R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 39
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12 and 13, was based on the assumption that a relation of the formp/p = constant held throughout the fluid. W e now learn that when ashock occurs, the value of p/p changes for each particle, and the amountof change will not, in general, be the same for all particles. In other words,the results derived in the two preceding articles need not hold in the region of thex,t-plane beyond a shock front, except in the case that all particles undergothe same change in entropy. In Art. 15 some consequences of this situationwill be discussed.yyI t has already been indicated that the actual change in the value ofp/p is in most cases not considerable.
T o compute this difference, we mayagain use the expression for p /pi from (11), and we findy2(is)= 21 [£•(eiY -PI-EiP27pyPi\_P\1 ] = 21 [ A V z i\P2/JpiLh 2rξ_χ!JN o w h = 6, so that for ξ = 3 (half the maximum value of ξ), for example,the bracket has the value 0.217.
If the bracket is developed in a power seriesin ξ — 1, we find2(19)2i^=Li> ft _ i)» [1 - I ft _ 1) + . . . ] .* See Fig. 148 in which λ = exp [(Si -S )/gR].214.4A L G E B R A OF T H ESHOCK205CONDITIONSW i t h ξ = 1.2, corresponding to p /pi = 1.29, using the first two termsgives 0.00063 as compared with the exact value 0.00068 found from (18).I t is seen that if ξ — 1 is not too large, i.e., for not too strong a shock, theassumption that the value of p/p does not change is not too bad an approximation.2y464.
The algebra of the shock conditionsT h e shock conditions (2a), (2b), and (2c) are three algebraic equationsin the seven variables Ui , pi , pi , u , p , P 2 , and c. T h e y determine a fourdimensional variety (hypersurface) in a space of seven dimensions.
A complete study of this algebraic variety will not be attempted here. Equations(10) present a comparatively simpler case, in which two sets of three variables ΐίί,ρι,ρι and u' ,p ,p are related by three equations; these maybe considered as determining a point-to-point transformation of a threedimensional space onto itself.First of all, we note that the transformation is an involution, i.e., theequations do not change if the subscripts 1 and 2 are interchanged. W e maysolve for one set, say u p ,p , using m = piu[, and obtain, in addition tothe trivial solution u = u[ , p = p\ , p = pi , the relations22222j22222u=2(20)2p2(7 -27 - +mV7 + 1= — j —[-(7 -+DmDu[Dpi +2mu[]7 + 1P2=(7%ΎΡι +(7 -2l)mu['T h e first of the following equalities is obtained by multiplying the firstequation of (20) by h u[ , and the second follows by an interchange of subscripts:2/ n i\1.2(21)h UiU''2,/2=Ui27pi+—7 — 12yii=u2+p2—.7 — 1pP!2In general, the square of a velocity divided by 2g is called a velocity head,and in the adiabatic case yp/(y — l)pg is the pressure head (see Sec.
2.5).If we disregard the influence of gravity, the sum of velocity head and pressure head is the " t o t a l h e a d " Η which occurs in the Bernoulli equation.*Using H (to indicate that this refers to the relative velocities), we canwrite (21) in the formf(22)J/J =ffj=Λ*φ?.* T h e total head is constant for a particle only when the motion is steady,Sec. 2.5.47see206III.ONE-DIMENSIONALFLOWT h e equality of H[ and Η\ , which is actually (2c), lends itself to the (misleading) interpretation that the shock transition behaves like the steadyflow of an inviscid fluid.A dimensionless formulation of the conditions (10) is obtained by introducing the M a c h numbers M[ and Mas defined in (12) and the ratios2P2(23)—Then (10c) becomes(24a)η=piUi-717iPi=Uif =-1-==Γf-ξP2tffa-1),and (10b) may be written as~11 -f-1V(24b)or as(24c)yM' *.21-Equation (24a) is the equation of an equilateral hyperbola in the f ,77-plane,having the asymptotes f = 1/h and η =2at f = h (Fig.
75) ,248—1/h and intersecting the f-axis2T h e part of the hyperbola for which f or η is negativehas no physical significance. Under our assumption that the subscript 1refers to the values before the shock, we have f <shock determines a point Ρι(ζ,η)1, η >1; a particularto the left of A on the hyperbola, where-4(1,1) corresponds to zero shock. Equation (24b) is the equation of aand passing through Pi . Ifstraight line through A with slope — yM[Pi is given, then the slope is determined; if the slope is given, then Pi isdetermined as the intersection of the straight line with the hyperbola.^slope-yM!slope-/M'22intersectionat £ « h * 62ζTp2PiF I G .
75. Hyperbola of — = η versus - = |\PiP214.4T h e point P2ALGEBRAOF T H E SHOCK207CONDITIONSwith coordinates 1/f, Ι/η corresponding to a given shock Pialso lies on the hyperbola (24a), but to the right of A. I t may easily be located on the graph by using the fact that OP2makes the same angle withthe f-axis as OPi does with the 77-axis. From E q . (24c) we see that the slopeof AP2is-yM .2Solving Eqs.
(24a) and (24b) for η and f, which amounts to finding thepoint at which the line through A with slope — yM[intersects the hyperbola, we find, of course, η = ζ = 1 and as the second point of intersection17 +Pi(25)ξς«ί1-[2yM?7 +Ρ2(Ύ-1)] =1 [Μ?2yτι,τ'+ Ϊ7-7 +ΊΚΎ12Μ 1ψ1 Μί*+Κ*Analogous equations, with the subscripts 1 and 2 interchanged and withξ, τ;, and f replaced by their reciprocals, are obtained from (24a) and (24c).T h e first equation (25) and its analogue are identical with (13). B y meansof Eqs.
(24) and (25) each of the quantities 77, f, ξ ( = 1/f), M[*, andM2can be expressed in terms of any one of the others as a linear or a linearfractional function. A l l five variables have the value 1 in the case of zeroshock, where no change occurs in the values of u'p, and p.
T h e deviationyfrom unity of any of the quantities above can be taken as a measure of thestrength of the shock.If Ι/ξ is used in place of f in the first equation (24), we find(26)h\vwhich is known as the Hugoniot-ξ)=ξη- 1 ,equation.™ In Fig. 76 is shown that part5v£ ( a p p r o a c h e s h3M15,22(approaches102 e6as ηfY7 as η - * > ω )15F I G . 76. Graphs of AfJ , M?, and ξ versus η.2co)20208III.
ONE-DIMENSIONALFLOWof the equilateral hyperbola (26) for which ξ > 1, η > 1, together with thetwo curves giving M[ and Mas functions of η, according to the equations (13).2If the first equation (25) is multiplied by its analogue, we obtain a symmetrical relation between the relative Mach numbers, namely,.-(^-.--έΧ^-ί·-^)or(27)2yΜΪΜ'= 2 +22(y -1)(M[+2M\2which is the same as the relation (11.30') between the actual Mach numbers in the steady case.Practical problems are often such that some data concerning the actualvelocities u u are known in advance. In these cases we must use relationswhich do not involve the relative velocities u[ u .One such relation is theHugoniot equation (26), which can be written in the formh2y(26')V2-V1_P2 —+ ViP2yPi2Ρ2 +PiOf course, this follows directly from (15) without first introducing thedimensionless variables.
If an expression for m not involving c is obtainedfrom (10a) and substituted in (10b), one finds(28)p, -pi = - 2 2 P2 —(«, -u,)\PiEliminating p or pi from Eqs. (26') and (28) leads t o *2(pi -PiY = ^ {u-2ui) [(y2+l)p2+(7 -l)pj(29)= ξ (w 2UiYKy + l )P l+(7-Another important relation of this type, linking the change in temperature(i.e., in p/p) to (u — Ui) will be derived in the next section.22f5. Representation of a shock in the speedgraph planeIn Sec. 12.3 we saw that in one-dimensional flow the state of a movingparticle at any moment is determined by two quantities: the velocity uand the quantity ν defined in (12.19), which represents the density (and* The curve of p versus u , for fixed state 1, is sometimes used in place of the"shock c u r v e " given in the next section.2214.5REPRESENTATION IN THESPEEDGRAPH209PLANEpressure) and likewise has the dimensions of a velocity. I t is assumed, ofcourse, that a (p,p)-relation holds for the particle.
In the speedgraphplane (w,y-plane) the continuous motion of a particle is represented by acurve. In the case of a simple wave this curve is one and the same ± 4 5 °line for all particles.When a particle passes through a shock front, its representative point(u,v) in the speedgraph plane undergoes a sudden jump. W e shall now studythe conditions governing this discontinuous transition. W e shall restrictourselves to the case when the sound velocity a is related to ρ and ρ bythe equation a = yp/p where y is constant throughout. This is admissibleif before and after the shock the motion is supposed strictly adiabatic andthe fluid an inviscid perfect gas. T h e relation between ν and p/p isy(30)v2=4a(_-Tl)\y(7 -2ρl) p2see (12.19'), and the inequality (9) is expressed by(300v ^ vi .2T h e relation between the pairs of coordinates u Vi and u ,v , corresponding to the states before and after a shock, is determined by the threeshock conditions (2a), (2b), and (2c).
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