# Numerical simulations of attachment-line boundary layer in hypersonic flow,
Part II: the features of three-dimensional turbulent boundary layer

Youcheng  Xi1<EMAIL_ADDRESS>Bowen  Yan2  Guangwen Yang2  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

###### Abstract

In this study, we investigate the characteristics of three-dimensional
turbulent boundary layers influenced by transverse flow and pressure
gradients. Our findings reveal that even without assuming an infinite sweep, a
fully developed turbulent boundary layer over the present swept blunt body
maintains spanwise homogeneity, consistent with infinite sweep assumptions. We
critically examine the law-of-the and temperature-velocity relationships,
typically applied two-dimensional turbulent boundary layers, in three-
dimensional contexts. Results show that with transverse velocity and pressure
gradient, streamwise velocity adheres to classical velocity transformation
relationships and the predictive accuracy of classical temperature-velocity
relationships diminishes because of pressure gradient. We show that near-wall
streak structures persist and correspond with energetic structures in the
outer region, though three-dimensional effects redistribute energy to align
more with the external flow direction. Analysis of shear Reynolds stress and
mean flow shear directions reveals in near-wall regions with low transverse
flow velocity, but significant deviations at higher transverse velocities.
Introduction of transverse pressure gradients together with the transverse
velocities alter the velocity profile and mean flow shear directions, with
shear Reynolds stress experiencing similar changes but with a lag increasing
with transverse. Consistent directional alignment in outer regions suggests a
partitioned relationship between shear Reynolds stress and mean flow shear:
nonlinear in the inner region and approximately linear in the outer region.

###### keywords:

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##  1 Introduction

Boundary layer flows in practical engineering problems are generally turbulent
and three-dimensional. However, to date, most theoretical, experimental, and
numerical simulation studies on turbulent boundary layers have been confined
to two-dimensional boundary layers. These studies have provided fundamental
characteristics of turbulent boundary layers and have guided the development
of turbulence models, such as Reynolds-Averaged Navier-Stokes (RANS) models.

Although there have been some studies on three-dimensional turbulent boundary
layers, such as the experimental study of a three-dimensional supersonic
turbulent boundary layer induced by a curved fin(Konrad & Smits, 1998), the
three-dimensional shock-wave/turbulent boundary-layer interactions(Gaitonde &
Adler, 2023), the three-dimensional turbulent boundary layer over a model
lifting body(Dong et al., 2024), these investigations are mostly case-
specific, and the relationships between three-dimensional and two-dimensional
turbulent boundary layers remain unclear. Recently, we conducted simulations
of the attachment-line boundary layer of a hypersonic leading edge. We
realized that this configuration serves as a natural model to bridge two-
dimensional and three-dimensional turbulent boundary layers: In the
attachment-line region, the flow approximates a two-dimensional turbulent
boundary layer, whereas as the flow develops downstream along the chordwise
direction, away from the attachment-line region, its three-dimensional
characteristics gradually become more pronounced. This progression evolves the
flow from a two-dimensional to a three-dimensional turbulent boundary layer.
Therefore, the swept blunt body model can be considered a standardized model
for studying three-dimensional turbulent boundary layers.

This paper is the second part of our study on the numerical simulations of
attachment-line boundary layer in hypersonic flow. In the first part, the
roughness-induced subcritical transitions of attachment-line boundary layer
for a swept blunt body have been studied. 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 behind 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. Following the
transition of the boundary layer to a fully developed turbulent state, we
continued our computations further downstream. The analysis of this fully
developed turbulent boundary layer in the downstream region the core of this
study.

In this paper, we describe the general characteristics of three-dimensional
turbulent boundary layers in this configuration, focusing on the properties of
the mean flow and low-order turbulence statistics. This paper is organized as
follows. The methodology and models of this study are also presented for the
sake of completeness of the paper in section 2. The major results of the
three-dimensional turbulent boundary are shown in section 3 and the
conclusions are given in section 4.

##  2 Methodology

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 specific gas constant and $\gamma=1.4$ 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 in this section.

\begin{overpic}[width=433.62pt]{Schematics.pdf} \end{overpic} Figure 1:
Schematics of the swept blunt leading edge used for numerical simulations.

The computational model is designed based on the recent experimental test and
the schematics of the model are shown in figure 1. The computational model can
be likened to a sandwich-like configuration, where the top and bottom layers
consist of semicircles with a radius of $R_{1}=17.5\text{mm}$, 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 $z=L_{2}=40\text{mm}$, at the center of the
leading plate. The radius of the roughness is $R_{2}=2\text{mm}$. The length
of the whole model is designed as $L_{1}=200\text{mm}$. Based on experiments,
the surface temperature is set to $T_{w}^{*}=370\text{K}$. The blunt leading
edge is placed in a hypersonic flow of $M_{\infty}=6.0$, the swept angle is
$\Lambda=45^{o}$ and the temperature of the incoming flow is set to $56.58K$
according to experimental conditions. Grids that used in the present
simulations are presented in table 1, based on wall units. 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 turbulent region. More details about the numerical methods,
the strategies, the computational grids and boundary conditions used for
performing simulations have already been presented in the first part of this
series of study. The H0200 cases in the first part of this series of study is
used for further analysis.

It is important to note that our calculations are based on results obtained
from ILES, rather than DNS, some specific quantitative trends or small scale
features may exhibit deviations. However, the first-order statistical
quantities and mean-flow results are considered to be reliable. Based on our
experience, the grid requirements for DNS of general supersonic three-
dimensional boundary layers are as follows: The grid spacing in the two
directions parallel to the wall, based on wall units, should ideally be
isotropic and satisfy the criteria $\Delta\xi\approx\Delta\zeta\leqslant 4$;
The grid spacing in the wall-normal direction should meet the specified
requirement $\Delta\eta\leqslant 0.5$. These lead to the condition that
effective mesh spacing
$\Delta\approx\left(\Delta\xi\times\Delta\zeta\times\Delta\eta\right)^{1/3}$
is always smaller than two times the local Kolmogorov length scale. Also,
there should be at least $100-200$ points within the boundary layer. These
conditions ensure the grid convergence of high-order turbulence statistics
(with errors less than 0.1%). Given the conditions of the present
configuration, we need approximately 8 times more grids than we currently have
to resolve all the scales, which exceeds our existing computational
capabilities.

\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 | $6.35\times 10^{9}$ | 8.5 | 0.4 | 18.2
Table 1: Grid points and maximum grid sizes in wall units at the turbulent
boundary region.

To analyze the characteristics of the turbulent boundary layer flow, we
extracted a fully developed turbulent section of the previously computed
three-dimensional transitional boundary layer along the spanwise direction for
detailed analysis. To facilitate the analysis investigating the mean field
characteristics, both Reynolds-averaged and Favre-averaged mean quantities are
employed, following the approach of Huang et al. (1995). The Reynolds-averaged
mean of an arbitrary variable $f$ is denoted by $\overline{f}$, while the
Favre-averaged mean is denoted by $\tilde{f}=\overline{\rho
f}/\overline{\rho}$. In addition, the fluctuations around the Reynolds and
Favre averages are represented by single and double primes, respectively. That
is, $f^{\prime}=f-\overline{f}$ and $f^{\prime\prime}=f-\tilde{f}$. Given the
absence of homogeneous directions in the configurations under consideration,
achieving ideally averaged states necessitates a considerable amount of time.
The statistical analysis of the flows was conducted over a span of 1800 time
units, encompassing approximately 900,000 steps. This duration is roughly
sixfold the time required for a flow to evolve from the inlet to the spanwise
flow outlet. To ascertain the independence of flow statistics from the
duration of statistical analysis, we performed an additional test. This test
involved comparing flow statistics derived from two distinct averaging
intervals, with one interval containing 50% more statistical steps than the
other, to ensure that any discrepancies of the flow statistics for major
variables between the two intervals were small enough, not exceeding 0.1%.

Some common variables are defined along the whole three-dimensional surface.
Here, as the usual boundary layers in previously, we define the velocity
$\overline{u}^{+}$, based on inner scale as

$\left.\begin{aligned}
\overline{u}^{+}&=\frac{|\overline{u}_{p}|}{\overline{u}_{\tau}},\quad|\overline{u}_{p}|=\sqrt{\overline{u}_{\xi}^{2}+\overline{w}^{2}},\quad
h^{+}=\frac{\overline{\rho}_{w}\overline{u}_{\tau}h}{\overline{\mu}_{w}},\\\
\overline{u}_{\tau}&=\sqrt{\frac{\overline{\tau}_{w}}{\overline{\rho}_{w}}},\quad\overline{\tau}_{w}=\frac{\overline{\mu}}{Re}\left.\frac{\partial\overline{u}_{p}}{\partial
h}\right|_{h=0},h^{*}=\frac{\overline{\rho}\sqrt{\overline{\tau_{w}}/\rho}h}{\overline{\mu}}\end{aligned}\right\\},$
(8)

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

$C_{f}=\frac{2\overline{\mu}_{w}^{*}}{\rho_{\infty}^{*}U_{\infty}^{*2}}=\frac{2\overline{\mu}_{w}}{Re}\frac{\partial\overline{u}_{p}}{\partial
h}=2\overline{\tau}_{w},{\theta}_{tw}=-\left|\kappa\nabla\overline{T}\cdot\bm{n}\right|.$
(9)

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 $\bm{n}$. The thickness of
boundary layer $\delta_{99}$ is mainly defined based on spanwise velocity as
the wall-normal distance from the wall to the location where
${\overline{w}}=0.99{\overline{w}_{e}}$.

##  3 Results

###  3.1 Validation of the numerical results

Before conducting the related analysis, it is crucial to first verify the
reliability of the data obtained from the current simulation. This is
particularly challenging in compressible flows. Unlike incompressible flows,
compressible flows lack sufficient experimental data support, mainly because
accurate measurements in the supersonic regime are very difficult(Smits &
Dussauge, 2005; Gatski & Bonnet, 2013). For general high Mach number three-
dimensional compressible boundary layers, this is undoubtedly even more
challenging. Fortunately, in our current designed case, we can somewhat refer
back to two-dimensional boundary layers, at least along the attachment line,
where the flow reverts to a statistically two-dimensional boundary layer.
Consequently, validation of the present data is conducted by comparing the
characteristics of the boundary layer along the attachment line. For
compressible two-dimensional boundary layers, Morkovin’s hypothesis(Morkovin,
1962) is generally considered valid, which posits that compressibility effects
do not alter the time and length scales of turbulence. By considering the
density variations in the turbulent boundary layer region, compressible
turbulence can be analogized to incompressible turbulence through local
parameters. Then, the scaled Reynolds stress can be written as

$\left.\begin{aligned}
\left(u_{i}^{*}\right)^{2}&=\frac{\overline{\rho}}{\overline{\rho}_{w}}\frac{\widetilde{u_{i}^{\prime\prime
2}}}{\overline{u}_{\tau}^{2}},\quad i=1,2,3\\\
\left(u_{i}u_{j}\right)^{*}&=\frac{\overline{\rho}}{\overline{\rho}_{w}}\frac{\widetilde{u_{i}^{\prime\prime}u_{j}^{\prime\prime}}}{\overline{u}_{\tau}^{2}},\quad
i\neq j\end{aligned}\right\\},$ (10)

which should collapse the data to the incompressible ones. Here,
$\overline{u}_{\tau}$ is the friction velocity and subscript $w$ represents
the variables at the wall surface. The results of a incompressible turbulent
boundary layer at $Re_{\tau}=445$ simulated by Jiménez et al. (2010) and a
compressible turbulent boundary layer at $Re_{\tau}=453$ simulated by Cogo et
al. (2022) are used as references. The comparison for major components of
Reynolds stress is shown in figure 2. All calculated profiles are presented as
a function of the semilocal scaling $h^{*}$, which is based on $\tau_{w}$ and
local properties, proposed by Huang et al. (1995). In the figure, the black
solid line represents the spanwise-transformed Reynolds stress component
$(w^{*})^{2}$. The red and blue solid lines indicate the Reynolds stress
components in the $x-$direction $(u^{*})^{2}$ and the wall-normal
$y-$direction $(v^{*})^{2}$, respectively. The black dashed line represents
the shear Reynolds stress component $(wv)^{*}$. As can be viewed, the computed
and statistically analyzed Reynolds stresses at the exact attachment line
generally align with the results obtained from direct numerical simulations
for both incompressible and compressible flat-plate boundary layers. The peak
values and shapes of the Reynolds stresses in the streamwise direction are
consistent with the corresponding reference values. This agreement validates
the resolution of our current computational setup. However, the Reynolds
stresses in the wall-normal and chordwise directions are smaller than the
reference values. This discrepancy is likely due to the influence of the
leading-edge shock wave and the chordwise pressure gradient in the conditions
of this study. At the leading edge, the height of the shock wave from the wall
is approximately 13mm, which is about sixteen times the turbulent boundary
layer thickness at that location. This suppresses the development of
fluctuations in this direction, resulting in lower corresponding Reynolds
stresses $(v^{*})^{2}$. Along the chordwise direction at the leading edge, the
condition $\partial/\partial x=0$ is satisfied only at the exact attachment
line($x=0$), hence the Reynolds stresses $(u^{*})^{2}$ in this direction are
also somewhat suppressed.

\begin{overpic}[width=216.81pt]{Attachment_line_Reynolds_Stress.eps}
\end{overpic} Figure 2: Comparison of the scaled Reynolds stress obtained from
the current simulation with the reference values at the spanwise location
$z=145.5$.

###  3.2 General features of the turbulent boundary layer

####  3.2.1 Mean field

The laminar and transitional boundary layer have been documented in the
previous study. Here, we focus on the turbulent section. The basic features of
turbulent boundary layer are reflected in their profiles along the wall normal
directions. The profiles for major variables along the wall normal direction,
at several locations, are shown in figure 3 and 4. It can be observed that as
the attachment-line boundary layer develops from the upstream position
$(z=136.3)$ to the downstream position $(z=163.6)$, the boundary layer does
not thicken as expected for a well-known flat plate boundary layer. Instead,
the thickness of the boundary layer remains almost constant. This indicates
that the fully developed turbulent attachment-line boundary layer can also
reach a corresponding asymptotic state, in the present cases. By broadening
our perspective further, it becomes evident that this assumption remains valid
even in regions where crossflow effects are significant, as demonstrated in
figure 4. All these phenomena indicate that, under the conditions of our
study, the development of the turbulent boundary layer in the wall-normal
$\eta-$direction as well as the spanwise $z-$direction are inhibited, and the
boundary layer does not thicken in the spanwise direction.

\begin{overpic}[width=433.62pt]{CompareProfiles_AT.eps} \end{overpic} Figure
3: The wall-normal profiles of mean-flow variables at the attachment-line
plane. \begin{overpic}[width=433.62pt]{CompareProfiles_Cross.eps}
\end{overpic} Figure 4: The distributions for mean-flow variables with wall
normal direction at the chordwise location $s_{\xi}=8.71,s_{\xi}=13.50$ and
$s_{\xi}=18.61$.

Away from the attachment-line, unlike traditional two-dimensional boundary
layers, the presence of pressure gradients in three-dimensional boundary
layers induces a crossflow component. The three-dimensional boundary layer are
most conveniently described in a local streamline coordinate system as shown
in figure 5$(a)$, the main streamwise and the crossflow directions are defined
based on the angle of external streamlines
$\theta_{s}=\arctan(\overline{u}_{\xi,e}/\overline{w}_{e})$ at the edge of the
turbulent boundary layer. The streamwise and the crossflow velocities are
defined as

$\left.\begin{aligned}
u_{s}&=\overline{w}\cos\theta_{s}+\overline{u}_{\xi}\sin\theta_{s},\\\
u_{c}&=-\overline{w}\sin\theta_{s}+\overline{u}_{\xi}\cos\theta_{s}.\end{aligned}\right\\}.$
(11)

\begin{overpic}[width=260.17464pt]{LocalStreamline.pdf} \put(3.0,76.0){$(a)$} \put(95.0,14.0){$z$} \put(20.0,75.0){$\xi$} \put(50.0,80.0){External streamline} \put(67.0,79.0){\vector(1,-2){7.0}} \put(75.0,26.0){$\overline{w}$} \put(70.0,45.0){${u}_{s}$} \put(60.0,34.0){$\theta_{s}$} \put(45.0,50.0){$\overline{u}_{\xi}$} \put(31.0,43.0){${u}_{c}$} \end{overpic} | \begin{overpic}[width=154.8pt]{CrossCoordinate.pdf} \put(3.0,90.0){$(b)$} \put(37.0,28.4){\vector(3,-1){18.0}} \put(54.0,24.0){$\overline{\tau}_{w}$} \put(37.0,28.4){\vector(-3,-2){18.0}} \put(12.0,12.0){External streamline direction} \end{overpic}
---|---
Figure 5:  $(a)$ Schematic of a local streamline coordinate system and $(b)$
Three dimensional velocity profiles of the present simulations at chordwise
location $s_{\xi}=13.50$.

As shown in figure 5$(b)$, when the wall surface is approached, the magnitude
of the crossflow velocity $u_{c}$ (the red lines) increases from small value
at the edge of boundary layer, reaches a maximum value, and then decreases
back to zero at the wall surface to satisfy the no-slip condition. In
contrast, the behaviour of the streamwise velocity $u_{s}$ (the blue lines) is
qualitatively similar to that in two-dimensional boundary layers. The
direction of the wall shear stress vector $\overline{\tau}_{w}$ determines the
tangent to the limiting streamline at the wall. In general, the angle between
the tangents to the external streamline and the limiting streamline at the
wall surface is non-zero. As a result, when acrossing the hight of the
boundary layer, the velocity vector rotates through an angle at the wall (the
transparent grey lines). Similarly, the shear stress vector tends to rotate
away from its direction at the wall as the distance from the surface
increases.

\begin{overpic}[width=433.62pt]{SurfacePressure.eps} \put(72.0,15.0){$(a)$}
\put(10.0,13.0){$s_{\xi}=4.22$} \put(26.0,12.0){$s_{\xi}=8.71$}
\put(38.0,10.0){$s_{\xi}=13.50$} \put(55.0,7.0){$s_{\xi}=18.61$} \end{overpic}
---
\begin{overpic}[width=433.62pt]{Coordinate_Cf_Change.pdf}
\put(3.0,30.0){$(b)$} \put(6.0,0.0){$s_{\xi}=4.22$} \put(21.0,30.0){$(c)$}
\put(24.0,0.0){$s_{\xi}=8.71$} \put(44.0,30.0){$(d)$}
\put(40.0,0.0){$s_{\xi}=13.50$} \put(72.0,30.0){$(e)$}
\put(70.0,0.0){$s_{\xi}=18.61$} \end{overpic}
Figure 6: $(a)$ The pressure distributions along the surface. $(b-e)$ Three
dimensional velocity profiles of the present simulations at several chordwise
location, ranging from the attachment-line region to chordwise downstream. The
specific locations corresponding to the three-dimensional velocity profiles
are sequentially marked with red circles in $(a)$.

As shown in figure 6, when the flow develops from the attachment line to
chordwise downstream, the pressure gradient increases (figure 6$(a)$), leading
to stronger crossflow velocities (figure 6$(b-e)$). Consequently, the boundary
layer becomes thicker, and the rotation of the velocity vectors within the
boundary layer becomes more pronounced.

####  3.2.2 fluctuation field

Then, we establish an intuitive visualization of the three-dimensional
turbulent boundary layer flow by presenting the distribution of instantaneous
or fluctuated variables contours at different sections. The instantaneous
density contours in several selected cross sections are shown in figure
7$(c)$, ranging from attachment line to chordwise exit. The exact locations of
these slices are shown in figure 7$(a)$ and $(b)$. The contour exhibite the
typical structure, already found in flat-plate turbulent boundary layers
(Smith & Smits, 1995; Pirozzoli & Bernardini, 2011; Duan et al., 2014; Cogo et
al., 2022), dominated by large-scale patterns inclined at some angle.
Therefore, simply observing the characteristics on a single cross-section
makes it difficult to distinguish between two-dimensional and three-
dimensional boundary layers. As a supplement, some nearly wall-parallel slices
for instantaneous density contours are shown in figure 8. From the attachment
line to the chordwise exit, the density perturbations become sparser as the
density amplitude decreases (changing from red to blue). With increasing
distance from the wall (from Figure 8$(a)$ to Figure 8$(c)$), the
characteristic structures of the perturbations grow larger. Near the wall, the
disturbance structures exhibit a clear preferential orientation, and it can be
faintly observed that the disturbance structures gradually shift from the
spanwise direction to the chordwise direction.

To obtain clearer images of the disturbances themselves, we subtract the mean
flow to observe the fluctuations. Figures 9 and 10 show contours of the
instantaneous spanwise velocity fluctuations $w^{\prime}$ and temperature
fluctuations $T^{\prime}$ for the present cases in nearly wall-parallel slices
taken at three different locations, standing for both the inner $(h^{+}=15.5)$
and outer region $(h/\delta_{99}=0.18/0.53)$ at the selected stations.

As seen in figure 9$(a)$, the spanwise velocity field in the inner layer
presents the typical streaky pattern. These streaky patterns are distributed
along the spanwise $z-$direction in the attachment line region. As the
chordwise distance $s_{\xi}$ increases, these structures tend to align with
the direction of the external streamline. Therefore, all these patterns
exhibit a certain radiating state, appearing as skirt-like structures
extending from the attachment line along the chordwise direction. Similar
visualizations of near-wall streaks in flat-plate boundary layer simulations
were also reported by (Duan et al., 2010; Pirozzoli & Bernardini, 2011; Duan
et al., 2011; Cogo et al., 2022). These findings suggest that the near-wall
streak structures in the three-dimensional turbulent boundary layer resemble
those in a two-dimensional flat-plate turbulent boundary layer, both aligning
with the direction of the external streamline. The alternating distribution of
high and low momentum zones forms the velocity streak structures depicted in
this figure. This formation is associated with ejection and sweep events.
These large-scale structures can extend several boundary layer thicknesses in
the direction of the external streamline and contain a significant portion of
the turbulent kinetic energy. As the slice move away from the surface (figure
9$(b)$), a qualitatively similar pattern found in figure 9$(a)$ are also
presented on a much larger scale. These structures can be identified as outer-
layer streaks, which can be identified as large scale coherent structures or
”superstructures” in outer region(Ganapathisubramani et al., 2006; Hutchins &
Marusic, 2007). From this perspective, the near-wall small-scale structures
appear to be the footprints of the large-scale structures from the outer layer
in the near-wall region. This observation is also consistent with what has
been previously understood in flat-plate boundary layers. The scenario changes
significantly as the nearly surface-parallel plane moves further away, the
flow becomes extremely intermittent, as present in figure 9$(c)$.

\begin{overpic}[width=433.62pt]{GXi_Slice_Locations.pdf} \put(1.0,57.0){$(a)$}
\put(1.0,33.0){$(b)$} \put(38.0,57.0){$(c)$} \put(41.0,51.5){$h$}
\put(41.0,39.0){$h$} \put(41.0,28.0){$h$} \put(41.0,16.0){$h$}
\put(41.0,5.0){$h$} \put(69.5,0.5){$z$} \end{overpic} Figure 7: $(a)$
Schematics of the swept blunt leading edge with the average surface skin
friction $\overline{\tau}_{w}$ and the selected region. $(b)$ the enlarged
figure for the region selected in $(a)$. $(c)$ the instantaneous density of
the selected cross-sections in $(a)$ and $(b)$.
\begin{overpic}[width=433.62pt]{GEta_Slice_Locations.pdf}
\put(5.0,25.0){$(a)$} \put(39.0,25.0){$(b)$} \put(73.0,25.0){$(c)$}
\end{overpic} Figure 8: $(a-c)$ stand for instantaneous density $\rho$
distributions at spanwise cross sections at grid index
$\eta=33(h^{+}=15.5),101(h/\delta_{99}=0.18)$ and $201(h/\delta_{99}=0.53)$,
respectively. $h^{+},h$ and $\delta_{99}$ are defined based on the variables
at exact attachment-line plane. $\delta_{99}\approx 0.8$ at the present case.
\begin{overpic}[width=433.62pt]{PertB_Wp.pdf} \put(-1.0,38.0){$(a)$}
\put(32.0,38.0){$(b)$} \put(64.0,38.0){$(c)$} \put(-1.5,20.0){$s_{\xi}$}
\put(15.0,-1.0){$z$} \put(48.0,-1.0){$z$} \put(82.0,-1.0){$z$} \end{overpic}
Figure 9: $(a-c)$ stand for spanwise fluctuations $w^{\prime}$ at the cross
sections at grid index $\eta=33,101$ and $201$, respectively.
\begin{overpic}[width=433.62pt]{PertB_Tp.pdf} \put(0.0,38.5){$(a)$}
\put(33.0,38.5){$(b)$} \put(65.0,38.5){$(c)$} \put(-1.5,19.0){$s_{\xi}$}
\put(15.0,-1.0){$z$} \put(48.0,-1.0){$z$} \put(82.0,-1.0){$z$} \end{overpic}
Figure 10: $(a-c)$ stand for temperature fluctuations $T^{\prime}$ at the
cross sections at grid index $\eta=33,101$ and $201$, respectively.

The temperature field in the same layers (Figure 10$(a-c)$) also reveals a
similar picture. The figures $(a)$ and $(b)$ exhibit clear streak structure,
which is qualitatively similar to that of spanwise velocity fluctuations
$w^{\prime}$. Upon closer inspection, one can observe a close correspondence
between zones with positive temperature fluctuations (red regions) and low-
speed streaks (blue regions), and vice versa. This is a typical manifestation
of the well-known tendency for velocity and temperature fluctuations in shear
flows to be negatively correlated. In this case, it is a consequence of the
fact that outward wall-normal motions transport negative velocity fluctuations
and positive temperature fluctuations from the inner, low-speed, high-
temperature layers to the upper layers. Similar observations have been
elucidated in previous studies in flat-plate boundary layer by Pirozzoli &
Bernardini (2011).

All these observations tell us that the typical near-wall disturbance
structures in two-dimensional and three-dimensional turbulent boundary layers
are very similar. However, due to the modulation effect of the mean flow,
these flow structures exhibit a strong tendency to align with the direction of
the external flow.

###  3.3 Law-of-the-wall and velocity-temperature relationships

####  3.3.1 Law-of-the-wall

All prior researches on mean velocity transformations in compressible
turbulent boundary layers was mainly confined to statistically two-dimensional
boundary layers, hence their performance in actual three-dimensional boundary
layers remains largely unknown. Here, we will conduct a study of velocity
transformations, in an attempt to further elucidate the impact of three-
dimensional effects and pressure gradients on the effectiveness of these
compressibility transformations.

As analyzed in the previous mean flow analysis, in a typical three-dimensional
boundary layer, the velocity vector parallel to the wall can point in any
direction, and this direction changes with the height away from the wall.
Therefore, directly transforming the mean velocity parallel to the wall, as
done in a two-dimensional flat-plate boundary layer, is not meaningful, since
the direction of the velocity vector is not uniform. Thus, we need to
decompose this velocity vector into two different directions to ensure the
uniformity of the velocity vector direction. One can simply decompose and
project the velocity vector onto the spanwise $z-$ direction and the chordwise
$\xi-$direction. The reason to chose these two directions is that, in the
current three-dimensional boundary layer, the pressure gradient exists only in
the $x-y$ plane and not in the spanwise $z-$direction. By using this
decomposition, one can decouple the pressure gradient effects from the flow in
a specific direction. Therefore, in each chordwise location, except the exact
attachment line, two velocities $w$ and $u_{\xi}$ along the $z$ and $\xi$
directions, are used to assess various compressibility scaling relations in
case of three dimensional flows. However, from a statistical perspective, in
regions very close to the attachment line, the values of $u_{\xi}$ are very
small over a substantial area, thus showing limited significance. Instead, we
chose the mainstream velocity direction $u_{s}$ parallel to the external
streamline as the second velocity vector to characterize the three-dimensional
flow

The van Driest transformation(Van Driest, 1951) is widely used to transform
the mean velocity profile in a compressible boundary layer to a profile from
an incompressible boundary layer by taking the variations of mean density into
account. In general three-dimensional boundary layer, the velocity vector
parallel to the wall surface can be In the present three dimensional boundary
layer, the similar form in the present study can be taken as:

$w_{V_{D}}^{+}=\int_{0}^{\overline{w}^{+}}\sqrt{\frac{\overline{\rho}}{\overline{\rho}_{w}}}d{\overline{w}^{+}},\quad
u_{sV_{D}}^{+}=\int_{0}^{\overline{u_{s}}^{+}}\sqrt{\frac{\overline{\rho}}{\overline{\rho}_{w}}}d{\overline{u_{s}}^{+}}$
(12)

This transoformation works well in collapse velocity profiles in normal
adiabatic boundary layer flows. However, in diabatic boundary layers, the
transformation fails and can not collapse into the incompressible law of the
wall. Trettel & Larsson (2016) proposed a new transformation based on the
stress-balance conditions and can be written as

$\left.\begin{aligned}
w_{T_{L}}^{+}&=\int_{0}^{\overline{w}^{+}}\sqrt{\frac{\overline{\rho}}{\overline{\rho}_{w}}}\left[1+\frac{1}{2\overline{\rho}}\frac{\partial\overline{\rho}}{\partial
h}h-\frac{1}{\overline{\mu}}\frac{\partial\overline{\mu}}{\partial
h}h\right]d\overline{w}^{+},\\\
u_{sT_{L}}^{+}&=\int_{0}^{\overline{u}_{s}^{+}}\sqrt{\frac{\overline{\rho}}{\overline{\rho}_{w}}}\left[1+\frac{1}{2\overline{\rho}}\frac{\partial\overline{\rho}}{\partial
h}h-\frac{1}{\overline{\mu}}\frac{\partial\overline{\mu}}{\partial
h}h\right]d\overline{u}_{s}^{+},\\\
h^{*}&=\frac{\overline{\rho}}{\overline{\mu}}\sqrt{\frac{\overline{\tau}_{w}}{\overline{\rho}}}h\end{aligned}\right\\}$
(13)

The Trettle & Larsson transformation is functioning well in collapsing the
profiles for a non-adiabatic turbulent boundary layer to an equivalent
incompressible profile, especially for channel flow. Their performances in
boundary layers are not that good. To improve the overall collaspe of the mean
velocity profile in boundary layers, Volpiani et al. (2020) developed a new
transformation based on data-driven approaches, this transformation can be
written as

$\displaystyle
w_{V}^{+}=\int_{0}^{\overline{w}^{+}}\sqrt{\frac{\overline{\rho}/\overline{\rho}_{w}}{\overline{\mu}/\overline{\mu}_{w}}}d\overline{w}^{+},u_{sV}^{+}$
$\displaystyle=\int_{0}^{\overline{u}_{s}^{+}}\sqrt{\frac{\overline{\rho}/\overline{\rho}_{w}}{\overline{\mu}/\overline{\mu}_{w}}}d\overline{u}_{s}^{+},h_{V}^{+}=\int_{0}^{\overline{h}^{+}}\sqrt{\frac{\overline{\rho}/\overline{\rho}_{w}}{(\overline{\mu}/\overline{\mu}_{w})^{3}}}dh^{+}$
(14)

Griffin et al. (2021) further introduced a new transformation, which involves
scaling the viscous stress through semi-local non-dimensionalization within
the viscous sublayer, and addressing the Reynolds shear stress to preserve the
approximate balance between turbulence production and dissipation in the log-
law region. This transformation has been demonstrated to effectively unify the
velocity profiles across a diverse array of flows, encompassing heated,
cooled, and adiabatic boundary layers, as well as fully developed channel and
pipe flows. The transformation is referred to the total-stress-based
transformation, can be written as

$\left.\begin{aligned}
w_{T_{S}}^{+}&=\int_{0}^{w^{+}}\frac{S_{eq,w}^{+}}{1+S_{eq,w}^{+}-S_{T_{L},w}^{+}}dh^{*},S_{eq,w}^{+}=\frac{\overline{\mu}_{w}}{\overline{\mu}}\frac{\partial
w^{+}}{\partial
h^{*}},S_{T_{L},w}^{+}=\frac{\overline{\mu}}{\overline{\mu}_{w}}\frac{\partial
w^{+}}{\partial h^{+}},\\\
u_{s,T_{S}}^{+}&=\int_{0}^{u^{+}_{s}}\frac{S_{eq,u_{s}}^{+}}{1+S_{eq,u_{s}}^{+}-S_{T_{L},u_{s}}^{+}}dh^{*},S_{eq,u_{s}}^{+}=\frac{\overline{\mu}_{w}}{\overline{\mu}}\frac{\partial
u_{s}^{+}}{\partial
h^{*}},S_{T_{L},u_{s}}^{+}=\frac{\overline{\mu}}{\overline{\mu}_{w}}\frac{\partial
u_{s}^{+}}{\partial h^{+}},\\\ \end{aligned}\right\\}.$ (15)

\begin{overpic}[width=186.45341pt]{MeanProfiles/MP_W_Xi1201.eps} \put(10.0,75.0){$(a)$} \end{overpic} | \begin{overpic}[width=186.45341pt]{MeanProfiles/MP_W_Xi1401.eps} \put(10.0,75.0){$(b)$} \end{overpic}
---|---
\begin{overpic}[width=186.45341pt]{MeanProfiles/MP_W_Xi1601.eps} \put(10.0,75.0){$(c)$} \end{overpic} | \begin{overpic}[width=186.45341pt]{MeanProfiles/MP_W_Xi1801.eps} \put(10.0,75.0){$(d)$} \end{overpic}
\begin{overpic}[width=186.45341pt]{MeanProfiles/MP_W_Xi2001.eps} \put(10.0,75.0){$(e)$} \end{overpic} | \begin{overpic}[width=186.45341pt]{MeanProfiles/MP_W_Xi2201.eps} \put(10.0,75.0){$(f)$} \end{overpic}
\begin{overpic}[width=186.45341pt]{MeanProfiles/MP_W_Xi2301.eps} \put(10.0,75.0){$(g)$} \end{overpic} | \begin{overpic}[width=186.45341pt]{MeanProfiles/MP_W_Xi2401.eps} \put(10.0,75.0){$(h)$} \end{overpic}
Figure 11: Spanwise mean-velocity profiles, transformed according to several transformations, at different chordwise locations. $(a)-(h)$ stand for the location at chordwise $s_{\xi}=0.00,~{}4.22,~{}8.71,~{}13.50,~{}18.61,~{}24.10,~{}26.99$ and $30.00$, respectively. \begin{overpic}[width=186.45341pt]{MeanProfiles/MP_Us_Xi1201.eps} \put(10.0,75.0){$(a)$} \end{overpic} | \begin{overpic}[width=186.45341pt]{MeanProfiles/MP_Us_Xi1401.eps} \put(10.0,75.0){$(b)$} \end{overpic}
---|---
\begin{overpic}[width=186.45341pt]{MeanProfiles/MP_Us_Xi1601.eps} \put(10.0,75.0){$(c)$} \end{overpic} | \begin{overpic}[width=186.45341pt]{MeanProfiles/MP_Us_Xi1801.eps} \put(10.0,75.0){$(d)$} \end{overpic}
\begin{overpic}[width=186.45341pt]{MeanProfiles/MP_Us_Xi2001.eps} \put(10.0,75.0){$(e)$} \end{overpic} | \begin{overpic}[width=186.45341pt]{MeanProfiles/MP_Us_Xi2201.eps} \put(10.0,75.0){$(f)$} \end{overpic}
\begin{overpic}[width=186.45341pt]{MeanProfiles/MP_Us_Xi2301.eps} \put(10.0,75.0){$(g)$} \end{overpic} | \begin{overpic}[width=186.45341pt]{MeanProfiles/MP_Us_Xi2401.eps} \put(10.0,75.0){$(h)$} \end{overpic}
Figure 12: Streamwise mean-velocity profiles, transformed according to several
transformations, at different chordwise locations. $(a)-(h)$ stand for the
location at chordwise
$s_{\xi}=0.00,~{}4.22,~{}8.71,~{}13.50,~{}18.61,~{}24.10,~{}26.99$ and
$30.00$, respectively.

The transformed spanwise and streamwise velocity profiles at several chordwise
locations are shown in figures 11 and 12, ranging from the exact attachment
line to downstream chordwise direction. Firstly, we look at the profiles at
the exact attachment line (Figure 11$(a)$ and Figure 12$(a)$). Excluding the
Trettel & Larsson transformation, the other three transformations yielded
satisfactory results. Based on previous research, the Trettel & Larsson
transformation is expected to perform well in fully developed, non-spatially
varying boundary layers such as channel flow; however, in the conditions
investigated in this study, it inaccurately predicted the logarithmic region.
It is well-known that the Van Driest transformation generally performs poorly
in compressible, heat-exchanging flat-plate turbulent boundary layer flows.
Nevertheless, under the conditions of this study, the Van Driest
transformation produced acceptable results. As the position gradually moves
away from the attachment line to the point of maximum transverse pressure
gradient, different velocity transformations continue to exhibit similar
patterns to those observed at the attachment line. This indicates that in the
present fully developed three-dimensional turbulent compressible boundary
layer, even in the presence of transverse pressure gradients, the mainstream
velocity component still largely conforms to the velocity transformations
previously established for flat-plate boundary layers. Meanwhile, it is
important to note that, under the conditions studied, the transverse velocity
components gradually increase from zero along the chordwise direction starting
from the attachment line. Therefore, the chordwise velocities at the locations
shown in figures $(b-d)$ are not very large. As the transverse velocity
increases, shown in figures $(e-h)$, the slopes and intercepts of various
transformations for spanwise velocity profiles begin to deviate from the
classical relationships. Meanwhile, the transformed streamwise velocity
profiles still exhibite a clear log-law region, as shown in figure 12. This
indicate that along the streamwise directions, the streamwise mean velocities
still exhibit some feature similar to the two-dimensional turbulent boundary
layer. When the transverse velocity increases further to a magnitude
comparable to the spanwise velocity, the regularity of the entire
transformation no longer holds for spanwise velocity. However, the streamwise
velocities still show a common behaviour and regions similar to the log-law of
the two-dimensional flat-plate boundary layer can also be identified. It
should be noted that the results presented here are merely illustrative.
Whether a general law-of-the-wall, similar to that in two-dimensional flat-
plate boundary layers, exists in three-dimensional boundary layers remains an
open question that requires further in-depth investigation.

####  3.3.2 velocity-temperature relationship

The temperature profiles also play a vital role in compressible boundary,
therefore, the widly used relationship between the mean velocity and mean
temperature is also shown to assess the scaling relations in three-dimensional
boundary layers. The mean temperature-velocity relation model given by Zhang
et al. (2014), which is also known as the generalized reynolds analogy, is
defined as

$\left.\begin{aligned}
\frac{\overline{T}}{T_{e}}&=\frac{T_{w}}{T_{e}}+\frac{T_{rg}-T_{w}}{T_{e}}\frac{\overline{u}_{p}}{\overline{u}_{p,e}}+\frac{T_{e}-T_{rg}}{T_{e}}\left(\frac{\overline{u}_{p}}{\overline{u}_{p,e}}\right)^{2},\\\
T_{rg}&=T_{e}+\frac{r_{g}\overline{u}_{p,e}^{2}}{2C_{p}},r_{g}=\frac{2C_{p}(T_{w}-\overline{T}_{e})}{\overline{u}_{p,e}^{2}}-2Pr\frac{\theta_{tw}}{\overline{u}_{p,e}\overline{\tau}_{w}},\end{aligned}\right\\},$
(16)

where the subscript $e$ stands for the variables at the edge of boundary layer
and the velocity $\overline{u}_{p}$ represents the velocity parallel to the
local surface. It should be noted that, in this study, the mean temperature
$T_{e}$ and velocity $\overline{u}_{p,e}$ outside the boundary layer change
with the chordwise position $s_{\xi}$. The relationship between the mean
temperature and mean velocity at different chordwise locations is illustrated
in the figure 11. Figure $(a)$ stands for the mean temperature profiles
against mean velocity at the exact attachment line $s_{\xi}=0$ and the
chordwise exit $s_{\xi}=30$. It reveals that the relation proposed by Zhang et
al. (2014) yields an acceptable results. In fact, the accurate prediction
results at the exact attachment line are as expected, since it statistically
returns to a zero-pressure-gradient flat-plate boundary layer. However,
achieving good predictive results at the chordwise exit was surprising. This
can be attributed to two potential reasons: in the current flow study, the
governing spanwise momentum equation for the spanwise velocity $w$ is actually
weakly coupled with the other equations, as the boundary layer seems to be
homogeneous in spanwise direction. Secondly, at the exit location, although
there is a strong crossflow velocity, there is no pressure gradient along
either the chordwise or spanwise directions. As a result, the energy generated
by the pressure work accumulated upstream is entirely absorbed into the mean
flow temperature and velocity.

\begin{overpic}[width=208.13574pt]{TemperatureVelocity1.eps} \put(12.0,15.0){$(a)$} \end{overpic} | \begin{overpic}[width=208.13574pt]{TemperatureVelocity2.eps} \put(12.0,15.0){$(b)$} \put(40.0,30.0){\vector(1,1){30.0}} \put(72.0,60.0){$s_{\xi}$ increase} \end{overpic}
---|---
Figure 13: Panels (a) and (b) show the mean temperature profile against mean
velocity at several chordwise locations, compared with relation of Zhang et
al. (2014).

Then we look at the profiles at the region with both crossflow and pressure
gradient effect (figure 13$(b)$). The figure shows the results from small
$s_{\xi}$ to further chordwise downstream, as pointed by the vector. It can be
observed that when the pressure gradient and velocity in the transverse
direction are relatively small, the temperature-velocity relationship proposed
by Zhang et al. (2014) still aligns well with the computational results. As
the transverse pressure gradient and transverse velocity increase, the
original temperature-velocity relationship begins to deviate in the outer
boundary layer region. As the flow further develops downstream, the transverse
pressure gradient starts to decrease (figure 6$(a)$), although the transverse
velocity continues to increase. The deviation between the original
temperature-velocity relationship and the computational results in the outer
boundary layer region gradually diminishes, eventually returning to the trend
indicated by the blue line in figure 13$(a)$ at the chordwise exit. These
phenomena indicate that the traditional temperature-velocity relationship is
significantly influenced by the transverse pressure gradient in three-
dimensional boundary layers. Attention must also be given to the observation
that, in the chordwise direction downstream from the attachment line, the
overall turbulence intensity exhibits a decreasing trend. Additionally, the
chordwise velocity in certain regions remains at relatively low levels
compared to the spanwise velocity. Therefore, this flow condition might thus
differ from typical three-dimensional turbulent boundary layers.

###  3.4 Reynolds stresses

In this part, we further look at the features of Reynolds stresses of the
present simulations. The scaled Reynolds stresses at the exact attachment line
are shown in previous and are found in good agreements with the results from
incompressible and compressible flat-plate boundary layer. As the transverse
pressure gradient begin to increase, the scaled Reynolds stress are shown in
figure 14. Two more scaled Reynolds stresses are defined based on the
direction of surface coordinate as follows

$\left.\begin{aligned}
\left(u_{\xi}^{*}\right)^{2}&=\frac{\overline{\rho}}{\overline{\rho}_{w}}\frac{\widetilde{u_{\xi}^{\prime\prime}u_{\xi}^{\prime\prime}}}{\overline{u}_{\tau}^{2}},\widetilde{u_{\xi}^{\prime\prime}u_{\xi}^{\prime\prime}}=\bm{t}_{x}^{2}\widetilde{u^{\prime\prime}u^{\prime\prime}}+\bm{t}_{y}^{2}\widetilde{v^{\prime\prime}v^{\prime\prime}}+2\bm{t}_{x}\bm{t}_{y}\widetilde{u^{\prime\prime}v^{\prime\prime}},\\\
\left(u_{n}^{*}\right)^{2}&=\frac{\overline{\rho}}{\overline{\rho}_{w}}\frac{\widetilde{u_{n}^{\prime\prime}u_{n}^{\prime\prime}}}{\overline{u}_{\tau}^{2}},\widetilde{u_{n}^{\prime\prime}u_{n}^{\prime\prime}}=\bm{n}_{x}^{2}\widetilde{u^{\prime\prime}u^{\prime\prime}}+\bm{n}_{y}^{2}\widetilde{v^{\prime\prime}v^{\prime\prime}}+2\bm{n}_{x}\bm{n}_{y}\widetilde{u^{\prime\prime}v^{\prime\prime}}\end{aligned}\right\\},$
(17)

where $\bm{t}=(\bm{t}_{x},\bm{t}_{y})$ and $\bm{n}=(\bm{n}_{x},\bm{n}_{y})$
stands for the unit surface tangent and normal vectors, in the $x-y$ plane.
From these figures, the evolution of Reynolds stress with chordwise location
is clearly presented. When the pressure gradient first affects the turbulence
in the attachment line region, there is a slight increase in turbulence
intensity, reflected in the slight rise in spanwise Reynolds stress and
turbulent kinetic energy (as shown in figures 14$(a)$ and $(d)$).
Subsequently, under the influence of the chordwise favorable pressure
gradient, the fluid not only experiences shear in the spanwise direction but
also begins to be affected in the chordwise direction. This results in the
peak Reynolds stress in the $x-y$ plane shifting towards the wall (figures
14$(b,c)$). With further acceleration of the fluid in the chordwise direction,
chordwise fluctuations become significantly enhanced (figure 14$(e)$). This
indicates that the pressure gradient can, to some extent, induce a
redistribution of turbulent energy. However, as the fluid continues to
accelerate and develops downstream, fluctuations in all directions show a
notable decreasing trend. A noticeable phenomenon is that when the fluctuation
intensity shows a significant decrease, the primary Reynolds stress along the
chordwise direction exhibits a distribution pattern very similar to that of
the primary Reynolds stress in a flat-plate boundary layer (figure 14$(e)$).
Additionally, the fluctuations in the wall-normal direction do not seem to be
noticeably affected by the pressure gradient, apart from the last station
(figure 14$(f)$). The variations in general reflect the combined effects of
shear caused by the three-dimensional nature of the flow field and the
favorable pressure gradient along the chordwise direction.

\begin{overpic}[width=208.13574pt]{RS_WW.eps} \put(-1.0,75.0){$(a)$} \end{overpic} | \begin{overpic}[width=208.13574pt]{RS_UU.eps} \put(-1.0,75.0){$(b)$} \end{overpic}
---|---
\begin{overpic}[width=208.13574pt]{RS_VV.eps} \put(-1.0,75.0){$(c)$} \end{overpic} | \begin{overpic}[width=208.13574pt]{RS_K.eps} \put(-1.0,75.0){$(d)$} \end{overpic}
\begin{overpic}[width=208.13574pt]{RS_UxiUxi.eps} \put(-1.0,75.0){$(e)$} \end{overpic} | \begin{overpic}[width=208.13574pt]{RS_UnUn.eps} \put(-1.0,75.0){$(f)$} \end{overpic}
Figure 14: Comparison of the scaled Reynolds stress and kinetic energy at
different chordwise locations $s_{\xi}$. $(a-f)$ stand for the
$(w*)^{2},(u*)^{2},(v*)^{2}$, scaled turbulent kinetic energy
$2\overline{\rho}K/(\rho_{w}u_{\tau}^{2})$, $(u_{\xi}^{*})^{2}$ and
$(u_{n}^{*})^{2}$, respectively.

###  3.5 Anisotropy tensor

Here, in this section, we try to understand the state of turbulence by using
anisotropy tensor of Reynolds stress. The anisotropy tensor
$\widetilde{b}_{ij}$, which is defined as

$\widetilde{b}_{ij}=\frac{\overline{\rho
u_{i}^{\prime\prime}u_{j}^{\prime\prime}}}{\overline{\rho
u_{i}^{\prime\prime}u_{i}^{\prime\prime}}}-\frac{\delta_{ij}}{3}=\frac{\widetilde{u_{i}^{\prime\prime}u_{j}^{\prime\prime}}}{\widetilde{u_{i}^{\prime\prime}u_{i}^{\prime\prime}}}-\frac{\delta_{ij}}{3}$
(18)

is used to understand the state of turbulence at different chordwise location.
The invariants of this tensor can be written as(Simonsen & Krogstad, 2005)

$I_{1}=\widetilde{b}_{ii}=0,\quad
II_{2}=-\frac{1}{2}\widetilde{b}_{ij}\widetilde{b}_{ji},\quad
III_{3}=det(\widetilde{b}_{ij}).$ (19)

The state of the anisotropy stress tensor can be shown in an anisotropy
invariant map, as shown in figure 15. The blue cross stands for the location
of the first grid away from the wall surface and the red cross represents the
101th grid away from the surface. All the points, along the wall-normal
directions, inside the boundary layer are shown in the figures. The points
close to the wall, as $h\to 0$, are also marked, which are very close to the
two-component turbulence, as the wall significantly restricts turbulence
fluctuations in the direction normal to the wall in the near-wall region. As
the position moves away from the surface, the reynolds stress anisotropy
tensor exhibits a tendency to one-component turbulence. Subsequently, as the
position moves further away from the wall, the wall’s constraint on the
turbulence progressively weakens. The Reynolds stress begins to exhibit
characteristics of rod-like turbulence and, with increasing distance from the
wall, tends to become more isotropic. As the position moves away from the
attachment-line boundary layer (from figures 15 $(b)$ to $(e)$), the
distribution characteristics of the anisotropic Reynolds stress tensors remain
consistent with those observed at the attachment line. This indicates that the
anisotropic characteristics of the Reynolds stress tensors are not
significantly affected by the three-dimensional effects, which could provide
guidance for subsequent modeling of three-dimensional turbulent boundary
layers.

\begin{overpic}[width=208.13574pt]{Anisotropy1201.eps} \put(15.0,70.0){$(a)$} \put(25.0,73.0){ One component} \put(25.0,60.0){ Two component} \put(62.0,74.0){\vector(1,0){28.0}} \put(40.0,58.0){\vector(0,-1){14.0}} \put(35.0,20.0){Isotropic} \put(33.0,21.0){\vector(-1,-1){10.0}} \end{overpic} | \begin{overpic}[width=208.13574pt]{Anisotropy1401.eps} \put(15.0,70.0){$(b)$} \put(55.0,20.0){rod-like turbulence} \put(15.0,62.0){disk-like turbulence} \put(55.0,25.0){\vector(-1,1){12.0}} \put(26.0,61.0){\vector(-1,-4){10.0}} \end{overpic}
---|---
\begin{overpic}[width=208.13574pt]{Anisotropy1601.eps} \put(15.0,70.0){$(c)$} \end{overpic} | \begin{overpic}[width=208.13574pt]{Anisotropy1801.eps} \put(15.0,70.0){$(d)$} \end{overpic}
\begin{overpic}[width=208.13574pt]{Anisotropy2001.eps} \put(15.0,70.0){$(e)$} \end{overpic} | \begin{overpic}[width=208.13574pt]{Anisotropy2401.eps} \put(15.0,70.0){$(f)$} \end{overpic}
Figure 15: Invariant mapping of Reynolds stress anisotropy in three-
dimensioanl boundary layer at different chordwise locations $s_{\xi}$. $(a-f)$
stand for chordwise locations at $s_{\xi}=0.00,4.22,8.71,13.50,18.61$ and
$30.00$, respectively.

###  3.6 Turbulent kinetic energy budgets

Further investigation of the turbulence in this three dimensional boundary
layer is conducted by analysing the turbulent kinetic energy equation. The
transport equation of the turbulent kinetic energy can be expressed as

$\frac{\partial\overline{\rho}K}{\partial
t}=\mathbf{C}+\mathbf{P}+\mathbf{\Pi}+\bm{\epsilon}+\mathbf{M}+\mathbf{D},$
(20)

where

$\left.\begin{aligned}
\mathbf{C}&=-\frac{\partial\overline{\rho}K\widetilde{u}_{i}}{\partial
x_{i}},\mathbf{P}=-\overline{\rho}\widetilde{u_{i}^{\prime\prime}u_{k}^{\prime\prime}}\frac{\partial\widetilde{u_{i}}}{\partial
x_{k}},\mathbf{\Pi}=\overline{p^{\prime}\frac{\partial
u_{i}^{\prime}}{\partial x_{i}}},\\\
\bm{\epsilon}&=-\overline{\tau_{ik}^{\prime}\frac{\partial
u_{i}^{\prime}}{\partial
x_{k}}},\mathbf{M}=\overline{u_{i}^{\prime\prime}}\left(\frac{\partial\overline{\sigma}_{ik}}{\partial
x_{k}}-\frac{\partial\overline{p}}{\partial x_{i}}\right),\\\
\mathbf{D}&=-\frac{\partial}{\partial
x_{k}}\left[\frac{\overline{\rho}\widetilde{u_{i}^{\prime\prime}u_{i}^{\prime\prime}u_{k}^{\prime\prime}}}{2}+\overline{p^{\prime}u_{i}^{\prime}}\delta_{ik}-\overline{\tau_{ik}^{\prime}u_{i}^{\prime}}\right],\end{aligned}\right\\},$
(21)

stands for the convection term($\mathbf{C}$), production term($\mathbf{P}$),
pressure-dilatation term($\mathbf{\Pi}$), dissipation rate
term($\bm{\epsilon}$), mass flux term($\mathbf{M}$) and transport-diffusion
term($\mathbf{D}$), respectively.

\begin{overpic}[width=173.44534pt]{TurbB1201.eps} \put(65.0,70.0){$(a)$} \end{overpic} | \begin{overpic}[width=173.44534pt]{TurbB1401.eps} \put(15.0,70.0){$(b)$} \end{overpic}
---|---
\begin{overpic}[width=173.44534pt]{TurbB1601.eps} \put(15.0,70.0){$(c)$} \end{overpic} | \begin{overpic}[width=173.44534pt]{TurbB1801.eps} \put(15.0,70.0){$(d)$} \end{overpic}
\begin{overpic}[width=173.44534pt]{TurbB2001.eps} \put(15.0,70.0){$(e)$} \end{overpic} | \begin{overpic}[width=173.44534pt]{TurbB2201.eps} \put(15.0,70.0){$(f)$} \end{overpic}
Figure 16: The energy budget distributiones along the wall normal direction in
three-dimensioanl boundary layer at different chordwise locations $s_{\xi}$.
$(a-f)$ stand for chordwise locations at $s_{\xi}=0.00,4.22,8.71,13.50,18.61$
and $24.10$, respectively.

Figure 16 shows the basic results of energy budget distributiones along the
wall normal direction at several chordwise position. From the distribution
shown in the figures 16$(a-e)$, it is evident that the dominant terms are the
production term $\mathbf{P}$, dissipation term $\bm{\epsilon}$, and the
transport-diffusion term $\mathbf{D}$, consistent with the analysis in
traditional two-dimensional boundary layers. This indicates that even though
three-dimensional effects significantly alter the flow characteristics
(introducing crossflow, pressure gradient effects, etc.), their impact on the
turbulence energy balance mechanism is minimal. However, as shown in figure
16$(f)$, the convection term (the black line) become obvious. Although the
proportion of the convection term in the energy budget analysis increased,
this does not imply an absolute intensification of the convection. In fact,
under the influence of the chordwise pressure gradient, the corresponding
turbulence is suppressed, leading to a reduction in the production,
dissipation and transport-diffusion terms, which in turn makes the convection
term more pronounced. This implies that the core modeling terms for turbulence
in three-dimensional boundary layer flows remain consistent with those in two-
dimensional boundary layers.

###  3.7 Analysis of Shear Direction Discrepancies

In the previous discussion, we briefly introduced some basic characteristics
of the computed three-dimensional turbulent boundary layer. In this section,
we will delve deeper into the shear effects within the three-dimensional
boundary layer. Traditionally, in two-dimensional turbulent boundary layers
(assuming the mainstream and wall-normal velocities are $u_{\xi}$ and
$u_{n}$), the focus of shear effects is primarily on
$\widetilde{u_{\xi}^{\prime\prime}u_{n}^{\prime\prime}}$ and the corresponding
meanflow shear $du_{\xi}/dh$. However, in three-dimensional cases, the shear
effects should actually encompass both
$\widetilde{u_{\xi}^{\prime\prime}u_{n}^{\prime\prime}}$ and
$\widetilde{w^{\prime\prime}u_{n}^{\prime\prime}}$, as well as the
corresponding $du_{\xi}/dh$ and $dw/dh$. From the previous analyses, we
observed that the direction of the mean velocity within the boundary layer
changes with the distance from the wall. Consequently, the direction of the
resultant shear Reynolds stress also varies accordingly. This naturally
prompts the question: does the direction of the shear Reynolds stress still
align with the direction of the resultant mean flow shear under these
conditions?

In general, the Reynolds shear stresses can be expressed as

$\left.\begin{aligned}
\widetilde{u_{\xi}^{\prime\prime}u_{n}^{\prime\prime}}&=\bm{t}_{x}\bm{n}_{x}\widetilde{u^{\prime\prime}u^{\prime\prime}}+\left(\bm{t}_{x}\bm{n}_{y}+\bm{t}_{y}\bm{n}_{x}\right)\widetilde{u^{\prime\prime}v^{\prime\prime}}+\bm{t}_{y}\bm{n}_{y}\widetilde{v^{\prime\prime}v^{\prime\prime}},\\\
\widetilde{w^{\prime\prime}u_{n}^{\prime\prime}}&=\bm{n}_{x}\widetilde{u^{\prime\prime}w^{\prime\prime}}+\bm{n}_{y}\widetilde{v^{\prime\prime}w^{\prime\prime}}\end{aligned}\right\\},$
(22)

and we define the angle
$\theta_{rs}=\arctan{\left(\widetilde{w^{\prime\prime}u_{n}^{\prime\prime}}/\widetilde{u_{\xi}^{\prime\prime}u_{n}^{\prime\prime}}\right)}$
between the resultant shear Reynolds stress and the chordwise $\xi-$direction,
and the angle $\theta_{ms}=\arctan{\left(\overline{\partial{w}/\partial
h}/\overline{\partial u_{\xi}/\partial h}\right)}$ between the mean shear
direction and the chordwise direction. The variation of the directional angles
of shear Reynolds stress and mean-flow shear at different chordwise locations
are shown in figure 17 and some mean-flow velocity components are also shown
for better understanding the behaviours.

\begin{overpic}[width=208.13574pt]{Xi1401_RsMs.eps} \put(35.0,60.0){$(a_{1})$} \put(65.0,60.0){$(a_{2})$} \end{overpic} | \begin{overpic}[width=208.13574pt]{Xi1601_RsMs.eps} \put(35.0,60.0){$(b_{1})$} \put(65.0,60.0){$(b_{2})$} \end{overpic}
---|---
\begin{overpic}[width=208.13574pt]{Xi1801_RsMs.eps} \put(35.0,60.0){$(c_{1})$} \put(65.0,60.0){$(c_{2})$} \end{overpic} | \begin{overpic}[width=208.13574pt]{Xi2001_RsMs.eps} \put(35.0,60.0){$(d_{1})$} \put(65.0,60.0){$(d_{2})$} \end{overpic}
Figure 17: $(a_{1}-d_{1})$ Variation of the directional angles ($\theta_{rs}$
and $\theta_{ms}$ in degree) of shear Reynolds stress and mean flow shear with
distance from the wall, at several chordwise locations. $(a_{2}-d_{2})$ The
corresponding mean velocity profiles at relative spanwise locations. $(a-d)$
stand for chordwise locations at $s_{\xi}=4.22,8.71,13.50$ and $18.61$,
respectively. Some velocity component is enlarged for clarity.

At different chordwise stations, the distribution of the angles
$(\theta_{rs},\theta_{ms})$ is generally similar and can be divided into three
regions along the wall-normal direction. First is the near-wall region, where
both $\theta_{rs}$ and $\theta_{ms}$ exhibit positive angles. There is a
significant discrepancy between the two at the wall due to the shear Reynolds
stress being zero there, while the mean shear gradient has a substantial
value. As the distance from the wall increases, the two angles rapidly
converge. In regions with negligible transverse pressure gradients and
velocity, $\theta_{rs}$ and $\theta_{ms}$ approximate equality, indicating a
linear relationship between the shear Reynolds stress and the mean flow shear
gradient (figure 17$(a_{1})$). As the transverse velocity increases, the
difference between the two angles in this region also grows, suggesting a
nonlinear relationship between the Reynolds shear stress and the mean flow
shear when the transverse components ($\overline{u}_{\xi}$ in figures
17$(a_{2}-d_{2})$) become sufficiently large (figures 17$(b_{1}-d_{1})$). When
the wall-normal distance further increases, a sudden change in the angles is
observed. This is attributable to the pressure gradient causing the velocity
profile to overshoot, resulting in a reversal of the mean flow gradient from
positive to negative. A similar phenomenon occurs in the shear Reynolds
stress, but its change lags behind the mean flow shear, with the extent of
this lag increasing with higher transverse velocities. After these angular
transitions, both $\theta_{rs}$ and $\theta_{ms}$ quickly return to similar
levels, indicating that in the outer region of the boundary layer, at a
certain distance from the wall, the shear Reynolds stress and mean flow shear
again appear to exhibit a linear relationship. This observation indicates
that, in the case of three-dimensional flows, the relationship between the
shear Reynolds stress and the mean flow shear is inherently nonlinear.
Consequently, a simple linear approximation would introduce certain
inaccuracies.

##  4 Conclusions

Overall, this study represents the second part of our series on hypersonic
swept blunt-body flows. In this segment, we have examined the characteristics
of three-dimensional turbulent boundary layers in the presence of transverse
flow and pressure gradients. The main findings of this study are as follows:

  * •

In the flow over a swept blunt body discussed in this paper, even without the
assumption of infinite sweep, if the boundary layer reaches a fully developed
turbulent state, the subsequent flow state also satisfies the infinite sweep
assumption—the boundary layer is homogeneous along the spanwise direction.

  * •

We examined the law-of-the-wall and the temperature-velocity relationship,
which have been widely applied in two-dimensional turbulent boundary layers,
within the context of three-dimensional boundary layers. Our findings indicate
that, when the transverse velocity is not excessively high, the spanwise
velocity still adheres to some classical velocity transformation
relationships. Excluding the Trettel & Larrson transformation, existing
transformation forms generally yield good predictive results, with the
transformations proposed by Griffin et al. (2021) and Volpiani et al. (2020)
aligning best with incompressible scaling. However, as the transverse velocity
further increases, the spanwise velocity no longer conforms to traditional
scaling laws. Instead, the streamwise velocity $u_{s}$, in the present
condition, conforms to traditional scaling laws and classical transformations
can collapse the data to the log-law of the incompressible ones. We further
examined the applicability of the temperature-velocity relationship proposed
by Zhang et al. (2014) under such three-dimensional conditions. The study
found that, in the absence of pressure gradients or when the transverse
velocity is relatively low, the traditional temperature-velocity relationship
still holds well. However, when the pressure gradient and transverse velocity
are significant, the predictive accuracy of the temperature-velocity
relationship markedly diminishes. The potential reason for this deviation is
that in boundary layers with pressure gradients, the velocity profile along
the direction of the pressure gradient exhibits an overshoot, which
significantly alters the corresponding relationship between temperature and
velocity, leading to an overestimation in the predictive results.

  * •

Analysis of fluctuating velocities, temperature, Reynolds stresses, Reynolds
stress anisotropy tensors, and turbulent kinetic energy in three-dimensional
boundary layers indicates that the fundamental turbulent characteristics
remain qualitatively consistent with those in two-dimensional turbulent
boundary layers. Typical near-wall streak structures still appear in three-
dimensional turbulent boundary layers and correspond with the energetic
structures in the outer region. However, due to the three-dimensional effects,
turbulent energy is redistributed within the three-dimensional turbulent
boundary layer. This redistribution causes the turbulent energy to align more
with the three-dimensional mean flow direction, resulting in the streak
structures tending to orient along the direction of the external streamlines.

  * •

By analyzing and comparing the directions of the shear Reynolds stress and the
mean flow shear, the study found that in the near-wall region, when the
transverse flow velocity is relatively low, the two align well. However, when
the transverse velocity is high, there is a significant deviation between
their directions. Additionally, with the introduction of a pressure gradient
in the transverse direction, the velocity profile in the corresponding
direction exhibits an overshoot, which alters the mean flow shear direction.
The shear Reynolds stress also undergoes a similar change, but with a certain
lag, and this lag increases with higher transverse velocity. Finally, in the
outer region, some distance away from the wall, the directions of the shear
Reynolds stress and mean flow shear are essentially consistent. This indicates
that in three-dimensional boundary layers with transverse flow, the
relationship between shear Reynolds stress and mean flow shear is partitioned:
it is nonlinear in the inner region and can be approximated to return to a
linear relationship in the outer region.

Finally, we would like to say that the leading edge of the swept configuration
is an ideal model for studying three-dimensional turbulent boundary layers. It
effectively bridges the traditional, unified concepts of two-dimensional
turbulent boundary layers with the unique effects inherent to three-
dimensional turbulent boundary layers.

[Supplementary data]

[Acknowledgements]We acknowledge Yancheng MetaStone Tech. Co. for prociding us
with all the computational resources required by this work. Useful discussions
with Professor Chunxiao Xu, Professor Qibing Li of Tsinghua University,
Professor Jie Ren of Beijing Institute of Technology, Dr. Jian Liu of China
Aerodynamics Research and Development Center and Dr. Yi Huang of China Academy
of Launch Vehicle Technology are gratefully acknowledged.

[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
and the National Key Project GJXM92579 and the Grants 20231001 in
Supercomputing Center in Yancheng.

[Declaration of interests]The authors report no conflict of interest.

[Data availability statement]The full data set of the simulations is of the
order of thousands of gigabytes. By contacting the authors, a smaller subset
can be made available.

[Author ORCIDs]

Youcheng Xi, https://orcid.org/0000-0002-6484-0231;

Bowen Yan, https://orcid.org/0009-0002-0655-9414;

Guangwen Yang, https://orcid.org/0000-0002-8673-8254

Song Fu, https://orcid.org/0000-0003-2052-7435

[Author contributions] Youcheng Xi: Funding acquisition, Conceptualization,
Data curation, Formal analysis, Coding, Investigation, Methodology,
Validation, Writing original draft. Bowen Yan & Guangwen YANG: Software and
optimization on high performance computing, Computational Resources. Song Fu:
Funding acquisition, Supervision, Resources, Writing-review & editing.

## References

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Engel_Ralph

16884 DESY-PROC-2017-XX Patras 2017

# Search for hidden-photon Dark Matter with FUNK

R<EMAIL_ADDRESS>D. Veberič1, C. Schäfer1, A.
Andrianavalomahefa1, K. Daumiller1, B. Döbrich2, J. Jaeckel3, M. Kowalski4,5,
A. Lindner4, H.-J. Mathes1, J. Redondo6, M. Roth1, T. Schwetz-Mangold1, and R.
Ulrich1
1Institute for Nuclear Physics  Karlsruhe Institute of Technology (KIT)
Germany
2Physics Department  CERN  Geneva  Switzerland
3Institute for Theoretical Physics  Heidelberg University  Germany
4Deutsches Elektronen Synchrotron DESY  Hamburg  Germany
5Department of Physics  Humboldt University  Berlin  Germany
6Department of Theoretical Physics  University of Zaragoza  Spain

###### Abstract

It has been proposed that an additional U(1) sector of hidden photons could
account for the Dark Matter observed in the Universe. When passing through an
interface of materials with different dielectric properties, hidden photons
can give rise to photons whose wavelengths are related to the mass of the
hidden photons. In this contribution we report on measurements covering the
visible and near-UV spectrum that were done with a large, 14 m2 spherical
metallic mirror and discuss future dark-matter searches in the eV and sub-eV
range by application of different electromagnetic radiation detectors.

For the introduction to the hidden-photon physics and related extension of the
Standard Model see [1, 2]. For results of a similar experiment, although with
a smaller mirror see [3, 4].

##  1 Experimental setup

For this experiment a mirror composed of 36 segments is used. For more details
on the setup see [5, 6]. The experiment is set-up in a light-tight window-less
room with concrete walls of at least 2 m thickness. The inner area (see Fig.
1), encompassing the camera and the mirror, is additionally light insulated
with a thick black curtain and a 120 $\upmu$m layer of opaque polyethylene
foil.

As the light detector a 29 mm diameter photomultiplier (PMT) ET 9107BQ with
very low dark-current properties was chosen. The PMT has a blue-green
sensitive bialkali photocathode with the quantum efficiency extended into the
ultra-violet range with the peak quantum efficiency of around 28% and
excellent single electron and pulse-height resolution, suitable for the photon
counting. The PMT camera is placed on a motorized linear-stage that can drive
it (perpendicularly to the mirror axis) in and out of the center of the
spherical mirror. The PMT front is additionally equipped with a motorized
shutter that can obscure the entrance of photons.

Signals from the PMT were digitized with the PicoScope 6404D digital
oscilloscope. In Fig. 2 two examples of triggered traces are given. A single-
photon (SP) pulse can be observed on the left and a trace containing several
pulses in the 1.6 $\upmu$s trigger window is shown on the right. Traces with
multiple pulses were discarded since they can be produced only by cosmic-ray
showers. Based on measurements of the SP charge spectrum with an LED flasher,
Fig. 3-left, a cut on the allowed range of observed charges was also applied,
as seen in Fig. 3-right. The efficiency of the latter cut on SP traces is
estimated to be 75%.

##  2 Preliminary limit on mixing parameter

Figure 1: Schematic (left) and photo (right) of the experimental setup.

Figure 2: Typical examples of captured traces with single pulse (left) and
many pulses within a short time span (right).

Figure 3: _Left:_ Charge distribution for a flasher run with very low power
setting and, therefore, composed mostly by single photo-electrons. _Right:_
Charge distribution observed in one of the measurement runs. Note that in both
cases $q=\lg(Q/\text{e})$ where $Q$ is the charge of a pulse.

The selected photon counts for a 30-day run performed in February and March
2017 can be seen in Fig. 4-left. The data was taken in cycles of four 60 s
measurements performed in all four possible combinations of the two positions
of the PMT camera (_in_ and 8 cm _out_ of the center), and the two positions
of the shutter (_open_ and _close_), as schematically shown in Fig. 4-right.
The average rate of the whole run is $R=0.535$ Hz and the relative differences
$\Delta R$ in the four different configurations are shown in Fig. 4-right.

The difference of the count rates between _open_ and _close_ for the PMT being
_in_ the radius point is proxy for the dark-matter signal. With the shutter
_open_ there are $\Delta R=0.0032\pm 0.0014$ Hz more counts registered than
with _closed_. Ignoring for a moment any possible systematic effects, we
obtain the limit shown in Fig. 5 denoted with _FUNK sensitivity_. To determine
possible systematic uncertainties that might be related to temperature changes
or the limited accuracy of the measurement time, we also compare the rates
with the _closed_ PMT _in_ and _out_ of the radius point. The two count rates
agree within the statistical uncertainty ($\Delta R=0.0007\pm 0.0014$ Hz).
Nevertheless, the comparison of the count rates for the _open_ PMT _in_ and
_out_ of the radius point we found significantly larger count rate for the PMT
_open_ and _out_ of the radius point, possibly related to the different
imaging properties of the setup in the two positions. Additional measurements
are in progress to better understand this systematic behavior. For now we
treat this difference ($\Delta R\approx 0.025$ Hz) as an upper limit of the
overall systematic uncertainty of the measurement and use it to derive a
preliminary upper limit [1] on the magnitude of the mixing parameter $\chi$ in
the sensitivity range of the PMT, see the line denoted _FUNK preliminary
(sys)_ in Fig. 5.

Summary. No significant signal was found. The detailed analysis of the data is
still ongoing, thus here we are reporting only preliminary results with a
maximally conservative estimate of possible systematic uncertainties.

Future plans. We are planning further searches for possible hidden-photon dark
matter with measurements extended into the MHz, GHz, and THz range.

Acknowledgments. We gratefully acknowledge partial support from the Helmholtz
Alliance for Astroparticle physics (HAP), funded by the Initiative and
Networking Fund of the Helmholtz Association.

Figure 4: _Left:_ observed pulse rate in one of the measurement runs. _Right:_
A measurement is composed of many event cycles, where in each cycle four
different 60-second measurements are performed. The schematic show these four
different combinations obtained with open/closed shutter and with camera
in/out.

Figure 5: Preliminary limits on the mixing parameter $\chi$ derived from
systematic dominated (blue) and statistics dominated (red) assumptions. The
right figure is an enlargement of the right plot around the dark-matter mass
window corresponding to the optical emission of the visible photons. The
previous result obtained with a CCD camera [5] is also shown (light-blue).

## References

  * [1] B. Döbrich _et al._ , “Hidden Photon Dark Matter Search with a Large Metallic Mirror,” Proc. 10th Patras Workshop 2014, Geneva, Switzerland, [arXiv:1410.0200].
  * [2] B. Döbrich _et al._ , “The FUNK search for Hidden Photon Dark Matter in the eV range,” Proc. Photon 2015 Conf., Novosibirsk, Russia, [arXiv:1510.05869].
  * [3] J. Suzuki, T. Horie, Y. Inoue, and M. Minowa, “Experimental Search for Hidden Photon CDM in the eV mass range with a dish antenna,” JCAP 09, 042 (2015), [arXiv:1504.00118].
  * [4] J. Suzuki, Y. Inoue, T. Horie, and M. Minowa, “Hidden photon CDM search at Tokyo,” Proc. 11th Patras Workshop 2015, Zaragoza, Spain, [arXiv:1509.00785].
  * [5] D. Veberič _et al._ , “Search for dark matter in the hidden-photon sector with a large spherical mirror,” Proc. 34th Int. Cosmic Ray Conf. 2015, The Hague, The Netherlands, PoS ICRC2015, 1191 (2016) [arXiv:1509.02386].
  * [6] D. Veberič _et al._ , “Search for hidden-photon dark matter with the FUNK experiment,” Proc. 35th Int. Cosmic Ray Conf. 2017, Busan, Korea, PoS ICRC2017, 880 (2017).


We define $\beta': \V{\ZOrientation{T}{}} \to 2^{\V{\DirM{G}{M}}}$ and $\gamma: \E{\ZOrientation{T}{}} \to 2^{\V{\DirM{G}{M}}}$.
First, for every edge $e = tt'$ we define a guard as follows.

\begin{align*}
\Fkt{\gamma}{e} &\coloneqq \Set{e' \in M \mid \Abs{e' \cap X_{t'}} = 1 \text{ and } \Abs{e' \cap X_{t}} = 1}
\end{align*}

This is well-defined since the edge also intersects $\Compl{X_{t_i}}$ in exactly one vertex.
Second, for every $t$ we define $\beta$ as follows.

\begin{align*}
\Fkt{\beta'}{t} &\coloneqq \Set{e \in M \mid e \subseteq X_t}
\end{align*}

Now the edges that occur in the tight cuts are not yet in any of the bags $\beta'$ defines.
So for every such edge $m$ pick any vertex $x \in \{t \mid e \in \Fkt{\gamma}{t}\text{ and for all $t'$  with } t' >_{\ZOrientation{T}{}} t \text{ holds } e \cap X_t = \emptyset\}$, where $<_{\ZOrientation{T}{}}$ denotes the natural order of $\Fkt{V}{T}$ induced by $\ZOrientation{T}{}$.
Then we add $m$ to its bag $\Fkt{\beta'}{x}$ obtaining the map $\beta: \V{\ZOrientation{T}{}} \to 2^{\V{\DirM{G}{M}}}$ that actually forms a partition of $\V{\DirM{G}{M}}$.

Now, we need to show that $\Brace{\ZOrientation{T}{},\beta,\gamma}$ is a directed tree decomposition.
Since the cuts are tight, every guard has size exactly $1$.
Every edge $e \in M$ is contained in exactly one bag and the bags have size $2$ or at most $3$ (which is case only if they contain the guard-vertex), thus the width of the decomposition is at most $2$.

At last suppose we have a directed walk leaving some subtree and returning without meeting the guard.
Then we get a directed cycle in $\DirM{G}{M}$ passing one of the tight cuts.
But in $G$ this corresponds to an even cycle having two non-matching edges in the tight cut.
Consider the matching $M'$ such that this cycle is an $M$-$M'$-conformal cycle and otherwise $M$ and $M'$ are the same, then $M'$ has two edges in the cut contradicting its tightness.

§ CONCLUSION

Since the $M$-width of a bipartite matching covered graph is entirely determined by the $M$-width of its braces, results similar to <ref> should be possible for higher values of $k$.
There should not be too many $k$-extendable braces of $M$--width exactly $k$.
However, suppose we know all allowed braces for $M$--width $k-1$ there might be several braces of -width $k-j$ for small $j$ with $M$--width equal to $k$.
It would be interesting to see if one can develop a method to determine those braces, at least for $k\in\Set{4,6}$.
Characterising the bipartite matching covered graphs of $M$--width $k$ immediately yields characterisations for digraphs of cycle-width $k$ as we have seen in <ref> and might therefore be of higher interest than braces of higher -width.

We believe the reason for the existence of nice results like the matching elimination width for braces of -width $2$ lies in the linear structure of their optimal decompositions.
For braces of higher width the decompositions quickly become non-linear and therefore much harder to handle.
Similar to other width-parameters one could restrict the structure of the decomposition tree to one such that the spine of the spine is a path.
It would be interesting to see whether every brace of bounded linear -width has a linear decomposition with properties similar to those of the optimal width-$2$-decompositions discussed in this paper.
In other words, does bounded linear -width imply bounded matching elimination width?

We wish to acknowledge Nishad Kotharis help on this paper.
Especially the section on cubic trees and the discovery of the linear nature of optimal width $2$ -decompositions for braces are due to his own curiosity and many fruitful discussions over the course of a month.

[1]
Stefan Arnborg.
Efficient algorithms for combinatorial problems on graphs with
bounded decomposability—a survey.
BIT Numerical Mathematics, 25(1):1–23, 1985.

[2]
Richard A Brualdi and Hazel Perfect.
Extension of partial diagonals of matrices i.
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