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I know that the von Kármán constant ($k\approx0.4$) is useful in turbulence modeling. Something that bothers me is my lack of understanding where $k$ comes from or the physical meaning behind it. If $k$ is mainly used as a proportionality in the log-wind profile, why is it within the formulation of the Monin-Obukhov length? What is the physical meaning behind $k$? Is it a universal value (i.e. valid on extraterrestrial planets)? Why is $k$ considered important enough to be eponymous, while I see $\frac{1}{k}$ more often in formulations than $k$?

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K is in fact just an empirical proportionality value without any significant meaning. In fact each of the tracers (such as temperature, humidity and ...) should have had their own K BUT we dump that coefficient into the (star)* version of that tracer, for example T* (near surface temperature) or Hum* (near surface humidity), so we can still use K in their respective formulation.

Monin-Obukhov length has K in it because that length is related to u*. That length comes from dividing the Shear production and buoyancy procution/dissipation term in turbulence kinetic energy (TKE) equation. As you know Shear production is a function of momentum transfer and u*. Since there is a u* there is a k in there too.

Unlike other stuff in turbulence, the story of K is unfortunately boring.

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