POTENTIAL TURBULENCE MODEL PREDICTIONS OF FLOW PAST A TRIANGULAR CYLINDER USING AN UNSTRUCTURED STAGGERED MESH METHOD

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1 POTENTIAL TURBULENCE MODEL PREDICTIONS OF FLOW PAST A TRIANGULAR CYLINDER USING AN UNSTRUCTURED STAGGERED MESH METHOD Xg Zhang Blair Perot Department of Mechanical and Industrial Engeerg, University of Massachusetts, Amherst Amherst, MA 13, U.S.A. <xzhang@ecs.umass.edu> ABSTRACT Numerical simulation of the turbulent flow around a triangular cylder at Reynolds number of 45, is presented this paper. A body force potential model is used to model the turbulent motion. This approach is able to model non-equilibrium turbulence accurately at a cost and complexity comparable to k- models. The numerical method used this calculation is an unstructured staggered mesh scheme. The Strouhal number and timeaveraged velocity profiles obtaed from this simulation agree with experiments. INTRODUCTION The flow around a triangle provides an example of bluff body flow with fixed separation pots. If the Reynolds number is not too small the flow is herently unsteady and a Von Karman vortex street appears with a well-defed frequency. If the Reynolds number is sufficiently high the flow will be turbulent and a turbulence model must be cluded to model the turbulent fluctuations. The large-scale motions of the vortices are not turbulence, so they should be resolved by the numerical scheme and only the small-scale fluctuations are modeled. LDA measurement by Sjunnesson et al. [1] of vortex sheddg flow past a triangular cylder a duct at Re D =45, is a useful test case for unsteady turbulent flow of this kd. Their experimental study was motivated by the application to flame holders. Johansson et al [] carried out numerical simulation of this flow usg a k- model. Durb [3] (1994) carried out a simulation usg a k-- v model. In some similar simulations by Franke et al. [4], they compared the ability of different models to predict turbulent vortex sheddg from a rectangular cylder. Franke s conclusion is that some k- models do not predict the right sheddg frequency and Reynolds stress transport models can produce results good agreement with the experiments. It is hypothesized that this is because the turbulence is not equilibrium with the mean flow. The proposed turbulent potential model is a simplified Reynolds stress transport model, which has the ability of modelg non-equilibrium turbulence with the computg cost and complexity comparable to k- model. EQUATIONS Mean Flow Equations r u rr r r T + ( uu) = p+ ν ( u + u ) t u r = r R where u r is the mean velocity vector, p is the mean pressure/density, ν is the kematic viscocity, and R r is the Reynolds stress tensor. Turbulence Modelg In the turbulence vortex sheddg problem, the turbulence motion is not usually equilibrium with the mean flow. Prior evidence dicates that the commonly used Boussesq hypothesis (lear eddy viscosity) is not a good approximation for this problem due to the assumption of the turbulent equilibrium. In the past, for this kd of problem to be effectively predicted, the modeled transport equations for the Reynolds stress tensor itself had to be solved. However,

2 the Reynolds stress transport equations are significantly more difficult to solve than twoequation models. Recently, a body force potential model has been developed. This model is an alternative approach to modelg the Reynolds stress tensor. The primary quantities are the scalar and vector potentials of the divergence of the Reynolds stress tensor. The proposed model does not suffer from these difficulties. It has fewer variables than Reynolds stress transport models, the equations are not strongly coupled and are therefore not as numerically stiff, and no realizability conditions are needed. The governg equations of the turbulent potential model are discussed detail Perot [5] (1999). The Reynolds stress is related to the scalar and vector potentials of the turbulent body force by, r r R = φ+ ψ The governg equations for the turbulence quantities are: Dk υ t = ( υ + ) k + P Dt σ k D υ ˆ = ( υ + t ) + ( C 1P C ) Dt σ k Dφ υ 1/ t ˆ υ ( φ φ = ( υ+ ) φ α + Dt σk k φ (1 α ) φ + 1 (k /3 φ ) + P+ k1+ C /Re k pw ψ ψ φ α 1 ( P + 3) υt C 1+ pw/re k r 1/ Dψ υt v υ k φ υ ψ ˆ ( = ( + ) α + 1/ Dt σk k ( kφ ) v v (1 α) v ψ v + φω 1 ψ+ P φω + k 1+ w /Re k vv v ψ ψ ψ α 1 ( + ) P 3 υt C 1+ pw /Re k 1/ 1/ ) + φ k υt 1+ C p4 υ ) 1 r + ψ k υt 1+ 4 υ where v v φk 1 k P = ψ ω, υ t = C µ, α =, Re = φ υ k ˆ = υ k k υ φ φ and C,C,C, σσ,,c,c,c,c,c are constants. 1 µ k p1 p p3 p4 pw CODE Numerical Scheme The D ensemble-averaged unsteady Navier-Stokes equations are solved numerically with an unstructured staggered mesh method. A two-dimensional triangular (Delaunay) mesh is generated for this computation. For this kd of mesh, there is a Veronoi dual mesh associated with it. The faces of the two meshes are always locally orthogonal. The orthogonality of these dual mesh can be used to develop discretization operators that closely mimic their contuous counterparts. These discretization operators are ideally suited to representations of the Navier-Stokes equation based on the vorticity. A stream function formulation is used to elimate the pressure term the momentum equation. For spatial discretization, a limited gradient method (a second order upwd scheme) is used for the convection term. As for the time advancement, the convection term is explicit Figure 1. An example of D unstructured triangular mesh.

3 and a three-step second order Runge-Kutta scheme is used, the diffusion term is implicit. The symmetric semi-positive defite algebraic system resultg from the discetization is solved usg a conjugate gradient iterative method. The detailed description of this Navier-Stokes solver is presented Perot & Zhang [6]. The property that this method conserves ketic energy both locally with cells and globally makes it a good choice for performg turbulence modelg. Doma and Boundary Conditions In order to compare with the experimental data, we select a computational doma that is the same as the configuration of Sjunnesson s experiment. The mesh is generated by TRIANGLE [7] an automatic D Delaunay mesh-maker. There are approximately, triangles our calculation (see Figure.) In the present calculation, the let mean stream-wise velocity is a constant value, the vertical velocity is zero. For turbulent ketic energy and dissipation rate, we use the same conditions described Johnasson s paper. U = 15.5 m/s This let velocity is evaluated based on the total mass flow of their experiment. These values are also used as the itial value for the whole doma. l is the height of the duct. A zero gradient boundary condition is used for all the variables at the outlet. A no-slip boundary condition is used for the duct wall. RESULTS Calculation of D unsteady turbulent flow around a triangle cylder with Reynolds number U H / ν = 45, is presented, where H is the height of the triangle. A Von Karmann vortex street is formed behd the triangle. No special triggerg measure is taken to start the vortex sheddg, the unsteadess the computational result evolved naturally. To illustrate the periodicity of the flow, the stream function of a pot about one triangle height behd the triangle near the centerle is shown Figure 3. It can be seen that an almost perfect periodicity exists. The sheddg frequency is (s -1 ). The correspondg Strouhal number defed by, Sr = fh U k = (.5U ) 3.16k =.l Figure. Computational doma and mesh.

.6.4 Stream Function. -. -.4.4.8.1.16. Time Figure 3. The stream-function of one pot about one cylder height behd the triangle near the centerle. is.

4 .6.4 Stream Function Time Figure 3. The stream-function of one pot about one cylder height behd the triangle near the centerle. is.58, which should be compared with experimental data of.5 and the computed value of.7 Johnnasson (1991). The stream-wise velocity contour is shown figure 4. The length of recirculation zone is about 5mm. Figure 5 shows an stantaneous velocity vector plot, we can see that the center of a vortex is rolled up at the lower edge and a new vortex is begng to roll up at the upper edge. Although the stantaneous flow is asymmetric, the time-averaged fields are always symmetric or anti-symmetric. Figure 5. Instantaneous velocity vector plot. Figure 6 shows the stream-wise velocity at different locations behd the triangle. The calculated velocity profiles are reasonable agreement with the experiment. However, far down stream the wake zone, the velocity profiles do not recover as quickly as the experiments. The explanation for this is that we can not resolve the boundary layer this computation due to the limit of our computg capacity. In the far field, the real wall turbulent boundary layer is much thicker than what was obtaed the calculation. Due to the conservation of mass, the experimental velocities near the centerle are thus larger than the computed ones. Figure 4. Streamwise velocity Contours.

5 (a) (b) (c) (d) (e) Figure 6. Mean stream-wise velocity behd the triangle:, calculations; *, experiments. (a) 15mm, (b) 38mm, (c) 61mm, (d) 15mm, (e) 376mm

6 CONCLUSIONS In this paper, numerical simulation of flow past triangular cylder at high Reynolds number (45,) is presented. The stantaneous flow situation is very complex due to the presence of vortex sheddg and turbulence. The calculation was performed usg an unstructured staggered mesh scheme. A turbulent potential model is used to model the small-scale fluctuation motion. The capability of the turbulent potential model to predict turbulent vortex sheddg has been demonstrated this calculation. Computed Strouhal number and mean velocity profiles down stream of the triangular cylder are reasonable agreement with experiment data. It is hypothesized that this agreement is a result of the non-equilibrium nature of the underlg turbulence model. various turbulence models. Proc. 8th Symp. On Turbulent Shear Flows, Munich, 1991, pp [5] J. B. Perot, Turbulence Modelg Usg Body Force Potentials. Physics of Fluids, 11 (9), 1999 [6] J. B. Perot and X. Zhang, Reformulation of the unstructured staggered mesh method as a classic fite volume method. Second International Symposium on Fite Volumes for Complex Applications: Problems and Perspectives, July 19-,1999, Duisberg, Germany [7] Jonathan Richard Shewchuk, Triangle: Engeerg a D Quality Mesh Generator and Delaunay Triangulator, First Workshop on Applied Computational Geometry (Philadelphia, Pennsylvania), pages , ACM, May ACKNOWLEDGMENTS The fancial support of the Office of Naval Research (grant number: N ) is gratefully acknowledged REFERENCES [1] A. Sjunnesson, C. Nelsons and E. Max, LDA measurement of velocities and turbulence a bluff body stabilized flame. Volvo Flygmotor AB, Trollhattan, 1991 [] Stefan H. Johansson, Lars Davidson and Erik Olsson, Numerical Simulation of Vortex Sheddg Past Triangular Cylders at High Reynolds Number Usg a k- Turbulence Model. International Journal for Numerical Methods Fluids, VOL. 16, (1993) [3] P. A. Durb, Turbulence modelg for separated flow. Annual Research Briefs-1994, Center for Turbulence Research, Stanford University [4] R. Franke and W. Rodi, Calculation of vortex sheddg past a square cylder with

TURBULENT VORTEX SHEDDING FROM TRIANGLE CYLINDER USING THE TURBULENT BODY FORCE POTENTIAL MODEL

Proceedgs of ASME FEDSM ASME 2 Fluids Engeerg Division Summer Meetg June 11-15, 2 Boston, Massachusetts FEDSM2-11172 TURBULENT VORTEX SHEDDING FROM TRIANGLE CYLINDER USING THE TURBULENT BODY FORCE POTENTIAL