R. von Mises - Mathematical theory of compressible fluid flow, страница 12

Consider a circuit(3, i.e.,a simple closed curve in space together with a given sense of description(indicated in Fig. 18 by an arrow). On e, each element of arc can then beconsidered as an infinitesimal vector al, having the direction of the tangentto 6. A s usual, q denotes the instantaneous velocity at each point. If thescalar product of q and di is integrated around the circuit, the line integral(1)is called the circulationaround e. A l l values of q are to be taken for thesame value of t.2T h e circulation is additive in the following sense.

Suppose we " b r i d g e "the circuit C by some path ABABDAand BAEB(see Fig. 19) and give the t w o new circuitsthe same sense of description as 6. Then the circulationsΓι and Γ , respectively, along the new circuits satisfy2Γ =(2)Γι +Γ .2In fact, the definition (1) shows that Γι is the integral of q » a l along thepath ABDA,which can be broken up into A Β plus BDA.the integral along the path Β A plus AEB.Similarly, Γ2 isIn the sum Γι + Γ , the integrals2along A Β and Β A cancel, since q is the same, while al has opposite direc­tions, along the two paths. Therefore the sum reduces-to integrals alongBDAand AEB,which is exactly the integral around (B, i.e., Γ.T h e relation (2) can be generalized.

Suppose G, is any open two-sidedsurface spanning e, i.e., having <B as its rim. From one side of G, the senseof description of C appears counterclockwise, and normals to & will alwaysbe drawn out from this side. On Q, draw t w o sets of curves forming a net­work, as in Fig. 20.

Each mesh of the network is a closed circuit, the sense5556II. GENERALTHEOREMSof description being taken counterclockwise as viewed from the normal tothe surface, and has a value of the circulation corresponding to i t : Γ ι , Γ ,• · · , etc. B y repeated application of (2) we find2(3)Γ =Γι +Γ2+· * * +Tm,where m is the number of meshes in the network. W e now increase thenumber of " bridges'' in such a way that the network becomes more denseand all meshes become smaller, while the number of terms in Eq.

(3) in­creases. Let a function y be defined at each point Ρ of & as the limit of thequotient of the circulation along the contour of a mesh around Ρ by thearea of the mesh, for meshes about that point becoming steadily smallerin all directions. For the first mesh, the circulation Γ ι is then approximatelygiven by 7 1 ddi, where dd\ is the area of the mesh and 7 1 the value of 7F I G . 18. Vector along circuit.F I G .

19. Illustration of additivity of circulation.F I G . 20. Circulation as surface integral.6.157CIRCULATIONat some point in the mesh, the approximation becoming better as dQ\gets smaller; and similarly for the other meshes. Thus, as the number ofterms increases indefinitely, the right-hand member of (3) yields the surfaceintegral of y over a, or(4)=/Τda.From the definition of 7, it is obvious that the value of this function atany point of <2 depends upon the distribution of the velocity q in the neigh­borhood of this point.

In computing this relationship let us choose curveson Q which always cross at right angles. Then at any point Ρ we can set up arectangular coordinate system, taking the 2-axis in the direction of the nor­mal to α at Ρ and the x- and ^/-directions tangent to the two curves throughΡ so as to form a right-handed coordinate system. Then an infinitesimalmesh starting at Ρ is of the type illustrated in Fig. 2 1 . In computing theline integral (1) for this mesh, the path may be broken up into four in­finitesimalelements, and q dl evaluated for each part.

Along PPilcontribution is q dx, along P\Pi it is [qx— [qx +y(dq /dy)dy]dx,xand along P^P+along P Pz(dq /dx)dx]dy,yit is —q dy.y2T h e sum oftheit istheseterms gives the integral along the whole path, and the circulation alongthe contour of this infinitesimal mesh is thereforedT-(=dqvdxSince dx dy is the area of this mesh, the function 7 must have the valuedq /dx — dq /dy at P.

N o w this quantity is exactly the ^-component of theyxvector known as the curl of q, defined by(5)curl η =q\dy^dz 'do* _dQzdzdx 'Jh _dxddq*\dy J'I t is to be noted that this definition of curl q is valid in any rectangularright-handed coordinate system. Since the ^-direction is here that of the3F I G . 21.

Circulation around infinitesimal mesh.58II.G E N E R A LTHEOREMSnormal to the surface 0t, it appears that y has the value of the component ofcurl q normal to the surface:y = (curl q ) ;nThus, (4) may be written as(6)Γ =[(curl q )nda,and this formula is independent of the coordinate system used. If a vectorda be introduced, having magnitude da and the direction of the normal tothe surface, E q . (6) may also be written as(60Γ =f(curl q ) . r f 5When the two expressions (1) and ( 6 ' ) are combined, the result isj> q - d l = j(curl q ) » d 5This vector formula is known as Stokes9Theorem.4When q is the velocityof flow, it states that the circulation along any circuit is given b y the sur­face integral of curl q over any surface spanning the circuit.I t is obvious that ( 6 ) , or ( 6 ' ) , can be applied only if it is possible to findsome surface that has the given circuit as rim and on which curl q is de­fined everywhere. For example, in the case of a flow around an infinitecylindrical obstacle, no such surface can be found for any circuit whichsurrounds the cylinder.

E v e n here the theorem can be applied, to give asomewhat different result. A s shown in Fig. 22, two such circuits, 6 i and6 2 , can, b y a bridge AB, be combined into a single circuit for which a suit-F I G . 22. Circuits surrounding obstacle.6.2M E A N59ROTATIONable spanning surface exists. Then the integral (6) extended over this sur­face gives Γι — Γ , since (B is given a reverse orientation and the contribu­22tions from A Β cancel.

In particular, if curl q =irrotational flow (see A r t . 7 ) , then Γι — Γ20 in the domain of flow,= 0, or Γι =Γ2: the circula­tion is equal for all circuits surrounding the obstacle.2. Mean rotationT h e vector curl q defined by (5) can be given a simple kinematic inter­pretation.L e t Ρ be a point of the moving mass, and let Q be a neighboring point.In rigid-body rotation with angular velocity ω about an axis through P ,the curl of the velocity vector at Q is the same, namely 2ω, no matter whereQ is situated.

This is not so in the case of a fluid or of any deformable mass.W e start by computing the angular velocity of PQ about an arbitrarygiven axis through P . Taking Ρ as the origin, we choose a right-handedrectangular coordinate system such that the ^-direction is that of the axis.L e t PQ = dr and let Q be the projection of Q onto the x, 2/-plane (see Fig.r23), Θ the angle between the z-axis and PQbetween the x-axis and PQ'to Q is PQf=(colatitude), and φ the angle(longitude). Then the distance from the z-axissin θ dr, and the rectangular coordinates of Q, relative toP , are given by dx = cos φ sin θ dr, dy = sin φ sin Θ dr, and dz = cos θ dr.Therefore, if the velocity at Ρ is q, the velocity vector at Q (relative to P ) iscos φ sin θ + - ~ sin φ sin Θ +cos θ dr.dydzJT h e angular velocity of the segment PQ about the z-axis is obtained bydividing the distance from the z-axis to Q, i.e., P Q ' , into the componentof the velocity at Q in the direction which is perpendicular to PQ' and tothe z-axis.

T h e angles which this direction makes with the χ-, y-, and z-axesζΡXF I G . 23. Computation of mean rotation.60II. GENERAL THEOREMSare φ +90°, φ, and 90°, respectively, and the cosines of these angles are— sin φ, cos φ, and 0. Thus the required component is[picos φ \_dx2θ± in φ dy2S~\~|_dz( Jn \dxdcos φ —dzp>) sin φ cos φ ] sin θ drdy JJsin φJcos 0 dr.Division by sin θ dr then gives the angular velocity of PQ about the z-axis,the value depending, in general, on the coordinates θ and φ of Q.N o w we compute the average angular velocity for all points Q on thesame circle of latitude θ = constant, on the sphere dr = constant, by firstintegrating with respect to φ from 0 to 2ir and then dividing by 2π. Allintegrals vanish except those of the first two terms, giving(7)TM f-£)= ^ (curl q ) .zThis result being independent of θ and dr, the same value is obtained forthe average or mean angular velocity about the 2-axis of the whole infini­tesimal sphere at P .

Also, the ^-direction could be any direction, so theabove result shows that at any point Ρ of the moving fluid, the vector ^curl qrepresents the (instantaneous)mean angular velocity or mean rotation for allsegments PQ within an infinitesimal sphere of center P . W e call it brieflythe mean rotation or mean angular velocity of the fluid element around P .5Excluding the case curl q = 0, E q . (5) defines at each moment t at eachpoint of the fluid a vector curl q which is twice the mean angular velocityof the fluid element around P. This vector is usually called the vortex vector.A n y line within the fluid which at each of its points has the same directionas curl q is called a vortex line.

All vortex lines passing through the pointsF I G . 24. Vortex tube with various circuits.6.3KELVIN'S61THEOREMof a closed curve e, not itself a vortex line, form a vortex tube. A vortextube of infinitesimal cross section is called a vortex filament. T h e lateral sur­face of the tube will be called its mantle*For any circuit, such as d in Fig. 24, which lies on the mantle of a vortextube but does not encircle the tube, the circulation must be zero. This followsfrom (6) since a surface & spanning this circuit can be taken on themantle, where the normal component of curl q is everywhere zero. This isno longer true for a circuit which encircles the tube, as β or6 2 in Fig.