R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 35
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ά β=h\v0±+Ci].184III. ONE-DIMENSIONAL FLOWFrom Eqs. (6) and (7) we can deduce the following property of eitherset of curves, particle lines or cross-characteristics. The integral curves, particle lines or cross-characteristics,constant, Co, Co +equidistantc, Co +corresponding2c, ·to equidistantvalues of the·, intersect any straight characteristic atpoints.2.
Centered waves33A particularly simple case occurs when the envelope Ε reduces to a singlepoint, i.e., when the set of straight characteristics consists of radii emanating from one point. A flow pattern of this kind is called a centered wave.If we suppose that the center lies at the origin, the quantity x in Eq. (1)0vanishes and (1) reduces to β = x/t.
For polytropic flow Eq. (60 now gives/(8)x\~ith2ort = k[vo±-\± x = - v0\2 / u + 1 )t + k { ^ )as the equation for the particle lines, and Eq. (70 gives(9)Jt = hi [vo ± -or(J-J± x = -vot +fcias the equation for the cross-characteristics.
These equations include allfour cases of centered waves (forward or backward, compression or rarefaction). T h e constants k and k\ , which single out individual curves, havepositive values in the case of a rarefaction wave and negative values (together with negative t) for a compression wave. T h e velocity u = dx/dt canbe found by differentiating (8) or, alternatively, by solving for u in the relations ( 2 ) , with β =(10)u =where α0=x/t:κ +(κ — l)vo/2 is the sound velocity at a stagnation point (whichoccurs when the particle line has a vertical tangent).
From (10) and υ =Vo ±u there followsίΛ Λ\(11)I*^ \^υ = db——- - +κ + 1ίκ + 1=*—rκ +Since dx/dt = u becomes infinite as we approach the origin along any particle line, as can be noted from (8) and (10), we see that the center of thewave corresponds in a certain sense to the state \u \ =therefore ρ =oo and a =, ν =oo (ando o ) .
On the other hand, since on each ray throughthe center u and ν have constant values, the center is mapped into thewhole line Γ of the speedgraph.13.2CENTERED185WAVESB y means of (11), Eqs. (8) and (9) can be written as*t = constant-v~=h2to ( - )on a particle line,1')(V2V2 / 2£ = constant · ζ Γ= toI )on a cross-characteristic.Then, from (10) we obtain, for later use,λχ -ut = ~ (x2/2± v t) = ^~2^0I\= ±a t I^J(ll")0 0v t2/U-l )(3-Κ)/2(Κ-1) on a particle line,on a cross-characteristic.Since i; is proportional to a and a is proportional to p * , one concludesfrom (11') that, for a given particle, ρ is proportional to the —2/(κ + 1)power of \t\, where | t | is the time elapsing between the actual state andthe state \u \ = P O , ρ = o o . Consequently the density ρ changes monotonically, decreasing in rarefaction waves, increasing in compression waves.2- 1Figure 66 shows the particle lines for various values of the constant kand the speedgraph of a forward rarefaction wave, with VQ = 5, κ = 1.4.
Allother waves of the same type are obtained by affine transformations, whilereflections in the x- or /-axis or both supply waves of the other three kinds(see Fig. 67). A s has been mentioned, the " c e n t e r " point on each particleline corresponds to the state \u \ = ° o , ρ = °° (and to the point at infinityin the speedgraph); by setting / = oo in (11'), we see that the point at infinity on each particle line corresponds to u = Ψν = F2o /(^ — 1),and ν = 0 in the speedgraph.
A physically possible centered wave canbe represented only by a portion of one of the four regions indicated inFig. 67.0=0If C is the constant in the polytropic relation, i.e., C = p/p , then Eqs.(12.190 and (11) giveK2 / U - l)( 1 2 )' -G^g")"'""" =* For Eqs. (11') and (11") U is a parameter which varies with the particle line orcross-characteristic; it is the time at which u = 0 on the curve considered. On theother hand, VQ and ao are constant for the whole flow.186III. ONE-DIMENSIONALFLOWF I G . 66. Example of a centered forward rarefaction w a v e : vo = 5.
Shown areparticle lines for equidistant values of the constant k.Together, (12) and (10) determine the derivatives θψ/θχ = ρ and θψ/dt =— pu of the particle function ψ(χ,ί) of the centered wave. One can easilyverify by differentiation that(is)M)= ±ltfV^r"-"(vo ±On the other hand, the potential function Φ(χ ί)respect to χ and}- ϋ2-Ρjj'.has u as derivative withu=2κ -1Κ22—412Vas derivative with respect to t. These conditions are fulfilled by the expression(14)Φ(Χ,$ =(τ ^ a*)^-τ ° ·α2It is seen that the centered simple wave is one of the examples discussed inSec.
7.4 [example (a) with c = 0] and in Sec. 12.2 [the example of a solutionfor (12.17c)].For each point (x,t) the Mach number is given by2(15)Μ_ l«l _=2Iu I— I t )=2I χ =F a4\κ — 1Vot db X13.3O T H E R E X A M P L E S OF S I M P L E W A V E S187F I G . 67. T h e four types of centered simple w a v e . Corresponding changes in thetwo planes.with the upper signs for a forward wave and the lower signs for a backwardwave.3.
Other examples of simple wavesL e t us assume that the envelope Ε of the straight characteristics consistsof two arcs AB and CD (see Fig. 68), both tangent to a straight line BC.DυFIG. 68. Limited simple-wave disturbance.188III. ONE-DIMENSIONALFLOWW e suppose further that the tangents at A and D are parallel and haveslope α,ο, the sound velocity at u = 0, measured from the /-axis. Then fora forward wave the image of that part of the x,/-plane covered by thetangents to Ε is the segment A'B' of the 45° line, where A' (= D') has thecoordinates u = 0, ν = v , and the endpoint B' ( = C") with coordinatesu v say, may be found by reversing the usual construction: draw theparallel to BC through Μ in the speedgraph plane and then the line withslope —5 meeting the 45° line at B'.
T h e flow pattern in the x,/-plane isuniquely determined in the region mapping into A'B', i.e., in the region between the parallel tangents at A and D, with the exception of the tworegions AAB and DDC in which tangents to Ε intersect each other. Sincethe tangents at A and D are characteristics, the solution in the adjoiningregion may be analytically quite different provided merely that the combined solution is continuous along these lines.
Thus we may assume thatu = 0, ν = v outside these parallels. T h e map in the speedgraph of theseregions is then the point A'.In the time interval determined by the two horizontals through Β andC, say / = Z_2 and / = Z , the motion is determined for all x. T h e pajticlelines can be constructed easily by means of the speedgraph or can be computed according to the formulas developed in Sec. 1. W e wish however tostudy how the w-values, considered as functions of x, change in time (seeFig. 69).0hh02A t the time / = Z , the midpoint of the interval mentioned above, thevelocity is zero for all points to the left of A , it increases as we cross thestraight characteristics from zero at A to the maximum value u, at Ο000• xFIG.
69. Distortion of the velocity profile.13.3OTHEREXAMPLESOF S I M P L E189WAVES(which maps into Β'), and then drops to zero again at D . If the arcs A Βand CD are symmetric with respect to the point 0, the curve of u versus χwill be symmetric with respect to the line χ = constant through 0 (see themiddle diagram of Fig. 69). N o w the u-values are constant along a straightcharacteristic. Thus, if we examine the w-values along t = ti, the samew-values are found, but further to the right, corresponding to greater valuesof x. I t is clear, however, that the rate of movement to the right is largestfor the maximum value u\ and least for the end values.
T h e analogous phenomenon, only with shifting to the left, takes place if we pass from / = toto t = L.i (see Fig. 69).0This flow pattern obviously represents the behavior of a limited disturbance within a fluid otherwise at rest and it shows clearly that the expression " w a v e " is appropriate. T h e two endpoints of the disturbanceprogress at the rate a while the culmination point moves at a higher speed,(see DoD and AD in Fig. 69). Thus, the front part of the wave is steepened,while the rear part is flattened out.
This result may be compared with theresults of Sec. 4.2 for a small perturbation where we found that each partof the initial disturbance progresses, unchanged in shape, at the soundspeed ao · There, however, in omitting the terms of higher order in thebasic equations, we made the problem in the x,/-plane a linear one, so thatthe characteristics were the same for all solutions and, in fact, the parallelsof slope ± a . T h e exact solution, here considered, shows that the characteristics are divergent to the rear of the flow and convergent to the front;this brings about a distortion of the curves of u versus x.
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