**4. Conclusions**

Local correlation-based transition models in the sense of empirical correlations incorporated into Reynolds-averaged Navier-Stokes methods have been discussed. A logical path for the development of such models is highlighted such that a variety of combinations of turbulence and transition equations lead to different modeling alternatives. For instance, the pioneering work by Menter et al. [1] two-equation γ-Reθ transition model sums up to a total of four-equation model by the incorporation of the two-equation k-ω SST turbulence model of Menter et al. [27]. In the same line of development but in a leaner approach, Walters and Cokljat [14] developed a three-equation k-kL-ω model. Similarly, Medida [31] developed a three-equation S-A-γ-Reθ transition model that is a sum of the Menter et al. [1] twoequation γ-Reθ transition model and the one-equation S-A turbulence model [28].

In fact, in a recent work, Menter [22] reached to the conclusion that the Re<sup>θ</sup> equation was rather redundant. Without any loss of accuracy, Menter produced a leaner three-equation k-ω SST-γ transition model by incorporating a novel oneequation intermittency transport γ-model [22] with the two-equation k-ω SST turbulence model of Menter et al. [27]. In the same line of thought, Nagapetyan-Agarwal constructed the so-called two-equation transition model of WA-γ [24] by incorporating the Wray-Agarwal (WA) wall-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]. These two models paved the way for developing yet another leaner transition model by Bas et al. [19] with the introduction of the algebraic Bas-Cakmakcioglu (B-C) model by incorporating an algebraic γ-function with the one-equation S-A turbulence model [28].

The Bas-Cakmakcioglu (B-C) [19] model qualifies as a 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 again the pragmatic idea of further reducing the total number of equations. Thus, rather than deriving extra equations for intermittency convection and diffusion, already

present convection and diffusion terms of the underlying turbulence model could have been used. From a philosophical point of view, the transition, as such, is just a phase of a general turbulent flow. In a sense, addition of artificially manufactured transition equations may appear to be rather redundant. Yet, for most of industrial flow types, there is experimental evidence that a close relation between the scaled vorticity Reynolds number and the momentum thickness Reynolds number exists. This fact stands out as the primary reason for the success of the class of so many intermittency transport equation models following the Menter's pioneering two-equation γ-Reθ model [1]. Using the present B-C model, a number of twodimensional test cases including flat plates, airfoils, turbomachinery blades, and three-dimensional low speed wind turbine and high-speed transport plane wing were simulated with quite successful results. These results may be regarded to vindicate this leaner approach of using even lesser equations for industrial design aerodynamics problems.
