Mixing length model

Mixing length model
The mixing length is a distance that a fluid parcel will keep its original characteristics before dispersing them into the surrounding fluid. Here, the bar on the left side of the figure is the mixing length.
law of the wall, horizontal velocity near the wall with mixing length model

In fluid dynamics, 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]

Contents

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, \ \xi' , 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, \ T, 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 \ T'. So \ T' can be seen as the temperature deviation from its surrounding environment after it has moved over this mixing length \ \xi'.

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:

 T = \overline{T} + T', [6]

where \ \overline{T}, is the slowly varying component and \ T' is the fluctuating component.

In the above picture, \ T' can be expressed in terms of the mixing length:

\ T' = -\xi' \frac{\part \overline{T}}{\part z}.

The fluctuating components of velocity, \ u', \ v', and \ w', can also be expressed in a similar fashion:

\ u' = -\xi' \frac{\part \overline{u}}{\part z}. \ v' = -\xi' \frac{\part \overline{v}}{\part z}. \ w' = -\xi' \frac{\part \overline{w}}{\part z}.

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, \ w' must be in a neutrally stratified fluid.

Taking the product of horizontal and vertical fluctuations gives us:

\ \overline{u' w'} = \overline{\xi' ^2} \left | \frac{\part \overline{w}}{\part z}\right| \frac{\part \overline{u}}{\part z}.

The eddy viscosity is defined from the equation above as:

\ K_m=\overline{\xi'^2} \left| \frac{\part \overline{w}}{\part z}\right|,

so we have the eddy viscosity, \ K_m expressed in terms of the mixing length, \ \xi'.

References

  1. ^ [|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. 
  2. ^ Prandtl, L. (1925). "Z. angew". Math. Mech. 5 (1): 136–139. 
  3. ^ Bradshaw, P. (1974). "Possible origin of Prandt's mixing-length theory". Nature 249 (6): 135–136. Bibcode 1974Natur.249..135B. doi:10.1038/249135b0. 
  4. ^ 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. 
  5. ^ Prandtl, L. (1926). "Proc. second int. Congr. appl. Mech.". Zurich. 
  6. ^ "Reynolds Decomposition". Florida State University. 6 December 2008. http://www.eng.fsu.edu/~dommelen/courses/flm/flm00/topics/turb/node2.html. Retrieved 2008-12-06. 

]

See also

  • Law of the wall

Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Model photosphere — Some essential steps in model atmosphere analysis for determining stellar abundances (Figure by Bengt Gustafsson, Astronomical Observatory, Uppsala). The photosphere denotes those solar or stellar surface layers from which optical radiation… …   Wikipedia

  • Model yachting — This article is concerned with model vessels that are operated, either though programatic means or by remote control. For models intended for static display see ship model San Francisco Model Yacht Club X class (1000 square inches, few other… …   Wikipedia

  • RGB color model — RGB redirects here. For other uses, see RGB (disambiguation). A representation of additive color mixing. Projection of primary color lights on a screen shows secondary colors where two overlap; the combination of all three of red, green, and blue …   Wikipedia

  • Mixture model — See also: Mixture distribution In statistics, a mixture model is a probabilistic model for representing the presence of sub populations within an overall population, without requiring that an observed data set should identify the sub population… …   Wikipedia

  • CMYK color model — Color p …   Wikipedia

  • High (New Model Army album) — Infobox Album | Name = High Type = Album Artist = New Model Army Released = August 2007 Recorded = State of the Ark, Surrey Genre = Rock Length = 49:15 Label = Attack Attack Producer = Chris Kimsey Reviews = Last album = Carnival (2005) This… …   Wikipedia

  • Radio-controlled model — A radio controlled model (or RC model) is a model that is steerable with the use of radio control. All types of vehicles imaginable have had RC systems installed in them, including cars, boats, planes, and even helicopters and scale railway… …   Wikipedia

  • Plug flow reactor model — The plug flow reactor (PFR) model is used to describe chemical reactions in continuous, flowing systems. The PFR model is used to predict the behaviour of chemical reactors, so that key reactor variables, such as the dimensions of the reactor,… …   Wikipedia

  • Computational fluid dynamics — Computational physics Numerical analysis  …   Wikipedia

  • Occam's razor — For the aerial theatre company, see Ockham s Razor Theatre Company. It is possible to describe the other planets in the solar system as revolving around the Earth, but that explanation is unnecessarily complex compared to the modern consensus… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”