In this chapter, the coupling of IBMs with turbulence and wall models is discussed to provide the reader with a guideline to apply these methods to high Reynolds number flows. In fact, is this context, the small thickness of the flow boundary layer, combined with the impossibility to benefit from a wall-normal mesh refinement, challenges the use of IBMs unless additional models are used at the wall.

The possibility to resort to adaptive wall refinement is presented, although it is also shown that it can be combined only with RANS models.

As the textbook is concerned with the application of immersed boundary methods for complex flow simulations, some general preliminary considerations are necessary in order to make the book self-consistent.

Basic concepts about fluids, their governing equations and the fundamentals relating to numerical integration are introduced and discussed.

Using a simple numerical example of the flow around a square cylinder, the relation between spatial numerical resolution and smallest flow scale is introduced and explained in connection with the successive requirements of immersed boundary methods.

A final discussion of the concepts of verification and validation of a numerical model closes the chapter.

When the flow and immersed object dynamics are two-way coupled, the problem is a fluid-structure interaction and additional changes are necessary to implement immersed boundary methods. Depending on the coupling between flow and structure solvers (loose or strong), the nature of the structure (rigid or deformable body) and the specific solution algorithms, several possibilities are available and this chapter aims at providing insights to guide the choice.

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.

This chapter covers the basics of cubature rules. Starting from the definition of polynomial spaces and cubature rules, it discusses interpolatory cubature rules, Tchakaloff’s theorem on positive cubature rules, and Sobolev’s theorem on invariant cubature rules. It provides product-type cubature rules on several regular domains, such as product domains, balls, and simplexes, as well as invariant cubature rules on these domains, and a brief section on constructing cubature rules numerically.

The chapter explores a connection between cubature rules and the discrete Fourier transform of exponential functions defined via lattice translation. Such discrete Fourier analysis yields cubature rules for exponential functions for the integral over the spectral set of the lattice, which become cubature rules on the fundamental domain of the spectral set for generalized cosine and sine functions, defined as certain symmetric and antisymmetric exponential sums. Furthermore, under appropriate transformation, these generalized trigonometric functions define Chebyshev polynomials that inherit the orthogonality of generalized sine and cosine functions, which lead to cubature rules for algebraic polynomials on the range of the fundamental domain under the transformation.