Decoupled transient schemes for viscoelastic fluid flow with inertia

A decoupling time-marching algorithm for the transient simulation of viscoelastic fluids with inertia is proposed. The method is an extension of the second-order implicit stress formulation previously reported for inertialess problems [D’Avino G, Hulsen MA. Decoupled second-order transient schemes for the flow of viscoelastic fluids without a solvent contribution. J Non-Newton Fluid Mech 2010;165:1602–12]. At each time step, the momentum and continuity equations are solved by using a time-discretized but space-continuous form of the constitutive equation, based on a first-order, semi-implicit Euler scheme. The advantage of using such a form for the force term in the momentum balance is the possibility to use the decoupled procedure even when the solvent viscosity is small or absent. In the next substep, the stress unknowns are computed by using the calculated velocity field. For the time discretization of the momentum equation, four schemes, i.e. Gear, Crank–Nicolson, Gear mixed explicit–implicit and Gear with velocity predictor, have been compared. A second-order, semi- implicit Gear scheme is adopted for the discretization of the constitutive equation, where the convection term is taken implicitly whereas all the other terms are explicit. Three test problems have been considered to validate the proposed algorithm. The method is demon- strated to be second-order accurate in time. The Crank–Nicolson scheme is found to give numerical oscil- lations for any time step size. The Gear-based discretizations are stable and the choice of the specific scheme is related to the investigated problem and the set of parameters. New results are also given for a falling sphere in a Giesekus fluid without solvent contribution.