2021 Virtual Undergraduate Research Symposium
2021 Virtual Undergraduate Research Symposium
Simulating Nonlinear Waves on Vortex Rings in Ideal Fluids
Simulating Nonlinear Waves on Vortex Rings in Ideal Fluids
PROJECT NUMBER: 4 | AUTHOR: Kaleigh Rudge, Applied Mathematics and Statistics
MENTOR: Scott Strong, Applied Mathematics and Statistics
ABSTRACT
Vortex filaments are a fundamental structure in low-temperature fluid dynamics. Understanding their motion and the helical waves on the vortex filament is crucial to understanding how quantum turbulence relaxes in nature. Building these geometric curves can be approached from multiple viewpoints. Using the circular trigonometric functions, we can define helical modes, which are the components of nonlinear waves, on a closed track. Using the hyperbolic trigonometric functions, we can apply the work of Hinedori Hasimoto, which considers a soliton wave on an infinite line that can be spliced onto a closed curve. Finally, using the Jacobi elliptic functions we can build the nonlinear waves on the vortex line while eliminating the piecewise component of the curve, allowing for smoothness. Understanding the curve geometry of these vortex filaments, specifically the curvature, allows us to predict how these vortex filaments travel through space. Once we have built our static model of the vortex line, we use it as our starting condition for dynamical simulations of these lines. These dynamical simulations ultimately give us predictions of how the vortex filaments behave and how quantum liquids dissipate turbulent energy.
PRESENTATION
AUTHOR BIOGRAPHY
Kaleigh is a junior majoring in computational and applied mathematics with a McBride honors minor in public affairs. She currently does research in the AMS department under the guidance of Dr. Scott Strong looking at modeling nonlinear waves on vortex filaments. Her research has been focused on developing a set of mathematical equations that can be used to model the nonlinear waves, while still having control over the geometry of the vortex filaments. Kaleigh hopes to continue this and have more developed dynamical simulations and further explore the partial differential equations that also appear in these vortex filaments.
Hi Kaleigh, fantastic project and poster! Could you explain a bit more on how the torsion and curvature of the vortex line might give intuition into this dynamical model?
Hi Zoe! When we understand the torsion and curvature, at each point along the curve, we can apply it to the Frenet-Serret equation. From there, we can get the normal, binormal, and tangent vectors at each point on the curve. Once we move into the dynamical setting, using the Biot-Savart law, we find that the speed at which the vortex filament travels through space is in the binormal direction and its magnitude is proportional to the curvature. So having knowledge of the curvature and torsion can lead us to understand how the filament is traveling through space.
Thanks, Kaleigh! Could you take a bit of a 10,000 foot view and identify how the limitations of current efforts to understand vortex filaments fall short and the real-world impact of those limitations?
Hi Dylan! The current limitations are in two different spots right now. First, using only the circular trig functions, we are limited to the types of waves we can put on the curve, as they are a specific case of the elliptic functions, and do not allow the waves we build to have hyperbolic properties on the filament. The second limitation is when we use the hyperbolic soliton on the filament (work by Hasimoto). With this one, we have to use a piecewise function to get the soliton connected to the rest of the track to build the filament, but Hasimoto’s soliton is designed as an infinite line, where our track is circular. This means that even if we can match the endpoints, we can still experience if the tangent, normal and binormal vectors do not all match up, which has been a challenge. These limitations have made it difficult for us to understand the curvature and binormal directions of the filaments, which is inhibiting us from fully understanding how they dynamically evolve. In terms of the real-world impact beyond this, I am not 100% sure how much these limitations impact the real-world applications at the moment.
Nice work, Kaleigh! What are your plans for the next steps here?
Thank you! Currently, the next steps are working on reworking Hasimoto’s soliton to make it in terms of the elliptic functions and run our dynamical simulations through that model (which would be a more realistic fit). Additionally, this problem of understanding the nonlinear waves on the vortex filaments can be approached through solving PDEs, specifically Schrodinger’s equation. Now that I have learned a bit more and gained a foundation in PDEs, our plan is to also shift some of our focus to working through that approach.