R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 45

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(12.22).This suggests investigating other conditions under which one or anotherof the pairs of equations (34) possesses an integral. M o r e explicitly, we wishto find those functions λ(ψ,ρ) for which there exists a function W(\l/,p£,u,t)such that W = constant holds by virtue of, say, the first pair of equations(34) and Eq. (27). This question can be answered quite generally, but werestrict ourselves here to those functions λ which correspond to the case ofa perfect gas, see (33). W e find that δ(ψ) must either be constant, cor­responding to the isentropic case discussed above, or else have the form(35)δ{ψ) = Ηφ -where k and(36)φο) - ,ν 2are arbitrary constants.

T h e corresponding integral isWx= ν,φ + tp -ξ -,(v k(t)1) \ p /1N= constant,where ψ has been suppressed. This is easily checked, for0dWi = (udt+ tdp -d& + (*du+ ^ r - d p ) + (p dt -k^-r# )= 0along the characteristic, since there the first bracket vanishes by Eq. (27)and the second and third by the first pair of equations in (34). For thisδ(ψ), the corresponding distribution of entropy from particle to particle isgiven by(37)Fit) =A = iyk?)\For any isentropic motion, ν + u remains constant on each (u + a ) characteristic.

Flows for which the constant does not change from charac­teristic to characteristic are called simple waves. Correspondingly, we mayequate Wi, given by Eq. (36), to an over-all constant and thereby generatea class of solutions of Eq. (31) which may be called nonisentropic simplewaves. I t should be emphasized that for a perfect gas these solutions aredefined only for the entropy distribution given by (37). W e may easily15.8NONISENTROPIC SIMPLE235WAVESverify that this procedure does generate solutions of Eq.

(31). For withu and t defined by (28), so that (27) holds, we have on equating dW\ tozero:ψάυ,+k*—dpp+ pdt-άφ = 0.p"v1Using (28) once more and remembering that from (35): λ = k\f/~ /p\ weobtain2[*$ Κ4- )] Κ4 ) »$]*=°'+λ#++λ +which holds identically in ψ and p. Equating the coefficients of άψ and dpto zero and eliminating ψ and ρ themselves from the resulting equations,we findWdp~ JΛ~2=°'in agreement with Eqs. (31) and (32). Thus we have shown that any solu­tion of the first-order equation Wi = constant—with u and t as in Eqs.

(28)— i s also a solution of the original second-order equation (31). In the classi­cal theory of the Monge-Ampfere equation such first-order equations arecalled intermediate integrals.A similar discussion can be made for the second pair of equations in (34).Again δ(ψ) is either a constant or given by Eq. (35). N o w Wi is replaced byand equating this to a constant throughout the flow leads to anisentropicsimple waves of a second kind, in the same way that equating ν — u to anover-all constant does in the isentropic case.W e may introduce new variables a and β by means of the equationsa= ψι,β =W,2and for flows other than simple waves these may be taken as new inde­pendent variables.

T h e two families of curves a = constant, β = constant,in the ψ,ρ-plane are the characteristics, so that the variables a and βplay a similar role to that of ξ and η in A r t . 12. In these new variables it maybe shown that 1/ψ, I/ρ, u, and t all satisfy linear second-order equations, ofthe formθ Ζη2da θβ,Πa -ί dZnβ \dadZ \nδβ )=Q23GIII. ONE-DIMENSIONAL FLOWFor 7 = | we have η = 3 for znzn= u, and η =(ΛΓ +— 3 for zn=1/ψ, η =— 4 for zn=1/p, η = 4 for= t. For general y we have η = Ν, — (Ν + 1),1), — Ν*, respectively, where Ν=(y +1)/2(7 — 1).

T h e analogywith Eq. (12.34) becomes evident if that equation is rewritten with thecharacteristic variables ξ =ν +u and η =ν — u as new independentvariables [see Eq. (12.43)]. Thus, for the entropy distributiontions governing the motion may, by appropriate(37) the equa­change of independentvari­ables, be replaced by a linear equation in l/ψ, 1/p, u, or t of the type consideredin Sec. 12.4.Alternatively the same conclusions concerning the characteristics, Eqs.(34), can be reached by considering Eqs.

(17) and (18) as a planar system ofthree homogeneous first-order equations for u, p, and p. For this purpose,E q . (17) is written out asaccording to E q . (9.25). T h e general theory of characteristics, as developedin A r t . 9, will then yield the characteristic conditions (34).CHAPTER IVPLANE S T E A D Y POTENTIAL F L O WArticle 16Basic Relations1. Direct approachW e deal in this article with the plane, steady, irrotational flow of an elas­tic, inviscid fluid, neglecting the influence of gravity.

T h e basic hypothesisfor steady plane flow is that all quantities are independent of ζ and t,and that q , the z-component of the velocity vector q, vanishes. T h e com­ponents of q are then functions of the coordinates χ and y. T h e relationsvalid for this type of motion are contained in, or can be derived from, var­ious discussions in Chapters I and I I .

For the convenience of the reader,however, we start here by setting up the main equations of our problemwithout making use of previous results, except, of course, for the very firstprinciples.T h e fact that a motion in the x,?/-plane is steady gives the equation ofcontinuity ( l . I I ) the formz(1)d(pg*) _j_ d(pq )ydx=0dyT h e fact that it is irrotational is expressed byψ(2)- ψdx= 0.dyBoth equations can be transformed in such a way that they become inde­pendent of the choice of the coordinate axes.

Denote by d/ds differentiationin the direction of the velocity vector q, and by d/dn differentiation in thedirection which makes an angle of + 9 0 ° with the first. If we identify, at apoint P , the x- and ^/-directions in (1) and (2) with the s- and n-directions,respectively, (1) and (2) yield* dsμdsμdn9237dsdn238IV. P L A N E S T E A D Y P O T E N T I A LFLOWηq+^ds streamlineFIG. 87.

Streamline and normal.since q = q, q = 0 at P . Although q = 0, the derivatives of q = qare not zero. If θ denotes the angle between q and any fixed direction, it isseen from Fig. 87 that the increment of q isqdd; thus,x(3)yvds_dq _ dqnydxyny6Θq_dqndqy~ ds ~ R'=dn ~dyqdd_dn'qwhere 1/R = dd/ds denotes the curvature, and R the radius of curvatureof the streamline. T h e equations of continuity and irrotationality, (1) and( 2 ) , respectively, then take the formsd(pq) ,-aT+pqdddq ,dp ,^ ^ d sds dnp(4)dq—+qdd+pq=Λ^d6— q —dnH=0,dsor, equivalently,(40d(\og ρ) , θ (log q) , Μdsdsdn=θ (log q)Qdd=an a«T h e variable ρ still appears in ( 4 ) , but may be eliminated as follows.

T h efluid, assumed elastic, satisfies a (p,p)-relation. Thus there is a derivativedp/dp, which we have called a, where a is the sound velocity, and(5)dp _ dp dpdsdp dsi_apa as'2T h e s-component of the N e w t o n equation—which in the absence of gravityand viscosity is equivalent to the differential form (2.210 of the Bernoulliequation—is(6)dpdsdqdtdq16.1 D I R E C T239APPROACHThen, if we introduce the Mach number Μ = q/a, the Eqs.

(4) readdq _qds ~ MUΘΘdq _— lfrn'2d0dn' ds'qHere the first equation is independent of the assumption of irrotationality.However, Μ involves a as well as q. T o eliminate a we use the Bernoulliequation in the integrated form [see Eq. (8.3)]:(8)= constant,t+[^.2Jρwhere, in the irrotational flow of an elastic fluid, the constant is the samefor the entire flow, as shown in Sec. 6.5.The explicit computation will be carried out only for a polytropic fluid:ρ = Cp , where κ is a constant > 1. ThenK= —,ρaf — = — —- - + constant =•'ρκ — 1ρκ —aKI+ constant,and the Bernoulli equation is22q _|_22aκ -_*aικ- νwhere a denotes the value of the sound velocity at a stagnation point.Thus,22Κ —1 28(9)a = a, -——q,and2(10)Μ= — =aκ +2—012 2q - dsM 1 2κ — 12α^2~~The equations (7), with Μ — 1 expressed by (10), and a scale factor achosen, are two differential equations of first order for the unknowns q andΘ. The set of Eqs.

(7) and (10) includes all the information on which theanalysis of the flow pattern under discussion will be based* If, in addition tothe velocity distribution, the values of pressure, density, and absolutetemperature are required, these follow from the Bernoulli equation inconnection with the (p,p)-relation and the equation of state. (See Art. 8,and in particular Sec. 8.4, where in Table I, numerical values of the vari­ables are given for the polytropic case with κ = y = 1.4.)22——κ—1 = - 2Q2κ —22α ββ2* In the general non-polytropic case Eq.

(8) takes the place of (10).98240IV. P L A N E S T E A D Y P O T E N T I A L F L O WReviewing our earlier results we note that the fact that ρ and ρ can beexpressed in terms of q justifies the use of q as a dependent variable. Thenthe other natural variable to determine q is 0, i.e., the angle that q makeswith some fixed direction; moreover, 0 determines the streamline pat­tern, which is fixed for steady flow.

I t is then likewise natural to introducethe rate of change along a streamline, d/ds, and the rate of change in thedirection normal to the s-direction.*From (9) it is seen that a decreases steadily as q increases from zero, andthat a vanishes for q = q wherem(11)This value of q cannot be exceeded. Introducing qnoulli equation in the formma =(9)(q-min (9) gives the Ber­q).On the other hand, from ( 9 ) , q equals a if both have the value(ΙΙ')ϊ, = α (y'^fj*.This value, defined by the condition q = a, has been called the transitionvelocity, also sonic or transonic speed.

T h e poly tropic relation in the formPi\Pi/where p , pi, ai are some corresponding values of ρ, ρ, a, shows that p, p,and (since κ > 1) also p/p vanish for q = q .From the polytropic relation and (9') the relation between pressure andvelocity [see Eq.

(8.28)] takes the formxm>/<κ-1)J^ = [~l-^~*(12)PsLQrn _where p is the value of ρ at a stagnation point, f T h e relation betweendensity and velocity corresponding to (12) is28(120-=1 - Λ·LQm _\In the following we shall let p = 1, unless otherwise specified.Ps28* If the equations are considered in a region rather than at a single point, we areactually using curvilinear coordinates, namely, the system of streamlines and theirorthogonal trajectories.t In Sec. 8.1 the relation between ρ and q was discussed for a general (p,p)-rela­tion, while in Sec. 8.3 a polytropic relation was assumed.24116.2 P O T E N T I A L A N D S T R E A M F U N C T I O N SDimensionless variables can be introduced in various ways, dividing qeither by qm, or by a , or by a8= qt .

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