Taylor-Green vortex sheet, reduced units

The Taylor-Green vortex sheet is a solution to the 2D Navier-Stokes equations for an incompressible Newtonian fluid:

\frac{d \mathbf{u}}{d t}= \nu \nabla^2 \mathbf{u} - \nabla p/\rho ,

where \mathbf{u} is the velocity field, p is the pressure, \nu=\mu/\rho is the kinematic viscosity, and \rho is the fixed density of the fluid. The time derivative is a total derivative:

\frac{d \mathbf{u}}{d t} =  \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u}

It is common to choose parameters that simplify the equations, but that can obscure the role of the different parameters. In the following, I provide expressions with all relevant parameters included, with their physical dimensions. I later pass to dimensionless, or reduced, units, in terms of the Reynolds and Courant numbers.

 

Continue reading