This chapter is devoted to numerical examples and applications intended as tutorials for the interested reader. The possibility to download and use a computer code together with the book is given, and some of the described examples can be replicated using the provided code. The examples are of increasing complexity and they range from simple two-dimensional flows up to complex three-dimensional problems with fluid-structure interaction.

A detailed description of the computer code is also included in order to allow the readers to quickly get acquainted with the method and allow them to modify it according to their needs.

This chapter begins with a motivation to use computational models in scientific and technical applications. An overview of the advantages and drawbacks of numerical simulations with respect to laboratory experiments is given and advancements in various fields are discussed.

After this general introduction, a historical overview of the subject is presented and the present state of the art is discussed. In particular, it is shown that immersed boundary methods are being used in all fields of computational science and the number of scientific publications per year has been increasing with a constant acceleration over the past two decades: This has resulted in an exploding research field in which a reference textbook is still missing.

Finally, the objective of the book and the plan of the various chapters is given.

As IBMs have gained popularity, their use has expanded to multiphysics problems in which the Navier-Stokes equations are only one among many other possibilities. In this chapter, a list of advanced applications is described in which IBMs are used to solve heat transfer, phase change and chemical reaction problems. These examples are intended as suggestions to extend the application of immersed boundary methods to complex physics problems.

The various forcing strategies to be implemented in the governing equations are described in this chapter. Two big categories are first introduced, namely continuous forcing and discrete forcing methods. The various techniques are then detailed and the steps needed to implement them into an existing flow solver are described.

As any immersed boundary method has to be coupled with a solution algorithm for the governing equations, pseudo-compressibility and fractional-step methods are described in detail and some issues related to their combination with IBMs illustrated.

With this chapter, the technical part of immersed boundary methods is initiated. Here it is explained how to define in the most convenient way a complex geometry object and how, after having immersed it in a computational grid, it is possible to determine the position (tagging) of the Eulerian nodes with respect to the boundary of the body.

Several computational geometry theorems are used to design an efficient computational algorithm which makes possible the tagging step within limited CPU time even when the computational grid contains tens of millions of nodes and the immersed object is described by hundreds of thousands of elements. This efficiency is key in problems involving moving bodies, deformable objects or fluid-structure interaction problems.

This chapter is devoted to the application of IBMs to problems with moving boundaries. Specific adaptations of the algorithms are needed in order to cope with the Eulerian nodes at the interface that change position from inside to outside the body within one time step.

In turn, the boundary reconstruction of the solution is also affected and the necessary changes to the method are described.

In this chapter it is explained how to compute the hydrodynamic loads produced by pressure and viscous stresses over an immersed surface. Several procedures are illustrated that entail different computational costs and degree of precision. The choice depends on whether only the resultant of the forces is needed or if the local values of the loads are needed. Finally, a simple validation of the discussed methods for a body with prescribed kinematics is shown.

We introduce solutions to the diffusion equation (Fick’s second law), which arises from Fick’s first law and continuity. Diffusion into semi-infinite half spaces as well as problems in finite spaces and the approach to equilibrium are addressed and solutions are given. The second part of the chapter describes fundamental, atomic scale aspects of diffusion in the solid state.

The powerful methods of dimensional analysis are introduced via the pi-theorem. The reader discovers that many of the results obtained in Chapters 3 and 4 can be arrived at using dimensional analysis alone. These include drag and pipe flow. Dynamical similarity is explained.

In this chapter we study the idealised, inviscid fluid. The central formula is Bernoulli’s equation, and its consequences are explored in a number of examples. Next we look at flow which is irrotational (vortex free) and develop potential theory, which in two dimensions can be treated very elegantly using complex analysis and the Cauchy–Riemann equations.

This chapter is mostly about solid mechanics: Cauchy stress, finite and infinitesimal strain, rotation. Velocity and acceleration are developed in both inertial and non-inertial fames. This is central to the education of the physicist and engineer, but the development leads to a derivation of the Navier–Stokes equations, which are central to fluid dynamics.

The equations of fluid dynamics and energy balance are arrived at from the starting point of the powerful Reynolds transport theorem. After writing down the four conservation laws – mass, energy, linear and angular momentum – their consequences when inserted into the transport equation are revealed, in particular Cauchy’s equations of motion, Navier–Stokes equations and the equation of energy balance. A number of prevalent examples are given, including Stokes’s formulae and the Darcy law. The chapter concludes with the theory of the boundary layer.

Here we begin fluid dynamics with the science of fluids at rest. This includes planetary science aspects of atmospheric and oceanic pressure, the forced and free vortex. Here also are introduced the three basic differential operators: grad, div and curl, which will be used throughout the book.