R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 73
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T h e latter corresponds to η = 0 or normal shock. In both cases the resulting deflectionis zero.T h e symmetry allows attention to be focussed on the range 0 :§τι^cot α? , which corresponds to the top half of the shock polar. I t is clear2from Fig. 149 that for each value of Μι , e has a maximum in this range.This is in agreement with the statement about δ which was made at thebeginning of this section. T h e value of η for which this maximum occurs is found from (32) by setting de/άτιroot of the equation c ( l — C)TI +AT*+equal to zero.
I t is therefore a(3c — 2cd — d)n2Λτ,2+C =0,-\)M?+ d(l— d) = 0 orwhereA= (γ +\)ΜΐΒ=(y +1)Λ/=- ( i l / , - l)[(y-C2Χ4+2,+2(yDMf++4,2].The roots of this equation which correspond to real τι are given in terms22.6T H EDEFLECTIONOF AS T R E A M L I N EB YA393SHOCKof Mi by,T\— cot[(γ +21)M*σι — — —(34)+2[(γ +2(71) M? +D M Γ + 4]2]+ l ) [ ( y + l)Jf « + 8(γ -, Mj* V(yt2[(T+l)Mt+21)Μ! ΤΊ6]22]and the root in question is the positive square root of this expression. Asomewhat simpler expression is obtained for the sine of the angle σι atwhich maximum deflection occurs:sin σι=[(+7l) Μ ι(35),4]-V(y+Ι)[( Ύ + i ) ^ !4+8(-7DM?+16]For each given Mi the maximum possible deflection, 5, of the streamline may be determined by first computing the position σι of the shockfrom (34) or ( 3 5 ) , and substituting the result in (32) to obtain the corresponding € = tan δ. A graph of 5 x versus Mi is given in Fig.
150, and afew corresponding values in Table V I (Sec. 7 ) .I t is clear (see Fig. 149) that the maximum deflection increases monotonically with Mi . Letting Μ ι —• oo we have from (34), n —> 1/h and from(32), e —> (A — 1)/2A = l/-\/y — 1. Thus a streamline cannot be deflectedby more than arc sin (I/7) by means of a shock, whatever the incidentMach number may be.
For 7 = 7/5 this angle is 45° 35'.m a xma2222T h e formula (32) can also be used to find rThus since δ = σι — σ it yields= cot σ in terms of η .222T2τι1 +Tir(cn +- c)n +2(122d)n(1-d)'50/S 40ε60appro aches 4 5° 3 5 'as2010035Μ,79F I G . 1 5 0 . Graph of maximum deflection angle versus incident Mach number.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 S394TM,«cc2u/(cot αi.cot 0,)/ /™41_tΛ-1>M,»1I1- f>c/(-cot' av-cot ° i > //F I G . 151. Relation between shock angles for fixed incident Mach number,so that(36)(πT2 =-whereCI=—c21)TITIc - dW +en +(1 -(7 +l)Mi '4+di2(1 -d)dnd+4'(7 - 1 ) M + 2(7 + 1 ) M*X= 1 -22X2For each value of M i there is a corresponding curve in the ri,r -plane2relating the inclinations σ ,χσ of the streamline to the shock before and2after the transition, see Fig.
151. A s Mi increases from 1 to 0 0 , ci decreasesmonotonically from 2/(7 +1) to 0, and di from 1 to (7 — l)/(7 +1).Thus for each positive τι , (36) shows that r increases monotonically with2Μ ι fromMi = 1:T2Μ ι = oo ;T(7 + l ) n2n + (7 + 1) '2to(7+I)2-7(7 -7T TI ,1)and vice versa for n < 0. E v e r y curve (36) which corresponds to a physi-22.6T H E D E F L E C T I O N OF A S T R E A M L I N E B Y A S H O C K395eally possible transition lies between these last t w o .
Each such curve intersects the line r = η at the origin and the t w o points (zbcot ai, dbcot αχ).2Only the segment of the curve which lies between these last t w o points isof interest since for a physically possible shock the inequality (33) must besatisfied.T h e formula (36) was obtained by algebraic manipulation of Eqs. ( 3 ) .N o w interchange of the subscripts 1 and 2 leaves these equations unaltered.I t follows therefore that interchange of these subscripts in (36) will yield anew formula, which is also a consequence of Eqs. ( 3 ) . Thus we obtainC T222+«2where=°+ 1)Μ7= ( τ ~ 1)M+( + 1)Mβ2(222'22272This may be considered as an equation determining M22*at any point of the2η , r -plane.2Consider the family of curves in the n , T - p l a n e given by (37) with un2restricted parameter M .2This family is the reflection in the line rof the family given by E q .
(36) with the parameter Mi2=ηunrestricted.N o w we have just seen that only those points lying between the linesr2=τι and r2=(7 +1) 7-1/(7 — 1) correspond to physically possibleshocks. Accordingly, only that segment of each of the curves (37) whichlies in this wedge-shaped region is of physical interest. T h e bounds on σ2given in Sec. 3 can now be verified b y considering the values of r at the2ends of this segment.In some problems it is the pressure ratio η across the shock, and not theM a c h number Mi in front, which is known (see Sec. 23.4). T h e deviationof η from 1 is a measure of the strength of the shock front, as also are thedeviations of the equivalent quantities ξ, f, Mi nand M. From Fig.
143b2nwe haveητ2_ cotσιcot σ2_ v/uiby ( T I +since σ2=u2Ui2which provides a relation betweeen σι,b y f. N o w_v/u_'σ and the strength represented2σι — δ, we may replace τe ) / ( l — c-τι). Then on solving for e we have2in this equation396V. I N T E G R A T I O N T H E O R Y A N D€SHOCKSapproach 0as Ty -*coiii1^>ΊF I G . 152. Relation between deflection and shock angle for fixed shock strength,where=51Ζξ - 1=1-f=^ +(Λ*- 1)^-1)'1For each shock strength s, (38) represents a curve in the e,ri-plane,see Fig. 152.
A s η increases from 1 to o o , s decreases monotonically from ooto h /(h - 1) = (7 + l ) / 2 . Thus for fixed positive η , (38) shows that eincreases monotonically with η from22η = 1 :6 = 0,to'V6( 7 + l ) n2+ ( 7 - l ) 'All points lying between these two curves correspond to physically possibletransitions. Notice that the curve η = oo is the same as Mi = oo in Fig. 149.Equation (38) can also be obtained by setting Μ= M( n + 1) inthe coefficients c and d of Eq.
(32) and using (21).22in27. Strong and weak shocksW e have seen that for each given supersonic state pi, p i , qi, 0i there is asimple infinity of states p , P 2 , qi, 02 which satisfy the shock conditions.Thus at least one further condition is required in order to fix the state 2.In many problems (see Sees. 23.2 and 23.3) this condition consists in prescribing the deflection δ = 0 — 0i of the streamline. W e assume, for definiteness, that the given value of δ is positive and also, of course, that it isno larger than 5 x .22m a22.7STRONG A N D WEAK397SHOCKSWhen Mi and δ are known, (32) becomes a cubic equation for η . Thiscubic has just two roots lying in the range given by the inequality (33),(see Fig.
149). If δ = 5 x these roots are coincident and there is just onepossible inclination σι of the shock. In this case the state 2 is uniquely determined. If however δ < othese roots are distinct and there are twopossible inclinations for the shock. T o each there corresponds a differentstate 2 behind the shock. T h e transition corresponding to the larger root(smaller σ ι ) will be called the weak shock for the given deflection, and thatcorresponding to the smaller root (larger σ ι ) , the strong shock. T o justifythese names we notice that as η increases, Mil = Μι/(τι+ 1) decreases,and therefore the pressure ratio p /pi decreases, see (21).
In Fig. 145a theweak shock is given by the position of Q nearer to Qi and the strong shockby the position nearer to A. A s δ —> 0 the strong one tends to the normalshock and the weak to the zero shock.mam a x3022In general the flow behind the strong shock is subsonic and that behindthe weak shock supersonic, see Fig. 149. However this distinction fails ifδ is too close to £.
T o prove this it is sufficient to show that for anyMi > 1 the positive value, δ, of δ for which the flow is sonic behind theshock (i.e., M = 1), is not equal to o. W e shall do this by showing thatthe value of η corresponding to δis in fact always larger than that corresponding to δ· Thus we shall also have proved that the flow behind thestrong shock is always subsonic, see Fig. 149.m a x3 θ ηm a x28 θ ηι η 8 ΧAccording to (37), when M= 1 the angles σι and σ are related by222rT l22+(+71)*But r can always be expressed in terms of η and Mi2by means of (36).Thus on eliminating r we find22n2+(7 +Hence for each Mi^l)(ciT!+ dif2-(7 +l)(cin2+ d ) = 0.x1, the value of η for which δ = δ8 0 ηis a root of theequationΆτι+ΕτιC =+0,whereA = 2,Β = (γ +C =-(M1)M!t2-4-1)[((3 T-y)M,2l)Mt2+4,+ 2].T h e roots of this equation which correspond to real η are given by398V.T l2 = otC2I N T E G R A T I O N=σ ι- 1 [(γ +(39)T H E O R Y-Ϋ,ΜΐA N D(3 -SHOCKS+ 4]y)M?2+ ^-V(y+l)[(y+\)MJ2(3 --+ (γ + 9 ) ] ,y)MJ4and the root in question is the positive square root of this last expression.Equation (39) should be compared with (34), which gives the value offor which δ = o[(γ +.
2Smη. Also the formulam a x-1)Μι(3 -7)]Wσ ιV ( 7 + 1)[(7 + l)Mi - 2(3 - 7) MJ + (τ + 9)]4+4γΜι2compares with (35).T o show that the value of η given by (39) is always larger than thatgiven by (34) we consider the behavior of the functions f(x)C and g(x)Bx +~ Ax+2C for χ > 0. Clearly f(x)Bx +both positive for sufficiently large a;, and f(x)— ^(x) = (7 +Ax+and g(x)=are\)M x(x22+1)is positive for χ > 0. Hence 0 ( 2 ) is negative when χ takes on the positiveroot of fix)of g(x)= 0, but eventually becomes positive. Thus the positive root= 0, given by (39), is larger than that oif(x)Table V I gives o= 0, given by (34)., δ η and the corresponding values of σι for various80m a xvalues of Mi and 7 = 7/5.
T h e corresponding curves in the ri,e-plane areindicated in Fig. 149. T h e difference between the t w o values of σι neverexceeds 4°30'. Similarly om a x— δis never greater than 30'.8 θ ηA s an example we consider a uniform stream with Mibe deflected through an angle δ == 2 which is to10° by means of a straight shock lineOS through a given point 0 , see Fig. 153. This value of δ is less than bothT A B L ETHEDEFLECTIONS6m a xVALUESANDVI5OF σι8 0n(yWITH=CORRESPONDING7/5)Maximum DeflectionM2= 1Mi0*11.01.52.02.53.03.54.0οο90°66°36'64°40'64°48'65°15'65°41'66° 3'67°48/Omax0"1Oson0°12° 6'22 58'29°48'34° 4'36°52'38°47'45°35'90°62°15'βΐ^'62°39'63°46'64°37'65°15'67°48'0°11°41'22°43'29°40'34° 1'36°50'38°45'45°35'023.1COMPARISONOFSHOCKSA N DSIMPLEWAVES399F I G .
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