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PVP2017-65019

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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
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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:
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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)
Mx  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.
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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
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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
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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
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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
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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
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(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
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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
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[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.
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