# 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.[1][2]

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.

## References

1. Chorin, A. J., Numerical solution of the Navier-Stokes equations, Math. Comp., 22, 745-762 (1968).
2. Kim, J. and Moin, P., Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59, 308-323 (1985).

Some part of the content is copypasted from http://en.wikipedia.org/wiki/Taylor–Green_vortex