Compressible vortex sheets separating from solid boundaries

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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

Abstract
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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 and Continuous Dynamical Systems, 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), 82-141. doi: 10.1016/0022-0396(85)90101-9.
[2]	G. Batchelor, An Introduction to Fluid Dynamics, Second paperback edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1999.
[3]	P. Berg, The existence of subsonic helmholtz flows of a compressible fluid, Comm. Pure Appl. Math., 15 (1962), 289-347. doi: 10.1002/cpa.3160150302.
[4]	L. Bers, Existence and uniqueness of a subsonic flow past a given profile, Comm. Pure. Appl. Math., 7 (1954), 441-504. doi: 10.1002/cpa.3160070303.
[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), 1711-1727. doi: 10.1007/s00033-013-0312-6.
[6]	A. Chorin and J. Marsden, A Mathematical Introduction to Fluid Mechanics, 3rd edition, Springer, 1992.
[7]	J. d'Alembert, Essai d'une nouvelle théorie de la résistance des fluides, 1752.
[8]	V. Elling, A possible counterexample to well-posedness of entropy solutions and to {Godunov} scheme convergence, Math. Comp., 75 (2006), 1721-1733. doi: 10.1090/S0025-5718-06-01863-1.
[9]	V. Elling, Existence of algebraic vortex spirals, in Hyperbolic problems. Theory, Numerics and Applications., vol. 1 of Ser. Contemp. Appl. Math. CAM, 17, World Sci. Publishing, Singapore, 2012, 203-214.
[10]	R. Finn and D. Gilbarg, Asymptotic behaviour and uniqueness of plane subsonic flows, Comm. Pure Appl. Math., 10 (1957), 23-63. doi: 10.1002/cpa.3160100102.
[11]	L. Föppl, Wirbelbewegung hinter einem Kreiszylinder, Sitz. K. Bayr. Akad. Wiss., 7-18.
[12]	H. Helmholtz, Über discontinuirliche Flüssigkeits-Bewegungen, Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 215-228.
[13]	G. Kirchhoff, Zur Theorie freier Flüssigkeitsstrahlen, J. Reine Angew. Math., 70 (1869), 289-298. doi: 10.1515/crll.1869.70.289.
[14]	C. D. Lellis and L. S. Jr., On admissibility criteria for weak solutions of the Euler equations, Arch. Rat. Mech. Anal., 195 (2010), 225-260. doi: 10.1007/s00205-008-0201-x.
[15]	T. Levi-Civita, Scie e leggi di resistenzia, Rend. Circ. Mat. Palermo, 23 (1907), 1-37.
[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), 225-237. doi: 10.1051/m2an:2006012.
[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), 1-88.
[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, Physical and Engineering Sciences, 365 (1979), 105-119. doi: 10.1098/rspa.1979.0009.
[19]	L. Prandtl, Über Flüssigkeitsbewegung bei sehr kleiner Reibung, in Verhandlungen des III. Internationalen Mathematiker-Kongresses, Heidelberg}, 1904.
[20]	D. Pullin, On similarity flows containing two-branched vortex sheets, in Mathematical aspect of vortex dynamics (ed. R. Caflisch), SIAM, 1989, 97-106.
[21]	L. Rayleigh, On the resistance of fluids, Phil. Mag., 11 (1876), 430-441.
[22]	A. Roshko, Perspectives on bluff body aerodynamics, Journal of Wind Engineering and Industrial Aerodynamics, 49 (1993), 79-100. doi: 10.1016/0167-6105(93)90007-B.
[23]	P. Saffman, Vortex Dynamics, Cambridge University Press, 1992.
[24]	V. Scheffer, An inviscid flow with compact support in space-time, J. Geom. Anal., 3 (1993), 343-401. doi: 10.1007/BF02921318.
[25]	M. Shiffman, On the existence of subsonic flows of a compressible fluid, J. Rat. Mech. Anal., 1 (1952), 605-652.
[26]	A. Shnirelman, On the nonuniqueness of weak solutions of the Euler equation, Comm. Pure Appl. Math., 50 (1997), 1261-1286. doi: 10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.0.CO;2-6.
[27]	J. Smith, Behaviour of a Vortex Sheet Separating from a Smooth Surface, Technical Report 77058, Royal Aircraft Establishment, 1977.
[28]	V. Sychev, Asymptotic theory of separation flows, Fluid Dynamics, 17 (1982), 20-30.
[29]	M. van Dyke, An Album of Fluid Motion, The Parabolic Press, Stanford, California, 1982.

show all references

References:

[1]	H. Alt, L. Caffarelli and A. Friedman, Compressible flow of jets and cavities, J. Diff. Eqs., 56 (1985), 82-141. doi: 10.1016/0022-0396(85)90101-9.
[2]	G. Batchelor, An Introduction to Fluid Dynamics, Second paperback edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1999.
[3]	P. Berg, The existence of subsonic helmholtz flows of a compressible fluid, Comm. Pure Appl. Math., 15 (1962), 289-347. doi: 10.1002/cpa.3160150302.
[4]	L. Bers, Existence and uniqueness of a subsonic flow past a given profile, Comm. Pure. Appl. Math., 7 (1954), 441-504. doi: 10.1002/cpa.3160070303.
[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), 1711-1727. doi: 10.1007/s00033-013-0312-6.
[6]	A. Chorin and J. Marsden, A Mathematical Introduction to Fluid Mechanics, 3rd edition, Springer, 1992.
[7]	J. d'Alembert, Essai d'une nouvelle théorie de la résistance des fluides, 1752.
[8]	V. Elling, A possible counterexample to well-posedness of entropy solutions and to {Godunov} scheme convergence, Math. Comp., 75 (2006), 1721-1733. doi: 10.1090/S0025-5718-06-01863-1.
[9]	V. Elling, Existence of algebraic vortex spirals, in Hyperbolic problems. Theory, Numerics and Applications., vol. 1 of Ser. Contemp. Appl. Math. CAM, 17, World Sci. Publishing, Singapore, 2012, 203-214.
[10]	R. Finn and D. Gilbarg, Asymptotic behaviour and uniqueness of plane subsonic flows, Comm. Pure Appl. Math., 10 (1957), 23-63. doi: 10.1002/cpa.3160100102.
[11]	L. Föppl, Wirbelbewegung hinter einem Kreiszylinder, Sitz. K. Bayr. Akad. Wiss., 7-18.
[12]	H. Helmholtz, Über discontinuirliche Flüssigkeits-Bewegungen, Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 215-228.
[13]	G. Kirchhoff, Zur Theorie freier Flüssigkeitsstrahlen, J. Reine Angew. Math., 70 (1869), 289-298. doi: 10.1515/crll.1869.70.289.
[14]	C. D. Lellis and L. S. Jr., On admissibility criteria for weak solutions of the Euler equations, Arch. Rat. Mech. Anal., 195 (2010), 225-260. doi: 10.1007/s00205-008-0201-x.
[15]	T. Levi-Civita, Scie e leggi di resistenzia, Rend. Circ. Mat. Palermo, 23 (1907), 1-37.
[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), 225-237. doi: 10.1051/m2an:2006012.
[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), 1-88.
[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, Physical and Engineering Sciences, 365 (1979), 105-119. doi: 10.1098/rspa.1979.0009.
[19]	L. Prandtl, Über Flüssigkeitsbewegung bei sehr kleiner Reibung, in Verhandlungen des III. Internationalen Mathematiker-Kongresses, Heidelberg}, 1904.
[20]	D. Pullin, On similarity flows containing two-branched vortex sheets, in Mathematical aspect of vortex dynamics (ed. R. Caflisch), SIAM, 1989, 97-106.
[21]	L. Rayleigh, On the resistance of fluids, Phil. Mag., 11 (1876), 430-441.
[22]	A. Roshko, Perspectives on bluff body aerodynamics, Journal of Wind Engineering and Industrial Aerodynamics, 49 (1993), 79-100. doi: 10.1016/0167-6105(93)90007-B.
[23]	P. Saffman, Vortex Dynamics, Cambridge University Press, 1992.
[24]	V. Scheffer, An inviscid flow with compact support in space-time, J. Geom. Anal., 3 (1993), 343-401. doi: 10.1007/BF02921318.
[25]	M. Shiffman, On the existence of subsonic flows of a compressible fluid, J. Rat. Mech. Anal., 1 (1952), 605-652.
[26]	A. Shnirelman, On the nonuniqueness of weak solutions of the Euler equation, Comm. Pure Appl. Math., 50 (1997), 1261-1286. doi: 10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.0.CO;2-6.
[27]	J. Smith, Behaviour of a Vortex Sheet Separating from a Smooth Surface, Technical Report 77058, Royal Aircraft Establishment, 1977.
[28]	V. Sychev, Asymptotic theory of separation flows, Fluid Dynamics, 17 (1982), 20-30.
[29]	M. van Dyke, An Album of Fluid Motion, The Parabolic Press, Stanford, California, 1982.

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