A novel sensor-based blended wall-modeling approach for large-eddy simulation of flows with relaminarization

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Content A NOVEL SENSOR-BASED BLENDED WALL-MODELING APPROACH FOR LARGE-EDDY
SIMULATION OF FLOWS WITH RELAMINARIZATION
by
Naili Xu
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
([MECHANICAL ENGINEERING])
May 2025
Copyright 2025 Naili Xu

Acknowledgements
I would like to start by expressing my deepest gratitude to my advisor, Prof. Ivan Bermejo-Moreno, for
his unwavering support, patience, and guidance throughout my Ph.D. journey. I have been incredibly
fortunate to benefit from his expertise and mentorship. His belief in my potential and his constant support
for my academic and professional growth have been truly invaluable.
I am also sincerely grateful to Prof. Carlos Alejandro Pantano-Rubino, Prof. Geoffrey Spedding, Prof.
Mitul Luhar, and Prof. Mihailo Jovanovic for serving on my qualifying exam and defense committees.
Their insightful feedback and guidance have played a crucial role in shaping my research. Additionally,
I extend my appreciation to my undergraduate research advisor, Prof. Hang Xu, for sparking my early
interest in fluid mechanics and inspiring me to pursue a research career.
My deepest gratitude goes to my parents, whose unwavering love and support have made everything
possible. My mother, in particular, has been my greatest source of strength. Her unconditional love,
encouragement, and sacrifices have given me the confidence to pursue my dreams. Even from miles away,
she has always been there for me, offering endless emotional support and reminding me of my resilience.
I would not be where I am today without her.
To my labmates—thank you for making this journey so much more enjoyable beyond research. Your
camaraderie, laughter, and support have made even the longest days in the lab more memorable and rewarding.
ii

Finally, I acknowledge financial support from NASA (grants #80NSSC18M0148, #80NSSC22M0297) and
NSF (#2143014), as well as computational resources provided by ALCF at ANL (Theta cluster) through an
INCITE Program allocation and USC’s CARC (Endeavour cluster).
iii

Table of Contents
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Relaminarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Overview of WMLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Classifying wall models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Laminar wall model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.3 Grid resolution requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.4 Challenges and developments in WMLES . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Gaussian “speed bump” configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 NASA hump configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Objectives and scope of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Chapter 2: Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1 LES formalism and governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Equilibrium wall model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Sensor-based blended wall model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.1 Sensors for the detection of pressure-driven relaminarization . . . . . . . . . . . . 28
2.3.2 Blending factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.3 Hysteresis scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Chapter 3: Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1 Gaussian “speed bump” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 NASA hump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Chapter 4: Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1 Gaussian-shaped bump configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1.1 Flow over a Gaussian-shaped bump at ReL = 106 with relaminarization . . . . . . 42
4.1.1.1 Incoming turbulent boundary layer . . . . . . . . . . . . . . . . . . . . . 45
4.1.1.2 Streamwise distribution of pressure and skin friction coefficients . . . . 45
4.1.1.3 Streamwise variation of boundary layer properties . . . . . . . . . . . . 54
iv

4.1.1.4 Wall-normal profiles of mean streamwise velocity and Reynolds stresses 59
4.1.1.5 Instantaneous flow fields and relaminarization . . . . . . . . . . . . . . . 62
4.1.2 Flow over a Gaussian-shaped bump at ReL = 2 · 106
and 4 · 106 without
relaminarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.3 Reynolds number effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1.4 Sensitivity analyses of sensor-based blended WMLES results to model parameters . 78
4.1.4.1 Sensor reference value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.1.4.2 Hysteresis time scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2 NASA hump configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2.1 Effect of exchange heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2.2 Effect of grid resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.3 Effect of delay time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Chapter 5: Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Chapter 6: Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.1 Assessment of existing laminar-to-turbulent transition sensors to predict relaminarization 101
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
v

List of Tables
4.1 Grid parameters for simulation cases at ReL = 106
. δ0 is the reference boundary
layer thickness. (s, n, z) are the wall-parallel, wall-normal, and transverse (spanwise)
topological directions, respectively. N indicates the number of cells. Grid sizes given in
millions (M) of cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Mesh resolution for simulation cases at ReL = 2 · 106
(top two rows) and 4 · 106
(bottom row). δ0 is the reference boundary layer thickness. (s, n, z) are the wall-parallel,
wall-normal, and transverse (spanwise) directions, respectively. Ly is the domain height.
N indicates the number of cells. Grid sizes given in millions (M) of cells. . . . . . . . . . . 64
4.3 Summary of wall model exchange height configurations . . . . . . . . . . . . . . . . . . . 87
4.4 Summary of grid resolution configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
vi

List of Figures
3.1 Geometric configuration and computational domain in the xy plane. The geometry is
extruded 0.04L in the spanwise direction, z. . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Geometric configuration and computational domain of NASA hump in the xy plane. The
geometry is extruded 0.3L in the spanwise direction, z. . . . . . . . . . . . . . . . . . . . . 39
4.1 Grid spacing profiles along the streamwise (a), wall-normal (b), and spanwise (c) directions
used by the WMLES conducted at different grid resolution for the ReL = 106
case. For
clarity, markers are plotted every 500, 1, and 50 points in (a), (b), and (c), respectively,
with offsets in (a) and (c) for each case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Wall-normal profiles of the incoming turbulent boundary layer at the reference location
(x/L = −0.6) for the ReL = 106
case, comparing the wall-modeled LES with an
equilibrium wall model (dotted green line) with the DNS data of Uzun & Malik (2020).
(a) Mean streamwise velocity, normalized by the freestream velocity, as a function of the
wall-normal distance in outer scaling (n/δ). The inset shows the same quantity in inner
scaling (U
+ = U/uτ as a function of n
+ = n/ℓv), with the viscous and log-law theoretical
estimates represented by dashed and dash-dotted lines, respectively. (b) Reynolds stresses,
⟨u
′
iu
′
j
⟩, normalized by the square of the freestream velocity, as a function of the wallnormal distance in outer scaling, n/δ. Gray areas represent the wall-modeled region in
the WMLES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Streamwise profiles of pressure (a,c) and friction (b,d) coefficients from WMLES of the
ReL = 106
case (denoted by the 1M label prefix) performed using an equilibrium wall
model (Eq) (a,b) and a blended wall model based on the acceleration sensor (Acc) (c,d)
with different grid resolutions (T: coarsest, C: coarse, M: medium, F: fine), compared with
DNS data (black solid with symbols) by Uzun & Malik (2020). . . . . . . . . . . . . . . . . . 47
4.4 Streamwise profiles of grid spacing in outer (a,b,c) and inner (d,e,f) units for the streamwise
(a,d), wall-normal first cell centroid (b,e), and spanwise (c,f) coordinate directions, obtained
from WMLES of the ReL = 106
case performed using a blended wall model based on the
acceleration sensor with grid resolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
vii

4.5 Streamwise profiles of pressure (a) and friction (b) coefficients from WMLES of the
ReL = 106
case conducted on the fine grid (F), performed using an equilibrium wall
model (Eq, dotted green), a blended wall model based on the acceleration sensor (Acc,
solid red), and a blended wall model based on the relaminarization sensor (Rel, dashed
blue), compared with DNS data (black solid with symbols) by Uzun & Malik (2020). Cf
values in (b) scaled by a factor of 103
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.6 Streamwise profiles of the normalized sensors (a) and blending factors (b) from WMLES
of the ReL = 106
case performed using a blended wall model based on the acceleration
sensor (solid red) and the relaminarization sensor (dashed blue). . . . . . . . . . . . . . . . 52
4.7 Streamwise profiles of boundary layer displacement thickness (a), momentum thickness
(b), edge velocity (c), and Rotta-Clauser parameter (d) obtained from WMLES of the
ReL = 106
case performed using an equilibrium wall model (dotted green), a blended wall
model based on the acceleration sensor (solid red), and a blended wall model based on the
relaminarization sensor (dashed blue), compared with DNS data (black solid) by Uzun &
Malik (2020). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.8 Wall-normal line probes extracted at different streamwise locations (left to right) of mean
(time- and spanwise-averaged) wall-parallel velocity and Reynolds stresses (top to bottom)
for WMLES of the ReL = 106
case performed using an equilibrium wall model (dotted
green), a blended wall model based on the acceleration sensor (solid red), and a blended
wall model based on the relaminarization sensor (dashed blue), compared with DNS data
(black solid) by Uzun & Malik (2020). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.9 Contours of instantaneous fluctuations of the local skin friction coefficient on the wall (a),
instantaneous streamwise velocity fluctuation extracted on wall-shaped surfaces located
a wall-normal distance of 0.1δ0 (b), 0.2δ0 (c), δ0 (d), 2δ0 (e), and instantaneous streamwise
velocity on a vertical plane (f), for an acceleration-based blended wall-modeled LES of the
ReL = 106
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.10 Grid spacing profiles along the streamwise (a), wall-normal (b), and spanwise (c) directions
used by the WMLES conducted at different grid resolution for the ReL = 2 · 106
and
ReL = 4 · 106
cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.11 Streamwise profiles of grid spacing in outer (a,b,c) and inner (d,e,f) units for the streamwise
(a,d), wall-normal first cell centroid (b,e), and spanwise (c,f) coordinate directions, obtained
from WMLES of the ReL = 2 · 106
and 4 · 106
cases performed using a blended wall model
based on the acceleration sensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
viii

4.12 Wall-normal profiles of the incoming turbulent boundary layer at the reference location
(x/L = −0.5) for the ReL = 2 · 106
case, comparing the wall-modeled LES with an
equilibrium wall model (dotted green line) with the DNS data of Uzun & Malik (2022).
(a) Mean streamwise velocity, normalized by the freestream velocity, as a function of
the wall-normal distance in outer scaling (n/δ). The inset shows the same quantity in
inner scaling (U
+ = U/uτ as a function of n
+ = n/ℓv), with the viscous and log-law
theoretical estimates represented by dashed and dash-dotted lines, respectively. (b)
Reynolds stresses (⟨u
′
iu
′
j
⟩) normalized by the square of the freestream velocity, as a
function of the wall-normal distance in outer scaling (n/δ). Gray areas represent the
wall-modeled region in the WMLES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.13 Streamwise profiles of pressure coefficient (a,c) and friction coefficient (b,d) obtained from
WMLES of the ReL = 2 · 106
(a,b) and ReL = 4 · 106
(c,d) cases, performed using an
equilibrium wall model (dotted green), a blended wall model based on the acceleration
sensor (solid red), and a blended wall model based on the relaminarization sensor (dashed
blue). For ReL = 2 · 106
(a,b), WMLES results are compared with DNS data by Uzun
& Malik (2022) (black solid, solid circles), and experiments by Gray et al. (2022) (hollow
circles) and Robbins et al. (2021) (hollow triangles). For the ReL = 4 · 106
case (c,d),
reference data includes results from experiments by Gray et al. (2022) (circles) at two Mach
numbers (0.1, hollow; 0.2, solid) and Williams et al. (2021)(solid triangles). Cf values in
(b,d) scaled by a factor of 103
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.14 Streamwise profiles of the time- and spanwise-averaged blending factor from WMLES
performed using blended wall models based on the acceleration (solid red) and relaminarization (dashed blue) sensors for the ReL = 2 · 106
(a) and ReL = 4 · 106
(b)
cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.15 Wall-normal line probes extracted at different streamwise locations (left to right) of mean
(time- and spanwise-averaged) wall-parallel velocity and Reynolds stresses (top to bottom)
for WMLES of the ReL = 2 · 106
case performed using an equilibrium wall model (dotted
green), a blended wall model based on the acceleration sensor (solid red), and a blended
wall model based on the relaminarization sensor (dashed blue), compared with DNS data
(black solid) by Uzun & Malik (2022). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.16 Contours of instantaneous fluctuations of the local skin friction coefficient on the wall (a),
instantaneous streamwise velocity fluctuation extracted on wall-shaped surfaces located
a wall-normal distance of 0.1δ0 (b), 0.2δ0 (c), δ0 (d), 2δ0 (e), and instantaneous streamwise
velocity on a vertical plane (f), for the ReL = 2 · 106
case. . . . . . . . . . . . . . . . . . . 73
4.17 Contours of instantaneous fluctuations of the local skin friction coefficient on the wall (a),
instantaneous streamwise velocity fluctuation extracted on wall-shaped surfaces located
a wall-normal distance of 0.1δ0 (b), 0.2δ0 (c), δ0 (d), 2δ0 (e), and instantaneous streamwise
velocity on a vertical plane (f), for the ReL = 4 · 106
case. . . . . . . . . . . . . . . . . . . 74
4.18 Streamwise profiles of pressure (a) and friction (b) coefficients obtained from WMLES
performed using a blended wall model based on the acceleration sensor at different
Reynolds numbers: ReL = 106
(solid), 2 · 106
(dotted), 4 · 106
(dashed). Cf values in (b)
scaled by a factor of 103
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
ix

4.19 Streamwise profiles of boundary layer displacement thickness (a), momentum thickness
(b), and edge velocity (c) obtained from WMLES performed using a blended wall model
based on the acceleration sensor at different Reynolds numbers: ReL = 106
(solid), 2 · 106
(dotted), 4 · 106
(dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.20 Streamwise profiles of pressure coefficient (a,c) and friction coefficient (b,d) from WMLES
of the ReL = 106
(a,b) and ReL = 2 · 106
(c,d) cases performed using an equilibrium
wall model (dotted green), a blended wall model based on the acceleration sensor with
different reference (normalization) values: 0.018 (dotted red), 0.025 (solid red), and 0.030
(dashed red), compared with DNS data by Uzun & Malik (2020, 2022) (black solid), and
experiments by Gray et al. (2022) (hollow circles) and Robbins et al. Robbins et al. (2021)
(hollow triangles). Cf values in (b,d) scaled by a factor of 103
. . . . . . . . . . . . . . . . . 79
4.21 Streamwise profiles of the blending factor from WMLES of the ReL = 106
(a) and
ReL = 2 · 106
(b) cases performed using a blended wall model based on the acceleration
sensor with different reference values: 0.018 (dotted red), 0.025 (solid red), and 0.030
(dashed red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.22 Streamwise profiles of pressure (a) and friction (b) coefficients from WMLES of the
ReL = 106
case performed using a blended wall model based on the acceleration sensor
with different delay time factors (α = 0, 500, 1000), compared with DNS data (black solid
with symbols) by Uzun & Malik (2020). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.23 Comparison of surface pressure coefficient (Cp) and skin friction coefficient (Cf )
distributions for different exchange heights in the NASA hump simulation with the
equilibrium wall model. The inset within the skin friction coefficient plot shows the
streamwise profile of wall-model exchange height normalized by the reference boundary
layer thickness, hwm(x)/δref. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.24 Comparison of surface pressure coefficient (Cp) and skin friction coefficient (Cf )
distributions for different exchange heights in the NASA hump simulation with a blended
wall model based on the acceleration sensor. The inset within the skin friction coefficient
plot shows the streamwise profile of wall-model exchange height normalized by the
reference boundary layer thickness, hwm(x)/δref. . . . . . . . . . . . . . . . . . . . . . . . 90
4.25 Streamwise profiles of the time- and spanwise-averaged blending factor from WMLES
performed using blended wall models based on the acceleration sensors. . . . . . . . . . . 91
4.26 Streamwise profiles of grid spacing in inner units for the streamwise, wall-normal first
cell centroid, and spanwise coordinate directions, obtained from WMLES of the case
performed usingan equilibrium wall model with different grid resolutions. . . . . . . . . . 91
4.27 Comparison of surface pressure coefficient (Cp) and skin friction coefficient (Cf )
distributions for different grid resolution in the NASA hump simulation with the
equilibrium wall model. The inset within the skin friction coefficient plot shows the
streamwise profile of wall-model exchange height normalized by the reference boundary
layer thickness, hwm(x)/δref. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
x

4.28 Comparison of surface pressure coefficient (Cp) and skin friction coefficient (Cf )
distributions for different grid resolution in the NASA hump simulation with the
equilibrium wall model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.29 Comparison of surface pressure coefficient (Cp) and skin friction coefficient (Cf )
distributions for different delay time in the NASA hump simulation using the blended
wall model with acceleration sensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.1 Streamwise profiles of pressure coefficient (a) and friction coefficient (b) obtained from
WMLES of the ReL = 106
case performed using the sensor-based wall models of Bodart
& Larsson (solid purple) and Mettu & Subbareddy (dash-dotted yellow), compared with
DNS data (black solid) by Uzun & Malik (2020). The fraction of time that the equilibrium
wall model is being applied for each method is shown in (c) with the corresponding time
filtered turbulence kinetic energy distribution (d). Cf values in (b,d) scaled by a factor of
103
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2 Streamwise profiles of the sensor value (a) and time-filtered kinetic energy (b) from
WMLES of the ReL = 106
case performed using the sensor developed by Bodart &
Larsson with different filtering scales: 1 (solid purple), 10 (dotted purple), 100 (dashed
purple), and time-averaged (dash-dotted purple). . . . . . . . . . . . . . . . . . . . . . . . . 104
6.3 Streamwise profiles of the sensor value (a) and zoomed in the relaminarization region (b),
time-filtered friction velocity (c) and time-filtered pressure gradient (d) from WMLES of
the ReL = 106
case performed using the acceleration sensor with different filtering time
scales, Tf = γL/U∞, with γ ≈ 0.01 (solid yellow), 0.1 (dotted red) and 1.0 (dashed green). 105
xi

Abstract
Wall-bounded flows undergoing pressure-driven relaminarization present significant challenges for equilibrium wall models in WMLES, as these models assume a fully turbulent boundary layer and struggle to
accurately predict skin friction. To overcome this limitation, we introduce a sensor-based blended wallmodeling approach with two sensors designed to detect relaminarization and retransition processes and
the incorporation of a hysteresis timescale that accounts for a delayed response of the boundary layer to
the pressure gradient effects. Based on the local sensor value, a blending strategy is employed to dynamically transition between an equilibrium wall model and either a no-slip boundary condition or a laminar
wall model. The effectiveness of this approach is assessed in two flow configurations: a Gaussian-shaped
speed bump and the NASA wall-mounted hump.
For the Gaussian-shaped speed bump, numerical investigations are conducted at three Reynolds numbers: ReL = 106
, at which the flow partially relaminarizes, ReL = 2 · 106
, and ReL = 4 · 106
, all with
a freestream Mach number M = 0.2. The proposed wall-modeling approach is validated by comparing
spanwise-periodic WMLES results with experimental data from Williams et al. (2020, 2021) and Gray et
al. (2022), as well as DNS results from Uzun & Malik (2020-2022). In addition, the performance of previously developed laminar-to-turbulent transition sensors, such as those by Bodart & Larsson (2012) and
Mettu & Subbareddy (2018), is assessed. The predictive capabilities and robustness of the newly proposed
approach are evaluated for all three Reynolds numbers. A posteriori analyses suggest that the sensors correctly identify the relaminarization present in the ReL = 106
case, with the blended wall model improving
xii

flow predictions, while still being applicable without any change in the model parameters to the higher
Reynolds number flows, which do not exhibit relaminarization.
To further evaluate the robustness of this approach, we investigate relaminarizing flow over the NASA
wall-mounted hump at a freestream Mach number of M = 0.2 and a Reynolds number Reθ ≈ 7, 000,
based on the upstream momentum thickness. The results are compared against experimental data from
Greenblatt et al. (2006) and Naughton et al. (2006), as well as WRLES simulations by Uzun and Malik (2017).
A systematic analysis of the effects of wall-model exchange height (hwm) and grid resolution demonstrates
their significant impact on flow predictions. Reducing the exchange height in the favorable pressure gradient (FPG) region near the leading edge of the hump enhances the accuracy of quasi-relaminarization
predictions. Likewise, in the separated flow region, a lower exchange height improves skin friction predictions, underscoring its critical role in accurately capturing flow separation and reattachment. The effect
of the delay time parameter is examined, revealing that while moderate values can improve predictions
of the skin friction coefficient in the relaminarization region, larger delay values can lead to significant
underprediction.
Although the focus of this study is on relaminarization, pressure-driven separation and reattachment
are also analyzed in the flows under consideration. The parametric analyses conducted quantify the sensitivity of the predictions to modeling choices and identify opportunities for future extensions of the proposed wall modeling approach for application to a wider range of non-equilibrium effects beyond relaminarization, through automated adaptation of the wall-model exchange height and grid refinement strategies.
xiii

Chapter 1
Introduction
1.1 Motivation
Computational fluid dynamics (CFD) has evolved into a cornerstone of modern engineering, driven by
breakthroughs in numerical methods and high-performance computing. The advancement of CFD technology has been instrumental in reducing product design costs while also facilitating the development
of innovative platforms and systems, which plays a vital role across all stages of the design and validation process (Slotnick et al., 2014). In the conceptual design phase, simplified CFD models help optimize
key performance metrics such as drag, fuel consumption, and thrust, though they introduce conservatism
due to computational trade-offs. During the preliminary and detailed design phase, CFD becomes indispensable for refining complex configurations in aerospace and gas turbine applications, reducing reliance
on physical testing despite computational cost limitations. For space exploration and hypersonic flight,
where ground testing is limited, CFD often serves as the primary tool for aerodynamic analysis. In the
product validation and certification phase, CFD is used to confirm test results, refine designs, and address
unexpected technical challenges, minimizing costly physical tests and enabling engineers to explore new
design spaces with greater confidence. Improving the accuracy and efficiency of CFD remains a key goal,
enhancing predictive capabilities and reducing reliance on expensive experimental validation.
1

Despite significant advancements in computational fluid dynamics (CFD), accurately predicting complex turbulent flows, particularly those involving transition and separation, remains a major challenge
for Reynolds-Averaged Navier-Stokes (RANS) methods (Slotnick et al., 2014). These difficulties arise from
the inherent limitations of time-averaged turbulence modeling, which fails to capture unsteady flow characteristics and non-equilibrium effects. Most RANS models assume fully developed turbulence, leading
to poor predictions in transitional and relaminarizing flows where turbulence production and dissipation
vary significantly. Additionally, RANS struggles with adverse pressure gradients (APG), often overpredicting turbulent mixing and failing to capture the correct onset of separation and reattachment. The empirical
nature of RANS models further limits their adaptability to complex or high-speed flows, as they are typically calibrated for specific scenarios. These challenges motivate the need for higher-fidelity approaches
such as Large Eddy Simulation (LES) (Leonard, 1975), which offer improved predictive accuracy by resolving large-scale turbulent structures while reducing computational cost compared to direct numerical
simulation (DNS).
The primary advantage of LES over RANS lies in its ability to capture a significant portion of the turbulence kinetic energy, leading to improved accuracy in turbulent flow simulations. While LES resolves
the larger, energy-carrying eddies, it models the smaller, universal eddies, reducing computational costs
in comparison to Direct Numerical Simulation (DNS), where all scales of motion must be resolved. Unlike
DNS, which requires matching the Kolmogorov scales, LES requires a grid size small enough to resolve
the integral scale of motion, making it more efficient (Piomelli, 2014). The flow field in LES is filtered to
separate the large and small eddies, with large-scale motions directly resolved and smaller scales modeled through subgrid-scale (SGS) models, such as the Smagorinsky model. This model approximates the
effects of unresolved scales by adding an eddy-viscosity term to the governing equations (Smagorinsky,
1963). More sophisticated SGS models, like dynamic models, adapt the model constants based on local flow
conditions, which further enhances simulation accuracy (Piomelli & Balaras, 2002).
2

Given its ability to resolve complex turbulent flows at Reynolds numbers where DNS becomes computationally prohibitive, LES has become an indispensable tool in modern computational fluid dynamics. It
is particularly useful in scenarios where traditional turbulence models, such as RANS, fall short, especially
when simulating intricate phenomena like relaminarization and separation in turbulent flows. Accurate
predictions of these phenomena are essential for improving drag reduction, enhancing flow stability, and
optimizing heat transfer efficiency, which directly impacts performance and fuel efficiency in industries
like aerospace and automotive. Furthermore, LES helps mitigate structural damage, optimize designs, and
ensure system reliability by minimizing undesirable flow behaviors like turbulence and instability. These
benefits also translate into significant economic advantages, such as reduced operational costs and extended component lifespan.
Building on the ability of LES to resolve large-scale turbulent motions, it is also crucial to address transitional effects in LES simulations. To capture these effects, certain subgrid-scale (SGS) models account
for the vanishing of eddy viscosity as the flow transitions to a laminar state (Germano et al., 1991; Vreman, 2007). Simulating transitional flows at high Reynolds numbers using LES can be computationally
demanding due to the high resolution requirements near the wall. For a turbulent boundary layer developed over a flat plate of streamwise length Lx, the grid size (number of grid points) scales with Re13/7
Lx
,
where ReLx = U∞Lx/ν is the Reynolds number based on Lx and U∞ is the freestream velocity (Choi &
Moin, 2012). More stringent near-wall grid resolution requirements arise in transitional flows to accurately
resolve the energy-carrying and anisotropic scales and capture the underlying instability mechanisms governing the transition process (Sayadi & Moin, 2012; Lardeau et al., 2012). In contrast to laminar-to-turbulent
transitional flows, the LES grid resolution requirements for relaminarizing flows have been deemed less
demanding (Piomelli & Scalo, 2009), while still showing its capability to accurately predict relaminarization (Bader et al., 2017).
3

To address the computational challenges associated with high Reynolds number wall-bounded flows,
wall-modeled LES (WMLES) has been proposed (Piomelli & Balaras, 2002). In WMLES, the inner layer near
the wall is modeled, while the outer layer is resolved (Bose & Park, 2018). Wall models provide closure
for the turbulent stresses and handle the interactions between the modeled inner layer and resolved outer
layer (Larsson et al., 2016). This approach allows for computationally feasible simulations by reducing the
grid resolution requirements in the near-wall region compared to (wall-resolved) LES. However, during
transition and relaminarization, the flow undergoes significant changes in that near-wall, modeled region,
which must therefore be accounted for by the wall model.
Relaminarization, the intriguing process in which a turbulent boundary layer reverts to a laminar state,
has been the subject of extensive study due to its relevance in various engineering applications, including
aircraft design, low-pressure turbines, rocket nozzles, and hypersonic scramjet engines (Viswanath et al.,
2006; van Dam et al., 1993; Jagannathan et al., 2012; Smith, 1988; Walberg, 1991; Konopka et al., 2012; Teramoto et al., 2017). While initially explored for scientific curiosity, there has been a renewed interest in
understanding and modeling this phenomenon in these practical contexts. However, the complex interaction between turbulent and laminar behaviors in quasi-relaminarized flows presents significant challenges
in both comprehending and accurately simulating such flows.
One of the primary challenges in simulating relaminarization is the unfavorable scaling of wall-resolved
large-eddy simulations (WRLES), which are computationally prohibitive at high Reynolds numbers. For
Reynolds numbers around 105
, WRLES requires massive computational resources, such as clusters with
thousands of processors, making it impractical for many engineering applications (Piomelli & Balaras,
2002). Furthermore, for Reynolds numbers typical in aerodynamic and geophysical studies (ranging from
106
to 109
), wall-resolved LES is not feasible due to excessive computational demands. In contrast, wallmodeled LES (WMLES) mitigates this issue by using wall models to approximate near-wall turbulence,
4

allowing for coarser grid resolutions near the wall and making simulations of high Reynolds number flows
more computationally efficient (Piomelli & Balaras, 2002).
However, existing wall-modeled LES approaches face limitations in accurately capturing transition and
relaminarization, particularly because the wall models often assume fully-developed turbulence, which
results in an overprediction of wall shear stress (τw) in laminar and quasi-laminar regions. This leads to
inaccuracies in boundary layer growth and dynamics (Lozano-Durán et al., 2018; Mettu & Subbareddy,
2022).
To address these challenges, this thesis proposes a novel sensor-based blended wall-modeling approach
designed to accurately simulate wall-bounded flows undergoing pressure-driven relaminarization in WMLES. Traditional equilibrium wall models, which assume fully turbulent boundary layers, struggle to predict skin friction in relaminarizing flows. This work overcomes this limitation by incorporating a dynamic
sensor that blends an equilibrium shear-stress wall model with a laminar wall model/no-slip boundary condition, accounting for hysteresis effects along fluid pathlines. The effectiveness and robustness of this new
approach are evaluated through numerical simulations at various Reynolds numbers, specifically for the
Gaussian-shaped speed bump and NASA wall-mounted hump configurations, using experimental and DNS
data for validation. These configurations are chosen due to their relevance in capturing relaminarization
effects and representing complex aerodynamic and pressure-driven flow phenomena. The performance of
the proposed approach is also compared with previously developed transition sensors, and its adaptability across different flow conditions, Reynolds numbers, and grid resolutions is assessed. Ultimately, the
goal is to provide a reliable, computationally efficient wall-modeling method that can be applied to flows
with significant relaminarization and to explore its potential for broader applications in aerodynamic and
geophysical flow simulations.
5

1.2 Relaminarization
Relaminarization refers to the process in which a turbulent boundary layer reverts back to a laminar state.
Often, clear turbulent-laminar demarcation is absent and turbulent fluctuations do not vanish completely,
and the boundary layer remains in a quasi-laminar state. Previous studies have identified several factors
that contribute to relaminarization, including the dissipation of turbulence kinetic energy by viscosity,
external forces (e.g., buoyancy, magnetic fields), flow acceleration under highly favorable pressure gradients (FPG), flow curvature and rotation, and surface mass flux (Narasimha & Sreenivasan, 1979). A precise
criterion for identifying relaminarization remains elusive, as well as the full understanding of the state of
a quasi-relaminarized flow, which poses significant modeling challenges.
Determining the precise criterion for the occurrence of relaminarization has posed challenges, as it
is a gradual rather than a discontinuous process. In Launder’s study (Launder, 1964), a non-dimensional
pressure gradient parameter was introduced as an indicator for the initiation of the laminarizing boundary
layer. This parameter, later denoted as K (see Kline et al., 1967) and termed both as acceleration and relaminarization parameter by different studies (e.g., Kreskovsky et al., 1974; Spalart, 1986; Uzun & Malik, 2020),
is defined as K = (ν/U2
e
)(∂Ue/∂s), where s represents the streamwise direction, Ue is the boundary layer
edge velocity in that direction, and ν denotes the kinematic viscosity, serves as a measure of the relaminarization. Critical values of K ranging from 2 · 10−6
to 3.7 · 10−6 have been reported as thresholds beyond
which relaminarization occurs (Launder, 1964; Patel & Head, 1968; Schraub, 1965; Spalart, 1986). Alternative combinations of the relaminarization parameter K and the skin friction coefficient Cf in the form of
KC−n
f
have been proposed by Back et al. (1964) and Launder & Stinchcombe (1967), where the exponent n
ranges between 1
2
and 3
2
. This mechanism suggests that K becomes dominant at moderate Reynolds numbers, as the skin friction coefficient, Cf , varies slowly with Reynolds number in a turbulent flow. Patel &
Head (1968) recast K/C3/2
f
as a non-dimensional streamwise pressure gradient ∆p = −[ν/(ρu3
τ
)](∂p/∂s),
6

where uτ is the friction velocity and ρ is the fluid density. Additionally, Patel introduced another parameter, ∆τ = −[ν/(ρu3
τ
)](∂τ /∂n), derived from the non-dimensionalization of the wall-normal shear stress,
where ν is the kinematic viscosity, τ is the wall-normal shear stress, and n is the wall-normal coordinate.
In their study, the presence of favorable pressure gradients was found to lead to significant departures (in
the form of an ‘overshoot’) of the wall-normal profiles of mean streamwise velocity from the logarithmic
law (that characterizes fully developed turbulent boundary layers in equilibrium, see Bradshaw & Huang,
1995) conducive to relaminarization when ∆p < −0.0235 and ∆τ < −0.009. Bradshaw (1969) later corrected the latter threshold to ∆τ < −0.013 when developing a general eddy-Reynolds-number criterion
for relaminarization. Narasimha & Sreenivasan (1973) acknowledged the significance of parameters like
∆p and ∆τ in indicating deviations from the standard constant-pressure laws. However, they cautioned
that these departures alone may not be sufficient to accurately identify the reversion phenomenon. They
argued that the suggested critical values for ∆p and ∆τ are encountered and exceeded before apparent
reversion occurs. Based on their observations, they proposed that relaminarization occurs when pressure
forces dominate over the nearly frozen Reynolds stress. To explain this behavior, they introduced the concept of a quasi-laminar limit, particularly for large values of Λ, defined as Λ = −(δ/τ0)(∂p/∂s), where δ
represents the boundary layer thickness, and τ0 is a characteristic Reynolds stress.
1.3 Overview of WMLES
WMLES involves simulating high-energy eddies in the outer layer and modeling smaller energetic eddies
in the inner layer. This is crucial because turbulence kinetic energy is carried by eddies of varying sizes in
layers close to and away from the wall (Jiménez, 2012). Research by Jimenez & Hoyas (2008) has demonstrated that the turbulence resolved in the outer layer is both generated and dissipated within that region,
mirroring the patterns of Reynolds stress production and dissipation observed in high-Reynolds-number
7

wall turbulence. As a consequence, the complex dynamics of the inner layer, including streaks and quasistreamwise vortices, are simplified by the wall model to a single value of wall shear stress, τw, at each point
on the wall surface (Larsson et al., 2016). In addition to predicting τw, wall models can also provide the wall
heat flux, qw, or wall temperature, Tw, which are crucial for compressible flow simulations in aerospace
and heat transfer applications. This simplification significantly reduces the complexity of wall-turbulence
dynamics, leading to substantial computational efficiencies.
1.3.1 Classifying wall models
WMLES approaches can be classified into two main categories: hybrid RANS/LES and stress-based wallmodeled LES. The primary distinction between these methods lies in how the LES region is coupled with
the near-wall region (Larsson et al., 2016). In hybrid RANS/LES methods, the LES region exists only above
a defined “interface” in the flow domain. Below this interface, the governing equations transition to a
RANS formulation. This approach allows for a combination of the resolved LES and the modeled RANS
approaches to capture both large-scale turbulent structures and the near-wall effects (Piomelli, 2008). On
the other hand, stress-based WMLES extends the LES equations all the way down to the wall itself. This
requires the wall shear stress τw to be known as a boundary condition. To estimate the wall shear stress, a
wall model is employed, which solves a model within a thin layer of thickness hwm near the wall (Kawai &
Larsson, 2012). This model utilizes the resolved flow variables obtained from the LES region to approximate
the wall shear stress.
Stress-based WMLES can be further categorized as equilibrium and non-equilibrium wall models. Nonequilibrium wall models incorporate non-equilibrium effects such as unsteady advection and pressure gradients by resolving simplified Navier-Stokes equations on an embedded thin mesh (Park & Moin, 2014),
whose major disadvantage is the requirement of a separate near-wall grid with full connectivity in all
8

coordinate directions, remaining problematic in complex geometries. Equilibrium wall models can be implemented in practice using different methods. One common approach is to algebraically solve the log-law
equation to determine the friction velocity uτ . The log-law equation establishes a relationship between
the mean velocity profile, the friction velocity, and the wall-normal coordinate. By solving this equation
algebraically, the friction velocity can be obtained, allowing for the estimation of the wall shear stress and
other near-wall quantities (Spalding et al., 1961). Alternatively, numerical methods can be employed to
solve the ordinary differential equation (ODE) derived from the log-law equation. This ODE characterizes the behavior of the mean velocity profile and is typically solved iteratively along with the governing
equations of the flow (Bodart & Larsson, 2011). Griffin et al. (2023) proposed a wall model for turbulent
wall-bounded flows with heat transfer, which extends an ODE model by incorporating compressibility
effects and closing with an algebraic temperature–velocity relation. The model accurately estimates nearwall profiles of temperature and velocity, making it suitable for use as a boundary condition in LES or
RANS solvers.
1.3.2 Laminar wall model
For external flows, the thin laminar boundary layer imposes stringent mesh resolution requirements, resulting in prohibitively high computational costs at practical Reynolds numbers. To address this challenge,
laminar wall models have been developed to reduce the need for fine mesh resolution by leveraging local self-similar solutions of the boundary layer equations. These models enable efficient simulation of
near-wall flows, particularly in high-Reynolds-number scenarios, without compromising accuracy.
9

A notable advancement in laminar wall modeling is the utilization of the Falkner-Skan similarity solution (White & Majdalani, 2006) to approximate the behavior of boundary layers in external flows. Gonzalez et al. (2021) proposed a Falkner-Skan-based wall model (FSWM) to compute wall shear stress (τw) and
boundary layer profiles, which is governed by the third-order ordinary differential equation (ODE):
f
′′′ + ff′′ +
2m
1 + m
h
1 − (f
′
)
2
i
= 0,
with the boundary conditions: f(0) = f
′
(0) = 0, f′
(∞) = 1, where f(η) is the streamfunction in
similarity form, and its derivatives describe the velocity profile within the boundary layer. The similarity
variable η is defined as η = y/
(m + 1)Ue(x)/(2νx)

where y is the wall-normal coordinate, ν is the
kinematic viscosity, Ue(x) is the edge velocity, and x is the streamwise coordinate. The non-dimensional
velocity components are expressed as u/Ue(x) = f
′
(η) and v/Ue(x) = f(η) −
m−1
m+1 ηf′
(η) in the streamwise and wall-normal directions, respectively. Thus, f
′
(η) represents the normalized streamwise velocity,
and f
′′(η) corresponds to the normalized streamwise velocity gradient in the wall-normal direction. At
the wall (η = 0), the boundary conditions ensure that both the streamwise and wall-normal velocity
components satisfy the no-slip condition. The far-field condition f
′
(∞) = 1 enforces that the velocity
asymptotically approaches the free-stream velocity.
The wall shear stress is then given by
τw = µ
m + 1
2
Ue(x)
3
νx
f
′′(0)
where f
′′(0) is obtained numerically. Since shear stress at the wall is determined by the near-wall velocity
gradient, this formulation allows for an accurate prediction of skin friction in attached boundary layers.
The parameters m and k, defining the edge velocity as Ue(x) = kxm, are obtained through least-squares
fitting of the outer flow, enabling the generation of high-resolution baseflows for parabolized stability
10

equation (PSE) analyses. The parameter m is directly linked to the Falkner-Skan pressure-gradient parameter β = 2m/(1 + m) which quantifies the influence of the pressure gradient on the boundary layer. A
positive β indicates a favorable pressure gradient (FPG) with accelerating flow, β = 0 corresponds to a zero
pressure gradient (ZPG) like the Blasius boundary layer, and negative values (−0.198 ≤ β < 0) signify
an adverse pressure gradient (APG), which can promote boundary layer separation. The implementation
of the FSWM also incorporates exception-handling routines to ensure numerical stability. For instance,
the wedge angle is calculated as the angle between the wall-tangent and freestream velocity vector, with
validity ensured by checks on the distance from the stagnation point and the applicability of β within the
range −0.198 < β < 2.0. For β ≤ −0.198, where separated velocity profiles occur, the flow is treated as
separated, and a no-slip boundary condition is imposed. Additionally, to mitigate the computational cost
of solving the Falkner-Skan equations for each computational cell, lookup tables can be employed.
By utilizing the Falkner-Skan similarity solution, this model achieves a substantial reduction in computational cost compared to direct numerical simulation (DNS). Its scaling behavior follows Rex
L
, in contrast
to the more demanding Rex
37/14
L
scaling required for DNS, making it an efficient approach while still capturing key instabilities in laminar regions. As a result, the FSWM serves as a valuable tool for predicting
transition in high-Reynolds-number flows, such as those occurring near airfoil leading edges, providing
a practical alternative to more computationally expensive methods. Gonzalez et al. (2021) demonstrated
the model’s high accuracy even on under-resolved grids, effectively replicating DNS results for stagnation
flow and pressure-gradient boundary layer cases.
Although laminar wall models have proven successful, their implementation presents challenges related to the placement of probing points. As highlighted by LAM (2023), for laminar wall models, probing
points should ideally be located outside the boundary layer to capture the local edge velocity, in contrast
to equilibrium wall models, which position probing points within the logarithmic region. If the mesh is
too fine, the edge velocity may be underestimated, while a coarse mesh may place the probing point too
11

far from the boundary layer. To address this, LAM (2023) suggested using an inviscid approximation of
the edge velocity, Ue, based on Bernoulli’s principle. While this approach is effective in many cases, it is
sensitive to the choice of reference point for x and can significantly differ from viscous flow solutions in
complex geometries.
Prior approaches for laminar/transitional wall modeling include no-slip models, which impose standard boundary conditions on coarse grids but suffer from overpredicted wall shear stress and errors in
disturbance growth rates (Slotnick et al., 2014). Stability-informed methods, such as coupling parabolized
stability equations (PSE) with LES (Lozano-Durán et al., 2018), track instabilities via linearized disturbance
evolution. Integral boundary layer (IBL) methods solve simplified momentum equations for boundary
layer parameters but lack fidelity in non-equilibrium flows. Dynamic slip models (Cabot, 1995) adjust wall
stresses based on local flow states, while hybrid RANS-LES frameworks (Park & Moin, 2014) blend turbulence closures but struggle with transitional regions. These methods often trade generality for efficiency
or rely on empirical assumptions, limiting their predictive accuracy in pre-transitional zones. In contrast,
the Falkner-Skan wall model (FSWM) leverages similarity solutions to generate physics-based baseflows,
balancing accuracy and computational efficiency for high-Reynolds-number applications.
1.3.3 Grid resolution requirements
In an extensive analysis conducted by Chapman (1979), a thorough investigation into the grid resolution
requirements for LES considering the multi-scale nature of a turbulent boundary layer was performed.
Within the outer layer (approximately y/δ > 0.2), Chapman estimated that the required number of grid
points to effectively resolve a domain with dimensions Lx × δ × Lz is proportionally related to Re0.4
. A
suitable grid configuration would encompass
∆x/δ, ∆z/δ
≈ (0.08, 0.05) and have ∆y/δ vary linearly
from 0.02 at y/δ = 0.2 to 0.05 at the boundary layer edge. Conversely, for the viscous sublayer (around
y
+ = y/lv < 50), the necessary points scale as
NxNyNz

vs
∝ Re1.8
, a scaling adjusted to Re13/7

by Choi & Moin (2012). An appropriate LES grid would adopt
∆x
+, ∆z
+

≈ (40, 20) and maintain a
wall-normal grid spacing that shifts approximately linearly from ∆y
+ ≈ 1 to ∆y
+ ≈ 5 (Larsson et al.,
2016). At significantly high Reynolds numbers, even though the computational cost of the outer layer
remains unaffected by Reτ (Pao, 1965), both the viscous and logarithmic layers incur costs scaling as
O

Re2
τ

. As a result, LES converges to a computational cost nearly comparable to DNS for boundary
layer flows (Chung & Pullin, 2009).
In evaluating the computational cost of wall-modeled large-eddy simulation (WMLES), achieving the
appropriate grid resolution for both the inner and outer layers of the flow is essential. While the outer
layer generally requires the same resolution as standard LES, determining the required grid resolution for
the inner layer presents a greater challenge, particularly in stress-based WMLES. Hybrid LES/RANS methods simplify the inner layer resolution, resembling traditional RANS approaches, by imposing less strict
requirements on wall-parallel grid spacings (Piomelli & Balaras, 2002). In contrast, stress-based WMLES,
where the LES extends closer to the wall, requires refined grids for both the inner and outer regions to
ensure grid independence. The recommended grid resolution of
∆x/δ, ∆y/δ, ∆z/δ
≈ (0.08, 0.02, 0.05)
(Kawai & Larsson, 2012) strikes a balance between maintaining simulation accuracy and minimizing computational cost for turbulent boundary layers in mechanical equilibrium.
1.3.4 Challenges and developments in WMLES
Wall-modeled large-eddy simulation (WMLES) faces several challenges in advancing its accuracy and applicability. One significant challenge is modeling separated and non-equilibrium flows, particularly in the
context of shock/boundary-layer interactions, where discrepancies in experimental parameters and uncertainties in skin friction profiles complicate validation. Approaches modeling the convection/pressuregradient terms such as those by Yang et al. (2015) offering potential improvements. Additionally, multiphysics effects like chemical reactions, passive scalar variance, and radiation near the wall require further

development, as current models oversimplify these interactions, often assuming zero wall-normal gradients in the log-layer (Larsson et al., 2015). The need for adaptive or automated modeling also presents
a challenge, especially in defining the wall model exchange heights (hwm) without prior knowledge of
boundary layer thickness (δ), which is critical for optimizing grid resolution. Iterative solutions and adaptive methods, akin to LES-RANS blending functions proposed by Abe (2014), are necessary to address these
issues and improve WMLES for complex flows.
1.4 Gaussian “speed bump” configuration
The Gaussian “speed bump” configuration, initially introduced by Slotnick (2019) and designed by Boeing,
has the potential to provide valuable insights into relaminarization and retransition phenomena, complementing previous studies that have mainly focused on smooth-body flow separation (Simmons et al.,
2017, 2018; Rizzetta & Garmann, 2022; Larsson et al., 2023). Recent experimental work on the speed-bump
configuration by Williams et al. (2020, 2021) examined inflow boundary conditions and investigated the
sensitivity of Reynolds number variation in characterizing separation bubbles. Gray et al. (2022) enhanced
skin friction measurements through modified oil film interferometry, and the results were compared with
a spanwise-periodic direct numerical simulation, demonstrating the method’s effectiveness. Several research groups have performed spanwise periodic DNS to investigate the turbulent flow over the bump
geometry (Uzun & Malik, 2020; Balin & Jansen, 2021; Shur et al., 2021) at different Reynolds numbers. Numerical simulations have demonstrated the occurrence of relaminarization in regions with strong acceleration, and incipient flow separation in regions characterized by adverse pressure gradients at a Reynolds
number ReL = 106
, based on freestream conditions and a length scale L that characterizes the extent of
the bump.
At higher Reynolds numbers, the suppression of relaminarization is observed, accompanied by more
pronounced flow separation in adverse pressure gradient regions. Rizzetta & Garmann (2023) conducted
14

an extensive set of wall-resolved large-eddy simulations (WRLES) over a Gaussian bump configuration to
identify the required bump height and Reynolds number capable of inducing flow separation. Iyer & Malik
(2020) performed WMLES of the flow at Reynolds numbers of ReL = 106
and ReL = 3.6 · 106
to evaluate
the effects of grid resolution, topology, and wall models. However, their study failed to accurately capture
the separation region observed in DNS. In subsequent studies by Iyer & Malik (2023a), they put forth
the notion that grid resolution should be contingent on the Reynolds number to effectively capture the
separation characteristics inherent in the flow. Prakash et al. (2022) carried out an assessment of WMLES
with a specific focus on the IDDES-SST variant for the flow over the specific configuration, systematically
analyzing the influence of grid resolution, elevation constants, and the location of the RANS-LES interface.
To better predict the separation bubble, Agrawal et al. (2022) developed a sensor, whose robustness
was tested at two Reynolds numbers ReL = 2 · 106
and ReL = 3.4 · 106
. Additionally, Balin et al. (2020)
used both the Spalart-Allmaras (SA) and the SST k-ω RANS closures in an improved delayed detached
eddy simulation (IDDES) model for wall-modeled LES, but the model poorly predicted the effects of the
pressure gradient, indicating that RANS is not a reliable near-wall model for this configuration. Whitmore
et al. (2021) applied the approaches of equilibrium wall-modeled LES and slip wall-modeled LES, and found
non-monotonic convergence for simulations with the equilibrium wall-stress model and the Neumann
velocity boundary condition. Therefore, it remains challenging to capture the near-wall physics of the
flow over this “speed bump” configuration, not only for low-fidelity simulations like RANS, but also for
higher-fidelity modeling approaches like WMLES. Meanwhile, less emphasis is placed on the prediction of
relaminarization characteristics of this flow at ReL = 106
in previous studies.
1.5 NASA hump configuration
The NASA hump, also referred to as the curved wall-mounted hump, represents a prototypical flow configuration designed to study flow separation and reattachment phenomena under adverse pressure gradients.
15

The flow over the hump features a range of complex behaviors, including quasi-relaminarization, flow separation, and reattachment.
Seifert & Pack (2002) conducted experiments in a cryogenic pressurized wind tunnel at chord Reynolds
numbers ranging from 2.4 · 106
to 26 · 106
, simulating a Glauert-Goldschmied airfoil’s upper surface. This
study demonstrated the effectiveness of active separation control at high Reynolds numbers while bypassing laminar-to-turbulent transition. Greenblatt et al. (2006) further examined low-speed flow separation
using steady suction, creating a comprehensive dataset of surface pressure, velocity fields, and wall shear
stress, which have since served as benchmarks for computational validation. Additionally, Naughton et al.
(2006) applied oil-film interferometry to measure skin friction distributions over the hump, emphasizing
the method’s capability to capture reverse-flow and high-gradient regions. Despite its utility, discrepancies
remained between computational and experimental predictions of reattachment locations.
Direct numerical simulations (DNS) have played a critical role in resolving detailed flow structures for
both baseline and controlled cases. Postl & Fasel (2006) presented DNS results for unforced and steady
suction scenarios, achieving good agreement with experimental data. Notably, improvements in domain
width enhanced predictions of the recirculation region, though discrepancies near the reattachment point
persisted. Large eddy simulation (LES) has emerged as a powerful tool for modeling separated flows over
the NASA hump, offering enhanced predictive capabilities compared to other methods. You et al. (2006)
employed dynamic subgrid-scale models to simulate baseline and synthetic jet-controlled cases, achieving
greater accuracy than DES, URANS, and constant-coefficient LES models. Franck & Colonius (2010) explored the effects of compressibility on flow separation using compressible LES, documenting changes in
separation bubble size and velocity profiles. Wall-modeled LES (WMLES) studies by Park (2016) and Iyer
& Malik (2016) demonstrated the potential of wall models in capturing mean flow features over coarse
grids, though predictions remained sensitive to grid resolution and wall model-exchange location. Kahraman & Larsson (2020) introduced a postprocessing algorithm that systematically determines the exchange
16

location for an equilibrium wall model, using flow over a wall-mounted hump as a validation case. The
algorithm successfully identifies exchange locations, with thinner wall-modeled layers in nonequilibrium
regions and thicker layers in equilibrium regions, highlighting its potential for simulating complex geometries. Implicit LES (ILES) studies by Sekhar et al. (2015) and Morgan et al. (2007) highlighted the technique’s
capability to capture separated flow physics with coarse grids, further advancing the field.
Detached eddy simulation (DES), which combines RANS and LES methodologies, has also been applied
to the NASA hump. Krishnan et al. (2006) and Bozinoski et al. (2012) demonstrated that DES could predict
reattachment length and pressure coefficients more accurately than RANS models, successfully capturing
the three-dimensional structure of separated flows in the wake. Lattice Boltzmann methods (LBM) have
gained attention for high-fidelity simulations of separated flows. Noelting et al. (2008) and Duda & Fares
(2016) employed LBM-based very large eddy simulation (VLES) to study baseline and controlled cases,
achieving excellent agreement with experimental data. Their findings underscored the importance of accurately modeling endplate geometry to avoid artificial periodicity, a critical aspect of realistic simulations.
Reynolds-averaged Navier–Stokes (RANS) simulations, while widely used for industrial applications
due to their computational efficiency, often struggle to accurately predict separated flows. Studies by Iaccarino et al. (2004), Balakumar (2005) and He et al. (2007) applied RANS with various turbulence models
(k −ϵ, k −ω and SA) to simulate separation control effects. While RANS captured general flow features, it
frequently underpredicted turbulent shear stress and overpredicted separation bubble size. Morgan et al.
(2006) incorporated high-order spatial discretization into RANS simulations, achieving improved accuracy
compared to second-order schemes. Rumsey (2007) summarized findings from the NASA Langley CFDVAL2004 workshop, which focused on validating computational methods for synthetic jets and turbulent
separation control. This workshop highlighted significant discrepancies between RANS and LES results,
underscoring the need for hybrid approaches.
17

Wall-resolved Large-Eddy Simulations have shown that the flow acceleration over the front portion
of the hump is sufficiently strong to exceed the relaminarization criterion of Narasimha & Sreenivasan
(1973), but only for a short streamwise region. While the flow does not fully relaminarize, the skin friction
variation shows a plateau, which is often overpredicted by lower-fidelity methodologies. Previous studies
have primarily focused on separation characteristics, with less attention given to (quasi-)relaminarization
in such flow configurations. This highlights a significant gap in the current understanding, as existing
models often fail to accurately capture the nuanced behaviors of turbulent boundary layers near the relaminarization threshold. The limited focus on (quasi-)relaminarization underscores the need for advanced
simulation methods that can more accurately represent these complex behaviors, particularly in flows subjected to varying pressure gradients.
1.6 Objectives and scope of the thesis
The present study introduces a novel WMLES approach intended to improve predictions for flows undergoing relaminarization due to favorable pressure gradients. The near-wall modeling approach blends two
commonly used boundary conditions: an ODE-based equilibrium wall model and a no-slip (or, potentially,
a laminar wall model) boundary condition. The blending is dictated by a local sensor that departs from
traditional sensors used in transitional flows (based on estimates of turbulence kinetic energy), and relies instead on relaminarization indicators, and incorporates a delay to account for history effects of the
boundary layer in the relaminarization process.
The proposed wall modeling approach is used to simulate turbulent boundary layer flow over a Gaussianshaped bump geometry at a freestream Mach number of M = 0.2, across three different Reynolds numbers,
as well as over the NASA hump configuration at a Reynolds number of Reθ ≈ 7, 000.
This thesis is structured as follows. Chapter 2 outlines the numerical framework, introduces the sensorbased blended WMLES methodology, and details the suite of sensors evaluated in later analyses. Chapter 3
18

presents the computational configurations for turbulent boundary layer flows over a Gaussian-shaped
bump and a NASA hump, including geometric and boundary condition specifications. Chapter 4 analyzes
simulation outcomes, benchmarking results against existing experimental and high-fidelity numerical data
while assessing sensitivity to model parameters. Finally, Chapter 5 synthesizes the core findings, discusses
their implications for turbulence modeling, and highlights the efficacy of the proposed sensors in capturing
critical flow physics over the studied geometries. The appendix provides a comparative analyses of the new
sensors against established laminar-to-turbulent transition criteria from prior literature.
19

Chapter 2
Methodology
This chapter presents the methodology used in this study, with a particular focus on the proposed sensorbased blended wall-modeling approach. The primary objective of this approach is to enhance the prediction
of relaminarizing flow regions, which pose challenges for conventional wall models that typically assume
fully turbulent boundary layers. Section 2.3 introduces the newly developed wall-modeling framework,
incorporating sensor-based blending to dynamically adapt between different wall treatments based on local
flow characteristics. The benchmark configurations used for model evaluation, including the Gaussian
speed bump and the NASA wall-mounted hump, are described in Section 3. The numerical setup, including
inflow conditions, boundary treatments, and turbulence generation techniques, is also detailed.
All simulations are conducted using an in-house developed compressible flow solver that implements
a finite-volume, cell-centered formulation to discretize and integrate the (LES) low-pass filtered NavierStokes equations. The solver operates on unstructured, body-fitted meshes, using a second-order spatial
discretization and a fourth-order Runge-Kutta time stepper. All meshes considered in the present simulations are comprised of hexahedral cells. A calorically perfect gas fluid model is assumed with a power-law
temperature dependence of the viscosity and a constant Prandtl number, P r = 0.7. The explicit subgrid
scale model proposed by Vreman (2004) is employed with a fixed model coefficient of 0.07 to account
20

for unresolved turbulent scales. The flow solver has been validated in previous studies of compressible
turbulent boundary layers (Hoy & Bermejo-Moreno, 2022).
2.1 LES formalism and governing equations
In LES, resolved quantities corresponding to fluid motions at large, energy-containing scales are obtained
by filtering flow fields using a spatiotemporal filter kernel operator denoted as G. This operator acts on
any given field, f, to obtain the filtered quantity, denoted as f (Leonard, 1975). The form of the filter kernel
operator depends on the filtering technique used in LES, like spatial or spectral filtering. Mathematically,
this filtering operation can be represented as:
f ≡ F [f] ≡
Z
D
G(
⃗ξ − ⃗ξ
′)f(
⃗ξ
′)d
⃗ξ
′
(2.1)
where the integral is extended over the entire simulation domain. In general, ⃗ξ = {x;t}, and the filtering
operation is performed in space and time. Additionally, the filter-α-derivative commutation operator,
Cα [f], is defined as
Cα [f] = F

∂f
∂α
−
∂F [f]
∂α =
Z
D
G(
⃗ξ − ⃗ξ
′)
∂f(
⃗ξ
′)
∂α d
⃗ξ
′ −
∂
∂α Z
D
G(
⃗ξ − ⃗ξ
′)f(
⃗ξ
′)d
⃗ξ
′
(2.2)
which is zero for a filter that commutes with the α-derivative operator.
Introducing Favre (i.e., density weighted) filtered quantities
˜f =
f ρ
ρ
(2.3)
21

the filtered conservation laws of mass, linear momentum, and total energy for a compressible flow, in the
absence of external forces and assuming that commutation errors can be neglected, can be expressed in
differential form, as (Bermejo-Moreno, 2009):
∂ρ
∂t +
∂ρu˜j
∂xj
= 0 (2.4)
∂ρu˜i
∂t +
∂ρu˜iu˜j
∂xj
= −
∂p
∂xj
+
∂ ˘dij
∂xj
−
"
∂ρugiuj
∂xj
−
∂ρu˜iu˜j
∂xj
#
+
"
∂dij
∂xj
−
∂ ˘dij
∂xj
# (2.5)
∂ρe˜T
∂t +
∂ρe˜T u˜j
∂xj
= −
∂ρu˜j
∂xj
+
∂ ˘diju˜i
∂xj
−
∂q˘j
∂xj
−
"
∂pe]T uj
∂xj
−
∂ρe˜T u˜j
∂xj
#
−
"
∂puj
∂xj
−
∂pu˜j
∂xj
#
+
"
∂dijui
∂xj
−
∂ ˘diju˜i
∂xj
#
−
"
∂qj
∂xj
−
∂q˘j
∂xj
#
(2.6)
where Einstein summation convention is implied: ρ is the density, ui(i = 1, 2, 3)is the velocity component,
eT = e +
1
2
ρuiui
is the total energy per unit mass (e is the internal energy per unit mass), qj is the heat
flux given by Fourier’s law as qj = −κ∂T /∂xj (T is the temperature and κ is the thermal conductivity),
and σij is the stress tensor, defined as
σij = −pδij + dij (2.7)
where p is the thermodynamic (or equilibrium) pressure, δij is the Kronecker delta (1 if i = j, 0 otherwise)
and, for a Newtonian fluid:
dij = 2µSij +

µv −
2
3
µ

∂uk
∂xk
δij . (2.8)
µv is the bulk viscosity, µ is the dynamic viscosity which is computed by Sutherland’s law, and Sij =

∂jui + ∂iuj

/2 is the strain rate tensor. In general, µ = µ(T), κ = κ(T). Thus dij = dij (T,u), qj =
2

qj (T). An equation of state relates p to two other state variables of the fluid; for an ideal gas p = ρRT,
where R is a gas constant. The enthalpy, h, is defined as h = e + p/ρ.
For a calorically perfect gas, as considered in the present study, the specific heat capacities at constant
pressure and volume (cp and cv respectively) and their ratio γ = cp/cv are constants (i.e., do not depend on
other thermodynamic variables), so that the internal energy and enthalpy can be expressed as e = cvT =
p/
(γ − 1) ρ

and h = cpT = γe.
In equations 2.4- 2.6,
˘f refers to the formal expression of f with all constituent variables replaced with
their Favre-filtered counterparts, from which it differs:
˘dij ≡ dij
T , ˜ u˜

= µ

T˜

∂u˜i
∂xj
+
∂u˜j
∂xi
!
+

µv

T˜

−
2
3
µ

T˜

∂u˜k
∂xk
δij ̸= ˜dij (2.9)
q˘j ≡ −κ˘
∂T˜
∂xj
= −κ

T˜
∂T˜
∂xj
= qj

T˜

̸= ˜qj (2.10)
The filtered thermodynamic pressure can be obtained through the filtered equation of state; for an ideal
gas: p = RρT = RρT˜ = (γ − 1) ρe˜. If the total energy equation is used, this brings modeled quantities
into the calculation of the filtered pressure (and temperature):
p
γ − 1
= cvρT˜ = ρe = ρeT −
1
2
ρuiui = ρe˜T −
1
2
ρu˜iu˜i −
1
2
ρ

ugiui − u˜iu˜i

(2.11)
Terms in square brackets on the right-hand side of equations 2.5 and 2.6 involve unresolved quantities
and thus require modeling. In the present simulations, only the first term in square brackets of each
equation is modeled, while all other square-bracket terms are neglected. In particular, Vreman’s subgridscale model (Vreman, 2004) is used to express the unknown turbulent stress tensor τ
S
ij = ρ(ugiuj −u˜iu˜j ) as
a combination of resolved velocity gradients and an eddy viscosity term −2µeS˘
ij + τ
S
kkδij/3. Considering
2

the resolved velocity gradients in different directions to capture the anisotropic effects of turbulence, the
following eddy viscosity was proposed by Vreman (2004):
µe = cρ
s
Bβ
αijαij
(2.12)
with
αij = ∂iu˜j =
∂u˜j
∂xi
(2.13)
βij = ∆2
mαmiαmj (2.14)
Bβ = β11β22 − β
2
12 + β11β33 − β
2
13 + β22β33 − β
2
23 (2.15)
∆m is the grid spacing in the m-th direction, which is taken as the cubic root of the cell volume in the
present simulations that consider unstructured meshes. The model constant c is related to the Smagorinsky
constant Cs by c ≈ 2.5C
2
s = 0.07. To model the turbulent heat flux in the filtered total energy equation, a
constant turbulent Prandtl number P rt = 0.7 is used in this numerical study, to obtain the subgrid-scale
(eddy) thermal diffusivity as αe = (µe/ρ)/P rt
.
The filtered conservation laws of Eqs. 2.4–2.6 in integral form at each mesh cell c with volume δVc and
bounding surface area δAc are defined as follows:
d
dt
Z
δVc
qdV +
Z
δAc
FdA =
Z
δAc
GdA (2.16)
where q = [ρ, ρuei
, ρeeT ]
T
is the state vector of conserved variables, F =

ρun, ρueiun + pni
,(ρeeT + p)un
T
is the convective (Euler) flux vector and G =
h
0,(
ˇdij + d
S
ij )nj ,
ˇdijueinj + (ˇqj + q
S
j
)nj
iT
is the diffusive
flux vector. The component of the relative flow velocity normal to the surface of the (cell) control volume
24

is un = ueini
, where ni
is the unitary surface normal. In semi-discrete form, Eq. 2.16 can be expressed for
each cell c as:
d
dt
qc =
1
δVc
X
f∈δAc
(−F
f
c + Gf
c
)A
f
c
(2.17)
where qc is the state vector at the cell centroid, whereas F
f
c and G
f
c are the convective and viscous fluxes
at face f of the cell c.
2.2 Equilibrium wall model
The equilibrium wall model formulation of Kawai & Larsson (2012) is considered first. It is a wall-stress
model (Larsson et al., 2016) which solves the following system of equilibrium boundary layer ordinary
differential equations (ODE) in a near-wall, refined grid at each boundary face:
d
dn
"
(µ + µt,wm)
du||
dn
#
= 0 (2.18)
d
dn

(µ + µt,wm)u||
du||
dn
+ cp

µ
P r
+
µt,wm
P rt,wm !
dT
dn

 = 0 (2.19)
with the eddy dynamic viscosity of the wall-model, µt,wm, obtained from a mixing-length model with a
van Driest damping function (Van Driest, 1956)
µt,wm = κρuτn

1 − exp
−
n
+
A+
!

2
(2.20)
where u|| denotes the wall-parallel velocity, T is the temperature, n is the local wall-normal coordinate,
uτ =
p
τw/ρ is the friction velocity, which is a function of the unknown wall shear stress magnitude
τw (the wall shear stress vector is assumed parallel to the wall-parallel velocity). The molecular fluid
properties are the dynamic viscosity, µ, the specific heat capacity at constant pressure, cp, and the Prandtl
number, P r. The superscript ‘+’ denotes inner units, such that n
+ = n/ℓv, where lv = µ/(ρuτ ) is the
25

viscous length scale. The model constants κ = 0.41, A+ = 17, and P rt,wm = 0.9 are chosen for the
current simulations.
The discretization of Eqs. 2.18 and 2.19 is performed using a second-order finite-volume method following (Larsson et al., 2016). The ODE system is solved in a separate wall-model grid, for each boundary
face, that extends from the wall up to a specified wall-model (exchange) height, hwm. The wall-model grid
overlaps with the coarser LES grid, which also extends to the wall. At the exchange location, n = hwm, the
wall model retrieves the LES temperature T, pressure p, and wall-parallel velocity u|| as inputs to estimate
the instantaneous magnitude of the wall stress vector τw and the wall temperature Tw (for the case of an
adiabatic wall as considered in this study).
The wall-model height hwm is typically chosen between 10% and 20% of the boundary layer thickness (Kawai & Larsson, 2012). In the present simulations, both spatially uniform and spatially varying
distributions of the wall-model height are considered for different simulations. The uniform distributions
prescribe a value of hwm = 0.1δ0, where δ0 represents the boundary layer thickness at a reference location.
As noted in §3 for the speed bump configuration, based on prior DNS and experiments, and depending on
the grid resolution, δ0/L ≈ 0.0046 at xref/L = −0.8 and δ0/L ≈ 0.013 at xref/L = −0.5 are considered
for the ReL = 106
case. For higher Reynolds number cases, a boundary layer thickness of δ0/L ≈ 0.0072
at xref/L = −0.65 is considered. In the NASA hump configuration, the reference boundary layer thickness
is taken as δ0/L ≈ 0.0726 at xref/L = −2.14.
2.3 Sensor-based blended wall model
As will be shown in the results presented in §4.1.1.2, application of the previously introduced equilibrium
wall model (§2.2) in regions of relaminarization can significantly overpredict the friction coefficient and,
consequently, other quantities of interest of the boundary layer. Previous work in WMLES of laminar-toturbulent transitional flows has introduced wall-modeling strategies that locally switch on and off either
26

the entire wall model (resorting to a no-slip boundary condition within the LES that estimates the velocity
gradient based on the first cell adjacent to the wall) (Bodart & Larsson, 2012) or the eddy viscosity in
the wall-model formulation (setting the right-hand side of Eq. 2.20 to zero) (Mettu & Subbareddy, 2018).
In those methods, the switch of boundary conditions is based on whether the value of a local sensor
(computed for each wall boundary face) is below or above a specified threshold. Different sensors (and
corresponding thresholds) have been proposed to distinguish laminar and turbulent regions in the context
of laminar-to-turbulent transition, commonly relying on relative measures of turbulence kinetic energy at
the wall-model exchange height.
Whereas these prior approaches have shown improved prediction of forward transitional flows (i.e.,
from laminar to turbulent states), their adequacy for flows with relaminarization (reverse transition), such
as the flow over a Gaussian bump considered in this study, is compromised (see appendix 6.1). One main
reason is that, as discussed in §1, in a relaminarizing flow, turbulence fluctuations are not necessarily dissipated, and may remain in a nearly ‘frozen’ state, rendering those transition sensors inadequate. Therefore, other sensors are needed to detect relaminarization. A second, more subtle reason why previous
approaches successful in the prediction of (forward) transitional flows may not be suited for application to
relaminarizing flows is the delay in the response of the boundary layer between the action of the agent triggering relaminarization (e.g., pressure gradient) and the manifestation of relaminarization (also discussed
in §1, and references therein).
In this work, we propose a wall-modeling approach that is inspired by those existing sensor-based
transition wall-models (Bodart & Larsson, 2012; Mettu & Subbareddy, 2018) but introduces three key differences. The first is in the definition of new sensors that are suited to identify relaminarization. Secondly,
rather than a binary switch between two boundary conditions (i.e., fully activating or deactivating the
equilibrium wall model or its eddy viscosity depending on whether the sensor exceeds a threshold), we
27

apply a local blending of the two boundary conditions, with the blending factor dependent on the sensor value. Lastly, we introduce a local time delay that accounts for a lag in the response of boundary
layer quantities during the relaminarization process. Each of these elements of the methodology will be
explained next.
2.3.1 Sensors for the detection of pressure-driven relaminarization
Two related sensors are considered to identify the onset of relaminarization in the presence of a favorable
pressure gradient (FPG) and the retransition to turbulence once the FPG subsides. The applicability of these
sensors is investigated in §4 through a posteriori analyses for the simulation of the flow over a Gaussian
bump, which is statistically homogeneous in the spanwise direction and statistically stationary in time.
The suitability of each sensor for on-the-fly calculations from the data available to the wall model in
more general, complex flows at different flow regimes, and particularly in the context of finite-volume
unstructured-mesh based flow solvers is also discussed here.
The formation and break-up of low-speed streaks, as explained by Kline et al. (1967), are crucial in
understanding the production and transport of turbulence kinetic energy, as it also pertains to relaminarization and transition processes. These streaks are formed by the streamwise vorticity, which gradually
lifts low-speed fluid away from the wall through the secondary vorticity (Nychas et al., 1973; Kim et al.,
1971). Bursting occurs when rapid and sudden instabilities take place. Pressure gradients influence the
ejection process of fluid between the inner and outer layers. Positive pressure gradients intensify and
increase the frequency of energy, momentum, and vorticity transfer, while negative pressure gradients
reduce the rate of bursting. Several experiments have found that bursting and, thus, streak break-up are
suppressed when the relaminarization parameter K = (νw/U2
e
)(∂Ue/∂s) = −[νw/(ρeU
3
e
)](∂p/∂s) exceeds a threshold value between 2 · 10−6
and 3.7 · 10−6
(Launder, 1964; Moretti & Kays, 1965; Kline et al.,
1967; Patel & Head, 1968).
28

To study relaminarizing flows, a different non-dimensionalization of the streamwise pressure gradient,
referred to as ∆p = −[νw/(ρwu
3
τ
)](∂p/∂s) in Patel & Head (1968), and commonly known as the acceleration parameter (Uzun & Malik, 2020), has been proposed in prior work. It is worth noting that K is also
referred to as the “acceleration parameter" in some references (Narasimha & Sreenivasan, 1979; Bourassa &
Thomas, 2009). Patel & Head experimentally observed that when ∆p exceeds 0.018, the near-wall velocity
profile deviates from the logarithmic law, marking the onset of relaminarization. Recent DNS of the flow
over a Gaussian bump conducted by Uzun & Malik (see Figure 12 in Uzun & Malik, 2020) found the acceleration parameter and the suggested threshold of 0.018 suitable to identify the region of relaminarization
in the flow. In contrast, for the same DNS, the use of the relaminarization parameter, K, required lowering
its threshold below its standard values to identify the same region of relaminarization as with ∆p (and to
modify the threshold differently at the onset than at the retransition locations). The acceleration (∆p) and
relaminarization (K) parameters are related through the friction coefficient by ∆p ∝ K/C3/2
f
(see Patel
& Head, 1968).
Inspired by the parameters K and ∆p, to account for the physics of relaminarization and retransition induced by the aforementioned effects of pressure gradients on the suppression of the bursting of
low-speed streaks and driving departures of the near-wall velocity from the log-law, we introduce the
two sensors considered in this study, termed ‘relaminarization sensor’, σR, and ‘acceleration sensor’, σA,
defined as:
σR = −
νw
ρe⟨ue⟩
3

∂pw
∂s
(2.21)
σA = −
νw
ρw⟨uτ ⟩
3

∂pw
∂s
(2.22)
29

where the subscripts ‘e’ and ‘w’ denote (local) edge and wall quantities, respectively. The friction velocity
uτ =
p
τw/ρw depends on the wall shear stress, τw. The streamwise pressure gradient, ∂pw/∂s, is approximated by the value obtained at the wall-model exchange height as the dot product of the pressure gradient
vector and the unitary vector in the wall-parallel velocity direction for the corresponding boundary face,
and it is thus well defined even on walls with geometric discontinuities. A filtering operation, denoted by
angular brackets ⟨·⟩, can be applied to the calculation of the sensors to remove local spatial oscillations
and high-frequency temporal unsteadiness. Depending on the characteristics of the flow, different choices
of filters can be considered, such as spatial filters along directions of statistical homogeneity and temporal filters, either global (for statistically stationary flows) or local (e.g., based on an exponential moving
average of Meneveau et al., 1996), with a suitable filter time scale that captures the unsteadiness of energycontaining motions (see examples in Bodart & Larsson, 2011; Mettu & Subbareddy, 2018). In our present
simulations, we utilize temporal filters implemented with an exponential moving average employing the
weight function W(t) = exp(−t/T) (see Bodart & Larsson, 2011), where the time scale T is defined as
T = h
2
wm/νsgs or, alternatively, T = const, with the constant chosen as approximately 0.1L/U∞. A desirable property of the sensors and the corresponding variables entering their definition is their robustness
to the choice of filtering time scale. The sensitivity of the proposed sensors to this filtering time scale will
be assessed in appendix 6.1, and contrasted with that of surrogates of the turbulence kinetic energy found
in laminar-to-turbulent transition sensors previously developed in the literature (Bodart & Larsson, 2011;
Mettu & Subbareddy, 2018).
Both sensors are based on local quantities of the boundary layer. From an implementation standpoint,
the acceleration sensor, σA, depends only on local wall or near-wall quantities that can be readily calculated
at each wall boundary face within the wall-model solver (including wall quantities and data retrieved at
the exchange height, hwm). Thus, σA is generally applicable to flows in arbitrarily complex geometries,
as it does not require the retrieval of any additional off-wall boundary layer quantities. In contrast, the
30

relaminarization sensor, σR, not only depends on near-wall quantities but also on boundary layer edge
quantities, which poses additional challenges to its implementation and application in complex geometries.
For the flow over a Gaussian-shaped bump at Mach number 0.2 under consideration in this study, the
edge density and edge velocity entering the formulation of the relaminarization sensor are approximated,
respectively, by the freestream density and by the potential velocity (Launder, 1964), up = U∞
p
1 − Cp,
defined in terms of the local pressure coefficient, Cp = (p − p∞)/q∞, where q∞ = ρ∞U
2
∞/2 is the
freestream dynamic pressure.
2.3.2 Blending factor
At each boundary face, the wall-modeled quantities (i.e., the wall shear stress, τw, and either the wall
heat flux, qw, for isothermal walls, or the wall temperature, Tw, for adiabatic walls) are obtained as a
linear combination (blending) of the corresponding quantities derived from the application of two separate
boundary conditions: an equilibrium wall model (by solving the ODE system as described in §2.2) and
either a no-slip wall (i.e., by calculating at the boundary face the wall-normal derivative of the resolved
wall-parallel velocity and temperature in the LES grid) or, more generally, a laminar wall model (e.g., by
setting the turbulent eddy viscosity of the equilibrium wall model to zero and lowering the exchange height
to the centroid of the first cell adjacent to the wall boundary face). The latter (laminar wall model) can
mitigate the grid resolution requirements imposed by the no-slip condition in regions of relaminarization.
Denoting by Qbwm any wall quantity obtained by the blended wall model, Qns the wall quantity retrieved by the no-slip boundary condition on the LES grid (or, alternatively, the laminar wall model), and
Qewm the wall quantity calculated by the equilibrium wall model from the LES data exchanged at n = hwm,
then the blending process is formulated as:
Qbwm = βQns + (1 − β)Qewm (2.23)
31

where β ∈ [0, 1] is the local blending factor. The extrema of the blending factor correspond to the full
application of either the equilibrium wall model (for β = 0) or the no-slip (alternatively the laminar wall
model) wall boundary condition (for β = 1).
The local blending factor, β, is defined in terms of the chosen sensor, σ, as
β = min[max(σ/σ0, 0), 1], (2.24)
where σ0 is a reference value of the sensor characterizing the physical phenomenon under consideration
(e.g., relaminarization) transitioning between limiting states (turbulent and laminar boundary layer flow)
most accurately described by each individual boundary condition (‘ewm’ and ‘ns’). The min/max functions
are included in the functional dependence to ensure that β ∈ [0, 1].
This blending approach generalizes existing methods (e.g., as applied to transitional flows, see Bodart &
Larsson, 2011; Mettu & Subbareddy, 2018) that rely on a local binary (rather than gradual) switch between
two boundary conditions based on whether the local sensor, σ, exceeds the reference value (threshold),
σ0, by defining a different, binary blending factor of the form
βbinary =



0 if σ < σ0
1 otherwise
(2.25)
Due to the significant variation in the predicted wall shear stress between the two boundary conditions,
the use of a binary switch between boundary conditions can lead to oscillations near the onset of the
relaminarization region. Furthermore, DNS results of the vorticity field indicate that the near-wall region
of the boundary layer becomes intermittent and does not exhibit a clear demarcation between turbulent
and laminar regions (Balin & Jansen, 2021). This suggests that applying a strict sensor threshold and a
binary switch alone may not accurately capture the gradual relaminarization process. Additionally, in
32

flows such as the one considered in this study, it has been previously observed that a fully laminar state is
not achieved by the end of the favorable pressure gradient region (Balin et al., 2020). Thus, even for cases
with incomplete relaminarization, a modification of the boundary layer properties can still be accounted
for by the smooth blending of boundary conditions (when the sensor is still below the reference value).
2.3.3 Hysteresis scale
Previous studies have observed that relaminarization is not completed at the location where indicators
such as the acceleration and relaminarization parameters peak, but further downstream (Narasimha &
Sreenivasan, 1973, 1979; Uzun & Malik, 2020). Whereas commonly defined thresholds for these parameters
found in the literature successfully identify the onset of relaminarization, subsequent reversion (retransition) from quasi-laminar to turbulent flow does not occur at the location where these indicators fall below
such thresholds either, but also some distance downstream. A phenomenon of hysteresis thus arises, which
has been explained by a delay between the flow acceleration imposed externally and the response of the
Reynolds stress correlation (Bourassa & Thomas, 2009).
To account for this hysteresis in the modeling of backward and forward transition processes, we consider a hysteresis time scale defined as
T = α
νw
u
2
τ
(2.26)
where α is a time scale coefficient. The hysteresis time is factored in as a spatial delay, d ≡ uτT = ανw/uτ ,
when calculating the blending factor at any wall point, x. The resulting blending factor at x is obtained
as the value of β given by Eq. 2.24 for the sensor value that corresponds to a streamwise location given
by x − ds(x), where s is the streamwise unitary vector at x, and thus, a distance d upstream, resulting
β(x) = β[σ(x − ds(x))].
The effect of pressure gradient and Reynolds number on turbulent boundary layer structures is taken
into consideration when defining the hysteresis scale and justifying the associated parameter α. Monty
33

et al. (2011) conducted a parametric study on adverse pressure gradient turbulent boundary layers, considering the pressure gradient parameter, Reynolds number, and acceleration parameter. The profiles obtained
at a constant pressure gradient parameter reveal that the behavior of the logarithmic region is independent
of the Reynolds number. This suggests that the decrease in mean velocity below the log-law region is primarily influenced by the pressure gradient. In a subsequent study by Harun et al. (2013), the modifications
of large-scale features in non-equilibrium boundary layers under adverse, favorable, and zero pressure
gradients were investigated. An energy spectrum analysis demonstrated the varying effects of pressure
gradients on the turbulence structures in the inner and outer regions of the boundary layer. In the case of
zero pressure gradient, intense turbulent fluctuations were observed in the inner region near 15 viscous
units from the wall, corresponding to a streamwise wavelength of approximately 103 wall units. In the
outer region, a second peak was observed near 300 viscous units from the wall, representing a streamwise length scale of approximately 3400 wall units, roughly equivalent to the boundary layer thickness.
The inner peak, associated with near-wall elongated streaks, is expected to exhibit a rapid response to
changes in pressure gradient and remained nearly identical in the three pressure gradient cases. However,
the secondary peak related to superstructures displayed notable differences. These scales are suppressed
relative to the wall shear in the presence of a favorable pressure gradient and enhanced under an adverse
pressure gradient. Therefore, in the present study, α = 103
is chosen as a suitable value to reflect the
response of a boundary layer to non-equilibrium conditions for all Reynolds number cases considered.
The sensitivity of the results to this parameter is assessed in section §4.1.4. The degree of applicability of
the proposed hysteresis time scale, T, and choice of α in other flow configurations is left as future work.
In particular, the proposed hysteresis time scale relies on the assumption of attached flow, making it less
suitable (unnecessarily large) in separated flow regions. Nonetheless, in such separated flow regions, the
assumptions behind the equilibrium wall model are also compromised, but generally have a lesser impact
34

in the prediction of quantities of interest such as the friction coefficient (much lower than in attached flow
regions).
35

Chapter 3
Problem setup
This study employs two benchmark configurations to assess the performance of the proposed sensor-based
blended wall-modeling approach: a turbulent boundary layer over a Gaussian-shaped bump and the NASA
wall-mounted hump. These cases are selected due to their well-documented pressure-gradient effects
and the availability of experimental and high-fidelity numerical data for validation. Both configurations
are widely used in turbulence modeling studies, as they exhibit complex flow phenomena such as quasirelaminarization, separation, and reattachment.
At specific Reynolds numbers, both cases experience flow acceleration sufficient to induce quasirelaminarization, making them valuable for evaluating the model’s ability to capture this process. The
proposed sensor-based, blended wall-modeling approach is applied to conduct simulations of the flow
over a Gaussian-shaped bump at three different Reynolds numbers (ReL = 106
, 2 · 106
, and 4 · 106
). The
favorable and adverse pressure gradients imposed by the wall shape produce departures from equilibrium
conditions in the incoming turbulent boundary layer. For the lowest Reynolds number under consideration, ReL = 106
, the flow exhibits partial relaminarization upstream of the bump apex, whose prediction
is the main focus of the new modeling strategy. For the other two, higher Reynolds numbers, ReL = 2·106
and 4 · 106
, relaminarization is absent, but improvements in the prediction of the flow in the accelerated
flow regions and the robustness of the proposed blended wall model will be assessed. Different degrees of
36

flow separation on the leeward side of the bump are present for the three Reynolds numbers under consideration. Whereas the prediction of flow separation is not the primary objective of the newly introduced
wall model, it will be discussed in relation to history effects in the boundary layer stemming from upstream relaminarization. In the NASA hump case, the flow acceleration over the front portion of the hump
exceeds the relaminarization criterion proposed by Narasimha & Sreenivasan (1973) over a short distance,
resulting in a plateau in the skin-friction coefficient profile without achieving full relaminarization. The
following subsections provide the details of the two configurations, including their geometric definitions
and computational setups.
3.1 Gaussian “speed bump”
The three-dimensional bump configuration introduced by Slotnick (2019) in the form of a streamwise
Gaussian distribution profile tapered along the spanwise direction is defined analytically by
y(x, z) = h
2

1 + erf L
2 − 2z0 − |z|
z0
!
 exp "
−

x
x0
2
#
, (3.1)
where x, y, and z denote the axial, vertical, and spanwise directions, respectively. The constants are
chosen as x0 = 0.195L, z0 = 0.06L, and h = 0.085L, where L is the spanwise width. The present
simulations consider a span-uniform version of the original three-dimensional configuration defined by
y(x) = h exp
−(x/x0)
2

within a reduced spanwise domain of width Lz = 0.04L. A side view of the
computational domain is shown in Figure 3.1.
To evaluate the effectiveness of the newly proposed sensor-based blended wall-modeling approach,
simulations are carried out across three different Reynolds numbers: ReL = 106
, 2 · 106
, and 4 · 106
, based
on the freestream velocity, density and viscosity, and the length L used in Eq. 3.1. Time-averaged statistics
are gathered over a period Tavg ≥ 10L/U∞, after an initial transient Tavg ≥ 5L/U∞.
37

−1.0 −0.5 0.0 0.5 1.0 1.5 2.0
x/L
−0.2
0.0
0.2
0.4
y/L
Lin/L Lss/L Lsp/L
H/L
sponge
y = h exp h
−(x/x0)
2
i
Flow
Figure 3.1: Geometric configuration and computational domain in the xy plane. The geometry is extruded
0.04L in the spanwise direction, z.
The inflow to the computational domain is positioned at x/L = −0.98, where synthetic turbulence
is generated using a digital filtering technique (Klein et al., 2003; Xie & Castro, 2008; Touber & Sandham,
2009), with the necessary input of mean and turbulence wall-normal profiles extracted from prior DNS of a
turbulent boundary layer by Hoyas & Jiménez (2008) at commensurate Reynolds numbers to those from the
DNS and experiments targeted by the present WMLES. A sponge region is applied by means of streamwise
grid stretching and source terms of the form Si ∝ [(x − xs)/ℓs]
2
(ϕi − ϕi
), where ϕi
is the instantaneous
field (density, momentum density components, and total energy density), ϕi
is the corresponding target
field obtained by matching the mean outflow condition, xs, is the streamwise coordinate corresponding
to the start of the sponge region, and ℓs is the streamwise extent of the sponge, which is intended to
dampen turbulence fluctuations before reaching the outlet. A characteristic outflow boundary condition
is imposed at the outlet of the computational domain, with a uniform pressure matching the incoming
freestream pressure at M = 0.2. For the ReL = 106
case, this configuration yields a mean inflow boundary
layer thickness of δ0/L ≈ 0.0046 at xref/L = −0.8 and a friction Reynolds number of Reτ,ref ≈ 530,
matching the inflow conditions from the DNS by Uzun & Malik (2020). For the case with ReL = 2 · 106
, a
reference boundary layer thickness of δ0/L ≈ 0.0072 with Reτ,ref ≈ 639 at xref/L = −0.65 is targeted,
in alignment with the experimental reference data of Williams et al. (2020).
38

−3 −2 −1 0 1 2 3 4
x/L
−0.5
0.0
0.5
1.0
y/L
Lin/L Lsn/L Lst/L H/L
Flow NASA Hump stretching
Figure 3.2: Geometric configuration and computational domain of NASA hump in the xy plane. The
geometry is extruded 0.3L in the spanwise direction, z.
At the top boundary, a slip-wall condition is enforced, differing from the method used in the DNS
by Uzun & Malik (2020, 2022), which, for the ReL = 106
case imposed pressure and velocity profiles at
y/L = 0.45 extracted from a precursor RANS simulation conducted in a taller domain of H/L = 0.5,
whereas for the ReL = 2 · 106
case imposed a non-reflecting characteristic boundary condition at y/L =
1.0. On the bottom wall, different wall-modeling strategies will be considered in the present study. First,
the existing equilibrium wall model of Kawai & Larsson (2012) is briefly described, with the choice of
parameters used in the present simulations. Afterward, the proposed sensor-based blended wall-modeling
approach is introduced in §2.3, with two different sensors. The wall is considered adiabatic in this study,
but the introduced wall-modeling approaches are similarly applicable in the context of isothermal walls.
The domain outlet is placed at a streamwise location of x/L = 2.0.
3.2 NASA hump
The computational setup of NASA hump closely follows the experimental conditions documented by
Greenblatt et al. (2006), as illustrated in Figure 3.2. A prescribed turbulent boundary layer is applied at
the inlet, corresponding to a Reynolds number based on the hump chord length of Rec ≈ 936000. The
freestream Mach number upstream of the hump is set to 0.2, ensuring subsonic conditions. To capture
three-dimensional effects while maintaining computational efficiency, a periodic spanwise domain is used
39

with a spanwise extent of 0.3L. The reference boundary layer, chosen at x/L = −2.14, has a thickness of
δref = 0.0726L, where the momentum-thickness Reynolds number reaches Reθ = 7, 000, aligning well
with experimental measurements. A sponge zone is implemented downstream of the primary region of
interest to minimize reflections. The outflow boundary of the computational domain is positioned at the
end of the sponge zone, located at x/L = 3.9. Characteristic boundary conditions are imposed at the
outflow to ensure a non-reflective exit.
The use of a spatially uniform wall-model exchange height is common in the literature, even for flows
with spatially evolving turbulent boundary layers. One primary reason is that the boundary layer thickness
is generally not known a priori throughout the domain before running the simulation. Since the recommended exchange height for equilibrium turbulent boundary layers (Kawai & Larsson, 2012; Larsson et al.,
2016) is typically set at 10 − 20% of the boundary layer thickness, prior knowledge of this thickness is
required. However, for non-equilibrium boundary layers, determining an appropriate percentage is less
straightforward. An alternative approach, widely adopted in WMLES studies, is to set the exchange height
at the first cell centroid adjacent to the wall for each boundary face, inherently linking it to the grid resolution. While this simplifies the parallel implementation of data exchange between the wall model and
the LES, it can introduce significant numerical errors, such as log-layer mismatch (Kawai & Larsson, 2012).
Despite these potential inaccuracies, this method has remained popular due to its computational simplicity. Additionally, some studies suggest that in flows experiencing acceleration due to favorable pressure
gradients, secondary inner boundary layers may develop within the original boundary layer. Determining
the appropriate exchange height in such scenarios remains an open question. To address this, a systematic
study is conducted to examine the effects of wall-model exchange height and grid resolution on WMLES
predictions.
40

Chapter 4
Results
4.1 Gaussian-shaped bump configuration
Results obtained from simulations with the blended wall-model will be compared in this section with
those obtained from application of an equilibrium wall model everywhere (i.e., without sensors nor blending), and, in appendix 6.1, with simulations employing previously introduced sensor-based wall-modeling
approaches for laminar-to-turbulent transition prediction (Bodart & Larsson, 2012; Mettu & Subbareddy,
2018). Whenever available, data from experimental studies by Williams et al. (2020, 2021), Gray et al.
(2021, 2022), prior DNS results from Uzun & Malik (2020, 2022), and WMLES simulations from Balin et al.
(2020), Iyer & Malik (2020), Whitmore et al. (2021) and Agrawal et al. (2022) will also be utilized for these
comparisons. In the present simulations, the freestream Mach number is set to M = 0.2, whereas it was
set to 0.1 in the experimental tests conducted by Gray et al. (2022) for the ReL = 2·106
case. In a prior numerical study, Iyer & Malik (2023b) found that the Mach number has a negligible effect on their simulation
results.
The determination of the boundary layer thickness as the wall-normal distance where the flow velocity has reached approximately 99% of the freestream velocity becomes challenging in configurations
that involve curved surfaces or significant pressure gradients. The problem is due in part to the lack of
a uniform streamwise velocity region away from the bottom wall. Several methods have been proposed
41

to overcome these challenges (Vinuesa et al., 2016; Griffin et al., 2021). In the present study, a simplified
criterion inspired by the diagnostic plot method introduced in Vinuesa et al. (2016) is used, which estimates the boundary layer thickness as the wall-normal distance where the root-mean square streamwise
velocity fluctuation reaches 2% of the freestream velocity, U∞. Other methods have been suggested in the
literature that rely, for example, on the wall-normal weighted spanwise vorticity falling below a certain
fraction (e.g., 2%) of its peak value along the wall-normal direction (Uzun & Malik, 2020). Whereas the
latter method may be suitable for DNS studies, in which the near-wall peak of spanwise vorticity can be
accurately characterized, it can be more prone to numerical errors in the context of WMLES, where the
peak spanwise vorticity must be estimated using the wall model, and the vorticity calculation is less accurate due to the use of coarser grids. Despite its advantages, the method based on the turbulence intensity
also presents several shortcomings, as noted in Griffin et al. (2021), such as the dependence of the estimate of turbulence intensity on grid resolution in LES, and the inability to be used in fully laminar flows
(where the turbulence intensity is zero throughout the boundary layer). However, for quasi-laminar states
in which the upstream turbulence remains nearly frozen, as in the lower ReL case here considered, this
method based on a measure of the turbulence intensity remains applicable, and will be used in the present
study, with the same threshold of 2% proposed in Vinuesa et al. (2016).
4.1.1 Flow over a Gaussian-shaped bump at ReL = 106 with relaminarization
Wall-modeled LES presents challenges associated with the selection of a suitable exchange height, hwm,
where information from the LES is transferred to the wall model as its top boundary condition and with
convergence under grid refinement. Even for spatially-evolving boundary layer flows, as noted in §2.2,
the exchange height is often chosen as a uniform value throughout the wall boundary, corresponding to
about 10-20% of the reference boundary layer thickness obtained at a representative location where such
information is known or can be estimated. Besides its simplicity, another reason for this uniform choice is
42

that lack of boundary-layer thickness information without a precursor simulation or detailed experimental
measurements prevents the determination of a spatially-varying exchange height. On the other hand,
certain wall-model implementations can only specify the exchange height as a given grid point/cell above
the wall (in some cases, restricted to the first cell adjacent to the wall), which makes the resulting exchange
height intrinsically tied to the grid resolution. In the present work, our wall-model implementation can
specify an arbitrary spatially-variable exchange height, transferring information to the wall model from
the cell centroid of the LES grid closest to the specified local target exchange height for each boundary
face. Simulations with variable and uniform wall-model exchange heights will be presented. The effects
of grid resolution will also be assessed in the present work, by conducting simulations on increasingly
refined grids, and addressing the degree of convergence (or lack thereof) in different regions obtained
with a purely equilibrium wall model and with sensor-based blended wall models. The two topics are
related, and there is ongoing active research on the development of automated grid adaptation algorithms
that can incorporate the wall-model exchange height within the context of WMLES (Kahraman & Larsson,
2020).
Four grids are considered for the ReL = 106
case, denoted as coarsest (T), coarse (C), medium (M),
and fine (F), with increasing resolution, respectively. All grids use a uniform streamwise grid spacing
from the inflow location (x/L = −0.98) up to x/L = 1 and then stretched until the outlet, located at
x/L = 2, where ∆s/L = 0.0011. For the coarsest, coarse, and medium grids (T, C, M), the streamwise
grid spacing is set as ∆s/δ0 = 0.08, with the reference boundary layer thickness taken at x/L = −0.5
as δ0/L = 0.013, from the DNS data (Uzun & Malik, 2020). The fine grid (F) considers the same uniform
streamwise grid spacing ratio of ∆s/δ0 = 0.08 but with the reference boundary layer thickness taken
at x/L = −0.8 as δ0/L = 0.0046 from the DNS data (Uzun & Malik, 2020). A uniform spanwise grid
spacing is used for all grids, equal to ∆z = 0.05δ0 = 6.5 · 10−4L for the coarsest and coarse grids, half
that value for the medium (M) grid ∆z = 3.25 · 10−4L, and ∆z = 0.05δ0 = 2.3 · 10−4L for the fine grid.
43

Resolution δ0
L hwm
∆s
δ0

∆s
L

out

∆n
δ0

wall

∆n
δ0

n=2δ0

∆n
L

top
∆z
δ0 Ns Nn Nz Grid size
1M-Coarsest 0.0130 0.1δ0 0.08 0.0011 0.02 0.08 0.10 0.050 2837 69 62 12 M
1M-Coarse 0.0130 variable 0.08 0.0011 0.02 0.08 0.05 0.050 2837 86 55 15 M
1M-Medium 0.0130 variable 0.08 0.0011 0.02 0.08 0.05 0.025 2837 86 124 30 M
1M-Fine 0.0046 0.1δ0 0.08 0.0011 0.02 0.08 0.10 0.050 6871 70 174 84 M
Table 4.1: Grid parameters for simulation cases at ReL = 106
. δ0 is the reference boundary layer thickness.
(s, n, z) are the wall-parallel, wall-normal, and transverse (spanwise) topological directions, respectively.
N indicates the number of cells. Grid sizes given in millions (M) of cells.
−1 0 1 2
x/L
0.0004
0.0006
0.0008
0.0010
∆s/L
1M-T
1M-C
1M-M
1M-F
10−4 10−3 10−2 10−1
n/L
10−4
10−3
10−2
10−1
∆n/L
0.00 0.04
z/L
0.0003
0.0004
0.0005
0.0006
∆z/L
(a) (b) (c)
Figure 4.1: Grid spacing profiles along the streamwise (a), wall-normal (b), and spanwise (c) directions
used by the WMLES conducted at different grid resolution for the ReL = 106
case. For clarity, markers
are plotted every 500, 1, and 50 points in (a), (b), and (c), respectively, with offsets in (a) and (c) for each
case.
Table 4.1 summarizes the grid spacing settings. The wall-normal grid spacing varies from ∆n/δ(x) = 0.02
at the wall to ∆n/δ(x) = 0.08 at n = 2δ(x), further stretched toward the top boundary. The coarsest
and fine grids consider δ(x) = δ0 (uniform), whereas the coarse and medium grids have a variable δ(x) =
δ0

1.0 − 0.7 exp(−(x/L/0.195)2
)

+0.002(x/L+0.5) based on the available DNS data for the streamwise
variation of the boundary layer thickness (Uzun & Malik, 2020), which first increases, then decreases in
the acceleration region near the bump apex, and then continues to increase further downstream. The wallmodel exchange height is set to hwm(x) = 0.1δ(x). Therefore, the coarsest and fine grids incorporate a
uniform wall model exchange height, whereas the coarse and medium grids have a variable wall-model
exchange height. Figure 4.1 shows grid spacing profiles in terms of L for these four grids along each
coordinate direction. The resulting grid spacing profiles in outer (δ99) and inner (viscous) units will be
shown later in Figure 4.4.
44

4.1.1.1 Incoming turbulent boundary layer
The characteristics of the incoming turbulent boundary layer at the location x/L = −0.6 (upstream of
the bump) are evaluated first, considering results from a simulation in which the equilibrium wall model is
applied everywhere, and comparing with DNS reference data. This characterization is also representative
of the incoming boundary layer for most of the other simulations conducted in this study at this Reynolds
number, since the equilibrium wall model is generally applied in the region upstream of the bump (i.e., the
blending factor is zero or nearly zero in that region).
Figure 4.2 presents wall-normal profiles of mean streamwise velocity and Reynolds stresses at the
x/L = −0.6 reference station. The boundary layer thickness based on the local turbulence intensity
reaching 2% of U∞ is estimated as 0.012L. The grayed out background indicates the wall-modeled region.
An inset within Figure 4.2(a) represents the mean streamwise velocity in inner scaling, effectively capturing
the logarithmic layer of the velocity profile. This inner scaling exhibits characteristics consistent with the
von Kármán constant κ = 0.41 and an intercept constant of C = 5.2.
4.1.1.2 Streamwise distribution of pressure and skin friction coefficients
We present in Figure 4.3 time- and spanwise-averaged streamwise profiles of pressure coefficient, Cp,
and the skin friction coefficient Cf = τw/q∞ = 2τw/(ρ∞U
2
∞) obtained from simulations conducted on
grids of increasing resolution with the equilibrium wall model and the blended wall model based on the
acceleration sensor, along with DNS results from Uzun & Malik (2020). Simulations using the equilibrium
wall model everywhere (Figure 4.3a,b) significantly overestimate the skin friction coefficient, Cf , near
the apex of the bump (x/L ≈ 0), and fail to predict a second local peak immediately downstream of the
apex, both features corresponding to the processes of relaminarization and retransition, respectively. The
predicted Cp profiles align well with the DNS reference data up to the medium grid. For the fine grid,
the Cp prediction downstream of the bump apex is compromised by the excessively large extent of the
45

0.0 0.2 0.4 0.6 0.8 1.0 1.2
n/δ(x)
0.0
0.2
0.4
0.6
0.8
1.0 U/U∞
0.0 0.2 0.4 0.6 0.8 1.0 1.2
n/δ(x)
−2.5
0.0
2.5
5.0
7.5
10.0
12.5
15.0
hu
0u
0i/U2
∞
hv
0v
0i/U2
∞
hw
0w
0i/U2
∞
hu
0v
0i/U2
∞
Reynolds stresses
DNS
1M-Eq-F
100 101 102 103 n
0 +
10
20
U
+
n
+
ln n
+
0.41 + 5.2
(a) (b)
Figure 4.2: Wall-normal profiles of the incoming turbulent boundary layer at the reference location (x/L =
−0.6) for the ReL = 106
case, comparing the wall-modeled LES with an equilibrium wall model (dotted
green line) with the DNS data of Uzun & Malik (2020). (a) Mean streamwise velocity, normalized by the
freestream velocity, as a function of the wall-normal distance in outer scaling (n/δ). The inset shows the
same quantity in inner scaling (U
+ = U/uτ as a function of n
+ = n/ℓv), with the viscous and loglaw theoretical estimates represented by dashed and dash-dotted lines, respectively. (b) Reynolds stresses,
⟨u
′
iu
′
j
⟩, normalized by the square of the freestream velocity, as a function of the wall-normal distance in
outer scaling, n/δ. Gray areas represent the wall-modeled region in the WMLES.
predicted separation (where the friction coefficient becomes negative, x/L ≈ 0.1–0.5), which lowers the
Cp well below the DNS values, before it recovers to the correct values downstream of reattachment. On
the other hand, simulations on the coarsest, coarse, and medium grids fail to predict the mild mean flow
separation observed in the DNS. Grid convergence of the results is mostly achieved upstream of the bump
apex (x/L < −0.2), but no convergence is obtained in the relaminarization, retransition, and separated
flow regions for the grid resolutions considered. In fact, the simulation on the fine grid shows reduced
accuracy in the prediction of skin friction coefficient near the bump apex, with a significant overprediction
relative to the DNS data. This highlights, first, the inadequacy of applying an equilibrium wall model in
regions of relaminarization and retransition, as the assumption of fully-turbulent boundary layer in the
equilibrium wall model is not suitable in such regions, and, second, the sensitivity to grid resolution in
those flow regions.
46

−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
−1.50
−1.25
−1.00
−0.75
−0.50
−0.25
0.00
0.25
Cp
1M-Eq-T
1M-Eq-C
1M-Eq-M
1M-Eq-F
DNS
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
0
2
4
6
8
10
12
Cf
× 10
3
−0.1 0.0 0.1
5.0
7.5
10.0
12.5
(a) (b)
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
−1.50
−1.25
−1.00
−0.75
−0.50
−0.25
0.00
0.25
Cp
1M-Acc-T
1M-Acc-C
1M-Acc-M
1M-Acc-F
DNS
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
0
2
4
6
8
10
Cf
× 10
3
−0.1 0.0 0.1
5.0
7.5
10.0
(c) (d)
Figure 4.3: Streamwise profiles of pressure (a,c) and friction (b,d) coefficients from WMLES of the ReL =
106
case (denoted by the 1M label prefix) performed using an equilibrium wall model (Eq) (a,b) and a
blended wall model based on the acceleration sensor (Acc) (c,d) with different grid resolutions (T: coarsest,
C: coarse, M: medium, F: fine), compared with DNS data (black solid with symbols) by Uzun & Malik (2020).
47

In contrast, simulations conducted using the proposed blended wall model with the acceleration sensor
on the same grids with increasing resolution (Figure 4.3c,d) clearly resolve the overprediction of the friction
coefficient near the bump apex and capture the presence of a second peak in Cf due to retransition, for all
but the coarsest grid. Also, a high degree of grid-convergence is attained as the grid resolution increases,
which was not observed when only the equilibrium wall model was applied. Despite the improved prediction in the relaminarization and retransition regions, no mean separation is predicted with the blended
wall model for any of the grid resolutions considered, indicative of history effects of the relaminarizing
and retransitioning boundary layer, as will be described later in more detail. Accurate prediction of the
separated flow regions may require additional sensing capabilities to incorporate a different blending of
boundary conditions, or modifications of the grid resolution and wall-model exchange height, which are
not addressed in the present study.
Profiles of grid spacing in outer (i.e., scaled with the local boundary layer thickness) and inner units
(i.e., scaled with the local viscous length) along the streamwise direction are presented in Figure 4.4 for
simulations using the blended wall model with the acceleration sensor. Similar profiles (not shown) are
obtained for simulations using the equilibrium wall model everywhere, with mild differences in the relaminarization, retransition and separated flow regions, due to the solution-dependent local scaling factors
(e.g., viscous length obtained from the local wall shear stress), which differ between simulation strategies
as shown in Figure 4.3.
The streamwise grid spacing in outer units (∆s/δ99) for the coarsest, coarse, and medium grids exceeds (i.e., is coarser than) the 0.08 standard WMLES value (Larsson et al., 2016) upstream of x/L = −0.5,
and remains above 0.05 until Cf drops toward incipient separation downstream of the bump apex. Similarly, the spanwise grid spacing in outer units (∆z/δ99) for the coarsest and coarse grids remains above
the standard 0.05 value used in WMLES upstream of x/L = −0.5, decreasing further downstream. For
the medium and fine grids, the spanwise grid spacing is finer than that standard WMLES value. The
48

0.00
0.05
0.10
0.15
0.20
∆s/δ99
1M-Acc-T
1M-Acc-C
1M-Acc-M
1M-Acc-F
0.00
0.01
0.02
∆
n1/δ99
0.00
0.05
0.10
∆z/δ99
−1.0 −0.5 0.0 0.5 1.0
x/L
20
40
60
80
∆
s
+
−1.0 −0.5 0.0 0.5 1.0
x/L
2
4
6
8
10
∆
n
+
1
−1.0 −0.5 0.0 0.5 1.0
x/L
10
20
30
40
∆
z
+
(a) (b) (c)
(d) (e) (f)
Figure 4.4: Streamwise profiles of grid spacing in outer (a,b,c) and inner (d,e,f) units for the streamwise
(a,d), wall-normal first cell centroid (b,e), and spanwise (c,f) coordinate directions, obtained from WMLES
of the ReL = 106
case performed using a blended wall model based on the acceleration sensor with grid
resolutions.
low Reynolds number of this relaminarizing flow makes the grid spacing in inner units (Figure 4.3d,e,f)
approach the ranges of wall-resolved LES (Choi & Moin, 2012; Larsson et al., 2016) for the finer grids considered, in order to still comply with the WMLES grid-spacing requirements in outer units. Although the
wall-normal grid-spacing remains above one (thus coarser than in wall-resolved LES and DNS), in regions
where significant blending with the no-slip boundary condition occurs (i.e., where relaminarization and
retransition occur) values of ∆n
+ ≲ 5 and ∆z
+ ≲ 20 are seemingly required for accurate predictions of
Cf with the blended wall model. Suitable blending with laminar wall models (instead of with the no-slip
condition) may impose less stringent resolution requirements. The coarsest grid presents ∆n
+ ≈ 10 and
∆z
+ ≈ 50 near the bump apex, resulting in inaccurate results for this flow.
To compare the results obtained with the two different sensors (acceleration and relaminarization) proposed in §2.3.1 we present in Figure 4.5 time- and spanwise-averaged streamwise profiles of the pressure
coefficient Cp and the skin friction coefficient Cf = τw/q∞ = 2τw/(ρ∞U
2
∞) from simulations conducted
49

−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
−1.50
−1.25
−1.00
−0.75
−0.50
−0.25
0.00
0.25
Cp
1M-Eq-F
1M-Rel-F
1M-Acc-F
DNS
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
0
2
4
6
8
10
Cf
× 10
3
−0.1 0.0 0.1
4
6
8
10
12
(a) (b)
Figure 4.5: Streamwise profiles of pressure (a) and friction (b) coefficients from WMLES of the ReL =
106
case conducted on the fine grid (F), performed using an equilibrium wall model (Eq, dotted green),
a blended wall model based on the acceleration sensor (Acc, solid red), and a blended wall model based
on the relaminarization sensor (Rel, dashed blue), compared with DNS data (black solid with symbols)
by Uzun & Malik (2020). Cf values in (b) scaled by a factor of 103
.
on the fine grid with each sensor, alongside results from a simulation using the equilibrium wall model
everywhere on the same fine grid, and reference DNS results from Uzun & Malik (2020). Within the relaminarization region, results with the blended wall models using either the acceleration or the relaminarization
sensor effectively follow the DNS results, successfully resolving the overestimation of Cf observed when
an equilibrium wall model is applied everywhere. The primary peak of the skin friction coefficient, Cf ,
found at x/L ≈ −0.055, is correctly predicted by both sensor-based blended wall models. However, a
small overprediction in the minimum of the pressure coefficient distribution, Cp, is observed when using
the newly proposed blended approach with both sensors. The immediate decrease in Cf following the
primary peak happens while the flow is still accelerating upstream of the bump apex and continues as the
flow enters the adverse pressure gradient region. As noted in previous DNS studies (Balin & Jansen, 2021;
Uzun & Malik, 2020), the favorable pressure gradient (FPG) generated upstream of the adverse pressure
gradient (APG) region stabilizes the flow, but its effect diminishes once the flow passes the bump apex. At
this stage, the flow undergoes a deceleration process initiated by the APG. Although the effects of this APG
are in the early stages, the flow manages to preserve sufficient momentum, which enables the surviving
50

instabilities to induce a rapid transition back to a turbulent state. The presence of a secondary peak in
Cf at x/L ≈ 0.05, as observed in the DNS data, indicates the occurrence of a retransitional flow region.
However, previous WMLES results from Iyer & Malik (2020) and Balin et al. (2020) poorly predicted this
secondary peak. The relaminarization sensor results in a better prediction of the local minimum, followed
by an advanced rise to the local maximum. On the other hand, the acceleration sensor successfully captures the secondary peak, although it exhibits a deeper local minimum upstream. This secondary peak is
dependent on the cumulative effects of upstream pressure gradients on the boundary layer. An analysis
of the sensitivity of the predictions to the sensor-based blended parameters is presented in 4.1.4. Following the secondary peak, another sharp decrease in Cf occurs. This decrease is caused by the continued
presence of the adverse pressure gradient in the downstream region, which further decelerates the flow.
As already noted, despite the improved agreement in the relaminarization and retransition regions, the
proposed sensor-based blended wall models at this grid resolution do not detect the incipient separation
in the strong adverse pressure gradient region found in the DNS results, predicting instead fully attached
flow. Nonetheless, the prediction of the pressure coefficient distribution in this region downstream of the
bump apex is significantly improved (slightly more so with the relaminarization sensor), relative to the
simulation that applies the equilibrium wall model everywhere at the same fine grid resolution, for which
separation is largely overpredicted. Interestingly, previous WMLES studies of this configuration found
in the literature showed a reduced tendency of the flow to separate instead (Iyer & Malik, 2020; Balin
et al., 2020). For the present simulations with blended wall models, we attribute the reduced tendency
for flow separation to the fuller wall-normal profiles of streamwise velocity that result from retransition
shortly downstream of the bump apex, compared with those obtained when the equilibrium wall model
is applied everywhere (and thus, despite the assumption of a turbulent boundary layer throughout), as
will be shown in Figure 4.8. Therefore, history effects of the boundary layer during relaminarization and
retransition play an important role in its subsequent ability to sustain adverse pressure gradients before
51

−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
−5
−4
−3
−2
−1
0
1
s/s
0
1M-Eq-F
1M-Rel-F
1M-Acc-F
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
0.0
0.2
0.4
0.6
0.8
1.0
β
(a) (b)
Figure 4.6: Streamwise profiles of the normalized sensors (a) and blending factors (b) from WMLES of the
ReL = 106
case performed using a blended wall model based on the acceleration sensor (solid red) and
the relaminarization sensor (dashed blue).
separation. Several open questions remain on the applicability of wall models for accurate prediction of
separation and reattachment. Such questions include whether the wall-model exchange location should
be reduced near separated flow regions, and what the optimal grid resolution should be in those regions.
Although not explored in the present study, other specialized sensors (see, e.g., Agrawal et al., 2022) could
be utilized to detect departures from the validity of equilibrium wall-model assumptions to improve the
treatment of boundary conditions in the vicinity of separated flow regions, which could be integrated into
a similar blending formalism to the one introduced in §2.3, with a multi-sensor basis.
We plot in Figure 4.6 the time- and spanwise-averaged normalized acceleration and relaminarization
sensors and corresponding blending factors obtained from each simulation, which offer valuable insights
into the sensor’s effectiveness at identifying the onset and end of the relaminarization region. For each
simulation, the sensor is normalized by its chosen reference value, σ0, so that comparisons between different sensors can be made more easily. Both sensors show a qualitatively similar behavior upstream of
the bump apex, first decreasing below zero and then increasing toward their peak value in the region of
relaminarization, before decreasing to zero at the bump apex. Despite the qualitative similarities, the relaminarization sensor peaks upstream of the acceleration sensor, which exhibits a steeper decrease nearing
52

the bump apex. Whereas the relaminarization sensor exceeds the chosen reference value(i.e., σ/σ0 > 1)
in a region upstream of the bump, the acceleration sensor remains always below its reference value. This
translates into the streamwise distribution of the blending factor, β, reaching unity for the simulation with
the relaminarization sensor, while for the simulation with the acceleration sensor the blending factor remains always below one, albeit quite close to reaching one in the relaminarization region. We recall that a
local blending factor of zero indicates that the equilibrium wall model is applied at that location, whereas
a value of one means that the no-slip LES boundary condition is instead applied at that location. For simulations of this configuration with either sensor, the region around relaminarization presents high β values
(i.e., near unity) for both sensors. Furthermore, the hysteresis scale applied (as described in §2.3.3) translates into the blending factor recovering a zero value slightly downstream of the bump apex (in contrast to
the sensor distribution). This spatial delay is critical in the accurate prediction of the secondary peak of the
friction coefficient that results from retransition of the boundary layer to a turbulent state, as previously
explained when discussing Figure 4.5.
Compared with simulation results obtained using the equilibrium wall model applied everywhere, a
slight overprediction of the friction coefficient near the onset of the relaminarization region is observed
for the two simulations with blended wall models in Figure 4.5(b), as well as small oscillations found as the
flow reaches the windward part of the bump. The shift in the location of the local minimum of the friction
coefficient distribution before retransition predicted by the blended wall model with the relaminarization
sensor (see inset of Figure 4.5b) may be explained by a slightly lower blending factor near the apex of the
bump when compared to the acceleration sensor (Figure 4.6).
Downstream of the bump apex, both acceleration and relaminarization sensors become negative, with
the acceleration sensor reaching much lower normalized values. Both sensors recover positive values
around x/L ≈ 0.2, which marks the onset of separation in the DNS reference data. As a result of the
positive sensor values, the (averaged) blending factor departs from zero, reaching peak values around
53

x/L ≈ 0.5 before slowly decreasing farther downstream. Throughout the entire region downstream of
the bump apex, the magnitude of the acceleration sensor reaches larger values than the relaminarization
sensor. The reason is that the acceleration sensor, σA, is inversely proportional to the cube of the friction
velocity, whereas the relaminarization sensor, σR, relies on the edge velocity. The friction velocity reaches
lower values than the edge velocity near separation. Correspondingly, the blending factor in that downstream region is approximately twice larger in the simulation with the acceleration parameter (peaking
around β ≈ 0.5) than for the relaminarization sensor, and experiences a faster rise as the friction coefficient
decreases toward its smallest value. In that downstream region of the flow, the acceleration sensor and
corresponding blending factor are more oscillatory than the relaminarization sensor and blending factor
counterparts, resulting from the nature of the friction velocity, compared to the edge velocity.
Similar prediction of the skin friction coefficient are obtained in that downstream region despite the
different values of the sensors (and associated blending coefficients), attributed to several factors. First, the
boundary layer thickness has grown significantly in the downstream region (x/L > 0.2), implying that the
near-wall layer is better resolved, since the grid spacing is based on the upstream boundary layer thickness
for these grids. Second, the boundary layer in that downstream region is recovering from a region with
a significant adverse pressure gradient that has reduced the friction coefficient to its lowest values. The
predictions obtained by the two sensors and blending coefficients upstream (accounting by the cumulative
effect of the pressure gradient) have left the boundary layer in similar states for both simulations. As a
result, differences between the predictions obtained by the wall-model or the no-slip boundary conditions
in that region have a reduced effect in the resulting Cf .
4.1.1.3 Streamwise variation of boundary layer properties
Figure 4.7 illustrates the streamwise variation of four parameters characterizing the boundary layer throughout the flow: the displacement thickness, δ
∗
(x) = R δ(x)
0
[1 − ρu||/(ρeu||,e)]dn, the momentum thickness,
54

−0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0
x/L
0
10
20
30
δ
∗/L (×103)
−0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0
x/L
0.0
2.5
5.0
7.5
10.0
θ/L (×103)
−0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0
x/L
1.0
1.2
1.4
Ue/U∞
1M-Eq-F
1M-Rel-F
1M-Acc-F
DNS
−0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0
x/L
−10
0
10
20
30
40
50 βRC
(a) (b)
(c) (d)
Figure 4.7: Streamwise profiles of boundary layer displacement thickness (a), momentum thickness (b),
edge velocity (c), and Rotta-Clauser parameter (d) obtained from WMLES of the ReL = 106
case performed
using an equilibrium wall model (dotted green), a blended wall model based on the acceleration sensor
(solid red), and a blended wall model based on the relaminarization sensor (dashed blue), compared with
DNS data (black solid) by Uzun & Malik (2020).
55

θ(x) = R δ(x)
0
[(ρu||)/(ρeu||,e)](1 − u||/u||,e)dn, the edge velocity normalized by the freestream velocity,
Ue/U∞, and the Rotta-Clauser parameter, βRC =

δ
∗/τw
∂pw/∂s
. In the previous expressions for δ
∗
and θ, the integrals are taken along the local wall-normal direction, n, from the wall (n = 0) up to the
boundary layer edge, denoted by the subscript e (corresponding to n = δ(x), where δ(x)is the local boundary layer thickness, based on Vinuesa et al., 2016), considering time- and spanwise-averaged density, ρ, and
wall-parallel velocity, u||. The Rotta-Clauser parameter is expressed in terms of the time- and spanwiseaveraged wall shear stress, τw, streamwise wall pressure gradient, ∂pw/∂s, and the displacement thickness,
δ
∗
, and can be recast in terms of the skin friction and pressure coefficients as βRC = (δ
∗/Cf )(∂Cp/∂s).
All wall-modeling methods under consideration show slight underprediction of δ
∗
and θ before the
region of flow relaminarization. The intense pressure gradient associated with flow acceleration causes
the boundary layer to thin until just before the bump apex. This thinning effect results in a decrease in
both δ
∗
and θ in the relaminarization region, showing good agreement across all WMLES at this grid resolution. As the flow decelerates beyond the bump apex, both δ
∗
and θ experience significant growth. The
simulation with the equilibrium wall model applied everywhere aligns well with DNS results for δ
∗
and θ
before the separation region. However, due to the large overprediction of the separated flow region, the
simulation results significantly deviate from the reference DNS data, consequently affecting the prediction
of the post-separation, reattachment, and recovery regions. Simulations with the proposed sensor-based
blended wall models greatly improve the prediction of these boundary layer quantities, although a slight
overprediction is still observed further downstream in the recovery region. Consistent with observations
made in previous DNS studies (Uzun & Malik, 2020; Balin & Jansen, 2021), a region is found downstream of
the start of the recovery (x/L ≈ 0.3) where the displacement thickness δ
∗ decreases and the momentum
thickness θ continues to increase until the pressure gradient becomes mildly favorable again (x/L ≈ 0.4).
This region is correctly identified by the sensor-based, blended wall-modeled simulations; in contrast, the
simulation with the equilibrium wall model applied everywhere overpredicts the peak of displacement

thickness by nearly a factor of two and incorrectly predicts the location of the momentum thickness peak.
Further downstream, both δ
∗
and θ plateau as the pressure gradient approaches zero, at values slightly
overpredicting the DNS results for the blended wall-modeled simulations (compared to a larger overprediction for an equilibrium-only wall-model), and then start to increase again, indicative of the recovery of
near-equilibrium boundary layer conditions.
Similar conclusions can be drawn regarding the improved prediction of the shape factor, H = δ
∗/θ
(not shown) and the normalized edge velocity, Ue/U∞, using the sensor-based blended wall models. The
edge velocity (Figure 4.7c) consistently follows the inverse of the variation seen in the pressure coefficient
(Figure 4.5a): near the bump apex, the favorable pressure gradient causes a rapid increase in Ue until it
reaches its maximum value. However, in the downstream section of the bump, where adverse pressure gradients dominate, Ue decreases significantly, and overprediction is observed in the separation region for the
simulation applying the equilibrium wall model everywhere. The departure from equilibrium conditions
in a turbulent boundary layer, characterized by transition from a small to a large mean velocity defect, can
be quantified using the Rotta–Clauser pressure-gradient parameter, βRC, which remains constant in flows
exhibiting self-similar velocity profiles, such as in an equilibrium turbulent boundary layer (Maciel et al.,
2006), is null in the case of zero-pressure gradient turbulent boundary layers, and becomes singular at separation and reattachment locations (as τw crosses zero). As shown in Figure 4.7(d), the boundary layer flow
over the Gaussian bump at ReL = 106
considered here exhibits significant departures from equilibrium
conditions along the entire bump, with βRC only asymptotically approaching a near-zero, constant value
downstream of the bump (beyond x/L ≈ 0.6), where equilibrium conditions are recovered. Overall, the
application of sensor-based blended wall models significantly enhances the accuracy of βRC prediction,
aligning closely with the DNS reference data, except in the incipient separation region.
57

0.5 1.0
0
1
2
n/δref
x/L = −0.4
0.5 1.0
0
1
2
3 x/L = −0.2
1.0 1.2
0
1
2
3 x/L = −0.1
1.0 1.5
0
1
2
3 x/L = 0.0
0 1
0
1
2
3 x/L = 0.1
0 1
0
2
4
6 x/L = 0.2
0 1
0
5
x/L = 0.6
0 10
0
1
2
n/δref
0 10
0
1
2
3
0 10
0
1
2
3
0 10
0
1
2
3
0 10
0
1
2
3
0 20
0
2
4
6
0 20
0
5
0 5
0
1
2
n/δref
0 5
0
1
2
3
0 5
0
1
2
3
0 5
0
1
2
3
0 5
0
1
2
3
0 10
0
2
4
6
0 10
0
5
0 5
0
1
2
n/δref
0 5
0
1
2
3
0 5
0
1
2
3
0 5
0
1
2
3
0 10
0
1
2
3
0 10
0
2
4
6
0 10
0
5
0 1
0
1
2
n/δref
0 2
0
1
2
3
0 2
0
1
2
3
0 2
0
1
2
3
0 5
0
1
2
3
0 5
0
2
4
6
0 5
0
5
U/U∞ 100
u
0
rms/U∞ 100
v
0
rms/U∞ 100
w
0
rms/U∞
−1000 hu
0v
0i/U
2∞
DNS Eq Acc Rel
Figure 4.8: Wall-normal line probes extracted at different streamwise locations (left to right) of mean
(time- and spanwise-averaged) wall-parallel velocity and Reynolds stresses (top to bottom) for WMLES
of the ReL = 106
case performed using an equilibrium wall model (dotted green), a blended wall model
based on the acceleration sensor (solid red), and a blended wall model based on the relaminarization sensor
(dashed blue), compared with DNS data (black solid) by Uzun & Malik (2020).
58

4.1.1.4 Wall-normal profiles of mean streamwise velocity and Reynolds stresses
In Figure 4.8, we compare wall-normal line probes extracted at several streamwise locations of mean (timeand spanwise-averaged) wall-parallel velocity and Reynolds stresses for ReL = 106
simulation cases on
the fine grid. At the beginning of the FPG region (x/L = −0.4), the wall-normal profiles coincide for the
three WMLES, closely matching the DNS results for the mean streamwise velocity, and showing a small
overprediction in the magnitude of normal and shear Reynolds stresses. The shape of the mean streamwise
velocity along the FPG region (up to the bump apex at x/L = 0) is increasingly affected, with departures
in the logarithmic and wake regions of the boundary layer. The wake is largely reduced and, when plotted
in inner scaling (not shown), deviations of the mean streamwise velocity from the log-law also become
significant, consistent with a first laminarescent and then relaminarizing boundary layer (Sreenivasan,
1982). Significant changes in the shape of the velocity profiles occur as the accelerated flow approaches
the bump apex. At x/L = −0.2, only a small overlap exists between the logarithmic layer equation for
equilibrium boundary layers and the actual velocity profile (not shown), indicating a significant deviation
from the logarithmic law, which validates the use of the acceleration parameter proposed by Patel & Head
(1968) as a means to identify the breakdown of the logarithmic layer and distinguish the relaminarization
region.
The normal Reynolds stresses in the FPG region present only minor differences between simulations
with the equilibrium wall model applied everywhere and the sensor-based blended wall models. Consistent
with the observations of Narasimha & Sreenivasan (1979), these normal Reynolds stresses respond slowly
to the flow acceleration in the outer region of the boundary layer, and remain nearly ‘frozen’ without
significant dissipative decay, thus resulting in similar profile shapes and peak values along that FPG region
of the bump, but with an increased anisotropy of the turbulence. On the other hand, the Reynolds shear
stress profiles experience larger changes as the flow accelerates, with an initial increase of the peak value
59

(from x/L = −0.4 to −0.2), followed by a reduction and flattening of the wall-normal profile (implying a
decorrelation between the streamwise and wall-normal velocity fluctuations, u
′
and v
′
).
Compared with the DNS data, the velocity and Reynolds stress profiles show generally good agreement
up to x/L ≈ −0.2 for all wall models shown in Figure 4.8, but differences begin to arise at that location
between the sensor-based blended wall models and the equilibrium wall model near the exchange height
(hwm), with the latter overpredicting the Reynolds stresses. These differences are exacerbated as the flow
reaches the bump apex (x/L = 0) and continue along the leeward side of the bump, where the purely equilibrium wall model departs significantly from the DNS results, whereas sensor-based blended wall-model
predictions more closely follow the reference data. In particular, the mean streamwise velocity is largely
underpredicted near the wall with the equilibrium wall model, resulting in a momentum deficit that triggers earlier and larger separation compared to the DNS results. The magnitude and wall-normal location
of the peaks of Reynolds stresses are also mispredicted by the equilibrium wall model downstream of the
bump apex and into the recovery region. The two sensor-based blended wall models tested follow more
closely the DNS results in that downstream region, although an overprediction of wall-normal Reynolds
stresses can be observed.
The formation of an internal layer within the incoming turbulent boundary layer in the strong FPG
region is manifested in Figure 4.8 by the emergence of near-wall peaks of Reynolds stresses and associated inflection (‘knee’) points between two (inner and outer) peaks (e.g., see streamwise, spanwise, and
shear Reynolds stress profiles at x/L = −0.1) (Webster et al., 1996; Parthasarathy & Saxton-Fox, 2023).
The internal layer (which, as it forms, is mostly immersed below the wall-model exchange height) grows
thicker in the APG region downstream of the bump apex (x/L > 0), becoming dominant in the turbulence
statistics and primarily governing the friction coefficient. Sudden changes in wall boundary conditions,
such as pressure gradients (Tsuji & Morikawa, 1976) and surface curvature discontinuities (Baskaran et al.,
60

(a)
(b)
(c)
(d)
(e)
(f)
Figure 4.9: Contours of instantaneous fluctuations of the local skin friction coefficient on the wall (a),
instantaneous streamwise velocity fluctuation extracted on wall-shaped surfaces located a wall-normal
distance of 0.1δ0 (b), 0.2δ0 (c), δ0 (d), 2δ0 (e), and instantaneous streamwise velocity on a vertical plane (f),
for an acceleration-based blended wall-modeled LES of the ReL = 106
.
1987; Wu & Squires, 1998), are recognized as the primary factors contributing to the development of internal layers. In the current setup, which employs a smooth Gaussian shape and maintains stable local wall
boundary conditions, the transition between adverse and favorable pressure gradients plays a key role in
the emergence and later growth of the internal layer (Uzun & Malik, 2022). Furthermore, the inner and
outer layers become nearly independent of each other, with the outer layer behaving similarly to a free
shear layer (Uzun & Malik, 2020). As the internal layer thickens and eventually becomes dominant in the
boundary layer, the first originated peaks of the inner layer in the streamwise and spanwise stress profiles
become more prominent. Thus, the internal layer is primarily influenced by FPG effects, while the outer
layer responds to a combination of streamline curvature and pressure gradient.
61

4.1.1.5 Instantaneous flow fields and relaminarization
Contours of instantaneous fluctuations of the local skin friction coefficient on the wall, instantaneous
streamwise velocity fluctuation extracted on wall-parallel surfaces at different wall-normal distances, and
instantaneous streamwise velocity on a vertical plane at ReL = 106
are shown in Figure 4.9, corresponding
to a simulation using the blended wall model based on the acceleration sensor on the fine grid. During
the relaminarization induced by acceleration, four distinct stages can be clearly distinguished (Bader et al.,
2017; Narasimha & Sreenivasan, 1979). Close to the wall, typical streamwise streaks are visible in the flow
upstream of the relaminarization region (x/L ≲ −0.3). As the peak favorable pressure gradient (FPG)
region is approached (x/L ≈ [−0.25, −0.05]), flow acceleration leads to noticeable straightening, growth
in the width and length of the streaks, with a reduced lateral interaction (Figure 4.9b,c). In this accelerated
region, corresponding to the second stage, stretched structures dominate in the near-wall layer, stabilizing
and thinning the boundary layer; a noticeable decrease of the intensity of velocity fluctuations and size
of turbulence structures in the outer layer is observed at wall-parallel surfaces extracted at n/δ0 = 1
and 2 in Figure 4.9(d) and (e) starting at x/L ≈ −0.15, indicative of the thinning of the boundary layer:
those wall-parallel surfaces are at a relatively higher location for the thinned boundary layer, resulting in
the decreased levels of (nearly-frozen) turbulence fluctuations. Concurrently, smaller structures emerge
near the wall, manifested by an increased spottiness of the contour of friction coefficient fluctuations
(Figure 4.9a), indicating a transition of the near-wall boundary layer towards a quasi-laminar state (Bader
et al., 2017).
At the time instant considered in Figure 4.9, larger patches of decreased shear stress fluctuations are
visible in the x/L ∈ [−0.08, 0.01] region. Whereas the relaminarization process is gradual, lacking a
distinct demarcation point for the onset of intermittency, retransition from the quasi-laminar state to a
turbulent one (third stage) appears more sharply demarcated on the contours of wall shear stress and
near-wall velocity fluctuations at x/L ≈ 0.05. This sharper retransition, compared to the more gradual
62

relaminarization, is consistent with the steeper decrease of the acceleration sensor (and blending factor)
shown in Figure 4.6. In the retransition process, smaller-scale turbulence structures are seen to emerge
throughout the boundary layer, strengthening it against separation in the remaining adverse pressure
gradient (APG) that follows. These spots visually resemble those observed in the middle stages of natural
laminar-turbulent transition, as noted by Elder (1960). As the boundary layer returns to a fully turbulent
state, these small-scale structures increase in size and reach their peak intensity due to intense turbulent
activity, coinciding with the small local maximum in the skin friction (Figure 4.5b) profile (Balin & Jansen,
2021). The outer layer (Figure 4.9d,e) takes longer to recover (until x/L ≈ 0.1-0.15) than the near-wall
internal layer (Figure 4.9b,c). As the turbulent boundary layer thickens evolving in the APG (fourth stage),
the size and organization of turbulence structures is observed to differ significantly from those found
in the incoming boundary layer (prior to relaminarization). Although the blended wall-model LES does
not predict mean flow separation, instantaneous contours of the streamwise velocity evidence incipient
separation (Figure 4.9f). Larger turbulence structures with an increased isotropy are observed in that
downstream region of the bump, relative to the streamwise streaky structures of the incoming boundary
layer found near the wall.
4.1.2 Flow over a Gaussian-shaped bump at ReL = 2·106 and 4·106 without relaminarization
As the Reynolds number is doubled and quadrupled, previous experiments (Robbins et al., 2021; Gray et al.,
2022; Williams et al., 2021) and numerical studies (Uzun & Malik, 2022) have shown two main differences in
the characteristics of the flow over the Gaussian-shaped bump configuration under consideration. Firstly,
relaminarization is absent in the FPG upstream of the bump apex, but the flow acceleration still imposes
departures from the equilibrium state of the boundary layer. Secondly, the flow in the APG exhibits a
larger mean separated flow region. To assess the adequacy and robustness of the proposed sensor-based
blended wall modeling approach, simulations are conducted at two higher Reynolds numbers with the
63

same model parameters as in the lower Reynolds number case, and the results are compared with those
obtained with a standard equilibrium wall model applied everywhere, along with available experimental
and DNS datasets.
For the ReL = 2 · 106
case, results are presented from simulations performed on two meshes of
increasing resolution, denoted as coarse and medium grids, with a domain height Ly/L = 1.0, following
the DNS of Uzun & Malik (2022). The coarse grid has a uniform streamwise and spanwise spacings of
∆s = ∆z = 0.08δ0, with δ0 determined as 0.0072L at x/L = −0.65 based on the experimental setup
of Gray et al. (2022). The fine grid increases the streamwise resolution in the bump region, with also a
finer uniform spanwise spacing of (∆s)bump = ∆z = 0.05δ0. The wall-normal grid spacing remains the
same between the coarse and fine grids for this Reynolds number. For the ReL = 4 · 106
case, results on a
medium-resolution grid will be presented, following similar grid spacing profiles to the medium-resolution
grid of the ReL = 2 · 106
case, but with a domain height Ly/L = 0.45 (as in the ReL = 106
case), which
accounts for the different mesh size between the medium-resolution meshes for these two higher Reynolds
numbers. Table 4.2 and Figures 4.10, 4.11 show further details on the grids and corresponding spacing
profiles for these higher Reynolds numbers.
Wall-normal profiles of mean streamwise velocity and Reynolds stresses in the incoming turbulent
boundary layer for the case ReL = 2 · 106
case at a reference location of x/L = −0.5 are presented
in Figure 4.12. These profiles are extracted from results of the simulation that applies the equilibrium
wall model everywhere, but are equally representative of the results obtained with the two sensor-based,
Resolution δ0
L
hwm
δ0

∆s
δ0

bump

∆s
δ0

out

∆n
δ0

wall

∆n
δ0

n=2δ0

∆n
L

top
∆z
δ0 Ly Ns Nn Nz Grid size
2M-Coarse 0.0072 0.1 0.08 0.239 0.025 0.08 0.1 0.08 1.00 4128 94 67 26 M
2M-Medium 0.0072 0.1 0.05 0.239 0.020 0.08 0.1 0.05 1.00 6397 94 112 67 M
4M-Medium 0.0072 0.1 0.05 0.239 0.020 0.08 0.1 0.05 0.45 6397 70 112 50 M
Table 4.2: Mesh resolution for simulation cases at ReL = 2 · 106
(top two rows) and 4 · 106
(bottom row).
δ0 is the reference boundary layer thickness. (s, n, z) are the wall-parallel, wall-normal, and transverse
(spanwise) directions, respectively. Ly is the domain height. N indicates the number of cells. Grid sizes
given in millions (M) of cells.
64

−1 0 1 2
s/L
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
∆s/L
2M-Coarse
2M-Medium
4M-Medium
10−4 10−3 10−2 10−1 100
n/L
10−3
10−2
10−1
∆n/L
0.00 0.04
z/L
0.00035
0.00040
0.00045
0.00050
0.00055
0.00060
∆z/L
(a) (b) (c)
Figure 4.10: Grid spacing profiles along the streamwise (a), wall-normal (b), and spanwise (c) directions
used by the WMLES conducted at different grid resolution for the ReL = 2 · 106
and ReL = 4 · 106
cases.
0.000
0.025
0.050
0.075
0.100
∆s/δ99
2M-Coarse
2M-Medium
4M-Medium
0.000
0.005
0.010
0.015
∆
n1/δ99
0.000
0.025
0.050
0.075
0.100
∆z/δ99
−1.0 −0.5 0.0 0.5 1.0
x/L
20
40
60
80
∆
s
+
−1.0 −0.5 0.0 0.5 1.0
x/L
0
10
20
∆
n
+
1
−1.0 −0.5 0.0 0.5 1.0
x/L
20
40
60
80
∆
z
+
(a) (b) (c)
(d) (e) (f)
Figure 4.11: Streamwise profiles of grid spacing in outer (a,b,c) and inner (d,e,f) units for the streamwise
(a,d), wall-normal first cell centroid (b,e), and spanwise (c,f) coordinate directions, obtained from WMLES
of the ReL = 2 · 106
and 4 · 106
cases performed using a blended wall model based on the acceleration
sensor.
65

0.0 0.2 0.4 0.6 0.8 1.0 1.2
n/δ(x)
0.0
0.2
0.4
0.6
0.8
1.0 U/U∞
0.0 0.2 0.4 0.6 0.8 1.0 1.2
n/δ(x)
−2
0
2
4
6
8
10
12
hu
0u
0i/U2
∞
hv
0v
0i/U2
∞
hw
0w
0i/U2
∞
hu
0v
0i/U2
∞
Reynolds stresses
DNS
2M-Eq-M
100 101 102 103 104 n
0 +
10
20
U
+
n
+
ln n
+
0.41 + 5.2
(a) (b)
Figure 4.12: Wall-normal profiles of the incoming turbulent boundary layer at the reference location
(x/L = −0.5) for the ReL = 2 · 106
case, comparing the wall-modeled LES with an equilibrium wall
model (dotted green line) with the DNS data of Uzun & Malik (2022). (a) Mean streamwise velocity, normalized by the freestream velocity, as a function of the wall-normal distance in outer scaling (n/δ). The
inset shows the same quantity in inner scaling (U
+ = U/uτ as a function of n
+ = n/ℓv), with the viscous
and log-law theoretical estimates represented by dashed and dash-dotted lines, respectively. (b) Reynolds
stresses (⟨u
′
iu
′
j
⟩) normalized by the square of the freestream velocity, as a function of the wall-normal distance in outer scaling (n/δ). Gray areas represent the wall-modeled region in the WMLES.
blended wall models, since the region upstream of the bump also applies the equilibrium wall model for
all simulations, as dictated by the calculated local sensors and blending factors (as shown later when discussing Figure 4.14). WMLES results are compared with the DNS reference data obtained by Uzun & Malik
(2022) at the same Reynolds number. A slight overprediction of the normalized mean streamwise velocity
is observed, as well as a small underprediction of wall-normal and spanwise Reynolds stresses. In the inset
of Figure 4.12(a), the van Driest transformed mean streamwise velocity profile shows good agreement with
DNS data in the log-layer (where the wall-model exchange height is placed).
The streamwise evolution of the time- and spanwise-averaged pressure and friction coefficients is
shown in Figure 4.13. For the ReL = 2 · 106
case, in the upstream region of the bump, characterized by a
mild APG, the predicted pressure coefficient Cp from both the equilibrium wall model and the sensor-based
blended WMLES aligns well with experimental data and DNS results. Subsequently, the pressure gradient
shifts to a strongly favorable state, between x/L ≈ −0.29 and the apex of the bump, x/L = 0, causing
66

−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
−1.25
−1.00
−0.75
−0.50
−0.25
0.00
0.25
Cp
2M-Eq-M
2M-Rel-M
2M-Acc-M
DNS
Exp R
Exp G
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
0
2
4
6
8
10
Cf
× 10
3
−0.1 0.0
4
6
8
10
(a) (b)
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
−1.50
−1.25
−1.00
−0.75
−0.50
−0.25
0.00
0.25
Cp
4M-Eq-M
4M-Rel-M
4M-Acc-M
Gray M = 0.1
Gray M = 0.2
Exp W
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
0
2
4
6
8
10
Cf
× 10
3
−0.1 0.0
4
6
8
10
(c) (d)
Figure 4.13: Streamwise profiles of pressure coefficient (a,c) and friction coefficient (b,d) obtained from
WMLES of the ReL = 2 · 106
(a,b) and ReL = 4 · 106
(c,d) cases, performed using an equilibrium wall
model (dotted green), a blended wall model based on the acceleration sensor (solid red), and a blended
wall model based on the relaminarization sensor (dashed blue). For ReL = 2 · 106
(a,b), WMLES results
are compared with DNS data by Uzun & Malik (2022) (black solid, solid circles), and experiments by Gray
et al. (2022) (hollow circles) and Robbins et al. (2021) (hollow triangles). For the ReL = 4 · 106
case (c,d),
reference data includes results from experiments by Gray et al. (2022) (circles) at two Mach numbers (0.1,
hollow; 0.2, solid) and Williams et al. (2021)(solid triangles). Cf values in (b,d) scaled by a factor of 103
.
67

flow acceleration that steepens the velocity gradient near the wall and increases Cf . Interestingly, the peak
of the skin friction coefficient is found slightly downstream when using the blended wall model with sensors compared to the equilibrium wall model. The former shows better agreement with experimental data,
although a minor overprediction near the apex remains. The magnitude of the pressure coefficient at the
suction peak is also slightly overpredicted. These overpredictions might stem from differences in the top
boundary condition between WMLES (slip wall) and DNS (non-reflecting characteristic free air). Nonetheless, predictions with the current WMLES approach improve upon previous WMLES results by Iyer & Malik (2023a) and Whitmore et al. (2021), which underpredicted the friction coefficient using the equilibrium
wall model and slip-wall boundary conditions, respectively. Upon reaching the bump apex, the pressure
gradient turns adverse, leading to flow deceleration. This deceleration causes significant flow separation,
evident from the negative distribution of Cf . The equilibrium wall model shows a broad region of reversed
flow, although it appears slightly delayed compared to the DNS data. In this separation region, the equilibrium wall model aligns better with DNS reference data for Cp, while both sensor-based models exhibit
improved agreement with experimental data compared to DNS between x/L ≈ 0.25–0.5. However, they
do mispredict the onset of the separation region. Flow reattachment happens slightly earlier in the sensorbased blended wall modeling approaches, which provides better alignment with experiments compared to
DNS results. The prediction of the separation bubble size improves compared to prior IDDES-SST simulations by Prakash et al. (2022) where mean separation was absent. Furthermore, the Cp results exhibit good
agreement with DNS in the post-separation recovery region. The plateau observed in the pressure coefficient Cp distribution from the apex to the tail of the bump indicates the presence of separated flow and
serves as a clear indication of its extent. In summary, while the sensor-based models exhibit discrepancies
in predicting certain aspects of the flow behavior, such as the separation region and the slope of the skin
friction coefficient Cf , they still provide valuable insights and demonstrate reasonable agreement in other
68

regions. The effect of grid resolution in the prediction of separation is still an open topic of study in the
WMLES literature.
For the ReL = 4·106
case (Figure 4.13c,d) the WMLES results are compared with the experimental data
from Gray et al. (2022) at two Mach numbers (0.1 and 0.2, the latter matching more closely the Reynolds
number of the WMLES, but with less Cf data available near the bump apex). Qualitatively, the trends
observed in the WMLES results generally align with those seen in the experimental dataset. However, in
quantitative terms, the WMLES appear to overpredict the magnitude of the pressure coefficient near the
bump apex. Concerning the skin friction coefficient Cf , WMLES demonstrate good agreement with the
experiment up to the onset of the separation region. Notably, the sensor-based blended wall modeling
approaches provide a slightly better prediction compared to the equilibrium wall model. Beyond the onset
of the separation region, WMLES tend to overestimate the size of the separation bubble, leading to a
delayed reattachment. Earlier work by Iyer & Malik (2023a) suggested that the Gaussian-shaped speed
bump represents a Reynolds number-independent regime based on assessments involving three predictions
(experimental data, WMLES, and RANS). However, in the present study, Cp displays more pronounced
sensitivity to the Reynolds number, ReL, near the bump apex and within the separation region, as will be
detailed in a subsequent section.
Figure 4.14 illustrates the streamwise profiles of the blending factor for the ReL = 2 · 106
and ReL =
4 · 106
cases. While relaminarization is not observed (consistent with DNS and experiments), significant
changes of mean streamwise velocity are found within the logarithmic layer of the flow, signifying departures from equilibrium. As a result, some degree of blending of boundary conditions in the wall model is
applied in the FPG region, although this is considerably less than what is observed in the lower ReL = 106
case (compared with Figure 4.6b). Thus, the blending factor applied in the FPG region decreases with increasing Reynolds number (from values near unity at the lowest Reynolds number, to approximately half
and one-third at the ReL = 2 · 106
and ReL = 4 · 106
cases, respectively). The application of such partial
69

−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
0.0
0.2
0.4
0.6
0.8
1.0
β
2M-Eq-M
2M-Rel-M
2M-Acc-M
(a)
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
0.0
0.2
0.4
0.6
0.8
1.0
β
4M-Eq-M
4M-Rel-M
4M-Acc-M
(b)
Figure 4.14: Streamwise profiles of the time- and spanwise-averaged blending factor from WMLES performed using blended wall models based on the acceleration (solid red) and relaminarization (dashed blue)
sensors for the ReL = 2 · 106
(a) and ReL = 4 · 106
(b) cases.
blending generally contributes to better agreement with reference data for the skin friction coefficient Cf .
In the separated flow region downstream of the bump apex, there is a noticeable divergence of the blending
factor between both sensor-based models, with the acceleration sensor translating into significantly larger
blending factors than the relaminarization sensor. The low value of the friction coefficient in the separated
flow region, however, implies that the impact of the blending factor on the prediction of wall quantities
in that separated flow region is relatively limited. In contrast, for the ReL = 2 · 106
, the equilibrium wall
model exhibits different characteristics in the separated region, indicating that the prediction of separation
and recovery might be more influenced by the history effects of the boundary layer stemming from the
upstream conditions prior to separation (i.e., FPG followed by the initial APG). A similar conclusion can
be drawn from the analysis of the ReL = 4 · 106
case, for which the simulations with the relaminarization
and acceleration sensors show larger differences at the peak of Cf near the bump apex (higher peak for the
relaminarization sensor), translating into a different extent of the separation bubble. For the ReL = 4 · 106
case the simulation with the relaminarization sensor applies a lower fraction of the no-slip boundary condition in the wall-model blending upstream of the bump apex than the simulation with the acceleration
sensor (in contrast with the lower Reynolds number cases), resulting in a higher friction coefficient peak
70

attained before the bump apex, where the adverse pressure gradient starts. Across the three Reynolds
numbers tested, prediction of a larger Cf peak generally produces larger subsequent separation (e.g., by
the equilibrium wall model at ReL = 106
and 2 · 106
, and by the relaminarization-sensor blended wall
model at ReL = 4 · 106
).
Wall-normal line probes of mean (time- and spanwise-averaged) wall-parallel velocity and Reynolds
stresses for the Re = 2 · 106
case are presented in Figure 4.15. These probes are extracted at various
streamwise locations, comparing WMLES utilizing an equilibrium wall model everywhere, a blended wall
model based on the acceleration sensor, and a blended wall model based on the relaminarization sensor. The
WMLES results exhibit favorable agreement with the DNS reference data (Uzun & Malik, 2022), particularly
before the separation region. At x/L = −0.4, both the equilibrium WMLES and sensor-based blended
WMLES slightly overpredict the DNS results. A slightly lower intercept constant is captured, which can
be attributed to the presence of a gentle APG region as the flow approaches the bump. Although no
relaminarization is present at this Reynolds number (ReL = 2·106
), consistent with DNS results, noticeable
alterations are observed within the logarithmic layer. A shift in the velocity profile above the standard
logarithmic layer (not shown) might account for the slight overestimation in the skin friction coefficient
Cf (Figure 4.13b). Notably, there are significant dissimilarities observed in these profiles at the bump apex
between the ReL = 106
and ReL = 2 · 106
cases. This discrepancies likely contribute to the varying sizes
of the separation bubble in the downstream region. When comparing the streamwise Reynolds normal
stress in the internal layer, WMLES demonstrates better agreement with DNS than RANS and IDDESSST (Prakash et al., 2022). Additionally, concerning the wall-normal Reynolds normal stress, the tested
WMLES approaches effectively address the overprediction in the internal layer observed in pure RANS
and hybrid RANS-LES (Prakash et al., 2022). Near the peak of FPG (x/L ≈ −0.1), the blended WMLES
tend to slightly underpredict the magnitude of the Reynolds shear stress in the newly forming internal layer
71

0.5 1.0
0.0
0.5
1.0
n/δ(x)
x/L = −0.4
0.5 1.0
x/L = −0.2
1.0 1.2
x/L = −0.1
1.00 1.25
x/L = 0.0
0 1
x/L = 0.1
0 1
x/L = 0.2
0 1
x/L = 0.6
0 10
0.0
0.5
1.0
n/δ(x)
0 10 0 10 0 10 0 10 0 20 0 10
0.0 2.5
0.0
0.5
1.0
n/δ(x)
0 5 0 5 0 5 0 5 0 10 0 10
0 5
0.0
0.5
1.0
n/δ(x)
0 5 0 5 0 5 0 10 0 10 0 10
0 1
0.0
0.5
1.0
n/δ(x)
0 2 0 1 0 2 0.0 2.5 0 5 0 5
U/U∞ 100
u
0
rms/U∞ 100
v
0
rms/U∞ 100
w
0
rms/U∞
−1000 hu
0v
0i/U
2∞
DNS Eq Acc Rel
Figure 4.15: Wall-normal line probes extracted at different streamwise locations (left to right) of mean
(time- and spanwise-averaged) wall-parallel velocity and Reynolds stresses (top to bottom) for WMLES of
the ReL = 2 · 106
case performed using an equilibrium wall model (dotted green), a blended wall model
based on the acceleration sensor (solid red), and a blended wall model based on the relaminarization sensor
(dashed blue), compared with DNS data (black solid) by Uzun & Malik (2022).
72

(a)
(b)
(c)
(d)
(e)
(f)
Figure 4.16: Contours of instantaneous fluctuations of the local skin friction coefficient on the wall (a),
instantaneous streamwise velocity fluctuation extracted on wall-shaped surfaces located a wall-normal
distance of 0.1δ0 (b), 0.2δ0 (c), δ0 (d), 2δ0 (e), and instantaneous streamwise velocity on a vertical plane (f),
for the ReL = 2 · 106
case.
(identifiable by the presence of a ‘knee’ point in the profiles), likely due to the partial imposition of a noslip boundary condition, as the equilibrium WMLES aligns better with the reference data in the upstream
location. Despite those differences, the use of the blended wall models for these higher Reynolds number
cases without relaminarization provides predictions of comparable accuracy to those obtained with the
equilibrium wall model applied everywhere without any change in parameters relative to those used for
relaminarizing flow conditions at lower Reynolds numbers, for which predictions are largely improved.
4.1.3 Reynolds number effects
When comparing instantaneous turbulence structures through contours of friction and velocity fluctuations extracted on surfaces parallel to the wall at different heights, the higher-Reynolds number cases
(Figures 4.16 and 4.17 for ReL = 2 · 106
and 4 · 106
, respectively) show significant qualitative differences
with the relaminarizing case at ReL = 106 described earlier (Figure 4.9). The range of friction coefficient
73

(a)
(b)
(c)
(d)
(e)
(f)
Figure 4.17: Contours of instantaneous fluctuations of the local skin friction coefficient on the wall (a),
instantaneous streamwise velocity fluctuation extracted on wall-shaped surfaces located a wall-normal
distance of 0.1δ0 (b), 0.2δ0 (c), δ0 (d), 2δ0 (e), and instantaneous streamwise velocity on a vertical plane (f),
for the ReL = 4 · 106
case.
fluctuations attained in the boundary layer region leading to the FPG by the wall model is reduced, relative
to the lower ReL = 106
case. This reduction is interpreted as a consequence of the nature of the wallmodeled LES, which does not resolve the finest spatial and temporal scales of the turbulent flow below the
exchange location. The friction coefficient fluctuations observed in these WMLES are thus attributed to
the modulation of larger-scale structures in the logarithmic layers, rather than to the very-near-wall smallscale turbulence structures. Previous experiments and DNS studies on turbulent boundary layers (Wang
et al., 2020) found a logarithmic scaling of wall shear stress fluctuations with the friction Reynolds number
(τ
+
w,rms ∝ ln Reτ ), with the near-wall low-speed streaks dominating the most energetic fluctuations of
shear stress (not captured in a WMLES), and the larger-scale motions (originating in the log and outer
layers) having a weaker, modulating effect on those fluctuations. As already mentioned for the ReL = 106
case, the thinning of the boundary layer driven by the accelerated flow in the FPG region of the bump is
evidenced by the apparent decrease in velocity fluctuations seen in the contours extracted at wall-parallel
74

heights of n/δ0 = 1 and 2 (plots d and e in Figures 4.9, 4.16 and 4.17), which extend into the region downstream of the bump apex, where the growth of the boundary layer induced by the APG extends again onto
those heights.
Near the peak FPG region (x/L ∈ [−0.1, −0.05]), larger streamwise coherent streaks of friction coefficient fluctuations emerge, more clearly visible at the highest Reynolds number (Figure 4.17a). These
streaks are directly modulated by the corresponding streaks of streamwise velocity fluctuations found at
n/δ0 = 0.1, and have a spanwise width comparable to the boundary layer thickness at that location. In
contrast to the relaminarization process that takes place for the ReL = 106
case, manifested by a dampening of friction coefficient oscillations (Figure 4.9a), for the higher ReL cases without relaminarization, an
increased level of friction fluctuations is seen to start upstream of the bump apex, near the peak of mean
friction coefficient. Smaller-scale structures start to develop in that region up to x/L ≈ 0.1 which marks
the onset of separation. The spanwise homogeneity of these regions of formation of small-scale structures
found in the higher ReL cases contrasts with the inhomogeneous character of the retransition process
found in the ReL = 106
case, with larger patches of high friction coefficient fluctuations surrounded by
other equally large, quieter patches.
In the regions of mean flow separation that result for the ReL = 2·106
and 4·106
cases, large spanwise
coherence is found in structures of velocity fluctuations on extracted surfaces closer to the wall (n/δ0 = 0.1
and 0.2). The spanwise coherence (e.g., x/L ≈ 0.2) is larger for the flow at the higher Reynolds numbers
with mean separation than for the lower ReL = 106
case, for which only incipient separation was predicted. The spanwise coherence is broken farther downstream within the separation bubble, but recovers
near the mean reattachment location around x/L ≈ 0.4. The influence of the spanwise domain width was
not considered in the present study, but it is expected to have an influence in the three-dimensionality of
the large-scale structures within the separation bubble.
75

−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
−1.50
−1.25
−1.00
−0.75
−0.50
−0.25
0.00
0.25
Cp
1M-Acc
2M-Acc
4M-Acc
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
0
2
4
6
8
Cf
× 10
3
−0.1 0.0
4
6
8
10
(a) (b)
Figure 4.18: Streamwise profiles of pressure (a) and friction (b) coefficients obtained from WMLES performed using a blended wall model based on the acceleration sensor at different Reynolds numbers:
ReL = 106
(solid), 2 · 106
(dotted), 4 · 106
(dashed). Cf values in (b) scaled by a factor of 103
.
The effects of increasing Reynolds number on the wall and boundary layer time- and spanwise-averaged
quantities are assessed in Figures 4.18 and 4.19 from WMLES performed using the acceleration sensorbased blended wall model. The pressure coefficient, Cp, exhibits some sensitivity to an increase in Reynolds
number, ReL, with the most significant differences occurring around the bump apex and in the separation
region downstream. The friction coefficient, Cf , exhibits the expected monotonic decrease with increasing
ReL both upstream of the FPG peak and in the downstream recovery region. Whether the flow relaminarizes or not imposes a drastic difference in the Cf profile, with a double peak structure associated with
relaminarization and retransition upstream and downstream of the bump apex. For cases without relaminarization (ReL = 2·106
and 4·106
), a reduced sensitivity on the Reynolds number (within the range under
consideration) is observed between the end of the FPG upstream of the apex and the separated flow region
downstream. Despite the lower Cf of the incoming turbulent boundary layer upstream of the acceleration
region with increasing Reynolds number, the peak Cf attained near the bump apex for the ReL without
relaminarization surpasses that of the lowest ReL in which relaminarization occurs. The larger peak Cf
of the higher ReL cases leads to a subsequent steeper negative gradient producing earlier separation, but
whose location is nearly insensitive to the Reynolds number between the ReL = 2 · 106
and 4 · 106
cases.
76

−0.5 0.0 0.5 1.0
x/L
0
10
20
30
δ
∗/L × 103
−0.5 0.0 0.5 1.0
x/L
0.0
2.5
5.0
7.5
10.0
θ/L × 103
−0.5 0.0 0.5 1.0
x/L
1.0
1.2
1.4
Ue/U∞
1M-Acc
2M-Acc
4M-Acc
(a) (b) (c)
Figure 4.19: Streamwise profiles of boundary layer displacement thickness (a), momentum thickness (b),
and edge velocity (c) obtained from WMLES performed using a blended wall model based on the acceleration sensor at different Reynolds numbers: ReL = 106
(solid), 2 · 106
(dotted), 4 · 106
(dashed).
Subsequent reattachment is delayed as the Reynolds number increases, recovering lower Cf values in the
recovery region. This slight increase in the size of the separation bubble observed for the higher Reynolds
number case, ReL = 4 · 106
, is in line with experimental trends (Gray et al., 2022).
The behavior of the edge velocity, the displacement and momentum boundary layer thicknesses follows
a similar trend at higher Reynolds numbers as those observed in the ReL = 106
case. Flow acceleration
results in a thinner boundary layer (and thus δ
∗
and θ), a trend that is reverted on the APG region. The
growth rate and subsequent peak of δ
∗
in that APG region increases with ReL, with the peak location
being much less sensitive. The momentum thickness, θ, however, exhibits decreasing growth rates in
the APG with increasing ReL, with the peak attained at locations further downstream, leading to local
divergence of the shape factor (not shown). Near the end of the bump, there is a brief plateau in the
boundary layer thickness. In situations with higher Reynolds numbers, large structures are generated in
the presence of flow separation. The reattachment of the separated flow plays a crucial role in shaping and
further developing a thick boundary layer in the recovery zone. It is important to note that the methods
employed in this study to assess boundary layer thicknesses and edge velocity do not differentiate between
attached and separated flows. Flow separation introduces complex flow patterns and vortical motions that
77

disrupt the typical structure of the boundary layer, which introduces uncertainty in the concept of a welldefined boundary layer thickness or edge velocity. However, Vinuesa et al. (2016) offers an advantage by
providing a smooth transition in estimating boundary layer quantities as the flow undergoes separation
and subsequent reattachment.
4.1.4 Sensitivity analyses of sensor-based blended WMLES results to model parameters
In this section, the sensitivity of the predictions obtained from WMLES using the newly proposed sensorbased blended wall-modeling approach is evaluated with respect to model parameters: the reference sensor
value, σ0, used in the calculation of the blending coefficient, and the hysteresis time scale. Additionally, in
appendix 6.1, the effect of the time-filtering scale entering the definition of the acceleration sensor will be
assessed, in comparison with time filtering applied to transition sensors previously defined in the literature.
The suitability of these parameters in other flow configurations will be evaluated in future studies.
4.1.4.1 Sensor reference value
Different threshold values have been proposed in the literature to define the onset of relaminarization
based on dimensionless pressure gradient measures such as the acceleration (∆p) and relaminarization
(K) parameters (Narasimha & Sreenivasan, 1979). The choice of a corresponding sensor reference value
(σ0) used in the proposed blended wall modeling strategy has implications in the resulting blending factor
(see §2.3), which weighs the relative contribution of each boundary condition type for the estimation of
the local wall shear stress. To assess the sensitivity of the blended WMLES results to the choice of the
sensor reference value, we conduct simulations using the acceleration sensor for three different values of
σ0 equal to 0.018, 0.025, and 0.030. Figure 4.20 presents Cp and Cf streamwise distributions at ReL = 106
and ReL = 2 · 106
for those three reference values.
78

−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
−1.50
−1.25
−1.00
−0.75
−0.50
−0.25
0.00
0.25
Cp
1M-Acc-F, σ0 = 0.018
1M-Acc-F, σ0 = 0.025
1M-Acc-F, σ0 = 0.030
DNS
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
0
2
4
6
8
Cf
× 10
3
−0.1 0.0 0.1
4
6
8
(a) (b)
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
−1.25
−1.00
−0.75
−0.50
−0.25
0.00
0.25
Cp
2M-Acc-M, σ0 = 0.018
2M-Acc-M, σ0 = 0.025
2M-Acc-M, σ0 = 0.030
DNS
Exp R
Exp G
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
0
2
4
6
8
10
Cf
× 10
3
−0.1 0.0
4
6
8
10
(c) (d)
Figure 4.20: Streamwise profiles of pressure coefficient (a,c) and friction coefficient (b,d) from WMLES of
the ReL = 106
(a,b) and ReL = 2 · 106
(c,d) cases performed using an equilibrium wall model (dotted
green), a blended wall model based on the acceleration sensor with different reference (normalization)
values: 0.018 (dotted red), 0.025 (solid red), and 0.030 (dashed red), compared with DNS data by Uzun
& Malik (2020, 2022) (black solid), and experiments by Gray et al. (2022) (hollow circles) and Robbins et
al. Robbins et al. (2021) (hollow triangles). Cf values in (b,d) scaled by a factor of 103
.
79

For the ReL = 106
case, reasonable agreement with the reference data is found for the three sensor
normalization values until the peak of the skin friction coefficient Cf , accurately capturing both the suction
peak of pressure and the maximum skin friction. Quantitatively, the peak Cf is better predicted with a
sensor reference value of 0.025, with slight over- and under-predictions for σ0 values of 0.03 and 0.018,
respectively. A σ0 = 0.025 value successfully captures the secondary peak but exhibits a deeper local
minimum immediately upstream. On the other hand, σ0 = 0.03 accurately predicts the local minimum,
but it is followed by an advanced rise to the local maximum. In the recovery region downstream the
second Cp peak, σ0 values of 0.025 and 0.03 show comparable agreement with DNS, while predictions
with σ0 = 0.018 are significantly less accurate. For the ReL = 2 · 106
case, similar observations apply to
the recovery region, with σ0 = 0.018 slightly overpredicting the extent of separation and leading to an
underprediction of Cf downstream of reattachment. Elsewhere, the accuracy of the prediction of Cp and
Cf are much less dependent on the chosen sensor reference value for this higher ReL case, owing to the
lack of relaminarization and retransition. Predictions of Cf in the first half of the bump generally align
well with DNS results. However, there is a slight overprediction at the bump apex for all sensor reference
values. This difference might be related to different top boundary condition between the present WMLES
and the DNS. Downstream of the bump peak, all σ0 values exhibit better agreement with experimental
data for Cp compared to DNS between x/L ≈ 0.25–0.5 in the separation region. Nonetheless, a delay in
the predicted separation point by the blended WMLES can be observed for all σ0 values relative to DNS
and experiments. Thresholds of 0.025 and 0.03 result in a slightly smaller separation bubble, while 0.018
provides a separation region size comparable to DNS data but with a delayed reattachment. In the postseparation recovery region, the Cp results are in good agreement with DNS for all σ0 values. However,
sensor reference values of 0.025 and 0.03 are quantitatively superior to 0.018 in terms of Cf .
Figure 4.21 illustrates the streamwise profiles of blending factor for the ReL = 106
and ReL = 2 · 106
cases for the three reference values considered. The starting and final points where the blending approach
80

−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
0.0
0.2
0.4
0.6
0.8
1.0
β
1M-Acc-F, σ0 = 0.018
1M-Acc-F, σ0 = 0.025
1M-Acc-F, σ0 = 0.030
(a)
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
0.0
0.2
0.4
0.6
0.8
1.0
β
2M-Acc-M, σ0 = 0.018
2M-Acc-M, σ0 = 0.025
2M-Acc-M, σ0 = 0.030
(b)
Figure 4.21: Streamwise profiles of the blending factor from WMLES of the ReL = 106
(a) and ReL = 2·106
(b) cases performed using a blended wall model based on the acceleration sensor with different reference
values: 0.018 (dotted red), 0.025 (solid red), and 0.030 (dashed red).
becomes effective in predicting relaminarization and retransition in the ReL = 106
case are less sensitive
to the choice of sensor reference value. However, there is a more noticeable divergence in the maximum
values of the blending factor with varying σ0. As is noted in DNS results, a fully laminar state is not
achieved by the end of the favorable pressure gradient region. When σ0 = 0.018 is employed, it can result
in an overly aggressive application of the no-slip LES boundary condition in a significant fraction of the
FPG region. This excessive application is responsible for the underprediction observed in the peak of the
skin friction coefficient Cf (Figure 4.20b). Conversely, when σ0 = 0.03 is used, it leads to a greater weight
assigned to the wall model when blending boundary conditions, causing a slight overprediction in the
magnitude of the pressure and friction coefficients. For the ReL = 2 · 106
case, the resulting blending
factor is largely decreased (nearly halved) in the FPG region upstream of the bump apex for all sensor
reference values. Interestingly, the blending factor in the separation and recovery regions reaches values
exceeding those found in the lower ReL = 106
case, with the 0.025 and 0.03 reference values producing
closer blending factor distributions in the downstream flow regions, compared to the larger blending factor
generally obtained for the σ0 = 0.018.
81

In summary, the selection of the acceleration sensor reference value has a direct impact on the agreement with DNS results at ReL = 106
, particularly concerning the peak of the skin friction coefficient
Cf distribution and the its local minimum and maximum corresponding to retransition. This impact is
primarily attributed to the cumulative effects of pressure gradients on the boundary layer in the upstream
region. A σ0 = 0.025 reference value seems to strike a balance by accurately capturing the secondary
peak while exhibiting a deeper local minimum upstream (between the two peaks). However, for higher
Reynolds numbers, the sensitivity of the acceleration sensor to the reference value is not as pronounced.
4.1.4.2 Hysteresis time scale
The effect of the coefficient α in the definition of the hysteresis time scale given in Eq. 2.26 is evaluated
by conducting simulations for the ReL = 106
case on the medium-resolution grid with the acceleration
sensor blended wall model for three different values of α set to zero (i.e., no hysteresis), the nominal 1000
and an intermediate value of 500. The results, in terms of streamwise profiles of pressure and skin-friction
coefficients, are shown in Figure 4.22. The hysteresis time scale impacts the prediction of retransition from
the quasi-laminar state of the boundary layer attained during the relaminarization in the favorable pressure
gradient region upstream of the bump apex to the turbulent boundary layer. This effect is manifested by
the location and intensity of the second, local peak of skin friction coefficient. When no hysteresis (delay)
is considered (α = 0), retransition occurs too quickly after the bump apex, such that the second peak in
Cf is found upstream of the location predicted by the DNS and the attained value of Cf at that second
peak is too large, relative to the DNS reference data. As the hysteresis time scale is increased, retransition
is moved downstream, and the second Cf peak decreases in intensity. It is thus found that the process of
retransition in the presence of an adverse pressure gradient entails a delay time.
Two other relevant observations regarding the prediction of retransition can be made when comparing
the predictions obtained with the nominal value of α = 1000 between the medium and fine grids, shown
82

−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
−1.50
−1.25
−1.00
−0.75
−0.50
−0.25
0.00
0.25
Cp
1M-Acc-M, α = 1000
1M-Acc-M, α = 500
1M-Acc-M, α = 0
DNS
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
0
2
4
6
8
Cf
× 10
3
−0.1 0.0 0.1
4
6
8
(a) (b)
Figure 4.22: Streamwise profiles of pressure (a) and friction (b) coefficients from WMLES of the ReL = 106
case performed using a blended wall model based on the acceleration sensor with different delay time
factors (α = 0, 500, 1000), compared with DNS data (black solid with symbols) by Uzun & Malik (2020).
by the solid lines in Figures 4.22 and 4.20, respectively. First, an increased grid resolution improves the
prediction of Cf immediately downstream of the first peak and at the second peak, whose strength shows
certain dependence on the grid resolution between the medium and fine meshes under consideration.
Second, the local minimum between the two Cf peaks is seen to underpredict the DNS data for both grids.
Better predictions that decrease the dependency on the grid resolution in the relaminarization region might
be achieved by blending with suitable laminar wall models (instead of with a no-slip boundary condition).
Except in the retransition region near the bump apex, the impact of choice of the α value in the prediction of Cp and Cf is small, particularly upstream of the bump apex, where almost no variation in the
prediction of these quantities is observed for the three α values considered. Mild differences exist in the
recovery region downstream of the bump x/L ≳ 0.5, which also corresponds to a region requiring longer
to converge the statistics.
83

4.2 NASA hump configuration
The NASA hump simulation demonstrates the intricate interaction between quasi-relaminarization, separation, and reattachment. As the flow accelerates over the convex leading surface of the hump, it experiences a strong favorable pressure gradient, which reduces turbulence intensity. This results in quasirelaminarization, where turbulent structures are dampened, and the boundary layer exhibits laminar-like
characteristics. However, as the flow moves further along the hump, it encounters an adverse pressure
gradient, leading to boundary layer separation and the formation of a separation bubble. The separation
point is typically located near the crest of the hump, where the adverse pressure gradient induces boundary
layer detachment. The separated shear layer rolls up into large-scale vortical structures, which interact
with the downstream recirculating region.
In this section, results from simulations using the blended wall model will be compared against those
obtained with a globally applied equilibrium wall model (i.e., without sensors or blending). The focus is on
evaluating the effects of the wall-model exchange height (hwm) and grid resolution on the accuracy of the
predictions. By systematically varying these parameters, this study aims to assess their impact on the accuracy of predictions of relaminarization, separation and reattachment resulting from pressure gradients.
Whenever possible, numerical results will be benchmarked against experimental data from Greenblatt et al.
(2006) and wall-resolved LES (WRLES) simulations by Uzun & Malik (2018). The findings show that while
variations in the wall-model exchange height have minimal impact on the pressure coefficient (Cp), they
significantly influence the skin friction coefficient (Cf ), particularly in regions of quasi-relaminarization
and separation. Specifically, lowering the exchange height closer to the wall improves the model’s predictions in the favorable pressure gradient and separation regions. As the flow moves past the separation
point, turbulence is regenerated, allowing the flow to reattach and overcome the adverse pressure gradient. This reattachment is characterized by an increase in skin friction coefficient (Cf ) and wall shear
stress, in line with the observations of Kahraman & Larsson (2020), who found that adapting the exchange
84

location enhances model accuracy in non-equilibrium flow regions. These modifications to the exchange
height enable the model to more accurately capture the transition and separation, further reinforcing the
necessity of dynamically adjusting the wall-modeling interface.
The exchange height hwm defines the location where velocity and temperature are extracted from the
LES solution and provided as input to the wall model. To evaluate its effect on flow predictions, four different formulations of hwm are considered. The exchange height is expressed as a fraction of the reference
boundary layer thickness (δref), ensuring consistency across different flow conditions. The baseline case
uses a constant hwm/δref = 0.1, a commonly adopted value for equilibrium wall models. In the variable
exchange height cases, modifications are introduced to examine their influence on quasi-relaminarization
and separation.
Two key streamwise locations are used in defining the variable hwm cases: xs ≈ 0.656L, corresponding
to the approximate separation point, and xr ≈ 1.13L, marking the reattachment location. These positions
serve as transition points where hwm is either reduced or restored, ensuring that changes in the wallmodeling interface align with key flow features. The tested profiles aim to assess the impact of the exchange
height on flow predictions and provide insights for future efforts in automatically determining optimal
local values of hwm, though such an implementation is beyond the scope of this study.
The four exchange height configurations considered are:
• HWM-U (Uniform): A constant exchange height across the entire domain, set to 10% of the reference boundary layer thickness:
hwm = 0.1δref
85

• HWM-V1 (Variable 1): A gradual reduction in hwm within the favorable pressure gradient region,
reaching zero at the separation point xs, followed by reactivation at xr:
hwm =



0.1δref, x < 0
0.1δref ·
x
xs
, 0 ≤ x < xs
0, xs ≤ x < xr
0.1δref, x ≥ xr
• HWM-V2 (Variable 2): Similar to HWM-V1, but with a reduced scaling factor in the transition
region:
hwm =



0.1δref, x < 0
0.05δref ·
x
xs
, 0 ≤ x < xs
0, xs ≤ x < xr
0.1δref, x ≥ xr
• HWM-V3 (Variable 3): The exchange height is set to zero for all x ≥ 0:
hwm =



0.1δref, x < 0
0, x ≥ 0
A summary of these configurations is provided in Table 4.3, facilitating comparison between cases.
It is worth noting that a staircase pattern will be observed in the hwm(x) function in simulation, which
can be attributed to the use of the nearest cell centroid method for determining the wall-model exchange
location. Despite the smooth target function provided, the grid discretization and the nearest-cell approximation cause the exchange height to exhibit abrupt changes at discrete grid points. This results in the
86

Table 4.3: Summary of wall model exchange height configurations
Case Label Exchange Height Definition
-U Uniform hwm/δref = 0.1 (baseline)
-V1 Gradual reduction to zero at xs, restored at xr
-V2 Same as V1 but with a smaller scaling factor in the transition region
-V3 Abrupt reduction to hwm = 0 at x = 0
staircase-like pattern observed in the hwm(x) function. While the pattern is discrete, the overall trend
of the wall-model exchange height still follows the intended smooth behavior set by the target function,
highlighting the effect of grid resolution on the accuracy of the wall-modeling interface placement. Future
improvements to the implementation of the current wall model could be made to use interpolation rather
than the nearest cell centroid for the quantities being exchanged from the LES grid to the wall model,
which may help to reduce these discretization effects.
In addition to variations in hwm, two grid configurations are used as base cases to further assess the
effect of resolution on the accuracy of the simulations. The coarse grid maintains a uniform streamwise
grid spacing of ∆s/δref = 0.08 from the inflow location (x/L = −3.0) up to x/L = 1, followed by gradual
stretching toward the outlet (x/L = 3.9). The spanwise spacing remains uniform at ∆z/δref = 0.08, while
the wall-normal spacing varies from ∆n/δref = 0.02 at the wall to ∆n/δref = 0.08 at n = 2δref, with
further stretching toward the top boundary.
The fine grid maintains the same spacing from the inflow to the start of the hump but refines the
streamwise spacing to ∆s/δref = 0.02 on the hump to improve resolution in the adverse pressure gradient
region. The spanwise spacing is reduced to ∆z/δref = 0.05, and the wall-normal spacing varies from
∆n/δref = 0.02 at the wall to ∆n/δref = 0.04 at n = 2δref, with further stretching toward the top
boundary. These grid configurations are summarized in Table 4.4.
Table 4.4: Summary of grid resolution configurations
Grid Label Streamwise Spacing ∆s/δref Spanwise Spacing ∆z/δref Wall-Normal Spacing ∆n/δref
-C 0.08 (inflow to x/L = 1), stretched after 0.08 0.02 (wall) to 0.08 (n = 2δref), stretched to 0.1
-F 0.08 (inflow to hump), 0.02 (hump), stretched after 0.05 0.02 (wall) to 0.04 (n = 2δref), stretched to 0.1
87

4.2.1 Effect of exchange heights
Figure 4.23 compares the surface pressure coefficient (Cp) and skin friction coefficient (Cf ) distributions
for the NASA hump simulation, assessing the impact of different exchange heights using the equilibrium
wall model. The Cp distribution indicates that while all cases capture the overall pressure trend along the
hump, variations appear in the separation and recovery regions downstream compared with the reference
data. However, the effect of exchange height on Cp remains minimal. In contrast, the right panel displays
the Cf distribution, which demonstrates a stronger sensitivity to exchange height selection. The separation and reattachment locations shift across different cases, with lower exchange heights leading to an
extended separation bubble. This results in a delayed reattachment and improved agreement with experimental data in the separation region. The inset figure shows the exchange location (hwm), highlighting
distinct variations across cases that directly impact the near-wall resolution. These results underscore the
significant influence of exchange height selection in wall-modeled large-eddy simulations of relaminarized
and separated flows. Optimizing exchange height selection can enhance flow predictions, particularly in
regions with strong favorable and adverse pressure gradients. Further exploration of adaptive exchange
height strategies may improve the accuracy of simulations for complex wall-bounded flows.
Figure 4.24 shows the surface pressure coefficient (Cp) and skin friction coefficient (Cf ) for the NASA
hump simulation using the sensor-based blended wall model. The left plot shows the Cp distribution along
the surface, demonstrating overall agreement between models, with minor deviations observed near the
reattachment region. However, the right plot reveals more noticeable differences in the Cf distribution,
particularly in the quasi-relaminarization region. The blended wall model significantly improves the prediction of this region by better capturing the transition from turbulent to quasi-laminar flow, which is
critical in accurately predicting skin friction in the presence of flow acceleration.
The inset provides a detailed view of the variation in the wall-model exchange height, underscoring
its impact on predictive accuracy. These findings are consistent with the results of Kahraman & Larsson
88

−0.5 0.0 0.5 1.0 1.5
x/L
−0.8
−0.6
−0.4
−0.2
0.0
0.2
0.4
Cp
Experiments
Uzun et al.
Eq-C-V1
Eq-C-V2
Eq-C-V3
Eq-C-U
−0.5 0.0 0.5 1.0 1.5
x/L
−2
0
2
4
6
Cf
× 10
3
0 1
0.00
0.05
0.10
hwm/δref
x/L
Figure 4.23: Comparison of surface pressure coefficient (Cp) and skin friction coefficient (Cf ) distributions
for different exchange heights in the NASA hump simulation with the equilibrium wall model. The inset
within the skin friction coefficient plot shows the streamwise profile of wall-model exchange height normalized by the reference boundary layer thickness, hwm(x)/δref.
(2020), who introduced an algorithm for dynamically adapting the wall-model exchange location in nonequilibrium flow regions. Similar to their work, the present study shows that adjusting the wall-model
height in regions of strong pressure gradients, such as the favorable pressure gradient region at the start of
the hump, leads to a better prediction of the quasi-relaminarization process. In particular, the acceleration
in this region induces partial relaminarization, making it crucial to lower the wall-model height for accurate
predictions of the transition. The sensor-based blended wall model captures this transition more effectively
than the equilibrium model, resulting in improved predictions for the quasi-relaminarization regions.
Figure 4.25 shows the streamwise profiles of the blending factor. Although full relaminarization is not
observed (consistent with WRLES), a plateau in the skin friction coefficient (Cf ) indicates deviations from
equilibrium. Consequently, partial blending of boundary conditions is applied in the favorable pressure
gradient (FPG) region of the wall model, leading to better agreement with reference Cf data. Downstream
of the bump apex, in the separated flow region, the acceleration sensor results in significantly high blending
factors due to its formulation. However, the minimal variation in the friction coefficient in this region
suggests that the blending factor has a limited effect on wall quantities (and, thus, the overall application
89

−0.5 0.0 0.5 1.0 1.5
x/L
−0.8
−0.6
−0.4
−0.2
0.0
0.2
0.4
Cp
Experiments
Uzun et al.
Acc-F-V1
Acc-F-V2
−0.5 0.0 0.5 1.0 1.5
x/L
−2
0
2
4
6
Cf
× 10
3
0 1
0.00
0.05
0.10
hwm/δref
x/L
Figure 4.24: Comparison of surface pressure coefficient (Cp) and skin friction coefficient (Cf ) distributions
for different exchange heights in the NASA hump simulation with a blended wall model based on the
acceleration sensor. The inset within the skin friction coefficient plot shows the streamwise profile of
wall-model exchange height normalized by the reference boundary layer thickness, hwm(x)/δref.
of the wall-model or no-slip boundary condition) in the separated flow area. In this case, the prediction
of separation and recovery appears to be more influenced by the selection of exchange heights and grid
resolution.
4.2.2 Effect of grid resolution
In this section, two variable exchange height configurations are tested: one starts at zero at the beginning
of the hump, increases gradually, resets to zero near the separation region, and then adopts a uniform value
further downstream; the other sets the exchange height to zero immediately after the start of the hump.
Figure 4.26 presents the grid spacing profiles in inner units along each coordinate direction for these grids.
Figure 4.27 compares the surface pressure coefficient (Cp) and skin friction coefficient (Cf ) distributions for the NASA hump simulation, evaluating the impact of different grid resolutions. In simulations
with a variable wall-model exchange height over the hump, independently doubling the grid resolution
in the streamwise, wall-normal, and spanwise directions had minimal effect in the attached flow regions.
The right panel shows the Cf distribution, emphasizing the sensitivity of skin friction predictions to grid
refinement. While all resolutions capture the separation location reasonably well, discrepancies arise in
90

−0.5 0.0 0.5 1.0 1.5
x/L
−2.0
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
2.0
s/s
0
Acc-F-V1
Acc-F-V2
−0.5 0.0 0.5 1.0 1.5
x/L
0.0
0.2
0.4
0.6
0.8
1.0
β
(a) (b)
Figure 4.25: Streamwise profiles of the time- and spanwise-averaged blending factor from WMLES performed using blended wall models based on the acceleration sensors.
−2 0 2 4
s/L
100
200
300
400
500
∆
s
+
Eq-C
Eq-C-z2
Eq-C-y2
Eq-C-x2
−2 0 2 4
s/L
5
10
15
20
25
30
∆
n
+
1
−2 0 2 4
s/L
50
100
150
200
250
300
∆
z
+
(a) (b) (c)
−2 0 2 4
s/L
0
100
200
300
400
500
∆
s
+
Eq-F
Eq-F-y2
−2 0 2 4
s/L
2
4
6
8
10
12
14
∆
n
+
1
−2 0 2 4
s/L
60
80
100
120
140
160
180
∆
z
+
(d) (e) (f)
Figure 4.26: Streamwise profiles of grid spacing in inner units for the streamwise, wall-normal first cell
centroid, and spanwise coordinate directions, obtained from WMLES of the case performed usingan equilibrium wall model with different grid resolutions.
91

the predicted magnitudes. Although adequate near-wall resolution is crucial for accurate flow predictions
in complex scenarios, further grid refinement does not necessarily lead to improved results.
−0.5 0.0 0.5 1.0 1.5
x/L
−0.8
−0.6
−0.4
−0.2
0.0
0.2
0.4
Cp
Experiments
Uzun et al.
Eq-C
Eq-C-z2
Eq-C-y2
Eq-C-x2
−0.5 0.0 0.5 1.0 1.5
x/L
0
2
4
6
Cf
× 10
3
0 1
0.00
0.05
0.10
hwm/δref
x/L
Figure 4.27: Comparison of surface pressure coefficient (Cp) and skin friction coefficient (Cf ) distributions
for different grid resolution in the NASA hump simulation with the equilibrium wall model. The inset
within the skin friction coefficient plot shows the streamwise profile of wall-model exchange height normalized by the reference boundary layer thickness, hwm(x)/δref.
Figure 4.28 examines the influence of grid resolution on simulations using an equilibrium wall model,
where the exchange height is set to the first cell centroid over the hump. Doubling the wall-normal grid
resolution impacts the friction coefficient prediction in the hump region, which remains inaccurate. This
suggests that the accuracy of the prediction is influenced by the combined effects of grid resolution and
exchange height, highlighting the limitations of the equilibrium wall model.
Ichimiya et al. (1998) examined the behavior of a turbulent boundary layer over a flat plate undergoing relaminarization due to flow acceleration in a converging sectional area, using the VITA technique.
The study found that while relaminarization alters ejection and sweep events, it does not significantly
suppress bursting in the inner layer. Additionally, the mean bursting frequency distribution in the outer
layer during relaminarization differs notably from that observed in the developing and retransition phases
92

−0.5 0.0 0.5 1.0 1.5
x/L
−0.8
−0.6
−0.4
−0.2
0.0
0.2
0.4
Cp
Experiments
Uzun et al.
Eq-F
Eq-F-y2
−0.5 0.0 0.5 1.0 1.5
x/L
−2
0
2
4
6
Cf
× 10
3
0 1
0.00
0.05
0.10
hwm/δref
x/L
Figure 4.28: Comparison of surface pressure coefficient (Cp) and skin friction coefficient (Cf ) distributions
for different grid resolution in the NASA hump simulation with the equilibrium wall model.
of a turbulent boundary layer, suggesting that relaminarization initiates in the outer layer. This observation may clarify why the selection of exchange height has a greater impact on predictions in the quasirelaminarization region than grid resolution. Since the equilibrium wall model is unsuitable for predicting
quasi-relaminarization, gathering input data from the outer layer, where relaminarization begins, may lead
to greater deviations.
4.2.3 Effect of delay time
Figure 4.29 examines the effect of the delay time parameter α on the surface pressure coefficient (Cp) and
skin friction coefficient (Cf ) in the NASA wall-mounted hump simulation. The left panel presents the Cp
distribution along the surface, where only minor differences are observed across different values of α. All
cases generally match experimental data, except in the regions near separation and reattachment. In contrast, the right panel, showing the Cf distribution, highlights a more pronounced effect of α, particularly in
the plateau region. As α increases, there is a noticeable reduction in the accuracy of Cf predictions in the
initial favorable pressure gradient (FPG) region. Specifically, the friction coefficient is significantly underpredicted in this region, suggesting that the hysteresis time is too long to properly capture the boundary
layer’s response to flow acceleration. While α has a limited effect on Cp, the accuracy of Cf predictions is
93

highly sensitive to the choice of α, especially when capturing the dynamics of relaminarization and retransition in the boundary layer. This suggests that selecting an appropriate timescale is critical to improving
model performance in these regions.
−0.5 0.0 0.5 1.0 1.5
x/L
−0.8
−0.6
−0.4
−0.2
0.0
0.2
0.4
Cp
Experiments
Uzun et al.
α = 1000
α = 2500
α = 5000
α = 10000
−0.5 0.0 0.5 1.0 1.5
x/L
0
2
4
6
Cf
× 10
3
0 1
0.00
0.05
0.10
hwm/δref
x/L
Figure 4.29: Comparison of surface pressure coefficient (Cp) and skin friction coefficient (Cf ) distributions
for different delay time in the NASA hump simulation using the blended wall model with acceleration
sensors.
Nagano et al. (1998) conducted experimental studies on turbulent boundary layers under sustained adverse pressure gradients, revealing that fluctuating velocity waveforms in the near-wall region exhibit prolonged temporal elongation compared to zero-pressure-gradient flows, with time scales escalating alongside the pressure gradient parameter. Notably, this increase deviated from proportionality with the classical
viscous time scale ν/u2
τ
. Their analysis identified the Taylor time scale as the most effective descriptor for
the intrinsic dynamics of non-equilibrium flows under adverse pressure gradients. We propose its potential
utility in optimizing delay time scale selection for future investigations.
94

Chapter 5
Conclusions and future work
5.1 Conclusions
In this thesis, we introduce a novel sensor-based blended wall-modeling approach that integrates local
relaminarization sensors, which quantify the dimensionless pressure gradient experienced by the boundary layer. The sensor value locally dictates the degree of blending of the equilibrium wall model and the
no-slip/laminar boundary conditions, allowing to better capturing the gradual nature of flow relaminarization and transition processes driven by curvature-induced pressure gradients. Two sensors are considered,
based on the acceleration and relaminarization parameters previously introduced in the literature for the
study of flows with reverse transition. Hysteresis effects in the response of the boundary layer to relaminarization and retransition are built into the proposed blending factor that weighs the degree of application
of each boundary condition in the proposed wall model by accounting for a spatial delay that is calculated
locally based on the friction velocity and viscosity at the wall. Whereas sensor-based wall-modeling approaches have been introduced in earlier studies (Bodart & Larsson, 2012; Mettu & Subbareddy, 2018) for
improved predictions of laminar-to-turbulent transition, those sensors (focused on transition) and wallmodeling strategies (with a binary switch between boundary conditions, rather than a blend) are found to
be inadequate for flows with relaminarization. The main reason stems from the fact that turbulence fluctuations (used by transition sensors) do not decay but remain nearly ‘frozen’ (Narasimha & Sreenivasan,
95

1979) in pressure-driven gradually relaminarizing flows, such as the one considered in the present study.
The performance of this newly introduced approach is assessed in two flow configurations: a Gaussianshaped speed bump and the NASA wall-mounted hump.
Wall-modeled large-eddy simulations (WMLES) of a turbulent boundary layer over a Gaussian-shaped
bump are conducted at three Reynolds numbers: ReL = 106
, 2 · 106
, and 4 · 106
. At ReL = 106
, the flow
undergoes relaminarization and stabilization in the acceleration region, whereas this effect is suppressed
at higher Reynolds numbers. In the ReL = 106
case, we identify limitations of the equilibrium wall model,
which assumes a fully turbulent boundary layer. This assumption leads to an overestimation of friction
and momentum losses in the relaminarization zone, highlighting the need for improved modeling in such
transitional flow regions.
Application of the proposed wall modeling approach to the ReL = 106 flow over the Gaussian bump
significantly improves the prediction of the friction coefficient in the relaminarization and retransition
regions, compared to the equilibrium wall model and previously existing sensor-based wall models. The
acceleration and relaminarization sensors effectively identify the gradual process of (partial) relaminarization and the sharper retransition to a turbulent boundary layer. Combined with the blending of boundary
conditions, and the crucial addition of the hysteresis effects, the decreased maximum value of friction coefficient upstream of the bump apex is accurately predicted, along with a secondary peak induced by the
retransition found shortly downstream (in the initial stages of the adverse pressure gradient imposed by
the wall shape). While instantaneous separation in the leeward side of the bump was still predicted by
the proposed blended wall models, the mild mean separation found in the reference DNS results was not
captured (in contrast to the largely overpredicted separation predicted when the equilibrium wall model
is applied everywhere). This result highlights the influence of history effects of the boundary layer in
96

the prediction of separation, which could require additional sensors (Agrawal et al., 2022) and modifications of the wall modeling approach to further improve predictions (along with different grid resolution
requirements) in such separated flow regions.
When tested on the higher Reynolds number cases (ReL = 2 · 106
and 4 · 106
), the proposed sensorbased blended wall modeling strategy correctly predicts the absence of relaminarization, without requiring
any change in the model parameters used to predict relaminarization at the lower Reynolds number. The
blended wall models provide comparable accuracy in the non-relaminarizing, high-ReL cases to the equilibrium wall model, with slight improvements in some of the predictions of the friction coefficient near the
bump apex and in the separation bubble. The blending factor is observed to decrease at higher ReL, thus
expecting a degree of convergence toward the equilibrium wall model prediction, as intended. Further
predictive improvements at these higher ReL might be accomplished by incorporating non-equilibrium
turbulent boundary layer effects into the wall model formulation (Kawai & Larsson, 2013; Park & Moin,
2014; Kamogawa et al., 2023), not explored in the present work. The proposed blended wall models successfully demonstrate applicability to a range of Reynolds numbers that encompasses flows with and without
relaminarization driven by pressure gradients. Targeting more complex flow configurations at different
regimes of compressibility in future studies could help to further assess the robustness and general applicability of the proposed wall-modeling method.
The simulation results obtained at the three different ReL considered in this study for the Gaussian
bump configuration are also used to explore Reynolds number effects on the predicted time- and spanwiseaveraged boundary layer quantities, as well as to identify three-dimensional flow features that can be
(partially) captured by the wall model, in relation to the internal layer that develops in the FPG region
within the boundary layer, and how it evolves through the relaminarization, retransition, separation and
reattachment processes. While the prediction of instantaneous fluctuation levels of the wall shear stress is
97

naturally limited by the information fed to the wall model from the LES at the exchange location (generally placed within the log layer), and thus missing small-scale temporal and spatial fluctuations that arise
nearer the wall, the proposed wall model could be augmented with existing models of inner-outer scale
interactions (Marusic et al., 2010; Mathis et al., 2013) to recover better predictions of fluctuating quantities
near the wall.
To further evaluate the generality of the proposed sensor-based blended wall-modeling approach, we
also examine the flow over the NASA hump configuration at a freestream Mach number of 0.2. The
proposed acceleration sensor successfully identifies the quasi-relaminarization region, and the blended
approach enhances prediction accuracy compared to a standard equilibrium wall model.
One key finding is that the wall-model exchange height plays a critical role in capturing the transition
and separation of the flow. A systematic analysis of the effects of exchange height and grid resolution
reveals that grid resolution has a more pronounced impact at higher exchange heights, emphasizing the
need for sufficient near-wall resolution in non-equilibrium regions. Lowering the exchange height in the
favorable pressure gradient (FPG) region near the start of the hump improves the prediction of quasirelaminarization, ensuring a better transition from turbulent to quasi-laminar flow. This observation aligns
with the findings of Kahraman & Larsson (2020), who demonstrated that adapting the exchange location
enhances accuracy in non-equilibrium boundary layers. Similarly, in the separated flow region, lowering
the exchange height results in improved skin friction predictions, highlighting its influence in capturing
flow separation and recovery.
The impact of the delay time parameter α is also examined. As α increases, the accuracy of the skin
friction coefficient (Cf ) prediction in the initial FPG region declines, leading to a significant underprediction of Cf . This suggests that overly long hysteresis timescales prevent the model from capturing the rapid
transition from turbulent to quasi-laminar flow in this region. However, α has a relatively minor effect on
98

the pressure coefficient (Cp), indicating that its influence is more pronounced on near-wall quantities than
on the overall pressure distribution.
Additionally, while the blending factor plays a crucial role in adjusting the wall-modeling interface, its
influence on wall quantities varies across different flow regions. In the FPG regions, the blending approach
significantly enhances prediction accuracy. However, in the separated flow downstream of the bump apex,
the impact of the blending factor is less pronounced, suggesting that separation and reattachment dynamics
are more strongly governed by exchange height and grid resolution rather than by the blending factor itself.
Ultimately, these simulations highlight the importance of dynamically adapting wall-modeling strategies to evolving flow conditions, particularly in non-equilibrium regions. The ability to adjust the wallmodel exchange height and appropriately tune the delay time parameter is critical for improving predictive
accuracy in capturing complex transition and separation phenomena. These findings reinforce the effectiveness of sensor-based blended wall models and provide valuable guidance for their future application
in high-fidelity large-eddy simulations of turbulent flows with strong pressure gradients effects.
5.2 Future work
Future research could explore strategies to improve the modeling of quasi-laminar regions, which remain under-resolved at typical WMLES grid resolutions and significantly impact flow predictions. Approaches such as physics-based wall models leveraging Pohlhausen polynomials (Majdalani & Xuan, 2020)
or Falkner-Skan similarity solutions (Gonzalez et al., 2021) could enhance predictive accuracy without requiring excessive computational resources. Additionally, extending sensor-based blended methods to detect and adapt to other flow phenomena, such as separated flow regions, could enable a more dynamic and
responsive wall-modeling approach that accounts for non-equilibrium effects beyond relaminarization.
Further improvements could focus on integrating sensor-based blending with automated algorithms
for determining the optimal wall-model exchange height hwm and performing grid adaptation—such as
99

those proposed by Kahraman & Larsson (2020). Given the intricate and subtle interplay between exchange
height selection, grid resolution, and flow physics, a more cohesive framework that adapts both wallmodeling parameters and grid refinement in tandem could further improve predictive accuracy in complex
turbulent flows.
Additionally, applying the sensor-based blended wall model to other flow configurations could provide
further insight into its effectiveness. Potential cases include flows over a 3D infinite swept wing, highly
accelerated turbulent boundary layers in wind tunnel experiments, and adverse pressure gradient turbulent
boundary layers with upstream favorable pressure gradients (Rumsey & Spalart, 2009; Bourassa & Thomas,
2009; Parthasarathy & Saxton-Fox, 2023). These investigations could help assess the model’s performance
across a broader range of flow conditions and refine its applicability to complex aerodynamic simulations.
100

Chapter 6
Appendix
6.1 Assessment of existing laminar-to-turbulent transition sensors to
predict relaminarization
We consider two existing sensors proposed in the literature for WMLES of transitional flows by Bodart
& Larsson (2012), denoted by the subscript BL, and by Mettu & Subbareddy (2018), denoted by the MS
subscript. Both sensors are based on local estimates of the turbulence kinetic energy:
σBL =
⟨
p
u
′
i
u
′
i
/2⟩
⟨uτ ⟩
, (6.1)
σMS = 0.15
⟨ρ⟩⟨k⟩
⟨µ⟩⟨|S|⟩, (6.2)
where u
′
i = ui − ⟨ui⟩ is the wall-parallel velocity fluctuation, ρ is the density, k = ⟨uiui⟩ − ⟨ui⟩⟨ui⟩ is the
turbulence kinetic energy, µ denotes the dynamic viscosity, and |S| =
p
2SijSij corresponds to the norm
of the strain-rate tensor, Sij , all of which are taken at the wall-model exchange height. uτ =
p
τw/ρw is
the estimated local friction velocity and ⟨·⟩ indicates time-filtered quantities, using an exponential moving
average with weight W(t) = exp(−t/T) and a defined filtering time scale T(t) = (SijSij )
−1/2
. For each
sensor, a threshold value was defined, σBL,0 = 1.4 and σMS,0 = 0.25. If the sensor exceeds the threshold
value, the equilibrium wall model is applied. Otherwise, laminar boundary layer conditions are assumed,
101

−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
−1.50
−1.25
−1.00
−0.75
−0.50
−0.25
0.00
0.25
Cp
DNS
BL
MS
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
0
2
4
6
8
10
Cf
× 10
3
−0.1 0.0 0.1
4
6
8
10
(a) (b)
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
0.0
0.2
0.4
0.6
0.8
1.0
f
BL
MS
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
0.000
0.005
0.010
0.015
0.020
0.025
0.030
hki
(c) (d)
Figure 6.1: Streamwise profiles of pressure coefficient (a) and friction coefficient (b) obtained from WMLES
of the ReL = 106
case performed using the sensor-based wall models of Bodart & Larsson (solid purple)
and Mettu & Subbareddy (dash-dotted yellow), compared with DNS data (black solid) by Uzun & Malik
(2020). The fraction of time that the equilibrium wall model is being applied for each method is shown
in (c) with the corresponding time filtered turbulence kinetic energy distribution (d). Cf values in (b,d)
scaled by a factor of 103
.
such that Bodart & Larsson’s method switches to an LES-nearest-cell linear approximation of momentum
and energy fluxes (i.e., no-slip LES boundary condition), whereas Mettu & Subbareddy’s approach switches
off (i.e., zeroes) the eddy viscosity of the wall model. A zero eddy viscosity implies a linear approximation
of the velocity profile from the exchange height to the wall in an equilibrium wall model formulation,
which can lead to significant inaccuracies depending on the resolution of the LES grid.
102

For a more meaningful comparison with the newly proposed sensor-based blended wall modeling approach, the simulations here conducted following Bodart & Larsson’s and Mettu & Subbareddy’s methodologies enforced the application of the equilibrium wall model in the region x/L < −0.8, so that the
prediction of wall quantities (including Cp and Cf ) matched the other simulations of the present study
upstream of the bump. Additional tests (not shown) in which the switching of boundary conditions based
on the σBL and σMS sensors was enabled in the full domain (including x/L < −0.8), resulted in worsened
predictions by the transitional sensor methods, particularly of Cf , throughout the domain. The reason for
these worsened predictions is that the equilibrium wall model was not fully applied in the region upstream
of the bump (as the sensor fell below the threshold) where a turbulent boundary layer should have been
detected.
Predictions of Cp and Cf distributions with these existing sensor-based wall models are presented in
Figure 6.1, along with the time- and spanwise-averaged fraction of time in which the equilibrium wall
model is being applied, f, and the time-filtered turbulence kinetic energy, ⟨k⟩. A unitary value of f(x)
indicates the activation of the equilibrium wall model across the entire spanwise domain for all time steps
at the specific streamwise location (x), whereas a zero value indicates the deactivation of the equilibrium
wall model (for BL) or its eddy viscosity (for MS) at that particular location. The results demonstrate
that the σBL sensor, with the threshold proposed in the literature, struggles to distinguish turbulent and
quasi-laminar regions in this flow configuration (mostly identifying a laminar flow throughout the shown
domain), while the σMS sensor fails to precisely predict the onset of relaminarization. A strong correlation
between f and ⟨k⟩ is found, as expected, also suggesting a dependency on the filtering time scale used
to estimate ⟨k⟩. Bodart & Larsson’s method presents a significant delay in predicting the effects of the
APG downstream of the bump apex, where a sharp decrease in the skin friction coefficient Cf results in
a deeper local minimum. Subsequently, there is a steeper and delayed rise to secondary local maximum,
resulting in fully attached flow downstream of the bump. It is notable that the threshold initially proposed
103

−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
0
1
2
3
4
5
σBL
Scale 1
Scale 10
Scale 100
Averaged
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
0.00
0.02
0.04
0.06
0.08
0.10
0.12
hki
(a) (b)
Figure 6.2: Streamwise profiles of the sensor value (a) and time-filtered kinetic energy (b) from WMLES
of the ReL = 106
case performed using the sensor developed by Bodart & Larsson with different filtering
scales: 1 (solid purple), 10 (dotted purple), 100 (dashed purple), and time-averaged (dash-dotted purple).
for σBL is found unsuitable for predicting relaminarization in this specific flow configuration. In contrast,
Mettu & Subbareddy’s approach tends to overestimate Cf near the peak due to an excessive application
of the equilibrium wall model, and fails to capture the secondary peak around x/L = 0.05.
To gain further insight into the reasons behind the poor performance of both sensors in this particular
flow type involving relaminarization and retransition, the impact of filtering characteristic time scale in
the estimation of the turbulence kinetic energy is evaluated in Figure 6.2. As the time scale increases, the
spanwise-averaged sensor value increases, approaching its time-averaged (rather than temporally-filtered)
counterpart. This dependency comes from a high sensitivity in the estimation of the local turbulence
kinetic energy to the chosen time scale, as shown in Figure 6.2(b). The use of other flow-dependent sensor
thresholds and time-filtering scales might improve the predictions for these sensors.
In contrast to the large sensitivity of the filtered turbulence kinetic energy to the filtering time scale
observed for σBL and σMS, a reduced sensitivity is found for the quantities entering the definition of
the acceleration sensor, σA. This sensitivity is assessed in Figure 6.3 by examining the instantaneous
streamwise profiles of the σA sensor, the time-filtered friction velocity, ⟨uτ ⟩, and time-filtered pressure
gradient,

∂p/∂s
, across various filtering time scales. For each case, the filtering time scale is defined as
104

−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
−0.04
−0.02
0.00
0.02
0.04
σAcc
γ ≈ 0.01
γ ≈ 0.1
γ ≈ 1.0
−0.4 −0.3 −0.2 −0.1 0.0
x/L
−0.04
−0.02
0.00
0.02
0.04
σAcc
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
0.01
0.02
0.03
0.04
0.05
0.06
hu
τ i
−0.50 −0.25 0.00 0.25 0.50 0.75 1.00
x/L
−10
−5
0
5
10
15
h∂p∂si
(a) (b)
(c) (d)
Figure 6.3: Streamwise profiles of the sensor value (a) and zoomed in the relaminarization region (b), timefiltered friction velocity (c) and time-filtered pressure gradient (d) from WMLES of the ReL = 106
case
performed using the acceleration sensor with different filtering time scales, Tf = γL/U∞, with γ ≈ 0.01
(solid yellow), 0.1 (dotted red) and 1.0 (dashed green).
105

Tf = γL/U∞. For short filtering time scales (γ ≈ 0.01), increased fluctuations are observed in the sensor
value; however, no shift in mean values is observed with filtering scale, unlike sensors that rely on estimates
of the turbulence kinetic energy (Bodart & Larsson, 2012; Mettu & Subbareddy, 2018). Consequently, the
terms utilized in the acceleration sensor demonstrate significantly lower sensitivity to variations in timefiltering scales for the flow considered in this application. Although not shown, similar conclusions are
found for the relaminarization sensor.
106

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Asset Metadata

Creator Xu, Naili (author)

Core Title A novel sensor-based blended wall-modeling approach for large-eddy simulation of flows with relaminarization

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School Andrew and Erna Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Degree Conferral Date 2025-05

Publication Date 04/07/2025

Defense Date 03/10/2025

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

Tag OAI-PMH Harvest,relaminarization,retransition,turbulent boundary layers,wall-modeled large eddy simulation

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Advisor Bermejo-Moreno, Ivan (committee chair), Luhar, Mitul (committee member), Spedding, Geoffrey (committee member), Jovanovic, Mihailo (committee member)

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Abstract (if available)

Abstract Wall-bounded flows undergoing pressure-driven relaminarization pose challenges for wall-modeled large-eddy simulations using equilibrium wall models. To improve predictions in regions of relaminarization and retransition, this thesis introduces a sensor-based blended wall-modeling approach, incorporating sensors that detect departures from equilibrium, along with a hysteresis timescale to account for delayed boundary layer response to pressure gradient effects. A blending strategy dynamically combines an equilibrium wall model with either a no-slip boundary condition or a laminar wall model based on local sensor values.

The proposed modeling approach is tested on two flow configurations: a Gaussian-shaped speed bump at Reynolds numbers based on the bump length Re_L=1e6, 2e6, and 4e6, and the NASA wall-mounted hump at a momentum-thickness Reynolds number Re_θ≈7000. The freestream Mach number is M=0.2 in both configurations. The speed bump case is validated against data from experiments and direct numerical simulations, demonstrating that the proposed sensors correctly identify the relaminarization present for ReL=106, and the blended wall model improves predictions for all Reynolds numbers.

For the NASA hump, comparisons with experimental and wall-resolved large-eddy simulation data explore the sensitivity of predictions to the wall-model exchange height (h_wm) and grid resolution. A lower h_wm in the favorable pressure gradient region improves relaminarization predictions, while reducing h_wm in the separated flow region enhances skin friction accuracy. The impact of the delay time modeling parameter is also examined in both configurations.