Vortex Theorems for the Ideal Fluid
In connection with the study of wing wakes, separation, and related phenomena, it is of value to study the properties of the field vorticity vector f, (1-16). The reader is assumed to be familiar with ways of describing the field of the velocity vector Q and with the concept of an instantaneous pattern of streamlines, drawn at a given time, everywhere tangent to this vector. A related idea is the “stream tube, ” defined to be a bundle of streamlines sufficiently small that property variations across a normal section are negligible by comparison with variations along the length of the tube. Similar concepts can be defined for any other vector field, in particular the field of f. Thus one is led to the idea of a vortex line and a vortex tube, the arrows along such lines and tubes being directed according to the right-hand rule of spin of fluid particles.
Because f is the curl of another vector, the field of vortex lines has certain properties that not all vector fields possess. Two of these are identified by the first two vortex theorems of Helmholtz. Although these theorems will be stated for the vorticity field, they are purely geometrical in nature and are unrelated in any way to the physics or dynamics of the fluid, or even to the requirement of continuity of mass.
1. First Vortex Theorem. The circulation around a given vortex tube (“strength” of the vortex) is the same everywhere along its length.
This result can be proved in a variety of ways, one simple approach being to apply Stokes’ theorem to a closed path in the surface of the vortex tube constructed as indicated in Fig. 1-4.
Fig. 1-4. Two cross sections of a vortex tube.
We turn to (1-15) and choose for S the cylindrical surface lying in a wall of the tube. Obviously, no vortex lines cross S, so that
n • f = n • (V X Q) = 0. (1-23)
Hence the circulation Г around the whole of the curve C vanishes. By examining C, it is clear that
0 = Г= Гв~-Га + (two pieces which cancel each other). (1-24)
Hence Г л = Гд. Sections A and В can be chosen arbitrarily, however, so the circulation around the vortex is the same at all sections.
Incidentally, the circulation around the tube always equals JJn • f dS, where the integral is taken over any surface which cuts through the tube but does not intersect any other vortex lines. It can be concluded that this integral has the same value regardless of the orientation of the area used to cut through the tube. A physical interpretation is that the number of vortex lines which go to make up the tube, or bundle, is everywhere the same.
2. Second Vortex Theorem. A vortex tube can never end in the fluid, but must close onto itself, end at a boundary, or go to infinity.
Examples of the three kinds of behavior mentioned in this theorem are a smoke ring, a vortex bound to a two-dimensional airfoil spanning across from one wall to the other in a two-dimensional wind tunnel, and the downstream ends of horseshoe vortices representing the loading on a threedimensional wing. This second theorem can be quite easily deduced from the continuity of circulation asserted by the first theorem; one simply notes that assuming an end for a vortex tube leads to a situation where the circulation is changing from one section to another along its length.
The first two vortex theorems are closely connected to the fact that the field of f is solenoidal, that is,
V • f = 0. (1-25)
When this result is inserted into Gauss’ theorem, (1-14), we see that just as many vortex lines must enter any closed surface as leave it.
There is a useful mathematical analogy between the f-field and that of the magnetic induction vector B. The latter satisfies one of the basic Maxwell equations,
V • В = 0. (1-26)
Although no such analogy generally exists with the fluid velocity vector field, it does so when the density is constant, which simplifies the continuity equation to
V • Q = 0. (1-27)
Hence, flow streamlines cannot end, and the volume flux through any section is the same as that through any other section at a given instant of time. One may examine in the same light the field of tubes of the vector pQ in a steady compressible flow.
3. Third Vortex Theorem. We now proceed to derive the third vortex theorem, which is connected with the dynamical properties of the fluid. Following Milne-Thompson (1960, Section 3.53), we start from the vector identity
The second step is to take the curl of (1-28), noting that the curl of a gradient vanishes,
VXa=VX§ + 0- VX(QX?). (1-29)
The operations Vx and d/dt can be interchanged, so that the first term on the right becomes df/cU. For any two vectors A and B,
V X (A X В) = (B • V)A – (A • V)B – B(V • A) + A(V ■ B). (1-30)
VX(QXf) = (T V)Q — (Q • V)r – f (V • Q) + 0. (1-31)
Substituting into (1-29), we have
V X a = ^ – a • V)Q + f(V ■ Q). (1-32)
So far, our results are purely kinematical. We next introduce the conservation of mass, (1-1), second line:
whence, after a little manipulation, (1-32) may be made to read
Under the special conditions behind (1-11), a is the gradient of another vector and its curl vanishes. Thus for inviscid, barotropic fluid in a conservative body force field, the foregoing result reduces to
This last is what is usually known as the third vortex theorem of Helmholtz. In the continuum sense, it is an equation of conservation of angular momentum. If the specific entropy s is not uniform throughout the fluid, one can determine from a combination of dynamical and thermodynamic considerations that
V X a = V X (TVs). (1-36)
When inserted into (1-34), this demonstrates the role of entropy gradients in generating angular momentum, a result which is often associated with the name of Crocco.
To examine the implications of the third vortex theorem, we shall look at three special cases, in increasing order of complexity. First consider an initially irrotational flow, supposing that at all times previous to some given instant f = 0 for all fluid particles. In the absence of singularities or discontinuities, it is possible to write for this initial instant, using (1-35),
Since the quantity f/p is an analytic function of space and time, Taylor’s theorem shows that it vanishes at all subsequent instants of time. Hence, the vorticity vector itself is zero. We can state that an initially irrotational, inviscid, barotropic flow with a body force potential will remain irrotational. This result can also be proved by a combination of Kelvin’s and Stokes’ theorems, (1-12) and (1-15).
Next examine a rotational but two-dimensional flow. Here the vorticity vector points in a direction normal to the planes of flow, but derivatives of the velocity Q in this direction must vanish. We therefore obtain
Once more, by Taylor’s theorem, f/p remains constant. This is equivalent to the statement that the angular momentum of a fluid particle of fixed mass about an axis through its own center of gravity remains independent of time. In incompressible liquid it reduces to the invariance of vorticity itself, following the fluid.
For our third example we turn to three-dimensional rotational flow. Let us consider an infinitesimal line element ds which moves with the fluid and which at some instant of time is parallel to the vector f/p. That is,
where e is a small scalar factor. Since the line element is attached to the fluid particles, the motion of one end relative to the other is determined by the difference in Q between these ends. Taking the ж-component, for instance,
§t (ds) = (ds • V)Q = e ^ • v) Q. (1-41)
Comparing this last result with the third vortex theorem, (1-35), we are led to
= 0, or e = const. (1-42)
It follows that (1-39) holds for all subsequent instants of time, and the vector f/p moves in the same way as the fluid particles do in three dimensions.
The proportionality of the length of a small fluid element to the quantity f/p can be interpreted in terms of conservation of angular momentum in the following way. As implied by (1-39), this length is directed along the axis of spin. Hence, if the length increases, the element itself will shrink in its lateral dimension, and its rate of spin must increase in order to conserve angular momentum. To be precise, the quantity f/p is that which increases, because angular momentum is not directly proportional to angular velocity for a variable density particle, but will decrease as the density increases, the density being a measure of how the mass of the fluid particle clusters about the spin axis.
For a constant-density fluid, of course, the vorticity itself is found to be proportional to the length of the fluid particle. In general, we can conclude that vortices are preserved as time passes, and that they cannot decay or disappear except through the action of viscosity or some other dissipative mechanism. Their persistence is revealed by many phenomena in the atmosphere. For example, one often sees the vortex wake, visualized through the mechanism of condensation trails, remaining for many miles after an airplane has passed.
As a final remark, we point out that in flows where only small disturbances from a fixed uniform stream condition occur, it can be proved that the vorticity f is preserved (to first order in the small perturbation) in the same way that it is in a constant-density fluid, since the effect of variations in density on the quantity f/p is of higher order.