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.

Teaching a first course in fluid mechanics is always challenging because of the numerous new concepts that were not used in previous engineering courses. The source of the problem is in the highly nonlinear nature of the governing equations, which cannot be solved analytically for a general case. Over the years, various flow regimes were treated by neglecting large portions of the governing equations, leading to approximate solutions, which are unique to that limited problem. Consequently, many introductory fluid mechanics courses focus on surveying those localized solutions and not dealing directly with the governing equations. This approach perhaps is less mathematically intense, but it may confuse the novice student who cannot connect the various partial solutions to construct a more comprehensive understanding of the fluid mechanics discipline.

The present approach is aimed at easing the learning process by providing an early overview of the field and identifying a roadmap for understanding the different flow regimes. This is accomplished by presenting the governing equations (in their simplest laminar form) early on. In each of the following chapters, students are reminded how the particular subject of the chapter relates to the original roadmap (i.e., hydrostatics is a case in which the velocity in the governing equations is zero). Another benefit is that some similar cases are combined and students do not feel that each case needs a new and different mathematical approach.

Introduction

One approach in teaching introductory fluid mechanics is to avoid the presentation of complex fluid dynamic equations. This is done for very good reasons, including the lack of preparation in partial differential equations and the overall complexity of the problem. Although avoiding the introduction of complex equations is welcome by the average student, the negative and long-term outcome of this approach is that there is no clear rationale and a connecting string among the various chapters that follow. Therefore students are asked to be patient and “suffer quietly” at the beginning and the benefits of a clear roadmap will surface with the systematic approach that follows.

The mechanisms controlling fluid motion may include elements of basic mechanics, heat transfer, phase change, chemical reactions, and even molecular mechanics. Limiting the discussion to the simple mechanics of fluids, usually leads to principles such as the conservation of mass momentum and energy. In this text, however, we concentrate on the conservation of mass (continuity equation) and the conservation of momentum and assume a simple Newtonian fluid without heat transfer. Hence our objective in this chapter is to derive these two conservation laws, the conservation of mass and momentum.

Introduction

The science of fluid mechanics has matured over the last 200 years, but even today we do not have complete and exact solutions to all possible engineering problems. Although the governing equations (called the Navier–Stokes equations) were established by the mid-1800s, solutions did not follow immediately. The main reason is that it is close to impossible to analytically solve these nonlinear partial differential equations for an arbitrary case. Consequently, the science of fluid mechanics has focused on simplifying this complex mathematical model and on providing partial solutions for more restricted conditions. Therefore the different chapters on classical fluid mechanics are based on retaining different portions of the general equation while neglecting other lower-order terms. This approach allows the solution of the simplified equation, yet preserves the dominant physical effects (relevant to that particular flow regime). Finally, with the enormous development of computational power in the 21st century, numerical solutions of the fluid mechanic equations have become a reality. However, in spite of these advances, elements of modeling are still used in these solutions, and the understanding of the “classical” but limited models is essential for successfully using these modern tools.

This first chapter provides a short introduction on the historical evolution of fluid mechanics and a brief survey of fluid properties. After this introduction, the fluid dynamic equations are developed in the next chapter.

Introduction

The fluid dynamic equations developed in Chapter 2 are complex and cannot be solved analytically for an arbitrary case. Up to this point, the classical approach was presented in which major simplifications allowed some partial solutions. In recent years, however, numerical techniques and computational power have improved significantly. This facilitated the solution of the nonlinear fluid dynamic equations, which now can be added to the growing number of practical engineering tools.

There are two major advantages to the numerical approach when one is attempting to solve the fluid dynamic equations. The first is the possibility of solving these complex equations, which cannot be solved analytically. The process begins with a numerical approximation for the fluid dynamic equations. The fluid domain is then discretized into small cells or into a grid, where the equations are applied. By specifying the equations at each cell or point, we reduce the partial differential equations to a set of algebraic relations. Thus the second major advantage of the numerical approach is the ability to replace the nonlinear partial differential equations with a set of algebraic equations, which are usually solved by iterative methods. There are a large number of methods for approximating the equations, for grid generations, and for solution methodology. This chapter attempts to explain the generic principles of the numerical approach and the process leading to the numerical solutions.

Introduction

The need to channel water flow and other fluids must have originated in the early civilizations, and one of the better-known inventions, the Archimedes screw, dates back to the third century b.c.e. The Archimedes screw, or screwpump, shown in Fig. 11.1, was used to transfer water from lower reservoirs into higher irrigation ditches.

Over the years, many inventions focused on developing various machines either for pumping fluids or using fluid energy to drive other machinery (e.g., turbines). These machines may be classified as positive-displacement or continuous-flow machines. Some mechanical solutions using these two types of hardware are shown schematically in Fig. 11.2.

For example, the most basic configuration is a piston sliding inside a cylinder, as shown in Fig. 11.2(a). If this schematic is considered a pump, then while the piston is moving to the left it is pushing the fluid out of the cylinder through an open valve. The pumping operation of fresh fluid can continue when the piston is moving backward, closing this (exhaust) valve and opening the intake valve, creating a reciprocating cycle. This type of machine is called a positive-displacement machine because a fixed volume of fluid is captured in the cylinder and then transferred across the pump. Another example is the rotating-gear pump, shown in Fig. 11.2(b). This is also a positive-displacement machine, because there are fixed volumes of fluid between the outer wall and the gears.

Fluid mechanics is a fascinating but complex science and some problems cannot be solved by simple intuition. The reason behind this is the complex nonlinear differential equations, which cannot be solved analytically. The approach that evolved over recent centuries is to develop simple models for specific flow regions so that engineering calculations and predictions become possible. Unfortunately, even these simple models rely on complex mathematics, which makes introductory courses on this subject extremely difficult and sometimes confusing to students.

On the other hand, numerical solutions have matured recently and generating a solution for a given geometry can be achieved by a simple “run” command. The approach of many users is to run a large number of cases and develop their own “learning curve” of the problem, exactly as is done by experiments. The ease of generating attractive, colorful solutions creates the illusion (for many students) that further study of the subject is unnecessary.

The first objective of this introductory text is to familiarize students (and many will be exposed to only one course on fluids) with the basic elements of fluid mechanics. Therefore, if their future work relies on occasional numerical solutions, they will be familiar with the jargon of the discipline and with the expected results. At the same time, this book can serve as a long-term reference text, contrary to the oversimplified approach occasionally used for such introductory courses.

Introduction

In the previous chapter, the effects of pressure in a fluid were isolated (because the fluid was not moving) and the resulting forces were investigated. If we use the Navier–Stokes equations as a roadmap for gradually increasing the complexity of the models, then the next level requires the addition of the inertia terms in the momentum equations. For simplicity, it is assumed that the effects of viscosity are negligible, and the examples focus on cases for which such a simplification is acceptable. The addition of the viscosity term (and resulting friction) is discussed in the next chapter.

This is the first chapter in which solutions for fluid in motion are studied. By starting with the simple 1D model the basic principles can be easily demonstrated. The conservation of momentum, for example, closely resembles the classical mechanical formulation and easily can be explained. At first, however, we must clarify the meaning of the 1D flow assumption. Let us start by observing the velocity distribution inside a stream of fluid leaving a pipeline (Fig. 4.1). It is likely that the exiting flow velocity will not be uniform, and the size of the arrows in the figure describes the velocity distribution. If this velocity distribution has an axisymmetric shape then we can call the flow 2D because the velocity u will have a distribution u = u(r), where r is the radial direction (e.g., we must consider both the x and r variables).

Introduction

We have seen in the previous chapter that for specific cases certain terms of the fluid dynamic equations can be neglected, yet these simplified solutions still contain the dominant physical elements. The flow in pipes was an excellent example for developing exact solutions for the low-Reynolds-number case and then the formulation was extended to the higher-Reynolds-number cases, based on experimental observations. Also, the examples presented in the previous chapter can be considered as internal flows. The discussion in this chapter extends the modeling capability to include external flows as well. One of the objectives of this chapter is to demonstrate that neglecting certain terms in the governing equations can be done systematically. The use of dimensional analysis allows a logical approach for simplifying the governing equations and provides relative scaling for the various terms. A secondary objective of this chapter is to introduce a flow regime called the high-Reynolds-number flow (discussed in Chapters 7 and 8) and to explain the success of incompressible flow models. In spite of the complex equations, a reasonable solution for the flow over bodies (and the resulting forces such as lift and drag) can be obtained and their physical origins explained. This approach of treating high-Reynolds-number flows provides the basis for modern aerodynamics and hydrodynamics.

The process of simplifying the fluid dynamic equations (and neglecting certain terms) is not arbitrary. In fact, it is based on rigorous assumptions.

Introduction

In Chapter 3 the effects of pressure in a fluid were isolated (because the fluid was not moving), and in Chapter 4 the inertia terms were added. The inclusion of viscosity, its effects, and the resulting velocity distribution are discussed here. For example, the velocity distribution for the laminar flow inside a pipe is formulated and the average velocity is calculated. This provides the relation between the simple 1D average velocity model (of Chapter 4) and the more complex (and realistic) 2D or 3D flows.

The solutions presented early in this chapter are often called exact solutions. This means that, for a few limited cases, a set of logical assumptions leads to simplification of the fluid dynamic equations, which allows their solution (for laminar flow)! Also, the cases presented in this chapter (e.g., the flow in pipes) is often termed as internal flows. The discussion on external flows is delayed to the following three chapters.

The second part of this chapter demonstrates the approach that evolved during the past 200 years for solving fluid dynamic problems (because there is no closed-form analytic solution to the complete fluid dynamic equations). According to this approach, to develop a practical engineering solution, we must start with a simple but exact solution that determines the major parameters and the basic trends of the problem (e.g., the pressure drop in a circular pipe versus the Reynolds number).

Introduction

The concept of high-Reynolds-number flows was discussed in Chapter 6 and it was concluded that near a solid-body surface for an attached flow a thin boundary layer exists. In Chapter 7 this boundary layer was investigated, and the small-thickness assumption was verified. It was also concluded that the pressure distribution around a vehicle could be obtained by the solution of the inviscid flow outside the thin boundary layer. These modeling conclusions, along with some general features of such flow fields, is summarized in Fig. 8.1.

In term of forces, the boundary layer solution provides the skin-friction estimate and the resulting skin-friction-related drag-force component. However, viscous effects, in addition to the boundary layer, can be present in the wake and in areas of flow separation. For example, we can see the effects of viscous flowmomentum loss by comparing the velocity distribution ahead and behind the vehicle (as shown in the figure). Clearly, in the wake the flow is slower and there is a loss of linear momentum (which is the drag, as was shown in the previous chapter).

Solution of the flow outside this viscous layer should provide information on the velocity and pressure distributions, as depicted by the centerline pressure distribution shown in the upper part of Fig. 8.1 (recall that there is no change in the pressure across the boundary layer). In the case in which the flow is attached, we can define an irrotationl flow model and solve for the velocity distribution.