R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 17
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W e start with the equation forplane nonsteady motion derived from Eq. ( 1 4 ) :21^ 2 /_ 2\~2,/2\dxa Jdy\a )2\222__ _^2,adx dy2\ 3Φ _a2dt2^ ftdΦ _a dx dt22ftdj>=0a dy dt2Again introducing polar coordinates [by means of Eqs. (32)] we obtain, ifΦ is independent of 0:86II. GENERALTHEOREMSthe case of nonsteady flow with cylindrical symmetry. Also generalizingEq. (19) to nonsteady flow or working directly from Eq. (14), we obtainfor spherical symmetry:!±dd_ ? A _J_?!i _i2r- ^+2=o0dr' \a')α ata dr dt^ r *Comparing these last two equations with E q . (24), we see that we may222qwrite{4ό)dr'\a')a' dV\'drdtr+9rwhere ν = 0, 1, or 2 stands for nonsteady parallel flow, nonsteady flow withcylindrical symmetry, or with spherical symmetry, respectively. Againthere are only two independent variables, namely, r and t.
W e find fromEq. (16) that in each casea-<* - Uβ1}+27·If in the sense of A r t . 4 we replace a by O o , the sound velocity of thefluid at rest, omit the terms of higher order, and replace r by x, we obtainthe generalized one-dimensional wave equation22^ -(45)α^ -οαο^=0'with the above meaning of v. T h e general solution for ν = 2 isΦ = -[fi(x -aot) +fi(x+orf)],Xwhere fi and / are arbitrary functions of one variable; this may be comparedto d'AlemberVs solution for ν = 0.
T h e theory of cylindrical waves, ν = 1,is more difficult than, and essentially different from, that for ν = 0 orν = 2. This has an analogy in the general theory of one-, two-, and threedimensional waves (see end of A r t . 4, where we found that the second casediffers essentially from the first and third ones).2Article 8S t e a d y Flow Relations1. General relations among q, p, p, and ΤWhen the flow of a compressible fluid is steady, i.e., independent of time,and a (p, p)-relation holds on each streamline, then each of the four quanti-8.1GENERAL87RELATIONSties q, ρ, p, and Τ (velocity magnitude, pressure, density, and absolutetemperature) can be expressed, on a given streamline, as a function of anysingle one of them.
In fact, the equation of state, the (p, p)-relation, andthe Bernoulli equation supply three relations among these four variables.If the same (p, p)-relation holds on all streamlines (elastic fluid) and if theflow is steady and irrotational (steady potential flow), then the relationships among q, p, p, and Τ are the same for the entire flow, as will beshown presently.When gravity is neglected, the Bernoulli equation in differential form is(2.21'):(1)+ ^ = 0,ρffdgwhere the differentials refer to changes in q and ρ along a streamline.
Onintegrating, and introducing the stagnation pressure p , the value which ρassumes at a point of the streamline where q = 0, we find8This equation gives q as a function of ρ for all points of the streamline, thefunction depending on the parameter p . Since p, p, and q are nonnegativequantities it follows that ρ ^ p everywhere along the streamline, themaximum value being attainable only at stagnation points.I n the case of an elastic fluid we may consider the function Ρ of ρ (Sec. 2.5)whose derivative is 1/p, say Ρ = fp^dp/p where p is some reference pressure. Then Ρ is a monotonically increasing function of p.
In this case Eq.880(2) may be written(2')which gives again ρ ^£ = Pip.)-P(p),p.8When gravity is neglected, the Bernoulli function Η (Sec. 2.5) is definedby(3)I+ =ΡgH.For steady irrotational flow this quantity Η was shown in Sees. 6.5 and 7.1to be constant throughout the whole fluid mass.
From Eq. ( 3 ) , this constantvalue is also the value of P/g at all stagnation points; thus P, and consequently p, has the same value at all stagnation points. When ρ is known,the value of ρ may be determined by means of the (p, p)-relation, and thevalue of Τ follows from the values of ρ and ρ by means of the equation ofstate. Thus we conclude: In the steady irrotational flow of an elastic fluid, the88II.
GENERALTHEOREMSstagnation values p , p , T of ρ, ρ, Τ are the same on all streamlines. Underthese conditions, E q . ( 2 ) , giving q as a function of p, is also the same for allstreamlines. W e now study this relationship.888Whether the (p, p)-relation holds for a single streamline only or for thewhole fluid mass, it is assumed that ρ = 0 corresponds to ρ = 0 and thatρ increases monotonically with ρ so that from ρ ^ p follows ρ ^ p on thestreamline. W e even assume strictly monotonical increase, i.e., dp/dp =a does not vanish, except possibly at ρ = 0. Then a graph of ρ versus 1/pwill have the form shown in the right half of Fig.
35. T o the left, the integral of 1/p from ρ to p is plotted as the abscissa corresponding to theordinate p; each curve corresponds to a particular value of the parameterp , which appears as the p-intercept of the curve. From E q . (2) it followsthat the horizontal axis to the left is also the (positive) axis of q/2. I n Fig.35 it has been assumed that the integral f dp/p converges for ρ = 0, which8s2s8pF I G .
35. R i g h t half: ρ versus 1/p according to (p,p)-relation.L e f t half: ρ versusq /2 according t o q /2 = P(p )— P(p) for various values of p , where Ρ =j dp/p.22a8pP<F I G . 36. ρ versus q for various values of p and corresponding values of q,D o t t e d line shows relation for incompressible flow.88.1GENERAL89RELATIONSimplies in particular that dp/dp = a vanishes for ρ = 0. W i t h this assumption, the curves at the left meet the horizontal axis at finite values of q /2;in the contrary case, all curves would have this axis as an asymptote.2Considering q, rather than q /2, we obtain graphs of ρ versus q for variousvalues of the parameter p (Fig.
36). From Eq. (1), we find28%-<*thus each curve has a horizontal tangent at q = 0, ρ = p and another atρ = 0 (ρ = 0 ) , with q finite or infinite. Each curve must therefore have aninflection point for some intermediate point (q, p).* Differentiating Eq. ( 4 ) ,and using dp/dp = a , we haves2dpdp2W-=p~qddp dp=q-p-TqVd Tqq"=p+p^or, in terms of the Mach number, Μ = q/a,<>g - - | w - ^ - D .5T h e product pq is the flux per unit area and can be called the flow intensity.Then the conclusions that can be drawn from Eq. (5) may be expressed bythis statement: The curve of ρ versus q has an inflection point when the Μachnumber equals 1 (at the sonic point); the flow intensity pq increases with thevelocity q in subsonic flow, reaches a maximum at the sonic point, and decreasesas q increases in supersonic flow.
To each value of the parameter p (or of por of T ) there corresponds a certain transition velocity q (abscissa of theinflection point) where Μ = 1, and a certain maximum velocity q whereρ = ρ = 0. If the value of q is finite then the M a d number Μ— >C O asq ~ qm , since a —> 0.
For exceptional (p, p)-relations where the above mentioned fdp/p, does not converge as ρ —> 0 and q is infinite (which thenoccurs for all values of the parameter), it can happen that Μ remains finiteas q —> ο©.88stmm>2mIn the case of an incompressible fluid, when ρ = p , we have fp.dp/p(ρ — ρ )/p , and the curves of ρ versus q are the parabolass8=sg2= 2P' -p.PsThese curves for various values of p are the dotted lines in Fig. 36. I t isclear from the figure that the compressible fluid behaves in the subsonic8* W e shall assume that there is only one such inflection point. I t can be shownthat this will certainly be the case if d p/dp > 0.2290II.
G E N E R A LTHEOREMSrange, very much like an incompressible fluid while the flow has an entirely different character when the velocities are supersonic.T h e above statement concerning flow intensity may be given anotherinterpretation. In any infinitesimal stream tube the fact that mass is preserved and the flow is steady means that the flux (which equals pq timesthe cross section) is the same across any cross section of the tube; thus theflow intensity pq must be inversely proportional to the cross section of thetube.
Then decreasing cross section corresponds to increasing q—and viceversa—in subsonic flow, exactly as in the incompressible case. For supersonicflow, however, increasing cross section corresponds to increasing velocityof flow: the minimum cross section corresponds to Μ = 1. This behavior isillustrated in the radial flow studied in Sec. 7.3: in the subsonic flow the velocity decreases from q = q to q — 0, while the cross section of the channelincreases; in the supersonic flow the velocity increases from q to q withincreasing cross section.ttm22If an overall (p,p)-relation is given, each possible steady irrotational flowpattern is characterized, so far as the relationship between ρ and q (or ρand q, or Τ and q) is concerned, by the value of a single parameter.
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