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

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


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

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.], click

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