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We start with a definition:
CFD (computational fluid dynamics) is a set of numerical methods applied to obtain
approximate solutions of problems of fluid dynamics and heat transfer.
According to this definition, CFD is not a science by itself but a way to
apply the methods of one discipline (numerical analysis) to another (heat
and mass transfer). We will deal with details later. Right now, a brief
discussion is in order of why exactly we need CFD.
A distinctive feature of the science of fluid flow and heat and mass
transfer is the approach it takes toward description of physical processes.
Instead of bulk properties, such as momentum or angular momentum of
a body in mechanics or total energy or entropy of a system in thermodynamics, the analysis focuses on distributed properties. We try to determine
entire fields such as temperature T (x , t) velocity v(x , t), density ρ(x , t),
etc.1 Even when an integral characteristic, such as the friction coefficient
or the net rate of heat transfer, is the ultimate goal of analysis, it is derived
from distributed fields.
The approach is very attractive by virtue of the level of details it provides. Evolution of the entire temperature distribution within a body can
1 Throughout the book, we will use x = (x , y, z ) for the vector of space coordinate and t
for time.
be determined. Internal processes of a fluid flow such as motion, rotation,
and deformation of minuscule fluid particles can be taken into account.
Of course, the opportunities come at a price, most notably in the form of
dramatically increased complexity of the governing equations. Except for
a few strongly simplified models, the equations for distributed properties
are partial differential equations, often nonlinear.
As an example of complexity, let us consider a seemingly simple task of
mixing and dissolving sugar in a cup of hot coffee. An innocent question
of how long or how many rotations of a spoon would it take to completely
dissolve the sugar leads to a very complex physical problem that includes
a possibly turbulent two-phase (coffee and sugar particles) flow with a
chemical reaction (dissolving). Heat transfer (within the cup and between
the cup and surroundings) may also be of importance because temperature
affects the rate of the reaction. No simple solution of the problem exists.
Of course, we can rely on the experience acquired after repeating the
process daily (perhaps more than once) for many years. We can also
add a couple of extra, possibly unnecessary, stirs. If, however, the task
in question is more serious—for example, optimizing an oil refinery or
designing a new aircraft—relying on everyday experience or excessive
effort is not an option. We must find a way to understand and predict the
Generally, we can distinguish three approaches to solving fluid flow
and heat transfer problems:
1. Theoretical approach —using governing equations to find analytical
2. Experimental approach —staging a carefully designed experiment
using a model of the real object
3. Numerical approach —using computational procedures to find a
Let’s look at these approaches in more detail.
Theoretical approach. The approach has a crucial advantage of providing exact solutions. Among the disadvantages, the most important is that
analytical solutions are only possible for a very limited class of problems,
typically formulated in an artificial, idealized way. One example is the
Poiseuille solution for a flow in an infinitely long pipe (see Figure 1.1).
The steady-state laminar velocity profile is
U (r) =
r 2 − R 2 dp
4μ dx
Figure 1.1 Laminar flow in an infinite pipe.
where U is the velocity, R is the pipe radius, dp/dx is the constant
pressure gradient that drives the flow, and μ is the dynamic viscosity of the
fluid. On the one hand, the solution is, indeed, simple and gives insight into
the nature of flows in pipes and ducts, so its inclusion into all textbooks of
fluid dynamics is not surprising. On the other hand, the solution is correct
only if the pipe is infinitely long,2 temperature is constant, and the fluid is
perfectly incompressible. Furthermore, even if we were able to build such
a pipe and find a useful application for it, the solution would be correct
only at Reynolds numbers Re = URρ/μ (ρ is the density of the fluid) that
are below approximately 2,000. Above this limit, the flow would assume
fully three-dimensional and time-dependent turbulent form, for which no
analytical solution is possible.
It can also be noted that derivation of analytical solutions often requires
substantial mathematical skills, which are not among the strongest traits
of many modern engineers and scientists, especially if compared to the
situation of 30 or 40 years ago. Several reasons can be named for the deterioration of such skills, one, no doubt, being development of computers
and numerical methods, including the CFD.
Experimental approach. Well-known examples are the wind tunnel
experiments, which help to design and optimize the external shapes of
airplanes (also of ships, buildings, and other objects). Another example
is illustrated in Figure 1.2. The main disadvantages of the experimental approach are the technical difficulty (sometimes it takes several years
before an experiment is set up and all technical problems are resolved)
and high cost.
Numerical (computational) approach. Here, again, we employ our ability to describe almost any fluid flow and heat transfer process as a solution
of a set of partial differential equations. An approximation to this solution
is found in the result of a computational procedure. This approach is not
problem-free, either. We will discuss the problems throughout the book.
2 In
practice, the solution is considered to be a good approximation for laminar flows in
pipes at sufficiently large distance (dependent on the Reynolds number but, at least few
tens of diameters) from the entrance.
Figure 1.2 The experiment for studying thermal convection at the Ilmenau
University of Technology, Germany (courtesy of A. Thess). Turbulent convection
similar to the convection observed in the atmosphere of Earth or Sun is simulated
by air motion within a large barrel with thermally insulated walls and uniformly
heated bottom.
The computational approach, however, beats the analytical and experimental methods in some very important aspects: universality, flexibility,
accuracy, and cost.
The history of CFD is a fascinating subject, which, unfortunately, we
can only touch in passing. The idea to calculate approximate solutions of
differential equations describing fluid flows and heat transfer is relatively
old. It is definitely older than computers themselves. Development of
numerical methods for solving ordinary and partial differential equations
started in the first half of the twentieth century. The computations at
that time required use of tables and dull mechanical work of dozens,
if not hundreds, of people. No wonder that only the most important
(primarily military-related) problems were addressed and only simple,
one-dimensional equations were solved.
Invention and subsequent fast development of computers (see
Figure 1.3) opened a wonderful possibility of performing millions—and
then millions of millions—of arithmetic operations in a matter of
seconds. This caused a rapid growth of the efforts to develop and apply
methods of numerical simulations. Again, military applications, such as
modeling shock waves from an explosion or a flow past a hypersonic
Peak Floating Point Performance [flops]
Blue Gene/L
Cray T3E(64 proc.)
Cray I
Mark I
Figure 1.3 Development of high-performance computers. The speed measured as
the number of floating operations per second grows approximately tenfold every
five years.
jet aircraft were addressed first. In fact, development of faster and
bigger computers until 1980s was largely motivated by the demands of
military-related CFD. First simulations of realistic two-dimensional flows
were performed in the late 1960s, while three-dimensional flows could
not be seriously approached until the 1980s.
In the last 20 to 30 years, the computer revolution has changed the field
of CFD entirely. From a scientific discipline, in which researchers worked
on unique projects using specially developed codes, it has transformed
into an everyday tool of engineering design, optimization, and analysis.
The simulations are routinely used as a replacement of or addition to
prototyping and other design techniques. The problem-specific codes are
still developed for scientific purposes, but the engineering practice has
almost entirely switched to the use of commercial or open-source CFD
codes. The market is largely divided between a few major brands, such as
FLUENT, STAR-CD, CFX, OpenFOAM, and COMSOL. They differ in
appearance and capabilities but are all essentially the numerical solvers of
partial differential equations with attached physical and turbulence models,
as well as modules for grid generation and post-processing the results.
This book is intended as a brief but complete introduction into CFD.
The focus is not on development of algorithms but on the fundamental
principles, formulation of CFD problems, the most basic and common
computational techniques, and essentials of a good CFD analysis. The
book’s main task is to prepare the reader to make educated choices while
using one of the ready CFD codes. A reader seeking deeper and more
detailed understanding of specific computational methods is encouraged
to use more advanced and more specialized texts, references to some of
which are presented at the end of each chapter.
A comment is in order regarding the bias of the text. All CFD texts
are, to some degree, biased in correspondence with the chosen audience
and personal research interests of the authors. More weight is given
to some of the methods (finite difference, finite element, spectral, etc.)
and some of the fields of application (heat transfer, incompressible fluid
dynamics, or gas dynamics). The preferences made in this book reflect
the choice of mechanical, chemical, and civil engineers as the target
audience and the intended use for applied CFD instruction. The focus is
on the finite difference and finite volume methods. The finite element and
spectral techniques are introduced, but only briefly. Also, more attention
is given to numerical methods for incompressible fluid dynamics and
heat transfer than for compressible sub- and supersonic flows.
The book contains 13 chapters. We are already at the end of Chapter 1.
The remaining chapters are separated into three parts: “Fundamentals,”
“Methods,” and “Art of CFD.” Part I deals with the basic concepts of
numerical solution of partial differential equations. It starts with Chapter
2 introducing the equations we are most likely to solve: the governing
equations of fluid flows and heat transfer. We consider various forms
of the equations used in CFD and review common boundary conditions.
Necessary mathematical background and the concept of numerical approximation are presented in Chapter 3. Chapter 4 discusses the basics of the
finite difference method. We also introduce the key concepts associated
with all CFD methods, such as the truncation error and consistency of
numerical approximation. The principles and main tools of the finite volume method are presented in Chapter 5. Chapter 6 is devoted to the
concept of stability of numerical time integration. Some popular and
important (both historically and didactically) schemes for one-dimensional
model equations are presented in Chapter 7. The material summarizes the
discussion of the fundamental concepts and can be used for a midterm
programming project.
Part II, which includes Chapters 8 through 10, contains a compact
description of some of the most important and commonly used CFD techniques. Methods of solution of systems of algebraic equations appearing
in the result of the CFD approximation are discussed in Chapter 8. Chapter
9 presents some schemes used for nonsteady heat conduction and compressible flows. The discussion is deliberately brief for such voluminous
subjects. It is expected that a reader with particular interest in any of them
will refer to other, more specialized texts. Significantly more attention
is given to the methods developed for computation of flows of incompressible fluids. Chapter 10 provides a relatively broad explanation of
the issues, presents the projection method, and introduces some popular
Part III consists of Chapters 11 to 13 and deals with subjects that
are not directly related to the numerical solution of partial differential
equations, but nevertheless are irreplaceable in practical CFD analysis.
They all belong to a somewhat imprecise science in the sense that the
approach is often decided on the basis of knowledge and experience rather
than exact knowledge alone. The subjects in question are the turbulence
modeling (Chapter 11), types and quality of computational grids (Chapter
12), and the complex of issues arising in the course of CFD analysis, such
as uncertainty and validation of results (Chapter 13). The discussion is, by
necessity, brief. A reader willing to acquire truly adequate understanding
of these difficult but fascinating topics should consult the books listed at
the end of each chapter.
http://www.top500.org/— Official Web site of the TOP500 project providing
reliable and detailed information on the world most powerful supercomputers.
http://www.cfd-online.com/— A rich source of information on CFD: books,
links, discussion forums, jobs, etc.
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