R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 83
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Μ = 0.51,7 = 1.405.to solve the mathematical problem of analytic continuation. (This wasdone in S^c. 21.3 by means of an integral representation; other proceduresof analytic continuation are also available.*)Flows which for q —> oo reduce to flow around a circle, Co, have beenconsidered by several authors and by means of more or less different methods. Cherry has carried out such a solution in detail.
H e assumes a freestream Mach number M= Mi = 0.51 ( τ = η = 0.05) and the flow iscontinued beyond the sonic line, Μ1 (r< = 0.17), the maximum Machnumber of the flow being Μ = 1.39, r = 0.28 (Fig. 166). T h e figure showsthat for small values of Μ, the contour C (the streamline ψ = 0 ) is veryclose to a circle Co ; at about Μ = 0.86 ( τ = 0.13) the contour C startsto deviate from the circle by exhibiting a smaller ordinate than the circle.T h e flow obtained is thus a flow past a symmetric oval C, which stays closeto a circle.m510 000This flow around C does not encounter a limit line. If, however, the continuation of the solution inside C—which has no physical meaning—is computed, a limit line appears at about ψ = —0.06, with cusp well off the contour ψ = 0.
Cherry has also obtained, with the same condition at infinity,the flow past a slightly cambered cylinder (contour C is no longer symmetric to the x-axis, but slightly thicker above and thinner below thex-axis as compared to C). In this case, the maximum Mach number rises52* See e.g. papers cited in Note 52.444V. I N T E G R A T I O N T H E O R Y A N DSHOCKSto Μ = 1.56 and the eusp of the limit line comes quite close to C: at the cuspthe Mach number is approximately 1.34.
Of course, if the cusp were tobreak through the contour, there would no longer be a physically validsolution. For M0 0= 0.6 a flow (with Co a circle) has been computed showinga limit line which has actually penetrated the curve from the inside, whilefor parts of the contour where Μ^0.5 the curve obtained differs littlefrom a circle and no other limit line appears in the flow. I t should be keptin mind that we consider here for one and the same given contour P0afamily of flows (with the free-stream Mach number M°° as parameter);each of these reduces to flow about P0as q—> <*>; however, the shape ofmP , i.e. of the profile determined, depends upon M , so that in varying M°°0 0we do not study various flows about the same profile.
(This is also important for the evaluation of the so-called limit line conjecture. See Sec. 4.)Cherry has also shown how to proceed in the problem of flow past a circleif circulation is present. LighthhTs method in this case meets with certaintheoretical difficulties.53Lighthill, as well as Cherry, has indicated a general method for continuingthe flow around a contour into the supersonic region. (For solutions in thesubsonic region see Sees. 21.3 and 21.4). A s in the example of the circlethese methods are based on series expansions of W o ( f ) and systematic procedures are given for finding the analytic continuation starting with anappropriate branch of the hodograph flow.
Considerable practical difficulties are encountered.Apart from these difficulties let us reconsider the basis of our investigations (cf. also beginning of Sec. 21.1). W e started out to find a compressibleflow past a given closed contour P0; in other words, we wanted a solutionof the compressible flow equations with Pas a streamline. W e obtained,0however, a family of flows depending on Μas parameter, which, only as00some representative quantity tends to a limit, has the given P0as a streamline.
T h e shape actually obtained for the contour Ρ in the #,?/-plane dependson the value ofand varies with it. One can conclude from the results ofBergman, Cherry, and Lighthill that for values of M0 < Μ0 0< Μι,0 0in a certain intervala purely subsonic flow around a closed profile P(Af°°) =PMin the physical plane obtains. One can even assert that for values of M°° in aninterval beyond Μι,say Μι< M°° < Μ2<1 there exists a flow past aΡ Μ , with an imbedded supersonic region. This value M2very close to Mi.may however beT h e size of the supersonic region as well as the value ofthe maximum Mach number in that region depends on the value of M°°[in the interval (Mi,M )].2On the other hand, when a hodograph solutionis constructed, even by a completely correct analytic continuation, so long25.1SOMEADDITIONAL445PROBLEMSas we are not certain of the value of Μ (and therefore whether the chosenM°° is less than i t ) , we cannot be sure that the final profile Pin the£,//-plane will be closed, without double point, etc.2MI n practice, if one works out a solution to the end, actually determiningthe physical-plane flow and the curve Ρ Μ (as in Fig.
166), and if this Ρ Μis closed, is not reached by a limit line, and is sufficiently close to P , thenof course one will have found an approximate solution of the originalproblem.0(b) Remarks on channel flow. Consider a channel (we use the words ductand nozzle in the same sense), symmetric with respect to a straight axis,or center line, which we take as the rc-axis. In the so-called de Laval nozzle,there is a minimum cross section at the throat, at χ = 0, say; on each sideof the throat the cross sections increase symmetrically.
T h e contour isthus formed by two lines converging through the " e n t r y " section to thethroat, then diverging from the throat through the " e x i t " section. T w omain types of flow can be distinguished, which we describe in the simplifiedone-dimensional or "hydraulic" way, where it is assumed (a) that we mayneglect the deviation of q from the horizontal direction, and ( b ) that over across section normal to the center line the speed q and consequentlypressure, density, etc., are the same. Then the flow may be either symmetrical with respect to the throat with subsonic velocities on both sides ofit, and subsonic or sonic speed at the throat, or it may be asymmetrical with subsonic speed on one side of the throat and supersonic speedon the other (at the throat the speed is sonic).In the symmetric type the flow starts from a state of high pressure withvelocity zero at χ = — oo.
I t then accelerates while expanding throughoutthe converging entry section, reaching its maximum velocity at the throat.Then it decelerates while being compressed and takes on zero velocityagain at infinity. This is the simplified one-dimensional description. A c tually, the velocity is not the same throughout a cross section. For givencontour of a channel it is possible that the speed remains subsonic throughout, even across the whole throat section; or, for the same contour, it mayremain subsonic along the x-axis, the axis of the channel, while two supersonic " p o c k e t s " symmetric to both x- and ?/-axes form next to the wall,with the greatest speed reached at the wall.
Figure 167 shows curves ofconstant speed, the shaded regions representing the supersonic enclosures.54A nonsymmetric type of channel flow occurs when the ratio of entrancepressure to exit pressure is above a certain limit. T h e velocity rises fromzero at χ = — oo to a supersonic value to the right of the throat, while the gasis expanding.
(Notice the previously mentioned fact that subsonic flow iscompressed and supersonic flow expanded in a diverging section.) T h e lines55446V. 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 Sof constant speed are now quite different from those in the preceding case,as indicated in Fig. 168. A t the wall the sonic speed is reached upstream ofthe throat, and at the center line downstream of it.This flow exhibits a singularity so far not encountered in our examplesof plane flow, namely a branch line. This singularity, typical of asymmetricchannel flow, has been investigated by Lighthill and by Cherry.
T h ephenomenon may be qualitatively understood if we follow in the hodographthe velocities along a transonic streamline. W i t h the axis of the channelas the x-axis, q = 0 everywhere along the center line of the channel sothat the g -axis in the hodograph is the streamline φ = 0 (Fig. 169). Consider a streamline in the upper half of the channel, y > 0; in the subsonicregion q < 0 on this streamline; but on this same streamline in the supersonic region eventually q > 0.
Thus, each streamline intersects the lineψ = 0 at a second point, in addition to the hodograph origin from which allstreamlines start. These streamlines intersect each other and they have anenvelope. W e know from the general theory, A r t . 19, that this envelope is56yzyyM>1M>1F I G . 167. Symmetric channel flow with supersonic enclosures (shaded); lines ofconstant speed are shown.yF I G . 168. Asymmetric channel flow with lines of constant speed.
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