This discussion has devolved into a debate on what Anderson and Eberhardt’s (A&E) article ‘Understanding Flight' really means. Having read it two or three times now it seems to me that it is a bit like the curate’s egg – ‘Good in parts’
They go to some pains to debunk the familiar “equal transit times” explanation which is fair enough because it doesn’t hold water, but that is because the basic assumption of equal transit times is wrong, not because of the subsequent attempt to link the undoubted fact that air flows faster over a wing upper surface than the lower generates a differential pressure which can be (in principle) calculated using Bernouilli’s equation. Their arguments on the invalidity of Bernouilli when applied to flow around a wing are tilting at strawmen I think.
Another fundamental problem with this description is that the air’s pressure and speed are not related by the Bernoulli equation for a real wing in flight! The Bernoulli equation is a statement of the conservation of energy. For it to be applied the system must be in equilibrium and no energy added to the system. As you will see in the discussion below, a great deal of energy as added to the air. Before the wing came by the air was standing still. After the passage of the wing there is a great deal of air in motion. A 250-ton jet at
cruise speed is doing a lot of work to stay in the air. Much of the fuel that is burned is adding energy to the air to create lift. Thus the Bernoulli equation is not applicable
I disagree. The energy added is used to overcome a side effect of lift generation – drag.
Picture in your mind several wings: an asymmetric wing in normal flight, the same wing in inverted flight, a symmetric wing and a flat plate. For each one, an orientation into the wind can be found which gives zero lift. We call this orientation the zero effective angle of attack. Now if one were to measure the lift of these wings as a function of the effective angle of attack, the results for all of them would be similar
Nobody will argue with that. Put another way it says that the lift curve slope of a wing of infinite aspect ratio is independent of camber.
But the consequence of using this definition of AoA is that each and every wing section has a different datum, making it impossible, or at least very difficult, to compare the characteristics of various sections – not a great idea!
There is another mistaken description of lift, which we will call the wrong-Newtonian description of lift, although those that teach it just call it the Newtonian description of lift. This description of lift states that diverting air down produces lift, and that lift is a reaction force. This part is true. Unfortunately, in the wrong-Newtonian description of lift the air is diverted down by impact with the bottom of the wing. .....
Although there is a little of this kind of lift for most wings, it is minimized for efficient wings. The amount of air impacted by the bottom of the wing is far too small to account for the lift.
No problem there either
Yet another common description of lift is that of circulation theory. Here the air is seen to rotate around the wing. This is sometimes used to explain the acceleration of the air over the top to the wing. There is a great deal of jargon, such as "starting vortex" and "bound vortices", associated with this description. Circulation theory is a mathematical abstraction useful and accurate for aerodynamic calculations. Mathematically, circulation is a non-zero curl in the airflow in a closed line integral around a wing, which
is simply a statement that the wing bends the air.
For sure circulation theory is usually expressed in complicated math, but it can also be explained in plain English (or even American!). Way back in post #33 of this thread I gave a url. for such an explanation. It does a lot more than just state that the wing bends the air.
In brief, the lift of a wing is a reaction force and is proportional to the amount and vertical velocity of air is diverted from the horizontal to the vertical, with almost all of the air diverted from above the wing.
Again, no problem with this as an overall explanation, but it doesn’t really tell us much about how it all happens. A&E go on to say that
Lift = mdot * vv
Where mdot is a mass flow rate and vv is the vertical downwash velocity imparted by the wing measured in the air’s rest frame. That is a trivial statement unless we can understand a bit more about the two terms.
They suggest:
We would first like the reader to view the wing as a kind of "virtual scoop" as illustrated in figure 5. The amount of air intercepted by the wing is related to the lift distribution along the wing. The shape of the virtual scoop is half of an ellipse with the major axis equal to the wingspan and the minor axis proportional to the chord length (distance from leading to trailing edges) of the wing. The air intercepted is diverted down with the highest downward velocity near the wing and the deflection speed tapering to zero as the distance above the wing increases, as shown in the figure. This is not intended to imply that there is a real, physical scoop with clearly defined boundaries, and uniform flow. But this visualization aid does allow for a clear understanding of how the amount diverted air is affected by speed and density
The amount of air intercepted by the scoop, mdot, is proportional to the
• area of the wing
• wing’s speed
• air’s density
To a good approximation, neither the angle of attack nor the load on the wing affects the
amount of intercepted air.
In one respect this is wrong. mdot is the flow leaving the TE and is proportional to wing span not wing area. We still have no idea how much air is affected though. A&E calculate that the deflected air might be drawn from as much as a semispan above the wing, but this is based on some assumptions rather than any definite scheme of things.
So far as I can see, apart from saying that it is proportional to AoA and airspeed they give no guidance on how the vertical velocity is generated – a fundamental piece of knowledge so far as our understanding of lift generation is concerned.
Moving on to their strictures on the (mis)use of Bernouilli, they are of course correct when they say that the general application should be:
Static pressure + 0.5 rho*V^2 = Total pressure
Their argument is that if energy is added to the flow then total pressure will increase and Bernouilli’s equation will be invalidated. Equally true of course if energy is extracted from the flow. But let us look a little deeper.
There is abundant evidence from wake survey experiments that total pressure is not constant behind a wing producing lift, so in broad terms they are right, although not because energy is added – rather it is subtracted. However, it is also true that this loss of total pressure is confined to a small area just behind the TE – in the wing wake in fact.
The NASA site that Italia 458 referenced to define streamlines says:
A
streamline is a path traced out by a massless particle as it moves with the flow. ........ Since there is no normal component of the velocity along the path, mass cannot cross a streamline. ............. We can use
Bernoulli's equation to relate the pressure and velocity along the streamline. .............. Since no mass passes through the surface of the airfoil (or cylinder), the surface of the object is a streamline.
The wing surface may be a streamline, but since the streamwise velocity is zero everywhere on the wing surface applying Bernouilli there would be silly. Close to the wing the streamwise velocity increases steadily through the boundary layer, but in a turbulent boundary layer (as exists over 90% plus of the wing’s surface) there is a constant, if random, exchange of mass from the high energy outer regions towards the regions close to the surface. It is this energy transfer that permits turbulent boundary layers to accept higher adverse pressure gradients before separation. However, this exchange of mass means there can be no streamlines inside the boundary layer and consequently Bernouilli’s equation cannot be applied there.
At the outer edge of the boundary layer (where there is no more mass transfer) there will be a bounding streamline and from this point out Bernouilli may be applied. This is confirmed by all those wake surveys, which show that outside the wing wake the total pressure is constant.
What does this mean for the application of Bernouilli to the flow around a lifting wing? It means that it can be used to calculate pressures and velocities around a shape that is close to, but not exactly the same as, the basic wing. This does not mean though that the equal transit time explanation can be retained!
I think I can safely say that none of this bothers practising aerodynamicists who are perfectly happy to use pressures measured on the wing surface and go from there via 0.5rhoV^2 to get to wing loading.
Then there is the bit:
In other words, the pressure difference drives the acceleration of the air, not the other way around.
Yup! But I don’t see it as any pressure difference along a streamline. Think in their air-rest frame. If the air is obliged to follow a curved path as the wing passes through it there must be a centripetal force making it do so. This has to be some sort of pressure differential. But the total pressure of the air at rest is equal to the ambient static pressure. The pressure differential therefore has to come from a drop in pressure at the wing surface (or more strictly I suppose at the outer edge of the boundary layer). Since Bernouilli applies at the outer edge of the boundary layer this drop in static pressure will be accompanied by an increase in velocity. Pressure difference across streamlines is the driving force.
Rather grudgingly, A&E say:
Although circulation theory can be used for accurate calculations of lift, it does not give a simple, intuitive description of the lift on the wing. We have also shown that the pressure and velocity of the air over a real wing in flight at not related by Bernoulli’s equation. Newton's laws hold without exception for both subsonic and supersonic flight, and can be used to yield an understanding of many concepts without complicated mathematics
From my pov. Newton’s laws as expressed by A&E do give a simple quantification of the lift on a wing, but it is if anything oversimple. It tells us nothing of how lift is actually generated nor how it might be distributed over the wing. If that is all you want, then fine, but if you want a bit more depth then I can only refer you back to the Arvin Gentry article I referenced in post#33 – plain English but realistic and informative.