How do you calculate the wing moment coefficient knowing the moment at each spanwise station?

Question

A lot of theoretical formulas exist for finding the moment coefficient for airfoils, but I have been confused by how to get the moments for a wing. I came up with three related questions that have been confusing me a lot:

1. If I know the moment coefficient at each spanwise station for the wing (so the moment coefficient for each airfoil used along the span), how can I get the moment coefficient for the whole wing? Is it just a weighted integral? If so, what's the formula look like?

2. How does one find the moment coefficient for a straight tapered wing?

3. How does one find the moment coefficient for a swept wing?

Answer 1

You can use the Lifting Line Theory to first calculate the downwash at each spanwise station, which are then converted to the induced AOA ($\alpha_i$).

The sectional moment coefficient and lift coefficient are:

$$c_m = c_{m_0}+c_{m_\alpha}(\alpha-\alpha_i)$$

$$c_l=c_{l_0}+a(\alpha-\alpha_i)$$

To get the total moment of a straight tapered wing (about the root chord leading edge for simplicity), it's a simple integration away, noting that the sectional moment is usually about the 1/4c of the local chord:

$$M_{LE} = q_\infty \int_{-b/2}^{b/2}{\left[ c(y)c_m(y)-\left(x_{LE}(y)+\frac{1}{4}c(y)\right)c_l(y) \right]c(y)dy}$$

where $c$ is the local chord length, $x_{LE}$ is the distance from the root LE to the local leading edge, and $q_\infty$ is the free-stream dynamic pressure. You'll have to normalize against a reference chord (e.g. MAC) to get the moment coefficient.

Finite large sweep wings have crossflow gradient that would render Lifting-Line Theory unreliable. You'd have to use a vortex surface method, such as Vortex Lattice, to derive accurate predictions.