Mixing length model

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, $\backslash \; \backslash xi\text{'}$, before mixing with the surrounding fluid. Prandtl described that the mixing length,{{cite book | last = Prandtl | first = L. | title = Proc. Second Intl. Congr. Appl. Mech. | location = Zürich | date = 1926 }}

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

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:{{cite news|url=https://web1.eng.famu.fsu.edu/~dommelen/courses/flm/flm00/topics/turb/node2.html |title=Reynolds Decomposition | publisher=Florida State University | date=6 December 2008 | access-date=2008-12-06}}

$$T\; =\; \backslash overline\{T\}\; +\; T\text{'},$$

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

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

$$T\text{'}\; =\; -\backslash xi\text{'}\; \backslash frac\{\backslash partial\; \backslash overline\{T\}\}\{\backslash partial\; z\}.$$

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

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

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

Taking the product of horizontal and vertical fluctuations gives us:

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

The eddy viscosity is defined from the equation above as:

$$K\_m=\backslash overline\{\backslash xi\text{'}^2\}\; \backslash left|\; \backslash frac\{\backslash partial\; \backslash overline\{w\}\}\{\backslash partial\; z\}\backslash right|,$$

so we have the eddy viscosity, $K\_m$ expressed in terms of the mixing length, $\backslash xi\text{'}$.

• Law of the wall
• Reynolds stress equation model

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Category:Oceanography

Category:Turbulence