A high-order accurate five-equations compressible multiphase approach for viscoelastic fluids and solids with relaxation and elasticity

Abstract

A novel Eulerian approach is proposed for numerical simulations of wave propagation in viscoelastic media, for application to shocks interacting with interfaces between fluids and solids. We extend the five-equations multiphase interface-capturing model, based on the idea that all the materials (gases, liquids, solids) obey the same equation of state with spatially varying properties, to incorporate the desired constitutive relation; in this context, interfaces are represented by discontinuities in material properties. We consider problems in which the deformations are small, such that the substances can be described by linear constitutive relations, specifically, Maxwell, Kelvin–Voigt or generalized Zener models. The main challenge lies in representing the combination of viscoelastic, multiphase and compressible flow. One particular difficulty is the calculation of strains in an Eulerian framework, which we address by using a conventional hypoelastic model in which an objective time derivative (Lie derivative) of the constitutive relation is taken to evolve strain rates instead. The resulting eigensystem is analyzed to identify wave speeds and characteristic variables. The spatial scheme is based on a solution-adaptive formulation, in which a discontinuity sensor discriminates between smooth and discontinuous regions. To compute the convective fluxes, explicit high-order central differences are applied in smooth regions, while a high-order finite-difference Weighted Essentially Non-Oscillatory (WENO) scheme is used at discontinuities (shocks, material interfaces and contacts). The numerical method is verified in a comprehensive fashion using a series of smooth and discontinuous (shocks and interfaces), one- and two-dimensional test problems.