R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 85

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In the absence of a mathematicalexistence proof, we begin with a survey of the properties of these knownsolutions, including also the somewhat analogous case of channel flow inwhich the walls of the channel replace the profile.** However, in contrast to the problem of the profile the available space in ttiechannel is limited.25.3MATHEMATICALEVIDENCE ANDEXPERIMENT451F I G . 172. T w o transonic nozzles derived from Ringleb's flow.In Ringleb's flow, Fig. 132, we may consider two streamlines as fixedwalls, as in Fig. 172. W e obtain in this way examples of smooth transonicchannel flow; each nozzle shows a supersonic enclosure and there is aspecific maximum speed obtained at one of the walls.

For Fig. 172: q^l* is ob­tained at the vertex of streamline 3, while q^l* appears at the vertex ofstreamline 4, and q^l* > q^l* ; on all streamlines q° = 0.* Actually we ob­tain in this way a whole family of smooth transonic flows with varying g ,the corresponding maximum value of Μ being well above one.

W e maychoose as walls of our channels, smooth streamlines not reached by thelimit line (as in Fig. 172a). Between such smooth walls the theoretical flowaccelerates continuously from a subsonic through sonic to supersonicspeed and smoothly decelerates again.m a xA n exact transonic flow solution, not of the pocket type, but rather ofthe type of the nonsymmetric subsonic-supersonic channel flow, is themixed spiral flow between two streamlines (Sec.

17.4). W e may also men­tion the vortex flow between two concentric circles, separated by the circlewhere Μ = 1. In this flow there is no limit line.In Sec. 21.3 we constructed in detail a flow past a profile similar to a* N o t e that in this case q° (or ΛΓ ) would not be an appropriate parameter.0452V. I N T E G R A T I O N T H E O R Y A N DSHOCKScircle and in Sec.

1 we discussed the work of Cherry and his students, whocomputed such flows with supersonic enclosures and also flows past pro­files similar to other prescribed shapes (see N o t e 52). W e also know thatto a given profile Ρ there exists a speed qi, such that for g < qi only apurely subsonic flow past the given Ρ results.From general considerations and the results of Lighthill and Cherry wecan add that, for each given P , there exists a subsonic q > qi such thatfor any q in the interval between qi and q a transonic flow with super­sonic enclosures is obtained about some smooth profile, P(q°°) = P .However, in all these examples of flows defined so as to reduce to flow pasta given profile P , as q —> °° the actual profile P , found by the indirecthodograph method, changes in shape with q°°.

Hence, these results do notafford examples of flows past a given fixed profile Ρ for varying free streamspeeds. (In the channel interpretation of Ringleb's flow the effect of chang­ing profile is also observed, inasmuch as the shapes of the walls are changedas we consider regions of increasing g ) .W e mention in this connection interesting results found by S. Tomotikaand K. Tamada by means of a procedure specially adapted to the region oftransonic flow. T h e profiles they obtain vary with q°° (or M°°); they found,however, that for varying (subsonic) values of M°° ( M = 0.717, 0.745 and0.752) the resulting profiles Pwere almost identical.

One may thus con­sider their flows as an approximation to flow about a fixed profile forvarying values of the parameter.Finally, flows past a given profile P , as well as various types of channelflow, have been computed by some of the usual approximation methods.W e shall give a few results obtained by such methods inasmuch as theyare of interest for the present problem. G. I .

Taylor has computed a sym­metric nozzle flow with supersonic pockets by means of a series expansionin the physical plane. Another method, of which the idea is due to Prandtl,consists in an iteration procedure with the "linearized f l o w " (PrandtlGlauert method) as first approximation. Arranged, by H . Gortler, in theform of an expansion in powers of a thickness parameter, it likewise pro­duced smooth transonic flow patterns past given profiles; no obstacle seemsto appear in the computations. Finally, H . Emmons successfully appliedthe classical method of replacing the original differential equations in thephysical plane by difference equations and solving the resulting system ofa finite number of algebraic equations by means of an iteration procedure(a method denoted today as "relaxation m e t h o d " ) .

Again, the numericalwork could be carried through and some examples of potential flows withbounded supersonic regions were o b t a i n e d .0002002Q0m9qm a x620 0M636465Thus we see: Exact transonic flows (spiral flow, Ringleb flow, etc.) existfor certain specific contours; flows with supersonic enclosures have likewise25.4LIMIT-LINECONJECTURE453been found (by means of hodograph methods) past contours determined aApproximate solutions past given contours have been found byvarious direct methods.posteriori.There seems, however, to be little experimental confirmation for theexistence of such smooth mixed potential flows with imbedded supersonicregions.

Smooth acceleration from subsonic to supersonic flow has been ob­served. But the observed deceleration from supersonic to subsonic flow isnot such as might be expected from the mathematical evidence. In general,the deceleration takes place in a nonisentropic way by means of a shock,although deceleration with no observed shocks has also been reported forvalues of q° not much larger than #i. There is no mathematical counter­part in the work discussed above (based on potential flow, which is revers­ible) to the observed stormy deceleration, and to the general lack ofsmoothness in observed transonic flows..

Thus there seems to emerge adefinite discrepancy between certain theoretical results and the observa­tions.66However, the experimental observations are made on a given fixed profilewith some appropriate parameter, saybeing varied so as to generatea family of flows past the same profile*; whereas in our theoretical ex­amples the profile Pchanges along with M . Comparing observations onflows about a fixed profile Ρ with mathematical results regarding flowsabout a sequence of varying profiles Ρ presupposes the unproved assump­tion that the essential features are the same for the two different situa­tions.

I t is possible that the difference between the two situations may beof minor importance, in view of results such as those of Tomotika andTamada where almost identical profiles Ρ were found for different valuesof M°°. However, as long as we have no more definite information on thispoint we cannot strictly speak of a discrepancy between observations andtheory, since they do not refer to the same situation.0 0MMM4. Limit-line conjectureT h e general lack of experimental counterpart to the computed examplesof smooth transonic flows points to a discrepancy between theory and ob­servation, although it is true that in the computations we do not strictlyreproduce the experimental situation. On the other hand, we note thatsome remarkable singularities are found in the mathematical study offlows, namely, limit lines.

Can this furnish an explanation?Comments and suggestions of v. Karman, Tsien, K u o , and others,prompted to a considerable extent by Ringleb's example, deal with thispossibility. After enumerating some of the singular properties of limit lines(infinite acceleration, infinite pressure gradient, streamlines "turning back"* Similar comments apply to channel flow.454V. I N T E G R A T I O N T H E O R Y A N DSHOCKSat the limit line and causing " a quite impossible flow pattern , etc.) v.Karman concludes that of the fundamental physical assumptions, con­tinuity and irrotationality, which underly our analysis, one must be un­tenable. "Since the continuity cannot be violated, it must be assumed thatthe flow becomes rotational." This cannot happen in an inviscid steadycompressible flow free of shocks.

Thus the appearance of a limit line isconsidered by v. Karman as the (mathematical) criterion for the (physical)breakdown of steady isentropic inviscid (irrotational) flow.*,,67This idea, which is at the basis of the limit-line conjecture, has not beenformulated as a precise statement but rather as a general conception whichpostulates for the problem under consideration a mutual dependence betweenobserved physical shocks and computed limit lines in the flow field. (Otherexplanations based on quite different ideas have also been proposed by thesame authors, in particular by Tsien.) Historically, the limit-line conjec­ture is strongly connected with the actual investigation of the mathe­matical properties of such lines; since these properties turned out to beof a very singular nature implying the breakdown of the smooth mathe­matical potential flow, a relation suggested itself between this breakdownand the physical phenomenon of shocks.

I t was then thought that theconverse was also true, namely, that a physical shock implied a mathe­matical "shock-solution", exhibiting a limit line.In more precise terms the conjecture may be described as follows. In thecase of flow past a fixed profile, we know that if the Mach number atinfinity is less than a certain Mi then all velocities are subsonic. I t is thengenerally assumed that there is a value M > Mi (which may be very closeto Mi) such that for M between Mi and M smooth mixed flows resultwhich under certain circumstances should be of the "pocket t y p e " . f I tis assumed on one hand that the physical flow, for these values of M ,is without shocks and on the other hand, that the Jacobian / = d(x, y)/d(q, Θ) (or an equivalent determinant) is different from zero everywherein the flow field.

(For subsonic velocities this Jacobian is negative, exceptperhaps at isolated points.) If then M°° is further increased to M°° =M,shocks will become manifest and at the same time, in the mathematicaldescription of the problem, 1 = 0 will be found along some infinitesimallysmall arc of curve. This limit line will become more pronounced if Μ°° isstill further increased.20 020 02* Ringleb says that at the limit line there occurs a Stroemungsstoss and he callsthe limit line Stosslinie, " S t o s s " being the German word for shock.f I t seems that this last point is generally assumed by analogy with the sim­ilar fact which holds for flows past varying profiles and on account of results ob­tained by approximation methods (Sec.

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