R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 55
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105), by φ7 the angle from thex-axisto the" i n i t i a l " C~ (i.e., the C~ which corresponds to the sonic point of Γο"), andby 2ξ the given constant which singles out the particular vt,as in E q .(2) or in Eq. (16.43). Then, under the assumption of a polytropic gas, wehave the following relations.In a forward wave (Fig. 105)(3)Q -θ=a+ ha -Θ = 2f,θ -a= φ ,and consequently, using the first line of Table I I I (Sec. 16.6), or Fig. 105,2£ = 90° -6=t-φ7 ,Φ~ Φ(4)Q +0=a + A<r+0= 2v,Φ7 = ha,ha — 2ξ.θ + a =φ,+and consequently2η = 90° + St = φ ί ,φ ! = —ha,Q = 90° -2η — ha.(0 -0,),initial direction^initial C "F I G . 105.
Forward wave in physical plane and hodograph plane.18.2 S T R E A M L I N E S A N D C R O S S M A C HLINES291FIG. 106. Backward wave in physical plane and hodograph plane.T h e last equation (4') [and similarly (3')]angle of inclination φ+of any straight C+gives the relation between theand the velocity q along it, sinceb y E q . (16.39'), σ depends only on q.* Another useful relation isθ-(5)6 = ±(hat-a'),where a' = 90° — a is the angle, introduced in Sec. 16.6, between the direction of a Γ-characteristic at a point P' and the radius vector O'P'.2.
Numerical data. Streamlines and cross Mach linesW e now take κ = y = 1.4 in a polytropic (p,p)-relation. Consider theparticular forward wave with 2£ = 90° which corresponds to the Γο~:θ = a + ha — 90°. Remembering that a = 0° and a = 90°, we seethat Q = 0°, and that φ~ = θ - a = ha - 90°.tttLikewise, for the T~-wave with 2η = 90°, or θ = 90° — a — ha, wehave 6 = 0°, φ = θ + a = 90° - ha.+tWhile the velocity varies from q to q and the M a c h number Μ from 1to oo, the M a c h angle a varies from 90° to 0° and a from 0° to 90°. Takingthe above waves, both with d = 0°, as representative, we see from the formulas that, for the forward wave, φ~ turns from —90° to (h — 1) X 90° =130.45°, i.e., through 220.45°, and θ from 0° to 130.45°; for the backwardwave, φ turns from 90° to - (A - 1) X 90° = - 1 3 0 .
4 5 ° , i.e., through- 2 2 0 . 4 5 ° , and θ from 0° to - 1 3 0 . 4 5 ° . These facts are collected in Table I V .tmt+* v. Mises, [26], uses λ = 90° ± φ rather than φ, the angle λ being shown in ourfigures; in terms of λ the relation corresponding t o the second equation in each ofthe sets (30 and (40 is thenha =bwith our previous sign convention.e = λ,t292IV.
P L A N ESTEADYPOTENTIALFLOWTABLE IVT H E VARIATION OF SIGNIFICANT QUANTITIES IN FORWARDAND BACKWARD SIMPLE W A V E SForwardΓ : θ = a +0+ha -Backward WaveWave90°, φ~ = θ -θ =a-a.-ha + 90°, φ+= θ +αqQΜm1->0°90°0°0°90°130.45°130.45°0-90°0°0090°90°220.45°+90°--130.45°0°--130.45°F I G .
107. Forward and backward simple waves with 0 = 0.tFigure 107 shows in one and the same figure the hodographs of the ΓΪ"and the Γο~, with 6 = 0° for both, as well as the pairs of straight characteristics in the physical plane, Ct, Ct, and CJ ,. T h e backward waveΓ^" extends from the dashed line Cf (which corresponds to the point P ),turning clockwise by 220.45° towards the dashed line Ct, ; the completeforward wave begins at the solid line CT (likewise corresponding to P )and turns in the counterclockwise sense b y 220.45° towards.tttIn Fig.
108 the deflection angle θ versus Mach number M , and likewiseθ versus a, is plotted.For any wave, with arbitrary 6 , the first six lines of Table I V remain thesame, while in the seventh line we must then write as entry φ — φ =F 90°tt18.2 N U M E R I C A L293DATAI20rF I G . 108. Deflection angle θ versus Mach number Μ and versus Mach angle a.instead of φ, and in the last line 0 — 0* instead of 0; in (3') and (4') we stillhave φ\ = θι + 90° = 2η and φΤ = B - 90° = - 2 £ , respectively.tFor supersonic flow and κ = y = 1.4, T a b l e V gives a tabulation of various quantities which characterize a simple w a v e : 0 — 6 , Μ, α, φ — φ ,ρ/ρί and Q.
T h e relations between the various angles are given by Eqs.(3) and ( 4 ) . Of course, we could add to the table corresponding values ofp/pt, T/T ,etc. I n T a b l e I (Sec. 8.4) we tabulated p/p , p/p and T/TandttιsV/Vt = (p/p*)/0.5283,(p/pi) = (p/p )/0.6339,T/T5t=ss(7 /7 )/0.8333.TTsW e have seen in the preceding section that the velocity distribution overthe set of straight M a c h lines is determined if we know the epicycloid inthe hodograph which is the image of the simple wave (that is, its " label''2£ or 2η) and the inclination φ of each straight M a c h line. Consider, forexample, 2η = 120°, i.e., the backward wave Q + θ = 2η = 120°. W e wantthe velocity vector q along the straight M a c h line of slope, say, tan 28°.Here φ« = 120°, B = 120° - 90° = 30°, and the range of φ-values in thephysical plane is from 120° to —100.45°; hence φ = 28° is in this range.Corresponding to | φ — φι \ = 92° we find in the table the values Μ =2.132, θι - θ = 30° (for in a backward wave 0 ^ θ), and a = 27.97°.Hence, since 0, = 30°, 0 = 0°, and for Μ = 2.132 we find, e.g., fromT a b l e I , q/q = 0.6897; thus the velocity vector along that M a c h lineis determined.ttmIV.
P L A N E S T E A D Y P O T E N T I A L294TABLEFLOWVVARIOUS (QUANTITIES WHICH CHARACTERIZE A SIMPLE W A V E (WHEN κ =\e - e |tΜvlvta- Φι I(deg)IΦ(deg)(deg)1.4)Q(deg)01.0001.0009009011.0820.90767.5723.439121.1330.85162.0030.009231.1770.80558.1834.729341.2180.76255.2038.809451.2570.72352.7442.2695101.4350.56644.1855.82100151.6050.44238.5566.45105201.7750.34234.2975.71110251.9500.26130.8584.15115302.1340.19727.9592.05120352.3290.14525.4399.57125402.5380.10423.21106.79130452.7650.07421.21113.79135503.0130.05119.39120.61140553.2870.03417.71127.29145603.5940.02216.15133.85150653.9410.01314.70140.30155704.3390.00813.32146.68160754.8010.00512.02152.98165805.3480.00210.78159.22170856.0070.0019.58165.42175906.8190.0018.43171.57180957.8510.0007.32177.681851009.2100.0006.23183.7719010511.0910.0005.17OO0.000130.450189.83195220.45220.45N e x t let us consider streamlines and cross-characteristics. If we want tofind the equation of either family we have t o give the particular family oflines which form the straight characteristics.
L e t it be given in the form(6)y = βχ + yoifi).So far, such information was not needed since the relation between the slopeof a straight characteristic and the flow variables along it depends only onthe label 2£ or 2η.L e t (6) define C^-lines β — tan (0 + a); since Q(q) + Θ = 2η, with ηgiven, a and θ are known as functions of q, and hence of β. T h e differentialequations of streamlines and cross-characteristics are§= tan Θ =k(0)and^= tan (0 -a) =18.2 S T R E A M L I N E SA N DCROSSM A C HLINES295respectively.
B y differentiation of (6)dy = χ άβ + β dx +y (fi)dfi0results, and substituting this in the equations of the streamlines and crosscharacteristics, one obtains(7)—KάβU=X+'°(®Κβ)-—=andyβanaXάβ+'°WyΗβ)-β1respectively.Each of these is a linear differential equation of first order for χ = χ (β),which upon integration provides, together with ( 6 ) , a parametric representation of the respective family of curves.
T h e constant of integration determines the particular streamline or cross-characreristic. In case of a wavecentered at the origin, y (fi)Q= 0 in ( 6 ) .I t seems more practical, however, t o give the family of straight M a c hlines in a way better adapted to the'problem, and to use a kind of generalized polar coordinates. I t is easily seen that (in addition to the knowledge of2η or 2ξ) the family of straight M a c h lines is determined if merely onestreamline or one cross-characteristic is known in the a?,2/-plane. In fact,knowing one streamline in the flow plane, we know θ at each point (x,y) ofthis line; then, in the case of a backward wave, Q — 2η — θ determines qand a(q).θ +Therefore, at each point of the given streamline, the directiona of the C+through this point is known.
T h e set can be determinedanalogously from the knowledge of one cross-characteristic, C~; now Θ — ais known along this C~, and Q +and finally θ +a =2η — (θ— a) provides q. T h e n a,a follow at all points of the C~.Consider now the first case where one streamline is given: χ =y =a(t),b(t), db/da = tan θ, where t is a parameter (see Fig. 109). ConsiderFIG.
109. A property of the streamlines in a simple wave.200IV. P L A N E S T E A D Y P O T E N T I A LFLOWagain a backward wave. T h e straight characteristics are then the C . For+apointP(x,?/) on t h e C t h r o u g h P , χ = a + r cos φ++0and?/ = b + r sin φ ,+where r = P P . If we consider a, 6, r, and φ"*" = φ as functions ofthe0equation of the streamline through Ρ is(db + r cos φ άφ + dr sin φ) cos 0— (da — r sin φ άφ +and, since db cos 0 — da sin 0 =cos φ ) sin 0 = 0,0, it follows, with φ — 0 = a, thatr άφ cos α + dr sin α = 0, or(8)L ^dr=_tana.W e note that this is the same equation which can be written immediatelyfor a centered wave, using ordinary polar coordinates (/·,φ).
T o integrate(8) we use Eq. (16.390: tan a =Hence,(9)(1/Ji) cot σ == h tan ( — - — Π d0,(1/A) cot[(2iy -r = r (cos0^—jφ)/Λ]..I t is seen that for φ = 2η = φι ,r = r , while for the other extreme, | φ — φ* |tending to Λ X 90° ^ 220°, the distance between the streamlines tends toward infinity.0In the same way, we find for the cross-characteristics the differentialequation(10)Iff* = _drtan 2a,with the integralHere r tends towards infinity when φ — φ* —> 0 as well as when φ — φ —>- 220.45° (see Fig. 110a, b ) .
*From Eqs. (9) and (11) the following property of both the set of allstreamlines and of all cross-characteristics follows: If the constant r ischanged to 2 r , 3 r , · · · , the respective curves intersect any straight characteristic at equidistant points (see Fig. 109 and the end of Sec. 13.1).ι000In adapting the preceding discussion to a forward wave, we must change2η to - 2 { , etc.* We note that, even if we take an extremely small value for r , we can only sketcha small part of the complete streamline since r increases rapidly with | φ — φ | .0ι18.2 S T R E A M L I N E S A N D C R O S S M A C HLINES297F I G . 110.
(a) Streamline in a centered simple wave, (b) Cross-characteristic in acentered simple wave.298IV. P L A N E STEADY P O T E N T I A LFLOWX0χF I G . 111. Flow around a convex corner.3. Examples of simple waves(a) T h e most important example of a simple wave solution is the flowaround a convex corner (see Fig. 111). W e assume that the oncoming supersonic flow for which the straight wall XA is a streamline is uniform withgiven velocity q i , where 0i equals the angle of XA with the x-axis.
T h e angle0 of the velocity q is known along the streamline A Y. [We note that theseare not complete Cauchy data as discussed in Chapters I I and I I I and inSec. 16.4, since only 0 is known along AY.] T h e flow is completely determined, as a uniform flow, to the left of the characteristic CT through A,and this flow can change to a simple wave across a characteristic. T h e onlycondition beyond A is that AY be Ά streamline.
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