R. von Mises - Mathematical theory of compressible fluid flow, страница 10

L e t Ρ = (χ, y 0) be a fixed point in the x,?/-plane.yTheintegral intheformula(16), for f(x,y, 0, t),extendedover asphere S of radius R = aot about P, may be expressed as an integral in thea;,?/-plane. If d§> is an element of area at the point (ξ, η, f ) on S, and dAdenotestheprojectionofd§> ontothe£,2/-plane,thend§:dA=46I. I N T R O D U C T I O NR:\/R2— r where r22= (χ — ξ)2+(y — η) .

T h e projection of S onto the2x,?/-plane is the interior of the circle C of radius R about P , covered twice, sothatFor given / and g, the two-dimensional solution u(x, y, t) may be givenin the form (23), with the definition (16) of / replaced by(28)Since the integrand in (28) is zero, except in the intersection of C andCo, it is again true that at any time t there will be no perturbation at anypoint ofthe x,2/-plane whose distance fromthe initial disturbance isgreater than a t. I t cannot, however, be concluded that any region will0become free of perturbation after a certain time has elapsed.Article 5Subsonic and Supersonic Motion. Mach Number, Mach Lines1.

Small perturbation of a state of uniform motionT h e theory, developed in the preceding article, for a small perturbationof an elastic fluid initially in a state of rest can easily be extended to thecase of an elastic fluid initially in a state of uniform motion. In the firstplace, constant values of velocity, pressure, and density, q = q , ρ =0po,ρ = po, are compatible with the basic equations (1.1) to ( l .

I I I ) , providedmerely that the values p and po satisfy ( l . I I I ) and that the effect of gravity0is negligible. Further, the equations of motion are not different when re­ferred to a coordinate system moving with constant velocity(inertiasystem). I t follows, therefore, that for an observer moving with constantvelocity q , all phenomena will be exactly as described in A r t . 4: a small0disturbance at time t = 0 at a point 0' of the inertia system is perceptibleafter time t on the surface S of a sphere of radius R = aot and center 0'.A n observer at rest, how ever, will at time t perceive the point 0' to be dis­rplaced by an amount q t from its initial location Ο at t = 0.

T o him the0spheres S successively reached by the perturbation will form a pencil of5.1S M A L L P E R T U R B A T I O N OF U N I F O R M47MOTIONspheres with centers progressing along a line in the direction of q at therate g while their radii increase at the rate Oo .00T h e nature of this pencil depends upon the relative sizes of q and a .In Figs.

11 to 13 are shown the three possible configurations of the pencilof spheres, with spheres corresponding to t = 1, 2, 3. T h e plane of thedrawing is any plane containing Ο and q ; each circle represents the inter000F I G . 11. Disturbance originated at Ο spreading in t = 1, 2, 3, · · · seconds withsound speed a while gas moves with uniform horizontal speed q < a , subsonicflow.00F I G .

12. See Legend to F i g . 11. q > a , supersonic flow.00F I G . 13. See Legend to F i g . 11. q «0a,0sonic flow.048I.INTRODUCTIONsection of a sphere S with this plane. T h e circle corresponding to time t hasthe radius a t and center at the abscissa qot. Thus the right-hand intersec­tion of S with the horizontal axis has abscissa qot + a t = (qo + ao)t and isalways to the right of 0, moving to the right indefinitely as t increases.

T h eleft-hand intersection has abscissa (qo — ao)t.00For <?o < «o (Fig. 11), the left-hand intersection falls to the left of 0 ,and farther to the left for larger t. All spheres include the source of the origi­nal disturbance, so that the perturbation is propagated in all directionsfrom the source (although the speed of propagation is not the same on allrays emanating from this point) and eventually reaches all points of space.Thus, there is not too much difference between this phenomenon and theone discussed in the preceding article, where q = 0.If q > ao (Fig.

12), the left-hand intersection with abscissa (q — aot)falls to the right of Ο and moves to the right as t increases; no sphere in­cludes the source of the disturbance. All spheres are interior and tangent toa circular half-cone whose semivertex angle ao is given by000ωsinαοao =—.QoIn this case, the perturbation is not propagated in all directions from the source,but only in such directions as lie interior to the half-cone determined byEq. ( 1 ) . This is a situation entirely different from that occurring in thecases qo = 0 and qo < a .Finally, suppose q = ao (Fig. 13). T h e half-cone given by E q .

( 1 ) de­generates to the cone corresponding to a = 90°, i.e., the whole space to theright of the plane through Ο normal to the direction of qo . Here, the sur­face of any sphere S touches the boundary plane at the point O. T h e per­turbation is propagated in all directions pointing to the right of this plane.0002. TerminologyT h e deeply rooted difference in the behavior of a small perturbation inthe cases q J ao has led to a terminology generally adopted today in thetheory of compressible fluids. T h e ratio qo/a , flow velocity to sound veloc­ity, is called the Mach number, Μ ο, of the flow (before perturbation), thehalf-cone determined by Eq. ( 1 ) a Mach cone, and the angle ao the Machangle, all named after Ernst Mach, who was the first to observe and de­scribe this type of phenomenon.

For uniform flow, the three cases0025(2)Mo = ^ < 1,aoM =l,0andMo > 1are referred to as subsonic, sonic, and supersonic flows, respectively. AMach cone exists only in the case of supersonic flow; for subsonic flow, no5.349P R O P A G A T I O N OF P E R T U R B A T I O Nreal angle a corresponds to Eq. ( 1 ) . If g0= 0, the Mach number is zero,0but this is also true for any q if a = QO , i.e., in the case of an incompressible00fluid (dp = 0, dp/dp =QO). Thus the M a c h number is, in a certain sense,also a measure of the comparative deviation of the actual behavior of acompressible fluid from that of an incompressible one.For reasons that will become clear later, these definitions are also used ina wider sense. If q, p , and ρ are values of the velocity, pressure, and densityat any point of a fluid in nonuniform motion in which a ( p , p)-relation isdefined, and if(3)a =4/ p ,\M =dp'and-=qA/f,qsin a =* V dp'a=a1,qM'then a is called the local sound velocity (see Sec 4.2), Μ the local Μach number,and α (when it exists) the local Mach angle.A n y region of a fluid in motion in which Μ< 1 is described as subsonic,and in which Μ > 1 as supersonic.

Points where Μ= 1 are known as sonicpoints. In the same sense we speak of subsonic, supersonic, or sonic speed,or flow. Sometimes the expression transonic(or transsonic) is used to de­scribe a region in which I — Μ changes its sign, or a flow in which Μ isclose to 1 everywhere. For the sake of completeness, it may be mentionedalso that regions for which the value of Μ is exceptionally high are oftenreferred to as hypersonic.I t is to be noted that all these definitions presuppose the definition of thederivative dp/dp. T h e y certainly apply in the case of an elastic fluid, whereby hypothesis a one-to-one ( p , p)-relation holds throughout the medium.In all other cases, the symbols α, M, etc., may be used only in connectionwith an ad hoc definition of dp/dp. For example, it is possible to define(4)dpdpdpdt=II dpdtwhere d/dt has the same meaning as in (1.4); for the case of a steady mo­tion, this definition reduces to(4a)*dpl 4 a j=ds/ds'In particular, it may be that a ( p , p)-relation holds for each particle, al­though not the same relation for all particles.

An example of such behav­ior is a strictly adiabatic flow of a perfect inviscid gas, where the entropyc log ( p / p ) changes from one particle to another (see Sec 1.5).v73. Propagation of the perturbation according to directionFor a fluid initially at rest, the perturbation is propagateduniformlyin all directions, as seen in Sec. 4.5. Let us return to the special case where50I.INTRODUCTIONu = U initially within a sphere S of radius c; it was seen that u is different0from zero at time t only in the concentric spherical shell of thickness 2cand average radius aot, and that the integral(5)over the spherical shell has a constant value, J = 4TUC/3,for all values oft. In this section we need only the case where c is very small and U largeso that the perturbation, at each time t, may be considered as concentratedon the surface of the sphere of radius aot about the small sphere So, whichmay be considered to coincide with its center 0.

In any case, the perturba­tion u is the same at all points of any sphere about 0, so that the strength Iis uniformly distributed as to direction: to any bundle of rays through 0filling a solid angle da corresponds the perturbation strength(Ι/±π)άα.This may be expressed by saying that the intensity of propagation in anydirection is 7/47Γ.In the case of a moving fluid, however, the intensity of propagation is nolonger the same for all directions from the initial point 0, and, if the mo­tion is supersonic, even vanishes for some directions, as seen in Sec. 1.Nevertheless, there is still symmetry about the direction q : the directionslying on a cone with vertex 0 and axis parallel to qo are indistinguishablewith respect to intensity.

If the semivertex angle of this cone is β, then allthese directions make the angle β with the direction q . In computing howthe perturbation strength varies with β, the appropriate element da is theangular space between the two circular cones with semivertex angles βand β + άβ, respectively. For a cone with semivertex angle β, the solidangle at the vertex is a = 27r(l — cos β); thus, for the angular space betweenthe two cones,00da =2π sin β άβ.A t any time t, the perturbation is uniformly distributed over the sphericalsurface S with center C (where OC = q t) and radius R = aot.

If OP and OQare rays on the two neighboring cones (see Fig. 14a), where Ρ and Q lie onS and OP = I, say, then the area d§> cut off between the two cones is given by0dS = 2π(ΡΛ0 arc PQ = 2πΙ sin βl άβwhere δ is the angle between PQ and the normal to OP. T h e angle δ maybe computed from the triangle OCP by the law of sines:sin δqotaot.sin β = Mo sin β.5.351P R O P A G A T I O N OF P E R T U R B A T I O Nsupersonic casesubsonic case(b)(a)F I G . 14. Part of spherical surface (center C , radius a t) cut off by two neighboringcones (center O, side I).QThis formula shows that the triangles O C P , for varying t, are similar tri­angles.