Classical Riemann Problem Research Articles

A mathematical model for flow field with condensation discontinuity: One‐dimensional Euler equations with singular sources

Condensation induces both heat addition and mass consumption effects on the carrier flow field, leading to complex wave patterns. In this study, we aim to elucidate the structures of flow field under the influence of condensation. To achieve this, we present a mathematical model, which is formulated using Euler equations with a singular source term. Given the difficulties posed by both weak solution theory and the Dal Maso–LeFloch–Murat (DLM) theory (J. Math. Pures Appl. 483‐548 (1995)) in defining solutions to the Riemann problem, we propose a novel approach. This involves the coupling of weak solutions within two subregions, taking into consideration the discontinuity of source term. Firstly, we identify two stationary discontinuities originating from the singular source term—namely, stationary waves and composite waves. Admissibility criterions are developed for these discontinuities to facilitate the selection of physically meaningful solutions. Secondly, we employ a double classical Riemann problems (CRPs) framework to analyze the structures of Riemann solution. Our analysis reveals that the Riemann solution may exhibit seven structures with stationary wave and four structures with composite wave. Finally, the proposed model and its theoritical results are applied to validate the structure of the flow field with condensation discontinuities. For the pure heat addition problem, we successfully demonstrate all wave patterns of the flow field, perfectly matching the outcomes of numerical experiments.

The Riemann problem for a generalized Burgers equation with spatially decaying sound speed. I Large‐time asymptotics

In this paper, we consider the classical Riemann problem for a generalized Burgers equation, u t + h α ( x ) u u x = u x x , $$\begin{equation*} u_t + h_{\alpha }(x) u u_x = u_{xx}, \end{equation*}$$ with a spatially dependent, nonlinear sound speed, h α ( x ) ≡ ( 1 + x 2 ) − α $h_{\alpha }(x) \equiv (1+x^2)^{-\alpha }$ with α > 0 $\alpha >0$ , which decays algebraically with increasing distance from a fixed spatial origin. When α = 0 $\alpha =0$ , this reduces to the classical Burgers equation. In this first part of a pair of papers, we focus attention on the large-time structure of the associated Riemann problem, and obtain its detailed structure, as t → ∞ $t\rightarrow \infty$ , via the method of matched asymptotic coordinate expansions (this uses the classical method of matched asymptotic expansions, with the asymptotic parameters being the independent coordinates in the evolution problem; this approach is developed in detail in the monograph of Leach and Needham, as referenced in the text), over all parameter ranges. We identify a significant bifurcation in structure at α = 1 2 $\alpha =\frac{1}{2}$ .

Riemann problem for constant flow with single-point heating source

This work focuses on the Riemann problem of Euler equations with global constant initial conditions and a single-point heating source, which comes from the physical problem of heating one-dimensional inviscid compressible constant flow. In order to deal with the source of Dirac delta-function, we propose an analytical frame of double classical Riemann problems (CRPs) coupling, which treats the fluids on both sides of the heating point as two separate classical Riemann problems and then couples them. Under the double CRPs frame, the Riemann solution is self-similar, and three types of structures are found. The theoretical analysis is also supported by the numerical simulation. Furthermore, the uniqueness of the Riemann solution is established with some restrictions on the Mach number of the initial condition.

ADER Methods for Hyperbolic Equations with a Time-Reconstruction Solver for the Generalized Riemann Problem: the Scalar Case

The ADER approach to solve hyperbolic equations to very high order of accuracy has seen explosive developments in the last few years, including both methodological aspects as well as very ambitious applications. In spite of methodological progress, the issues of efficiency and ease of implementation of the solution of the associated generalized Riemann problem (GRP) remain the centre of attention in the ADER approach. In the original formulation of ADER schemes, the proposed solution procedure for the GRP was based on (i) Taylor series expansion of the solution in time right at the element interface, (ii) subsequent application of the Cauchy–Kowalewskaya procedure to convert time derivatives to functionals of space derivatives, and (iii) solution of classical Riemann problems for high-order spatial derivatives to complete the Taylor series expansion. For realistic problems the Cauchy–Kowalewskaya procedure requires the use of symbolic manipulators and being rather cumbersome its replacement or simplification is highly desirable. In this paper we propose a new class of solvers for the GRP that avoid the Cauchy–Kowalewskaya procedure and result in simpler ADER schemes. This is achieved by exploiting the history of the numerical solution that makes it possible to devise a time-reconstruction procedure at the element interface. Still relying on a time Taylor series expansion of the solution at the interface, the time derivatives are then easily calculated from the time-reconstruction polynomial. The resulting schemes are called ADER-TR. A thorough study of the linear stability properties of the linear version of the schemes is carried out using the von Neumann method, thus deducing linear stability regions. Also, via careful numerical experiments, we deduce stability regions for the corresponding non-linear schemes. Numerical examples using the present simplified schemes of fifth and seventh order of accuracy in space and time show that these compare favourably with conventional ADER methods. This paper is restricted to the one-dimensional scalar case with source term, but preliminary results for the one-dimensional Euler equations indicate that the time-reconstruction approach offers significant advantages not only in terms of ease of implementation but also in terms of efficiency for the high-order range schemes.

Bounded and compact weighted essentially nonoscillatory limiters for discontinuous Galerkin schemes: Triangular elements

Two new classes of compact weighted essentially nonoscillatory (WENO) polynomial limiters are presented for second-, third-, fourth-, and fifth-order discontinuous Galerkin (DG) schemes on irregular simplex elements. The presented WENO-DG procedures are extensions of the high-order WENO finite-volume and finite-difference schemes of Zhu and Shu (2017) [25], (2019) [26] to high-order unstructured DG schemes. A compact positivity preserving limiter is applied to the solutions to ensure pressure and density remain within physical ranges at all time. It is then verified that the bounded WENO-DG maintains the formal order of accuracy of the underlying DG schemes in the smooth regions. The performance of the proposed WENO-DG is also demonstrated with inviscid test cases including the classical Riemann problems, shock-turbulence interaction, scramjet, blunt body flows, and the double Mach Reflection problems.

Monodisperse particle-laden exchange flows in a vertical duct

We study buoyancy-driven exchange flow of two mixtures in a vertical narrow duct (two-dimensional channel as well as pipe) theoretically. While the light mixture is assumed always to be a pure fluid, the heavy mixture can be selected as either a pure or a particle-laden fluid. A small width-to-length ratio considered for the duct ($\unicode[STIX]{x1D6FF}\ll 1$) has been used as the perturbation parameter in developing a lubrication model (negligible inertia). In particular, we have adopted the methodology of Zhou et al. (Phys. Rev. Lett., vol. 94, 2005, 117803) for free-surface particle-laden film flows and extended it to a lock exchange system in confined geometry under the Boussinesq approximation. The resulting model is in the form of the classical Riemann problem and has been solved numerically using a robust total variation diminishing finite difference scheme. Both pure and particle-laden cases are investigated in detail. It is observed that the interface between the two fluids takes a self-similar shape at long times. In the case that both heavy and light fluids are pure, the dynamics of the flow is governed by two dimensionless quantities, namely the Reynolds number, $Re$, and the viscosity ratio, $\unicode[STIX]{x1D705}$, of the light and heavy fluids. The interpenetration of the heavy and light layers increases with $Re$ but decreases with $\unicode[STIX]{x1D705}$. Also, the heights of the heavy and light fronts change with $\unicode[STIX]{x1D705}$ but remain unchanged with $Re$. In the case of the particle-laden flow, however, four additional dimensionless parameters emerge, namely the initial volume fraction of particles, $\unicode[STIX]{x1D719}_{0}$, the ratio of particle diameter to duct width, $r_{p}$, and the density ratios of particles to carrying fluid, $\unicode[STIX]{x1D709}$, and of light fluid to carrying fluid, $\unicode[STIX]{x1D702}$. The effect of these parameters on the dynamics of the flow has been quantified through a systematic approach. In the presence of solid particles, the interface between the heavy and light layers becomes more curved compared to the case of pure fluids. This modification occurs due to the change of heavy mixture viscosity alongside the duct. Novel particle-rich zones are further discovered in the vicinity of the advancing heavy and light fronts. These zones are associated with different transport rates of the fluid and solid particles. The degree of particle enrichment remains the same with $Re$, is enhanced by $\unicode[STIX]{x1D705}$, $r_{p}$ and $\unicode[STIX]{x1D702}$, and is slightly diminished with $\unicode[STIX]{x1D719}_{0}$ and $\unicode[STIX]{x1D709}$. On the other hand, the stretched exchange zone between the heavy and light fronts grows with $r_{p}$, $\unicode[STIX]{x1D702}$ and $Re$, but decays with $\unicode[STIX]{x1D719}_{0}$, $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D709}$.

A family of HLL-type solvers for the generalized Riemann problem

The generalized Riemann problem (GRP) is the initial value problem for a conservation law with piecewise smooth, but discontinuous initial data. We provide a new method for solving the GRP approximately, that can be used as a building block for high order finite volume or discontinuous Galerkin methods. Our new GRP solvers use the approximate states and wave speeds obtained through a HLL-type Riemann solver and use this information to build an approximation of the state in the GRP of any order. What is new about this approach compared to most previous solvers is that we no longer need to solve a classical Riemann problem exactly. We give a detailed explanation of this strategy for HLL and HLLC solvers for the Euler equations, as well as for the HLLD solver for MHD equations. We demonstrate the performance of the solvers from this new family of GRP solvers for a broad range of test problems.

When shock is shocked: Riemann problem dynamics at pulse ionization of a shock wave

We study the dynamics of the gas flow discontinuities after pulse ionization of a half space in front of a flat shock wave moving in a channel. Pulse volumetric electric discharge initiated in the vicinity of the shock concentrates in front of the shock and heats the gas there. The heating is shown to be very rapid. We use the shadow imaging technique and a high speed camera to study the flow pattern evolution after the discharge. The pattern consists of two shocks separated by a contact surface. This structure corresponds to the classical Riemann problem formulation. Based on the observed pattern, we estimate the amount of discharge energy converted to heat during the discharge time: the rate of temperature increase is in the order of several degrees K per nanosecond.

High Order Accurate Direct Arbitrary-Lagrangian-Eulerian ADER-MOOD Finite Volume Schemes for Non-Conservative Hyperbolic Systems with Stiff Source Terms

AbstractIn this paper we present a 2D/3D high order accurate finite volume scheme in the context of direct Arbitrary-Lagrangian-Eulerian algorithms for general hyperbolic systems of partial differential equations with non-conservative products and stiff source terms. This scheme is constructed with a single stencil polynomial reconstruction operator, a one-step space-time ADER integration which is suitably designed for dealing even with stiff sources, a nodal solver with relaxation to determine the mesh motion, a path-conservative integration technique for the treatment of non-conservative products and ana posterioristabilization procedure derived from the so-called Multidimensional Optimal Order Detection (MOOD) paradigm. In this work we consider the seven equation Baer-Nunziato model of compressible multi-phase flows as a representative model involving non-conservative products as well as relaxation source terms which are allowed to become stiff. The new scheme is validated against a set of test cases on 2D/3D unstructured moving meshes on parallel machines and the high order of accuracy achieved by the method is demonstrated by performing a numerical convergence study. Classical Riemann problems and explosion problems with exact solutions are simulated in 2D and 3D. The overall numerical code is also profiled to provide an estimate of the computational cost required by each component of the whole algorithm.

Implicit, semi-analytical solution of the generalized Riemann problem for stiff hyperbolic balance laws

We present a semi-analytical, implicit solution to the generalized Riemann problem (GRP) for non-linear systems of hyperbolic balance laws with stiff source terms. The solution method is based on an implicit, time Taylor series expansion and the Cauchy–Kowalewskaya procedure, along with the solution of a sequence of classical Riemann problems. Our new GRP solver is then used to construct locally implicit ADER methods of arbitrary accuracy in space and time for solving the general initial–boundary value problem for non-linear systems of hyperbolic balance laws with stiff source terms. Analysis of the method for model problems is carried out and empirical convergence rate studies for suitable tests problems are performed, confirming the theoretically expected high order of accuracy.

The Riemann problem in the quasi-one-dimensional approximation

The classical one-dimensional Riemann problem is generalized to the quasi-one-dimensional case. A plane slotted channel with a discontinuous cross section is considered. The resulting exact self-similar solution is compared with numerical results obtained for a system of quasi-onedimensional and two-dimensional equations. It is shown that they are in good qualitative agreement and, for certain parameters, also agree well quantitatively.

A high order approximation of hyperbolic conservation laws in networks: Application to one-dimensional blood flow

We present a methodology for the high order approximation of hyperbolic conservation laws in networks by using the Dumbser–Enaux–Toro solver and exact solvers for the classical Riemann problem at junctions. The proposed strategy can be applied to any hyperbolic system, conservative or non-conservative, and possibly with flux functions containing discontinuous parameters, as long as an exact or approximate Riemann problem solver is available. The methodology is implemented for a one-dimensional blood flow model that considers discontinuous variations of mechanical and geometrical properties of vessels. The achievement of formal order of accuracy, as well as the robustness of the resulting numerical scheme, is verified through the simulation of both, academic tests and physiological flows.

The generalized Riemann problems for compressible fluid flows: Towards high order

The generalized Riemann problems (GRP) for nonlinear hyperbolic systems of balance laws in one space dimension are now well-known and can be formulated as follows: Given initial data which are smooth on two sides of a discontinuity, determine the time evolution of the solution near the discontinuity. In particular, the GRP of (k+1)th order high-resolution is based on an analytical evaluation of time derivatives up to kth order, which turns out to be dependent only on the spatial derivatives up to kth order. While the classical Riemann problem serves as a primary “building block” in the construction of many numerical schemes (most notably the Godunov scheme), the analytic study of GRP will lead to an array of “GRP schemes”, which extend the Godunov scheme. Currently there are extensive studies on the second-order GRP scheme, which proves to be robust and is capable of resolving complex multidimensional fluid dynamic problems (Ben-Artzi and Falcovitz, 2003 [4]). In this paper, we provide a new approach for solving the GRP for the compressible flow system towards high order accuracy. The derivation of second-order GRP solver is more concise compared to those in previous works and the third-order GRP (or quadratic GRP) is resolved for the first time. The latter is shown to be necessary through numerical experiments with strong discontinuities. Our method relies heavily on the new treatment of rarefaction waves. Indeed, as a main technical step, the “propagation of singularities” argument for the rarefaction fan, is simplified by deriving the linear ODE systems for the “evolution” of “characteristic derivatives”, in the x–t coordinates, for generalized Riemann invariants. The case of sonic point is incorporated into a general treatment. The accuracy of the derived GRP solvers is justified and numerical examples are presented for the performance of the resulting schemes.

Lie group analysis and Riemann problems for a 2 × 2 system of balance laws

The aim of this paper is to present two different approaches, both based on Lie symmetry analysis, allowing to obtain solutions of the Riemann problem with non-classical discontinuous initial data for a system of balance laws. The first method uses the invariance with respect to a Lie group of exact point symmetries, whereas the second one is based on the approximate Lie symmetries. The two methods are applied to a particular 2 × 2 system of balance laws describing rate-type materials. Moreover, some numerical simulations are performed in order to validate the approximate solutions and check that the exact solution, as a parameter involved in the initial conditions tends to zero, approaches the solution of the classical Riemann problem.

A multislope MUSCL method on unstructured meshes applied to compressible Euler equations for axisymmetric swirling flows

A finite volume method for the numerical solution of axisymmetric inviscid swirling flows is presented. The governing equations of the flow are the axisymmetric compressible Euler equations including swirl (or tangential) velocity. A first-order scheme is introduced where the convective fluxes at cell interfaces are evaluated by the Rusanov or the HLLC numerical flux while the geometric source terms are discretizated to provide a well-balanced scheme i.e. the steady-state solutions with null velocity are preserved. Extension to the second-order space approximation using a multislope MUSCL method is then derived. To test the numerical scheme, a stationary solution of the fluid flow following the radial direction has been established with a zero and nonzero tangential velocity. Numerical and exact solutions are compared for classical Riemann problems where we employ different limiters and effectiveness of the multislope MUSCL scheme is demonstrated for strongly shocked axially symmetric flows like in spherical bubble compression problem. Two other tests with axisymmetric geometries are performed: the supersonic flow in a tube with a cone and the axisymmetric blunt body with a free stream.

Adaptive multiresolution or adaptive mesh refinement? A case study for 2D Euler equations

We present adaptive multiresolution (MR) computations of the two-dimensional com- pressible Euler equations for a classical Riemann problem. The results are then compared with respect to accuracy and computational eciency, in terms of CPU time and memory requirements, with the corresponding nite volume scheme on a regular grid. For the same test case, we also perform compu- tations using adaptive mesh renement (AMR) imposing similar accuracy requirements. The results thus obtained are compared in terms of computational overhead and compression of the computational grid, using in addition either local or global time stepping strategies. We preliminarily conclude that the multiresolution techniques yield improved memory compression and gain in CPU time with respect to the adaptive mesh renement method.

Shock Solutions for Particle-Laden Thin Films

We derive a lubrication model describing gravity-driven thin film flow of a suspension of heavy particles in viscous fluid. The main features of this continuum model are an effective mixture viscosity and a particle settling velocity, both depending on particle concentration. The resulting equations form a $2 \times 2$ system of conservation laws in the film thickness $h(x,t)$ and in $\phi h$, where $\phi(x,t)$ is the particle volume fraction. We study flows in one dimension under the constant flux boundary condition, which corresponds to the classical Riemann problem, and we find the system can have either double-shock or singular shock solutions. We present the details of both solutions and examine the effects of the particle settling model and of the microscopic length scale b at the contact line.

Non‐reflecting boundary conditions for the compressible Euler and Navier‐Stokes equations in the finite volume method

AbstractThe paper is concerned with the numerical implementation of the inlet and outlet boundary conditions in the finite volume method for the solution of the 3D Euler and Navier‐Stokes equations. The explicit time marching procedure is described. The classical Riemann problem is modified for physically relevant boundary conditions with the aim to keep conservation laws. This technique was used in [2]. The initial condition in the Riemann problem is replaced by the suitable one‐sided boundary condition. This results in the acceleration of the numerical method itself. On the inlet the pressure and the density and the angle of attack or velocity vector and the entropy are prescribed. On the outlet the pressure or normal component of the velocity or temperature or mass flow are investigated in such a way to obtain the unique solution of the modified Riemann problem. Various combinations of inlet and outlet boundary conditions are investigated. This results in the sufficiently precise approximation of real flow boundary conditions. Numerical examples illustrating the usefulness of the proposed approach for cascade flow are presented. Another numerical example is shown in [3]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On some generalizations of the classical Riemann problem

AbstractThe special class of the homogeneous vector boundary Riemann problems on the finite sequence of algebraic surfaces is investigated completely. Its coefficients are the noncommutative permutative matrices of the arbitrary but not prime order, and boundary conditions are given on the system of open contours. The constructive solution procedure and definite structure of the canonical solution matrix are obtained and present some generalizations of the classical Riemann problem. Simultaneously the corresponding class of algebraic equations for the appropriate covering surfaces is formed explicitly too. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Generalized Riemann Problem of Linear Conjugation for Non-Homogeneous Polyanalytic Equations of Order $n$ in $W_{n,p}(D)$

We consider a non-homogeneous polyanalytic partial differential equation of order n in a simply-connected domain D with smooth boundary \partial D in the complex plane \mathbb C . Initially we transform the given equation into an equivalent system of integro-differential equations and then find the general solution of the former in W_{n,p} (D) . Next we pose and prove the solvability of a generalized Riemann problem of linear conjugation to the differential equation. This is effected by first reducing the Riemann problem to a corresponding one for a polyanalytic function. The latter is solved by first transforming it into n classical Riemann problems of linear conjugation for n holomorphic functions expressed in terms of the analytic functions which define the polyanalytic function. The solution of the classical Riemann problem is available in the literature.