Taylor–Green vortex

Incompressible Navier–Stokes equations

The incompressible Navier–Stokes equations in the absence of body force, and in two spatial dimensions, are given by

${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }v}{\mathrm{\partial }y}=0,}$

${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }u}{\mathrm{\partial }y}=-\frac{1}{\rho }\frac{\mathrm{\partial }p}{\mathrm{\partial }x}+\nu \left(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}\right),}$

${\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }v}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }v}{\mathrm{\partial }y}=-\frac{1}{\rho }\frac{\mathrm{\partial }p}{\mathrm{\partial }y}+\nu \left(\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{y}^2}\right).}$

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 ${\displaystyle 0\le x,y\le 2\pi }$, the solution is given by

${\displaystyle u=\sin x\cos yF(t),v=-\cos x\sin yF(t),}$

where ${\displaystyle F(t)=e^{-2\nu t}}$, ${\displaystyle \nu }$ being the kinematic viscosity of the fluid. Following the analysis of Taylor and Green for the two-dimensional situation, and for ${\displaystyle A=a=b=1}$, gives agreement with this exact solution, if the exponential is expanded as a Taylor series, i.e. ${\displaystyle F(t)=1-2\nu t+O(t^2)}$.

The pressure field ${\displaystyle p}$ can be obtained by substituting the velocity solution in the momentum equations and is given by

${\displaystyle p=\frac{\rho }{4}\left(\cos 2x+\cos 2y\right)F^2(t).}$

The stream function of the Taylor–Green vortex solution, i.e. which satisfies ${\displaystyle \mathbf{v}=\mathrm{\nabla }\times \mathit{\psi }}$ for flow velocity ${\displaystyle \mathbf{v}}$, is

${\displaystyle \mathit{\psi }=\sin x\sin yF(t)\hat{\mathbf{z}}.}$

Similarly, the vorticity, which satisfies ${\displaystyle \mathit{\omega }=\mathrm{\nabla }\times \mathbf{v}}$, is given by

${\displaystyle \mathit{\omega }=2\sin x\sin yF(t)\hat{\mathbf{z}}.}$

The Taylor–Green vortex solution may be used for testing and validation of temporal accuracy of Navier–Stokes algorithms.

A generalization of the Taylor–Green vortex solution in three dimensions is described in.