Taylor–Green vortex

Editing Taylor–Green vortex exact solution of the Navier-Stokes equation characterized with periodicity, the two periods are \(\{2\pi,0,0\}\) and \(\{0,2\pi,0\}\). The velovities \(u\) as functions of coordinates \(x\) are defined as follows:

u_1 = \sin(x_1)~ \cos (x_2)~ F(t) \qquad \qquad u_2 = -\cos(x_1)~ \sin(x_2) ~F(t) $$

where $$F(t) = e^{-2\nu t}$$; $$\nu$$ being the kinematic viscosity of the fluid. The pressure field $$p$$ can be obtained by substituting the velocity solution in the momentum equations and is given by



p = \frac{\rho}{4} \left( \cos(2x_1) + \cos(2x_2) \right)~ F(t)^2 $$

The stream function of the Taylor–Green vortex solution, i.e. which satisfies $$ \mathbf{v} = \nabla \times \boldsymbol{\psi}$$

is

\psi = \{0,0,\sin(x) \sin(y) F(t) \}. $$

Similarly, the vorticity, which satisfies $$ \mathbf{\omega} = \nabla \times \mathbf{v} $$, is given by

\vec{\omega} = \{0,~0,~2\sin(x)\sin(y) F(t)\}. $$

The Taylor–Green vortex solution may be used for testing and validation of temporal accuracy of Navier-Stokes algorithms.

More solutions of the Navier-Stokes equation can be obtained with the invariant transformation of the Taylor-Green vortex; in particular, with the appropriate rotations, translations and scaling.