Taylor-Green vortex#

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

\begin{array}{r} \begin{aligned} u (x, y, t) & = - \sin (π x) \cos (π y) e^{- 2 π^{2} t / Re}, \\ v (x, y, t) & = \cos (π x) \sin (π y) e^{- 2 π^{2} t / Re}, \\ p (x, y, t) & = \frac{1}{4} (\cos (2 π x) + \cos (2 π y)) e^{- 4 π^{2} t / Re}, \\ ω (x, y, t) & = - 2 π \sin (π x) \sin (π y) e^{- 2 π^{2} t / Re} . \end{aligned} \end{array}

The domain, either periodic or not, is typically given as $Ω = [0, 2]^{2}$ . The above analytical solutions are used for initial and boundary conditions. The simulation, for example, runs from $t = 0$ to $t = 1$ and errors are measured at $t = 1$ .

For a phyem implementation of the shear layer rollup using the dual-field method introduced in [Dual-field NS, Zhang et al., JCP (2022)], click phyem_df2_TGV.py.

↩️ Back to 🌊 Navier-Stokes equations.