# Characteristic length

In physics, a characteristic length is an important dimension that defines the scale of a physical system. Often, such a length is used as an input to a formula in order to predict some characteristics of the system.

Examples:

In computational mechanics, a characteristic length is defined to force localization of a stress softening constitutive equation. The length is associated with an integration point. For 2D analysis, it is calculated by taking square root of the area. For 3D analysis, it is calculated by taking cubic root of the volume associated to the integration point.[1]

## Calculation

The general formula for the characteristic length is the volume of a system divided by its surface[2]:

${\displaystyle L_{c}=V_{body}/A_{surface}}$

A typical use-case is calculating flow through circular and non-circular tubes, in order to examine flow conditions (i.e. the Reynolds number). In those cases, the characteristic length is the diameter of the pipe, or in case of non-circular tubes its hydraulic diameter ${\displaystyle D_{h}}$:

${\displaystyle D_{h}=4A_{c}/p}$

Where ${\displaystyle A_{c}}$is the cross-sectional area of the pipe and p its wetted perimeter. It is defined such that it reduces to a circular diameter of D for circular pipes.

For flow through a square duct with a side length of a, the hydraulic diameter ${\displaystyle D_{h}}$is:

${\displaystyle D_{h}=4a^{2}/4a=a}$

For a rectangular duct with side lengths a and b:

${\displaystyle D_{h}={\frac {4ab}{2(a+b)}}={\frac {2ab}{a+b}}}$

For free surfaces (such as in open-channel flow), the wetted perimeter includes only the walls in contact with the fluid.[3]

## References

1. ^ J. Oliver, M. Cervera, S. Oller, Isotropic damage models and smeared crack analysis of concrete. Proceedings of SCI-C 1990 (1990) 945–958.
2. ^ "Characteristic Length - calculator". fxSolver. Retrieved 2018-07-08.
3. ^ Çengel, Yunus A.; Cimbala, John M. (2014). Fluid mechanics : fundamentals and applications (3rd ed.). New York: McGraw Hill. ISBN 9780073380322. OCLC 880405759.