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Description
The dynamics of a fluid flow inside 2D square and 3D cubic cavities

under various configurations were simulated and analyzed using a

spectral code I developed.

This code was validated against known studies in the 3D lid-driven

cavity. It was then used to explore the various dynamical behaviors

close to the onset

The dynamics of a fluid flow inside 2D square and 3D cubic cavities

under various configurations were simulated and analyzed using a

spectral code I developed.

This code was validated against known studies in the 3D lid-driven

cavity. It was then used to explore the various dynamical behaviors

close to the onset of instability of the steady-state flow, and explain

in the process the mechanism underlying an intermittent bursting

previously observed. A fairly complete bifurcation picture emerged,

using a combination of computational tools such as selective

frequency damping, edge-state tracking and subspace restriction.

The code was then used to investigate the flow in a 2D square cavity

under stable temperature stratification, an idealized version of a lake

with warmer water at the surface compared to the bottom. The governing

equations are the Navier-Stokes equations under the Boussinesq approximation.

Simulations were done over a wide range of parameters of the problem quantifying

the driving velocity at the top (e.g. wind) and the strength of the stratification.

Particular attention was paid to the mechanisms associated with the onset of

instability of the base steady state, and the complex nontrivial dynamics

occurring beyond onset, where the presence of multiple states leads to a

rich spectrum of states, including homoclinic and heteroclinic chaos.

A third configuration investigates the flow dynamics of a fluid in a rapidly

rotating cube subjected to small amplitude modulations. The responses were

quantified by the global helicity and energy measures, and various peak

responses associated to resonances with intrinsic eigenmodes of the cavity

and/or internal retracing beams were clearly identified for the first time.

A novel approach to compute the eigenmodes is also described, making accessible

a whole catalog of these with various properties and dynamics. When the small

amplitude modulation does not align with the rotation axis (precession) we show

that a new set of eigenmodes are primarily excited as the angular velocity

increases, while triadic resonances may occur once the nonlinear regime kicks in.
ContributorsWu, Ke (Author) / Lopez, Juan (Thesis advisor) / Welfert, Bruno (Thesis advisor) / Tang, Wenbo (Committee member) / Platte, Rodrigo (Committee member) / Herrmann, Marcus (Committee member) / Arizona State University (Publisher)
Created2019
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Description
Chebfun is a collection of algorithms and an open-source software system in object-oriented Matlab that extends familiar powerful methods of numerical computation involving numbers to continuous or piecewise-continuous functions. The success of this strategy is based on the mathematical fact that smooth functions can be represented very efficiently by polynomial

Chebfun is a collection of algorithms and an open-source software system in object-oriented Matlab that extends familiar powerful methods of numerical computation involving numbers to continuous or piecewise-continuous functions. The success of this strategy is based on the mathematical fact that smooth functions can be represented very efficiently by polynomial interpolation at Chebyshev points or by trigonometric interpolation at equispaced points for periodic functions. More recently, the system has been extended to handle bivariate functions and vector fields. These two new classes of objects are called Chebfun2 and Chebfun2v, respectively. We will show that Chebfun2 and Chebfun2v, and can be used to accurately and efficiently perform various computations on parametric surfaces in two or three dimensions, including path trajectories and mean and Gaussian curvatures. More advanced surface computations such as mean curvature flows are also explored. This is also the first work to use the newly implemented trigonometric representation, namely Trigfun, for computations on surfaces.
ContributorsPage-Bottorff, Courtney Michelle (Author) / Platte, Rodrigo (Thesis director) / Kostelich, Eric (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2016-05
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Description
Using object-oriented programming in MATLAB, a collection of functions, named Fourfun, has been created to allow quick and accurate approximations of periodic functions with Fourier expansions. To increase efficiency and reduce the number of computations of the Fourier transform, Fourfun automatically determines the number of nodes necessary for representations that

Using object-oriented programming in MATLAB, a collection of functions, named Fourfun, has been created to allow quick and accurate approximations of periodic functions with Fourier expansions. To increase efficiency and reduce the number of computations of the Fourier transform, Fourfun automatically determines the number of nodes necessary for representations that are accurate to close to machine precision. Common MATLAB functions have been overloaded to keep the syntax of the Fourfun class as consistent as possible with the general MATLAB syntax. We show that the system can be used to efficiently solve several differential equations. Comparisons with Chebfun, a similar system based on Chebyshev polynomial approximations, are provided.
ContributorsMcleod, Kristyn Noelle (Author) / Platte, Rodrigo (Thesis director) / Gelb, Anne (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / School of International Letters and Cultures (Contributor)
Created2014-05
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Description
The three-dimensional flow contained in a rapidly rotating circular

split cylinder is studied numerically solving the Navier--Stokes

equations. The cylinder is completely filled with fluid

and is split at the midplane. Three different types of boundary

conditions were imposed, leading to a variety of instabilities and

complex flow dynamics.

The first configuration has a strong

The three-dimensional flow contained in a rapidly rotating circular

split cylinder is studied numerically solving the Navier--Stokes

equations. The cylinder is completely filled with fluid

and is split at the midplane. Three different types of boundary

conditions were imposed, leading to a variety of instabilities and

complex flow dynamics.

The first configuration has a strong background rotation and a small

differential rotation between the two halves. The axisymmetric flow

was first studied identifying boundary layer instabilities which

produce inertial waves under some conditions. Limit cycle states and

quasiperiodic states were found, including some period doubling

bifurcations. Then, a three-dimensional study was conducted

identifying low and high azimuthal wavenumber rotating waves due to

G’ortler and Tollmien–-Schlichting type instabilities. Over most of

the parameter space considered, quasiperiodic states were found where

both types of instabilities were present.

In the second configuration, both cylinder halves are in exact

counter-rotation, producing an O(2) symmetry in the system. The basic state flow dynamic

is dominated by the shear layer created

in the midplane. By changing the speed rotation and the aspect ratio

of the cylinder, the flow loses symmetries in a variety of ways

creating static waves, rotating waves, direction reversing waves and

slow-fast pulsing waves. The bifurcations, including infinite-period

bifurcations, were characterized and the flow dynamics was elucidated.

Additionally, preliminary experimental results for this case are

presented.

In the third set up, with oscillatory boundary conditions, inertial

wave beams were forced imposing a range of frequencies. These beams

emanate from the corner of the cylinder and from the split at the

midplane, leading to destructive/constructive interactions which

produce peaks in vorticity for some specific frequencies. These

frequencies are shown to be associated with the resonant Kelvin

modes. Furthermore, a study of the influence of imposing a phase

difference between the oscillations of the two halves of the cylinder

led to the interesting result that different Kelvin

modes can be excited depending on the phase difference.
ContributorsGutierrez Castillo, Paloma (Author) / Lopez, Juan M. (Thesis advisor) / Herrmann, Marcus (Committee member) / Platte, Rodrigo (Committee member) / Welfert, Bruno (Committee member) / Tang, Wenbo (Committee member) / Arizona State University (Publisher)
Created2017