University of Southern California Dissertations and Theses

Flow and thermal transport at porous interfaces

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Flow and thermal transport at porous interfaces

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Content FLOW AND THERMAL TRANSPORT AT POROUS INTERFACES
by
Shilpa Vijay
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirement for the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)
December 2023
Copyright 2023 Shilpa Vijay

Acknowledgements
This thesis represents the culmination of a lifetime of cultivating a curious mindset and thirst
for knowledge, with the guidance and support of many individuals. I am deeply grateful to
each and every person who has provided me with unwavering support. I apologize if I have
overlooked anyone in this acknowledgment, and offer my sincere thanks to all who have
helped me along the way.
I would like to express my sincere gratitude to my advisor, Dr. Mitul Luhar, for his
invaluable guidance and support throughout my academic journey. I first worked with him
in the summer after my junior year and was immediately drawn to his unbridled enthusiasm
and dedication to experimental fluid mechanics. His mentorship and encouragement inspired
me to pursue my PhD under his guidance. I am particularly grateful for his patience and
thoughtful guidance over the years. I thoroughly enjoyed all our weekly discussions, and his
positive attitude and cheerfulness always left me feeling motivated and enthusiastic about
my progress. I also had the privilege of taking AME 530b and serving as a TA for AME
309, where I witnessed firsthand how effective teaching practices can enhance the overall
learning experience. Dr. Luhar’s kindness and empathy towards his students have taught
me valuable lessons on how to be a supportive mentor. During the COVID-19 pandemic,
Dr. Luhar’s unwavering support was especially crucial during some challenging months. His
willingness to accommodate my constraints and prioritize my well-being has taught me that
being successful is not just about doing good work but also about being open-minded and
caring toward students, colleagues, and the broader community. I am truly grateful for his
mentorship and could not have asked for a better advisor.
ii

I am deeply indebted to my qualifying exam and defense committee members, namely
Dr. Satwinder Sadhal, Dr. Felipe De Barros, Dr. Paul D. Ronney, and Dr. Ivan BermejoMoreno, for their invaluable insights and critical advice on my work. Additionally, I am
grateful for the valuable inputs provided by Sasindu Pinto and Dr. Louis Cattafesta from the
Illinois Institute of Technology for the final part of my thesis. I also enjoyed taking courses
with Dr. Sadhal (AME 515) and Dr. Bermejo-Moreno (AME 511), which considerably
strengthened my foundational knowledge. I am especially thankful to Dr. Ronney for the
career advice he gave me, which inspired me to consider academia as a potential career path.
To other faculty members at USC AME that have significantly influenced my experience
here: Special thanks must go to Dr. Geoff Spedding for introducing me to the wonderful
world of flight while TAing AME 105 for him and for honing effective presentation skills. He
was also instrumental in shaping my philosophy toward research by teaching me how to be a
neutral observer of my work. I am also extremely thankful to Dr. Alejandra Uranga for her
unwavering support when I was initially struggling in her class, AME 535a. Her kindness
and encouragement over the years have been invaluable to me. I am also incredibly fortunate
to have taken AME 524 with Dr. Firdaus Udwadia. Dr. Udwadia introduced me to the book
’Men of Mathematics’ by Eric Temple Bell which opened my eyes to the personal journeys of
scientists and allowed me to place their discoveries in a historical and contextual framework.
I would also like to extend many thanks to the wonderful staff members here at the AME
department - Melissa, Irice, Natalie, Amanda, Chrissy, Gilberto, Ellecia, and Silvana - for
their prompt communication and administrative support. I’d also like to recognize the
efforts of Seth Weiman from the Graduate Machine Shop for all his help in manufacturing
my experimental setups.
I want to express my heartfelt appreciation for the F3 fluids community here at USC
AME - fondly also known as the Fluids Supergroup. Through our weekly group meetings,
many lunches, and many more happy hours, I have had the opportunity to broaden my
research horizon, bounce off ideas, and seek camaraderie and support. I would especially
iii

like to acknowledge Chris for this quote: ’Enjoy and appreciate that you have the room to
feel stupid’. It has been a privilege to share that room with Chris, both in a literal and a
figurative sense. Many such interactions with her have kept me grounded and motivated. I
greatly appreciated being able to discuss anything and everything with her while being officemates for most of our PhDs. To my fellow graduate students - Christoph, Saakar, Yohanna,
Emma, Trystan, Arturo, Michael, Andrew, Mark, Idan, Chase, Madeleine, Morgan, Jocelyn,
JP, Ben, James, and Vamsi - I want to express my gratitude for being an amazing support
system. Throughout the years, I was also incredibly fortunate to have worked with a fantastic bunch of high-school students through SHINE (Sonal, Madeline, Asa, Noah, Jacklyn,
Sarah, Brody), undergraduates (Stara, Bryce, Michelle, Julia, Gerald), and Masters students
(Yixuan, Taylor, Bo-wei, Upasana) that contributed tangibly to this thesis. I have learned
a lot from you.
My stay here at USC was made especially enjoyable by the members of the Vidushak
Improv Comedy Group - Vishnu, Sanmukh, Hiteshi, Pratyusha, Rimita, Anamika (and many
others). I would look forward to Friday evenings for fun and laughter. More importantly, it
pushed me out of my comfort zone every week and taught me ’Yes-And’ - a concept that was
instrumental in my research and encouraged me to keep exploring new ideas. I also thank
the Trojan Cricket Club for allowing me to participate in their leagues. The game of (test)
cricket has provided me with role models who instilled in me the importance of a great work
ethic and the motivation to push myself even after a tough day.
I am immensely grateful to the vibrant and diverse city of Los Angeles. This remarkable
city is characterized by dynamic energy and cultural richness, which has helped shape not
only my academic journey but also my personal growth. I extend my heartfelt appreciation
to Los Angeles for being a catalyst in my life. The fact that I ended up in Los Angeles
was completely unexpected and happened by chance, yet it marked a pivotal chapter in
my life as it became the first place I called home away from my family. In navigating the
challenges, particularly with housing, I learned valuable lessons about the profound impact
iv

of culture and environment on one’s success. These experiences underscored the holistic
nature of success and its dependence on the confluence of various factors. Over the years,
I have had the privilege of encountering a diverse array of individuals, whether casually in
a local coffee shop or within the academic community. These interactions have enriched
my perspective and contributed significantly to my overall growth. The city’s stimulating
environment, coupled with its year-round (almost) perfect weather, has made it an ideal
backdrop for my academic studies. I am truly thankful for the influence of this city on my
journey, and I carry with me the lessons learned and memories forged as I move forward in
my academic and personal endeavors.
I am grateful to my mentors and friends at my undergraduate institution, the College
of Engineering Pune (COEP). They provided me with the support and motivation I needed
to apply for and be accepted into the S.N Bose Scholars Program in my junior year. I am
especially indebted to Indo-US Science and Technology Forum (IUSSTF) for providing me
with this opportunity and for the 2015 cohort of scholars for their friendships during my
summer at USC and beyond. It laid a solid foundation for me to apply and succeed in the
PhD program at USC. Special mention here to M. Prakash Academy during my high school
years for fostering my curiosity and abstract thinking abilities. I originally joined them with
the goal of doing well in the entrance exams for college, but it was here that I learned to go
beyond the textbook and solve problems just for the sake of curiosity. I especially cherish
the friendships with Pooja, Neeraja, and Vismaya from that time. Conversations with them
have provided me with grounded perspectives throughout my PhD.
This endeavor would not have been possible without my family. Special thanks to my
father for allowing me to dream big and for his innumerable sacrifices that have led me to be
here. I am especially inspired by my mother, who pursued her PhD in her 40s, for showing
me that anything is possible with adequate strength and resilience. I thank my brother for
his steady encouragement, and for teaching me to think outside of the box; and to all my
extended family for their support. I cannot adequately acknowledge their contribution, but
v

suffice it to say that they have shaped me into who I am. I would also like to extend my
sincere thanks to the Kaul family for their boundless enthusiasm and love, and for embracing
my dreams as their own.
Finally, I extend my heartfelt gratitude to my husband, Ishan, for his unyielding support,
encouragement, and unwavering belief in my abilities. His presence has brought genuine joy
into my life every day, and his intellectual and emotional support has been instrumental in
shaping this journey. I thank him for being my pillar of strength and for walking with me
together on this remarkable journey.
Financial support from USC (Viterbi Graduate Fellowship, Teaching Assistantships),
Air Force Office of Scientific Research award FA9550-19-1-7027, and the National Science
Foundation grant no. 1943105 are gratefully acknowledged.
vi

Table of Contents
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
1 Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Turbulent flow over complex surfaces . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Porous substrates for passive turbulence control . . . . . . . . . . . . . . . . 2
1.2.1 Permeability of a porous medium . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Passive scalar transport . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Drag reduction over porous surfaces . . . . . . . . . . . . . . . . . . . 6
1.3 Contribution and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Chapter 2: Convective heat transfer in partially-porous channels . . . . . . . . . . 10
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Porous medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.3 Heat transfer measurements . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.4 PIV measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.1 Mean flow statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.2 Heat transfer characteristics . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.3 Pumping power efficiency . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Chapter 3: Thermo-physical properties of 3D-printed porous materials . . . . . . 29
3.1 Moving towards designed porous media . . . . . . . . . . . . . . . . . . . . . 29
3.2 Porous lattice design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.1 Constraints on achievable geometry . . . . . . . . . . . . . . . . . . . 34
3.3 Permeability characterization . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.1 Model to predict principle components of permeability . . . . . . . . 37
vii

3.3.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.3 ANSYS Fluent setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 Thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.3 ANSYS Fluent Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 Chapter 4: Passive flow control with anisotropic porous walls . . . . . . . . . . . . 59
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1.1 Drag reduction mechanism . . . . . . . . . . . . . . . . . . . . . . . . 60
4.1.2 Previous experiments with anisotropic porous substrates . . . . . . . 64
4.1.3 Contribution of this study . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.1 Porous substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.3 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3.1 Smooth wall reference measurement . . . . . . . . . . . . . . . . . . . 72
4.3.2 Porous wall friction factor measurements . . . . . . . . . . . . . . . . 73
4.3.3 Observations over isotropic substrates . . . . . . . . . . . . . . . . . . 77
4.3.4 Observations over anisotropic substrates . . . . . . . . . . . . . . . . 77
4.3.5 Departure from hydraulically smooth regime . . . . . . . . . . . . . . 79
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5 Chapter 5: Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
A Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
A.1 Constraints on achievable geometry . . . . . . . . . . . . . . . . . . . . . . . 95
A.2 Model to predict principle components of permeability . . . . . . . . . . . . 97
viii

List of Tables
2.1 Thermo-physical properties of the metal foams used in the experiments . . . 14
2.2 Average friction velocities for the two foams as a function of Reb . . . . . . . 23
3.1 Here are the details of the parameter space based on the cubic lattice structure
proposed in 3.2. Note that here ’H’ corresponds to the highest chosen nominal
rod spacing (s = 3) mm, ’M’ corresponds to s = 2 mm, ’L’ corresponds to
s = 1.5 mm and ’T’ corresponds to (s = 0.8) mm. . . . . . . . . . . . . . . 33
3.2 This table lists measurements of pore sizes and rod sizes of 3D printed samples.
Note that the uncertainties reported here are ±2σ to provide a conservative
estimate in the variation of dimensions . . . . . . . . . . . . . . . . . . . . . 45
3.3 Permeability predictions from ANSYS, model and experiment are compared
here. The range for the model predictions are derived from (d, sx, sy, sz) as
listed Table 3.2. The range permeability estimates are obtained from analysis
in 3.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.1 Parameter space used for the experiments. In each case, the first row indicates
the nominal sizes, while the values in parentheses show the measured sizes.
The range for the permeability values represents the maximum and minimum
values predicted by the (3.20) based on the measured sizes. . . . . . . . . . . 67
ix

4.2 This table lists values of a and b as estimated from the linear curve fitting
process for each tile. Note that as Reb increases, f asymptotes to C1 =
2aHf /ρ. Reˆ
b = 1.5fs is evaluated as the Reynolds number at which flow has
transitioned from the hydraulically smooth regime. . . . . . . . . . . . . . . 81
x

List of Figures
2.1 Photographs showing 10 ppi and 40 ppi foams used in the experiment (from
left to right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Schematic of the experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Zoomed-in schematic of the test section . . . . . . . . . . . . . . . . . . . . . 19
2.4 Mean velocity profiles for (a) 10 ppi, (b) 40 ppi . . . . . . . . . . . . . . . . 22
2.5 Mean Reynolds shear stress profiles for (a) 10 ppi, (b) 40 ppi . . . . . . . . . 23
2.6 Friction factor vs Bulk Reynolds Number . . . . . . . . . . . . . . . . . . . . 24
2.7 Nusselt Number vs Bulk Reynolds Number plotted for both 10 ppi and 40
ppi foams. The smooth wall data is obtained from Boomsma, Poulikakos, and
Zwick [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.8 Stanton Number vs Bulk Reynolds Number . . . . . . . . . . . . . . . . . . 26
2.9 Pumping power vs thermal resistance . . . . . . . . . . . . . . . . . . . . . . 27
3.1 CAD renderings of anisotropic lattice geometry. Here we see (a) isometric
view for HHL lattice. (b) the view in direction with higher planar porosity
(HH) and (c) the view in the direction with smaller planar porosity (HL) . . 32
3.2 Photographs of 3D printed cubes (side = 25 mm) of anisotropic porous material 32
3.3 The portion highlighted in red shows a unit cell identified from the lattie with
rod size d and rod-spacings sx, sy, and sz in the three directions . . . . . . . 34
3.4 Figure here shows the unit cell divided into two zones . . . . . . . . . . . . . 38
xi

3.5 Achievable limits for (a) directional porosities (b)permeabilities for 3D printed
porous geometries as proposed in while considering the printing constraint
(sx, sy, sz) ∈ [1, smax] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6 Schematic for measurement of permeability . . . . . . . . . . . . . . . . . . . 41
3.7 Comparison between model predictions for permeability and unit cell Stokes
flow simulations in ANSYS fluent for (a)x-direction, and (b) y-direction for
anisotropic porous lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.8 Figure above shows the comparison of pressure gradient with superficial (bulkaveraged) velocity through the sample for isotropic cases (MMM,LLL) and
anisotropic case (MML). Here MM and ML represent the rod spacings in the
direction perpendicular to the flow. . . . . . . . . . . . . . . . . . . . . . . . 43
3.9 Linear fit estimates and 95% prediction interval for the ML face of MML . . 44
3.10 Metal foam unit cell modeled as a (a) 2D Hexagonal unit cell with blobs at
the intersection [3], (b) 3D Octet lattice [16] . . . . . . . . . . . . . . . . . . 49
3.11 Schematic of the thermal conductivity experiment . . . . . . . . . . . . . . . 49
3.12 Variation of normalized saturated thermal conductivity Kef f /Ks with volumetric porosity ϵ (a) (0 < ϵ < 1), (b)zoomed-in to show 0.6 < ϵ < 1 . . . . . 53
3.13 Above figures show thermal conductivity estimates with volumetric porosity by varying spacing while keeping planar porosity in the direction of heat
conduction constant z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.1 Schematic side view and top view of the experimental setup (not to scale) . . 68
4.2 Figure above shows the variation of pressure gradient dP/dx over a smooth
wall in between different pressure ports for two values of Reb . . . . . . . . . 70
xii

4.3 (a) Variation of pressure gradient with bulk velocity for a smooth wall. We see
a laminar flow region until Ub ≈ 0.2 m/s as shown by the linear fit after which
the flow transitions to turbulence. b) Variation of friction factor with bulk
Reynolds number compared with empirical predictions for a smooth laminar
duct f
∗ and correlations developed by Cheng [12] . . . . . . . . . . . . . . . 73
4.4 (a) shows the variation of pressure gradient with bulk velocity for HHH. (b)
demonstrates the process of linear fitting to estimate the value of friction factor. 74
4.5 (a) Pressure gradient measurements for HHH and smooth wall (SW) reference.
(b) Variation of friction factor with Reynolds number for HHH and SW. Note
that the shaded region indicates 95% confidence interval in predictions. . . 75
4.6 Comparison of friction factor estimates for different isotropic and anisotropic
materials. The curves here indicate quadratic fits obtained via equation 4.10
with the shaded region indicating 95% prediction interval . . . . . . . . . . 76
4.7 Variation of f with √κyy
+. The symbols and colors used in this figure are
consistent with those in Fig. 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . 79
xiii

Nomenclature
Acronyms
ppi pores per inch
Symbols
α thermal diffusivity
δν = ν/uτ viscous length scale
V˙ volumetric flow rate
W˙ pumping power
ϵ volumetric porosity
ϵx planar porosity normal to the streamwise direction
ϵy planar porosity normal to the wallnormal direction
ϵz planar porosity normal to the spanwise direction
κxx streamwise permeability
κyy wall-normal permeability
κzz spanwise permeability
κ second order permeability tensor
u velocity vector
µ dynamic viscosity of the fluid
ν kinematic viscosity of the fluid
ϕxy = κxx/κyy streamwise to wall normal permeability anisotropy ratio
ρ density of the fluid
τ shear stress
τp shear stress at the porous wall
c Forchheimmer correction factor
Cf skin friction factor
Cp specific heat capacity of the fluid
d rod-size
Dx, Dy, Dz hydraulic diamter in streamwise, wallnormal and spanwise direction
xiv

Da Darcy Number
f Darcy-Weisbach friction factor
Gr Grashof Number
h convective heat transfer coefficient
Hf height of the fluid region
Hp height of the porous region
L length of the test section
l
+
t normalized spanwise slip length
l
+
U normalized streamwise slip length
Nu Nusselt Number
P pressure
P r Prandtl Number
Q thermal power output
q
′′ heat flux
Rth thermal resistance
Reb bulk-averaged Reynolds number
sx, sy, sz rod-spacings in streamwise, wall-normal and spanwise directions
St Stanton Number
T temperature
t time
Tint temperature estimated at the porous medium interface
Tin temperature of the fluid at the inlet
Tout temperature of the fluid at the outlet
Tpl average temperature of the plate
u, v, w, streamwise, wall normal, spanwise velocity
Ub bulk-averaged velocity
uτ friction velocity
W width of the test section
x, y, z streamwise, wall-normal, spanwise coordinate
Kf thermal conductivity of fluid
Ks thermal conductivity of solid
Kef f fluid-saturated thermal conductivity of the porous medium
Subscripts
() b bulk averaged value
xv

() f in reference to fluid region
() s in reference to solid region
() ef f effective value
Superscript
() ′ fluctuating component
() + normalized by viscous length scale ν/uτ
() p
evaluated at the porous wall
() s
evaluated at the smooth wall
xvi

Abstract
This thesis aims to systematically investigate the impact of porous substrates on fluid flow
and heat transfer with potential applications in mitigating drag and enhancing thermal
transport. For porous and patterned surfaces, porosity and surface microstructure hav˚ae
been shown to modify the structure and dynamics of near-wall turbulence. It is important
to understand how the microstructure of these surfaces influences momentum and scalar
transport to facilitate the design of passive flow control strategies. In addition to leveraging
additive manufacturing to design porous substrates that have the potential to enhance heat
transfer or reduce drag in channel flow, this effort seeks to improve our understanding of
how porous medium microstructure affects flow and thermal dispersion.
The effects of permeability on turbulent heat transfer across a porous interface were
experimentally investigated in a partially-porous channel flow setup with commercially available aluminum foams. Temperature measurements at various locations in the channel were
used to characterize heat exchanger performance for a range of bulk Reynolds numbers
(Reb ≈ 800−2500). Particle Image Velocimetry measurements at a subset of these Reynolds
numbers were used to characterize the shear stress at the interface and gain insight into
the flow structure. Increasing permeability of the aluminum foams significantly alters the
flow structure and dynamics, resulting in the emergence of large vortex structures associated
with Kelvin-Helmholtz (K-H) instability. These structures enhance interfacial thermal dispersion. However, this increase comes at the cost of increased pumping power requirements:
an increase in permeability also leads to an increase in friction.
Previous numerical simulations have demonstrated drag reduction over streamwise prefxvii

erential substrates, which yield larger effective slip lengths for the streamwise mean flow
compared to the turbulent cross-flows. A deterioration in performance is usually observed
when the normalized wall normal permeability (√κyy
+) is higher than 0.4, likely due to
the presence of large-scale motions associated with the K–H instability. To determine if the
trends observed in the numerical simulations are applicable to physically realizable materials,
a family of anisotropic periodic lattices was created using 3D printing. This allowed rod size
and spacing in different directions to achieve different ratios of streamwise, wall-normal, and
spanwise bulk permeabilities (κxx, κyy, κzz). We investigated the thermophysical properties
of these 3D-printed materials to establish a connection between the microstructure and the
saturated thermal conductivity and permeability. Our results show that empirical models
for stochastic foams or isotropic lattices are inadequate for predicting the thermophysical
properties of anisotropic porous lattices. To address this, we developed phenomenological
models to predict the principal components of the permeability tensor and effective thermal conductivity. Comparisons between the predictions and results from experiments and
ANSYS Fluent simulations show that small variations in rod spacing and size due to manufacturing tolerances can lead to significant discrepancies between measured and predicted
thermophysical properties, highlighting the limitations of physically realizable materials.
Finally, we experimentally investigated the effect of anisotropic permeability on the
drag response for the 3D-printed porous substrates. The porous materials were mounted in
a benchtop water channel. Pressure drop measurements were taken in the fully developed
region of the flow to characterize drag as a function of bulk permeability for materials with
xviii

Chapter 1: Introduction
1.1 Turbulent flow over complex surfaces
Turbulent flows in engineering often involve complex surfaces with patterns, roughness, and
porosity that modify the structure and dynamics of near-wall turbulence. Nature provides us
with excellent examples of surfaces that alter near-wall turbulence resulting in drag reduction
and turbulent mixing. For example, the dermal denticles found on shark skin are a key
feature that contributes to drag reduction is a quasi-preiodic stucture of troughs and valleys
which are aligned in the streamwise direction. Further experiments with riblets, inspired by
shark denticles, have demonstrated up to 10% drag reduction, while other methods, such
as superhydrophobic coatings and anisotropic permeable substrates, also show promise in
reducing drag [63, 19]. Similar to riblets, the experiments using seal fur [26] - which also
displays a high streamwise and low transverse compliance - demonstrated a drag reduction
of about 12%.
Similarly, the presence of aquatic vegetation (which can be considered a rough, porous
substrate) significantly alters the mean and turbulent flow field over several length scales
and assumes a critical role in the uptake of nutrients within various ecosystems [43, 11].
The presence of vegetation interacts with the flow at various length scales. For example, the
uptake of nutrients by an individual blade depends on the boundary layer on that blade.
However, the displacement of larvae and pollen from a seagrass meadow or kelp forest depends
on the flow structure at the scale of the community of plants, called the meadow or canopy.
The examples listed above highlight how complex surfaces can yield favorable outcomes
in nature such as reduction in turbulent drag or an increase in turbulent mixing. Similar
engineered patterned surfaces can also be used to increase turbulent mixing and reduce drag,
which could be beneficial for various heat transfer applications ranging from solar heaters
1

to electronics cooling ([34],[28][27]). Indeed, an extensive range of technologies involve flow
and heat transfer over complex surfaces. Examples include flows in fuel cells, packed bed
energy storage systems, battery thermal management, cooling in the gas turbine, and metal
foam heat sinks for electronic cooling.
However, complex surfaces interacting with fluid flows can also have adverse impacts
on engineering systems. To illustrate, bio-fouling on ship hulls can lead to a 10% increase in
drag and a corresponding performance reduction. Similarly, within a compact heat exchanger
constructed from a porous metal foam, the blockage of pores can trigger a substantial increase
in pressure drop, potentially reaching orders of magnitude higher than that of clean foams.
Thus, it becomes inperative to understand how the attributes of these surfaces influence
near-wall turbulence dynamics to facilitate the design of innovative passive flow control
strategies.
In this chapter, we will first discuss permeability as a significant intrinsic characteristic
of porous media. We will then provide a brief overview of its effects on porous and patterned
surfaces in terms of reducing drag and increasing passive thermal transport. Finally, we will
present open questions and discuss the contributions of this thesis.
1.2 Porous substrates for passive turbulence control
1.2.1 Permeability of a porous medium
A porous medium is typically characterized by its microstructure properties, porosity ϵ, and a
second order permeability tensor κ. The volumetric porosity represents the fraction of empty
volume to the total volume of the material, and the permeability tensor κ characterizes the
resistance of flow through the material. The permeability was first used by Henry Darcy in
1856, who postulated an empirical relationship, known as the Darcy’s law to characterize
viscous flow in porous media, as follows:
2

∇P = −µκ−1U. (1.1)
Here ∇P is the pressure gradient for a fluid of dynamic viscosity µ with a velocity vector
velocity of U = (u, v, w) through a porous medium of permeability κ. The pressure gradient
across a porous medium is directly proportional to the superficial velocity and dynamic
viscosity and inversely proportional to the permeability of the medium. This relationship
particularly holds for Stokes flow without inertial effects. κ is a second-order tensor expressed
as below:
κ =






κxx κxy κxz
κxy κyy κyz
κxz κyz κzz






. (1.2)
The diagonal components (κxx, κyy, κzz ) relate components of the pressure gradient with
flow in the same direction, and the off-diagonal terms relate the velocities to the pressure
gradient in the respective perpendicular directions. If the diagonal components are equal,
κxx = κyy = κzz, and off-diagonal components are zero, then the permeability is said to
be isotropic. In this case, the permeability can be expressed as a scalar. If the diagonal
components are not equal, then the permeability is said to be anisotropic. Permeability
values are typically a function of porosity ϵ, pore size, and ligament size and vary with
porous microstructure.
Flow through a porous medium can be characterized by the Darcy-Brinkman equation,
where an extra Laplacian term (Brinkman term) is added to the classical Darcy equation
1
ρ
∇P = −νκ−1u + ˜ν∇2u. (1.3)
Modifying the surface microstructure of porous walls has proven to be a promising
method for multifunctional passive flow control. For example, high porosity stochastic
isotropic foams made of metals such as Copper and Aluminum have received significant
3

interest as they offer potential as a lightweight structure for thermal management, energy
absorption and acoustic control. Also known as open-celled metal foams, they offer a high
specific surface area ratio and a tortous path for the fluid and have been shown to enhance
convective heat transfer in applications such as cryogenic heat exchangers, and compact
heat sinks for electronics cooling. Previous experimental and numerical efforts [6, 57, 9, 15]
show that the permeability of such foams significantly alters the structure and dynamics of
turbulence and can lead to the formation of Kelvin-Helmholtz (K-H) like rollers which can
enhance thermal exchange across the interface.
1.2.2 Passive scalar transport
The mixing of a scalar in wall-bounded turbulence has implications for a variety of engineering and environmental applications. Heat transfer is enhanced in turbulent flow compared
to laminar flow, resulting in higher surface temperatures for combustion technologies, jet
turbine blades downstream of combustion events, and re-entry vehicles. Additionally, the
mixing of particulates is of interest in environmental flows, as turbulence can transport pollutants across continents and oceans. A passive scalar is one that does not affect the velocity
field when included in the flow. For example. the inclusion of heat can be considered passive
if the heating is mild enough that the gravitational forces are insignificant compared to any
other forces.
A passive scalar obeys the advection diffusion equation. For the case where temperature
(T) is a passive scalar, the governing equation becomes
∂T
∂t + ∇ · (uT) = ∇ · (α∇T), (1.4)
where α is the thermal diffusivity. The thermal diffusivity α is related to the thermal
conductivity K of the medium as α =
K
ρCp
where Cp is the specific heat capacity of the
fluid. For a fluid-saturated porous medium, thermal conductivity is typically reported as
Kef f which considers the combined effect of solid and fluid medium.
4

Passive scalar transport over a complex wall is of great engineering interest because
the presence of a wall leads to a considerable enhancement of momentum, mass, and scalar
transport. However, its performance must be weighed against the energy required to operate
the system, which typically comes from minimizing drag, and hence fuel cost to achieve the
desired output.
Recently, the impact of turbulence on thermal transport at porous interfaces has been
studied. It has been demonstrated that wall permeability alters the structure and dynamics
of turbulence [6]. Permeability length scales normalized with viscous units √
κ
+ =
√
κuτ/ν
have been shown to be an important length scale in characterizing turbulence over porous
surfaces. Throughout this document, length scales presented using (+) have been made
dimensionless using the viscous length scale given by δν = ν/uτ . Here ν is the kinematic
viscosity of the fluid and uτ is the friction velocity which is determined from the shear stress
τ measured at the wall as uτ =
p
τ/ρ. The friction Reynolds number is given as the ratio
between the characteristic length scale h (typically channel half-height or boundary layer
thickness) and the viscous length scale.
Effectively, √
κ
+
can be interpreted as the ratio of the effective pore diameter to the typical length scale of the viscous sublayers over the individual wall elements. Wall permeability
is expected to have only an influence on the turbulent flow when this ratio is sufficiently
large. For small √
κ
+
, the discontinuous viscous sublayers over the wall elements are relatively thick and they coalesce and form a continuous classical viscous sublayer covering the
complete wall.
Experiments by Suga and colleagues [57] indicate that porous surfaces enhance the
onset of turbulence. For the same porosity, the transition to turbulence occurs at a lower
Reynolds number as permeability increases. Turbulence intensities are generally augmented
over the porous medium. Given this enhancement of turbulence intensities (and potentially,
mixing) there has been a sustained effort to comprehend passive thermal exchange over a
porous interface. Numerical solutions suggest an increase in turbulent kinetic energy at
5

the porous-fluid interface compared to the solid wall, which is associated with the presence
of large-scale vortical structures observed in several studies [6, 44]. Chandesris et al., [9]
extended this study to include passive turbulent heat transfer. Kuwata et al., [31] conducted
direct numerical simulation of conjugate heat transfer of porous wall in a duct flow for
Reynolds number 3500. Large-scale streamwise perturbation occurs over the porous wall
due to the K-H instability, intensifying turbulence.
The permeability of porous materials can impact the saturated effective thermal conductivity, known as Kef f . This can affect the spread of a passive scalar, as seen in Equation
(1.4). The Prandtl number, which measures the ratio of kinematic viscosity to thermal diffusivity (P r = ν/α), is also influenced by changes in the porous microstructure. A Prandtl
number of 1 indicates that momentum and heat diffuse at the same rate, resulting in similar scalar and velocity fields. Changes in the porous microstructure can also directly affect
thermal diffusion at the interface.
1.2.3 Drag reduction over porous surfaces
Typically, an enhancement in passive scalar transport such as thermal disperson comes at the
cost of increasing drag and pumping power requirements for the working fluid. Hence, it is of
engineering interest to design porous interfaces that not only offer significant enhancements
for heat transfer, but also mitigate drag and hence pumping power requirements. As with
passive scalar transport, momentum transport over a porous interface is also closely linked
with the permeabililty and microstructure of the porous medium.
Previous numerical work by Abderrahaman-Elena and Garc´ıa-Mayoral [1] extended the
virtual origin concept of Luchin, Manzo, and Pozzi [39] to anisotropic porous substrates
using the Darcy-Brinkman equations. In this analysis, they considered materials for which
the permeability matrix took the following form κ = diag(κxx, κyy, κzz) and the off-diagonal
terms as zero. They showed that for flow conditions in which the turbulence length scales
were much larger than the size of porous-scale structure the streamwise and spanwise slip
6

lengths can be approximated as l
+
U ∝
√
κxx
+
and l
+
t ∝
√
κzz
+
. Further DNS simulations
by Gomez-de-Segura and Garcia-Mayoral [22] found that such substrates reduced friction
drag by ≈ 25%. Linear stability analyses suggest that there is a limit to the drag-reducing
capabilities of permeable surfaces, with a deterioration in performance typically observed
when the normalized wall-normal permeability (√κyy
+) is greater than 0.4. This is thought
to be due to the onset of a Kelvin-Helmholtz type instability, which limits the maximum
achievable drag reduction [1, 22].
1.3 Contribution and outline
Due to its applications and benefits, momentum and passive scalar anisotropic porous walls
have received significant attention recently. However, most previous studies focused heat
exchanger characterization in channels completely occluded with porous foams [4, 38, 29,
55]. Previous numerical and modeling efforts have shown that porous walls change the
structure and dynamics of turbulence near the surface offer possibilities for developing flow
control strategies. Although previous numerical simulations demonstrate the possibility of
designing multifunctional porous substrates, these observations are yet to be observed with
physically realizable materials.
There are several challenges in developing porous walls for thermal management and drag
reduction. Some of these challenges are outlined below.:
• It is unclear how the permeability of the porous medium affects thermal transport
at the interface, and specifically how it influences global heat transfer measures when
using a porous wall as a heat exchanger.
• The impact of anisotropic porous microstructure on permeability and thermal diffusivity remains uncertain, and the validity of established relationships for isotropic foams
has not been verified.
• The drag response of porous materials with anisotropic bulk permeabilities has not
been characterized.
7

Keeping these challenges in mind, this thesis aims to develop tunable porous media
for multifunctional passive flow control, specifically for heat transfer enhancement and drag
reduction.
In Chapter 2, we investigate the influence of permeability on convective heat transfer
over a porous wall with isotropic permeability. We first analyze the performance of a partially
porous channel flow as a heat exchanger and compare it to that of a smooth wall channel flow.
Additionally, we use Particle Image Velocimetry (PIV) to visualize the flow near the porous
walls and observe the effect of permeability in intensifying near-wall turbulence. We also
consider the implications of increasing wall permeability for achieving higher heat transfer
and this motivates the design of tailored porous microstructures to achieve multifunctional
turbulence control by also reducing drag.
In Chapter 3, we consider the design of porous media with anisotropic permeabilities
using stereolithographic 3D printing. Specifically, we propose a cubic lattice with a tunable
microstructure. We utilize a combination of experiments and ANSYS Fluent simulations to
characterize diagonal elements of the permeability tensor κ and saturated effective thermal
conductivity Kef f . We observe that previous models for isotrpic foams are inadequate in
predicting κ and Kef f for custom-designed anisotropic cubes. We propose simple phenomological models and characterize the variation porous microstructure. We also show that
physica limitations in manufacturing can cause up to an order of magnitude disparity in the
prediction of permeability.
Chapter 4, describes the experimental effort to characterize the drag response of
anisotropic porous walls. We utilize geometries proposed in Chapter 3 to manufacture
porous walls with κxx
κyy
∈ [0.15, 6.8] and characterize pressure drop across a fully developed
section in channel flow arrangement. We demonstrate that streamwise preferential substrates κxx
κyy
> 1 show the lowest relative increase in drag. Furthermore, we demonstrate that
√κyy
+ plays an important role in drag increase. All porous substrates tested show departure
from the hydraulically smooth laminar flow regime at a Reynolds number lower than that
8

for a reference smooth wall, which is indicative of an earlier transition to turbulence. These
constitute first set of friction factor measurements over a wide range of Reynolds numbers
in the transition to turbulence regime.
9

Chapter 2: Convective heat transfer in partially-porous
channels
2.1 Introduction
Heat transfer and fluid flow at the interface between a porous medium and an unobstructed
fluid is important in several applications, including electronics cooling [34], solar heaters [27],
and packed-bed heat exchangers [5, 67, 55]. These applications, coupled with advances in
the development of structured porous materials with desirable thermophysical properties,
have led to an increasing interest in understanding thermal exchange at porous interfaces in
recent years [9, 45].
Thermal transport at a porous interface depends on not only the thermophysical properties of the porous medium but also its thickness, placement, configuration, and interactions
with the adjacent fluid. Integral measures of performance in a convective heat exchanging
device include the Nusselt number and the Stanton number [2, 40]:
Nu =
hH
K
=
q
′′
∆T
H
K
; St =
h
ρCpUb
=
q
′′
∆T ρCpUb
. (2.1)
Here h = q
′′/∆T is the convective heat transfer coefficient based on the heat flux per unit
area q
′′ and a reference temperature difference ∆T. H is a characteristic length scale and Ub
is a characteristic velocity scale. The fluid density, thermal conductivity, and specific heat
capacity are ρ, K, and Cp, respectively. Note that Stanton number is related to the Nusselt
number as follows:
St =
Nu
ReP r
, (2.2)
where P r = ν/α is the Prandtl number of the fluid and Re = UbH/ν is the Reynolds number.
10

Here, ν is the kinematic viscosity of the fluid and α = K/(ρCp) is the thermal diffusivity.
Previous theoretical studies of fully developed forced convection in a partially porous channel
have showed that Nusselt number is dependent on the thickness of the porous medium, a
non-dimensional measure of permeability Darcy number Da = κ/H2
, and the effective fluidsaturated thermal conductivity Kef f of the porous medium for a given Reynolds number
[49, 33]. Despite the potential applications, relatively little is known about the fundamental
relationship between turbulent and heat exchange at the porous medium-fluid interface. A
key limitation is that few experimental and numerical datasets exist for heat transfer over a
porous interface over a range of permeabilities (κ) and Reynolds numbers (Re) over a limited
streamwise development length [40].
Permeability of the porous substrate has been shown to change the structure and dynamics of turbulence [6, 48, 57, 56, 32]. In particular, Suga et al. [57] used three ceramic
foams of same porosity (≈ 0.8) but different values of permeability as porous substrates in a
turbulent channel flow with bulk Reynolds number (Reb) ranging from 1000 to 10200. It was
observed that the porous substrate enhances the onset of turbulence as compared to a solid
wall, with transition to turbulence observed at bulk Reynolds number as low as Reb = 1300
for the highest permeability case. Furthermore, the wall normal fluctuations are enhanced
near the permeable wall due to weakening of wall blocking effects, which results in large
turbulent wall shear stress at the porous walls.
Previous numerical studies also indicate an increase in turbulent kinetic energy at the
porous-fluid interface as compared to a solid wall which is associated with the existence of
large-scale vortical structures associated with the Kelvin-Helmholtz type instability [6, 44].
Chandesris et al. [9] performed a direct numerical simulation on an isotropic porous wall in a
turbulent channel flow configuration at bulk Reynolds number Reb = 5500, the influence of
such large-scale structures was detected even at a low Prandtl number of P r = 0.1. Due to
large-scale pressure waves that penetrate inside the porous medium, significant temperature
fluctuations are induced when the temperature difference is large, resulting in a strong peak
11

of root mean square (RMS) temperature inside the porous medium. Similar behavior was
not observed for velocity fluctuations due to drag effects. This suggests that large-scale
vortical structures are responsible for increasing turbulent heat flux over the porous region
interface compared to a solid wall.
Although heat transfer over porous interfaces has significant applications, experimental
characterization of thermal transport over porous interfaces over a wide range of Reynolds
numbers is limited, particularly in the transition to turbulence in turbulent flow regimes.
Previous experimental studies have investigated open-celled metal foams extensively in configurations for which the metal foams fully occlude the channel. Usually manufactured out
of a highly conductive material with the solid phase thermal conductivity values ranging
from K ≈ 150 W/m-K(Graphite) to K ≈ 200 W/mK (Aluminum), and K ≈ 400 W/m-K
(Aluminum), the open-celled metal foam structure offers a high specific surface area ratio,
good thermally conducting solid phase and tortuosity, which enhances mixing and promotes
convective heat transfer [38, 5, 42, 67, 29]. Experiments from Boomsma, Poulikakos, and
Zwick [5] show that the presence of aluminum foams result in a consistently higher convection (Nu upto 140) as compared to no foams (Nu upto 15) for the range of flow rates tested,
while showing a comparative decrease in thermal resistance (Rth). Moreover, each foam of
different permeability showed a different response, highlighting the effect of porous microstructure on the convective heat transfer. A further detailed review of thermal-hydraulic
transport in metal foams as a function of thermo-physical properties can be found in [67].
Thermal transport over a porous interface depends on not only the thermophysical properties of the porous medium but also its thickness, placement, configuration, and interactions
of the fluid and porous medium at the interface. However, the fundamental understanding
of thermal transport in such systems is limited. Therefore, the focus of this study is to
perform a laboratory experiment investigation of thermal transport over a porous interface
in a channel flow arrangement using commercially available Aluminum foams.
Particularly, for the same value of porosity, the effects of pore size and permeability
12

are characterized using two different foams of nominal pore sizes of 10 and 40 pores per
inch (ppi). The foams are attached to a heater block and placed in a forced convection
arrangement adjacent to an unobstructed channel. Temperature measurements made at
several locations along the channel are used to evaluate heat transfer efficiency as a function
of pumping power requirements for bulk Reynolds numbers from 500-2500. Particle Image
Velocimetry measurements are made at a subset of these Reynolds numbers at three different
locations along the porous medium to estimate the interfacial shear stress. Heat transfer
performance is discussed in terms of Nusselt number, Stanton number and pumping power
requirements.
2.2 Methods
2.2.1 Porous medium
Experiments were conducted adjacent to two different types of open-celled metal foams
made of Aluminum (ERG Aerospace), as shown in Fig. 2.1. Metal foams have shown several
desirable qualities of a high-performance compact heat exchanger. Particularly, the opencelled metal foam structure offers a high specific surface area to volume ratio, good thermally
conducting solid phase, and tortuosity, which enhances mixing and promotes convective heat
transfer [38, 5, 42, 67, 29]. Usually manufactured out of a highly conductive material such
as Graphite (K ≈ 150 W/m-K), Aluminum (K ≈ 200 W/mK ), and Copper (K ≈ 400
W/m-K), they have demonstrated promise in a variety of applications. More recently, the
progressively decreasing size of electronics components and requirements of high processing
power has led to high heat fluxes at the chip level, and thermal management of such systems
has become a critical bottleneck for power densification. For example, hybrid and fuel cell
vehicles utilize the Si-based IGBT (Integrated Gate Bipolar Transistor) controller, which
must dissipate about 100 W/cm2and maintain a temperature below 1250C. Utilizing metal
foams with pin-fin structures in IGBT modules has shown to enhance heat transfer for such
13

high-performance thermal management solutions for electric vehicles, data centers, and other
high-power semiconductor applications [34].
The metal foams used in this experiment are shown below in Figure 2.1. The nominal
pore sizes corresponded to 10 and 40 pores per inch (ppi). The thermophysical properties of
these foams have been adopted from Calmidi and Mahajan [8] and Boomsma and Poulikakos
[4] for respective pores per inch (ppi) and porosities, with water as the working fluid. Length
scale arising from permeability κ is also expected to be a crucial parameter, based on which
the Darcy number is evaluated as Da = κ/H2
p
. Here, Hp is the thickness of the porous
medium as illustrated in Fig. 2.3a. Specifically, √
κ determines the length scale at which
shear penetrates into the porous medium [6]. These two foams were primarily chosen such
that for a constant porosity, different values of permeability were obtained. This choice
will particularly help isolate the heat transfer response of the foam with respect to porous
medium microstructure, characterized in terms of the Da. A complete list of all geometrical
and thermophysical properties is listed in Table 2.1.
Figure 2.1: Photographs showing 10 ppi and 40 ppi foams used in the experiment (from left
to right)
Foam Kef f [W/m-K] κ [10−10 m2
] Da =
κ
H2
p
10 ppi 7.6 3600 9.6 × 10−3
40 ppi 5.4 710 1.8 × 10−3
Table 2.1: Thermo-physical properties of the metal foams used in the experiments
14

2.2.2 Setup
The goal of this experiment was to measure the heat exchanger of the open-celled aluminum
foams in a partially porous convective flow arrangement. This was achieved by directing
water as a coolant through a rectangular channel whose top wall was replaced with a porous
aluminum foam attached to a heater via a heater spreader plate. The heat was first conducted into the foams and then convected across the channel into the coolant stream. The
characterization of a partially porous channel as a heat exchanger included measuring temperatures of the heater spreader plate, the coolant temperature at inlet and outlet, the power
delivered to the heater block. Particle Image Velocimetry measurements were also performed
to characterize shear stress at the interface and understand the influence of permeability of
the porous substrate on the flow structure and subsequently on the heat exchanger performance in this system.
A custom test section was machined from acrylic with a cutout of length L = 320
mm designed to hold the porous aluminum foam of thickness Hp = 6.35 mm. The width
of the test section was W = 50 mm, and the height of the unobstructed fluid region was
Hf = 9.35 mm, resulting in Hf /Hp = 1.48. A heater was attached to the aluminum foam via
a aluminum heater spreader plate through which the heat was conducted and then eventually
convected into the coolant stream through the aluminum foams. Two sets of experiments are
performed. First, the thermal performance of this system was characterized by coolant inlet
and outlet temperatures and the temperatures at various locations on the heater spreader
plate. Subsequently, these results are coupled with Particle Image Velocimetery measurements at a subset of these Reynolds numbers to characterize the shear stress and provide
more insight about the flow structure at the porous interface. A schematic of the experimental setup is shown in 2.2. The test section assembly and procedure for each set of experiments
is discussed in detail in Sections 2.2.3 and 2.2.4 respectively. ,
15

Figure 2.2: Schematic of the experiment
2.2.3 Heat transfer measurements
A system of T-type thermocouples (Copper/Constantan) of 0.13 mm diameter (5TC-TT-T36-72 from Omega Inc.) were used to make point temperature measurements at the inlet,
outlet as well as over various locations on the heater spreader plate. Three equidistant slots
were drilled into the heater spreader plate at x = 30, 160, 290 mm for the thermocouples,
which were sealed using a thermally conductive paste (Arctic Silver Ceramique). An average
of these three temperatures gave the measure of the average plate temperature Tpl. This
measurement is motivated by experiments done by Boomsma, Poulikakos, and Zwick [5] for
a channel fully occluded with aluminum foams and will allow a comparison of convective
heat transfer measurements in both configurations. The inlet Tin and outlet Tout temperatures were also monitored using thermocouples throughout each run. This is illustrated
in Fig. 2.3a. The data from the thermocouples were measured by a USB data acquisition
device (TC-08, Omega.Inc) This device enabled the real-time measurement,recording, and
display of the temperatures with an accuracy of ±0.5
◦C.
Aluminum foams were glued to an aluminum heater spreader plate using a thermally
16

conductive epoxy (MG Chemicals). A custom-made Aluminium heating block was attached
to the heater spreader plate on top using Arctic Silver Ceramique thermal conductive paste.
Five holes, equally spaced across the length, each of 12.5 m diameter were drilled into the
block to hold cartridge heaters rated at 330 W (O.E.M. Heaters). For this experiment only
two equi-spaced cartridge heaters were used. A set of plug-in power meters was used to
monitor the power delivered to each cartridge heater. The entire heating arrangement was
insulated in an acrylic box (K ≈ 0.1) covered with a thermally reflecting tape sheet to
minimize environmental losses.
A 2.1 kW recirculating chiller was used to pump water through the system, which
regulated a constant inlet temperature of 23◦ ± 0.8
◦ C. Using a electronic proportioning
valve, the chiller achieved bulk flow velocities Ub in the range of 0.07 m/s to 0.28 m/s, which
translated to bulk Reynolds number Reb = UbHf /ν based on the height of the unobstructed
channel Hf to be 650 to 2600. For this range, the flow rates (V˙ ) were measured using a
washdown flowmeter (McMaster Carr) with an accuracy of ±2% of Full Scale (FS). Bulk
velocity was subsequently computed from the flowrate using the unobstructed fluid height Hf
as Ub = V /ρH ˙
fW. This corresponded to lower end of Reb ranges considered by Suga et al.
[57] in experiments to characterize the effect of permeability of ceramic foams on turbulence.
The two different open-celled metal foams were tested for thermal performance characterization using identical procedure. Each foam was mounted onto the cutout along with
the heater and insulation. Water was then pumped into the channel using the chiller for
about 30 minutes at the maximum attainable flowrate, which varied for each foam, in order
to remove any air bubbles for the foam and ensure that the inlet temperature of the water
was maintained at 23◦ ± 0.8
◦ C. After this stabilization period, the power to the cartridge
heaters was turned on and the entire system was allowed to reach a thermal steady-state.
The temperatures at various locations were recorded at 1 second intervals in real-time using
the USB data acquisition device and stored in a PC. Thermal equilibrium was said to be
achieved when each temperature did not vary by more than 10% over a 10 minute period.
17

Since the chiller had a cyclic heating-cooling cycle with a variation of 0.8
◦ C from the set
temperature of 23◦ C, the data was recorded for a further 12 minutes or 600 sample points
to ensure at least two cycles of variation even at lowest flow rate. The flowrate was adjusted
to a lower value using the electronic proportioning valve and the system was given about 20
minutes to stabilize and reach thermal equillibrium for the measurement of next data point.
For each flowrate, the heat transfer rate to the coolant across the test section was
evaluated by the following energy balance equation:
q = ρAUbCp(Tout − Tin) (2.3)
Here A = W L is the area of cross-section of the foam exposed to the coolant and Ub is the
bulk velocity of the coolant. The inlet and outlet temperatures (Tin, Tout) were measured
150 mm before and after the test section in the coolant stream. The heat transfer rate was
checked back with the power supplied to the heaters and did not vary by more than 10% of
supplied power at any given time.
2.2.4 PIV measurements
To provide further insight into flow structures in the partially porous channel and also to
characterize the shear stress at the porous interface, PIV measurements were pursued at
three locations along the length of the channel. These tests were performed at a subset of
Reynolds numbers from Section 2.2.3, Reb = 1200, 1700, 2300. These correspond to the lower
end of Reb ranges considered by Suga et al. [57]. Furthermore, recent turbulent boundary
layer experiments over high-porosity isotropic foams show that the flow adjusts to a new
substrate in the streamwise direction over a distance of ≈ 30−40Hp, where Hp is the porous
layer thickness [15]. Hence, to characterize the along the streamwise direction, each test
window began at 150, 210, 270 mm (≈ x/Hp = 23.6, 33.0, 42.5) from the leading edge and
extended 25 (≈ 3.94Hp) mm downstream, as shown in Fig. 2.3b. PIV measurements at the
18

320 mm Hp = 6.34 mm Hf = 9.34 mm Inlet Outlet Thermocouple qw = 660 W Heater block Porous medium Tin = 230 C
Tplate,in Tplate,m Tplate,out
(a) Test section showing geometry of the porous and fluid region, various thermocouple locations and heating assembly
150 mm210 mm 270 mm
(b) Test section showing the locations where PIV data was collected along the
length of the channel
Figure 2.3: Zoomed-in schematic of the test section
three different streamwise locations allow us to characterize bulk convective heat transfer
properties of systems with limited development length.
A 5W continuous wave laser with integrated optics was used to generate a laser sheet at
each test location. A high-speed camera (Phantom VEO-410L) along with AF-S NIKKOR
300mm F2.8G lens was used whose memory storage capacity allowed capturing 16200 images
at 1kHz for Reb = 1200, 1700 and at 2kHz for Reb = 2200. The total duration of the
measurements was approximately 180 turnover times for Reb = 1200, 2200 and approximately
270 turnover times for Reb = 1700, where the turnover time is estimated as (Hf /U b and
the bulk-averaged velocity is defined as Ub = V /˙
[W(Hf )]. Here V˙
is the flow rate measured
by the flowmeter. The images were processed in PIVlab [61, 60] using the Fast-Fourier
transform routine with a minimum box size of 16 pixels and 50 % overlap, which yielded
19

about 52 (vertical) x 126 (horizontal) data points in the unobstructed section. Based on the
friction velocities measured (Table 2.2), the resolution was ∆y
+ = ∆x
+ = 1−4 in inner units.
For each case, the porous wall location (y/Hf = 0) and smooth wall location (y/Hf = 1) is
determined by averaging the pixel intensities of a PIV snapshot in the streamwise direction
and identifying the two peak values in the wall normal direction. This is based on the
observation that as the laser sheet passes through the acrylic and reaches the aluminum
foam, it is reflected back in the same direction resulting in the peak intensities at the porous
interface as well as the smooth wall.
2.3 Results and Discussion
In this section, first experiment measurements for mean flow, turbulence statistics and flow
structure are presented for both the 10 ppi and 40 ppi foams. We then present heat transfer
characterization for these foams in a partially porous channel configuration. Specifically, we
report Nusselt number (Nu) and Stanton number (St). Finally, we quantify the convective
heat exchanger performance of such system by weighing it against energy required to run
the system in terms of the pumping power requirements required to run the system.
2.3.1 Mean flow statistics
Fig. 2.4 shows the mean velocity profiles for the channel for experiments for both types of
foams. The velocity distribution across the channel is normalized by the bulk velocity Ub,
calculated from the flowrate measured using the flowmeter and the area of cross-section of
the unobstructed region. For each foam, the measurements were taken at three locations
in the streamwise (x) direction as discussed previously in Section 2.2.4 for three Reynolds
numbers Reb = 1200, 1700, 2200. In the unobstructed fluid region, y/Hf = 0 corresponds to
the porous interface and y/Hp = 1.48 corresponds to the adiabatic smooth wall. The region
in y/Hp > 0.88 is affected by optical distortions and hence is not reported here. These
20

statistics were computed by averaging both in time and in the streamwise directions.
As seen in Fig. 2.4, there is evidence of flow development as we move further in streamwise direction. For x/Hp = 23.6, the mean velocity profile remains relatively symmetric,
however as x/Hp increases, we see that the velocity profile distribution becomes asymmetric
and deviates from parabolic. More specifically, we can see for increasing streamwise distance,
the bulk of the flow is shifted towards the smooth wall. For x/Hp = 23.6, we see a significant
slip velocity (Uw ≈ 0.35Ub) with a Umax/Ub < 1 . This can be attributed to an increase
in flow through the porous region as the flow develops. For a further increase in streamwise distance, the velocity profile deviates from parabolic and the bulk of the flow is shifted
towards the smooth wall even at lower most Reynolds number Reb = 1200. For example,
for Reb = 2300, at x/Hp = 23.6, the Umax/Ub = 1.2 which increases to Umax/Ub = 1.2 at
x/Hp = 42.5. This implies that the flow tends to become turbulent at x/Hp = 42.5. This
tendency is enhanced in the foam with a higher permeability (10 ppi) as compared to the 40
ppi and consistent with observations from Suga et al. [57]. The variation in mean velocity
profiles shows that the flow is not hydrodynamically developed in the the test section.
Profiles for Reynolds shear stress are plotted in Fig. 2.5 for all cases. The profiles show
the presence of an (almost) linear region in the middle of the unobstructed channel. To
evaluate the friction velocities at the porous and smooth walls (u
p
τ
and u
s
τ
), the total shear
stress (i.e. Reynolds shear stress plus viscous shear stress) is extrapolated from this linear
region to y/Hp = 0 and y/Hp = 1.48 respectively [6]. To estimate the viscous shear stress
mean velocity gradients from the measured mean velocity profiles were utilized. However,
near wall velocity gradients were expected to be inaccurate due to lower wall normal resolution (∆y
+ ≈ 1 − 4). Hence, the friction velocities are estimated using a linear fit to the
total stress region between the maximum and the minimum values of Reynolds shear stress.
Consistent with mean velocity profiles, the intercept of the (almost) linear region on the
porous wall increases with streamwise distance. For a given foam and Reynolds number,
the friction velocity was averaged over all three locations. The resulting average friction
21

velocity estimates are shown in Table 2.2a and Table 2.2b. For each case, u
p
τ > us
τ
consistently,delineating the effects of permeability. More specifically, for a given Reynolds number
10-ppi foam consistently showed higher friction velocities than 40-ppi foam.
Using the shear stress estimates obtained from friction velocities at the porous wall, a
friction factor for each case is estimated as follows
Cf =
τp
1
2
ρU2
b
, (2.4)
where τp = ρ(u
p
τ
)
2
. These values are plotted in Fig. 2.6. 10-ppi foam shows a consistently
higher Cf value than 40-ppi foam. For 10-ppi foam, Cf is constant within experimental
uncertainty. This is consistent with the results of Suga et al. [57, 56], who showed that as
the wall permeability increases, the wall normal component of velocity fluctuation increases
resulting in a higher turbulent shear stress near the porous wall.
Figure 2.4: Mean velocity profiles for (a) 10 ppi, (b) 40 ppi
22

Figure 2.5: Mean Reynolds shear stress profiles for (a) 10 ppi, (b) 40 ppi
Reb u
s
τ
[m/s] u
p
τ
[m/s]
1200 0.008 0.0094
1700 0.011 0.015
2200 0.016 0.020
(a) 10 ppi foam
Reb u
s
τ
[m/s] u
p
τ
[m/s]
1200 0.005 0.008
1700 0.009 0.012
2200 0.012 0.016
(b) 40 ppi foam
Table 2.2: Average friction velocities for the two foams as a function of Reb
2.3.2 Heat transfer characteristics
For the two types of foams, the thermocouple measurements were used to calculate the
Nusselt number and Stanton number:
Nu =
hH
K
=
q′′
∆T
H
K
; (2.5)
St =
h
ρCpUb
=
q
′′
∆T ρCpUb
. (2.6)
23

1000 1500 2000
Reb
0
0.01
0.02
0.03
Cf;p
10 ppi
40 ppi
Figure 2.6: Friction factor vs Bulk Reynolds Number
Figure 2.7: Nusselt Number vs Bulk Reynolds Number plotted for both 10 ppi and 40 ppi
foams. The smooth wall data is obtained from Boomsma, Poulikakos, and Zwick [5]
To evaluate the heat exchanger performance of the partially porous channel, the temperature
difference ∆T is to be chosen such that it gave a good measure of the thermal performance
of the test section. Since it was not possible to reliably measure the temperature at the
porous medium-fluid interface, an estimate of the interface temperature (Tint) was made as
follows. Previous experiments [57] for flow over a porous substrate concluded that only a
24

small percentage of mass flowrate (< 5%) penetrates into the porous medium. Note that this
experiment was performed at a higher bulk Reynolds number ranges (Reb = 1000 − 10200)
as compared to this study (Reb = 600−2600) for Hp = Hf . The primary heat transfer mechanism in the porous medium can assumed to be conduction as a first approximation. Using
the one-dimensional heat conduction equation and the fluid-saturated thermal conductivity
of the porous medium, a temperature Tint is estimated at the porous medium-fluid interface
as per below:
q
′′ = Kef f
Tpl − Tint
Hp
(2.7)
Kef f are obtained from Table 2.1 for respective foams. The interface temperature Tint is
then used to estimate a Nusselt number for the unobstructed fluid region as per Eq. 2.5.
The Nusselt number and Stanton number for the two different types of foams is plotted
against various bulk Reynolds numbers in Fig. 2.7 and Fig. 2.8. Note that the Nusselt
number is correlated with the Stanton number as per Eq. 2.2. There is limited prior work
characterizing the dependence of Nusselt number on bulk Reynolds number in porous walls.
However, as per smooth wall correlations for Nusselt number with Reynolds number, the
Nusselt number is expected to vary with the friction coefficient. Since the permeability also
plays a role in thermal transport, in the present experiment the Nusselt number will also
depend on the Darcy number provided Hf /Hp remains the same. This is observed in Fig. 2.7.
At the lowest Reynolds number tested (Reb = 650), 10 ppi foams (Da = 9.6 × 10−3
) show
a Nusselt number of Nu = 8.4 as compared to that of 40 ppi foams (Da = 1.9 × 10−3
)
with an Nu = 4.2. As the Reynolds number increases, the Nusselt number for 10 ppi foam
consistently remains higher than that of 40 ppi foam. Also since the friction coefficient
remains constant with Reynolds number as seen in Fig. 2.6, the Nusselt number for each
of the foam also remains constant within the experimental uncertainty for the range of
Reynolds numbers tested. This is more clearly seen in Fig. 2.8 as the Stanton number
appears to remain constant after Reb = 1200.
In addition, Nusselt number for smooth wall (no foam) from Boomsma, Poulikakos, and
25

Zwick [5] are also plotted. It is seen that even though the Nusselt number for the smooth
wall steadily increases with bulk Reynolds number, it remains lower that the Nusselt number
obtained in this experiment by ≈ 70% for the 10 ppi foam and by 50% for the 40 ppi foam
even at Reb = 2400. Moreover, as compared to the results for channels fully occluded with
aluminum foam, the Nusselt number obtained in the current experiment are much lower. This
is expected as a fully occluded channel will result in more tortous coolant paths, enhancig
mixing and hence convective heat transfer.
Figure 2.8: Stanton Number vs Bulk Reynolds Number
2.3.3 Pumping power efficiency
Finally, for an optimized heat exchanger design, the convective heat transfer capacity must
not come at the cost of energy required to run the system, which is the pumping power in
this configuration. For a given flowrate V˙ and pressure drop ∆P across the channel of length
L, the pumping power W˙ can be defined as
W˙ = ∆PV . ˙ (2.8)
This can be compared against the thermal resistance Rth offered by the channel, defined as
26

Rth =
Tint − Tin
q
′′
=
Tint − Tin
mC˙ p(Tout − Tin)
. (2.9)
The power supplied by the cartridge heater was compared with the temperature difference
of the fluid across the heater. It was observed that only a maximum ≈ 20% heat energy
was lost to the environment. Overall, minimizing both the thermal resistance and pumping
power will increase convection heat transfer across the porous interface. W˙ vs Rth is plotted
in Fig. 2.9 for both the foams. As noted previously, the 10ppi foam consistently performs
better than the 40 ppi foam in terms of the convective heat transfer metrics but requires
a higher pumping power requirement. In comparison, 40 ppi foam offers lower convective
heat transfer but does not come at the cost of pumping power requirements. In this limited
dataset, the most optimal performance is by 40 ppi foam operating at Reb = 1700 as it
collectively optimizes both W˙ and Rth.
Figure 2.9: Pumping power vs thermal resistance
2.4 Conclusion
In a study, two foams with the same porosity value but different nominal pore sizes (10
ppi and 40 ppi) and permeability were analyzed for convective heat transfer in a partially
27

porous channel flow setup. The Darcy number (Da) plays a crucial role in not only the
transition of turbulence but also in enhancing heat transfer at the interface. However, this
also leads to an increase in friction and hence, pumping power requirements. The next
step is to find a relationship between the friction factor (Cf ) and the non-dimensional heat
transfer coefficients (Nu and St) for a porous wall, just like it exists for a smooth wall. These
relationships follow the formula St = f(Cf , P r) where Cf = f(Re) for a smooth wall.
The experiments assumed Hf /Hp to be constant, but previous numerical studies have
shown that the ratio of the porous medium thickness to the unobstructed fluid medium is
an essential parameter in characterizing the heat transfer coefficient. Moreover, based on
dimensional analysis, for a porous wall of permeability κ with finite thickness Hp placed
adjacent to a fluid layer Hf , given a Reynolds number Reb, the Nusselt number can be
expressed as:
Nu = f(Reb, Da, Hf /Hp, P r) (2.10)
It is important to note that this study has some limitations as the experimental datasets
produced here provide limited parameter space to quantify the relationships proposed in
Eq. 2.10. Moreover, these data have been characterized in a hydrodynamically and thermally
developing flow regime. However, this study can be extended to fully developed flow and
for various values of Hf /Hp for isotropic foams. Nevertheless, this study is the first set of
experiments that emphasizes the importance of Da. Notably, it highlights that the length
scale √
κ arising from permeability plays an essential role in convective heat transfer dynamics
at the porous medium-fluid interface.
28

Chapter 3: Thermo-physical properties of 3D-printed
porous materials
3.1 Moving towards designed porous media
Metal foams have an isotropic stochastic structure by design and are generally manufactured
at higher porosities (ϵ > 0.8). Also, the high thermal conductivity of solid-phase limits
the variation in the effective thermal conductivity of saturated metal foam. Under the
constraint of pumping power requirements, an excessive increase in the interstitial area
causes an increase in drag. A substantial part of the interstitial area may not actively
participate in heat exchange. Vafai [62] theoretically showed the importance of channeling
effect in variable porosity media. Heterogeneity in permeability and porosity in the porous
medium will affect scalar transport by introducing variability in fluid residence time, hence
affecting both heat transfer and momentum exchange.
LFMs were first investigated as structural components with high compressive strength
and resistance to plastic buckling, before being considered as potential convective heat transfer media . Various studies have compared LFMs against other options such as prismatic
cores, woven metal textiles, metal foams, and traditional louvered fins [38]. For load-bearing
heat exchangers, LFMs and prismatic core structures have been found to outperform all
other options in terms of heat transferred per unit temperature difference and pressure drop
[50]. LFMs have also been found to have superior mechanical and convective transport characteristics compared to stochastic metal foams [16]. For example, lattice-Framed Structures
(LFM) have been found to be highly effective in increasing forced convective heat transfer,
as demonstrated by studies from Seo Young Kim et al. [54], Zhao [67], Ekade and Krishnan
[16], and Son et al. [55]. In particular, Ekade and Krishnan [16] found that octet lattices
29

have a normalized permeability that is 20 − 30% higher than stochastic foam with a similar
porosity and lower friction factor values. With the advent of additive manufacturing, it has
become easier to create lattice structures with a range of porosities and unit cells, which
can achieve necessary multifubnctionality with desirable mechanical, thermal, and hydraulic
properties. 3-D printing allows for the manufacturing of lattices with systematically varied
porosity, making it possible to incorporate anisotropy.
While past work has analyzed the influence of microstructural variations over a limited
range, the successful adoption of LFMs for multi-functional design requires an understanding
of the particular application needs can be tailored. For instance, multifunctional dragreducing and effective heat dissipation capabilities have the potential to reduce the volume
and weight of heat exchangers, making LFMs a viable alternative to stochastic metal foams
for heat-exchange applications. Previous experiments and numerical simulations [1, 22, 53,
10] have shown that drag reduction is a function of the difference between streamwise (κxx)
and spanwise (κzz) permeabilities. Hence, as we further move ahead to design anisotropic
porous substrates for a specific application, further research is needed to understand how
the porous medium microstructure affects flow and thermal dispersion.
Specifically, this chapter focuses on the characterization of thermophysical properties
for a family of simplified anisotropic cubic micro-structure lattices in order to capture a relationship between the microstructure properties and saturated thermal conductivity (Kef f )
and permeability (κ). First, porous lattice geometry is introduced and limitations on manufacturability are explained. Next section introduces permeability, and a model is proposed
to predict permeability just based on geometric parameters. Predictions of the model are
compared with experimental results and ANSYS Fluent simulations. In the next section,
experiments and numerical simulation results for thermal conductivity are compared with
existing lumped parameter models for prediction.
30

3.2 Porous lattice design
This work extends the implementation of the cubic unit cell lattice structure first proposed
in [10] using stereolithography printing to generate tunable lattices with anisotropic permeability and thermal conductivity. Rapid advances in additive manufacturing have made it
possible to manufacture highly porous (ϵ > 0.8, where ϵ is the volumetric porosity) lattices
with small rod-sizes and large openings. To achieve a tunable microstructure, a cubic lattice
geometry with a square ligament cross-section is proposed and shown in the unit cell below.
The porous structure is uniquely governed by four independent parameters such that
ϵ = f(d, sx, sy, sz), (3.1)
where d is the rod-size (or the ligament thickness), and (sx, sy, sz) is the rod spacing in
(x, y, z) direction. The average volumetric porosity is given by the equation
ϵ = 1 −
d
2
(sx + sy + sz) − 2d
3
(sxsysz)
. (3.2)
The current set of porous geometries are 3D printed using the state of the art Formlabs Form2
and Form3 printers. For more details on the printing process, please refer to Efstathiou [14].
31

(a) (b) (c)
Figure 3.1: CAD renderings of anisotropic lattice geometry. Here we see (a) isometric view
for HHL lattice. (b) the view in direction with higher planar porosity (HH) and (c) the view
in the direction with smaller planar porosity (HL)
(a) HHH (b) MMM (c) LLL
Figure 3.2: Photographs of 3D printed cubes (side = 25 mm) of anisotropic porous material
32

Sample d (mm) sx (mm) sy (mm) sz (mm) ϵ
HHH 0.4 3 3 3 0.95
MMM 0.4 2 2 2 0.89
LLL 0.4 1.5 1.5 1.5 0.83
TTT 0.4 3 0.8 3 0.50
HHM 0.4 2 3 3 0.94
HHL 0.4 1.5 3 3 0.92
HHT 0.4 0.8 3 3 0.87
MMH 0.4 3 3 2 0.92
MML 0.4 3 3 2 0.88
MMT 0.4 3 3 1.5 0.80
LLH 0.4 3 3 1.5 0.88
LLM 0.4 3 3 1.5 0.85
LLT 0.4 3 1.5 3 0.74
LMH 0.4 3 1.5 3 0.90
Table 3.1: Here are the details of the parameter space based on the cubic lattice structure
proposed in 3.2. Note that here ’H’ corresponds to the highest chosen nominal rod spacing (s = 3) mm, ’M’ corresponds to s = 2 mm, ’L’ corresponds to s = 1.5 mm and ’T’
corresponds to (s = 0.8) mm.
33

3.2.1 Constraints on achievable geometry
Consider the cubic lattice geometry shown in Figure 3.3, characterized by rods of crosssection d × d and spacings (sx, sy, sz) in the x, y, and z directions. For this geometry, we
can calculate the directional porosities corresponding to the minimum opening area along a
given axis, as:
ϵx =
(sy − d)(sz − d)
sysz
; ϵy =
(sx − d)(sz − d)
sxsz
; ϵz =
(sx − d)(sy − d)
sxsy
. (3.3)
Figure 3.3: The portion highlighted in red shows a unit cell identified from the lattie with
rod size d and rod-spacings sx, sy, and sz in the three directions
Defining the dimensionless parameters, ˆsx = sx/d etc. we have:
ϵx =
(ˆsy − 1)(ˆsz − 1)
(ˆsysˆz)
; ϵy =
(ˆsx − 1)(ˆsz − 1)
(ˆsx)ˆsz
; ϵZ =
(ˆsx − 1)(ˆsy − 1)
(ˆsx)ˆsy
. (3.4)
The above equations show that directional porosities depend on reduced lengths, rx, ry, rz
as below:
rx =
sx − d
sx
= 1 − sˆ
−1
x
; ry =
sy − d
sy
= 1 − sˆ
−1
y
; rx =
sz − d
sz
= 1 − sˆ
−1
z
. (3.5)
Specifically, we have
34

ϵx = ryrz; ϵy = rzrz; ϵz = ryrx. (3.6)
Rearranging the terms gives us
rx =
rϵyϵz
ϵx
; ry =
rϵxϵz
ϵy
; rz =
rϵxϵy
ϵz
, (3.7)
and since ˆsx = (1 − rx)
−1 and so on, we have
sˆx =

1 −
rϵyϵz
ϵx
−1
, sˆy =

1 −
rϵxϵz
ϵy
−1
, sˆz =

1 −
rϵxϵy
ϵz
−1
. (3.8)
Equation (A.6) can be used to identify the dimensionless spacings ( ˆsx, sˆy, sˆz) to generate
(ϵx, ϵy, ϵz). Furthermore, it allows us to identify which limits over achievable anisotropies and
directional porosities. For example, (A.6) indicates that for directional porosities such as
ϵyϵz > ϵx would lead to ˆsx < 0. Assuming that the desired porosity values arranged in
decreasing order of magnitude (ϵx > ϵy > ϵz), then the porosities must satisfy:
ϵxϵy < ϵz. (3.9)
This along with manufacturing constraints using the SLA printer limits the maximum achievable anisotropy. With the formlabs Form2/Form3 printers, we can consistently print lattices
with rod size d = 0.4 mm. Printing at d = 0.2 mm is possible, but less reliable. The maximum recommended unsupported span for the printers is s = 5 mm, though larger spans
are possible for supported geometries. Thus we can reasonably expect to generate lattices
with (ˆsx, sˆy, sˆy) ∈ [1, sˆ]max where ˆsmax = 20. This translates to rmax = 1 − (1 − sˆmax) = 0.95
and (ϵx, ϵy, ϵz) ∈ [0, ϵmax] with ϵmax = r
2
max = 0.9025. This gives rise to more restrictive
constraints:
ϵzϵy ≥ ϵxϵmax. (3.10)
The above constraint also implies that if the maximum directional porosity is set to ϵx = ϵmax,
35

the other two porosity values are equal: ϵy = ϵz.
3.3 Permeability characterization
The concept of permeability (κ) was initially introduced by Henry Darcy in 1856 to describe
the level of resistance that porous materials pose to the flow of fluid. Through his famous
column experiments, Darcy proposed an empirical relationship, now known as Darcy’s Law,
which modeled fluid flow through sand filters. According to this relationship, the velocity of
fluid is directly proportional to the gradient in hydraulic head across the material and the
material’s permeability. As outlined in 1.2, the generalized version of Darcy’s empirical law
is commonly used to characterize the flow through a porous medium, and is represented by
the equation ∇p + µκu = 0. This equation includes a permeability tensor in R
3 as given
below:
κ =






κxx κxy κxz
κxy κyy κyz
κxz κyz κzz






. (3.11)
This relationship particularly holds for Stokes flow without inertial effects; as the flow velocity
increases, this relationship is better expressed with an additional Forchheimmer term. For
instance, in the x direction, the modified Darcy-Forchheimmer equation can be expressed
as:
dP
dx = −
µ
κ
Ub +
c
√
κ
U
2
b
. (3.12)
Here √c
κ
U
2
is the Forchheimmer correction term, with c being the Forchheimmer coefficient.
Non-dimensionally, the permeability is usually expressed in the form of the Darcy number
Da = κ/D2 where D is a characteristic length scale.
Lu, Stone, and Ashby [38] developed a simple model linking temperature profile with
properties of the porous medium such as porosity and permeability. Previous experiments to
36

characterize permeability have focused in stochastic isotropic foams such as open-celled metal
foams [3, 16, 47, 62]. A more detailed review of pressure drop studies in open-celled metal
foams can be found in [30]. Overall, porosity, permeability, and Forcheimmer coefficient
influence the overall thermal efficiency of a metal foam.
Previous experimental efforts to measure anisotropic permeability have been motivated
by soil engineering, hydraulics, and petroleum engineering, dealing with low porosity granular samples. Moreover, very few studies measure more than the principal component of
permeability. Liakopolous [37], first showed that the permeability matrix is a symmetric
tensor and measured the 1D permeability of anisotropic sandstone sample taken at 0, 30, 60◦
and his measured values matched the permeability predictions by rotating the measured
permeability matrix in the 0◦ direction. More recently, Lei [36] developed a test to measure the 2D tensor permeability matrix by measuring pressure gradient in two directions
and corresponding velocities in fractured anisotropic media. According to recent research
conducted by Pinto and colleagues, there is potential for enhancing permeability predictions
for periodic anisotropic lattices. The study highlights the need for improvements in models,
particularly in selecting a characteristic length scale for accurately predicting anisotropic
permeabilities using the Darcy-Forchheimer framework.
3.3.1 Model to predict principle components of permeability
In this section, we introduce a phenomenological model for the prediction principal components of the permeability tensor for this simplified lattice geometry. This model assumes
that the permeability in any given direction across the unit cell can be estimated based on
the effective hydraulic diameters. That is
κxx ∝ D
2
x
. (3.13)
Dx represents the total hydraulic diameter in the x direction. Hence, for 1D Darcy’s law,
we can express the pressure drop in the x direction in terms of the hydraulic diameter:
37

dP
dx = Cµ 1
D2
x
ux, (3.14)
where C is an empirical constant. This hydraulic diameter varies along the axis of interest
for the cubic lattice. As an example, we first derive the principle component of permeability
tensor in the x-direction Kxx as per below.
The unit cell can be split into two different zones in Figure 3.4. The first zone is of
length d and the second zone is of length sx − d.
Figure 3.4: Figure here shows the unit cell divided into two zones
For zone 1, the hydraulic diameter is
Dx1 =
4(sy − 1d(sz − d)
1[(sy − d) + (sz + d)] =
2d( ˆsy − 1)( ˆsz − 1)
( ˆsy − 1) + (ˆsz − 1). (3.15)
Similarly, for zone 2, the hydraulic diameter is given by
Dx2 =
4(sysz − d
2
)
4d
= d(ˆsysˆz − 1). (3.16)
We now introduce a weighting factor wx based on the cross-sectional areas of both zones.
From conservation of mass, we get
m˙ = UxbA = Ax1Ux1N = Ax2Ux2N, (3.17)
where ˙m is the average mass flow rate through the porous sample with N unit cells, Uxb is
38

the bulk (or superficial) velocity in the x direction, and Ax1, Ax2 represent the area of crosssection of the fluid to go through zones 1 and 2 with a velocity of Ux1 and Ux2 respectively.
Hence, pressure gradient in the x-direction can be given as
∆P = Cµ
1 − wx
Ax1D2
x1
+
wx
Ax2D2
x2

A
N
Uxb. (3.18)
Since the lattice structure is periodic, A/N = Ax = sysz for a given unit cell. We determine
the weighting factor wx based on the relative distance travelled in the direction of the flow
in each zone. Thus, wx = d/sx and hence (1 − wx) = (sx − d)/sx. And hence, pressure drop
along x-axis across the unit cell can be expressed as
dP
dx = −µC
d
sx
sˆysˆz
(ˆsy − 1)(ˆsz − 1)
1
D2
x1
+
sx − d
sx
sˆysˆz
(ˆsy − 1)(ˆsz − 1)
1
D2
x2

Uxb. (3.19)
From (A.10) and (A.15) and Dracy’s law, we can express
kxx =
1
C

d
sx
sˆysˆz
(ˆsy − 1)(ˆsz − 1)
1
D2
x1
+
sx − d
sx
sˆysˆz
(ˆsy − 1)(ˆsz − 1)
1
D2
x2
−1
. (3.20)
Here, C is an empirical constant derived from the approximation for laminar flow through
rectangular ducts (cite Kakac book) as below
fRe = 24(1 − 1.3553α ∗ +1.9467α ∗
2 −1.7012α ∗
3 +0.9564α ∗
4 −0.2537α∗
5
), (3.21)
where α∗ = min((sy − d)/(sz − d),(sz − d)/(sy − d)). Similarly, kyy and kzz can also be
derived.
Fig. 3.5a depicts the porosity ratios that are permissible in various directions for the
cubic lattice geometry. The achievable porosity ratios when spacings (sx, sy, sz) fall within
[1, smax] are represented by black dots. On the other hand, the shaded regions indicate the
39

(a) (b)
Figure 3.5: Achievable limits for (a) directional porosities (b)permeabilities for 3D printed
porous geometries as proposed in while considering the printing constraint (sx, sy, sz) ∈
[1, smax]
limits established by equation (3.9). The current cubic lattice geometry also restricts the
possible range of permeability values, similar to directional porosities. The allowable permeability ranges, while considering the printing constraint (sx, sy, sz) ∈ [1, smax], are presented
in Fig. 3.5b. It is important to note that the different components of permeability are interdependent, meaning that there is limited ability to adjust κxx, κyy, and κzz independently.
3.3.2 Experimental Setup
An in-house setup has been developed to measure the 1D permeability as shown in Figure
3.6. A 25 × 25 × 25 mm sample is placed in a test section through which water is pumped
in using a submersible pump connected to a programmable valve. Three high resolution
pressure transuder (PX409 -001DV from Omega Inc.) with a combined linearity, hysteresis
and repeatability specification of 0.08% of full scale are connected in all three directions to
measure the streamwise pressure drop ∂P
∂x as well as ∂P
∂y ,
∂P
∂z . The procedure to measure the
40

permeability matrix then follows from Darcy’s law.
Figure 3.6: Schematic for measurement of permeability
3.3.3 ANSYS Fluent setup
To ascertain the permeability tensor κ for anisotropic porous materials, we adopted the
methodology outlined by Zampogna and Bottaro [65]. Leveraging the ANSYS Fluent software package, we computed the individual forced Stokes flow problems for a unit cell of
the cubic lattice. Given that the microstructures assessed were periodic in design, we only
needed to compute two Stokes flow problems to establish the permeability components, κxx
and κyy. κzz is identical to either κxx or κyy, depending on the given configuration.
The Figure 3.3 the identified unit cell. In order to solve two Stokes flow problems, a
consistent body forcing with a unit amplitude was implemented towards the direction of
the evaluated permeability component. Darcy’s law was used to determine the permeability
from the volume-averaged velocity obtained. The solid boundaries were treated as no-slip
conditions, while periodic boundary conditions were applied to the unit cell boundaries.
Time-marching schemes were adopted for each of these simulations. The simulation was
considered to have reached a steady state when the residual in the permeability was below
10−6
. Our results were confirmed to be reliable as a mesh independence study demonstrated
41

convergence of the grid.
3.3.4 Results and Discussion
Comparison of model estimates with ANSYS
(a) (b)
Figure 3.7: Comparison between model predictions for permeability and unit cell Stokes flow
simulations in ANSYS fluent for (a)x-direction, and (b) y-direction for anisotropic porous
lattices
Predictions from the model for anisotropic geometries are compared against results
obtained from ANSYS Fluent. Fig. 3.7 shows that this simple model generates reasonable
predictions for permeability for a few representative geometries. Model predictions are within
±25% across all cases, with an average absolute error of less than 7%.
Permeability estimates from experiment
A subset of samples from Table 3.1 were 3D printed and subjected to permeability testing
using the procedure described in 3.3.2. To estimate permeability in the principle direction
of the permeability tensor (κxx, κyy, κzz), the cube was rotated to create variable directional
42

porosity in the direction of flow. Figure 3.8 illustrates how anisotropy can affect the pressure
gradient experienced by the sample. LLL shows a significantly higher pressure gradient than
MMM, while for MML, we can observe the difference between the two directions. Here, MM
and ML refer to the rod spacings in the plane that is perpendicular to the flow direction.
Figure 3.8: Figure above shows the comparison of pressure gradient with superficial (bulkaveraged) velocity through the sample for isotropic cases (MMM,LLL) and anisotropic case
(MML). Here MM and ML represent the rod spacings in the direction perpendicular to the
flow.
To estimate the permeability from pressure drop measurements using 1D Darcy’s law,
(3.12) is divided by Ub to get
1
Ub
dP
dx = −
µ
κ
+
c
√
κ
Ub. (3.22)
A linear fit for in (3.22) is estimated such that
1
Ub
dP
dx = aUb + b. (3.23)
43

Using the intercept b, permeability is calculated as κ = −µ/b. Note that the uncertainty
in permeability estimates is evaluated using the 95% prediction interval of the intercept.
We expect the relationship in (3.22) to be linear, however, we observe some scatter as seen
in the Figure 3.9 below. This scatter is due large uncertainty in pressure estimates at low
velocities. To determine a prediction interval for permeability, we can calculate the value of
b and estimate its 95% prediction interval. This process is repeated for all tested samples.
Additionally, a non-zero slope indicates that a > 0, demonstrating the presence of the
Forchheimer term. This is due to the increased importance of inertial effects at higher flow
rates.
Figure 3.9: Linear fit estimates and 95% prediction interval for the ML face of MML
44

Sample d (mm) sx (mm) sy (mm) sz (mm)
MMM 0.45 ± 0.06 2.06 ± 0.16 2.06 ± 0.16 2.06 ± 0.16
LLL 0.46 ± 0.04 1.23 ± 0.17 1.23 ± 0.17 1.23 ± 0.17
HHM 0.43 ± 0.06 2.96 ± 0.16 3.00 ± 0.16 2.09 ± 0.20
HHL 0.37 ± 0.07 2.90 ± 0.4 2.8 ± 0.4 1.5 ± 0.4
HHT 0.37 ± 0.07 3.13 ± 0.34 3.12 ± 0.34 0.76 ± 0.14
MML 0.44 ± 0.06 2.03 ± 0.26 2.09 ± 0.22 1.64 ± 0.30
Table 3.2: This table lists measurements of pore sizes and rod sizes of 3D printed samples.
Note that the uncertainties reported here are ±2σ to provide a conservative estimate in the
variation of dimensions
Comparison of experimental measurements with model and ANSYS Fluent predictions
The table 3.3 presents the constraints that physically realizable materials have. While the
model and ANSYS predictions are in agreement within a reasonable uncertainty, experimental predictions fall at the lower end of the range of model predictions obtained from
measurements in Table 3.2. This suggests that, although mean rod sizes and spacings can
be used to estimate, the variation in sizes to 95% prediction interval indicates the limitations
in 3D printing to print idealized geometries. This variation in sizes and clogging of pores
can result in lower values of permeability than those predicted from idealized models. Additionally, model predictions suggest that even a ≈ 20% variation in both d and (sx, sy, sz)
can lead to permeability values that are an order of magnitude higher at the upper limit
compared to the lower limit. Therefore, these limitations must be considered while designing
and predicting permeability values using additive manufacturing.
45

Sample Method √
κxx(mm)
√κyy (mm) √
κzz (mm)
MMM Model [0.17,0.45] [0.17,0.46] [0.17,0.45]
Experiment 0.32 0.32 0.32
LLL Model [0.05,0.4] [0.05,0.4] [0.05,0.4]
Experiment 0.28 0.28 0.28
HHM Model [0.40,0.71] [0.40,0.71] [0.48,0.79]
Experiment 0.27 0.27 0.28
HHL Model [0.16,0.94] [0.15,0.94] [0.31,1.09]
Experiment 0.18 0.18 0.32
HHT Model [0.03,0.27] [0.03,0.27] [0.28,0.57]
Experiment 0.07 0.07 0.4
MML Model [0.16 0.44] [0.16 0.44] [0.2 0.46]
Experiment 0.12 0.12 0.15
Table 3.3: Permeability predictions from ANSYS, model and experiment are compared here.
The range for the model predictions are derived from (d, sx, sy, sz) as listed Table 3.2. The
range permeability estimates are obtained from analysis in 3.3.4
46

3.4 Thermal conductivity
3.4.1 Background
Saturated effective thermal conductivity (Kef f ) of isotropic porous foams such as metal foams
have been shown to depend on microstructure properties such as pore size, pore-density,
ligament size, and porosity( [51, 30]). However, characterization of Kef f for structured
anisotropic porous lattices is limited.
For a simplified one-dimensional conduction case, the upper and lower bounds of the
effective steady-state thermal conductivity of the solid-fluid medium can be derived using
the electrical resistance analogy and are given as follows:
Kef f,max = Ks(1 − ϵ) + Kf ϵ (3.24)
Kef f,min =

ϵ
Kf
+
1 − ϵ
Ks
−1
(3.25)
The thermal conductivity of packed beds with porosities between 0.3 and 0.5 can be
accurately predicted using the upper bounds of Kf and Ks. Nevertheless, these expressions
are not as reliable when the porosity and thermal conductivity of the solid phase are high.
Furthermore, previous research has been restricted to packed beds of spheres or stochastically
isotropic foams.
Kef f , has been experimentally investigated with different solid-fluid interfaces for high
porosity metal foams. Calmidi and Mahajan [8] reported the measurements of thermal conductivity with Aluminium foams with air and water both as the fluid medium at porosities
ϕ > 0.9 and also developed a simplified 2D model considering the foam to be hexagonal.
They observed no systematic effect of pore density variation on the effective thermal conductivity. Bhattacharya [3], extended this model to include circular blobs of metal at the
intersection. Boomsma and Poulikakos [5] developed an analytical model considering a three47

dimensional cell geometry of the foam using a tetrakaidecahedron. In all these cases, the
authors concluded that for a higher solid-fluid thermal conductivity ratio, increasing the
thermal conductivity of the solid phase is essential to see a marked increase in the effective
thermal conductivity of the composite system. In the case of metal foam heat exchangers,
the thermal conductivity of the solid phase Ks > 200W/m − K is much higher than that of
the fluid phase Kf < 1W/m − K, making the contribution of the fluid phase in the overall
conduction negligible.
There have been many attempts to create a model for thermal conductivity based on
the microstructure of porous materials [23, 25, 24, 46, 51]. Some efforts have been made to
model the microstructure of metal foams as a unit cell. For high porosity foams (ϵ > 0.9),
popular choices include two-dimensional hexagonal unit cells, representative cubic unit cells
with nodes, and 3D tetrakaidecahedron structures (see Fig. 3.10). In addition, recent efforts
have focused on modeling the unit cell of metal foams using lattice framed structures [50, 16].
The Octet lattice has demonstrated potential as a model for high porosity open cell metal
foam. It has been proposed that the Octet truss geometry can be viewed as an organized
form of metal foam geometry, offering both fluid flow and heat transfer properties as well as
structural advantages. Thanks to advancements in additive manufacturing techniques, it is
now feasible to 3D print the required unit cell geometry instead of constructing it as a unit
cell for a complex design. As a result, establishing a link between Kef f and microstructure
parameters is highly desirable.
Subsequent subsections are organized as follows. We will first introduce the experimental
method and ANSYS Fluent simulations we used to evaluate the thermal conductivity of
anisotropic foams. We then compare the values obtained against previous predictions for
lattice structures and suggest avenues for improvements.
48

Figure 3.10: Metal foam unit cell modeled as a (a) 2D Hexagonal unit cell with blobs at the
intersection [3], (b) 3D Octet lattice [16]
3.4.2 Experimental setup
Fluid-saturated thermal conductivity was experimentally characterized for all samples in
Table 3.1 for a constant d = 0.4 mm. These were then compared against values obtained from
ANSYS Fluent simulations, obtained both in the Z and X direction. Although experiments
were only conducted with water (Kw/Ks ≈ 3) as the working fluid, Air (Kair ≈ 0.6), and
Aluminum Kal ≈ 200 was also considered in the Fluent simulations to study the effect of
solid-to-fluid thermal conductivity ratios on the effective thermal conductivity.
Figure 3.11: Schematic of the thermal conductivity experiment
A custom-made experimental setup is used to estimate effective thermal conductivity
49

of such saturated porous samples with a cross-sectional area perpendicular to the heat flux
A = 76.2mm × 50.8mm having a thickness ∆x = 6.4 mm as shown in Fig. 3.11. These are
placed in an insulated cavity and saturated with water. The sample is attached to copper
plates on the top and bottom using a thermally conductive paste (Arctic Silver Ceramique)
to ensure an even heat flux distribution. The top plate is heated from the above using a
10 W Silicon Rubber Heater (SRFG-203/10 from Omega Inc), whose heat flux is controlled
externally using a DC Power supply (GW Instek PSP-405). The lower copper plate is
immersed in a large water bath and maintained at a constant temperature. The temperature
difference across the copper plates was measured using miniature Type-K thermocouples of
0.13mm diameter (5TC-GG-K-36-36 from Omega Inc.). The thermocouples are attached
to a Data Acquisition System (TC-08, Omega Inc), allowing live monitoring of individual
temperatures.
Effective thermal conductivity measurements were generated as follows. The sample
material was attached between the copper plates using thermally conductive paste, and the
patch heater was used to provide near-uniform heat input, Q, to the upper copper plate.
The temperature difference across the samples, ∆T, was measured by three thermocouples
attached to the top and bottom plates. The thermal conductivity of the sample was calculated based on the temperature difference, the dimensions of the sample, and the power of
heater applied as discussed below.
Before measuring thermal conductivity, the porous samples were soaked in water for
over 3 hours to remove any air bubbles in the pores. Then, the sample slot was filled with
water and the sample was placed in it. To ensure minimal thermal resistance in the intended
heat transfer direction, the bottom plate temperature was stabilized at around 10◦C. Once
equilibrium was reached, the heater was turned on and the data acquisition system (DAQ,
Omega TC-08) started recording. It took 20-40 minutes for the system to reach a steady
state for each measurement (i.e., a nearly constant temperature difference). Temperature
readings for the upper and lower plates were collected for 10 minutes at a rate of 1 Hz
50

(600 data points). Throughout this interval, the temperature difference remained relatively
consistent within ±0.5
◦C.
To determine the effective thermal conductivity, Kef f , two methods were employed.
The first method involved using the heat conduction equation directly:
Q =
Kef fA∆T
h
, (3.26)
where Q represents the power output, A is the cross-sectional area of the sample (i.e., the
area over which conduction takes place), h is the length of the sample in the direction of heat
transfer, and ∆T is the temperature difference between the upper and lower surfaces. The
power input was controlled by controlling the voltage input, V , to the heater. An estimate
of Q was obtained using Ohm’s law:
Q =
V
R2
, (3.27)
where R is the known resistance of the patch (78.4 Ω). For each sample, temperature
differences were measured for 3 to 6 different heat inputs, and the resulting effective thermal
conductivity was estimated. The overall effective thermal conductivity of the sample was
obtained by averaging the values acquired from each heat input.
An alternative approach is to plot ∆T against Q for three to six different measurements,
and then carry out a linear regression analysis. The regression line slope, m, can be translated
into an effective conductivity estimate using:
Kef f =
Hp
Am. (3.28)
Here Hp is the height of the porous medium The results of both methods were compared
and were consistent with one another within uncertainty.
Finally, buoyancy effects can be estimated via the Grashof number in equation 3.29,
51

which represents the ratio of buoyancy forces to viscous forces in the porous material:
Gr =
gβ∆T D3
ν
2
. (3.29)
In this study, the working fluid was water, which has a coefficient of expansion of β ≈
0.21 × 10−3 K−1 and kinematic viscosity ν ≈ 10−6 m2/s. As noted below, the characteristic
pore-scale length scale is D ≈ 1 mm and the temperature difference is expected to be
∆T ≈ 10 K. The acceleration due to gravity is g = 9.81 m/s2
. For these conditions, we
expect Gr ≈ 20, which is below the threshold that gives rise to natural convection. In
addition, all the experiments were carried out with the heated surface on top, i.e., with
stable density stratification. As such, we do not anticipate buoyancy-driven convection to
arise in the experiments.
3.4.3 ANSYS Fluent Setup
The unit cell in Fluent is simulated as follows. For thermal conductivity in the Z direction,
the top Z plane is set a heat temperature of Tt = 320 K, and the bottom is set to a constant
temperature boundary condition of Tb = 300K. The X and Y planes are given periodic
boundary conditions. The average heat flux in x direction is determined, and using the
one-dimensional steady-state heat equation; the thermal conductivity is evaluated.
3.4.4 Results
In this section, we will be presenting the thermal conductivity measurements for various
porous materials. Initially, we measured the thermal conductivity of the solid sample, which
was found to be Ks = 0.32 ± 0.03 Wm−1K−1
. Moving forward, we will be normalizing
the effective thermal conductivity by Ks to present the results in dimensionless form. We
will first discuss the results for isotropic cases followed by anisotropic cases. We will also
compare the experimental results with the numerical predictions made using ANSYS Fluent
52

and empirical correlations for metal foams and octet lattices ([8, 16, 64]).
Isotropic foams
(a) (b)
Figure 3.12: Variation of normalized saturated thermal conductivity Kef f /Ks with volumetric porosity ϵ (a) (0 < ϵ < 1), (b)zoomed-in to show 0.6 < ϵ < 1
The measured thermal conductivities for the isotropic samples (LLL, MMM, HHH) are
shown in Fig. 3.12. In general, these measurements show that the normalized effective
conductivity increases with increasing porosity. This is consistent with expectations since
the fluid medium (water, Kf ≈ 0.6 Wm−1K−1
) is more conductive than the solid phase in
these experiments (resin, Ks ≈ 0.3 Wm−1K−1
). Specifically, the dimensionless conductivity
increases from Ke/Ks = 1.6 ± 0.03 for the lowest porosity LLL sample with ϵ = 0.72 ± 0.01
to Ke/Ks = 1.78 ± 0.04 for the highest porosity HHH sample with ϵ = 0.91 ± 0.01.
Our obtained measurements are consistent with both the upper and lower bounds in
equations 3.24 and 3.24. It’s worth noting that our measurements align well with the predictions from the empirical model presented by Calmidi and Mahajan [8] and Wang et al. [64].
However, it’s important to keep in mind that these models were both derived empirically for
53

high-porosity metal foams and isotropic lattices, respectively. As a result, we can observe
that the empirical predictions for the model by Calmidi and Mahajan [8] make predictions
lower than the theoretical lower bound for ϵ ≤ 0.85, and predictions made by Wang et al.
[64] make predictions lower than the theoretical lower bound for ϵ ≤ 0.65. Physically, this
is not possible, so these models have limitations in the range of porosities that they can be
applied to.
While the lumped parameter model (denoted in figure 3.12 by sx = sy = sz) captures
the overall trends, such as increasing conductivity with porosity, there is some discrepancy
in magnitude. This can be attributed to differences in geometry between the CAD and
nominal specifications and the actual printed model. In particular, the as-printed specimens
had lower porosity (ϵ ≈ 0.7 − 0.9) than the designed geometries (ϵ ≈ 0.8 − 0.95) due to
printing tolerances. This directly leads to a reduction in effective conductivity.rical model
presented by Calmidi and Mahajan [8]. The lumped parameter model captures the overall
trends reasonably well (i.e., increasing conductivity with porosity) though there is some
disagreement in magnitude. This can be attributed to differences in geometry between
the CAD and nominal specifications and the as-printed model. Specifically, the as-printed
specimens generally had much lower porosity (ϵ ≈ 0.7 − 0.9) compared to the designed
geometries (ϵ ≈ 0.8 − 0.95) due to printing tolerances. This contributes directly to reduced
effective conductivity.
Anisotropic foams
The findings for the anisotropic cases are presented in Figure 3.13. In each of these cases, the
planar porosity in the direction of heat transfer remains constant, indicating that they have
identical in-plane spacing. For example, for the planar configuration LL, we examined cases
LLL, LLM, LLH, and so on. There is a noticeable trend with porosity, but unfortunately,
the experimental results do not match the ANSYS Fluent simulations since we obtained
simulation results from nominal pore sizes. The geometry of the as-printed materials had
54

uncertainties in measurements due to tolerance accounting, resulting in a lower measured
porosity value compared to its nominal size.
For all samples with planar configuration LL (Figure 3.13a), there is a reasonable agreement with the predictions from Calmidi and Mahajan [8] and Wang et al. [64]. This is
consistent with the assumption that these models are accurate for high porosity foams.
However, as we decrease the planar porosity in the cases of MM and LL (Figures 3.13b
and 3.13c), the models are no longer accurately able to predict thermal conductivity values.
Furthermore, we see that ANSYS Fluent predictions match closely with lumped parameter
predictions which do not match with experimental predictions. Overall, this suggests a need
for the development of more models that can be used to predict thermal conductivity.
55

(a) Planar configuration HH (b) Planar configuration MM
(c) Planar configuration LL
Figure 3.13: Above figures show thermal conductivity estimates with volumetric porosity by
varying spacing while keeping planar porosity in the direction of heat conduction constant z
56

3.5 Conclusion
In this chapter, we introduce a family of anisotropic lattices that can be easily 3D printed.
By varying four independent parameters (d, sx, sy, sz), we can systematically create the desired anisotropy to form a periodic unit cell. We also discuss the limitations of this geometry
in terms of achievable planar porosities. To accurately predict the permeability and thermal conductivity of these lattices, we utilize a combination of experiments, ANSYS Fluent
simulations, and lumped models.
In order to predict permeability for various lattices, we propose a straightforward model
based on the hydraulic diameter of the unit cell. Our predictions are validated through
ANSYS Fluent simulations, which show that they are accurate within approximately 20%
for all geometries tested. Additionally, we construct a custom experimental setup to measure
permeability values. Our findings indicate that anisotropy affects the pressure gradient
experienced by the cube. While the model and ANSYS predictions are in agreement with
reasonable uncertainty, experimental predictions tend to be lower than the model predictions
obtained through measurement. This emphasizes the limitations of physically realizable
materials when characterizing permeability values, which must be taken into account when
designing and predicting permeability values using additive manufacturing.
In order to determine the effective thermal conductivity of a unit cell, we used a specific
set of 3D printed anisotropic cells. Our approach involved a combination of experimental
testing and ANSYS Fluent predictions, which we then compared to models for high-porosity
metal foams and isotropic lattices found in existing literature. To conduct our experiments,
we developed a setup that measured steady-state thermal conductivity using 3D-printed
materials (specifically, formlabs clear resin) and water as the working fluid.
Porosity emerges as a major factor in determining the thermal conductivity of the
material. Generally, an increase in porosity leads to an increase in thermal conductivity
for the material-fluid combination used in this study, including 3D-printed samples made
57

from resin and saturated with water, where the thermal conductivity of the solid phase is
lower than that of the fluid phase. This is in agreement with previous research, which found
that the thermal behavior of metal foams can be accurately predicted based on porosity.
The lumped parameter model used in this study did not produce predictions as accurate as
the empirical model, likely due to the fact that the empirical model uses measured porosity
values of the printed samples, while the lumped parameter model only uses the CAD-specified
geometry.
The printed samples had slightly different spacing in various dimensions, resulting in
lower porosity values than specified in the geometries. Additionally, manufacturing tolerances may vary depending on the print direction. Future studies could focus on further investigating the impact of printing parameters on the thermal behavior of additivelymanufactured materials, such as different printing orientations and the potential of new
printing techniques and materials to achieve enhanced thermal properties. Additionally, further studies could examine the impact of other factors on the thermal conductivity of porous
materials, including the shape and size of the pores as well as the relative conductivity of
the solid and fluid phases.
58

Chapter 4: Passive flow control with anisotropic porous
walls
4.1 Introduction
Patterned and porous surfaces have shown significant promise as a method of passive turbulence control. For instance, streamwise aligned riblets have demonstrated drag reductions
upto 10% in laboratory experiments ([20, 21]). Recent advances in additive manufacturing
have enabled the manufacturing of porous geometries offering control over the design of the
microstructure. It has previously been shown that [16] the Octet truss geometry can be
treated as an ordered form of metal foam geometry for fluid flow and heat transfer properties while simultaneously offering structural advantages. Moreover, the periodicity of these
structures allows for the optimization of geometry for a specific set of applications.
Here, we investigate the effect of anisotropy on the drag response of porous substrates.
Recent theoretical and numerical simulations indicate that materials with streamwise preferential anisotropy – substrates where the permeability is higher in the streamwise direction
(κxx) as compared to the spanwise (κzz) and wall normal (κyy) directions – have the potential to lower drag. Previous simulations results have predicted that as much as 25% drag
reduction can be achieved by utilizing streamwise preferential porous substrates through
a physical mechanism similar to that observed over riblets ([1, 53]). Furthermore, formation of KH-type instability is predicted to be a mechanism for degradation of performance.
However, it remains to be seen if the trends observed in the numerical simulations hold for
physically-realizable materials.
In this chapter, we start by briefly discussing the theory behind the drag reduction observed over anisotropic porous substrates. Guided by this theory, we design and manufacture
59

porous substrates with varying levels of anisotropy using additive manufacturing. Next, we
present our experimental results for friction factor on each of these substrates over a range
of bulk Reynolds numbers. This experiment also provides insight into the departure from
laminar behaviour for various anisotropies. My colleague Christoph Eftathiou designed and
manufactured the benchtop water channel facility. I tested the facility for flow development
over the smooth wall, created and manufactured the porous substrates, and collected and
analyzed the experimental data.
4.1.1 Drag reduction mechanism
Functional surfaces like those inspired by sharkskin, including riblets and denticles, represent
some of the most effective methods tested for reducing turbulent skin friction. These surfaces,
when appropriately designed in terms of shape and size, have demonstrated the capability to
reduce drag by as much as 10% in controlled laboratory experiments and up to 2% in realworld scenarios ([39, 20]). The underlying principle behind their drag-reducing effectiveness
lies in their anisotropic nature: they offer significantly less resistance to the mean flow in the
streamwise direction compared to turbulent cross-flows [39]. Within the grooves of riblets,
the mean flow in the streamwise (x) direction experiences minimal impedance, resulting
in a high interfacial slip. Conversely, cross-flows in the wall-normal (y) and spanwise (z)
directions originating from turbulence are obstructed by the riblets and forced farther away
from the wall.
This obstruction effect weakens the quasi-streamwise vortices associated with the energetic near-wall (NW) cycle [52], consequently reducing turbulent mixing and momentum
transfer above the riblets [13]. Initially, skin friction reduction increases as riblet spacing
and height increase. However, beyond a certain threshold size, performance deteriorates
significantly. Early studies attributed this decline to the NW vortices becoming trapped
within the riblet grooves [13, 35]. More recent research by Garcia-Mayoral and Jimenez [20]
has suggested that a Kelvin-Helmholtz (KH) instability may also contribute to the decline
60

in performance.
Anisotropic porous substrates have the potential to reduce drag in wall-bounded turbulent flows through a mechanism similar to riblets. This mechanism can be explained using
the virtual origin framework proposed by Luchini, Manzo, and Pozzi [39]. A general description of this framework is given below. Within this framework, a streamwise slip length
(l
+
U
) and a transverse slip length (l
+
T
) are defined to determine the virtual origin perceived
by the mean flow and turbulent cross-flow below the porous surface. The superscript +
denotes normalization with respect to the friction velocity (uτ ) and kinematic viscosity (ν).
The virtual origin for turbulent cross-flow can also be interpreted as the location where the
quasi-streamwise near-wall (NW) vortices perceive a non-slipping wall. According to Luchini, Manzo, and Pozzi [39], the virtual origins of mean flow and turbulent cross-flow are
set by those of streamwise and spanwise velocities, respectively, which means that l
+
U ≈ l
+
x
and l
+
T ≈ l
+
z
. The friction coefficient f for a wall-bounded channel flow can be defined as
f =
τw
1
2
ρU2
b
(4.1)
Here, τw is the wall shear stress and Ub is the bulk-averaged velocity of the flow in the
channel. U
+
b = Ub/uτ in the bulk-averaged velocity normalized by friction velocity. Drag
reduction (DR) can then be quantified as:
DR = −
f − fs
fs
(4.2)
Here, the subscript ’s’ indicates a reference smooth channel.
For vanishingly small surface textures, it has been shown that DR is essentially caused
by an offset between the positions of the virtual walls perceived by the streamwise mean
flow and the overlaying turbulence, ∆+
l = l
+
U − l
+
T
. Physically, when ∆+
l
is positive, the
surface is able displace turbulent cross-flows arising from the NW vortices into a region of
lower mean shear. This limits energy transfer from the mean flow to the turbulence and
61

suppresses the NW vortices. Evidence from previous numerical simulations and experiments
has shown that riblets reduce drag by damping cross-flow fluctuations and weakening the
effects of quasi-streamwise vortices. This occurs due to the higher spanwise resistance than
the streamwise resistance produced by riblets.
For small features, drag reduction can be expressed as
DR = −
f − fs
fs
∝ l
+
U − l
+
T
(4.3)
Previous experiments and numerical simulations show that ∆U
+ = m(l
+
U − l
+
T
), where ∆U
+
is called the Hama roughness function which represents the shift in the logarithmic region
of the mean velocity profile relative to smooth wall conditions [18, 21, 19]. Positive values
of ∆U
+ represent an increase in the mean velocity for fixed uτ values, and are therefore
indicative of drag reduction. Specifically, for small changes in friction, the drag reduction
can be quantified as
DR = p
2f0∆U
+ = m
p
2f0(l
+
U − l
+
T
), (4.4)
Here, f0 is the friction factor experienced by as nominally smooth wall. AbderrahamanElena and Garc´ıa-Mayoral [1] and G´omez-De-Segura and Garc´ıa-Mayoral [22] extended the
virtual origin framework to anisotropic permeable substrates and showed that streamwise
and spanwise permeabilities can be related to slip the lengths as l
+
x ≈
√
κxx
+
and l
+
z ≈
√
κzz
+
for deep substrates, i.e., for H+ ≫ l
+
U where H is the thickness of the porous substrate. This
implies that ∆U
+ ∝
√
κxx
+ −
√
κzz
+
and drag reduction can be achieved with anisotropic
materials with streamwise to spanwise anisotropy ratio, ϕxy = κxx/κyy > 1. This is obtained
by assuming that the idealized Darcy-Brinkman framework governs the flow within the
porous layer in response to the overlaying shear.
Although these models do not resolve the flow to the pore scale, they suggest that the
drag reduction is achieved when the material is highly streamwise anisotropic in order to
obtain a large √
κxx
+ −
√
κzz
+
. Previous numerical simulations have shown up to 25% drag
62

reduction over streamwise preferential porous substrates that yield larger effective slip lengths
for the streamwise mean flow compared to the turbulent cross-flows. Hence, slip length-based
models generate useful predictions for linear drag reduction regimes over anisotropic porous
substrates.
Linear stability analyses suggest that there is a limit to the drag-reducing capabilities
of permeable surfaces. A deterioration in performance is typically observed when the normalized wall-normal permeability (√κyy
+) is greater than 0.4. This is thought to be due to
the onset of a Kelvin-Helmholtz type instability, which limits the maximum achievable drag
reduction (Abderrahaman-Elena and Garc´ıa-Mayoral [1] and G´omez-De-Segura and Garc´ıaMayoral [22]). Similar to riblets, when permeable substrates are present, the linear drag
regime can break down due to the appearance of KH-like rollers, caused by a relaxation in
the wall-normal permeability. Numerical simulations indicate that the maximum drag reduction occurs at √κyy
+ ≈ 0.4, after which drag increases. Flow visualizations do not show
the signature streaky structure associated with smooth-walled flows, instead, streamwise periodic spanwise constant rollers resembling Kelvin-Helmholtz vortices are present. This is
broadly consistent with previous literature which suggests that flows over porous materials
are susceptible to a KH instability that gives rise to spanwise-coherent energetic rollers [6,
15]. Furthermore, resolvent analysis has also predicted the emergence of energetic spanwise
rollers [10] with streamwise wavelength λ
+ ≈ 150 as the wall-normal permeability increases
beyond √κyy
+ ≳ 0.4.
These large-scale rollers can have a significant impact on interfacial scalar and momentum transport. Chandesris et al. [9] and Nishiyama, Kuwata, and Suga [45] have demonstrated that the streamwise permeability has a more significant effect on the heat transfer
performance of a channel than the spanwise permeability. Chandesris et al. [9] also discovered that the large-scale temperature fluctuations in the porous medium region were caused
by the combined effects of large-scale velocity fluctuations due to the K–H instability and a
large temperature gradient. Nishiyama, Kuwata, and Suga [45]) demonstrated that the co63

herent large-scale structures in temperature fluctuations over the porous wall in cases with
streamwise preferential anisotropy are accompanied by the large-scale velocity structures
induced by the K–H instability. Thus, being able to account for – and control – the emergence of KH rollers is a vital step towards passively controlling turbulent flows for thermal
management and drag reduction.
4.1.2 Previous experiments with anisotropic porous substrates
It has been established that permeability modifies the structure and dynamics of turbulence
near the surface. Most prior studies of turbulent flows over porous materials have considered
beds of spherical particles or open-cell reticulated foams which are approximately isotropic.
There are few studies on anisotropic porous foams, and fewer so in wall-bounded channel
flow.
Suga et al. [59] carried out Particle Image Velocimetry over anisotropic porous media in
channel flow at bulk Reynolds number Reb ∈ [900, 13600]. They focused on materials with
higher wall-normal permeability as compared to streamwise permeability, i.e., materials with
anisotropy ratios ϕxy varying from 1/190 to 1/1.5. In all cases, a relative increase in friction
velocity was observed at the porous wall as compared to the smooth wall. Moreover, for the
case with lowest permeability, they observe that although the flow is laminar at Reb = 900,
from Reb = 1300 the mean velocity profile becomes asymmetric, indicating that transition
to turbulence occurs in this interval. All other cases were tested at Reb > 3400 where the
flow is already fully turbulent. Therefore, there is a dearth of data quantifying friction factor
for anisotropic porous substrates in the transitional regime.
Chavarin et al. [10] conducted experiments on turbulent flow over porous materials with
streamwise preferential permeability. These small-scale experiments examined the effect of
3D printed porous materials with both ϕxy < 1 and ϕxy > 1 at Reτ ≈ 120. The material
with higher streamwise permeability did not show a significant difference from smooth wall
conditions. However, as anticipated, the material with higher wall normal permeability
64

caused the emergence of Kelvin-Helmholtz type rollers and resulted in a substantial increase
in drag.
Efstathiou [14] performed experiments on 3D-printed materials with streamwise preferential permeability on turbulent boundary layer flows at Reτ ≈ 360 . For fully developed conditions, friction estimates show that the 3D-printed porous substrate led to a small
(< 5%) increase in drag. Further, PIV-based measurements of turbulence statistics and velocity spectra indicate that the observed drag increase can be attributed to the emergence of
energetic spanwise rollers resembling Kelvin-Helmholtz vortices. This translates into structures with streamwise wavelength λ
+
x ≈ 250 − 400, consistent with the range identified in
DNS simulations ([22]).
Recently, Morimoto et al. [41] conducted channel flow experiments on substrates with
high streamwise preference and ϕxy ≈ 18.3. Experiments were carried out at bulk Reynolds
numbers ranging from Reb ∈ [5000, 15000]. As predicted by DNS results, a drag increase was
observed when √κyy
+ > 0.48, corresponding to Reb ≥ 5000. However, no drag reduction
observed at Reb = 3000, which corresponds to √κyy
+ = 0.33. This lack of drag reduction
was attributed to an increase in wallnormal fluctuations, even at Reb = 3000. This suggests
that in addition to bulk properties such as permeability, the impact of anisotropic porous
materials on turbulent flows also depends on the specific interfacial geometry. Nevertheless,
these prior studies indicate that √κyy
+ is the dominant scale governing drag increase.
4.1.3 Contribution of this study
Previous experimental datasets are limited in the sense that friction velocity is not directly
obtained from an independent measure but inferred from other turbulent quantities such as
mean velocity or Reynolds shear stress profiles. This limitation does not apply to prior numerical simulations. However, numerical simulations have been restricted to low to moderate
Reynolds numbers, where the scale separation between outer and inner length scales might
not be sufficient. In this study, we leverage findings from Chapter 3 to systematically design
65

and manufacture anisotropic porous materials with varying streamwise, spanwise, and wallnormal permeability. We test these materials in a benchtop channel flow setup with direct
pressure drop measurements, which can be used to estimate changes in drag.
4.2 Methods
4.2.1 Porous substrates
Rapid advances in additive manufacturing techniques such as stereolithography 3D printing
techniques have made it possible to manufacture porous lattices with tunable microstructure.
Similar to Chapter 3, this work uses the cubic lattice geometry described by Chavarin et al.
[10] and Efstathiou [14] to generate anisotropic materials. Specifically, we vary rod spacing
in the streamwise (sx), wall-normal (sy) and spanwise (sz) directions to create a family of
anisoptropic materials. Fabrication constraints (printing resolution, allowable unsupported
lengths, and resin drainage) dictated the minimum rod spacing. Maximum spacing and pore
size was limited by the maximum allowable overhang (or unsupported length) for the 3Dprinted parts before sagging was observed, causing the geometry to deviate from its designed
dimensions. The goal was to independently capture the effects of variation of pore size in
each direction.
The parameter space shown in Table 4.1 is considered in the lab experiments. Each 3Dprinted geometry is referenced using a 3-letter combination that represents spacings in the
x, y, and z dimensions. The letter H (High) corresponds to the largest streamwise spacing
possible (s = 3 mm), M (Medium) and L (Low) correspond to intermediate spacings (s = 2
mm and s = 1.5 mm, respectively) and T (Tiny) corresponds to the lowest spacing possible
(s = 0.8 mm) before individual pores started fusing. For example, a substrate with nominal
spacings (sx, sy, sz) = (3, 1.5, 3) in mm is labeled as HLH. For all cases, the nominal rod size
is kept constant at d = 0.4 mm.
66

d (mm) sx (mm) sy (mm) sz (mm) √
κxx (mm) √κyy (mm) √
κzz (mm) Marker
HHH 0.4 3 3 3 0.79 0.79 0.79
(0.49 ± 0.12) (3.06 ± 0.28) (2.98 ± 0.26) (3.1 ± 0.26) [0.62,0.82] [0.62,0.83] [0.61,0.83]
MMM 0.4 2 2 2 0.37 0.37 0.37
(0.44 ± 0.04) (1.86 ± 0.13) (1.89 ± 0.12) (1.88 ± 0.12) [0.17,0.45] [0.17,0.45] [0.17,0.45]
LLL 0.4 1.5 1.5 1.5 0.2 0.2 0.2
(0.48 ± 0.07) (1.56 ± 0.17) (1.63 ± 0.17) (1.51 ± 0.17) [0.05,0.40] [0.05,0.40] [0.05,0.40]
MHH 0.4 2 3 3 0.62 0.55 0.55
(0.43 ± 0.03) (2.09 ± 0.11) (3.00 ± 0.08) (2.96 ± 0.08) [0.48,0.79] [0.40,0.71] [0.40,0.72]
LHH 0.4 1.5 3 3 0.59 0.44 0.44
(0.37 ± 0.07) (1.5 ± 0.2) (2.9 ± 0.2) (2.8 ± 0.2) [0.31,1.09] [0.16,0.94] [0.15,0.94]
THH 0.4 0.8 3 3 0.41 0.13 0.13
(0.44 ± 0.11) (0.84 ± 0.09) (3.02 ± 0.10) (3.09 ± 0.10) [0.28,0.57] [0.03,0.27] [0.03,0.27]
HHM 0.4 3 3 2 0.55 0.55 0.62
(0.43 ± 0.03) (2.96 ± 0.08) (3.00 ± 0.08) (2.09 ± 0.11) [0.40,0.72] [0.40,0.71] [0.48,0.79]
HHT 0.4 3 3 1.5 0.13 0.13 0.41
(0.44 ± 0.11) (3.09 ± 0.10) (3.02 ± 0.10) (0.84 ± 0.09) [0.03,0.27] [0.03,0.27] [0.28,0.57]
HLH 0.4 3 1.5 3 0.44 0.59 0.44
(0.45 ± 0.03) (3.02 ± 0.08) (1.51 ± 0.09) (3.06 ± 0.09) [0.16,0.94] [0.31,1.09] [0.15,0.94]
HTH 0.4 3 0.8 3 0.13 0.41 0.13
(0.44 ± 0.04) (3.09 ± 0.11) (0.87 ± 0.09) (3.09 ± 0.09) [0.03,0.27] [0.28,0.57] [0.03,0.27]
MLM 0.4 2 1.5 2 0.28 0.32 0.29
(0.44 ± 0.03) (2.09 ± 0.11) (1.65 ± 0.15) (2.03 ± 0.13) [0.16,0.44] [0.2,0.46] [0.16,0.44]
Table 4.1: Parameter space used for the experiments. In each case, the first row indicates
the nominal sizes, while the values in parentheses show the measured sizes. The range for the
permeability values represents the maximum and minimum values predicted by the (3.20)
based on the measured sizes.
4.2.2 Setup
The aim of this experiment was to analyze the influence of substrate anisotropy on friction
factors and to quantify the relative drag increase or decrease compared to smooth surfaces.
67

Figure 4.1: Schematic side view and top view of the experimental setup (not to scale)
For this, a benchtop channel flow setup was constructed where the test section allowed for
permeable substrates to be flush-mounted adjacent to an unobstructed flow. The friction
factor was characterized by measuring the pressure drop across the substrates in the fully
developed region of the channel for a range of Reynolds numbers. A schematic of the
experiment is shown in Fig. 4.1.
The benchtop channel comprised an upstream flow conditioning section with flow
straighteners and a 2:1 contraction to ensure uniform flow in the test section. An acrylic
test section was created with a cut-out of length L = 700mm to ensure the substrate was
flush with the upstream smooth wall. The width of the test section was W = 80 mm and
the height of the unobstructed channel was Hf = 6.34 mm, providing an aspect ratio of
W/Hf ≈ 12. Seven pressure taps, each measuring 2 mm in diameter, were bored into the
housing along the centerline, with the first pressure tap placed 94 mm from the test section
entrance and subsequent pressure taps being 101 mm apart from each other. These pressure
taps allowed the measurement of pressure drop at various locations in the test section to
68

identify the region of the test section with fully developed flow, i.e., the region characterized
by a linear decrease in pressure with distance.
Flow in the channel was generated using a submersible pump placed in a water tank.
The volumetric flow rate was manually adjusted using a ball valve and measured with a
washdown flowmeter (McMaster-Carr) with an accuracy of 2%. The volumetric flow rate
varied from V˙ = 50 cm3
s
−1
to V˙ = 250 cm3
s
−1
. The bulk velocity was calculated using the
unobstructed channel height as Ub = V /H ˙
fW. This resulted in bulk Reynolds numbers in
the range of Reb = UbHf /ν ∈ [500, 4000]. These bulk Reynolds numbers span both laminar
and turbulent flow conditions.
4.2.3 Experimental procedure
Identification of fully-developed region
According to guidelines for smooth-wall development, flow is considered fully developed at a
distance of more than 50 times the hydraulic diameter (≈ Hf in this case) [7, 66]. However,
there is limited information on development length over porous substrates. In previous
experiments on anisotropic porous walls in a channel flow, Suga, Nakagawa, and Kaneda
[58] conducted PIV experiments at ≈ 100Hf from the entrance and observed flow conditions
that were fully developed for an aspect ratio of W/Hf ≈ 10. On the other hand, Efstathiou
and Luhar [15] observed that flow can be considered fully developed over high porosity
isotropic foams in a flat-plate boundary layer at approximately ≈ 44Hp from the entrance,
as was also observed for boundary layer flows over streamwise preferential anisotropic porous
walls [14]. As seen in Chapter 2, flow is still developing at L/Hf ≈ 22 over high porosity
isotropic metal foams. Furthermore, in the benchtop channel flow experiments performed by
Chavarin et al. [10] for L/Hf ≈ 30 , flow appears fully developed within the PIV window.
Therefore, it is crucial to identify the region of the test section for which the flow can be
considered fully developed while ensuring that the pressure gradient can be measured with
reasonable experimental uncertainty.
69

Figure 4.2: Figure above shows the variation of pressure gradient dP/dx over a smooth wall
in between different pressure ports for two values of Reb
To determine an appropriate measurement region, we measure the pressure gradient
between the pressure port closest to the exit (port 7) and all other ports for a reference smooth
wall, as shown in Figure 4.1. In Figure 4.2, ’1-7’ represents the pressure gradient measured
between pressure port 1 and pressure port 7, ’2-7’ represents the pressure gradient measured
between ports 2 and 7, and so on. There is a noticeable variation in pressure gradient between
’1-7’, ’2-7’, and ’3-7’ for both Reynolds numbers tested. However, for ports ’4-7’, ’5-7’, and
’6-7’, the measured pressure gradient converges to within ≈ 10%. Note that the pressure
gradient between ’6-7’ has relatively measurement uncertainties due to the smaller pressure
drop expected between closely-placed ports. Note that port 4 is located approximately 60Hf
downstream of the entrance. This provides sufficient development length as compared to
70

smooth wall guidelines and is double the development length of Chavarin et al. [10]. Hence,
all subsequent pressure gradients reported in this chapter for the different porous samples
are based on pressure differences measured between pressure port 4 and pressure port 7.
Experimental procedure
All substrates listed in Table 4.1 were tested using the same procedure. The substrates were
placed flush into the cutout and fluid was pumped through the test section. The flow rate
was adjusted manually using the ball valve, beginning with the lowest measurable velocity
and increasing systematically in intervals until the maximum flow rate was reached. This
was then repeated in the direction of decreasing flow rate. Approximately 80 points were
collected for each flow rate. For each flowrate, the pressure difference was measured for
30 seconds at a frequency of 4166 Hz and logged to a PC workstation using a National
Instruments Data Acquisition Device after a 30 second settling period. A pressure difference
was measured at zero flowrate before and after each run to account for any Zero Offset in
the transducer. The Zero Offset did not vary by more than 2% across all samples tested.
Uncertainty quantification
There are two main sources of uncertainty for the friction factor measurements: pressure drop
and flow rate measurements. The variation in porous microstructure due to fabrication limitations is another source of potential uncertainty since permeability can vary substantially
as a function of microstructure. To determine the uncertainty in mean pressure transducer
measurements, we used the maximum of two values: (1) the standard error of each 30-second
test run, consisting of 120,000 samples taken at 4166 Hz and (2) the maximum uncertainty
of the pressure transducer PX-409, which is 5.5 Pa. For each run, the uncertainty in flow
rate measurement was 2%. Using the standard Klein-McClintock method for uncertainty
propagation, we estimated the absolute uncertainty in friction factor. Additionally, there is
a significant source of error due to the deviation of desired microstructure dimensions from
71

the designed ones, as shown in Table 4.1. This means that the permeability estimates used
to interpret some of the results presented below suffer from significant uncertainty.
4.3 Results and Discussion
4.3.1 Smooth wall reference measurement
To obtain a reference measurement, a smooth, solid wall is placed flush into the cutout.
Figure 4.3a shows the pressure gradient across ’4-7’ for the smooth wall at various values of
bulk averaged velocity. The flow transitions from laminar flow across the range of velocities
tested, as evidenced by the linear region in the pressure gradient below Ub ≈ 0.2 m/s
(Reb ≈ 1600) and a near-quadratic trend beyond this value. The pressure gradient is used
to compute a Darcy-Weisbach friction factor:
f
∗
s =
dP
dx Dl
1
2
ρU2
b
. (4.5)
In this equation, Dl
is the characteristic length scale (e.g., hydraulic diameter). For rectangular ducts, a laminar equivalent diameter was proposed by Jones in 1976 to ensure geometric
similarity between circular and rectangular ducts. A modified Reynolds number Re∗
b
is defined using this equivalent diameter to allow the relationship f
∗ = 64/Re∗
b
, expected for ducts
with circular cross-sections, to be easily applied to laminar flows in ducts with non-circular
cross-sections. For the present analysis, Dl
is defined as follows:
Dl =

2
3
+
11
24
Hf
W

2 −
Hf
W
Dh. (4.6)
Here, Dh = 4HfW/2(Hf + W) is the hydraulic diameter.
Figure 4.3b compares friction factors estimated from the pressure drop measurements
against model predictions for f
∗
. For low Reynolds numbers, the measurements agree with
the theoretical prediction f
∗ = 64/Re∗
b within uncertainty. The flow transitions to turbulence
72

(a) (b)
Figure 4.3: (a) Variation of pressure gradient with bulk velocity for a smooth wall. We
see a laminar flow region until Ub ≈ 0.2 m/s as shown by the linear fit after which the
flow transitions to turbulence. b) Variation of friction factor with bulk Reynolds number
compared with empirical predictions for a smooth laminar duct f
∗ and correlations developed
by Cheng [12]
for Reb > 1600 and for higher Reb values, the friction factor asymptotes to a value that aligns
well with the empirical correlation proposed by Cheng [12]. Together, these measurements
confirm that the benchtop channel and pressure measurements system can reproduce prior
results for smooth-wall conditions.
4.3.2 Porous wall friction factor measurements
Each porous sample listed in Table 4.1 was flush-mounted into the cutout and the pressure
gradient was measured across ’4-7’. A representative plot of the pressure gradient with bulkaveraged velocity for HHH is displayed in Figure 4.4a. The friction factor for each datapoint
can be determined using the following equation:
f =
dP
dx Hf
1
2
ρU2
b
(4.7)
73

(a) (b)
Figure 4.4: (a) shows the variation of pressure gradient with bulk velocity for HHH. (b)
demonstrates the process of linear fitting to estimate the value of friction factor.
where Hf is used as the characteristic length scale. This is consistent for all subsequent
calculations as the laminar equivalent diameter is only applicable to channel flow with smooth
walls; its validity for porous walls is uncertain. Figure 4.4b shows that there is a linear
relationship between 1
Ub
dP
dx and Ub. With this transformation, the quadratic curve fitting
problem to predict the pressure gradient
dP
dx = aU2
b + bUb (4.8)
turns into a linear curve fitting problem that yields the constants a and b:
1
Ub
dP
dx = aUb + b. (4.9)
We can combine equation (4.7) with (4.8) to show that
f =
2Hf
ρ

a +
b
Ub

= C1 +
C2
Reb
(4.10)
74

where C1 = 2Hfa/ρ is an estimate for the asymptotic value of the Darcy-Weisbach friction
factor for large Reb. The second term C2/Reb represents the initial (laminar) relationship
between pressure drop and bulk velocity, with C2 = 2H2
f
b/µ.
Figure 4.5 compares the pressure gradient and friction factor for the isotropic, highspacing HHH case with the reference smooth wall (SW) measurement. HHH consistently
shows a higher pressure gradient, resulting in a higher friction factor for each Reynolds
number. Note that the shaded region in Fig. 4.5b represents the 95% confidence interval for
the friction factor obtained by the curve fit. Additionally, all the smooth wall data also uses
Hf as the characteristic length scale for consistency. Details of all anisotropic porous foams
are discussed in the following subsection.
(a) (b)
Figure 4.5: (a) Pressure gradient measurements for HHH and smooth wall (SW) reference.
(b) Variation of friction factor with Reynolds number for HHH and SW. Note that the shaded
region indicates 95% confidence interval in predictions.
75

(a) Isotropic cases (b) Varying rod spacings in x.
(c) Varying rod spacings in y. (d) Varying rod spacings in z.
(e) Varying microstructure orientation. (f) Role of wall-normal permeability
Figure 4.6: Comparison of friction factor estimates for different isotropic and anisotropic
materials. The curves here indicate quadratic fits obtained via equation 4.10 with the shaded
region indicating 95% prediction interval
76

4.3.3 Observations over isotropic substrates
Figure 4.6a illustrates the changes in friction factor with bulk Reynolds number for all
isotropic permeability cases (HHH, MMM, LLL) in comparison to smooth wall predictions.
As the Reynolds number decreases, all three cases reach a state of laminar behavior. However,
all cases show a deviation from laminar values at Reynolds numbers lower than the smooth
wall, which is indicative of an earlier transition to turbulence. HHH and MMM consistently
generate higher friction factors than LLL for all tested Reynolds numbers. Therefore, for
these isotropic cases, an increase in permeability leads to higher drag for a given Reynolds
number. For these substrates, the rod spacing varies from 1.5 mm (LLL) to 3.0 mm (HHH).
Thus, as the substrate pore size, and hence permeability, increases, the ratio of pore size to
spacing (Hp/s) approaches unity. At this large pore limit, the pore-scale Reynolds number
is large (s
+ =
uτ s
ν > 60) and the substrate thickness becomes comparable to pore-size
(Hp ≈ 2s). As a result, the substrate essentially acts as a layer of drag-producing largescale roughness and there is very limited separation between the pore scale and the outer
length-scale (i.e., Hf ) in the flow.
4.3.4 Observations over anisotropic substrates
When the spacing of the rods is reduced in the streamwise direction (sx) while maintaining
other dimensions, the spanwise and wall-normal permeabilities decrease. This results in
greater anisotropy. The measurements obtained here show that, for approximately constant
streamwise permeability, this reduction in spanwise and wall-normal permeabilities leads to
a decrease in friction.
Figure 4.6b shows that the friction factor curves for HHH, MHH, and LHH approximately converge for high Reynolds numbers. However, the THH curve remains significantly
lower than the curves for the remaining materials and is only marginally higher than the SW
reference case. The drag generated by THH is the lowest, followed by LHH and then MHH.
77

Decreasing sx restricts the pore size in the wall-normal and spanwise directions, i.e., reduces
κyy and κzz, while approximately maintaining κxx. Therefore, THH is the most ”streamwise
preferential” material and shows the smallest drag increase. For LHH, and MHH, an increase
in sx, leads to an increase in κyy and κzz. Despite being streamwise preferential, the THH
material does not yield drag reduction. This is consistent with previous observations made in
experiments utilizing similar porous materials to the THH case [10], where no drag reduction
was observed compared to a smooth wall. The increase in drag for the THH case could be
potentially due to two causes: (1) √κyy
+ is greater than the threshold for K-H rollers, and
(2) the geometry at the interface consists of a thick layer of horizontal rods, which is likely
to restrict the effective mean slip length, l
+
U
. The first effect is considered below. However,
the impact of interfacial geometry on slip and drag reduction is not evaluated in great detail.
Now consider Figure 4.6d, where we vary spacing in the spanwise direction. This figure
shows that the HHT material generated lower drag than the HHM and HHH materials,
which have larger values for the spanwise spacing sz. Limiting sz results in limiting pore size
for wall normal and streamwise flow. Consequently, HHT has the lowest κxx and κyy of the
materials shows in this figure. In principle, a low value of κxx is not a preferable configuration
to reduce drag. However, a low value of √κyy
+ limits the drag penalty. Moreover, for small
sz values, the dense upper layer of rods is aligned in the direction of the flow, which can be
beneficial in terms of the interfacial slip.
Figure 4.7 further illustrates the effect of √κyy
+. For all values of Reb tested, the
minimum possible √κyy
+ is ≈ 4 which is much greater than √κyy
+ ≈ 0.4. To calculate
√κyy
+ =
uτ
√κyy
ν
, we estimate uτ based on the momentum balance across an elemental
length dx in the unobstructed channel as ρu2
τ =
dP
dx Hf . Drag measurements for each case also
indicate no evidence of drag reduction as compared to reference smooth wall values. Based
on these observations, it may be the case that all the cases considered here are susceptible to
the emergence of spanwise rollers. In general, samples with a higher κyy, and hence a higher
√κyy
+, show a higher friction factor.
78

We can observe the effect of microstructure orientation on the drag response in Figure
4.6e. The streamwise preferential material (THH with κxx > κyy = κzz) has the lowest drag
penalty, as expected. HHT follows with the next best performance, likely due to its low κyy.
The HTH case experiences the greatest drag increase, which is mainly attributed to its large
sx and sz and thus large openings for the wall-normal flow (large κyy). In general, the drag
increases with increasing √κyy
+.
Figure 4.7: Variation of f with √κyy
+. The symbols and colors used in this figure are
consistent with those in Fig. 4.6
4.3.5 Departure from hydraulically smooth regime
There has been a sustained effort to model onset of transition over rough walls. Tools such as
the Moody diagram provide consolidated results that can be used by practitioners. However,
there is currently limited information available regarding the onset of transition across porous
walls. Suga et al. [57] have demonstrated that increasing wall permeability for stochastic
isotropic foams increases turbulence, and generally leads to a lower Reynolds number for
transition. More recent experiments on anisotropic porous foams [59] have shown that the
mean velocity profile remains parabolic at a bulk Reynolds number of 900, but complete
79

turbulence transition occurs at a Reynolds number of 3600 for foams with a porosity of
ϕ = 0.7 and permeability ratio ϕxy ≈ 0.006. The friction factor data obtained in this study
falls within the range of 500 < Reb < 4000, which is at the lower end of the turbulence regime.
As Reb ≈ 500 approaches, the fitted friction factor curves curves asymptotically approach the
hydraulically smooth condition for nearly all the materials tests. At the higher end of the Reb
range, the friction factor data asymptote towards a constant value that is higher than than
the smooth wall reference Using the estimates obtained from equation (4.9), as Ub approaches
infinity, f → 2aHf /ρ. To generate an approximate estimate for the Reynolds number, Reˆ
b, at
which flow over the porous walls transitions away from hydraulically smooth (laminar) regime
and towards turbulence, we identify the threshold for which f > 1.5fs, where fs represents
the smooth wall friction factor. Estimated values for Reˆ
b are listed in table 4.2. Though
1.5fs is an arbitrary cutoff, the Reˆ
b estimates show trends consistent with expectation.
Specifically, cases THH and HHT which have the lowest wall-normal permeability show the
highest Reˆ
b, i.e., the transition occurs later for these low κyy cases. The earliest transition
is observed for the high-permeability HLH and HHH cases. This is consistent with previous
studies which show earlier transition to turbulence for high-permeability foams [57].
Further research into anisotropic porous materials for drag reduction, thermal management, and noise control may benefit from a diagram similar to the Moody chart to predict
friction factor and onset of transition. These data can also provide insight into the choice of
Reb for future DNS simulations. Capturing flow structure in the transitional regime would
be particularly interesting.
4.4 Conclusion
This chapter presents the first dataset of measurements of friction factor over a wide range
of anisotropy for physically realizable (3D-printed) porous materials. Previous experiments
and numerical simulations have suggested the use of streamwise preferential substrates for
drag reduction. Models developed in these prior studies indicate that materials with high
80

a(÷103
) b(÷103
) Reˆ
b
HHH 4.98 0.46 540
MMM 5.16 0.33 680
LLL 4.35 0.33 940
MHH 4.87 0.37 660
LHH 5.30 0.40 580
THH 3.40 0.03 1300
HHM 4.74 0.40 650
HHT 3.50 0.22 1200
HLH 5.00 0.54 400
HTH 3.74 0.40 800
MLM 6.40 0.14 1000
Table 4.2: This table lists values of a and b as estimated from the linear curve fitting process
for each tile. Note that as Reb increases, f asymptotes to C1 = 2aHf /ρ. Reˆ
b = 1.5fs is
evaluated as the Reynolds number at which flow has transitioned from the hydraulically
smooth regime.
streamwise permeability and low spanwise permeability are promising for passive drag reduction. High wall-normal permeabilities can also compromise performance by triggering rollers
resembling KH vortices. However, prior numerical simulations have primarily considered
low to moderate Reynolds numbers and used the idealized Darcy-Brinkman equation in the
porous medium. Similarly, prior experimental datasets are limited in that friction velocities
are not directly obtained from an independent measure but inferred from other turbulent
quantities such as Reynolds shear stress measurements. Moreover, there are few experimental datasets available that systematically examine the impact of anisotropic porous materials
on drag. This chapter addresses this limitation by leveraging 3D-printing techniques to create a family of anisotropic materials and generates direct drag estimates for these materials
via pressure drop measurements.
For isotropic substrates, an increase in permeability (and pore size) leads to higher drag
for a given Reynolds number. At the large pore limit, the substrate acts as a layer of dragproducing large-scale roughness with limited separation between the pore scale and the outer
length-scale in the flow. While we do not observe drag reduction, porous substrates with
high streamwise permeability and low spanwise and wall-normal permeabilities resulted in
81

the least amount of drag compared to a reference smooth wall measurement. Specifically, the
THH material is perhaps the most ”streamwise preferential” case considered here (nominal
ϕxy > 3) and shows the smallest increase in drag. Increasing sx leads to an increase in κyy
and κzz for LHH and MHH. These materials further increase drag compared to a smooth
wall.
We estimate that the wall normal permeabilities for all tested cases exceeded the threshold (√κyy
+ > 0.4) for the emergence of energetic spanwise rollers similar to Kelvin-Helmholtz
vortices. This is consistent with the results that drag reduction was not observed in the current dataset. In addition, the specific interfacial geometry can also play a role by decreasing
the effective slip length for the streamwise mean flow. Our measurements indicate that
porous walls exhibit a departure from hydraulically smooth behavior at different values for
bulk Reynolds numbers depending on the geometry. Streamwise-preferential materials depart from smooth-wall behavior at higher values for bulk Reynolds number. Materials with
the highest wall-normal permeabilities exhibited departure from smooth-wall behavior at the
lowest Reb values.
The results of this chapter suggest that streamwise-preferential porous materials could
be a viable solution for reducing drag in turbulent wall-bounded flows. It is important to
be mindful, however, that the interfacial geometry should not lead to a decrease in slip.
Additionally, when the relative foam thickness s/h ∼ O(1), roughness and inertial effects
may contribute to an increase in friction. Furthermore, such materials may have potential
applications in improving heat transfer while minimizing frictional losses.
82

Chapter 5: Closing remarks
5.1 Conclusion
The goal of this thesis is to generate the foundational knowledge that will enable the development of structured porous materials for multi-functional passive flow control, particularly
in terms of heat transfer and drag reduction. Previous numerical and modeling efforts have
shown that porous walls change the structure and dynamics of turbulence near the surface,
offering possibilities for developing flow control strategies. For instance, previous numerical
simulations indicate that streamwise-preferential porous substrates may reduce drag in wallbounded turbulent flows. Yet, these observations have not been verified in experiments with
physically realizable materials.
The challenges addressed in this thesis are:
• Investigate how the permeability of a porous wall can enhance heat transfer at a systemscale.
• Examine how the micro-structure of porous materials affects both their permeability
and thermal diffusivity.
• Analyze the drag response of porous materials with anisotropic bulk permeabilities.
In Chapter 2, we tackle the first challenge by conducting experiments to study the
effect of isotropic wall permeability on heat transfer in partially porous channel flow. We
conducted a bench-top experiment involve heat transfer across commercially-available metal
foams over a range of Reynolds numbers in the transitional and turbulent flow regimes,
and used a system-scale approach to analyze the global heat exchanger performance of the
configuration. Our observations show that porous substrates with higher permeability exhibit
consistently higher Nusselt numbers. We utilized Particle Image Velocimetry to visualize
flow near porous walls and correlate heat exchange performance with the intensification in
83

near-wall turbulence due to permeability. Furthermore, we considered the implications of
increasing permeability to achieve higher heat transfer rates in terms of efficiency. We found
that for isotropic foams, increasing permeability comes at the cost of increasing friction and
pumping power requirements. This motivated us to design tailored porous microstructures
that achieve multifunctional turbulence control while minimizing drag.
In Chapter 3, we designed and fabricated porous media with anisotropic permeabilities
using stereolithographic 3D printing. Specifically, we propose a cubic lattice with a tunable
microstructure. We utilized a combination of experiments and ANSYS Fluent simulations to
characterize diagonal elements of the permeability tensor κ and saturated effective thermal
conductivity Kef f for these cubic lattice materials. Previous models for isotropic foams were
shown to be inadequate in predicting κ and Kef f for the custom anisotropic materials as
these models only rely on volumetric porosity. We propose and tested simple phenomological
models for permeability and thermal conductivity, and characterized the impact of the porous
microstructure on these thermophysical properties. Our findings indicate that variations in
porous microstructure due to manufacturing tolerances can result in order-of-magnitude
differences in the prediction of permeability.
In Chapter 4, we present the first set of friction factor measurements over anisotropic
porous walls in channel flow. We utilized geometries from Chapter 3 to manufacture porous
walls with a microstructure designs motivated by numerical studies that predicted dragreduction of streamwise preferential materials. Though we did not observe any drag reduction, we found that streamwise preferential substrates with κxx
κyy
> 1 had the lowest relative
increase in drag, which is consistent with prior results. High wall-normal permeabilities
√κyy
+ were a major factor in the observed drag increases. It is worth noting that previous numerical simulations predicting drag reduction utilized the idealized Darcy-Brinkman
formulation inside the porous medium, rather than explicitly accounting for the porous material microstructure. Designing porous materials that yield effective bulk properties and
interfacial slip coefficients that reduce drag in numerical simulations remains a challenge.
84

Despite this, our results provide a starting point for predicting the drag performance of
anisotropic materials based on bulk properties. The bulk Reynolds numbers considered in
the experiments fell in the transitional regime. As a result, the friction factor measurements
were used to characterize the departure from the laminar hydraulically smooth regime for
the different materials. These findings can guide future efforts on the design and evaluation
of porous materials for passive flow control, and inform the development of physical models
that can capture the effect of anisotropic materials on transitional and turbulent flows.
5.2 Outlook
With the aim of building a framework suitable for implementation in engineering systems,
there are still many unanswered questions in the development of passive flow control strategies using porous walls for drag reduction and enhanced thermal transport. While empirical
global measures like Nusselt number or friction factors for porous walls can offer initial design guidance, the findings suggest a physics-based foundation rooted in interactions between
porous substrates and turbulent boundary layers. Hence, the of development passive heat
transfer enhancement strategies over porous walls should include both: a deeper exploration
of system-scale behaviour using bulk properties such as permeability and saturated effective
thermal conductivity, to be necessarily complemented by perspectives on the structure and
dynamics of turbulence over a porous boundary layer. Some specific research directions along
these lines include:
Flow and thermal dispersion at a microscale level:
Additional research is needed to characterize the impact of pore-scale geometry on bulk
properties such as permeability, Forchheimer resistance, and effective thermal conductivity, particularly for anisotropic materials. Furthermore, fully resolved simulations of the
microscale flow at the pore scale are required to confirm whether the Darcy-Brinkman description is valid within the porous wall.
85

Turbulent boundary layers over anisotropic porous substrates:
A sustained experimental effort is needed to delineate the impact of permeability and interfacial roughness (or geometry). At present, it remains uncertain whether turbulent boundary
layers over porous media can be compared to those over rough walls. There is also growing evidence that a porous wall maybe treated as a permeable wall with surface roughness
[17]. The interfacial geometries play a particularly important role for low-porosity materials.
Most prior work focused on the development of passive turbulence control using porous or
patterned surfaces relies on slip length formulations. Thus it is impoerative to characterize
how these effective slip lengths for the overlying flow depend on bulk properties such as
permeability and the specific interfacial geometry.
Heat transfer over anisotropic porous substrates:
There is currently a lack of experimental data characterizing global heat exchange performance over anisotropic porous walls. System-level measurements are needed to estimate
quantities such as the Nusselt number and thermal efficiency so that they may be compared
with previous numerical predictions (for e.g. Nishiyama, Kuwata, and Suga [45]) However,
there are few manufacturing techniques that can be readily used to fabricate thermallyconductive anisotropic materials. Metal 3D printing is one possibility though it still suffers
from resolution limitations and it can be prohibitively expensive to generate 3D-printed
porous metal components on a large scale. Typically, the smallest ligament or feature size is
limited to ≈ 1 mm. The resulting large pore sizes necessitate scaling up the experiment. This
also motivates research in additive manufacturing techniques to explore the development of
thermally conductive filaments and resins that can enable consistent, low-cost printing of
heat transfer materials with feature sizes ∼ O(1) mm.
86

Bibliography
[1] Nabil Abderrahaman-Elena and Ricardo Garc´ıa-Mayoral. “Analysis of anisotropically
permeable surfaces for turbulent drag reduction”. In: Physical Review Fluids 2.11
(2017). issn: 2469990X. doi: 10.1103/PhysRevFluids.2.114609.
[2] Adrian Bejan. Convection Heat Transfer: Fourth Edition. 2013. doi: 10 . 1002 /
9781118671627.
[3] A. Bhattacharya, V.V. Calmidi, and R.L. Mahajan. “Thermophysical properties of
high porosity metal foams”. In: International Journal of Heat and Mass Transfer 45.5
(Feb. 2002), pp. 1017–1031. issn: 00179310. doi: 10.1016/S0017-9310(01)00220-4.
url: https://linkinghub.elsevier.com/retrieve/pii/S0017931001002204.
[4] K. Boomsma and D. Poulikakos. “The effects of compression and pore size variations
on the liquid flow characteristics in metal foams”. In: Journal of Fluids Engineering,
Transactions of the ASME 124.1 (2002), pp. 263–272. issn: 00982202. doi: 10.1115/
1.1429637.
[5] K. Boomsma, D. Poulikakos, and F. Zwick. “Metal foams as compact high performance
heat exchangers”. In: Mechanics of Materials 35.12 (Dec. 2003), pp. 1161–1176. issn:
01676636. doi: 10.1016/j.mechmat.2003.02.001.
[6] W. P. Breugem, B. J. Boersma, and R. E. Uittenbogaard. “The influence of wall
permeability on turbulent channel flow”. In: Journal of Fluid Mechanics 562 (Sept.
2006), pp. 35–72. issn: 00221120. doi: 10.1017/S0022112006000887.
[7] J Byme, A P Hatton, and P G Marriott. TURBULENT FLOW AND HEAT THE
ENTRANCE REGION OF WALL PASSAGE TRANSFER IN A PARALLEL. Tech.
rep.
[8] V. V. Calmidi and R. L. Mahajan. “The Effective Thermal Conductivity of High Porosity Fibrous Metal Foams”. In: Journal of Heat Transfer 121.2 (May 1999), pp. 466–471.
issn: 0022-1481. doi: 10.1115/1.2826001. url: https://asmedigitalcollection.
asme.org/heattransfer/article/121/2/466/415714/The-Effective-ThermalConductivity-of-High.
[9] M. Chandesris et al. “Direct numerical simulation of turbulent heat transfer in a fluidporous domain”. In: Physics of Fluids 25.12 (Dec. 2013), p. 125110. issn: 1070-6631.
87

doi: 10 . 1063 / 1 . 4851416. url: http : / / aip . scitation . org / doi / 10 . 1063 / 1 .
4851416.
[10] Andrew Chavarin et al. “Resolvent-based design and experimental testing of porous
materials for passive turbulence control”. In: International Journal of Heat and Fluid
Flow 86 (Dec. 2020). issn: 0142727X. doi: 10 . 1016 / j . ijheatfluidflow . 2020 .
108722.
[11] Zhengbing Chen, Chunbo Jiang, and Heidi Nepf. “Flow adjustment at the leading
edge of a submerged aquatic canopy”. In: Water Resources Research 49.9 (Sept. 2013),
pp. 5537–5551. issn: 19447973. doi: 10.1002/wrcr.20403.
[12] Nian-Sheng Cheng. “Formulas for Friction Factor in Transitional Regimes”. In: (). doi:
10.1061/ASCE0733-94292008134:91357.
[13] Haecheon Choi, Parviz Moin, and John Kim. “Direct numerical simulation of turbulent flow over riblets”. In: Journal of Fluid Mechanics 255 (1993), pp. 503–539. issn:
14697645. doi: 10.1017/S0022112093002575.
[14] Christoph Efstathiou. Developing Tuned Anisotropic Porous Substrates for Passive
Turbulence Control. Tech. rep. 2020.
[15] Christoph Efstathiou and Mitul Luhar. “Mean turbulence statistics in boundary layers
over high-porosity foams”. In: J. Fluid Mech 841 (2018), pp. 351–379. doi: 10.1017/
jfm.2018.57. url: https://doi.org/10.1017/jfm.2018.57.
[16] Pawankumar Ekade and Shankar Krishnan. “Fluid flow and heat transfer characteristics of octet truss lattice geometry”. In: International Journal of Thermal Sciences
137 (Mar. 2019), pp. 253–261. issn: 12900729. doi: 10.1016/j.ijthermalsci.2018.
11.031.
[17] L B Esteban et al. Mean flow of turbulent boundary layers over porous substrates.
Tech. rep.
[18] Karen A. Flack and Michael P. Schultz. “Roughness effects on wall-bounded turbulent
flows”. In: Physics of Fluids 26.10 (Sept. 2014). issn: 10897666. doi: 10 . 1063 / 1 .
4896280.
[19] R. Garc´ıa-Mayoral, G. G´omez-De-Segura, and C. T. Fairhall. “The control of near-wall
turbulence through surface texturing”. In: Fluid Dynamics Research 51.1 (2019). issn:
01695983. doi: 10.1088/1873-7005/aadfcc.
88

[20] Ricardo Garc´ıa-Mayoral and Javier Jim´enez. “Drag reduction by riblets”. In: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering
Sciences 369.1940 (Apr. 2011), pp. 1412–1427. issn: 1364503X. doi: 10.1098/rsta.
2010.0359.
[21] Ricardo Garc´ıa-Mayoral and Javier Jim´enez. “Hydrodynamic stability and breakdown
of the viscous regime over riblets”. In: Journal of Fluid Mechanics 678 (July 2011),
pp. 317–347. issn: 14697645. doi: 10.1017/jfm.2011.114.
[22] G. G´omez-De-Segura and R. Garc´ıa-Mayoral. “Turbulent drag reduction by anisotropic
permeable substrates-analysis and direct numerical simulations”. In: Journal of Fluid
Mechanics 875 (2019). issn: 14697645. doi: 10.1017/jfm.2019.482.
[23] C T Hsu, P Cheng, and K W Wong. A Lumped-Parameter Model for Stagnant Thermal Conductivity of Spatially Periodic Porous Media. Tech. rep. 1995. url: http :
//heattransfer.asmedigitalcollection.asme.org/.
[24] C. T. Hsu, K. W. Wong, and P. Cheng. “Effective stagnant thermal conductivity of
wire screens”. In: Journal of Thermophysics and Heat Transfer 10.3 (1996), pp. 542–
545. issn: 15336808. doi: 10.2514/3.825.
[25] C.T. Hsu, P. Cheng, and K.W. Wong. “Modified Zehner-Schlunder models for stagnant
thermal conductivity of porous media”. In: International Journal of Heat and Mass
Transfer 37.17 (Nov. 1994), pp. 2751–2759. issn: 00179310. doi: 10 . 1016 / 0017 -
9310(94 ) 90392 - 1. url: https : / / linkinghub . elsevier . com / retrieve / pii /
0017931094903921.
[26] Motoyuki Itoh et al. “Turbulent drag reduction by the seal fur surface”. In: Physics of
Fluids 18.6 (2006). issn: 10706631. doi: 10.1063/1.2204849.
[27] Nima Fallah Jouybari and T. Staffan Lundstr¨om. “Performance improvement of a solar
air heater by covering the absorber plate with a thin porous material”. In: Energy 190
(Jan. 2020), p. 116437. issn: 03605442. doi: 10.1016/j.energy.2019.116437. url:
https://linkinghub.elsevier.com/retrieve/pii/S0360544219321322.
[28] Nima Fallah Jouybari, T. Staffan Lundstr¨om, and J. Gunnar I. Hellstr¨om. “Investigation of thermal dispersion and intra-pore turbulent heat flux in porous media”. In:
International Journal of Heat and Fluid Flow 81 (Feb. 2020). issn: 0142727X. doi:
10.1016/j.ijheatfluidflow.2019.108523.
89

[29] James Klett. High Thermal Conductivity Mesophase Pitch-Derived Graphitic Foams.
Tech. rep.
[30] Prashant Kumar and Frederic Topin. State-of-the-Art of Pressure Drop in Open-Cell
Porous Foams: Review of Experiments and Correlations. Nov. 2017. doi: 10.1115/1.
4037034.
[31] Y. Kuwata, K. Tsuda, and K. Suga. “Direct numerical simulation of turbulent conjugate heat transfer in a porous-walled duct flow”. In: Journal of Fluid Mechanics
(2020). issn: 14697645. doi: 10.1017/jfm.2020.669.
[32] Yusuke Kuwata and Kazuhiko Suga. “Transport Mechanism of Interface Turbulence
over Porous and Rough Walls”. In: Flow, Turbulence and Combustion 97.4 (Dec. 2016),
pp. 1071–1093. issn: 15731987. doi: 10.1007/s10494-016-9759-9.
[33] A V Kuznetsov. “Analytical Study of Fluid Flow and Heat Transfer during Forced
Convection in a Composite Channel Partly Filled with a Brinkman-Forchheimer Porous
Medium”. In: Flow, Turbulence and Combustion 60 (1998), pp. 173–192. doi: 10.1023/
A:1009998703180.
[34] Jooyoung Lee et al. “Liquid cooling module incorporating a metal foam and fin hybrid
structure for high power insulated gate bipolar transistors (IGBTs)”. In: Applied Thermal Engineering 173 (June 2020). issn: 13594311. doi: 10.1016/j.applthermaleng.
2020.115230.
[35] S.-J Lee and S.-H Lee. Flow field analysis of a turbulent boundary layer over a riblet
surface. Tech. rep. 2001.
[36] G. Lei et al. “Calculation of full permeability tensor for fractured anisotropic media”. In: Journal of Petroleum Exploration and Production Technology 5.2 (June 2015),
pp. 167–176. issn: 21900566. doi: 10.1007/s13202-014-0138-6.
[37] A. C. Liakopoulos. “Darcy’s coefficient of permeability as symmetric tensor of second rank”. In: International Association of Scientific Hydrology. Bulletin 10.3 (1965),
pp. 41–48. issn: 00206024. doi: 10.1080/02626666509493405.
[38] T J Lu, H A Stone, and M F Ashby. Application of Heat Transfer in Microelectronics
Packaging. Tech. rep. 1997.
90

[39] Paolo Luchini, Fernando Manzo, and Amilcare Pozzi. “Resistance of a grooved surface
to parallel flow and cross-flow”. In: Journal of Fluid Mechanics 228 (1991), pp. 87–109.
issn: 14697645. doi: 10.1017/S0022112091002641.
[40] Shadi Mahjoob and Kambiz Vafai. “A synthesis of fluid and thermal transport models for metal foam heat exchangers”. In: International Journal of Heat and Mass
Transfer 51.15-16 (July 2008), pp. 3701–3711. issn: 0017-9310. doi: 10 . 1016 / J .
IJHEATMASSTRANSFER.2007.12.012.
[41] Mahiro Morimoto et al. “Measurements for characteristics of turbulence over a streamwise preferential porous substrate”. In: (2023). doi: 10.21203/rs.3.rs-2695305/v1.
url: https://doi.org/10.21203/rs.3.rs-2695305/v1.
[42] Arun Muley et al. “Foam Heat Exchangers: A Technology Assessment”. In: Heat Transfer Engineering 33.1 (Jan. 2012), pp. 42–51. issn: 0145-7632. doi: 10.1080/01457632.
2011.584817. url: http://www.tandfonline.com/doi/abs/10.1080/01457632.
2011.584817.
[43] Heidi M. Nepf. “Flow and transport in regions with aquatic vegetation”. In: Annual
Review of Fluid Mechanics 44 (2011), pp. 123–142. issn: 00664189. doi: 10.1146/
annurev-fluid-120710-101048.
[44] M. E. Nimvari, M. Maerefat, and M. K. El-Hossaini. “Numerical simulation of turbulent flow and heat transfer in a channel partially filled with a porous media”. In:
International Journal of Thermal Sciences 60 (Oct. 2012), pp. 131–141. issn: 12900729.
doi: 10.1016/j.ijthermalsci.2012.05.016.
[45] Yudai Nishiyama, Yusuke Kuwata, and Kazuhiko Suga. “Direct numerical simulation
of turbulent heat transfer over fully resolved anisotropic porous structures”. In: International Journal of Heat and Fluid Flow 81 (Feb. 2020), p. 108515. issn: 0142727X.
doi: 10 . 1016 / j . ijheatfluidflow . 2019 . 108515. url: https : / / linkinghub .
elsevier.com/retrieve/pii/S0142727X19301420.
[46] I Nozad, R G Carbonellt, and S Whitaker. HEAT CONDUCTION IN MULTIPHASE
SYSTEMS-I THEORY AND EXPERIMENT FOR TWO-PHASE SYSTEMS. Tech.
rep. 5. 1985, pp. 84–855.
[47] J W Paek et al. Effective Thermal Conductivity and Permeability of Aluminum Foam
Materials 1. Tech. rep. 2. 2000.
91

[48] Dubravka Pokrajac and Costantino Manes. “Velocity measurements of a free-surface
turbulent flow penetrating a porous medium composed of uniform-size spheres”. In:
Transport in Porous Media 78.3 SPEC. ISS. (2009), pp. 367–383. issn: 01693913. doi:
10.1007/s11242-009-9339-8.
[49] D. Poulikakos and M. Kazmierczak. “Forced Convection in a Duct Partially Filled With
a Porous Material”. In: Journal of Heat Transfer 109.3 (Aug. 1987), pp. 653–662. issn:
0022-1481. doi: 10.1115/1.3248138. url: https://asmedigitalcollection.asme.
org/heattransfer/article/109/3/653/412706/Forced-Convection-in-a-DuctPartially-Filled-With.
[50] Z. G. Qu et al. “A theoretical octet-truss lattice unit cell model for effective thermal conductivity of consolidated porous materials saturated with fluid”. In: Heat and
Mass Transfer/Waerme- und Stoffuebertragung 48.8 (Aug. 2012), pp. 1385–1395. issn:
09477411. doi: 10.1007/s00231-012-0985-y.
[51] Paola Ranut. “On the effective thermal conductivity of aluminum metal foams: Review and improvement of the available empirical and analytical models”. In: Applied
Thermal Engineering 101 (May 2016), pp. 496–524. issn: 13594311. doi: 10.1016/
j . applthermaleng . 2015 . 09 . 094. url: https : / / linkinghub . elsevier . com /
retrieve/pii/S1359431115010194.
[52] Stephen K Robinson. COHERENT MOTIONS IN THE TURBULENT BOUNDARY
LAYER. Tech. rep. 1991, pp. 601–640. url: www.annualreviews.org/aronline.
[53] Marco E. Rosti, Luca Brandt, and Alfredo Pinelli. “Turbulent channel flow over an
anisotropic porous wall - Drag increase and reduction”. In: Journal of Fluid Mechanics
842 (2018). issn: 14697645. doi: 10.1017/jfm.2018.152.
[54] Seo Young Kim, Jae-Mo Koo, Andrey V. “EFFECT OF ANISOTROPY IN PERMEABILITY AND EFFECTIVE THERMAL CONDUCTIVITY ON THERMAL PERFORMANCE OF AN ALUMINUM FOAM HEAT SINK”. In: Numerical Heat Transfer, Part A: Applications 40.1 (July 2001), pp. 21–36. issn: 1040-7782. doi: 10.1080/
10407780121436.
[55] Karen N. Son et al. “Design of multifunctional lattice-frame materials for compact
heat exchangers”. In: International Journal of Heat and Mass Transfer 115 (2017),
pp. 619–629. issn: 00179310. doi: 10.1016/j.ijheatmasstransfer.2017.07.073.
92

[56] K. Suga, M. Mori, and M. Kaneda. “Vortex structure of turbulence over permeable
walls”. In: International Journal of Heat and Fluid Flow 32.3 (June 2011), pp. 586–
595. issn: 0142727X. doi: 10.1016/j.ijheatfluidflow.2011.02.016.
[57] K. Suga et al. “Effects of wall permeability on turbulence”. In: International Journal
of Heat and Fluid Flow 31.6 (Dec. 2010), pp. 974–984. issn: 0142727X. doi: 10.1016/
j.ijheatfluidflow.2010.02.023.
[58] Kazuhiko Suga, Yuka Nakagawa, and Masayuki Kaneda. “Spanwise turbulence structure over permeable walls”. In: Journal of Fluid Mechanics 822 (July 2017), pp. 186–
201. issn: 14697645. doi: 10.1017/jfm.2017.278.
[59] Kazuhiko Suga et al. “Anisotropic wall permeability effects on turbulent channel flows”.
In: Journal of Fluid Mechanics 855 (Nov. 2018), pp. 983–1016. issn: 14697645. doi:
10.1017/jfm.2018.666.
[60] William Thielicke and Ren´e Sonntag. “Particle Image Velocimetry for MATLAB: Accuracy and enhanced algorithms in PIVlab”. In: Journal of Open Research Software
9.1 (May 2021), p. 12. issn: 2049-9647. doi: 10.5334/jors.334.
[61] William Thielicke and Eize J. Stamhuis. “PIVlab – Towards User-friendly, Affordable
and Accurate Digital Particle Image Velocimetry in MATLAB”. In: Journal of Open
Research Software 2 (Oct. 2014). issn: 2049-9647. doi: 10.5334/jors.bl.
[62] Kambiz Vafai. “Convective flow and heat transfer in variable-porosity media”. In:
Journal of Fluid Mechanics 147 (1984), pp. 233–259. issn: 14697645. doi: 10.1017/
S002211208400207X.
[63] M. WALSH. “Turbulent boundary layer drag reduction using riblets”. In: American
Institute of Aeronautics and Astronautics (AIAA), Jan. 1982. doi: 10.2514/6.1982-
169.
[64] Nanqiao Wang et al. “Prediction of effective thermal conductivity of porous lattice structures and validation with additively manufactured metal foams”. In:
Applied Thermal Engineering 187 (Mar. 2021). issn: 13594311. doi: 10 . 1016 / j .
applthermaleng.2021.116558.
[65] Giuseppe A. Zampogna and Alessandro Bottaro. “Fluid flow over and through a regular
bundle of rigid fibres”. In: Journal of Fluid Mechanics 792 (Apr. 2016), pp. 5–35. issn:
14697645. doi: 10.1017/jfm.2016.66.
93

[66] E. S. Zanoun, M. Kito, and C. Egbers. “A study on flow transition and development
in circular and rectangular ducts”. In: Journal of Fluids Engineering, Transactions of
the ASME 131.6 (June 2009), pp. 0612041–06120410. issn: 00982202. doi: 10.1115/
1.3112384.
[67] C. Y. Zhao. “Review on thermal transport in high porosity cellular metal foams with
open cells”. In: International Journal of Heat and Mass Transfer 55.13-14 (2012),
pp. 3618–3632. issn: 00179310. doi: 10.1016/j.ijheatmasstransfer.2012.03.017.
94

Appendices
This appendix provides the derivation for phenomological model for prediction of bulk permeability K as a function of sx, sy, sz, and d based on zonal weighting of hydraulic diameter.
A.1 Constraints on achievable geometry
Consider the cubic lattice geometry shown in Figure 1 and Figure 3, characterized by rods
of cross-section d×d and spacings (sx, sy, sz) in the x, y, and z directions. For this geometry,
we can calculate the directional porosities corresponding to the minimum opening area along
a given axis, as:
ϵx =
(sy − d)(sz − d)
sysz
, ϵy =
(sx − d)(sz − d)
sxsz
, ϵz =
(sx − d)(sy − d)
sxsy
(A.1)
Defining the dimensionless parameters, ˆsx = sx/d etc. we have:
ϵx =
(ˆsy − 1)(ˆsz − 1)
(ˆsysˆz)
, ϵy =
(ˆsx − 1)(ˆsz − 1)
(ˆsx)ˆsz
, ϵZ =
(ˆsx − 1)(ˆsy − 1)
(ˆsx)ˆsy
(A.2)
The above equations show that directional porosities depend on reduced lengths,
rx, ry, rz as below:
rx =
sx − d
sx
= 1 − sˆ
−1
x
, ry =
sy − d
sy
= 1 − sˆ
−1
y
, rx =
sz − d
sz
= 1 − sˆ
−1
z
(A.3)
Specifically, we have
ϵx = ryrz, ϵy = rzrz, ϵz = ryrx, (A.4)
95

Rearranging the terms gives us
rx =
rϵyϵz
ϵx
, ry =
rϵxϵz
ϵy
, rz =
rϵxϵy
ϵz
, (A.5)
and since ˆsx = (1 − rx)
−1 and so on, we have
sˆx =

1 −
rϵyϵz
ϵx
−1
, sˆy =

1 −
rϵxϵz
ϵy
−1
, sˆz =

1 −
rϵxϵy
ϵz
−1
(A.6)
Equation (A.6) can be used to identify the dimensionless spacings ( ˆsx, sˆy, sˆz) to generate
(ϵx, ϵy, ϵz). Furthermore, it allows us to identify which limits over achievable anisotropies and
directional porosities. For example, (A.6) indicates that for directional porosities such as
ϵyϵz > ϵx would lead to ˆsx < 0. Assuming that the desired porosity values arranged in
decreasing order of magnitude (ϵx > ϵy > ϵz), then the porosities must satisfy:
ϵxϵy < ϵz (A.7)
This along with manufacturing constraints using the SLA printer limits the maximum
achievable anisotropy. With the formlabs Form2/Form3 printers, we can consistently print
lattices with rod size d = 0.4 mm. Printing at d = 0.2 mm is possible, but less reliable. The
maximum recommended unsupported span for the printers is s = 5 mm, though larger spans
are possible for supported geometries. Thus we can reasonably expect to generate lattices
with (ˆsx, sˆy, sˆy) ∈ [1, sˆ]max where ˆsmax = 20. This translates to rmax = 1 − (1 − sˆmax) = 0.95
and (ϵx, ϵy, ϵz) ∈ [0, ϵmax] with ϵmax = r
2
max = 0.9025. This gives rise to more restrictive
constraints:
ϵzϵy ≥ ϵxϵmax (A.8)
The above constraint also implies that if the maximum directional porosity is set to
ϵx = ϵmax, the other two porosity values are equal: ϵy = ϵz
96

A.2 Model to predict principle components of permeability
We can also develop a phenomenological model for the principal components of the permeability tensor for this simplified geometry. This model assumes that the permeability in
any given direction across the unit cell can be estimated based on the effective hydraulic
diameters. That is
Kxx ∝ Dxx
2
(A.9)
Dxx represents the total hydraulic diameter in the x direction.
Hence, for 1D Darcy’s law, we can express the pressure drop in the x direction in terms
of the hydraulic diameter:
dP
dx = Cµ 1
D2
x
ux (A.10)
where C is an empirical constant.
This hydraulic diameter varies along the axis of interest for the cubic lattice. As an
example, we first derive the principle component of permeability tensor in the x-direction
Kxx as follows: The unit cell can be split into two different zones (see Figure 3). The first
zone is of length d and the second zone is of length sx − d.
For zone 1, the hydraulic diameter is
Dx1 =
4(sy − 1d(sz − d)
1[(sy − d) + (sz + d)] =
2d( ˆsy − 1)( ˆsz − 1)
( ˆsy − 1) + (ˆsz − 1) (A.11)
Similarly, for zone 2, the hydraulic diameter is given by
Dx2 =
4(sysz − d
2
)
4d
= d(ˆsysˆz − 1) (A.12)
97

We now introduce a weighting factor wx based on the cross-sectional areas of both zones.
From conservation of mass, we get
m˙ = UxbA = Ax1Ux1N = Ax2Ux2N (A.13)
where ˙m is the average mass flow rate through the porous sample with N unit cells,
Uxb is the bulk (or superficial) velocity in the x direction, and Ax1, Ax2 represent the area
of cross-section of the fluid to go through zones 1 and 2 with a velocity of Ux1 and Ux2
respectively.
Hence, pressure gradient in the x-direction can be given as
∆P x = Cµ
1 − wx
Ax1D2
x1
+
wx
Ax2D2
x2

A
N
Uxb (A.14)
Since the lattice structure is periodic, A/N = Ax = sysz for a given unit cell.
We determine the weighting factor wx based on the relative distance travelled in the
direction of the flow in each zone. Thus, wx = d/sx and hence (1 − wx) = (sx − d)/sx.
And hence, pressure drop along x-axis across the unit cell can be expressed as
dP
dx = −µC
d
sx
sˆysˆz
(ˆsy − 1)(ˆsz − 1)
1
D2
x1
+
sx − d
sx
sˆysˆz
(ˆsy − 1)(ˆsz − 1)
1
D2
x2

Uxb (A.15)
From (A.10) and (A.15) and Dracy’s law, we can express
Kxx =
1
C

d
sx
sˆysˆz
(ˆsy − 1)(ˆsz − 1)
1
D2
x1
+
sx − d
sx
sˆysˆz
(ˆsy − 1)(ˆsz − 1)
1
D2
x2
−1
(A.16)
Here, C is an empirical constant derived from the approximation for laminar flow
through rectangular ducts (cite Kakac book) as below:
98

fRe = 24(1 − 1.3553α ∗ +1.9467α ∗
2 −1.7012α ∗
3 +0.9564α ∗
4 −0.2537α∗
5
) (A.17)
where α∗ = min((sy − d)/(sz − d),(sz − d)/(sy − d))
Similarly, Kyy and Kzz can also be derived.
99

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Creator Vijay, Shilpa (author)

Core Title Flow and thermal transport at porous interfaces

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Degree Conferral Date 2023-12

Publication Date 12/12/2023

Defense Date 10/03/2023

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