R. von Mises - Mathematical theory of compressible fluid flow, страница 16

1.2), which are the lines in the x,<-planedefined by dx/dt = q and which trace the history of each particle; however,the pressure distribution along these lines does depend on c . For thijs ex­ample the particle lines are given by22x2x2χ =Cll ± i *κ — 1+At " \m+1where A is the parameter; this may be verified by differentiation.

Alongeach particle line, a is £ **times a constant that is a simple func­tion of c and A. N o w ρ = constant·ρ"> so that a = dp/dp = constantp * . Thus, along each particle line, ρ is proportional t o f~and ρ t o£-2κ/(«+ΐ). ^t t of proportionality are again functions of C2 and A.- 2 (_ 1 ) / (+ 1 )2-12KK+1)ec o n sa ns1(29)(b)« -ra5A—*,,GX = j + ct -',1a2γ= (κ -—C,3—2K-2)CR"[(« -1 ) * + ct ~'\.27.5STEADY PLANE MOTIONχctκ — IF I G . 30. Particle lines — hequidistant values of d =t~lK79— constant = k, f o r e = —0.3, κ = 1.4, fork ~.Klul)Here the particle lines are given by(30)j +~~ τ t ~ = constant = fc;ικ — 1lKsome of these curves are shown in Fig. 30, with c = — 0.3 and κ — 1.4.From Eqs.

(29) and (30) we have a = C/C(K — 1)(κ — 2)t ~ along anyparticle line, so that ρ is here proportional to 1/t. For constant t and vari­able k, a is a multiple of /c, and ρ is proportional toAlso when 2is constant, dx = tdk from Eq. (30), so that on computing the fluid massincluded between two positions at a given time, we obtainlΓpdx= constant · Γfc1/u_1)dk = constant · [k~KKK2l)K-ki ~ ].l{KT h e particle lines in Fig. 30 are drawn for equidistant values of k* ~l{Kl)l)= d,namely for d = 0, 2, 4, 6, 8.

Hence, the mass between any two successivelines is the same.5. Steady plane motionSince a substantial part of this book will be devoted to problems ofsteady plane potential flow, only a preliminary discussion will be given here.B y hypothesis q = 0 and ΘΦ/dt = constant, so that Eq. (14) reduces to2W i t h the polytropic (p, p)-relation, the formula for a again takes the form80II.

GENERALTHEOREMS(17), where a = (1 — κ)ΘΦ/ΘΙ is the square of the stagnation value of thesound velocity.I t is often advantageous to use polar coordinates r, 0; then q is decom­posed into a radial component q and a circumferential component qe. Takingthe x-axis along a radius, we have at any point on this radius2rΘΦ ___ ΘΦΘΦ _dx"(32)^ =^ ==dxdxdr dr '2^dy22dxdr_1 ΘΦ-yqr dr dd- r d 6 'ed^ ldj .qdy r ΘΘ==11 θΦ _2ydx dyq2_ dq _ dqe _ 1 <3ΦΘΦ_Y y -1l^d% .\3Φr ΘΘr dr=r2Θ Φ _ dq _2r ddxdy dx22dqqerdy1r ddr'T o derive the last of these formulas, a figure generalizing Fig. 28 can beused.

T h a t is, we note that the ^-derivative of the ^/-component of q isq /r, the same as in the discussion in Sec. 3. Here, however, we must alsotake account of the ^/-component of qe, whose ^/-derivative is (1/r) dqe/dd.Equations (32) are the usual formulas for the derivatives of Φ in polar co­ordinates, written for θ = 0.Equation (31) can now be written asrr2(33)(ldr \_ <?Λ _a2 J2qrqe(1d-%1a \r dr ddθ Φ >\r dd /22+ \r dd22^ r drj\a J2'or in terms of velocity components asfoA\(34)Qr , dq .

q1 Γ2 dq(dqe , dq \<2 dq^-f- + -|-• + i - = - \q -f- + q q I -f - + - | - 1 + qedrr dd ra |_ dr\drr dd/r ddd9rr2rrreThe latter equation is actually the polar form ofdq>dx 0 - & " ) - ~ ( £+£ )+£ θ - 3 θ -βrather than of E q . (31). Thus, if E q . (34) is used, the condition (1) for theexistence of a potential function must be added; for plane motion E q . (1)reduces to<*Sl - OS? = 0dx dyin rectangular coordinates, or/or\dqedq . q _reΛSTEADY PLANE7.5MOTION81in polar coordinates, as is seen from (32).

Equations (34) and (35) combinedare equivalent to Eq. (33).T h e case of radial motion, qe = 0 and q a function of r only, has alreadybeen discussed at the end of Sec. 3. N o w we consider a more general caseof what may be called an axially symmetric flow : q and qe are independentof 0, but q does not vanish. T h e condition (35) then readsr19re(36)^ + ?° = 0,drrorrq=econstant.N o w 2wrqe is the value of the circulation Γ on the circle of radius r aboutthe origin, so we have here an illustration of the case mentioned previously(Sec. 6.1), where Γ is constant (but different from zero) on circuits surround­ing an infinite cylindrical obstacle. T h e "obstacle" here is a certain circularcylinder, r = ri (see Sec.

6 ) . T h e differential equations apply only to theregion outside the corresponding circle with center at 0 . In this (doublyconnected) region a regular potential flow with 2rrq= constant = Γexists. I t will be seen presently that the immediate neighborhood of r = 0is without interest for us.BOnce qe has been found, Eq. (34) can serve to determine q . This equa­tion, with the ^-derivatives omitted, reads after multiplication by rr20Here q is written for q + q .

N o w , considering a general elastic fluid,we set dH/dr = d(% q + P)/dr = 0 (see end of Sec. 1), or22re2+^ = 0 ,ρ drand see that the right member of Eq. (37) equals*1 dp- .ρ dr-rqrTThus, after dividing by the factor rq , we can integrate both sides to obtainT(39)log (rq )r=— log ρ + constant,rq=rC Q n s t , a n tΡThis, in conjunction with (36), solves the problem. For if we call the twoconstants in Eqs. (36) and (39) C( = Γ/2π) and k respectively, the equa­tions squared and added show that/ η\Λ(40), k2 2rq=C+2or-r=C—k+.r—- .9Pq(pq)"* This result also follows directly from the equation of continuity (3).2282II. GENERALTHEOREMST h e Bernoulli equation in its integrated form establishes, as we know, arelation between q and p.

If this relation is used to eliminate ρ from(40) the latter equation links r to q , and since qe is already known, as afunction of r, we have finally a relation between q and r. A n examinationof these relationships reveals two important phenomena, which will bediscussed in the following section.

(See also end of Sec. 17.4.)222226. Transition between subsonic and supersonic flow. Limit lineT h e relation between ρ and q established by the Bernoulli equation willbe discussed in some detail in A r t . 8. Here it will suffice to know that ρ de­creases monotonically when q increases, while the product pq first increasesfrom zero at q = 0, reaches a maximum at the sonic point q = α, Μ = 1,and then decreases towards zero as q increases through supersonic values.T h e function k /(pq) is plotted against q in Fig. 31; the abscissa OE ofthe minimum point is, as just stated, q = a . T h e graph shows also thehyperbola with the ordinates C /q .

T h e ordinates of the heavily drawncurve in Fig. 31 are the sums of the ordinates of the other two curves, andthese sums must equal r according to E q . ( 4 0 ) . Since one curve has aminimum at the sonic point and then increases without bound, while theother decreases monotonically, it is apparent that whatever the (positive)constants C and k are, the resultant curve must have a minimum F withan ordinate different from zero and an abscissa OFq > OE , that is, lyingin the supersonic region. Denote this minimum ordinate FoF by r .2222202222202If some value r = OA is given, the graph shows that two different values2subsonicψ | supersonicF I G . 31.

Spiral flow obtained by addition of vortex flow and radial flow.7.6T R A N S I T I O NFLOW.L I M I TL I N E83of q correspond to it when r > r ] if r < r no q can be found that willsatisfy Eq. (40). This means: There exist for each pair of constants C and ktwo different axially symmetric potential flows, both extending over theregion from r = ri to r = oo, with the same velocity at r = n . One flow,corresponding to the branch of the curve to the right of F, is entirely super­sonic, while the other one includes subsonic as well as supersonic velocities.A t the circle r = η the flow ends; it has here a "natural l i m i t " .

Somethingsimilar was found in Sec. 3 in the discussion of radial flow. But in the radialflow, the limit line coincided with the line on which Μ = 1 and thuscould have been attributed to the fact that the sound velocity had beenreached. W e now learn that the natural flow limit has nothing to do withthe border line between a subsonic and a supersonic region.

On the contrary,the present example shows that a " m i x e d " potential flow is possiblewithout any singularity or irregularity occurring at the border between theregions where Μ < 1 and Μ > 1.222222I n Figs. 32 and 33 are indicated the streamlines corresponding to the twosolutions determined by Fig. 31. T o find these streamlines, one has onlyF I G .

32. Channel in mixed spiral flow.F I G . 33. Channel in supersonic spiral flow.84II. GENERALTHEOREMSΟF I G . 34. Showing angle between velocity and radius vector in spiral flow.to take from Fig. 31, for various positions of the point A on the verticalaxis, the magnitudes AB = q and BC or BD for q .

Then the ratio qe'-qigives the slope of the streamline, namely, tan 0, with respect to the radiusvector (see Fig. 34). Graphical or numerical integration supplies the stream­lines.22erFigure 32 refers to the mixed flow, with the smaller values of q . In thiscase the velocity is zero at infinity, it increases monotonically up to thesound velocity at the sonic circle, and then to supersonic values in theannular region between the sonic and the limit circles. T h e limit at r = <*>of the slope of the streamline with respect to the radius vector is, from Eqs.2(36) and (39)lim » = lim l = lim ^τ·-» qq-*oqq^okqr=r=kconstant.In this case, then, the streamlines approach logarithmic spirals as r —•> o o .If the two heavily drawn curves in Fig.

32 represent the walls of a channel,the flow inside the channel can follow the pattern shown in the sketch upto the limit circle.Figure 33 shows the streamlines in the case of the completely supersonicmotion. Here the velocity at the limit line is the same as in the case ofFig. 32. But now the velocity increases as we go outward. Since q% tends tozero with increasing r, the flow becomes more and more radial. T h e direc­tion of the flow can be inward, as indicated in the figures, or outward inboth cases.7. Other particular cases of the general potential equationIn the case of steady motion the general equation (14) becomes(41)QxQyα2dΦdx dy_qyq,da2Φdy dzqqd%a dz dxzx20O T H E R C A S E S OF T H E P O T E N T I A L E Q U A T I O N7.785where from Eq.

(16)— 1/ 2 ,— — (g +K22a= a-sx2 ,ft2\+ft).This same equation in cylindrical coordinates r, 0, and -ε, with q = ΘΦ/dr,rqe=(42)(1/r) ΘΦ/ΘΘ takes the form^2 ·,· /2\Vα /dr22_~2^ /2\-2^ /2\r <90 \α /<^ \a /,22ftft2ra22 ~ ^~ - ^ - 2ra <90 dzdr d0222~ ~M r / "f - + £ ( ira dz drr \22+2= 0.a /2N o t e that from (42) for a —> °o we obtain the polar form of the Laplaceequation, namely,d%1 θΦ3Φ1 3Φ _drr d0d-εr dr222222QFor ft = 0 in Eq. (42) we find Eq.

(33) again. Another case with only twoindependent variables is that of an axialsymmetry where Φ depends onlyon ζ and r and not on 0, and Eq. (42) reduces to^ * Λ - ^ ) ^ ( ΐ - ^ ) - 2 Μ « ^ | + Β Γ» 0 .·+dr \a)dz \a /a dr dzrIf here ζ and r are replaced by χ and y respectively, so that the x-axis is the1121axis of revolution, this equation differs from Eq. (31) only by the termQv/y- One may introduce the equationp ( -«()+g»(i-?{)-2 *g(43)9t 1dx\2a )2dy\2r+ '- -0qea)a22dx dyywhere ν = 0 for the case of plane flow, ν = 1 for that of rotational sym­metry.N e x t we consider nonsteady motions.