A fully conservative model for compressible two‐fluid flow (original) (raw)
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A novel five-equation model for inviscid, non-heat-conducting, compressible two-fluid flow is derived, together with an appropriate numerical method. The model uses flow equations based on conservation laws and exchange laws only. The two fluids exchange momentum and energy, for which source terms are derived from fundamental physical laws. The Riemann invariants of the governing equations are derived, and used in the construction of an Osher-type approximate Riemann solver. A consistent finite-volume discretization of the source terms is proposed. The source terms have distinct contributions in the cell domain and at the cell faces. For the source-term evaluation at the cell faces, the Riemann solver is elegantly exploited. Numerical results are presented for shock-tube and shock-bubble-interaction problems. The resemblance with experimental results is very good. Free-surface pressure oscillations do not occur, without any precaution. The paper contributes to state of the art in computing two-fluid flows.
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We consider a two-fluid model of two-phase compressible flows. First, we derive several forms of the model and of the equations of state. The governing equations in all the forms contain source terms representing the exchanges of momentum and energy between the two phases. These source terms cause unstability for standard numerical schemes. Using the above forms of equations of state, we construct a stable numerical approximation for this two-fluid model. That only the source terms cause the oscillations suggests us to minimize the effects of source terms by reducing their amount. By an algebraic operator, we transform the system to a new one which contains only one source term. Then, we discretize the source term by making use of stationary solutions. We also present many numerical tests to show that while standard numerical schemes give oscillations, our scheme is stable and numerically convergent.
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A simple, robust and accurate HLLC-type Riemann solver for two-phase 7-equation type models is built. It involves 4 waves per phase, i.e. the three conventional rightand left-facing and contact waves, augmented by an extra "interfacial" wave. Inspired by the Discrete Equations Method (Abgrall and Saurel, 2003), this wave speed (I u) is assumed function only of the piecewise constant initial data. Therefore it is computed easily from these initial states. The same is done for the interfacial pressure I P. Interfacial variables I u and I P are thus local constants in the Riemann problem. Thanks to this property there is no difficulty to express the nonconservative system of partial differential equations in local conservative form. With the conventional HLLC wave speed estimates and the extra interfacial speed I u , the four-waves Riemann problem for each phase is solved following the same strategy as in Toro et al. (1994) for the Euler equations. As I u and I P are functions only of the Riemann problem initial data, the two-phase Riemann problem consists in two independent Riemann problems with 4 waves only. Moreover, it is shown that these solvers are entropy producing. The method is easy to code and very robust. Its accuracy is validated against exact solutions as well as experimental data.
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The present paper is devoted to the computation of two-phase flows using the two-fluid approach. The overall model is hyperbolic and has no conservative form. No instantaneous local equilibrium between phases is assumed, which results in a two-velocity two-pressure model. Original closure laws for interfacial velocity and interfacial pressure are proposed. These closures allow to deal with discontinuous solutions such as shock waves and contact discontinuities without ambiguity with the definition of Rankine–Hugoniot jump relations. Each field of the convective system is investigated, providing maximum principle for the volume fraction and the positivity of densities and internal energies are ensured when focusing on the Riemann problem. Two-finite volume methods are presented, based on the Rusanov scheme and on an approximate Godunov scheme. Relaxation terms are taken into account using a fractional step method. Eventually, numerical tests illustrate the ability of both methods to ...
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ERCOFTAC Series, 2012
In many situations, the equations of state (EOS) found in the literature have only a limited range of validity. Besides, different types of EOS are required for different fluids of compressible multi-fluid flows. These inspire us to investigate compressible multi-fluid flows with different types of equation of state (EOS). In this paper, the oscillation-free adaptive method for compressible two-fluid flows with different types of equation of state (EOS) is proposed. By using a general form of EOS instead of solving the non-linear equation, the pressure of the mixture can be analytically calculated for compressible multi-fluid flows with different types of EOS. It is proved that it preserves the oscillation-free property across the interface. To capture the interface as fine as sharp interface, the quadrilateral-cell based adaptive mesh is employed. In this adaptive method, the cells with different levels are stored in different lists. This avoids the recursive calculation of solution of mother (non-leaf) cells. Moreover, the edges are separated stored into two lists for leaf edges and non-leaf edges respectively. Hence, there is no need to handle the handing nodes and no special treatment at the interface between the finer cell and the coarse cell. Thus, high efficiency is obtained due to these features. To examine its performance in solving the various compressible two-fluid flow problems with two different types of EOS, the interface translation and bubble shock interaction case with different types of EOS are employed. The results show that it can adaptively and accurately solve these problems and especially preserve the oscillation-free property of pressure and velocity across the material interface.
A Numerical Method for the Solution of Compressible and Incompressible Fluid Flows
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Numerical solution of Navier-Stokes equations for two-dimensional viscous compressible flows
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A Simple Method for Compressible Multi-uid Flows
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