Vol. 4, No. 1, 2009

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ISSN: 2157-5452 (e-only)
ISSN: 1559-3940 (print)
A higher-order upwind method for viscoelastic flow

Andrew Nonaka, David Trebotich, Gregory Miller, Daniel Graves and Phillip Colella

Vol. 4 (2009), No. 1, 57–83
Abstract

We present a conservative finite difference method designed to capture elastic wave propagation in viscoelastic fluids in two dimensions. We model the incompressible Navier–Stokes equations with an extra viscoelastic stress described by the Oldroyd-B constitutive equations. The equations are cast into a hybrid conservation form which is amenable to the use of a second-order Godunov method for the hyperbolic part of the equations, including a new exact Riemann solver. A numerical stress splitting technique provides a well-posed discretization for the entire range of Newtonian and elastic fluids. Incompressibility is enforced through a projection method and a partitioning of variables that suppresses compressive waves. Irregular geometry is treated with an embedded boundary/volume-of-fluid approach. The method is stable for time steps governed by the advective Courant–Friedrichs–Lewy (CFL) condition. We present second-order convergence results in L1 for a range of Oldroyd-B fluids.

Keywords
viscoelasticity, Oldroyd-B fluid, Godunov method, Riemann solver, projection method, embedded boundaries
Mathematical Subject Classification 2000
Primary: 65N06, 76D05
Milestones
Received: 7 August 2008
Revised: 22 May 2009
Accepted: 25 May 2009
Published: 11 June 2009
Authors
Andrew Nonaka
Center for Computational Sciences and Engineering
Lawrence Berkeley National Laboratory
Mail Stop 50A-1148
1 Cyclotron Road
Berkeley, CA 94720-8142
United States
http://https://seesar.lbl.gov/ccse/index.html
David Trebotich
Applied Numerical Algorithms Group
Lawrence Berkeley National Laboratory
Mail Stop 50A-1148
1 Cyclotron Road
Berkeley, CA 94720-8142
United States
Gregory Miller
Department of Applied Science
University of California, Davis
1 Shields Ave.
Davis, CA 95616-8254
United States
Daniel Graves
Applied Numerical Algorithms Group
Lawrence Berkeley National Laboratory
Mail Stop 50A-1148
1 Cyclotron Road
Berkeley, CA 94720-8142
United States
Phillip Colella
Applied Numerical Algorithms Group
Lawrence Berkeley National Laboratory
Mail Stop 50A-1148
1 Cyclotron Road
Berkeley, CA 94720-8142
United States