R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 67
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(20.8) f. [ W e note that/(n, n)nif θ isare at — η=ne~nsidoes not contribute additional poles and corresponding residues as Chaplygin's original [ ^ n ( n ) ]would. ] T h e limit of the sum of the residues at16these poles is equal to a series — L, where(-ΐΓ£ = Jrv Σ(17)Γ ( 5 )"= 2+ 1Γ(η + 2)Γ(-η -Hence, the analytic2is W2n ("+ , e )+ 1n=2 1 · 3 · 5 · · · (2n +R|)C^„(r)e1)continuationof W\ in the region corresponding to— L and we recognize that E q .
(15) alone could not give the desired continuation since the W2in (15) does not take care of all relevantpoles. N o matter what our choice of /(n, r i ) , the poles of ψ (τ)—asa funcηtion of η—must be taken into consideration. In a similar way we obtain— Wi — 2L and — W2— L in the regions corresponding to Rzand # 4This solution is not symmetric about the ?/-axis. W e obtain a symmetricsolution if we take Wi + L as the solution in Ri ; it can be seen that L —> 0as qm—> °o, as we expect. T h e whole solution in the regions Ri,R, # 3 ,2R4 is then:(18)Wi +L,W—Wii9— L,—W .2T h e continuation of — W into the first region is again W\ + L, demonstrat2ing that ψ is single-valued in the physical plane.B y means of asymptotic estimates of | ψ (τ)η\ such as (10), which aredifferent for subsonic and supersonic τ-values, it follows that Wi convergesfor τ < τι,W converges for η < r ^ 1 — e ( e a positive number, appear2* This corresponds to the fact that the argument of — 1 in (16) must be taken as— π and π respectively for the integrals along the semicircles to tend to zero.+ηΓ ( α ) Γ ( η - b + 1)ηnν, .,nrι ,(09.(cf.
Lighthill quoted N o t e I V . 50). This asymptotic formula is obtained by application of Stirling's formula.300V. 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 Sing in the estimate of | ψ (τ)|), and L converges for all r. Thus, we haveηindeed obtained the continuation of W\ +L over the whole field of flow,including the supersonic region.This example demonstrates both the principle of the method and itsdifficulties.4. General solution for the subsonic region*For subsonic r a compact and explicit solution is available.In addition to the above-mentioned properties of ψ (τ)7the main tool isηthe partial fraction expansion (with respect to η for fixed r ) of ψ (τ)holdingηfor 0 ^ r < T and complex n:TΓΐΨη(τ) = ens— i —m + η+ η Σm-2Lm +=omC e ^ (r)]mmmJηwhere in the last form of this expansion we define Co =n/(n+ ra) =1, C\ =1 for η = m = 0.
Equation (19) expresses ψ (τ)η0, andfor complex η in terms of the ψ™(τ), where m is a positive integer.fFrom (19) we obtain, with r^η(τ)β~(20)ηβ1Cm^ (T)e ,=mM= η Σ™=oand since, by (1.0), 7 ( r ) =(21)7(r) =—^— eη + mΠ τ η ^ ι ^ φ (τ)β~ \ηηΣr ewm ( e-m81Σ=S l )m=0( n + m ) ( s-S l )we also haveC e V (r).mwmm=0From Eqs. (20) and (21) follows the desired result, namely, the continuationof expansion(2) throughout the subsonic region.
Using (20), E q .(2) can be written*(τ,β)(22)Σnee-*"ΓΣr^e*"*= * ΓLn-o= dL»»=o- « ΣΣ - ϊ η- β ^ Ηr»-o m +Σ«<«+«><—>-«>]n-o mϊτηθ-JΛ βΒ_+Β1 ~JηΐΊ/oo(Σ^ηΓ_/0\n=0\+ η _ 1)^/+C0* The considerations of this section, due to Lighthill, are not elementary in character. T h e main result is that the integral representation (23) is valid in the whole subsonic region.t Equation (19) is the result of applying Mittag-Leffler's theorem to e ^ V ^ r ) .21.4G E N E R A L SOLUTION FOR T H E SUBSONIC361REGIONFrom now on to the end of the section we write simply w rather than WQFrom ( 2 0 . 1 3 ) ,dwA-Tz =Σ,ηο ξ-lnηη=οας,and E q .
(22 ) may be written:( 2 3)° * .J=ί { Σ C^ W e " "1Jj™/ "'oΓd w ( f ) ] + u,(0)} .W e may now verify that ψ satisfies the conditions postulated in Sec.1 .First, as qm—> oo, all r — >0 , except rmlim e ~8Sl0=1; also from ( 5 ) := ^ = g,and we see that the right side of E q . ( 2 3 ) reduces to/dw + w(0)=0[w(t)].One likewise verifies directly that (23 ) satisfies the stream-function equation.W e shall now show how this representation of the solution is defined forall subsonic r .
F r o m dw/dz = w (z)= f, we conclude that ζ is an analyticffunction of f in the hodograph of the incompressible flow. This, however, is not a simple plane but a Riemann surface, R (consisting of twosheets with a branch point at ξ= 1in our example of the circular cylinder),and ζ{ξ) is regular on R. T h e integral in Eq. (23 ) can be written(24)f"'= f *' "r S* = f *' V <fe(f).+,rdwdt)Jo+,αςJQJoIf ζ = z corresponds to f = 0 ,0Ym(z)=Γr+1dzJz0is a regular function of ζ in the 2-plane outside the body if we assume thatSince we know, in addition, that ζ(ξ) is regular on R itthere is no circulation.follows that Z (X)— Y \z{$)\ is also regular on R.
T h e last term in (24 )mmcan be written as Z (e ~ ~ ).ms8ll9In the absence of circulation this is thereforesingle-valued on a Riemann surface,which corresponds to the incompressible hodograph surface R as locus of the points ( r , Θ) for which ~~ ~esSl%elies on /£. T h e R* is the hodograph of the subsonic part of the compressibleflow and (23 ) is regular on R* if it converges.362V.
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 SW e now show thatψ = S Γ Σ(23')re >n*' Γ("im+ 1<fe(T)lconverges everywhere if the flow is purely subsonic? In fact, if we integratealong a path, of length I in the physical plane, joining f = 0 and e~* ~lon R%ethen, since at any point s < σ, its sonic value,f r ^(r)+1On the other hand rm6~ /2πηι2στη7and ψτη(τ)^ le= C \l/ (T)emm~(σ- )(,η+1)8ιwhere, for large m (p.
359): Cm81m~Ve > hencema2πτηThus we see that for large enough m a term of (23') is comparable with(Vle ~ /2Tm)e~ ~ ,aam((X8)which assures the convergence.*Thus for subsonic flow without circulation, the problem has been solvedin the explicit form of the integral representation (23).W e turn now to a presentation of Bergman's method, and at the end ofthe article, we shall indicate the relation between Bergman's and LighthilPsmethods.5. Bergman's integration methodIn an attempt to continue and improve Chaplygin's pioneer work, which,while highly successful in many ways, failed for the problem of flow pastan obstacle, S.
Bergman began to apply a general mathematical idea to thisproblem. T h e purpose was to establish a correspondence between analytic9functionsofa complex variable(i.e. solutions of incompressibleflowproblems) and solutions of linear partial differential equations of elliptictype (such as the equation for ψ). This is achieved b y means of an integralrepresentation of the solution from which properties of the solution can bededuced by means of complex function theory. ( I n particular, solutions ofthe compressible flow equations can be obtained which are multivaluedand have singularities of the type needed in the flow problems under consideration.
I n fact, as seen before, even in the incompressible flow arounda circle the stream function cannot be represented b y a single convergentseries of single-valued functions).L e t Po be the contour of an obstacle in the physical plane; denote b yYo(qfi)the stream function of the incompressible flow problem, so thatφο = 0 on P ,Qand Δψο = 0 outside P .0Just as in the methods of Chaply-* Actually there is uniform convergence. Also, the differentiated series convergeuniformly, etc., and we may verify that ψ satisfies the stream-function equation.21.5BERGMAN'S INTEGRATION363METHODgin and Lighthill, we wish to associate with ψ a φ satisfying the compressible hodograph equation, which is not too different from ψ as long asa representative Mach number is small, and which reduces to ψο in thelimit q —> oo. Of course ψ will not vanish on P : it can however be assumedand has been proved under certain circumstances that φ = 0 on a streamlineΡ close t o Po .00m010First we rewrite E q . (20.5), in the same w a y as before, b y introducinginstead of q the new variable λ of E q .
( 3 ' ) and obtain E q . (6) with eithers or λ as independent variable. N e x t we introduce ψ* by E q . (7) or by(25)α = Vψ* = <χφ,,Τ = — —,αwhere primes denote differentiation with respect t o λ, and we obtain E q .( 8 ) , where in the polytropic case,( 2 6 )- ~ί -F[ 1 6""4 ( 3~2 k ) m 2~( 3 kl ) M t ]-Since both F and λ are given in terms of M, F is given in terms of λ .W e try to integrate E q . (8) b y settingψ* = <7ο(λ, θ) +(27)ΣG (\)g (\n11θ)nη=1(note that this is not a separation of variables).
Each g in E q . (27) is aharmonic function of λ, θ. Using the symbol Δ = d /d\ + d /d0 andsubstituting, we obtain with Go = 112n2+Ft*=Σ[A(G g )n+n222FG g ]nnn=0= Σ (Gl'ffn +(28)2G'n% +n=0G Ag +nFG g )nnnσλ= Σ[+ FG )ng+n2G'n+Fg .9W e shall see that E q . (8) is satisfied if we put(29)^(29')G'n+1=-1^( n = l , 2 , ···)( η = 0, 1, · · · )= G" + FGnnwhere g is arbitrary and Go = 1. T h e right side of E q . (28) becomes:0Σn=lIgnG'n+i -g -iGn]n+Fg .0304V. 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 SThis series equals the limit for η —» °o of g G' +in— goG[ .
This is so far anformal computation. I t will he shown later, using a majorant method, thatthe series in (27) and the necessary derivatives converge uniformly in acertain region and that not only g Gnbut also g G +innn—> 0 as η —> oo. Thusthe right-hand side of E q . ( 2 8 ) reduces to —goG[ + Fg0cause of E q . ( 2 9 0 for η = 0, G[ = FG=0= 0, since, beF.T h e G are determined by ( 2 9 ' ) if we add the conditionn(29")<?.(-«>) = 0(n =1,2...),and it is seen that the sequence Go, G\, G , · · · depends only on2F(\).Hence it is uniquely determined for a given (p, p)-relation, it can be computed once and for all, and can be tabulated.On the other hand, the sequence g , gi , g ,02· · · depends on the arbitraryfunction g .
W e introduce now the complex variable0(30)where σ =Ζ =A -withi$Λ = λ +σ +log qm—1.17, as defined in Eqs. ( 4 ) and ( 5 ) . Since τ =q/qm, thesecond E q . ( 5 ) can be written(31)= log q -hm (λ + log q )mand it follows that with ζ = qe~%9(31/)σ,denoting again the complex velocitylim Ζ = log q -ιθ = log f.T h e variable Ζ is the same as LighthilFs variable s — Si — id, and it willserve a similar purpose.F o r a fixed qm13the £τ (λ, θ) of E q . ( 2 7 ) which are harmonic in (λ, θ)ηare also harmonic in (Λ, Θ). W e then define a sequence of functions of· · · Avhere/ (Z) is an arbitrary analytic function of Ζ andZ\ fo(Z),fi(Z),0where/I = — i f - i (w = 1, 2, · · · ) by puttingn(32)f (Z)=nwhich implies/„(0) = 0 for η =ρ (\θ)(33)η-Λ=4[f (Z)]n(n = 0, 1, .
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