R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 44

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14.1, the concept of regions ofideal fluid flow separated by lines, across which the shock conditions hold,is only an expedient for representing, to a certain degree of approximation,the solutions of the equations (10) for viscous flow with small μ . If it werepossible to find exact integrals of (10) satisfying given boundary conditions,for the empirical values of μ , such integrals would supply more realisticflow patterns which would include narrow regions within which the vari­ables u, p, and ρ change rapidly.

Thus one would expect an approximateintegral of (10) derived by means of (13) and (14) to yield a good approxi­mation to the flow pattern. In an attempt to find an approximate integralof (10), however, the following difficulty arises.00When a differential quotient, such as du/dx, is replaced by a differencequotient (u— u )/(x +\ — x ), a reasonable approximation is obtainedonly if the intervals (x +\ — x ) are small compared to any interval in whichu experiences a sizable change (see Fig.

86). This means that at leastseveral avvalues must fall within the transition region. But we saw in Sec.11.3 that under average conditions the thickness of a transition layer is ofthe order of magnitude of 0.1 mm or less. N o w it is hardly possible to operatev+ivvvvv* Probably this method could be resurrected now that the high-speed computingmachines can do the calculations involved.15.6INVISCID FLOW B E H I N D CURVED SHOCKLINE229U4FIG. 86.

Approximationof differentialcoefficientby differencequotient.with intervals measured in hundredths of millimeters, even with the bestcomputing machines available today.One way out of this difficulty that has been found convenient is to in­crease the value of μ appearing in Eqs. (13).062If, for example, μ is taken one0hundred times larger than the experimental value, one still finds that thegreater part of the flow is hardly affected by the viscosity terms. On theother hand, there appear transition regions with a thickness of the order ofmagnitude of 10 mm, which are well represented by a computation basedon intervals of the order of 1 mm. I t is important to notice that, for smallvalues of μ , the relation between the initial and final values of the transi­0tion are effectively the shock conditions, and these are independent of μ .06.

The inviscid flow behindα curved shock lineIn the simple case of a straight shock line, a flow that is uniform beforethe shock is still isentropic after the shock. W e shall now consider the moregeneral case where, in a one-dimensional flow, a discontinuity point movesat a nonconstant velocity c. This corresponds, in the £,£-plane, to a curvedshock line.

I t is assumed that the motion prior to the shock is uniform, butit is no longer isentropic after the shock, since the magnitude of thesudden change of entropy at the shock transition depends, for each particle,on the instantaneous velocity c of the shock point at the moment when theparticle reaches the transition line.T h e theory for inviscid flow developed in Arts. 12 and 13 was based on theassumption that the specifying equation for the flow was an over-all rela­tion between ρ and p. This is no longer true of the flow after the shock, sincedifferent particles have different values of entropy. T h e condition of strictlyadiabatic flow for the region behind the shock leads, as was seen in Sec. 1.5,only to the condition(17)230III.

ONE-DIMENSIONALFLOWwhere S, the entropy or a given function of it, is a known function of ρand p. T o study the inviscid flow behind the shock—or any inviscid flowfor which ρ is a given function of ρ only for each particle line—we must goback to E q . (11.2) and the first component of Newton's equation, namely,(18+,and proceed as in A r t . 12, but this time with the less restrictive condition(17) in place of the over-all relation.Exactly as in Sec. 12.2, the first equation can be satisfied by introducingthe particle function ψ(χ,ί):( 1 9 )pTx>=pu-Tf=When ψ(χ,ί) is introduced into the second equation (18), the left-handmember becomes, as in Sec. 12.2,*\2 Ιr\2 ι(*++ 2uaIVu2^but this time the expression for the right-hand side is not so simple.T h e specifying equation (17) expresses the fact that £(p,p) remainsconstant along each particle line.

Thus S(p,p) is a function of ψ determinedby the boundary conditions (here the conditions along the shock line).From S(p,p) = F(\p) we derive(20)—?2 + —fy= F'Wdp dxdp dxdx=ip'pHere F' is the derivative of F and is therefore also a known function.If S were an over-all constant, we would have F' = 0. I t is customary to in­troduce the notion of sound velocity, a, even in this case, where dp/dp isnot defined (its value depending on dx/dt).

A n acceptable definition (seeSees. 5.2 and 9.6) is/ x2_2 1KdS/dpdS/dpJ=dp/dtdp/dt'T h e second equality is an immediate consequence of (17). Thus a is a knownfunction of ρ and ρ as soon as S(p,p) is known, and can be expressed in termsof φ and ρ if F(\//) is also known. Solving for dp/dx in (20) and using a forthe quotient of the derivatives of S, we have(22)^ = a ?£ + ^ := ag +dxdxdS/dpdx222'dS/dp'p F15.7ASECOND231APPROACHThus the second equation (18) becomesT h e left-hand member of (23) is identical with that of E q .

(12.11'), butthere the right-hand side was zero.In (23) the right-hand side includes p, which equals θψ/dx; and F',which is a known function of ψ once F(\p) is determined; and finally dS/dp,a given function of ρ and ρ which, by virtue of S(p,p) = Fty), can be ex­pressed as a function of ψ and p. Since u is given by ( — dyp/dt): (θψ/dx),we see that the coefficients in (23), as well as the right-hand side, dependonly on ψ and its first-order derivatives. Thus: Eq. (23) is a planar* nonhomogeneous differential equation of second order in ψ, differing from (12.11')only by the term on the right.For a perfect gas the entropy is proportional to the logarithm of p/p .Choosing S to be a simple function of the entropy, rather than the entropyitself, we may writeyο/\S(p p)=Z-,yΡdS1τ - = - ,pydppydSypΤ - = - - ^ Ϊ >dpρΎ + 12αρ= 7 - ,ρand in this case the right-hand member of (23) is pF'ty).T h e principal result which we can derive from (23) is the following.

Sincethe characteristics of the differential equation depend only on the secondorder terms, they are the same in the present case as in that consideredin Arts. 12 and 13. Thus, the characteristic lines in the x,t-plane are the lineswith the slopes u + a and u — a, where a is the same function of p,p as before.(See also comments at the end of Sec. 24.2.) Further conclusions analogousto those of Arts. 12 and 13, however, cannot be drawn. T h e interchange ofdependent and independent variables, the use of the speedgraph, etc., areno longer of avail, since Eq.

(23) is nonhomogeneous.y+1I t was shown in Sec. 14.3 that the actual change in entropy across a shockis in most cases very small. Thus, if the flow is uniform before the shock,if the shock is not too strong and if, at the same time, the variation of slopealong the shock line is slight, the derivative F' will be small.

In these cir­cumstances one may, as a rule, consider the flow after the shock to beisen tropic.637. A second approach64There is an alternative approach to this problem of nonisentropic flow.If u times the first equation in (18) is added to the second, the latter is re* T h a t is, linear in the derivatives of highest order, see Sec. 9.4.232III.

ONE-DIMENSIONALFLOWplaced by+4-(pdx% (PU)dt+ PU )=20.This equation allows us to introduce a new function l(x,t)(24)d\ = pudx-(p +such thatpu)dt,just as the first equation in (18) permits the introduction of yp(x,t) such that(25)d\p = ρ dx — pu dt.Substituting from this into Eq. (24) we have(26)d\ = udt-ρ dt.W e have seen that in strictly adiabatic flow the entropy S(p,p) is a func­tion of ψ alone, F(\p), determined by the boundary conditions.

Prescriptionof F therefore provides an algebraic relation between ρ,ρ,ψ throughoutthe flow. If any two of these three variables are selected as new independentvariables in place of χ and t, then the third may be considered a known func­tion of these two for any given problem. Moreover, Eq. (26) can be rewrit­ten as(27)άξ = udt+ t dp,where £ = I + pt. W e are thus led to select ψ and ρ as the two new inde­pendent variables in place of χ and t, and to replace I by ξ. T h e functionsu(yp,p) and t(\p,p) are then given by<28>and Eq. (25) yields for'-%•χ(ψ,ρ):dx _dt_a * "(29)dxη11=* * " ; 'dtudpΤ - = 0.dpSubstitution from (28) in (29) givesdx,,(30)=d\pdid£2ζΨ+1d\f/ d\f/dpρ 'Fdxd£ d\dp ~dtdp 'and if χ = χ(·ψ, ρ) is eliminated between these two equations, the followingMonge-Ampere equation is obtained for £ = ξ ( ψ , ρ ) :=215.8(31)NONISENTROPIC SIMPLE**Θψ βρ22(Λ-\\θψθρ/233WAVES= — (1)dp\p/Here, since ρ is a known function of ψ and p, the right-hand side is a knownfunction of the same variables.

Once a suitable solution ξ(ψ,ρ) of this equa­tion is determined, the remaining variables u, t, and χ are given as functionsof φ and ρ by Eqs. (28) and, according to (30), by*=/[(i4 :wiS4+respectively.Unlike Eq. (23) the Monge-Ampere equation (31) is not planar. However,it has a very simple form in that its right-hand member is a function of theindependent variables alone. This member may be rewritten as — 1/p timesdp/dp, and since this derivative is to be taken w ith ψ (or S) fixed, we maywrite2Tdp __ dS/dp _ 1dp ~dS/dp ~a'2according to E q .

(21). Hence(32)£(ΐ)=_'<0,dp \p/λ = ap1 .λConversely, if λ is given, corresponding functions *S(p,p) and F ( ^ ) can beobtained by integrating this last equation and expressing the result in theform S ( p , ) =PFor a perfect gas we may take, as before, S = p/p , so that 1/p =p~ \F{^)}andyllyl,y, = 1±1,δ(ψ) = - y i [FW]m\2yV7In the case of isentropic motion, ρ is a function of ρ alone whether or not thegas is perfect. Then the right-hand side of E q . (31), and hence λ, depends onρ alone. For a perfect gas in isentropic motion both are proportional topowers of ρ by (33) since δ (ψ) is then an over-all constant.(33)λ = ρ-'δ(*),8.

Nonisentropic simple waves. LinearizationT h e Monge-Ampere equation (31) possesses two families of characteris­tics, defined, as before, to be lines in the plane of the independent variablesacross which the analytic character of a solution may change. Along acharacteristic either(34)\άψ - dt = o,λ dp + du = 0,or+ dt = 0,λ dp — du = 0,λάφ234III. ONE-DIMENSIONALFLOWdepending on the family to which the characteristic belongs. T h e top equa­tions in (34) reduce, by virtue of Eq. (25), to±dx = (ua) dt,respectively, in agreement with the result concerning the characteristics ofEq. (23). For isentropic motion we have seen that λ is a function of ρ alone,in which case the bottom equations in (34) may be integrated to giveν zt u = constant,which is in agreement with E q .

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