Numerical simulation of the flow over a tubercled wing

The objective of the present study is to carry out a numerical study of the flow around a NACA0021 modified wing by the incorporation of sinusoidal tubercles on its leading edge at a Reynolds number equal to 225,000. The SST k- turbulence model is used as closure to the incompressible governing equations. Runs have been performed for several attack angles. Results show that for lower angles of attack, tubercles reduce the drag coefficient with a slight increase in lift.


Introduction
Fauna and flora have always inspired humankind in their inventions and problem solving. Several technologies in all fields of science have been developed by the observation of nature. This approach, known also as biomimetic, has been widely adopted in the design of submarines and marine propellers in order to reduce drag. In particular, the performance of airfoils can be improved by generating streamwise vortices in boundary layer using tubercled leading edge inspired by prior studies carried out by marine biologists on the morphology of humpback whales pectoral flippers [1][2]. They demonstrated that the high aspect ratio with large sinusoidal tubercles along the flipper leading edge can cause a considerable increase in the stall angle of attack and the maximum lift coefficient. Several studies have been performed on tubercled leading edge wings during the last years. In particular, experimental studies have demonstrated their benefits. Generally, in most studies, the utilization of airfoils with tubercled leading edge showed improved performance in terms of lower drag, higher lift and shorter separation region. However, in some studies, it is found an opposite trend. Experimental studies aimed to measure lift, drag and pitching moments for various attack angles in wind tunnels. Results are generally analyzed by varying amplitude and wavelength of tubercles. Table 1 summarizes previous experimental studies on tubercled wings. Recent applications of CFD to solve the Navier-Stokes equations for tubercled wings are summarized in Table  2. In most of studies, in-house research codes that are rarely available for researchers have been used. It is more convenient to perform studies in this topic using available commercial CFD codes such as Fluent. The latter can solve laminar and turbulent, incompressible and compressible, 2D and 3D, steady and unsteady flows.
The conflicting findings observed in the literature regarding the effect of tubercles on the performance of wings make this topic a challenging subject. Further studies are necessary and the present paper constitutes one.

Mathematical model
Two NACA 0021 wings designed, built and tested by Bolzon et al. [21] have been used in the present study. One wing had a smooth leading edge, whereas the other wing had tubercles with amplitude of 10.5 mm and a wavelength of 60 mm along its entire leading edge as depicted in Figure 1  The three-dimensional, incompressible and steady Reynolds averaged Navier-Stokes equations are given below: The Reynolds stress tensor ij R is approximated using one of the different turbulence models provided in the CFD commercial software Fluent. A careful analysis of the literature reveals that there is no agreement on the selection of turbulence models in the analysis of the flow around tubercled wings. However, it seems that SST k- provides better performance in predicting the flow structure around a tubercled wing. The computational domain has been created according to experiments by Bolzon et al. [21] in a wind tunnel. The domain has an inlet section of 0.5×0.5 m and a length of 2.2 m. The wing is placed at a distance of 0.45 m from the inlet. The domain has been meshed using a multi block technique. A block represented by a cylinder containing the wing to allow its rotation in order to obtain the desired angle of attack and a second block for the remaining domain. Figure 2 depicts an overview the computational domain.

Fig. 2. Computational domain meshing
The steady Reynolds averaged Navier-Stokes equations, the turbulence model equations and the corresponding boundary conditions have been numerically solved using the pressure based solver of the commercial CFD code Fluent. A second order upwind scheme has been selected to discretize momentum and turbulence terms. The algorithm PRESTO is used to compute the coupling between the pressure and the velocity field. The runs were assumed to reach convergence once the residuals fall below the value of 10 −6 for all variables.

Results
The results of the simulations are presented as follows.
First, a mesh independency test has been performed in order to ensure that the numerical solution is independent to the size of the grid used. This operation is then followed by a detailed analysis of the flow around the wings in terms of lift coefficient, drag coefficient and streamlines. The results have been plotted at Reynolds number of 0.225 million for angles of attack ranging from 0 to 20°.

Mesh independency test
A mesh independency test has been performed. Three grids of 1 million, 2 million and 4 million have been used for angles of attack ranging from 0 to 20° at a Reynolds number of 0.225 million. Figure 3 shows a comparison of lift and drag coefficients obtained using the three grids. It is observed that similar values are obtained using grids 2 and 3. Thus, results are obtained using a mesh of 2 million cells. a.

Lift and drag coefficients
The effect of the angle of attack on the lift and drag coefficients of wings with smooth and tubercled leading edges is presented in Figure 4. Both lift and drag coefficients increase as the angle of attack increases. For both experimental and numerical data, at lower angles of attack, leading edge tubercles produce lift coefficient slightly greater than smooth leading edge. The difference increases as the angle of attack increases. As stated by many researchers, this trend can be attributed to the formation of a laminar separation bubble on the section side of the wings [10][11][12][13][14][15][16][17][18][19][20]45]. However, there is no widespread agreement on this finding. The laminar separation bubble on the section side of wings have not been observed by other researchers [4,6]. For lower angles of attack ( < 8 (for experimental measurement) and  < 12 (for numerical data)), tubercles reduce the drag coefficient as illustrated in Figure 3.b. However, for angles of attack greater than these values, tubercles increase the drag coefficient.

Conclusions
The flow around two swept wings, one smooth and one tubercled, has been numerically analyzed using the commercial code Fluent. Turbulence has been modeled using SST k- model, the most widely used model for this kind of flows. It is found that for lower angles of attack, tubercles reduce the drag coefficient with a slight increase in lift. The discrepancy between numerical results and experimental measurements at attack angles greater than 5 cannot be attributed to experimental errors as it is observed for both wings. Indeed, under these conditions, laminar separation bubble on the section side of the wings observed in many previous experimental studies appears and SST k- model falls in predicting correctly the airfoil's pressure distribution.