Generalized Riemann problem-based upwind scheme for the vorticity transport equations

• A staggered numerical scheme is constructed for the vorticity transport equations.
• A term is added to make the hyperbolic system well-conditioned.
• A generalized Riemann solver is used to determine the numerical flux.
• The performance is assessed on vortex dynamics and turbulent flows.

We construct an efficient finite volume scheme to solve the three-dimensional vorticity–velocity form of the equations governing incompressible flows. The vorticity transport equations are recast in a form that allows for the development of an accurate upwind scheme. The resulting scheme employs a staggered arrangement of vorticity and velocity and utilizes generalized Riemann problems to determine the numerical flux at the cell faces. The resulting scheme is found to account for convection and stretching of vorticity in a stable manner without the need to perform any additional correction steps to control the divergence of vorticity. Several test cases including isolated and interacting vortices as well as turbulent flow are investigated to assess the effectiveness of the numerical method.