Anti-diffusion interface sharpening technique for two-phase compressible flow simulations
Introduction
The impact of a shock wave on two-phase compressible flows is a fundamental topic in science and engineering. To better understand the instability phenomena that are important for the evolution of such flows, basic configurations, such as shock-bubble interactions [26] in two-phase compressible flows, are studied to investigate the Richtmyer–Meshkov instability [4], the Kelvin–Helmholtz instability and the mixing mechanisms in shock-accelerated flows [40]. Flows of this type are present in many engineering applications including supersonic mixing and combustion systems [43] and extra-corporeal shock-wave lithotripsy [26]. Two-phase compressible flows interacting with a shock wave have been studied extensively during the last decades, where in particular detailed numerical simulations help to elucidate the essential phenomena and interaction mechanisms [24], [29], [43], [31], [21].
In simulations the phase interface location is tracked explicitly or captured implicitly, where the approaches are front-tracking methods [39], level-set methods [7] and volume-of-fluid (VOF) methods [8]. Front-tracking methods follow explicitly the interface by means of marker points. Level-set methods capture the interface position by the zero-contour of the level-set function. VOF methods use the volume fraction to locate one phase in another phase. The interface position is captured by a diffused-interface representation, or located by an interface reconstruction algorithm. Different numerical methods were developed for different interface representation methods: among others the front-tracking/ghost-fluid method [39] for front-tracking representations; discontinuous Galerkin methods [44] or weighted essentially non-oscillatory (WENO) schemes [7], [13], [11] for level-set representations; volume tracking by means of an interface reconstruction [35], interface-compression method [33], and the anti-diffusive scheme [16] for VOF representations. Methods also worth noting for two-phase compressible flow simulations include the localized artificial diffusivity method for mass-fraction representations [38], [15], the stratified flow model and AUSM+-up scheme [5], the γ-model [1], [12], [20], and the arbitrary Lagrangian–Eulerian scheme which involves a grid evolution [18].
The VOF volume-capturing method possesses the advantage of exact conservation properties, but suffers from numerical diffusion which causes two-fluid interfaces to smear. Specific numerical schemes to suppress or counter-act the numerical diffusion, and to maintain the interface sharpness in the course of simulations are thus desirable and essential for VOF methods. Previous works on maintaining a sharp interface without an interface reconstruction include the interface compression technique by [33] which originates from the interface compression technique for two-phase incompressible flows by [23], and the anti-diffusive numerical scheme based on a limited downwind strategy where stability and consistency criteria are proposed to make use of the downwind contribution [16].
In this paper, we propose an interface sharpening technique for two-phase compressible flow simulations based on VOF methods. The idea of sharpening the two-fluid interface is to provide a correction algorithm which can be applied as post-processing to the volume-fraction field after each time step. For such a purpose, an anti-diffusion equation is solved for counter-acting the numerical diffusion resulting from the underlying discretization. The interface-sharpening technique is modular and can be applied to general underlying VOF discretizations. No interface reconstruction is required to track the interface position. The technique originally has been developed and verified for two-phase incompressible flow simulations [37]. Compressible flows pose the particular challenge of ensuring consistency among the flow variables. It is the objective of this paper to present a further development of the method for two-phase compressible flows with numerical validation results and some applications as feasibility demonstration.
The paper is organized as follows. First, the governing equations for two-phase compressible flows adopted in the paper are described. Second, the underlying Riemann solver, the numerical method for the volume-fraction transport equation and in particular the numerical method for interface sharpening by solving an anti-diffusion equation are detailed. Special consideration is given to modifications of the incompressible formulation [37] necessary for compressible flows. Validation cases for different equations of state (EOS) and in 1 and 2 dimensions are presented to illustrate the improvement obtained by the interface sharpening method. Finally, an application of the method to a complex interaction problem is given for illustration.
Section snippets
Governing flow equations
Various mathematical models for two-phase compressible flow simulations have been developed with different sets of governing equations [14], [32], [17]. In this paper we consider a basic conservative formulation of the Euler equations assuming a single velocity and pressure equilibrium. The two phases are represented by the respective volume fractions, where the formulation of the volume-fraction transport equations of [8] is adopted. This volume-fraction transport equation formulation has been
Riemann solver
We point out that the particular choice of Riemann solver serves as example for using the interface-sharpening method which can be formulated for other Riemann solvers in a straight-forward way.
The HLL Riemann solver [41] is adopted for calculating the numerical flux at cell face, FHLL,where U = (α, β, αρα, βρβ, ρu, ρE) is the vector of the cell-averaged conserved variables, F is the cell-average flux, S is the bound of the fastest signal
Numerical results
First, a 1-dimensional air-helium shock-tube problem and a 1-dimensional molybdenum-MORB shock-contact problem are considered to verify the interface sharpening method with the ideal-gas EOS and the Mie–Grüneisen EOS respectively. Then, 2-dimensional shock-bubble interactions based on the experiments of [9], and a 2-dimensional shock-contact problem are considered to verify the interface sharpening method in multiple-dimensions, and to illustrate the small-scale interface structures recovered
Concluding remarks
In this paper an interface sharpening method based on solving an anti-diffusion equation is presented for two-phase compressible flow simulations. A conservative formulation of the Euler equations with the volume-fraction equations of [8] which are capable of simulations with the ideal-gas EOS and the Mie–Grüneisen EOS is employed as the flow governing equations. The HLL Riemann solver and the numerical method for solving the volume-fraction transport equations are described as example for the
Acknowledgment
The first author gratefully acknowledges the support by the TUM IGSSE (Technische Universität München International Graduate School of Science and Engineering) for this work within the Project 3.01.
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