Velocity-vorticity-helicity formulation and a solver for the Navier-Stokes equations

For the three-dimensional incompressible Navier-Stokes equations, we present a formulation featuring velocity, vorticity and helical density as independent variables. We find the helical density can be observed as a Lagrange multiplier corresponding to the divergence-free constraint on the vorticity variable, similar to the pressure in the case of the incompressibility condition for velocity. As one possible practical application of this new formulation, we consider a time-splitting numerical scheme based on an alternating procedure between vorticity – helical density and velocity – Bernoulli pressure systems of equations. Results of numerical experiments include a comparison with some well-known schemes based on pressure-velocity formulation and illustrate the competitiveness on the new scheme as well as the soundness of the new formulation.