R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 49

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Then from (44)(45)Αθ =6 =*({,)m-θ&ηο) =(ηι-f)-(ηο -ξ)=ηι -ηο .Thus this angle depends only upon the two fixed C~-lines, and not on thevariable ξ. Also, interchanging the role of the two families, we obtain(45')Δ'β =e' =*(&,„)-θ(ξ ,η)0=|o -ξιSimilarly for the angle Q(q), which is a function of q, M, p, or p:(45")Δ<? = Q F C 1,0 -Q(l;, ηο) =IH -η„ =-€,Thus, it is seen that the angle between the flow directions atcorrespondingpoints of two fixed characteristics remains the same all along these characteris­tics) and an analogous constancy relation holds for the differences in Q.sIn Sec. 4 we indicated how the net of M a c h lines is determined in a cer­tain domain by specific types of boundary conditions. T o construct thisnet we used the compatibility relations (24) in the form of finite differenceqqF I G .

96. Constancy of ΔΘ along characteristics.16.7259N E T S OF C H A R A C T E R I S T I C Sequations (25) and the direction conditions (23). This most direct proce­dure, essentially due to J. Massau, becomes a practical approximationmethod if the compatibility relations are used in the form (43) together with(23) and with a tabulation of Θ (or Q) as a function of q along a M a c h line(see end of preceding section). A first approximation thus found can beimproved b y iteration. I n the following we describe a few other usefulprocedures, both numerical and graphical, for the determination of theMach lines, if the net of Γ-characteristics is considered known.T o fix ideas, consider the characteristic boundary-value problem where weknow that two arcs OA, OB in the #,?/-plane are parts of a £-line and an17-line, respectively. T h e case where the whole or part of one of these curvesis a straight line will be considered in A r t .

18 where we shall study so-calledsimple waves; here they are assumed to be curved. Consider a subdivision£0, £ 1 , · * · , £m , * · · on OA and η ,· · · , η , · · · on OB and a corres­ηι,0ηponding lattice where a general nodal point has the coordinates (ξπι,ηη)or briefly (m,n).Then from Eqs. (44), 0(46)—6mnθτηΟ —m n= 77 — £ , we havenθθη +m000 =0.Similarly(460Qmn~QmO~Qon+QoO= 0.Hence knowing 0 and Q at the subdivision points on OA, OB, we can deter­mine them at all nodal points.W e still need, however, the coordinates x ymnin the physical plane cor­mThresponding to ( £ , η ) to which qmηm nbelongs. T o find these coordinates, i.e.,our original independent variables, we use a step-by-step procedure.

T h esimplest is the recurrence procedureχ„t/mny-m—l ,n— t»anVmnVm,n—1—Φτη-Ι,η+Φ mn+Φ mn~/\%mn\Xm—1 ,n)j(47)xwhere the x o ,y ,o,m>mpoints along OAΦτη,η-l—tan/\\XmnXm,n—l)jΧο,η , 2/o.n are the known coordinates of the subdivisionand OB,respectively; the φ^η, Φ^η are known at allnodal points since 0 and q are known there. Thus (47) are two simultaneouslinear equations which determine x ,mny -i,nmym noncex ,n-i,my n-\mtand x _ i , ,mnhave been found. According to the quasi-linear character of ourproblem the original unknowns 0, q are found first through the linear com­patibility relations, or in other words through our knowledge of the net inthe hodograph, while for the independent variables χ and y we need a step­wise procedure based on the direction conditions.

T h e indicated procedure,200IV.PLANESTEADYPOTENTIALFLOWwhich is, of bourse, essentially a rearrangement of that in Sec. 4, can like­wise be made more accurate by means of iterations.T h e best known graphical procedure for constructing the Mach net isbased on the property that the C , C~ through Ρ are perpendicular to theΓ~, Γ through P' in the hodograph. Consider a C~ in the physical planewith points P i , P , P3, · · · on it and known velocity vectors q i , q ,q , · · · .

If we plot these in the hodograph, their endpoints P[,P ,P 3 , · · · lie on the corresponding Γ". Then the tangent at P[ to this Γis perpendicular to the characteristic C t through P (Fig. 97). This is usedsystematically in a procedure due to A . Busemann (see Fig. 98). Denotelattice points in the hodograph by P'. Then, starting from the givenboundary values, the Mach net of points Pis constructed successivelyaccording to the rule++2232-tmnmnΡmnPm,n-\-\ -LPmnPm—1 ,n >Ρ m n P m - f l ,n -LΡmnPm, π—1 ·In this way the r~-curves correspond to the C~, and the Γ to the C ;dotted line segments in the hodograph which have r e d i r e c t i o n are per­pendicular to dotted segments in the flow plane which have C -direction,+++F I G .

0 7 . Graphical construction of Mach lines.F I G . 98. Reciprocal lattices in flow plane and hodograph.17.1 D I F F E R E N T I A L E Q U A T I O N S F O R L E G E N D R E T R A N S F O R M S261and similarly for Γ and C~. If in the physical plane each mesh is denotedby the two numbers equal to the smallest pair of subscripts of the fourcorners P ,it is seen that the mesh (i,k) corresponds to the point P\ in thesense that the four sides around the mesh (i,k) are respectively normal tothe four sides through P' (reciprocal lattices).+ikkikArticle 17Further Discussion of the H o d o g r a p hMethod1. Differential equations for the Legendre transformsIn the preceding article we started with the two nonlinear partial differ­ential equations of first order, (16.1) and (16.2), which express continuityand irrotationality. Transforming these to natural or intrinsic coordinates,we obtained the basic nonlinear equations (16.7) for q and 0, which to­gether with Eq.

(16.10) defined the problem completely. In Sec. 16.2 thepotential φ and stream function ψ were introduced by means of (16.15),(16.17), and (16.18). T h e equations (16.18) with ψ and ψ as dependent, χand y as independent variables, or their natural form (16.18') again con­stitute a pair of nonlinear partial differential equations of first order. These,although simple in appearance, are quite complicated since ρ has to be ex­pressed in terms of q, and q in terms of φ. N e x t we derived a nonlinearsecond-order equation for φ alone and one for ψ alone, Eqs. (16.14) and(16.21), and the corresponding intrinsic forms (16.16') and (16.20'). InSec. 16.5 the hodograph transformation was introduced and it led to variouslinear equations.

Using now q and 0 (the previous dependent variables) asindependent variables, we obtained the basic linear equations of first order(16.31), and also the equations of second order (16.32) or (16.32'), (16.33)or (16.33') for ψ alone or φ alone, respectively.In this and the following section we shall give still other useful equationsof the problem. I t is left to the reader to consider and determine variousanalogies with the equations of Chapter I I I .One pair of nonlinear equations in the physical plane for the velocitycomponents q = u, q = ν consists of Eqs. (16.2) and (16.14):xyIV.262PLANESTEADYPOTENTIALFLOWThese equations can be linearized by the direct interchange of variablesintroduced in Sec. 10.6 and used in Chapter I I I .

If the Jacobian. _ du dvJ~ dxdydu dv _d(u,v)dydx~d(x,y)is different from zero we obtain the linear system/ 22s(α -u)dy ,(dxuv[ — +(3)dy\,J + (a -2\dx2v) —Λ= 0,- Ου= οduT o satisfy the second of these equations we introduce a function Φ(μ,ν)such that** = χduthen the first Eq. (3) yields(4)W** = νdv'X )/r\/ 2(5)(α2x 5 Φ .θφ. /2u ) — + 2ui; — — + (α 2-yΟ^Φ2\20ν) —Λ= 0.T h e Φ introduced in this way is the Legendre transformof the potential φ.9From Eqs. (4) and (16.15) we have(4')άΦ = χ du + y dv,a\p = udx+ ν dy,and by integration we obtainΦ = f(x du + y dv) = J d(xu + yv) — J(u dx + ν dy)= xu + yv — φ.Hence, between φ considered as a function of χ and y and its transform Φconsidered as a function of u and v, the relations,.d<p .

dtpΦ = xu+yv — φ = x — + y(6)8,<p = ux +XdφV_dΦ ,dΦvy — Φ = u1- νdudv.Φhold.Using q, Θ as independent variables instead of u, ν and introducing theabbreviations(7)χ cos θ + y sin θ = X,y cos 0 — χ sin 0 =F17.1 D I F F E R E N T I A LEQUATIONS FOR LEGENDRE TRANSFORMS263for the components of the radius vector r in the direction of the flow andnormal to it, we obtain from ( 6 ) : Φ = q (x cos Θ + y sin θ) — φ = qX —φ, so that<«·> M'^and(6')= q^L dqφφ,Φ = Χ ^ - φ.dsT h e following relations, derived from Eqs.

(16.17) and ( 4 ) , serve t o deter­mine the stream function ψ, if Φ(η, V) is known:, v0dxj/(d% ,d% \θψq θθ)'dd ~~θΦ(,2d%\ordq"\dqd6PPV dqdd )'2Likewise, a Legendre transform Ψ of the stream function ψ may be defined.If w e interchange the variables x, y for the variables pu, pv, the continuityequation becomesdxd(pw)4-yd(pv)d=πThis equation may be satisfied by introducing a function Ψ of pw and pi>such that(9)a(pw)= y,ΓΤ—τ =θ(ρν)- ζalso,-y—d(p?)=F.B y integration of ( 9 ) , the following relations, analogous to (6) and ( 6 ' ) ,obtain:ψ = puy -pvx -ψ = x -+ y--ψ = Υ - - φ ,(10),dVdYΨ = pu - — - + py -j—- Ψ = pq —— d(pw)a(py)a(pg)W e now derive α pair o/ first-order equations for Φ and Ψ analogous to Eqs.(16.31).

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