In the previous chapters, with an exception of GPG, the finite element formulations are based on the Galerkin methods in which test functions are chosen to be the same as the trial functions. This is not required in the weighted residual methods.

Weighted residual methods other than the Galerkin methods include spectral element methods (SEM), least square methods (LSM), moment methods, or collocation methods, in which the test functions or weighting functions are not necessarily the same as the trial functions. In spectral element methods (SEM), polynomials in terms of nodal values of the variables are combined with special functions such as Chebyshev or Legendre polynomials. For least square methods, the test functions are constructed by the derivative of the residual with respect to the nodal values of the variables. Some arbitrary functions are chosen as test functions for the moment and collocation methods. Recently, the weighted residual concept has been used in meshless configurations, known as the finite point method (FPM), partition of unity method, meshless cloud method, or element-free method.

This revised second edition of Computational Fluid Dynamics represents a significant improvement from the first edition. However, the original idea of including all computational fluid dynamics methods (FDM, FEM, FVM); all mesh generation schemes; and physical applications to turbulence, combustion, acoustics, radiative heat transfer, multiphase flow, electromagnetic flow, and general relativity is maintained. This unique approach sets this book apart from its competitors and allows the instructor to adopt this book as a text and choose only those subject areas of his or her interest.

The second edition includes new sections on finite element EBE-GMRES and a complete revision of the section on the flowfield-dependent variation (FDV) method, which demonstrates more detailed computational processes and includes additional example problems. For those instructors desiring a textbook that contains homework assignments, a variety of problems for FDM, FEM, and FVM are included in an appendix. To facilitate students and practitioners intending to develop a large-scale computer code, an example of FORTRAN code capable of solving compressible, incompressible, viscous, inviscid, 1-D, 2-D, and 3-D for all speed regimes using the flowfield-dependent variation method is available at http://www.uah.edu/cfd.

Our explorations on the methods of finite differences and finite elements have come to an end. In Chapter 1, it was intended that the reader recognize the analogy between these two methods in one dimension. In fact, such an analogy exists for linear problems in all multidimensional geometries as long as the grid configurations are structured. In structured grids, with adjustments of the temporal parameters in generalized Galerkin methods and both temporal and convection diffusion parameters in generalized Petrov-Galerkin methods, the analogy between finite difference methods (FDM) and finite element methods (FEM) can be shown to exist also.

Traditionally, FEM equations are developed in unstructured grids as well as in structured grids. The FEM equations written in unstructured grids have global nodes irregularly connected around the entire domain, thus resulting in a large sparse matrix system, but the data management can be handled efficiently by using the element-by-element (EBS) assembly as discussed in Sections 10.3.2 and 11.5. FDM equations cannot be written in unstructured grids unless through FVM formulations. Thus, the FDM equations written only in structured grids cannot be directly compared with FEM equations written in general unstructured grids. Thus, the notion of FEM being more complicated, requiring more computer time than FDM, is an unfortunate comparison. For fair comparisons, FEM equations must be written in structured grids as in FDM.

For fluid dynamics associated with nonlinearity and discontinuity, there have been significant developments in the last two decades both in finite difference methods (FDM) and finite element methods (FEM). Concurrent with upwind schemes in space and Taylor series expansion of variables in time for FDM formulations with various orders of accuracy, numerous achievements have been made in FEM applications since the publication of an earlier text [Chung, 1978]. These new developments include generalized Galerkin methods (GGM), Taylor-Galerkin methods (TGM) [Donea, 1984], and the streamline upwind Petrov-Galerkin (SUPG) methods [Heinrich et al., 1977; Hughes and Brooks, 1982], alternatively referred to as the streamline diffusion method (SDM) [Johnson, 1987], and Galerkin/least squares (GLS) methods [Hughes and his co-workers, 1988–1998]. In the sections that follow, it will be shown that computational strategies such as SUPG or SDM and other similar methods can be grouped under the heading of generalized Petrov-Galerkin (GPG) methods. Recent developments include unstructured adaptive methods [Oden et al., 1986; Löhner, Morgan, and Zienkiewicz, 1985], characteristic Galerkin methods (CGM) [Zienkiewicz and his co-workers, 1994–1998], discontinuous Galerkin methods (DGM) [Oden and his co-workers, 1996–1998], and flowfield-dependent variation (FDV) methods [Chung and his coworkers, 1995–1999], among others. On the other hand, the concepts of FDM and FEM have been utilized in developing finite volume methods in conjunction with unstructured grids [Jameson, Baker, and Weatherill, 1986]. It appears that FDM and FEM continue to co-exist and develop into a mature technology, mutually benefitting from each other.

We begin in this chapter with the general discussion of boundary conditions for the nonlinear momentum equations, followed by Taylor-Galerkin methods (TGM) and generalized Petrov-Galerkin (GPG) methods as applied to Burgers’ equations. Some special topics such as Newton-Raphson methods and artificial viscosity are also discussed in this chapter. Applications to the Navier-Stokes system of equations characterizing incompressible and compressible flows are presented in Chapters 12 and 13, respectively.

General

Relativistic theory is divided into two categories: special relativity and general relativity. In special relativity, we follow Einstein’s postulate establishing the universality of the speed of light, c, relative to any unaccelerated observer, regardless of the motion of the light’s source from the observer. General relativity arises as an extension to special relativity to describe the motion of particles evolving under the presence of gravitational fields. In order to take into account the effect of gravitation, however, we must abandon the Eulerian coordinates used in Newtonian fluid dynamics. Instead, it is necessary to invoke a curvilinear four-dimensional manifold (the spacetime) to represent particle’s trajectories.

Many of the problems encountered in astrophysics are involved in the numerical solution of special or general relativistic fluid dynamics equations. Active research in this subject area has been in progress for the past 40 years. Earlier studies include structure and evolution of stars [Chandrasekhar, 1942; Aller and McLaughlin, 1965, among others]. Black hole accretion flows have been studied extensively as evident from numerous publications [Paczynski and Wiita, 1980; Katz, 1980; Eggum et al., 1988; Hawley et al., 1984a,b; Clarke et al., 1985; Stella and Vietri, 1997; Bromley et al., 1998; Font et al., 1998a,b; Koide et al., 1999; Font et al., 1999]. Some of the recent activities include Gamma ray bursts [Meszaros and Rees, 1993; Sari and Piran, 1998; Fishman and Meegan, 1995; and Panattescu and Meszros, 1998], explosive and jet phenomena [Norman, 1997], and astrophysical turbulence flows and instability [Bulbus and Hawley, 1998].

In this chapter, finite element analyses of both inviscid and viscous compressible flows are examined. Traditionally, computational schemes for compressible inviscid flow are developed separately from compressible viscous flows, governed by Euler equations and Navier-Stokes system of equations, respectively. However, it is our desire in this chapter to study numerical schemes capable of treating a compressible flow with or without the effect of viscosity or diffusion. Furthermore, it would be desirable to develop a scheme that can handle all speed regimes – not only the compressible flow, but the incompressible flow as well. To accomplish this goal, the most suitable governing equations to use are the Navier-Stokes system of equations written in conservation form in terms of conservation variables. Advantages of transforming the conservation variables into entropy variables and primitive variables will be explored. One of the most prominent features in compressible flow calculations is the ability of numerical schemes to resolve shock waves or discontinuities in high-speed flows. Furthermore, compressible viscous flows at high Mach numbers and high Reynolds numbers lead to significant numerical difficulties. We shall address these and other issues in this chapter.

To this end, we begin with the general description of the governing equations in Section 13.1, followed by the Taylor-Galerkin methods (TGM), generalized Galerkin methods (GGM), generalized Petrov-Galerkin (GPG) methods, characteristic Galerkin methods (CGM), and discontinuous Galerkin methods (DGM) in Sections 13.2 through 13.4. Finally, the flowfield-dependent variation (FDV) methods introduced in FDM and discussed earlier in Section 6.5 will be presented for FEM applications (Section 13.6). This subject will be treated again in Chapter 16, where many of the methods in both FDM and FEM can be shown to be the special cases of FDV methods.

This book is intended for the beginner as well as for the practitioner in computational fluid dynamics (CFD). It includes two major computational methods, namely, finite difference methods (FDM) and finite element methods (FEM) as applied to the numerical solution of fluid dynamics and heat transfer problems. An equal emphasis on both methods is attempted. Such an effort responds to the need that advantages and disadvantages of these two major computational methods be documented and consolidated into a single volume. This is important for a balanced education in the university and for the researcher in industrial applications.

Finite volume methods (FVM), which have been used extensively in recent years, can be formulated from either FDM or FEM. FDM is basically designed for structured grids in general, but is applicable also to unstructured grids by means of FVM. New ideas on formulations and strategies for CFD in terms of FDM, FEM, and FVM continue to emerge, as evidenced in recent journal publications. The reader will find the new developments interesting and beneficial to his or her area of applications. However, the subject material is often inaccessible due to barriers caused by different training backgrounds. Therefore, in this book, the relationship among all currently available computational methods is clarified and brought to a proper perspective.

Introduction

So far we have focused mainly upon the dynamics of waves in uniform and stationary ambient fluids. In the case of interfacial waves, we assumed that horizontal boundaries, if present, were flat so that the depth of the ambient was constant. Since interfacial wave speeds are a function of depth the question arises as to how the waves propagate in a medium that, for example, gets shallower approaching a beach. Likewise, for internal waves we have usually assumed that the background stratification is uniform. But we have seen that the stratification varies vertically in both the atmosphere and ocean. Except in the consideration of unstable shear flows, for the most part we have also neglected the presence of background winds and currents. In this chapter we examine how waves propagate in media that are non-uniform in the sense that the depth or stratification and the background flow varies.

We begin with the general treatment of small-amplitude wave propagation in non-uniform, but slowly varying media. This is known as ray theory. A simplified set of equations arising from ray theory assumes the motion is two-dimensional and steady, and that the background varies in only one spatial dimension. The theory is applied to study surface and interfacial waves approaching a beach and to examine internal waves approaching critical and reflection levels. In a somewhat different mathematical approach, the study of tunnelling examines the partial transmission and reflection of small-amplitude internal waves through weakly stratified regions.

Why write a book on internal gravity waves when so many other books cover the subject already? The textbooks listed in the appendix include at least some discussion of internal gravity waves. Some focus upon interfacial waves, which are internal gravity waves at interfaces; some focus upon internal waves, which exist in continuously stratified fluid. Different books emphasize different dynamics such as mechanisms for generation, propagation in non-uniform media, nonlinear evolution and stability. Textbooks on geophysical fluid dynamics (e.g. Gill (1982), Vallis (2006)) understandably devote only a chapter to the subject because, although internal waves are non-negligible in their influence upon global weather and ocean circulation patterns, they are by no means dominant. Internal waves are noise, if sometimes irritatingly loud. Textbooks on the theory of waves and instability (e.g. Whitham (1974), Lighthill (1978), Drazin and Reid (1981), Craik (1985)) examine how non-uniform media and nonlinearity affect the evolution of interfacial and internal waves. But these books can be daunting to graduate students lacking strong mathematical backgrounds. Textbooks on stratified fluid dynamics (e.g. Turner (1973), Baines (1995)) help to provide physical insight into the dynamics of internal gravity waves through a combination of theory and laboratory experiments, though sometimes without providing the mathematical details. Some textbooks are devoted to the subject of internal gravity waves (e.g. Miropol'sky (2001), Nappo (2002), Vlasenko et al. (2005)), but these focus either on atmospheric or oceanic waves.

Introduction

In this chapter we examine the various mechanisms through which interfacial waves and internal waves can be created. Broadly speaking, the waves can be generated either by solid bodies or by disturbances within the fluid such as convection, imbalance of large-scale flows, gravity currents and turbulence. Here we focus upon generation by solid bodies, the theory for which is better established.

In a uniformly stratified fluid, we have already seen that an oscillating body creates a cross-shaped pattern of waves. Here, we study this problem in more detail by examining the structure of the wave beams that emanate from a vertically oscillating cylinder and sphere. Although this mechanism has little bearing on geophysical flows, it is amenable to study in laboratory experiments which can test the limitations of theory applied to this seemingly simple circumstance.

More realistic is the study of uniform and tidal flow over topography. This subject has been well studied in theory and by way of experiments, and the results have been compared with ample observational data. Because much of this work is discussed elsewhere, only the salient points will be presented here.

Oscillating bodies

In Section 3.3 we saw that any body oscillating with frequency ω in uniformly stratified fluid with buoyancy frequency N0 creates a cross-shaped pattern of internal waves if ω < N0. Here we examine how the structure of the wave beams is affected by the shape of the body in two and three dimensions.

Introduction

Outside of theoretical interest in their peculiar properties, one of the main motivations for understanding the dynamics of internal gravity waves is that they occur naturally in the atmosphere and oceans. In the atmosphere, through transporting horizontal momentum from the ground upwards, internal waves influence wind speeds and consequently the thermal structure of the atmosphere. By contrast, internal waves in the ocean are primarily important as they affect mixing through the transport of energy. Although internal waves do not play a dominant role in the evolution of weather and climate, their influence is non-negligible: numerical simulations that do not include the effects of internal waves predict wind speeds and temperature in the atmosphere incorrectly and they do not account for the observed levels of turbulent diffusivity in the oceans. At the mesoscale in the atmosphere internal wave breaking is a source of clear-air turbulence and in the ocean, internal solitary waves influence biological activity over continental shelves through mass transport and mixing.

This chapter begins with a brief introduction to stratified fluids and internal gravity waves and then gives an overview of the structure of the atmosphere and oceans with mention of internal gravity wave phenomena in these fluids. In the following sections, we derive the equations describing the motion and thermodynamics of fluids and then make approximations relevant to internal gravity wave dynamics. The derivations are sometimes heuristic, aiming to provide physical intuition rather than emphasizing rigour.

Introduction

If a fluid's potential density decreases continuously with height then it can support internal waves that, like interfacial waves, move up and down due to buoyancy forces but which are not confined to an interface: they can move vertically through the fluid. This chapter focuses upon the dynamics of small-amplitude internal waves in uniformly stratified, stationary fluid. It also examines some effects of shear in non-uniform stratification, with a more general treatment given in Chapter 6.

The key quantity that determines the temporal evolution and spatial structure of internal waves is the buoyancy frequency. Heuristic methods based on fluid-parcel arguments are used to define the buoyancy frequency. These help to anticipate the dispersion relation that describes internal waves that are not influenced by rotation. The dispersion relation for Boussinesq and non-Boussinesq waves is then derived for liquids and gases. The results are then extended by including the influence of boundaries, shear and rotation.

The buoyancy frequency

In our consideration of interfacial waves we assumed that the density changed discontinuously across the interface between, say, warm and cold or fresh and salty fluid. In reality, such interfaces are not infinitesimally thin. They are thick, and the density varies continuously from one side of the interface to the other. We can justifiably assume the interface thickness is negligibly small only if the horizontal scale of interfacial disturbances is relatively large. If this is not the case, then the effects of continuous vertical density variations must be considered.