How is the lift generated by a number of out-of-plane staggered wings calculated?

Question

I want to calculate the lift generated by a number of out-of-plane staggered wings but I don't think a valid approach is to simply use superposition and add the sum of the lift of all the wing's in isolation.

I have heard of Munk's stagger theorem but am unsure of it's implications.

I think the implication of Munk's theorem is that there is a limit to the amount of downwash and thus lift that can be generated by staggered wings.

Lets do an example:

I can calculate the lift force generated by a single planar wing as:

$$L=\frac{1}{2}\rho V^2SC_L$$

Where $\rho$ is air density, $V$ is free-stream velocity and $C_L$ is the finite wing lift coefficient, which is a function of the wing geometry.

Now, if I have say 5 staggered wings, is this statement:

$$L=5\times\frac{1}{2}\rho V^2SC_L$$

valid?

Now I know that the pressure distributions over the different wings will interact wing each other, but my question assumes that the wings are separated sufficiently far away from each other that this effect is negligible. Or is this interaction of pressure distribution a result of the stagger theorem?

I will appreciate any light being shed on this subject.

Answer 1

It depends on the vertical distance between the wings. Munk's theorem only covers the effect of their horizontal distance (in flow direction) and says it doesn't matter (in inviscid flow).

Your statement is correct if this distance is infinitely high (and the atmosphere has constant density over altitude). Only then there is no interference between the separate wings and their combined lift is the sum of the lift of every isolated wing.

On the other end of the scale, all wings together will just have the lift of a single wing if they are not separated at all. This is trivially easy to prove, but not practical.

For distances between the two extremes, lifting line theory gives you a good answer when the distance is not small. Below that, viscous effects will start to dominate the picture, and if the wings are separated only by something similar to the wing's thickness, they will impede the flow between the wings, making the combination appear as one blunt object to the outer flow, with consequences like flow separation and the collapse of lift.

Once you move the wings apart by more than their chord length, you will get practical solutions where the wings will produce more lift than a single one and even have lower induced drag than a single wing creating the same lift force. However, a distance sufficient to make interference negligible will mean that the struts between the wings will become long and heavy, so their mass cannot be neglected. I would expect that there is an optimum of lift minus mass which is somewhere around a vertical distance of one to three chord lengths.

Unfortunately, literature on the topic is more concerned with reducing drag, so the closest I could find is a work by Ilan Kroo on non-planar lifting systems. Using the solutions for biplanes will not give an answer to your question, because every wing interfers with every other, so adding a fourth and fifth wing to a triplane will reduce the lift of the triplane, but will in sum increase lift.

Answer 2

Munk's theorem describes that when moving streamwise direction staggered wings AND keeping the circulation distribution (so changing incidence) the induced drag is kept constant.

It is not saying anything about lift (although can be obtained from circulation) and other drag components. What you are describing can't be derived from Munk's theory.

The difficulty is that one lifting surface modifies the pressure field around itself, having another wing affected by that pressure field will modify the pressure field on both wings affecting each other.

That situation is a complex aerodynamic situation, and very likely (so far I don't know) there is not analytical solution. You will need to use semi-empirical models like this ESDU or a simple panel method calculation. But it is not straight forward to calculate lift.