R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 74
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153. Deflection of a uniform stream by strong and weak shocks.and δfor the given Mi . From (32) we find that the strong shockhas an inclination of σι = 83°42' to the incident stream and the weak shockan inclination of σι = 39° 19'. These are indicated in the figure. T h e corresponding values of Mare 1.9879 and 1.2671, and hence from Eqs.
(21)and (22) we obtainOmax8 θ ηin= 0.5794and0.8032,= 4.4438and1.7066,^ = 2.6487and1.4584,M2n^PiPiT h e two values of σ are 83°42' - 10° = 73°42' and 39°19' and the corresponding values of M are 0.6037 and 1.6405.210° = 2 9 Ί 9 ' ,2Article 23Examples Involving Shocks1. Comparison of deflections caused by shocks and simple wavesW e first derive a result concerning the similarity of transitions throughshocks and simple waves, which is useful in numerical problems involvingthese phenomena.
Consider a compression through a simple wave in whichan incident stream with Mi = 2 is deflected by 10°. From Sec. 18.2 we findfor the state 2 after the deflection: p /pi = 1.7052, p /pi = 1.4640 andM = 1.6514. These values are almost the same as those given at the endof the last article for a deflection of the same amount through a weakshock. This can be understood from the following considerations.222W e wish to compare the transition through a weak shock front from afixed state 1 to a variable state 2, with that through a simple wave.
For400V. 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 Seither of the transitions we may consider the state 2 to be a function of thevariable deflection δ =0— 0i · Three similar properties of these two2types of transition have been found.(a) T o the second order of the density difference p — P i , the quan2tity p/premains constant across a shock. Through aysimplewave it remains exactly constant.(b) T h e same relation, (22.3c') with fixed state 1, holds betweenp,2p , and q for the two phenomena.22(c) If a weak shock for which δ < 0 is compared with a forward wave,and a weak shock for which δ >0 with a backward wave, thenthe coefficients of the first two terms in the expansion of q — qi2in powers of 6 — 0i are the same for the two phenomena in each2case.T h e first property was discussed in Sec.
14.3 and mentioned again in Sec.22.3; the second follows from the fact that (22.3c') is both a shock condition and an expression of Bernoulli's equation for the simple wave. T h ethird is a restatement of the osculatory property of the shock polar and thetwo epicycloids through its corner given in Sec. 22.4.I t follows from (a) and ( b ) that the expansion property (c) holds alsofor p2— pi and p — pi . Furthermore if F(p, p, g, 0) is any function of the2state variables p, p, q, and 0 which has a Taylor series at pi,then the same expansion property holds for F— F2xp i , qi, 0 i ,.
Thus, for instance, ifthe transition takes place through a weak shock for which δ < 0 or througha forward wave, we may w riter(1)F 2F,= F[(d-20 ) +Xi<(02-0χ) +2O(0-20 ) ,X3the two phenomena differing only in the term O ( 0 — 0 i ) . Here F' and F"32are the first and second derivatives of F with respect to 0 taken along aΓ ^characteristic; thusp'=(l V ^ j .
^ d F . d ^ d F . d I ^~a^dpa^d^dedqda~fwhere dp/άθ, dp/dd, and dq/dd signify the rates of change of pressure,density and speed with polar angle along a r -characteristic in the hodo+graph plane. Hence="^ Ι- Λνρ α^ρ)( 2 )ηαρ+++βθ'—pqdq and on a r -characteristicsince from Bernoulli's equation dp =dq =βξ\+q tan a dd. Likewise, if the transition takes place through a weakshock for which δ > 0 or through a backward wave(3)F 2F, = ' Λ ( 02-00 +¥F {e*x-0O + O(0 220O323.1C O M P A R I S O N OF S H O C K S A N D S I M P L E W A V E S401where now the differentiation is along a Γ -characteristic, i.e.,/λ\Γ4.'Pf,d F1dF\T h e function F may, of course, contain pi,dFdF~\, 0i as parameters. Also,pi,qifrom (2) and (4) it follows that if F (as a function of p, p, q, 0) is even in0 — 0i then 'Fi =— F[ and "F; this result is a consequence of the= F"xsymmetry of the pair ofΓ-characteristics through the point q i , in thehodograph, since this ensures that F varies with 0 — 0 on ΓX+in the sameway as with 0i — 0 on Γ~~.For example, consider the component of velocity after the transition inthe direction of the initial velocity qi , which in Sec.
.22.3 was designatedby U. Then F =U = q cos (0 — 0i) is an even function of the deflection0 — 0 i . Applying the differentiation in (2) to this function we have= q [tana cos(0 -F'F"= 2q tan a i l L\0i) -sin(0 - 0 i ) ] ,. " j " ) tana cos(0 sm 2a'0i) -7sin(0 -0i).J2On putting 0 = 0 in these equations and substituting the results into ( 1 ) ,:we obtainU2~Qi00 += a(0 -g l26(0 20i)+ O(0 -20O ,32wherea = t a n «b =d-y + i)sin 2α/t^a i^i—= v1ϊ>< -=32t *~ °)M+8M4(Mi2-l)42This formula applies if the transition is through a shock for which δ < 0or a forward wave.
For the other case, the only difference is that a is to bereplaced by —a.Suppose that now we restrict our attention to forward waves and shocksfor which the deflection is negative. Let the final state be denoted by 2,1when the transition is a shock, and by 2,2 when it is a simple wave. N o w(1) gives F ,2 as a function of 0 along the Γ ^characteristic through (qi , 0 i ) .22Hence, since this characteristic(#2,2, 02.2)is also ther -characteristic+we may differentiate (1) to giveFl*= F[ +F?(0/<= Fi0(02 -2-00 + 0 ( 0 2 -(5)2+0i).0i) ,2through402V.But F' ,i,2I N T E G R A T I O N T H E O R YA N DSHOCKSdiffer from F , , F , 2 , respectively, b y terms O ( 0F'li2222— 0i)3since F' and F" are themselves functions of the state variables, see ( 2 ) , andthe states 2,1 and 2,2 differ by this amount.
Hence Eqs. ( 5 ) will hold alsofor F'2,1 and / ^ ' ι .If now the flow passes through a second transition to a third state 3, wehave(6)F-aF= F (d22282where, if the first transition is a shock F2= F^.i, F"2a simple waveF 2 = F , 2 , F "3= F^'i, and if it is= F ^ · In either.case2F= F[ + F x ( 0 -Fl= F'[ +2Θ2) + 0(03 - 0 ) ,θ ) + JF?(0 --z00 +2O(0 -0 ) ,22X(7)0(02 -0i).From Eqs.
( 1 ) , ( 6 ) , and ( 7 ) we now obtainF8-(Fa -Fi =(0=F i3F ) +(F2-0l) +2-FO\Fl(h0l) +"20 ( 03 -02 , 02 "0l) .3Clearly the same argument may be extended to any number of transitions.If a fixed initial state 1 is connected to a variable final state 2 by a series oftransitions, each of which is either a weak shock with negative deflection or aforward wave, then for any function(8)F2-Fx =F((0-2p, q, 0) :F(p,00 +^ ( 0 2 -310O +0(Δ ),32where Δ is the biggest deflection (regardless of sign) caused by the transitions,and a prime denotes the differentiationin ( 2 ) .
// each of the transitionsiseither a weak shock with positive deflection or a backward wave, then a similarformulaF f replaced by F\,holds with F[,f" F i respectively, a prime nowdenoting the differentiation in ( 3 ) . Moreover, if F is an even function of 0 — 0 i ,then F[=- F i and F? =,"Fx .2.
Supersonic flow along a partially inclined wallW e consider now a limiting case of the problem discussed in Sec. 22.1.First the point A is chosen as the origin and the point Β removed toinfinity. T h e fluid is therefore assumed to be passing horizontally acrossthe positive ?/-axis at constant pressure p and density p , and with uniform00supersonic speed q . These boundary conditions, namely0a = α ,0q = q > a,000 = 0on χ = 0 for all y > 0,tare realized if the coordinate system is moved in the negative ^-directionwith speed </o into the fluid at rest.Secondly, we consider a wall which slopes upwards at an angle δ for a023.2FLOWA L O N GAP A R T I A L L Y I N C L I N E D403W A L Ldistance I to the point D , and then runs horizontally, see Fig. 154.
T h ewall introduces the additional boundary conditionsθ = δ > 0ony — x tan δforθ = 0ony = I sinfor000 < χ <I cosδ ,0(9)δ0χ >I cosδ .0This represents the limiting case of walls which have small curvature except in the neighborhood of two points A and Z>, where the curvature becomes very large; such walls were considered in Sec.
22.1.T h e boundary conditions on χ = 0 determine a unique continuous solution above the line AC which is inclined at an angle a = arc sin (a /(?o) tothe horizontal. This solution is represented in Fig. 154 b y the set of horizontal streamlines, and the figure is drawn under the assumption that δis larger than the M a c h angle a . I n this case it is immediately evidentthat this continuous solution is inconsistent with the boundary conditionson AD. Thus a flow pattern which includes a shock line must be sought.0000I n Sec.
22.7 it was shown that provided δ is less than a certain maximum(depending on Μ ) there are two possible positions for a straight shock A S(Fig. 155) which deflects the incident flow abruptly into the directionθ = δ . W e shall not consider here a shock with subsonic flow behind it.Thus our discussion will be limited to values of δ ^ δβοη and furthermore to the weak shock for such a value.000032In the hodograph plane (Fig. 155) the point P with polar coordinates,q = qo, Θ = 0 represents the whole region of the physical plane in whichq and θ remain constant.
T h e end point P i of the shock transition lies onor outside the sonic circle on the upper half of the shock polar with cornerat P , and has the polar angle θ = δ . T h e straight shock front A S in thephysical plane is perpendicular to P o P i , and after passing through it thefluid particles move parallel to the wall AD.T h e deflected streamlines may be given a second deflection, equal andopposite to the first, by means of a simple wave centered at D, see Sec. 18.3.T w o such waves are possible, a forward and a backward one, correspondingto the segments of Γ - and r~-characteristics, P i P * andP1P2 , joining00χ0+F I G . 154.
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