R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 79

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I t is a consequence of Newton's equation and the FirstLaw of Thermodynamics. W e shall now investigate Eqs. (2) and (3) quitegenerally. Later the discussion will be restricted to steady plane flow andin particular to the problem formulated at the beginning of this section.In strictly adiabatic steady motion Eq. (1) reads q « g r a d S = 0.

Thusthe vectors curl q X q and grad S are each perpendicular to q, and hence,see ( 3 ) , the same is true for grad gH. In this case the derivative of Η alonga streamline is zero, and Η is therefore constant along the line. N o w thetotal head Η differs from Η only in having / replaced by P , and as we sawat the end of Sec. 2.5 the latter differ, in strictly adiabatic motion, by' atmost a constant for each particle. Hence Η remains constant along a stream­line and we recapture Bernoulli's equation (2.20').41For isentropic motion S = constant, and there is an over-all ( ρ , p)-rela­tion. Then P , which differs from / by at most a constant for each particle,will differ from it by at most an over-all constant. In (2) we may then putgrad S = 0 and replace Η by H, so as to recover Eq.

(6.17), which is* validfor an arbitrary elastic fluid. T h e latter equation has already been dis­cussed in Sec. 6.5.24.1A D I A B A T I C F L O W OF I N V I S C I DFLUID425Since Η may be used in place of Η whenever the motion is strictlyadiabatic, it may be called the total head or Bernoulli function in this case.W e now consider steady flow which is not necessarily strictly adia­batic, and the vanishing first of grad Η and then of curl q in Eq.

( 3 ) . If Ηis constant throughout the flow, then the equation becomes(4)curl q X q = Τ grad S.This means that grad S is perpendicular to both q and curl q. Thus: Inthe steady motion of an inviscid fluid throughout which Η is constant, thesurfaces on which the entropy has constant values are composed of streamlinesand vortex lines. A n important consequence of (4) is the following.

If alsocurl q = 0 throughout the flow then grad S = 0, and we can state: In thesteady irrotational motion of an inviscid fluid throughout which Η is constant,all particles have the same entropy. In Sec. 6.5 we saw that the converse isnot necessarily true.Returning to Eq. (3), we now suppose that the motion is irrotational:curl q = Ο. Then Τ grad S = grad gH, so thatcurl (T grad S) = grad Τ X grad S = 0.Thus either the flow is isothermal: grad Τ = 0, or it is isentropic: grad S =0, or the surfaces on which Τ has constant values coincide with those onwhich S has constant values.

In strictly adiabatic flow the first and thirdalternatives imply that both S and Τ remain constant on streamlines. Sinceaccording to Eq. (2.12) S and Τ are not functionally related, this meansthat the pressure ρ and density ρ also remain constant on the streamlines.T h e second alternative implies, by (3), that Η is also constant throughoutthe flow. In any irrotational, strictly adiabatic, steady motion of an invis­cid fluid, either the particles carry constant values of ρ and p, or the flow isisentropic with constant total head.*2For a perfect gas I = U + p/p = yp/(y — l)p, according to Eqs.

(2.13)and (2.23). Hence Eq. (22.19) should more strictly read Hi = Η2, sincethere is not necessarily any connection between the expressions for Ρ on thetwo sides of the shock. Thus if in the steady motion in front of a plane shockΗ = constant, as in the case when the motion is uniform, then Η = con­stant after the shock for strictly adiabatic flow. If in addition S is notconstant after the shock, as occurs in the case mentioned when the shockis not straight, then the motion is rotational after the shock. This followsfrom the second result above. For uniform incident flow the strictly adiabatic,steady, plane motion behind a curved shock is rotational™In strictly adiabatic steady flow both // and S are constant on eachstreamline, and we may assume that their variation from streamline tostreamline is determined by the boundary conditions in any particular426V.

I N T E G R A T I O N T H E O R Y A N DSHOCKSproblem. If the motion is not only steady but also plane, we may introduce(as in Sec. 16.2) a stream function ψ(χ,ν) to satisfy the equation of con­tinuity:(5)pg, --q,P=v- -.N o w we have seen that (5) implies that φ remains constant along stream­lines.

Thus S and gH are functions of ψ alone, say F ( ^ ) and G ( ^ ) , which aredetermined by the boundary conditions. Also for plane motion, the vortexvector curl q has just a ^-component, ω say. Equation (3) therefore re­duces to a scalar equation along the normal to the streamline, namelyq\dnq\d\f/9dn / 'd\p / dn 'Thus finally we have*ω = p(TF'(6)-G'),where primes signifiy differentiation with respect to ψ, and we have usedEq. (16.19): pq = θφ/θη.This result, essentially due to L.

Crocco, has the following interpreta­tion. A material filament initially perpendicular to the z,?/-plane remainsperpendicular and is therefore at each instant a vortex filament. B y con­tinuity its cross-sectional area is inversely proportional to p, and hence itsvorticity is proportional to ω/ρ. Thus from (6) we may state: In a strictlyadiabatic steady plane flow the vorticity varies linearly with temperature onany streamline.

For isentropic flow, F = 0, the vorticity is constant, inagreement with Helmholtz' second vortex theorem (Sec. 6.4).44r2. Equation for the stream functionW e may now deduce an equation for ψ to replace (16.21) when the flow isstrictly adiabatic but not necessarily elastic. Referring to Eq. (16.20) wesee that dp/dn can no longer be replaced by adp/dn since there is no over-all(p, p)-relation.

Instead we must proceed as in Sec. 15.6. Thus from S(p p) =F(\f/) we deriveydSdpdp dn+dSd_dp dnR=F'ty)WJ— =dnpqF\'μι* For a perfect gas, this shows that the mean rotation ω is a linear function ofρ and ρ on each streamline.24.2EQUATIONFORT H ESTREAMFUNCTION427so thatdp(7),qF'PdndS/dp'where as beforedS/dp(8)=dp/dtdS/dp ~ dp/dt 'according to ( 1 ) .A change must also be made in (10.200. T h e term dq /ds cannot benreplaced by dq/dn since the irrotationality condition(16.7) no longerholds. N o w since dq /ds — dq/dn = ω, we must writenand use the expression in (6) for ω.When these two changes are made, the final equation for φ becomes('s^)-'r+wAs in Sec.

16.2 the left-hand side of this equation may be written in Car­tesian form to give(9)T h e left-hand member of (9) is identical with that of Eq. (16.21) but therethe right member was zero.T h e right-hand side of (9) includes F' and G', which are known functionsof ψ, as well as T, a and dS/dp, given functions of ρ and ρ which, by virtueof S(p, p) = F ( ^ ) , can be expressed as functions of ρ and ψ. Thus we stillhave to express p, q , and q as functions of ψ and its derivatives. From thedefinition of Η we have, on neglecting the gravity term, the Bernoulli equa­tionxy(10)ig2+I = gH =0(ψ),where / is a known function of ρ and ρ which can be expressed as a functionof ρ and ψ from S(p, p) = F ( ^ ) .

This equation, together with ( 5 ) , serves todetermine p, q , and q as functions of ψ, d^/dx and θψ/dy. Thus: Eq. (9)xy428V. I N T E G R A T I O N T H E O R YANDSHOCKSis a planar* nonhomogeneous differential equation of the second order in φ.I t plays a role similar to that of Eq. (15.23) in the one-dimensional nonsteady case.A s an illustration of these calculations, consider the case of a perfect gas,for which the entropy S isfs-f-e*αϊ)(y -iog£,l)ypsee Eq. (1.7). ThendSdpgR(y -dS\)p'=_ygRdp(y -2=\)p'TPρand in this case the right-hand member of (9) isIn addition / = yp/(y — l ) p = a /(y — 1), where in terms of ρ and ψ2(13)exp [ (a = yp"'217-\)F/gR\.Hence (10) becomes(14)U2P~+1λ7 — 1In order to determine p, q ,βχρ[(Ύ -WgR]= G.q as functions of ψ, d^/dx, d\p/dy, we firstxyexpress ρ in terms of q from ( 5 ) :and use this to eliminate ρ from (14).

T h e resulting equation for q as a func­tion of ψ, dyp/dx, d\p/dy is(16)Aq2+ J L= 1,where1y exp [ ( γ -DFffl/gR]Γ/^2^·'-\ ^~2ι)Ι2Once q is determined from this equation, the density follows from (15) andthen the velocity components from (5) and the speed of sound from (13).W e return now to the general equation (9) and show how the correspond* T h a t is, linear in the derivatives of highest order, see Sec.

9.4.t Here we must use the entropy S itself, and not a function of it as in Sec. 15.6.24.2E Q U A T I O N FOR T H E S T R E A M429FUNCTIONing equation for the rotational motion of an elastic fluid can be deduced fromit. Let the (p, p)-relation be written in the form S(p, p) = 0, and considerAS(P, p) to be the entropy function of the gas.* Then since the motion isisentropic, (9) will hold with F = 0:fFor isentropic motion / may be replaced by Ρ in the Bernoulli equation(10), so that it becomes(18)\q + P = gH =GM.This equation, together with ( 5 ) , serves to determine p, q and q as func­xytions of ψ, θφ/θχ, θψ/dy.

W e see that when Ρ is expressed as a functionof p, the latter satisfies(180~+P(P) =D,plwhereOnce ρ is determined, q and q follow from ( 5 ) and a from its expressionas a function of p. In particular, if the motion is irrotational then G =constant in (18), and (17) reduces to (16.21). Even in this case the coeffi­cients are not known explicitly as functions of ψ and its derivatives [seeremark after (16.21)].f This completes the discussion of Sec. 16.2 and ex­tends it to the case when irrotationality is not granted.xyAn important conclusion which we can draw from (9) is the following.Since the characteristics of the differential equation depend only on thesecond-order terms, they are the same for (9) as for (16.21).

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