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,

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

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

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