Favre averaging

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 scientist A. J. Favre in 1965, ^{[1] [2]} although Osborne Reynolds has also already introduced the density-weighted averaging in 1895. ^[3] The averaging results a simplistic form for the nonlinear convective terms of the Navier-Stokes equations, at the expense of making the diffusion terms complicated.

Favre averaged variables

Favre averaging is carried out for all dynamical variables except the pressure. For the velocity components, ${\displaystyle u_{i}}$, the Favre averaging is defined as

${\displaystyle {\widetilde {u_{i}}}={\frac {\overline {\rho u_{i}}}{\overline {\rho }}}}$

where the overbar indicates the typical Reynolds averaging, the tilde denotes the Favre averaging and ${\displaystyle \rho (\mathbf {x} ,t)}$ is the density field. The Favre decomposition of the velocity components is then written as

${\displaystyle u_{i}={\widetilde {u_{i}}}+u_{i}''}$

where ${\displaystyle u_{i}''}$ is the fluctuating part in the Favre averaging, which satisfies the condition ${\displaystyle {\widetilde {u_{i}''}}=0}$, that is to say, ${\displaystyle {\overline {\rho u_{i}''}}=0}$. The normal Reynolds decomposition is given by ${\displaystyle u_{i}={\overline {u_{i}}}+u_{i}'}$, where ${\displaystyle u_{i}'}$ is the fluctuating part in the Reynolds averaging, which satisfies the condition ${\displaystyle {\overline {u_{i}'}}=0}$. 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

${\displaystyle {\widetilde {u_{i}}}={\overline {u_{i}}}\left(1+{\frac {\overline {\rho 'u_{i}'}}{{\overline {\rho }}\,{\overline {u_{i}}}}}\right).}$

The advantage of Favre-averaged variables are clearly seen by taking the normal averaging of the term ${\displaystyle \rho u_{i}u_{j}}$ that appears in the convective term of the Navier-Stokes equations written in its conserved form. This is given by ^{[4] [5]}

${\displaystyle {\begin{aligned}{\overline {\rho u_{i}u_{j}}}&={\overline {\rho }}\,{\overline {u_{i}}}\,{\overline {u_{j}}}+{\overline {\rho }}{\overline {u_{i}'u_{j}'}}+{\overline {u_{i}}}{\overline {\rho 'u_{j}'}}+{\overline {u_{j}}}{\overline {\rho 'u_{i}'}}+{\overline {\rho 'u_{i}'u_{j}'}}\\&={\overline {\rho }}{\widetilde {u_{i}}}{\widetilde {u_{j}}}+{\overline {\rho u_{i}''u_{j}''}}.\end{aligned}}}$

As we can see, there are five terms in the averaging when expressed in terms of Reynolds-averaged variables, whereas we have only two terms when it is expressed in terms of Favre-averaged variables.

References

1. Favre, A. J. (1965). The equations of compressible turbulent gases. Aix-Marseiller University (France) Inst. De Mecanique Statistique De La Turbulence.
2. Favre, A. J. (1969). Statistical equations of turbulent gases, in Problems of Hydrodynamics and Continuum Mechanics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 231-266.
3. Reynolds, O. (1895). IV. On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Philosophical transactions of the royal society of london.(a.), (186), 123-164.
4. Libby P.A., Williams, F. A., (1980), Turbulent reaction flows, pp. 14-16.
5. Bilger, R. (1975). A note on Favre averaging in variable density flows. Combustion Science and Technology, 11(5-6), 215-217.