# Mixing length model

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

## 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:[6]

${\displaystyle T={\overline {T}}+T',}$

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

In the above picture, ${\displaystyle T'}$ can be expressed in terms of the mixing length considering a fluid parcel moving in the z-direction:

${\displaystyle T'=-\xi '{\frac {\partial {\overline {T}}}{\partial z}}.}$

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

${\displaystyle u'=-\xi '{\frac {\partial {\overline {u}}}{\partial z}},\qquad \ v'=-\xi '{\frac {\partial {\overline {v}}}{\partial z}},\qquad \ w'=-\xi '{\frac {\partial {\overline {w}}}{\partial z}}.}$

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, ${\displaystyle w'}$ must be in a neutrally stratified fluid.

Taking the product of horizontal and vertical fluctuations gives us:

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

The eddy viscosity is defined from the equation above as:

${\displaystyle K_{m}={\overline {\xi '^{2}}}\left|{\frac {\partial {\overline {w}}}{\partial z}}\right|,}$

so we have the eddy viscosity, ${\displaystyle K_{m}}$ expressed in terms of the mixing length, ${\displaystyle \xi '}$.

5. ^ Prandtl, L. (1926). Proc. Second Intl. Congr. Appl. Mech. Zürich.{{cite book}}: CS1 maint: location missing publisher (link)