The free (open) boundary condition (FBC) in viscoelastic flow simulations

The Free (or Open) Boundary Condition (FBC, OBC) was proposed by Papanastasiou et al. (A New Outflow Boundary Condition, Int. J. Numer. Meth. Fluids, 1992; 14:587–608) to handle truncated domains with synthetic boundaries where the outflow conditions are unknown. In the present work, implementation of the FBC has been extended to viscoelastic fluids governed by explicit differential constitutive equations. As such we consider here the Criminale-Ericksen-Filbey (CEF) model, which also reduces to the Second-Order Fluid (SOF) for constant material parameters. The Finite Element Method (FEM) is used to provide numerical results in simple Poiseuille flow where analytical solutions exist for checking purposes. Then previous numerical results are checked against Newtonian highly non-isothermal flows in a 4:1 contraction. Finally, the FBC is used with the CEF fluid with data corresponding to a Boger fluid of constant material properties. Particular emphasis is based on a non-zero second normal-stress difference, which seems responsible for earlier loss of convergence. The results with the FBC are in excellent agreement with those obtained from long domains, due to the highly convective nature of viscoelastic flows, for which the FBC seems most appropriate. The FBC formulation for fixed-point (Picard-type) iterations is given in some detail, and the differences with the Newton–Raphson formulation are highlighted regarding some computational aspects.