Skip to main content
SpringerLink
Log in

Existence and Stability of Multidimensional Transonic Flows through an Infinite Nozzle of Arbitrary Cross-Sections

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We establish the existence and stability of multidimensional steady transonic flows with transonic shocks through an infinite nozzle of arbitrary cross-sections, including a slowly varying de Laval nozzle. The transonic flow is governed by the inviscid potential flow equation with supersonic upstream flow at the entrance, uniform subsonic downstream flow at the exit at infinity, and the slip boundary condition on the nozzle boundary. Our results indicate that, if the supersonic upstream flow at the entrance is sufficiently close to a uniform flow, there exists a solution that consists of a C 1,α subsonic flow in the unbounded downstream region, converging to a uniform velocity state at infinity, and a C 1,α multidimensional transonic shock separating the subsonic flow from the supersonic upstream flow; the uniform velocity state at the exit at infinity in the downstream direction is uniquely determined by the supersonic upstream flow; and the shock is orthogonal to the nozzle boundary at every point of their intersection. In order to construct such a transonic flow, we reformulate the multidimensional transonic nozzle problem into a free boundary problem for the subsonic phase, in which the equation is elliptic and the free boundary is a transonic shock. The free boundary conditions are determined by the Rankine–Hugoniot conditions along the shock. We further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain. We also prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bers L. (1954) Existence and uniqueness of subsonic flows past a given profile. Comm. Pure Appl. Math. 7, 441–504

    MATH  MathSciNet  Google Scholar 

  2. Canić S.,Keyfitz B.L., Lieberman G. (2000) A proof of existence of perturbed steady transonic shocks via a free boundary problem. Comm. Pure Appl. Math. 53, 484–511

    Article  MathSciNet  Google Scholar 

  3. Chen G.-Q.,Feldman M. (2003) Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type. J. Amer. Math. Soc. 16, 461–494

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen G.-Q.,Feldman M. (2004) Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations. Comm. Pure Appl. Math. 57, 310–356

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen G.-Q.,Feldman M. (2004) Free boundary problems and transonic shocks for the Euler equations in unbounded domains. Ann. Sc. Norm. Super. Pisa Cl. Sci. 3(5): 827–869

    MATH  MathSciNet  Google Scholar 

  6. Chen S.-X. (2001) Existence of stationary supersonic flows past a point body. Arch. Ration. Mech. Anal. 156, 141–181

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen S.-X. (1998) Asymptotic behavior of supersonic flow past a convex combined wedge. Chinese Ann. Math. Ser. B 19, 255–264

    MATH  MathSciNet  Google Scholar 

  8. Costabel M.,Dauge M. (1993) Construction of corner singularities for Agmon-Douglis-Nirenberg elliptic systems. Math. Nachr. 162, 209–237

    MATH  MathSciNet  Google Scholar 

  9. Costabel M.,Dauge M. (1994) Stable asymptotics for elliptic systems on plane domains with corners. Comm. Partial Differential Equations 19, 1677–1726

    MATH  MathSciNet  Google Scholar 

  10. Courant R., Friedrichs K.O., (1948) Supersonic Flow and Shock Waves. Springer-Verlag, New York

    MATH  Google Scholar 

  11. Dafermos C.M., (2005) Hyperbolic Conservation Laws in Continuum Physics 2nd Ed. Springer-Verlag, Berlin

    MATH  Google Scholar 

  12. Dauge M.: Elliptic Boundary Value Problems in Corner Domains–Smoothness and Asymptotics of Solutions. Lecture Notes in Math. 1341, Springer-Verlag, Berlin, 1988

  13. Dong G.: Nonlinear Partial Differential Equations of Second Order. Transl. Math. Monographs, 95, AMS. Providence, RI, 1991

  14. Embid P.,Goodman J., Majda A. (1984) Multiple steady state for 1-D transonic flow. SIAM J. Sci. Statist. Comput. 5, 21–41

    Article  MATH  MathSciNet  Google Scholar 

  15. Finn R.,Gilbarg D. (1957) Three-dimensional subsonic flows, and asymptotic estimates for elliptic partial differential equations. Acta Math. 98, 265–296

    Article  MATH  MathSciNet  Google Scholar 

  16. Fletcher C.A.J., (1991) Computational Techniques for Fluid Dynamics Vols I-II. Springer-Verlag, Berlin

    Google Scholar 

  17. Gilbarg D., Trudinger N., (1983) Elliptic Partial Differential Equations of Second Order 2nd Ed. Springer-Verlag, Berlin

    MATH  Google Scholar 

  18. Glaz H., Liu T.-P. (1984) The asymptotic analysis of wave interactions and numerical calculations of transonic nozzle flow. Adv. in Appl. Math. 5, 114–146

    Article  MathSciNet  Google Scholar 

  19. Glimm J., Majda A., (1991) Multidimensional Hyperbolic Problems and Computations. Springer-Verlag, New York

    MATH  Google Scholar 

  20. Glimm J., Marshall G., Plohr B. (1984) A generalized Riemann problem for quasi-one-dimensional gas flow. Adv. Appl. Math. 5, 1–30

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. Grisvard P., (1985) Elliptic Problems in Nonsmooth Domains Monographs and Studies in Math 24. Pitman, London

    Google Scholar 

  22. Gu C.-H. (1962) A method for solving the supersonic flow past a curved wedge (in Chinese). Fudan J. 7, 11–14

    Google Scholar 

  23. Han Q.,Lin F.: Elliptic Partial Differential Equations. Courant Lecture Notes in Mathematics 1. Courant Institute/AMS, New York/Providence, 1997

  24. John F. Nonlinear Wave Equations, Formation of Singularities. University Lecture Series 2, AMS, Providence, 1990

  25. Kikuchi M., Shibata Y. (1989) On the mixed problem for some quasilinear hyperbolic system with fully nonlinear boundary condition. J. Differential Equations 80, 154–197

    Article  MATH  MathSciNet  Google Scholar 

  26. Koch H. (1993) Mixed problems for fully nonlinear hyperbolic equations. Math. Z. 214, 9–42

    MATH  MathSciNet  Google Scholar 

  27. Li T.-T. (1980) On a free boundary problem. Chinese Ann. Math. 1, 351–358

    MATH  MathSciNet  Google Scholar 

  28. Li T.-T.,Yu W.-C.: Boundary Value Problems for Quasilinear Hyperbolic Systems. Duke University Mathematics Series, V. Durham, NC 27706, Duke University, Mathematics Department. X, 1985

  29. Lieberman G. (1986) Mixed boundary value problems for elliptic and parabolic differential equations of second order. J. Math. Anal. Appl. 113, 422–440

    Article  MATH  MathSciNet  Google Scholar 

  30. Lieberman G. (1987) Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions. Ann. Mat. Pura Appl. (4) 148, 77–99

    Article  MathSciNet  Google Scholar 

  31. Lieberman G. (1988) Oblique derivative problems in Lipschitz domains II: Discontinuous boundary data. J. Reine Angew. Math. 389, 1–21

    MATH  MathSciNet  Google Scholar 

  32. Lieberman G., (1996) Second Order Parabolic Differential Equations. World Scientific Publishing Co Inc, River Edge NJ

    MATH  Google Scholar 

  33. Lien W.-C., Liu T.-P. (1999) Nonlinear stability of a self-similar 3-dimensional gas flow. Comm. Math. Phys. 204, 525–549

    Article  MATH  ADS  MathSciNet  Google Scholar 

  34. Liu T.-P. (1983) Nonlinear stability and instability of transonic gas flow through a nozzle. Comm. Math. Phys. 83, 243–260

    Article  ADS  Google Scholar 

  35. Liu T.-P. (1987) Nonlinear resonance for quasilinear hyperbolic equation. J. Math. Phys. 28, 2593–2602

    Article  MATH  ADS  MathSciNet  Google Scholar 

  36. Morawetz C.S.: On the non-existence of continuous transonic flows past profiles I–III. Comm. Pure Appl. Math. 9, 45–68 (1956); 10, 107–131 (1957); 11, 129–144 (1958)

    Google Scholar 

  37. Morawetz C.S. (1985) On a weak solution for a transonic flow problem. Comm. Pure Appl. Math. 38, 797–817

    MATH  MathSciNet  Google Scholar 

  38. Schaeffer D.G. (1976) Supersonic flow past a nearly straight wedge. Duke Math. J. 43, 637–670

    Article  MATH  MathSciNet  Google Scholar 

  39. Shiffman M. (1952) On the existence of subsonic flows of a compressible fluid. J. Ration. Mech. Anal. 1, 605–652

    MathSciNet  Google Scholar 

  40. Whitham G.B., (1974) Linear and Nonlinear Waves. John Wiley & Sons, New York

    MATH  Google Scholar 

  41. Xin Z., Yin H. (2005) Transonic shock in a nozzle: two-dimensional case. Comm. Pure Appl. Math. 58, 999–1050

    Article  MATH  MathSciNet  Google Scholar 

  42. Zhang Y. (1999) Global existence of steady supersonic potential flow past a curved wedge with a piecewise smooth boundary. SIAM J. Math. Anal. 31, 166–183

    Article  MATH  MathSciNet  Google Scholar 

  43. Zheng Y. (2003) A global solution to a two-dimensional Riemann problem involving shocks as free boundaries. Acta Math. Appl. Sinica (English Ser.) 19, 559–572

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL, 60208-2730, USA

    Gui-Qiang Chen

  2. Department of Mathematics, University of Wisconsin, 480 Lincoln Dr, Madison, WI, 53706-1388, USA

    Mikhail Feldman

Authors
  1. Gui-Qiang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mikhail Feldman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gui-Qiang Chen.

Additional information

Communicated by C.M. Dafermos

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, GQ., Feldman, M. Existence and Stability of Multidimensional Transonic Flows through an Infinite Nozzle of Arbitrary Cross-Sections. Arch Rational Mech Anal 184, 185–242 (2007). https://doi.org/10.1007/s00205-006-0025-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-006-0025-5

Keywords

Search

Navigation