R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 71

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Hence ρ /ρι = 1,so that Μ\ cannot be less than 1 and Mcannot be greater than 1. Onlyin the case of zero shock can either be equal to 1. In addition, as in (14.14)^22n2nΎ~2y1^ M22 n^ M l^xCO.For a physically possible shock, the component of velocity normal to theshock front is supersonic before and subsonic after the shock. The density ratioP2/P1 cannot exceed /i ( = 6 in air) and the square of the normal Mach numberafter the shock cannot be less than (7 — l)/2 7 ( = 1 / 7 ) , the extreme values cor­responding to an infinite pressure ratio p /pi = 0 0 . N o t e that since quthe Mach number Mi cannot be less than 1, nor can M be less than\/(7 — l)/27- I n addition since (9) gives a ^ ai we have M = M+v /a ^ Μ i + ν /a — M .

Thus the speed qi is supersonic, whereas q may22=2222n2222n2382V.INTEGRATIONTHEORYA N DSHOCKSbe subsonic or supersonic withΊ-ΖΛ2y< M22^ Μχ ^oo.2From the fact that Ui is supersonic it follows that sin σι = u\/qi > a\jq =sin a\. Similarly since w is subsonic we have sin σ < sin a , wheneverΜ > 1. Hence the acute angle between streamline and shock is no smallerthan the Mach angle αχ before the shock, and, when M ^ 1, no larger than theMach angle a after theshock. Moreover since M s i n V = M^ (τ — 1)/2γ,the angle after the shock is never smaller than arc sin \/2y/M \/y— 1.x2222222222n2(d) On crossing a shock line a fluid particle experiences an increase of en­tropy (and therefore of p/p ),the amount of increase depending on thevalues of ρ and ρ before and after the shock transition.

On the other hand,the theory of steady plane flow developed in Arts. 16 through 21, as ap­plied to a perfect fluid, is based on the assumptions that a relation p/p =constant holds throughout the fluid and that the flow is irrotational. I nother words, the results derived in the six preceding articles need not hold inthe region o] the x,y-plane beyond a shock line. T h e case in which all particlesundergo the same change in entropy is the exception. For then the flowbehind the shock will again be isentropic and, as we shall see in Sec. 24.1,irrotational.yyHowever, it may be recalled from Sec.

14.3 that the change in entropy isin most cases small. I n fact it is of the third order in (£ — 1), ξ = ρ /ρι,so that for shocks of moderate strength it is not a bad approximation toassume that the value of p/p does not change. Hence the conclusionsdrawn in Arts. 16 through 21 are approximately valid behind weakshocks.2y24(e) T h e shock conditions ( 3 a ) , ( 3 b ) , (3c) are three relations between thet w o sets of variables Ui, pi, pi and u , p , p . T h e remaining condition (3d)states that the tangential velocity components V i , v must be equal. Eachof the four conditions is unaltered when the subscripts 1 and 2 are inter­changed.

From the first three relations u , p , p may be expressed interms of U\, ρ , pi, thus in analogy to (14.20):2222222γU2 = -±—\2y^+7 + 1 |_2 = — ^ - r [-(τ7 + 1(17)V(72(/) If Ηn(y -mDpi-l)iJJf+ 2mtiJ,+ Dm2ypi + (7 —2l)mui'is the sum of the velocity head, corresponding to the normal com-22.3A N A L Y S I S OF T H E SHOCKponent u, and the pressure head, i.e. gH(18)H= Hln(u/2) + yp/(y — l ) p , then=n= h™22n383CONDITIONSAccording to condition (3d) this implies that(19)ffx=H,2where Η is the "total h e a d " which occurs in the Bernoulli equation.

Thus wesee that the total head of a par tick is unaltered on crossing the shock line.This conclusion could have been read off directly from (3c').W e have assumed in Sec. 2 that the motion on either side of the shock isstrictly adiabatic. N o w we have seen in Sec. 2.5 that for such a steadymotion gH is a constant on any streamline. This constant is q /2 = h q /2,where q is the maximum speed and q the sonic speed. Hence the maximumspeed and the sonic speed on any given streamline are the same before and afterthe shock. Moreover since gH = gH + v /2, E q . (18) may be written22mm2n(20)2tulU2= I(qj-f)= qt-If the shock is a normal shock, i.e., ν = 0 and σι = σ = 90°, then thislast relation reduces to2tfi?2 =q?.In this case the sonic speed on a streamline is the geometric mean of theparticle speeds before and after the shock.

Since qi ^ q it follows that fora normal shock qi is always supersonic and q subsonic. This conclusion isalso covered by the first result in ( c ) , since the normal component ofvelocity is now the speed itself.2522(g) If the dimensionless parametersPiη'Uip2£are introduced, then the points Pi(f, η) and P ( l / f , 1/17) lie on the hyperbolaof Fig. 75, and represent the given shock.

T h e Mach numbers Mand Mappear in the slopes — yM il and —yMof the lines A Pi and AP , re­spectively; these lines form equal angles with OA.T h e five quantities 77, f, ξ, M i and Mare determined when any one ofthem is given. This follows from the fact that any one of the five fixes thecorresponding points Pi and P on the hyperbola, and once these pointsare known each of the five can be read off from the figure by means of alength or a slope. Algebraic relations between them are given b y Eqs.(14.24) through (14.27) with M' replaced each time by M . Thus in par2in22 n22n22n2n384V.I N T E G R A T I O N T H E O R YA N DSHOCKSticular from Eqs. (14.25)P22y1η = — = — — Μ in — —,Pi7 + 1hM22(21)1=tξU2=pi=U,ς217 +1 ^ln=P2, 1_A '22and from (14.27)(22)(^M^-l^M^-l)^^.T h e Hugoniot equation, given by (14.26):(23)h\v- 0 = ^ - 1 ,f = *η=Piis unaltered.^,Pi26W i t h the same change, Fig.

76 gives the graphs of £, Mil,M2nversus η,for η ^ 1.4. Representation of a shock in the hodograph planeA t the beginning of A r t . 8 we saw that in steady flow the state of a mov­ing particle at any moment is determined by its velocity q. I t was assumedthat a (p, p)-relation holds on the streamline traced b y the particle andthat the Bernoulli constant (Sec. 2.5) is known.

For plane flow the velocityis represented by a point in the hodograph plane, and the continuous mo­tion of a particle by a curve in this plane.When a particle passes through a shock front its representative point(qXjq ) in the hodograph plane undergoes a sudden jump. W e shall nowystudy the conditions governing this discontinuous transition. For this pur­pose, see Fig. 143b, it is better to use the speed q and the angle σ (betweenthe streamline and shock front) to characterize the velocity, rather thanthe velocity components u = q sin σ and υ — q cos σ.W e assume that the position of the shock is a priori unknown, but thatthe state before the shock, i.e., the set of values p\, p i , qi, and 0i is given.N o w when pi,p i , q are known the shock conditions (3a) through (3d)xmay be considered as four equations governing the five quantities p , P 2 ,2(?2, <*\, <?2 · Since there are five unknowns and only four equations, there isa single infinity of solutions.

But to each set of values q ,σχ, σ22therecorresponds just one velocity vector behind the shock; its magnitude is q2and its direction is 0i +from a fixed initialσι — σ . Thus the end points of shock transitions2point lie on a curve in the hodograph plane. On this curvewe may take σι as parameter.L e t the points Qi,Q represent the velocity vectors q i , q in the hodo2222.4 R E P R E S E N T A T I O NIN T H E HODOGRAPHgraph plane, and take Cartesian axes O'U,O'VP L A N Ewith O'U along O'Qi,Fig.

144. T h e condition (3d) then implies that QiQ2385as inis perpendicular to theline O'S which represents the direction of the shock. Thus the coordinatesof the point Q are given by2U = ν cos σι +sin σι ,u2(24)γ=(U) cot σ ι .-qiT h e quantities u and ν occurring in these equations may be expressed in2terms of the given qi and the parameter σι by means of the relation (20)and the equation ν = q cos σι . Thusxq™ — qi cos σι..h qi sin σι1 / 22x= --— (q- v) =h UiT h e first of Eqs.

(24) now becomesu22m2(25)U = ( UB-2COSVI +U)AUA,where(26)UA^ ,UB=U,+q( l -iFrom the second of Eqs. (24) we have cotVi = V /(qi2cotVi = cosVi/(l -cosVi) = (U--UA)/(UB1^ .-U) , and from (25)U).On equating these2two expressions for cotVi we find that the locus of the point Q is the Car­2tesian leaf (Folium of Descartes)(27)F(β! -2~UuyuBU-<νthis curve is sketched in Fig. 145a.F I G . 144. t/,F-axes in the hodograph plane.386V.

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 S(a)(b)F I G . 145. Folium of Descartes and its construction.T h e fact that U g qi ^ U (or that the points A, Qi, Β lie in the ordershown) follows immediately from the conditions that qi be supersonic andless than q . Thus since the sonic speed q is q /h we have q ^ q /h orqi ^ q /h qi = U ; from q ^ q we have gi ^ qj/h + q (l - 1/h )or qi S (Qm /h qi) + qi(l — 1/A ) = U > Furthermore, it is easily verifiedfrom (26) that the points A and Qi are inverse with respect t o the soniccircle U + V = q /h . T h e folium is symmetric about the f/-axis,has a double point at Q i , and a double asymptote U = U .

This diagramis referred to as the shock polar, and was introduced b y A . Busemann.ABm2t2mA22222m2m222m222m222B2B27Equation (27), defining the shock polar, contains only distances meas­ured from the points A, B,Q , and the line joining these points. I t is pos­sible to give a geometrical construction of the shock polar based only onthese three points. L e t C be the circle on A Β as diameter, and Ρ any pointof C (Fig.

145b). Then the point of intersection, Q , of the line PD perpen­dicular t o AQiB, and the line QiE perpendicular to AP, lies on the shockpolar. A s Ρ traces out the circle C, the point Q traces out the polar. Forthe similar triangles Q DQi and AD Ρ yield Q D:QJ)= AD:PD,and forthe circle C we have (PD)= ADDB.HenceX22222/<W>Y (AD\*=VQiD/\PDjADDB'which is an alternative way of writing (27).T h e slope and curvature of each branch of the shock polar at the pointQi are of particular interest.

T h e first t w o derivatives of V with respect t o22.5SHOCK D I A G R A M A N D PRESSURE387HILLSU are easily computed from ( 2 7 ) ; on the branch with negative slope at Qi ,their values at this point are dV/dil=— Λ/(< ΖΙ — U )/(U— qi) andd V/dU= -((7B U )/V(qi- U )(U- ?i) . T h e correspondingradius of curvature is thereforeA22AA- C^XEJB - C/Λ).R=(28)B3BFrom (26) and #= 2<?#ι, the slope is found t o be — \/Mi — 1 =— cot «i where ai is the M a c h angle of the incident flow.

Thus this branchof the shock polar is inclined at an angle — (90° — a\) to the velocity vec­tor qi and hence is tangent at Qi to the characteristic epicycloid Γ~ throughthat point, see (16.37). Similarly, the other branch of the shock polar istangent at Qi to the r -epicycloid through Q . N o w the shock directionO'S in Fig.

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