Numerical Study of Natural Unsteadiness Using Wall-Distance-Free urbulence Models Yi-Lung Yang* and Gwo-Lung Wang Department of Mechanical Engineering, Chung Hua University No. 707, Sec 2, Wufu Road, Hsin Chu, aiwan * Email:yylung@chu.edu.tw Abstract. he periodic vortex shedding past a cylinder is presented. he program was validated by the solution of one-dimensional Stokes second problem first. hen, a two-dimensional steady flow through a cylinder at Reynolds number.54 was used to compare with the streamlines with experiment and also at Reynolds number 28.4, to check the separation point and the length of wake. At Reynolds number 280 where the wake is laminar, the solution becomes un-symmetric after 25 cycles of iterations automatically and becomes fully periodic flow after 56 cycles of iterations. For Reynolds number 000 with a turbulent wake, the numerical solution starts to un-symmetric after 2 cycles of iterations, and becomes fully periodic after 3 cycles of iterations. he streamlines of the periodic solution are compared very well with Arnone s result. Using 3500 steps in a cycle, the error on the Strouhal number is 9%. However, increasing to 70000 steps in a cycle, the error on the Strouhal number is less than %. Wilcox s high and low Reynolds number κ-ω models, Jones-Launder κ-ε model and Launder-Sharma κ-ε model were used to study the structure of turbulent vortex street at Reynolds number 2000. he high Reynolds number κ-ω model gives the largest eddy viscosity among these turbulent models. On the other hands, the solution using Jones-Launder κ-ε model was similar to laminar one. he spreading angle and the distance between trailing vortex are better simulated by using high Reynolds number κ-ω model, Besides, based on the accuracy of the predicted Strouhal number, the Wilcox s high Reynolds number κ-ω models is suggested. Keywords:Stokes second problem, Strouhal number, wall-distance-free turbulence models. Introduction he unsteady vortex shedding behind a body causes hysteric forces on the structure. If the frequency of the unsteady loading closes to the natural frequency of the structure, the vortex could induce severe structure vibration. he catastrophe of acoma Bridge is a famous example. However, the relationship between the natural unsteadiness and the evolution of a symmetric numerical solution into a cyclic solution is not well posed in the past. Hence, in this study a numerical program to reproduce the natural unsteadiness is an important demonstration of a robust scheme. Lienhard [] has summarized the vortex pattern behind a cylinder. For Reynolds number below 5, no separation is occurred. For Reynolds number between 5 and 40, there is a fixed pair of vortices in the wake. For Reynolds number between 40 and 300, the Karman vortex street is laminar. For Reynolds number between 300 and 3x0 5, the vortex street is fully turbulent. For Reynolds number between 3x0 5 and 3.5x0 6, the laminar boundary layer has undergone turbulent transition. he wake is narrower and disorganized. No vortex street is apparent. For Reynolds number above 3.5x0 6, the boundary layer is turbulent and the wake is thinner. Kimura and sutahara [2] have arranged a groove on the cylinder to control the cylinder separation. Placing the groove at 84 degree from the leading edge can minimize the drag coefficient of the cylinder. Bearman and Harvey [3] also studied the effect of dimples on the cylinder to trickle the boundary from laminar to turbulent. An effective reduction on the drag coefficient of the cylinder between Reynolds number 4x0 4 and 3x0 5 has been achieved. he numerical simulation of unsteady flow field has been addressed extensively. Jameson [4] extended the multigrid methodology to allow practical time steps with the use of an explicit code. Arnone et al. [5] expanded the Jameson s subiterative technique into a dual time stepping. A good prediction on the Strouhal number was shown in their work. Rumsey et al. [6] have explored the accuracy and efficiency of two different types of subiterations in both explicit and implicit codes for unsteady flows. he pseudo time subiterations removes the time-step limitations in practice. A pseudo time version of the code may be used to speed up the solution. Because the primary focus of this paper is on the turbulent modeling. he efficiency of the subiterative time-stepping method is not performed for the circular cylinder flow. he solution is marched by local time-stepping at the beginning till the flow field becomes unsymmetric. hen a prescribed time step is used for every cell, several cycles of calculations is proceeded to ensure the flow field is periodic [0].
he large region of separated flow behind the cylinder gives a challenge on the turbulent modeling. he wall-distance-free turbulent models may be more promising for calculations in complex flow field. In this paper, results with the Jones and Launder [7] κ-ε model, Launder and Sharma [8] κ-ε model, and Wilcox s high and low Reynolds number [9] κ-ω models are compared for the unsteady wake behind the cylinder. he unsteady hysteresis effect is reproduced computationally using these turbulence models. 2. Mathematical modeling he two-dimensional unsteady Reynolds-Averaged Navier-Stokes equations in conservative form are written as: U Ec Fc Ev Fv + + = + () t x y x y where ρ ρu ρv ρu 2 U = u p E ρ + ρuv ρv c = F ρuv c = 2 ρv + p ρ E ρue ( + p/ ρ) ρve ( + p/ ρ ) 0 0 τ xx τ xy E v = τ xy F v = τ yy uτ xx + vτ xy + K x uτ xy vτ yy K + + y he pressure is obtained by the equation of state for a perfect gas: 2 2 p = ( γ ) ρ[ E ( u + v ) / 2] (2) he stress tensor has been modeled using the Boussinesq approximation: τ = ( μ+ μ )[ u x + u x 2/3 u x δ ] 2/ 3ρk δ ij t ( i j j i) k k ij ij (3) and μt is the eddy-viscosity. A time-marching solver has been employed for the solution of the equations. he solver uses a fourth-order Runge-Kutta scheme. he cell-centered averaged scheme is used for the discretisation of the inviscid and viscous fluxes. A fouth-order artificial viscosity is added to reduce numerical oscillation. he solution is marched by local time-stepping at the beginning till the flow field becomes un-symmetric. hen a prescribed time step is used for every cell, several cycles of calculations is proceeded to ensure the flow field is periodic. he accuracy of the numerical solution can be checked by its Strouhal mumber nd St = (4) U where n is the frequency of the vortex shedding D is the diameter of the cylinder and U is the free stream velocity. he structure of the unsteady wake is more easily to present through the entropy distribution, the definition of the entropy is P s = C p ln Rln (5) P where C is the specific heat at constant pressure. p 3. urbulence models In this paper, four different wall-distance-free turbulent models of κ-ε and κ-ω have been assessed. hese are: () Jones-Launder (JL) κ-ε (2) Launder-Sharma (LS) κ-ε (3) Wilcox high-reynolds number (κ-ωh) κ-ω (4) Wilcox low-reynolds number (κ-ωl) κ-ω All models employ the boundary condition for the turbulent kinetic energy at solid boundaries k w = 0 (6) For the JL and LS models the boundary condition for the turbulent dissipative rate is ε w = 0 (7) In the case of κ-ωh and κ-ωl models, a zero gradient condition for specific dissipative rate on the solid wall was used. At the far field of the computational domain, the turbulent kinetic energy, turbulent dissipative rate, and specific dissipative rate are propagated outward. 4. Results he numerical program was validated by a one-dimensional Stokes second problem first. Figure (a) shows the velocity distribution at the beginning of the sine wave (t/=0) using three different numbers of grids with 40 intervals in a cycle. he maximum error from the analytical solution is less than 0.04% by using 2 grid points and only 0.023% by using 8 grid points. Figure (b) gives the velocity distribution at the quarter of the sine wave (t/=0.25). he agreement is very well again. able and able 2 give the maximum numerical error at t/=0 and t/=0.25 respectively. From this study, the accuracy of the solution is more sensitive to grid resolution. Figure 2(a) shows the computational O-grid of the circular cylinder flow and Figure 2(b) plots the two symmetrical monitor points on the back of the cylinder. here are 20 grids around the cylinder and2 grids in the normal direction. he grid point extends 5 diameters upstream and 0 diameters downstream. 2
he first calculation is performed for a Reynolds number of.54. Figure 3 shows the streamlines between the calculation and the experiment []. here is no separation in this case. he agreement in the streamlines is very good. Another validation of the program has been performed for the Reynolds number at 28.4. Figure 4 plots the instantaneous pressure variation of the monitor points. No un-symmetric solution was found for this Reynolds number. Figure 5 shows the streamlines of the re-circulating region between numerical solution and experiment []. he agreement on the separation points and the length of laminar wake are also very close. he natural vortex shedding past a cylinder is reproduced in the numerical solution. he successive pressure variation behind the cylinder is plotted in Fig. 6 for Reynolds number 280. he solution becomes un-symmetric after 25 cycles of iterations and becomes fully periodic vortex shedding after 56 cycles of iterations. he solution is marched by local time-stepping at the beginning till the flow field becomes un-symmetric. hen a constant time step is used to march the periodic solution. he predicted error in Strouhal number is 0% by using 70000 steps in a cycle and reduces to 7% by using 40000 steps in a cycle. For Reynolds number 000, the vortex street is fully turbulent. Figure 7 plots the horizontal and vertical velocity cyclic variation at the monitor point. he numerical solution is well periodic as shown by this plot. Figure 8 compares the instantaneous streamlines in nine instants over a cycle between the current solution and Arnone s results for laminar calculation. he agreement of the shedding of the vortex between these two solutions is very evident. he mechanism of vortex formation and merging is clearly shown in these plots. here are two different grid points used to study the grid resolution on the solution. he Reynolds number 000 was used. he calculation of 20x8 meshes requires a much more iterations than 20x2 meshes before the solution become un-symmetric as shown in Fig. 9. he instantaneous entropy contours between these two meshes are shown in Fig. 0. he finer the mesh, the better resolution of the wake is obtained. he predicted Strouhal number is quite similar in both cases. he evolution of pressure using four different time steps is shown in Fig.. he computed Strouhal number is normalized by the experimental value of 6 0.2. Using 3500 iterations ( Δ t = 2 x0 ) in a cycle, the accuracy of Strouhal number is only 8%. 7 A time step of Δ t = x0 improves the accuracy of Strouhal number to 99%. he boundary layer is laminar over the front, separate, and breaks up into a turbulent wake above Reynolds number 300. A Reynolds number of 2000 was used for turbulent calculation in this study. he separation points moving forward as the Reynolds number reaches 2000, the boundary layer has now attained their upstream limit ahead of maximum thickness. he instantaneous entropy contour is presented for different turbulence models in Fig. 2. he calculation using κ-ωh model gives a much stronger distribution of eddy viscosity. he spreading of the wake structure is smallest among these four cases. he distance between the downstream vortices by using the κ-ωh model is also largest among these models. he low Reynolds number κ-ωl model gives a slighter less eddy viscosity. Hence, the spreading of the wake becomes wider. he JL and LS models give an even larger spreading angle. Also, the speed of vortex sheet is large so the trailing vortices are narrowly packed. he evolution of pressure using the four different turbulence models is shown in Fig. 3. he predicted Strouhal number is better by using the high Reynolds number κ-ω model. 5. Conclusions A Rung-Kutta integration method coupled with wall-distance-free turbulent models was used to study the periodic vortex shedding past a cylinder. he solution was validated by the streamlines for Reynolds number.54 and 28.4. Numerical comparison between the current results and Arnone s results at Reynolds 000 was made for in nine instants over a cycle. he finer the mesh, the more iteration is required to form vortex sheet. he accuracy of Strouhal number requires a much smaller time step in the calculation. he use of different turbulence model in the codes has been explored for unsteady rowing up of vortex sheet. Only the wall-distance-free turbulence models were used to calculate the rowing up of vortex sheet. heκ-ωh gives a much larger eddy viscosity in the result. he distance between the vortices is increased. he resolution of vortex formation and rowing up is very similar to laminar solution by using JL turbulence model. he prediction of the Strouhal number is better by using the High Reynolds κ-ω turbulence model. More numerical test cases and maybe more advanced wall-distance-free turbulence models should be included to gain confidence in its predictive capability. 6. Acknowledgment he authors would like to thank the Chung Hua University (Contract No. CHU-94-M-05) for financially supporting this research. 7. References [] Lienhard, J. H., Synopsis of Lift, Drag and Vortex Frequency Data for Rigid Circular Cylinders, Washington State University, College of Engineering, Research Division Bulletin, pp. 300, 966. 3
[2] Kimura,., and sutahara,., Fluid Dynamic Effects of Grooves on Circular Cylinder Surface, AIAA Journal, Vol. 29, No. 2, pp. 2062-2068, 99. [3] Bearman, P. W., and Harvey, J. K., Control of Circular Cylinder Flow by the Use of Dimples, AIAA Journal, Vol. 3, No. 0, pp. 753-756, 993. [4] Jameson, A., ime Dependent Calculations Using Multigrid with Applications to Unsteady Flow Past Airfoils and Wings, AIAA-9-596, 99. [5] Arnone, A., Liou, M. S. and Povinelli, L. A., Integration of Navier-Stokes Equations Using Dual time Stepping and a Multigrid Method, AIAA Journal, Vol. 33, No. 6, pp. 985-990, 995. [6] Rumsey, C. L., Sanetrik, M. D., Biedron, R.., Melson, N. D., and Parlette, E. B., Efficiency and Accuracy of ime-accurate urbulent Navier-Stokes Computations, Computers & Fluids, Vol. 25, No. 2, pp. 27-236, 996. [7] Jones, W. P., and Launder, B. E., he Prediction of Laminarization with a wo-equation Model of urbulence, International Journal of Heat and Mass ransfer, Vol. 5, pp. 30-34, Feb. 972. [8] Launder, B. E., and Sharma, B. I., Application of the Energy Dissipation Model of urbulence to the Calculation of Flow Near a Spinning Disc, Letters in Heat and Mass ransfer, Vol., No. 2, pp. 3-38, 974. [9] Wilcox, D. C., Comparison of wo-equation urbulence Models for Boundary Layers with Pressure Gradient, AIAA Journal, Vol. 3, No. 8, pp. 44-42, 993. [0] Wang, G. L., Numerical Calculation of Natural Unsteadiness in Laminar and urbulent Flows, M.S. Dissertation, Department of Mechanical Engineering, Chung Hua University, Hsin Chu, aiwan, 2006 [] Van Dyke, M., An Album of Fluid Motion, Department of Mechanical Engineering, Stanford University, he Parabolic Press, Stanford, California, Fourth printing, pp. 20-30, 988. u/uo (40 X 20) (40 X 40) (40 X 80) analytical 0.8-0.03 0.6 0.4 0.2 0 0 2 4 6 8 0 Fig.(a) Solution of Stokes second problem at t/=0 u/uo 0.25 0.2 0.5 0. 0.05-0.04-0.05-0.06-0.07.8 2 2.2 2.4 2.6 2.8 3 3.2 0.35 (20 X 40) (40 X 40) (80 X 40) 0.3 analytical 0.35 0.3 0.25 0.2 0.5 η 0. -0.5 0 0.5.5 2 2.5 0 0 2 4 6 8 0 Fig.(b) Solution of Stokes second problem at t/=0.25 η 4
Fig. 3 he streamlines comparison at Re=.54. P P2.08.06.04.02 P, P2 Fig. 2(a) he computational grid for the cylinder 0.9 0.88 0 2 4 6 8 0 2 Fig. 4 he evolution of pressure at the two monitor points (Re=28.4) Fig. 2(b) he monitor points behind the cylinder Fig. 5 he streamlines comparison at Re=28.4 5
.08 P P2.06.04.02 time = 4 P, P2 0.9 0.88 0.992 0.99 0.99 9 8 7 6 25 25.5 26 26.5 27 0 0 20 30 40 50 60 70 Fig. 6 he evolution of pressure at the two monitor points (Re=28.4) 0.99 9 8 7 6 5 55.5 56 56.5 57 57.5 58 time = 5 time = 6 0.3 Y-COMPONEN OF VELOCIY 0.2 0. 0-0. -0.2 time = 7 time = 8-0.3-0.2-0.5-0. -0.05 0 0.05 0. 0.5 0.2 X-COMPONENOFVELOCIY Fig. 7 Instantaneous u and v velocity variation time = 9 Fig. 8 Comparison of evolution of streamlines between the current solution (left) and Arnone s solution (right).06 ( 20 x 8 ) ( 20 x 2 ).04 time =.02 P 0.9 time = 2 0.88 0.86 0.0.0 2.0 3.0 4.0 Fig. 9 he evolution of pressure at the monitor point using different meshes (Re=000) time = 3 6
20x2 κ-ωh 20x8 κ-ωl Fig. 0 Entropy plots by two different meshes on the wake structure (Re=000) D = 2.D-6 D =.D-6 D = 5.D-7 D =.D-7 P 0.9 0 0.5.5 2 2.5 3 3.5 4 Fig. he instantaneous pressure variation at the monitor point using four different time steps (Re=000) LS 7
JL Fig. 2 Instantaneous Entropy plots by different turbulence models (Re=2000) Low Reynolds Wilcox High Reynolds Wilcox Jones-Launder Launder-Sharma Laminar P 0 2 3 4 Fig. 3 he instantaneous pressure variation at the monitor point using four different time steps (Re=2000) able Maximum error for Stokes second problem at t/=0 ime Space 20 40 80 20 0.30% 0.04% 0.097% 40 0.053% 0.050% 0.049% 80 0.024% 0.023% 0.02% able2 Maximum error for Stokes second problem at t/=0.25 20 40 80 ime Space 20 0.058% 0.056% 0.053% 40 0.032% 0.03% 0.029% 80 0.026% 0.09% 0.08% 8