Direct numerical simulation of turbulent spots

A group of high-order finite-difference schemes for incompressible flow was implemented to simulate the evolution of turbulent spots in channel flows. The long-time accuracy of these schemes was tested by comparing the evolut ion of small disturbances to a plane channel flow against the growth rate p redicted by linear theory. When the perturbation is the unstable eigenfunct ion at a Reynolds number of 7500, the solution grows only if there are a co mparatively large number of (equispaced) grid points across the channel. Fi fth-order upwind biasing of convection terms is found to be worse than seco nd-order central differencing. But, for a decaying mode at a Reynolds numbe r of 1000, about a fourth of the points suffice to obtain the correct decay rate. We show that this is due to the comparatively high gradients in the unstable eigenfunction near the walls. So, high-wave-number dissipation of the high-order upwind biasing degrades the solution especially. But for a w ell-resolved calculation, the weak dissipation does not degrade solutions e ven over the very long times (O(100)) computed in these tests. Some new sol utions of spot evolution in Couette flows with pressure gradients are prese nted. The approach to self-similarity at long times can be seen readily in contour plots. (C) 2001 Elsevier Science Ltd. All rights reserved.