Author: Alex Wuschner
Mentor: Dr. Beverley McKeon
Co-Mentor: Maysam Shamai
Editor: Sophie Ding
Introduction: For many years, scientists and engineers have studied the phenomenon of turbulence intensely, aiming to learn how it’s governed, how it can be predicted, and how we can help effectively engineer the world with it in mind. In the NOAH Water Tunnel Laboratory, the newly installed Captive Trajectory System (CTS) has been utilized to study just that. Over the course of the summer of 2017, the capabilities of this system were explored and demonstrated through examples of increasing complexity. As a programmable cyber-mechanical machine capable of moving and rotating attached objects within the NOAH Water Tunnel, by the end of the summer the CTS was used to study how airfoils moved and behaved when subject to a turbulent phenomenon known as vortex shedding. Vortex shedding, a turbulent phenomenon characterized by a line of vortices alternating in orientation, is depicted in Figure 1 below.
Vortex shedding is a common phenomenon that is easily reproduced, such as by flowing water at a high enough speed past a cylinder. After observing how an airfoil’s motion was influenced by vortex shedding by allowing it to move freely in the water tunnel, the CTS was programmed in attempts to replicate the observed motion. By closely replicating how the CTS moved to the observed motion, the principles that guided the turbulent interactions with the airfoil could be further understood.
Since turbulence has many seemingly unpredictable properties, the CTS was programmed not in a way that would repeat the same exact path of motion with each trial, but rather in a way that would respond and react in approximately real time to the forces acting on the airfoil due to its interaction with the stream of vortices. This method of capturing an object’s path of motion, known as a captive trajectory, was implemented in multiple “example cases” to replicate situations, such as a object moving while attached to a spring or an object orbiting another according to the laws of gravity, as a means of proving the accuracy and precision with which the CTS was capable of operating. Using a force sensor and gathering position and velocity data from the CTS at very small time steps (about 200 Hz), the CTS would constantly adapt its trajectory in response to the small changes in the state of the system. Using these measurements, small time steps, and any other factors implemented in the programming of the CTS, enabling the CTS to produce captive trajectories allowed us to accurately mimic the type of motion predicted by theory or observation.
Developing a Mass-Spring-Damper Captive Trajectory: Before its application to complex, multi-dimensional systems, the CTS was first used to model simpler, well-understood physical systems. The first captive trajectory developed modeled a mass-spring-damper system restricted to one degree of freedom. Since this model is well understood, the response of the CTS was compared to theoretical expectations to assess the captive trajectory’s accuracy.
Both the measured applied forces as well as virtual forces were included in the design of this captive trajectory. Taking into account a restoring spring force proportional to the displacement from equilibrium and a damping force proportional to the velocity, the equation governing the motion of the CTS due to virtual forces is given as:
mx”= -kx-bx’ (1)
in which k and b are virtually defined coefficients, and m is the virtually defined mass of the test body.
The measured applied forces were received directly from the force sensor. Including the measured forces, F(t), the overall equation governing the motion of the CTS was:
mx”= -kx-bx’+F(t) (2)
An explicit Euler method was used to calculate the subsequent position and velocity at each time-step.
After collecting and analyzing the trajectory data, as displayed in Figure 2, it was clear that the amplitudes and periods of the oscillation matched those predicted by theory very well.
Aside from its good agreement with theoretical results, this first implementation of a captive trajectory also served to verify the responsiveness of the CTS. For the more complex experiments the CTS will eventually be used for, hydrodynamic forces are spatially and temporally dependent.
Observing Figure 3, a comparison of the commanded position and velocity with the actuated position and velocity, it can be seen that the differences between the commanded and implemented outputs were very small, and thus the CTS proved it could closely match the trajectory it was simulating.
Developing a Gravitational Model Captive Trajectory: Once the mass-spring-damper trajectory was successfully designed and implemented, the next step involved the implementation of a captive trajectory with multiple degrees of freedom. As a result of the previously described benefits of modelling a well-known system, a gravitational model was chosen as the type of motion that this captive trajectory would replicate. After defining a large, virtual, stationary mass located at the approximate center of the tunnel, the CTS was programmed to govern the motion of a smaller mass elsewhere in the coordinate system based on gravitational forces.
In the absence of measured forces, the equation governing the virtual force used in the CTS is given by:
in which M is the virtually defined mass of the central stationary body, m is the virtually defined mass of the test body, r is the distance between the two, and k is a constant.
The virtually defined forces separated into components are given as:
Each of these components are numerically integrated to yield the commanded velocity and position in each direction.
After designing the trajectory, it was implemented on the CTS in the form of a captive trajectory program. The initial conditions of the trajectory could be varied to produce circular, elliptical, and hyperbolic orbits, as predicted by theory. Examples of the orbits implemented on the CTS are shown below in Figure 4.
When the trajectories were manually manipulated by forces applied during the middle of the trajectory, the CTS immediately responded to both the direction and magnitude of the applied force. In Figure 5, an example is shown in which initial conditions were set defining a circular trajectory, but after a couple orbits a force was applied that pushed the CTS into an elliptical orbit.
Like the mass-spring-damper trajectory, the gravitational trajectory demonstrated the system’s potential to quickly and accurately respond to changes in both measured and virtual forces.
Modelling Unsteady Aerodynamics: During the spring of 2017, Morgan Hooper and Ben Barthel used the NOAH Laboratory to observe the free-response of an airfoil in the wake of a cylinder with vortex shedding.4 While vortex shedding has been heavily documented, its interactions with airfoils has not. Thus, it was considered an interesting case to study, and would serve as a way to compare the behavior of the CTS to a corresponding physical system.
The experimental setup allowed for the observation of a NACA 0018 airfoil that was free to exhibit heaving motion along the spanwise axis as well as pitching motion about the vertical z-axis.4 All other degrees of freedom were restricted. Due to reasons described in Hooper and Barthel (2017), the experimental setup was slightly revised during the current study to reduce friction by using a second linear motion cart, improved rotary bearing, and other minor improvements. A Sharp GP2Y0A21YK0F IR Distance sensor, and a Vishay HE-351 Hall-Effect Rotary Encoder were used to measure the translation and pitch of the airfoil during experiments.4 The free-response airfoil assembly is shown in Figure 6.
A 10.1 cm diameter PVC cylinder was placed upstream of the airfoil in the center of the test section in order to generate vortices. The airfoil mounting assembly was placed downstream, and for each run the airfoil was set to start at the center of the test sections with no angle of attack relative to the flow. The experimental setup is shown in Figure 7. The free-response of the airfoil was studied at multiple tunnel speeds.
A combination of applied and virtually defined forces were considered very carefully when designing the captive trajectory program. The attempts to simulate this type of motion differed from those of the mass-spring-damper and the gravitational trajectories in multiple ways, the most important of which being that in this captive trajectory, forces were measured on an actual test object. In order to do so, a NACA 0018 airfoil was mounted to the CTS, and remained inside the water tunnel during the course of the trajectory. This addition required the consideration of multiple other factors when attempting to predict the airfoil’s motion. A diagram of the mounted airfoil assembly is shown in Figure 8.
In the captive trajectory model, hydrodynamic forces were assumed to be the mechanism controlling the airfoil’s motion. In the force sensor coordinate system, forces measured in both the x and y-axes, as shown in the following equation:
As would be intuitively expected, larger masses and moments of inertia responded to changing hydrodynamic forces in a less dramatic manner than tests with a smaller virtual mass and moment of inertia.
For the majority of parameters, the captive trajectory program produced a response that shared many similarities to that of the free-response setup. Oscillations consistent with the frequency of the vortex shedding were observed in both the heaving motion along the y-axis and pitching motion. The relationships between the heaving and pitching oscillations showed similarities to the pattern observed in the experimental setup. As the mass and moment of inertia were lowered to the values of the real parameters, the trajectories became more and more similar to those observed in the experimental trials. Figure 9 shows a plot of the airfoil’s heaving and pitching position.
As the virtual parameters were reduced even further to approach the actual values, the CTS was no longer able to predict and actuate the correct trajectory of the airfoil. This was determined to be a caused by the physical, not virtual, restraints of the CTS and the force sensor.
When force data from the sensor is observed, it becomes evident that given the typical forces produced by the vortices on the airfoil, no vortices produce a force that would accelerate the airfoil beyond the acceleration bounds of the CTS’ motors. However, as noted previously, the force sensor is susceptible to noise, and it seems that this sensor noise is most likely the cause of the unrealistically large accelerations.
Before each trajectory began, a delay time was included in which the force sensor was able to read initial forces on the force sensor. At this time, the airfoil was in its starting position and the forces being read were those of the alternating vortices being shed on the stationary airfoil.
Figure 10: The pitching torques read on the airfoil before and after the initiation of the trajectory at about t = 65 seconds. Initially, as the stationary airfoil experiences vortex shedding, the force sensor picks up measurements with little relative noise. As the trajectory initiated and the airfoil began to move, the noise significantly amplified measurements despite the vortices being of the same strength as before.
While the airfoil was stationary, the noise in the force sensor data was noticeable, but relatively insignificant compared to the overall fluctuation of forces with each passing vortex. However, as the airfoil began its trajectory, the level of noise read by the force sensor dramatically increased to the point at which the noise amplitude surpassed force fluctuations produced by the vortices, leading to choppy movement of the airfoil. This can be seen in Figure 10.
It is evident that the increased noise stemming from the choppy movement of the airfoil caused the captive trajectory program to predict acceleration values that exceed acceleration constraints.
As can be observed from its motion, the airfoil vibrates throughout its trajectory. This is likely due to the noise in the force sensor: the small fluctuations in the force readings are directly translated to fluctuations in acceleration, which can produce a shaky type of motion. In order to solve this issue, the logical next step was to attempt to filter out the force sensor noise so that the force readings translating into motion would appear much smoother. A low pass filter of the force sensor readings was implemented, as this type of filter is effective in removing the effects of the small, random fluctuations of the sensor readings.
Multiple considerations were accounted for when choosing a filter. As mentioned before, the filter needed to be a low pass filter and it also needed to have minimal phase lag, otherwise known as the time delay between changes in the actual forces and changes in force measurement readings. If the phase lag of the filter is too large, the airfoil will not immediately respond to the actual forces (as the force readings lag in time) and will produce a trajectory that fails to model the system.
To assess the effect of a filter on the airfoil vibrations, a simple moving average filter was implemented in the captive trajectory program. Figure 11 shows the changes in the raw force sensor readings as the size of the filter was increased to average over a greater number of past force values.
As the filter size increases, the force sensor readings show that the vibration is noticeably decreased. However, for large filter sizes, the airfoil motion noticeably lags behind the varying forces caused by the vortices, and thus this simple moving average filter is not sufficient. Future work must include the development of a low pass filter that is able to effectively reduce the system vibration without introducing significant phase lag.
Conclusions: The progress made on the development of captive trajectories in the NOAH Laboratory has laid a foundation with which the CTS may be utilized as a tool for the study of free-response aerodynamic behavior. The mass-spring-damper and the gravitational models successfully proved the CTS’ ability to respond to forces in real-time to produce captive trajectories. While the captive trajectory modelling the airfoil in the wake of a cylinder with vortex shedding is still being refined, it has also provided a lot of insight into the complexities involved in attempting to replicate a system without an analytical solution to describe the motion of the model. Future work will be directed towards the design of a low pass filter to properly filter out the high frequency, zero mean noise from the force sensor while minimizing phase lag. This will allow the CTS to better simulate systems with lower virtual mass or large force fluctuations.
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Acknowledgments: The opportunity to conduct the research presented in this paper was made possible by Dr. Beverley McKeon and the McKeon Research Group in GALCIT as a part of the Caltech SURF Program. A special thanks goes to Maysam Shamai for guidance and assistance throughout the project, as well as to Morgan Hooper and Ben Barthel for their contributions to the design of the free-response experimental setup in NOAH Water Tunnel. I would also like to thank Will Dickson for his contributions towards the design and construction of the CTS, as well as his assistance throughout the various stages of the project. Final appreciation goes to the Caltech Associates for contributing towards the funding of this research and the opportunity this project has allowed me.