An efficient semi-implicit compressible solver for large-eddy simulations

In most combustion devices, the Mach number is small, and the most efficient computational approach is the solving of the low-Mach number formulation of the Navier-Stokes equations. In this approach, the acoustic waves propagate at an infinite speed. Nevertheless, when the acoustic speed plays an important role, as in the formation of instabilities, the compressible Navier-Stokes equations must be solved. If these equations are solved explicitly, the acoustic CFL condition imposes a drastic limit on the time-step for flows of low convective CFL number. Then, a semi-implicit approach, in which acoustic waves are solved implicitly, potentially offers significant efficiency gains using larger time-steps. Implicit compressible solvers able to deal with low-Mach number flows fall into two basic categories. The first category consists of the density-based solvers, which are originally designed for high-Mach number flows. When the Mach number goes to zero, the algebraic system of these solvers becomes ill-conditioned (Turkel et al. 1997; Guillard & Viozat 1998). This issue is classically overcome by using preconditioning techniques (Turkel 1987) or by performing a Taylor-series expansion in Mach number (Choi & Merkle 1993). The second category is that of the pressure-based solvers. These solvers were originally developed for incompressible flows. In this case, the pressure gradient in the momentum equation acts as a source term needed to maintain the incompressibility constraint. This dynamic pressure comes from the solution of a Poisson equation. Using this type of approach ensures that in contrast to non-preconditioned density-based systems, the pressure variations remain finite in the low-Mach limit. Many pressure-based implicit compressible solvers have been designed since the pioneering work of Harlow & Amsden (1968). Unfortunately, these methods are either first-order in time (Karki & Patankar 1989), non-mass conservative (Yoon & Yabe 1999), or they necessitate a very costly inner loop to couple the energy (or enthalpy) equation to the other equations (Zienkiewicz et al. 1999; Wall et al. 2002). A suitable method for the computation of acoustic instabilities in a combustion device needs to be mass conservative and efficient, which precludes the use of an inner loop to converge the energy equation, and it has to have low-dissipation. In this paper, a method that fulfills these requirements is proposed. It is a fractional-step method (Kim & Moin 1985) based on a characteristic splitting. This method consists of an advection and a pressure-correction step. In the pressure-correction step, a Helmholtz equation is solved implicitly to remove the acoustic CFL condition. The characteristic splitting allows the decoupling of the acoustic waves from the advection, and a second-order spatial and temporal convergence for linear acoustics can be obtained without inner loop. When this method is used in combination with a kinetic-energy conserving scheme for the advection