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.
This project is a synthesis of the author's learning over the semesters in working with the CFD Group at Arizona State University. The incompressible Navier-Stokes equations are overviewed, starting with the derivation from the continuity equation, then non-dimensionalization, methods of solving and computing quantities of interest. The rest of this document is expository analysis of solutions in a confined fluid flow, building toward a parametrically forced regime that generates complex flow patterns including Faraday waves. The solutions come from recently published studies Dynamics in a stably stratified tilted square cavity (Grayer et al.) and Parametric instabilities of a stratified shear layer (Buchta et al).
The transition to three-dimensional and unsteady flow in an annulus with a discrete heat source on the inner cylinder is studied numerically. For large applied heat flux through the heater (large Grashof number Gr), there is a strong wall plume originating at the heater that reaches the top and forms a large scale axisymmetric wavy structure along the top. For Gr ≈ 6 × 109, this wavy structure becomes unstable to three-dimensional instabilities with high azimuthal wavenumbers m ∼ 30, influenced by mode competition within an Eckhaus band of wavenumbers. Coexisting with some of these steady three-dimensional states, solution branches with localized defects break parity and result in spatio-temporal dynamics. We have identified two such time dependent states. One is a limit cycle that while breaking spatial parity, retains spatio-temporal parity. The other branch corresponds to quasi-periodic states that have globally broken parity.
The dissertation starts with a study of a volumetric expansion driven drainage flow of a viscous compressible fluid from a small capillary and channel in the low Mach number limit. An analysis based on the linearized compressible Navier-Stokes equations with no-slip condition shows that fluid drainage is controlled by the slow decay of the acoustic wave inside the capillary and the no-slip flow exhibits a slip-like mass flow rate. Numerical simulations are also carried out for drainage from a small capillary to a reservoir or a contraction of finite size. By allowing the density wave to escape the capillary, two wave leakage mechanisms are identified, which are dependent on the capillary length to radius ratio, reservoir size and acoustic Reynolds number. Empirical functions are generated for an effective diffusive coefficient which allows simple calculations of the drainage rate using a diffusion model without the presence of the reservoir or contraction.
In the second part of the dissertation, steady viscous compressible flow through a micro-conduit is studied using compressible Navier-Stokes equations with no-slip condition. The mathematical theory of Klainerman and Majda for low Mach number flow is employed to derive asymptotic equations in the limit of small Mach number. The overall flow, a combination of the Hagen-Poiseuille flow and a diffusive velocity shows a slip-like mass flow rate even through the overall velocity satisfies the no-slip condition. The result indicates that the classical formulation includes self-diffusion effect and it embeds the Extended Navier-Stokes equation theory (ENSE) without the need of introducing additional constitutive hypothesis or assuming slip on the boundary. Contrary to most ENSE publications, the predicted mass flow rate is still significantly below the measured data based on an extensive comparison with thirty-five experiments.
In this research, computational methods were developed to accurately simulate phase interfaces in compressible fluid flows with a focus on targeting primary atomization. Novel numerical methods which treat the phase interface as a discontinuity, and as a smeared region were developed using low-dissipation, high-order schemes. The resulting methods account for the effects of compressibility, surface tension and viscosity. To aid with the varying length scales and high-resolution requirements found in atomization applications, an adaptive mesh refinement (AMR) framework is used to provide high-resolution only in regions of interest. The developed methods were verified with test cases involving strong shocks, high density ratios, surface tension effects and jumps in the equations of state, in one-, two- and three dimensions, obtaining good agreement with theoretical and experimental results. An application case of the primary atomization of a liquid jet injected into a Mach 2 supersonic crossflow of air is performed with the methods developed.
The library is written to let the user determine where to refine and coarsen through custom refinement selector functions for static mesh generation and dynamic mesh refinement, and can handle smooth fields (such as level sets) or localized markers (e.g. density gradients). The library was parallelized with the use of the Zoltan graph-partitioning library, which provides interfaces to both a graph partitioner (PT-Scotch) and a partitioner based on Hilbert space-filling curves. The partitioned adjacency graph, mesh data, and solution variable data is then packed and distributed across all MPI ranks in the simulation, which then regenerate the mesh, generate domain decomposition ghost cells, and create communication caches.
Scalability runs were performed using a Leveque wave propagation scheme for solving the Euler equations. The results of simulations on up to 1536 cores indicate that the parallel performance is highly dependent on the graph partitioner being used, and differences between the partitioners were analyzed. FARCOM is found to have better performance if each MPI rank has more than 60,000 cells.