A Hybrid Staggered/Semistaggered Finite-Difference Algorithm for Solving Time-Dependent Incompressible Navier-Stokes Equations on Curvilinear Grids

Abstract

An accurate and efficient finite-difference method for solving the incompressible Navier-Stokes equations on curvilinear grids is developed. This method combines the favorable features of the staggered grid and semistaggered grid approaches. All components of velocity are stored at the corner vertices, and pressure is stored at the grid cell centers. All components of the momentum equations are discretized at cell vertices, facilitating a consistent discretization of the diffusive and convective terms as the boundaries are approached. The Christoffel symbol does not appear in the transformed equations and the cost of computation is comparable to that of the staggered-grid approach. A projection method is used to effectively evolve the discrete system in time, while ensuring a divergence-free velocity field. The discrete divergence and gradient operators of the projection step are constructed on a staggered gird layout leading to exact satisfaction of the discrete continuity. The solution of the Poisson-Neumann equation in the projection step is free of any spurious eigenmodes. The validity of the method is demonstrated on four benchmark problems. The Taylor-Green vortex problem is solved on a curvilinear grid with highly skewed cells and a second-order convergence in l _∞-norm is observed. The mixed convection in a lid-driven cavity is solved on a highly curvilinear grid and excellent agreement with literature is obtained. The results for flow past a cylinder and pusatile flow in a 90° bend are compared with the existing experimental data in the literature. The predictions agree well with the measured data, and validate the approach used.

Acknowledgments

This work was supported by a grant from AFOSR and a grant from the DEPSCoR-AFOSR programs. This support is gratefully acknowledged. Support for the second author (SA) from the National Science Foundation during the preparation of this manuscript is also acknowledged.