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