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Exact coherent structures in stably stratified plane Couette flow. (English) Zbl 1430.76288
Summary: The existence of exact coherent structures in stably stratified plane Couette flow (gravity perpendicular to the plates) is investigated over Reynolds-Richardson number \((Re-Ri_b)\) space for a fluid of unit Prandtl number \((Pr=1)\) using a combination of numerical and asymptotic techniques. Two states are repeatedly discovered using edge tracking – EQ7 and EQ7-1 in the nomenclature of J. F. Gibson and E. Brand [“Spanwise-localized solutions of planar shear flows”, ibid. 745, 25-61 (2014; doi:10.1017/jfm.2014.89)] – and found to connect with two-dimensional convective roll solutions when tracked to negative \(Ri_b\) (the Rayleigh-Bénard problem with shear). Both these states and M. Nagata’s [“3-Dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity”, ibid. 217, 519-527 (1990; doi:10.1017/S0022112090000829)] original exact solution feel the presence of stable stratification when \(Ri_b=O(Re^{-2})\) or equivalently when the Rayleigh number \(Ra:=-Ri_bRe^2Pr=O(1)\). This is confirmed via a stratified extension of the vortex wave interaction theory of P. Hall and S. Sherwin [ibid. 661, 178-205 (2010; Zbl 1205.76085)]. If the stratification is increased further, EQ7 is found to progressively spanwise and cross-stream localise until a second regime is entered at \(Ri_b=O(Re^{-2/3})\). This corresponds to a stratified version of the boundary region equations regime of K. Deguchi et al. [ibid. 721, 58-85 (2013; Zbl 1287.76099)]. Increasing the stratification further appears to lead to a third, ultimate regime where \(Ri_b=O(1)\) in which the flow fully localises in all three directions at the minimal Kolmogorov scale which then corresponds to the Osmidov scale. Implications for the laminar-turbulent boundary in the \((Re-Ri_b)\) plane are briefly discussed.

MSC:
76F45 Stratification effects in turbulence
76F06 Transition to turbulence
76E05 Parallel shear flows in hydrodynamic stability
37G99 Local and nonlocal bifurcation theory for dynamical systems
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
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