R. von Mises - Mathematical theory of compressible fluid flow, страница 14

If curl qvanishes at all points, then (18) shows that grad Η = 0, i.e., the Ber­noulli function has one and the same value everywhere and we can state:In the steady irrotational flow of an inviscid elastic fluid the Bernoullifunction,or the total head, has the same value on all streamlines. T h e converse is nottrue in general. I t can happen that the streamlines and vortex lines coin­cide, in which case the vector product curl q X q vanishes and Η is con­stant everywhere, although the motion is not irrotational. This case,however, is a very particular type of motion and cannot occur, for ex­ample, in a plane motion: q = 0, and d/dz = 0 where (5) shows that curlq is perpendicular to the z,2/-plane and therefore cannot coincide anywherewith q.13zI n the case of a nonsteady irrotational motion, Eq. (17) leads to(19)^=_grad{gHXdtwhich should be noted for later use.6.

Helmholtz derivation of the vortex theorems114Hermann von Helmholtz derived his two vortex theorems directly fromEq. (17) without using the concept of circulation. A n outline of his argu­ment, which, however, does not lead to a rigorous proof, follows.I t is well known that the curl of a gradient is zero. In fact, if for some func­tion φ a vector a satisfies a = grad φ thenda _ddydyx(θφ\dadx)dxyetc., and (5) shows that curl a vanishes identically.

Thus, on taking thecurl of both sides of Eq. (17), we getcurl ^5 _fdtc u ri (c u ri q χ q) = 0.If we interchange the order of differentiation in the first term, and writeλ for the vector curl q, this becomesg : + curl ( λ X q) = 0.(20)dtUsing the definition (5) of curl and that of vector product, we writethe ^-component of the second term on the left as(21)d— (\ qdyxy-λ^χ) -d— (\ qdzzx-\ q ).xz6.6H E L M H O L T Z ' D E R I V A T I O N OF V O R T E X T H E O R E M S67T h e terms resulting from the differentiation of the products are listedin the first two columns below:d\.xd\xdy-d\yqxdy.d\xdzdxd\q — dzz-xd\—dxrqx(210Λχ —ΓΓΑχ—dy.~λχ —dzdqdxdqxAy~Γdqx--A——zx—Λχdydzdxwhile the added terms in the third column cancel each other. T h e terms ineach line combine to a simple expression, and (21) can be replaced by(21")g^?dsgi.div*+X , d i v q - X ^da>where d/da denotes differentiation in the direction of λ (along the vortexline).

T h e second of the terms (21") is zero, since the divergence of a curlalways vanishes, as may be seen from Eq. (5). In the third term, the factordiv q can be replaced by — (l/p)dp/dt,from the equation ofcontinuity( l . I I ' ) . Thus ( 2 1 " ) becomesd\λ dpdsρ atxχ.

dqxdawhich is exactly the x-component ofdsρ atdaWhen (22) is substituted for the second term of Eq. (20), and the terms col­lected according to the Euler rule of differentiation (1.4), we obtainor, dividing through by ρ and setting Λ = λ/ρ,This is Helmholtz' equation, which can be interpreted so as to give the twovortex theorems.Let Ρ and Q (Fig. 27) be two neighboring points lying on the same vortex68II. GENERALTHEOREMSF I G . 27. Interpretation of Helmholtz' equation.line. Then PQ = A e for a suitable small e. After time dt, the particle initiallyat Ρ has moved to P' and the particle at Q to Q'', wherePPQQ' = ( q + |5 A e ) Λ .=qdt,7Then P ' Q ' is given byp7Q7 =_pp>+P Q +QQ>=(A+|5 A ( f t ) €.But then Eq.

(23') gives(24)FQ=7( A + ^d<) e = A ' e ,since the parenthesis is exactly the value A ' of A obtaining at the position Pafter time dt. From (24) follows, first, that P' and Q' again lie on a vor­tex line, up to terms of first order, since A gives the direction of thevortex line at P'\ this is in accord with the first vortex theorem. Secondly,Eq. (24) shows that the change in the distance PQ is proportional to thechange in A. N o w , during the transition from PQ to P'Q'', the mass of theparticle does not change; therefore, if dui and d&' are the normal crosssections of the vortex filament before and after the displacement, and ρand ρ the corresponding densities, we must havef;(pda)-(PQ)=(ρ'da')'(P'Q')from which(25)pA da = ρ'Λ'da'.N o w pA is, according to the definition of A, the length of the vector curl q,so that Eq. (25) expresses the second vortex theorem for a vortex filament:the product of cross section and vortex magnitude remains constant.

Thisleads to the analogous theorem for a vortex tube of finite cross section.69POTENTIAL7.1Article 7Irrotational M o t i o n151. PotentialFrom the vortex theory of the preceding article, it follows that in an in­viscid elastic fluid, a particle that at one time has no mean rotation cannotsubsequently acquire rotation.16If at some time t = 0 the entire fluid massunder consideration is irrotational, then it remains so.

Thus flow patternscan exist withcurl q = 0(1)for all t.A transition to rotational flow can occur only if viscosity becomes effective,or if the fluid ceases to be elastic, etc. In particular, if a flow originatesfrom a region where q, p, and ρ are constant (for example, from a state ofrest) the flow problem is irrotational everywhere and at all time, whetherit is steady or nonsteady.Mathematically, the condition (1) is equivalent to the statement thatq is a gradient, i.e., that there exists a function Φι(χ,?/,z,t) whose gradientis q :(2)q = gradft;q.

= — ,qy= —,q. =—.In addition to some given (p,p)-relation, the flow is subject to (a) the equa­tion of continuity, which we take in the form ( 1 . Ι Γ ) :and ( b ) the N e w t o n equation of motion, which we take in the form (6.19)holding for an inviscid elastic fluid when the flow is irrotational:(4)|5 =- g r a d (gH)- g r a d ( | + gh +=W i t h the use of q = grad Φι and_ grad Φ1=grad—,Eq. (4) may be written(5)grad+ gH)= 0.P).70II. GENERAL THEOREMSConsequently the sum σΦι/dt + gH must be a function of t only, sayIf F(t)is the indefinite integral of f(t),<)» +(5f(t).then„*(.,_„.._„,.rSince Ρ is independent of x, y, and z, the function Φ = Φι — F has thesame space derivatives as Φι and may, therefore, be used in place of Φι in( 2 ) .

Then Φ satisfies the four conditions,„ΘΦ( 6 )βΊsΘΦ=^'6y=ΘΦ">qTz=>qHere the six integrability conditionsd%d%dx dydy dx'9d%ΘΦdz dtdt dz'2are satisfied on account of Eqs. (1) and ( 4 ) , and Φ is thus determined, fora given flow pattern, to within an additive constant. T h e function Φ iscalled the potentialof the irrotational fluid flow. T h e reader is familiarwith the fact that grad Φ is normal to the surfaces Φ = constant; thus, thevelocity vector q is perpendicular to these equipotentialsurfaces, or morebriefly potential surfaces.

T h e magnitude of the component of q in any di­rection equals the directional derivative of Φ in that direction and, inparticular q equals (ϊΦ/ds where d/ds means differentiation in direction ofthe streamlines. Once Φ(χ,?/,ζ,0 is known, the flow is completely deter­mined, since the first three equations (6) give q, and then the last equa­tion determines P , which, together with the (p,p)-relation, determines ρand ρ as functions of x,y,z, and t. Obviously an additive constant in Φ hasno significance.I n the case of steady motion, q and Ρ independent of t, it follows from thefirst three equations (6) that d^/dt is independent of x, y, and z, becaused(d&/dt)/dx= θ(&Φ/βχ)/θί= 0, etc.; and from the last equation it fol­lows that dΦ/dt is not a function of t either.

Thus dΦ/dt is constant,everywhere and at all times, and is, in fact, equal to — gH by E q . ( 6 ) ;this is in agreement with the conclusion in Sec. 6.5 that Η is constant insteady irrotational flow.2. Equation for the potentialI n the derivation of Eqs. (6) only Eqs. (1) and (4) were used. Thus, an ar­bitrary function Φ(χ,2/,2,2), together with a (p,p)-relationand ( 6 ) , de­termines a distribution of values of q, p, and ρ w hich satisfy the N e w t o nr7.271EQUATION FOR P O T E N T I A Lequation, ( 4 ) , but which will not, in general, satisfy the equation of con­tinuity, ( 3 ) .

This condition will be fulfilled if Φ is a solution of the differ­ential equation which results when Φ is substituted in E q . ( 3 ) .T h e left-hand member of E q . (3) isττ(7)ι^. θΦδΦ. <9Φ22div q = div grad Φ = —+ _2+ _= ΔΦ,where the symbol Δ (Laplace operator) is used exactly as in A r t . 4. T h esound velocity may be defined by/o\2( 8 )-adpνas in Sec. 4.1, whenever there exists a (p,p)-relation; then the right-handmember of Eq. (3) is_1 dp __ 1 dp dp ___ 1 dp _ρ dtρ dp dta p dtdPa2dt29since by definition Ρ = / dp/ρ (Sec.

2.5). Thus Eq. (3) becomesΔΦ(9)-— -77.=2a2dtW e shall find [Eqs. (10) and ( 1 6 ) ] that both a and dP/dt are expressible2in terms of derivatives of Φ.In the case of an incompressible fluid, when a =co, the right-hand sideof Eq. (9) is zero; the potential Φ must therefore be a solution of Laplace'sequation, ΔΦ = 0, and all the classical results on Laplace's equation areapplicable.

Further, the equation does not involve t, which means that Φ(and therefore the whole pattern of flow) is determined at each momentonly by the boundary conditions holding at that moment: in the irrotational flow of an incompressible inviscid fluid, there is no "after-effect".T h e situation is much more complex when the fluid is compressible.From the last equation ( 6 ) , — ΡEuler's rule of differentiationdrm\( 1 0 )d- MP*_L= W lH\2)θφ+ίΛ=A.+ΘΦ/dt +q /2 +2d {ΘΦ . q \F s \ T t 2 )2q+gh, so that byθΦA+dh'In differentiating the function A, use has been made of dh/dt = 0 and ofthe fact that dh/ds is the cosine of the angle between the directions ofq ( = grad Φ) and grad h. Hence q dh/ds is the component of grad Φ inthe direction normal to h = constant; or q dh/ds — θΦ/θ/ι.

Using here Φ ,χ72II. G E N E R A LTHEOREMSΦ^, Φ , and ΦΊ , for the first partial derivatives ofΦ , we write out theΖsecond and third terms on the right of Eq. ( 1 0 ) :<2/dt \2 /,.2,2N, θΦΘΦΦζ ) = Φχ —+ Φρ ^ Γ Γa« * * dt,(Φ*+ ΦΊ ,+θ*2χν+ΦζΘΦζΖdt1ΘΦιdzdxdydzHence these two terms are equal. B y the use of Eq. (10), Eq. (9) can now bewritten asto-L^^L(12)a9 ( * *ds\dt2dt2a22ft2/+ΘΦl+a2dh'Since gravity is comparatively unimportant in problems of gas flow, thelast term in (12) will be dropped from now on; retaining this term would notgreatly increase the complexity of the equation.T o express Eq.