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Near wake characteristics of towed bodies in a stably stratified fluid

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Near wake characteristics of towed bodies in a stably stratified fluid

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Content Near Wake Characteristics of Towed Bodies in a Stably Stratified Fluid
by
Trystan Madison
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements of the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)
May 2022
Copyright 2022 Trystan Madison
Dedication
To my parents and brothers, for all your support
ii
Acknowledgements
Completing this dissertation has proven to be one of the most challenging and
rewarding experiences I have ever undertaken. I have been very fortunate to meet
and work with so many incredibly brilliant people over my time at USC. While it
may have been a winding and unconventional road to reach the finish line I have
valued the experiences that the PhD process has afforded me and I owe my success
to many individuals.
I’dfirstliketothankmyadviser, GeoffSpedding. Iholdatremendousamountof
respect for the way he approaches research and I thank him for shaping me into the
engineer I am today. The freedom I was offered to explore any idea that interested
me was important to me and his advice was absolutely appreciated. His focus on
making sure I was set up for success was instrumental in my career development.
He was always available to go over not only research materials, but also aid in
interview presentations. His mentorship has meant so much to me and I owe a
staggering amount of thanks to him over the course of my PhD career.
I also owe significant thanks to my lab colleagues who were always willing to
talk, or get a coffee, and discuss problems both research and life oriented. Those
moments are some of my more treasured PhD memories. Xinjiang Xiang is an
incredibly bright researcher and really helped me understand significant aspects of
stratified flows. Chris Ohh has been a great friend and colleague in the stratified
lab and her exuberance was always a welcome occurrence in the lab. I’d also like
to thank most of the USC super fluids group, though it may not have been called
iii
that for a majority of my time, for insight into random research topics, lunches
that extended longer than they probably should have, and an overall a sense of
community. Realizing I was not alone in this PhD process made the experience
much easier and I thank you all for it.
Lastly, I’d like to thank my parents and brothers for their unwaivering support
while progressing through this program. They have been champions of mine from
the very beginning and it means so much to me to have had that support for so
long.
iv
Table of Contents
Dedication ii
Acknowledgements iii
List of Tables viii
List of Figures ix
Abstract xiii
Chapter 1 Introduction 1
1.1 The importance of stratified flows . . . . . . . . . . . . . . . . . . . 1
1.2 Turbulence in stratified fluids . . . . . . . . . . . . . . . . . . . . . 4
1.3 Experimental Studies on Stratified Wakes . . . . . . . . . . . . . . . 6
1.3.1 Stratified Grid Turbulence . . . . . . . . . . . . . . . . . . . 6
1.3.2 Bluff Body Wakes . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Computations of Stratified Flows . . . . . . . . . . . . . . . . . . . 15
1.5 Parameterizing Stratified Near Wakes . . . . . . . . . . . . . . . . . 18
1.6 Claims of Universality . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6.1 Unstratified Wakes . . . . . . . . . . . . . . . . . . . . . . . 19
1.6.2 Stratified Wakes . . . . . . . . . . . . . . . . . . . . . . . . . 21
Chapter 2 Dissertation overview 26
Chapter 3 The turbulent wake of a towed grid in a stratified fluid 29
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
v
3.1.1 Turbulent Patches and Wakes . . . . . . . . . . . . . . . . . 30
3.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.1 Physical setup . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.2 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.3 Refractive-index-matching (RIM) . . . . . . . . . . . . . . . 36
3.3.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.5 Density profile filtering . . . . . . . . . . . . . . . . . . . . . 40
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.1 Instantaneous and time-averaged wake structure . . . . . . . 41
3.4.2 Mean and turbulence profiles . . . . . . . . . . . . . . . . . 45
3.4.3 Density Profiles . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.5.1 Initial conditions in stratified wakes . . . . . . . . . . . . . . 77
3.5.2 Initial 3D regime? . . . . . . . . . . . . . . . . . . . . . . . . 78
3.5.3 Buoyancy Reynolds number in laboratory experiments . . . 79
3.5.4 Parametrization and comparison with numerical simulation . 80
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Chapter 4 Laboratory and numerical experiments on the near
wake of a sphere in a stably stratified ambient 83
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.1.1 The sphere wake as a model problem . . . . . . . . . . . . . 84
4.1.2 The influence of initial conditions . . . . . . . . . . . . . . . 86
4.1.3 Scaling and turbulence in stratified wakes . . . . . . . . . . 87
4.1.4 Sphere wake regimes at low Re-Fr . . . . . . . . . . . . . . 90
4.1.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.2.1 Numerical method . . . . . . . . . . . . . . . . . . . . . . . 92
4.2.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . 94
4.2.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
vi
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.3.1 Wake structure . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.3.2 Time averaged wake properties . . . . . . . . . . . . . . . . 104
4.3.3 Fluctuating velocities and the buoyancy Reynolds number . 118
4.3.4 Parameterizing the near wake . . . . . . . . . . . . . . . . . 123
4.4 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . 129
Chapter 5 The effect of body geometry on stably stratified wakes131
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.2.1 Physical setup . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.2.2 Drag coefficients and effective diameter . . . . . . . . . . . . 137
5.2.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.3.1 Wake structure . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.3.2 Lateral Vorticity . . . . . . . . . . . . . . . . . . . . . . . . 144
5.4 Mean wake quantities . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.4.1 Streamwise velocity profiles . . . . . . . . . . . . . . . . . . 147
5.4.2 Centerline velocity . . . . . . . . . . . . . . . . . . . . . . . 147
5.4.3 Wake length scales . . . . . . . . . . . . . . . . . . . . . . . 149
5.4.4 Lee wave characteristics . . . . . . . . . . . . . . . . . . . . 150
5.5 Fluctuating velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.5.1 Streamwise fluctuating velocity . . . . . . . . . . . . . . . . 152
5.5.2 Crossfluctuation profiles . . . . . . . . . . . . . . . . . . . . 153
5.5.3 Strouhal number . . . . . . . . . . . . . . . . . . . . . . . . 155
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Chapter 6 Outlook 159
References 164
vii
List of Tables
3.1 Re and Fr of all towed grid experiments with corresponding symbols 35
4.1 Tow speed, U (cm/s), sphere diameter, D (cm), and buoyancy fre-
quency, N (rad/s) for each experimental configuration. The naming
convention RxFy will be used for Re = x00, Fr = y. . . . . . . . . . 95
4.2 Power law coefficients for wake height, L
v
, for all Re and Fr found
from simulation results . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.1 Tow speed, U (cm/s), sphere diameter, D (cm), and buoyancy fre-
quency,N (rad/s) for each experimental configuration.m, is the mesh
spacing of each grid and d is the bar diameter of the grid . . . . . . 136
5.2 Drag coefficients and ratio of effective diameter to body diameter for
each body geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . 138
viii
List of Figures
1.1 Regimes of stratified turbulence parameterized by Re and Fr . . . 5
1.2 Decay of fluctuating velocity components for a grid array . . . . . 8
1.3 Wake height and width growth rates for self-propelled body . . . . 10
1.4 Flow regimes in the early wake of a sphere based on Re and Fr . 11
1.5 Regimes for stratified wakes based on mean defect velocity . . . . 12
1.6 Scaling of flow quantities in stratified wakes for Re > 4000 and
Fr> 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1 Sketch of towed grid experiment . . . . . . . . . . . . . . . . . . . 33
3.2 Fr-Re parameter space . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 ω
y
/N and ω
y
/N for Fr = 1.3,Re = 2700 . . . . . . . . . . . . . . 42
3.4 Instantaneous and mean velocity field components . . . . . . . . . 44
3.5 ω
y
/N at specific Nt, for Fr = 1.3,Re = 2700 . . . . . . . . . . . . 46
3.6 L
z
(Nt), with least squares sine series fit . . . . . . . . . . . . . . . 47
3.7 w/U at the wake edge . . . . . . . . . . . . . . . . . . . . . . . . 48
3.8 Lee wave properties . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.9 Centerline stream-wise velocity relative to the grid . . . . . . . . . 50
3.10 Strouhal number of the wake-edge vortices . . . . . . . . . . . . . 52
3.11 Components of ω
y
for R2F.6 and R11F9 . . . . . . . . . . . . . . 53
3.12 Rescaled vorticity decay at the wake edge for Re = 11000 . . . . . 55
3.13 Rescaled vorticity decay at the wake edge for Fr = 2.4 . . . . . . . 55
3.14 Shear components in vertical slices, for Fr = 1.3,Re = 2700 . . . . 57
3.15 Evolution of nondimensional mean square vertical shear integrated
vertically over the wake region . . . . . . . . . . . . . . . . . . . . 58
ix
3.16 Rescaled vertically averaged mean square vertical shear . . . . . . 59
3.17 Time evolution of density profiles at Re = 11000 and Fr = 4 . . . 60
3.18 Local N
2
at Nt = 3 profile for Re = 11000, Fr = 4 . . . . . . . . 61
3.19 Thorpe sorted profile and corresponding Thorpe displacements at
Re = 5400, Fr = 4, Nt = 3 . . . . . . . . . . . . . . . . . . . . . . 62
3.20 Physical descriptions of overturning lengthscales, L
h
, L
Tmax
, and L
v
64
3.21 Density and bouyancy frequency profile patterns to define L
v
. . . 65
3.22 Comparison of overturn lengthscales L
v
, L
Tmax
, and L
h
. . . . . . 66
3.23 Local minimum Richardson number against Nt . . . . . . . . . . 68
3.24 Horizontal Froude number Fr
H
, and Re
H
Fr
2
H
, against Nt . . . . . 69
3.25 Local Richardson numbers at varied Fr and Re . . . . . . . . . . 71
3.26 Comparison of Ri
loc,max
and global Ri . . . . . . . . . . . . . . . . 72
3.27 Fluctuating quantities for Fr = 4.7,Re = 5700 . . . . . . . . . . . 74
3.28 w
0
max
/|U| in the wake region . . . . . . . . . . . . . . . . . . . . . 75
3.29 w
0
max
/|u|
max
in the wake region . . . . . . . . . . . . . . . . . . . . 77
4.1 Computational domain for near sphere wake simulations . . . . . 93
4.2 Laboratory setup for sphere wake experiments. . . . . . . . . . . . 96
4.3 Ensemble averaging example for towed body experiments . . . . . 98
4.4 ω
z
(x,y) for Re = (200, 500, 1000) and Fr = (0.5, 1, 8) from simula-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.5 ω
y
(x,z) for Re = (200, 500, 1000) and Fr = (0.5, 1, 8) from simula-
tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.6 {Re, Fr} regime diagram for current experiments (coloured shapes)
and from CH93 (background). . . . . . . . . . . . . . . . . . . . . 103
4.7 u/U(x,z) for Fr = 1 and Re = (200, 300, 500, 1000) . . . . . . . . 105
4.8 Contour plots of streamwise velocity Re = 500 and Fr = (1, 2, 4, 8) 105
4.9 Rescaled streamwise velocity in the vertical and horizontal center-
planes for Re = 1000 and Fr = 8. . . . . . . . . . . . . . . . . . . 107
4.10 Rescaled streamwise velocity Re = 1000 and Fr = 8. . . . . . . . 108
4.11 Evolution of the wake height, L
v
, for each Fr . . . . . . . . . . . 109
x
4.12 Minimum L
v
/D as a function of Fr . . . . . . . . . . . . . . . . . 111
4.13 Lee wavelength,λ/D vs Fr for R10 simulations, experiments, and
from Meunier et al. (2018) . . . . . . . . . . . . . . . . . . . . . . 113
4.14 Wakewidth,
L
h
R
from experiment and simulation for Fr = 1, 2, 4, 8 . 114
4.15 U
0
/U for Re = (200, 500) and Fr = (1, 8) . . . . . . . . . . . . . . 115
4.16 Rescaled centerline velocity, (U
0
/U)Fr
−2/3
, for Re = (300, 1000) . 117
4.17 Fluctuating components for sphere wake at Re = 1000, Fr = 8. . 119
4.18
w
0
0
u
0
0
for Re = (300, 1000) . . . . . . . . . . . . . . . . . . . . . . . . 120
4.19 Fr
h
vsG for R10 wakes . . . . . . . . . . . . . . . . . . . . . . . . 122
4.20 Velocity distributions in z for R10F8 at x/D = (3, 6, 8) . . . . . . 124
4.21 Velocity distributions in y for R10F8 at x/D = (3, 6, 8) . . . . . . 125
4.22 Normalized centerline velocity for Re = 200 and Re = 1000 . . . . 127
4.23 Fit model of wake half height, L
v
(x) for Fr = (1, 2) . . . . . . . . 128
4.24 Comparison of vertical length scales with Orr et l (2015) and Pal
et al (2017) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.1 (a) Sketch of tow tank for disk and grids . . . . . . . . . . . . . . 135
5.2 Time series of (ω
z
/N)(x,y), for large mesh spacing grid (LG) at
Fr = 1,Re = 1000. The x coordinate can be represented through
Nt since x/D =Nt.Fr/2 . . . . . . . . . . . . . . . . . . . . . . . 141
5.3 Time series of (ω
z
/N)(x,y), for small mesh spacing grid (SG) at
Fr = 1,Re = 1000 . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.4 Time series of (ω
z
/N)(x,y), for the solid disk at Fr = 1,Re = 1000. 142
5.5 Timeseriesof (ω
z
/N)(x,y),forthespherewakeatFr = 1,Re = 1000.143
5.6 Time series of, (ω
y
/N)(x,z), for LG in vertical centerplane atFr =
1,Re = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.7 Time series of, (ω
y
/N)(x,z), for SG in vertical centerplane atFr =
1,Re = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.8 Time series of,ω
y
/N, for the Disk in vertical centerplane atFr = 1
and Re = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
xi
5.9 Time series of, ω
y
/N, for sphere in vertical centerplane at Fr = 1
and Re = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.10
u
U
0
in the horizontal plane at Fr = 1 for a) sphere, b) disk, c)LG,
and d)SG. Profiles taken at x/D = [2, 10, 15] and are represented
by solid, dashed, and dotted lines, respectively. . . . . . . . . . . . 146
5.11 Centerline velocity U
0
/U for a) Fr = 1, (b) Fr = 2, and (c) Fr = 4 148
5.12 Wake half width, L
h
/D, for a) Fr = 1, b) Fr = 2, and c) Fr = 4.
Line styles are same as Figure 5.11 . . . . . . . . . . . . . . . . . 149
5.13 Wake half height, L
v
/D, for a) Fr = 1, b) Fr = 2, and c) Fr = 4.
Line styles are same as Figure 5.11. . . . . . . . . . . . . . . . . . 150
5.14 (a) Wavelength of the lee wave and (b) amplitude of the lee wave
for each body geometry, Fr pair. Open squares show results from
Meunier et al. (2018). Red circles in (b) are results from Chomaz
et al. (1993a). + are grid experiments conducted by Xiang et al.
(2015). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.15 Streamwise centerline fluctuating velocity,u
0
0
/U for (a)Fr = 1, (b)
Fr = 2, and (c)Fr = 4. Line styles for each geometry same as fig
5.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.16 Crossfluctuationprofilesu
0
v
0
/A, inthehorizontalplaneandFr = 1.
Solid line is at x/D = 5 and dashed lines are taken at x/D = 15 . 154
5.17 Downstream evolution of u
0
c
for (a) Fr = 1, (b) Fr = 2, and (c)
Fr = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.18 Downstream evolution of St for each geometry at Fr = 1. The
error bars show the standard deviation of 6 repeated runs. The
solid line shows St Nt
−1/3
. . . . . . . . . . . . . . . . . . . . . . . 156
5.19 Downstream evolution of St
eff
for each geometry at Fr = 1. . . . 156
6.1 Interpolation of mean streamwise velocity, u/U, at x/D = (1, 6) . 161
6.2 Centerline velocity for vertical and horizontal plane experiments as
well as hybrid experiment/simulation . . . . . . . . . . . . . . . . 162
xii
Abstract
The wake generated by a moving body in a stably stratified ambient is of par-
ticular interest due to its practical geophysical and naval applications. For most
applications, wakes are initially turbulent and eventually local length and veloc-
ity scales will evolve such that the wake will become dominated by low Froude
number dynamics, where distinct patches of vertical vorticity form irrespective of
the generating conditions. While many studies have focused on understanding the
late wake structure, fewer studies have focused on the near wake region, where the
background density gradient first starts to influence wake development. When and
how information persists from the initial conditions to the late time stages of flow
development is still an open question. The studies presented here investigate the
near wakes of several different body geometries across a range of Re and Fr in
order to provide some of the first quantitative descriptions in the near wake, and
ultimately provide a framework for understanding how information from the initial
conditions may be traced in the development of the stratified wake.
First a porous grid is used as a means to create a turbulent wake for 2700≤
Re≤ 11000 and 0.6≤ Fr≤ 9.1. Refractive index matching techniques are used
to gain optical access for particle imaging velocimetry measurements of the wake
quantities. The temporally averaged results show a distinct internal wave motion as
wellasKelvin-Helmholtz-generatedvortices,whichoccurpredominantlyatthewake
edge due to the strong vertical shear. After 10 buoyancy periods the wakes start to
xiii
become dominated by low Froude number dynamics. The near wake however shows
dependencies on both initial Re and Fr
Secondly, laboratory and numerical experiments are performed on a sphere wake
acrossRe = [200, 300, 500, 1000] andFr = [0.5, 1, 2, 4, 8]. In all cases the early wake
is affected by the presence of the density gradient, primarily in the form of the body
generated lee waves. Mean and fluctuating quantities do not reach similar states
and as such their evolution cannot be described by any universal scaling. Five
distinguishable regimes are observed across the parameter space and retain their
distinguishing features up to buoyancy times of 20, well into the intermediate wake
region.
Lastly, a series of experiments is conducted with varying body geometry at a
single Re = 1000 and Fr = [1, 2, 4] to investigate the effect of body geometry on
the near wake in a stratified fluid. Each wake can be identified based on the time
at which horizontal vortices begin to form. Despite perceived agreement in wake
averaged velocity profiles, differences in each wake can be found in both the mean
length scales and cross fluctuation profiles. At Fr = 1 the wake is dominated by
the body generated lee wave and the wakes can be thought of as a small distur-
bance superimposed on the lee wave motions. The body geometry sets the lee wave
amplitude, showing a direct demonstration of how the initial conditions can alter
the near wake properties.
xiv
Chapter 1
Introduction
1.1 The importance of stratified flows
Stratified flows are more ubiquitous than may first appear. Whether looking at a
flow in the atmosphere, where density changes as a function of height due to vari-
ations in pressure and temperature, or in the oceanic thermocline, where density
increases with depth due to salinity and temperature, the background density gra-
dient will play an important role in characterizing the flow behavior. The practical
implications of such flows are numerous ranging from underwater vehicles up to
large scale phenomena such as the flow over and around islands and other topology.
The ambient density gradient significantly affects the late time evolution of a
freely evolving turbulent patch. Eventually the stratified flow will form large scale
coherent structures that have a large ratio of horizontal to vertical scales. The
emergence of such patterns from seemingly disordered origins suggests a degree of
universality to stratified flows. Despite the appearance of universality, there may
1
still be information of the original pattern creator that is carried through to the
latest stages of flow development. Understanding how these structures initially
form is important for predictive modeling in geophysical studies as well as to aid in
the detection of pattern creators (Spedding, 2014). Long lasting fluid mechanical
signatures produced from animals of varying sizes, submersibles, and geographical
features has broad significance. How stratified fluids specifically attain order from
different initial conditions is not entirely well understood.
Stratified flows for geophysical and engineering applications can be best de-
scribed in terms of a Reynolds, Froude, and Rosby number,
Re =
UL
ν
, Fr =
U
NL
, Ro =
U
fL
, (1.1)
which represent the relative importance between inertial to viscous, buoyancy,
and Coriolis effects respectively for a flow of characteristic speed, U, and length
scale, L, with kinematic viscosity ν, and rotation frequency, f. The Brunt-Väisälä
frequency, N, is the natural oscillation frequency defined as N =
q
−
g
ρ
0
∂ρ
∂z
, where
g is the acceleration due to gravity, ρ
0
is a prescribed reference density, and ∂ρ/∂z
is the density gradient parallel to g. Holding all parameters fixed and increasing
the density gradient, increases the strength of stratification since N is directly
related to the density gradient. Increasing N will then decrease Fr. Lower values
of Fr represent flows that will be more strongly influenced by the background
density gradient. The internal Froude number can be thought of as a relationship
between convective time scales L/U and an internal wave period 1/N. Similarly,
2
the Rossby number expresses the importance of planetary rotation timescales, 1/f,
where f = 2Ωsinφ with planetary rotation rate Ω at latitude φ.
There is a class of oceanic and atmospheric flows, called mesoscale flows, where
length and time scales are not large enough to be impacted by planetary rotation
but are still impacted by the background density gradient. Thus, Re and Fr are
sufficient in characterizing these types of flows. One can imagine a flow over an
island with U = 10 ms
−1
and L = 10 km. In the atmosphere the natural buoyancy
frequency is on the order of N = 10
−2
rad s
−1
, which gives Re≈ 7× 10
9
and
Fr≈ 10
−1
. In contrast a small scale unmanned underwater vehicle operating in
the ocean can have velocity and length scales of U = 1 ms
−1
and L = .15 m with
an oceanic buoyancy frequency of N = 10
−3
rad s
−1
. This gives Re≈ 10
5
and
Fr≈ 10
3
. For the specific case of the submerged vehicle the natural buoyancy
time scale is three orders of magnitude lower than the initial time scale (L/U). As
a result it is not immediately apparent how stratification will impact such a flow.
Based onFr≈ 10
3
one may anticipate that stratification will not play a significant
roleintheevolutionofthewake. Iftheflowisallowedtoevolveintheabsenceofany
additional energy input , then the local length scale,L, will increase while the local
velocity scale, U, will decrease. As this occurs the convective time scale becomes
the same order of magnitude as the buoyancy time scale, resulting in Fr = 1. This
implies that all flows in a stratified environment, no matter their initial conditions
can and must be eventually affected by the background density gradient. The only
remaining concern then is how the flow, as it enters into this strongly stratified
3
state, can be characterized by its initial conditions including body geometry and
Re.
1.2 Turbulence in stratified fluids
For initially small Re, viscosity immediately dampens any small scale turbulent
fluctuations and there are important differences between initially viscous controlled
motion and inertially driven but buoyancy controlled flows. A buoyancy Reynolds
number, defined asRe
b
=ReFr
2
has been used to indicate the differences between
initially viscously controlled motion and buoyancy controlled flows. Figure 1.1
shows how the different flow regimes are classified based on Re and Fr. The tran-
sition from viscous dominated motions to strongly or weakly stratified turbulence
motion is based on the criterion, Re
b
= 1 (defined by the dotted line in figure 1.1),
and a number of experiments as well as simulations have focused on transitions
from one regime or the other with minimal overlap. Several studies have noted
that there is a distinct dynamical difference between low and high Re, strongly
stratified turbulence (Riley & de Bruyn Kops, 2003; Godoy-Diana et al., 2004;
Hebert & de Bruyn Kops, 2006b; Brethouwer et al., 2007; Deloncle et al., 2008).
For example the scaling analysis of Riley et al (2002) suggested a criterion where
ReFr
2
> 1 is required for a stratified flow to be turbulent , which also agreed with
theoretical analysis performed by Billant & Chomaz (2001). It was also shown that
whenReFr
2
< 1 strong shear was the predominate cause of energy dissipation rate
4
10
0
10
5
10
10
Re
10
-6
10
-4
10
-2
10
0
10
2
Fr
Weakly stratified turbulence
Strongly stratified turbulence
Kolmogorov turbulence
Viscosity-affected stratified flow
ocean
atmosphere
Figure 1.1: Stratified turbulence regimes based on the buoyancy reynolds number,
Re
b
= ReFr
2
. Viscously controlled motion is defined by Re
b
< 1 denoted by the
dotted line. Freely evolving turbulence in the ocean and atmosphere has very small
Fr. Plot replotted from Brethouer (2007)
(Hebert & de Bruyn Kops, 2006a,b). It has typically been difficult to directly com-
pare results found from simulations, laboratory experiments and field data. The
difficulty has been tied to the difference in types of stratified flow regimes, where
experiments and simulations quickly drop below a value of Re
b
= 1 while atmo-
spheric data had Re
b
>> 1 (Waite & Bartello, 2004; Waite, 2014). Turbulence in
the ocean and atmosphere is strongly stratified and understanding the trajectory,
as well as when and how such flows transition from one regime to the next is critical
in understanding the role initial conditions may play in the late time stages of flow
development.
5
1.3 Experimental Studies on Stratified Wakes
1.3.1 Stratified Grid Turbulence
The wakes produced in almost all engineering applications are highly turbulent.
Stratified grid turbulence is a canonical example of how a spatially evolving patch
of turbulence is influenced by the background density gradient.
The literature on stratified grid turbulence has spanned a wide range of initial
conditions in Re, Fr and geometry by varying the mesh spacing, M. Early exper-
iments began by towing grid arrays either horizontally (Lin & Veenhuizen, 1974;
Britteret al., 1983) or vertically (Dickey & Mellor (1980); Fernando (1988); De Silva
& Fernando (1992), the latter two generated a turbulent patch by oscillating a bi-
plane grid in the direction of the density gradient) through stably stratified fluids,
or by using closed-loop channels to cycle flow past a stationary array of cylinders
or rods Stillinger et al. (1983); Itsweire et al. (1986); Lienhard & Van Atta (1990);
Yoon & Warhaft (1990). Typically the grid array spanned the whole tank, a feature
that has some distinct differences to a traditional bluff body wake. This was done to
create an initial disturbance that was isotropic. As the turbulent motions evolved
under the background density ground the flow became anisotropic.
Figure 1.2 shows the fluctuating velocity components in the streamwise,u
0
, lat-
eral,v
0
, and vertical,w
0
, directions as a function of downstream distance normalized
bythemeshspacingreplottedfromthegridexperimentsofLin&Veenhuizen(1974)
6
atFr = 20 andRe = 4.5× 10
4
. Forx/M < 10 the the fluctuating velocity compo-
nentshavesimilarmagnitudes. Theflowbecomesincreasinglyanisotropicasvertical
motion is suppressed with u
0
/w
0
= 11 at x/M = 100 compared to u
0
/w
0
= 1.8 at
x/M = 10 (Lin & Veenhuizen, 1974). The development of anisotropy in a stratified
fluid was further studied by Dickey & Mellor (1980), who showed that the turbulent
kinetic energy (TKE) of a stratified turbulent patch decayed at the same rate to
that of a unstratified patch, up until a value of x/M≈ 275 or Nt≈ 6. Stillinger
et al. (1983) found a similar breakpoint between unstratified and stratified cases
corresponding to Nt≈ 3. A key difference between the former two studies was
the direction of the forcing, Dickey & Mellor (1980) towed a bi-plane grid verti-
cally while Stillinger et al. (1983) towed the grid array horizontally with respect to
the density gradient, indicating that anisotropy in the flow would develop despite
changing the orientation of the towed grid array.
The emergence of anisotropy in stratified grids was also found in studies on oscil-
lating grids (Browand et al., 1987; Hopfinger, 1987; Liu et al., 1987). The observed
suppression of vertical motion was described as a point of turbulence ’collapse’,
somewhat misnamed as the collapse had more to do with physical patch heights ei-
ther showing no growth and even decreasing in size rather than a lack of turbulent
motions. Once the turbulence reached this later time state large scale structures
with large aspect ratio, termed ’pancake’ vortices, would begin to form (Lin & Pao,
1979). Further quantitative studies of grid turbulence utilizing particle imaging
velocimetry (PIV) by Yap & van Atta (1993); Fincham et al. (1996); Praud et al.
7
10
1
10
2
x/M
10
-4
10
-3
u'/U, v'/U, w'/U
v'/U = .009(x/M)
-1
u'/U = .008(x/M)
-0.8
w'/U = .022(x/M)
1.5
Figure 1.2: Decay rate of fluctuating velocity components for a grid in a stratified
fluid at Fr = 20 and Re = 4.5× 10
4
. Circle, triangle, and square are u
0
, v
0
, w
0
,
respectively. Data replotted from Lin and Veenhuizen, 1974
(2005) found that the late time flow dynamics seemed to be independent of bothRe
and Fr and although the later time flow evolved in mostly two dimensions, there
were fundamental differences from two-dimensional turbulence Praud et al. (2005).
The horizontal vortices filled the experimental tanks but were uncorrelated in the
vertical. It was found that the shearing motion between vertically uncorrelated
layers accounted for nearly 90% of the kinetic energy dissipation.
The emerging consensus was that a turbulent patch generated in a stratified
ambient would evolve just like a nonstratified patch for the first few buoyancy peri-
ods. Eventually stratification would influence the flow and inhibit vertical motion
until eventually the flow evolved in mostly two dimensions.
8
1.3.2 Bluff Body Wakes
Stratified bluff body wakes are important not only for their practical nautical ap-
plications but as another example of a turbulent patch as it evolves in an otherwise
quiescent ambient. Some of the earliest studies were able to span a range of initial
Fr, 20≤Fr≤ 565 by looking at self propelled bodies (Lin et al., 1974). A similar
onset of anisotropy, like what was observed during grid turbulence experiments, was
also observed. For example, Figure 1.3 shows the evolution of the wake height, L
v
and widthL
h
replotted from Lin et al. (1974) as a function of the non dimensional
buoyancy time scale,Nt. Initially both the wake height and width grow asNt
−1/4
.
However after Nt/2π = .1, (the authors defined N = 1/2π
q
−
g
ρ
0
∂ρ
∂z
, with units of
rads
−1
.) the wake width growth would grows as Nt
0.4
while the wake height re-
mained relatively constant. The streamwise velocity fluctuations scaled withNt
−3/4
while the vertical fluctuations scaled as Nt
−1
. The initial self propelled studies
painted a fairly clear picture that the presence of a background density gradient
suppressed vertical motion.
Spheres have long been used as an example for wake studies and early stratified
sphere wake studies include those by Lin et al. (1992a); Chomaz et al. (1993b);
Bonneton et al. (1996). Early sphere studies were able to establish several flow
configurations were found in the early wake of a sphere, for .25≤ Fr = U/NR≤
12.7 and 150≤ Re≤ 50000 (Chomaz et al., 1993b) as well as different separation
points on the lee of the sphere have been observed, demonstrating the different
initial conditions that a stratified wake can display. Figure 1.4 illustrates these
9
Figure 1.3: Wake height, L
v
, and width,L
h
, growth rates. Near Nt≈ .1 the wake
height and width show drastically different behavior. Replotted from Lin et al.
(1974)
various regimes based on phenomenological details at early times with an emphasis
on systematically varyingFr. ForFr< 1.5 the flow was primarily two-dimensional
(2D label on figure 1.4), indicating buoyancy forces immediately impacted the flow,
a similar result to those found by Lin et al. (1992a,b). For 1.5≤ Fr≤ 4 there
is a transition from a completely lee wave dominated, or saturated, flow (SLW) to
the formation of Kelvin Helmholtz instabilities (T). The last regime occurs when
Fr > 4 and resembles the unstratified wake deemed the 3D regime. The near
wake of bluff bodies has been relatively untested quantitatively with most studies
mentioned being primarily qualitative in nature. It is clear specific flow regimes can
be established, though it is perhaps less clear how these specific flow regimes evolve
far downstream.
10
Figure1.4: FlowregimesfromChomazetal.(1993b)with2D:twodimensionalwake
regime at lowFr, SLW: ’Saturated’ lee wave regime, T: Transition regime that can
occur with (KH) or without (SKH) Kelvin-Helmholtz instability formation, 3D:
three-dimensional regime.
A natural extension of the early sphere experiments was to obtain quantitative
sphere wake data at comparable Re and Fr. Such studies were performed using
PIV techniques by Spedding et al. (1996a,b); Spedding (1997, 2002b). From the
turbulent sphere wake experiments a general model emerged based upon temporal
variation of power laws that were associated with the decay of the mean defect
velocity, U
0
. In this model an initial three dimensional (3D) turbulence adjusts to
the background density gradient through an intermediate non-equilibrium (NEQ)
regime which later evolves into mostly horizontal motions, or quasi-two-dimensional
(Q2D) motions, that decay slowly and keep a very stable geometry of vortical
structures. Figure 1.5 shows the general power law decay of the defect velocity and
when, in terms of Nt, the onset of each regime occurs.
11
Figure 1.5: Proposed universal curve of rescaled mean defect velocity U
0
using Fr
taken from Spedding (1997) illustrating the onset and decay rates of three dimen-
sional (3D), non equilibrium(NEQ), and quasi-two-dimensional(Q2D) regimes.
In the 3D regime it is proposed that the wake evolves similarly to an unstratified
wake where the mean defect velocity decays with a U
0
= (Nt)
−2/3
power law.
Primarily, this was based on early experiments of the grids such as those shown in
figure 1.2, in the absence of any evidence to the contrary. The early quantitative
sphere wake studies started at Nt≈ 10, due to limitations of optical methods,
and it seemed based on the literature that wakes would initially evolve like an
unstratified wake until bouyancy began to alter the wake. The decay of U
0
is also
accompanied by a half-width and height growth rate of (Nt)
1/3
. Early experiments
Browand et al. (1987), simulations (Itsweire et al, 1993), and theoretical analysis
(Gibson, 1980, Riley & Lelong, 2000) of decaying stratified turbulence suggest that
the buoyancy forces impact the flow as early as Nt≈ 1. An important distinction
is that this early time 3D regime may exist only for a weakly stratified wake, one
12
with high initial Fr. For example Lin et al. (1992a,b) found that for flows with
initial Fr≤ 2, vertical growth rates were immediately suppressed in the lee of the
sphere.
AtNt≈ 2 the dynamics of the wake transition into the non-equilibrium (NEQ)
regime, where the flow begins to be altered by buoyancy. The NEQ regime begins
to show evidence of anisotropy, with a decrease in the growth rate of vertical length
and velocity scales as buoyancy continues to effect the evolving wake. This regime
is characterized by a reduction in the decay rate of the mean defect velocity, with
U
0
≈ (Nt)
−0.25
. The decrease in the decay rate of the mean centerline velocity has
been described by restratification effects, which are attributed to the transfer of
gravitational potential energy to kinetic energy, where heavy and light parcels of
fluid mover toward their original positions (Spedding, 1997). Recent simulations
suggest that turbulent kinetic energy enhancement is due to increased production
effected by buoyancy forcing the turbulent motions to become increasingly more
two dimensional where the lateral reynolds shear stress,−u
0
v
0
, dominates (Redford
et al., 2015). Redford et al. (2015) also shows a similar mean defect decay rate
similar to those found in laboratory experiments and contends that it may be that
processes that affect the mean streamwise velocity, rather than the turbulent kinetic
energy, thatarecriticaltounderstandingthetransitionfromthe3DtoNEQregime.
While vertical motion begins to be suppressed the wake width continues to grow
as (Nt)
1/3
still consistent with that of a homogeneous wake (Spedding, 1997). The
wake height was found to remain constant through the NEQ and was able to scale
13
empirically with Fr
0.6
(Spedding, 2002b). It is during the NEQ regime that large
coherent patterns of vertical vorticity,ω
z
, begin to emerge and whose longevity and
persistence are characterized by the far wake.
Another transition in the mean defect velocity with U
0
≈ (Nt)
−0.76
, occurs at
Nt≈ 80 signifying the onset of the quasi two dimensional (Q2D) regime. In the
Q2D regime the wake width continues to grow with the same (Nt)
1/3
power law. A
result that is somewhat surprising as one may expect the horizontal growth rates
to increase due to suppression of vertical growth heights. The wake height begins
to grow again scaling as (Nt)
1/2
. This regime lasts for very large measurable
times and the development of the ’pancake’ vortices continues. These structures
are Fr independent, where Chomaz et al. (1993b) showed that for a towed sphere
the vortices developed for both weakly and strongly stratified fluids, and are thus
an inherent property of all stratified flows. Density measurements behind both a
laminar and turbulent sphere wake confirmed that the structure of the eddies where
independent of initial Fr and even Re (Bonnier & Eiff, 2002).
The downstream scalings of wake centerline velocity, U
0
, wake width, L
y
, wake
height,L
z
, and Strouhal number, defined asSt =D/λ
x
whereλ
x
is the streamwise
spacing of vortices, could be generalized for all turbulent stratified wakes. The
scalings were found to be independent ofFr in the NEQ and Q2D regimes. Figure
1.6 shows a general summary of wake quantity evolution for stratified sphere wakes
for Re > 4000 and 4≤ Fr ≤ 240. The similarity of statistical profiles of the
sphere wakes, and the lack of influence of Fr suggests some degree of universality
14
Figure 1.6: General sketch of self similar evolution in stratified sphere wakes. (a)
Wake width, L
y
, which grows similar to that of the unstratified wake. (b) Wake
height, L
z
, initially does not grow as vertical motion is suppressed then begins to
grow as (Nt)
1/2
. (c) Centerline velocity, U
0
has a reduced decay rate and then
begins to decay as the homogeneous case (dotted) near Nt≈ 80. (d) Strouhal
number evolves as (Nt)
−1/3
and continues this evolution through NEQ and Q2D
evolution
in sphere wakes but the mechanisms by which information from initial conditions
is transmitted to the NEQ and even Q2D regime to achieve such an ordered state
is still an open question.
1.4 Computations of Stratified Flows
A number of studies have concerned themselves with the evolution of wake-like
turbulent patches and whether or not the same long lived structures exist with
varying start up conditions. There was some initial uncertainty on whether the
15
observed wakes from laboratory experiments were specific to spheres, or some other
symmetry on a towed body, but direct numerical simulations of large Re wake-like
conditions, in particular where turbulent fluctuations are super imposed on a mean
flow, showed that the ’pancake’ eddies developed at late times without a body to
drive any symmetry or pattern (Gourlay et al., 2001; Dommermuth et al., 2002;
Diamessis et al., 2005, 2011). A self preservation model by Meunier et al (2006)
as well as a theory for fossilized turbulence (Gibson, 1980) suggests that the initial
Re has an impact on intermediate to late wake dynamics. Diamessis et al. (2011)
confirmed this for Re = 10
5
where the NEQ regime was prolonged when compared
to the Re = 5000 case. Although far field quantities such as mean centerline decay
rate,andwakelengthscales,appearedtomatchreasonablywellbetweensimulations
andlaboratoryexperimentsthedifferinginitializationproceduresmadeitextremely
difficult to compare near and intermediate wake quantities. Dommermuth et al.
(2002) suggests that achieving appropriate initial turbulence properties of the wake
is essential for accurately simulating the near to intermediate wake, which up to
this point has been rather limited in its quantitative detail.
There has been a concentrated effort to conduct simulations of sphere wakes
with the body included in the computational domain, as opposed to some mean
profile and fluctuation stand in. This has allowed simulations to gain access to near
wake flow quantities as it transitions into the NEQ regime. Early studies focused
on small Reynolds numbers, Re < 200, and found similar regimes by decreasing
Fr to those found in experiments (Hanazaki, 1988). More recent studies have been
16
able to increase the Reynolds number to more dynamical interesting regimes, while
systematically varying initial Fr. Simulations at Re = 1000 on the near wake of a
sphere found that differences from the homogeneous case were caused by stratifica-
tion effects in the early wake. For example the decay rate of the centerline velocity
decreased when Fr decreased (Orr et al., 2015). Other body inclusive simulations
have focused on small Fr cases and have shown that stratification serves as a sta-
bilizer to the wake until Fr = 1. Further decreasing Fr causes the flow to become
unstable again and can show a regeneration of turbulent fluctuations (Pal et al.,
2016, 2017; Chongsiripinyo et al., 2017). Even more recent body inclusive simu-
lations have started looking at the turbulent wake behind a disk (Chongsiripinyo
& Sarkar, 2020) These body inclusive simulations now make it possible to directly
compare with near wake laboratory data a key benchmark in not only understand-
ing nuances between laboratory and computational experiments but as a way of
understanding the origins of patterns in stratified fluids.
An attractive feature of simulations is their ability to attain high Reynolds
numbers, and by extension Re
b
, that are significantly larger than those produced
in the laboratory. As Re increases there is an increase in the vertical shear of
quasi-horizontal layers which forms secondary Kelvin-Helmholtz instabilities, and
a significant body of research has been devoted to whether or not the secondary
instabilities destabilize the larger scale coherent vortices (Riley & de Bruyn Kops,
2003; Waite & Bartello, 2004; Augier & Billant, 2011; Diamessis et al., 2011).The
17
secondary instabilities do not significantly alter the larger vortices, and observa-
tions of cloud formations in the wake of islands that can persist for hundreds of
kilometers provide some proof to the longevity of the structures Spedding (2014).
The surprising fact that persistent stable vortical pattens develop regardless of how
initial conditions are set provides further evidence that stratified wakes may evolve
toward some prescribed universal state.
1.5 Parameterizing Stratified Near Wakes
There have been attempts both theoretical and empirical to model the entire life-
time of a stratified wake. Meunier et al (2006) proposed a theoretical model for
stratified wakes that was able to recover growth rates of vertical and horizontal
length scales for the NEQ and Q2D regimes. However the model already presup-
poses that the near wake will act as though it would if it were the homogeneous
case. This contradicts empirical evidence found by Orr et al. (2015). Orr et al.
(2015) provided empirical functions that could be used to predict the centerline
velocity decay as well as the wake height, depending on the downstream location.
These functions were also used to help predict the spatial evolution of mean and
fluctuating velocities. The mean and fluctuating quantities could be characterized
by Equation 1.2 where, q/q
0
is a flow quantity normalized by its centerline value,
18
L
z
is the normalized wake half height, and B and C are empirical coefficients that
vary based on quantity and Fr.
q
q
0
= exp
−
(Lz−B)
2
C
2
−
B
2
)
C
2
+ exp
−
(Lz+B)
2
C
2
−
B
2
)
C
2
(1.2)
The generality of the empirical functions suggest that there is a systematic way
to characterize the near wake. However, these fits have yet to be tested experimen-
tally or at lower Fr.
1.6 Claims of Universality
1.6.1 Unstratified Wakes
The idea of a universal state is not unique to stratified wakes. In a series of papers
(Townsend, 1976) developed a hypothesis that the structure of turbulence in all self-
similar turbulent shear flows of a particular type (i.e. jet or wake) will eventually
reach the same state that depends only on the dynamics of the turbulent energy
balance. As a result, it is implied that when turbulence reaches this state there
will be no memory of the conditions which originally created it and development
of the flow will only depend on the net force on the fluid. Bevilaqua & Lykoudis
(1978) tested this hypothesis by measuring velocity and turbulence quantities in
the wake of a porous disk and a sphere, each having the same drag and therefore a
similar net force on the fluid. They found that the sphere developed a self-similar
19
structure much sooner than the porous disk and turbulence intensities were three
times larger for the sphere than in the wake of the disk for downstream distances up
to x/D≈ 110. It was also noted that there was a distinct difference in each wake,
related to initial turbulence quantities, but it could not be ruled out that each wake
may asymptote to the universal case proposed by Townsend.
AsimulationcounterparttoBevilaqua&Lykoudis(1978),conductedbyRedford
et al (2012), provided wake characteristics significantly farther downstream (x/D≈
1000). It was found that two different initialization procedures, on where a line
of vortex rings and the other where a mean flow is superimposed with turbulent
fluctuations, did eventually reach a similar state. However, at this stage the local
velocity,U, was approximately one percent of the initial velocity, and the extremely
long times it takes to reach such a state implies that true universal turbulence may
be impractical.
Plots of normalized mean velocity versus normalized cross stream length scale,
usually by the wake half width or half height, have been used as evidence that wakes
reach an asymptotic state that is independent of initial conditions, but the cross
stream length scale used to rescale the velocity can vary significantly from each
experiment (Johansson et al, 2003; George et al, 2004). A similarity analysis shows
that while mean velocity profiles and properly scaled reynolds stress, u
0
v
0
, profiles
wereindependentofupstreamconditions, theinitialconditionsdocontrolproperties
such as the growth rate, scale parameters, and other moment profiles (George et al
2004). If the initial conditions set such paramaeters, then there should be distinct
20
time frames in axisymmetric wakes where remnants of initial conditions are not
only relevant in the development of the wake but may even be traced back to the
generator.
1.6.2 Stratified Wakes
Stratified wakes appear to approach a universal state much more quickly than sim-
ilar unstratified wakes and while systematic variation in Fr seemed unimportant
in the development of the wake at moderate to late times an argument could be
made that remnants of initial conditions exist based on varying geometry. Meunier
& Spedding (2004) conducted experiments using a disk, hemisphere, cube and a 6:1
prolate spheroid. As may be expected bodies with sharper edges had comparatively
thicker and more energetic wakes than those of more streamlined bodies. In fact
it was found that the mean centerline velocity defect could be rescaled using the
amount of entrained fluid which is based entirely on the drag of the body.
The amount of entrained fluid by an object is directly related to the drag force
on the object (Tennekes & Lumley, 1972), and can be defined as
F
drag
=ρ
0
πD
2
eff
4
U
2
B
(1.3)
whereD
eff
istheeffectivediameteroftheentrainedfluid,travelingwithvelocity,
U
B
. The drag force on any object has been well studied and can also be related to
a drag coefficient, C
D
.
21
F
drag
=
1
2
C
D
ρ
0
πD
2
4
(1.4)
Combining Equation 1.3 and Equation 1.4 gives a simplified definition of ef-
fective diameter, based only on the drag coefficient.
D
eff
=D
s
C
D
2
(1.5)
Lastly an effective Froude number can be defined using the effective diameter,
Fr
eff
=
2U
B
ND
eff
(1.6)
Normalizing the centerline velocity, U
0
, the horizontal wake width, L
y
, and
Strouhal number, St by Fr
e
ff would show that each wake would eventually scale
as:
U
0
U
B
(Fr
eff
)
2/3
≈ 6.6(Nt)
−.76
(1.7)
L
y
D
eff
≈ 0.35(
x
D
eff
)
.35
(1.8)
(St
eff
)≈ 0.65(
x
D
eff
)
−.35
(1.9)
This suggested that there was no memory of initial conditions in the late wake.
One may expect differences in the turbulence quantities for the various geometries
22
similar to those found by Bevilaqua & Lykoudis (1978). However, even the cross
fluctuation profiles,u
0
v
0
, were self-similar as early asx/D≈ 170 and calculation of a
turbulent Reynolds number, defined as R
T
= (U
2
0
)/(u
0
2
0
) where U
0
is the centerline
velocity and u
0
0
is amplitude of the cross fluctuation profile, showed some initial
variation but eventually reached a single value of R
T
≈ 15.
Though drag wakes can be treated similarly at late times and have an enormous
amount of applications, perhaps more practical to nautical application are those
of momentumles or self-propelled wakes. Momentumless wakes in stratified fluids
have been reasonably well studied (Schooley,1962; Lin, 1974; Meunier & Spedding,
2006; Brucker & Sarkar, 2010; Destadler & Sarkar, 2012). While axisymmetric
wakes with momentum in an unstratitifed flow show centerline velocity decay near
(x/D)
−2/3
and a wake height growth near (x/D)
1/3
, a wake that is momentumless
has a centerline velocity decay of (x/D)
−4/5
and a wake height growth closer to
(x/D)
1/5
and so it may be expected that a momentumless wake free to evolve in
the presence of stratification would behave differently from those of drag wakes
(Tennekes and Lumley, 1972). This is the case when drag is equal to thrust. There
was no consistent Froude number scaling that collapsed the velocity and velocity
fluctuation profiles, and different curves were observed for different sized propellers
attached to the same body (Meunier & Spedding, 2006). However attaining the
exactmomentumlessconditionisextremelydifficultandanargumentcouldbemade
that such wakes are unlikely in practice. All other wakes that were either over or
23
underthrusted by more than 2% achieved the same self similar curve as with drag
wakes.
Following the studies of varying body geometry self propelled wakes a set of
general laws emerged for all stratified wakes. Specifically a length scale for any
wake can be determined based on a momentum thickness,
D
mom
=D
q
C
D
/2
q
|1− (U
2
C
)/(U
2
B
)| (1.10)
whereU
C
isthecriticalvelocitywhenthewakewillbemomentumless. Notethatfor
a drag wakeU
C
= 0 and then the effective diameter from Equation 1.5 is recovered.
Thecharacteristicwakewidth, meandefectvelocityandStrouhalnumberthenscale
empirically as,
L
y
D
mom
Fr
−0.35
mom
= 0.275(Nt)
0.35
(1.11)
U
0
U
B
Fr
−0.35
mom
= 6.6(Nt)
−0.76
(1.12)
St
mom
Fr
−0.34
mom
= 0.823(Nt)
−0.35
(1.13)
where the momentum Fourde number is,
Fr
mom
=
2U
B
ND
mom
(1.14)
24
It then appears that there is a disconnect between an early time region that
is dependent on the initial conditions and a late time wake that is seemingly void
of any information related to the startup conditions. The main objective then is
to determine when and how information that derives from the initial conditions
persists as the wake evolves towards this uniform state.
25
Chapter 2
Dissertation overview
The objectives of this work serve several purposes. The primary focus is to deter-
mine how details of an initial disturbance, some initially turbulent, can persist as
a stratified wakes evolves. This is done by systematic variation of Reynolds num-
ber and Froude number as well as the geometric configuration of the towed body.
Comparisons between laboratory and numerical experiments serve as an example
of how relatively simple empirical functions can be used to characterize early wake
quantities.
Chapter 3 investigates the wake generated by a circular grid for 2700≤ Re≤
11000 and 0.6≤ Fr≤ 9. The use of refractive index matching allows for optical
access in the near wake and provides some of the first quantitative information in
proposed three dimensional regimes. The results show how averaged and instanta-
neous quantities can be obtained using this refractive index matching technique and
also illustrates in some general sense the effect thatRe andFr have on a turbulent
patch that has some wake-like characteristics. The geometry of the wake is severely
26
impacted by lee wave motion and suggests that a near wake can not be universal
or 3D contrary to a generalized model of stratified sphere wakes.
Chapter 4 refocuses on the sphere wake, providing quantitative wake quantities
in the near wake. The study makes direct comparisons between laboratory and
numerical experiments across a range of 0.5≤ Fr≤ 8 and 200≤ Re≤ 1000. At
such low Re the wakes are not turbulent and one may expect that influence from
the initial conditions could persist without the presence of turbulence. The results
show that relatively simply simulations can be used to show systematic variation
with both Re and Fr and are in agreement with the experiments.
Chapter 5 highlights another set of experiments utilizing varied body shape
aims to determine what quantities should be measured in order to understand the
formation of patterns in stratified wakes. Four different geometries are studied: a
sphere, a disk, a large mesh spacing grid, and a small mesh spacing grid, across
three Fr = [1, 2, 4] and a single Re = 1000. Results show that for all Fr tested
buoyancy impacts the flow, showing that for all tested initial conditions tested
(Re, Fr, and body shape) buoyancy affected the flow in some way. This makes
it unlikely that a model of an initial 3D regime is a good descriptor of the flow
mechanics at these particular Re and Fr combinations. This study also makes
some effort to generalize flow quantities in an effort to create parameterizations
that can vary with Fr. This has strong implications for initialization procedures
for future numerical simulations.
27
Lastly some thoughts are written on the future of stratified wakes research in
particular how understanding the origin of coherent patterns could lie within the
use of hybrid laboratory and numerical experiments. In these experiments, infor-
mation from experiments deriving from PIV measurements can be mapped onto a
computational grid and used as inlet conditions. These types of experiments allow
for accurate near wake modelling seeing as how the initial conditions themselves
derive from experimental data.
28
Chapter 3
The turbulent wake of a towed grid in a
stratified fluid
X. Xiang, T. J. Madison, P. Sellappan, G. R. Spedding
Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles,
CA 90089, USA
Sections of this chapter appear in Journal of Fluid Mechanics 775: 149-177, June 2015.
DOI: https://doi.org/10.1017/jfm.2015.299
In a stable background density gradient, initially turbulent flows eventually evolve
into a state dominated by low Froude number dynamics and frequently also con-
tain persistent pattern information. Much empirical evidence has been gathered on
these latter stages, but less on how they first got that way, and how information
on the turbulence generator may potentially be encoded into the flow in a robust
and long-lasting fashion. Here an experiment is described that examines the initial
stages of evolution in the vertical plane of a turbulent, grid-generated wake in a
29
stratified ambient. Refractive-index-matched fluids allow optically-based measure-
ment of early (Nt< 2) stages of the flow, even when there are strong variations in
the local density gradient field.
Suitably-averaged flow measures show the interplay between internal wave mo-
tions and Kelvin-Helmholtz-generated vortical modes. The vertical shear is dom-
inant at the wake edge, and the decay of horizontal vorticity is observed to be
independent of Fr. Stratified turbulence, originating from K-H instabilities, devel-
opsuptonon-dimensionaltimeNt≈ 10, andthescaleseparationbetweenOzmidov
and Kolmogorov scales is independent of Fr at higher Nt.
The detailed measurements in the near wake, with independent variation of both
Reynolds and Froude number, while limited to one particular case, are sufficient to
show that the initial turbulence in a stratified fluid is neither three-dimensional,
nor universal. The search for appropriately-generalisable initial conditions may be
more involved than hoped for.
3.1 Introduction
3.1.1 Turbulent Patches and Wakes
A number of studies have concerned themselves with the evolution of turbulent
patches or wakes which are then free to evolve in an otherwise quiescent ambient.
Early experimental studies on wakes of bluff bodies include those by Lin et al.
30
(1992a); Chomaz et al. (1993b); Bonneton et al. (1996); Spedding et al. (1996a);
Spedding(1997,2002b). Ageneralpictureemergedofthreeregimes, whereaninitial
3D turbulence adjusts to the background buoyancy forcing through an intermediate
non-equilibrium (NEQ) regime, which later evolves in mostly horizontal motions
(Q2D) that decay slowly and maintain a very stable geometry of coherent vortical
structures. Their slow decay is quite consistent with the earlier computations and
predictions of Riley et al. (1981). While plausible that these NEQ-Q2D transitions
are consistent with the instability mechanisms successfully generalized by Billant
(2010) and Billant et al. (2010), detailed examination is difficult because the initial
conditions themselves are not prescribed with great precision.
There was some initial uncertainty about whether the observed wakes were pe-
culiar to spheres, or other conditions and symmetries on the body, but early di-
rect numerical and large-eddy simulations of high Re wake-like initial conditions
(where turbulence is superimposed upon a given mean flow profile) showed that
satisfactorily-similar phenomena could be generated with no body at all (Gourlay
et al., 2001; Dommermuth et al., 2002; Diamessis et al., 2005). Experiments on the
spread of turbulent patches in a stratified background were broadly consistent with
the late-wakes descriptions but do not show the same regular Q2D vortex arrays
(Fernando, 1988; De Silva & Fernando, 1992) since there are no mean shear initial
conditions to set it up.
Though Meunier & Spedding (2004, 2006) contend that there are universal as-
pects of these momentum wakes that cross all likely initial conditions, quantitative
31
and detailed experiments have not been made to test how useful the idea may be in
decaying and spreading grid turbulence, with no special body or initial conditions
to drive particular symmetries.
As DNS can extend to higher Re
b
, the possibility has been raised that energetic
secondaryinstabilitiesofthedevelopingshearlayersinstrongly-stratifiedturbulence
(Riley & de Bruyn Kops, 2003; Hebert & de Bruyn Kops, 2006b; Diamessis et al.,
2011) provide a further barrier to extrapolating laboratory experiments to higher
Re, and so it is interesting to consider experimental devices that may eventually
reach high Re
b
regimes (Augier et al., 2014). The different initial conditions can
also be compared in similar frameworks, experimental or computational (e.g. Pal
et al. (2013)).
3.2 Objectives
The detailed and fine-grain purpose of this study is to use a towed grid to generate
a disturbance that has also some wake characteristics, and various initial turbu-
lence intensities, so their evolution with distance from the grid (or wake age) may
be traced to provide an example of a turbulent field adjusting to the background
density gradient. The grid has a circular cross section and rectangular meshes
with equal spacing in the horizontal and vertical directions so the initial distur-
bance/forcing is also of equal magnitude in{y,z}. The experiments are performed
in a refractive-index matched fluid and special care is taken to resolve mean and
fluctuating quantities in both instantaneous and time-averaged measures.
32
tow wire
FOV support wire
grid
15R
FOV
grid
R
5.5R
4.75R
4.5R 4.25R
5R
Figure 3.1: Sketch of towed grid experiment. Top view on left and side view on the
right
The general idea is that the detailed, and quantitative study of the first stages
of stratified grid wake can shed some light as to how the well-studied late-wake
stages emerge, and how well this one example conforms to a general model of initial
turbulence in a stably-stratified background.
3.3 Experimental methods
3.3.1 Physical setup
Experiments were conducted by towing a circular grid of radius R = 4 cm and
square mesh spacing, M = 3.2 mm on 0.45 mm diameter bars (so the solidity is
26%) in a 0.80
3
m tank. The grid orientation was maintained using support and
tow wires (of 1.14 mm diameter) as shown in figure 3.1. The tow and support wires
were attached to the grid at diametrically opposite points on its circumference to
33
10
4
10
0
10
1
Re
Fr
Figure 3.2: Fr-Re parameter space. Straight lines from top to bottom indicate
Fr-Re pairs with N ={0.2, 0.4, 0.8, 1.5, 2.8}, respectively. The symbols represent
experiments listed in table 3.1. These symbol conventions will be used consistently
throughout the paper.
assuresteadytranslationmotiononly. Thecentremeasurementplaneisnotdirectly
disturbed by the wires. The fluid depth H = 35 cm, so the grid was 4.5R from the
free surface and 4.25R from the bottom of the tank. The grid was towed at speeds,
U ={6.8, 14.3, 27.6}cm/s, throughatotaldistanceof 15R, enteringthefieldofview
(FOV) after 5R so as to avoid transients from startup. Subsequent measurements of
statistics in mean and turbulence quantities over left (new) and right (old) parts of
the flow field, either time- or space-averaged, revealed no significant or systematic
differences. The temperature of the fluid was maintained at 22
◦
C and the average
kinematic viscosity was ν = 1.005× 10
−6
m
2
/s. Re ranged from 2700 to 11000 and
for N between 0.2≤N≤ 2.8 rad/s, Fr varied over 0.6 - 9.1. Figure 3.2 shows the
resulting{Fr−Re} parameter space.
34
Reference Re Fr Symbol Figure Number
R2F.6 2700 .6 F 6-11, 20, 21
R2F1 2700 1.3 3-10, 14
R2F2 2700 2.4 6, 8-10, 13, 15-18, 20, 21
R2F4 2700 4.7 J 9, 10, 15-18, 20, 21
R2F9 2700 9.1 C 10, 15-18, 20, 21
R5F1 5700 1.3 6-8, 10
R5F2 5700 2.4 6, 8-10, 13, 15-18, 20, 21
R5F4 5700 4.7 N 6, 8, 10, 15-19
R5F9 5700 9.1 M 10, 15-18
R11F2 11000 2.4
6-10, 12, 13, 15-18, 20, 21
R11F4 11000 4.7 I 6, 8, 10, 12, 15-18, 20, 21
R11F9 11000 9.1 B 6, 8, 10-12, 15-18, 20, 21
Table 3.1: Re and Fr of all experiments, with symbols corresponding to figure
3.2. The last column lists the figure in which each run appears. The symbols are
consistent throughout.
3.3.2 Data acquisition
Threecomponentsofvelocityinatwo-dimensionalplane(3C2D)wereestimatedus-
ing a stereoscopic Particle Imaging Velocimetry system from LaVision. Two match-
ing cameras (LaVision-Imager sCMOS) with a resolution of 2560×2160 pixels were
driven at a 40 Hz base rate. The measurement plane was illuminated by an Nd:YAG
laser (LaVision NANO L100-50PIV), pulsed also at 40 Hz. The laser beam is spread
intoasheetwithathicknessofabout2mminthetestsection. Inmostcasestheflow
was seeded with 20μm diameter polyamide particles (Dantec Dynamics PSP-20)
with an average density of 1.03 g/cm
3
. At the highest stratification whenN = 2.8,
35
titanium dioxide particles with average density 4.23 g/cm
3
and diameter 15μm were
substituted. Images were processed with the multi-pass cross-correlation algorithm
with window deformation and shift of 50%, and with interrogation window sizes
varying from 128× 128 to 32× 32 pix.
3.3.3 Refractive-index-matching (RIM)
The standard two-tank method was used to establish a linear density gradient, and
the refractive index was held constant at any density by systematically varying the
concentrations of both evaporated salt (Ca and Mg free) and ethanol (200 proof)
solutions. A modified version of the RIM technique (summarised by Daviero et al.
(2001)) was implemented for a broad variation of densities. An empirical rela-
tionship between density and concentration at 20
◦
C for salt solution was provided
by Weast & Lide (1989), and that for ethanol solution by Perry et al. (1997). They
can be approximated by quadratic fits as
Salt: ρ
s
= 2.46× 10
−5
C
2
s
+ 0.0070C
s
+ 0.9982 (3.1)
Ethanol: ρ
e
=−7.9× 10
−6
C
2
e
− 0.0013C
e
+ 0.9982 (3.2)
whereC
s
andC
e
areconcentrationsofsaltandethanol, andρ
s
andρ
e
aredensitiesof
salt and ethanol solutions, respectively. WhenC
e
≤ 0.35, there is an approximately
linear response in refractive index to changes in concentration. In this range, the
36
refractive index dependence on concentration is provided by Weast & Lide (1989),
and approximated using a linear fit as,
Salt: n
s
= 1.77× 10
−3
C
s
+ 1.3330 (3.3)
Ethanol: n
e
= 6.81× 10
−4
C
e
+ 1.3330 (3.4)
where n
s
and n
e
are refractive indices of salt and ethanol solutions, respectively.
Basedontheserelations,saltandethanolsolutionsthataretheoreticallyrefractive-
-index-matched were prepared with a specific density difference. Further verifica-
tion and refinement of RIM was achieved by observing the deflection of a small
laser pointer beam after passing through a triangular, transparent container with
the two solutions, and/or their mixture. The relative uncertainty in refractive index
Δn/n < 1× 10
−4
. The heavy-side tank of the two tanks was filled with only salt
solution, and the light-side tank was filled with the required mixture of salt and
ethanol to yield the required density difference and corresponding N.
There can be quite large variations in kinematic viscosity in the strongest strat-
ifications, with δν/ν≈ 0.2 over a vertical distance of 4R around the wake center
when N = 2.8. The prediction and estimation of ν is quite uncertain at high
concentrations when μ(C
e
) curves can be nonlinear and non-monotonic, and the
chemistry of their combinations is not necessarily simple. The maximum difference
in Re depends on δρ as well as δν and is about 10% across the wake, which does
37
not typically move the data from one significantly different regime to another (ex-
periments are compared across factors of 2 or 4 in Re). Furthermore, local wake
averages use both upper and lower wake edges to derive the quantitative statistics.
Finally, we note that the refractive-index matching considerably complicates
many standard methods of density estimation, and renders all current optical,
whole-field methods unusable. Methods for overcoming these difficulties are be-
ing tested, but there are no local (for example, at the wake edge, or its interior)
measures of density in the current data, only bulk estimates from samples with-
drawn before and after experiments.
3.3.4 Analysis
Velocity components{u,v,w} are available over{x,z} in the centreplane y = 0
and at 1/20 s intervals. In the lee of the towed grid, for small x/R and smallx/M,
there is no good reason to search for, or assume self-similar characteristics and one
searches for local measures that are appropriate for preserving local pattern and
geometry if it exists in some reference frame. As a start, one may consider either
spatial,
hqi(
− →
x
0
,t
l0
) =
1
ΔL
Z
L
2
L
1
q(
− →
x = (x,z
0
),t
l0
)dx, (3.5)
or temporal,
q(
− →
x
0
) =
1
Δτ
Z
τ
2
τ
1
q(
− →
x = (x
0
,z
0
),t
l
)dt
l
. (3.6)
38
averaging so the variableshqi, q, q are spatially and temporally averaged, and
instantaneous flow field quantities, respectively.
− →
x
0
is downstream position with
coordinates(x
0
,z
0
), withoriginatthecentreofthetowedbody. Averagingtimeand
length scales are Δτ =τ
2
−τ
1
and ΔL =L
2
−L
1
, respectively, andx
0
= (L
2
+L
1
)/2.
Note thatt
l
is laboratory time, whereast is defined locally for every vertical slice as
the time elapsed since the grid passed that vertical slice. ThereforeNt corresponds
to x/R through the relation x/R =Nt·Fr.
For spatial averaging to be valid, the averaging distance ΔL must be small com-
pared with characteristic stream-wise length scales in the flow, which is reasonable
at large x
0
or, equivalently, Nt.
ΔL
x
0
=
ΔL/R
Nt·Fr
1. (3.7)
While this approximation is widely used to describe the late wake (Spedding et al.,
1996a), it is not appropriate for the near wake, so all averaged results described
here (and denoted with an overbar) are temporal averages in a frame of reference
that moves with the grid.
The root mean square (r.m.s.) fluctuating quantity based on temporal averaging
is:
q
0
(
− →
x
0
) = [
1
Δτ
Z
τ
2
τ
1
[q(
− →
x
0
,t
l
)−q(
− →
x
0
)]
2
dt
l
]
1
2
(3.8)
Finally, to assure sufficient samples for convergence of both mean and fluctuating
quantities, temporal averaging is performed on the data from each run and then
39
ensemble averaged over 16 runs. The combined result is for column-wise statistics
that have up to 900 elements (the exact number varies with x).
3.3.5 Density profile filtering
Vertical density profiles were obtained at constantNt, varyingx/R for differentFr,
stations at the center of the wake. The process for extracting density from conduc-
tivity measurements is relatively straightforward, and changes in output voltage of
the conductivity prove are linearly related to the conductivity by,
V
0
=Gσ +V
off
(3.9)
where V
0
is the output voltage, V
off
is the voltage output of the probe in air, σ is
the conductivity of the solution and G is the gain. The gain is determined using the
same equation using a solution of known conductivity that was prepared before an
experimental run. The relationship between concentration of salt and conductivity
was empirically represented as (Haynes, 2009),
C
s
= 1.74× 10
−4
σ
2
+ 4.87× 10
−
2σ + 2.53× 10
−
2 (3.10)
CombiningEquations 3.1,3.9, 3.10directlydeterminesthedensityfromtheoutput
voltage of the probe.
The noise present in the conductivity measuring device can be problematic es-
pecially when the need to obtain accurate spatial gradients is required, as is needed
40
for calculation of local values of N. A two step filtering process was used to ob-
tain smooth density profiles while still maintaining the general features seen in the
profiles. Profiles are first filtered using a second order, low pass filter. The filter-
ing frequency was chosen by looking at a profile of the background density profile,
where no overturns should be present. Perceived overturns in those regions are due
to electrical noise. The density profiles were then filtered until the false overturns
were no longer present, corresponding to a wavelength of 0.2 cm. Once filtered,
smoothing splines were then fit to the profiles where the coefficients were used to
determine the vertical density gradient.
3.4 Results
3.4.1 Instantaneous and time-averaged wake structure
Figure 3.3 shows the normalized lateral vorticity, ω
y
/N(x,z), at 4 different times
withthemeasurementwindowfixedinthelaboratoryreferenceframe. Notehowthe
Nt =x/R = 0 origin moves with the grid. ω
y
develops in shear layers at the wake
edge, while the flow in the interior wake region is relatively uniform. The trailing
vortices drift slowly upstream, growing larger and weaker through diffusion and/or
merging and then begin to lose coherence as they occupy the wake interior. The
wake contracts in the vertical direction, reaching a minimum height at 3≤Nt≤ 4
before expanding again. This wake contraction is observed at all {Re, Fr}. As Fr
increases, the minimum wake height occurs at a largerx/R, but still at 3≤Nt≤ 4.
41
Nt
z/R
−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5
−1.5
−1
−0.5
0
0.5
1
1.5
−2 −1 0 1 2 3
x/R
C D E
A
B
Nt
z/R
0 0.5 1 1.5 2 2.5 3 3.5 4
−1.5
−1
−0.5
0
0.5
1
1.5
0 1 2 3 4 5
x/R
G
F
A
B
C
D
E
Nt
z/R
1.5 2 2.5 3 3.5 4 4.5 5 5.5
−1.5
−1
−0.5
0
0.5
1
1.5
2 3 4 5 6 7
x/R
F+G
A B
C
D
E
Nt
z/R
3 3.5 4 4.5 5 5.5 6 6.5 7
−1.5
−1
−0.5
0
0.5
1
1.5
4 5 6 7 8 9
x/R
F+G
A
B
D+E
C
Nt
z/R
1 2 3 4 5 6
−1.5
−1
−0.5
0
0.5
1
1.5
1 2 3 4 5 6 7
x/R
(a) (b)
(c) (d)
(e)
Figure 3.3: (a-d) 4 snapshots ofω
y
/N for Fr = 1.3,Re = 2700 (R2F1,). Minimum
contour levels, |c
min
| = ±1, and contour spacing, Δc = 1. The evolution and
interaction of individual vortices is given by letter labels, based on the full time
resolution series. The shaded region is not accessible to both cameras. (e) A time-
average, ω
y
/N. (|c
min
| =±0.5, Δc = 0.5.)
42
The local flow characteristics can be discerned from the evolution of the indi-
vidual vortices, which are tracked and marked by letters A to G in the figure. The
initial shear layer evolves into discrete vortices similar to A, B and C. D and E,
as well as F and G, show the pairing-merging process. In the grid-based reference
frame, any given vortex appears at larger Nt or x/R as wall-clock time increases.
The time-averaged, lateral vorticity, ω
y
/N therefore preserves the overall wake ge-
ometry and its contraction, which is fixed with respect to the grid, but not the
individual vortex histories, which are not.
The dynamics of the near wake can also be illustrated by the various velocity
component fields separately. Instantaneous and time-averaged streamwise velocity
fields are given in figure 3.4(a,b). The pinching of the iso-contours at the location
of highest ∂u/∂z coincides with the position of the first vortices in figure 3.3(a),
commensurate with their Kelvin-Helmholtz origin.
The wake contraction is clear in both mean and instantaneous fields, but the
time-averaged fields have more broadly-spread contours after it, as the locations
of maximum ∂u/∂z vary in z. The contraction itself is associated with a mean
upwards/downwards inflow in both instantaneous (c) and mean (d) vertical velocity
fields, from which it is clear that the vertical disturbances have moved much further
from the wake centre than have the streamwise disturbances. The mean vertical
flows do not average out to zero but maintain a coherent structure, which therefore
is moving at the speed of the grid. The remaining fluctuations in w (e) stay at the
wake edge, characterized by alternating-sign contours, associated with the vortex
43
Nt
z/R
0 0.5 1 1.5 2 2.5 3 3.5 4
−1.5
−1
−0.5
0
0.5
1
1.5
0 1 2 3 4 5
x/R
(a) (b)
(c) (d)
(e)
Nt
z/R
0 0.5 1 1.5 2 2.5 3 3.5 4
−1.5
−1
−0.5
0
0.5
1
1.5
0 1 2 3 4 5
x/R
Nt
z/R
2 4 6 8 10
−1.5
−1
−0.5
0
0.5
1
1.5
2 4 6 8 10 12 14
x/R
Nt
z/R
0 0.5 1 1.5 2 2.5 3 3.5 4
−2
−1.5
−1
−0.5
0
0.5
1
1.5
0 1 2 3 4 5
x/R
Nt
z/R
2 4 6 8 10
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2 4 6 8 10 12 14
x/R
Nt
z/R
0 0.5 1 1.5 2 2.5 3 3.5 4
−1.5
−1
−0.5
0
0.5
1
1.5
0 1 2 3 4 5
x/R (f)
Figure 3.4: Instantaneous and mean velocity field components: (a) u/U, (b)
u/U, (c) w/U, (d) w/U, (e) (w− w)/U, and (f) v/U for Fr = 1.3,Re =
2700 (R2F1, ). (|c
min
| = ±{0.1, 0.06, 0.05, 0.02, 0.04, 0.06} and Δc =
{0.1, 0.06, 0.05, 0.02, 0.02, 0.06}.)
44
structures there, and do not propagate outwards. The instantaneous out-of-plane
(or lateral) component of velocity, v/U is mainly incoherent and drawn in coarse
detail only in panel(f). The association of coherent regions of lateral motion with
wake-edge vortices begins only towards the later stages where Nt> 4.
3.4.2 Mean and turbulence profiles
Wake height and lee waves
To form appropriate averaged measures in the near wake, the wake geometry and
length scales must be specified and measured in terms that do not presume self
similarity of mean or turbulent velocity profiles. The most clear signal comes from
ω
y
(shown in figure 3.3(e)), and profiles of ω
y
(z) (shown for a range of Nt in
figure 3.5). The decay of the peak ω
y
can be observed by tracking the amplitude
of crests, and their diffusion is noted in the increasing width of the crests. The
peaks occur closer to the wake centre as Nt increases up to Nt = 4 and the trend
is reversed for Nt ≥ 4. The crests roughly correspond to the locations of the
maximum velocity gradient as can be seen in figure 3.5(c)(d). The redistribution
of turbulent kinetic energy from wake edge to centre (see Fig. 3.3) is responsible
for the changing shape of u, as the relatively irregular shape evolves into a more
Gaussian-like profile at late time.
The crests of ω
y
may be used to define the edge of the wake, and the distance
between the top and bottom crests defines a wake height, 2h(Nt). Though this
procedure will eventually fail when the vortices dissipate due to viscosity, it is
45
(a) (b)
(c) (d)
−5 −4 −3 −2 −1 0 1 2 3 4 5
−1.5
−1
−0.5
0
0.5
1
1.5
ω
y
/N
z/R

Nt = 0.4
Nt = 1
Nt = 2
Nt = 4
−2 −1 0 1 2
−1.5
−1
−0.5
0
0.5
1
1.5
ω
y
/N
z/R

Nt = 4
Nt = 6
Nt = 8
−0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1
−1.5
−1
−0.5
0
0.5
1
1.5
u/|U|
z/R

Nt = 0.4
Nt = 1
Nt = 2
Nt = 4
−0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1
−1.5
−1
−0.5
0
0.5
1
1.5
u/|U|
z/R

Nt = 4
Nt = 6
Nt = 8
Figure 3.5: ω
y
/N at specific Nt, for Fr = 1.3,Re = 2700 (R2F1,).
robust in the near wake region. The dimensionless wake half-height is defined as
L
z
=h/R, and shown as a function of Nt in figure 3.6.
The minimum values, L
z,min
, for all curves lie between 3≤Nt≤ 4 independent
of Fr and Re. If λ is a wavelength in x, it may be written as
λ
R
=
U·Nt
λ
NR
= Fr·Nt
λ
(3.11)
whereNt
λ
is the wavelength inNt. There is a linear relationship betweenλ/R and
Fr, with slope Nt
λ
. The wave pattern is more pronounced when plotted locally at
46
0 2 4 6 8 10
0.7
0.8
0.9
1
1.1
Nt
L
z
Figure 3.6: L
z
(Nt), with least squares sine series fit. Solid lines, dashed line, and
dotted line, are for Re ={2700, 5700, 11000}; different symbols correspond to cases
shown in table 3.1. Solid line with corresponds to R2F1 shown in figure 3.5. Data
points are subsampled for clarity.
the wake edge as in figure 3.7. By measuring the ΔNt between the first trough and
first crest, λ/R is calculated using equation 3.11 and shown in figure 3.8(a). The
results agree well with the linear theory prediction, Nt
λ
=N· 2π/N = 2π. AsNt
λ
is associated with the wake contraction then we may infer it to be a consequence of
a steady lee wave behind the grid, with wavelength independent of Re. Such a result
is also consistent with experiments in Chomaz et al. (1993b) and the simulations in
Hanazaki (1988) for sphere wakes with Fr> 1.
By inspection of figure 3.6, the wave amplitude, plotted in figure 3.8(b), appears
to have no systematic dependence on either Re or Fr. This contrasts sharply with
the sphere case shown by Chomaz et al. (1993b), where the scaled amplitude ζ/R
47
0 5 10 15 20
−0.1
−0.05
0
0.05
0.1
0.15
Nt
w/|U|

R2F.6
R2F1
R5F1
R11F2
Figure 3.7: w/U at the wake edge.
increased linearly in the near wake with Fr for Fr∈ [0.3, 0.8], had maximum ampli-
tudeζ/R = 1 for Fr∈ [0.8, 1.5], and decreased slowly with Fr after that. Here, the
wave amplitude is disconnected from the outer length scale, R, of the grid because
fluid particles in the midline, for example, are never obliged to turn over the grid,
instead passing straight through. This is a straightforward demonstration of how
near wake geometry can and must depend on the initial and boundary conditions.
The lee waves propagate into the exterior and isophase lines can be drawn from
the zero-crossings of the vertical velocity component. These lines are shown for
the lower half-plane in figure 3.8(c). Similar isophase lines were also reported
by Chomaz et al. (1993b) for the sphere wake.
The mix of grid permeability and vertical wave motions induced by the partial
blockage leads to some interesting effects on the wake. When λ is small, at low Fr,
48
(a) (b)
10
0
10
0
10
1
Fr
λ/(2R)

10
0
10
0
Fr
ζ/R
0 5 10 15 20
−2.5
−2
−1.5
−1
−0.5
0
Nt
z/R
0 2 4 6 8 10 12
x/R (c)
Figure 3.8: (a)λ/Rvs. Fr. + and×, experimental results of Chomazet al. (1993b),
with Re = 2177, 4514, respectively; solid line, λ/R = 2πFr; dashed line, numerical
simulation of Hanazaki (1988); other symbols, selected results from this study, see
table 3.1 for Re and Fr; (b) ζ/R vs. Fr. Solid line, 2ζ/R = Fr; other symbols are
from the same data source as panel (a); (c) Lee wave isophase curves (every π) in
the vertical centre plane, for R2F.6(F).
even the inner wake region is influenced by the lee wave as it reduces the effective
wake area, somewhat akin to a nozzle in a jet. Figure 3.9(a) shows the mean
stream-wise centre-line velocity, u
0
, relative to the grid, for varying Fr at constant
Re. (Since the grid is towed from right to left, both u
0
and U are negative.) For
49
0 2 4 6 8 10 12 14 16 18 20
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
Nt
(u
0
−U)/|U|

R2F.6
R2F1
R2F2
R2F4
(a) (b)
0 2 4 6 8 10 12 14 16 18 20
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Nt
(u
0
−U)/|U|

R2F2
R5F2
R11F2
Figure 3.9: Centerline stream-wise velocity relative to the grid. (a) For Re = 2700
and varying Fr; . (b) For Fr = 2.4 and varying Re.
all Fr, there is an initial decrease in the relative centre-line velocity for Nt≤ 2,
most likely due to the blockage effect of the grid. The flow then accelerates to a
local maximum at Nt∼ 3.5 for low Fr = 0.6 and 1, where the streamlines pinch
inwards in figure 3.4(b) and at the first minimum in figure 3.6. With increasing Fr,
the distance from the grid to the first troughx/R = Fr·Nt increases, and the local
acceleration is less pronounced. The initial dip in u
0
is less evident with higher Re
as shown in figure 3.9, likely as the grid blockage becomes less effective.
This difference in the initial dip affects the local wake behaviour for cases with
the same Fr but varying Re. When the nominal Re approximately doubles, the
towing speedU doubles, and so doesN to keep the nominal Fr constant. However,
the local flow speedu
0
behind the grid increases by a factor less than 2 (as shown in
figure3.9(b)). ThereforethelocalFrrightbehindthegridisnolongerthesame, and
50
instead, cases with higher nominal Re will have a lower local Fr. Similarly, the local
Re is increased by a factor less than 2. An important and somewhat paradoxical
consequence is that wakes with higher nominal Re may be less turbulent because
the local Fr is lower. (Further evidence will be found later in Fr
H
in figure 3.24(a).)
These results suggest that a universal decay law may not exist for the centre-line
stream-wise velocity in the early wake of a towed grid, at least for low Fr. The lee
wave is also dominant at low Fr in the early wake of a towed sphere, where it is
likely to affect the centreline velocity also. The universal decay law for the early
wake suggested by Spedding (1997) was entirely conjectural based on a presumed
non-stratified initial wake, and may not be applied to low Fr. If there are universal
conditions, they can only exist at sufficiently high Fr, at least > 4.
Free shear layers or wake edge
The eddies at the wake edge may originate from Kelvin-Helmholtz instability of
each layer. A Strouhal number can be estimated using St = k·θ, where k is a
spectroscopic wavenumber and θ is computed from
θ =
1
2
Z
+∞
−∞
u
U

1−
u
U

dz (3.12)
as for a stationary grid in a free stream with streamwise velocityU. k is determined
from the highest peak in the power spectrum of the wake edge fluctuating vertical
velocity. St is shown in figure 3.10, and varies between 0.2 and 0.3, with an average
St≈ 0.24. If St is computed by St = k·R, the resulting range St∈ [0.9, 1.7]
51
0 2 4 6 8 10
0
0.05
0.1
0.15
0.2
0.25
0.3
Fr
St

Figure 3.10: Strouhal number of the wake-edge vortices. The symbols correspond
to the cases listed in table 3.1.
agrees well with the similar measure reported by Chomaz et al. (1993b) for the
homogeneousspherewakeinthisRerange. Howeverthereisnoobviousdependence
on either Re or Fr, in contrast with the Re
0.5
dependence for the sphere, perhaps
because the grid shear layer does not start from a growing boundary layer, but
always begins abruptly at the same corner point.
Initially wake-edge vortices merge with neighbors (figure 3.3), growing in size
but decaying in intensity. The early-time mean vorticity has two primary sources:
the vertical shear due to the streamwise velocity gradient, and the horizontal shear
due to the lee wave. The mean lateral vorticityω
y
, the horizontal shear∂w/∂x, and
the vertical shear ∂u/∂z are shown in figure 3.11 for Fr = 0.6 and 9.0. In the early
wake, though w is about the same magnitude as u (evident in figure 3.7 and 3.9),
the horizontal shear is much smaller than the vertical shear. Even at low Fr = 0.6,
when the early vertical motion is expected to be vigorous, the vertical shear ∂u/∂z
52
0 5 10 15 20
−2
0
2
4
6
8
Nt
ω
y
,−∂u/∂z,∂w/∂x

0 2 4 6 8 10 12
x/R
(a) (b)
0 1 2 3 4 5 6
0
2
4
6
8
Nt
ω
y
,−∂u/∂z,∂w/∂x

0 2 4 6 8 10 12
x/R
Figure 3.11: Horizontal vorticity ω
y
(solid line), vertical shear−∂u/∂z (dashed
line), and horizontal shear ∂w/∂x (dotted line), for (a) R2F.6 (F), and (b) R11F9
(B).
still predominates and accounts for most w
y
at late times, when vertical motion is
suppressed. For high Fr, w is initially small, and ∂u/∂z remains a close measure
of ω
y
(figure 3.11(b)). This implies that the decay of ω
y
at the wake edge is not
strongly influenced by Fr even at early times.
To investigate the effect of Fr and Re on the decay of wake-edge vorticity, an
appropriate scaling is sought. Because the major source of ω
y
is the vertical shear,
it should be nondimensionalized by U/R. However, unlike the well-studied late
wake of a sphere, where the velocity distribution is Gaussian, the early wake of the
grid is not fully developed (see figure 3.5(c)(d)). Therefore the global length scale
R does not properly represent the length scale of the vortices themselves, which
only occupy the edge of the wake, and the tow speed U does not represent the
characteristic shearing velocity. Instead, one can define a modified local vorticity
thickness δ
ω
for this shear layer problem, as the distance between the nearest two
53
locations where ω
y
= 0.2ω
y,peak
, and one may also define a shearing velocity u
s
as
the difference between streamwise velocities at those two locations. Taking the first
available δ
ω
and u
s
as the initial scales, ω
y
may be scaled by u
s
/δ
ω
.
The rescaled mean vorticity is plotted against Nt and x/R for Re = 11000
and varying Fr in figure 3.12. For low Fr, more buoyancy periods are required
for the rescaled vorticity to reach the same amplitude, compared with the high Fr
case. This is counterintuitive if one imagines buoyancy to play an important role in
damping ω
y
by suppressing vertical motion. When ω
y
is plotted against x/R, the
curves collapse fairly well (figure 3.12(b)), regardless of Fr. The observation is on
the mean vorticity, and Fr could still affect the vertical shear through fluctuating
quantities and through ∂v/∂z. This possible influence on the flow instability will
be discussed further in the following two sections.
The effect of Re is shown in figure 3.13(a) for Fr = 2.4. ω
y
decays more slowly
at higher Re because viscous forces weaken compared with inertia-related forces as
Re increases. A simple Re
−1
scaling as shown in figure 3.13(b) collapses the curves
quite well.
Once a vertical structure is set, then the ensuing vertical gradients of the hori-
zontal velocity slow down primarily by viscosity. The strong influence of viscosity
and comparative unimportance of stratification parameters is quite consistent with
the expected phenomenology in a strongly stratified regime with low buoyancy
Reynolds number. It is this regime to which most previous laboratory experiments
on strongly stratified turbulence gravitate (Fincham et al., 1996; Spedding, 1997;
54
(a) (b)
10
−1
10
0
10
1
10
0
Nt
ω
y
/(u
s
/δ
ω
)

R11F2
R11F4
R11F9
10
0
10
1
10
2
10
0
x/R
ω
y
/(u
s
/δ
ω
)

R11F2
R11F4
R11F9
Figure 3.12: Rescaled vorticity decay at the wake edge for Re = 11000, (a) against
Nt; (b) against x/R.
(a) (b)
10
0
10
1
10
0
x/R
ω
y
/(u
s
/δ
ω
)

R2F2
R5F2
R11F2
10
−4
10
−3
10
0

1
Re
·
x
R
ω
y
/(u
s
/δ
ω
)
R2F2
R5F2
R11F2
Figure 3.13: Rescaled vorticity decay at the wake edge for Fr = 2.4, (a) against
x/R; (b) against
1
Re
·
x
R
.
Praud et al., 2005). The estimation of local buoyancy Reynolds number is consid-
ered later.
55
The magnitude and scaling of shear components
The importance of (∂u/∂z)
2
in determining the kinetic energy dissipation is con-
sistent with grid turbulence experiments by Fincham et al. (1996) and Praud et al.
(2005). There is a closer resemblance to the sphere wakes of Spedding (2002b) be-
cause of the influence of the mean flow which breaks the expected rough equivalence
of (∂u/∂z)
2
with (∂v/∂z)
2
. Figure 3.14 shows selected shear components available
in the experiments at Nt = 5 and Nt = 10. Though (∂u/∂z)
2
>> (∂v/∂z)
2
at
early times, the difference between them decreases.
(∂w/∂x)
2
is expected to be much smaller than (∂v/∂x)
2
(Spedding, 2002b) since
the stratification suppresses the horizontal gradient of vertical velocity, but, due to
the lee-wave-generated mean flow, (∂w/∂x)
2
is similar in magnitude to (∂v/∂x)
2
at early times, becoming smaller as the amplitude of lee wave decreases in the far
wake.
Thestrongverticalshearisthoughttobeanimportantmechanismfortriggering
secondaryturbulence(Lilly,1983;Hebert&deBruynKops,2006b), eveninstrongly
stratified flows (Riley & de Bruyn Kops, 2003; Diamessis et al., 2011), and therefore
has received significant attention in stratified turbulence research. The averaged
mean-squared vertical shear S
2
is
S
2
=

∂u
∂z
!
2
+

∂v
∂z
!
2
. (3.13)
56
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
z/R
(∂(v,w)/∂x)
2
,(∂(u,v)/∂z)
2

(∂u/∂z)
2
(∂w/∂x)
2
(∂v/∂x)
2
(∂v/∂z)
2
Figure 3.14: Shear components in vertical slices, for Fr = 1.3,Re = 2700 (R2F1,).
The black and grey lines show Nt = 5, 10, respectively. Solid lines, dashed lines,
dash-dot lines, and dotted lines show (∂u/∂z)
2
, (∂w/∂x)
2
, (∂v/∂x)
2
, and (∂v/∂z)
2
,
respectively.
As (∂u/∂z)
2
dominates, non-negligibleS
2
occurs primarily at the wake edge where
themostvigorousKelvin-Helmholtzinstabilitiesoccur, andhere, aswehavealready
seen, the geometry and location of these layers are strongly influenced by the lee
waves. The evolution ofhS
2
i
z
/(U/R)
2
(h·i
z
is a temporal average and between
[−L
z
,L
z
]) is shown in figure 3.15. Note because we take the mean vertical shear
over the entire wake region, the global length scaleR and velocity scaleU are used.
57
10
−1
10
0
10
1
10
−2
10
−1
10
0
10
1
Nt
hS
2
i
z
/(U/R)
2

R2F2
R2F4
R2F9
R5F2
R5F4
R5F9
R11F2
R11F4
R11F9
Figure 3.15: Evolution of nondimensional mean square vertical shear integrated ver-
tically over the wake region. The solid lines, dashed lines, and dotted lines, denote
Re = 2700, 5700, 11000, respectively. Black lines, dark gray lines, and light gray
lines, denote Fr ={2.4, 4.7, 9.1}, respectively. Symbol is added to the beginning of
each line for further clarity.
At fixed Re, increasing Fr leads to lowerhS
2
i
z
/(U/R)
2
, as expected. In Sped-
ding (2002b) the Fr-dependence of (∂u/∂z)
2
was removed when only fluctuating u
components were included. However, at fixed Fr, higher Re is also associated with
a smallerhS
2
i
z
/(U/R)
2
. For this particular experiment, this means at the sameNt
(or equivalently, same x/R, since Fr is fixed), though the towing speed is doubled
from case to case, the average vertical shear is increased by a multiplier less than
2. The consequences of this will be discussed later.
58
(a) (b)
10
−1
10
0
10
1
10
−4
10
−3
10
−2
Nt
hS
2
i
z
/Re

R2F4
R5F4
R11F4
10
−1
10
0
10
1
10
−4
10
−3
10
−2
10
−1
Nt
hS
2
i
z
Fr
0.7
/Re

R2F2
R5F2
R11F2
R2F4
R5F4
R11F4
R2F9
R5F9
R11F9
Figure 3.16: Vertically averaged mean square vertical shear, (a) rescaled with 1/Re,
against Nt, for Fr = 4.7; (b) rescaled with Fr
0.7
/Re, against Nt, for the same data
set as figure 3.15.
Diamessis et al. (2011) found that a scaling ofhS
2
i/(U/R)
2
(Fr
2
/Re), whereh·i
denotesaveragingoverthevolumeofthewake, collapsedthelatewakedatafor 30<
Nt< 200. The Re-contribution comes from arguments in Riley & de Bruyn Kops
(2003) who suggested thathS
2
i/(U/R)
2
/Re, which is a part of the full dissipation
term, would be independent of Re once truly turbulent-like behavior is developed.
Noting the differences above in how Re and Fr are varied, the same re-scaling
fails to collapsehS
2
i
z
(mean vertical shear over the vertical centerplane only) here.
However, in the early wake of a grid, the velocity profile is not fully developed
and not Gaussian-like, and the global scales of R and U may not be the best
scaling parameter forhS
2
i. Regardless of nondimensionalization, a scaling of Re
−1
collapses the dimensional shear with moderate success in figure 3.16(a). If Fr is
varied systematically, then Fr
−2
overcompensates in early wakes. Arguing that
59
Figure 3.17: Density profiles atNt = [0, 1, 2, 3] forRe = 11000 andFr = 4. Profiles
are offset for clarity.
a late-wake, Fr-independent state has not yet been reached, one may empirically
search for a different scaling for Fr, as shown in figure 3.16(b), a result that is
purely empirical, and may depend on this particular grid geometry. We return to
this point later.
3.4.3 Density Profiles
Time evolution and local buoyancy frequency
Figure 3.17 shows instantaneous density profiles atNt = (0, 1, 2, 3) forRe = 11000
andFr = 4. Nt = 0 corresponds to the ambient density profile. Initially atNt = 1
small perturbations and overturning motions, where heavier fluid is on top of lighter
fluid, form through the wake region. As the density field continues to evolve from
60
Figure 3.18: Local buoyancy frequency profile,N
2
for last profile sounding in Figure
3.17. Note negative values of N
2
, indicating when lighter fluid is on top of heavier
fluid.
the initially strong shear, the overturning motion becomes more prominent. This is
seen atNt = 3 where small amplitude perturbations still permeate the center of the
density profile but larger amplitude density fluctuations concentrate near the wake
edge. The large amplitude fluctuations do not occur exactly at z/R = 1 instead
shifting toward the center of the grid which is consistent with the presence of the
lee wave that forces a contraction of the wake. the more pronounced structure near
the wake edge is also consistent with Kelvin-Helmholtz instabilities along the wake
edge that initially form and grow through pairing and merging.
Fromthedensityprofiles, thelocalbuoyancyfrequencycanbedeterminedwhere
N
2
loc
=−gρ
0
(∂ρ(z)/∂z). Figure 3.18showstheresultingbuoyancyfrequencyforthe
sameNt = 3 profile in Figure 3.17. It is convenient to look at N
2
as opposed toN
61
Figure 3.19: (a). Instantaneous (solid) and sorted (dashed) density profiles for
Re = 5400, Fr = 4. (b). Thorpe displacements measured at each position for the
profiles in (a).
since regions where the density gradient is positive, i.e. where overturning occurs,
will cause N to be imaginary. The minimum value of N
2
loc
corresponds near the
center of the larger overturns because the density gradient has the largest positive
value at this point.
Overturning length scales
Several overturning length scales can be investigated in order to determine the
relative size of overturning motion in the early wake. A general measure for the
vertical size of an overturn can be related to the Thorpe displacement, d
i
, and
Thorpescale,L
T
(Thorpe,1977). TheThorpedisplacementisdefinedasthevertical
displacement a fluid particle must travel in order to reach a gravitational stable
sorted profile. In practice this requires sorting the instantaneous density profile so
62
lighter fluid is always on top of heavier fluid where at each z position the Thorpe
displacement is given by, d
i
= z
i
−z
0
. The Thorpe scale is then the root mean
square of the thorpe displacements defined as L
T
=hd
2
i
i
1/2
, wherehi denotes an
appropriate vertical average over an overturning region. Figure 3.19(a) shows
a density profile taken at Nt = 3 for Re = 5400 and Fr = 4 along with its
gravitationally stable, sorted profile. Figure 3.19(b) is the corresponding Thorpe
displacements. Generally, thethorpescalehasbeenusedtodeterminethemeansize
of overturns in turbulent patches, however the presence of through flow for the wake
of a towed grid complicateds the selection of a patch size. Larger overturns exist
at the wake edge and will have larger Thorpe displacements. Figure 3.19(b) shows
the maximum Thorpe displacement, L
Tmax
defined as (d
i
)
max
, occurs at the wake
edge and an average of this value, at the wake edge, is likely a more representitive
measure of determining size of an overturn than an rms value calculated over the
entire wake region.
A local horizontal overturning length scale, L
h
, is computed from the zero-
crossing of the autocorrelation functions of the fluctuating lateral vorticity, ω
0
y
, and
fluctuating vertical velocity, w
0
. At the wake edge, ω
0
y
and w
0
can be reconstructed
into a time series. Then the time difference between the first positive zero-crossing
and first negative zero-crossing is computed from the autocorrelation functions,
separately for each time series and denoted as δt
1
and δt
2
the local overturning
timescale is then taken asδt
lh
= (δt
1
+δt
2
)/2. L
h
is then computed byL
h
=δt
lh
u
loc
,
where u
loc
is the local mean streamwise velocity with respect to the grid at the
63
A
B
C
D
E F
Figure 3.20: Density contours found in a typical Kelvin Helmholtz billowtrain.
Original plot from Riley & Lelong (2000). Definition of overturning length scales
are as follows: L
h
is measured from E to F, L
Tmax
is measured from B to C, and
L
V
is measured from A to D
wake edge. This definition is valid for the early wake when the wake edge is clearly
defined and vertical motion has not been completely suppressed by the ambient
stratification.
The definitions of each length scale are illustrated by Figure ??. The horizontal
overturning length scale, L
h
can be thought of as the horizontal size of an eddy at
the wake edge. At points E and F the vertical velocity is 0. If one were to plot the
vertical velocity then between E and F then a sinusoidal shape would be obtained
with a 0 velocity also at the center of the structure. The wavelength of that sinusoid
is thenL
h
. It is also clear by observation of the density contours that the maximum
thorpe scale,L
Tmax
, is the length scale from B to C as this is the only region where
64
Figure 3.21: Sketch showing the definition of L
v
. L
v
can be measured even if no
overturns are present, where as in the example shown here L
Tmax
= 0.
overturning is present. Another vertical length scale, L
v
, can be defined based on
the idea that any deviation from the background density gradient in the shear layer
is caused by eddies at the wake edge and can be measured as the spatial scale from
pointAtopointDinFigure 3.20. StartingfrompointAandmovingtowardpointD
there is an initial pinching of the density contours, which corresponds to an increase
in the density gradient relative to the background. After the overturn region (B to
C) there is another pinching of density contours corresponding to a more negative
density gradient with respect to the initial background. L
v
is also associated with
a very specific pattern in the buoyancy frequency profile N
2
. Figure 3.21 shows
the N
2
profile for such a case. L
v
is found by finding the vertical distance between
the first and fourth zeros of the corresponding N
loc
−N
2
profile corresponding to
the top shear layer. This process is then also repeated for the bottom shear layer
65
Figure 3.22: Comparison between lengthscales (a) L
Tmax
against L
h
. Solid black
line is L
Tmax
≈.5L
h
(b) L
v
against L
h
. Solid black line is L
v
≈.8L
h
. Black sym-
bols, grey symbols and open symbols correspond to Fr ={2.5, 4, 9}, respectively.
Triangle, circle, and diamond correspond toRe ={2500, 5000, 11000}, respectively.
using the last and fourth to last zero. This process is automated for each individual
run. L
v
and L
h
are attempts at measuring the height and width of a vortex, while
L
Tmax
only captures the largest scale where overturning has occurred within that
vortex.
Figure 3.22 shows a comparison of L
Tmax
and L
v
with L
h
for Fr ={2.5, 4, 9}
and Re ={2500, 5000, 11000}. In general as Fr increases the overturning length
scales tend to also increase. The same is also true for increasingRe. This is because
at low Fr there is a strong resistance to overturning motion due to the strong
stratification, and as such the vertical extent that a KH billow train occupies is
comparatively smaller than a higher Fr case. As a result the maximum overturn
66
regions must also be smaller for smallerFr andRe based on the definition in Figure
3.20. There is a linear trend for each of the lengthscales whereL
Tmax
scales as.5L
h
and L
v
scales as .8L
h
.
Global Richardson number
The strength of the vertical shear and its likely role in promoting local overturning
instability can be measured by a minimum Richardson number (Ri) criterion. Ri
was introduced by ?? to describe the ratio of contributions to the energy of tur-
bulence from the vertical velocity gradient and density gradient. He showed that
turbulence could develop from laminar flow when Ri was less than the ratio of eddy
viscosity to eddy conductivity, which he assumed to be unity. In oceanography, Ri
has a more general form that takes linear stratification into account, and it local
value Ri
loc
can be defined as
Ri
loc
=
N
2
S
2
. (3.14)
In this study, Ri
loc
is more of a semi-local number becauseS has a local value, vary-
ing across the wake, whileN is just the globalN. This is an enforced simplification
because there are no data on local ρ(x,z), or its spatial gradients. In practice, one
may expect quite strong initial variations in ρ(x,z) in a turbulent stratified wake
(Xu et al., 1995). With the resulting wide variation in local N, it might be hard in
any event to compute a meaningful Ri
loc
. Doing so in this experiment must await
planned density profile measurements.
67
10
−2
10
−1
10
0
10
1
10
2
10
−4
10
−3
10
−2
10
−1
10
0
10
1
Nt
Ri
loc,min

R2F2
R5F2
R11F2
R2F4
R5F4
R11F4
R2F9
R5F9
R11F9
Ri
loc,min
= 1
Ri
loc,min
= 0.25
(Nt)
2
Figure 3.23: Local minimum Richardson number, againstNt, for the same data set
as figure 3.15.
At any given Nt, since N is constant, Ri
loc,min
will occur at the wake edge. Its
time evolution is given in figure 3.23. All cases begin at a value much smaller than
the nominal critical value of
1
4
. The initial values have a systematic dependence on
both Fr and Re. For cases of the same Re, the smaller the Fr, the larger the initial
Ri
loc,min
, meaningthe verticaloverturningis moresuppressed in thebeginning. This
dependence is the same as reported by Diamessis et al. (2011). For Fr fixed, larger
Re corresponds to larger initial Ri
loc,min
, perhaps counter-intuitively, and in contrast
with the simulations. The values of Ri
loc,min
presented here are estimates using a
constant buoyancy frequency. One can also look at local changes inN to determine
local changes and evolution of Ri
68
(a) (b)
10
−1
10
0
10
1
10
−1
10
0
10
1
10
2
10
3
10
4
Nt
Re
H
Fr
2
H
(Nt)
−2
10
−1
10
0
10
1
10
−2
10
−1
10
0
10
1
Nt
Fr
H

R2F2
R5F2
R11F2
R2F4
R5F4
R11F4
R2F9
R5F9
R11F9
(Nt)
−1
Figure 3.24: (a) Horizontal Froude number Fr
H
, and (b) Re
H
Fr
2
H
, against Nt, for
the same data set as figure 3.15.
Aftertheinitialgradualincrease,thefurtherincreaseinRi
loc,min
isFr-independent
and scales as (Nt)
2
. For smaller Fr, this late-time limit is reached at a relatively
larger Nt. The same Fr-independence was also observed by Spedding (2002b)
and Diamessis et al. (2011) for the wake of sphere. Though Re-independence is
also observed here, the contrasting Re-dependence is observed in Diamessis et al.
(2011) when ν is changed, and we can only claim Re-independence for ν fixed.
In figure 3.23, Ri
loc,min
crosses unity at Nt≈ 10 for all conditions. This value
is the same as reported by Diamessis et al. (2011) for the low Re = 5000 runs. The
criterion of Ri
loc
< 1 for development of shear instability, and therefore stratified
turbulence, may be argued to be equivalent to Re
H
Fr
2
H
> 1 (Riley & de Bruyn Kops
(2003); Hebert & de Bruyn Kops (2006a,b)), where Re
H
=u
H
l
H
/ν is the horizontal
Reynolds number and Fr =u
H
/(l
H
N) is the horizontal Froude number.
69
Figure 3.24 shows the evolution of Fr
H
and Re
H
Fr
2
H
for different cases. For
constant Fr, the smaller Re leads to a relatively larger local Fr
H
at early times,
which is consistent with the analysis in section 3.4.2. Fr
H
drops below unity at very
smallNt for all cases, and further decrease of Fr
H
at later time scales approximately
with (Nt)
−1
, and is Fr-independent (and given ν fixed, Re-independent). Similar
collapse of data is also observed in Re
H
Fr
2
H
, but scales approximately with (Nt)
−2
.
Local Richardson Number
The strongest density gradients will occur at overturning locations. One can define
a local maximum and minimum Richardson number, Ri
loc,min
and Ri
locmax
, in the
shear layer as
Ri
loc,max
=
hN
2
max
i
S
2
max
;Ri
loc,min
=
hN
2
min
i
S
2
max
(3.15)
For each density profile the local buoyancy frequency can be calculated (Figure
3.18). Additionally the vertical length of the shear layer is defined by, δ. S
max
is
the maximum value of the shear within this region. hN
2
min
i andhN
2
max
i are the
maximum and minimum values of N
2
in the shear layer region averaged over all
experimental runs at a given Re and Fr. Figure 3.25(a) shows the time evolution
of Ri
loc,max
and Ri
loc,min
compared to a global Ri from Figure 3.23 for Re = 5000
and varying Fr. Ri
loc,max
and Ri
loc,min
represent the range of Richardson numbers
that occur within the shear layer. Negative values ofRi indicate regions where there
is a positive density gradient, where heavier fluid is on top of lighter fluid. Initially
70
Figure 3.25: Time evolution of Ri
loc,max
and Ri
loc,min
: (a) For constant Re = 5000
and Fr = 2, 4, 9. (b) For constant Fr = 4 and Re = 2500, 5000, 11000. In each
plot the solid black line is Ri using the background density gradient, N. Each
group of data points has been slightly offset for clarity. Black, grey, and open
symbols correspond to Fr = 2.5, 4, 9. Triangle, circle, and diamonds correspond to
Re = 2500, 5000, 10000, respectively.
there is a distinct Fr dependence similar to the trend found in Figure 3.23, where
Ri decreases as Fr increases. At Nt≈ 1.5 the maximum and minimum values of
Ri become Fr-independent. Figure 3.25(b) shows the time evolution of Ri
loc,max
and Ri
loc,min
at Fr = 4 and varying Re. For this case there does not appear to be
anyRe dependence even as early asNt≈.5. In both the constantFr andRe cases
the values of local values of Ri tend to keep diverging from the global Ri which is
consistent with the growing overturns. It is important to note that the local values
of Ri depict a snapshot of overturning motion but says nothing of whether or not
overturning motion will continue.
71
Figure 3.26: Ri
loc,max
for all Fr and Re cases versus time. The solid black line is
Ri
min
using global N. Dashed black line is the critical value Ri
c
= 1/4. Symbols
are the same as Figure 3.25
Figure 3.26 shows Ri
loc,max
for all cases as a function of time along with the
global Ri, based on the background buoyancy frequency, N. Initially the values of
Ri
loc,max
have a large spread but approach a similar value near Nt≈ 3. The local
values ofRi also scale with (Nt)
2
, similar to the scaling of the globalRi. The max-
ima tend to approach the critical value at a lower Nt than the global case. Several
cases have already crossed the critical value of Ri
c
= 1/4 atNt = 2 and byNt = 3
all cases have crossed the threshold. However, atNt = 3 overturning is still present
given the large range of Ri seen in Figure 3.25. Although overturns are growing
in time, due to pairing and merging mechanisms from KH instabilities along the
wake edge, causingRi
locmax
to increase pas the critical value, the critical value itself
gives no indication of whether or not the production of stratified turbulence and
72
shear instabilities has ceased, only that some mixing event has previously occurred.
This is also consistent with the idea that an overturning event is simply a remnant
of some initial turbulent disturbance (Gibson, 1980). This would suggest that the
critical value of 1/4 may not be the most reliable measure of whether or not strati-
fied turbulence has ceased and the topic of a threshold for Richardson number has
been contested previously (Galperin et al 2007).
Fluctuating quantities
The primary local source of turbulence is at the wake edge, where shear layers
induce Kelvin-Helmholtz instability and eddies form. Then the eddies evolve by
a pairing-merging mechanism and spread into interior wake. In this section we
examine the characteristics of the turbulent quantities.
Figure 3.27(a) and 3.27(b) show the vertical profiles ofu
0
/U andw
0
/U at differ-
ent Nt for Fr = 4.7,Re = 5700. Before Nt = 3, the shape and magnitude of u
0
/U
and w
0
/U are closely matched, with peak values at z/h =±1. The magnitude and
width of the peaks of both u
0
/U and w
0
/U increase with Nt. Note that the latter
peaks are slightly broader in z as waves propagate to the exterior. After Nt = 3,
the maxima in u
0
/U have grown in the midline, while (w
0
/U)
max
remains around
z/h =±1. Figure 3.27(c) shows the centerline u
0
0
/U and w
0
0
/U. The shapes of
the two curves are perhaps similar, increasing at early times as the wake becomes
more turbulent, and then dissipating due to viscosity, but the amplitudes are very
73
different. Not only is the initial forcing not isotropic in{x,z}, but wave modes
propagate outwards, away from the centerline.
−2 −1 0 1 2
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
z/h
u
′
/|U|

(a) (b)
(c) (d)
−2 −1 0 1 2
0
0.01
0.02
0.03
0.04
0.05
0.06
z/h
w
′
/|U|

0 2 4 6 8
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Nt
u
′
0
/|U|,w
′
0
/|U|
−2 −1 0 1 2
−1.5
−1
−0.5
0
0.5
1
x 10
−3
z/h
−(u−u)(w−w)
U
2

Figure 3.27: Fluctuating quantities for Fr = 4.7,Re = 5700 (R5F4,N). (a)u
0
/U in
vertical slices; (b) w
0
/U in vertical slices; (c) u
0
0
/U (solid line) and w
0
0
/U (dashed
line) in wake centre; (d) Reynolds stress in vertical slices. In (a), (b), (d), z is
nondimensionalised by local half wake height; black solid line, dashed line, and
dotted line for Nt = {1, 2, 3}, respectively; gray solid line and dashed line for
Nt ={4, 6}.
The Reynolds stress in vertical slices is plotted in figure 3.27(d) for the same
case. The peak values are located around z/h =±1, with magnitude increasing
74
before Nt = 3, and decreasing after Nt = 4. The inner wake region becomes quite
turbulent afterNt = 4 as another opposite-sign peak occurs between the wake edge
and center, and grows even larger in magnitude than the wake edge peak atNt = 6.
(a) (b)
(c) (d)
0 2 4 6 8 10
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Nt
w
′
max
/|U|

R2F.6
R2F2
R2F4
R2F9
0 2 4 6 8 10 12
0.02
0.04
0.06
0.08
0.1
0.12
0.14
x/R
w
′
max
/|U|
0 2 4 6 8 10 12
0
0.01
0.02
0.03
0.04
0.05
0.06
Nt
w
′
max
/|U|

R11F2
R11F4
R11F9
0 2 4 6 8 10 12
0
0.02
0.04
0.06
0.08
0.1
Nt
w
′
max
/|U|

R2F2
R5F2
R11F2
Figure 3.28: w
0
max
/|U| in the wake region. (a) For constant Re = 2700, varying
Fr, agains Nt; (b) Same data as (a), but against x/R; (c) Constant Re = 11000,
varying Fr, against Nt; (d) Constant Fr = 2.4, varying Re, against Nt.
At early times, the fluctuating quantities reflect the simultaneous dynamics of
KH modes with the lee wave at the wake edge. Figure 3.28(a) shows the maximum
75
fluctuating vertical velocity against Nt for Re = 2700 but varying Fr. As Fr in-
creases, so doesw
0
max
and the peak occurs at smallerNt. When Fr = (1/N)/(R/U)
is high, R/U is small compared with 1/N and a maximum after a fixed number
of overturns appears at smaller Nt. If plotted against x/R, the maxima occur at
x/R∈ [3, 5] as shown in figure 3.28(b).
At higher Re, the trend that w
0
max
peaks at smaller Nt for larger Fr persists
but the amplitude does not increase systematically. Here Re may be large enough
so that initial turbulent dissipation has removed kinetic energy from the eddy mo-
tions before there is any significant interaction with the exterior. If that is so,
then it presumably contrasts with low Re cases where this does not happen, and
Figure 3.28(a) and (c) have results from the same physical experiment, but in two
rather different dynamical regimes, distinguished by Re
H
·Fr
2
H
= Re
b
.
Because Re
H
·Fr
2
H
is a local parameter, to be consistent, w
0
max
is nondimension-
alized by the maximum|u| at the wake edge behind the grid, which occurs at Nt
roughly equal to that of the initial dip in figure 3.9. The results for similar cases
are shown in figure 3.29. The peak locations and magnitudes match reasonably
well with those shown in figure 3.24(b). For Re = 2700, Fr = 9.1 and Fr = 4.7
have Re
b,max
> 200, Fr = 2.4 has Re
b,max
≈ 30, and Fr = 0.6 has Re
b,max
be-
low unity, so the peak magnitudes of w
0
max
/|u|
max
differ considerably. The three
cases with Re = 11000 have Re
b,max
∈ [20, 100], and the three cases with Fr = 2.4
have Re
b,max
∈ [10, 30], indicating they are likely in the same dynamical regime,
respectively, so peak magnitudes of w
0
max
/|u|
max
are similar.
76
(a) (b)
(c) (d)
0 2 4 6 8 10
0.05
0.1
0.15
0.2
0.25
Nt
w
′
max
/|u|
max

R2F.6
R2F2
R2F4
R2F9
0 2 4 6 8 10 12
0.05
0.1
0.15
0.2
0.25
x/R
w
′
max
/|u|
max
0 2 4 6 8 10 12
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Nt
w
′
max
/|u|
max

R11F2
R11F4
R11F9
0 2 4 6 8 10 12
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Nt
w
′
max
/|u|
max

R2F2
R5F2
R11F2
Figure 3.29: w
0
max
/|u|
max
in the wake region, where|u|
max
is the maximum mean
streamwise velocity at the wake edge behind the grid, for the same data set as
figure 3.28.
3.5 Discussion
3.5.1 Initial conditions in stratified wakes
As noted in the introduction, independently, for stratified experiments in general,
simulations in boxes, and for both types of investigations of localized, stratified
77
turbulence„ there has been continuing discussion on the generality of any given
configuration. Even when turbulent stratified wakes can be shown to be indistin-
guishable at late times, given a proper rescaling by the initial horizontal momentum
flux (Meunier & Spedding, 2004), the conjectured similarity at early times has not
been demonstrated, only inferred. Here, detailed examination of one particular
case is sufficient to show that there cannot be a universal or general set of initial
conditions. In figure 3.8, the lee wavelength, λ agrees satisfactorily with previous
experiments, both laboratory and numerical, on a sphere wake. The variation of
wave amplitude with Fr however, is quite different, and is because of the vary-
ing (and sometimes counter-intuitive) influence of Fr and Re on the effective grid
obstruction itself. This is only one case, but it is sufficient to dismiss a general
claim.
3.5.2 Initial 3D regime?
Theexperimentandanalysismethodsdescribedhereweredeliberatelyconfiguredto
allow measurements at early times, close to the grid. Figure 3.4 shows how instanta-
neous and mean data are available atx/R< 1/2 andNt< 1/2, limited only by the
stereoscopic access of the camera pair. These small x/R are sufficient, however, to
provide data at supposed 3D regimes (Nt< 2 Spedding (1997)). The velocity field,
whenever measurable, is always affected by the presence of the background density
gradient. For example, in figure 3.9, the centerline velocity is greatly affected by
the lee wave at low Fr. The initial flow is not isotropic, but the peak local values of
78
w
0
max
/|u|
max
in figure 3.29 also always have some systematic variation with Fr. The
difference in peak values is more pronounced at low Re, and at high Re one might
claim some general similarity for Nt < 2. There is no truly 3D turbulent field in
this experiment. If there is a minimum Fr required for some effectively unstratified
early motion, then it must be at least Fr> 9, the maximum tested.
3.5.3 BuoyancyReynoldsnumberinlaboratoryexperiments
A condition for true turbulence in the presence of stratification is often set on
Re
H
Fr
2
H
. IfRe
H
Fr
2
H
isequivalenttobuoyancyReynoldsnumber(Hebert&deBruynKops,
2006b), then the late-time scale separation between the Ozmidov and Kolmogorov
scales is also independent of Fr. In figure 3.24 (b), the experimentally-derived
Re
H
Fr
2
H
cross unity atNt≈ 10 for all cases. This agrees with the result inRi
loc,min
(figure 3.23), supporting the idea of their equivalence in predicting/measuring strat-
ified turbulence. Before Nt≈ 10, because Fr
H
< 1 and Re
H
Fr
2
H
> 1, the flow in
this experiment can be asserted to be mostly in the stratified turbulence regime , is
weakly viscous, and dissipation at large scales is negligible. AfterNt≈ 10, because
Fr
H
< 1 and Re
H
Fr
2
H
< 1, the flow starts to enter the viscous-affected stratified flow
regime , the stratified turbulence is strongly damped by viscosity, and dissipation
occurs predominantly at large scales. The results of the towed grid are in good
agreement with prevailing notions of a minimum Re
H
Fr
2
H
for true turbulence in a
stratified medium. As in previous experiments (Fincham et al., 1996; Praud et al.,
79
2005;?) there is a short period of time available before no range of scales for turbu-
lence is available. Though at late times, the flow in the vertical centreplane tends
to evolve in a similar way, and is viscous-stratification dominated, the trajectory
for the flow to enter such a state varies. These trajectories are not general and vary
in sometimes curious ways with Re and Fr.
3.5.4 Parametrization and comparison with numerical sim-
ulation
There are a number of occasions in this study where Re-dependence and param-
eterizations from experiment and simulation do not agree, or are not equivalent.
For example, the observed Re-dependence ofhS
2
i
z
/(U/R)
2
in figure 3.15 is not the
same as presented in Diamessis et al. (2011). There are two sets of considera-
tions that are different in the experiments and in the simulations. In numerical
experiment the nominal Re is varied by modifying ν, with U constant, and so an
Re-dependence of the nondimensional shear is the same as that of the dimensional
shear, which actually has the same Re-dependence as this experiment. Second, as
previously noted in section 3.4.2, the streamwise velocity behind the grid does not
preserve scale similarity, thus making the local Fr and Re relations different from
nominal. For example, in the simulations,U is constant and thereforeN is constant
for a fixed Fr, but N in laboratory experiments scales with U to keep Fr constant.
Therefore, in experiment, when Re doubles, N also doubles, but the local vertical
shear is increased by a multiplier less than 2. Therefore quantities that ought to
80
scale onU must use a localU, found from experiment. This value is peculiar to this
experiment, and likely to this grid permeability. It is why for fixed Fr, higher Re is
associated with a smallerhS
2
i
z
/(U/R)
2
, and it is why Ri
loc,min
apparently increases
for larger Re.
Doubling Re or Fr may thus not be equivalent in laboratory or numerical ex-
periment, because each is making transitions between dynamically different regimes
at different points. Dynamical equivalence is only assured for geometrically-similar
flows, and the local geometry of the grid forcing here is not the same as the wake
behind a sphere, or for turbulence superimposed on a mean profile.
3.6 Conclusions
The turbulent wake of a grid was studied partly as a means of generating turbulence
of varying intensity to observe the interaction with the stable density background,
and partly as a means of producing a patch of wake free to evolve in space and
time. At the same time, one may look for similarities and differences with the
much-studied sphere wake.
In the absence of evidence to the contrary, in fact in the absence of any evidence
at all from near-wake turbulence in stratified fluids, it has been conjectured (e.g.
Spedding (1997)) that there is an initial stage in stratified wakes where the flow
evolves much as it does in the absence of stratification. General and universal
models have even been proposed for all wake defect magnitudes and signs provided
they are not very close to zero (the momentumless condition) (Meunier & Spedding,
81
2004, 2006) and simple analytical models have been constructed starting with this
base condition over a wide range of Re and Fr (Meunier et al., 2006). The results
presented here show that there can be no such universal initial condition. For Fr
∈{0.6, 10} and Re∈{2700, 11000} there are no conditions where the flow can be
claimed to be either 3D or universal.
There is an interesting interplay between the dynamics of lee wave motions and
of shear-layer instabilities. The balance between them is reasonably captured by an
internal Froude number, Fr = (1/N)(U/R) where 1/N is a characteristic timescale
of the lee wave andR/U is an eddy turnover time. In fact the shear layer evolution
can proceed almost oblivious of Fr at either low or high Fr and so there are different
trajectories for the flow to arrive at a late-time regime. These trajectories can be
described in terms of Re
H
· Fr
2
H
and the experiments described here cover a range
where some may be claimed to be turbulent for some number of overturns, and
over some range of scales, while others are constrained from the start by either
stratification or viscosity, or both.
Though this is a rather detailed study of one grid geometry over a range of
{Re,Fr}, the consistent time-averaging and sampling in a reference frame attached
to the body sets the stage for further studies that can compare the specific effect
of various geometries on early wake parameters, some of which emerge to set the
conditions for the persistent late wakes.
This work is supported by ONR Contract N00014-14-1-0422, under the management of Dr. R.
Joslin.
82
Chapter 4
Laboratory and numerical experiments
on the near wake of a sphere in a stably
stratified ambient
T.J. Madison, X. Xiang, G. R. Spedding
Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles,
CA 90089, USA
Sections of this chapter appear in Journal of Fluid Mechanics, Accepted November 2021.
The flow around and behind a sphere in a linear density gradient has served as
a model problem for both body-generated wakes in atmospheres and oceans, and
as a means of generating a patch of turbulence that then decays in a stratified
ambient. Here, experiments and numerical simulations are conducted for 20 values
of Reynolds number, Re, and internal Froude number, Fr, where each is varied
independently. In all cases, the early wake is affected by the background density
83
gradient, notablyintheformofthebody-generatedleewaves. Meanandfluctuating
quantities do not reach similar states, and their subsequent evolution would not be
collapsible under any universal scaling. There are five distinguishable flow regimes,
which mostly overlap with previous literature based on qualitative visualisations
and, in this parameter space, they maintain their distinguishing features up to and
including buoyancy times of 20. The possible relation of the low{Re,Fr} flows to
their higher{Re,Fr} counterparts is discussed.
4.1 Introduction
4.1.1 The sphere wake as a model problem
The evolution of a sphere wake in a uniform density stratification is a convenient
model problem to investigate the fluid dynamics of practical geophysical and nau-
tical applications, such as the wakes generated by islands or the flow around sub-
merged bodies. In addition to the Reynolds number, Re = ρUL/μ, a second di-
mensionless group characterizes the relative strength of stratification, known as the
internal Froude number, Fr = U/NL, where N
2
=−(g/ρ
0
)∂ρ/∂z. U and L are
characteristic velocity and length scales in the flow,ρ
0
is a reference density,∂ρ/∂z
is the density gradient, g is the acceleration due to gravity in the direction of the
density gradient, andμ is the dynamic viscosity of the fluid. For the particular case
of the sphere wake,U andL are taken as the body speed and diameter, respectively.
84
The internal Froude number is Fr = 2U/ND, based on radius, D/2, so the value
Fr = 1 denotes the maximum resonance between a convective time and a buoyancy
time, as explained in §2.3.
After some time, initially turbulent wakes evolving in a background density gra-
dient develop a regular pattern of alternating vortices whose origin can be traced
back to instabilities in the developing shear layers (Spedding, 2002b). The coherent
structures emerge during what has been termed the non-equilibrium regime (NEQ)
as the density gradient increasingly exerts an influence on the fluctuating and then
mean vertical velocities. This period of adjustment of the wake to its background
begins at about Nt = 2, and ends at about Nt = 50, depending on the initial
conditions. During this time, suppression of the fluctuating vertical velocity com-
ponents reduces the associated Reynolds stress components and so decreases the
kinetic energy dissipation rate (Brucker & Sarkar, 2010; Redford et al., 2015). The
vertical velocity fluctuations are diminished partly by the work required to lift fluid
parcels from their equilibrium position, and partly because they can be removed
from the wake as wake-generated internal waves. The energy contained in these
waves, and removed from the wake, is an increasing fraction of the total as Re
increases (Abdilghanie & Diamessis, 2013; Rowe et al., 2020) and high-resolution
simulations in Watanabe et al. (2016) demonstrated that the energy loss from wave
radiation can be equal to the energy dissipated in turbulence in the wake itself.
85
4.1.2 The influence of initial conditions
After NEQ, the vortex motions persist. Streamwise averages of the mean and fluc-
tuating velocities then yield statistically similar profiles in both lateral and vertical
directions, and can be re-scaled based only on an effective drag coefficient (Meunier
& Spedding, 2004, 2006). In that case, the only amplitude variations come from the
net horizontal momentum flux, and in all other respects the particular shapes have
no far-wake influence. Instead, the dynamics are self-similar, much as the original
hypothesis proposed by Townsend (1976) for three-dimensional turbulent wakes.
However, even in unstratified turbulent wakes, the hypothesis has been called into
question in experiment (Bevilaqua & Lykoudis, 1978; George, 1989) and in compu-
tations (Redford et al., 2012), which showed that differences in the wake structure
fardownstreamcouldbetracedbacktodifferencesintheinitialconditions, affecting
the evolution of mean and fluctuating quantities in differing ways.
Experiment and simulations agree on some approximate measures of the evolu-
tion of various length and velocity scales (Gourlay et al., 2001; Dommermuth et al.,
2002; Brucker & Sarkar, 2010; Diamessis et al., 2011), but it is not easy to find gen-
eral rules and predictions for when and how special features from initial conditions
will prevent agreement. Early numerical simulations (Métais & Herring, 1989) in-
dicated that stably-stratified turbulence-in-a-box would retain memory of different
initial conditions (such as balances of vortical vs. internal wave motions). On the
other hand the widespread emergence of late-wake coherent structures and simi-
lar wake growth/decay rates in simulations having widely differing initializations
86
(Gourlay et al., 2001; Dommermuth et al., 2002; Diamessis et al., 2011; Redford
et al., 2015) suggests that the particular route to the late wake may well be unim-
portant.
4.1.3 Scaling and turbulence in stratified wakes
A scale similarity in strongly stratified wakes has been demonstrated analytically
by Billant & Chomaz (2001) who showed that when Fr
h
<< 1, a vertical length
scalewouldbel
v
=U/N, whenenergyisapproximatelyequallypartitionedbetween
kinetic and potential energy. In that case a local Froude number based on vertical
length scales and local fluctuating horizontal velocity, u
h
, Fr
v
= u
h
/Nl
v
= O(1).
Lindborg (2006) showed that the Fr
v
= 1 invariance can alternatively be demon-
strated as a consequence following the stipulation of energy equipartition. The
dynamics of stratified turbulence is ultimately determined by the relative mag-
nitudes of turbulent, buoyancy and viscous dynamics.Spedding (1997) proposed
that turbulent stratified wakes evolve through three distinct regimes: a presumed
three-dimensional regime (3D) is initially unaffected by stratification, followed by
a non-equilibrium (NEQ) interval when buoyancy begins to significantly modify
the dynamics then transitions into a flow state (Q2D) consisting mostly of two-
dimensional motion, where the vertical velocity component, w, is small compared
with {u,v}. The wake regime sequence of 3D – NEQ – Q2D is based on wake-
scale and mean quantities, which themselves are outcomes from local turbulence
dynamics, as well as the initial forcing atU andD. If we define local Reynolds and
87
Froude numbers based on horizontal integral length scales, l
h
, so Re
h
=u
h
l
h
/ν and
Fr
h
=u
h
/Nl
h
, then a combination of these yields a buoyancy Reynolds number,
R = Re
h
Fr
2
h
, (4.1)
the magnitude of which measures the range of scales over which horizontal turbu-
lent motions can occur, even while being constrained by the condition of strong
stratification, when l
v
<< l
h
and Fr
v
= 1. Lindborg (2006) showed that this kind
of strongly-stratified turbulence produces a forward energy cascade with horizontal
wavenumber scaling ofk
−5/3
h
, as indeed measured in the atmosphere (Nastromet al.,
1984; Nastrom & Gage, 1985). Brethouwer et al. (2007) extended the analysis to
show that flows forR >> 1 are little affected by viscosity, and furthermore that
withinthatregimetherearetwodistinctsub-regimes. IfwedefineanOsmidovscale,
l
O
= (/N
3
)
1/2
as a scale where overturning motions are damped by stratification,
and a Kolmogorov scale, l
K
= (ν
3
/)
1/4
, as a small scale where viscosity consumes
turbulent fluctuations, then there is a range l
K
≤ l
t
≤ l
O
where turbulence can
proceed mostly unaffected by either stratification or viscosity, and another range
l
t
>l
O
where energetic turbulence is nevertheless strongly influenced by buoyancy.
In many laboratory experiments, especially at late times, size limitations lead to
R<< 1 and Godoy-Dianaet al. (2004) proposed that the observed independence of
late-time dynamics onFr can be understood as a limit when flows reach a viscous-
dominated attractor, though relevant scaling laws for aR>> 1 state to reach the
viscous regime could depend on the initial state.
88
Turbulence can occur in scales from l
K
up to l
O
, and de Bruyn Kops & Riley
(2019) have drawn attention to a parameter we shall termG, following early work
by Gibson (1980) and Gargett et al. (1984). Writing l
O
/l
K
=/N
2
ν, then since an
rms turbulent length scale can be written as l
t
=u
3
t
/, the scale ratio is
G =

N
2
ν
= Re
t
Fr
2
t
, (4.2)
whereRe
t
andFr
t
arebasedonthermsturbulentvelocity,u
t
andl
t
.G wasdescribed
as an activity parameter in de Bruyn Kops & Riley (2019) and together with a local
Froude number, its value can be used to separate dynamically dissimilar regimes
as an initial turbulence decays against a stratified background. The expressions in
eqs.(4.1, 4.2) depend on l
h
,u
h
and l
t
,u
t
, respectively, where u
h
is an rms velocity
scale in the horizontal and u
t
is the total rms velocity. The ratio ofG/R varies
with time, and with Re (de Bruyn Kops & Riley, 2019). The relationships between
a dissipation rate, and other length and velocity scales is readily supported in
strong, box-filling turbulence, but less clear in weaker turbulence that evolves in
highly anisotropic fashion from its start. Wakes are also non-uniform in space,
and measures such asG andR will be non-uniformly distributed across the wake,
declining to zero outside it. Zhou & Diamessis (2019) considered these matters in
some detail in temporal simulations that ran up to Re = 4× 10
5
.
There is interest on how a turbulent stratified wake makes the transition from
Strongly Stratified Turbulence (SST) to viscous dominated and at what critical
values ofR
c
andG
c
the transition may occur. Rigorous estimates ofR andG
89
require accurate information at scales approaching l
K
(Riley & de Bruyn Kops,
2003; de Bruyn Kops & Riley, 2019), and these authors explored how such criteria
for different dynamical regimes could apply to experiments and simulations where
only larger-scale information is known.
In a similar spirit, Zhou & Diamessis (2019) considered the regime R > 1
and Fr
h
<< 1, (where the requirement onR is less stringent than the original
R>> 1 prescription) and interrogated simulations where Re = [5k, 100k, 400k] and
Fr = [4, 16, 64]. Intervals of Nt from 50 to 200 were identified where Fr
v
≈ 1, but
the measure declined steadily in all cases. If the conditionsR> 1 and Fr
h
<< 1 are
usedtodefineSST,thenathresholdvaluebasedoninitialparametersofReFr
−2/3
≥
5× 10
3
was proposed. Body-inclusive simulations are reaching higher Re, with
regions in the wakes that can claim to be fully turbulent and that then transition
to SST states. Chongsiripinyo & Sarkar (2020) ran Large Eddy Simulations to
simulate a disk wake at Re = 5× 10
4
, and described successive transitions from
weakly-stratified, to intermediate-stratified, to strongly-stratified (WST – IST –
SST) turbulence. This succession of regimes could be identified in localised regions
of the disk wake and could be reached because the initial Re was comparatively
high.
4.1.4 Sphere wake regimes at low Re-Fr
Early experiments on the wake of a sphere show a range of distinct shedding and
wave regimes at comparatively low Re and Fr (Lin et al., 1992a; Chomaz et al.,
90
1993a). It is clear specific flow regimes can be established, though it is perhaps
less clear how these specific flow regimes evolve far downstream. In particular,
the claims of geometry independence (Spedding, 1997; Meunier & Spedding, 2004)
can only be made for flows where initial Re≥ 5000 and Fr≥ 4 are high enough
so that some scale independent turbulent motions can exist. Therefore, wakes
with lower initial Re and Fr may then contain some details specific to the initial
conditions downstream of the body. Xiang et al. (2015) have already shown that
there are no universal characteristics in the near wake of a towed grid over a range
2700≤ Re≤ 11000 and 0.6≤ Fr≤ 9. Such a conclusion might be anticipated when
theRe,Fr values extend so low. Numerical simulations that include the body (Orr
et al. (2015); Pal et al. (2016, 2017); Chongsiripinyo et al. (2017); Chongsiripinyo
& Sarkar (2020); Ortiz-Tarin et al. (2019), the latter two are for a circular disk
and a prolate spheroid, respectively) now reach Re = 5× 10
4
, and much focus
is on extending techniques to increase Re further. Here we refocus on the sphere
wake at moderateRe andFr, using computational and experimental methods that
match in parameter space to measure wake characteristics near the sphere and then
extending to downstream distances where stratification begins to dominate.
4.1.5 Objectives
The purpose of this study is to systematically cover a region of {Re−Fr} parameter
space that covers a number of distinct regimes depending on the relative dominance
of Fr or Re-dependent effects. The space Re∈ [200, 1000], Fr∈ [0.5, 8] contains
91
completely laminar flows and those with irregular motion that are the first signs of
turbulence. The parameter space also has flows that are strongly constrained by
body-generated lee waves to those where the near wake is fully separated. There is
explicit overlap with the flow visualisation experiments of Lin et al. (1992a) (LI92)
and Chomaz et al. (1993a) (CH93), and with the numerical simulations of Orr
et al. (2015) (OR15). Here, the focus is on independent and systematic variation of
bothRe andFr in this parameter range, and numerical simulations and laboratory
experiments were run together, under the same nominal conditions. The goal is to
compare descriptions of the varying flow regimes within this study and with existing
literature, and moreover to give quantitative descriptions of characteristic features.
Ultimately we seek to predict when and if the quantitative data can be used to
extract wake generator information from the wake signatures themselves.
4.2 Methods
4.2.1 Numerical method
Numerical experiments of the stratified sphere wake were solved using a finite vol-
umesolverinOpenFOAM.Givenaninitialstratificationinthevertical(z)direction,
the density and pressure fields are decomposed into mean and fluctuating compo-
nents given by Equations 4.3 and 4.4, where x is the 3D coordinate in space and t
is time.
ρ(x,t) =ρ(z) +ρ
0
(x,t) (4.3)
92
Near wake = 0.025D
Outer = 0.25D
Intermediate = 0.037D
Figure 4.1: Computational domain for near wake sphere simulations. Results are
taken from a smaller domain outlined in red with dimensions 4D× 4D× 15D, in
the (y,z,x) directions respectively.
p(x,t) =p(z) +p
0
(x,t) (4.4)
Equations 4.5, 4.6, 4.7 are the continuity, momentum, and density evolution
equations, solved under the Boussinesq approximation along with a hydro-static
balance term (∂ρ/∂z =−gp) where u is the velocity, g is the gravitational acceler-
ation in the direction of the density gradient, α is the thermal diffusivity, and ν is
the kinematic viscosity.
∇·u = 0; (4.5)
∂u
∂t
+u·∇u =−∇p
0
+ν·∇
2
u +ρ
0
g (4.6)
∂ρ
0
∂t
+u·∇ρ
0
=w·
dρ
dz
+α·∇
2
ρ
0
(4.7)
SimulationswereconductedatRe = [200, 300, 500, 1000]andFr = [0.5, 1, 2, 4, 8].
BothU andRaremaintainedataconstantvalueof1foreachsimulation, sochanges
to Fr are made by altering dρ/dz and changes to Re are made by altering ν. In a
93
thermally-stratified water column, the ratio of momentum to thermal diffusivity, as
measured by the Prandtl number, Pr = ν/α = 7, and in a salt stratification with
molecular diffusivity, D
s
, the equivalent Schmidt number, Sc = ν/D
s
= 700. The
small-scaleresolutionrequirementsinthesimulationsthusriseaccordingly, andhere
we setPr = 1. The computations will not be expected to resolve the small scales of
scalargradients. Temporalandspatialderivativesaresecondorderaccurate. Figure
4.1 shows the computational domain and observation window. The computational
domain was 16D× 16D× 70D in the (y,z,x) directions respectively. To focus
on the near wake properties, and to avoid possible boundary effects, measurements
are only taken up to x/D = 15. A coarse mesh with 3 M cells was used for
Re = (200, 300, 500). A finer mesh with approximately 17 M cells is used for
Re = 1000. The mesh is always finer close to the body: the near wake has 1.7 M
cells in the coarse mesh and 11.8 M for the fine mesh. In order to reduce reflections
from internal waves at the domain boundaries, zero gradient boundary conditions
are adopted. The sphere is oscillated back and forth in each direction once to break
flow symmetry (Lee, 2000).
4.2.2 Experimental setup
Experiments were conducted in a 1 m× 1 m× 2.5 m tow tank. Stratification and
opticalaccesswasachievedusingarefractiveindexmatchedtwo-tankfillingmethod
(Xiang et al., 2015). The sphere was towed from right to left and was suspended
from a translation stage with three thin wires of diameter, d = 0.5 mm as shown in
94
R2 R3 R5 R10
U D N U D N U D N U D N
F.5 0.37 5.5 0.29 0.40 7.8 0.21 0.70 7.8 0.37 0.95 11.1 0.34
F 1 0.53 3.9 0.29 0.55 5.5 0.21 0.99 5.5 0.37 1.41 7.8 0.37
F 2 0.74 2.8 0.29 0.80 3.9 0.21 1.41 3.9 0.37 1.97 5.5 0.37
F 4 1.03 2.0 0.29 1.16 2.8 0.21 1.91 2.8 0.37 2.81 3.9 0.37
F 8 1.41 1.4 0.29 1.61 2.0 0.21 2.68 2.0 0.34 3.95 2.8 0.37
Table 4.1: Tow speed,U (cm/s), sphere diameter,D (cm), and buoyancy frequency,
N (rad/s) for each experimental configuration. The naming convention RxFy will
be used for Re = x00, Fr = y.
figure 4.2. Re and Fr were kept near nominal values of Re = [200, 300, 500, 1000]
and Fr = [0.5, 1, 2, 4, 8], matching the simulations. Each experimental run was
performed a minimum of six times.
Table 4.1 shows the experimental parameters for each Re and Fr tested.The
kinematic viscosity, ν = 1.0005× 10
−6
m
2
/s. The temperature of the water was
maintained near 25±1
◦
C. Refractive index matching can cause variations in the
kinematic viscosity. For N = 0.37 rad/s the salinity of the water at the bottom
of the tank was 35 g NaCl/kg H
2
O. A transition from fresh water to a salinity of
35 g/kg at 25
◦
C corresponds to a Δν/ν≈ 0.1 over the height of the fluid (Nayar
et al., 2016) for a possible 10% variation inν. The variations ofν over the diameter
of the largest sphere are less than 2%. Though flow transitions for sphere wakes,
both stratified and unstratified, are sensitive to Re for Re≤ 1000, the difference in
ΔRe from Δν did not move the data from one flow regime to another in these tests.
95
U
FOV
2.5D
2.25D
20D
3.6D
3.6D
ρ
Figure 4.2: Experimental setup for sphere wake experiments. Distances shown are
with respect to the largest diameter sphere tested, D = 11.1 cm.
The tank was filled with water/salt/alcohol to a height, H = 80 cm, which ensured
the center of the largest diameter sphere was 3.6D away from both the free surface
and the bottom of the tank. In this configuration the sphere traveled 10D before
data acquisition. Data acquisition began when the center of the sphere was in the
middle of the field of view, which was set to a streamwise location to postpone
contamination from start- and end conditions. The startup transients in the wake
propagate upstream, dragged there by the wake itself. When the sphere stops, the
wake collides with the sphere, and any bow-wave type conditions bounce off the
front wall,and back into the sphere and wake. These effects limit the observation
time window available. That time window depends on operating conditions, and is
determined from flow visualization experiments and then checked when practicable
by ensuring statistical similarity at different x positions within the FOV.
Two-components of velocity were obtained in the horizontal (xy) and vertical
(xz) centerplanes using a planar particle imaging velocity (PIV) system. The tank
was filled with titanium dioxide particles with an average density of 4.23 g/cm
3
96
and diameter of 15 μm. The image plane was illuminated with an Nd:YAG laser
operated at a wavelength of 532 nm and a repetition rate of 20 Hz. Images were
processed using a multipass algorithm with initial interrogation box size of 64× 64
pix with 50% overlap to a final box size of 32× 32 pix. The box resolution therefore
ranged from 0.05D to 0.22D in the horizontal plane.
4.2.3 Analysis
The averaged results from experiments are ensemble averages over all repeated
experimental runs. Wake quantities are averaged with respect to the moving body
for each downstream position, x. In a vertical slice, the mean wake quantity can be
written
q(x) =
1
K
K
X
i=1
q
i
(x, 0,z) (4.8)
whereq is the ensemble averaged wake quantity at downstream location x,q
i
is the
instantaneous value at x = (x, 0,z). Subscripti designates the laboratory reference
frame of the data. For example q
1
(c, 0) is the first instance when wake data were
available at downstream location (c, 0). K is the total number of ensembles for a
single run. K = 100 for the smallest diameter sphere and 54 for the largest. The
ensemble averages for each experimental run are then averaged together over all
runs. Figure 4.3 shows an example of three ensembles at downstream locationx = 0
97
Figure 4.3: Example of wake quantity, q, ensembles at downstream locations x = 0
andx =c as the sphere moves through the field of view in a fixed laboratory frame
of reference. q could be a directly estimated quantity, such as{u,v,w}, or a derived
measure from spatial derivatives
and x = c for three laboratory reference frames. The same averaging technique is
used in the horizontal plane,
q(x) =
1
M
M
X
i=1
q
i
(x,y, 0) (4.9)
where,q(x), is the wake averaged quantity at coordinatesx = (x,y, 0), andM is the
total number of ensembles available in the horizontal plane for position x. In the
horizontal plane M varied between 124 and 287 for the largest and smallest sphere
cases, respectively. The root-mean-square(r.m.s) fluctuating quantities based on
98
temporal averaging in the vertical and horizontal plane are
q
0
(x) =

1
K
K
X
i=1
(q
i
(x = (x, 0,z))−q
i
(x))
2

1/2
(4.10)
q
0
(x) =

1
M
M
X
i=1
(q
i
(x = (x,y, 0))−q
i
(x))
2

1/2
(4.11)
Vertical and horizontal half wake heights and widths, L
v
and L
h
respectively, are
calculated based on distance from the centerline to the point where the local time
averaged velocity is 15% of the centerline velocity,u = 0.15U
0
(matching the criteria
in Orr et al. (2015)) with which we make comparison.
The buoyancy frequency, N, has units of rad/s, so a buoyancy timescale in
seconds is t
b
= 2π/N. The time required for a neutrally buoyant particle at mid-
equator to move on a semi-circle around a sphere of diameter D is t
c
=πD/2U. A
maximum resonance between buoyancy-induced internal waves and displacement of
fluid over the body occurs when the buoyancy timescale, t
b
, is twice the convective
time scale, t
c
. Their ratio is then
t
b
2t
c
=
2π
N
U
πD
=
2U
ND
, (4.12)
and a Froude number based on D/2 will equal 1 when t
b
= 2t
c
.
99
Figure 4.4: ω
z
(x,y) for Re = [200, 500, 1000] and Fr = [0.5, 1, 8] from simulations.
Each plot has 10 evenly spaced contours over±|ω
z
/N|
max
. The reference colorbar
in the figure is for F8R10. Note that the vertical scale in y is expanded.
4.3 Results
4.3.1 Wake structure
The wake vorticity field will be described from the simulation results, since simu-
lation and experimental results will later be shown to be similar. Figures 4.4 and
4.5 show the instantaneous vertical, ω
z
, and lateral, ω
y
, vorticity in the horizontal
and vertical centerplane respectively, at Re = [200, 500, 1000] and Fr = [0.5, 1, 8].
All snapshots were taken at the same simulation tie step t
∗
=Ut/D = 75.
In figure 4.4 there are distinct, and different, variations in the wake geometry
for variations in both Re and Fr. The middle column, for Fr = 1, varies the least
with Re. At this maximum resonance condition (eq.4.12, t
b
= 2t
c
) the lee waves
control the conditions on the sphere and the wake is symmetric, at all Re, in both
100
Figure 4.5: ω
y
(x,z) for Re = [200, 500, 1000] and Fr = [0.5, 1, 8] from simulations.
Each plot has 10 evenly spaced contours over±|ω
y
/N|
max
horizontal and vertical centerplanes (Fig. 4.5, center column). As the stratification
is relaxed (Fr = 8, right column), the centerline symmetry is maintained in both
horizontal and vertical planes at low Re = 200, but is broken for Re≥ 500 (cases
R5, R10 in the rightmost F8 column). In F8R10 there are strong gradients in both
ω
z
andω
y
, and evidence of a number of smaller scales of motion behind the sphere
and in the developing wake.
When Fr = 1, the flow in the vertical centerplane travels around the sphere
edge, rejoining almost at the equatorial plane. The dominant wavelength in the
streamwise direction does not vary with Re (middle column, Fig. 4.5). In the left
column of Figs 4.4,4.5,t
b
≈t
c
and the preferred internal wavelength is shorter than
a half circumference, so the flow departs from the sphere at an azimuth angle of
approximately 2π/3. In the horizontal centerplane, the shear layers extend further
101
downstream as the flow is forced to travel around the sphere, rather than above and
below it. At allRe, the wake now destabilises in the horizontal, with high amplitude
(ξ >R) excursions. The wavelength decreases with increasingRe. Pal et al. (2016)
and Chongsiripinyo et al. (2017) also found a rebirth of turbulent fluctuations with
decreasingFr below 0.5 (at fixedRe = 3700), and thoughRe here is insufficient to
produce turbulence, the strong gradients in ω
z
and in ω
y
for F.5R10 (bottom left
cornerofFigs4.4, 4.5)showthesame, andsomewhatcounter-intuitive, consequence
of the quasi-two-dimensional forcing.
ThequalitativelydifferentwakesinFigs4.4,4.5areplacedina{Re-Fr}diagram
and compared with the observations from CH93 in Fig. 4.6. A classification of
the wakes from both experiment and simulation is shown by coloured symbols.
For Fr≤ 1, variations in Re were not important and the wakes transition from
steady,planarsymmetric(red)tounsteadyvortexshedding(purple)asFr decreases
from 1. The regimes from CH93 are saturated lee wave, and two-dimensional, the
latter since the centerplane and nearby layers generate vortex wakes much as a 2D
cylinder would. For Fr = 2, 4 the CH93 classification was of a transitional regime
(T) without, and then with Kelvin-Helmholtz (KH) instability with increasing Fr.
In the current study, there is a variation with Re at these intermediate Fr, and
the wakes at Fr = 2 can be planar symmetric (red), vertically asymmetric but
horizontally symmetric (yellow) or planar oscillating (green, the equivalent of KH
in T). The planar oscillation is seen at higher Fr, lower Re also, in regions that
lie outside the experiments of CH93. Multiple unsteady modes appear at higher
102
10
0
10
1
10
2
10
3
10
4
2D
SLW
T
3D
Figure 4.6: {Re, Fr} regime diagram for current experiments (coloured shapes) and
from CH93 (background). CH93 divided the space mainly by Fr into four main
subregions: two-dimensional (2D), saturated lee wave (SLW), transition (T), and
three-dimensional(3D). The vertical dashed lines bound each subregion in Fr, Re
space. The symbols denote wake structure across Fr, Re for the present study:
horizontal vortex street (purple circles), steady, planar symmetric (red squares),
vertical asymmetric (yellow triangle), planar oscillation (green diamonds) and mul-
tiple unsteady modes (blue right facing triangle).
Fr (blue), at lower Fr for higher Re. The detailed studies of LI92 and CH93
paid close attention to conditions on the sphere, especially at low {Re, Fr}. Here
we are mostly concerned with the pattern and geometry in the intermediate wake
(omitting details of recirculation and separation zones) that then may, or may not
persist into later times. In this respect, we note that the wake decay at largex could
be enhanced in Fig 4.4, 4.5 through numerical diffusion in the low order method.
The regimes identified are qualitatively consistent with existing literature, and the
flow fields and their parametric variations can now be examined quantitatively
103
4.3.2 Time averaged wake properties
Streamwise velocity comparison
Figure 4.7 shows the time averaged streamwise velocity, u/U, for simulations (left
column) and experiments (right column) in the vertical centerplane for constant
Fr = 1 and varyingRe. The most prominent feature at Fr = 1 is the lee wave which
causes an initial decrease in wake height just behind the sphere. The amplitude of
the contraction increases with increasing Re. The contraction occurs with a local
minimum in the streamwise centerline velocity and has been observed at Re >
1000 for spheres (Bonnier & Eiff, 2002; Pal et al., 2017) and behind towed grids
(Xiang et al., 2015). The minimum u occurs at x/D = 1.5 for all Re simulations
and at x/D = 1.3± 0.1 for the experiments. The spacing and amplitude of the
wake pulsations is unaffected by Re. The background level in experiments is lower
(darker) because very small pixel displacements are on average drawn back to 0 due
to peak-locking.
Figure 4.8 shows a similar comparison at fixed Re = 500 and varying Fr. As
Fr increases, so does the wavelength, λ, of the lee waves. x/D can be related to
a buoyancy time, Nt through x/D = NtFr/2, and the minimum in u occurs at
Nt = 3.3± 0.3 andNt = 3.2± 0.3 for simulations and experiments, respectively at
Fr = [1, 2, 4]. At Fr = 8, the minimum would be expected at x/D = 25, which is
outside the observation windows in Fig. 4.8. The location ofu
min
agrees with towed
grid experiments whereu
min
was found between 3≤Nt≤ 4 for 2700≤ Re≤ 11000
104
Figure 4.7: u/U(x,z) for Fr = 1 and Re = [200, 300, 500, 1000]. Simulations are in
the left column, laboratory experiments are in the right. Each case has 10 evenly
spaced contours between u
max
and u
min
Figure 4.8: u/U(x,z) for Re = 500 and Fr = [1, 2, 4, 8]. Conventions are same as in
Figure 4.7
105
and 0.6≤ Fr≤ 9. The wavelength λ is expected to be a linear function of Fr,
λ =πFr as verified later.
Mean Velocity Profiles
Figure 4.9 shows the mean streamwise velocity profiles at several downstream loca-
tions rescaled by the local centerline value,u/U
0
, in both the vertical and horizontal
centerplanes for Re = 1000, Fr = 8. The vertical and horizontal coordinates are
normalized by the local half height and width, L
v
and L
h
, respectively. In the
vertical centerplane there is good agreement between the simulations and exper-
iments, though for positive values of z/L
v
the profiles from experiments do not
fall to zero. The defect is caused by the wakes of the tow wires, which otherwise
appear to be superimposed upon the mean wake with no other effect. The mean
wake profile shapes are not otherwise distinguishably different from each other, over
x/D = [4−−15].
Initially the horizontal plane velocity profiles in Figure 4.9(b), are more varied,
and the peak values of u/U
0
may be off centerline, as also found in OR15 for
Re = 1000,Fr≥ 4. The horizontal profiles reach a similar state by the end of
the simulation domain. For this particular case x/D = 8 corresponds to Nt = 2,
whenbuoyancyeffectsarethoughttobecomesignificant, andverticalandhorizontal
profiles, though similar, are not the same.
Figure 4.10 shows rescaled velocity profiles from experiment further downstream
atx/D = [15, 20, 25, 30] for the horizontal and vertical centerplanes. Profiles in the
106
3 -2 -1 0 1 2 3
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-3 -2 -1 0 1 2 3
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
(b) (a)
Figure 4.9: Rescaled streamwise velocity in the vertical (a) and horizontal (b) cen-
terplanes for Re = 1000 and Fr = 8. Black and red lines correspond to simulations
and experiments, respectively.
vertical plane retain their shape even at x/D = 30. At the furthest x/D observed
the horizontal plane profiles have a similar shape to the velocity profiles in the
vertical plane. The similar profile shapes encourage a search for regularities that
govern the time evolution of length and velocity scales.
Wake length scales
Figure 4.11 shows the variation in vertical length scales with Re and Fr, and for
experiment and simulation. The agreement between experiment and simulation is
satisfactory, as the amplitudes and wavelengths together with variations in govern-
ing parameters are the same. At the lowestFr = 1, local wake heights are strongly
shaped by lee waves initiated at the body. For each {Re, Fr} pair, the internal
107
-3 -2 -1 0 1 2 3
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-3 -2 -1 0 1 2 3
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
(a) (b)
Figure 4.10: Rescaled streamwise velocity profiles farther downstream in measure-
ments from horizontal and vertical centerplane experiments for Re = 1000 and
Fr = 8.
wave oscillation is superimposed on a gradual growth in L
v
. At lower Re, L
v
al-
most doubles over x/D≈ 15. Higher Re wakes have smaller L
v
and they grow
less rapidly in x, as seen qualitatively in Fig. 4.7. At higher Re, kinetic energy
can be driven towards small-scale mixing, rather than slow laminar growth in an
internal-wave dominated near-field. The vertical growth is strongly limited by the
background stratification when everywhere Fr≤ 1.
As Fr increases, the wavelength and amplitude of the wake disturbances in-
crease. At Fr = 8 the Re trend abruptly reverses as now higher Re leads to higher
L
v
. AtRe = 1000,L
v
increases at first, and then stays almost constant. This region
of constant L
v
was found by Spedding (2002b) where the first available measure-
ments were at x/D = 10. This almost constant L
v
(x/D), even decreasing slightly
at first, can also be seen in Diamessis et al. (2011), where the initial height increases
108
0 5 10 15
0
0.5
1
1.5
2
0 5 10 15
0
0.5
1
1.5
2
0 5 10 15
0
0.5
1
1.5
2
0 5 10 15
0
0.5
1
1.5
2
F8
F2
F1
F4
(c) (d)
(b)
R2
R5
R3
R10
(a)
Figure 4.11: Downstream evolution of
Lv
R
for simulations (Black lines) and experi-
ments (Red lines). (a) F 1 (b) F 2 (c) F 4 (d) F 8.
with Re, and in Dommermuth et al. (2002), Brucker & Sarkar (2010) and Zhou &
Diamessis (2019). Reports ofL
v
(x) vary considerably in detail in the literature (ops
cit.), with variations from both Fr and Re. Differing physical mechanisms could
be behind similarL
v
observations, and Meunier et al. (2006), for example, contend
that the low-growth region is only a consequence of the slow transition between
turbulent and viscous scaling regimes.
It has been noted before (Spedding et al., 1996b) that a lower limit on Fr that
allows turbulence over some range of scales can be estimated through the Ozmidov
scale, l
0
, the largest overturning scale allowed by the stratified ambient
l
0
∼

N
3
1
2
, (4.13)
109
where is the kinetic energy dissipation rate. In homogeneous turbulence∼u
03
/l,
where u
0
is a fluctuating velocity and l an integral length scale so
l
0
∼

u
03
lN
3
!1
2
. (4.14)
For turbulent scales of l to be initially unaffected by the stratification, l
0
≥l and

l
0
l
!
=

u
0
U
!3
2

l
D
!
−
3
2

U
ND

3
2
≥ 1 (4.15)
Inunstratifiedspherewakes(Gibsonetal.,1968;Uberoi&Freymuth,1970),u
0
/U≈
0.3 and l/D≈ 0.4 so

U
ND

3
2
≥ 5, Fr≥ 3. (4.16)
These arguments are concerned with when a minimum range of length scales for
turbulentenergeticscouldbeexpected, hencewhenturbulentdynamicsmayexplain
wake length scales. Here the {Re-Fr} range covers a region mostly in laminar and
stratification dominated wakes, and the strong influence of the body-generated lee
wave can be seen throughout Fig 4.11. Figure 4.12 shows the first minimum half
wake height (L
v
/D)
min
as a function of Fr for all the simulations. When Fr < 4,
(L
v
/D)
min
decreasesasReincreases. ForFr≥ 4theverticalscalescanbeinfluenced
by the first signs of turbulence, and the differences in (L
v
/D)
min
with Re begin to
shrink. Computations at higher Re∈ [1, 4× 10
5
] (Zhou & Diamessis, 2019) show
110
0 2 4 6 8 10
Fr
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
R10
R5
R3
R2
Figure 4.12: (L
v
/D)
min
for simulations as a function of Fr. Re are given by the
symbols in the legend.
that Fr is less influential in determining vertical scales as Re increases, perhaps
because the evolving flow has spent longer atR> 1.
The mean wake height (minus lee-wave induced oscillations) increases with
downstream distance, and given the similar forms of the velocity profiles found
in figures 4.9 and 4.10 we may find a power law of the form L
v
/D≈ α(x/D)
β
.
Table 4.2 shows the fitting coefficients, α and β, for the simulations and experi-
ments. The average growth rate, β, for the simulations is 0.29±.05 and for the
experiments is 0.27±.06. These wake growth rates do not show a clear depen-
dence on either Fr or Re. They have been measured at x/D much smaller than
previous experimental vertical stratified wakes data (Spedding, 2002b). Pal et al.
(2017) have measured L
v
in a sphere-inclusive simulation but did not attempt a
parametrization. Chongsiripinyo et al. (2017) have given detailed descriptions of
the vortex dynamics and structures, focusing on Fr < 1, pointing out that the
111
Re = 200 Re = 300 Re = 500 Re = 1000
Fr α β α
e
β
e
α β α
e
β
e
α β α
e
β
e
α β α
e
β
e
1 .85 .27 .52 .22 .76 .79 .31 .25 .68 .23 .68 .30 .59 .17 .68 .29
2 .74 .30 .90 .23 .66 .28 .67 .28 .58 .27 .74 30 .42 .32 .60 .32
4 .78 .27 .75 .26 .71 .24 .60 .30 .58 .26 .62 .27 – – .36 .26
8 .62 .35 .57 .25 .50 .37 .50 .27 .51 .31 .56 .21 – – .6 .32
Table 4.2: Power law coefficients for wake height, L
v
, for all Re and Fr found from
simulation results. α
e
and β
e
are coefficients for the experiments.
near-wake structures would be essential in carrying information from the near wake
into the later stages of development.
It is clear that vertical length scales in the stratified wake can show significant
variation with both Re and Fr, and that the relative importance of turbulent
motions then also delineates regimes whereR is large or small. Furthermore, it
is also likely, given the different literature findings, that initial conditions play an
as-yet unexamined role. We shall return to this topic in the discussion section.
The lee wavelength, λ, was determined from the streamwise distance between
extrema in x of the half wake height. Figure 4.13 shows the normalized lee wave-
length as a function of Fr for the R10 simulations and experiments. In a linear
density gradient, the natural cyclic buoyancy frequency is N/2π, and the wave-
length is then λ = 2πU/N or λ/D = πFr, independent of Re. This relationship
112
0 1 2 3 4 5 6 7 8 9
0
5
10
15
20
25
30
35
R10 Sim
R10 Exp
Meunier et al (2018)
Figure 4.13: λ/D vsFr for R10 simulations, experiments, and from Meunier et al.
(2018)
accounts well for the observations from experiment and simulation, and from mea-
surements from Meunier et al. (2018) for Fr≤ 2. When Fr > 2, the experiments,
simulations and Meunier et al. (2018) still agree but are slightly larger than the
values predicted by linear theory.
Figure 4.14 shows the downstream evolution of the half wake width,L
h
/R. The
L
h
(x) curves are qualitatively different, depending on Fr. At low Fr≤ 4 the same
footprint of lee waves as seen in Fig. 4.11 occurs inL
h
(c.f. Figs 4.7,4.8). Only when
Fr = 8 dothe curves show amoregradualincrease. Thereare significant differences
between experiment and computations over the intermediate ranges Fr = 2, 4 and
Re = 500, 1000. The F2R10 case has been re-run and investigated closely, and
the differences remain. The L
h
measure is affected by the unsteadiness in the
simulation, which is not present in experiment, or in the simulations of OR15.
113
0 5 10 15
0.5
1
1.5
2
2.5
0 5 10 15
0.5
1
1.5
2
2.5
0 5 10 15
0.5
1
1.5
2
2.5
3
3.5
0 5 10 15
0.5
1
1.5
2
2.5
3
3.5
R2
R5
R3
(a) (b)
(c) (d)
R10
F1 F2
F8 F4
Figure 4.14:
L
h
R
from experiment and simulation for Fr = 1, 2, 4, 8 in (a-d). Simu-
lations are in black, experiments in red.
The unsteady solution appears to be affected by the startup conditions where the
sphere is accelerated back and forth, sending this borderline case to the KH regime
rather than SKH. The F8R5 curves differ because the grid resolution was too low
as the emerging turbulence generates smaller scales. In general, the solutions for
experiment and simulation match closely.
Centerline velocity
Figure 4.15 shows the mean centerline velocity evolution, (U
0
/U) vs. (x/D) for
various Re and Fr combinations in simulation and experiment. At the lower Re
(Fig. 4.15a)andlowerFr (Fig. 4.15c),thereisnostrongdependenceoneitherFr or
Re. At Fr = 1 (Fig. 4.15c), the lee waves are the dominant influence, andU
0
cycles
up and down out of phase with the product of L
h
L
v
, as required for continuity.
114
0 5 10 15
10
-1
10
0 R2
(a)
0 5 10 15
10
-1
10
0
R5
(b)
0 5 10 15
10
-1
10
0
F8
(d)
0 5 10 15
10
-1
10
0
F1
(c)
F8
F4
F2
F1
R10
F5
R3
R2
Figure 4.15: Downstream evolution of the normalized mean centerline velocity,
U
0
/U for (a) Re = 200 (b) Re = 500 (c) Fr = 1 (d) Fr = 8. Simulations in black,
experiments in red.
These fluctuations at lower Re, Fr have been referred to as wake collapse (c.f.
Bonnier & Eiff (2002); Xiang et al. (2015)) although nothing is collapsing (figures
4.11, 4.14) and the oscillatory function lies on top of a gradual and uniform decay.
At higher Re and higherFr there is a single rise in U
0
and then a decay with x/D
that is more pronounced at higher Fr and Re, respectively.
There is good agreement between experiment and simulation for U
0
(x), better
than for the L
h
measure, supporting the idea that the physics computed and mea-
suredarethesame, butsomeaveragingprocessesarelessrobustinlimitedaveraging
domains.
115
Early work on stratified sphere wakes had no means of reliably measuring wake
characteristics at early times because the apparatus could not readily deploy refrac-
tive index matching, and the earliest measurement times were at Nt≈ 10. From
these times onward, it was notable, and curious, that the mean and turbulence
quantities then seemed to evolve according to power laws which could be written in
terms as though the influence of stratification (through Fr) were negligible. Then,
sincex =Ut, we can make the equivalencex/D = (Nt)Fr/2 and expressions of the
form
U
0
U
=α

x
D

β
U
0
U
Fr
−β
=α (Nt)
β
(4.17)
can be written. The exponent β does not change between the two forms, and
the only difference is the position in x/D or Nt for the data, which depends on
Fr. In stratified flows that decay and eventually come under the leading influence
of buoyancy, one might expect Fr-independent, three-dimensional dynamics first
and then when buoyancy enters, it may do so at some fixed Nt. Instead, Fr-
independent power laws were measured even at late times. Fr-independence in
strongly stratified turbulence was explained and predicted by Billant & Chomaz
(2001), when l
v
= U/N. Here we are interested in when the initial early wake
transitions into NEQ and Q2D states, so it is useful to plot a rescaled U
0
(Nt).
116
10
-1
10
0
10
1
10
-1
10
0
10
1
(a)
10
-1
10
0
10
1
10
-1
10
0
10
1
(b)
Figure 4.16: Rescaled centerline velocity, (U
0
/U)Fr
−2/3
for (a) Re = 300 and
(b)Re = 1000 versus buoyancy timescale, Nt, for Fr = 1, 2, 4, 8. Solid, dash,
dash-dot, and dotted lines correspond to Fr = 8, 4, 2, 1, respectively
Figure 4.16 shows the the evolution in Nt of the centerline velocity for Re =
300, 1000 with varying Fr. Here, the wake defect is just emerging into the Nt do-
main (Nt≈ 3) where previous experiments made their first measurements (Sped-
ding, 2001). The data do not show a low decay rate that has classically described
the NEQ regime, but at low and highRe emerge at aboutNt = 10 with a decay law
that is similar to -2/3 (−0.66±.14 and−0.63±.11) for simulation and experiment,
respectively. Spedding et al. (1996a) found NEQ-Q2D scaling only for Re≥ 4000,
so the rather moderate success in collapsing U
0
at Re = 1000 is not in conflict. If
the early signs of -2/3 decay are correct, then, if applicable, they must describe a
regime that precedes the previously measured wakes. Though the the decay rates
are the same as for a turbulent, unstratified wake, the non-negligible influence of
the stratification has been evident in all the preceding data.
117
4.3.3 FluctuatingvelocitiesandthebuoyancyReynoldsnum-
ber
The maximum Re here, based on the sphere diameter and tow speed, is 1000, and
the flow can hardly be considered turbulent. However, all decaying wakes, globally,
or locally, will eventually enter a stage in whichRe is small, so there is some interest
in what fluctuating quantities look like even at small Re. Then the questions arise
as to how this state was achieved and whether the origin from a higher energy state
is relevant.
In this experiment, fluctuating quantities as defined in eqs.4.10 and 4.11 can
come from two sources: one is from unsteady activity in the wake at large scales,
where coherent structures pass through measurement points, or volumes. The sec-
ond is from fluctuations that derive from turbulent dynamics. Referring to Figs 4.4
and 4.5, only the lower right corner could qualitatively be described as turbulent,
thoughitiseasytoseehownon-negligiblefluctuationswilloccurinothersituations,
such as for Fr = 0.5. The measurements ofu
0
,v
0
,w
0
described here are therefore not
measures of turbulence, but are diagnostic measures of these early wakes in their
various {Re-Fr} states.
Figure 4.17 compares {u
0
,v
0
,w
0
} at three downstream locations,x/D = [3, 5, 10]
for Re = 1000 and Fr = 8. At each downstream location the simulations and
experiments agree well. The magnitude of the peaks of the streamwise fluctuations
decreases with downstream distance while the width of the peaks increases as the
wake width increases. u
0
(y) has a double-peaked profile, as would be obtained
118
0 0.1 0.2
-2
0
2
(a)
0 0.1 0.2
-2
0
2
(b)
0 0.1 0.2
-2
0
2
(c)
Figure 4.17: (a)u
0
(y), (b) v
0
(y) and (c) w
0
(z) for Re = 1000,Fr = 8. Simulations
and laboratory experiments are represented by solid and dashed lines, respectively
when fluctuating quantities are associated with mean shear. v
0
(y) has a similar
shape though with smaller amplitude peaks. w
0
(z) has only a single maximum,
which decays faster withx/D than the horizontal components. The first turbulence
profiles for a stratified wake (Spedding, 2001) were measured at x/D = 6 (orNt =
9) and all three fluctuating velocity components were initially double-peaked. The
difference comes from Re, which was 5000 in the previous study, and 1000 here.
The dependence of the velocity fluctuation magnitudes on Re and Fr is shown
in figure 4.18. At Re = 300w
0
0
falls almost immediately to approximately 1% ofu
0
0
,
for all Fr. At Re = 1000 the Fr = 4, 8 cases remain at w
0
0
/u
0
0
≈ 1 up to x/D = 15.
There are fluctuations in the ratio with wavelength equal to the lee wavelengths.
The averages are temporal averages so the lee waves themselves, which are steady in
119
0 5 10 15
x
D
10
-2
10
0
w
′
0
u
′
0
(a)
Fr = 8
Fr = 4
Fr = 2
Fr = 1
Fr = 0.5
0 5 10 15
x
D
10
-2
10
0
w
′
0
u
′
0
(b)
Figure 4.18: Ratio of centreline fluctuations
w
0
0
u
0
0
for (a) Re = 300 and (b) Re = 1000
for varying Fr, as given in legend of (a).
the sphere reference frame, do not directly contribute to w
0
0
, but some components
of this wave-induced field clearly impact the measurement. In this respect then, all
wakes are buoyancy-influenced at all times.
It is useful to locate the sphere wakes discussed here in Fr
h
−G space described
by de Bruyn Kops & Riley (2019). The wake properties are not uniform so we
make local measures as explored and detailed in Xiang et al. (2015) and Zhou &
Diamessis (2019). The latter authors proposed a criterion of ReFr
−2/3
≥ 5× 10
3
,
based on initial wake parameters, D and U, could be used to indicate activity in a
strongly-stratified regime (eq.4.2 and after). At the highest {Re, Fr} combination
here ReFr
−2/3
= 250, far below the suggested criterion. The relationship is counter-
intuitive,however,astheproductReFr
−2/3
decreasesforincreasingFr andfixedRe.
ThusforFr = 1, theproductis1000, whileforFr = 32, itis100. Thisisbecausethe
120
quantityG aims to give relations when stratification is strongly felt, yet Re is still
high, and blindly following this prescription would indicate that the R10F1 case is
closer to SST than is R10F8. This cautionary note notwithstanding, we may make
some local calculations, close to the wake edge, in order to estimate bothG and
Fr
h
. The reasoning follows the arguments of Billant & Chomaz (2001) who show
that there is a limit where a strongly-stratified flow adjusts so that Fr
v
= 1, when
l
v
=U/N. The length scale evolution is then indicated by Fr
h
=u
0
/Nl
h
, where l
h
is the local horizontal length scale. This measure is derived from autocorrelations
of time series of w
0
. The autocorrelation is plotted versus the lag number and the
distance to the first zero crossing of the autocorrelation function is multiplied by the
time interval, expressed as t
loc
. Then the associated local horizontal length scale,
l
h
=u
0
t
loc
. The process is similar to that used by Diamessis et al. (2010) and Xiang
et al. (2015), and is repeated for each experimental run and then averaged over all
runs. If we retain the localu
0
as a measure of the available kinetic energy in velocity
fluctuations, and we keep l
h
as an indicator of turbulent length scales, then
Fr
t
=
u
0
Nl
h
Re
t
=
u
0
l
h
ν
,
(4.18)
andG can be calculated from eq.(4.2) without direct recourse to assumptions about
the isotropy of , for example. We may note that variations in ρ and its gradient
∂ρ/∂z are neglected, and a truly local measure may take these into account, though,
121
10
-2
10
-1
10
0
10
1
10
2
10
-2
10
-1
10
0
10
1
Viscous effects dominate
Weak bouyancy effects
Nt = 1.0
Nt = 1.0
Stratified turbulence
Nt = 5.0
Nt = 4.0
Nt = 3.0
Nt = 2.0
Nt = 1.5
Nt = 1.0
R10F2
R10F1
R10F8
Figure 4.19: Fr
h
vsG for R10 wakes. F1 and F2 wakes immediately start in the
viscous dissipation regime.
again,theseareapproximateindicatorsofastatewhichindetailrequirestheNavier-
Stokes equations. Estimates ofG and Fr
h
are plotted in figure 4.19. The highest
valueofG, forR10F8, islessthan30. ThoughtheconditionG > 1ismet, recallthat
G represents the range of scales between an Ozmidov scale l
O
and a Kolmogorov
scale, l
K
, and their ratio, l
O
/l
K
∼G
4/3
(de Bruyn Kops & Riley, 2019). The small
range of scales implied byG≤ 30 is not sufficient for turbulent dynamics to exist
or be sustainable. Two lowerFr cases are also plotted for the earliestNt available,
and even ignoring the dubious calculation assumptions at these low {Re-Fr}, the
data lie firmly in the viscous dominated regime, as they ought. Chongsiripinyo &
Sarkar (2020) constructed a similar plot, albeit for a disk wake, and withR on the
abscissa. The initial condition of Re = 5× 10
4
allowed a starting point in weak
122
buoyancy effects which then transitioned through all stages, WST – IST – SST,
before significant viscous damping.
4.3.4 Parameterizing the near wake
Spatial paramaterization
Regardless of whether the flow qualifies as turbulent, the similar forms of u(y),
u(z), w(y) and w(z) suggest that generic equations could be fit to the data. The
exercise of doing so may be a convenient and quantitative means of summarising
the profile shapes and evolution over a range of {Re-Fr} and also provides early
wake functions that can be used to seed a bodyless computation or prediction. We
follow the example of OR15 who proposed that a wake quantity, q, normalized by
its centerline value, q
0
, can be expressed as a weighted sum of Gaussian functions
and derivatives,
q(z)
q
0
= exp
"
(z/L
v
−B
z
)
2
C
z
2
+
B
z
2
C
z
2
#
+ exp
"
(z/L
v
+B
z
)
2
C
z
2
+
B
z
2
C
2
z
#
. (4.19)
q(y)
q
0
= exp
"
(y/L
h
−B
y
)
2
C
y
2
+
B
y
2
C
y
2
#
+ exp
"
(y/L
h
+B
y
)
2
C
y
2
+
B
y
2
C
y
2
#
, (4.20)
The y and z scales are normalised by average L
h
and L
v
, and their independent
evolution with time allows asymmetric conditions to be written. The constants
B
i
, i =y,z and C
i
are fit from the data.
123
-2 -1 0 1 2
0
0.2
0.4
0.6
0.8
-2 -1 0 1 2
0
0.1
0.2
0.3
-2 -1 0 1 2
0
0.1
0.2
0.3
(c)
(a) (b)
Figure 4.20: Velocity distributions in z for R10F8 at x/D = [3, 6, 8] of (a) U(z)/U
(b) u
0
/U and (c) w
0
/U for simulations, experiments, and predictions from OR15
using cited coefficients and eq.(4.19). Simulations, experiments and predictions are
given by dashed, solid, and red lines respectively
124
-2 -1 0 1 2
0
0.2
0.4
0.6
0.8
-2 -1 0 1 2
0
0.1
0.2
0.3
-2 -1 0 1 2
0
0.1
0.2
0.3
(b) (a)
(c)
Figure 4.21: Velocity distributions in y for R10F8 at x/D = [3, 6, 8] of (a) U(y)/U
(b) u
0
/U and (c) v
0
/U for simulations, experiments, and predictions. Conventions
as in previous figure.
Figures 4.20 and 4.21 compare mean and fluctuating quantities in the horizontal
and vertical centerplanes for the simulations, laboratory experiments, and eq. (4.19
& 4.20) at Re = 1000 and Fr = 8. Differences in the maximum mean streamwise
velocity are less than 5%. Differences in the fluctuating components are larger,
and peak magnitudes differ by 15% initially and as the wake evolves downstream
are closer to 10%. The laboratory and numerical experiments both have slightly
smalleramplitudesthanpredictedthoughthelaboratoryandnumericalexperiments
themselves are in good agreement. These coefficients B and C do not depend
systematically on either Re orFr, and with one or two exceptions the numbers do
not differ beyond uncertainty.
125
Centerline paramaterizations
The spatial parameterizations in Section 4.3.4 are useful provided both the center-
line wake quantity and the corresponding wake length scales, L
v
or L
h
are known.
OR15 found that normalising the centerline velocity with the wake half height,
(U
0
/U)/(L
v
/D) =U
∗
/L
∗
v
, was successful in collapsing curves of u
0
(x) over a range
of Fr. U
∗
/L
∗
v
can also be thought of as the ratio of the vertical Froude number,
Fr
v
defined by the half wake height, to the global Fr,
U
0
/U
L
v
/D
=
U
0
NL
v
ND
U
= 2
Fr
v
Fr
. (4.21)
This exercise is tested in figure 4.22. The normalization is moderately successful at
collapsing not just the early wake but also further downstream at Re = 200. An
empirical power law fit of the R2 wakes givesU
∗
/L
∗
v
(x/D)
−0.91±.05
. For lowFr and
Re,L
v
scales as (x/D)
1/3
(table 4.2), andU scales as (x/D)
−2/3
(figure 4.16), then
U
∗
/L
∗
v
≈ (x/D)
−1
.
For the Re = 1000 wakes (figure 4.22b), the normalization seems reasonable for
the larger Fr = 4, 8. The exponent of the power law fit is−2.4±.20, indistinguish-
able from the -2.39 value found by OR15. For eitherRe, the lowerFr wakes do not
scale conveniently, and departures are likely due to the strong lee waves at lower
Fr. OR15 suggest that U
∗
/L
∗
v
represents the available mean kinetic to potential
energy and then the collapse shows that this ratio scales directly with internal Fr.
126
5 10 15
0
2
4
6
(b)
Fr = 8
Fr = 4
Fr = 2
Fr = 1
OR15
0 5 10 15
0
0.5
1
1.5
2
2.5
(a)
Fr = 8
Fr = 4
Fr = 2
Fr = 1
2.8(x/D)
-.91
Figure 4.22: Normalized centerline velocity for (a) Re = 200 and (b) Re = 1000.
Here U
∗
/L
∗
v
is the dimensionless parameter in Eq. (4.21). The red dotted line in
(a) is an empirical fit based on an average over all Fr. The red dotted line in (b)
is the fit found by OR15 for Re = 1000 and Fr≥ 4. Black lines are for simulation
results.
The overall growth/decay laws can be disentangled from the wave-induced fluc-
tuations at low Fr, and a statement of that can be made in
L
v
D
=a

x
D

b
+a
1
(x
−c
) sin

d
x
D
+φ

. (4.22)
The coefficients (a-d) can be determined through a combination of known lee
wave behavior (as verified in Fig. 4.10) and empirical fits. From table 4.2 the wake
height growth exponent is not different from a 1/3 growth rate such as found in the
turbulent wakes of Spedding (1997), and b is set to 0.33. The coefficient d = λ,
the wavelength of the lee wave, and since 2πλ = 2π(2/Fr), d = 2/Fr. φ is the
offset of the first trough of the lee wave. Figure 4.23 compares this fit function with
127
0 5 10 15
0
0.2
0.4
0.6
0.8
1
0 5 10 15
0
0.2
0.4
0.6
0.8
1
R10
R5
R3
R2
(a) (b)
Figure 4.23: Wake half height, L
v
(x), for (a) F1 and (b) F2. Simulations shown as
black lines while red lines are from eq. 4.22 and eq 4.23
the simulation data for Fr = [1, 2]. There is good agreement between the idealized
model and the simulations. The coefficients can be rewritten as functions of Re
and Fr, as in eq. 4.23. These relations provide a framework for modelling mean
quantities, in particular the wake height, for wakes under strong stratification.
a =−.0002Re + 2.7Fr− 0.7
a
1
=−.00008Re + 1.9Fr + 0.6
c =−.001Re + 4.5Fr− 2.5
(4.23)
128
4.4 Discussion and conclusions
This work presents the first quantitative and systematic study through independent
variations ofRe andFr of the canonical stratified sphere wake problem. It focuses
on the domain of moderate values of both parameters (Re ∈ [200, 1000], Fr ∈
[0.5, 8]) where the competing influences of inertial, buoyancy and viscous terms
result in a number of qualitatively different flow regimes. The {Re-Fr} parameter
space is covered equally for experiments and simulations. By design it also overlaps
with certain cases in OR15, and the R10F1 comparisons are shown in figure 4.24(a).
When Fr = 1, the lee waves have their strongest influence on the near wake, and
there is reasonable agreement up to x/D = 5. After that, the OR15 values are
greater. Figure 4.24(b), for R10F4, shows that the differences may be caused at
least in part by numerical resolution, as the higher resolution results agree much
more closely at large x/D. Figure 4.24(a) also includes the equivalent result from
Pal et al. (2017) which comes from a significantly higher Re run.
The reasonable agreements between experiment and simulation here are not
completely trivial, since the early wake length and velocity scales depend on details
of the separation location. Nevertheless, we would not advocate pushing these sec-
ondordermethodstohigherRe, assignificantlymoresophisticatednumericalmeth-
ods are required (e.g. Pal et al. (2017); Zhou & Diamessis (2019); de Bruyn Kops
& Riley (2019)) to deal with complex boundaries and/or initial highRe turbulence
over a range of scales. One should also note that extension to lowerFr also requires
significant resolution to resolve the fine structures that re-emerge when Fr < 0.5
129
0 5 10 15
0
0.2
0.4
0.6
0.8
1
R10F1
OR15
Pal et al (2017), Re3700 Fr1
0 5 10 15
0
0.2
0.4
0.6
0.8
1
R10F4
OR15, low res
OR15, high res
(a) (b)
Figure 4.24: Vertical length scale for (a)R10F1 simulations, OR15, and Pal et al.
(2017) and (b) R10F4, OR15 low and high resolutions. Experiments excluded for
clarity, as they have been shown to be similar to simulations in the present study.
(Pal et al., 2016; Chongsiripinyo et al., 2017), the lowest Fr covered here. Both
simulations and experiments will aim to explore higher Re domains, and existing
work on the importance of the activity parameter denoted hereG will be helpful in
understanding the location in turbulence-buoyancy interactions.
130
Chapter 5
The effect of body geometry on stably
stratified wakes
Turbulent stratified wakes have been shown to reach a uniform end state, indepen-
dent of the initial conditions, provided that the flow is rescaled by a momentum
thickness based only on the drag coefficient of the body. How, and when, differ-
ent geometries may reach that uniform end state is less clear. Experiments are
conducted for four different geometrical configurations:a disk, sphere, large mesh
spacing grid, and small mesh spacing grid, across a low to moderate range of Froude
numbers, Fr = [1, 2, 4], and a single Reynolds number, Re = 1000. At all points
stratification influences the evolution of the wakes. Distinct characteristics of the
body geometry are observed through the downstream evolution of mean and fluctu-
ating quantities, suggesting that in these low Reynolds number domains, conditions
entering the universal decay stages are not themselves universal, so variations in
initial conditions will be present in the later wake.
131
5.1 Introduction
Satelliteandship-basedmeasurements(Changetal.,2019)andmesoscalenumerical
modelling (Hasegawa et al., 2009; Perfect et al., 2018) show coherent island wake
structures for which Re = O(10
9
) and Fr = 0(1). There are strong anisotropies
and scale separations in oceanic and geophysical flows and the influence of small
scales on the larger coherent motions can be expressed through an eddy viscosity
(Apel, 1987; Dong et al., 2007; Perfect et al., 2018), ν
t
, when Re
t
=O(10
3
), which
quite closely supports observed shedding frequencies and structure coherence. A
typical marine application may have U = 10 m/s and L = 10 m for Re = O(10
8
)
and Fr = O(10
3
). The high Fr suggests a very minor influence of buoyancy, but
at later times in the decaying wake one might have U = 10
−1
m/s and L = 10
2
m
for Re = O(10
7
) and Fr = O(1). Practical investigation of stratified flows clearly
requiresabroadrangeofFr andbeforeanymodelingcoefficientscanbeproposed, a
large range ofRe, with particular importance in resolving turbulent flows at higher
Re.
Initial experiments on stratified wakes focused on those created by a simple ge-
ometry such as a sphere (Lin & Pao, 1979; Lin et al., 1992a; Chomaz et al., 1993a;
Spedding et al., 1996a) and the success of early numerical experiments (Gourlay
et al., 2001; Dommermuth et al., 2002; Brucker & Sarkar, 2010; Diamessis et al.,
132
2011; Pasquetti, 2011) in re-creating the qualitative and quantitative features pro-
vided evidence that details of the wake creator were unimportant. Experimental
findings of the similarity of the late wakes of bluff, sharp-edged and streamlined
bodies (Meunier & Spedding, 2004) added further support.
There are three issues outstanding: first, there is evidence in experiment and
computation that turbulent wakes at high Re may never reach the ideal universal,
self-similar state (George, 1989; Johansson et al., 2003; Redford et al., 2012), and
that mean and turbulence quantities relax to their final asymptotic decay laws at
different times/downstream distances. Second, results from laboratory experiments
at relatively low Reynolds number (Re = 5×10
3
are common) may not extrapolate
readily to higher Re because qualitatively different flow states may be possible
at higher Re. Zhou & Diamessis (2019) and Rowe et al. (2020) show that when
Re increases to 4× 10
5
, secondary instabilities and self-sustaining turbulence can
generate and regenerate small-scale, energetic eddies that are absent at lower Re,
and also that the energy budget is strongly affected because the higher Re wakes
eventually radiate more of their energy in wake-driven internal waves. Third, a
declaration of the influence of initial conditions may depend on the relevance and
resolution of the measurement, and similarity of some time-averaged quantity may
not rule out the possibility of differences in some other measure.
As body-inclusive simulations reach higherRe (Orr et al., 2015; Pal et al., 2016,
2017; Chongsiripinyo et al., 2017; Chongsiripinyo & Sarkar, 2020), one would like
to have predictive and generalizable models instead of running a new experiment
133
or simulation for every single case. Here we make a systematic comparison of the
stratified wakes behind 4 different body shapes. It is an experimental study so Re
cannot usefully be varied but Fr = [1, 2, 4] allows the difference between relatively
strong and weak stratification to be investigated.
5.2 Methods
5.2.1 Physical setup
Four different geometric configurations, a large mesh spacing grid (LG), a small
mesh spacing grid (SG), a disk, and a sphere were tested atFr = 2U/ND = [1, 2, 4]
and Re =UD/ν = 1000.
Experiments were conducted in a 1 m× 1 m× 2.5 m tow tank. The density
gradient,∂ρ/∂z, was achieved using a standard two-tank filling method and optical
access was achieved using a refractive index matched technique outlined by (Xiang
et al., 2015). A pulley system attached to a translation stage was used to drive
the bodies from right to left in the tow tank, as illustrated in Figure 5.1. For the
disk and grids a guide wire was placed diametrically opposite to the tow wire to
ensure steady translation through the field of view. Both the tow and guide wires
had diameter, d
wire
= 0.5 mm to ensure that there was minimal influence in either
vertical or horizontal centerplane. The sphere was suspended by 3 tow wires, also
0.5 mm in diameter, and towed right to left. Each experimental run was repeated
a minimum of 6 times. The disks and grids each had a diameter of 8 cm and were
134
(c) (b)
(d) (e)
Figure 5.1: Sketch of tow tank experimental setup for disk and grids.Sketches of
towed bodies shown in (b) through (e). Sphere(b), Disk (c), Large mesh spacing
grid (LG) (d), and Small mesh spacing grid (SG) (e).
135
Fr = 1 Fr = 2 Fr = 4
U D N U D N U D N m d
LG 1.48 8.0 .33 1.40 8.0 .19 1.37 8.0 .09 1.40 0.22
SG 1.40 8.0 .33 1.40 8.0 .19 1.40 8.0 .09 0.70 0.11
Disk 1.39 8.0 .33 1.40 8.0 .20 1.69 8.0 .10 – –
Sphere 1.41 7.8 .37 1.97 5.5 .37 2.81 3.9 .37 – –
Table 5.1: Tow speed,U (cm/s), sphere diameter,D (cm), and buoyancy frequency,
N (rad/s) for each experimental configuration.m, is the mesh spacing of each grid
and d is the bar diameter of the grid
towed at U = 1.4 cm/s. N was varied between 0.09≤ N≤ 0.37 rad/s to match
the target Fr. U varied between 1.4 and 2.8 cm/s for the sphere. LG had a bar
diameter, d, of 0.22 cm with a mesh spacing of m = 1.4 cm. SG had bar diameter,
d = 0.11 cm, and mesh spacing, m = 0.70 cm. The sphere diameter was varied
between 3.9 and 7.8 cm and N was kept constant at N = 0.37 rad/s to attain
the correct Fr. Table 5.1 shows U, D, and N for each experiment. The tank was
filled to a height, H = 80 cm, ensuring that each body was at least 5D away from
top and bottom of the tank. Each body travelled into the middle of the field of
view, approximately 12.5D, before data collection began. In a typical experiment,
transients in the startup conditions eventually appear in the observation window
as the wake-induced forward motion brings them there. Similarly, disturbances in
front of the body reflect back from the front wall and also disturb the wake. These
effects limit the maximum observation time in any single experiment.
136
Planar particle imagine velocimetry (PIV) was used to obtain two components
of velocity in the horizontal (x,y) and vertical (x,z) centerplanes. u and v com-
ponents of velocity were obtained from horizontal plane images. u and w velocity
components were obtained from the vertical plane images. The tank was filled with
titanium dioxide particles with an average density of 4.23 g/cm
3
and average di-
ameter of 15μm. The image plane was illuminated using an Nd:Yag double pulsed
laser, with wavelength of 532 nm fired at 10 Hz. Post processing was done using
a multi pass correlation algorithm with initial box size of 64× 64 and decreasing
down to 32× 32. The final box size is equivalent to 0.04D× 0.04D.
5.2.2 Drag coefficients and effective diameter
It has been shown that stratified late wake quantities of any origin can be re-scaled
with an effective diameter, D
eff
, based on the drag coefficient of the body Meunier
& Spedding (2004):
D
eff
=D
s
C
D
2
(5.1)
The drag coefficients for the sphere and disk are C
D
= [0.40, 1.1] respectively
(?), who also defines a loss coefficient,K
p
, for mesh biplane grids as the ratio of the
pressure difference across the grid to the dynamic pressure,
K
p
=
ΔP
1
2
ρU
2
. (5.2)
137
Sphere Disk LG SG
C
D
, K
p
0.40 1.10 0.92 0.92
D
eff
/D 0.45 0.74 0.68 0.68
Table 5.2: Drag coefficients and ratio of effective diameter to body diameter for
each body geometry.
K
p
for a grid is similar to a drag coefficient for a solid body, and is determined by
the grid geometry by
K
p
=
β(1−α
2
)
α
2
. (5.3)
β = 1.3 for grids with Re < 2000 and α is the ratio of solid to open areas, so for
the mesh spacing, m, and bar diameter, d,
α = (1−
d
m
)
2
. (5.4)
The effective diameter for the grids then can be written
D
eff
=D
s
K
p
2
. (5.5)
From the mesh measurements in table 5.1, eqs(5.2) and (5.4) yield a loss coefficient,
K
p
= 0.92. Table 5.2 shows C
d
, K
p
and D
eff
/D for each geometry.
138
5.2.3 Analysis
Wake quantities are ensemble-averaged with respect to the moving body at each
downstream location, x. In a vertical slice, the mean wake quantity, q, can be
written
q(x) =
1
K
K
X
i=1
q
i
(x, 0,z) (5.6)
so q is summed over all K snapshots at each downstream location{x,y}, q
i
is the
instantaneous value at x = (x, 0,z). Subscripti designates the laboratory reference
frame of the data.
Similarly, wake quantities in the horizontal center plane, can be written
q(x) =
1
M
M
X
i=1
q
i
(x,y, 0). (5.7)
K varied from 171 to 300 and M between 150 and 297 over all experiments.
Fluctuatingquantitiesintheverticalandhorizontalcenterplanesaredetermined
from root mean squares
q
0
(x) =

1
K
K
X
i=1
(q
i
(x = (x, 0,z))−q(x))
2

1/2
, (5.8)
and
q
0
(x) =

1
M
M
X
i=1
(q
i
(x = (x,y, 0))−q(x))
2

1/2
. (5.9)
139
The half wake heights and widths, L
v
and L
h
, were calculated based on the
vertical or lateral distance from the centerline to the point where the local time
averaged velocity is 15% of the centerline velocity, u = 0.15U
0
5.3 Results
5.3.1 Wake structure
Figure 5.2 shows five snapshots ofω
z
(x,y), forFr = 1 andRe = 1000 for the large
meshspacinggrid,(LG).Initially,justbehindthegridsmall-scalewakedisturbances
lie in the interior wake region (Figure 5.2(a)). The disturbances diffuse and the
remainingω
z
is concentrated along the wake edge. Ridges of high magnitudeω
z
are
inclined inwards towards the wake center (b). As the separate shear layers coalesce,
a large scale undulation begins (c),(d). These undulations then collect ω
z
, much
as Kelvin-Helmholtz instabilities do (Figure 5.2(e), (f). The SG wake develops in
much the same way (Figure 5.3), as undulations become noticeable atNt≈ 25, and
concentrationsofω
z
developinsidethem. Theoverallgeometrylooksverysimilarto
the LG wake, but the smaller-scale initial fluctuations appear to influence details of
the distribution ofω
z
(x,y), and the wavelength of the large-scale, K-H like structure
in the final tile is longer than for the LG equivalent.
The downstream evolution of ω
z
/N behind the disk contrasts sharply with the
porous grids. Since the flow cannot permeate the body, a large pressure difference is
maintained between the front and back faces. The separation points are fixed at the
140
Figure 5.2: Time series of (ω
z
/N)(x,y), for large mesh spacing grid (LG) at Fr =
1,Re = 1000. The x coordinate can be represented through Nt since x/D =
Nt.Fr/2
.
Figure 5.3: Time series of (ω
z
/N)(x,y), for small mesh spacing grid (SG) at Fr =
1,Re = 1000
.
141
Figure 5.4: Time series of (ω
z
/N)(x,y), for the solid disk at Fr = 1,Re = 1000.
disk edge and the high intensity shear layers are drawn together to the wake center
where they interact and develop a sinuous instability with much smaller wavelength
than in the equivalent grid frames. There is evidence of a blockage effect extending
in front of the body, and of small-scale turbulence in the near wake before the
coherent structures assemble in the final frame. Though D
eff
differs by less than
10% between disk and porous grid, the qualitative differences in in wakes are clear.
The sphere wake in figure (5.5) is a very different case, and has been described
in some detail in Madison et al. (2021). The sphere is an axisymmetric bluff body,
with no fixed separation point at sharp edges. Instead, the separation point can
move in complicated ways in response to the relative magnitudes ofRe andFr. For
this case,Fr = 1, there is a maximum resonance between flow convective times and
the buoyancy timescale, and the near wake is strongly influenced by the presence
of a lee wave that suppresses many other motions close to the sphere. The upper
and lower shear layers originate at slightly different locations on the sphere and in
142
Figure 5.5: Time series of (ω
z
/N)(x,y), for the sphere wake at Fr = 1,Re = 1000.
any given run, this asymmetry projects onto the horizontal centerplane as a wake
that tilts off-axis. In the near wake, the vertical vorticity is spread over a broader
area and the two opposite-signed sheets join at Nt = 5 with a wavelength that is
determined by the lee wave (Madison et al., 2021). In the range Nt = [6, 14] the
ridges of peak vertical vorticity are inclined with respect to the mean wake, as they
were for LG and SG, and this Nt interval is when the second lee wave modulates
the wake width and height. The lee wave is fixed in the body reference frame, but
eventually is free to depart from the wake, which is quite uniform in x atNt≈ 28,
but which then begins to develop undulations that increase in amplitude with time.
The influence of the lee waves around the different bodies can be seen in the
vertical centerplane cuts.
143
Figure 5.6: Time series of, (ω
y
/N)(x,z), for LG in vertical centerplane at Fr =
1,Re = 1000.
Figure 5.7: Time series of, (ω
y
/N)(x,z), for SG in vertical centerplane at Fr =
1,Re = 1000.
5.3.2 Lateral Vorticity
Time series of (ω
y
/N)(x,z) for LG and SG are given in figures 5.6 & 5.7. The wake
iscomposedoftwoshearlayerswhichcontractbehindthegridatNt≈ 3, coinciding
with the reduction in amplitude of theω
z
layers in the horizontal centerplane views
of figures 5.2(a) and 5.3(a). The next maximum wake height is reached at Nt =
[6− 7], which is when the tilted ridges of ω
z
(x,y) are seen in the horizontal plane.
The two shear layers remain separate, but symmetric, showing no obvious signs of
interaction.
144
The extrema of wake contraction and subsequent expansion at Nt = 3&6 are
seen also for the disk (figure 5.8(a, b)), but the impermeable body and the pres-
sure and lee-wave induced wake contraction bring the opposing shear layers closer
together, and the high-amplitude coherent shear layers rapidly destabilize to form
small-scale secondary patches of vorticity. The small-scale vortices of the same sign
then merge so the intermediate wake shear layers (figure 5.8c) for Nt > 10 grow
slowly in x (or Nt) similar to the grid wakes.
The lee-wave induced wake contraction is stronger for the sphere (figure 5.9(a))
than for the other geometries and is accompanied again by the generation of small-
scale motions which decrease the amplitude of the coherent shear-layer velocity
gradients in the z direction. By the end of the timeseries, however, the opposite-
signed vorticity layers grow slowly by diffusion, and no further strong interactions
are observed. In a very coarse-grained view, the vertical shearing motions remain
the strongest but the shear layer geometries are similar, the differences between
body shapes having been resolved in the near wake. On the other hand the wake
undulations in the horizontal plane do appear to keep a signature of the body
geometry and the locations and magnitudes of the separation shear layers.
145
Figure 5.8: Time series of,ω
y
/N, for the Disk in vertical centerplane atFr = 1 and
Re = 1000.
Figure 5.9: Time series of, ω
y
/N, for sphere in vertical centerplane at Fr = 1 and
Re = 1000.
-2 -1 0 1 2
0
0.5
1
-2 -1 0 1 2
0
0.5
1
-2 -1 0 1 2
0
0.5
1
-2 -1 0 1 2
0
0.5
1
(c) (d)
(b)
(a)
LG
Disk
SG
Sphere
Figure 5.10:
u
U
0
in the horizontal plane at Fr = 1 for a) sphere, b) disk, c)LG, and
d)SG. Profiles taken at x/D = [2, 10, 15] and are represented by solid, dashed, and
dotted lines, respectively.
146
5.4 Mean wake quantities
5.4.1 Streamwise velocity profiles
The profiles of u(y) and their evolution in x/D show both the early imprint of the
different body shapes and then the disappearance of this information at largerx/D
as they all evolve into a smooth Gaussian-type shape. The sphere wake (fig 5.10a)
initially is double-peaked, the disk wake (fig 5.10b) has a more triangular shape,
and the grid wakes are flat-topped. Each one of these different shapes can readily be
reconciled with the vorticity distributions described in the previous section. Traces
of the individual bar wakes are especially clear in the x/D = 2 profile of LG (fig
5.10c), but byx/D = 10 all velocity profiles look similar. Here is the disappearance
of the influence of upstream conditions, at least in this aspect of the wake signature,
even at low Fr and comparatively low Re.
5.4.2 Centerline velocity
The main effect of stratification on the mean centerline vlocity (figure 5.11) is
to strongly modulate the gradual decay of U
0
(x) through the lee waves, which
serve to alternately contract and expand the wake, and at all Fr∈ [1, 2, 4] the
initial wake contraction and a recirculation region behind the solid bodies leads to
U
0
/U > 1. Though there are differences in the streamwise profiles up to x/D = 5,
by x/D = 15 they are indistinguishable. There are some characteristic features
that are not shared by all geometries. At Fr = [2, 4] (fig 5.11b,c) the centerline
147
0 5 10 15
0
0.5
1
1.5
0 5 10 15
0
0.5
1
1.5
0 5 10 15
0
0.5
1
1.5
Sphere
LG
Disk
SG
(a) (b)
(c)
Figure 5.11: Centerline velocity U
0
/U for a) Fr = 1, (b) Fr = 2, and (c) Fr = 4
velocity for SG initially increases, not unlike the sphere and disk wakes. The small
mesh spacing creates a blockage effect that is not apparent in the LG wakes at
the same Fr. A similar blockage effect was also observed for turbulent grid wakes
at 2700 ≤ Re ≤ 11000 and 0.6 ≤ Fr ≤ 10 Xiang et al. (2015). The sphere,
disk and SG wake centerline velocities subsequently decay rapidly and overlap by
x/D≈ 10 at Fr = [2, 4]. U
0
/U for LG decays more slowly than for the other
three geometries. For a given downstream distance, U
0
/U remains larger for LG at
both Fr = 2, 4, although the trajectory of U
0
/U approaches the sphere, disk, and
SG. The measurable differences in the two grid wakes arise despite their having the
same solid area, and demonstrate how changing geometry, in this instance the mesh
spacing, can change the evolution of mean quantities in the near wake.
148
0 5 10 15
0
0.5
1
1.5
0 5 10 15
0
0.5
1
0 5 10 15
0
0.5
1
(a) (b)
(c)
Figure 5.12: Wake half width, L
h
/D, for a) Fr = 1, b) Fr = 2, and c) Fr = 4.
Line styles are same as Figure 5.11
5.4.3 Wake length scales
The strong stratification at Fr = 1 (figure 5.12(a)) affects solid and porous bodies
differently. Behind solid bodies the wake contracts sharply as the lee wave impinges
on the wake immediately behind the sphere and disk. The influence is smaller on
the grid flows which can come through the grid as well as around the edge. Small
L
h
are associated with large U
0
in the early wake (previous figure 5.11). The effect
appears to be local as all curves exit the field of view at x/D = 15 with similar
L
h
. Already at Fr = 2, the order has changed and the disk has the highest L
h
, its
variation tracking the earlier development of undulations in the wake (figure 5.8).
AsFr based on body diameter increases to 4 (figure 5.12(c)) the roles are reversed
and the fluid that has passed above and below around the radius is not immediately
constrained by lee of the lee wave, and can expand and grow with increasingx. The
local Fr of the grid bars is small, and presents less of an energetic obstacle to flow
passing through the grid. Now, at higher Fr, there are differences in L
h
. At larger
149
0 5 10 15
0
0.5
1
0 5 10 15
0
0.5
1
0 5 10 15
0
0.5
1
(a) (b)
(c)
Figure 5.13: Wake half height, L
v
/D, for a) Fr = 1, b) Fr = 2, and c) Fr = 4.
Line styles are same as Figure 5.11.
x, one expects the wakes to grow, but no clear sign or rate of this growth can be
seen here.
The wake half-height, L
v
is largely determined by the lee waves at Fr = 1, 2
(figure 5.13a,b), and the influence of the body geometry is quite minor. It is only
when the dominance of the stratification is released somewhat at Fr = 4 when
body geometry differences emerge, foremost in the disk wake which grows rapidly
up to x/D≈ 7 but then diminishes to settle at about L
v
/D≈ 0.5, alongside the
others. The expected lee wavelength increases withFr so even this growth is inside
the envelope provided by the lee wave itself.
5.4.4 Lee wave characteristics
Theleewavelength,λ,wascalculatedbymeasuringthestreamwisedistancebetween
extrema in L
v
. λ/D is plotted in Figure 5.14(a) for each body geometry and as a
150
0 1 2 3 4 5
0
0.5
1
1.5
0 1 2 3 4 5
0
2
4
6
8
10
12
14
16
(a) (b)
Sphere
Disk
SG
LG
Meunier et al
(2018)
Figure 5.14: (a) Wavelength of the lee wave and (b) amplitude of the lee wave for
eachbodygeometry,Fr pair. OpensquaresshowresultsfromMeunieret al.(2018).
Red circles in (b) are results from Chomaz et al. (1993a). + are grid experiments
conducted by Xiang et al. (2015).
function of Fr. The cyclic buoyancy frequency for a stratified fluid is N/2π, and
the wavelength is then λ = 2πU/N or λ/D =πFr. The wavelength of the lee wave
created in a stratified fluid is independent of both Re and the specific geometry
that created it. The wavelengths for the present experiments qualitatively agree
well with the results from Meunier et al. (2018).
Figure 5.14 plots the lee wave amplitude,ζ/D as a function ofFr for each wake.
ζ is determined by measuring the vertical distance between first minimum and first
maximum of L
v
.
At Fr = 4 all 4 wakes have an amplitude ζ/R = 0.4. Decreasing Fr shows a
clear difference between the solid bodies and the grids. At Fr = 1, the sphere and
disk amplitudes are near 1. The amplitude for the solid bodies is consistent with
the results found by Chomaz et al. (1993a), shown as red circles in figure 5.14(b).
151
0 5 10 15
0
0.1
0.2
0.3
0 5 10 15
0
0.1
0.2
0.3
0 5 10 15
0
0.1
0.2
(c)
(b)
(a)
Figure 5.15: Streamwise centerline fluctuating velocity, u
0
0
/U for (a) Fr = 1, (b)
Fr = 2, and (c)Fr = 4. Line styles for each geometry same as fig 5.12
The grid wakes tend to stay near ζ/R = 0.5 regardless of Fr. Previous grid
wake experiments also found that the lee wave amplitude did not appear to depend
on Fr Xiang et al. (2015). The amplitude of those experiments is also plotted in
figure 5.14(b). The grids in the present study have larger lee wave amplitudes, this
is likely because the present grids have a solid to open area ratio of 0.71, while the
grids in Xiang et al. (2015) were 0.26. Less fluid passes through the grid region here
resulting in slightly larger lee wave amplitudes.
5.5 Fluctuating velocity
5.5.1 Streamwise fluctuating velocity
Figure 5.15 shows the normalized streamwise centerline fluctuating velocity, u
0
0
/U
for each Fr. At Fr = 1 (Figure 5.15a), velocity fluctuations are associated mainly
152
with the lee-wave induced flows. For the sphere, the lee wave is steady and fluctu-
ations are low. In other geometries the lee wave does not round the body closely
and the streamwise fluctuations are measurably higher, but by x/D = 15 all have
converged to u
0
0
/U≈ 0.04.
AtFr = 4,differencesinu
0
0
persistuptox/D = 15,andu
0
0
/U = [.07,.04,.04,.03]
for the disk, sphere, SG and LG respectively. Meunier & Spedding (2004) found
that forFr> 4Re> 5400 a similarity of the fluctuating components occurred near
Nt = 30, which here is equivalent to x/D = 60, well into the intermediate wake.
Studies of both stratified (Redford et al. (2015)) and unstratified (Bevilaqua &
Lykoudis (1978)) wakes suggest that differences in fluctuating or turbulent quanti-
ties can persist up to thousands of diameters downstream. Because the wakes here
all have low Re, u
0
comes primarily from unsteady motion of large scale structures
passing through a fixed measurement point at constant x/D, as opposed to a truly
turbulent velocity scale.
5.5.2 Crossfluctuation profiles
Following the analysis Meunier & Spedding (2004), cross-stream profiles of the
Reynolds stress may be expected to take a double-peaked form, as would be pre-
dicted if the turbulent fluctuations themselves are proportional to the mean shear.
Figure 5.16 shows u
0
v
0
(y) for each wake in the horizontal centerplane at x/D =
[5, 15]. The near wakes at x/D = 5 take on a number of forms but by x/D = 15
153
-2 -1 0 1 2
-1
-0.5
0
0.5
1
-2 -1 0 1 2
-1
-0.5
0
0.5
1
-2 -1 0 1 2
-1
-0.5
0
0.5
1
-2 -1 0 1 2
-1
-0.5
0
0.5
1
(c) (d)
(b)
(a)
Sphere
Disk
LG SG
Figure 5.16: Crossfluctuation profiles u
0
v
0
/A, in the horizontal plane and Fr = 1.
Solid line is at x/D = 5 and dashed lines are taken at x/D = 15
a double-peaked average shape is recovered, and can be fit with a function of the
form
u
0
v
0
(y) =−u
0
2
c
y
L
h
e
−y
2
/2L
2
h
(5.10)
where u
0
c
is the maximum amplitude and the characteristic width has been set to
L
h
. (Since L
h
is known, then u
0
c
can be solved for directly.) Though eq.(5.10) is
decreasingly close to the actualu
0
v
0
(y) profile moving back into the near wake, this
fit function is retained as a consistent amplitude definition throughout, and Figure
5.17 shows the evolution of its maximum amplitude, u
0
c
, for each body geometry
and Fr.
u
0
c
varies greatly between body shapes, especially at low Fr, similar to the u
0
0
fluctuations themselves in figure(5.16). The velocity fluctuations and the cross-
fluctuations are higher in the disk wake than in the grid wakes and then the sphere
154
0 5 10 15
10
-3
10
-2
10
-1
0 5 10 15
10
-3
10
-2
10
-1
0 5 10 15
10
-3
10
-2
10
-1
(a)
(c)
(b)
Figure 5.17: Downstream evolution of u
0
c
for (a) Fr = 1, (b) Fr = 2, and (c)
Fr = 4.
has the lowest values. At higher Fr = 4 the differences at x/D = 15 are the
smallest.
5.5.3 Strouhal number
The wakes are well-resolved in time and characteristic frequencies can be estimated.
At the edge of each wake, defined by the 0.15U
0
threshold, a time series of v
0
was
assembledfromindividualensembles, andthemeanperiodofoscillation,T wasused
to estimate a frequency f = 1/T, and hence Strouhal number, St =fD/U. Figure
5.18 shows St for the wakes at low Fr = 1. A decay of StNt
−1/3
is also plotted in
figure 5.18 as measured and explained in Speddinget al. (1996b); Spedding (2002a);
Meunier&Spedding(2004). Thedataarequitescatteredwithnoobviousadherence
to any one decay rate. The influence of different body shapes was normalized in
155
10
0
10
1
0.2
0.4
0.6
0.8
1
Figure 5.18: Downstream evolution of St for each geometry at Fr = 1. The error
barsshowthestandarddeviationof6repeatedruns. ThesolidlineshowsStNt
−1/3
.
10
0
10
1
0.2
0.4
0.6
0.8
1
Figure 5.19: Downstream evolution of St
eff
for each geometry at Fr = 1.
Meunier & Spedding (2004) through D
eff
, estimated for the bodies here in Table
5.2. Then St
eff
=fD
eff
/U and St
eff
is plotted against x/D
eff
in figure 5.19.
The solid black line is the relation St
eff
= 0.65(x/D
eff
)
−.34
found by Meunier
& Spedding (2004). Most of the data here fall below this line but a least squares
fit gives the same slope St
eff
= 0.50(x/D
eff
)
−.34
. The basic Nt
−1/3
power law is a
very robust result, seen for spheres at Fr∈ [2, 240],Re∈ [5, 10× 10
3
in Spedding
(2002a) and over different body shapes for Fr = 8, 32,Re = 5× 10
3
in Meunier
156
& Spedding (2004). The data in figure 5.19 do not collapse, but the variation at
x/D > 10 is less, and the previous literature results were not only for a minimum
Re = 5× 10
3
but also taken for x/D≥ 100.
5.6 Summary
The results presented here are some of the first direct comparisons of near wake
data for differing initial conditions in a stratified ambient. At Fr = 1, the wake
length scales and mean velocity for each geometry start to overlap around 15D
downstream. The overlap at Fr = 1 is likely a result of the lee wave dominated
wake. If the onset of universality is simply a function of the length of time a wake
is free to evolve in a stratified medium then the lowest Fr wakes presented here
have evolved for almost 20 buoyancy periods. At Fr = 1 whenx/D = 10,Nt = 20
whereas at Fr = 4, Nt = 5.
Despite perceived agreement in some of the mean wake quantities, for example
atU
0
/U andFr = 1, eachwakecanbeidentifiedbythetimeatwhichthehorizontal
vortices start to form. For example the sharper edged bodies, like the disk, show
coherent eddies at Nt = 16, while the sphere wake shear layers only started to
become unstable at Nt = 50, and no clear vortex formation.
The cross fluctuation profiles also show how in certain measures, the wakes can
be distinguished from one another. The disk always has the largest u
0
v
0
max
for all
Fr tested, because of the increased shear created at the body. The grids have the
157
next largest values of u
0
v
0
max
, with the sphere having the lowest magnitude in the
near wake.
A universal state for stratified wakes has really only been tested for turbulent
wakes with Re ≥ 5000 and Fr ≥ 4. At Re = 1000, where the flows are not
turbulent, and small scale mixing of the fluid cannot erase information from the
body, one might expect distinct differences to persist into the intermediate and
even far wake.
The experiments presented here primarily focus on the near wake with average
quantities up to x/D = 15. It is unlikely to expect much uniformity in any of the
wakes that are this close to the body. The fact that any of the wake quantities
at Fr = 1 start to converge with each other can be attributed to the dominant
effect of the lee waves, and the geometry-dependent wakes can be thought of as a
small disturbance superimposed on the lee wave motions. Nevertheless, the body
geometry does determine the lee wave amplitude in the near wake.
158
Chapter 6
Outlook
The near wake of a moving body in a stratified fluid has been much less studied than
the far wake and it has been only recently that there has a been a concerted effort
to understand near wake properties. The introduction of refractive index matched
experimentsandbodyinclusivesimulationshavenowmadeitpossibletogainaccess
to wake data at the earliest stages of flow development. Continuing improvements
to PIV algorithms and experimental setups will help propel the advancement and
understanding of stratified wakes. Improvements to developing models that can
take into account specifics of the initial conditions will be paramount in furthering
our understanding of stratified near wakes. Ultimately the aim is to determine
definitively if information from the original pattern creator persists into the far
wake.
The experiments presented here are an effort to understand how initial condi-
tions: Re, Fr, and body geometry, will alter the wake and how the wakes trend
toward some prescribed later time state. These studies can act as benchmarks for
159
future researchers and the empirical models prescribed can be tested and used as a
means of direct comparison both for future experiments or simulations.
Increasing Re and hybrid laboratory/numerical experiments are two ways in
which the work presented here could be expanded upon. The majority of the ex-
periments discussed have been conducted at lowFr and low to moderateRe where
in some instances the flow can be said to be turbulent but not in others. Revisiting
the near wakes of initially turbulent wakes, by increasing Re, will be essential in
understanding the true notion of universality. If true universality does exist then
perhaps it is a consequence of the flow being initially fully turbulent. The exper-
iments conducted here have shown that there is no universal initial condition and
the wakes observed in the present studies are always influenced by the background
density gradient. There is always some wake quantity that is different between each
wake, as might be expected when looking for wake data so close to the body.
Secondly, obtaining good agreement between laboratory and simulation data is
crucial to understanding the mechanisms that are involved in creating the late time
patterns that form in stratified wake. It is difficult for simulations to both resolve
wake information near the body and extend toward extremely far downstream dis-
tances. Instead simulations have opted for either small downstream distances with a
body inclusive simulation or very large downstream distances with LES/DNS simu-
lations that presuppose a gaussian mean flow. The body-less LES/DNS simulations
are initialized with a gradually increasing stratification which can be a difficult art.
On th other hand the body inclusive simulations begin with the desired density
160
Figure 6.1: Interpolation of experimental data onto a computational grid for mean
streamwise velocity, u/U, at Fr = 4,Re = 3700 at (a) x/D = 1 and (b) x/D = 6
gradient always present. There is some work being done on the use of a body in-
clusive DNS that can be used as an input for a larger LES simulation (VanDine
et al., 2018). Since simulations and experiments are finally able to make direct
comparisons in the near wake, and with reasonable agreement, it is plausible that
experimental data could be used as a direct input for future simulations.
Some preliminary results suggest that this type of hybrid experiment and sim-
ulation may be possible. Experiments were conducted at every 0.1D from−.6D≤
y/D,z/D≤ 0.6 in the horizontal and vertical centerplanes of a sphere atRe = 3700
and Fr = 4. This gives data on 13 planes in both directions. Once the mean
profile was interpolated onto the computational grid the simulation was ran us-
ing the interpolated profile as an inlet condition. Simulations were conducted by
the computational fluid dynamics group at UC-San Diego, who also provided the
computational grid dimensions.
161
Figure 6.2: Downstream evolution of centerline velocity for experiments and hybrid
lab experiment/simulation at Re = 3700 and Fr = 4
Mean velocity profiles were obtained for each plane and were then interpolated
onto a computational grid at a constant x/D location. Figure 6.1 shows the inter-
polated mean profile at x/D = 1 and x/D = 6. The mean interpolated profiles are
consistent with previous experimental results. Just behind the sphere the mean flow
takes on a circular cross section. By x/D = 6 the wake has a larger aspect ratio in
the horizontal. The x/D = 6 mean profile was used as the inlet to the simulation.
Figure 6.2 shows the mean streamwise centerline velocity for the horizontal and ver-
tical plane experiments as well as the hybrid experiment/simulation. The hybrid
experiment/simulation starts at x/D = 6 and agrees well with both horizontal and
vertical laboratory experiments. This is an encouraging sign that the methodology
can work. However more work is needed to make ensure that turbulence quanti-
ties can also agree. Body inclusive DNS is computationally expensive and if this
162
methodology can work there will be a new way to initialize simulations that will
be inherently derived from the wake problem being studied. The ultimate goal is
then to be able to use this hybrid experiment/simulation to initialize near wakes
without a body that could then be extended to much farther downstream locations
and even larger Re.
163
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Abstract (if available)

Abstract The wake generated by a moving body in a stably stratified ambient is of particular interest due to its practical geophysical and naval applications. For most applications, wakes are initially turbulent and eventually local length and velocity scales will evolve such that the wake will become dominated by low Froude number dynamics, where distinct patches of vertical vorticity form irrespective of the generating conditions. While many studies have focused on understanding the late wake structure, fewer studies have focused on the near wake region, where the background density gradient first starts to influence wake development. When and how information persists from the initial conditions to the late time stages of flow development is still an open question. The studies presented here investigate the near wakes of several different body geometries across a range of Re and F r in order to provide some of the first quantitative descriptions in the near wake, and ultimately provide a framework for understanding how information from the initial conditions may be traced in the development of the stratified wake. ❧ First a porous grid is used as a means to create a turbulent wake for 2700 ≤ Re ≤ 11000 and 0.6 ≤ F r ≤ 9.1. Refractive index matching techniques are used to gain optical access for particle imaging velocimetry measurements of the wake quantities. The temporally averaged results show a distinct internal wave motion as well as Kelvin-Helmholtz-generated vortices, which occur predominantly at the wake edge due to the strong vertical shear. After 10 buoyancy periods the wakes start to become dominated by low Froude number dynamics. The near wake however shows dependencies on both initial Re and Fr. ❧ Secondly, laboratory and numerical experiments are performed on a sphere wake across Re = [200, 300, 500, 1000] and Fr = [0.5, 1, 2, 4, 8]. In all cases the early wake is affected by the presence of the density gradient, primarily in the form of the body generated lee waves. Mean and fluctuating quantities do not reach similar states and as such their evolution cannot be described by any universal scaling. Five distinguishable regimes are observed across the parameter space and retain their distinguishing features up to buoyancy times of 20, well into the intermediate wake region. ❧ Lastly, a series of experiments is conducted with varying body geometry at a single Re = 1000 and Fr = [1, 2, 4] to investigate the effect of body geometry on the near wake in a stratified fluid. Each wake can be identified based on the time at which horizontal vortices begin to form. Despite perceived agreement in wake averaged velocity profiles, differences in each wake can be found in both the mean length scales and cross fluctuation profiles. At Fr = 1 the wake is dominated by the body generated lee wave and the wakes can be thought of as a small disturbance superimposed on the lee wave motions. The body geometry sets the lee wave amplitude, showing a direct demonstration of how the initial conditions can alter the near wake properties.

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Asset Metadata

Creator Madison, Trystan (author)

Core Title Near wake characteristics of towed bodies in a stably stratified fluid

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Degree Conferral Date 2022-05

Publication Date 01/06/2022

Defense Date 12/17/2021

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag geophysical flows,OAI-PMH Harvest,stratified flow,wakes

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Spedding, Geoffrey (committee chair), Luhar, Mitul (committee member), Nutt, Steven (committee member)

Creator Email tmadison@usc.edu,trystanmadison@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC110455710

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Source 20220112-usctheses-batch-907 (batch), University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

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