Abstract
In this paper, we study the well-posedness problem on transonic shocks for steady ideal compressible flows through a two-dimensional slowly varying nozzle with an appropriately given pressure at the exit of the nozzle. This is motivated by the following transonic phenomena in a de Laval nozzle. Given an appropriately large receiver pressure P r , if the upstream flow remains supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the flow is compressed and slowed down to subsonic speed, and the position and the strength of the shock front are automatically adjusted so that the end pressure at exit becomes P r , as clearly stated by Courant and Friedrichs [Supersonic flow and shock waves, Interscience Publishers, New York, 1948 (see section 143 and 147)]. The transonic shock front is a free boundary dividing two regions of C2,α flow in the nozzle. The full Euler system is hyperbolic upstream where the flow is supersonic, and coupled hyperbolic-elliptic in the downstream region Ω+ of the nozzle where the flow is subsonic. Based on Bernoulli’s law, we can reformulate the problem by decomposing the 3 × 3 Euler system into a weakly coupled second order elliptic equation for the density ρ with mixed boundary conditions, a 2 × 2 first order system on u2 with a value given at a point, and an algebraic equation on (ρ, u1, u2) along a streamline. In terms of this reformulation, we can show the uniqueness of such a transonic shock solution if it exists and the shock front goes through a fixed point. Furthermore, we prove that there is no such transonic shock solution for a class of nozzles with some large pressure given at the exit.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alt H.W., Caffarelli L.A., Friedman A.: Compressible flows of jets and cavities. J. Differ. Eqs. 56, 82–141 (1985)
Azzam A.: On Dirichlet’s problem for elliptic equations in sectionally smooth n-dimensional domains. SIAM. J. Math. Anal. 11(2), 248–253 (1980)
Azzam A.: Smoothness properties of mixed boundary value problems for elliptic equations in sectionally smooth n-dimensional domains. Ann. Polon. Math. 40, 81–93 (1981)
Bers, L.: Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Surveys in Applied Mathematics, vol. 3. Wiley, New York; Chapman and Hall, London, 1958
Canic S., Keyfitz B.L., Lieberman G.M.: A proof of existence of perturbed steady transonic shocks via a free boundary problem. Comm. Pure Appl. Math. LIII, 484–511 (2000)
Chen G., Feldman M.: Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type. J.A.M.S. 16(3), 461–494 (2003)
Chen S.: Stability on transonic shock fronts in two-dimensional Euler systems. Trans. Am. Math. Soc. 357(1), 287–308 (2005)
Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Interscience Publishers, New York, 1948
Fletcher, C.A.J.: Computational Techniques for Fluid Dynamics, vol. I–II. Springer, Berlin, 1991
Friedman A., Vogel T.I.: Cavitational flow in a channel with oscillatory wall. Nonlinear Anal. 7, 1175–1192 (1983)
Gilbarg D.: Hörmander, L.: Intermediate Schauder estimates. Arch. Rational Mech. Anal. 74(4), 297–318 (1980)
Gilbarg, D., Tudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Grundlehren der Mathematischen Wissenschaften, vol. 224. Springer, Berlin, 1983
John, F.: Nonlinear Wave Equations, Formation of Singularities. University Lecture Series, 2. American Mathematical Society, Providence, 1990
Li, J.,Xin, Z.,Yin, H.: On transonic shocks in a nozzle with variable pressure. Comm. Math. Phys. (2009, to appear)
Li, J., Xin, Z., Yin, H.: The uniqueness of a three-dimensional transonic shock in a curved nozzle with variable end pressure, Preprint, 2007
Lieberman G.M.: Mixed boundary value problems for elliptic and parabolic differential equation of second order. J. Math. Anal. Appl. 113(2), 422–440 (1986)
Lieberman G.M.: Oblique derivative problems in Lipschitz domains II. J. Reine Angew. Math. 389, 1–21 (1988)
Luping H., Taiping L.: A conservative, piecewise-steady difference scheme for transonic nozzle flow, Hyperbolic partial differential equations, III. Comput. Math. Appl. Part A 12(4–5), 377–388 (1986)
Majda, A.: One Perspective on Open Problems in Multidimensional Conservation Laws. Multidimensional hyperbolic problems and computation, IMA 29, pp. 217–237. Springer, Berlin, 1990
Morawetz C.S.: Potential theory for regular and Mach reflection of a shock at a wedge. Comm. Pure Appl. Math. 47, 593–624 (1994)
Morawetz, C.S.: On the nonexistence of continuous transonic flows past profiles, I. Comm. Pure Appl. Math. 9, 45–68 (1956); III 11, 129–144 (1958); 17, 357–367 (1964); 10, 107–132 (1975)
Nazarov, S.A., Plamenevsky, B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries, De Gruyter Expositions in Mathematics, Walter de Gruyter, Berlin, New York, 1994
Xin, Z., Yin, H.: Transonic shock in a nozzle I, two-dimensional case. Comm. Pure Appl. Math., LVIII, 999–1050 (2005)
Xin Z., Yin H.: Transonic shock in a curved nozzle, 2-D and 3-D complete Euler systems. J. Differ. Equ. 245(4), 1014–1085 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Bressan
This research was supported in part by the Zheng Ge Ru Foundation when Yin Huicheng was visiting The Institute of Mathematical Sciences, The Chinese University of Hong Kong. Xin is supported in part by Hong Kong RGC Earmarked Research Grants CUHK-4028/04P, CUHK-4040/06P, and Central Allocation Grant CA05-06.SC01. Yin is supported in part by NNSF of China and Doctoral Program of NEM of China.
Rights and permissions
About this article
Cite this article
Xin, Z., Yan, W. & Yin, H. Transonic Shock Problem for the Euler System in a Nozzle. Arch Rational Mech Anal 194, 1–47 (2009). https://doi.org/10.1007/s00205-009-0251-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-009-0251-8