R. von Mises - Mathematical theory of compressible fluid flow, страница 11

For the same reason, the ratio I/dot is independent of t and dependsonly on β and on the given ratio q /a = Μ ο. Then the strength of pertur­bation corresponding to da is0Id§>4ττ R(6)2?rZ sin β άβI24ττ (dot)202Vl-Mo sin β22in the subsonic case, M < 1. In the supersonic case, M > 1, t w o suchterms must be combined if β < α , since the rays in da meet the sphere Stwice, corresponding to the t w o possible triangles OCP and O C P ' , for thegiven values of β and Mo (Fig.

14b), and t w o possible values of the ratiol/aot. If ao < β, the strength of perturbation corresponding to da is zero.000In all cases the largest and smallest values of I, as β varies, are q t + a4and I q t — dot |, respectively, so that the ratio (l/dot) varies at most be­tween ( M — l ) and (M + l ) .

Thus, if M < 1, the intensity of propa­gation [that is, the coefficient of da in (6)] lies between finite positivelimits, the difference between these limits tending to zero for M = 0 (caseof a fluid at rest). In the supersonic case, M > 1, the factor \ / l — M sin βin the denominator of (6) will vanish as β attains the value α , sincesin a = I/Mo, and will be imaginary for all larger values of β. T h e intensityof propagation thus tends to infinity for rays adjacent to the M a c h cone,and in this sense it is often said that small perturbations in a fluid movingat supersonic speed are essentidlly propdgdted dlong the surfdce of the Machcone.02002Q20002000252I.INTRODUCTIONFormulas giving the total strength of perturbation for any cone coaxialwith the Mach cone can be obtained by integrating Eq.

( 6 ) .4. Steady motion in two dimensions. Mach linesAs seen in Sec. 1.2, in any steady flow there exist fixed streamlines, i.e.,curves fixed in space which are pathways of the fluid particles, and there­fore have at each point the direction of the velocity vector. In a twodimensional steady flow, the streamlines form a family of plane curves notintersecting each other, except at singular points.

I t is assumed also thatthe fluid is elastic, or, if it is not, that a suitable definition of dp/dp at eachpoint is given.In a supersonic region of such a flow there is determined at each point notonly a value of the sound velocity, a, and Mach number, Μ, but also aMach angle, a, with sin a = 1/M, and therefore at each point Ρ twodirections forming the angles +a and — a with the streamline through Ρ(see Fig. 15). A t sonic points, where Μ = 1 and a = 90°, the two directionsare opposite to each other. As long as Μ is finite, a is different from zero,and the two directions cannot coincide. Thus, in any portion of the planein which the motion is supersonic and Μ finite, two direction fields aredefined which, under the usual continuity assumptions, determine two setsof curves.

These curves are called Mach lines. If the angle between thevelocity vector q and a fixed axis of reference be denoted by 0, then theangles for the Mach lines are 0 + a and 0 — a, respectively. In the particu­lar case of a uniform supersonic flow, with qo in the x-direction, the Machlines are the two sets of parallel straight lines forming the angles +a and— a , respectively, with the x-axis.In general, the Mach lines form a network of curvilinear quadrangles,since two lines pass through each point. In any suitably restricted region ofsupersonic flow, two sets of Mach lines can be distinguished, such thatevery line of the one set intersects each line of the other set and no line ofits own set. T h e angle of intersection is 2a, and this angle is bisected by thestreamline passing through the point of intersection. In general, the stream­lines are not diagonal lines in the network of Mach lines.F I G .

15. Streamline and Mach lines in two-dimensional steady supersonic flow.5.4STEADY PLANE MOTION. MACH LINES53Analytically, the Mach lines can in some cases be represented by twoequations, each containing a parameter. For example, in the case of uniformmotion parallel to the x-axis, the Mach lines are given byy— χ tan a0=C\ ,y+ χ tan a0= c ,2where a is constant, and C\ and c are variable parameters. In other cases,20a single equation with a single parameter holds for both sets. For example,all tangents to the unit circle,χ cos φ +ysin φ = 1,where φ is the parameter, form a network of the required type in the regionexterior to the circle.Since a is acute (except at sonic points) the Mach lines can be given adefinite orientation: the positive direction on any Mach line will be taken sothat the line points toward the same side of the normal to the streamlineas does q. T a k e counterclockwise angles as positive; then the line formingthe angle +a with q will be called the C , or plus Mach line, while theother will be called the C~, or minus Mach line.

Thus an observer passingalong a streamline will at each point have the C M i n e to his left and theC~-line to his right. A t Ρ (Fig. 15) the C -line forms the angle 0 + a, andthe C~-line the angle θ — a, with the x-axis.++c+c~F I G . 16. Spreading of perturbations.ΒAB,XF I G . 17. Range of influence of disturbance at Αι , and interval of dependence.of solution at C.54I.INTRODUCTION5. Significance of the Mach linesT h e full significance of the M a c h lines will become clear only as theanalytical basis of the theory is developed in the succeeding chapters.

Someinteresting properties, however, can be discussed even at this stage.Consider again a steady two-dimensional supersonic flow of an elasticfluid. In a sufficiently small neighborhood of a point Ρ at which the velocityis q, the flow can be considered as a uniform flow of velocity q. N e a r P ,therefore, a small perturbation at Ρ can spread only in the angle betweenthe tangents t o the C -line and C~-line at P . Having reached a point P'in this neighborhood, the perturbation can spread further only between thetwo M a c h directions at P' (see Fig. 16), and similarly for successive pointsP " , P " · · · .

Since M a c h lines of the same set do not intersect each other,this means that a small perturbation at Ρ cannot spread to any point outsidethe region between the original C and C~, and can have no effect on theflow exterior t o this region.+f+N o w , let us consider a curve 3C: ΑχΑΒΒι, which is crossed in the samesense (at a nonzero angle) b y all Mach lines, C andf C~, through its points.T h e Mach lines are shown in Fig. 17. A s we have just seen, a small dis­turbance at Αχ cannot have any effect outside the region between the t w oMach lines through A\ . Also the (7"-lines through Ai and A cannot inter­sect, etc. Therefore, in particular, a disturbance at Ai cannot influencethe flow at any point interior to the quadrangle ACBD. T h e same is truefor Bi and for any point of 3C outside the arc AB.

Thus the flow inside thequadrangle ACBD, formed by the four M a c h lines through A and B, isindependent of what happens on 3C outside the quadrangle. If we assumethat the differential equations of the flow, together with a knowledge ofconditions along 3C, are sufficient to determine the flow in some area adjacentto JC, it follows from the above that the solution inside ACBD is deteraiined b y values of the flow variables along AB, and is independentof values on 3C t o the left of A and to the right of B. A n analogous sit­uation does not obtain in the case of an incompressible fluid or a com­pressible fluid in subsonic flow, where M a c h lines do not exist.

For thesecases, a disturbance spreads in all directions; the flow in one region isnever independent of what happens in another.+These very incomplete preliminary remarks on the role of the M a c h linesare intended to give a rough idea of how different in their physical natureare supersonic flow and the more familiar subsonic flow. I t is thus notsurprising that specific mathematical methods have to be developed forsolving flow problems in the t w o cases.

On the other hand, the readershould not conclude that there is no common ground in the theory for thetwo cases of compressible fluid motion, or that some conspicuous phe­nomenon marks each transition of a particle from subsonic to supersonicflow.CHAPTER IIGENERAL THEOREMSArticle 6Vortex Theory of Helmholtz a n d Kelvin11. CirculationA kinematic notion useful in many problems of hydrodynamics is thatof circulation, which may be defined as follows.