Data-driven large eddy simulation of forced 2D turbulence
The direct numerical simulation (DNS) solves the dimensionless governing equations in the vorticity ($\omega$) and streamfunction ($\psi$) formulation in a doubly periodic square domain with length $L=2\pi$:
\[ \frac{\partial \omega}{\partial t} + \mathcal{N}(\omega,\psi)=\frac{1}{Re}\nabla^2\omega - f -r\omega, \qquad \qquad \nabla^2\psi = -\omega. \]
Here, $\mathcal{N}(\omega,\psi)$ represents the nonlinear advection term:
\[ \mathcal{N}(\omega,\psi)=\frac{\partial \psi}{\partial y}\frac{\partial \omega}{\partial x} - \frac{\partial \psi}{\partial x}\frac{\partial \omega}{\partial y}, \]
and $f$ represents a deterministic forcing:
\[ f(x,y)= k_f[\cos{(k_fx)} + \cos{(k_fy)}]. \]
Filtering $\overline{(\cdot)}$ the DNS equations gives the large eddy simulation (LES) equations:
\[ \frac{\partial \overline{\omega}}{\partial t} + \mathcal{N}(\overline{\omega},\overline{\psi})=\frac{1}{Re}\nabla^2\overline{\omega}-\overline{f}-r\overline{\omega}+\underbrace{\mathcal{N}(\overline{\omega},\overline{\psi}) - \overline{\mathcal{N}({\omega},{\psi})}}_{\Pi}, \]
\[ \nabla^2\overline{\psi} = -\overline{\omega}. \]
In this work, we use a fully convolutional neural network (CNN) to model the closure term $\Pi$.
