A combined momentum-interpolation and advection upstream splitting pressure-correction algorithm for simulation of convective and acoustic transport at all levels of Mach number

Abstract

A pressure-correction algorithm is presented for compressible fluid flow regimes. It is well-suited to simulate flows at all levels of Mach number with smooth and discontinuous flow field changes, by providing a precise representation of convective transport and acoustic propagation. The co-located finite volume space discretization is used with the AUSM flux splitting. It is demonstrated that two ingredients are essential for obtaining good quality solutions: the presence of an inertia term in the face velocity expression; a velocity difference diffusive term in the face pressure expression, with a correct Mach number scaling to recover the hydrodynamic and acoustic low Mach number limits. To meet these two requirements, a new flux scheme, named MIAU, for Momentum Interpolation with Advection Upstream splitting is proposed.

Conservative finite-volume framework and pressure-based algorithm for flows of incompressible, ideal-gas and real-gas fluids at all speeds

“…A contact discontinuity is a linearly degenerate wave and represents the main source of error in terms of convergence of the applied finite-volume method under mesh refinement [81,86], with the contact discontinuity progressively smoothing over the course of the simulation [87,88]. To test the accuracy of the proposed finitevolume framework in predicting contact discontinuities, a moving contact discontinuity in a one-dimensional domain with a length of 1 m is simulated, as considered in previous studies [59,63]. The contact discontinuity is initially located at x 0 = 0.5 m, with the initial conditions of the left and right states given as Table 1 and the frequency given in Table 2.…”

“…As a first test, the propagation of acoustic waves in a one-dimensional domain is simulated. The formation and propagation of acoustic waves is an important feature of compressible flows and predicting acoustic waves reliably is known to be challenging [43,44,63,82]. In these simulations, the acoustic waves are generated at the domain inlet by a sinusoidal velocity perturbation with amplitude ∆u 0 .…”

“…When developing numerical methods for flows at all speeds, it is important to recognise the numerical implications associated with the flow speed U , represented by the Mach number M = U/a, where a = 1/(ρβ s ) is the speed of sound, and with the isentropic compressibility of the fluid, β s = {dρ/(ρ dp)} s , that relates changes in pressure p and density ρ of a fluid at constant entropy. While pressure and density are strongly coupled for large flow speeds (M > 0.1), in particular for supersonic flows (M > 1), the pressure-density coupling from an incompressible flow to the propagation of strong shock waves, stationary and high-speed discontinuous waves as well as the propagation of linear acoustic waves [54,56,59,63].…”

“…In order to improve the performance for low Mach number flows, recent work has been focusing on combining density-based methods with segregated pressurecorrection algorithms [54][55][56][57][58] and/or reformulating the energy equation as an equation for pressure [55,[59][60][61][62]. These density-based algorithms have been applied successfully to a wide range of flows, including flows ranging from an incompressible flow to the propagation of strong shock waves, stationary and high-speed discontinuous waves as well as the propagation of linear acoustic waves [54,56,59,63].…”

Numerical modelling of shock-bubble interactions using a pressure-based algorithm without Riemann solvers

“…Traditionally, tailored pre-conditioning techniques have been applied to extend density-based methods to low-Mach number flows (Turkel, 2006;Turkel et al, 1993), which are however computationally very expensive for transient problems. This has been motivating recent work on combining density-based methods with segregated pressure-correction algorithms (Fuster and Popinet, 2018;Moguen et al, 2019Moguen et al, , 2012Xiao, 2004) and hybrid density/pressurebased algorithms (Park and Munz, 2005;van der Heul et al, 2003), in which the continuity equation is solved for density but the energy equation is reformulated as an equation for pressure. An additional difficulty for interfacial flows associated with density-based methods is that the pressure field has to be reconstructed based on the applied thermodynamic closure model, which has proved to be a considerable difficulty in interfacial cells where two bulk phases coexist (Abgrall and Karni, 2001;Allaire et al, 2002;Murrone and Guillard, 2005).…”