We investigate fully developed turbulence in stratified plane Couette flows using direct numerical simulations similar to those reported by Deusebioet al.(J. Fluid Mech., vol. 781, 2015, pp. 298–329) expanding the range of Prandtl number $Pr$ examined by two orders of magnitude from 0.7 up to 70. Significant effects of $Pr$ on the heat and momentum fluxes across the channel gap and on the mean temperature and velocity profile are observed. These effects can be described through a mixing length model coupling Monin–Obukhov (M–O) similarity theory and van Driest damping functions. We then employ M–O theory to formulate similarity scalings for various flow diagnostics for the stratified turbulence in the gap interior. The midchannel gap gradient Richardson number $Ri_{g}$ is determined by the length scale ratio $h/L$ , where $h$ is the half-channel gap depth and $L$ is the Obukhov length scale. As $h/L$ approaches very large values, $Ri_{g}$ asymptotes to a maximum characteristic value of approximately 0.2. The buoyancy Reynolds number $Re_{b}\equiv \unicode[STIX]{x1D700}/(\unicode[STIX]{x1D708}N^{2})$ , where $\unicode[STIX]{x1D700}$ is the dissipation, $\unicode[STIX]{x1D708}$ is the kinematic viscosity and $N$ is the buoyancy frequency defined in terms of the local mean density gradient, scales linearly with the length scale ratio $L^{+}\equiv L/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ , where $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ is the near-wall viscous scale. The flux Richardson number $Ri_{f}\equiv -B/P$ , where $B$ is the buoyancy flux and $P$ is the shear production, is found to be proportional to $Ri_{g}$ . This then leads to a turbulent Prandtl number $Pr_{t}\equiv \unicode[STIX]{x1D708}_{t}/\unicode[STIX]{x1D705}_{t}$ of order unity, where $\unicode[STIX]{x1D708}_{t}$ and $\unicode[STIX]{x1D705}_{t}$ are the turbulent viscosity and diffusivity respectively, which is consistent with Reynolds analogy. The turbulent Froude number $Fr_{h}\equiv \unicode[STIX]{x1D700}/(NU^{\prime 2})$ , where $U^{\prime }$ is a turbulent horizontal velocity scale, is found to vary like $Ri_{g}^{-1/2}$ . All these scalings are consistent with our numerical data and appear to be independent of $Pr$ . The classical Osborn model based on turbulent kinetic energy balance in statistically stationary stratified sheared turbulence (Osborn,J. Phys. Oceanogr., vol. 10, 1980, pp. 83–89), together with M–O scalings, results in a parameterization of $\unicode[STIX]{x1D705}_{t}/\unicode[STIX]{x1D708}\sim \unicode[STIX]{x1D708}_{t}/\unicode[STIX]{x1D708}\sim Re_{b}Ri_{g}/(1-Ri_{g})$ . With this parameterization validated through direct numerical simulation data, we provide physical interpretations of these results in the context of M–O similarity theory. These results are also discussed and rationalized with respect to other parameterizations in the literature. This paper demonstrates the role of M–O similarity in setting the mixing efficiency of equilibrated constant-flux layers, and the effects of Prandtl number on mixing in wall-bounded stratified turbulent flows.

Continuously forced, stratified shear flows occur in many geophysical systems, including flow over sills, through fjords and at the mouths of rivers and estuaries. These continuously forced shear flows can be unstable and drive turbulence, which can enhance the rate of mixing. In this study, we analyse three-dimensional direct numerical simulations of an idealized stratified shear flow that is continuously forced by weakly relaxing both the buoyancy and streamwise velocity towards prescribed mean profiles. We explore a range of large and small Richardson numbers, for constant Reynolds and Prandtl numbers (). After a turbulent steady state develops, three regimes are observed: (i) a weakly stratified, overturning regime, (ii) a strongly stratified, scouring regime and (iii) an intermediately stratified, intermittent regime. The overturning regime exhibits partially formed overturning billows that break down into turbulence and broaden the velocity and buoyancy interfaces. Conversely, the scouring regime exhibits internal gravity waves propagating along the strongly stratified buoyancy interface, while turbulence on either side of the buoyancy interface reinforces the stratification. The intermediate regime quasi-periodically alternates between behaviours associated with the overturning and scouring regimes. For each case, we quantify an appropriate measure of the efficiency of mixing and examine its dependence on relevant parameters including appropriate definitions of the buoyancy Reynolds number, gradient Richardson number and horizontal Froude numbers. Using a framework involving sorted buoyancy coordinates as introduced by Nakamura (J. Atmos. Sci., vol. 53, 1996, pp. 1524–1537) and Winters & D'Asaro (J. Fluid Mech., vol. 317, 1996, pp. 179–193), we examine the underlying physical mechanisms leading to broadening and thinning of the buoyancy interface.

We report an idealized numerical study of a melting and freezing solid adjacent to a turbulent, buoyancy-affected shear flow, in order to improve our understanding of topography generation by phase changes in the environment. We use the phase-field method to dynamically couple the heat equation for the solid with the Navier–Stokes equations for the fluid. We investigate the evolution of an initially flat and horizontal solid boundary overlying a pressure-driven turbulent flow. We assume a linear equation of state for the fluid and change the sign of the thermal expansion coefficient, such that the background density stratification is either stable, neutral or unstable. We find that channels aligned with the direction of the mean flow are generated spontaneously by phase changes at the fluid–solid interface. Streamwise vortices in the fluid, the interface topography and the temperature field in the solid influence each other and adjust until a statistical steady state is obtained. The crest-to-trough amplitude of the channels is larger than approximately 10 the viscous length scale, but is much larger and more persistent for an unstable stratification than for a neutral or stable stratification. This happens because a stable stratification makes the cool melt fluid buoyant such that it shields the channel from further melting, whereas an unstable stratification makes the cool melt fluid sink, inducing further melting by rising hot plumes. The statistics of flow velocities and melt rates are investigated, and we find that channels and keels emerging in our simulations do not significantly change the mean drag coefficient.

Significance: The Gulf Stream front separates the North Atlantic subtropical and subpolar ocean gyres, which have very distinct physical, biological, and chemical properties. Exchange across the front is believed to be important for both ocean circulation and biological productivity; however, the physical mechanisms responsible are not fully understood. Here, we present direct observational evidence of enhanced mixing across the north wall of the Gulf Stream. High-resolution simulations suggest that the presence of the sharp density front, not typically resolved in ocean-circulation models, is central to generating the observed mixing. Interpreting the observations in terms of the flux of freshwater and nutrients across the Gulf Stream north wall suggests an important mechanism of exchange that has not been fully explored.

We investigate numerically transient linear growth of three-dimensional perturbations in a stratified shear layer to determine which perturbations optimize the growth of the total kinetic and potential energy over a range of finite target time intervals. The stratified shear layer has an initial parallel hyperbolic tangent velocity distribution with Reynolds number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}=U_0 h/\nu =1000$ and Prandtl number $\nu /\kappa =1$ , where $\nu $ is the kinematic viscosity of the fluid and $\kappa $ is the diffusivity of the density. The initial stable buoyancy distribution has constant buoyancy frequency $N_0$ , and we consider a range of flows with different bulk Richardson number ${\mathit{Ri}}_b=N_0^2h^2/U_0^2$ , which also corresponds to the minimum gradient Richardson number ${\mathit{Ri}}_g(z)=N_0^2/(\mathrm{d}U/\mathrm{d} z)^2$ at the midpoint of the shear layer. For short target times, the optimal perturbations are inherently three-dimensional, while for sufficiently long target times and small ${\mathit{Ri}}_b$ the optimal perturbations are closely related to the normal-mode ‘Kelvin–Helmholtz’ (KH) instability, consistent with analogous calculations in an unstratified mixing layer recently reported by Arratia et al. (J. Fluid Mech., vol. 717, 2013, pp. 90–133). However, we demonstrate that non-trivial transient growth occurs even when the Richardson number is sufficiently high to stabilize all normal-mode instabilities, with the optimal perturbation exciting internal waves at some distance from the midpoint of the shear layer.