Consistent splitting schemes for incompressible viscoelastic flow problems

Abstract

Viscoelastic fluids are highly challenging from the rheological standpoint, and their discretization demands robust, efficient numerical solvers. Simulating viscoelastic flows requires combining the Navier–Stokes system with a dynamic tensorial equation, increasing mathematical and computational demands. Hence, fractional-step methods decoupling the calculation of the flow quantities are an attractive option. In consistent fractional-step schemes, the splitting of the equations is derived from the continuous level, so that neither mass nor momentum balance is sacrificed. Thus, no corrections or velocity projections are needed, resulting in fewer algorithmic steps than other classical approaches. Moreover, consistency guarantees the absence of both numerical boundary layers and splitting errors, enabling high-order accuracy in space and time. This article introduces the first consistent splitting methods for incompressible flows of viscoelastic fluids, for which arbitrary constitutive laws are allowed. We present the formulation and the algorithm, along with various numerical examples testing their accuracy. First-, second- and third-order backward-differentiation schemes are numerically tested, and optimal convergence is confirmed for several spatial and temporal discretizations. Furthermore, good numerical stability is verified in challenging benchmark problems, including steady and time-dependent solutions.