# Numerical simulations of attachment-line boundary layer in hypersonic flow,
Part I: roughness-induced subcritical transitions

Youcheng  Xi1<EMAIL_ADDRESS>Bowen  Yan2  Guangwen Yang2  Xinguo
Sha3  Dehua  Zhu3  Song  Fu1  1School of Aerospace Engineering, Tsinghua
University, 100084 Beijing, China 2Institute of High Performance Computing,
Department of Computer Science and Technology, Tsinghua University, 100084
Beijing, China 3China Academy of Aerospace Aerodynamics, 100074 Beijing, China

###### Abstract

The attachment-line boundary layer is critical in hypersonic flows because of
its significant impact on heat transfer and aerodynamic performance. In this
study, high-fidelity numerical simulations are conducted to analyze the
subcritical roughness-induced laminar-turbulent transition at the leading-edge
attachment-line boundary layer of a blunt swept body under hypersonic
conditions. This simulation represents a significant advancement by
successfully reproducing the complete leading-edge contamination process
induced by surface roughness elements in a realistic configuration, thereby
providing previously unattainable insights. Two roughness elements of
different heights are examined. For the lower-height roughness element,
additional unsteady perturbations are required to trigger a transition in the
wake, suggesting that the flow field around the roughness element acts as a
disturbance amplifier for upstream perturbations. Conversely, a higher
roughness element can independently induce the transition. A low-frequency
absolute instability is detected behind the roughness, leading to the
formation of streaks. The secondary instabilities of these streaks are
identified as the direct cause of the final transition.

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##  1 Introduction

The subcritical transition of leading-edge boundary layer near the attachment
line of swept wings plays an important role in aerodynamic, which means the
boundary layer may undergo transition to turbulence below the critical
Reynolds number predicted by linear stability theory(LST). This phonomenon is
especially critical because turbulent flow that starts at the leading edge of
a swept wing can propagate downstream, affecting extensive regions of the
wing’s chord and compromising its overall aerodynamic performance.

As the actual flow is three-dimensional in nature, to simplify the problem, it
is common to employ the swept Hiemenz boundary layer past a flat
plate(Rosenhead1963; Schlichting2017) as an approximation model for the actual
three-dimensional boundary layer(Poll1979; Hall1984; Theofilis1998;
Theofilis2003) around the leading edge. Based on this model, the LST performed
by Hall1984 gives a linear critical Reynolds number of $Re_{crit}\approx
583.1$, which is in good agreement with the previous experimental finding
(Poll1979) as well as the numerical simulation of Spalart1988. However, in
many experimental tests(Gaster1967; Poll1979; Arnal1997), transitions are
often observed at a significantly lower value $Re_{tr}\approx 250$, if the
boundary layer is subject to sufficiently large external disturbulences. In
order to understand the discrepancy between linear stability results and
experimental findings, finite amplitude perturbations and nonlinear processes
have to be taken into account. Obrist2012 and John2014; John2016 carried out
direct numerical simulations on a swept Hiemenz boundary layer with a pair of
stationary counter-rotating streamwise vortex-like structures with finite
amplitude. A bypass transition scenario has been identified, which can explain
the occurrence of subcritical transition in experiments. The initial pair of
stationary counter-rotating vortex-like structures lead to the transient
growth of streaks according to the lift-up effect, and then the damped primary
vortices and streaks interacts with unsteady secondary perturbations, causing
secondary instabilities and leading to the final transition to turbulence.

However, the aforementioned conclusions are based on simplified model in
incompressible flow only. When compressible effects (such as Mach number,
shock waves, wall temperature, etc.) are taken into account, the problem
becomes significantly more complex. Based on previous studies(Theofilis2006;
LiFei2008; Mack2008; Xi2021a; Xi2021b; Fedorov2022), for large sweep Mach
numbers, the attachment-line mode is inviscid in nature, while for lower sweep
Mach numbers, the attachment-line instability exhibits the behaviours of
viscous Tollmien–Schlichting waves. Detailed reviews for these research have
been included in our previous studies(Xi2021a; Xi2021b) and the connection
between the linear stability features of the flow and the issues discussed in
this study is not particularly direct. Therefore, we will not elaborate on
them here.

In fact, experimental investigations of high-speed attachment-line flow date
back to 1959. Initially, Beckwith1959 focused on the effects of sweep angles
and heat flux along the attachment line in supersonic conditions. They
detected the transition of attachment-line flow in their Mach 4.15
experiments, studying the effect of sweep angles over a relatively wide range.
Later, Creel1986 and Chen1991 conducted experiments with a free-stream Mach
number of 3.5 and various sweep angles, also detecting transition along the
attachment line and finding transition Reynolds numbers around 650 (based on
the boundary layer length scale at the leading edge). Skuratov1991 performed
similar tests to validate Creel et al.’s results. Murakami1996 studied
hypersonic attachment-line flow in a Ludwieg-tube wind tunnel. In some
conditions, the bypass scenario is the most possible reasons for the
transition. During the experiments, without the end plates and trip wires, the
attachment-line boundary layer can keep laminar along the entire attachment
line.

\begin{overpic}[width=238.49231pt]{Exp.pdf} \end{overpic} Figure 1: Infrared
measurements of the temperature distribution along the leading edge of the
swept blunt body. Dots indicate the positions of the pressure sensors; pink
represents high-temperature regions, while blue indicates low-temperature
regions.

Recently, experimental tests are performed over a swept blunt leading edge,
with a swept angle of $45^{o}$, in the FD-07 Mach 6.0 hypersonic wind tunnel
of the China Academy of Aerospace Aerodynamics. During the experiment, despite
high levels of external perturbations, in some cases, the attachment-line
boundary layer remained laminar. When pressure sensors are mounted at the
attachment-line position on the leading edge of a swept blunt body model, the
surface of the model is no longer smooth. Due to unavoidable installation
errors during the experiment, effective roughness elements, such as small
protrusions or depressions, form along the attachment line. Experiments with
this configuration have shown that disturbances induced by these roughness
elements can effectively trigger the transition of the attachment-line
boundary layer to turbulence, as shown in figure 1.

Previous studies indicate that the phenomenon of subcritical transition is
highly significant in the context of attachment-line flows. However, most of
these studies have been limited to incompressible flows or confined to linear
analysis, leaving a significant gap in the understanding of subcritical
compressible flows. Furthermore, even in incompressible flow scenarios,
existing computational analyses have often employed simplified models or
introduced artificial disturbances to facilitate numerical studies, raising
questions about their validity under actual conditions. Therefore, it is
imperative to conduct numerical investigations of the three-dimensional
boundary layer at the leading edge of compressible blunt bodies under relative
realistic conditions. In this study, we perform numerical simulations of
transitional high-speed attachment-line boundary layers that develop from
finite amplitude initial disturbances caused by roughness element. These
simulations correspond to experimental investigations of roughness-induced
transition over a real blunt configuration, without assuming an infinite span.
Unlike typical transitional studies, we have calculated the complete
transition to turbulence over a real configuration. Our primary aim is to
investigate the physical mechanisms of transition induced by roughness
elements in three-dimensional attachment-line boundary layers at the leading
edge. This paper is organized as follows. In section 2, the governing
equations are introduced as well as the details for numerical simulations. The
results for transitional three-dimensional boundary layers are presented in
section 3 and the conclusions and some discussions are given in section
LABEL:sec4.

##  2 Methodology

###  2.1 Governing equations

The governing equations for all simulations in this work are the dimensionless
compressible Navier–Stokes(NS) equations for a Newtonian fluid, which can be
written as:

$\frac{\partial Q}{\partial t}+\frac{\partial F_{j}}{\partial
x_{j}}+\frac{\partial F_{j}^{v}}{\partial x_{j}}=0,$ (1)
$Q=\left[\rho,{\rho{u_{1}}},{\rho{u_{2}}},{\rho{u_{3}}},{{E_{t}}}\right]^{T},$
(2) ${F_{j}}=\left[{\begin{array}[]{c}{\rho{u_{j}}}\\\
{\rho{u_{1}}{u_{j}}+p{\delta_{1j}}}\\\ {\rho{u_{2}}{u_{j}}+p{\delta_{2j}}}\\\
{\rho{u_{3}}{u_{j}}+p{\delta_{3j}}}\\\
{\left({{E_{t}}+p}\right){u_{j}}}\end{array}}\right],{F_{j}^{v}}=\left[{\begin{array}[]{c}0\\\
{{\tau_{1j}}}\\\ {{\tau_{2j}}}\\\ {{\tau_{3j}}}\\\
{{\tau_{jk}}{u_{k}}-{q_{j}}}\end{array}}\right].$ (3)

Throughout this work the coordinates $x_{i},(i=1,2,3)$ are referred to as
$x,y,z$, respectively, with corresponding velocity components
$u_{1}=u,u_{2}=v,u_{3}=w$. $F_{j}$ and $F_{j}^{v}$ stand for the inviscid and
viscous flux. The total energy $E_{t}$ and the viscous stress $\tau_{ij}$ are
given as, respectively,

$\displaystyle E_{t}$
$\displaystyle=\rho\left(\frac{T}{\gamma(\gamma-1)M^{2}_{\infty}}+\frac{u_{k}u_{k}}{2}\right),$
(4) $\displaystyle\tau_{ij}$
$\displaystyle=\frac{\mu}{Re_{\infty}}\left(\frac{\partial u_{i}}{\partial
x_{j}}+\frac{\partial u_{j}}{\partial
x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial u_{k}}{\partial x_{k}}\right).$

The pressure $p$ and heat flux $q_{i}$ are obtained from:

$p=\frac{\rho T}{\gamma M_{\infty}^{2}},\quad
q_{i}=-\frac{\mu}{(\gamma-1)M_{\infty}^{2}Re_{\infty}Pr}\frac{\partial
T}{\partial x_{i}}.$ (5)

The viscosity is calculated using the Sutherland law

$\mu=T^{3/2}\frac{T_{\infty}+C}{T\cdot T_{\infty}+C},$ (6)

with $C=110.4K$. The free-stream Reynolds number $Re_{\infty}$, Mach number
$M_{\infty}$ and Prandtl number $Pr$ are defined as

$Re_{\infty}=\frac{\rho_{\infty}^{*}U_{\infty}^{*}l_{0}^{*}}{\mu_{\infty}^{*}},\quad
M_{\infty}=\frac{U_{\infty}^{*}}{\sqrt{\gamma R_{g}^{*}T_{\infty}^{*}}},\quad
Pr=0.72,$ (7)

where $\rho_{\infty}^{*}$, $U_{\infty}^{*}$, $T_{\infty}^{*}$ and
$\mu_{\infty}^{*}$ stand for the freestream density, velocity, temperature and
viscosity, respectively. $R_{g}^{*}=287\text{J}/(\text{K}\cdot\text{Kg})$
represents the gas constant and $\gamma$ stands for the ratio of specific
heat. The length scale $l_{0}^{*}$ is chosen as $1$ millimeter in this
research. The $*$ denotes dimensional flow parameters.

###  2.2 Numerical method

Two solvers have been employed in this study. The first code we use to perform
computations is the high-order finite difference code developed recently at
Tsinghua University. A shock-fitting (S-F) method (Zhong1998) is used to
compute steady hypersonic viscous flow together with the high-order accurate
non-compact finite differences methods. The 5th-order upwind scheme (for
inviscid flux $F_{j}$) and the 6th-order centre scheme (for viscous flux
$F_{vj}$) are used to compute the flow field. A 4th-order Runge-Kutta method
is applied for the time integration, and the simulations are performed until
the maximum residual reaches a small value on the order of $10^{-15}$. A full
implicit scheme can also be used for fast convergence. Validations of the code
and some applications for calorically perfect gas and thermal-chemical non-
equilibrium flow can be found in our previous studies(Xi2021a; Xi2021b;
Chen2022). The solver is used mainly to determing the location of the leading
shock and give a high quality initial field.

The second code, used in this study, is a well-validated fluid dynamic shock
capture (S-C) solver OPENCFD, developed by Li2008, which is mainly used to
simulate the whole transition/turbulent processes. The code has been validated
and verified in previous studies(Li2008; Liang2010; Li2010). For three-
dimensional calculations presented in this study, a hybrid high-order finite
difference scheme, including the seventh-order upwind scheme, fifth-order and
seventh-order WENO schemes(Jiang1996), together with a shock sensor(Dang2022)
is used for the inviscid flux in the characteristic form. Based on that
formular, during the calculation, more than 98% of the regions use the linear
seventh-order upwind scheme, only a few regions corresponding to
discontinuities use the nonlinear WENO schemes, which greatly increase the
calculation efficiency. A standard sixth-order central difference scheme is
used for viscous flux. Time advancement is achieved by the explicit third-
order total variation diminishing Runge-Kutta scheme.

###  2.3 Models

\begin{overpic}[width=433.62pt]{Schematics.pdf} \end{overpic} Figure 2:
Schematics of the swept blunt leading edge used for numerical simulations.

The computational model comes from recent experimental tests in the FD-07 Mach
6.0 hypersonic wind tunnel of the China Academy of Aerospace Aerodynamics. The
experimental model is the front part of a delta wing with a swept angle
$\Lambda$ of 45 degrees. The thickness of the wing is $2L_{4}=40$. The
spanwise length along the attachment line is $425\text{mm}$. An asymptotic
state can be reached at around half the position of the model. The front part
of the wing is polished and can be seen as a plate. In this study, we have
established a coordinate system, as in figure 2 wherein the $z$-axis aligns
with the leading edge of the swept blunt model, coinciding with the attachment
line and extending in the corresponding spanwise direction. The normal
direction on the corresponding attachment line and swept blunt body is defined
as the $y$-axis. Finally, the $x$-axis is defined to complete the typical
Cartesian coordinate system in conjunction with these two axes. As usual, a
body fitted coordinate $(\xi,\eta,z)$ is also established with the same
spanwise direction as the Cartesian coordinate system, while the $\xi$-axis is
defined along the chordwise direction and the $\eta$-axis is defined along the
surface normal directions.

Based on that geometry, a computational model is designed as in figure 2. The
computational model can be likened to a sandwich-like configuration, where the
top and bottom layers consist of semicircles with a radius of $R_{1}=17.5$,
and the intermediate layer is a flat plate with a width of $5\text{mm}$.
Together, these three layers form the complete swept blunt body configuration.
The roughness elements is located at $z=L_{2}=40$, at the center of the
leading plate. The radius of the roughness is $R_{2}=2$. The length of the
whole model is designed as $L_{1}=200$. Based on experiments, the surface
temperature is set to $T_{w}^{*}=370\text{K}$ , other relative flow parameters
are listed in table 1.

Flow conditions | $M_{\infty}$ | $Re_{\infty}$ | $T_{\infty}^{*}$ | $T_{w}^{*}$ | $\Lambda$ | $\gamma$
---|---|---|---|---|---|---
| 6.0 | $1.8\times 10^{4}$ |  $56.58$K |  $370$K | $45^{o}$ | 1.4
Parameters for roughness elements | $S_{r}$ | $k_{h}$ | $k_{h}/\delta_{bl}^{*}$ | $d$ | $Re_{kk}$ | $N_{k}$
case H0100 | 1.0 | 0.1mm | $\approx 0.5$ | 4mm | $\approx 678$ | $\approx 87$
case H0200 | 1.0 | 0.2mm | $\approx 1$ | 4mm | $\approx 2776$ | $\approx 125$
Table 1: Basic parameters for flow and roughness at basic grid. $N_{k}$ is the
number of wall normal points for $0\leqslant y\leqslant k_{h}$.
$\delta_{bl}^{*}=0.2\text{mm}$ is the thickness of the laminar boundary layer
at the attachment-line boundary layer.

As previous analysis around the attachment-line boundary layers, we define the
sweep Mach number $M_{s}$ and the sweep Reynolds number $Re_{s}$ as

$Re_{s}=\frac{w^{*}_{\infty}\delta_{al}^{*}}{\nu^{*}_{r}}\approx
714,M_{s}=\frac{w^{*}_{\infty}}{c^{*}_{s}}\approx 2$ (8)

based on the length scale $\delta_{al}^{*}=\sqrt{\nu^{*}_{r}{\partial
u^{*}_{e}}/{\partial x^{*}}}$ at exact attachment line $x^{*}=0$ and the
variables outside of the attachment-line boundary layer. $c^{*}_{s}$ is the
sound speed after the leading shock, $\nu^{*}_{r}$ stands for the kinematic
viscosity at recovery temperature $T^{*}_{r}\approx 433K$ and the temperature
at the edge of the leading attachment-line boundary layer is
$T_{at,e}^{*}\approx 260K$. The value ${\partial u^{*}_{e}}/{\partial x^{*}}$
at exact attachment line $x^{*}=0$ is not known a priori for the present case,
the potential flow around a circular cylinder with equivalent radius $L_{4}$
is thus used to evaluate the derivative. By using the linear stability theory
over two dimensional domains, the neutral surface of the most dangerous
discrete mode are presented in figure 3, over a $Re_{s}-M_{s}-\beta$
coordinate. Detailed settings and calculations can be found in many previous
studies(Theofilis2006; Gennaro2013; Xi2021a). Here, $\beta$ is the normal
spanwise wave numbers. The red line in figure 3 indicates the case we focus on
in this study. It is found that the critical Reynolds number increases with
increasing sweep Mach number, which indicates that the leading viscous mode
(Görtler-Hämmerlin mode) is supressed by the compressible effects. Also, it is
clearly shown that the present case (the red line) is locate at the stable or
subcritical region, which means that the transition to turbulence at the
present case is not triggered by a linear instability.

\begin{overpic}[width=390.25534pt]{NeutralSurface.png}
\put(50.0,56.0){Unstable} \put(84.0,36.0){Stable} \end{overpic} Figure 3: The
neutral surface of the most dangerous discrete temporal mode over the
$Re_{s}-M_{s}-\beta$ plane. The growth rate space is divided into stable and
unstable regions by the neutral surface.

###  2.4 Roughness elements

\begin{overpic}[scale={0.126}]{3D_Roughness.png} \put(2.0,85.0){$(a)$}
\end{overpic}

\begin{overpic}[scale={0.131}]{3D_Roughness2.png} \put(2.0,95.0){$(b)$}
\put(50.0,49.0){$x$} \put(5.0,75.0){$y$} \put(50.0,-1.0){$z$}
\put(5.0,25.0){$y$} \end{overpic}

Figure 4: $(a)$ The grid distributions arount the roughness with the roughness
height $k_{h}=0.1$mm in full resolution. $(b)$ The shape of roughness in two
cross sections

The roughness element, shown in figure 4, designed to simulate a pressure
sensor with a circular disk configuration during the experiment, is
characterized by the function expressed in polar coordinates $(r,\phi)$, with
the shape of the shoulders being defined by a hyperbolic tangent function in
similar ways as in previous studies(Kurz2014; Kurz2016). The function is
defined as

$h(r,\phi)=\frac{k_{h}}{2}+\frac{k_{h}}{2}\tanh\left[\frac{S_{r}}{k_{h}}\left(\frac{d}{2}-r\right)\right],$
(9)

with $k_{h}$ and $d$ being the height and diameter of the roughness. The slope
factor $S_{r}$ is set to 1.0 for all cases in the present study. In general,
the center of the roughness is locate at the points $(x_{c},z_{c})=(0,40)$,
the diameter is $d=2R_{2}=4$.

Another important parameter for roughness induced transition is the Roughness
reynolds number $Re_{kk}$, characterised based on the height $(k_{h})$ and the
velocity $(w)$ in the undisturbed laminar flow with respect to the position of
the roughness. This roughness reynolds number is defined as a function of

$Re_{kk}=\frac{\rho(k_{h})w(k_{h})k_{h}}{\mu(k_{h})},$ (10)

and listed in table 1 based on the laminar boundary layer.

###  2.5 Simulation strategy

\begin{overpic}[width=433.62pt]{CalculationProcess.pdf} \put(5.0,52.0){$(a)$
step 1} \put(25.0,52.0){$(b)$ step 2} \put(45.0,52.0){$(c)$ step 3}
\end{overpic} Figure 5: The outline of calculation processes. S-F: Shock
Fitting, S-C: Shock Capture

The calculation process for this kind of problem is divided into three steps,
with two kinds of compressible solvers(Li2008; Xi2021b) as shown in figure 5.
Assuming the incoming flow has reached an asymptotic state, a two-dimensional
calculation with infinity span assumption $(\partial/\partial z=0)$ is
performed first using a shock-fitting solver. With the exact location of the
shock revealed, alignment and clustering of the mesh along the bow shockwave
can be easily achieved for the following shock-capture calculation. To
diminish the numerical perturbations between the two solvers, a three-
dimensional domain is further designed for pre-calculation with a periodic
boundary condition along the spanwise direction, with the newly built grid and
the initial field from the fitting solver. When the calculation is converged,
the solution from the middle slice of the periodic three-dimensional domain is
used for the fully three-dimensional calculation. As the boundary layer
develops along the attachment line and the chordwise direction, non-reflection
outlet boundary conditions are used further downstream along the attachment
line and the chordwise direction. Away from the surface, as the shock is
embeded in the computational domain, freestream boundary conditions are used
at the outside.

To closely mimic the conditions observed during experimental tests, the
generation of unsteady perturbations is implemented in two distinct phases. In
the first phase, random velocity perturbations, with maximum amplitude
constituting approximately $2\%$ of the free-stream velocity, are introduced
upstream of the leading shock waves. This procedure aims to replicate the
perturbations measured in wind tunnel experiments. In the second phase, to
simulate the disturbances inherent to upstream boundary layers along the
attachment line, random wall normal blowing and suction are executed via a
hole on the wall. These disturbances also possess an maximum amplitude of
roughly $2\%$ of the free-stream velocity. The specified hole is positioned at
coordinates $(z_{c},x_{c})=(30,0)$ and defined with a radius of $2$.

In the computational analyses conducted within the scope of this study, two
distinct cases were examined. In the first scenario, characterized by a
roughness element height of 0.1, unsteady perturbations were deliberately
introduced to facilitate the onset of transition. Otherwise, the transition
would not occur within the wake flow induced by the roughness elements.
Conversely, the scenario involving a roughness element height of 0.2,
presented a fundamentally different dynamic. The inherent absolute instability
associated with this configuration led to a spontaneous disruption of flow
symmetry. This natural progression towards asymmetry effectively initiates the
transition process, obviating the need for the introduction of external
perturbations.

###  2.6 Computational grids

####  2.6.1 Grid distributions for shock-fitting solver

The basic grid number used for shock-fitting simulations is $N_{\xi}\times
N_{\eta}=801\times 401$. The distribution of the grid points in the wall
normal direction is controlled through a function that provides clustering
towards the wall, with two parameters $h_{im}$ and $\sigma_{s}$. The
distribution function, which maps $\eta$ to $h$, can be expressed as

$\displaystyle h=H_{shk}\frac{a_{y}(1+Y)}{(b_{y}-Y)},b_{y}=1+2a_{y},\quad
a_{y}=\frac{h_{im}}{1-2h_{im}},Y=2\frac{\left[1-\tanh{(\sigma_{s})}\right]\frac{1+\eta}{2}}{1-\tanh{\left(\sigma_{s}\frac{1+\eta}{2}\right)}}-1,$
(11)

where $H_{shk}$ is the local shock height and is solved as a dependent
variable with the flow field in shock-fitting methods. $\eta$ is a uniform
grid distribution along the region $\left[-1.0,1.0\right]$, $h$ is the actual
wall normal grid distributions. The values of $h_{im}$ and $\sigma_{s}$ are
chosen to be $0.3$ and $0.95$ for the fitting simulations presented in this
paper. Along the chordwise direction, at the wall surface, the surface grid
$s(x,y)$ is clustered at the round head with the function

$\frac{s}{S}=\frac{a_{\xi}(1-\xi)}{b_{\xi}-\xi},b_{\xi}=1+2a_{\xi},a_{\xi}=\frac{s_{im}}{1-2s_{im}},$
(12)

where $s_{im}$ is chosen to be $0.2$ for the simulations based on shock-
fitting methods, $s$ is the local surface curve length along the model
surface, and $S$ is the total model surface curve length. Note that the
surface grid does not change during the calculation.

####  2.6.2 Grid distributions for shock-capture solver

The basic grid numbers used for shock-capture simulations are $N_{\xi}\times
N_{\eta}\times N_{z}=801\times 401\times 8$ and $N_{\xi}\times N_{\eta}\times
N_{z}=2401\times 401\times 4401$. More grids points with $N_{\xi}\times
N_{\eta}\times N_{z}=3001\times 601\times 6601$ are used to validate the
independence of the solutions, as shown in figure 7. Also, the independence of
the solutions have be further verificated by the Reynold stress as well as the
balance of the kinetic energy in the full developed turbulent region. Unlike
the fitting solver, for the capture solver, the wall normal grid needs
clustering towards the wall as well as the shock region. Therefore, the wall
normal grid is divided into three parts. The first part is at the region
$\left[0,0.105H_{shk}\right]$, where $H_{shk}$ is the local shock height. The
second part is a transition region which connects the near-wall region and the
shock region and is located at the region
$\left[0.105H_{shk},0.945H_{shk}\right]$. The final region stands for the part
used to capture the shock and is located at the region
$\left[0.945H_{shk},1.05H_{shk}\right]$. As the local shock height $H_{shk}$
is solved by the fitting solver in advance, the grid lines are adapted well to
both the body and the shock shape. Unlike the usual calculations of the
transitional/turbulent boundary layer, the shock regions in the present
simulation have been taken into account with enough accuracy.

\begin{overpic}[width=260.17464pt]{gridNormal.eps} \put(1.0,62.0){$(a)$}
\put(1.0,32.0){$(b)$} \end{overpic} Figure 6: Basic grid distributions with
401 grid points along the wall normal directions. $(a)$ The scaled grid
distributions. $(b)$ The scaled grid spacing along the wall normal direction.

For basic grid, the first region has 201 grid points, with the distributions
$h_{1}$ as

$h_{1}=\frac{21}{1600}\frac{1+\eta_{1}}{3.5-\eta_{1}}H_{shk},$ (13)

where $\eta_{1}$ is the uniform distributions in region $\left[-1,1\right]$.
The third region has 101 grid points with a uniform distribution. The second
region has 101 grid points and is used to link grids with different grid
spacings using a Hermite function obtained by imposing C3 continuity of the
resulting stretching function. The grid distributions and relative grid
spacing is shown in figure 6. For the step 2 calculation of the periodic box,
8 grids are used to cover a spanwise region of length $20$. For the final full
three-dimensional calculation, 4401 grids are distributed uniformly along the
spanwise direction. The roughness element and unsteady hole are captured with
101 grids points.

By using the above approach, for the basic grid numbers, around the
attachment-line boundary layer, $125$ grid points are located inside the
boundary layer(based on laminar boundary layer thickness) in the wall normal
$\eta$-axis direction. About $230$ grid points are located at the leading-edge
plate along the $x$-axis. As the boundary layers develop along the chordwise
direction as well, $136$ grid points are located inside the boundary layer at
the chordwise outlet boundary. Detailed grid numbers that are used to resolve
roughness are shown in table 1. Comparing with the existing
researches(Mayer2011; Tullio2013; Tullio2015; Groskopf2016; Giovanni2018;
Hader2019; Shrestha2019) on numerical simulations of high-speed boundary layer
transitions, these grid numbers and distributions can reveal the detailed
behaviour of linear and nonlinear waves. To further identify the possible
features of the turbulence at the further downstreams, grids points along the
wall normal directions are increased to 601 in the final simulations presented
in this paper. The grid sizes in wall units in the turbulent region are shown
and listed in Table 2. Based on that grid size, as there are no additional
stress and heat flux terms in the present simulations, resolutions for an
implicit large-eddy simulations (ILES) can be recognized for the possible
turbulent region.

\begin{overpic}[width=411.93767pt]{i1201_GridIndependentStudy.png}
\end{overpic} Figure 7: Regions of flow separations in a cross-cut plane
through the centre of a roughness for case H0100, marked by contours of
$w=-0.001$, for basic and high resolutions. The axis is stretched for clarity.
\begin{overpic}[width=238.49231pt]{inner_scales_grids.eps} \end{overpic}
---
| $\xi$ | $\eta$ | $\zeta$ | Total points | $\Delta\xi_{max}^{+}$ | $\Delta\eta_{max}^{+}$ | $\Delta\zeta_{max}^{+}$
---|---|---|---|---|---|---
2401 | 601 | 4401 | $6.4\times 10^{9}$ | 8.5 | 0.4 | 18.2
Table 2: Grid points and maximum grid sizes in wall units at the turbulent
boundary region.

##  3 Results

###  3.1 Verification of the asymptotic assumption

In the current study, as the three-dimensional boundary layer develops along
the spanwise direction, the commonly employed asymptotic assumption
($\partial/\partial z=0$) for attachment-line and crossflow analyses can be
verified by examining the profiles upstream of the roughness. Taking the case
H0100 as an example, the profiles of major variables such as density $\rho$,
spanwise velocity $w$, and temperature $T$ in the attachment-line region
$x\in[-2.5,2.5]$ are depicted in figure 8 and 9. A perfect alignment of these
profiles is observed, which confirms two critical points. Firstly, the exact
correspondence of results obtained from the S-C and S-F solvers corroborates
the precision of the solvers utilized in this research. Secondly, and more
significantly, because the agreements of the profiles at inlet and at
$z=18.2\text{mm}$, the asymptotic assumptions are still applicable to a
spatially-developing attachment-line boundary layer.

\begin{overpic}[width=433.62pt]{RhoWT_AtX_0.eps} \put(-2.0,45.0){$(a)$}
\put(28.0,45.0){$(b)$} \put(52.0,45.0){$(c)$} \put(76.0,45.0){$(d)$}
\end{overpic} Figure 8: Profiles of major variables at $x=0$ plane, for case
H0100.  \begin{overpic}[width=433.62pt]{RhoWT_AtX_025.eps}
\put(-2.0,45.0){$(a)$} \put(28.0,45.0){$(b)$} \put(52.0,45.0){$(c)$}
\put(76.0,45.0){$(d)$} \end{overpic} Figure 9: Profiles of major variables at
$x=2.5$ plane, for case H0100.

Upon broadening our perspective further, it becomes apparent that even in
regions where crossflow effects are pronounced, this assumption remains valid,
as shown in figure 10. It is imperative to emphasize, however, that the
upstream flow has reached an asymptotic state, and that this assumption is
applicable to fully developed laminar boundary layers, provided that the
spanwise extent is sufficiently large.

\begin{overpic}[width=433.62pt]{RhoWT_CrossRegion_Compare.eps}
\put(-2.0,43.0){$(a)$} \put(28.0,43.0){$(b)$} \put(52.0,43.0){$(c)$}
\put(76.0,43.0){$(d)$} \end{overpic} Figure 10: Profiles of major variables at
three locations ($30^{o},60^{o},90^{o}$ ) over cylinder surface, for case
H0100.

We further compare the profiles (at $x=2.5$), a little bit away from the
attachment line at $x=0$, as presented in figure 11. As the flow develops
along the chordwise direction, the basic feature of the variables are kept the
same, but the boundary layer has become slightly thinner.

\begin{overpic}[width=433.62pt]{RhoWT_AtX_Compare.eps} \put(-2.0,45.0){$(a)$}
\put(28.0,45.0){$(b)$} \put(52.0,45.0){$(c)$} \put(76.0,45.0){$(d)$}
\end{overpic} Figure 11: Comparison of variables at $x=0$ and $x=2.5$ planes,
for case H0100.

###  3.2 General features of the flow fields

\begin{overpic}[width=433.62pt]{CalculationResults1_H0100s.pdf} \end{overpic}
Figure 12: Instantaneous iso-surface of $\lambda_{2}=-0.035$, colour indicates
$w$, for the first part of case H0100.
\begin{overpic}[width=433.62pt]{CalculationResults2_H0100s.pdf} \end{overpic}
Figure 13: Instantaneous iso-surface of $\lambda_{2}=-0.035$, colour indicates
$w$, for the second part of case H0100.
\begin{overpic}[width=433.62pt]{CalculationResults1_H0200s.pdf} \end{overpic}
Figure 14: Instantaneous iso-surface of $\lambda_{2}=-0.035$, colour indicates
$w$, for the case H0200.

The general features of the whole flow fields are shown in figure 12 and
figure 13, with the iso-surfaces of $\lambda_{2}=-0.035$ transient fields. The
iso-surfaces are colored with spanwise velocity $w$. The whole flow fields can
be divided into three parts. The first part is the roughness region, in which
the initial laminar flow is perturbed by the surface deformation. Typical
vortex structures are formed behind the roughness. The breakdowns of the
vortex structures lead to typical turbulent structures, along the attachment
line, and small vortexes structures are shown. The second part is the
transitional region, in which the initial turbulences at the attachment line
develop along the spanwise direction as well as the chordwise direction. At
the very beginning, the turbulences are located around the attachment line and
the turbulent region expands along the chordwise direction slightly. Then, as
the flows develop further downstream, the turbulent structures are flushed
from attachment-line region to chordwise outlet. The final part is the fully
turbulent region, where the fully developed turbulences cover the whole region
of the leading-blunt body, ranging from attachment line to chordwise outflow.

When the height of the roughness element is increased from 0.1 to 0.2, there
are significant differences in the vortex structures formed behind the
roughness, as shown in figure 12 and 14. These differences can be seen more
directly in the contours of surface average heat fluxes $\theta_{tw}$ and skin
frictions $\overline{\tau}_{w}$, as depicted in figures 15 and 16. These
metrics essentially serve as ”footprints” of the boundary layer dynamics,
providing insights into the complex interactions and flow structures present
within the boundary layers. Here, as the usual boundary layers in previously,
we define the velocity $\overline{u}^{+}$, based on inner scale as

$\left.\begin{aligned}
h^{+}&=\frac{\overline{\rho}_{w}\overline{u}_{\tau}h}{\overline{\mu}_{w}},\quad\overline{u}^{+}=\frac{|\overline{u}_{p}|}{\overline{u}_{\tau}},\quad|\overline{u}_{p}|=\sqrt{\overline{u}_{\xi}^{2}+\overline{w}^{2}},\\\
\overline{u}_{\tau}&=\sqrt{\frac{\overline{\tau}_{w}}{\overline{\rho}_{w}}},\quad\overline{\tau}_{w}=\frac{\overline{\mu}}{Re}\left.\frac{\partial\overline{u}_{p}}{\partial
h}\right|_{h=0},\end{aligned}\right\\},$ (14)

where, $\overline{u}_{p}$ is the velocity parallel to the surface. The skin-
friction coefficient $C_{f}$ and surface heat-flux ${\theta}_{tw}$ for this
kind of flow are defined as

$C_{f}=\frac{2\overline{\mu}_{w}^{*}}{\rho_{\infty}^{*}U_{\infty}^{*2}}=\frac{2\overline{\mu}_{w}}{Re}\frac{\partial\overline{u}_{p}}{\partial
h}=2\overline{\tau}_{w},{\theta}_{tw}=-\left|\kappa\nabla\overline{T}\cdot\bm{n}\right|.$
(15)

The derivatives of surface normals, denoted as ${\partial}/{\partial h}$, for
arbitrary variables $f_{\psi}$, are determined through a two-step process.
Initially, the gradients of the variables $f_{\psi}$ are computed utilizing
the identical scheme adopted for the calculation of viscous fluxes during the
simulations. Subsequently, the derivatives of the surface normals ${\partial
f_{\psi}}/{\partial h}$ are obtained by projecting the calculated gradients
$\nabla f_{\psi}$ onto the surface normal vectors $\bm{n}$.

\begin{overpic}[width=208.13574pt]{SurfaceHeatFlux_H01.png} \put(1.0,30.0){$(a)$} \end{overpic} | \begin{overpic}[width=208.13574pt]{SurfaceHeatFlux_H02.png} \put(1.0,30.0){$(b)$} \end{overpic}
---|---
Figure 15: The surface heat fluxes ${\theta}_{tw}$ distributions for two cases. $(a)$ for H0100, $(b)$ for H0200. \begin{overpic}[width=208.13574pt]{SurfaceSkinFriction_H01.png} \put(1.0,30.0){$(a)$} \end{overpic} | \begin{overpic}[width=208.13574pt]{SurfaceSkinFriction_H02.png} \put(1.0,30.0){$(b)$} \end{overpic}
---|---
Figure 16: The surface skin friction $\overline{\tau}_{w}$ distributions for
two cases. $(a)$ for H0100, $(b)$ for H0200.

The magnitude contour of average density gradients for case H0200, at the
attachment-line plane, is depicted in figure LABEL:H02_Average_SliceX0. This
illustration provides a comprehensive view of the general flow field
characteristics for both cases. The presence of surface roughness induces a
shock slightly ahead of the roughness. As this shock evolves away from the
surface and progresses downstream, it shapes into a curved shock surface under
the influence of the incoming flow. The interaction of this induced shock with
the leading shock of the blunt body, followed by its reflection back into the
boundary layer downstream, results in a noticeable deformation of the leading
shock. Subsequent to the roughness-induced shock, the compressed fluids
undergo expansion and acceleration, leading to the formation of a
recompression shock at the roughness’s tail along the $z-$direction.
Meanwhile, a shear layer develops behind the roughness, and the recompression
shock once again impinges on the leading shock, reflecting back into the
boundary layers. When the flow evolves further downstream, the high-shear
region at the outside of the boundary layer becomes much weak, as reflected as
the decrease of magnitudes for the density gradient $\left|\nabla\rho\right|$.

\begin{overpic}[width=433.62pt]{X_slice0.png} \put(1.0,45.0){$$}\end{overpic}
\@@toccaption{{\lx@tag[ ]{{17}}{Density gradient magnitude contours of the
case H0200, at attachment-line plane $x=0$. The red line stands for the
computational domain.}}}\@@caption{{\lx@tag[: ]{{Figure 17}}{Density gradient
magnitude contours of the case H0200, at attachment-line plane $x=0$. The red
line stands for the computational domain.}}} \end{figure} \par The figure
\ref{Average_W_CenterLine} and \ref{Average_T_CenterLine} show the
distributions of mean velocity and temperature along the attachment line in
the wall-normal direction. Additionally, the size and specific location of the
corresponding separation bubbles are indicated by blue lines in the figures.
It is evident from the figure that the separation bubbles induced by small
roughness elements are lower in height compared to those induced by large
roughness elements, but extend farther downstream. Conversely, the separation
bubbles induced by larger roughness elements extend farther upstream. The
corresponding velocity and temperature profiles highlight the approximate
location of the shear layer and illustrate the process through which low-speed
fluid, due to the lift-up effect, is elevated away from the wall.
\par\begin{figure}\centering\begin{overpic}[width=433.62pt]{H01_Figure_MeanW.eps}
\put(-2.0,22.0){$(a)$} \end{overpic}
\begin{overpic}[width=433.62pt]{H02_Figure_MeanW.eps} \put(-2.0,22.0){$(b)$}
\end{overpic} \@@toccaption{{\lx@tag[ ]{{18}}{Line plots of average spanwise
velocity $\overline{w}$ around the roughness for $(a)$ H0100 and $(b)$ H0200
cases. The red and blue lines stand for the wall surfaces and seperation
bubbles, respectively.}}}\@@caption{{\lx@tag[: ]{{Figure 18}}{Line plots of
average spanwise velocity $\overline{w}$ around the roughness for $(a)$ H0100
and $(b)$ H0200 cases. The red and blue lines stand for the wall surfaces and
seperation bubbles, respectively.}}} \@add@centering\end{figure}
\par\begin{figure}\centering\begin{overpic}[width=433.62pt]{H01_Figure_MeanT.eps}
\put(-2.0,22.0){$(a)$} \end{overpic}
\begin{overpic}[width=433.62pt]{H02_Figure_MeanT.eps} \put(-2.0,18.0){$(b)$}
\end{overpic} \@@toccaption{{\lx@tag[ ]{{19}}{Line plots of average
Temperature $(\overline{T}-T_{w})/(T_{\infty}-T_{w})$ at the attachment line
around the roughness for $(a)$ H0100 and $(b)$ H0200 cases. The red and blue
lines stand for the wall surfaces and seperation bubbles,
respectively.}}}\@@caption{{\lx@tag[: ]{{Figure 19}}{Line plots of average
Temperature $(\overline{T}-T_{w})/(T_{\infty}-T_{w})$ at the attachment line
around the roughness for $(a)$ H0100 and $(b)$ H0200 cases. The red and blue
lines stand for the wall surfaces and seperation bubbles, respectively.}}}
\@add@centering\end{figure}
\par\begin{figure}\centering\begin{tabular}[]{cc}\begin{overpic}[width=208.13574pt]{H01_Seperation.png}
\put(1.0,42.0){$(a)$}
\end{overpic}&\begin{overpic}[width=208.13574pt]{H02_Seperation.png}
\put(1.0,42.0){$(b)$} \end{overpic}\\\
\begin{overpic}[width=208.13574pt]{H01_StreamLines_Tec.png}
\put(1.0,76.0){$(c)$}
\end{overpic}&\begin{overpic}[width=208.13574pt]{H02_StreamLines_Tec.png}
\put(1.0,76.0){$(d)$} \end{overpic}\\\
\begin{overpic}[width=208.13574pt]{Spatial_SL_H01.png} \put(1.0,52.0){$(e)$}
\end{overpic}&\begin{overpic}[width=208.13574pt]{Spatial_SL_H02.png}
\put(1.0,52.0){$(f)$} \end{overpic}\end{tabular} \@@toccaption{{\lx@tag[
]{{20}}{$(a)$ and $(b)$ stand for the iso-surface of spanwise velocity
$\overline{w}=0$ for H0100 and H0200 cases, respectively. $(c)$ and $(d)$
represent the limiting streamlines along the surfaces for H0100 and H0200
cases, respectively. $(e)$ and $(f)$ stand for the spatial streamlines around
the roughness for H0100 and H0200 cases, respectively. The colors of the lines
are used to distinguish the different height of the streamlines, upstream. The
seeds of the white lines are locate at $h=0.02$, while those of the light blue
lines are locate at $h=0.05$.}}}\@@caption{{\lx@tag[: ]{{Figure 20}}{$(a)$ and
$(b)$ stand for the iso-surface of spanwise velocity $\overline{w}=0$ for
H0100 and H0200 cases, respectively. $(c)$ and $(d)$ represent the limiting
streamlines along the surfaces for H0100 and H0200 cases, respectively. $(e)$
and $(f)$ stand for the spatial streamlines around the roughness for H0100 and
H0200 cases, respectively. The colors of the lines are used to distinguish the
different height of the streamlines, upstream. The seeds of the white lines
are locate at $h=0.02$, while those of the light blue lines are locate at
$h=0.05$.}}} \@add@centering\end{figure} \par Since the flow field induced by
roughness elements is inherently a complex three-dimensional flow,
characteristics along the attachment line alone are insufficient to adequately
reflect the corresponding flow features. To capture the fundamental three-
dimensional nature of the flow, we present the distribution of limiting
streamlines near the roughness elements and the iso-surfaces (Figure
\ref{Average_3D_Features}) where the average spanwise velocity $\overline{w}$
is zero, which are used to characterize the three-dimensional flow properties.
Upstream of the roughness elements, a distinct recirculation region is
observed. For lower roughness elements, this recirculation region forms a
relatively complete separation bubble, as evidenced by the upstream
recirculation streamlines. For higher roughness elements, in addition to a
more pronounced primary separation region, a secondary separation line is
clearly visible in the limiting streamlines near the roughness elements. As
the flow progresses downstream, different degrees of flow convergence and
vortex characteristics appear behind the roughness elements. The downstream
streamlines indicate that the wake of the lower roughness element quickly
returns to a more orderly state, which is also reflected by the absence of
distinct vortex structures in the wake. The corresponding spatial streamlines
in figure \ref{Average_3D_Features}$(e)$ become very orderly, with no
noticeable twists or entanglements. In contrast, the higher roughness element
exhibits completely different characteristics, with the formation of a pair of
counter-rotating vortices in the wake, clearly reflected in the corresponding
separation lines on the wall in the limiting streamlines in
\ref{Average_3D_Features}$(d)$. Also, noticeable twists or entanglements of
the spatial streamlines can be found in \ref{Average_3D_Features}$(f)$.
\par\par\@@numbered@section{subsection}{toc}{Mechanisms of the roughness
induced transition} \par\par\@@numbered@section{subsubsection}{toc}{One-point
statistics of the transitional flow field} In this section, we try to
understand the transition mechanisms of these flows by using long time one-
point spectra statistics. At the beginning, all sampled points are located at
the attachment-line plane($x=0$). Figure \ref{H01_Slices_TM} and
\ref{H02_Slices_TM} show the flow fields at the exact attachment-line plane
for the two cases, together with the statistical results of the selected
points. The regions of the revease flow near the roughness are pointed by the
white lines. As mentioned previous, the two distinct cases deliver two
different dynamics with respect to the transition processes. In the first
scenario, case H0100, the transition would not occur within the wake flow
induced by the roughness elements, without the unsteady perturbations, even
though the roughness Reynolds number $Re_{kk}$ or the height of the roughness
$k_{h}$ are beyound the critical values in normal plate boundary
layers\cite[citep]{(\@@bibref{AuthorsPhrase1Year}{Bernardini2012,Bernardini2014,Estruch2017}{\@@citephrase{,
}}{})}. Contrariwise, as the height of roughness element increases to 0.2, the
transition will occur without employing forcing of any kind, which suggest a
self-sustaining mechanism that causes the flow to transition.
\par\begin{figure}\centering\begin{overpic}[width=433.62pt]{H01_Middle_Slice_Density_TargetPoints.pdf}
\put(-3.0,54.0){$(a)$} \put(-3.0,33.0){$(b)$} \put(-3.0,16.0){$(c)$}
\end{overpic} \@@toccaption{{\lx@tag[ ]{{21}}{$(a)$ The density contour at the
attachment-line plane $x=0$ for case H0100. $(b)$ The temporal evolutions of
instantaneous chordwise velocity perturbations $u^{\prime}$ of the selected
points. $(c)$ The amplitudes for different Fourier modes $|F(u^{\prime})|$ for
different frequency at the selected points.}}}\@@caption{{\lx@tag[: ]{{Figure
21}}{$(a)$ The density contour at the attachment-line plane $x=0$ for case
H0100. $(b)$ The temporal evolutions of instantaneous chordwise velocity
perturbations $u^{\prime}$ of the selected points. $(c)$ The amplitudes for
different Fourier modes $|F(u^{\prime})|$ for different frequency at the
selected points.}}} \@add@centering\end{figure} \par This is also reflected in
the corresponding measurement point signals of $u^{\prime}$. In H0100 case, we
introduced random perturbations, resulting in the signal detected at point
$P_{1}$ exhibiting typical broadband characteristics, with energy distributed
relatively uniformly across a range of frequencies. Additionally, there is a
slight increase in perturbation amplitude at the frequency around 100 KHz. As
the flow continues to develop downstream from $P_{2}$ to $P_{5}$, the
perturbations gradually increase, and disturbances of the frequency around 100
KHz become the dominant perturbations. This leads to the frequency amplitude
distribution characteristics evolving towards typical turbulent features.
\par\begin{figure}\centering\begin{overpic}[width=433.62pt]{H02_Middle_Slice_Density_TargetPoints.pdf}
\put(-3.0,54.0){$(a)$} \put(-3.0,33.0){$(b)$} \put(-3.0,16.0){$(c)$}
\end{overpic} \@@toccaption{{\lx@tag[ ]{{22}}{$(a)$ The density contour at the
attachment-line plane $x=0$ for case H0200. $(b)$ The temporal evolutions of
instantaneous chordwise velocity perturbations $u^{\prime}$ of the selected
points. $(c)$ The amplitudes for different Fourier modes $|F(u^{\prime})|$ for
different frequency at the selected points.}}}\@@caption{{\lx@tag[: ]{{Figure
22}}{$(a)$ The density contour at the attachment-line plane $x=0$ for case
H0200. $(b)$ The temporal evolutions of instantaneous chordwise velocity
perturbations $u^{\prime}$ of the selected points. $(c)$ The amplitudes for
different Fourier modes $|F(u^{\prime})|$ for different frequency at the
selected points.}}} \@add@centering\end{figure} \par When the height of the
roughness element increases to 0.2 mm, prominent instability waves are
observed at points $P_{1},P_{2}$ and $P_{3}$. The figures indicate that the
disturbances, which are amplified as they travel downstream from $P_{1}$ to
$P_{3}$, not only grow in a convective manner but also exhibit characteristics
of absolute instability, as they are amplified over time at fixed locations.
Based on the results of the discrete Fourier transformation, the frequency of
the most representative perturbations is around 10 KHz, referred to as low-
frequency perturbations in this study. As these perturbations evolve
downstream, some high-frequency components are gradually amplified. This
amplification is evidenced by the increasing amplitude of disturbances in the
region beyond 400 KHz. Simultaneously, the low-frequency perturbations appear
to reach a saturation state as they evolve downstream, with their amplitude
showing minimal growth. This is evidenced by the nearly constant amplitude of
low-frequency disturbances from points $P_{2}$ to $P_{4}$ in the figures. As
the flow continues to develop further downstream to point $P_{5}$, the overall
disturbance spectrum exhibits typical broadband characteristics which
indicates that the flow are stepping into full turbulent. \par To better
understand the mechanisms and identify possible nonlinear coupling features
during the transition processes, higher-order spectral (HOS) analysis is
employed. Specifically, bispectral analysis is utilized here to examine
nonlinear signals. This method enables the detection and quantification of
possible nonlinear interactions between different frequency components of the
signals. The bispectrum $\mathscr{B}(\omega_{1},\omega_{2})$ for a signal
$f(t)$ is defined as
\begin{equation}\mathscr{B}(\omega_{1},\omega_{2})=E\left[F(\omega_{1})F(\omega_{2})F^{c}(\omega_{1}+\omega_{2})\right],\end{equation}
where $F(\omega)$ is the Fourier transform of the temporal signal $f(t)$ and
$\omega$ is the frequency. $E[.]$ stands for an expected value. The
superscript $c$ represents the complex conjugate.
\par\begin{figure}\centering\begin{tabular}[]{ccc}\begin{overpic}[width=138.76157pt]{Bispectrum/H0100/P1_Points.pdf}
\put(18.0,75.0){$(a)$}
\end{overpic}&\begin{overpic}[width=138.76157pt]{Bispectrum/H0100/P2_Points.pdf}
\put(18.0,75.0){$(b)$}
\end{overpic}&\begin{overpic}[width=138.76157pt]{Bispectrum/H0100/P3_Points.pdf}
\put(18.0,75.0){$(c)$} \end{overpic}\end{tabular}
\begin{tabular}[]{cc}\begin{overpic}[width=138.76157pt]{Bispectrum/H0100/P4_Points.pdf}
\put(18.0,75.0){$(c)$}
\end{overpic}&\begin{overpic}[width=138.76157pt]{Bispectrum/H0100/P5_Points.pdf}
\put(18.0,75.0){$(d)$} \end{overpic}\end{tabular} \@@toccaption{{\lx@tag[
]{{23}}{The normalized bispectrum $|\mathscr{B}|$ of the perturbations
$u^{\prime}$ at the points $P_{1}$ to $P_{5}$ for the H0100 case. Panels $(a)$
to $(d)$ correspond to points $P_{1}$ to $P_{5}$,
respectively.}}}\@@caption{{\lx@tag[: ]{{Figure 23}}{The normalized bispectrum
$|\mathscr{B}|$ of the perturbations $u^{\prime}$ at the points $P_{1}$ to
$P_{5}$ for the H0100 case. Panels $(a)$ to $(d)$ correspond to points $P_{1}$
to $P_{5}$, respectively.}}} \@add@centering\end{figure}
\par\begin{figure}\centering\begin{tabular}[]{ccc}\begin{overpic}[width=138.76157pt]{Bispectrum/H0200/P1_Points.pdf}
\put(18.0,75.0){$(a)$}
\end{overpic}&\begin{overpic}[width=138.76157pt]{Bispectrum/H0200/P2_Points.pdf}
\put(18.0,75.0){$(b)$}
\end{overpic}&\begin{overpic}[width=138.76157pt]{Bispectrum/H0200/P3_Points.pdf}
\put(18.0,75.0){$(c)$} \end{overpic}\end{tabular}
\begin{tabular}[]{cc}\begin{overpic}[width=138.76157pt]{Bispectrum/H0200/P4_Points.pdf}
\put(18.0,75.0){$(c)$}
\end{overpic}&\begin{overpic}[width=138.76157pt]{Bispectrum/H0200/P5_Points.pdf}
\put(18.0,75.0){$(d)$} \end{overpic}\end{tabular} \@@toccaption{{\lx@tag[
]{{24}}{The normalized bispectrum $|\mathscr{B}|$ of the perturbations
$u^{\prime}$ at the points $P_{1}$ to $P_{5}$ for the H0200 case. Panels $(a)$
to $(d)$ correspond to points $P_{1}$ to $P_{5}$, respectively. Note that
certain features have been magnified for enhanced readability and
clarity.}}}\@@caption{{\lx@tag[: ]{{Figure 24}}{The normalized bispectrum
$|\mathscr{B}|$ of the perturbations $u^{\prime}$ at the points $P_{1}$ to
$P_{5}$ for the H0200 case. Panels $(a)$ to $(d)$ correspond to points $P_{1}$
to $P_{5}$, respectively. Note that certain features have been magnified for
enhanced readability and clarity.}}} \@add@centering\end{figure} \par The
bispectrum of selected points in figure \ref{H01_Slices_TM} and
\ref{H02_Slices_TM} are shown in figure \ref{H01_Bispectrum_TM} and
\ref{H02_Bispectrum_TM}, respectively. For both cases, the most representative
interactions between two waves with different frequencies are shown. Based on
the definition of bispectrum \eqref{eq_bispectrum}, it measures the nonlinear
interactions between frequencies $\omega_{1}$ and $\omega_{2}$, as well as
their sum $\omega_{1}+\omega_{2}$. For a purely linear signal, the bispectrum
theoretically should be zero or very close to zero. For general nonlinear
signal, the diagonal elements (if $\omega_{1}=\omega_{2}=\omega_{0}$) of the
bispectrum reflect the interactions between a frequency $\omega_{0}$ with
itself and its double frequency $2\omega_{0}$. Significant values along the
diagonal often indicate the presence of harmonic components in the signal. The
off-diagonal elements (if $\omega_{1}\neq\omega_{2}$) illustrate the nonlinear
interactions between distinct frequencies $\omega_{1}$ and $\omega_{2}$, and
their sum frequency $\omega_{1}+\omega_{2}$. The obvious values off the
diagonal suggest the nonlinear coupled phenomena between different frequency
components. \par The basic behaviour of the perturbations, shown in figure
\ref{H01_Bispectrum_TM} and \ref{H02_Bispectrum_TM}, are the same as described
before. For the H0100 case, the results indicate that regions with larger
magnitudes of low-frequency disturbances are primarily located on or near the
diagonal. This suggests that in this condition, the excitation of higher-order
harmonics plays a significant role in the nonlinear evolution of the
corresponding disturbances. As the disturbances propagate downstream, the
spectral distribution of disturbances at points $P_{2}$ through $P_{4}$
appears to remain relatively unchanged. This indicates that the composition of
the disturbances remains nearly constant, suggesting that the disturbances
have grown to a certain extent and have reached a nearly saturation stage.
\par For the H0200 case, the bispectrum exhibits characteristics that are
markedly different from the previous condition. At the first three signal
recording points ($P_{1}$, $P_{2}$, and $P_{3}$), the dominant disturbances in
the overall disturbance spectrum appear only in the low-frequency region,
consistent with the results from the previous power spectrum analysis. As the
disturbances further develop to point $P_{4}$, some high-frequency
disturbances begin to emerge, roughly within the
$\left[400\text{KHz},800\text{KHz}\right]$ range. These high-frequency
disturbances exhibit significant nonlinear interactions with the low-frequency
disturbances as well as the zero-frequency disturbances, with the frequencies
of the low-frequency disturbances remaining consistent with those recorded at
the previous three points. This suggests that the low-frequency disturbances
have evolved sufficiently, and the detected high-frequency disturbances
correspond to a secondary instability arising in the already saturated low-
frequency disturbances combined with the mean field.
\par\begin{figure}\centering\begin{tabular}[]{c}\begin{overpic}[width=424.94574pt]{Select_Surface_PointsH01.png}
\put(1.0,55.0){$(a)$} \put(14.0,9.0){\vector(-1,-1){4.0}}
\put(5.0,2.0){$P_{s_{1},h_{1}}$} \put(79.0,47.0){\vector(2,-3){3.0}}
\put(80.0,38.0){$P_{s_{14},h_{1}}$} \put(20.5,19.0){\vector(-1,1){4.0}}
\put(12.0,25.0){$P_{s_{15},h_{1}}$} \put(69.0,48.0){\vector(-1,1){3.0}}
\put(60.0,53.0){$P_{s_{24},h_{1}}$} \put(23.5,13.8){\vector(1,-2){3.0}}
\put(28.0,5.0){$P_{s_{4},h_{1}}$} \put(32.5,19.5){\vector(1,-1){5.0}}
\put(40.0,11.0){$P_{s_{7},h_{1}}$} \end{overpic}\\\
\begin{overpic}[width=424.94574pt]{Select_Surface_PointsH02.png}
\put(1.0,55.0){$(b)$} \put(23.5,23.0){\vector(-1,1){3.0}}
\put(15.0,28.0){$P_{s_{25},h_{2}}$} \put(65.0,46.0){\vector(-1,1){5.0}}
\put(55.0,52.0){$P_{s_{32},h_{2}}$} \end{overpic}\end{tabular}
\@@toccaption{{\lx@tag[ ]{{25}}{The selected points for two cases. $(a)$ for
H0100, $(b)$ for H0200. The corresponding points are sequentially recorded as
$s_{1},s_{2},\cdots,s_{32}$ along the $z$-axis from upstream to downstream,
starting from the attachment line to chordwise downstream. The subscripts
$h_{1}$ and $h_{2}$ are used to distinguish the points in different
cases.}}}\@@caption{{\lx@tag[: ]{{Figure 25}}{The selected points for two
cases. $(a)$ for H0100, $(b)$ for H0200. The corresponding points are
sequentially recorded as $s_{1},s_{2},\cdots,s_{32}$ along the $z$-axis from
upstream to downstream, starting from the attachment line to chordwise
downstream. The subscripts $h_{1}$ and $h_{2}$ are used to distinguish the
points in different cases.}}} \@add@centering\end{figure}
\par\begin{figure}\centering\begin{tabular}[]{c}\begin{overpic}[width=286.19078pt]{H01_Surface_Points.eps}
\put(8.0,47.0){$(a)$} \put(63.0,47.0){$(b)$} \end{overpic}\\\
\begin{overpic}[width=429.28616pt]{H02_Surface_Points.eps}
\put(8.0,32.0){$(c)$} \put(42.0,32.0){$(d)$} \put(76.0,32.0){$(e)$}
\end{overpic}\end{tabular} \@@toccaption{{\lx@tag[ ]{{26}}{The spectra
$E_{\rho^{\prime}}$ of perturbations density $\rho$ at the selected points in
figure \ref{Srf_SF_Select_Points}. $(a)$ and $(b)$ stand for the two groups of
selected points in H0100 case. $(c-e)$ represent the three groups of selected
points in H0200 case. From light to dark blue dashed lines, the spectra
represent points from spanwise upstream to downstream.}}}\@@caption{{\lx@tag[:
]{{Figure 26}}{The spectra $E_{\rho^{\prime}}$ of perturbations density $\rho$
at the selected points in figure \ref{Srf_SF_Select_Points}. $(a)$ and $(b)$
stand for the two groups of selected points in H0100 case. $(c-e)$ represent
the three groups of selected points in H0200 case. From light to dark blue
dashed lines, the spectra represent points from spanwise upstream to
downstream.}}} \@add@centering\end{figure} \par To provide a more
comprehensive understanding of the disturbance evolution process during
transition, we shift our focus slightly away from the wake of roughness
elements at the attachment-line region to analyze the characteristics of
disturbance evolution around the entire roughness element, including the
upstream and horseshoe vortex regions. In order to trace the evolution
characteristics of disturbances in the three-dimensional flow field, we select
and record the fluctuating density on the three-dimensional surfaces around
the roughness elements for analysis. The selected points for H0100 and H0200
cases are shown in figure \ref{Srf_SF_Select_Points} together with the surface
skin friction. The selected points can roughly be divided into three groups.
The first group is located along the exact attachment line, extending from
upstream to downstream of the roughness element, labeled sequentially as
$s_{1},s_{2},\cdots,s_{14}$. The second group is located on the side of the
roughness element, also extending from upstream to downstream, labeled as
$s_{15},s_{16},\cdots,s_{24}$. The third group, labeled as
$s_{25},s_{26},\cdots,s_{32}$, is also on the side of the roughness element
but is further from the second group of detection points. The first two groups
are the same for both cases, while the third group appears only in the H0200
case to track the evolution of the corresponding horseshoe vortex.
\par\par\par The spectra of those points are shown in figure
\ref{Srf_SF_Select_Points_Spectra}. In the scenarios of lower height roughness
element, the incoming flow ahead of the roughness is subjected to the
additional perturbations. Based on the spectra(the red lines in figure
\ref{Srf_SF_Select_Points_Spectra}$(a)$ and $(b)$), no dominant frequency
could be observed. As the flow gradually approaches the roughness element, the
overall amplitude of disturbances increases progressively (as indicated by the
red dashed line in Figure \ref{Srf_SF_Select_Points_Spectra}$(a)$). After the
fluid passes over the roughness element, the disturbances are somewhat
suppressed due to the favourable pressure gradient resulting from the
expansion effects of the high-pressure region induced by the shock wave ahead
of the roughness. This suppression leads to a decrease in the amplitude of the
disturbances (as indicated by the black dashed line in Figure
\ref{Srf_SF_Select_Points_Spectra}$(a)$). However, once the flow moves past
the roughness element, the disturbances exhibit a marked tendency to increase
again. It is important to note that the frequency range of disturbances that
first starts to grow significantly is approximately 100 KHz, which is
consistent with previous analyses. As for the vortex structures formed on both
sides of the roughness element, they also exhibit similar patterns of change
(as indicated in Figure \ref{Srf_SF_Select_Points_Spectra}$(b)$). However, the
corresponding amplitudes have not grown significantly large (i.e., a distinct
plateau region appears in the mid-to-low frequency range), which correlates
with the previous observation that no significant transition phenomena were
observed on both sides. \par In scenarios involving a higher roughness
element, different phenomena are observed (as shown in Figures
\ref{Srf_SF_Select_Points_Spectra}$(c-e)$). Due to the absence of additionally
introduced artificial disturbances under this condition, the amplitude of
disturbances upstream of the roughness element is relatively low, especially
in the mid-to-high frequency range. As the disturbances approach the roughness
element, the amplitude of low-frequency components is generally amplified.
Additionally, within the separation bubbles both upstream and downstream of
the roughness element, some peaks appear in the high-frequency range
(indicated by the red dashed line and black dashed line, where the red dashed
line represents the upstream separation bubble and the black dashed line
represents the downstream separation bubble), reflecting the high-frequency
characteristics of the separation bubbles. Similar to the lower roughness
element, disturbances exhibit a certain degree of suppression after passing
over the roughness element (the amplitude corresponding to the black dashed
line is relatively low). As the flow further moves downstream, disturbances in
the high-frequency range gradually increase, resulting in some peaks (these
peaks correspond to the high-frequency range $[400KHz,800KHz]$ identified in
previous analyses), eventually leading to fully developed disturbances and the
transition to turbulent flow. On the side stimulated by the horseshoe vortex
generated by the higher roughness element, similar characteristics are
observed, as shown in Figure \ref{Srf_SF_Select_Points_Spectra}$(e)$. As
indicated by the previous analysis of wall heat flux and skin friction, there
exists a relatively 'quiet' zone between the two transition peaks (as shown in
Figure \ref{Srf_SF_Select_Points_Spectra}$(d)$). The corresponding disturbance
amplitude in this region is smaller compared to the regions on either side.
The disturbances in this zone only begin to grow once the disturbances on both
sides have fully developed. \par\begin{figure}\begin{center}
\begin{tabular}[]{cc}\begin{overpic}[width=208.13574pt]{CSE1_H01.pdf}
\put(-3.0,74.0){$(a)$}
\end{overpic}&\begin{overpic}[width=208.13574pt]{CSE2_H01.pdf}
\put(-3.0,74.0){$(b)$} \end{overpic}\end{tabular} \end{center}
\@@toccaption{{\lx@tag[ ]{{27}}{The spanwise evolutions of perturbations
$\rho^{\prime}$ with different frequencies, around the roughness element for
case H0100. $(a)$ and $(b)$ represent the perturbations along the surface
lines of the first and second groups of selected points, respectively. The
roughness is locate at the region $[38,42]$.}}}\@@caption{{\lx@tag[: ]{{Figure
27}}{The spanwise evolutions of perturbations $\rho^{\prime}$ with different
frequencies, around the roughness element for case H0100. $(a)$ and $(b)$
represent the perturbations along the surface lines of the first and second
groups of selected points, respectively. The roughness is locate at the region
$[38,42]$.}}} \end{figure} \par\begin{figure}\begin{center}
\begin{tabular}[]{cc}\begin{overpic}[width=208.13574pt]{CSE1_H02.pdf}
\put(-2.0,74.0){$(a)$}
\end{overpic}&\begin{overpic}[width=208.13574pt]{CSE2_H02.pdf}
\put(-2.0,74.0){$(b)$} \end{overpic}\end{tabular}
\begin{overpic}[width=208.13574pt]{CSE3_H02.pdf} \put(-2.0,74.0){$(c)$}
\end{overpic} \end{center} \@@toccaption{{\lx@tag[ ]{{28}}{The spanwise
evolutions of perturbations $\rho^{\prime}$ with different frequencies, around
the roughness element for case H0200. $(a),(b)$ and $(c)$ represent the
perturbations along the surface lines of the first, second and third groups of
selected points, respectively. The roughness is locate at the region
$[38,42]$.}}}\@@caption{{\lx@tag[: ]{{Figure 28}}{The spanwise evolutions of
perturbations $\rho^{\prime}$ with different frequencies, around the roughness
element for case H0200. $(a),(b)$ and $(c)$ represent the perturbations along
the surface lines of the first, second and third groups of selected points,
respectively. The roughness is locate at the region $[38,42]$.}}} \end{figure}
\par The evolution of amplitude along the spanwise direction for different
frequency disturbances over the wall surface under two operating conditions is
illustrated in the figure \ref{SpPRho_H01} and \ref{SpPRho_H02}. In the
figures, all resolved frequencies are represented by semi-transparent grey
lines, and the evolution processes of disturbances at several typical
frequencies are prominently marked. These images provide a clearer depiction
of the disturbance behavior described in our previous discussions. \par Based
on our existing data, it can be roughly determined that the triggering of
transition downstream of smaller roughness elements is initially caused by
disturbances around 100 KHz. These disturbances originate from the interaction
between incoming flow disturbances and the separation bubbles at the leading
and trailing edges induced by the roughness elements. Subsequently,
disturbances corresponding to a frequency of around 200 KHz (second harmonic)
grow rapidly, ultimately leading to turbulence formation. For higher roughness
elements, the separation bubble at the trailing edge of the roughness element
induces a distinct dominant frequency (Kelvin-Helmholtz instability of the
shear layer). This dominant frequency, in the wake region, leads to the
formation of corresponding low-frequency streaks. The secondary instability of
these streaks further induces the generation of high-frequency secondary
unstable waves, with components in the $[400KHz,800KHz]$ range directly
growing to saturation, causing transition. Additionally, the horse-shoe vortex
structures induced by the leading-edge separation bubble and upstream flow for
higher roughness elements develop downstream, forming corresponding sider
streak structures. The secondary instability of these streaks is also one of
the reasons for inducing transition. The high-frequency disturbance components
corresponding to the leading-edge separation bubble (located between
$[1MHz,2MHz]$) also grow rapidly downstream. These higher frequency
disturbances and the secondary instability of the streaks collectively induce
the early occurrence of side-edge transition.
\par\par\@@numbered@section{subsubsection}{toc}{Mode decompositions of the
transitional flow fields} In this section, we aim to reveal the basic
structures for later stage transition of these flows using two- and three-
dimensional mode decompositions. Given the extensive computational grid
involved, documenting every instantaneous signal throughout the entire flow
domain is virtually impossible. Therefore, to analyze specific flow
characteristics, we strategically focused on capturing variable signals within
targeted regions to reveal the featured flow structures. Even though regarding
the analysis of three-dimensional flow fields, it is important to acknowledge
that the significant disk storage requirements for time-sequential data make
it impractical to achieve the same level of precision across a broad frequency
range(especially the low frequency region) as that obtained from one-point
statistics analysis. Therefore, we intend to primarily utilize modal
decomposition analysis to investigate the potential occurrence of high-
frequency disturbances and the three-dimensional structural characteristics of
later stage transition. The sub-block regions shown in figure \ref{SubBlocks}
are used to record the instantaneous signal of basic variables. The
resolutions $(Ns_{\xi}\times Ns_{\eta}\times Ns_{\zeta})$ of the blocks for
case H0100 and H0200 are $201\times 301\times 801$ and $401\times 301\times
701$, respectively. Additionally, $600$ time samples are recorded at a time
step of $\Delta t=0.1$, covering a total of 60 basic time units.
\par\begin{figure}\centering\begin{tabular}[]{cc}\begin{overpic}[width=212.47617pt]{H01_SubBlocks.png}
\put(1.0,50.0){$(a)$}
\end{overpic}&\begin{overpic}[width=212.47617pt]{H02_SubBlocks.png}
\put(1.0,50.0){$(b)$} \end{overpic}\end{tabular} \@@toccaption{{\lx@tag[
]{{29}}{The sub-block regions used for spectra analysis. $(a)$ for H0100,
$(b)$ for H0200. The surface skin-frictions are shown for
clarity.}}}\@@caption{{\lx@tag[: ]{{Figure 29}}{The sub-block regions used for
spectra analysis. $(a)$ for H0100, $(b)$ for H0200. The surface skin-frictions
are shown for clarity.}}} \@add@centering\end{figure} \par Peoper Orthogonal
Decomposition (POD) and Dynamic Mode Decomposition
(DMD)\cite[citep]{(\@@bibref{AuthorsPhrase1Year}{Rowley2009,Schmid2010,Mezic2013,Schmid2022}{\@@citephrase{,
}}{})} are used to identify the representative 3D mode structures of the core
regions for the three-dimensional transitional boundary layers, as depicted in
figure \ref{SubBlocks}. For the completeness of the article, we briefly
introduce the corresponding processes here. Both methods are started with the
data sequence of the matrix form
\begin{equation}\bm{V}_{1}^{N}=\left[\bm{v}_{1},\bm{v}_{2},\cdots,\bm{v}_{N}\right]^{T},\end{equation}
with data samples $(\bm{v}_{i},i=1,2,\cdots,N)$ at different temporal instants
with a uniform time interval, $N$ is the number of temporal data samples and
superscript $T$ represents the transpose. In this paper, $\bm{v}_{i}$ is a
vector which can be any variable whthin the three-dimensional flow fields and
matches exactly the resolution sizes of sub-block regions. Reduced Singular
value decomposition (SVD)
\begin{equation}\bm{V}_{1}^{N}=\bm{\varPhi}\bm{\Sigma}\bm{W}^{T},\end{equation}
is the basic methods to determine the decomposition. The matrices
$\bm{\varPhi}=\left[\Phi_{1},\Phi_{2},\cdots,\Phi_{N}\right]$ and $\bm{W}$
contain the left and right singular vectors, and matrix $\bm{\Sigma}$ holds
the singular values $(\sigma_{1},\sigma_{2},\cdots,\sigma_{N})$ along its
diagonal. The columns of matrix $\bm{\varPhi}$ stands for the spatial
distributions of the POD modes, and the average energy of the modes can be
recoved by $\sigma_{i}^{2}/M_{n}$ ($M_{n}$ is the grid numbers). \par As the
present POD modes generally contain multiple frequencies, to isolate the
structure with a single characteristic frequency,
DMD\cite[citep]{(\@@bibref{AuthorsPhrase1Year}{Rowley2009,Schmid2010,Mezic2013,Schmid2022}{\@@citephrase{,
}}{})} is used to find the structure, amplitude and dynamics. In DMD, the
collected snapshots are divided into two groups $\bm{V}_{1}^{N-1}$ and
$\bm{V}_{2}^{N}$. If a linear mapping matrix $\mathcal{M}_{k}$ connects two
groups of snapshots is assumped, we have
\begin{equation}\bm{V}_{2}^{N}=\mathcal{M}_{k}\bm{V}_{1}^{N-1}.\end{equation}
Replacing the snapshot matrix $\bm{V}_{1}^{N-1}$ by the singular values and
vectors from SVD leads the relation
\begin{equation}\bm{V}_{2}^{N}=\mathcal{M}_{k}\bm{U}\bm{\Sigma}\bm{V}^{T}.\end{equation}
Then multiplying both sides by $\bm{U}^{T}$ produces
\begin{equation}\bm{U}^{T}\mathcal{M}_{k}\bm{U}=\bm{U}^{T}\bm{V}_{2}^{N}\bm{V}\bm{\Sigma}^{-1}\equiv\tilde{\bm{S}},\end{equation}
which is low-rank approximation of $\mathcal{M}_{k}$. The eigenvalues
$\bm{\mu}$ and the eigenvectors $\bm{\Phi}$ of $\mathcal{M}_{k}$ can be
approximately obtained by solving the eigenvalue problem:
\begin{equation}\tilde{\bm{S}}\bm{Y}=\bm{\mu}\bm{Y},\bm{\mu}=\text{diag}\left(\mu_{1},\mu_{2},\mu_{3},\cdots\right),\bm{Y}=\left(\bm{Y}_{1},\bm{Y}_{2},\bm{Y}_{3},\cdots\right),\end{equation}
and the eigenvectors can be expressed as
\begin{equation}\bm{\Phi}=\bm{U}\bm{Y}\end{equation} The amplitudes $\alpha$
of the DMD modes are dertermined by solving a convec optimization problem
\begin{equation}\underset{\alpha}{\operatorname{minimize}}\quad
J(\alpha):=\left\|\bm{\Sigma}\bm{V}^{T}-\bm{Y}\bm{D}_{\alpha}\bm{V}_{and}\right\|_{F}^{2},\end{equation}
where $\left\|\cdot\right\|_{F}^{2}$ stands for the Frobenius norm of the
matrix. $\bm{V}_{and}$ is the Vandermonde matrix
\begin{equation}\bm{V}_{and}=\left[\begin{array}[]{cccc}1&\mu_{1}&\cdots&\mu_{1}^{N-1}\\\
1&\mu_{2}&\cdots&\mu_{2}^{N-1}\\\ \vdots&\vdots&\ddots&\vdots\\\
1&\mu_{k}&\cdots&\mu_{k}^{N-1}\end{array}\right],\end{equation} represents the
temporal evolution of the dynamic modes. $\bm{D}_{\alpha}$ is the diagonal
matrix
\begin{equation}\bm{D}_{\alpha}=\left[\begin{array}[]{llll}\alpha_{1}&&&\\\
&\alpha_{2}&&\\\ &&\ddots&\\\ &&&\alpha_{k}\end{array}\right]\end{equation} of
the amplitude. The most important modes are selected using the sparsity
promoting algorithm of \cite[citet]{\@@bibref{Authors
Phrase1YearPhrase2}{Jov2014}{\@@citephrase{(}}{\@@citephrase{)}}}. For
specific modes $k$, its temporal growth rate and frequency can be represents
by the real and imaginary parts of ${\log{(\mu_{k})}}/{\Delta t}$,
respectively.
\par\begin{table}\centering\begin{tabular}[]{cccccccc}\hline\cr\hline\cr
Case&Observations&$Ns_{\xi}$&$Ns_{\eta}$&$Ns_{\zeta}$&Dof&No.
(snapshots)&Memory (TB)\\\ \hline\cr
H0100&$\lambda_{2}$&201&301&801&$0.48\times 10^{8}$&600&0.22\\\ \hline\cr
H0200&$\lambda_{2}$&401&301&701&$0.85\times 10^{8}$&600&0.4\\\
\hline\cr\hline\cr\end{tabular} \@@toccaption{{\lx@tag[ ]{{3}}{Basic
parameters of domains and the observations used for the decomposition. Degree
of freedom(Dof) stands for the variables numbers per snapshots. Memory
represents the memory requirements for storing all input
data.}}}\@@caption{{\lx@tag[: ]{{Table 3}}{Basic parameters of domains and the
observations used for the decomposition. Degree of freedom(Dof) stands for the
variables numbers per snapshots. Memory represents the memory requirements for
storing all input data.}}}
\<EMAIL_ADDRESS>\put(0.0,68.0){$(a)$} \put(0.0,28.0){$(b)$} \end{overpic}
\@@toccaption{{\lx@tag[ ]{{30}}{The spectrum and selected modes of DMD for
H0100 case in three-dimensional region. The $\lambda_{2}$ is used as
observation variable. $(a)$ shows the spectrum. $(b)$ shows the frquency
$\Im(\log(\mu)/\Delta t)$ and amplitude $|\alpha|$.}}}\@@caption{{\lx@tag[:
]{{Figure 30}}{The spectrum and selected modes of DMD for H0100 case in three-
dimensional region. The $\lambda_{2}$ is used as observation variable. $(a)$
shows the spectrum. $(b)$ shows the frquency $\Im(\log(\mu)/\Delta t)$ and
amplitude $|\alpha|$.}}} \@add@centering\end{figure}
\par\begin{figure}\centering\begin{overpic}[width=433.62pt]{H01_DMD_Modes_Lambda2.pdf}
\end{overpic} \@@toccaption{{\lx@tag[ ]{{31}}{Spatial structures of the
selected DMD modes with iso-surfaces of $\lambda_{2}=-5\times 10^{-5}$, for
case H0100.}}}\@@caption{{\lx@tag[: ]{{Figure 31}}{Spatial structures of the
selected DMD modes with iso-surfaces of $\lambda_{2}=-5\times 10^{-5}$, for
case H0100.}}} \@add@centering\end{figure} \par In approximating the Koopman
eigenfunctions of continuous systems associated with nonlinear NS equations
through dynamic modes, selecting appropriate variables becomes crucial for
generating significant spatio-temporal patterns. Koopman theory suggests that
good observables might better capture the dynamics of nonlinear systems.
Consequently, this study employs the $\lambda_{2}$, a variable derived from
vortex identification, serving as a dynamic indicator for structures. The
decomposition's fundamental parameters are detailed in Table
\ref{BasicParametersForDMD}. Due to the prohibitive size of the input
datasets, necessitating distributed memory high-performance computing, this
work adopts and adapts a parallelized algorithm, as described by
\cite[citet]{\@@bibref{Authors
Phrase1YearPhrase2}{Sayadi2016}{\@@citephrase{(}}{\@@citephrase{)}}}, to
facilitate DMD. \par The DMD results for case H0100 are shown in figure
\ref{DMD_Lambda2_H01_Spectrum} and \ref{DMD_Lambda2_H01_Modes}. The most
important modes are shown in the spectrum. Excluding the associated mean flow
mode, the selected dominant modes all exhibit distinct streak characteristics.
The corresponding disturbance structures are primarily distributed downstream
of the roughness elements, highlighting the feature that in the later stages
of transition and in turbulent states, higher-frequency disturbances gradually
become more prominent. Additionally, the decomposed Modes 1, 2, and 3 all
display significant disturbance characteristics near the roughness elements.
This indicates that the transition is not simply and spontaneously occurring
at a certain downstream distance from the roughness elements, but is instead
connected with the upstream disturbances near the leading-edge of the
roughness elements.
\par\begin{figure}\centering\begin{overpic}[width=433.62pt]{H02_DMD_Spectrum_Lambda2.pdf}
\put(0.0,65.0){$(a)$} \put(0.0,30.0){$(b)$} \end{overpic}
\@@toccaption{{\lx@tag[ ]{{32}}{The spectrum and selected modes of DMD for
H0200 case in three-dimensional region. The $\lambda_{2}$ is used as
observation variable. $(a)$ show the spectrum. $(b)$ shows the frquency
$\Im(\log(\mu)/\Delta t)$ and amplitude $|\alpha|$.}}}\@@caption{{\lx@tag[:
]{{Figure 32}}{The spectrum and selected modes of DMD for H0200 case in three-
dimensional region. The $\lambda_{2}$ is used as observation variable. $(a)$
show the spectrum. $(b)$ shows the frquency $\Im(\log(\mu)/\Delta t)$ and
amplitude $|\alpha|$.}}} \@add@centering\end{figure}
\par\begin{figure}\centering\begin{overpic}[width=433.62pt]{H02_DMD_Modes_Lambda2.pdf}
\end{overpic} \@@toccaption{{\lx@tag[ ]{{33}}{Spatial structures of the
selected DMD modes with iso-surfaces of $\lambda_{2}=-5\times 10^{-5}$, for
case H0200.}}}\@@caption{{\lx@tag[: ]{{Figure 33}}{Spatial structures of the
selected DMD modes with iso-surfaces of $\lambda_{2}=-5\times 10^{-5}$, for
case H0200.}}} \@add@centering\end{figure} \par The results of case H0200 are
presented in figure \ref{DMD_Lambda2_H02_Spectrum} and
\ref{DMD_Lambda2_H02_Modes}, which exhibit distinct differences compared to
case H0100. The Mode 1 exhibits typical streak structures, however, unlike
previous conditions, in addition to the streaks at the central position
(around $x=0$), there are also streak regions on either side. These lateral
regions correspond to the vortex structures formed at the edges of the larger
roughness elements, continuing their downstream development. Modes 2, 3, and 4
reflect the flow structures known as hairpin vortices in the typical
transitional boundary layer induced by roughness elements. These structures
represent the intense momentum mixing processes occurring during transition.
These modes clearly illustrate that at certain distances from the roughness
elements, the flow streaks evolving from the upstream horseshoe vortex enter a
strong nonlinear phase earlier than those along the centerline, forming
corresponding vortex structures.
\par\par\@@numbered@section{section}{toc}{Conclusions and Discussions} In
summary, this study presents simulations of the complete transition process
from laminar flow to turbulence in a hypersonic three-dimensional swept
leading-edge boundary layer over an experimental configuration. Based on the
results from linear stability theory and the experimental observations, the
subcritical state of the possible transition along the attachment line is
confirmed. Two roughnesses based on the experimental tests are modeled and
designed to trigger transitions. The main findings of this study can be
summarized as the following points \begin{itemize} \par\itemize@item@In the
flow over a swept blunt body discussed in this paper, even without the
assumption of infinite sweep, if the incoming boundary layer reaches an
asymptotic state (for laminar flow), the subsequent flow state also satisfies
the infinite sweep assumption—the boundary layer is homogeneous along the
swept direction. \par\par\itemize@item@The two different heights of roughness
elements in the configuration studied in this paper result in completely
different transition characteristics. For lower-height roughness element, the
element alone cannot directly induce the corresponding boundary layer
transition. Certain random perturbations need to be introduced during the
simulation. For higher-height roughness element, it can directly induce
boundary layer transition by themselves without the need for additional
perturbations. The flow near the roughness elements resembles that of a flat
plate boundary layer, where vortex structures are triggered in their vicinity.
In the transition phenomena induced by external disturbances and lower-height
roughness elements, the transition mainly occurs directly downstream of the
wake of the roughness elements, but the wake vortices induced by the roughness
elements do not directly destabilize and lead to the final transition. In the
transition simulation with higher-height roughness element, the horseshoe
vortices generated by the roughness elements form corresponding streaks, which
preferentially destabilize and lead to the transition to turbulence, while the
wake vortices directly behind the roughness elements destabilize and
transition further downstream. \par\par\itemize@item@One-point power spectral
analysis and bispectral analysis, together with the DMD analysis, are used to
identify the detailed transition mechanism. For the case of small roughness
element, the wake flow induced by the roughness act as a disturbance selector
and amplifier, selecting and amplifying the incoming disturbances from
upstream. This causes disturbances with frequencies around 100 KHz to
preferentially grow and lead to the final transition. Through DMD analysis, we
can infer that the high frequency instability is strongly linked to the
separation bubble upstream of the roughness elements. For larger roughness
elements, there exists a low-frequency absolute instability in the wake
induced by the roughness elements. This low-frequency disturbance, around 10
KHz, generates corresponding low-frequency streaks, and the high frequency
secondary instability of these low-frequency streaks is the primary reason for
the transition in the wake. The high-frequency disturbance components
corresponding to the leading-edge separation bubble (located between [1 MHz, 2
MHz]) also grow rapidly downstream. These higher frequency disturbances and
the secondary instability of the streaks collectively induce the early
occurrence of side-edge transition. \par\end{itemize}
\par\par\par\backsection[Supplementary data]{}
\par\backsection[Acknowledgements]{We acknowledge Yancheng MetaStone Tech. Co.
for prociding us with the computational resources required by this work.
Useful discussions with Professor Qibing Li of Tsinghua University and
Professor Jie Ren of Beijing Institute of Technology are gratefully
acknowledged.} \par\backsection[Funding]{This work received support from the
NSFC Grants 12202242, 12172195 and 12388101. The authors are also grateful for
the support from the National Key Research and Development Plan of China
through project no. 2019YFA0405201, the National Key Project GJXM92579 and the
Grants 20231001 in Supercomputing Center in Yancheng.}
\par\backsection[Declaration of interests]{The authors report no conflict of
interest.} \par\backsection[Data availability statement]{The full data set of
the simulations is of the order of 80 thousands of gigabytes. By contacting
the authors, a smaller subset can be made available.} \par\backsection[Author
ORCIDs]{ \par Youcheng Xi, https://orcid.org/0000-0002-6484-0231; \par Bowen
Yan, https://orcid.org/0009-0002-0655-9414; \par Guangwen Yang,
https://orcid.org/0000-0002-8673-8254 \par Song Fu,
https://orcid.org/0000-0003-2052-7435} \par\backsection[Author contributions]{
Youcheng Xi: Funding acquisition, Conceptualization, Data curation, Formal
analysis, Coding, Investigation, Methodology, Validation, Writing original
draft. Bowen Yan \& Guangwen YANG: Funding acquisition, Software and
optimization on high performance computing, Computational Resources. Xinguo
Sha \& Dehua Zhu: Model designing, Wind Tunnel Testing, Experimental
Investigation. Song Fu: Funding acquisition, Supervision, Resources, Writing-
review \& editing.} \par\par\par \thebibliography
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Juillen, Reneaux \& Gasparian}{Arnal {\em et~{}al.\/}
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