Lift and Drag
Chet Goerzen and Adam Paul; Spring 2020
1. Introduction
In this lab the formation of starting vortices was explored. The effects of lift and drag as a function of airspeed were also explored.
1.1 Vortices
Starting vortices are a by-product of Kelvin’s Circulation Theorem, which states that circulation is conserved (Kundu et al., 2008). When a foil moves through the air it will create circulation which allows lift to be produced. This is an effect of the Kutta–Zhukhovsky Lift Theorem, which states that \(L = \rho U \Gamma\) This is true for arbitrary bodies in an ideal fluid. With these restrictions the drag is identically zero. This is obviously not true for real fluid flows. In a real fluid, the drag comes from pressure forces at the leading and trailing edge of the body, as well as skin drag. Skin drag is the drag occurring at the surface of a body due to frictional forces. Due to the Kutta-Zhukovsky Lift Theorem, if the lift is increased by increasing the angle of attack, then the circulation must also increase. In order for circulation to be conserved, an equal and opposite vortex must be created at the trailing edge of the foil. This is known as the starting vortex. The following video is a Computational Fluid Dynamics (CFD) simulation of a starting and stopping foil (Agromayor et al., 2017).
Stopping vortices are also visible in the simulation. Once the foil has stopped moving it is interesting to note that the remaining vortices are of equal and opposite strength, so that the total circulation of the system is zero. In a real fluid viscous dissipation would quickly cause the vortices to dissipate, with the energy dispersed as heat.
1.2 Foils
Two very important characteristics of a foil are lift and drag. Lift is the upward force given by the Kutta–Zhukhovsky Lift Theorem mentioned above. In the Kutta-Zhukhovsky Lift Theorem, $\Gamma$ is given by: \(\Gamma = \oint U \cdot ds\) From the above relationship it is interesting to note that : \(L \propto U^{2}\) Drag is the force that opposes motion, and there are two main kinds of it. There is skin drag, which is drag produced along the surface of a body by frictional effects, and there is form drag, which is drag produced by pressure forces. In an ideal fluid with no vorticity the lift and drag are both zero. This is not true for real fluids though.
2. Procedure
2.1 Vortices
To demonstrate starting vortices we obtained a small foil and propelled it through a wave tank. We found that using a high angle of attack created a larger starting vortex, which allowed the vortex to be more easily seen. The addition of blue ink at the trailing edge of the foil allowed for better visualisation as well.
2.2 Foils
The apparatus consisted of a small foil, wind tunnel, smoke machine, lift and drag sensor, and a laser. The foil was connected by means of a small rod to the lift and drag sensor. The lift was measured by measuring the upward force transmitted through the rod. The drag force created a torque on the rod which was also measured by the lift drag sensor. The smoke machine nozzle was placed about 10 cm in front of the foil in the wind tunnel. The laser was placed behind the foil and faced towards the foil. The laser illuminated the smoke produced by the smoke machine, and allowed for better visualisation of the flow around the foil. The data collected by the lift and drag sensor was saved to a text file and was then graphed using Logger Pro. In order to visualise the flow the wind tunnel was turned on, with a high flow speed. The laser was turned on and the smoke was then dispensed from the smoke machine. This allowed a nice visualisation of the flow around the foil. We found that reducing the background lighting improved this visualisation, as well as having a dark background. Once the visualization part of the experiment was completed, we performed the lift and drag measurements. This was done by slowly varying the flow speed of the wind tunnel while taking lift and drag data. Once the experiment was completed it was apparent that both lift and drag increased with increasing flow speed. Using data collected in a previous experiment, the lift to drag coefficient of this foil was calculated. This was found to be about 7.6. For comparison, the lift to drag ratio of the Wright flyer was about 8.3. An interesting aspect of the lift to drag ratio is that it is numerically equivalent to the glide ratio, which is the horizontal distance an aircraft could cover for a given loss in altitude. This means that our foil could glide for 7.6 km if it started from an altitude of 1 km. We also investigated the flow around a foil at a high angle of attack. We had to devise another stand so that we could adjust the angle of attack of a foil. In this experiment it was apparent that there was some flow separation at the trailing edge of the foil.
3. Results
3.1 Starting Vortex
The starting vortex experiment worked quite well. One can easily see the vortex shoot off towards the bottom of the page. We were unable to obtain a visualisation of the stopping vortex. We found that increasing the angle of attack and speed of the foil through the water helped in creating a visible vortex. It was observed that the size of the vortex was dependent on the angle of attack and flow speed, and therefore must have been proportional to lift.
3.2 Foils
Note that the streamlines stay in contact with the foil throughout the foil. This means that flow separation did not occur and thus the foil was not past it’s critical angle. This is expected as the angle of the foil appears to be close to horizontal. Lift and drag measurements were made, however without flow speed measurements there are not very many interpretations to be made. During the experiment it was qualitatively observed that the drag was non-zero, and increased with flow speed.
4. Discussion
4.1 Starting Vortex
There were a few things that we could have improved upon in this lab. It would have been nice to develop an apparatus for dragging a foil through a fluid. This would allow us to control the velocity of the foil as well as the angle of attack. With these capabilities we would be able to investigate the strength and size of the starting vortex with varying velocity and angle of attack with a more quantitative approach. With our current experimental setup, we were able to qualitatively observe that the strength of the vortex increased with both angle of attack and flow velocity. It may also be possible to see a stopping vortex if one was able to place a blob of dye at a known stopping point. This would allow us to create an experiment similar to the CFD simulation shown above. It would be interesting to demonstrate that the total vorticity of the tank remains at zero. In our experiment it was observed that the vortex rapidly dissipated with time. This is consistent with the fact that water has a non-zero viscosity, and thus dissipative forces come into play.
4.2 Foils
Another improvement would be to modify the foil in the wind tunnel so that the angle of attack could be adjusted. This would allow us to investigate lift and drag with changing angle of attack. This would also allow us to more easily view flow separation once the angle of attack had been increased past the critical angle. It may be interesting to show that lift decreases after the foil has reached the critical angle of attack. It would also be useful to record the flow speed while lift and drag data are taken during the experiment. This would allow us to show the positive relationship between flow speed with lift and drag. This would also allow us to investigate if $L \propto U^{2}$ for a real fluid flow. In our experiment we were able to observe the streamlines staying in contact with the foil. This was expected as the foil was at a low angle of attack, and unlikely to be anywhere near the critical angle. The lift was qualitatively found to be proportional to flow speed. This is consistent with the Kutta–Zhukhovsky Lift Theorem. The drag was non-zero which was not consistent with the theory for an ideal, non-viscous flow. Because air has some viscosity, there is form drag due to pressure forces as well as skin drag due to frictional forces on the surface of the foil.
5. References
Agromayor, R., Rúa, J., & Kristoffersen, R. (2017, September). Simulation of Starting and Stopping Vortices of an Airfoil. In Proceedings of the 58th Conference on Simulation and Modelling (SIMS 58) Reykjavik, Iceland, September 25th–27th, 2017 (No. 138, pp. 66-75). Linköping University Electronic Press.
Kundu, P. K., & Cohen, I. M. (2001). Fluid mechanics. Elsevier Academic Press.