R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 84
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Sonic speedreached at wall upstream of the throat and at axis downstream of it. Vertical linescorrespond to one-dimensional theory.25.1SOME A D D I T I O N A L447PROBLEMSa characteristic Γ . A symmetric family of streamlines exists with the symmetric Γ as envelope. These two characteristics form an edge and itsimage in the .r,?/-plane is the branch line.
T h e region (in the hodograph)between the Γ and T~ is covered three times (Fig. 170). Through a pointΡ in the region with positive q passes (a) a streamline which has cut theT~ (the £-line in Fig. 170) before reaching P, and then contacts the Γ ;(b) one which crosses the Γ , reaches Ρ and contacts the Γ ; and (c) onewhich crosses thecontacts the Γ and then reaches P .+-+y++-+Thus it is not possible to use g, θ as independent variables in an analytictreatment; however φ and ψ may replace them, and we may then expandq(«p, yp) and Θ(φ, ψ) in series in the neighborhood of a sonic point and substi-F I G .
169. Hodograph with streamlines and branch line Γflow.I+in transonic channelsonic l i n e \ c )-ζF I G . 170. Triply-covered region between branch lines in characteristic £,r;-planefor transonic channel flow.448V. I N T E G R A T I O N T H E O R Y A N DSHOCKStute these expansions in a system obtained by interchanging variables in(16.31). Figure 171 shows the region near the sonic point on the axis in thephysical plane, as investigated by Lighthill and Cherry.
T h e branch lineconsists of the characteristics Cb and C&~ curving toward the right fromthe point of contact with the sonic line. T h e continuation C of Cwhich(with an inflection point) goes down towards the left, is not a branch line[i.e.
the Jacobian d(q, 0)/d(x, y) ^ 0], and the same holds for the symmetriccontinuation C~ of the C ~. T o each point Ρ on the branch line C& belongsa q and a Θ (of course subject to the compatibility relation), and there issome point Q which has the same q and 0, therefore satisfying the samecompatibility relation (Fig. 171 shows the line of constant q and that ofconstant θ through Ρ and Q); the same arguments apply to the CT and C~.T o the left of the C+ ( C ~ ) , the correspondence between x,y and q,6 is oneto one. In the hodograph the C& and C are both mapped into one Γ ,the Cb~ and C~ into one Γ~, which appear respectively as one η-line, andone £-line, in the ^,^-plane. ( I t may be useful to visualize the mappinginto the hodograph of a closed curve, such as a circle with its center atthe sonic point of the x-axis.
Outside the two branches of the edge themapping is one to one. T h e image of the curve we are considering crossesthe Γ , continues between the two branches towards the Γ , goes backagain to the Γ , then down again crossing the Γ~, and continues outsidethe two branches, forming a closed curve.)+++b+b++++-+I n regard to the problem of transonic channel flow as compared withthe situation considered in A r t .
21, the essential difference is the following. There we could try to construct a flow along the lines of the corresponding incompressible flow, a flow which had the incompressible flow— -•axis* straightstreamlinelines ofconstant speedF I G . 171. Sonic point of straight streamline as double branch point in transonicchannel flow.25.2E X I S T E N C E OF F L O W P A S TPROFILE449as its limit. In the present problem, however, there is no limit case to guideus. Cherry has successfully overcome this difficulty by constructing a flowwhich exhibits typical channel-flow properties as explained above. For thiswork we refer to the original paper.572.
Problem of existence of flow past a profile58In preceding articles we have pointed out repeatedly that the nonlinearity of our basic equations was a source of essential difficulties. Suchdifficulties occur even if the problem is known to be entirely subsonic(elliptic) or entirely supersonic (hyperbolic). A second difficulty, the possibility of a partly elliptic, partly hyperbolic problem, is not restricted to nonlinear equations..This may be seen in the example of the linear Tricomiequation, y(d u/dx ) + (d u/dy ) = 0, which is elliptic for y > 0 and hyperbolic for y < 0. (Chaplygin's equation (17.24) can be approximated byTricomi's equation for values of σ close to zero, i.e., values of q close toq .) However, since this equation is linear we can indicate in advance theregions of elliptic and of hyperbolic behavior and the transition line, namelyy = 0.2222tT h e deep-seated complication typified by the mixed flow of a compressible fluid lies in the combination of transition and nonlinearity whichmakes the hyperbolic or elliptic character depend on the solution considered,so that the transition line also varies, depending on the solution.
Furthermore, a particular solution is to be singled out by appropriate boundaryconditions; but the boundary conditions appropriate to an elliptic problemare quite different from those for a hyperbolic problem. Thus we anticipatea new situation which has not yet been clarified, and which is at the basisof the difficulty of indicating correct boundary conditions for mixed problems, such as channel flow, flow past a profile, etc.T o fix the ideas, consider the flow past a profile and specifically the following situation: a steady uniform flow parallel to the a:-axis is disturbedby the presence of a body which has, in the z,?/-plane, a (convex) contour Ρwith continuously changing tangent and curvature; we suppose given a(p, p)-relation and a scale factor q or a relative to which the velocityq* of the undisturbed flow is subsonic* Then, in analogy to the corresponding boundary-value problem for incompressible flow past a smooth profile,we might ask: Does there exist a smooth potential flow (that is, a flowwithout viscosity and heat conduction, and also without shock discontinuities) past the given profile Ρ which coincides at infinity with the givenundisturbed flow? Specifically, does there exist a potential function <p(x,y)with continuous first and second derivatives which satisfies Eq.
(16.14)everywhere outside the profile Ρ together with the conditions θφ/θη = 060m83* We consider here qmas fixed and q°° (or M°°) as a varying parameter.V. 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 S450along Ρ (where d/dn denotes differentiation in the direction normal to P)and, as ζ = χ + iy —> oo, the condition θφ/θχ —> q°, θφ/dy —> 0?Let gdenote the greatest velocity attained in the compressible flowunder consideration; then with q now fixed, gdepends on both q° andΡ and satisfies qq . If g< qt, then the entire flow is subsonic;however, if gis supersonic, the flow is necessarily mixed, i.e., transonic,since the velocity is subsonic at infinity, by hypothesis, and also at stagnation points on the profile.m a xm a xmmaxmm a xm a xIn the first case, that of purely subsonic flow, a potential flow alwaysexists, just as in the case of the analogous incompressible problem.
This wasshown independently by L. Bers and by M . Shiffman. M o r e precisely theyhave shown that to a given profile Ρ of the type described above therecorresponds a subsonic velocity qi < q such that, for q°° < q\, there is aunique solution of the boundary-value problem formulated above; moreover, that this solution is purely subsonic: #< q , and, as q°° varies overthe open interval from 0 to qi, the maximum velocity gvaries over therange from 0 to q . (If the profile Ρ is symmetric with respect to the x-axisit can even be shown that gis a monotonic function of q°°.) If q* = qi,then the velocity must be sonic at some point of the profile and if q" > q\,local supersonic speeds cannot be excluded; both the existence and uniqueness proofs fail.A t present our main concern is the case of a subsonic q° > gi .
W e askwhether a solution exists, in the form of a mixed flow including supersonicregions (where there may occur the type of discontinuities which we havefound possible for hyperbolic problems) which is still a potential flow.M o r e precisely, we ask whether, as q° varies over an appropriate intervalbeyond qi, there exist mixed flows past a given smooth convex profile P ,in which the supersonic regions form enclosures, or "pockets'' adjacentto the profile. (See Fig.
175, disregarding for the moment the lines OTiand OT . See also Fig. 166.) This last requirement is based upon previousexamples and on the fact that, in subsonic flow, the maximum velocity isattained on the profile.61tm a xtm a xtm a x323. Apparent conflict between mathematical evidence and experiment.Certainly, smooth transonic flows can exist; in fact, we have studiedseveral concrete examples of this type.
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