# Scaling law or order estimation for turbulent viscosity in a rectangular cavity

Is there any recognized scaling law for the eddy viscosity inside a rectangular cavity? Indeed, that should mean some sort of homogenized value, since the quantity is spatially dependent.

I have made several RANS simulations of the pipe flow over a set of rectangular cavities (generally of slightly different geometries, not a regular array) under low Mach number and with the typical length scale of couple centimetres. The eddy viscosity distribution was of course not the same but the typical values were always of order $$10^{-3}$$ in SI units and since the fluid was air, there is not much difference between the dynamic and kinematic quatntity. Is there any way to show that this observation is supported by some sort of educated guess?

Here is a naive attempt based on the Boussinesq hypothesis. Let the dimensions of a 2D cavity be $$H$$ and $$W$$ and the typical flow speed inside the cavity $$U$$. Then we have

$$U^2 \approx \nu_t \left( \frac{U}{H} + \frac{U}{W} \right)-\frac{2}{3}\frac{1}{2} U^2$$

which gives

$$\nu_t \approx \frac{4}{3}\frac{HW}{W+H}U$$