R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 87
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W e consider a pieceof the contour along which the velocities are supersonic, i.e. in thepocket, and two points Λ, Β on it. Through each of them we draw both=458V. I N T E G R A T I O N T H E O R Y A N D S H O C K Scharacteristics and obtain the points of intersection of these characteristicswith the sonic line, e.g., in the order Ai Bi A$ B .* Consider the hodograph2image of the arc A Β and of the four points of intersection.
Because of themonotonicity law the images A[,B[, A ,2B must lie on the sonic circle in2the hodograph in the same order as their originals in the physical plane.N e x t the contour is deformed in such a w a y that, within the pocket astraight segment ^4*J5* is inserted. This can be done in such a manner thatany number of derivatives of the function which determines the contourremain continuous. Along A*B* we have θθ/ds = 0, and hence dq/dn = 0.Consider first the case that the speed q increases (or decreases) along A*B*or that there is a subsegment of A*B* along which q is monotonic.
Then thehodograph image A'*B'*of the (sub)segment A*B* will be a segment of aradius through the origin 0 ' . Clearly the points of intersection of the fourepicycloids through A *, Β * with the sonic circle will be in the order (omittingthe stars) AiB B Ax22(or ΒχΑιΑ Β ),22and hence not in the order requiredby the monotonicity law. Thus there is a contradiction.72N o w assume that q is constant along A*B*.
T h e hodograph image is asingle point A* and clearly the above contradiction cannot be derived.However, we can conclude that adjacent t o the straight segment alongwhich q and θ are constant, there is a small triangle in the supersonicpocket, bounded b y that segment and two intersecting straight characteristics through its end points, in which q = constant. W e call this triangleA*B*D. Adjacent to the straight characteristic A*D there must then be inthe pocket a simple wave W~, say, with straight characteristics C , and+cross-characteristics C~. This, however, leads to a contradiction, since wehave seen (Art.
18, p. 296) that the distance between two cross-characteristics measured along the straight characteristics tends to infinity as theMach angle a tends to 90°. Thus it is impossible that a simple wave contained in the finite pocket extends all the w a y to the sonic line. On theother hand, as is easily seen, it is also impossible that before the sonic lineis reached, the C~ pass out of the simple wave region.
Hence we again reacha contradiction.W e have thus proved that a flattened segment of a contour within asupersonic pocket is incompatible with our assumptions. Hence in theneighborhood of admissible profiles there are certainly profiles for whichno neighboring solution exists. Therefore(unlessflattenedprofiles areexcluded from the considerations) the original problem is not a correctlyset one.f* This order can be ensured b y taking AB sufficiently small; for all arcs withinsuch an A Β the same order will then appear.
Another possible order is Α A B B .t W e must in principle admit the possibility that b y flattening a piece of contourthe whole flow pattern changes abruptly so that the supersonic pocket moves awayand no longer contains the flattened piece. Such an abrupt change of flow correspondγ2x225.6CONJECTURES ON EXISTENCE I N T H E459LARGEOn the other hand take the classical problem of a Laplace potential flow.I t is certainly true that in the infinitesimal neighborhood of an admissiblecontour (i.e., one for which the incompressible flow problem past the contour has a solution) there are contours with corners, inadmissible becauseof infinite velocities.
Nevertheless, there is a significant difference betweenthe t w o situations: the compressible flow seems to be much more "sensit i v e " than the Laplace potential flow to a variation of the contour. A contour may be flattened without introducing a corner, i.e., a flattened contour can still have continuous curvature and actually as many continuousderivatives as we please; in this case there would be no trouble for Laplace'sequation. Interest in these considerations thus lies in the observation thateven so slight a discontinuity as that introduced by the flattening can makethe contour "inadmissible".
This sensitivity may be considered as pointing towards an explanation of why smooth transonic potential flow of thetype considered is rarely observed in nature.73Frankl, Guderley, and Busemann have discussed the problem of possible general lack of neighboring flows by means of suggestive arguments,partly physical, which make it plausible that the slighest irregularity ofthe contour leads to breakdown of potential flow.746. Conjectures on existence and uniqueness in the largeW e recall the problem that was taken as point of departure, and askourselves why we actually thought that an existence theorem might holdfor transonic flow.
T h e answer is obviously that it was formulated inanalogy to the classical incompressible flow problem (a linear problem) andis supported by the theorem for compressible subsonic flow (a nonlinearproblem). However, certain results holding for linear but mixed problemspoint towards a negative rather than a positive answer; the correspondingconjectures concerning our problem are particularly suggestive. Only afew hints can be given here.In 1923 Tricomi studied the equation of mixed typementioned at the beginning of Sec. 2. M o r e generally, writing uetc., the equationxx(2')A(x,y)uxx+ B(x, y)uxy+C(x, y)uyy= d u/dx ,22= F(x, y, u u , u )is an equation of mixed type if the function Δ ( # , y) = By2xy— 4AC changessign across a curve without vanishing identically.ing to an arbitrarily delicate change in contour would imply that the problem is notcorrectly set..460V.
I N T E G R A T I O N T H E O R Y A N D S H O C K SReturning to E q . (2) we shall show why a correctly posed mixed boundary-value problem may differ essentially from the analogous elliptic problem (incompressible or compressible subsonic flow past a profile is elliptic).Consider (Fig. 173) a region bounded by an arc C in the elliptic halfplane, y > 0, and two characteristics SiT, S T.
T h e equation of thecharacteristics is easily found, since (see A r t . 9, p. 108) y dy + dx = 0or dx = ±( — yYdy, whence022(x - c) + iy2z2= 0,with c an arbitrary constant. These characteristics, real only for y S 0,are semicubic parabolas. I t has been proved by Tricomi that a solution uof (2) is determined in the region above b y boundary values along C andalong one of the characteristics alone, say SiT, while values along S Tcannot be prescribed arbitrarily.For us the following extension (due to Frankl) of this problem is ofimportance.
Consider (Fig. 174) a region bounded by an arc C o betweenSi and S in the elliptic region, by two arcs of characteristics issuing froman arbitrary point Ο on the transition line and by two arbitrary noncharacteristic arcs issuing from Si and S , the latter intersecting the characteristics through Ο in 7\ and T respectively (the "arbitrary" arcs mustlie as shown in the figure, with SiTi intersecting every characteristic ofthe family containing OTi only once and similarly for S T ). T h e figure alsoshows the characteristics TiR and T R, as well as the characteristics through51 and S . T h e value of u is given along TiSiS T (where we go from Si to5 along C o ) , but not along any arc T T in OTiT R, nor along OTi and OT ™T h e function u is then determined firstly in OTiSiS T 0, and then in thequadrangle TiOT R (and actually beyond this region, in the whole characteristic triangle bounded by the horizontal line SiS and the t w ocharacteristics SiT and S T issuing from Si and S , respectively; this,however, is not needed for the following).
But if u is uniquely determinedin the region RTiSiS T Rby the values along T SiS T , then obviously0275229222222X222222222222x22χΤF I G . 173. Illustrating Tricomi problem.ΤF I G . 174. Illustrating Frankl problem.25.6CONJECTURES ON EXISTENCE I N T H ELARGE461F I G . 175. Illustrating conjecture that data cannot be prescribed everywhere onarbitrary contour.values of u cannot be given arbitrarily along T\T . Thus, the problem in whichu is prescribed along a closed contour which lies partly in the elliptic,partly in the hyperbolic region is incorrectly posed.2Following Busemann, Frankl, and particularly Guderley, we now formulate a similar problem of flow past a profile with supersonic enclosure(Fig.
175). W e consider only the upper part of the profile which, for convenience, is supposed to be symmetric; let Π denote the upper half planeoutside the contour, Η the supersonic part of Π between SiS and the sonicline, which is here the transition line S\OS . T h e flow is subsonic (elliptic)in the part of Π extending to infinity outside the sonic line, and supersonic(hyperbolic) in SiTiT S OSi. This problem differs from the Frankl problem in the fact that the elliptic region, the subsonic part, extends to infinity and in the fact that the differential equation for φ is nonlinear;consequently, we do not know a priori the position of the sonic (transition)line, nor of T\ and T , as one does in FrankFs problem. (The equation forφ is likewise nonlinear; to use it we would need an analogue of the Franklproblem with θφ/dn rather than φ given along the boundary.) If we couldassume that a similar uniqueness* theorem holds for this much more difficult problem we could conclude as before: Assume that a solution existsin Π; for this solution, ψ = 0 along AT\T B;it is however uniquely determined by the boundary value ψ = 0 along ATi and BT alone.
Hence, ifthe arc T\T of the contour is deformed (no matter how slightly) so thatit is no longer part of the level line ψ = 0 of the solution, then no solutionwill exist for the deformed contour, f This would mean that the problemwe took as point of departure is not correctly set in the sense explainedat the beginning of the preceding section.22222222Thus under the assumption that in the more general mixed and nonlinearcase and with the elliptic region extending to infinity, a result similar to* N o t e that only a uniqueness theorem is required in this case,t N o t e that neither this conclusion nor that regarding the nonexistence fortened profiles applies if the contours are assumed analytic.flat462V.
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