A hodograph approach to the rotational compressible flow of an ideal fluid
Author:
Gianfranco Chiocchia
Journal:
Quart. Appl. Math. 47 (1989), 513-528
MSC:
Primary 76M05; Secondary 76H05, 76N10
DOI:
https://doi.org/10.1090/qam/1012274
MathSciNet review:
MR1012274
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Abstract: The classical hodograph equation for the study of the irrotational compressible flow of an ideal gas is here extended to the rotational case in which an entropy gradient normal to the streamlines is present. The formulation is in terms of Crocco’s stream function and leads to an equation differing from that valid for the irrotational case because of an additional nonlinear te[ill]m. In spite of this nonlinearity an exact solution can be found, describing a class of rotational vortices. Moreover, perturbative solutions can be found for the case in which the entropy gradient and the vorticity are small: one of these is here presented and describes the superposition of a constant vorticity distribution to the transonic region of the Ringleb flow.
L. Crocco, Una nuova funzione di corrente per lo studio del moto rotazionale dei gas, Rendiconti R. Acc. Naz. Lincei XXIII (2), (1936)
C. Ferrari and F. Tricomi, Transonic Aerodynamics, Academic Press, New York, and London, 1968
- G. I. Taylor, Recent Work on the Flow of Compressible Fluids, J. London Math. Soc. 5 (1930), no. 3, 224–240. MR 1574079, DOI https://doi.org/10.1112/jlms/s1-5.3.224
- Friedrich Ringleb, Exakte Lösungen der Differentialgleichungen einer adiabatischen Gasströmung, Z. Angew. Math. Mech. 20 (1940), 185–198 (German). MR 3140, DOI https://doi.org/10.1002/zamm.19400200402
- Antony Jameson, Iterative solution of transonic flows over airfoils and wings, including flows at Mach $1$, Comm. Pure Appl. Math. 27 (1974), 283–309. MR 375942, DOI https://doi.org/10.1002/cpa.3160270302
- P. Germain and R. Bader, Solutions élémentaires de certaines équations aux dérivées partielles du type mixte, Bull. Soc. Math. France 81 (1953), 145–174 (French). MR 58834
- G. Chiocchia and B. Gabutti, A new transformation for computing hypergeometric series and the exact evaluation of the transonic adiabatic flow over a smooth bump, Comput. & Fluids 17 (1989), no. 1, 13–23. Special issue in honour of Gino Moretti on the occasion of his 70th birthday (Farmingdale, NY, 1987). MR 977559, DOI https://doi.org/10.1016/0045-7930%2889%2990004-2
- Graham F. Carey (ed.), Special issue on inverse design concepts in engineering sciences, John Wiley & Sons, Ltd., Chichester, 1986. Papers from the international conference held at the University of Texas, Austin, Tex., October 1984; Internat. J. Numer. Methods Engrg. 22 (1986), no. 2. MR 829547
L. Crocco, Una nuova funzione di corrente per lo studio del moto rotazionale dei gas, Rendiconti R. Acc. Naz. Lincei XXIII (2), (1936)
C. Ferrari and F. Tricomi, Transonic Aerodynamics, Academic Press, New York, and London, 1968
G. I. Taylor, Recent work on the flow of compressible fluids, J. London Math. Soc. 5, 224 (1930)
F. Ringleb, Exakte Lösungen der Differentialgleichungen einer adiabatischen Gasströmung, Z. Angew. Math. Mech. 20 4, 185–198 (1940)
A. Jameson, Iterative solution of transonic flows over airfoils and wings, including flows at Mach 1, Comm. Pure Appl. Math. XXVII, 283–209 (1974)
P. Germain and R. Bader, Solutions élementaires de certaines équations aux derivées partielles du type mixte, Bull. Soc. Math. de France 81, 145–174 (1953)
G. Chiocchia and B. Gabutti, A new transformation for computing hypergeometric series and the exact evaluation of the transonic adiabatic flow over a smooth bump, Comput. & Fluids 17 1, 13–23 (1989)
L. Zannetti, A natural formulation for the solution of two-dimensional or axisymmetric inverse problems, Int. J. Num. Math. Eng. 22, 451–453 (1986)
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Article copyright:
© Copyright 1989
American Mathematical Society
