DIFFUSIVELY DRIVEN SHEAR FLOWS IN STRATIFIED FLUIDS
George Perry Harabin
A dissertation submitted to the faculty at the University of North Carolina at Chapel Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of
Mathematics in the College of Arts and Sciences.
Chapel Hill 2016
ABSTRACT
George Perry Harabin: Diffusively Driven Shear Flows in Stratified Fluids (Under the direction of Roberto Camassa and Richard M. McLaughlin)
It is only recently that O.M. Phillips [10] showed that in the presence of an impermeable, non-vertical plane inserted into a stratified fluid creates a parallel shear flow up the plane. The present dissertation combines theoretical, numerical, and experimental extensions of Phillips’ work on diffusively driven flows such as time dependence, three-dimensional effects, and optimal shapes. We first develop the governing equations for time-dependent three-dimensional diffusively driven flows in cylindrical pipe geometries. Using Phillips’ time independent solutions we non-dimensionalize this system and identify the Schmidt number as the single free parameter. We then consider a simple extension of Phillips’ solutions to the time dependent case; the Laplace transform is used to recast the governing equations in the frequency domain, where the role of the Schmidt number can be understood. Long time asymptotics for the time dependent system are derived and a principal temporal frequency p1
Sc is identified. A numerical integration of the governing equations reveals
that in systems with small Schmidt numbers, this oscillations at this frequency are prominent in the time evolution.
We then consider the time independent problem for non-planar geometries where the governing system becomes a pair of coupled Poisson’s equations. Two methods of solution are developed for this system: first, the system is recast as a pair of uncoupled Fredholm equations of the second kind. The solution to these equations may be expressed as a Neumann series, which converges under certain criteria. In the case that the cross section of the domain is a circle, we are able to separate the solution into radial and angular components which allows us to solve the system exactly, as well as express the general term in the series solution. The qualitative features of both solutions are investigated and their comparison this discussion sheds light on the convergence criteria for the series solution.
system— one important quantity which depends on the geometry of the pipe is the mass flux. We investigate the geometric optimization of integral functionals depending on the velocity, density. A criterion is derived for a geometry to be critical (a possible optimizer), and a gradient descent technique based on this criterion is developed and applied to the mass flux integral. We also show that for a certain "golden" radius, the circular cylinder is a critical pipe geometry for the mass flux under the constraint of a fixed cross sectional area.
ACKNOWLEDGEMENTS
First and foremost I would like to thank my advisors Rich and Roberto. Over the past five years they have taught me the value of true excellence in all my work. They have pushed me to achieve my potential, and I can say without a doubt that I would not be half as successful in the things I have done were it not for their guidance. I would also like to thank Leandra Vicci who always made intelligent suggestions to improve my research and whose discussions were invaluable.
I would like to thank my mom for inspiring my curiosity in science as a kid, and my dad for always supporting me in my studies without question, and both for providing me all the help I needed to overcome any obstacle.
I would like to give a great number of thanks to Pierre-Yves Passagia; I would not have the whole fourth chapter of this thesis were it not for his help and suggestions. I want to thank him for serving on my committee, and more importantly for seeing potential in my work when my future seemed the most uncertain. I would also like to thank Jeremy Marzuola who made a number of helpful suggestions about how to improve my work and was very kind in researching and sending me literature relevant to these suggestions. Finally I would like to thank Mark Williams for taking the time to serve on my committee and thoroughly understand my work.
I would like to thank the undergraduate students who helped in my work, most notably Grace McLaughlin and Tyler Kress, whose help and patience was invaluable in the early days of this project.
TABLE OF CONTENTS
LIST OF FIGURES . . . viii
LIST OF TABLES . . . x
CHAPTER 1: INTRODUCTION . . . 1
1.1 Motivation and previous work . . . 1
1.2 Setup of the problem . . . 3
1.3 Two dimensional shear solutions . . . 5
1.3.1 Non-dimensionalization . . . 6
CHAPTER 2: THE TIME DEPENDENT PROBLEM . . . 9
2.1 Setup . . . 9
2.2 Inverse Laplace Transform . . . 12
2.2.1 Branch structure and singularities . . . 12
2.3 Long Time Asymptotics . . . 14
2.4 Oscillatory evolution and the case Sc = 1. . . 19
2.4.1 Sc = 1 . . . 23
2.5 Discussion . . . 24
CHAPTER 3: THREE DIMENSIONAL DOMAINS . . . 26
3.1 Setup . . . 26
3.2 Fredholm Theory . . . 28
3.3 The interior problem for the circular cylinder . . . 30
3.4 Exact solutions for the circular cylinder . . . 33
3.4.1 Qualitative features of the exact solutions . . . 35
CHAPTER 4: SHAPE OPTIMIZATION . . . 50
4.1 Setup . . . 50
4.2 The shape derivative . . . 51
4.3 The shape derivative for mass flux in circular pipe domains . . . 54
4.4 The gradient descent algorithm . . . 56
CHAPTER 5: EXPERIMENTAL STUDIES . . . 67
5.1 Experimental equipment . . . 67
5.2 Rayleigh-Benard and double diffusive convection . . . 68
5.3 Diffusively driven flows . . . 69
5.4 Buoyancy considerations and time scales . . . 69
5.5 Experiments . . . 72
CHAPTER 6: CONCLUSION . . . 82
APPENDIX A: DOUBLE DIFFUSIVE CONVECTION . . . 85
A.1 Introduction . . . 85
A.2 Setup . . . 85
A.2.1 O( ) . . . 86
A.3 Non-Dimensionalization . . . 87
A.4 Instability Conditions . . . 88
A.5 Free Surface . . . 88
APPENDIX B: COEFFICIENTS . . . 90
APPENDIX C: MATHEMATICA CODE . . . 91
LIST OF FIGURES
1.1 Isopycnals of a stratified fluid meeting a wall inclined at an angle ↵. . . 2
1.2 Pipe geometry for the general problem. . . 4
1.3 Non-dimensional velocity and density perturbation plotted in the domain of the problem. 7 1.4 Non-dimensional velocity and density perturbation plotted versus ⌘. . . 8
2.1 Branch structure for . . . 11
2.2 Branch structure for + . . . 11
2.3 The deformed Bromwich contour. . . 14
2.4 Deformed contour for application of Watson’s Lemma. . . 16
2.5 Sc= 500 . . . 20
2.6 Sc= 7 . . . 21
2.7 Sc=.1 . . . 22
2.8 Branch structure for uand f in the case Sc = 1. . . 23
2.9 Sc=.1 . . . 25
3.1 An arbitrary example of the pipe’s cross-section, and the domain for the PDE system. 27 3.2 The coordinate system used in the cylindrical geometry. . . 31
3.3 Interior velocity r0 = 1 . . . 36
3.4 Interior velocity r0 = 10 . . . 37
3.5 Interior density r0 = 1 . . . 38
3.6 Interior density r0 = 10 . . . 39
3.7 Exterior velocity r0 = 1 . . . 40
3.8 Exterior velocity r0 = 10 . . . 41
3.9 Exterior density r0 = 1 . . . 42
3.10 Exterior density r0 = 10 . . . 43
3.11 Truncation error in the series for the radial component of the velocity vs. r0 . . . . 46
3.15 . . . 49
4.1 A plot of the pre-factor in the shape derivative as a function of r0 . . . 57
4.2 Deformation of a circle of cross sectional area larger than the critical circle . . . 57
4.3 Deformation of a circle of cross sectional area smaller than the critical circle . . . 58
4.4 L2 error in vn plotted against maximum mesh area with a least squares fit line in red. 60 4.5 L2 error in vn plotted against maximum mesh area with a least squares fit line in red. 60 4.6 Domain of cross-sectional area ⇡(1.5383)2 after descent for m= 2 . . . 61
4.7 Domain of cross-sectional area ⇡(1.5383)2 after descent for m= 3 . . . 62
4.8 Domain of cross-sectional area ⇡(1.5383)2 after descent for m= 4 . . . 63
4.9 Domain of cross-sectional area ⇡(1.5383)2 after descent for m= 5 . . . 64
4.10 Domain of cross-sectional area ⇡(6.1532)2 after descent for m= 2 . . . 65
4.11 Domain of cross-sectional area ⇡(6.1532)2 after descent for m= 3 . . . 66
4.12 Domain of cross-sectional area ⇡(6.1532)2 after descent for m= 4 . . . 66
4.13 Domain of cross-sectional area ⇡(6.1532)2 after descent for m= 5 . . . 66
5.1 A small sphere in Phillips’ planar setup. . . 70
5.2 Planar experiment . . . 73
5.3 Cylinder experiment side view . . . 75
5.4 Cylinder experiment front view . . . 76
5.5 Low concentration experiment . . . 77
LIST OF TABLES
3.1 Critical radius vs. n . . . 45
CHAPTER 1 Introduction
1.1 Motivation and previous work
The spontaneous appearance of flows in stratified fluids settling on sloping boundaries are extremely important to understand, not only due to their novel applications [1], but also due to the ubiquity of both of the generating features of these flows in nature. These flows can be found in fluid filled rock fissures which are density stratified either due to the presence of a salt gradient (as would be found in the ocean), or by a thermal gradient present due to heating at the surface. Some of the important applications of these flows include the dispersion of contaminants in fluid filled rock fractures, diagenesis [7], ocean boundary mixing [13], and salt and nutrient transport in rock fissures [10]. There are also novel applications of these flows, such as the diffusively driven motion of a neutrally buoyant wedge in a stratified fluid, which was studied by Allshouse and Peacock ([1]).
The original motivation for this thesis may be traced back to O.M. Phillips’ pioneering paper in which he derives an exact analytical model for shear flow solutions with tilted impermeable planar half-space and channel geometries like the one depicted in Fig (1.1). Phillips identified these flows as buoyancy driven boundary layer flows resulting from an incompatibility between the impermeable condition at the wall and a hydrostatic state in the stratified fluid. In addition to understanding the mechanism for these flows, Phillips produced a characteristic length scale for the boundary layer along with other important dimensional quantities. Around the same time, Carl Wunsch studied studied thermally stratified flows through a boundary layer analysis in his paperOn oceanic boundary mixing [13], and showed that the shear flow assumption made by Phillips becomes invalid when the tilt angle of the wall is such that sin(↵) =O(Ra 1/4), whereRa is the Rayleigh number of the flow.
Though Wunsch studies thermally stratified fluids, he chooses an insulating boundary condition for the wall, bringing his results into perfect analogy with Phillips’.
g
Increasing
Density
Wall
α
dealing with thermally stratified fluids, in which the boundaries have a certain thermal conductivity; this thesis falls firmly in the former category. Among the authors who work with concentration gradients, V.G. Baydulov, Yu. D. Chashechkin and A.V. Kistovich [6, 2] stand out for their numer-ous extensions of Phillips work which include an analysis of the time dependent case (through a spatial Fourier decomposition) and extensions of these flows to non-flat geometries. In light of their important work on these problems, the novelty of this thesis lies in its new approach to the time dependent problem which uncovered oscillatory behavior in time evolution of the system, as well as its extension of the theory to certain three-dimensional geometries.
1.2 Setup of the problem
In this section I will derive the governing equations of motion for the system under the shear flow ansatz, as well as presenting a non-dimensional form of the equations; this equations will serve as the starting point for all discussion in subsequent chapters.
Consider a linearly stratified fluid with density profile
⇢(z) =⇢0 ⇢0N 2
g z
which is contained in a pipe tilted with respect to gravity at an angle ↵ (such as the one shown in Fig. (1.2)). It is important that the pipe be uniform and have infinite extent in the ⇠ direction for the shear flow ansatz to be made. In this case we will define the⇠ˆ direction to be the uniform direction of infinite descent, ⇣ˆ = ˆg⇥ˆ⇠ and finally ⌘ˆ = ˆ⇣⇥ˆ⇠. We will then impose a Cartesian coordinate system (⇣,⌘,⇠) with axes parallel to the respective unit vectors. In three dimensional systems, the origin of the coordinate system is left unspecified and will be chosen on a case by case basis. We will model this system through the incompressible Navier-Stokes equations coupled to an advection-diffusion equation for the density.
ut+u·ru= r p
⇢ +⌫ u+g (1.1)
⇢t+u·r⇢= ⇢ (1.2)
Figure 1.2: Pipe geometry for the general problem.
At the boundary we will impose a no-slip condition on the velocity and a no-flux condition on the density
u|@⌦ = 0 (1.4)
@⇢
@n @⌦ = 0. (1.5)
Phillips showed that by assuming a shear flow and a special density ansatz the governing equations become a coupled set of linear ODEs. We will impose a similar ansatz in this thesis: namely we assume that the full density and velocity fields take the form:
⇢=⇢0
⇢0N2
g (⇠sin(↵) +⌘cos(↵)) +⇢0f(⇣,⌘, t)
Under this ansatz and the Boussinesq approximation, the system (1.1), (1.2) reduces to the form
@tu(x, t) ⌫ u(x, t) = gsin(↵)f(x, t) (1.6)
@tf(x, t) f(x, t) =
N2sin(↵)
g u(x, t). (1.7)
where xis a vector representing the cross sectional variables (⇣,⌘). This set of equations will be the basis of discussion for the entirety of this thesis. It is worth mentioning that in the time independent case, the Boussinesq approximation is not needed to reduce the full system to the above form. In the section below, I will detail O.M. Phillips derivation of the two dimensional solution and discuss relevant features such as the natural boundary layer length scale which appears in the solution, and will be used to recast (1.6) and (1.7) in non-dimensional variables.
1.3 Two dimensional shear solutions
Consider a setup in which the domain is semi-infinite half space bounded by a tilted plane wall such as in (1.1). The time independent problem has a dramatically simplified form owing to translation invariance in⇣ and ⇠; the governing equations reduce to the form
u00(⌘) = gsin(↵)
⌫ f(⌘) (1.8)
f00(⌘) = N
2sin(↵)
g u(⌘). (1.9)
Supplemented with the two boundary conditions u(0) = 0, and f0(0) = N2cos(g ↵), the system may easily be solved to yield the solution pair
u(⌘) = 2 cot(↵)e ⌘sin( ⌘)
f(⌘) = N
2cos(↵)
g e
⌘cos( ⌘)
where a new spatial frequency has been defined as
=
✓
N2sin2(↵) 4⌫
These solutions provide a characteristic length, velocity, and density perturbation scale which are 1/ ,2 cot(↵), and N2cos(g ↵) respectively. Renormalizing each quantity by its characteristic scale, and plotting against a non-dimensional normal wall coordinate ⌘ gives a good idea of the behavior of these solutions and is shown in Fig. (1.3) and Fig. (1.4). One feature which is immediately apparent is the decaying oscillatory nature of the solutions, which die out after approximately two oscillations. The average flow, which will prove to be useful in experimental measurements is defined as:
hui= 2⇡
Z 2⇡
0
u(⌘)d⌘ = e
⇡sinh(⇡) cot(↵)
⇡ ⇡.15885 cot(↵)
Phillips calculates the mass flux of the system to be
M =
Z 1
0
⇢ud⌘=⇢0cot(↵)
5 4⇢0
N2cos2(↵) g sin(↵) ⇢0
N2
g ⇠cos(↵).
In addition to its parametric dependencies, in the three dimensional case the mass flux will depend on the geometry of the domain; this will be the subject of study in Chapter 4.
1.3.1 Non-dimensionalization
With exact solutions for the two-dimensional flat plane problem in hand, we are able to put equations (1.6) and (1.7) into non-dimensional form. We will make the following non dimensional substitutions with no change in notation:
t! t
2 2⌫
x! px 2
f ! N
2cos(↵)
p
2 g f
Wall
(
a
)
u(η)
f(η)
Wall
(b)
u(η)
f(η)
Figure 1.4: Non-dimensional velocity and density perturbation plotted versus ⌘.
In terms of non-dimensional variables, equations (1.6) and (1.7) become
@tu(x, t) u(x, t) = f(x, t) (1.10)
Sc@tf(x, t) f(x, t) =u(x, t). (1.11)
along with their respective boundary conditions
u|@⌦= 0
@f
@n @⌦ = ˆ⌘·nˆ.
CHAPTER 2
The Time Dependent Problem
The time dependent problem in a semi-infinite planar geometry represents a simple but important extension of Phillips’ work. This work was first taken up by Kistovich and Chaschechkin in their paper The structure of transient boundary flow along an inclined plane in a continuously stratified medium [6] from the perspective of a spatial Fourier decomposition. The authors identify two temporal length scales in the problem whose ratio is given by the Schmidt number of the system — the one free parameter in the non-dimensionalized system (1.10),(1.11). A different perspective on the evolution of the time dependent problem can be found through a Laplace transform representation of the solution; this is the approach I will take up in this chapter. This time the Schmidt number takes on a different role; I will show that in the small Schmidt number limit the system exhibits oscillatory behavior in its evolution— a discovery which remains unmentioned in the literature.
2.1 Setup
Suppose that a flat plane given by the equation ⌘ = 0 is inserted into a linearly stratified fluid. This is the simplest geometry one can study to understand the time evolution of the system (1.10),(1.11). On account of the simple geometry, the quantitiesu andf should also display a simple
spatial dependence; to be specific, we will assume that they only depend on ⌘ andt. Under these
assumptions, the system (1.10),(1.11) becomes
@u(⌘, t) @t
@2u(⌘, t)
@⌘2 = f(⌘, t) (2.1)
Sc@f(⌘, t) @t
@2f(⌘, t)
@⌘2 = u(⌘, t). (2.2)
We will assume that the system starts from rest, sou(⌘,0) = 0andf(⌘,0) = 0, and that the system
ˆ
u andfˆrespectively, the system (2.1),(2.2) becomes
suˆ(⌘, s) @
2uˆ(⌘, s)
@⌘2 = fˆ(⌘, s) (2.3)
Scsfˆ(⌘, s) @
2fˆ(⌘, s)
@⌘2 = ˆu(⌘, s). (2.4)
For each fixedsthis is a second order ODE in⌘ and the solutions may be found in the form
ˆ
u(⌘, s) =c1(s)e +(s)⌘+c2(s)e (s)⌘
ˆ
f(⌘, s) =c3(s)e +(s)⌘+c4(s)e (s)⌘
where + and can be determined through (2.3),(2.4) as
+(s) = q
(1 + Sc)s+ps2(Sc 1)2 4
p
2
(s) =
q
(1 + Sc)s ps2(Sc 1)2 4
p
2 .
To impose the spatial decay condition for these functions one must carefully consider the branch structure of +(s) and (s); to satisfy the boundary condition we must require that the real parts
of these functions are always positive. The branch structure for +(s)and (s)is shown in figures
(2.1) and (2.2). The branch cut for the inner square root ps2(Sc 1)2 4 is drawn in blue, while
the branch cut for the outer square root is drawn in black, and has been chosen to give the principal branch of the complex square root. For there is a elliptical component of the cut which will give
u andf oscillatory behavior.
With the branches of +(s), (s) having been chosen to satisfy the condition at ⌘! 1, we
can choosec1(s),c2(s),c3(s), and c4(s) to satisfy the condition at⌘= 0. Doing so gives
c1(s) =
1
s( (s) +(s))(Scs+ +(s) (s))
�� �
�� �
���
�+��
�-�� ��
�
�-�� -�
�-��
Figure 2.1: Branch structure for
�� �
�� �
�
�-��
-�
�-��
c3(s) =
2
+(s) s
s( (s) +(s))(Scs+ +(s) (s))
c4(s) =
s 2(s)
s( (s) +(s))(Scs+ +(s) (s))
2.2 Inverse Laplace Transform
In order to recover u andf from their respective Laplace transforms, we can use the Bromwich
Integral:
u(⌘, t) = 1 2⇡i
Z
C
ˆ
u(⌘, s)estds (2.5)
f(⌘, t) = 1 2⇡i
Z
C
ˆ
f(⌘, s)estds. (2.6)
where C is the Bromwich contour, consisting of a straight line going from i1 to +i1 where >0. The standard way to proceed is by deforming the contour to wrap around the branch cuts and singularities of the integrand meanwhile calculating each of their contributions; we must take care in doing this because fˆanduˆ do not inherit all the branch structure of +(s) and (s), and each have singularities in addition to the branch structure.
2.2.1 Branch structure and singularities
We will first look at the branch cut structure for uˆ andfˆ. Notice that each of these functions
has a symmetry in +(s)and (s); that is swapping one for the other does not change the value of
the function. This is exactly the change that occurs when one passes through the branch cut for the inner square root — evidently neitheruˆor fˆneed branch cuts along this line segment.
There are three other singularities which come from the denominator ofc1(s) throughc4(s). The
first occurs whens= 0 and is a simple pole; the residue at this pole will return the steady state solution which was found by O.M. Phillips [?]. It is a straightforward calculation to show that
Res
s=0uˆ(⌘, s) =
p
2e ⌘/p2sin
✓
⌘
p
2
Res
s=0
ˆ
f(⌘, s) = p2e ⌘/p2cos
✓
⌘
p
2
◆
The second set of singularities occur when +(s) = (s), and are located ats=±|1 Sc2 |. These
singularities do not contribute to the integral, however, as a simple application of the M-L bounding inequality will show that as the contour is tightened around these singularities the respective integrals vanish. There is one more potential singularity whenScs= +(s) (s). By carefully considering
the branch structure of the problem, one can show that this singularity is a simple pole which can only occur ifSc >1 and is located at s= 1
((Sc 1)Sc)1/2; denote this point s . It will be useful to
calculate the residue of uˆ(⌘, s)est anduˆ(⌘, s)est at this point. In order to do this, note that:
lim
✏!0 +(s +i✏) =
1 2 ✓ Sc 1 Sc ◆1/4 +i r
3Sc + 1 4 ✓ 1 (Sc 1)Sc ◆1/4 lim
✏!0 (s +i✏) =
1 2 ✓ Sc 1 Sc ◆1/4 i r
3Sc + 1 4 ✓ 1 (Sc 1)Sc ◆1/4 and also
(Scs+ +(s ) (s ))0|s=s = Sc 1.
With these identities in hand, it is a relatively straightforward calculation to show that
Res
s=s uˆ(⌘, s) = 2 ✓
Sc3
(Sc 1)(3Sc + 1)2 ◆1/4
e⇠re⌘+s tsin (⇠ im⌘)
Res
s=s
ˆ
f(⌘, s) = 2
r
Sc 3Sc + 1e
⇠re⌘+s t(⇠
imcos (⇠im⌘) ⇠resin (⇠im⌘))
with⇠re= 12 Sc 1Sc 1/4
, and⇠im= 12 ⇣
(3Sc+1)2 Sc(Sc 1)
⌘1/4
.
With all this information, one may deform the Bromwich contour in the complexsplane to wrap
�� �
�� �
��
�� �� ��
��
Figure 2.3: The deformed Bromwich contour.
express the integrals defining u andf as
u(⌘, t) = 1 2⇡i
5 X
i=1 Z
Ci
estuˆ(⌘, s) + Res
s=s uˆ(⌘, s) + Ress=0uˆ(⌘, s)ds (2.7) f(⌘, t) = 1
2⇡i 5 X
i=1 Z
Ci
estfˆ(⌘, s) + Res
s=s
ˆ
f(⌘, s) + Res
s=0
ˆ
f(⌘, s)ds. (2.8)
2.3 Long Time Asymptotics
Having understood the steady state solution which comes from the pole at 0as Phillips’ original solution, one can start to analyze the transient behavior of the full solution which is defined as the difference of the full solution from the stationary state. We expect that the long time behavior of the transient parts ofuˆ and fˆwill be determined by the branch cut structure near the imaginary axis.
compared to the exponentially decaying pieces. To make this reasoning precise I will show that:
utrans(⌘, t) ⇠ 1
2⇡i ✓Z
C2
estuˆ(⌘, s)ds+
Z
C4
estuˆ(⌘, s)ds ◆
(2.9)
ftrans(⌘, t) ⇠
1 2⇡i
✓Z
C2
estfˆ(⌘, s)ds+
Z
C4
estfˆ(⌘, s)ds ◆
(2.10)
by analyzing the long time asymptotics of each of these integrals. To put each of these integrals into a simpler form for asymptotic analysis I will relate the integrand over the inner piece of the elliptical contour to the one over the outer piece. This is accomplished in a straightforward manner by defining
c+(s) = +
(s)(sSc 2(s)) s( 2(s) 2
+(s))(s2Sc(Sc 1) 1)
c (s) = (s)(sSc
2 +(s)) s( 2(s) 2
+(s))(s2Sc(Sc 1) 1)
Notice that:
Z
C2
estuˆ(s,⌘)ds=
Z
C2
est⇣c (s)e +(s)⌘ c
+(s)e (s)⌘ c (s)e (s)⌘ ⌘
ds Z
C2
estfˆ(s,⌘)ds=
Z
C2
est⇣( 2+(s) s)c (s)e +(s)⌘ ( 2(s) s)c
+(s)e (s)⌘
( 2(s) s)c (s)e (s)⌘⌘ds
with a similar simplification holding for the integrals overC4. This simplification is owing to the fact
that neither +(s)nor 2(s) have branch cuts insideC2 or C4 so the respective integrals vanish. If
one relates the integrand on the inside of the elliptical branch cut to the one on the outside another simplification can be made to the integration contour.
Z
C2
estuˆ(⌘, s)ds= 2
Z
Coutside 2
est⇣c (s)e +(s)⌘ +c
+(s) sinh( (s)⌘) c (s) cosh( (s)⌘) ⌘
ds Z
C2
estfˆ(⌘, s)ds= 2
Z
Coutside 2
est⇣( 2+(s) s)c (s)e +(s)⌘+ ( 2(s) s)c
+(s) sinh( (s)⌘)
( 2(s) s)c (s) cosh( (s)⌘)⌘ds.
Through the addition of an exponentially decaying term, and an application of Cauchy’s theorem one may deform the contoursCoutside
�� �
�� �
�
������
�����Figure 2.4: Deformed contour for application of Watson’s Lemma.
in Fig. (2.5). As long as the decay of the integrals overCupper andClower is algebraic, as remains to
be shown, we can add exponentially decaying terms with impunity. To assist in simplifying notation, define:
ˆ
uaux(⌘, s) = 2 ⇣
c (s)e +(s)⌘+c
+(s) sinh( (s)⌘) c (s) cosh( (s)⌘) ⌘
ˆ
faux(⌘, s) = 2 ⇣
( 2+(s) s)c (s)e +(s)⌘+ ( 2(s) s)c
+(s) sinh( (s)⌘)
( 2(s) s)c (s) cosh( (s)⌘)⌘
As it stands
1 2⇡i
Z
C2[C4
estuˆ(⌘, s)ds ⇠ 1
2⇡i Z
Cupper[Clower
estuˆaux(⌘, s)ds (2.11)
1 2⇡i
Z
C2[C4
estfˆ(⌘, s)ds ⇠ 1
2⇡i Z
Cupper[Clower
We are now in a position to apply Watson’s lemma to each of the integrals, however, it is useful to note that
Z
Cupper
estuˆaux(⌘, s)ds = Z
Clower estuˆ
aux(⌘, s)ds (2.13) Z
Cupper
estfˆaux(⌘, s)ds = Z
Clower estfˆ
aux(⌘, s)ds (2.14)
with the over-bar denoting complex conjugation— it suffices to compute only one of these integrals in order to obtain our desired result since by this reasoning,
1 2⇡i
Z
C2[C4
estuˆ(⌘, s)ds ⇠ 1
⇡Im
Z
Cupper
estuˆaux(⌘, s)ds (2.15)
1 2⇡i
Z
C2[C4
estfˆ(⌘, s)ds ⇠ 1
⇡Im
Z
Cupper
estfˆaux(⌘, s)ds. (2.16)
Finally, it will be useful to observe that
+ ✓ i p Sc ◆ = s
Sc + 1 2pSc(1 +i)
(s) =
s✓ s pi
Sc
◆ l (s)
wherel (s) is analytic andl ⇣pi Sc
⌘
= Sc+12Sc ; in addition, the following evaluations will be useful
(Scs 2(s))
s( 2(s) 2
+(s))(s2Sc(Sc 1) 1) s=pi Sc
= i
p
Sc (Sc + 1)
(Scs 2+(s))
s( 2(s) 2
+(s))(s2Sc(Sc 1) 1) s=pi Sc
= i
With these results in place, we are able to apply Watson’s lemma to the integrals over the upper contour, giving the result
Z
Cupper
estuˆaux(⌘, s)ds⇠
p
2⇡ 1
t3/2(Sc + 1) 0 B B @
eit/pSc ✓
e q
Sc+1
2pSc(1+i)⌘ e Sc⌘2
2(Sc+1)t ◆
p
Sc + 1
Sc3/4ei ⇣ t p Sc+ ⇡ 4 ⌘
e Sc⌘ 2 2(Sc+1)t⌘
1 C C A Z Cupper
estfˆaux(⌘, s)ds⇠
p
2⇡ i
t3/2(Sc + 1) 0 B B @
eit/pSc ✓
Sce qSc+1
2pSc(1+i)⌘+e Sc⌘2
2(Sc+1)t ◆
p
Sc(Sc + 1)
+Sc1/4ei ⇣ t p Sc+ ⇡ 4 ⌘
e Sc⌘ 2 2(Sc+1)t⌘
1 C C A.
The long time asymptotics for utrans(⌘, t)and ftrans(⌘, t) follow directly from equations (2.15), and
(2.16).
utrans(⌘, t)⇠ r
2 ⇡
1
t3/2(Sc + 1) 0 B @ e
qSc+1 2pSc⌘sin
⇣ t p Sc q Sc+1 2pSc⌘
⌘
e Sc⌘ 2 2(Sc+1)tsin
⇣ t p Sc ⌘ p
Sc + 1
Sc3/4e Sc⌘ 2 2(Sc+1)t⌘sin
✓ t p Sc + ⇡ 4 ◆1 C A (2.17) and
ftrans(⌘, t)⇠ r
2 ⇡
1
t3/2(Sc + 1) 0 B @ Sce q Sc+1 2pSc⌘cos
⇣ t p Sc q Sc+1 2pSc⌘
⌘
+e Sc⌘ 2 2(Sc+1)t cos
⇣ t p Sc ⌘ p
Sc(Sc + 1)
+Sc1/4e Sc⌘ 2 2(Sc+1)t⌘cos
2.4 Oscillatory evolution and the case Sc = 1
500 1000 1500 2000 Time(non-dimensional) 0.1
0.2 0.3 0.4
(a)
u( π 2 2 ,t) uasy( π
2 2,t)
(a) Velocity evolution for Sc= 500
500 1000 1500 2000 Time(non-dimensional)
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
(b)
f(0,t)
fasy(0,t)
(b) Density evolution for Sc= 500
10 20 30 40 50 Time(non-dimensional) 0.35
0.40 0.45
(a)
u( π
2 2 ,t) uasy( π
2 2,t)
(a) Velocity evolution for Sc= 7
10 20 30 40 50 Time(non-dimensional) -1.40
-1.35 -1.30 -1.25 -1.20
(
b
)
f(0,t) fasy(0,t)
(b) Density evolution for Sc= 7
0 2 4 6 8 10 12 14 Time(non-dimensional) 0.2
0.4 0.6 0.8 1.0
(a)
u( π
2 2 ,t) uasy( π
2 2,t)
(a) Velocity evolution for Sc=.1
2 4 6 8 10 12 14 Time(non-dimensional)
-4
-3
-2
-1
(b)
f(0,t)
fasy(0,t)
(b) Density evolution for Sc=.1
2.4.1 Sc = 1
In the case that the Schmidt number is exactly 1, the analysis of the system must be changed owing to singular behavior in +(s)and (s). In this case, the characteristic roots take the much
simpler form:
+(s) =ps+i
(s) =ps i.
The branch cut structure for these roots consist of two lines starting on the imaginary axis at ⌥i
respectively and extending parallel to the negative real axis all the way out to 1. Both u andf
inherit this branch structure, as long as one simple pole at the origin whose residue is the same as in the previous case. The structure of the singularities for uand f is shown in 2.8.
�
-
�
�� �
�� �
Figure 2.8: Branch structure foru and f in the caseSc = 1.
density perturbation fields.
utrans(⌘, t)⇠
1
p
2⇡t3/2 0
@e ⌘sin (t ⌘) e ⌘2
4 sin(t) p
2 e
⌘2
4 ⌘sin ⇣
t+⇡ 4
⌘1
A (2.19)
ftrans(⌘, t)⇠
1
p
2⇡t3/2 0
@e ⌘cos (t ⌘) +e ⌘2
4 cos(t) p
2 +e
⌘2
4 ⌘cos ⇣
t+⇡ 4
⌘1
A. (2.20)
These asymptotic formulae are also compared against the exact solution obtained through numerical integration in Fig. (2.9) and. As one would expect, the oscillatory behavior is intermediate in magnitude compared to theSc = 7 and Sc =.1cases.
2.5 Discussion
The presence of oscillations in the development of the flow is certainly an interesting feature, however, it is natural to ask about the stability of these low Schmidt number flows in order to ascertain whether they could actually be observed. For O.M. Phillips solution, which is a stratified shear flow, it seems natural to apply the Miles-Howard stability criterion which states that a necessary condition for instability is that the local Richardson number of the flow, defined asRi=N2/(du/dz)2
0 5 10 15 20 25 30 Time(non-dimensional) 0.2
0.4 0.6 0.8 1.0
(a)
u( π
2 2 ,t) uasy( π
2 2,t)
(a) Velocity evolution for Sc= 1
5 10 15 20 25 30 Time(non-dimensional)
-0.1 0.1 0.2 0.3
(
b
)
f(0,t) fasy(0,t)
(b) Density evolution for Sc= 1
CHAPTER 3
Three Dimensional Domains
In this chapter I will develop the theory for the time independent three-dimensional problem. Starting with the governing partial differential equations, I will develop two different methods of solution for the system. The first approach is to integrate the partial differential equations and rely on the well developed theory of Fredholm equations to write the solution as a Neumann series. The advantage of this approach is its ability to be applied to domains of arbitrary cross sectional shape. The limitation of this approach lies in the convergence of the series, which will only be guaranteed when the domain is small relative to the boundary layer length. The other method of solution is through separation of variables, however, this method will only work if the cross section of the domain is circular; nonetheless this solution will provide a valuable benchmark for both the Fredholm solution and numerical solutions which will be discussed in the next chapter. Finally, though it is not taken up in this chapter, it is worth noting that all the work presented here can be extended to the time dependent case in a manner exactly analogous to the one used in chapter 2. For the time dependent case, the fundamental solutions will be Bessel functions rather than exponentials which leads to a relatively difficult computation. For the circular-cylinder domains, it is also possible to bound the Richardson number below by a constant (depending on the radius of the domain) times Schmidt number, leading to the same considerations for instability as discussed in chapter 2.
3.1 Setup
Consider the time independent version of the system (1.6),(1.7):
u(x) =f(x) (3.1)
Ω
∂Ω
Figure 3.1: An arbitrary example of the pipe’s cross-section, and the domain for the PDE system.
The natural domain for this system is the transverse cross section of the pipe-domain, which is pictured as the cut open section of the pipe in figure (1.2), and will be assumed to be smooth to guarantee existence of a classical solution. The boundary conditions for u andf are given by:
u|@⌦= 0
@f
@n @⌦ = ˆ⌘·nˆ.
Uniqueness of the solution to this system is easy enough to prove: assuming another solution pair u˜, f˜existed, the difference between the solutions u = u u˜ and f =f f˜would satisfy
equations (3.1) and (3.1) with homogeneous boundary conditions. One can compute the dissipation energy of the solution pair:
Z
⌦|r
u (x)|2+|rf (x)|2dx=
Z
⌦
u (x) u (x) +f (x) f (x)dx
=
Z
⌦
f (x)u (x) u (x)f (x)dx= 0.
(3.3)
3.2 Fredholm Theory
In this section I will derive the restatement of equations (3.1), (3.2) as Fredholm equations of the second kind. It will be assumed throughout the section that the domain⌦is bounded, otherwise
the norms of the operators defined in this section will be unbounded in norm and will violate the hypothesis needed to guarantee a convergent series solution. It is also necessary to discuss the definitions of the Neumann and Dirichlet solution operators which will be used in this section. Define the Dirichlet solution operator to be the linear operator GD : L2(⌦) ! L2(⌦) such that given a
function f 2L2(⌦),GD[f]solves weakly
GD[f] =f with GD[f] = 0 on @⌦
The Neumann solution operator, on the other hand, requires a bit of care to define, as a solution to the Neumann Poisson problem is only defined up to a constant. In order to specify a unique solution, one more constraint must be satisfied; I will impose a constraint on the mean of the solution. Define the Neumann solution operator to be the linear operator GN :L2(⌦) ! L2(⌦) such that
given a function f 2L2(⌦),G
N[f]solves weakly
GD[f] =f with
@GN[f]
@n = 0 on @⌦
and satisfies the constraint
Z
⌦
GN[f]dx= 0.
That these operators can be constructed with the desired properties is the consequence of an application of the Lax-Milgram theorem to the weak statement of these problems and may be found in Evans’Partial Differential Equations, and it follows that these operators are bounded onL2(⌦).
With these operators having been defined, let us rewrite equations (3.1) and (3.2) in this language:
u=GD[f]
f = GN[u] +⌘+hf ⌘i
where ⌘ is the transverse coordinate shown in (1.2), and the bracket represents the average of these quantities over the domain ⌦. It will be useful to note that, for domains which have a vertical
reflection symmetry where the origin of the coordinate system is placed along the line of symmetry, this average will be 0 due to the anti-symmetry of f and ⌘ over the line of symmetry; this fact will be used in the next section where we consider a circular domain.
The system (3.4) may be decoupled by inserting one equation into the other, yielding
u= GD⇤GN[u] +GD[⌘+hf ⌘i]
f = GN⇤GD[u] +⌘+hf ⌘i
(3.5)
or equivalently
(I+GD⇤GN) [u] =GD[⌘+hf ⌘i]
(I+GN ⇤GD) [f] =⌘+hf ⌘i.
(3.6)
with the ⇤symbol denoting the product or composition of the operators. Bothu andf can therefore
be defined through Fredholm equations of the second kind. It is a well known fact that the operators (I +GD⇤GN) and (I +GN ⇤GD) are invertible through the following series (typically called a
Neumann series):
u=
1 X
n=0
( GD⇤GN)n[GD[⌘+hf ⌘i]]
f =
1 X
n=0
( GN ⇤GD)n[⌘+hf ⌘i]
(3.7)
which will converge provided that kGD ⇤GNk<1 and kGN⇤GDk<1.
It is often more convenient to reformulate equation (3.7) through the PDE which each term solves. Denoting
un= ( GD⇤GN)n[GD[⌘+hf ⌘i]]
it is simple to show that the following recursion relationship holds
2u
n(x) = un 1(x) 2f
n(x) = fn 1(x).
(3.8)
The boundary conditions may also be inferred from the definition:
un|@⌦= 0
@ un
@n |@⌦ = 0
@fn
@n|@⌦ = 0 fn|@⌦= 0.
In this way, the equations defining the series foruandf are completely decoupled, and each term in
the series 3.7 may be solved for iteratively— this is the chief advantage over equations (3.1), (3.2). 3.3 The interior problem for the circular cylinder
In general, a solution of equation (3.6) is hard to solve through an eigenfunction expansion. A heuristic reason can be provided: for an eigenfunction expansion the first in the Neumann series will be a double series for most domains, the second term a quadruple series and so on. All of this is owing to the operatorsGD and GN not sharing a common set of eigenfunctions. The one pipe
geometry where this is not the case is the interior of a circular cylinder. For this geometry it is convenient to switch to polar coordinates with the origin at the center of the circle, which will be taken to have radiusr0, the coordinate system is pictured in Fig. 3.2. In this coordinate system we
will seek series solutions of the form (3.7) by solving the iterative PDEs which define each term in the series. It is easy to obtain the first term in these series by noting that
f0(r,✓) =rsin(✓)
and u0 solves the equation
r0 θ
r
Figure 3.2: The coordinate system used in the cylindrical geometry.
The solution to this equation may easily be obtained by separating variables and assuming that u0
takes the formu0(r,✓) = ˜u0(r) sin(✓). This gives the solution
u0(r,✓) = ✓
r3 r2 0r
8
◆
sin(✓).
The next term takes more work to obtain and will be listed below:
u1(r,✓) = ✓
r7 6r2
0r5+ 24r40r3 19r60r
9216
◆
sin(✓)
f1(r,✓) = ✓
r5 3r20r3+ 4r40r
192
◆
sin(✓).
The separation of variables used in the first two terms can be applied to subsequent terms; inserting the ansatz
un(r,✓) = ˜un(r) sin(✓)
into the systems given by (3.7) yields equations for u˜n and f˜n:
L2r[˜un] = ˜un 1
L2r[ ˜fn] = ˜fn 1
whereLr is the differential operator:
Lr= d2 dr2 +
1 r d dr 1 r2
the corresponding boundary conditions are:
˜
un(0) =Lr[˜un]|r=0 = 0 u˜n(r0) = d
drLr[˜un]r=r0
= 0
˜
fn(0) =Lr[ ˜fn]|r=0 = 0 f˜n0(r0) =Lr[ ˜fn] r=r0
= 0.
The process of integrating these equations may be greatly simplified by realizing thatf˜n andu˜n are both polynomials in r. The formula for the nth term can then be stated: defining the coefficient
cn= 8 > > < > > :
Pn 1
i=1 4n i (n+1cii) (n+2 i) if nis even. Pn 1
i=1
(2n+1 2i)ci
4n i (n+1 i) (n+2 i) if nis odd
withc1= 1, we have
un(r,✓) = ( 1)n 2n X
i=0
c2n+1 ir2i 1r2(2n i0 1)
4i 1 (i) (i+ 1) sin(✓)
fn(r,✓) = ( 1)n+1 2n X
i=0
c2n i+1r2i+1r02(2n i)
4i (i+ 1) (i+ 2) sin(✓)
3.4 Exact solutions for the circular cylinder
Encouraged by the separation of variables used in the Neumann series, one might guess that the full problem is susceptible to the same attack. In this section I will solve both the interior and exterior circular cylinder problems through the same separation of variables used in the Neumann series. Starting with the interior problem, we make the ansatz
uint(r,✓) = ˜uint(r) sin(✓)
fint(r,✓) = ˜fint(r) sin(✓)
and inserting this into (3.1) and (3.2) yields
Lr[˜uint] = ˜fint
Lr[ ˜fint] = u˜int.
(3.9)
with the boundary conditions
˜
uint(0) = 0 u˜int(r0) = 0
˜
fint(0) = 0 df˜
int
dr (r0) = 1.
Since these equations are second order linear ordinary differential equations, the general solution is:
˜
uint(r) =cint1 ber1(r) +cint2 bei1(r)
˜
fint(r) =cint2 ber1(r) cint1 bei1(r)
(3.10)
where the coefficients c1 and c2 are determined through the boundary conditions to be
cint1 = 2
p
2bei1(r0) cint(r
0)
cint2 = 2
p
2ber1(r0) cint(r
and
cint(r0) =bei0(r0)(ber1(r0) +bei1(r0)) bei1(r0)(ber0(r0) ber2(r0))
+bei2(r0)) ber1(r0)( ber0(r0) +ber2(r0) +bei2(r0))
The exact solution for the interior problem holds great utility as a benchmark for convergence of the Neumann series as well as numerical solutions (which will be used for shape optimization in the next chapter). The exterior problem may be solved in a similar manner: letting
uext(r,✓) = ˜uext(r) sin(✓)
fext(r,✓) = ˜fext(r) sin(✓)
the radial functions also solve (3.10), however, the boundary conditions must be changed to impose decay at spatial infinity
˜
uext(r0) = 0 lim r!1u˜
ext(r) = 0
df˜ext
dr (r0) = 1 rlim!1
˜
fext(r) = 0.
With these boundary conditions the radial solutions take the form
˜
uext(r) =cext1 ker1(r) +cext2 kei1(r)
˜
fext(r) =cext2 ker1(r) cext1 kei1(r)
cext1 = 2
p
2kei1(r0) cext(a)
cext2 = 2
p
2ker1(r0) cext(a)
and
c(a) =kei0(r0)(ker1(r0) +kei1(r0)) kei1(r0)(ker0(r0) ker2(r0))
3.4.1 Qualitative features of the exact solutions
(a) The interior velocity field for a pipe of radius 1. The red color in the bottom half of the pipe represents an upward flow, while the blue color on the top half represents a downward flow.
0.0 0.2 0.4 0.6 0.8 1.0
-0.05 -0.04 -0.03 -0.02 -0.01 0.00
r
u
˜ int
(
b
)
(a) The interior velocity field for a pipe of radius 10.
0 2 4 6 8 10
-0.5
-0.4
-0.3
-0.2
-0.1 0.0
r
u
˜ int
(
b
)
(b) The interior radial velocity field for a pipe of radius 10 plotted against the radius of the domain. The black line at the right indicates the wall of the domain.
(a) The interior density field for a pipe of radius 1. The red color in the top half of the pipe represents a lighter than ambient density, while the blue color on the bottom half represents a heavier than ambient density.
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
r
f
˜ int
(
b
)
(a) The interior density field for a pipe of radius 10.
0 2 4 6 8 10
0.0 0.5 1.0 1.5
r
f
˜ int
(b)
(b) The interior radial density field for a pipe of radius 10 plotted against the radius of the domain. The black line at the right indicates the wall of the domain.
(a) The exterior velocity field for a pipe of radius 1. The red color in the top half of the pipe represents an upward flow, while the blue color on the bottom half represents a downward flow.
2 4 6 8 10
0.00 0.05 0.10
r
u
˜ ext
(
b
)
(a) The exterior velocity field for a pipe of radius 10.
6 8 10 12 14
0.0 0.1 0.2 0.3
r
u
˜ ext
(b)
(b) The exterior radial velocity field for a pipe of radius 10 plotted against the radius of the domain. The black line at the left indicates the wall of the domain.
(a) The exterior density field for a pipe of radius 1. The red color in the bottom half of the pipe represents a lighter than ambient density, while the blue color on the top half represents a heavier than ambient density.
0 2 4 6 8 10
-0.6 -0.4 -0.2 0.0
r
f
˜ ext
(
b
)
(a) The exterior density field for a pipe of radius 10.
6 8 10 12 14
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0
r
f
˜ ext
(
b
)
(b) The exterior radial density field for a pipe of radius 10 plotted against the radius of the domain. The black line at the left indicates the wall of the domain.
3.5 Comparison of the Neumann series with the exact solution
With both exact and series solutions in hand, it is useful to revisit the question of convergence for the series solution, which is only guaranteed if the norms of the operators satisfy the bounds
kGD ⇤GNk < 1 and kGN ⇤GDk < 1. From the domain monotonicity of the eigenvalues of the
Laplacian, one expects the norms of these operators to depend monotonically on the area of the domain. To support this intuition, we will show that in the case of the circular cylinder the non-dimensional radiusr0 controls the norm of the solution operators: for smaller domains the series
solution will converge without issue, however, past a certain critical radius the L2 error of the series
solutions grows no matter how many terms are retained. The critical radius will be determined numerically in this section, and the qualitative features of the series solution on either side of the critical radius will be investigated, thus providing a good idea of when and how the series solution breaks down.
To begin, we plot the L2 error of the series solution against the non-dimensional radius of the
pipe for various truncations of the series; this is done in Figs. (3.11) and (3.12). One immediately apparent feature is the point atr0⇡2.8where the RMS error of the various truncated series intersect:
past this critical point adding more terms to the series does not decrease the L2 error. To determine
this value more precisely, one may look at the ratios of the norms of the terms in the series (3.7).
Run= k un+1k
kunk
Rfn= k fn+1k
kfnk
For fixedn these ratios are functions of the non-dimensional radiusr0. The critical radius can be
determined by first solving the equations
Run= 1
Rfn= 1
worth mentioning that this table was created by solving the equation Run= 1, however, a similar
calculation can be made forRfn and the limiting radius agrees precisely with the one made from Run.
It is also interesting to approach the question of convergence from a more qualitative standpoint: what features of the full solution can the series solution accurately capture? By plotting the radial velocity for the exact and series solutions for various pipe radii, we find that the series solution fails to capture the boundary layer behavior present in the full solution— this is done in Figs. 3.13 and 3.14 with the series being truncated after five terms. It is immediately apparent that the magnitude of the series solutions grows quickly after one passes the critical radius, however, a more subtle feature is missing as well: the boundary layer behavior. From this analysis, it seems likely that the series solution will diverge if the cross sectional area of the pipe is on the order of the boundary layer length.
n Critical r0
1 2.84819 2 2.87009 3 2.87117 4 2.87122 5 2.87122
Table 3.1: Critical radius vs. n
3.6 Exterior solution versus flat plane solution
1.0 1.5 2.0 2.5 3.0 0.0
0.5 1.0 1.5 2.0 2.5 3.0
r0
L
2E
rror 1 Term
2 Terms 5 Terms
Figure 3.11: Truncation error in the series for the radial component of the velocity vs. r0
1.0 1.5 2.0 2.5 3.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
r0
L
2E
rror 1 Term
2 Terms 5 Terms
0.0 0.5 1.0 1.5 2.0
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05 0.00
r
(a)
Series Solution Exact Solution
(a) The exact interior radial velocity compared against the series solution for a pipe of radius 2.
0 1 2 3 4
-500
-400
-300
-200
-100 0
r
(
b
)
Series Solution Exact Solution
(b) The exact interior radial velocity compared against the series solution for a pipe of radius 4.
0.0 0.5 1.0 1.5 2.0 0.0
0.5 1.0 1.5
r
(
a
)
Series Solution Exact Solution
(a) The exact interior radial density compared against the series solution for a pipe of radius 2.
0 1 2 3 4
-1500
-1000
-500 0
r
(
b
)
Series Solution Exact Solution
(b) The exact interior radial density compared against the series solution for a pipe of radius 4.
0 2 4 6 8 10 0.0
0.1 0.2 0.3 0.4
(
a
)
r0=1
r0=10
r0=100
Flat Plane Solution
(a) The exterior radial velocity plotted against the flat plane solution for various radii
0 2 4 6 8 10
-1.0
-0.5 0.0
(
b
)
r0=1
r0=10
r0=100
Flat Plane Solution
(b) The exterior radial density plotted against the flat plane solution for various radii
CHAPTER 4 Shape Optimization
The introduction of geometry in three-dimensional setups gives a new method of controlling the behavior of the solutionsu andf: rather than focussing on parametric dependencies, in this chapter
I will focus on the optimization of functionals depending on the state of the system. The study of how geometry affects the mass flux functional can be found as far back as [10] where Phillips notes that the convective mass flux in large channel geometries is be many times greater than the diffusive mass flux. Heitz, Peacock and Stocker [5, 9] have also considered parametric studies of the volume flux of the system to determine the optimal setup for dye transport. Both the mass flux and the volume flux can be defined through integrals of the state variables u andf over the
cross section of the domain, and it becomes a natural question for the three-dimensional problem whether general integral quantities can be optimized by varying the geometry of the pipe. In this chapter I will undertake a principled study of this question, deriving the shape derivative a general integral functional and then applying it to the case of the horizontal cross sectional mass flux. I will show that for this mass flux functional the shape derivative of the cylindrical domain vanishes for a particular radius of the pipe. I will also increase the magnitude of the mass flux for symmetric star-shaped pipes of larger and smaller cross-sectional areas through a numerical gradient descent algorithm.
4.1 Setup
Consider again the three-dimensional time-independent system in a bounded pipe domain of arbitrary cross section
u=f
f = u.
(4.1)
These coupled partial differential equations act as constraints which define the state functions u
and f, where the arguments of the state functions have been omitted. One may consider a general
integral functional depending on these states (and perhaps varying spatially as well):
J(⌦) =
Z
⌦
j(u, f,x)dx. (4.2)
The dependence of J on⌦ is displayed explicitly.
The method of finding optimal domains for the functional is highly reminiscent of optimizing ordinary functions, and to this end, one should hope to find the stationary geometries of (4.2) by defining a "shape derivative, differentiating the functional with respect to the domain, and deriving necessary conditions for a domain to be an optimizer. The crux of the method lies in defining the variation ofJ(⌦) with respect to a domain deforming diffeomorphism, while keeping in mind that the statesu andf must satisfy the constraint equation (4.1). Again, throughout this chapter ⌦will
be assumed to be smooth except where otherwise noted.
4.2 The shape derivative
Consider a deformation of ⌦determined by a one parameter family of smooth diffeomorphisms
Tt:R2!R2 fort 0such that T0 is the identity map; we will denote ⌦t=Tt(⌦). Now define a
vector field onR2 through the formulav(x, t) =T 1
t⇤ dtdTt(x) with the lower star indicating the
push forward of the map. We will define the shape derivative of J(u, f,⌦)in the direction v(x, t)at time0 through the formula
dJ(⌦;v) = d
dtJ(⌦t) t=0 (4.3)
if it exists. It is shown in Sokolowski and Zolesio’sIntroduction to Shape Optimization [12] that the shape derivative of (4.2) may be expressed as:
dJ(⌦;v) =
Z
⌦
@uj(u, f,x)u0(⌦;v) +@fj(u, f,x)f0(⌦;v)dx+ Z
@⌦
j(u, f,x)(v·n)ˆ dS. (4.4)
where u0(⌦;v)andf0(⌦;v) are the shape derivatives of u andf. Shape differentiating u andf is
Lemma 1. Suppose that u and f are defined as in equation (4.1) such that the shape derivatives
exist: then the shape derivatives ofu and f satisfy the boundary value problem:
u0(⌦;v) =f0(⌦;v)
f0(⌦;v) = u0(⌦;v).
(4.5)
with accompanying boundary conditions
u0(⌦;v)|@⌦=
@u
@nvn
@f0(⌦;v) @n @⌦=
@ @s
✓ vn
✓
@f
@s
@⌘ @s
◆◆ .
Here vn= (v·n)ˆ , and @@s represents the unit tangent derivative operator on @⌦.
With this result in hand, it is possible compute the shape derivative ofJ(⌦)from equation (4.4), however, this form of the shape derivative has the weakness that one must pick the vector field
vn before the computation; in the optimization of J(⌦) one wishes to infer the vn corresponding
to steepest ascent or descent. To correct this fault, we seek to rewrite (4.4) as the pairing of a distribution living on the boundary of⌦integrated againstvn. Before we can do this, it is necessary
to define the adjoint system:
Definition. Given the state functions u and f and cost function j(u, f,x), defined as above, the adjoint states u⇤ andf⇤ are defined to solve the system:
u⇤ = f⇤+@uj(u, f,x)
f⇤ =u⇤+@fj(u, f,x).
(4.6)
with the boundary conditions
u⇤|@⌦= 0
@f⇤
@n @⌦= 0.
The adjoint system is precisely what is needed to rewrite equation (4.1). Starting from (4.4) and using the definitions of the adjoint system to rewrite the terms @uj(u, f,x)and@fj(u, f,x), the
shape derivative can be written
After integrating by parts twice and using the definition of the shape derivatives of u and f this
integral can be written as one over the boundary of the domain, that is:
dJ(⌦;v) =
Z
@⌦ ✓
j(u, f,x)vn
@u⇤
@n
@u
@nvn f ⇤ @ @s ✓ vn @f @s @⌘ @s ◆◆ dS.
One more integration by parts on the last term (assuming that the boundary is closed) yields the desired result:
dJ(⌦;v) =
Z
@⌦ ✓
j(u, f,x) @u⇤ @n
@u
@n+
@f⇤
@s ✓ @f @s @⌘ @s ◆◆
vndS. (4.7)
The form of equation (4.7) mimics that of a gradient; solving for the states u andf and their
adjoints u⇤ and f⇤ gives the steepest ascent direction
vn=j(u, f,x)
@u⇤
@n
@u
@n+
@f⇤
@s ✓ @f @s @⌘ @s ◆ . (4.8)
One may also chose to impose additional constraints on the domain: the specification of the area of the domain is commonly desired and is relatively simple to impose in the following manner. The condition that area be preserved in the deformation of the domain is equivalent to requiring
Z
@⌦
vndS= 0,
whch implies that
Z
@⌦ ⌧
j(u, f,x) @u⇤ @n
@u
@n +
@f⇤
@s ✓ @f @s @⌘ @s ◆ @⌦
vndS = 0
withh...i@⌦ representing the mean of the function on the boundary of the domain. With this identity,
the required modification to the shape derivative becomes:
dJ(⌦;v) =
Z
@⌦ ✓
j(u, f,x) @u⇤ @n
@u
@n+
@f⇤
@s ✓ @f @s @⌘ @s ◆◆ ⌧
j(u, f,x) @u⇤ @n
@u
@n +
@f⇤
@s ✓ @f @s @⌘ @s ◆ @⌦ !
vndS.
4.3 The shape derivative for mass flux in circular pipe domains
Consider the mass flux due to convective motion through a horizontal cross section of the circular cylinder pipe of non-dimensional radiusr0. This mass flux may be defined in dimensional form as
M =p2 cot(↵)
Z
z=c
⇢(⇣,⌘,⇠)u(⇣,⌘)dA
where z denotes the vertical cartesian coordinate aligned with gravity and uis the non-dimensional
velocity. Inserting the definition of⇢
M =p2 cot(↵)
Z
z=c ✓
⇢0
⇢0N2 g c+
N2cos(↵)⇢0
p
2 g f(⇣,⌘) ◆
u(⇣,⌘)dA
= ⇢0N
2cot(↵) cos(↵)
g
Z
z=c
f(⇣,⌘)u(⇣,⌘)dA
with the second line holding because the mean of over the cross section of the pipe uis0(as may be proved from the system (4.1)). Through a change of variables, this integral may be related to one over the normal cross section⌦, giving:
M(⌦) = ⇢0N
2cot2(↵)
g
Z
⌦
u(x)f(x)dx.
This integral may be studied parametrically, with an eye towards optimizing the tilt angle or basic stratification, however, with the machinery developed in the previous section one may also search for optimal domain geometries — this is the approach we will take up in this section.
Consider the functional
J(⌦) =
Z
⌦
u(x)f(x)dx
which is the mass flux stripped of any parametric dependencies in the pre-factor. Through the use of equation (4.1) one can show that
J(⌦) =
Z
⌦ |r
u(x)|2dx
which is always negative.
cross section of the pipe (⇠ being constant) which leads to a positive quantity for the normal mass flux. The reason that the study of the horizontal cross section was undertaken is due to the tendency in the optimization of the domain for the normal cross sectional mass flux, for the domain to become unbounded in the ⌘ direction (even if an area constraint is imposed) which is unphysical .This is also interesting on account of the relation of the above integral to kinetic energy and its dissipation, therefore, in optimizing the geometry for mass flux, we also optimize the geometry for this integral quantity. From these considerations, I will seek to minimizeJ while preserving the cross sectional
area of the domain.
Following the procedure from the previous section, one writes the adjoint system:
u⇤(x) = f⇤(x) +f(x)
f⇤(x) =u⇤(x) +u(x).
(4.10)
with the boundary conditions
u⇤|@⌦= 0
@f
@n @⌦ = 0.
The solution to the state equation (4.1) has been derived in chapter 3, quoting the result:
u(r,✓) = ˜u(r) sin(✓)
f(r,✓) = ˜f(r) sin(✓)
with
˜
u(r) =c1ber1(r) +c2bei1(r)
˜
f(r) =c2ber1(r) c1bei1(r).
(4.11)
These solutions can be fed into the adjoint system, which may also be separated into an angular and radial part. To be precise:
u⇤(r,✓) = ˜u⇤(r) sin(✓)
whereu˜⇤(r) and f˜⇤(r)solve the ordinary differential equations
Lr[˜u⇤] = f˜⇤+ ˜f
Lr[ ˜f⇤] = ˜u⇤+ ˜u.
(4.12)
with the boundary conditions
˜
u⇤(0) = 0 u˜⇤(r0) = 0.
˜
f⇤(0) = 0 f˜⇤0(r0) = 0.
The radial adjoint equations can be solved through variation of parameters to give:
˜
u⇤(r) =c⇤1ber1(r) +c⇤2bei1(r) +r(c2 c1)ber0(r)
2p2
r(c1+c2)bei0(r)
2p2 ˜
f⇤(r) =c⇤2ber1(r) c⇤1bei1(r) +
r(c2 c1)bei0(r)
2p2 +
r(c1+c2)bei0(r)
2p2 .
(4.13)
Where c⇤1 and c⇤2 are given in the appendix. Finally, one may use the state and adjoint functions to
express the area preserving shape derivative given by equation (4.8) as
dJ(⌦;v) = r0 2
✓
f⇤(r0)(f(r0) r0) r2
0
+u⇤0(r0)u0(r0) ◆ Z 2⇡
0
cos(2✓)vnd✓. (4.14)
This expression is interesting on two counts. First, it says that the steepest descent direction is given by vn/cos(2✓). Secondly, if one plots the pre-factor as a function of the non-dimensional
radius of the pipe the shape derivative vanishes for r0 ⇡3.0766; this is shown in Fig. 4.1 with the
critical radius highlighted in red.
4.4 The gradient descent algorithm
With one critical geometry having been identified it is natural to ask what critical geometries for smaller and larger cross sectional area pipes might be. One clue comes from (4.14) which says that the steepest descent direction is proportional to cos(2✓). Taking a radial vector field on R2 whose magnitude is given by 1
2 ⇣
f⇤(r0)(f(r0) r0) r2
0 +u
⇤0(r
0)u0(r0) ⌘
1 2 3 4 r0
-0.2 -0.1 0.1 0.2 0.3 0.4
Figure 4.1: A plot of the pre-factor in the shape derivative as a function ofr0
area less than the critical circular geometry will have a positive proportionality factor in vn, and
will be stretched vertically, whereas those with larger cross sectional area will be stretched horizontally.
