Reynolds-averaged Navier-Stokes equations

The Reynolds-averaged Navier–Stokes (RANS) equations are time-averaged

equations of motion for fluid flow. They are primarily used while dealing with turbulent flows. These equations can be used with approximations based on knowledge of the properties of flow turbulence

Turbulence

In fluid dynamics, turbulence or turbulent flow is a fluid regime characterized by chaotic, stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time. Flow that is not turbulent is called laminar flow...

 to give approximate averaged solutions to the Navier–Stokes equations.
For a stationary

Stationary process

In the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...

, incompressible flow of 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:...

, these equations can be written as:
The left hand side of this equation represents the change in mean momentum of fluid element owing to the unsteadiness in the mean flow and the convection by the mean flow.

The Reynolds-averaged Navier–Stokes (RANS) equations are time-averaged

equations of motion for fluid flow. They are primarily used while dealing with turbulent flows. These equations can be used with approximations based on knowledge of the properties of flow turbulence

Turbulence

In fluid dynamics, turbulence or turbulent flow is a fluid regime characterized by chaotic, stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time. Flow that is not turbulent is called laminar flow...

 to give approximate averaged solutions to the Navier–Stokes equations.
For a stationary

Stationary process

In the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...

, incompressible flow of 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:...

, these equations can be written as:
The left hand side of this equation represents the change in mean momentum of fluid element owing to the unsteadiness in the mean flow and the convection by the mean flow. This change is balanced by the mean body force, the isotropic stress owing to the mean pressure field, the viscous stresses, and apparent stress owing to the fluctuating velocity field, generally referred to as Reynolds stresses

Reynolds stresses

In fluid dynamics, the Reynolds stresses is the stress tensor in a fluid due to the random turbulent fluctuations in fluid momentum. The stress is obtained from an average over these fluctuations.-Averaging and the Reynolds stress:To illustrate, here we use Cartesian vector index notation...

.

Derivation of RANS equations

The basic tool required for the derivation of the RANS equations from the instantaneous Navier–Stokes equations is the Reynolds decomposition

Reynolds decomposition

In fluid dynamics and the theory of turbulence, Reynolds decomposition is a mathematicaltechnique to separate the average and fluctuating parts of a quantity.For example, for a quantity the decomposition would be...

. Reynolds decomposition refers to separation of the flow variable (like velocity ) into the mean (time-averaged) component and the fluctuating component .
Thus,

where is the position vector.

The following rules will be useful while deriving the RANS. If and are two flow variables (like density , velocity , pressure , etc.) and is one of the independent variables then,
Now the Navier–Stokes equations of motion for an incompressible Newtonian fluid are:

Substituting,

, etc.

and taking a time-average of these equations yields,

The momentum equation can also be written as,
On further manipulations this yields,

where,

is the mean rate of strain tensor.

Finally, since integration in time removes the time dependence of the resultant terms, the time derivative must be eliminated, leaving: