R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 38

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It is as­sumed that the original differential equations for the motion of an ideal fluid:the equation of continuity, Newton's equation, and the specifying condition, arevalid at all points of the x,t-plane with the exception of certain shock lines";across these lines the state variables are discontinuous, the sudden changes beinggoverned by rules derived from the theory of viscous and/or heat-conductingu* A somewhat analogous situation presents itself in the theory of an incompressiblefluid with regard to the boundary layer solution of the Navier-Stokes equations.198fluids.III. ONE-DIMENSIONAL FLOWIn more general problems shock surfaces in a;,i/,z,£-space take theplace of shock lines.2.

The shock conditions for α perfect gasIn Sees. 11.3 and 11.4 it was seen that in the steady flow of a viscousfluid a rapid change from the state Ui,p pito another state u ,p ,ph22is2possible only if these six values satisfy the three relations (11.9'), (11.22),and (11.23):p\Ui =(la)(lb)pi +doί+—22+mux =• p2-λ=7 — 1 pi2pw,mu ,24+—ywhere m is an abbreviation for piUi or p u .2,-°1pIf this same flow is viewed from22—2a coordinate system moving in the negative ^-direction at constant speed cand if Ui, u now represent velocities with respect to the moving coordinate2system, Eqs.

(1) read(2a)(2b)(2c)pi +~ c)22pi(ui— c)m(ui-2c) = p2Τ+== p (uPi=2— c) =m,+m(u-c),(u27 — 1 pi2-c)22+γP27 — 1 p '2T h e same situation arises, viewed from a fixed coordinate system, if thetransition progresses at constant speed c toward the right. Thus, Eqs. (2)hold also for a particular kind of nonsteady motion. W e shall now showthat Eqs. (2) are the limiting transition conditions even in the most generalcase of nonsteady flow; we start from the general equations of Sec.

11.1.Consider a small segment of the x-axis, say from x\ to x , which progresses2at speed c, where c may be positive or negative. Then the material deriva­tive, which is d/dt = u(d/dx) + d/dt in the fixed x,£-coordinate system, maybe written(3)jdt={u- c ) ? dx+-,ddtwhere d'/dt means the derivative with respect t o time at a point fixed inthe moving segment, and χ now refers to coordinates on the moving seg­ment, while u and c are velocities relative to the original coordinate system.** This corresponds to the change of coordinates considered in Sec. 24.5.14.2SHOCK C O N D I T I O N S FOR A P E R F E C T GAS199Then, since c is independent of x, the continuity equation (11.2) can bewrittenf(4)dxc)] + §= 0.dt[p(u -Likewise, Newton's equation (11.3) takes the formiu-c)g(5)+P, ° £± ( p - d - 0 ,+dxdtdxand if we add Eq.

(4) multiplied by u, this becomes(5')f[ u(uP- c ) +P- a '} +xd- ^ = 0 .dxdtFinally, the specifying condition (11.4'), which states that the flow is simply(but not strictly) adiabatic, is now,χd (u,gRT\.d' (u,gRT\k[*-*-'£\-<>-+Adding E q .

( 4 ) , this time multiplied by u/2+ gRT/(y— 1), we obtain(6')+dtdN o w Τ may be replaced by its value from the equation of state (11.4); thenEqs. ( 4 ) , ( 5 ' ) , and (6') are three equations determining u, p, and ρ as func­tions of χ and t.Each equation is of the form+ ^J? =Addxοdtand when integrated over the interval xi to x supplies a relation of the form2(7)Α(χύ- A(xi) +ΓJXI^dx = 0.atW e now consider solutions for which the time derivatives d'/dt of the statevariables remain bounded as μ and k tend to zero.

Then E q . (7) holds also0for the limit flow, with the integral tending to zero as x2approaches Xi .200III. ONE-DIMENSIONALFLOWThus, if 1 and 2 refer to adjacent points on either side of the shock line, thedifference A — A, must vanish. In order to have x, and x approach eachother from opposite sides of the shock, with (7) remaining valid throughout,it is, of course, necessary that c be exactly the velocity of the shock front (i.e.,the slope of the shock line).

Introducing successively for A the three expres­sions from (4), ( 5 ' ) , and ( 6 ' ) , we obtain three conditions:22[p(u -c)]\ = 0,[pu(u-c) + ρ -σ' }\ = 0,χT h e first of these is exactly (2a). Since we assume that the fluid behaveslike an ideal fluid on either side of the shock, the viscous stress σ and theheat flux k(dT/dx) must vanish at 1 and 2. Then subtracting c times the firstrelation (8) from the second, we have (2b). T h e third relation givesχ,(«? ) + up!- c ) ( £ +Ji7 — 1 ρ/\ 2= 0.Here we must subtract c times the second equation and simplify by meansof the first to obtain Eq.

(2c).Thus the three equations (2a), (2b), and (2c) represent necessary condi­tions relating the initial and final values of an abrupt transition. One re­striction must still be added to the conditions. I t was seen that the flow ofA r t . 11, which in the limit supplies a special case, at least, of such a transi­tion (namely one with d''/dt = 0 for all variables), is not reversible: it alwaysgoes from lower to higher values of θ [see Eq. (11.13) and Fig.

54] and there­fore of Τ or ρ I p. Thus, the program indicated at the end of the precedingsection may be formulated more precisely as follows.We consider flow patterns in the x,t-plane which fulfill the differential equa­tions of ideal fluid theory everywhere except on certain curves (of unknownshape), while along these shock lines occur discontinuities, in u, p, and p,which satisfy the three conditions (2) and the inequalityu(9)11£Hi,P2Piwhere for any particle, state 1 precedes state 2.Flow patterns of this type are often called "discontinuous solutions of theideal fluid equations".

I t should be remembered, however, that the discon­tinuity conditions (2) cannot be derived without taking viscosity into, ac­count. In fact, if it is assumed that the flow is inviscid even in the transi4314.3SOME P R O P E R T I E S OFtion zone, then the value of p/p(2) are compatible with201SHOCKScannot change for any particle.

But Eqs.yγP2F=y,Pias will be seen below in Sec. 3.T h e first to study shock problems of compressible fluid flow was themathematician B. Riemann (1860). H e did not think in terms of viscosity,and, on the basis of observation, he took for granted the possibility of dis­continuities. His shock conditions included (2a) and (2b), but he used= P I / P I rather than (2c). Although this procedure is not justified,Pi/Pi7the numerical results for ordinary conditions do not differ considerably fromthose obtained by the proper method (see Sec. 3 ) .

T h e correct shock condi­tions were first given by W . Rankine (1870) and then, independently, byH . Hugoniot (1889).44Moreover, the essential point is not the derivation of necessary conditionsto be fulfilled at a surface of discontinuity. T h e only justification for admit­ting solutions of the type considered here (regions of continuity separatedby shock lines) is supplied by the existence of viscous flow solutions exhibit­ing transition regions whose width tends to zero simultaneously with theviscosity coefficient μ.

Flow patterns including shock lines are not "discon­tinuous solutions of the ideal fluid equations" (see also Sec. 15.2), butrather asymptotic solutions of the viscous fluid equations for the limit caseμ^Ο.3. Some properties of shocksT h e shock conditions consist of the three equations (2a), (2b), (2c), andthe inequality ( 9 ) .45Before we work with these equations, some limitingcases will be mentioned. If ux= u2= c, then m = 0, while (2b) gives pi =p . In this case the third condition (2c) is fulfilled for an arbitrary value of2Pi = p .

This possibility is not usually included in the concept of shock,2since no particle crosses the line of discontinuity. Another limiting case istii — u2?± c. Then, as before, from (2b) it follows that pi = p ,2while thethird condition leads to pi = p . N o actual discontinuity occurs, and this2case will be referred to as zero shock. T h e same conclusion follows if weknow only that pi = p ,2or that pi = p , provided that the particles ac­2tually cross the shock line.T o bring the shock conditions into a more suitable form, we first introducethe velocities relative to the shock front, which moves at velocity c:U\ =U\— c,u2= u2— c.202III. ONE-DIMENSIONALFLOWThen Eqs.

(2a) and (2b) become(10a)= ra,piu[ = p u2(10b)pi — p2= m(u2— u[).2Here u[, u , and ra are assumed to be different from zero. Equation2(2c)may be written as/ο '\'(2c )Ui'2-u2^72/p=pA2I — - — J.7—1 \P2pi/T h e factor u[ — u may be replaced from (10b) by (p — pi)/m. Further­more, we may write u''/ra for 1/p , by (10a). After multiplying through byra, we obtain22(p2 — pi)(u[+ u)==2or, with the usual abbreviation h=2(10c)Uip u[22y7 - 1(7 +— p\U = h (p u2222(p u2—2piu[)l)/(7 — 1),— p\u[).I n addition, we have the inequality ( 9 ) . From (2c') we therefore have— u > 0, or2(11)\u[ I >\ u2I .B y (10a), the quantities u[ and u have the same sign.

Thus if Ui > c,we have u[ > 0, u > 0, and (11) becomes u — c > u — c or U\ > u \if Ui < c, then we have u[ < 0, u < 0, and (11) becomes c — Ui > c — uor Ui < u . In (10a), the inequality (11) gives pi < p ; and in (10b), sincera has the same sign as u[ and u , it gives pi < p . Finally, the tempera­ture Τ is proportional to p/p, so that (9) gives directly the inequality 7\ <T.22x222222222Thus, since we have assumed that state 1 precedes state 2 in time, i.e.,that a particle enters the moving shock in state 1, we have the followingresult: A physically possible shock (that is, a rapid transition governed byviscous fluid theory) is always a "compression shock"; pressure, density, andtemperature increase, while the absolute value of the relative velocity decreases.T h e two possibilities u < c and u > c are illustrated in Fig. 77.

I t wasshown in Sec. 3.4 that in the strictly adiabatic flow of a viscous fluid theentropy of a particle cannot decrease. W e shall see later that, in an actualcompression shock, the entropy does in fact increase.xxAnother interesting fact can be derived if we consider the relative Machnumbers M[ , M corresponding to the relative velocities u[, u before and2214.3203SOME P R O P E R T I E S OP SHOCKSafter the shock. A s usual, we define the sound velocity a and the M a c hnumber Μ in strictly adiabatic inviscid flow by(12)a = y*,ρM'2= ^a2=^ =ypP2™.ypThen, if v! in (10b) and (10c) is replaced by ypM' /m,these equations be­2comeypiMιPiQipi-yp M2+ p )M[Solving these for M'and M' ,2-22Since p /pi2222p,x= 0.+ pi)M2we obtain22y-2p (h pΜ ί ' - ϊ + I a + XJLi,(13)= p2pi» = I + J & + 1^2.M2y2y^ 1, it is seen that M[p2y2cannot be less than 1 and M'22cannotbe greater than 1; they can equal 1 only in the case of zero shock.

M o r e ­over, M2cannot be less than (7 — 1)/2γ, the value corresponding toP1/P2 = 0 (infinite compression). Thus^-P-(14)1ύ M'ύ 1 ύ M[222^00.2yIf E q . (10c) is solved for p , we find2h p2 — piPi T~2,iii — h uft pi — P2using (10a). Thus, as pi/p decreases from 1 to 0, the ratio p /pi increasesfrom 1 to h . Thus we have learned: In a physically possible shock,the velocity relative to the shock front is supersonic before and subsonicafter the shock. The density ratio p /pi cannot exceed h ( = 6 in air) andthe square of the relative Mach number after the shock cannot be less than(7 — l)/27 ( = y ) , the extreme values corresponding to an infinite pressureratio p /pi = <». N o t e that the actual velocities u , u , and c may allbe subsonic.N e x t we use Eq. (15) to show that the entropy increases during an actualshock transition. Since the entropy is essentially the logarithm of p/p , it issufficient to show that( Λ Kxu— hui2(!5)P2 =Pl--Γ-,=2222222x2y(16)* ! > i iorP?_(<!?y>o.Ρ2PiPi\Pl/T h e inequality is certainly true in the case p /pi =Ύ72° ° , where p /pi =2h.2204III.

ONE-DIMENSIONALW e consider therefore the casep /pi2FLOWh . If we take the expression for<2Vz/Ρι from (15), we find^ P 2 — pi _h pi — pΈΐ — (Pi\pi/=2/P2Vh (p /pi) — 1 _ / p 2h — (p /pi)\pi2=Vpi/2222•Since p / p i < h , the denominator of the first term on the right cannot benegative. If suffices therefore to consider only the numerator of the com­bined fraction, i.e., to show that the function22(17)*({) = h ± -1 -2-(h2where {=p /2P l,has positive values for 1 < ξ where ξ < h . B y differentiation,2(17')«'(*) = H -(i7")2YH'R2"(ξ) =-γ(17-(7 ++n/^r- 2l ) i T = H [l2+ 7(Ύ +Dr"-TiT1= 7(7 ++(7 -1)Π,Dr .ix* --2From these it appears that ζ, z', and z" vanish at ξ = 1 (corresponding tono increase of density, i.e., zero shock) and that z" is positive for ξ > 1,from which it follows that ζ (ξ) is positive for ξ > 1. Thus we have shownthat the entropy is greater after the shock than before; the amount of theincrease depends on the values of u\ p, and ρ before and after the shocktransition.*T h e theory of the one-dimensional flow of a perfect fluid, as developed inArts.

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