×

Kármán vortex street in incompressible fluid models. (English) Zbl 1434.35077

Summary: This paper aims to provide a robust model for the well-known phenomenon of Kármán vortex street arising in fluid mechanics. The first theoretical attempt to model this pattern was given by Th. v. Kármán [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1911, 509–517 (1911; JFM 42.0800.01); ibid. 1912, 547–556 (1912; JFM 43.0854.01)]. He considered two parallel staggered rows of point vortices, with opposite strength, that translate at the same speed. Following the ideas of P. G. Saffman and J. C. Schatzman [SIAM J. Sci. Stat. Comput. 2, 285–295 (1981; Zbl 0484.76032)], we propose to study this phenomenon in the Euler equations by considering two infinite arrows of vortex patches. The key idea is to desingularize the point vortex model proposed by von Kármán. Our construction is flexible and can be extended to more general incompressible models.

MSC:

35Q31 Euler equations
35Q35 PDEs in connection with fluid mechanics
35B32 Bifurcations in context of PDEs
35B36 Pattern formations in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abramowitz M and Stegun I A 1965 Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables(Courier Corporation) (New York: Dover)
[2] Aref H 1995 On the equilibrium and stability of a row of point vortices J. Fluid Mech.290 167-81 · Zbl 0853.76011 · doi:10.1017/S002211209500245X
[3] Beichman J and Denisov S 2017 2D Euler equation on the strip: stability of a rectangular patch Commun. PDE42 100-20 · Zbl 1362.76006 · doi:10.1080/03605302.2016.1258576
[4] Bertozzi A L and Constantin P 1993 Global regularity for vortex patches Commun. Math. Phys.152 19-28 · Zbl 0771.76014 · doi:10.1007/BF02097055
[5] Burbea J 1982 Motions of vortex patches Lett. Math. Phys.6 1-16 · Zbl 0484.76031 · doi:10.1007/BF02281165
[6] Castro A, Córdoba D and Gómez-Serrano J 2016 Existence and regularity of rotating global solutions for the generalized surface quasi-geostrophic equations Duke Math. J.165 935-84 · Zbl 1339.35234
[7] Castro A, Córdoba D and Gómez-Serrano J 2016 Uniformly rotating analytic global patch solutions for active scalars Ann. PDE2 34 · Zbl 1397.35020 · doi:10.1007/s40818-016-0007-3
[8] Castro A, Córdoba D and Gómez-Serrano J 2019 Uniformly rotating smooth solutions for the incompressible 2D Euler equations Arch. Ration. Mech. Anal.231 719-85 · Zbl 1405.35147 · doi:10.1007/s00205-018-1288-3
[9] Constantin P, Majda A J and Tabak E 1994 Formation of strong fronts in the 2D quasigeostrophic thermal active scalar Nonlinearity7 1495 · Zbl 0809.35057 · doi:10.1088/0951-7715/7/6/001
[10] Córdoba A, Córdoba D and Gancedo F 2018 Uniqueness for SQG patch solutions Trans. Am. Math. Soc. Ser. B 5 1-31 · Zbl 1390.35244 · doi:10.1090/btran/20
[11] Córdoba D, Fontelos M A, Mancho A M and Rodrigo J L 2005 Evidence of singularities for a family of contour dynamics equations Proc. Natl Acad. Sci. USA102 5949-52 · Zbl 1135.76315 · doi:10.1073/pnas.0501977102
[12] Chemin J-Y 1993 Persistance de structures géometriques dans les fluides incompressibles bidimensionnels Ann. Sci. Ecole Norm. Super.26 1-26 · Zbl 0781.13002 · doi:10.24033/asens.1665
[13] Dritschel D G, Hmidi T and Renault C 2019 Imperfect bifurcation for the quasi-geostrophic shallow-water equations Arch. Ration. Mech. Anal.231 1853-915 · Zbl 1409.76036 · doi:10.1007/s00205-018-1312-7
[14] De la Hoz F, Hmidi T, Mateu J and Verdera J 2016 Doubly connected V-states for the planar Euler equations SIAM J. Math. Anal.48 1892-928 · Zbl 1342.35239 · doi:10.1137/140992801
[15] Deem G S and Zabusky N J 1978 Vortex waves: stationary ‘V-states’, interactions, recurrence, and breaking Phys. Rev. Lett.40 859-62 · doi:10.1103/PhysRevLett.40.859
[16] Gancedo F 2008 Existence for the α-patch model and the QG sharp front in Sobolev spaces Adv. Math.217 2569-98 · Zbl 1148.35099 · doi:10.1016/j.aim.2007.10.010
[17] Gancedo F and Patel N 2019 On the local existence and blow-up for generalized SQG patches (arXiv:1811.00530)
[18] García C, Hmidi T and Soler J 2018 Non uniform rotating vortices and periodic orbits for the two-dimensional Euler equations (arXiv:1807.10017)
[19] Gryanik V, Borth H and Olbers D 2004 The theory of quasi-geostrophic von Kármán vortex streets in two-layer fluids on a beta-plane J. Fluid Mech.505 23-57 · Zbl 1067.76094 · doi:10.1017/S0022112004008122
[20] Hassainia Z and Hmidi T 2015 On the V-states for the generalized quasi-geostrophic equations Commun. Math. Phys.337 321-77 · Zbl 1319.35188 · doi:10.1007/s00220-015-2300-5
[21] Hassainia Z, Masmoudi N and Wheeler M H 2019 Global bifurcation of rotating vortex patches Commun. Pure Appl. Math. (https://doi.org/10.1002/cpa.21855) · Zbl 1452.76038
[22] Jimenez J 1987 On the linear stability of the inviscid Kármán vortex street J. Fluid Mech.178 177-94 · Zbl 0633.76040 · doi:10.1017/S0022112087001174
[23] Helms L L 2009 Potential Theory (London: Springer) · Zbl 1180.31001 · doi:10.1007/978-1-84882-319-8
[24] Hmidi T and Mateu J 2017 Existence of corotating and counter-rotating vortex pairs for active scalar equations Commun. Math. Phys.350 699-747 · Zbl 1360.35157 · doi:10.1007/s00220-016-2784-7
[25] Hmidi T, Mateu J and Verdera J 2013 Boundary regularity of rotating vortex patches Arch. Ration. Mech. Anal.209 171-208 · Zbl 1286.35201 · doi:10.1007/s00205-013-0618-8
[26] von Kármán T 1911 Über den Mechanismus des Widerstands, den ein bewegter Korper in einer Flüssigkeit erfährt Göttinger Nachr., Math. Phys. Kl.1 509-17 · JFM 42.0800.01
[27] von Kármán T 1912 Über den Mechanismus des Widerstands, den ein bewegter Korper in einer Flüssigkeit erfährt Göttinger Nachr., Math. Phys. Kl.2 547-56 · JFM 43.0854.01
[28] Kirchhoff G R 1876 Vorlesungenber Mathematische Physik. Mechanik (Leipzig: Teubner) · JFM 08.0542.01
[29] Kiselev A, Yao Y and Zlatos A 2015 Local regularity for the modified SQG patch equation Pure Appl. Math.70 1253-315 · Zbl 1371.35220 · doi:10.1002/cpa.21677
[30] Kress R 2014 Linear Integral Equations (New York: Springer) · Zbl 1328.45001 · doi:10.1007/978-1-4614-9593-2
[31] Sir Horace Lamb 1932 Hydrodynamics (Cambridge: Cambridge University Press) · JFM 58.1298.04
[32] Lieb E and Loss M 1997 Analysis (Providence, RI: American Mathematical Society)
[33] Majda A and Bertozzi A 2002 Vorticity and Incompressible Flow (Cambridge: Cambridge University Press) · Zbl 0983.76001
[34] Matsumoto M 1999 Vortex shedding of bluff bodies: a review Fluids Struct.13 791-811 · doi:10.1006/jfls.1999.0249
[35] Newton P K 2001 The N-Vortex Problem(Analytical Techniques) (New York: Springer) · Zbl 0981.76002 · doi:10.1007/978-1-4684-9290-3
[36] Pierrehumbert R T 1980 A family of steady, translating vortex pairs with distributed vorticity J. Fluid Mech.99 129-44 · Zbl 0473.76034 · doi:10.1017/S0022112080000559
[37] Plotka H and Dritschel D G 2012 Quasi-geostrophic shallow-water vortex-patch equilibria and their stability Geophys. Astrophys. Fluid Dyn.106 574-95 · Zbl 07649665 · doi:10.1080/03091929.2012.674128
[38] Polvani L M 1988 Geostrophic vortex dynamics PhD Thesis MIT/WHOI WHOI-88-48
[39] Polvani L M, Zabusky N J and Flierl G R 1989 Two-layer geostrophic vortex dynamics. Part 1. Upper-layer V-states and merger J. Fluid Mech.205 215-42 · Zbl 0676.76093 · doi:10.1017/S0022112089002016
[40] Rayleigh L 1879 Acoustical obervation II Phil. Mag.7 149-62 · doi:10.1080/14786447908639584
[41] Rosenhead L 1929 Double row of vortices with arbitrary stagger Math. Proc. Camb. Phil. Soc.25 132-8 · JFM 55.0475.01 · doi:10.1017/S030500410001865X
[42] Saffman P G and Schatzman J C 1981 Properties of a vortex street of finite vortices SIAM J. Sci. Stat. Comput.2 285-95 · Zbl 0484.76032 · doi:10.1137/0902023
[43] Saffman P G and Schatzman J C 1982 Stability of the Kármán vortex street J. Fluid Mech.117 171-85 · Zbl 0514.76017 · doi:10.1017/S0022112082001578
[44] Schatzman J C 1981 A model for the von Kármán vortex street PhD Thesis California Institute of Technology
[45] Serfati P 1994 Une preuve directe d’existence globale des vortex patches 2D C. R. Acad. Sci., Paris I 318 515-8 · Zbl 0803.76022
[46] Strouhal V 1878 Uber eine besondere art der tonnerregung Ann. Phys., Lpz.5 216-51 · doi:10.1002/andp.18782411005
[47] Vallis G K 2008 Atmospheric and Oceanic Fluid Dynamics (Cambridge: Cambridge University Press)
[48] Yudovich Y 1963 Nonstationary flow of an ideal incompressible liquid Zh. Vychisl. Mat.3 1032-66 · Zbl 0129.19402
[49] Watson G N 1944 A Treatise on the Theory of Bessel Functions (Cambrige: Cambrige University Press) · Zbl 0063.08184
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.