On stationary solution of the problem of an incompressible viscous fluid at high Reynolds numbers

A. A. Fomin ; L. N. Fomina

Matematičeskoe modelirovanie i čislennye metody (2015), pp. 92-109

Résumé

The research explored questions of the convergence of iterative processes and correctness of the solutions on the example of the problem about a steady-state flat square lid-driven cavity flow of incompressible viscous liquid. The problem is solved for Reynolds numbers of $15\,000 Re 20\,000$ and steps of grid $1/128 > h > 1/2048$. The findings of the research illustrate that not for all relationships between Re and h the convergence of iterative processes is stable and the resulting steady-state solutions are qualitatively correct. We conducted a qualitative analysis of the solutions of the problem in the coordinate system $(Re, 1/h)$ in terms of the convergence of iterative process, solution correctness and the required computing time. According to the literature and the results of systematic calculations we conclude that the stability of the convergence of iterative process on the coarse grid depends on the degree of influence of the artificial viscosity and/or the condition number of the matrix of difference elliptical linear algebraic equations, and on the detailed grid it depends on the grid Reynolds number. At high Reynolds numbers steady calculations can be carried out either on very coarse grids, or on very detailed ones. The width of the zone of instability in terms of parameter $1/h$ increases with increasing Reynolds number. Since the coarse grid solution is incorrect, and the use of detailed grid leads to very high costs of computer time, the further increase of the Reynolds number in the problem is associated with increasing the order of approximation of the differential equations.

Détail

Keywords: Navier–Stokes equations, a lid-driven cavity flow, a convergence of iterative process.