**2.2 k-ω models**

While the k-ε model has experienced success in computational modeling, it has deficiencies in some situations. In particular, the k-ε model performs suitably away from walls, in the main flow. However, it has issues in the boundary layer zone, particularly with low Reynolds numbers. Here, Reynolds numbers refer to local Reynolds numbers that decrease as one moves closer to the wall and the no-slip condition exerts its influence (rather than to the Reynolds number based on macroscopic dimensions such a pipe diameter or plate length).

A significant development in CFD was brought forward by the development of k-ω model that replaced the transport equation for ε with a specific rate of turbulence dissipation, ω [10]. The new equations are:

$$
\rho \frac{\partial (u\_i k)}{\partial \mathbf{x}\_i} = \rho \mathbf{P}\_k + \rho \mathbf{P}\_b - \rho \rho \mathbf{\beta} \alpha k + \frac{\partial}{\partial \mathbf{x}\_i} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_k} \right) \frac{\partial k}{\partial \mathbf{x}\_i} \right] \tag{13}
$$

$$
\rho \frac{\partial (u\_i \alpha)}{\partial \mathbf{x}\_i} = \frac{\partial}{\partial \mathbf{x}\_i} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_{ao}} \right) \frac{\partial \alpha}{\partial \mathbf{x}\_i} \right] + \frac{a \alpha}{k} P\_k - \beta \rho \alpha^2 \tag{14}
$$

With a turbulent viscosity calculated as:

$$
\mu\_t = \rho \frac{\kappa}{\alpha} \tag{15}
$$

#### **2.3 Shear stress transport family of models**

Recognizing that the k-ε and k-ω model each have strengths and weaknesses, a new model was proposed that uses both of these approaches in a way that harnesses their strengths [11]. This new approach, termed the Shear Stress Transport model (SST), smoothly transitions from the k-ω model near the wall to the k-ε model in the main flow. With the SST model, the governing equation for turbulent dissipation is recast into an ω form. The governing equations are:

$$\frac{\partial(\rho u\_i \mathbb{k})}{\partial \mathbf{x}\_i} = P\_k - \beta\_1 \rho k \alpha \rho + \frac{\partial}{\partial \mathbf{x}\_i} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_k} \right) \frac{\partial k}{\partial \mathbf{x}\_i} \right] \tag{16}$$

$$\frac{\partial(\rho u\_i \alpha)}{\partial \mathbf{x}\_i} = a\_3 \frac{\alpha}{\kappa} P\_\kappa - \beta\_2 \rho \alpha^2 + \frac{\partial}{\partial \mathbf{x}\_i} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_w} \right) \frac{\partial \alpha}{\partial \mathbf{x}\_i} \right] + 2(1 - F\_1) \rho \frac{1}{\sigma\_{w2} \alpha} \frac{\partial k}{\partial \mathbf{x}\_i} \frac{\partial \alpha}{\partial \mathbf{x}\_i} \tag{17}$$

and the turbulent viscosity is found from

$$\mu\_{\rm t} = \frac{a\rho k}{m\omega \, (a\rho, SF\_2)}\tag{18}$$

As before, Pk is the production of turbulent kinetic energy and ω reflects the specific rate of turbulent destruction. As noted earlier, the σ terms are turbulent Prandtl numbers associated with their subscript. The function F1 is the aforementioned blending function that transfers the k-ω model near the wall to the k-ε model away from the wall from the wall. The S term is the magnitude of the shear strain rate.

While ostensibly, the SST model is used for fully turbulent flows, it has shown ability to capture both laminar and turbulent flow regimes [12]. However, in the next section we discuss a set of modifications to the SST models that are specifically designed to handled laminar/transitional/turbulent flow regimes that are recommended.

## **2.4 SST transitional models**

The already discussed turbulent models were largely developed based on correlations of canonical fully turbulent flow situations (such as flows over flat plates, airfoils, Falkner-Skans flows, and flows in tubes and ducts). Of course, researchers and engineers often experience situations where the flow is partially turbulent or other situations where the flow changes so that for part of the time it is laminar and other times turbulent. Consider for example pulsatile flow wherein the fluid velocity changes sufficiently so that for parts of the flow period, different flow regimes occur. There are a number of approaches to handle these situations but with respect to the RANS models, the approaches generally utilize the concept of turbulent intermittency. Intermittency was originally defined as the percentage of time that a flow was turbulent. However, more recently, turbulent intermittency has been used as a multiplier on the rate of turbulent kinetic production [13–15].

Here we will set forth two current transitional models, both based on the SST turbulence approach. The first method involves two extra transport equations. One for the intermittency, γ, which is a multiplier to the turbulent production. The transport equation for turbulent intermittency is:

$$\frac{\partial(\rho \chi)}{\partial t} + \frac{\partial(\rho u\_i \chi)}{\partial \mathbf{x}\_i} = P\_{\gamma,1} - E\_{\gamma,1} + P\_{\gamma,2} - E\_{\gamma,2} + \frac{\partial}{\partial \mathbf{x}\_i} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_\gamma} \right) \frac{\partial \chi}{\partial \mathbf{x}\_i} \right] \tag{19}$$

The P and E terms are, respectively, production and dissipation of intermittency. An additional transport equation is required for the transitional momentum thickness Reynolds number. This added equation is:

$$\frac{\partial(\rho \text{Re}\_{\partial t})}{\partial t} + \frac{\partial(\rho u\_i \text{Re}\_{\partial t})}{\partial \mathbf{x}\_i} = P\_{\partial t} + \frac{\partial}{\partial \mathbf{x}\_i} \left[ \sigma\_{\partial t} (\mu + \mu\_t) \frac{\partial \text{Re}\_{\partial t}}{\partial \mathbf{x}\_i} \right] \tag{20}$$

Together, solution to Eqs. (19) and (20) determine the local state of turbulence. They result in an intermittency that takes values between 0 and 1. For fully laminar flow, γ = 0 and the model reverts to a laminar solver. When γ = 1, the flow is fully turbulent. The turbulent production then is then multiplied by the local value of the intermittency, γ. Interested readers are invited to review the development of this model, including implementation for problems that involve heat transfer [16–22].

Recently, the above two-equation model was modified to reduce the two transitional transport equations to a single Equation [23] and that approach was later adapted by [24] to accurately solve for situations in confined pipe/duct/tube flows. Essentially, Eqs. (19) and (20) are replaced by a single intermittency equation which is:

$$\frac{\partial(\rho u\_i \gamma)}{\partial \mathbf{x}\_i} = P\_\gamma - E\_\gamma + \frac{\partial}{\partial \mathbf{x}\_i} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_\gamma} \right) \frac{\partial \gamma}{\partial \mathbf{x}\_i} \right] \tag{21}$$

#### *Applications of Computational Fluid Dynamics Simulation and Modeling*

As with the two-equation approach, the intermittency factor γ will take on values between 0 and 1. Also, as before, The P and E terms represent, respectively, the production and destruction in local value of intermittency.

For these intermittency models, the onset of turbulence is calculated by a series of correlation functions. In particular, a local value of the critical Reynolds number is determined from

$$\operatorname{Re}\_{\vartheta\varepsilon} = \mathbf{C}\_{TU1} + \mathbf{C}\_{TU2} \exp\left[-\mathbf{C}\_{TU3} T \boldsymbol{\mu}\_L \mathbf{F}\_{\rm PG}(\lambda\_{\vartheta L})\right] \tag{22}$$

Eq. (22) is used to identify the location of laminar-turbulent transition. It is based on the local value of the momentum layer thickness. The C terms are correlation constants and are based on comparison of numerically simulated results with experimentation. An important term in Eq. (22) is the local value of the mid-boundarylayer turbulence intensity (TuL). This value is attained at the midpoint of the boundary layer as an output from an empirical formulation based on experimentation.

Local production of intermittency is calculated from:

$$P\_{\gamma} = F\_{\text{length}} \cdot \rho \cdot \mathbb{S} \cdot \gamma \cdot F\_{\text{onset}} \cdot [\mathbf{1} - \gamma] \tag{23}$$

As we have already noted, the term S is the shear strain rate. A new term that appears in Eq. (23) is the so-called onset transition term (Fonset) which is calculated using the following set of equations.

$$F\_{\text{onet}} = \text{MAX}\left[\min\left(\frac{\text{Re}\_V}{2.2 \cdot \text{Re}\_{\partial t}}, 2\right) - \max\left(1 - \left(\frac{R\_T}{3.5}\right)^3, 0\right), 0\right] \tag{24}$$

$$Re\_V = \rho d\_w^2 \mathbb{S}/\mu \tag{25}$$

$$\text{Re}\,\_{T} = \rho \kappa / \mu \text{w} \tag{26}$$

Similarly, the local rate of destruction of intermittency is found by:

$$E\_{\gamma} = 0.06 \cdot \rho \cdot \Omega \cdot \gamma \cdot F\_{turb} \cdot [50\gamma - 1] \tag{27}$$

$$F\_{turb} = e^{-(\mathcal{R}\_T/2)^4} \tag{28}$$

$$P\_{\kappa} = \mu\_t \cdot \mathbb{S} \cdot \mathfrak{Q} \tag{29}$$

We have already noted that these transitional turbulence models were initially developed for external boundary layer flows (flat plate boundary layers, airfoil flows, Falkner-Skans flows, etc.). Insofar as we have adopted them for internal flow, some modification was required. We recommend, at least for flows through pipes, tubes, and ducts, that the initial constants determined in [23] be replaced by alternative values from [24].

While we recommend the above approach for solving transitional flow problems, this area of research is also heavily studied by other researchers who have provided alternative approaches to handle such flows. We cite them here for readers who are interested in those alternative but complementary viewpoints [25–33].

#### **2.5 Reynolds-stress models**

Reynolds stress models (RSM) are quite different from the RANS approach that was just discussed. For RSMs, transport equations are used for all components of the Reynolds stress tensor and an eddy viscosity is not utilized. These models are expected to be superior for situations with non-isotropic turbulence and flows with *Turbulence Models Commonly Used in CFD DOI: http://dx.doi.org/10.5772/intechopen.99784*

significant components of transport in three directions. There are a number of RSM versions, some of which will be discussed here. The so-called SSG-RSM model employed here utilizes the following momentum transport equation:

$$\rho \frac{\partial u\_j}{\partial t} + \rho \frac{\partial}{\partial \mathbf{x}\_i} \left( u\_i u\_j \right) = -\frac{\partial p'}{\partial \mathbf{x}\_j} + \mu \frac{\partial}{\partial \mathbf{x}\_i} \left( \frac{\partial u\_j}{\partial \mathbf{x}\_i} + \frac{\partial u\_i}{\partial \mathbf{x}\_i} \right) - \rho \frac{\partial}{\partial \mathbf{x}\_i} \left( \overline{u\_i u\_j} \right) + B\_j \tag{30}$$

The second-to-last term on the right-hand side represents the Reynolds stresses. There is a pseudo-pressure term p' that is calculated from the local static pressure p and local velocity gradient from the following expression.

$$p' = p + \frac{2}{3}\mu \frac{\partial u\_k}{\partial \mathbf{x}\_k} \tag{31}$$

The Reynolds stresses are calculated by a collection of six equations for all directional possibilities. The transport equations for Reynolds stresses are:

$$
\rho \frac{\partial \overline{u\_j u\_i}}{\partial t} + \rho \frac{\partial}{\partial \mathbf{x}\_k} \left( u\_k \overline{u\_i u\_j} \right) - \frac{\partial}{\partial \mathbf{x}\_k} \left[ \left( \mu + \frac{2}{3} \rho \mathbf{C}\_\varepsilon \frac{k^2}{\varepsilon} \right) \frac{\partial \overline{u\_i u\_j}}{\partial \mathbf{x}\_k} \right] = P\_{\vec{\eta}} - \frac{2}{3} \delta\_{\vec{\eta}\vec{\rho}} \rho \varepsilon + \Phi\_{\vec{\eta}} + P\_{\vec{\eta}, k} \tag{32}
$$

We note that a turbulence dissipation term, ε, appears in Eq. (32) and it has to be solved from its own transport equation. We refer readers to [34, 35] for more details.

A modification to the above is realized from the Baseline RSM (BSL RSM) model. It differs from the SSG RSM in that the transport equation for ε is replaced by a transport equation for ω. The new equation is:

$$\begin{split} \rho \frac{\partial \boldsymbol{\alpha}}{\partial t} + \rho \frac{\partial (\boldsymbol{u}\_{k} \boldsymbol{\alpha})}{\partial \mathbf{x}\_{k}} &= \frac{\partial}{\partial \mathbf{x}\_{k}} \left[ \left( \mu + \frac{\mu\_{t}}{\sigma\_{\alpha 3}} \right) \frac{\partial \boldsymbol{\alpha}}{\partial \mathbf{x}\_{k}} \right] + \boldsymbol{\alpha}\_{3} \frac{\partial \boldsymbol{\alpha}}{\boldsymbol{k}} \boldsymbol{P}\_{k} - \beta\_{3} \rho \boldsymbol{\alpha}^{2} + (\boldsymbol{1} - \boldsymbol{F}\_{1}) \frac{\boldsymbol{\mathcal{Q}} \boldsymbol{\alpha}}{\sigma\_{\alpha 2}} \frac{1}{\boldsymbol{\alpha}} \frac{\partial \boldsymbol{k}}{\partial \mathbf{x}\_{k}} \frac{\partial \boldsymbol{\alpha}}{\partial \mathbf{x}\_{k}} \\ &\tag{33} \end{split} \tag{34}$$

This approach blends between two different models that are used near the wall and alternatively away from the wall. The modeling is accomplished using a weighting function, similar to the SST:

$$
\phi\_3 = F\_1 \cdot \phi\_1 + (1 - F\_1) \cdot \phi\_2 \tag{34}
$$

Where the symbols ϕ correspond to any particular transport variable in the near wall and far wall regions. Various constants change their values in the two regions, so that:

The constants near the wall:

$$
\sigma\_k = \sigma\_\alpha = 2, \beta = 0.075, \alpha = 0.553 \tag{35}
$$

The constants away from the wall:

$$
\sigma\_k = \mathbf{1}, \sigma\_\alpha = \mathbf{1}.\mathbf{1}68, \beta = \mathbf{0}.0828, \alpha = \mathbf{0}.44\tag{36}
$$

The last RSM version to be discussed is the Explicit Algebraic RSM (EARSM). This approach includes a non-linear relationship between the local values of the Reynolds stresses and the vorticity tensors. It is focused on flows with secondary

motions and curvature [36]. The local values of the Reynolds stresses are calculated using an anisotropy tensor which is based on algebraic equations [36]. This is contrasted with RSM approaches that solve for the Reynolds stress components using differential transport equations. The approach is to use higher order terms for many of the flow phenomena. It was designed to handle secondary flow situations and flows with extensive curvature and rotation. The governing equations are complex and lengthy and for brevity sake, we refer interested readers to [36].
