R. von Mises - Mathematical theory of compressible fluid flow, страница 96

This transformation, applied to gas dynamics, is found in S. D . POISSON,cit. N o t e 1.23. T h e derivation may be found in textbooks on partial differential equa­tions, e.g.,H . B A T E M A N , PartialDifferentialEquationsof MathematicalN e w Y o r k : D o v e r , 1944, or [ 2 ] . See also G. H A M E L , Mechanikgart: Teubner, 1956, p.

108.der Kontinua,T h e following is of interest for us. If to an equation such as ( 5 ) ,Physics,Stutt­484NOTES A N DADDENDAArticle 17#2φa(u.v)^2φ£2φ— = 0,h c(u,v)h 26(u.v)du'dudv'dvwe apply the Legendre transformation defined by Eqs. ( 4 ) , ( 4 ' ) , ( 6 ) , we obtain theequation2i^Pa2<h\ dV _ ^\dx'dy)dya2id*pd<p\ d <p\dxdy) dxdy^ /d<p d<p\ d <p _ ^22\dxdy)dx ~C2The fixed characteristics of the first of these equations are given bya dv 22b dudv(dv\( — )\du1/i,2 = a-+ c du = 0,2(b ±.\/b2-ac) ,and the characteristics in the x,y-plane, which depend on the solution <p(x,y),given by/dx\( - )=\dy/u2a dx + 2b dx dy + c dy = 0,22-1- (b =F ^/ba2-areac).Hence/ dv\Idx\\duj 2,1\dy)\,2and this is the same orthogonality relation as in (9.15) and in (16.37).

W e realizethat this orthogonality is a general mathematical fact, which does not depend on par­ticular properties of our mechanical equations.10. W e may obtain the equation for the Legendre stream function by applying theformal Legendre transformation to the equation for the stream function, viz.

Eq.(16.21) which we write as in N o t e 1, with p = 1. W e now put d^/dx = r, d\p/dy = tand apply the contact transformation, as in Eq. (10): V(r,t)= xr + yt — \J/(x,y). B ythe rule formulated in N o t e 9, we then obtainQ* (a prr2-2r) 22* rtrt+ *tt(a p22-t ) = 0.2Here ap is a function of r + t \ the variables t = pu, —r — pv are the componentsof the flux vector pq. Hence this is a mapping not onto the hodograph plane, i.e.the q-plane, but onto the pq-plane; see SAUER [29], p.

157.11. R. SAUER, cit. N o t e III.16, investigates the (p,p)-relationsfor which E q . (21)becomes a "Darboux e q u a t i o n " , in which case a general integral depending on twoarbitrary functions is known.12. The inversion formulas of this section are given, for example, in Chaplygin'spaper, cit. N o t e 5. T h e y are also derived in detail in a paper by F. Ringleb, which weshall discuss extensively in A r t . 20: F. RINGLEB, " E x a k t e Losungen der Differentialgleichungen einer adiabatischen Gasstromung", Z. angew.

Math. Mech. 20 (1940), pp.85-198.13. Bibliographical data have been given in the Notes to A r t . 7. Compare also thetreatment by Ringleb in the paper quoted in N o t e 12. A further exact solution, notdiscussed by us, which belongs in this group has been defined and studied by W .TOLLMIEN, " Z u m Uebergang von Unter- zu Uberschallstromung", Z. angew. Math.Mech.

17 (1937), pp. 117-136.14. For Figs. 100b and 101b, the drawings (Figs. 39 and 40) in [24], Chapter V by W .G. BICKLEY, have been used in part.22CHAPTER485IVArticle 1715. See S. A . CHAPLYGIN, cit. N o t e 5, P a r t V . His method was modified, elaboratedand used in many ways by v . Kdrman and H . S.

Tsien, see T . v. K A R M A N , cit. N o t eIV.30 and H . S. T S I E N , "Two-dimensional subsonic flow of compressiblefluids",/. Aeronaut. Sci. 6 (1939) pp. 399-407.16. See S. A . CHAPLYGIN, cit. N o t e 5, p. 97 of translation.17. W e have thus seen that for a gas with (p,p)-relation:ρ = A — B/p, theChaplygin equation θ ψ/θσ + Κ(σ)θ φ/θθ= 0 reduces to the Laplace equation.F o r another particular pressure-density relation the Chaplygin equation reducesto the Tricomi equation, namely, θ ψ/θσ + σ(θ ψ/θθ )= 0. This gas is called the" T r i c o m i g a s " . Compare for example F.

TRICOMI " C o r r e n t i fluide transoniche edequazioni a derivate parziali di tipo m i s t o " , Rend. Seminar. Mat. Torino 12 (1953),pp. 37-52.222222218. I t can be shown that for this (p,q)-relationplane, viz.,_a /d*>\dx V dx)Θ+/2the potential equation in the x,y-ΘΛdy V dy)_"°'is always elliptic and is in fact the differential equation of minimal surfaces.19. Compare also F. H . CLAUSER, "Two-dimensional compressible flows havingarbitrarily specified pressure distributions for gases with gamma equal to minuso n e " , Sijmposium on Theoretical Compressible Flow, White Oak, Maryland(1949), pp.1-32.20.

C. C. L I N , " O n an extension of the von Kdrmdn-Tsien method to two-dimen­sional subsonic flows with circulation around closed profiles", Quart. Appl.Math.4 (1946), pp. 291-297, reprinted in [20]. See also L. BERS, " O n a method of constructingtwo-dimensional subsonic compressible flows around closed profiles", NACATech.Notes 969 (1945), and A . GELBART, " O n subsonic compressible flows by a method ofcorrespondence", NACATech.

Notes 1170 (1945). A simple method due to K . JAECKEL["Verallgemeinerung des Tsien'schen Verfahrens" (unpublished)] is described byLighthill in [24], p. 226.21. Compare for example L. M . MILNE-THOMPSON, TheoreticalAerodynamics,N e w Y o r k : van Nostrand, 1947, p.

128 ff. T h e original papers are: R. v. M I S E S , " Z u rTheorie des Tragflachenauftriebes, erste M i t t e i l u n g " , Z. Flugtech.Motorluftschiffahrt,8 (1917), pp. 157-163, and " z w e i t e M i t t e i l u n g " , ibid. 11 (1920), pp. 68-73 and 87-89.22. See for example [31], p. 505. T h e original investigations are by H . S. T S I E N ,cit. N o t e 15.23. See [31], p. 509, and H . S. T S I E N , cit.

N o t e 15.24. L. BERS, " A n existence theorem in two-dimensional gas d y n a m i c s " , Proc.Symp. Appl. Math. (A.M.S.)1 (1949), pp. 41-46.Article 1825. Two-dimensional simple waves were studied by L. PRANDTL, " N e u e Untersuchungen uber die stromende Bewegung der Gase und D a m p f e " . Physik.

Z. 8 (1907),pp. 23-30. T h e systematic description is due to T . M E Y E R " U b e r zweidimensionaleBewegungsvorgange in einem Gas, das mit Uberschallgeschwindigkeit s t r o m t " ,Forschungsh.Ver. deut. Ing. 62 (1908), pp. 31-67 (included in [20]). T h e name fre­quently used is Prandtl-Meyer flow. Compare also N o t e 111.31, and papers quoted inN o t e 29.26. In a simple wave the state variables are constant along straight lines; in otherwords, they depend only on the angle φ which such a line makes with a fixed direction.486NOTES A N DADDENDAArticle 18This forms a counterpart to the flows considered in A r t . 17 (vortex flow, radial flow,spiral flow) where the state variables depend only on radial distance from an origin.27.

Conical flow, i.e., a flow in which the state variables are constant on concentricrays, may be considered as a generalization of a centered simple-wave flow (see forexample [27], p. 262). If, in addition to this property, axial symmetry also prevails,then the surfaces of constant state are circular cones.

See G. I . T A Y L O R and J . W .MACOLL, cit. N o t e 11.21, and J . W . MACOLL, " T h e conical shock wave formed by a conemoving at high s p e e d " , Proc. Roy. Soc. A169 (1937), pp. 459-472.In a paper by J . H . GIESE, "Compressible flows with degenerate hodographs",Aberdeen Proving Ground, Ballistic Research Rept. 657 (1948), a systematic investigation of steady flows with degenerate hodographs is carried out. T h e author discusses two-dimensional flows with one-dimensional hodographs, or three-dimensionalflows with one- or two-dimensional hodographs. Those with one-dimensional hodographs are designated as simple waves, the others as double waves. Compare alsothe nonisentropic simple waves of Sec.

15.7.28. T h e numerical examples are given in more detail than usual in order to clarifyand illustrate the conditions under which one, t w o , or no solutions exist. A similaraim prompts our comments on Cauchy's data, p. 302. Compare also Sec. 20.4 wherewe discuss other completely different compressible flows around corners. T h e y satisfy,of course, different boundary conditions.29. T h e choice of examples in this section is similar to that in [21] p.

282 ff. A l lfigures have been newly constructed here by W . Gibson.Compare also the paper by J . H . GIESE, cit. N o t e 27, the paper by L. STEINBERG," T h e geometry of the envelope of Mach lines forming a compression w a v e " , O N RContract 562(07), Grad. D i v . Appl. M a t h . , Brown Univ., Tech. R e p t . 4 (1955), pp.1-41, and R. E. M E Y E R , " O n waves of finite amplitude in ducts.