R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 65

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T h e formula which gives WQ in terms of f is1Wo =φο +ίψο=log^~ -(26)(1A-^= ? log 2f π+ Σ ^) ·f ) = 9. ( l o g 2f« log (1 -2πi n /π \Here Q = 2^i denotes the flux through the slit, per second, and the velo­city on the free boundaries BC and EC has been put equal to unity: qi= 1. T h e expansion in (26) is valid for | f | < 1.

T h e value of Q is connec­ted with the asymptotic width, 2b, of the jet: Q = 2bp qi = 2bpo in the incom­pressible case. From (26)0Σ^8ίη2ηΛ= --(θ+0/.( 2 7 )φο.. -= 9. f l o g 2? + Σπ \*— cos 2ηθ\i n/0.745-1(-) -1IT7Γ + 2/!r «O.I6 7 |tΌ0.020.040.060.080.100.120.14ΤιF I G . 137. Contraction ratio b/a versus η .Μ0.160.1820.6W e see that for θ = Ο,ψο = 0. For q = qi =οΣ 9± - sin· 2ηθ349SUBSONIC JET1, 0πsin 2ηθΛ- - 0.2Hence if 0 > 0, ψ = -(QM(0 + (π/2) - 0) = -Q/2 o n f i C , while if(9 < ο_,φ = -(QM(0 - (π/2) - 0) = Q/2 on BC, as it should be. On ABand ^45, for 0 = Τττ/2, the sum of the series is zero; hence on ΑΒ,ψ=Q/2, on AB, ψο = —Q/2.

Thus the boundary conditions are satisfied.N o w consider the compressible jet. T h e velocity q will still be constant onthe free boundaries; we call it qi and assume it equal to one, η = l / # . A c ­cording to the rule of Sec. 2 we then formιη= 2^0, the seriesιη=db000m(28)2n2nel.* =riπ LinJψ (τι)ηW e remember and may check that for q —» oo ( n —> 0) this reduces to theψο of (27). Chaplygin has proved that the above converges together withall its necessary derivatives if q < qi < q , i.e., τ < η <Moreover, aswe have seen, (28) satisfies the boundary conditions exactly.

In fact for 0 =±π/2, φ = =F(?/2; for τ = τι , i.e., for q = q\ , the right side of (28) isexactly the same as the right side of (27) for q = 1. Hence, indeed, alongABC, ψ = Q/2, and along ABC, ψ = -Q/2. Therefore (28) is the exactsolution of the subsonic jet problem.T h e velocity potential φ corresponding to ψ in (28) can be determined aspreviously explained; then χ and y are obtained as infinite series in r and 0;each series contains the flux Q as a factor. T h e final aim is the velocitydistribution throughout the field of flow, i.e., r,0 in terms of x,y. Sincethe problem is subsonic, there are no singularities of branch or limit type;the numerical computation, however, becomes very involved.Some typical questions of jet flow can be dealt with directly by meansof the hodograph solution.

Such a question is—as in the incompressiblecase—to find the form of the jet (i.e., the shape of the free streamlines) and.in particular, its asymptotic width. If we call the asymptotic width 2b (seeFig. 136), while 2a is the given width of the slit, we wish to find λ = b/a.Since we know χ and y in terms of τ and 0, we can find the values of χ andy on the jet boundary, i.e. for r = τχ . In particular the expression for y isof the form: y/Q equal to a known function of τ , θ, and Q is proportional to2b. N o w as 2a is the width of the slit, the value of y on the jet boundary,for 0 = π/2, is equal to —a; hence one obtains λ in terms of τι (see Fig.137 which corresponds to κ = 1.4).

In the incompressible case, i.e. for η —*0, the value of λ has been found by Kirchhoff: λ = π/(π + 2) = 0.611. T h emtλ350IV. P L A N E S T E A D Y P O T E N T I A LFLOWother extreme appears for η = 0.167 (sonic value), for which λ = 0.745.T h e values of λ for various τ\, were computed by Chaplygin.T h e method of this section is applicable to more general steady-flow jetproblems as long as the speed q on the free streamlines is everywhere belowthe sonic speed.

In terms of pressure, this will be so if the outside pressurePi in the receiver is higher than the "critical" pressure corresponding toΜ = 1. If pi is lower than the critical pressure, the jet is wholly or partlysupersonic, and the Chaplygin method does not apply.55CHAPTER VINTEGRATION THEORY A N D S H O C K SArticle 21Development of Chaplygin's Method1. The problemIn this article we shall describe two different approaches aimed at obtain­ing solutions of certain boundary-value problems. T h e jet problem of thepreceding section was a boundary problem that could be solved exactly.T h e main reason for this was that the boundary conditions appeared in anatural way in the hodograph plane; in addition, the problem dealt withwas entirely subsonic.In the important problems of flow past a body and of channel flow, theboundary conditions are given in the physical plane and therefore the ap­plication of the hodograph method to these problems meets with greatdifficulties.

In the first part of the present article we shall discuss methodsdue mainly to M . J. Lighthill. T h e work of Τ . M . Cherry is independentof that of Lighthill; it goes in the same direction and in certain respectsfurther. However, it does not seem appropriate to discuss here both LighthilPs and Cherry's work, and the latter is somewhat more difficult topresent in a small space.

T h e second part of the article will deal with S.Bergman's method and some of his results. W e shall also point out themathematical relation between the t w o approaches.In Sees. 20.1 and 20.2 Chaplygin's method for the construction of com­pressible flows was described. A compressible flow was constructed cor­responding to an incompressible flow, according to definite rules. In theproblem of flow past an obstacle the application of Chaplygin's method runsas follows. First, we attempt to determine the complex potential w(z) =<p(%, y) +y) of the given boundary-value problem for incompressibleflow. Then, with dw/dz = q — iq = qe~ = f, the complex potentialw{z) is expressed as a function of f, namely, w(z) = W o ( f ) = <Po(q, Θ)+ i\pv{q, Θ).

A s explained in Sec. 2 of the preceding article W o ( f ) is next ex­panded into a Taylor series in the neighborhood of a stagnation point f = 0,and using r = q /q ? we associate with the incompressible \po(q, Θ) a com­pressible stream function ψ(τ, θ) by means of the rule contained in Eqs.x2yn351lB352V. I N T E G R A T I O N T H E O R Y A N DSHOCKS(20.10) and (20.11); we accept this ψ(τ, θ) as an approximate solution of theoriginal boundary-value problem on the basis of the fact that ^ ( τ , Θ) tendsto ψ (ς θ) as q —» .T w o basic difficulties regarding this conception are immediately recog.nized.

First, there is no reason to assume that ψ(τ,θ),if reverted to thephysical plane, will satisfy the original boundary conditions for Μ > 0.T h e streamline in the z,?/-plane which in the compressible flow corre­sponds to the given contour, for which we have solved the incompressibleproblem, will not coincide with this contour; its shape will depend on a(dimensionless) parameter, e.g. on η , the " v e l o c i t y " of the undisturbedflow. W e shall have to be satisfied if, for a certain range of values of η ,this curve is close to the given contour.01m00W e may also look at this difficulty from the following point of view.

W eare unable to find a solution which satisfies both the equations of motionand the given boundary conditions. I n the approach just mentioned, wesatisfy the differential equations exactly and the boundary conditions ap­proximately.* Another possibility, well known in applications, is to try tosatisfy the boundary conditions exactly and the differential equations onlyapproximately. This is the approach of methods in which simplified differ­ential equations in the physical plane are substituted for the exact equa­tions. (If these equations are linear, then the boundary-value problem canbe solved in many cases, at least in principle.)Second, the construction of ψ{τβ) clearly works only in a region ofconvergence of both ypoiqft) and ψ(τ,θ). Since the incompressible flow has asingularity in the hodograph, the T a y l o r expansion of w (£) (about a cer­tain stagnation point) will have a limited circle of convergence; a separatepower series is required for each region of convergence.

These series mustbe analytic continuations of each other across the boundaries of these re­gions, and we assume here that this problem in the theory of functionscan be solved. However, even if such a solution of the incompressible flowproblem has been found and we associate with these series the correspond­ing Chaplygin series, as in Eqs.

(20.10) and (20.11), it cannot be assertedthat the latter need be analytic continuations of each other, and we shallsee that actually they are not (Sec. 3 ) .QT h e developments which we shall discuss in this article concern only thissecond difficulty, for which decisive results have been obtained. T h e firstdifficulty is not tackled in any of them.

This holds for the investigations ofCherry, Lighthill, and others working in a similar direction, as well as forthose of Bergman and his collaborators.W e shall present in the following three sections Lighthill's method andmain result. In so doing it seems useful, after some preparatory work to be* This applies even for the simplified approach discussed in Sees. 17.5 and 17.6.21.2R E P L A C E M E N T OF C H A P L Y G I N ' S F A C T O R353given in the next section, to start with the consideration of a concrete ex­ample, namely the compressible circulation-free flow about a circular cylin­der, and to explain the principle of the solution and the essential difficultiesencountered, without going into too many mathematical details.Thestudy of this problem, which involves supersonic as well as subsonic veloci­ties, will thus contribute to an understanding of the general situation.

T h esolution presented in the example is based throughout on series expansionsof the (incompressible) hodograph potential W o ( f ) . This could be avoided inthe subsonic region (see Sec. 4 ) , but for the continuation to supersonicspeeds one has at any rate to start from the series expansion of the hod­ograph potential.Ageneral solution for the subsonic region will then be discussed inSec.

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