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Designing a more nonlinearly stable laminar flow via boundary manipulation

Published online by Cambridge University Press:  04 December 2013

S. M. E. Rabin ,
C. P. Caulfield  and
R. R. Kerswell
S. M. E. Rabin
Affiliation:
Department of Applied Mathematics & Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
C. P. Caulfield*
Affiliation:
Department of Applied Mathematics & Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK BP Institute, University of Cambridge, Madingley Rise, Madingley Road, Cambridge CB3 0EZ, UK
R. R. Kerswell
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
*
†Email address for correspondence: c.p.caulfield@bpi.cam.ac.uk
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Abstract

We show how a fully nonlinear variational method can be used to design a more nonlinearly stable laminar shear flow by quantifying the effect of manipulating the boundary conditions of the flow. Using the example of plane Couette flow, we demonstrate that by forcing the boundaries to undergo spanwise oscillations in a certain way, it is possible to increase the critical disturbance energy for the onset of turbulence by 41 %. If this is sufficient to ensure laminar flow (i.e. ambient noise does not exceed this increased threshold), nearly four times less energy is consumed than in the turbulent flow which exists in the absence of imposed spanwise oscillations.

JFM classification

Instability: Nonlinear instability Instability: Transition to turbulence Mathematical Foundations: Variational methods
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 738 , 10 January 2014 , R1
Copyright
©2013 Cambridge University Press 

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References

Baron, A. & Quadrio, M. 1996 Turbulent drag reduction by spanwise wall oscillations. Appl. Sci. Res. 55, 311–326.Google Scholar
Bewley, T. R. 2001 Flow control: new challenges for a new renaissance. Prog. Aerosp. Sci. 37, 21–58.CrossRefGoogle Scholar
Bottin, S. & Chate, H. 1998 Statistical analysis of the transition to turbulence in plane Couette flow. Eur. Phys. J. B 6 (1), 143–155.CrossRefGoogle Scholar
Cherubini, S. & De Palma, P. 2013 Nonlinear optimal perturbations in a Couette flow: bursting and transition. J. Fluid Mech. 716, 251–279.Google Scholar
Cherubini, S., De Palma, P., Robinet, J. Ch. & Bottaro, A. 2010 Rapid path to transition via nonlinear localized optimal perturbations in a boundary-layer flow. Phys. Rev. E 82, 066302.Google Scholar
Cherubini, S., De Palma, P., Robinet, J.-C. & Bottaro, A. 2011 The minimal seed of turbulent transition in the boundary layer. J. Fluid Mech. 689, 221–253.CrossRefGoogle Scholar
Duguet, Y., Monokrousos, A., Brandt, L. & Henningson, D. S. 2013 Minimal transition thresholds in plane Couette flow. Phys. Fluids 25 (8), 084103.CrossRefGoogle Scholar
Duque-Daza, C. A., Baig, M. F., Lockerby, D. A., Chernyshenko, S. I. & Davies, C. 2012 Modelling turbulent skin-friction control using linearised Navier–Stokes equations. J. Fluid Mech. 702, 403–414.Google Scholar
Högberg, M., Bewley, T. R. & Henningson, D. S. 2003 Linear feedback control and estimation of transition in plane channel flow. J. Fluid Mech. 481, 149–175.CrossRefGoogle Scholar
Jovanović, M. R. 2008 Turbulence suppression in channel flows by small amplitude transverse wall oscillations. Phys. Fluids 20, 014101.Google Scholar
Jung, W. J., Mangiavacchi, N. & Akhavan, R. 1992 Suppression of turbulence in wallbounded flows by high frequency spanwise oscillations. Phys. Fluids A 4, 1605–1607.CrossRefGoogle Scholar
Juniper, M. P. 2011 Triggering in the horizontal Rijke tube: non-normality, transient growth and bypass transition. J. Fluid Mech. 667, 272–308.Google Scholar
Kasagi, N., Suzuki, Y. & Fukagata, K. 2009 Systems-based feedback control of turbulence for skin friction reduction. Annu. Rev. Fluid Mech. 41, 231–251.Google Scholar
Kawahara, G. 2005 Laminarization of minimal plane Couette flow: going beyond the basin of attraction of turbulence. Phys. Fluids 17, 041702.Google Scholar
Kim, J. & Bewley, T. R. 2007 A linear systems approach to flow control. Annu. Rev. Fluid Mech. 39, 383–417.CrossRefGoogle Scholar
Laadhari, F., Skandaji, L. & Morel, R. 1994 Turbulence reduction in a boundary layer by local spanwise oscillating surface. Phys. Fluids 6, 3218–3220.Google Scholar
Lieu, B. K., Moarref, R. & Jovanović, M. R. 2010 Controlling the onset of turbulence by streamwise travelling waves. Part 2. Direct numerical simulation. J. Fluid Mech. 663, 100–119.Google Scholar
Manneville, P. 2008 Understanding the sub-critical transition to turbulence in wall flows. Pramana 70, 1009–1021.CrossRefGoogle Scholar
Moarref, R. & Jovanović, M. R. 2010 Controlling the onset of turbulence by streamwise travelling waves. Part 1. Receptivity analysis. J. Fluid Mech. 663, 70–99.Google Scholar
Moarref, R. & Jovanović, M. R. 2012 Model-based design of transverse wall oscillations for turbulent drag reduction. J. Fluid Mech. 707, 205–240.Google Scholar
Monokrousos, A., Bottaro, A., Brandt, L., DiVita, A. & Henningson, D. S. 2011 Nonequilibrium thermodynamics and the optimal path to turbulence in shear flows. Phys. Rev. Lett. 106 (13), 134502.Google Scholar
Mullin, T. & Kerswell, R. R. 2005 IUTAM Symposium on Laminar-Turbulence Transition and Finite Amplitude Solutions. Springer.Google Scholar
Peixinho, J. & Mullin, T. 2007 Finite-amplitude thresholds for transition in pipe flow. J. Fluid Mech. 582, 169–178.Google Scholar
Pringle, C. C. T. & Kerswell, R. R. 2010 Using nonlinear transient growth to construct the minimal seed for shear flow turbulence. Phys. Rev. Lett. 105 (15), 154502.Google Scholar
Pringle, C. C. T., Willis, A. P. & Kerswell, R. R. 2012 Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos. J. Fluid Mech. 702, 415–443.Google Scholar
Quadrio, M. 2011 Drag reduction in turbulent boundary layers by in-plane wall motion. Phil. Trans. R. Soc. Lond. A 369, 1428–1442.Google Scholar
Rabin, S. M. E., Caulfield, C. P. & Kerswell, R. R. 2012 Triggering turbulence efficiently in plane Couette flow. J. Fluid Mech. 712, 244–272.Google Scholar
Romanov, V. A. 1973 Stability of plane-parallel Couette flow. Funct. Anal. Applics. 7, 137–146.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Taylor, J. R. 2008 Numerical simulations of the stratified oceanic bottom boundary layer. PhD thesis, University of California, San Diego.Google Scholar