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Influence of turbulence modelling and grid discretization on the simulation of flow-forces on tubes in cross-flow
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
409
Influence of turbulence modelling and grid discretization on the simulation of flow-forces on tubes in cross-flow K. Schr~Sder and H. Gelbe Institute of Process and Plant Technology, Technical University Berlin, Stral3e des 17. Juni 135, D-10623 Berlin, Germany Two-dimensional CFD-simulations of single-phase cross-flow around a single fixed tube are carried out and compared with experimental data. Several turbulence models (ke-, kin-, Reynolds stress and large eddy model) are applied in combination with different grid discretizations using the programs STAR-CD and CFX. The grid discretization together with the turbulence model has a great influence on the resulting drag and lift forces and on the frequency of vortex shedding. These values are essential in order to simulate the flow-induced vibration excitation in tube bundles, which is the object of this investigation. First results on threedimensional and unsteady simulations of the flow-induced vibration excitation of flexible tubes, tube-rows and tube bundles are published in [5]. 1.
INTRODUCTION
The CFD-analysis of the tube vibration induced by cross-flow was the subject of some investigations during the last years. Ichioka et al. [ 1] applied a Finite-Difference scheme on a bodyfitted moving grid to solve the unsteady Navier-Stokes equations without using a turbulence model. Their model is restricted to low Reynolds-numbers. Kassera et al. [2, 3] simulated the flow induced vibration for a single flexible tube and for full flexible tube bundles and Kassera [4] presented the three-dimensional simulation of the resonant vortex shedding vibration for a single tube. They used the Finite-Volume Method on a cartesian grid (see Fig. 3) and applied different turbulence models (kin-model, Large Eddy Simulation and a Reynolds stress model) to take the turbulent nature of the flow into account. Schr6der et al. [5] used for the first time a commercial CFD-program and demonstrate the problems arising in unsteady simulations of forces and tube-motions in tube bundles. The grid discretization together with the turbulence model has a great influence on the resuiting drag and lift forces and the frequency of the vortex shedding. Therefore in the investigation presented here, some turbulence models and different grid discretizations were tested for a rigid tube. 2. DISCRETIZATION OF THE F L O W FIELD Different discretizations of the geometry were compared. Five of the used grids can be seen in the Figures 1 to 3. The grid B1 in Fig. la has 216 cells in peripheral direction in the near wall region. This high resolution decreases with increasing radial distance from the wall to 54 cells by the usage of cell matching methods. The second grid B2 in Fig. lb is similar to the discretization B 1, but has 432 cells in peripheral direction near the wall. In Fig. 2a the grid
410
II . N ~ IIJNNl liar1 IIAltt~l
iir'~.4. II
!! Fig. 1. a) Grid B1 with 216 cells in peripheral direction in 2 cell layers. b) Grid B2 with 432 cells in peripheral direction in 5 cell layers.
INIIILIII IN.Ill INN_Ill ~ l l l
J~-qlll Ill
.!!!
Fig. 2. a) Grid B4 with 216 cells in peripheral direction in 50 cell layers. b) Grid B5 with 72 cells in peripheral direction in 18 cell layers.
B4 is shown. The grid has in comparison with grid B 1 also 216 cells in peripheral direction but lllll Illll [ in 50 cell layers in radial direction; this fine partition reduces to 54 cells in peripheral direction. For comparison a discretization of 72 cells in peripheral direction is used for grid B5, but with iZ a fine partition in radial direction and no cell matching in the solution domain. Fig. 3. Grid B3 with 48 cells in periphThe grid B3 shown in Fig. 3 used by eral direction. Kassera et al. [2, 3, 4] is also tested and discussed in the present work. In this case only 48 cells in the near wall region of the tubes are used for the peripheral discretization. IIIII IIlll
[ _
.
.
.
.
.
.
.
3. C O M P A R I S O N OF D I F F E R E N T T U R B U L E N C E M O D E L S Comparing different turbulence models for the computation of the flow field some two equations turbulence models, Reynolds stress models and a Large-Eddy model with different subgridscale models were tested for a single rigid tube. The computations were carried out with the CFD-programs STAR-CD and CFX.
3.1.
Computations for a Reynolds-number of 140000 The experimental data of Cantwell and Coles [6] for a Reynolds number Re = uooda/v =140000 were used for a comparison. The same parameters as in the experiments: tube diameter d = 0.10137 m, freestream velocity uoo = 21.2 m/s and the viscosity and density of air were taken in the simulation. The turbulence intensity in the upstream region for this experiment was less than 0.1%, so a laminar separation could be expected and the transition from laminar to turbulent flow lies in the boundary layer. This is an important fact for the turbulence modelling, because the transition can not be well predicted by k~-model [7]. The computations were carried out with the time step size of At = 0.0001 s, the QUICK differenc-
411 ing scheme and the grid discretization B 1 (see Fig. 1). The results for the time averaged pressure distribution obtained for three different turbulence models can be seen in Fig. 4. The pressure coefficient Cp = (p-p=)/(0.5.p.u 2) is plotted over the angle around the tube surface, beginning at the stagnation point. - - - kco-model (Wilcox) The computations with the standard ke1 "~ . . . . ke-model (standard) model and the two-layer ke-model of Norris and 0.5 k~-2-1ayer-model Reynolds [8] can not predict the pressure distri. . . . k~-model (nonlinear) bution. The computed pressure for t~ > 120 ~ in rJ~ 0 x exp. Cantwell & C o l e s the flow separation area is too high, so the re-0.5 , "" . . . . . . . . . suiting drag coefficients are too low. Franke [9] obtained much better results with a modified -1 ~ ~, ;' two-layer ke-model and a finer partition in ra-1.5 ~ dial tube direction for the pressure distribution -2 J Re=140000 and 144 cells in peripheral direction. The results da~O.10~a7m obtained with the k~o-model by Wilcox [7] can ,
I
,
I
,
I
,
,
.
describe the observed experimental pressure 0 30 60 90 120 150 180 oc distribution quite well. The pressure coefficient Fig. 4. Computed pressure distributions for the minimum is acceptable with a deviation around the tube surface for different turof less than 20 %. In opposition to the standard bulence models compared with experiand the two-layer ke-model, the pressure distrimental data by Cantwell and Coles [6]. bution computed with the quadratic nonlinear ke-model [10] is in good agreement to the experimental for a > 90 ~ This turbulence model computes the components of the Reynolds-stress tensor with algebraic equations and so takes into account the anisotropic nature of turbulence [7]. The amplitude for the lift coefficient computed with the kt~-model and the nonlinear kemodel is more than 5-times higher than the amplitudes for the standard and the two-layer kemodel. The resulting average drag coefficient CD = 1.17 is comparable to the measured value of 1.25. Using grid B 1, the k0~-model enables an accurate prediction of the Strouhal-number Sr = fvda/uoo = 0.197 for the test case with a relative error of less than 10 % compared to the measured Strouhal-number of 0.179. All ke-models fail in the prediction of the Strouhalnumber with a relative error of more than 30 %; one reason may be the inaccurate determination of the laminar separation point by the ke-models in this special test case with a high Reynolds number and a very low turbulence level in the upstream flow. The nonlinear ke-model predicts an average drag coefficient of 0.91 with an error of 27%, whereas the computed value for the standard ke-model is only 0.3. The computed shedding frequencies by Franke [9] with the modified two-layer ke-model and a differential Reynolds-stress model in conjunction with the two-layer ke-model showed also a relative error of more than 30%.
3.2. Computations for a Reynolds-number of 6450 The experimental data of Gog [11] for a Reynolds number of 6450 were used for an additional comparison. The upstream turbulence level was about 1%, so a laminar separation could not be expected. The selected time step size At = (1/100) Zvortex-shedding = 0.0007 S seems to be appropriate for an accurate time discretization of the vortex shedding excitation. The same parameters as in the experiments: tube diameter d =0.04 m, freestream velocity uoo = 2.47 m/s and the viscosity and density of air were taken in the simulation. All computations were carried out with the grid discretization B5 (see Fig. 2b). The cells in peripheral
412 direction were increased to 108 cells. ke-model (standard) 1 "'\ ___ ke-model (quad. nonlinear Fig. 5 shows the time averaged pressure 0.5 ~ ..... ke-model (cubic nonlinear distribution obtained for different ke turbulence x exp. Gog (1982) models. The MARS differencing scheme was O 0 applied for the simulations. The pressure coef-0.5 ficient is plotted over the angle around the tube surface. For this Reynolds number the standard x . . . . . . . ,.___. ke-model can predict the pressure distribution -1.5 quite well. The predicted pressure coefficients -2 Re=6450 in the separation area are nearby the measured da=0.04m values. The vortex shedding frequency could be -2.5 ' ' ' ' ' ' ' ' ' ' ' calculated with an error of less than 5%. In op0 30 60 90 120 150 180 o~ position to the standard and the quadratic nonFig. 5. Pressure d i s t r i b u t i o n s a r o u n d the linear ke-model, the pressure distribution comtube surface for different turbulence moputed with the cubic nonlinear ke-model [10] is dels computed with STAR-CD and comin good agreement to the experimental data in pared with experimental data by Gog [11 ]. the total range of o~. The computed vortex shed....... algebraic-Reynoldsstress ding frequency has an error of less than 0.5%. 1 ~ . . . . k~-model (Low-Reynolds) Additionally computations were also car0.5 [-- ~ , . differential Reynoldsstress ded out with the CFD-program CFX on the O~ 0 9 x exp. Gog (1982) same grid layout. The difference to the grid -0.5 used for the STAR-CD calculations is, that the -1 "\x x x x grid is divided in four structured blocks. For the -1.5 \\\ \ ~~.,,,,~ velocities the QUICK differencing scheme and ", -,Jfor the turbulence equations a HYBRID differ-2.5 \ encing scheme was applied. The results for the -3 \ ,-". . . . . . R'e'--"6~.'S"6"" computed pressure distribution can be seen in -3.5 ' ' '' da='0"04m Fig. 6. In comparison to the STAR-CD results 0 30 60 90 120 150 180 (z the pressure coefficients calculated with CFX Fig. 6. Pressure d i s t r i b u t i o n s a r o u n d the are lower. One reason may be the different caltube surface for different turbulence culation of the pressure field. For the CFX calmodels computed with CFX. culation a SIMPLEC algorithm was used instead of the PISO algorithm applied for the STAR-CD calculations. Comparing the results obtained with the algebraic Reynolds-stress ke-model to the differential Reynolds-stress model, greater differences can be observed for o~ > 120 in the separation area. The calculated vortex shedding frequencies of 14.8 Hz for the differential stress model and 13.4 Hz for the quadratic nonlinear ke-model are in good agreement to the measured frequency of 14 Hz. The calculated pressure coefficients for the Low-Reynolds kcomodel with a viscous damping function for the near wall cells are too low compared to the algebraic and differential Reynolds stress models. The calculated vortex shedding frequency of 20.2 Hz is much higher than the observed one. One reason for this inadequate results may be the grid spacing near the wall, so the conditions for Low-Reynolds calculation near the wall are not valid. STAR-CD offers two LES-models with different subgrid scale models (SGS), namely that of Smagorinsky and a two equations kl-model. The computed pressure coefficients in Fig. 7 are lower than the measured values for (x > 60 ~ with a relative error of up to 80% I
413 compared to the measured values. The calculated vortex shedding frequency of 13.5 Hz for the Smagorinsky and 15 Hz for the SGS N-model are acceptable. Breuer [ 12] carried out a large eddy simulation for a tube in cross-flow and a Reynolds-number of 3900. His results were in good agreement with the measurements. The reasons for the better results are twofold: the SGS-model was modified near the wall and the grid and time discretization was much finer than for the presented simulations in this paper. For acceptable results with a large eddy simulation the numerical costs will be very high.
1 " .... 0.5 O" 0 -0.5
~ ~,
-1
~\
-2 ~ -2.5
L E S ( S m a g o r i n s k y SGS) L E S (kl-model SGS) ke-model (cubic nonlinear) x exp. Gog (1982)
t 0
Re=6450 da= 0. 04m x
.'~2..-" ,
"
',
, , I , I , I , , . 30 60 90 120 150 180 (z
Fig. 7. Pressure distributions around the tube surface for LES simulations computed with STAR-CD and compared to [ 11]. 4. C O M P A R I S O N OF D I F F E R E N T GRID D I S C R E T I Z A T I O N S
The measurements of Cantwell and Coles 1 ~ I---kin-model (grid B1) I [6] for a single rigid tube were also used for a 0.5 ]- ~ I .... kin-model(grid B2) I comparison of different discretizations out[ '~ Jo----e kin-model(gridB3) J I_ 9 ] - - - - ko~-model (grid B4) r} lined in section 2. The ko)-model was applied 0 l ~ 1--- kco-model(grid B5) ]] with the QUICK differencing scheme and the -0.5 ~1~[ x exp. cantweu&colesq same time step size of At = 0.0001 s was used. o~ -1 F The results for the pressure distribution are ' -presented in Fig. 8. -1.5 The fine grid B4 with 18054 cells and a computation time of 41.10 seconds per time -2 step yields the best result for the pressure coefficient in the total range of the peripheral an0 30 60 90 120 150 180 gle. Especially the computed position of the o~ pressure minimum at c~ = 71~ and the value of Fig. 8. Computed pressure distribution for Cp =-1.4 gives a good agreement with the exdifferent grid discretizations with the experimental data. An excellent result can be perimental data by Cantwell and Coles [6]. obtained for the drag coefficient cD = 1.24. The computed Strouhal-number of Sr = 0.182 confirm, that the kco-model enables results with high accuracy. The results obtained with the grids B 1 (7686 cells and a computation time of 9.52s per time step) and B2 (9558 cells and 11,08s per time step) are good compromises obtaining satisfying results (Sr = 0.197) within an acceptable computational time. The simple discretization B3 (6950 cells and 7.63s per time step), which has the lowest calculation time, is in good agreement with the measurements between 85 and 180 degrees. The separation point is fixed by the edge of the grid at about 74 degree, so the grid cannot predict the pressure distribution in the range of 50 < ~ < 80 and the calculated Strouhal-number of 0.232 is 30% higher than the observed value of 0.179 by Cantwell and Coles. Testing the error of cell matching, grid B5 with a high resolution in radial and a low resolution in peripheral direction (see Fig. 2b) was applied with no cell matching in the solution domain; the computed pressure distribution is acceptable in comparison with the results for the grid B3, the computed Strouhal number 0.184 for B5 is quite well.
414 5. CONCLUSIONS A comparison between the simulation and experimental results for a rigid single tube in cross-flow with a Reynolds-number of 140000 show, that the best results for the pressure distribution at the tube surface, the frequency of vortex shedding and the lift and drag forces can be obtained with the implemented kor-turbulence model. The results computed with the standard ke-turbulence and the ke-2-1ayer model are not satisfying. Simulations for a Reynolds-number of 6450 with different turbulence models and different CFD-programs compared to experimental results demonstrate the strong influence of CFD-code and turbulence model. The best results can be obtained for the cubic nonlinear kemodel. The pressure coefficients calculated with the CFD-program CFX differ from the resuits calculated with STAR-CD. The differences in the simulation results for the differential stress model and the nonlinear ke-model are small. The large eddy simulations carried out with STAR-CD could not predict the measured pressure distribution. The reason for the inaccurate results are the grid and time discretization and the wall treatment for the subgrid scale implemented in STAR-CD.
REFERENCES [ 1] T. Ichioka, Y. Kawata, H. Izumi, T. Nakamura, K. Fujita, Two-dimensional flow analysis of fluid structure interaction around a cylinder and a row of cylinders, Symposium on Flow Induced Vibration, Minneapolis, 1994, ASME PVP-Vol. 273, pp. 283-288. [2] V. Kassera, L. Kacem-Hamouda, K. Strohmeier, Numerical simulation of flow induced vibration of a tube bundle in uniform cross flow, Symposium on Flow Induced Vibrations, Honolulu, 1995, ASME PVP-Vol. 298, pp. 37-43. [3] V. Kassera and K. Strohmeier, Simulation of tube bundle vibrations induced by cross flow, Journal of Fluids and Structures (1997) 11, pp. 909-928. [4] V. Kassera, Three dimensional CFD-analyses of tube vibrations induced by cross flow, ASME AD-Vol. 53-2 Vol. II, 4 th Int. Symp. on FSI, A, FIV & N, Dallas, 1997, pp. 137-143. [5] K. Schr6der and H. Gelbe, Two- and three-dimensional CFD-simulation of flow-induced vibration excitation in tube bundles, Chem. Eng. and Proc. 38 (1999), pp. 621-629. [6] B. Cantwell and D. Coles, An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder, J. of Fluid Mechanics, 136 (1983),pp. 321-374. [7] D.C. Wilcox, Turbulence Modelling for CFD, DCW Industries Inc., La Canada, California, 1994. [8] L.H. Norris and W.C. Reynolds, Turbulent channel flow with a moving wavy boundary, Report No. FM-10, Department of Mechanical Engineering, Stanford University, USA, 1975. [9] R. Franke, Numerische Berechnung der instationaren Wirbelabl6sung hinter zylindrischen Kt~rpern, Dissertation, Universit~it Karlsruhe (TH), 1991. [ 10] T.H. Shih, J. Zhu and J.L. Lumley, A realizable reynolds stress algebraic equation model, NASA TM-105993, 1993. [ 11 ] W. Gog, Untersuchungen der Erregermechanismen am Einzelrohr und am querangestr6mten Rohrbtindel, Dissertation D 83, Technische Universit~it Berlin, 1982. [ 12] M. Breuer, Large Eddy Simulation of the Sub-Critical Flow Past a Circular Cylinder: Numerical and Modeling Aspects, Int. J. for Numerical Methods in Fluids, Wiley, Chichester, 1998.
409
Influence of turbulence modelling and grid discretization on the simulation of flow-forces on tubes in cross-flow K. Schr~Sder and H. Gelbe Institute of Process and Plant Technology, Technical University Berlin, Stral3e des 17. Juni 135, D-10623 Berlin, Germany Two-dimensional CFD-simulations of single-phase cross-flow around a single fixed tube are carried out and compared with experimental data. Several turbulence models (ke-, kin-, Reynolds stress and large eddy model) are applied in combination with different grid discretizations using the programs STAR-CD and CFX. The grid discretization together with the turbulence model has a great influence on the resulting drag and lift forces and on the frequency of vortex shedding. These values are essential in order to simulate the flow-induced vibration excitation in tube bundles, which is the object of this investigation. First results on threedimensional and unsteady simulations of the flow-induced vibration excitation of flexible tubes, tube-rows and tube bundles are published in [5]. 1.
INTRODUCTION
The CFD-analysis of the tube vibration induced by cross-flow was the subject of some investigations during the last years. Ichioka et al. [ 1] applied a Finite-Difference scheme on a bodyfitted moving grid to solve the unsteady Navier-Stokes equations without using a turbulence model. Their model is restricted to low Reynolds-numbers. Kassera et al. [2, 3] simulated the flow induced vibration for a single flexible tube and for full flexible tube bundles and Kassera [4] presented the three-dimensional simulation of the resonant vortex shedding vibration for a single tube. They used the Finite-Volume Method on a cartesian grid (see Fig. 3) and applied different turbulence models (kin-model, Large Eddy Simulation and a Reynolds stress model) to take the turbulent nature of the flow into account. Schr6der et al. [5] used for the first time a commercial CFD-program and demonstrate the problems arising in unsteady simulations of forces and tube-motions in tube bundles. The grid discretization together with the turbulence model has a great influence on the resuiting drag and lift forces and the frequency of the vortex shedding. Therefore in the investigation presented here, some turbulence models and different grid discretizations were tested for a rigid tube. 2. DISCRETIZATION OF THE F L O W FIELD Different discretizations of the geometry were compared. Five of the used grids can be seen in the Figures 1 to 3. The grid B1 in Fig. la has 216 cells in peripheral direction in the near wall region. This high resolution decreases with increasing radial distance from the wall to 54 cells by the usage of cell matching methods. The second grid B2 in Fig. lb is similar to the discretization B 1, but has 432 cells in peripheral direction near the wall. In Fig. 2a the grid
410
II . N ~ IIJNNl liar1 IIAltt~l
iir'~.4. II
!! Fig. 1. a) Grid B1 with 216 cells in peripheral direction in 2 cell layers. b) Grid B2 with 432 cells in peripheral direction in 5 cell layers.
INIIILIII IN.Ill INN_Ill ~ l l l
J~-qlll Ill
.!!!
Fig. 2. a) Grid B4 with 216 cells in peripheral direction in 50 cell layers. b) Grid B5 with 72 cells in peripheral direction in 18 cell layers.
B4 is shown. The grid has in comparison with grid B 1 also 216 cells in peripheral direction but lllll Illll [ in 50 cell layers in radial direction; this fine partition reduces to 54 cells in peripheral direction. For comparison a discretization of 72 cells in peripheral direction is used for grid B5, but with iZ a fine partition in radial direction and no cell matching in the solution domain. Fig. 3. Grid B3 with 48 cells in periphThe grid B3 shown in Fig. 3 used by eral direction. Kassera et al. [2, 3, 4] is also tested and discussed in the present work. In this case only 48 cells in the near wall region of the tubes are used for the peripheral discretization. IIIII IIlll
[ _
.
.
.
.
.
.
.
3. C O M P A R I S O N OF D I F F E R E N T T U R B U L E N C E M O D E L S Comparing different turbulence models for the computation of the flow field some two equations turbulence models, Reynolds stress models and a Large-Eddy model with different subgridscale models were tested for a single rigid tube. The computations were carried out with the CFD-programs STAR-CD and CFX.
3.1.
Computations for a Reynolds-number of 140000 The experimental data of Cantwell and Coles [6] for a Reynolds number Re = uooda/v =140000 were used for a comparison. The same parameters as in the experiments: tube diameter d = 0.10137 m, freestream velocity uoo = 21.2 m/s and the viscosity and density of air were taken in the simulation. The turbulence intensity in the upstream region for this experiment was less than 0.1%, so a laminar separation could be expected and the transition from laminar to turbulent flow lies in the boundary layer. This is an important fact for the turbulence modelling, because the transition can not be well predicted by k~-model [7]. The computations were carried out with the time step size of At = 0.0001 s, the QUICK differenc-
411 ing scheme and the grid discretization B 1 (see Fig. 1). The results for the time averaged pressure distribution obtained for three different turbulence models can be seen in Fig. 4. The pressure coefficient Cp = (p-p=)/(0.5.p.u 2) is plotted over the angle around the tube surface, beginning at the stagnation point. - - - kco-model (Wilcox) The computations with the standard ke1 "~ . . . . ke-model (standard) model and the two-layer ke-model of Norris and 0.5 k~-2-1ayer-model Reynolds [8] can not predict the pressure distri. . . . k~-model (nonlinear) bution. The computed pressure for t~ > 120 ~ in rJ~ 0 x exp. Cantwell & C o l e s the flow separation area is too high, so the re-0.5 , "" . . . . . . . . . suiting drag coefficients are too low. Franke [9] obtained much better results with a modified -1 ~ ~, ;' two-layer ke-model and a finer partition in ra-1.5 ~ dial tube direction for the pressure distribution -2 J Re=140000 and 144 cells in peripheral direction. The results da~O.10~a7m obtained with the k~o-model by Wilcox [7] can ,
I
,
I
,
I
,
,
.
describe the observed experimental pressure 0 30 60 90 120 150 180 oc distribution quite well. The pressure coefficient Fig. 4. Computed pressure distributions for the minimum is acceptable with a deviation around the tube surface for different turof less than 20 %. In opposition to the standard bulence models compared with experiand the two-layer ke-model, the pressure distrimental data by Cantwell and Coles [6]. bution computed with the quadratic nonlinear ke-model [10] is in good agreement to the experimental for a > 90 ~ This turbulence model computes the components of the Reynolds-stress tensor with algebraic equations and so takes into account the anisotropic nature of turbulence [7]. The amplitude for the lift coefficient computed with the kt~-model and the nonlinear kemodel is more than 5-times higher than the amplitudes for the standard and the two-layer kemodel. The resulting average drag coefficient CD = 1.17 is comparable to the measured value of 1.25. Using grid B 1, the k0~-model enables an accurate prediction of the Strouhal-number Sr = fvda/uoo = 0.197 for the test case with a relative error of less than 10 % compared to the measured Strouhal-number of 0.179. All ke-models fail in the prediction of the Strouhalnumber with a relative error of more than 30 %; one reason may be the inaccurate determination of the laminar separation point by the ke-models in this special test case with a high Reynolds number and a very low turbulence level in the upstream flow. The nonlinear ke-model predicts an average drag coefficient of 0.91 with an error of 27%, whereas the computed value for the standard ke-model is only 0.3. The computed shedding frequencies by Franke [9] with the modified two-layer ke-model and a differential Reynolds-stress model in conjunction with the two-layer ke-model showed also a relative error of more than 30%.
3.2. Computations for a Reynolds-number of 6450 The experimental data of Gog [11] for a Reynolds number of 6450 were used for an additional comparison. The upstream turbulence level was about 1%, so a laminar separation could not be expected. The selected time step size At = (1/100) Zvortex-shedding = 0.0007 S seems to be appropriate for an accurate time discretization of the vortex shedding excitation. The same parameters as in the experiments: tube diameter d =0.04 m, freestream velocity uoo = 2.47 m/s and the viscosity and density of air were taken in the simulation. All computations were carried out with the grid discretization B5 (see Fig. 2b). The cells in peripheral
412 direction were increased to 108 cells. ke-model (standard) 1 "'\ ___ ke-model (quad. nonlinear Fig. 5 shows the time averaged pressure 0.5 ~ ..... ke-model (cubic nonlinear distribution obtained for different ke turbulence x exp. Gog (1982) models. The MARS differencing scheme was O 0 applied for the simulations. The pressure coef-0.5 ficient is plotted over the angle around the tube surface. For this Reynolds number the standard x . . . . . . . ,.___. ke-model can predict the pressure distribution -1.5 quite well. The predicted pressure coefficients -2 Re=6450 in the separation area are nearby the measured da=0.04m values. The vortex shedding frequency could be -2.5 ' ' ' ' ' ' ' ' ' ' ' calculated with an error of less than 5%. In op0 30 60 90 120 150 180 o~ position to the standard and the quadratic nonFig. 5. Pressure d i s t r i b u t i o n s a r o u n d the linear ke-model, the pressure distribution comtube surface for different turbulence moputed with the cubic nonlinear ke-model [10] is dels computed with STAR-CD and comin good agreement to the experimental data in pared with experimental data by Gog [11 ]. the total range of o~. The computed vortex shed....... algebraic-Reynoldsstress ding frequency has an error of less than 0.5%. 1 ~ . . . . k~-model (Low-Reynolds) Additionally computations were also car0.5 [-- ~ , . differential Reynoldsstress ded out with the CFD-program CFX on the O~ 0 9 x exp. Gog (1982) same grid layout. The difference to the grid -0.5 used for the STAR-CD calculations is, that the -1 "\x x x x grid is divided in four structured blocks. For the -1.5 \\\ \ ~~.,,,,~ velocities the QUICK differencing scheme and ", -,Jfor the turbulence equations a HYBRID differ-2.5 \ encing scheme was applied. The results for the -3 \ ,-". . . . . . R'e'--"6~.'S"6"" computed pressure distribution can be seen in -3.5 ' ' '' da='0"04m Fig. 6. In comparison to the STAR-CD results 0 30 60 90 120 150 180 (z the pressure coefficients calculated with CFX Fig. 6. Pressure d i s t r i b u t i o n s a r o u n d the are lower. One reason may be the different caltube surface for different turbulence culation of the pressure field. For the CFX calmodels computed with CFX. culation a SIMPLEC algorithm was used instead of the PISO algorithm applied for the STAR-CD calculations. Comparing the results obtained with the algebraic Reynolds-stress ke-model to the differential Reynolds-stress model, greater differences can be observed for o~ > 120 in the separation area. The calculated vortex shedding frequencies of 14.8 Hz for the differential stress model and 13.4 Hz for the quadratic nonlinear ke-model are in good agreement to the measured frequency of 14 Hz. The calculated pressure coefficients for the Low-Reynolds kcomodel with a viscous damping function for the near wall cells are too low compared to the algebraic and differential Reynolds stress models. The calculated vortex shedding frequency of 20.2 Hz is much higher than the observed one. One reason for this inadequate results may be the grid spacing near the wall, so the conditions for Low-Reynolds calculation near the wall are not valid. STAR-CD offers two LES-models with different subgrid scale models (SGS), namely that of Smagorinsky and a two equations kl-model. The computed pressure coefficients in Fig. 7 are lower than the measured values for (x > 60 ~ with a relative error of up to 80% I
413 compared to the measured values. The calculated vortex shedding frequency of 13.5 Hz for the Smagorinsky and 15 Hz for the SGS N-model are acceptable. Breuer [ 12] carried out a large eddy simulation for a tube in cross-flow and a Reynolds-number of 3900. His results were in good agreement with the measurements. The reasons for the better results are twofold: the SGS-model was modified near the wall and the grid and time discretization was much finer than for the presented simulations in this paper. For acceptable results with a large eddy simulation the numerical costs will be very high.
1 " .... 0.5 O" 0 -0.5
~ ~,
-1
~\
-2 ~ -2.5
L E S ( S m a g o r i n s k y SGS) L E S (kl-model SGS) ke-model (cubic nonlinear) x exp. Gog (1982)
t 0
Re=6450 da= 0. 04m x
.'~2..-" ,
"
',
, , I , I , I , , . 30 60 90 120 150 180 (z
Fig. 7. Pressure distributions around the tube surface for LES simulations computed with STAR-CD and compared to [ 11]. 4. C O M P A R I S O N OF D I F F E R E N T GRID D I S C R E T I Z A T I O N S
The measurements of Cantwell and Coles 1 ~ I---kin-model (grid B1) I [6] for a single rigid tube were also used for a 0.5 ]- ~ I .... kin-model(grid B2) I comparison of different discretizations out[ '~ Jo----e kin-model(gridB3) J I_ 9 ] - - - - ko~-model (grid B4) r} lined in section 2. The ko)-model was applied 0 l ~ 1--- kco-model(grid B5) ]] with the QUICK differencing scheme and the -0.5 ~1~[ x exp. cantweu&colesq same time step size of At = 0.0001 s was used. o~ -1 F The results for the pressure distribution are ' -presented in Fig. 8. -1.5 The fine grid B4 with 18054 cells and a computation time of 41.10 seconds per time -2 step yields the best result for the pressure coefficient in the total range of the peripheral an0 30 60 90 120 150 180 gle. Especially the computed position of the o~ pressure minimum at c~ = 71~ and the value of Fig. 8. Computed pressure distribution for Cp =-1.4 gives a good agreement with the exdifferent grid discretizations with the experimental data. An excellent result can be perimental data by Cantwell and Coles [6]. obtained for the drag coefficient cD = 1.24. The computed Strouhal-number of Sr = 0.182 confirm, that the kco-model enables results with high accuracy. The results obtained with the grids B 1 (7686 cells and a computation time of 9.52s per time step) and B2 (9558 cells and 11,08s per time step) are good compromises obtaining satisfying results (Sr = 0.197) within an acceptable computational time. The simple discretization B3 (6950 cells and 7.63s per time step), which has the lowest calculation time, is in good agreement with the measurements between 85 and 180 degrees. The separation point is fixed by the edge of the grid at about 74 degree, so the grid cannot predict the pressure distribution in the range of 50 < ~ < 80 and the calculated Strouhal-number of 0.232 is 30% higher than the observed value of 0.179 by Cantwell and Coles. Testing the error of cell matching, grid B5 with a high resolution in radial and a low resolution in peripheral direction (see Fig. 2b) was applied with no cell matching in the solution domain; the computed pressure distribution is acceptable in comparison with the results for the grid B3, the computed Strouhal number 0.184 for B5 is quite well.
414 5. CONCLUSIONS A comparison between the simulation and experimental results for a rigid single tube in cross-flow with a Reynolds-number of 140000 show, that the best results for the pressure distribution at the tube surface, the frequency of vortex shedding and the lift and drag forces can be obtained with the implemented kor-turbulence model. The results computed with the standard ke-turbulence and the ke-2-1ayer model are not satisfying. Simulations for a Reynolds-number of 6450 with different turbulence models and different CFD-programs compared to experimental results demonstrate the strong influence of CFD-code and turbulence model. The best results can be obtained for the cubic nonlinear kemodel. The pressure coefficients calculated with the CFD-program CFX differ from the resuits calculated with STAR-CD. The differences in the simulation results for the differential stress model and the nonlinear ke-model are small. The large eddy simulations carried out with STAR-CD could not predict the measured pressure distribution. The reason for the inaccurate results are the grid and time discretization and the wall treatment for the subgrid scale implemented in STAR-CD.
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