An Efficient Algorithm for the Multicomponent Compressible Navier-Stokes Equations in Low and High Mach Number Regimes

The goal of this study is to develop an efficient numerical algorithm applicable to a wide range of compressible multicomponent flows, including nearly incompressible low Mach number flows, flows with strong shocks, multicomponent flows with high density ratios and interfacial physics, inviscid and viscous flows, as well as flows featuring combinations of these phenomena and various interactions between them. Although many highly efficient algorithms have been proposed for simulating each type of the flows mentioned above, a construction of universal solvers is known to be challenging. Extreme cases, such as incompressible and highly compressible flows, or inviscid and viscous flows, require different numerical treatments in order to maintain efficiency, stability, and accuracy of the method. Linearized block implicit (LBI) factored schemes (see e.g. [1], [2]) are known to provide an efficient way of solving the compressible Navier-Stokes equations implicitly, which allows avoiding stability restrictions at low Mach number and high viscosity. However, their splitting error has been shown to grow as the Mach number goes to zero and dominate physical fluxes (see [3]). In this paper, a splitting error reduction technique is proposed to solve the issue. A shock-capturing algorithm from [4] is reformulated in terms of finite differences, extended to the stiffened gas equation of state (SG EOS) and combined with the LBI factored scheme to stabilize the method around flow discontinuities at high Mach numbers. A novel stabilization term is proposed for low Mach number applications. The resulting algorithm is shown to be efficient in both low and high Mach number regimes. The algorithm is further extended to the multicomponent case using an interface capturing strategy (see e.g. [5]) with surface tension as a continuous surface force (see [6]). Special care is taken to avoid spurious oscillations of pressure and generation of artificial acoustic waves in the numerical mixture layer by including specific functions of the EOS parameters as flow variables. Numerical tests are presented to verify the performance and stability properties for a wide range of flows.