Why KZ Is Incorrect

Incorrect unphysical explanation of lift by Kutta-Zhukovsky

1. Arguments against KZ

The Kutta-Zhukovsky lift theory with lift generated by large scale circulation around the wing section determined by the Kutta condition of zero velocity at the trailing edge, does not describe actual physics:

  1. The KZ solution has zero drag with high pressure at separation, which is not observed in real flow.
  2. The circulation in real flow varies with the contour and shows to be small for a contour close to the wing surface.
  3. The argument that the Kutta condition is enforced by viscosity lacks physical rationale.
  4. The Kutta condition is in fact both unphysical and unmathematical.
  5. The KZ solution does not include stall and is thus dangerous to use.

KZ theory is thus a “mathematical trick” without physical correspondence. This was clear from start and KZ has survived only because a correct physical explanation of lift has been lacking.  When now this is available KZ has no longer any role to play.

Below you find a further discussion of 1. – 4. with the following standard text books as background:

The authors confess that they find the KZ theory perplexing and without scientific logic, but present it anyway because something has to be presented and KZ theory is the best they can come up with.

KZ theory represents at least two of Aristotle’s logical fallacies: Affirming the Consequent and Argument from Ignorance, as shown in detail below.

2. The Penguin Logic of KZ

KZ theory is an example of Penguin Logic which has infected modern science according to the following reasoning showing Aristotle’s logical fallacy of Affirming the Consequent and deducing the assumption:

  • If we assume that (A) there is large scale circulation then (B) lift is generated.
  • We observe (B) and conclude that (A) is true, that is that there is circulation.
  • We now have a theory explaining lift from circulation, which is also confirmed by experiment, without having to explain from where circulation is coming.

We may compare with the following “theory” which is also common, but less sophisticated:

  • If (A) there is lower pressure above than below the wing, then (B) lift is generated.,
  • We observe (B), and conclude (A), that is that there is lower pressure above.
  • We now have a theory explaining lift from pressure difference, which is also confirmed by experiment, without having to explain why the pressure is lower above which is the real problem.

3. Experimental Verification of KZ is Shaky

The key experimental evidence put forward to support the KZ circulation theory is still today the 1926 article An Investigation of the Flow of Air around and Airfoil of Infinite Span, by Bryant and Williams. The case considered is close to stall with a large wake and it is noted that the contour to determine the circulation cannot be chosen close to the airfoil and should cut the wake perpendicularly. The evidence is thus very shaky and the results are obviously twisted to confirm KZ as a given truth.

4. Contradictions of KZ Circulation Theory (John D. Anderson)

We quote from and comment on the essential Chapter 4 Incompressible Flow Over Airfoils from John D Anderson’s reference text Fundamentals of Aerodynamics:

4.1 2d Flow around Airfoil Sections

  • In the period 1912-1918, the analysis of airplane wings took a giant step forward when Ludwig Prandtl and his colleagues at Gottingen, Germany, showed that the aerodynamic considerationof wings could be split into two parts: (1) the study of the section of a wing-an airfoil-and (2) the modification of such airfoil properties to account for the complete, finite wing. 
  • This approach is still used today; indeed, the theoretical calculation and experimental measurement of modern airfoil properties have been a major part of the aeronautics research carried out by the National Aeronautics and Space Administration (NASA) in the 1970s and 1980s.

Anderson here presents the basic assumption of the Old Theory: 2d flow around an airfoil section can be used as a meaningful simplification of real flow around a real 3d airfoil.

We show in our work that this is not the case since the real flow is 3d with in particular a completely crucial  3d separation pattern which cannot be captured by restriction to 2d. This is the fundamental error of The Old Theory which has led to the invention of a circulation theory of flight without connection to reality and thus without real predictive value. The New Theory on the other hand offers a wealth of predictions and thus measured against Popper’s falsifiability criterion is a scientific theory, while the Old Theory is not.

4.2 Introduction of Vortex Sheet

  • The concept of replacing the airfoil surface in Fig. 4.10 with a vortex sheet is more than just a mathematical device; it also has physical significance. In real life, there is a thin boundary layer on the surface, due to the action of friction between the surface and the airflow (see Fig. 1.28). 
  • This boundary layer is a highly viscous region in which the large velocity gradients produce substantial vorticity; i.e., V’ xV is finite within the boundary layer. (Review Sec. 2.12 for a discussion of vorticity.) Hence, in real life, there is a distribution of vorticity along the airfoil surface due to viscous effects, and our philosophy of replacingthe airfoil surface with a vortex sheet (such as in Fig. 4.10) can be construed as a way of modeling this effect in an inviscid flow.

We here see the effect of a no-slip boundary layer generating transversal vorticity distributed around the wing section. Here is a typical picture of cross-section of a vortex sheet with transversal vorticity distributed around the wing surface supposed generated by friction in a boundary layer:

4.3 The  Kutta Condition: Unphysical and Unmathematical

  • Reflecting on Figs. 4.12 and 4.13, we emphasize again that in establishing the steady flow over a given airfoil at a given angle of attack, nature adopts that particular value of circulation (f2 in Fig. 4.12) which results in the flow leaving smoothly at the trailing edge. This observation was first made and used in a theoretical analysis by the German mathematician M. Wilhelm Kutta in 1902. Therefore, it has become known as the Kutta condition.

The Kutta condition demands zero flow velocity at the trailing edge which requires a strong pressure gradient to bring the flow to stagnation and thus a high pressure at the trailing edge, but the physics of this high pressure is missing, and accordingly cannot be observed experimentally.

The Kutta condition is thus unphysical, and it is also unmathematical by posing a Dirichlet zero velocity condition for an Euler equation with slip elsewhere. Thus both physics and mathematics give in protests to the celebrated Kutta condition.

4.4 The Mechanism: Starting Vortex

  • This brings us to the summary as well as the crux of this section. As the flow over an airfoil is started, the large velocity gradients at the sharp trailing edge result in the formation of a region of intense vorticity which rolls up downstream of the trailing edge, forming the starting vortex. This starting vortex has associated with it a counterclockwise circulation. Therefore, as an equal-and opposite reaction, a clockwise circulation around the airfoil is generated. As the starting process continues, vorticity from the trailing edge is constantly fed into the starting vortex, making it stronger with a consequent larger counterclockwisecirculation. In turn, the clockwise circulation around the airfoil becomes stronger,making the flow at the trailing edge more closely approach the Kutta condition, thus weakening the vorticity shed from the trailing edge. Finally, the starting vortex builds up to just the right strength such that the equal-and-oppositeclockwise circulation around the airfoil leads to smooth flow from the trailing edge (the Kutta condition is exactly satisfied). When this happens, the vorticity shed from the leading edge becomes zero, the starting vortex no longer grows instrength, and a steady circulation exists around the airfoil.

Here large scale circulation is explained as Nature’s way of satisfying Kelvin’s theorem dictate of zero total vorticity.

4.5 Non-Physical Pressure Distribution

In the picture below we see the pressure distribution on the upper and owed surface of a wing section at an angle of attack of 9 degrees. We see high pressure at the trailing edge which is not observed in real flow: The KZ solution has zero drag and is therefore non-physical.

Compare with experimental measurements presented by NASA:

We see that (taking scaling into account) that the pressure at the trailing edge of the KZ solution is about 1, while measurements show 0.1 – 0.2. KZ is completely wrong at the trailing edge and it is what happens there which makes flight possible.

4.6 Without Friction Could We Have Lift? (Section 4.5.1 in book)

  • We noted that lift on an airfoil is primarily due to the surface pressure distribution…. Hence pressure is the dominant player in the generation of lift, and shear stress has a negligible effect on lift. It is for this reason that the lift of an airfoil below stall can be accurately predicted by inviscid theories…
  • However, if we lived in a perfectly inviscid world, an airfoil could not produce lift. Indeed, the  presence of friction is the very reason why we have lift. These sound like strange, even contradictory statements. What is going on here? The anwer is that in real life, the way that nature insures that the flow will leave smoothy at the trailing edge….is that the viscous boundary layer remains attached to the surface all the way to the trailing edge. Nature enforces the Kutta condition by friction. If there were no boundary layer (i.e. no friction), there would be no physical mechanism in the real world to achieve the Kutta condition. 

We see here the contradictory logic of the Kutta-Zhukovsky circulation theory of lift: Viscosity is both inessential and completely essential. Anderson’s question What is going on here? is fully adequate and clearly expresses the non-physical character of 2d circulation theory of lift.

Anderson’s argument that viscosity is neceassry to enforce the Kutta condition is an example of Aristotle’s logical fallacy of Argument from Ignorance: Since the Kutta condition must be satisfied (which is not the case by the way), there must be some cause to this effect and since we cannot come up with anything which is not viscosity, it must be viscosity.

(Notice that Section 4.5.1 is added in the 2001 edition of the book, apparently in an effort to meet an obvious and pertinent question).

4.7 Unsuccessful Interview with John D. Anderson

Under Interviews you find an unsuccessful attempt of mine to ask John D some perfecty innocent questions connecting to his statement in Introduction to Flight:

  • As Curator of Aerodynamics at the Smithonian’s National Air Space Museum the author is frequently asked by visitors how a wing produces lift – a natural and perfectly innocent question. Unfortunately there is no satisfactory one-liner for an answer. Even a single paragraph does not suffice. After a hundred years since the Wright Flyer, different people take different points of view about what is the most fundamental mechanism that produces lift, some pressing their views with almost religious fervor.

My question if lift is really generated by viscosity, is left without answer by Anderson.

5. Understanding Flight by David F. Anderson and S. Eberhardt

We quote from Understanding Flight by David F. Anderson and Scott Eberhardt:

  • For those with some familiarity with aerodynamics, there may be some confusion with the connection of viscosity and lift. Many simulations in aerodynamics are done with zero viscosity or, more accurately, in the limit of zero viscosity. 
  • Viscosity is introduced implicitly with the Kutta-Joukowski condition, which requires that the air come smoothly off atthe trailing edge of the wing. So, in reality these “zero viscosity” calculations reintroduce viscosity via the Kutta-Joukowski condition. In a fluid without viscosity, such as superfluid helium, a wing could not fly.
  • Also, in most mathematical descriptions of lift the boundarylayer is considered so small that it is ignored. Many erroneously claim that ignoring the boundary layer is equivalent to having zero viscosity.This is incorrect because viscosity is implicit in the condition that air follows thecurvature of the wing.
We see here the ideas that (i) viscosity enforces the KZ condition and (ii) viscosity makes the flow stick to the upper surface, both of which are incorrect. An interview with the authors of the book is recorded under Interviews.  
6. Aerodynamics of the Airplane by Schlichting and TruckenbrodtWe quote from Aerodynamics of the Airplane by H. Schlichting and E. Truckenbrodt:

  • It is seen that the viscosity of the fluid, after all, causes the formation of circulation and, therefore, the establishment of lift. In an inviscid fluid, the original flow without circulation and, therefore, with flow around the trailing edge, would continue indefinitey. No starting vortex would form and, consequently, there would be no circulation about the wing, and no lift. 
  • We want to emphasize that decisive progress has been made not through accumulation of large numbers of experimental results, but rather through synthesis of theoretical considerations with a few basic experimental results. Through numerous detailed examples, we have endeavored to enhance the reader’s comprehension of the theory.

We see how an incorrect idea (lift from viscosity) is sold by reference to authority with the shakiness of the the argument covered up by inflation of shaky theory to overshadow experimental observation, as an expression of 20th century collapse of scientific logic.

7. Aerodynamics, Aeronautics and Flight Mechanics by MvCormick

We quote from Aerodynamics, Aeronautics and Flight Mechanics by McCormick:

  • Because of viscosity, the flow at the trailing edge cannot continue to turn the sharp edge to flow upstream… The resulting flow pattern causes the fluid particles to accelerate and decelerate over the lower surface. Hence from Bernoulli’s equation there is a decrease of air pressure above the plate and increase below it. This pressure difference acting on the airfoil produces lift.

8. The Role of Viscosity

The crucial element of the Old Theory as KZ Circulation Theory of Lift, is the role given to viscosity to ensure that the flow separates at the sharp trailing edge by preventing

  1. flow around the trailing edge like inviscid potential flow
  2. separation on the upper surface before the trailing edge.

Viscosity is thus given the double effect of 1. not allowing the flow to stick around the trailing edge and 2. making the flow stick to the upper surface. Viscosity is thus given the role to both stick and not stick, without reason of course since the effects are contradictory.

The truth is that a flow with a viscous boundary layer will separate on the crest of the wing and thus viscosity has no effect of stickiness.

The truth that inviscid potential flow without boundary layer sticks to the upper surface because it can only separate at stagnation and until stall this does not occur before the trailing edge. Opposite to the conventional truth, it is thus non-viscosity with slip which creates lift, not viscosity with no-slip. This argument is presented in more detail as Why KZ Lift Theory is Wrong. See also Lift Not Generated Circulation.

9. Posts on KZ

5 thoughts on “Why KZ Is Incorrect

  1. This is a most interesting argument. I can understand Anderson’s comment about religious
    fervour.

    The KZ condition is incorporated in many successful aerodynamic calculation procedures, such as VGK (Viscous Garbedian and Korn) which is available from ESDU (Google ESDU, if you wish).

    Such procedures incorporate boundary-layer calculation procedures, laminar and turbulent. They have good predictive capability, and have been used successfully to design aerofoils. They embody therefore a satisfactory theoretical basis. Until a better theory comes along the KZ hypothesis will do. There is an analogy with classical mechanics and special relativity. Newton was ok until speeds comparable with the speed of light had to be considered. Einstein supplanted Newton in those instances. But not many engineers need bother with anything other than classical mechanics.

    Obviously, KZ does not do a good job when there is a lot of separated flow, for then there is nothing approximating smooth flow past the trailing edge. But that does not invalidate KZ for those most interesting cases where the flow is approximately smooth.

  2. Even a physically incorrect theory can sometimes give partly correct results, but an incorrect theory is dangerous since it in new situations can give completely wrong results. A wrong conclusion from KZ is that a trailing edge must or should be sharp.

  3. Dear Dr. Johnson:
    After reviewing your website I have to respectfully disagree with your rejection of firmly established (and very useful) aerodynamics principles, such as the Kutta-Zhukovsky Theorem and the Prandtl Lifting Line Theory…
    Surely you are aware of Newton’s humble admission: “If I have seen further than others, it is by standing on the shoulders of giants.”
    I find your approach completely lacking in humility; in fact, your combative and bombastic assertions that everyone is wrong, show complete disrespect to the joint undertaking of the progress of the sciences—where building on the foundations previously laid is the most rational way to advance new ideas.
    While I would like to understand your “new” theory, I find that your material does not communicate anything effectively. If asked to describe this new theory to someone, I would be at a complete loss, not able to describe coherently a single thought.
    From what little I can make sense of, you seem to be advocating numerical methods of solving the Navier-Stokes equations. Well this is certainly nothing new, but as a mathematician, surely you must know that numerical methods such as finite element, are only approximate solutions to discretized pieces of the partial differential equations of Navier-Stokes.
    If you have a mathematical solution to the Navier-Stokes problem, I would direct you to submit your entry to the Millennium Prize, where this problem constitutes one of the six remaining unsolved problems in mathematics. In fact, the question of turbulent fluid flow remains one of the most daunting unsolved problems in physics.
    Werner Heisenberg, according to an apocryphal story, when asked what he would ask God, replied: “When I meet God, I am going to ask him two questions: Why relativity? And Why turbulence? I really believe he will have an answer to the first.”
    In summary, I would like to first understand what it is you are trying to convey, but you seem incapable of organizing your thoughts and arguments in any way that I can make sense of.
    Regards,
    Gordon Arnaut.

  4. As far as flow is concerned, electricity is a fluid. Electrical flow in various electrical circuits at a fixed voltage is determined by their electric (leak) resistance and reactance. Of course, fluid flow in various fluid circuits such as in pipes, on objects and in joints at a fixed pressure is determined by their fluid (leak) resistance and reactance. Both the prior hydromechanics and the prior relevant international standards of seals were developed on the basis of not knowing such concepts and theories, and must be problems.
    Xu’s Sealing and Flowing Theories of Fluids are a theory based on the newly discovered physical quantities, such as leak resistance and fluid reactance, converting mechanisms of fluid pressure (energy) and kinetic energy, and self-sealing mechanism of sealing rings. For detailed explication, please see the second and third papers of The Special Issue of Xu’s Sealing and Flowing Theory of Fluids:
    http://www.sciencepublishinggroup.com/specialissue/164024

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