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 formula_1 at time formula_2 specified by
formula_3
formula_4
formula_5
The continuity equation formula_6 determines that formula_7. 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. The incompressible Navier–Stokes equations in the absence of body force, and in two spatial dimensions, are given by
formula_8
formula_9
formula_10
The first of the above equation represents the continuity equation and the other two represent the momentum equations. In the domain formula_11, the solution is given by
formula_12
where formula_13, formula_14 being the kinematic viscosity of the fluid. Following the analysis of Taylor and Green for the two-dimensional situation, and for formula_15, gives agreement with this exact solution, if the exponential is expanded as a Taylor series, i.e. formula_16. The pressure field formula_17 can be obtained by substituting the velocity solution in the momentum equations and is given by
formula_18
The stream function of the Taylor–Green vortex solution, i.e. which satisfies formula_19 for flow velocity formula_20, is
formula_21
Similarly, the vorticity, which satisfies formula_22, is given by
formula_23
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.
Taylor–Green vortex solution
In fluid dynamics, the Taylor–Green vortex is an unsteady flow of a decaying vortex, which has an exact closed form solution of the incompressible Navier–Stokes equations in Cartesian coordinates.