A coarse-constrained multiscale method for accelerating incompressible flow computations

Abstract

We present a coarse-constrained multiscale (CCM) method for accelerating incompressible flow computations. Reducing the number of degrees of freedom of the Poisson solver by powers of two in the primitive variable fractional-step method, or the vorticity-stream function formulation of the problem accelerates these computations while, for the first level of coarsening, retaining the same level of accuracy in the fine-resolution velocity field variables. CCM is a modular approach that facilitates data transfer with simple interpolations and uses black-box solvers for the Poisson and advection-diffusion equations in the flow solver. Here, we implement a particular CCM method, CCMRK3, that uses the full weighting operation for mapping from fine to coarse grids, the third-order Runge-Kutta method for time stepping, and finite differences for the spatial discretization. After solving the Poisson equation on a coarsened grid, bilinear interpolation is used to obtain the fine data for consequent time stepping on the full grid. We compute several benchmark flows: the Taylor-Green vortex, a vortex pair merging, a double shear layer, decaying turbulence, the Taylor-Green vortex on a distorted grid, and laminar flow over a circular cylinder. In all cases we use either FFT-based or V-cycle multigrid linear-cost Poisson solvers. We favor the FFT-based Poisson solver for Cartesian grid applications due to its robustness. A linear acceleration rate is obtained for all the cases we consider, with reduction factors in computational time between 2 and 42. We also find that the computational savings increases with increasing distortion ratio on non-Cartesian grids, which makes the CCM method a useful tool for generalized curvilinear incompressible flow solvers.