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The purpose of this paper is to investigate the two-phase flow problems involving gas–liquid mixture.

The governed equations consist of a range of conservation laws modeling a classification of two-phase flow phenomena subjected to a velocity nonequilibrium for the gas–liquid mixture. Effects of the relative velocity are accounted for in the present model by a kinetic constitutive relation coupled to a collection of specific equations governing mass and volume fractions for the gas phase. Unlike many two-phase models, the considered system is fully hyperbolic and fully conservative. The suggested relaxation approach switches a nonlinear hyperbolic system into a semilinear model that includes a source relaxation term and characteristic linear properties. Notably, this model can be solved numerically without the use of Riemann solvers or linear iterations. For accurate time integration, a high-resolution spatial reconstruction and a Runge–Kutta scheme with decreasing total variation are used to discretize the relaxation system.

The method is used in addressing various nonequilibrium two-phase flow problems, accompanied by a comparative study of different reconstructions. The numerical results demonstrate the suggested relaxation method’s high-resolution capabilities, affirming its proficiency in delivering accurate simulations for flow regimes characterized by strong shocks.

While relaxation methods exhibit notable performance and competitive features, as far as we are aware, there has been no endeavor to address nonequilibrium two-phase flow problems using these methods.

Partially‐decoupled upwind‐based total‐variation‐diminishing (TVD) finite‐difference schemes for the solution of the conservation laws governing two‐dimensional non‐equilibrium vibrationally relaxing and chemically reacting flows of thermally‐perfect gaseous mixtures are presented. In these methods, a novel partially‐decoupled flux‐difference splitting approach is adopted. The fluid conservation laws and species concentration and vibrational energy equations are decoupled by means of a frozen flow approximation. The resulting partially‐decoupled gas‐dynamic and thermodynamic subsystems are then solved alternately in a lagged manner within a time marching procedure, thereby providing explicit coupling between the two equation sets. Both time‐split semi‐implicit and factored implicit flux‐limited TVD upwind schemes are described. The semi‐implicit formulation is more appropriate for unsteady applications whereas the factored implicit form is useful for obtaining steady‐state solutions. Extensions of Roe's approximate Riemann solvers, giving the eigenvalues and eigenvectors of the fully coupled systems, are used to evaluate the numerical flux functions. Additional modifications to the Riemann solutions are also described which ensure that the approximate solutions are not aphysical. The proposed partially‐decoupled methods are shown to have several computational advantages over chemistry‐split and fully coupled techniques. Furthermore, numerical results for single, complex, and double Mach reflection flows, as well as corner‐expansion and blunt‐body flows, using a five‐species four‐temperature model for air demonstrate the capabilities of the methods.

This paper aims to numerically study the compositional flow of two- and three-phase fluids in one-dimensional porous media and to make a comparison between several upwind and central numerical schemes.

Implicit pressure explicit composition (IMPEC) procedure is used for discretization of governing equations. The pressure equation is solved implicitly, whereas the mass conservation equations are solved explicitly using different upwind (UPW) and central (CEN) numerical schemes. These include classical upwind (UPW-CLS), flux-based decomposition upwind (UPW-FLX), variable-based decomposition upwind (UPW-VAR), Roe’s upwind (UPW-ROE), local Lax–Friedrichs (CEN-LLF), dominant wave (CEN-DW), Harten–Lax–van Leer (HLL) and newly proposed modified dominant wave (CEN-MDW) schemes. To achieve higher resolution, high-order data generated by either monotone upstream-centered schemes for conservation laws (MUSCL) or weighted essentially non-oscillatory (WENO) reconstructions are used.

It was found that the new CEN-MDW scheme can accurately solve multiphase compositional flow equations. This scheme uses most of the information in flux function while it has a moderate computational cost as a consequence of using simple algebraic formula for the wave speed approximation. Moreover, numerically calculated wave structure is shown to be used as a tool for a priori estimation of problematic regions, i.e. degenerate, umbilic and elliptic points, which require applying correction procedures to produce physically acceptable (entropy) solutions.

This paper is concerned with one-dimensional study of compositional two- and three-phase flows in porous media. Temperature is assumed constant and the physical model accounts for miscibility and compressibility of fluids, whereas gravity and capillary effects are neglected.

The proposed numerical scheme can be efficiently used for solving two- and three-phase compositional flows in porous media with a low computational cost which is especially useful when the number of chemical species increases.

A new central scheme is proposed that leads to improved accuracy and computational efficiency. Moreover, to the best of authors knowledge, this is the first time that the wave structure of compositional model is investigated numerically to determine the problematic situations during numerical solution and adopt appropriate correction techniques.

A new approach for solving steady incompressible Navier‐Stokes equations is presented in this paper. This method extends the upwind Riemann‐problem‐based techniques to viscous flows, which is obtained by applying modified artificial compressibility Navier‐Stokes equations and fully discrete high‐order numerical schemes for systems of advection‐diffusion equations. In this approach, utilizing the local Riemann solutions the steady incompressible viscous flows can be solved in a similar way to that of inviscid hyperbolic conservation laws. Numerical experiments on the driven cavity problem indicate that this approach can give satisfactory solutions.

The purpose of this work is to present the implementation of weighted essentially non-oscillatory (WENO) wavelet methods for solving multiphase flow problems. The particular interest is gas–liquid two-phase mixture with velocity non-equilibrium. Numerical simulations are carried out on different scenarios of one-dimensional Riemann problems for gas–liquid flows. Results are validated and qualitatively compared with solutions provided by other standard numerical methods.

This paper extends the framework of WENO wavelet adaptive method to a fully hyperbolic two-phase flow model in a conservative form. The grid adaptivity in each time step is provided by the application of a thresholded interpolating wavelet transform. This facilitates the construction of a small yet effective sparse point representation of the solution. The method of Lax–Friedrich flux splitting is used to resolve the spatial operator in which the flux derivatives are approximated by the WENO scheme.

Hyperbolic models of two-phase flow in conservative form are efficiently solved, as shocks and rarefaction waves are precisely captured by the chosen methodology. Substantial computational gains are obtained through the grid reduction feature while maintaining the quality of the solutions. The results indicate that WENO wavelet methods are robust and sufficient to accurately simulate gas–liquid mixtures.

Resolution of two-phase flows is rarely studied using WENO wavelet methods. It is the first time such a study on the relative velocity is reported in two-phase flows using such methods.

We present results of a mathematical model for the gas flow in an internal combustion engine consisting of a single cylinder with an inlet and outlet pipe. In order to achieve optimal performance of the engine the dependence of the gas flow on physical parameters such as pipe dimensions and valve geometry need to be understood. A system of ordinary differential equations (in time t) with discontinuous right‐hand side describes the gas properties in the cylinder, whereas the gas flow in each pipe is modelled by the Euler equations, a system of hyperbolic partial differential equations. The explicit method of Euler and a TVD scheme are used for solving these equations. However, since the coupling of the pipe equations with the o.d.e. system in the cylinder on one side and atmospheric gas properties on the other appeared to be a main problem, we concentrate on appropriate coupling conditions. The numerical techniques involve discretization in space and time, and we present different methods of discrete coupling. As a main result we show that the various coupling methods lead to quite different numerical solutions. Therefore, a careful treatment of the coupling conditions is crucial.

The purpose of this paper is to present a new stabilised low-order finite element methodology for large strain fast dynamics.

The numerical technique describing the motion is formulated upon the mixed set of first-order hyperbolic conservation laws already presented by Lee et al. (2013) where the main variables are the linear momentum, the deformation gradient tensor and the total energy. The mixed formulation is discretised using the standard explicit two-step Taylor-Galerkin (2TG) approach, which has been successfully employed in computational fluid dynamics (CFD). Unfortunately, the results display non-physical spurious (or hourglassing) modes, leading to the breakdown of the numerical scheme. For this reason, the 2TG methodology is further improved by means of two ingredients, namely a curl-free projection of the deformation gradient tensor and the inclusion of an additional stiffness stabilisation term.

A series of numerical examples are carried out drawing key comparisons between the proposed formulation and some other recently published numerical techniques.

Both velocities (or displacements) and stresses display the same rate of convergence, which proves ideal in the case of industrial applications where low-order discretisations tend to be preferred. The enhancements introduced in this paper enable the use of linear triangular (or bilinear quadrilateral) elements in two dimensional nearly incompressible dynamics applications without locking difficulties. In addition, an artificial viscosity term has been added into the formulation to eliminate the appearance of spurious oscillations in the vicinity of sharp spatial gradients induced by shocks.

A mathematical gas dynamic model for semiconductor devices is numerically analysed. The well‐known ballistic diode problem is taken as an example.

The discontinuous Galerkin finite element method (DGFEM) is very suited for realizing high order resolution approximations on unstructured grids for calculating the hyperbolic conservation law. However, it requires a significant amount of computing resources. Therefore, this paper aims to investigate how to solve the Euler equations in parallel systems and improve the parallel performance.

Discontinuous Galerkin discretization is used for the compressible inviscid Euler equations. The multi-level domain decomposition strategy was used to deal with the computational grids and ensure the calculation load balancing. The total variation diminishing (TVD) Runge–Kutta (RK) scheme coupled with the multigrid strategy was employed to further improve parallel efficiency. Moreover, the Newton Block Gauss–Seidel (GS) method was adopted to accelerate convergence and improve the iteration efficiency.

Numerical experiments were implemented for the compressible inviscid flow problems around NACA0012 airfoil, over M6 wing and DLR-F6 configuration. The parallel acceleration is near to a linear convergence. The results indicate that the present parallel algorithm can reduce computational time significantly and allocate memory reasonably, which has high parallel efficiency and speedup, and it is well-suited to large-scale scientific computational problems on multiple instruction stream multiple data stream model.

The parallel DGFEM coupled with TVD RK and the Newton Block GS methods was presented for hyperbolic conservation law on unstructured meshes.

In this paper, the flow of multiphase fluids in a one-dimensional homogeneous porous media involving the gravity effects is numerically studied using the dominant wave method. The paper aims to discuss these issues.

The numerical scheme used for solving the pressure equations, obtained for the black-oil model, is a backward Euler scheme while the hyperbolic mass conservation equations, derived for both black-oil and Buckley-Leverett models, are solved using the dominant wave method. Higher-order schemes are achieved using either variable derivatives along with the minmod limiter or a MUSCL type interface construction scheme using the Fromm's limiter. The mass conservation equations are solved using the first-order forward Euler method in time. Harten's entropy correction procedure is employed to avoid non-physical expansion shocks.

It was found that the dominant wave method can accurately solve multiphase flow equations involving gravity effects. Numerical experiments also show that both minmod and Fromm's limiters can be successfully used to construct higher-order schemes while the minmod limiter gives slightly more diffuse solutions.

The flow models considered here include two- and three-phase Buckley-Leverett and the black-oil models and the capillary effects are neglected.

The proposed scheme can be efficiently used for solving problems involving non-convex flux functions especially those experienced during gravity drainage process in hydrocarbon reservoirs.

To the best of authors knowledge, this is the first time that the dominant wave method has been used to tackle multiphase flow problems involving gravity effect.