Drop some ink in a glass of water. It will slowly spread through the whole glass, moving in a manner known as diffusion. This process is so common that it gets its own chapter. We will describe the basics of diffusion, as captured by the heat equation, before understanding how diffusion comes about from an underlying randomness. We will see this through the eyes of the Langevin and Fokker-Planck equations.

Take anything in the universe, put it in a box, and heat it up. Regardless of what you start with, the motion of the substance will be described by the equations of fluid mechanics. This remarkable universality is the reason why fluid mechanics is important.

The key equation of fluid mechanics is the Navier-Stokes equation. This textbook starts with the basics of fluid flows, building to the Navier-Stokes equation while explaining the physics behind the various terms and exploring the astonishingly rich landscape of solutions. The book then progresses to more advanced topics, including waves, fluid instabilities, and turbulence, before concluding by turning inwards and describing the atomic constituents of fluids. It introduces ideas of kinetic theory, including the Boltzmann equation, to explain why the collective motion of 1023 atoms is, under the right circumstances, always governed by the laws of fluid mechanics.

Take water and push it through a pipe. If the flow is slow, then everything proceeds in a nice, orderly fashion. But as you force the water to move faster and faster, it starts to wobble. And then those wobbles get bigger until, at some point the flow loses all coherence as it tumbles and turn, tripping over itself in an attempt to push forwards. This is turbulent flow.

Understanding turbulence remains one of the great outstanding questions of classical physics. Why does it occur? How does it occur? How should we characterise such turbulent flows? The purpose of this chapter is to take the first tiny steps towards addressing these questions.

Many of the most interesting things in fluid mechanics occur because simple flows are unstable. If they get knocked a little bit, the fluid curls up into interesting shapes, or dissolves into some messy turbulent flow. In this chapter, we start to understand how these processes can happen.

The basic geometric parameters that define airfoil and wing shapes are presented prior to the basic aerodynamic forces and moments, as well as the nondimensional coefficients, that are used for airfoils and wings. A general description of the impact of airfoil geometry on the resulting aerodynamics, including the effects of camber and thickness, is presented. This includes how flow around a wing is different from flow around an airfoil, as well as methods to estimate the impact of wing geometry on lift and drag. Finally, the chapter concludes with the contributing factors to airplane drag and the methods to estimate zero-lift drag. coefficient of an airplane

Analysis capabilities are developed that include the impact of compressibility on the derivation of equations that govern subsonic compressible and transonic flows, including the relations to transform incompressible experimental data or geometry to subsonic compressible Mach numbers. Transonic flow is defined, including explanations for the flow characteristics that distinguish this flow regime from other flow regimes. Estimation techniques are developed for the critical and drag-divergence Mach numbers. The impact of transonic flow on aircraft design is discussed, including the impact of wing sweep on airplane aerodynamics and the role of supercritical airfoils. Finally, the transonic area rule is discussed, including the impact on transonic and supersonic aircraft.

Understand the physical concepts that apply to supersonic wing aerodynamics, such as knowing the difference between a subsonic and supersonic leading edge and how that impacts the airfoils used for the wing. Information is also presented that expands on why subsonic and supersonic drag-due-to-lift components are caused by different physical phenomenon. Supersonic 2D and 3D flow theories can then be used to analyze the forces and moments acting on a supersonic wing, including conical flow theory for calculating wing-tip effects. The reader should then be able to explain how supersonic airplanes make a compromise between subsonic and supersonic aerodynamic performance, and how that impacts airfoil and wing design parameters. Fuselage shapes then use slender body theory for analysis, and details about how boattails are used for reducing base drag.

Aerodynamic design decisions are rarely made without considering multidisciplinary design factors, which leads to compromises between aerodynamics and other disciplines. Readers will learn how to increase lift on an airplane, and how to modify an airplane in order to achieve aerodynamic improvements. Drag reduction will then be discussed, including ways to reduce drag and the impact of drag reduction on aircraft design. The chapter ends with a study of aircraft from the past and how aerodynamic considerations were included in their design.

Chapter 4 helps the reader develop a basic understanding of boundary layers and their impact on aerodynamic flows, and be able to obtain solutions for basic laminar flows and use the results to estimate properties, such as boundary layer thickness, shear stress, and skin friction. Descriptions are included for the characteristics of turbulent boundary layers, and how they compare to laminar boundary layers. Readers will be presented with information on how drag is impacted by laminar and turbulent boundary layers, including friction and separation. Analysis approaches will be presented to estimate turbulent boundary layer properties, such as boundary layer thickness, shear stress, and skin friction, including how to complete a control volume analysis of a boundary layer flow. Information is also presented for how to calculate the heat transfer and heat-transfer rate for a constant-property flow.

Basic concepts are developed for supersonic airfoils, including why supersonic airfoils have low lift-to-drag ratios compared to subsonic airfoils. Supersonic airfoil flows are discussed, leading to a basic understanding of the flow around supersonic airfoils and the shock/expansion wave patterns that develop. Three supersonic airfoil theories are then developed, including linear theory, second-order theory, and shock-expansion theory.

Learn about the primary ways to determine the aerodynamics of a vehicle, including semi-empirical methods, as well as various fidelity levels for computational approaches to predicting aerodynamics. Readers should be able to determine which levels of computational aerodynamic tools are appropriate for determining various aerodynamic characteristics (e.g., stall, cruise drag, cruise lift). Know the advantages of ground-based experimental testing, as well as the limitations and inaccuracies, as well as flight testing. Understand why the integrated triad of ground test, flight test, and computational simulation are important.

The chapter begins with the basic thermodynamic concepts that form the basis of high-speed flow theory, including a basic physical understanding of the second law of thermodynamics. This results in the ability to use the isentropic flow relationships in analyzing the properties of a compressible flow field, which results in the ability to analyze flow in a stream tube, and understand how a converging–diverging nozzle works. The basic relations for determining the change in flow properties across shock waves and expansion fans are developed, which make it possible to analyze flow fields using shock and expansion calculation methods. The basic relations for viscous flow are developed, leading to the relations for calculating the local skin-friction coefficient for a compressible boundary layer. The reader will then be able to understand the cause and effect of shock–boundary layer and shock–shock interactions. Finally, concepts for how flight vehicles are tested in wind tunnels are developed, which explains why it is difficult to fully model full-scale flight characteristics.