HLLC-type Riemann solver with approximated two-phase contact for the computation of the Baer-Nunziato two-fluid model HLLC-type Riemann solver with approximated two-phase contact for the computation of the Baer-Nunziato two-fluid model (original) (raw)
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A simple HLLC-type Riemann solver for compressible non-equilibrium two-phase flows
Computers & Fluids, 2015
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.
Numerical approximation for a Baer–Nunziato model of two-phase flows
Applied Numerical Mathematics, 2011
We present a well-balanced numerical scheme for approximating the solution of the Baer-Nunziato model of two-phase flows by balancing the source terms and discretizing the compaction dynamics equation. First, the system is transformed into a new one of three subsystems: the first subsystem consists of the balance laws in the gas phase, the second subsystem consists of the conservation law of the mass in the solid phase and the conservation law of the momentum of the mixture, and the compaction dynamic equation is considered as the third subsystem. In the first subsystem, stationary waves are used to build up a well-balanced scheme which can capture equilibrium states. The second subsystem is of conservative form and thus can be numerically treated in a standard way. For the third subsystem, the fact that the solid velocity is constant across the solid contact suggests us to compose the technique of the Engquist-Osher scheme. We show that our scheme is capable of capturing exactly equilibrium states. Moreover, numerical tests show the convergence of approximate solutions to the exact solution.
On the HLLC Riemann solver for interface interaction in compressible multi-fluid flow
Journal of Computational Physics, 2009
In this work, the HLLC Riemann solver, which is much more robust, simpler and faster than iterative Riemann solvers, is extended to obtain interface conditions in sharp-interface methods for compressible multi-fluid flows. For interactions with general equations of state and material interfaces, a new generalized Roe average is proposed. For single-phase interactions, this new Roe average does not introduce artificial states and satisfies the Uproperty exactly. For interactions at material interfaces, the U-property is satisfied by introducing ghost states for the internal energy. A number of numerical tests suggest that the proposed Riemann solver is suitable for general equations of state and has an accuracy comparable to iterative Riemann solvers, while being significantly more robust and efficient.
Journal of Computational Physics, 2020
A new Riemann solver is built to address numerical resolution of complex flow models. The research direction is closely linked to a variant of the Baer and Nunziato (1986) model developed in Saurel et al. (2017a). This recent model provides a link between the Marble (1963) model for two-phase dilute suspensions and dense mixtures. As in the Marble model, Saurel et al. system is weakly hyperbolic with the same 4 characteristic waves, while the system involves 7 partial differential equations. It poses serious theoretical and practical issues to built simple and accurate flow solver. To overcome related difficulties the Riemann solver of Linde (2002) is revisited. The method is first examined in the simplified context of compressible Euler equations. Physical considerations are introduced in the solver improving robustness and accuracy of the Linde method. With these modifications the flow solver appears as accurate as the HLLC solver of Toro et al. (1994). Second the two-phase flow model is considered. A locally conservative formulation is built and validated removing issues related to nonconservative terms. However, two extra major issues appear from numerical experiments: The solution appears not self-similar and multiple contact waves appear in the dispersed phase. Building HLLC-type or any other solver appears consequently challenging. The modified Linde (2002) method is thus examined for the considered flow model. Some basic properties of the equations are used, such as shock relations of the dispersed phase and jump conditions across the contact wave. Thanks to these ingredients the new Riemann solver with internal reconstruction (RSIR), modification of the Linde method, handles stationary volume fraction discontinuities, presents low dissipation for transport waves and handles shocks and expansion waves accurately. It is validated on various test problems showing method's accuracy and versatility for complex flow models. Its capabilities are illustrated on a difficult two-phase flow instability problem, unresolved before.
Nuclear Engineering and Design, 2019
A new method is proposed to solve the two-phase two-fluid six-equation model. A Roe-type numerical flux is formulated based on a very structured Jacobian matrix. The Jacobian matrix with arbitrary equation of state is formulated and simplified with the help of a few auxiliary variables, e.g. isentropic speed of sound. Because the Jacobian matrix is very structured, the eigenvalue and eigenvector can be obtained analytically. An explicit Roetype numerical solver is constructed based on the analytical eigenvalue and eigenvector. A critical feature of the method is that the formulation of the solver does not depend on the form of the equation of state. The proposed method is applicable to realistic two-phase problems. It is applied to the BWR Full-size Fine-mesh Bundle Test (BFBT) benchmark. Considering simplified physical models, the solutions are in very good agreement with those from both existing codes and experiment data. The numerical solver using analytical eigenvalue and eigenvector is shown to be stable and robust.
Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers
Numerical Methods for Partial Differential Equations, 2002
We report here on our numerical study of the two-dimensional Riemann problem for the compressible Euler equations. Compared with the relatively simple 1-D configurations, the 2-D case consists of a plethora of geometric wave patterns that pose a computational challenge for high-resolution methods. The main feature in the present computations of these 2-D waves is the use of the Riemann-solvers-free central schemes presented by Kurganov et al. This family of central schemes avoids the intricate and timeconsuming computation of the eigensystem of the problem and hence offers a considerably simpler alternative to upwind methods. The numerical results illustrate that despite their simplicity, the central schemes are able to recover with comparable high resolution, the various features observed in the earlier, more expensive computations.
A New Approximate Riemann Solver Applied to HLLC Method
Proceedings of 10th World Congress on Computational Mechanics, 2014
In solving Euler equations applying finite volume techniques, the calculations of numerical fluxes across cell interfaces, have become an essential item. The numerical scheme exactitude, the ability of handle discontinuities and the correct prediction of the propagating waves velocity, are strongly dependent on such numerical fluxes. The pioneer work of Godunov [1] was the starting point to solve the Euler equations by means of Riemann solvers. The excellent results obtained with Godunov technique, motivated several lines of research with the purpose of extending it to three dimensional flows, to achieve higher order of accuracy, etc. All calculating schemes that incorporate Riemann solvers are very precise, but unfortunately, computational demands are intense because of the non linear system of algebraic equations which must be solved in an iterative manner. An alternative which will demand less computational effort, could be provided by the use of approximate Riemann solvers, although less accurate and also, less robust. In this paper, an approximate Riemann solver which does not require iterations, possesses a high degree of accuracy and a lower computational demand in solving the Euler equations, is described. It is based on the use of dimensional analysis to reduce the number of independent variables needed to outline the physics of the problem. The scheme here presented is compared in accuracy as well as in computational effort with an exact iterative solver and with three well known approximated solvers: the Two Rarefactions Riemann Solver, the Two Shocks Riemann Solver, and an Adaptive version of these two. Substantially smaller mean errors have been found with the approximation here presented than those found with the best of all the above mentioned approximated solvers. Finally, a finite volume computer code to solve one-dimensional Euler equations using the Harten, Lax and van Leer Contact (HLLC) scheme, was developed. Results obtained solving the Shock Tube problem with the HLLC scheme, have shown no significant differences in accuracy and robustness when either the new approximate Riemann solver or the exact solver, are used. From the point of view of computer resources, the new approximate solver offers advantages.
Numerical Modeling of Two-Phase Flows Using the Two-Fluid Two-Pressure Approach
Mathematical Models and Methods in Applied Sciences, 2004
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 ...
A Non Homogeneous Riemann Solver for Two-phase Shallow Water Flows
The purpose of this research is to develop a simulation method for two-phase flows using shallow water equations. The hydraulics is modeled by the two-dimensional shallow water flows with variable horizontal density. The variation of density in the water flows can be attributed to the variation of thermal and salinity properties of the water. As an example of two-phase shallow water flows is the inclusion of the salty water from the sea into the fresh water of a river. Driving force of the phase separation and the mixing is the gradient of the density. For the numerical solution procedure we propose a non-homogeneous Riemann solver in the finite volume framework. The proposed method consists of a predictor stage for the discretization of gradient terms and a corrector stage for the treatment of source terms. The gradient fluxes are discretized using a modified Roe's scheme using the sign of the Jacobian matrix in the coupled system. A well-balanced discretization is used for the treatment of source terms. The efficiency of the solver is evaluated by several test problems for two-phase shallow water flows. The numerical results demonstrate high resolution of the proposed non-homogeneous Riemann solver and confirm its capability to provide accurate simulations for two-phase shallow water equations under flow regimes with strong shocks.
Computers & Fluids, 2014
Computation of compressible two-phase flows with single-pressure single-velocity two-phase models in conjunction with the moving grid approach is discussed in this paper. A HLLC-type scheme is presented and implemented in the context of Arbitrary Lagrangian-Eulerian formulation for solving the fiveequation models. In addition, the extension to multicomponent cases is also examined. The method is first assessed on a variety of Riemann problems including both fixed and moving grids applications showing its simplicity and robustness. The method is also tested on 2-D moving mesh applications including fluid-structure interactions. The heat and mass transfer modeling is finally examined for two-phase mixtures. Computations using a fractional step approach of water hammer and fast depressurization with flashing are performed. Good agreement is obtained with available experimental data. All computations are performed with the Europlexus fast transient dynamics software.