View source for Taylor-Green vortex

[[Taylor-Green vortex]] is [[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:
:<math>
u_1 = \sin(x_1)~ \cos (x_2)~  F(t) \qquad \qquad u_2 = -\cos(x_1)~ \sin(x_2) ~F(t)
</math>

where <math>F(t) = e^{-2\nu t}</math>; <math>\nu</math> being the [[Viscosity#Kinematic viscosity|kinematic viscosity]] of the fluid. The pressure field <math>p</math> can be obtained by substituting the velocity solution in the momentum equations and is given by

:<math>
p = \frac{\rho}{4} \left( \cos(2x_1) + \cos(2x_2) \right)~ F(t)^2
</math>

The [[stream function]] of the Taylor–Green vortex solution, i.e. which satisfies <math> \mathbf{v} = \nabla \times \boldsymbol{\psi}</math>
<!-- for flow velocity <math>\mathbf{v}</math>, 
!-->
is
:<math>
\psi = \{0,0,\sin(x) \sin(y) F(t) \}.
</math>

Similarly, the [[vorticity]], which satisfies <math> \mathbf{\omega} = \nabla \times \mathbf{v} </math>, is given by
:<math>
\vec{\omega} = \{0,~0,~2\sin(x)\sin(y) F(t)\}.
</math>

The Taylor–Green vortex solution may be used for testing and validation of temporal accuracy of Navier-Stokes algorithms.<ref>
[[Alexandre Chorin|Chorin, A. J.]], ''Numerical solution of the Navier-Stokes equations'', Math. Comp., 22, 745-762 (1968).</ref><ref>[[John Kim (professor)|Kim, J]]. and [[Parviz Moin|Moin, P.]], ''Application of a fractional-step method to incompressible Navier-Stokes equations'', J. Comput. Phys., 59, 308-323 (1985). </ref>

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==
<references/>

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

[[Category:Navier-Stokes]]
[[Category:Exact solution]]
[[Category:Periodic functions]]
[[Category:Fluid dynamics]]
[[Category:Articles in English]]