*2.3.2.2 Walters and Cokljat k-kL-ω model*

Walters and Cokljat's three-equation k-kL-ω model [14] is proposed by the introduction of a transport equation for the laminar kinetic energy (kL) into the conventional k-ω turbulence model and is used for natural and bypass transitional flows. This model is based on the understanding that the freestream turbulence is the cause of the high amplitude streamwise fluctuations in the pretransitional boundary layer, and these fluctuations are quite distinctive from the classic turbulence fluctuations. Also, growth of the laminar kinetic energy correlates with low frequency wall-normal fluctuations of the freestream turbulence. In this model, the total kinetic energy is assumed to be the sum of the large-scale energy which contributes to laminar kinetic energy and the small-scale energy which contributes to turbulence production. Thus, the transport equation for laminar kinetic energy (kL) is solved in conjunction with the turbulent kinetic energy (kT). Since the k-kL-ω model uses a fully local formulation, it is suitable for the modern CFD codes and appears to be the first local model to specifically address pretransitional growth mechanism that is responsible for bypass transition [14].

### *2.3.2.3 Menter one-equation γ model*

Menter's one-equation γ transition model [22] is a simplified version of the two-equation γ-Reθ transition model [10, 11]. In the new model, the Reθ equation is avoided, and the experimental correlations for transition onset is embedded into the γ equation in a simplified fashion. In effect, the simplified one-equation γ model still possesses the same level of predictive capabilities as the original model. Menter et al. [22] summarize the advantages and the key changes to the model as follows: the new model is still fully local with new correlations valid for nearly all types of transition mechanisms, solves for one less equation, which is computationally cheaper; it is Galilean invariant; it has less coefficients that makes the model easier to fine-tune for specific application areas; and the new model would be coupled to any turbulence model that has viscous sublayer formulation. Menter et al. tested their model against most of the test cases which they previously used for the twoequation model. The results show that the new one-equation model is quite successful, and it would be a viable replacement for the original model.

#### *2.3.2.4 Nagapetyan and Agarwal two-equation WA-γ transition model*

Following the trend for reducing the number of transition equations, a novel method was developed by integrating the recent Wray-Agarwal (WA) walldistance-free one-equation turbulence model [23] based on the k-ω closure, with the one-equation intermittency transport γ-equation of Menter et al. [22] to construct the so-called two-equation Nagapetyan-Agarwal transition model WA-γ [24]. An

**107**

 \_\_\_\_ ∂ ν*<sup>T</sup>* <sup>∂</sup>*<sup>t</sup>* <sup>+</sup> \_\_\_<sup>∂</sup> ∂*xj*

+ *Cb*<sup>2</sup> \_\_\_\_

(ν*<sup>T</sup> uj*) = γ*BC Cb*1*S*ν*<sup>T</sup>* − *Cw*<sup>1</sup> *fw* (

∂ ν*<sup>T</sup>* ∂*xj* \_\_\_\_ ∂ ν*<sup>T</sup>* <sup>∂</sup>*xj*} \_\_ ν*T d* ) 2 + \_\_1 σ{ \_\_\_∂ ∂*xj*[

The γBC function works in such a way that the turbulence production is damped (γBC = 0) until some transition onset criteria is fulfilled. After a point at which the onset criteria is ensured, the damping effect of the intermittency function γBC is checked,

(ν*<sup>L</sup>* + ν*T*) \_\_\_ ∂ν

∂*xj*]

(5)

*Transition Modeling for Low to High Speed Boundary Layer Flows with CFD Applications*

important difference between the one-equation turbulence model derived earlier from k-ω models and the baseline turbulence model is the addition of a new cross diffusion term and a blending function between two destruction terms [23]. It was reported that the presence of destruction terms enables the Wray-Agarwal (WA) model to switch between a one-equation k-ω or one equation k-ε model. The new two-equation model was quite successfully validated for computing a number of two-dimensional benchmark experiments such as the transitional flows past flat plates in zero and slowly varying pressure gradients, flows past airfoils such as the

Bas and Cakmakcioglu (B-C) model [20] is an algebraic or zero-equation model that solves for an intermittency function rather than an intermittency transport (differential) equation. The main approach behind the B-C model follows the pragmatic idea of further reducing the total number of equations. Rather than deriving extra equations for intermittency convection and diffusion, already present convection and diffusion terms of the underlying turbulence model could be used. From a philosophical point of view, the transition, as such, is just a phase of a general turbulent flow. Addition of, in a sense, artificially manufactured transition equations appear to be rather redundant. Yet, for most of industrial flow types, the experimentally evidenced close relation between the scaled vorticity Reynolds number and the momentum thickness Reynolds number stood out as the primary reason for the success of so many intermittency transport equation models following the Langtry and Menter's original two-equation γ-Reθ model [11]. In the application, the production term of the underlying turbulence model is damped until a considerable amount of turbulent viscosity is generated, and the damping effect of the transition model would be disabled after this point. The Spalart-Allmaras (S-A) turbulence model [28] is used as the baseline turbulence model, and rather than using an intermittency equation, just an intermittency function is proposed to control its production term. To this end, the B-C model is also a local correlation transition model that can be easily implemented for both 2-D and 3-D flows with reduced number of equations. For instance, for a 3-D problem, the B-C model solves for six equations (1 continuity + 3 momentum + 1 energy + 1 turbulence), whereas the two-equation γ-Reθ model solves for nine equations (1 continuity + 3 momentum + 1 energy + 2 turbulence + 2 transition). In addition, in the B-C model formulation, the freestream turbulence intensity parameter is only present in the critical momentum thickness Reynolds number function that makes the calibration of the model quite easy for different problems. The details of the B-C model formulation are presented in the following. The S-A one-equation turbulence model is used as the underlying turbulence model for the B-C model. The S-A model solves for a transport equation for a new working variable νT, which is related to the eddy viscosity. The B-C model's transition effects are included into the turbulence model is provided by multiplying the intermittency distribution function (γBC) with the production term of the S-A equation given as:

*DOI: http://dx.doi.org/10.5772/intechopen.83520*

S809, Aerospatiale-A, and NLR-7301 two-element airfoils.

*2.3.2.5 Bas and Cakmakcioglu algebraic transition model*

*Transition Modeling for Low to High Speed Boundary Layer Flows with CFD Applications DOI: http://dx.doi.org/10.5772/intechopen.83520*

important difference between the one-equation turbulence model derived earlier from k-ω models and the baseline turbulence model is the addition of a new cross diffusion term and a blending function between two destruction terms [23]. It was reported that the presence of destruction terms enables the Wray-Agarwal (WA) model to switch between a one-equation k-ω or one equation k-ε model. The new two-equation model was quite successfully validated for computing a number of two-dimensional benchmark experiments such as the transitional flows past flat plates in zero and slowly varying pressure gradients, flows past airfoils such as the S809, Aerospatiale-A, and NLR-7301 two-element airfoils.

#### *2.3.2.5 Bas and Cakmakcioglu algebraic transition model*

Bas and Cakmakcioglu (B-C) model [20] is an algebraic or zero-equation model that solves for an intermittency function rather than an intermittency transport (differential) equation. The main approach behind the B-C model follows the pragmatic idea of further reducing the total number of equations. Rather than deriving extra equations for intermittency convection and diffusion, already present convection and diffusion terms of the underlying turbulence model could be used. From a philosophical point of view, the transition, as such, is just a phase of a general turbulent flow. Addition of, in a sense, artificially manufactured transition equations appear to be rather redundant. Yet, for most of industrial flow types, the experimentally evidenced close relation between the scaled vorticity Reynolds number and the momentum thickness Reynolds number stood out as the primary reason for the success of so many intermittency transport equation models following the Langtry and Menter's original two-equation γ-Reθ model [11].

In the application, the production term of the underlying turbulence model is damped until a considerable amount of turbulent viscosity is generated, and the damping effect of the transition model would be disabled after this point. The Spalart-Allmaras (S-A) turbulence model [28] is used as the baseline turbulence model, and rather than using an intermittency equation, just an intermittency function is proposed to control its production term. To this end, the B-C model is also a local correlation transition model that can be easily implemented for both 2-D and 3-D flows with reduced number of equations. For instance, for a 3-D problem, the B-C model solves for six equations (1 continuity + 3 momentum + 1 energy + 1 turbulence), whereas the two-equation γ-Reθ model solves for nine equations (1 continuity + 3 momentum + 1 energy + 2 turbulence + 2 transition). In addition, in the B-C model formulation, the freestream turbulence intensity parameter is only present in the critical momentum thickness Reynolds number function that makes the calibration of the model quite easy for different problems. The details of the B-C model formulation are presented in the following.

The S-A one-equation turbulence model is used as the underlying turbulence model for the B-C model. The S-A model solves for a transport equation for a new working variable νT, which is related to the eddy viscosity. The B-C model's transition effects are included into the turbulence model is provided by multiplying the intermittency distribution function (γBC) with the production term of the S-A equation given as:

$$\begin{aligned} \frac{\partial \boldsymbol{\nu}\_{T}}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_{j}} \{\boldsymbol{\nu}\_{T} \boldsymbol{\mu}\_{j}\} &= \boldsymbol{\chi}\_{BC} \mathbf{C}\_{b1} \mathbf{S} \boldsymbol{\nu}\_{T} - \mathbf{C}\_{w1} \mathbf{f}\_{w} \left(\frac{\boldsymbol{\nu}\_{T}}{d}\right)^{2} + \frac{1}{\overline{\sigma}} \left\{ \frac{\partial}{\partial \mathbf{x}\_{j}} \left[ (\boldsymbol{\nu}\_{L} + \boldsymbol{\nu}\_{T}) \frac{\partial \boldsymbol{\nu}}{\partial \mathbf{x}\_{j}} \right] \right. \\ &\quad \star \mathbf{C}\_{b2} \frac{\partial \boldsymbol{\nu}\_{T}}{\partial \mathbf{x}\_{j}} \frac{\partial \boldsymbol{\nu}\_{T}}{\partial \mathbf{x}\_{j}} \Bigg\} \end{aligned} \tag{5}$$

The γBC function works in such a way that the turbulence production is damped (γBC = 0) until some transition onset criteria is fulfilled. After a point at which the onset criteria is ensured, the damping effect of the intermittency function γBC is checked,

*Boundary Layer Flows - Theory, Applications and Numerical Methods*

Reynolds number is reported [1] to be around 10%.

details of the model are available in the literature [1, 11].

*2.3.2.2 Walters and Cokljat k-kL-ω model*

*2.3.2.3 Menter one-equation γ model*

are proportional in a Blasius boundary layer. For most of the flow types, the relative error between the scaled vorticity Reynolds number and momentum thickness

The model solves for two additional equations besides the underlying two-equation k-ω SST turbulence model, an intermittency equation (γ) that is used to trigger the turbulence production term of the k-ω SST turbulence model and a momentum thickness Reynolds number transport equation (Reθ) that includes experimental correlations that relates important flow parameters such as turbulence intensity, freestream velocity, pressure gradients etc. and supplies it to the intermittency equation. The

Walters and Cokljat's three-equation k-kL-ω model [14] is proposed by the introduction of a transport equation for the laminar kinetic energy (kL) into the conventional k-ω turbulence model and is used for natural and bypass transitional flows. This model is based on the understanding that the freestream turbulence is the cause of the high amplitude streamwise fluctuations in the pretransitional boundary layer, and these fluctuations are quite distinctive from the classic turbulence fluctuations. Also, growth of the laminar kinetic energy correlates with low frequency wall-normal fluctuations of the freestream turbulence. In this model, the total kinetic energy is assumed to be the sum of the large-scale energy which contributes to laminar kinetic energy and the small-scale energy which contributes to turbulence production. Thus, the transport equation for laminar kinetic energy (kL) is solved in conjunction with the turbulent kinetic energy (kT). Since the k-kL-ω model uses a fully local formulation, it is suitable for the modern CFD codes and appears to be the first local model to specifically address

pretransitional growth mechanism that is responsible for bypass transition [14].

Menter's one-equation γ transition model [22] is a simplified version of the two-equation γ-Reθ transition model [10, 11]. In the new model, the Reθ equation is avoided, and the experimental correlations for transition onset is embedded into the γ equation in a simplified fashion. In effect, the simplified one-equation γ model still possesses the same level of predictive capabilities as the original model. Menter et al. [22] summarize the advantages and the key changes to the model as follows: the new model is still fully local with new correlations valid for nearly all types of transition mechanisms, solves for one less equation, which is computationally cheaper; it is Galilean invariant; it has less coefficients that makes the model easier to fine-tune for specific application areas; and the new model would be coupled to any turbulence model that has viscous sublayer formulation. Menter et al. tested their model against most of the test cases which they previously used for the twoequation model. The results show that the new one-equation model is quite success-

ful, and it would be a viable replacement for the original model.

*2.3.2.4 Nagapetyan and Agarwal two-equation WA-γ transition model*

Following the trend for reducing the number of transition equations, a novel

distance-free one-equation turbulence model [23] based on the k-ω closure, with the one-equation intermittency transport γ-equation of Menter et al. [22] to construct the so-called two-equation Nagapetyan-Agarwal transition model WA-γ [24]. An

method was developed by integrating the recent Wray-Agarwal (WA) wall-

**106**

and the remaining part of the flow is taken to be fully turbulent (γBC = 1). For this purpose, an exponential function of the form (1-e<sup>−</sup><sup>x</sup> ) is proposed for the γBC as follows:

$$\gamma\_{BC} = \mathbf{1} - \exp\left(-\sqrt{Term\_1} - \sqrt{Term\_2}\right) \tag{6}$$

where Term1 and Term2 are defined as:

where  $\text{Term}\_1$  and  $\text{Term}\_2$  are defined as: 
$$Term\_1 = \frac{\max\left(Re\_\mathsf{B} - Re\_{\mathsf{A}\mathsf{c}}, 0.0\right)}{\chi\_1 Re\_{\mathsf{A}\mathsf{c}}}, \\ Term\_2 = \frac{\max\left(\mathsf{V}\_{\mathsf{B}\mathsf{C}} - \chi\_2, 0.0\right)}{\chi\_2} \tag{7}$$

and,

$$Re\_0 = \frac{Re\_v}{2.193} \quad \text{and} \quad Re\_v = \frac{\rho d\_w}{\mu} \,\Omega, \quad \nu\_{BC} = \frac{\nu\_t}{U d\_w} \tag{8}$$

In the above, ρ is the density, μ is the molecular viscosity, dw is the distance from the nearest wall, νBC is a proposed turbulent viscosity-like nondimensional term where νt is the turbulent viscosity, U is the local velocity magnitude, dw is the distance from the nearest wall, and χ1 and χ2 are calibration constants. Reθ<sup>c</sup> is defined as the critical momentum thickness Reynolds number, which is a correlation that is based on a range of transition experiments. In effect, Term1 checks for the transition onset point by comparing the locally calculated Reθ with the experimentally obtained critical momentum thickness Reynolds number Reθc. As soon as the vorticity Reynolds number Rev exceeds a critical value, Term1 becomes greater than zero and the intermittency function γBC begins to increase. However, the vorticity Reynolds number Rev relation above is a function of the square of the wall distance dw; therefore, it takes a very low value inside the boundary layer where the wall distance is quite low. Because of this, Term1 alone is not enough for intermittency generation inside the boundary layer. To remedy this, Term2 is introduced. Inspecting the Term2 equation with the νBC relation shows that the regions close to wall is inversely related and the damping effect of the transition model would be disabled inside the boundary layer. In effect, Term2 checks for the viscosity levels inside the boundary layer, and the turbulence production is activated wherever νBC exceeds a critical value χ2. In order to determine the calibration constants' χ1 and χ<sup>2</sup> values, the well-known zero pressure gradient flat plate test case of Schubauer and Klebanoff [29] is used. This test case represents a natural transition process due to the wind tunnel used in the experiment generates a freestream Tu around 0.2%. The model calibration is done by numerical experimentation; setting χ1 and χ2 such that the transition occurs at the same location as in the experiment. As a result, the χ1 and χ2 values are set to be 0.002 and 5.0, respectively.

Any experimental Reθc correlation could be used in the model. However, it should be noted that, since the S-A turbulence model does not solve for the local turbulent kinetic energy, local turbulence intensity values cannot be calculated. Due to this reason, the turbulence intensity Tu is assumed, for now, to be constant in the entire flow domain as Suluksna et al. [30] and Medida [31] have also suggested. For this lack of ability for calculating the local Tu values, the B-C model has some deficiency in this respect that it cannot handle some physical effects compared with the models that can dynamically calculate the local Tu levels. Whereas this deficiency makes the B-C model rather limited, there are quite a few aerodynamic flows for which the model is still viable. The transition onset correlation that was also used in the original two-equation γ-Reθ model [1] is given by:

*Re<sup>c</sup>* = 803.73 (*Tu*<sup>∞</sup> + 0.6067)−1.027 (9)

**109**

*Transition Modeling for Low to High Speed Boundary Layer Flows with CFD Applications*

As mentioned before, any transition onset correlation would be incorporated into the B-C model. For instance, a class of potential transition onset correlations along with the one preferred in the present B-C model is shown in **Figure 1**. Currently, the B-C model is available in the SU2 (Stanford University

Unstructured) v6.0, an open-source CFD solver by the ADL of Stanford University [32]. The SU2 can solve two- and three-dimensional incompressible/compressible

Some outstanding test cases that make a good platform for measuring novel transition model performances are simulated by the foregoing transition models. These cases cover a wide range of flows from low speed two-dimensional flat plate and airfoil test cases to three-dimensional wind turbine blade and aircraft wing test

Well-known benchmark experiments such as the Schubauer and Klebanoff natural transition flat plate experiment [29] and the ERCOFTAC T3 series flat plate experiments by Savill [33] are used. The T3 series flat plate experiments consist of three zero pressure flat plate cases (T3A, T3B, and T3A-) and five variable pressure flat plate cases (T3C1, T3C2, T3C3, T3C4, and T3C5), in which the pressure gradients are generated using an adjustable upper tunnel wall. In all ERCOFTAC T3 test cases, the free stream turbulence intensities vary between 0.1 and 6%. **Table 1** summarizes the upstream conditions of the Schubauer and Klebanoff and the ERCOFTAC T3 flat

**Figure 2** shows the numerical and experimental skin friction coefficients of the zero pressure gradient test cases of S&K, T3A, T3B and T3A-, respectively. The figures include numerical predictions of several researchers, including for instance Suzen and Huang [9], Langtry and Menter [11], Walters and Cokljat [14], Menter et al. [22], Nagapetyan and Agarwal [24], and Medida [31]. In the S&K calibration case, the B-C model displays a good agreement with the experiment for the transition onset point similar to other methods. For the T3A and T3B cases, the B-C model shows rather late transition onset, whereas the other models predict some early or late onset points. Specifically, Nagapetyan and Agarwal [24] show a very good agreement with the experiment as to the transition onset and rapid skin-friction

**3. Two- and three-dimensional test cases for low to high speeds**

Euler/RANS equations using linear system solver methods.

cases from low to high speeds.

**Figure 1.**

plate experiments.

**3.1 Low speed flat plate test cases**

*DOI: http://dx.doi.org/10.5772/intechopen.83520*

*Transition onset correlations compared with experiments.*

*Transition Modeling for Low to High Speed Boundary Layer Flows with CFD Applications DOI: http://dx.doi.org/10.5772/intechopen.83520*

**Figure 1.** *Transition onset correlations compared with experiments.*

*Boundary Layer Flows - Theory, Applications and Numerical Methods*

purpose, an exponential function of the form (1-e<sup>−</sup><sup>x</sup>

γ*BC* = 1 − *exp*(−√

where Term1 and Term2 are defined as: *Term*<sup>1</sup> <sup>=</sup> *max*(*Re*<sup>θ</sup> <sup>−</sup> *Re<sup>c</sup>*,0.0) \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ <sup>χ</sup>1*Re<sup>c</sup>*

χ2 values are set to be 0.002 and 5.0, respectively.

the original two-equation γ-Reθ model [1] is given by:

*Re*<sup>θ</sup> <sup>=</sup> \_\_\_\_ *Rev*

and,

and the remaining part of the flow is taken to be fully turbulent (γBC = 1). For this

2.193 and *Rev* <sup>=</sup> <sup>ρ</sup>*dw*

In the above, ρ is the density, μ is the molecular viscosity, dw is the distance from the nearest wall, νBC is a proposed turbulent viscosity-like nondimensional term where νt is the turbulent viscosity, U is the local velocity magnitude, dw is the distance from the nearest wall, and χ1 and χ2 are calibration constants. Reθ<sup>c</sup> is defined as the critical momentum thickness Reynolds number, which is a correlation that is based on a range of transition experiments. In effect, Term1 checks for the transition onset point by comparing the locally calculated Reθ with the experimentally obtained critical momentum thickness Reynolds number Reθc. As soon as the vorticity Reynolds number Rev exceeds a critical value, Term1 becomes greater than zero and the intermittency function γBC begins to increase. However, the vorticity Reynolds number Rev relation above is a function of the square of the wall distance dw; therefore, it takes a very low value inside the boundary layer where the wall distance is quite low. Because of this, Term1 alone is not enough for intermittency generation inside the boundary layer. To remedy this, Term2 is introduced. Inspecting the Term2 equation with the νBC relation shows that the regions close to wall is inversely related and the damping effect of the transition model would be disabled inside the boundary layer. In effect, Term2 checks for the viscosity levels inside the boundary layer, and the turbulence production is activated wherever νBC exceeds a critical value χ2. In order to determine the calibration constants' χ1 and χ<sup>2</sup> values, the well-known zero pressure gradient flat plate test case of Schubauer and Klebanoff [29] is used. This test case represents a natural transition process due to the wind tunnel used in the experiment generates a freestream Tu around 0.2%. The model calibration is done by numerical experimentation; setting χ1 and χ2 such that the transition occurs at the same location as in the experiment. As a result, the χ1 and

Any experimental Reθc correlation could be used in the model. However, it should be noted that, since the S-A turbulence model does not solve for the local turbulent kinetic energy, local turbulence intensity values cannot be calculated. Due to this reason, the turbulence intensity Tu is assumed, for now, to be constant in the entire flow domain as Suluksna et al. [30] and Medida [31] have also suggested. For this lack of ability for calculating the local Tu values, the B-C model has some deficiency in this respect that it cannot handle some physical effects compared with the models that can dynamically calculate the local Tu levels. Whereas this deficiency makes the B-C model rather limited, there are quite a few aerodynamic flows for which the model is still viable. The transition onset correlation that was also used in

*Re<sup>c</sup>* = 803.73 (*Tu*<sup>∞</sup> + 0.6067)−1.027 (9)

\_\_\_\_\_\_ *Term*<sup>1</sup> − √

2

\_\_\_\_ *<sup>μ</sup>* Ω, <sup>ν</sup>*BC* <sup>=</sup> \_\_\_\_ <sup>ν</sup>*<sup>t</sup>*

\_\_\_\_\_\_

) is proposed for the γBC as follows:

*Term*2) (6)

(8)

,*Term*<sup>2</sup> <sup>=</sup> *max*(ν*BC* <sup>−</sup> <sup>χ</sup>2,0.0) \_\_\_\_\_\_\_\_\_\_\_\_\_\_ <sup>χ</sup><sup>2</sup> (7)

*Udw*

**108**

As mentioned before, any transition onset correlation would be incorporated into the B-C model. For instance, a class of potential transition onset correlations along with the one preferred in the present B-C model is shown in **Figure 1**.

Currently, the B-C model is available in the SU2 (Stanford University Unstructured) v6.0, an open-source CFD solver by the ADL of Stanford University [32]. The SU2 can solve two- and three-dimensional incompressible/compressible Euler/RANS equations using linear system solver methods.
