# Parametrization of wing lift and drag in 3D

Lift and drag of 2D airfoil section is described by polars, which are functions of angle of attack $\alpha$. But I cannot find something similar for 3D wing.

For computation (on computer) most convenient would be some representation in terms of projections ($c_A$,$c_B$,$c_C$) of air velocity vector $\vec v$ and and orthonormal vector which describe orientation of the wing $\vec a$ (left), $\vec b$ (up), $\vec c$ (forward)

$c_A = ({\vec v}.{\vec a})/|\vec v|$

$c_B = ({\vec v}.{\vec b})/|\vec v|$

$c_C = ({\vec v}.{\vec c})/|\vec v|$

One intuitive solution is to project $\vec v$ to plane spanned by ($\vec c$,$\vec b$) where I can employ 1D polar of wing section (modified by effect of finite aspect ratio but thats an other story). But that does not give me any reasonable lift&drag when $\vec v$ is close to perpendicular to that plane ($|c_A| \rightarrow 1$.).

background

At the end I want to use it for game aircraft simulator with rigid body dynamics. I would prefer to represent the aerodynamic data in compact way (using 1D polar) and than just modify it by that of-the-plane component $c_A$. But at the same time I would like to capture behaviour of the wing in non standard regimes of flight where $|c_A| \rightarrow 1$.

I don't want reinvent the wheel, probably this is some basic knowledge, just I'm not so familiar with the subject (I'm just ethusiast, I did not study aerodynamics systematically).

## 2 Answers

3D wing aerodynamics are too complicated to describe in terms of a polar using only the parameter $\alpha$ since they are also largely a function of side-slip angle $\beta$ and Mach number $M$.

In industry these data are represented by use of 3D lookup tables and interpolated between intermediate values or $\alpha$, $\beta$, and $M$.

You can use the 2D polars and 3D wing theory to estimate the 3D wing aerodynamic tables using integrated distributions along the wing at various angles of attack, side slip, and Mach number and interpolate between them trilinearly:

Input ($\alpha, \beta, M$) -> Output ($C_L, C_D, C_S$)

where $C_S$ is the out of plane side force coefficient.

Background: Former Aerophysics engineer dealing with developing external aerodynamics data for launch vehicles.

• OK, I'm thinking about similar interpolated look-up table, although since my target application is prety memory and CPUtime limited I cannot afford 3D data grid, and I prefer to use dot-protucts rather than angles (to ommit trigonometric and inverse trigonometric function performance cost). But thanks for info how it is going in aviation industry. – Prokop Hapala Nov 21 '16 at 10:16

Your case of |$C_A$| ≠ 0 is called sideslip by aerodynamicists. Your source for calculating wings of finite aspect ratio gives already the equations for swept wings - just treat the whole wing as a single, uniformly swept wing when you have a sideways flow component. This will modify the lift in proportion to the cosine of the sideslip angle. The logical consequence is that for |$C_A$|=1 the wing provides no lift, and only the fuselage and thrust prevent the aircraft from falling out of the sky. Since induced drag is proportional to lift squared, this part of drag will change with the square of the cosine. Viscous drag will roughly stay the same.

However, sideslip will cause lift asymmetries, side forces and additional moments in roll and yaw that were not present in straight flight. Now the fuselage and the vertical tail will influence the aircraft dynamics, and normally an airplane wing has dihedral, so a sideslip causes an opposite variation in the local angle of attack of each wing in proportion to the cosine of the dihedral angle.

If you head over to Aviation SE, you will find many answers which deal with those effects. You will also find information which helps you to understand lift and drag better. The claim in your linked source that tip vortices cause induced drag is misleading (to say the least).