Approximate Riemann Solution Research Articles

A fully conservative sharp-interface method for compressible multiphase flows with phase change

In this paper, a fully conservative sharp-interface method is developed for compressible multiphase flows with phase change. To enforce conservation, the coupling between two different phases is achieved via the interfacial fluxes, which are determined by solving a general Riemann problem with phase change. In addition, thermodynamical consistency is guaranteed by taking account of latent heat in the phase-change Riemann problem. This is efficiently and robustly solved by a novel approximate Riemann solver, where the complex eight-dimensional root-finding procedure required in the exact Riemann solver is simplified to solely iterating the mass flux. Compared to the existing methods, the present approximate Riemann solution consists of four waves, at which the jump conditions are imposed strictly to enforce the conservation of mass, momentum, and energy. Furthermore, for the evaluation of the mass flux, through numerical simulations and theoretical analysis, we have demonstrated that the adjacent states of a phase interface should be used to ensure numerical consistency, which, to our knowledge, has not been reported in the literature. Various numerical examples, including the phase-change Riemann problem, the oscillating droplet problem, and the shock-droplet interaction problem, are simulated to validate the present method. The numerical results are in good agreement with the exact solutions and the previous simulations. It is shown that, with the present method, the phase change effects in compressible multiphase flows are well resolved with conservation strictly preserved.

A graphics processing unit-based robust numerical model for solute transport driven by torrential flow condition

Solute transport simulations are important in water pollution events. This paper introduces a finite volume Godunov-type model for solving a 4×4 matrix form of the hyperbolic conservation laws consisting of 2D shallow water equations and transport equations. The model adopts the Harten-Lax-van Leer-contact (HLLC)-approximate Riemann solution to calculate the cell interface fluxes. It can deal well with the changes in the dry and wet interfaces in an actual complex terrain, and it has a strong shock-wave capturing ability. Using monotonic upstream-centred scheme for conservation laws (MUSCL) linear reconstruction with finite slope and the Runge-Kutta time integration method can achieve second-order accuracy. At the same time, the introduction of graphics processing unit (GPU)-accelerated computing technology greatly increases the computing speed. The model is validated against multiple benchmarks, and the results are in good agreement with analytical solutions and other published numerical predictions. The third test case uses the GPU and central processing unit (CPU) calculation models which take 3.865 s and 13.865 s, respectively, indicating that the GPU calculation model can increase the calculation speed by 3.6 times. In the fourth test case, comparing the numerical model calculated by GPU with the traditional numerical model calculated by CPU, the calculation efficiencies of the numerical model calculated by GPU under different resolution grids are 9.8–44.6 times higher than those by CPU. Therefore, it has better potential than previous models for large-scale simulation of solute transport in water pollution incidents. It can provide a reliable theoretical basis and strong data support in the rapid assessment and early warning of water pollution accidents.

3D Cell-centered hydrodynamics with subscale closure model and multi-material remap

We extend a higher-order finite volume cell-centered hydrodynamic (CCH) formulation to include an interface-aware subscale closure model and a multi-material remap for simulating 3D compressible hydrodynamic problems within an arbitrary Lagrangian-Eulerian (ALE) framework. This CCH formulation involves a multidirectional approximate Riemann solution using quadratic polynomial reconstructions of the stress tensor and the velocity. At the subscale level, we determine pair-wise material interactions by solving a distinct approximate Riemann problem at the common interface, using the volume of fluids (VOF) method to find the interface. Material interactions are constrained to ensure smooth pressure equilibration among materials. The accuracy and robustness of the ALE method is demonstrated by simulating a suite of 3D Cartesian multi-material problems covering both gas and solid dynamics, where each test case has two or more materials.

On the use of tabulated equations of state for multi-phase simulations in the homogeneous equilibrium limit

The use of complex multi-parameter equations of state in computational fluid dynamics is limited due to their expensive evaluation. Tabulation methods help to overcome this limitation. We propose in this work a tabulation approach for real equations of state that is also suitable for multi-component flows and multi-phase flows with phase transition based on the homogeneous equilibrium method. The tabulation method is based on piecewise polynomials and allows adaptive refinement in state space, a local variation of the degree of the polynomials and even cut-cells to represent saturation lines. A detailed analysis of the benefits of the chosen tabulation approach is provided showing results of benchmark problems for nitrogen under sub- and trans-critical conditions. The tables are used in numerical simulations based on the finite volume approach. To provide reference solutions, an exact Riemann solver is constructed for the tabulated real EOS. Several numerical flux calculations based on approximate Riemann solutions and flux vector splitting approaches are modified for the tabulated equation of state. Besides the comparison for benchmark problems, two applications are depicted, a shock–bubble interaction and an injection of a trans-critical jet, which demonstrate the applicability of this approach to more complex flow simulations.

Performance Study of MUSCL Schemes Based on Different Numerical Fluxes

The numerical flux is an important element in the numerical model for solving shallow water flow problems in MUSCL scheme. It is used to determine the normal transport through the boundary of the control body. With the TVD numerical flux, because of the simple form of Lax–Friedrichs, it is adopted in many articles. In fact, there are many different numerical fluxes based on exact Riemann solutions or approximate Riemann solutions, and these numerical fluxes can be combined with the MUSCL scheme. For the one-dimensional and two-dimensional shallow water equations, starting from the accuracy and discontinuous capture ability, a series of numerical examples are simulated based on the six different numerical fluxes with MUSCL scheme. Through the result analysis, the paper find out the numerical fluxes which are more suitable for shallow water flow combined with MUSCL scheme.

Numerical Study of Hydrodynamic and Solute Transport with Discontinuous Flows in Coastal Water

This paper aimed to develop a depth-averaged explicit model for flow and pollutant transport in coastal waters based on the shallow water equations and the mass advection-diffusion equation. The proposed model was discretized using the finite volume method (FVM) with triangular cells. Then, it applied Roe’s approximate Riemann solution to compute the water momentum flux on the grid interfaces. This model enabled the higher accuracy in capturing the dry-wet moving fronts (discontinuous problems for flow and solute). The high-resolution scheme was evaluated to solve the advection and diffusion terms for mass transport. The model was verified by comparing the predictions of analytical solutions, laboratory tests, and other simulations for Gironde estuary with good computational accuracy. The developed model was also used to calculate the circulation and the motion of chemical oxygen demand (COD) pollutants from the sewage outfalls in the Zhuanghe coastal water with dry and wet moving boundaries. The research results showed that the residual current directions of spring and neap tides were basically the same in the Zhuanghe coastal water. However, the tide residual current of spring tide was slightly greater than that of the neap tide. In addition, there were tide residual currents from the northeast to the southwest in nearshore water and from the southwest to the northeast outside the banks, respectively. The tidal flows in the alongshore direction were strong, resulting in highly spread concentration distributions. In particular, the COD concentration reached some parts of the southern water. It could be seen that the excessive pollutant discharge from the sewage outfalls located at Zhuanghe district would cause serious pollution in aquaculture water near Shicheng island.

Numerical study on pollutant transport in Dalian bay based on hydrodynamic model

Based on the depth-averaged two-dimensional shallow water and pollutant transport equation, the coupling model of water flow and water quality with explicit scheme is developed in this study. The unstructured triangular grid is adopted to locally refine the mesh around sewage outlet or in high-gradient regions of terrain change for the coupling model. The finite volume method is applied to ensure the conservation of mass for each element. This hydrodynamic model applies the Roe solver approximate Riemann solution with second-order accuracy to compute the water momentum flux on the grid interface. Taking Dalian Bay as the research object, the numerical model established is used to simulate the hydrodynamic characteristics and pollutant transport process. The computed results of the tide level, flow current and flow direction agree well with the measured data in Dalian Bay. The spatial and temporal distribution of pollutant in water are analyzed and discussed in this study. Simulated results show that the two-dimensional hydrodynamic and pollutant transport model can accurately simulate the mass transport in coastal waters, and it can provide a scientific basis on coastal water environment protection for the research water.

Parallel Computation of a Dam-Break Flow Model Using OpenACC Applications

AbstractTwo key factors in dam-break modeling are accuracy and speed. Therefore, high-performance calculations are of great importance to the simulation of dam-break events. In this study, we develop a two-dimensional hydrodynamic model based on the finite volume method to simulate the dam-break flow routing process. Roe’s approximate Riemann solution is adopted to solve the interface flux of grid cells and accurately simulate the discontinuous flow. A graphics processing unit (GPU)-based parallel method, OpenACC, is used to realize parallel computing. Because an explicit discrete technique is used to solve the governing equations, and there is no correlation between grid calculations in a single time step, the parallel dam-break model can be easily realized by adding OpenACC directives to the loop structure of the grid calculations. To analyze the performance of the model, we considered the Pangtoupao flood storage area in China using a Nvidia Tesla K20c card and four different grid division schemes. By ...

Spurious Numerical Oscillations in the Preissmann Slot Method: Origin and Suppression

AbstractThis paper investigates the source of the spurious numerical oscillations often observed in simulations using the well-known Preissmann slot method and proposes a nonoscillatory numerical fix that can efficiently suppress the numerical oscillations. The root of these oscillations is identified by comparing the orbits calculated by a first-order Godunov type model with an ideal numerical orbit in the phase plane. It is found that in the very thin layer in the vicinity of the conduit roof the numerical model has insufficient numerical viscosity to avoid the often-observed oscillations. In order to remove these spurious oscillations, an approximate Riemann solution is proposed that automatically enhances the numerical viscosity whenever the water level is in the vicinity of the conduit roof and the pressurization of the conduit is proximate. A comparison of results from the proposed model with both experimental data and analytical solutions show that it can provide nonoscillatory solutions over a wid...

Parallel computation of a dam-break flow model using OpenMP on a multi-core computer

High-performance calculations are of great importance to the simulation of dam-break events, as discontinuous solutions and accelerated speed are key factors in the process of dam-break flow modeling. In this study, Roe’s approximate Riemann solution of the finite volume method is adopted to solve the interface flux of grid cells and accurately simulate the discontinuous flow, and shared memory technology (OpenMP) is used to realize parallel computing. Because an explicit discrete technique is used to solve the governing equations, and there is no correlation between grid calculations in a single time step, the parallel dam-break model can be easily realized by adding OpenMP instructions to the loop structure of the grid calculations. The performance of the model is analyzed using six computing cores and four different grid division schemes for the Pangtoupao flood storage area in China. The results show that the parallel computing improves precision and increases the simulation speed of the dam-break flow, the simulation of 320 h flood process can be completed within 1.6 h on a 16-kernel computer; a speedup factor of 8.64× is achieved. Further analysis reveals that the models involving a larger number of calculations exhibit greater efficiency and a higher rate of acceleration. At the same time, the model has good extendibility, as the speedup increases with the number of processor cores. The parallel model based on OpenMP can make full use of multi-core processors, making it possible to simulate dam-break flows in large-scale watersheds on a single computer.

A Non-oscillatory Preissmann Slot Method Based Numerical Model

Abstract The Preissmann Slot Method concept is implemented by a first-order Godunov type numerical model. In order to remove the spurious numerical oscillations, an approximate HLL Riemann solution is proposed to automatically augment the numerical viscosity whenever the water level is in the vicinity of the conduit roof and the pressurization of the conduit is proximate. The comparison of the model's results with analytical solutions show that the proposed model can provide non-oscillatory solutions over a wide range of the wave velocities between 10 to 1000 ms -1 . The results confirm that the proposed model also limits data smearing to a reasonable level.

A Lagrangian staggered grid Godunov-like approach for hydrodynamics

Much research in Lagrangian staggered-grid hydrodynamics (SGH) has focused on explicit viscosity models for adding dissipation to a calculation that has shocks. The explicit viscosity is commonly called “artificial viscosity”. Recently, researchers have developed hydrodynamic algorithms that incorporate approximate Riemann solutions on the dual grid [28,29,35,30,2,3]. This approach adds dissipation to the calculation via solving a Riemann-like problem. In this work, we follow the works of [28,29,35,30] and solve a multidirectional Riemann-like problem at the cell center. The Riemann-like solution at the cell center is used in the momentum and energy equations. The multidirectional Riemann-like problem used in this work differs from previous work in that it is an extension of the cell-centered hydrodynamics (CCH) nodal solution approach in [7]. Incorporating the multidirectional Riemann-like problem from [7] into SGH has merits such as the ability to resist mesh instabilities like hourglass null modes and chevron null modes. The approach is valid for complex multidimensional flows with strong shocks. Numerical details and test problems are presented.

Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes

This work is concerned with the numerical capture of stiff viscous shock solutions of Navier-Stokes equations for complex compressible materials, in the regime of large Reynolds numbers. After [2] and [6], a relevant numerical capture is known to require the satisfaction of an extended set of non classical Rankine-Hugoniot conditions due to the non conservation form of the governing PDE model. Here, we show how to enforce their validity at the discrete level without the need for solving local non linear algebraic problems. Non linearities are bypassed when introducing new averaging techniques which are proved to satisfy all the desirable stability properties when invoking suitable approximate Riemann solutions. A relaxation procedure is proposed to that purpose with the benefit of a fairly simple overall numerical method.

An iterative Riemann solver for systems of hyperbolic conservation laws, with application to hyperelastic solid mechanics

In this paper, we present a general iterative method for the solution of the Riemann problem for hyperbolic systems of PDEs. The method is based on the multiple shooting method for free boundary value problems. We demonstrate the method by solving one-dimensional Riemann problems for hyperelastic solid mechanics. Even for conditions representative of routine laboratory conditions and military ballistics, dramatic differences are seen between the exact and approximate Riemann solution. The greatest discrepancy arises from misallocation of energy between compressional and thermal modes by the approximate solver, resulting in nonphysical entropy and temperature estimates. Several pathological conditions arise in common practice and modifications to the method to handle these are discussed. These include points where genuine nonlinearity is lost, degeneracies, and eigenvector deficiencies that occur upon melting.

Numerical Solution of the Polymer System by Front Tracking

The paper describes the application of front tracking to the polymer system, an example of a nonstrictly hyperbolic system. Front tracking computes piecewise constant approximations based on approximate Riemann solutions and exact tracking of waves. It is well known that the front tracking method may introduce a blowup of the initial total variation for initial data along the curve where the two eigenvalues of the hyperbolic system are identical. It is demonstrated by numerical examples that the method converges to the correct solution after a finite time, and that this time decreases with the discretization parameter.

An investigation of random choice method for three-dimensional steady supersonic flows

In this paper, an unsplit random choice method (RCM) is developed and applied to numerically solve three-dimensional supersonic steady flow problems. In order to keep the contacts (slip surfaces) crisply resolved, a new Lagrangian formulation is employed. Due to the lack of exact solutions to 3D Riemann problems, approximate Riemann solutions in the weak sense are adopted. The RCM is thus as efficient as the deterministic TVD schemes, and yields almost identical results in the model problems. Copyright © 1999 John Wiley & Sons, Ltd.

Flux difference splittings and limiters for the resolution of contact discontinuities

Harten, Lax, and van Leer developed a theory of approximate Riemann solutions and used it to derive numerical methods for systems of conservation laws. In the first part of this paper, we reformulate this theory so that it can be used to compare various flux difference splitting formulae. In particular we show how to compare the resolution of contact discontinuities by various methods with that of the Roe method. Since the Roe method can resolve an isolated steady contact perfectly, it is a good standard for comparison. In the second part of the paper, we show that Harten's subcell resolution method is the ultimate compressive limiter. We discuss the differences between compressive and noncompressive limiters and show why it is important to choose the limiter to suit the problem at hand.

On Godunov-Type Schemes for Lagrangian Gas Dynamics

In this paper, Godunov-type schemes are considered for the equations of gas dynamics using Lagrangian coordinates. A Roe linearization is constructed for a general equation of state. It does not coincide with that for Eulerian coordinates. It is shown that this linearization fails in the vicinity of strong compressions, in the sense that the approximate Riemann solution contains unphysical states of negative specific volume. An algorithm to calculate a priori bounds for the smallest and largest signal velocity is obtained by correcting the signal velocities of this Roe linearization. These bounds are used within a very simple Godunov-type scheme which captures strong compressions very well. Numerical results are shown for several test problems.

Approximate Riemann solutions of the two-dimensional shallow-water equations

A finite-difference scheme based on flux difference splitting is presented for the solution of the two-dimensional shallow-water equations of ideal fluid flow. A linearised problem, analogous to that of Riemann for gasdynamics, is defined and a scheme, based on numerical characteristic decomposition, is presented for obtaining approximate solutions to the linearised problem. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second-order scheme which avoids non-physical, spurious oscillations. An extension to the two-dimensional equations with source terms, is included. The scheme is applied to a dam-break problem with cylindrical symmetry.

Simplified Second-Order Godunov-Type Methods

Harten, Lax, and van Leer showed how to construct a simple approximate Riemann solution which contains only one intermediate step. This construction assumes that you have a priori bounds on the smallest and largest signal velocities in the exact Riemann solution.Here we propose a number of algorithms for obtaining these bounds. Heuristic arguments are presented to support our choice of bounds. We derive first-order schemes which use these approximate Riemann solutions and show their relationship to known finite difference schemes.Next, we use the approach of van Leer et al. [1] to construct second-order schemes based on these approximate Riemann solutions. Of particular interest is a central difference scheme requiring no upwind switches. This scheme is only slightly more complex than standard predictor–corrector finite difference schemes. Preliminary numerical results are presented which show that these schemes are nonoscillatory, have good shock resolution and produce results which are competitive with those produced by more complex second order Godunov-type schemes.