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Scaling of small vortices in stably stratified shear flows

Published online by Cambridge University Press:  25 May 2017

Kengo Deguchi Open the ORCID record for Kengo Deguchi [Opens in a new window]
Kengo Deguchi*
Affiliation:
School of Mathematical Sciences, Monash University, VIC 3800, Australia
*
†Email address for correspondence: [email protected]
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Abstract

The present paper treats the large Reynolds number scaling of coherent structures in stably stratified flows sheared between two horizontally placed walls. Three-dimensional steady solutions are used to confirm the theoretical scaling. For small values of the Richardson number, the previously known scaling based on the vortex–wave interaction/self-sustaining process is found to give excellent predictions of the numerical results. When the Richardson number is increased, the maximum size of the vortices is limited by the Ozmidov scale. The largest possible Richardson number to sustain the vortices is predicted to be of order unity when the typical length scale of the vortices reaches the Kolmogorov scale. The minimum-scale vortices are governed by unit Reynolds number Navier–Stokes equations.

JFM classification

Aerodynamics: High-speed flow Instability: Nonlinear instability Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 821 , 25 June 2017 , pp. 582 - 594
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Copyright
© 2017 Cambridge University Press 

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References

Beaume, C., Chini, G. P., Julien, K. & Knobloch, E. 2015 Reduced description of exact coherent states in parallel shear flows. Phys. Rev. E 91, 043010.Google ScholarPubMed
Blackburn, H. M., Hall, P. & Sherwin, S. 2013 Lower branch equilibria in Couette flow: the emergence of canonical states for arbitrary shear flows. J. Fluid Mech. 726, R2.Google Scholar
Brand, E. & Gibson, J. F. 2014 A doubly localized equilibrium solution of plane Couette flow. J. Fluid Mech. 750, R3.CrossRefGoogle Scholar
Brethouwer, G., Billant, P., Lindborg, E. & Chomaz, J.-M. 2007 Scaling analysis and simulation of strongly stratified turbulent flows. J. Fluid Mech. 585, 343–368.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Clever, R. M. & Busse, F. H. 1992 Three-dimensional convection in a horizontal fluid layer subjected to a constant shear. J. Fluid Mech. 234, 511–527.CrossRefGoogle Scholar
Clever, R. M., Busse, F. H. & Kelly, R. E. 1977 Instabilities of longitudinal convection rolls in Couette flow. Z. Angew. Math. Phys. 28, 771–783.Google Scholar
Deguchi, K. 2015 Self-sustained states at Kolmogorov microscale. J. Fluid Mech. 781, R6.CrossRefGoogle Scholar
Deguchi, K. & Hall, P. 2014 Canonical exact coherent structures embedded in high Reynolds number flows. Phil. Trans. R. Soc. A 372, 20130352.Google ScholarPubMed
Deguchi, K. & Hall, P. 2015 Asymptotic descriptions of oblique coherent structures in shear flows. J. Fluid Mech. 782, 356–367.Google Scholar
Deguchi, K. & Hall, P. 2016 On the instability of vortex–wave interaction states. J. Fluid Mech. 802, 634–666.Google Scholar
Deguchi, K., Hall, P. & Walton, A. G. 2013 The emergence of localized vortex–wave interaction states in plane Couette flow. J. Fluid Mech. 721, 58–85.Google Scholar
Eaves, T. S. & Caulfield, C. P. 2015 Disruption of SSP/VWI states by a stable stratification. J. Fluid Mech. 784, 548–564.CrossRefGoogle Scholar
Gibson, J. F., Halcrow, J. & Cvitanović, E. 2009 Equilibrium and traveling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 243–266.CrossRefGoogle Scholar
Hall, P. & Horseman, N. J. 1991 The linear inviscid secondary instability of longitudinal vortex structures in boundary layers. J. Fluid Mech. 232, 357–375.Google Scholar
Hall, P. & Sherwin, S. 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178–205.Google Scholar
Hall, P. & Smith, F. T. 1991 On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641–666.Google Scholar
Itano, T. & Generalis, S. C. 2009 Hairpin vortex solution in planar Couette flow: a tapestry of knotted vortices. Phys. Rev. Lett. 102, 114501.Google Scholar
Kawahara, G., Uhlmann, M. & Van Veen, L. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44, 203–225.Google Scholar
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519–527.Google Scholar
Olvera, D. & Kerswell, R. R.2017 Exact coherent structures in stably-stratified plane Couette flow. J. Fluid Mech. (submitted).Google Scholar
Ozmidov, R. V. 1965 On the turbulent exchange in a stably stratified ocean. Izv. Acad. Sci. USSR Atmos. Ocean. Phys. 1, 861–871.Google Scholar
Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135–167.Google Scholar
Portwood, G. D., de Bruyn Kops, S. M., Taylor, J. R., Salehipour, H. & Caulfield, C. P. 2016 Robust identification of dynamically distinct regions in stratified turbulence. J. Fluid Mech. 807, R2.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.Google Scholar
Wang, J., Gibson, J. F. & Waleffe, F. 2007 Lower branch coherent states: transition and control. Phys. Rev. Lett. 98, 204501.Google Scholar