R. von Mises - Mathematical theory of compressible fluid flow, страница 98

Sci. Fennicae 19 (1893), pp. 3-31 [Eq. (11), p. 13]. T h e logarithmic term is multi­plied by τ~ and since η is negative, the product tends to zero as r —> 0 and the expan­sion still tends to 1. L i n d e l o f s formula in explicit form and with notation adaptedto the aerodynamic problem is given, including tables, in I . E. GARRICK and C.K A P L A N , " O n the flow of a compressible fluid by the hodograph method I I — F u n d a ­mental set of particular flow solutions of the Chaplygin differential e q u a t i o n " ,Ν AC A Kept.

No. 790 (1944). See also V . HUCKEL, " T a b l e s of hypergeometric func­tions for use in compressible-flow t h e o r y " , Ν AC A Tech. Notes 1716 (1948).In a useful N o t e V . O ' B R I E N ["Remarks on Chaplygin Functions", J. Aeronaut.Sci. 23 (1956), pp. 894-895; also N O r d 7386 The Johns Hopkins University, Appl.Phys.Lab., CM-871 (1956), pp. 1-6] points out that those n, for which not only c but also aor b is a negative integer, yield special cases for the logarithmic solution, and shegives the formula which in this case replaces Garrick and Kaplan's formula. T h e firstvalues of η for which this happens are η = —2, —5, —12.

T h e entries correspondington = —2, —5, —12 in the tables of V. Huckel are substantially wrong, as shown byMiss O'Brien. ( T h e solution for η = - 2 , a = 270°, given by H. J. Davies, see N o t e 50,is correct and independent of these shortcomings.)50. For α = 270° we have, for κ = y = %: η = —2, a = H,b = —5, c = — 1; thusnot only c but also 6 is a negative integer (cf. preceding N o t e ) .

H . J. Davies, usingearlier work by Temple and Yarwood and by Lighthill, has computed this flow and2η490NOTES A N D A D D E N D AArticle 20has tabulated the hypergeometric functions required in the computation. See H . J .D A VIES, " T h e two-dimensional irrotational flow of a compressible fluid around ac o r n e r " , Quart.J. Mech.Appl.Math.6 ( 1 9 5 3 ) , pp. 7 1 - 8 0 . T h e paper b y T e m p l e andY a r w o o d which was used b y Davies is quoted in N o t e 30. T h e expansion for F incase c — — 1 (n = — 2 ) is given by M .

J . LIGHTHILL, " T h e hodograph transformationin trans-sonic flow. I I . Auxiliary theorems on the hypergeometric functions ^ ( r ) " ,nProc.Roy. Soc. A 1 9 1 ( 1 9 4 7 ) , p p . 3 4 1 - 3 5 1 .51. See H . K R A F T and C. G . D I B B L E , " S o m e two-dimensional adiabatic compres­sible flow p a t t e r n s " , J. Aeronaut.Sci. 1 1 ( 1 9 4 4 ) , p p .

2 8 3 - 2 9 8 .52. See C. C. CHANG and V . O ' B R I E N , "Some exact solutions of two-dimensionalflows of compressible fluid with hodograph m e t h o d " , Ν AC A Tech. Notes 2886 ( 1 9 5 3 ) .53. J . W . CRAGGS, " T h e compressible flow corresponding t o a lineQuart.Appl.Math.doublet",10 ( 1 9 5 2 ) , p p .

8 8 - 9 5 .54. See Chaplygin's Memoir cit. N o t e 5, Part I I I , p. 50 ff. H e gives the theory andsolves various problems related t o his j e t . Figure 137 is based on values b y D . F.FERGUSON a n d M . J. LIGHTHILL, " T h e hodograph transformation in trans-sonic flow.I V . T a b l e s " , Proc. Roy. Soc. A192 (1947), pp. 135-142; i t would, however, differ verylittle if Chaplygin's values were used.55. For orientation see the section b y A . FERRI in [31], p. 700 ff.

See also F. I .F R A N K L , " T h e flow of a supersonic j e t from a vessel with plane w a l l s " , Doklady Akad.NaukS.S.S.R.58 ( 1 9 4 7 ) , pp. 381-384 [Grad.Div. Appl.Math.,BrownUniv.,Trans.A 9 - T - 3 2 ( 1 9 4 9 ) ] , D . C. PACK, " O n the formation of shock waves in supersonic gasj e t s " , Quart.J. Mech.Appl.Math.1 ( 1 9 4 8 ) , pp. 1-17, and the work b y G . B I R K H O F Fand Ε. H . ZARANTONELLO, Jets, Wakes, and Cavities, N e w Y o r k : Academic Press, 1957.CHAPTER VArticle 211.

M . J . LIGHTHILL, " T h e hodograph transformation in trans-sonic flow. I I I .Flow around a b o d y " . Proc. Roy. Soc. A 1 9 1 (1947), p p . 352-369, and his presentationin [24], Chapter V I I , §§ 4, 5, 7, 10. F o r the properties of the hypergeometric functioncf. the literature cited in N o t e I V . 4 2 , and particularly LighthilPs paper, c i t .

N o t eI V . 5 0 . T h e Chaplygin-Cherry-Lighthill theory is also presented in D . C . P A C K , " H o d ­ograph methods in gas d y n a m i c s " , Inst, for Fluidof Maryland,Rept.Dynamicsand Appl.17 (1951/52).2. Working out further the formula for Τ we obtainΤ =κ+12Μ(1 -4Μ )23 / 2'F o r the V of Eq. ( 7 ) we haveT h e equation for the potential φ which corresponds to ( 6 ) readsΘφ2d <pθφ2—4 - —4 - 7 — = 0;ds ^ θθ ^ds722Math.,Univ.CHAPTER491VArticle 21the equation for φ* where φ* = V<p, which corresponds to (8) iswithΡ = —K+ l16MA(1 -M)23 [16 -4(3 -2κ) M2+(3 -k)M\These formulas are taken from Bergman's paper, cit. N o t e IV.44, p. 465.3.

T h e f a c t o r / ( n , n ) = e " i is adequate for our purpose. Lighthill, in the papercit. N o t e 1, as well as in his article in [24], p. 238, specifies the properties which sucha normalizing factor f(n,n)must have in the case of flow with circulation, wheree^n'i is not adequate (see Eq. (81) in [24]).

W e do not consider flow with circula­tion. F o r proof of E q . (10) see M . J. LIGHTHILL, cit. N o t e IV.50.n e4. W e present the work of S . GOLDSTEIN, M . J. LIGHTHILL, and J. W . CRAGGS, " O nthe hodograph transformation for high-speed flow. I . A flow without circulation",Quart. J. Mech.

Appl. Math., 1 (1948), pp. 344-57.5. E. W . BARNES, " A new development of the theory of the hypergeometric func­t i o n " , Proc. London Math. Soc, Ser. 2, 6 (1908), pp. 141-177. Compare W H I T T A K E R WATSON [8], §§ 14.5, 14.51. In [8], the properties of the gamma function used here mayalso be found.6. T h e poles of ψη(τ)/ψ (τι)are points z (m = 2, 3, · · · ) , — m < z < — (m — 1),where ψ (τι)= 0; the corresponding residues are quite complicated.

See Appendixto M . J. LIGHTHILL, cit. N o t e 1.ηmmΖιη7. See M . J . LIGHTHILL, cit. N o t e 1. A paper on the same subject, though not cor­rect in certain respects (see Lighthill's criticism in the Appendix to his paper), isH . S. T S I E N and Υ . H . K u o , cit. N o t e IV.30.

A presentation of this work is given inK u o ' s article in [31].8. See M . J. LIGHTHILL, cit. N o t e 1, p. 366.9. S. BERGMAN, " Z u r Theorie der Funktionen, die eine lineare partielle Differentialgleichung befriedigen. I " , Rec. math. New Ser. 2 (44) (1937), pp.

1169-1198, and" T h e approximation of functions satisfying a linear partial differential e q u a t i o n " ,Duke Math. J. 6 (1940), pp. 537-561. See also "Operatorenmethoden in der Gasd y n a m i k " , Z. angew. Math. Mech. 32 (1952), pp. 33-45, which contains a fairly com­prehensive list of references, and " N e w methods for solving boundary value prob­l e m s " , Z.

angew. Math. Mech. 36 (1956), pp. 182-191. T h e theory is also presented inBERGMAN-SCHIFFER [1].10. A proof of the continuity of the mapping of P onto Ρ is difficult to derive.However, for sufficiently small velocities and sufficiently smooth P an estimate forthe maximum deviation can be given. In adition it can be proved under certaincircumstances that II is closed for a closed P (see BERGMAN-SCHIFFER [1], pp. 151152).11.

W e obtain for F in terms of λ :000F = \~ [ao + α ! ( - λ ) 2 / 3 + « ( - λ ) '2243+ · · · ] ,« ο = He,« ι = 0.W e add the following result, see BERGMAN-SCHIFFER [1], p. 146 ff.: E q . (8) with theF of (26) multiplied by a parameter, has no eigenvalues, so that the "first boundaryvalue p r o b l e m " for this equation has a solution.12. In this and the next section we follow in general the presentation in P a r t Iof the paper R .

v . M I S E S and M . SCHIFFER, " O n Bergman's integration method in492NOTES A N DADDENDAArticle 21t w o dimensional compressible flow", Advances in Appl. Mech. 1 (1948), pp. 249-285.Most of the results reported in this paper were given in S. BERGMAN, " A formula forthe stream function of certain flows", Proc. Natl. Acad.

Set. U.S. 29 (1943), pp. 276281, after having been presented in the lectures: S. BERGMAN, " T h e hodographmethod in the theory of compressible fluids", Supplements to MISES-FRIEDRICHS [25].Tables for the G (\), introduced in Eq. (27), for the polytropic gas κ = y = 1.405,are given by S. BERGMAN and B . E P S T E I N , "Determination of a compressible fluidflow past an ovalshaped obstacle", J. Math, and Phys. 6, (1947) pp. 195-222.13.

T h e use of the variable Ζ — Λ — iθ which takes the place of Bergman's ζ —λ + ιθ, was suggested by Ludford, who contributed much towards the simplified pres­entation of Bergman's method given in our text. T h i s variable Ζ enables us to recoverthe incompressible stream function to(q,d)from the compressible ^ ( # , 0 ) as in E q .(39).

This was not attempted in Bergman's papers where the stress is on the trans­formation (36). However, in our conception of the problem this recovery is essential.nI t should be kept in mind that two different passages to the limit are to be dis­tinguished:1) q fixed, q varying, i.e., we consider the same velocity for varying degrees of com­pressibility. As q increases, the flow becomes less and less compressible. Then as q —>mmmoo: gr — •oo , Μ —> 0, τ —> 0, λ —>— o o , Λ — > log q.t2) q fixed, q varying from q to zero, or, if we consider subsonic flow, from q tozero. This is the more usual consideration of a flow pattern for some fixed degree ofcompressibility. Then as q —> 0: q — q /V6,Μ —> 0, τ —> 0, λ —>— o o , Λ = λ +(σ + log q ) — •- oo.mmttmmW e see that Μ —• Ο,τ —• 0, λ—•— o o appear in both situations.