Taylor–Green vortex

In the original work of Taylor and Green, a particular flow is analyzed in three spatial dimensions, with the three velocity components \mathbf{v}=(u,v,w) at time t=0 specified by : u = A \cos ax \sin by \sin cz, : v = B \sin ax \cos by \sin cz, : w = C \sin ax \sin by \cos cz. The continuity equation \nabla \cdot \mathbf{v}=0 determines that Aa+Bb+Cc=0. The small time behavior of the flow is then found through simplification of the incompressible Navier–Stokes equations using the initial flow to give a step-by-step solution as time progresses. An exact solution in two spatial dimensions is known, and is presented below. ==Incompressible Navier–Stokes equations==

The incompressible Navier–Stokes equations in the absence of body force, and in two spatial dimensions, are given by : \frac{\partial u}{\partial x}+ \frac{\partial v}{\partial y} = 0, : \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} = -\frac{1}{\rho} \frac{\partial p}{\partial x} + \nu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right), : \frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} = -\frac{1}{\rho} \frac{\partial p}{\partial y} + \nu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\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 0 \le x,y \le 2\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\nu t}, \nu being the kinematic viscosity of the fluid. Following the analysis of Taylor and Green A generalization of the Taylor–Green vortex solution in three dimensions is described in . ==References==