Mixing length model
![](http://upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Mixing_length.jpg/400px-Mixing_length.jpg)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Law_of_the_wall_%28English%29.svg/400px-Law_of_the_wall_%28English%29.svg.png)
In fluid dynamics, the mixing length model is a method attempting to describe momentum transfer by turbulence Reynolds stresses within a Newtonian fluid boundary layer by means of an eddy viscosity. The model was developed by Ludwig Prandtl in the early 20th century.[1] Prandtl himself had reservations about the model,[2] describing it as, "only a rough approximation,"[3] but it has been used in numerous fields ever since, including atmospheric science, oceanography and stellar structure.[4]
Physical intuition[]
The mixing length is conceptually analogous to the concept of mean free path in thermodynamics: a fluid parcel will conserve its properties for a characteristic length, , before mixing with the surrounding fluid. Prandtl described that the mixing length,[5]
may be considered as the diameter of the masses of fluid moving as a whole in each individual case; or again, as the distance traversed by a mass of this type before it becomes blended in with neighbouring masses...
In the figure above, temperature, , is conserved for a certain distance as a parcel moves across a temperature gradient. The fluctuation in temperature that the parcel experienced throughout the process is . So can be seen as the temperature deviation from its surrounding environment after it has moved over this mixing length .
Mathematical formulation[]
To begin, we must first be able to express quantities as the sums of their slowly varying components and fluctuating components.
Reynolds decomposition[]
This process is known as Reynolds decomposition. Temperature can be expressed as:
, [6]
where , is the slowly varying component and is the fluctuating component.
In the above picture, can be expressed in terms of the mixing length:
The fluctuating components of velocity, , , and , can also be expressed in a similar fashion:
although the theoretical justification for doing so is weaker, as the pressure gradient force can significantly alter the fluctuating components. Moreover, for the case of vertical velocity, must be in a neutrally stratified fluid.
Taking the product of horizontal and vertical fluctuations gives us:
.
The eddy viscosity is defined from the equation above as:
,
so we have the eddy viscosity, expressed in terms of the mixing length, .
References[]
- ^ Holton, James R. (2004). "Chapter 5 – The Planetary Boundary Layer". Dynamic Meteorology. International Geophysics Series. 88 (4th ed.). Burlington, MA: Elsevier Academic Press. pp. 124–127.
- ^ Prandtl, L. (1925). "Z. angew". Math. Mech. 5 (1): 136–139.
- ^ Bradshaw, P. (1974). "Possible origin of Prandt's mixing-length theory". Nature. 249 (6): 135–136. Bibcode:1974Natur.249..135B. doi:10.1038/249135b0.
- ^ Chan, Kwing; Sabatino Sofia (1987). "Validity Tests of the Mixing-Length Theory of Deep Convection". Science. 235 (4787): 465–467. Bibcode:1987Sci...235..465C. doi:10.1126/science.235.4787.465. PMID 17810341.
- ^ Prandtl, L. (1926). "Proc. Second Intl. Congr. Appl. Mech". Zurich.
- ^ "Reynolds Decomposition". Florida State University. 6 December 2008. Retrieved 2008-12-06.
See also[]
- Law of the wall
- Reynolds stress equation model
- Oceanography
- Turbulence