R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 61
Текст из файла (страница 61)
W e consider ΜD=^1 (see end of Sec. 2 for Μ222infinite. Here, therefore, we have only to investigate the loci h (x,y)xh2(x,y)1). N o w<pqh\h cos a cannot become infinite unless either hi or h becomes=± oof= =b oo ; in other words, there are only branch lines Bi and B ,2andno analogue of the <£*. W e now consider a (locally) single-valued solutionin the x,y-p\txne and lines (or points) along which hi or h considered as a2function of x, y becomes infinite. T h e discussion becomes similar to thatfor limit lines if we invert the Eqs. ( 3 ) , which were basic in our previousinvestigation. Then=and from (3) and (4) we obtainn d(x,y)/d(£,q)y%t = Vyhih sin 2a = hi cos φ~, or η2= cos φ~~/h sin 2a. This suggests thatυ2we introduce(13)h =_}_ _,hi sin 2athn1h sin 2a"2If we assume a ^ 90°, 0°, then the k are equivalent to l/h, and we obtain(from (3)d£..3η+— = ki s i n φ ,—dxdo;(14)d£— =dy(14')T h e locus ki(x,y)^+7—ki c o s φ ,j = kik sin 2a =2dry=—^ =dy._—k s i n φ ,Ί2k COS φ ,2(hih sin 2a)2.1= 0 is called a branch line in the £,2/-plane, briefly aand its image in the £,77-plane an edge bi.
W e assume now k2Bi,τ* 0, so thatdt\ 9^ 0 on locus (then the flow in the physical plane is not a simple wave or326IV.P L A N ESTEADYPOTENTIALFLOWa uniform flow). From ki = 0, we see by (14) that d£ = 0; hence £ = constant. Therefore &i is an 77-line in the £,7j-plane, and Bi is a C+in the flowplane. I t follows as in the previous investigations that all lines in the x,yplane, with the exception of those lines which have the ξ-direction at theirpoint of intersection with Β γ, appear in the £,r?-plane tangent to thestraight vertical 77-line, b\, the edge.36Among these are the lines φ — constant and ψ = constant.
Hence the edge bi is an envelope of streamlinesand equipotential lines. T h e ^-direction, i.e., the C~-direction, is the excepAssuming also dki/dsi 9^ 0, it can be shown (seetional direction at Bi.similar proofs in a previous section) that the image of a line which has the^-direction at its intersection with Βχ, and in particular of a C~, must havea cusp at 6 1 . Since the images of the C~ are the straight horizontal £-lines,they return at the edge (see Fig. 130). For a C~-lined(0 -a)=θ(θ -a)d^and since άξ changes sign at an intersection with bi, the same must be trueybranchlin eφ * constan tY*constantχ0Vedge.b,//ψ* constan tΡζ0'F I G .
130. Branch line and edge.19.6 B R A N C H L I N E Sϊοτά(Θ — a)unless, exceptionally, 1 +327a /Q' = 0; hence in general a C~ hasan inflection point at its intersection with the Bx,37Since the lines of constant speed and those of constant direction arelines η + ξ = constant and η — ξ = constant in the £,?7-plane, they intersectthe vertical edge at ± 4 5 ° and thus are not tangent to i t ; therefore in the flowplane they have the exceptional direction at Bi. Hence, at each point ofB,xthe C~, the line of constant speed and that of constant direction touch eachother, while the streamlines and equipotential lines bisect the angles between Βχ and the inflection tangent of the C~~, and cross the branch linewithout singularity.I t also follows that at Βχ the direction of the vector grad q is perpendicular to the C-direction:dq/dsi = 0 (the same holds for the vector grad 0 ) .I t is easily seen that in steady irrotational flow (also in three dimensions) the direction of grad q coincides with that of dq/dt; hence at anypoint of Bi the acceleration vector is perpendicular to the C~-direction there.T h e acceleration at Bi is finite.
I n factb = ~ = q^= q sin a tan a (ki +dtdsk ).22This is finite for fci = 0, k finite, and zero if k also vanishes. A point where2ki = k = 0 is called a double branch22point.A branch line has physical reality. I n fact, its characteristic property ofdividing t w o flow regions in which the same velocity q occurs for different ris in no way extraordinary. One and the same (single-valued) hodographsolution r =r(q), however, cannot represent such a flow; hence a seriesexpansion of the hodograph solution must break down at the edge in thehodograph.Branch lines appear (and have been described by M .
J. Lighthill and byΤ . M . Cherry ) in flow in a symmetrical channel, which accelerates from38zero velocity at one end to supersonic velocity at the other (see Sec. 25.1).For reasons of symmetry there is not one but two branch lines, Bi andwith images b\ (kY= 0) in the rj-direction, and b (k22T h e image of their point of intersection is a sonic double branch point, hk2=B,2= 0) in the ^-direction.=0.W e review the main properties of a branch line obtained here: Branchlines exist only in supersonic flow. A branch line Bi in the flow plane, k\(x, y) =0, is a C*-characteristic.At each point Ρ of B\, where k20, bk\/ds\0,the C~, the curve of constant speed, and that of constant direction through Ρtouch each other-, streamlines and potential lines bisect the angles between theB\ and the CT at Ρ which has there an inflectionpoint.not infinite at Ρ and has the direction perpendicularThe accelerationisto the C~.
The image b\328IV. P L A N E S T E A D Y P O T E N T I A LFLOWof Β ι , the edge, is a (vertical, straight) η-line. It is the envelope of streamlinesand potential lines, and the images of all curves having the exceptionalCT-direction at their intersection with B\ have cusps on bi .W e further review: In case of a limit line, the edge of the fold separatesdifferent sheets in the x,?/-plane; in case of a branch line, the separating edgeis in the £,7?-plane.
T h e £,??-plane in the vicinity of the critical curve in thefirst case, the x,y-p\&ne in the vicinity of the branch line in the second case,are each covered once, by hypothesis. T h e difference in properties found forlimit line and edge (critical curve and branch line), which have analogousmathematical definitions, is due to the difference between the x,?/-plane andthe £,7?-plane; the values ofare directly related to the flow variables,the x,y are not.7. Final remarks(a) Conditions at the sonic line. A s noted before, there is no analogue tothe sonic limit line since D cannot tend toward infinity unless hi or h or2both do so. However, at a =necessary that not only hi(h )90°, for a point to be a branch point, it isbut also hi cos a (h cos a) tends to infinity;22then h'i —-» 0 (/r —> 0 ) , φ$ —> °o (φ —> oo).
A double branch point is charac2terized by both ki and kη2being zero. This result may be added to the results established in Sec. 5 for the singularities at a = 90°.( b ) Remarks on some particularsolutions. T h e simplest example with alimit line, radial flow (Sec. 17.4), shows a sonic limit line, where some ofthe properties established for ordinary limit lines are modified. T h e spiralflow (Sec.
17.4) shows an ordinary limit line, hi = 0, with no cusps. In thenext article, we shall find in a flow studied by Ringleb a limit line with twocusps. This flow features a double limit point at infinity and a sonic pointof a straight streamline*. A limit line with a double limit point hi = h = 0,2will likewise be found in the next article in the example of the compressibledoublet (which likewise has a sonic limit point on a straight streamline).For a simple wave, j(x,y)=0 in a two-dimensional region; the wholehodograph image reduces to one 77-line—in the case of a forward w a v e — a n dthe transition to hodograph equations is not possible.
Some features of thegeometry of simple waves can be easily discussed from the present pointof view.Consider a forward wave, Q — θ = 2ξ = constant, for which therefored£ = 0 (we assume a ^ 0°, 90° for simplicity); Eqs. (14) then show thatki = 0, hi =00 and each cross-characteristic, C ,y+is a branch line ki =0.T h e two-dimensional flow region is covered by these branch lines. Onthe other hand the straight characteristics, C~, have in general an en* Such a point has some particular features, based on άψ = 0, άθ = 0—and hence\f/ = 0, θ = 0—on the straight streamline.qφ32920.1 S E P A R A T I O N O F V A R I A B L E Svelope which is easily seen to have all the properties of a limit line £ , al2though there is no hodograph solution to serve as a starting point.
Eachstraight characteristic C~ is simultaneously a line of constant speed and ofconstant direction and the C , the streamlines, and the equipotentials are+all cusped at the envelope. Along this limit line the Jacobian j(x,y)indeterminate: fci = 0, k2is= =b <*>.(c) Remark on the (x,t)-problem.Branch lines and limit lines exist likewise in the one-dimensional nonsteady flow of an ideal fluid considered inChapter I I I . Limit lines appear as envelopes of either family of characteristics, and in particular as envelopes of the straight characteristics in simplewaves. A n example of a branch line was found in Sec.
12.5 and illustratedin Fig. 62. I t is the (u — a)-characteristic BD in the physical plane, whichintersects the characteristics of the other family at their inflection points.T h e theory can be worked out along the lines of this article. T h e(u +a)-characteristics (i.e., the lines dx/dt = u +a) are the 77-lines andthe (u — a)-characteristics the ξ-lines; we know that ν — u and ν +u remain constant along these ξ-lines and 77-lines, respectively. There are twotypes of limit lines, £ 1 and £ 2 , and of branch lines, B\ and # .239Article 20Chaplygin's H o d o g r a p h Method1. Separation of variables40In A r t .
16 we derived the linear equations (16.32") for the stream functionY(q,d) in polytropic flow:qV2a*Jdq* ^2\a.*/ dff*(1)Following Chaplygin we now introduce, instead of q, a new variable r bysetting2(2)r =!= * ~12«-T h e Bernoulli equation then takes the form(3)r +(j-) =1ora = o.(l -r) ,!330IV. P L A N E S T E A D Yand since, with pa=POTENTIAL1 as before, (α/α )8=2FLOWp*~\ the following relationshold:Ρ =(30.(1 -1τ)1 / (- ,V =υ2rιΛτ2-=-,As q goes from 0 to q1 -1 "Λ "=-τΥ ^1κ -,11 .= ——=ρ;κ— 11 — τ1 — τκ -+- 1Arthe subscripts β, m and Ζ in Eqs. ( 2 ) , ( 3 ) , ( 3 ' ) have their usual meaning.ΜΜP.Urt, τ varies from 0 to 1; to the sonic value qmt=atcorresponds the above value r which equals \ for κ = y = 1.4. T h e firsttorder equations (16.31) become( 4 )P(r)Tr=W 'Q(T)60'=with«•) TO - , , ) : : T - 1 )2((( 1+-'"""""')CW- ? R = Wand Eq.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
