Solution of transient viscoelastic flow problems approximated by a term-by-term VMS stabilized finite element formulation using time-dependent subgrid-scales

Abstract

Some finite element stabilized formulations for transient viscoelastic flow problems are presented in this paper. These are based on the Variational Multiscale (VMS) method, following the approach introduced in Castillo and Codina et al. (2019), for the Navier–Stokes problem, the main feature of the method being that the time derivative term in the subgrid-scales is not neglected. The main advantage of considering time-dependent sub-grid scales is that stable solutions for anisotropic space–time discretizations are obtained; however other benefits related with elastic problems are found along this study. Additionally, a split term-by-term stabilization method is discussed and redesigned, where only the momentum equation is approached using a term-by-term methodology, and which turns out to be much more efficient than other residual-based formulations. The proposed methods are designed for the standard and logarithmic formulations in order to deal with high Weissenberg number problems in addition to anisotropic space–time discretizations, ensuring stability in all cases. The proposed formulations are validated in several benchmarks such as the flow over a cylinder problem and the lid-driven cavity problem, obtaining stable and accurate results. A comparison between formulations and stabilization techniques is done to demonstrate the efficiency of time-dependent sub-grid scales and the term-by-term methodologies.