R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 19
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1.5)—Eq. (11) depends on one parameter: the value of theconstant. A s seen in Sec. 1, the function ρ of q depends, once the (p, p ) relation is given, on the value of the parameter p .Therefore, the function8ρ of q involves the two independent parameters p and p .
Actually it turns88out that, if the (p, p)-relation is given by E q . (11), and in this case alone,these parameters appear as scale factors only, i.e., for the dimensionlessvariables p/p8or p/p (rather than ρ or p) in terms of the M a c h number Μ8(rather than q), a single functionis to be computed for each variable.On carrying out the integration indicated in Eq.
( 2 ) , or using the valueof Ρ from (2.22c) with p= ρ,0we findβ(12)On the other hand, differentiation of E q . (11) yieldsdp— = αdp2ρ η κ-\= κ— ρρDividing (12) by (13), we get(14)or, solving for p/p in Eq. (14),a(15)κ8Ρ* (ρ\= κ— I — )ρ \ρ /κ8896II. GENERALFrom Eqs. (11)and (15)THEOREMSwe obtain also/^V/*/\-i/U-nι(1.6), assumingand finally, from the equation of statethe fluid to be aperfect gas,£ =& = ( ^ μ + 1 y .(π)2pp\ 2/and (17) express the result mentioned at the end ofI sEquations (15),i(16),sthe preceding paragraph.In expressing the velocity q in dimensionless form, we take as scale factorthe stagnation value a of the sound velocity; by means of Eqs.
(13),8and (17),s(18)a= κ*-8( J= a=τ~·aW e note that the equality between the first and last members of (18)consistent with the result a2«a=M*22a(£-)22s(20)and (16),^Lpas=M= M(2\Ps/sN e x t , from Eqs. (19)give/ \(κ-1)/κ- = Ma(sis= KgRT following from the definitions of aand of a perfect gas. Then Eqs. (18) and (15)(19)(15),a is given by/K\\-lΛ- ^ M22+ 1)/.the flow intensity pq satisfies/ι- ^ M\ ΔK+2\-(«+l)/2(«-l)l)Jand the so-called dynamic pressure pq /2 is given by2(21)^__ = Μ ( ^ Μ2pas\zs2Δ+ 1)/If we denote the values of all quantities at a sonic point, Μ = 1, by thesubscript t, Eqs. (15) to (17) and (19) to (21) give the relations(22)and22 ^22* * + ι=α/- -I- I\-U+D/2(K-1)(κ +7P i ff< =K( 2 3 )Pefle\~2— /, U +,P#T=pa88l\ι V" "-»(1—-— 1I t was seen in Sec. 1 that p q is the maximum value of pq.tt8.4ADIABATICT h e maximum velocity q97AIRFLOW, corresponding to ρ =m0, may be determined from Eqs.
(12) and ( 1 8 ) :(24)2qFor all κ >p2K=£msκ — 1 p.,1 the maximum value qmκ — 1is finite. All the preceding equationshold also in the isothermal case, κ =particular, qm1, if limits are taken as κ - > 1; inis then infinite for all values of the parameters p and p .8sCombining Eq. (24) and the first equation (23), we derive the important relation(25)q=mκ —7Iqt ·I t can be verified, from Eqs. (15) or (19), that for ρ = 0, q = qmnumber is infinite if κ >, the Mach1. From (17) the corresponding temperatureis seen to be zero, from which it is clear that the limit q = qmTmcan never bereached in an actual flow.I t is to be emphasized, again, that all formulas derived in this section arevalid for:(a) Steady flow along a single streamline (stream tube) if E q . (11)holds at all points of the streamline;( b ) Steady irrotational flow in one, two, or three dimensions, if Eq.(11) holds throughout the fluid.4.
Adiabatic (irrotational) airflowD r y air can be considered as approximating to a diatomic perfect gas,for which the theoretical value of the adiabatic exponent y = c /c is7/5 = 1.4. Experiments lead to a slightly higher value, not above 1.405.T h e choice of one of these values rather than the other makes little difference in the results obtained, in view of the fact that the whole theory isapproximate. For example, the ratio of the pressures at sonic and stagnation points, given by (22), ispV + Λ"21 = hPs-7/(7-D2=0.5283= 0.5274Jfory=v1.400,for τ = 1-405.For the sake of simplicity, all numerical data in this book concerningadiabatic airflow will be based on the assumption that the value of κ in thepreceding formulas is 1.4.
In particular, this means that the ratio of theareas of the two circles C and C in the hodograph plane, using Eq. (25),is (1.4 + 1)/(1.4 — 1) = 6, while the corresponding velocity ratios, fromEqs. (25) and (24), arem(26)q- = V 6 = 2.45,qttq- = V 5 = 2.24,as*α3= \ / \ = 0.91.V Ό98II. GENERAL THEOREMSρΡΤρρ,1,qpqptqtF I G . 40.
— , - , —-, — , — , —βqmpqtversus Mach number. Use scale at right forpqtlast curve.In Fig. 40, the ratios ρ/ρ ,8M a c h number M,P/P«, T/T ,aand q/qmare plotted against thewith the use of Eqs. (15) to (17) and (19). Afurthercurve represents the ratio ptqt/pq, which may be computed from Eq. (20)and the second equation (23) as„(27)£«'pq1/91* (!LZ_1*«M \ K + 1+\U+L)/2(«-L)2\κ + 1/Since cross sectional area is inversely proportional to flow intensity, theratio (27) is also that of the cross section at any point of the stream tubeto the cross section at a sonic point, i.e., to the minimum cross sectionof the tube. T h e remaining curve shows the flow intensity pq/ptqt, whichhas its maximum value a t M = 1.
Numerical values for the first five functions are given in the Table on the following page (Table I ) .T h e (p,g)-relation of Fig. 36 may now be graphed exactly, since the(p,p)-relation is given explicitly in E q . (11), and this curve is given in Fig.41. T h e formula expressing this relation i s25This formula was obtained previously, in the particular case of purely radialflow, as Eq. (7.23'). Except for scale factors, this curve represents themeridian of the pressure hill for all values of the parameters p and p ,ae268.4ADIABATICTABLEDEPENDENCEJ0000000000011123468101520250001020304051246802500000000010000000000000000000IOF F L O W V A R I A B L E S ON M A C H N U M B E RP/PP/PaT/Taa0999939997299937998889982599303972508956178400656025282841238272401278002722006590006330001020000236———100000000000000000000999959998099955999209987599502980289242784045739996339453114394982300507623027660051900141000493———1000000000000000000000099AIRFLOW099998999929998299968999509980099206968999328488652833337764068966555563571423810121950724604762021730123500794ΜFOR A D I A B A T I CQ/Qmp /pqtqtOO0.00.004470.008940.013420.017890.022360.044680.089090.176090.259160.336860.408250.472870.557090.666670.801780.872870.937040.963090.975900.989070.993810.9960257.87428.94219.30014.48111.5915.82182.96351.59011.18821.03821.01.03041.17621.68754.234610.71953.180190.11535.943755.21537746305ΡM-0qF I G .
41. ρ versus q for adiabatic airflow.AIRFLOWII. GENERAL100THEOREMSArticle 9Theory of Characteristics1. IntroductionIn order to explain the concept of characteristics of partial differentialequations, to the extent that is needed in the theory of compressible fluidflow, we start with a preliminary examination of the relatively simple caseof steady potential flow in two dimensions.
I t has been seen (Sec. 7.5) thatin this case the potential function Φ(χ,ν) must satisfy the second-orderpartial differential equation( - i ) f t - ^ / l + ( -^)ft = i?2Li()\a / dxa dx dy\a J dyHere q and q are the first partial derivatives of Φ with respect to χ and yrespectively, and a the square of the sound velocity; a can also be expressed in terms of first-order derivatives of Φ, e.g., by Eq.
(7.17) in thecase of a poly tropic (p,p) -relation. Equation ( 1 ) therefore falls within thegeneral class of equations of the form2x2122y22(2) A * -?+2 B - * - *-+ C * - ? = F ,dxdx dy1dy2where A, B, C, and F are functions of Φ and its first-order derivatives, andpossibly also of χ and y N o w we ask: Of what significance is the fact thatΦ satisfies an equation of type (2)?27Suppose that the values of Φ and βΦ/dx are known for all points of acertain straight line parallel to the ?/-axis, say χ = x . These values thendetermine for all points of the line χ = x , all derivatives with respect to yof both these quantities, in particular βΦ/dy, d%/dx dy, and d%/dy . T h eonly second-order derivative of Φ not determined on χ = x by the givenvalues is d%/dx , and this derivative can be computed from E q .
( 2 ) , in theform00202(2')—dx= - —'_A dx dyθ2φC^A dyd+2FA'whenever A is different from zero. T h e given values also determine Φ,except for terms of higher order, on any nearby parallel line, say χ = χ =Xo + dx:Λ(3)Φ(χι ,y)=Φ(χ ,2/)0+—(xo ,y) dx ;9.1101INTRODUCTIONand we can even obtain|(xo ,y) + ^ - f (xo ,y) dx,dxdxprovided the coefficient of dx can be determined by Eq. ( 2 ' ) .
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