R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 52
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Radial flow, (a) T w o radial flows and limit line, (b) Density and velocityversus distance in plane radial flow.27517.4 E X A C T H O D O G R A P H S O L U T I O N Sq=Tdenoted the velocity in the direction of increasing r.) T h e result±qwas: T h e curve q = q(r)has t w o branches, with no real point for r lessthan a certain value r=minr,twhere Μ=1. T h e upper branch of thelower curve in Fig.
101b represents the velocity q/qmin a purely supersonicflow (of the source or sink t y p e ) , with velocities ranging from sonic speedq at r = r to maximum velocity qttmat r =<». T h e lower branch representsthe velocity in a subsonic flow (of source or sink t y p e ) , with velocities goingfrom q = 0 at infinity to sonic speed q at r = r ; there the t w o flows meet.ttAs q increases from q = 0 to q = qt, the density decreases from its stagnation value ρ = p = 1 to its sonic value ρ = p , and as q goes from q = qt8toqt, the density decreases from p to zero.
T h e same conclusions are= qmtreached by direct consideration of the flow intensity pq in (34') [see Eq.(8.5) and Fig. 40]. Introducing sonic values, (34') can also be writtenpq 'rtA t the sonic circle, r =r,twhere the t w o flows meet, theJacobian0) vanishes since, from (33) and (27),d(<P, t)/d(q>Dβ{φ,φ)==Μd(qfi)Also the acceleration b becomes infiniteb = dq/dt = q(dq/ds)-2lk2pqthere. In fact,=±q(dq/dr);using (34') and the differentiation formula (8.5) for pq, we obtain(35)h=KM*-1)'which becomes infinite for Μ = 1. Across the limit circle r = r the flowstcannot be continued and in the vicinity of this line they must be regarded as physically impossible.
Our solution offers the simplest example ofthe fact that an extremely simple single-valued solution in the hodograph(0i^θ ^0)2may lead to different physical flows which meet along alimiting line.(c) Superposition.Spiral flow. Since the equations for φ, ψ, Φ, Ψ in thehodograph are linear, a linear combination of two solutions of each of theseequations is again a solution. N e x t , χ and y follow from Φ or Ψ by the rela^(#,0)Φ(#,0)tions (4) or ( 9 ) , and from φ, ψ by (25'). W e conclude that if \f/i(q,S) and2are t w o solutions of the stream-function equation, orare two solutions of the equation for Φ then ψ=Ci^i+Φι(<?,0)2ΟιΦχ + ο Φ are likewise solutions of the respective equations. T h e new22coordinates which correspond, e.g., to φ are given byandc ^ or Φ =22276IV. P L A N E S T E A D Y P O T E N T I A L FLOWwhere χι ,yi and x ,y22x(q,B)= CiXi(q,ff) +y(qf)= ayiiqJ)c x {qj),22+ 4# (?,0),2are the coordinates corresponding to φι andψ,2respectively.
Applying this principle to the two preceding solutions discussed in (a) and (b) we obtain (with C >0, k >0) by means of (29)and (34), the new solutionCχ =sin 0 +— cos 0 = - \Cqpqq\(37)+ - q ),ΡIyx2q7/ =COS(Ζ0 + — sin0 =P9< >°3 rCqx+rp2This last equation is the same as Eq. (7.40) and the discussion of thelatter given in Sec. 7.6 and illustrated in Fig. 31 applies.
T o see that theflows are the same we write Eqs. (37) in the form:χ = r cos (0 — 0),y = r sin (0 — 0),where r is given in (37') and/ccos 0 = — ,pqr. Csin 0 = — ,qrso that 0 is the angle that the streamlines make with the radius vector.Then with the usual meaning of q and qe, we findren• ήqeCsin 0 = — = —orήqrkcos0 = — = —qpqror",= C,Ίrqe.rq p = k.ri.e., the equations (7.36) and (7.39). Hence the flows are identical, and wemerely add a few remarks.T h e main result obtained in Sec. 7.6 was that there exist for each pair ofconstants C, k two different flows, both extending over the same regionfrom r = r to r — *>, with the same supersonic velocity ^ at r , one enfttirely supersonic, the other of the " m i x e d " type. T o compute qi we differentiate (37') and obtain( 3 8 )?+( ^(1_M2)P=0,°R^2= I +f"2>1'27717.4 E X A C T H O D O G R A P H S O L U T I O N Sfromthis MiDd(<p,\l/)/d(q,e).=from φιJacobianFor the stream function ψι of the vortex flow we findC6, by Eqs.
(16.31): θψι/θθ=Ave compute thethe value qi follows. N e x t ,=0, θψι/dq=Cp/q. Therefore,using (33), we obtain for the present flowand from (27)and this is zero for the value Μ= Μιof (38). W e also see that the valueof the acceleration(40)'^ = qA =dtdsdvq * A = > A lθφθθ Ddqd=(f kDtends to infinity at the limit circle r = r .zAlso from (37") and (38), Mi\/Mιcos θι =1 follows. Hence, from cos θι == sin a , we conclude that the M a c h lines of one of the two familieslare tangent to the limit circle (see Sec. 19.3), and that the streamlines ofeither flow meet the limit circle under the angle a .
Since the component oftthe velocity normal to a characteristic equals a, it follows that the radialvelocity q is sonic at the limit circle.rAll streamlines of the supersonic flow are congruent and so are those ofF i g . 102. Spiral flow: Typical streamlines and limit line.278IV. P L A N E STEADY P O T E N T I A LFLOWthe mixed flow. Figure 102 shows one streamline of each flow. These are tobe considered as two streamlines, each for a different flow, not as onestreamline with a cusp; the two flows should not be thought of as continuations of each other.
They appear simultaneously in our example, where asingle stream function \p(qfi) yields two different flows which meet atr = ri with the same q = qi. Let us add, however, the following remark.The reader should not be left with the impression that the appearance of alimiting line is an essentially mathematical phenomenon, introduced by theuse of the hodograph representation. He should rather remember that thesesame flows exhibiting the same singularities were obtained in Art.
7 bymeans of considerations exclusively in the physical plane.5. The Chaplygin-Karman-Tsienapproximation(a) Definitions. In this book, approximations have not been dealt with;we mean by an approximation a modification of the basic differentialequations with the purpose of simplifying the mathematics of a problemwithout changing too much its physical meaning. In this and the next section we shall take up a method which in relation to the basic equations ofsteady potential flow must be considered an approximation in the abovesense.
It is however, likewise possible to interpret the procedure as theexact theory of a gas with a particular specifying equation. This specifyingequation will turn out to be the linearized (p,p)-relation ρ = A — B/p,Β > 0 mentioned already in Eqs. (1.5c) and (2.17d). The method, while it isof an elementary character, is interesting from various points of view; oneis that it can be considered as a simplified model of the general planeproblem posed in Art. 16.W e start with the equations (19)(1 -d<p _d\Μ )*3ψΘφ _2ρθθ'βρin powers of M(1 2!M )* = ( l +*-L±2yields, with κ = y == 1 -i^tJIt is thus seen that (p./p) (1 -Μγ2'\1/C«-1)ΜΛ(1 -ΜΫ21.4••• = 1 -M*+Mθλ'1.
Expansion for polytropic gas of/J2θθwhere 1/p stands for ρ /ρ with ρ =β(1-Μ γθψ= (!//>)(! -0.3M +4Μγ2···.differs from unity17.5 T H E C H A P L Y G I N - K A R M A N - T S I E Nonly by terms of the order of M .KθφΘΧ}NextΚ=consider(1 -by unity, Eqs. (19) simplify t o2(41)Μ )/ρ22_θφΘΘ=3φθθ1this279If—pending further discussion—weA( l / p ) ( l — M )*approximateAPPROXIMATION=θψθλ'^inrelationapproximationtothevariablesand a = ft* pdq/q [see (22) and (24')] introduced andused by Chaplygin. Clearly Κ then becomes equal to unity and Eqs. (23)reduce to(41')^θφθσ}(hpθθ'=θφ _ΘΘθψθσ'which are the same equations as (41) except that — a takes the place ofX. Indeed the definitions (17) and (22) of λ and a, namely,(42)show that dX =d\Vl -dqq—da if \/lM29— M— M /pT o justify the replacement of y/lq /q :*22m1 -h r2Ρ16b y unity we may also, follow2Μ—1 =ρq'is put equal t o p.2ing Chaplygin, write Κ in terms of r =„Κdadq,,where h =22=Γ 2τ)(1 -2,Ηκ+1·-« -1Then,Tr" (1 - r)dKhr2h2+19and this shows that Κ (τ), which decreases with increasing r , varies so slowlyfor small r-values that it is practically constant and may be put equal tounity.16Finally, consider the expression of ρ in terms of M ,2namely,If the right side is to equal(1 — M )\The approximation= p, which we shall briefly-call the Chaplyginapproximation,\/l— M22is obtained by taking forrelation, the value κ =κ must equal — 1 .
W eκ, the exponent in the—I. In this case, dX =review:polytropic—da where X and a are givenby Eqs. (42), and the basic equations (16.31) become Cauchy-Riemannequa-* This variable will be widely used later, particularly in A r t s . 20 and 21. Formulasfor Μ and ρ in terms of τ are given in E q . (20.3')·IV. P L A N E280STEADYPOTENTIALFLOWHons (41) or (41') in the Cartesian coordinates (λ, — 0), or (σ,0). Consequently φ + ιψ is an analytic function of λ — τθ (or σ + ιθ), and the Laplaceequation holds for both φ and ψ.17T h e assumption κ = — 1 can be interpreted to mean that we consider thepressure-density relation ρ = — Β/ρ + A, Β > 0 (note that a constant canbe added to any polytropic relation) as an approximation to the usualrelation ρ = Cp , κ = 1.4.
T h e choice of A and Β is still arbitrary, and it isin the choice of these constants that von Karman and Tsien deviate fromChaplygin. Chaplygin chose A and Β so as to make the line ρ =—B/p-\-A(Fig. 103) tangent to the usual isentrope of a gas at a stagnation point,KΡF I G . 103. Tangent approximations to isentrope.i.e., he substituted for the usual (p,p)-relation, the relation ρ — ps— κρρ (1/ρ8=— l / p ) .
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