The intermittency boundary in stratified plane Couette flow

Abstract

We study stratified turbulence in plane Couette flow using direct numerical simulations. Two external dimensionless parameters control the dynamics, the Reynolds number $\mathit{Re}=Uh/{\it\nu}$ and the bulk Richardson number $\mathit{Ri}=g{\it\alpha}_{V}Th/U^{2}$, where $U$ and $T$ are half the velocity and temperature difference between the two walls respectively, $h$ is the half channel depth, ${\it\nu}$ is the kinematic viscosity and $g{\it\alpha}_{V}$ is the buoyancy parameter. We focus on spatio-temporal intermittency due to stratification and we explore the boundary between fully developed turbulence and intermittent flow in the $\mathit{Re}{-}\mathit{Ri}$ plane. The structures populating the intermittent flow regime show coexistence between laminar and turbulent patches, and we demonstrate that there are qualitative differences between the previously studied low-$\mathit{Re}$ low-$\mathit{Ri}$ intermittent regime and the high-$\mathit{Re}$ high-$\mathit{Ri}$ intermittent regime. At low-$\mathit{Re}$ low-$\mathit{Ri}$, turbulent regions span the entire gap, whereas at high-$\mathit{Re}$ high-$\mathit{Ri}$, turbulence is confined vertically with complex dynamics arising from interacting turbulent layers. Consistent with a previous investigation of Flores & Riley (Boundary-Layer Meteorol., vol. 129 (2), 2010, pp. 241–259), we present evidence suggesting that intermittency in the asymptotic regime of high-$\mathit{Re}$ Couette flows appears for $L^{+}<200$, where $L^{+}=Lu_{{\it\tau}}/{\it\nu}$, with $L$ being the Monin–Obukhov length scale, $L=u_{{\it\tau}}^{3}/C_{{\it\kappa}}q_{w}$, $q_{w}$ the wall heat flux, $C_{{\it\kappa}}$ the von Kármán constant and $u_{{\it\tau}}=\sqrt{{\it\tau}_{w}/{\it\rho}_{0}}$ the friction velocity determined from the wall shear stress ${\it\tau}_{w}$, where ${\it\rho}_{0}$ is the constant background density. We also consider the mixing as quantified by various versions of the flux Richardson number $\mathit{Ri}_{f}$, defined as the ratio of the conversion rate from kinetic to potential energy to the turbulent kinetic energy injection rate due to shear. We investigate how laminar and turbulent regions separately contribute to the overall mixing. Remarkably, we find that although fluctuations are greatly suppressed in the laminar regions, $\mathit{Ri}_{f}$ does not change significantly compared with its value in turbulent regions. As we observe a tight coupling between the mean temperature and velocity fields, we demonstrate that both Monin–Obukhov self-similarity theory (Monin & Obukhov, Contrib. Geophys. Inst. Acad. Sci. USSR, vol. 151, 1954, pp. 163–187) and the explicit algebraic model of Lazeroms et al. (J. Fluid Mech., vol. 723, 2013, pp. 91–125) predict the mean profiles well. We thus use these models to trace out the boundary between fully developed turbulence and intermittency in the $\mathit{Re}{-}\mathit{Ri}$ plane.