how much do these type of flaps increase drag?

lukus09

by how much will single-slotted Fowler-type flaps increase the drag coefficient of an aerofoil?

Filip Larsen

McCormick [1] mentions, that the increment in wing drag [itex]\Delta C_D[/itex] for a slotted flap as function of flap deflection angle [itex]\delta[/itex] is given approximately by
[tex]
\Delta C_D = 0.9 (c_f / c)^{1.38} (S_f / S) sin^2 \delta
[/tex]
where c_f/c is the flap to wing cord ratio and S_f/S is the flap to wing surface ratio. He does not specify the source of this approximation, but I presume it is based on theoretical and practical analysis of various combinations of NACA airfoils and flap types.

[1] Aerodynamics, Aeronautics and Flight Mechanics, 2nd ed., McCormick, Wiley, 1995.

lukus09

thanks, just what im looking for...
is sin theta the flap angle?
it says the change in drag coeffcient is equal to, so do i add this on to the original drag of the aerofoil?

Filip Larsen

Theta is the deflection angle, with theta = 0 meaning fully retracted. The maximum angle depend on the precise construction but the given approximation should be valid up to 90 degrees, I believe. McCormick notes that the optimal (maximum C_L for wing+flap) deflection angle is around 40 degrees for single slot flaps. For other types the value is different.

Yes, you add the flaps drag to the total drag coefficient you have for the airfoil wing, or plane in question.

I may also add, that McCormick gives similar drag expression for plain flaps where only the initial factor is different (1.7 instead of 0.9). This suggests that you in general can model drag for a specific flap construction by fitting that factor to measured drag (like if you have C_D for a plane at various flap deflection angles).

lukus09

if the flaps are fully retracted and theta is equal 0 that would mean the equation is equal to 0? is sin^2(theta) a trig identity? why am i get syntax error on calculator?

Filip Larsen

Note, that [itex]\sin^2 \theta[/itex] or [itex] \sin^2(\theta)[/itex] is short-hand notation for [itex] (\sin \theta )^2[/itex] or [itex] (\sin(\theta))^2[/itex]. Its a very common notation, especially with trigonometric functions.

Chronos

You need to know fuselage aerodynamics before calculating flap affects.

Brian_C

How can that equation be valid at such large angles? I would expect flow separation to come into play long before you reach 90 degrees of deflection.

Filip Larsen

I can only comment that McCormick have plots of measured and theoretical values up to 90 degrees with no (visible) discontinuities and that this approximation is for the drag coefficient, not the lift which would be a different matter. That said, I too would be surprised if the approximation were to be just as accurate at 90 degree as at, say, 10, but that is just a hunch. And who would want to allow flap deflection over, say, 40 degrees anyway.

Cyrus

Typically it is the drag curves that are in error, not the lift (because of the estimation of drag due to shear stresses). So if the drag agrees well, the lift is probably very good too. The equation you provided is interesting. It appears to apply to a wing, not a wing section, but doesn't account for wingspan, or how far outboard the flaps extend. So, use with caution.

Cyrus

Yes and no. We need to clear up some general misconceptions here in several of the posts. There is an aerodynamic interaction effect between the wing and fuselage. As a result, the total drag can be written as:


Note, I have assumed a linear model structure for the simplicity, which may not be valid. We can break down the wing drag as:


where the first term is the wing in a clean configuration. Again, I assumed the model is linear for the sake of this example. So one then needs to know two things:

-(a) How well is the linear perturbation assumption in Eq. [2] valid?

-(b) Does the interaction drag in Eq. [1] change significantly due to the perturbation?

Note also, that while not explicitly shown, the values in Eqs. [1,2] are generally a function of angle of attack, sideslip, and airspeed.

If the aspect ratio of the wings are large, then one can approximate the increase in drag based on sectional data from any good source, such as Ref. [1], and assume the perturbation found in the table applies to the entire wing.

References
[1] Theory of Wing Sections, Abbot and Von Doenhoff, Dover Press, 1959.

Filip Larsen

By "lift ... would be a different matter" I was referring to C_L for simple unflapped airfoils not having an accurate analytical model for AoA above the AoA for maximum C_L. There you would not expect such analytical models to be valid for high AoA.

Another point to make to argue for the validity of high deflection angles in the approximation, could be to say, that the flap deflection angle is not the same as angle of attack and that high deflection angles not necessarily imply equally high AoA for the flap airfoil since it lives in the downstream from the wing airfoil.

I must admit, after thinking about it I am not sure what the surface area ratio S_f/S signifies that is different from the chord ratio, as the surface area ratio is not present in any other analytical or measured relationships McCormick presents in his section on flaps whereas the cord ratio surely is.

It seems to me that this approximation is somehow extended to apply to a wing sections that include both flapped and unflapped parts, but if that then means the surface area ratio should be set equal to the chord ratio when applying the approximation for a flapped airfoil, I don't know. And this also seems to imply that flapped sections induce drag on unflapped sections which, in general, doesn't make sense.

I see that Theory of Wing Sections (which McCormick references for some of his airfoil data) has some drag data that maybe could be used to verify this drag approximation.

Cyrus

Ah, I see what you meant now - quite right.

That is an excellent point.