Introduction to Theoretical and Computational Fluid Dynamics

C Pozrikidis

Oxford University Press, 1997

ISBN: 0195093208

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Errata

The Second Edition of this book is available

Why this book?

This book discusses the fundamental principles and equations governing the flow of incompressible Newtonian fluids, while simultaneously illustrating the use of numerical methods from computing solutions to a broad range of problems. Sample topics include the computation of stationary interfacial shapes, the derivation of exact solutions to the equation of motion, hydrodynamic stability, flow at low Reynolds numbers, vortex motion, boundary-integral methods for potential and Stokes flow, and finite-difference methods for Navier-Stokes flow. A unique synthesis of the theoretical and computational aspects of the discipline, this work serves as an ideal resource and a reference for professionals and researchers in various fields of science and engineering, including physics, chemical, mechanical, biomechanical and aerospace engineering, applied mathematics, and computational science.

Contents

Chapter 1: Kinematics of a Flow
1.1. Fluid velocity and motion of fluid parcels
1.2. Lagrangian labels
1.3. Properties of parcels, conservation of mass, and the continuity equation
1.4. Material lines, material vectors, and material surfaces
1.5. Differential geometry of surfaces
1.6. Description of a material surface in Eulerian form
1.7. Streamlines, stream tubes, path lines, and streak lines
1.8. Vorticity, vortex lines, vortex tubes, and circulation around loops
1.9. Line vortices and vortex sheets

Chapter 2: Analysis of Kinematics
2.1. Irrotational flows and the velocity potential
2.2. The reciprocal relation for harmonic functions, and Green's functions of Laplace's equation
2.3. Integral representation and further properties of potential flow
2.4. The vector potential for incompressible flow
2.5. Representation of an incompressible flow in terms in the vorticity
2.6. Representation of a flow in terms of the rate of expansion and vorticity
2.7. Stream functions for incompressible flow
2.8. Flow induced by vorticity
2.9. Axisymmetric flow induced by vorticity
2.10. Two-dimensional flow induced by vorticity

Chapter 3: Stresses, the Equation of Motion,
and the Vorticity Transport Equation

3.1. Forces acting in a fluid, traction, the stress tensor, and the equation of motion
3.2. Constitutive relations for the stress tensor
3.3. Traction, force, torque, energy dissipation, and the reciprocal theorem for incompressible Newtonian fluids
3.4. Navier-Stokes', Euler's and Bernoulli's equation
3.5. Equations and boundary conditions governing the motion of an incompressible Newtonian fluid
3.6. Traction, vorticity, and flow kinematics on rigid boundaries, free surfaces, and fluid interfaces
3.7. Scaling of the Navier-Stokes equation and dynamic similtude
3.8. Evolution of circulation around material loops and dynamics of the vorticity field
3.9. Computation of exact solutions to the equation of motion in two dimensions
based in the vorticity transport equation

Chapter 4: Hydrostatics
4.1. Pressure distribution within a fluid in rigid body motion
4.2. The Laplace-Young equation
4.3. Two-dimensional interfaces
4.4. Axisymmetric interfaces
4.5. Three-dimensional interfaces

Chapter 5: Computing Incompressible Flows
5.1. Steady unidirectional flows
5.2. Unsteady unidirectional flows
5.3. Stagnation-point flows
5.4. Flow due to a rotating disk
5.5. Flow in a corner due to a point source
5.6. Flow due to a point force

Chapter 6: Flow at Low Reynolds Numbers
6.1. Equations and fundamental properties of Stokes flow
6.2. Local solutions in corners
6.3. Nearly-unidirectional flows
6.4. Flow due to a point force
6.5. Fundamental solutions of Stokes flow
6.6. Stokes flow past or due to the motion of rigid bodies and liquid drops
6.7. Computation of singularity representations
6.8. The Lorentz reciprocal theorem and its applications
6.9. Boundary integral representation of Stokes flows
6.10. Boundary-integral-equation methods
6.11. Generalized Faxen's relations
6.12. Formulation of two-dimensional Stokes flow in complex variables
6.13. Effects of inertia and Oseen flow
6.14. Unsteady Stokes flow
6.15. Computation of unsteady Stokes flow past or due to the motion of particles

Chapter 7: Irrotational Flow
7.1. Equations and computation of irrotational flow
7.2. Flow past or due to the motion of three-dimensional body
7.3. Force and torque exerted on a three-dimensional body
7.4. Flow past or due to the motion of a sphere
7.5. Flow past or due to the motion of non-spherical bodies
7.6. Flow past of due to the motion of two-dimensional bodies
7.7. Computation of two-dimensional flow past or due to the motion of a body
7.8. Formulation of two-dimensional flow in complex variables
7.9. Conformal mapping
7.10. Applications of conformal mapping to flow past two-dimesensional bodies
7.11. The Schwarz-Christoffel transformation and its applications

Chapter 8: Boundary Layers
8.1. Boundary-layer theory
8.2. The boundary layer on a semi-infinite flat plate
8.3. Boundary layers in acclerating and decelerating flow
8.4. Computation of boundary layers around two-dimensional bodies
8.5. Boundary layers in axisymmetric and three-dimensional flows
8.6. Unsteady boundary layers

Chapter 9: Hydrodynamic Stability
9.1. Evolution equations and forumulation of the linear stability problem
9.2. Solution of the initial-value problem and normal-mode analysis
9.3. Normal-mode analysisof unidirectional flows
9.4. General theorems of the temporal stability of inviscid shear flows
9.5. Stability of a uniform layer subject to spatially periodic disturbances
9.6. Numerical solution of the Orr-Sommerfeld and Rayleigh equations
9.7. Stability of certain classes of unidirectional flows
9.8. Stability of a planar interface in potential flow
9.9. Viscous interfacial flows
9.10. Capillary instability of a curved interface
9.11. Inertial instability of rotating fluids

Chapter 10: Boundary-Integral Methods for Potential Flow
10.1. The boundary-integral equation
10.2. Boundary-element methods
10.3. Generalized boundary-integral representations
10.4. The single-layer potential
10.5. The double-layer potential
10.6. Investigation of integral equations of the second kind
10.7. Regularization of integral equations of the second kind
10.8. Completed double-layer representation for exterior flow
10.9. Iterative solution of integral equations of the second kind

Chapter 11: Vortex Motion
11.1. Invariants of the motion
11.2. Point vortices
11.3. Vortex blobs
11.4. Two-dimensional vortex sheets
11.5. Two-dimensional flows with distributed vorticity
11.6. Two-dimensional vortex patches
11.7. Axisymmetric flow
11.8. Three-dimensional flow

Chapter 12: Finite-Difference Methods for the Convection-Diffusion Equation
12.1. Definitions and procedures
12.2. One-dimensional diffusion
12.3. Diffusion in two and three dimensions
12.4. One-dimensional convection
12.5. Convection in two and three dimensions
12.6. Convection-diffusion in one dimension
12.7. Convection-diffusion in two and three dimensions

Chapter 13: Finite-Difference Methods for Incompressible Newtonian Flow
13.1. Methods based on the vorticity transport equation
13.2. Velocity-pressure formulation
13.3. Implementation of methods in primitive variables
13.4. Operator splitting, projection, and pressure-correction methods
13.5. Methods of modified dynamics or false transients

Appendix A: Index Notation, Differential Operators, and Theorems of Vector Calculus
A.1. Index Notation
A.2. Vector and matrix products, differential operators in Cartesian coordinates
A.3. Orthogonal curvilinear coordinates
A.4. Differential operators in cylindrical and plane polar coordinates
A.5. Differential operators in spherical polar coordinates
A.6. Integral theorems of vector calculus

Appendix B: Primer of Numerical Methods
B.1. Linear algebra equations
B.2. Computation of eigenvalues
B.3. Nonlinear algebraic equations
B.4. Function interpolation
B.5. Computation of derivatives
B.6. Function integration
B.7. Function approximation
B.8. Integration of ordinary differential equations
B.9. Computation of special functions