R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 50
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Using the last of E q s . (9) and the relation dp/dq = —pq/a , wefind immediatelyφ = puy -vx -P2264IV. P L A N E STEADY P O T E N T I A LFLOW3ΨdqδΨ _δθ~d(pq)dqδΨd(pu)δΨd(pv)d(pu)δθd(pv)δθ_=χP<1Hence,δΨ(9.,Tr/,„ 2xδΨrg-,r<i-*\£--«x,and substituting from ( 4 " ) we obtain the desired equations:dqqdd3ΘdqElimination of Φ or Ψ in the usual way, yields the linear second-order equations:/(0x2)1,qθΦ , θΦΓ ^ Μ - ^ + ^+2#Φ^ = ° '22,,,νgd( 1 3 )Π Γ Μ*^22* , d+2θ*d Γ* ,+p ?a*Tq °=LP(I-rfgΊgM )]2Comparing the second-order equations for φ, ψ, Φ, and Ψ, we note that thosefor ψ and Φ, namely, Eqs.
(16.32') and (12), are simpler than the othertwo.2. Other linear differential equations(a) Equations for X and Ζ = qY. If Φ(#,0) is a solution of (12), then,since all coefficients in (12) depend only on q, any derivative δΦ/δθ, δ Φ/δθ ,22- - - will likewise satisfy (12). In particular, since δΦ/δθ = qY, the function(14)Ζ = qY = q(y cos θ — χ sin 0)satisfies the same equation as Φ:/1 r\(,5)QdZdZ22.dZT=Tpat d* aj ++q=0From Eqs. ( 4 " ) and (15) we obtain for X and Ζ the linear systemnUvrb)dZ _ d XdZ _6^ "δθd0 'tfdX2~ W = l~δζ "qX'Elimination of X leads back to (15), and elimination of Ζ to an equationfor X alone.
T h e equation (15) is as simple as Eqs. (12) or (16.320. Equations (16) compare with Eqs. (11) and (16.31).17.2 O T H E R L I N E A R D I F F E R E N T I A L265EQUATIONS(b) Equations with independent variables Q, θ or £, η. If we consider thevarious linear equations of second order (16.32'), (16.33'), (12), (13), and(15), and for the moment denote by F any of the dependent variables, wesee that, with respect to the second-order terms, all these equations are ofthe formtfE-_ M2dq1 d /*2q27ΘΘ " '2~2'This is to be expected since these terms determine the fixed characteristicsin the </,0-plane (real and distinct only where Μ > 1), which are the sameno matter which function of q we use to describe the flow.
From each ofthem we find again Eq. (16.35). T h e same result follows, of course, fromeach of the pairs of equations (11), (16), or (16.31), by means of the technique of A r t . 10.T h e various differential equations become simpler, if instead of q, thevariable Q, defined in Eq.
(16.41) for supersonic flow, is used; more generally, in the subsonic and supersonic cases, respectively, we define thenew variables λ and Q by(17)d= V ^Q~2dqq1d\dq1=Vl -M2qT h e second-order terms in the respective equations of second order are thensimply<?F _d F,2dQd^FΘΘ2θλ22d^FΘΘ '+2while the first-order equations (16.31) take the more symmetric form:(18)*=VM£\iH,dQ( 1 9 )^=ρΘΘ9_ Vl ~ΘΧρM* Wθθ'3^ΘΘνΜ^16φρ==Vl -ΘΘM>dQJΜθψΘΧ'2ρM{From Eqs. (18) and (19) follow equations of second order for φ and for ψ,as, e.g., Eq. (21.6) for ψ derived from (19).In the supersonic case we may also use the characteristic variables ξ,ηgiven by Eqs.
(16.43), (16.44). T h e equations (18) then take the form(20)θφθξ-\/M2ρ1 θψθξ 'θφθη\/M-2ρ1 θψθηThese equations have the nature of compatibility relations. T h e \^ariablesQf λ, ξ, η will be much used in Arts. 19 and 21. In the second-order equations266IV. P L A N ESTEADYPOTENTIALFLOWderived from Eq. (20) the second-order terms reduce to the mixed derivative with respect to £ and η, and if we deal similarly with the Legendretransforms we obtain, for example,θ'Φ(21)_ q + q" /θΦΘΦ\θξ θηwhere q' = dq/dQ = q tan α and q, q', q" are to be expressed in terms ofξ, 77. From a solution Φ(ξ,?7) of (21) we obtain with the notation (7)11(21')X- L ( ^ + ^ \_ ΐ / θ Φ _ θ Φ \7from which χ and y follow in terms of ξ, η.
An equation of the same simpleform as (21) holds for the stream function φ.*For some purposes it is advantageous (see Art. 20) to use, instead of thevariables q, 0, the variables g , 0. This will be considered when needed.2(c) Equations with independent variables σ and 0. W e mention one moreimportant transformation of the basic equations due also to Chaplygin.Let us introduce a new variable(22)σ=Γ^?, ^=-^.JqqdqqIt is immediately seen that Eqs. (16.31) becomeand the second-order equation (16.32) takes the formwhereΚ =(24')1-Z#p2is a complicated function of σ. (See Fig.
99.) It is seen from (22) that σdecreases with increasing q and that for q —> 0, σ —> <» as —log q\ also σ = 0for q = q and σ is negative for supersonic g-values, positive for subsonicq-values. The function K, depending on σ, tends towards unity as σ —• +<»,q —> 0; it equals zero for σ = 0, q = g , Μ = 1; and tends to — oo asq-^> q and as σ tends to its negative minimum value.
(The second derivative of σ as function of q, that is, the first derivative of — p/q equals(p/q)(l + M ), and is therefore always positive.)tfm2* Other linear equations with ζ, η as independent variables are E q s . (19.7).17.3T R A N S I T I O NF R O MH O D O G R A P HTOPHYSICALP L A N E267This transformation is used in two different contexts. On the one hand,an expansion of \/K = \ / l — Μ /p in powers of Μ shows that, for apolytropic fluid with κ = 1.4, this expression differs from unity only byterms of the order of M , i.e., y/K = 1 — 0.3 M · · · .* This suggests theapproximation Κ = 1, invented by Chaplygin and later elaborated in the v.Karman-Tsien method (see Sees.
5 and 6 ) . On the other hand, the simpleform of (24) with Κ | 0 for subsonic, sonic, supersonic flow is a convenient starting point for the study of transonic flow, the flow in the neighbor2AAKF I G . 99. Κ — (\ — Μ )/ρ2hood of Μ=2as function of σ = fVpdq/q.1 (see A r t . 25).3. Transition from the hodograph to the physical planeI n Sec. 16.5 we remarked that it would not be correct to say that by thehodograph transformation the original nonlinear problem has been linearized. This can be done only by having recourse to approximations.
W ehave merely split off one portion of the total problem which can be treatedby methods of linear analysis. Once a solution in the hodograph plane hasbeen found we still have to transfer it back to the physical plane.* W e obtain y/K= (1/Ρ·)(1 - 0 . 3 1 * · · · ) , but, as before, p. = 1.268IV. P L A N E S T E A D Y P O T E N T I A LFLOWSuppose that we know a solution yp{q,B) of E q .
(16.32') in some regionin the hodograph plane. Then the functions d\p/dq, θψ/θθ12and from (16.31) the functions θφ/dq, θφ/θθ.q dx= άφ,+ q dyxycan be computed,N e x t , we have+ pq dy = άψ—pq dxyxand from these, if pq ^ 0, dx and dy follow:2(25) dx = - cos θ άφ — - sin θ άψ ,2 LPJdy = - sin θ άφ + - cos θ άψ<ZLΡJIanddx _ cos 0 d<p(25')dq1 sin θ d-ψ~q~dq~~pqdx _ cos θ δφdq'ΘΘdy _ sin θ dφ , 1 COS θ d\pT~dtfdgq~ dddy _ sin θ dφP~q~dq'qθθ'1 COS 0 d ^ .<Z~d0dd1 sin θ d\pρρqd0 'the coordinatesin the physical plane may then be found as functionsof qfi by quadratures. If, in a formal simplification, 2 = χ + iy is usedwe obtain(25")dz= l (ed<(>+i ^ )>and if the derivatives of φ are expressed in terms of those of ψ b y (16.31),the result may be given in condensed form b yT h e streamlines in the £,?/-plane are the images of the curves yp(q,6) =constant = k in the hodograph.
T h e computation of the streamlines maybe arranged so that the necessary integration is simplified, since alonga streamline Eqs. (25) take the form(26)dx =G^ l l άφdy =qAlso dq =Dάφ,άψ= 0.qalong a streamline, and therefore with—[(d\l//de)/(d\f//dq)]de= d(^)/d(<7,0),άφsince q=q(6,k)functions x(6,k),=-Όάθ=A(qfi)άθ =along the streamline \p(q,d)y(6,k),=B(fi,k)k.Ifdd,B(6,k)9^ 0, thewhich define the streamlines, are then obtained17.3 T R A N S I T I O NFROM HODOGRAPHTO PHYSICALPLANE269by integrating(26')dx =cos 0q(B,k)B(d,k) dB,dy =sin 0B(e,k) dd.qifljk)T h e whole computation involves considerable work, and is based on the assumption that ψ^,θ) = k can be solved for q.If we know a solution Φ(μ,ν) or Φ(<?,0) of Eqs. (5) or (12), the determination of χ = x(qfi), y = yiqfi) by Eqs.
(4) in the first case, and byEqs. ( 4 " ) and (7) in the second case, is easier. However, in order to find thestream function yp{qfi), a quadrature is needed. W e may, e.g., find ΘΨ/dq,ΘΨ/ΘΘ from Eqs. (11), then Ψ by a quadrature, and ψ from the second Eq.(10); or we may use Eqs. (8) or (80 to find derivatives of ψ, and then determine ψ itself by a quadrature.A situation where Eqs. (15) and (16) are useful arises, e.g., in the case ofa Cauchy problem.
If along a curve X (noncharacteristic) we know qfi, thenwe know the image 3C' of X in the hodograph and x,y along JC'; therefore weknow X , F, and qY = Z, i.e., we have for the pair of linear equations (16)the Cauchy data Ζ, X along 3C' and this determines a solution within acharacteristic quadrangle. Then, from X and Ζ in terms of q and 0, weknow x(q,B) and y(q,B). A similar approach applies to the characteristicboundary-value problem.In this connection let us recall that the transition from the hodograph tothe physical plane was already considered in Sec.
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