Stratified Couette flow

In regular plane Couette flow, a viscous fluid is situated between two infinite, parallel plates and usually the top plate moves at some constant velocity generating a linear velocity profile in steady state (§4.2-4.3) G. K. Batchelor (2000).

In stratified Couette flow, the plates are additionally held at fixed temperatures to provide a stable stratification. It is a canonical geometry in which to investigate the dynamics of stratified shear flows as it exhibits a transition to turbulence, intermittent turbulence, coherent structures, and diapycnal mixing among other rich behavior Qi Zhou , John R. Taylor , C. P. Caulfield (2017).

In this test we follow Catherine A. Vreugdenhil , John R. Taylor (2018) who use stratified Couette flow as a test of their large-eddy simulation model. We will compare our results to theirs and the direct numerical simulation results of Enrico Deusebio , C. P. Caulfield , J. R. Taylor (2015) and Qi Zhou , John R. Taylor , C. P. Caulfield (2017).

Simulation setup

In our setup replicating that of Catherine A. Vreugdenhil , John R. Taylor (2018), a domain of size $(L_x, L_y, L_z)/h = (4\pi, 2\pi, 2)$ is bounded in the vertical $z$-direction by two infinite, parallel plates or walls at $z = \pm h$ with horizontal velocity $u = \pm U_w$ and constant temperature $\theta = \pm \Theta_w$ where $\theta$ denotes temperature in this section. The resulting flow is governed by the Reynolds, Richardson, and Prandtl numbers

\[\text{Re} = \frac{U_w h}{\nu}, \quad \text{Ri} = \frac{\alpha g \Theta_w h}{U_w^2}, \quad \text{Pr} = \frac{\nu}{\kappa} \, ,\]

where $\nu$ is the kinematic viscosity, $\kappa$ is the thermal diffusivity, $\alpha$ is the thermal expansion coefficient, and $g$ is the gravitational acceleration.

Defining horizontal averages for velocity $U = \langle u \rangle$ and $\Theta = \langle \theta \rangle$ we can define the friction velocity $u_\tau$ and friction temperature $\theta_\tau$ at the wall by

\[u_\tau^2 = \frac{\tau_w}{\rho_0} = \nu \left\vert \frac{\partial U}{\partial z} \right\vert_{z = \pm h}, \quad \theta_\tau = \frac{q_w}{u_\tau} = \frac{\kappa}{u_\tau} \left\vert \frac{\partial\Theta}{\partial z} \right\vert_{z = \pm h} \, ,\]

where $\tau_w$ is the wall stress and $q_w = u_\tau \theta_\tau$ is the wall heat flux.

From here the friction Reynolds number and the Nusselt number can be defined

\[\text{Re}_\tau = \frac{u_\tau h}{\nu}, \quad \text{Nu} = \frac{q_w h}{\kappa \Theta_w} \, ,\]

which can be computed and compared.

Velocity and temperature

$\mathrm{Re}_\tau$ and $\mathrm{Nu}$ values

LES viscosity and diffusivity