Mixing length model

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,   ξ ′ , before mixing with the surrounding fluid. Prandtl described that the mixing length,

In the figure above, temperature,   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   T ′ . So   T ′ can be seen as the temperature deviation from its surrounding environment after it has moved over this mixing length   ξ ′ .

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:

T = T ¯ + T ′ ,

where   T ¯ , is the slowly varying component and   T ′ is the fluctuating component.

In the above picture,   T ′ can be expressed in terms of the mixing length:

  T ′ = − ξ ′ ∂ T ¯ ∂ z .

The fluctuating components of velocity,   u ′ ,   v ′ , and   w ′ , can also be expressed in a similar fashion:

  u ′ = − ξ ′ ∂ u ¯ ∂ z ,   v ′ = − ξ ′ ∂ v ¯ ∂ z ,   w ′ = − ξ ′ ∂ w ¯ ∂ 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 ′ must be in a neutrally stratified fluid.

Taking the product of horizontal and vertical fluctuations gives us:

  u ′ w ′ ¯ = ξ ′ 2 ¯ | ∂ w ¯ ∂ z | ∂ u ¯ ∂ z .

The eddy viscosity is defined from the equation above as:

  K m = ξ ′ 2 ¯ | ∂ w ¯ ∂ z | ,

so we have the eddy viscosity,   K m expressed in terms of the mixing length,   ξ ′ .