R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 69
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( 7 ) .// the generator K(zi, z , t), where z\, z , t are independent complex22ables, satisfies Eq. (47), namely L (K)vari= 0, and the side condition (48), then+v(zi, z ) defined by (49) satisfies the same equation, namely L (v)=then we put c = F/4 and restrict the complex variables Z\,z toA — i0,+22Ζ= A +ιθ then K(Z,Z, t) and u(Z,finally, we assume for Κ the conditionas in (53), the imaginaryf(t)stream functionas qmZ)willΖ =both satisfy0. 7/Eq.
( 8 ) . //(52) and choose the previouslyarbitrarypart of u(Z, Z) will reduce to the incompressible—> oo..The discussion was made in the domain of complex Λ ι , θχ on account ofits simplicity[Eq. (47) being hyperbolic, (48) being a condition along acharacteristic, etc.]. However, it is easily checked that the generator Κneed only satisfy—for real Λ, θ—Eqs. ( 5 1 ) , (52), and the side condition(480^-= 0,i^-fort = λ -id.A generating function satisfying all these conditions is the one given in(35), where w e now write G{t\ λ, θ) = K(Z,(56)K(Z,Z,t)Z, t) or£ J - G (\)(tn=»o 2 n !=n-Z)\nwhere (?ή = G'n-i + FG -i.W e find b y straightforward computation thatnthe right side of (56) satisfies E q .
(51). Hence, not only the \p*(q, Θ) definedby Eq. (36) but also the generator G(t; \, Θ) of Eq. (35) satisfies Eq. ( 8 ) . W ehave seen before that (52) holds for the present Κ of (56), and we canverify that (480 holds.16A s a second example consider LighthilPs generating function, in E q . (23).Here the notation is different: Ζ = s — Si — ιθ, t = log f, or e = f; thetw in E q . (23) is the hodograph potential w of the present notation, and0with a = V~ we consider the generatorl(57)K(Z,Z,t)= α Σm=0C ^(r)em+ ΪΘ+1)m ( 8 ]This function satisfies the differential equation since this is so for eachterm separately.
Also condition (48) holds true. W e have, in fact,ds1ΘΘ=Σm=oC emm ( 8 l + i e + t )Γ# (atJ[_as+mc^ ]Jmfor t = Z.372V. I N T E G R A T I O N T H E O R Y A N DSHOCKSAccording to Eq. (21), this equals (d/ds)(aV) = 0, since V = 1/a; thatcondition (52) holds was verified before in Sec. 4.W e thus possess a general principle which allows us to transform ananalytic function, a solution of an incompressible problem, into a solution ofthe stream-function equation, by means of a generator K, and we possesstwo examples of such generators.In order to be useful such a generator must have certain properties: thenew compressible flow pattern should be similar to the incompressible pattern as long as some representative Mach number is small. T h e generatorshould be explicitly given, e.g., by means of a well-converging expansion,and convergence should hold in as large a region as possible. LighthilPsand Bergman's operators have these properties, though, each with differentemphasis in these respects.8.
Relation of the two methods17W e shall now consider in more detail the relation between themethods. W e take as a starting point LighthilPs formula (23)and rewrite it aswhere R (s)m=rem. T h e termis an analytic function of s — id and therefore(58)is harmonic in s and 0. Thus (23) can be expressed as:00(59)ψ(8, θ) =ΣRm(s)h (s,mor with ψ* = αψm=0where(61)H (s)m=a(s)R (s).mθ)two21.8373R E L A T I O N OF T H E T W O M E T H O D SFrom (21), remembering that a = 1/V, we obtain(62)=1,m=0and see that there is an identically valid relation between theH (s).mThus, comparing E q .
(60) with E q . (27) and remembering that, exceptfor a constant, s is the same as λ, we see that the same equation is solvedby the two authors by the same type of expansion and that (60) leadsnecessarily to the identity (28) with h and Hnntaking the place of g andnG.nN o w , however, the difference becomes manifest: the requirement for theg is different from that for the h . Bergman chooses in ( 2 9 )nn_Qndιa7 "* »~ '9lwhile we find from ( 5 8 )-d h nds-nh+n*[w'(e -*)].M1Consequently from ( 2 8 ) the following formula for the Hn(63)Σ(Hn -2nH'=+ FH )hnnn-*[ti/(eM l-")].2obtains:Σn=0Here, according to (62), the right side is zero and (63) will be satisfied if(64)HiNow ΗHi2nH'n+n+ FH= 0.nn snn-2nH'= C « ( s ) ^ n ( s ) e ; using F == a(s)R (s)n-= α [ Λ » — (Τ +FHn= αΟ β \ψ:ηη-2n)R'nΤφ'η-+nVn)-(a"/a)we findnTR ]n= 0because of ( 6 ' ) · Hence (64) holds.
Equation (64), namely, HiFHnFGn= 0 ( η = 0 , 1 , 2 , · · · ) compares with E q . ( 2 9 0 , namely, Gn— 2nH'n+- <?»+i += 0 (n = 1, 2, ...),<?o = 1.I t is an advantage of LighthilPs method that the functions H (s)nareconnected with the hypergeometric functions, whose properties have beendealt with extensively. As seen in Sec. 4 the convergence is guaranteed inthe whole subsonic domain.Bergman's G (s),ndefined b y the recurrence ( 2 9 0 , are not connected witha well-known system of functions. This is a consequence of his definitionof the g which in turn constitutes an asset of his method.
His recurrencenformula (29) for the gnfunctions Re[(\+ιθ) ]ηis, in fact, the recurrence formula for harmonicwith proper normalization. Due to this corre-374V. I N T E G R A T I O N T H E O R Y A N D S H O C K Sspondence the operator defined in E q . (37), which transforms an analyticfunction of a complex variable into a solution of E q . ( 8 ) , preserves manyproperties of the former.18Article 22Shock Theory1. Nonexistence of solutionsJust as in the case of one-dimensional nonsteady flow, see Sec. 14.1, itcan be shown that the differential equations governing the steady planeflow of an ideal fluid admit no solution satisfying certain boundary conditions which can be enforced by simple physical arrangements.
T h e equations in question are the x- and ^/-components of Newton's equation (1.1),with gravity omitted:dqx(1)_=dtwhere d/dt(16J):q d/dxx(2)_dp__dx'dq_^ _~dt ~y=P_dpdyq d/dy, together with the equation of continuity+yέ{ p q x )+ Tyi p 9 v )°'=and the specifying condition: dS/dt — 0 (see Sec. 2.3).Assume that between the points A and Β on the ?/-axis (see Fig. 141)the fluid is passing uniformly with supersonic velocity parallel to the #-axis:yFIG.exists.141. Enforceableboundaryconditionsforwhichno inviscidsolutionN O N E X I S T E N C E22.1OF375SOLUTIONSQx — Qo, q = 0, ρ = p , and a = ao < qo, where Qo ,Ρο, and ao are constants.
Through the points A and Β the t w o straight lines inclined atangles ± arc sin (a /<?o) are drawn, and the horizontal streamlines, representing the uniform flow q = q , q — 0, and ρ = p within the triangleABC, are inserted. Then this solution satisfies the differential equationsgiven above as well as the boundary conditions on χ = 0. N o w the fluid iselastic by virtue of the entropy being the same for all particles as they crossAB. IV^oreover, see Sec. 6.5, any solution of these equations satisfyingthe boundary conditions must be irrotational, since the Bernoulli function is prescribed by these boundary conditions to be constant throughout the corresponding flow.
But according to the theory of an elastic inviscid and nonconducting fluid in irrotational motion, given in Sec. 16.4,the solution is uniquely determined in the characteristic triangle ABC bythe conditions along AB. This means that the differential equations of aninviscid and nonconducting fluid have no solution in ABC consistent withthe given boundary conditions other than the solution q = q , q = 0everywhere in ABC.yQ0x0y0x0yI t is certainly possible to subject an actual fluid mass passing between A and Β to some additional conditions. T h e argument is based ontwo observable properties of fluids in supersonic motion.
First, a fluidmass moving uniformly can be deflected without appreciably disturbingconditions upstream, provided the deflection is gradual enough. Secondly,if a given deflection is gradual enough for one M a c h number, then it is alsofor all larger M a c h numbers. These facts will be reflected in the theory presented in the last t w o sections of this article.19Suppose that in the present example, Fig. 141, the fluid is bounded frombelow by a wall which has one end at the point A and is horizontal at thatend. Provided that it does not slope upward too abruptly beyond A, sucha wall will not disturb the existing conditions between A and B. I n Fig. 141the wall is represented by a curve AD, and this curve must necessarily bea streamline for the fluid particles passing through A. N o w as the M a c hnumber M increases, the lines AC and BC become more nearly horizontal.Thus, however slightly the curve AD slopes upward, the point D fallswithin the triangle ABC for sufficiently large M ,and a contradictionarises.
T h e inclination Θ of the streamline at D must be that determined bythe tangent to the curve AD at D, and not Θ = 0 determined by the solution given above.00A s in A r t . 14 the way to overcome this contradiction is to take intoaccount viscosity and/or heat conduction. I n a fluid of low viscosity thenoticeable influence of these on the flow pattern is assumed to be localizedin extremely thin layers within which occur rapid changes in the statevariables.
Throughout the rest of the flow the equations of ideal fluid theory376V. I N T E G R A T I O N T H E O R Y A N DSHOCKSare supposed to describe the actual (viscous) flow adequately. A flow pattern of this combined type is observed in cases, such as the example above,for which there exists no solution based only on the theory of ideal fluids.A workable theory of steady plane flows containing such thin layers orshocks can be based on the following principles. It is assumed that the original differential equations for the motion of an ideal fluid, namely theequation of continuity, Newton's equation, and the specifying condition, arevalid at all points of the x,y-pfane with the exception of certain shock lines",whose positions are a priori unknown] across these lines the state variables arediscontinuous, the sudden changes being governed by rules derived from thetheory of viscous and/or heat-conducting fluids.20uW e must now determine the form these shock conditions take in the caseof steady plane flow.
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