Turbulent Vortex Shedding from Triangle Cylinder Using the Turbulent Body Force Potential Model

Numerical simulation of the turbulent flow around a triangular cylinder at a Reynolds number of 45,000 is presented in this paper. Both steady and unsteady vortexshedding results are presented. 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-ε models. The numerical method used in this calculation is an unstructured staggered mesh scheme. The property that this method conserves kinetic energy both locally within cells and globally makes it a good choice for performing turbulence modeling. For the unsteady solution, the Strouhal number and time-averaged velocity profile agree well with experiments. However, the steady solution that was obtained by using a symmetric boundary condition at the centerline leads to poor predictions of the time-averaged mean velocity profile. INTRODUCTION The flow around a triangle provides an example of bluff body flow with fixed separation points. If the Reynolds number is not too small the flow is inherently unsteady and a Von Karman vortex street appears with a well-defined frequency. If the Reynolds number is sufficiently high the flow will be turbulent and a turbulence model must be included to model the turbulent fluctuations. In the case of turbulent vortex shedding, we have the option of including the large scale vortex shedding in the turbulence model and calculating a steady mean flow, or of solving for the large scale vortex shedding by numerical scheme while only including the small scale turbulence in the model. The former approach is less expensive, but we show here less likely to give accurate predictions. This is hypothesized to be due to the fact that the large-scale vortex structures do not behave like equilibrium turbulence. Sjunnesson (1991) measured the flow of a triangular cylinder in a duct. Their experimental study was motivated by the application to flame holders. Johansson et al. (1993) carried out numerical simulation of this flow using a k-ε model. Durbin (1994) also carried out the same simulation using a k-ε2 v model. Franke et al. (1991) compared the ability of different models to predict turbulent vortex shedding from a rectangular cylinder. Franke’s conclusion is that some k-ε models do not predict the right shedding frequency and Reynolds stress transport models can produce results in good agreement with the experiments. The turbulent potential model is a simplified Reynolds stress transport model, which has the ability of modeling non-equilibrium turbulence with the computing cost and complexity comparable to k-ε model. TURBULENCE MODEL AND NUMERICAL SCHEME The primary difficulty of modeling unsteady turbulent vortex shedding is thought to be that the turbulence is not in equilibrium with the mean flow, because the large vortices move and decay at the same time-scale as the turbulence. The most common constitutive relation, the eddy viscosity hypothesis (or linear Boussinesq hypothesis) is probably incorrect in this case. 1 Copyright © 2000 by ASME In the past, avoiding an algebraic constitutive relation for the Reynolds stresses required solving coupled transport equations for the Reynolds stress themselves. Recently, a new modeling approach, the turbulence potential model has been developed. It is capable of modeling the complex turbulent physics associated with separation and unsteady flow. This turbulent potential model is well suited to vortex shedding problem because it does not require a constitutive relation relating the Reynolds stress tensor to the mean flow. The model hypothesizes evolution equations for the scalar and vector potentials of the turbulent body force (the divergence of the Reynolds stress tensor). It has the accuracy of a Reynolds stress model, at a cost comparable to modern two equation models. The governing equations of the turbulence potential model will not be presented here. Their initial development is described in Perot (1999). Our numerical method uses an unstructured staggered mesh scheme which can conserve mass, momentum, and kinetic energy to machine precision. The turbulence quantities are advected using an unwinding scheme to guarantee positivity constraints. The model integrates up to the wall, so wall functions are not used, but the first grid point should be in the laminar sub-layer to obtain accurate predictions. The details for this numerical method, including accuracy analysis and conservation property are discussed in Perot & Zhang (1999). FLOW OVER A TRIANGLE In order to compare with the experimental data, we select a computational domain that is the same as the configuration of Sjunnesson’s experiment. The mesh is generated by TRIANGLE – an automatic 2D-Delaunay mesh maker. There are approximately 25,000 triangles in our calculation (see Figure 1). In the present calculation, the inlet mean stream-wise velocity is a constant value, the vertical velocity is zero. For turbulent kinetic energy and dissipation rate, we use the same conditions described in Johnasson’s paper. Figure 1. Computational domain and mesh. 0 . 17 = in U m/s 2 ) 05 . 0 ( in in U k =