## Introduction to Theoretical and Computational Fluid Dynamics

### C. Pozrikidis

## Table of 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