R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 86
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3 ) . A t any rate, this assumption should notbe considered as part of the "limit-line conjecture".25.4LIMIT-LINE455CONJECTUREW e shall see at the end of this section that a careful consideration of thevarious examples which were discussed in the preceding sections in factdoes not support the limit-line conjecture. However, historically, thedecisive criticism was of a purely analytical nature. K . O. Friedrichs hasinvestigated the mathematical question (implied in the above-describedconjecture): Is it possible that / < 0 everywhere in the flow field forΜ< M , but that for M°° = M we find 1 = 0 somewhere in the supersonic region (i.e., in the interior of the pocket or on the sonic line or onthe contour of the body) ? His answer, derived under rather strong assumptions regarding the nature of the flow, is negative.
(Friedrichs' work waspreceded by results in the same direction due to A . A . Nikolskii and G. I .T a g a n o v which are of interest in several respects.) His result and proofwere improved by A . R. Manwell and later by C. S. Morawetz and I. I .Kolodner who were able to reduce Friedrichs' assumptions. H e had assumed solutions x(q,6) y{qfi) where χ and y were analytic functions of g, Θdepending continuously on M°°.
In the last-named paper the authors require only, in addition to the continuous dependence of the solution onthat the streamfunction ^(qfi) have continuous second derivatives.T h e result is based on a lemma which can be applied to either of the twosituations in which we are interested: the case of flow past a fixed profilestudied for varying values of M , and the case where the profile itselfvaries with M . Their main results are essentially as follows, (a) Considera potential flow past a given fixed profile with bounded curvature.
Assumethat the flow depends continuously on M°° and that, for a certain value ofM°°, the flow is mixed and has supersonic pockets. Then no limit line canappear in this supersonic pocket, either in its interior or on the boundary,(b) If by means of the hodograph method a set of profiles P in the physicalplane is constructed, depending continuously on M°° and if for an M= Ma limit line appears, the corresponding profile Ρcan no longer be everywhere of bounded curvature.68002269700 00 0M0 02MW e now ask: W h a t is the relation between the mathematical resultsreviewed in the preceding section and the above results of Friedrichs andother authors? Regarding all these theoretical results, we can see thatinasmuch as they have the same subject they clearly support each other.W e see that (a) can offer neither support nor contradiction since we haveno mathematical example of a family of transonic flows past a fixed profile.Further, let us confront statement (b) with what the study of hodographexamples has taught us.
W e studied flows past profiles Ρ which vary as aparameter M°° is varied in such a way that an originally subsonic flow(where M< M i ) becomes transonic for M ° ° > Mi . For these flows pastprofiles of bounded curvature, the limit line was found inside the profile(see Sec. 1), i.e., outside the flow region. As M was further increased, in aM0 00 0456V.
I N T E G R A T I O N T H E O R Y A N DSHOCKSrange Μι < Μ< M , a cusp of the limit line approached and finallyreached the changing profile Pwhich was thereby distorted; in particularat a common point of contour and limit line there is infinite curvature ofthe contour (as seen in Art. 19). All this is in complete agreement with theresults of Friedrichs and his followers: limit lines do not appear in the flowfield, i.e., outside of the profile or on it, and if a limit line meets the profileits curvature cannot be everywhere bounded.002MOn the other hand, we may wish to compare the mathematical resultsincluding those of Friedrichs and others with the experimental evidence,and in particular, to check the limit-line conjecture. T h e statement (a)of Kolodner and Morawetz which relates to a fixed profile may be comparedwith observations on fixed smooth profiles; we know that shocks have beenobserved in connection with deceleration but, according to (a) for a contourwith bounded curvature a limit line cannot appear in the flow field or onthe contour.
Thus, this comparison points against the conjecture "that theultimate collapse of potential transonic flow is due to a limit line in the flowfield".This is also confirmed by a careful consideration of the examples describedin Sec. 1. There are transonic flows past smooth profiles, showing supersonic enclosures with Mach numbers so far beyond one that experimentswould produce shocks—and no limit line appears in the flow field. Hence(insofar as the comparison between these mathematical results regardingvarying profiles and observations on fixed profiles is valid) the limitline conjecture does not work here. Also, at any rate, wherever limit lineswere obtained, they did not and could not show a relation to the abovementioned asymmetric behavior where shocks appear associated primarilywith transonic deceleration rather than acceleration.
W e repeat: the resultsof our computations and our examples (Sec. 17.4, A r t . 20, Sees. 21.3 and25.1) are in agreement and mutual confirmation with the mathematical results reported in the preceding pages. T h e limit-line conjecture even inthe general sense of some kind of parallelism between physical shocksand mathematical limit lines does not seem to hit the real problem.5.
The local approachT h e situation described in Sees. 2-4, with its contrast between physicalobservations and mathematical results, would be the more disturbing ifwe were in possession of a mathematical existence and uniqueness theoremregarding the problem explained at the beginning of Sec. 2. Since this isnot so, efforts have been made in the other direction with the aim of proving, or at least of making it plausible, that the known examples of flows arenot typical but rather exceptional.I t will be useful for what follows to introduce the concept of a "well s e t "25.5T H ELOCAL457APPROACHor "correctly s e t " problem, in the sense of J.
Hadamard. A boundaryvalue problem for a partial differential equation is said to be correctly setor correctly posed if a solution exists, is unique, and depends continuouslyon the given boundary data. T h e precise nature of the continuous dependence on the data must of course be specified in each problem, as wellas the class of functions among which one is looking for a solution. As-anexample of a correctly set problem, we mention the existence theorem forsubsonic flow, explained in Sec.
2 where the solution can be shown to varycontinuously with q° and with P.71Returning to our present subject, we shall show that in the neighborhoodof a profile Ρ past which a transonic potential flow with supersonic pocketsexists, other profiles can be found for which no such neighboring flow exists,although they satisfy all assumptions made at the beginning of Sec. 2.W e have seen in Sec. 18.3, Fig. 118, that in a specific instance we destroysmooth flow by introducing an arbitrarily small concave arc into the givencontour.
Convexity of the profile was, however, one of the hypotheses ofour problem (Sec. 2 ) . But it can be shown that a flow of the type we consider becomes impossible even if a part of the contour, within a supersonicpocket, is straightened along an arbitrarily small segment.This interesting result was first proved by A . A . Nikolskii and G. I.Taganov (see N o t e 69); we give a brief but complete proof, deriving firsta simple and basic inequality. If we put Μ = 1 in the first of Eqs.
(16.7)we obtain dd/dn = 0 or dd/dy = tan0 (dd/dx); then, the second Eq. (16.7)may be written as— sin 0 — — cos 0 + —dx- — = 0.cos θ dxdyN o w take the //-direction normal to the sonic line S, and pointing towardsubsonic velocities, and the ^-direction tangent to S so that we turn frompositive χ to positive y by + 9 0 ° ; then, we write d/da for d/dx and d/dvfor d/dy, and θ\ for the angle between sonic line and flow direction. Sincedq/da = 0 along S, we obtain immediately(ι)Λ3dv=qdecos 0i da '2From dq/dv = 0, it follows that dd/da0 and we obtain the "monotonicity l a w " : // a point moves along the sonic line S so that the subsonicregion is to the left, then the polar angle 0 of the velocity vector q at the pointcannot increase.W e now apply this result to our present problem.
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