The Riemann problem for a hyperbolic model of two-phase flow in conservative form

This article is to continue the present author's work (International Journal of Computational Fluid Dynamics (2009) 23 (9), 623–641) on studying the structure of solutions of the Riemann problem for a system of three conservation laws governing two-phase flows. While existing solutions are limited and found quite recently for the Baer and Nunziato equations, this article presents the first instance of an exact solution of the Riemann problem for two-phase flow in gas–liquid mixture. To demonstrate the structure of the solution, we use a hyperbolic conservative model with mechanical equilibrium and without velocity equilibrium. The Riemann problem solution for the model equations comprises a set of elementary waves, rarefaction and discontinuous waves of various types. In particular, such a solution treats both the wave structure and the intermediate states of the two-phase gas–liquid mixture. The resulting exact Riemann solver is fully non-linear, direct and complete. On this basis then, we use locally the exact Riemann solver for the two-phase flow in gas–liquid mixture within the framework of finite volume upwind Godunov methods. In order to demonstrate the effectiveness and accuracy of the proposed solver, we consider a series of test problems selected from the open literature and compare the exact and numerical results by using upwind Godunov methods, showing excellent oscillation-free results in two-phase fluid flow problems.     

Delta Shock Wave Solution of the Riemann Problem for the Non-homogeneous Modified Chaplygin Gasdynamics

The motivation of the present study is to derive the solution of the Riemann problem for modified Chaplygin gas equations in the presence of constant external force. The analysis leads to the fact that in some special circumstances delta shock appears in the solution of the Riemann problem. Also, the Rankine–Hugoniot relations for delta shock wave which are utilized to determine the strength, position and propagation speed of the delta shocks have been derived. Delta shock wave solution to the Riemann problem for the modified Chaplygin gas equation is obtained. It is found that the external force term, appearing in the governing equations, influences the Riemann solution for the modified Chaplygin gas equation.

     

Explicit solution to the exact Riemann problem and application in nonlinear shallow‐water equations

The Riemann solver is the fundamental building block in the Godunov‐type formulation of many nonlinear fluid‐flow problems involving discontinuities. While existing solvers are obtained either iteratively or through approximations of the Riemann problem, this paper reports an explicit analytical solution to the exact Riemann problem. The present approach uses the homotopy analysis method to solve the nonlinear algebraic equations resulting from the Riemann problem. A deformation equation defines a continuous variation from an initial approximation to the exact solution through an embedding parameter. A Taylor series expansion of the exact solution about the embedding parameter provides a series solution in recursive form with the initial approximation as the zeroth‐order term. For the nonlinear shallow‐water equations, a sensitivity analysis shows fast convergence of the series solution and the first three terms provide highly accurate results. The proposed Riemann solver is implemented in an existing finite‐volume model with a Godunov‐type scheme. The model correctly describes the formation of shocks and rarefaction fans for both one and two‐dimensional dam‐break problems, thereby verifying the proposed Riemann solver for general implementation. Copyright © 2007 John Wiley & Sons, Ltd.     

Solution of Riemann problem for dusty gas flow

A direct approach is used to solve the Riemann problem for a quasilinear hyperbolic system of equations governing the one dimensional unsteady planar flow of an isentropic, inviscid compressible fluid in the presence of dust particles. The elementary wave solutions of the Riemann problem, that is, shock waves, rarefaction waves and contact discontinuities are derived and their properties are discussed for a dusty gas. The generalised Riemann invariants are used to find the solution between rarefaction wave and the contact discontinuity and also inside rarefaction fan. Unlike the ordinary gasdynamic case, the solution inside the rarefaction waves in dusty gas cannot be obtained directly and explicitly; indeed, it requires an extra iteration procedure. Although the case of dusty gas is more complex than the ordinary gas dynamics case, all the parallel results for compressive waves remain identical. We also compare/contrast the nature of the solution in an ordinary gasdynamics and the dusty gas flow case.     

Riemann solution for ideal isentropic magnetogasdynamics

In the present paper, we study the Riemann problem for quasilinear hyperbolic system of partial differential equations governing the one dimensional ideal isentropic magnetogasdynamics with transverse magnetic field. We discuss the properties of rarefaction waves, shocks and contact discontinuities. Differently from single equation methods rooted in the ideal gasdynamics, the new approach is based on the system of two nonlinear equations imposing the equality of total pressure and velocity, assuming as unknowns the two values of densities, on both sides of the contact discontinuity. Newton iterative method is used to obtain densities. The resulting exact solver is implemented with the examples of general applicability of the proposed approach. For comparisons with exact solution we also shown numerical results obtained by the total variation diminishing slope limiter centre scheme. It is shown that both analytical and numerical results demonstrate the broad applicability and robustness of the new Riemann solver.     

A mode split,Godunov‐type model for nonhydrostatic,free surface flow

A previously developed model for nonhydrostatic, free surface flow is redesigned to improve computational efficiency without sacrificing accuracy. Both models solve the Reynolds averaged Navier–Stokes equations in a fractional step manner with the pressure split into hydrostatic and nonhydrostatic components. The hydrostatic equations are first solved with an approximate Riemann solver. The hydrostatic solution is then corrected by including the nonhydrostatic pressure and requiring the velocity field to obey the incompressibility constraint. The original model requires the solution of a Riemann problem at every cell face for each vertical layer of cells, which is computationally expensive. The redesigned model instead solves the shallow water (long wave) equations for the hydrostatic solution. Vertical shear is computed by subtracting the shallow water equations from the full three dimensional equations, which removes the hydrostatic thrust terms. Therefore, the required fluxes may be more efficiently computed with velocity based upwind differencing rather than solving a Riemann problem in each vertical layer of cells. This approach is termed mode splitting and has been used in hydrostatic coastal and ocean circulation models, but not surf zone models. Numerical predictions are compared with analytical solutions and experimental data to show that the mode split model is as accurate as the original model, but requires significantly less computational effort especially for large numbers of cell layers. Published 2014. This article is a U.S. Government work and is in the public domain in the USA.     

Flux difference splitting for open-channel flows

A finite difference scheme based on flux difference splitting is presented for the solution of the one-dimensional shallow-water equations in open channels, together with an extension to two-dimensional flows. A linearized problem, analogous to that of Riemann for gas dynamics, is defined and a scheme, based on numerical characteristic decomposition, is presented for obtaining approximate solutions to the linearized 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. The scheme is applied to a one-dimensional dam-break problem, and to a problem of flow in a river whose geometry induces a region of supercritical flow. The scheme is also applied to a two-dimensional dam-break problem. The numerical results are compared with the exact solution, or other numerical results, where available.     

SHEAR WAVE SCATTERING FROM A PARTIALLY DEBONDED PIEZOELECTRIC CYLINDRICAL INCLUSION

I. INTRODUCTION Owing to the intrinsic coupling characteristics between electric and elastic behaviors, piezoelectricmaterials have been used widely in technology such as transducers, actuators, sensors, etc. Studieson electroelastic problems of a piezo…     

An exact Riemann solver for detonation products

Numerical methods based upon the Riemann Problem are considered for solving the general initial-value problem for the Euler equations applied to real gases. Most of such methods use an approximate solution of the Riemann problem when real gases are involved. These approximate Riemann solvers do not yield always a good resolution of the flow field, especially for contact surfaces and expansion waves. Moreover, approximate Riemann solvers cannot produce exact solutions for the boundary points. In order to overcome these shortcomings, an exact solution of the Riemann problem is developed, valid for real gases. The method is applied to detonation products obeying a 5th order virial equation of state, in the shock-tube test case. Comparisons between our solver, as implemented in Random Choice Method, and finite difference methods, which do not employ a Riemann solver, are given.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

A linearized Riemann solver for the steady supersonic Euler equations

A very simple linearization of the solution to the Riemann problem for the steady supersonic Euler equations is presented. When used locally in conjunction with the Godunov method, computing savings by a factor of about four relative to the use of exact Riemann solvers can be achieved. For severe flow regimes, however, the linearization loses accuracy and robustness. We then propose the use of a Riemann solver adaptation procedure. This retains the accuracy and robustness of the exact Riemann solver and the computational efficiency of the cheap linearized Riemann solver. Numerical results for two- and three-dimensional test problems are presented.     

Generalized and exact solutions for oblique shock waves of real gases with application to real air

Calculation of the oblique shock wave of real gases is a difficult and time consuming problem because it involves numerical solution of a set of 10 equations, two of which (i.e., the equation of state and enthalpy function)—if available—are of a very complicated algebraic form. The present work presents a generalized method for calculating oblique shock waves of real gases, based on the Redlich-Kwong equation of state. Also described is an exact method applicable when the exact equation of state and enthalpy function of a real gas are available. Application of the generalized and the exact methods in the case of real air showed that the former is very accurate and at least twenty times faster than the latter. An additional contribution of the study is the derivation of real gas oblique shock wave equations, which are of the same algebraic form as the well known ideal gas normal shock wave relations.     

Density–velocity equations with bulk modulus for computational hydro-acoustics

This paper reports a new set of model equations for Computational Hydro Acoustics (CHA). The governing equations include the continuity and the momentum equations. The definition of bulk modulus is used to relate density with pressure. For 3D flow fields, there are four equations with density and velocity components as the unknowns. The inviscid equations are proved to be hyperbolic because an arbitrary linear combination of the three Jacobian matrices is diagonalizable and has a real spectrum. The left and right eigenvector matrices are explicitly derived. Moreover, an analytical form of the Riemann invariants are derived. The model equations are indeed suitable for modeling wave propagation in low-speed, nearly incompressible air and water flows. To demonstrate the capability of the new formulation, we use the CESE method to solve the 2D equations for aeolian tones generated by air flows passing a circular cylinder at Re = 89,000, 46,000, and 22,000. Numerical results compare well with previously published data. By simply changing the value of the bulk modulus, the same code is then used to calculate three cases of water flows passing a cylinder at Re = 89,000, 67,000, and 44,000.     

MUSTA schemes for multi‐dimensional hyperbolic systems: analysis and improvements

We develop and analyse an improved version of the multi‐stage (MUSTA) approach to the construction of upwind Godunov‐type fluxes whereby the solution of the Riemann problem, approximate or exact, is not required. The new MUSTA schemes improve upon the original schemes in terms of monotonicity properties, accuracy and stability in multiple space dimensions. We incorporate the MUSTA technology into the framework of finite‐volume weighted essentially nonoscillatory schemes as applied to the Euler equations of compressible gas dynamics. The results demonstrate that our new schemes are good alternatives to current centred methods and to conventional upwind methods as applied to complicated hyperbolic systems for which the solution of the Riemann problem is costly or unknown. Copyright © 2005 John Wiley & Sons, Ltd.     

Impact of the interplanetary magnetic field on thewave flow pattern in the neighborhood of the Earth’s bow shock under sharp variations in the solar wind dynamic pressure

The impact of the interplanetary magnetic field on transformation and disintegration of the Earth’s bow shock into a system of magnetohydrodynamic (MHD) shock waves, rotational discontinuities and rarefaction waves under the action of abrupt variations in the solar wind dynamic pressure is simulated in the three-dimensional non-plane-polarized formulation within the framework of the ideal magnetohydrodynamic model using the solution of the MHD Riemann problem of breakdown of an arbitrary discontinuity. This discontinuity arises when a contact discontinuity, on which the solar wind density increases or decreases suddenly and which travels together with the solar wind, impinges on the Earth’s bow shock and propagates along its surface. The interaction pattern is constructed in the quasisteady- state formulation as a mosaic of exact solutions obtained on computer using an original MHD Riemann solver. The wave flow patterns are found for all elements of the surface of the bow shock as functions of their latitude and longitude for various jumps in the density on the contact discontinuity and characteristics parameters of the solar wind and interplanetary magnetic field at the Earth’s orbit. It is found that when the solar wind dynamic pressure increases, a fast MHD shock wave, that first penetrates into the magnetosheath, is always formed. When the solar wind dynamic pressure decreases, the influence of the interplanetary magnetic field can lead to the development of the leading fast MHD shock wave in certain zones on the surface of the Earth’s bow shock. The solution obtained can be used to interpret measurements on spacecraft in the solar wind at the libration point and in the neighborhood of the Earth’s magnetosphere.     

Higher order Boussinesq-type equations for water waves on uneven bottom

Higher order Boussinesq-type equations for wave propagation over variable bathymetry were derived. The time dependent free surface boundary conditions were used to compute the change of the free surface in time domain. The free surface velocities and the bottom velocities were connected by the exact solution of the Laplace equation. Taking the velocities on half relative water depth as the fundamental unknowns, terms relating to the gradient of the water depth were retained in the inverse series expansion of the exact solution, with which the problem was closed. With enhancements of the finite order Taylor expansion for the velocity field, the application range of the present model was extended to the slope bottom which is not so mild. For linear properties, some validation computations of linear shoaling and Booij' s tests were carried out. The problems of wave-current interactions were also studied numerically to test the performance of the enhanced Boussinesq equations associated with the effect of currents. All these computational results confirm perfectly to the theoretical solution as well as other numerical solutions of the full potential problem available.     

Ideal gas flow behind a finite-amplitude shock wave

The equations of one-dimensional (with a plane of symmetry) adiabatic motion of an ideal gas are transformed to a form convenient for studying flows between a moving piston and a shock wave of variable intensity. The solution is found for the equations of a motion containing a shock wave which propagates through a quiescent gas with variable initial density and constant pressure. This solution contains four arbitrary constants and, in a particular case, gives an example of adiabatic shockless compression by a piston of a gas initially at rest.     

An exact analytic method applied to nonhomogeneous ring- and stringer-stiffened cylindrical shells

In this paper, the nonlinear axial symmetric deformation problem of nonhomogeneous ring- and stringer-stiffened shells is first solved by the exact analytic method. An analytic expression of displacements and stress resultants is obtained and its convergence is proved. Displacements and stress resultants converge to exact solution uniformly. Finally, it is only necessary to solve a system of linear algebraic equations with two unknowns. Four numerical examples are given at the end of the paper which indicate that satisfactory results can be obtained by the exact analytic method.     

Analytical Solution for Isothermal Flow in a Shock Tube Containing Rigid Granular Material

Analytical solution of shock wave propagation in pure gas in a shock tube is usually addressed in gas dynamics. However, such a solution for granular media is complex due to the inclusion of parameters relating to particles configuration within the medium, which affect the balance equations. In this article, an analytical solution for isothermal shock wave propagation in an isotropic homogenous rigid granular material is presented, and a closed-form solution is obtained for the case of weak shock waves. Fluid mass and momentum equations are first written in wave and (mathematical) non-conservation forms. Afterwards by redefining the sound speed of the gas flowing inside the pores, an analytical solution is obtained using the classical method of characteristics, followed by Taylor’s series expansion based on the assumption of weak flow which finally led to explicit functions for velocity, density and pressure. The solution enables plotting gas velocity, density and pressure variations in the porous medium, which is of high interest in the design of granular shock isolators.     

An Approximate Roe-Type Riemann Solver for a Class of Realizable Second Order Closures

A realizable, objective second-moment turbulence closure, allowing for an entropy characterisation, is analyzed with respect to its convective subset. The distinct characteristic wave system of these equations in non-conservation form is exposed. An approximate solution to Ihe associated one-dimensional Riemann problem is constructed making use of approximate jump conditions obtained by assuming a linear path across shock waves. A numerical integration method based on a new approximate Riemann solver (flux-difference-splitting) is proposed for use in conjunction with either unstructured or structured grids. Test calculations of quasi one-dimensional flow cases demonstrate the feasibility of the current technique even where Euler-based approaches fail.