R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 47
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89) and draw the characteristicsthrough them, or rather, straight lines in the characteristic( K n o w i n g q and 0, we know their directions 0 ±directions*a). T h e y have an intersection, 3, on the upper side of AB, and an intersection, 4, on the lowerCDFIG. 8 9 . Cauchy problem.side.
Considering all distances 12, 13, and 23 as infinitesimal and neglectingterms of higher order, we conclude from (24) that, with an obvious notation,ft — gi =+(03 -0i)gi tanqz — 02 =— (03 — 02)02 tanai(25)a.2For the point 4 the opposite signs hold in Eqs. (25). W i t h respect to theunknown qz and 0 , the determinant of these equations is3(qi tan a\ + q tan a ),22which is different from zero since tan a cannot change sign; thus both g3and 0 can be computed.
In this way, starting from a sequence of neighbor3ing points along AB, one can derive from the given values of q and 0 on ABthe values along a second row of points, A'B'.Continuing in the samemanner, q- and 0-values are found for all lattice points within a curvilineartriangle ABC,where AC and BC are characteristics, one belonging to the* This section is, mathematically, a straightforward application of the considerations of Sec. 10.3 to our flow problem.
In order to avoid repeated interruption of thepresentation we refer the reader to A r t . 10 for .more careful formulations, for comments and N o t e s .248IV. P L A N E S T E A D Y P O T E N T I A L FLOWplus set and one to the minus set—provided the procedure does not breakdown earlier, which can happen if the direction of a cross line such asΑ'Β',A"B",· · · approaches somewhere a characteristic direction (seeSec. 10.3). However, due to the noncharacteristic nature of AB,and thecontinuity of all functions involved, this can happen only at a finite distancefrom AB. All these conclusions apply, of course, also to the triangleon the lower side ofABDAB.As a second case, consider data given along two intersecting lines A Β andAC, one of them a characteristic.
Assume that A Β is a minus characteristicand that the noncharacteristic arc ACis in one of the angular spacesbetween AB and the plus characteristic through A. Values of q and 0 alongA Β must be given in such a way that at each point its angle with the rr-axisis φ~ =0 — a, and such that the second equation (24) holds. It followsthat if the geometric shape of A Β is given, we may prescribe only the valueof either q or 0 at one point of AB;then q and 0 are determined along AB.W e further suppose that either 0 or q (or some component of q) is givenalong AC.From the data along A Β the initial elements of the plus characteristicsat all points of A Β can be derived (Fig.
90), and we assume that they areplotted in the direction toward AC.If the point 1 is adjacent to A, thecharacteristic element through 1 will cut the line ACFrom a given value at 3 and from qz — qi =in some point 3.(03 — 0ι)#ι tan αϊ , both thequantities 03 and q can be derived. This enables us to find the beginningz34 of the minus characteristic through 3, and the compatibility relationsapplied to the segments 34 and 24 give the values q^, 04 · In this way,step by step, the whole quadrangle ABDC,where CD and BD are characteristics, can be filled by a net of points at which q is known.A slight modification of the procedure has to take place if A C is likewisea characteristic, here a plus characteristic.
If we know that of two intersecting arcs, given geometrically, one is a C+and the other a C~, thenneither 0 nor q can be arbitrarily prescribed anywhere along these twocurves. Their values follow from the two direction conditions and the twocompatibility conditions, with both q and θ determined by the two direcAΑtion conditions at A. The stepwise construction of the net inside the char-C"FIG.
90. Data along two intersecting lines, one of them a characteristic.16.5249HODOGRAPH.0ACθFIG. 9 1 . Plane duct.acteristic quadrangle ABDCprocedure in A r t . 10.is analogous to the preceding one and to theAs an example, consider the duct ABCD of Fig. 91. T h e entrance velocities along A Β determine, independently of the shape of the walls, a supersonic flow in triangle I.
Since the walls are streamlines, we know θ alongthem. T h e values of q and θ found for A Ε with the known 0-values alongAD determine q in AEF, region I I , where AE and EF are characteristics.In the same way, knowing 0 along BC leads to the velocity distribution inBEG, region I I I . From the values of q and θ along EF and EG (two characteristics) follow those in region I V , and so on. All this holds, of course, onlyin the case of a purely supersonic motion. In a subsonic flow it is not possible to attribute the flow pattern in any partial region to the influence ofspecific data.5. Hodograph5In Sec.
8.2 the notion of hodograph has been introduced as follows. In asteady two-dimensional flow a point-to-point correspondence was madebetween a point P(x,y) in the x,y-p\&ne, at which the velocity is q, and apoint P in a q ,q -p\sn\e with rectangular coordinates q , q or polar coordinates q,6. This mapping of the points Ρ onto the points P' is known asthe hodograph transformation.
In general, a streamline in the physical plane,or plane of flow, is mapped onto a line in the <?,0-plane, the hodographplane, which we again call a streamline. Our first task is to find the differential equations which the stream function and the potential function haveto satisfy as functions of q and q , or of q and Θ, rather than of χ and yor other coordinates in the flow plane. T h e passage to the hodograph plane,that is, the introduction of the previously dependent variables q and θ asindependent variables, is equivalent to the interchange of the pairs x,y andqfi as studied in Sec. 10.6.fxyxxyyL e t / be a differentiable function of q, and consider its value at the pointΡ of the physical plane and at a neighboring point P i .
L e t d/dl designatedifferentiation in the direction Ρ Pi ; then(26)dfdl=dfdgdfdedq dld6 dl'250IV. P L A N ESTEADYPOTENTIALFLOWW e apply this formula four times, taking φ and ψ for / and the directionsds and dn (Fig. 87) for dl. Then recalling (15') and (19),(27)dipds2-·= o,dnwe find the following two pairs of linear equations for dq/ds, dB/ds anddq/dn, dB/dn, as the respective unknowns:9(28)_ d<p dq , d<p dB~ dqdsdB ds'0 =^~"dq ds4- tyfOLQ_d<p dqd<p dd~ dqdndBdn'pqdB ds '=HHd\p dq , dip dB—- Η — - — .dq dndB dn—Each of these two pairs has the determinantdtp d\f/ __ d(<p,\f/)δθ dq ^ d(q, Θ) 'dip dipD = dq dd(29)the Jacobian of φ,ψ with respect to q,B.
Supposing that D is different fromzero, we obtaindq _ 1 dipD dB>ds(30)A =ddn—11dBdsQd<pD de>pqdBdn1βψθφIf these values are introduced in the two basic equations (7), the factor1/D drops out, and we obtain the fundamental equations in the hodographplane(31)d<pdq=Μpq2ΙδψdBdipqctyd$pdq'When (31) is compared with equations (18) and (18'), which follow immediately from the definitions of potential and stream function, its superiority is obvious. Since ρ and Μ are given functions of q, and q is anindependent variable in (31), these equations are linear.B y anojther differentiation we can easily eliminate either φ or φ and obtain a single second-order equation for the other:16.5 H O D O G R A P H251Carrying out one differentiation in (32) we findaV _qM2dq22ρ d_ (q\βψqdq \p) dq1ΘΘ2N o w using (from the Bernoulli equation)d /l\ _1 dp dp _p dp dqdq\pjwe obtain Chaplygin'sequationOf)L^.
|V^2+dqq21 qpa2i+dd2( 1+21M 2 )^=q20.dqThis equation is true for an arbitrary elastic fluid. If, for a polytropicgas, Mis expressed in terms of q by means of (10), the result is2Λ _2„(32")V9_2\22,A aVκ_-^Λ2_2\2 ,*_+_i A aVι\a Jdq ^2//1Λα , 7 θ022Here the first and third parentheses can be written as (1 — q /q )2(1 — q /q )22f2mandrespectively.In a similar way we find4- 1 -M2?2d <p2<*021 -M2ΡΪ_d_ (pq<fy \1 -\=0d°If we carry out the differentiation, a term dM/dq or da/dg will eventuallyremain. In the case of a polytropic gas we obtainq\l - M ) 0 + (1 - M ) 1 | + ?(1 + *Μ<) * = 0,(33")222where M may be replaced by means of (10). Equations (32) for ψ are simpler than (33) for φ.For an incompressible fluid, p = p = 1, α —> <», Μ —> 0, the equations(31) reduce to the Cauchy-Riemann equations in the variables log q and—0, and both (32') and (33') reduce to Laplace's equation in these variables.In this case a complex potential w(z) = φ + ίψ, where ζ = χ + iy is introduced.
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