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 Re = Uh/ν and the bulk Richardson number Ri = gα 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, ν is the kinematic viscosity and gα 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 Re-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-Re low-Ri intermittent regime and the high-Re high-Ri intermittent regime. At low-Re low-Ri, turbulent regions span the entire gap, whereas at high-Re high-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-Re Couette flows appears for L + < 200, where L + = Lu τ /ν, with L being the Monin-Obukhov length scale, L = u 3 τ /C κ q w , q w the wall heat flux, C κ the von Kármán constant and u τ = √ τ w /ρ 0 the friction velocity determined from the wall shear stress τ w , where ρ 0 is the constant background density. We also consider the mixing as quantified by various versions of the flux Richardson number 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, 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 Re-Ri plane.

Early Warning Signals for Regime Transition in the Stable Boundary Layer: A Model Study

“…In our simulations this correlation becomes weaker as stability grows. The relatively small domain size causes slow temporal fluctuations, which correspond to spatial fluctuations in a larger domain (see also Deusebio et al 2015). Similar to Tsukahara et al (2006) and Deusebio et al (2015), the temporally averaged first-and second-order statistics are almost insensitive to the domain size.…”

“…The relatively small domain size causes slow temporal fluctuations, which correspond to spatial fluctuations in a larger domain (see also Deusebio et al 2015). Similar to Tsukahara et al (2006) and Deusebio et al (2015), the temporally averaged first-and second-order statistics are almost insensitive to the domain size. Domain independence is verified more extensively for the neutral case and one stably stratified case by additional runs using double horizontal domains, which confirms this insensitivity (not shown).…”

“…A possible explanation for the absence of such an increase inσ (σ w ) is that fluctuations of the turbulent intensity occur naturally, and these may not be easily distinguished from wave activity. Additionally, long-term fluctuations may occur in the small-size system used here (Deusebio et al 2015).…”

Exact coherent structures in stably stratified plane Couette flow

“…At a given point in the parameter plane (Re, Ri b , 1), 10 simulations were performed with random initial conditions with the point assigned to the turbulent region if 50% or more of these runs remained turbulent after a time 1000 h/U and otherwise to the relaminarisation region. Figure 1 shows that the turbulent part of parameter space extends to higher Ri b at a given Re for the larger domain with the boundary extending as Re → ∞ for some Ri b (Re) (see Deusebio et al (2015) for a similar plot but at higher Re and larger domain size). Also shown for completeness is the region Ri b < 0 (unstable stratification) where the problem is equivalent to Rayleigh-Benard convection with imposed shear: see appendix B which shows how the usual Rayleigh number Ra = −Ri b Re 2 P r and the neutral stability line is given by Ra = 1708/16 (the extra factor of 16 because the boundary separation has been non-dimensionalised to 2 rather than 1).…”

“…Ignoring the Prandtl number (set to unity throughout this study) still leaves 2 parameters and then a 1-dimensional laminar-turbulent boundary as opposed to the 0-dimensional situation in unstratified pCf (e.g. see figure 1 of Deusebio et al (2015)). Understanding exactly how this boundary behaves for large Re is an important open problem in stratified turbulence which has obvious implications for parametrising turbulence in ocean, atmosphere and climate models.…”