R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 59
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9 and 16 the line elements of Mach lines are here denotedby dsi and ds^ .f T h e sign of Ri (of R ) is chosen positive if the center of curvature of a £-line(77-line) lies in the direction of increasing η (increasing £).219.2 S O M E B A S I C F O R M U L A S . S U B S O N I C CASES315of the £- and 77-lines, respectively, and the corresponding curvaturesR2κι, K , as well as the curvature κ of a streamline, we find:2(5)K1κι = —Ri=*2 = 77 =R2-θ(θ -α)θ(θ -dsiθ(θ +α)Aid£'θ (θ +a)dS2= — =(*L +ds2 cos α \θβιa)fh θη1ds /'Λ_2 cos a \h2=ΐ Λ12hj'Denoting by a prime differentiation with respect to q, we find (always withQ =η +f, β =η -ξ) that(6)B y equating in (3) the mixed derivatives of χ and t/, respectively, and thensimplifying, we obtain a pair of linear equations for hi, h :2^sin 2a +^Ai cos 2a) = 0,-1^ (Λ +-l ) (hi + h2 cos 2a) = 0,2(7)^2sin 2a +with Q' = l/(q tan a ) .
f While these equations do not seem to offer any particular advantages with respect to the general integration problem, ascompared to other pairs of linear equations (see Sees. 17.1 and 17.2), theywill help in our present discussion.33W e finish this preparatory section by considering the vanishing of representative Jacobians in the subsonic cases. Dealing first with the limit typesingularities, we consider the Jacobiand\Q*,qv)QPpq* N o t e that Ri = 0 does not follow from hi — 0 if, exceptionally, 1 + ot /Q = 0;fin the polytropic case this happens for the exceptional Μ « 2/\/S — κ).t I n polytropic flow:j«'Q'* + 12 cos a29which shows how the expression becomes infinite as a —> 90°fIV. P L A N E S T E A D Y P O T E N T I A L316FLOWwhich vanishes only if*« =*and0,= 0,and this implies, according to (16.31),*Hence, all four derivatives \f/q=0.<p , φβ vanish.
Then also x ,qqy,qΧΘ , yevanish. Clearly, such a singularity is isolated. Incidentally it can then beshown that this singular point is a saddle point for both (p(q,6) and ^(#,0)unless all second-order derivatives of φ and ψ vanish there. I n fact, from(16.31) we conclude thatiq^M(8)2Ρ2= —I t is also seen from (16.31) that if <p e = 0, b o t h ^qg gand ψββ vanish.
HenceUqjpee — Ψ β < 0 unless all three second-order derivatives of ψ vanish; a52similar argument holdsfor<p(q,6).T o study the branch type singularities for subsonic flow, we consider forq 7± 0 the JacobianI t can vanish only if dq/dn = 0, dq/ds = 0, and it follows from Eqs. (16.7)that then, in addition θθ/ds =θθ/θη = 0. W e may show again that thesingular point is a saddle point for q and θ unless all second-order derivatives are zero.
Hence, at any rate, these singularities occur only at isolatedpoints.3. Limit lines £ iand£2N e x t , let us consider the supersonic region of a flow using the hodographcoordinates £,η. W e shall assume in this and the next section that a isneither 90° nor 0 ° . In Sec. 5 the case a — 90° will be studied.W e consider for a given solution in the £,r;-plane the locus M£>*?) = 0,and in particular the mapping onto the x,?/-plane in the neighborhood ofthis locus. L e t ρ be a point* with coordinates £0 ,νο, where /&i(£o ,yo) = 0 ,(dhi/drizQt1IQV=?± 0.
Then by the implicit function theorem there is a curve#(£)> i t h τ/ο = g(£o) on which /ΐι(ξ,τ/) = 0. W e call it a criticalwcurveand denote it by U , and we call the curve £ 1 in the flow plane correspond* We use here p, m, q · · · , for points in the £,»7-plane rather than, as in previous articles, Ρ', M'Q...}f19.3 L I M I T L I N E S £ 1 A N D £317ing to h a limit line. A point on £ 1 , or rather a point Ρ such that hi = 0at its image p, will be termed a ZzratZ pomZ.W e consider first points m on the critical curve where both dhi/θξ anddhi/θη are different from zero.
F r o m the first of these conditions we concludethat h does not have the ^-direction at ra, and from the second, using (7)and a ?± 0°, τ* 90°, that h ιέ 0 at ra. T h e n (1) shows that at M, the image2of ra, the limit line £i has the ^-direction, i.e., £i is tangent t o the characteristic C+at M.T h e same conclusion holds for any curve 6 throughΜwhose image c, say η = / ( ξ ) , does not have the ξ-direction at ra. A l lsuch curves β are tangent to £i atM.For example, the images in the ξ,^-plane of lines of constant speed or oflines of constant direction do not have at ra the exceptional direction,the ξ-direction.
I n fact, for the latter, dd vanishes, and if a ^i(dQ+i.e.,90°, di\ =άθ) 9* 0; a similar conclusion holds for the former. Hence theselines are tangent to £i at M.A consequence is that the direction of thevector grad q at Μ is normal to £i, and hence to the C+through Μ.If, however, the curve c has the exceptional direction atra, i.e., άη/άξ =/'(£) = 0, we can no longer conclude that the corresponding C has the ηdirection at M. W e prove that in this case β has a cusp at M.In fact,along eG= *F+ /'(£)*„so that since both χ and|= </*+/'vanish with hi a n d / ' ( ξ ) =ζ0, both dx/άξ anddy/d£ are zero at M. On the other hand,dx„dy. .//and not both right sides vanish since, using ( 3 ) ,XttVr,-=^2sin2α^0.Hence the singular point of e at Μ is a cusp. Since the £-line at ra has theexceptional direction, the C~ at ilf has a cusp there, its tangent making theangle 2a with the direction of £.
W e know that streamlines and equi-Apotential lines bisect the angles of the C~-and C -directions. Since they+are therefore not tangent to £i at M,we infer that their images in the£,?7-plane must have the exceptional £-direction at ra and that therefore at Μ they must have cusps. This may also be seen directly. From hi = 0and ( 4 " ) , ψξ = 0 and φ =ξ0 follow; hence at ra, for a streamline, d\f/ =\{/ άη = 0, and as ψ 9^ 0 (since hηη2^ 0 ) , άη = 0; hence at ra the streamline has the ^-direction. T h e same conclusion holds for potential lines.318IV.
P L A N E S T E A D Y P O T E N T I A LN e x t consider the acceleration at Μ.(q tan a ) " we findFLOWUsing again Q =η +£, Q'1(9)_dqb =_ dqndtds_2COSa \dsids J22 cos a2h/2which is thus seen to be infinite at a limit line, as found before in particularexamples. T h e same holds true for the velocity gradient.A l l our results, of course, have their counterparts for points of an £ :h = 0, at which dh /d£ -A 0, ΘΗ /Θη ^ 0. W e review: Consider in the characteristic plane the locus Λι(ξ,τ?) = 0, i.
e., the critical curve h , and pointsof k at which dhi/θξ0, dhi/θη0. Its image, <£i, the limit line, is theenvelope of plus Mach lines C , of curves of constant speed and of curves of2222+y_C, 17-lineψ« constantC " f-line2a,p.p'^•constantF I G . 126. L i m i t line and critical curve.31919.4 S P E C I A L P O I N T S O F T H E L I M I T L I N Econstant direction] it is the locus of cusps of the C~~ Mach lines, of streamlinesand of equipotential lines, all of which have in the ξ,η-pfone the ^direction attheir intersection with l\. For the critical curve, JH = 0, the roles of C and(Γ, of η and ξ, are reversed.+W e have seen in examples that t w o physical solutions meet at a limitline.
W e now consider the correspondence between the x,y- and £,77planes near such a line (see Fig. 126). Consider a point Ρ in the physical plane. I t is the image of two points ρ and p . I n fact through Ρ pass twocharacteristics of the nonexceptional kind, two C -lines, which touch the£i-line at Μ and Μ', respectively, with corresponding points m and m' onk .
On the ?7-line through m lies p, on that through m' lies p \ the two pointsρ, p' are on opposite sides of the critical curve k since η varies in the opposite sense along Ρ Μ and along PM'. Likewise two ξ-lines (C~-characteristics), PQ and PQ', pass through Ρ making angles 2ai and 2a with the(^-characteristics PM and ΡΜ', respectively.
T h e point q correspondingto the point Q lies on the ξ-line through ρ and on k , while the point q'lies on the ξ-line through p' and on k .f+f2W e obtain a one-to-one correspondence in the usual way by consideringseparately the two sheets in the physical plane, which are each the imageof one side of the critical curve hi = 0. On each sheet in the physical planethere is one flow.344. Special points of the limit line(a) Cusps of the limit line. Our discussion is not y e t finished. So far wehave assumed that at the point of the curve Μ ξ , η ) = 0 considered, bothdhi/θξ and dhi/θη are different from zero.
N o w we drop this assumption.Κ Mi,*?) = 0 and dhi/θη = 0, it follows from (7), since (a'/Q')- 1 ^ 0 ,that h = 0; hence hi = h = 0, and from (3) that all four derivativesθχ/θξ, · · · vanish. Such a point will be called a double limit point T h e pointof intersection of two critical curves hi = 0, h = 0 is such a point; thecorresponding point in the flow plane is the point of intersection of twolimit lines £ 1 and <£ [see subsection ( b ) below and example in Sec.
20.5.]W e lay aside this case for the moment.2222N o w , let us consider a point m of hi = 0 where dhi/θη0, 3hi/θξ = 0,θ\/θ£ 5* 0; the critical curve has the ^-direction. If the relation Ηι(ξ,η) = 0is written in the form η = g(£), theng' = άη/άξ = ( — θ^/.βξ)/(3hi/θη) = 0at m, while g"(£) = (-θ%/θξ )/(θ^/θη)j± 0. A s before, at the point M,the image of m, dx/άξ = Χξ + x^g'(£) = 0 from hi = 0 and g = 0, andlikewise dy/d£ = 0, while22fax.
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