R. von Mises - Mathematical theory of compressible fluid flow, страница 101

Τ . M . CHERRY, " F l o w of a compressible fluid about a c y l i n d e r " , Proc. Roy.Soc. A192 (1947), pp. 45-79. A textbook presentation of Cherry's work may be foundin K u o ' s article in [31], p. 521 ff. In this article, p. 529 ff., an account of Tsien andK u o ' s own work, cit. N o t e 7, is given.52. Τ . M . CHERRY, " N u m e r i c a l solutions for transonic flow", Proc. Roy.

Soc. A196(1949), pp. 32-36. H . C. L E V E Y , using Cherry and Lighthill's method investigated" H i g h s p e e d flow of gas past an approximately elliptic c y l i n d e r " , Proc.CambridgePhil. Soc. 46 (1950), pp. 479-491.Tables adapted to Lighthill and Cherry's method include the previously men­tioned ones by D . F.

FERGUSON and M . J. LIGHTHILL, N o t e IV.54, thoseby V . HUCKEL, N o t e IV.49 and also Τ . M . CHERRY, " T a b l e s and approximate formu­lae for hypergeometric functions of high order, occurring in gas flow t h e o r y " , Proc.Roy. Soc. A217 (1953), pp. 222-234.W e mention, in addition, the following interesting paper which for reasons ofspace has not been discussed in our t e x t : Τ . M . CHERRY, " A transformation of thehodograph equation and the determination of certain fluid m o t i o n s " , Phil.Trans.Roy.

Soc. London A245, (1953), pp. 583-624. T h e method described is applied to uni­form flow past cylinders as well as to channel flow.53. Τ . M . CHERRY, " F l o w of a compressible fluid about a cylinder. I I . Flow withcirculation", Proc. Roy. Soc. A196 (1949), pp. 1-31. See M . J. LIGHTHILL, cit. N o t e 1,and M . J. LIGHTHILL, " O n the hodograph transformation for high speed flow. I I .

Aflow with circulation", Quart. J. Mech. Appl. Math. 1 (1948), pp. 442-450.54. G . I . T A Y L O R , " T h e flow of air at high speeds past curved surfaces", ARCRepts. & Mem. 1381 (1930). See also papers quoted in N o t e 11.17, and N o t e 11.21. T h ehydraulic treatment for both types of flow is worked out, for example in [29] and[32]. N o t e that there is also an everywhere supersonic symmetric type of channel flowwith supersonic (minimum) speed at the throat and increasing velocities towards theleft and the right.55.

See the dissertation of T . M E Y E R , cit. N o t e IV.25. H . GORTLER, " Z u m Dbergang von Unterschall- zu Uberschallgeschwindigkeiten in D u s e n " , Z. angew. Math.Mech. 19 (1939), pp. 325-337, investigates the possibility of a transition from thesymmetric " T a y l o r - t y p e " flow to the nonsymmetric " M e y e r - t y p e " flow.56. See K . O. FRIEDRICHS, " T h e o r e t i c a l studies on the flow through nozzles andrelated problems", ΝDRCAppl. Math. Rept.

No. S2.1R (1944), M . J. LIGHTHILL, cit.N o t e IV.38, and Τ . M . CHERRY, " E x a c t solutions for flow of a perfect gas in a t w o dimensional L a v a l n o z z l e " , Proc. Roy. Soc. A203 (1950), pp. 551-571; see [31], pp. 532ff.See also N o t e 62 on the work of T o m o t i k a and Tamada. (Our Fig. 174 is essentiallythe same as Lighthill's Fig. 1 in the paper quoted above; also Fig. 2 of Cherry's paperhas been used.)57. W e mention also recent work of S. BERGMAN, " O n representation of streamfunctions of subsonic and supersonic flows of compressible fluids", J. RationalMech.Anal. 4 (1955), pp.

883-905, where he gives explicit formulas for subsonic flows in aregion bounded by segments of straight lines and free boundaries. T h e method may498NOTES A N DADDENDAArticle 25be considered as a counterpart of that for the Schwarz-Christoffel problem and it hasto overcome difficulties typical of that problem.58. T h e following five sections, which conclude the book, are inspired by R . v.M I S E S , "Discussion on transonic flow", Communs. Pure Appl.

Math. 7 (1954), pp.145-148.59. F. G. TRICOMI, "Sulle equazioni lineari alle derivate parziali di 2° ordine, ditipo m i s t o " , Atti accad. nazl. Lincei, Mem. Classe sci.fis. mat. e nat., Ser. 5, 14 (1923),pp. 133-217 [translation: Grad. Div. Appl. Math., Brown Univ., Trans.

A9-T-26 (1948)].See also F. G. TRICOMI, cit. N o t e 11.27, particularly pp. 387-478.60. See R . v. MISES, cit. N o t e 58. Compare the valuable article by L. BERS, " R e ­sults and conjectures in the mathematical theory of subsonic and transonic gas flow",Communs. Pure Appl. Math. 7 (1954), pp. 79-104. See also the less mathematical re­view article by W . R . SEARS, "Transonic potential flow of a compressible fluid", J.Appl.

Phys. 21 (1950), pp. 771-778.61. L. BERS, "Existence and uniqueness of a subsonic compressible flow past agiven profile", Communs. Pure Appl. Math. 7 (1954), pp. 441-504. M . SHIFFMAN, " O nthe existence of subsonic flows of a compressible fluid", J. Rational Mech. Anal. 1(1952), pp. 605-652. Both papers are very technical. Comments on the existence proofsfor the subsonic problem may be found in L. BERS, cit. N o t e 60. Bers' existence prooffor the Chaplygin-Kdrman-Tsien gas (see Sees.

17.5,6) is quoted in N o t e IV.24. Com­pare also D . GILBARG, "Comparison methods in the theory of subsonic flows", J.Rational Mech. Anal. 2 (1953), pp. 233-251, and R. F I N N and D . GILBARG cit. N o t eIV.44, where a uniqueness theorem is proved (with respect to all other flows, eithersubsonic or mixed) in a more elementary way and under slightly weaker conditionsthan in Bers* paper.In our text we assume a smooth profile and zero circulation.

If the otherwisesmooth profile has a protruding corner Τ the subsonic flow is uniquely determinedby its free-stream velocity if at Τ the "Kutta-Joukowski c o n d i t i o n " holds, (thisbeing equivalent to knowledge of the circulation). Compare the papers by Bers andby Finn and Gilbarg.62. T h e papers by S.

TOMOTIKA and K . TAMADA are: "Studies on two-dimensionaltransonic flows of compressible fluid—Part I " , Quart. Appl. Math. 7 (1950), pp. 381397; also Part I I , ibid. 8 (1950), pp. 127-136, and P a r t I I I , ibid. 9 (1951), pp. 129-147.T h e authors apply their method also to transonic channel flow. Compare the pre­sentation in K u o ' s article in [31], p. 540 ff.

(channel flow) and p. 546 ff. (flow past aprofile).63. G. I . TAYLOR, cit. N o t e 54.64. T h e linearized method for subsonic flow is due to L. PRANDTL, " U b e r Stromungen, deren Geschwindigkeit mit der Schallgeschwindigkeit vergleichbar s i n d " ,J. Aero. Research Inst. Univ. Tokyo 6 (1930), p.

14 ff; H . GLAUERT, " T h e effect ofcompressibility on the lift of an a i r f o i l " , Proc. Roy. Soc. A118 (1928), pp. 113-119;J. ACKERET, "Uber Luftkrafte bei sehr grossen Geschwindigkeiten, insbesondere beiebenen Stromungen", Helvetica Physica Acta, 1 (1928), pp. 301-322. (See also Notes1.20 and V.24.)T h e iteration method proposed by L. PRANDTL ["Allgemeine Uberlegungen iiberdie Stromung zusammendruckbarer Flussigkeiten", FondazioneAllessandroVolta,Atti dei Convegni 5 Roma (1935), pp. 169-197 (reprinted without the appendix, in Z.angew.

Math. Mech. 16 (1936), pp. 129-142)] has been applied to transonic problemsby H . Gortler, who computed flow past a w a v y wall, and by C. K a p l a n : H . GORTLER,"Gasstromungen mit Ubergang von Unterschall- zu Uberschallgeschwindigkeiten",Z. angew. Math. Mech. 20 (1940), pp. 254-262, C. K A P L A N , " T h e flow of a compressibleCHAPTERV499Article 25fluid past a curved surface", Ν AC A Tech. Report 768 (1943). T h e convergence of thepertinent series has not been proved.65. H . W .

EMMONS, " F l o w of a compressible fluid past a symmetrical airfoil in awind tunnel and in free a i r " , Ν AC A Tech. Notes 1746 (1948); and regarding channelflow: H . W . EMMONS, " T h e theoretical flow of a frictionless, adiabatic, perfect gasinside of a two-dimensional hyperbolic n o z z l e " , Ν AC A Tech. Notes 1003 (1946).66. Compare, for example, statements in the paper by SEARS, cit. N o t e 60. On theother hand, experiments have been reported which, within the limits of observation,show no evidence of shocks: e. g., H . W . LIEPMANN, H . ASCHKENAS, and J.

D . COLE,"Experiments on Transonic F l o w " , Contract W 33-038 ac 1717 (11592), GuggenheimAeronaut. Lab., California Inst. Technol. (1947).67. T . v. K A R M A N , H . S. T S I E N , and H . S. TSIEN and Υ . H . K u o , all cit. N o t eIV.30. Similar ideas are expressed by M . J. LIGHTHILL in [24], p.

251. T h e idea of link­ing the appearance of shocks with a mathematical breakdown appears actually alsoin other forms. As one example compare C. Kaplan, cit. N o t e 64, who suggests thatit is reasonable to assume that the value of Μfor which his expansion of q in powersof a given parameter starts diverging "marks the limit of irrotational potential flowand also probably indicates the first appearance of a compression shock at the solidboundary."0068. K . O.

FRIEDRICHS and D . A . FLANDERS, " O n the non-occurrence of a limitingline in transonic flow", Communs. Pure Appl. Math. 1 (1948), pp. 287-301. See alsoH . S. TSIEN'S review of this article: Appl. Mechanics Revs. 3 (1950), N o . 753, andthe ensuing controversy.69. A .

A . N I K O L S K I I and G. I . TAGANOV, " G a s motion in a local supersonic regionand conditions of potential-flow breakdown", Prikl. Mat. Meh. 10 (1946), pp. 481502 [translation: Ν AC A Tech. Mem. 1213 (1949)].70. A . R . M A N W E L L , " A note on the hodograph transformation", Quart.Appl.Math.