Vorticity-Velocity formulation for the stationary Navier-Stokes equations: The three-dimensional case

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. A...

The stationary three-dimensional Navier-Stokes Equations with a non-zero constant velocity at infinity

Spectral discretization of the vorticity, velocity and pressure formulation of the Navier-Stokes equations

We consider the Navier–Stokes equations in a twoou three-dimensional domain provided with non standard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity. We write a variational formulation of this problem with three independent unknowns: the vorticity, the velocity and the pressure, and prove the existence of a solution for thi...

The Stationary Navier-Stokes Equations

Having considered the linear Stokes equations, we will now bring back the nonlinear term and consider the nonlinear version of the Stokes system. The solution of these equations can be viewed as the limit to which the solution of the full N-S equations tends as t tends to infinity. Of course no one knows at this point if the solutions of the N-S equations do tend to some limit as t tends to inf...

Vorticity alignment results for the three-dimensional Euler and Navier-Stokes equations

We address the problem in Navier-Stokes isotropic turbulence of why the vorticity accumulates on thin sets such as quasi-one-dimensional tubes and quasi-two-dimensional sheets. Taking our motivation from the work of Ashurst, Kerstein, Kerr and Gibson, who observed that the vorticity vector ω aligns with the intermediate eigenvector of the strain matrix S, we study this problem in the context of...