Skip to main content
SpringerLink
Log in

Enhanced Dissipation and Inviscid Damping in the Inviscid Limit of the Navier–Stokes Equations Near the Two Dimensional Couette Flow

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

In this work we study the long time inviscid limit of the two dimensional Navier–Stokes equations near the periodic Couette flow. In particular, we confirm at the nonlinear level the qualitative behavior predicted by Kelvin’s 1887 linear analysis. At high Reynolds number Re, we prove that the solution behaves qualitatively like two dimensional Euler for times \({{t \lesssim Re^{1/3}}}\), and in particular exhibits inviscid damping (for example the vorticity weakly approaches a shear flow). For times \({{t \gtrsim Re^{1/3}}}\), which is sooner than the natural dissipative time scale O(Re), the viscosity becomes dominant and the streamwise dependence of the vorticity is rapidly eliminated by an enhanced dissipation effect. Afterwards, the remaining shear flow decays on very long time scales \({{t \gtrsim Re}}\) back to the Couette flow. When properly defined, the dissipative length-scale in this setting is \({{\ell_D \sim Re^{-1/3}}}\), larger than the scale \({{\ell_D \sim Re^{-1/2}}}\) predicted in classical Batchelor–Kraichnan two dimensional turbulence theory. The class of initial data we study is the sum of a sufficiently smooth function and a small (with respect to Re −1) L 2 function.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bahouri H., Chemin J.-Y., Danchin R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 343. Springer, Heidelberg, 2011

  2. Bajer K., Bassom A.P., Gilbert A.D.: Accelerated diffusion in the centre of a vortex. J. Fluid Mech. 437, 395–411 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Balmforth N.J., Morrison P.J.: Normal modes and continuous spectra. Ann. N. Y. Acad. Sci. 773(1), 80–94 (1995)

    Article  ADS  Google Scholar 

  4. Balmforth N.J., Morrison P.J., Thiffeault J.-L.: Pattern Formation in Hamiltonian Systems with Continuous Spectra; A Normal-Form Single-Wave Model (2013, preprint)

  5. Bardos C., Titi E.S., Wiedemann E.: The vanishing viscosity as a selection principle for the Euler equations: the case of 3D shear flow. C. R. Math. 350(15), 757–760 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bassom A.P., Gilbert A.D.: The spiral wind-up of vorticity in an inviscid planar vortex. J. Fluid Mech. 371, 109–140 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Batchelor G.K.: Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids 12(12), II–233 (1969)

    Article  Google Scholar 

  8. Beck M., Wayne C.E.: Metastability and rapid convergence to quasi-stationary bar states for the two-dimensional Navier–Stokes equations. Proc. R. Soc. Edinb. Sec. A Math. 143(05), 905–927 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bedrossian J., Masmoudi N., Mouhot C.: Landau damping: paraproducts and Gevrey regularity (2013). arXiv:1311.2870

  10. Bedrossian J., Masmoudi N.: Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations. Publ. Math. de l’IHÉS 1–106 (2013)

  11. Berestycki H., Hamel F., Nadirashvili N.: Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena. Commun. Math. Phys. 253(2), 451–480 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Bernoff A.J., Lingevitch J.F.: Rapid relaxation of an axisymmetric vortex. Phys. Fluids 63717–3723 (1994)

  13. Bony J.M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non lináires. Ann. Sci. E. N. S. 14, 209–246 (1981)

    MathSciNet  MATH  Google Scholar 

  14. Bouchet F., Morita H.: Large time behavior and asymptotic stability of the 2D Euler and linearized Euler equations. Phys. D 239, 948–966 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Boyd J.P.: The continuous spectrum of linear Couette flow with the beta effect. J. Atmos. Sci. 40(9), 2304–2308 (1983)

    Article  ADS  Google Scholar 

  16. Briggs R.J., Daugherty J.D., Levy R.H.: Role of Landau damping in crossed-field electron beams and inviscid shear flow. Phys. Fluid 13(2), 421–432 (1970)

  17. Buckmaster T., De Lellis C., Székelyhidi Jr L.: Dissipative euler flows with Onsager-critical spatial regularity (2014). arXiv:1404.6915

  18. Caglioti E., Maffei C.: Time asymptotics for solutions of Vlasov–Poisson equation in a circle. J. Stat. Phys. 92(1/2), 301–323 (1998)

  19. Cardoso O., Tabeling P.: Anomalous diffusion in a linear array of vortices. EPL (Europhysics Letters) 7(3), 225 (1988)

    Article  ADS  Google Scholar 

  20. Chapman S.J.: Subcritical transition in channel flows. J. Fluid Mech. 451, 35–98 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Constantin P., Kiselev A., Ryzhik L., Zlatoš A.: Diffusion and mixing in fluid flow. Ann. Math. (2) 168(2), 643–674 (2008)

    Article  MATH  Google Scholar 

  22. De Lellis C., Székelyhidi Jr., L.: Dissipative Euler flows and Onsager’s conjecture (2012). arXiv preprint arXiv:1205.3626

  23. De Lellis C., Székelyhidi L. Jr.: The h-principle and the equations of fluid dynamics. Bull. Am. Math. Soc. 49(3), 347–375 (2012)

    Article  MATH  Google Scholar 

  24. Drazin P.G., Reid W.H.: Hydrodynamic stability. Cambridge University Press, Cambridge, 1981

  25. Ellingsen T., Palm E.:: Stability of linear flow. Phys. Fluids 18, 487 (1975)

    Article  ADS  MATH  Google Scholar 

  26. Faou E., Rousset F.: Landau Damping in Sobolev Spaces for the Vlasov-HMF model (2014). arXiv:1403.1668

  27. Foias C., Temam R.: Gevrey class regularity for solutions of the Navier–Stokes equations. J. Funct. Anal. 87, 359–369 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gevrey M.: Sur la nature analytique des solutions des équations aux dérivées partielles. Premier mémoire. Ann. Sci. École Norm. Sup. (3) 35, 129–190 (1918)

    MathSciNet  MATH  Google Scholar 

  29. Gilbert A.D.: Spiral structures and spectra in two-dimensional turbulence. J. Fluid Mech. 193, 475–497 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Gilbert A.D.: A cascade interpretation of lundgren’s stretched spiral vortex model for turbulent fine structure. Phys. Fluids A Fluid Dyn. 5, 2831 (1993)

    Article  ADS  MATH  Google Scholar 

  31. Grenier E., Guo Y., Nguyen T.: Spectral instability of characteristic boundary layer flows (2014). arXiv:1406.3862

  32. Haynes P.H., Vanneste J.: Dispersion in the large-deviation regime. Part 1, shear flows and periodic flows–J Fluid Mech 745321–350 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  33. Hörmander L.: The Nash–Moser theorem and paradifferential operators. Anal. et cetera 429–449 (1990)

  34. Hwang H.J., Velaźquez J.J.L.: On the existence of exponentially decreasing solutions of the nonlinear Landau damping problem. Indiana Univ. Math. J 58(6), 2623–2660 (2009)

  35. Isett P.: Hölder continuous Euler flows in three dimensions with compact support in time (2012). arXiv preprint arXiv:1211.4065

  36. Iyer G., Novikov A.: Anomalous diffusion in fast cellular flows at intermediate time scales (2014). arXiv:1406.3881

  37. Kelvin L.: Stability of fluid motion-rectilinear motion of viscous fluid between two parallel plates. Philos. Mag. 5(24), 188 (1887)

  38. Kolmogorov A.N.: Dissipation of energy in locally isotropic turbulence. Dokl.Akad. Nauk SSSR 32, 16–18 (1941)

    ADS  MATH  Google Scholar 

  39. Kraichnan R.H.: Inertial ranges in two-dimensional turbulence. Phys. Fluids 10(7), 1417–1423 (1967)

  40. Kukavica I., Vicol V.: On the radius of analyticity of solutions to the three-dimensional Euler equations. Proc. Am. Math. Soc. 137(2), 669–677 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. Landau L.: On the vibration of the electronic plasma. J. Phys. USSR 10(25) (1946)

  42. Latini M., Bernoff A.J.: Transient anomalous diffusion in Poiseuille flow. J. Fluid Mech. 441, 399–411 (2001)

    Article  ADS  MATH  Google Scholar 

  43. Levermore D., Oliver M.: Analyticity of solutions for a generalized Euler equation. J. Differ. Equ. 133, 321–339 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Li Y.C., Lin Z.: A resolution of the Sommerfeld paradox. SIAM J. Math. Anal. 43(4), 1923–1954 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  45. Lin Z., Zeng C.: Inviscid dynamic structures near Couette flow. Arch. Ration. Mech. Anal. 200, 1075–1097 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  46. Lindzen R.: Instability of plane parallel shear flow (toward a mechanistic picture of how it works). PAGEOPH 126(1), 103–121 (1988)

  47. Lundgren T.S.: Strained spiral vortex model for turbulent fine structure. Phys. Fluid 25, 2193 (1982)

    Article  ADS  MATH  Google Scholar 

  48. Malmberg J., Wharton C., Gould C., O’Neil T.: Plasma wave echo. Phys. Rev. Lett. 20(3), 95–97 (1968)

    Article  ADS  Google Scholar 

  49. Morrison P.J.: Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70(2), 467–521 (1998)

    Article  ADS  MATH  Google Scholar 

  50. Mouhot C., Villani C.: On Landau damping. Acta Math. 207, 29–201 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  51. Nirenberg L.: An abstract form of the nonlinear Cauchy–Kowalewski theorem. J. Differ. Geom. 6, 561–576 (1972)

    MathSciNet  MATH  Google Scholar 

  52. Nishida T.: A note on a theorem of Nirenberg. J. Differ. Geom. 12, 629–633 (1977)

    MathSciNet  MATH  Google Scholar 

  53. Orr W.: The stability or instability of steady motions of a perfect liquid and of a viscous liquid, Part I: a perfect liquid. Proc. R. Irish Acad. Sec. A Math. Phys. Sci. 27, 9–68 (1907)

    Google Scholar 

  54. Rayleigh L.: On the stability, or instability, of certain fluid motions. Proc. Lond. Math. Soc. S1–S11(1), 57 (1880)

  55. Rhines P.B., Young W.R.: How rapidly is a passive scalar mixed within closed streamlines? J. Fluid Mech. 133, 133–145 (1983)

  56. Ryutov D.D.: Landau damping: half a century with the great discovery.Plasma Phys. control. Fusion 41(3A), A1 (1999)

  57. Schecter D.A., Dubin D., Cass A.C., Driscoll C.F., Lansky I.M., et. al.: Inviscid damping of asymmetries on a two-dimensional vortex. Phys. Fluid 12, 2397–2412 (2000)

  58. Taylor G.: Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 219(1137), 186–203 (1953)

    Article  ADS  Google Scholar 

  59. Trefethen L.N., Embree M.: Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, New Jersey, 2005

  60. Trefethen L.N., Trefethen A.E., Reddy S.C., Driscoll T.A.: Hydrodynamic stability without eigenvalues. Science 261(5121), 578–584 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  61. Vanneste J.: Nonlinear dynamics of anisotropic disturbances in plane Couette flow. SIAM J. Appl. Math. 62(3), 924–944 (electronic) (2002)

  62. Vanneste J., Morrison P.J., Warn T.: Strong echo effect and nonlinear transient growth in shear flows. Phys. Fluids 10, 1398 (1998)

    Article  ADS  Google Scholar 

  63. Villani C.: Hypocoercivity. American Mathematical Society, Providence, 2009

  64. Yaglom A.M.: Hydrodynamic Instability and Transition to Turbulence, vol. 100. Springer, Berlin, 2012

  65. Young W., Pumir A., Pomeau Y.: Anomalous diffusion of tracer in convection rolls. Phys. Fluids A Fluid Dyn. (1989–1993) 1, 462–469 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  66. Yu J.H., Driscoll C.F.: Diocotron wave echoes in a pure electron plasma. IEEE Trans. Plasma Sci. 30(1), 24–25 (2002)

  67. Yu, J.H., Driscoll, C.F., O‘Neil, T.M.: Phase mixing and echoes in a pure electron plasma. Phys. Plasmas 12(055701) (2005)

Download references

Author information

Authors and Affiliations

  1. University of Maryland, College Park, USA

    Jacob Bedrossian

  2. Courant Institute of Mathematical Sciences, New York, USA

    Nader Masmoudi

  3. Princeton University, Princeton, USA

    Vlad Vicol

Authors
  1. Jacob Bedrossian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nader Masmoudi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vlad Vicol
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jacob Bedrossian.

Additional information

Communicated by V. Šverák

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bedrossian, J., Masmoudi, N. & Vicol, V. Enhanced Dissipation and Inviscid Damping in the Inviscid Limit of the Navier–Stokes Equations Near the Two Dimensional Couette Flow. Arch Rational Mech Anal 219, 1087–1159 (2016). https://doi.org/10.1007/s00205-015-0917-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-015-0917-3

Keywords

Search

Navigation