Author(s): Qadeer, Saad | Advisor(s): Wilkening, Jon A | Abstract: In 1831, Michael Faraday observed the formation of standing waves on the surface of a vibrating fluid body Subsequent experiments have revealed the existence of a rich tapestry of patterned states that can be accessed by varying the frequency and amplitude of the vibration and have spurred vast research in hydrodynamics and pattern formation These include linear analyses to determine the conditions for the onset of the patterns, weakly nonlinear studies to understand pattern selection, and dynamical systems approaches to study mode competition and chaos Recently, there has been some work towards numerical simulations in various three-dimensional geometries These methods however possess low orders of accuracy, making them unsuitable for nonlinear regimesWe present a new technique for fast and accurate simulations of nonlinear Faraday waves in a cylinder Beginning from a viscous potential flow model, we generalize the Transformed Field Expansion to this geometry for finding the highly non-local Dirichlet-to-Neumann operator (DNO) for the Laplace equation A spectral method relying on Zernike polynomials is developed to rapidly compute the bulk potential We prove the effectiveness of representing functions on the unit disc in terms of these polynomials and also show that the DNO algorithm possesses spectral accuracy, unlike a method based on Bessel functions The free surface evolution equations are solved in time using Picard iterations carried out by left-Radau quadrature The results are in perfect agreement with the instability thresholds and surface patterns predicted for the linearized problem The nonlinear simulations reproduce several qualitative features observed experimentally In addition, by enabling one to switch between various nonlinear regimes, the technique allows a precise determination of the mechanisms triggering various experimental observations

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Simulating Nonlinear Faraday Waves on a Cylinder

Parametrically forced gravity waves in axisymmetric mode in a circular cylinder filled with FC-72 with large liquid depth have been studied numerically. The instability threshold and wave breaking thresholds are plotted from the simulated results which show good agreement with the reported experimental and theoretical results. A notable observation is the presence of different time scales of wave amplitude modulations at different regimes. The wave amplitude response exhibits amplitude modulations, period tripling and period quadrupling without breaking of waves. Inertial collapse of the wave trough causes a high velocity jet ejection has also been observed when forcing amplitude crosses the breaking limit.

https://iopscience.iop.org/article/10.1088/1742-6596/822/1/012073/pdf

D Krishnaraja

Papers

Numerical simulation of parametrically forced gravity waves in a circular cylindrical container