R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 72
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144 is perpendicular to QiQyS which in the limit Q —+ Qi becomes the tangent at Qi . Hence for very weak shocks (q /qi close to 1) theangle between the shock front and the incident streamline is approximatelythe Mach angle en ,m22+x2228A s Q approaches Qi in Fig. 145b, the line A Ρ approaches the normaldirection to the negatively inclined branch of the shock polar at Qi · Moreover, in the limit D coincides with Q , and the length of the segment A Ρis then, according to (28), the radius of curvature R. Therefore the centerof curvature Ζ is the intersection of the line through Qi parallel to the limitposition of AP and the tangent to the circle C at A. Thus, the projectionof Ζ on O'Qi is A, the inverse of Q\ with respect to the sonic circle.
Fromthe construction given in Sec. 16.5 for the center of curvature Ζ of thecorresponding characteristic epicycloid, it can be shown that Ζ has thesame property. Hence Ζ = Ζ and the two curves actually osculate at Qi.2xT h e Cartesian leaf is determined b y the values qi and q alone. Each pointQ on it represents a transition satisfying conditions (3a) through ( 3 d ) .However, these equations do not determine which of the points Qi, Qirepresents the state in front of the shock. But we have seen in the precedingsection that in a physically possible transition the speed must decrease.Thus Q can represent conditions behind the shock only if it lies on theclosed loop of the folium. I n the following, the term shock polar will referonly to this closed loop.
T h e remainder of the folium inside the maximumcircle represents those states from \vhich the state Qi can be reached b ytransition through a shock.m225 . Shock diagram and pressure hillsApart from size and orientation in the q , # -plane the shock polar is determined b y the ratio q\/q or, according t o (16.10) and (16.11), b y theMach number Mi of the incident stream. Hence all polars corresponding tothe same value of Mi are similar.
This remark enables us to confine our atxmy388V. I N T E G R A T I O N T H E O R Y A N DSHOCKSVuF I G . 146. T h e shock diagram.tention for the rest of this section to those polars which correspond to afixed direction of q i , and q = 1. Alternatively the velocity q in the following may be considered to represent the dimensionless velocity q/q ofany particular case.mmFor q = 1 the sonic speed q is 1/h, so that qi may vary between 1/hand 1. A s qi increases from 1/h to 1, U decreases from 1/h to 1/h , and inthe U, F-plane the corresponding shock polar expands from an infinitesimal circuit around the point (1/h, 0) to the circle with the points (1/h , 0)and ( 1 , 0) at the ends of a diameter, see Fig. 146. A l l shock polars lie between these extreme curves and to each point Q within the circle there corresponds one possible shock: 0 ' Q is the velocity vector q , the corner Qiof the polar through Q gives the velocity vector qi = O'Qi, and the perpendicular to Q i Q determines the shock direction O'S.
T h e graph of thisfamily of polars is called the shock diagram.mt2A222222So far the pressure and density have entered only through the Bernoulliconstant q /2 which is the same on either side of the shock line and equalto each member of (3c'). A s we have seen in (8.24), this constant is a multiple of p /p and therefore of the stagnation temperature T . Thus thestagnation temperature has the same value on either side of the shock.However, the stagnation pressure p (and hence the stagnation density p )does change on crossing the shock.2m88sssFor any given stagnation pressure p , the pressure itself may be introduced by erecting the pressure hill (see Sec. 8.2) on the £/,F-plane of theshock polar, i.e., on the hodograph plane.
For the polytropic case this hill isgiven by (16.12). On setting κ = y and q = 1 this equation becomes8m(29)ρ = p«[l -2p/(T-l)q]q2=U*+V\22.5389SHOCK D I A G R A M A N D PRESSURE HILLST h e hill, which is a surface of revolution, is shown in Fig. 147a. Its base isthe circle q = q = 1 and its summit lies on the p-axis at the point ρ = p .Consider now the intersections of the plane perpendicular to the U,Vplane whose trace in that plane is QiQ , with the family of pressure hillsobtained by varying p . For any two such curves the ordinates are constantmultiples of each other.
In particular select the two intersections whichcorrespond to p = p \ , p 2, the stagnation pressures on the two sides ofthe shock (Fig. 147b). Let P i and P be the points on these two curves corresponding to the pressures pi and p (i.e., those points whose projectionsare Qi and Q respectively). Then the line PiP is tangent to both curves.For when the velocity vector q corresponds to points on the line Q1Q2 inFig.
147a, it may be resolved into a constant component ν along O'S and avariable component u perpendicular to this direction. N o w (29) is derivedfrom the Bernoulli equation, of which (2.21') is the differential form. Hencema2sssS2222dp=— pqdq=— p[u du + ν dv] =— pu duon either of the curves in Fig. 147b. Thus the slope of the first curve at= —piUi and that of the second at P is (dp/du)= — pu.These are equal by the first shock condition (3a).
Moreover, the slope ofthe line P i P is (pi — p )/(ui — u ) which, according to the second shockcondition (3b), is the same as the two previous slopes. Hence P i P is thecommon tangent of the two curves.As Q moves along a given polar, P moves in the tangent plane at P ito the pressure hill through P i , this hill being determined by p i . For anygiven position of Q , the point P is the intersection of the vertical linethrough Q with this tangent plane. Once P is determined, there is a uniquepressure hill of the family (29) passing through P , or in other words aP i is (dp/du)i2222222222s22222ρ(a)F I G . 147. Section of the pressure hill,(b) Common tangent.(b)(a) Vertical section with trace Q1Q2 ,390V.
INTEGRATION THEORY AND SHOCKS1.0λ0.5°15101520VF I G . 148. Graph of stagnation pressure ratio versus pressure ratio.unique value of ps2·29W e conclude this section by obtaining explicit formulas for the ratio p /p is2.8Since T is the same on the t w o sides of the shock and the flow is adiabatic8there, we haveVp/i/W/\PsJ\Pi7\W'HenceχPel - 1 / ( 7 - 1 ) ^ 7 / ( 7 - 1λ = — = ηp2where, as in Sec.
3, η = p /pi and ξ =s2)ξ,p/pi .*2Equations (21) through (23) now allow us to express λ in terms of £, 77,f, Minor M2nalone. In particular, we havef*m\\(30)λ= ,- i / ( 7- i ) \h\+Iv+Wjΐ"|Ύ/(Ύυ'and Fig. 148 gives the graph of λ versus η.6. The deflection of α streamline by a shockT h e deflection δ of a streamline on crossing a shock is given by the angleQzO'Qi in Fig. 144, i.e., σι — σ . A s Q moves along the polar in Fig. 145a22from Qi to A this angle increases from zero to a maximum, and then decreases to zero again. N o w we have seen that the shock polar is determinedexcept for size and orientation by the M a c h number Mxof the incidentstream. Thus we conclude: For each value of M\, there is a maximum deflection which can be produced by means of a shock.
This is analogous to what wasfound in Sec. 18.3 for the deflection of a stream by means of a simple w a v e .On a given polar there is a relation between the deflection δ and the in* From Eq. (1.7), λ = exp [(>Si — S )/gK]in entropy.2so that it also represents the change22.6THE DEFLECTION OF Λ STREAMLINE BY A SHOCK391clination σι of the shock to the incident streamline. Thus from Eqs. (24),(25), and Fig. 145 we findtan δV— 1^ cot σιU|"(gi ~=ALIf we writeτι =+U)Ua(gi -+UHUb)COtVi]COtAr,Jcot σι .= tan δ,cot σι(31)(7 +(2 =ι_=ι_=JbL21)Λίι2 '- ι)( *Μthen this formula becomes(en +2(32)(1 -+cWd)n(1 -d)'For each value of M i there is a corresponding curve in the n,€-planerelating the deflection δ to the shock inclination σ ι , see Fig.
149. As Miincreases from 1 to <x>, c increases monotonically from —2/{y+l)toOand d from 0 to 2/(y + 1). Thus for fixed positive η , Eq. (32) shows thatI ;••441ftocus offt.^-i >-cot"—Μ,-ωL.MeM,-2>ftft.-^^^ -*cot (ft.Ν»F I G . 149. Relation between deflection and shock angle for fixed incident Machnumber.392V.I N T E G R A T I O N T H E O R Ye increases monotonically with MiMr=A N DSHOCKSfrom1:(7+3)r,2(y ++1)'toΜι =oo :€2τι=(+7l)T l* +(7_1) 'and vice versa for η < 0. E v e r y curve (32) which corresponds to a physically possible transition lies between these last two.
Each such curve intersects the η-axis at the origin and at the t w o pointsη = ±V~d/c=±Λ/ ΜΙ2-1 = ± c o t αι ,where ai is the M a c h angle corresponding to Mi.T h e origin correspondsto the point A on the shock polar, and the other t w o points to the doublepoint Qi . Hence for physically possible shocks our attention must be restricted to the range^(33)cot αϊ ;the end points correspond to zero shock, for which σ = αϊ , 180° — αι asχwas seen before. Thus, as we found in Sec. 3, for each Mi the smallest possible angle between the shock line and incident streamline is the M a c hangle c*i = arc sin 1/M\ ; the largest possible angle is 90°.
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