Circulation Theory

I want to begin by saying that I am not an aeronautical engineer and I make no claim of being an expert, but I see what appears to me to be several serious discrepancies in the circulation theory. I’m sure that there are some details of the theory that I don’t understand or that I simply have wrong and look forward to the clarification on those details.
As I understand it, the circulation theory is the application of the Magnus theory of circulation to a wing. In the Magnus theory, a spinning cylinder drags a whirlpool of air around with it (circulation) that interacts with a flow to create pressure difference. Where the circulation is in the same direction as the flow the resulting velocity is higher and where the circulation is in the opposite direction of the flow the resulting velocity is lower. The difference in velocity is responsible for the pressure difference as per Bernoulli’s equation.
As I recall, the Magnus theory was proven wrong several times, most notably in 1904 when the boundary layer was discovered. That is not to say the Magnus effect doesn’t exist but spinning cylinder does not drag a whirlpool of air around with it. It only drags a very very thin boundary layer of air with it. I’m not asking anyone to take my word on that. Anyone with access to a lathe and a piece of round stock can easily prove this for them self. Just leave 6 or more inches of material hanging out of the chuck so that the pumping action of the chuck doesn’t interfere with the experiment. With 1.5 inch stock spinning at 4000 RPM I was not able to detect any circulation around the rod with my hand or a piece of paper. According to the circulation theory and C.A. Marchaj’s book Aero-Hydrodynamics of Sailing third edition page 186, the velocity of the circulation ½” from the surface of the cylinder should be 1.5in x Pi in/rev x 4000 RPM x 60 min/hr / (12 in/ft x 5280 ft/mile) x .75in / 1in = 13.38 mile per hour. That is quite a stout wind and should be easy to detect, if it were there.
The flow lines depicted in the Magnus theory show a smooth and perfectly symmetrical flow that bends up in front of the cylinder and back down behind the cylinder placing the flow lines back at the exact same level and angle as before the cylinder. I am sure that someone will point out that these depict the flow of a perfect fluid and not a real fluid but on page 185 figure 2.6 of C.A. Marchaj’s book Aero-Hydrodynamics of Sailing third edition the depiction shows what the flow of a real fluid should look like according to the circulation theory. The actual smoke traces look completely different. They show flow lines that are not at all symmetrical and very little if any upwash in front of the cylinder. This high speed photograph of a baseball is an excellent example. http://www.100.nist.gov/photos.htm
Actual experimental data from measuring lift on a spinning cylinder frequently does not match the theoretical lift. I’m not talking about minor differences. I’m talking about significant negative lift being produced when the lift equation predicts positive lift. An example can be seen in C.A. Marchaj’s book Aero-Hydrodynamics of Sailing third edition page 193 figure 2.8.
The Magnus theory also states that in a perfect fluid there would be no drag in creating Magnus lift. I assume by perfect fluid we mean that the fluid has no viscosity. If we assume that the Magnus theory was correct and that lift is the result of the circulation interacting with flow, how could the circulation be produces in a perfect fluid? With no viscosity, the spinning cylinder could not transfer that motion to the fluid so there could not be any circulation and therefore no lift.
From what I have read and my observations, the flow over a spinning cylinder is deflected as it passes the cylinder because of the boundary layers separating asymmetrically from the cylinder. Assuming in this case the surface velocity of the cylinder is greater than the flow velocity. Where the cylinder rotation and the flow are in opposite directions, the flow very near the surface is retarded very quickly causing the boundary layer to became unstable and detach more easily. Where the cylinder rotation and the flow are in the same direction, the flow very near the surface is accelerated making the boundary layer more stable and allows the flow to stay attached longer. This theory actual agrees with the observed flow around a spinning cylinder and I believe is the correct explanation of the Magnus effect.
Once we realize that the lift on a spinning cylinder is not produced in the way that Magnus had theorized and that the fluid flow is deflected slightly by the cylinder, then it is easy to prove that there is an associated drag with the lift. Any one wanting to follow along can find the exact detail in Marchaj’s Aero-Hydrodynamics of Sailing third edition starting about page 371. The angle of the local deflection of flow is referred to as the induced angle or induced angle of incidence. Any time the induced angle is greater than zero, the lift vector is no longer perpendicular to Vo. The lift vector is angled back and therefore has a drag component to it. This is induced drag.
I can’t seem to locate the exact text I am looking for but I believe that there is a statement that “If there is lift and no drag then there must be circulation” to justify applying the circulation theory to a wing. I assume that this statement also implies an infinite wing in a fluid with no viscosity. This is a strange statement to justify the application of circulation to a wing since there is no attempt to prove that a wing can produce lift without drag and since we have already proven that a spinning cylinder can not do this, the statement is completely meaningless. If the statement is meant to imply that a wing produces lift in the same way as the spinning cylinder, then I agree but we have already proven the explanation given in the Magnus theory was wrong. There was no circulation of air around the spinning cylinder so how can there be circulation around a wing? The spinning cylinder created lift by changing the direction of fluid flow as it past the cylinder, a force was applied to the mass of fluid resulting in an acceleration, F=MA. The cylinder experienced an equal but opposite force which we called lift. Because the acceleration of the fluid resulted in a directional change of the flow the induced angle is greater than zero. If the induced angle is greater than zero then the lift vector has a drag component. Lift is created but there is always an accompanying induced drag, even on an infinite wing in a perfect fluid.
The attempt to apply the circulation theory to a wing creates mathematical problems and formulas that contradict the basic concept of circulation. If we refer back to Marchaj’s Aero-Hydrodynamics of Sailing third edition beginning on page 371 the induced angle is discussed as well as the angle of incidence. On page 372 about mid page there is a sentence “For a foil of infinite aspect ratio, i.e. in two dimensional flow, the induced angle of incidence is zero. This would have to be true for it to have no induced drag which is the basis for the circulation theory. On page 373 near the bottom we find the sentence “Applying the action and reaction principal one may find that lift generated by a foil can only be provided by a downward acceleration to the fluid particles affected by the presents of the foil.” We go on to derive the formula for lift on the next page. As you can see the lift is proportional to the induced angle. Now we have a contradiction. We can not produce lift without drag because the angle of incidence is zero on an infinite wing and lift is a function of the angle of incidence so lift would have to be zero also. The only time induced drag is zero is when lift is zero. The fact that no lift could occur on an infinite wing according to the circulation theory shouldn’t be a surprise. If we look at the 2D explanation of the circulation theory, the upwash in front is exactly equal to the downwash behind the wing so no net force can occur.
We can simulate an infinite wing in a wind tunnel by allowing the wings to extend to the walls of the wind tunnel. When we do this I would bet my last dollar the wing will create lift, the induced angle will be greater than zero, and there will be induced drag. Yet according to the circulation theory none of these things should happen.
In C.A. Marchaj’s book Aero-Hydrodynamics of Sailing third edition page 370 figure 2.98 we have a drawing of how the distribution of down wash should look behind a wing according to the circulation theory. You can see from the drawing that the greatest downwash is near the tips of the wing. On page 364 photo 2.28 we have some actual photos of actual downwash behind a wing using aluminum powder to make it visible. From the photos it is obvious that the greatest downwash is not at the tips but in the middle of the wing. Clearly the theoretical downwash according to the circulation theory looks nothing like the actual downwash behind a wing.
Any one or two of these discrepancies wouldn’t bother me so much but considering the number of discrepancies I have to ask if this is going to be just like the classical theory of lift with the unequal length paths. In that theory if two particles are side by side at the leading edge and separate, one traveling over the wing and the other under the wing, they will have to meet back up at the trailing edge. The one traveling over the top of the wing travels a longer distance so it must be traveling faster. That theory was proven wrong over 100 years ago but it is still showing up in text book today. There is absolutely no reason to think that the two particles will have to meet back up at the trailing edge.
I realize that the circulation theory is like a religion to some people and that my blasphemy is unforgivable but if anyone is going to convince me that the circulation theory is correct it will have to be with facts and proof not emotional arguments. I realize that the circulation theory is being taught in every fluids class and is in almost every technical book as the correct explanation of lift but try to remember that just a few years ago the classical theory of lift was being taught as the correct explanation of lift. I, like most people, swallowed the classical theory hook, line and sinker, without ever considering it could be wrong. Not this time. I need all the loose ends tied up and all the discrepancies explained before I buy into this one.
One other observation I had was, if circulation creates lift, why would so many jets have the engines mounted below the wing? Wouldn’t mounting the engines under the wing accelerate the air under the wing more than on top of the wing? Wouldn’t this be opposite the direction of circulation? Wouldn’t that reduce or completely destroy circulation and therefore lift?

 

Well, on lift created by wings here is the Newton's theory fighting back Circulation's and Bernouilli's ones:
http://www.aa.washington.edu/faculty/eberhardt/lift.htm

Here a nice explanation on Bernouilli's and Circulation:
http://www.av8n.com//how/htm/airfoils.html

And here a discussion on Bernoulli's vs Newton's:
http://www.grc.nasa.gov/WWW/K-12/airplane/bernnew.html

From this last:
"Arguments arise because people mis-apply Bernoulli and Newton's equations and because they over-simplify the description of the problem of aerodynamic lift. The most popular incorrect theory of lift arises from a mis-application of Bernoulli's equation. The theory is known as the "equal transit time" or "longer path" theory which states that wings are designed with the upper surface longer than the lower surface, to generate higher velocities on the upper surface because the molecules of gas on the upper surface have to reach the trailing edge at the same time as the molecules on the lower surface. The theory then invokes Bernoulli's equation to explain lower pressure on the upper surface and higher pressure on the lower surface resulting in a lift force. The error in this theory involves the specification of the velocity on the upper surface. In reality, the velocity on the upper surface of a lifting wing is much higher than the velocity which produces an equal transit time. If we know the correct velocity distribution, we can use Bernoulli's equation to get the pressure, then use the pressure to determine the force. But the equal transit velocity is not the correct velocity. Another incorrect theory uses a Venturi flow to try to determine the velocity. But this also gives the wrong answer since a wing section isn't really half a Venturi nozzle. There is also an incorrect theory which uses Newton's third law applied to the bottom surface of a wing. This theory equates aerodynamic lift to a stone skipping across the water. It neglects the physical reality that both the lower and upper surface of a wing contribute to the turning of a flow of gas.

The real details of how an object generates lift are very complex and do not lend themselves to simplification. For a gas, we have to simultaneously conserve the mass, momentum, and energy in the flow. Newton's laws of motion are statements concerning the conservation of momentum. Bernoulli's equation is derived by considering conservation of energy. So both of these equations are satisfied in the generation of lift; both are correct. The conservation of mass introduces a lot of complexity into the analysis and understanding of aerodynamic problems. For example, from the conservation of mass, a change in the velocity of a gas in one direction results in a change in the velocity of the gas in a direction perpendicular to the original change. This is very different from the motion of solids, on which we base most of our experiences in physics. The simultaneous conservation of mass, momentum, and energy of a fluid (while neglecting the effects of air viscosity) are called the Euler Equations after Leonard Euler. Euler was a student of Johann Bernoulli, Daniel's father, and for a time had worked with Daniel Bernoulli in St. Petersburg. If we include the effects of viscosity, we have the Navier-Stokes Equations which are named after two independent researchers in France and in England. To truly understand the details of the generation of lift, one has to have a good working knowledge of the Euler Equations."

 

I'm a long way from fully understanding circulation theory but I've found a book that was recommended to me by an aerodynamicist: "Introduction to Aerodynamics-Basic Aerodynamics in Newtonian Terms" by Gale Craig.
From the introduction:" False equal transit time explanation of lift is not taught to college level students of aerodynamics who must calculate real lift forces. Unfortunately, the texts involved in these teachings are generally too complex and abstract for concise learning of basic princibles,while the lower level simplistic and false teachings of equal transit time are useless or worse. Thus there is a need for basic aerodynamics description in Newtonian terms. This effort is intended for application to that need."
witten in 2002 ISBN 0-9646806-3-7

 

From Wikipedia:
"The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are a set of equations that describe the motion of fluid substances such as liquids and gases. These equations establish that changes in momentum (acceleration) of fluid particles are simply the product of changes in pressure and dissipative viscous forces (similar to friction) acting inside the fluid. These viscous forces originate in molecular interactions and dictate how sticky (viscous) a fluid is. Thus, the Navier-Stokes equations are a dynamical statement of the balance of forces acting at any given region of the fluid.

They are one of the most useful sets of equations because they describe the physics of a large number of phenomena of academic and economic interest. They are used to model weather, ocean currents, water flow in a pipe, motion of stars inside a galaxy, flow around an airfoil (wing). They are also used in the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of the effects of pollution, etc. Coupled with Maxwells Equations they can be used to model and study magnetohydrodynamics.

The Navier-Stokes equations are differential equations which describe the motion of a fluid. These equations, unlike algebraic equations, do not seek to establish a relation among the variables of interest (e.g. velocity and pressure), rather they establish relations among the rates of change or fluxes of these quantities. In mathematical terms these rates correspond to their derivatives. Thus, the Navier-Stokes equations for the most simple case of an ideal fluid with zero viscosity states that acceleration (the rate of change of velocity) is proportional to the derivative of internal pressure.

This means that solutions of the Navier-Stokes equations for a given physical problem must be sought with the help of calculus. In practical terms only the simplest cases can be solved in this way and their exact solution is known. These cases often involve non turbulent flow in steady state (flow does not change with time) in which the viscosity of the fluid is large or its velocity is small (small Reynolds number).

For more complex situations, such as... lift in a wing, solutions of the Navier-Stokes equations must be found with the help of computers. This is a field of sciences by its own called computational fluid dynamics."

Tom Speer may explain you, much better than I could do, the intrincancies on these matters.

Cheers

 

Desely; you are correct..it has logical problems. It is used because, for the most part, it works in most situations. But it must always be remebered that circulation theory is only a mathematical contrivance...read Lanchester (1906), Prandtl (1911-18), and Lerbs (1952)...that falls out of the Kutta-Jaukowski condition; it is not a true representation of physics. Just pull the thread to the end on how  (capital Gamma) is calculated.

Edit to add, some of the problems that you are having with your examples in attempting to look at the Magnus effect are Renyolds based, and cannot be seen in real applications beyond the critical Renyolds numbers.. i.e. the massless, inviscid, irrotational assumption does not hold and you have to look to the experimental corrections.

 

This fellow says that if there is no viscosity there is no lift.
http://www.arvelgentry.com/origins_of_lift.htm