Numerical Modelling Of The Riemann Problem
For A Mathematical Two-phase Flow Model
Author(s)
D. Zeidan
Abstract
We report preliminary results obtained using Godunov methods of a centred type
for hyperbolic systems of conservation laws for which the analytical solution of the
Riemann problem is too difficult to develop. For this purpose,we consider amathematical
model used for modelling an unsteady compressible non-equilibrium mixture
two-phase flow which results in a well-posed initial value problem in a conservative
form. The mathematical two-phase flow model consists of six first-order
partial differential equations that represent one-dimensional mass, momentum and
energy balances for a mixture of gas-liquid and gas volume concentration, gas
mass concentration and the relative velocity balances. This system of six partial
differential equations has the mathematical property that its six characteristic roots
are all real with a complete set of independent eigenvectors which form the basic
structure of the solution of the Riemann problem. The construction of the solution
of the Riemann problem for the model poses several difficulties. Since the model
possesses a large number of non-linear waves, it is not easy to consider each wave
separately to derive a single non-linear algebraic equation for the unknown region,
the star region, between the left and right waves.We propose numerical techniques
of a centred type specifically developed for high speed single-phase gas flows. A
main feature of centred methods is that they do not explicitly require the solution
of the Riemann problem. This is a desirable property which guarantees that the
methods can handle the solution of the Riemann problem numerically and resolve
both rarefaction and shock waves for the model in a simple way with good accuracy.
Finally, we present some numerical simulations for the gas-liquid two-phase
flow Riemann problem to illustrate the efficiency of the proposed schemes.
Keywords: gas-liquid two-phase flow, thermodynamically compatible system, mathematical
model, well-posed initial value problem, Riemann problem, Godunov
methods, TVD SLIC scheme.
Keywords
gas-liquid two-phase flow, thermodynamically compatible system, mathematical
model, well-posed initial value problem, Riemann problem, Godunov
methods, TVD SLIC scheme.
Related Book
Other papers in this volume
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