R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 36

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Further, ν — u isconstant for the whole flow, so that the distortion of the curves for v, andtherefore for ρ and p, is similar.00If we extend the time value t to the lower or upper limit of the time in­terval (t-2, t ) in which the flow pattern is determined for all x, we findthat du/dx becomes infinite on one side of the crest of the wave.Thesolution breaks down completely for values of t outside the time interval(t-2, t ). For instance, at t = L_ and t = / there is more than one value ofu determined by some values of x*234233Another example of a simple wave solution is shown in Fig.

70. In a tubeextending to infinity in both directions the fluid is initially at rest; thevalue of a is the same everywhere, a ; and ν is constant, say ν = v . Sup­pose that at t = 0 a piston starts moving toward the right from the positionχ = 0 according to the law χ = x\(t), represented by the curve C.

T h etangent to C at the origin is vertical, so that the velocity of the piston is0035* T h e breakdown is not due to the discontinuity in slope of the ΐί,χ-curve, andclearly occurs in every case of a limited simple wave disturbance. In fact, it is eas­ily seen that, for increasing t, du/dx first becomes infinite at a point correspondingto that of maximum negative slope on the initial w,z-curve.190III.ONE-DIMENSIONALFLOWcontinuous at t = 0.

T h e curve C is the particle line for all particles adjacentto the piston on either side. T h e initial conditions u = 0, ν — vQuniquelydetermine the state of rest u = 0, ν = vo at all points within any character­istic quadrangle determined by a segment of the x-axis to the right or leftof the origin. T h e characteristics for this state of rest are the straight lineswhose slopes, measured from the f-axis, are =ba . Thus the t w o straight0lines OA and OB with slopes Ψαο limit the outside regions in which thevelocity is identically zero. T h e flow pattern between OA and OB is de­termined as follows. Each point Ρ on the Xi-curve has a velocity u, repre­sented by the slope of the tangent at P .

Then Ρ is mapped in the speedgraph into one of the points P' or P" with abscissa u and lying on the + 4 5 °or —45° line through 0' with coordinates 0, v . Since OA is a (u — ^ - c h a r ­0acteristic and OB a (u +a)-characteristic, the area between OA and C iscovered by a backward wave and maps into the —45° line, while the areabetween OB and C is covered by a forward wave, mapped onto the + 4 5 °line.

Using P' for the right and P"for the left part of the flow, one can con­struct the two straight characteristics at P, using the lines with slopes ± 5in the speedgraph. Along each of these lines u and υ are constant, so thatone can proceed to the construction of the particle lines.T h e pressure values (determined by the v-values) are different on thetwo sides of the piston.

T h e difference in v-values is given by the distancetFIG. 70. Construction of the two centered simple waves in the case of a disturbancecaused by a piston moving in a gas at rest along a path C.13.4C O M B I N A T I O N OF S I M P L E191WAVESP'P".T o bring about this flow, it is necessary to apply to the piston an ex­ternal force toward the right, gradually increasing it to give the appropriatedifference P ' P " .T h e flow to the right is seen to be a compression wave.

This type ofsolution of a flow problem breaks down after a finite time interval. W e shalldiscuss this in A r t . 14.T h e rarefaction wave to the left continues indefinitely. When the pistonreaches the speed u = v (point P" in the speedgraph), the particles adja­cent to the piston have attained maximum speed; any further increase inthe velocity of the piston leaves a cavity, in which ρ = 0, between theseparticles and the piston, while the velocities of all other particles tendasymptotically to the value u = v .004. Combination of simple wavesIf the values of u and ρ are constant in any finite region of the #,£-plane,the flow in any adjacent neighborhood can only be a simple wave. This isseen immediately from the fact that a region of constant u,p is mappedinto a single point in the speedgraph plane. Then, any adjacent neighbor­hood cannot map into an area in the speedgraph plane and must correspondto an element of arc passing through that point.* (Of course, the notion ofsimple wave includes the limiting case of uniform flow.)A simple wave can form the transition between any uniform state: u =U\, ρ = p i , and another such state: u = u , ρ = p , provided that eitheru + ν or u — ν has the same value in both states.

(Since ν cannot benegative, u — U\ must be less than Vi or greater than — v respectively). B ycombining a forward and a backward wave, and inserting a uniform motionin between, any final state can be reached. A n example is shown in Fig. 71.Here u = 0 at the beginning and at the end of the motion, but pressureand density have increased corresponding to the change in ν from A' to C.T h e flow in the x,£-plane is not unique, however, since both envelopes E\and E of the straight characteristics can be chosen arbitrarily.

T h e onlyrestriction is that the critical regions, bounded for each wave by the twoextreme characteristics and the envelope, be outside the flow region. Forthe first simple wave the two extreme straight characteristics in questionare the ones which are parallel to the line segments MA\and MBiinthe speedgraph; and for the second, the ones parallel t o MBandMC .In our example v and v , that is, the ordinates of A' and C", have the ratio1:1.2; with κ = 1.4, this corresponds to a compression ratio of (1.2) :1 ~5:2.

This ratio appears clearly in the figure as the concentration of thevertical particle lines at the end of the flow compared to that at the begin222h220225* A more formal proof follows lines similar to those for the plane case (Sec. 18.1).192III.ONE-DIMENSIONALFLOWuFIG. 71. The use of a pair of simple waves to effect a given compression withoutresultant motion.ning of the flow.

T o induce the flow indicated in Fig. 71, two pistons mustbe used, having, respectively, the motions of the first and last particle lines.In order to choose, from among all possible flows in the x,/-plane, onewhich achieves a given compression as quickly as possible, we use centeredwaves with the centers located as shown in Fig. 72.

( T h e subscripts 0, 1,and 2, when attached to u, v, or a, will signify the values of that quantityin the initial, intermediate, and final uniform flow regions, respectively.)Here the left piston is first accelerated up to the velocity u =(vxand then movedatconstantspeed, with v-valueVi =— v )/22(v20+Vo)/2,until the desired compression is attained; the right piston remains atrest until the time t =h =k/ao, when the first wave reaches it, and itmust then start moving with speed U\ and be decelerated down to u = 0.2T h e compression ratio is k/h . T h e time interval t (see Fig.

72) can be com2puted by applying Eq. (8) to the second wave, and the time required forttilt4 -I10— JFIG. 72. Effecting a given compression in the shortest time.13.4C O M B I N A T I O N OF S I M P L E193WAVESνGΕADu•χθFIG. 73. Penetration of two simple waves. AEFCregions, and CFGFthe region of penetration.and BE^F\Care the simple waveXthe whole process turns out t o b e36(16)W i t h a combination of a forward and a backward wave any compressionratio can be realized. I t will be seen later (Sec.

14.6) that no solution ofsimilar type exists if the right-hand piston is kept entirely at rest.A s another example we study the penetration of t w o simple waves. I t isassumed (see Fig. 73) that at t = 0 the interval AB contains gas at rest:u = 0, ν = VQ . This uniform state then obtains throughout the triangleABC, where AC and BC have slopes =ba measured from the /-axis. I n thespeedgraph, the entire triangle ABC maps onto one point A'.

T o the left,along AD, and to the right, along BDi, some disturbance is imposed. T h egiven values of u and υ along these segments map them in the speedgraphonto t w o curves A'D' and A'D[ . (B coincides with A', as does C.) B ymeans of the methods described in Sec. 12.5 and Sec. 12.7, the flow patterncorresponding to the areas A'D'E' and A'D[E[ can be developed. In theareas adjacent to AC and BC the flows must be simple waves mapping ontothe characteristics A'E' and A'E\, respectively. T h e areas in which thesesimple waves occur are bounded by straight characteristics EF and E\F\ ,which are mapped into the points E' and E[, respectively, and by thecross-characteristics CF and CFi , which must map into A'E' and A'E[.A t C the t w o simple waves meet and the problem to be solved is the sub­sequent flow in the penetration region CFGFi , which maps into the rec­tangle A'E'G'E'i in the speedgraph.0f37T h e data for this problem include: the curves CF and CFi in the physicalplane and the compatible u,v-v&\ues along these curves, with ν — u = valong the first and ν + u = v along the second.

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