Saffman $k-\omega^2$

Saffman’s $k-\omega^2$ turbulence model, initiated by Saffman’s research, plays a role in the two-equation models dedicated to turbulence research since the time of Kolmogorov in the 1940’s.

The basics of Saffman’s model is shown in the portrayal of a statistically steady or ‘slowly varying’ inhomogeneous turbulence field alongside the mean velocity distribution. This model states that turbulence can be described by ‘densities’ adhering to nonlinear diffusion equations. These equations account for a spectrum of phenomena, including convection by mean flow, amplification due to interaction with a mean velocity gradient, dissipation from turbulence interaction, and self-interaction diffusion.

Central to the Saffman model are two key equations: the energy equation and the $\omega^2$ equation. The energy equation integrates terms for the amplification of energy owing to the mean velocity gradient and dissipation attributable to vorticity, coupled with a diffusion term governed by eddy viscosity. This eddy viscosity also facilitates the diffusion of mean momentum by turbulent fluctuations. The $\omega^2$ equation, which governs the changes of vorticity density within the turbulent field, stands as the definitive feature, setting the Saffman model apart by explicitly considering the behavior of vorticity density. This is a bit different than specific dissipation ($\omega$).

Saffman’s model is further demonstrated through analytical and numerical solutions for a variety of flow scenarios, including Couette flow, plane Poiseuille flow, and free turbulent flows. The model’s capacity to predict phenomena such as the von Kármán constant in the law of the wall, via estimations of dimensionless constants within the equations. This is interesting given the early 1970’s publication date. The Saffman $kω^2$ is historically important for these reasons for early developments of two-equation $k-\omega$ models.