Nonlinear fluid dynamics in complex geometries

A high-order spectral element method was developed to solve the 2D Navier-Stokes equations in convoluted geometries. The method is based on the hybridizable discontinuous Galerkin (HDG) method of Nguyen, et al., 2011 combined with spectral elements and isoparametric mapping for the geometry. The HDG method is further combined with a Rosenbrock time-stepping algorithm for unsteady problems, yielding high order spatial and temporal accuracy. The discretized incompressible Navier-Stokes equations, combined with the continuity equation, yield a system of differential algebraic equations (DAE) which are efficiently solved with the Rosenbrock algorithm given an appropriate set of stage coefficients attuned to the index of the DAE system. Computational performance data are presented demonstrating hp spectral convergence in spatial resolution and 3rd order convergence in time discretization. Scaling analysis and timing data are presented to select optimal parameters for computational efficiency with the three major components of CPU time including (1) a global sparse linear solve, (2) local elemental inverses, and (3) elemental back substitution for local variables. Nonlinear flows in highly convoluted geometries are relevant to applications in a wide array of chemical engineering problems; examples include flows in packed beds and other porous media, microfluidics, and meandering river dynamics. Applying isoparametric mapping to a spectral element mesh provides a high order of accuracy and stability without the need for lengthy re-gridding protocols as with standard finite element meshes. Furthermore, with the local solver based on the HDG method of Nguyen, the large coupled system is broken into global and local subproblems which allow local variables to be solved independently and efficiently on each element. Numerous examples are presented for both steady state and unsteady Navier-Stokes flows over a range of highly distorted flow channel geometries which exhibit many flow characteristics of interest. We observe separated flows and multiple steady states as well as oscillatory driving functions with possibly non-oscillatory system dynamics. We also observe departure from quasi-steady behavior with different flow configurations at given system states (Reynolds numbers) which differ from steady state results. This combined isoparametric mapped HDG-Rosenbrock platform is appropriate for extension to many areas of interest to chemical engineers, including microfluidics, empirical turbulent flow models, and non-Newtonian flows.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Chemical Engineering

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2023

Author dc:creator

• Biagioli, Madeleine George

Contributors dc:contributor

• Higdon, Jonathan J.L.
• Rao, Christopher V
• Peters, Baron G
• Kuzan, John D

• Fluid Dynamics
• Hybridizable Discontinuous Galerkin Method
• Hdg
• High Performance Computation
• Rosenbrock
• Navier-stokes
• Convection
• Reynolds Number
• Convective Diffusion
• Complex Geometries
• Extended Domains
• Isoparametric Mapping
• Nonlinear Flow
• Nonlinear Differential Equations
• Differential Algebraic Equations
• Discontinuous Galerkin
• Galerkin
• Spectral Elements
• High Order

Statement dc:rights

• © 2023 Madeleine George Biagioli

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/120197

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F120197