A Higher-Order Upwind Method for Viscoelastic Flow in Irregular Geometries

We present a high-resolution numerical method to capture elastic shear waves in incompressible viscoelastic fluids. The viscoelastic fluid is described by the Oldroyd-B constitutive equation for the polymer stress coupled to the incompressible Navier-Stokes equations. In our approach, we leverage the hyperbolic nature of the equations to make use of higher-order Godunov methods which have been previously used with much success in capturing shocks in compressible flow. The hyperbolic step also utilizes a new exact and efficient Riemann solver. Incompressibility is enforced through a projection method and a special partitioning of variables which leads to proper characteristic properties in the hyperbolic step. An embedded boundary method / volume-of-fluid method allows irregular geometries to be represented on Cartesian grids whereby the boundary cuts regular cells into irregular control volumes requiring special discretization stencils; away from boundaries standard finite differences are used. We demonstrate second-order accuracy for both highly elastic flows of a Maxwell fluid and steady-state flow of an Oldroyd-B fluid in the Newtonian limit.