Taylor-Green vortex

In fluid dynamics, the Taylor-Green vortex is a 2-dimensional, unsteady flow of a decaying vortex, which has the exact closed form solution of incompressible Navier-Stokes equations in Cartesian coordinates. It is named after the British physicists and mathematicians Geoffrey Ingram Taylor and George Green.

Incompressible Navier-Stokes equations

The incompressible Navier-Stokes equation in the absence of body force is given by:$frac\{partial\; u\}\{partial\; x\}+\; frac\{partial\; v\}\{partial\; y\}\; =\; 0$

:$frac\{partial\; u\}\{partial\; t\}\; +\; ufrac\{partial\; u\}\{partial\; x\}\; +\; vfrac\{partial\; u\}\{partial\; y\}\; =-frac\{1\}\{\; ho\}\; frac\{partial\; p\}\{partial\; x\}\; +\; u\; left(\; frac\{partial^2\; u\}\{partial\; x^2\}\; +\; frac\{partial^2\; u\}\{partial\; y^2\}\; ight)$

:$frac\{partial\; v\}\{partial\; t\}\; +\; ufrac\{partial\; v\}\{partial\; x\}\; +\; vfrac\{partial\; v\}\{partial\; y\}\; =-frac\{1\}\{\; ho\}\; frac\{partial\; p\}\{partial\; y\}\; +\; u\; left(\; frac\{partial^2\; v\}\{partial\; x^2\}\; +\; frac\{partial^2\; v\}\{partial\; y^2\}\; ight)$The first of the above equation represents the continuity equation and the other two represent the momentum equations.

Taylor-Green vortex solution

In the domain $0\; le\; x,y\; le\; pi$, the solution is given by

:$u\; =\; sin\; x\; cos\; y\; F(t)\; qquad\; qquad\; v\; =\; -cos\; x\; sin\; y\; F(t)$

where $F(t)\; =\; e^\{-2\; u\; t\}$, $u$ 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

:$p\; =\; frac\{\; ho\}\{2\}\; left(\; cos\; 2x\; +\; sin\; 2y\; ight)\; F^2(t)$

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

References

ee also

* Navier-Stokes equations
* Vortex