TwoDTurb Module

\[\newcommand{\J}{\mathsf{J}}\]

Basic Equations

This module solves two-dimensional incompressible turbulence. The flow is given through a streamfunction $\psi$ as $(u,\upsilon) = (-\partial_y\psi, \partial_x\psi)$. The dynamical variable used here is the component of the vorticity of the flow normal to the plane of motion, $q=\partial_x \upsilon- \partial_y u = \nabla^2\psi$. The equation solved by the module is:

\[\partial_t q + \J(\psi, q) = \underbrace{-\left[\mu(-1)^{n_\mu} \nabla^{2n_\mu} +\nu(-1)^{n_\nu} \nabla^{2n_\nu}\right] q}_{\textrm{dissipation}} + f\ .\]

where $\J(a, b) = (\partial_x a)(\partial_y b)-(\partial_y a)(\partial_x b)$. On the right hand side, $f(x,y,t)$ is forcing, $\mu$ is hypoviscosity, and $\nu$ is hyperviscosity. Plain old linear drag corresponds to $n_{\mu}=0$, while normal viscosity corresponds to $n_{\nu}=1$.

Implementation

The equation is time-stepped forward in Fourier space:

\[\partial_t \widehat{q} = - \widehat{J(\psi, q)} -\left[\mu k^{2n_\mu} +\nu k^{2n_\nu}\right] \widehat{q} + \widehat{f}\ .\]

In doing so the Jacobian is computed in the conservative form: $\J(f,g) = \partial_y [ (\partial_x f) g] -\partial_x[ (\partial_y f) g]$.

Thus:

\[\mathcal{L} = -\mu k^{-2n_\mu} - \nu k^{2n_\nu}\ ,\]

\[\mathcal{N}(\widehat{q}) = - \mathrm{i}k_x \mathrm{FFT}(u q)- \mathrm{i}k_y \mathrm{FFT}(\upsilon q)\ .\]