Reynolds-averaged Navier–Stokes equations

The Reynolds equations or Reynolds - averaged Navier -Stokes equations (after Osborne Reynolds ) are a simplification of the Navier -Stokes equations, which are used in computational fluid dynamics for the approximation of turbulent flows. Because of the English term Reynolds - Averaged Navier -Stokes equations are also referred to as RANS equations.

As for turbulent flows with industrially relevant Reynolds numbers, the Navier -Stokes equations can not be solved numerically with reasonable cost (see Direct numerical simulation), the variables are divided into a mean value and a fluctuation value. Here, the average value is selected so that the variation of the mean value is then zero. One possibility is the Reynolds averaging, wherein the averaged over a small period of time or the ensemble averaging for transient currents. This dip in the equations additional terms, which must then be described by a turbulence model.

The incompressible Navier -Stokes equations, the instantaneous values ​​of the speed components and the pressure can be replaced by the appropriate addition of the mean and statistical fluctuation. Density and viscosity variations are neglected. In concrete terms for the momentum equation of the Navier -Stokes equations in Einstein summation convention shear

In replacement of the mentioned variables, and:

The difference resulting from the averaging term follows from the non-negligible velocity correlation. This tensor is called the Reynolds stress tensor (RST ).

Wherein the compressible Navier -Stokes equations addition the so-called Favre averaging is used in order to prevent products from averages. Here is additionally obtained in addition to the Reynolds stress tensor, the turbulent kinetic energy as a further unknown Term