Discontinuous Galerkin simulator of shallow vortical flow with turbulence

Abstract

Shallow vortical flow can often occurs past (un)submerged topographies, prevailing in quasi-steady states with turbulence. Practically, vortical flow is represented by the two-dimensional (2D) Reynolds-Averaged Navier–Stokes equations, including the two-equation k-ε turbulent model (RANS-k-ε), and are commonly resolved by finite difference/volumes second-order accurate solvers. Such RANS-k-ε solvers, in addition to needing a fine resolution, require adding artificial treatments–extrinsic (unlocalised) reconstructions of wet-dry fronts with slope-limiting–that can impact the vortical eddy predictions. The second-order discontinuous Galerkin (DG2) solver intrinsically integrates the wet-dry fronts and uses localised slope-limiting; resulting in an implicit large eddy simulator with the shallow water equations (DG2-SWE) that can only simulate uncompounded eddies. A novel DG2 solver of RANS-k-ε (DG2-RANS-k-ε) is devised for simulating a wider range of vortical eddies, by: first, transforming the 5×5 advective-diffusive RANS-k-ε system into a 13×13 advection-dominated system; second, extending the DG2 formulation to the 13×13 system, with adaptation of its robustness treatments for the mean-flow variables; and, last, adding a new combination of stability/positivity-preserving treatments for turbulent-flow quantities. The DG2-RANS-k-ε solver is evaluated for simulating five experimental benchmarks using coarse, medium and fine resolutions. Results show that DG2-RANS-k-ε can reproduce compound eddies from the medium resolution, and that DG2-RANS (without k-ε) can better reproduce laminar wakes. Using the medium resolution reduces runtimes by 7-fold and running on the GPU further reduce runtimes by 2-to-6-fold. The code, including simulation setup files, is open-source within a new release of the LISFLOOD-FP hydraulic modelling environment (https://doi.org/10.5281/zenodo.7628739), with documentation and demonstration videos (https://www.seamlesswave.com/DG2_RANS).