A Numerical Simulation Comparing the Efficiencies of Tubercle Versus Straight Leading Edge Airfoils for a Darrieus Vertical Axis Wind Turbine By: Ross Neal
Abstract: The efficiencies of sinusoidal and straight leading edge airfoils (NACA 0015) in a three straight wing Darrieus vertical axis wind turbine (VAWT) were compared using a fluid numerical solver (CFX). The airfoils and surrounding air modeled were meshed with body sizing and inflation layer methods and a cylinder mesh containing the surfaces of the airfoils was meshed separately as the mesh would be rotated during numerical solve. To save computing time, only a portion of the height of the VAWT was modeled as the streamlines should be repeating along the vertical leading edge of the airfoils. The numerical solver computed the torque applied to the rotated cylinder containing the airfoil surfaces at speeds from 15 to 35 rad/s for both designs at 10 m/s inlet wind speed of air with 1.185 kg/m 3 density for 4 complete rotations. The resulting torques from the time series data were averaged and converted to units of power (W) and coefficient of power and were graphed over rotation speed and tip speed ratio respectively to compare both airfoil designs. Introduction: Now that gasoline prices have greatly decreased, it is becoming harder for green energy, such as wind energy, to compete for investments over cheap, traditional, and polluting power sources such as coal. Therefore, it is more important now more than ever to make green energy more attractive by making them more efficient at producing power. One such solution came from a company called Whale Power based in Toronto. The company used biomimetics to design more efficient airplane wings, vacuum pumps, hydroelectric turbines and horizontal wind turbines. The company believed that the unique shape of the tubercles on the leading edge of the pectoral fins of
humpback whales allowed the whales to swim more efficiently than having smooth on fins allowing the animal to swim with much less effort. In fact, there have been several papers, experiments, prototypes (shown in Figure 1), and even products (shown in Figure 2) demonstrating that the tubercle leading edge airfoils consistently outperform their leading edge smooth counterparts 1 2 3. The tubercles on the leading edge of the airfoils combine the best of swept forward and backward wing designs to redirect streamlines into bands (shown on Figure 3) which reduces the turbulent vortices/cavitations and laminar flow separation that would normally cause stalls on traditional wings by increasing the angle of attack (the relative angle between the airspeed vector and the chord of the airfoil blade) shown in Figure 4. The stalling robustness of the tubercle airfoil allows for 1 Saadat, Haj Hariri, and Fish, Explanation of the Effects of Leading Edge Tubercles on the Aerodynamics of Airfoils and Finite Wings, The American Physical Society Division of Fluid Dynamics 21 Nov. 2010, <http://meetings.aps.org/meeting/dfd10/event/132841>. 2 Murray, Gruber, and Fredriksson, Effect of Leading Edge Tubercles on Marine Tidal Turbine Blades, The American Physical Society Division of Fluid Dynamics 22 Nov. 2010, <http://meetings.aps.org/meeting/dfd10/event/133206>. 3 Custodio, Henoch, and Johari, Separation Control on a Hydrofoil Using Leading Edge Protuberances, The American Physical Society Division of Fluid Dynamics 19 Nov. 2006, <http://meetings.aps.org/meeting/dfd06/event/53973>.
higher angles of attack of the airfoil without stalls or as much drag and therefore more lift is created due to increased fluid pressure under the airfoil (more cross sectional area of the airfoil is directly exposed to the fluid). This technology could be applied to airplane wings that could save on fuel due to less drag caused by the wings and to blades of wind/hydroelectric turbines that would generate more torque at lower fluid speeds leading to more power and electricity generated. However, most research involving this technology has been put into fan blades and airplane wings, but not in vertical axis wind turbines (VAWT) like the Darrieus design shown in Figure 5. Theoretically, if the modified tubercle airfoil increases the efficiency of lift for wings on airplanes and on fan blades, the same idea could be applied to vertical axis wind turbines. The tubercle technology would be especially advantageous for Darrieus VAWTs over more popular horizontal axis wind turbines (HAWT) because it can accentuate the advantages VAWTs have
over HAWTs; VAWTs can operate at especially low wind speeds and therefore at lower elevations and this technology would theoretically increase the power generated at those speeds significantly. Also, since not as much power is needed to rotate the tubercle VAWT, less starting electricity would be needed/wasted to start rotation of the VAWT (VAWTs are not self starting from standstill unlike HAWTs). Plus, since the wind speed is the same along the entire length of the airfoil on VAWTs unlike HAWTs, the tubercles size/shapes can be more easily optimized for the average wind speed experienced at a particular location leading to a relatively simpler shape to manufacture compared to HAWTs where the cross sectional airfoil shape has to be constantly optimized at different radial speeds experienced along the blade in addition to the tubercles. To validate the claims of increased efficiencies of the tubercle leading edge airfoils over traditional straight edge designs and to test if the technology would equally apply Darrieus VAWTs, I decided to numerically model and compute the power generated by tubercle and straight leading edge Darrieus VAWTs. Modeling: I decided to use a well understood simple popular airfoil for the VAWT as a National Advisory Committee for Aeronautics (NACA) 0015 airfoil show in Figure 6. The four digit number represent constants in the symmetrical four digit airfoil NACA equation:
where c is the chord length, x and y t are the normalized horizontal and vertical coordinates of the edge of the airfoil respectively, and t is the maximum thickness normalized to the fraction size of the chord length (last two digits of the NACA is 100*t). The overall design was three NACA 0015 equidistant airfoils around a circle of 1 m diameter with 20 cm chords (shown in Figure 7 with force vectors). The pitch angle Θ was chosen to be 10 o (close to the NACA 0015 stall angle shown in Figure 8 for a given Reynolds number calculated later) for both the tubercle and non tubercle airfoils so that I could directly compare both airfoils and accentuate the theoretical advantages of the low drag at higher angle of attack for the tubercle airfoil. The Reynolds number can be calculated for an airfoil (fluid
dynamics) as: Re=V*c/v where Re is the Reynolds number, V is the flight/wind speed, c is the chord length, and v is the kinematic viscosity of the fluid (air in this case). For a chosen 10 m/s wind speed, the Reynolds number ranges from around 120,000 to 140,000 for elevations between 1700 m and sea level respectively which means the magenta line in Figure 8 most closely matches the simulation. To model the tubercles on the leading edge of the airfoil, the following sinusoidal equation was used in Figure 9. To get a 3D model of a tubercle airfoil, the base NACA 0015 shape was vertically extruded with the sine wave as a leading edge guideline. Then, a second NACA 0015 shape was extruded normally (straight up vertically) and both 3D objects were intersected and joined to get a straight tailing edge and a sinusoidal leading edge shown in Figure 10. Then, three copies of the airfoil were arranged as previously shown in Figure 7 with the center of rotation inside a 8x3x0.3 m rectangular prism located 2 m from one end and centered
along the 3 m side. To reduce element count and facilitate meshing and to allow rotation of airfoils, two more cylinders were centered on axis of rotation of diameter 0.75 m and 1.5 m with height of 0.3 m as shown in Figure 11. The outer box and the innermost cylinder were meshed (using ANSYS software) with a body sizing method of 5 mm and a multi zone method with hexa/prism shapes with free mesh tetrahedrons while the outer cylinder minus the inner cylinder and the airfoils (airfoils aren t meshed) were meshed with a body sizing method of 2 mm shown in Figure 12. All the
interfaces between shapes/regions touching have 20 inflation layers at a 1.2 growth rate including the surface of the airfoils shown zoomed in on Figure 13. After meshing the model, a fluid dynamics numerical solver was used (CFX) to calculate the drag and lift forces on the airfoils. The boundary conditions were set up as shown in Figure 14. The inlet wind (air at 25 o C with 1.185 kg/m 3 density which is about 350 m of elevation above
sea level) conditions is set to 10 m/s at 5% intensity to simulate a median wind speed. The red symmetry boundaries allow for the VAWT to be taller if need be since the airflow should be roughly the same at any height along the airfoil. In order to get a power curve data to properly calculate torque generated by airfoils, the wind speed in the inlet remained the same 10 m/s, while the rotational speed of the airfoils varied between 15 and 35 rad/s in 5 rad/s increments (the entire cylinder mesh region was rotated about the big blue arrow in Figure 14 to rotate the airfoil surfaces). To balance accurate results and be computationally shorter, 400 timesteps and 100 maximum loops per timestep at.0001 variance (100 per revolution or approximately 3 o rotation increments) for each simulation was chosen. Therefore, the entire simulation length was calculated by 4*2π*(rad/s) for 4 revolutions and the time step (dt) is 1/400 the entire simulation length which values are shown on Figure 15. Since the Darrieus VAWT blades were strictly vertical and not a twisted helical blades (twisted helical blades would have required more height of the airfoil to be calculated whereas the simulations took a full month of computer time already), the torque applied to the airfoil varies over time as each of the three airfoil go from the optimum angle (leading edge of the
airfoil is close to facing directly the wind speed vector) to generate the most useful lift at different time throughout the rotation cycle. Therefore, in order to compare the efficiencies of both smooth and tubercle airfoils in Darrieus VAWTs, the torque applied to the blades (the rotating cylinder) had to be averaged over the entire simulation time (shown as an example in Figure 16). The first revolution of torque data on the airfoils had to be discarded due to the time it takes for the streamlines of the wind of the inlet to fully penetrate the entire VAWT simulation. Results: The results were very encouraging as the tubercle leading edge airfoil VAWT consistently and significantly outperformed the smooth leading edge airfoil VAWT in average torque and power generated; the minimum improvement of power generated by the tubercle was just under twice (up to a maximum of 6 times more) than that over the traditional design. The average power generated was calculated by multiplying the rotational speed to the average torque generated. The coefficient of power was also calculated by the following equation: where Cp is the coefficient of power, P is the power generated (W) calculated from the torque, p is the density of the fluid (air at 1.185 kg/m 3 ), A is the area swept by the airfoils (0.3 m height with 1 m diameter makes 0.3 m 2 area swept), and V is the velocity of the fluid (air at 10
m/s). The tip speed ratio (TSR) was calculated from multiplying the rotation speed (rad/s) and the radius of the rotation (0.5 m) divided by wind speed (10 m/s). All the calculated averages of generated torques, powers, coefficients of power as well as the percent improvement of the tubercle over normal airfoil designs are shown in Figure 17 and was entered in two separate graphs with different units yet show the same percentage improvements shown in Figure 18.
The Tubercle airfoil generates some incredible efficiencies approaching a coefficient of power greater than 0.6 or maybe 0.7 for higher TSRs whereas the normal airfoil has a localized stall at 25 rad/s (TSR of 1.25) and the coefficient of power may be plateauing between 0.3 and 0.4 for TSRs higher than 1.75. Conclusion: Given more computational power and resources, a more complete coefficient of power versus TSR graph could be made to show optimum TSR for coefficient of power. In addition, a more complex helical airfoils could be introduced to mitigate varying torques generate over each cycle of rotation which could also reduce mechanical stresses and failures due to reduced vibrations. Plus, a scaled down experiment could be made in a wind tunnel to see how well the numerical simulation matches experimentation. Also, a separate study could be completed on just the on the equation for the leading edge tubercle airfoil (NACA shape airfoil and pitch could be also be optimized for tubercle technology) to see if varying the amplitude and frequency could be optimized for certain wind speeds since there hasn t been much research on that particular subject matter. The added efficiencies of adding a simple sinusoidal leading edge airfoil were found to be between 2 6 times for efficient compared to the traditional straight edged airfoils in a Darrieus VAWT. The relative stalling robustness of the tubercle airfoil over the traditional straight edged airfoil allowed for greater percentage of the rotation cycle to have useable lift and increase overall efficiency. The sheer efficiency improvement should be more than enough to justify any additional manufacturing costs of the propellers especially with the relatively smaller blades Darrieus VAWTs typically have compare to modern giant HAWTs.