**2.6 Scale adaptive models**

So far, we have presented RANS-based models that perform conservation calculations at each grid element. If turbulence is present, the impact of turbulence appears via the eddy viscosity. Traditionally users either *a priori* specify that the flow is laminar (so no eddy viscosity included) or the flow is turbulent (in which case an eddy viscosity is determined and applied throughout the flow field). The recent development of transitional modeling frees the researcher from having to *a priori* predict the level of turbulence. With transitional modeling, the numerical code automatically reverts to laminar flow in areas with low Reynolds numbers and also automatically becomes a turbulent model in areas where the Reynolds number is larger.

Regardless of the method that is selected, the coupled equations are solved for each computational element and the turbulent viscosity is applied to the fluid in the element under consideration.

In contrast to this approach, there is another major group of computational techniques that are termed "scale adaptive models". These are models that resolve part of the turbulent motions but model flow features that are smaller than the element size. Since there is less modeling and more actual resolution of fluid motion, one might expect the scale-adaptive models to be more accurate than RANS; and there are cases where that is so (particularly for free shear flows, swirling flows, boundary layer separation, and jets). However, the RANS approach can be more accurate than scale-adaptive methods in some situations, including wall bounded flows. Also, RANS is less computationally expensive because the eddy viscosity provides the link to the time-averaged flow field and the local turbulence with a very simple calculation. In fact, for even problems of modest complexity, scale adaptive models are more time consuming.

There are a number of established and new Scale-Adaptive Models that are used in CFD simulations. We will not be exhaustive in this section by covering all the existing models, rather we will focus on some of the models we think are most useful and representative. Interested readers are directed to an excellent comprehensive discussion provided by [34, 37].

#### *2.6.1 Scale-adaptive SST models*

One of the primary decisions that models are faced with is whether to perform calculations in steady or unsteady mode. Typically with numerical simulation, unsteadiness is driven by either timewise changes in boundary conditions or it is related to unsteady phenomena that occur in an otherwise steady scenario. A classic example is the Karmen Vortex Street that occurs in a wake region of a blunt object. **Figure 1**, shown below, illustrates this phenomenon.

Researchers have often conjectured that if a RANS model is performed with sufficiently small elements and time steps, the unsteady features of the flow would naturally be resolved. But in fact, this is not true. It is important to note that steady state calculations using RANS models will often provide very accurate information

## *Turbulence Models Commonly Used in CFD DOI: http://dx.doi.org/10.5772/intechopen.99784*

#### **Figure 1.**

*Unsteady wake region, even though oncoming flow is steady state.*

about averaged quantities (like drag), these simulations will miss details in the rapidly fluctuating downstream wake region. This issue was explored in depth in [35] where time-averaged results of drag obtained from unsteady RANS simulations were compared with calculations from steady RANS calculations (using the SST transitional model that was previously described). It was found that the steady state calculations were able to accurately capture drag forces but were only partially adept at capturing vortex movement in the downstream wake region.

With this discussion as background, it is now time to turn attention to the governing equations of scale-adaptive RANS models. The model to be discussed here uses the SST approach for the underlying governing equations (in the literature it is often termed the SAS-SST model). The scale-adaptive approach modifies the ω transport equation based on [37]. In particular, a new transport equation is presented that incorporates the turbulent length scale L and is set forth here:

$$
\Phi = \sqrt{k}L \tag{37}
$$

and

$$\frac{\partial \Phi}{\partial t} + \rho \frac{\partial U\_i \Phi}{\partial \mathbf{x}\_i} = \frac{\Phi \mathbf{P}\_k}{k} \left( C\_1 - C\_2 \left( \frac{L\_t}{L\_k} \right)^2 \right) - \rho C\_3 k + \frac{\partial}{\partial \mathbf{x}\_i} \left( \frac{\mu\_t}{\sigma\_\Phi} \frac{\partial \Phi}{\partial \mathbf{x}\_i} \right) \tag{38}$$

Values of the various constants can be found in [34, 37] and are not repeated here for brevity. The term Lt is a novel modification; it refers to the von Karmen length scale. **Figures 2** and **3** are provided that show a comparison of downstream wake regions for an unsteady RANS calculation using the SST model (**Figure 2**) and a simulation using the scale-adaptive SST modification. Results are obtained from [34]. It can be seen that the standard SST model does capture a periodic release of eddies from the downstream side of a circular cylinder (shown in blue). In both images, the flow is left-to-right. The color legend is keyed to the local values of the turbulent length scale. Clearly the scale-adaptive approach provides a much wider range of turbulent eddy sizes.

#### **Figure 2.**

*Calculations of turbulent length scale for flow over a circular cylinder, based on an unsteady SST model.*

#### **Figure 3.**

*Calculations of turbulent length scale for flow over a circular cylinder, based on a scale-adaptive unsteady SST model.*

#### *2.6.2 LES WALE model*

Another common approach to dealing with these types of problems is based on the so-called "large eddy simulation". To the best knowledge of the authors, the first articulation of a LES model was [38] and the models have been updated in the intervening decades. Here we focus on one popular and current LES method (the Wall-Adaptive Local Eddy, or WALE LES model). The general processes of LES modeling are the same, regardless of which variant is used. LES models involve the filtering of eddies that are smaller than the size of the computational elements. The algorithm incorporates an eddy viscosity for flow scales that are not resolved.

For this model, the tensor-form of the Navier Stokes equations is:

$$\frac{\partial}{\partial t}(\overline{\rho}\overline{u}\_i) + \left(\frac{\partial \overline{\rho}\overline{u}\_i \overline{u}\_j}{\partial \mathbf{x}\_i}\right) = -\frac{\partial \overline{p}}{\partial \mathbf{x}\_j} + \frac{\partial}{\partial \mathbf{x}\_i} \left(\mu \left(\frac{\partial \overline{u}\_i}{\partial \mathbf{x}\_j} + \frac{\partial \overline{u}\_j}{\partial \mathbf{x}\_i}\right)\right) + \frac{\partial \mathbf{r}\_{\overline{\eta}}}{\partial \mathbf{x}\_i} \tag{39}$$

where *τij* is the small-scale stress defined as

$$
\sigma\_{i\bar{j}} = -\overline{\rho u\_i u\_j} + \overline{\rho u}\_i \overline{u}\_j = 2\mu\_{\text{sg}} \overline{\mathbf{S}}\_{i\bar{j}} + \frac{1}{3} \delta\_{i\bar{j}} \tau\_{kk} \tag{40}
$$

And the *Sij* term indicates the strain rate tensor for large scale motions. The small-scale eddy viscosity *μsgs* is found from

$$\mu\_{\rm gv} = \rho \left(\mathbf{C}\_{\rm w}\Delta\right)^{2} \frac{\left(\mathbf{S}\_{\rm \vec{\boldsymbol{\eta}}}^{d}\mathbf{S}\_{\rm \vec{\boldsymbol{\eta}}}^{d}\right)^{3/2}}{\left(\overline{\mathbf{S}}\_{\rm \vec{\boldsymbol{\eta}}}\overline{\mathbf{S}}\_{\rm \vec{\boldsymbol{\eta}}}\right)^{5/2} + \left(\mathbf{S}\_{\rm \vec{\boldsymbol{\eta}}}^{d}\mathbf{S}\_{\rm \vec{\boldsymbol{\eta}}}^{d}\right)^{5/4}} \tag{41}$$

The term Cw is a constant and the symbol Δ = (element volume)1/3. The tensor Sij <sup>d</sup> is calculated from the strain-rate and vorticity tensors, as shown here

$$\mathbf{S}\_{ij}^d = \overline{\mathbf{S}}\_{ik}\overline{\mathbf{S}}\_{kj} + \overline{\mathbf{Q}}\_{ik}\overline{\mathbf{Q}}\_{kj} - \frac{1}{3}\delta\_{ij} \left( \overline{\mathbf{S}}\_{mn}\overline{\mathbf{S}}\_{mn} - \overline{\mathbf{Q}}\_{mn}\overline{\mathbf{Q}}\_{mn} \right) \tag{42}$$

And the vorticity tensor Ω*ij* is defined as

$$\overline{\mathfrak{Q}}\_{ij} = \frac{1}{2} \left( \frac{\partial \overline{u}\_i}{\partial \mathbf{x}\_j} + \frac{\partial \overline{u}\_j}{\partial \mathbf{x}\_i} \right) \tag{43}$$
