R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 41
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14.6, where, however, theshock was moving from right to left). If Β represents the point at which theshock reaches the right end of the tube, where the velocity must be 0 for allvalue of t, then the particle line through Β must remain vertical. A solutionsatisfying this condition is obtained by assuming another shock line BCwith slope c' (c' < 0) along which the inclined particle lines again changedirection, but now in the opposite sense, so as to become vertical again.This phenomenon, where a shock front moves forward and then back, isknown as shock reflection. T h e expression "reflection" suggests the symmetΒAχ0ν2FIG. 8 1 . Reflection of a shock front at a fixed wall.216III.
ONE-DIMENSIONALFLOWtry condition, c = — c. W e shall see, however, that this relation is not fulfilled even approximately, except for very weak shocks.fThis same flow can be considered in a two- or three-dimensional space,rather than in a tube, all particles with the same χ having the same velocity,etc. Then the " e n d " of the tube is represented by a wall normal to thear-direction.In the speedgraph (Fig, 81) the fluid state prior to the first shock isrepresented by a point 1 on the f-axis. T h e point 2, corresponding to thestate between the two shocks, must lie somewhere on the shock curvewith its corner at 1, and in fact on the right half of this curve, since u[ < 0.If the vertical axis of the shock diagram is then shifted so as to pass through2, there is one curve with its vertex at 2, and on this must lie the point 3representing the fluid state after the second shock. Since we know that thefluid is at rest after the second shock, the point 3 must also lie on the f-axis,so that 3 is uniquely determined once 2 is known.
Thus, given the state 1,there exists a simply infinite number of possible transitions 1-2-3, eachcorresponding to a different increase of ν or p/p (temperature). As the Machnumbers Mi and M are zero, we can characterize each possible transitionby the value of M , the Mach number of this intermediate state. N o w Mhas the value 2/(y — 1) times the slope u /v (measured from the *;-axis)of the radius vector from the origin to the point 2.
Hence Mmay rangebetween 0 and 2/7(7 — 1), the maximum value corresponding to the casewhen the point 2 is at infinity on the shock curve, with the radius vectorbecoming the common asymptote of the shock curves (see Sec. 14.5). Ourtask is now to compute the ratio c'/c and the pressure and density ratiosPz/pi and p /pi as functions of M .32222232T o this end we apply Eq. (14.21); for the first shock we have u[ = — c,u = u — c. Dividing all members by u and introducing M=p u /yp ,we find2222ΦΛcu2\c\uJ2_ " ,2cu22222PiΎ27 — 1 P\U227 -1M '22For the second shock we must first replace the subscripts 1 and 2 in E q .(14.21) by 2 and 3 and then use ui = u - c', u = -c'. If we divideagain by u , the equations read2z22u \2uj\2= — +u227 - I Mu)27yP z—1 p W2 *322215.1SHOCK217REFLECTIONA comparison of the first and third members of (1) with the first and secondmembers of (2) shows that c/u2and c /u,satisfy the same quadratic equa2tion.
Remembering that c is positive and c' negative and that h = (7 + 1)/2(7 — 1), we find(3)^u=A2^,+ s-.-^/(ΦΤ^t.i^-s,2 'so that*(3')As M2c3 -7 -ASc3 -7 +AS 'goes to zero, S becomes infinite, and the ratio c''/c has the limit — 1.= 2 / 7 ( 7 — 1), the value of S is (37 — l ) / 4 , so thatA t the upper limit, M2c'/c =— 2/h , which is — ^ in the case 7 = ^. As is to be expected, the for2mula for c'/uagrees with the first result of (14.37), found in the simpler2case studied in Sec.
14.6.Using the ratios c/u and c'/u ,2in terms of M , we can derive the density22increase from the first shock condition, E q . (14.10a), applied to the twoshocks:— pic = p (u(4)2p (u— c);22— c')2— p c'.=3If the values from (3) are introduced into the last expression, the resultcan be reduced toU'\ρ{PiΛ =}4+^( 7 +24 + M \y4S)AS)'= 0 and the value 7 ( 7 +This quotient has the value 1 for M2( = 21 for 7 = I)1 +1 -+2at the upper limit of M1)/(γ — l )T o compute the pressure increase we can solve in Eq.
(1) forin terms of c/u and in (2) for pz/pzu2in terms of c'/u22.2pi/piU2. In combination with2(4) this leads to,.χV*=c— u2^ 2cc — uPi— (7 +2c — (7 +21)^2l)u 12which, by the use of Eqs. (3), may be reduced to(50pzVl_ 4 +yM (yA + 7^2222(7+1 +4S)+1 -AS)* From (3) also follows (c — U2)(u2 — c') = a . Hence the Mach numbers of thestate 2 relative to the incident and reflected shocks are reciprocals.22218III. ONE-DIMENSIONALΟFLOW1ΌΟ1.89«wF I G . 82. Pressure, density, and shock speed ratios, produced by reflection of ashock front, as functions of the Mach number Μ2 of the intervening flow.This ratio increases from 1 to 00 as Μ increases from 0 to its maximumvalue.In Fig. 82 the quantities — c/c', p /pi , and p*/pi are shown as functionsof Μ2 in the case 7 = 1.4.23In a similar way the pressure ratio pz/p2, across the reflected shock alone,may be determined.
W e find4P2which increases from 1 for very weak incident shocks (Af small) to(37 — l ) / ( 7 — 1) = 8 for very strong ones (Af near its limiting value).Alternatively this formula may be combined with (5') to give the ratio ofthe pressure increments across the two shocks,22Pz -P2P2 -pi=4S +(γ +1)4S -(7 +1) "This ratio approaches 1 for very weak incident shocks—corresponding tothe acoustic case—but increases to 2 7 / ( 7 — 1) = 7 for very strong shocks.Similar results are found for the density.For very small M the Taylor developments of (3'), (4'), and (5') throughterms of first order give2(6)-'c-1+^M2,ΔThe principal results are that^ ~ 1 + 2 M2,^- ~1+2 M .3pithe72pireflectedshockcanhavea velocityof15.2D I S C O N T I N U O U S S O L U T I O N S OF I D E A L F L U I D E Q U A T I O N Spropagationas small as one-third that of the incoming219shock and that veryconsiderable increases of pressure and density may be caused by the reflection.2.
Discontinuous solutions of the equations for an ideal fluidI t was stressed in Sec. 14.2 that flow patterns which include shock frontscannot rightly be considered " discontinuous solutions of the differentialequations for an ideal fluid". These equations are, of course, satisfied bythe w,p,p-distributions in the spaces between shock lines; but across a shockline the conditions of inviscid flow are violated. For example, in the adiabatic case inviscid flow requires that each particle keep its entropy value,but the value is changed as a particle crosses a shock line.
This contradiction cannot be eliminated by any sophistry about the " nature of discontinuous transitions". T h e correct theoretical basis for admitting flowpatterns which include shocks is the fact that these flows are asymptoticsolutions, for μ —> 0, of the equations for viscous (and/or heat-conducting)fluids.Nevertheless, as in other branches of continuum mechanics (elasticitytheory, incompressible fluid theory), there also exist true discontinuoussolutions of the partial differential equations for an ideal fluid. T h e occurrence of such solutions was discussed previously (Arts.
9 and 10):across the so-called characteristic lines certain first-order derivatives of thedependent variables may change abruptly,* without any of the differentialequations being violated. Essentially, this depends upon the fact that theseequations- do not lead to any conclusions about these derivatives across thecharacteristic. N o t e that shock lines are not characteristics.W e saw in Sec. 12.2 that in the one-dimensional flow of an ideal fluidthere occur characteristics which have the slope u + a or u — a, measured from the J-axis. Across such a characteristic line the variables u, p,and ρ remain continuous, but their derivatives normal to the line maychange abruptly; the integration theory developed in Arts.
12 and 13 depended essentially on the use of these characteristics. These characteristics,when considered in the general theory of characteristics given in A r t . 9,correspond to the vanishing of the second factor of E q . (9.26). I t was mentioned, however, at the end of A r t . 9, that in all cases of ideal fluidmotion another type of characteristic exists, corresponding to the vanishingof the first factor in (9.26). In the one-dimensional case this factor is(u\i + λ ) , indicating that the normal to the characteristic has the slope4* A discontinuity is usually called of zero order if the variables themselves undergoa sudden change, of nlh order if the discontinuity appears first in the nth derivatives.Since, however, in fluid mechanics one may take as dependent variables either thevelocity components, etc., or such functions as particle function and potential, thisclassification is not always unambiguous.220III.ONE-DIMENSIONALFLOWλι/λ = —l/u and the characteristic itself the slope u.
Thus, all particlelines in the x,t-plane are characteristics in this sense. W e shall now discussthe possible discontinuities which may occur across the particle lines; theymay appear in incompressible, as well as in compressible, flow.453T h e first to consider flow problems of this kind, in the case of incompressible fluids, was H . von Helmholtz. In a famous paper of 1868 he studiesparticularly the free jet problem: a steady two-dimensional flow is boundedby two streamlines, along each of which the pressure has a constant value.In the absence of gravity or other external forces this flow is not affectedby the presence of fluid at rest, bordering these streamlines, for clearly inthis case both the continuity equation and Newton's equation are stillsatisfied for each fluid element.
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