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.

References

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