Turbulence modeling

The simulation program can be run in URANS mode, solving the Unsteady Reynolds-averaged Navier-Stokes equations using statistical turbulence models, as well as in a mode for Large-Eddy-Simulation (LES) or combinations of both.

In previous URANS investigations with a variety of different one- and two-equation turbulence models as well as Explicit Algebraic Reynolds-stress Models (EARSM), the LLR $k$-$\omega$ model by Rung [10] exhibited the best overall performance for airfoil flows with large separation [11]. It represents an improved two-equation eddy-viscosity model formulated with special respect to the realizability principle.

Additionally, hybrid URANS/LES simulations have been performed which are also based on Rung's LLR $k$-$\omega$ model. Here, the destruction term in the transport equation for the turbulent kinetic energy $k$ is replaced by a formulation based on the turbulent length-scale $L_t$:

$$\frac{D k}{D t} - {\rm Diff}(k) = {\rm Prod}(k) - \beta_k \underbrace{\frac{k^{3/2}}{L_t}}_{{\rm Diss}(k)}$$ (1)

This turbulent length-scale is used to switch between the URANS and the LES mode, according to the concept of Detached Eddy Simulation (DES) [12]:

$$L_t = \mbox{min} \left( L_{t,URANS}, L_{t,LES} \right)$$

$$= \mbox{min} \left( \frac{\sqrt{k}}{c_\mu \omega}, C_{DES} \Delta \right)$$ (2)

with the specific dissipation $\omega$. In the LLR model, the anisotropy parameter $c_{\mu}$ is not constant. In the LES mode, the length-scale $L_{t,LES}$ is given by the definition of Strelets [13] using the local resolution of the mesh: $\Delta = \mbox{max} \left( \Delta x, \Delta y, \Delta z \right)$ and the constant $C_{DES}=0.78$ obtained from calibration of decaying isotropic turbulence [14].

Compared to LES, the DES approach offers the possibility to yield results of high resolution and physical quality without the demand of extensive numerical effort. Usually the near wall mesh for DES computations can be created similarly to those of RANS simulations. In the last years, several promising applications of such approaches have been published [12,13,15]. One goal of the present study is to verify the applicability of statistical turbulence models for this kind of flow and to prove that all important flow features can be captured by a comparison to the results of a DES.