CFD Analysis of a SD 7003 Airfoil with a Local Correlation Based Transition and Turbulence Model

The SD 7003 has been investigated quite extensively in the past at different Reynolds numbers. At moderate Reynolds numbers and low angles of attack its upper surface is dominated by a large laminar separation bubble which provides a demanding test case for RANS-based transition models. In this article, a widely used open-source Computational Fluid Dynamics (CFD) toolbox has been used mainly to investigate the (i) aerodynamics coefficients and (ii) boundary layer transitional behaviors of a SD 7003 airfoil using a local correlation-based γ – Reθt SST [1] transition and turbulence model. Several researchers have proposed different correlations which can be used in the γ – Reθt SST. In this article, five such correlations have been investigated for the SD 7003 at low Reynolds number which is particularly challenging for CFD analysis.


Introduction
Airfoils are used in dierent engineering applications like wind turbines, airplanes etc. In the operating conditions, airfoils typically encounter transitions from laminar to turbulent ow in the boundary layer (BL). The SD 7003 airfoil has been investigated quite extensively in the past at dierent Reynolds numbers through diversied computation techniques and experimental investigations. It has been found that at moderate Reynolds numbers and low angles of attack, the upper surface of SD 7003 is dominated by a large laminar separation bubble (LSB). According to [2] -"At these low Reynolds numbers, the ow may remain laminar over a signicant portion of the airfoil rendering it susceptible to separation from even mild adverse pressure gradients. For moderate incidence, separation leads to the formation of a closed laminar separation bubble (LSB) which reattaches through transition into turbulence." CFD analysis of airfoils at low Reynolds number that is associated with LSB is a challenging task for computationally inexpensive RANS-based transition models. Diversied approaches can be used to model dierent types of BL transitions. Two popular transition and turbulence models in Computational Fluid Dynamics (CFD) analysis are: (i) The local correlation-based transitional model called γ − Re θ which was originally proposed by Menter et al. [1]; and (ii) The phenomenological k − k L − ω model [3].
Over the years, dierent researchers [48] have proposed dierent correlations for calculating the two undisclosed proprietary parameters required by the original γ −Re θ model. In the present work, CFD analysis of SD 7003 airfoil has been performed at a low Reynolds number of 60, 000. To tackle the transition modeling, γ − Re θ model is used in OpenFOAM [9] using ve dierent correlations, proposed by [48]. It should be noted that the present work is an extension of the earlier research activities with k − k L − ωSST model which was presented in [10]. Selected nal results from the CFD analysis have been compared with other computational (DNS & LES) and experimental datasets for verication and validation.

The OpenFOAM Case
To perform the CFD analysis, a widely used open-source CFD toolbox called OpenFOAM has been used in the present work. To verify the results, mesh sensitivity analysis has been conducted with 3 mesh sizes (termed as Coarse, Medium and Fine). In Table 1, the salient features of these three mesh sizes are presented. It should be noted that the same three mesh sizes were previously investigated in the previous research [10] using the k − k L − ω model. It should be noted that the SD 7003 has sharp trailing edges for which the C-type mesh is preferred and thus used in the present work. More discussions about the computational domain and the mesh were included in [10].
In Figure 1, the mesh around SD 7003 is illustrated. Whereas, the mesh distribution around the leading and trailing edges of the airfoil are depicted in Figures 2 and 3 respectively. For the present analysis, a steady-state solver called "simpleFoam", which is available in OpenFOAM and employs the SIMPLE algorithm to solve the incompressible Navier-Stokes equations, was used. The Reynolds Number (Re) for all the cases is about 60,000 and all the CFD analyses were conducted at 4 o angle of attack.

Mesh Sensitivity Analysis
To verify the CFD results, the variation of the pressure coecients (C p ) around the surface of SD 7003 airfoil has been investigated using the γ − Re θt SST [1] model using the correlation proposed by Tomac et al. [8], which is shown in Figure 4. The C p distributions for the three mesh sizes are quite close to each other for the entire surface. These results have been compared with the ILES results obtained by [2] and it can be seen that there are noticeable dierences around certain locations. The variation of aerodynamic coecients for 3 mesh sizes are presented in Table 2, which shows that further renement of mesh sizes is not needed.

Skin Friction Coecients
In Figure 5, the variations of skin friction coecients (C f ) around the SD 7003 airfoil that are obtained from the CFD analysis using the ve correlations [48] with the γ − Re θt SST [1] model are shown. These results are compared with Implicit Large Eddy Simulation (ILES) [2] and XFOIL [14].
Both the transition from laminar to turbulent ow and the laminar separation bubble are situated between the separation point x separation (where the C f becomes negative near x/c ≈ 0.23 for the ILES result) and reattachment point x reattachment (where the C f value becomes positive again near x/c ≈ 0.65 for the ILES result). So, the distance between x separation and x reattachment is indicative of the length of the bubble size. The two major observations from this gure are: • All the RANS-based correlations are over-predicting the reattachment points signicantly when compared with the ILES result.
• Among the ve correlations [48], the the distance between x separation and x reattachment is maximum for [7] and minimum for [8]. Five dierent correlations for γ − Re θ Semicond .Sci .Technol . model [1], proposed by [48], have been investigated through CFD analysis of SD 7003 airfoil at low Reynolds number in the present study. It has been found that all the correlations have under-predicted the C l values and overpredicted the C d values. However, aerodynamic coecients (C m ,C l , and C d ) obtained from the k −k L −ω model [3] using the Falkner-Skan based modication [10] are closer to the experimental results than Tomac et al. [8]. Also, the correlation by Tomac et al [8] is predicting the separation and reattachment points, for the ow around SD 7003 at low Reynolds number, better than the other four correlations [47].