Hydrodynamic Fluctuations in Laminar Fluid Flow. II. Fluctuating Squire Equation

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We use fluctuating hydrodynamics to evaluate the enhancement of thermally excited fluctuations in laminar fluid flow using plane Couette flow as a representative example. In a previous publication (J. Stat. Phys. 144:774, 2011) we derived the energy amplification arising from thermally excited wall-normal fluctuations by solving a fluctuating Orr-Sommerfeld equation. In the present paper we derive the energy amplification arising from wall-normal vorticity fluctuation by solving a fluctuating Squire equation. The thermally excited wall-normal vorticity fluctuations turn out to yield the dominant contribution to the energy amplification. In addition, we show that thermally excited streaks, even in the absence of any externally imposed perturbations, are present in laminar fluid flow.
© Springer Science+Business Media, LLC 2012. The authors acknowledge stimulating discussions with Andreas Acrivos, Bruno Eckhardt and Mihailo Jovanovic. The research was supported by the Spanish Ministry of Science and Innovation through research grant FIS2008-03801.
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1. Dorfman, J.R., Kirkpatrick, T.R., Sengers, J.V.: Generic long-range correlations in molecular fluids. Annu. Rev. Phys. Chem. 45, 213–239 (1994) 2. Ortiz de Zárate, J.M., Sengers, J.V.: Hydrodynamic Fluctuations in Fluids and Fluid Mixtures. Elsevier, Amsterdam (2006) 3. Landau, L.D., Lifshitz, E.M.: Fluid Mechanics. Pergamon, London (1959). 2nd revised English version (1987) 4. Fox, R.F., Uhlenbeck, G.E.: Contributions to non-equilibrium thermodynamics. I. Theory of hydrodynamical fluctuations. Phys. Fluids 13, 1893–1902 (1970) 5. Farrell, B.F., Ioannou, P.J.: Stochastic forcing of the linearized Navier-Stokes equations. Phys. Fluids A 5, 2600–2609 (1993) 6. Farrell, B.F., Ioannou, P.J.: Variance maintained by stochastic forcing of non-normal dynamical systems associated with linearly stable shear flows. Phys. Rev. Lett. 72, 1188–1191 (1994) 7. Bamieh, B., Dahleh, M.: Energy amplification in channel flows with stochastic excitation. Phys. Fluids 13, 3258–3269 (2001) 8. Jovanovic, M.R., Bamieh, B.: Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145–183 (2005) 9. Eckhardt, B., Pandit, R.: Noise correlations in shear flows. Eur. Phys. J. B 33, 373–378 (2003) 10. Tremblay, A.M.S., Arai, M., Siggia, E.D.: Fluctuations about simple nonequilibrium steady states. Phys. Rev. A 23, 1451–1480 (1981) 11. Lutsko, J.F., Dufty, J.W.: Hydrodynamic fluctuations at large shear rate. Phys. Rev. A 32, 3040–3054 (1985) 12. Lutsko, J., Dufty, J.W., Das, S.P.: Fluctuations and dissipation of a fluid under shear: linear dynamics. Phys. Rev. A 39, 1311–1324 (1989) 13. Lutsko, J.F., Dufty, J.W.: Long-ranged correlations in sheared fluids. Phys. Rev. E 66, 041206 (2002) 14. Wada, H., Sasa, S.I.: Anomalous pressure in fluctuating shear flow. Phys. Rev. E 67, 065302(R) (2003) 15. Ortiz de Zárate, J.M., Sengers, J.V.: Transverse-velocity fluctuations in a liquid under steady shear. Phys. Rev. E 77, 026306 (2008) 16. Ortiz de Zárate, J.M., Sengers, J.V.: Nonequilibrium velocity fluctuations and energy amplification in planar Couette flow. Phys. Rev. E 79, 046308 (2009) 17. Sengers, J.V., Ortiz de Zárate, J.M.: Velocity fluctuations in laminar fluid flow. J. Non-Newton. Fluid Mech. 165, 925–931 (2010) 18. Ortiz de Zárate, J.M., Sengers, J.V.: Hydrodynamic fluctuations in laminar fluid flow. I. Fluctuating Orr-Sommerfeld equation. J. Stat. Phys. 144, 774–792 (2011) 19. Drazin, P.G., Reid, W.H.: Hydrodynamic Stability, 2nd edn. Cambridge University Press, Cambridge (2004) 20. Kubo, R.: The fluctuation-dissipation theorem. Rep. Prog. Phys. 29(1), 255–284 (1966) 21. Schmid, P.J., Henningson, D.S.: Stability and Transition in Shear Flows. Springer, Berlin (2001) 22. Hwang, Y., Cossu, C.: Amplification of coherent streaks in the turbulent Couette flow: an input-output analysis at low Reynolds numbers. J. Fluid Mech. 643, 333–348 (2010) 23. Courant, R., Hilbert, D.: Methods of Mathematical Physics. Wiley, New York (1953). Wiley Classics Library edition (1996) 24. Romanov, V.A.: Stability of plane-parallel Couette flow. Dokl. Akad. Nauk SSSR 196, 1049–1051 (1971) 25. Gustavsson, L.H., Hultgren, L.N.: A resonance mechanism in plane Couette flow. J. Fluid Mech. 98, 149–159 (1980) 26. Albright, J.R.: Integrals of products of Airy functions. J. Phys. A, Math. Gen. 10(4), 485–490 (1977) 27. Boon, J.P., Yip, S.: Molecular Hydrodynamics. McGraw-Hill, New York (1980). Dover edition (1991) 28. Hansen, J.P., McDonald, I.R.: Theory of Simple Liquids, 2nd edn. Academic Press, London (1986) 29. Gayme, D.F., McKeon, B.J., Bamieh, B., Papachristodoulou, A.: Amplification and nonlinear mechanisms in plane Couette flow. Phys. Fluids 23, 065108 (2011) 30. Dolph, C.L., Lewis, D.C.: On the application of infinite systems of ordinary differential equations to perturbations of plane Poiseuille flow. Q. Appl. Math. 16, 97–110 (1958) 31. Waleffe, F.: Transition in shear flows. Nonlinear normality versus non-normal linearity. Phys. Fluids 7, 3060–3066 (1995) 32. Hof, B., van Doorne, C.W.H., Westerweel, J., Nieuwstadt, F.T.M., Faisst, H., Eckhardt, B., Wedin, H., Kerswell, R.R., Waleffe, F.: Experimental observation of nonlinear travelling waves in turbulent pipe flow. Science 305, 1594–1598 (2004) 33. Komminaho, J., Lundbladh, A., Johansson, A.V.: Very large structures in plane turbulent Couette flow. J. Fluid Mech. 320, 259–285 (1996) 34. Duguet, Y., Schlatter, P., Henningson, D.S.: Formation of turbulent patterns near the onset of transition in plane Couette flow. J. Fluid Mech. 650, 119–129 (2010) 35. 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(19) (2006)