Archive | 2019
Turbulent Flow Simulation for a Sharp Immersed Boundary Method
Abstract
In this paper, a wall function is combined with the sharp immersed boundary method (IBM) for turbulent flow simulation on Cartesian grid. The wall function derived directly from Spalart-Allmaras (S-A) turbulence model or its modified version is employed to determine proper tangential velocity boundary condition. Several two-dimensional (2-D) cases, such as non-inclined / inclined flat plate, zero attack NACA 0012 airfoil, are investigated to check the accuracy of the present IBM. Although the skin friction coefficients are very difficult to predict, the surface pressure coefficient of the airfoil can be calculated accurately even with a coarse grid, in which the y + of the reference point defined at leading edge is about 501.9. Compared to the original wall function, the modified wall function achieves a better grid convergence. The validation cases of plate and NACA 0012 have shown the primary ability of the present IBM to treat attached flow. INTRODUCTION Although body-conformal grids, structured or unstructured, have been widely used in Computational Fluid Dynamics (CFD) to simulate flows around arbitrary bodies, the automatic grid generation or reconstruction is still a big challenge when dealing with complex or moving geometry. Therefore, the Cartesian grid methods, which can generate the grid automatically and rapidly around complex solid boundaries, have come back to the lime-light. In the simplest Cartesian grid method, boundaries are generally approximated as a series of staircase-like steps, which would introduce deviations into the solution (Ferziger and Peric, 2012). In order to reproduce the wall boundary smoothly and accurately in Cartesian grid solvers, three primary methods have been applied: the hybrid grid method (Charlton, 1997, Wang, 1998), the cut-cell method (Lee and Ruffin, 2007, Harada et al., 2017) and the immersed boundary methods (Mittal and Iaccarino, 2005, Du et al., 2014, Du and Sun, 2015). When the layer grids are used along with the Cartesian grids in a hybrid grid method, the boundary layer is easily treated as commonly used body-conformal grids. However, the laborious grid generation and complicated coordinate transformation and interpolation are required again (Lee and Ruffin, 2007). The cut-cell methods cut the cells that intersect with wall surfaces. The cut-cell method can track the boundary as exact as body-conformal grids, whereas the calculation of the intersection is complicated, and special treatments such as cell-merged techniques are required to handle the small cells in the vicinity of the geometry (Udaykumar et al., 2001). On the other hand, the IBM uses only orthogonal cells, and sharp interface can also be achieved straightforward (Du et al., 2014, Majumdar et al., 2001), it is a simple and robust method. Adaptive grid refinement technique has been proved to be a promising approach to enhance the local grid resolution only at interested regions, such as near the boundary or with high gradient, and have been validated in incompressible / compressible laminar flow. With the help of local grid refinement, Tullio et al. (de Tullio et al., 2007) have also proved that simulation of a turbulent boundary layer is possible if the minimum cell size is smaller than the height of the viscous sublayer. However, the isotropic property of Cartesian grid will significantly increase near wall grid density, which will become fatal when the Reynolds number becomes higher. Therefore, two distinct wall treatments have been proposed to alleviate the grid resolution constraint of High-Reynolds simulation. The first treatment is to take advantage of the wall function methods to bridge the near wall region to core region of the turbulence with analytical expressions, which are derived from turbulent boundary layer function. Ghosh et al. (Ghosh et al., 2010) used a power-law expression along with a discrete continuity equation to reconstruct the velocity vector at near wall grid. Lee and Ruffin (Lee and Ruffin, 2007) used Spalding’s formulation, which yields a unified form valid for the log law layer and the viscous sublayer as well as the buffer layer, to determine the tangential velocity at the ghost cell. Kalitzin et al. (Kalitzin et al., 2005) suggested to build adaptive wall functions based on wellresolved numerical look-up tables for each individual turbulent model to ensure consistency between the eddyviscosity and the velocity gradient. Recent years, in the researches team of University of Tokyo (Tamaki et al., 2017, Harada et al., 2017), following the advice of Kalitzin, a new SA wall model (Allmaras and Johnson, 2012) is served as a