Monday, February 24, 2014

Understanding Lift by Wings: The Lift-To-Drag Ratio

Some ways to generate lift can be categorized as follows:
-Mass Ejection (e.g. rockets)
-Buoyancy (e.g. dirigibles)
-Deflection (e.g. airplanes)

We will talk about wings, which falls under the category of Deflection.

There is a distinction to be made between mechanism and result. The finer details of fluid dynamics explain the necessary phenomena and mechanisms (i.e. laminarity, separation, etc.) for the wing to successfully act as a deflector. Force production (lift, drag) can be explained by the conservation of momentum and Newton's Third Law. We will be looking at the force production in a wing.

Wings work by deflection of the freestream air. You can treat the wing as tilting the net momentum vector of the portion of the oncoming freestream that it influences. So, the momentum vector of the air before it reaches the wing points in one direction and the momentum vector after it has passed the wing is tilted by some angle Φ. Keep in mind, this angle is not necessarily equivalent to angle of attack!

The conservation of momentum states that the momentum vector does not change in magnitude and is only redirected, as the wing does not produce energy or momentum. You can represent the new components in the horizontal and vertical directions of the tilted momentum vector by simple trigonometry. By Newton's Third Law (for every action there is an opposite and equal reaction), finding these components tell you the drag and lift.

The difference in these components is what makes flight by wings (deflection) so efficient. In effect, it is a mechanical advantage by a trigonometric trick! Instead of generating lift directly, we only have to overcome the drag necessary to sustain the momentum deflection.

As you may know, as you increase an angle from 0 degrees the sine component changes relatively rapidly, whereas the cosine component does not. This is reflected in the so-called small angle approximations. Thus we can expect that small angles of momentum deflection will be more efficient, or in other words, have a higher lift-to-drag ratio.

The equations to represent the relative lift and drag components are simple. The functions for normalized lift and drag are represented as L and D, respectively and are function of the momentum tilt angle, Φ. The resulting lift-to-drag ratio is represented as R.

There is something wrong with the above ratio equation. As Φ approaches 0, the equation becomes indeterminate (0/0). We know that in fact it should be 0. There actually should be a skin friction drag term in the denominator, because we know that even when the wing is not producing lift it still generates drag. We will denote this as f, and though it is a small number that becomes negligible as Φ increases, it is a function of Φ. The following is a corrected ratio equation.

From this we basically get the maximum L/D ratios possible for a wing to achieve. Below I have plotted the equation for several constant skin friction values (f=constant).

The lift-to-drag ratio converges at sufficiently high freestream deflection angle, and the skin friction term is important for the lower momentum deflection angles under the assumption of constant skin friction values. We see that this method of analysis reveals the range of maximum lift-to-drag ratios we are accustomed to seeing for typical airfoils.

Remember that this analysis is for when the wing behaves as it should; a deflector of freestream momentum. A wing cannot act as an effective deflector in the case of flow separation.

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