Exact coherent structures in stably stratified plane Couette flow
2017; Cambridge University Press; Volume: 826; Linguagem: Inglês
10.1017/jfm.2017.447
ISSN1469-7645
Autores Tópico(s)Fluid Dynamics and Vibration Analysis
ResumoThe 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 Gibson & Brand ( J. Fluid Mech. , vol. 745, 2014, pp. 25–61) – 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 Nagata’s ( J. Fluid Mech. , vol. 217, 1990, pp. 519–527) original exact solution feel the presence of stable stratification when $Ri_{b}=O(Re^{-2})$ or equivalently when the Rayleigh number $Ra:=-Ri_{b}Re^{2}Pr=O(1)$ . This is confirmed via a stratified extension of the vortex wave interaction theory of Hall & Sherwin ( J. Fluid Mech. , vol. 661, 2010, pp. 178–205). 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 Deguchi, Hall & Walton ( J. Fluid Mech. , vol. 721, 2013, pp. 58–85). 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.
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