Reynolds-averaged Navier-Stokes equations

The Reynolds-averaged Navier–Stokes equations (or RANS equations) are time-averaged
equations of motion for fluid flow. The idea behind the equations is 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 \scriptstyle u the decomposition would be...

, whereby an instantaneous quantity is decomposed into its time-averaged and fluctuating quantities, an idea first proposed by Osborne Reynolds

Osborne Reynolds

Osborne Reynolds FRS was a prominent innovator in the understanding of fluid dynamics. Separately, his studies of heat transfer between solids and fluids brought improvements in boiler and condenser design.-Life:...

. The RANS equations are primarily used to describe 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 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...

to give approximate time-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 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 in Einstein notation

Einstein notation

In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae...

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

Mean flow

In fluid dynamics, the fluid flow is often decomposed into a mean flow – and deviations from the mean. The averaging can be done either in space or in time....

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 the Reynolds stress

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:...

. This nonlinear Reynolds stress term requires additional modeling to close the RANS equation for solving, and has led to the creation of many different turbulence models. The time-average operator

is a Reynolds operator

Reynolds operator

In fluid dynamics and invariant theory, a Reynolds operator is a mathematical operator given by averaging something over a group action, that satisfies a set of properties called Reynolds rules...

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

. Reynolds decomposition refers to separation of the flow variable (like velocity

) into the mean (time-averaged) component (

) and the fluctuating component (

). Because the mean operator is a Reynolds operator

Reynolds operator

In fluid dynamics and invariant theory, a Reynolds operator is a mathematical operator given by averaging something over a group action, that satisfies a set of properties called Reynolds rules...

, it has a set of properties. One of these properties is that the mean of the fluctuating quantity being equal to zero(

). Thus,

, where

is the position vector. Some authors prefer using

instead of

for the mean term (since an overbar is sometimes used to represent a vector). In this case, the fluctuating term

is represented instead by

. This is possible because the two terms do not appear simultaneously in the same equation. To avoid confusion, the notation

will be used to represent the instantaneous, mean, and fluctuating terms, respectively.
The properties of Reynolds operator

Reynolds operator

In fluid dynamics and invariant theory, a Reynolds operator is a mathematical operator given by averaging something over a group action, that satisfies a set of properties called Reynolds rules...

s are useful in the derivation of the RANS equations. Using these properties, the Navier–Stokes equations of motion, expressed in tensor notation, are (for an incompressible Newtonian fluid):

where

is a vector representing external forces.

Next, each instantaneous quantity can be split into time-averaged and fluctuating components, and the resulting equation time-averaged,

to yield:

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:

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