R. von Mises - Mathematical theory of compressible fluid flow, страница 95

169-186, H . A . BETHE, " T h e theoryof shock waves for an arbitrary equation of s t a t e " , OSRD Rept. A o. 545 (1942), andH . W E Y L , "Shock waves in arbitrary fluids", Communs. Pure Appl. Math. 2 (1949),pp. 103-122.T46. This fact is often made the basis of an approximation in which the results ofArts. 12 and 13 are again used behind the shock. See for example K . O. FRIEDRICHS,"Formation and decay of shock w a v e s " , Communs. Pure Appl. Math.

1 (1948), pp.211-245, and A . F. P I L L O W , cit. N o t e 35. See also N o t e V . 24 and M . J . LIGHTHILL,cit. N o t e 3.47. For this case (14.22) is equivalent to Prandtl's relation, see N o t e V . 2 5 .48. This is referred to as the Hugoniot curve. If the state 2 were connected to thestate 1 by an inviscid adiabatic process the hyperbola would be replaced by ηΡ = 1,a curve which is asymptotic to the f- andr7 -axes and osculates the hyperbola at A.49. See H . HUGONIOT, N o t e 32, and previous N o t e .50. This remark plus the fact that the entropy change across a shock is of the thirdorder in p —pi gives a theorem similar to the one for steady plane flow in Sec. 23.1,the only changes being that 0 is replaced by u, and F is now a function of p,p, andu, with derivatives2F' = padFdphρ dFa dp1dFduΛandF =(-IaP\dFdpρ dF\dF\-- — ) + —.a dp Jdu51.

See B . R I E M A N N , N o t e 11.23, where the problem of an initial discontinuityseparating regions of uniform flow is treated. For a corrected treatment see Η . W E B E R[33], pp. 522-531. T h e two initial discontinuities in this example are equivalent tospecial cases of those treated by Riemann (reflection of the initial conditions aboutχ = 0,1). An extended treatment of Riemann's problem has been given by R. COURANTand K . O.

FRIEDRICHS, " I n t e r a c t i o n of shock and rarefaction waves in one-dimen­sional m o t i o n " , OSRD Rept. No. 1567 (1943).Article 1552. This reflection problem was treated by H . HUGONIOT, cit. N o t e 32, who alsoconsidered successive reflections from a uniformly moving piston and fixed wall inturn. M o r e recently, it has been discussed by H . PFRIEM, "Reflexionsgesetze furebene Druckwellen grosser Schwingungsweite", Forsch. Gebiete Ingenieurwesens12(1941), pp. 244-256.53. This result is due to H . HUGONIOT, cit. N o t e 32 (p.

9 4 ) , who discussed and usedthe corresponding discontinuity.54. H . v . HELMHOLTZ, cit. N o t e 11.23.55. See the section on the Lanchester-Prandtl wing theory in v. M I S E S and FRIED­RICHS [25]. A short history has been given by R. v. M I S E S in the notes to ChapterI X in [16].56. T h e occurrence of contact discontinuities in problems involving shocks wasfirst emphasized by J. v. N E U M A N N , " T h e o r y of shock w a v e s " , OSRD Rept. No. 1140(1943). See also N o t e 53.57. T h e reflection treated in Sec.

1 is equivalent to a special case of the presentproblem in which the two shocks have equal strength. A second type of interactionoccurs when the two shocks move in the same direction so that one overtakes theother. A study of interactions of shocks and rarefaction waves in one-dimentionalflows was made by R . COURANT and K . ( ) . FRIEDRICHS, cit. N o t e 51. Theory and ex-482NOTES A N DADDENDAArticle 15periment are compared in I . I .

GLASS and G. N . PATTERSON, " A theoretical and ex­perimental study of shock-tube flows", J. Aeronaut. Sci. 22 (1955), pp. 73-100.58. T h e theorem of N o t e 50 predicts that, to the second order, u = ui + u (seesimilar result in Sec. 2 3 . 6 ) . Hence in this example (ui + u )/v= 2 / 5 y/ΐ = 0.1512 isthe estimated value of x, which is good agreement considering the strengths of theshocks.22059.

T h e idea of the method presented in this section is due to R . v . Mises. I t wasworked out by G. S. S. LUDFORD, H . POLACHEK, and R . J . SEEGER, ' O n unsteadyflow of compressible viscous fluids", J. Appl. Phys. 24 (1953), pp. 490-495.60. See J . v. N E U M A N N , "Proposal and analysis of a new numerical method for thetreatment of hydrodynamic shock problems", NDRCAppl.

Math. Rept. No. 108.1R(1944). H e found that the particles acquire small oscillations, superimposed on theirtrue paths, after passing through the location of the shock, and he interpreted thisin terms of internal energy.61. See for example H . GEIRINGER " O n numerical methods in wave interactionproblems", Advances in Appl. Mech. 1 (1948) pp. 201-248, and remarks of K . O.FRIEDRICHS in [31], pp.

50-58.62. See the paper quoted in N o t e 59. A second way out, having a similar effect,is to change the viscosity law; see J . v . NEUMANN and R. D . RICHTMEYER, " A methodfor the numerical calculation of hydrodynamic shocks", J. Appl. Phys. 21 ( 1 9 5 0 ) ,pp. 232-237, and R . v . M I S E S , cit. N o t e 1.6.63. See Notes 46 and V . 4 5 .64. This is due to Μ . H . M A R T I N , " T h e propagation of a plane shock into a quietatmosphere", Can. J. Math. 5 (1953), pp. 3 7 - 3 9 .

T h e approach was developed byG. S. S. LUDFORD and Μ . H . M A R T I N , "One-dimensional anisentropic flows", Com>muns. Pure Appl. Math. 7 (1954), pp. 4 5 - 6 3 , and G. S. S. LUDFORD, "GeneralisedRiemann invariants", Pacific J. Math. 5 (1955), pp. 441-450.CHAPTER IVArticle 161. If Po denotes a reference density, and ψ , ψ the partial derivatives of ψ, E q .( 2 1 ) is replaced byχυHere ( p a / p ) is a given function of ψ + Φν (Sec.

24.2).W i t h the same notation, the second-order equation for steady flow with axial sym­metry is022*- -2χ-2φμ+-- fey °-+y=Here ( p a / p ) is a given function of (l/y )(t+ ψ ).Equation (16.21) is replaced by Eq. (24.9) if the flow is strictly adiabatic but notnecessarily elastic.2. Even apart from the fact that the (x,t)-problem of Chapter I I I is always hyper­bolic it is mathematically much easier than the present problem.

In A r t . 12 we could0222x2υ483CHAPTER IVArticle 16indicate explicit general solutions. N o t h i n g similar exists here.3. W i t h respect to the characteristics as Mach lines, cf. N o t e 1.25.4. T w o mathematically interesting special cases may be considered: ( a ) thecharacteristics in the x,T/-plane are rectilinear, and ( b ) those in the w,t>-plane arerectilinear, (see SAUER [7], p.

90 ff.). I n our problem both cases can be realized by con­sidering particular (p,p)-relations: the first, with the "Pe>es-Munk (p,p)-relation",ρ = A + Β [arc tan ρ — p/(l + p ) ] ; the second, with the raletion ρ = A — B/p(Sees. 17.5 and 17.6) generalized t o supersonic flow (cf. e.g. [71, p. 100 ff.); physically,this generalization is controversial. Compare also R . SAUER, cit. N o t e 11.25. T h e hodograph method as explained in this section (cf.

also Sees. 8.6, 10.6,and 10.7 is due to Chaplygin ( 1 8 6 9 - 1 9 4 4 ) : S. A . CHAPLYGIN, ' O n gas j e t s " , Sci. Mem.MoscowMem.Univ.Math.Phys.Sec. 21 ( 1 9 0 2 ) ,1063 ( 1 9 4 4 ) . Compare also N o t ep p . 1-121 (translation:Ν AC ATech.11.23..Chaplygin quotes as predecessors P. MOLENBROEK, " U e b e r einige Bewegungeneines Gases bei Annahme eines Geschwindigkeitspotentials", Arch. Math. Phys. 9( 1 8 9 0 ) , pp. 157-195 (included in [ 2 0 ] ) , and the work of HELMHOLTZ of 1868, cit.

N o t e11.23. W e also mention A . STEICHEN, " B e i t r a g e zur Theorie der zweidimensionalenBewegungsvorgange in einem Gase, das mit Uberschallgeschwindigkeit s t r o m t " ,Dissertation,Gottingen, 1909. D . RIABOUCHINSKY ["Mouvement d'un fluide com­pressible autour d'un o b s t a c l e " Compt.rend.194 ( 1 9 3 2 ) , pp. 1215-1216] has drawnattention to the importance of Chaplygin's work. A n early account of the theorywas also given b y B. DEMTCHENKO, " S u r les mouvements lents des fluides compross i b l e s " , Compt.rend. 194 ( 1 9 3 2 ) , pp. 1218-1222. Extensive bibliographies on the hodo­graph method are in [20], p.

263 ff. and in [21], p. 441 ff.6. M a n y of the properties of the fixed characteristics and their relation to the Machlines are contained in the paper by L. PRANDTL and A . BUSEMANN, cit. N o t e 11.24.I n connection with the orthogonality of Mach lines and fixed characteristics cf. Sec.9.4, particularly E q .

( 9 . 1 5 ) .7. Essentially the same definition of characteristic coordinates (our E q . ( 4 3 ) ) isfound in the paper cit. N o t e 6 in which the graphical procedure of p. 260 is also intro­duced. T h e method has been further developed b y G. GUDERLEY, " D i e Charakteristikenmethode fur ebene und achsensymmetrische Uberschallstromungen", Jahresber.deut.

Luftfahrtforsch.1 ( 1 9 4 0 ) , pp. 522-535. See also K . OSWATITSCH, " U e b e r die Char-akteristikenverfahrender H y d r o m e c h a n i k " ,Z.angew.Math.Mech.25/27(1947),pp. 195-200, 264-270, and W . T O L L M I E N , " S t e a d y two-dimensional rotationally sym­metric supersonicf l o w s " , Grad.Div.Appl.Math.,BrownUniv.,Trans.AG-T-1(1946).8. This rather obvious theorem is directly analogous to one of the " s l i p line the­o r e m s " , due to H . Hencky and L. Prandtl, well-known in the theory of a plane per­fectly plastic body.Article 179. T h e usual derivation is t o start with a second-order equation such as Eq. (16.14)and t o apply t o it the "contact transformation" associated with the name of L e gendre.