Riemann Problem For Gas Dynamics Research Articles

An extension of the TV-HLL scheme for multi-dimensional compressible flows

This paper investigates a very simple method to numerically approximate the solution of the multi-dimensional Riemann problem for gas dynamics, using the literal extension of the Toro Vazquez-Harten Lax Leer (TV-HLL) scheme as its basis. Indeed, the present scheme is obtained by following the Toro–Vazquez splitting, and using the HLL algorithm with modified wave speeds for the pressure system. An essential feature of the TV-HLL scheme is its simplicity and its accuracy in computing multi-dimensional flows. The proposed scheme is carefully designed to simplify its eventual numerical implementation. It has been applied to numerical tests and its performances are demonstrated for some two-dimensional and three-dimensional test problems.

English

This work deals with the implementation of reduced differential transform method (RDTM) for solving the Riemann problem for gas dynamics in one dimension. The RDTM is an analytical method that can be applied to many linear and nonlinear partial differential equations and is capable of reducing the size of computations. Using this method, the solution is calculated in the form of convergent power series with easily computable components. The definition and basic properties of RDTM are investigated. Some new generalized formulas of reduced differential transforms are derived. The Riemann problem that describes the isentropic flow of an inviscid gas is considered to demonstrate the effectiveness and promising of the proposed algorithm.   Key words: Reduced differential transform method, gas dynamics, isentropic flow of an inviscid gas equations, conservation law.

A simple two-dimensional extension of the HLL Riemann solver for hyperbolic systems of conservation laws

We derive a simple method to numerically approximate the solution of the two-dimensional Riemann problem for gas dynamics, using the literal extension of the well-known HLL formalism as its basis. Essentially, any strategy attempting to extend the three-state HLL Riemann solver to multiple space dimensions will by some means involve a piecewise constant approximation of the complex two-dimensional interaction of waves, and our numerical scheme is not the exception. In order to determine closed form expressions for the involved fluxes, we rely on the equivalence between the consistency condition and the use of Rankine–Hugoniot conditions that hold across the outermost waves. The proposed scheme is carefully designed to simplify its eventual numerical implementation and its advantages are analytically attested. In addition, we show that the proposed solver can be applied to obtain the edge-centered electric fields needed in the constrained transport technique for the ideal magnetohydrodynamic (MHD) equations. We present several numerical results for hydrodynamics and magnetohydrodynamics that display the scheme's accuracy and its ability to be applied to various systems of conservation laws.

Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids

Numerical simulations [2-D Riemann problem in gas dynamics and formation of spiral, in: Nonlinear Problems in Engineering and Science—Numerical and Analytical Approach (Beijing, 1991), Science Press, Beijing, 1992, pp. 167–179; Discrete Contin. Dyn. Syst. 1 (1995) 555–584; 6 (2000) 419–430] for the Euler equations for gas dynamics in the regime of small pressure showed that, for one case, the particles seem to be more sticky and tend to concentrate near some shock locations, and for the other case, in the region of rarefaction waves, the particles seem to be far apart and tend to form cavitation in the region. In this paper we identify and analyze the phenomena of concentration and cavitation by studying the vanishing pressure limit of solutions of the full Euler equations for nonisentropic compressible fluids with a scaled pressure. It is rigorously shown that any Riemann solution containing two shocks and possibly one-contact-discontinuity to the Euler equations for nonisentropic fluids tends to a δ-shock solution to the corresponding transport equations, and the intermediate densities between the two shocks tend to a weighted δ-measure that, along with the two shocks and possibly contact-discontinuity, forms the δ-shock as the pressure vanishes. By contrast, it is also shown that any Riemann solution containing two rarefaction waves and possibly one-contact-discontinuity to the Euler equations for nonisentropic fluids tends to a two-contact-discontinuity solution to the transport equations, and the nonvacuum intermediate states between the two rarefaction waves tend to a vacuum state as the pressure vanishes. Some numerical results exhibiting the processes of concentration and cavitation are presented as the pressure decreases.

A Riemann problem in gas dynamics with bifurcation

We construct the solution of the Riemann problem for 2 × 2 isothermal gas dynamics in a duct with discontinuous diameter. Besides shocks and rarefaction, there are standing waves. The solution exists globally. It is obtained as an asymptotic solution for an appropriate Cauchy problem with continuous data. In certain cases bifurcation occurs and there are three solutions, one of which is unstable. This is an example of a Riemann problem whose solution depends discontinuously on the initial data.

Efficient solution algorithms for the Riemann problem for real gases

A procedure for constructing solutions to the Riemann problem for gas dynamics with a general convex equation of state is given. Approximate procedures, involving a local parametrization of the equation of state, are introduced in order to calculate numerical fluxes in conservative finite difference schemes. This leads to difference schemes which are as accurate and almost as fast as the analogous schemes for polytropic gases. Numerical results in one and two space variables are presented.

Perturbation of the Riemann problem in gas dynamics

Perturbation of the Riemann problem in gas dynamics

The Riemann Problem in Gas Dynamics

We consider the Riemann problem (R.P.) for the 3 x 3 system of gas dynamics equations in a single space variable. We assume that the specific internal energy e = e(v, s) (s = specific entropy, v = specific volume) satisfies the usual hypotheses, p 0,?p > 0 (p =-e, pressure); we also assume some reasonable hypotheses about the asymptotic behavior of e. We call functions e satisfying these hypotheses energy functions. THEOREM 1. For any initial data (Ul, Ur) (Ul = (v1, Pi, ul), Ur = (Vr, Pr' Ur), u = flow velocity), the R. P. has a solution. We introduce two conditions: P p2 apw(v, e) 0), (I) a e(v,p) > P (v,p>O)(II) THEOREM 2. (I) is necessary and sufficient for uniqueness of solutions of the R. P. Nonuniqueness persists under small perturbations of the initial data. (I) is implied by the known condition a e(v,p) >O0 (v, p >O0),(* av which holds for all usual gases. (I) implies (II). We construct energy functions e that violate (II), that satisfy (II) but violate (I), and that satisfy (I) but violate (*). In all solutions considered, the shocks satisfy the entropy condition and the Lax shock conditions.

The Riemann problem in gas dynamics

We consider the Riemann problem (R.P.) for the 3 × 3 3\, \times \, 3 system of gas dynamics equations in a single space variable. We assume that the specific internal energy e = e ( v , s ) e = e(v,\,s) (s = specific entropy, v = specific volume) satisfies the usual hypotheses, p v > 0 , p v v > 0 , p s > 0 ( p = − e v = {p_v}\, > \,0,\,{p_{vv}}\, > \,0,\,{p_s}\, > \,0\,(p\, = \, - \,{e_v}\, = pressure); we also assume some reasonable hypotheses about the asymptotic behavior of e. We call functions e satisfying these hypotheses energy functions Theorem 1. For any initial data ( U l , U r ) ( U l = ( v l , p l , u l ) ({U_l},\,{U_r})\,({U_l}\, = \,({v_l},\,{p_l},\,{u_l}) , U r = ( v r , p r , u r ) {U_r}\, = \,({v_r},\,{p_r},\,{u_r}) , u = flow velocity), the R. P. has a solution. We introduce two conditions: \[ ( (I) ) ∂ ∂ v p ( v , e ) ⩽ p 2 2 e a m p ; ( v , e > 0 ) , \begin {array}{*{20}{c}}\tag {$\text {(I)}$} {\frac {\partial } {{\partial v}}\,p(v,\,e) \leqslant \frac {{{p^2}}} {{2e}}} & {(v,\,e\, > \,0),} \\ \end {array} \] \[ ( (II) ) ∂ ∂ v e ( v , p ) ⩾ − p 2 a m p ; ( v , p > 0 ) . \begin {array}{*{20}{c}}\tag {$\text {(II)}$} {\frac {\partial }{{\partial v}}\,e(v,\,p)\, \geqslant - \frac {p} {2}} & {(v,\,p\, > \,0).} \\ \end {array} \] Theorem 2. (I) is necessary and sufficient for uniqueness of solutions of the R. P. Nonuniqueness persists under small perturbations of the initial data. (I) is implied by the known condition ( ( ∗ ) ) ∂ ∂ v e ( v , p ) > 0 ( v , p > 0 ) , \begin{equation}\tag {$(\ast )$} {\frac {\partial } {{\partial v}}e(v,p) > 0} \qquad (v,p > 0), \end{equation} which holds for all usual gases. (I) implies (II). We construct energy functions e that violate (II), that satisfy (II) but violate (I), and that satisfy (I) but violate (*). In all solutions considered, the shocks satisfy the entropy condition and the Lax shock conditions.

On the influence of initial conditions on the motion of water confined within a tank

The aim of this paper is to investigate the influence of initial values on the motion of waves in shallow water confined between two vertical walls of a tank with a polygonal bottom. A method introduced by G.S.S. Ludford to study a Riemann problem in gas dynamics is modified and used to obtain an estimate of the time of breakdown of an initially smooth but arbitrary wave motion on the surface of the water.