Proceedings of the ASME 2017 Pressure Vessels and Piping Conference PVP2017 July 16-20, 2017, Waikoloa, Hawaii, USA PVP2017-65019 RESEARCH ON CROSS FLOW INDUCED VIBRATION OF FLEXIBLE TUBE BUNDLE Zhipeng FENG Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China Chengdu, Sichuan, China Wenzheng ZHANG Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China Chengdu, Sichuan, China Yixiong ZHANG Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China Chengdu, Sichuan, China Fenggang ZANG Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China Chengdu, Sichuan, China Huanhuan QI Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China Chengdu, Sichuan, China Xuan HUANG Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China Chengdu, Sichuan, China ABSTRACT When the elastic deformation of the tube bundle is considered, the interaction between the flow field and the structure becomes more complicated. In order to investigate the flow induced vibration (FIV) problems in flexible tube bundle, a numerical model for fluid-structure interaction system was presented firstly. The unsteady three-dimensional Navier-Stokes equation and LES turbulence model were solved with the finite volume approach on structured grids combined with the technique of dynamic mesh. The dynamic equilibrium equation was discretized according to the finite element theory. The configurations considered are tubes in a cross flow. Firstly, the flow-induced vibration of a single flexible tube under uniform turbulent flow are calculated when Reynolds number is 1.35×104. The variety trends of lift, drag, displacement, vertex shedding frequency, phase difference of tube are analyzed under different frequency ratios. The nonlinear phenomena of lockedin, phase-switch are captured successfully. Meanwhile, the limit cycle and bifurcation of lift coefficient and displacement are analyzed using trajectory, phase portrait and Poincare sections. Secondly, the mutual interaction of two in-line flexible tubes is investigated. Different behaviors, bounded by critical distances between the tubes, critical velocity, and wake vortex pattern are highlighted. Finally, four tube bundle models were established to study the effect of the number of flexible tube on the FIV characteristics. Thanks to several calculations, the critical velocity of instability vibration and the effect of tube bundle configurations on fluid forces and dynamics were obtained successfully. It is therefore expected that further calculations, with model refinements and other validation studies, will bring valuable informations about bundle stability. Further comparisons with experiment are necessary to validate the behavior of the method in this configuration. 1 INTRODUCTION Operation experience and scientific research show that flow induced vibration (FIV) and its related wear and fretting fatigue are the main causes of the heat transfer tube rupture. The mechanism is vortex shedding, fluid elastic instability, acoustic resonance and turbulent buffeting [1]. Many scholars have done a lot of research on the FIV of single tube or single cylinder (see, Placzek et al. [2]; Simoneaua et al. [3]; Goverdhan et al. [4]). The earlier research on FIV mainly relies on experiment (such as, Feng [5], Griffin [6], Khalak and Williamson [7], Govardhan and Williamson [8]). The FIV characteristic of cylinder with high mass ratio was conducted by Feng [5] who undertook one of the first comprehensive experimental studies of this problem. Feng’s data have only two branches (Initial and Lower branch). For lower mass ratio cylinder system, fairly comprehensive reviews on this FIV problem can be found in the article by Williamson (Khalak and Williamson [7]; Govardhan and Williamson [8]). They found that, at lower Reynolds numbers (3500-10 000), the FIV system has three branches (Initial, Upper, and Lower), a much larger peak amplitude, and a broader synchronization range. Besides experimental study, much progress has been 1 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use made numerically toward the understanding of the dynamics of FIV. In this complicated problem, a lot of methods were used to solve Navier-Stokes equations, involving computational fluid dynamics method [9], finite element method [10], vortex element methods [11], time-marching technique [12], etc. Among these methods, The CFD method is the mostly used. The circular cylinder is generally simulated by an equivalent massspring-damper to investigate the dynamics of flow induced cylinder vibration and the influence of cylinder oscillation on flow field. Placzek [2] studied the FIV characteristics of circular cylinder and wake vortex structure. Evangelinos [9] did a three dimensional DNS study (1-DOF) at Re=1000 for flexible cylinders. Li et al. [13] used the space-time finite element method to investigate the FIV of a two-dimensional elastic mounted circular cylinder under the uniform flow when Reynolds number is 200. Gabbai and Benaroya [14] reviewed the mathematical models used to investigate FIV of circular cylinders and a variety of issues concerning the flow-induced vibration were discussed. Compared with a large number of studies on FIV of single tube, relatively few studies have been done on the flow-structure coupling vibration of tube bundles [15]. Price [16] introduced almost all theoretical models describing the fluid elastic instability of tube bundles. Due to the mutual interaction between the wake flow of tube bundle, the vibration induced by wake flow and vortex will be enhanced. Singh et al. [17] first used direct simulation method to solve the N-S equation to study the static stability of a row of tubes. Ichioka et al. [18] studied the FIV of a row of tubes using the finite difference method to solve the two-dimensional N-S equation. Longattea et al. [19], Omar et al. [20] proposed a numerical model of FIV using ALE method, but the model can only predict the vibration frequency of elastically supported rigid tube in cross flow, without considering the tube deformation. Though much progress has been made during the past decades, both numerically and experimentally, the complex interaction between structure and fluid is not completely understood yet and remains to be discovered. Meanwhile, flowinduced vibration of an elastic cylinder is of strongly nonlinear quality [13]. However, there are few nonlinear analyses [20]. On the other hand, due to the complexity of the FIV of three dimensional flexible tube, the existing research is mainly aimed at two-dimensional elastic support rigid tube [20]. Therefore, it cannot consider the interaction between elastic distortion of structure and fluid flow. When the elastic deformation of the tube bundle is considered, the interaction between the flow field and the structure becomes more complicated. The characteristics of the vibration and flow field are usually related to the arrangement and flow velocity of the tube bundle. With the equipment process parameters (flow rate, temperature, etc.) becoming higher, it is necessary to construct more accurate physical models to analyze the interactions between fluid and structure as well as their nonlinear response characteristics. The object of this paper is to study the FIV characteristics of flexible tube (bundle) under cross flow. An application on the single tube case is first described and compared to the literature based on the coupled approach. Secondly, the nonlinearity of single flexible tube under turbulent flow are then investigated. The Poincare section and limit cycle of lift coefficient and lateral displacement are as examples to illustrate its nonlinear characteristics. The "lock-in" phenomenon, phase switch, phase portrait and limit cycle can be constructed of the present fluidflexible tube coupling system. The bifurcation can be also investigated. Thirdly, the mutual interaction of two in-line flexible tubes is investigated. The critical pitch between the tubes, critical velocity, and wake vortex pattern are highlighted. Finally, the behavior of four flexible tube bundles is studied numerically and some results are also compared to some available experiments. The influence of tube bundle arrangement and flow velocity upon FIV characteristics can be also investigated. Such an approach is expected to help assessing some vibration for designers. 2 NUMERICAL MODEL The finite volume method is used to solve the threedimensional, viscous, transient, incompressible N-S equations, and the turbulent flow field is solved by the large eddy simulation method. The finite element method is used to discrete the heat transfer tube structure, and the Newmark integral method is used to solve the transient dynamic equilibrium equation to obtain the displacement and velocity response of the structure. Considering the large deformation of the structure and the deformation problem of flow field grid caused by the large deformation, the diffusion method based on diffusion smoothing is adopted to control the mesh update of the moving boundary. The flow-structure coupling interface is used to transfer data between solid domain and fluid domain. Finally, the fluid structure interaction model is established. 2.1 THE CFD MODEL The general conclusion is that even advanced RANS (Reynolds averaged Navier–Stokes) models such as non-linear realizable and RNG types of k-ε models severely underestimate the high turbulent kinetic energy levels observed in densely packed tube bundles. The LES results on the fine mesh are comparable to a DNS and experiments and reasonable agreement is still achieved with a coarse mesh [21]. The LES model can obtain satisfactory results in the turbulent flow field which RANS cannot. Thus, the fluid domain is calculated by LES in the present work. The governing equations employed for LES are obtained by filtering the time-dependent Navier-Stokes equations in physical space. Filtering the continuity and momentum equations, one obtains: 2 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use ui 0 xi (1) ij ui ui u j 1 p ui ( ) xi x j x j t x j x j (2) the fluid loads are calculated by computational fluid dynamics. Hence the algorithm is a single loop one: (1) Solving of displacements and velocity via the structural dynamic equations using Newmark algorithm and transferring the displacement to dynamic mesh solver. (2) Updating the fluid domain grid due to the current tube displacement. (3) Computation of velocity and pressure field and obtaining fluid forces acting on structure, in addition , transferring them to structure dynamics solver. (4) Returning to step (1) and the calculation can be processed again. where ρ is fluid density, μ is dynamic viscosity, t is time, p is pressure. ui (i=1,2,3) is velocity components and is Cartesian coordinates. p and ui arethe filtered variable of p and ui respectively. ij ui u j ui u j is the subgrid-scale stress defined by Algebraic Wall-Modeled LES (WMLES) approach. 2.2 THE STRUCTURAL ANALYSIS MODEL The flexible tube is discretized according to the finite element theory [22] and for each rod a mass matrix (M) and a stiffness matrix (K) are generated. The Newmark method is used for integrating the dynamics equilibrium equations over time. (3) Mx Cx Kx F(t) where M and K are mass matrix and stiffness matrix respectively, C being the damping matrix, expressed as a proportional Rayleigh damping C=αM+βK. x , x and x are displacement, velocity and acceleration of node. F(t) coming from fluid computation by CFD model takes the form of a loading vector on nodes. The initial conditions in velocity and displacement are taken to be nil for the whole structures. 3 A SINGLE FLEXIBLE TUBE UNDER CROSS FLOW 3.1 MODELING The computational model is shown in Fig. 1. All the structured grids were generated using ICEM CFD. The distance between inlet and tube center is 5D, between outlet and tube center 15D. The upper and lower sides to tube center are 5D respectively. In Fig. 1, boundary conditions are a specified fluid inlet for the upstream border (left side in Fig. 1) and a fixed pressure at the downstream one (right side in Fig. 1). Other boundaries are symmetry and wall. Tube wall is the fluidstructure interface, and set as dynamic mesh condition. 2.3 THE DYNAMIC MESH MODEL Diffusion-based smoothing method is used to update a dynamic mesh. The mesh motion is governed by the diffusion equation as in equation (4): (u s ) 0 (4) where, us is the mesh displacement. On deforming boundaries, the boundary conditions are such that the mesh motion is tangent to the boundary. The diffusion coefficient γ in equation (4) can be used to control how the boundary motion affects the interior mesh motion and is a function of the cell volume V. The form is γ=1/Vα，here, α is the control parameter. The equation (4) is discretized using finite volume method. The cell centered solution for the displacement velocity us is interpolated onto the nodes using inverse distance weighted averaging, and the node positions are updated according to: x new x old us t Fig. 1 Schematic of computational model Fluid parameter: density ρ=998.2kg/m3, dynamic viscosity μ=0.001003pa·s. Non-dimensional inflow velocity Ur=U/(fnD) =0.5~10, where U is the upstream velocity and fn is the natural frequency of the tube. Tube parameter: length L=0.5m, outside diameter D＝0.01 m, inner diameter Di＝0.0095 m, elastic modulus E＝1010 Pa, Poisson ratio υ ＝ 0.3, density ρs ＝ 6500 kg/m3, damping coefficient α＝5.098, β＝2.15×10-4. Time control: The time step both in structural dynamics computation and fluid dynamics computation is 0.00025s. (5) 2.4 THE COUPLING BETWEEN FLUID-STRUCTURE Both calculations (fluid and structure dynamics) are using the same time step to implement the fully coupling between fluid and structure by iteration. Data transfer of structure and fluid is achieved via fluid-structural-interaction interface. The deformation of fluid domain is defined by displacement of structure obtained via computational structural dynamics and 3.2 CHECKING OF THE NUMERICAL MODEL Introducing the following non-dimensional quantities for describing briefly in Table 1. 3 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Table 1 Non-dimensional quantities Cl=Fl/(0.5ρAU2) Lift coefficient ClRMS=FlRMS/(0.5ρAU2) r.m.s. of lift coefficient Cd=Fd/(0.5ρAU2) Drag coefficient 2 Mean drag coefficient Cd=Fd/(0.5ρAU ) 2 CdRMS=FdRMS/(0.5ρAU ) r.m.s. of drag coefficient x/D Streamwise displacement y/D Lateral displacement Ax/D=xRMS/D Streamwise amplitude Ay/D=yRMS/D Lateral amplitude St=fvsD/U Vortex shedding frequency Ur=U/(fnD) Reduced velocity Gap velocity Upr＝Up/fnD t*=Ut/D Dimensionless time Table 2 Details of grids used in mesh-independence tests and their fluid force Nc Nr y+ CdRMS St Grid A 128 65 0.293 0.872 0.248 Grid B 68 65 0.293 0.875 0.229 Grid C 128 33 0.293 0.918 0.229 Grid D 84 17 1.467 0.906 0.229 Norberg [23] --- --- --- 0.99±0.05 0.215±0.005 Grid A where, Up＝UP/(P－D); ρs and ρ are tube density and fluid density; ζ is the damping ratio; FdRMS and FlRMS are the r.m.s. of drag and lift; Fd is the mean of drag; xRMS and yRMS are the r.m.s. of streamwise and lateral displacement; U is inflow velocity; D is tube diameter; fn is the natural frequency of tube; A is the projective area of computation; P is the pitch between adjacent two tubes; t is time. A uniform flow with Re=3800 is calculated first to assess the performance of the method. Structured, non-uniform, boundary-fitted grids were generated for the solution domain as shown in Fig. 2. All the structured grids were generated using ICEM CFD. The O-type grid is generated around the tube to ensure good quality meshes. The grid expands away from tube boundary in radial-direction with the geometric expansion factor 1.08 within O-block. Away from the O-block, the grid expands with the geometric expansion factor 1.4. Table 2 provides some details of grid, lift, drag, and St, also including the numbers of nodes on the surface of the tube and in the radial-direction, and the maximum values in the domain of the standard y+. Table 2 indicates the four grids have a small standard y+(y+≈1). It should be noted that, Grid A is the finest grid, Grid B tests the influence of circumference grid nodes, Grid C tests the influence of radial-direction grid nodes, Grid D is the mesh adopted after investigating the influence of grid resolution on flow filed characteristics. The comparison of main parameters is also shown in Table 2. It can be seen that the present result is compared to the experimental data and existing models in the literature [23]. That shows the present grids are all reasonable. Furthermore, in order to further validate the numerical model, a case of FIV for a three-dimensional flexible tube is computed and analyzed. Figure 3 shows the variation of the frequency ratios fex/fn, response frequency fex to natural frequency fn, and lateral amplitude Ay/D versus reduced velocity Ur. As can be shown in Fig. 3, the numerical results are compared to experimental data [24]. That is further confirmed the present numerical model is reasonable. Grid C Grid B Grid D Fig. 2 Four computational grids (a) frequency (b) amplitude Fig. 3 Response verse Ur of single flexible tube 3.3 FREQUENCY RATIO EFFECT ON FIV The influences of frequency ratio on FIV characteristics and nonlinearity of single flexible tube are to be discussed by making the inflow velocity fixed and decreasing the natural frequency fn of the single flexible tube. The Reynolds number is 1.35×104 based on inflow velocity. 3.3.1 FLUID FORCES AND AMPLITUDES Figure 4(a) shows the effect of frequency ration fn/fst on CdRMS, CdMAX, ClRMS, ClMAX. It shows that, the maximum and minimum peak value of drag coefficients occurs at fn/fst=1.25 and fn/fst=0.56 respectively. The maximum and minimum peak value of lift coefficients occurs at fn/fst=1.67 and fn/fst=0.45 respectively. The trends of lateral amplitude Ay/D is shown in Fig. 4(b). Ay/D reaches its peak value at fn/fst=0.56. It obviously shows that, the maximum lateral responses appears at the minimum drag coefficient, and not at the maximum drag coefficient as ordinarily considered. The relationship between 4 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use fvs/fst and fn/fst is presented in Fig. 4(c), where fvs represents the vertex shedding frequency of the vibrating tube, fst represents the vortex shedding frequency corresponding to the stationary tube, and fn is the natural frequency of the tube. The maximum of fvs/fst appears at about fn/fst=1.25~2, which indicates, in this range, the interaction between tube and fluid is the most intense. In summary, when fn/fst=0.56~2.5, the “lock-in” occurs. The onset of “lock-in” occurs at the minimum drag coefficient (slightly larger than the minimum lift force coefficient) for various frequency ratios. In the range of “lock-in”, the lateral amplitude decreases with frequency ratio increasing. (a) Fluid forces Fig. 5 phase angle versus fn/fst of single tube fn/fst=0.15 fn/fst=0.45 fn/fst=0.5 fn/fst=0.56 fn/fst=1.67 fn/fst=5 (b) Amplitudes (c) Vortex shedding frequency Fig. 4 Main parameters versus fn/fst of single tube 3.3.2 PHASE DIFFERENCE The FIV characteristics for a single flexible tube can be well characterized by plotting the phase between lift force and lateral displacement. Figure 5 shows the phase difference versus frequency ratio fn/fst. When the frequency ratio is between 0.45 and 0.5, the phase between the lift force and the lateral displacement undergoes a suddenly change from out-phase to in-phase mode. This jump phenomenon of phase difference is called the ‘‘phase-switch’’, which is a typical nonlinear phenomenon. The phase angle , found as a function of time by using the Hilbert transform, are shown in Fig. 6. These figures show that, when fn/fst=0.15, the phase difference is 180°. While in the transition stage, the phase angle “slips” periodically through 360°and its time history becomes disorderly. On the other hand, as fn/fst≥0.56, the phase difference remains close to 0°. That is to say, phase difference between the lift force coefficient and lateral displacement is in-phase mode. Fig. 6 Time history of phase difference at different fn/fst 3.3.3 PHASE PORTRAIT AND LIMIT CYCLE Phase portrait is a very useful tool to analyze the dynamics of fluid-tube system, and limit cycle is one of the most important characteristics of nonlinear vibration. The limit cycles of lateral displacement and lift coefficient when fn/fst=0.15~5 are shown in Fig. 7 and Fig. 8 respectively. Figure 9 shows the Poincare section of lift coefficient. Cl′ represents the derivative of lift coefficient and it was calculated by the difference method with second order accurate. The period of the Poincare map is the vortex shedding period. When the frequency ratio fn/fst=0.15 or 0.56≤fn/fst<2.5, the shape of the lateral displacement limit cycle is an ellipse. However, the shape of the lift coefficient limit cycle changes from the simple ellipse to a complex geometric figure. Meanwhile, there is only one point in Poincare section. As frequency ratio fn/fst is any other values, the phase portrait curves of lift coefficient and lateral displacement become very complex. There are a lot of points in Poincare section which forms a complex situation. However, it is not the chaos motion 5 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use as it is independent of the initial condition. Thanks to several detailed analyses of limit cycle and Poincare map, it can be found that, under turbulent flow, there is no bifurcation of periodic solution for three-dimensional flexible tube within the frequency ratio range from fn/fst=0.15 to fn/fst=5. That is different from the case of two-dimensional circular cylinder under cross flow at Reynolds number is 200 which the bifurcation of periodic solutions could occur [13]. fn/fst=0.5 Fig. 9 Poincare section of lift coefficient at different fn/fst fn/fst=1.67 fn/fst=0.15 4 MUTUAL INTERACTION OF TWO IN-LINE TUBES In this section, the objective is to study the interaction of two in-line flexible tubes. Tube 2 is set downstream in the wake of tube 1 and the tube pitch Px is 1.2D~4D, that is, the pitch ratio Px/D=1.2~4. Figure 10 shows the configuration. Px fn/fst=0.5 U tube 1 tube 2 Fig. 10 Configuration of in-line tubes Fig. 7 Limit cycle of lateral displacement at different fn/fst fn/fst=0.15 4.1 CRITICAL TUBE PITCH From the variation of CdRMS and ClRMS, Ax/D and Ay/D versus Px/D, and the comparison between tube1, tube2 and the single-tube, as shown in Fig. 11 and Fig. 12, it can be found: The fluid forces and amplitudes are closely related to the pitch ratio. The trend of the amplitudes with pitch ratio is the same as that of the fluid forces with pitch ratio. When Px/D≤2, the transverse variables(ClRMS、Ay/D) of tube1 and tube2 are small, close to the single-tube. When Px/D>2, the transverse variable of tube 2 is far greater than that of tube 1 and the single-tube. That is to say, the critical pitch ratio of in-line tubes is 2. fn/fst=1.67 fn/fst=0.5 Fig. 8 Limit cycle of lift coefficient at different fn/fst fn/fst=0.15 fn/fst=1.67 Fig. 11 CdRMS, ClRMS vs. Px/D of in-line tubes 6 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Fig. 12 Ax/D, Ay/D vs. Px/D of in-line tubes (a) Ax/D 4.2 CRITICAL VELOCITY The vibration characteristics of in-line tubes will be changed at critical pitch ratio. The following takes the cases with pitch ratio Px/D=1.6 and Px/D=3 as an example to analyze the relationship between flow velocity and vibration. Figure 13 (a), Figure 13 (b) show the changes of CdRMS, ClRMS with Upr. CdRMS decreases firstly with the increase of Upr and then increases to a constant value. ClRMS of the in-line tubes with Px/D=3 decreases firstly and then increases with the increase of Upr, but it decreases dramatically after reaching the maximum value. For the in-line tubes with Px/D=1.6, the ClRMS increases with Upr increasing, and it also decreases dramatically after reaching the maximum value. Figure 14 (a) and Figure 14 (b) show the changes of Ax/D and Ay/D with Upr. Figure 13 and Figure 14 reveal that, CdRMS, ClRMS, Ax/D, Ay/D of tube 1 and ClRMS, Ay/D of tube 2 begin to increase rapidly at Upr=3. Only the streamwise variables (CdRMS and Ax/D) of tube 2 increase at the velocity Upr=4. CdRMS of the in-line tubes with Px/D=1.6 takes the minimum value at Upr =4. ClRMS, Ax/D, Ay/D increase rapidly at Upr=4. Therefore, the critical velocity of the in-line tubes with Px/D=3 is Upr=3, and the critical velocity for Px/D=1.6 is Upr=4. (b) Ay/D Fig. 14 Amplitudes vs. Upr of in-line tubes 4.3 WAKE VORTEX PATTERN When the pitch ratio is lower than the critical value, the vortex pattern of two in-line tubes is shown in Fig. 15 (a, b). Under the critical pitch ratio(such as Px/D=1.6), the flow field of in-line tubes is similar to that of the single tube. The vortex spacing is small and the intensity is low. The wake vortex size increases with flow velocity increasing. There is no stable vortex shedding behind the tube 1. When the pitch ratio is greater than the critical pitch ratio, the vortex shedding occurs after both two tubes, and the vortex structure will change significantly at the critical velocity. Figure15 (c, d) show the wake vortex structures of the in-line tubes with Px/D=3 at Upr=2.2 and Upr=3.8. At a lower velocity, the vortex structure in the wake of tube 2 is similar to the single tube’s, but at a higher velocity, two parallel rows vortices appeare behind the tube 2. The lateral space between the two rows vortices increases obviously. (a) CdRMS (a) Upr=2.7 (b) Upr=4.0 (c) Upr=2.2 (d) Upr=3.8 Fig. 15 Vortex pattern under P/D=1.6 (a, b) and P/D=3 (c,d) (b) ClRMS Fig. 13 Fluid forces vs. Upr of in-line tubes 7 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use 5 BEHAVIOR OF FLEXIBLE TUBE BUNDLE 5.1 MODELING The aim of this section is to model the interactions within a small bundle set of 9 tubes in a square array. Fig. 16 shows the local mesh and computational domain of the flexible tube bundle, the pitch diameter ratio P/D=Px/D=Py/D=1.5. The computational domain and boundary conditions are similar to that of Fig. 1. To facilitate the presentation, the tube bundles are labeled as tube 1, tube 2, tube 3, tube 4, tube 5, tube 6, tube 7, tube 8, and tube 9, as shown in Fig. 17. column tubes. The drag value of tube 4 located in the middle of the first column is greater than that of tube 1 and tube 7. The drag of the second and the third column tubes are almost the same. The tube 5, located in the center of tube bundle, has the minimum drag. For lift coefficient, the difference between these tubes is not significant. When Upr > 3, the lift coefficient of tube 4 increases sharply. Fig. 18 Deformed grid and vorticity magnitude contour Fig. 16 Local mesh of 3 × 3 tube bundle First Second Third column column column First row 1 2 3 Second row 4 5 6 Third row 7 8 9 Fig. 17 Tube numbering 5.2 RESULTS, COMPARISON WITH LITERATURE IN 3×3 FLEXIBLE TUBE BUNDEL Firstly, the FIV characteristics of 3×3 flexible tube bundle are studied and some results are compared with the data reported by Schowalter et al. [24]. Figure 18 shows a zoom on the tube bundle, at a time when tubes are deformed, and a contour plot of vorticity magnitude at the same time and highlights the coupling between these tubes. Figure 19 shows the trajectory of each tube. By comparing Fig. 18 and Fig. 19 reveal that for each row tubes, the motion of the two adjacent tubes is always opposite, such as tube 1 and tube 4, tube 4 and tube 7. Figure 20 shows the variation of transverse amplitude versus Upr in 3×3 flexible tube bundle. It can be shown the numerical results are compared to experimental data [24]. That shows the present numerical model is reasonable. Figure 21 shows the variation of drag coefficient and lift coefficient of each tube versus Upr in 3×3 flexible tube bundle. For drag coefficient, the maximum value appears in the first Fig. 19 Trajectory of each tube in 3×3 flexible tube bundle Fig. 20 Amplitude versus Upr of 3×3 flexible tube bundle 8 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use (c) 5-tube model (d) 5-tube-II model Fig. 22 Tube bundle models (a) Drag coefficient In order to compare the predicted results of these four tube bundle models with different numbers of flexible tube, the tube 5 in the middle of the tube bundle is chosen as the research object. Figure 23 and Fig. 24 show the comparison of fluid forces and amplitudes of these models respectively. The drag of the 5-tube-II model is far greater than the other three models. It is mainly because that its arrangement is a staggered tube array, different from the other three aligned tube arrays. That reveals that the arrangement of tube array has a great influence on fluid forces. The variation of lift coefficient versus Upr of the singletube model can best illustrate the unstable vibration behavior of tube bundle. The variation of streamwise amplitude is similar to that of drag coefficient. The streamwise amplitude of the 5-tubeII model is the largest among the four tube bundle models, and the other three models give results without great difference. For transverse amplitude, at a lower flow velocity, the transverse amplitude is very small, but when the flow velocity exceeds a certain value, its value increases rapidly. The corresponding velocity is the critical velocity. From the figures we can conclude that the single-tube model gives the largest critical velocity. The critical velocity predicted by the 5-tube model and the 3×3-tube model are basically identical. The 5-tube-II model represents a staggered tube array so the predicted critical velocity has a great difference compared with the 3×3 fully flexible tube bundle. To sum up, the 5-tube model can reflect the vibration characteristics of the fully flexible tube bundle. The single-tube model predicts a larger critical velocity, this is consistent with the literature [20] but other comparisons are awaited. (b) Lift coefficient Fig. 21 Fluid forces versus Upr of 3×3 flexible tube bundle 5.3 COMPARISON AMONG DIFFERENT TUBE BUNDLES The objective of this section is to discuss the vibration characteristics of different tube bundles. Additionally, the critical velocity of instability vibration and the effect of tube configurations on fluid force and dynamics will be obtained. Four kinds of tube bundle model are established, as shown in Fig. 22. The shaded part represents the flexible tube and the rest are rigid fixed tubes. The tube numbering is the same as Fig. 17. The first model is a 3×3 fully flexible tube bundle, see Fig. 22(a); the second model is the single-tube model, see Fig. 22(b). The model is widely used in theoretical study of fluid elastic instability. It is considered that only tube 5 is flexible in the 3×3 tube bundle, and the other tubes are rigid and fixed; the third model is the 5-tube model, see Fig. 22(c). It is considered that tube 1, tube 3, tube 7, tube 9 in the tube bundle are fixed and rigid; the fourth model is the 5-tube-II model, see Fig. 22(d), which is obtained by removing tube 1, tube 3, tube 7, tube 9 of the 3×3 flexible tube bundle. (a) 3×3-tube model (b) single-tube model (a) Drag coefficient (b) Lift coefficient Fig. 23 Fluid forces versus Upr of four tube bundle models 9 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use REFERENCES [1] Weaver, D. S., Ziada, S., and Au-Yang M. K., et al., 2000, “Flow-Induced Vibrations in Power and Process Plant Components-Progress and Prospects,” Journal of Pressure Vessel Technology, 122, pp. 339-348. [2] Placzek, A., and Sigrist, J. F., 2009, “Numerical Simulation of an Oscillating Cylinder in a Cross-Flow at Low Reynolds Number, Forced and Free Oscillations,” Computers & Fluids, 38, pp. 80-100. [3] Simoneaua, J. P., Thomas, S., and Moussallam, N., et al., 2011, “Fluid Structure Interaction between Rods and a Cross Flow-Numerical Approach,” Nuclear Engineering And Design, 241, pp. 4515-4522. [4] Goverdhan, R., and Williamson, C. H. K., 2004, “Vortex Induced Vibrations,” Annual Review Fluid Mechanics, 36, pp. 413-455. [5] Feng, C. C., 1968, “The Measurement of Vortex-Induced Effect in the Flow Past Stationary and Oscillating Circular Cylinder and D-Section Cylinders,” Vancouver, University of British Columbia. [6] Griffin, O. M., 1980, “Vortex-Excited Cross Flow Vibrations of a Single Circular Cylinder,” ASME Journal of Pressure Vessel Technology, 102, pp. 258-166. [7] Khalak, A., and Williamson, C. H. K., 1999, “Motions, Forces and Mode Transitions in Vortex-Induced Vibrations at Low Mass-Damping,” Journal of Fluids and Structures, 13(7-8), pp. 813-851. [8] Govardhan, R., and Williamson, C. H. K., 2000, “Modes of Vortex Formation and Frequency Response for a Freely Vibrating Cylinder,” Journal of Fluid Mechanics, 420, pp. 85-130. [9] Evangelinos, C., Lucor, D., and Karniadakis, G. E., 2000, “DNS-Derived Force Distribution on Flexible Cylinders Subject to Vortex-Induced Vibration,” Journal of Fluids and Structures, 14(3), pp. 429-440. [10] Mittal, S., and Kumar, V., 1999, “Finite Element Study of Vortex-Induced Cross-Flow and In-Line Oscillations of a Circular Cylinder,” International Journal for Numerical Methods in Fluids, 31, pp. 1087-1120. [11] Zhou, C. Y., So, R. M., and Lam, K., 1999, “VortexInduced Vibrations of an Elastic Circular Cylinder,” Journal of Fluids and Structures, 13(2), pp. 165-189. [12] Jadic, I., So, R. M. C., and Mignolet, P., 1998, “Analysis of Fluid-Structure Interactions Using a Time-Marching Technique,” Journal of Fluids and Structures, 12(6), pp. 631-654. [13] Li, T., Zhang, J. Y., and Zhang, W. H., 2011, “Nonlinear Characteristics of Vortex-Induced Vibration at Low Reynolds Number,” Commun Nonlinear Sci Numer Simulat, 16, pp. 2753-2771. [14] Gabbai, R. D., and Benaroya, H., 2005, “An Overview of Modeling and Experiments of Vortex-Induced Vibration (a) Streamwise amplitude (b) Lateral amplitude Fig. 24 Amplitudes versus Upr of four tube bundle models 6 CONCLUSIONS In this paper, the main work presented is focused on threedimensional flexible tube subjected to cross flow, including a single flexible tube, two in-line tubes, and four tube bundle models. The nonlinearity of single flexible tube under turbulent flow, the mutual interaction of two in-line flexible tubes, and the behavior of flexible tube bundles are investigated. Conclusions are drawn as follows: 1) There is no bifurcation of lift and lateral displacement occurred in single flexible tube submitted to uniform turbulent flow within the frequent ratio fn/fst=0.15~5. That is different from the case of two-dimensional circular cylinder under cross flow at Reynolds number is 200 which the bifurcation of periodic solutions could occur [13]. In the peak of drag and lift force coefficient versus frequency ratio, the “lock-in” occurs. The phase angle reaches zero under “lock-in”, and the dynamic behavior is a periodic motion. 2) The critical pitch ratio of two in-line tubes is 2. The critical velocity of in-line tubes depends on pitch ratio where the critical value of in-line tubes with P/D=3.0 is Upr=3.0 and with P/D=1.6 is Upr=4.0. The wake vortex pattern of the downstream tube is independent vortex street within the critical velocity and critical pitch. Beyond the critical velocity and critical tube pitch, two parallel rows vortex streets will be formed behind the downstream tube. 3) The arrangement of tube bundle has a great influence on the fluid forces and vibration response. The 5-tube model can basically reflect the vibration characteristics of the fully flexible tube bundle. The critical velocity predicted by the single-tube model is rather large, this is consistent with the literature [20] but other comparisons are awaited. 4) Further refinements of models and other validation studies are indeed required using available literature (Hassan, Gerber, and Omar [25]; Bouzidi et al. [26]), but such an approach is expected to help assessing some vibration configurations occurring in reactor components. ACKNOWLEDGMENTS This work is supported by the National Science Foundation of China (No. 51606180). 10 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use [15] [16] [17] [18] [19] [20] of Circular Cylinders,” Journal of Sound and Vibration, 282, pp. 575-616. Feng, Z. P., Zang, F. G., and Zhang, Y. X., 2014, “Numerical Simulation of Fluid Structure Interaction in Two Flexible Tubes,” Atomic Energy Science and Technology, 48(8), pp. 1428-1434. Price, S. J., 1995, “A Review of Theoretical Models for Fluid-Elastic Instability of Cylinder Arrays in CrossFlow,” Journal of Fluids and Structures, 9, pp. 463-518. Singh, P., Causssignac, P. H., and Fortes, A., et al., 1989, “Stability of Periodic Arrays of Cylinders across the Stream by Direct Simulation,” Journal of Fluid Mechanics, 205, pp. 553-571. Ichioka, T., Kawata, Y., and Izumi, H., et al., 1994, “TwoDimensional Flow Analysis of Fluid Structure Interaction around a Cylinder and a Row of Cylinders,” ASME Journal of PVP, 273, pp. 33-41. Longattea, E., Bendjeddoub, and Z., Soulib, M., 2003, “Methods for Numerical Study of Tube Bundle Vibrations in Cross-Flows,” Journal of Fluids and Structures, 18, pp. 513-528. Omar, H., Hassan, M., and Gerber, A., 2009, “Numerical Estimation of Fluidelastic Instability in Staggered Tube Arrays,” Proceedings Of The ASME 2009 Pressure Vessels And Piping Division Conference, Prague, Czech, pp. 1-10. Feng, Z. P., Jiang, N. B., and Zang, F. G., et al., 2016, “Nonlinear Characteristics Analysis of Vortex-Induced [21] [22] [23] [24] [25] [26] 11 Vibration For a Three Dimensional Flexible Tube,” Commun Nonlinear Sci Numer Simulat, 34, pp. 1-11. Benhamadouche, S., and Laurence, D., 2003, “LES, Coarse LES, and Transient RANS Comparisons on the Flow across a Tube Bundle,” International Journal of Heat and Fluid Flow, 24, pp. 470-479. Wang, X. C., 2003, “The Finite Element Method,” Beijing, Tsinghua University Press, pp. 468-495. Norberg, C., 2003, “Fluctuating Lift on a Circular Cylinder: Review and New Measurements,” Journal of Fluids and Structures, 17, pp. 57-96. Schowalter D., Ghosh I., Kim S. E., Haidari A., 2006, “Unit-Tests Based Validation and Verification of Numerical Procedure to Predict Vortex-Induced Motion,” Proceedings of Omae2006, 25th International Conference on Offshore Mechanics and Arctic Engineering, Hamburg, Germany, pp. 184-187. Hassan, M., Gerber, A., and Omar, H., 2010, “Numerical Estimation of Fluidelastic Instability in Tube Arrays,” Journal of Pressure Vessel Technolog, 132, pp. 1-11. Bouzidi, S. E., Hassan, M., and Fernandes, L. L., 2014, “Numerical Characterization of the Area Perturbation and Time Lag for a Vibrating Tube Subjected to Cross-Flow,” Proceedings of the ASME 2014 Pressure Vessels and Piping Division Conference, Anaheim, USA, pp. 1-11. Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/28/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use

1/--страниц