BASICS OF FLUID MECHANICS AND INTRODUCTION TO COMPUTATIONAL FLUID DYNAMICS

Transcription

1 BASICS OF FLUID MECHANICS AND INTRODUCTION TO COMPUTATIONAL FLUID DYNAMICS

2 Numerical Methods and Algorithms Volume 3 Series Editor: Claude Brezinski Université des Sciences et Technologies de Lille, France

3 BASICS OF FLUID MECHANICS AND INTRODUCTION TO COMPUTATIONAL FLUID DYNAMICS by TITUS PETRILA Babes-Bolyai University, Cluj-Napoca, Romania DAMIAN TRIF Babes-Bolyai University, Cluj-Napoca, Romania Springer Library of Congress Cataloging-in-Publication Data

4 ebook ISBN: Print ISBN: Springer Science + Business Media, Inc. Print 2005 Springer Science + Business Media, Inc. Boston All rights reserved No part of this ebook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Springer's ebookstore at: and the Springer Global Website Online at:

5 Contents Preface 1. INTRODUCTION TO MECHANICS OF CONTINUA 1 Kinematics of Continua 1.1 The Concept of a Deformable Continuum 1.2 Motion of a Continuum. Lagrangian and Eulerian Coordinates 1.3 Euler Lagrange Criterion. Euler s and Reynolds (Transport) Theorems 2 General Principles. The Stress Tensor and Cauchy s Fundamental Results 2.1 The Forces Acting on a Continuum 2.2 Principle of Mass Conservation. The Continuity Equation 2.3 Principle of the Momentum Torsor Variation. The Balance Equations Constitutive Laws. Inviscid and real fluids 3.1 Introductory Notions of Thermodynamics. First and Second Law of Thermodynamics The Cauchy Stress Tensor The Cauchy Motion Equations Principle of Energy Variation. Conservation of Energy General Conservation Principle xiii Constitutive (Behaviour, Stresses-Deformations Relations) Laws 32 Inviscid (Ideal) Fluids 34 Real Fluids

6 vi 3.5 Shock Waves The Unique Form of the Fluid Equations DYNAMICS OF INVISCID FLUIDS Vorticity and Circulation for Inviscid Fluids. The Bernoulli Theorems Some Simple Existence and Uniqueness Results Irrotational Flows of Incompressible Inviscid Fluids. The Plane Case Conformal Mapping and its Applications within Plane Hydrodynamics 4.1 Helmholtz Instability Principles of the (Wing) Profiles Theory 5.1 Flow Past a (Wing) Profile for an Incidence and a Circulation a priori Given Profiles with Sharp Trailing Edge. Joukovski Hypothesis Theory of Joukovski Type Profiles Example An Iterative Method for Numerical Generation of Conformal Mapping Panel Methods for Incompressible Flow of Inviscid Fluid 6.1 The Source Panel Method for Non-Lifting Flows Over Arbitrary Two-Dimensional Bodies 6.2 The Vortex Panel Method for Lifting Flows Over Arbitrary Two-Dimensional Bodies 6.3 Example Almost Potential Fluid Flow Thin Profile Theory Mathematical Formulation of the Problem Solution Determination Unsteady Irrotational Flows Generated by the Motion of a Body in an Inviscid Incompressible Fluid 9.1 The 2-Dimensional (Plane) Case 9.2 The Determination of the Fluid Flow Induced by the Motion of an Obstacle in the Fluid. The Case of the Circular Cylinder 9.3 The 3-Dimensional Case

7 Contents vii 9.4 General Method for Determining of the Fluid Flow Induced by the Displacement of an Arbitrary System of Profiles Embedded in the Fluid in the Presence of an A Priori Given Basic Flow 10 Notions on the Steady Compressible Barotropic Flows Immediate Consequences of the Bernoulli Theorem The Equation of Velocity Potential (Steichen) Prandtl Meyer (Simple Wave) Flow Quasi-Uniform Steady Plane Flows General Formulation of the Linearized Theory Far Field (Infinity) Conditions The Slip-Condition on the Obstacle The Similitude of the Linearized Flows. The Glauert Prandtl Rule Mach Lines. Weak Discontinuity Surfaces Direct and Hodograph Methods for the Study of the Compressible Inviscid Fluid Equations A Direct Method [115] Chaplygin Hodograph Method. Molenbroek Chaplygin equation VISCOUS INCOMPRESSIBLE FLUID DYNAMICS The Equation of Vorticity (Rotation) and the Circulation Variation Some Existence and Uniqueness Results The Stokes System Equivalent Formulations for the Navier Stokes Equations in Primitive Variables Pressure Formulation 4.2 Pressure-Velocity Formulation Equivalent Formulations for the Navier Stokes Equations in Non-Primitive Variables Navier Stokes Equations in Orthogonal Generalized Coordinates. Stream Function Formulation A Coupled Formulation in Vorticity and Stream Function The Separated (Uncoupled) Formulation in Vorticity and Stream Function 152

8 viii 5.4 An Integro-Differential Formulation Similarity of the Viscous Incompressible Fluid Flows The Steady Flows Case Flows With Low Reynolds Number. Stokes Theory The Oseen Model in the Case of the Flows Past a Thin Profile Flows With High (Large) Reynolds Number Mathematical Model The Boundary Layer Equations Probabilistic Algorithm for the Prandtl Equations Example Dynamic Boundary Layer with Sliding on a Plane Plaque INTRODUCTION TO NUMERICAL SOLUTIONS FOR ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS Introduction Discretization of a Simple Equation Using the Finite Difference Method Using the Finite Element Method Using the Finite Volume Method Comparison of the Discretization Techniques The Cauchy Problem for Ordinary Differential Equations 3.1 Examples 4 Partial Differential Equations 4.1 Classification of Partial Differential Equations The Behaviour of Different Types of PDE Burgers Equation Stokes Problem The Navier Stokes System FINITE-DIFFERENCE METHODS Boundary Value Problems for Ordinary Differential Equations Supersonic Flow Past a Circular Cylindrical Airfoil Discretization of the Partial Differential Equations The Linear Advection Equation Discretization of the Linear Advection Equation 257

9 Contents ix 3.2 Numerical Dispersion and Numerical Diffusion Lax, Lax Wendroff and MacCormack Methods Diffusion Equation Forward-Time Scheme Centered-Time Scheme Backward-Time Scheme Increasing the Scheme s Accuracy Numerical Example Burgers Equation Without Shock Lax Scheme Leap-Frog Scheme Lax Wendroff Scheme Hyperbolic Equations Discretization of Hyperbolic Equations Discretization in the Presence of a Shock Method of Characteristics Elliptic Equations Iterative Methods Direct Method Transonic Flows Stokes Problem Compact Finite Differences The Compact Finite Differences Method (CFDM) Approximation of the Derivatives Fourier Analysis of the Errors Combined Compact Differences Schemes Supercompact Difference Schemes Coordinate Transformation Coordinate Stretching Boundary-Fitted Coordinate Systems Adaptive Grids FINITE ELEMENT AND BOUNDARY ELEMENT METHODS Finite Element Method (FEM) Flow in the Presence of a Permeable Wall PDE-Toolbox of MATLAB Least-Squares Finite Element Method (LSFEM) First Order Model Problem 356

10 x 2.2 The Mathematical Foundation of the Least-Squares Finite Element Method Div-Curl (Rot) Systems Div-Curl (Rot)-Grad System Stokes Problem Boundary Element Method (BEM) Abstract Formulation of the Boundary Element Method Variant of the Complex Variables Boundary Element Method [112] The Motion of a Dirigible Balloon Coupling of the Boundary Element Method and the Finite Element Method THE FINITE VOLUME METHOD AND THE GENERALIZED DIFFERENCE METHOD ENO Finite Volume Schemes ENO Finite Volume Scheme in One Dimension ENO Finite Volume Scheme in Multi-Dimensions Generalized Difference Method Two-Point Boundary Value Problems Second Order Elliptic Problems Parabolic Equations Application SPECTRAL METHODS Fourier Series The Discretization Approximation of the Derivatives Orthogonal Polynomials Discrete Polynomial Transforms Legendre Polynomials Chebyshev Polynomials Spectral Methods for PDE Fourier Galerkin Method Fourier-Collocation Chebyshev-Tau Method Chebyshev-Collocation Method The Calculation of the Convolution Sums Complete Discretization 460

11 Contents xi 4 Liapunov Schmidt (LS) Methods Examples Stokes Problem Correction in the Dominant Space 479 Appendix A Vectorial-Tensorial Formulas 483 References 487 Index 497

12 Preface The present book through the topics and the problems approach aims at filling a gap, a real need in our literature concerning CFD (Computational Fluid Dynamics). Our presentation results from a large documentation and focuses on reviewing the present day most important numerical and computational methods in CFD. Many theoreticians and experts in the field have expressed their interest in and need for such an enterprise. This was the motivation for carrying out our study and writing this book. It contains an important systematic collection of numerical working instruments in Fluid Dynamics. Our current approach to CFD started ten years ago when the University of Paris XI suggested a collaboration in the field of spectral methods for fluid dynamics. Soon after preeminently studying the numerical approaches to Navier Stokes nonlinearities we completed a number of research projects which we presented at the most important international conferences in the field, to gratifying appreciation. An important qualitative step in our work was provided by the development of a computational basis and by access to a number of expert softwares. This fact allowed us to generate effective working programs for most of the problems and examples presented in the book, an aspect which was not taken into account in most similar studies that have already appeared all over the world. What makes this book special, in comparison with other similar enterprises? This book reviews the main theoretical aspects of the area, emphasizes various formulations of the involved equations and models (focussing on optimal methods in CFD) in order to point out systematically the most utilized numerical methods for fluid dynamics. This kind of analysis leaving out the demonstration details takes notice of the convergence

13 xiv and error aspects which are less prominent in other studies. Logically, our study goes on with some basic examples of effective applications of the methods we have presented and implemented (MATLAB). The book contains examples and practical applications from fluid dynamics and hydraulics that were treated numerically and computationally most of them having attached working programs. The inviscid and viscous, incompresible fluids are considered; practical applications have important theoretical outcomes. Our study is not extended to real compresible fluid dynamics, or to turbulence phenomena. The attached MATLAB 6 programs are conceived to facilitate understanding of the algorithms, without optimizing intentions. Through the above mentioned aspects, our study is intended to be an invitation to a more complete search: it starts with the formulation and study of mathematical models of fluid dynamics, continues with analysis of numerical solving methods and ends with computer simulation of the mentioned phenomena. As for the future, we hope to extend our study and to present a new more complete edition, taking into account constructive suggestions and observations from interested readers. We cannot end this short presentation without expressing our gratitude to our families who have supported us in creating this work in such a short time, by offering us peace and by acquitting us from our everyday duties. The authors