R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 51
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16.4 and 16.7, when weattempted in various ways to derive the Mach lines from the Γ-lines, corresponding to certain given initial data. Since the transition there, however,was based on the use of characteristics, the considerations were limited tothe supersonic case. Also—in contrast to the present approach—themethods of those sections were numerical or graphical methods.Let us now comment on the final step, the determination of q,B in termsof x,y. W e return to our starting point where a particular solution 4/{q,6) ofEq. (16.320 was known in some region R of the g,0-plane, where q ^ q .Our objective is to find, if possible, the velocity q over the x,7/-regioncorresponding to R.
W e saw that, if pq ^ 0, the four derivatives dx/dq,• ·, dy/θθ are determined by (250,terms of d<p/dq, · · · , Θψ/ΘΒ, andintegration gives χ = x(q,B) y = y(q,B).N o w , these last equations must be inverted. T h e flow in the physicalplane, in the region corresponding to R, is determined if q and 0 are givenas single-valued functions of x,y, i.e. ifm2m}d(x,y)d{qfi)=d(x,y) 3(φ,φ)3{φ,ψ) d(q,B)270IV. P L A N E S T E A D Y P O T E N T I A LFLOWwhere, from (16.31),Since ρ and Μ are given functions of q, the vanishing of D depends only onthe given solution ypiqfl) in β . Tnese circumstances will be discussed indetail in Art.
19. One result, however, is obvious, namely, that in subsonic flow, / can vanish only if all four derivatives θψ/dq, · · · , Θφ/ΘΘvanish; then, also dx/dq, · · · , dy/θθ vanish. I t can be seen that this canhappen only at isolated points. (For a more complete discussion see Sec.19.2.)On the other hand, if Μ > 1 the determinant D, considered as a functionof q and Θ, can vanish as the difference of two positive terms. T h e locusD(qfl)= 0 defines in general a curve in the hodograph whose image inthe flow plane we call a limit line.
W e found such lines in certain instances in Chapters I I and I I I . I t was seen that in the neighborhood ofsuch a limiting line there are two flows which meet there; by that we meanthat both flows have there the same q and 0. These circumstances, and alsothe situation for Μ = 1, will become clear when we study more examplesand the general theory. A t any rate, as long as there is no singularity inthe mapping of the flow plane onto the hodograph and vice versa there isequivalence between results in these planes, but whenever the mapping issingular, we need a study of the mapping in the neighborhood of thesingularity.In actual problems one is faced with additional difficulties.
So far wemade the assumption that a particular solution, say tiqfl),of the hodograph equation had been found, and w e found a difficulty in the possibleoccurrence of limit lines. Important practical problems, however, such asflow in a duct or around a profile, are boundary-value problems, and herenew circumstances arise. Only a few indications are in order at this time.First, in such problems, the boundary conditions are given in the physical plane, and in general it is not possible to derive from them boundaryconditions in the hodograph, which would determine the correspondinglinear hodograph problem.
W e may think of flow around a profile, with thetypical boundary condition that the contour in the physical plane forms astreamline i.e. that φ = 0 along the given contour. Since we cannot derivefrom these data the velocity distribution q along the given contour, we donot know the image of the contour in the hodograph and cannot set upthe corresponding linear hodograph problem. Moreover, and this is perhaps an even more fundamental difficulty, in many cases we are not certainwhether we are in possession of correct boundary conditions even in thephysical plane, correct in the sense that a uniquely determined solution of17.4 E X A C T H O D O G R A P HSOLUTIONS271the problem exists which satisfies the differential equations and the boundary conditions, and depends, in an appropriate sense, continuously Uponthe boundary data.
Much of this difficulty follows from the nonlineajity:in fact the elliptic or hyperbolic character of our nonlinear equationsdepends upon the solution under consideration, which, in turn, should besingled out by means of the boundary conditions; and it is well-knownthat in the t w o cases of elliptic and hyperbolic problems quite different setsof boundary conditions must be given in order to determine a solution.(See A r t .
25).Consider, e.g., a duct with supersonic flow at the entrance section (seeend of Sec. 16.4). I n this case the flow will be uniquely determined as longas it remains supersonic. If, however, the flow does not remain supersonic,then it is no longer determined b y the conditions at the entrance. Or consider the flow around a profile: if we know or assume that the problem isentirely subsonic, the analogy t o incompressible problems leads t o theformulation of boundary conditions in the physical plane, for which definitemathematical results have recently been reached (see Sec.
25.2). If, however, we cannot exclude the possibility of a mixed problem, we do notknow whether the assumed boundary conditions, suggested by physicalintuition and mathematical analogy, determine a solution (i.e., whetherwe are not prescribing either too little or too much), quite apart from thetask of obtaining the solution.Given this state of affairs we shall to a great extent have recourse toindirect methods.
Starting with examples of exact solutions, one may thencharacterize a posteriori the physical situation to which the flow defined b ythis exact solution actually corresponds. These examples are interesting assuch; in addition, they illustrate certain general theoretical facts that maybe typical. This holds for the simple examples which will be considered inthe following section as well as for the problems which we shall study inA r t . 20 after having added to our knowledge of the basic theory. Actually,even the more powerful methods to be presented in A r t .
21 constitute onlyan indirect approach, although there the object is to solve boundary valueproblems.4. Radial flow, vortex flow, and spiral flow obtained as exact solutions in thehodographW e shall now consider as first examples of exact hodograph solutions,some of the flows which have already been obtained in the physical planein A r t . 7. This will lead to new insight into certain properties of theseflows.(a) Compressible vortex flow.
This is a particular case of the axially symmetric flow of Sec 7.5, with radial velocity q = 0. Here we take as a starting13r272IV. P L A N E S T E A D Y P O T E N T I A L FLOWpoint equation (12) and the particular solution(28)Φ =(C > 0 ) .-ΟΘFrom Eqs. (7) and ( 4 " ) we findθΦ— = 0 = χ cos 0 +dqθΦ— =duy sin 0,—C = q{y cos 0 — χ sin 0),and consequently,_ C sin 0 _ Ctfj/9 θ Λq_C cos 0 _q_Cg*q2q2I t is easy to invert these equations and to obtain(30)*=%from which follows(31)_^ = tan0 = ^ =dxQx,or+xy=—yQas the equation of the streamlines, which are here concentric circles withconstant value of the speed q along each circle. T o the maximum valueq = qmcorresponds, by (31), a minimum valuerim n =r of the radius r,xand C = q r\ .
T h e circulation has the same value Γ for all streamlines:m(32)j> q.rfl =Φ q.rfl =Γ =j>φ qrdd= 2*rq= 2TTC,orΓand η =(30)r/2Kqmr= τΓ. Also-=ny1+( — I ) 3 P=v i-ρ-* ·W e see that on the circle r = r\, where g = g , density and pressure arewzero. As r increases indefinitely, q decreases towards q = 0, and ρ and ρincrease from zero toward their stagnation values (see Fig. 100a,b).the circle r = hri the velocity is sonic, Μ=14On1; hence there is supersonicflow in the ring between r = η and r = hn and subsonic flow outside. FromEq. (6') we find φ =—Φ = (70, which confirms that the equipotential linesare the orthogonal trajectories of the streamlines.(b) Radialmotion.Source and sink. N e x t , we wish to consider radial17.4 E X A C T H O D O G R A P H S O L U T I O N S273motion, i.e., a flow that goes radially from or toward a center (see Sec.
7.3).T h e solution of (16.32')(33)ψ = ke,where k is an arbitrary constant, corresponds to straight hodograph streamlines through 0'. B y means of (16.31) and (25') we find the coordinatefunctionskχ = — cos 0,(34)ky = — sin 0.pqpqand see that the physical streamlines are indeed straight lines through thesubsonic Tea ionr* hr,,M = 1=r,, M=co(a)1.00.80.60.40.20I2h3r/r,45(b)F i g . 100. Compressible vortex flow, (a) Circular streamlines, (b) Density, pressure, velocity versus distance r.274IV. P L A N E STEADY P O T E N T I A LFLOWorigin (see Fig. 101).
From (34)(34')r, k=z±=—='pqm2wpq'where ± m is the strength per unit length of the source, and the ± signs apply according as t o whether k ^ 0. T h e circles: r = constant, are linesof constant q\ they are perpendicular to the radial streamlines; hencethey are the equipotential lines which are here concentric circles. Also,substituting in (34') for ρ or q, respectively, we obtain, with p = 1,s(34")2r= *(l--£Y=±k± - - ( 1-ρ*" )1T h e relation ( 3 4 " ) between q and r has been discussed in Sec. 7.3. (Theresupersonic streamlinesubsonic streamlinei.o0.6\M<1^^^^0.40.2^—M>1M<1r/r"t(b)FIG. 101.
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