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Parametric Forcing of Confined and Stratified Flows

Description

A continuously and stably stratified fluid contained in a square cavity subjected to harmonic body forcing is studied numerically by solving the Navier-Stokes equations under the Boussinesq approximation. Complex dynamics

A continuously and stably stratified fluid contained in a square cavity subjected to harmonic body forcing is studied numerically by solving the Navier-Stokes equations under the Boussinesq approximation. Complex dynamics are observed near the onset of instability of the basic state, which is a flow configuration that is always an exact analytical solution of the governing equations. The instability of the basic state to perturbations is first studied with linear stability analysis (Floquet analysis), revealing a multitude of intersecting synchronous and subharmonic resonance tongues in parameter space. A modal reduction method for determining the locus of basic state instability is also shown, greatly simplifying the computational overhead normally required by a Floquet study. Then, a study of the nonlinear governing equations determines the criticality of the basic state's instability, and ultimately characterizes the dynamics of the lowest order spatial mode by the three discovered codimension-two bifurcation points within the resonance tongue. The rich dynamics include a homoclinic doubling cascade that resembles the logistic map and a multitude of gluing bifurcations.

The numerical techniques and methodologies are first demonstrated on a homogeneous fluid contained within a three-dimensional lid-driven cavity. The edge state technique and linear stability analysis through Arnoldi iteration are used to resolve the complex dynamics of the canonical shear-driven benchmark problem. The techniques here lead to a dynamical description of an instability mechanism, and the work serves as a basis for the remainder of the dissertation.

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Created

Date Created
  • 2019

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Rotating split-cylinder flows

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

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.

Contributors

Agent

Created

Date Created
  • 2017

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Network Based Models of Opinion Formation: Consensus and Beyond

Description

Understanding the evolution of opinions is a delicate task as the dynamics of how one changes their opinion based on their interactions with others are unclear.

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Agent

Created

Date Created
  • 2021