1. Fundamentals of Computational Fluid
Dynamics
Harvard Lomax and Thomas H. Pulliam
NASA Ames Research Center
David W. Zingg
University of Toronto Institute for Aerospace Studies
August 26, 1999
10. Chapter 1
INTRODUCTION
1.1
Motivation
The material in this book originated from attempts to understand and systemize numerical solution techniques for the partial di erential equations governing the physics
of uid ow. As time went on and these attempts began to crystallize, underlying
constraints on the nature of the material began to form. The principal such constraint
was the demand for uni cation. Was there one mathematical structure which could
be used to describe the behavior and results of most numerical methods in common
use in the eld of uid dynamics? Perhaps the answer is arguable, but the authors
believe the answer is a rmative and present this book as justi cation for that belief. The mathematical structure is the theory of linear algebra and the attendant
eigenanalysis of linear systems.
The ultimate goal of the eld of computational uid dynamics (CFD) is to understand the physical events that occur in the ow of uids around and within designated
objects. These events are related to the action and interaction of phenomena such
as dissipation, di usion, convection, shock waves, slip surfaces, boundary layers, and
turbulence. In the eld of aerodynamics, all of these phenomena are governed by
the compressible Navier-Stokes equations. Many of the most important aspects of
these relations are nonlinear and, as a consequence, often have no analytic solution.
This, of course, motivates the numerical solution of the associated partial di erential
equations. At the same time it would seem to invalidate the use of linear algebra for
the classi cation of the numerical methods. Experience has shown that such is not
the case.
As we shall see in a later chapter, the use of numerical methods to solve partial
di erential equations introduces an approximation that, in e ect, can change the
form of the basic partial di erential equations themselves. The new equations, which
1
11. CHAPTER 1. INTRODUCTION
2
are the ones actually being solved by the numerical process, are often referred to as
the modi ed partial di erential equations. Since they are not precisely the same as
the original equations, they can, and probably will, simulate the physical phenomena
listed above in ways that are not exactly the same as an exact solution to the basic
partial di erential equation. Mathematically, these di erences are usually referred to
as truncation errors. However, the theory associated with the numerical analysis of
uid mechanics was developed predominantly by scientists deeply interested in the
physics of uid ow and, as a consequence, these errors are often identi ed with a
particular physical phenomenon on which they have a strong e ect. Thus methods are
said to have a lot of arti cial viscosity" or said to be highly dispersive. This means
that the errors caused by the numerical approximation result in a modi ed partial
di erential equation having additional terms that can be identi ed with the physics
of dissipation in the rst case and dispersion in the second. There is nothing wrong,
of course, with identifying an error with a physical process, nor with deliberately
directing an error to a speci c physical process, as long as the error remains in some
engineering sense small". It is safe to say, for example, that most numerical methods
in practical use for solving the nondissipative Euler equations create a modi ed partial
di erential equation that produces some form of dissipation. However, if used and
interpreted properly, these methods give very useful information.
Regardless of what the numerical errors are called, if their e ects are not thoroughly understood and controlled, they can lead to serious di culties, producing
answers that represent little, if any, physical reality. This motivates studying the
concepts of stability, convergence, and consistency. On the other hand, even if the
errors are kept small enough that they can be neglected (for engineering purposes),
the resulting simulation can still be of little practical use if ine cient or inappropriate
algorithms are used. This motivates studying the concepts of sti ness, factorization,
and algorithm development in general. All of these concepts we hope to clarify in
this book.
1.2
Background
The eld of computational uid dynamics has a broad range of applicability. Independent of the speci c application under study, the following sequence of steps generally
must be followed in order to obtain a satisfactory solution.
1.2.1 Problem Speci cation and Geometry Preparation
The rst step involves the speci cation of the problem, including the geometry, ow
conditions, and the requirements of the simulation. The geometry may result from
12. 1.2. BACKGROUND
3
measurements of an existing con guration or may be associated with a design study.
Alternatively, in a design context, no geometry need be supplied. Instead, a set
of objectives and constraints must be speci ed. Flow conditions might include, for
example, the Reynolds number and Mach number for the ow over an airfoil. The
requirements of the simulation include issues such as the level of accuracy needed, the
turnaround time required, and the solution parameters of interest. The rst two of
these requirements are often in con ict and compromise is necessary. As an example
of solution parameters of interest in computing the ow eld about an airfoil, one may
be interested in i) the lift and pitching moment only, ii) the drag as well as the lift
and pitching moment, or iii) the details of the ow at some speci c location.
1.2.2 Selection of Governing Equations and Boundary Conditions
Once the problem has been speci ed, an appropriate set of governing equations and
boundary conditions must be selected. It is generally accepted that the phenomena of
importance to the eld of continuum uid dynamics are governed by the conservation
of mass, momentum, and energy. The partial di erential equations resulting from
these conservation laws are referred to as the Navier-Stokes equations. However, in
the interest of e ciency, it is always prudent to consider solving simpli ed forms
of the Navier-Stokes equations when the simpli cations retain the physics which are
essential to the goals of the simulation. Possible simpli ed governing equations include
the potential- ow equations, the Euler equations, and the thin-layer Navier-Stokes
equations. These may be steady or unsteady and compressible or incompressible.
Boundary types which may be encountered include solid walls, in ow and out ow
boundaries, periodic boundaries, symmetry boundaries, etc. The boundary conditions
which must be speci ed depend upon the governing equations. For example, at a solid
wall, the Euler equations require ow tangency to be enforced, while the Navier-Stokes
equations require the no-slip condition. If necessary, physical models must be chosen
for processes which cannot be simulated within the speci ed constraints. Turbulence
is an example of a physical process which is rarely simulated in a practical context (at
the time of writing) and thus is often modelled. The success of a simulation depends
greatly on the engineering insight involved in selecting the governing equations and
physical models based on the problem speci cation.
1.2.3 Selection of Gridding Strategy and Numerical Method
Next a numerical method and a strategy for dividing the ow domain into cells, or
elements, must be selected. We concern ourselves here only with numerical methods requiring such a tessellation of the domain, which is known as a grid, or mesh.
13. CHAPTER 1. INTRODUCTION
4
Many di erent gridding strategies exist, including structured, unstructured, hybrid,
composite, and overlapping grids. Furthermore, the grid can be altered based on
the solution in an approach known as solution-adaptive gridding. The numerical
methods generally used in CFD can be classi ed as nite-di erence, nite-volume,
nite-element, or spectral methods. The choices of a numerical method and a gridding strategy are strongly interdependent. For example, the use of nite-di erence
methods is typically restricted to structured grids. Here again, the success of a simulation can depend on appropriate choices for the problem or class of problems of
interest.
1.2.4 Assessment and Interpretation of Results
Finally, the results of the simulation must be assessed and interpreted. This step can
require post-processing of the data, for example calculation of forces and moments,
and can be aided by sophisticated ow visualization tools and error estimation techniques. It is critical that the magnitude of both numerical and physical-model errors
be well understood.
1.3
Overview
It should be clear that successful simulation of uid ows can involve a wide range of
issues from grid generation to turbulence modelling to the applicability of various simpli ed forms of the Navier-Stokes equations. Many of these issues are not addressed
in this book. Some of them are presented in the books by Anderson, Tannehill, and
Pletcher 1] and Hirsch 2]. Instead we focus on numerical methods, with emphasis
on nite-di erence and nite-volume methods for the Euler and Navier-Stokes equations. Rather than presenting the details of the most advanced methods, which are
still evolving, we present a foundation for developing, analyzing, and understanding
such methods.
Fortunately, to develop, analyze, and understand most numerical methods used to
nd solutions for the complete compressible Navier-Stokes equations, we can make use
of much simpler expressions, the so-called model" equations. These model equations
isolate certain aspects of the physics contained in the complete set of equations. Hence
their numerical solution can illustrate the properties of a given numerical method
when applied to a more complicated system of equations which governs similar physical phenomena. Although the model equations are extremely simple and easy to
solve, they have been carefully selected to be representative, when used intelligently,
of di culties and complexities that arise in realistic two- and three-dimensional uid
ow simulations. We believe that a thorough understanding of what happens when
14. 1.4. NOTATION
5
numerical approximations are applied to the model equations is a major rst step in
making con dent and competent use of numerical approximations to the Euler and
Navier-Stokes equations. As a word of caution, however, it should be noted that,
although we can learn a great deal by studying numerical methods as applied to the
model equations and can use that information in the design and application of numerical methods to practical problems, there are many aspects of practical problems
which can only be understood in the context of the complete physical systems.
1.4
Notation
The notation is generally explained as it is introduced. Bold type is reserved for real
physical vectors, such as velocity. The vector symbol ~ is used for the vectors (or
column matrices) which contain the values of the dependent variable at the nodes
of a grid. Otherwise, the use of a vector consisting of a collection of scalars should
be apparent from the context and is not identi ed by any special notation. For
example, the variable u can denote a scalar Cartesian velocity component in the Euler
and Navier-Stokes equations, a scalar quantity in the linear convection and di usion
equations, and a vector consisting of a collection of scalars in our presentation of
hyperbolic systems. Some of the abbreviations used throughout the text are listed
and de ned below.
PDE
ODE
O E
RHS
P.S.
S.S.
k-D
~
bc
O( )
Partial di erential equation
Ordinary di erential equation
Ordinary di erence equation
Right-hand side
Particular solution of an ODE or system of ODE's
Fixed (time-invariant) steady-state solution
k-dimensional space
Boundary conditions, usually a vector
A term of order (i.e., proportional to)
16. Chapter 2
CONSERVATION LAWS AND
THE MODEL EQUATIONS
We start out by casting our equations in the most general form, the integral conservation-law form, which is useful in understanding the concepts involved in nite-volume
schemes. The equations are then recast into divergence form, which is natural for
nite-di erence schemes. The Euler and Navier-Stokes equations are brie y discussed
in this Chapter. The main focus, though, will be on representative model equations,
in particular, the convection and di usion equations. These equations contain many
of the salient mathematical and physical features of the full Navier-Stokes equations.
The concepts of convection and di usion are prevalent in our development of numerical methods for computational uid dynamics, and the recurring use of these
model equations allows us to develop a consistent framework of analysis for consistency, accuracy, stability, and convergence. The model equations we study have two
properties in common. They are linear partial di erential equations (PDE's) with
coe cients that are constant in both space and time, and they represent phenomena
of importance to the analysis of certain aspects of uid dynamic problems.
2.1 Conservation Laws
Conservation laws, such as the Euler and Navier-Stokes equations and our model
equations, can be written in the following integral form:
Z
V (t2 )
QdV ;
Z
V (t1 )
QdV +
Z t2 I
t1
S (t)
n:FdSdt =
Z t2 Z
t1
V (t)
PdV dt
(2.1)
In this equation, Q is a vector containing the set of variables which are conserved,
e.g., mass, momentum, and energy, per unit volume. The equation is a statement of
7
17. 8
CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS
the conservation of these quantities in a nite region of space with volume V (t) and
surface area S (t) over a nite interval of time t2 ; t1 . In two dimensions, the region
of space, or cell, is an area A(t) bounded by a closed contour C (t). The vector n is
a unit vector normal to the surface pointing outward, F is a set of vectors, or tensor,
containing the ux of Q per unit area per unit time, and P is the rate of production
of Q per unit volume per unit time. If all variables are continuous in time, then Eq.
2.1 can be rewritten as
d Z QdV + I n:FdS = Z PdV
(2.2)
dt V (t)
S (t)
V (t)
Those methods which make various numerical approximations of the integrals in Eqs.
2.1 and 2.2 and nd a solution for Q on that basis are referred to as nite-volume
methods. Many of the advanced codes written for CFD applications are based on the
nite-volume concept.
On the other hand, a partial derivative form of a conservation law can also be
derived. The divergence form of Eq. 2.2 is obtained by applying Gauss's theorem to
the ux integral, leading to
@Q + r:F = P
(2.3)
@t
where r: is the well-known divergence operator given, in Cartesian coordinates, by
!
@ +j @ +k @ :
(2.4)
r: i @x @y @z
and i j, and k are unit vectors in the x y, and z coordinate directions, respectively.
Those methods which make various approximations of the derivatives in Eq. 2.3 and
nd a solution for Q on that basis are referred to as nite-di erence methods.
2.2 The Navier-Stokes and Euler Equations
The Navier-Stokes equations form a coupled system of nonlinear PDE's describing
the conservation of mass, momentum and energy for a uid. For a Newtonian uid
in one dimension, they can be written as
@Q + @E = 0
(2.5)
@t @x
with
2 3
2
3
u 3 2
0
6
6
Q=6
6
6
4
7
7
u7
7
7
5
e
E
6
6
=6
6
6
4
7 6
7 6
2+p 7;6
u
7 6
7 6
5 4
u(e + p)
4 @u
3 @x
4
3
u @u +
@x
@T
@x
7
7
7
7
7
5
(2.6)
18. 2.2. THE NAVIER-STOKES AND EULER EQUATIONS
9
where is the uid density, u is the velocity, e is the total energy per unit volume, p is
the pressure, T is the temperature, is the coe cient of viscosity, and is the thermal
conductivity. The total energy e includes internal energy per unit volume (where
is the internal energy per unit mass) and kinetic energy per unit volume u2=2.
These equations must be supplemented by relations between and and the uid
state as well as an equation of state, such as the ideal gas law. Details can be found
in Anderson, Tannehill, and Pletcher 1] and Hirsch 2]. Note that the convective
uxes lead to rst derivatives in space, while the viscous and heat conduction terms
involve second derivatives. This form of the equations is called conservation-law or
conservative form. Non-conservative forms can be obtained by expanding derivatives
of products using the product rule or by introducing di erent dependent variables,
such as u and p. Although non-conservative forms of the equations are analytically
the same as the above form, they can lead to quite di erent numerical solutions in
terms of shock strength and shock speed, for example. Thus the conservative form is
appropriate for solving ows with features such as shock waves.
Many ows of engineering interest are steady (time-invariant), or at least may be
treated as such. For such ows, we are often interested in the steady-state solution of
the Navier-Stokes equations, with no interest in the transient portion of the solution.
The steady solution to the one-dimensional Navier-Stokes equations must satisfy
@E = 0
(2.7)
@x
If we neglect viscosity and heat conduction, the Euler equations are obtained. In
two-dimensional Cartesian coordinates, these can be written as
@Q + @E + @F = 0
(2.8)
@t @x @y
with
2 3 2 3
2
2
q1
u 3
v 3
6 7 6 7
6 2+ 7
6
7
Q = 6 q2 7 = 6 u 7 E = 6 u uv p 7 F = 6 v2uv p 7
6 7 6 7
6
7
6
(2.9)
4 q3 5 4 v 5
4
5
4
5
+ 7
q4
e
u(e + p)
v ( e + p)
where u and v are the Cartesian velocity components. Later on we will make use of
the following form of the Euler equations as well:
@Q + A @Q + B @Q = 0
(2.10)
@t
@x
@y
@E
@F
The matrices A = @Q and B = @Q are known as the ux Jacobians. The ux vectors
given above are written in terms of the primitive variables, , u, v, and p. In order
19. 10
CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS
to derive the ux Jacobian matrices, we must rst write the ux vectors E and F in
terms of the conservative variables, q1 , q2, q3 , and q4 , as follows:
E
F
2
3
6
7 6
7 6
7 6
7 6
7 6
7=6
7 6
7 6
7 6
7 6
7 6
5 6
6
4
2
6
6
6
6
6
=6
6
6
6
6
6
4
E1
2
6
6
6
6
6
=6
6
6
6
6
6
4
3 2
F1 7 6
7 6
7 6
6
F2 7 6
7 6
7=6
7 6
7 6
F3 7 6
7 6
7 6
5 6
4
F4
E2
E3
E4
3
7
7
7
2
2 7
3; q2 ; ;1 q3 7
; 1)q4 + 2 q1 2 q1 7
7
7
7
q3 q2
7
7
q1
7
7
7
5
3
2q
q4 q2 ; ;1 q2 + q3 2
2
2
q1
2 q1
q1
(2.11)
3
7
7
7
7
7
7
7
7
2 7
;1 q2 7
2 q1 7
7
7
5
3
q3
2
q1
(2.12)
q2
(
q3
q3 q2
q1
( ; 1)q4 + 3;
2
q4 q 3
q1
;
2
q3
q1
2q
;1 q2 2 3
2
q1
;
+
We have assumed that the pressure satis es p = ( ; 1) e ; (u2 + v2)=2] from the
ideal gas law, where is the ratio of speci c heats, cp=cv . From this it follows that
the ux Jacobian of E can be written in terms of the conservative variables as
A=
2
6
6
6
6
6
@Ei = 6
6
6
@qj 6
6
6
6
6
4
0
1
0
0
a21
(1 ; ) q3
q1
q2
q1
;1
q
; q2 q1
q1 3
(3 ; ) q2
q1
q
q3
1
a41
a42
a43
q2
q1
where
a21 =
!
; 1 q3 2 ; 3 ;
2 q1
2
q2
q1
!2
0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
5
(2.13)
20. 2.2. THE NAVIER-STOKES AND EULER EQUATIONS
2
; 1)4
+ q3
q1
!2
q4 ; ; 1 43 q2
a42 =
q1
2
q1
!
!
q2 q3
a43 = ;( ; 1) q q
1
1
and in terms of the primitive variables as
!2
a41 = (
q2
q1
!3
2
!
2
6
6
6
6
6
A=6
6
6
6
6
6
4
0
1
q2
q1
!3
5;
+ q3
q1
!2 3
5
0
0
;uv
v
u
0
a41
a42
a43
u
a21 =
q4
q1
!
q2
q1
!
(2.14)
a21 (3 ; )u (1 ; )v ( ; 1)
where
11
3
7
7
7
7
7
7
7
7
7
7
7
5
(2.15)
; 1 v2 ; 3 ; u2
2
2
a41 = ( ; 1)u(u2 + v2) ; ue
a42 =
e ; ; 1 (3u2 + v2 )
2
a43 = (1 ; )uv
(2.16)
Derivation of the two forms of B = @F=@Q is similar. The eigenvalues of the ux
Jacobian matrices are purely real. This is the de ning feature of hyperbolic systems
of PDE's, which are further discussed in Section 2.5. The homogeneous property of
the Euler equations is discussed in Appendix C.
The Navier-Stokes equations include both convective and di usive uxes. This
motivates the choice of our two scalar model equations associated with the physics
of convection and di usion. Furthermore, aspects of convective phenomena associated with coupled systems of equations such as the Euler equations are important in
developing numerical methods and boundary conditions. Thus we also study linear
hyperbolic systems of PDE's.
21. 12
CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS
2.3 The Linear Convection Equation
2.3.1 Di erential Form
The simplest linear model for convection and wave propagation is the linear convection
equation given by the following PDE:
@u + a @u = 0
(2.17)
@t @x
Here u(x t) is a scalar quantity propagating with speed a, a real constant which may
be positive or negative. The manner in which the boundary conditions are speci ed
separates the following two phenomena for which this equation is a model:
(1) In one type, the scalar quantity u is given on one boundary, corresponding
to a wave entering the domain through this in ow" boundary. No boundary condition is speci ed at the opposite side, the out ow" boundary. This
is consistent in terms of the well-posedness of a 1st-order PDE. Hence the
wave leaves the domain through the out ow boundary without distortion or
re ection. This type of phenomenon is referred to, simply, as the convection
problem. It represents most of the usual" situations encountered in convecting systems. Note that the left-hand boundary is the in ow boundary when
a is positive, while the right-hand boundary is the in ow boundary when a is
negative.
(2) In the other type, the ow being simulated is periodic. At any given time,
what enters on one side of the domain must be the same as that which is
leaving on the other. This is referred to as the biconvection problem. It is
the simplest to study and serves to illustrate many of the basic properties of
numerical methods applied to problems involving convection, without special
consideration of boundaries. Hence, we pay a great deal of attention to it in
the initial chapters.
Now let us consider a situation in which the initial condition is given by u(x 0) =
u0(x), and the domain is in nite. It is easy to show by substitution that the exact
solution to the linear convection equation is then
u(x t) = u0(x ; at)
(2.18)
The initial waveform propagates unaltered with speed jaj to the right if a is positive
and to the left if a is negative. With periodic boundary conditions, the waveform
travels through one boundary and reappears at the other boundary, eventually returning to its initial position. In this case, the process continues forever without any
22. 2.3. THE LINEAR CONVECTION EQUATION
13
change in the shape of the solution. Preserving the shape of the initial condition
u0(x) can be a di cult challenge for a numerical method.
2.3.2 Solution in Wave Space
We now examine the biconvection problem in more detail. Let the domain be given
by 0 x 2 . We restrict our attention to initial conditions in the form
u(x 0) = f (0)ei x
(2.19)
where f (0) is a complex constant, and is the wavenumber. In order to satisfy the
periodic boundary conditions, must be an integer. It is a measure of the number of
wavelengths within the domain. With such an initial condition, the solution can be
written as
u(x t) = f (t)ei x
(2.20)
where the time dependence is contained in the complex function f (t). Substituting
this solution into the linear convection equation, Eq. 2.17, we nd that f (t) satis es
the following ordinary di erential equation (ODE)
df = ;ia f
(2.21)
dt
which has the solution
f (t) = f (0)e;ia t
(2.22)
Substituting f (t) into Eq. 2.20 gives the following solution
u(x t) = f (0)ei (x;at) = f (0)ei( x;!t)
(2.23)
where the frequency, !, the wavenumber, , and the phase speed, a, are related by
!= a
(2.24)
The relation between the frequency and the wavenumber is known as the dispersion
relation. The linear relation given by Eq. 2.24 is characteristic of wave propagation
in a nondispersive medium. This means that the phase speed is the same for all
wavenumbers. As we shall see later, most numerical methods introduce some dispersion that is, in a simulation, waves with di erent wavenumbers travel at di erent
speeds.
An arbitrary initial waveform can be produced by summing initial conditions of
the form of Eq. 2.19. For M modes, one obtains
u(x 0) =
M
X
m=1
fm (0)ei mx
(2.25)
23. 14
CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS
where the wavenumbers are often ordered such that 1 2
M . Since the
wave equation is linear, the solution is obtained by summing solutions of the form of
Eq. 2.23, giving
u(x t) =
M
X
m=1
fm (0)ei m(x;at)
(2.26)
Dispersion and dissipation resulting from a numerical approximation will cause the
shape of the solution to change from that of the original waveform.
2.4 The Di usion Equation
2.4.1 Di erential Form
Di usive uxes are associated with molecular motion in a continuum uid. A simple
linear model equation for a di usive process is
@u = @ 2 u
(2.27)
@t
@x2
where is a positive real constant. For example, with u representing the temperature, this parabolic PDE governs the di usion of heat in one dimension. Boundary
conditions can be periodic, Dirichlet (speci ed u), Neumann (speci ed @u=@x), or
mixed Dirichlet/Neumann.
In contrast to the linear convection equation, the di usion equation has a nontrivial
steady-state solution, which is one that satis es the governing PDE with the partial
derivative in time equal to zero. In the case of Eq. 2.27, the steady-state solution
must satisfy
@2u = 0
(2.28)
@x2
Therefore, u must vary linearly with x at steady state such that the boundary conditions are satis ed. Other steady-state solutions are obtained if a source term g(x)
is added to Eq. 2.27, as follows:
@u =
@t
" 2
@u
@x2
; g(x)
#
(2.29)
giving a steady state-solution which satis es
@ 2 u ; g(x) = 0
@x2
(2.30)
24. 2.4. THE DIFFUSION EQUATION
15
In two dimensions, the di usion equation becomes
"
#
@u = @ 2 u + @ 2 u ; g(x y)
(2.31)
@t
@x2 @y2
where g(x y) is again a source term. The corresponding steady equation is
@ 2 u + @ 2 u ; g(x y) = 0
(2.32)
@x2 @y2
While Eq. 2.31 is parabolic, Eq. 2.32 is elliptic. The latter is known as the Poisson
equation for nonzero g, and as Laplace's equation for zero g.
2.4.2 Solution in Wave Space
We now consider a series solution to Eq. 2.27. Let the domain be given by 0 x
with boundary conditions u(0) = ua, u( ) = ub. It is clear that the steady-state
solution is given by a linear function which satis es the boundary conditions, i.e.,
h(x) = ua + (ub ; ua)x= . Let the initial condition be
u(x 0) =
M
X
m=1
fm(0) sin m x + h(x)
(2.33)
where must be an integer in order to satisfy the boundary conditions. A solution
of the form
M
X
u(x t) = fm(t) sin mx + h(x)
(2.34)
m=1
satis es the initial and boundary conditions. Substituting this form into Eq. 2.27
gives the following ODE for fm :
dfm = ; 2 f
(2.35)
m m
dt
and we nd
m
fm(t) = fm(0)e; 2 t
(2.36)
Substituting fm(t) into equation 2.34, we obtain
u(x t) =
M
X
m=1
m
fm (0)e; 2 t sin m x + h(x)
(2.37)
The steady-state solution (t ! 1) is simply h(x). Eq. 2.37 shows that high wavenumber components (large m) of the solution decay more rapidly than low wavenumber
components, consistent with the physics of di usion.
25. CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS
16
2.5 Linear Hyperbolic Systems
The Euler equations, Eq. 2.8, form a hyperbolic system of partial di erential equations. Other systems of equations governing convection and wave propagation phenomena, such as the Maxwell equations describing the propagation of electromagnetic
waves, are also of hyperbolic type. Many aspects of numerical methods for such systems can be understood by studying a one-dimensional constant-coe cient linear
system of the form
@u + A @u = 0
(2.38)
@t
@x
where u = u(x t) is a vector of length m and A is a real m m matrix. For
conservation laws, this equation can also be written in the form
@u + @f = 0
(2.39)
@t @x
@f
where f is the ux vector and A = @u is the ux Jacobian matrix. The entries in the
ux Jacobian are
@f
aij = @ui
(2.40)
j
The ux Jacobian for the Euler equations is derived in Section 2.2.
Such a system is hyperbolic if A is diagonalizable with real eigenvalues.1 Thus
= X ;1AX
(2.41)
where is a diagonal matrix containing the eigenvalues of A, and X is the matrix
of right eigenvectors. Premultiplying Eq. 2.38 by X ;1 , postmultiplying A by the
product XX ;1, and noting that X and X ;1 are constants, we obtain
@X ;1u
z }| {
@ X ;1AX X ;1u
=0
(2.42)
@t +
@x
With w = X ;1u, this can be rewritten as
@w + @w = 0
(2.43)
@t
@x
When written in this manner, the equations have been decoupled into m scalar equations of the form
@wi + @wi = 0
(2.44)
i
@t
@x
1
See Appendix A for a brief review of some basic relations and de nitions from linear algebra.
26. 2.6. PROBLEMS
17
The elements of w are known as characteristic variables. Each characteristic variable
satis es the linear convection equation with the speed given by the corresponding
eigenvalue of A.
Based on the above, we see that a hyperbolic system in the form of Eq. 2.38 has a
solution given by the superposition of waves which can travel in either the positive or
negative directions and at varying speeds. While the scalar linear convection equation
is clearly an excellent model equation for hyperbolic systems, we must ensure that
our numerical methods are appropriate for wave speeds of arbitrary sign and possibly
widely varying magnitudes.
The one-dimensional Euler equations can also be diagonalized, leading to three
equations in the form of the linear convection equation, although they remain nonlinear, of course. The eigenvalues of the ux Jacobian matrix, or wave speeds, are
q
u u + c, and u ; c, where u is the local uid velocity, and c = p= is the local
speed of sound. The speed u is associated with convection of the uid, while u + c
and u ; c are associated with sound waves. Therefore, in a supersonic ow, where
juj > c, all of the wave speeds have the same sign. In a subsonic ow, where juj < c,
wave speeds of both positive and negative sign are present, corresponding to the fact
that sound waves can travel upstream in a subsonic ow.
The signs of the eigenvalues of the matrix A are also important in determining
suitable boundary conditions. The characteristic variables each satisfy the linear convection equation with the wave speed given by the corresponding eigenvalue. Therefore, the boundary conditions can be speci ed accordingly. That is, characteristic
variables associated with positive eigenvalues can be speci ed at the left boundary,
which corresponds to in ow for these variables. Characteristic variables associated
with negative eigenvalues can be speci ed at the right boundary, which is the inow boundary for these variables. While other boundary condition treatments are
possible, they must be consistent with this approach.
2.6 Problems
1. Show that the 1-D Euler equations can be written in terms of the primitive
variables R = u p]T as follows:
@R + M @R = 0
@t
@x
where
2
3
u
0
M = 6 0 u ;1 7
4
5
0 p u
27. 18
CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS
Assume an ideal gas, p = ( ; 1)(e ; u2=2).
2. Find the eigenvalues and eigenvectors of the matrix M derived in question 1.
3. Derive the ux Jacobian matrix A = @E=@Q for the 1-D Euler equations resulting from the conservative variable formulation (Eq. 2.5). Find its eigenvalues
and compare with those obtained in question 2.
4. Show that the two matrices M and A derived in questions 1 and 3, respectively,
are related by a similarity transform. (Hint: make use of the matrix S =
@Q=@R.)
5. Write the 2-D di usion equation, Eq. 2.31, in the form of Eq. 2.2.
6. Given the initial condition u(x 0) = sin x de ned on 0 x 2 , write it in the
form of Eq. 2.25, that is, nd the necessary values of fm (0). (Hint: use M = 2
with 1 = 1 and 2 = ;1.) Next consider the same initial condition de ned
only at x = 2 j=4, j = 0 1 2 3. Find the values of fm(0) required to reproduce
the initial condition at these discrete points using M = 4 with m = m ; 1.
7. Plot the rst three basis functions used in constructing the exact solution to
the di usion equation in Section 2.4.2. Next consider a solution with boundary
conditions ua = ub = 0, and initial conditions from Eq. 2.33 with fm(0) = 1
for 1 m 3, fm (0) = 0 for m > 3. Plot the initial condition on the domain
0 x . Plot the solution at t = 1 with = 1.
8. Write the classical wave equation @ 2 u=@t2 = c2 @ 2 u=@x2 as a rst-order system,
i.e., in the form
@U + A @U = 0
@t
@x
where U = @u=@x @u=@t]T . Find the eigenvalues and eigenvectors of A.
9. The Cauchy-Riemann equations are formed from the coupling of the steady
compressible continuity (conservation of mass) equation
@ u+@ v =0
@x @y
and the vorticity de nition
@v
! = ; @x + @u = 0
@y
28. 2.6. PROBLEMS
19
where ! = 0 for irrotational ow. For isentropic and homenthalpic ow, the
system is closed by the relation
= 1 ; ; 1 u2 + v2 ; 1
2
1
;1
Note that the variables have been nondimensionalized. Combining the two
PDE's, we have
@f (q) + @g(q) = 0
@x
@y
where
v
q= u
f = ;v u
g = ;u
v
;
One approach to solving these equations is to add a time-dependent term and
nd the steady solution of the following equation:
@q + @f + @g = 0
@t @x @y
(a) Find the ux Jacobians of f and g with respect to q.
(b) Determine the eigenvalues of the ux Jacobians.
(c) Determine the conditions (in terms of and u) under which the system is
hyperbolic, i.e., has real eigenvalues.
(d) Are the above uxes homogeneous? (See Appendix C.)
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