Taylor-Green vortex

An exact solution to the two-dimensional incompressible Navier-Stokes equations is given by Geoffrey Ingram Taylor , Albert Edward Green (1937) describing the unsteady flow of a vortex decaying under viscous dissipation. The viscous terms balance the time derivatives while the nonlinear advection terms balance the pressure gradient term. We use the doubly-periodic solution described by (p. 310) Jan S Hesthaven , Tim Warburton (2007)

\[\begin{aligned} u(x, y, t) &= -\sin(2\pi y) e^{-4\pi^2\nu t} \\ v(x, y, t) &= \sin(2\pi x) e^{-4\pi^2\nu t} \\ p(x, y, t) &= -\cos(2\pi x) \cos(2\pi y) e^{-8\pi^2\nu t} \end{aligned}\]