Numerical stability and convergence of iterative low-Mach variable density time-integration schemes

Low-Mach variable-density flows are frequently encountered in turbulent reactive flows and buoyant turbulence. The stiffness of the low-Mach governing equations poses challenges for numerical stability, especially when an equation of state (EOS) relating the density and transported scalars must be satisfied. A common solution is to use an iterative time integration scheme with a pressure projection step. A computational study is performed to determine the numerical stability and convergence in two different iterative schemes: one where the EOS and scalar boundedness are satisfied but the primary conservation of the scalar is affected by iteration error, and one where the primary conservation is satisfied but the EOS and scalar boundedness are affected instead. First, the 1D governing equations are used to derive approximate analytical expressions for the convergence rates of both schemes for a binary mixture of light and heavy fluids in terms of global flow variables. This expression is confirmed in 1D direct numerical simulations (DNS). The expressions indicate lower convergence rates for the EOS-violating scheme at similar Courant-Friedrichs-Lewy (CFL) number. In 3D DNS of a turbulent planar jet of light fluid through heavy fluid, the EOS-violating case is shown to be more unstable and slower converging than the conservation-violating case, also at similar CFL.