Wings, Round Cylinders and Circulation

The following involves conjecture and conclusions of the author, based on established facts plus observations. A principal objective is to provide an alternative explanation of circulation development, differing from the classical explanation of circulation being a reaction to the starting vortex. (This is a work in progress) The reader is invited to evaluate the following with an open mind, which may or may not reach similar conclusions.

Aerodynamics texts usually mention, with little explanation, the "Karman vortex street," and describe it as a trail of alternating vortices downstream of a round cylinder in passing flow, as indicated in the following figure, and as photographed by Ludwig Prandtl(1) many decades ago.

Associated with each vortex created by the cylinder is transverse deflection of passing flow, and in accordance with Newton's laws, there must be transverse force generated, of direction opposite to that of flow deflection. Such transverse force was noted by the author, at an age of about eight years, when riding in the back seat of a rowboat having a floating minnow bucket tethered behind on a string. As the minnow bucket was being displaced alternatingly right and left in oscillating motion the author reached back to stop the oscillation and was surprised by the force generated. Thus it is clear that a round cylinder can spontaneously generate lift in passing flow, even though of alternating direction. This brings into question the popular belief, that wing lift develops in reaction to viscous effects at its sharp trailing edge. It is also noteworthy that a wing or airfoil leaves a "starting vortex" behind simultaneously with development of lift, much like a round form leaves a vortex behind with each lift reversal. The similarities would seem to invite investigation.

Oscillations generally involve amplification and feedback mechanisms. Whether or not oscillations occur is related to magnitudes of these effects and possible limiting factors. For example, a radio receiver of WW1 vintage photographed at the Dayton, Ohio Air Force Museum, and shown below, has a tuning knob below the dial and a knob labeled "Regeneration" to the right of it. This knob controlled a small amount of signal fed back from output to input of an amplifer stage. With a bit of signal fed back so as to add to incoming signal the amplification was increased. It was a delicate adjustment, for the difference between sufficient feedback for reception and excess feedback leading to oscillations was small.

Since oscillations generally involve feedback it seems reasonable to expect feedback may be involved in cylinder lift oscillations, and if that indeed occurs, then it may also be true that feedback is also involved in wing lift.

As indicated in the following illustration, and photographed(2) by Ludwig Prandtl (1875-1953), flow passing a non-rotating cylinder at low speed (low Reynolds number) divides into opposite paths of equal velocity and path length over the cylinder surface. Any assymetry is opposed by the stabilizing effect of greater viscous drag in the longer and faster path. With equal flows, pressure reduction associated with centrifugal effects in flows curvature would be equal at opposite surfaces, resulting in no net lift.

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Any rotation of the cylinder, even of low speed, will change the balance of viscous drag and promote faster flow over the rearward-moving surface. This will lengthen the faster path and shorten the opposite path, as photographed by Prandtl(3), and indicated in the next figure. Lift toward the longer path of lower pressure is then created.

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Gradients of increased pressure below and decreased pressure above accelerate oncoming flow upward, thereby lowering the "stagnation line" where oncoming flow divides between above-cylinder and below-cylinder, as indicated. In addition, the rear stagnation line, from where merged upper and lower flows depart, is lowered by dynamic pressure of increased upper velocity. Because pressure differences promote oncoming flow upward movement and increased path lengths difference, pressure difference promotes more pressure difference. This is a regenerative feedback effect. The overall effects of slow rotation and feedback as indicated are: upward movement in oncoming air ahead, greater flow velocity above, reduced velocity below and downward departure direction of merging flows at the rear. In total, these effects can be regarded as a rotational movement superimposed on passing flow. We can refer to this rotational addition as "circulation," analogous to "circulation" of a lifting wing.

A regenerative effect of pressure gradients would also exist for the non-rotating cylinder, if difference in pressure were spontaneously initiated, as by random disturbance. However, at low Reynolds number the degenerative effect of viscous drag would tend to restore flows equality. At higher Reynolds numbers (increased flow velocity) increased boundary layer thickness can displace passing flows away from the stabilizing effect of viscous surface drag, as depicted in Prandtl's photographs (4), and as illustrated in the next figure.

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When boundary layer effects displace flows to sufficiently reduce the stabilizing degenerative effect of surface drag, the regenerative feedback effect can become dominant. Then a random disturbance of flows balance can be regeneratively amplified, rapidly increasing the unbalance, circulation and lift until stall occurs. Stall begins when lagging boundary layer accumulation in the faster flow and adverse aft pressure gradient becomes sufficient to prevent further increase of circulation. In stall, residual boundary layer accumulation and pressure gradient initiate a reversed direction of circulation and lift. The new direction of circulation and lift increases rapidly until stall and reversal again occur in continuously repeating cycles.

Accompanying each reversal of circulation an aft vortex is carried rearward in the flow, creating the trail of vortices referred to as a "Karman vortex street." The cylinder transmits no angular momentum to the flow. Equal angular momenta of circulation and aft vortices are created by pressure gradients which form equal opposite-direction moment couples so that circulation and its coincident aft vortex are generated with equal and opposite angular momentum change. This satisfies the "law of conservation of momentum" of physics. Integrated from any position, the equal and opposite momenta add to zero change.

In the case of wing lift, circulation is initiated with pressure difference accompanying redirection of merging aft flows toward the direction pointed by the wing. As in the case of the round cylinder, pressure difference produces upwash which produces more pressure difference as upwash momentum is intercepted. Thus circulation of a wing also involves regeneration. Unlike the cylinder, however, circulation is limited to the level corresponding to flow departure direction as pointed by the wing.

Excess circulation producing flow departure angle greater than pointed by the airfoil would produce an opposing pressure increment manifested in a reversed "starting vortex" having angular momentum equal to angular momentum lost from circulation. Thus the direction pointed by the airfoil both initiates circulation and and controls its magnitude.

Pressure gradients around an airfoil create forward and rearward moment couples. The forward couple tends to accelerate oncoming flow into circulation while the rearward couple is in opposition to circulation.

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In mature circulation the equal couples exert equal and opposite moments on circulation angular momentum, which remains constant. In the beginning, however, circulation growth is driven by flow departure direction. As the forward couple is accelerating oncoming air into circulation, the rearward couple is producing the "starting vortex," as photographed by Prandtl(5) and illustrated here. Created in pressure differences, with no angular momentum coupled from the airfoil, circulation and starting vortex are created with equal and opposite in angular momenta.

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When circulation matures the rearward couple recovers circulation kinetic energy at a rate equal to that imparted to oncoming flow ahead by the forward couple. The removed kinetic energy at the rear is recycled forward through Bernoullian process. Thus the lift process is energy conserving. When circulation is reduced, as by reduction in angle of attack, which reduces forward and rearward couples, excess angular momentum is left behind. In particular, when a lifting airfoil is quickly stopped in forward motion, the entire circulation is shed into a vortex at the rear, as photographed by Prandtl(6).

The foregoing effort to describe circulation and lift in Newtonian terms of action-reaction can undoubtedly be improved. However, even as is, it may be considered more logical than popular explanations based on "equal transit time" or "induction" like that of electromagnetism and the vague explanation that circulation is a reaction to the starting vortex. Circulation and starting vortex are created in balanced equal and opposite pressure couples, in which opposing inertial effects of circulation and starting vortex contain the pressure differences. Thus, in a sense, circulation and starting vortex are each created through inertial opposition of the other, in Newtonian action-reaction, so that each is, in a sense, a reaction to the other.

(1 ) Prandtl and Tietjens, Applied Hydro and Aeromechanics, Figure 60, Dover, ISBN 0-486-60375-X
(2) op. cit., Figure 1.
(3) op. cit., Figure 14
(4) op.cit., Figure 4
(5) op.cit., Figure 54
(6) op.cit. Figure 55

(C) 2006 Gale M. Craig