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

Compressible vortex sheets separating from solid boundaries

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.
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.

[1]

Anna Kaźmierczak, Jan Sokolowski, Antoni Zochowski. Drag minimization for the obstacle in compressible flow using shape derivatives and finite volumes. Mathematical Control and Related Fields, 2018, 8 (1) : 89-115. doi: 10.3934/mcrf.2018004

[2]

Young-Pil Choi. Compressible Euler equations interacting with incompressible flow. Kinetic and Related Models, 2015, 8 (2) : 335-358. doi: 10.3934/krm.2015.8.335

[3]

Shuxing Chen, Gui-Qiang Chen, Zejun Wang, Dehua Wang. A multidimensional piston problem for the Euler equations for compressible flow. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 361-383. doi: 10.3934/dcds.2005.13.361

[4]

Gui-Qiang Chen, Bo Su. A viscous approximation for a multidimensional unsteady Euler flow: Existence theorem for potential flow. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1587-1606. doi: 10.3934/dcds.2003.9.1587

[5]

Volker W. Elling. Shock polars for non-polytropic compressible potential flow. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1581-1594. doi: 10.3934/cpaa.2022032

[6]

Joseph E. Paullet, Joseph P. Previte. Analysis of nanofluid flow past a permeable stretching/shrinking sheet. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4119-4126. doi: 10.3934/dcdsb.2020090

[7]

Myoungjean Bae, Hyangdong Park. Three-dimensional supersonic flows of Euler-Poisson system for potential flow. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2421-2440. doi: 10.3934/cpaa.2021079

[8]

Yunfei Su, Lei Yao, Mengmeng Zhu. Exponential decay for 2D reduced gravity two-and-a-half layer model with quantum potential and drag force. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022040

[9]

Zineb Hassainia, Taoufik Hmidi. Steady asymmetric vortex pairs for Euler equations. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1939-1969. doi: 10.3934/dcds.2020348

[10]

Ludovic Godard-Cadillac. Vortex collapses for the Euler and Quasi-Geostrophic models. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3143-3168. doi: 10.3934/dcds.2022012

[11]

Jean-françois Coulombel, Paolo Secchi. Uniqueness of 2-D compressible vortex sheets. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1439-1450. doi: 10.3934/cpaa.2009.8.1439

[12]

Tai-Ping Liu, Zhouping Xin, Tong Yang. Vacuum states for compressible flow. Discrete and Continuous Dynamical Systems, 1998, 4 (1) : 1-32. doi: 10.3934/dcds.1998.4.1

[13]

Thomas H. Otway. Compressible flow on manifolds. Conference Publications, 2001, 2001 (Special) : 289-294. doi: 10.3934/proc.2001.2001.289

[14]

Najwa Najib, Norfifah Bachok, Norihan Md Arifin, Fadzilah Md Ali. Stability analysis of stagnation point flow in nanofluid over stretching/shrinking sheet with slip effect using buongiorno's model. Numerical Algebra, Control and Optimization, 2019, 9 (4) : 423-431. doi: 10.3934/naco.2019041

[15]

Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Phase transition and separation in compressible Cahn-Hilliard fluids. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 73-88. doi: 10.3934/dcdsb.2014.19.73

[16]

Jian Zhai, Jianping Fang, Lanjun Li. Wave map with potential and hypersurface flow. Conference Publications, 2005, 2005 (Special) : 940-946. doi: 10.3934/proc.2005.2005.940

[17]

Michael Renardy. Backward uniqueness for linearized compressible flow. Evolution Equations and Control Theory, 2015, 4 (1) : 107-113. doi: 10.3934/eect.2015.4.107

[18]

Jianwei Yang, Ruxu Lian, Shu Wang. Incompressible type euler as scaling limit of compressible Euler-Maxwell equations. Communications on Pure and Applied Analysis, 2013, 12 (1) : 503-518. doi: 10.3934/cpaa.2013.12.503

[19]

Qing Chen, Zhong Tan. Time decay of solutions to the compressible Euler equations with damping. Kinetic and Related Models, 2014, 7 (4) : 605-619. doi: 10.3934/krm.2014.7.605

[20]

Jianwei Yang, Dongling Li, Xiao Yang. On the quasineutral limit for the compressible Euler-Poisson equations. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022020

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (123)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]