Can the circulation around a wing be predicted (roughly) without the help of a computer? [duplicate]

Question

This site already has several questions about how airplanes fly. Some of the answers do give useful insights, but the only real explanation I've found involves using a computer to solve the Navier-Stokes equations with a no-slip boundary condition (ref 1). By "real explanation," I mean one that actually predicts the amount of lift, starting from basic physics.

References 1 and 2 both suggest that an intuitive explanation might not be possible without significant input from computer calculations, but their definition of "intuitive" might be more restrictive than mine. I don't mind an explanation that relies on some advanced math, and I don't even mind if it only gives a very rough estimate, as long as it isn't computer-assisted.

To be more specific, consider the Kutta-Joukowski theorem, which already has a simple intuitive explanation. The K-J theorem relates the lift to the net circulation around the wing (under certain simplifying conditions), so if we had a non-computer-assisted way to estimate the net circulation, then we could put these pieces together to get a decent explanation of how airplanes fly.

Question: Can we estimate the circulation around an approximately flat thin airfoil$^\dagger$ under typical steady-state flying conditions, without any help from computers, using the no-slip condition and the basic principles that are used to deduce the Navier-Stokes equations?

Assuming the Kutta condition (that the flow leaves the surface only at the trailing edge) is fine. An argument that appeals to the starting-vortex might be okay (page 345 in ref 2), as long as it leads to a quantitative estimate valid for steady-state flying conditions.

$^\dagger$ ^{I'm focusing on an approximately flat thin airfoil because this seems like the simplest case. Other conditions (like allowing a slightly rounded leading edge) are also welcome if they help simplify the argument.}

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References:

1. McLean (2014), Understanding Aerodynamics: Arguing from the Real Physics, Wiley (https://onlinelibrary.wiley.com/doi/book/10.1002/9781118454190)

2. Anderson (2017), Fundamentals of Aerodynamics (sixth edition), McGraw Hill (https://www.mheducation.com/highered/product/fundamentals-aerodynamics-anderson/M9781259129919.html)

Here's a possibly related Big List question, but without any obvious matches in the list:

• Exact Solutions to the Navier-Stokes Equations

Answer 1

Historically, aerodynamicists used complex analysis to estimate the circulation past airfoils. Here's the process:

Assume that the flow around an airfoil is roughly two-dimensional, and that the cross-section of the airfoil around the "extra" dimension is uniform. In that scenario, the circulation around the airfoil is uniform along that "extra" dimension, and you only need to consider a 2-D flow around the airfoil cross-section to obtain any relevant physics.

Consider converting the 2-D real plane under consideration into the complex plane ($z = x+ iy$), and construct a "complex velocity" of the form $v' = u - iv$, where $u$ and $v$ are the $x$- and $y$-components of the velocity respectively. If the flow is also assumed to be inviscid and incompressible, there is always a complex scalar function f such that:

$$\frac{df}{dz} = v'$$

In addition, any conformal mapping of such a flow/function is also a valid flow/function of this type. This leads to a nifty trick—if I can find a conformal mapping that maps a circle/cylinder to an airfoil cross-section, I can use that same conformal mapping to convert the inviscid/incompressible flow past a circle/cylinder (which is simple to find) to the flow past that airfoil cross-section. That, combined with the Kutta condition, provides a prediction of the circulation past the airfoil.

Luckily, there are many such conformal mappings—the most famous is perhaps the Zhukovsky transform, which is just:

$$w = z + 1/z$$

A direct prediction for the circulation past a Zhukovsky airfoil can be found here. A visualization of the Zhukovsky transformation of flow past a circle to flow past an airfoil cross-section can be found here.