Riemann Solver Research Articles

High-order methods for hypersonic flows with strong shocks and real chemistry

We compare high-order methods including spectral difference (SD), flux reconstruction (FR), the entropy-stable discontinuous Galerkin spectral element method (ES-DGSEM), modal discontinuous Galerkin methods, and WENO to select the best candidate to simulate strong shock waves characteristic of hypersonic flows. We consider several benchmarks, including the Leblanc and modified shock-density wave interaction problems that require robust stabilization and positivity-preserving properties for a successful flow realization. We also perform simulations of the three-species Sod problem with simplified chemistry with the chemical reaction source terms introduced in the Euler equations. The ES-DGSEM scheme exhibits the highest stability, negligible numerical oscillations, and requires the least computational effort in resolving reactive flow regimes with strong shock waves. Therefore, we extend the ES-DGSEM to hypersonic Euler equations by deriving a new set of two-point entropy conservative fluxes for a five-species gas model. In this paper, hypersonic Euler equations refer to the multi-species Euler equations for which the internal energy and thermodynamic properties are computed using the Rigid-Rotor Harmonic-Oscillator model. Stabilization for capturing strong shock waves occurs by blending high-order entropy conservative fluxes with low-order finite volume fluxes constructed using the HLLC Riemann solver. The hypersonic Euler solver is verified using the non-equilibrium chemistry Sod problem. To this end, we adopt the Mutation++ library to compute the reaction source terms, thermodynamic properties, and transport coefficients. We also investigate the effect of real chemistry versus ideal chemistry, and the results demonstrate that the ideal chemistry assumption fails at high temperatures, hence real chemistry must be employed for accurate predictions. Finally, we consider a viscous hypersonic flow problem to verify the transport coefficients and reaction source terms determined by the Mutation++ library.

Simulating the dusty gas flow model via NHRS scheme

We introduce some vital solutions for the dusty gas flow. Specifically, we implement the nonhomogeneous Riemann solver (NHRS) scheme in order to investigate the unsteady planar flow of an isentropic, inviscid compressible fluid in the presence of dust particles. The principle of conservation of mass, momentum, and energy governs the dynamical behavior of a fluid. The Rusanov scheme is also implemented to examine the dusty gas flow model for comparison and validation. One of the important characteristics of the NHRS technique is its capability to compute the numerical flux corresponding to the real states of solution in the absence of a Riemann solution. The numerical results exhibit that the NHRS system outperforms the Rusanov scheme and validates the ability to simulate the dusty gas flow model with strong shocks and rarefaction waves accurately. Finally, more sophisticated models in applied science can be solved using the NHRS scheme.

A shock-capturing meshless method for solving the one-dimensional Saint-Venant equations on a highly variable topography

Abstract The Saint-Venant equations are numerically solved to simulate free surface flows in one dimension. A Riemann solver is needed to compute the numerical flux for capturing shocks and flow discontinuities occurring in flow situations such as hydraulic jump, dam-break wave propagation, or bore wave propagation. A Riemann solver that captures shocks and flow discontinuities is not yet reported to be implemented within the framework of a meshless method for solving the Saint-Venant equations. Therefore, a wide range of free surface flow problems cannot be simulated by the available meshless methods. In this study, a shock-capturing meshless method is proposed for simulating one-dimensional (1D) flows on a highly variable topography. The Harten–Lax–van Leer Riemann solver is used for computing the convective flux in the proposed meshless method. Spatial derivatives in the Saint-Venant equations and the reconstruction of conservative variables for flux terms are computed using a weighted least square approximation. The proposed method is tested for various numerically challenging problems and laboratory experiments on different flow regimes. The proposed highly accurate shock-capturing meshless method has the potential to be extended to solve the two-dimensional (2D) shallow water equations without any mesh requirements.

On the computation of compressible multiphase flows with heat and mass transfer in elastic pipelines

The present work is motivated by the engineering need to simulate the multiphase flows in the course of the start-up of a long-distance crude oil pipeline. We propose a model for compressible multiphase flows with heat and mass transfer and diffusion processes in elastic pipelines. The model is derived with two approaches, i.e., (a) the spatial averaging procedure of a single phase model and (b) the Arbitrary Lagrangian Eulerian (ALE) formulation of the Baer-Nunziato model with quasi-1D approximation. The diffusion processes include the viscous dissipation, wall friction, heat conduction, and heat exchange with external environment. In particular, the wall friction consists of steady friction term calculated by Darcy-Weisbach formula, and the unsteady friction determined by the instantaneous-acceleration-based (IAB) model. The unsteady friction modifies the characteristic structure of the hyperbolic part of the model. For the solution of the hyperbolic part, we propose a three-wave approximate Riemann solver incorporating the unsteady friction term. Mass and heat transfer are realized via instantaneous relaxations of the chemical potential and temperature, respectively. Efficient iterative relaxation procedures for N-phase flows have been proposed. We have validated the proposed model and numerical methods against some benchmark multiphase problems and applied the model to calculate the start-up of a realistic liquid pipeline with intermediate pump station boundary condition.

A family of well-balanced WENO and TENO schemes for atmospheric flows

We herein present a novel methodology to construct very high order well-balanced schemes for the computation of the Euler equations with gravitational source term, with application to numerical weather prediction (NWP). The proposed method is based on augmented Riemann solvers, which allow preserving the exact equilibrium between fluxes and source terms at cell interfaces. In particular, the augmented HLL solver (HLLS) is considered. Different spatial reconstruction methods can be used to ensure a high order of accuracy in space (e.g. WENO, TENO, linear reconstruction), being the TENO reconstruction the preferred method in this work. To the knowledge of the authors, the TENO method has not been applied to NWP before, although it has been extensively used by the computational fluid dynamics community in recent years. Therefore, we offer a thorough assessment of the TENO method to evidence its suitability for NWP considering some benchmark cases which involve inertia and gravity waves as well as convective processes. The TENO method offers an enhanced behavior when dealing with turbulent flows and underresolved solutions, where the traditional WENO scheme proves to be more diffusive. The proposed methodology, based on the HLLS solver in combination with a very high-order discretization, allows carrying out the simulation of meso- and micro-scale atmospheric flows in an implicit Large Eddy Simulation manner. Due to the HLLS solver, the isothermal, adiabatic and constant Brunt-Väisälä frequency hydrostatic equilibrium states are preserved with machine accuracy.

An entropy consistent and symmetric seven-equation model for compressible two-phase flows

We propose new seven-equation model and related solution methods for compressible two-phase flows. The model is built on new closure laws for the interaction between the coexisting two phases. It takes a symmetric form for the two phases and shares some favorable properties with the well-known Baer-Nunziato model [4]: the equations system is unconditionally hyperbolic and is conservative for the total momentum and energy; the interfacial interaction between phases is an isentropic process for each phase. Moreover, it permits to obtain a linear degenerate field associated with the wave which separates the mixture, and the full set of the associated Riemann invariants can be derived explicitly. The pressure and velocity relaxation terms are included for the additional interaction between phases. Numerical approximation of the entire model is performed in an operator splitting manner – two Godunov-type schemes utilizing respectively a HLL solver and a composite approximate Riemann solver are used to discretize the homogeneous system; stable implicit algorithms are explored for the non-instantaneous pressure and velocity relaxations. Several examples are provided to illustrate the property of the new model and the capability of the solution methods.

A two-phase density-based solver for simulating wet steam flows with non-equilibrium condensation. I. Approach and verification

This paper presents a new two-phase density-based solver based on the finite difference method for simulating high-speed wet steam flows with non-equilibrium condensation. The solver employs the Eulerian–Eulerian approach to model the wet steam flow using a fifth-order accurate weighted compact nonlinear scheme. The phase change of wet steam involves droplet growth in a non-equilibrium condensation process based on the internally consistent classical theory. To speed up computation, a tabulated equation of state approach with curvilinear grids is developed, and an improved Harten-Lax–van Leer-contact-type Riemann solver is used to compute inviscid fluxes. Furthermore, the automatic differentiation technique is applied to avoid manually deriving complicated derivatives when computing flux Jacobian matrices and thermodynamic properties. A numerical investigation is conducted on flow via various kinds of nozzles and blades, and the results demonstrate that the numerical model accurately predicts the behaviors observed in the experiments.

A cell-centered spatiotemporal coupled method for the compressible Euler equations

A cell-centered spatiotemporal coupled method is developed to solve the compressible Euler equations. The spatial discretization is performed using an improved weighted essentially non-oscillation scheme, where the Harten–Lax–van Leer–contact approximate Riemann solver is used for computing the numerical fluxes. A two-stage fourth-order scheme is adopted to carry out time advancement for unsteady problems. The proposed method is featured by spatiotemporal coupling time-stepping that can be generalized without using the case-dependent generalized Riemann problem solver. A number of one- and two-dimensional test cases are presented to demonstrate the performance of the proposed method for solving the compressible Euler equations on structured grids. The numerical results indicate that the novel method can achieve relatively large Courant–Friedrichs–Lewy (CFL) number compared to other studies that implement the two-stage fourth-order scheme, and that it is more capable of capturing small-scale flow structures than the Runge–Kutta (RK) method.

Multidimensional Generalized Riemann Problem Solver for Maxwell’s Equations

Approximate multidimensional Riemann solvers are essential building blocks in designing globally constraint-preserving finite volume time domain and discontinuous Galerkin time domain schemes for computational electrodynamics (CED). In those schemes, we can achieve high-order temporal accuracy with the help of Runge–Kutta or ADER time-stepping. This paper presents the design of a multidimensional approximate generalized Riemann problem (GRP) solver for the first time. The multidimensional Riemann solver accepts as its inputs the four states surrounding an edge on a structured mesh, and its output consists of a resolved state and its associated fluxes. In contrast, the multidimensional GRP solver accepts as its inputs the four states and their gradients in all directions; its output consists of the resolved state and its corresponding fluxes and the gradients of the resolved state. The gradients can then be used to extend the solution in time. As a result, we achieve second-order temporal accuracy in a single step. In this work, the formulation is optimized for linear hyperbolic systems with stiff, linear source terms because such a formulation will find maximal use in CED. Our formulation produces an overall constraint-preserving time-stepping strategy based on the GRP that is provably L-stable in the presence of stiff source terms. We present several stringent test problems, showing that the multidimensional GRP solver for CED meets its design accuracy and performs stably with optimal time steps. The test problems include cases with high conductivity, showing that the beneficial L-stability is indeed realized in practical applications.

Accurate and Robust Hybrid HLLC Riemann Solver on Triangular Grids

The HLLC scheme is an accurate, approximate Riemann solver. However, it suffers from the carbuncle phenomenon and numerical shock instability problems on both rectangular and triangular grids. In this paper, a hybrid HLLC scheme () is developed by adding some extra computations to the original HLLC scheme in order to achieve a stable and accurate scheme. The idea of the hybrid scheme is straightforward as it uses a simple shock-capturing function with parameter to detect regions where pressure oscillations such as shock waves are significant. In these regions, a less diffusive version of the scheme () is activated with a new weighting function with parameter . This new weighting function is designed to control the stability and accuracy of the hybrid scheme via the ratio of the square of the speed of sound of two adjacent cells. A linear perturbation analysis of an odd–even decoupling problem is used to analyze the effectiveness of the proposed hybrid scheme in damping perturbations. Several numerical examples are given to demonstrate that the new scheme () can obtain accurate solutions and that it is simple and efficient and requires comparable computational time with the original scheme.

Tightly coupled hyperbolic treatment of buoyant two-phase flow and transport in porous media

Buoyant incompressible multiphase flow and transport in porous media typically is simulated by coupling an elliptic flow equation with a hyperbolic transport description. The tight coupling between flow and transport either requires a fully implicit solution algorithm or very small time steps, if solved sequentially. In any case, however, a large linear system has to be solved each time step.Here a new solution approach is devised, which relies on solving a coupled hyperbolic system of conservation laws with an explicit finite volume method. Consequently, only local operations have to be performed, which is a great advantage in case of massive parallel simulations and if GPUs are employed.The devised method is based on the isothermal Euler equations with momentum source terms accounting for resistance due to the porous medium and buoyancy. In this system the pressure is proportional to the density and in case of very small Mach numbers and relative density variation, the computed velocity field is divergence free and converges towards the mean total volumetric flow in the porous domain. To account for saturation transport, the system was augmented by an additional hyperbolic equation. In order to obtain the numerical fluxes, a characteristic based approximative Riemann solver was developed and 2nd order accuracy in space and time is achieved by piecewise linear reconstruction.Two-phase flow test cases with buoyant plumes and a lock exchange problem demonstrate that the devised hyperbolic approach recovers the correct incompressible solutions. Numerical experiments also demonstrate that the method is accurate and efficient.Thus the devised method is very promising, and conceptually it also can be applied for flow and transport in heterogeneous media; but it remains to be investigated how well it performs for such cases.

An improved SPH scheme for the 3D FEI (Fluid-Elastomer Interaction) problem of aircraft tire spray

The problem of aircraft tire spray produced by taking off or landing on wet runways is widely concerned. When simulating this kind of FEI(Fluid-Elastomer Interaction) problem with traditional SPH((Smoothed Particle Hydrodynamics) method, there are some drawbacks in both fluid and elastomer calculation. For the elastomer simulation, the TL-SPH(Total Lagrangian-SPH) is used to deal with the deformation of tire under complicated motion and load. For the fluid simulation, a new SPH scheme of Riemann solver combined with transport velocity is proposed to solve the problem of particle clumping when the free surface is suffering from violent splash. Then the tire spray model by unified SPH is established, in which both the tire deformation and the spray pattern are validated by experiments. Finally, based on the large-scale tire spray simulation, the phenomenon of tire hydroplaning is studied.

Numerical Simulation of Nonlinear Wave Propagation from Deep to Shallow Water

Herein, a numerical model is proposed to simulate the nonlinear wave propagation from deep to shallow water and wave breaking phenomena. In the numerical model, the governing equations selected, in which the momentum equations were added to the eddy-viscous breaking and bottom friction terms to simulate the wave breaking phenomenon, are suitable for the wave propagation from deep to shallow water. The spatial derivations of the governing equations are discretized with the hybrid scheme, combining the finite-difference and finite-volume methods. To numerically simulate the nonlinear wave propagation in waters with various depths accurately, the non-conservative governing equations are reorganized as conservative to facilitate a total variation diminishing (TVD) type scheme using a Riemann solver. Extensive numerical tests of nonlinear wave propagation have been realized in waters with large relative water depths and varying water depths. The comparisons between numerical and analytical or experimental results indicated that the numerical results are reasonable and reliable, and the present numerical model can effectively simulate the wave-breaking phenomenon.

FluxNet: A physics-informed learning-based Riemann solver for transcritical flows with non-ideal thermodynamics

Traditional Riemann solvers fall into two broad categories: exact solvers, which require multiple iterations to achieve high accuracy, and approximate linearized solvers, which achieve fast runtime at the expense of reduced accuracy. Here we explore learning-based Riemann solvers, termed FluxNets, as a new approach that balances accuracy and computational efficiency especially for transcritical and supercritical flow problems involving non-ideal thermodynamics. Various designs and training strategies are applied to fully connected neural networks. By comparing data-driven versus physics-informed loss functions, as well as neural networks of varying size, the results show that order-of-magnitude reductions in error compared to the Roe solver can be achieved with compact architectures. Numerical validation on 1D and 2D test cases reveals that a physics-informed FluxNet promotes smoothness, numerical stability, and generalizability, and is capable of producing entropy-stable results free of non-physical expansion shocks as compared to the Roe solver. Parallelization can be leveraged to accelerate inference such that the gains in accuracy are achieved at one quarter the runtime of exact solvers. The trade-off in accuracy versus efficiency may be justified in the case of non-ideal flows where even minor errors can result in spurious oscillations and instabilities.

The initial boundary value problem for second order traffic flow models with vacuum: Existence of entropy weak solutions

We study the IBVP for a non-strictly hyperbolic 2 × 2 system of conservation laws on an interval with characteristic boundaries, modelling traffic dynamics including vacuum states on a road stretch. After giving a detailed characterization of the admissible states at the boundary in terms of Riemann solver and entropy conditions, we prove existence of entropy weak solutions for data of bounded variation in the Riemann invariant coordinates by convergence of wave-front tracking approximations.

A novel Eulerian SPH shallow water model for 2D overland flow simulations

A 2D Eulerian meshless shallow water model based on smoothed particle hydrodynamics (SPH) is proposed to study overland flow. Rain on a dry bed to generate a runoff flow is one common hydraulic phenomenon. However, it is difficult for existing Lagrangian SPH shallow water models to present the flow because no computational particle exists on a dry bed. To overcome this difficulty, the Eulerian form of the 2D shallow water equations is used to describe flow behavior in the proposed model. An HLLC approximate Riemann solver is applied to calculate fluxes between two neighboring particles to smear out the instabilities caused by the discretization of the nonlinear convective term and the discontinuity of wet-dry beds. Additionally, we use a surface reconstruction method (SRM) to ensure that the proposed model has well-balanced and positivity-preserving properties. Bed friction and infiltration effects are also analytically considered. Eight case studies involving various overland flows with wet-dry transitions, rainfall, infiltration, bed friction and complex topography are adopted to test the proposed model capacity. The first-order convergence rate is obtained in the convergence analysis. Against the exact and measured results, in the analytical case studies, the relative root-mean-square errors of depth and velocity are approximately 0.02% to 0.3% and 1.3% to 1.7%, respectively. In the experimental case studies, the relative root-mean-square errors of depth and discharge (velocity) are approximately 3.8% and 1.2% to 6.1%, respectively. The good agreement demonstrates between the results shows the reliability of the proposed 2D Eulerian SPH shallow water model for real-world overland flows.

A unified consistent source term computational algorithm for the [formula omitted]-based compressible multi-fluid flow model

Using the criterion of one-fluid preservation, we developed a consistent algorithm for the source term to solve the γ-based compressible multi-fluid flow model with three approximate Riemann solvers, namely the Lax–Friedrichs (LxF), Kurganov, and Harten–Lax–van Leer contact (HLLC) solvers. The consistent algorithm comprises a standard Godunov solver with a high-order reconstruction and a consistent source term integration part. We prove that the present algorithm is consistent with Abgrall's criterion of the moving-material-interface property in the finite volume method framework. The cell boundary velocity for the source term discretization is found to be the same as HLLC solvers from five-equation model, but for the first time for LxF and Kurganov Riemann solvers. We develop 12 compressible multi-fluid solvers by combining four reconstruction schemes and three approximate Riemann solvers together with their consistent-source-term integration algorithms. All 12 solvers can maintain the one-fluid preservation and moving-material-interface properties numerically in several one- and two-dimensional example cases. The simulation of an underwater explosion demonstrates that a boundary-variation-diminishing reconstruction predicts an interface with width controlled within approximately three cells which is independent of the Riemann solver. The simulation of an explosion near a free surface demonstrates that the proposed model can simulate a severe compressible multi-fluid flow involving an interface across which there are large differences in density, pressure and parameters in the equation of state. In conclusion, the proposed consistent algorithm provides a unified framework for one kind of non-conservative hyperbolic system into a conservative hyperbolic system and a source term with velocity divergence, where the former can be computed by classical Godunov-type algorithms and the latter can be solved by the proposed consistent algorithm.

A robust and contact resolving Riemann solver for the two-dimensional ideal magnetohydrodynamics equations

This paper presents a new cell-centered numerical method for ideal magnetohydrodynamics (MHD) that can be used in Lagrangian or Eulerian discretization. A two-dimensional Riemann solver based on the HLLC-type MHD method is established, which can be viewed as an extension from the HLLC-2D in hydrodynamics. The main feature of the algorithm is to introduce a nodal contact velocity and ensure the compatibility between edge fluxes and the nodal flow intrinsically. It transforms naturally from Lagrangian setting to the Eulerian setting in terms of grid nodal velocity, and gains benefits of the Lagrangian nature of the scheme. In the Lagrangian approach, the finite volume scheme itself can keep the magnetic field divergence-free strictly, while in the Eulerian case, a special constrained transport (CT) algorithm is constructed from the discontinuous fluxes on cell interfaces to ensure solenoidal nature again. Numerical tests are presented to demonstrate the performance of this new solver and compare the difference between the Lagrangian and Eulerian methods.

A GPU-accelerated adaptive particle refinement for multi-phase flow and fluid-structure coupling SPH

In this paper, a graphical processing unit (GPU) implementation of adaptive particle refinement (APR) in the Smoothed Particle Hydrodynamics (SPH) framework is used to simulate the multi-phase flow and fluid-structure coupling problems. The multi-phase flow and fluid-structure coupling SPH models based on the Riemann solver are presented and the local particle refinement technique based on the axis-aligned square refinement pattern is adopted to split particles. A particle shifting technique constructing virtual fine particles (VFP-PST) is proposed to regularize the particle distribution for numrical accuracy and stability. To facilitate the device memory throughput efficiently, a dynamic resource management algorithm is used to implement GPU-accelerated adaptive particle refinement. Precision analysis shows the effect of the smooth length ratio of fine and coarse particles on numerical accuracy. Different tests have demonstrated that the proposed GPU-accelerated IAPR is accurate and stable and provides a promising approach to simulate the multi-phase flow and fluid-structure problems with low computational cost and comparable precision when compared with uniform particles.

Convergence of first‐order finite volume method based on exact Riemann solver for the complete compressible Euler equations

AbstractRecently developed concept of dissipative measure‐valued solution for compressible flows is a suitable tool to describe oscillations and singularities possibly developed in solutions of multidimensional Euler equations. In this paper we study the convergence of the first‐order finite volume method based on the exact Riemann solver for the complete compressible Euler equations. Specifically, we derive entropy inequality and prove the consistency of numerical method. Passing to the limit, we show the weak and strong convergence of numerical solutions and identify their limit. The numerical results presented for the spiral and the Richtmyer‐Meshkov problem are consistent with our theoretical analysis.