Shear layer rollup

The shear layer rollup is a 2-dimensional ideal incompressible flow (Euler flow). And the external body force is zero.

The flow is in a periodic domain \(\Omega = [0, 2\pi]^2\), and components of the initial velocity \(\boldsymbol{u}^0 = \begin{bmatrix}u^0 & v^0\end{bmatrix}^{\mathsf{T}}\) are

\[\begin{split}u^0 = \left\lbrace\begin{aligned} &\tanh\left(\dfrac{y-\frac{\pi}{2}}{\delta}\right)\quad \text{if } y\leq\pi\\ &\tanh\left(\dfrac{\frac{3\pi}{2}-y}{\delta}\right)\quad \text{else} \end{aligned}\right.,\end{split}\]

and

\[v^0 = \epsilon\sin(x),\]

where \(\delta=\pi/15\) and \(\epsilon=0.05\).

Fig. 13 The vorticity field of the shear layer rollup flow at \(t\in\left\lbrace0,4,8\right\rbrace\) with contour lines for \(\left\lbrace\pm 1,\pm 2, \cdots, \pm 6\right\rbrace\).

It is seen that two vorticity layers gradually roll up due to the initial perturbation in the velocity field.

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_slr.py.

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