R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 63

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I t is seen from (16)that the streamlines in the hodograph form a system of circles through theorigin with centers on the g^-axis; their equation isJ(16')zq = C sin Θ,* Regarding the case η = 1, see p. 341.P2(C > 0 ) .336IV. P L A N E STEADY P O T E N T I A LFLOW( W e make the restriction to C > 0, to avoid double covering of the physicalplane.) This shows that for C ^ qthe maximum speed on each stream­mline equals C and is reached for 0 = 90°, while the maximum velocity equalsqmfor C > q· If 0 is substituted from (16') into (17), a parametric repre­msentation of the streamlines in the x,y-p\sme is obtained with q as parameter.Thus, if Τ = T(q)denotes the integral in (17), we obtainFor comparison we consider the analogous incompressible flow, whichcan be derived by introducing ρ = p and a = oo in the preceding formulas,and obtain instead of (17)0_ , cos 20_ .

sin 20where we now assume that k is positive. Orχ = r cos 20,y = r sin 20,_kφ = \/kpo(r — x);hence for the streamlines of the incompressible flowr -kχ = —-C poor?/ =ά( * έ)·2+For various values of C the streamlines form a family of confocal parabolasin the physical plane and of circles in the hodograph (see Fig. 131). T h estreamline for C —> oo is the strip of the £-axis from χ = 0 to χ = +,and the flow may be considered as flow around this edge. T h e larger the val­ues of C, the more nearly the parabolas approach the edge. T h e velocityhas a constant value q on concentric circles about the origin, with radiusequal to k/2p q .0002T h e compressible flow cannot be considered as flow around the edge sincesome streamlines will reach a limit line before completing the turn. T o findthis line, <£, we compute from (16) d(<p,\l/)/d(q,e) = 0, using Eq.

(17.27), andfind(«•-ι)(«)'-ί(*)'-^«·«?.-.)-aHence the equation of the limit line in the hodograph, i.e., of the criticalcurve, is simply(18)cos0 =± ^thus consisting of two branches. Since we have assumed C > 0, we have20.3A FLOW W I T H IMBEDDED SUPERSONIC REGION337F I G . 131. Streamlines for incompressible flow around an edge.restricted ourselves to considering the hodograph solution in the upperhalf-plane q ^ 0. T h e branch to the right (0 ^ θ ^ 90°) corresponds tothe upper sign in (18), or θ = 90° - a, that to the left (90° ^ θ ^ 180°)to the lower sign in (18), or θ = 90° + a. W e next compute θφ/θξ and3φ/3η which for a < 90° are equivalent to hi and h ,y2θφk / .(sin Θ tan a + cos 0),— =qd£δφk , .,(sin 0 tan a — cos 0),Λ— =λΝqdr;and see that on the branch where θ = 90° — α, βφ/θξ ^ 0 θφ/θη = 0, whilefor θ = 90° + α, d^/<9£ = 0, di^/dr; ^ 0.

Hence the branch to the rightis an / , that to the left an l . T h e limit line in the hodograph is tangentto both the sonic circle and the maximum circle; in the polytropic case it isan ellipse (see the second Fig. 132 graphed for κ = y = 1.4). Combining(18) and (16') we find, for the polytropic case,2xιC 0 S(i1M2" q2ο,2~κ— 1 _~ ~~2aκ—12sC sin θ22or(19)sin θ 4-±AKsin 0 + ^ = 0 ,22~'338IV.P L A N ESTEADYPOTENTIALFLOWF I G . 132.

Ringleb flow in physical plane, and hodograph plane.as the relation between the constant C of a particular streamline and theinclination θ — θι of this streamline at the limit line. T h e analogous rela­tion between C and q = qi iswo?+Writing θι and qi in Eqs. (16') and (19) and solving we obtain(19")q, = C sin β,,sin θ, =2-±±K± i j / (K+1)*Hence the streamline corresponding to C reaches £ only if [(κ +1)/4] C22^20.3339A FLOW W I T H IMBEDDED SUPERSONIC REGIONa ; or with κ = 1.4, if C ^ |α = 1.67α .

I t is thus seen that limit singu­larities occur only on streamlines for whichβ8β(20)C^ia .8T o the smallest C-value, which equals f a , corresponds sin θι =(κ + l ) / 4 = f, M , = 4/(3 - JC) = 2.5, M = 1.58, q = ( 2 / ν Τ + Ί ) α . =1.29 a, ; the largest M -value on this streamline is M= 16/(3 — κ) =^r, ^f= M= 2.5. Equation (19") shows that those streamlines forwhich (20) holds with the inequality sign meet the limit line in the physicalplane in four real points, symmetric in pairs with respect to the z-axis, theaxis of symmetry; the general theory of Sec. 19.3 or direct computationshows that these are cusps of the streamlines (see first Fig. 132).T h e equation of the limit line in the z,y-plane is found in parametric formb y replacing in (17), θ from E q .

( 1 8 ) :2e1lt2max(is')m a x222y=* - * + * ! · ,± Zz .kVJpq Mpq MThis limit line is symmetric with respect to the z-axis. I t consists of twobranches: one on the positive side of the z-axis, corresponding to the rightbranch in the hodograph, and one on the negative side, corresponding tothe left one.

Each branch has a cusp and then extends towards infinity. T h eUpper branch, £ , goes from A to the cusp D and then to infinity; and thelower branch <£i is symmetric with respect to the z-axis. For Μ —> oo :cos θ = 0, a = 0, to which corresponds χ —> oo y —•» oo ; there hi = A = 0.T h e point A of «£ on the z-axis is a sonic point: Μ = 1; it is the image ofthe two different hodograph points A[ and A ' . W e see that at both A[ andA : yp = 0, ψβ ^ 0 (see A r t . 19.5).

T h e point A in the physical plane isnot a double limit point but the sonic point of an £ and of an <£i whichmap (exceptionally) into the same point A. This point is also the sonicpoint of a straight streamline. T h e streamline direction at A coincides withthat of the line of constant Θ.222222922q2W e have seen that a limit line generally has cusps (Sec. 19.3) which playan important role. W e determine them now for this particular problem.Computing from (18'), dx/dq and dy/dq and equating both derivatives tozero, we obtainq2= 2a -aq^,dqq =^ —V o— κ= 1.6a,cos θ = =fc - = ± 0.63,qcorresponding to the points D', D' in Fig.

132.N e x t we consider the curves of constant velocity. I t is seen from (17)that these curves are circles, as in the incompressible case, with centers onthe z-axis but no longer concentric. Thus, in particular, the sonic line in the340IV. P L A N E S T E A D Y P O T E N T I A LFLOWphysical plane is a circle. For the circle of velocity q, the abscissa of thecenter is k[T + (l/2pg )] and the radius is k/2pq, as in the incompressiblecase with ρ = p ; for q = q , the center recedes towards infinity.20mFrom what precedes, it is seen that there are three types of stream­lines in this example:(a) Streamlines on which the speed is entirely subsonic and which re­semble the streamlines of the incompressible flow around the edge, suchas streamline 1 in Fig.

132.(b) Smooth streamlines which are not entirely subsonic, such as stream­lines 2 and 3. Their maximum Mach number, however, is less than 2.5. T h e yhave zero speed at infinity; the speed increases and becomes sonic whenthe streamlines enter the circle of constant sonic velocity; it reaches a maxi­mum at the x-axis, and decreases again towards subsonic values. Their hodo­graph intersects the sonic circle but remains entirely inside the limit line.T h e streamline 4 corresponding to M= 2.5, separates the streamlines(b) from the next group.

In the hodograph this streamline is tangent to thelimit line, and in the physical plane it passes through the two symmetriccusps D and D of the limit line. A t these points it has infinite curvature.m a x(c) Streamlines on which the value of the maximum velocity is greaterthan ^a (with corresponding maximum Mach number greater than 2.5);they have cusps at their intersections with the limit line.

There are twotypes of such streamlines. T h e first, ( d ) , such as streamline 5 for instance,intersects the hodograph limit line at four points, which correspond to fourcusps in the physical plane. Such a line actually has three parts; the firstone (solid) extends in the hodograph from left to right through 0 ' , the sec­ond one (dashed) is outside of the hodograph limit line and the third one(dotted) back inside. T h e corresponding parts are shown in the physicalplane. T h e other type, ( c ) , e.g. streamline 7, intersects the hodograph limitline only twice and has accordingly two (symmetric) cusps. Streamline 6separates these two types.82T h e group (b) gives an example of streamlines leading from subsonic tosupersonic velocities and back again. This happens in a continuous way andwithout shock. W e may consider any two streamlines of type (b) as wallsof a channel; it is thus seen that passage through sonic speed in this type ofchannel flow is possible in isentropically accelerating and decelerating flow.This is an example of a smooth transonic flow.* (See also Sec.

25.3 ff.).4. Further comments and generalizationsT h e solution (8), which holds for η ^ — 1, — 2, · · · , has the propertythat F is analytic and —> 1 as τ —> 0, and there is just one such solution of* ^The meeting of streamlines such as 1, 2, 3, 4 with the outer branch of the limitline is only apparent.34120.4 F U R T H E R C O M M E N T S A N D G E N E R A L I Z A T I O N S( 7 ' ) in general. In the exceptional case η =— 1, there is, however, a secondsolution which has this property. W e may verify directly that/=(1 -τ)' */ (1 }satisfies ( 7 ' ) .

Thus in addition to ( 1 5 ) ,(21)φ = Ar~\l-sin 0,τ) -κΚκ ι)φ = Ar~h(l +TCOS 0)— 1 . This solution has featuresis one more solution corresponding t o n =quite similar to ( 1 5 ) . T h e flow has again a limit line with t w o cusps. A s in(15) there are in this flow smooth streamlines along which a subsonic flowbecomes sonic, then supersonic, and decelerates again to subsonic velocities.T h e streamline which passes through the cusps separates the regular smoothstreamlines from streamlines with cusps at the limit line.If in (15) and (21) we put Aq= C, where C remains fixed as qmm—> oo,it is seen that to these two compressible flows there corresponds the same incom­pressible flow ψ =(C/q) sin 0, φ =T h e t w o functions ψ-ι(τ)(C/q)cos 0, discussed in Sec.

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