Abstracts - KTH Mechanics

136
A Simple Model for Gas-Grain Two Phase Isentropic Flow
and a High Resolution Scheme for the Numerical Solution of
the Governing Equations
J. Hudson andD.Harris ∗
This contribution presents a simple model for the isentropic two phase flow of a
solid granular material dispersed in a gas obeying the perfect gas law. The dispersed
and continuous phases are both treated as continua and an Eulerian description of
the flow is adopted.
The first key physical quantity in the model is the solids volume fraction and the
model comprises continuity equations and balances of linear momentum for the gas
and solid phases.
The second key physical quantity is the fluctuation energy for the solids phase,
otherwise known as the granular temperature. The fluctuation energy arises in the
theory of the statistical mechanics of granular materials and dense gases and was
later introduced into continuum models. The model is complete by the fluctuation
energy equation. The presence of the solids fluctuation energy gives rise to a solids
pressure in the balance of momentum for the solids phase. We assume the simplest
possible equation of state for the solids pressure, namely the analogy of the perfect
gas law. In the absence of viscosity the classical equal pressures model is ill-posed,
but we demonstrate numerically that the presence of the solids pressure due to the
solids fluctuation energy gives rise to regimes in which the model is indeed well-posed.
The hyperbolicity of the system of equations is investigated numerically and the
numerical solution of the equations in conservation form is accomplished using a
high-resolution scheme 1 in the hyperbolic regime. Careful attention is given to the
discretisation of the inhomogeneous terms. Three one-dimensional test cases are considered,
namely a simple advection test problem, a square pulse test problem and a
steady state problem and we use these to obtain quantitative and qualitative insight
into the predictions of the model.
∗ School of Mathematics, University of Manchester.
1 Hudson et al, submitted to J. Comp. Phys.