Taylor-Green vortex#

The Taylor-Green vortex (TGV) is a series of analytical solutions of NS equations. In 2-dimensions, the TGV analytical solution is usually of the form,

\[\begin{split}\begin{aligned} u(x, y, t) &= - \sin(\pi x) \cos(\pi y) e^{-2\pi^2 t /\mathrm{Re}},\\ v(x, y, t) &= \cos(\pi x) \sin(\pi y) e^{-2\pi^2 t /\mathrm{Re}},\\ p(x, y, t) &= \frac{1}{4} \left(\cos(2\pi x) + \cos(2\pi y)\right)e^{-4\pi^2 t /\mathrm{Re}},\\ \omega(x, y, t) &= -2\pi \sin(\pi x)\sin(\pi y) e^{-2\pi^2 t /\mathrm{Re}}. \end{aligned}\end{split}\]

The domain, either periodic or not, is given as \(\Omega=[0,2]^2\).

↩️ Back to 🌊 Navier-Stokes equations.