Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T21:31:54.632Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonexistence of Coherent Structures in Two-dimensional InviscidChannel Flow

Published online by Cambridge University Press:  29 February 2012

H. Kalisch
H. Kalisch*
Affiliation:
University of Bergen, Department of Mathematics, P.O. Box 7800, 5020 Bergen, Norway
*
⋆ Corresponding author. E-mail: henrik.kalisch@math.uib.no
Get access

Abstract

Two-dimensional inviscid channel flow of an incompressible fluid is considered. It isshown that if the flow is steady and features no horizontal stagnation, then the flow mustnecessarily be a parallel shear flow.

Keywords

inviscid channel flowhodograph transformmaximum principlesteady flow
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 2: Solitary waves , 2012 , pp. 77 - 82
Copyright
© EDP Sciences, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bjørkavåg, M., Kalisch, H.. Wave breaking in Boussinesq models for undular bores. Phys. Lett. A, 375 (2011), 157–1578. CrossRefGoogle Scholar
Constantin, A., Ehrnström, M., Wahlén, E.. Symmetry of steady periodic gravity water waves with vorticity. Duke Math. J., 140 (2007), 591–603. CrossRefGoogle Scholar
Constantin, A., Strauss, W.. Exact steady periodic water waves with vorticity. Comm. Pure Appl. Math., 57 (2004), 481–527. CrossRefGoogle Scholar
Craig, W.. Non-existence of solitary water waves in three dimensions. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 360 (2002), 2127–2135. CrossRefGoogle ScholarPubMed
P.G. Drazin, W.H. Reid. Hydrodynamic stability. Cambridge University Press, Cambridge, 2004.
Dubreil-Jacotin, M.-L.. Sur la détermination rigoureuse des ondes permanentes périodiques d’ampleur finie. J. Math. Pures Appl., 13 (1934), 217–291. Google Scholar
Dubreil-Jacotin, M.-L.. Sur les théorèmes d’existence relatifs aux ondes permanentes périodiques ’a deux dimensions dans les liquides hétérogènes. J. Math. Pures Appl., 16 (1937), 43–67. Google Scholar
Ehrnström, M.. A note on surface profiles for symmetric gravity waves with vorticity. J. Nonlinear Math. Phys., 13 (2006), 1–8. CrossRefGoogle Scholar
Ehrnström, M.. Uniqueness for steady periodic water waves with vorticity. Int. Math. Res. Not., 2005 (2005), 3721–3726. CrossRefGoogle Scholar
Fraenkel, L.E.. On Kelvin-Stuart vortices in a viscous fluid. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366 (2008), 2717–2728. CrossRefGoogle Scholar
Friedlander, S., Strauss, W., Vishik, M.. Nonlinear instability in an ideal fluid. Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 187–209. CrossRefGoogle Scholar
D. Gilbarg, N.S. Trudinger. Elliptic partial differential equations of second order. Grundlehren der Mathematischen Wissenschaften 224, Springer, Berlin-New York, 1977.
Goubet, O.. A relation between the pressure gradient and the flux for the general channel flow problem. Appl. Math. Optim., 34 (1996), 361–365. CrossRefGoogle Scholar
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 Traveling Waves in Turbulent Pipe Flow. Science, 10 (2004), 1594–1598. CrossRefGoogle Scholar
Hur, V.M, Lin, Z.. Unstable surface waves in running water. Comm. Math. Phys., 282 (2008), 733–796. CrossRefGoogle Scholar
Ibragimov, N.H., Aitbayev, R, Ibragimov, R.N.. Three-dimensional nonlinear rotating surface waves in channels of variable depth in the presence of formation of a small perturbation of atmospheric pressure across the channel. Commun. Nonlinear Sci. and Numer. Simul., 14 (2009), 3811–3820. CrossRefGoogle Scholar
Ibragimov, R.N., Pelinovsky, D.E.. Three-dimensional gravity waves in a channel of variable depth. Commun. Nonlinear Sci. and Numer. Simul., 13 (2008), 2104–2113. CrossRefGoogle Scholar
Kalisch, H.. Periodic traveling water waves with isobaric streamlines. J. Nonlinear Math. Phys., 11 (2004), 461–471. CrossRefGoogle Scholar
Kalisch, H.. A uniqueness result for periodic traveling waves in water of finite depth. Nonlinear Anal., 58 (2004), 779–785. CrossRefGoogle Scholar
Thomson, W.. On disturbing infinity in Lord Rayleigh’s solution for waves in a plane vortex stratum. Nature, 23 (1880), 45–46. Google Scholar
H. Lamb. Hydrodynamics. Cambridge University Press, London, 1924.
Moin, P., Kim, J.. Numerical investigation of turbulent channel flow. J. Fluid Mech., 118 (1982), 341–377. CrossRefGoogle Scholar
Trefethen, A.E., Trefethen, L.N., Schmid, P.J.. Spectra and pseudospectra for pipe Poiseuille flow. Comput. Methods Appl. Mech. Engrg., 175 (1999), 413–420. CrossRefGoogle Scholar
Wahlén, E.. Steady water waves with a critical layer. J. Differential Equations, 246 (2009), 2468–2483. CrossRefGoogle Scholar
Waleffe, F.. Homotopy of exact coherent structures in plane shear flows. Phys. Fluids, 15 (2003), 1517–1534. CrossRefGoogle Scholar
Åsén, P.O., Kreiss, G.. On a rigorous resolvent estimate for plane Couette flow. J. Math. Fluid Mech., 9 (2007), 153–180. CrossRefGoogle Scholar