9.4 RANS Models
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Fig. 9.10. Time averaging for a statistically steady flow (left) and ensemble averaging for an unsteady flow (right)
these averaging processes; applying it to the Navier-Stokes equations yields
the Reynolds-averaged Navier-Stokes (RANS) equations.
From Eq. (9.27), it follows that
= 0. Thus, averaging any linear term
in the conservation equations simply gives the identical term for the averaged
quantity. From a quadratic nonlinear term we get two terms, the product of
the average and a covariance:
The last term is zero only if the two quantities are uncorrelated; this is rarely
the case in turbulent flows and, as a result, the conservation equations contain
terms such as p- ,
called the Reynolds stresses, and p m , known as the
turbulent scalar flux, among others. These cannot be represented uniquely in
terms of the mean quantities.
The averaged continuity and momentum equations can, for incompressible flows without body forces, be written in tensor notation and Cartesian
coordinates as:
where the Tij are the mean viscous stress tensor components:
Finally the equation for the mean of a scalar quantity can be written: