8.2.1 Vorticity Method

Take the rotation of

$$\frac{\partial \mbox{$\bf v$}}{\partial t} + (\mbox{$\bf v$}... ...box{$\bf v$}= - \nabla \bar{p} + \nu \nabla^{2}\mbox{$\bf v$}$$

to find

$$\frac{\partial \mbox{$\bf w$}}{\partial t} + (\mbox{$\bf v$} \cdot \nabla )\mbox{$\bf w$}= \nu \nabla^{2}\mbox{$\bf w$}$$ (8.55)

with the vorticity $\mbox{$\bf w$}\equiv \nabla \times \mbox{$\bf v$}$.
$\Longrightarrow$ Vorticity is transported both by an advective process ( $\mbox{$\bf v$} \cdot \nabla \, \mbox{$\bf w$}$) and by viscous diffusion.

Write the (divergence-free) velocity as the rotation of a streaming function $\mbox{$\bf u$}$:

$$\mbox{$\bf v$}\equiv \nabla \times \mbox{$\bf u$}$$ (8.56)

This does not determine $\mbox{$\bf u$}$ uniquely; $\Longrightarrow$require that $\nabla \cdot \mbox{$\bf u$}=0$.

Basic equations for the vorticity method:

$$\frac{\partial \mbox{$\bf w$}}{\partial t} + (\mbox{$\bf v$} \cdot \nabla )\mbox{$\bf w$} = \nu \nabla^{2}\mbox{$\bf w$}$$ (8.57)

$$\nabla^{2}\mbox{$\bf u$} = -\mbox{$\bf w$}$$ (8.58)

$$\mbox{$\bf v$} = \nabla \times \mbox{$\bf u$}$$ (8.59)

Two-dimensional case: $\mbox{$\bf u$}$ and $\mbox{$\bf w$}$ have only $z$-components:

$$\frac{\partial w}{\partial t} = \nu \nabla^{2}w -\left(v_{x}\partial_{y}-v_{y}\partial_{x}\right)w$$ (8.60)

$$\nabla^{2}u = -w$$ (8.61)

$$\mbox{$\bf v$} = u \nabla \times \mbox{$\bf e$}_{z} = \mbox{$\left( \begin{array}{r} \partial _{y}u \\ \vspace{-9 pt}\\ -\partial_{x}u \end{array} \right)$}$$ (8.62)

$\Longrightarrow$Apply Lax-Wendroff to these equations:



Stability is again governed by the CFL condition

$$\Delta t \leq \frac{2 \Delta l}{\sqrt{2}v_{max}}$$ (8.63)

In addition, the presence of diffusive terms implies the restriction

$$\Delta t \leq \frac{(\Delta l)^{2}}{\nu}$$ (8.64)