Multi-dimensional finite volume scheme for the vorticity transport equations

A finite-volume scheme is developed for the three-dimensional, incompressible vorticity transport equations (VTE) using multi-dimensional upwinding, with the goal of efficient computations of vortex-dominated flows. By modifying the VTE with a term proportional to the divergence of the vorticity, a stable hyperbolic PDE system with a simple structure is revealed. The structure of the resulting eigen-system, including the vortex stretching term, makes the formulation and solution of the generalized Riemann problem and multi-dimensional upwinding more natural, when compared to the Euler equations. To reduce the computational costs of determining the transverse fluxes, a flux-based wave propagation approach is employed. In this approach, the transverse fluxes are computed via direct manipulation of the one-dimensional generalized Riemann problem with no additional Riemann problem solutions needed. The numerical scheme is implemented within an adaptive mesh refinement framework and evaluated on a series of canonical vortex-dominated flows. A translating vortex flow reveals that the multi-dimensional upwinding substantially outperforms one-dimensional schemes in terms of accuracy and computational time, especially when the vortex propagates oblique to cell surfaces. By including transverse fluxes in simulations of propagating vortex rings, key physical attributes including propagation velocity and impulse can be captured with better accuracy compared to one-dimensional schemes. Further vortex ring simulations demonstrate that the proposed multi-dimensional scheme can preserve vorticity with relatively coarse grids compared to simulations employing the incompressible Euler equations. Integrated quantities from leapfrogging vortex ring problems and energy spectra from decaying turbulent flows confirm that the multi-dimensional scheme can accurately reproduce flows with complex vortex stretching and multiple length scales. (C) 2018 Elsevier Ltd. All rights reserved.