Data-driven subgrid momentum and heat fluxes modeling of thermal convection
The direct numerical simulation (DNS) solves the dimensionless governing equations in the vorticity ($\omega$), streamfunction ($\psi$), and temperature fluctuation ($\theta$) formulation in a Fourier-Chebyshev grid with aspect ratio $\Gamma = 3\pi$:
\[ \frac{\partial \omega}{\partial t} + \mathcal{N}(\omega,\psi)=\nu\nabla^2\omega - \frac{\partial\theta}{\partial x}, \qquad \qquad \nabla^2\psi = -\omega, \]
\[ \frac{\partial\theta}{\partial t} + \mathbf{u}\cdot\nabla\theta - u_y=k\nabla^2\theta. \]
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 velocity vector $\mathbf{u}=(u_x,u_y)$, where
\[ u_x = \frac{\partial \psi}{\partial y}, \qquad u_y = -\frac{\partial \psi}{\partial x}, \]
Filtering $\overline{(\cdot)}$ the DNS equations gives the large eddy simulation (LES) equations:
\[ \frac{\partial \overline{\omega}}{\partial t} + \mathcal{N}(\overline{\omega},\overline{\psi})=\nu\nabla^2\overline{\omega}-\frac{\partial\overline{\theta}}{\partial x}+\underbrace{\mathcal{N}(\overline{\omega},\overline{\psi}) - \overline{\mathcal{N}({\omega},{\psi})}}_{\Pi_\omega}, \qquad \qquad \nabla^2\overline{\psi} = -\overline{\omega}, \] \[ \frac{\partial\overline{\theta}}{\partial t} + \mathbf{\overline{u}}\cdot\nabla\overline{\theta} - \overline{u_y}=k\nabla^2\overline{\theta} + \underbrace{\mathbf{\overline{u}}\cdot\nabla\overline{\theta} - \overline{\mathbf{u}\cdot\nabla\theta}}_{\Pi_\theta}. \]
In this work, we use two fully convolutional neural networks (CNN) with identical structure to model the closure terms $\Pi_\omega$ and $\Pi_\theta$.
