I want to model turbulent flow in fully developed pipe flow using an eddy diffusion coefficient K [m^2/s]. Using gradient diffusion theory, K can be obtained as
K = u * x
where u [m/s] denotes a characteristic velocity scale, which I choose as the cross-sectional average flow velocity, and x denotes the turbulence length scale. x could be chosen as the size of the largest eddies in turbulent flow in my scenario.
In a CFD Wiki, I found that x can be obtained as
x = 3.8% * d
where d [m] denotes the pipe diameter. My question is, where do the 3.8% come from? My guess is this stems from measurement data. However, I can't find a source for it. Any reference to this number or a similar model would be appreciated!