We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We present a deep probabilistic convolutional neural network (PCNN) model for predicting local values of small-scale mixing properties in stratified turbulent flows, namely the dissipation rates of turbulent kinetic energy and density variance, 
$\varepsilon$ and 
$\chi$. Inputs to the PCNN are vertical columns of velocity and density gradients, motivated by data typically available from microstructure profilers in the ocean. The architecture is designed to enable the model to capture several characteristic features of stratified turbulence, in particular the dependence of small-scale isotropy on the buoyancy Reynolds number 
$Re_b:=\varepsilon /(\nu N^2)$, where 
$\nu$ is the kinematic viscosity and 
$N$ is the background buoyancy frequency, the correlation between suitably locally averaged density gradients and turbulence intensity and the importance of capturing the tails of the probability distribution functions of values of dissipation. Empirically modified versions of commonly used isotropic models for 
$\varepsilon$ and 
$\chi$ that depend only on vertical derivatives of density and velocity are proposed based on the asymptotic regimes 
$Re_b\ll 1$ and 
$Re_b\gg 1$, and serve as an instructive benchmark for comparison with the data-driven approach. When trained and tested on a simulation of stratified decaying turbulence which accesses a range of turbulent regimes (associated with differing values of 
$Re_b$), the PCNN outperforms assumptions of isotropy significantly as 
$Re_b$ decreases, and additionally demonstrates improvements over the fitted empirical models. A differential sensitivity analysis of the PCNN facilitates a comparison with the theoretical models and provides a physical interpretation of the features enabling it to make improved predictions.
We study the inertial migration of a torque-free neutrally buoyant sphere in wall-bounded plane Couette flow over a wide range of channel Reynolds numbers 
$Re_c$ in the limit of small particle Reynolds number (
$Re_p\ll 1$) and confinement ratio (
$\lambda \ll 1$). Here, 
$Re_c = V_{wall}H/\nu$, where 
$H$ denotes the separation between the channel walls, 
$V_\text {wall}$ denotes the speed of the moving wall, and 
$\nu$ is the kinematic viscosity of the Newtonian suspending fluid. Also, 
$\lambda = a/H$, where 
$a$ is the sphere radius, with 
$Re_p=\lambda ^2 Re_c$. The channel centreline is found to be the only (stable) equilibrium below a critical 
$Re_c$ (
$\approx 148$), consistent with the predictions of earlier small-
$Re_c$ analyses. A supercritical pitchfork bifurcation at the critical 
$Re_c$ creates a pair of stable off-centre equilibria, located symmetrically with respect to the centreline, with the original centreline equilibrium becoming unstable simultaneously. The new equilibria migrate wall-ward with increasing 
$Re_c$. In contrast to the inference based on recent computations, the aforementioned bifurcation occurs for arbitrarily small 
$Re_p$ provided that 
$\lambda$ is sufficiently small. An analogous bifurcation occurs in the two-dimensional scenario, that is, for a circular cylinder suspended freely in plane Couette flow, with the critical 
$Re_c$ being approximately 
$110$.
We study the effects of buoyancy, surface-tension gradients and phase boundary on the stability of a layer of water that is confined between air at the top and a layer of ice at the bottom. The temperature of the overlying air and flux condition at the free surface of the water layer are such that the layer is susceptible to both thermal and thermocapillary instabilities. We perform a linear stability analysis to identify these modes of instability and investigate the effects of the phase boundary on them. We find that with increasing thickness of the ice layer, the critical Rayleigh and Marangoni numbers for the instabilities are found to first decrease and then asymptote to constant values for ice thicknesses much larger than the thickness of the water layer. In the case of thermocapillary instability, we find that the thickness of the ice layer has negligible influence on the stability threshold for dimensionless wavenumber 
$k \gg 1$, and that the presence of an unstably stratified liquid layer significantly alters the stability threshold for 
$k = O (1)$. Furthermore, the inclusion of Marangoni stresses reduces the stability threshold of the thermal instability.
