Mixing length model

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:

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

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, ${\displaystyle \xi '}$.