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]
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 .
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: [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 considering a fluid parcel moving in the z-direction:
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, .
{{
cite book}}
: CS1 maint: location missing publisher (
link)
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]
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 .
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: [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 considering a fluid parcel moving in the z-direction:
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, .
{{
cite book}}
: CS1 maint: location missing publisher (
link)