Turbulence modelling

In previous investigations of unsteady turbulent flows, improved one- and two-equation models exhibited the best performance. Therefore three different eddy-viscosity turbulence models are applied to the present case: Spalart-Allmaras (SA) one-equation model [9], Wilcox $k$-$\omega$ model [10] and the LLR $k$-$\omega$ model [11]. The latter is an improved two-equation model with special respect to the realizability conditions.

In simulations of unsteady turbulent flows by Reynolds-averaged approaches, the treatment of turbulent time-scales always requires special attention. An important assumption in the derivation of statistical turbulence models is that time-averaging can be used instead of ensemble-averaging. Therefore the applicability of these models depends on the existence of a spectral gap of one or two orders between the resolved time-scales and the modelled scales. Otherwise a formal conflict can arise from an overlapping of resolved and modelled motions. The turbulence model will transfer engergy from the large-scale motion into dissipation. But a RANS approach cannot provide a counteracting mechanism (back-scatter).

It is difficult to get a reliable estimation for the existence of a spectral gap in advance. To check the applicability of RANS simuations Rung [12] suggests an approximation for the ratio between resolved time-scale $T_m$ and modelled turbulent time-scale $T_t$:

$$\frac{T_m}{T_t} \approx \gamma \frac{Re^{1/5}}{St}$$ (2)

$$\mbox{boundary-layer:} \ \gamma \in [1,10] ; \ \mbox{free shear layer:} \ \gamma \in [0.1,1] \ .$$

Finally the entire flow field needs to be checked for the smallest occuring turbulent time-scales. The resolved time-scales of the large-scale motions have to be significantly larger than the high-frequency small-scale motions that are captured by the turbulence model. Problems can occur if parts of the spectrum are modelled as well as resolved.