On the use of moving least squares for pressure discretization in low mach number flows

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Traditionally, two families of finite volume schemes have been developed to compute both compressible and incompressible flows. Thus, density-based solvers are used for the computation of flows when compressibility effects are important (mainly transonic, supersonic and hypersonic flows), whereas pressure-based solvers are designed to compute incompressible flows. However, it is preferable the development of solvers useful for all the regimes of a flow, this is not only for user’s convenience, but also because the importance of flows where low and high Mach regions are present (for example flow past an aerodynamic profile at high angle of attack), or when compressibility effects are important, even in low Mach number flows. Thus, the modification of density or pressure-based solvers to compute all-speed flows is a current active area of research. Godunov-like schemes are among the most widely used methods for CFD. When Godunov schemes are used in a density-based solver for low-Mach number computations, it is needed a central discretization for the pressure in order to obtain the adequate scaling of pressure with the square of the Mach number. However, the most widely used numerical schemes lead many times to non-physical solutions. Moreover many of the pressure discretization techniques used by the schemes developed for low Mach number, allow the existence of four-field solutions for the pressure (checkerboard) . In this work we present a modification in the pressure discretization of low-Mach numerical schemes. We propose using Moving-Least Squares (MLS) approximations to the discretization of the pressure flux in the numerical schemes developed for low-Mach number flows. This simple modification avoids all the problems related with checkerboard and it obtains a very accurate representation of the pressure field. The centered character of MLS approximations assures the correct scaling of the pressure with the square of the Mach number [1, 2, 3]. The high-order is achieved using MLS for the computation of the derivatives in the reconstruction step of a finite volume scheme [4]. We present here the results of the application of the proposed pressure flux discretization to the AUSM-family schemes (in particular we test the cases of the AUSM+-up [5] and SLAU [6] schemes).