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] Also, Ali and Dey^[5] hypothesized an advanced concept of mixing instability.

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, ${\displaystyle \ \xi '}${\displaystyle \ \xi '}, before mixing with the surrounding fluid. Prandtl described that the mixing length,^[6]

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

To begin, we must first be able to express quantities as the sums of their slowly varying components and fluctuating components.

This process is known as Reynolds decomposition. Temperature can be expressed as:^[7]

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

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

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

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

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

$${\displaystyle u'=-\xi '{\frac {\partial {\overline {u}}}{\partial z}},\qquad \ v'=-\xi '{\frac {\partial {\overline {v}}}{\partial z}},\qquad \ w'=-\xi '{\frac {\partial {\overline {w}}}{\partial z}}.}$${\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, ${\displaystyle w'}${\displaystyle w'} must be in a neutrally stratified fluid.

Taking the product of horizontal and vertical fluctuations gives us:

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

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

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

• Law of the wall
• Reynolds stress equation model

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• Turbulence

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