On Mr. Bernoulli...aerodynamics 101
G'day all,
Time to weigh in on this, with a slightly theoretical bias, so for what it's worth...
Bernoulli's theorem is not really that difficult a concept to grasp. Bernoulli found that the 'total pressure' in an incompressible flow along a given streamline is constant! There, that wasn't so hard! But what does it mean?
In technical terms: for a given 'streamline', 'static pressure' + 'dynamic pressure' + a gravitational potential term = 'total pressure', which is a constant.
As applied to the 'low speed' aerodynamics of an aircraft (less than 1/2 the speed of sound), compressibility effects of air can be ignored, as can the gravitational term. This simplifies then to: static pressure + dynamic pressure = constant.
Knowing this means that we can calculate, for instance, IAS from a pitot static system, which is fortunate.
But can we use Bernoulli's theorem to calculate the lift (pressure distribution) on a wing? Short answer, no. However, to say that Bernoulli's theorem doesn't apply to the lift generated on an aerofoil is, imho, a non-sense. It certainly gives us some insight as to why the pressure varies accross an aerofoil. Where the velocity of the airstream changes, then the 'total pressure tango' between dynamic pressure and static pressure occurs, hence the possibility to produce a net lifting force if we are clever enough. Remember, the only force transmitted to the wing from the airflow is by means of 'pressure' acting on an 'area' of that wing. Force = pressure x area, so if we sum up all the respective regions of 'pressure x area', we end up with a resultant force, which in aviation we arbitrarily express as 'lift' (perpendicular to the 'freestream' airflow), and 'drag' (parallel to the 'freestream' airflow).
What we need is to be able to calculate these lift and drag forces. The whole concept of engineering is about producing 'models' to enable us to make a reasonable prediction of cause and effect. Bernoulli's theorem is just one such model, within its limitations. Mr. Newton's laws represent another handy and reasonably reliable model, within their limitations.
One way to calculate the lift and drag forces would be to map the pressure distribution around the aerofoil, and sum up the component forces to find the resultant. Unfortunately, Bernoulli's equation can't do that for us. Bernoulli's theorem only applies to streamlines absent significant 'viscous effects'. This 'inviscid flow' assumption breaks down in the boundary layer, and in the wake.
To get around this, we measure the lift and drag resultants (and pitching moment)directly in a wind tunnel, and establish 'coefficients' (CL, CD and Cm) to be used in calculations. This model works reasonably well because it bypasses the need to understand exactly what is happening in the boundary layer, and models only the measured outputs based on the measured inputs.
Another method might be to estimate the change in momentum in the wake (Newton's laws). This will tell us the 'reaction' necessary in the wing to produce the change in momentum of the airflow, but the reaction force on the wing is still the sum of varying pressures (Bernoulli again) acting on their respective areas. The change in momentum is an inescapable consequence of lift production... think of a propellor, or a helicopter rotor downdraft. The momentum method tells us only the total reaction, and nothing about the nitty gritty e.g. centre of pressure.
As stated earlier in this thread, lift can be estimated by either model, with the results in close agreement. The models don't change what actually happens, but are just our attempt to get a handle on it, so that we can design aircraft that fly!
Also... There has been some discussion about the 'fidelity' of adjacent air molecules i.e. do they meet up again at the trailing edge, or have a fling? I've never liked the 'longer path' explanation for increased velocity, and hence reduced pressure, because it presumes 'fidelity' (to carry on the analogy). Without doubt, in a three dimensional airflow around an aerofoil producing lift, the molecules do not meet up. The opposing spanwise flows on the upper and lower surfaces of the wing act to keep the previously chummy molecules apart.
As far as I know, aerodynamic theory does not require such a condition that the same molecules meet up at the trailing edge either. A gentleman named Mr. Cutta established a theory (model) known as the 'Cutta condition', which represents the only possible steady state solution for an airflow around an aerofoil. This has to do with the theoretical concept of 'circulation' superimposed into the airflow to account for the observed facts. Initially the rear stagnation point in an aerofoil (set to produce lift in the normal, upright sense) is forward of the trailing edge, on the upper surface. This creates what is called a 'starting vortex' at the trailing edge, forcing a unique 'circulation' value into the airflow around the aerofoil just sufficient to move the stagnation point to the trailing edge i.e. the upper and lower surface streamlines meet happily at the trailing edge. At this point, the transient starting vortex is cast off into the wake never to be seen again, and circulation is maintained resulting in a stable airflow. While there are numerous other constraints, there is no intrinsic requirement for the previously adjacent air molecules to meet up again at the trailing edge.
Any questions???
Regards,
Harro