Shock Waves

pp 1–18 | Cite as

A high-order accurate AUSM\(^+\)-up approach for simulations of compressible multiphase flows with linear viscoelasticity

  • M. RodriguezEmail author
  • E. Johnsen
  • K. G. Powell
Original Article


An Eulerian approach for simulations of wave propagation in multiphase, viscoelastic media is developed in the context of the Advection Upstream Splitting Method (AUSM). We extend the AUSM scheme to the five-equation model for simulations of interfaces between gases, liquids, and solids with constitutive relations appropriately transported. In this framework, the solid’s deformations are assumed to be infinitesimally small such that they can be modeled using linear viscoelastic models, e.g., generalized Zener. The Eulerian framework addresses the challenge of calculating strains, more naturally expressed in a Lagrangian framework, by using a hypoelastic model that takes an objective Lie derivative of the constitutive relation to transform strains into velocity gradients. Our approach introduces elastic stresses in the convective fluxes that are treated by generalizing AUSM flux-vector splitting (FVS) to account for the Cauchy stress tensor. We determine an appropriate discretization of non-conservative equations that appear in the five-equation multiphase model with AUSM schemes to prevent spurious oscillations at material interfaces. The framework’s spatial scheme is solution adaptive with a discontinuity sensor discriminating between smooth and discontinuous regions. The smooth regions are computed using explicit high-order central differences. At discontinuous regions (i.e., shocks, material interfaces, and contact surfaces), the convective fluxes are treated using a high-order Weighted Essentially Non-Oscillatory (WENO) scheme with \(\hbox {AUSM}^+\)-up for upwinding. The framework is used to simulate one-dimensional (1D) and two-dimensional (2D) problems that demonstrate the ability to maintain equilibrium interfacial conditions and solve challenging multi-dimensional and multi-material problems.


Five-equation model \(\hbox {AUSM}^+\)-up Multi-component Linear viscoelasticity Zener model 

List of symbols


Advection Upstream Splitting Method


Flux-vector splitting


Weighted Essentially Non-Oscillatory


Flux-difference splitting

\(\rho \)


\(\alpha ^{(k)}\)

kth component volume fraction


Number of materials


Velocity vector




Total energy

\(\sigma _{ij}\)

Cauchy stress tensor


Heat flux

\(\kappa \)

Thermal conductivity


Number of relaxation frequencies


Internal energy

\(e^{(\mathrm {e})}\)

Elastic energy




Noble-Abel Stiffened-Gas


Equation of state


NASG EOS material properties

\(\dot{\epsilon }_{ij}\)

Strain-rate tensor

\(\dot{\epsilon }^{(\mathrm {d})}_{ij}\)

Deviatoric component of \(\dot{\epsilon }_{ij}\)

\(\tau ^{(\mathrm {d})}_{ij}\)

Deviatoric component of \(\sigma _{ij}\)

\(\tau ^{(\mathrm {v})}_{ij}, \tau ^{(\mathrm {e})}_{ij}\)

Viscous and elastic contribution of \(\tau _{ij}\)

\(\mu _{\mathrm {b}}, \mu _{\mathrm {s}}\)

Bulk and shear viscosities

\(\lambda _{\mathrm {r}}\)

Relaxation time


Underrelaxed shear modulus

\(G_{\mathrm {r}}\)

Relaxed shear modulus


Heaviside function

\(\psi \)

Shear relaxation function

\(\varsigma ^{(l)}\)

lth relaxation shear coefficient

\(\theta ^{(l)}\)

lth relaxation frequency

\(\xi ^{(l)}\)

lth memory variable


kth component speed of sound

\(\zeta _{\mathrm {max}}\)

Maximum wavespeed

\(\nu \)

Courant number

\(\nu _\mu , \nu _\kappa \)

von Neumann numbers

\(F_{i\pm 1/2}\)

Interface flux

\(\phi , \varphi \)

Normal and tangent convective flux

\(\eta _{kk}\)

Normal Cauchy stress tensor flux

\(\eta _{kl}\)

Tangent Cauchy stress tensor flux


Normal interface velocity


Tangent interface velocity


Normal and tangent Mach number

\(\mathcal {M}\)

Split Mach number function

\(\mathcal {P}\)

Split pressure function

\(\kappa _{\mathrm {p}}, \kappa _{\mathrm {u}}\)

\(\hbox {AUSM}^+\)-up coefficients

\(\mathcal {A}\)

AUSM discretization operator


Twin regular reflection–refraction

\(\sigma _{\mathrm {von}}\)

von Mises stress



The authors thank Shahaboddin Alahyari Beig for insightful conversations in the development of this work. This research was inspired by the work of Meng-Sing Liou, who will be missed by the community.


This work was supported in part by the Ford Foundation Dissertation Writing Fellowship by ONR Grants N00014-12-1-0751 (under Ki-Han Kim) and N00014-18-1-2625 (under Timothy Bentley) and by NSF Grant Number CBET 1253157.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentUniversity of MichiganAnn ArborUSA
  2. 2.Aerospace Engineering DepartmentUniversity of MichiganAnn ArborUSA

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