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This book opens with a discussion of the vorticity-dynamic formulation of the low Mach number viscous flow problem. It examines the physical aspects of the velocity and the vorticity fields, their instantaneous relationship, and the transport of vorticity in viscous fluids for steady and unsteady flows. Subsequently, using classical analyses it explores the mathematical aspects of vorticity dynamics and issues of initial and boundary conditions for the viscous flow problem. It also includes the evolution of the vorticity field which surrounds and trails behind airfoils and wings, generalizations of Helmholtz’ vortex theorems and the Biot-Savart Law. The book introduces a theorem that relates the aerodynamic force to the vorticity moment and reviews the applications of the theorem. Further, it presents interpretations of the Kutta-Joukowski theorem and Prandtl’s lifting line theory for vorticity dynamics and discusses wake integral methods. The virtual-mass effect is shown to be the seminal event in unsteady aerodynamics and a simple approach for evaluating virtual-mass forces on the basis of vorticity dynamics is presented.

The book presents a modern viewpoint on vorticity dynamics as the framework for understanding and establishing the fundamental principles of viscous and unsteady aerodynamics. It is intended for graduate-level students of classical aerodynamics and researchers exploring the frontiers of fully unsteady and non-streamlined aerodynamics.

### Chapter 1. Introduction

The science of aerodynamics is often defined as the branch of dynamics, which treats of the motion of air and other gases and of the forces acting on bodies in motion through air or on fixed bodies in a current of air (or other gases).
James C. Wu

### Chapter 2. Theorems of Helmholtz and Kelvin

Helmholtz’ (1858) vortex theorems paved the way for the legendary Ludwig Prandtl (1921) to invent the lifting line theory, a crowning advancement in theoretical aerodynamics.
James C. Wu

### Chapter 3. Vorticity Kinematics

Kinematics is a branch of dynamics defined as the science concerned with motions in themselves, apart from their causes (Webster 1953). Vorticity kinematics is a branch of vorticity dynamics concerned with the instantaneous relationship between the velocity field v and the vorticity field ω. The fields v and ω in unsteady flows are time-dependent. In studies of vorticity kinematics, however, only the instantaneous relationship between v and ω, and not their change with time, is of concern. It is thus not necessary to view v and ω as time-dependent fields. For the following discussions, v and ω are treated as functions only of r, the position vector; their time-dependencies are not stated explicitly.
James C. Wu

### Chapter 4. Vorticity Kinetics

Newton’s second law of motion, as applied to the fluid medium, yields the well-known momentum theorem that states: The time rate of change of the total momentum of a fluid system of fixed identity is equal to the force acting on the system.
James C. Wu

### Chapter 5. Vorticity-Moment Theorem

The vorticity-moment theorem of aerodynamics discussed in this chapter is valid for viscous and unsteady flows. Earlier derivations of the theorem (Wu 1978, 1981) are updated and important features of the theorem are revisited in the present study. The theorem is shown to encompass much of the classical inviscid steady theories of aerodynamics , including the lifting-line theory, and is a rigorous mathematical consequence of the incompressible continuity and Navier–Stokes momentum equations:
James C. Wu

### Chapter 6. Classical Aerodynamics

The vorticity-moment theorem contains the following mathematical statements.
James C. Wu