MODELING AND MEASUREMENT OF SURFACE PRESSURE FLUCTUATION 
IN
 
AN IMP
INGING
 
JET
 
 
By
 
 
Nasem
 
A. Aukla
 
 
 
 
 
 
 
 
 
 
A THESIS
 
 
Submitted to
 
Michigan State University
 
in partial fulfillment of the requirements
 
for the degree of
 
 
Mechanical Engineering 

 
Master of
 
Science
 
 
2018
 
 
 
 
 
 
 
 
ABSTRACT
 
MODELING AND MEASUREMENT OF SURFACE PRESSURE FLUCTUATION 
IN
 
AN 
IMPING
ING
 
JET
 
 
By
 
Nasem
 
A. Aukla
 
Imping
ing
 
jets 
are used in
 
many
 
engineering 
and industrial 
applications,
 
including
 
heating, 
cooling
,
 
drying, 
food
 
processing
, and
 
s
urface cleaning, among others
. 
The
 
present thesis work is 
focused on modeling and studying the unsteady wall
-
pressure signature produced by a jet 
impinging normally on a flat surface
.
 
This study 
is 
divided into two main parts
:
 
A theoretical part, to establ
ish the 
beginning step
 
towards building a physics
-
based, 
mathematical model to calculate the surface
-
pressure fluctuation on the impingement surface. The 
mathematical model is used to explore the effects of changing the flow and jet
-
vortices parameters, 
on
e at a time, on the characteristics of the surface
-
pressure fluctuation. Three main parameters are 
examined: vortex
-
passage frequency, jet R
eynolds number, and vortex circulation. 
 
An experimental part, to measure the unsteady surface pressure fluctuation 
on the 
impingement surface for an axisymmetric jet at
 
normal
 
incidence.  Measurements 
are done,
 
for 
Reynold numbers 





8272 and 24818 (based on the jet diameter (

) and jet exit velocity), 
using a microphone array extending radially from the stagnation point (





) into the wall
-
jet 
zone (







). 
 
Comparison of the model and the experimental results sho
ws that, despite of the model 
simplicity, certain qualitative features of the unsteady wall pressure are similar within the 
stagnation zone. This outcome establishes confidence to continue further development of the 
model in the future. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Copyri
ght by 
 
NASEM
 
A. 
AUKLA
 
2018 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
iv
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
DEDICATED TO THE MEMORY OF MY FATHER AND MOTHER
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
v
 
 
ACKNOWLEDGMENTS
 
 
 
 
I would like to 
sincerely
 
acknowledge my family
, advisor, committee members, friends, 
and colleagues, because without 
their help and support this work could not have 
been done
. 
Many 
people have
 
helped me to come 
at
 
this 
point,
 
and I cannot go further before expressing my 
most 
profound
 
gratitude to them.
 
Firstly, to Prof. Dr. Ahmed Naguib for giving me an opportunity to be
 
one of his 
students
 
and to work on an 
exciting
 
and challenging topic. His advice, direction, and encouragement were 
much
 
supportive 
of
 
my thesis.
 
My profound gratitude to my committee members Prof.
 
Dr. Farhad Jaberi and Prof. Dr. 
Ricardo Mejia
-
Alvarezwhos
e 
for being 
on 
my 
committee
. 
 
Many thanks to Dr. K
e
 
Zhang who helped me with my experimental setup. 
 
My profound gratitude to Dr. Sami Al
-
Araji, Mr. Dhia Al
-
Ashaqr, and Mr. Nazar Hussain 
who helped me to achieve my goal, without their support I would not h
ave accomplished this work.
 
Finally, Many thanks to my colleague and best 
friend,
 
Fady Hindo. The honest and purest 
gentleman who I met in my entire life.  Thank you for your 
tru
e
 
friendship. 
 
 
 
 
 
 
 
 
vi
 
 
TABLE OF CONTENTS
 
 
 
 

.

.
 
..v
iii
 

.

 
...
i
x
 
KEY TO SYMBOLS


..







.
 
 
xi
v
 
CHAPTER
 
1: Introduction
 

...

.


 

.1
 
1.1
 
Free Jet
s
 





 
..1
 
1.1.1
 



 
.
1
 
1.1.2
 
Literature Review



 
1.2
 
Impinging
 



 

4
 
1.2.1
 
   


.
...
 
...4
 
1.2.2
 
   
Literature Review



.

...5
 
1.2.3       



 
.
9
 
1.3
 
 




 
.
12
 
            
1.4        Objective
s


..


 
.
13
 
CHAPTER
 



...
 

15
 





 
15
 
            




 
17
 
           
2.2.1
 
 



17
 
           
2.2.2
 
 
Wall (Vortex Rings Images) 
...



19
 
           
2.2.3      Advection Field (Stagnation Point Potential Flow) 
...



 
22
 
           
2.2.4     
 
Vortex Advection and Flow
-



 
22
 
           
2.2.5      Wall
-



.

 
26
 



 
28
 
           
2.3.1      Computation of the Time
-
Depend
ent Stream Function and Velocity Field
 
                          



29
 
           



29
 



30
 
           
2.4.
1      Impinging
-


..
31
 
 

31
 
 
2.4.1.2     
Validation of the Wall
-
Pressure
-
Grid Resolution 

 
 
2.4.1.3     
Validation of the Time Step Size
 





37
 
       
    
2.4.2      Application to A Line Vortex Above A Flat Wall

 
CHAPTER
 



4
5
 
  


  


49
 
  


..



.
64
 
  
3.3      Circulation 
Effect
 


..

.
 
.
66
 
 
CHAPTER
 

..

...


 
7
1
 
  
4.1   
 
 


..




71
 
  



..


 
72
 
 
vii
 
 
4.3     Hot
-
wire Setup and Calibration 

 
76
 



83
 



89
 
4.6     
Procedure to Acquire 
the 
Microphone Signal
s
 



89
 
 
CHAPTER
 
5: Jet Characteristics and Wall
-



..

 
90
 
5.1     Initial Shear Layer 
Self
-
Similarity
 



90
 



94
 
5.3     Time 



...
.
99
 
          


.
99
 
          


  
103
 
          



107
 
5.4     S



112
 
5.5     
Comparison between Experiment and Mathematical Model
 

...

 
.
123
 
 
CHAPTER
 
6: Conclusion
s
 
and Recommendation
s






.

 
1
32
 
 

 
.137
 
 


.

.

142
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
viii
 
 
LIST OF TABLES
 
 
 
 
Table 2.1 Comparison of high
-
fidelity and present model features



..

16
 
 
Table 3.1 Model parameters for frequency 



, Reynold number 





, and circulation 



 


...


49
 
Table 4.1 Sample of 
the
 
microphones

 
sensitivity
 

..

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
ix
 
 
L
IST OF FIGURES
 
 
 
 
Figure 1.1 Schematic of a free
-
jet flow, depicting various flow development zones 

 
..2
 
Figure 1.
2
 
Schematic of the impinging jet at normal incidence 
(taken from 
Al
-
Aweni [7]

 
5
 
Figure 2.1 Schematic drawing of the 
full mathemati
cal model of the impinging jet

 
18
 
Figure 2.
2
 
Radial wall
-
pressure distribution at 
the time instant when
 
the negative pressure spike 
(minimum pressure) associated with vortex passing 
is located
 
at 
r=5 
cm
 
location.
 
Different color 
lines represent
 
differ
ent
 
radial
 
domain size
s
 
used for computing the wall pressure (as given by 
the legend). 
The inset
 
shows 
the 
overall distribution, while 
the 
main plot depicts a magnified 
view to show the change between the different cases.

 
32
 
Figure 2.
3
 
N
ormalized Root Mean Square Error (
NRMSE
) resulting from comparing the results 
of domain sizes 10 and 15 
cm
 
(blue) and 15 and 20 cm (red). The results are shown up to the 
computation time corresponding to the time instant where the negative pressure spike a
ppears at 
r 
= 5 
cm
.

 
33
 
Figure 2.
4
 
Radial wall
-
pressure distribution at 
the time instant when
 
the negative pressure spike 
(minimum pressure) associated with vortex passing 
is located
 
at 




.
 
Different color lines 
re
present a
 
different
 
number of wall grid points used for computing the wall pressure (as given 
by the legend). 
The inset
 
shows 
the 
overall distribution, while 
the 
main plot depicts a magnified 
view to show the change between the different cases.


 
3
5
 
Figure 2.
5
 
Normalized Root Mean Square Error (
NRMSE
) resulting from comparing the results 
of grid resolution 300 and 600 points (blue) and 600 and 1200 points (red). The results are shown 
up to the computation time corresponding to the time i
nstant whe
n
 
the negative pressure spike 
appears at 






 
36
 
Figure 2.
6
 
Radial wall
-
pressure distribution at 
the instant when
 
the negative pressure spike 
(minimum pressure) associated with vortex passing 
is located
 
at 




.
 
Different color lines 
represent a
 
different
 
number of computational time steps in the period between injecting two 
successive vortices (as given by the legend).
 
The inset
 
shows 
the 
overall distribution, while 
the 
main plot depicts a magnified view 
to show the change between the different cases

 
38
 
Figure 2.
7
 
Normalized Root Mean Square Error (
NRMSE
) resulting from comparing the results 
of time step 4.066e
-
06 
s
 
(1000 
PPC
) and 2.033e
-
06 
s
 
(2000 
PPC
) (blue) and 2.033e
-
06 
s
 
(2000 
PPC
) and 1.0161e
-
06 
s
 
(4000 
PPC
) (red). The results are shown up to the computation time 
corresponding to the time instant whe
n
 
the negative pressure spike appears at 





 
39
 
Figure 2.
8
 
Schematic of a line vortex centered at (
x
o
, y
o
) above an infinite flat wall at 
y 
= 0. An 
image
 
vortex centered at (
x
o 
,
-
y
o
) models the presence of the wall


 
41
 
 
 
x
 
 
Figure 2.
9
 
Computational and analytical results of
 
the wall pressure and associated streamlines 
for a line vortex above a wall. Different subplots represent different times (as indicated beneath 
the plots). 
The streamlines 
are shown
 
in a frame of reference
 
convecting with the vortex

..
 
43
 
 
Figure 2.
10
 
N
ormalized Root Mean Square Error (NRMSE) between 
a
nalytical and numerical
 
values of the wall
-
pressure distribution beneath a line vortex above a flat wall. The time duration 
shown is from the start of the vortex motion until the vortex center is located at
 
r
 

 
44
 
 
Figure 3.1 
Streamlines
 
and normalized 
pressure
 
at different time step
s
 
for frequency 





, 
Re
=5000, and circulation 










 
51
 
 
Figure 3.2 
Stremlines
 
and normalized 
pressure
 
at different time step for
 
frequency 
f
= 980 
Hz
, 
Re
=5000, and circulation 










 
52
 
 
Figure 3.3 Effect of vortex passage frequency on the radial distribution of the mean pressure 
coefficient
. The green broken lines outline the boundary of the sta

 
54
 
 
Figure 3.4 Effect of vortex passage frequency on the radial distribution of the 
RMS
 
pressure 
coefficient

 
56
 
 
Figure 3.5 Pressure time series at various
 
r/D
 
locations within the stagnation zone, 
Re
=5000, 









, 
and 




















. The small 
discontinuities in the time series for 





 
correspond to the time instant when a new 
vortex is injected at the top of the computational domain. Because the d
iscontinuities are small, 
no attempt was made to get rid of them by using a larger wall
-
normal extent for the 
computational domain
 

 
58
 
 
Figure 3.6 Pressure time series at various frequencies
,
 
Re
=5000, 









, 
and
 























. The small discontinuities in the time series for the two larger 
frequencies correspond to the time instant when a new vortex is injected at the top of the 
computational domain. Because the disc
ontinuities are small, no attempt was made to get rid of 
them by using a larger wall
-
normal extent for the computational domain.
 

 
.62
 
 
Figure 3.7 
Induced v
elocity 
by an infinite line
-
vortex array along a line parallel to the array. 
Results are sho
wn over one vortex spacing (

) with the vortex located at the center (



). 

 
63
 
.
 
Figure 3.8
 
Individual and collective induced wall velocity by three
-
line vortices placed relative 
to a wall a

-
frequency case when the 
leading vortex is located at 






 
..64
 
 
Figure 3.9 Effect of 
Reynolds number
 
on the radial distribution of the mean pressure 
coefficient


 
. 65
 
 
Figure 3.10 Effect of 
Reynolds number
 
on the radial distribution of the 
RMS
 
pressure coefficient
 

.66
 
 
xi
 
 
 
Figure 3.1
1
 
Effect of vortex 
circulation
 
on the radial distribution of the mea
n pressure coefficient
 

67
 
 
Figure 3.1
2
 
Effect of vortex 
circulation
 
on the radial distribution of the 
RMS
 
pressure coefficient
 


 
 
Figure 3.1
3
 
Pressure time series at various 
r/D
 
locations within the stagnation zone, 
Re
=5000, 









, 
and 





 

 

69
 
 
Figure 3.1
4
 
Pressure time series 
for
 
various 
vortex 
circulation
s
,
 
Re
=5000, 





, 
and 

























 
..70
 
 
Figure 4.1 F
low configuration for 
normal
-
incidence 
impingement jet
 

72
 
 
Figure 4.2 Experiment
al
 
facility
 

 
73
 
 
Figure 4.3   An image of the impingement plate and the sliding ta
ble attached to the Velmex 
manual traverse system
 

 
74
 
 
Figure 4.4 Block diagram of 
the 
setup used 
for
 
calibration and measurements of the hot
-
wire

 
76
 
 
Figure 4.5 
An
 
image
 
of the motorized 
system for 
traversing 
the 
hot
 
wire
 

 
77
 
 
Figure 4.6 Block diagram of the setup for the square wave test
 

 
79
 
 
Figure 4.7 An image of the oscilloscope screen showing a typical square wave test result
 

 
79
 
 
Figure 4.8 An image of the h
ot
-
wire, temperature sensor, nozzle, impingement plate, and the 
conditioning box

 
.81
 
 
Figure 4.9 Sample of hot
-
wire calibrations before and after an experiment 

 
83
 
 
Figure 4.10 Front and cross section 
(
B
-
B
)
 
view 
(top and bottom respectively) 
of
 
the
 
microphone 
array configuration used in the present work
 

 
84
 
 
Figure 4.1
1
 
Block diagram of the calibration setup for the microphone array
 

 
86
 
 
Figure 4.1
2
 
An image of the calibration 
setup
 
for the microphone array
 

 
.87
 
 
Figure 4.1
3
 
A sample of 
the 
microphone calibration 
results:
 
sensitivity (top) and phase 
(bottom)

 
88
 
 
Figure 5.1 Shear
-
layer mean velocity profile at 
X/D
=0.2 
for
 
various 
Reynolds num
bers 

.92
 
 
 
xii
 
 
Figure 5.2 
Shear
-
layer fluctuati
ng
-
velocity root
-
mean
-
square profile
 
at 
X/D
=0.2 
for
 
various 
Reynolds numbers
 

 
92
 
 
Figure 5.3 Shear
-
layer mean velocity profile at various 
X/D
 
locations and 





 
93
 
 
Figure 5.4 
Shear
-
layer fluctuati
ng
-
velocity root
-
mean
-
square profile
 
at various 
X/D
 
locations and 





 
.94
 
 
Figure 5.5 Normalized Root Mean Square Pressure 







 
versus 



 
for different 



 
and 
two Reynolds numbers: (a) 
Re
D
 
= 8272, and (b) 
Re
D
 
= 24818

 
96
 
Figure 5.6 Normalized Root Mean Square Pressure 







 
versus 



 
at
 
Re
D
 
= 8272 and 
24818
,
 
for
:  



















and
 








  

 
.98
 
Figure 5.7 Sample normalized Pressure signals 









 
at
 
Re
D
 
= 8272
,
 
for
 











 
for:  





















 
and 








  

 
100
 
 
Figure 5.8 Sample normalized Pre
ssure signals 









 
at
 
Re
D
 
= 24818
, for
 











 
for:  





















 
and 








  

 
102
 
 
 
Figure 5.9 Probability Density Function 



 
for the pressure signal at 
Re
D
 
= 8272
,
 
for 











, and:  




















and 








  
..

 
.105
 
  
 
Figure 5.10 Probability Density Function 



 
for the pressure signal at 
Re
D
 
= 24818
,
 
for 











, and:  




















and 








  





 

 
106
 
 
Figure 5.11 Radial distribution of skewness for 








and 



and: 








, and 










 
109
 
 
Figure 5.12 Radial distribution of Kurtosis for 








and 



and: 








, and 










 
..111
 
 
Figure 5.13 Normalized 
PSD
 
contour plots at 





 
for:  

















, 
and 









  
..

 
115
 
 
 
Figure 5.14 Normalized 
PSD
 
contour plots at 





 
for:  

















, 
and 










 
.117
 
 
Figure 5.15 Normalized 
PSD
 
at 




 
for
,
 








and 



at: 





















  


 
118
 
 
Figure 5.16 Normalized 
PSD
 
at 





 
for 








and 



at: 





















  

 
.119
 
 
Figure 5.17 Normalized 
PSD
 
at  








 

 






 
for:  

















and 









 
.121
 
 
xiii
 
 
 
Figure 5.18 Normalized 
PSD
 
at  








 

 








 
for:  

















and 









 
..122
 
 
Figure 5.1
9
 
Samp
le time series from microphone measurements (top) and the vortex
-
array model 
(bottom) at the stagnation point (





) and end of the stagnation zone (





). 
Experimental data are shown for the 





 
and 





, and model results for the 
refe

 
..
125
 
 
Figure 5.
20
 
Sample wall
-
pressure signature characteristic of that produced by vortex
-
induced 
separation from microphone measurements at 





, 





 
and 






 
126
 
 
Figure 5.
21
 
Comparison of the effect of varying 



 
on RMS wall
-
pressure fluctuation between 
the experimental (top) and the model (bottom) results at 




 
. In the model, increasing 
frequency corresponds to decreasing 



. The br
oken green lines outline the end of the 

 
128
 
 
Figure 5.
22
 
Comparison of the effect of varying 



 
on 
RMS
 
wall
-
pressure fluctuation between 
the experimental (top) and the model (bottom) results 
at 






 
(experiment) and 



 
(model)
. The broken green lines outline the end of the stagnation zone

 
..131
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
xiv
 
 
KEY TO SYMBOLS
 
 
 
 



 
Hot
-
wire calibration coefficient
s
 

 
a
 
Constant 
in the potential stagnation
-
point flow eq
uations
 
C
 
Volumetric constant related to 
the volume
 
of the 
vortex ring
 
c
 
Sound speed
 


 
Mean
-
pressure coefficient 
 


 
Root
-
mean
-
square
-
pressure coefficient 
 
D
 
Jet diameter
 
E
 
Hot
-
wire output v

 


 
Temperature
-
c
orrected
 
h
ot
-
wire 
output 
voltage
 




 
Complete elliptic integral of the second kind
 


 
Measured 
h
ot
-
wire voltage
 


 
Strain
-
rate tensor 
 
f
 
Frequency
 
H
 
Distance between the impingement plate and the jet exit
 




 
Complete elliptic integral of the fir
st 
kind
 
l
 
Side length of the plane wave tube 
 
N
 
Number of samples in 
a 
time series
 
P
 
Mean wall pressure
 


 
D
ynamic pressure 
of the jet 












 
 
PSD
 
Power spectral density 
of
 
the wall
-
pressure fluctuation
 

 
 
  
Wall 
pressure 
 
 
xv
 
 


 
  
Wall
-
pressure fluctuation
 


 
  
Root mean square 
of
 
the wall pressure fluctuation
 
q
 
  
Flow s
ource 
of 
pressure
 


 
  
Jet 
Reynolds number based on the jet diameter and the jet exit velocity
 


 
  
Vortex
 
core
 
radius
 


 
  
Initial vor
tex core radius
 
r, 

 
  
Impingement
-
plate polar coordinate
s
 



 
  
Strouhal number based on jet diameter 
 


 
  
Temperature of the fl
ow
 
during 
h
ot
-
wire calibration 
 


 
  
Temperature of the fl
ow
 
during 
h
ot
-
wire measurements
 


 
  
Hot
-
w
ire temperature
 



 
  
Radial component of 
velocity induced by the image on the real vortex ring
 



 
         
Dimensionless radial component of 
velocity induced by the image on the real vortex
   
 
                    
ring                    
   
 
    
 


 
  
Normal component of 
velocity induced by the image on the real vortex ring
 



 
  
Dimensionless normal component of 
velocity induced by the image on the real 
 
                    
vortex
 
Ring
 
 



 
  
Radial 
component of se
lf
-
induced velocity
 


 
  
Normal component of self
-
induced velocity
 



 
  
Dimensionless normal component of self
-
induced velocity
 


 
  
Jet exit v
elocity
 


 
  
Radial v
elocity 
component
 
(parallel to the plate)
 


 
  
Axial v
elocity
 
component
 
(normal to the plate)
 


 
  
Azimuthal v
elocity 
component
 
 
xvi
 
 





 
  
Initial 
coordinates
 
of the vortex ring

 
in polar coordinate
s
 






 
  
Initial 
coordinates
 
of the vortex
 

 
in Cartesian coor
dinate
 
x, y, z
 
  
Cartesian
 
coordinate
 
system
 
with origin at 
the 
center of the 
impingement 
plate 
 

 
  
C
artesian coordinate
s of the pressure source
 
t
 
  
Time
 

 
  
Velocity p
otential 
 


 
  
Rotation rate 
tensor 
 

 
  
Vortex 
Circulation
 


 
  
Circulation around a contour that encompasses the shear laye
r
 

 
  
V
orticity
 

 
  
Wavelength
 

 
  
Momentum thickness of the jet shear layer
 

 
  
Fluid density
 

 
  
Fluid dynamic viscosity
 

 
  
Fluid kinematic viscosity
 


 
  
Velocity difference across t
he shear layer
 

 
  
Stream function
 


 
  
Dimensionless stream function
 
 
1
 
 
CHAPTER 
1: Introduction
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Experimental, computational and analytical research has been used to study impinging jets 
for the past decades because of their significance t
o many applications, including cooling, heating, 
drying, air conditioning, and ventilation. The present research is concerned with modeling
,
 
and 
understanding the physics of wall
-
pressure generation in impinging jets. This knowledge is 
significant for appl
ications involving flow
-
induced noise and vibration. To motivate the current 
research, a summary of relevant previous literature is necessary. However, some understanding of 
the basic characteristics and flow features of free (non
-
impinging) jets is essent
ial. Therefore, the 
present discussion starts with an overview of the latter. This is followed by a 
brief 
summary of the 
relevant work on impinging jets and their wall
-
pressure characteristics. Finally, the motivation and 
the specific objectives of this th
esis are outlined at the end of the chapter.
 
1.1
 
Free Jets
 
1.1.1 Background
 
Jets can 
be classified
 
into different categories according to: the nozzle exit shape (e.g. 
circular, square, triangular, lobed, etc.), the nozzle contour (e.g. smooth versus sharp
-
edge
d), the 
pipe length from which the jet emerges (long 
or
 
short; if any), and the initial discharge condition 
(
free,
 
wall
 
and surface jet). The most 
basic
 
type
 
of 
jets
 
is the axisymmetric free 
jet,
 
driven by 
pressure to emerge at the end of a contoured nozzl
e into a 
quiescent
 
ambient. For a free jet, after 
exiting the nozzle, 
a 
free shear layer with uniform pressure 
surrounds the jet
. As the flow develops 
farther 
downstream, mixing of the jet fluid with the ambient causes an increase in the mass flow 
rate in 
the jet stream (i.e. via flow entrainment), and, in conjunction with viscous and turbulence 
effects, leads to 
jet
 
spreading and decreasing of the flow speed to conserve momentum. 
 
 
2
 
 
Depending on the 
characteristics of the 

 
jet
 
development 
may be divided into three zones as 
depicted 
in Figure 1.1
. see Shih
-
I Pai [1]
 

 
Zone 1: in this zone
,
 
there is a potential core in the central part of the jet and a mixing 
 
zone is sandwiched between the potential core and the surrounding me
dium. The 
potential core has a centerline velocity equal to the jet exit velocity
.
 
This 
zone
 
extends 
up to 
4 

 
6 jet diameters
;
 

 
Zone 2: represents the 
transition where the velocity profile gradually changes until self
-
similarity 
is 
established
. 
This 
zone
 
e
xtends 
from 
the end of zone 1 
to 20 jet diameters
; 
 

 
Zone 3: represents the self
-
similarity zone where the 
transverse
 
mean 
velocity profile 
is similar at 
different
 
axial
 
distances when normalized using the local centerline 
velocity and jet width. 
 
 
Figure 
1.1 Schematic of a free
-
jet flow, depicting various flow development zones 
 
 
3
 
 
1.1.2 Literature Review
 
The 
primary
 
and fundamental interest in studying free 
jets
 
has been focused on the 
instability of the shear layer downstream from the nozzle exit, and the t
urbulence development 
farther downstream. Grant [2] 
theoretically 
studied 
the shear layer instability in axisymmetric jets
. 
This study shows the formation of vortex structures from an instability wave originating 
at the 
beginning of
 

mmetric shear layer. The initially weak instability wave 
amplifies with downstream distance, ultimately leading to
 
creation of the
 
eddies. 
 
Popiel and Trass
 
[3] visualized free and impinging round jets using a smoke wire
. 
For the 
free jet, 
t
hey 
showed the 
vortices to form in the potential
-
core region within the free shear layer. 
The downstream
 
merging of these vortices causes the creation of larger eddies. These vortices 
increase mixing and enhance the entrainment rate
.
 
Also, the axial symmetry 
of
 
the near 
field of 
the nozzle exit is found to be created by the generation of the toroidal vortices, because they cause 
a significant upstream interaction
. 
Zaman and Hussain [4] studied the natural 
large
-
scale
 
structures 
in the axisymmetric mixing layer surrounding
 
the jet. They found the flow from the jet exit to the 
end of the potential core to be 
controlled
 
by two characteristic length scales: the jet diameter and 
the initial shear layer thickness. Near the jet 
exit,
 
the momentum thickness controls the flow 
struc
ture. Initially, the momentum thickness is 
thin,
 
and as the shear
-
layer 
rolls
-
up into vortical 
structures,
 
the momentum thickness 
grows
. The vortices
 
resulting from the roll
-
up
 
start to interact 
and 
create
 
larger eddies. After a distance 
x
, which is compar
able to the 
jet
 
diameter, the diameter 
length scale controls the flow structure. As the jet Reynolds number
 
(









; where 


 
is 
the jet exit velocity, 

 
the jet diameter, and 

 
the fluid kinematic viscosity)
 
increases, the ratio of 
the 
shear
 
layer thickness at the nozzle 
lip to the diameter of the jet decreases, which leads to 
a 
smaller in
itial instability wavelength, relative to the jet diameter, and larger number of vortex 
 
4
 
 
pairings before the vortices size becomes comparable to the jet diameter, at the end of the potential 
core.  
 
1.2
 
Impinging Jet 
 
1.2.1 Background
 
In an impinging jet, the f
low is incident on a, typically, flat wall at a distance 
H
 
from the 
jet exit. Though the angle of incidence of the jet relative to the wall
-
normal direction may change, 
the present work is only concerned with normal incidence; i.e. where the jet symmetry a
xis is 
perpendicular
 
to the wall. The flow field for an impinging jet may be divided into three zones (see 
Figure 1.
2
): 
 

 
Free
-
 
jet
 
zone: which 
represents
 
the domain stretching from the nozzle exit to the point where 
the existence of the plate does not infl
uence the flow. Within this zone, the jet flow and 
associated flow features are as discussed in the previous section; 
 

 
Stagnation zone: which 
represents
 
the domain where the mean flow direction changes from 
being normal to being parallel to the plate. This
 
zone, which
 
extends up to 





 
in the 
radial direction, has the maximum mean pressure of the flow 
(
at the stagnation point on the 
wall
)
;
 

 
Wall
-
jet zone: 
w
hich 
corresponds
 
to the domain 





. Unsteady separation of the 
boundary layer is known to o
ccur in this region due to the interaction of the jet vortices with 
the wall; See 
Didden and Ho [5] and 
Landreth and Adrian [6]
.
 
 
5
 
 
 
 
Figure 1.
2
 
Schematic of the impinging jet at normal incidence 
(taken
 
from 
Al
-
Aweni [7
]
)
 
1.2.2 Literature Review
 
A brief revi
ew is provided here of the current knowledge on wall
-
pressure fluctuation in 
impinging jets. Similar to the current work, focus will be on 
H/D
 
values extending to the end of 
the potential core (





 
). To characterize the strength of the pressure fluctuation, typically 
the radial distribution of the root mean square of the fluctuation (


) is examined. The recent 
work of Krishna [8] provides the most comprehensive 


 
data, co
vering the Reynolds number 
range 











, and 







. Krishna showed that for 





, the largest 
pressure fluctuation are found in the wall
-
jet zone in the range 











 
for all Reynolds 
numbers. A second peak within the stagnati
on zone emerged, near 







, when 



 
reached 
25,000 and the impingement wall was sufficiently far from the jet (





). When present, this 
secondary peak magnitude was higher than that in the 
wall
-
jet
 
zone.
 
 
6
 
 
Consistent with the work of Krishna 
[8], several earlier studies have identified the wall
-
jet
-
zone peak of the pressure fluctuation. These include the work of Hall and Ewing [9] and [10], who 
found the peak at 







 
for 










 
and 









, El
-
Anwar 
et al.
 
[11], at 







 
for 





 
and 





, and Al
-
Aweni [7],  







 
for 





 
and 





.
 
The variation in the exact 
location of
 
the peak between the different studies 
might be due to Reynolds number, the jet initial condition, or t
he radial spacing between the 
measurements. 
The 
RMS
 
level of the pressure fluctuation associated with this peak is quite large, 

dynamic pressure (based on jet exit velocity), but this level, along 
with the overall level of 
pressure fluctuation in the wall
-
jet zone, decrease with increasing both 



 
and 



. This Reynolds number trend is seen in the data of Krishna [8], while the 



 
dependence is reported in all aforementioned studies.
 
The
 
 



 
peak in the stagnation zone is most clearly seen in the data of Krishna [8], which 
e
xtend into a high
-
enough 



 
range for the peak to be observed. The presence of this peak is also 
implied in the data of Hall and Ewing [9
,
10] at 





, although there were not sufficient 
measurement points to ascertain the specific locatio
n of the peak inside the stagnation zone. 
Furthermore, Hall and Ewing employed a jet that exits at the end of a fully
-
developed turbulent 
pipe flow, which exhibits significant pressure fluctuations at the stagnation point (relative to jets 
exiting from a c
ontoured nozzle) due to the absence of a potential core.
 
Within the stagnation zone, 


 
increases with increasing 



; opposite to the trend in 
the wall
-
jet region. This behavior is generally associated in the literature with the growing 
influence of the vortical structures on the stagnation zone as the vortices grow to a size co
mparable 
to the jet diameter through successive pairings (e.g. see Al
-
Aweni [7]). On the other hand, the 
 
7
 
 
Reynolds number influence is found to decrease the stagnation
-
zone pressure fluctuation (Krishna 
[8]). No specific explanation for this trend 
is 
known 
at this point.
 
Frequency spectra analysis of the wall
-
pressure time series measured in the studies 
referenced above show that the frequencies of the spectral peaks are consistent with the passage 
frequency of the jet vortices. In addition to depending on t
he initial conditions and Reynolds 
number of the jet, the latter frequency is predominantly affected by the distance between the jet 
exit and the impingement plate. As the distance increases from 
H/D=
2 to the end of the potential 
core, the number of vortex
 
ring parings ahead of impingement increases, decreasing the passage 
frequency of the vortices. Al
-
Aweni [7] used simultaneous time
-
resolved flow visualization and 
wall
-
pressure measurement, employing a microphone array
,
 
for a jet Reynolds number of 7970, 
and 
H/D=
2,3, and 4. He showed that for 
H/D=
2
,
 
the first vortex merging occurred within the wall
-
jet zone, as the vortices traveled parallel to the impingement wall. For 





, the first pairing 
was complete
d
 
before reaching the impingement plate, while f
or 
H/D=4
, the second merging took 
place ahead of the plate. When pairing happened ahead of reaching the wall, each merging resulted 
in halving the fundamental frequency in the pressure spectra. When merging happened while the 
vortices traveled past, and in
teracted with the wall (





), the spectrum contained the original 
vortex formation frequency (or Strouhal number 







) and its sub
-
harmonic (







). 
The lowest frequency observed, at 





 
after two pairings, corresponded to 







. The 
drop of the
 
dominant pressure
-
fluctuation Strouhal number with 



 
and the overall order of 
magnitude of the Strouhal number values reported in Al
-
Aweni [7] is consistent with the findings 
of Hall and Ewing [9] and [10], El
-
Anwar 
et al.
 
[11] and Krishna [8].
 
The ea
rliest explanation for the large pressure fluctuation associated with the 


 
peak in 
the wall
-
jet zone came from the work of Didden and Ho [5].
 
These authors studied a normally 
 
8
 
 
impinging axisymmetric forced air jet at Reynolds number of 19000, and
 
H/D=
4. Using phase 
averaged pressure and hot
-
wire measurements, they showed that the high level of pressure 
fluctuation in the wall
-
jet region is associated with the unsteady boundary layer separation, and 
subsequent formation of a
n
 
opposite
-
signed 
second
ary
 
vortex, when a jet (primary) vortex 
interact
ed
 
with the wall. Such an unsteady boundary
-
layer separation process
,
 
and associated 
phenomena, was first noted by Harvey and Perry [12] in relation to trailing wing
-
tip vortices 
interacting with the ground. 
In impinging jets, formation of secondary vortices was also 
reported 
in the work of
 
Landreth and Adrian [6]
,
 
for Reynolds number of 6500 using particle image 
velocimetry, and the flow visualization of Popiel and Trass [3].
 
Didden and Ho [5]
 
showed that 
the
 
boundary layer separation was associated with the adverse pressure gradient imposed by the 
jet vortices on the boundary layer
.
 
Naguib and Koochesfahani [13] used whole
-
field velocity data of an isolated axi
-
symmetric vortex ring interacting with a flat wa
ll to understand the fundamental surface
-
pressure 
generation mechanism associated with vortex
-


-
pressure 
generating sources
 
and wall
-
pressure signature from the velocity
-
field data. In addition to 
connecting negative wall
-
pressure peaks with the primary and secondary vortices, they were able 
to identify an important source of positive pressure fluctuation, not known before the
n. 
Specifically, they showed that the high strain zone associated with the separation of the boundary 
layer (induced by the main vortex ring) was an important source of positive pressure fluctuation. 
The results
,
 
however
,
 
did not show a strong negative pre
ssure peak, as was found in the work of 
Didden and Ho [10] beneath the separated zone. This may have been due to the limited spatial 
resolution of the experimental data within the separating boundary layer. 
 
 
9
 
 
Al
-
Aweni [
7
] conducted a comprehensive study of 
the wall
-
pressure fluctuation and 
associated generation mechanisms in impinging jets. The work utilized both experimental data 
(from simultaneous time
-
resolved flow visualization and wall
-
pressure sensor
-
array data) and axi
-
symmetric laminar 
CFD
 
calculatio
n of isolated vortices interacting with a flat wall. An interesting 
and new finding from this study is that pressure fluctuation where the wall
-
jet 


 
peak 
is 
observed are especially strong
 
when vortex
-
wall interaction happens while two vortices are in the 
process of pairing. He showed that during such pairing, which happened at 






, the resulting 
secondary vortex is much stronger than wh
en a single vortex interacts with the wall
. In particular, 
he saw that, 
occasionally, vortices may pass without pairing, in which case, the pressure spikes 
were not as strong. The switch between pairing/no
-
pairing seemed to happen randomly in time
. 
The 
nea
r
-
wall pairing
 
produced 
strong negative pressure spikes that reach
ed
 
a magnitude 
comparable with the dynamic pressure
,
 
based on the jet exit velocity. Al
-
Aweni
 
also found that 
the positive
-
pressure source identified earlier by Naguib and Koochesfahani [13]
, which is 
associated with the high strain rate within the separating boundary layer, has a significant influence 
on the shape and evolution of the strong negative spikes. 
 
1.2.3 Governing 
Equations
 
To get better understanding of the physical mechanisms le
ading to wall
-
pressure 
fluctuation,
 
it is insightful to connect the pressure 
fluctuation
 
generation to the 
vortical
 
structure
s
. 




 
in 
incompressible
 
turbulent flow: 
 
 

























 
(1.1)
 
 
10
 
 
Where, 







 
is
 
the velocity gradient 
tensor 
and Einstein

 
tensor 
notation 
is
 
used
. The forcing term 
in equati
on (1.1) (
right
-
hand
 
side) can 
be 
divided
 
into two parts: symmetric (in terms of the strain 
rate 


), 
and antisymmetric (in terms of the rotation tensor 


); see Bradshaw and Koh [14]:
 
 






















 
(1.2)
 
Where,
 
 




















 
(1.3)
 
 




















 
(1.4)
 
 
It is known that the rotation tensor is connected to the vorticity vector through: 
 
 













 
(1.5)
 
Thus, equation (1.2) can 
be written as
 
 


























 
(1.6)
 
From equation (1.6), the pressure source strength (
q
) is given by: 
 
 















 
(1.7)
 
Equation (1.6) may be written in vector form as follows,
 
 


























 
(1.8)
 
 
 
11
 
 

 
function can be used to solve equation (1.8) (e.g. 
Blake
 
[15]) to get the pressure on a solid 
wall 
beneath an unsteady flow, where the wall
-
normal
 
coordinate 



, for pressure 
source 
distribution 











 
within the flow
. More specifically, 
 




















































































































 
(1.9)
 
                                               
 
The volume integral
 
(first term)
 
in equation (1.9) 
rep
resents
 
the contribution to the wall 
pressure by the flow structure within the body of the flow
, while the surface integral is computed 
over the wall beneath the flow
. 
For a flat wall, the second term is negligible (from boundary layer 
approximation perspe
ctive). 
The 
volume integrand
 
shows that there are two wall
-
pressure 
generation mechanisms: one related to strain rate, and the other to vorticity. The former results in 
the generation of positive, and the latter in negative wall pressure. Thus, flow featur
es where 
rotation dominates strain effects (e.g. in the core of vortices), generate negative pressure, while 
those associated with dominant strain (e.g. the zone in between interacting vortices) results in 
positive pressure generation; e.g. see Naguib and 
Koochesfahani [13]. Another important feature 
of the volume integrand in equation (1.9) is that the source term 
effect
 
is inversely proportional 
to
 
the distance between the 
point of 
wall
-
pressure observation
 
and the pressure source location (
as
 
seen 
from
 
t
he numerator of the integrand). 
Thus, as this
 
distance 
increases
, the 
observed 
wall 
pressure decreases. 
Therefore
, the pressure observed at a point on the wall is a 
global
 
quantity
,
 
related to all features within the flow with the net pressure being the re
sult of the integrated effect 
of the strength and proximity of the different sources.
 
 
12
 
 
1.3
 
Motivation
 
The motivation for this research is to understand the physics of, and to predict the unsteady 
surface pressure generation in impinging jets due to their signif
icance in flow
-
induced noise and 
vibration. As described earlier in this chapter, the basic connection between the dominant flow 
features in impinging jets and wall
-
pressure generation is fairly well understood. However, there 
is practically no effort that
 
capitalizes on this understanding to develop physics
-
based (also known 
as structure
-
based) models to compute the wall pressure in impinging jets. Such models, if 
sufficiently accurate, could be valuable as engineering design tools for flow
-
induced noise a
nd 
vibration applications, since the models are much more efficient to run than direct numerical 
simulations, and they are more robust than non
-
physics based turbulence models. 
 
Additionally, physics
-
based models could be used to understand the underlying 
flow 
physics from a point of view that is not possible with experiments or numerical simulations. 
Specifically, in real flows, or their simulations, it is difficult to vary certain flow or structure 
parameters one at a time because of the interdependence o
f these parameters. With a mathematical 
model, such variations are possible, which could lead to a clearer understanding of the effect of 
individual parameters on quantities of interest; the wall
-
pressure fluctuation in the present work. 
An example of such
 
model
-
based insights may be found in the work of 
Monnier 
et al.
 
[16], where 
a Gaussian
-
core vortex array was used to model the wake of a harmonically pitching airfoil. The 
model helped to understand the connection between the parameters describing the wak
e
-
vortex 
configuration (vortex streamwise and cross
-
stream spacing, vortex circulation, and core radius) 
and the mean thrust acting on the airfoil.
 
 
 
 
13
 
 
1.4 Objectives
 
The objectives of this study may be summarized as follows: 
 
1
-
 
Developing a 
simple
 
mathematica
l model, which represents the 
first step
 
towards a high
-
fidelity model, for predicting wall
-
pressure fluctuations in normally impinging jet flows. 
The model will be used to 
explore
 
the effect of vortex passing frequency, 
vortex
 
circulation, and Reynold num
ber on the unsteady wall
-
pressure characteristics, while 
varying one parameter at a time. In addition, the model results, which are only applicable 
to the stagnation zone due to inherent 
simplicity/
limitations of the model, will be compared 
to experimental
 
data obtained in the present study.
 
2
-
 
To conduct measurements of the unsteady surface pressure in a normally impinging jet 
using a microphone array. The measurements are done for two Reynolds numbers:  




 
8272 and 24818 for 
H/D
 
values of 2, 3 and 4 (reaching to the end of the potential core). 
The measurements will be used to examine the influence of the Reynolds number on the 
characteristics of the wall pressure flucution (primarily the radial di
stribution of the root 
mean square pressure fluctuation, the probability density function of the pressure 
fluctuation and the power spectral density)
. The lower Reynolds number is selected to 
match this of the earlier study by Al
-
Aweni [
7
] in the same jet 
facility. Al
-
Aweni utilized 
time
-
resolved flow visualization and microphone
-
array measurements to connect the flow 
features to the wall
-
pressure characteristics. Thus, combining Al
-

those from the present work, it is possible to 
infer
 
the effect of Reynolds number o
n
 
the 
sources of wall
-
pressure generation. 
 
The remainder of the thesis is organized to show the mathematical model details in Chapter 2, the 
mathematical model results in 
Chapter
 
3, the experimental apparatus and procedure i
n Chapter 4, 
 
14
 
 
the experiment results and their comparison with the model results in Chapter 5, and conclusions 
and recommendations for future work in Chapter 6.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
15
 
 
CHAPTER 
2: Mathematical Model
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2.1. Modeling Backgroun
d
 
Prior to
 
describing
 
the features of the physics
-
based model developed here, a recap of the 
flow details that should be captured by 
a good
 
model 
is described
. Near the jet exit, vortices 
are 
generated
 
due to
 
the instability of the shear layer and its subs
equent roll
-
up into axisymmetric 
vortex rings. 
As the vortex rings travel downstream
,
 
the 
vortex size 
and strength 
increases by 
merging.
 
Several such 
mergings
 
may
 
occur, with the number of successive 
pairing
s
 
depending on 
Reynolds number and distance betwe
en the nozzle and the impingement plate. 
When the Reynolds 
number decreases, the coherence of the flow structure increases, and the number of successful 
pairings increases.
 
Nearing impingement, within the stagnation zone
, the flow changes its direction 
to 
be parallel to the plate. Further downstream as the flow advects through the 
wall
-
jet zone, 
boundary layer separation leads to secondary vortex formation and maximum wall
-
pressure 
fluctuation. 
Based on the 
literature
 
review 
of
 
Chapter 1, surface pressure f
luctuation in impinging 
jets 
is caused
 
by both inviscid phenomena (direct influence of vortices in stagnation and wall
-
jet 
zones) and viscous phenomena (vortex
-
wall and vortex
-
vortex
-
wall interaction in the wall
-
jet 
zone). 
 
The above description highlights
 
that a complex model is needed to capture the impinging 
jet flow features and associated wall
-
pressure fluctuation. More specifically, elements of a high
-
fidelity model should consist of 
 

 
A viscous vortex ring model
;
 

 
Ability to model vortex pairing
;
 

 
Advec
tion velocity field consistent with the actual mean jet velocity field
;
 
 
16
 
 

 
An impingement
-
plate boundary layer model
;
 

 
A vortex
-
boundary layer interaction model
;
 

 

-
pressure from the velocity field
.
 
However, develop
ing such high
-
fidelity model is an 
ambitious
 
goal that requires several stages of 
development. This work focuses on the starting step by considering only the simplest possible 
model. Table (2.1) 
shows
 
a 
comparison
 
of the features of a high
-
fidelity and the
 
present model.
 
Feature
 
High
-
Fidelity Model
 
Current Model
 
Vortex ring model
 
Viscous
-
core vortex model
 
Potential vortex model
 
Vortex
-
vortex interaction
 

 
Mutual induction
 

 
Pairing model
 
Not 
m
odeled
 
Advection field
 
Based on impinging
-
jet mean 
flow; 
e.g.,
 
fro
m CFD or 
experiment
 
Potential stagnation
-
point 
flow
 
Impingement
-
plate boundary 
layer
 
Modeled
 
Not 
m
odeled
 
Vortex
-
boundary layer 
interaction
 
Modeled
 
Not 
m
odeled
 
Wall
-
pressure calculation
 

 

equation
 
Table 2.
1 Comparison
 
of high
-
fidelity and present model features
 
Table (2.1) demonstrates that the 
present
 
model is 
very simple
 
and ignores all viscous and vortex
-
vortex interactions effects. However, evidence suggests that within the stagnation
-
zone wall
-
 
17
 
 
pressure
 
fluctuations are primarily generated via potential flow mechanisms for 






. 
Therefore, the present model may be 
successful
 
in at least reproducing the same qualitative 
features of stagnation
-
zone wall
-
pressure fluctuations. One of the 
main
 
goals of this work is to 
assess the ability of the present simple model to do so.
 
2.2 Model Details
 
2.2.1. Vortex Rings
 
The full 
inviscid 
mathematical model for the impingement
-
jet problem is depicted 
schematically in figure (2.1). 
A cylindrical coordinate 







 
system 
i
s 
used to describe the 
problem mathematically. However, becau
se of symmetry, the model equations have no 
dependence on 



. 
Each of the jet vortices is modeled employing a potential vortex ring that has 
an axis of symmetry perpendicular to the impingement wall. Each ring has circulation 



,  radius 



, zero cor
e radius 




, and 
vertical core coordinate 



. 
In the absence of the wall, the ring is 
in free space and translate
s
 
downwards (for the sense of circulation depicted in figure 2.1
)
 
due to 
self
-
induction effects. The stream function for the ring is gi
ven by Helmholtz equation [17] 
 
 
18
 
 
 
Figure 2.1 Schematic drawing of the 
full mathematical model of the impinging jet
 
 






































 
(2.1)
 
where
 
 
 


















 
(2.2)
 




 
represe
nts complete elliptic integral of the first 
kind,
 
and 




 
represents complete elliptic 
integral of the second kind:
 
 
 
19
 
 
 

























 
(2.3)
 
 
























 
(2.4)
 
To make equation 2
.1 dimensionless
, the initial vortex
-
ring radius 


 
is used as a length scale. 
S
ince in jet flow, the jet vortices have a radius approximately equal to the jet opening radius, this 
length scale is equivalent to half the jet diameter, 
D
/2
. The velocity
 
scale is taken as the velocity 
of the advection field (potential stagnation flow, at the top of the computational domain
,
 








), which is given by 




 
(see
 
section 2.2.3)
. This velocity scale 
correspond
s
 
to 
the 
mean 

exit 
jet veloci

 
(2.1)
 
is:
 
 










































 
(2.5)
 
The above equations are for zero
-
core radius potential vortex ring. The 
infinitesimal
 
core leads to 
an infinite induced
 
velocity
 
at the core
, which makes it impossible to track the vortex movement. 
To overcome this problem, a finite, but small, vortex core radius is assumed 






. Following 
the work of Walker 
et al
. [18], the velocity distribution inside the core is
 
assumed to be uniform. 
This leads to the following self
-
induced velocity of the ring:
 




















 
(2.6)
 
2.2.2. Wall (
V
ortex 
R
ings 
I
mages)
 
The equations discussed in section 2.2.1 are valid for a vortex in free space
. These 
equations
 
need to 
be modified
 
for the presence of the impingement wall; 
i.e.,
 
by enforcing a zero 
wall
-
normal velocity, or no
-
penetration, condition at the wall. 
This
 
is done by adding to the model 
an image vortex, relative to the wall 





 
with 
equal but opposite circulation 




 
to the real 
 
20
 
 
vortex (see figure 2.1). this velocity induced by the image on the real vortex ring has radial and 
wall
-
normal components 









. Thus, as the real vortex ring approaches the wall, the 
image vortex ring will affect the real ring in two ways: by stretching the ring radially outwards and 
reducing the approach velocity of the ring toward the wall. The radial stretching of the ring causes 
the ring radius to increase. According to Helmholtz, 
for inviscid incompressible flow, the vortex 
lines move with the fluid particles, and the vortex ring must have constant volume, i.e.
 
 













 
(2.7)
 
Where
 

 
is constant. Thus, in the presence of the wall, 

 
increases and 


 
decreases with time. 
Both of these quantities affect the self
-
induced velocity (see equation 2.6). The resulting self
-
induced velocity componen
ts 









, using equations (2.6 and 2.7) respectively are: 
 
 




 
(2.8)
 
 




















 
(2.9)
 
 



























 
 
The stream function for 
the image vortex is given by 
 














































 
(2.10)
 















































 
where
 



















 
(2.11)
 
 
21
 
 
And as
 
before, 





  
and 





 
represent the complete elliptic integrals of the first and the second 
kind
 
respectively
. 
 
The velocity induced by the image on the real vortex is obtained from the stream function 
due to the 
image
 
by applying equations (2.12
 
and 2.13) and setting 






 
and 





; i.e.
, the 
coordinates of the real vortex core.
 
 










 
(2.12)
 
 















 
 
 











 
(2.13)
 
 
















 
 
This
 
lead
s
 
to the velocity components produced by the image vortex ring on the real vortex 
(see 
Walker 
et al
. 
[
18
]
)
:
 
 















































 
(2.14)
 
 

















































 
 

























 
(2.15)
 
 
22
 
 
 



























 
2.2.3. Advection Field (Stagnation Point Potential Flow)
 
The next 
phase
 
of
 
the model involves adding steady flow to represent advection of the 
vortex ring
s
 
by the jet mean flow. To this end, potential point
-
stagnation flow 
is selected
. 
The 
stream function of the 
flow
 
is given by [19]
 
as
 
 





 
 
(2.16)
 
 









 
 
 
Applying equations (2.12 and 2.13)
 
to (2.16) 
leads to 
(see Naguib 
et al
 
[
20]
)
:
 
 




 
(2.17)
 
 








 
 






 
(2.18)
 
 










 
where 



 
is a constant with dimension 





.
 
Equation
 
2.1
7
 
shows that the mean flow radial velocity increases linearly and unboundedly 
with 

. In reality, this kind of variation is expected to hold only near the stagnation point with 
deviation from the model increasing with the radial distance. Ther
efore, we anticipate that this 
crude, yet very simple, model to be 
reasonable
 
only within the stagnation zone. 
 
2.2.4. Vortex 
A
dvection and 
F
low
-
F
ield 
E
volution in 
T
ime
 
 
For a single vortex convecting through the 









 
computational dom
ain 
shown in figure 2.1, 
the 
velocity field at any time instance consists of three components due to:
 
1.
 
t
he 
real vortex
;
 
 
23
 
 
2.
 
t
he 
image vortex
;
 
3.
 
t
he 
stagnation flow
.
 
The stream function of the system is the sum of the three corresponding stream functions; 
specific
ally
 














































































 
(2.19)
 






















































































 
The velocity components of the 
resulting flow
 









 
are found using equations (2.12 and 
2.13) 
 



































































































































































 
(2.20)
 
 
24
 
 






























































































































































































 


























































































































































 
 
(2.21)
 


























































































































































































 
 
where
 
 


















 
(2.22)
 
 
25
 
 


















 
(2.23)
 
The summation in (2.20) and (2.21) is over the number of vortices 

 
present in the domain 
at any given time instant. 
At the start of the computation, 
a
 
vortex ring 
is pla
ced
 
at the top of the 
domain at 

















 
. as time progresses, the ring core coordinates 





 
change with time according to 
 
 




















 
(2.24)
 

















































 






















































 
 
 





















 
(2.25)
 















































 
























































 
To model the periodic formation of vortice
s in a jet, as the leading vortex moves toward 
the impingement plate
, subsequent vortices 
are added
 
at the top of
 
the domain at a 
selected 
frequency
. All vortices approach the wall, gradually changing their dominant travel to be radially 
outwards. The comp
utation 
is continued
 
until the first 
vortex
 
passes the end of the domain (
i.e.,
 



)
.
 
 
26
 
 
2.2.5. Wall
-
Pressure Calculation and Evolution in Time
 

equation.
 
 

















 
(2.26)
 
Where
, 

 
is the velocity potential, 

 
is gr
avity acceleration, and 




 
is a function of time. 
Evaluating the LHS on the wall and 




 
at a suitable reference point, and ignoring gravity effect
,
 
equation (2.26) becomes 
 
 


























 
(2.27)
 
Or 
 
 




























 
 
(2.28)
 
Equation (2.28) shows that to obtain the wall pressure at a given point at any instant in time 
requires:
 
1
-
 
Knowledge of the radial v
elocity component at the wall 


. This information is readily available 
from the velocity field equations (2.20) and (2.21)
;
 
2
-
  
Selection of a reference point where: the velocity magnitude 


 
is known and the pressure 


is fixed so 
it can be used as a reference pressure. The first of these conditions is easily satisfied 
using equations (2.20) and (2.21). The second condition is more difficult to satisfy and
,
 
as will be 
seen below
,
 
meeting this condition sets a lower limit on the radi
al computational domain size 

 
and an upper limit on the computation
al
 
time. 
 
 
27
 
 
3
-
  
Calculation of the time rate of change of the velocity potential difference between the point of 
interest and the reference point. At every time instant, the velocity potential can 
be computed
 
by 
inte
grating the velocity field. 
This
 
leads to two potential issues. First, the integration yields the 
potential
 
with
 
an unknown additive integration function of time. This 
issue
 
is not problematic since 
the time function 
is subtracted
 
out when computing the po
tential 
difference
. Second, though the 
velocity field is known analytically, its form is not easy to integrate. Therefore, integration is done 
numerically to get the potential difference at each time instant. The resulting fields are 
subsequently different
iated numerically in time to arrive at the first term on the 
RHS
 
of equation 
(2.28). Further details follow.
 
As depicted in figure (2.1), the reference point 
is selected
 
on the wall at the end of the 
computational domain 





. 
This point rem
ains unaffected by the vortex rings until the first 
vortex ring convects through the entire domain, reaching near
 




. Thus, the pressure 


 
is 
steady up to the point of arrival of the first vortex ring, at which point the computation is st
opped. 
This imposes a limit on the computational time, leading to a tradeoff process. Given the periodic 
influence of the vortices, the computation must be run until at least two vortices pass by a given 
point on the wall in order to obtain a full period o
f pressure fluctuations. Thus, 


 
must be large 
enough for the computation time to be larger than the lowest period of vortex passage of interest. 
On the other hand, 


 
cannot be made arbitrarily large since the radial domain must be discretized 
finely for the integration leadi
ng to the calculation of the velocity potential difference, which could 
lead to a prohibitively long integration time. Therefore, 


 
is set only as large as necessary by 
allowing enough running time for the core of at least the second vortex injected 
into the domain to 
reach the largest radial location of interest. As discussed previously, it is expected that the 
applicability of the present model is limited to the stagnation zone. Accordingly, the extent of the 
 
28
 
 
radial domain of interest 


 
(see f
igure 2.1) is 






, where 





 
is the 
equivalent of the 
jet diameter. The unsteady potential function difference 









, is calculated using:
 
 




















 
(2.29)
 
which leads to 
 
 
 
 


















,
 
(2.30)
 
 
Where,
 

 
is a dummy 
variable for integration along
 
the radial coordinate
. The integral (2.30) is 
evaluated numerically, where the integrand is obtained by setting 



 
in equation (2.2
0
).
 
Notably, since it is the time derivative of (2.30) that is required for evaluating the wall pressure, 
the stagnation flow component, which is steady, does not affect the unsteady potential difference 
term.
 
2.3. Numerical Details 
 
The model was implemented
 
numerically using Matlab. The implementation consisted of 
two main tasks. The first one involved periodically seeding vortices at the entrance of the domain 
and tracking the core centers of these vortices as they advect through the computational 
domain
. 
K
nowledge of the core centers locations at each time instant enabled computation of the 
instantaneous stream function and velocity field. This information is used to visualize the flow 
field concurrently with the wall pressure. The second task employed the 
distribution of the radial 
velocity component at the wall to compute the distribution of the wall
-
pressure at each time instant. 
The specifics of the numerical implementation of these two tasks 
are given
 
in the two following 
sub
-
sections.
 
 
 
29
 
 
2.3.1. Computati
on of the Time
-
Dependent Stream Function and Velocity Field
 
Given the initial core coordinates (
R
o
, Z
o
) of a vortex at the top of the computational 
domain, 
subsequent
 
locations of the core (
R, Z
) at each time step 
was determined
 
by solving 
equations (2.24 
and 2.25) using 
fourth
-
order
 
Runge
-
Kutta method for the two variables 
R
(
t
)
 
and 
Z
(
t
)
. Once (
R, Z
) was 
determined
, the stream function 
was calculated
 
by evaluating equation (2.19) 
on 100 × 100
-
point grid using symbolic math tools in Matlab. 
The solution was 
independent of the 
time step and grid resolution, as demonstrated in Section 2.4.
 
 
 
2.3.2. Calculation of 
t
he Wall Pressure
 

at any radial location relative to the pr
essure at the reference point requires knowledge of the radial 
velocity at the same radial location and at the reference point, as well as the rate of change of the 
velocity potential difference between the pressure observation and the reference point.
 
The
 
velocity information was straightforward to obtain. 
Specifically, once the vortex core locations 
were determined 
at 
every time instant in the first task, the radial velocity distribution on the wall 
could be determined by setting 
z
 
= 0 in the analytical e
quation (2.20).
 
The reference velocity 


 
was further determined by setting 
r
 
= 
r
ref
 
in the resulting equation. The radial velocity distribution, 
and subsequently the wall pressure, was evaluated at 600 points on the wall spanning from the axis 
of symmetry of the domain to the reference
 
point. This number of points was sufficiently large for 
the calculated wall pressure to be independent of the number of grid points (see Section 2.4 for 
details). 
 
On the other hand, to find the
 
unsteady term 














 
on the wa
ll 

equation (2.28), 








 
was determined by numerically computing the integral (2.30). 
 
30
 
 
Specifically, the integral was discretized on the 600
-
point wall grid using the method of rectangles, 
leading to:
 
 


















 
(2.31)
 
 



























 
Where
 
i
 
is an index of the wall grid points starting from the 
point
 
next to the reference point and 
increasing towards the axis of symmetry, 
n
 
is th
e number of grid points (600), and 


 
is the radial 
resolution of the grid. The negative sign on the right
-
hand side reflects the fact that the integration 
is in the direction of decreasing 
r
 
coordinate.
 
To evaluate the summation in (2.31)
, the radial vel
ocity 
u
ri
 
was 
evaluated
 
by substituting 
for 
z
 
= 0 and 
r 
= 
r
i
 
in equation (2.20). This enabled evaluation of 








 
using (2.31) for 
all time instants. Subsequently, the time derivative of the potential difference was computed using 
forward
 
finite differencing. The solution was independent of the time step and grid resolution, as 
demonstrated in Section 2.4.
 
2.4 
 
Validation of the Computational Approach 
 
Validation tests 
were done
 
to ensure that the model results are independent of:
 
-
 
Radial c
omputational domain size;
 
-
 
Wall
-
pressure resolution (number of points on the wall to calculate 
the 
pressure);
 
-
 
Time step.
 
In addition
, the outcomes of these validation tests, 
in terms of
 
the domain size, radial wall
-
grid resolution and time step, were employ
ed to numerically compute the unsteady surface 
pressure associated with a line vortex advecting parallel to a flat wall. 
This
 
was done
 
since:
 
(1) the 
advecting line vortex problem has certain similarities with the current vortex
-
rings problem; and 
 
31
 
 
(2) the 
wall
-
pressure is known analytically for the line vortex problem, enabling further 
verification of the numerical solution approach and implementation.  
 
2.4.1
 
Impinging
-
Jet Model 
 
2.4.1.1 Validation of the Computational Domain Size
 
As described in Section 2.2, th
e main influence of the radial domain size is related to the 
basic assumption that the pressure at the reference point, which is placed at the radial end of the 
domain, remains steady throughout the computation. Thus, a larger domain allows longer 
computat
ion time since it takes the first vortex injected into the domain longer to reach the end 
and affect the pressure at the reference point. To ensure that the selected domain size is appropriate, 
three
 
different computational 
domains
 
having a radial extent o
f 
10, 15, and 20 
cm
 
are compared
. 
These values correspond to approximately 20, 30 and 40 times the initial vortex radius.
 
For each 
of these domains, the wall pressure 
is computed
 
and the resulting radial distribution at a time 


is located
 
at 
r 
= 5 
cm
 
is considered (Figure 2.
2
). The selected location
 





 
is 
10
 
times the initial vortex radius; 
well beyond what would be 
considered
 
the stagnation zone of the jet (
r
 
= one jet diameter 

 
twice 
the initial vortex ring radius), which is the 
main
 
focus of the present work. The three pressure 
distribution
s from the different domains are compared in pairs at the selected time instant using a 
Normalized Root Mean Square error (
NRMSE
), defined as follows: 
 
 
 





























 
(2.32)
 
 
whe
re
 
subscripts domain 1 and domain 2 denote the two 
domains
 
under comparison, 




 
is the magnitude of the negative pressure spike (minimum pressure) produced by the vortex, 
and the summation is over all wall grid points; 
n
 
= 600. Other than the do
main radial extent, other 
 
32
 
 
model parameters are the same as used for the results presented in Chapter 3.
 
It is noteworthy that 
the computational time for the three different cases varie
d
 
from
 
about
 
10
 
to 
45
 
minutes 
for the 
smallest
 
and 
the largest
 
domains 
r
espectively for 
the reference case which is define in table 3.1
.
 
 
 
 
 
Figure 2.
2
 
Radial wall
-
pressure distribution at 
the time instant when
 
the negative pressure 
spike (minimum pressure) associated with vortex passing 
is located
 
at 
r=5 
cm
 
location.
 
Diffe
rent color lines represent
 
different
 
radial
 
domain size
s
 
used for computing the wall 
pressure (as given by the legend). 
The inset
 
shows 
the 
overall distribution, while 
the 
main plot 
depicts a magnified view to show the change between the different cases.
 
 
 
 
33
 
 
Figure 2.
3
 
depicts the 
NRMSE
 
% as a function of time up to the time corresponding to 
Figure 2.
2
. The maximum pressure 
magnitude 




 
at each time step 
is 
used 
in calculating 
NRMSE
. As seen from the figure, the effect of increasing the domain size beyond 10 cm is
 
practically zero
 
(well below
 
0.
1%
)
. Therefore, a 10
-
cm 
wide
 
domain 
was utilized
 
for all the 
computations conducted
 
here. 
 
 
 
 
 
 
 
 
Figure 2.
3
 
Normalized Root Mean Square Error (
NRMSE
) resulting from comparing the results 
of domain sizes 10 and 15 
cm
 
(blue) and 15 and 20 cm (red). The results are shown up to the 
computation time corresponding to the time instant where t
he negative pressure spike appears at 
r 
= 5 
cm
.
 
 
 
 
 
 
34
 
 
2.4.1.2
 
    
Validation of the Wall
-
Pressure
-
Grid Resolution 
 
This step targets the validation of the 
resolution
 
of the wall grid used to find the wall
-
pressure fluctuations. The higher the 
resolution,
 
the better t
he accuracy. However, increasing the 
resolution
 
is expensive, consuming more computer resources and time. 
To validate the selected 
wall resolution
, three wall resolutions 
were examined
 
for a radial domain extent of 10 cm
: 0.33
mm
 
(300 points on the wall), 0
.16
mm
 
(600 points on the wall), and 0.0833
mm
 
(1200 
points
 
on the 
wall). 
These resolutions correspond to approximately 0.07, 0.034 and 0.017 of the initial vortex 
ring radius 


. 
All other model parameters are as described in Chapter 3. It is noteworthy that the 
computational time for the three different cases 
varie
d
 
from
 
about
 
10
 
to 40 minutes 
for the 
resolution 
0.
33
mm
 
and
 
0.
0833mm
 
respectively
 
for the reference case which
 
is define in table 3.1.
 
The pressure
 
distributions for the three cases considered are shown in figure 2.
4
 
for 
the
 
time instant when the pressure spike 
is located
 
at
 





 
(
t
 
= 0.0039 
s
)
.
 
The difference between 
the different cases is quantified using 
NRM
SE
, which is
 
shown in figure 2.
5
 
up to the time 
when
 
the negative pressure spike appears 
at 




 
.
 
Noticable from the plot, the 
maximum 
error 
decreased from 
approximately 
10% for the case between 300 and 600 points to 
less than 1
% for 
the case between 600
 
and 1200 points
. Therefore, 600 grid points on the wall were used for all 
wall
-
pressure results reported in Chapter 3.
 
 
35
 
 
 
Figure 2.
4
 
Radial wall
-
pressure distribution at 
the time instant when
 
the negative pressure 
spike (minimum pressure) associated with 
vortex passing 
is located
 
at 



.
 
Different 
color lines represent a
 
different
 
number of wall grid points used for computing the wall 
pressure (as given by the legend). 
The inset
 
shows 
the 
overall distribution, while 
the 
main plot 
depicts a magnified view to show the change between 
the different cases.
 
 
 
 
 
36
 
 
 
Figure 2.
5
 
Normalized Root Mean Square Error (
NRMSE
) resulting from comparing the 
results of grid resolution 300 and 600 points (blue) and 600 and 1200 points (red). The results 
are shown up to the computation time corresponding 
to the time instant whe
n
 
the negative 
pressure spike appears at 





.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
37
 
 
2.4.1.3
 
    
Validation of the Time Step Size
 
The final validation step is to 
verify
 
the size of the time step to produce results with an 
acceptable accuracy while minimizing time and computer resources. For this evaluation, 
three
-
time
 
s
teps
 
are 
used
 
corresponding to 1000, 2000 and 4000
-
time 
steps
 
during the period between 
injecting two successive vortices in the domain. The corresponding time step is 4.06, 2.03, and 
1.016 

s
, respectively, for the vortex passage frequency considered
 
(det
ails of how the physical 
value of 
frequency
 
is determined will 
be given
 
in Chapter 3)
. All other model parameters are as 
described in Chapter 3. 
The computational time for the three different cases 
varie
d
 
from 
approximately 10 
minutes for 1000 
points/cycle
 
to 
40 
minutes 
f
or 4000 
points/cycle
 
for
 





.
 
Pressure distributions for the three
-
time steps utilized 
are shown
 
in
 
figure
 
2.
6
 
at 
the
 
instant 
when the pressure spike 
is located
 
at 





 
(
t
 
= 0.0039 
s
)
.
 
As before, 
t
he 
difference between 
the different cases is quantified using 
NRMSE
 
in figu
re 2.
7
.
 
As seen
 
from the plot, the 
maximum 
error decreased from 3% 
when comparing
 
between 1000 and 2000 point per cycle (
PPC
) to 1.65% 
between 2000 and 4000 point per cycle (
PPC
)
. All computations done here utilized 2000 
PPC
.
 
 
38
 
 
 
Figure 2.
6
 
Radial wall
-
pres
sure distribution at 
the instant when
 
the negative pressure spike 
(minimum pressure) associated with vortex passing 
is located
 
at 



.
 
Different color 
lines represent a
 
different
 
number of computational time steps in the period between injecting 
two successive vortices (as given by the legend).
 
The inset
 
shows 
the 
overall distribution, 
while 
the 
main plot depicts a magnified view 
to show the change between the different cases.
 
 
 
 
39
 
 
 
Figure 2.
7
 
Normalized Root Mean Square Error (
NRMSE
) resulting from comparing the 
results of time step 4.066e
-
06 
s
 
(1000 
PPC
) and 2.033e
-
06 
s
 
(2000 
PPC
) (blue) and 2.033e
-
06 
s
 
(2000 
PPC
) and 1.0161e
-
06 
s
 
(4000 
PPC
) (red). The results are shown up to the computation 
time corresponding to the time instant whe
n
 
the negative pressure spike appears at 



.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
40
 
 
2.4.2 Application to 
a
 
Line Vortex Above 
a
 
Flat Wall
 
To validate the overall unsteady
-
wall
-
pressure computational approach and the 
implemented algorithms
, it was desired to apply the method/algorithms to a closely related 
problem
, where an analytical solution for the unsteady wall pressure is known.  The 
problem
 
of a 
line vortex above a wall seemed to be appropriate. Similar to the vortex
-
ring 
problem
, the presence 
of the line vortex above a flat 
wall
, as depicted schematically in
 
figure 2.
8
, is modeled using an 
imaginary image vortex placed symmetrically relative to the wall. The presence of this image 
vortex causes advection of the vortex pair in the positive streamwise direction (for the given sense 
of circulation), producing an
 
unsteady wall
-
pressure imprint on the wall. Physically, this scenario 
is similar to that of the vortex ring once it changes its advection direction from being predominantly 
towards, to being approximately parallel to the wall. In both the line and ring vo
rtex cases, the 
unsteady pressure field at a given point on the wall 
is caused
 
by the passage of vortices above the 
point
. Therefore, the ability to reproduce the 
unsteady
 
wall
-
pressure 
field
 
of the line vortex 
accurately using the tools developed for the 
impinging jet problem provides further confidence in 
the implementation of the model developed in the present work. 
 
 
41
 
 
 
Figure 2.
8
 
Schematic of a line vortex centered at (
x
o
, y
o
) above an infinite flat wall at 
y 
= 0. An 
image
 
vortex centered at (
x
o 
,
-
y
o
) m
odels the presence of the wall.
 
 
The velocity, 
and velocity 
potential for the line vortex problem are given by [21]:
 























 
(2.31)
 



































 
(2.32)
 
 
The corresponding wall
-
pressure distribution at a given time instant is computed using eq
uations 
(2.31) and (2.32) in conjunction with equation (2.28)
. 
The
 
numerical implementation
 
is the same 
as 
described in Section 2.3. For this calculation, the vortex and computational parameters 
are kept
 
the same as for the vortex ring case (without the in
clusion of the stagnation flow). Specifically, the 
parameter
s
 
values 
a
re
:
 
 
 
42
 
 
-
 
Domain size: 

 
-
 
Time step: 













 
-
 
Circulation: 










 
-
 
Wall resolution: 





















 
-
 
Vortex
 
location above the plate 






 
Figure 2.
9
 
depicts the wall
-
pressure signature obtained an
alytically and computationally, 
concurrently with the streamlines at selected time steps. The streamlines are 
depicted
 
in a frame 
of reference convecting with the 
vortex
 
but placed at the appropriate 
x
 
location as the vortex travels 
in the positive 
x
-
direc
tion

pressure peak 
is found
; as expected. Two additional positive, but substantially weaker, peaks 
are 
found
 
ahead and behind the vortex center. Overall, excellent agreement 
is found
 
betwe
en the 
computed and analytical wall pressure distribution even up to the time when the vortex is 
fairly
 
close to the reference point (subplot f in figure 2.
9
). This agreement 
is quantified
 
by calculating 
the 
NRMSE
, which 
is displayed
 
in figure 2.
10
 
versus 
time. For this plot, the time show
n
 
is only up 
to the point when the vortex center reaches 
x
 
= 5 
cm
. As seen from the figure, the 
NRMSE
 
does 
not exceed 0.3% over the entire duration depicted.
 
 
 
 
43
 
 
 
 
(a)
 
Time=0.00278 s
 
(b)
 
0.00627 s
 
 
 
(c)
 
0.01324 s
 
(d)
 
0.02022 s
 
 
 
(e)
 
0.0237 s
 
(f)
 
0.02719 s
 
Figure 2.
9
 
Computational and analytical results of the wall pressure and associated streamlines 
for a line vortex above a wall. Different subplots represent different times (as indicated beneath 
the plots). 
The streamlines 
are shown
 
in 
a frame of reference
 
convecting with the vortex.
 
 
44
 
 
 
Figure 2.
10
 
Normalized Root Mean Square Error (NRMSE) between 
a
nalytical 
and numerical
 
values of the wall
-
pressure distribution beneath a line vortex above a flat wall. The time 
duration shown is from the
 
start of the vortex motion until the vortex center is located at 
r
 
= 5 
cm.
 
 
 
 
 
 
 
 
 
 
 
 
45
 
 
CHAPTER 
3: Mathematical Model Results
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
This chapter will focus on the interpretation of the mathematical model results including 
the mean and Root Mean Squa
re (
RMS
) of the fluctuating pressure. The analysis was done for 
three main cases. For each 
case,
 
the solution was obtained when changing only one parameter 
while the other 
parameters
 
remained unchanged. This facilitates understanding the effect of 
changing
 
each of these 
parameters
 
on the wall
-
pressure characteristics. In fact, varying one 
parameter at a time is one advantage of the model because in the actual flow it is generally not 
possible to change these parameters independent of one another. The three 
parameters examined 
are:  vortex passage frequency (
f
), jet Reynolds number 


, and vortex circulation
 



. Though 

(vortex ring diameter) and initial jet velocity (stagnation flow velocity at top of the computation 
domain) in ord
er to facilitate comparison with actual jet flow results.
 
The nominal values of the model parameters were selected to be representative of an actual 
jet flow at the end of the potential core. The details of determining these values follow. Crow and 
Champag
en [14] showed that the jet unsteadiness at the end of the potential core is dominated by 
the
 
preferred, or column, mode with a Strouhal number of 





. 
Thus,
 
 







 
(3.1)
 
In the model, the jet diameter 

 
is reasonably equivalent to twice the vortex ring radius; i.e. 





, and the jet velocity 


 
to the centerline velocity of the stagnation (advection) flow at the top 
o
f the computational domain; i.e. 







. Making these substitutions in equation (3.1):
 
 
46
 
 
 










 
(3.2)
 
In equation (3.2), both 


 
and 


 
are known a priori. 


 
is the height of the computational domain, 
and 


 
is the initial vortex ring radius (equivalent to the 
jet 
exit radius), which is arbitrarily selected 
without loss of generality since it is also chosen as the length scale for making quantities non
-
dimensional. To determine 

, and hence the vortex pas
sing frequency via equation (3.2), use is 
made of 



:
 
 





















 
(3.3)
 
where, 

 
is the kinematic viscosity. Equation (3.3) leads to:
 
 














 
(3.4)
 
Hence, by specifying the ri
ng radius, jet Reynolds number and knowing the computational domain 
height it is possible to compute the nominal vortex passage frequency (corresponding to the jet 
column, or preferred mode). Because the jet preferred mode frequency corresponds to the vort
ex 
passage frequency at the end of the potential core, the frequency calculated as outlined above 
represents vortices near the end of the potential core (approximately 


 
downstream of the jet 
exit). To model vortex passage at locations closer to the exit of the jet, use is made of the fact that 
the terminal vortex passage frequency of the preferred mode is the result of successive pairing
s
 
of 
vortices, which
 
initially 
form as an instability of the jet shear layer, rather 
than
 
the jet column. 
With each pairing, the vortex passage frequency drops by a factor of 2. Hence, to represent vortex 
passage at distances closer to the jet, the nominal frequency is increased by a fa
ctor of 2, 4, etc. 
 
To connect the nominal vortex circulation 



 
value to the jet flow, use is made of the 
approximation of the circulation for a shear layer (Koochesfahani and Dimotakis [22]):
 
 
47
 
 
 






 
(3.5)
 
where, 


 
is the velocity difference across the shear layer, 

 
is the wavelength between vorti
ces, 
and 


 
is the circulation around a contour that encompasses the shear layer and extends a length 

 
in the streamwise direction. Equation (3.5) assumes all vorticity is concentrated in the shear layer 
vortices and is equivalent to computing the
 
velocity jump across a zero
-
thickness potential vortex
-
sheet from the circulation density (circulation per unit length) along the sheet. For the jet flow 





 
and 







 
[23], thus:
 
 






























 
(3.6)
 
For all 
cases studied,
 
parameters
-
maintained
 
constant 
are
 
-
 

 
radial extent: 








 
(see figure 2.1 for definition). 


 
is 
selected to be much larger than 


 
and to ensure invariance of the results with further 
in
crease in 


, as discussed in Chapter 2;
 
-
 
Initial vortex ring radius: 








;
 
-
 
Initial wall
-
normal distance between the vortex ring and the impingement plate (also wall
-
normal extent of the computational domain 


): 










 
m
. This value was 
not 
critical
. 
If
 


 
is increased
, the vortices will 
travel a longer distance towards the wall before 
they have 
any effect on the wall. 
Since the focus of the model is on the wall
-
pressure, 


 
needs to be larger than or equal to
 
the height at which the vortex presence affects the wall 
pressure. 
The choice of five times the ring radius fulfills this requirement for all cases 
examined here;
 
-
 
Volumetric constant (

) in equation (2.7):













. This value was 
taken to 
be the same as used
 
by Walker
 
et al
 
[18]
. The specific value of C is not critical since it only 
 
48
 
 
affects the self
-
induced velocity of the vortex, which in turn only affects how quickly the 
vortex ring lineally approaches the wall. 
 
-
 
Air kinematic viscosity (
at 


): 














. Air is selected as the fluid 
since the results of the model are compared to the measurements in an air jet in Chapter 5.
 
With 


 
and 


 
values fixed, the selection of 



, sets 

 
(equation 3.4), which in turn se
ts 

 
and 

 
(equations 3.2 and 3.6, respectively). A 
reference
 
case was arbitrarily selected to correspond 
to 







, resulting in 







 
and 










. This reference case was 
repeated in three series of parametric investigations ai
med at examining the influence of 

, 



 
and 

 
on the wall pressure. Each series contained three cases, including the reference and involved 
varying only one parameter, while all other parameters remained fixed in order to isolate the 
influence of 
each parameter. This one parameter at a time variation is only possible using a 
mathematical model since in the real jet 

, 



 
and 

 
are generally interdependent. Indeed, this 
independence is even reflected in equations (3.2), (3.4) and (3.6), whi
ch attempt to mimic the real 
jet flow in a simplistic way.   
 
The parameter values for all cases investigated are summarized in table 3.1. For the frequency 
series, two other frequencies representing doubling and quadrupling the reference
-
case frequency 
ar
e examined. As discussed earlier these higher frequencies represent vortex passage prior to the 
second and first vortex pairing, respectively, in the real jet (which corresponds to shorter jet to 
impingement plate distance). For the Reynolds number cases, 
the three values utilized are 







. Finally, the three cases in the circulation 



 
series are those from the 
reference case 










,
 
and the others taken to be half and double of the reference 
circulation
. For each series,
 
the reference case is highlighted in 
table 3.1 using green color. 
 
 
 
49
 
 
case
 
Reynolds 
number 



 
Frequency 




 
Jet velocity 







 
Circulation 







 
Constant 







 
Frequency 



 
5000
 
245.9
 
7.8698
 
0.1798
 
163.954
 
5000
 
491.8
 
7.8698
 
0.1798
 
163.954
 
5000
 
983.6
 
7.8698
 
0.1798
 
163.954
 
Reynolds 
number 





 
5000
 
245.9
 
7.8698
 
0.1798
 
163.954
 
15000
 
245.9
 
23.6094
 
0.1798
 
491.862
 
25000
 
245.9
 
39.3490
 
0.1798
 
819.77
 
Circulation 



 
5000
 
245.9
 
7.8698
 
0.0899
 
163.954
 
5000
 
245.9
 
7.8698
 
0.1798
 
163.954
 
5000
 
245.9
 
7.8698
 
0.3596
 
163.954
 
Table 3.1 Model parameters for frequency 



, Reynold number 





, and circulation 



 
cases. Rows highlighted in green depict the parameters for the reference case
 
3.1 Frequency Effect  
 
Figure 3
.1 and 3.2 show snapshots of the streamlines of the computed flow and associated 
wall
-
pressure for the lowest and highest frequency respectively. In each figure, the point
-
vortex 
location is indicated with an asterisk. In both figures, a given vortex appro
aches the wall moving 
vertically downwards without much change in the radial location, until the vortex is very close to 
the wall where the main travel direction switches from towards to parallel to the wall. The main 
difference between the two frequency c
ases is that multiple vortices are seen within the 
computational domain in the high
-
frequency case (figure 3.2). The spacing between the vortices 
initially decreases, as they approach the wall, resulting in packing them rather densely within the 
stagnation
 
zone. Subsequently, as the vortices travel parallel to the wall, their spacing increases 
 
50
 
 
substantially. The change in the vortex spacing is related to their convection velocity. Initially, as 
they move towards the wall, their convection velocity decreases
 
(due to the decrease in the 
stagnation flow wall
-
normal velocity and the opposing influence of the image vortex). Once they 

by the image vortex both com

within the wall
-
jet zone the spacing of the vortices is increased substantially.
 
In the wall
-
jet zone, b
ecause 
the vortices are spaced far apart, regardless of their frequency, 
their wa
ll
-
pressure signature is similar
 
in character
 
in both the low
-
 
and high
-
frequency cases. This 
signature takes the form of a negative pressure spike immediately beneath the vortex (where the 
induced velocity by the vortex is highest on the wall) surrounded 
by two small positive peaks up 
and downstream of the negative peak. This signature
, which is consistent with the expected 

focusing

 
of the stream lines beneath a 
vortex (see figure 2.9), 
can 
be observed best at the last 
time instant (largest 




 
vortex location) in figures 3.1 and 3.2. The negative spike first appears 
when the vortex gets close enough to the wall (as will be seen later, this takes place within the 
stagnation zone, between 











), then it monotonically increase
s in magnitude as the 
vortex travels radially outward
s
. This monotonic increase is caused by the continuous decrease in 
the vortex distance to the wall with increasing time. Since potential vortices are employed in the 
present model, as the wall nears the 
vortex center, the induced velocity on the wall becomes higher 
and higher, and therefore the corresponding pressure is expected to become lower and lower.
 
 
 
51
 
 
 
t=0 s
 
 
t=0.
0004
 
s
 
 
t=0.
0008
 
s
 
 
 
t=0.
0011
 
s
 
 
 
t=0.
0015
 
s
 
 
 
t=0.
0019
 
s
 
 
 
t=0.
0022
 
s
 
 
 
t=0.
0
026
 
s
 
 
t=0.
003
 
s
 
Figure 3.1 
Streamlines
 
and normalized 
pressure
 
at different time step
s
 
for frequency 





, 
Re
=5000, and circulation 









 
 
52
 
 
 
 
t=0.
0
 
s
 
 
 
t=0.
0004
 
s
 
 
 
t=0.
0008
 
s
 
 
 
t=0.
0012
 
s
 
 
 
t=0.
0015
 
s
 
 
 
t=0.
0021
 
s
 
 
 
t=0.
0023
 
s
 
 
 
t=0.
0026
 
s
 
 
t=0.
003
 
s
 
Figure 3.2 
Stremlines
 
and normalized 
pressure
 
at different time step for frequency 
f
= 980 
Hz
, 
Re
=5000, and circulation 









 
 
53
 
 
 
To examine the wall
-
pressure characteristics quantitatively, radial profiles of the m
ean and 
RMS
 
pressure are computed. Both of these quantities are calculated over one period of oscillation 
(



) after the initial transients of starting the calculation have passed. The mean pressure 
distribution is shown in figure 3.3
 
in the form of a me
an
-
pressure coefficient 












; 
where 


 
is the dynamic pressure based on 


, 
for a truncated radial domain that focuses on the 
stagnation zone. The latter is delineated in these figures, as well as similar ones later in the
 
chapter, 
using broken green lines. As seen in figure 3.3, the mean pressure has its maximum at the 
stagnation point (as expected). It should be clarified that the stagnation point pressure coefficient 
exceeds unity because of t
hree
 
reasons. First, the dyn
amic pressure 
used for normalization 
is 
computed using a velocity scale that is representative of the stagnation (advection) flow only. 
Therefore, any added mean streaming velocity towards the wall due to the presence of the vortices 
does not affect the ve
locity scale. The added velocity would increase the stagnation pressure 
beyond that of the stagnation flow alone, causing the pressure coefficient to go beyond unity. 
The 
difference between the overall pressure and that due to the stagnation flow alone can
 
be seen in 
figure 3.3, where 


 
of the stagnation flow alone is also plotted (indicated with 
SP
 
in the legend). 
As expected, this distribution does not vary with frequency. Second, the reference pressure 


 
used 
in defining 


 
is taken from a different spatial location (









 
than that where 


,
 
and 
hence 


, 
is calculated
 
(









. 
Third
, the flow is unsteady, and therefore, generally 
speaking, the stagnation pressure coefficient need not be unity.
 
 
 
54
 
 
 
Figure 3.3 Effect of vortex passage fr
equency on the radial distribution of the mean pressure 
coefficient
. The green broken lines outline the boundary of the stagnation zone
 
 
Figure 3.3 shows that 


 
at the stagnation point increases with increasing frequency. This effect 
can be understo
od as being caused by the net streaming velocity induced by the vortices towards 
the wall. As the frequency increases, more vortices are present simultaneously within the 
stagnation zone (e.g. compare figures 3.1 and 3.2), and the superposition of their in
duced velocity 
increases with the number of vortices, leading to larger stagnation pressure. Another interesting 
effect is the development of a local negative peak just outside the stagnation zone, which is seen 
clearly for the highest frequency. This effe
ct is also understood to be caused by the net streaming 
velocity induced on the wall by the collective effect of the vortices, which becomes stronger with 
increasing frequency. Unlike
 
the streaming effect towards the wall at 





, which 
understandably p
roduces a positive pressure peak on the wall at the stagnation point, when the 
flow velocity is forced to zero, the stronger the wall
-
parallel velocity induced by the vortices, the 
 
 
55
 
 
lower the pressure. The presence of a localized peak at a given radial loca
tion, for large enough 
frequency, is indicative of the location where the collective influence of the vortices is highest. As 
discussed in figure 3.2, as the vortices approach the wall, they get packed densely resulting in 
maximum collective effect. Howeve
r, subsequently the vortex spacing increases substantially 
reducing this effect. The negative peak in the mean pressure distribution is believed to be a 
manifestation of this behavior when enough vortices are present in the simulation (i.e. when the 
vortex
 
passing frequency is high enough).
 
T
he negative peakon figure 3.3 can be explained 
alternatively based on figures 3.1. and
 
3.2. More tightly packed vortices means a larger "induced" 
flow.
 
Since this flow cannot bypass the streamlines, the velocity is
 
incr
eased in regions where the 
spacing between streamlines is
 
reduced. That is the place where the vortices first approach the 
wall.
 
Since vortex injection frequency is higher, streamlines tend to stay
 
closer to each other almost 
permanently at that point.
 
The
 
radial distribution of 








 
is depicted in figure 3.4. Two different trends 








), the level of pressure fluctuation decreases with increasing frequency, wh
ile the opposite takes 
place for larger 



 




 
near the stagnation point. As 



 
increases beyond this flat zone, two different behaviors are 
seen, depending on frequ
ency. At low frequency, a decrease in the 
RMS
 
level is observed with the 
increase in 



, before a monotonic increase is observed. The initial decrease with 



 
becomes 
smaller with increasing frequency, eventually disappearing at the highest frequency
, where the 
RMS
 
level increases monotonically with 



 
beyond the initial flat distribution near the stagnation 
point. 
 
 
 
56
 
 
 
Figure 3.4 Effect of vortex passage frequency on the radial distribution of the 
RMS
 
pressure 
coefficient
. The green broken lines o
utline the boundary of the stagnation zone
 
 
Possible physical reasoning for the trends observed in the 


 
distributions could be 
developed via inspection of sample pressure times series.
 
Figure 3.5 depicts pressure signals for 
one cycle of vort
ex passage at various 
r/D
 
locations within the stagnation zone. Two plots are 
included in the figure for the lowest and highest frequencies; 





 
(top) and 





 
(bottom), respectively. For both cases, the general features of the signals are quali
tatively similar, 
depicting a fundamental difference in the signal shape between radial locations that are below 
approximately 








 
and those that at larger radial locations. For 



 
locations larger than 
0.5, an energetic pressure signature that i
s characterized by a strong negative peak is observed. As 
discussed in figures 3.1 and 3.2, the strong/narrow negative pressure peak is found directly beneath 
the vortex closest to the wall, and is produced by vortex passage. Once seen at the wall, this pe
ak 
continuously increases in magnitude with 



 

 
 
57
 
 
decreases as the vortex convects radially outwards. This increase in negative
-
peak strength is seen 
in figure 3.5 between 











, for both frequencies. 
 
Below 







, the pressure s

it does not exhibit the strong negative peak characteristic of vortex passage. Since the only 
unsteadiness in the model originate from vortex passage, there is no doubt that the unsteady 
pressure is
 
still connected to the vortices; it is just that the unsteadiness does not reflect the strong 
local effect, near the vortex core. Therefore, hereafter the pressure unsteadiness is characterized as 
remote 
for 







, and 
local
 
for 







. The remote effects are more harmonic and weak 
in nature, and the local effect are strong and only felt at the wall for sufficiently large 



. For 
real (viscous core) vortices, the distinction between local and remot
e is expected to depend on how 
far is the point of observation from the vortex
-
core center relative to the vortex core radius. Since 
the vortex core size increases with each merging, one would expect that local effects would extend 
farther and farther with
 
merging. This physical hypothesis cannot be tested with the present model, 
which is based on zero
-
core
-
size potential point vortices.
 
 
 
 
 
 
 
 
 
 
 
58
 
 
 
 
Figure 3.5 Pressure time series at various 
r/D
 
locations within the stagnation zone, 
Re
=5000, 









, 
and 




















. The small 
discontinuities in the time series for 





 
correspond to the time instant when a new 
vortex is injected at the top of the computational domain. Because the discont
inuities are 
small, no attempt was made to get rid of them by using a larger wall
-
normal extent for the 
computational domain.
 
 
 
 
59
 
 
The discussion of figure 3.5 provides insight regarding the effect of 



 
on 


 
in the 
stagnation zone. To analyze t
he influence of frequency, pressure time series at the same radial 
location but different frequencies are shown in figure 3.6. Two radial locations are chosen, at the 
start and end of the stagnation zone
: 





 
(top) and 





 
(bottom), respectivel
y. These 
two locations are chosen because one of them exhibits decrease in pressure fluctuation level with 
f
requency (





), and the other, the opposite. Focusing first at the stagnation point, the 
decrease in the amplitude of the sinusoidal
-
like pressu
re variation with increasing frequency is 
obvious. 
This trend may be clarified 
as follows: per earlier discussion of figures 3.1 and 3.2
, as the 
frequency increases, the vortices become packed densely near the wall within the stagnation zone
. 
 
As a result,
 
the 
induced velocity 
on
 
the stagnation 
stream
line
 
(i.e. the flow approaching the 
stagnation point) 
by the individual vortices
 
become increasingly
 
overlapping. This reduces the 
vortex
-
to
-
vortex 
velocity 
fluctuation, and leads to more of a steady streaming 
flow with increasing 
frequency.
 
 
To substantiate this physical picture
 
just described
, 
figure 3.7 shows the induced velocity 
along a line that is parallel 
to an array of line vortices with varying inter
-
vortex spacing: 10 mm, 
50 mm
, and 
500 mm
. This simple
 
situation is intended to 
emulate
 


wall are parallel to the stagnation streamline
 
in the present model
).
   
The results in fig
ure 3.7 are 
plotted for only one wavelength (i.e. for a length equal to the spacing between the vortices), since 
outside the shown range, the induced velocity would repeat periodically for an infinite array. For 
all vortices in the array, the circulation 
i
s
 
the same. As seen from figure 3.7, for the largest 
wavelength (smallest packing of vortices), the vortices are spaced so far away that th
e induced 
velocity 
from the neighboring vortices has no influence within the wavelength shown (as seen 
 
60
 
 
from the veloc
ity practically decaying to zero at the ends of the wavelength). 
As the vortices 
become more packed (




), the effect of neighboring vortices overlaps more, causing an 
overall rise in the mean velocity (i.e. increasing streaming effect), and a reductio
n in the velocity 
fluctuation (since the induced velocity amplitude does not fall too much before the influence of a 
neighboring vortex is felt). This trend is particularly evident for the highest vortex packing (




), where
 
the induced velocity is p
ractically steady, having the strongest streaming 
component and no fluctuation.
 
 
On the other hand, within the zone where the local vortex effects are felt (bottom plot in 
figure 3.6), the pressure amplitude increases only slightly with increasing frequenc
y, suggesting 
that this increase is not responsible for the strong increase in 
RMS
 
with frequency seen in figure 
3.4. Instead, a substantial broadening of the pressure peaks is found to take place with increasing 
frequency. This shows that the monotonic in
crease in 


 
with frequency for 







 
is a 

-

broadening is again a result of the increase in the number of packed vortices affecting the press
ure 
as the frequency increases. However, unlike 







, 
the effect of vortex packing relates to the 
induced velocity along a line that is practically normal, rather than parallel, to the vortex array, 
which leads to a different influence of vortex packi
ng.
 
To further clarify the effect of vortex spacing on the induced velocity on the wall, a simple 
situation is considered where three line vortices are placed at the same positions relative to a wall 
as
 
the vortices in the model
 
when the leading vortex cen
ter is at 





 
(i.e. the same location 
for which the time series are shown in figure 3.6).
 
The induced velocity on the wall by each of 
these vortices is plotted in figure 3.8 versus distance along the wall, together with the velocity 
induced by all of them. If th
e vortex spacing is so large such that only one vortex passes through 
 
61
 
 
the domain at a time
 
(as in the low
-
frequency case of the present model)
, the induced velocity 
signature at the time when the vortex is located at 






, would be the same as given by
 
the 
red line in figure 3.8. On the other hand, if the vortex spacing is so small such that when the leading 
vortex passes 





, two other vortices are 
trailing 
in close proximity (with relative spacing 
similar to the high
-
frequency case of the model), 
substantial broadening of the induced velocity is 
seen due to the effect of the trailing vortices. This should influence the wall pressure in the same 
way, clarifying the influence of frequency on the time series in figure 3.6.
 
 
 
 
 
 
 
 
 
 
62
 
 
 
 
Figure 3.6 Pre
ssure time series at various frequencies
,
 
Re
=5000, 









, 
and 























. The small discontinuities in the time series for the 
two larger frequencies correspond to the time instant when a new vortex is injected at the top 
of the computational d
omain. Because the discontinuities are small, no attempt was made to 
get rid of them by using a larger wall
-
normal extent for the computational domain.
 
 
 
 
 
63
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 3.7 
Induced v
elocity 
by an infinite line
-
vortex array along a line parallel to t
he array. 
Results are shown over one vortex spacing (

) with the vortex located at the center (



). 
Different colors indicate different spacing (i.e. different vortex packing density).
 
 
 
64
 
 
 
Figure 3.8
 
Individual and collective induced wall velocity by 
three
-
line
 
vortices placed relative 
to a wall at the sa

-
frequency case when the 
leading vortex is located at 





.
 
3.2 Reynolds Number Effect
 
Figure 3.
9
 
and 3.
10
 
show the 
radial distribution for the 
mean 
and 
RMS 
wall 
pressure
 
when 
the jet Reynolds number i
s varied
. 
Since the velocity scale is defined as that based on the stagnation 
(advection) flow at the top of the computational domain, increasing 



 
corresponds to increasing 
the strength/velocity of the stagnation flow. In addition, 
because the mod
el is inviscid, essentially 

jet relative to that induced by the vortices since all vortex parameters remain unvaried. 
 
Figure 3.9 shows that the stagnatio
n
-
point (
SP
) flow 


 
does not vary with Reynolds 
number. This is expected given the normalization by a velocity scale based on the stagnation flow. 
In contrast, the overall 


 
exhibits a deviation from 


 
of 
SP
 
flow that decreases
 
with increasing 
 
 
65
 
 



. Th
is should not be too surprising given that the deviation is produced by the streaming flow 
induced by the vortices, which should remain invariant since the vortex characteristics are 
unchanged between Reynolds numbers. This invariance when normalized with 
an increasing 
dynamic pressure causes the deviation from 
SP
 
mean pressure to decrease with increasing 



. 
Similarly, the decrease of the 
RMS
 
pressure coefficient with Reynolds number (figure 3.10) can 
be attributed to an invariance in the unsteady pressure from vortex passage that produces smaller 
coefficient of pressure when normalized by the incre

 
 
Figure 
3.9 Effect
 
of 
Reynolds number
 
on the radial distribution of the mean pressure 
coefficient
 
 
 
66
 
 
 
Figure 3.10 Effect of 
Reynolds number
 
on the radial distribution of the 
RMS
 
pressure 
coefficient
 
 
3.3 Circulation 
Effect
 
 
Figures 3.
1
1
 
and 3.
1
2
 
depict
 
the 
mean
 
and 
RMS
 
pressure distributions for various vortex 
circulation magnitudes (i.e. vortex strengths). The case with the intermediate circulation value is 
the reference case (the same as the 



 
in section 3.1), where one vortex at a time is 
affecting the wall pressure. The other two cases are similar with the exception of the vortex 
strength, which is weaker in one case and stronger in the other.
 
 
The overall features of the mean pressur
e profile stay the same with varying circulation 
strength. Quantitatively, the stagnation pressure 
increases,
 
and a local minimum develops with 
increasing circulation. The same trends were observed to take place with increasing frequency in 
section 3.1. In
 
that case, these changes were attributed to the increase in induced velocity due to 
the increasing number of vortices with increasing frequency. Here, the effect is similar but since 
 
 
67
 
 
the number of vortices does not change (the frequency is the same for al
l three cases), the induced 
velocity increases due to circulation.
 
 
 
Figure 3.
1
1
 
Effect of vortex 
circulation
 
on the radial distribution of the mean pressure 
coefficient
 
 
 
68
 
 
 
Figure 3.
1
2
 
Effect of vortex 
circulation
 
on the radial distribution of the 
RMS
 
p
ressure 
coefficient
 
 
In regard to
 
the 
RMS
 
distribution (figure 3.
1
2
), the overall shape remains invariant with 
changing circulation (consult figure 3.4 to see the details of the reference case/intermediate 
circulation more clearly) but the 
RMS
 
level incre
ases monotonically with increasing circulation 
(consistent with the stronger vortices). Overall, the pressure signal shapes within the stagnation 
zone remain similar with increasing circulation, while the magnitude of the pressure peaks 
increases (leading 
to the larger 
RMS
 
level). This may be seen by comparing figure 3.
1
3
 
for the 
highest
-
circulation case to the top plot in figure 3.5 for the reference case. A more direct 
comparison between the time series shapes of the lowest and highest circulation cases a
t the 
stagnation point and end of the stagnation zone is given in figure 3.
1
4
. The results clearly 
demonstrate the increasing strength of the pressure fluctuations without change in the pressure 
signal shapes.
 
 
 
 
69
 
 
 
Figure 3.
1
3
 
Pressure time series at variou
s 
r/D
 
locations within the stagnation zone, 
Re
=5000, 









, 
and 





 
 
 
 
 
70
 
 
 
 
Figure 3.
1
4
 
Pressure time series 
for
 
various 
vortex 
circulation
s
,
 
Re
=5000, 





, 
and 























.
 
 
 
 
 
71
 
 
CHAPTER 
4: Experiment Setup and Apparatus
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Thi
s 
chapter
 
exhibit
s
 
the experimental setup that was used to measure the unsteady wall 
pressure and the 
initial 
jet velocity
 
profiles
. The coordinate system and the configuration of the 
flow 
utilized
 
in the experiment will be described. The measurement techn
iques and procedures 
used to measure 
the 
velocity
 
and
 
the 
fluctuating pressure will be demonstrated. Moreover, the 
stepper motor and 
2D
 
traversing system used to 
traverse
 
the hot
-
wire velocity probe over the 
measurement domain, the hardware and software us
ed to coll
ect
 
experimental data, and the 
assembly of the experimental components will 
be explained
. 
 
4.1 
Coordinate System and Flow Configuration
 
This research is focused on the 
normal
 
incidence impinging jet flow. The flow 
configuration is depicted in fig
ure 4.1. Two coordinate systems 
are used
. 
The first
 
system
 
is
 








cartesian
 
system with origin at the jet exit centerline, and the second is a 
p
olar
 
system 







 
originating at the center of the impingement plate and at 
(shown in figure 4.1 for the plane 



).
 
 
The impinging jet facility used in the curre
nt experiments resides in the Flow Physics and 
Control Laboratory at Michigan State University. The facility underwent a major renovation and 
subsequent characterization by Al
-
Aweni [
7
].  The jet 
discharges
 
through a 
round exit
 
with 
diameter 




, at th
e end of a contoured nozzle, resulting in an initial condition of a top
-
hat velocity profile.  The flow from the jet imping
es
 
on a circular flat plate with 
diameter



. 
The disc diameter is 
greater
 
than the jet diameter 
by an order of
 
magnitude to reduce
 
the effect of 
the edge of the disc on the flow. The 
mean 
flow 
doe
s not chang
e
 
in the 

 
direction and it
 
i
s assumed 
 
72
 
 
to be axisymmetric
 
(as verified in Al
-
Aweni [7]), and
 
the distance between the jet exit and the 
impingement plate 

 
is adjustable.
 
 
Figu
re 4.1 F
low configuration for 
normal
-
incidence 
impingement jet
 
 
4.2 Experiment General Assembly
 
The experimental facility 
is presented
 
in 
figure
 
4.2,
 
and it consists of a centrifugal blower, 
type Dayton 4C108, 
driven by
 



 
HP 
DC
 
motor. The flow rate th
rough the facility is adjusted by 
changing the speed of the motor
. The 
blower
 
provides provide air to the jet through a 
PVC
 
pipe 
with a of diameter 





. The exit pipe 
diameter
 
of the blower is smaller than
,
 
and it does not 
touch the inner diameter of th
e 
PVC
 
pipe to reduce the vibration effect from the 
blower
.
 
The air passes into a flow conditioning chamber with dimensions 

















,
 
before entering a nozzle with an area contraction ratio of 80, to decrease
 
the turbulence 
intensity. The tur
bulence intensity was measured at the nozzle exit using a hot
-
wire anemometer, 
 
 
 
73
 
 
and 
was found 
to be less than 1% for the jet velocity of 





 
(based on the streamwise velocity 
fluctuations) 
 
 
Figure 4.2 Experiment
al
 
facility
 
 
Referring to figure 4.3, th
e impingement disc 
is embedded flush in 
a vertical square plate 
with dimensions of 











. The circular 
disc
 
is fitted with 30 microphones for 
measuring the unsteady wall pressure, including eight microphones, employed in the current work, 
that ar
e arranged as a line array along the radial direction (see section 4.4 for more details). The 
square plate is placed on a sliding table that can be moved in the 
x
-
direction using a manual 
traverse
 
system
 
(
as shown in figure 4.3
)
, 
model
 
Velmex A1506P40
-
S1.5
-
TL, which allows to change the 
distance between the impingement plate and the nozzle exit (
H
). The Velmex traverse system can 
move total distance of 115 
mm
 
with an accuracy of 0.0254 
mm
. 
 
 
 
74
 
 
 
Figure 4.3   An image of the impingement plate and the slid
ing table attached to the Velmex 
manual traverse system
 
A steel frame 
is used to hold
 
the conditioning chamber, the nozzle, and the impingement 
plate. The frame is mounted on a table different from the one 
utilized
 
to support the blower to 
isolate the fac
ility from the vibration generated by the 
blower
. 
 
In the course of this study, a stepper
-
motor
-
driven Velmex traversing system was added to 
the facility. The system was used to traverse 
a single
 
hot
-
wire probe to characterize the jet velocity 
profiles nea
r the nozzle exit (i.e. the initial condition)
. 
The hot
-
wire 
wa
s attached to the carriage 
of the Velmex system via a 
custom
-
made
 
arm (see figure 4.4
 
and 4.5
). The 
traverse
 
system has 
three
-
degrees of freedom
 
(
DOF
)
: two
-
line
a
r and one rotation
al
 
(
model
s
 
MB4
027K2J
-
56 and 
 
Square
 
plate
 
Circular
 
impingement
 
disc
 
Velmex 
manual 
travers
e
 
system
 
Hole 
with 
embedded
 
Microphone
s
 
Slid
ing 
table
 
 
75
 
 
B4872TS respectively
)
. Only the linear 
DOF
 
are used in the present work to move the velocity 
probe in 
X
 
and 
Y
 
directions.  The traverse system is controlled by Velmex controller, type 
VP9000
, 
with the 
ability
 
to control the 
system
 
manually th
rough a 
joystick,
 
or programmatically through a
n
 
RS
-
232
 
interface with a desktop 
PC
 
computer. 
 
The jet mean exit velocity was determined from measurement of the 
difference between 
the 
stagnation pressure in the setting chamber
 
and the ambient pressure
 
(fur
ther details are given 
in section 4.3)
. Data acquisition was 
accomplished using National Instruments NI PCI
-
6024E 
PC
-
based analog to digital (
A/D
) converter card. The 
A/D
 
card
 
was coupled with National Instruments 
BNC2080 analog breakout board
 
to facilitat
e signal connections via coaxial BNC cables
.
 
The card 
has 12
-
bit
 
resolution and input range that is changeable between 




 
to 




, and it can 
sample 
up 
to 16 single
-
ended multiplexed channels at 
a 
maximum rate of  

  
using
 
LabVIEW version 8.2
.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
76
 
 
4.3 Hot
-
wire Setup and Calibration 
 
 
A single hot
-
wire probe was used to measure the streamwise velocity p
rofile of the jet near 
the exit of the nozzle. Figure
s
 
4.4 and 4.5 depict 
a 
schematic of the hot
-
wire setup used 
during
 
calibration and measurements
,
 
and an image showing the motorized traverse system respectively. 
 
 
 
Figure 4.4 Block diagram of 
the 
setu
p used 
for
 
calibration and measurements of the hot
-
wire
 
 
 
 
77
 
 
 
 
 
Figure 4.5 
An
 
image
 
of the motorized system for 
traversing 
the 
hot
 
wire
 
The hot
 
wire was 
built
 
using tungsten with a sensing length and diameter of 1 
mm
 
and 


 
respectively, which provide a length to dimeter ratio of 200. The
 
wire was used to measure the 
mean and the fluctuating 
streamwise 
velocity
 
while operated using 
a Constant Temperature 
Anemometer (
CTA
)
,
 
model TSI 1750, at an overheat ratio 






,
 
defined as:
 
 
 









 
(4.1)
 
Vertical 
movemen
t
 
Horizontal 
movement
 
Hot
-
wire 
probe
 
Jet exit
 
 
78
 
 
Where, 


 
represents the
 
operating
 
(heated) 
hot
-
wire resistance, and 


 
the hot
-
wire resistance at 
room temperature
. The overheat ratio
 
can 
be 
as high as
 
1
 
but 
it 
is generally kept
 
between 
0
.6
-
0
.8
.
 
The higher the value,
 
the better the 
velocity sensitivity and 
the smaller the temperature sensitivity. 
Too high of a value
,
 
however, could cause
 
oxidation
, and hence drift in the response
 
of the wire
.
 
The bandwidth 
cut
-
off frequency 


 
of the hot
-
wir
e 
was found
 
using 
the following equation
 
 








 
(4.2)
 
Where, 

 
represents a 
response
 
time
, which was obtained from 
a
 
square wave test. Figures 4.6 and 
4.7 depict the diagram for the square wave test and the result respectively.
 
During the tes
t, 





 
square wave with frequency of 




 
was fed from Agilent WAVETEK function 
generator to the square wave input of the 
CTA
. The output of the 
CTA
 
was captured on a Tektronix 
TDS 1002B digital oscilloscope, as shown in figure 4.7. the figure also d
emonstrates how 

 
was 
determined to be 


 
yielding a bandwidth of 



.
 
 
 
 
 
79
 
 
 
Figure 4.6 Block diagram of the setup for the square wave test
 
 
Figure 4.7 An image of the oscilloscope screen showing a typical square wave test result
 
 
 
 
80
 
 
The hot
-
wire wa
s calibrated in
-
situ by placing the wire near the center of the jet within the 
potential core where the jet velocity can be obtained from the difference between the stagnation 
and the ambient pressure. This pressure difference was measured using either hig
h a (10 torr) or a 
low (1 torr) pressure transducer Baratron model 223BD
-
00010ACU or 223BD
-
00001ACU 
respectively. The pressure transducers have sensitivity of 0.75 
mV/Pa
 
and 7.5 
mV/Pa
 
respectively. 
The 
positive
-
input
 
side of 
the 
pressure transducer was 
con
nected
 
via Tygon tubing to a pressure 
tap in the wall of the setting chamber, just upstream of the nozzle. The other, low pressure, side of 
the pressure transducer was left open to ambient pressure. Temperature of the air flow 
was 
measured
 
sing a thermisto
r, type Omega DP
-
25
-
TH, 
with
 
a sensitivity of 






. The 
measured temperature was used to correct the hot
-
wire output for the variation of the flow 
temperature
 
from that of the calibration
. Figure 4.8 depicts an image of the hot
-
wire, temperature 
se
nsor, stagnation pressure tap
,
 
nozzle, impingement plate, and the setting chamber.
 
 
81
 
 
 
Figure 4.8 An image of the hot
-
wire, temperature sensor, nozzle, impingement plate, and the 
conditioning box
 
 
To acquire the pressure, the temperature, and the hot
-
wire signal, a LabVIEW program was 
developed for this purpose. Typically, the signals 
were acquired
 
for eight different velocities 
depending on the velocity range in the experiments which varied between 
5 to 
15
 
m/s
. All 
hot
-
wire 
voltages were 
corrected
 
for
 
the variation of the temperature during the period of calibration using
 
 




















 
(4.3)
 
 
 
Hot
-
wire
 
Setting
 
chamber
 
Temp. 
sensor
 
Impingement 
plate
 
Stagnation 
pressure
 
tap
 
 
82
 
 
Where, 


 
represents the corrected hot
-
wire voltage, 


 
the measured hot
-
wire voltage, 


 
The 
hot
-
wire temperature calculated using equation 
(
4.4
)
 
below, 


 
the
 
flow
 
temperature measured 
during the acquisition of data, 


 
the average temperature during the calibration process. 
The 
hot
-
wire
 
temperature 




 
was
 
found using 
 
 











 
(4.4)
 
Where, 

 
represents the resistance
-
temperature coefficient (for tungsten 0.0045 
º
C
-
1
)
,
 
and 


 
is the 
ambient 
temperature.
 
For the calibration, the j
et 
velocity
 


 
was found using
 

 
 
 







 
(4.5)
 
 





































 
(4.6)
 
Where, 

 
is 
the air density, 


 
the atmospheric pressure, 

 
the 
ideal 
gas constant
 
for air
 










, 


 
the 
a
mbient 
air temperature
, measured
 
using 
the 
temperature sensor, and 


 
is the 
stagnation pressure in the settling chamber, 
measured using 
the 
pressure transducer
.  
Note that 





 
is one over the
 
contraction 
area 
ratio (1/80), which is neglec
ted in the above calculation.
 
The data pairs of the jet velocity and the corrected voltage 
of the
 
hot
-
wire sensor were 
fitted
 
with equation 
(
4.7
)

-
squares
:
  
 
 









 
(4.7)
 
Where, 

 
r
epresents the corrected voltage of the hot
-
wire, 

 
the measured velocity using the 
pressure transducer,
 
and
 
n
 
(typically 
in the range 

 
=
 
0.4 to 0.45
)
, 





 
are 
the equation 
constants found using
 
the
 
least
-
squares method
.
 
 
83
 
 
The
 
calibration was done
 
before and after the experiments. Figure 4.9 depicts a 
sample
 
of 
typical calibrations after and before the 
experiments
, which agree 
within
 
a 
maximum difference of 
0.8 %.
 
 
Figure 4.9 Sample of hot
-
wire calibrations before and after an experiment 
 
4.4 Mi
crophone Setup and Calibration
 
The fluctuating pressure was 
measured
 
using eight electret Panasonic WM
-
61A 
microphones 
embedded
 
in the impingement plate
,
 

sensing hole and package diameter are 2 
mm
 
(0.08
D
) and 6 
mm
 
(0.24
D
) respectively. The 
microphones have a 
flat 
frequency response between 20
-
20,000 
Hz
 
and 
a 
typical 
manufacturer
-
provided 
sensitivity
 
of
 





 
@ 
1kHz
 
(corresponding to 11.22
 
to
 
28.18 
mV/Pa
,) with 
DC
 
supply 
voltage in the range 2
V
 
and 10
V.
 
The eight
-
microphone array 
was
 
arranged starting from 
the center of the disc (
r/D
=0) along the radial direction with a 
spacing of
 
0.33
D
 
between the centers 
 
 
84
 
 
of each successive microphones. Figure 4.10 depicts the configuration of the microphones on the 
impingement p
late. The microphones were connected to a
 
homemade
 
16
-
channel microphone 
circuit powered by 9 
DC
 
volts to provide power to the microphone and connect the microphone to 
the NI
-
6024E board
.
 
 
 
Figure 4.10 Front and cross section 
(
B
-
B
)
 
view (top and bottom r
espectively) 
of
 
the
 
 
microphone array configuration used in the present work
 
Since the
 
sensitivity of the microphones
 
provided by the 
manufacturer is nominal
, the
 
individual microphones had to
 
be calibrate
d before each experiment. This 
also accounts for 
p
ossible change in 
sensitivity due to variation in temperature, humidity, dirt, and installation. The 
eight microphones 
were calibrated, one at a time
 
by using a plane wave tube (
PWT
) and a reference 
microphone
 
that has known sensitivity. The reference micr
ophone used for 
the 
calibration was 
 
 
85
 
 
Brüel and Kjær (
B&K
) 




 
model
 
4923
-
A
-
001 
with
 
sensitivity
 
of
 








at
 
a 
polariz
ation
 
voltage of 


. The microphone has a
 
bandwidth
 
of
 





. 
 
The 
PWT
 
was built by Al
-
Aweni [
7
] using a 











PVC
 
square tube, which 
fastens
 
to the impingement plate
, by using two clamps
, as shown in figures 4.1
1
 
and 4.1
2
. The 
tube
 
has eight holes for mounting
 
the
 
B&K
 
microphone at the same 
radial
 
locations of the 
Panasonic 
microphones. On the other hand, the 
wall
 
of the tube that is in contact with the 
impingement plate 
was removed allowing the Panasonic micrphone array to be embedded in the 
PWT
.
 
Acoustic waves 
were generated
 
in the tube using Agilent model HP
-
33120A function 
generator coupled to a Hafler
-
P100
0 
amplifier,
 
which drives a Dayton model RS150S
-
8 audio 
speaker
.
 
The speaker
 
was placed
 
at one end of the plane wave tube
, generating
 
w
hite 
noise acoustic 
waves in order to excite all frequencies of interest simultaneously inside the 
PWT
.
 
According to 
[24]
, for example, if an acoustic wave with a 
wavelength
 




 
(
corresponding
 
to
 
a frequency 






; where
 







 
represent the tube 
cross
-
section 
side 
length, the sound frequency, and the speed of sound respectively) propagates in a square so
lid duct, 
then the wave will remain planar at any given cross section. This means, the phase and the 
magnitude of pressure will remain constant over the cross section at the given plane. So, the 
reference microphone (
B&K
) and the microphone to be calibrate
d (Panasonic) will be exposed to 
the same pressure magnitude and phase because they are located at the same cross section. 
 
During calibration, 
each microphone was calibrated individually 
by
 
inserting the reference 
microphone into the 
PWT
 
at the same cross
 
section of the 
Panasonic 
microphone
. 
One hundred
 
records, each having
 
4096
 
point data 
points sampled 
at 50 
k
Hz
, 
were acquired from two 
A/D
 
channels
,
 
to which the microphones 
were
 
connected. 
Applying the analysis shown in Al
-
Aweni 
 
86
 
 
[7], the 
acquired
 
data pr
oduced the 
magnitude and phase response of the Panasonic microphones
; 
as exemplified in figure 4.
1
3
.
 
 
 
Figure 4.1
1
 
Block diagram of the calibration setup for the microphone array
 
 
 
87
 
 
 
Figure 4.1
2
 
An image of the calibration 
setup
 
for the microphone a
rray
 
Table 4.1 show
s
 
a 
sample of the eight microphones

 
sensitivit
ies, where microphone 1 is 
located at the stagnation point and microphone 8 is placed at the end of the measurement domain
. 
 
 
The values given, which fall 
within the 
manufacturer
-
reported nom
inal 
range
,
 
are found by 
averaging the magnitude response over the 
frequenc
y range
 
100
-
5000 
Hz
.  
 
 
Microphone
 
1
 
2
 
3
 
4
 
5
 
6
 
7
 
8
 
Sensitivity 
mV/Pa
 
18.625
 
18.447
 
23.789
 
19.623
 
18.837
 
22.652
 
25.666
 
25.624
 
Table 4.1 Sample of 
the
 
microphones

 
sensitivity
 
obtai
ned from calibration
.
 
 
 
B & K 
microphone
 
Plane wave 
tube
 
Speaker
 
Nozzle
 
Impingement 
plate
 
 
88
 
 
 
 
 
Figure 4.1
3
 
A sample of 
the 
microphone calibration 
results:
 
sensitivity (top) and phase 
(bottom)
 
 
 
 
89
 
 
4.5 Procedure to Acquire Velocity Profile 
 
After calibration, the hot
-
wire was placed near to the edge of the shear layer. The 
latter
 
was 
found 
approximately 
by 
monitoring the hot
-
wire signal on
 
an oscilloscope 
as
 
the wire 
was 
traversed across
 
the shear layer.  Once the edge of the shear layer 
was 
found
, the wire was traversed 
to 50 to 100 different positions (with a resolution of
 


), depending on the 
X
 
location, such that 
measurements were conducted across the entire shear layer. The movement and the data 
acquisition were automated using a LabView program. At each location, time series containing
 
409600
 
data point
s
 
of 
jet veloci
ty
, 
jet 
temperature, and 
hot
-
wire
 
signal w
ere
 
acquired at 5000 
samples/second
. 
The duration of the acquisition 


 
is such that almost 5000 vortices travel across 
the measurement point at the lowest frequency of vortex passage (
St
 
= 0.3 and the low
est velocity 
of 








). The sampling rate is also selected to be much larger than any frequency of interest 
(







, based on the highest velocity).
 
4.6 Procedure to Acquire 
the 
Microphone Signal
s
 
 
After 
calibration of the microphones, 
the 
PWT
 
was removed and the impingement plate 
was fixed at a desired distance (
H/D
). Before acquiring the microphone signal
s
, the 
mean jet
 
velocity was fixed 
to provide
 
the desired 
Reynolds number. Once the jet reached steady state,
 
800 
data records contai
ning 
512
-
point 
per record
 
were acquired 
at a rate of 5000 
samples/s
 
from all 
microphones in the array 
using 
a 
LabVIEW program. 
 
 
 
 
 
 
 
 
90
 
 
CHAPTER 
5: Jet Characteristics and Wall
-
Pressure Measurements 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Before investigating the pressure measuremen
ts, it is important to demonstrate the 
jet
 
characteristics
 
as it emerges from the nozzle; i.e. the initial condition. This chapter will explore 
the 
characteristics
 
of the 
jet
 
and the wall
-
pressure measurements.  The 
jet
 
characteristics
 
were 
found
 
by measur
ing the streamwise flow velocity component using a single hot wire, traversed 
across the shear layer at a few streamwise locations. The data were used to demonstrate the self
-
similarity 

 
in agreement with the literature.  T
he pressure 
measurements from this study were compared with previous studies using different statistical 
quantities, including power spectra
,
 
root 
mean
 
square of the pressure fluctuation, probability 
density functions (
pdf
), skewness, and kurtosis.
 
5.1 Ini
tial Shear Layer 
Self
-
Similarity
 
 
 
It is
 
important to examine the 
characteristics
 
of the shear layer and the 
initial flow
 
conditions. To achieve this goal, the mean 
and fluctuating 
streamwise velocit
y
 
were 
measured,
 
and the data were used to obtain the cor
responding cross
-
stream velocity profiles across the shear 
layer. For these measurements, the hot
-
wire was initially moved using 
relatively large
 
steps, with 
a resolution of 0.5 
mm/step
, in the 
transverse
 
direction and the wire output signal was monitored 
on the oscilloscope to approximately locate the edges of the shear layer. Subsequently, the 
movement resolution was 
refined
 
gradually to verify the location of the 
edges
 
of the shear layer. 
Once the edges were 
found
, the probe was controlled to traverse ac
ross the whole shear layer with 
even higher resolution (0.01 
mm/step
) to properly resolve the high
-
shear zone 
within
 
the shear 
layer. The data were recorded at Reynolds numbers of 8272, 165454, and 33090 which represented 
velocities of 
5 
m/s
, 10 
m/s
, and 2
0 
m/s
,
 
respectively. Profiles were obtained at 
three different
 
 
91
 
 
streamwise locations (
X/D
= 0.2, 0.4, and 0.8). The self
-
similarity of the profiles obtained at 
different Reynolds numbers 
was verified
 
using normalized mean and fluctuati
ng
 
velocity plots, as 
s
hown in figure
s
 
5.1 and 5.2 respectively. For these plots, the origin of the cross
-
stream coordinate 
is taken at the shear layer centerline; defined as the 
y
 
location where the mean velocity is half of 
the jet exit velocity. The 
y
 
coordinate is normalized 
by the momentum thickness 



,
 
which is 
found using
 
 



































 
(5.1)
 
Where 




 
is the mean streamwise velocity profile, and 


 
is the jet exit velocity. To minimize 
the erro
r resulting from hot
-
wire data near the shear layer outer edge, where reverse velocity may 
occur due to the energetic shear
-
layer vortices
, the momentum thickness was calculated by 
truncating the integration limit to the location where 




 
is 10% of the
 
jet velocity; i.e. the lower 
integral limit
 
in equation 5.1 










.  As seen from figure
s
 
5.1 and 5.2, the velocity profiles 
for the various Reynolds numbers at the same streamwise location (
X/D
=0.2) collapse well. The 
collapse of the veloci
ty profiles 
demonstrates
 
the initial self
-
similarity of the jet. 
 
 
92
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 5.1 Shear
-
layer mean velocity profile at 
X/D
=0.2 
for
 
various 
Reynolds numbers 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 5.2 
Shear
-
layer fluctuati
ng
-
velocity root
-
mean
-
square profile
 
at 
X/D
=0.2
 
for
 
various 
Reynolds numbers
 
 
 
 
 
93
 
 
To 
check
 
the self
-
similarity at different 
X/D
 
locations,
 
mean and fluctuati
ng
-
velocity 
profiles of the shear layer 
were 
measured
 
at three streamwise locations (
X/D
= 0.2, 0.4, and 0.8) 
for Reynolds number of 8272. The results 
are 
depicted
 
in figure
s
 
5.3 and 5.4 respectively,
 
which 
demonstrate that the velocity profiles collapse for the distance 
X/D
=0.2
-
0.8, further confirming 
self
-
similarity.
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 5.3 Shear
-
layer mean velocity profile at various 
X/D
 
locations and
 




 
 
 
94
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 5.4 
Shear
-
layer fluctuati
ng
-
velocity root
-
mean
-
square profile
 
at various 
X/D
 
locations and 




 
5.2 The Root Mean Square of the Fluctuating Pressure  
 
The root mean square pressure is calculated from the pressu
re time series using
 
 

















 
(5.2)
 
Where, 



is the instantaneous pressure, 

 
is the mean pressure, 
i
 
indicates the time index, or 
sample number in the digitized pressure time series, and 
n
 
is the total number 
of samples in the 
time series. The number of samples is 409600 samples with 
a 
sampl
ing
 
frequency
 
of
 
5000 
samples/sec
. Note that since microphones are used for measuring the pressure, they are incapable 
of capturing
 

. However, the measured time series typ
ically have a small offset voltage error, 

 
 
 
95
 
 
Figure 5.5.a depicts the radial distribution of the root mean square results for the pressure 
fluctuation 




 
for Reynolds number 82
72.  The fluctuati
ng
-
pressure 
RMS
 
value is normalized 
by the dynamic pressure 













 
and the radial location (
r
) is normalized by the jet 
dimeter (
D
).  Results are shown for different 
H/D
 
values, represented by different colors. The 
re
sults show that, though the overall shape of the root mean square pressure distribution is similar 
for all 
H/D
 
values, the 



magnitude depends on 
H/D
. A peak is noticed for all three 
H/D
 
= 2, 3 
and 4 cases at 
r/D
 

 
1.33. The magnitude of this pea
k, and 



in the 
wall
-
jet zone
 
in general, 
decreases with increase in 
H/D
. Also, the location of the maximum 



seems to shift towards 
smaller 
r/D
 
with increase in 
H/D

of
 
the maximum cannot be determined with the present measurement resolution. 
 
On the other hand, within the 
stagnation zone
, the trend with 
H/D
 
is reversed, where the 
magnitude of the root mean square pressure increases, rather than decreases, with increasin
g 
H/D
. 
For example, the magnitude of 



at the stagnation point (
r/D =
 
0) increases to 54% of the 
maximum for 
H/D=
4, in contrast to 14% at 
H/D=
2
. Also noteworthy
, for all 
H/D
 
values, the 
normalized  


 
drops rapidly for the range 







 
to a magnitude of around 5% by the end 
of the measurement domain. 
 
Figure 5.5.b depicts 





 
for the higher Reynolds number of 24818. The distribution 
has the same behavior as for 
Re
D
 
= 8272, except for 
H/D
=4, where a second peak emerges wi
thin 
the stagnation zone, at 
r/D
=0.67. Significantly, unlike all other cases, at the higher Reynolds 
number and 
H/D
 
= 4, the maximum 


 
is found in the stagnation, instead of the wall
-
jet zone. 
The magnitude of 


 
at the stagnation point 
increases to 42% of the maximum for 
H/D
=4, in 
contrast to 12% at 
H/D
=2.
 
 
 
 
 
96
 
 
 
 
 
 
 
 
 
 
 
 
 
 
( a )
 
 
 
 
 
 
 
 
 
 
 
 
 
 
( b )
 
Figure 5.5 Normalized Root Mean Square Pressure 







 
versus 



 
for different 



 
and 
two Reynolds numbers: (a) 
Re
D
 
=
 
8272, and (b) 
Re
D
 
= 24818
 
 
 
 
97
 
 
To better 
examine
 
the effect of Reynolds number, Figure 5.6.a depicts a comparison 
between the 


 
radial distributions for 
Re
D
 
= 8272 and 24818 when 
H/D
=2. The figure shows 
that as Reynolds number increases, the  




 
profile does not change, for all practical purposes. 
For both cases, the minimum pressure is at the stagnation point, where 
r/D
 
= 0, and the maximum 
pressure is measured at 
r/D
=1.33.  Figures 5.6.b and 5.6.c depict a similar comparison for 
H/D
=3 
and 4
, respectively. For both cases, increasing the Reynolds number, results in decreasing the level 
of pressure fluctuations within both the stagnation and wall
-
jet zones. In addition, as noted 
previously, a second local 


 
peak emerges at 
r/D
 
= 0.67 
for the higher Reynolds number
 
and 





. It is unclear if this peak reflects a change in the physics of wall
-
pressure generation 
within the stagnation zone at the higher 
Re
D
, or simply that the peak becomes observable due to 
reduction on the level of pr
essure fluctuations in the wall
-
jet zone.
 
Overall, the characteristics of the 
RMS
 
profiles and how they change with 



 
and 



 
is 
very consistent with those reported in the literature, and summarized in section 1.2.2.
 
 
 
 
 
 
98
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure
 
5.6 Normalized Root Mean Square Pressure 







 
versus 



 
at
 
Re
D
 
= 8272 and 
24818
,
 
for
:  



















and 








  
 
 
 
 
 
 
 
 
 
 
( a )
 
 
( b )
 
 
( c )
 
 
 
99
 
 
5.3 Time Series Analysis  
 
5.3.1 Time Series 
 
Figure 5.7.a depicts sample normalized
-
pres
sure signals
 
at 
Re
D
 
= 8272 at the stagnation 
point (





) and the location of maximum
 


 
(







) for  






 
As expected 
from the 


 
distribution (figure 5.5
.
a), the pressure signal at 
r/D
=0 is low compared to that at 
r/D
=1.
33. Aside from this, inspecting the time series enables extraction of additional interesting 
information. Specifically, Figure 5.7.a demonstrates that the signal shape is completely different 
between the stagnation point and
 
r/D
 
= 1.33. At the former locat
ion, the signal is relatively 
symmetric around zero level and looks like a distorted sinusoid. In contrast, at 
r/D
 
= 1.33, the 

dynamic pressure! In his study
 
in the same jet facility at a similar Reynolds number and 
H/D
, Al
-
Aweni [7] used simultaneous time
-
resolved flow visualization and wall
-
pressure measurements 
and numerical simulations to demonstrate that these very strong pressure spikes are a result of t
he 
interaction of the jet vortices with the wall and the formation of secondary vortices
 
(see section 
1.2.2)
. 
 
Similar strong negative spikes can also be seen at the same 
r/D 
= 1.33 location but the 
larger 
H/D
 
values of 3 and 4 (figures 5.7.b and 5.7.c res
pectively). However, a significant 
difference between the signals observed at the three 
H/D
 
values is that the average time period 
between spikes increases with increasing 
H/D
. Based on the flow visualization of Al
-
Aweni [7], it 
is known that this increase
 
is due to two successive pairings of the jet vortices before reaching the 
impingement wall: one pairing taking place between 
H/D
 
= 2 and 3, and the other between 
H/D
 
= 
3 and 4. Interestingly, the magnitude of the spikes in figure 5.7 is almost unaffected 
by the 
H/D
 
values, suggesting that the strength of the pressure produced from the vortex
-
wall interactions is 
 
100
 
 
maintained with increasing 
H/D
. These observations lead to the following explanation for the 
decrease in the maximum 


 
(and likely 



 
in general for the whole wall
-
jet region as well) 
with increasing 
H/D
 
(see figure 5.5): with the strength of the spikes remaining invariant with 
H/D
 
but becoming less frequent, the 


 
value must decrease. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 5.7 S
ample normalized Pressure signals 









 
at
 
Re
D
 
= 8272
,
 
for
 











 
for:  





















 
and 








  
 
 
 
 
( a )
 
 
( b
 
)
 
 
( c )
 
 
 
101
 
 
Figure 5.8 
displays plots similar to those in figure 5.7 for the higher Reynolds number. 
Generally spea
king, observations similar to those made in relation to figure 5.7 can be made from 
figure 5.8. However, there are also some notable differences. One of these relate to the strength of 
the negative pressure spikes. Unlike the time series for 
Re
D
=8272, the 
strength of the negative 
pressure spikes at 
r/D
 
= 1.33 decreases noticeably with increasing 
H/D
 
for 
Re
D
=24818. This 
decrease is associated with the signal becoming more irregular and the appearance of high
-
frequency fluctuation. This is consistent with the
 
decrease in the 


 
level with increasing 
Re
D
 
observed in figures 5.6.b and 5.6.c at 
H/D
 
= 3 and 4 respectively. The signal forms in figure 5.8.b 
and figure 5.8.c suggest that this decrease is associated with weakening of the pressure spikes at 
th
e higher Reynolds number and larger 
H/D
, which might be related to the jet vortices breaking 
up and becoming irregular/turbulent with increasing Reynolds number (as implied from the 
irregularity of the signal and appearance of high frequency fluctuations).
 
 
 
 
 
 
 
 
 
 
102
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 5.8 Sample normalized Pressure signals 









 
at
 
Re
D
 
= 24818
, for
 











 
for:  





















 
and 








  
 
 
 
 
 
 
 
( c )
 
 
( 
b 
)
 
 
( 
a
 
)
 
 
 
103
 
 
5.3.2 Probability Density Function (pdf)  
 
The t
ime series analysis in the previous section shed
s
 
light on the general characteristics of 
the pressure signal at the stagnation point, as well as where 


 
is highest in the wall
-
jet zone. 
However, these observations are based on short, randomly se
lected time series samples. Therefore, 
to ensure that the observations made in section 5.3.1 are statistically relevant, Probability Density 
Function (
pdf
) 
results are examined. The 
pdf 
was estimated
 
by finding the maximum and minimum 
fluctuating
-
pressure
 
values
 
in a given time series. The range bound by these values was
 
then 
divid
ed 
into 30 equal
-
width bins
. Finally, 
the number of data points falling in each bin was divided 
by the total number of points in the time series and the bin width to obtain the pr
obability of the 
pressure value occurring within a given bin per bin width, or the 
pdf
.
 
 
Figure 5.9 depicts the 
pdf
 
for the pressure time series at 
Re
D
 
= 8272 and 











 
for all three 



 
values. By examining the plots, we can see that a
t 






, the 
pdf
 
is 
approximately symmetric and narrow, which is consistent with the symmetric, low
-
level character 
of the corresponding signals observed in figure 5.7. On, the other hand, at 







, the signal 
has a pronounced negative skewness
, as reflected in the long negative tail and consistent with the 
strong negative pressure spikes noted earlier in figure 5.7. In addition, the 
pdf
 
is substantially wider 
than that at 





, consistent with the smaller 





 
at 





 
(
figure 5.5.a). 
 
Another interesting observation at 
H/D
 
= 2, is the presents of a plateau with a hint of a peak 
at 










. This suggest the presence of a bi
-
model phenomenon. Interestingly, Al
-
Aw
e
ni[7] found the vortex structures to either
 
merge as they convect past 








 
or to pass 
without merging. This may explain the subtle bi
-
modal feature of the 

 
at 








 
in figure 
5.9.a
 
 
104
 
 
At 





, the width of the 
pdf
 
increases with increasing 
H/D
, demonstrating the 
increasing lev
el of 
 





 
at the stagnation point with larger 
H/D
 
(see figure 5.5.a).
 
At 







, the pdf remains negatively skewed with increasing 
H/D
, approximately reaching 
negative pressure values as high as 

P
d
 
for all 
H/D
 
values. However, the p
df appears to become 
overall narrower with increasing 
H/D
, consistent with the corresponding reduction in 
 





 
(see figure 5.5.a).
 
The narrowing of the pdf primarily manifests itself in the reduction in the 
pdf
 
value for the large negative 
pressure spikes (

 
approximately less than 







), which 
reinforces the idea discussed in section 5.3.1 of these spikes becoming less frequent at larger 



 
due to vortex pairing.
 
 
 
 
 
 
 
 
 
105
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 5.9 Probability Densit
y Function 



 
for the pressure signal at 
Re
D
 
= 8272
,
 
for 











, and:  




















and 








  
 
Figure 5.10 depicts the 

 
results for 
Re
D
 
= 24818 at 











, and all 



 
values. By exami
ning the
se
 
plots, we can see the same general behavior as seen for 
Re
D
 
= 8272. 
However, there are some notable differences. Overall, the 
pdf
 
at 
r/D
 
= 1.33 is not as strongly 
skewed as for the lower Reynolds number case. Additionally, the long negative tail
 
of the 
pdf
 
 
 
 
( 
a
 
)
 
 
( 
b
 
)
 
 
( c )
 
 
 
106
 
 
extends to smaller negative
-
pressure magnitudes with increasing 
H/D
; in contrast to reaching 
approximately the same value for 
Re
D
 
= 8272. The reduction in the magnitude of the negative 
spikes with increasing 
H/D
 
was also noted earlier from the 
time series plots in figure 5.8.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 5.10 Probability Density Function 



 
for the pressure signal at 
Re
D
 
= 24818
,
 
for 











, and:  




















and 








  





 
 
 
 
 
 
( 
a
 
)
 
 
( 
b
 
)
 
 
( c )
 
 
 
107
 
 
5.3.3 Skewne
ss and Kurtosis  
 
The overall features of the 
pdfs
 
presented in section 5.3.2 may be expressed in terms of 
their skewness and kurtosis. The skewness provides a measure of the symmetry of the 
pdf
. A 
skewed 
pdf
 
exhibits a long negative tail (if negatively s
kewed) or a positive one (if positively 
skewed). On the other hand, the kurtosis indicates how flat (as opposed to 
having a prominent 
peak
) a 
pdf
 
distribution is. A 
pdf
 
with large kurtosis tends to have long tails. For reference, a 
Gaussian 
pdf
 
has a skewn
ess of zero (due to its symmetry) and a kurtosis of 3. In the present work, 
the skewness and kurtosis were calculated as follows
 
 



















 
(5.3)
 
 




















 
(5.4)
 
 
where, 


 
is the instantaneous pressure, 

 
is the mean pressure, 
i
 
indicates the time index, or sample 
number in the digitized pressure time series, 

 
is 
the 
standard deviation
 
of the pressure time series 
points
, and 
n
 
is the total number o
f samples in the time series
 
Figure 5.11.a depicts the variation of skewness over the measurement domain (
r/D
 
range 
0 to 2.33) for a Reynolds number of 8272 at 
H/D
 
= 2, 3, and 4. The plots show that at 
H/D
=2, the 
pressure signal has positive skewness in th
e stagnation zone (





), but it switches sign and 
becomes negative in the wall
-
jet zone. For 
H/D
=3 and 4, 
the switch from positive to negative 
skewness happens 
near the end of the stagnation zone
.
 
The negative skewness is particularly strong 
at 
r/D
 
=
 
1.33 and 1.67. This is consistent with the long negative tail of the 
pdfs
 
and the strong 
negative pressure spikes in the time series discussed earlier for 
r/D
 
= 1.33. However, as 
r/D
 
increases further, the skewness magnitude decreases monotonically; thoug
h it remains negative.
 
 
108
 
 
Figure 5.11.b depicts the skewness results for the higher Reynolds number of 24818. With 
the exception of one apparently errant data point (at 
r/D
 
= 0.33 and 
H/D
 
= 4), the general qualitative 
behavior of the skewness distribution is 
similar to the lower Reynolds number. However, at 
Re
D
 
= 
24818, the largest negative skewness is not as large in magnitude as for 
Re
D
 
= 8272. Also, the 
skewness becomes zero (or low valued) by 
r/D
 
= 2.0; implying the 
pdfs
 
reach symmetry by the 
end of the me
asurement domain.
 
 
 
 
 
 
 
 
 
 
 
 
109
 
 
 
 
 
 
 
 
 
 
 
 
 
 
( a )
 
 
 
 
 
 
 
 
 
 
 
 
 
 
( b )
 
Figure 5.11 Radial distribution of skewness for 








and 



and: 








, and 









 
 
 
 
110
 
 
Figure 5.12.a depicts the kurtosis results for Reynolds number of 8272. W
ithin the 
stagnation zone (





), the kurtosis is similar to that of a normal (Gaussian) distribution with 
a magnitude near 3. In the wall
-
jet zone, the kurtosis initially increases substantially (up to the 
radial location 
r/D
 
= 1.67) then it decays mon
otonically. For the higher Reynolds number, the 
kurtosis behavior (figure 5.12.b) is surprisingly different. In this case, the kurtosis is highest at the 
stagnation point and decays monotonically with increasing 
r/D
. The reason for the fundamental 
change i
n kurtosis behavior with Reynolds number is not clear. Measurements at intermediate 
Reynolds numbers would be recommended in order to observe if this change is gradual or abrupt
,
 
and attempt to understand the reasons behind it.
 
The large magnitude of negat
ive skewness and kurtosis near the location of maximum 


 
in the wall jet 
zone indicates the presents of strong pressure spikes. This is consistent with the 
strong negative peaks found in the work of 
Didden and Ho [5], Hall and Ewing [
9,10
] and Al
-
Aw
e
ni [7]. The latter study connected the formation of the spikes to the secondary vortex formation 
(as also Didden and Ho [5] and Hall and Ewing [
9,10
]), and the formation of 
high
 
strain zone in 
the boundary layer beneath the jet vortex.
 
 
 
 
 
 
 
 
 
 
 
111
 
 
 
 
 
 
 
 
 
 
 
 
 
 
( a )
 
 
 
 
 
 
 
 
 
 
 
 
 
 
( b )
 
Figure 5.12 Radial distribution of Kurtosis for 








and 



and: 








, and 









 
 
 
 
112
 
 
5.4 Spectra Analysis  
 
Analyzing spectra of the pressure fluctuations is considered one of the useful 
tool
s
 
to get 
information about the frequency content 
of
 
the pressure signal. In this work, spectral information 
is presented as power spectral density (
PSD
). The 
PSD
 
is 
calculate
d as an average of the 
PSDs
 
obtained from
 

 
different pressure data records, as f
ollows:
 
 
 




























 
(5.5)
 
 
Where, 



 
is the 
Fourier transform of the 
i
th
 
pressure
 
data record, 

 
is the number of points in the 
data record
, 

 
is an index indicating frequency 
and 
 
f
s
 
is 
the 
sam
pling frequency
. The physical 
frequency corresponding to each 

 
value is given by 









 
(where 


 
is the frequency 
resolution of the 
PSD
).
 
 
To compute the 
PSD
, 409600 
samples
 
of a given pressure signal were acquired at a 
sampling frequency o
f 5000 
Hz
. 
The resulting
 
data 
were divided
 
into 800 
records
,
 
each containing 
512 points. The 
PSD
 
was obtained for each record by taking its Fast Fourier Transform (
FFT
), 
multiplying the transform by its conjugate, and dividing the result by the square of t
he number of 
points (512 points) and the spectrum frequency resolution (


 
= 5000/512 =
 
9.77 Hz). The 
resulting spectrum 
random 
uncertainty is 3.5%.
 
Figure 5.13 shows the normalized 
PSD
 
in the form of contour plots versus 
r/D
 
and Strouhal 
number (
St
D
 
= 
fD
/U
j
) at 





and all 



 
values. This way of presenting the spectra 
provides a global perspective of the entire measurement domain, but it does not allow clear 
observation of some of the less dominant spectral features. To see these, the spect
ra are presented 
using line plots at 
r/D
 
= 0 and 1.33 in figure 5.15.  The contour plots in Figure 5.13 show a 
 
113
 
 
dominant spectral peak 
at 






 
(the actual value varies between 0.64 for 
H/D
 
= 2 and 3, and 
0.58 for 
H/D
 
= 4, which is approximately wit
hin the spectrum 



 
resolution of 0.05). From the 
work of Al
-
Aweni [7]
, it is known that the 







 
corresponds to that of the jet vortices after 
the first pairing. 
 
As discussed in section 5.3.1, Al
-
Aweni also found that for approximately th
e same 
Reynolds number, the first vortex pairing took place as the vortices traveled parallel to the wall, 
within the wall
-
jet zone, at 
H/D
 
= 2, and ahead of reaching the impingement plate, at 
H/D
 
= 3. A 
second pairing was observed before the vortices reac
hed the plate at 
H/D 
= 4. These conclusions 
suggest that the spectrum should be dominated by fluctuations at 




 
0.6 at 
H/D
 
= 2 and 3, and 







 
at 
H/D
 
= 4. The former expectation is consistent with the results in figure 5.13.a and 
5.13.b. H
owever, the dominance of 




 
0.6, instead of 




 
0.3 at 
H/D
 
= 4 (figure 5.13.c) 
seems inconsistent with Al
-

fact that the dominant spectrum peak in figure 5.13.c is seen within
 
the wall
-
jet zone (
r/D
> 1). 
Inspecting a sample of the corresponding time series (figure 5.7.c), it is evident that the time series 
is highly irregular and dominated by strong negative spikes with varying strength (i.e. modulation) 
from one spike to anoth
er. The spectrum of such a signal is not expected to yield a clean peak at 
the dominant frequency, but rather a broader spectrum of multiple peaks. Indeed, although 







 
is dominant in figure 5.13.c, another, barely visible peak is seen at 




 
0.3 (pointed to by a
 
white
 
arrow
 
in figure 5.13.c
). This peak can also be observed more clearly in the line plots of figure 
5.15.b. Although the observed peak is weaker than that at 







, this could be caused by the 
vortices being less coheren
t after the second merging (which was observed in the flow 
visualization videos of Al
-
Aweni). However, this reasoning cannot be ascertained at this stage. 
 
 
114
 
 
A better indication of the presence of pressure fluctuation at the second
-
pairing frequency 
(




 
0.3) at 
H/D
 
= 4 may be seen in the stagnation zone, particularly at 
r/D
 
= 0, in figure 5.13.c. 
Due to the simplicity of the signal at the stagnation point (figure 5.7.c), the corresponding spectrum 
has a more straight forward interpretation. The contou
r plot in figure 5.13.c and the line plots in 
figure 5.15.a show that indeed when 
H/D
 
= 4, the dominant spectral peak shifts to 




 
0.3 at 
r/D
 
= 0. 
 
 
 
 
 
 
 
 
 
 
 
 
 
115
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 5.13 Normalized 
PSD
 
contour plots at 





 
for
:  

















, 
and 









  
 
Figure 5.14 shows normalized 
PSD
 
contour plots similar to those in figure 5.13 but for 




. Consistent with the lower Reynolds number, the spectrum peak at 







 
is 
dominant within t
he wall
-
jet zone at 
H/D
 
= 2 and 3. At 
H/D
 
= 4, the dominant peak clearly shifts 
to 







 
within the stagnation zone at 







. Recall that this is also the radial location 
 
 
 
( 
a
 
)
 
 
( 
b
 
)
 
 
( c )
 
 
 
116
 
 
where the maximum 


 
is found for the larger Reynolds number an
d 
H/D
 
= 4 (figure 5.5.b), 
which is different from all other cases. These results reinforce earlier observations regarding the 
overall weakening of the pressure fluctuation in the wall
-
jet zone with increasing 
H/D
 
and 
Reynolds number. Since it is known from
 
the work of Al
-
Aweni that the vortex
-
wall interactions 
dominate the pressure fluctuations in the wall
-
jet zone at low
 
Reynolds number, the present results 
suggest that the ability of these interactions to generate unsteady pressure weakens with increasing
 
Reynolds number, leading to the observed dominance within the stagnation zone.
 
Another notable characteristic of the spectra at the higher Reynolds number is the rise of 
low
-
frequency fluctuations. These 
are
 
seen clearly in both the contour plot in figure
 
5.14.c and the 
line plots in figure 5.16. The latter plots also show the general shift of the dominant frequency 
towards low values with increasing 
H/D
 
(figure 5.16.a).
 
 
 
 
 
 
 
 
 
 
 
117
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 5.14 Normalized 
PSD
 
contour plots at 





 
for:  

















, 
and 









  
 
 
 
 
 
 
 
 
 
 
 
 
( 
a
 
)
 
 
( 
b
 
)
 
 
( c )
 
 
 
118
 
 
 
 
 
 
 
 
 
 
 
 
 
 
( a )
 
 
 
 
 
 
 
 
 
 
 
 
 
 
( b )
 
Figure 5.15 Normalized 
PSD
 
at 




 
for
,
 








and 



at: 





















  
 
 
 
 
119
 
 
 
 
 
 
 
 
 
 
 
 
 
 
( a )
 
 
 
 
 
 
 
 
 
 
 
 
 
 
( b )
 
Figure 5.16 Normalized 
PSD
 
at 





 
for 








and 



at: 





















  
 
 
 
 
120
 
 
Figures 5.17 and 5.18 depict a direct comparison of the normalized 
PSD
 
at the two different 
Reynolds number ( 









. These plots are provided to facilitate understanding 
of the Reynolds number effect on the spectra. Overall, both figures show that
 
the Strouhal number 
band of the pressure fluctuations is consistent for both Reynolds number. However, 
the 
PSD
 
has 
hi
gher level and sharp peaks at the lower Reynolds number; in comparison to being broader and 
having lower level with increasing Reynolds number. This suggests that the basic wall
-
pressure 
generating mechanisms remain the same with increasing Reynolds number
, but they become 
weaker and more stochastic in nature. In addition, at the largest 
H/D
 
of 4, and the high Reynolds 
number
,
 
the vortex
-
wall interaction effectiveness in generating pressure in the wall jet zone 
weakens substantially.
 
 
 
 
 
 
 
121
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 5.17 Normalized 
PSD
 
at  








 

 






 
for:  

















and 








 
 
 
 
 
 
( 
a
 
)
 
 
( 
b
 
)
 
 
( c )
 
 
 
122
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 5.18 Normalized 
PSD
 
at  








 

 








 
for:  

















and 








 
 
 
 
 
 
 
 
 
 
 
 
( 
a
 
)
 
 
( 
b
 
)
 
 
( c )
 
 
 
123
 
 
5.5 Comparison between Experiment and Mathematical Model
 
 
In the present section, a comparison is conducted between the results of the experiments 
and the model. The purpose of this comparison is to assess t
he degree by which the model is 
successful in capturing the underlying physics of wall
-
pressure generation in impinging jets. It is 
emphasized here that, given the fairly crude nature of the model, the comparison is focused on 
qualitative features and tren
ds and is constrained to the stagnation zone of the impinging jet. 
 
The discussion in section 3.1 demonstrated that the character of the modeled wall
-
pressure 
time series is different depending on whether the radial location is near the stagnation point (







) or the end of the stagnation zone (







). Near the stagnation point, the pressure signal 
was weaker and characterized with sinusoidal like variation, and the end of the stagnation zone, 
the signal featured a prominent negative pressure peak 
and a strong, but less prominent positive 
peak. The pressure variation (which are replicated at the bottom of figure 5.19) were connected to 













) effects of vortex passage. Comparing similar 
signals obtained expe
rimentally at the lower Reynolds number and 





 
(top of figure 5.19), 
we see very similar qualitative features, suggesting that he basic physics of wall
-
pressure 
generation in the stagnation zone are captured by the present model, notwithstanding its h
igh level 
of simplicity.
 
Some of the differences between the experimental and model results in figure 5.19 include 
the broadness of the positive and negative peaks at 





. The experimental time series clearly 
exhibit much broader peaks, which is not su
rprising given that the real jet vortices have a finite 
core size, in comparison to the point vortices employed in the present model. Thus, one of the 
important future improvements of the model is to utilize finite
-
core vortices; for example, Oseen 
type, h
aving Gaussian vorticity distribution. This point is also important from the perspective that 
 
124
 
 

depend on how far is the point from the vortex center, relative
 
to the vortex core size
.
 
 
 
 
 
 
 
 
 
 
 
 
125
 
 
 
 
Figure 5.1
9
 
Sample time series from microphone measurements (top) and the vortex
-
array model (bottom) at the stagnation point (





) and end of the stagnation zone (





). Experimental data are shown for the 





 
and 





, and model results for the 
reference case (emulating the conditions at the end of the potential core).
 
 
 
 
 
126
 
 
Though shown over a short time period, the experimental time series in figure 5.19 is 
characteristic of the remainder of the
 
time series, with the shown signature repeating quasi
-
periodically. However, at 





,
 
there are moments in time were the signature character is 
different. This is exemplified in figure 5.20 for the same time
-
window size as the top of figure 
5.19. The s
ignal shape during such periods is very similar to that associated with the vortex
-
induced separation of the boundary layer. This conclusion can be made based on the work of Al
-
Aweni [7]. Such vortex
-
boundary
-
layer interactions cannot be captured by the pr
esent model, 
which is both inviscid and does not include any modeling of boundary layer effects.
 
 
Figure 5.
20
 
Sample wall
-
pressure signature characteristic of that produced by vortex
-
induced 
separation from microphone measurements at 





, 






 
and 





 
 
Another point concerning the comparison in figure 5.19 is that it is done using the 
experimental data at the lower Reynolds number. At the higher Reynolds number, as discussed 
previously, small
-
scale pressure fluctuations start to appear,
 
implying the formation of small
-
scale 
 
 
127
 
 
turbulence. Such turbulence, which obviously cannot be reproduced by the present model, makes 
it more difficult at times to identify the vortex
-
passing signature in the time series.
 
Another interesting qualitative fea
ture of wall
-
pressure fluctuation in the stagnation zone 
that is captured by the vortex
-
array model relates to the effect of 



 
on
 


. The reader is 
reminded that the effect of increasing 



 
in the model is simulated by decreasing the vortex
 
passing frequency; i.e. emulating the reduction in frequency via vortex pairing. This frequency 
effect, which was presented in figure 3.4 and is reproduced in the bottom of figure 5.21, is 
qualitatively similar to that seen in the experimental results (fi
gure 5.21 top). At high frequency 
(small 



), both model and experimental results show the pressure fluctuations to be lowest at 
the stagnation point and rise to be highest at the end of the stagnation zone. As the frequency 
decreases (



 
increases),
 
the level of pressure fluctuations increases at the stagnation point. Also 
both the model and the experimental results show that within the zone 







, the pressure 
fluctuations increase with 



 
at highest
-
frequency (smallest 



), while they dec
rease with 



 
for the middle and lowest frequency (









). For all cases, the 
RMS
 
pressure level 
increases with 



 
when 







 
(within the stagnation zone) for both the model and the 
measurement results.
 
 
 
 
 
128
 
 
 
 
Figure 5.
21
 
Comparison 
of the effect of varying 



 
on RMS wall
-
pressure fluctuation 
between the experimental (top) and the model (bottom) results at 




 
. In the model, 
increasing frequency corresponds to decreasing 



. The broken green lines outline the end 
of 
the stagnation zone
 
 
 
 
129
 
 
Figure 5.21 also show consistency between the calculation and the measurement in that the 
trend of increasing 
RMS
 
level with increasing 



 
(decreasing frequency) at the stagnation point, 
reverses by the end of the stagnation zone. T
he switching point, which happens at a specific 



 
location in the model results, 
happens at different 



 
locations in the measurements, depending 
on which two 



 
cases are considered. This difference in the location of switching might be due 
to t
he increasing vortex core size with increasing 



 
(due to vortex pairing); an effect that is not 
captured in the present model but need to be included in future development of the model.
 
Finally, f
igure 5.
22
 
demonstrates the general consistency between 
the experimental (top
 
plot
) and 
the 
model (bottom
 
plot
)
 
results regarding Reynolds number effect
. 
In both cases,
 
as 
Reynolds number increases
,
 
the fluctuati
ng
 
pressure 
level 
decreases
. Based on the discussion of 
the model results in section 3.2, the overal
l decrease in the normalized 
RMS
 
pressure fluctuation 
with Reynolds number is predominantly due to the increase of the normalization scale (the jet 
dynamic pressure) while the level of pressure fluctuation remaining invariant. Of course in the real 
jet it 




 
without affecting the 
strength of the vortices, and hence the wall
-
pressure fluctuation. However, a more realistic 
interpretation of the clue from the model results is that perha
ps the strength of the jet vortices 
increases with 



 

hypothesis: the vortex strength (vorticity/circulation) is expected to increase in proportion to the 
shear in th
e separating shear layer, which is expected to scale as the viscous shear stress in the 
separating boundary layer. The latter is expected to be proportional to 






, based on laminar 
boundary layer theory (Al
-
Aweni [7] showed that the boundary lay
er at the jet exit is laminar over 
the Reynolds number investigated). In contrast, the dynamic pressure should increase as 




. 
Thus, if the wall pressure increases in proportion to the square of vorticity/circulation (based on 
 
130
 
 
the pressure source 
term in equation 1.7) of the jet vortices, the 
normalized
 
pressure would decay 
as 





. This could explain the decay of the normalized pressure fluctuation with Reynolds 
number.
 
 
 
 
 
 
 
 
 
 
 
131
 
 
 
 
Figure 5.
2
2
 
Comparison of the effect of varying 



 
on 
RMS
 
wall
-
pressure fluctuation 
between the experimental (top) and the model (bottom) results 
at 






 
(experiment) and 





 
(model)
. The broken green lines outline the end of the stagnation zone
 
 
 
 
132
 
 
CHAPTER 
6: Conclusion
s
 
and Recommendation
s
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
The 
present 
investigation 
is focused on studying the unsteady surface pressure fluctuation 
in jets impinging normally on a flat wall. The
 
study 
is 
divided into two parts
:
 
t
he first part is 
concerned with 
developing 
a
 
simple 
physics
-
based 
mathem
atical model
 
of the impinging
-
jet wall 
pressure, and the second part involves measurements of the unsteady wall pressure in an existing 
impinging jet facility. 
 
The intent of developing the mathematical model is twofold: 
 
(I)
 
as a first step in a multi
-
step pr
ocess of developing a high
-
fidelity, efficient model that may 
be used as a design tool for predicting wall
-
pressure fluctuation 
for problems involving
 
flow
-
induced noise and vibration 
by
 
impinging jets; 
 
(II)
 
to utilize the model for understanding the connectio
n between the characteristics of the jet 
vortex structures and the surface pressure
 
by varying the main jet and vortex parameters 
one at a 
time, and
 
investigating the influence of this variation on the wall
-
pressure 
characteristics.
 
As the 
first
-
step
 
in th
e development process, the present model is very simple, consisting 
of an array of potential, point
-
vortex rings that are advected under their own influence and that of 
their image vortices (due to the presence of the impingement wall), in addition to a st
eady 
advection field (emulating the mean
-
jet flow) consisting of potential stagnation point flow. This 

at each time instant as the vortex rings advect periodica
lly towards the wall then radially outwards.
 
In comparison to the real jet, the present model does not account for several significant 
phenomena. These include the viscous core of real vortices, vortex pairing, vortex
-
boundary layer 
interaction, and the sp
ecific mean advection field of the jet flow. As such, the model is not expected 
 
133
 
 
to be useful, in quantitative or qualitative sense, in characterizing the pressure fluctuations outside 
the stagnation zone of the impinging jet, where vortex
-
wall interaction 
is known from literature to 
play a significant role in wall
-
pressure generation. However, the model results are expected to be 
qualitatively consistent with the characteristics of the wall
-
pressure fluctuation within the 
stagnation zone. Establishing this 
point, which is done here using experimental data from the 
second part of the investigation, would provide the necessary initial confidence to continue the 
development of the model in the future.
 
The model was utilized to examine
 
the effects of changing 
th
e vortex
-
passing
 
frequency, 
jet 
Reynolds number, and the 
vortex 
circulation on the fluctuati
ng
 
wall
-
pressure 
in
 
the stagnation 
zone. 
The effect of varying the frequency effectively corresponded to varying the spacing between 
the jet exit and the impingemen
t plate (



). Overall, the model results revealed that the 
stagnation zone pressure fluctuation are either low
-
level, sinusoidal
-
like and rather symmetric for 







, or energetic, featuring strong positive and negative peaks, with the negative peak 
being 
more prominent, for 









influences remained fairly invariant with 



, local effe
cts increased with increasing 



 
because 
of the increased proximity of the vortices to the wall as they convect radially outwards. 
 
For 







, a
s the frequency increase
d (



 
decreased)
, 


 
was found to 
decrease 
due to 
the increasing
 

pac
king 

of vortices near the wall within the stagnation zone
. The 
corresponding decrease in the inter
-
vortex spacing caused the induced
 
flow
 
along the stagnation 
streamline (which is parallel to the vortex array)
 
by the individual vortices to overlap
,
 
reducing
 
the 
vortex
-
to
-
vortex 
velocity and pressure 
fluctuation
.
 
An opposing frequency (



) trend was found 
on locally produced fluctuation (







), where increased vortex packing/passage frequency 
 
134
 
 
resulted in increasing 


. This opposite
 
effect was connected to the broadening of the induced 
velocity peaks 
on the wall under
neath (which is normal, rather than parallel to the array) with the 
packing density of vortices.
 
 
When changing the vortex 
circulation
 
(

)
 
and the Reynolds number (



)
, the overall 
radial distribution of 



remained qualitatively the same. The main influence of increasing these 
two parameters was to either increase (


 
or decrease (



) the strength of the pressure fluctuation 
relative to the dynam
ic pressure of the jet.
 
 
In t
he second 
(experimental) 
part
 
of the investigation,
 
the unsteady 
wall pressure
 
was
 
measured at
 
two Reynolds number
s
 
(









;
 
based on exit jet velocity 




 
and 
the jet diameter 



)
 
for 
a jet at 
norm
al impingement incidence. 
The
 
pressure 
was measured 
using 
an 
array of eight microphones
 
over a radial domain
 
range 











 
for
 
three separation
 
distances
 
between the
 
jet 
exit and
 
the i
mpi
ngement wall
:
  












 
The result
s
 
yielded
 
radial distributions of
 


 
that were consistent, both in their shape as 
well as their trends with 



 
and 



, with the literature
. 
Within the stagnation zone, several 
observations showed good qualitative agreement with the mathematical m
odel results. These 
include, the wall
-
pressure time series features at 







 
versus 







, and the trends in 
the 


 
distribution with both 



 
and 



. On the other hand, it appeared that discrepancy in 
finer qualitative details be
tween the experimental and the model results may be accounted for by 
including a vortex viscous
-
core in the model.
 
 
Analysis of the experimental data alone, using
 
time series 
and power spectra
, 
suggest
ed
 
that the 
basic
 
wall
-
pressure generating mechanisms r
emain the same with increasing Reynolds 
number. 
However, the overall weakening of the level of pressure fluctuation with Reynolds 
number was 
inferred 
to be due to two reasons. First, the slower increase of the pressure source 
 
135
 
 
strength (square of vorticity/
vortex circulation) with 



 

pressure (










 
respectively).  
Second
,
 
the weakening of 
the vortex
-
wall interaction 
effectiveness in generating pressure in the wall jet zone
. This was hypothe
sized to
 
be related to the 
jet vortices breaking up and becoming irregular/turbulent with increasing Reynolds number
. This 
was implied from the appearance of small
-
scale random fluctuations in the pressure time series, 
and the irregularity of the negative 
pressure spikes at the higher Reynolds number (particularly at 





)
 
.
 
Another interesting observation from the experimental data was the fact that the strong 
negative pressure spikes in the wall
-
jet zone did not weaken with increased 



, notwithstan
ding 
that the overall pressure 
RMS
 
level decreased with 



. This was observed in both time series as 
well as in the wall
-
pressure probability density functions and skewness results. Since it is well 
understood in the literature that these spikes are pro
duced during vortex
-
wall interaction, and given 
that the passage frequency of vortices decreases with 



 
due to vortex pairing, it was concluded 
here that the decrease in 
RMS
 
is due to the reduction in frequency of the spikes. 
 
Overall, the results of t
he present investigation, in addition to providing some new insights 
into the connection between the unsteady wall pressure and the jet 
vortical structures
, establishes 
an encouraging first step towards developing a physics
-
based model of impinging jets wa
ll
-
pressure fluctuation
. However, 
as a first step, the present model is very simple and it
 
does not 
include
 
phenomena that are known to be important for wall
-
pressure generation:
 
vortex
-
vortex 
interaction, impingement
-
plate boundary layer, vortex
-
boundary 
layer interaction, viscous
-
core 

calculate the wall
 
pressure. 
Th
ese elements should 
be added in
 
future 
development of the
 
model
. The present study suggests that, as far as the 
stagnation zone is concerned, the n
ext two highest priority elements would be the inclusion of 
 
136
 
 
viscous
-
core vortices and the use of a more realistic advection field. Both of these items are 
expected to not only enhance the qualitative agreement with physical observations in the stagnation 
z
one, but to possibly also lead to reasonable quantitative comparisons.
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
137
 
 
 
 
 
 
 
 
 
 
APPENDIX
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
138
 
 
Appendix:
 
Uncertainty Calculation
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
-
 
Finding the uncertainty for the root mean square pressure  




[
(see 28)]
 
















 














 












 

















 
Where 

 
represent
s
 
the uncertainty and 

 
represent
s 
the
 
number of independent sample
s
 
and was 
found 
assuming 

 
to be equal to the number of vortices passing during the measurement period. 
Thus,
 























 
Where 

 
represent
s
 
the time 
period between t
he passage of successive vortices, and
 














 
 
 
 
139
 
 
2
-
 
Finding 
the 
uncertainty for 
the third order
 
moment (
s
kewness
)
,
 
assuming Gaussian 
variation
 
[(see 28)]
 
































































 


















 





















 












































 








































 





































































 





























 
































 
More general
ly
 
 









































 
Finding 










 
by a
ssum
ing
 
 
140
 
 

















 
































 
















 
Where 
 











































 
Finding 



















 
and 
assuming Gaussian variation
 
 






















 



















 

















































 


































 






















































 
 
 
141
 
 
3
-
 
Finding uncertainty for 
the fourth order moment
 
(
k
urtosis
)
, assuming Gaussian variation
 
and 
following
 
the same procedu
re for 
calculating the uncertainty of skewness
 
[(see 28)]
 
































































 






















 






















































 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
142
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
BIBLIOGRAPHY
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
143
 
 
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