Semi-Implicit Runge–Kutta Schemes: Development and Application to Compressible Channel Flow

Abstract

Two semi-implicit six-stage Runge–Kutta algorithms are developed for the simulation of wall-bounded flows. Using these schemes, time integration is implicit in the wall-normal direction, and explicit in the other directions, to relax the time step constraint due to the fine mesh near the wall. The explicit subscheme is a six-stage fourth-order low-storage Runge–Kutta scheme. Based on analysis in Fourier space and results obtained for propagation test cases, the semi-implicit schemes are shown to be of order 3 and, for waves discretized by a number of points per period between 4 and 16, to be as accurate as, or more accurate than, the standard explicit fourth-order Runge–Kutta algorithm in terms of dissipation and dispersion. The large-eddy simulation of a compressible turbulent channel flow at a friction Reynolds number of 360 and a Mach number of 0.1 is then carried out with one of the proposed algorithms. The computational time is reduced by a factor 1.33 with respect to a large-eddy simulation using the explicit subscheme in all directions. Wall-pressure and velocity spectra from the large-eddy simulation are presented to give insights into the flow turbulent structures. In particular, wave number–frequency spectra are calculated. Acoustic components appear to be identified in these spectra.