R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 64
Текст из файла (страница 64)
3.in ( 1 5 ) and ( 2 1 ) must each satisfy E q . ( 7 ) forη =+ 1 also. If we consider directly the case η =ai =1, 6i =-ι 3/ψκ1 we have from ( 8 ) ,— 1/(κ — 1), andι2K - 1 2 !>= ^[1 -T(1 -K/5/1\- l V - l)23!τ)""-"].T h u s ^ i ( r ) appears as the sum of t w o terms, which are respectively equalto the ψ _ ι ( τ ) in ( 1 5 ) and ( 2 1 ) .F ( ΐ , - ^ Τ ^ τ )47For the corresponding F(ai,= -=-±K[τ"1-r-'il-h, 2 ; r ) :r)"*- '],1and one may easily check that the expression to the right tends toward1 as r tends to zero.W e turn now to another example. T h e flow studied in detail in Sec.
3 hasbeen considered as the compressible counterpart of incompressibleflowaround an edge, i.e., a " corner" of 0 ° opening. I t is natural to try to generalize in a similar way the well-known incompressible flow around a convexcorner (see Fig. 133). I t is known that the hodograph streamlines corresponding to the incompressible flow around a corner of angle 360° — a are afamily of lemniscates (actually one loop of each lemniscate), all within theangle a — 180°; their common tangents form the hodograph streamline ψ =0, which is the image of the two legs of the angle in the flow plane. These342IV. P L A N E S T E A D Y P O T E N T I A LFLOWlemniscates play the role of the circles in the edge flow. Compressible counterparts of these and similar flows may be constructed by means of themethod considered in this article.
A few results are the following.T h e principal features regarding the limit line, etc., remain unchanged.Consider, e.g., a — 270° (see Fig. 134). T h e limit line in the physical planeF I G . 133. Incompressible flow around a 90° corner.F I G . 134. Compressible flow around a 90° corner.20.4 F U R T H E R C O M M E N T S A N DGENERALIZATIONS343has again two cusps. There is again one well-defined streamline, S (the analogue of streamline 4 in the edge flow), which separates the smoothstreamlines ( b ) which do not meet the limit line at all from those streamlines(c) which feature two or four cusps (one on each of the four branches ofthe limit line). T h e sonic line is now no longer a circle but resembles a loopof a lemniscate.Consider now the complex potential w of the incompressible flow:.mw = ζ ,f = qeW e eliminate ζ and obtain, with A =A .
m/(m-l)m/(m-l)Λm—1—id= mzm~/m / ( m _ 1 ),7Π..λΎΠ' ( cos0 — % sin0 1.\m — 1m — I /I n this notation the Ringleb flow corresponds to m = §, and the flow of thelast example t o r n = f. Let m be between J and 1; the incompressible flow48is a flow around a convex corner of angle β = 2π — α, a = π/τη (in radians);the angle α is between 2π and π, and β between 0 (Ringleb flow) and π.'T o the incompressible stream functionw = Αξ.= Aqm/(m-l)A-Ληmsin= Aq» / ( « — « )0 = Aq·^Λsin0π — αm — 1corresponds, in general, the compressible flow(8')ψ = Agx/(T- F ( a , 6 , c ; r ) sin π — aa)Here π/(π — a) stands for the η of E q . (8) and we havea + b = η(8 )α6 =η(η +2=κ — 11)κ — Iπ — α=-Ζ(κ — 1)κ — 1c = η +1 =2π — ατ— α—(π — ay.If here the angle a is of the forma = ^ ^ T Tρ +1(p =1.2.3.·.)(as, e.g., the angle α = 270° of Fig.
134) then c = (2ττ - α)/(ιτ - a ) = - p ,i.e a negative integer; it is thus seen that with β = 2π — α, for β-valuessuch as β = π/2, 2π/3, 3π/4, etc., a solution, φ, of the preceding form ( 8 );IV. P L A N E S T E A D Y P O T E N T I A L344FLOWdoes not exist and another expansion must be used.a = 270° of Fig. 134 has been computed explicitly.T h e exceptional case4950For angles not of the above form, e.g., for ail angles β < 90°, the abovesolution exists; in particular forβ = 60°, c =51— |, and for—\,n=52β =46.8° the solutions have been studied in detail. In the last case the infinitehypergeometric series reduces to a polynomial of degree four.
These casesshow no new feature compared with the cases β = 0°, 90° above (Figs. 132 and134).T h e examples of the last two sections can be adapted to general elasticfluids. T h e restriction to polytropic flow (with the value of κ taken as 1.4 incomputations) is made in order to obtain concrete results for the most important case. Each of these particular solutions can be regarded a posteriori as a solution of a boundary-value problem, e.g., by considering ineach case certain streamlines as fixed boundaries of the flow.5. Compressible doubletW e pass now to an example that is distinguished by a particularly interesting limit line. W i t h the notation of p. 343 we consider the case m =— 1, η =i.e., we consider a compressible analogue of the doublet™. Here/r»o\(22)w =1- - ,dw-ΐθ1f = = qe= - ,.i .
θ4>o = ? s i n - .!2T h e corresponding compressible stream function (for κ =1.4) is, according to ( 8 ) ,Φ= Vq T FM\Zq T f(j)(α,6, | ; τ ) sin - =lMsin-,k(23)a +b =1 _21κ -__91 "h'a_° ~_3_8(κ -1) "15ϊϋ·F o r / ( τ ) one has the expansion/(r) =1 -fτ+fWr2This converges uniformly for τ ^ 1. Near τ = 1 an expansion in 1 — τ maybe used (see footnote p. 331).Corresponding to this ψ the φ may be determined; the coordinates χ, yare then given by (17.250· If r is used instead of q and (30 is noted, theformulas (17.250 are replaced by20.59roCOMPRESSIBLE DOUBLETV r - ^ = ^ c o s 0 - ^ ( l - r ) - " < ' - "orοτστ345sineand three similar ones. Integration givesVq~m-'> χ = 2r/'(r) (cos | -r, / 4(l -r)r, , 4(lr ) " - ' j , = 2r/'(r) (sin | -, / (\ cos | ) + / cos-,θ(24)-11ί sin | ) + / sin | .W e shall now discuss the singularities of this transformation.
There isno branch line. Indeed it is seen from (23) and from the fact that both /and/' are finite, that neither θψ/dq nor θψ/θθ can become infinite except atthe point q = 0 for which χ —> oo and y —> <*>; this is an isolated branchpoint.Thelimit singularities arehere of some interest (see Sec. 19.4). From (23)characteristicDF I G . 135. Limit line for compressible doublet flow.346IV. P L A N E S T E A D Y P O T E N T I A Lwe compute q (θψ/dg) dz \/MFLOW— 1 (θψ/θθ), which are essentially the same2as θψ/θη and 3ψ/3ξ. This gives(25)orThis line in the hodograph, the critical curve, has a double point d forθ = 180°, where 1 + 4τ/'// = 0, i.e., r ^ 0.45. A t this point both 3φ/3£ = 0and 3ψ/3η = 0; consequently hi = h = 0, and, by (19.7), dhi/θη =3h /3£ = 0.
N e x t we compute at the double point the second derivativesof φ and find that 3 <p/3q = 3 φ/3θ = 0, d <p/dqdd ^ 0. From this weconclude as at the end of Sec. 19.4 that the point is an extremum for ψand a saddle point for φ: no streamline, ψ = constant, passes through thispoint; the streamlines encircle it (see Fig. 128).222222T h e critical curve is a double loop curve touching the sonic circle atθ = 0° and the maximum speed circle at θ = 180° (see Fig. 135).
T h e firstof these two points is the sonic point of a straight streamline (similar topoint A in the Ringleb flow).In the physical plane the limit line consists of the lines hi = 0 and h = 0,which are both cusped at their common point Z). T h e cusp tangents at Dhave characteristic directions. In the neighborhood of D the flow is confinedto the obtuse angle (covered four times) above the cusps. This is an interesting example of a limit point of higher order, hi = h = 0, where the criticalcurves intersect.226.
Subsonic jetW e turn now to the consideration of a boundary-value problem: Chaplyg i n ^ method applied to a subsonic jet.* T h e problem of a gas escapingthrough a slit between plane walls adapts itself particularly well to Chaplygin's method (Sec. 2 ) , since this is essentially a problem in the hodographplane. Here the shape of the escaping fluid is not known beforehand: theboundaries of the jet are/ree boundaries. Since the flow is assumed steady,the boundaries, both fixed walls and free boundaries, do not change in timeand are streamlines.
Hence along the boundary, ψ must be piecewiseconstant. Along a straight wall the angle θ is a given constant; hence thehodograph image of such a line is radial through O' with known slope.64* See also comments in Sec. 15.2.20.6SUBSONIC JET347Along a free boundary the pressure is constant. T h e influence of gravityand other external forces being neglected, it follows from Bernoulli's equation, in either the incompressible or compressible case, that on a free boundary surface the velocity has some constant value, say qi . Thus, the freestreamlines are mapped onto arcs of circles about 0' of radius qi.
Hence,for such a problem we know the boundary and the boundary values in thehodograph for the incompressible as well as for the compressible case; theseboundary conditions coincide if the shape of the vessel, the total fluxthrough the orifice and speed at the jet boundary are the same in the twocases.T h e particular jet problem considered by Chaplygin is shown in the figure(see Fig.
136). There is a particularly simple disposition of walls, in whichthe vertical wall A Β is a continuation of AB. T h e distance Β Β = 2α is given.W e assume that the half plane to the left of ABBA is filled with fluid; theflow starts with zero velocity at infinity to the left and converges t o a parallel jet, with constant velocity q on the free boundaries of the jet, which arestreamlines. Along the horizontal center streamline we take φ = 0; andφ = φι, say, along ABC while φ = —φι along ABC.xT h e boundary ABC is mapped onto A'B'Cin the hodograph plane andthe same holds for ABC and A'B'C.All streamlines in the hodograph gofrom A' to C.
Along B'C'B' the velocity q equals qi .F I G . 136. Chaplygin's subsonic j e t .348IV.PLANESTEADYPOTENTIALFLOWIn this problem the method of Helmholtz-Kirchhoff-Joukowskifurnishes directly what we need for the application of Chaplygin s method, namely the (incompressible) hodograph potential Wo(£). T h e method can be applied to muchmore general data than the ones considered here. In our case the result issimple and well-known.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
