### Abstract

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 D 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} DO.Re..2/or equivalently when the Rayleigh number Ra VD ..Ri_{b}Re_{2}Pr D 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} D 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} D 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.

Original language | English |
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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 Sep 2017 |

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### ASJC Scopus subject areas

- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering

### Cite this

*Journal of Fluid Mechanics*,

*826*, 583-614. https://doi.org/10.1017/jfm.2017.447

**Exact coherent structures in stably stratified plane Couette flow.** / Olvera, D.; Kerswell, R. R.

Research output: Contribution to journal › Article

*Journal of Fluid Mechanics*, vol. 826, pp. 583-614. https://doi.org/10.1017/jfm.2017.447

}

TY - JOUR

T1 - Exact coherent structures in stably stratified plane Couette flow

AU - Olvera, D.

AU - Kerswell, R. R.

PY - 2017/9/10

Y1 - 2017/9/10

N2 - The existence of exact coherent structures in stably stratified plane Couette flow (gravity perpendicular to the plates) is investigated over Reynolds-Richardson number (Re-Rib) space for a fluid of unit Prandtl number .Pr D 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 Rib (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 Rib DO.Re..2/or equivalently when the Rayleigh number Ra VD ..RibRe2Pr D 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 Rib D 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 Rib D 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.

AB - The existence of exact coherent structures in stably stratified plane Couette flow (gravity perpendicular to the plates) is investigated over Reynolds-Richardson number (Re-Rib) space for a fluid of unit Prandtl number .Pr D 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 Rib (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 Rib DO.Re..2/or equivalently when the Rayleigh number Ra VD ..RibRe2Pr D 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 Rib D 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 Rib D 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.

UR - http://www.scopus.com/inward/record.url?scp=85020017776&partnerID=8YFLogxK

U2 - 10.1017/jfm.2017.447

DO - 10.1017/jfm.2017.447

M3 - Article

VL - 826

SP - 583

EP - 614

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -