# Mixing Length Hypothesis Statement

In fluid dynamics, the **mixing length model** is a method attempting to describe momentum transfer by turbulenceReynolds stresses within a Newtonian fluidboundary 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[edit]

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[edit]

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

### Reynolds decomposition[edit]

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, .

## See also[edit]

## References[edit]

## See also[edit]

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