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May  2022, 21(5): 1581-1594. doi: 10.3934/cpaa.2022032

Shock polars for non-polytropic compressible potential flow

Institute of Mathematics, Academia Sinica, 6F Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Rd., Taipei 106319, Taiwan

Received  September 2021 Published  May 2022 Early access  February 2022

Fund Project: This paper is based upon work supported by Taiwan MOST grant 108-2115-M-001-002-MY2 and 110-2115-M-001-005-MY3

We consider compressible potential flow for general equations of state. Assuming hyperbolicity and convex equation of state, we prove that shock polars have a unique critical point (in each half), as well as a unique sonic point, with critical and strong shocks always on the subsonic side. We also show existence of normal and oblique shocks, as well as monotonicity of density, enthalpy, pressure along each half-polar, with Mach number monotone only along the subsonic part.

Citation: 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
References:
[1]

M. BaeGui-Qiang Chen and M. Feldman, Regularity of solutions to regular shock reflection for potential flow, Invent. Math., 175 (2009), 505-543.  doi: 10.1007/s00222-008-0156-4.

[2]

M. Bae, Gui-Qiang Chen and Mikhail Feldman, Prandtl-Meyer reflection configurations, transonic shocks and free boundary problems, Preprint, arXiv: 1901.05916v2.

[3]

M. Bae and Wei Xiang, Detached shock past a blunt body, Preprint, arXiv: 1909.13281.

[4]

H. Bethe, On the Theory of Shock Waves for an Arbitrary Equation of State, Technical Report PB-32189, Clearinghouse for Federal Scientific and Technical Information, U.S. Dept. of Commerce, Washington, D.C., 1942.

[5]

A. Busemann, Handbuch der Experimentalphysik, volume IV, Akademische Verlagsgesellschaft, Leipzig, 1931.

[6]

J. ChenC. Christoforou and K. Jegdic, Existence and uniqueness analysis of a detached shock problem for the potential flow, Nonlinear Anal., 74 (2011), 705-720.  doi: 10.1016/j.na.2010.08.041.

[7]

S. X. Chen, A free boundary problem of elliptic equation arising in supersonic flow past a conical body, Z. Angew. Math. Phys., 54 (2003), 387-409.  doi: 10.1007/s00033-003-2111-y.

[8]

S. X. Chen, Stability of a Mach configuration, Commun. Pure Appl. Math., 59 (2005), 1-35.  doi: 10.1002/cpa.20108.

[9]

S. X. Chen and D. N. Li, Conical shock waves for isentropic euler system, Proc. Roy. Soc. Edinburgh Sct. A, 135 (205), 1109-1127.  doi: 10.1017/S0308210500004297.

[10]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, 1948.

[11]

C. De Lellis and L., Jr. Székelyhidi, The Euler equations as a differential inclusion, Ann. Math., 170 (2009), 1417-1436.  doi: 10.4007/annals.2009.170.1417.

[12]

C. De LellisL. Székelyhidi and 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.

[13]

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.

[14]

V. Elling, Counterexamples to the sonic criterion, Arch. Rat. Mech. Anal., 194 (2009), 987-1010.  doi: 10.1007/s00205-008-0196-3.

[15]

V. Elling, Instability of strong regular reflection and counterexamples to the detachment criterion, SIAM J. Appl. Math., 70 (2009), 1330-1340.  doi: 10.1137/080724769.

[16]

V. Elling, Existence of algebraic vortex spirals, In Hyperbolic problems. Theory, Numerics and Applications., volume 1 of Ser. Contemp. Appl. Math. CAM, 17, pages 203–214. World Sci. Publishing, Singapore, 2012.

[17]

V. Elling, Relative entropy and compressible potential flow, Acta Math. Sci. (ser. B), 35 (2015), 763-776.  doi: 10.1016/S0252-9602(15)30020-5.

[18]

V. Elling, Self-similar 2d Euler solutions with mixed-sign vorticity, Commun. Math. Phys., 348 (2016), 27-68.  doi: 10.1007/s00220-016-2755-z.

[19]

V. Elling, Barotropic Euler shock polars, To appear in Z. Angew. Math. Phys., 2021.

[20]

V. Elling, Shock polars for ideal and non-ideal gas, J. Fluid Mech., 916(A51), 2021. doi: 10.1017/jfm.2021.147.

[21]

V. Elling and T. P. Liu, Physicality of weak Prandtl-Meyer reflection, In RIMS Kokyuroku, volume 1495, pages 112–117. Kyoto University, Research Institute for Mathematical Sciences, May 2006.

[22]

V. Elling and T. P. Liu, Supersonic flow onto a solid wedge, Commun. Pure Appl. Math., 61 (2008), 1347-1448.  doi: 10.1002/cpa.20231.

[23]

V. Elling, Triple points and sign of circulation, Phys. Fluids, 31 (2019), 126106. 

[24]

B. X. Fang, Stability of transonic shocks for the full Euler system in supersonic flow onto a wedge, Math. Methods Appl. Sci., 29 (2006), 1-26.  doi: 10.1002/mma.661.

[25]

J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Commun. Pure Appl. Math., 18 (1965), 697-715.  doi: 10.1002/cpa.3160180408.

[26]

L. F. Henderson and R. Menikoff, Triple-shock entropy theorem and its consequences, J. Fluid Mech., 366 (1998), 179-210.  doi: 10.1017/S0022112098001244.

[27]

P. D. Lax, Hyperbolic systems of conservation laws II, Commun. Pure Appl. Math., 10 (1957), 537-566.  doi: 10.1002/cpa.3160100406.

[28]

M.C. Lopes-FilhoJ. LowengrubH. J. Nussenzveig Lopes and Y. X. Zheng, Numerical evidence of nonuniqueness in the evolution of vortex sheets, ESAIM:M2AN, 40 (2006), 225-237.  doi: 10.1051/m2an:2006012.

[29]

Th. Meyer, Ueberzweidimensionale Bewegungsvorgänge in einem Gas, das mit Ueberschallgeschwindigkeit strömt, Forschungsheft des Vereins Deutscher Ingenieure (VDI), 62: 31–67, 1908.

[30]

D. Pullin, On similarity flows containing two-branched vortex sheets, In R. Caflisch, editor, Mathematical aspect of vortex dynamics, pages 97–106. SIAM, 1989.

[31]

V. Scheffer, An inviscid flow with compact support in space-time, J. Geom. Anal., 3 (1993), 343-401.  doi: 10.1007/BF02921318.

[32]

A. Shnirelman, On the nonuniqueness of weak solutions of the Euler equation, Commun. Pure Appl. Math., 50 (1997), 1261-1286.  doi: 10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.3.CO;2-4.

[33]

A. Shnirelman, Weak solutions with decreasing energy of the incompressible Euler equations, Commun. Math. Phys., 210 (2000), 541-603.  doi: 10.1007/s002200050791.

[34]

V. M. Teshukov, On the shock polars in a gas with general equations of state, J. Appl. Math. Mech., 50 (1986), 71-75. 

[35]

V. M. Teshukov, Stability of regular shock wave reflection, Prikl. Mekhanika I Tech. Fizika, 30 (1989), 26-33.  doi: 10.1007/BF00852163.

[36]

H. Weyl, Shock waves in arbitrary fluids, Commun. Pure Appl. Math., 2 (1949), 103-122.  doi: 10.1002/cpa.3160020201.

[37]

Y. Q. Zhang, Global existence of steady supersonic potential flow past a curved wedge with a piecewise smooth boundary, SIAM J. Math. Anal., 31 (1999), 166-183.  doi: 10.1137/S0036141097331056.

show all references

References:
[1]

M. BaeGui-Qiang Chen and M. Feldman, Regularity of solutions to regular shock reflection for potential flow, Invent. Math., 175 (2009), 505-543.  doi: 10.1007/s00222-008-0156-4.

[2]

M. Bae, Gui-Qiang Chen and Mikhail Feldman, Prandtl-Meyer reflection configurations, transonic shocks and free boundary problems, Preprint, arXiv: 1901.05916v2.

[3]

M. Bae and Wei Xiang, Detached shock past a blunt body, Preprint, arXiv: 1909.13281.

[4]

H. Bethe, On the Theory of Shock Waves for an Arbitrary Equation of State, Technical Report PB-32189, Clearinghouse for Federal Scientific and Technical Information, U.S. Dept. of Commerce, Washington, D.C., 1942.

[5]

A. Busemann, Handbuch der Experimentalphysik, volume IV, Akademische Verlagsgesellschaft, Leipzig, 1931.

[6]

J. ChenC. Christoforou and K. Jegdic, Existence and uniqueness analysis of a detached shock problem for the potential flow, Nonlinear Anal., 74 (2011), 705-720.  doi: 10.1016/j.na.2010.08.041.

[7]

S. X. Chen, A free boundary problem of elliptic equation arising in supersonic flow past a conical body, Z. Angew. Math. Phys., 54 (2003), 387-409.  doi: 10.1007/s00033-003-2111-y.

[8]

S. X. Chen, Stability of a Mach configuration, Commun. Pure Appl. Math., 59 (2005), 1-35.  doi: 10.1002/cpa.20108.

[9]

S. X. Chen and D. N. Li, Conical shock waves for isentropic euler system, Proc. Roy. Soc. Edinburgh Sct. A, 135 (205), 1109-1127.  doi: 10.1017/S0308210500004297.

[10]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, 1948.

[11]

C. De Lellis and L., Jr. Székelyhidi, The Euler equations as a differential inclusion, Ann. Math., 170 (2009), 1417-1436.  doi: 10.4007/annals.2009.170.1417.

[12]

C. De LellisL. Székelyhidi and 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.

[13]

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.

[14]

V. Elling, Counterexamples to the sonic criterion, Arch. Rat. Mech. Anal., 194 (2009), 987-1010.  doi: 10.1007/s00205-008-0196-3.

[15]

V. Elling, Instability of strong regular reflection and counterexamples to the detachment criterion, SIAM J. Appl. Math., 70 (2009), 1330-1340.  doi: 10.1137/080724769.

[16]

V. Elling, Existence of algebraic vortex spirals, In Hyperbolic problems. Theory, Numerics and Applications., volume 1 of Ser. Contemp. Appl. Math. CAM, 17, pages 203–214. World Sci. Publishing, Singapore, 2012.

[17]

V. Elling, Relative entropy and compressible potential flow, Acta Math. Sci. (ser. B), 35 (2015), 763-776.  doi: 10.1016/S0252-9602(15)30020-5.

[18]

V. Elling, Self-similar 2d Euler solutions with mixed-sign vorticity, Commun. Math. Phys., 348 (2016), 27-68.  doi: 10.1007/s00220-016-2755-z.

[19]

V. Elling, Barotropic Euler shock polars, To appear in Z. Angew. Math. Phys., 2021.

[20]

V. Elling, Shock polars for ideal and non-ideal gas, J. Fluid Mech., 916(A51), 2021. doi: 10.1017/jfm.2021.147.

[21]

V. Elling and T. P. Liu, Physicality of weak Prandtl-Meyer reflection, In RIMS Kokyuroku, volume 1495, pages 112–117. Kyoto University, Research Institute for Mathematical Sciences, May 2006.

[22]

V. Elling and T. P. Liu, Supersonic flow onto a solid wedge, Commun. Pure Appl. Math., 61 (2008), 1347-1448.  doi: 10.1002/cpa.20231.

[23]

V. Elling, Triple points and sign of circulation, Phys. Fluids, 31 (2019), 126106. 

[24]

B. X. Fang, Stability of transonic shocks for the full Euler system in supersonic flow onto a wedge, Math. Methods Appl. Sci., 29 (2006), 1-26.  doi: 10.1002/mma.661.

[25]

J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Commun. Pure Appl. Math., 18 (1965), 697-715.  doi: 10.1002/cpa.3160180408.

[26]

L. F. Henderson and R. Menikoff, Triple-shock entropy theorem and its consequences, J. Fluid Mech., 366 (1998), 179-210.  doi: 10.1017/S0022112098001244.

[27]

P. D. Lax, Hyperbolic systems of conservation laws II, Commun. Pure Appl. Math., 10 (1957), 537-566.  doi: 10.1002/cpa.3160100406.

[28]

M.C. Lopes-FilhoJ. LowengrubH. J. Nussenzveig Lopes and Y. X. Zheng, Numerical evidence of nonuniqueness in the evolution of vortex sheets, ESAIM:M2AN, 40 (2006), 225-237.  doi: 10.1051/m2an:2006012.

[29]

Th. Meyer, Ueberzweidimensionale Bewegungsvorgänge in einem Gas, das mit Ueberschallgeschwindigkeit strömt, Forschungsheft des Vereins Deutscher Ingenieure (VDI), 62: 31–67, 1908.

[30]

D. Pullin, On similarity flows containing two-branched vortex sheets, In R. Caflisch, editor, Mathematical aspect of vortex dynamics, pages 97–106. SIAM, 1989.

[31]

V. Scheffer, An inviscid flow with compact support in space-time, J. Geom. Anal., 3 (1993), 343-401.  doi: 10.1007/BF02921318.

[32]

A. Shnirelman, On the nonuniqueness of weak solutions of the Euler equation, Commun. Pure Appl. Math., 50 (1997), 1261-1286.  doi: 10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.3.CO;2-4.

[33]

A. Shnirelman, Weak solutions with decreasing energy of the incompressible Euler equations, Commun. Math. Phys., 210 (2000), 541-603.  doi: 10.1007/s002200050791.

[34]

V. M. Teshukov, On the shock polars in a gas with general equations of state, J. Appl. Math. Mech., 50 (1986), 71-75. 

[35]

V. M. Teshukov, Stability of regular shock wave reflection, Prikl. Mekhanika I Tech. Fizika, 30 (1989), 26-33.  doi: 10.1007/BF00852163.

[36]

H. Weyl, Shock waves in arbitrary fluids, Commun. Pure Appl. Math., 2 (1949), 103-122.  doi: 10.1002/cpa.3160020201.

[37]

Y. Q. Zhang, Global existence of steady supersonic potential flow past a curved wedge with a piecewise smooth boundary, SIAM J. Math. Anal., 31 (1999), 166-183.  doi: 10.1137/S0036141097331056.

Figure 1.  Supersonic flow onto a slender wedge. The strong-type shock does not usually appear
Figure 2.  Shock polar (full Euler, polytropic $ \gamma = 7/5 $, $ M_0 = 3 $)
Figure 3.  $ v^x, v^y $ plane shock polar for polytropic potential flow, $ \gamma = 5/3 $ and $ M_0 = 3.5 $
Figure 4.  $ j,\theta $ plane shock polar for polytropic potential flow, $ \gamma = 5/3 $ and $ M_0 = 3.5 $
Figure 5.  Mass flux reaches a maximum at critical speed ($ v = c $; the overused term "critical" means "sonic" in this context, in contrast to the shock polar context)
Figure 6.  $ h $ is strictly convex in $ -1/2 \varrho^2 $. Tangents to its graph have slope $ ( \varrho c)^2 $, the chord from up- to downstream state represents $ ( j^n)^2 $. The Lax condition $ \varrho_0c_0< j^n_0 = j^n< \varrho c $ is obvious here
Figure 7.  Shock-velocity angles $ \beta $ and turning angle $ \theta $ in relation to shock, shock normal, mass flux and velocity vectors; $ \varrho_0 = 1 $ chosen to make $ {\mathbf{v}}_0 = {\mathbf{j}}_0 $ for the diagram. Decreasing $ \beta_0 $ while holding $ \theta $ fixed increases both $ v $ and $ j $, which requires $ M<1 $, explaining why critical-type shocks are transonic since their $ \theta $ is fixed to first order
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