**1. Introduction**

Industrial design aerodynamics heavily depends on development of new CFD methods that can be only as good as their experimental database. All these industrial design CFD codes, as they may be called, are constantly in search of better physical modeling starting with appropriate transition and turbulence modeling. To this end, although numerical representation of turbulence has reached the acceptable levels of accuracy for computational aerodynamics, transition modeling has yet to reach the level of turbulence modeling capability for routine calculations. Therefore, transition modeling as part of turbulence has always been standing as the crux of the matter with regard to turbulence modeling. Today, state of the art

Reynolds Averaged Navier-Stokes (RANS) solvers are widely available for numerically predicting fully turbulent part of flow fields by frequent use of, for instance, one- or two-equation turbulence closure models. However, none of these models are adequate to handle flows with significant transition effects due to the lack of practical transition modeling. Menter et al. [1] state that some of the main requirements for pragmatic transition modeling are the following: calibrated prediction of the onset and length of transition, allow inclusion of different mechanisms, allow local formulation, and allow a robust integration with background turbulence models.

Nevertheless, transition modeling as applied to CFD methods has followed certain line of evolution covering a range of methods starting from simple linear stability methods such as the eN method [2, 3] to almost or fully predictive methods such as LES and DNS that are very costly for engineering applications [1]. The eN method is the lowest level transition model based on linear stability theory. This method has found quite wide application in numerical boundary layer methods [4], but translating this into RANS methods has proven quite demanding as it requires a high-resolution boundary layer code that must work hand in hand with the RANS method. Also, this method is also dependent on the empirical factor-n that is not universal and depends on the type of flow.

Following the eN method, a better level of complexity that is compatible with the CFD methods is the low Reynolds number turbulence models [5]. Yet, they do not reflect real flow physics and lack the true predictive capability. These methods take advantage of the fortuitous ability of the wall damping terms mimicking some of the effects of transition. Next in the line of increasing complexity comes the class of the so-called correlation-based transition models [1]. These models are based on the fundamental approach of blending the laminar and the turbulent regions of the flow field by introducing intermittency equations to the turbulence equations. In this line, based on the boundary layer methods, there are three similar examples of intermittency equation approach that was introduced by Dhawan and Narasimha [6], Steelant and Dick [7], and Cho and Chung [8]. First, Dhawan and Narasimha [6] used a generalized form of intermittency distribution function in order to combine the laminar and the turbulent flow regions. Second, Steelant and Dick [7] proposed an intermittency equation that behaves like an experimental correlation. Third, Cho and Chung [8] introduced the k-ε-γ model which was formulated by an additional transport equation-γ to the well-known k-ε turbulence model. Finally, Suzen and Huang [9] significantly improved intermittency equation approach for flow transition prediction by combining the last two methods with a model that simulates transition in both streamwise and cross-stream directions. However, these models all rely on nonlocal flow data, and it was difficult to embed these models into practical CFD codes. These models require calculating the momentum thickness Reynolds number-Reθ, which is an integral parameter, and comparing it with a critical momentum thickness Reynolds number. For this reason, these early models are "nonlocal" methods that require exhausting search algorithms for flows with complex geometries.

After the success of the "nonlocal" transition models that use intermittency transport equations including experimental correlations, a range of new methods [10, 11] has been developed, called as the local correlation-based transition models (LCTM) by Menter et al. [1] that are compatible with the modern CFD codes. This compatibility has been achieved by the experimental observation that a locally calculated parameter called as the vorticity Reynolds number (Rev) is proportional to the momentum thickness Reynolds number (Reθ) in a Blasius boundary layer. This observation is also shown to be quite effective for a wide class of flow types with moderate pressure gradients. This is due to the fact that the relative error between

**103**

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

the two parameters is less than 10% for such flows [1]. Therefore, the vorticity Reynolds number-Rev would be used in order to avoid all the troublesome work that

Following the success of the γ-Reθ two-equation transition model of Menter et al. [1], some other two-, or three-equation models are proposed, such as the near/freestream intermittency model by Lodefier et al. [12], variations of the k-kL-ω models of Walters and Leylek [13] and Walters and Cokljat [14], and the k-ω-γ model of Fu and Wang [15] with super/hypersonic flow applications. In addition, some researchers proposed extensions to local correlation-based transition models (LCTM) in order to take more physical phenomena into account. To this end, cross-flow instability effects by Seyfert and Krumbein [16], surface roughness effects by Dassler et al. [17], and compressibility effects by Kaynak [18] were included. Meanwhile, Bas et al. [19] proposed a very pragmatic approach by introducing an algebraic or a zero-equation model called later as the Bas-Cakmakcioglu (B-C) model [20]. Herein, it was shown that an equivalent level of prediction compared with the two- and three-equation models could be achieved with less equations provided that physics was correctly modeled. In parallel, Kubacki et al. [21] proposed yet another algebraic transition model with a good level of success vindicating this line of approach. Similarly, Menter et al. [22] proposed a new one-equation γ-model which is the simplification of their earlier two-equation γ-Reθ model [11] without the Reθ-equation that produced equal level of results as in the original model. Following this logical trend for reducing the total number of equations, the Wray-Agarwal (WA) wall-distance-free oneequation turbulence model [23] was complemented with the Menter et al. [22] one-equation intermittency transport-γ model to obtain the so-called two-equation Nagapetyan-Agarwal WA-γ transition model [24]. In the following, a brief review of the transition modeling is made that covers the practical applications of a range of models that are currently used in the industrial design aerodynamics. Based on the present authors' recent experiences, the Bas-Cakmakcioglu model [20] will be covered in some detail to display the viability of the algebraic intermittency equation approach vis-a-vis the one- and two-equation local correlation-based

The well-known eN method is based on the linear stability theory [25], and it is developed by assuming that the flow is two-dimensional and steady, the boundary layer is thin and the level of disturbances in the flow region is initially very low. In this method, the Orr-Sommerfeld eigenvalue equations are solved by using the previously obtained velocity profiles over a surface in order to calculate the local instability amplification rates of the most unstable waves for each profile. By taking the integral of those rates after a certain point where the flow first becomes unstable along each streamline, an amplification factor is calculated. Transition is said to occur when the value of the amplification factor exceeds a threshold N value.

In the low Reynolds number turbulence models, the wall damping functions are modified in order to capture the transition effects [5]. To be able to predict the

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

existed in the nonlocal models.

transition models (LCTM).

**2.1 eN Method**

**2. Review of transition models**

Typical values of N vary between 7 and 9.

**2.2 Low Reynolds number turbulence models**

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

the two parameters is less than 10% for such flows [1]. Therefore, the vorticity Reynolds number-Rev would be used in order to avoid all the troublesome work that existed in the nonlocal models.

Following the success of the γ-Reθ two-equation transition model of Menter et al. [1], some other two-, or three-equation models are proposed, such as the near/freestream intermittency model by Lodefier et al. [12], variations of the k-kL-ω models of Walters and Leylek [13] and Walters and Cokljat [14], and the k-ω-γ model of Fu and Wang [15] with super/hypersonic flow applications. In addition, some researchers proposed extensions to local correlation-based transition models (LCTM) in order to take more physical phenomena into account. To this end, cross-flow instability effects by Seyfert and Krumbein [16], surface roughness effects by Dassler et al. [17], and compressibility effects by Kaynak [18] were included. Meanwhile, Bas et al. [19] proposed a very pragmatic approach by introducing an algebraic or a zero-equation model called later as the Bas-Cakmakcioglu (B-C) model [20]. Herein, it was shown that an equivalent level of prediction compared with the two- and three-equation models could be achieved with less equations provided that physics was correctly modeled. In parallel, Kubacki et al. [21] proposed yet another algebraic transition model with a good level of success vindicating this line of approach. Similarly, Menter et al. [22] proposed a new one-equation γ-model which is the simplification of their earlier two-equation γ-Reθ model [11] without the Reθ-equation that produced equal level of results as in the original model. Following this logical trend for reducing the total number of equations, the Wray-Agarwal (WA) wall-distance-free oneequation turbulence model [23] was complemented with the Menter et al. [22] one-equation intermittency transport-γ model to obtain the so-called two-equation Nagapetyan-Agarwal WA-γ transition model [24]. In the following, a brief review of the transition modeling is made that covers the practical applications of a range of models that are currently used in the industrial design aerodynamics. Based on the present authors' recent experiences, the Bas-Cakmakcioglu model [20] will be covered in some detail to display the viability of the algebraic intermittency equation approach vis-a-vis the one- and two-equation local correlation-based transition models (LCTM).
