In this paper we present numerical techniques suitable for a direct numerical simulation in the spatial setting. We demonstrate the application to the simulation of compressible flat plate flow instabilities. We compare second and fourth order accurate spatial discretization schemes in combination with explicit multistage time stepping for the simulation of the 2D Navier-Stokes equations. We consider Mach numbers 0.5 and 4.5. In the vicinity of the outflow boundary, an efficient buffer domain treatment is introduced, which is suitable in conjunction with an explicit time integration scheme. This treatment requires only a short buffer domain to damp wave reflections at the outflow boundary. Results for the instability of Tollmien-Schlichting (T-S) waves are compared with two instability theories, linear stability theory (LST) and linear parabolized stability equations (PSE). The growth rates of T-S waves for parallel base flow at both Mach numbers compare well with LST results. Moreover, the growth rates of T-S waves for nonparallel base flow compare well with results obtained by solving the PSE at Mach number 0.5. The second order discretization scheme requires, however, considerably higher grid resolution than the fourth order method to achieve accurate results. High amplitude disturbances were also considered to activate nonlinear terms. The nonlinearity strongly affects the form of the T-S waves and the growth rate of the disturbances. The results obtained here support the use of these numerical techniques in flow simulations with increasing complexity such as flat plate flow simulations up to the turbulent regime and with separation regions in 3D. The results also encourage the use of perturbations derived from the compressible PSE as inlet perturbations for nonparallel flow.
Wasistho, B., Geurts, B. J., & Kuerten, J. G. M. (1997). Simulation techniques for spatially evolving instabilities in compressible flow over a flat plate. Computers and fluids, 26(26), 713-739. https://doi.org/10.1016/S0045-7930(97)00021-2