Penalization of the Spalart-Allmaras turbulence model without and with a wall function: Methodology for a vortex in cell scheme

Immersed boundary methods (IBM) are alternative methods to simulate fluid flows around complex geometries. The grid generation is fast as it does not need to conform to the fluid-solid interface. However, special treatments are needed in the flow equations to properly take into account the wall proximity. The penalization method is a particular case of the IBM in which the wall boundary conditions are imposed via continuous forcing terms into the governing equations. Reynolds Averaged Navier-Stokes (RANS) equations completed with a turbulence model are still the most common way to model turbulence in engineering applications. However, RANS turbulence model implementation with penalization into a vortex formulation is not straight forward, in part because of the variable turbulent viscosity and partly because of the boundary conditions. This paper extends the penalization technique to turbulent flows. The objective of this paper is to validate the use of the Spalart-Allmaras turbulence model in the context of penalization and vortex formulation. Details of the resolution using a Vortex In Cell (VIC) numerical scheme are given. The proposed scheme is based on the advection of particles of vorticity and particles of turbulent viscosity. A Lagrangian framework is chosen to solve the advection part. The remaining parts of the system of equations are solved with an Eulerian framework using a Cartesian uniform grid. To avoid fine meshes near the wall, a wall function compatible with the penalization method and the vortex formulation is proposed. The formulation and the coding are validated against the well-known periodic channel flow. Velocity profiles are computed without and with the wall function. Results agree with analytic law of the wall solutions, showing that RANS simulations can be conducted with VIC schemes and penalization.