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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 to give approximate averaged solutions to the Navier-Stokes equations. For an incompressible flow of Newtonian fluid, these equations can be written as
Additional recommended knowledge
The left hand side of this equation represents the change in mean momentum of fluid element due 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 due to the mean pressure field, the viscous stresses, and apparent stress due to the fluctuating velocity field, generally referred to as Reynolds stresses.
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 refers to separation of the flow variable (like velocity u) 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 f and g are two flow variables (like density (ρ), velocity (u), pressure (p), etc.) and s is one of the independent variables (x,y,z, or t) 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 of strain tensor.
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Reynolds-averaged_Navier-Stokes_equations". A list of authors is available in Wikipedia.|