R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 18

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A n y oneof the following may be used as parameter: the stagnation values p , ρ ,or T ; the value of the transition velocity q ; the value of the maximumvelocity q (provided this value is finite); or the Bernoulli constant H,which is equal to the stagnation value of P/g or to the square of q dividedby 2g, if in the definition of Ρ the reference pressure is zero.8sβtmm2. Hodograph representation23I n steady flow of any type there is a definite velocity vector q, independ­ent of t, for each point P. If these vectors are plotted with a fixed origin 0 ' ,so that q = O'P', then to each point Ρ of the flow pattern there will cor­respond some point P' (see Fig.

37). Then, for example, to all points of onestreamline there will correspond a curve formed by the respective pointsP'. This correspondence, or mapping, is known as the hodograph transfor­mation. W e shall also speak of the hodograph space in which the points Plie, as contrasted with the physical space in which the flow actually occurs.N o t e that each point Ρ is mapped into exactly one point P', although theconverse is not true: one point P' can represent several points Ρ of thephysical space—namely, points with the same q.tIf a streamline £ of physical space is mapped into a curve <£' of the hodo­graph space, the tangent to £ ' at P' has the direction of the instantaneousrate of change of q, i.e., the direction of the acceleration vector.

N o w , inthe absence of gravity and viscosity, the acceleration vector has, by E q .(1.1), the direction of —grad p\ thus the isobar (surface of constant pres-8.2HODOGRAPH91REPRESENTATIONsure p ) through any point Ρ of the physical space is perpendicular to thetangent to <£' at P'.T h e results of Sec. 1 show that, in the case of a steady irrotational flow,all points P' must lie on or within the sphere with center 0' and radiusq (maximum v e l o c i t y ) : all of physical space is mapped into the interiorand boundary of this sphere in the hodograph space. For exceptional (p, p ) relations, this sphere may be infinite.

All stagnation points (where q = 0and Μ = 0) map into the center 0 ' ; points with q = q and hence (in thenonexceptional case)Af =<Χ > correspond to points on the boundary of thesphere; and the sonic points of physical space (where Μ = l ) map onto thesonic sphere with center 0' and radius q . T h e subsonic region of flow mapsinto the interior of the sonic sphere and the supersonic domain into theshell between the spherical surfaces q = q and q = q .mmttmT h e hodograph transformation is used extensively in the study of planemotion.

In this case, all points P , as well as all points P ' , lie in a plane, andso we speak of the physical plane and of the hodograph plane. T h e spheresare now replaced by concentric circles C and C with center 0 and radiiq and q , respectively (Fig. 37); the subsonic part of the flow maps intothe interior of the sonic circle C , the supersonic part into the annularregion between C and the maximum circle C , stagnation points map into0 ' , sonic points into points on the circle C , and points with q = q intopoints of CttmrmttmtmCorresponding to each point P' within C , there is a value of the pres­sure p, determined by the distance O'P' = q and E q . ( 2 ) . If these p-valuesmyο<F I G .

37. Physical plane and hodograph plane. Streamline £ in physical plane and<£' in hodograph. Sonic circle and maximum circle.92II. GENERALTHEOREMSΡF I G . 38. Pressure hill with subsonic and supersonic region.are plotted above the hodograph plane as points Q, such that the perpen­dicular distance to the plane is P'Q = p, then the points Q correspondingto a given streamline lie on the bell-shaped surface of revolution obtainedby rotating one of the ρ,^-curves of Fig.

36 about the p-axis. In the case ofan elastic fluid in irrotational motion the same surface holds for all stream­lines. A sketch of this surface, sometimes called the pressure hill, is givenin Fig. 38. Each position of the generating curve is called a meridian of thesurface of revolution; the path of any point of the generating curve is aparallel. T h e surface is divided into an upper and a lower part by theparallel circle whose projection is C ; the upper part resembles an ellipsoidof revolution in the neighborhood of one vertex, while the lower portionhas the character of a hyperboloid (of revolution) of one sheet.tIn the terminology of differential geometry* the upper portion of thepressure hill consists of elliptic points, while the points below the criticalcircle are hyperbolic points.

This means that if the surface is cut by a planeparallel to the tangent plane at Q and sufficiently close to it, the curve ofintersection approximates an ellipse in the first case, and a hyperbola in.the second. T h e axes of this conic (called Dupin's indicatrix), are parallelto the "principal directions on the surface at Q; in the case of a surfaceof revolution these are the directions of the tangent to the meridian throughQ and the tangent to the parallel circle. If coordinates in these two direc­tions are called χ and y respectively, the equation of the indicatrix has theform,,2(0)2IT + ΊΓ = constant.HiRi* A reader who is unfamiliar with the elements of differential geometry may omitthe remainder of this section.HODOGRAPH8.293REPRESENTATIONHere, | Ri | and | R | are the radii of curvature of the two plane curves cutoff on the surface b y normal sections through the two principal directionsat Q.

T h e signs of Ri and R are to be taken the same if both centers ofcurvature lie on the same side of the tangent plane, and opposite in thecontrary case.22For a surface of revolution, one principal section gives the meridianthrough Q. Thus \ Ri\ is the radius of curvature at Q of the generatingcurve denned by Eq. ( 2 ) . Using Eq. (5) and(7)^ = tan (90° + 0) =dq-cot 0,where 0 is the angle between the normal to the p,g-curve and the g-axis,we obtain from the formula for the radius of curvaturedp21 _dq1= ρ sin 0 {Μ32-1)[-(ITand K\ (Fig.

39) is the center of curvature. T h e second principal sectionis the normal section through the tangent to the parallel circle. T h e theoremof Meunier (proved in elementary differential geometry), when applied tothis case, shows that the center of curvature K of this section lies on theaxis of revolution. Hence2I R1 =(9)2cos 0'since 0 is also the angle between the horizontal plane of the parallel circleand the normal section, while q is the radius (of curvature) of the parallelcircle.For points Q on the lower portion of the pressure hill, R and R haveopposite signs, as in Fig. 39a; also, 0 is acute and M > 1.

Thus, with the useof Eqs. (8) and (9), Eq. (6) becomesx222(60xp sin 0 ( M32— 1) — - cos 0 = constant.qT h e angle β between the x-axis and the asymptotes of the hyperbola (6')is determined bytan β = -Λρ sin 0 (Μcos 0232-1).If the axes and asymptotes of ( 6 ' ) , all in the tangent plane at Q, areprojected onto the horizontal plane (Fig. 39b), then Q projects onto P\ the94II. GENERALTHEOREMSx-axis (first principal direction) onto the radial direction O'P',2/-axis onto the (dotted) line normal to O'P'asymptotes make an angle β' with O'P',where-««-·«·-»·= 5 ? $ £ τ ϊFrom Eqs.

(4) and ( 7 ) , pq = cot 0, and therefore tan β' = Μ2the sine of the Mach angle a (Sec. 5.2) is l/M,(10)tan β' = M22and theat P'. T h e projections of the— 1 = cot a,0' = 90° -22— 1. N o w ,so thata.On a surface with hyperbolic points, the lines which at each point havethe direction of the asymptotes of the indicatrix are called the asymptoticlines of the surface; these lines are indicated in Fig. 38. Equation (10) thusΡ(a)On.(b)F I G .

39. Principal centers of curvature Κι and Kof an asymptotic direction on pressure hill.2for hyperbolic point. Projection8.395CASE OF P O L Y T R O P I C R E L A T I O Nshows that the asymptotic lines on the pressure hill are projected onto thelines in the hodograph plane making the angle ± ( 9 0 ° — a) with the direc­tion of q.I t will be shown later (Sec. 16.6) that, in any steady plane irrotationalflow, the hodograph transformation maps the Mach lines (Sec. 5.2) of thephysical plane into curves in the hodograph plane which meet the raysthrough 0 ' at the angles d= (90° — a).

Thus, the result expressed by Eq. (10)can also be stated as follows: The Mach lines of a steady planeirrotationalflow of an elastic fluid are mapped into curves in the hodograph plane whichare the projections of the asymptotic lines on the pressure hill. This result wasfirst given by L. Prandtl and A . Busemann.243. Case of polytropic (p,p)-relationT h e relation between ρ and q (or between ρ and q or Τ and q) is particu­larly simple if the (p p)-relation has the polytropic form}— = constant = — ·(11)For given κ—and we shall usually take κ = 7, where y is the adiabaticexponent (Sec.

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