In a strange turn of events, friction comes into play to rescue the situation: The flow around the trailing edge is not possible because, in reality, the air that has slid along the two sides (particularly the lower surface) of the airfoil has been slowed by friction and thus robbed of the energy necessary to turn the corner at the trailing edge. If the air could travel to the (non-existent) stagnation point on the upper surface it would be going from a region of low static pressure at the trailing edge into a region of the higher pressure at the stagnation point. That is very difficult as we shall see. Instead, the air simply gives up by going in the direction given by the trailing edge angle. Thus, one could say that friction causes the flow to follow the angle dictated by the trailing edge. Two scientists came up with a rule as follows: The trailing edge assures that the wing circulation arises in just the amount required to have the flow leaving the trailing edge in its direction. The German mathematician Martin Kutta and the Russian scientist Nikolai Joukowski came up independently with that rule through observation of the flow physics in the early 20th Century. Another step of progress.
In our aerodynamic world, the friction experienced by the flow on its lower surface is largely responsible for the establishment of the vorticity that provides lift. We might wish we could live without the consequences of friction but, the reality is that we could not. Just picture yourself on a smooth ice sheet trying to go for a walk!
Figure 3.1 is an opportunity to introduce a little more nomenclature. The curved shape of the airfoil mean line is normally characterized by the maximum distance (called the camber) between the straight chord (dashed line) and the mean line, expressed as a few percent of the chord. The airfoil’s angle of attack is the angle of the chord relative to the freestream direction, that is, the direction of the flow if the airfoil was not present.
Our understanding is at the point where we (a good mathematician) can determine the vortex strength distribution along any airfoil shape, provided it is reasonable. The calculated vorticity distribution along the chord allows the determination of the velocity distribution everywhere in the field and along the airfoil’s two surfaces. At positive angles of attack, velocities on the upper surface will be larger than the incoming flow (the freestream) and those below are lower. The total of the distributed vorticity (i.e., the circulation) is directly related to the lift exerted by the airfoil at the angle of attack chosen (Fig. 3.2).
In summary, the trailing edge installs vorticity on the airfoil and distributes it along the mean line. The distributed vorticity, together with the freestream flow, determine the local flow velocity everywhere in the field, including on both sides of the mean line. A wing section with finite thickness would naturally impose small changes in velocities associated with the thickness, but because the wing is typically thin, the local velocity adjustments for thickness are quite small and we will ignore them here.
How good is this model? For substantiation of what we might measure in a wind tunnel or on an airplane, we must be able to relate the local velocity to the local pressure. Pressure is easier to measure than velocity itself, especially near the surfaces of the airfoil section. The relation between velocity and pressure turns out not to be simple and it will be addressed in due time. Assuming, for the time being, that such capability is in hand, the lift can be calculated for an airfoil at modest angles of attack using our vorticity model and compared to a wind-tunnel measurement. The variation of lift with angle of attack is given very closely as measured in the laboratory, even when airfoil thickness is considered. Another aspect of the model validity is that it correctly provides the location on the airfoil where the force is effectively applied. The engineer calls this the aerodynamic center which for most airfoil designs is, in theory and in practice, at the quarter chord location on the airfoil. There, the pitching moment of the wing is independent of angle of attack and can be easily calculated using the methodology used to determine lift. The pitching moment is a torque about a point on the mean line of an airfoil or, in the case of a wing, about a line parallel to the span that must be countered by a torque from the horizontal tail surface.
So far so good, but the model is not perfect. It ceases to be adequate at large angles of attack because the flow fails to follow the prescription imposed by its shape. In ordinary aerodynamic terms, we encounter stall of the wing at high angles of attack. The model also says nothing about the resistive force called drag.
The trailing edge function seems pretty straightforward in light of the arguments made above: it must be sharp and point downward. It may be noted that the local vorticity at the trailing edge must be zero, according to the Kutta-Joukowski rule so that the air velocities along the upper and lower sides of the airfoil are identical, in speed and direction.
At this point in the evolution of our understanding of aerodynamics we have the capability to analyze shapes (the airfoil section) we might consider to be the basis for constructing a three-dimensional wing. We are, however, a long way from having the ability to specify the lift that is desired and, with a mathematical model, determine the right airfoil or wing shape. We have not yet even considered that resistive forces, otherwise known as drag forces, will be in play. The road to a practical and well performing wing was not going to be easy! More breakthroughs in the necessary understanding will be made and we will tackle them.