Large-scale influence of numerical noises as artificial stochastic disturbances on a sustained turbulence

Abstract

We investigate the large-scale influence of numerical noises as tiny artificial stochastic disturbances on a sustained turbulence. Using two-dimensional (2-D) turbulent Rayleigh–Bénard convection (RBC) as an example, we solve numerically the Navier–Stokes equations, separately, by means of a traditional algorithm with double precision (denoted RKwD) and the so-called clean numerical simulation (CNS). The numerical simulation given by RKwD is a mixture of the ‘true’ physical solution and the ‘false’ numerical noises that are random and can be regarded as a kind of artificial stochastic disturbances; unfortunately, the ‘true’ physical solution is mostly at the same level as the ‘false’ numerical noises. By contrast, the CNS can greatly reduce the background numerical noise to any a required level so that the ‘false’ numerical noises are negligible compared with the ‘true’ physical solution, thus the CNS solution can be used as a ‘clean’ benchmark solution for comparison. It is found that the numerical noises as tiny artificial stochastic disturbances could indeed lead to large-scale deviations of simulations not only in spatio-temporal trajectories but also even in statistics. In particular, these numerical noises (as artificial stochastic disturbances) even lead to different types of flows. The shearing convection occurs for the RKwD simulations, and its corresponding flow field turns to a kind of zonal flow thereafter; however, the CNS benchmark solution always sustains the non-shearing vortical/roll-like convection during the whole process of simulation. Thus we provide rigorous evidence that numerical noises as a kind of small-scale artificial stochastic disturbances have quantitatively and qualitatively large-scale influences on a sustained turbulence, i.e. the 2-D turbulent RBC considered in this paper.