R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 60
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nay.»320IV.P L A N ES T E A D YPOTENTIALFLOWHere x# = 0, y# = 0, g" ^ 0 and x , y cannot both vanish since h ^ 0.nv2Hence, not both derivatives of second order are zero, and £i is seen to havea cusp at M. W e further conclude (since h29^ 0) that again the C+at Μtouches <£i (see Fig.
127), i.e., has the cusp tangent there, but does notitself have a cusp, and that the same holds for any curve through Μ whoseimage does not have the ^-direction at ra, e.g., for the curves of constantspeed and of constant direction at M. On the other hand, the image of theC~~ has the ξ-direction at m; this is also true of the images of the streamlineand the equipotential line at M , since neither has the <£i-direction atM.A s before, this last fact can also be seen directly from ψζ =0,0, ψη^etc.(or points) where li shows an extremum and <£i a cuspSuch a point Μplays an important role; in fact, a streamline that passes through a cusp of£>i separates those streamlines that do not encounter £ i at all from thosestreamlines that do encounter it (and do so in general at two points) andconsequently have cusps at <£i .
I n the characteristic plane, streamlinesthat intersect h in the ξ-direction (and in general at two points) and thosethat do not intersect it at all are separated by a streamline that contacts hat m in the £-direction. T h e region between the branches of £i is coveredthree times (see Fig. 127); otherwise the mapping is one to one (see alsoFig. 118).
Of course similar considerations apply to an <£ · All this will2find its illustration in an example to be considered in detail in Sec. 20.3.(b) Intersectionof limit lines. W e now consider a point, a, in theory-planewhere hi = h = 0, and consequently dhi/θη = dh /d% = 0. In this case all22four derivativesy vanish at a. W e assume that both dhi/θξv^0, dh ldr\ 9^ 0. Hence, in the vicinity of this point we can write for hi(£,r\) = 02the explicit form ξ = g(rj) with g'(y{) = 0 at a. Also h {£,y)2ystreamline0streamlineF I G .
127. Cusp of limit line.χ= 0 may be19.4 S P E C I A L P O I N T S OF T H E L I M I T321LINEwritten as η = / ( f ) , where/'(£) = 0 at α. N e x t , differentiating Eqs. (3) wefind at the point adh+= — cos φ ,dhi .— sin0όξ*tt = — cos 0όξ2Jdh . , +sin φ ,ση—2Consider the curve £ i corresponding to λι(£,ι?) = 0. W e prove that it hasa cusp at Λ(.τ,?/), the point corresponding to α(ξ,η) (see Fig. 128). In fact,at A,dx= χ + χ$=ηάη0,dyάη=Vn +υΰ'=ο,ixάη2T h e second term to the right in the last equation is zero unless g" = oo at A;but g" = (-θ%/θη )/(θ^/θξ)at 4 , where dhi/δξ ^ 0 by hypothesis, anddifferentiation of ( 7 ) shows that θ%ι/θη remains finite at A. Hence at A:ofx/άη = χ , (Fy/άη = y and dy/dx = y lx= tan φ .
W e concludethat the curve £i has a cusp at A with the tangent in the C -direction. I nexactly the same way we find that the curve <£ corresponding to h = 0has a cusp at A with the C~-direction as tangent. I t can then be seen thatthe inner angle in the χ y-plane is covered four times.22ηηvvm+m+2yF I G . 128. Neighborhood of a double limit point.2322IV. P L A N E S T E A D Y P O T E N T I A LFLOWFrom ( 4 " ) and the properties at the point under consideration we findthat φξ = ψξη = 0, thatη2n&n= ίθΛί dl%22cosι~d£~fy>aan.τζζτψι =ddhi dk22 · 2.2sm-pqHence, according to whether (θΗι/3ξ)(θΗ /θη)^ 0 the point is either asaddle point (upper sign) or an extremum for^>, and the other way for ψ.For the first case, the extremum of ψ, a streamline near the point a inthe hodograph will be closed, and its image in the physical plane (see Fig.128) demonstrates the fourfold covering of the angle at A.
I n the next article we shall find an interesting illustration of this situation.W e do not discuss singularities of "higher order".25. Limit singularities for Μ=1I n our previous derivations, a ^ 90° and α ^ 0° were assumed andactually used in many conclusions. If a = 90°, i.e., at the sonic line, a partlynew situation presents itself which we shall now study with respect tolimit-type singularities.35For a = 90° the Jacobians J and D no longer exhibit the same behavior and it is now the vanishing of D which will serve as a criterion. Using ( 4 " ) , we haveD = J —-4^— cos a = pqh\h cos a2 sin α2do)=cot a . ,—=-ρ- w , -W e see that if J < » then D = 0, and if J =nonzero (finite or infinite).Writing <ρ , φ in terms of ψ , ψ we haveξ(11)βηφι = - [ Ρ<*>, D may be zero orθq\f/ + ψ cot α],q2θ<ρ = - [q\p + ψ cot a].ΡηqβA s a —> 90°, ^ cot a —> 0, unless ^ —•(which we exclude).Laying aside the case D —• ± o o which corresponds to branch typesingularities (see Sec.
6 ) , we call an ordinary sonic point, in contrast tolimit point Or branch point, a point where \p j± 0; then both φ$ and <ρ remain nonzero. For example, in the case of spiral flow (see Sec. 17.4) thesonic line (a circle) is not a limit line. Straight computation shows that with0 0qη19.5LIMIT SINGULARITIES FOR Μ = 1323the notation of A r t . 17, where C, k > 0 :.ψ9=Cp,—,Φβ = k,7Ίcot a7-C,Qcot a . ~+C<ρ = kηΡ— C tan a),)hi = - (a\p„<ρξ = kFor a —> 90°, ^Ρb =A = - ( - + C tan a ) ,a\p/'2kf>qZk cot a 22225^ 0 and hence all points are ordinary.ςA new type of limit point (and limit line) which we shall call sonicpoint is characterized b y ΜφβCp'= 1, \[/ = 0. A t such a point either ψqςlimit= 0,0, or φ = 0, φβ = 0.
T h e first case can clearly happen along a wholeςarc of curve which we then denote as a sonic limit line, £ .tI n the secondcase the point is isolated, or as we shall see presently, it may be the common point of an £ and an <£i, for instance.tT h e simplest example of an £appears in the radial flow (Sec. 17.4).tAlong the line Μ = 1, which in this case is a circle, we find, using the aboveformulas with C = 0 :ιΛ/77cot α:Ρhi =A2 =—,bapτ PQ tan o:.2=kThere is no other limit line in this example, since both hi and h are every2where nonzero.*From φι = <ρ = 0 a sonic limit line is an equipotential line.
Hence, atηevery point of a sonic limit line, the streamline direction is perpendicular tothe £ , and since a = 90°, it follows that both the Cand C~ are enveloped+tb y the £t(see F i g . 129). Since at the £ ,φt9= 0, φ ^ 0, άφ = φ dq +θαφβ άθ = φβ άθ, we see that if άφ = 0 also άθ = 0, and vice versa; hence theline θ = constant, the isocline, is likewise perpendicular t o the £ .tM o r e generally, and in analogy t o our study of the £ 1 and £ 2 , we concludefrom Eqs.
(17.25), since άφ = 0, ψ = 0, φ ^ 0 at the £ :9dx = - — sin θ άθ,tθdy = — cos θ άθ.Thence(12)^=- c o t Θ,if άθ0.dx* N o t e that in the spiral flow hi—* — 0 0 , Λ —> oo along the ordinary sonic line,while in the radial flow hi and hi are finite along the sonic limit line.2324IV. P L A N E STEADY P O T E N T I A LFLOWHence an element of any curve in the hodograph on which dd ^ 0 maps intoan element with slope dy/dx = —cot θ in the x,?/-plane, i.e., normal to thestreamline, or tangential to the £ .
N e x t , considering an exceptional element with dd = 0, and using q as parameter, we find from Eq. (17.25') thatdx/dq = 0, dy/dq = 0, but that d x/dq\ d y/dq cannot vanish simultaneously, and we conclude that the curves which at the sonic line in the hodograph have the exceptional (radial) direction will map into cusped curves.W e also note that the characteristics C , C~ lie on different sides of theisocline (and streamline) forming an angle of 180° with each other since inthe hodograph they are separated by the isocline.W e conclude fromt22+that the acceleration, which has been found infinite along an £ i or <£ , isalso infinite at any sonic limit point where ψβ ^ 0.2In the radial flow example the lines of constant direction are the radii(which are perpendicular to the <£*). T h e radii are also the streamlines forboth flows, and at each point of the sonic limit circle the two streamlinesmay be considered to form a degenerate cusp.A sonic limit point could be the point of intersection of an £ with an £ i ,or with an £ , or with both* (double sonic limit point).
I t could also bethe sonic point of an <£i or of an <£ (an example will be found in Sec. 20.3)or of both (again double sonic limit point). If the point is the point ofintersection of an £ and an <£i, say, then ψβ must also vanish there. For,since ip = 0 along the sonic line q = q , we have from Taylor's formulaapplied to \j/ that \p = 0(q — q ) while tan a = 0(q — q )~\ so that\p tan a = 0(q — q Y; hence ^ = q\l/ tan a — ψ tends to the value of— ψβ at q = q irrespective of the path of approach.
But ψξ = 0 everywheret22tqtqqqtttqθtysonic limit line,£t0F I G . 129. Sonic limit line.* For an example see the last of the papers quoted in N o t e 35.19.6 B R A N C H L I N E S325along £1 ; hence ψ = 0 at such a point of intersection. This also shows that9a line £i along which Μψθ=1 everywhere is not possible. T h e conclusion= 0 is not valid if the point under consideration is the sonic point of an£i without lying on an £ (see examples in Sees.
20.3 and 20.5 where ψβ ^ 0 ) .tW e review: The sonic limit line £ ,characterized by \p = 0 along a sonictline, is a (piecewise)equipotentialqline and a line of constant velocity; at apoint where ψβ ^ 0, it is the envelope of both familiesof characteristics and,more generally, of all curves whose images in the hodograph plane do not havethe exceptional radial direction. Α11 curves whose images have the radial direction at the sonic circle have cusps at the £. On the £ttthe acceleration becomes infinite.W e do not study here or in the following the case a = 0°, since lines withρ = 0 will in any case appear only as boundaries of flow regions.6. Branch linesW e now study the vanishing of the reciprocals d and j of Jacobians D andJ, respectively.
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