Time integration of variable-density flows in the low-Mach limit: Convergence and stability for buoyancy and shear-generated turbulence

The low-Mach limit presents unique numerical challenges in flows where the density is variable in time and space, such as turbulent reactive flows and buoyant turbulence. An iterative time-integration scheme, combined with a pressure projection step, has been shown to improve numerical stability while avoiding the costly solution of an implicit system of equations. In the past, the authors have conducted tests of such a scheme to determine the effects of numerical parameters on stability and convergence properties. However, these tests were limited to shear-driven turbulence, and a comprehensive parameter sweep has yet to be completed and analyzed. To this end, direct numerical simulations are conducted for both a variable-density shear mixing layer and reduced-order configurations of the Rayleigh-Taylor instability while sweeping through the parameters of an iterative time integration scheme. These parameters include: the Courant-Friedrichs-Lewy number, whether a semi-Lagrangian or Eulerian scheme is used to transport the scalar, the guesses used to initialize the subiterations at each timestep, and whether the equation of state or the scalar transport equation is exactly satisfied at the end of each subiteration. Using these results, optimal sets of parameters are determined, for both shear and buoyancy-generated turbulence, to minimize conservation errors and instabilities while maximizing the convergence rate.