Much work has been done on low-dimensional models of turbulence and fluid systems in the 16 years since the first edition of this book appeared. In preparing the second edition, we have not attempted a comprehensive review: indeed, we doubt that this is possible, or even desirable. Rather, we have added one chapter and several sections and subsections on some new developments that are most closely related to material in our first edition. We have also made minor corrections and clarifications throughout, and added comments in several places, as well as correcting a number of errors that readers have pointed out. Here, to orient the reader, we outline the major changes.

Clancy Rowley (the new member of our team) has contributed a chapter on balanced truncation, a technique from linear control theory that chooses bases that optimally align inputs and outputs. Over the past ten years this has led to the method of balanced proper orthogonal decomposition (BPOD), which is especially useful for systems equipped with sensors and actuators. Since low-dimensional models provide a computational means for studying control of turbulence, we feel that BPOD has considerable potential. This new chapter (5) now closes the first part of the book (readers familiar with the first edition must therefore remember to add 1 to correctly identify the following eight chapters). The only other entirely new sections are 7.5, a discussion of traveling modes in translation-invariant systems, 12.6, a review of work on coherent structures in internal combustion engines, and 12.7, which gathers a miscellany of recent results.

This chapter and the following one provide a review of some aspects of the qualitative theory of dynamical systems that we need in our analyses of low-dimensional models derived from the Navier–Stokes equations. Dynamical systems theory is a broad and rapidly growing field which, in its more megalomaniacal forms, might be claimed to encompass all of differential equations (ordinary, partial, and functional), iterations of mappings (real and complex), devices such as cellular automata and neural networks, as well as large parts of analysis and differential topology. Here our aim is merely the modest one of introducing, with simple examples, some tools for analysis of nonlinear ordinary differential equations that may not be as familiar as, say, perturbation and asymptotic methods.

The viewpoint of dynamical systems theory is geometric, and invariant manifolds play a central rôle, but we do not assume or require familiarity with differential topology. In the same way, symmetries are crucial in determining the behavior, and permitting the analysis, of the low-dimensional models of interest, but we avoid appeals to the subtleties of group theory in our introduction to symmetric bifurcations. Thus, it should be clear that these two chapters cannot substitute for a serious course (or, more likely, courses) in dynamical systems theory. The makings of such a course can be found in the books of Arnold [15,17], Guckenheimer and Holmes [144], Arrowsmith and Place [18], or Glendinning [129], and in other references cited below. In particular we omit entirely any discussion of partial differential equations, which may seem scandalous, since this book ostensibly treats turbulence as described by the Navier–Stokes equations.

Turbulence

Turbulence is the last great unsolved problem of classical physics. Although temporarily abandoned by much of the community in favor of particle physics, the current popularity of chaos and dynamical systems theory (as well as funding problems in particle physics) is now drawing the physicists back. During the interim and up to the present, turbulence has been avidly pursued by engineers.

Turbulence has enormous intellectual fascination for physicists, engineers, and mathematicians alike. This scientific appeal stems in part from its inherent difficulty – most of the approaches that can be used on other problems in fluid mechanics are useless in turbulence. Turbulence is usually approached as a stochastic problem, yet the simplifications that can be used in statistical mechanics are not applicable – turbulence is characterized by strong dependency in space and in time, so that not much can be modeled usefully as a simple Markov process, for example. The nonlinearity of turbulence is essential – linearization destroys the problem. Many problems in fluid mechanics can be approached by supposing that the flow is irrotational – that is, that the vorticity is zero everywhere. In turbulence, the presence of vorticity is essential to the dynamics. In fact, the nonlinearity, rotationality, and the dimensionality interact dynamically to feed the turbulence – hence, to suppose that a realization of the flow is two-dimensional also destroys the problem. There is more, but this is enough to make it clear that one faces the turbulence problem stripped of the usual arsenal of techniques, reduced to hand-to-hand combat. One is forced to find unexpected chinks in its armor almost by necromancy, and to fabricate new approaches from whole cloth. This is its fascination.

As we have described in Part One, attempts to build low-dimensional models of truly turbulent processes are likely to involve averaging or, more generally, modeling to account for neglected modes that are dynamically active in the sense that their states cannot be expressed as an algebraic function of the modes included in the model. Such models are in turn likely to involve probabilistic elements. Here, “neglected modes” may refer to (high wavenumber) modes in the inertial and dissipative ranges or to mid-range, active modes whose wavenumbers might be linearly unstable. They also may refer to spatial locations that are omitted, in selecting a subdomain of a large or infinite physical spatial extent. The boundary layer model of Chapter 10, for example, contains a forcing term representing a pressure field, unknown a priori, imposed on the outer edge of the wall region. While estimates of this term can be obtained from direct numerical simulations (e.g. [244]), a natural simplification is to replace it with an external random perturbation of suitably small magnitude and appropriate power spectral content. More generally, many processes modeled by nonlinear differential equations involve random effects, in either multiplicative form (coefficient variations) or additive form, and it is therefore worth making a brief foray into the field of stochastic dynamical systems to sample some of the tools available.

In this chapter we give a very selective and cursory description of how one can analyze the effect of additive white noise on a system linearized near an equilibrium point.

In numerical simulations of turbulence, one can only integrate a finite set of differential equations or, equivalently, seek solutions on a finite spatial grid. One method that converts an infinite-dimensional evolution equation or partial differential equation into a finite set of ordinary differential equations is that of Galerkin projection. In this procedure the functions defining the original equation are projected onto a finite-dimensional subspace of the full phase space. In deriving low-dimensional models we shall ultimately wish to use subspaces spanned by (small) sets of empirical eigenfunctions, as described in the previous chapter. However, Galerkin projection can be used in conjunction with any suitable set of basis functions, and so we discuss it first in a general context.

After a brief description of the method in Section 4.1, we apply it in Section 4.2 to a simple problem: the linear, constant-coefficient heat equation in both one- and two-space dimensions. We recover the classical solutions, which are often obtained by separation of variables and Fourier series methods in introductory applied mathematics courses. We then consider an equation with a quadratic nonlinearity, Burgers’ equation, which was originally introduced as a model to illustrate some of the features of turbulence [65]. The remainder of the chapter is devoted to the Navier–Stokes equations. In Section 4.3 we describe Fourier mode projections for fluid flows in simple domains with periodic boundary conditions, paying particular attention to the way in which the incompressibility condition is addressed. The final Section 4.4 focuses on the use of empirical eigenfunctions and introduces some issues that arise in making the drastic truncations necessary to obtain low-dimensional models.

Physical systems often exhibit symmetry: we have already remarked on the symmetries of spanwise translation and reflection in boundary layers and shear layers and of rotations in circular jets. One could cite many more such cases. Of course, symmetric systems do not always, or even typically, exhibit symmetric behavior, and the study of spontaneous symmetry breaking is an important field in physics. These physical phenomena have their analogs in dynamical systems and in particular in ODEs, as we describe in this chapter.

The theory of symmetric dynamical systems and their bifurcations relies heavily on group theory and especially the notions of invariant functions and equivariant vector fields. The major references are the two volumes by Golubitsky and Schaeffer [134] and Golubitsky et al. [136]. In this chapter, as in the last, we attempt to sketch relevant parts of the theory using simple examples and without undue reliance on abstract mathematical ideas.

In this chapter we describe the qualitative structure, in phase space, of some of the low-dimensional models derived in the preceding chapter. We also discuss the physical implications of our findings. Drawing on the material introduced in Chapters 6–9, we solve for some of the simpler fixed points (steady, time-independent flows and traveling waves) and discuss their stability and bifurcations under variation of the loss parameters αj introduced in Section 10.1. We focus on the five mode model (N = 1, K1 = 0, K3 = 5) introduced in the original paper of Aubry et al. [22], and referred to there as the “six mode model,” the k3 = 0 mode being implicitly included in the model of the slowly varying mean flow. The full range of dynamical behavior of even such a draconian truncation as this is bewilderingly complex and still incompletely understood, but we are able to give a fairly complete account of a particular family of solutions – attracting heteroclinic cycles – which appear especially relevant to understanding the burst/sweep cycle which was described in Section 2.5.

In Sections 11.1 and 11.2 we use the nesting properties of invariant subspaces, noted in Section 10.5, to solve a reduced system, containing only two (even) complex modes, for fixed points. We exhibit the bifurcation diagram and discuss the stability of a particular branch of fixed points corresponding to streamwise vortices of the appropriate spanwise wavenumber. Due to the spanwise translation invariance (Section 10.3), circles of such equilibria occur in phase space.

Nonlinear continuum mechanics is one of the fundamental subjects that form the foundation of modern computational mechanics. The study of the motion and behavior of materials under different loading conditions requires understanding of basic, general, and nonlinear kinematic and dynamic relationships that are covered in continuum mechanics courses. The finite element method, on the other hand, has emerged as a powerful tool for solving many problems in engineering and physics. The finite element method became a popular and widely used computational approach because of its versatility and generality in solving large-scale and complex physics and engineering problems. Nonetheless, the success of using the continuum-mechanics-based finite element method in the analysis of the motion of bodies that experience general displacements, including arbitrary large rotations, has been limited. The solution to this problem requires resorting to some of the basic concepts in continuum mechanics and putting the emphasis on developing sound formulations that satisfy the principles of mechanics. Some researchers, however, have tried to solve fundamental formulation problems using numerical techniques that lead to approximations. Although numerical methods are an integral part of modern computational algorithms and can be effectively used in some applications to obtain efficient and accurate solutions, it is the opinion of many researchers that numerical methods should only be used as a last resort to fix formulation problems. Sound formulations must be first developed and tested to make sure that these formulations satisfy the basic principles of mechanics. The equations that result from the use of the analytically correct formulations can then be solved using numerical methods.

This book is focused on presenting the nonlinear theory of continuum mechanics and demonstrating its use in developing nonlinear computer formulations that can be used in the large displacement dynamic analysis. To this end, the basic concepts used in continuum mechanics are first presented and then used to develop nonlinear general finite element formulations that can be effectively used in the large displacement analysis. Two nonlinear finite element dynamic formulations will be considered in this book. The first is a general large-deformation finite element formulation, whereas the second is a formulation that can be used efficiently to solve small-deformation problems that characterize very and moderately stiff structures. In this latter case, an elaborate method for eliminating the unnecessary degrees of freedom must be used to be able to efficiently obtain a numerical solution. An attempt has been made to present the materials in a clear and systematic manner with the assumption that the reader has only basic knowledge in matrix and vector algebra as well as basic knowledge of dynamics. The book is designed for a course at the senior undergraduate and first-year graduate level. It can also be used as a reference for researchers and practicing engineers and scientists who are working in the areas of computational mechanics, biomechanics, computational biology, multibody system dynamics, and other fields of science and engineering that are based on the general continuum mechanics theory.

In the preceding chapters, the general nonlinear continuum mechanics theory was presented. In order to make use of this theory in many practical applications, a finite dimensional model must be developed. In this model, the partial differential equations of equilibrium are written using approximation methods as a finite set of ordinary differential equations. One of the most popular approximation methods that can be used to achieve this goal is the finite element method. In this method, the spatial domain of the body is divided into small regions called elements. Each element has a set of nodes, called nodal points, that are used to connect this element with other elements used in the discretization of the body. The displacement of the material points of an element is approximated using a set of shape functions and the displacements of the nodes and possibly their derivatives with respect to the spatial coordinates. In this case, the dimension of the problem depends on the number of nodes and number and type of the nodal coordinates used.

In the literature, there are many finite element formulations that are developed for the deformation analysis of mechanical, aerospace, structural, and biological systems. Some of these formulations are developed for small-deformation and small-rotation linear problems, some for large-deformation and large-rotation nonlinear analysis, and the others for large-rotation and small-deformation nonlinear problems. Several numerical solution procedures and computational algorithms are also proposed for solving the resulting system of finite element differential equations.

Matrix, vector, and tensor algebras are often used in the theory of continuum mechanics in order to have a simpler and more tractable presentation of the subject. In this chapter, the mathematical preliminaries required to understand the matrix, vector, and tensor operations used repeatedly in this book are presented. Principles of mechanics and approximation methods that represent the basis for the formulation of the kinematic and dynamic equations developed in this book are also reviewed in this chapter. In the first two sections of this chapter, matrix and vector notations are introduced and some of their important identities are presented. Some of the vector and matrix results are presented without proofs because it is assumed that the reader has some familiarity with matrix and vector notations. In Section 3, the summation convention, which is widely used in continuum mechanics texts, is introduced. This introduction is made despite the fact that the summation convention is rarely used in this book. Tensor notations, on the other hand, are frequently used in this book and, for this reason, tensors are discussed in Section 4. In Section 5, the polar decomposition theorem, which is fundamental in continuum mechanics, is presented. This theorem states that any nonsingular square matrix can be decomposed as the product of an orthogonal matrix and a symmetric matrix. Other matrix decompositions that are used in computational mechanics are also discussed. In Section 6, D’Alembert’s principle is introduced, while Section 7 discusses the virtual work principle. The finite element method is often used to obtain finite dimensional models of continuous systems that in reality have infinite number of degrees of freedom. To introduce the reader to some of the basic concepts used to obtain finite dimensional models, discussions of approximation methods are included in Section 8. The procedure for developing the discrete equations of motion is outlined in Section 9, while the principle of conservation of momentum and the principle of work and energy are discussed in Section 10. In continuum mechanics, the gradients of the position vectors can be determined by differentiation with respect to different parameters. The change of parameters can lead to the definitions of strain components in different directions. This change of parameters, however, does not change the coordinate system in which the gradient vectors are defined. The effect of the change of parameters on the definitions of the gradients is discussed in Section 11.

The kinematic and force equations developed in the preceding two chapters are general and applicable to all types of materials. The mechanics of solids and fluids is governed by the same equations, which do not distinguish between different materials. The definitions of the strain and stress tensors, however, are not sufficient for describing the behavior of continuous bodies. The force–displacement relationship or equivalently the stress–strain relationship is required in order to be able to distinguish between different materials and solve the equilibrium equations. The continuum displacements depend on the applied forces, and the force–displacement relationship depends on the material of the continuum. To complete the specification of the mechanical properties of a material, one needs additional set of equations called the constitutive equations, which serve to distinguish one material from another. The form of the constitutive equations of a material should not be altered in the case of a pure rigid-body motion. These equations, therefore, must be objective, and should not lead to change in the work and energy of the stresses under an arbitrary rigid-body motion. Using the constitutive equations, the partial differential equations of equilibrium obtained in the preceding chapter can be expressed in terms of the strains. Using the strain–displacement relationships, these equilibrium equations can be expressed in terms of displacements or position coordinates and their time and spatial derivatives. If the continuum density is considered as an unknown variable, as it is the case in some fluid applications, the continuity equations can be added to the resulting system of partial differential equations in order to have a number of equations equal to the number of unknowns.

If the constitutive equations of a material depend only on the current state of deformation, the behavior is said to be elastic. If the stresses can be derived from a stored energy function, the material is termed hyperelastic or called Green elastic material. A more general class of materials, for which the stresses cannot be derived from a stored energy function, is called Cauchy elastic material. For hyperelastic materials, the work done by the stresses during a deformation process is path independent. That is, the work done depends only on the initial and final states. For such systems, the continuum returns to its original configuration after the load is released. For viscoelastic materials, on the other hand, the work done during a deformation process is path dependent due to the dissipation of energy during the deformation process. The constitutive equations of viscoelastic materials are formulated in terms of rate of deformation measures in order to account for the energy dissipation.

The analysis of plastic deformation is important in many engineering applications including crashworthiness, impact analysis, manufacturing problems, among many others. When materials undergo plastic deformations, permanent strains are developed when the load is removed. Many materials exhibit elastic–plastic behaviors, that is, the material exhibits elastic behavior up to a certain stress limit called the yield strength after which plastic deformation occurs. If the stress of elastic–plastic materials depends on the strain rate, one has a rate-dependent material; otherwise the material is called rate independent. In the classical plasticity analysis of solids, a nonunique stress–strain relationship that is independent of the rate of loading but does depend on the loading sequence is used (Zienkiewicz and Taylor, 2000). In rate-dependent plasticity, on the other hand, the stress–strain relationship depends on the rate of the loading.

The yield strength of elastic–plastic materials can increase after the initial yield. This phenomenon is known as strain hardening. In the theory of plasticity, there are two types of strain hardening, isotropic and kinematic hardening. In the case of isotropic hardening, the yield strength changes as the result of the plastic deformation. In the case of kinematic hardening, on the other hand, the center of the yield surface experiences a motion in the direction of the plastic flow. The kinematic hardening behavior is closely related to a phenomenon known as the Bauschinger effect, which is the result of a reduction in the compressive yield strength following an initial tensile yield. The kinematic hardening effect is important in the case of cyclic loading.