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December  2016, 36(12): 6781-6797. doi: 10.3934/dcds.2016095

Compressible vortex sheets separating from solid boundaries

Volker Elling 1,

1. 

Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, United States

Received  January 2016 Revised  April 2016 Published  October 2016

We prove existence of certain compressible subsonic inviscid flows with vortex sheets separating from a solid boundary. To leading order, any perturbation of the upstream boundary causes positive drag. We also prove that if a region of irrotational inviscid flow bounded by a vortex sheet and slip condition wall is enclosed in an angle less than $180^\circ$, then the velocity is zero.
Keywords: separation, Vortex sheet, Prandtl-Glauert, drag, corner, Euler., compressible, potential flow.
Mathematics Subject Classification: Primary: 76N15; Secondary: 76B1.
Citation: Volker Elling. Compressible vortex sheets separating from solid boundaries. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6781-6797. doi: 10.3934/dcds.2016095
References:
[1]

H. Alt, L. Caffarelli and A. Friedman, Compressible flow of jets and cavities,, J. Diff. Eqs., 56 (1985), 82. doi: 10.1016/0022-0396(85)90101-9. Google Scholar

[2]

G. Batchelor, An Introduction to Fluid Dynamics,, Second paperback edition. Cambridge Mathematical Library. Cambridge University Press, (1999). Google Scholar

[3]

P. Berg, The existence of subsonic helmholtz flows of a compressible fluid,, Comm. Pure Appl. Math., 15 (1962), 289. doi: 10.1002/cpa.3160150302. Google Scholar

[4]

L. Bers, Existence and uniqueness of a subsonic flow past a given profile,, Comm. Pure. Appl. Math., 7 (1954), 441. doi: 10.1002/cpa.3160070303. Google Scholar

[5]

G.-Q. Chen, V. Kukreja and H. Yuan, Well-posedness of transonic characteristic discontinuities in two-dimensional steady compressible euler flows,, Zeitschrift für angewandte Mathematik und Physik, 64 (2013), 1711. doi: 10.1007/s00033-013-0312-6. Google Scholar

[6]

A. Chorin and J. Marsden, A Mathematical Introduction to Fluid Mechanics,, 3rd edition, (1992). Google Scholar

[7]

J. d'Alembert, Essai d'une nouvelle théorie de la résistance des fluides,, 1752., (). Google Scholar

[8]

V. Elling, A possible counterexample to well-posedness of entropy solutions and to {Godunov} scheme convergence,, Math. Comp., 75 (2006), 1721. doi: 10.1090/S0025-5718-06-01863-1. Google Scholar

[9]

V. Elling, Existence of algebraic vortex spirals,, in Hyperbolic problems. Theory, (2012), 203. Google Scholar

[10]

R. Finn and D. Gilbarg, Asymptotic behaviour and uniqueness of plane subsonic flows,, Comm. Pure Appl. Math., 10 (1957), 23. doi: 10.1002/cpa.3160100102. Google Scholar

[11]

L. Föppl, Wirbelbewegung hinter einem Kreiszylinder,, Sitz. K. Bayr. Akad. Wiss., (): 7. Google Scholar

[12]

H. Helmholtz, Über discontinuirliche Flüssigkeits-Bewegungen,, Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, (): 215. Google Scholar

[13]

G. Kirchhoff, Zur Theorie freier Flüssigkeitsstrahlen,, J. Reine Angew. Math., 70 (1869), 289. doi: 10.1515/crll.1869.70.289. Google Scholar

[14]

C. D. Lellis and L. S. Jr., On admissibility criteria for weak solutions of the Euler equations,, Arch. Rat. Mech. Anal., 195 (2010), 225. doi: 10.1007/s00205-008-0201-x. Google Scholar

[15]

T. Levi-Civita, Scie e leggi di resistenzia,, Rend. Circ. Mat. Palermo, 23 (1907), 1. Google Scholar

[16]

M. Lopes-Filho, J. Lowengrub, H. N. Lopes and Y. Zheng, Numerical evidence of nonuniqueness in the evolution of vortex sheets,, ESAIM:M2AN, 40 (2006), 225. doi: 10.1051/m2an:2006012. Google Scholar

[17]

V. Maz'ya and B. Plamenevskii, Estimates in $L_p$ and in Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points,, Amer. Math. Soc. Transl. (2), 123 (1984), 1. Google Scholar

[18]

D. Moore, The spontaneous appearance of a singularity in the shape of an evolving vortex sheet,, Proceedings of the Royal Society of London A: Mathematical, 365 (1979), 105. doi: 10.1098/rspa.1979.0009. Google Scholar

[19]

L. Prandtl, Über Flüssigkeitsbewegung bei sehr kleiner Reibung,, in Verhandlungen des III. Internationalen Mathematiker-Kongresses, (1904). Google Scholar

[20]

D. Pullin, On similarity flows containing two-branched vortex sheets,, in Mathematical aspect of vortex dynamics (ed. R. Caflisch), (1989), 97. Google Scholar

[21]

L. Rayleigh, On the resistance of fluids,, Phil. Mag., 11 (1876), 430. Google Scholar

[22]

A. Roshko, Perspectives on bluff body aerodynamics,, Journal of Wind Engineering and Industrial Aerodynamics, 49 (1993), 79. doi: 10.1016/0167-6105(93)90007-B. Google Scholar

[23]

P. Saffman, Vortex Dynamics,, Cambridge University Press, (1992). Google Scholar

[24]

V. Scheffer, An inviscid flow with compact support in space-time,, J. Geom. Anal., 3 (1993), 343. doi: 10.1007/BF02921318. Google Scholar

[25]

M. Shiffman, On the existence of subsonic flows of a compressible fluid,, J. Rat. Mech. Anal., 1 (1952), 605. Google Scholar

[26]

A. Shnirelman, On the nonuniqueness of weak solutions of the Euler equation,, Comm. Pure Appl. Math., 50 (1997), 1261. doi: 10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.0.CO;2-6. Google Scholar

[27]

J. Smith, Behaviour of a Vortex Sheet Separating from a Smooth Surface,, Technical Report 77058, (7705). Google Scholar

[28]

V. Sychev, Asymptotic theory of separation flows,, Fluid Dynamics, 17 (1982), 20. Google Scholar

[29]

M. van Dyke, An Album of Fluid Motion,, The Parabolic Press, (1982). Google Scholar

show all references

References:
[1]

H. Alt, L. Caffarelli and A. Friedman, Compressible flow of jets and cavities,, J. Diff. Eqs., 56 (1985), 82. doi: 10.1016/0022-0396(85)90101-9. Google Scholar

[2]

G. Batchelor, An Introduction to Fluid Dynamics,, Second paperback edition. Cambridge Mathematical Library. Cambridge University Press, (1999). Google Scholar

[3]

P. Berg, The existence of subsonic helmholtz flows of a compressible fluid,, Comm. Pure Appl. Math., 15 (1962), 289. doi: 10.1002/cpa.3160150302. Google Scholar

[4]

L. Bers, Existence and uniqueness of a subsonic flow past a given profile,, Comm. Pure. Appl. Math., 7 (1954), 441. doi: 10.1002/cpa.3160070303. Google Scholar

[5]

G.-Q. Chen, V. Kukreja and H. Yuan, Well-posedness of transonic characteristic discontinuities in two-dimensional steady compressible euler flows,, Zeitschrift für angewandte Mathematik und Physik, 64 (2013), 1711. doi: 10.1007/s00033-013-0312-6. Google Scholar

[6]

A. Chorin and J. Marsden, A Mathematical Introduction to Fluid Mechanics,, 3rd edition, (1992). Google Scholar

[7]

J. d'Alembert, Essai d'une nouvelle théorie de la résistance des fluides,, 1752., (). Google Scholar

[8]

V. Elling, A possible counterexample to well-posedness of entropy solutions and to {Godunov} scheme convergence,, Math. Comp., 75 (2006), 1721. doi: 10.1090/S0025-5718-06-01863-1. Google Scholar

[9]

V. Elling, Existence of algebraic vortex spirals,, in Hyperbolic problems. Theory, (2012), 203. Google Scholar

[10]

R. Finn and D. Gilbarg, Asymptotic behaviour and uniqueness of plane subsonic flows,, Comm. Pure Appl. Math., 10 (1957), 23. doi: 10.1002/cpa.3160100102. Google Scholar

[11]

L. Föppl, Wirbelbewegung hinter einem Kreiszylinder,, Sitz. K. Bayr. Akad. Wiss., (): 7. Google Scholar

[12]

H. Helmholtz, Über discontinuirliche Flüssigkeits-Bewegungen,, Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, (): 215. Google Scholar

[13]

G. Kirchhoff, Zur Theorie freier Flüssigkeitsstrahlen,, J. Reine Angew. Math., 70 (1869), 289. doi: 10.1515/crll.1869.70.289. Google Scholar

[14]

C. D. Lellis and L. S. Jr., On admissibility criteria for weak solutions of the Euler equations,, Arch. Rat. Mech. Anal., 195 (2010), 225. doi: 10.1007/s00205-008-0201-x. Google Scholar

[15]

T. Levi-Civita, Scie e leggi di resistenzia,, Rend. Circ. Mat. Palermo, 23 (1907), 1. Google Scholar

[16]

M. Lopes-Filho, J. Lowengrub, H. N. Lopes and Y. Zheng, Numerical evidence of nonuniqueness in the evolution of vortex sheets,, ESAIM:M2AN, 40 (2006), 225. doi: 10.1051/m2an:2006012. Google Scholar

[17]

V. Maz'ya and B. Plamenevskii, Estimates in $L_p$ and in Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points,, Amer. Math. Soc. Transl. (2), 123 (1984), 1. Google Scholar

[18]

D. Moore, The spontaneous appearance of a singularity in the shape of an evolving vortex sheet,, Proceedings of the Royal Society of London A: Mathematical, 365 (1979), 105. doi: 10.1098/rspa.1979.0009. Google Scholar

[19]

L. Prandtl, Über Flüssigkeitsbewegung bei sehr kleiner Reibung,, in Verhandlungen des III. Internationalen Mathematiker-Kongresses, (1904). Google Scholar

[20]

D. Pullin, On similarity flows containing two-branched vortex sheets,, in Mathematical aspect of vortex dynamics (ed. R. Caflisch), (1989), 97. Google Scholar

[21]

L. Rayleigh, On the resistance of fluids,, Phil. Mag., 11 (1876), 430. Google Scholar

[22]

A. Roshko, Perspectives on bluff body aerodynamics,, Journal of Wind Engineering and Industrial Aerodynamics, 49 (1993), 79. doi: 10.1016/0167-6105(93)90007-B. Google Scholar

[23]

P. Saffman, Vortex Dynamics,, Cambridge University Press, (1992). Google Scholar

[24]

V. Scheffer, An inviscid flow with compact support in space-time,, J. Geom. Anal., 3 (1993), 343. doi: 10.1007/BF02921318. Google Scholar

[25]

M. Shiffman, On the existence of subsonic flows of a compressible fluid,, J. Rat. Mech. Anal., 1 (1952), 605. Google Scholar

[26]

A. Shnirelman, On the nonuniqueness of weak solutions of the Euler equation,, Comm. Pure Appl. Math., 50 (1997), 1261. doi: 10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.0.CO;2-6. Google Scholar

[27]

J. Smith, Behaviour of a Vortex Sheet Separating from a Smooth Surface,, Technical Report 77058, (7705). Google Scholar

[28]

V. Sychev, Asymptotic theory of separation flows,, Fluid Dynamics, 17 (1982), 20. Google Scholar

[29]

M. van Dyke, An Album of Fluid Motion,, The Parabolic Press, (1982). Google Scholar

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