A divergence-free semi-implicit finite volume scheme for ideal,viscous, and resistive magnetohydrodynamics

In this paper, we present a novel pressure-based semi-implicit finite volume solver for the equations of compressible ideal, viscous, and resistive magnetohydrodynamics (MHD). The new method is conservative for mass, momentum, and total energy, and in multiple space dimensions, it is constructed in such a way as to respect the divergence-free condition of the magnetic field exactly, also in the presence of resistive effects. This is possible via the use of multidimensional Riemann solvers on an appropriately staggered grid for the time evolution of the magnetic field and a double curl formulation of the resistive terms. The new semi-implicit method for the MHD equations proposed here discretizes the nonlinear convective terms as well as the time evolution of the magnetic field explicitly, whereas all terms related to the pressure in the momentum equation and the total energy equation are discretized implicitly, making again use of a properly staggered grid for pressure and velocity. Inserting the discrete momentum equation into the discrete energy equation then yields a mildly nonlinear symmetric and positive definite algebraic system for the pressure as the only unknown, which can be efficiently solved with the (nested) Newton method of Casulli et al. The pressure system becomes linear when the specific internal energy is a linear function of the pressure. The time step of the scheme is restricted by a CFL condition based only on the fluid velocity and the Alfvén wave speed and is not based on the speed of the magnetosonic waves. Being a semi-implicit pressure-based scheme, our new method is therefore particularly well suited for low Mach number flows and for the incompressible limit of the MHD equations, for which it is well known that explicit density-based Godunov-type finite volume solvers become increasingly inefficient and inaccurate because of the more and more stringent CFL condition and the wrong scaling of the numerical viscosity in the incompressible limit. We show a relevant MHD test problem in the low Mach number regime where the new semi-implicit algorithm is a factor of 50 faster than a traditional explicit finite volume method, which is a very significant gain in terms of computational efficiency. However, our numerical results confirm that our new method performs well also for classical MHD test cases with strong shocks. In this sense, our new scheme is a true all Mach number flow solver.     

Study of a collocated Lagrange‐remap scheme for multi‐material flows adapted to HPC

This paper describes a collocated numerical scheme for multi‐material compressible Euler equations, which attempts to suit to parallel computing constraints. Its main features are conservativity of mass, momentum, total energy and entropy production, and second order in time and space. In the context of a Eulerian Lagrange‐remap scheme on planar geometry and for rectangular meshes, we propose and compare remapping schemes using a finite volume framework. We consider directional splitting or fully multi‐dimensional remaps, and we focus on a definition of the so‐called corner fluxes. We also address the issue of the internal energy behavior when using a conservative total energy remap. It can be perturbed by the duality between kinetic energy obtained through the conservative momentum remap or implicitly through the total energy remap. Therefore, we propose a kinetic energy flux that improves the internal energy remap results in this context. Copyright © 2016 John Wiley & Sons, Ltd.     

A finite volume solver for 1D shallow‐water equations applied to an actual river

This paper describes the numerical solution of the 1D shallow‐water equations by a finite volume scheme based on the Roe solver. In the first part, the 1D shallow‐water equations are presented. These equations model the free‐surface flows in a river. This set of equations is widely used for applications: dam‐break waves, reservoir emptying, flooding, etc. The main feature of these equations is the presence of a non‐conservative term in the momentum equation in the case of an actual river. In order to apply schemes well adapted to conservative equations, this term is split in two terms: a conservative one which is kept on the left‐hand side of the equation of momentum and the non‐conservative part is introduced as a source term on the right‐hand side. In the second section, we describe the scheme based on a Roe Solver for the homogeneous problem. Next, the numerical treatment of the source term which is the essential point of the numerical modelisation is described. The source term is split in two components: one is upwinded and the other is treated according to a centred discretization. By using this method for the discretization of the source term, one gets the right behaviour for steady flow. Finally, in the last part, the problem of validation is tackled. Most of the numerical tests have been defined for a working group about dam‐break wave simulation. A real dam‐break wave simulation will be shown. Copyright © 2002 John Wiley & Sons, Ltd.     

Stability of a Crank–Nicolson pressure correction scheme based on staggered discretizations

In the context of LES of turbulent flows, the control of kinetic energy seems to be an essential requirement for a numerical scheme. Designing such an algorithm, that is, as less dissipative as possible while being simple, for the resolution of variable density Navier–Stokes equations is the aim of the present work. The developed numerical scheme, based on a pressure correction technique, uses a Crank–Nicolson time discretization and a staggered space discretization relying on the Rannacher–Turek finite element. For the inertia term in the momentum balance equation, we propose a finite volume discretization, for which we derive a discrete analogue of the continuous kinetic energy local conservation identity. Contrary to what was obtained for the backward Euler discretization, the dissipation defect term associated to the Crank–Nicolson scheme is second order in time. This behavior is evidenced by numerical simulations. Copyright © 2013 John Wiley & Sons, Ltd.     

Remarks on the numerical solution of the adjoint quasi‐one‐dimensional Euler equations

We examine the numerical solution of the adjoint quasi‐one‐dimensional Euler equations with a central‐difference finite volume scheme with Jameson‐Schmidt‐Turkel (JST) dissipation, for both the continuous and discrete approaches. First, the complete formulations and discretization of the quasi‐one‐dimensional Euler equations and the continuous adjoint equation and its counterpart, the discrete adjoint equation, are reviewed. The differences between the continuous and discrete boundary conditions are also explored. Second, numerical testing is carried out on a symmetric converging–diverging duct under subsonic flow conditions. This analysis reveals that the discrete adjoint scheme, while being manifestly less accurate than the continuous approach, gives nevertheless more accurate flow sensitivities. Copyright © 2011 John Wiley & Sons, Ltd.     

固定床化学链燃烧氧化反应的数值模拟

基于Chimera网格采用有限体积法模拟了450个颗粒随机填充固定床中的化学链燃烧的氧化反应过程,并采用三维瞬态N-S方程,结合压力Poisson方程方法,详细分析了床层入口Re=5时的颗粒内部和外部的传质传热过程．模拟结果揭示了在大颗粒的固定床中,颗粒内部有效扩散系数对颗粒内部的传质起着决定性作用,而且颗粒表面的浓度梯度决定了总反应速率;另外,有惰性芯的结构化颗粒能有效地改善颗粒内部总的反应速率,颗粒的转化速率,并且能使床层很快地达到热平衡．模拟结果能更好地帮助我们认识固定床化学链反应器中的反应和组分传递机理．     

A new weighted upwind finite volume element method based on non‐standard covolume for time‐dependent convection–diffusion problems

In modern numerical simulation of problems in energy resources and environmental science, it is important to develop efficient numerical methods for time‐dependent convection–diffusion problems. On the basis of nonstandard covolume grids, we propose a new kind of high‐order upwind finite volume element method for the problems. We first prove the stability and mass conservation in the discrete forms of the scheme. Optimal second‐order error estimate in L₂‐norm in spatial step is then proved strictly. The scheme is effective for avoiding numerical diffusion and nonphysical oscillations and has second‐order accuracy. Numerical experiments are given to verify the performance of the scheme. Copyright © 2013 John Wiley & Sons, Ltd.     

A combined finite volumes - finite elements method for a low-Mach model

In this paper, we develop a finite volumes - finite elements method based on a time splitting to simulate some low-Mach flows. The mass conservation equation is solved by a vertex-based finite volume scheme using a τ-limiter. The momentum equation associated with the compressibility constraint is solved by a finite element projection scheme. The originality of the approach is twofold. First, the state equation linking the temperature, the density, and the thermodynamic pressure is imposed implicitly. Second, the proposed combined scheme preserves the constant states, in the same way as a similar one previously developed for the variable density Navier-Stokes system. Some numerical tests are performed to exhibit the efficiency of the scheme. On the one hand, academic tests illustrate the ability of the scheme in term of convergence rates in time and space. On the other hand, our results are compared to some of the literature by simulating a transient injection flow as well as a natural convection flow in a cavity.     

Numerical simulation of dielectric bubbles coalescence under the effects of uniform magnetic field

In this research, the co-axial coalescence of a pair of gas bubbles rising in a viscous liquid column under the effects of an external uniform magnetic field is simulated numerically. Considered fluids are dielectric, and applied magnetic field is uniform. Effects of different strengths of magnetic field on the interaction of in-line rising bubbles and coalescence between them were investigated. For numerical modeling of the problem, a computer code was developed to solve the governing equations which are continuity, Navier–Stokes equation, magnetic field equation and level set and reinitialization of level set equations. The finite volume method is used for the discretization of the continuity and momentum equations using SIMPLE scheme where the finite difference method is used to discretization of the magnetic field equations. Also a level set method is used to capture the interface of two phases. The results are compared with available numerical and experimental results in the case of no-magnetic field effect which show a good agreement. It is found that uniform magnetic field accelerates the coalescence of the bubbles in dielectric fluids and enhances the rise velocity of the coalesced bubble.     

A non‐diffusive,divergence‐free,finite volume‐based double projection method on non‐staggered grids

Second‐order accurate projection methods for simulating time‐dependent incompressible flows on cell‐centred grids substantially belong to the class either of exact or approximate projections. In the exact method, the continuity constraint can be satisfied to machine‐accuracy but the divergence and Laplacian operators show a four‐dimension nullspace therefore spurious oscillating solutions can be introduced. In the approximate method, the continuity constraint is relaxed, the continuity equation being satisfied up to the magnitude of the local truncation error, but the compact Laplacian operator has only the constant mode. An original formulation for allowing the discrete continuity equation to be satisfied to machine‐accuracy, while using a finite volume based projection method, is illustrated. The procedure exploits the Helmholtz–Hodge decomposition theorem for deriving an additional velocity field that enforces the discrete continuity without altering the vorticity field. This is accomplished by solving a second elliptic field for a scalar field obtained by prescribing that its additional discrete gradients ensure discrete continuity based on the previously adopted linear interpolation of the velocity. The resulting numerical scheme is applied to several flow problems and is proved to be accurate, stable and efficient. This paper has to be considered as the companion of: 'F. M. Denaro, A 3D second‐order accurate projection‐based finite volume code on non‐staggered, non‐uniform structured grids with continuity preserving properties: application to buoyancy‐driven flows. IJNMF 2006; 52 (4):393–432. Now, we illustrate the details and the rigorous theoretical framework. Copyright © 2006 John Wiley & Sons, Ltd.     

Global volume conservation in unsteady free surface flows with energy absorbing far‐end boundaries

A wave absorption filter for the far‐end boundary of semi‐infinite large reservoirs is developed for numerical simulation of unsteady free surface flows. Mathematical model is based on finite volume solution of the Navier–Stokes equations and depth‐integrated continuity equation to track the free surface. The Sommerfeld boundary condition is applied at the far‐end of the truncated computational domain. A dissipation zone is formed by applying artificial pressure on water surface to dissipate the kinetic energy of the outgoing waves. The computational scheme is tested to verify the conservation of total fluid volume in the domain for long simulation durations. Combination of the Sommerfeld boundary and dissipation zone can effectively minimize reflections and prevent cumulative changes in total fluid volume in the domain. Solitary wave, nonlinear periodic waves and irregular waves are simulated to illustrate the numerical developments. Earthquake excited surface waves and nonlinear hydrodynamic pressures in a dam–reservoir are computed. Copyright © 2009 John Wiley & Sons, Ltd.     

Spurious numerical refraction

The purpose of this paper is to investigate the effect of a non-uniform mesh in two dimensions (2D). A change in mesh size will, in general, result in spurious refraction (and reflection) which is entirely numerical (rather than physical) in origin. To facilitate the analysis, the mesh geometry has been highly simplified in that only a single change in mesh size is considered. The analysis is based on a finite element wave model. The domain consists of two conterminous regions discernible only by their different nodal spacings in the x-direction. The interface between the two regions is internal to the mesh and is a straight line. The model is based upon the Crank-Nicolson linear finite element scheme applied to the second order wave equation. The results of the analysis are confirmed by numerical experiments. It is shown that under particular numerical conditions total internal reflection may occur and when this is the case, the transmitted wave is evanescent. An analysis of the energy flux associated with the incident, reflected and trasmitted waves shows that energy is conserved across the interface between the two regions.     

A staggered control volume scheme for unstructured triangular grids

The purpose of this work is to introduce and validate a new staggered control volume method for the simulation of 2D/axisymmetric incompressible flows. The present study introduces a numerical procedure for solving the Navier–Stokes equations using the primitive variable formulation. The proposed method is an extension of the staggered grid methodology to unstructured triangular meshes for a control volume approach which features ease of handling of irregularly shaped domains. Two alternative elements are studied: transported scalars are stored either at the sides of an element or at its vertices, while the pressure is always stored at the centre of an element. Two interpolation functions were investigated for the integration of the momentum equations: a skewed mass-weighted upwind function and a flow-oriented exponential shape function. The momentum equations are solved over the covolume of a side or of a vertex and the pressure–velocity coupling makes use of a localized linear reconstruction of the discontinuous pressure field surrounding an element in order to obtain the pressure gradient terms. The pressure equation is obtained through a discretization of the continuity equation which uses the triangular element itself as the control volume. The method is applied to the simulation of the following test cases: backward-facing step flow, flow over a two-dimensional obstacle and flow in a pipe with sudden contraction of cross-sectional area. All numerical investigations are compared with experimental data from the literature. A grid convergence and error analysis study is also carried out for flow in a driven cavity. Results compared favourably with experimental data and so the new control volume scheme is deemed well suited for the prediction of incompressible flows in complex geometries. © 1997 John Wiley & Sons, Ltd.     

Evaluation of an energy relaxation method for the simulation of unsteady,viscous, real gas flows

This study investigates a new energy relaxation method designed to capture the dynamics of unsteady, viscous, real gas flows governed by the compressible Navier–Stokes equations. We focus on real gas models accounting for inelastic molecular collisions and yielding temperature‐dependent heat capacities. The relaxed Navier–Stokes equations are discretized using a mixed finite volume/finite element method and a high‐order time integration scheme. The accuracy of the energy relaxation method is investigated on three test problems of increasing complexity: the advection of a periodic set of vortices, the interaction of a temperature spot with a weak shock, and finally, the interaction of a reflected shock with its trailing boundary layer in a shock tube. In all cases, the method is validated against benchmark solutions and the numerical errors resulting from both discretization and energy relaxation are assessed independently. Copyright © 2004 John Wiley & Sons, Ltd.     

A new numerical model for simulations of wave transformation,breaking and long‐shore currents in complex coastal regions

In this paper, we propose a model based on a new contravariant integral form of the fully nonlinear Boussinesq equations in order to simulate wave transformation phenomena, wave breaking, and nearshore currents in computational domains representing the complex morphology of real coastal regions. The aforementioned contravariant integral form, in which Christoffel symbols are absent, is characterized by the fact that the continuity equation does not include any dispersive term. A procedure developed in order to correct errors related to the difficulties of numerically satisfying the metric identities in the numerical integration of fully nonlinear Boussinesq equation on generalized boundary‐conforming grids is presented. The Boussinesq equation system is numerically solved by a hybrid finite volume–finite difference scheme. The proposed high‐order upwind weighted essentially non‐oscillatory finite volume scheme involves an exact Riemann solver and is based on a genuinely two‐dimensional reconstruction procedure, which uses a convex combination of biquadratic polynomials. The wave breaking is represented by discontinuities of the weak solution of the integral form of the nonlinear shallow water equations. The capacity of the proposed model to correctly represent wave propagation, wave breaking, and wave‐induced currents is verified against test cases present in the literature. The results obtained are compared with experimental measures, analytical solutions, or alternative numerical solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

Cahn–Hilliard/Navier–Stokes Model for the Simulation of Three-Phase Flows

In this article, we describe some aspects of the diffuse interface modelling of incompressible flows, composed of three immiscible components, without phase change. In the diffuse interface methods, system evolution is driven by the minimisation of a free energy. The originality of our approach, derived from the Cahn–Hilliard model, comes from the particular form of energy we proposed in Boyer and Lapuerta (M2AN Math Model Numer Anal, 40:653–987,2006), which, among other interesting properties, ensures consistency with the two-phase model. The modelling of three-phase flows is further completed by coupling the Cahn–Hilliard system and the Navier–Stokes equations where surface tensions are taken into account through volume capillary forces. These equations are discretized in time and space paying attention to the fact that most of the main properties of the original model (volume conservation and energy estimate) have to be maintained at the discrete level. An adaptive refinement method is finally used to obtain an accurate resolution of very thin moving internal layers, while limiting the total number of cells in the grids all along the simulation. Different numerical results are given, from the validation case of the lens spreading between two phases (contact angles and pressure jumps), to the study of mass transfer through a liquid/liquid interface crossed by a single rising gas bubble. The numerical applications are performed with large ratio between densities and viscosities and three different surface tensions.     

Time-dependent solutions of viscous incompressible flows in moving co-ordinates

A time-accurate solution method for the incompressible Navier-Stokes equations in generalized moving coordinates is presented. A finite volume discretization method that satisfies the geometric conservation laws for time-varying computational cells is used. The discrete equations are solved by a fractional step solution procedure. The solution is second-order-accurate in space and first-order-accurate in time. The pressure and the volume fluxes are chosen as the unknowns to facilitate the formulation of a consistent Poisson equation and thus to obtain a robust Poisson solver with favourable convergence properties. The method is validated by comparing the solutions with other numerical and experimental results. Good agreement is obtained in all cases.     

Phase field model for coupled displacive and diffusive microstructural processes under thermal loading

A non-isothermal phase field model that captures both displacive and diffusive phase transformations in a unified framework is presented. The model is developed in a formal thermodynamic setting, which provides guidance on admissible constitutive relationships and on the coupling of the numerous physical processes that are active. Phase changes are driven by temperature-dependent free-energy functions that become non-convex below a transition temperature. Higher-order spatial gradients are present in the model to account for phase boundary energy, and these terms necessitate the introduction of non-standard terms in the energy balance equation in order to satisfy the classical entropy inequality point-wise. To solve the resulting balance equations, a Galerkin finite element scheme is elaborated. To deal rigorously with the presence of high-order spatial derivatives associated with surface energies at phase boundaries in both the momentum and mass balance equations, some novel numerical approaches are used. Numerical examples are presented that consider boundary cooling of a domain at different rates, and these results demonstrate that the model can qualitatively reproduce the evolution of microstructural features that are observed in some alloys, especially steels. The proposed model opens a number of interesting possibilities for simulating and controlling microstructure pattern development under combinations of thermal and mechanical loading.     

Hybrid finite element/volume method for 3D incompressible flows with heat transfer

An implicit hybrid finite element (FE)/volume solver has been extended to incompressible flows coupled with the energy equation. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix-free implicit cell-centred finite volume (FV) method. The pressure Poisson equation is solved by the node-based Galerkin FE method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. Our current staggered-mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centres and the auxiliary variable at vertices. The Generalized Minimal Residual (GMRES) matrix-free strategy is adapted to solve the governing equations in both FE and FV methods. The presented 2D and 3D numerical examples show the robustness and accuracy of the numerical method.     

On conservative finite element formulations of the inviscid Boussinesq equations

The finite element discretization of the inviscid Boussinesq equations is studied with particular emphasis on the conservation properties of the discrete equations. Methods which conserve the total energy, total temperature and total temperature squared, or two of the above mentioned quantities, are presented. The effect of time discretization, and other numerical errors, on the conservation laws is considered. Finally, the theory is supported and illustrated by several numerical experiments.