In fluid dynamics

Fluid dynamics

In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, mixing length model is a method attempting to describe momentum

Momentum

In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

 transfer by turbulence

Turbulence

In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time...

 Reynolds stresses

Reynolds stresses

In fluid dynamics, the Reynolds stress is the stress tensor in a fluid obtained from the averaging operation over the Navier-Stokes equations to account for turbulent fluctuations in fluid momentum.-Definition:...

 within a newtonian fluid

Newtonian fluid

A Newtonian fluid is a fluid whose stress versus strain rate curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.-Definition:...

 boundary layer

Boundary layer

In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity of the fluid are considered in detail. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal...

 by means of an eddy viscosity. The model was developed by Ludwig Prandtl in the early 20th century. Prandtl himself had reservations about the model, describing it as, "only a rough approximation,"
but it has been used in numerous fields ever since, including atmospheric science, oceanography

Oceanography

Oceanography , also called oceanology or marine science, is the branch of Earth science that studies the ocean...

 and stellar structure

Stellar structure

Stars of different mass and age have varying internal structures. Stellar structure models describe the internal structure of a star in detail and make detailed predictions about the luminosity, the color and the future evolution of the star....

.

Physical intuition

The mixing length is conceptually analogous

Analogy

Analogy is a cognitive process of transferring information or meaning from a particular subject to another particular subject , and a linguistic expression corresponding to such a process...

 to the concept of mean free path

Mean free path

In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...

 in thermodynamics

Thermodynamics

Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

: a fluid parcel will conserve its properties for a characteristic length, , before mixing with the surrounding fluid. Prandtl described that the mixing length,
In the figure above, temperature

Temperature

Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

, , is conserved for a certain distance as a parcel moves across a temperature gradient

Gradient

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

. 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

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:

,
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

Pressure gradient force

The pressure gradient force is not actually a 'force' but the acceleration of air due to pressure difference . It is usually responsible for accelerating a parcel of air from a high atmospheric pressure region to a low pressure region, resulting in wind...

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, .