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 Title of Thesis Boundary Elements Methods for Compressible Fluid Flow Problems Author(s) Muhammad Mushtaq Institute/University/Department DetailsDepartment of Mathematics / University of Engineering & Technology, Lahore Session2011 SubjectMathematics Number of Pages257 Keywords (Extracted from title, table of contents and abstract of thesis)Methods, System, Compressible, Fluid, Technique, Compressible, Boundary, Flow, Calculation, Problems, Potential, Elements, Dimensional AbstractThe boundary element method is a numerical technique which consists of sub– dividing the boundary of the body into a series of discrete elements, over which the function can vary.This technique offers important advantages over domain type solutions such as finite elements and finite differences.One of the most important features of this method is the much smaller system of equations and the considerable reduction in data required to run a program. Furthermore, the method is well–suited to problems with an infinite domain.Boundary element methods can be formulated using two different approaches called the ‘direct’ and the ‘indirect’ methods. In this thesis, the direct and indirect boundary element methods have been used for the calculation of the compressible fluid flow around arbitrary bodies.The results obtained using boundary element methods have been compared with analytical solutions.These methods have been implemented with the help of computer programming using FORTRAN 77. In chapter (1), statement of the problem, literature survey and brief description of the method of solution are discussed.In chapter (2) the fundamental equations for compressible fluid are given and in chapter (3) mathematical formulation of the equation for inviscid compressible fluid flow is given.In chapter (4) the constitutive equations for the boundary element methods for the interior and exterior domains are derived, and in chapter (5) the various types of boundary elements, the matrix formulation for the equation of the direct method.The numerical evaluation of the integrals over an element and the solution of the resultant system of equations are discussed.In chapter (6) the direct and indirect boundary element methods are applied to the three – dimensional potential flow problems and the comparisons of the analytical and computed results are presented.In chapter (7) both the direct and indirect boundary element methods have been used to calculate two dimensional inviscid compressible flow past a symmetric and Joukowski aerofoils.It has been seen that the computed results in all the above mentioned cases are in good agreement with the analytical results. Finally, conclusions and suggestions for future work are given at the end.
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 S. No. Chapter Title of the Chapters Page Size (KB) 1 0 CONTENTS vii 88 KB 2 1 INTRODUCTION 1.1 Statement of the Problem 1.2 Literature Survey 1.3 Brief Description of the Method of Solution 1 126 KB 3 2 2.1 Introduction 2.2 Equation of Continuity for a Compressible Fluid 2.3 Stress – Strain Rate Relationship for a Newtonian Fluid 2.4 Navier – Stokes’ Equations of Motion for a Compressible Viscous Fluid 2.5 The Energy Equation 2.6 Equation of State for a Perfect Gas 17 120 KB 4 3 3.1 Introduction 3.2 Concepts of Thermodynamics 3.3 Ratio of Specific Heats 3.4 Specific Entropy 3.5 Compression and Expansion of Gases 3.6 Types of Processes 3.7 General Relationships Between Pressure , Specific Volume , and Temperature 3.8 Bernoulli’s Equations for Steady Inviscid Flow 3.9 Energy Equation 3.10 Velocity of Sound 3.11 Mach Number 3.12 Types of Compressible Flows 3.13 Isentropic and Non – Insentropic Inviscid Compressible Flows 3.14 Irrotational Isentropic Flow 3.15 Mathematical Formulation of the Equation for Irrotational Isentropic Flow 24 163 KB 5 4 MATHEMATICAL FORMULATION OF THE EQUATIONS FOR BOUNDARY ELEMENT METHODS 4.1 Introduction 4.2 Mathematical Formulation of the Equation for the Direct Boundary Element Method 4.3 Mathematical Formulation of the Equation for the Indirect Method 43 217 KB 6 5 MATRIX FORMULATION OF THE BOUNDARY INTEGRAL EQUATION 5.1 Introduction 5.2 Types of Boundary Elements 5.3 Matrix Formulation for the Direct Boundary Element Method 5.4 Matrix Formulation for Indirect Boundary Element Method Equation 5.6 Evaluation of Integrals 5.7 Analytical Evaluation of Coefficient of Integrals for Two-Dimensional Case 5.8 Solution of the System of Equations 59 256 KB 7 6 BOUNDARY ELEMENT METHODS APPLIED TO POTENTIAL FLOW PROBLEMS 6.1 Introduction 6.2 Unbounded Flow Field Calculations 6.3 Mathematical Modeling of a Wake 103 598 KB 8 7 BOUNDARY ELEMENT METHODS FOR COMPRESSIBLE FLUID FLOW PROBLEMS 7.1 Introduction 7.2 Theory of Aerofoils 7.3 Symmetric Aerofoil 7.4 Unbounded Flow Field Calculations Using Boundary Element Methods 7.5 Direct Boundary Element Method for Constant Element Approach 7.6 Matrix Formulation for Direct Constant Boundary Element Approach 7.7 Direct Boundary Element Method for Linear Element Approach 7.8 Indirect Boundary Element Method 7.9 Indirect Boundary Element Method for Linear Element Approach 7.10 Joukowski’s Aerofoil 7.11 Flow Past a Joukowski Aerofoil 7.12 Indirect Boundary Element Method 148 517 KB 9 8 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK 241 84 KB 10 9 REFERENCES 243 106 KB