On the spectral instability of parallel shear flows

Emmanuel Grenier[1]; Yan Guo[2]; Toan T. Nguyen[3]

  • [1] tabacckludge ’Equipe Projet Inria NUMED INRIA Rhône Alpes Unité de Mathématiques Pures et Appliquées UMR 5669, CNRS et École Normale Supérieure de Lyon 46, allée d’Italie 69364 Lyon Cedex 07 France
  • [2] Division of Applied Mathematics Brown University 182 George street Providence RI 02912 USA
  • [3] Department of Mathematics Penn State University State College, PA 16803 USA

Séminaire Laurent Schwartz — EDP et applications (2014-2015)

  • page 1-14
  • ISSN: 2266-0607

Abstract

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This short note is to announce our recent results [2,3] which provide a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924), C.C. Lin and Tollmien (1940s), among other prominent physicists. Precisely, we construct growing modes of the linearized Navier-Stokes equations about general stationary shear flows in a bounded channel (channel flows) or on a half-space (boundary layers), for sufficiently large Reynolds number R → ∞ . Such an instability is linked to the emergence of Tollmien-Schlichting waves in describing the early stage of the transition from laminar to turbulent flows.

How to cite

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Grenier, Emmanuel, Guo, Yan, and Nguyen, Toan T.. "On the spectral instability of parallel shear flows." Séminaire Laurent Schwartz — EDP et applications (2014-2015): 1-14. <http://eudml.org/doc/275800>.

@article{Grenier2014-2015,
abstract = {This short note is to announce our recent results [2,3] which provide a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924), C.C. Lin and Tollmien (1940s), among other prominent physicists. Precisely, we construct growing modes of the linearized Navier-Stokes equations about general stationary shear flows in a bounded channel (channel flows) or on a half-space (boundary layers), for sufficiently large Reynolds number $R \rightarrow \infty $. Such an instability is linked to the emergence of Tollmien-Schlichting waves in describing the early stage of the transition from laminar to turbulent flows.},
affiliation = {tabacckludge ’Equipe Projet Inria NUMED INRIA Rhône Alpes Unité de Mathématiques Pures et Appliquées UMR 5669, CNRS et École Normale Supérieure de Lyon 46, allée d’Italie 69364 Lyon Cedex 07 France; Division of Applied Mathematics Brown University 182 George street Providence RI 02912 USA; Department of Mathematics Penn State University State College, PA 16803 USA},
author = {Grenier, Emmanuel, Guo, Yan, Nguyen, Toan T.},
journal = {Séminaire Laurent Schwartz — EDP et applications},
language = {eng},
pages = {1-14},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {On the spectral instability of parallel shear flows},
url = {http://eudml.org/doc/275800},
year = {2014-2015},
}

TY - JOUR
AU - Grenier, Emmanuel
AU - Guo, Yan
AU - Nguyen, Toan T.
TI - On the spectral instability of parallel shear flows
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2014-2015
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
SP - 1
EP - 14
AB - This short note is to announce our recent results [2,3] which provide a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924), C.C. Lin and Tollmien (1940s), among other prominent physicists. Precisely, we construct growing modes of the linearized Navier-Stokes equations about general stationary shear flows in a bounded channel (channel flows) or on a half-space (boundary layers), for sufficiently large Reynolds number $R \rightarrow \infty $. Such an instability is linked to the emergence of Tollmien-Schlichting waves in describing the early stage of the transition from laminar to turbulent flows.
LA - eng
UR - http://eudml.org/doc/275800
ER -

References

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  1. P. G. Drazin, W. H. Reid, Hydrodynamic stability. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University, Cambridge–New York, 1981. 
  2. E. Grenier, Y. Guo, and T. Nguyen, Spectral instability of symmetric shear flows in a two-dimensional channel, http://arxiv.org/abs/1402.1395. Zbl06548164
  3. E. Grenier, Y. Guo, and T. Nguyen, Spectral instability of characteristic boundary layer flows, http://arxiv.org/abs/1406.3862. 
  4. W. Heisenberg, Über Stabilität und Turbulenz von Flüssigkeitsströmen. Ann. Phys. 74, 577–627 (1924) 
  5. W. Heisenberg, On the stability of laminar flow. Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, pp. 292–296. Amer. Math. Soc., Providence, R. I., 1952. 
  6. C. C. Lin, The theory of hydrodynamic stability. Cambridge, at the University Press, 1955. 
  7. W. Orr, Stability and instability of steady motions of a perfect liquid and of a viscous fluid, Parts I and II, Proc. Ir. Acad. Sect. A, Math Astron. Phys. Sci., 27 (1907), pp. 9-68, 69-138. 
  8. Lord Rayleigh, On the stability, or instability, of certain fluid motions. Proc. London Math. Soc. 11 (1880), 57–70. Zbl12.0711.02
  9. H. Schlichting, Boundary layer theory, Translated by J. Kestin. 4th ed. McGraw–Hill Series in Mechanical Engineering. McGraw–Hill Book Co., Inc., New York, 1960. Zbl0096.20105
  10. A. Sommerfeld, Ein Beitrag zur hydrodynamischen Erklärung der turbulent Flussigkeitsbewe-gung, Atti IV Congr. Internat. Math. Roma, 3 (1908), pp. 116-124. 
  11. W. Wasow, The complex asymptotic theory of a fourth order differential equation of hydrodynamics. Ann. of Math. (2) 49, (1948). 852–871. Zbl0031.40202
  12. W. Wasow, Asymptotic solution of the differential equation of hydrodynamic stability in a domain containing a transition point. Ann. of Math. (2) 58, (1953). 222–252. Zbl0051.06602
  13. W. Wasow, Linear turning point theory. Applied Mathematical Sciences, 54. Springer-Verlag, New York, 1985. ix+246 pp. Zbl0558.34049

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