$\textbf{1. Introduction:}$¶
$\textbf{1.1 Starting Vorticies and Flow Separation:}$¶
The circulation (denoted by $\Gamma$) bound to an airfoil dictates the location of stagnation points, where incoming streamlines attatch to and separate from the surface of the airfoil. When an airfoil accelerates from rest, the fluid around it undergo significant changes. As the airfoil gains momentum, the fluid on its lower surface moves toward the stagnation point on the upper surface. In cases where the airfoil possesses a sharp trailing edge, the fluid may lack the energy necessary to fully curve around to the upper stagnation point. This results in a separation of the fluid from the airfoil at the trailing edge forming a starting vortex in the airfoil's wake. Over time, the bound circulation around the airfoil increases enough to push the upper stagnation point to the tip of the trailing edge. Here, the lower surface fluid can separate without significant curvature, halting the generation of circulation ($\Gamma =0$) and thus stopping the formation of vortices in the wake. This crital point is known as the Kutta condition.
The characteristics of starting vortices are a direct consequence of Kelvin’s circulation theorem which states the total circulation in a closed system remains constant. Therefore, the circulation generated around the airfoil's trailing edge necessitates an equal and opposite circulation in the form of a starting vortex. Changing the velocity ($\frac{d\textbf{u}}{dt}$) during the airfoil's movement will consequenctly produce additional starting vorticies during the airfoil's movement. Similarily, when the airfoil comes to a stop (deccelerates) a stopping vortex is generated with opposite circulation to the starting vortex.
$\textbf{1.2 Lift and Drag:}$¶
The presence of these shed vortices is crucial for generating lift on an airfoil. Lift requires a pressure difference between the airfoil's upper and lower surfaces, dictated by Bernoulli’s principle which relates increased flow speed with decreased pressure. This is highlighted in the Kutta-Zhorkovsky lift theorem for irrotational, ideal flows:
$L = \rho U \Gamma$
Where the lift is denoted by $L$, and the fluid density and flow speed are denoted by $\rho$ and $U$, respectively. The circulation around the airfoil is defined as $\Gamma = \oint \textbf{u} \cdot d\textbf{s}$ which can also be expressed as $\Gamma = \oint \textbf{$\omega$} \cdot d\textbf{A}$. This alludes to a squared dependance of the lift on the flow speed, $L \propto U^{2}$.
Angle of attack¶
The angle of attack (AoA) of an airfoil is the relative angle between the chord line of the airfoil with repect to the direction of the of the background flow. The chord line is a common term used in aviation. It is an imaginary straight line that can be drawn between the leading and trailing edge of the airfoil.
Drag¶
The drag force opposes the motion of the airfoil as it moves through the mean flow. There are a couple of types of drag that act on an airfoil. The first being the form drag which is also known as the pressure drag, which arises from pressure differences around the airfoil. As the AoA increases the pressure drag increases. The second component of drag is the induced drag which comes from the lift.
$\textbf{Methods:}$¶
Materials for starting vortex experiment:¶
- 3D printed airfoil
- Glass tank
- Blue dye
- Pipette
First the glass tank was filled with water. Using the pipette, a small amount of the blue dye was deposited in the tank. The dye was used to help visualize the starting vortex. To create the starting vortex, the airfoil was inserted and dragged across the tank with a certain angle of attack. The higher the magnitude of the AoA, the stronger the starting vortex will be. This relationship is demostrated in the following video which can be viewed using this link: https://vimeo.com/924470163?share=copy
Materials for streamline experiment:¶
- Wind tunnel
- Smoke machine
- Red laser
- Adjustable airfoil
First the smoke machine was turned on to allow the temperature to increase. The airfoil in the wind stunnel was mounted onto a support that allowed for adjustment to what the angle of attack was. A laser sitting at the open end of the wind tunnel pointed at the airfoil was used to help visualize the streamlines around the airfoil. The video for the experiment can be viewed using the following link: https://vimeo.com/924495700?share=copy
$\textbf{Discussion:}$¶
Starting vortex in the tank¶
The starting vortex occurs because of flow separation around the airfoil and conservation of circulation. As the airfoil moves through the water it will generate a trapped vortex around it from the circulation of water around the airfoil. The strength of this circulation is relative to the angle of attack. Circulation around the airfoil is created from pressure differences around the airfoil as it moves the medium. Conservation of angular momentum tells us that there should be no net vorticity within the box (tank). If the Kutta condition is met, that would imply that there would be no starting vortex due to the stagnation points being perfectly aligned with the leading edge and the tip of the trailing edge and there is no flow separation. This occurs when the angle of attack is neutral in the tank demonstration. In the video, there is still a bit of circulation generated due to human error but it is clear that there is no distinct starting vortex in the tank. If the $\omega$ < 0 around the airfoil then the starting vortex which gets created will have $\omega$ > 0 meaning it will spin in a counter clockwise motion. The opposite will occur if the trapped vortex have positive vorticity. As shown in the video, if the angle of attack is small then the strength of the starting vortex will also be small. As the angle of attack increases so will the strength of the starting vortex until you hit what is called the stall angle. When the angle of attack reaches the stall angle there will also be flow separation at the leading edge of the airfoil and with the starting vortex off the trailing edge creating two vortices off the airfoil.
Visualizing flow over a cambered airfoil was successfully demonstrated utilizing the low-speed air tunnel, a smoke machine, and a laser incident on the trailing edge. At a neutral angle of attack, the streamlines (flow) were observed to approximately split with half moving over the airfoil and half underneath, appearing to attach themselves to the airfoil. Therefore, the flow was assumed to generally be perpendicular to the mean camber line with a stagnation point assumed to be at the tip of the leading edge. A positive angle of attack resulted in flow separation occurring at the trailing edge. Minor turbulence was observed, but this may have been the result of inhomogeneous smoke clouds passing over the foil. Finally, the negative angle of attack displayed obvious flow separation, with minimal smoke at the trailing edge and a stall vortex resulting from the leading edge.
$\textbf{Tips for next time:}$¶
At first the spout of the smoke machine was pointing into the wind tunnel which was making it very hard to see the streamlines around the airfoil. We ended up turning the the spout to be so that it was directly in line with the airfoil and got better visualization of the streamlines that way. If for some reason, the spout gets moved again then making this small adjustment should make a big impact on your results.
I made the mistake of moving the airfoil a little too fast through the tank when trying to take the videos which made it a little tricky to get nicely formed starting vortices. A tip for next time would be to move the airfoil slower than you would initially think and the tank visualization should come out a little nicer.