The Joukowski transformation is a great way to study basic airfoils using potential flow. It gives a nice way to derive the Kutta condition, as well as the classic lift slope dC_L/d\alpha \approx 2 \pi. But in all those demonstrations, I never saw the lift and drag curves of the Joukowski airfoil! Luckily, I have XFOIL to remedy that problem. I’m comparing the symmetric NACA to Joukowski airfoils at a Reynolds number of 1 million, for thicknesses of 10%, 15%, and 20%.
First, have a look at the 20% NACA vs Joukowski airfoils:
The Joukowski airfoil’s max thickness is behind that of the NACA airfoil, and afterward thins quickly, reaching a cusped trailing edge. Time to look at the forces!
The thinner airfoils stall sooner, as expected, whereas thicker airfoils can maintain the pressure gradient. It looks like the Joukowski airfoils have a higher L/D generally. Neat. They also have better behavior when it comes to the moment: no big discontinuities.
Overall, there isn’t a huge difference. Of course, a cusped trailing edge (like the Joukoski airfoil’s) is hard to build and maintain in reality. Still, it’s nice to take a look at the standard theoretical tools and see how they compare to the engineering baselines.
6 replies on “Joukowski Airfoil Performance”
Hello, i am a 3rd year Aerospace Engineering student. While researching for my report assignment i found this study was curious of how exactly did you plot the cl vs alpha and other graphs for Joukowski airfoils using xfoil
To get the data, I created coordinates for the Joukowski airfoil in a text file, then loaded them into XFOIL. Then, similar to the example session log here, I simulated the performance over a range of angles (with PACC and ASEQ), saved the data to a text file (PWRT), then plotted them in MATLAB.
Sorry for the slow response, my notifications get lost sometimes.
I used to have a pair of FPV helicopter rotor blades that seemed to have the cusped trailing shape. I lost that pair of blades and reordered them from Horizon, they should be delivered to my place today. I am testing a few blades for my FPV 250 gram helicopter that generally flies for half an hour with a dive and punch-out stunt move, all in one flight. Will be interesting to see how the cusped blade performs in different Reynold number regions compared to naca0015. I see your plots are for a high Reynold number, like 1 million. My FPV helicopter is in the 100 thousand region.
I received the cusped shaped helicopter blades yesterday. It is made by Horizon Hobby, part number BLH5801. It is a 10% airfoil based on NACA0010. I used to reject all 10% airfoils, thats why I discarded my first purchase and now buying it again. The cL versus alpha is basically identical to NACA. I am still skeptical how good the stall character is. Your chart shows that this cusped 10% airfoil is better than NACA0010 in stall character, but your chart also shows that NACA0015 airfoil is still better than cusped 10% airfoil.
I tested the Joukowski 10 airfoil made by Horizon Hobby, BLH5801 helicopter blades, over the weekend. The Joukowski 10 is modified based on NACA0010, which has optimal alpha for L/D at 4 degrees. Your L/D chart only includes up to 8 degrees of data for the Joukowski 10 airfoil and doesn’t include the 10-degree data point, which is what worries me. I have three test alpha settings’ FPV videos: for the 6-degree fuel economy, the 8-degree performance, and the 10-degree extraneous mode. They all performed nearly identically to NACA0012 airfoil blades, manufactured by Microheli, which produces aftermarket substitute parts for Horizon helicopters. These are as predicted in your chart.
The cusped blades have 3% more endurance (fuel economy) than conventional NACA0012 blades, but it is well within the statistical margin of error. Also, the cusped blades do not have a hard performance decline or stall at 10 degrees of alpha.
Due to the peculiar vertical rotor fluid dynamics, the optimal alpha for cruising with the NACA airfoil is about 1 degree or more higher than that of the static test. NACA0012’s static, optimal alpha for fuel economy is 5 degrees, so the endurance test was done with 6 degrees of alpha. Since Joukowski 10 is nearly identical to NACA0012 for my practical tests, 6 degrees of alpha was used.
For the 10% thickness foils at higher angles, they don’t have data at high angles because the flow becomes too separated, and the simulation doesn’t converge.
Very interesting to know! Thank you for the information.