Effects of numerical anti-diffusion in closed unsteady flows governed by two-dimensional Navier-Stokes equation

Abstract

Numerical methods producing acceptable results for a long time abruptly blow up, without providing any indication of localized onset of sudden numerical instability. This has been identified as focusing problem in literature. It is noted that the scale selection of error does not depend on the relevant excited physical space-time scales. While this has been encountered in weather prediction studies, it is not widely reported from the solution of Navier-Stokes equation (NSE). Recently, in “Focusing phenomenon in numerical solution of two-dimensional Navier-Stokes equation, In: Pirozzoli S., Sengupta T. (eds) High-Performance Computing of Big Data for Turbulence and Combustion, CISM International Centre for Mechanical Sciences (Courses and Lectures), vol 592. Springer, Cham (2019)”, focusing was demonstrated for a steady fluid flow and its mechanism was identified from global spectral analysis (GSA) of 2D convection diffusion equation (CDE). Focusing was shown to be due to the anti-diffusion caused by the discretization of diffusion term for the chosen numerical scheme. The present work consolidates the one-to-one correspondence between numerical anti-diffusion of 2D CDE and focusing for unsteady flows by solving flow inside a 2D lid driven cavity (LDC) for the Reynolds number of 10,000. We also present a method to remove numerical anti-diffusion using multi-dimensional filters. Detailed analysis of space-time discretization with filters is also provided to explain the cure of focusing.