Self-similar mixing in stratified plane Couette flow for varying Prandtl number

We investigate fully developed turbulence in stratified plane Couette flows using direct numerical simulations similar to those reported by Deusebio et 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 arc observed. These effects can he described through a mixing length model coupling Monin-Obukhov (M-O) similarity theory and van Driest damping functions. Wc 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 equivalent to epsilon / (nu N-2), where epsilon is the dissipation, nu 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+ equivalent to L/delta(nu), where delta(nu). is the near-wall viscous scale. The flux Richardson number Ri(f) equivalent to -B/P, where B is the buoyancy flux and P is the shear production, is found to be proportional to Rig This then leads to a turbulent Prandtl numher Pr-t equivalent to nu(t)/kappa(t) of order unity, where nu(t) and kappa(t) are the turbulent viscosity and diffusivity respectively, which is consistent with Reynolds analogy. The turbulent Froude number Fr-h equivalent to epsilon / (NU'(2)), where U' 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 he independent of Pr. The classical Osborn model based on turbulent kinetic energy balance in statistically stationary stratified sheared turbulence (Osborn, Phys. Oceanogr, vol. 10, 1980, pp. 83-89), together with M-O scalings, results in a parameterization of kappa(t),/nu similar to nu(t.)/nu similar to 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-hounded stratified turbulent flows.