## Date of Award

2009

## Degree Type

Thesis

## Degree Name

Doctor of Philosophy

## Program

Mechanical and Materials Engineering

## Supervisor

Prof. J.M. Floryan

## Abstract

This dissertation presents implicit spectrally-accurate algorithms based on the concept of immersed boundary conditions (IBC) for solving a range of computational fluid dynamics (CFD) problems where the physical domains involve boundary irregularities. Both fixed and moving irregularities are considered with particular emphasis placed on the two-dimensional moving boundary problems. The physical model problems considered are comprised of the Laplace operator, the biharmonic operator and the Navier-Stokes equations, and thus cover the most commonly encountered types of operators in CFD analyses. The IBC algorithm uses a fixed and regular computational domain with flow domain immersed inside the computational domain. Boundary conditions along the edges of the time-dependent flow domain enter the algorithm in the form of internal constraints. Spectral spatial discretization for two-dimensional problems is based on Fourier expansions in the stream-wise direction and Chebyshev expansions in the normal-to-the-wall direction. Up to fourth-order implicit temporal discretization methods have been implemented. The IBC algorithm is shown to deliver the theoretically predicted accuracy in both time and space.

Construction of the boundary constraints in the IBC algorithm provides degrees of freedom in excess of that required to formulate a closed system of algebraic equations. The ‘classical IBC formulation’ works by retaining number boundary constraints that are just sufficient to form a closed system of equations. The use of additional boundary constraints leads to the ‘over-determined formulation’ of the IBC algorithm. Over- determined systems are explored in order to improve the accuracy o f the IBC method and to expand its applicability to more extreme geometries. Standard direct over-determined solvers based on evaluation of pseudo-inverses of the complete coefficient matrices have been tested on three model problems, namely, the Laplace equation, the biharmonic equation and the Navier-Stokes equations. In all cases tested the over-determined

formulations based on standard solvers were found to improve the accuracy and the range o f applicability o f the IBC method.

Efficient linear solvers suitable for the spectral implementation of the IBC method have been developed and tested in the context of two-dimensional steady and unsteady Stokes flow in the presence of fixed boundary irregularities. These solvers can work with the classical as well as the over-determined formulations of the method. Significant acceleration of the computations as well as significant reduction of the memory requirements have been accomplished by taking advantage of the structure of the coefficient matrix resulting from the implementation of the IBC algorithm. Performances of the new solvers have been compared with the standard direct solvers and are shown to be of up to two orders of magnitude better. It has been determined that the new methods are by at least an order of magnitude faster than the iterative methods while removing restrictions based on the convergence criteria and thus expanding the severity of the geometries that can be dealt with using the IBC algorithm. The performance of the IBC method combined with the new solvers has been compared with the performance of a method based on the generation of the boundary conforming grids, and is found to be better by at least two orders of magnitude. Application of the new solvers to the unsteady problems also results in performance improvement of up to two orders of magnitude. The specialized solvers applied to the over-determined formulation is shown to be at least two orders of magnitude faster than their standard counterparts while capable of extending the range of applicability of the IBC algorithm by 50%-70% for the Stokes flow problem. The concept of the specialized solvers has been extended to solve two-dimensional moving boundary problems described by the Navier-Stokes equations, where the new solver has been shown to result in a significant acceleration of computations as well as substantial reduction in memory requirements.

The conceptual aspects of extending the IBC algorithm for solving three-dimensional problems have been presented using the vorticity-velocity formulation of the three- dimensional Navier-Stokes equations. Test results on the implementation of the IBC algorithm for three-dimensional problems are discussed in the context of heat diffusion

IV

problems in the presence of fixed as well as moving boundaries. The algorithm is shown to be spectrally-accurate in space and capable of delivering theoretically predicted accuracy in time for the different test problems. Given a potentially large size of the resultant linear algebraic system, various methods that take advantage of the special structure of the coefficient matrix have been explored in search for an efficient solver, including two versions of the specialized direct solver as well as serial and parallel iterative solvers. Both versions of the specialized direct solver have been shown to be more computationally efficient than the other solution methods.

Possible applications of the IBC algorithm for analyzing physical problems have also been presented. The advantage of using IBC algorithm is illustrated by considering its application to two physical problems, which are - i) analysis of the effects of distributed roughness on friction factor and ii) analysis of traveling wave instability in wavy channels. These examples clearly show the attractiveness of the IBC algorithm for studying effects of a large array of boundary geometries on the flow field.

## Recommended Citation

Husain, Syed Zahid, "IMMERSED BOUNDARY CONDITIONS METHOD FOR COMPUTATIONAL FLUID DYNAMICS PROBLEMS" (2009). *Digitized Theses*. 4230.

https://ir.lib.uwo.ca/digitizedtheses/4230