When an airfoil is accelerated from rest, it creates a region of high pressure around its lower surface and a
region of lower pressure around its upper surface as the fluid is displaced. As a result of the pressure
gradient, fluid will flow up and around the trailing edge of the foil, resulting in a vortex being shed in the
foil’s wake. This vortex is called the starting vortex. Vortices are shed whenever a foil is accelerated,
however, only the vortex shed from a foil accelerating from rest is called a starting vortex.
The existence of these shed vortices is required in order for lift to be generated by an airfoil. The simplest
definition of lift is the net upwards pressure force acting on an object, in other words, it is required that
pressure at the upper surface of the foil be lower than the pressure at the bottom surface in order to generate
lift. This is equivalent to saying that the flow speed above the foil is greater than the flow speed below, as
Bernoulli’s principle states that an increase in flow speed results in a decrease in pressure. The higher flow
speed about the upper surface implies a nonzero circulation about the entire surface of the foil. Circulation
(Γ) is defined as the line integral around a contour enclosing the object of interest:
\oint_C^{}\mathbf{U}\cdot\text{d}\mathbf{S} , where U is the flow field. Since vorticity is a conserved
quantity, in order for there to be circulation about the airfoil, an equal and opposite vortex must be shed.
Therefore, for any increase in lift, circulation must increase and an opposite vortex must be shed.
As per the Kutta-Zhorkovsky lift theorem, lift (a force per unit length perpendicular to airflow) can be
represented as the product of the uniform air density (ρ), free stream air velocity, and clockwise circulation:
\text{L}=\rho\text{U}\Gamma. Kutta-Zhorkovsky's theorem originated from the simple case of a long cylinder in an
inviscid airflow, but its approximation can also mimic the uninterrupted airflow around other objects (such as
an airfoil) with accuracy. A key assumption made in ensuring the flow around an airfoil follows this theorem
involves airflow around the two stagnation points. These are precise zero-velocity points on either side of the
airfoil, from which the air separates between the upper and lower portion of the wing on the incident side, and
rejoins smoothly (as in, with no turbulence) on the trailing side. This smooth flow is referred to as satisfying
the Kutta condition, which brings together important factors of lift generation, such as air traveling
faster over the upper side of the airfoil, and a finite, non-zero circulation of air about the airfoil.
Alternately, when the airflow separates from the airfoil, eliminating the streamline (such as during a high
angle of attack, known as the critical angle), the Kutta condition is no longer satisfied. If it is
unable to re-establish itself as a streamline in this new orientation, lift will no longer be generated.
A rectangular bin was filled with water. A drop of blue dye was placed on the trailing edge of an airfoil. The airfoil was inserted into the water and moved across the length of the bin. The resulting starting vortex was observed with assistance from the dye. The size, speed of rotation and direction of rotation of the starting vortices were compared for varying angles of attack and foil speeds.
An airfoil with a chord of 8.0cm and thickness of 0.9cm was placed into a wind tunnel, just short of the nozzle from a smoke machine. With the wind tunnel on, a red LED was pointed to illuminate the flow of smoke around the airfoil. Head-on, high (~15°), and low (~-35°) angles of attack were recorded and observed.
For a zero angle of attack, it can be seen that a clockwise starting vortex forms. For increasing angles of attack, it can be seen that larger, faster clockwise starting vortices formed. For negative angles of attack, the vortices formed rotate counterclockwise, with greater speed for larger angles of attack.
In this video, we can see that slower, smaller clockwise vortices are formed in the wake of the foil. It is also worth noting that in the slowest example, many small vortices can be seen in the dye drop as a result of small hand motions. This illustrates the fact that any acceleration results in a shed vortex.
During the head-on trial, streamlines were observed to notably attach themselves to the airfoil about the stagnation points, implying that the Kutta condition is satisfied. At a high angle of attack, flow separation appears to occur towards the trailing end, with notable turbulence occurring. This is amplified during the low angle of attack trial, from which very obvious flow separation occurs, and no smoke is seen at the trailing edge of the airfoil.
As was seen in the first starting vortex video, a clockwise vortex was formed in the wake of the cambered
airfoil at a zero angle of attack. This implies that there is a counterclockwise circulation about the foil and
that lift is being generated in the direction of the cambered edge of the foil, illustrating how lift can be
generated at a zero angle of attack. Additionally, when the angle of attack and foil speed is increased, the
starting vortex rotation speed increases, implying that more lift is being generated in these situations.
However, this isn’t always true, as at large angles of attack (>~15°), there will be a stall and lift will be
reduced. The counter clockwise vortices seen in the wake of the foil at a negative angle of attack imply that
lift is being generated in the direction of the flat side of the foil.
Based off of the introduced Kutta-Zhorkovsky lift theorem, streamlined flow from the head-on angle of
attack would result in a lift force accelerating the stationary airfoil upwards. During the high and low angles
of attack, the critical (or stall) angle was clearly exceeded, such that the airfoil would feel no lift due to
the flow separation. As the force sensor was inoperable during this experiment, future analysis would benefit
from these measurements to confirm what forces were felt on the airfoil at various angles of attack and wind
speeds. As well, further experiment could utilize various airfoil shapes (similar to popular shapes used in
aviation, such as symmetrical airfoils used by aerobatic planes).