Two-dimensional Riemann Problem Research Articles

A discrete unified gas kinetic scheme with sparse velocity grid for rarefied gas flows

In this paper, a discrete unified gas kinetic scheme (DUGKS) with a sparse grid method applied in velocity space (DUGKS-SG) is proposed for simulating rarefied gas flows. The DUGKS-SG decomposes the computationally demanding problem into smaller and independent subproblems, thereby reducing the computational costs and exhibiting good parallelism. Several numerical tests, including the two-dimensional Riemann problem and the lid-driven microcavity flow, have been conducted to validate the performance of the DUGKS-SG. Comparisons with the original DUGKS and the Direct Simulation Monte Carlo (DSMC) method demonstrate that DUGKS-SG can provide satisfactory results with improved efficiency. Specifically, a maximum speedup of 9.486 for a 2D case with 7 CPU cores and 13.035 for a 3D case with 8 CPU cores can be achieved. These results suggest that the proposed DUGKS-SG can serve as an efficient numerical method for rarefied gas flow simulations.

A genuinely three-dimensional Riemann solver based on self-similar variables in curvilinear coordinates

A genuinely three-dimensional Riemann solver named MuSIC-3DC (Multidimensional, Self-similar, strongly-Interacting, Consistent, Three-dimensional, Curvilinear coordinates) in curvilinear coordinates is proposed to simulate three-dimensional complex engineering problems. Following Balsara's idea, a three-dimensional Riemann problem at each grid vertex is studied. In order to accurately capture contact discontinuities, the linear variations are assumed in the strongly-interacting zone. Also, the self-similar variables are adopted to simplify the derivation according to the self-similar evolution of the Riemann problem, and the strongly-interacting state could be determined by combining the contributions of all the two-dimensional Riemann problems at the surfaces bounded the strongly-interacting state. Besides, the one-dimensional Riemann problem at the center of each cell interface is solved by the one-dimensional HLLE scheme. After these, the numerical flux at each cell interface is obtained by assembling the three-dimensional fluxes at the vertices and the one-dimensional flux at the center of the cell interface. Several numerical test cases are simulated to validate the proposed solver. The spherical blast wave and three-dimensional Riemann problems indicate that the MuSIC-3DC scheme is capable of capturing three-dimensional self-similar flow structures accurately and suppressing the mesh imprinting phenomenon effectively. The transonic case demonstrates the MuSIC-3DC scheme's high resolution in capturing shock waves. In the hypersonic cases, the MuSIC-3DC scheme is more accurate in predicting stagnation and reattachment heating loads. Moreover, the MuSIC-3DC scheme is capable of obtaining more physical surface heating distribution by considering the information traveling both normal and transverse to the cell interfaces.

Weighted compact nonlinear scheme based on smooth scale separation for self-adaptive turbulence eddy simulations

Multiscale compressible turbulent flow is widely present in the aerospace field, and the co-existence of shock wave discontinuities and multi-scale turbulence in the flow field poses significant challenges for high-fidelity numerical simulation. To design high-order nonlinear scheme to improve the accuracy of complex turbulence simulation with stable shock wave capture, a weighted compact nonlinear scheme – targeted essentially non-oscillatory (WCNS-T) scheme is developed based on targeted essentially non-oscillatory scheme within the WCNS framework. Also, an improved variant of WCNS-T with smooth scale separation (denoted as WCNS-ST) is proposed from the perspective of both numerical convergence and accuracy. The numerical properties of different nonlinear weights are discussed in classical numerical problems, including spectral properties analysis, and two-dimensional Riemann problem. Then, the newly developed WCNS-ST scheme based on smooth scale separation was used to combine a unified turbulence model called Self-Adaptive Turbulence Eddy Simulation (SATES), which is applied to simulate subsonic flow past a 30P30N airfoil and supersonic base flow. The good convergence and numerical results are obtained using the newly developed WCNS-ST scheme, whereas WCNS-T shows worse convergence. In addition, adaptive dissipation control is further introduced for the WCNS-ST scheme, and the new variant (denoted as WCNS-STA) still shows good residual convergence and predicts more accurate results due to low numerical dissipation. The above results indicate that the proposed smooth scale separation and its WCNS-STA scheme have a good performance in SATES simulations of multiscale flow problems, making it more potential for complex engineering applications.

Hybrid compressible lattice Boltzmann method for supersonic flows with strong discontinuities

Within the framework of the hybrid recursive regularized lattice Boltzmann (HRR-LB) model, we propose a novel hybrid compressible LB method to ensure the conservation of total energy in simulating compressible flows with strong discontinuities. This method integrates a LB solver to handle the mass and momentum conservation equations via collision-streaming steps on standard lattices, while a finite volume method (FVM) is employed for the conservation of the total energy equation. The flux reconstruction in the FVM is achieved through a momentum coupled method (MCM). The interface momentum, crucial for reconstructing the convective fluxes and determining the upwind extrapolation of passive scalar quantities in MCM, is derived from the LB method. The validity and accuracy of the proposed method are evaluated through six test cases: (I) isentropic vortex convection in subsonic and supersonic regimes; (II) non-isothermal acoustic pulse; (III) one-dimensional Riemann problems; (IV) two-dimensional Riemann problem; (V) double Mach reflection of a Mach 10 shock wave; and (VI) shock–vortex interaction. Numerical results demonstrate that this method surpasses the previous HRR-LB model by Guo et al. [“Improved standard thermal lattice Boltzmann model with hybrid recursive regularization for compressible laminar and turbulent flows,” Phys. Fluids 32, 126108 (2020)] in terms of accuracy and robustness when dealing with strong shock waves.

On the impact of dimensionally-consistent and physics-based inner products for POD-Galerkin and least-squares model reduction of compressible flows

Model reduction of the compressible Euler equations based on proper orthogonal decomposition (POD) and Galerkin orthogonality or least-squares residual minimization requires the selection of inner product spaces in which to perform projections and measure norms. The most popular choice is the vector-valued L2(Ω) inner product space. This choice, however, yields dimensionally-inconsistent reduced-order model (ROM) formulations which often lack robustness. In this work, we try to address this weakness by studying a set of dimensionally-consistent inner products with application to the compressible Euler equations. First, we demonstrate that non-dimensional inner products have a positive impact on both POD and Galerkin/least-squares ROMs. Second, we further demonstrate that physics-based inner products based on entropy principles result in drastically more accurate and robust ROM formulations than those based on non-dimensional L2(Ω) inner products. As test cases, we consider the following problems: the one-dimensional Sod shock tube, a two-dimensional Riemann problem, a two-dimensional Kelvin-Helmholtz instability, and two-dimensional homogeneous isotropic turbulence.

A Study of Traveling Wave Structures and Numerical Investigation of Two-Dimensional Riemann Problems with Their Stability and Accuracy

The Riemann wave system has a fundamental role in describing waves in various nonlinear natural phenomena, for instance, tsunamis in the oceans. This paper focuses on executing the generalized exponential rational function approach and some numerical methods to obtain a distinct range of traveling wave structures and numerical results of the two-dimensional Riemann problems. The stability of obtained traveling wave solutions is analyzed by satisfying the constraint conditions of the Hamiltonian system. Numerical simulations are investigated via the finite difference method to verify the accuracy of the obtained results. To extract the approximation solutions to the underlying problem, some ODE solvers in FORTRAN software are applied, and outcomes are shown graphically. The stability and accuracy of the numerical schemes using Fourier’s stability method and error analysis, respectively, to increase the reassurance are investigated. A comparison between the analytical and numerical results is obtained and graphically provided. The proposed methods are effective and practical to be applied for solving more partial differential equations (PDEs).

Two-Dimensional Riemann Problems: Transonic Shock Waves and Free Boundary Problems

Two-Dimensional Riemann Problems: Transonic Shock Waves and Free Boundary Problems

A parallel discrete unified gas kinetic scheme on unstructured grid for inviscid high-speed compressible flow simulation

The discrete unified gas kinetic scheme (DUGKS) is a recently devised approach to simulate multiscale flows based on the kinetic models, which also shows distinct features for continuum flows. Most of the existing DUGKS are sequential or based on structured grids, thus limiting their scope of application in engineering. In this paper, a parallel DUGKS for inviscid high-speed compressible flows on unstructured grids is proposed. In the framework of the DUGKS, the gradients of the distribution functions are calculated by a least-square method. To parallelize the method, a graph-based partitioning method is employed to guarantee the load balancing and minimize the communication among processors. The method is validated by several benchmark problems, i.e., a two-dimensional (2D) Riemann problem, 2D subsonic flows passing two benchmark airfoils, a 2D regular shock reflection problem, 2D supersonic flows (Mach numbers are 3 and 5) around a cylinder, an explosion in a three-dimensional (3D) box, a 3D subsonic flow around the Office National d'Etudes et de Recherches Aérospatiales M6 wing, a 3D hypersonic flow (Mach number is 10) around a hemisphere, and a supersonic flow over the Northrop YF-17 fighter model. The numerical results show good agreement with the published results, and the present method is robust for a wide range of Mach numbers, from subsonic to hypersonic. The parallel performance results show that the proposed method is highly parallel scalable, where an almost linear scalability with 93% parallel efficiency is achieved for a 3D problem with over 55 × 106 tetrahedrons on a supercomputer with up to 4800 processors.

Riemann problems for a hyperbolic system of nonlinear conservation laws from the Liou–Steffen pressure system

This paper is devoted to a hyperbolic system of nonlinear conservation laws, that is, the pressure system independent of density and energy from the Liou–Steffen flux-splitting scheme on the compressible Euler equations. First, the one-dimensional Riemann problem is solved with eight kinds of structures. Second, the two-dimensional Riemann problem is discussed; the solution reveals a variety of geometric structures; by the generalized characteristic analysis method and studying the pointwise interactions of waves, we construct 29 kinds of structures of solution consisting of shocks, rarefaction waves and contact discontinuities; the theoretical analysis is confirmed by numerical simulations.

A class of structurally complete approximate Riemann solvers for trans- and supercritical flows with large gradients

The wave structure of approximate Riemann solvers has a significant impact on the accuracy and computational requirements of finite volume codes. We propose a class of structurally complete approximate Riemann solvers (StARS) and provide an efficient means for analytically restoring the expansion wave to pre-existing three-wave solvers. The method analytically restores the expansion, is valid for arbitrary thermodynamics, and has comparable complexity to the popular Harten-Hyman entropy fix. The StARS modification is applied to a Roe scheme, resulting in Roe-StARS with noticeable improvements in unsteady transcritical and supercritical conditions with large flow gradients. A novel scaling analysis is performed on the flow conditions that cause rarefaction fluxes and the magnitude of errors if the rarefaction is omitted. Four test cases are examined: a transcritical shock tube, a shock tube with periodic bounds resulting in interfering shocks and rarefactions, a two-dimensional Riemann problem, and a “gradient” Riemann problem—a variant on the traditional Riemann problem featuring an initial gradient of varying slope rather than an initial step function. The results highlight the complex causes and effects of entropy violations, and encourage further study of StARS-type solvers for modern flow problems in which high flow speeds, large gradients, and non-ideal thermodynamics are increasingly common.

Restoring the conservativity of characteristic-based segregated models: Application to the hybrid lattice Boltzmann method

A general methodology is introduced to build conservative numerical models for fluid simulations based on segregated schemes, where mass, momentum, and energy equations are solved by different methods. It is especially designed here for developing new numerical discretizations of the total energy equation and adapted to a thermal coupling with the lattice Boltzmann method (LBM). The proposed methodology is based on a linear equivalence with standard discretizations of the entropy equation, which, as a characteristic variable of the Euler system, allows efficiently decoupling the energy equation with the LBM. To this extent, any LBM scheme is equivalently written under a finite-volume formulation involving fluxes, which are further included in the total energy equation as numerical corrections. The viscous heat production is implicitly considered thanks to the knowledge of the LBM momentum flux. Three models are subsequently derived: a first-order upwind, a Lax–Wendroff, and a third-order Godunov-type schemes. They are assessed on standard academic test cases: a Couette flow, entropy spot and vortex convections, a Sod shock tube, several two-dimensional Riemann problems, and a shock–vortex interaction. Three key features are then exhibited: (1) the models are conservative by construction, recovering correct jump relations across shock waves; (2) the stability and accuracy of entropy modes can be explicitly controlled; and (3) the low dissipation of the LBM for isentropic phenomena is preserved.

A high resolution numerical method for solving atmospheric models

We develop a numerical method to solve atmospheric model equations, which is a hybrid of the exponential time differencing (ETD) and the equation solving solution gradient (ESSG) method. The Coriolis force term due to geostrophic effects is handled by the ETD method and the convective flux balance terms are implemented by the ESSG method. In the ESSG method, gradients of solution variables are solved by their corresponding difference equations instead of interpolating from nearby solution values as in the conventional finite volume method. In this way, significant features of the model equations can be preserved and correctly considered to provide a reliable simulation. Feasibility of the proposed method is verified in a number of applications and shows it is more accurate than other common schemes, including the first-order upwind (UD) scheme and the second-order minimum modulus (minmod) scheme. We apply the method to a one-dimensional shock tube, a two-dimensional Riemann problem, and an atmospheric vortex flow. Our studies indicate that the exponential time differencing together with the equation solving solution gradient method provides very accurate simulations for the atmospheric flow model equations.

On a Two-Dimensional Riemann Problem for Hyperbolic System of Nonlinear Conservation Laws

On a Two-Dimensional Riemann Problem for Hyperbolic System of Nonlinear Conservation Laws

High resolution central scheme using a new upwind slope limiter for hyperbolic conservation laws

In this paper, a new high resolution central-upwind scheme for hyperbolic conservation laws is proposed, based on a new upwind biased slope limiter. The new scheme is defined as a correction of KT central scheme by applying the new slope limiter. The new limiter uses more upwind information for the piecewise linear interpolations, so that it has an upwind nature. The main features of the new scheme are high resolution and nonoscillatory. At the same time, it retains the efficiency and simplicity of central scheme. The total-variation diminishing (TVD) property and maximum principle of the new scheme are proved. The numerical experiments, including one-dimensional, two-dimensional Riemann problems and double Mach reflection problem for Euler equations, show the desired resolution and robustness of the new scheme.

Improvement of the genuinely multidimensional ME-AUSMPW scheme for subsonic flows

Since proposed, the genuinely multidimensional Riemann solvers have been attracting more attentions because it considers the waves normal and parallel to the cell interfaces at the same time. However, all these methods are built upon the hyperbolic property of the compressible Euler equations. Therefore, they can not be applied to solve the Euler equations in steady subsonic cases which are with the elliptic property. In this study, a novel multidimensional Riemann solver called MULE-AUSMPWM (Multidimensional E-AUSMPW Modified) is proposed. It is built upon the Zha–Bilgen splitting procedure and adopts the Balsara's multidimensional wave model as the ME-AUSMPW scheme. In addition, it improves the ME-AUSMPW scheme's accuracy at subsonic speeds by avoiding simulating the convective fluxes in fully upwind manners. Extensive numerical tests, such as the two dimensional Euler problem, the one dimensional moving contact discontinuity case, the two-dimensional double Mach reflection case, the two-dimensional Riemann problem, and the turbulent flow past a 2d NACA0012 airfoil case, are conducted. Results illustrate that the MULE-AUSMPWM scheme proposed is capable of simulating compressible complex flows and improving the existing multidimensional Riemann solvers' accuracy at subsonic speeds remarkably.

SPH-ALE Scheme for Weakly Compressible Viscous Flow with a Posteriori Stabilization

A highly accurate SPH method with a new stabilization paradigm has been introduced by the authors in a recent paper aimed to solve Euler equations for ideal gases. We present here the extension of the method to viscous incompressible flow. Incompressibility is tackled assuming a weakly compressible approach. The method adopts the SPH-ALE framework and improves accuracy by taking high-order variable reconstruction of the Riemann states at the midpoints between interacting particles. The moving least squares technique is used to estimate the derivatives required for the Taylor approximations for convective fluxes, and also provides the derivatives needed to discretize the viscous flux terms. Stability is preserved by implementing the a posteriori Multi-dimensional Optimal Order Detection (MOOD) method procedure thus avoiding the utilization of any slope/flux limiter or artificial viscosity. The capabilities of the method are illustrated by solving one- and two-dimensional Riemann problems and benchmark cases. The proposed methodology shows improvements in accuracy in the Riemann problems and does not require any parameter calibration. In addition, the method is extended to the solution of viscous flow and results are validated with the analytical Taylor–Green, Couette and Poiseuille flows, and lid-driven cavity test cases.

Rarefaction Wave Interaction and Shock-Rarefaction Composite Wave Interaction for a Two-Dimensional Nonlinear Wave System

In order to construct global solutions to two-dimensional (2D for short) Riemann problems for nonlinear hyperbolic systems of conservation laws, it is important to study various types of wave interactions. This paper deals with two types of wave interactions for a 2D nonlinear wave system with a nonconvex equation of state: Rarefaction wave interaction and shock-rarefaction composite wave interaction. In order to construct solutions to these wave interactions, the authors consider two types of Goursat problems, including standard Goursat problem and discontinuous Goursat problem, for a 2D self-similar nonlinear wave system. Global classical solutions to these Goursat problems are obtained by the method of characteristics. The solutions constructed in the paper may be used as building blocks of solutions of 2D Riemann problems.

Global solutions of a two-dimensional Riemann problem for the pressure gradient system

<p style="text-indent:20px;">We are concerned with a two-dimensional Riemann problem for the pressure gradient system that is a hyperbolic system of conservation laws. The Riemann initial data consist of four constant states in four sectorial regions such that two shocks and two vortex sheets are generated between the adjacent states. The solutions keep the four constant states and four planar waves outside the outer sonic circle in the self-similar coordinates, while the two shocks keep planar until meeting the outer sonic circle at two different points and then generate a diffracted shock to connect these points, whose location is <em>apriori</em> unknown. Then the problem can be formulated as a free boundary problem, in which the diffracted transonic shock is the one-phase free boundary to connect the two points, while the other part of the sonic circle forms a fixed boundary. We establish the global existence of a solution and the optimal Lipschitz regularity of both the diffracted shock across the two points and the solution across the outer sonic boundary. Then this Riemann problem is solved globally, whose solution contains two vortex sheets and one global shock containing the two originally separated shocks generated by the Riemann data.</p>

Development of accurate and robust genuinely two-dimensional HLL-type Riemann solver for compressible flows

When using the conventional direction splitting method to calculate multidimensional high speed gasdynamical flows, Riemann solvers capable of resolving contact surface and shear wave accurately will suffer from different forms of shock instability, such as the notorious carbuncle phenomenon. The stability analysis shows that the lack of dissipation in the direction transverse to the shock front leads to the shock instability of low-dissipation HLLEM solver. To overcome this defect, an accurate and carbuncle-free genuinely two-dimensional HLL-type Riemann solver is proposed. Using Zha-Bilgen splitting method, the flux vector of two-dimensional Euler equations is split into the convective flux and pressure flux. An algorithm similar to AUSM+ scheme is adopted to calculate the convective flux and the pressure flux is calculated by the low-dissipation HLLEM scheme. Following Balsara’s idea, the genuinely two-dimensional properties of the new solver are achieved by solving the two-dimensional Riemann problem that considers transversal features of the flow at each vertex of the cell interface. Numerical results of benchmark tests demonstrate that the new solver has higher resolution and better robustness than the conventional HLLEM solver implemented in dimension by dimension.

Semi-Lagrangian lattice Boltzmann method for compressible flows.

This work thoroughly investigates a semi-Lagrangian lattice Boltzmann (SLLBM) solver for compressible flows. In contrast to other LBM for compressible flows, the vertices are organized in cells, and interpolation polynomials up to fourth order are used to attain the off-vertex distribution function values. Differing from the recently introduced Particles on Demand (PoD) method [Dorschner, Bösch, and Karlin, Phys. Rev. Lett. 121, 130602 (2018)PRLTAO0031-900710.1103/PhysRevLett.121.130602], the method operates in a static, nonmoving reference frame. Yet the SLLBM in the present formulation grants supersonic flows and exhibits a high degree of Galilean invariance. The SLLBM solver allows for an independent time step size due to the integration along characteristics and for the use of unusual velocity sets, like the D2Q25, which is constructed by the roots of the fifth-order Hermite polynomial. The properties of the present model are shown in diverse example simulations of a two-dimensional Taylor-Green vortex, a Sod shock tube, a two-dimensional Riemann problem, and a shock-vortex interaction. It is shown that the cell-based interpolation and the use of Gauss-Lobatto-Chebyshev support points allow for spatially high-order solutions and minimize the mass loss caused by the interpolation. Transformed grids in the shock-vortex interaction show the general applicability to nonuniform grids.