Comparative Study of the Accuracy of the Fundamental Mesh Structures for the Numerical Solution of Incompressible Navier-Stokes Equations in the Two-Dimensional Cavity Problem

Abstract

The accuracies of the staggered, semistaggered, vertex collocated, and cell-center collocated meshes are compared for the incompressible Navier-Stokes equations in primitive variables using the two-dimensional lid-driven cavity test problem in both the traditional form with uniform lid velocity and in the regularized form without corner discontinuities. Central differencing is used in all discretizations. The momentum equations are integrated explicitly after the solution of a Poisson pressure equation that enforces mass conservation. Richardson extrapolation is employed to estimate the correct solutions in cases without precise reference solutions. For both forms of the problem, the staggered and the cell-center collocated meshes are shown to produce comparably accurate results, while the vertex collocated mesh shows poorer accuracy. The almost unknown semistaggered mesh, which is too sensitive to the velocity discontinuity in the traditional cavity problem, appears as the most accurate mesh in the regularized cavity problem.

Comparative Study of UNIFAES and other Finite-Volume Schemes for the Discretization of Advective and Viscous Fluxes in Incompressible Navier-Stokes Equations, Using Various Mesh Structures

“…The pressure errors are more visible than the velocity errors; curiously, the pressure errors associated with the semistaggered mesh are opposite to the pressure errors of the other meshes. For Re ¼ 1,000, the semistaggered mesh presents superb results, overcoming all other meshes, and showing accelerating errors when the UNIFAES is employed, analogous to its behavior with central differencing in [25]. The pressure errors are also in opposite direction with respect to those of the other meshes.…”

“…Figures 8-13 present the profiles of the velocity components and pressure along the centerlines in the regularized cavity flow problem for Re ¼ 100, 1,000, and 10,000 with the four types of meshes using the UNIFAES. Again, the corresponding results for central differencing for the lower Re values are presented in [25]. Predictably, the smoother boundary conditions of the regularized form of the cavity problem lead to solutions much closer to the converged solution than the discontinuous conditions of the uniform lid velocity problem.…”

“…Reference values due to Ghia et al [31] are also included. For the lower values, the corresponding solutions with central differencing are shown in [25].…”

“…This article also completes the comparison of these meshes as performed in a companion paper [25], since the UNIFAES allows the accurate solution of problems with high Reynolds numbers, which could not be solved there by the central differencing. Like the article referred, the present article employs the classic test problems of the 2-D lid-driven cavity flow in the standard form (with uniform lid velocity) and in the regularized form (which removes the singularity at the lid corners).…”

“…Like the article referred, the present article employs the classic test problems of the 2-D lid-driven cavity flow in the standard form (with uniform lid velocity) and in the regularized form (which removes the singularity at the lid corners). The reader is also referred to [25] for details on the governing equations, on the method of solution, and on the methodology of Richardson extrapolation.…”