I understand that people use the RANS/LES equations to calculate how air flows above a wing. But what is the exact formula that people use to derive coefficients like drag / lift after doing these calculations?
Do people do these calculations just to look at pretty pictures?
Is fully simulating turbulence overkill if you're just trying to get 2 coefficients (lift / drag) for a wing?
You're correct, most CFD is overkill. Most engineers are not taught when a tool is overkill -- they're just taught to use the 'best' tool they have.
CFD produces pretty pictures that can give insight into a flow -- which can sometimes help diagnose and fix problems with the flow. Beyond that, it often boils down to a smaller set of numbers that are calculated. As you pointed out, lift and drag coefficients are two important ones. CFD will calculate all six major coefficients -- three forces and three moments. All six are worth having. CFD can also calculate the load distribution across the wing, which is important to feed over to the structures folks. Of course, these things can be calculated with much simpler methods.
Most aircraft operate at quite high Reynolds numbers. The boundary layers are relatively thin, and the pressure distribution around the aircraft can be treated as inviscid. A panel code, or even a vortex lattice calculation will do a good job at estimating the six forces and moments (and load distributions) except for viscous contributions to drag, stall, and transonic effects. An Euler code will be able to handle transonic effects and other complexities of compressible flow.
Basic viscous drag can be estimated with a wetted area / form factor buildup. This won't tell you details of where problems occur, but it will get you a very close approximation of the cruise drag.
All of these are approximations, a good engineer understands them and uses good judgement to know when to use which tool. CFD is not only an approximation because of the mesh resolution and the simplification of turbulence. But most CFD geometry models leave out a lot of details of the aircraft -- finite trailing edge angles, gaps and seams, fasteners, antennae and other protuberances, flap tracks, control surface hinges, bonding straps, etc. etc. To capture these things, the geometry model would be much more complex, the meshing effort would surely double, the mesh size would probably double, and the computation time would increase accordingly.
Most CFD is run with control surfaces un-deflected. This means the aircraft is not trimmed in pitch. Most CFD is run without propulsion effects of the engines -- the aircraft is not trimmed in Thrust/Drag and the effect of the giant vacuum cleaner is ignored.
To get around these limitations, you need to run a bunch of CFD cases -- say at a bunch of angles of attack and with a few control surface deflections. Then, you can do some offline calculations to approximate what the drag of the trimmed solution would be if you were to run it. Likewise, you end up doing a handbook base drag buildup to add in the drag of all the things you left out of the CFD model.
That is just for an ordinary cruising aircraft. For complex models like an eVTOL aircraft with many rotors that possibly tilt as it goes through transition -- trim is a complex 6DOF problem and there are many redundant control effectors. You're interested in many angles of attack and climb angles at many velocities through transition. This flow is inherently unsteady and CFD calculations take a very long time. In the end, a single CFD solution is nearly useless -- to get something useful, you need hundreds of such solutions.
Again, this is where lower fidelity tools (some of which can trim on the fly) really shine.
There are times when CFD is essential, but often it is over-used. Master the spectrum of tools and you'll be way ahead.
It seems to me that fully simulating turbulence is overkill
For a modern jetliner one can expect its boundary layer to be turbulent on 90 to 95% of its surface. Due to the lower Reynolds numbers, that range can be better for a GA airplane made of smooth carbon fiber - but the most part of its surface is going to have a turbulent boundary layer too.
Being able to simulate turbulence is therefore of paramount importance in order to calculate with confidence the aerodynamic characteristics of the airplane.
what is the exact formula that people use to derive coefficients like drag / lift after doing these calculations?
It's enough to use the definition of those coefficients:
$\begin{cases} C_l=\frac{L}{½ \rho V² S}\\ C_d=\frac{D}{½ \rho V² S}\\ C_m=\frac{M}{½ \rho V² Sc} \end{cases}$
Fully simulating turbulence around a whole aircraft is overkill. It is also impossible for real-word Reynolds numbers for any foreseeable future even on supercomputers. You mention LES (large eddy simulation) and RANS (Reynolds-averaged Navier-Stokes). Neither fully simulates turbulence.
In RANS, all turbulence is modeled by approximate models, no turbulence is simulated.
In LES, the large turbulent structures are simulated, but due to the necessity to use affordable computational grids, the small structures are modeled by approximate models.
In turbulent flow, the turbulence also affects the mean flow, through the presence of the turbulent stresses in the Reynolds-averaged Navier-Stokes equations. Therefore often, you have to somehow simulate the effects of turbulence, at least by modelling the turbulence by approximate models in RANS, to get the right flow shape, and especially flow separation.
In nice attached flow, you can often get away with Euler equations or even by potential flow. That does NOT mean that CFD is overkill! Both of these approaches ARE CFD, if they are used for a numerical simulation on a computer on a grid (but analytical computations of Joukowski profiles using conformal conformal projection with a pen and paper or anything similar with other equations is not CFD). Even the small disturbance potential equation simulations are still CFD.
When flow separates, it is often necessary to model (or simulate, if you can afford it) turbulence as precisely as possible to get the flow shape and the Reynolds number dependence right. Getting reasonable computational result for a lift coefficient is quite easy. What is much harder is a good result for the drag coefficient. Especially the skin friction contribution is tricky and requires resolving the boundary layers well even for attached flow.
Also do not forget that an airplane is not just the wing but also other structures, where the flow is much more complicated and harder to compute with analytical methods.
What is the exact formula for the lift and drag coefficients? Another answer gives you their definitions. But that is really just scaling to different dimensionless units.
The main point you have to do is to compute those forces!
To compute the forces you need to extract tje forces caused by the pressure distribution along the body and the forces caused by viscous friction along the body. You have to integrate those across the whole surface. The pressure force is perpendicular to the surface and the viscous force is parallel to the surface in the direction of the local flow speed.
The more difficult one, as already mentioned, are the viscous forces (skin friction). You need properly resolved boundary layers including the viscous sublayers. Just using the a wall function with the law of the wall in some form will cause significant inaccuracies.
