A staggered projection scheme for viscoelastic flows

¹ Institut National Polytechnique de Toulouse (INPT), Toulouse, France
² TotalEnergies E&P, CSTJF, Pau, France
³ Institut de Mécanique des Fluides de Toulouse (IMFT), CNRS & Université de Toulouse, Toulouse, France
⁴ Institut de Radioprotection et de Sûreté Nucléaire (IRSN), Fontenay-aux-Roses, France

^* Corresponding author: jean-claude.latche@irsn.fr

Received: 24 October 2021
Accepted: 22 February 2023

Abstract

We develop a numerical scheme for the flow of viscoelastic fluids, including the Oldroyd-B and FENE-CR constitutive models. The space discretization is staggered, using either the Marker-And-Cell (MAC) scheme for structured nonuniform grids, or the Rannacher and Turek (RT) nonconforming low-order finite element approximation for general quadrangular or hexahedral meshes. The time discretization uses a fractional-step algorithm where the solution of the Navier–Stokes equations is first obtained by a projection method and then the transport-reaction equation for the conformation tensor is solved by a finite volume scheme. In order to obtain consistency, the space discretization of the divergence of the elastic part of the stress tensor in the momentum balance equation is derived using a weak form of the MAC scheme. For stability and accuracy purposes, the solution of the transport-reaction equation for the conformation tensor is split into pure convection steps, with a change of variable to the log-conformation tensor, and a reaction step, which consists in solving one ODE per cell via an Euler scheme with local sub-cycling. Numerical computations for the flow in the lid-driven cavity at Weissenberg numbers above one and the flow around a confined cylinder confirm the efficiency of the scheme.