There is a rather clever way that aerodynamicists group information about airfoils. We have seen that lift changes linearly with area, density, camber, and small angles, and as the square of the velocity. We have also seen that lift has a complex dependence on the airfoil geometry.
The thickness and camber are geometric properties of the airfoil cross-section, the area depends on the geometric chord and the span of the wing, the airspeed and density (altitude) depend on flight conditions, and the angle of attack sort of connects the airfoil geometry to the flight conditions.
Suppose that we collect all the previous information into one equation:
The constant here would be a collection of all the previous constants.
We know how the flight stuff changes :
The geometric stuff (thickness and camber) will not change when we change flight conditions--it will still be the same shaped airfoil regardless of altitude or speed.
So let's combine the geometric stuff and the angle of attack term into a new variable called the lift coefficient --> Cl. The original equation then looks like:
The value of Cl will depend on the geometry and the angle of attack.
The "density x velocity squared" part looks exactly like a term in Bernoulli's equation of how pressure changes in a tube with velocity:
The term "0.5 x density x velocity squared" is called the dynamic pressure.
We can use this idea in our lift equation by re-defining the value of the constant.
Now one last trick--let's just include the constant in the definition of Cl.
To summarize:
SO WHAT GOOD IS ALL THIS??
Well, if we know the Cl for an airfoil and we know the flight conditions (dynamic pressure) and the area, we can determine the lift of the wing. (Designs the weight of an aircraft)
Or, the reverse, for a known Cl and dynamic pressure we can determine how big a wing (area) we need to lift the weight of an aircraft. (Designs the size of an aircraft)
Or, for a known Cl and wing size (area) and weight of the aircraft, we can determine the dynamic pressure. This would tell us the speed to fly for a given altitude or the altitude we could attain for a given speed. (Sets flight conditions)
And finally, if we could measure the lift of a wing and knew the area and dynamic pressure, we could determine the Cl.
This last example seems a bit obscure--so let's try a little experiment.
Let's set the Angle to 5 degrees, Thickness = 0.5, Camber = 0.2.
Now let's say we have a large airliner flying at 250 mph, at 25,000 ft, with a wing area of 1000 sq ft.
WHAT IS THE LIFT GENERATED?
Looks
like about 88,250 pounds.
Now, what is the Cl for this
wing?
Get the density from the simulator (Density = 0.00107)
WHAT VALUE OF Cl DO YOU GET?
Now let's compare this to a radio-controlled model airplane flying at 35 mph, at 100 ft, with a 10 ft wing area.
WHAT IS THE LIFT AND THE DENSITY (NEEDED FOR DYNAMIC PRESSURE)?
WHAT IS THE Cl FOR THIS MODEL AIRPLANE?
Hey, that's the same number that we got for the full sized airplane at a different altitude and speed!!!
Let's try a small wind tunnel model--50 mph Speed, 0 ft Altitude, 1 sq ft Wing Area.
WHAT IS THE LIFT AND THE DENSITY (NEEDED FOR DYNAMIC PRESSURE)?
WHAT IS THE Cl FOR THIS WIND TUNNEL MODEL?
The same number again!! What's going on here? We have different size objects, at different speeds, at different altitudes, and different dynamic pressures. But they all have the same Cl .
While we have been changing the size of the airplane, the speed, and the altitude, we have not changed the basic geometry of the airfoil. For all three cases, the Angle was 5 degrees, the Thickness = 0.5, and the Camber = 0.2.
Remember that we defined the Cl to include geometry information and the angle of attack information. So the Cl for an airfoil remains the same for all kinds of flight conditions and sizes of aircraft.
WHY IS THIS IMPORTANT?
This rather simple fact makes wind tunnel testing possible for aircraft lift.
We have seen that we can determine the Cl at wind tunnel speeds, altitudes, and sizes and get the same answer that we would get for a full size aircraft at real flight conditions. Once we determine the Cl, we can set new flight conditions or design other sized aircraft and know what the lift will be. The conditions which we picked for this problem contain a combination of camber, thickness, and angle of attack. Remember in the previous lesson that it was difficult to determine how an airfoil's lift varied with this combination of variables (sometimes the lift increases with thickness; sometimes it decreases depending on the camber). Now, if we can determine the Cl, either through wind tunnel testing or analysis, we can describe this relationship.
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