fredag 29 november 2019

The (Unphysical) Kutta Condition

Kutta and Zhukovsky named Fathers of Modern Aerodynamics saved fluid dynamics from collapse after the Wright brothers with their Flyer in 1903 had shown that powered human flight was possible, in blatant contradiction to the theoretical prediction that it is not possible. The great idea of Kutta and Zhukovsky was to add large scale circulation to potential flow according to this generic picture:

We see potential flow (left) with zero lift and drag from cancelling low (L) and high (H) pressure and flow separation before the trailing edge,  modified by large scale circulation (middle) around the wing section into flow with lift and separation at the trailing edge (right). Kutta a Zhukovsky claimed that the large circulation was generated by a sharp trailing edge preventing the flow from turning around the edge, as in potential flow creating high pressure on top of the wing destroying lift.

The argument was that it was the singularity of the sharp trailing edge that was powerful enough to generate the large scale circulation around the section, with the effect of creating lift. A wing thus had to have a sharp trailing and so the concept of airfoil was born as a wing with a sharp trailing edge and to help the design of airplanes, a database with 1600 wing sections was created, all airfoils with sharp trailing edge.

But there was one caveat: It was early on observed that wings with rounded trailing edges worked just as fine as wings with sharp trailing edge. Rounded edge of diameter up to 2% of the chord gave the same lift and drag as with sharp edge, and the same lift but a bit larger drag for up to 10%, that is of the same diameter as the leading edge, see Trailing Edge Geometry.The conclusion could only be that lift was not an effect of a sharp trailing edge.

But that did not prevent the KZ circulation theory to serve as the salvation from collapse all through the modern era of aviation. After all, the data base only listed airfoils with sharp trailing edges and so the conclusion was that it had some effect, albeit somewhat mysterious.

But the KZ theory is an example of Aristotle's logical fallacy of confirming the consequent of an assumption. The argument started with the correct implication that if there is circulation, then there is lift, and concluded from observing lift (the consequent) that there must be circulation (the assumption). This type of argument is common in science as a technique to affirm an assumption, but the logic is missing and science with incorrect logic is non-science, that is nonsense.

The singularity of the sharp trailing edge was thus used to explain lift and it also came to be a crucial element of computational fluid mechanics CFD: The presence of the singularity allowed prescribing the velocity in a potential flow CFD code at the trailing edge and thus moving the separation back to the trailing edge from its position in potential flow. The argument appeared to be that from a singularity anything can happen. The effect was to artificially introduce strong suction (or blowing) on top of the wing thus causing circulation around the wing as in KZ theory and lift. The trick to prescribe the velocity in a CFD code at the trailing edge (easy to do) was celebrated as the Kutta condition. 

Standard CFD codes such as RANS or LES thus implement the Kutta condition and so they are able to give reasonable predictions to lift for airfoils with sharp trailing edge, but not to drag because circulation does not change the zero drag of potential flow.

On the other hand, DFS Direct Finite Element Simulation computes lift and drag of wings with rounded trailing edges without any trick of artificially specifying the velocity at the trailing edge, all in close accordance with observation.

A relevant question is then what standard CFD would give for wings with rounded trailing edges? Results are sparse because airfoils are supposed to have sharp trailing edges and so standard CFD comes with the Kutta condition. Or the other way around, without the Kutta condition standard CFD gives completely wrong lift.

However the arcticle Numerical Study Comparing RANS and LES Approaches on a Circulation Control Airfoil by Rumsey and Nishino offers information. The study concerns the flow around a wing subject to a mechanism of blowing on the leading edge which creates circulation and thus enhances lift. The interesting thing is that the trailing edge is rounded allowing us to study the performance of RANS and LES without the singularity of a sharp trailing edge. The reason it is rounded is to not prevent circulation like a sharp trailing edge.

The figures below show the wing section with blowing mechanism at leading edge (right) and rounded trailing edge (left). We see a pressure distribution with unphysical (not observed) high pressure at the trailing edge connecting to a (not observed) separation pattern. We thus see that RANS and LES without sharp trailing edge and Kutta condition gives incorrect pressure distribution.

On the other hand, DFS shows the observed pressure distribution of separation without pressure rise.

Altogether, standard CFD comes with the Kutta condition, which artificially creates circulation and lift, which means that standard CFD is unphysical.

DFS does not use any Kutta condition and is physical because it is based on first principle physics.

Connecting to the discussion on no-slip vs slip, recall that standard no-slip CFD flow without the Kutta condition, will separate on the crest of the wing and then give little lift.

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