger wall value for ωw and a lower skin-friction coeﬃcient compared to Wilcox’s method. As implied by the discussion in Section 7.1.2, a decrease in wall shear gives rise to an increased u+ distribution (Figure 8.10). Table 8.4: Inﬂuence of ωw on cf at Reθ = 2900 (case FPBL) ωw /(L p∞ /ρ∞ ) cf × 103 - 3.362 – k, ω 98 (WBC) 28280 3.6128 +7.5% k, ω 98 (RBC) 628233 3.3345 −0.8% Source Experiment (DeGraaﬀ) ∆ It was found by explicitly increasing the wall value of ω that the inﬂuence on the skin friction decreases with increasing ωw . However, the order of magnitude of ωw that was necessary to achieve a virtually ωw -independent solution is very sensitive to the grid resolution. On ﬁne grids, ωw had to be much larger to yield a cf that was independent of ωw than on coarse grids. This favors the application of Menter’s procedure for determining ω at the wall since it is grid dependent (Equation 6.6). In addition, variations of ωw had negligible inﬂuence on the ﬂow solutions for boundary layers under strong pressure gradients. It was further found that extremely large values of ωw cause numerical diﬃculties and can prevent iterative convergence. It is therefore recommended to compute a ﬂow solution with, say, Rudnik’s method and then successively increase ωw until the variation in the skin-friction coeﬃcient with ωw is below some predeﬁned limit. 154 8 Numerical Issues 25 Experiment (DeGraaff) kω 98 RBC kω 98 WBC 15 u + 20 10 5 0 0 10 101 y+ 102 103 Figure 8.10: Inﬂuence of ωw on the computed velocity proﬁle of the ﬂat-plate boundary layer (WBC stands for Wilcox’s surface boundary condition, RBC denotes Rudnik’s surface boundary condition). 8.5.2 Dependence on Free-Stream Value of ω It is mentioned in the literature (see, for example, Kok, 2000; Menter, 1992) that results obtained with Wilcox’s k, ω models are sensitive to the value of ω speciﬁed in the farﬁeld. In the current study, this sensitivity was noticed only if the farﬁeld boundary was located very close to the body. For example, for the ﬂat-plate computations presented in Subsection 7.1.2 the closest farﬁeld boundary was located one half plate length away from the body. To evaluate the inﬂuence of ω∞ two computations were performed with ω∞ = 0.565 and ω∞ = 5650. Note that k∞ was varied correspondingly so that µT in the free stream was not aﬀected. (In both computations, k∞ was very small compared to u2∞ .) Despite the larger diﬀerence of ω∞ speciﬁed in the two computations, the skin-friction coeﬃcients obtained diﬀer less than 1.5 percent (Table 8.5). For airfoil A, the farﬁeld boundary was located 18 chords away from the body; in one computation ω∞ = 1.331 and in the other case ω∞ = 13310 was 8.5 155 Boundary-Value Dependences speciﬁed. Virtually no eﬀects of changing ω∞ is encountered (Table 8.6). (In the computations of the cylinder ﬂows there was no boundary where farﬁeld values for ω had to be speciﬁed.) Table 8.5: Inﬂuence of ω∞ on cf at Reθ = 2900 (case FPBL) ω∞ /(L p∞ /ρ∞ ) cf × 103 Experiment (DeGraaﬀ) - 3.362 – k, ω 88 (RBC) low ω∞ 0.565 3.4890 +3.8% k, ω 88 (RBC) high ω∞ 5650 3.4383 +2.3% Source ∆ From looking at the results presented above, the following conclusions regarding the free-stream value of ω are drawn: Speciﬁcation of ω∞ is not crucial for boundary layer ﬂows in the framework of full Navier-Stokes computations if the farﬁeld boundary is suﬃciently far away. Suﬃciently in this context means at least one characteristic length scale of the body. However, for boundary-layer methods where ω∞ is speciﬁed at the boundary-layer edge the value of ω∞ can have a signiﬁcant inﬂuence on computational results. Table 8.6: Inﬂuence of ω∞ on drag of airfoil A due to friction ω∞ /(L p∞ /ρ∞ ) cD × 102 due to friction k, ω 98 (WBC) low ω∞ 1.331 0.6469 k, ω 98 (WBC) high ω∞ 13310 0.6468 Source 9 Summary and General Conclusions Following Hirschel (1999), ﬂow topology was used as a guideline to turbulence modeling for separated ﬂows. This approach was investigated on the basis of a separated airfoil ﬂow. It was found that consideration of ﬂow topology of the mean velocity ﬁeld of separated ﬂows permits identifying the topological ﬂow structures of possible importance for turbulence modeling. However, modiﬁcations of the turbulence model conﬁned to the recirculation zone of separated ﬂows do not improve predictive accuracy of the computation. The key issue for the accurate prediction of turbulent separation is correctly capturing boundary-layer development upstream of the primary separation and the boundary-layer state at separation. It was also found that the nominally two-dimensional ﬂow around the airfoil showed strong evidence of three-dimensionality. In addition, it was argued that separated airfoil ﬂows in general are problematic for the assessment of turbulence models due to the strong interaction between primary separation and pressure gradient via the circulation around the airfoil. In the following, assessment of turbulence-model performance was pursued for the prediction of boundary-layer development with strong adverse pressure gradient. For this purpose, a comparative study of eleven modern eddy-viscosity turbulence models was performed, with special regard to turbulent boundary-layer separation caused by an adverse pressure gradient. Four ﬂow cases with increasing physical complexity were selected and a classiﬁcation of the cases based on the topology of the mean velocity ﬁelds was performed. The cases consisted of a ﬂat-plate boundary layer (case FPBL), a non-equilibrium boundary layer on an axial cylinder with strong pressure gradient without separation (case BS0), a non-equilibrium boundary layer on an axial cylinder with pressure-induced separation (case CS0), and again the separated airfoil ﬂow (case AAA). For each case extensive experimental data including Reynolds stresses are available. Computational results were compared with experimental data. A large variation in predictive accuracy of the models was encountered even for the simplest ﬂow case. In addition, predictive accuracy obtained with a given model for a given ﬂow case varied signiﬁcantly between the variables evaluated. For the separated boundary-layer ﬂow CS0 for example, it happened that with the 1998 k, ω model of Wilcox good results were obtained for streamwise velocity while poor results were obtained for skin friction. This kind of behavior was not consistent throughout the ﬂow cases 157 considered. Moreover, the “best” turbulence model changed from ﬂow case to ﬂow case. The performance of the turbulence models investigated in this work is summarized in Tables 9.1 and 9.2. Containing the ﬂow classes deﬁned, the tables are also meant to serve as an attempt for recommending a turbulence model for a given ﬂow class. For example, the 1998 k, ω model of Wilcox may be recommended for boundary-layer ﬂows with strong adverse pressure gradient without separation. However, many more computations of ﬂows belonging to a given class must be performed before this kind of recommendation can be made on a thorough basis. For the separated airfoil ﬂow, it was found that turbulence models yield higher predictive accuracy if they limit the eddy viscosity in boundary layers using the assumption of constant ratio between turbulent kinetic energy and Reynolds shear stress. However, this ﬁnding does not hold for the two nonequilibrium boundary-layer cases BS0 and CS0. For example, the 1998 k, ω model of Wilcox does not build on this assumption but it yielded the best agreement of computational results with measurements for case BS0. For the ﬂow cases with adverse pressure gradient, models using transport equations for the eddy viscosity typically showed better response to pressure gradient than the algebraic model of Baldwin & Lomax (1978). In addition, predictions of the one-equation models were generally much closer to each other than those of the k, ω models. Regarding transitional ﬂows, it was found that user-speciﬁed transition location and the rate at which turbulence models approach fully-turbulent state can have major impact on computational results. In particular, transportequation models generally show too slow transition when compared with experiment or DNS. For the airfoil ﬂow, which featured a transitional bubble, delay of transition and/or specifying transition too far downstream prevented convergence to steady state and yielded unrealistic ﬂow solutions. The reason was that the turbulence models did not introduce suﬃcient levels of Reynolds averaging in the transition region in order to suppress unsteadiness resulting from transition mechanisms. Several numerical issues were also addressed in this work. For example, grid convergence of computational results was veriﬁed based on systematic evaluation of grid-convergence indices. Regarding the application of the numerical scheme utilized for this work, the use of local preconditioning for low Mach numbers proved to be mandatory. Numerical tests showed that standard artiﬁcial damping must be reduced in boundary layers to minimize the 158 9 Summary and General Conclusions eﬀect of numerical diﬀusion on the laminar part of the boundary layers and to prevent spurious momentum loss. For models based on k and ω, dependence of computational results on the speciﬁcation of ω at the wall was investigated: Extremely large values of ω at the surface have to be prescribed in order to minimize solution dependence on ωw . Regarding the free-stream value of ω, only a very small inﬂuence of ω∞ on the computation was seen for the cases considered. As a closing note, the current work shows also that a uniﬁed approach consisting of combined and coordinated eﬀorts of experiment, DNS, LES and RANS is necessary to attack the problem of turbulence modeling. In addition, it is highly desirable that experiments supply detailed information on possible three-dimensional eﬀects in nominally two-dimensional ﬂows, on unsteadiness and the procedures applied to average the measured data, and on transition locations. Regarding the transition process, detailed information not only about the skin-friction coeﬃcient but also about the change of momentum thickness in the transition region are important for the validation of turbulence models. Following Hirschel (2003), experiments should also give conclusive results for determining the ﬂow class which the ﬂow under consideration belongs to. Table 9.1: Summary of performance of turbulence models investigated in the present work for the separated ﬂow over the Aerospatiale-A airfoil (case AAA) Flow class (see Table 1.1) Test case See Section Good performance Medium performance Poor performance 1.1 a) AAA separated airfoil ﬂow 7.4 – Johnson-King; k, ω SST; SALSA; Edwards-Chandra k, ω SST modiﬁed; Spalart-Allmaras; k, ω TNT; Baldwin-Lomax; k, ω 1998; Wallin; k, ω 1988; k, ω LLR 159 Table 9.2: Summary of performance of turbulence models investigated in the present work for the turbulent boundary-layer cases (cases FPBL, BS0 and CS0) Flow class (see Table 7.2) Test case BL 1.1 FPBL ﬂat plate BL 2.1 BS0 ≥0 ∂p ∂x BL 3.1 CS0 ≥ 0 with separation ∂p ∂x See Subsection 7.1 7.2 Good performance k, ω SST mod. k, ω TNT Spalart-Allmaras SALSA k, ω 1998 k, ω SST mod. Edwards-Chandra k, ω SST Spalart-Allmaras SALSA Wallin 7.3 Medium performance k, ω LLR Baldwin-Lomax Edwards-Chandra k, ω 1998 Wallin k, ω LLR k, ω TNT k, ω SST mod. Wallin k, ω SST k, ω 1998 k, ω LLR SALSA Spalart-Allmaras Edwards-Chandra Poor performance k, ω 1988 k, ω SST k, ω 1988 Baldwin-Lomax k, ω 1988 k, ω TNT Baldwin-Lomax – 160 10 10 Outlook Outlook Computational results for separated ﬂows presented in this work demonstrate that the turbulence models investigated are not a reliable tool for computing such ﬂows. In addition, it is greatly to be feared that this statement will, more or less, hold for every turbulence model based on classical Reynolds averaging. This gives rise to the fundamental question whether separated turbulent ﬂows are generally amenable to the concepts of Reynolds averaging. In order to correctly predict mean variables of separated turbulent ﬂow ﬁelds, the state of turbulent boundary layer at primary separation must be accurately reproduced by the computational method. A richer description of the boundary-layer ﬂow in the vicinity and upstream of separation than oﬀered by RANS might be necessary to perform this task. In principle, two approaches can be utilized in this regard: Large-eddy simulation or direct numerical simulation. Yet, as with DNS, LES of aerodynamic ﬂows at high Reynolds numbers including large regions of thin boundary layers is not feasible today or in the foreseeable future (Spalart, 2000). However, great potential is seen for hybrid methods combining RANS and LES to substantially reduce computational costs compared to performing LES for the entire ﬂow domain. Compared to pure RANS, hybrid methods can yield a better description of turbulence in regions where necessary. A partially new idea of RANS/LES coupling will be proposed in Section A of the appendices. It is intended to form the basis for future work. Part III Appendices 162 A A.1 A RANSLESS – A New Approach to RANS/LES Coupling RANSLESS – A Partially New Approach to Coupling RANS and LES for Turbulent Flows Brief Review of Turbulence Physics at Turbulent Separation Before proceeding with the discussion of a computational approach oﬀering an alternative to classical RANS for separated ﬂows, a brief summary of turbulence physics of boundary-layer separation is given in the following. Simpson and his co-workers have performed extensive experimental and theoretical investigations of separating turbulent boundary layers (see Wetzel & Simpson, 1998; Chesnakas & Simpson, 1997; Simpson, 1996; Chesnakas & Simpson, 1996; Simpson, 1991; Agarwal & Simpson, 1990; Simpson et al., 1977). Recapitulating their main ﬁndings, one can say that large-scale turbulent structures rapidly grow in all directions when approaching separation of a turbulent boundary layer. Due to the large eddies intermittent ﬂow is observed far away from the wall at the position of maximum Reynolds shear stress in the separated shear layer immediately downstream of separation. The eddies supply high levels of turbulence to the backﬂow region underneath the separated shear layer. This means that in the backﬂow region velocity ﬂuctuations such as u and v are large, but eﬀective Reynolds shear stress like −u v is low. Moreover, ﬂuid elements in the backﬂow do not come from far downstream as is suggested by mean streamlines. Instead, they are intermittently transported to and away from the wall through turbulent diffusion mechanisms induced by the large eddies in the separated shear layer. Hence, mean streamlines do not represent mean path lines of ﬂuid elements in the backﬂow region. In summary, large-scale turbulent structures, high levels of turbulence and slow mean motion characterize turbulent separation. Therefore, modeling Reynolds stresses by means of mean velocity gradients is not appropriate in this ﬂow region. Given this premise, it is believed that well-resolved LES is better suited to compute turbulent separation and will yield higher predictive accuracy compared to RANS. A.2 RANS/LES Coupling for Separated Flows Several researchers have introduced methods for RANS/LES coupling or related techniques. For an overview, the reader is referred to Sagaut (2002); A.2 RANS/LES Coupling for Separated Flows 163 Friedrich & Rodi (2000); Chung & Hyung (1997). In the present work, a hybrid RANS/LES approach is considered where turbulent separation and a small part of the boundary layer upstream of separation will be computed by LES. The relatively small LES domain, in turn, will be surrounded by conventional RANS. The latter will be utilized to compute the remaining ﬂow ﬁeld and thereby provide mean ﬂow variables used as a basis for the necessary boundary conditions for the LES. This concept will be called RANSLESS which stands for RANS-surrounded LES Scenario. Figure A.1 shows a schematic overview of the current approach for a mildly separated airfoil ﬂow. Two possible LES domains are shown in the ﬁgure because it must be determined what shape and size of LES domains are necessary to achieve suﬃciently accurate results. Location, shape and size of the LES domain will be speciﬁed by the user based on ﬂow solutions obtained from pure RANS. This means that in a ﬁrst step conventional RANS will be used to yield an approximate solution of the ﬂow ﬁeld. In the long run, this process will happen in a self organizing manner. possible LES domains surrounded by RANS Figure A.1: Overview of RANS/LES coupling for mildly separated airfoil ﬂow. Standard RANS and LES methods will be employed in the corresponding domains and coupling between the RANS and LES codes will take place by exchange of ﬂow variables evaluated at the interfaces of the domains. To obtain a physically consistent and tightly coupled ﬂow-ﬁeld solution proper boundary conditions for both numerical methods must be speciﬁed at the interfaces. Hence, the key issue in the proposed work will be the development of methods for generating physically and numerically accurate boundary conditions at the interfaces. We propose an approach, novel to the knowledge 164 A RANSLESS – A New Approach to RANS/LES Coupling of the author, for specifying inﬂow boundary conditions for LES from mean velocities supplied by a surrounding RANS. This is discussed in the following. A.2.1 Inﬂow Conditions for LES One of the most delicate issues in RANS/LES coupling is the application of proper inﬂow conditions for LES on boundaries where the ﬂow enters the LES domain coming from the RANS region. The reason is that LES requires unsteady inﬂow conditions containing turbulent ﬂuctuations and, by deﬁnition, the latter are not available from RANS. Flow variables computed by RANS are mean values, and turbulent ﬂuctuations must be “artiﬁcially” created. These must be superposed on the RANS solution to yield unsteady inﬂow boundary conditions for the LES, with mean values matched to corresponding variables of the upstream RANS. Often, random ﬂuctuations were employed for this purpose (Lund et al., 1998). However, it is important to create physically meaningful turbulent ﬂuctuations because inﬂuence of poor inﬂow conditions can persist over a long downstream distance (Chung & Hyung, 1997). A new method is proposed for creating turbulent inﬂow conditions from RANS data. It is derived from a procedure for generating turbulent inﬂow data for simulations of spatiallydeveloping boundary layers proposed in Lund et al. (1998). It is believed that the method outlined in this work is able to generate ﬂuctuations with amplitude and phase information closely linked to realistic turbulent structures. Consequently, the method reduces inﬂuence of the inﬂow interface to a minimum and minimizes the length of the section necessary for the development of organized turbulent motion. Figure A.2 shows a schematic overview of coupling LES with RANS at the LES inﬂow boundary; the following discussion relates to the ﬁgure. The idea is to link RANS and LES by an overlap region. The upstream end of the overlap is referred to as “LES inﬂow boundary” and the downstream end is termed “RANS outﬂow boundary”. Inside the overlap, a recycle station and a forcing region will be present. The latter is discussed below because it is used for prescribing outﬂow boundary conditions for the RANS computation. (This subsection is concerned with inﬂow conditions for LES.) As mentioned above, ﬁrst, a RANS solution for the entire ﬂow ﬁeld will be performed. Then, a mean velocity proﬁle will be extracted from the (steady) RANS solution at the LES inﬂow boundary and random ﬂuctuations will be superposed on this mean proﬁle. The amplitudes and covariances of the random ﬂuctua- A.2 165 RANS/LES Coupling for Separated Flows tions will be constructed to satisfy the Reynolds-stress tensor of the RANS computation at the LES inﬂow boundary. The unsteady velocities obtained in this manner will be prescribed as inﬂow boundary conditions for the LES computation. It is noted that random ﬂuctuations serve only to “induce” ﬂuctuations within the LES domain and to trigger transition. Next, velocity ﬂuctuations obtained with LES at the recycle station will be re-scaled and subsequently superposed on the mean velocity proﬁle of the RANS solution at the LES inﬂow boundary. Thus, re-scaled ﬂuctuations from inside the LES domain replace random ﬂuctuations used for startup. This re-scaling and superposition process will be constantly repeated and after an initial transient, realistic turbulent structure will be obtained at the recycle station as well as at the LES inﬂow boundary. shear layer edge RANS Overlap LES turbulent boundary layer u’in= γ u’rec forcing region ψ=0 Detachment 1111111111111111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000000000000000 LES inflow boundary recycle station RANS outflow boundary Figure A.2: Generation of turbulent ﬂuctuations at LES inﬂow boundary (schematic view). Scaling Laws To apply the LES inﬂow procedure outlined above appropriate scaling laws for re-scaling turbulent ﬂuctuations are required. A simple method patterned after the work of Lund et al. (1998) will be utilized for this purpose. Fluctuations at the recycle stations will be simply multiplied by a scaling factor: ui |LES inﬂow = χui |recycle station . (A.1) 166 A RANSLESS – A New Approach to RANS/LES Coupling χ can be determined in diﬀerent ways. Lund et al., for example, suggested using the ratio of friction velocities at the recycle station and the LES inﬂow boundary: uτ |recycle station χ= . (A.2) uτ |LES inﬂow However, in derivation of Equation A.2 Lund et al. assumed an equilibrium boundary layer. Since we are envisaging separation where the ﬂow is not in equilibrium new prescriptions for computation of χ might prove to be necessary. For this purpose, ratios of maximum shear stresses and maximum kinetic energies will be investigated: χ= u v m |recycle station u v m |LES inﬂow , χ= kmax |recycle station . kmax |LES inﬂow (A.3) It is believed that these or similar relations will lead to promising results in non-equilibrium situations. Superposition In addition to the development of new scaling laws special care must be taken for the superposition of mean and ﬂuctuating velocities. As a ﬁrst test, mean velocities computed by RANS and velocity ﬂuctuations obtained with Equation A.1 will be summed. The inﬂow boundary condition for the LES domain then reads: ui,LES |LES inﬂow = Ui,RANS |LES inﬂow + ui |LES inﬂow . (A.4) ui and Ui denote unsteady velocity components and Reynolds-averaged velocity components, respectively. This procedure is called inﬂow-velocity modulation. However, Kaufmann et al. (2002) performed an analysis of inﬂow velocity modulation in terms of acoustic waves. They showed that specifying inﬂow conditions for LES of gas burners based on velocity modulation can lead to uncontrolled pressure waves and resonances. In their work, Kaufmann et al. presented a new method called inﬂow-wave modulation. Amplitudes of ingoing waves were imposed without interacting with outgoing waves, and undesired reﬂections and interactions between entering and leaving waves were suppressed. Therefore, the inﬂow-wave modulation procedure of Kaufmann et al., eventually modiﬁed, is proposed for imposing turbulent ﬂuctuations on RANS velocities. A.3 167 Closing Note A.2.2 Outﬂow Conditions for LES Speciﬁcation of LES outﬂow condition will be closely patterned after the work of Schlüter & Pitsch (2001) and is therefore not further discussed. A.2.3 Inﬂow Conditions for RANS On boundaries where the mean ﬂow moves from the LES to the RANS domain, time averaged LES data evaluated at the interface will be speciﬁed as inﬂow boundary conditions for the RANS. As pointed out in Qéméré et al. (2000) and Schlüter et al. (2003) some special care must be taken for prescribing turbulence variables for the turbulence model employed. Because applying inﬂow conditions for RANS computations was successfully performed for computing the cylinder ﬂows discussed in Subsections 7.2 and 7.3, application to RANS/LES coupling is deemed to be straightforward. A.2.4 Outﬂow Conditions for RANS Outﬂow conditions for RANS have to account for upstream inﬂuence of the mean LES solution. For this purpose, a forcing region inside the overlap of the LES and RANS domains will be used (Figure A.2). In the forcing region, body forces will be applied to the RANS computation in order to drive RANS velocity proﬁles to match mean LES proﬁles in the forcing region. The body forces will be formulated such that they are functionally dependent on diﬀerences inside the forcing region between mean LES velocity components and corresponding velocities components in the RANS computation: fi ∝ 1 (Ui,RANS − Ui,LES ) . tc (A.5) tc is some characteristic time scale. It can, to ﬁrst approximation, be determined from the following relation: tc = A.3 length of the forcing region . U∞ Closing Note It is yet to see whether RANSLESS will improve predictive accuracy of computations of pressure-induced turbulent separation. However, by concept, it oﬀers the following advantages over existing methods: 168 A RANSLESS – A New Approach to RANS/LES Coupling • Physically meaningful turbulent ﬂuctuations are prescribed at the LES inﬂow. • Unlike in detached-eddy simulation (DES), no transition region between RANS and LES with an undeﬁned turbulent state of the ﬂow exists. • No extra LES or DNS need to be performed to create a database of ﬂow variables used for speciﬁcation of inﬂow conditions for the LES domain. • Application of RANSLESS is possible to boundary-layer ﬂows as well as to free shear layers or swirl ﬂows. RANSLESS will be applied ﬁrst to a simple ﬂat-plate boundary layer to test the interface treatments proposed. After successful validation of RANSLESS in this ﬂow the same separated ﬂow cases which were used in this work for studying turbulence-model performance will be computed. In order to investigate the gain of predictive accuracy accomplished by RANSLESS, results obtained will be compared to results from RANS computations and to experimental results. Maturing of RANSLESS for application to general complex ﬂows is the long-term goal. 169 B Details of the Johnson-King Model In order to solve the Johnson-King model equations, an initial distribution of (−u v )1/2 is required. For this purpose, two preliminary steps are performed: 1. A converged ﬂow ﬁeld solution is computed with the help of a simple algebraic turbulence model, for example the Cebeci-Smith or BaldwinLomax model. 2. Next, the mean strain rate and eddy viscosity obtained from step one are used to evaluate (−u v m )1/2 employing the following relation: ) µt ∂u ∂v 1/2 = (B.1) + (−u v m ) . ρ ∂y ∂x m (−u v m )1/2 is then inserted into Equations 3.10 – 3.13 to yield a ﬁrst approximation of the so-called equilibrium eddy viscosity µt, eq . For this purpose, the non-equilibrium parameter σ(x) is set constant and equal to unity, σ(x) = 1. Through Equation B.1, µt, eq gives rise to a new value of (−u v m )1/2 . The ﬁnal value of µt, eq is iteratively found such that it satisﬁes the equilibrium form of the model summarized as follows: µt, eq = µto , eq 1 − e(−µti , eq /µto , eq ) , (B.2) µti , eq = ρD2 κy(−u v m, eq )1/2 , −y(−u v m, eq )1/2 D = 1 − exp , νA+ µto , eq = 0.0168ρue δv∗ FKleb ) µt , eq ∂u ∂v 1/2 = (−u v m, eq ) + ρ ∂y ∂x (B.3) (B.4) (B.5) . (B.6) m The iteration of µt, eq can, for example, be performed by a simple Aitken method. From comparing deﬁnitions of µto and µto , eq (Equations 3.13 and B.5) it follows that the parameter σ(x) is deﬁned as the ratio of non-equilibrium and equilibrium outer eddy viscosity: σ(x) = µto /µto , eq . 170 B Details of the Johnson-King Model It is further noted that the ﬂow solution is held ﬁxed during the iteration for µt, eq . Once the desired value for µt, eq is found, a new ﬂow ﬁeld is computed. The procedure pertaining to step 2 is repeated until a converged solution of the ﬂow ﬁeld is obtained. To solve the non-equilibrium Johnson-King model, results from step two, above, with σ(x) = 1, are used as starting conditions. If, however, the solution process has already proceeded to the stage where a non-equilibrium solution is available, then the latter is used as the starting condition instead. Solving the non-equilibrium model means iteratively establishing a distribution for σ(x) so that values of (−u v m )1/2 independently obtained from Equation 3.14 and Equation B.1 coincide. This can be accomplished by the following steps: 1. From the starting conditions, compute an equilibrium eddy viscosity µt, eq from Equations B.2 – B.6, as discussed in step two above but without updating the ﬂow solution. Of course, at the very ﬁrst step in the solution of the Johnson-King model, this eddy viscosity will be identical to the one obtained in step 2 above. It diﬀers, however, from the latter when a new (−u v )1/2 distribution is available. 2. Similarly, compute a non-equilibrium eddy viscosity µt using Equations 3.10 – 3.13 and B.1. Again, at the beginning, this viscosity will have the same value as µt, eq since σ(x) = 1. However, in the course of the computation σ(x) = 1 for non-equilibrium ﬂows and µt will, in general, diﬀer from µt, eq . 3. The rate equation for the streamwise development of the maximum Reynolds shear stress is solved next. More precisely, the square root of the maximum Reynolds shear stress divided by the density is obtained but for sake of brevity (−u v m )1/2 is referred to as the maximum Reynolds shear stress. Inserting (−u v m )1/2 into Equation B.1 yields an additional eddy viscosity which is denoted by µ̃t, m : µ̃t, m = ρ(−u v m ) ∂u ∂v + ∂x ∂y . m 4. In order to make the maximum Reynolds shear stress obtained from the rate equation and the one obtained from Equation B.1 coincide, which is equivalent to µ̃t, m = µt, m , an appropriate value for σ(x) must 171 be found. Since σ(x) acts on the outer eddy viscosity, this can be accomplished by iterating µto , m such that µt, m = µ̃t, m . σ(x) can then be determined by the ratio of the value of µto , m when µt, m = µ̃t, m to the initial value of µto , m . This procedure can be expressed by a simpliﬁed Newton method: ! f (µto , m ) = µt, m (µto , m ) − µ̃t, m = 0 , (n) (n+1) (n) µ to , m = µ to , m − f (µto , m ) (n) f (µto , m ) , with f (µto , m ) = (n) (n) ∂f (µto , m ) (n) ∂µto , m (n) ≈ 1 − e(−µti /µto ) . (1) With µ̃t, m from step 3, µt, m and µti , m from step 2 at hand, the simpliﬁed Newton algorithm for the iteration of µto , m reads: (n+1) (n) µ to , m = µ to , m (n+1) µt, m (n+1) = µ to , m µ̃t, m (n) µt, m , (n+1) 1 − e−µti , m /µto , m , with n = 1, 2 . Finally, the single-step update for σ(x) is given by: (3) σ(x)new = σ(x)old µ to , m (1) µ to , m . 5. The last step is to solve the rate equation for (−u v m )1/2 with the newly obtained σ(x) distribution and to compute the ﬁnal non-equilibrium eddy viscosity from Equations 3.10 – 3.13 using the new value for (−u v m )1/2 . 172 C C.1 C Graphs of Computational Results Graphs of Computational Results Boundary Layer with Adverse Pressure Gradient (Case BS0) The rest of this page has been deliberately left blank U/Ur U/Ur 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 c) a) 10 10 30 Experiment Spalart-Allmaras Edwards-Chandra SALSA 30 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 d) b) 10 10 20 y [mm] 20 y [mm] Figure C.1: Velocity proﬁles for case BS0 at x/D = −1.08857. 20 y [mm] 20 y [mm] Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC U/Ur U/Ur 30 Experiment kω SST MBC kω SST modified 30 Experiment kω TNT RBC Wallin RBC kω LLR RBC C.1 Boundary Layer with Adverse Pressure Gradient (Case BS0) 173 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 c) a) 10 10 30 Experiment Spalart-Allmaras Edwards-Chandra SALSA 30 40 40 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 d) b) 10 10 20 y [mm] 20 y [mm] Figure C.2: Velocity proﬁles for case BS0 at x/D = −0.544286. 20 y [mm] 20 y [mm] Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC U/Ur U/Ur U/Ur U/Ur 30 Experiment kω SST MBC kω SST modified 30 Experiment kω TNT RBC Wallin RBC kω LLR RBC 40 40 174 C Graphs of Computational Results U/Ur U/Ur 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 c) a) 10 10 30 Experiment Spalart-Allmaras Edwards-Chandra SALSA 30 40 40 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 d) b) 10 10 20 y [mm] 20 y [mm] Figure C.3: Velocity proﬁles for case BS0 at x/D = −0.0907143. 20 y [mm] 20 y [mm] Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC U/Ur U/Ur 30 Experiment kω SST MBC kω SST modified 30 Experiment kω TNT RBC Wallin RBC kω LLR RBC 40 40 C.1 Boundary Layer with Adverse Pressure Gradient (Case BS0) 175 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 c) a) 10 10 30 Experiment Spalart-Allmaras Edwards-Chandra SALSA 30 40 40 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 d) b) 10 10 20 y [mm] 20 y [mm] Figure C.4: Velocity proﬁles for case BS0 at x/D = 0.0907143. 20 y [mm] 20 y [mm] Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC U/Ur U/Ur U/Ur U/Ur 30 Experiment kω SST MBC kω SST modified 30 Experiment kω TNT RBC Wallin RBC kω LLR RBC 40 40 176 C Graphs of Computational Results U/Ur U/Ur 0 0.2 0.4 0.6 0 0.2 0.4 0.6 c) a) 10 10 40 50 50 0 0.2 0.4 0.6 0 0.2 0.4 0.6 d) b) 10 10 30 y [mm] 20 30 y [mm] 20 Figure C.5: Velocity proﬁles for case BS0 at x/D = 1.08857. 30 y [mm] 20 40 Experiment Spalart-Allmaras Edwards-Chandra SALSA 30 y [mm] 20 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC U/Ur U/Ur 40 Experiment kω SST MBC kω SST modified 40 Experiment kω TNT RBC Wallin RBC kω LLR RBC 50 50 C.1 Boundary Layer with Adverse Pressure Gradient (Case BS0) 177 0 0.2 0.4 0.6 0 0.2 0.4 0.6 c) a) 10 10 30 y [mm] 30 y [mm] 50 40 50 Experiment Spalart-Allmaras Edwards-Chandra SALSA 40 0 0.2 0.4 0.6 0 0.2 0.4 0.6 d) b) 10 10 20 20 30 y [mm] 30 y [mm] Figure C.6: Velocity proﬁles for case BS0 at x/D = 1.63286. 20 20 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC U/Ur U/Ur U/Ur U/Ur 50 40 50 Experiment kω SST MBC kω SST modified 40 Experiment kω TNT RBC Wallin RBC kω LLR RBC 178 C Graphs of Computational Results 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur 0 -0.5 -1 -1.5 -2 0 -0.5 -1 -1.5 -2 c) a) y [mm] 20 20 y [mm] 30 Experiment Spalart-Allmaras Edwards-Chandra SALSA 30 0 -0.5 -1 -1.5 -2 0 -0.5 -1 -1.5 -2 d) b) 10 10 20 y [mm] 20 y [mm] 30 Experiment kω SST MBC kω SST modified 30 Experiment kω TNT RBC Wallin RBC kω LLR RBC Figure C.7: Reynolds-shear-stress proﬁles for case BS0 at x/D = −1.08857. 10 10 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur C.1 Boundary Layer with Adverse Pressure Gradient (Case BS0) 179 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur 0 -0.5 -1 -1.5 -2 0 -0.5 -1 -1.5 c) a) y [mm] 20 20 y [mm] 30 Experiment Spalart-Allmaras Edwards-Chandra SALSA 30 40 40 0 -0.5 -1 -1.5 -2 0 -0.5 -1 -1.5 -2 d) b) 10 10 20 y [mm] 20 y [mm] 30 Experiment kω SST MBC kω SST modified 30 Experiment kω TNT RBC Wallin RBC kω LLR RBC Figure C.8: Reynolds-shear-stress proﬁles for case BS0 at x/D = −0.544286. 10 10 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur -2 40 40 180 C Graphs of Computational Results 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur 0 -0.5 -1 -1.5 -2 0 -0.5 -1 -1.5 -2 c) a) y [mm] 20 20 y [mm] 40 30 40 Experiment Spalart-Allmaras Edwards-Chandra SALSA 30 0 -0.5 -1 -1.5 -2 0 -0.5 -1 -1.5 -2 d) b) 10 10 20 y [mm] 20 y [mm] 30 Experiment kω SST MBC kω SST modified 30 Experiment kω TNT RBC Wallin RBC kω LLR RBC Figure C.9: Reynolds-shear-stress proﬁles for case BS0 at x/D = −0.0907143. 10 10 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur 40 40 C.1 Boundary Layer with Adverse Pressure Gradient (Case BS0) 181 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur 0 -0.5 -1 -1.5 -2 0 -0.5 -1 -1.5 -2 c) a) y [mm] 20 20 y [mm] 40 30 40 Experiment Spalart-Allmaras Edwards-Chandra SALSA 30 0 -0.5 -1 -1.5 -2 0 -0.5 -1 -1.5 -2 d) b) 10 10 20 y [mm] 20 y [mm] Figure C.10: Reynolds-shear-stress proﬁles for case BS0 at x/D = 0.0907143. 10 10 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur 40 30 40 Experiment kω SST MBC kω SST modified 30 Experiment kω TNT RBC Wallin RBC kω LLR RBC 182 C Graphs of Computational Results 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur 0 -0.5 -1 -1.5 -2 -2.5 0 -0.5 -1 -1.5 -2 -2.5 c) a) 30 y [mm] 20 30 y [mm] 20 50 40 50 Experiment Spalart-Allmaras Edwards-Chandra SALSA 40 0 -0.5 -1 -1.5 -2 -2.5 0 -0.5 -1 -1.5 -2 -2.5 d) b) 10 10 30 y [mm] 20 30 y [mm] 20 Figure C.11: Reynolds-shear-stress proﬁles for case BS0 at x/D = 1.08857. 10 10 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur 50 40 50 Experiment kω SST MBC kω SST modified 40 Experiment kω TNT RBC Wallin RBC kω LLR RBC C.1 Boundary Layer with Adverse Pressure Gradient (Case BS0) 183 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur 0 -0.5 -1 -1.5 -2 -2.5 0 -0.5 -1 -1.5 -2 c) a) 20 20 y [mm] 30 y [mm] 30 50 40 50 Experiment Spalart-Allmaras Edwards-Chandra SALSA 40 0 -0.5 -1 -1.5 -2 -2.5 0 -0.5 -1 -1.5 -2 -2.5 d) b) 10 10 20 20 30 y [mm] 30 y [mm] Figure C.12: Reynolds-shear-stress proﬁles for case BS0 at x/D = 1.63286. 10 10 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur -2.5 50 40 50 Experiment kω SST MBC kω SST modified 40 Experiment kω TNT RBC Wallin RBC kω LLR RBC 184 C Graphs of Computational Results C.2 C.2 Boundary Layer with Pressure-Induced Separation (Case CS0) 185 Boundary Layer with Pressure-Induced Separation (Case CS0) The rest of this page has been deliberately left blank 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0 c) a) y [mm] y [mm] 20 Experiment Spalart-Allmaras Edwards-Chandra SALSA 20 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0 d) b) 10 10 y [mm] y [mm] Figure C.13: Velocity proﬁles for case CS0 at x/D = −1.08857. 10 10 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC U/Ur U/Ur U/Ur U/Ur 20 Experiment kω SST MBC kω SST modified 20 Experiment kω TNT RBC Wallin RBC kω LLR RBC 186 C Graphs of Computational Results U/Ur U/Ur 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0 c) a) 20 y [mm] 20 y [mm] 30 Experiment Spalart-Allmaras Edwards-Chandra SALSA 30 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0 d) b) 10 10 20 y [mm] 20 y [mm] Figure C.14: Velocity proﬁles for case CS0 at x/D = −0.544286. 10 10 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC U/Ur U/Ur 30 Experiment kω SST MBC kω SST modified 30 Experiment kω TNT RBC Wallin RBC kω LLR RBC C.2 Boundary Layer with Pressure-Induced Separation (Case CS0) 187 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0 c) a) 10 10 y [mm] 30 y [mm] 30 50 40 50 Experiment Spalart-Allmaras Edwards-Chandra SALSA 40 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0 d) b) 10 10 20 20 30 y [mm] 30 y [mm] Figure C.15: Velocity proﬁles for case CS0 at x/D = −0.0907143. 20 20 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC U/Ur U/Ur U/Ur U/Ur 50 40 50 Experiment kω SST MBC kω SST modified 40 Experiment kω TNT RBC Wallin RBC kω LLR RBC 188 C Graphs of Computational Results U/Ur U/Ur 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0 c) a) 10 10 y [mm] 30 y [mm] 30 50 40 50 Experiment Spalart-Allmaras Edwards-Chandra SALSA 40 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0 d) b) 10 10 20 20 30 y [mm] 30 y [mm] Figure C.16: Velocity proﬁles for case CS0 at x/D = 0.0907143. 20 20 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC U/Ur U/Ur 50 40 50 Experiment kω SST MBC kω SST modified 40 Experiment kω TNT RBC Wallin RBC kω LLR RBC C.2 Boundary Layer with Pressure-Induced Separation (Case CS0) 189 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0 c) a) 10 10 y [mm] 30 y [mm] 30 50 40 50 Experiment Spalart-Allmaras Edwards-Chandra SALSA 40 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0 d) b) 10 10 20 20 30 y [mm] 30 y [mm] Figure C.17: Velocity proﬁles for case CS0 at x/D = 0.181429. 20 20 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC U/Ur U/Ur U/Ur U/Ur 50 40 50 Experiment kω SST MBC kω SST modified 40 Experiment kω TNT RBC Wallin RBC kω LLR RBC 190 C Graphs of Computational Results U/Ur U/Ur 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0 c) a) 10 10 y [mm] 30 30 y [mm] 50 60 40 50 60 Experiment Spalart-Allmaras Edwards-Chandra SALSA 40 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0 d) b) 10 10 20 20 30 y [mm] 30 y [mm] Figure C.18: Velocity proﬁles for case CS0 at x/D = 0.362857. 20 20 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC U/Ur U/Ur 50 60 40 50 60 Experiment kω SST MBC kω SST modified 40 Experiment kω TNT RBC Wallin RBC kω LLR RBC C.2 Boundary Layer with Pressure-Induced Separation (Case CS0) 191 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0 c) a) 10 10 30 40 40 y [mm] 30 y [mm] 60 50 60 Experiment Spalart-Allmaras Edwards-Chandra SALSA 50 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0 d) b) 10 10 20 20 30 40 40 y [mm] 30 y [mm] Figure C.19: Velocity proﬁles for case CS0 at x/D = 0.725714. 20 20 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC U/Ur U/Ur U/Ur U/Ur 60 50 60 Experiment kω SST MBC kω SST modified 50 Experiment kω TNT RBC Wallin RBC kω LLR RBC 192 C Graphs of Computational Results U/Ur U/Ur 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0 10 c) 10 a) 20 20 60 70 50 60 70 Experiment Spalart-Allmaras Edwards-Chandra SALSA 50 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0 10 d) 10 b) 20 20 40 y [mm] 30 40 y [mm] 30 Figure C.20: Velocity proﬁles for case CS0 at x/D = 1.08857. 40 y [mm] 30 40 y [mm] 30 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC U/Ur U/Ur 60 70 50 60 70 Experiment kω SST MBC kω SST modified 50 Experiment kω TNT RBC Wallin RBC kω LLR RBC C.2 Boundary Layer with Pressure-Induced Separation (Case CS0) 193 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0 10 c) 10 a) 20 20 60 70 50 60 70 Experiment Spalart-Allmaras Edwards-Chandra SALSA 50 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0 10 d) 10 b) 20 20 40 y [mm] 30 40 y [mm] 30 Figure C.21: Velocity proﬁles for case CS0 at x/D = 1.63286. 40 y [mm] 30 40 y [mm] 30 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC U/Ur U/Ur U/Ur U/Ur 60 70 50 60 70 Experiment kω SST MBC kω SST modified 50 Experiment kω TNT RBC Wallin RBC kω LLR RBC 194 C Graphs of Computational Results U/Ur U/Ur 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0 10 c) 10 a) 20 20 60 70 50 60 70 Experiment Spalart-Allmaras Edwards-Chandra SALSA 50 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0 10 d) 10 b) 20 20 40 y [mm] 30 40 y [mm] 30 Figure C.22: Velocity proﬁles for case CS0 at x/D = 2.17714. 40 y [mm] 30 40 y [mm] 30 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC U/Ur U/Ur 60 70 50 60 70 Experiment kω SST MBC kω SST modified 50 Experiment kω TNT RBC Wallin RBC kω LLR RBC C.2 Boundary Layer with Pressure-Induced Separation (Case CS0) 195 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur 0 -0.5 -1 -1.5 -2 0 -0.5 -1 -1.5 0 0 10 10 y [mm] y [mm] 20 Experiment Spalart-Allmaras Edwards-Chandra SALSA 20 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 0 -0.5 -1 -1.5 -2 0 -0.5 -1 -1.5 -2 0 0 d) b) 10 10 y [mm] y [mm] 20 Experiment kω SST MBC kω SST modified 20 Experiment kω TNT RBC Wallin RBC kω LLR RBC Figure C.23: Reynolds-shear-stress proﬁles for case CS0 at x/D = −1.08857. c) a) 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur -2 196 C Graphs of Computational Results 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur 0 -0.5 -1 -1.5 -2 0 -0.5 -1 -1.5 -2 0 0 c) a) y [mm] 20 y [mm] 20 30 Experiment Spalart-Allmaras Edwards-Chandra SALSA 30 0 -0.5 -1 -1.5 -2 0 -0.5 -1 -1.5 -2 0 0 d) b) 10 10 20 y [mm] 20 y [mm] 30 Experiment kω SST MBC kω SST modified 30 Experiment kω TNT RBC Wallin RBC kω LLR RBC Figure C.24: Reynolds-shear-stress proﬁles for case CS0 at x/D = −0.544286. 10 10 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur C.2 Boundary Layer with Pressure-Induced Separation (Case CS0) 197 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur 0 -0.5 -1 -1.5 -2 -2.5 -3 0 -0.5 -1 -1.5 -2 -2.5 0 0 c) a) 20 20 30 y [mm] y [mm] 30 50 40 50 Experiment Spalart-Allmaras Edwards-Chandra SALSA 40 0 -0.5 -1 -1.5 -2 -2.5 -3 0 -0.5 -1 -1.5 -2 -2.5 -3 0 0 d) b) 10 10 20 20 30 y [mm] 30 y [mm] 50 40 50 Experiment kω SST MBC kω SST modified 40 Experiment kω TNT RBC Wallin RBC kω LLR RBC Figure C.25: Reynolds-shear-stress proﬁles for case CS0 at x/D = −0.0907143. 10 10 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur -3 198 C Graphs of Computational Results 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur 0 -0.5 -1 -1.5 -2 -2.5 -3 0 -0.5 -1 -1.5 -2 -2.5 -3 0 0 c) a) 20 20 30 y [mm] 30 y [mm] 50 40 50 Experiment Spalart-Allmaras Edwards-Chandra SALSA 40 0 -0.5 -1 -1.5 -2 -2.5 -3 0 -0.5 -1 -1.5 -2 -2.5 -3 0 0 d) b) 10 10 20 20 30 y [mm] 30 y [mm] 50 40 50 Experiment kω SST MBC kω SST modified 40 Experiment kω TNT RBC Wallin RBC kω LLR RBC Figure C.26: Reynolds-shear-stress proﬁles for case CS0 at x/D = 0.0907143. 10 10 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur C.2 Boundary Layer with Pressure-Induced Separation (Case CS0) 199 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur 0 -0.5 -1 -1.5 -2 -2.5 -3 0 -0.5 -1 -1.5 -2 -2.5 0 0 c) a) 20 20 30 y [mm] y [mm] 30 50 40 50 Experiment Spalart-Allmaras Edwards-Chandra SALSA 40 0 -0.5 -1 -1.5 -2 -2.5 -3 0 -0.5 -1 -1.5 -2 -2.5 -3 0 0 d) b) 10 10 20 20 30 y [mm] 30 y [mm] 50 40 50 Experiment kω SST MBC kω SST modified 40 Experiment kω TNT RBC Wallin RBC kω LLR RBC Figure C.27: Reynolds-shear-stress proﬁles for case CS0 at x/D = 0.181429. 10 10 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur -3 200 C Graphs of Computational Results 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur -4 20 30 y [mm] y [mm] 30 50 40 50 Experiment Spalart-Allmaras Edwards-Chandra SALSA 40 -3 -1 -1.5 -2 -2.5 -3 -3.5 -4 0 -0.5 -1 -1.5 -2 -2.5 0 0 d) b) 10 10 20 20 30 y [mm] 30 y [mm] 50 40 50 Experiment kω SST MBC kω SST modified 40 Experiment kω TNT RBC Wallin RBC kω LLR RBC Figure C.28: Reynolds-shear-stress proﬁles for case CS0 at x/D = 0.362857. 0 10 20 -4 -3.5 0 c) 10 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC -0.5 0 0 a) -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur C.2 Boundary Layer with Pressure-Induced Separation (Case CS0) 201 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur -4 -3 20 30 40 y [mm] 40 y [mm] 30 60 50 60 Experiment Spalart-Allmaras Edwards-Chandra SALSA 50 -2 -1 -1.5 -2 -2.5 -3 -3.5 -4 0 -0.5 -1 -1.5 0 0 d) b) 10 10 20 20 30 40 40 y [mm] 30 y [mm] 60 50 60 Experiment kω SST MBC kω SST modified 50 Experiment kω TNT RBC Wallin RBC kω LLR RBC Figure C.29: Reynolds-shear-stress proﬁles for case CS0 at x/D = 0.725714. 0 10 20 -3 -2.5 0 c) 10 -4 -3.5 -0.5 0 0 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 0 -0.5 -1 -1.5 -2 -2.5 a) 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur -3.5 202 C Graphs of Computational Results 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur -4 20 40 y [mm] 30 40 y [mm] 30 60 70 50 60 70 Experiment Spalart-Allmaras Edwards-Chandra SALSA 50 -3 -1 -1.5 -2 -2.5 -3 -3.5 -4 0 -0.5 -1 -1.5 -2 -2.5 0 0 d) b) 10 10 20 20 40 y [mm] 30 40 y [mm] 30 50 50 Figure C.30: Reynolds-shear-stress proﬁles for case CS0 at x/D = 1.08857. 0 10 20 -4 -3.5 0 c) 10 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC -0.5 0 0 a) -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur 70 60 70 Experiment kω SST MBC kω SST modified 60 Experiment kω TNT RBC Wallin RBC kω LLR RBC C.2 Boundary Layer with Pressure-Induced Separation (Case CS0) 203 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur 0 -1 -2 -3 -4 -5 0 -1 -2 -3 -4 0 0 c) a) 10 10 30 30 40 y [mm] y [mm] 40 50 50 70 60 70 Experiment Spalart-Allmaras Edwards-Chandra SALSA 60 0 -1 -2 -3 -4 -5 0 -1 -2 -3 -4 -5 0 0 d) b) 10 10 20 20 30 30 40 y [mm] 40 y [mm] 50 50 Figure C.31: Reynolds-shear-stress proﬁles for case CS0 at x/D = 1.63286. 20 20 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur -5 70 60 70 Experiment kω SST MBC kω SST modified 60 Experiment kω TNT RBC Wallin RBC kω LLR RBC 204 C Graphs of Computational Results 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur 0 -1 -2 -3 -4 -5 0 -1 -2 -3 -4 -5 0 0 c) a) 10 10 30 30 40 y [mm] y [mm] 40 50 50 70 80 60 70 80 Experiment Spalart-Allmaras Edwards-Chandra SALSA 60 0 -1 -2 -3 -4 -5 0 -1 -2 -3 -4 -5 0 0 d) b) 10 10 20 20 30 30 40 y [mm] 40 y [mm] 50 50 Figure C.32: Reynolds-shear-stress proﬁles for case CS0 at x/D = 2.17714. 20 20 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 2 <u’v’>/0.001Ur 2 <u’v’>/0.001Ur 70 80 60 70 80 Experiment kω SST MBC kω SST modified 60 Experiment kω TNT RBC Wallin RBC kω LLR RBC C.2 Boundary Layer with Pressure-Induced Separation (Case CS0) 205 206 C Graphs of Computational Results a) <u’u’>/0.001Ur 2 14 10 8 6 4 2 0 0 2 10 20 30 40 y [mm] 50 60 b) 14 <v’v’>/0.001Ur Experiment Wallin RBC 12 12 Experiment Wallin RBC 10 8 6 4 2 0 0 10 20 30 40 y [mm] 50 60 Figure C.33: Proﬁles of normal Reynolds stresses for case CS0 at x/D = 0.725714. C.3 C.3 Separated Airfoil Flow (Case AAA) Separated Airfoil Flow (Case AAA) The rest of this page has been deliberately left blank 207 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0 c) a) 0.01 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.01 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0 d) b) z/c 0.005 z/c 0.005 Figure C.34: Velocity proﬁles for airfoil A at x/c = 0.3. z/c 0.005 z/c 0.005 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC u/U∞ u/U∞ u/U∞ u/U∞ 0.01 0.01 Experiment kω SST MBC kω SST modified Johnson-King Experiment kω TNT RBC Wallin RBC kω LLR RBC 208 C Graphs of Computational Results u/U∞ u/U∞ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0 c) a) 0.005 0.005 0.015 0.01 0.015 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.01 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0 d) b) 0.005 0.005 Figure C.35: Velocity proﬁles for airfoil A at x/c = 0.4. z/c z/c Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC u/U∞ u/U∞ z/c z/c 0.015 0.01 0.015 Experiment kω SST MBC kω SST modified Johnson-King 0.01 Experiment kω TNT RBC Wallin RBC kω LLR RBC C.3 Separated Airfoil Flow (Case AAA) 209 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0 c) a) 0.005 0.005 0.02 0.015 0.02 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.015 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0 d) b) 0.005 0.005 z/c 0.01 z/c 0.01 Figure C.36: Velocity proﬁles for airfoil A at x/c = 0.5. z/c 0.01 z/c 0.01 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC u/U∞ u/U∞ u/U∞ u/U∞ 0.02 0.015 0.02 Experiment kω SST MBC kω SST modified Johnson-King 0.015 Experiment kω TNT RBC Wallin RBC kω LLR RBC 210 C Graphs of Computational Results u/U∞ u/U∞ 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0 c) a) 0.005 0.005 0.015 z/c 0.015 z/c 0.025 0.02 0.025 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.02 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0 d) b) 0.005 0.005 0.01 0.01 Figure C.37: Velocity proﬁles for airfoil A at x/c = 0.6. 0.01 0.01 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC u/U∞ u/U∞ 0.015 z/c 0.015 z/c 0.025 0.02 0.025 Experiment kω SST MBC kω SST modified Johnson-King 0.02 Experiment kω TNT RBC Wallin RBC kω LLR RBC C.3 Separated Airfoil Flow (Case AAA) 211 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0 0.005 c) 0.005 a) 0.01 0.01 0.03 0.035 0.025 0.03 0.035 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.025 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0 0.005 d) 0.005 b) 0.01 0.01 0.02 z/c 0.015 0.02 z/c 0.015 Figure C.38: Velocity proﬁles for airfoil A at x/c = 0.7. 0.02 z/c 0.015 0.02 z/c 0.015 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC u/U∞ u/U∞ u/U∞ u/U∞ 0.03 0.035 0.025 0.03 0.035 Experiment kω SST MBC kω SST modified Johnson-King 0.025 Experiment kω TNT RBC Wallin RBC kω LLR RBC 212 C Graphs of Computational Results u/U∞ u/U∞ 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0 c) a) 0.01 0.01 z/c z/c 0.03 0.03 0.05 0.04 0.05 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.04 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0 d) b) 0.01 0.01 0.02 0.02 z/c z/c Figure C.39: Velocity proﬁles for airfoil A at x/c = 0.775. 0.02 0.02 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC u/U∞ u/U∞ 0.03 0.03 0.04 Experiment kω SST MBC kω SST modified Johnson-King 0.04 Experiment kω TNT RBC Wallin RBC kω LLR RBC 0.05 0.05 C.3 Separated Airfoil Flow (Case AAA) 213 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0 c) a) 0.01 0.01 0.02 0.02 0.05 0.06 0.04 0.05 0.06 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.04 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0 d) b) 0.01 0.01 0.02 0.02 z/c 0.03 z/c 0.03 Figure C.40: Velocity proﬁles for airfoil A at x/c = 0.825. z/c 0.03 z/c 0.03 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC u/U∞ u/U∞ u/U∞ u/U∞ 0.05 0.06 0.04 0.05 0.06 Experiment kω SST MBC kω SST modified Johnson-King 0.04 Experiment kω TNT RBC Wallin RBC kω LLR RBC 214 C Graphs of Computational Results u/U∞ u/U∞ 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0 c) a) 0.01 0.01 0.02 0.02 z/c 0.04 z/c 0.04 0.05 0.05 0.07 0.06 0.07 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.06 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0 0.01 d) 0.01 b) 0.02 0.02 0.03 0.03 0.04 z/c 0.04 z/c Figure C.41: Velocity proﬁles for airfoil A at x/c = 0.87. 0.03 0.03 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC u/U∞ u/U∞ 0.05 0.05 0.07 0.06 0.07 Experiment kω SST MBC kω SST modified Johnson-King 0.06 Experiment kω TNT RBC Wallin RBC kω LLR RBC C.3 Separated Airfoil Flow (Case AAA) 215 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0 z/c Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC z/c Experiment Spalart-Allmaras Edwards-Chandra SALSA 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0 z/c Experiment kω TNT RBC Wallin RBC kω LLR RBC z/c Experiment kω SST MBC kω SST modified Johnson-King 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 d) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 b) Figure C.42: Velocity proﬁles for airfoil A at x/c = 0.9. 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 c) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 a) u/U∞ u/U∞ u/U∞ u/U∞ 216 C Graphs of Computational Results u/U∞ u/U∞ 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0 z/c Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC z/c Experiment Spalart-Allmaras Edwards-Chandra SALSA 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0 z/c Experiment kω TNT RBC Wallin RBC kω LLR RBC z/c Experiment kω SST MBC kω SST modified Johnson-King 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 d) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 b) Figure C.43: Velocity proﬁles for airfoil A at x/c = 0.93. 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 c) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 a) u/U∞ u/U∞ C.3 Separated Airfoil Flow (Case AAA) 217 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0 c) a) 0.02 0.02 z/c z/c 0.06 0.06 0.08 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.08 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0 d) b) 0.02 0.02 0.04 0.04 z/c z/c Figure C.44: Velocity proﬁles for airfoil A at x/c = 0.96. 0.04 0.04 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC u/U∞ u/U∞ u/U∞ u/U∞ 0.06 0.06 0.08 Experiment kω SST MBC kω SST modified Johnson-King 0.08 Experiment kω TNT RBC Wallin RBC kω LLR RBC 218 C Graphs of Computational Results u/U∞ u/U∞ 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0 c) a) 0.025 0.025 0.1 0.075 0.1 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.075 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0 d) b) 0.025 0.025 z/c 0.05 z/c 0.05 Figure C.45: Velocity proﬁles for airfoil A at x/c = 0.99. z/c 0.05 z/c 0.05 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC u/U∞ u/U∞ 0.1 0.075 0.1 Experiment kω SST MBC kω SST modified Johnson-King 0.075 Experiment kω TNT RBC Wallin RBC kω LLR RBC C.3 Separated Airfoil Flow (Case AAA) 219 0.006 2 2 0.006 0.007 0 0.001 0.002 0.003 0.004 0.005 -<u’v’>/U∞ 0 0 d) b) z/c 0.005 z/c 0.005 Figure C.46: Reynolds-shear-stress proﬁles for airfoil A at x/c = 0.3. 0 0 z/c 0.001 0.01 0.002 0.001 0.005 0.003 0.005 0.006 0.007 0 0.002 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.01 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.003 c) z/c 0.005 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 0.004 0 0 a) 0.004 0.005 -<u’v’>/U∞ 2 -<u’v’>/U∞ 2 -<u’v’>/U∞ 0.007 0.01 Experiment kω SST MBC kω SST modified Johnson-King 0.01 Experiment kω TNT RBC Wallin RBC kω LLR RBC 220 C Graphs of Computational Results 0.004 0.005 2 2 0.004 0.005 0 0.001 0.002 0.003 -<u’v’>/U∞ z/c 0 0 d) b) 0.005 0.005 z/c z/c 0.01 0.01 Figure C.47: Reynolds-shear-stress proﬁles for airfoil A at x/c = 0.4. 0 0.015 0 0.01 0.001 0.004 0.005 0 0.001 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.015 0.002 0.005 z/c 0.01 0.001 0.002 0.003 0.004 0.005 0.002 c) 0.005 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 0.003 0 0 a) 0.003 -<u’v’>/U∞ 2 -<u’v’>/U∞ 2 -<u’v’>/U∞ 0.015 Experiment kω SST MBC kω SST modified Johnson-King 0.015 Experiment kω TNT RBC Wallin RBC kω LLR RBC C.3 Separated Airfoil Flow (Case AAA) 221 0.004 2 2 0.004 0.005 0 0.001 0.002 0.003 -<u’v’>/U∞ z/c 0 0 d) b) 0.005 0.005 z/c 0.01 z/c 0.01 Figure C.48: Reynolds-shear-stress proﬁles for airfoil A at x/c = 0.5. 0 0.02 0 0.015 0.001 0.004 0.005 0 0.001 0.01 0.02 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.015 0.002 0.005 z/c 0.01 0.001 0.002 0.003 0.004 0.005 0.002 c) 0.005 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 0.003 0 0 a) 0.003 -<u’v’>/U∞ 2 -<u’v’>/U∞ 2 -<u’v’>/U∞ 0.005 0.02 0.015 0.02 Experiment kω SST MBC kω SST modified Johnson-King 0.015 Experiment kω TNT RBC Wallin RBC kω LLR RBC 222 C Graphs of Computational Results 0.004 2 2 0.004 0 0.001 0.002 0.003 -<u’v’>/U∞ z/c 0.025 0 0 d) b) 0.005 0.005 0.01 0.01 0.015 z/c 0.015 z/c Figure C.49: Reynolds-shear-stress proﬁles for airfoil A at x/c = 0.6. 0.02 0.003 0.004 0 0 0.015 0.025 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.02 0 0.01 z/c 0.015 0.001 0.005 0.01 0.001 0.002 0.003 0.004 0.001 c) 0.005 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 0.002 0 0 a) 0.002 0.003 -<u’v’>/U∞ 2 -<u’v’>/U∞ 2 -<u’v’>/U∞ 0.025 0.02 0.025 Experiment kω SST MBC kω SST modified Johnson-King 0.02 Experiment kω TNT RBC Wallin RBC kω LLR RBC C.3 Separated Airfoil Flow (Case AAA) 223 2 2 0.004 0 0.001 0.002 0.003 -<u’v’>/U∞ z/c 0 0 d) b) 0.01 0.01 z/c 0.02 z/c 0.02 Figure C.50: Reynolds-shear-stress proﬁles for airfoil A at x/c = 0.7. 0.04 0 0.03 0.003 0.004 0 0 0.02 0.04 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.03 0.001 0.01 z/c 0.02 0.001 0.002 0.003 0.004 0.001 c) 0.01 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 0.002 0 0 a) 0.002 0.003 -<u’v’>/U∞ 2 -<u’v’>/U∞ 2 -<u’v’>/U∞ 0.004 0.04 0.03 0.04 Experiment kω SST MBC kω SST modified Johnson-King 0.03 Experiment kω TNT RBC Wallin RBC kω LLR RBC 224 C Graphs of Computational Results 0.004 0.005 2 2 0.004 0.005 0 0.001 0.002 0.003 -<u’v’>/U∞ z/c 0 0 d) b) 0.01 0.01 0.02 0.02 z/c z/c 0.03 0.03 Figure C.51: Reynolds-shear-stress proﬁles for airfoil A at x/c = 0.775. 0.05 0 0.04 0 0.004 0.005 0 0.001 0.03 0.05 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.04 0.001 0.02 z/c 0.03 0.002 0.01 0.02 0.001 0.002 0.003 0.004 0.005 0.002 c) 0.01 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 0.003 0 0 a) 0.003 -<u’v’>/U∞ 2 -<u’v’>/U∞ 2 -<u’v’>/U∞ 0.05 0.04 0.05 Experiment kω SST MBC kω SST modified Johnson-King 0.04 Experiment kω TNT RBC Wallin RBC kω LLR RBC C.3 Separated Airfoil Flow (Case AAA) 225 0.004 2 2 0.004 0.005 0 0.001 0.002 0.003 -<u’v’>/U∞ z/c 0 0 d) b) 0.01 0.01 0.02 0.02 z/c 0.03 z/c 0.03 0.04 0.04 Figure C.52: Reynolds-shear-stress proﬁles for airfoil A at x/c = 0.825. 0.06 0 0.05 0.004 0.005 0 0 0.04 0.06 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.05 0.001 0.03 0.04 0.001 0.02 z/c 0.03 0.002 0.01 0.02 0.001 0.002 0.003 0.004 0.005 0.002 c) 0.01 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 0.003 0 0 a) 0.003 -<u’v’>/U∞ 2 -<u’v’>/U∞ 2 -<u’v’>/U∞ 0.005 0.06 0.05 0.06 Experiment kω SST MBC kω SST modified Johnson-King 0.05 Experiment kω TNT RBC Wallin RBC kω LLR RBC 226 C Graphs of Computational Results 0.005 0.006 2 2 0.005 0.006 0 0.001 0.002 0.003 0.004 -<u’v’>/U∞ 0.06 0.07 0 0 d) b) 0.01 0.01 0.02 0.02 0.03 0.03 0.04 z/c 0.04 z/c Figure C.53: Reynolds-shear-stress proﬁles for airfoil A at x/c = 0.87. 0.05 0.004 0.005 0.006 0 0 z/c 0.07 0 0.04 0.06 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.05 0.001 0.03 z/c 0.04 0.001 0.02 0.03 0.002 0.01 0.02 0.001 0.002 0.003 0.004 0.005 0.006 0.002 c) 0.01 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 0.003 0 0 a) 0.003 0.004 -<u’v’>/U∞ 2 -<u’v’>/U∞ 2 -<u’v’>/U∞ 0.06 0.07 0.05 0.06 0.07 Experiment kω SST MBC kω SST modified Johnson-King 0.05 Experiment kω TNT RBC Wallin RBC kω LLR RBC C.3 Separated Airfoil Flow (Case AAA) 227 0.005 0.006 2 2 0.005 0.006 0.007 0 0.001 0.002 0.003 0.004 -<u’v’>/U∞ z/c 0 0 0 0 z/c Experiment kω TNT RBC Wallin RBC kω LLR RBC z/c Experiment kω SST MBC kω SST modified Johnson-King 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 d) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 b) Figure C.54: Reynolds-shear-stress proﬁles for airfoil A at x/c = 0.9. 0.001 0.001 z/c 0.002 0.002 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.003 0.005 0.006 0.007 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.004 c) Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 0.003 0 0 a) 0.004 -<u’v’>/U∞ 2 -<u’v’>/U∞ 2 -<u’v’>/U∞ 0.007 228 C Graphs of Computational Results 0.005 0.006 0.007 2 2 0.005 0.006 0.007 0 0.001 0.002 0.003 0.004 -<u’v’>/U∞ z/c 0 0 0 0 z/c Experiment kω TNT RBC Wallin RBC kω LLR RBC z/c Experiment kω SST MBC kω SST modified Johnson-King 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 d) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 b) Figure C.55: Reynolds-shear-stress proﬁles for airfoil A at x/c = 0.93. 0.001 0.001 z/c 0.002 0.002 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.003 0.005 0.006 0.007 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.004 c) Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 0.003 0 0 a) 0.004 -<u’v’>/U∞ 2 -<u’v’>/U∞ 2 -<u’v’>/U∞ C.3 Separated Airfoil Flow (Case AAA) 229 0.006 2 2 0.006 0.007 0 0.001 0.002 0.003 0.004 0.005 -<u’v’>/U∞ 0 0 d) b) 0.02 0.02 0.04 0.04 0.06 z/c 0.06 z/c Figure C.56: Reynolds-shear-stress proﬁles for airfoil A at x/c = 0.96. 0 0.1 0 0.08 0.001 0.06 0.002 0.005 0.006 0.007 0 0.001 z/c 0.1 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.08 0.002 0.04 z/c 0.06 0.003 0.02 0.04 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.004 c) 0.02 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 0.003 0 0 a) 0.004 0.005 -<u’v’>/U∞ 2 -<u’v’>/U∞ 2 -<u’v’>/U∞ 0.007 0.1 0.08 0.1 Experiment kω SST MBC kω SST modified Johnson-King 0.08 Experiment kω TNT RBC Wallin RBC kω LLR RBC 230 C Graphs of Computational Results 0.006 0.007 2 2 0.006 0.007 0 0.001 0.002 0.003 0.004 0.005 -<u’v’>/U∞ 0 0 d) b) 0.025 0.025 z/c 0.05 z/c 0.05 0.075 0.075 Figure C.57: Reynolds-shear-stress proﬁles for airfoil A at x/c = 0.99. 0 0.1 0 0.075 0.001 z/c 0.002 0.005 0.006 0.007 0 0.001 Experiment Spalart-Allmaras Edwards-Chandra SALSA 0.1 0.002 0.05 0.075 0.003 0.025 z/c 0.05 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.004 c) 0.025 Experiment Baldwin-Lomax kω 88 WBC kω 98 WBC 0.003 0 0 a) 0.004 0.005 -<u’v’>/U∞ 2 -<u’v’>/U∞ 2 -<u’v’>/U∞ 0.1 Experiment kω SST MBC kω SST modified Johnson-King 0.1 Experiment kω TNT RBC Wallin RBC kω LLR RBC C.3 Separated Airfoil Flow (Case AAA) 231 232 D D Overview of Algorithmic Accomplishments Overview of Algorithmic Accomplishments In the course of the work, the following major algorithmic issues were accomplished: • A ﬂow analysis tool, called newmono, was designed and implemented. With this tool, ﬂow quantities can be extracted from the ﬂow solution and plotted in coordinate systems that are locally normal and parallel, that is “monocline”, to selected topological skeleton lines. In particular, vectors and tensors are transfered to the locally monocline system by according transformation rules. For reasons of brevity, the details of newmono and its implementation are not presented in this thesis. • Local preconditioning for low Mach numbers was implemented into the MUFLO ﬂow solver (Subsection 8.2). (Several diﬀerent preconditioning methods were investigated; not reported here.) • The Johnson-King model was modiﬁed according to topological arguments (Subsection 4.3.4). • The 1998 k, ω model of Wilcox was implemented into the FLOWer code (Subsection 6.1). • Low-Reynolds-number modiﬁcations for the 1988 and 1998 k, ω models of Wilcox and the k, ω TNT model of Kok were implemented into the FLOWer code. In this vein, the implementation of the explicit algebraic Reynolds-stress model of Wallin in the FLOWer code was extended to yield the non-linear form of the model (Section 6). • The artiﬁcial damping terms in the FLOWer code were modiﬁed such that the damping is reduced in the direction normal to the wall in boundary layers (Subsection 8.4). • Local preconditioning in the FLOWer code was modiﬁed for airfoil ﬂows: In the vicinity of stagnation points, local preconditioning is reduced compared to other regions of the ﬂow. Maximum local preconditioning is achieved around trailing edges. This procedure proved to yield best possible results in two aspects: First, stability problems of the preconditioning method which are related to large ﬂow-angle changes in the vicinity of the leading-edge stagnation point are resolved by reducing preconditioning in this region. Secondly, pressure oscillations at the 233 trailing edge due to incorrect scaling of artiﬁcial damping terms at low Mach number are substantially reduced by maximal local preconditioning (Subsection 8.2). • A modiﬁed version of the k, ω SST model of Menter was implemented into the FLOWer code (Subsection 7.1.3). • An adaption technique for the exit pressure of internal ﬂows was designed and implemented into the FLOWer code (Subsection 7.2). • A new boundary treatment was implemented into the FLOWer code to allow for prescribing measured inﬂow conditions. 234 E E.1 E Typical FLOWer Input Decks Typical FLOWer Input Decks Typical FLOWer Input Deck for Case FPBL $$ $$ Input 116.10 $$ $$---------------------------------------------------------------------$$ Testcase $$ -------$$ STRING 2D flat plate DeGraaff & Eaton $$ $$---------------------------------------------------------------------$$ General Control Data $$ -------------------$$ I2D3D 2 ILAG 0 INCORE 1 RESTOL 1.0e-06 FOTOL 1.0e-13 PEXIT 1.0 ISTEPOUT 10 NSAVE 200 ITLNS 1 $$ $$---------------------------------------------------------------------$$ Flow Data $$ --------$$ MACH 0.02848 ALPHA 0.00 $$ Bezugslaenge fuer Re-Zahl L=7.0m -> Re RENO 4471085.12002 RELEN 1.00 TINF 296.4 $$ $$---------------------------------------------------------------------$$ Geometrical Data $$ ---------------$$ AREF 1.0000 E.1 Typical FLOWer Input Deck for Case FPBL 235 XYZREF 0.2500 0.0000 0.0000 CREF 1.0000 SREF 1.0000 $$ $$---------------------------------------------------------------------$$ Space and Time Discretization Data $$ ---------------------------------$$ TUSPACE 11 $$ $$---------------------------------------------------------------------$$ Boundary Treatment Control Data $$ ------------------------------$$ BCV 0 $$ $$---------------------------------------------------------------------$$ Turbulence Model Data $$ --------------------$$ $$----<Wallin EARSM Model>--ITURB 27 $$ $$----<nonlinear, 2D>--ITU27LIN 0 ITU27DIM 2 $$ $$----<Wilcox values for 1988 Model>--ITUKWSET 1 $$ $$----<Wilcox’ wall boundary condition for omega>--BCTURBKW 0 $$ $$----<use lo Reynolds-number modifications>---ILORENO 1 $$ $$----<contribution of k to stresses and energy>--$$----<KINFLU is newly defined since version 116.10> KINFLU 2 $$ KPRDLIM 1000000. $$ 236 E Typical FLOWer Input Decks $$---------------------------------------------------------------------$$ Transition Data $$---------------$$ NTRAN 1 XTRANU 0.04 XTRANL 0.04 ZTRAN 1.00 $$ $$---------------------------------------------------------------------$$ Multigrid Control Data $$ ---------------------$$ LEVEL 6 NGIT 3 3 ISTART 3 ITYPC 1 MAXLEV 3 NEND 12000 6000 15000 0 0 0 NDUM 1 1 1 1 1 1 EPSC 0.2 DTVI 0.00 $$ $$---------------------------------------------------------------------$$ RUNGE-KUTTA CONTROL PARAMETERS FOR MESH LEVEL $$ --------------------------------------------$$ LEVPAR 2 $$...................................................................... GRIDF CFL 6.50 CFLS 3.75 RVIS4 64. RVIS2 2. 2. 2. RVIS0 16. ZETA 0.8 FILTYPE 1 ISMOO 2 EPSXYZ 0.2 1.0 0.0 SMS 1 1 1 1 1 $$...................................................................... GRIDC E.1 Typical FLOWer Input Deck for Case FPBL 237 CFL 6.50 CFLS 3.75 RVIS4 64. RVIS2 2. 2. 2. RVIS0 16. ZETA 0.8 FILTYPE 1 ISMOO 2 EPSXYZ 0.2 1.0 0.0 SMS 1 1 1 1 1 $$...................................................................... GRIDEND $$ $$---------------------------------------------------------------------$$ Block Dependent Data $$ -------------------$$ MBLM 1 $$ BLOCK001 IVIS 40 IJKDIR 3 1 0 $$ BLOCKEND $$ $$---------------------------------------------------------------------$$ Time Accurate Data $$ -----------------STEPTYPE 0 TIMESTEP 0.2 NMAX2T 100 MAXIT2T 250 OUTWHAT 0 NOUTSURF -1 SURFVAL 1 1 20 $$ $$ $$---------------------------------------------------------------------$$ Preconditioning $$ --------------$$ IPREC 1 238 E Typical FLOWer Input Decks UPC EPSLOCM E.2 1.0 4.0 Typical FLOWer Input Deck for Case BS0 and CS0 $$ $$ Input 116.10 $$ $$---------------------------------------------------------------------$$ Testcase $$ -------$$ STRING Driver Cylinder case BS0 $$ $$ $$---------------------------------------------------------------------$$ General Control Data $$ -------------------$$ I2D3D 2 IROSY 1 ILAG 0 PHI 0.5 INCORE 1 RESTOL 3.0e-06 FOTOL 1.0e-13 ISTEPOUT 10 ISTEPADP 300 $$PEXIT 1.0024 PEXIT 1.0024874853 PADAPT 1 NSAVE 300 ITLNS 1 $$ $$---------------------------------------------------------------------$$ Flow Data $$ --------$$ MACH 0.08772 ALPHA 0.00 RENO 280000. E.2 Typical FLOWer Input Deck for Case BS0 and CS0 239 RELEN 1. TINF 291.00 $$ $$---------------------------------------------------------------------$$ Geometrical Data $$ ---------------$$ AREF 1.0000 XYZREF 0.2500 0.0000 0.0000 CREF 1.0000 SREF 1.0000 $$ $$---------------------------------------------------------------------$$ Space and Time Discretization Data $$ ---------------------------------$$ TUSPACE 11 $$ $$---------------------------------------------------------------------$$ Boundary Treatment Control Data $$ ------------------------------$$ BCV 0 $$ $$---------------------------------------------------------------------$$ Turbulence Model Data $$ --------------------$$ $$----<Wallin EARSM Model>--ITURB 27 $$ $$----<nonlinear, 2D>--ITU27LIN 0 ITU27DIM 2 $$ $$----<Wilcox values for 1988 Model>--ITUKWSET 1 $$ $$----<Rudnik wall boundary condition for omega>--BCTURBKW 1 $$ $$----<use lo Reynolds-number modifications>---- 240 E Typical FLOWer Input Decks ILORENO 1 $$ $$----<contribution of k to stresses and energy>--$$----<KINFLU is newly defined since version 116.10> KINFLU 2 $$ KPRDLIM 1000000. $$ $$---------------------------------------------------------------------$$ Transition Data $$ --------------$$ NTRAN 0 $$ $$---------------------------------------------------------------------$$ Multigrid Control Data $$ ---------------------$$ LEVEL 6 NGIT 3 3 ISTART 3 ITYPC 2 MAXLEV 5 NEND 12000 6000 10000 0 0 0 NDUM 1 1 1 1 1 1 EPSC 0.2 DTVI 8.0 $$---------------------------------------------------------------------$$ RUNGE-KUTTA CONTROL PARAMETERS FOR MESH LEVEL $$ --------------------------------------------$$ LEVPAR 2 $$...................................................................... GRIDF CFL 6.5 CFLS 3.75 CFLTU 6.5 RVIS4 64. RVIS2 2. 2. 2. RVIS0 16. ZETA 0.5 ISMOO 2 E.2 Typical FLOWer Input Deck for Case BS0 and CS0 241 EPSXYZ 0.2 1.0 0.0 FILTYPE 1 SMS 1 1 1 1 1 $$...................................................................... GRIDC CFL 6.5 CFLS 3.75 CFLTU 6.5 RVIS4 64. RVIS2 2. 2. 2. RVIS0 16. ZETA 0.5 ISMOO 2 EPSXYZ 0.2 1.0 0.0 FILTYPE 1 SMS 1 1 1 1 1 $$...................................................................... GRIDEND $$ $$---------------------------------------------------------------------$$ Block Dependent Data $$ -------------------$$ MBLM 1 $$ BLOCK001 IVIS 40 IJKDIR 0 0 0 $$ BLOCKEND $$ $$---------------------------------------------------------------------$$ Preconditioning $$ --------------$$ IPREC 1 UPC 1.0 EPSLOCM 2.0 242 E.3 E Typical FLOWer Input Decks Typical FLOWer Input Deck for Case AAA $$ $$ Input 116.10 $$ $$---------------------------------------------------------------------$$ Testcase $$ -------$$ STRING ONERA A-AIRFOIL 13.3 DEG AOA, F2 CASE $$ $$---------------------------------------------------------------------$$ General Control Data $$ -------------------$$ I2D3D 2 ILAG 0 INCORE 1 RESTOL 1.0e-06 FOTOL 1.0e-16 ISTEPOUT 10 NSAVE 500 ITLNS 1 $$ $$---------------------------------------------------------------------$$ Flow Data $$ --------$$ MACH 0.15 ALPHA 13.3 RENO 2.00E+6 RELEN 1.00 TINF 294.4 $$ $$---------------------------------------------------------------------$$ Geometrical Data $$ ---------------$$ AREF 1.0000 XYZREF 0.2500 0.0000 0.0000 CREF 1.0000 SREF 1.0000 $$ E.3 Typical FLOWer Input Deck for Case AAA 243 $$---------------------------------------------------------------------$$ Space and Time Discretization Data $$ ---------------------------------$$ TUSPACE 11 $$ $$---------------------------------------------------------------------$$ Boundary Treatment Control Data $$ ------------------------------$$ BCV 1 $$ $$---------------------------------------------------------------------$$ Turbulence Model Data $$ --------------------$$ $$----<New Wilcox 1998 Model>--ITURB 28 $$ $$----<Wilcox wall boundary condition for omega>--BCTURBKW 0 $$ $$----<USE lo Reynolds-number modifications>---ILORENO 1 $$ $$----<contribution of k to stresses and energy>--$$----<KINFLU is newly defined since version 116.10> KINFLU 2 $$ KPRDLIM 1000000. $$ $$---------------------------------------------------------------------$$ Transition Data $$---------------$$ NTRAN 1 XTRANU 0.1167 XTRANL 0.30 ZTRAN 1.00 $$ $$---------------------------------------------------------------------$$ Multigrid Control Data 244 E Typical FLOWer Input Decks $$ ---------------------$$ LEVEL 6 NGIT 4 2 ISTART 0 ITYPC 1 MAXLEV 4 NEND 0 10000 1000 500 1 0 NDUM 1 1 1 1 1 1 EPSC 0.2 DTVI 4.0 $$ $$---------------------------------------------------------------------$$ RUNGE-KUTTA CONTROL PARAMETERS FOR MESH LEVEL $$ --------------------------------------------$$ LEVPAR 2 $$...................................................................... GRIDF CFL 6.50 CFLS 3.75 RVIS4 64. RVIS2 2. 2. 2. RVIS0 16. ZETA 0.70 FILTYPE 1 ISMOO 2 EPSXYZ 0.2 1.0 0.0 SMS 1 1 1 1 1 $$...................................................................... GRIDC CFL 6.50 CFLS 3.75 RVIS4 64. RVIS2 2. 2. 2. RVIS0 16. ZETA 0.70 FILTYPE 1 ISMOO 2 EPSXYZ 0.2 1.0 0.0 SMS 1 1 1 1 1 $$...................................................................... E.3 Typical FLOWer Input Deck for Case AAA 245 GRIDEND $$ $$---------------------------------------------------------------------$$ Block Dependent Data $$ -------------------$$ MBLM 1 $$ BLOCK001 IVIS 40 IJKDIR 3 0 0 $$ BLOCKEND $$ $$---------------------------------------------------------------------$$ Time Accurate Data $$ -----------------STEPTYPE 0 TIMESTEP 0.2 NMAX2T 100 MAXIT2T 250 OUTWHAT 0 NOUTSURF -1 SURFVAL 1 1 20 $$ $$ $$---------------------------------------------------------------------$$ Preconditioning $$ --------------$$ IPREC 1 UPC 1.0 EPSLOCM 0.25 References Abid, R., Vatsa, V. 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Lebenslauf Persönliche Daten Name: Geburtsdatum: Geburtsort: Alan Celić 21.05.1971 Tübingen Beruf seit 09/03 Post-doctoral Research Engineer am Centre Européen de Recherche et de Formation Avancée en Calcul Scientiﬁque (CERFACS), Toulouse, Frankreich 04/1998 - 08/03 Wissenschaftlicher Mitarbeiter am Institut für Aerodynamik und Gasdynamik (IAG) der Universität Stuttgart Studium 10/1990 - 03/1998 Studium der Luft- und Raumfahrttechnik an der Universität Stuttgart mit den Vertiefungsfächern Flugzeugbau/Leichtbau und Strömungslehre Diplomarbeit am NASA Ames Research Center (Moﬀett Field, Kalifornien, USA) mit dem Titel Computational Study of Surface Tension and ” Wall Adhesion Eﬀects on an Oil Film Flow Underneath an Air Boundary Layer“ 09/1996 - 04/1997 NASA Ames Research Associate for CFD 10/1995 - 04/1996 Werksstudent bei der Firma Simons & Susslin Manufacturing Inc., San Jose, Kalifornien, USA Schule 08/1981 - 05/1990 Gymnasium Spaichingen 08/1977 - 07/1981 Schillerschule Spaichingen (Grundschule)

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