Q2P1 SPLITTiNG Finite Element Method for Large Eddy Simulation

Abstract

In A finite element code based on P2P1 tetra element has been developed for the large eddy simulation (LES) of turbulent flows around a complex geometry. Fractional 4-step algorithm is employed to obtain time accurate solution since it is less expensive than the integrated formulation, in which the velocity and pressure fields are solved at the same time. Crank-Nicolson method is used for second order temporal discretization and Galerkin method is adopted for spatial discretization. For very high Reynolds number flows which would require a formidable number of nodes to resolve the flow field, SUPG (Streamline Upwind Petrov Galerkin) method is applied to the quadratic interpolation function for velocity variables. Noting that the calculation of intrinsic time scale is very complicated when using SUPG for quadratic tetra element of velocity variables, the present study uses a unique intrinsic time scale proposed by Codina et al.[6] since it makes the present three-dimensional unstructured code much simpler in terms of implementing SUPG. In order to see the effect of numerical diffusion caused by using an upwind scheme (SUPG), those obtained from P2P1 Galerkin method and P2P1 Petrov-Galerkin approach are compared for the flow around a sphere at some Reynolds numbers. Both Smagorinsky model and dynamic model are adopted as subgrid scale models in the context of P2P1 finite element method. With the parallelization of the present code in mind, diagonal preconditioning is used for the momentum equation and ILU preconditioner is used for the pressure equation. As a bench mark problem for code validation, turbulent flow around a sphere has been studied at various Reynolds numbers. Drag coefficient has been obtained and compared with existing experimental. It is shown that the drag coefficient for the sphere agrees well with the existing experimental data when the Reynolds number is less than the critical value (~4x105). We also observe that the drag coefficient decreases as the Reynolds number goes beyond the transition value. Turbulent flow around MIRA model with wheels has been calculated at Re=2 x 106 using a relatively coarse grid for LES since the up-to-date serial computer can not afford to use a fine enough mesh to resolve very high Reynolds number flows around complex geometries. The computed drag coefficient is compared with the data from wind tunnel test. In order to study the mesh resolution effect on LES solution, the parallel version of the present code based on the domain decomposition and MPI (Message Passing Interface) is being developed and will be discussed.