In the original work of Taylor and Green, a particular flow is analyzed in three spatial dimensions, with the three velocity components at time specified by
The continuity equation determines that . 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.
Animation of a Taylor-Green Vortex using colour coded Lagrangian tracers
where , being the kinematic viscosity of the fluid. Following the analysis of Taylor and Green for the two-dimensional situation, and for , gives agreement with this exact solution, if the exponential is expanded as a Taylor series, i.e. .
The pressure field can be obtained by substituting the velocity solution in the momentum equations and is given by
The stream function of the Taylor–Green vortex solution, i.e. which satisfies for flow velocity , is
Similarly, the vorticity, which satisfies , is given by
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 in described in.