INFLUENCE OF SURFACE TOPOLOGY ON THE AERODYNAMICS OF RECTANGULAR 
CYLINDERS 

By 

Kian Kalan 

A DISSERTATION 

Submitted to 
Michigan State University 
in partial fulfillment of the requirements 
for the degree of 

Mechanical Engineering – Doctor of Philosophy 

2023 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
ABSTRACT 

Precision Airdrop Systems (PADS) face difficulties in controlling their landing accuracy 

when  flow-induced  vibrations  of  the  suspension  lines  occur.  Recent  research  has  identified  a 

previously unknown cause of these vibrations: galloping of the suspension cables. Galloping is a 

type  of  vibration  that  can  occur  in  cylinders  with  non-circular  cross-sections.  The  suspension 

cables  in  PADS  have  a  cross-section  that  is  approximately  rectangular  in  shape  with  rounded 

corners, but with the added complexity of surface topology (due to braiding of the lines). Using 

load measurements, recent experiments have shown that the presence of surface topology can alter 

the stability of rectangular cylinders to galloping; an effect that is dependent on Reynolds numbers. 

Knowledge  of  the  corresponding  topology  effect  on  the  flow  around  the  cylinders  is  presently 

lacking. Therefore, this study aims to investigate the impact of surface topology on the boundary 

layer and near-wake flow around a rectangular cylinder with a side-ratio of 2.5 and fully-rounded 

corners  (half-circular  leading  and  trailing  edges).  The  Reynolds  number  based  on  the  cylinder 

thickness (𝑑) is in the range  𝑅𝑒𝑑 = 800 − 2500. The surface topology is defined using spatial 

Fourier modes with an amplitude of 5% of 𝑑, applied along the perimeter only (2D geometry) and 

along both the perimeter and the span (3D geometry) of the cylinder. While not an exact replica, 

this surface topology represents the characteristics of the actual suspension cable reasonably well. 

The study also investigates the effects of different topology amplitudes by using cylinders with 

2.5% and 10% of 𝑑. Single-component molecular tagging velocimetry is employed to measure 

the streamwise velocity and visualize the flow field at various locations above the surface and in 

the wake of the cylinder.  

Mean and root-mean-square velocity profiles are analyzed to examine the development of 

the boundary layer and separated flow on the top and bottom surfaces of the cylinder. The mean 

 
 
separation  bubble  and  the  development  of  the  shear  layer  unsteadiness  over  the  surface  of  the 

cylinders are discussed at 𝛼 = 0° and at different Reynolds numbers. The results demonstrate the 

Reynolds  number-dependent  effect  of  the  surface  topology  cross-sectional  geometry  and  its 

variation along the span. An interpretation is provided of how these results could influence the 

galloping instability of the cylinder.  

The  wake  flow  is  investigated  to  help  better  understand  the  relationship  between  wake 

structures, surface topology, and the characteristics of the boundary layer. To achieve this, wake 

mean and rms velocity profiles are interrogated and the effect  of the geometry on the Strouhal 

number of the wake vortex shedding is analyzed. An examination is also conducted to investigate 

the  unsteady  flow  physics  of  the  boundary  layer  and  its  relationship  to  the  wake  flow.  This 

examination uses quantitative measures and flow visualization, and focuses on the smooth-surface 

cylinder. The analysis identifies and compares different Reynolds number dependent boundary-

layer  flow  regimes.  The  correlation  between  the  wake  vortex  shedding  structure  and  various 

boundary-layer regimes is examined and compared to established understanding in literature for a 

sharp-corner rectangular cylinder. 

The results reveal that the details of the topology near the leading edge of the cylinder are 

most significant in affecting the behavior of the boundary layer flow. For the particular topology 

wavelength used in the present study, the biggest effect is found when a topology peak is present 

at the leading edge for the 2D (2Dp) geometry.  In comparison to the smooth cylinder, the 2Dp 

topology  substantially  increases  the  separation  zone  thickness  and  the  separated  shear  layer 

unsteadiness. The ensuing wake flow, exhibits an increased wake closure length, slower recovery 

of  the  mean  centerline  velocity,  lower  vortex  shedding  Strouhal  number,  and  disrupted  wake 

vortex organization.  

 
 
  
Copyright by 
KIAN KALAN 
2023

 
 
 
 
 
 
 
 
 
ACKNOWLEDGMENTS 

To my 

Parisa 

parents, Mohammadreza and Robab 

siblings, Keihan and Aysan 

advisors: Dr. Ahmed Naguib and Dr. Manoochehr Koochesfahani 

committee: Dr. Mohsen Zayernoori and Dr. Phanikumar Mantha 

lab mates: Dr. Mark Feero, Dr. David Olson, Dr. Alireza Safaripour, Dr. Mitchell 

Albrecht, Dr. Basil Abdelmegied, Borhan Alhosseini, and Dr. Patrick Hammer  

undergraduate mentor: Dr. Behshad Shafii 

dearest friends 

and College of Engineering and Department of Mechanical Engineering, especially Dr. 

Katy Colbry 

It was impossible without you. 

Thank you! 

v 

 
 
 
 
 
 
TABLE OF CONTENTS 

Chapter 1. Introduction ................................................................................................................... 7 
1.1 Motivation ............................................................................................................................. 7 
1.2 Literature Review ................................................................................................................ 10 
1.3 Scope of the Current Study ................................................................................................. 14 
1.4 Outline ................................................................................................................................. 15 

Chapter 2. Experimental Techniques ............................................................................................ 17 
2.1 Flow Facility and Test Models ............................................................................................ 17 
2.2 Molecular Tagging Velocimetry ......................................................................................... 24 

Chapter 3. Boundary Layer Statistics - Mean and rms Velocity .................................................. 29 
3.1 Effect of the Cross-Sectional Geometry.............................................................................. 29 
3.2 Effect of the Surface Topology Amplitude ......................................................................... 45 
3.3 Effect of Surface Topology Variation Along the Span ....................................................... 57 
3.4 On Galloping ....................................................................................................................... 70 

Chapter 4. Characteristics of the Wake Flow ............................................................................... 89 
4.1 Mean and rms Velocity Characteristics .............................................................................. 89 
4.2 Vortex Shedding Frequency .............................................................................................. 112 

Chapter 5. On the Unsteady Behavior of the Flow ..................................................................... 122 
5.1 Nature of the Boundary Layer Unsteadiness..................................................................... 127 
5.2 Phase-Average Analysis of the Wake ............................................................................... 140 

Chapter 6. Conclusions ............................................................................................................... 151 

BIBLIOGRAPHY ....................................................................................................................... 155 

APPENDIX ................................................................................................................................. 158 

vi 

 
 
 
 
 
 
 
Chapter 1. Introduction 

1.1 Motivation 

Geometries  with  non-circular  cross-sections  are  susceptible  to  large-amplitude  self-

sustained  vibration  known  as  galloping,  which  arises  from  fluid  forces  induced  by  structural 

vibration  in  a  fluid  flow.  Transverse,  or  plunge,  galloping  occurs  when  the  orientation  of  the 

geometry to the oncoming flow changes, resulting in changes in the aerodynamic forces. Winter 

winds  can  induce  galloping  in  ice-coated  power  lines,  as  can  they  do  so  in  bridge  decks, 

skyscrapers,  etc.  Marine  structures  are  not  safe  from  galloping  in  the  ocean  current  and, 

specifically motivating the current study, suspension cables of US Army precision airdrop systems 

(PADS) are seen to suffer from this aeroelastic instability which deteriorates the controllability 

and the accuracy of the army airdrop cargo delivery [1, 2].  

An elastically mounted round-corner rectangular cylinder allowed to have an oscillatory 

motion in 𝑦 direction due to an environmental disturbance is shown in Figure 1. The oscillatory 

motion velocity (𝑦̇ ) in the transverse direction will change the effective angle of attack 𝛼, or AoA, 

with time, thereby changing the lift (𝐹𝐿) and drag (𝐹𝐷) forces. The corresponding change in the 

transverse force (𝐹𝑦) coefficient with 𝛼 can be calculated from the lift and drag coefficients from 

Equation (1). 

Figure 1. Schematic of the cross-section of the cylinder with zero amplitude topology and forces 
acting on it upon oscillating in the transverse (y) direction. 

7 

 
 
 
 
 
 
𝐶𝑦 =

𝐹𝑦
0.5𝜌𝑈∞

2𝑑𝑙 

=   −

1
cos2 𝛼

(𝐶𝐿 cos 𝛼 +   𝐶𝐷 sin 𝛼),    

                            (1) 

where the lift and drag coefficients (𝐶𝐿 and 𝐶𝐷, respectively) are calculated as below: 

𝐶𝐿   =

𝐹𝐿
0.5𝜌𝑈𝑟𝑒𝑙

2 𝑑𝑙

 ,  

𝐶𝐷   =

𝐹𝐷
0.5𝜌𝑈𝑟𝑒𝑙

2 𝑑𝑙

. 

                (2) 

                                                                (3) 

In the above equations, 𝑈∞ and 𝑈𝑟𝑒𝑙 are the steady freestream and the instantaneous oncoming 

(relative to the moving cylinder) velocities, respectively, where  

𝑈𝑟𝑒𝑙 = √𝑈∞

2 + 𝑦2̇ ,  

    (4) 

is the magnitude of the vector difference between the freestream velocity and the cylinder velocity 

𝑦̇ = 𝑑𝑦/𝑑𝑡, 𝜌 is the fluid density, 𝑑 is the cylinder width, and 𝑙 is the cylinder span. Galloping can 

occur when 𝐹𝑦 increases with increasing the angle of attack; hence, reinforcing the oscillation and 

making the structure unstable [1]. Considering objects facing air flows, which is most pertinent to 

the PADS application, the galloping oscillation frequency is typically sufficiently smaller than that 

of vortex shedding 𝑓𝑠, in which case galloping is assumed to be quasi-steady. Accordingly, the 

relevant  criterion for instability to  galloping is  based on the variation of  𝐹𝑦 with  𝛼  for a  static 

cylinder  [3].  This  quasi-steady  assumption  is  typically  valid  at  high  reduced  velocities  𝑈𝑟  (i.e. 

inverse of the non-dimensional oscillation frequency, which is the same as the structure natural 

frequency 𝑓𝑛): 

𝑈𝑟 =

𝑈∞
𝑓𝑛𝑑

> 20.  

   (5) 

Another prominent mechanism for flow-induced vibration is vortex induced vibration, which takes 

place at a reduced velocity in the vicinity of 𝑈𝑅, which is the reduced velocity at which the natural 

8 

 
 
 
 
 
 
 
 
 
 
   
   
 
 
 
 
  
 
 
 
 
 
 
 
                  
 
 
 
 
 
 
 
 
 
frequency of the elastic system equals the vortex shedding, or Strouhal frequency and is defined 

as: 

𝑈𝑅 =

1

𝑆𝑡

=

𝑈∞
𝑓𝑠𝑑

              (6) 

The Strouhal number, 𝑆𝑡, for the rectangular cylinders is reported by Parkinson [4] to be in the 

order of 10−1. More specifically related to the current study, 𝑆𝑡 falls within the range of (0.1-0.25) 

for the sharp and circular leading-edge cylinders based on the experimental studies of Okajima [5] 

and Parker&Welsh [6]. As a result, the reduced velocity corresponding to the vortex shedding will 

be in the range of (4-10).  

If the cylinder is modelled as a single-degree-of-freedom mass-spring-damper system, the 

equation of motion for the cylinder will be in the form  

𝑚𝑦̈ + 2𝑚𝜁𝑦𝜔𝑦𝑦̇ + 𝑘𝑦𝑦 = 𝐹𝑦 =

1

2

𝜌𝑈∞

2 𝑑𝐶𝑦,   

 (7) 

where  𝑚  is  the  mass  per  unit  span  of  the  cylinder,  𝜁𝑦  is  the  damping  factor  due  to  structural 

dissipation, 𝑘𝑦 is the spring constant, 𝜔𝑦 = (

1/2
)

𝑘𝑦
𝑚

= 2𝜋𝑓𝑦 is the circular natural  frequency of 

plunging, and 𝑓𝑦 = 𝑓𝑛 is the natural frequency in Hz. For small angles of attack, 𝛼, 𝑈𝑟𝑒𝑙 and 𝐶𝑦 

can be expanded in power series as: 

𝛼 = arctan (

𝑦̇
𝑈∞

) =

𝑦̇
𝑈∞

+ 𝑂(𝛼2),  

𝑈𝑟𝑒𝑙 = 𝑈∞ +  𝑂(𝛼2),   

𝐶𝑦(𝛼) = 𝐶𝑦|𝛼=0 +

𝜕𝐶𝑦
𝜕𝛼

|𝛼=0𝛼 + 𝑂(𝛼2). 

 (8) 

 (9) 

(10) 

At 𝛼 = 0, the lift direction is parallel to the galloping direction and the force coefficient in the 𝑦 

direction  is  equal  to  the  negative  lift  coefficient,  𝐶𝑦(𝛼 = 0) = −𝐶𝐿(𝛼 = 0);  see  Equation  (1). 

Substituting for this in Equation (10) and using the result, up to linear power in 𝛼, to replace 𝐶𝑦 in 

Equation (7), yields: 

9 

 
                                              
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
𝑚𝑦̈ + 2𝑚𝜔𝑦(𝜁𝑦 −

𝜌𝑈∞𝑑
4𝑚𝜔𝑦

𝜕𝐶𝑦
𝜕𝛼

|𝛼=0)𝑦̇ + 𝑘𝑦𝑦 = −

1

2

𝜌𝑈∞

2 𝑑𝐶𝐿|𝛼=0    

            (11) 

Setting the term in parentheses on the left-hand side of Equation (11) to 

𝜁𝑇 = 𝜁𝑦 −

𝜌𝑈∞𝑑
4𝑚𝜔𝑦

𝜕𝐶𝑦
𝜕𝛼

|𝛼=0,  

            (12) 

the net damping factor (including both structural and fluid damping), the solution to Equation (11) 

will be:  

𝑦 =

1
2

𝜌𝑈∞

2 𝑑𝐶𝐿|𝛼=0

𝑘𝑦

+ 𝐴𝑦𝑒−𝜁𝑇𝜔𝑦𝑡 sin [𝜔𝑦(1 − 𝜁𝑇
2)

1
2𝑡 + 𝜙], 

            (13) 

where, 𝐴𝑦 is the amplitude of oscillation and 𝜙 is the corresponding phase shift. As seen from the 

solution, the stability condition, determining whether the oscillations increase over time, depends 

on  the  net  damping  coefficient  sign  𝜁𝑇.  If  𝜁𝑇 > 0,  vibrations  would  decay  with  time  and  the 

stability condition is met. Equivalently, referring to Equation (12), the model is stable if  

𝜕𝐶𝑦
𝜕𝛼

|𝛼=0 < 0,    

           (14) 

and otherwise it can be unstable. Specifically, if 

𝜕𝐶𝑦
𝜕𝛼

> 0, a critical velocity can be found above 

which the net damping coefficient becomes negative, which marks the onset of plunge galloping 

instability. The critical reduced velocity is found by setting 𝜁𝑇 = 0 in Equation (12), leading to: 

𝑈𝑐𝑟𝑖𝑡
𝑓𝑦𝑑

=

4𝑚(2𝜋𝜁𝑦)
𝜌𝑑2

/

𝜕𝐶𝑦
𝜕𝛼

|𝛼=0.  

1.2 Literature Review 

(15) 

Suspension lines of precision airdrop systems have been recently shown to be susceptible 

to  galloping  [1,  2];  with  possible  consequent  negative  effect  on  the  performance  and  the 

controllability of PADS. The cross-section of these suspension lines is typically non-circular and 

resembles a rectangle with rounded corners, as given in Figure 1. Additionally, due to braiding, 

the surface of the lines is  not  smooth,  but  is  rather  characterized by topological  surface height 

10 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
variations [1, 2]. Thus, it is important to understand the effect of the surface topology of these 

suspension  lines  on  their  aeroelastic  behavior;  specifically,  their  static  𝐶𝑦 − 𝛼  characteristics. 

Siefers et al. [2] observed that a Dacron cable at low Reynolds numbers can show low-frequency 

vibrations, at a much lower frequency compared to that based on the vortex shedding Strouhal 

number they measured in the wake of the object. They hypothesized that galloping is the dominant 

form  of  instability  leading  to  the  suspension  line  vibration,  but  this  hypothesis  needed  to  be 

assessed by directly measuring the aerodynamic loads acting on the line.  

In a recent study, Feero et al. [7, 8] observed that the presence of the surface topology can 

change the galloping stability of cylinders of the same geometry, as shown in Figure 2, somewhere 

in the Reynolds numbers 𝑅𝑒𝑑 = 𝑈∞𝑑/𝜈 of 1100 to 2500 which is within the relevant range for 

PADS  (1000  to  10000).  These  conclusions  were  made  based  on  measurement  of  the  mean 

transverse force coefficient (𝐶𝑦) variation with AoA for the cylinders. More specifically, referring 

to Figure 2-left, at 𝑅𝑒𝑑=1100, the smooth cylinder shows negative slope of 𝐶𝑦 at 𝛼=0°; meaning 

that galloping could not occur. On the other hand, the cylinder with surface topology (Figure 2-

right) shows a positive slope at  the same Reynolds number;  i.e., susceptibility to  galloping. At 

𝑅𝑒𝑑 = 2500 and higher Reynolds numbers, both cylinders behave similarly, exhibiting stability 

to  galloping.  Thus,  the  effect  of  surface  topology  on  galloping  instability  of  the  cylinders  is 

Reynolds number dependent.  

11 

 
Figure 2. Galloping force coefficient characteristics of the smooth cylinder (left) and the cylinder 
with surface topology (right) reported in [7, 8] in a different test facility. Legend shows 𝑅𝑒𝑑 
values. 

It is well established in the literature that galloping instability of bluff bodies is closely 

connected  to  the  separation  and  reattachment  behavior  of  the  boundary  layer.  Rectangular 

cylinders  with  sharp  corners,  whose  galloping  characteristics  are  one  of  the  most  studied,  are 

prone to galloping when their side-ratio falls within 0.75 < 𝑐/𝑑 < 3 (where 𝑐 is the cylinder chord 

as defined in Figure 1). In a review paper, Parkinson [4] explains that the link between galloping 

instability and side-ratio for these cylinders lies in the body length downstream of the boundary 

layer separation point. Knowing that the shear layer separation occurs at the sharp corners of the 

leading  edge,  the  afterbody  of  the  cylinders  with  side-ratio  less  than  0.75  is  too  short  that  the 

separated shear layer and its vortices are not able to interact with the cylinder lateral surface and 

exert  pressure  distribution  favorable  for  galloping.  When  the  side-ratio  is  large  enough 

(approximately 𝑐/𝑑 ≥ 3), the separated shear layers reattach to the lateral surfaces of the cylinder 

before  finally  separating  at  the  sharp  corners  downstream  and  there  is  no  afterbody  for  the 

separated shear layer vortices to interact with the cylinder sides. Therefore, these cylinders show 

stability to galloping.  

Given the generally small number of studies for the cylinders at relatively low Reynolds 

numbers (𝑅𝑒𝑑 ≤ 104), Feero et al. [9] conducted a study on the rectangular cylinders with sharp 

12 

 
 
 
corners  and  side-ratio  𝑐/𝑑  range  of  1 − 3.  They  observed  that  when  the  Reynolds  number  is 

sufficiently  low,  in  their  case  below  5000,  even  the  cylinder  with  long  afterbody  is  prone  to 

galloping. From the surface pressure data, they observed that the reattachment point on the bottom 

side surface of the cylinder does not move with increasing 𝛼 from 0° to 5° where the force data 

indicates  positive  𝐶𝑦 − 𝛼  slope.  The  opposite  of  this  behavior  is  observed  at  higher  Reynolds 

numbers where the separation bubble on the bottom side is seen to shrink in length with 𝛼. They, 

therefore,  hypothesized  that  the  separated  shear  layer  reattachment  on  the  bottom  side  of  the 

cylinder is necessary but not sufficient to prevent galloping instability. For stability, the separation 

bubble length has to shrink with increasing 𝛼 as well. Interestingly, they also observed that below 

𝑅𝑒𝑑 = 5000, increasing the side ratio in the range of 1-3 results in higher instability to galloping 

which  is  deduced  from  higher  𝐶𝑦 − 𝛼  curve  slope  (corresponding  to  lower  critical  reduced 

velocity; see Equation 15).  

Carassale et al. [10] studied square cylinders (𝑐/𝑑 = 1) with sharp and round corners with 

𝑟/𝑑 of 0.067 and 0.133 at high Reynolds numbers 𝑅𝑒 > 104. They observed that above a critical 

Reynolds number and in a turbulent freestream, the separating shear layer behavior on the top side 

of  the  cylinder  with  the  largest  corner  radius  plays  a  significant  role  in  determining  the  lift 

coefficient. They found that the shear layer reattaches to the top and bottom surfaces of the cylinder 

at  small  𝛼  notwithstanding  the  very  short  afterbody  length.  In  addition  to  this  so-called 

supercritical behavior, surface pressure measurements also  indicate shrinkage of the separation 

bubble  on  the  bottom  side  with  𝛼,  resulting  in  more  pressure  recovery  and  lift  increase.  The 

separation bubble on the top surface, on the other hand, lengthens and exerts higher suction which 

contributes to even more lift. Therefore, considering that 𝐶𝐿 ≈ −𝐶𝑦 at small 𝛼, lift increasing with 

𝛼 is opposing to the favorable conditions to galloping and the geometry is stable. When 𝛼 is large 

13 

 
enough, the separated shear layer on the top side is not able to reattach and suction drops and the 

lift coefficient decreases with 𝛼 (i.e., 𝐶𝑦 increases with 𝛼). The latter behavior is similar to that at 

sub-critical Reynolds numbers starting from 𝛼 = 0 for the case of a laminar freestream.  

1.3 Scope of the Current Study 

Understanding the flow field associated with galloping instability of the PADS’ suspension 

lines  motivates  the  current  study.  The  main  focus  of  the  current  work  is  on  understanding  the 

boundary layer development on the cylinder surface, and how this development is affected by the 

surface  topology  and  Reynolds  number.  A  range  of  geometries  is  studied  to  systematically 

investigate  and  distinguish  the  effects  of  cross-sectional  geometry,  topology  amplitude,  and 

presence of topology variation along the span. This investigation is carried out in a water tunnel in 

the Reynolds number range 𝑅𝑒𝑑 = 800 − 2500, which is pertinent to the galloping problem of 

PADS. The same range  has been examined,  from force  characteristics  perspective only, in  the 

wind tunnel, as discussed above [7, 8]. However, it is important to note that the flow and the force 

characteristics in two different test facilities may be different notwithstanding matching of 𝑅𝑒𝑑. 

Such  differences  may  be  caused  by  discrepancies  in  freestream  turbulence  intensity,  cylinder 

aspect ratio, model end conditions, and test section blockage. For this reason, complementary force 

measurements were done by a collaborator on the same cylinder models and experimental setup 

employed in the present study [11]. These results, included here in Figure 3 for reference, show 

that similar to the data in Figure 2-right, the present cylinder model with topology is unstable to 

galloping at  𝑅𝑒𝑑 = 1100, but it is stable at  𝑅𝑒𝑑 = 2400. The crossover point appears to be at 

𝑅𝑒𝑑 = 1800.   

14 

 
Figure 3. Comparison of the 𝐶𝑦 − 𝛼 behavior for the cylinder with surface topology similar to 
[7, 8] for 𝑅𝑒𝑑 = 1100, 1800 and 2400 [11]. 

In addition to investigating the boundary layer, the wake of the cylinder is also examined 

at 𝛼 = 0° in the same Reynolds number range used for the boundary layer study. The main focus 

for the wake results presented here is to measure the vortex shedding frequency for the cylinders, 

with  versus  without  topology,  given  the  relevance  of  this  knowledge  to  the  possibility  of 

interaction of galloping with vortex induced vibrations (in which case quasi-steady analysis is not 

accurate).  The  connection  between  the  boundary  layer/separated  shear  layer  behavior  with  the 

wake, and the initial wake width with the shedding frequency are also investigated in this work. 

1.4 Outline 

The discussions from here on are organized in four chapters. Flow facility and experimental 

techniques are introduced in Chapter 2, where the procedure for 3D printing the test models are 

explained, the implementation of single-component molecular tagging velocimetry (1-c MTV) and 

the  characterization  of  uncertainties  are  outlined,  and  the  image  processing  approaches  are 

described.  

The primary results are categorized and discussed in three chapters. Chapter 3 focuses on 

the  streamwise-velocity  mean  and  rms  characteristics  of  the  boundary  layer  flow  over  the 

15 

 
 
 
 
cylinders. In this chapter, the effect of the complex surface topology on the boundary layer flow 

is  broken  into  sub-problems  by  separately  studying  the  effect  of  topology  amplitude,  cross-

sectional feature, and variation along the span. Furthermore, Chapter 3 investigates how the effect 

of different surface topology features on the boundary layer flow undergoes changes with varying 

the Reynolds number. Finally, based on the current understanding in literature of the connection 

between the boundary layer flow behavior and instability of the cylinders to galloping, the effect 

of different surface topology features and Reynolds number on the stability of the cylinders will 

be  analyzed.  Chapter  4  focuses  on  the  wake  flow  of  the  cylinders  and  how  the  mean  and  rms 

characteristics of the streamwise velocity in the wake of the cylinders are affected by the surface 

topology features. The vortex shedding frequency and how it is affected by the surface topology 

will be studied against Reynolds number in this chapter. In Chapter 5, the investigation will focus 

on  unsteady  flow  characteristics  in  an  effort  to  gain  a  deeper  understanding  of  the  connection 

between the boundary layer and wake flow. The study will utilize flow visualization techniques 

and conduct additional statistical analysis to characterize various flow regimes. Subsequently, the 

phase-averaged velocity in the wake will be analyzed for each flow regime. 

16 

 
 
 
Chapter 2. Experimental Techniques 

2.1 Flow Facility and Test Models 

The experiments are conducted in a closed-return, free-surface water tunnel (Engineering 

Laboratory Design, ELD, Model 502 6") in the Turbulent Mixing and Unsteady  Aerodynamics 

Laboratory  (TMUAL)  at  Michigan  State  University.  The  contraction  ratio  between  the  settling 

chamber and the test section of this tunnel is 6:1. The impeller circulating the water in the tunnel 

is driven by an electric motor, controlled by a TOSHIBA VF-S9 inverter which facilitates accurate 

setting of the tunnel flow speed. The test section dimensions are 152 mm × 152 mm × 457 mm. 

An additional flow management in the form of a 38.1 mm (1.5") long honeycomb with 3.2 mm 

(1/8") diameter cells is implemented downstream of the contraction section with its downstream 

face located about 13 mm (0.5") upstream of the start of the plexiglass test section. This was done 

to reduce the turbulence level of the tunnel below that achievable with flow conditioning upstream 

of the contraction alone. Installation of this extra honeycomb has shown to decrease the turbulence 

(rms) level of the tunnel and improve the uniformity of the freestream mean velocity profile across 

the  tunnel  (Kalan’s  M.S.  project  [12]).  The  tunnel  was  operated  for  several  hours  before  the 

measurements were done to make the water temperature as uniform and steady as possible. Due 

to possible room temperature variations, water temperature in the test section was measured with 

a calibrated Omega 450 ATH thermometer equipped with a T-type thermocouple with ±0.2° C 

accuracy before each experiment. The actual tunnel speed was then set to meet the target Reynolds 

number based on the kinematic viscosity value at the measured water temperature. It is noteworthy 

that  the  tunnel  temperature  was  again  measured  after  each  experiment  to  confirm  that  no 

considerable temperature difference had occurred during the measurements. In all cases, the tunnel 

temperature variation before and after the measurements was within the thermocouple accuracy. 

17 

 
 
 
Placement of the model in the test section results in a 9.8% geometrical blockage (at 𝛼 =

0°). The measurements are performed at 𝑅𝑒𝑑 = 800, 1100, 1400, 1800 and 2500, corresponding 

to  𝑈∞ = 4.7, 6.5, 8.3, 10.7,  and  14.8  cm/s  respectively,  at  water  temperature  of  25°C,  where 

freestream turbulence level of the tunnel is found to be below 1.7%, 1.5%, 1.4%, 1.2%, and 1% 

for  the  aforementioned  Reynolds  numbers,  respectively.  No  effort  is  made  to  decompose  the 

turbulence fluctuations into slushing and broadband components. 

The  models  are  3D  printed  cylinders  with  a  11.4  mm-diameter  (7/16")  tight-tolerance 

multipurpose  stainless-steel  rod  inserted  along  their  centerline  to  prevent  warping.  This  shaft 

extends beyond the cylinder’s span on one end, passing through a skimmer plate (Figure 4), and 

is clamped to a closed-loop Applied Motion Products TSM17Q-1RG stepper motor with encoder 

that controls the angular position of the cylinder, and hence the AoA of the model, with an angular 

resolution  of  0.02°.  The cylinder  mounting  setup  is  built  on  a  rigid  structure  isolated  from  but 

assembled around the water tunnel to prevent the transfer of any possible vibrations from the tunnel 

to the cylinder.  

A  Plexiglass  skimmer  plate,  spanning  76  mm  (5.1d)  upstream  and  downstream  of  the 

model’s center, is utilized to provide a well-defined wall boundary condition at the free surface of 

the test section. There is a 0.5 mm clearance between the ends of the cylinder and both the tunnel 

bottom  surface  and  the  skimmer  plate.  The  streamwise  position  of  the  cylinder  is  such  that  its 

centerline is 13.3d downstream of the test section entrance. 

18 

 
 
 
 
 
 
 
(a) 

(b) 
Figure 4. (a) 3D model of the experimental setup in the test section. (b) Schematic of the top 
view of the optical system for 1c-MTV measurements and the tunnel test section. Imaging fields 
of view (FOV) utilized for MTV measurements are indicated on the schematic. 

An idealized definition is used to prescribe the surface topology of the test cylinder, instead 

of using a replica of an actual PAD suspension line’s geometry. This is motivated by two factors: 

first,  the  present  study  aims  at  providing  general  basic  understanding  of  the  effect  of  surface 

topology independent of specific applications; second, the idealized topology is defined precisely 

using two-dimensional Fourier-mode synthesis. Specifically, the topology 𝜀 (see Figure 5), which 

is the deviation of the cylinder surface from the baseline smooth cylinder cross-section (depicted 

in Figure 1 and Section A-A in Figure 5), is given as follows: 

=

0

2

[cos [2π(

𝑠
𝜆𝑠

 + 

𝑧
𝜆𝑧

)] + cos[2π(

𝑠
𝜆𝑠

 - 

𝑧
𝜆𝑧

)]],  

(16) 

19 

 
 
 
 
 
 
 
 
 
 
where 𝑠 is the wall-tangential coordinate defined along the surface of the baseline (smooth) model 

(see Section A-A in Figure 5), 𝑧 is the coordinate along the span of the cylinder, 0 is the topology 

amplitude, and  𝜆𝑠 and  𝜆𝑧 are the topology wavelengths in  𝑠 and 𝑧 directions, respectively (see 

Figure 5). The wavelength along the 𝑠 direction is equal to 𝑃/𝑛 with 𝑃 and 𝑛 being the baseline 

model’s  perimeter,  and  the  integer  number  of  topology  wavelengths  around  the  perimeter, 

respectively. 

The  values  of  the  geometric  parameters  of  the  test  models  are  selected  such  that  they 

approximate the Dacron parachute suspension line used by Siefers [13]. This typical suspension 

line  has  the  following  geometrical  parameters:  𝑐/𝑑 ≈ 2.4,  𝑟/𝑑 ≈ 0.4, 0/𝑑 ≈ 5%,  and  a  24-

strand  braid.  For  the  cylinder  models  used  here,  the  cross-sectional  dimensions  of  the  baseline 

(smooth surface) model give a side ratio 𝑐/𝑑 = 2.5, and a fully-round LE and TE, 𝑟/𝑑 = 0.5 (see 

Figure  1  for  definition  of  terms).  𝜆𝑧/𝜆𝑠 = 1.5  and  𝑛 = 10  are  chosen  based  on  counting  the 

number of wavelengths around the perimeter of the Dacron line and measuring the peak to peak 

distance  along  the  span.  Different  topology  amplitudes  of  0/𝑑 = 2.5%, 5%  and  10%  are 

investigated, with the 0/𝑑 = 5% case being the closest to the actual Dacron cable.  

The test models are enlarged by a factor of 7.7 relative to the actual Dacron cable scale, 

where  the  width,  chord,  and  corner  radius  are  𝑑 = 15  mm,  𝑐 = 37.5  mm,  and  𝑟 = 7.5  mm, 

respectively. The span of the models is 𝑙 =  139.6 mm, which extends over the full height of the 

test section and results in an aspect ratio 𝑙/𝑑 of 9.3. 

20 

 
 
 
 
 
 
 
 
 
A-A 

A-A 

B-B 

𝜆𝑠 

𝑥 

𝑦 

B-B 

Figure 5. Top and cross-sectional views of the cylinder with 0/𝑑 =5%, 𝑛=10, and 𝜆𝑧/ 𝜆𝑠=1.5. 

 To systematically study the effect of the surface topology, several models are investigated, 

in addition to the main geometry shown in Figure 5. A model with zero topology amplitude (i.e. 

smooth cylinder) is employed as a baseline case. A second set of models is made with surface 

topology, where the cross-section is constant along the whole span (𝜆𝑧 = ∞). This set, which is 

referred  to  as  the  2D  set,  includes  two  different  cross-section  shapes:  one  with  a  peak  in  the 

topology at the LE, and the other with a valley. These “peak-leading” and “valley-leading” shapes 

are the same as cross-sections A-A and B-B, respectively, of the 3D cylinder geometry in Figure 

5. The 2D models are used to study the effect of the topology on the flow in the absence of flow 

three-dimensionality induced by the spanwise waviness of the topology.  Comparing results from 

the 2D models to those from the 3D model is expected to provide insight into the significance of 

21 

 
 
 
 
  
 
 
 
the  spanwise  variation  of  the  topology.  For  the  3D  geometry,  measurements  are  done  at  two 

spanwise planes corresponding to sections A-A and B-B in Figure 5. These cases are indicated as 

3Dp and 3Dv respectively, where the p and v in the notation signify the presence of a peak and a 

valley, respectively, at the cylinder’s leading edge. The corresponding 2D model cases are denoted 

as 2Dp and 2Dv, respectively.  

  Models following the prescribed surface topology are 3D printed by an Objet Connex350 

multi-material 3D printing system. The nominal resolution of this printer is 30 microns along 𝑦 

axis and 100 microns along 𝑥 and 𝑧 axes as defined in Figure 5. The material used to print the 

cylinders  is  Stratasys  RGD450  Rigur  which  is  a  rigid  opaque  white  material.  Given  that  the 

readable input file format to this 3D printer is ‘.stl’, the cylinders should be designed in a way that 

can be further translated to  this  format. The process  of doing so is that the surface topology is 

defined mathematically in MATLAB (see code in Appendix) and discrete points are generated to 

make a point cloud which is afterwards introduced to SolidWorks software in text ‘.txt’ format. In 

this software, this point cloud is transformed into a mesh by the ‘mesh prep wizard’ feature. It is 

possible then to save this mesh as a .stl file. However, no manipulation can be directly done on 

this mesh in SolidWorks and any change has to be done to the main point cloud. In the course of 

the  study,  manipulation  of  the  mesh  became  necessary  to  generate  a  through  hole  for  fitting  a 

metallic shaft in the 3D printed cylinders in order to reinforce them. These sorts of alterations are 

much easier to conduct in a CAD software in comparison to defining them mathematically in the 

process  of  point  cloud  generation.  To  go  around  this  issue,  the  ‘.stl’  file  was  loaded  into 

SpaceClaim software which is an Autodesk product. This software provides capability of making 

a concentric through hole inside the cylinders with the desired diameter.  

22 

 
 
Accuracy of the 3D printed objects is investigated using a KEYENCE VR3200 3D scanner 

device. The accuracy of this device in surface height measurements is ±2𝜇𝑚. The measurements 

are done on the surface of a cylinder with a 3D surface topology that is close but not identical to 

the  models  used  in  the  measurements.  Specifically,  some  parameters  such  as  ratio  of  the 

wavelengths and corner radius are slightly different, but the topology amplitude is the same. This 

cylinder is used in preliminary studies. The results shown in Figure 6 are samples of the measured 

surface profiles, on one of the sides and one of the edges of the cylinder, compared with the design 

topology profile produced by the mathematical definition. Since the resolution of the 3D printer is 

better along the 𝑦-axis, as seen in Figure 6, a better surface quality of the 3D printed cylinder is 

obtained on the side of the cylinder than along the edge. The rms deviation from the design profile 

is 2.3% and 10.5% of the design wave amplitude on the side and along the edge, respectively and 

it can be seen that, overall, the printed geometry is satisfactory and follows the intended profile.   

23 

 
 
 
 
 
 
3D 
Scanner 

3D 
3D 
Scanner 
Scanner 

Figure 6. Accuracy of the 3D printed object. Surface measured by the scanner is compared to the 
design sinusoidal wave.  

2.2 Molecular Tagging Velocimetry 

One-component molecular tagging velocimetry (1c-MTV) is utilized to measure the flow 

velocity. MTV is a whole-field non-intrusive measurement technique that relies on using a flowing 

medium premixed with molecules that can be turned into long-lifetime tracers upon excitation by 

photons of a particular wavelength [14, 15]. Typically, a pulsed laser is used to “tag” the regions 

of interest, and those tagged regions are interrogated at two successive times within the lifetime of 

the  tracer  to  make  an  image  pair  of  so  called  ‘delayed’  and  ‘undelayed’  images  as  seen  in 

Figure 7. 

24 

 
 
 
 
 
 
 
 
 
Flow 

Flow 

𝑡 = 𝑡0 

𝑡 = 𝑡0+5ms 

Figure 7. 1c-MTV images. 'delayed' image on the right is taken 5ms after capturing the 
'undelayed' image on the left. The lines have been colored to enhance their visibility. 

 The present implementation of MTV employs a phosphorescent supramolecule tracer [14] 

with chemical composition of 1 × 10−4 M of maltosyl-𝛽-cyclodextrin, 0.055 M of cyclohexanol, 

and ∼1 × 10−5 M of 1-bromonaphthalene, which assures its solution saturation. The tracers are 

excited by a Lambda Physik LPX 210i XeCl 308 nm UV excimer laser. The discharged rectangular 

laser beam passes through a pair of coated converging spherical lenses to form a travelling parallel 

beam, that is then steered through a set of coated mirrors toward the tunnel test section. Prior to 

entering the test section, the parallel laser beam passes through a pair of converging cylindrical 

lenses. This process creates a thin laser sheet with a thickness of less than 1 mm across the tunnel 

test section. Laser lines are formed after the laser sheet passes through a beam shaping object, as 

depicted Figure 4-b. The beam shaping object is a thin metal sheet with a custom pattern of vertical 

narrow slots to form the desired tagging pattern. For the present work, between 28 to 38 tagged 

lines are oriented to measure the streamwise velocity component with a spacing of 0.08𝑑 in the 

streamwise direction. The individual tagging lines, being as narrow as about 0.7 mm each (based 

on full width 1/𝑒2), enter the test section on the side where quartz window inserts are installed. 

Other  test  section  walls  are  made  out  of  Plexiglas,  which  enables  imaging  of  the  molecular 

emission while blocking the UV laser light  from exiting the test section. The optical  system is 

positioned  on  a  manually  movable  three-degree-of-freedom  traversable  optics  bench,  which  is 

mounted on the tunnel but kept separate from it to prevent the transmission of tunnel vibrations. 

25 

 
 
 
 
 
One-component MTV measures the Lagrangian displacement of the fluid particles in the 

direction normal to a tagging line at every pixel along the line, providing one component of the 

flow velocity at very high spatial resolution. The technique for calculating the line displacement 

using spatial correlation is described in [16]. For the present work, the tagged lines are oriented to 

measure the streamwise velocity component with a spacing of 40 μm (0.003𝑑) along each of the 

lines. This high spatial resolution makes 1c-MTV particularly suited for boundary-layer-resolved 

measurements.  Three  fields  of  view  (FOV)  are  adopted  for  data  acquisition  by  translating  the 

optical tray downstream (see Figure 4-b). The first one stretches between 𝑥/𝑑 = -1.15, near the 

leading edge (LE), to 𝑥/𝑑 = 1.15, near the trailing edge (TE) to measure the boundary layer flow. 

Its cross-stream dimension extends from 𝑦/𝑑 ≈ −0.5 to 1.9 focusing on the boundary layer flow. 

The second field of view, stretching between 𝑥/𝑑 ≈ 0.17 and 3.80 covers the downstream half 

and the immediate wake of the cylinder to allow simultaneous boundary layer and the near wake 

measurements. Finally, the third field of view covers the wake farther downstream from 𝑥/𝑑 ≈

2.97 up to 6. The implementation of overlapping fields of view enables precise alignment and 

assembly of the results. The latter two fields of view stretch between 𝑦/𝑑 ≈ −1.2 and 1.2 in the 

cross-stream direction to symmetrically capture the wake flow.  

There  are  different  sources  of  uncertainty  contributing  to  the  accuracy  of  the 

measurements. Among them, laser pulses inconsistency and possible vibrations of the setup, along 

with  MTV  settings  such  as  signal  to  noise  ratio  SNR,  correlation  window  size,  tagging  line 

thickness  and  spacing,  line  shape,  etc.  can  affect  the  subpixel  accuracy  [16].  Furthermore,  the 

delayed images can be correlated to a single undelayed image, as in the current study, which is 

usually  a  time-average  image  from  a  time  series  of  instantaneous  undelayed  images.  This 

assumption is only valid when the variations between the instantaneous and average undelayed 

26 

 
 
 
 
 
images  are  small.  Correlating  the  instantaneous  undelayed  images  with  their  temporal  average 

gives an estimation of the total uncertainty of the measurements, or subpixel accuracy, introduced 

by  the  above-mentioned  sources.  Such  analysis  is  conducted  with  the  instantaneous  undelayed 

images  acquired  at  a  SNR  representative  of  the  delayed  images  of  the  velocity  measurements. 

Subpixel accuracy of the measurements in terms of rms are calculated to be around 0.16 pixels. 

  The  undelayed  image  is  taken  1 𝜇𝑠  after  the  laser  pulse  to  avoid  capturing  the  florescence 

emission of the tracers. The time delay Δ𝑡 between the two MTV images (undelayed and delayed) 

changes between different Reynolds numbers from 8ms for 𝑅𝑒𝑑 = 800 to 3.5ms for 𝑅𝑒𝑑 = 2500. 

The exposure time of the images is set to 10% of the time delay Δt. The time delay is customized 

for each Reynolds number to ensure proper pixel displacement of the tagged region during the 

time delay. The exposure time is adjusted accordingly to maintain the highest possible SNR, and 

simultaneously minimize the pixel displacement during the exposure. 3072 images are acquired at 

the  lower  Reynolds  numbers  𝑅𝑒𝑑 = 800  and  1100,  and  for  the  remaining  Reynolds  numbers, 

2048 images are acquired at a sampling rate of 7 Hz. The duration of the time series corresponds 

to a convective time scale of 1375 𝑑/𝑈∞ for 𝑅𝑒𝑑 = 800 and 2977 𝑑/𝑈∞ for 𝑅𝑒𝑑 = 2500. Based 

on the shedding frequency measured at the far wake, by calculating power spectral density from 

velocity  time  series,  this  corresponds  to  about  290  and  610  shedding  cycles  at  the  lowest  and 

highest  Reynolds  numbers,  respectively.  A  pco.pixelfly  camera  equipped  with  a  Nikon  Nikkor 

58mm f/1.2 lens and an 11 mm Nikon extension ring is utilized for image acquisition.  

The camera has a 14-bit charge-coupled device (CCD) sensor with a resolution of 1392 × 1040 

pixels and a pixel size of 6.45 𝜇𝑚 × 6.45 𝜇𝑠. The maximum acquisition frequency at the maximum 

frame size without any pixel binning is about 7 frames per second.  

27 

 
  Statistical  convergence  uncertainty,  which  is  another  important  source  of  uncertainty,  is 

reported  in  Table  1,  for  the  largest  uncertainty  over  the  entire  measurement  domain.  The 

uncertainties are reported as % of 𝑈∞ for the mean velocity and % of the local 𝑢𝑟𝑚𝑠 for the rms 

velocity, calculated based on the work of Benedict and Gould [17]: 

𝜎𝑈̅ =

𝑢𝑟𝑚𝑠
√𝑁 

, 

𝜎𝑢𝑟𝑚𝑠 =

𝑢𝑟𝑚𝑠
√2𝑁

,                        

        (17) 

        (18) 

where 𝑁 is the number of the independent samples and is taken here to be the number of the vortex 

shedding cycles occurring in the duration of the time series.  

Table 1. Convergence uncertainty of mean and rms velocity measurements as percentage of 𝑈∞ 
and local 𝑢𝑟𝑚𝑠, respectively. 

𝑅𝑒𝑑   =  800  𝑅𝑒𝑑   =  1100  𝑅𝑒𝑑   =  1400  𝑅𝑒𝑑   =  1800  𝑅𝑒𝑑   =  2,500 

Mean velocity 

1.67% 

1.97% 

2.27% 

2.28% 

1.61% 

rms velocity 

4.73% 

4.01% 

4.46% 

4.13% 

3.17% 

Convergence uncertainty of the mean displacement and velocity due to the subpixel accuracy, are 

estimated to be 0.012 pixels and 0.1% of 𝑈∞, respectively. Convergence uncertainty of the rms 

displacement due to the subpixel accuracy is calculated to be 0.009 pixels which corresponds to 

0.075% of 𝑈∞.  

28 

 
  
  
 
    
 
 
 
 
    
 
 
 
 
 
 
 
 
 
 
Chapter 3. Boundary Layer Statistics - Mean and rms 
Velocity 

The  current  study  aims  to  reveal  the  underlying  flow  physics  related  to  the  complex 

geometry  of  the  rectangular  cylinders  with  round  corners  and  surface  topology.  This  should 

ultimately provide fundamental  understanding of the unique behavior of  these cylinders at  low 

Reynolds  numbers,  as  reported  in  the  previous  studies  [7,  8].  As  a  step  towards  this  goal,  this 

chapter covers results related to the streamwise velocity field in the boundary layer of the cylinders. 

In  the  following  sections,  mean  and  rms  velocity  profiles  and  color  contour  maps  from  MTV 

measurements  along  the  surface  of  the  cylinders  will  be  discussed.  The  focus  of  Sections  3.1 

through 3.3 will be on the boundary layer measurements carried out at the angle of attack 𝛼 = 0°. 

In order to distinguish the effects of the cross-sectional geometry on flow behavior from those due 

to shape variation along the span, Section 3.1 will focus on the 2D geometries, with the amplitude 

pertinent to the PADS application (𝜀/𝑑 = 5%), and how they compare with the baseline smooth 

geometry. Section 3.2 is focused on the study of the topology amplitude effects on the flow field 

by comparing cylinders with lower and higher topology amplitudes than the PADS application. In 

section 3.3, the effects of variation of the geometry along the span will be interrogated, and the 

cross-sections of the 3D geometry corresponding to the 2D cylinders will be discussed. In the final 

section of this chapter, observations pertinent to the galloping phenomenon will be presented by 

analyzing changes in the flow around the geometries with variation in the angle of attack. 

 3.1 Effect of the Cross-Sectional Geometry 

In this section, the smooth cylinder, along with the 2Dp and 2Dv cylinders with 𝜀/𝑑 = 5% 

at 𝛼 = 0° will be discussed in the Reynolds number range of 𝑅𝑒𝑑 = 800 − 2500, to analyze the 

effect of the cross-sectional geometry on the flow behavior relevant to the application of the PADS.  

29 

 
 
A sample set of the boundary layer mean and rms velocity profiles from MTV is depicted 

in  Figure  8.  The  boundary  layer  at  the  three  most  upstream  measurement  locations,  up  to  just 

downstream of the round corner, appear to be attached. While not clearly visible from Figure 8, at 

the next downstream position, the boundary layer separates. This is determined from the presence 

of  reversed  flow  near  the  wall.  The  cylinder  surface  glare  during  MTV  imaging,  plus  the 

geometries’ inherent curvature, does not allow accurate interrogation of a few (up to ten, depending 

on the specific location) pixels in the immediate vicinity of the wall. In addition, given the finite 

streamwise resolution of the measurements and the presence of a gap between the tagging lines, 

separation and reattachment locations can be better estimated from extrapolation of the separation 

zone boundary (to be discussed later), with an uncertainty equal to half the streamwise distance 

between the tagging lines which is about 4.5% of the baseline width (≈ 0.045𝑑).  

 In addition to the separation of the boundary layer, the velocity profile also develops an 

inflection  point,  where  the  cross-stream  velocity  gradient  is  highest  along  the  cross-stream 

direction. Above the reversed-flow region, this profile reflects the presence of a separated shear 

layer. From Figure 8-bottom, it can be seen that the unsteadiness grows in the shear layer as the 

rms velocity becomes more prominent with the downstream. The shear layer is characterized by 

comparing the spread of the region with the high rms velocity together with the strength of the 

𝑢𝑟𝑚𝑠  peaks,  and  will  be  discussed  later  in  this  chapter.  For  a  better  visual  presentation,  color 

contour maps of these velocity data are plotted and compared for different cylinder geometries in 

later figures.  

30 

 
Figure 8. Boundary layer mean (top) and rms (bottom) velocity profiles on the smooth cylinder 
at 𝑅𝑒𝑑 = 2500 . Flow is from left to right. Every other profile is shown to avoid clutter. 

Figure 9 shows the normalized mean (left) and rms (right) velocity color contour maps for 

the smooth cylinder. In this figure, the separated flow is tracked by marking the region with reverse 

flow (negative velocity) in cold colors (cyan to pink spectrum).  

With the current experimental resolution, the flow seems to remain fully attached at 𝑅𝑒𝑑 =

1100,  but  shows  a  minuscule  closed  separation  bubble  at  𝑅𝑒𝑑 = 800  and  1400,  with  the 

separation bubble’s thickness not exceeding 2% of the baseline width 𝑑. Magnified view of these 

contour  maps  is  depicted  in  Figure  10  to  better  show  the  separation  bubbles  near  the  surface. 

Increasing  the  Reynolds  number  to  𝑅𝑒𝑑 = 1800  is  associated  with  a  thin  but  open  separation 

bubble, with the separation point located downstream of the leading-edge round corner. Further 

increase in the Reynolds number causes the flow to reattach to the surface before separating again 

farther downstream of the trailing edge round corner into the wake. Therefore, a non-monotonic 

relationship between the separation/reattachment of the boundary layer and Reynolds number is 

observed for the smooth cylinder in the Reynolds number range of interest. Specifically, increasing 

the  Reynolds  number  first  develops  an  open  separation  bubble  then  leads  to  reattached  flow. 

31 

 
 
 
Overall,  the  flow  follows  the  contour  of  the  smooth  geometry  well  and  is  not  displaced 

significantly by the, generally thin, separation bubble from the cylinder’s surface. 

                                 𝑅𝑒𝑑 = 800 

                                 𝑅𝑒𝑑 = 1100 

                                 𝑅𝑒𝑑 = 1400 

                                 𝑅𝑒𝑑 = 1800 

                                 𝑅𝑒𝑑 = 2500 

Figure 9. Smooth cylinder. Mean (left) and rms (right) velocity color contour maps normalized 
by the freestream velocity 𝑈∞. Rows show different Reynolds numbers. Flow is from left to 
right. 

32 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
                                                                                                                                                   𝑅𝑒𝑑 = 800 

                                                                                                                                                   𝑅𝑒𝑑 = 1100 

                                                                                                                                                   𝑅𝑒𝑑 = 1400 

                                                                                                                                                   𝑅𝑒𝑑 = 1800 

                                                                                                                                                   𝑅𝑒𝑑 = 2500 

Figure 10. Smooth cylinder. Magnified view of the mean velocity color contour maps 
normalized by the freestream velocity 𝑈∞. The vertical scale is stretched, distorting the view, to 
make the separation zone visible. Rows show different Reynolds numbers. Flow is from left to 
right. 

As seen in Figure 9 (right), the boundary layer at 𝑅𝑒𝑑 = 800 − 1400 remains largely quiet with 

relatively low unsteadiness, with the maximum fluctuations being lower than 5% of the freestream 

velocity. Better details of the fluctuating velocity development in the streamwise direction may be 

observed  in  Figure  11.  In  this  figure,  the  maximum  rms  velocity  value  across  each  profile  is 

33 

 
 
 
 
 
 
 
 
 
 
 
extracted and plotted versus 𝑥/𝑑 at each Reynolds number. Figure 11 shows that fluctuations grow 

downstream for all Reynolds numbers. Excluding the regrowth of the fluctuation seen downstream 

of 𝑥/𝑑 ≈ 0.6  at  𝑅𝑒𝑑 = 2500, the unsteadiness  level  at 𝑅𝑒𝑑 = 1800 is  the highest  in the shear 

layer with a faster growth rate with the downstream distance than all other cases. This coincides 

with the fact that an open separation is observed at this Reynolds number only.  Figure 11 also 

shows  that  at  𝑅𝑒𝑑 = 800 − 1800,  the  𝑟𝑟𝑚𝑠,𝑝/𝑈∞ growth  is  monotonic.  At  𝑅𝑒𝑑 = 2500,  the 

maximum  𝑢𝑟𝑚𝑠  values  reach  a  peak  (“saturate”)  before  a  second  growth  stage  develops 

downstream  of  saturation.  The  separated  shear  layer  reattaches  to  the  surface  at  this  Reynolds 

number, and the first peak of the maximum 𝑢𝑟𝑚𝑠 values occurs slightly before the reattachment 

point. The second stage of growth is downstream of reattachment, where the shear layer from the 

reattached flow starts developing.  

Figure 11. Comparison between maximum 𝑢𝑟𝑚𝑠/𝑈∞ development along the surface of the 
smooth cylinder at different Reynolds numbers.   

A  similar  analysis  to  the  above  is  done  for  the  2Dp  cylinder  with  𝜀/𝑑 = 5%  using  the 

results given in Figure 12. The effect of this geometry on promoting flow separation is evident. 

The flow separates downstream of the topology peak at the front corner with the separation point 

34 

 
 
predominantly  insensitive  to  Reynolds  number.  The  separation  bubble  remains  open,  and  it 

probably reattaches only at the highest Reynolds number studied, 𝑅𝑒𝑑 = 2500. This trend with 

Reynolds number is expected, but given the current data resolution, the reattachment found at the 

highest Reynolds number is not certain. To be clear, Figure 13 displays a magnified view of the 

near-wall velocity profiles in the vicinity of the reattachment location. As seen from the figure, 

while a zero crossing is not observed in the velocity profile across the tagging line at x/d=0.96, the 

data point closest to the wall is about 5 pixels away from the surface, leaving a possibility of a 

zero-crossing closer to the wall. In either case, the separation zone boundary travels very close to 

the  surface  over  the  topology  peak  at  the  aft  corner  of  the  2Dp  cylinder  at  𝑅𝑒𝑑 = 2500.  Also 

noteworthy is that the inside of the topology valleys is filled with a “dead” cushion of fluid with 

very weak negative velocity, irrespective of the Reynolds number and separation bubble thickness. 

Considering 𝑢𝑟𝑚𝑠/𝑈∞ data, the separated shear layer at 𝑅𝑒𝑑 = 1100 shows a higher level 

of fluctuations compared to the lower and higher adjacent Reynolds numbers (Figure 12, right). 

Further  increase  of  the  Reynolds  number  to  𝑅𝑒𝑑 = 1800  exhibits  widening  over  a  short 

streamwise fetch of the region carrying high velocity fluctuations at around 𝑥/𝑑 = 0.4. Also, the 

highest level of fluctuations is observed at this Reynolds number. This widening of the high rms 

zone can be related to vortex formation from the shear layer roll-up, as will be seen from flow 

visualization results in Chapter 5 (Figure  65).  Increasing the Reynolds number to  𝑅𝑒𝑑 = 2500 

pushes the widening of the high rms region towards the upstream direction, and also brings this 

region closer to the surface. Therefore, it seems for this geometry at 𝑅𝑒𝑑 = 1800 and 2500, the 

separated shear layer  rolls  up and vortical  structures  start  developing above the model surface. 

This vortex formation, and moving closer to the surface at  𝑅𝑒𝑑 = 2500, is associated with the 

probable closure of the separation bubble noted from the mean velocity data.  

35 

 
                                 𝑅𝑒𝑑 = 800 

                                 𝑅𝑒𝑑 = 1100 

                                 𝑅𝑒𝑑 = 1400 

                                 𝑅𝑒𝑑 = 1800 

                                 𝑅𝑒𝑑 = 2500 

Figure 12. 2Dp-5% cylinder. Mean (left) and rms (right) velocity color contour maps normalized 
by the freestream velocity 𝑈∞. Rows show different Reynolds numbers. Flow is from left to 
right. 

Figure 13. Magnified view of the Mean velocity profiles over the topology peak at the aft corner 
of the 2Dp cylinder at 𝑅𝑒𝑑 = 2500. Red points mark the zero-crossing of the velocity profile.  

36 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 14. Comparison between maximum 𝑢𝑟𝑚𝑠/𝑈∞ development along the surface of the 2Dp 
cylinder at different Reynolds numbers.   

The maximum rms velocity value across each profile over the surface of the 2Dp cylinder 

is  Examined  in  Figure  14.  At  𝑅𝑒𝑑 = 800 − 1400  the  maximum  𝑢𝑟𝑚𝑠  values  increase 

monotonically  with  the  downstream.  This  monotonic  trend  is  disturbed  at  higher  Reynolds 

numbers,  when  vortex  formation  from  roll-up  of  the  shear  layer  is  observed  (see  Figure  65  in 

Chapter 5), and the fluctuation intensity reaches a peak. At 𝑅𝑒𝑑 = 2500, the peak in the maximum 

𝑢𝑟𝑚𝑠 values is observed at 𝑥/𝑑 = 0.9, right before the point of possible reattachment. A decaying 

trend follows afterward, reinforcing the expectation that the flow does reattach to the surface. It is 

observed in previous studies that the maximum fluctuations in the shear layer over airfoils, back-

step geometries, and bluff bodies plateau in the vicinity of the reattachment point and decay farther 

downstream; for example, see references [18-20]. 

The  separation  zone  boundary  for  the  2Dp  cylinder  geometry  at  different  Reynolds 

numbers  is  depicted  in  Figure  15.  This  boundary  is  obtained  by  identifying  the  zero-crossing 

location of each mean velocity profile at a given 𝑥 location to delineate the border surrounding the 

reverse-flow  region.  A  second-order  polynomial  fit  is  applied  to  these  locations  to  capture  the 

37 

 
 
shape of the separation zone boundary. This polynomial fits the data points very well for most 

cases.  However,  a  second-order  polynomial  cannot  fully  follow  a  complex  boundary  with,  for 

example,  both  concave  and  convex  curvatures,  such  as  seen  at  𝑅𝑒𝑑 = 2500  (see  the  transition 

between orange and light blue colors between  𝑥/𝑑 ≈ 0.25 and 1 in Figure 12 at this Reynolds 

number). Higher orders (up to sixth-order) polynomials are tested for this case, but did not show a 

considerable improvement over the second-order fit. Nonetheless, the second-order polynomial is 

sufficient to represent the separation zone boundary and therefore is used for the purpose of this 

study, unless otherwise noted.  

The  separation  zone  boundary  for  the  2Dp  cylinder,  shown  in  Figure  15,  supports  the 

earlier  observation  that  at  𝑅𝑒𝑑 = 2500  the  flow  reattaches  to  the  surface  just  upstream  of  the 

trailing edge. It also shows that the separation zone boundary is flatter and located farther from the 

surface  at  𝑅𝑒𝑑 = 1400  compared  to  𝑅𝑒𝑑 = 1100  and  1800,  which  shows  a  non-monotonic 

behavior. Figure 15 also shows more clearly that the separation point (the upstream end of the 

separation zone boundary) is similar for all Reynolds numbers. 

Figure 15. Comparison of the mean separation zone boundary for the 2Dp cylinder at different 
Reynolds numbers. Flow is from left to right. 

Similar to the 2Dp geometry, flow over the 2Dv cylinder, shown in Figure 16, separates 

right downstream of a topology peak. However, separation occurs farther downstream compared 

to the 2Dp case, and the separated flow remains much closer to the geometry surface such that as 

38 

 
 
it passes over the next topology bump, it almost reattaches to the surface. At 𝑅𝑒𝑑 = 800 − 1400, 

the flow reattaches to the surface upstream of a topology bump at 𝑥/𝑑 ≈ 0.55. Interestingly, this 

geometry shows low sensitivity to Reynolds numbers in the range of 𝑅𝑒𝑑 = 800 − 1400 in terms 

of the separation/reattachment behavior of the boundary layer. Increasing the Reynolds number 

further to 𝑅𝑒𝑑 = 1800 is associated with the opening up of the separation bubble, which thickens 

even further as the Reynolds number rises to 𝑅𝑒𝑑 = 2500. This is opposite to the 2Dp geometry 

behavior, but the smooth cylinder also developed an open separation bubble at 𝑅𝑒𝑑 = 1800. It is 

speculated that further increase in the Reynolds number would cause the separated flow to reattach 

to the 2Dv cylinder surface, similar to the other geometries. This can be a point of interest in future 

studies.   

39 

 
 
 
 
 
 
 
 
 
 
 
 
                                 𝑅𝑒𝑑 = 800 

                                 𝑅𝑒𝑑 = 1100 

                                 𝑅𝑒𝑑 = 1400 

                                 𝑅𝑒𝑑 = 1800 

                                 𝑅𝑒𝑑 = 2500 

Figure 16. 2Dv-5% cylinder. Mean (left) and rms (right) velocity color contour maps normalized 
by the freestream velocity 𝑈∞. Rows show different Reynolds numbers. Flow is from left to 
right. 

Figure 17 shows the mean separation zone boundary for the 2Dv cylinder. It can be noted 

that, similar to the 2Dp case, the separation point is similar over the studied range of Reynolds 

numbers. It is also evident how the separation zone boundary grows in lateral thickness at higher 

Reynolds numbers. However, the variation caused by the Reynolds number change is much more 

limited compared to the 2Dp geometry.  

40 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 17. Comparison of the mean separation zone boundary for the 2Dv cylinder at different 
Reynolds numbers. Flow is from left to right. 

Analysis of the maximum 𝑢𝑟𝑚𝑠 velocity for the 2Dv cylinder in Figure 18 shows that the 

fluctuation level in the shear layer increases monotonically with the downstream at all Reynolds 

numbers. The rate of this increase (i.e. the slope of the line), however, starts growing at  𝑅𝑒𝑑 =

1800 as the separation bubble opens up and the separation zone boundary starts thickening. The 

highest  fluctuation  level  is  observed  at  𝑅𝑒𝑑 = 2500  which  shows  the  thickest  separation  zone 

boundary.  

Figure 18. Comparison between maximum 𝑢𝑟𝑚𝑠/𝑈∞ development along the surface of the 2Dv 
cylinder at different Reynolds numbers.   

Overall, for the 2Dv cylinder, it appears that the separated flow merely fills the topology 

valleys and the effective body shape becomes close to that of the  smooth cylinder. Depicted in 

41 

 
 
 
Figure 19, the maximum 𝑢𝑟𝑚𝑠 values in the shear layer for the 2Dv cylinder are also very close to 

those from the smooth cylinder at 𝑅𝑒𝑑 = 800 − 1800, which suggests that the presence of a 2D 

topology with a valley at the leading edge does not alter the flow substantially from that over the 

baseline smooth geometry, especially when compared to the effect of the 2Dp geometry. It is at 

𝑅𝑒𝑑 = 2500 where the 𝑢𝑟𝑚𝑠 values deviate between the smooth and 2Dv cylinders, when the flow 

reattaches  to  the surface of the smooth  cylinder  but  remains separated over the 2Dv  cylinder’s 

surface. Moreover, the fluctuations are lower in the shear layer of the 2Dv cylinder than the 2Dp 

cylinder at all Reynolds numbers. This is consistent with the earlier observation that, excluding 

the  2Dp  cylinder  at  𝑅𝑒𝑑 = 2500,  the  maximum  𝑢𝑟𝑚𝑠,𝑝  level  shows  a  direct  relation  with  the 

thickness of the separation zone boundary. Figure 21 also shows that the fluctuations in the shear 

layer are amplified by the presence of the topology with a peak at the LE, compared to the baseline 

geometry.  

The domain of the shear layer containing the high rms fluctuations is highlighted for the 

smooth, 2Dp, and 2Dv cases in Figure 20. The loci of the maximum rms velocity are marked by 

circle markers, and the highlighted domain surrounds the rms fluctuations higher than 50% of the 

value of the 𝑢𝑟𝑚𝑠,𝑝 at each streamwise location. The color codes remain consistent with the plots 

shown so far, where the black, red, and green points and shaded regions are allotted to the smooth, 

2Dp, and 2Dv cases, respectively. 

42 

 
 
 
 
 
 
       𝑅𝑒𝑑 = 800 

       𝑅𝑒𝑑 = 1100 

       𝑅𝑒𝑑 = 1400 

       𝑅𝑒𝑑 = 1800 

       𝑅𝑒𝑑 = 2500 

Figure 19. Comparison between maximum 𝑢𝑟𝑚𝑠/𝑈∞ development along the surface of the 
smooth (black), 2Dp (red), and 2Dv (green) cylinders, at different Reynolds numbers.   

43 

 
 
 
 
 
 
 
 
  
                                      𝑅𝑒𝑑 = 800 

                                      𝑅𝑒𝑑 = 1100 

                                      𝑅𝑒𝑑 = 1400 

                                      𝑅𝑒𝑑 = 1800 

                                      𝑅𝑒𝑑 = 2500 

Figure 20. Domain of high rms velocity fluctuations, covering the region with rms velocity 
higher than 50% of the maximum rms velocity at each 𝑥/𝑑 location. Results for the smooth, 
2Dp, and 2Dv cases shown in black, red, and green respectively. Only the cross-section of the 
baseline geometry is shown for simplicity. Flow is from left to right. 

This  domain  is  thickest  for  the  2Dp  geometry  at  all  Reynolds  numbers,  but  remains 

comparable  between  the  smooth  and  the  2Dv  cases.  In  addition,  for  the  two  cases  of  the  2Dp 

cylinder at 𝑅𝑒𝑑 = 1800 and 2500, vortex formation from shear layer roll-up is observed on top 

44 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
of the cylinder surface which is associated with the rapid widening of the high rms domain (see 

Figure 65 in Chapter 5). 

3.2 Effect of the Surface Topology Amplitude 

Mean and rms velocity color contour maps for the 2Dp cylinder with lower (𝜀/𝑑 = 2.5%) 

and higher (𝜀/𝑑 = 10%) topology amplitudes compared to the cases considered so far are shown 

in Figure 21 and Figure 22, respectively, to examine the effect of the topology amplitude. From 

Figure 21 it can be seen that, similar to the 5% topology amplitude, the presence of a topology 

peak at the upstream corner of the cylinder causes an early separation of the flow, downstream of 

a  topology  peak  at  𝑥/𝑑 ≈ −0.78,  even  at  such  small  topology  amplitude.  In  addition  to  this 

similarity with the 2Dp-5% results, previously shown in Figure 12, the separation bubble remains 

open up to 𝑅𝑒𝑑 = 1800, and becomes very close to reattachment at 𝑅𝑒𝑑 = 2500. However, as 

before,  reattachment  cannot  be  ascertained  with  the  current  data  resolution.  Non-monotonic 

behavior in the separation zone boundary with Reynolds number is also seen for this case, where 

first, increasing Reynolds number to 𝑅𝑒𝑑 = 1100 makes the separation bubble thinner, but further 

increase up to 𝑅𝑒𝑑 = 1800 is associated with thickening of the separation bubble.  

45 

 
 
 
 
 
 
 
 
 
 
 
                                 𝑅𝑒𝑑 = 800 

                                 𝑅𝑒𝑑 = 1100 

                                 𝑅𝑒𝑑 = 1400 

                                 𝑅𝑒𝑑 = 1800 

                                 𝑅𝑒𝑑 = 2500 

Figure 21. 2Dp-2.5% cylinder. Mean (left) and rms (right) velocity color contour maps 
normalized by the freestream velocity 𝑈∞. Rows show different Reynolds numbers. Flow is from 
left to right. 

At the Reynolds number range  𝑅𝑒𝑑 = 800 − 1800, where open separation is observed, 

shear layer unsteadiness grows in strength downstream, similar to the case of 2Dp-5%. However, 

for the current  case, the  rapid expansion of the high unsteadiness  region,  which is  found to  be 

coincident with the shear layer roll-up, is not seen at 𝑅𝑒𝑑 = 1800 and is delayed to the higher 

Reynolds number of 2500. The shear layer is seen to roll up to form vortical structures at 𝑅𝑒𝑑 =

46 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2500, which is believed to be the reason for the separation bubble approaching the surface and the 

possibly reattaching. 

For  the  higher  amplitude  of  𝜀/𝑑 = 10%,  shown  in  Figure  22,  𝑅𝑒 = 2500  remains  the 

turning point for boundary layer separation/reattachment. As before for the same 2Dp geometry, 

below this Reynolds number, the separation bubble remains open.   

                                 𝑅𝑒𝑑 = 800 

                                 𝑅𝑒𝑑 = 1100 

                                 𝑅𝑒𝑑 = 1400 

                                 𝑅𝑒𝑑 = 1800 

                                 𝑅𝑒𝑑 = 2500 

Figure 22. 2Dp-10% cylinder. Mean (left) and rms (right) velocity color contour maps 
normalized by the freestream velocity 𝑈∞. Rows show different Reynolds numbers. Flow is from 
left to right. 

47 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Similar to the other amplitudes of the 2Dp geometry, for the 10% amplitude, there 𝑚𝑖𝑔ℎ𝑡 

be a late reattachment of the boundary layer at 𝑅𝑒𝑑 = 2500 that cannot be ascertained with the 

given resolution of the current data. The separation also occurs downstream of the topology peak 

at  the  front  corner  of  the  cylinder,  with  the  separation  point  showing  negligible  sensitivity  to 

Reynolds number. The separation bubble’s size does not vary significantly at 𝑅𝑒𝑑 = 800 − 1400, 

but starts thinning down at higher Reynolds numbers. Unsteadiness in the shear layer grows with 

downstream distance at 𝑅𝑒𝑑 = 800 − 1400 in the same manner, as other 2Dp geometries, before 

the signature of the vortex roll-up in widening the high 𝑢𝑟𝑚𝑠 zone starts emerging at 𝑅𝑒𝑑 = 1800 

and develop further at 𝑅𝑒𝑑 = 2500.  

The maximum rms velocity evolution in the streamwise direction is compared for all the 

2Dp geometries in Figure 23. In 𝑅𝑒𝑑 = 800 − 1400 range, the maximum rms velocity increases 

monotonically  for  all  2Dp  geometries.  The  results  for  the  5%  and  10%  amplitude  cases  match 

within the error bars at 𝑅𝑒𝑑 = 800 and 1400. The 2.5% amplitude case shows the highest 𝑢𝑟𝑚𝑠 

peak values in this Reynolds number range, with the difference compared to the other 2Dp cases 

becoming more significant as the Reynolds number increases. The results for all 2Dp cases are 

very close up to the mid-chord of the cylinders at 𝑅𝑒𝑑 = 1800, but farther downstream and at 0 <

𝑥/𝑑 < 1, the 10% amplitude case shows higher values than the other two cases. Downstream of 

𝑥/𝑑 = 1, the results for all 2Dp cases plateau, which coincides with  the  rapid widening of the 

domain of high velocity fluctuations for the 5% and 10% amplitude cases. However, such rapid 

widening of the high velocity fluctuations domain is interestingly not seen for the 2.5% amplitude 

case at this Reynolds number. At 𝑅𝑒𝑑 = 2500, the results for all 2Dp cases plateau over the surface 

and  decrease  with  further  increase  in  𝑥/𝑑,  which  happens  earlier  for  the  2.5%  amplitude  case 

compared to the other 2Dp cylinders. Rapid widening of the high velocity fluctuations domain is 

48 

 
seen and shear layer reattachment is expected for all 2Dp cylinders at this Reynolds number. It 

should be noted that the results for the 2.5% amplitude case are truncated at 𝑥/𝑑 = 0.75, since the 

𝑢𝑟𝑚𝑠,𝑝  results  are  extracted  at  the  location  of  the  first  rms  velocity  peak  developing  over  the 

surface.  At  this  streamwise  location,  the  peaks  do  not  exist  over  the  trajectory  of  the  first  rms 

velocity peak anymore, and the rms velocity peak starting to develop near the surface at 𝑥/𝑑 ≈

0.2 becomes the only prominent peak in the rms velocity profiles.  

49 

 
 
 
 
 
 
 
 
 
 
 
 
 
       𝑅𝑒𝑑 = 800 

       𝑅𝑒𝑑 = 1100 

       𝑅𝑒𝑑 = 1400 

       𝑅𝑒𝑑 = 1800 

       𝑅𝑒𝑑 = 2500 

Figure 23. Comparison between maximum 𝑢𝑟𝑚𝑠/𝑈∞ values along the surface of cylinders with 
2Dp geometry and different topology amplitudes at different Reynolds numbers. 

The effect of topology amplitude on the flow over the 2Dv geometry is analyzed in Figure 

24 and Figure 26. In Figure 24, mean and rms velocity color contour maps are shown for the 2Dv-

2.5% case. This geometry shows the least sensitivity to Reynolds number among the other studied 

50 

 
 
 
 
 
 
 
geometries.  There  are  two  isolated  closed  separation  bubbles  in  the  valleys  of  the  topology  at 

𝑅𝑒𝑑 = 800 − 1400.  At  higher  Reynolds  numbers,  these  two  bubbles  connect  to  form  a  single 

separation bubble. These details can be better seen in Figure 25, where a magnified view of the 

mean velocity profiles over the valleys on the surface of this cylinder are shown and the points of 

zero-crossing velocity are marked. Nevertheless, the flow separation does not considerably affect 

the overall flow over the surface, and the separation  bubbles act only as a “cushion” to fill the 

topology dips. The minimal effect of this geometry on the flow is further seen by monitoring the 

small unsteadiness levels in the shear layer throughout the studied range of Reynolds numbers. 

Moreover, the shear layer remains very close to the surface.  

51 

 
 
 
 
 
 
 
 
 
 
 
 
 
                                 𝑅𝑒𝑑 = 800 

                                 𝑅𝑒𝑑 = 1100 

                                 𝑅𝑒𝑑 = 1400 

                                 𝑅𝑒𝑑 = 1800 

                                 𝑅𝑒𝑑 = 2500 

Figure 24. 2Dv-2.5% cylinder. Mean (left) and rms (right) velocity color contour maps 
normalized by the freestream velocity 𝑈∞. Rows show different Reynolds numbers. Flow is from 
left to right. 

52 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
                                                                                                                                           𝑅𝑒𝑑 = 800 

                                                                                                                                           𝑅𝑒𝑑 = 1100 

                                                                                                                                           𝑅𝑒𝑑 = 1400 

                                                                                                                                           𝑅𝑒𝑑 = 1800 

                                                                                                                                           𝑅𝑒𝑑 = 2500 

Figure 25. Magnified view of the Mean velocity profiles over the valleys on the surface of the 
2Dv cylinder at different Reynolds numbers. Red points mark the zero-crossing of the velocity 
profile. 

The case with a higher amplitude of 𝜀/𝑑 = 10% is displayed in Figure 26. This is the only 

case where changing the topology amplitude is seen to affect the separation point. Here, separation 

occurs earlier than the 2Dv cylinders with 2.5% and 5% topology amplitudes, right downstream 

of  the  topology  peak  at  the  front  corner.  Nonetheless,  flow  reattaches  quickly,  and  the  closed 

separation bubble remains local within the first valley before the flow separates again downstream 

53 

 
                                                            
     
 
     
 
     
 
     
 
     
of  the  next  topology  peak.  The  latter  separation  point  is  comparable  to  that  seen  for  the  other 

amplitudes. Another effect of this higher topology amplitude is that, downstream of the second 

separation  point,  the  flow  remains  separated  over  the  remainder  of  the  surface  at  all  Reynolds 

numbers.  

                                 𝑅𝑒𝑑 = 800 

                                 𝑅𝑒𝑑 = 1100 

                                 𝑅𝑒𝑑 = 1400 

                                 𝑅𝑒𝑑 = 1800 

                                 𝑅𝑒𝑑 = 2500 

Figure 26. 2Dv-10% cylinder. Mean (left) and rms (right) velocity color contour maps 
normalized by the freestream velocity 𝑈∞. Rows show different Reynolds numbers. Flow is from 
left to right. 

From the rms velocity color contour maps in the right column of Figure 26, it can be seen 

that the unsteadiness grows in the shear layer with the downstream and Reynolds number. A rapid 

54 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
widening  of  the  high  rms  region  is  not  observed  for  any  2Dv  case  at  any  Reynolds  number, 

suggesting that the shear layer roll-up does not occur above the cylinder surface for this geometry 

in the studied range of Reynolds numbers. This observation correlates with the shear layer being 

very close to the surface.  

The maximum rms velocity at each streamwise location is extracted and compared for all 

the 2Dv geometries in addition to the smooth cylinder in Figure 27. The first streamwise location 

for this analysis is where a noticeable peak in the rms velocity profile is observed due to the shear 

layer  development.  In  the  Reynolds  number  range  of  𝑅𝑒𝑑 = 800 − 1800,  the  growth  in  the 

maximum rms velocity is monotonic. The 2.5% amplitude at 𝑅𝑒𝑑 = 2500 is the only case where 

the rms values reach a peak at around  𝑥/𝑑 = 0.25 before increasing again farther downstream. 

Overall, the lowest and highest topology amplitudes exhibit the lowest and the highest levels of 

shear layer velocity fluctuations, respectively. The 5% topology shows a close behavior to that of 

the 2.5% geometry in the range of 𝑅𝑒𝑑 = 800 − 1400. In this Reynolds number range, both of 

these  cylinders  show  a  closed  separation  bubble  on  the  cylinders’  surface.  At  𝑅𝑒𝑑 = 1800, 

reattachment does not occur for the 5% geometry anymore, which is similar to the case of 10% 

but opposite to the geometry with 2.5% amplitude. This is associated with a deviation in the level 

of fluctuations in the shear layer of the 5% geometry from that of the 2.5% geometry and shifts 

the 5% amplitude results towards the level of fluctuations seen in the 10% geometry, and they 

eventually collapse at 𝑅𝑒𝑑 = 2500. At the Reynolds number range 𝑅𝑒𝑑 = 800 − 1400 the values 

seem to approach those from the smooth cylinder with decreasing the topology amplitude. The 

smooth cylinder shows comparable 𝑢𝑟𝑚𝑠,𝑝 values to those from the 2Dv cylinders with 5% and 

10% amplitude at 𝑅𝑒𝑑 = 1800, but higher compared to the 2Dv-2.5% case. It was observed that 

the former cases show an open separation, whereas the latter case shows a thin closed separation 

55 

 
bubble over the surface. At 𝑅𝑒𝑑 = 2500, the results are very close for the smooth and the 2Dv-

2.5% cylinder, especially in that both show a non-monotonic behavior. These two cases, unlike 

the 2Dv-5% and 10% cases, show a closed separation bubble over the surface.  

         𝑅𝑒𝑑 = 800 

          𝑅𝑒𝑑 = 1100 

          𝑅𝑒𝑑 = 1400 

          𝑅𝑒𝑑 = 1800 

           𝑅𝑒𝑑 = 2500 

Figure 27. Comparison between maximum 𝑢𝑟𝑚𝑠/𝑈∞ values along the surface of cylinders with 
2Dv geometry and different topology amplitudes at different Reynolds numbers. Results are also 
compared against the smooth cylinder.  

56 

 
 
 
 
 
 
 
3.3 Effect of Surface Topology Variation Along the Span 

The effect of adding surface topology variation along the span on the velocity field over 

the cross-sectional geometries with a peak at the leading edge is examined using Figure 28. From 

the  mean  velocity  color  contour  maps  in  the  left  column  of  this  figure,  it  can  be  seen  that  the 

presence of three-dimensionality of the topology significantly alters the separation/reattachment 

behavior relative to the flow over the surface of the cylinders with 2Dp cross-sectional geometry. 

Overall,  the  separation  bubble  remains  very  thin  for  the  3Dp  case  and  predominantly  fills  the 

valleys of the topology over the surface; although the bubble gains lateral thickness with increasing 

the Reynolds number. 

Further  information  regarding  the  mean  boundary  layer  behavior  may  be  extracted  by 

characterizing the separation zone, as done in Figure 29. This helps estimate the separation and 

the  reattachment  location  and  characterize  the  overall  shape  of  the  separation  zone.  Figure  29 

compares  results from  the 2Dp and 3Dp configurations. The separation point for the 3Dp  case 

remains the same as that observed for the 2Dp cylinder. More specifically, the flow separates after 

the leading edge, downstream of a surface topology peak at the front corner of the cylinder. 

57 

 
 
 
 
 
 
 
                                 𝑅𝑒𝑑 = 800 

                                 𝑅𝑒𝑑 = 1100 

                                 𝑅𝑒𝑑 = 1400 

                                 𝑅𝑒𝑑 = 1800 

                                 𝑅𝑒𝑑 = 2500 

Figure 28. 3Dp-5% cylinder. Mean (left) and rms (right) velocity color contour maps normalized 
by the freestream velocity 𝑈∞. Rows show different Reynolds numbers. Flow is from left to 
right. 

The separation bubble for the 3Dp geometry remains open over the surface at 𝑅𝑒𝑑 = 800, 

although it is located very close to the surface and becomes extremely close to reattaching at the 

topology peaks. As the Reynolds number increases to 𝑅𝑒𝑑 = 1100 and 1400, the flow reattaches 

to  the  surface  upstream  of  a  topology  peak  and  ahead  of  the  trailing  edge.  Further  increase  of 

Reynolds number to 𝑅𝑒𝑑 = 1800 is seen to again move the separation bubble towards opening, 

although  it  remains  thin  and  very  close  to  the  surface.  The  overall  behavior  at  this  Reynolds 

58 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
number  range  of  𝑅𝑒𝑑 = 800 − 1800  is  more  comparable  to  that  seen  from  the  2Dv  case  and 

deviates  visibly  from  that  of  the  2Dp  cylinder  (as  seen  from  Figure  29).  The  separated  flow 

reattaches to the surface at 𝑅𝑒𝑑 = 2500 slightly after the mid-chord of the cylinder, upstream of 

a topology peak. The flow does not separate again until the topology peak at the aft corner of the 

cylinder. This is a considerable alteration of the reattachment behavior of the separated flow in 

comparison to the 2Dp geometry. More specifically, the reattachment point advances towards the 

leading edge compared to the 2Dp counterpart, the separation zone of which is shown in red in 

Figure  29. Overall, the  presence of topology three-dimensionality makes the separation bubble 

thinner and more likely to reattach at all Reynolds numbers.  

From the rms velocity shown in the right column of Figure 28, it can be seen that, except 

at 𝑅𝑒𝑑 = 2500, the unsteadiness levels keep growing with downstream distance. Also, the region 

with high rms velocity values remains close to the surface. The trend of fluctuations intensity does 

not show a monotonic behavior with respect to Reynolds number, where the shear layer remains 

fairly quiet at  𝑅𝑒𝑑 = 800, gains fluctuation intensity at higher Reynolds numbers up to  𝑅𝑒𝑑 =

1400, and again loses fluctuation intensity as the Reynolds number increases to 𝑅𝑒𝑑 = 2500. To 

further characterize the shear layer, the domain of high rms fluctuations is depicted in Figure 30. 

Comparing  the  3Dp  case  with  its  2D  counterpart,  it  can  be  seen  that  the  presence  of  three-

dimensionality in the topology pulls this domain closer to the surface and keeps it laterally more 

compact. 

59 

 
 
 
 
                                        𝑅𝑒𝑑 = 800 

                                        𝑅𝑒𝑑 = 1100 

                                        𝑅𝑒𝑑 = 1400 

                                        𝑅𝑒𝑑 = 1800 

                                        𝑅𝑒𝑑 = 2500 

Figure 29. Comparison of the mean separation zone boundary for the 2Dp and 3Dp cylinders, 
shown in red and blue lines, respectively. Rows show different Reynolds numbers. Flow is from 
left to right.   

60 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
                                        𝑅𝑒𝑑 = 800 

                                        𝑅𝑒𝑑 = 1100 

                                        𝑅𝑒𝑑 = 1400 

                                        𝑅𝑒𝑑 = 1800 

                                        𝑅𝑒𝑑 = 2500 

Figure 30. Domain of high rms velocity fluctuations, covering the region with rms velocity 
higher than 50% of the maximum rms velocity at each 𝑥/𝑑 location. 2Dp and 3Dp cases shown 
in red and blue, respectively.  Flow is from left to right. 

The spreading behavior of the high  𝑢𝑟𝑚𝑠 zone remains close between the 2Dp and 3Dp 

cylinders at the lower Reynolds number range of 𝑅𝑒𝑑 = 800 − 1400, where both cylinders show 

61 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
a trend consistent with the gradual growth of mixing layers. It is only at 𝑅𝑒𝑑 = 2500 where the 

3Dp geometry shows a rapid expansion of the highly unsteady domain around 𝑥/𝑑 = 0.4, where 

the fluctuations penetrate a topology valley, rather than merely sweeping above the peaks. This 

observation suggests the development of vortical structures downstream of 𝑥/𝑑 = 0.4, which is 

farther downstream compared to the 2D counterpart.  

The maximum rms velocity value across each profile is extracted and plotted versus 𝑥/𝑑 

at each Reynolds number, as seen in Figure 31. Unlike in the 2Dp case where 𝑢𝑟𝑚𝑠,𝑝 growth rate 

does not change much with 𝑥, the 3Dp case shows significant dependence with the growth rate 

generally  decaying  with  𝑥.  The  fluctuations  in  the  shear  layer  increase  more  rapidly  over  the 

surface of the 3Dp case up to about the mid-chord compared to the 2Dp cylinder. However, farther 

downstream, the 2D counterpart shows a faster unsteadiness growth. At some Reynolds numbers 

(i.e. 𝑅𝑒𝑑 = 1100, 1800, 2500) the maximum 𝑢𝑟𝑚𝑠 values for the 2Dp case surpass those of the 

3Dp  case  after  the  mid-chord.  At  𝑅𝑒𝑑 = 800  and  1,400,  the  unsteadiness  level  is  dominantly 

higher for the 2Dp and 3Dp, respectively. One interesting case is the 3Dp at 𝑅𝑒𝑑 = 2500, where 

the fluctuations even drop after the mid-chord and reach a minimum at 𝑥/𝑑 ≈ 0.9. For this case, 

the decreasing trend starts very close to where the flow reattaches to the surface and lingers in the 

reattached flow up to the whereabouts of the next separation point at the topology peak at the aft 

corner of the cylinder.  

62 

 
       𝑅𝑒𝑑 = 800 

       𝑅𝑒𝑑 = 1100 

       𝑅𝑒𝑑 = 1400 

       𝑅𝑒𝑑 = 1800 

       𝑅𝑒𝑑 = 2500 

Figure 31. Comparison between maximum 𝑢𝑟𝑚𝑠/𝑈∞ values along the surface of the 2Dp (red), 
and 3Dp (blue) cylinders, at different Reynolds numbers. 

The effect of adding surface topology variation along the span on the velocity field over 

the cross-sectional geometries with a valley at the leading edge is examined using Figure 32. From 

the mean velocity color contour maps on the left of this figure, it can be seen that the presence of 

63 

 
 
 
 
 
 
 
the surface topology variation  along the span has a small  effect  on the separation/reattachment 

behavior of the flow over the surface compared to the counterpart 2Dv cylinder.  

                                 𝑅𝑒𝑑 = 800 

                                 𝑅𝑒𝑑 = 1100 

                                 𝑅𝑒𝑑=1400 

                                 𝑅𝑒𝑑=1800 

                                 𝑅𝑒𝑑=2500 

Figure 32. 3Dv-5% cylinder. Mean (left) and rms (right) velocity color contour maps normalized 
by the freestream velocity 𝑈∞. Rows show different Reynolds numbers. Flow is from left to 
right. 

Overall, the separated flow remains fairly close to the surface, predominantly filling the 

topology  valleys  and  becoming  close  to  reattachment  at  the  topology  peaks.  The  reverse  flow 

region seems to gain lateral thickness downstream, which is the opposite of the behavior observed 

in  Figure  28  for  the  other  cross-section  of  this  3D  cylinder  with  a  peak  at  the  leading  edge. 

64 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Furthermore,  the  reverse  flow  region  seems  to  be  quite  insensitive  to  Reynolds  number  in  the 

studied range. 

The separation zone boundary for the 3Dv  case is characterized in  Figure  33 to  extract 

further  information  regarding  the  mean  boundary  layer  behavior.  The  separation  point  remains 

close to that of the 2D counterpart, located downstream of a topology peak at 𝑥/𝑑 ≈ −0.5 and not 

affected  by  Reynolds  number.  Flow  does  not  reattach  at  any  Reynolds  number,  although  the 

separation zone boundary remains close to the surface and becomes very close to the topology 

peaks. Increasing the Reynolds number increases the curvature of the mean separation boundary 

and  brings  it  closer  to  the  surface  farther  downstream.  Overall,  the  presence  of  the  three-

dimensionality for this cross-sectional geometry seems to make the reattachment more difficult 

and  the  separation  bubble  laterally  thicker  at  𝑅𝑒𝑑 = 800 − 1800.  It  is  only  at  a  high  enough 

Reynolds  number  𝑅𝑒𝑑 = 2500  that  the  separation  zone  boundary  becomes  more  curved  and 

approaches the surface sooner for the 3Dv compared to 2Dv. The second-order polynomial fit does 

not accurately represent the separation zone boundary for 3Dv at 𝑅𝑒𝑑 = 2500. Although the fit 

suggests an open separation for this case, the color map gives signs of possible reattachment to the 

surface upstream of the topology peak at the aft corner of the cylinder.  

65 

 
 
 
 
 
 
                                        𝑅𝑒𝑑 = 800 

                                        𝑅𝑒𝑑 = 1100 

                                        𝑅𝑒𝑑 = 1400 

                                        𝑅𝑒𝑑 = 1800 

                                        𝑅𝑒𝑑 = 2500 

Figure 33. Comparison of the mean separation zone boundary for the 2Dv and 3Dv cylinders, 
shown in green and brown lines, respectively. Rows show different Reynolds numbers. Flow is 
from left to right. 

Investigating the rms velocity shown in the right column of Figure 32, it can be seen that 

the overall unsteadiness level keeps growing with the downstream distance. Similar to most of the 

cases studied so far, the unsteadiness level does not show a monotonic behavior with Reynolds 

number. What is consistent is that the rms velocity values remain reasonably low over most of the 

surface and the high unsteadiness intensities show up only late above the downstream half of the 

surface. One peculiar and interesting observation is that at 𝑅𝑒𝑑 = 1400, the unsteadiness level for 

66 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
the 3Dv case is very low and limited to a narrow band, while the other cross-section of this cylinder 

with a peak at the leading edge (3Dp) shows its highest fluctuation rates at this Reynolds number.  

To  further  characterize  the  shear  layer  for  the  3Dv  cylinder,  the  domain  of  high  rms 

fluctuations  is  depicted  in  Figure  34,  where  it  is  compared  with  the  2D  counterpart.  At  lower 

Reynolds  numbers  of  𝑅𝑒𝑑 = 800 − 1400,  this  domain  remains  thinner  but  farther  from  the 

surface for the 3Dv case. As the Reynolds number increases, the domain for the 3Dv case grows 

laterally to become thicker than that of the 2D counterpart at 𝑅𝑒𝑑 = 2500, where it also moves 

closer to the surface than the domain for 2Dv case. At this Reynolds number, a rapid expansion of 

the domain is seen at around 𝑥/𝑑 = 0.65, downstream of a topology peak, which is believed to be 

an indication of the spanwise vortical structures’ formation and development.  

The maximum rms velocity value across each profile is extracted and plotted versus 𝑥/𝑑 

at each Reynolds number, as seen in Figure 35. Throughout the studied Reynolds number range, 

the maximum rms velocity values are significantly higher above the 3Dv surface compared against 

the 2D counterpart. The 𝑢𝑟𝑚𝑠,𝑝 growth rate does not change much with 𝑥 for the geometries with 

a  valley  at  the  leading  edge,  apart  from  𝑅𝑒𝑑 = 1400  and  2500  where  some  “saturation”  is 

observed in the growth. 

67 

 
 
 
 
 
 
 
 
 
                                        𝑅𝑒𝑑 = 800 

                                        𝑅𝑒𝑑 = 1100 

                                        𝑅𝑒𝑑 = 1400 

                                        𝑅𝑒𝑑 = 1800 

                                        𝑅𝑒𝑑 = 2500 

Figure 34. Domain of high rms velocity fluctuations, covering the region with rms velocity larger 
than 50% of the maximum rms velocity at each 𝑥/𝑑 location. 2Dv and 3Dv cases shown in green 
and brown, respectively. Flow is from left to right. 

68 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
       𝑅𝑒𝑑 = 800 

       𝑅𝑒𝑑 = 1100 

       𝑅𝑒𝑑 = 1400 

       𝑅𝑒𝑑 = 1800 

       𝑅𝑒𝑑 = 2500 

Figure 35. Comparison between maximum 𝑢𝑟𝑚𝑠/𝑈∞ values along the surface of the 2Dv (green) 
and 3Dv (brown) cylinders, at different Reynolds numbers. 

69 

 
  
 
 
 
 
 
 
3.4 On Galloping 

When studying galloping oscillations of the rectangular cylinders in the conditions valid 

for quasi-steady assumption, the classic understanding of the phenomenon is that, as the angle of 

attack increases, the separated shear layer on the bottom curves more towards the surface, hence 

generating  higher  suction  than  that  from  the  top  shear  layer  (see  Figure  36).  This  increasing 

pressure difference between the top and bottom lateral surfaces produces an increasing downward 

(positive, per axis definition in Figure 1) force 𝐹𝑦 that causes the cylinder to become unstable to 

galloping.  

Figure 36. Classic understanding of the galloping phenomenon, connecting the boundary layer 
separation behavior to pressure generation on the side surfaces and, hence, transverse galloping 
force (sketch is done following Naudascher & Rockwell [21]). 

The bottom shear layer reattaches to the surface at a high enough angle of attack, resulting 

in increasing suction on the bottom and reduction in 𝐹𝑦 with further increase of the angle of attack, 

70 

 
 
flipping the instability-promoting trend of 𝐹𝑦 with 𝛼. This reversed trend is related to the upstream 

movement of the reattachment point on the bottom surface with increasing angle of attack post 

reattachment of the shear layer. In other words, the literature indicates that a rectangular cylinder 

is  unstable  to  galloping  when  the  shear  layer  on  its  bottom  is  separated  and  stable  when  it 

reattaches. Given Equation 1 (see Chapter 1) 𝐹𝐿 ≈ −𝐹𝑦 at small angles of attack and the galloping 

behavior  can  be  predicted  using  the  lift  coefficient,  when  the  quasi-steady  assumption  holds. 

Therefore,  the  galloping  behavior  of  the  bluff  bodies  can  be  contrasted  against  that  of  the 

streamline  objects,  such  as  a  symmetric  airfoil,  by  considering  the  𝐶𝐿 − 𝛼  curve.  It  is  well 

stablished that for such bodies, the lift coefficient increases with increasing the angle of attack up 

to the critical angle of the stall. Up to stall, the flow on top of the airfoil either follows the surface 

closely  (at  high  Reynolds  number)  or  a  closed  separation  bubble  is  formed  (at  low  Reynolds 

number) and an attached flow is present on the bottom surface. Therefore, there is a net suction on 

the top and positive pressure on the bottom, resulting in a positive slope of the 𝐶𝐿 − 𝛼 curve up to 

stall. Hence, the transverse galloping phenomenon is primarily a bluff, rather than a streamline 

body problem.  

New studies [7-9], however, suggest that, at sufficiently low Reynolds number, there are 

factors  other  than  separation/reattachment  on  the  bottom  surface  that  contribute  to  galloping 

instability. Surface pressure measurements from Feero et. al [9] on a sharp-cornered cylinder with 

𝑐/𝑑 = 3 at 𝑅𝑒𝑑 = 1100 show that despite the reattachment of the shear layer to the bottom surface 

at 𝛼 = 0°, the cylinder is unstable to galloping. They relate this observation, which is contradictory 

to the classic understanding described above, to two factors. First, they find that the separation-

zone pressure on the top surface rises with increasing the angle of attack to cause a net overall 

increase in the average pressure on the surface, promoting an increase in  𝐹𝑦 with 𝛼 (which is a 

71 

 
destabilizing effect). This net pressure increase with AoA is opposite to that observed at higher 

Reynolds numbers of 𝑅𝑒𝑑 = 5000 and 10000. The second factor contributing to the unexpected 

instability to galloping of the cylinder, is the observation that unlike at 𝑅𝑒𝑑 = 5000 and 10000, 

at  𝑅𝑒𝑑 = 1100,  the  reattachment  point  on  the  bottom  surface  does  not  move  upstream  with 

increasing the angle of attack. Based on the discussion in the previous paragraph, the reattachment 

point not shifting upstream suggests that the pressure on the bottom surface does not increase with 

angle  of  attack,  and  therefore  there  is  no  mechanism  for  𝐹𝑦  to  decrease  with  𝛼  to  stabilize  the 

geometry.  Although  surface  pressure  data  are  not  available  for  the  current  cases  of  study,  this 

chapter  is  focused  on  analyzing  the  galloping  behavior  of  the  cylinders  based  on  the  links 

understood  in  literature  between  the  shear  layer  separation/reattachment  change  with  AoA  and 

galloping instability.  

The separation zone boundary of the smooth cylinder at three Reynolds numbers, 𝑅𝑒𝑑 =

800, 1400, and 2500, and five different angles of attack is shown in Figure 37. At 𝛼 = 0°, there 

is a very thin closed-separation bubble at all the Reynolds numbers. At higher angles of attack, 

separation  occurs  on  the  top  side  of  the  cylinder  at  all  Reynolds  numbers  downstream  of  the 

leading-edge round-corner. The separated shear layer  at  𝑅𝑒𝑑 = 800 stays reattached to  the top 

surface as the angle of attack is increased to 𝛼 = 1°, but becomes open at higher angles of attack. 

As the Reynolds number increases to 𝑅𝑒𝑑 = 1400, the separation bubble remains closed up to a 

lower angle 𝛼 = 0.5°. Further increase in the Reynolds number shows a closed separation bubble 

on the top surface only at 𝛼 = 0°. Therefore, it is observed that increasing the Reynolds number 

in  this  range  makes  the  separation  bubble  on  the  top  surface  of  the  smooth  cylinder  prone  to 

becoming open at a lower angle of attack.  

72 

 
On the bottom side of the cylinder, the flow remains attached at 𝑅𝑒𝑑 = 800 and 1400 as 

𝛼 increases, but shows a very thin closed separation bubble at  𝑅𝑒𝑑 = 2500 which shrinks as 𝛼 

increases. This behavior of the bottom shear layer should lead to an increase of the pressure on the 

bottom side with increasing AoA. Thus, the bottom surface is expected to contribute to increasing 

lift with AoA. A similar contribution to lift is possible from the top surface prior to opening of the 

separation  bubble,  with  the  latter  reversing  the  trend.  Moreover,  given  the  tendency  of  the 

separation bubble on the top  side of the cylinder to  become open  with  increasing  𝑅𝑒𝑑, the top 

surface contribution to positive lift is anticipated to decrease with 𝑅𝑒𝑑. 

𝛼 = 0° 

𝛼 = 0.5° 

Figure 37. Comparison of the mean separation zone boundary for the smooth cylinder at 
different angles of attack (different rows) and Reynolds numbers. 

73 

 
 
 
 
Figure 37 (cont’d) 

𝛼 = 1° 

𝛼 = 1.5° 

𝛼 = 2° 

74 

 
 
 
 
 
 
 
 
 
Noting  that  𝐶𝐿 ≈ −𝐶𝑦  for  small  𝛼,  the  above  inferences  are  consistent  with  the  𝐶𝑦 − 𝛼 

curves of the same cylinder geometry from Feero et al. [7, 8] (see Figure 2-left). Specifically, all 

curves exhibit a decrease of 𝐶𝑦 with angle of attack for small 𝛼 values (i.e increasing 𝐶𝐿 with 𝛼), 

which indicates stability to galloping.  

The separation zone boundary for the 2Dp cylinder is depicted in Figure 38. Focusing on 

the results at 𝛼 = 0° at the lower 𝑅𝑒𝑑 = 800 and 1400, the boundary layer separates downstream 

of the first bump located along the LE corner of the baseline geometry. Reynolds number seems 

to have a very small influence on the separation location. On the other hand, while the separated 

shear  layer  remains  open  at  the  low  Reynolds  numbers,  the  shear  layer  probably  reattaches 

upstream of the last lateral topology bump at 𝑅𝑒𝑑 = 2500.  

The  observations  in  the  previous  paragraph  were  discussed  earlier  in  Section  3.1.  To 

connect  the  features  of  boundary  layer  separation/reattachment  to  galloping  instability,  it  is 

important to examine how these features change with increasing the angle of attack. Referring to 

Figure  38,  the  separation  remains  open  on  the  top  and  the  bottom  of  the  2Dp  cylinder  at 

𝑅𝑒𝑑 = 800 and 1400 as 𝛼 is increased up to 1.5°. This behavior is reminiscent of that of the sharp-

corner  rectangular  cylinders  that  are  unstable  to  galloping  (Parkinson  [4]),  and  hence  one  may 

expect instability to galloping for this cylinder at 𝑅𝑒𝑑 = 800 and 1400. As the angle of attack is 

increased further to 2° (Figure 38-bottom), the shear layer reattaches on the bottom side, upstream 

of the topology peak at the aft corner of the cylinder, at 𝑅𝑒𝑑 = 800, suggesting pressure recovery 

on the bottom side and a switch of the 𝐶𝑦 − 𝛼 slope sign at 𝛼 ≈ 2°.  However, no reattachment is 

observed  at  𝑅𝑒𝑑 = 1400,  suggesting  that  the  cylinder  remains  unstable  to  galloping  over  the 

examined range of AoA at this Reynolds number.  

75 

 
Increasing Reynolds number to 2500 causes the separated shear layer to remain very close 

to reattaching to the top surface of the 2Dp cylinder up to 𝛼 = 1°, somewhere over the topology 

peak  at  the  aft  corner  of  the  cylinder.  Further  increase  to  𝛼 = 1.5°,  leads  to  opening  of  the 

separation bubble. In contrast, on the bottom side of the 2Dp case, the flow separates downstream 

of the topology peak at the front corner, and reattaches upstream of the topology peak at the aft 

corner at all measured angles of attack. The separation bubble on the bottom side is not shrinking 

significantly with  𝛼, which as stated  earlier, is  found in  [9] to  be important  for the stability to 

galloping.  On  the  other  hand,  considering  the  shear  layer  on  the  top  side  being  close  to 

reattachment  to  the  cylinder,  a  stronger  suction  pressure  (and  hence  higher  lift)  might  be 

developing with increasing AoA, which is a stabilizing effect. It is not known which of these two 

effects would be stronger, and hence is it uncertain if the 2Dp cylinder is stable or unstable at 𝑅𝑒 =

2,500. 

76 

 
 
 
 
 
 
 
 
 
𝛼 = 0° 

𝛼 = 0.5° 

𝛼 = 1° 

Figure 38. Comparison of the mean separation zone boundary for the 2Dp-5% cylinder at 
different angles of attack (different rows) and Reynolds numbers. 

77 

 
 
 
 
 
 
 
 
 
 
 
Figure 38 (cont’d) 

𝛼 = 1.5° 

𝛼 = 2° 

Results for the 2Dv are shown in  Figure 39, where the separation zone boundary is not 

depicted for any localized separation bubbles within the valleys of the topology. The separation 

bubble remains closed over the top surface at 𝑅𝑒𝑑 = 800 as the angle of attack increases to 𝛼 =

0.5°. With further increasing the angle of attack, the separation bubble becomes open, but remains 

very thin. The shear layer on the bottom surface forms two isolated separation bubbles, filling the 

topology valleys,  with  increasing the  angle of  attack (not plotted). Similar trend is  observed at 

𝑅𝑒𝑑 = 1400, but opening of the separation bubble on the top is promoted and is observed at 𝛼 =

0.5°. In addition, the separation bubble on the bottom is observed at higher angles of attack and 

78 

 
 
 
 
 
the isolated bubbles in the dips are seen as late as 𝛼 = 1°. At 𝑅𝑒𝑑 = 2500, the separated shear 

layer does not reattach to the top surface and moves farther from the surface with increasing  𝛼. 

Reattachment  of  the  separated  shear  layer  is  observed  to  the  bottom  surface,  upstream  of  last 

topology peak on the surface at 𝛼 = 2°.  

For the lower Reynolds numbers 𝑅𝑒𝑑 = 800 and 1400, the separation zone boundary on 

the bottom is generally very thin and the shear layer is reattached. Thus, the flow generally follows 

the baseline shape of the cylinder on the bottom side, which would likely lead to increased bottom 

surface pressure with AoA, contributing to stability to galloping. The reattachment of the top shear 

layer at 𝑅𝑒𝑑 = 800 and small angles of attack is expected to improve the stability to galloping 

(increasing 𝐶𝐿) for this geometry. It is difficult to judge the pressure variation on the top surface 

at 𝑅𝑒𝑑 = 1400, since the curvature of the top separation bubble only minimally changes at 𝛼 > 0. 

With the top shear layer failing to reattach as the Reynolds number increases to 𝑅𝑒𝑑 = 2500, the 

geometry  behaves  similar  to  a  sharp  corner  rectangular  cylinder  which  is  prone  to  galloping. 

Therefore,  increasing  the  Reynolds  number  in  the  studied  range  is  expected  to  make  the  2Dv 

cylinder more unstable.  

79 

 
 
 
 
 
 
𝛼 = 0° 

𝛼 = 0.5° 

𝛼 = 1° 

Figure 39. Comparison of the mean separation zone boundary for the 2Dv-5% cylinder at 
different angles of attack (different rows) and Reynolds numbers. 

80 

 
 
 
 
 
 
 
 
 
 
 
Figure 39 (cont’d) 

𝛼 = 1.5° 

𝛼 = 2° 

Proceeding to consider the 3D geometry, Figure 40 shows that for all angles of attack, the 

presence of topology variation along the span leads the separated shear layer on both sides of the 

3Dp cylinder to move closer to the surface compared to the 2Dp case (seen in  Figure 38). The 

separation bubble on the top surface becomes very close to reattachment over the topology peak 

at the aft corner of the cylinder at 𝛼 = 0° and 𝑅𝑒𝑑 = 800, but reattachment cannot be ascertained. 

With a slight increase in the angle of attack to 𝛼 = 0.5°, the separation bubble becomes clearly 

open and the separation zone boundary moves farther from the surface and has lower curvature 

with further increase of the angle of attack. This is expected to contribute to lower suction pressure 

81 

 
 
 
 
 
on the top, hence lower lift at higher 𝛼. On the bottom surface, the bottom shear layer reattaches 

to the surface at 𝛼 = 0.5°, upstream of the topology peak at the aft corner of the cylinder. With 

further increasing the angle of attack, three isolated separation bubbles form inside the topology 

valleys on the bottom surface. The bottom shear layer generally behaves more like an attached 

flow, following the overall shape of the body, and is therefore expected to contribute to stability 

against galloping. Overall, the 3Dp case at 𝑅𝑒𝑑 = 800 behaves like a rectangular cylinder with 

sharp corners at high Reynolds numbers, where the bottom shear layer is reattached and moves 

upstream with increasing the angle of attack, but the top shear layer remains fully separated. This 

is a classic galloping problem studied in the literature [4] and is expected to be stable to galloping. 

At the higher Reynolds numbers 𝑅𝑒𝑑 = 1400 and 2500, the top shear layer remains reattached to 

the surface as the angle of attack increases to  𝛼 = 1° and 1.5°, respectively. Therefore, the top 

shear layer is expected to impose higher suction pressure on the surface and contribute to higher 

lift, up to these angles of attack. Beyond 𝛼 = 1.5° at 𝑅𝑒𝑑 = 1400, and 𝛼 = 2° at 𝑅𝑒𝑑 = 2500, 

the top separated shear layer fails to reattach, and the suction pressure on the top surface is expected 

to drop. The separation bubble on the bottom is closed at these Reynolds numbers and the flow 

generally follows the baseline shape of the cylinder on the bottom side. The contribution of the 

bottom shear layer, therefore, is likely to increase the pressure recovery with angle of attack and 

stability to galloping. Without surface pressure measurements, it is difficult to determine whether 

the stabilizing behavior of the bottom shear layer will overcome the destabilizing effect of the top 

shear layer.  

The effect of Reynolds number is noted to be enhancing the reattachment of the separated 

shear  layer  on  the  top  surface,  hence,  improving  the  stability  to  galloping.  Overall,  due  to  the 

associated higher curvature of the shear layer on the top surface, higher suction pressure relative 

82 

 
to the 2Dp case is expected; i.e. promoting an increase in lift and stability to galloping. On the 

bottom side, the shear layer reattachment is expedited and a higher pressure recovery is expected 

for  the  3Dp  case.  This  behavior  is  anticipated  to  contribute  further  to  the  increase  in  lift  and 

stability to galloping for the 3Dp case compared to its 2D counterpart. It should be noted, however, 

that the behavior of the 3D cylinder as a whole cannot be solely determined by looking at the single 

cross-section corresponding to  the 2Dp  shape, since the cylinder cross-section varies along the 

span. 

𝛼 = 0° 

𝛼 = 0.5° 

Figure 40. Comparison of the mean separation zone boundary for the 3Dp-5% cylinder at 
different angles of attack (different rows) and Reynolds numbers. 

83 

 
 
 
 
 
 
 
 
 
 
Figure 40 (cont’d) 

𝛼 = 1° 

𝛼 = 1.5° 

𝛼 = 2° 

84 

 
 
 
 
 
 
 
 
 
Results from MTV measurement on the cross section of the 3D cylinder with a valley at 

the  leading  edge  are  presented  in  Figure  41.  The  top  shear  layer  separates  downstream  of  a 

topology peak and does not reattach at the lower Reynolds numbers 𝑅𝑒𝑑 = 800 and 1400. The 

separation point on the bottom surface seems to be similar to that on the top, being downstream of 

a topology peak. As the angle of attack increases, the separated shear layer on the bottom moves 

closer to the surface and becomes very close to reattachment at 𝛼 = 1° at the mentioned Reynolds 

numbers, but full reattachment is not certain until 𝛼 = 1.5°, which happens over the last topology 

peak on the surface. At 𝛼 = 2°, two isolated closed separation bubbles are observed at 𝑅𝑒𝑑 = 800, 

which are not shown in Figure 41. Although the stability of this cylinder to galloping cannot be 

ascertained by only examining one cross-section, the 3Dv case at 𝑅𝑒𝑑 = 800 and 1400 seems to 

behave similar to the rectangular cylinder with sharp corner which is unstable to galloping.  

𝛼 = 0° 

𝛼 = 0.5° 

Figure 41. Comparison of the mean separation zone boundary for the 3Dv-5% cylinder at 
different angles of attack (different rows) and Reynolds numbers. 

85 

 
 
 
 
 
Figure 41 (cont’d) 

𝛼 = 1° 

𝛼 = 1.5° 

𝛼 = 2° 

86 

 
 
 
 
 
 
 
 
 The  separation  zone  boundary  for  𝑅𝑒𝑑 = 2500  is  shown  using  the  zero-crossing  data 

points in Figure 41, since the second order polynomial fit does not do an adequate job for this case. 

In  fact,  at  𝑅𝑒𝑑 = 2500,  the  separation  bubble  on  the  top  surface  is  closed,  with  reattachment 

occurring upstream of the topology peak at the aft corner of the cylinder. Reattachment persists as 

the angle of attack increases to 𝛼 = 1° with the curvature of the separation zone boundary being 

higher  compared  to  the  lower  Reynolds  numbers.  This  is  expected  to  contribute  to  stability  to 

galloping. In addition, reattachment to the bottom surface is promoted to a lower angle of attack 

𝛼 = 1°, compared to the lower Reynolds numbers. The separation zone boundary on the bottom 

is  generally  very  thin  and  the  flow  generally  follows  the  baseline  shape  of  the  cylinder  on  the 

bottom side, which would likely lead to increased bottom surface pressure with AoA, contributing 

to  stability to  galloping.  Therefore, it is  expected that  unlike the 2Dv  geometry, increasing  the 

Reynold number enhances the stability to galloping for the 3Dv case in the examined Reynolds 

number range.  

To summarize, it is expected from MTV measurement on the surface of the 3Dp case, i.e. 

the cross-section of the 3D cylinder with a peak at the leading edge, that this case is slightly stable 

towards  galloping  at  the  lowest  Reynolds  number  studied,  but  becomes  more  stable  as  the 

Reynolds number increases. For the other cross-section with a valley at the leading edge (3Dv), a 

positive  contribution  to  stability  to  galloping  is  expected  only  at  the  highest  Reynolds  number 

studied. Similar trend was observed from force measurements on this cylinder in wind [7, 8] and 

water  [11]  tunnels,  where  the  cylinder  was  unstable  at  a  low  enough  Reynolds  number,  but 

increasing the Reynolds number was observed to improve the stability to galloping.  

Although the Reynolds numbers studied here have some commonality with those studied 

in the wind tunnel force measurements by Feero et al. [7, 8], it is important to note that the flow 

87 

 
and  the  force  characteristics  in  two  different  test  facilities  may  be  different  notwithstanding 

matching  of  𝑅𝑒𝑑.  Such  differences  may  be  caused  by  discrepancies  in  freestream  turbulence 

intensity, cylinder aspect ratio, model end conditions, and test section blockage. For this reason, 

complementary force measurements were done by a collaborator [11] on the same cylinder models 

and experimental setup employed in the present study, and the results are reported in Figure 3. As 

seen  from  the  figure,  these  results  convey  similar  conclusion  to  those  discussed  above  from 

characterization of the separation/reattachment characteristics of the boundary layer. Namely, the 

stability to galloping of the 3D cylinder improves with increasing Reynolds number. 

88 

 
 
 
Chapter 4. Characteristics of the Wake Flow 

Whole-field  1c-MTV  is  carried  out  in  the  wake  of  the  cylinders  to  study  the  effect  of 

surface topology on the wake development and characteristics. Studying the wake is of interest for 

understanding the connections between the wake structures and the surface topology and, more 

importantly,  finding  the  connection  to  the  boundary  layer  characteristics.  Secondly,  for  the 

particular  application  of  interest  for  this  study,  the  suspension  cables  of  PADS  are  likely  to 

experience  interactions  with  the  wakes  of  other  lines  based  on  their  configuration.  Therefore, 

studying  the  wake  features  of  an  isolated  cylinder  could  help  provide  boundary  conditions  for 

future  studies  of  the  instability  of  the  whole  system.  Furthermore,  knowledge  of  the  vortex 

shedding frequency in the wake of the cylinders gives insight into whether the cylinders are also 

susceptible  to  vortex-induced  vibrations  and  the  applicability  of  the  quasi-steady  analysis  of 

galloping. 

In the current study, the discussion of measurements in the wake of the cylinders at a range 

of Reynolds numbers, 𝑅𝑒𝑑 = 800, 1100, 1400, 1800, and 2500, is limited to the angle of attack 

α = 0°.  

4.1 Mean and rms Velocity Characteristics 

4.1.1 Smooth Cylinder 

Color contour maps of the wake mean and rms streamwise velocity for the smooth cylinder 

are shown in Figure 42. The effect of Reynolds number on the recovery of the wake is evident. 

The mean velocity contour map on the left column suggests that at 𝑅𝑒𝑑 = 800, a small reverse 

flow region forms over the round corners at the cylinder aft, but does not exceed the immediate 

wake. Moreover, there is a forward mean flow in the near wake before a secondary reverse flow 

region forms farther downstream. As the Reynolds number increases, the reverse flow region at 

89 

 
the  base  extends  farther  downstream.  The  secondary  reverse  flow  region  moves  upstream  and 

extends in size, so much that at 𝑅𝑒𝑑 = 1400, the two reverse flow regions join. Further increase 

in  Reynolds  number  shows  a  non-monotonic  trend,  where  the  reverse  flow  region  in  the  wake 

extends both in lateral and downstream direction at 𝑅𝑒𝑑 = 1800 but shrinks distinctly at 𝑅𝑒𝑑 =

2500. 

𝑅𝑒𝑑 = 800 

𝑅𝑒𝑑 = 1100 

𝑅𝑒𝑑 = 1400 

𝑅𝑒𝑑 = 1800 

𝑅𝑒𝑑 = 2500 

Figure 42. Color contour maps of the mean (left) and rms (right) streamwise velocity normalized 
by the freestream velocity (𝑢̅/𝑈∞ and 𝑢𝑟𝑚𝑠/𝑈∞, respectively) in the wake of the smooth cylinder 
in the Reynolds number range 𝑅𝑒𝑑 = 800-2500 in different rows. The trajectory of the peak 
𝑢𝑟𝑚𝑠 in the top shear layer is marked, up to the vortex formation location (maximum 𝑢𝑟𝑚𝑠 in the 
wake). Flow direction is from left to right. 

90 

 
 
 
 
 
 
 
 
 
 
 
Although  cross-stream  component  of  the  velocity  is  not  available  in  the  current 

measurements and a direct vorticity analysis cannot be accomplished, the trend observed in the 

current  study  bears  some  analogy  to  the  wake  flow  regimes  for  a  circular  cylinder  [22].  von 

Karman  vortex  street  is  observed  at  the  Reynolds  number  range  40 < 𝑅𝑒𝑑 < 60 − 100  for 

circular cylinders, where two flow regions with reverse flow exist: one attached to the cylinder aft 

surface, and the other located farther downstream in the wake where alternate vortex formation 

and shedding take place (Figure 43-top). As the Reynolds number increases to 60 − 100 < 𝑅𝑒𝑑 < 

200, the two reverse flow regions merge into one region that is attached to the aft surface. Similar 

behavior is observed in the current study for the smooth rectangular cylinder, where two distinct 

reverse  flow  regions  are  observed  at  𝑅𝑒𝑑 = 800 − 1100  which,  with  increasing  the  Reynolds 

number to 𝑅𝑒𝑑 ≥ 1400, merge into one.  

The  rms  color  maps  show  that  the  high  fluctuations  are  generally  concentrated  and 

contained inside small regions in the immediate wake of the smooth cylinder. As the Reynolds 

number  increases,  the  shear  layers  become  more  unsteady  and  the  fluctuations  in  the  wake 

intensify. The region where the maximum rms is found in the wake, which is associated with the 

vortex  formation[23-25],  moves  upstream  with  increasing  the  Reynolds  number  from  800  to 

1100, but remains relatively stationary at higher Reynolds numbers up to 𝑅𝑒𝑑 = 1800. Further 

increase to 𝑅𝑒𝑑 = 2500 shifts the region of highest unsteadiness farther upstream again, closer to 

the base of the cylinder. It is important to note here that the wake development is initiated from 

initial boundary layers that have steady and unsteady characteristics that could vary significantly 

between  case.  The  steady  boundary  layer  characteristics  were  discussed  in  Chapter  3,  and  the 

unsteady  characteristics  will  be  examined  in  Chapter  5,  where  it  will  be  discussed  that  𝑅𝑒𝑑 ≤

1400 exhibits wake formation from a reattached, “almost steady” boundary layer.  𝑅𝑒𝑑 = 1800 

91 

 
shows an unsteady transition between fully separated/reattached flow, although showing an open 

mean  separation  bubble,  and  reattachment  of  the  boundary  layer  occurs  again  at  𝑅𝑒𝑑 = 2500. 

Although  the  unsteady  behavior  will  be  discussed  in  Chapter  5  in  detail,  these  observations 

underline the complexity of the initial condition for the development of the wake and provide some 

context into why a non-monotonic trend is not surprising. 

Figure 43. von Karman vortex street (top) and alternative shedding (bottom) regimes in the wake 
of circular cylinders, following the discussion of Panton [22]. 

In  the  near-wake,  𝑥/𝑑  <  5,  several  features  may  be  examined:  the  location  and  the 

strength of the reverse flow, the wake closure length 𝑙𝑐 (i.e., the location where 𝑢̅𝑐/𝑈∞ = 0. See 

Figure 44 (top) for definition), and the “wake recovery” (the rate of increase in 𝑢̅𝑐/𝑈∞ downstream 

of the closure length). The vortex formation length definition can be seen in Figure 44 (bottom) 

and it is the distance between the cylinder trailing edge and the location of the peak 𝑢𝑟𝑚𝑠 in the 

wake. The wake mean velocity field is first characterized by examining 𝑙𝑐. This is accomplished 

by considering the streamwise evolution of the mean centerline velocity 𝑢̅𝑐. 

92 

 
 
 
 
Figure 44. Definition of the closure length 𝑙𝑐 on the mean velocity color contour map (top) as 
well as vortex formation length 𝑙𝑢 and wake width 𝛥𝑟𝑚𝑠 on the rms velocity color contour map 
(bottom). 

The mean centerline velocity  results for the smooth  cylinder  are shown in Figure  45  at 

different Reynolds numbers. From this figure it can be seen that as the Reynolds number increases 

from 𝑅𝑒𝑑 = 800 to 1100, the closure length becomes shorter as the vortex formation occurs closer 

to the aft of the body (see Table 2 for numerical values). Further increase of the Reynolds number 

to 𝑅𝑒𝑑 = 1400 does not alter the closure length, but at 𝑅𝑒𝑑 = 1800 the closure length increases 

again.  As discussed before in  Chapter 3, an open boundary layer separation is  observed at  this 

Reynolds number. Therefore, it takes longer for the shear layers from the two sides of the cylinder 

to interact and form vortices, hence the higher 𝑙𝑐. At 𝑅𝑒𝑑 = 2500 reattachment of the boundary 

layer  occurs  on  the  surface  and  the  closure  length  decreases  to  its  lowest  value.  The  vortex 

formation length is also shortest at this Reynolds number, judged by the peak 𝑢𝑟𝑚𝑠 location shown 

in Figure 42 (right).  

93 

 
 
 
 
Figure 45. Comparison between the streamwise evolution of the mean centerline velocity in the 
wake of the smooth cylinder in the Reynolds number range 𝑅𝑒𝑑 = 800 − 2500. The error bars 
for most cases are smaller than the marker size. 

Table 2. Summary of various wake parameters. The cases with open separation are highlighted in 
red, and if reattachment occurs at a higher Reynolds number, the case is highlighted in teal.  

𝑙𝑐/𝑑 

𝑙𝑢/𝑑  Δ𝑟𝑚𝑠/𝑑  Δ𝑠ℎ𝑒𝑎𝑟/𝑑  𝑆𝑡𝑑  𝑆𝑡Δrms  𝑆𝑡Δshear 

𝑅𝑒𝑑 = 800 
𝑅𝑒𝑑 = 1100 
𝑅𝑒𝑑 = 1400 
𝑅𝑒𝑑 = 1800 
𝑅𝑒𝑑 = 2500 
𝑅𝑒𝑑 = 800 
𝑅𝑒𝑑 = 1100 
𝑅𝑒𝑑 = 1400 
𝑅𝑒𝑑 = 1800 
𝑅𝑒𝑑 = 2500 
𝑅𝑒𝑑 = 800 
𝑅𝑒𝑑 = 1100 
𝑅𝑒𝑑 = 1400 
𝑅𝑒𝑑 = 1800 
𝑅𝑒𝑑 = 2500 
𝑅𝑒𝑑 = 800 
𝑅𝑒𝑑 = 1100 
𝑅𝑒𝑑 = 1400 
𝑅𝑒𝑑 = 1800 
𝑅𝑒𝑑 = 2500 

2.20 
1.66 
1.70 
1.96 
1.00 
2.23 
1.76 
2.64 
2.59 
1.26 
2.05 
1.57 
1.20 
1.10 
1.17 
2.23 
1.37 
1.94 
1.02 
0.59 

2.45 
1.80 
1.81 
1.12 
1.11 
1.58 
1.50 
2.22 
2.2 
1.50 
2.20 
1.68 
1.25 
1.16 
0.99 
2.28 
1.43 
1.63 
0.82 
0.57 

𝑅𝑒𝑑 = 800 

2.81 

2.84 

Smooth 

2Dp-
5% 

2Dv-
5% 

3Dp-
5% 

3Dv-
5% 

1.27 
1.20 
1.19 
1.29 
1.09 
1.65 
1.64 
1.79 
1.70 
1.28 
1.36 
1.31 
1.26 
1.24 
1.33 
1.38 
1.35 
1.41 
1.26 
0.98 

1.55 

94 

1.24 
1.21 
1.18 
1.22 
1.18 
1.63 
1.63 
1.74 
1.69 
1.45 
1.32 
1.29 
1.28 
1.31 
1.35 
1.31 
1.32 
1.36 
1.40 
1.31 

1.45 

0.23 
0.25 
0.27 
0.22 
0.22 
0.18 
0.17 
0.16 
0.15 
0.18 
0.22 
0.23 
0.24 
0.23 
0.21 
0.20 
0.22 
0.20 
0.20 
0.20 

0.29 
0.30 
0.32 
0.28 
0.24 
0.29 
0.27 
0.29 
0.25 
0.23 
0.30 
0.30 
0.30 
0.28 
0.28 
0.28 
0.29 
0.28 
0.25 
0.20 

0.21 

0.32 

0.29 
0.30 
0.32 
0.27 
0.26 
0.29 
0.28 
0.28 
0.25 
0.26 
0.29 
0.30 
0.31 
0.30 
0.28 
0.26 
0.29 
0.27 
0.28 
0.26 

0.30 

 
 
 
 
 
 
Table 2 (cont’d) 

3Dv-
5% 

𝑅𝑒𝑑 = 1100 
𝑅𝑒𝑑 = 1400 
𝑅𝑒𝑑 = 1800 
𝑅𝑒𝑑 = 2500 

1.15 
0.80 
0.76 
0.66 

1.38 
0.95 
0.68 
0.60 

1.34 
1.27 
1.22 
1.14 

1.37 
1.34 
1.35 
1.30 

0.22 
0.22 
0.20 
0.21 

0.29 
0.28 
0.25 
0.24 

0.30 
0.29 
0.27 
0.27 

Considering the reverse-flow magnitude, the maximum negative velocity slightly increases 

when  the  Reynolds  number  is  increased  from  𝑅𝑒𝑑 = 800  to  1100.  Further  increase  to  𝑅𝑒𝑑 =

1400 causes a slight decrease in the maximum negative velocity value which remains the same at 

the higher Reynolds number of 𝑅𝑒𝑑 = 1800. At 𝑅𝑒𝑑 = 2500, this value slightly decreases again. 

Therefore, there is not an obvious trend of the maximum negative velocity with respect to Reynolds 

number in the examined range. The streamwise location where the maximum reverse flow velocity 

takes place however, follows a more familiar trend. It shifts more upstream when Reynolds number 

increases from 𝑅𝑒𝑑 = 800 to 1100, remains similar at 𝑅𝑒𝑑 = 1400, before moving downstream 

again  at  𝑅𝑒𝑑 = 1800,  where  an  open  boundary  layer  separation  is  observed  on  the  cylinder’s 

surface.  Finally,  with  the  reattachment  of  the  shear  layer  to  the  surface  at  𝑅𝑒𝑑 = 2500,  the 

maximum reverse flow velocity occurs more upstream and closest to the cylinder trailing edge. 

This trend is similar to the trend of the closure length versus Reynolds number discussed before. 

Finally, the rate of centerline velocity recovery in the far wake at 𝑅𝑒𝑑 = 2500 indicates a 

wake closer to the asymptotic state 𝑢̅𝑐/𝑈∞ → 1 than the lower Reynolds numbers. It is unclear to 

what extent the far-field evolution might be affected by the blockage effects of the test section. 

This can be more extensively studied in future work.  

4.1.2 2Dp-5% Cylinder 

Figure 46 demonstrates the mean and rms velocity contour maps of the wake flow for the 

2Dp geometry. It is evident that the reverse flow region is considerably larger compared to the 

smooth geometry. The size of the reverse flow region does not follow a monotonic trend against 

95 

 
the Reynolds number. This region shrinks when the Reynolds number increases from 𝑅𝑒𝑑 = 800 

to 1100, but expands again at 𝑅𝑒𝑑 = 1400 and 1800. Increasing the Reynolds number further to 

𝑅𝑒𝑑 = 2500 shows a considerable reduction in the size of the reverse flow region, so much that it 

is the smallest amongst the studied range of Reynolds numbers. As it was seen in Chapter 3, the 

separated shear layer reattaches to the surface of this cylinder only at 𝑅𝑒𝑑 = 2500.  

The rms velocity contour maps depicted in the right column of  Figure 46 show that the 

region containing the high fluctuations occupies a region that is elongated more in the streamwise 

direction compared to the smooth cylinder. This elongation in the streamwise direction is more 

pronounced  as  the  Reynolds  number  increases.  The  rms  contour  maps  also  suggest  a  non-

monotonic  behavior  of  the  unsteadiness  intensity  with  respect  to  Reynolds  number.  The 

unsteadiness in the wake intensifies as the Reynolds number increases from 𝑅𝑒𝑑 = 800 before the 

trend reverses at 𝑅𝑒𝑑 = 1800. In fact, the lowest maximum normalized fluctuations are observed 

at 𝑅𝑒𝑑 = 2500. This case involves formation of vortex structures from instability of the separated 

shear layer above the cylinder side surface, which will be later discussed in Chapter 5 (see Figure 

65).  

The mean wake flow field of the 2Dp cylinder differs from the smooth cases as it does not 

exhibit  two  separated  reverse  flow  regions.  It  is  possible  that  this  occurs  at  a  lower  Reynolds 

number, but the current study does not provide sufficient data for a conclusive determination. 

96 

 
 
 
 
 
 
 
𝑅𝑒𝑑 = 800 

𝑅𝑒𝑑 = 1100 

𝑅𝑒𝑑 = 1400 

𝑅𝑒𝑑 = 1800 

𝑅𝑒𝑑 = 2500 

Figure 46. Color contour maps of the mean (left) and rms (right) streamwise velocity normalized 
by the freestream velocity (𝑢̅/𝑈∞ and 𝑢𝑟𝑚𝑠/𝑈∞, respectively) in the wake of the 2Dp-5% 
cylinder in the Reynolds number range 𝑅𝑒𝑑 = 800-2500 in different rows. The trajectory of the 
peak 𝑢𝑟𝑚𝑠 in the top shear layer is marked, up to the vortex formation length. Flow direction is 
from left to right. 

Figure 47 presents the mean centerline velocity in the wake of the 2Dp cylinder at different 

Reynolds  numbers.  The  closure  length  decreases  with  increasing  the  Reynolds  number  from 

𝑅𝑒𝑑 = 800 to 1100, but reverts trend and increases at  𝑅𝑒𝑑 = 1400. It remains very similar at 

𝑅𝑒𝑑 = 1800,  but  shrinks  considerably  as  the  separated  shear  layer  reattaches  to  the  surface  at 

97 

 
 
 
 
 
 
 
 
 
 
 
𝑅𝑒𝑑 = 2500. This non-monotonic trend of 𝑙𝑐 versus Reynolds number is similar to the smooth 

cylinder case (see Table 2 for numerical values).  

Figure 47. Comparison between the streamwise evolution of the mean centerline velocity in the 
wake of the 2Dp cylinder in the Reynolds number range 𝑅𝑒𝑑 = 800 − 2500. The error bars for 
most cases are smaller than the marker size. 

The  maximum  reverse  flow  velocity  slightly  increases  from  𝑅𝑒𝑑 = 800  to  1100,  but 

remains  fairly  similar  with  further  increasing  the  Reynolds  number,  before  a  distinct  decrease 

occurs  at  𝑅𝑒𝑑 = 2500. This  decrease can be  associated with  the reattachment of the  separated 

shear layer to the surface of the cylinder at  𝑅𝑒𝑑 = 2500, unlike all the other studied Reynolds 

numbers. The location of the maximum reverse flow velocity follows a more similar trend to the 

closure  length  versus  Reynolds  number.  It  first  moves  upstream  up  to  𝑅𝑒𝑑 = 1100,  reverts 

direction  and  moves  away  from  the  cylinder  towards  downstream  at  𝑅𝑒𝑑 = 1400,  and  reverts 

direction for a second time to move closer to the cylinder with further increasing the Reynolds 

number.  

The rate of centerline velocity recovery in the far wake of this cylinder shows a similar 

trend at all Reynolds numbers which is considerably deviated from an asymptotic state 𝑢̅𝑐/𝑈∞ →

1. In fact, unlike the case of the smooth cylinder, reattachment of the separated shear layer to the 

98 

 
 
surface at  𝑅𝑒𝑑 = 2500 does not  improve the wake recovery  towards the asymptotic state. The 

results also show that for this cylinder the wake recovery is noticeably slower than the smooth 

cylinder. 

4.1.3 2Dv-5% Cylinder 

The mean and rms velocity color contour maps in the wake of the 2Dv cylinder are shown 

in Figure 48. Overall, this cylinder has a closer behavior to the smooth cylinder. At 𝑅𝑒𝑑 = 1100, 

the mean velocity colormap suggests that there are two separate reverse flow regions in the near 

wake, and that the vortex formation  region  is  farther downstream  and not  yet  merged  with  the 

reverse flow region over the trailing edge. At 𝑅𝑒𝑑 = 800, the same observation holds true, with 

the exception that the reverse flow region connected to the surface is not observed. This absence 

is attributed to the measurement limitations in close proximity to the aft surface and inside the 

respective topology valley. Vortex formation region keeps moving closer to the trailing edge as 

the Reynolds number increases to 𝑅𝑒𝑑 = 1800, and is merged with the near trailing edge reverse 

flow region starting at 𝑅𝑒𝑑 = 1400. At 𝑅𝑒𝑑 = 2500, the reverse flow region grows compared to 

the lower Reynolds numbers, which is the opposite trend observed at the lower Reynolds numbers. 

It  is  noteworthy  to  recall  from  Chapter  3  that  an  open  separation  bubble  is  observed  over  the 

surface  of  this  cylinder  only  at  𝑅𝑒𝑑 = 1800  and  2500,  where  the  separation  zone  boundary 

remains  very  close  to  the  surface  at  the  former  Reynolds  number  and  moves  farther  from  the 

surface at the latter. Therefore, a systematic trend is seen in the size of the reverse flow region, 

until the full opening up of the separation bubble. 

The  rms  contour  maps  on  the  right  column  of  Figure  48  reveal  that  the  flow  in  the 

immediate wake is quiet at 𝑅𝑒𝑑 = 800. In other words, the rapid expansion of the region with high 

unsteadiness, which is  associated with  the vortex roll-up, takes place after about 1.5 diameters 

99 

 
downstream of the trailing edge. As the Reynolds number increases, the high unsteadiness region 

moves closer to the cylinder’s base, which agrees with the postulation that the vortex formation 

region merges with the reverse flow region at the trailing surface starting at 𝑅𝑒𝑑 = 1400.  

𝑅𝑒𝑑 = 800 

𝑅𝑒𝑑 = 1100 

𝑅𝑒𝑑 = 1400 

𝑅𝑒𝑑 = 1800 

𝑅𝑒𝑑 = 2500 

Figure 48. Color contour maps of the mean (left) and rms (right) streamwise velocity normalized 
by the freestream velocity (𝑢̅/𝑈∞ and 𝑢𝑟𝑚𝑠/𝑈∞, respectively) in the wake of the 2Dv-5% 
cylinder in the Reynolds number range 𝑅𝑒𝑑 = 800-2500 in different rows. The trajectory of the 
peak 𝑢𝑟𝑚𝑠 in the top shear layer is marked, up to the vortex formation length. Flow direction is 
from left to right. 

The highly unsteady region is confined in its streamwise extent, similar to the smooth cylinder, 

and  the  unsteadiness  intensifies  with  the  Reynolds  number  up  to  𝑅𝑒𝑑 = 1400.  Increasing  the 

100 

 
 
 
 
 
 
 
 
 
 
 
Reynolds number further is linked with a reverse trend, where the maximum unsteadiness level 

decays in the wake.   

Mean centerline velocity in the wake of the 2Dv cylinder is plotted in Figure 49. The vortex 

closure length keeps decreasing when Reynolds number increases from 𝑅𝑒𝑑 = 800 to 1800, but 

slightly increases again at 𝑅𝑒𝑑 = 2500. From Chapter 3, the separated shear layer reattaches to 

the surface at lower Reynolds numbers and becomes fully open at 𝑅𝑒𝑑 = 2500, which explains 

the change of trend for the 𝑙𝑐 versus 𝑅𝑒𝑑 (see Table 2 for numerical values). The maximum reverse 

flow velocity keeps monotonically increasing with Reynolds number which is a different behavior 

seen from smooth and 2Dp cylinders. The location of the maximum reverse flow velocity shows 

a  similar  trend  and  monotonically  moves  upstream  and  closer  to  the  cylinder  with  increasing 

Reynolds number.  

The  rate  of  centerline  velocity  recovery  in  the  far  wake  of  this  cylinder  shows  a  good 

deviation from an asymptotic state 𝑢̅𝑐/𝑈∞ → 1, where the recovery does not exceed about 70%. 

In this case, opening of the separation bubble on the surface at 𝑅𝑒𝑑 = 2500 does not deteriorate 

the wake condition away from an asymptotic state. Although not shown here directly, the rate of 

wake  recovery  for  this  case  at  all  Reynolds  numbers  except  𝑅𝑒𝑑 = 2500  is  comparable  to  the 

smooth  cylinder.  At  𝑅𝑒𝑑 = 2500,  the  recovery  of  the  wake  is  slower  for  the  2Dv  cylinder 

compared to the smooth cylinder. 

101 

 
Figure 49. Comparison between the streamwise evolution of the mean centerline velocity in the 
wake of the 2Dv cylinder at the Reynolds number range 𝑅𝑒𝑑 = 800 − 2500. The error bars for 
most cases are smaller than the marker size. 

4.1.4 3Dp-5% Cylinder 

Mean and rms velocity color contour maps are presented in Figure 50 for the wake of the 

cylinder with the three-dimensional geometry and in the plane of the cross-section with a peak at 

the  leading  edge  (3Dp).  At  the  lowest  Reynolds  number  𝑅𝑒𝑑 = 800,  the  mean  contour  map 

suggests that there is a confined and isolated reverse flow region in the wake which is not attached 

to the trailing end of the cylinder, and that vortex formation region is separated from the reverse 

flow  region  at  the  aft  of  the  body.  Interestingly,  this  behavior  is  very  different  than  the  2D 

counterpart, where the latter had an open separation bubble over the surface, translating into  a 

large  reverse  flow  region  in  the  wake.  As  the  Reynolds  number  increases  to  𝑅𝑒𝑑 = 1100,  the 

secondary reverse flow region merges with the reverse flow zone at the aft of the body to form a 

single reverse flow region. The size of this region peaks at 𝑅𝑒𝑑 = 1400, before starting to shrink 

again at higher Reynolds numbers.  

The rms velocity contour maps for 𝑅𝑒𝑑 = 800 on the right column of Figure 50 show the 

shear layers with higher unsteadiness level start rolling up (expansion of the high rms region) at 

102 

 
 
 
about 𝑥/𝑑  ≈ 2.75, and that vortex formation takes place relatively far from the tailing base of the 

cylinder. As the Reynolds number increases, the widening of the region with high rms velocity 

progresses towards upstream and closer to the cylinder’s base. Compared to the 2D counterpart, 

this case shows the high unsteady region is more confined, especially at 𝑅𝑒𝑑 = 2500. 

𝑅𝑒𝑑 = 800 

𝑅𝑒𝑑 = 1100 

𝑅𝑒𝑑 = 1400 

𝑅𝑒𝑑 = 1800 

𝑅𝑒𝑑 = 2500 

Figure 50. Color contour maps of the mean (left) and rms (right) streamwise velocity normalized 
by the freestream velocity (𝑢̅/𝑈∞ and 𝑢𝑟𝑚𝑠/𝑈∞, respectively) in the wake of the 3Dp-5% 
cylinder in the Reynolds number range 𝑅𝑒𝑑 = 800-2500 in different rows. The trajectory of the 
peak 𝑢𝑟𝑚𝑠 in the top shear layer is marked, up to the vortex formation length. Flow direction is 
from left to right. 

103 

 
 
 
 
 
 
 
 
 
 
 
Mean centerline velocity in the wake of the 3Dp cylinder shown in Figure 51 suggests that 

closure length trend with Reynolds number flip-flops twice in the range of study. Specifically, 𝑙𝑐 

first decreases when 𝑅𝑒𝑑 is increased from 800 to 1100, then increases at 𝑅𝑒𝑑 = 1400, before 

starting to decrease again at higher Reynolds numbers (see Table 2 for numerical values). This is 

similar to the 𝑙𝑐-𝑅𝑒𝑑 trend of the 2D counterpart.  

The maximum reverse flow velocity increases with increasing the Reynolds number up to 

𝑅𝑒𝑑 = 1800 and remains similar within the errorbars at the higher Reynolds number. The location 

of the maximum reverse flow velocity, however, follows a more similar trend to closure length. It 

moves upstream  with  increasing the Reynolds  number with  an exception of  𝑅𝑒𝑑 = 1400 case, 

where it occurs farther downstream. The 2Dp cylinder also showed a similar trend.  

The rate of centerline velocity recovery in the far wake of this cylinder shows a closer trend 

to an asymptotic state 𝑢̅𝑐/𝑈∞ → 1 at 𝑅𝑒𝑑 = 2500. The recovery at lower Reynolds numbers is 

also better compared to the 2D counterpart. Although not shown here directly, but the streamwise 

evolution of 𝑢̅𝑐 after the wake closure for this case, at all Reynolds numbers except 𝑅𝑒𝑑 = 2500, 

is lower than the smooth cylinder. At 𝑅𝑒𝑑 = 2500, this evolution is very similar to the smooth 

case.  

104 

 
 
 
 
Figure 51. Comparison between the streamwise evolution of the mean centerline velocity in the 
wake of the 3Dp cylinder at the Reynolds number range 𝑅𝑒𝑑 = 800 − 2500. The error bars for 
most cases are smaller than the marker size. 

4.1.5 3Dv-5% Cylinder 

Figure 52 shows the mean and rms velocity contour maps in the wake of the other cross-

section of the three-dimensional geometry, the 3Dv case. The mean velocity contour map at 𝑅𝑒𝑑 =

800 suggests the vortex formation region is separated from the reverse flow region attached to the 

trailing edge, which was also observed for the 3Dp case. However, the latter reverse flow region 

stretches farther downstream from the cylinder’s base. This is also true for the secondary reverse 

flow region  associated with the vortex formation region. At higher Reynolds numbers, the two 

reverse flow regions join together and there is a single reverse flow zone. Increasing the Reynolds 

number  also  results  in  a  monotonic  decrease  in  the  size  of  this  reverse  flow  region.  This 

monotonicity was not observed for any other case.  

The  rms  velocity  contour  maps  in  Figure  52  show  a  relatively  confined  region  of  high 

unsteadiness, after a rapid widening  of the shear  layer.  As the Reynolds  number increases, the 

widening of the shear layer monotonically shifts closer to the trailing base of the cylinder.  

105 

 
 
 
𝑅𝑒𝑑 = 800 

𝑅𝑒𝑑 = 1100 

𝑅𝑒𝑑 = 1400 

𝑅𝑒𝑑 = 1800 

𝑅𝑒𝑑 = 2500 

Figure 52. Color contour maps of the mean (left) and rms (right) streamwise velocity normalized 
by the freestream velocity (𝑢̅/𝑈∞ and 𝑢𝑟𝑚𝑠/𝑈∞, respectively) in the wake of the 3Dv-5% 
cylinder in the Reynolds number range 𝑅𝑒𝑑 = 800-2500 in different rows. The trajectory of the 
peak 𝑢𝑟𝑚𝑠 in the top shear layer is marked, up to the vortex formation length. Flow direction is 
from left to right. 

Mean centerline velocity in the wake of the 3Dv cylinder is presented in  Figure 53 and 

shows a monotonic decrease of closure length with increasing the Reynolds number (see Table 2 

for numerical values). The maximum reverse flow velocity remains constant within the error bars 

from  𝑅𝑒𝑑 = 800 − 1400  and  starts  increasing  as  the  Reynolds  number  further  increases.  The 

location of the maximum reverse flow velocity is shifted upstream in a monotonic fashion with 

106 

 
 
 
 
 
 
 
 
 
 
 
increasing the Reynolds number. In general, both the 2D and 3D geometries with a valley at the 

leading edge generally seem to show a more systematic trend of the reverse flow shrinkage with 

increase of the Reynolds number. 

As the Reynolds number increases, the rate of centerline velocity recovery in the far wake 

of this cylinder becomes closer to an asymptotic state. The recovery at lower Reynolds number 

range  𝑅𝑒𝑑 = 800 − 1400  is  comparable  to  the  2D  counterpart,  but  the  variation  of  surface 

topology along the span improves the recovery at the higher Reynolds numbers. In addition, the 

streamwise evolution of 𝑢̅𝑐 after the wake closure is generally faster at higher Reynolds numbers. 

Figure 53. Comparison between the streamwise evolution of the mean centerline velocity in the 
wake of the 3Dv cylinder at the Reynolds number range 𝑅𝑒𝑑 = 800 − 2500. The error bars for 
most cases are smaller than the marker size. 

4.1.6 Discussion 

Within this subsection, a comparison and discussion will be conducted on the results that 

have  been  presented  in  sections  4.1.1  through  4.1.5.  The  streamwise  evolution  of  the  mean 

centerline velocity and the mean separation zone boundary for different cylinders discussed before 

are summarized here in Figure 54 and Figure 55. It can be seen that, except for 𝑅𝑒𝑑 = 800, the 

presence of a topology peak at the leading edge of the 2D geometry overall increases the closure 

107 

 
 
length.  The  effect  of  surface  topology  on  the  wake  closure  length  is  more  appreciable  at  the 

intermediate Reynolds numbers 1400 and 1800. Based on the boundary layer measurements in 

Chapter 3 summarized in Figure 55, what is distinct about the 2Dp geometry is that the separation 

zone is laterally larger compared to the other geometries, with the exception of 𝑅𝑒𝑑 = 2500. This 

is consistent with the clear difference in the development of 𝑢̅𝑐/𝑈∞ for this geometry. In this case, 

the recovery of the mean centerline velocity downstream of the closure length is slower. Some 

cases of the geometries with a valley at the leading edge have also shown open separation bubbles 

over the surface in Chapter 3 (2Dv at 𝑅𝑒𝑑 = 2500 and 3Dv at all Reynolds numbers), but for these 

cases,  the  shear  layer  becomes  very  close  to  reattaching.  It  is  believed  that  thanks  to  this,  the 

development of the centerline mean velocity does not vary significantly from that of the geometries 

with attached flow and with a closed separation bubble on the surface, such as the smooth cylinder. 

Therefore, it indicates that the wake behavior is not substantially altered solely by the presence of 

open or closed separation, but rather the size of the separation zone and the proximity of the shear 

layer  to  the  cylinder  surface  emerge  as  critical  factors  affecting  the  wake  behavior.  Moreover, 

when considering the 3Dv cases, it should be considered that the separation bubble is closed at the 

other  spanwise  cross-section  corresponding  to  3Dp  at  all  Reynolds  numbers,  therefore  the 

separation bubble is not open along the entire span of the cylinder.  

108 

 
 
 
 
Smooth 

2Dp 

3Dp 

2Dv 

3Dv 

Figure 54. Collection of the streamwise evolution of the mean centerline velocity in the wake of 
different cylinders in the Reynolds number range 𝑅𝑒𝑑 = 800 − 2500. The error bars for most 
cases are smaller than the marker size. 

109 

 
 
 
 
 
 
Figure 55. Comparison of the mean separation zone boundary for the cylinders with a peak (left) 
and a valley (right) at the leading edge at different Reynolds numbers. Flow is from left to right. 

Interestingly,  the  two  cross-sections  of  the  3D  geometry  show  similar  trends  with  the 

Reynolds number to their 2D counterparts. Overall, the geometries with a peak at the leading edge 

first show a decrease in 𝑙𝑐 at 𝑅𝑒 = 1100, followed by an increase at 𝑅𝑒𝑑 = 1400, and another 

subsequent decrease with further increasing the Reynolds number. While the geometries with a 

valley at the leading edge show a consistent reduction in the closure length as the Reynolds number 

increases, indicating a faster early development of the wake. The only anomaly in this trend is the 

2Dv at the highest Reynolds number, which was seen to have an open boundary layer separation 

110 

 
 
 
in Chapter 3. The smooth cylinder follows a similar trend to 2Dp, where increasing the Reynolds 

number  causes  𝑙𝑐  to  decreases  twice  at  𝑅𝑒𝑑 = 1100  and  2500.  In  addition,  for  both  peak  and 

valley  geometries,  the  presence  of  the  topology  variation  along  the  span  decreases  the  closure 

length at all Reynolds numbers, with the exception of the valley configurations at 𝑅𝑒𝑑 = 800. In 

this case, 𝑙𝑐 is smaller for the 2D geometry in the absence of the variation along the span.  

Considering  the  reverse-flow  magnitude,  the  maximum  negative  velocity  is  nearly  the 

same  for  the  smooth  cylinder  and  keeps  increasing  with  the  Reynolds  number  within  the  data 

scatter for the other geometries. The only exception is the 2Dp cylinder, showing a decrease in the 

maximum negative velocity at 𝑅𝑒𝑑 = 2500. For this case, flow reattaches to the surface which 

does not happen at lower Reynolds numbers.  

The vortex formation length, 𝑙𝑢, for the smooth cylinder exhibits a consistent monotonic 

decrease, considering the measurement uncertainty, as the Reynolds number increases. This trend 

is also observed in geometries with a topology valley at the leading edge. However, for geometries 

with a topology peak at the leading edge, specifically 2Dp at 𝑅𝑒𝑑 = 1400 and 1800, and 3Dp at 

𝑅𝑒𝑑 = 1400, higher  𝑙𝑢 values are evident  at  the intermediate Reynolds numbers.  Furthermore, 

among  all  cases  at  𝑅𝑒𝑑 ≥ 1400,  2Dp  displays  the  highest  𝑙𝑢  value.  It  is  worth  noting  that  the 

presence of surface topology variation along the span appears to increase 𝑙𝑢 at the lowest Reynolds 

number, when compared to the 2D counterparts, but this trend reverses at 𝑅𝑒𝑑 ≥ 1100. 

Finally, the rate of centerline velocity recovery in the far wake for the smooth cylinder and 

that with 3D topology at 𝑅𝑒𝑑 = 2500 indicate very similar behaviors and a wake closer to the 

asymptotic  state  𝑢̅𝑐/𝑈∞ → 1 than  the  cylinders  with  2D  topology.  In  addition,  the  streamwise 

evolution of 𝑢̅𝑐 for 3D case at this Reynolds number is very close to the smooth cylinder, which is 

not the case for the 2D topology. Overall, variation of the surface topology along the span improves 

111 

 
the wake recovery closer to the asymptotic state, compared to the 2D counterparts. It is unclear to 

what extent the far-field evolution might be affected by the blockage effects of the test section. 

This can be more extensively studied in future work.  

In summary, the data obtained from this study reveals that surface topology can introduce 

minor or significant variations not only in the quantitative but also in the qualitative behavior of 

the  boundary  layer  and  wake.  However,  the  extent  of  these  variations  depends  on  the  specific 

characteristics  of  the  topology.  For  instance,  when  comparing  2Dv  and  2Dp  with  the  same 

amplitude  and  wavelength  but  different  "phase"  topology,  significantly  different  behaviors  are 

observed  in  the  boundary  layer  and  wake.  It  becomes  apparent  that  predicting  the  behavior  of 

intermediate  phases  or  other  wavelengths,  let  alone  more  generalized  topology,  is  challenging 

based solely on the current data. 

4.2 Vortex Shedding Frequency 

To  measure  the  vortex  shedding  frequency,  power  spectral  density  (PSD)  results  are 

obtained for the fluctuating velocity reported at the cross-stream location showing the highest 𝑢𝑟𝑚𝑠 

across the wake far downstream, at 𝑥/𝑑 ≈ 5.8. The results are depicted in Figure 56 plotted against 

the Strouhal number, 𝑆𝑡𝑑 = 𝑓𝑑/𝑈∞, based on the baseline width of the cylinder, where the 𝑆𝑡𝑑 

uncertainty is ±0.002. Evidence of vortex shedding can be found from the presence of spectral 

peaks in the power spectral density (additional evidence based on phase-averaged velocity results 

will  be  shown  in  Chapter  5).  Specifically,  a  clear  peak  in  the  spectrum  is  found  for  all  cases, 

although the peaks are weaker for some cases. 𝑆𝑡𝑑 shows a non-monotonic behavior for the smooth 

cylinder, where it increases with Reynolds number up to 𝑅𝑒𝑑 = 1400, before it possesses a lower 

but  constant  value  at  higher  Reynolds  numbers  (see  Table  2  for  numerical  values).  It  was 

previously  observed  that  the  boundary  layer  is  attached  in  the  800 − 1400  Reynolds  number 

112 

 
range. At 𝑅𝑒𝑑 = 1800, the peak is less prominent and distributed over a wider band, and this case 

showed an open boundary layer separation. At 𝑅𝑒𝑑 = 2500, which showed a reattached boundary 

layer, the peak regains prominence and becomes narrower.  

The 2Dp cylinder shows hardly prominent PSD peaks, and those observed are distributed 

over a wider band compared to the other geometries at the studied Reynolds numbers. The Strouhal 

number associated with the vortex shedding is reduced with increasing the Reynolds number up 

to 𝑅𝑒𝑑 = 1800, but recovers at 𝑅𝑒𝑑 = 2500. In Figure 56, it is challenging to distinguish a PSD 

peak for this geometry at 𝑅𝑒𝑑 = 1800, and the shedding Strouhal number is here determined at 

the  location  of  the  maximum  PSD  value.  As  discussed  before,  the  boundary  layer  remains 

separated over the surface of this cylinder at the former Reynolds number range and reattaches at 

𝑅𝑒𝑑 = 2500.  Overall,  𝑆𝑡𝑑  is  the  lowest  for  the  2Dp  cylinder  at  all  Reynolds  numbers.  It  is 

noteworthy to mention that a distinctive characteristic of this case is the location of the separation 

zone boundary, which is farthest from the surface, as observed in Chapter 3. 

The 2Dv cylinder shows prominent peaks distributed over a narrow band at all Reynolds 

numbers,  although  this  prominence  starts  decaying  after  𝑅𝑒𝑑 = 1800.  𝑆𝑡𝑑  also  increases  with 

increasing the Reynolds numbers up to 𝑅𝑒𝑑 = 1400, but starts decreasing as the Reynolds number 

further increases. The boundary layer over the 2Dv cylinder, as discussed in Chapter 3, was seen 

to reattach to the surface at Reynolds numbers 𝑅𝑒𝑑 = 800 − 1400, but failed to do so at higher 

Reynolds numbers and the separation zone boundary became farther from the surface.  

𝑆𝑡𝑑 is equal for the two cross-sections of the 3D cylinder within the uncertainty, where 

both show equal strength and narrower peaks at the higher Reynolds number. 𝑅𝑒𝑑 = 1400 is an 

exception, where the PSD peak is also strong and narrow for the 3Dv case, but it becomes weak 

and distributed over a broader band for the 3Dp case. It was also seen in Figure 54 and Table 2 

113 

 
that  the  mean  centerline  velocity  evolution  and  closure  length  are  significantly  different  (over 

1.1𝑑) between the two cross-sections at this Reynolds number.  

The Strouhal number associated with the vortex shedding in the wake of the two cross-

sections of the 3D cylinder falls in between the 𝑆𝑡𝑑 values for the 2Dp and 2Dv cases at all of the 

studied Reynolds numbers. It is also always lower than the value for the smooth cylinder. In fact, 

the  smooth  cylinder  shows  the  highest  Strouhal  number  compared  to  all  cases  with  surface 

topology  over  the  studied  Reynolds  number  range.  It  is  only  at  𝑅𝑒𝑑 = 1800,  where  the  𝑆𝑡𝑑 

becomes similar between the smooth and 2Dv cylinders within the uncertainty. At 𝑅𝑒𝑑 = 1400, 

the PSD peaks for the valley geometries are strong, while those for the peak geometries decay and 

become very weak. The  latter geometries also  have their respective maximum  𝑙𝑐 values at  this 

Reynolds number. Overall, the effect of surface topology on the trend of the most wake features 

is  Reynolds  number  dependent,  which  suggests  there  are  different  flow  regimes  each  cylinder 

undergoes over the studied range of Reynolds numbers.  

114 

 
                                𝑅𝑒𝑑 = 800 

                              𝑅𝑒𝑑 = 1800 

                               𝑅𝑒𝑑 = 1100 

                           𝑅𝑒𝑑 = 2500 

                               𝑅𝑒𝑑 = 1400 

Figure 56. Power spectral density vs. Strouhal number based on the baseline cylinder width for 
the Reynolds number range 𝑅𝑒𝑑 = 800 − 2500, measured at 𝑥/𝑑 ≈ 5.8 and location of the 
highest fluctuations along the cross-stream direction. 

The influence of surface topology on the 𝑆𝑡𝑑 values of the PSD peak for different cases, 

reported  in  Table  2,  is  evident.  The  spread  in  𝑆𝑡𝑑  across  the  geometries  at  different  Reynolds 

115 

 
 
  
 
 
 
 
numbers can be seen in Figure 57 (top), where many of the datapoints deviate from the average 

for all geometries by more than 10%. This convergence of the 𝑆𝑡𝑑 values is also highly sensitive 

to Reynolds number, which demonstrates the strong Reynolds number dependency of the surface 

topology effect. A distinct example is the 2Dp cylinder showing the lowest value of 𝑆𝑡𝑑 = 0.15 

overall. When the boundary layer characteristics were discussed in Chapter 3, this cylinder was 

observed  to  develop  an  open  separation  bubble  at  the  lower  𝑅𝑒𝑑,  with  the  shear  layer  located 

farthest  from  the  surface  compared  to  the  other  geometries.  With  reference  to  the  literature 

reviewed  below,  it  is  thus  concluded  that  one  way  that  the  surface  topology  influences  vortex 

shedding frequency is by displacing the top and bottom shear layers away from each other in the 

near  wake.  The  presence  of  surface  topology  and  whether  it  is  2D  or  3D  affects  the 

separation/reattachment of the boundary layer and, as a result, alters the spacing between the shear 

layers in the immediate wake of the cylinders. It is discussed in the existing literature that the latter 

has a substantial effect on the wake and subsequent vortex shedding frequency.  

116 

 
 
 
 
 
 
 
Figure 57. Vortex shedding Strouhal number comparison for all cylinders. 𝑆𝑡𝑑 (top), 𝑆𝑡𝛥𝑟𝑚𝑠 
(middle), and 𝑆𝑡𝛥𝑠ℎ𝑒𝑎𝑟 (bottom). The solid line in the plot represents the average value at each 
Reynolds number, while the broken lines indicate ±10% deviation from the average. 

Okajima [5],  in  his study  of rectangular  cylinders, infers that the shedding frequency is 

roughly  inversely  related  to  the  wake  width.  This  hypothesis  was  presented  earlier  by  Fage  & 

Johansen [26]. They suggest calculating the Strouhal number using the distance between the shear 

layers in the near wake when they become parallel before “rolling-up” as the proper length scale 

117 

 
 
 
 
instead of the body dimensions, and postulate that doing so helps correlate the vortex shedding 

frequency  and  leads  to  approaching  a  universal  Strouhal  number.  Roshko  [27]  developed  an 

inviscid model based on the free-streamline theory [28] to predict the drag of the bluff bodies and 

made use of the idea that the Strouhal number becomes universally a function of Reynolds number 

if scaled based on the separation between the shear layers. 

One  definition  of  the  center  of  the  shear  layer  used  in  the  literature  is  at  the  transverse 

location where the maximum streamwise velocity fluctuations occur [23-25]. Thus, the distance 

between the 𝑦 locations of the maximum 𝑢𝑟𝑚𝑠 across each 𝑥/𝑑 can be used as a way to calculate 

the distance between the shear layers 𝛥𝑟𝑚𝑠 (see Figure 44). The studies [23-25] also consider the 

vortex  formation  (roll-up)  region  for  circular  cylinders  to  be  the  immediate  wake,  up  to  the 

streamwise location of the maximum 𝑢𝑟𝑚𝑠, or up to the streamwise location where the shear layers 

have the narrowest cross-stream distance from each other. Following the literature, the trajectory 

of the shear layer center is here defined as the location of the maximum 𝑢𝑟𝑚𝑠 at each streamwise 

location and is extracted from the boundary layer and wake measurements, up to the formation 

length 𝑙𝑢 (see Figure 44), and subsequently superposed on the color contour maps of the fluctuating 

velocity 𝑢𝑟𝑚𝑠 in the wake as seen in Figure 42, Figure 46, Figure 48, Figure 50, and Figure 52. 

The shear layer separation, 𝛥𝑟𝑚𝑠, is then quantified as the cross-stream distance between the 𝑢𝑟𝑚𝑠 

maxima on the top and the bottom sides of the cylinder at the most upstream measurement location 

in wake measurements, 𝑥/𝑑 ≈ 1.2 (see Figure 44). The results are listed in Table 2. This location 

is well within the formation region, since it is located upstream of the location with the highest 

velocity fluctuations. The Strouhal number values based on 𝛥𝑟𝑚𝑠 are also listed in Table 2, and 

plotted in Figure 57 in red. 

118 

 
A qualitative comparison between the black and red datapoints in Figure 57 demonstrates 

a significant decrease in the spread of the Strouhal number of the PSD peak for different cylinders 

when  the  Strouhal  number  is  calculated  based  on  𝛥𝑟𝑚𝑠  at  the  lower  Reynolds  numbers.  Such 

convergence,  however,  is  not  seen  at  the  higher  Reynolds  number  of  𝑅𝑒𝑑 = 2500.  As  will  be 

further discussed in Chapter 5, the shear layer over the surface of the cylinders becomes unsteady 

at  this  Reynolds  number  and,  in  some  cases,  rolls  up  to  form  vortices  upstream  of  the  wake. 

Therefore, the vortex shedding mechanism needs to be distinguished for the cases with and without 

shear layer roll-up upstream of the trailing edge.  

In  another  attempt  to  further  reduce  the  variability  in  Strouhal  number  among  different 

geometries, the maximum separation between the centers of the top and bottom shear layers over 

the surface of the cylinders 𝛥𝑠ℎ𝑒𝑎𝑟 is used as the length scale to calculate 𝑆𝑡𝛥𝑠ℎ𝑒𝑎𝑟. The center of 

the shear layer in this analysis is defined as the location of the maximum 

𝑑𝑢̅

𝑑𝑦

 from the mean velocity 

profiles. Results of this analysis are plotted in blue datapoints in Figure 57 (bottom) and compared 

against the Strouhal numbers calculated using cylinder width 𝑑 and wake width 𝛥𝑟𝑚𝑠. A qualitative 

comparison demonstrates an improved collapse of the 𝑆𝑡𝛥𝑠ℎ𝑒𝑎𝑟 datapoints at all Reynolds numbers 

compared to 𝑆𝑡𝑑. However, calculating the Strouhal number this way results in a slightly worse 

collapse at the intermediate Reynolds numbers 𝑅𝑒𝑑 = 1400 and 1800.  

Comparison of the spread in 𝑆𝑡𝛥𝑟𝑚𝑠 and 𝑆𝑡𝛥𝑠ℎ𝑒𝑎𝑟 versus 𝑆𝑡𝑑 is quantitatively done for all cases, 

excluding the 3D geometry cases and is reported in Table 3. 3D cases are excluded since 𝛥𝑟𝑚𝑠 and 

𝛥𝑠ℎ𝑒𝑎𝑟 are found to be different in the two spanwise planes of measurement, i.e., 3Dp and 3Dv, 

and it is not known which of these values would be more relevant, or how to appropriately define 

a single shear layer separation distance for the cylinder in the presence of flow three dimensionality 

which is seen in the preliminary visualizations of the spanwise velocity.  

119 

 
Table 3. Comparison of the Strouhal number spread when calculated based on the baseline width 
𝑑, wake width 𝛥𝑟𝑚𝑠, and maximum separation of top and bottom shear layers 𝛥𝑠ℎ𝑒𝑎𝑟. 3D cases 
are excluded. 

(𝑆𝑡𝑑)𝑚𝑎𝑥 − (𝑆𝑡𝑑)𝑚𝑖𝑛
𝑆𝑡𝑑̅̅̅̅

  𝑆𝑡̅ 𝑑 

(𝑆𝑡Δ𝑟𝑚𝑠)𝑚𝑎𝑥 − (𝑆𝑡Δ𝑟𝑚𝑠)𝑚𝑖𝑛
̅̅̅̅̅̅̅̅
𝑆𝑡Δ𝑟𝑚𝑠

  𝑆𝑡̅ Δrms 

(𝑆𝑡Δ𝑠ℎ𝑒𝑎𝑟)𝑚𝑎𝑥 − (𝑆𝑡Δ𝑠ℎ𝑒𝑎𝑟)𝑚𝑖𝑛
̅̅̅̅̅̅̅̅̅̅
𝑆𝑡Δ𝑠ℎ𝑒𝑎𝑟

  𝑆𝑡̅ Δ𝑠ℎ𝑒𝑎𝑟 

𝑅𝑒𝑑

= 800 

𝑅𝑒𝑑

= 1100 

𝑅𝑒𝑑

= 1400 

𝑅𝑒𝑑

= 1800 

𝑅𝑒𝑑

= 2500 

23.8% 

0.21 

3.4% 

0.29 

0.1% 

36.9% 

0.22 

10.3% 

0.29 

6.8% 

49.5% 

0.22 

9.9% 

0.30 

13.2% 

40.0% 

0.20 

11.1% 

0.27 

18.3% 

19.7% 

0.20 

20.0% 

0.27 

7.5% 

0.30 

0.28 

0.30 

0.27 

0.30 

 The spread is seen to decrease significantly when using length scales representative of the wake 

width in comparison to the Strouhal number calculated based on the baseline cylinder width at all 

Reynolds numbers. The only exception is 𝑅𝑒𝑑 = 2500, where  𝑆𝑡̅ Δrms shows a comparable spread. 

As mentioned before, the wake width results in a better convergence at the intermediate Reynolds 

numbers  𝑅𝑒𝑑 = 1400  and  1800,  and  the  maximum  shear  layer  separation  results  in  the  best 

convergence at other Reynolds numbers.  

The  spread  for  each  Strouhal  number  definition  is  also  calculated  for  all  the  cylinders, 

excluding  the  3D  geometry  cases,  across  the  Reynolds  number  range.  It  is  determined  as  the 

difference  between  the  maximum  and  minimum  Strouhal  number  values,  expressed  as  a 

percentage  of  the  average  value,  to  assess  the  independence  or  sensitivity  of  the  definition  to 

Reynolds number. The results show that the spread is 11.1%, 19.0%, and 12.9% of the average 

values (0.21, 0.28, and 0.29) for 𝑆𝑡𝑑, 𝑆𝑡Δ𝑟𝑚𝑠, and 𝑆𝑡Δshear, respectively. Therefore, defining the 

120 

 
 
Strouhal number based on the wake width or the separation between the shear layers does not lead 

to a stronger independence of Reynolds number. 

Altogether,  these  results  suggest  that  changing  the  initial  wake  width  (shear  layers 

separation in the near wake) and the maximum separation of the top and bottom shear layers are 

important influences of surface topology on the vortex shedding frequency. This is caused by the 

topology effect on the boundary layer separation/reattachment behavior on the cylinder’s surface. 

However, altering the wake width and shear layers separation are not the sole influences, and there 

are other mechanisms through which the topology alters the shedding frequency. For instance, the 

maximum 𝑢𝑟𝑚𝑠 velocity over the surface of the 2Dp cylinder at 𝑅𝑒𝑑 = 2500 (see Figure 46) is 

larger than that in the wake. As noted earlier, there is a signature of vortex formation in the shear 

layer above the side surface of the 2Dp cylinder towards its trailing edge at this Reynolds number 

which will be discussed in detail in Chapter 5. In addition, the effect of topology on each of these 

length scales becomes more significant at certain Reynolds numbers. 

Using  the  wake  width  and  maximum  shear  layer  separation  as  the  length  scales  in 

calculating the Strouhal number for the 3D cases might be sufficient at lower Reynolds numbers 

when the flow is rather two dimensional. However, at higher Reynolds numbers, the flow is likely 

three-dimensional over the surface and in the wake of the 3D cylinder, and as a result, the behavior 

over separate cross-sections cannot be studied in isolation.  

121 

 
 
 
 
Chapter 5. On the Unsteady Behavior of the Flow  

The previous chapters covered analysis of the main statistics of the boundary layer and the 

wake  flow  of  the  rectangular  cylinders  without/with  surface  topology  in  a  range  of  Reynolds 

numbers. In this chapter, an attempt is made to complement this investigation to learn more about 

the flow physics by focusing on some of the key unsteady flow features. Moreover, the connection 

between  the  boundary  layer  and  wake  flow  unsteady  characteristics  is  explored  by  utilizing  a 

combination of quantitative analysis and qualitative flow visualization.  

The findings in Chapter 3 showed a Reynolds number dependent effect on the boundary 

layer unsteady characteristics (𝑢𝑟𝑚𝑠/𝑈∞). However, the question of what drives this unsteadiness 

remains unanswered. For example, it is unclear if transition of the separating shear layer, wake 

vortex shedding, some other flow feature, or combination of effects dominate the boundary layer 

unsteadiness. Chapter 4 demonstrated the effect of surface topology and Reynolds number on the 

wake  flow  and  the  vortex  shedding  frequency.  It  is  yet  to  be  determined,  however,  how  the 

boundary  layer  behavior  might  connect  to  the  wake  vortex  shedding  frequency,  and  the  wake 

vortex  structure  organization  and  phase-average  signature.  These  points  are  examined  in  this 

chapter. In the interest of conciseness, the focus of this examination is primarily on the smooth 

cylinder geometry.  

Before presenting results, a brief summary is made of the existing understanding of the 

flow physics over the rectangular cylinder geometry in order to help draw connections between 

the boundary layer and wake flow characteristics in later discussions. Unfortunately, the existing 

literature on the rectangular geometry with round corners is very limited. With the exception of 

the work of Parker & Welsh [6], the relevant information can only be found on the sharp-corner 

rectangular  cylinders.  Even  the  study  of  Parker  &  Welsh  does  not  provide  basis  for  a  direct 

122 

 
comparison, since it focuses on the effect of side ratio on the shedding frequency and boundary 

layer behavior at much higher Reynolds numbers compared to the present study. Moreover, the 

study  does  not  examine Reynolds  number  effect.  Therefore,  sharp-corner  rectangular  cylinders 

literature will be used as a background to help discuss the key elements of boundary layer and 

wake connections. While boundary layer separation is fundamentally different between the sharp- 

and round-corner cylinders (fixed versus non-fixed separation point), subsequent behavior of the 

separating  shear  layer  upstream  of  the  wake  has  common  qualitative  features  between  the  two 

geometries. This includes the formation of open or closed separation bubble, shear layer transition, 

and shear layer and wake vortex formation.  

Numerous  studies  have  been  done  on  the  flow  around  rectangular  cylinders  with  sharp 

corners.  One  important  geometrical  feature  of  these  cylinders  that  can  strongly  affect  the  flow 

behavior is the side ratio and the associated length of the afterbody, which is defined as the length 

of the cylinder’s surface downstream of the separation location of the shear layers that roll up into 

the wake vortices [4]. As illustrated in Figure 58 (top), for a sharp-corner cylinder, the afterbody 

normalized by 𝑑 can be either the same as 𝑐/𝑑 for open separation, or zero for closed separation 

bubble (i.e. the shear layer reattaches then separates again at the TE to form the wake vortices). 

For the round-corner cylinder, the length of the afterbody for open separation is less than 𝑐/𝑑 and 

depends on Reynolds number. For closed separation, the afterbody length will be larger than zero 

and depends on where separation occurs after reattachment (Figure 58, bottom).  

123 

 
 
 
Figure 58. Illustration of the afterbody length 𝑙𝑎𝑓𝑡 for a sharp- (top) and round-corner (bottom) 
cylinders. The broken blue lines indicate the trajectory of separated shear layers, the black arrow 
represents the separation point of the shear layer that rolls up into the wake vortices farther 
downstream. The extent of the afterbody is marked by the red arrows. 

Okajima [5] studied the velocity spectra in the wake of sharp-corner rectangular cylinders 

with  different  side  ratios  in  the  Reynolds  number  range  𝑅𝑒𝑑 = 70 − 20,000.  Measuring  the 

Strouhal  number  associated  with  the  dominant  frequency,  he  found  that  the  Strouhal  number 

remains  approximately  in  the  range  0.12 − 0.14  for  most  of  the  Reynolds  number  range  for  a 

square cylinder (𝑐/𝑑 = 1). A plot of his results is reproduced in Figure 59.  

Regime 2 

Figure 59. Strouhal number in the wake of rectangular cylinders with side ratio 𝑐/𝑑 = 1 
(square). Based on results from Okajima [5]. Sketch depicts the cylinder geometry with the 
separated shear layers indicated with broken lines. 

124 

 
 
 
 
 
 
The cylinders with side ratio 𝑐/𝑑 = 2 and 3 (Figure 60 and Figure 61, respectively) show an abrupt 

drop of the dominant Strouhal  number with  increasing Reynolds number. No additional abrupt 

jump of the Strouhal number is demonstrated with further increase in the Reynolds number for the 

cylinder with side ratio 𝑐/𝑑 = 2, as can be seen in Figure 60. However, the Strouhal number for 

the cylinder with 𝑐/𝑑 = 3 recovers at a high enough Reynolds number to a value close to the low-

Reynolds-number value before the drop (see Figure 61). 

Regime 1 

Regime 2 

Figure 60. Strouhal number in the wake of rectangular cylinders with side ratio 𝑐/𝑑 = 2. Based 
on results from Okajima [3]. Sketches depict the cylinder geometry with the separated shear 
layers indicated with broken lines. 

Regime 1 

Regime 2 

Regime 3 

Figure 61. Strouhal number in the wake of rectangular cylinders with side ratio 𝑐/𝑑 = 3. Based 
on results from Okajima [3]. Sketches depict the cylinder geometry with the separated shear 
layers indicated with broken lines. 

125 

 
 
 
Analyzing  flow  visualization  images,  Okajima  explains  the  connection  between  the 

boundary layer behavior and the Strouhal number of the wake vortex shedding. He reveals the 

existence of different flow regimes based on the geometry’s side ratio. It is well known that the 

boundary layer separation point is fixed on the sharp corners of the leading edge. For the cylinders 

with very small side ratio, there is not enough body length for the separated shear layer to reattach 

to the surface. Therefore, the open separation of the boundary layer exists over the full range of 

Reynolds numbers, and the wake is relatively wide and the Strouhal number is generally low and 

does not show an abrupt change (Figure 59).  

For the cylinder with 𝑐/𝑑 = 2, the separated shear layer reattaches to the surface at very 

low  Reynolds  numbers  (this  is  labeled  as  Regime  1  in  Figure  60),  but  the  separation  bubble 

lengthens  as the Reynolds number increases up  to  a point that the shear  layer fails to  reattach. 

Opening up of the separation bubble is associated with abrupt drop in the Strouhal number and 

widening  of  the  wake.  This  situation  corresponds  to  switching  from  Regime  1  to  Regime  2  in 

Figure 60. Power spectra of the fluctuating velocity however, reveal that in this switch there is a 

transient range of Reynolds numbers where both  the high and low frequencies in the wake co-

exist. As the Reynolds number increases in Regime 2, the lower frequency gains energy and the 

higher frequency gradually fades away. Okajima explains this by pointing out that in this Reynolds 

number  range,  the  boundary  layer  intermittently  switches  between  fully  open  separation  and 

reattachment over different cycles. In other words, although the mean velocity profile may suggest 

either reattached flow or open separation, instantaneously, the boundary layer may in fact reattach 

or fail to do so occasionally in this transient region. Further increase of the Reynolds number in 

the range studied by Okajima does not cause the separated shear layer to reattach again, hence the 

absence of a Strouhal number recovery. 

126 

 
The same observations hold for the cylinder with the higher side ratio 𝑐/𝑑 = 3, except that 

in addition to the Strouhal number associated with Regime 1, there are two PSD peaks observed 

when  transitioning  from  Regime  1  to  Regime  2.  Okajima  reports  as  many  as  three  Strouhal 

numbers at some Reynolds numbers. For this cylinder, at a high enough Reynolds number, the 

shear  layer  exhibits  transition  and  vortex  formation  above  the  cylinder’s  sidewall,  leading  to 

reattachment of the boundary layer and narrowing of the wake, which causes a recovery of the 

Strouhal number. This is labeled as Regime 3 in Figure 61. When the afterbody is long enough, 

which is the case for the cylinder with 𝑐/𝑑 = 4 in Okajima’s study, the maximum length of the 

separation bubble over the full range of Reynolds numbers is shorter  than 𝑐 and the separation 

bubble never becomes open. Therefore, no abrupt changes of the Strouhal number are observed 

with increasing Reynolds number (these results are not depicted here). Subsequent studies (e.g. 

Norberg [29] and references therein) confirmed the results of Okajima [3] and provided similar 

interpretations of the connections between the boundary layer behavior and the vortex shedding 

Strouhal number. It is notable that generally, previous studies have relied on flow visualization in 

interpreting the boundary layer behavior. The present work is unique in that it takes advantage of 

the high-resolution capability of 1c-MTV to analyze the boundary layer using velocity data. 

5.1 Nature of the Boundary Layer Unsteadiness 

Power spectral density is calculated to investigate the dominant frequencies of the velocity 

fluctuation in the boundary layer as the first step to analyze the boundary layer unsteadiness. PSD 

analysis  is  carried  out  at  two  streamwise  locations  of  𝑥/𝑑 = 0 and  0.71  above  the  cylinder’s 

surface,  and  the  cross-stream  location  of  the  maximum  fluctuations  (𝑢𝑟𝑚𝑠).  The  streamwise 

locations are picked to represent the boundary layer behavior at the mid-cord and just upstream of 

the trailing edge rounded corner (ahead of the start of the wake development), respectively. The 

127 

 
results are then compared against the PSD analysis in the “far” wake at 𝑥/𝑑 = 6 to investigate the 

extent by which the boundary layer fluctuations are influenced by the wake vortex formation. The 

results are shown in Figure 62 and Figure 63 using a logarithmic and a linear ordinate, respectively. 

𝑅𝑒𝑑 = 800 

𝑅𝑒𝑑 = 1800 

𝑅𝑒𝑑 = 1100 

𝑅𝑒𝑑 = 2500 

𝑅𝑒𝑑 = 1400 

Figure 62. Power spectral density vs. Strouhal number of the smooth cylinder in the Reynolds 
number range 𝑅𝑒𝑑 = 800 − 2500 measured at different streamwise locations and at the location 
of the highest fluctuations along the cross-stream direction. 𝑦-axis is logarithimc to make local 
peaks visible regardless of strength.  

128 

 
 
   
  
 
 
 
 
 
𝑅𝑒𝑑 = 800 

𝑅𝑒𝑑 = 1800 

𝑅𝑒𝑑 = 1100 

𝑅𝑒𝑑 = 2500 

𝑅𝑒𝑑 = 1400 

Figure 63. Power spectral density vs. Strouhal number of the smooth cylinder in the Reynolds 
number range 𝑅𝑒𝑑 = 800 − 2500 measured at different streamwise locations and at the location 
of the highest fluctuations along the cross-stream direction. 𝑦-axis is linear to better appreciate 
the relative importance of spectral peaks. 

Similar to the results from Chapter 4, Figure 62 and Figure 63 depict a clear and strong 

spectral  peak  at  𝑥/𝑑 = 6  corresponding  to  the  vortex  shedding  frequency  for  all  Reynolds 

numbers. At the lowest two Reynolds numbers, 𝑅𝑒𝑑 = 800 and 1,100, no peak is found at the 

129 

 
 
 
 
 
 
 
shedding frequency in the PSD of the boundary layer (𝑥/𝑑 = 0 and 0.71). This indicates that the 

boundary layer unsteadiness at these low Reynolds numbers is unaffected by vortex shedding. At 

𝑅𝑒𝑑 = 800,  the  vortex  formation  length  is  𝑙𝑢/𝑑 = 2.45  (see  Table  1  in  Chapter  4),  which  is 

significantly larger than the rest of the Reynolds numbers. It is possible that in this case, vortex 

formation is occurring too far downstream of the cylinder to practically affect the boundary layer 

flow. The same argument, however, cannot be used for  𝑅𝑒𝑑 = 1,100 since in this case 𝑙𝑢/𝑑 =

1.8, which is practically the same as 𝑙𝑢/𝑑 = 1.81 for 𝑅𝑒𝑑 = 1,400. The latter Reynolds number 

shows signs of the wake vortex formation affecting the boundary layer flow (Figure 62). Thus, the 

absence  of  the  wake  effect  at  𝑅𝑒𝑑 = 1100  is  likely  not  caused  by  the  remoteness  of  vortex 

shedding from the cylinder. However, as can be seen from Figure 62 and Figure 63, the strength 

of the PSD peak of vortex shedding (at 𝑥/𝑑 = 6) is higher at 𝑅𝑒𝑑 = 1400, compared to 1100, 

and the associated higher vortex strength may be the cause that vortex shedding unsteadiness is 

felt at 𝑅𝑒𝑑 = 1400 but not 1100 despite the similar vortex formation length.  

At  the  higher  Reynolds  numbers  of  𝑅𝑒𝑑 = 1800  and  2500,  the  wake  vortex  shedding 

frequency  is  clearly  visible  in  the  boundary  layer  frequency  content,  indicating  an  interaction 

between the wake vortex shedding and boundary layer behavior. In fact, at 𝑅𝑒𝑑 = 2500, the peak 

corresponding to vortex shedding is the largest in the PSD of the boundary layer. At this Reynolds 

number, vortex formation is the closest to the cylinder with 𝑙𝑢/𝑑 = 1.11. 

Irrespective of whether the vortex shedding frequency is present or absent in the PSD of 

the  boundary  layer,  with  the  exception  of  𝑅𝑒𝑑 = 2500,  the  boundary  layer  unsteadiness  is 

dominated by low frequency fluctuation. Close inspection of Figure 62, shows that the PSD peaks 

at low Strouhal number systematically shift towards a lower 𝑆𝑡𝑑 with increasing Reynolds number. 

Since 𝑅𝑒𝑑 is increased by increasing 𝑈∞, this effect could correspond to peaks that are at a fixed 

130 

 
physical  frequency in  Hz. This  point is  checked  in  Figure  64, where the  PSD for all Reynolds 

numbers and 𝑥/𝑑 = 0.71 are plotted versus frequency in Hz on the same graph over a frequency 

range that is focused at low frequencies. The results do indeed confirm the fixed nature of this 

frequency, which is tracked to the slushing frequency of the water tunnel (the same frequency is 

also found in the freestream). These fluctuations, however, are not significant and become smaller 

at higher Reynolds numbers (e.g. 1.9% and 1.6% of 𝑈∞ at 𝑅𝑒𝑑 = 800 and 1400, respectively at 

𝑥/𝑑 = 0.71) and we expect them to cause low-level low-frequency modulation of the flow without 

affecting  the  flow  behavior.  Thus,  in  an  ideal  facility  with  zero  freestream  fluctuation,  the 

boundary layer around the cylinder at 𝑅𝑒𝑑 = 800 and 1100 should be steady. As the Reynolds 

number  is  increased  to  𝑅𝑒𝑑 = 1400  and  beyond,  vortex  shedding  leads  to  boundary  layer 

unsteadiness  with  the  effect  becoming  stronger  with  increasing  𝑅𝑒𝑑,  at  least  within  the  range 

investigated. 

Figure 64. Demonstration of the signature of the sloshing frequency of the water tunnel on the 
measured velocity spectra. Power spectral density vs frequency of the smooth cylinder in the 
Reynolds number range 𝑅𝑒𝑑 = 800 − 2500 measured at 𝑥/𝑑 = 0.71 and at the location of the 
highest fluctuations along the cross-stream direction. Focusing on low frequencies to zoom on 
the trace of the tunnel sloshing. 

131 

 
 
In  the  Reynolds  number  range  𝑅𝑒𝑑 = 800 − 1400,  power  spectral  density  in  the  wake 

shows a clear sign of vortex shedding at a Strouhal number which keeps increasing with Reynolds 

number (black lines in Figure 62 and Figure 63). The second harmonic is also seen in the spectrum 

at twice the shedding frequency for 𝑅𝑒𝑑 = 800 and 1100. In Chapter 3, the boundary layer mean 

velocity profile indicated a reattached boundary  layer over the surface of this cylinder  at  these 

Reynolds numbers. These characteristics of a reattached boundary layer and increasing 𝑆𝑡𝑑 with 

𝑅𝑒𝑑 are consistent with Regime 1 behavior of the sharp-corner cylinder in Figure 60 and Figure 

61. 

As  the  Reynolds  number  increases  further  to  𝑅𝑒𝑑 = 1800,  the  boundary  layer  mean 

separation  bubble  becomes  open  (Chapter  3)  and  the  vortex  shedding  𝑆𝑡𝑑  drops  from  0.27  at 

𝑅𝑒𝑑 = 1400  to  0.22  at  𝑅𝑒𝑑 = 1800.  These  features  are  analogous  to  Regime  1-Regime  2 

transition in Figure 60 and Figure 61. It is interesting to note that the PSD in the wake at 𝑅𝑒𝑑 =

1400 shows the emergence of the lower 𝑆𝑡𝑑 peak at 0.22. At 𝑅𝑒𝑑 = 1800, that secondary peak 

at  lower  frequency  takes  over  and  becomes  dominant  while  the  higher  frequency  disappears. 

Additional increase in 𝑅𝑒𝑑 to 2500 maintains 𝑆𝑡𝑑 at 0.22, although at this Reynolds number the 

shear layer reattaches on the cylinder sidewall (Chapter 3).  

The switch from open to closed separation bubble with 𝑅𝑒𝑑 increase from 1800 to 2500 

seems to correspond to Regime 2- Regime 3 change for the sharp-corner cylinder in Figure 61. 

However, as discussed by Okajima [5], such a regime change is associated with Kelvin-Helmholtz 

vortex  formation  in  the  separated  shear  layer  above  the  cylinder  sidewall.  Inspection  of  flow 

visualization images in this study shows that such vortex formation does not occur for the smooth 

cylinder at any of the Reynolds numbers investigated. To demonstrate this point, Figure 65 shows 

a comparison between instantaneous flow visualization snapshots of the smooth cylinder at 𝑅𝑒𝑑 =

132 

 
2500 and the 2Dp-5% cylinder at the same Reynolds number. The latter geometry provides a case 

in which vortex formation occurs in the separated shear layer above the cylinder. Specifically, a 

vortex formation (highlighted by a red arrow in Figure 65-a) is observed over a surface topology 

peak,  as  inferred  from  the  distortion  of  the  MTV  lines  in  Figure  65-a.  This  vortex  travels 

downstream and remains close enough to the surface to impose reattachment just downstream of 

its location, as seen in Figure 65-b. Figure 65-c depicts the shedding of a single shear layer vortex 

into the wake. Similar MTV lines distortion is absent from the visualization images of the smooth 

cylinder in Figure 65. 

The  absence  of  vortex  formation  above  the  cylinder  sidewall  when  the  shear  layer  is 

reattached for the smooth cylinder at 𝑅𝑒𝑑 = 2500 is consistent with the flow behavior in Regime 

1  in  Figure  60  and  Figure  61  for  the  sharp-corner  cylinder.  This  suggests  that  the  rectangular 

cylinder with fully-round LE does not exhibit monotonic regime transitions with increasing 𝑅𝑒𝑑 

as the sharp-corner cylinder. Specifically, in terms of the boundary layer behavior, as the Reynolds 

number increases, we observe transition from Regime 1 (𝑅𝑒𝑑 = 800, 1100 and 1400) to Regime 

2 (𝑅𝑒𝑑 = 1800) then back to  Regime  1 (𝑅𝑒𝑑 = 2500). This  is believed to  be due to  the more 

complex nature of separation/reattachment on the cylinder with round LE, where the separation 

point is not fixed. Moreover, unlike the sharp-corner cylinder where separation occurs on the front 

face, for the round-LE cylinder separation is always observed on the flat part of the sidewall for 

the cases investigated here. This generally leads to significantly thinner separation zone for the 

round-LE cylinder compared to the sharp-corner cylinder. In turn, the difference in the separated 

shear layer trajectory between open- and closed separation bubble is not as significant, and hence 

the switch between the two states may be sensitive to various factors, such as movement of the 

separation  point,  proximity  of  wake  vortex  formation  location  to  the  cylinder,  etc.  It  is  also 

133 

 
noteworthy that although the boundary layer behavior at 𝑅𝑒𝑑 = 2500 corresponds to Regime 1 of 

the sharp-corner cylinder, the Strouhal number remains at 0.22 similar to 𝑅𝑒𝑑 = 1400 and does 

not exhibit the increase expected from narrowing of the wake associated with reattachment.  

(a) 

(b) 

 

 

 

(c) 
Figure 65. Snapshots from visualization of the boundary layer flow over the smooth cylinder 
(left) and 2Dp-5% cylinder (right) at 𝑅𝑒𝑑 = 2500. Red line marks MTV lines distortion 
implying presence of a vortex.  

To achieve insight into the instantaneous separation/reattachment flow behavior, forward 

flow probability (FFP) is calculated. FFP, which represents the probability of a positive velocity, 

is calculated at the closest valid pixel to the surface by counting the number of times a positive 

velocity occurs in a timeseries and dividing the result by the number of points in the time series. 

The  results  are  demonstrated  in  Figure  66,  plotted  versus  the  streamwise  location  over  the  flat 

134 

 
 
 
 
 
 
 
cylinder sidewall. A polynomial curve fit is applied to the data to help visualize the overall trend 

despite data scatter.  

FFP of 50% is sometimes used to indicate locations of separation and reattachment. This 

works well if the probability density function of the velocity is symmetric at these locations. In 

Figure 66, the mean separation/reattachment locations based on the mean velocity measurements 

(from Chapter 3) are superposed as vertical lines on the FFP. These agree well with the locations 

where the FFP plot crosses 50% level (indicated with the horizontal broken line). Inspecting Figure 

66,  the  FFP  is  lower  than  50%  over  a  certain  region  of  the  surface  at  𝑅𝑒𝑑 = 800  and  1400, 

indicating the presence of separation followed by reattachment. The case of 𝑅𝑒𝑑 = 1100 shows 

higher  than  50%  probability  over  the  whole  surface,  implying  the  absence  of  boundary  layer 

separation  on average.  These observations are  completely in  line with  the mean flow behavior 

discussed earlier in Chapter 3. Some deviation in the instantaneous behavior from this mean flow 

picture are however implied from the FFP. Specifically, at 𝑅𝑒𝑑 = 800, the FFP at the downstream 

end  of  the  flat  sidewall  exhibits  only  80%  FFP,  indicating  20%  probability  of  reverse  flow. 

Inspection  of  instantaneous  velocity  contour  plots  does  show  that  indeed  occasionally  the 

boundary layer exhibits open, but very thin separation bubble. Also, at 𝑅𝑒𝑑 = 1100, the flow near 

the middle of the cylinder exhibit less than 100% FFP but is bounded by 100% FFP on the upstream 

and  downstream  sides.  This  implies  a  relatively  rare  occurrence  of  a  closed  separation  bubble 

above the middle of the cylinder sidewall. 

135 

 
Figure 66. Forward flow probability over the surface of the smooth cylinder at different 
Reynolds numbers. The vertical lines represent the mean separation and reattachment locations 
from Chapter 3. 

The case of 𝑅𝑒𝑑 = 1800 had shown a fully separated flow without reattachment based on 

the mean velocity field analysis. However, it can be seen in Figure 66 that there is 45% probability 

of reattachment at the most downstream location 𝑥/𝑑 = 0.8. The implication of this observation 

is that, at this Reynolds number, the boundary layer reattaches to the surface almost half the time, 

despite of the absence of mean reattachment. Inspection of the velocity field reveals that the flow 

regime swings between fully separated and reattached states every few cycles, as demonstrated by 

the two snapshots shown in Figure 67. The top snapshot (Figure 67-a) demonstrates a time snap 

when the shear layer fails to reattach, corresponding to Regime 2. The bottom snapshot (Figure 

67-b) shows a reattached boundary layer on the surface of the cylinder. It is important to note that 

these snapshots are not taken during the same shedding cycle, but the change of the reattachment 

behavior of the boundary layer happens every once in a while. Both snapshots correspond to an 

instant where shedding of positive vorticity at the downstream end of the plot. This is based only 

on the streamwise velocity in the absence of cross-stream velocity measurements (𝜔𝑧   ≈  𝑑𝑢/𝑑𝑦).  

Therefore,  the  𝑅𝑒𝑑 = 1800  case  is  not  simply  a  case  of  an  open  separation  bubble,  but  an 

intermittent reattachment of the separated boundary layer.  

136 

 
 
(a) 

(b) 
Figure 67. Instantaneous color contour snapshots of the velocity field in the wake of the smooth 
cylinder at 𝑅𝑒𝑑 = 1800. Signatures of (a) separated and (b) reattached boundary layer. 

With  additional  increase  to  𝑅𝑒𝑑 = 2500,  the  separated  boundary  layer  fully  reattaches 

again to the surface, given that the FFP crosses the 50% threshold at about 𝑥/𝑑 = 0.5 and becomes 

very close to 100% before the end of the flat surface (gray line in Figure 66). This behavior may 

also be observed from the instantaneous snapshots in Figure 68, taken at two different phases of 

the shedding cycle, corresponding to shedding of positive and negative vorticity in the wake. While 

there is movement of the reattachment point between the two snapshots (the end of the separation 

zone, light blue color, is visible in the bottom but not the top snapshot), the boundary layer remains 

reattached on the flat surface. Therefore, the instantaneous behavior at 𝑅𝑒𝑑 = 2500 is consistent 

with  the separation  bubble remaining closed,  as implied from  the mean velocity  field  analysis. 

Although the bubble does fluctuate in size with the unsteadiness dominated by vortex shedding. 

137 

 
 
 
(a) 

(b) 
Figure 68. Instantaneous color contour snapshots of the velocity field in the wake of the smooth 
cylinder at 𝑅𝑒𝑑 = 2500. Signatures (a) clockwise and (b) counterclockwise vortices. 

To summarize the outcome of the above results, the smooth round-corner cylinder shows 

a  behavior  with  increasing  Reynolds  number  which  is  different  from  the  understanding  of  the 

sharp-corner  cylinder  in  the  literature.  The  sharp-corner  cylinder  with  a  similar  side  ratio 

experiences Regimes 1, 2, and 3 consecutively with increasing the Reynolds number. On the other 

hand,  with  the  range  investigated  of  800 ≤ 𝑅𝑒𝑑 ≤ 2500,  the  round-corner  cylinder  is  seen  to 

undergo  Regime  1  at  lower  Reynolds  numbers,  “flip-flop”  between  Regimes  1  and  2  at  an 

intermediate Reynolds number, and jump back to  Regime 1 as the Reynolds  number  is  further 

increased.  

There are significant differences between the flow over the smooth round-corner cylinder 

versus  that  of  a  rectangular  cylinder  with  sharp  corners.  Contrary  to  the  latter  case,  where  the 

138 

 
 
 
boundary layer separation point is fixed at the sharp corners, the separation point can move over 

the surface of the smooth cylinder. Therefore, the flow can separate anywhere over the leading-

edge curve, or the flat surface, or in the case of 𝑅𝑒𝑑 = 1100 even remain attached over the whole 

forward body (except for possible occasional occurrence of a small separation bubble near 𝑥/𝑑 =

0 as seen from the FFP results in Figure 66).  

It  is  well understood that the reattachment of the boundary layer over the surface of an 

object is greatly affected by the trajectory of the shear layer. Despite of separation, closed or open, 

the  separated  shear  layer  remains  very  close  to  the  cylinder  sidewall  for  round-corner  smooth 

cylinder (see Figure 9 in Chapter 3) compared to the sharp corner cylinder. The shear layer also 

does not undergo transition over the examined Reynolds number range, since flow visualization 

did not show any sign of vortical structures forming on the surface of the smooth cylinder even at 

the  highest  Reynolds  number  of  𝑅𝑒𝑑 = 2500.  Furthermore,  in  the  latter  case,  wake  vortex 

formation occurs so close to the cylinder base such that it now largely influences the shear layer 

unsteadiness over the surface (based on the spectra at 𝑥/𝑑 = 0 and 0.71 in Figure 62 and Figure 

63). Flow visualization and velocity snapshots additionally demonstrate the alternate behavior of 

the boundary layer flow in conjugation with wake vortex formation and shedding. Therefore, the 

reattachment  mechanism  for  this  regime  should  be  differentiated  from  that  of  a  sharp-corner 

cylinder, where the shear layer transition causes reattachment.  

Comparison of the sharp- and round corner-cylinders in a nutshell shows that the smooth 

cylinder  primarily  undergoes  Regime  1.  A  transition  to  Regime  2  is  also  observed  at  an 

intermediary Reynolds number, but it does not persist all the time. Although the Reynolds number 

range covered in this study does not capture a case where Regime 2 is fully dominant, it can well 

take place at a Reynolds number 1400 < 𝑅𝑒𝑑 < 1800 and/or 𝑅𝑒𝑑 > 2500. Similarly, Regime 3 

139 

 
could occur for sufficiently high Reynolds number. The sharp-corner cylinder, on the other hand, 

experiences monotonic transition between Regimes 1, 2, and 3 with increasing Reynolds number. 

It should be noted that for the sharp-corner cylinder the transition from Regime 2 to 3 for 𝑐/𝑑 =

2.5 takes place somewhere in the range 2000 < 𝑅𝑒𝑑 < 4000 (Norberg [29]). 

5.2 Phase-Average Analysis of the Wake 

As  observed  from  the  PSD  analysis,  Reynolds  number  affects  the  strength  and  the 

broadness of the spectral peak at the dominant frequency of the velocity fluctuations in the wake 

of the smooth cylinder. It is assumed that this frequency corresponds to vortex shedding, but this 

cannot be ascertained from spectral information alone. To extract more details regarding the flow 

structure corresponding to the spectral peak for different  flow regimes discussed above, phase-

averaged streamwise velocity profiles are obtained relative to an oscillation cycle at the spectrum-

peak frequency. These profiles are used to examine the presence of velocity signatures consistent 

with the shedding of counter-rotating spanwise vortices into the wake. To obtain the phase average 

velocity < 𝑢 >, data points in the velocity time series are phase-ordered in 16 phase-bins relative 

to their location in the oscillation cycle, where each bin contains about 80 datapoints. This location 

is indicated by the normalized phase 𝜙, where 𝜙 = 0 is at the start of the cycle and 𝜙 = 1 at the 

end.  

The presence of a velocity deficit in bluff body wakes makes it difficult to distinguish the 

signature  of  a  vortex  passing  in  the  phase-averaged  velocity  profile.  To  aid  the  reader  in 

understanding how the vortex signature is identified in the velocity profiles, an idealized velocity 

profile resulting from the superposition of profiles due to wake deficit 𝑢𝑑(𝑦) and a vortex 𝑢𝑣(𝑦) 

is demonstrated in Figure 69 for different relative strengths of the vortex (increasing strength from 

left to right). The top plots correspond to a clockwise vortex with a core center above the wake 

140 

 
centerline, at 𝑦𝑐/𝑑 = 0.25, while the bottom plots represent a counter-clockwise vortex below the 

centerline, at 𝑦𝑐/𝑑 = −0.25. The wake deficit profile is taken as a Gaussian function 

𝑢𝑤(𝑦) =

−𝑈𝑤
𝜎𝑤√𝜋

−

(𝑦)2
2𝜎𝑤

2 ,   

𝑒

(19) 

where, 𝑈𝑤 is the centerline wake deficit and 𝜎𝑤 is a wake width parameter. On the other hand, the 

vortex profile is based on Gaussian vorticity distribution in the vortex core 

𝑢𝑣(𝑦) =

𝜋𝑅𝑜

2𝜔𝑜
2𝜋(𝑦−𝑦𝑐)

−

(𝑦−𝑦𝑐)2
2
𝑅𝑜

], 

 [1 − 𝑒

(20) 

where,  𝑅𝑜  is  the vortex core radius  and  𝜔𝑜 is  the maximum  vorticity at  the vortex core  center 

(located at 𝑦𝑐). The profile parameters are chosen to provide profiles that are comparable to those 

measured with the increase of the vortex strength achieved by increasing 𝜔𝑜. 

Focusing on the velocity profiles on the top half of Figure 69, each plot contains the wake 

profile (red broken line), the vortex profile (blue broken line) and the superposed profiles (black 

line). Examining the case with the weakest vortex signature (left plot), it is clear that the overall 

signature of vortex presence distorts the wake profile very little, making it difficult to conclude 

that a vortex is present. This distortion, which can be seen more clearly for the strongest vortex 

case (right plot) is such that it produces certain asymmetries between the top and the bottom halves 

of the profile. Specifically, for the clockwise vortex with center above the wake centerline, the 

asymmetry is  such that the magnitude of the maximum  positive velocity gradient at  the vortex 

center (above the centerline) is larger than magnitude of the minimum velocity gradient below the 

centerline. Additionally, instead of the deficit profile smoothly approaching the zero asymptote 

away  from  the wake  centerline (red line), the overall profile (black line)  exhibits  an overshoot 

above the vortex core center and undershoot below (which laterally shifts the minimum profile 

velocity away from the wake centerline). Similar observations, but with the opposite sense, can be 

141 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
seen for the presence of a counter-clockwise vortex below the centerline (bottom plots in Figure 

69). 

Figure 69. Illustration of synthetic phase-averaged velocity profile constructed from the 
superposition of wake deficit (red) and Gaussian vortex (blue) streamwise velocity profiles. The 
vortex strength increases from left to right. The top and bottom rows depict clockwise and 
counterclockwise vortices with core centers above and under the wake centerline, respectively. 

Figure 70 shows the < 𝑢′ >=< 𝑢 > −𝑈∞ profiles for every other 𝜙 value with an increment 

of Δ𝜙 = 2 × 0.0625 for the smooth cylinder at 𝑅𝑒𝑑 = 800. 𝜙=0 is assigned to the instant with 

maximum  positive  velocity  gradient  𝑑 < 𝑢′ >/𝑑𝑦.  The  phase-average  analysis  is  done  at  a 

streamwise location 1𝑑 distance downstream of the vortex formation location for all cases. This 

case is discussed as a representative of flow Regime 1.  

142 

 
 
 
 
 
 
 
Following the orientation of Figure 69 , inspection of Figure 70 shows that indeed a < 𝑢′ >

 signature typical of the periodic passing of a counter-rotating vortex pair is evident. Specifically, 

at 𝜙 = 0, the velocity profile is qualitatively similar to that of a vortex with clockwise rotation 

present  above  the  wake  centerline.  At  this  phase,  the  positive  velocity  gradient  is  highest, 

consistent with the presence of the center of a clockwise vortex at the maximum gradient location 

(identified with a blue circle in Figure 70). Likewise, a similar profile, but with negative velocity 

gradient,  is  seen  in  the  middle  of  the  cycle  at  𝜙 = 0.5,  suggesting  the  passage  of  a  counter-

clockwise vortex below the wake centerline. The PSD corresponding to this case, along with other 

cases considered for the phase-average analysis, is depicted in Figure 71 and exhibits a strong peak 

that  is  more  than  two  orders  of  magnitude  above  other  fluctuations  in  the  spectrum.  The  PSD 

results  are  obtained  at  the  streamwise  location  of  the  phase  averaging,  and  the  cross-stream 

location of the maximum 𝑢𝑟𝑚𝑠.  

Figure 70. < 𝑢′ >/𝑈∞  at different phases measured 1𝑑 downstream of the formation length at 
𝑥/𝑑 = 4.7. Smooth cylinder at 𝑅𝑒𝑑 = 800. Maximum positive and negative velcoty gradient 
locations are marked at 𝜙 = 0 and 0.5. 

143 

 
 
Figure 71. Power spectral density vs. Strouhal number of the smooth cylinder for selected 
Reynolds numbers at the streamwise location of the phase-average analysis and at the cross-
stream location of the highest fluctuations. 

     To examine the cycle-to-cycle repeatability of the vortex signature, the velocity fluctuations in 

the  wake  of  the  cylinder  are  decomposed  into  two  components;  one  accounting  for  the  energy 

associated  with  the  phase-average  velocity,  and  the  other,  with  the  deviation  from  the  phase 

average. Together with the mean velocity this decomposition is known as triple decomposition, 

which was first introduced by Hussain and Reynolds [30]. Deviation from the phase average could 

be the result of variation in the vortex properties from cycle-to-cycle, random fluctuation (such as 

due to small-scale turbulence), and other possible structures. The decomposition leads to: 

𝑢𝑟𝑚𝑠

𝜙 2
2 = 𝑢′𝑟𝑚𝑠

𝑟
+ 𝑢𝑟𝑚𝑠

2     

              (21)  

where  𝑢′𝑟𝑚𝑠

𝜙   is  the  rms  velocity  calculated  from  the  phase-average  velocity  variation  over  one 

cycle, and 𝑢𝑟𝑚𝑠

𝑟

 is the rms associated with the remaining fluctuation. Figure 72 depicts this energy 

decomposition  for the smooth  cylinder at  𝑅𝑒𝑑 = 800. A qualitative interrogation of this figure 

shows that 𝑢′𝑟𝑚𝑠

𝜙  accounts for most of the fluctuating energy in the wake of the smooth cylinder 

at 𝑅𝑒𝑑 = 800. Therefore, Regime 1 is associated with a well-defined and repeatable wake vortex 

formation and shedding.  

144 

 
 
 
 
 
 
 
 
 
Figure 72. Decomposition of the fluctuating streamwise velocity energy for smooth cylinder at 
𝑅𝑒𝑑=800. Measured at 𝑥/𝑑 = 4.7. 

< 𝑢′ >  profiles  for  the  smooth  cylinder  at  𝑅𝑒𝑑 = 1800  are  shown  in  Figure  73  as  a 

representative of the flow flip-flopping between Regimes 1 and 2. A signature of periodic counter-

rotating vortex shedding can be distinguished in the < 𝑢′ > profiles, although it is much weaker 

than the signature observed for flow Regime 1. Specifically, the profiles for 𝑅𝑒𝑑 = 800 in Figure 

70  show  quite  a  bit  of  variation  across  the  phases.  In  contrast,  the  profiles  for  𝑅𝑒𝑑 = 1800 

predominantly look like wake deficit profiles that do not vary significantly over the phases and do 

not represent a flow that is dominated by regular vortex formation and shedding. However, the 

velocity profiles at 𝜙 = 0  and 0.5 show a small signature of passing of the vortices with clockwise 

and  counter-clockwise  rotation,  respectively.  The  velocity  profiles  at  these  phases  show  the 

maximum  positive  and  negative  velocity  gradients  𝑑 < 𝑢′ >/𝑑𝑦,  respectively.  Although  these 

gradients are not considerably different from that at the neutral phase 𝜙 = 0.25, they represent a 

small contribution from the organized vortex shedding on the velocity profiles. This observation 

that phase-averaging the velocity based on the shedding frequency does not result in an obvious 

vortex signature suggests that the vortex shedding is not highly regular and well-organized. The 

145 

 
 
PSD for this case shown in Figure 71 supports this conclusion by showing a broader and weaker 

peak than that of the Regime 1.   

Figure 73.< 𝑢′ >/𝑈∞ at different phases measured 1𝑑 downstream of the formation length at 
𝑥/𝑑 = 4.1. Smooth cylinder at 𝑅𝑒𝑑 = 1800. Maximum positive and negative gradient locations 
are marked at 𝜙 = 0 and 0.5. 

Cycle-to-cycle repeatability for the case of smooth cylinder at 𝑅𝑒𝑑 = 1800 is analyzed by 

decomposition of the fluctuating streamwise velocity energy and is represented in Figure 74. These 

results  convey  a  consistent  message  with  the  aforementioned  analysis  that  the  organized  von 

Karman vortex shedding frequency is weak and does not dominate the velocity fluctuations in the 

wake of the smooth cylinder at 𝑅𝑒𝑑 = 1800. 

146 

 
 
Figure 74. Decomposition of the fluctuating streamwise velocity energy for smooth cylinder at 
𝑅𝑒𝑑=1800 measured at 𝑥/𝑑 = 4.1. 

As discussed earlier, the case of the smooth cylinder at 𝑅𝑒𝑑 = 2500 is believed to have 

the characteristics of the flow Regime 1. Figure 75 demonstrates the < 𝑢′ > profiles for this case 

at every other phase bin. The periodic passing of a counter-rotating vortex pair is evident in this 

figure  where  the  velocity  profiles  at  phases  𝜙 = 0  and  0.5  qualitatively  resemble  the  velocity 

profiles  of  vortices  with  clockwise  and  counter  clockwise  rotation,  respectively.  The  PSD 

corresponding to this case shown in Figure 71 shows a narrow and strong harmonic peak, which 

along  with  the  phase  average  velocity  profiles,  suggest  a  recovery  of  the  von  Karman  vortex 

formation and shedding organization from Regime 2 after increasing the Reynolds number from 

1800 to 2500.    

147 

 
 
Figure 75.< 𝑢′ >/𝑈∞ at different phases measured 1𝑑 downstream of the formation length at 
𝑥/𝑑 = 3.4. Smooth cylinder at 𝑅𝑒𝑑 = 2500. Maximum positive and negative gradient locations 
are marked at 𝜙 = 0 and 0.5. 

The fluctuating streamwise velocity energy is decomposed and shown in Figure 76 for this 

case. This figure shows that the amount of the energy carried by the coherent von Karman vortex 

shedding  is  roughly  the  same  as  the  that  carried  by  the  remaining  fluctuations.  Although  the 

coherent component carries higher energy compared to the case of 𝑅𝑒𝑑 = 1800, the recovery of 

the vortex shedding organization is not as good as for the 𝑅𝑒𝑑 = 800 case.   

Figure 76. Decomposition of the fluctuating streamwise velocity energy for smooth cylinder at 
𝑅𝑒𝑑=2500 at 𝑥/𝑑 = 3.4. 

148 

 
 
 
It is also interesting to consider Regime 3 behavior as exemplified by the 2Dp-5% cylinder 

at 𝑅𝑒𝑑 = 2500 (see Figure 65). The phase-averaged velocity profiles at every other phase for this 

case are depicted in Figure 77. Almost no variation of the velocity profiles can be distinguished 

over different phases, and there is no distinct signature of a counter-rotating vortex shedding. The 

PSD for this case in Figure 71 shows a peak that is about an order of magnitude weaker than the 

other cases and is distributed over a wider frequency band. Therefore, it is not surprising that the 

phase-averaged velocity profiles do not vary significantly from the mean velocity profile, as the 

coherent structures are very weak.  

Figure 77. < 𝑢′ >/𝑈∞ at different phases measured 1𝑑 downstream of the formation length at 
𝑥/𝑑 = 3.8. 2Dp cylinder at 𝑅𝑒𝑑 = 2500. 

To  ascertain  this,  energy  decomposition  of  the  fluctuating  velocity  is  performed  and 

portrayed in Figure 78. It can be seen that the energy carried by the coherent structures associated 

with  von  Karman  vortex  shedding  is  negligible.  Therefore,  the  wake  seems  to  be  significantly 

turbulent in this regime and there is no persuasive sign of a coherent and organized wake vortex 

shedding structure.   

149 

 
 
  
Figure 78. Decomposition of the fluctuating streamwise velocity energy for 2Dp cylinder at 
𝑅𝑒𝑑=2500 at 𝑥/𝑑 = 3.8. 

The ratio of the phase-average to the total energy at the 𝑦 location of the maximum 𝑢𝑟𝑚𝑠

2

is tabulated for the different cases in Table 4. The higher this ratio is, the more repeatable is the 

vortex shedding from cycle to cycle. 

Table 4. Comparison of the ratio of the energy of the phase-average velocity to the total energy 
of the velocity fluctuations. 

smooth, 𝑅𝑒𝑑 = 800 

smooth, 𝑅𝑒𝑑 = 1800 

smooth, 𝑅𝑒𝑑 = 2500 

2Dp, 𝑅𝑒𝑑 = 2500 

𝜙 2
𝑢′𝑟𝑚𝑠

/𝑢𝑟𝑚𝑠

2% 

87% 

18% 

46% 

3% 

Inspection of Table 4 shows that Regime 1 has the highest cycle-to-cycle repeatability of 

the  phase  average,  meaning  that  the  attached/laminarly  reattached  flow  has  the  best  organized 

wake  vortex  shedding.  Wake  vortex  shedding  is  weakly  organized,  but  distinguishable  at  the 

transition regime between Regimes 1 and 2, where the separation bubble opens up without vortex 

formation above the cylinder sidewall, but closes occasionally. The worst repeatability belongs to 

Regime 3, where there is practically no sign of wake vortex formation. In this case, the shear layer 

vortex formation upstream of the wake seems to promote turbulent reattachment. However, these 

vortices seem to disrupt organized wake vortex formation. 

150 

 
 
 
 
  
 
 
Chapter 6. Conclusions  

This study experimentally investigates the effect of surface topology and Reynolds number 

on the fluid flow around a rectangular cylinder with round corners. The main motivation is the 

recent  discovery  of  the  underlying  cause  of  flow-induced  vibration  that  complicates  the 

controllability and compromises the landing accuracy of Precision Airdrop Systems (PADS). It is 

recently found that the suspension lines of PADS, the shape of which may be approximated as a 

rectangular cylinder with complex surface geometry, undergo galloping instability: a type of flow-

induced vibration seen in non-circular cross-section cylinders. To systematically investigate the 

effect of the surface geometry, a surface topology is mathematically defined using spatial Fourier 

modes  with  parameters  closely  representing  the  geometry  of  the  PADS  suspension  lines.  The 

topology is superposed onto a baseline shape of a smooth-surface rectangular cylinder with fully-

round leading and trailing edges. 

The  study  examines  the  influence  of  various  aspects  of  surface  topology,  including 

amplitude, variation along the span, and orientation relative to the oncoming flow, on the boundary 

layer and the near-wake flow around a rectangular cylinder. Models with surface topology include 

topology orientations with a peak or a valley at the leading edge but no variation along the span: 

these are referred to as 2Dp and 2Dv cases. Cylinders with similar topology but with variation 

along the span are also examined. In these cases, measurements are done at two planes along the 

spans, aligned with cross-sections corresponding to largest peak and valley at the leading edge 

(3Dp and 3Dv cases, respectively). The studied rage of Reynolds numbers, based on the cylinder 

thickness, is from 𝑅𝑒𝑑 = 800 − 2500. Utilizing single-component molecular tagging velocimetry 

(1c-MTV), the research is based on measurements of the streamwise velocity and visualization of 

the flow field at various locations above the surface and in the wake of the cylinder.  

151 

 
1c-MTV measurement of the mean and rms velocity profiles in the boundary layer of the 

cylinders  reveal  that  the  orientation  of  the  surface  topology  to  the  oncoming  flow  plays  a 

significant  role  in  determining  the  behavior  of  the  boundary  layer.  For  the  specific  topology 

wavelength utilized, the most notable impact occurs when the topology is associated with a peak 

at the leading edge of the cylinder. In this scenario, a considerably thicker separation zone forms, 

and  the  separated  shear  layer  becomes  thicker,  positioned  farther  away  from  the  cylinder. 

Additionally, the shear layer displays the highest level of unsteadiness compared to the smooth 

cylinder and the cylinder section where a valley is present at the leading edge. Moreover, the effect 

of surface topology on the separated shear layer behavior is Reynolds number dependent and this 

dependency  is  non-monotonic.  The  effect  of  topology  amplitude  is  also  studied  and  is  seen  to 

generally magnify the deviation of the boundary layer behavior from that of the baseline. Presence 

of the surface topology variation along the span decreases the separation zone thickness for the 

cross-sections with a peak at the leading edge, but does the opposite for the cross-sections with a 

valley at the leading edge. The latter case excludes the high end of the tested Reynolds numbers 

(𝑅𝑒𝑑 = 2500). A monotonic trend does not exist when comparing the fluctuating velocity in the 

shear layer for these cases.  

The separation zone boundaries on the top and bottom surfaces are further studied for the 

geometries with surface topology amplitude 𝜀/𝑑 = 5% at several angles of attack from 𝛼 = 0° −

2° to infer stability to galloping based on the quasi-steady assumption, and to compare the results 

against force measurements using similar geometries in other studies. In general, it is observed 

that the introduction of surface topology can lead to earlier separation of the boundary layer on the 

cylinder, occurring farther upstream compared to the smooth baseline geometry. This phenomenon 

is evident in both the 2Dp case and when measuring in a spanwise plane with a peak at the leading 

152 

 
 
edge in the 3D geometry. The present study draws on existing knowledge in the literature regarding 

galloping to interpret the identified boundary layer behavior and hypothesize that the promotion 

of earlier separation due to surface topology has a destabilizing effect on galloping. However, this 

destabilizing influence is counteracted by the stabilizing effect of the reattachment of the shear 

layer, which is promoted by the presence of spanwise variation in the topology (3D geometry) and 

increasing Reynolds number. The Reynolds number effect,  as inferred  from  the separation and 

reattachment characteristics of the top-surface shear layer, is consistent with the recent force data 

studies on similar geometries.  

Investigating  the  wake  flow  for  the  cylinders  with  smooth  surface  as  well  as  surface 

topologies with 5% amplitude at 𝛼 = 0° reveals that the case of 2Dp-5% has the biggest deviation 

in the mean and fluctuating wake flow characteristics from the other geometries. In addition, non-

monotonic Reynolds number dependency of the wake behavior is also seen across the geometries. 

Overall, altering the initial wake width and the maximum transverse separation of the shear layers 

plays  a  crucial  role  in  surface  topology's  influence  on  the  vortex  shedding  frequency.  It  is 

specifically  seen  that  the  vortex  shedding  frequency  scales  better  with  the  wake  than  with  the 

cylinder  width.  The  wake  width  variation  among  different  cases  stem  from  changes  in  the 

boundary layer separation/reattachment on the cylinder's surface.  

Finally, the unsteady behavior of the boundary layer and the wake of the smooth cylinder 

at  𝛼 = 0°  is  studied  at  different  Reynolds  numbers.  The  results  reveal  different  progression  of 

boundary layer separation/reattachment flow regimes with increasing Reynolds number compared 

to the widely studied sharp-corner cylinder. Despite this different progression, the boundary layer-

wake flow relationship is generally consistent with the literature understanding of the sharp-corner 

cylinder, where Strouhal number and organization of the wake vortex shedding is affected by the 

153 

 
 
 
boundary layer separation/reattachment characteristics. An exception to this agreement is found at 

the highest Reynolds number of 2500, where reattachment of the shear layer does not produce an 

increase in Strouhal number.  

Transition in the shear layer upstream of reattachment of the boundary layer is not observed 

for the smooth cylinder in the studied Reynolds number range, but is detected for some geometries 

with surface topology. As a sample result, boundary layer and near-wake flow visualization for 

the 2Dp-5% case reveals that shear layer vortex formation occurs above the side surface of the 

cylinder,  promoting  reattachment.  Phase-average  velocity  analysis  of  these  cases  demonstrate 

sensitivity  of  the  vortex  shedding  organization  to  different  flow  regimes.  Specifically,  the  best 

organized wake vortex shedding is observed for the attached/laminarly reattached boundary layer 

flow. Wake vortex shedding is weakly organized but identifiable for the regime with intermittent 

reattachment of the boundary layer, while reattachment due to transition in the shear layer (2Dp-

5% case) exhibits the least organized and barely observable wake vortex shedding. In this scenario, 

shear layer vortex formation  upstream  of the wake appears to  facilitate turbulent  reattachment. 

However, these vortices also appear to disrupt the organization of wake vortex shedding.  

For  future  research,  investigating  the  impact  of  surface  topology  amplitude  on  the 

galloping-related  boundary  layer  behavior  of  3D  surface  topologies  would  be  advantageous. 

Additionally, an analysis of the spanwise velocity can offer deeper insights into the flow behavior 

over  the  surfaces  of  the  cylinders  with  3D  surface  topology.  Furthermore,  building  upon  the 

findings of this study, it would be worthwhile to examine the flow regime over the cylinders at 

various other Reynolds numbers to explore the presence of different flow regimes. 

154 

 
 
 
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[2] 

K. Bergeron, D. Ecklebe, K. McClure, H. Johari, T. Curlett, and B. Pitman, "Parachute 
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T. Siefers, K. Greene, T. McLaughlin, and K. Bergeron, "Wind and water tunnel 
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[3] 

R. D. Blevins, "Flow-induced vibration," New York, 1977. 

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G. Parkinson, "Phenomena and modelling of flow-induced vibrations of bluff bodies," 
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R. Parker and M. C. Welsh, "Effects of sound on flow separation from blunt flat plates," 
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157 

 
 
 
 
 
The MATLAB code to generate the surface topology is enclosed here. It consists of a main script 

APPENDIX 

and a function. 

clear; 
clc; 
dx = .01;    %interval 
l =  139.6;      %span 

for z = 1:(l/dx)+1 
coor(z) = {topol2((z-1)*dx)}; 
end 

B = cell2mat (coor)'; 

csvwrite('rotate.txt',[B(:,1,:),B(:,2,:),B(:,3,:)]) 

p_mid = round(numel(B(:,1))/2); 

range = (B(:,2)>0); 

x_surface = B(:,1); 
y_surface = B(:,2); 
z_surface = B(:,3); 

x_s = reshape(x_surface,1900,[]); 
y_s = reshape(y_surface,1900,[]); 
z_s = reshape(z_surface,1900,[]); 

figure 
s=surf(x_s,z_s,y_s) 
s.EdgeColor = 'none'; 
axis on 
l = light('Position',[-50 -15 29]) 
set(gca,'CameraPosition',[208 -50 7687]) 
lighting phong 
shading interp 
colorbar EastOutside 

158 

 
 
  
  
 
  
 
  
  
 
 
 
 
function [finalcoordinates] = topol2(z) 

%topology amplitude 

sr = 2.5; %side ratio 
d = 1.5;  %chord 
c = d*sr; %diameter 
l = 139.6; % span 
r = 0.5*d; %corner radius 
es = 0.05*d; 
cycle = 10; %number of wavelengths around the perimeter 
p = 2*(c+d) + r*(2*pi-8); %perimeter of the baseline 
lamx = p/cycle; %wavelength around the perimeter 
lamz = lamx*1.5;   %wavelength along the span 
gama = atan (lamz / lamx); 
phi = 2*pi*z/lamz; %phase of the topology at 0 
dx = 0.0020;    %interval 
dt = dx / r; 

%% 1 

y1 = 0:dx:d/2-r;   
x1 = zeros(size(y1)) ; 
s1 = y1; 
cosarc = es * (cos(2*pi*cycle*s1/p + phi) + cos(2*pi*cycle*s1/p 
- phi)); 
n1=[zeros(size(y1))-1;zeros(size(y1))]; 
xt1 = -cosarc + x1 - c/2; 
yt1 = y1; 

eptest1 = sqrt( (xt1 - x1 +c/2).^2 + (yt1 - y1).^2); 

%% 2 
 t2 = pi-dt:-dt:pi/2; 
 x2 = r * cos(t2) ; 
 y2 = r * sin(t2) ; 
 s2 = d/2-r + r * (pi-t2); 
 n2 = [cos(t2);sin(t2)]; 
 a = [x2 ; y2]; 
 f2 = es *(cos(2*pi*cycle*s2/p + phi) + cos(2*pi*cycle*s2/p - 
phi)); 
 cosarc = (f2 .* n2) + a; 
 xt2 = cosarc(1,:) - c/2 + r; 
 yt2 = cosarc(2,:) + d/2 - r; 

 eptest2 = sqrt( ((xt2+ c/2 - r - x2).^2) + ((yt2-d/2 +r - 
y2).^2)); 

159 

 
 
  
  
  
  
  
 
 
%% 3 
x3 = -c/2+r +dx : dx :c/2-r   ; 
y3 = zeros(size(x3)); 
s3 = d/2-r +r*pi/2 + x3 + c/2 - r; 
cosarc3 = es * (cos(2*pi*cycle*s3/p + phi) + cos(2*pi*cycle*s3/p 
- phi)); 

n3 = [zeros(size(x3));zeros(size(x3))+1]; 

xt3 = x3; 
yt3 = cosarc3 + d/2; 
eptest3 = sqrt( (xt3 - x3).^2 + ((yt3-d/2 - y3).^2)); 

%% 4 
t4 = pi/2 -dt:-dt:0; 
 x4 = r * cos(t4) ; 
 y4 = r * sin(t4) ; 
 s4 = d/2-r +r*pi/2 +c - 2*r + r * (pi/2-t4);  
 n4 = [cos(t4);sin(t4)]; 
 a = [x4 ; y4]; 
 f4 = es * (cos(2*pi*cycle*s4/p + phi) + cos(2*pi*cycle*s4/p - 
phi)); 
 cosarc = (f4 .* n4) + a; 
 xt4 = cosarc(1,:) + c/2 - r; 
 yt4 = cosarc(2,:) + d/2 - r; 

 eptest4 = sqrt( ((xt4-c/2+r - x4).^2) + ((yt4-d/2+r - y4).^2)); 

%% 5 

y5 = d/2-r -dx:-dx:-d/2+r;        
x5 = zeros(size(y5)) ; 
s5 = d/2-r +r*pi +c - 2*r - y5 +d/2 -r;  
cosarc = es * (cos(2*pi*cycle*s5/p + phi) + cos(2*pi*cycle*s5/p 
- phi)); 
n5=[zeros(size(y5))+1;zeros(size(y5))]; 
xt5 = cosarc + x5 + c/2; 
yt5 = y5; 

eptest5 = sqrt( (xt5 - x5-c/2).^2 + (yt5 - y5).^2); 

%% 6 

t6 = 0-dt:-dt:-pi/2; 
 x6 = r * cos(t6) ; 
 y6 = r * sin(t6) ; 
 s6 = d/2-r +r*pi +c - 2*r +d -2*r - r * t6;  

160 

 
  
  
 
  
 
  
  
  
  
 n6 = [cos(t6);sin(t6)]; 
 a = [x6 ; y6]; 
 f6 = es * (cos(2*pi*cycle*s6/p + phi) + cos(2*pi*cycle*s6/p - 
phi)); 
 cosarc = (f6 .* n6) + a; 
 xt6 = cosarc(1,:) + c/2 - r; 
 yt6 = cosarc(2,:) - d/2 + r; 
 eptest6 = sqrt( (xt6-c/2+r - x6).^2 + (yt6+d/2 -r - y6).^2); 

%% 7 

x7 = c/2-r -dx: -dx :-c/2+r   ;  
y7 = zeros(size(x7)); 
s7 = d/2-r +r*pi*3/2 +c - 2*r +d -2*r - x7 + c/2 - r; 
cosarc7 = es * (cos(2*pi*cycle*s7/p + phi) + cos(2*pi*cycle*s7/p 
- phi)); 

n7 = [zeros(size(x7));zeros(size(x7))-1]; 

xt7 = x7; 
yt7 = -cosarc7 - d/2; 
eptest7 = sqrt( (xt7 - x7).^2 + ((yt7+d/2 - y7).^2)); 

%% 8 

t8 = -pi/2-dt:-dt:-pi; 
 x8 = r * cos(t8) ; 
 y8 = r * sin(t8) ; 
 s8 = d/2-r +r*pi*3/2 +c - 2*r +d -2*r + c - 2*r + r * (-pi/2 - 
t8); 
 n8 = [cos(t8);sin(t8)]; 
 a = [x8 ; y8]; 
 f8 = es * (cos(2*pi*cycle*s8/p + phi) + cos(2*pi*cycle*s8/p - 
phi)); 
 cosarc = (f8 .* n8) + a; 
 xt8 = cosarc(1,:) - c/2 + r; 
 yt8 = cosarc(2,:) - d/2 + r; 

 eptest8 = sqrt( ((xt8+c/2 - r - x8).^2) + ((yt8+d/2 - r - 
y8).^2)); 

%% 9 

y9 = -d/2+r+dx:dx:0;        
x9 = zeros(size(y9)); 
s9 = d/2-r +r*pi*2 +c - 2*r +d -2*r + c - 2*r + y9 + d/2 -r; 

161 

 
 
  
  
  
  
 
  
  
 
  
cosarc = es * (cos(2*pi*cycle*s9/p + phi) + cos(2*pi*cycle*s9/p 
- phi)); 
n9=[zeros(size(y9))-1;zeros(size(y9))]; 
xt9 = -cosarc - c/2; 
yt9 = y9; 

eptest9 = sqrt( (xt9 - x9+c/2).^2 + (yt9 - y9).^2); 

xtot=[xt1,xt2,xt3,xt4,xt5,xt6,xt7,xt8,xt9]; 
ytot=[yt1,yt2,yt3,yt4,yt5,yt6,yt7,yt8,yt9]; 
x_n = [x1,x2,x3,x4,x5,x6,x7,x8,x9]; 
y_n = [y1,y2,y3,y4,y5,y6,y7,y8,y9]; 
stotal = [s1,s2,s3,s4,s5,s6,s7,s8,s9]; 
stotal(1) 
stotal(end) 
ntotal = [n1,n2,n3,n4,n5,n6,n7,n8,n9]; 
epstesttotal = 
[eptest1,eptest2,eptest3,eptest4,eptest5,eptest6,eptest7,eptest8
,eptest9]; 
ftotal = es * cos(2*pi*cycle*stotal/p + phi); 
sur_face = (ftotal .* ntotal) + ntotal *r;  
ztot=zeros(size(xtot)) + z; 
finalcoordinates = [xtot;ytot;ztot]; 

end 

162