R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 48
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If w is differentiated with respect to ζ the complex velocity f2sζ = w (ζ)= — + ι—- = qdxdxx-iqy= qeis obtained, and w(z) is an analytic function of f, although in general notgiven by only one series in the whole field of flow.252IV. P L A N E S T E A D Y P O T E N T I A LFLOWT o the equations (31) through (33) we may apply the methods of integration valid in a linear problem, especially the method of combiningparticular solutions.
But it would not be correct to say that by transformation to the hodograph plane our original problem has been "linearized".(This can be done only by taking recourse to approximations.) W e have,however, split off one portion of the total problem that can be treated bymethods of linear analysis.As seen in Sec. 1 (for a polytropic fluid) and previously in Sec. 8.2, thehodograph mapping of the flow lies inside a circle of radius q , the supersonic part falling in the annular region between this circle and the soniccircle of radius q the subsonic part being inside the sonic circle. T h e ratioof the two radii is, for a polytropic fluid,mh(34)h = 6, h = 2.45 for κ = 1.4, and h = 2.437 for κ =agreement with the discussion in Sec. 8.3.26.
Characteristics in the hodograph plane1.405. All this is in6Since the transformed equations (31) are linear, there exist fixed characteristics, Γ, in the hodograph plane. B y the formulas given in A r t . 10,equations of these characteristics can be written down for the linear system(31) or read from (32') or (33') by formulas found in Art. 9:(35)===t-q tan a'dqqqEquations (35) also follow from the compatibility relations (24) [see Eq.(10.5) with a = b = 0 and the coefficients a; and hi (i = 1, · · · , 4) depending only on u and v]. Computing (35) in these two ways, by means ofthe linear equations (31) and from the equations (24), gives, incidentally,a proof of the fact (studied in full in Sec. 10.7) that the Γ-characteristics ofthe linear equations in the hodograph are the images of the ^characteristics in the x,?/-plane.For an independent and direct derivation of this fundamental relationwe consider the mapping of the characteristics in the physical plane, theMach lines, onto the hodograph plane.
W e ask: if the points P, Pi of thephysical plane are mapped onto P\ P\ of the hodograph, what is thedirection of ΡΊ\when PPmakes the angle ±a with the streamlinethrough P? T h e changes of q and 0 from Ρ to P\ areX16.6 C H A R A C T E R I S T I C S I N T H E H O D O G R A P H P L A N EUsing Eqs. (7) and M2— 1 = cot a, we have2\as(36)as/\as=b q tan a ^ J ds,dq = (~253± - cot a — J ds,qds/dd = (—from which Eq. (35) again follows, ordB^_q — = dz cot a .(37)T h e quotient q dd/dq is the tangent of the angle between the tangentto Γ at P ' and the radius vector O ' P ' (Fig.
92). Thus (37) states: T h echaracteristics Γ , Γ~ in the hodograph plane form the angles ± α ' , wherea = 90° — a, with the radius-vector direction (the direction of q). Inother terms, the C in the physical plane through Ρ is perpendicular tothe Γ in the hodograph through P', and similarly for C~ and Γ [see alsoEq. (24")].T h e essential property expressed in (35) or (37) is not, however, thismetric relation between the directions of the C- and Γ-characteristics butrather the fact that the slopes of the Γ-characteristics at P ' are determinedby the coordinates q and θ of P ' .
Therefore the hodograph characteristicscan be computed and drawn once and for all, without further use of theEqs. (7) or (18). T h e y depend only on the (p,p)-relation.In polytropic flow, cot a = \/Μ— 1 is given by the square root of thesecond expression of (10), and (37) becomes++-+2(38)(κ + 1) q — 2a2dd ==b_2a2-{κ -281) q \2hdq=q ~/h q2—2\ q2m-q \]2mq2)dqq 'By integration we obtain(39)θ = ± ( α + ha + constant),F I G . 92. Mach lines and hodograph characteristics.
Physical Plane. HodographPlane.254IV. P L A N E S T E A D Y P O T E N T I A LFLOWwhere σ is denned by— \*= *( (J^f?)'hcoW = Atana = ^ r ^ }(39')22as can be verified by differentiation, or, in one formula,(39")zt θ = arc tan ^ / j ^= + h arc tan2ί +constant.T h e expression t o the right can be written in various other equivalentforms.* A s q increases from q = q /h to q , Μ goes from 1 to » , σ from0° t o 90°, α from 90° to 0°, ot (the angle of the characteristic with theradius vector) from 0° to 90°, and 0 increases (upper sign) along a plustmmFIG. 93. T h e hodograph characteristics as epicycloids.characteristic or decreases (lower sign) along a minus characteristic by(A - 1) times 90°, which is 130.45° for κ = 1.4, and 129.32° for κ = 1.405.T h e geometrical nature of these characteristics is explained in Fig.
93.Let P' be an arbitrary point in the annular region corresponding to supersonic flow between the sonic circle C and the maximum circle C . L e t Cwith center Μ and radius (q — qi)/2 be that circle through P' tangent toboth C and C , for which the inclination of O'M is greater than 0, i.e., weconsider a r -characteristic. L e t B\ and B be the points of contact of Cwith C and C , respectively. B P' and ΟΆ are perpendicular t o thestraight line P'BiA. Then Β^Βχ/Ο'Βγ = h - 1; hence AP'/AB = h andtmmtm+2tm2Xq 2(0'A)a= *V-{0'A)\(O A)f2=-^±K(q2m- g) = a .22Hence O'P'A is the Mach angle a, and AP' is normal to the r -characteris+* This equation, which gives the relation between θ and Μ or between θ and qalong a characteristic line, is of course identical with the compatibility relation (24),which here could be integrated.16.6 C H A R A C T E R I S T I C S I N T H E H O D O G R A P H255PLANEFIG.
94. T w o epicycloids with horizontal initial direction.tic through P'. Also, cot AO'B = h tan O'P'A, and therefore AO'B is theangle a defined in (39'). From (39), 0 = a + ha - 90° + B , where 0,denotes the angle of the initial direction O'B to the fixed direction. T h earc ΒχΡ' equals 2a(q— qi)/2 = (h — l)q a, while the arc B^Bi equalsq [(6 — 0i) + (90° — a) — a] = q {ha — a) and the t w o arcs are equal.xxt0mtttT h e characteristic Γ is thus generated by a point P' fixed on the circleC as it rolls without slipping on the exterior of the circle C starting at Bo.Such curves are called epicycloids. Reversing the sense of the rolling, wegenerate the r~-characteristics. Both sets of characteristics are congruentepicycloids, and any curve of each set can be obtained from a fixed one byrotation around 0'.+tIncidentally, if h is an irrational number, indefinite continuation of the;rolling would finally generate epicycloids arbitrarily close t o all epicycloids(of both sets).
I t is also useful t o know that the center of curvature corresponding to the point P' is the point Ζ where the straight line from Oto Q, the second end point of the diameter P'M, crosses the normal P'A.rIn Fig. 94 are shown the t w o characteristics with horizontal initial direction, θι = 0. T h e y form an apparent cusp at the sonic circle where= 90°, i.e.
where the angle a between each characteristic and the horizontal radius vector is zero, and are tangent to the maximum circle where' = 90°, a = 0°, I 0 I = 130.5°, for κ = 1.4.A further helpful relation is the following. If we designate (Fig.
93) byu and ν the components AP' = q cos a and ΟΆ = q sin a of the velocityvector, normal and parallel to the tangent at P', then it is seen from thefigure thataaO'A = (O'Bi) cos a = q cos a = v,tand we obtainAP' = (0'B )2sin a = q sin a = u,m256IV. P L A N E S T E A D Y P O T E N T I A LFLOW(40)In this ellipse E, with semiaxes q , q , the radius vector q = O'P' makes theangle a with the w-axis (Fig. 95a).
B y using Ε the directions of the Machlines corresponding to a point P' in the hodograph can easily be found (seeFig. 95b).tmW e have seen that the right side of (39) depends on q only and can therefore be expressed in terms of any of the variables q, M, p, or ρ (except,of course, for a scale factor). T h e relations between these quantities weredeveloped in Sees. 8.3 and 8.4 and partly rederived in the present article,Sec. 1. Tables for different independent variables, as well as diagrams, areavailable (see N o t e 27 to Chapter I I ) .
A table serving as an illustrationrather than for technical purposes was given at the end of Sec. 8.4. Therewe tabulated, versus Μ, the magnitudes p/p , p/p , T/T , q/q , and p q lpq.In the following small table we show in addition some values of α, σ, and8ssmtν(b)(a)FIG. 95. Mach Ellipses.TABLEIIIΜa.σ1.01.52.02.53.03.54.090.0041.8130.0023.5819.4716.6014.480.000.0024.5335.2643.0849.1053.8557.6790.00Q =a +90.00101.91116.38129.12139.75148.53155.78220.45hat25716.7 N E T S O F C H A R A C T E R I S T I C SQ = OL +ha (measured in degrees) computed from the second formula of(23) and from (39') in terms of M, for κ =1.4.I t should he said that the geometrical fact that these hodograph characteristics happento he knowncurves, namely, epicycloids, is withoutmuch hearing on the problem.
T h e main point is that these fixed characteristics are known and can be plotted and tabulated. Incidentally, we recallthat another, actually more interesting, geometrical property of the Γ-lineshas been proved in Sec. 8.2, which in our present terminology can bestated thus: T h e fixed characteristics in the hodograph plane, here epicycloids, are the projections of the asymptotic lines of the pressure hill(see Fig. 38).7. The nets of characteristics in the physical and hodograph planesIn order to facilitate computations, it is convenient to adopt a coordinatesystem which will serve for the fixed characteristics in the hodograph plane,as well as for the M a c h lines. This can be done in various ways.
W e haveseen that the differential equation (35) of the Γ-curves can be explicitlyintegrated yielding (39). Guided by Eq. (35), we now introduce the function Q of q, defined by(41)dQ =-p--VM^l,*q tan aqor (see Table I I I ) with restriction to a polytropic fluid, by(4l')Q =f -p—+90° = α + /ΐσ.Jg q tan atThen the compatibility relations (24) and (24') along the C ,C~ on theone hand, and the equations of the fixed characteristics Γ ,Γ~ on the++other, may be written as follows:Q(q)— θ = constant,along Cand on Γ ,Q(q)+θ = constant,along Cand on Γ .++(42)N o w denote the M a c h lines C~, C+as ξ-lines and ?y-lines, respectively. If,accordingly, we introduce coordinates ξ,η by(43)Q(q)- 0= 2$,Q(q)+θ =2η,then from (42) the Mach lines C~ (and the r~-characteristics) are the linesη = constant (the ξ-lines), and the M a c h lines C+tics) are the lines ξ =+constant (the 77-lines); for each individual lineξ = ξ (or η = ηο) we have 2£ = 90° 0(and the r -characteris-00, (or 2η = 90° +00,), with 0, thevalue of 0, for this line, at the sonic circle where the r e d i r e c t i o n , the Γ -direction, and the flow direction coincide.7258IV.PLANESTEADYPOTENTIALFLOWFrom (43) follow the important relations(44)θ = v -ί,ν + £,Q =which, incidentally, show that the sum (η +f ) is constant along the concentric circles q = constant in the hodograph, while the difference (η — £)remains constant along the radial lines.
In the physical plane Θ = η — ξ =constant and Q = η + £ = constant designate the lines of constantinclination θ, and the lines of constant speed q, respectively.A s an immediate consequence of (42), we obtain the following propertyof the Mach lines (see Fig. 96). Consider two fixed M a c h lines of the samefamily, say the two £-lines, η = r ? , η = ηι ; let points on these two lines0having the same ξ be called "corresponding points", and let e denote theangle between the flow directions at corresponding points.
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