Favre averaging is the density-weighted averaging method, used in variable density or compressible turbulent flows, in place of the Reynolds averaging.
The method was introduced formally by the French physicist Alexandre Favre in 1965,[1][2] although Osborne Reynolds had also already introduced the density-weighted averaging in 1895.
[3] The averaging results in a simplistic form for the nonlinear convective terms of the Navier-Stokes equations, at the expense of making the diffusion terms complicated.
Favre averaging is carried out for all dynamical variables except the pressure.
For the velocity components,
, the Favre averaging is defined as:
ρ
ρ ¯
where the overbar indicates the typical Reynolds averaging, the tilde denotes the Favre averaging and
is the density field.
The Favre decomposition of the velocity components is then written as:
is the fluctuating part in the Favre averaging, which satisfies the condition
ρ
The normal Reynolds decomposition is given by
is the fluctuating part in the Reynolds averaging, which satisfies the condition
The Favre-averaged variables are more difficult to measure experimentally than the Reynolds-averaged ones, however, the two variables can be related in an exact manner if correlations between density and the fluctuating quantity is known; this is so because, we can write:
ρ ′
ρ ¯
The advantage of Favre-averaged variables are clearly seen by taking the normal averaging of the term
ρ
that appears in the convective term of the Navier-Stokes equations written in its conserved form.
ρ
ρ ¯
ρ ¯
ρ ′
ρ ′
ρ ′
ρ ¯
{\displaystyle {\begin{aligned}{\overline {\rho u_{i}u_{j}}}&={\overline {\rho }}\,{\overline {u_{i}}}\,{\overline {u_{j}}}+{\overline {\rho }}{\overline {u_{i}'u_{j}'}}+{\overline {u_{i}}}{\overline {\rho 'u_{j}'}}+{\overline {u_{j}}}{\overline {\rho 'u_{i}'}}+{\overline {\rho 'u_{i}'u_{j}'}}\\&={\overline {\rho }}{\widetilde {u_{i}}}{\widetilde {u_{j}}}+{\overline {\rho u_{i}''u_{j}''}}.\end{aligned}}}
As we can see, there are five terms in the averaging when expressed in terms of Reynolds-averaged variables, whereas we only have two terms when it is expressed in terms of Favre-averaged variables.