High order nonlinear filter methods for subsonic turbulence simulation with stochastic forcing

Abstract

Numerical simulations of forced turbulence in compressible fluids are challenging due to the multi-scale nature of the problem and conflicting requirements for numerical methods to accurately resolve the small scales and, at the same time, to handle shock waves and other discontinuities without generating spurious oscillations. Minimizing nonlinear instability and aliasing error while maintaining high accuracy before the simulation reaches the statistically stationary stage is computationally intensive. The goal of this work is to employ an efficient class of high order finite difference nonlinear filter methods for subsonic turbulence simulation with stochastic forcing. The 3D Euler equations for subsonic turbulence with temporally varying stochastic forcing at rms Mach numbers up to 0.6 are numerically solved using the Strang operator splitting of the homogeneous part of the Euler equations and the forcing terms. It was shown in Yee et al. (2013) that the Strang operator splitting is more stable than solving the full Euler equation with the source term included. The spatially seventh-order nonlinear filter methods with adaptive dissipation control developed by Yee & Sjögreen (2007, 2011) are used to solve the homogeneous system and an ODE solver is used to solve the forcing source terms. The nonlinear filter method includes a full time step of a spatially eighth-order central base method, using a third-order TVD Runge-Kutta time integration. The solution computed with the central base method is then nonlinearly filtered by an adaptive flow sensor and the dissipative portion of a seventh-order WENO with the Roe Riemann solver. In order to improve nonlinear stability of the base method without added numerical dissipation, the central base method discretizes the skew-symmetric split form of the inviscid flux derivatives. Both Ducros et al. and Kennedy-Gruber skew-symmetric split forms are tested. Numerical stability, computational efficiency, and effective spectral bandwidth of the nonlinear filter schemes are compared with those of second-order TVD and fifth- and seventh-order WENO methods. It is shown that the nonlinear filter method for this application is substantially more efficient, accurate and yields a superior spectral bandwidth compared to the standard TVD and WENO methods. The nonlinear filter method also demonstrates robust long-time integration for moderately compressible, statistically stationary turbulence with large-scale solenoidal forcing, including small-scale quantities such as enstrophy and mean-square dilatation.