TwoDNavierStokes Module

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, $\zeta=\partial_x \upsilon- \partial_y u = \nabla^2\psi$. The equation solved by the module is:

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

where $\mathsf{J}(a, b) = (\partial_x a)(\partial_y b)-(\partial_y a)(\partial_x b)$. 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:

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

In doing so the Jacobian is computed in the conservative form: $\mathsf{J}(a,b) = \partial_y [ (\partial_x a) b] -\partial_x[ (\partial_y a) b]$.

Thus:

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

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

AbstractTypes and Functions

Params

For the unforced case ($f=0$) parameters AbstractType is build with Params and it includes:

ν: Float; viscosity or hyperviscosity coefficient.
nν: Integer$>0$; the order of viscosity $n_\nu$. Case $n_\nu=1$ gives normal viscosity.
μ: Float; bottom drag or hypoviscosity coefficient.
nμ: Integer$\ge 0$; the order of hypodrag $n_\mu$. Case $n_\mu=0$ gives plain linear drag $\mu$.

For the forced case ($f\ne 0$) parameters AbstractType is build with ForcedParams. It includes all parameters in Params and additionally:

calcF!: Function that calculates the forcing $\widehat{f}$

Vars

For the unforced case ($f=0$) variables AbstractType is build with Vars and it includes:

zeta: Array of Floats; relative vorticity.
u: Array of Floats; $x$-velocity, $u$.
v: Array of Floats; $y$-velocity, $\upsilon$.
sol: Array of Complex; the solution, $\widehat{\zeta}$.
zetah: Array of Complex; the Fourier transform $\widehat{\zeta}$.
uh: Array of Complex; the Fourier transform $\widehat{u}$.
vh: Array of Complex; the Fourier transform $\widehat{\upsilon}$.

For the forced case ($f\ne 0$) variables AbstractType is build with ForcedVars. It includes all variables in Vars and additionally:

Fh: Array of Complex; the Fourier transform $\widehat{f}$.
prevsol: Array of Complex; the values of the solution sol at the previous time-step (useful for calculating the work done by the forcing).

calcN! function

The nonlinear term $\mathcal{N}(\widehat{\zeta})$ is computed via functions:

calcN_advection!: computes $- \widehat{\mathsf{J}(\psi, \zeta)}$ and stores it in array N.
calcN_forced!: computes $- \widehat{\mathsf{J}(\psi, \zeta)}$ via calcN_advection! and then adds to it the forcing $\widehat{f}$ computed via calcF! function. Also saves the solution $\widehat{\zeta}$ of the previous time-step in array prevsol.
updatevars!: uses sol to compute $\zeta$, $u$, $\upsilon$, $\widehat{u}$, and $\widehat{\upsilon}$ and stores them into corresponding arrays of Vars/ForcedVars.

Examples

examples/twodnavierstokes_decaying.jl: A script that simulates decaying two-dimensional turbulence reproducing the results of the paper by
McWilliams, J. C. (1984). The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech., 146, 21-43.
examples/twodnavierstokes_stochasticforcing.jl: A script that simulates forced-dissipative two-dimensional turbulence with isotropic temporally delta-correlated stochastic forcing.