R. von Mises - Mathematical theory of compressible fluid flow, страница 15

(12) in rectangular coordinates, we usef Q*\d(< >l*ιdQ*ιd(dQ*\\132dΦdx ',dxdy2mydΦdΦ*dxdz*'mand corresponding formulas for q and q . Applying Eqs. (11) and (13),writing out the terms of ΔΦ as in ( 7 ) , and omitting the gravity term fromEq. (12) we getyθ ΦΛdx V22q \α2/, <3 ΦΛθ*/ VαΛ22xτg, d% (W22z\α /2θχ d#- 2 ^ - ^ αθί2α2_ 2α1 d%q\Λθζaα2φdi/ dz- 2 ^ - ^ α di/ di2_2θί2 Μία22Φdz dxθ_ 2 ? i ^ = 0α dz θί2This is the general potential equation for compressible fluid flow, as com­pared to ΔΦ = 0 in the incompressible case. A s was mentioned above, ais a function of the first derivatives of Φ: by way of the (p,p)-relation, acan be expressed in terms of ρ or ρ or P, and according to Eq.

(6), whengravity is neglected,227.3STEADY RADIAL73FLOWIf the flow is steady and the velocity is sufficiently small, so that all seconddegree terms in velocity components may be neglected, the potential equa­tion again reduces to the classical one.

Without this approximation how­ever, Eq. (14) is nonlinear, and therefore the sum of two solutions need notsatisfy the equation: solutions cannot be superposed as in the case of lineardifferential equations.In the case of the polytropic (p,p)-relation p/p = (P/PO)", Eq. (8) givesa = p/p, while from Eq. (2.22c), Ρ = κρ/(κ - l ) p ; t h u s a = (κ - 1)P,so that Eq. (15) gives022K^ + i (grad Φ )σιΖ2W e may also verify from Eq.

(2.22d) that the same result holds for κ — — 1,corresponding to the linearized (p,p)-relation (see Sees. 1.4 and 2.5). For iso­thermal flow, p/p = constant, we have a = po/po = constant. In anyother case, the relation between a and Ρ is computed by means of the(p,p)-relation, and then Eq. (15) is used to eliminate P.T h e nonlinear partial differential equation (14), combined with (16)[or some other formula for a ] and with q = grad Φ, gives the condition thata function $(x,y,z,f) be the potential of a physically possible irrotationalflow of an inviscid elastic fluid. Very few examples are known of solutionsin terms of elementary functions. In the remainder of this article we con­sider some particular cases of Eq.

(14).2223 . Steady radial flow17T h e simplest example, other than uniform flow, which needs no furtherexplanation, is that of a steady motion along rays emanating from a fixedpoint 0. Since q is always directed along the radius from 0 , the potentialsurfaces must be concentric spheres with center 0 ; thus Φ is a function oft and r = (x + y + ζ ) only. If we denote by q the velocity in the direc­tion of increasing r, then22 ΛQrrdΦΙandτ=dr2q=rq2are functions of r only. In Sec. 1, it was seen that steady irrotational mo­tion requires that dΦ/dt be an absolute constant; then (16) yields the Bernoulliequation(17)a + ~~2~ l2K(=c o n s^a n^=αΛwhere a is the value of a corresponding to q = 0, the so-called stagnationvalue of the sound velocity.

( I t is assumed in this section that a polytropic(p,p)-relation holds.)s74II. GENERALTHEOREMSTaking the x-axis along the ray from Ο to an arbitrary point P , wehave q= q = 0, qyz= q at P , and since all derivatives of ΘΦ/dt vanish,xrEq. (14) reduces to(18)0-i)dx2^dy ^ dz22In the first term, θ Φ/θχ = dq /dx = dq /dr. T h e second term is dq /dy,22xryand Fig. 28 shows that dq /dy = g /r. T h e same is true of dqjdz,ryand Eq.(18) may be written as(19)^RFI_dr \^+?R2a J=o.r2When the value of a is substituted from Eq.

(17), the final differential2equation for q as a function of r readsrΛ ~α\dr(19')0 ~mΛ.22a2a8ι1\..2— (* + l)q— (κ — l ) grs2r_|_22— = 0rIn this equation the variables can be separated and the integration carriedout directly. I t is convenient, however, to make a change of variables fromr and q to the dimensionless quantities ξ and η defined byrand» -qrwhere r is an arbitrary constant. In terms of the new variables, Eq.

(19') is0άη _(19")ηI2 — (κ —2 -(κ +Ι)η2,2l)vJthe solution of which isl/U-D(20)±as can be verified by differentiation. A n y constant factor could be insertedin one member and Eq. (20) would still satisfy Eq. ( 1 9 " ) , but this is un-F I G . 28. Auxiliary relation for radial flow.7.3STEADYRADIAL75FLOW— r-1) upper1oteasympl\SPf01«*S/6/._e2M-»034iF I G . 29. Dimensionless velocity η — — versus dimensionless distance £ in steadya,rrradial flow where £ = — or = — for spatial or plane flow respectively.2necessary since an arbitrary (positive) factor r has already been includedin the definition of £.Before discussing this result let us find the relation between η and thelocal Mach number q/a; from E q .

(17) it follows that0(21)11M2In the solution, (20) for κ > 1, two values of η correspond to ξ = O O :η = 0 and η = [2/(κ — 1)]*, this last value being y/E for κ = 1.4. I t fol­lows from (21) that the corresponding values of Μ are 0 and O O . Figure29 (solid line) shows η as a function of ξ [using the positive sign in (20)]when κ = 1.4. There are two horizontal asymptotes: η = 0 and η =\ / 5 . T h e point (£ι, 771) corresponding to the minimum value of £ may befound by differentiation of £ as a function of η, giving(κ+1)/(«-!)When the value ηι is substituted in (21), the corresponding M a c h numberis seen to be Μ = 1.If this solution is considered for all rays in a certain solid angle, we havethe result: In a conically divergent channel two radial flows are possible, onesubsonic with velocity zero at 0 0 and the other supersonic with infinite Μ at976II.

GENERALTHEOREMSinfinity. This statement has been derived for η ^ 0, corresponding to an out­ward flow; if the minus sign is used in the solution (20), the graph isexactly the reflection of Fig. 29 in the £-axis, and the same statement istrue for an inward flow, where η ^ 0.T h e mass flux through unit solid angle in this cone is given by Q =pqr = α το ρξη. Since mass is conserved, Q must be the same for all r, andtherefore the same for all points on the £,*?-curve; thus Q may be computedon the lower branch of the curve, with ξ tending toand η to 0. FromEq. (20) it follows that £ η —•» 1; the limit of ρ may be called p , the stag­nation value, since it corresponds to η = q = 0.

Then2828(22)Q = α τ ρξη8= ar20820lim ρξη = a p r882.When Q is given, (22) determines r .0T h e fact that the flow does not extend to r = 0 is not surprising; this istrue even in the case of an incompressible fluid. T h e broken line in Fig. 29gives the velocity distribution in the case of an incompressible fluid for theflow with the same stagnation values p , p , and the same flux Q. Hereρ = p , so that Q = p qr = ρ α τ ζη. Compared with (22), this impliesthat the equation of the curve is £ η = 1. T h e left-hand endpoint is deter­mined by the condition that ρ may not become negative.

T h e Bernoulliequation (2.20), taken for an incompressible fluid and with gravity omit­ted, reads882s88s20from which it can be concluded that ρ goes through zero when q = 2p /pand η = q /a = 2p /p a .Since a is related to p , p by a =κρ /ρ ,the value of η at the critical point for the incompressible flow is, therefore,2/κ and, from ξη = 1, the value of ξ is κ/2. A s Μ decreases, the curve forthe compressible case is more and more nearly the same as the curve forincompressible flow.From (22) we obtain ρξη = p , which with E q .

(20) gives2222ss822228(23)and also, because p/p = constant,K(23')88288887.4NONSTEADY PARALLEL77FLOWAlternatively, E q . (23) could be derived from the Bernoulli equation inthe form1*+2V*=Viκ — 1p 'κ — l ρstogether with the (p,p)-relation. T h e main result (20) would then followfrom the continuity condition (22), with no reference to the potential.If a plane radial flow™ where r = (x + y )\ is studied, the only changeis that d%/dz is to be omitted in ( 1 8 ) ; then the factor 2 is missing fromthe second term in Eq. (19).

However, if £ is taken to be r/r , rather thanr /ro , the same differential equation ( 1 9 " ) results. Thus all conclusions,including Fig. 29, hold as before, provided £ is given the new interpreta­tion.2220224. Nonsteady parallel flowIf all particles move parallel to the x-axis, the equipotential surfaces areplanes perpendicular to this axis.

Thus, Φ depends only on χ and t, whileqxdΦ= —dx,and22q= q .xSince q = q = 0, E q . (14) reduces toyz( 4)- Ip ( i2dx2\a /2^ -t2« ; ^ L - o .a dt2a dx dt22A s before, a also involves derivatives of Φ; if the polytropic (p,p)-relation2is adopted, then E q . (16) gives(25,- « ( £+ £ ) .T h e problem characterized by Eqs.

(24) and (25) will be studied indetail in Chapter I I I , but a fairly general type of particular integral ofthese equations will be indicated here.AssumingχΦ(ζ, t) = a - + βχ + 7,$Φ= — = ax + ft2(26)qxwhere α, β, and y are functions of t only, can we choose α, β, and y so thatEqs. (24) and (25) are satisfied? When (26) is substituted in (25), a is givenas a quadratic function in x; also, d%/dx does not involve x.

Thus,when (24) is multiplied through b y a and (26) substituted in it, the lefthand member of the resulting equation is also a quadratic function of22278II.G E N E R A LT H E O R E M Sχ; the differential equation is satisfied if the coefficients of x2and χ andthe constant term, all functions of t, vanish identically. If these coeffi­cients are set equal to zero, we have the conditionsa" +(27)"βy"+"(+1αβ'+(κ +3) α α ' +β(21) αγ' +(« -+}(jt +α'+{ κ+11) a"}|3 ( ^ ^ « 02 )+==0,°'2/3') =0.These are three ordinary differential equations which can be solved suc­cessively for a, for β, and finally for 7.

T w o examples of solutions follow:2(a)1β = constant = C i ,a = —-—κ +1Ϊκ —1Ζ(28)23V,1 £κ ++1 ίκ+/1I n the particular case c = 0, both q and a are functions of x/t only; thismotion plays an important role in the theory of nonsteady one-dimensionalflow (see A r t . 13, "centered simple w a v e s " ) . Including a c -term has no ef­fect on the particle lines (see Sec.