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Layer formation and relaminarisation in plane Couette flow with spanwise stratification. (English) Zbl 1415.76277

Summary: In this paper we investigate the effect of stable stratification on plane Couette flow when gravity is oriented in the spanwise direction. When the flow is turbulent we observe near-wall layering and associated new mean flows in the form of large-scale spanwise-flattened streamwise rolls. The layers exhibit the expected buoyancy scaling \(l_z\sim U/N\) where \(U\) is a typical horizontal velocity scale and \(N\) the buoyancy frequency. We associate the new coherent structures with a stratified modification of the well-known large-scale secondary circulation in plane Couette flow. We find that the possibility of the transition to sustained turbulence is controlled by the relative size of this buoyancy scale to the spanwise spacing of the streaks. In parts of parameter space transition can also be initiated by a newly discovered linear instability in this system [G. Facchini et al., ibid. 853, 205–234 (2018; Zbl 1415.76225)]. When wall turbulence can be sustained the linear instability opens up new routes in phase space for infinitesimal disturbances to initiate turbulence. When the buoyancy scale suppresses turbulence the linear instability leads to more ordered nonlinear states, with the possibility for intermittent bursts of secondary shear instability.

MSC:

76F06 Transition to turbulence
76F45 Stratification effects in turbulence
76E05 Parallel shear flows in hydrodynamic stability

Citations:

Zbl 1415.76225
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References:

[1] Billant, P.; Chomaz, J. M., Theoretical analysis of the zigzag instability of a vertical columnar vortex pair in a strongly stratified fluid, J. Fluid Mech., 419, 29-63, (2000) · Zbl 0986.76021
[2] Billant, P.; Chomaz, J. M., Three-dimensional stability of a vertical columnar vortex pair in a stratified fluid, J. Fluid Mech., 419, 65-91, (2000) · Zbl 0986.76022
[3] Billant, P.; Chomaz, J.-M., Self-similarity of strongly stratified inviscid flows, Phys. Fluids, 13, 6, 16-45, (2001)
[4] Brethouwer, G.; Billant, P.; Lindborg, E.; Chomaz, J. M., Scaling analysis and simulation of strongly stratified turbulent flows, J. Fluid Mech., 585, 343-368, (2007) · Zbl 1168.76327
[5] Clever, R. M.; Busse, F. H., Three-dimensional convection in a horizontal fluid layer subjected to a constant shear, J. Fluid Mech., 234, 511-527, (1992) · Zbl 0744.76052
[6] Clever, R. M. & Busse, F. H.2000Tertiary and quaternary solutions for plane Couette flow with thermal stratification. In Physics of Rotating Fluids (ed. Egbers, C. & Pfister, G.), pp. 399-416. Springer.
[7] Couliou, M.; Monchaux, R., Large-scale flows in transitional plane Couette flow: a key ingredient of the spot growth mechanism, Phys. Fluids, 27, 3, (2015)
[8] Deguchi, K., Scaling of small vortices in stably stratified shear flows, J. Fluid Mech., 821, 582-594, (2017) · Zbl 1383.76104
[9] Deusebio, E.; Caulfield, C. P.; Taylor, J. R., The intermittency boundary in stratified plane Couette flow, J. Fluid Mech., 781, 298-329, (2015) · Zbl 1359.76147
[10] Duguet, Y.; Schlatter, P., Oblique laminar – turbulent interfaces in plane shear flows, Phys. Rev. Lett., 110, (2013)
[11] Facchini, G.; Favier, B.; Le Gal, P.; Wang, M.; Le Bars, M., The linear instability of the stratified plane Couette flow, J. Fluid Mech., 853, 205-234, (2018) · Zbl 1415.76225
[12] Falder, M.; White, N. J.; Caulfield, C. P., Seismic imaging of rapid onset of stratified turbulence in the South Atlantic Ocean, J. Phys. Oceanogr., 46, 4, 1023-1044, (2016)
[13] Flores, O.; Riley, J. J., Analysis of turbulence collapse in stably stratified surface layers using direct numerical simulation, Boundary-Layer Meteorol., 129, 241-259, (2010)
[14] Grimshaw, R. H. J., The modulation of an internal gravity-wave packet, and the resonance with the mean motion, Stud. Appl. Maths, 56, 3, 241-266, (1977) · Zbl 0361.76029
[15] Hall, P.; Sherwin, S., Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures, J. Fluid Mech., 661, 178-205, (2010) · Zbl 1205.76085
[16] Hall, P.; Smith, F. T., On strongly nonlinear vortex/wave interactions in boundary-layer transition, J. Fluid Mech., 227, 641-666, (1991) · Zbl 0721.76027
[17] Hamilton, J. M.; Kim, J.; Waleffe, F., Regeneration mechanisms of near-wall turbulence structures, J. Fluid Mech., 287, 317-348, (1995) · Zbl 0867.76032
[18] Holford, J. M. & Linden, P. F.1999aThe development of layers in a stratified fluid. In Mixing and Dispersion in Stably Stratified Flows (Dundee, 1996), pp. 165-179. Oxford University Press.
[19] Holford, J. M.; Linden, P. F., Turbulent mixing in a stratified fluid, Dyn. Atmos. Oceans, 30, 2, 173-198, (1999)
[20] Jacobitz, F. G.; Sarkar, S., The effect of nonvertical shear on turbulence in a stably stratified medium, Phys. Fluids, 10, 5, 1158-1168, (1998) · Zbl 1185.76681
[21] Jimènez, J.; Moin, P., The minimal flow unit in near-wall turbulence, J. Fluid Mech., 225, 213-240, (1991) · Zbl 0721.76040
[22] Kim, J.; Moin, P.; Moser, R., Turbulence statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech., 177, 133-166, (1987) · Zbl 0616.76071
[23] Kline, S. J.; Reynolds, W. C.; Schraub, F. A.; Runstadler, P. W., The structure of turbulent boundary layers, J. Fluid Mech., 30, 4, 741-773, (1967) · Zbl 1461.76274
[24] Leclercq, C.; Nguyen, F.; Kerswell, R. R., Connections between centrifugal, stratorotational, and radiative instabilities in viscous Taylor-Couette flow, Phys. Rev. E, 94, (2016)
[25] Leclercq, C., Partridge, J. L., Augier, P., Caulfield, C.-C. P., Dalziel, S. B. & Linden, P. F.2016b Nonlinear waves in stratified Taylor-Couette flow. Part 1. Layer formation. arXiv:1609.02885.
[26] Lefauve, A.; Partridge, J. L.; Zhou, Q.; Dalziel, S. B.; Caulfield, C. P.; Linden, P. F., The structure and origin of confined Holmboe waves, J. Fluid Mech., 848, 508-544, (2018)
[27] Lemoult, G.; Aider, J.-L.; Wesfreid, J. E., Turbulent spots in a channel: large-scale flow and self-sustainability, J. Fluid Mech., 731, (2013) · Zbl 1294.76165
[28] Lucas, D.; Caulfield, C. P.; Kerswell, R. R., Layer formation in horizontally forced stratified turbulence: connecting exact coherent structures to linear instabilities, J. Fluid Mech., 832, 409-437, (2017) · Zbl 1419.76345
[29] Mccomas, C. H.; Bretherton, F. P., Resonant interaction of oceanic internal waves, J. Geophys. Res., 82, 9, 1397-1412, (1977)
[30] Moin, P.; Mahesh, K., Direct numerical simulation: a tool in turbulence research, Annu. Rev. Fluid Mech., 30, 1, 539-578, (1998) · Zbl 1398.76073
[31] Molemaker, M. J.; Mcwilliams, J. C.; Yavneh, I., Instability and equilibration of centrifugally stable stratified Taylor-Couette flow, Phys. Rev. Lett., 86, 23, 5270-5273, (2001)
[32] Oglethorpe, R. L. F.; Caulfield, C. P.; Woods, A. W., Spontaneous layering in stratified turbulent Taylor-Couette flow, J. Fluid Mech., 721, (2013) · Zbl 1287.76018
[33] Olvera, D.; Kerswell, R. R., Exact coherent structures in stably stratified plane Couette flow, J. Fluid Mech., 826, 583-614, (2017) · Zbl 1430.76288
[34] Papavassiliou, D. V.; Hanratty, T. J., Interpretation of large-scale structures observed in a turbulent plane Couette flow, Intl J. Heat Fluid Flow, 18, 1, 55-69, (1997)
[35] Park, J.; Billant, P.; Baik, J.-J., Instabilities and transient growth of the stratified Taylor-Couette flow in a Rayleigh-unstable regime, J. Fluid Mech., 822, 80-108, (2017) · Zbl 1383.76133
[36] Park, J.; Billant, P.; Baik, J.-J.; Seo, J. M., Competition between the centrifugal and strato-rotational instabilities in the stratified Taylor-Couette flow, J. Fluid Mech., 840, 5-24, (2018) · Zbl 1419.76324
[37] Park, Y. G.; Whitehead, J. A.; Gnanadeskian, A., Turbulent mixing in stratified fluids: layer formation and energetics, J. Fluid Mech., 279, 279-311, (1994)
[38] Shalybkov, D.; Rüdiger, G., Stability of density-stratified viscous Taylor-Couette flows, Astron. Astrophys., 438, 2, 411-417, (2005) · Zbl 1085.85003
[39] Sutherland, B. R., Finite-amplitude internal wavepacket dispersion and breaking, J. Fluid Mech., 429, 343-380, (2001) · Zbl 1007.76024
[40] Taylor, J. R.2008 Numerical simulations of the stratified oceanic bottom boundary layer. PhD thesis, University of California, San Diego, CA.
[41] Taylor, J. R.; Deusebio, E.; Caulfield, C. P.; Kerswell, R. R., A new method for isolating turbulent states in transitional stratified plane Couette flow, J. Fluid Mech., 808, (2016) · Zbl 1383.76267
[42] Thorpe, S. A., Layers and internal waves in uniformly stratified fluids stirred by vertical grids, J. Fluid Mech., 793, 380-413, (2016) · Zbl 1382.76134
[43] Toh, S.; Itano, T., Interaction between a large-scale structure and near-wall structures in channel flow, J. Fluid Mech., 524, 249-262, (2005) · Zbl 1065.76553
[44] Tsukahara, T.; Kawamura, H.; Shingai, K., DNS of turbulent Couette flow with emphasis on the large-scale structure in the core region, J. Turbul., 7, (2006)
[45] Waleffe, F., On a self-sustaining process in shear flows, Phys. Fluids, 9, 4, 883-900, (1997)
[46] Zhou, Q.; Taylor, J. R.; Caulfield, C. P., Self-similar mixing in stratified plane Couette flow for varying Prandtl number, J. Fluid Mech., 820, 86-120, (2017) · Zbl 1383.76297
[47] Zhou, Q.; Taylor, J. R.; Caulfield, C. P.; Linden, P. F., Diapycnal mixing in layered stratified plane Couette flow quantified in a tracer-based coordinate, J. Fluid Mech., 823, 198-229, (2017) · Zbl 1422.76093
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