Reynolds Averaged Navier-Stokes (RANS) Equations

In this method, the instantaneous quantities in the governing equations for the mass, momentum and energy are decomposed into their mean and fluctuating components. For an incompressible flow, the instantaneous velocity obeys the following equation (in Cartesian tensor form)

(26.1)

The velocity components and scaler quantities such as pressure are decomposed following the Reynolds decomposition as

(26.2)

(26.3)

where and P are the time or ensemble average components and and the fluctuating components. By substituting and p of Eqs. (26.2) and (26.3) into Eq. (26.1) and time (or ensemble) averaging, the equations can be written in terms of the mean quantities as

(26.4)

The continuity equation in terms of mean quantities (for an incompressible flow) can be written as

(26.5)

Equations (26.4) are called the Reynolds-averaged Navier-Stokes equations. They have the same form as the laminar Navier-Stokes equations with the velocities and other variables representing time-averaged (or ensemble-averaged) values. However, an additional term appears in Eqn. (26.4) which represents the effect of turbulence and is called the Reynolds Stress Tensor: . This term introduces six unknown terms but matching equations are not available to close the system. Therefore, this term needs to be modeled in order to close the system of equations. Several approaches have evolved for this purpose. The commonly followed methodologies include

1. Eddy viscosity models and
2. Reynolds stress transport models.

All these approaches require a special treatment of turbulent flows near the wall. The special treatment on near wall flows has been described in a separate section.