A higher-order upwind method for viscoelastic flow

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 CourantFriedrichs-Lewy (CFL) condition. We present second-order convergence results in L 1 for a range of Oldroyd-B fluids.

A high-resolution finite-difference method for simulating two-fluid, viscoelastic gel dynamics

“…We review the details of this method since it will aid the description of the method we use for solving (14)- (15). For notational simplicity, we drop the subscript n from (13) in the description of the discretization below, and use θ to denote the network volume fraction.…”

“…The CTU scheme for advancing (13) in time from t to t + ∆t k at the (i, j) cell center can be written in conservative form as…”

“…The values of θ at the k + 1/2 time level and at the EW and NS edges of the cell centered at (i, j) are obtained by Taylor series expansions in which temporal derivatives are expressed in terms of spatial derivatives using (13). For each cell edge, this results in two approximations to θ, one from each of the two cells which share that edge.…”

“…13 We use the same predictor-corrector variant of the CTU algorithm as described in the previous section to approximate (43) and (44). That is, all terms in (43) and (44) but the last transverse derivatives are first computed, then the approximations at the cell edges are updated with the transverse derivatives.…”

“…We extend this method to handle the tensor equations for the viscoelastic stresses and elastic modulus. There are similarities of this method with previously developed techniques for treating single-phase viscoelastic fluids [8,13,14]. We use a second order finite-difference discretization of the momentum and incompressibility equations and adapt our iterative method from [9] to handle nonzero Reynolds number flows.…”