A Riemann problem solution methodology for a class of evolutionary mixture equations with an arbitrary number of components

• Published in: Applied numerical mathematics Vol. 76; pp. 145 - 165
• Main Authors: Crochet, M.W., Gonthier, K.A.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.02.2014
• Subjects: Algorithms; Combustion; Evolutionary; Explosions; Granular mixtures; Hyperbolic systems; Mathematical analysis; Mathematical models; Methodology; Multiphase flows; Nonconservative sources; Nonlinear algebraic equations; Order disorder; Riemann solver; Multiphase flows; Nonlinear algebraic equations; Nonconservative sources; Riemann solver; Hyperbolic systems; Granular mixtures
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/j.apnum.2013.09.004

The solution of the two-phase Riemann problem is a critical component of upwind finite-volume numerical schemes used to solve systems of evolutionary equations, which are routinely used to model compaction and combustion phenomena in gas–granular explosive mixtures. Extensions of a common two-phase model are currently being used to analyze the thermomechanics and combustion of explosive mixtures consisting of N components. Although a solution to the two-phase Riemann problem has been formulated, there is currently no available analogue for the N-phase system in the literature, due to the inherent difficulty of determining the correct wave ordering within the Riemann solver. The development of a solution for these systems is therefore an important step in the formulation of numerical schemes applied to N-phase mixtures. Here, an extension of the exact two-phase solution methodology is proposed for the N-phase case, which may be utilized in the construction of finite-volume schemes for multiphase systems, and can be used with general, convex equations of state. Finally, example problems for three-phase mixtures are considered to illustrate the accuracy of the solution compared to the results of a centered numerical scheme. These solutions also demonstrate the complexity of the possible wave configurations that arise when multiple solid phases are present, as well as the algorithmic challenges which must be addressed to provide a robust implementation.