R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 62
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( 1 ) becomesor, substituting from ( 4 ' ) and simplifying, we obtain00 4Λ. -,)* + (.- £ i ,)+,) £ -0.I t is seen that the coefficient of dfy/d0 changes sign from positive to negative as r increases through r .T o this last equation we apply the method of separation of variables,writing ψ(τ,θ) = Α (θ) Β (τ). Upon substitution one finds from well-knownconsiderations that A"/A must equal a constant.
If we put A"/A = — n ,we obtain, for η * 0, Α (θ) = Α (θ)= ae+ β β~= y sin (ηθ + δ ),with a , β or γ , δ as arbitrary constants. Following Chaplygin, we set2t2ηnηηnindηίηθnη41ηB(r)==UT)r*" / (r),2Bso that(6)φ(τ,θ)=ψ„(τ)4„(ί) =r %(rKa ennine+Λβ-'"·).* This notation is used by several authors. The Q(T) of (4') has nothing to do withthe angle Q(q) introduced in Art.
16.20.1 S E P A R A T I O N O F V A R I A B L E SThis gives for ψη(τ),(7)331and finally for / ( r ) , the equationsn~ '2and<?') r(l - ,)/." + [ ( » + 1) - ( „ - ^j)r]/i +- 0.Equation (7') is a hypergeometric equation(7")r(l~τ)/" +[cn-(α+Λ&„ +a &n/ =l)r]/' -0,ninvolving, however, only t w o parameters η and κ instead of the threeparameters in (7").A solution of ( 7 " ) regular for τ = 0 is given by the hypergeometric function F(a , b , c ; r ) , whose T a y l o r expansion isnnn*Y„h*· ϊ -"V Γ(αη+ ^)Γ(6 +Γ ( α ) Γ ( ο ) v=or ( c + j>)r ( c}ηΛΛny)/j/!provided c is not a negative integer or zero.
Hence we obtain the following solution of ( 7 ) , called Chaplygin Function or Chaplygin solution:42nMr)= r F[anl2l;r],b ,n +nn(8)F(a ,6nn)n+l,r) -1 + ——T+21(λ+1)(w+2 )* +·"·Here η is arbitrary except that it must not be a negative integer. I t is easilyseen that this series converges out to the singularity at r = 1, i.e., for r ^ 1;F is therefore an analytic function of τ which tends to 1 as r —^ 0.* Solutionsof (7) corresponding to the exceptional values of η, \ η \ > 1, will not be ofthe form (8) (see Sec.
4 ) . For η = — 1 we see from (8) that either a or bvanishes; in this case F = 1 satisfies ( 7 ' ) , and ψ-ι(τ)= r~* is still of theform ( 8 ) . (See Sec. 3, and beginning of Sec. 4 ) .n* Near τ =1 the following expansion may be used in general:F ( o , 6 , C ; t ) = F(a,6,c;l) F(a,6,a ++nF(c-a,c -&-6,c;l)(l -T h i s formula is used in Sec. 5.c+1;1-t)τ)^ -ψ(οα-a,c -6,c +1 -a -b;l -τ).«332IV. P L A N E S T E A D Y P O T E N T I A LFLOWI t is well-known, and we mention it for later use, that the ordinary differential equation of second order ( 7 " ) has a second independent solution,T~ F(an— η, 1 — η ; τ )— n,bnnwith the a and b as in ( 8 ) .
A n y solution of the equation ( 7 ' ) is a linearnncombination of these t w o particular solutions.*B y use of the principle of superposition a solution of (5) more generalthan (6) is obtained in the formφ(τ,θ)Σ Mr)(η)= αθ +(a en+ineβ β' )Αϊη9ηwhere the a , β are arbitrary constants and ψ (τ) is given by ( 8 ) . T h e rangeηnηof η will have to be set in each case; also, when there are infinitely manyterms, the convergence must be investigated.
T o each term ψ(τ,0)where Α (θ)Α (θ)ψη(τ),ηstands for yηsin (ηθ +na potentialδ ),η=φ(τ,θ)corresponds by ( 4 ) , namely,φ(τ,θ)(60= <2(τ)φ 'ίΑ (θ)ηάθ = φ (τ)ίΑ (θ)ηηάθ.η2. Relation to incompressible flow solutionsW e now want to relate these results t o corresponding results for an incompressible flow. T h e passage to the limit from compressible flow to incompressible flow may be made by letting qm—> oo. From (2) and ( 3 ' ) it isseen that this corresponds for fixed q and 0 t o r —> 0 and to Μ —> 0. If in(1) we let Μ —> 0, we obtain(9)qq> ^?*dq ^ ΘΘ+2q A=0,dqd+2Ha Laplace equation in polar coordinates q, 0. T h e equation has the particularsolutions q en±tne(for any n ) , and also the solution (a +d log q),bd) (c +where a, 6, c, d are arbitrary constants. We want to find compressible flowsolutions that reduce to solutions of (9) as qm—>0 0.
Such solutions can bedefined in many ways. T h e following is the correspondence proposed byChaplygin. W e consider q enlim ^ F(a ,b ,cT0nnn±tndand use the above-mentione$ fact that; r ) = 1. W e introduce now a reference speed q , whichxwill be kept constant when we let qmtend towards infinity. W e may assume qi = 1 without loss of generality; if τ is the corresponding value of r,1* T w o independent solutions of (7) for η = 0 are 1 and flfi' dr.f We use here ψ(τ, θ) also for the sum of terms (6).
T h e notation φ {τ) = φ isreserved for a solution of (7).1ηη20.2namely, τR E L A T I O N TO INCOMPRESSIBLE FLOW SOLUTIONS= l/q ,2χmthen (q/qi)or q = τ/τ= τ/τ21lim333. Hence,ι= lim (-Y' = q",T2and±i'n0 Ψη\τ)1·hm e— —= e±ίηθ ηq .W e therefore decide to associate with each term q eoccurring in anincompressible flow solution the term [ ψ ( τ ) / ι / ' η ( τ ) ] 6which satisfies ( 5 ' ) ·I t is seen that in this correspondence the argument θ of q remains unchanged.If then the stream function of an incompressible flow is given in the formnη(10)φο =αθ +Σ1q (<* en+inen±lne± ι η ββηβ'^),(η)we associate with it the ''corresponding" compressible flowΨ = αθ +(11)Σ(π)τ4^(«neβ β- ).+ineίηθηψηΚΤΐ)Before continuing we collect for reference solutions of (4) which correspond to the previously mentioned solutions of ( 9 ) :(a) φο = ΑΘ,<p =-(b) φο = Β log q,φο =ΒΘφο =—C q(c)ψ0=0Cqnnsin (ηθ+δ ),ηlog qAncos (ηθn+δ );ηand correspondingly(a)ψ =φ= ΑΑθ,ίΡ"1drJTJL(b)ψ= ΒίQ'1(c) Ψ = C p£LsinnΦη(τ )±φ = Βθdr,(ηθ+δ ),ηφ= - ~ Q c o sη(ηθ + δ ),ηψη(τ±)where ψ (τ) is given in ( 8 ) .
Each of the last three solutions satisfies ( 4 ) .Solutions (a) and ( b ) , which correspond to η = 0, are the source (sink)solution and the vortex solution of Sec. 17.4 respectively.W e now complete the explanation of Chaplygin's method. Consider thetwo first-order equations obtained from E q . (16.31) in the limit of incompressible flow:η44334IV. P L A N E S T E A D Y P O T E N T I A L FLOWFrom these, considered as Cauchy-Riemann equations, it is seen that φο + ίψοis an analytic function of (log q — id), hence of qe~x9= f. Chap= q — iqxylygin's method developed by him for the study of gaseous jets can then bedescribed as follows. Suppose that an incompressible flow problem has beensolved b y the method of the complex potential. W i t h ζ = χ + iy let w{z)be the complex potential, denote by dw/dz = f = qe~%eand call Wo(f) = w{z)= <po(q,6) +the complexvelocity,the hodograph potential.ιψο^,θ)transformation inverse to dw/dz = f (z), namely, ζ = ζ(ξ),Theexists providedthat f ' ( z ) 9^ 0.* N e x t , in the neighborhood of a stagnation point ξ = 0—provided this is a regular point—expand Wo(f) into a T a y l o r series in ξ so thatφο = 4 Γ Σ(13)Cnf lLn=04 Γ Σ=nCnq e~nL*-0J4 denoting "imaginary p a r t " , and form the corresponding series(14).-*-"].tLn-0ψη{τΐ)JT h e series (14) is, within its region of convergence, the stream functiona compressible flow and reduces to (13) as qmof—> <x>.
Following Chaplygin wethen consider (14) as an (approximate) compressible flow solution of theproblem whose incompressible flow solution is given b y Wo = φο + ΐψο.W e have to keep in mind, however, that the solution (14) need not be thecorrect solution of the compressible flow problem. T h e fact that (14) reduces to (13) as qm—>oo (and therefore in the limit of incompressible flowsatisfies both the differential equation and the given boundary conditions)does not imply that (14) satisfies these boundary conditions while q is finite.mT h e method, which as we shall see works without trouble in the case ofthe jet problem for which it was designed, meets great difficulties if appliedto other boundary-value problems.L e t us finally note that we are led to the same solutions ψ (τ)ηif (17.12)for the Legendre transform Φ, rather than the stream-function equation foiψ, is used.
W e then obtain an equation with the same cnbut an, bn instead of a , b , where a ' +n—n(nnnW= η +=η +1/(κ — 1), a 'bnn1,=— 1)/2(κ — 1). Hence the hypergeometric functions appear again,however with different dependence of an, W on the parameters η and κ.45* In general the function ζ(ξ) as an inverse function is not single-valued. In thecase of a multivalued solution z ( f ) , the hodograph potential w ( f ) will representonly one branch of the solution.t N o t e that for κ = — 1 this correspondence is not the same as that given inSec. 17.6.020.3 A F L O W W I T H I M B E D D E D S U P E R S O N I CREGION3353.
A flow with imbedded supersonic regionIn the remainder of this article we shall illustrate the above-explainedcorrespondence principle by several applications.W e have already reconsidered the source and sink flows. I n the presentsetup they correspond to η = 0. W e take next η = —I* and note thatEq. (70 is then satisfied by f = 1. Hence we obtain the solutionn(15)ψ(τ,θ)= Ar~hsin 0,<ρ(τ,θ) = At~\l-r)"cos 0,1 / u _ 1 )and, of course, a similar one with cos θ in ψ and sin 0 in φ.W e shall now study in some detail this simple exact solution, due to F.Ringleb, which will be found interesting from many aspects.
Using now qinstead of τ we consider for k real the hodograph solution [satisfying Eqs.46(16.31)]kψ = - sin 0,(16)kφ = — cos 0QPQwithwhere d(pq)/dq has been obtained from Eq. (8.5). W e find from E q . (17.25')dx _ k (cos 2Θdg ~ Ρ \qdy=fc /dq ~ p\cos θ\2a?q / 'zsin 2Θsin 2θ\2a q / '~q*2dx _sin 2Θ_kρθθq2dy _ k cos 2Θθθ ~ ~pq2'and, as may be verified by differentiation,(17)β-*Γ?«^ / *1β+L P<I«t pq A2qwhere the constants are chosen for reasons of symmetry.
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