In many fluid-dynamics models and quantities (Reynolds number, Strouhal number... generally various turbulent flow descriptions) there is a parameter (usually $L$ or $\ell$) descripted as characteristic linear dimension or characteristic length. What is that in intuitive and physical meaning? I understand the typical examples (hydraulic diameter etc.) but I can't see so far any systematic generalisation.
What is the definition? (Or, maybe: What is its role during the derivation?)
The characteristic length is the dimension that defines the length scale of a physical system. This implies that for any system, which my contain several length scales, there may only be one characteristic length scale. This is also generalizable to the other characteristic scales such as time, speed, etc.
As you may have read in my other answer, generally the smallest length scale is the characteristic length scale because the gradients there are usually the largest, generating most of the mass, heat or momentum transport. On the other hand, in dynamic systems where heat, mass or momentum hasn't been transported over a physical dimension (length, width, height) yet, the length scale is defined by the dynamics of the system. For example, in the case of semi-infinite diffusion, there is obviously no physical dimension to the system since it is infinite, but there is a (time-dependent) characteristic length defined by the growing of the diffusion boundary layer $\delta=\sqrt{\pi D t}$.
Not wanting to go into too much mathematical detail, a characteristic scale is a scale which when used to non-dimensionalize its respective physical quantity, all terms in the dynamic equations become $O(1)$ or less; in that sense a characteristic scale is the maximum value that quantity can have (within an order of magnitude). This can be useful if you need back-of-the-envelope estimates of some quantities but the dynamic equations are difficult (or impossible) to solve. An example of this can be found in this answer.
