Variationally derived discontinuity capturing methods: Fine scale models with embedded weak and strong discontinuities

Abstract

This paper presents a new stabilized method that is endowed with variationally derived Discontinuity Capturing (DC) features to model steep advection fronts and discontinuities that arise in multi-phase flows as well as in mixing flows of immiscible incompressible fluids. Steep fronts and discontinuities also arise in hypersonic compressible flows. The new method finds roots in the Variational Multiscale (VMS) framework that yields a coupled system of coarse and fine-scale variational problems. Augmenting the space of functions for the fine-scale fields with weak and/or strong discontinuities results in fine-scale models that naturally accommodate jumps in the fine fields. Variationally embedding the discontinuity enriched fine-scale models in the coarse-scale formulation leads to the Variational Multiscale Discontinuity Capturing (VMDC) method where stabilization tensors are naturally endowed with discontinuity capturing structure. In the regions with sharp gradients, these variationally projected fine-scale models augment the stability of the coarse-scale formulation to accurately capture sharply varying coarse solutions. Since the proposed method relies on local enrichment, it does not require either the complete or the dynamic enrichment algorithms that are invariably employed in methods that use global enrichment ideas. The scalar advection equation serves as a model problem to investigate the variational structure of the DC terms. The VMDC method is then applied to the Navier–Stokes equations and tested on problems involving two-phase flows with and without surface tension. These test problems highlight that fine-scale models not only stabilize the weak form, variationally derived fine models that are endowed with sharp discontinuities also augment the coarse scale solutions with features that are otherwise not adequately resolved by variational formulations that act only at the coarse-scale levels.