## TAU Best Practice Turbulence

### General

Turbulence is characterized by an irregular, three-dimensional, unsteady fluctuation of all flow quantities. These fluctuations cover a very large range of scales that cannot be completely resolved in practical applications. Turbulence models are therefore used to replace at least part of the details the fluctuation, in order to make complex flows tractable. The following philosophies are found:

- Direct Numerical Simulation (DNS): No modeling at all. The flow must be generally treated as unsteady and three-dimensional. Not applicable to technical problems.
- Large eddy simulation (LES): Resolve the large scales as far as possible and model the unresolved scales (sub grid scale models). The flow must be generally treated as unsteady and three-dimensional. The grid resolution requirements can be relaxed a bit with respect to DNS, but fairly isotropic grid cells are needed. Therefore the applicability to wall-boundaed flows at high Reynolds numbers is rather limited.
- (Unsteady) Reynolds-averaged Navier-Stokes models ((U)RANS): Solve time averaged equations for the mean flow and model the average effect of turbulence on these equations. If applied to unsteady mean flows, the mean flow frequencies must be clearly separated from the frequencies associated with the turbulent fluctuations. This is the state-of-the-art method in technical applications, allowing for reasonable mesh resolutions even at high Reynolds numbers. Various models are available for the so-called Reynolds-stress tensor, representing the mean effect of turbulence in terms of mechanical stresses.
- Hybrid RANS/LES methods: Treat part of the flow field with RANS (preferably the near wall region, in order to allow for stretched cells) and part of the flow field with LES. At least in the LEA part the flow must be treated as unsteady and three-dimensional. Coupling techniques are currently under investigation.

### Turbulence models in TAU

#### RANS models

There exist three different approaches to model the Reynolds stress tensor:

- Eddy viscosity models (EVM):

  The turbulence is assumed to increase the fluid friction, i.e. the molecular viscosity is increased by an eddy viscosity, depending on the flow (Boussinesq hypothesis). The eddy viscosity is provided by algebraic relations or transport equations for some turbulent quantities. In aerodynamic applications one-equation models of Spalart-Allmaras type and two-equation models of k-ω type are state-of-the art.

  Eddy viscosity models are usually numerically forgiving, allowing to treat flows around complex geometries. Their major drawback is that the Boussinesq hypothesis is strictly not valid. Therefore EVMs often show deficiencies respect to the prediction of separation, free vortices or shock positions.

  EVMs in TAU:

  - Spalart-Allmaras (SA) type models: SA type models use one transport equation for an eddy viscosity like quantity. The model type has been developed particularly for aeronautical flows and is therefore well suited for flows around airfoils, wings and aircraft like configurations. SA models usually behave numerically very well in various situations.

    Deficiencies have been observed when applied to separating flows, indicating that this type of models is too dissipative. The destruction term acts only in the near-wall region, so that free vortices are dissipating too rapidly. Some improvement can be achieved, using so-called vortex corrections. The original version (without trip term) is called SAO model in TAU.

    Nevertheless, the SAE model version is the default one-equation model in TAU, where the so-called Edwards modification has been introduced primarily for reasons of numerical stability.

    The SALSA model is a more advanced development of TU Berlin. There is little experience in its application, but it seems, that the SALSA model improves the prediction of shock locations in transonic flows.

    - K-ω models:

      K-ω models use two transport equations, one for the kinetic turbulence energy k and one for the specific dissipation rate ω. They have been found superior to the probably better known k-ε models due to their superior behavior in the viscous sub-layer at walls.

      The Wilcox k-ω model is the default two-equation model in TAU, behaving similar to the SA models, i.e. it can be well applied to all type of aeronautical configurations, but is too dissipative to predict well pressure induced separation. The model is known to be sensitive to the ω-value at the outer edge of the boundary layer (“free stream dependency”), but this has shown to be of little influence in technical applications.

      The Wilcox k-ω model with SST limitation additionally limits the eddy viscosity according to Menter’s SST model (see below), in order to sensitize the model to separating flow. This model variant is not documented in the literature, and there is little experience on the effect of the SST limitation here.

      The TNT k-ω model by Kok has been developed in order to remove the above mentioned freestream dependency of the Wilcox k-ω model. There have been only a few tests which do not show a significant difference to the standard Wilcox k-ω model.

      The Menter BSL k-ω model should be considered as an intermediate state in the development of the SST model described below. It is a blend of the Wilcox k-ω model near the wall and the standard k-ε model further apart, cast into k-ω formulation. As with the Kok TNT model, the Menter BSL model resolves the freestream dependency of the Wilcox k-ω model. It is not expected to differ significantly with respect to the Wilcox and Kok models.

      The Menter SST model additionally sensitizes the BSL model to separation, so that it is a recommended standard model particularly for separated flows. However, it has been observed that the SST model overestimates separation in some cases. As a remedy the strain rate instead of the vorticity may be used for the so-called SST limitation. This has been supposed in a more recent publication by Menter et al., but there is little experience yet with this option.

      The LEA model is the linear truncation of an EARSM, differing from the Wilcox k-ω model only in the definition of the eddy viscosity. It has shown to be superior to the Wilcox model with respect to the prediction of shock positions and is therefore recommended for transonic flows.

- Explicit Algebraic Reynolds Stress Models (EARSM) and nonlinear eddy viscosity models:

  The Reynolds stress tensor is represented by a series expansion in terms of the strain and rotation tensor, where the linear term is formally identical with the Boussinesq hypothesis. They rely on a background model that uses the same transport equations like the corresponding EVM.

  Nonlinear models allow the Reynolds stress tensor not to be parallel to the viscous stresses. Nevertheless there is little evidence for a benefit from the nonlinear terms in problems of external aerodynamics. The major improvement seems to be achieved by the modified definition of the eddy viscosity in the linear term (cf. LEA model (EVM)).

- (Differential) Reynolds stress models ((D)RSM):

  An individual transport equation is solved for each component of the Reynolds stress tensor.

  This class constitutes the highest level of turbulence modeling within the RANS framework. They have shown promising results, mainly with the DLR FLOWer code, for a fairly large variety of applications. Advantages have been shown with respect to shock locations, separation and the maintenance of free vortices in the flow field.

  The presumably higher reliability of the results requires higher computational effort due to the larger number of equations to be solved. Furthermore DRSMs have been observed to be more sensitive to the grid quality than other models.

  DRSMs consist of a variety of sub-models for the individual terms of the Reynolds stress transport equation.

  - Re-distribution terms

    This is the most important model which usually gives the name to the whole DRSM. The re-distribution term of the Wilcox RSM (stress-ω model) is taken from the Launder-Reece-Rodi model (LRR), omitting the so-called wall-reflection terms. In this from it can be combined with any ω-equation, allowing to treat wall-bounded flows.

    The SSG/LLR-ω model combines the Wilcox RSM near walls with the Speziale-Sarkar-Gatski model (SSG) further apart as do the Menter models within the context of eddy viscosity models (see above). This model requires to use the Menter BSL ω-equation. The SSG/LRR-ω model is the recommended RSM in TAU.

  - Diffusion terms

    There are basically two different diffusion models, one called “simple gradient diffusion hypothesis” (SGDH). Using a scalar diffusion coefficient, and one called “generalized gradient diffusion hypothesis” (GGDH), using a tensorial diffusion coefficient. With SGDH shock positions seem to be agree slightly better with the experiment, whereas GGDH seems to predict better incipient separation. It is therefore the recommended sub-model.

  - Dissipation models

    There is currently only one dissipation model available, namely isotropic dissipation.

  - Length scale equations There are four different ω-equations available with the DRSMs in TAU:
    - Wilcox ω-equation (1988)
    - Kok TNT ω-equation
    - Menter BSL ω-equation
    - Hellsten ω-equation

    The ω-equations seem to reflect the features of the corresponding k-ω models, where the best predictions are obtained with the Menter BSL ω-equation. With the SSG/LRR-w model this is the only possible choice. The Hellsten ω-equation gives unsatisfying results and should not be used.

#### Hybrid models

##### DES/XLES

#### Recommendations for application

The following table gives some indication, which model may be useful for which type of flow problem.

Flow problem | EVM | Nonlinear model | DRSM | Hybrid model
--- | --- | --- | --- | ---
Simple attached flows | any | (any) | (X) |
Transonic flows (shock position) | LEA (Menter SST) | Wallin-Johansson | (X) |
(Moderate) pressure induced separation (steady) | Menter SST |  | (X) |
Massive separation (unsteady) |  |  |  | (X)

Note, that higher predictive accuracy is often accompanied with lower numerical stability. In particular separation may lead to unsteady flow, preventing the convergence to steady state. In such a situation, switching to a more dissipative (less sensitive) model, like SAE or Wilcox k-ω, may help the convergence, but the result may be less accurate due to a separation zone that is missed. Problems occurring only in the transient phase might be overcome by restarting with a more sensitive model, e.g. the Menter SST, from a steady solution obtained with a less sensitive model, e.g. the Wilcox k-ω.