R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 77
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B y elimination of M = Μwe then obtain a relation betweenώ, ω and δ. T h e equation in M , δ, and ω is obtained from Eq. (22.37) bywriting η = cot ω and2τ2τ2r= cot (ω — δ) =261On solving for M ,2we have— €Τι,e = tan δ.414V. 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 ST h e second equation is obtained from this one b y changing the subscripts1 and 2 to 2 and Ϊ respectively, and at the same time e t o — e:(16M)_2Y(A2-1)(€ +5+21)2where clearly rTZ(T*_er )[er,(1 +2-21)(h? -l)r, +hh]9= cot ώ.T h e right-hand sides of Eqs. (15) and (16) become identical on settingτ2 = —τι.This possibility must be excluded since it corresponds to anequal and opposite transition immediately following the first, and such atransition would violate the inequality condition (22.16) for shocks.
Onequating these t w o right-hand members w e obtain a cubic equation for ras a function of η and e. One root of this equation is r2= —τ12and is to^e neglected; the t w o remaining roots satisfy(17)Ar+ Br? + C = 0,22T 2 = cot ώ,whereA = (h— 2)er2+2xΒ =-(hC =-h e(nl)n[(€-22(h-2l ) n - h e,1) (e --222l)Tl-2e],+ 1).2For each given pair of values ω, δ the t w o corresponding values of ώ aredetermined by this last equation.In order to express the coefficients A, B, C in (17) in terms of the originalparameters ω and 77, we eliminate e from them b y means of E q . (22.38):(1 8)* =_/ , • ,x7>θ(τι« + "ΐ) - Ί '"If a common factor — τ ι ( τ ι + l ) / [ s ( r iis ignored, they may then be written:2A(17')= s[(h2l ) e--—(Λ — l ) r , [ ( r iC =fc [s(r:S+222+!TZ2— τχτA t the other we have jj =(20)222+1) -1,1],« , for which (17) reduces toτ=0,2S21) -2°° or s = h /(hA„TTl1) - 1].A t one extreme we have η = 1 or s =(19)1) — l ] of the three coefficients+!(Λ» — i ) ( n — ί ) ·2)] ({h* -Β =2=S+Β τ?Μ2+= 0orn.— 1) for which it reduces toC .
= 0,23.4OBLIQUE SHOCKREFLECTION415whereA„= 2hWPoo =+-Ti[hWQi ++1),(2h21)],-= h [hW + 1].2T h e graphs of ώ versus ώ corresponding to Eqs. (19) and (20) for y = 7/5are indicated by broken lines in Fig. 160. T h e y intersect at the points:ω = ώ = 0°; ω = 0°, ώ = 90°; ω = ώ = ωο whereω= arc cot Λ / ( Τ +01)/(3 -y)= 39 14'.0For each value of s between the extreme values, the corresponding graphof ώ versus ω passes through these three points and lies in the region enclosed by the broken lines.
T w o such curves, for η =1.25 and η = 5, aredrawn in the figure.For each value of s there is a maximum value of ω (minimum ofn)beyond which (17) has imaginary roots, and a solution of the type considered does not exist. T h e condition for this maximum is the vanishing ofthe discriminant B2— 4 AC, and with ζ =Dz" -Ez2+Fz -(τ2+1) it becomesG = 0,ω (degrees)F I G .
160. Angle of reflection versus angle of incidence for selected incident shockstrengths.416V. I N T E G R A T I O N T H E O R Y A N DSHOCKSwhereD= (ft -i)Y,Ε= (ft -1 ) V + 4ft (ftF= 2(ft -G= (ft +2221) (3ft -22- l)s2l)s2--34ft (ft22(Λ -2Λ4-23)s +-2(ft2i) .22These maximum values of ω are plotted versus pi/p ( = I/77) in Fig. 161for 7 = \. T o any point which lies on or below this curve there correspondsa solution of the kind indicated in this section.25.
Properties of the reflectionT o find the pressure ratio η = ρ^/ρτ across the reflected shock we eliminate e between (18) and its analogue for the second shock, namelyTJ=* "sir,2+ 1) + 1 '_Sh + f\2="(Λ 2DO? -1) 'This last equation is obtained from (18) by changing the subscripts 1 and2 to 2 and I respectively, and at the same time e to — e. W e then obtainF I G . 161. Incident angle and pressure ratios for which reflection is possible.23.5P R O P E R T I E S OF T H E417REFLECTION0.710203040ω (degrees)5060F I G . 162. T o t a l pressure ratio versus incidence angle for η = 1.25, 5.which expresses s in terms of τ , s and the appropriate root r of (17). InFig. 162 the graphs of pi/pz = 1/ηη versus ω are plotted for the two valuesη = 1.25 and η = 5, with the direction of increasing ώ indicated by arrowheads.
T h e y show that as ώ increases, the total compression p^/pi acrossthe two shocks initially decreases slightly and then increases indefinitely.2λFor fixed ω, the Taylor developments of the larger root of (17) throughterms of the first order in (η — 1), see (17'), and of fj, see (21), throughterms of the second, give(Λ - l)n 2r2^η ~τι +Κ(η— 1),η + L(T? — l) ,2Κ=L = —2τι(Λ(η2-2+Ifi1)'l)(n + Λ)f(Λ* + lW(ri21+1)These formulas apply to the weak reflected shock.
According to Eq. (22.23)the corresponding development for the density ratio is7i)2T o an observer at rest with respect to the wall, the incident and reflectedshocks will have velocities of propagation c and c which satisfy(22)c cosec ω = c cosec ω,418V.I N T E G R A T I O NT H E O R YA N DSHOCKSthis being the condition that the point of intersection of the two shocksshould always lie at the wall.
Thus c/c = (η+ ΐγ/(τ+ 1)* has aTaylor development22-~i-^—(y,1).T h e equivalent formulas in the case of a strong reflected shock are left forthe reader to derive.T h e principal results concerning oblique shock reflection are as follows:The weak as well as the strong reflected shock can have an appreciably different inclination to the wall from that of the incident shock, and considerableincreases of pressure and density may be caused by the reflection.
T h e formerimplies also that the velocity of propagation of the reflected shock may beappreciably different from that of the incident shock, see Eq. (22). For each ηthere is just one value of ω for which the reflected shock can have the same inclination to the wall as the incident shock, and this value is independent of η.T h e results in Sec. 15.1 concerning the head-on reflection of a shock canbe recaptured by considering a limiting case. Thus as ω —» 0, i.e. η —> oo,we see from (17') that A = 0 ( n ) , 3 = O(n),C = 0 ( n ) , so that thelarger root of (17) tends to infinity with η such that2(23n -)1™\ -2( t f - D .
- f r . - 2)·-J)I n addition e —* 0, according to (18), in such a way that er tends to 1/s.xHence from (15), or (16), we see thatT o an observer who is at rest with respect to the wall, the flow behind theincident shock will be perpendicular to the shock, and its M a c h number,M say, different from M . However, since he moves in the ^/-direction,components of velocity perpendicular to the wall are unaltered.
HenceM cos co = M sin δ, so that M and M e tend to the same limit. If wedenote this limit by M itself, we find from (24)2222222From (22), (23), and (25) it follows thatcc2(h2~*2(h2-l)S+-1)S -(h2-(h? -2)2) *This result agrees with E q . (15.3') when c is replaced by — c' in the limit,23.6since clearly MINTERSECTIONis the M22419OF T W O SHOCKSof Sec.
15.1. W e leave it for the reader to verifythat the limiting pressure and density ratios are given by Eqs. (15.4') and(15.5')·One interesting result can be deduced without knowing these limit ratiosexplicitly. From (21) we find that in the limitThis is also the form taken by Eq. (21) when η = r2= h/\/h — 2 . Since2s and s are simple functions of η and η respectively, this means that foreach η the pressure ratio across the reflected shock is the same for the obliquereflection ω = ώ = ω as it is for head-on reflection.
For ώ < ω (see Fig. 162),00p-z/pi is less than that for head-on reflection, while for ώ > ω it is greater.06. Intersection of two shocksA simple example of the interaction of shocks occurs when t w o shocklines intersect. Assume that fluid crossing the segment A Β of the 2/-axis(Fig. 163) is in a state of uniform supersonic motion: q = q , 0 = 0, ρ = p ,00ρ = po, a = a < q . Consider two shocks located on oppositely inclined0and BC, the first causing the state (/ο,0,ρ ,Ροlines AC0where 0i >qifihPhPip ,p220to change to0, and the second causing it to change toq fi ,22where 0 < 0 . A t the line χ = x passing through C, the particles3920below C have speed qi, direction 0 , pressure pi and density p , while thoseΧabove C have q , 0 , p ,222xp .
In the hodograph plane, F i g . 163, these t w o2states are represented b y t w o points 1, 2 lying on the shock polar withcorner 0 at (q , 0 ) , with chords 01 and 02 perpendicular t o AC and BC0respectively.Under suitable conditions, we can find a flow pattern for χ > x0whichsatisfies all requirements by assuming that t w o new shock fronts (reflectedmaximumyΒAXF I G . 163.
Intersection of two shocks crossing a uniform stream.V. 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 S420shocks) form along appropriate lines CD and CE through C. T h e two statesof the particles after passing through these second shock fronts will berepresented by two points 3 and 4; the point 3 lies on the shock polar withcorner at 1 (the line 13 being perpendicular to CD),and 4 on the shockpolar with corner at 2 (the line 24 being perpendicular to CE).The twopoints 3 and 4 must satisfy the same two conditions as in Sec. 3, and againthe dividing streamline CF through C will, in general, be a contact discontinuity.I t is easy, in principle, to find the values of q, 0, ρ, ρ after the secondshocks, when the states I and 2 are given.
W e simply apply Eqs. (22.21)and (22.32) to the transitions across CD and CE. Using ρ and 0 to denotethe final values of the pressure and stream inclination, we obtainΛΛιχVP l(1 -cWd)ri++1]yL(7 + D i nd) '(1 -2 M?Γ( 2 6 >{CT\,Θ = 0i + arc tan/2+ 1)AJ2for the transition across CD, and(c'r0 =(2?02+ arc tan (1 -PL(TPi2+d')r2+(1 -d')WΓ)2c'W+1)(T22+i)for that across CE. In these equations η and r are the cotangents of the2inclinations of CD and CE to the streamlines in ACDand BCErespectively, the first being negative and second positive; the primes in E q . (27)indicate that Mi , in the definitions (22.31) of c and d must be replacedbyM.2B y eliminating η between the two equations (26) and rbetween the2two equations ( 2 7 ) , we obtain two relations between 0 and p, one for thetransition across CD and the other for that across CE.
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