The Vortex


Contents

  1. Introduction
  2. Velocity
  3. Vorticity
  4. Stream Function
  5. Examples of Vortices
  6. Some Vortex Lore
  7. References

Introduction

The Latin word vertex, verticis (m.) means "eddy" or "whirlwind," and also the top of the head, where the hair makes a small whirl, and by extension the top or summit of anything. "Vortex" is an alternative spelling, the one English has chosen to represent the literal meaning. The plural can be "vortices" or "vortexes" as you wish.

The results of a mathematical study of the vortex were published by Hermann von Helmholtz (1821-1894) in 1858, and by William Thomson, later Lord Kelvin (1824-1907), in 1869. The names of these two giants of classical physics are jointly associated with many of the principal concepts of vortex motion. The theory of vortex motion is not easy, but it has received a great deal of attention up to the present time.

Vortices are important in meteorology, since they are encounted in the vicinity of high and low pressure areas, and in whirls from dust devils to hurricanes, and quite vividly in tornadoes. They cause the humming of wires in the wind, and are shed from the wingtips of aircraft. The vortex that forms at the drain hole of a sink is an everyday experience. This article will explore some of these phenomena, using the mathematical analysis that is essential to understanding, but attempting qualitative explanations as often as possible.

Velocity

We will describe the motion of a fluid by its velocity field v(x,y,z,t), the three components of which are functions of the rectangular coordinates (x,y,z) as well as of the time t. In most cases we shall restrict ourselves to two dimensions (x,y) with no variation in the z-direction, because the analysis is much easier, and the atmosphere is approximately two-dimensional anyway. At any fixed time t, we can draw continuous lines that are always parallel to v, called streamlines, that never cross one another. As t varies, the streamlines will, in general, move unless the velocity field is steady--that is, unless the velocity at any point is constant.

If we concentrate on a small element of fluid dV, it will move as it is carried by the instantaneous velocity at the point that it occupies. If the flow is steady, it will move on a streamline, but this is not generally true. The time rate of change of any quantity relating to the element dV will be expressed by d/dt, called the substantial derivative. The acceleration, for example, will be dv/dt. We can express d/dt as the sum of the change at the point (x,y,z) as t varies, or ∂/∂t, and the change due to moving from point (x,y,z) to (x+dx,y+dy,z+dz) in unit time, or v·grad. [Browsers do not support the inverted delta, or del, commonly used to represent the gradient operator, so the older notation will be used here. The gradient is the vector directional derivative.] Therefore, d/dt = ∂/∂t + v·grad. For more information on this important point, consult the References.

The fluids we are considering can be gases or liquids. The important thing is that the stresses in them are either a hydrostatic pressure p(x,y,z,t) or a shear stress due to a velocity gradient, which is called viscosity. In most cases, our fluids can be considered as incompressible fluids of zero viscosity, or ideal fluids. This is as true of air as of water, if the conditions are right. A fluid can be considered approximately incompressible if the flow velocities are small compared to the speed of sound (340 m/s for air, 1500 m/s for water, at the earth's surface). Viscosity can be neglected if the dimensionless Reynolds number R = ρLU/μ is much larger than unity. Here, ρ is the density, μ the viscosity, L a length and U a velocity characteristic of the scale of variation in the motion.

It came as a surprise that many mathematical solutions for the flow of an ideal fluid, though quite correctly done, gave results unlike anything observed in practice. An excellent example is the steady motion of a sphere in a fluid at rest. The reason is that although most of the fluid did, indeed, act ideally, viscosity cannot be neglected near boundaries with solid surfaces. Therefore, the boundary conditions used in the solutions were unrealistic and gave unrealistic answers. This, of course, can be fixed, but it greatly complicates mathematical analysis.

The moving fluid, whether incompressible or compressible, must conserve mass, which puts very strict conditions on allowable velocity fields. This condition is the equation of continuity, ∂ρ/∂t + div (ρv) = 0. The divergence theorem can be used to show that this equation means that the rate of increase in mass in any volume V is the rate at which it enters through the surface S of the volume V. For an incompressible fluid, it reduces to simply div v = 0. (The divergence is del dot, the ratio of flux to density for an infinitesimal volume.)

For a nonviscous fluid, the pressure exerts a force of -grad p per unit volume. There is also a gravitational force ρg per unit volume. We'll write only the pressure force in what follows. It is easy to include any other body forces that may be required should we need them. The fluid obeys Newton's law of motion, so ρdv/dt = -grad p. This is the equation of motion. We generally use it not to determine the flow, but to determine the pressure when the flow is known.

The earth rotates with an angular velocity Ω = 7.292 x 10-5 rad/s, so it is not an inertial system. If we wish to assume that it is, then we must add the centrifugal force Ω x (Ω x r) and the Coriolis force 2Ω x v, each per unit mass, to the other forces acting. The centrifugal force is included in the gravitational force. The Coriolis force in the horizontal plane is 2Ωv sin φ, where φ is the latitude. It vanishes on the equator and is a maximum at the poles. North of the equator, it tends to turn a velocity to the right, and does the opposite to the south of the equator. A 2000 kg vehicle moving at 100 kmph at latitude 40° is acted upon by a Coriolis force of about 6 N, while its weight is 19,600 N. The centrifugal force, for comparison, is about 52 N. Although the Coriolis force acting on a discrete object, such as the vehicle, is negligible, the effects of the force acting throughout a large body of moving air are not, and tend to dominate the wind directions. The Coriolis force cannot speed up a wind, but it can change its direction. The dimensionless Rossby Number Ro = U/LΩ indicates the relative importance of the Coriolis force in a motion where U is a characteristic velocity and L a characteristic length. If the Rossby number is much less than unity, the Coriolis force will play an important role. For a typical low-pressure system in the atmosphere, U = 5 m/s, L = 500 km, which gives Ro = 0.137, so the Coriolis force is important. For a light aircraft, U = 80 m/s, L = 8 m, so Ro = 1.37 x 105, and the Coriolis force can be neglected.

Vorticity

Imagine that you have a small paddlewheel that you can stick in a flow without disturbing it too much, and can read its speed of rotation on its axis. In large-scale flows this can almost be done, but it is still mainly a thought experiment. It can even be imagined that the paddlewheel is carried along with the flow as you observe it, and small bits on the surface of water, for example, are a practical realization. If the paddlewheel rotates, then the fluid has vorticity at that point. By changing the orientation of the axis, the direction of maximum speed of rotation can be determined. It is, in principle, possible to find the vorticity field ω(x,y,z,t) in magnitude and direction at any point. Lines drawn parallel to ω are called vortex lines, and their density can express the strength of the rotation, just as streamlines define the velocity field and magnetic field lines define a magnetic field. They are not real, but greatly aid visualization.

The line integral of the component of velocity tangent to a closed curve is called the circulation, and clearly measures the amount of rotation. Let's take a small circle surrounding an area a = πr2 as the path of integration. If the angular velocity is ω, then the circulation will be 2πr x ωr = 2πωr2 = 2ωa. Thus, the circulation per unit area is directly proportional to the angular velocity of rotation.

There is a useful relation in vector calculus called Stokes's Theorem, which states that the circulation of a vector about any curve C is the surface integral of the curl (del cross) of the vector over the area enclosed by C. If this is applied to the present case, we find that curl v = 2ω, so that the rotation is half the curl of the velocity. Some very important results flow from this relation. Since the divergence of the curl of a vector is identically zero, div ω = 0. This means that if we consider a tube whose walls are parallel to ω, called a vortex tube, then this tube has the same strength, that is, product of the area and ω, at any point. Among other things, this means that the vortex tube cannot end within the fluid, and must either close or go to a boundary.

A very important theorem, due to Helmholtz and Kelvin, states that the substantial derivative of the circulation about any curve C in a fluid of zero viscosity (and some other conditions, of which incompressibility is good enough) vanishes. This applies to any curve C on the walls of a vortex tube, or on any surface parallel to the vorticity, and implies that vortex lines are carried with the fluid, the strength at any point remains constant, and, as we have already seen, a vortex line cannot end within the fluid.

This theorem is often used to establish the fact that if the initial state of a fluid to which it applies has no rotation, that is, that curl v = 0 everywhere, the fluid will remain irrotational as it moves. Conversely, it means that if rotation exists, it will persist for all time. Boundaries, at which the theorem does not apply, change this is practical cases. Vorticity can originate at surfaces, and can die out due to friction at surfaces. The Coriolis force due to the earth's rotation can also produce circulation, and is a very important source of vorticity.

Stream Function

It is not an easy matter to find velocity fields that satisfy the equation of continuity. If the motion is irrotational, curl v = 0, then we know that v can be derived from a potential function φ, v = -grad φ. If the fluid is incompressible, then the equation of continuity div v = 0 becomes div grad φ = 0. [div grad is the Laplacian, the sum of the second derivatives.] We know many ways of solving Laplace's equation, so they can be used here. This method is used in the case of the electrostatic field, with great success. Its greatest benefit is that only one scalar function φ must be determined, not three interrelated vector functions v.

We cannot use this method for rotational flow, since any velocity field derived from a potential φ is necessarily irrotational. Another method, for the special case of two-dimensional flows, will now be presented, the method of the stream function. It is analogous to the use of the vector potential in discussing the magnetic fields of currents. Consider a vector field A = kA(x,y). The function A(x,y) may also vary with the time, but we shall not include this, for simplicity. We suppose that v is derived from A by the rule v = curl A. Writing this out, we find that v = i(∂A/∂y) - j(∂A/∂x), so that vx = ∂A/∂y and vy = -∂A/∂x. Now, writing out the continuity equation div v = 0, we find that it is automatically satisfied for any function A!

To find the relationship between A and the vorticity, write out the z-component of curl v, to find that 2ω = ∂vy/∂x - ∂vx/∂y - -div grad A. In two-dimensional motion, the vorticity can only be parallel to the z-axis, since the velocity must lie in the xy-plane and is independent of z. The vector potential of a magnetic field satisfies the same equation, where the current takes the place of the vorticity. This is Helmholtz's equation, and methods are available for its solution, as in the case of Laplace's equation. The one scalar function A allows us to find the two interrelated components of the velocity.

A can be given a vivid interpretation. Consider the line integral of the normal component of the velocity to a curve from some point O, say the origin, to an arbitrary point P in the xy-plane. This integral will be the volume rate of flow of the fluid across the path of integration between O and P. Let us call the value of this line integral the quantity F, where dF = v sin θ ds, where ds is distance along the curve, and θ is the angle between v and ds, as shown in the figure. The integrand is then dF = vydx - vxdy = -(∂A/∂x)dx - (∂A/∂y)dy = -dA. Therefore, F = -A. The function F is called the stream function. It should be clear that the value of F depends only on the points O and P, since the flow will be the same across any line joining the two points. Or, what is the same thing, the flow across the closed path from O to P and back to O will be zero.

Consider two points P and P' on the same streamline, as shown at the left. The flow across any curve OP will be the same as across OP', so that the equation of the streamlines is F = const. If the flow is irrotational, then A will satisfy Laplace's equation, and solve the problem as well as the velocity potential φ In fact, A and φ are closely related; they are conjugate functions. In two dimensions, they are the real and imaginary parts of a complex analytic function. The streamlines A = const are orthogonal to the equipotentials φ = const.


Examples of Vortices

As the simplest example, consider a uniform flow v = vx = const. The streamlines are lines parallel to the x-axis, and the stream function F = -A = vy. If the velocity increases linearly with y, so that v = ay. Then F = -A = ay2/2. Now div grad A = -a = -2ω, or ω = a/2. This is a rotational flow with a constant vorticity density.

Suppose the fluid is rotating rigidly with angular velocity ω about the origin, as illustrated in the figure at the right. Then A - -ωr2/2, by integrating the flow across a radius. In cylindrical coordinates, div grad A = (1/r)(d/dr)(rdA/dr) = -2ω, exactly as might be expected. Again, the vorticity density is constant and equal to ω.

Now we consider a very important case, the circular rectilinear vortex. Let the vorticity density be constant within a radius ro of the origin, so that the velocity is ωr in this region. Outside ro, let the velocity be tangential and equal to C/r, where C is a constant. These velocity distributions obviously satisfy the equation of continuity, and could be found from Helmholtz's and Laplace's equations, respectively with the boundary conditions of zero velocity at the origin and at infinity, together with cylindrical symmetry. We find A = -C ln(r/ro) for the exterior region, and from this that div grad A = 0. For r > ro, the motion is irrotational. This may come as a surprise, but it is true. The little paddlewheel would not rotate in this region, though the fluid is moving in circles.

The two solutions must match at r = ro, which means that C = ωro2. The figure at the left shows the two regions of the vortex, the rotational region where the fluid rotates rigidly and its velocity increases linearly with r, and the irrotational region in which the velocity decreases inversely proportionally to r. We now know A everywhere; it increases quadratically in the rotational region, and decreases logarithmically in the irrotational region. The velocity anywhere can be obtained from its derivatives. Since it is a function of r only, the velocity is purely tangential.

Next, we want to find the pressure. To do this, we equate the centripetal acceleration to the pressure force: -ρv2/r = -dp/dr. The figure at the right shows that this equation can easily be found from first principles and Newton's Law, f = ma. The differential equation is easily integrated in the two regions of the vortex. In the irrotational region, p = p - (ρ/2)(voro/r)2, where p is the pressure at infinity. In the core of the vortex, we find dp/dr = ρω2r. When this is integrated, and the constant of integration chosen to make the pressures equal at the surface of the core, we find p = p - ρvo2 + ρv2/2. The pressure may fall to zero before the origin is reached. In this case, an empty "eye" is produced that extends from the origin out to this radius.

If the fluid is a liquid with a free surface, the free surface sinks as the pressure is reduced. The similarity to the vortex at a drain hole is close, but there are some important differences. In the actual vortex, fluid is spiralling in to the centre where it is discharged. An "eye" may be formed when the rotation becomes sufficient to draw the free surface down near the bottom. The depressed surface can resemble a tornado on a small scale.

In a hurricane, many other factors may enter, but the large-scale behavior is similar. A hurricane extends from the surface to the tropopause, 12-15 km high, while it is perhaps 1000 km in diameter, so the two-dimensional approximation is reasonably close. The vorticity is packed into the wall cloud surrounding the eye. As in the drain vortex, fluid is arriving on the surface, but instead of going down the drain is lifted abruptly in the wall cloud. The central area is surrounded by a large region of approximately irrotational flow. The pressure in the eye is not zero, however. Let's say that it is 900 mb, and see what velocities result. Now p - p = 104 Pa = ρvo2/2. Since ρ is about 1.1 kg/m3, we find vo = 135 m/s or 303 mph, a quite reasonable result.

Some Vortex Lore

Attempts have been made to use an idealized vortex line to understand certain vortex phenomena. A vortex line would have a finite strength, vorticity times area, but would have zero area, as a dipole has zero length. The result has unpleasant properties, however, since it tends to move at infinite speed unless absolutely straight. Another singular arrangement is a vortex sheet, which does behave more reasonably. Suppose there is a horizontal interface between horizontal winds of different velocity. If the change in velocity takes place over a small interval, this is a good approximation to a vortex sheet. Helmholtz and Kelvin showed that such a sheet was unstable to small perturbations. This instability has actually been observed, creating the disturbance called a Kelvin-Helmholtz "wave," although it is not really a wave. If there is a distinct interface between air masses with different properties, such as temperature, there may indeed be a surface wave excited by this instability.

A vortex pair consists of two equal and opposite vortices, separated by a certain distance, as shown at the left. The streamlines are exactly like the magnetic field of parallel current filaments carrying opposite currents, and are circles with their centres on the line joining the two filaments. This is the velocity field in a system in which the vortices are at rest. Adding a downward velocity so that the velocity at the midpoint is zero gives the velocities where the fluid is at rest. The pair tends to move together, in the same direction as their outer velocities, so that the absolute velocity halfway between them is zero. Their velocity of motion is the strength of one vortex divided by 2π times the distance between them. One should see this effect with a high and a low not far apart; they should move together making good progress.

Two vortices of the same strength rotate about a point halfway between them, while not making any forward progress. The case is shown in the figure at the right. Two lows in the northern hemisphere tend to revolve about each other in an anticlockwise sense. As I write this, two lows in the Atlantic off Newfoundland are executing such a dance, pictured below. Multiple vortices are sometimes seen in strong tornadoes, and these vortices appear to rotate about each other in this way. Parallel electric currents attract one another because of their magnetic interaction, but vortices do not behave similarly. There must be some tendency for vortices in the same direction to coalesce and strengthen, but I have not yet found any strong theoretical evidence for this. Practically, they do seem to attract in this way, but I plan to watch the behavior of closely neighboring lows in more detail.

The two lows mentioned above are shown in the figure. Each chart is 6 hours apart, beginning at 00Z on 6 March 2003. The centers of the lows are marked with circles with crosses in them. The two lows approach one another, with the line joining their centres rotating anticlockwise. The lows are deepening slowly as they approach. At 18Z they have coalesced into a single deep low.

A cylinder in a fluid flow sheds vortices of opposite senses alternately from its two sides, making a vortex street, also called a von Kármán street, behind it. These "streets" are seen on all scales, from flow in brooks to the atmosphere. Alternating transverse forces act on the cylinder, which can make it vibrate. This is the reason why wires "sing" in the wind. The situation is shown in the figure on the right, where the cylinder is stationary in a stream blowing from right to left. Behind the cylinder is a turbulent wake of slowed air. Two vortex sheets are formed on each side of the wake, and their instability results in the vortex streets for a certain range of Reynolds numbers. Vortices are formed in a Kelvin-Helmholtz instability in the same way. Analogous effects occur for a sphere, but here the vortices form rings.

The frequency at which vortices detach from the cylinder is given in Daugherty (see References) as f = 0.198(V/D)(1 - 19.7/Re), where V is the velocity of the flow, D the diameter of the cylinder, and Re the Reynolds number DV/ν. For air, ν = 0.15 cm2/s. The vortex street is formed for Re from about 120 up to 20,000, in the region of the laminar boundary layer. A wind velocity of 10 m/s (22 mph) would give a frequency of about 1 kHz. The intensity of the "singing" of a wire is greatly increased by resonance, of course.

An aircraft is supported by the reaction of the air forced downwards by its wings, which comes in from the sides. Two strong linear vortices are formed at each wingtip, as shown, which trail behind the aircraft and persist for some time, as vortices tend to do. These vortices can often be seen in moist air, which condenses in a helix along the vortex axes. Sometimes they appear in contrails, making two parallel lines of ice crystals. They also occur at propeller tips, often making a prominent helix as the aircraft moves forward.

A vortex line that makes a complete loop is often seen. Smoke rings are a very good example of this, and they show the dynamics of a ring vortex. A ring vortex moves through still air in the expected direction, and does not change much in size. Small effects, such as viscosity, do have an effect, and it seems that a dissipating smoke ring tends to increase in diameter. Two vortex rings moving in the same direction and close behind one another will affect the flows in both. The front ring will expand and slow down, while the rear ring will shrink and speed up, and may even move forward through the centre of the front ring. Then the rings have exchanged places, and the same evolution may occur, causing the rings to change places again. This leapfrogging is difficult, but delightful, to observe. Smokers who can blow smoke rings may want to try to realize it.

References

H. Lamb, Hydrodynamics, 6th ed. (New York: Dover, 1945). Chapter VII.

G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge: Cambridge University Press, 1970). Chapter 7.

L. Page, Introduction to Theoretical Physics, 3rd ed. (New York: D. Van Nostrand, 1952). Chapter V.

R. L. Daugherty and J. B. Franzini, Fluid Mechanics, 6th ed. (New York: McGraw-Hill, 1965). p. 428.


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Composed by J. B. Calvert
Created 6 March 2003
Last revised 24 July 2003