Bracket formulations and energy- and helicity-preserving numerical methods for the three-dimensional vorticity equation

The vorticity equation for three-dimensional viscous incompressible fluid flows is formulated within different bracket formalisms using the Poisson or Nambu bracket together with a dissipative bracket. The budgets of kinetic energy, helicity, and enstrophy derived from the bracket formulations are properly inherited by the finite difference equations obtained by invoking the discrete variational derivative method combined with the mimetic finite difference method. In particular, energy and helicity are conserved precisely in inviscid flow computations. The energy and enstrophy dissipate properly owing to viscosity in viscous flow computations, and the enstrophy is appropriately produced by the vortex stretching effect in both inviscid and viscous flow computations. The relationships between the stream function, velocity, and vorticity as well as the solenoidal conditions on the velocity and vorticity fields are also inherited. Numerical experiments on a periodic array of rolls that permits analytical solutions have been done to examine the properties and usefulness of the proposed method.