R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 76
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T h e remaining problem isto determine those values of Μ· and r for which Eqs. (11) and (12) yield2Αthe same value of δ and give ρ=όIf M3p\.and r are allowed to vary in Eqs. (11) and (12) respectively, then2the corresponding points (δ, p ) and (δ, p ) trace out two curves in the34δ,ρ-plane, see Fig. 157, whose intersection gives the common values ofdeflection and pressure.For each δ these curves have the general shape360shown in the figure, as may be seen by considering (11) as Mincreases fromzto oo, and (12) as r increases from —cot a to 0.
I t is easily shown thatMi22for δο sufficiently small (depending on M )the point Αι will lie between A02and At, so that the curves have just one point of intersection, and the latterwill be close enough to Afor it to correspond to sonic or supersonic state 4.2For in the limit δ - + 0 the distance A A%02yp (Mo20=2+— l)/(y1) > 0 and Αι —> Asection of the curves, with M4= M>22yp (M21)/(γ +-21) - >which is then the point of inter21. In addition Axlies above ^4 for2δ > 0 since pi > p > p . T h e result now follows from the continuity of002all functions involved with respect to δ .0Once this point of intersection is known, Μ ,3a3= arc sin ( 1 / M ) of BWito BC,3of BS ,2which gives the inclinationand r , which determines the position2may easily be found from Eqs. (11) and (12).
T h e streamlines inthe backward wave region SiBTiare then given byconstant;see (18.9), where r, φ are polar coordinates with pole at B.In practice one would obtain this intersection by taking various valuesfor δ in Eqs. (11) and (12), computing the corresponding values of p and3p , and then interpolating for equality. A first approximation is obtained4by noticing that on the lower side of the profile the transitions are a forward wave and a shock with negative deflection. Hence from ( 8 ) , withF = p, we have(13)-V ip„ = α(δ„ -6(δο -δ) +δ) + 0 ( δ , δ)203:where a, b are certain constants depending on the state 0 but not on δ or0δ.
Similarly(14)ρ3-ΡΟ =-ο(δο -δ) +&(δ 0δ) + 0(δ„, δ)23for the transitions on the upper side. Thusp - p = 2o(i - δ) + 0 ( δ „ , δ) ,4330so that the first approximation δ = δ (BC horizontal) will make p and p0differ by at most Ο ( δ ) .03373t410V.I N T E G R A T I O NT H E O R YA N DSHOCKSAs an example we consider the case M = 2, δ = 10°. T h e flow over theupper side of the profile for the first approximation δ = δ = 10° was discussed in the preceding section.
I n particular (the suffix 2 of that sectionbeing replaced by 3) we found000pz = 1.0026p ,0Μ ζ = 1.9884.Similarly for the flow on the lower side of the profile we find, with δ = 10°,ρ* = 0.9995p ,0MA= 1.9862.Thus the approximation δ = 10° gives a higher value of pz than of p .T h e true value of δ is therefore slightly larger, see Fig. 157. For δ = 10°6'the corresponding values are4pz = 0.9970p ,Μ ζ = 1.9920,pΜ04= 1.0052p ,0A= 1.9822,where now p exceeds pz.
B y linear interpolation we find the second approximation δ = 10°2'. A n y desired accuracy can be obtained by repeatingthis process with more accurate pressure values. Thus in this example eventhough the stream is initially deflected upwards, the dividing streamline BCis inclined downwards.4T h e corresponding values of the Mach numbers are Μ ζ = 1.9894 andMA = 1.9850, and the common value of the pressures p = ρ A = 1.0011p ·T h e reader should compute for himself the values of p , ρ , M , pz, p4 anddetermine the positions of the characteristic BTi and the shock line BS .I t will be found that the density ratio pi/pz is very close to unity so thatthe discontinuity across BC is in fact insignificant, even though we started0z2222ΡF I G .
157. Pressure-deflection figures for flows behind profile.23.4OBLIQUE SHOCKwith a sizable deflection δ0=411R E F L E C T I O N10°. T h e numerical values of the presentexample were used in constructing Fig. 156.T h e pressure force on the profile is(pi -p )Z = (1.7066 -0.5480)p Z = 1.1586pA20and it is directed downwards perpendicularto AB.Noticethatnoknowledge of the states 3 and 4 is required to determine this force.
In general we may write in analogy to Eqs. (13) and (14)P2 — Po = αδ + b5 +Pi -Po =Ο(δ ),2000+-aS06δ02+3Ο(δ ),03so that pi — p = — 2αδ + Ο ( δ ) . According to (2) the constant a is equalto —poQo tan cto = —yMopo/y/Mo— 1. Hence the pressure force on theprofile is200232VMT^-iPol+oiSo)-F o r the above example the first term in this expression gives the value1.1285ρ £, which is about 3 per cent too small.04. Behavior of a shock at a wall (oblique shock reflection)38In this section we shall discuss another example on the basis of the deflection theory developed in the preceding article.
T h e present example complements that of the head-on reflection of a shock at a wall which was treatedin Sec. 15.1.Assume (see Fig. 158) that a fixed plane wall is set at an oblique angle tothe direction of motion of a one-dimensional shock front whose strength isconstant and in front of which the gas is at rest. After the shock has passed,the gas moves towards the wall, whereas at the wall it must move parallelto it. T h e problem is to find a flow pattern behind the shock which satisfiesboth these boundary conditions.Since the velocity of the shock is constant the line of intersection ofshock and wall moves parallel to itself along the wall with constant velocity.In a system of coordinates moving with this velocity, the shock is effectivelyat rest and the wall is moving tangential to itself.
If the 2-axis is now takenalong the line of intersection, then since the boundary conditions are independent of both ζ and t, we may treat the problem as one of steady flowin the x,y-p\&ne.In this plane (see Fig. 159) the straight line OS represents the shockfront incident to the wall WOW at an angle ω.
T h e gas in front of the shockis now mo/ing along the direction of the wall, and the streamlines arevertical straight lines which continue below OS in a direction inclined412V. I N T E G R A T I O N T H E O R Y A N DSHOCKSyWF I G . 159. Shock reflection viewed from moving coordinate system.towards the wall. A t the wall the velocity must be vertical, and hence thestreamline through 0 , the point at which the shock front meets the wall,must remain vertical. Under suitable conditions a solution satisfying thisrequirement is obtained by assuming another shock line OS, making an23.4OBLIQUESHOCKR E F L E C T I O N413angle ώ with the downward vertical, across which the streamlines againchange direction so as to become vertical once more. Then according to Sec.22.7 there are two possible positions for this second shock and hence twovalues of ώ for each suitable ω.
In contrast to the example in the last sectionwe shall in the present case be able to specify precisely the conditions underwhich such a solution is valid.A n observer at rest with respect to the (now moving) wall first sees theshock OS moving obliquely towards the wall, and then the shock OS moving obliquely away from the wall. This phenomenon is known as obliqueshock reflection.
T h e term "reflection" suggests the symmetry conditionώ = ω. W e shall see however that this condition can be satisfied only forone particular value of ω. For very weak shocks it can be satisfied approximately for all values of ω.In the hodograph plane (Fig. 159), the fluid state above the first shockis represented by a point 1 on the g^-axis.
T h e point 2, corresponding to thestate after the first shock, lies at the intersection of the shock polar withits corner at 1 and the line through 1 making an angle 90° — ω with theg^-axis. T h e point 2 will also be denoted by Ϊ to indicate that it representsflow in front of the second shock. T h e point 2 (representing the state afterthe second shock) must lie on the shock polar with corner at Ϊ and inaddition on the g^-axis, since we require the streamlines to be vertical againafter the second shock. There are in general two possible positions for thepoint 2, one corresponding to a weak reflected shock and the other to astrong reflected shock.
T h e figure shows the ώ corresponding to the weakshock.T o fix the state 1, we note that the shock OS is characterized by itsstrength, which may be represented by η = ρ /ρι , and its angle of incidenceω to the wall. Then Eq. (22.21), with M= Mi sin ω, determines Mi andhence the point 1, so that the construction of the last paragraph can beeffected. Our task is now to compute ώ and the pressure ratio η across thesecond shock as functions of ω and η. T o this end we initially replace η byδ, the deflection of the streamlines caused by either shock in the x,?y-plane.Equation (22.38) gives the relationship between δ, η and ω.2inW e ηολν derive two equations, one connecting M , δ, ω and the otherΜ, δ, ώ.
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