Notes on Computational Fluid Dynamics: General Principles

7.15 Summary of turbulence modelling

The $k$ model solves transport equations for turbulent kinetic energy $\text{[math]}$ and dissipation rate $\text{[math]}$ , Sec. 7.1 .
It is the original of a family of two-equation models, which ultimately provide $\text{[math]}$ to calculate the turbulent stresses.
Initial and inlet values for $\text{[math]}$ and $\text{[math]}$ must be speciﬁed, which can be calculated from turbulent intensity $\text{[math]}$ and mixing length scale $\text{[math]}$ , respectively, Sec. 7.2 .
$\text{[math]}$ and $\text{[math]}$ are often estimated using functions that ﬁt experimental data for fully developed turbulent ﬂow, Sec. 7.3 .
The $\text{[math]}$ models replace $\text{[math]}$ by speciﬁc dissipation rate $\text{[math]}$ , with equivalent expressions for initial and inlet values, Sec. 7.10 .
The two “standard” $\text{[math]}$ models today are the $\text{[math]}$ SST model and $\text{[math]}$ 2006 model, Sec. 7.11 .
Models can provide a turbulent conductivity $\text{[math]}$ to calculate the turbulent heat ﬂux for thermal problems, Sec. 7.12 .

Turbulent boundary layers include a thin viscous sub-layer adjacent to the boundary with a linear velocity proﬁle and, further from the boundary, the inertial sub-layer with a log law proﬁle, Sec. 7.4 .
Proﬁles in temperature are similar to those for velocity, with equivalent linear and log law relationships, Sec. 7.13 .
Very thin cells are generally needed to resolve the viscous sub-layer to calculate the velocity gradient $\text{[math]}$ at the wall accurately, Sec. 7.5 .
Such thin cells within the boundary layer region can increase the mesh to a size which is prohibitively costly to run.

Wall functions permit much larger cells near the wall, by exploiting the universal character of the velocity distribution.
The functions increase $\text{[math]}$ at the wall to compensate for the under-prediction of $\text{[math]}$ with larger cells, to improve the prediction of the wall shear stress, Sec. 7.5 .
Thermal wall functions similarly increase $\text{[math]}$ at the wall to compensate for the under-prediction of $\text{[math]}$ , in order to improve the prediction of the wall heat ﬂux, Sec. 7.14 .
Standard wall functions make no adjustment to $\text{[math]}$ when the near wall cell centre falls below the transition within a buﬀer layer, Sec. 7.5 .
Other models include a continuous function of $\text{[math]}$ through the viscous sub-layer to the wall and adjustments for surface roughness, Sec. 7.6 .
Boundary conditions for turbulence ﬁelds with wall functions are based on observed proﬁles of those ﬁelds, Sec. 7.7 .

Turbulence models must predict the universal character of boundary layers when the viscous sub-layer is resolved with suﬃciently thin cells, Sec. 7.8 .
Models like the Launder-Sharma $k$ include source terms and damping functions to improve the predictions and to simplify boundary conditions, Sec. 7.9 .

Notes on CFD: General Principles - 7.15 Summary of turbulence modelling