R. von Mises - Mathematical theory of compressible fluid flow, страница 100

THOMAS, " A theory of the stability of shock w a v e s " , Proc. FirstMidwesternCon}. Fluid Dynamics, Urbana, Illinois (1950), pp. 109-120. His work is supported bythe plausible arguments of H . RICHTER, " D i e Stabilitat des Verdichtungsstosses ineine konkaven E c k e " , Z. angew. Math. Mech. 28 (1948), pp. 341-345.Article 2331. In the approximation method discussed in N o t e 24 most interest centers onthe pressure change p — pi (cf.

end of Sec. 3 ) . T h e third order terms in this case werefirst computed by Busemann, but his results are incorrect (see] for example [27], p.391). T h e corresponding theorem for the one-dimensional case is indicated in N o t e111.50.232. Compare N o t e 30.33. See for example the section "Supersonic flows with shock w a v e s " by A . FERRIin [31].34.

T h e analysis of this section is easily extended to a polygonal profile, see P . S.EPSTEIN, " O n the air resistance of projectiles", Proc. Natl. Acad. Sci. U.S. 17 (1931),pp. 532-547.35. See N o t e 33. An approximation to this flow pattern was given by M . J. LIGHT­HILL, " T h e conditions behind the trailing edge of a supersonic aerofoil", ARC Repts.& Mem.

1930 (1944). He found that the shocks A Si , BS2 continue as parabolas, asalready noticed by A . BUSEMANN, cit. N o t e 27. Errors in Lighthill's paper were latercorrected in his article cited in N o t e 24.36. On account of its shape (see F i g . 157) the graph of p^lpi versus δ is called a" h e a r t c u r v e " ( H e r z k u r v e ) . T h e family of such curves, obtained by varying the in­cident Mach number M is sketched in [27], p. 370, for example. Such a diagram is use­ful in problems involving a condition on the pressure (see also the end of Sec. 6 ) .237.

A . K A H A N E and L. LEES [ " T h e flow at the rear of a two-dimensional super­sonic a i r f o i l " , J. Aeronaut. Sci. 15 (1948), pp. 167-170] have shown that the differenceis actually of the fourth order: δ — δ = Κδ-f 0 ( δ ) , where Κ depends only on Mand is explicitly given. However, the approximation δ — δ = Κδis apparently oflimited application. For example, in the case considered next it gives δ — δο = 4',twice the correct value (cf. also the example on p. 122 of A . F E R R I , cit. N o t e I I .

26).04005004038. Here we treat only the so-called regular reflection. For a summary of theo­retical work, see H . POLACHEK and R. J . SEEGER, " O n shock wave phenomena: inter­action of shock waves in gases", Proc. Symp. Appl. Math. (A.M.S.)1 (1949), pp. 119144. An interesting expository article, in which theory and experiment are compared,has been given by W . B L E A K N E Y and A . H . T A U B , "Interaction of shock w a v e s " ,Revs. Modern Phys. 21 (1949), pp. 584-605. M o r e recent developments are summarizedin W . B L E A K N E Y , " R e v i e w of significant observations on the Mach reflection of shockw a v e s " , Proc.

Symp. Appl. Math. (A.M.S.)5 (1954), pp. 41-^7. For references onindependent German work see F. W E C K E N , "Stosswellenverzweigung bei Reflexion",Z. angew. Math. Mech., 28 (1948), pp. 338-341.39. Such an intersection of shocks can arise physically when a uniform supersonicstream is incident on two wedges in suitable neighboring positions. For photographssee [24], p. 139. A second type of intersection occurs when two shocks converge onone another from the same side of a uniform stream. T h e interaction of shock waveswas first considered by E. Mach in a series of papers (all in same journal) startingwith: E.

M A C H and J . W O S Y K A , " U b e r einige mechanische Wirkungen des elektrischenFunkens", Sitzber. Akad. Wiss. Wien, A b t . 11,72 (1876), pp. 44-52. Such problems, and496NOTES A N D ADDENDAArticle 23others, are discussed in the papers cited in the preceding note. See also F. W E C K E N ," G r e n z l a g e n gegabelter Verdichtungsstosse", Z. angew.Math.Mech.,29 ( 1 9 4 9 ) , p p .147-155.Article 2440.

This equation was first obtained (for the special case of strictly adiabatic mo­tion of a perfect gas, with Η = const.) b y L . CROCCO, " E i n e neue Stromfunktion furdie Erforschung der Bewegung der Gase mit R o t a t i o n " , Z. angew. Math. Mech. 17(1937), p p . 1-7. F o r an extensive discussion and bibliography of the material in thissection see C. A .

TRUESDELL, cit. N o t e I I . 1 .41. A flow for which Η = const, throughout is called homenergetic b y some au­thors (e.g. [24], p. 6 3 ) and isoenergetic b y others (e.g. [27], p. 2 0 1 ) . H o w e v e r , boththese terms can be misleading since gS is not the total energy per unit mass (cf. Sec. 2.2and end of Sec. 2.5, where it is pointed out that Ρ is not an energy).42. This result is only slightly weaker than the corresponding one for an elasticfluid (see Sec. 6.5) since a flow which satisfies the first alternative is necessarily helicoidal (if gravity is neglected), i.e., for suitable coordinates x, y, ζ the velocity po­tential has the form: φ = az + b arc tan y/x where a and b are constants.

Thisfollows from a paper by G. H A M E L , "Potentialstromungen mit konstanter Geschwind i g k e i t " , Sitzber.Preuss.Akad.Wiss.(1937), pp. 5-20.43. This was first discovered by J . HADAMARD, see [ 4 ] , p p . 362-369.44. C i t . N o t e 40. A similar result holds for axially symmetric flows (cf. Sec.1 6 . 2 ) ; the left members of Eqs. ( 5 ) and the right member of E q . ( 6 ) are then multi­plied b y y. Consequently a term —θψ/ydy is added t o the left member of E q . ( 9 ) ,and the right member is multiplied b y y . Corresponding changes must then be madein the equations which follow.245.

Compare Sec. 22.3 ( d ) . This qualitative statement concerning the effect ofentropy variation behind the shock has been examined b y C. A . TRUESDELL," T w o measures of v o r t i c i t y " , J. RationalMech.Anal.2 ( 1 9 5 3 ) , p p . 173-217.46. Μ . M U N K and R .

C. P R I M , " O n the multiplicity of steady gas flows having thesame streamline p a t t e r n " , Proc.Natl.Acad.Sci. U.S. 33 ( 1 9 4 7 ) , p p . 137-141. Theseauthors developed the principle for three-dimensional flow of a perfect gas (showingalso that it applied to flows containing shocks). T h e same principle is implied in apaper of B . L . H I C K S , P . E . GUENTHER and R . H . WASSERMAN, " N e w formulations ofthe equations for compressibleflow",Quart.Appl.Math.5 (1947), pp.

357-361.also R . C. P R I M , " S t e a d y rotational flow of ideal gases", J. RationalMech.SeeAnal.1( 1 9 5 2 ) , p p . 425-497.47. C i t . N o t e 40. Crocco also discussed the corresponding axially symmetriccase (see N o t e 4 4 ) .48. See Μ . H . M A R T I N , " S t e a d y , rotational, plane flow of a g a s " , Am. J. Math.72 (1950), pp.

465-484. T h e method has been exploited, for example, in A . G. HANSENand Μ . H . M A R T I N , "Some geometrical properties of plane flows", Proc.CambridgePhil. Soc. 47 (1951) p p . 763-776. F o r a third formulation of the equations of motion,which is not restricted t o two-dimensional cases, see the last t w o papers in N o t e 46.49. T h e next t w o sections form a development of ideas expressed by R . v . M I S E Sin the paper quoted in N o t e 1.6, and worked out in detail b y G. S.

S. LUDFORD, " T h eboundary layer natureof shock transition in a real fluid", Quart. Appl. Math. 10 (1952),pp. 1-16. F o r more information concerning asymptotic phenomena see N o t e 111.40.50. For an example in steady plane flow, see H . C. L E V E Y , " T w o dimensional sourceflow of a viscous fluid", Quart. Appl. Math 12 (1954), p p . 25-48. Other exact solu­tions of the equations of one-dimensional nonsteady viscous flow have been obtainedCHAPTERV497Article 24by K .

BECHERT, " E b e n e Wellen in idealen Gasen mit Reibung und Warmeleitung",Ann. Physik, Ser. 5, 40 (1941), pp. 207-248. Unfortunately none of these solutionsyields a shock in the limit μ —• 0. T h e situation is better for the boundary layer( N o t e 111.40), see §§ 42, 43 of S. GOLDSTEIN ( e d i t o r ) , Modern Developments in FluidDynamics, Vol. I, London and N e w Y o r k : Oxford U n i v . Press, 1938, and G . K U E R T I ,"Boundary layer in convergent flow between spiral w a l l s " ,Math, and Physics, 30(1951), pp. 106-115.0Article 2551.