Abstract
The existence of exact coherent structures in stably-stratified plane Couette
ow (gravity perpendicular to the plates) is investigated over Reynolds-Richardson number (Re-Rib) 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 (2014) - and found to connect with 2-dimensional convective roll solutions when tracked to negative Rib (the Rayleigh-Benard problem with shear). Both these states and Nagata's (1990) original exact solution feel the presence of stable stratification when Rib = O(Re-2) or equivalently when the Rayleigh number Ra := -RibRe2Pr = O(1). This is confirmed via a stratified extension of the Vortex-Wave-Interaction (VWI) theory of Hall & Sherwin (2010). If the stratification is increased further, EQ7 is found to progressively spanwise and cross-stream localise until a second regime is entered at Rib = O(Re-2/3). This corresponds to a stratified version of the Boundary Reduced Equation (BRE) regime of Deguchi, Hall & Walton (2013). Increasing the stratification further, appears to lead to a third, ultimate regime where Rib = 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-Rib) plane are briefly discussed.
ow (gravity perpendicular to the plates) is investigated over Reynolds-Richardson number (Re-Rib) 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 (2014) - and found to connect with 2-dimensional convective roll solutions when tracked to negative Rib (the Rayleigh-Benard problem with shear). Both these states and Nagata's (1990) original exact solution feel the presence of stable stratification when Rib = O(Re-2) or equivalently when the Rayleigh number Ra := -RibRe2Pr = O(1). This is confirmed via a stratified extension of the Vortex-Wave-Interaction (VWI) theory of Hall & Sherwin (2010). If the stratification is increased further, EQ7 is found to progressively spanwise and cross-stream localise until a second regime is entered at Rib = O(Re-2/3). This corresponds to a stratified version of the Boundary Reduced Equation (BRE) regime of Deguchi, Hall & Walton (2013). Increasing the stratification further, appears to lead to a third, ultimate regime where Rib = 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-Rib) plane are briefly discussed.
| Original language | English |
|---|---|
| Pages (from-to) | 583-614 |
| Number of pages | 32 |
| Journal | Journal of Fluid Mechanics |
| Volume | 826 |
| Early online date | 8 Aug 2017 |
| DOIs | |
| Publication status | Published - 10 Sept 2017 |