In general, the physical behavior of compressible flows is more complicated than in incompressible flows. Compressible flows may be viscous or inviscid, depending on flow velocities. Compressible inviscid flows are analyzed using the potential or Euler equations, whereas compressible viscous flows are solved from the Navier-Stokes system of equations. Shock waves may occur in compressible flows and require special attention as to the solution methods. Furthermore, shock wave turbulent boundary layer interactions in compressible viscous flows constitute one of the most important physical phenomena in computational fluid dynamics. Let us consider air flows at speeds greater than 100 m/s, which corresponds to a Mach number of approximately 0.3, but less than 1700 m/s, or approximately Mach 5. Air flows in this range (0.3 ≤M ≤5) may be considered as compressible and inviscid. This range is usually subdivided into regions identified as subsonic (0.3 < M <0.8), transonic (0.8 ≤ M ≤1.2), and supersonic (1.2<M?5). For M > 5, the flow is referred to as hypersonic. Hypersonic flows around a solid body are usually coupled with viscous boundary layers. Effects of dilatational dissipation due to compressibility, high temperature gradients, vortical motions within the secondary boundary layers, radiative heat transfer, vibrational and electronic energies, and chemical reactions are examples of some of the complex physical phenomena associated with hypersonic flows.

In order to take into account the compressibility and variations of density in highspeed flows, we utilize the conservation form of the governing equations, using the density-based formulation.

GENERAL

The finite element theory as applied to one-dimensional problems was discussed in Part One, Preliminaries. In general, finite element methods (FEM) are versatile in applications to multidimensional complex irregular geometries. Initial applications of FEM began with structural analysis in the late 1950s and primarily were based on variational principles. Duringthe early days of the development of FEM, applications were made for simple flow problems, beginning with Zienkiewicz and Cheung [1965], followed by Oden and Wellford [1972], Chung[1978], and Baker [1983], amongothers. Signifi-cant contributions in CFD began with the streamline upwind Petrov-Galerkin (SUPG) methods [Heinrich, Huyakorn, Zienkiewicz, and Mitchell, 1977; Hughes and Brooks, 1982; Hughes, Mallet, and Mizukami, 1986] or streamline diffusion methods (SDM) [Johnson, 1987], Taylor-Galerkin methods (TGM) [Donea, 1984; Löhner, Morgan, and Zienkiewicz, 1985], and hp adaptive methods [Oden and Demkowicz, 1991], among many other related works.

New approaches and various alternative methodologies are preponderant in the literature. Efforts are made in this book to simplify and unify some of the terminologies. For example, the original approaches of SUPG or SDM for convection-dominated flows have grown into GLS (Galerkin/least squares) when some changes in the formulation are introduced. It is suggested that all methods related to numerical diffusion test functions be called the generalized Petrov-Galerkin (GPG) methods. Hughes and his co-workers have contributed significantly in the past two decades to the GPG methodologies associated with the problems of convection-dominated flows and shock discontinuities.

Another example is the algorithm arising from the Taylor series expansion such as TGM. Zienkiewicz and his co-workers [Zienkiewicz and Codina, 1995] have applied for the past decade the concept of characteristic Galerkin methods (CGM) which produce results similar to TGM in dealing with convection-dominated problems for both compressible and incompressible flows.

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 inFDM 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.

The ultimate goal in computational fluid dynamics is to obtain desired solutions as accurately as possible while minimizing the requirement for computational resources. Thus, we ask: how do we achieve both “accuracy” and “efficiency”at the same time? Often we exercise a compromise where we may choose to sacrifice some accuracy for the sake of expediting a solution, or vice versa. Does an acceptable compromise exist? These are the types of questions that typically enter the minds of the CFD practitioner before undertaking a major project.

Given a fixed computational method and limited computer resources, one is confronted with a decision as to which direction to follow. The most feasible approach under these restricted circumstances will be to seek the best computational grid arrangement which will lend itself to the best possible accuracy and maximum efficiency. Adaptive methods are designed to achieve both accuracy and efficiency, with mesh refinements provided selectively only where needed.

The basic concept for adaptive methods consists of providing mesh refinements for efficiency (cost reduction) as dictated by predetermined criteria. The criteria for mesh refinements and unrefinements (coarsening) are determined by error indicators. The error indicators are usually represented by gradients of a suitable variable – the larger the gradient, the finer the mesh required.

In line with the two different grid generation schemes, structured and unstructured, two different adaptive methods are available, one for structured grids and another for unstructured grids. The structured adaptive methods are presented in Section 19.1, with the unstructured adaptive methods in Section 19.2.

STRUCTURED ADAPTIVE METHODS

Structured adaptive meshes may be constructed either by a control function approach or by a variational function approach. We shall discuss both of these methods next.

GENERAL

Multiphase flow is a common observation such as occurs in evaporation or condensation in which a liquid particle is transformed into a gas or vice versa. Other examples include phase changes involved in the boiling of liquids, tracking of free surfaces between gas and liquid, or rocket solid propellants which, upon ignition, change into a liquid phase and subsequently into a gas phase. Furthermore, rigid body motions of solids in the presence of gases or liquids such as in sedimentation and fluidized beds, and reactive laminar and turbulent flows in fluid-particle mixtures, are the complicated physical phenomena in multiphase flows.

Interfaces are present as a specified initial condition or as a result of phase changes through evaporation, condensation, melting, solidification, merging (coalescence), or breakup. Surface tension plays an important role in these interfaces. Interface kinematics dealing with interface tracking in the free surface flows may be described and solved in many different ways. Among them are: volume tracking methods, front tracking methods, level set methods, phase field formulations, continuum advection schemes, boundary integral methods, particle-based methods, and moving mesh methods. A brief review of these methods is given below.

Volume tracking methods, often known as the volume of fluid (VOF) method, were originated by Nichols and Hirt [1975] and Noh and Woodward [1976], and further extended by Hirt and Nichols [1981]. Since then, the VOF method has been improved significantly over the years [Rudman, 1997; Rider and Kothe, 1998].

Finite element methods (FEM) and topics related to finite element applications are presented in Part Three. We have seen in Chapter 1 that the finite element methods based on the standard Galerkin integral lead to results identical to those of the finite difference methods (FDM) for the examples of simple linear problems. In dealingwith nonlinear or convection-dominated flow problems in fluid dynamics, however, the standard Galerkin methods are no longer adequate. Various special strategies must be designed to assure stability and convergence, as we noted also in FDM. Dissipation and dispersion errors can be minimized with a high level of accuracy achieved in much the same way as in FDM. In this vein, the reader will see that finite element methods are analogous to finite difference methods in dealing with all aspects of the physics of fluids. Developments of both approaches in close alliance are shown to be complementary to each other. It is with this expectation that our journey begins.

GENERAL

We saw in Section 1.3 that finite element equations are obtained by the classical approximation theories such as variational or weighted residual methods. However, there are some basic differences in philosophy between the classical approximation theories and finite element methods. In the finite element methods, the global functional representations ofa variable consist ofan assembly oflocal functional representations so that the global boundary conditions can be implemented in local elements by modification of the assembled algebraic equations. The local interpolation (shape, basis, or trial) functions are chosen in such a manner that continuity between adjacent elements is maintained.

The finite element interpolations are characterized by the shape ofthe finite element and the order ofthe approximations. In general, the choice ofa finite element depends on the geometry ofthe global domain, the degree ofaccurac y desired in the solution, the ease ofintegration over the domain, etc.

In Figure 9.1.1, a two-dimensional domain is discretized by a series of triangular elements and quadrilateral elements. It is seen that the global domain consists ofmany subdomains (the finite elements). The global domain may be one-, two-, or three-dimensional. The corresponding geometries ofthe finite elements are shown in Figure 9.1.2. A one-dimensional element (as we have studied in Chapters 1 and 8) is simply a straight line, a two-dimensional element may be triangular, rectangular, or quadrilateral, and a three-dimensional element can be a tetrahedron, a regular hexahedron, an irregular hexahedron, etc. The three-dimensional domain with axisymmetric geometry and axisymmetric physical behavior can be represented by a two-dimensional element generated into a three-dimensional ring by integration around the circumference.

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.

In unstructured adaptive methods (Chapter 19), our assessments as to the merits and demerits of FDM versus FEM will be faced with a new challenge. This is because adaptive methods are instrumental in resolving many problems of numerical difficulties such as in shock waves and turbulence, making the fair comparison between FDM and FEM difficult.

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.

In the following sections, we shall describe a certain type of spectral element methods, least square methods, optimal control methods (OCM), and finite point methods (FPM). They are selected here for discussion because of their possible future potential for further developments.

SPECTRAL ELEMENT METHODS

The term “spectral” as used here implies a special function. Examples of such functions may be Chebyshev, Legendre, or Laguerre polynomials. These functions are expected to portray physical phenomena more realistically and precisely than other functions that have been discussed previously, leading to a greater solution accuracy. However, their applications are limited to simple geometries and simple boundary conditions.

Part Two presents the finite difference methods (FDM) and topics related to finite difference approximations. The subjects to be covered here include basic concepts of finite difference theory, various formulation strategies, and applications to incompressible and compressible flows. Finite volume methods (FVM) via FDM are also presented.

Although FDM as applied to CFD is widespread and many textbooks are available, the purpose of Part Two is to make detailed comparisons with other methods such as finite element methods (FEM) to be presented in PartThree (particularly in Chapter 16) for the benefit of the beginner and the practitioner alike. Historical developments, traditional treatments of finite difference methods, and some recent advancements are presented for this reason.

Chapter 3 discusses derivations of finite difference equations, followed in Chapter 4 by various finite difference schemes for solutions of elliptic, parabolic, hyperbolic, and Burgers' equations. General fluid dynamics problems of incompressible and compressible flows are presented in Chapters 5 and 6, respectively. Finally, finite volume methods (FVM) via FDM are discussed in Chapter 7.

Having studied various computational methods in Parts Two and Three and automatic grid generation, adaptive methods, and computing techniques in Part Four, we are now prepared to re-examine these methods and test our knowledge on some selected engineering problems of application. For the past four decades, many applications have been accumulated to such a great extent that it is impossible to review them all in this text. Rather, we limit our scope of study to the following areas:turbulence (Chapter 21), chemically reactive flows and combustion (Chapter 22), acoustics (Chapter 23), combined mode radiative heat transfer (Chapter 24), multiphase flows (Chapter 25), electromagnetic flows (Chapter 26), and relativistic astrophysical flows (Chapter 27).

The selection of computational methods depends on many factors such as types of flows, ranges of speeds, dimensions of domain, etc. A decision as to the choice of FDM, FEM, or FVM is now a matter of preference and judgments of the analyst in view of the information presented in the previous chapters.

In the following chapters, example problems and computational methods are chosen randomly, depending on availability of sources. Some of them are drawn from the student works at the University of Alabama in Huntsville, and others are from those available in the open literature. In each of the applications, the corresponding governing equations and associated physics are first introduced. This is then followed by the computational methods used, numerical results and evaluations, each example being self-contained as much as possible.

It is hoped that these examples serve as a reasonable guidance for the uninitiated reader toward his or her direction and destination in CFD research.

The dawn of the twentieth century marked the beginning of the numerical solution of differential equations in mathematical physics and engineering. Numerical solutions were carried out by hand and using desk calculators for the first half of the twentieth century, then by digital computers for the later half of the century. In Section 1.1, a brief summary of the history of computational fluid dynamics (CFD) will be given, along with the organization of text.

Before we proceed with details of CFD, simple examples are presented for the beginner, demonstrating how to solve a simple differential equation numerically by hand calculations (Sections 1.2 through 1.7). Basic concepts of finite difference methods (FDM), finite element methods (FEM), and finite volume methods (FVM) are easily understood by these examples, laying a foundation or providing a motivation for further explorations. Even the undergraduate student may be brought to an adequate preparation for advanced studies toward CFD. This is the main purpose of Preliminaries.

Furthermore, in Preliminaries, we review the basic forms of partial differential equations and some of the governing equations in fluid dynamics (Sections 2.1 and 2.2). These include nonconservation and conservation forms of the Navier-Stokes system of equations as derived from the first law of thermodynamics and are expressed in terms of the control volume/surface integral equations, which represent various physical phenomena such as inviscid/viscous, compressible/incompressible, subsonic/supersonic flows, and so on.

Typical boundary conditions are briefly summarized, with reference to hyperbolic, parabolic, and elliptic equations (Section 2.3). Examples of Dirichlet, Neumann, and Cauchy (Robin) boundary conditions are also examined, with additional and more detailed boundary conditions to be discussed later in the book.

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].T hese 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.

GENERAL

In heat transfer, there are three different modes conduction, convection, and radiation. We have included conduction and convection in the Navier-Stokes system of equations discussed in the previous chapters. Radiative heat transfer is another mode of heat transfer to be examined in this chapter. Heat transfer by radiation occurs in many engineering applications of nonparticipating and participating media. In this chapter, we study this subject as a separate mode of heat transfer first and then as a combined mode integrated into other modes.

In nonparticipating media, conduction and convection are absent. Here we are concerned with view factors, radiative boundary conditions, and radiative heat transfer in absorbing, emitting, and scattering media. Radiative heat transfer is associated with the radiative heat flux which involves integrals with respect to the wavelength, solid angle, and optical depth. The governing equation for radiative heat transfer, then, takes the form of integrodifferential equations. This aspect of the radiative heat transfer is unique and requires a special computational treatment.

Participating media combines the radiative heat transfer with conduction and/or convection. The most significant feature in the combined mode heat transfer is the fact that the radiative heat flux is always three-dimensional, even if the computational domain is chosen to be one- or two-dimensional. For this reason, special mathematical formulations and computational schemes must be developed. We discuss this subject in Sections 24.2.4 and 24.3.3.

For the sake of completeness and future reference, some basic definitions and formulas in radiative heat transfer are summarized below.