The lift coefficient is a number that aerodynamicists use to model
all of the complex dependencies of shape,
inclination, and some
flow conditions on lift. This equation is simply a rearrangement
of the lift equation where we solve for the
lift coefficient in terms of the other variables. The lift coefficient **Cl**
is equal to the lift **L** divided by the quantity: density **r** times
half the velocity **V** squared times the wing area **A**.

Cl = L / (A * .5 * r * V^2)

Here is a way to determine a value for the lift coefficient. In a controlled environment (wind tunnel) we can set the velocity, density, and area and measure the lift produced. Through division, we arrive at a value for the lift coefficient. We can then predict the lift that will be produced under a different set of velocity, density (altitude), and area conditions using the lift equation.

The lift coefficient contains the complex dependencies of object shape on lift. For three dimensional wings, the downwash generated near the wing tips reduces the overall lift coefficient of the wing. The lift coefficient also contains the effects of air viscosity and compressibility. To correctly use the lift coefficient, we must be sure that the viscosity and compressibility effects are the same between our measured case and the predicted case. Otherwise, the prediction will be inaccurate.

For very low speeds (< 200 mph) the compressibility effects are
negligible. At higher speeds, it becomes important to match Mach
numbers between the two cases. **Mach number** is the ratio of the
velocity to the speed of sound. So it is completely incorrect to
measure a lift coefficient at some low speed (say 200 mph) and apply
that lift coefficient at twice the speed of sound (approximately
1,400 mph, Mach = 2.0). The compressibility of the air will alter the
important physics between these two cases.

Similarly, we must match air viscosity effects, which becomes very
difficult. The important matching parameter for viscosity is the
Reynolds number. The **Reynolds number** expresses the ratio of
inertial forces to viscous forces. If the Reynolds number of the
experiment and flight are close, then we properly model the effects
of the viscous forces relative to the inertial forces. If they are
very different, we do not correctly model the physics of the real
problem and will predict an incorrect lift.

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*byTom
Benson
Please send suggestions/corrections to: benson@grc.nasa.gov *