An Efficient Semi-implicit Solver for Direct Numerical Simulation of Compressible Flows at All Speeds

We develop a semi-implicit algorithm for time-accurate simulation of compressible shock-free flows, with special reference to wall-bounded flows. The method is based on partial linearization of the convective fluxes in such a way to suppress, or at least mitigate the acoustic time step limitation. Together with replacement of the total energy equation with the entropy transport equation, this approach avoids the inversion of block-banded matrices involved in classical methods, which is replaced by much less demanding inversion of standard banded matrices. The method is extended to deal with implicit integration of viscous terms and to multiple space dimensions through approximate factorization, and used as a building block of a semi-implicit Runge–Kutta scheme which guarantees third-order of accuracy in time (Nikitin in Int J Numer Methods Fluids 51:221–233, 2006). Numerical experiments are carried out for isotropic turbulence, plane channel flow, and flow in a square duct. All available data support higher computational efficiency than existing methods, and saving of computer resources ranging from 85% under low-subsonic flow conditions (down to M0∼0.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_0 \sim 0.1$$\end{document}), to about 50% in supersonic flow.