UNIVERSITY OF SOUTHERN QUEENSLAND RADIAL-BASIS-FUNCTION CALCULATIONS OF HEAT AND VISCOUS FLOWS IN MULTIPLY-CONNECTED DOMAINS A dissertation submitted by LÊ CAO KHOA B.), Ho Chi Minh City University of Technology, VietNam, 2003 M.), Ho Chi Minh City University of Technology, VietNam, 2005 For the award of the degree of Doctor of Philosophy February 2011 Dedication To my family. Certification of Dissertation I certify that the idea, experimental work, results and analyses, software and conclusions reported in this dissertation are entirely my own effort, except where otherwise acknowledged. I also certify that the work is original and has not been previously submitted for any other award. KHOA LE-CAO, Candidate Date ENDORSEMENT A/Prof.
NAM MAI-DUY, Principal supervisor Date Prof. THANH TRAN-CONG, Co-supervisor Date Dr. CANH-DUNG TRAN, External supervisor (CSIRO) Date Acknowledgments I would like to express my deepest gratitude to A/Prof. Nam Mai-Duy and Prof.
Thanh Tran-Cong, my principal supervisors, not only for their invalu- able guidance throughout my research but also for their philosophical attitude which has inspired my research interest. Undoubtedly, without their continuing support and encouragement this thesis would not have been completed. In addition, I wish to express my sincere thanks to Dr. Canh-Dung Tran for acting his role as external supervisor, and to Prof.
David Buttsworth and Ms. Juanita Ryan for their kind supports. I gratefully acknowledge the financial support provided by the University of Southern Queensland (USQ) and the Commonwealth Scientific and Industrial Research Organisation (CSIRO), including a USQ Research Excellence scholar- ship, a Faculty of Engineering and Surveying (FoES) scholarship supplement, a Computational Engineering and Science Research Centre (CESRC) scholarship supplement, and a CSIRO Future Manufacturing Flagship Top Up scholarship. Finally, I would like to dedicate this work to my parents.
I am greatly indebted to my family for much unconditional support, understanding and love over the years and for endlessly encouraging me in academic pursuits. Notes to Readers To facilitate the reading of this thesis, a number of files are included on the attached CD to provide animation of some numerical results in this thesis. The contents of the CD include: 1.pdf: An electronic version of this thesis; 2. Chapter3-Circular-Circular-Annuli-velocity.wmv: An animation showing the evolution of velocity field of the buoyancy flow in a concentric circular- circular annulus using a Cartesian grid 36 × 36 (P r = 0.
Chapter3-Square-Circular-Annuli-velocity.wmv: An animation showing the evolution of velocity field of the buoyancy flow in a concentric square- circular annulus using a Cartesian grid 36 × 36 (P r = 0. Chapter4-Rotating-cylinder.wmv: An animation showing the evolution of the flow between a rotating circular cylinder and a fixed square cylinder using a Cartesian grid 26 × 26 (Section 4. Abstract This PhD research project is concerned with the development of accurate and ef- ficient numerical methods, which are based on one-dimensional integrated radial basis function networks (1D-IRBFNs), point collocation and Cartesian grids, for the numerical simulation of heat and viscous flows in multiply-connected domains, and their applications to the numerical prediction of the material properties of suspensions (i. In the proposed techniques, the employment of 1D-IRBFNs, where the RBFN approximations on each grid line are constructed through integration, provides a powerful means of repre- senting the field variables, while the use of Cartesian grids and point collocation provides an efficient way to discretise the governing equations defined on com- plicated domains.
Firstly, 1D-IRBFN-based methods are developed for the simulation of heat transfer problems governed by Poisson equations in multiply-connected do- mains. Derivative boundary conditions are imposed in an exact manner with the help of the integration constants. Secondly, 1D-IRBFN based methods are further developed for the discretisation of the stream-function - vorticity for- mulation and the stream-function formulation governing the motion of a New- tonian fluid in multiply-connected domains. For the stream-function - vorticity formulation, a novel formula for obtaining a computational vorticity bound- ary condition on a curved boundary is proposed and successfully verified.
For the stream-function formulation, double boundary conditions are implemented Abstract v without the need to use external points or to reduce the number of interior nodes for collocating the governing equations. Processes of implementing cross derivatives and deriving the stream-function values on separate boundaries are presented in detail. Thirdly, for a more efficient discretisation, 1D-IRBFNs are incorporated into the domain embedding technique. The multiply-connected domain is transformed into a simply-connected domain, which is more suitable for problems with several unconnected interior moving boundaries.
Finally, 1D- IRBFN-based methods are applied to predict the bulk properties of particulate suspensions under simple shear conditions. All simulated results using Cartesian grids of relatively coarse density agree well with other numerical results available in the literature, which indicates that the proposed discretisation schemes are useful numerical techniques for the analysis of heat and viscous flows in multiply-connected domains. Papers Resulting from the Research Journal Papers 1. Le-Cao and T.
Tran-Cong (2008) A Cartesian grid tech- nique based on one-dimensional integrated radial basis function networks for natural convection in concentric annuli, International Journal for Nu- merical Methods in Fluids, 57, p. Mai Duy and T. Tran-Cong (2009) An effective integrated- RBFN Cartesian-grid discretisation to the stream function-vorticity-temperature formulation in non-rectangular domains, Numerical Heat Transfer, Part B, 55, p. Published Papers Resulting from the Research vii Conference Papers 1.
Mai-Duy and T. Tran-Cong (2007) Radial basis function calculations of buoyancy-driven flow in concentric and eccentric annuli. The 16th Australasian Fluid Mechanics Conference, Gold Coast, QLD, Australia, 3-7 December. Proceedings of The 16th Australasian Fluid Mechanics Conference (CD), p.
The University of Queensland (ISBN 978-1-864998-94-8). Mai-Duy and T. Tran-Cong (2009) Direct simulation of two-dimensional particulate shear flows using radial basis functions. The 5th Australian-Korean Rheology Conference, Sydney, NSW, Australia, 1-4/Nov/2009.
Tran-Cong (2010) A new integrated-RBF-based domain-embedding scheme for solving fluid flow problems. The 9th World Congress on Computational Mechanics and 4th Asian Pacific Congress on Computational Mechanics (WCCM/APCOM 2010), Sydney, Australia, 19-23/Jul/2010. IOP Conference Series: Materials Science and Engineering, Vol. Tran-Cong (2010) Integrated- RBF calculations for direct simulation of shear suspension flows.
Inter- national Conference on Computational & Experimental Engineering and Sciences(ICCES MM’10), Busan, South Korea, 17-21/Aug/2010. IC- CES journal. Tech Science Press (ISSN: 1933-2815 (online)) (accepted, 30/Nov/2010) 5. Mai-Duy and T.
Tran-Cong (2010) Simulation Published Papers Resulting from the Research viii of fluid flows at high Reynolds numbers using radial basis function net- works. 17th Australasian Fluid Mechanics Conference, Auckland, New Zealand, 5-9/Dec/2010. Proceed- ings of 17th Australasian Fluid Mechanics Conference, Paper No 139, 4 pages. The University of Auckland (ISBN: 978-0-86869-129-9).
Contents Dedication ii Certification of Dissertation i Acknowledgments ii Notes to Readers iii Abstract iv Published Papers Resulting from the Research vi Acronyms & Abbreviations xv List of Tables xvi List of Figures xx Chapter 1 Introduction 1 Contents x 1.1 Governing equations and Discretisation methods .2 Viscous flows in multiply-connected domains .4 Outline of the Dissertation. 14 Chapter 2 1D-integrated-RBFN calculation of heat transfer in multiply-connected domains 17 2.1 Review of RBFN-based methods .1 Conventional direct/differential approach .2 Indirect/Integral approach .2 One-dimensional IRBFN method for heat transfer in multiply- connected domains. 40 Contents xi Chapter 3 1D-integrated-RBFN discretisation of stream-function - vorticity (ψ − ω) formulation in multiply-connected domains 42 3.3 The present technique .2 A new formula for computing vorticity boundary conditions 51 3.3 Numerical implementation of vorticity boundary conditions 54 3.1 Example 1: Circular shape domain .2 Example 2: Multiply-connected domain .3 Example 3: Concentric annulus between two circular cylin- ders .4 Example 4: Concentric annulus between a square outer cylinder and a circular inner cylinder. 74 Chapter 4 1D-integrated-RBFN discretisation of stream-function (ψ) formulation in multiply-connected domains 78 4.3 Brief review of 1D-integrated RBFNs .4 Proposed numerical procedure .1 Boundary values for stream function .1 Example 1: Steady flow between a rotating circular cylin- der and a fixed square cylinder .2 Example 2: Natural convection in an eccentric annulus between two circular cylinders .3 Example 3: Natural convection in eccentric annuli be- tween a square outer and a circular inner cylinder.
117 Chapter 5 1D-integrated-RBFN-based domain embedding tech- nique 119 5.2 Proposed domain-embedding technique .1 1D-IRBFN discretisation for extended domain .2 Imposition of the boundary conditions on the inner bound- aries. 139 Chapter 6 1D-integrated-RBFN calculation of particulate sus- pension flows 142 6.2 Governing equations and sliding frames concept .2 Sliding bi-periodic frames concept .3 Proposed numerical procedure .2 Sliding bi-periodic boundary conditions .3 Boundary conditions on the particles’ boundaries .1 Example 1: Sliding bi-periodic boundary conditions .2 Example 2: A rotating circular cylinder .3 Example 3: Shear suspension flow. 185 Acronyms & Abbreviations 1D-IRBFN One-Dimensional Indirect/Integrated Radial Basis Function Network BEM Boundary Element Method CFD Computational Fluid Dynamics DNS Direct Numerical Simulations DRBFN Direct/Differentiated Radial Basis Function Network FDM Finite Difference Method FEM Finite Element Method FVM Finite Volume Method IRBFN Indirect/Integrated Radial Basis Function Network MQ MultiQuadric ODE Ordinary Differential Equation PDE Partial Differential Equation SVD Singular Value Decomposition List of Tables 2.1 Example 1 (boundary value problem - Dirichlet boundary condi- tion - Case 1): Condition numbers of the IRBFN system matrix.2 Example 2 (boundary value problem - Dirichlet and Neumann boundary conditions): Overall accuracy of the solution T by the present technique. Condition numbers of the IRBFN system ma- trix are also included.
It is noted that a(b) represents a × 10b. It is noted that a(b) represents a × 10b .1 Example 1( circular shape domain): Errors by 1D-IRBFN-2s (Scheme 1) and 1D-IRBFN-4s (Scheme 2) in the computation of second derivatives of ψ at the boundary points. It is noted that a(b) represents a × 10b. 62 List of Tables xvii 3.
Condition numbers of the IRBFN system matrix are also included. It is noted that h is the spacing (grid size) and a(b) represents a × 10b .3 Example 2 (multiply-connected domain): Condition numbers of the system matrix and relative L2 errors of the solution. It is noted that h is the spacing (grid size) and a(b) represents a × 10b .4 Example 3 (circular - circular cylinders): Condition numbers of the 1D-IRBFN system matrix by the two formulations.5 Example 3 (circular - circular cylinders): Comparison of the av- erage equivalent conductivity on the inner and outer cylinders, keqi and keqo , between the present IRBFN technique using a grid of 52 × 52 and some other techniques for Ra in the range of 102 to 7 × 104. KG stands for Kuehn and Goldstein .6 Example 4 (square-circular cylinders): Comparison of the aver- age Nusselt number on the outer and inner cylinders, Nuo and Nui , for Ra from 104 to 106 between the present technique (grid 52 × 52) and some other techniques.1 Example 1 (rotating cylinder): Comparison of the stream-function values at the inner cylinder, ψw , for Re from 1 to 1000 between the present technique (grid of 52 × 52) and finite difference tech- nique.2 Condition numbers of the RBFN matrices associated with the harmonic and biharmonic operators.
104 List of Tables xviii 4.3 Example 2 (symmetric flow, concentric circular-circular annuli): Convergence of k̄eq with grid refinement for the flow at Ra = 102 .4 Example 2 (symmetric flow, concentric circular-circular annuli): Convergence of k̄eq with grid refinement for the flow at Ra = 103 .5 Example 2 (symmetric flow, concentric circular-circular annuli): Convergence of k̄eq with grid refinement for the flow at Ra = 3×103 .6 Example 2 (symmetric flow, concentric circular-circular annuli): Convergence of k̄eq with grid refinement for the flow at Ra = 6×103 .7 Example 2 (symmetric flow, concentric circular-circular annuli): Convergence of k̄eq with grid refinement for the flow at Ra = 104 .8 Example 2 (symmetric flow, concentric circular-circular annuli): Convergence of k̄eq with grid refinement for the flow at Ra = 5×104 .9 Example 2 (symmetric flow, concentric circular-circular annuli): Convergence of k̄eq with grid refinement for the flow at Ra = 7×104 .10 Example 2 (symmetric flow, eccentric circular-circular annuli): Comparison of the maximum stream-function values, ψmax , for two special cases ϕ = {−900 , 900} between the present technique and DQM technique.