A collocated, iterative fractional‐step method for incompressible large eddy simulation

Abstract

AbstractFractional‐step methods are commonly used for the time‐accurate solution of incompressible Navier–Stokes (NS) equations. In this paper, a popular fractional‐step method that uses pressure corrections in the projection step and its iterative variants are investigated using block‐matrix analysis and an improved algorithm with reduced computational cost is developed. Since the governing equations for large eddy simulation (LES) using linear eddy‐viscosity‐based sub‐grid models are similar in form to the incompressible NS equations, the improved algorithm is implemented in a parallel LES solver. A collocated grid layout is preferred for ease of extension to curvilinear grids. The analyzed fractional‐step methods are viewed as an iterative approximation to a temporally second‐order discretization. At each iteration, a linear system that has an easier block‐LU decomposition compared with the original system is inverted. In order to improve the numerical efficiency and parallel performance, modified ADI sub‐iterations are used in the velocity step of each iteration. Block‐matrix analysis is first used to determine the number of iterations required to reduce the iterative error to the discretization error of$O\left(\Delta t^2\right)$. Next, the computational cost is reduced through the use of a reduced stencil for the pressure Poisson equation (PPE). Energy‐conserving, spatially fourth‐order discretizations result in a 7‐point stencil in each direction for the PPE. A smaller 5‐point stencil is achieved by using a second‐order spatial discretization for the pressure gradient operator correcting the volume fluxes. This is shown not to reduce the spatial accuracy of the scheme, and a fourth‐order continuity equation is still satisfied to machine precision. The above results are verified in three flow problems including LES of a temporal mixing layer. Copyright © 2008 John Wiley & Sons, Ltd.