Linearized Riemann Solver Research Articles

On the Development of an Implicit Discontinuous Galerkin Solver for Turbulent Real Gas Flows

The aim of this work is to describe an efficient implementation of cubic and multiparameter real gas models in an existing discontinuous Galerkin solver to extend its capabilities to the simulation of turbulent real gas flows. The adopted thermodynamic models are van der Waals, Peng–Robinson, and Span–Wagner, which differ from each other in terms of accuracy and computational cost. Convective numerical fluxes across elements interfaces are calculated with a thermodynamic consistent linearized Riemann solver, whereas for boundary conditions, a linearized expression of the generalized Riemann invariants is employed. Transport properties are treated as temperature- and density-dependent quantities through multiparameter correlations. An implicit time integration is adopted; Jacobian matrix and thermodynamic derivatives are obtained with the automatic differentiation tool Tapenade. The solver accuracy is assessed by computing both steady and unsteady real gas test cases available in the literature, and the effect of the mesh size and polynomial degree of approximation on the solution accuracy is investigated. A good agreement with experimental and numerical reference data is observed and specific non-classical phenomena are well reproduced by the solver.

Mixing in turbulent compressible heated coaxial jets: A numerical study

The computation of compressible coaxial jets has moved into the field of interest not only in fundamental research but also in industrial applications, especially in chemical engineering. Numerical simulations of such flows are performed here, using a code specifically developed for gaseous turbulent flows which can also take into account chemical reactions. The coaxial jet can be regarded as a model of injection device in industrial applications; one can cite combustion and aeroacoustics technology. Three-dimensional numerical simulations in this configuration, already published by various authors in open literature are limited to incompressible isothermal flow. In our work, we have explicitly taken into account the temperature gradient effects on the dynamics and mixing mechanisms. Indeed, we have investigated a spatially developing compressible (isothermal and non-isothermal coaxial jet). The numerical model is based on time and space resolutions of compressible Navier-Stokes equations. The piecewise parabolic method (PPM) is combined with a linearized Riemann solver. This scheme adds non-linear dissipation intermittently just where and when needed in order to avoid spurious oscillations and guarantee monotonicity for the advection equation. The simulation can, therefore, be regarded as large-Eddy simulations: large scales are accurately solved with minimal viscosity and non-linear dissipation extracts energy out of the small scales in order to avoid non-physical oscillations. In order to study the mixing between Air-Air flows, we consider the mixture fraction f to track the mixing between two species seeded in the coaxial jets. A great attention is paid to the spatial-temporal evolution of the mixture fraction f, with a particular interest to probability density function. The comparison between the numerical results and experimental data is fairly good, with respect to the mean and turbulent fields. We found that the inner potential core length in the non-isothermal configuration reduced with respect to the isothermal coaxial jet due to the gradient of temperature. It was shown that in the non-isothermal coaxial jet, the temperature gradient leads to the rapid development of the inner Kelvin-Helmholtz vortices implying an efficient mixing of the species close to the exit of the computational domain.

On Hyperbolization of a Number of Continuum Mechanics Models

By using the Maxwell–Cattaneo approach, the equations of motion of a viscous heat conducting gas and of a heterogeneous medium are hyperbolized in one- and multivelocity approximations with relaxation laws of viscosity and thermal conductivity. The modified Godunov method with a linearized Riemann solver intended for integration of nondivergent systems is described.

Hyperbolic Model of a One-Velocity Viscous Heat-Conducting Medium

A modified, generalized-equilibrium model of a one-velocity heterogeneous medium is presented, which accounts for the viscosity and thermal conductivity of the mixture, as well as for interfractional heat transfer. A characteristic analysis of the model′s equations has been carried out, and their hyperbolicity is shown. A description is given of Godunov′s method with a linearized Riemann solver intended for calculating mixture flows on curvilinear grids, with the use of which a number of problems for a vapor–air mixture has been investigated.

Hyperbolic Model of a Single-Speed, Heat-Conductive Mixture with Interfractional Heat Transfer

A modified generalized equilibrium model of a single-speed, heat-conducting, heterogeneous mixture is proposed; it takes into account interfractional heat transfer. The model equations are analyzed, and their hyperbolicity is demonstrated. The Godunov method with the linear Riemann solver designed for the calculation of mixture flows on curvilinear grids is described.

Numerical Simulation of the Interaction of an Air Shock Wave with a Surface Gas–Dust Layer

Within the framework of the one-velocity and multivelocity models of a dust-laden gas with the use of the Godunov method with a linearized Riemann solver, the problem of the interaction of a shock wave with a dust-laden gas layer located along a solid plane surface has been studied.

New hyperbolic models of dusty gas flow

The paper presents hyperbolic models for dusty gas flow formulated for single- and multivelocity approximations with account for inter-fractional heat transfer. Characteristic analysis of equations for the model was performed. The Godunov method and linearized Riemann solver was applied for a solution on curvilinear mesh: the Prandtl−Meyer problem for air-droplet mixture was solved. The simulation results were compared with the self-similar solution.

An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

A vertex centred Jameson–Schmidt–Turkel (JST) finite volume algorithm was recently introduced by the authors (Aguirre et al., 2014 [1]) in the context of fast solid isothermal dynamics. The spatial discretisation scheme was constructed upon a Lagrangian two-field mixed (linear momentum and the deformation gradient) formulation presented as a system of conservation laws [2–4]. In this paper, the formulation is further enhanced by introducing a novel upwind vertex centred finite volume algorithm with three key novelties. First, a conservation law for the volume map is incorporated into the existing two-field system to extend the range of applications towards the incompressibility limit (Gil et al., 2014 [5]). Second, the use of a linearised Riemann solver and reconstruction limiters is derived for the stabilisation of the scheme together with an efficient edge-based implementation. Third, the treatment of thermo-mechanical processes through a Mie–Grüneisen equation of state is incorporated in the proposed formulation. For completeness, the study of the eigenvalue structure of the resulting system of conservation laws is carried out to demonstrate hyperbolicity and obtain the correct time step bounds for non-isothermal processes. A series of numerical examples are presented in order to assess the robustness of the proposed methodology. The overall scheme shows excellent behaviour in shock and bending dominated nearly incompressible scenarios without spurious pressure oscillations, yielding second order of convergence for both velocities and stresses.

Numerical scheme for accurately capturing gas migration described by 1D multiphase drift flux model

A numerical scheme is designed and implemented to solve a simplified set of equations modeling 1-D multi-phase flow based on drift flux model in an isothermal setup with phase dissolution. The difficulty in obtaining the analytical Jacobian of the fluxes leads to the difficulty in obtaining an efficient linearized Riemann solver which in turn affects the accuracy in capturing the contact wavefront/gas migration. To address this issue a fully explicit second order finite volume solver based on flux corrected transport (FCT) is implemented. The choice of variables used for limiting the fluxes affects the amount of numerical diffusion and an appropriate choice of the gradient in volume fraction is used. Practical test cases while drilling in the oil and gas industry, of gas injection inside a well annulus and shut-in of the vertical well are presented. The results conclude that the FCT solver is better and efficient for accurately capturing gas migration for multi-phase models with phase behaviour involving slip velocities given by algebraic relations.

Godunov Method for Calculating Multicomponent Heterogeneous Medium Flows

The modified Godunov method intended for integrating the nondivergent systems that describe a multivelocity heterogeneous mixture flow is presented. The linearized Riemann solver has been used in solving the Riemann problems.

An RKDG finite element method for the one-dimensional inviscid compressible gas dynamics equations in a Lagrangian coordinate

In this paper, Runge—Kutta Discontinuous Galerkin (RKDG) finite element method is presented to solve the one-dimensional inviscid compressible gas dynamic equations in a Lagrangian coordinate. The equations are discretized by the DG method in space and the temporal discretization is accomplished by the total variation diminishing Runge—Kutta method. A limiter based on the characteristic field decomposition is applied to maintain stability and non-oscillatory property of the RKDG method. For multi-medium fluid simulation, the two cells adjacent to the interface are treated differently from other cells. At first, a linear Riemann solver is applied to calculate the numerical flux at the interface. Numerical examples show that there is some oscillation in the vicinity of the interface. Then a nonlinear Riemann solver based on the characteristic formulation of the equation and the discontinuity relations is adopted to calculate the numerical flux at the interface, which suppresses the oscillation successfully. Several single-medium and multi-medium fluid examples are given to demonstrate the reliability and efficiency of the algorithm.

Efficient implementation of ADER schemes for Euler and magnetohydrodynamical flows on structured meshes – Speed comparisons with Runge–Kutta methods

ADER (Arbitrary DERivative in space and time) methods for the time-evolution of hyperbolic conservation laws have recently generated a fair bit of interest. The ADER time update can be carried out in a single step, which is desirable in many applications. However, prior papers have focused on the theory while downplaying implementation details. The purpose of the present paper is to make ADER schemes accessible by providing two useful formulations of the method as well as their implementation details on three-dimensional structured meshes. We therefore provide a detailed formulation of ADER schemes for conservation laws with non-stiff source terms in nodal as well as modal space along with useful implementation-related details. A good implementation of ADER requires a fast method for transcribing from nodal to modal space and vice versa and we provide innovative transcription strategies that are computationally efficient. We also provide details for the efficient use of ADER schemes in obtaining the numerical flux for conservation laws as well as electric fields for divergence-free magnetohydrodynamics (MHD). An efficient WENO-based strategy for obtaining zone-averaged magnetic fields from face-centered magnetic fields in MHD is also presented. Several explicit formulae have been provided in all instances for ADER schemes spanning second to fourth orders.The schemes catalogued here have been implemented in the first author’s RIEMANN code. The speed of ADER schemes is shown to be almost twice as fast as that of strong stability preserving Runge–Kutta time stepping schemes for all the orders of accuracy that we tested. The modal and nodal ADER schemes have speeds that are within ten percent of each other. When a linearized Riemann solver is used, the third order ADER schemes are half as fast as the second order ADER schemes and the fourth order ADER schemes are a third as fast as the third order ADER schemes. The third order ADER scheme, either with an HLL or linearized Riemann solver, represents an excellent upgrade path for scientists and engineers who are working with a second order Runge–Kutta based total variation diminishing (TVD) scheme. Several stringent test problems have been catalogued. [Display omitted]

An upwinded state approximate Riemann solver

Stability is achieved in most approximate Riemann solvers through ‘flux upwinding’, where the flux at the interface is arrived at by adding a dissipative term to the average of the left and right flux. Motivated by the existence of a collapsed interface state in the gas-kinetic Bhatnagar–Gross–Krook (BGK) method, an alternative approach to upwinding is attempted here; an interface state is arrived at by taking an upwinded average of left and right states, and then the flux is calculated as a function of this ‘collapsed’ interface state. This so called ‘state-upwinding’ approach gives rise to a new scheme called the linearized Riemann solver for the Euler and Navier–Stokes equations. The scheme is shown to be closely associated with the Roe scheme. It is, however, computationally less expensive and gives qualitatively comparable results over a wide range of problems. Most importantly, this scheme is found to preserve stationary contacts while not exhibiting the carbuncle phenomenon which plagues the Roe and other contact-preserving schemes. The scheme is therefore motivated as a new starting point to analyze the origin of the carbuncle phenomenon. Copyright © 2011 John Wiley & Sons, Ltd.

Accretion of a massive magnetized torus on a rotating black hole

We present numerical simulations of the axisymmetric accretion of a massive magnetized plasma torus on a rotating black hole. We use a realistic equation of state, which takes into account neutrino cooling and energy loss due to nucleus dissociations. The calculation are performed in the ideal relativistic MHD approximation using an upwind conservative scheme that is based on a linear Riemann solver and the constrained transport method to evolve the magnetic field. The gravitational attraction of the black hole is introduced via the Kerr metric in the Kerr–Schild coordinates. We simulate various magnetic field configurations and torus models, both optically thick and thin for neutrinos. We have found an effect of alternation of the magnetic field orientation in the ultrarelativistic jet formed as a result of the collapse. The calculations show evidence for heating of the wind surrounding the collapsar by the shock waves generated at the jet–wind border. It is shown that the neutrino cooling does not significantly change either the structure of the accretion flow or the total energy release of the system. The angular momentum of the accreting matter defines the time scale of the accretion. Due to the absence of the magnetic dynamo in our calculations, the initial strength and topology of the magnetic field determines the magnetization of the black hole, jet formation properties and the total energy yield. We estimate the total energy of accretion which transformed to jets as 1.3 × 10 52 ergs which was sufficient to explain hypernova explosions like GRB 980425 or GRB 030329.

A hybrid scheme for gas–dust systems stiffly coupled via viscous drag

We present a stable and convergent method for studying a system of gas and dust, coupled through viscous drag in both non-stiff and stiff regimes. To account for the effects of dust drag in the update of the fluid quantities, we employ a fluid description of the dust component and study the modified gas-dust hyperbolic system following the approach in Miniati and Colella [19]. In addition to two entropy waves for the gas and dust components, respectively, the extended system includes three waves driven partially by gas pressure and partially by dust drift, which, in the limit of vanishing coupling, tend to the two original acoustic waves and the unhindered dust streaming. Based on this analysis we formulate a predictor step providing first order accurate reconstruction of the time-averaged state variables at cell interfaces, whence a second order accurate estimate of the conservative fluxes can be obtained through a suitable linearized Riemann solver. The final source term update is carried out using a one-step, second order accurate, L-stable, predictor corrector asymptotic method (the α -QSS method suggested by Mott et al. [21]). This procedure completely defines a two-fluid method for gas-dust system. Using the updated fluid solution allows us to then advance the individual particle solutions, including self-consistently the time evolution of the gas velocity in the estimate of the drag force. This is done with a suitable particle scheme also based on the α -QSS method. A set of benchmark problems shows that our method is stable and convergent. When dust is modeled as a fluid (two-fluid) second order accuracy is achieved in both stiff and non-stiff regimes, whereas when dust is modeled with particles (hybrid) second order is achieved in the non-stiff regime and first order otherwise.

An all-speed relaxation scheme for interface flows with surface tension

We consider interface flows where compressibility and capillary forces (surface tension) are significant. These flows are described by a non-conservative, unconditionally hyperbolic multiphase model. The numerical approximation is based on finite-volume method for unstructured grids. At the discrete level, the surface tension is approximated by a volume force (CSF formulation). The interface physical properties are recovered by designing an appropriate linearized Riemann solver (Relaxation scheme) that prevents spurious oscillations near material interfaces. For low-speed flows, a preconditioning linearization is proposed and the low Mach asymptotic is formally recovered. Numerical computations, for a bubble equilibrium, converge to the required Laplace law and the dynamic of a drop, falling under gravity, is in agreement with experimental observations.

A parallel adaptive method for simulating shock-induced combustion with detailed chemical kinetics in complex domains

An adaptive finite volume approach is presented to accurately simulate shock-induced combustion phenomena in gases, particularly detonation waves. The method uses a Cartesian mesh that is dynamically adapted to embedded geometries and flow features by using regular refinement patches. The discretisation is a reliable linearised Riemann solver for thermally perfect gas mixtures; detailed kinetics are considered in an operator splitting approach. Besides easily reproducible ignition problems, the capabilities of the method and its parallel implementation are quantified and demonstrated for fully resolved triple point structure investigations of Chapman–Jouguet detonations in low-pressure hydrogen–oxygen–argon mixtures in two and three space dimensions.

Railgun experiment and computer simulation of hypervelocity impact of lexan projectile on aluminum target

Results of hypervelocity impact tests at an electromagnetic launcher with plasma armature are described. The tests were performed with a thick duralumin disk of 80–100 mm diameter and of 40 mm height impacted by a lexan cylinder of 1.5–3 g mass. The projectile velocity amounted to 4–5.5 km/s. The launcher operational characteristics are described. In the tests, various configurations of the target fracture were obtained including cratering and spall cracks, disks and cones. The computer simulations were accomplished by means of a 2D axisymmetrical SPH code using interparticle contact algorithms. In these algorithms, stresses and velocity at the contact points of Lagrangian particles are computed by the linear Riemann solver. For description of material failure, a model based on a threshold stress criterion was used. The main features of material distribution and target fracture are reproduced in simulations.

A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics

A new multi-state Harten–Lax–van Leer (HLL) approximate Riemann solver for the ideal magnetohydrodynamic (MHD) equations is developed based on the assumption that the normal velocity is constant over the Riemann fan. This assumption is same as that used in the HLLC (“C” denotes Contact) approximate Riemann solver for the Euler equations. From the assumption, it is naturally derived that the Riemann fan should consist of four intermediate states for B x ≠ 0, whereas the number of the intermediate states is reduced to two when B x = 0. Since the intermediate states satisfied with all jump conditions in this approximate Riemann system are analytically obtained, the multi-state HLL Riemann solver can be constructed straightforwardly. It is shown that this solver can exactly resolve isolated discontinuities formed in the MHD system, and hence named as HLLD Riemann solver. (Here, “D” stands for Discontinuities.) It is also analytically proved that the HLLD Riemann solver is positively conservative like the HLLC Riemann solver. Indeed, the HLLD Riemann solver corresponds to the HLLC Riemann solver when the magnetic field vanishes. Numerical tests demonstrate that the HLLD Riemann solver is more robust and efficient than the linearized Riemann solver, and its resolution is equally good. It indicates that the HLLD solver must be useful in practical applications for the ideal MHD equations.

A five equation reduced model for compressible two phase flow problems

This paper studies an Eulerian diffuse interface model for the simulation of compressible multifluid and two-phase flow problems. We first show how to derive this model from a seven equation, two pressure, two velocity model of Baer–Nunziato type using an asymptotic analysis in the limit of zero relaxation time. We then study the mathematical properties of the system, the structure of the waves, the expression of the Riemann’s invariants and the existence of a mathematical entropy. We also describe two different numerical approximation schemes for this system. The first one relies on a linearized Riemann solver while the second uses more heavily the mathematical structure of the system and relies on a linearization of the characteristic relations. Finally, we present some numerical experiments and comparisons with the results obtained by the two pressure, two velocity model as well as some test cases and comparisons with another five equation model recently proposed for interface computations between compressible fluids.