R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 58
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Figure 122 showsschematically a flow in a duct whose walls are partly straight, where regionsof constant flow, simple waves and general flows (shaded) alternate.(c) Particular cases. Assume that in a situation such as that consideredin ( b ) the C~ through A meets the opposite wall at a point where it isstill straight, i.e., to the left of A (see Fig.
123). T h e characteristic EG is across-characteristic of the simple wave AEG = W , and Wx drops out.ccccAF I G . 122. Succession of constant flows, simple waves, and general flows in a symmetric duct.310IV. P L A N E STEADY P O T E N T I A LFLOWBeyond EG the flow is no longer a simple wave and is determined as explained before. EG is sometimes called the reflected Mach line of AE.A further simplification arises if at the point Ε the wall has the shape of astreamline of the extended simple wave obtained from AEGthe straight C~-characteristicsby continuingacross EG; then this latter wave providesthe continuation of the flow beyond EG. This applies in the followingconstruction.( d ) Parallel exit out of a duct Consider a symmetric duct with otherwisegeneral supersonic flow (neither uniform nor simple w a v e ) in some regionlimited on the right b y the characteristics ED, ED (see Fig.
124). W e consider now the problem of determiningthe shape of the remaining part ofthe duct in such a way that the flow at the end is uniform and parallel t othe horizontal axis of symmetry. A simple wave, or rather a symmetricpair of simple waves, may provide the transition. Beyond D, we want thecharacteristic line ED to continue as a straight C+of W ,Aand similarlyfor ED and WA .
This will be achieved if the wall Ε A is constructed so thatit is a streamline of the simple wave WAXΕfor which ED is a cross-characteris-AF I G . 123. EG reflected Mach line ofAE.YgeneralflowΫF I G . 124. Parallel exit flow out of a duct.19.1 SINGULARITIES OF THE HODOGRAPH TRANSFORMATION311AFIG.
125. Parallel exit flow out of a duct with radial entrance flow.tic. Since q and θ are assumed to be known everywhere along ED, theconstant 2η = Θ + QB follows, as well as the velocity distribution alongall straight C crossing ED in given directions; in particular, the streamline through Ε can be computed or constructed. B y reason of symmetry,θ = 0 at D ; hence it is also zero at A and at A ; the walls beyond A andA can be continued horizontally, and the emerging flow will have constanthorizontal velocity to the right.Β+A particular case may be mentioned. Consider a duct where the generalflow (to the left of ED Ε in the preceding figure) is radial [see subsection(b) of Sec.
17.4]. This flow is explicitly known, including the shape of EDand the velocity values along it (see Fig. 125). The radii, such as OQ, arestreamlines, and since the values of q and 0 along the straight C throughQ remain constant, and Ε A is to be a streamline, it follows that the tangentto the wall at Ρ is parallel to OQ.+Other problems which include combinations of simple waves and shockswill be discussed in Art.
23.Article 19Limit Lines a n d Branch Lines1. Singularities of the hodograph transformationCertain singularities found first in some of the simple examples of Art. 7(radial flow, spiral flow) were reconsidered in Sec. 17.4 as singularities ofthe hodograph transformation. W e found them connected with the vanishing of certain functional determinants at points or rather along lines. Alongsuch a line in the physical plane, the limiting line or limit line (which inthese examples happened to be a circle), (see also Fig. 118c) the accelerationbecame infinite, the streamlines showed apparent cusps, etc. In the present312IV. P L A N E STEADY P O T E N T I A LFLOWarticle we shall study limiting lines, and their counterpart branch lines,in some detail, and again from the point of view of singularities of thehodograph transformation.30A physical solution assigns to a point Ρ in the flow plane one and onlyone q = q(r); in such a solution however, there is nothing to prevent thesame q from appearing at several points P.
If now instead of the flowQ = <l( )> which we want to determine, we obtain from one or other ofthe inverted linear hodograph equations a hodograph solution r = r(q),where r is a single-valued function of q, such a solution may not be everywhere equivalent to the desired flow q = q(r) since, roughly speaking, itmay give too much as well as too little: (a) a solution r = r(q) may wellassociate the same r with various q-values; (b) a solution q = q(r) inwhich the same q corresponds to different r-values cannot appear as asingle-valued function r = r(q).rA n illustration of the situation (a) is provided in the simple examples ofradial flow and spiral flow (Sec.
17.4). W e found that to each point in theflow field, except to points on the limit circle, there belonged two differentvelocities, which is physically impossible. W e had to distinguish betweentwo different solutions which meet at the limit line, i.e., which assume thesame q at every point of this line. W e saw in each case that the occurrenceof a limit line was connected with the vanishing of the JacobianD = d(<p,\f/)/d(q,e) or of an equivalent determinant; on the other hand, inSec.
17.3 we found D ^ 0 (D considered as a function of q and0) as the condition for deriving a solution q = q(r) from the hodograph solution r = r(q).If, however, d = D~ or an equivalent Jacobian is zero, then the transition from the physical plane to the hodograph plane is not possible. [Moregenerally, this implies that from equations (10.1), with right sides zero andthe coefficients dependent only on u and v, we cannot obtain equations(10.22).] W e found this situation in the case of a simple wave, where thesame q corresponds to infinitely many points r, and where the Jacobiand(q,6)/d(x,y) is zero in a two-dimensional region. A t any rate, the examplespoint to two types of singularities, corresponding to the remarks (a) and( b ) , each type connected with the vanishing of certain (essentially equivalent) Jacobians.A word about terminology may be added.
If in the mapping of an X,Yplane onto a £/,F-plane one calls M-lines the lines of the X , F - p l a n e alongwhich the Jacobian Δ = d(U,V)/d(X,Y)= 0 and ΑΓ-lines the lines of theX , F - p l a n e along which Δ becomes infinite, it is obvious, as a consequenceof [d(U,V)/d(X,Y)]-[d(X,Y)/d(U9V)] = 1, that the image of an M-line ofthe X , F - p l a n e is an iV-line of the C/,F-plane, and conversely. There aremathematically only t w o concepts. In our physical problem, however, thel31319.2 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 CASESphysical plane and the hodograph plane play very different roles, and fourdifferent names are in use. W e shall call a line in the flow plane along whichi = d(q e)/d(x,y) vanishes* a branch line of the flow plane, and a line inthe flow plane along which / = i~ = d(x, y)/d(q, θ) = 0 we shall call alimit line of the flow plane. If terms are needed for the hodograph imagesof these lines, we call the hodograph image of a limit line a critical curveand that of a branch line an edge?yx1A s has been done in previous articles, we shall denote by capitals theJacobians in which hodograph coordinates appear in the denominator, andby (corresponding) small letters those for which hodograph coordinatesappear in the numerator.
Hence, e.g.,d(q ,qv)xd(x,y)d(y,\p)='=d(x,y)Ddm.=d(x,y)aft,*)9'=d(q ,q )xyand, e.g.,jd(x y)d(qfi)=9=j12q tan a=J_pq 'D2Since d ( g , 0 ) / ( d ( £ , ? 7 ) = 2q tan a , the mapping of the supersonic region of thehodograph onto the ξ,^-plane is locally one to one, except when a = 90°or a = 0°, i.e., q = q or q = q . T h e determinants /,D and Ε areequivalent for supersonic flow and a ^ 0°, 90°. Both branch and limit linesare essentially supersonic phenomena. A brief consideration of the subsonic cases will be given in the next section before starting the main discussion.tm2. Some basic formulas.
Subsonic casesW e first collect some simple formulas which will be needed. W e have denoted the M a c h lines in the flow plane by C~ and C or by ξ-lines and q-lines,(Sec. 16.7). W e use this last term for both the M a c h lines in the #,2/-planeand the rectangular coordinate lines in the £,?7-plane. Denote by r the radiusvector of a point P , and by ξ,ή unit vectors which form angles —a and + a ,respectively, with the flow direction. Then θτ/3ξ and dr/θη will have theζ- and ^-directions respectively, and we may define functions /&ι(£,τ;) andfai^l) by the equations+(ί)I = hig = hn.* This Jacobian is named here as a representative of several, and in generalequivalent, Jacobians which we shall use according to the situation.314IV.
P L A N E S T E A D Y P O T E N T I A LFLOWHence, if dsi and ds denote* line elements of ξ-lines and ij-lines, respectively,2ξθ8ι'ds2andίο\(2ι)Λwith the notation φdsi,^ '=}Ηds ,2=ώ;'φ~ = θ — a we have= θ + α,+(3)ιdx,_— = Λι cos φ ,θχ,— = A cos φθνdy.+-f- = h sin 0 .στ;σξτ·J+2—= h sin φ ,Ί2Hence(4)J ==*Λωη2α.In the characterization of the singularities of the transformation it is preferable to use hi and h , and their reciprocals, rather than the Jacobian (4)since the vanishing of each of them has a distinct geometric meaning.W e note232o(qft)Also, with letter subscripts denoting partial differentiations,ybt = — pqhi sin a,^€ = qh\ cos a,^„ = pqh% sin a= qh cos a.2The hi, Λ are basic in the differential geometry of the net of characteristicsin the physical plane.Introducing, with a usual sign convention, f the radii of curvature R\,2* I n contrast t o A r t s .
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