xxx 2021 -idFLOxxxxx Wangsawijaya, D. D., Nicolai, C., & Ganapathisubramani,
B.

*Corresponding author. E-mail<EMAIL_ADDRESS>
# Time-averaged velocity and scalar fields of the flow surrounding a group of
cylinders

D. D. Wangsawijaya1∗ C. Nicolai1 and B. Ganapathisubramani1 Aerodynamics and
Fluid Mechanics Research Group, Faculty of Engineering and Physical Sciences,
University of Southampton SO17 1BJ, United Kingdom

(Received: XX 2021; Revised: XX XX 2021; Accepted: XX XX 2021)

###### Abstract

We examine the characteristics of mean flow and scalar concentration
characteristics of a turbulent boundary layer flow impinging on a cluster of
tall obstacles (which can also be interpreted as a porous obstruction). The
cluster is created with a group of cylinders of diameter $d$ and height $h$
arranged in a circular patch of diameter $D$.The solidity of the
patch/obstruction is defined by $\phi$ (the total planar area covered by
cylinders), which is systematically varied ($0.098\leq\phi\leq 1$) by
increasing the number of cylinders in a patch ($N_{c}$). A point source is
placed at ground level upstream of the patch and its transport around the
patch is examined. Time-averaged velocity and scalar fields, obtained from
simultaneous planar PIV-PLIF (particle image velocimetry-planar laser-induced
fluorescence) measurements, reveal that the characteristics of wake and flow
above porous patches are heavily influenced by $\phi$. In particular, we
observe that the horizontal and vertical extent of the wake and scalar
concentration downstream of the patches decreases and increases with $\phi$,
respectively. Here, the recirculation bubble is shifted closer to the trailing
edge of the patches as $\phi$ increases, limiting the flow from convecting
downstream, decreasing the scalar concentration and virtually ‘extending’ the
patch in streamwise direction. As the bubble forms in the trailing edge,
vertical bleeding increases and hence the concentration increases above patch
where the cylinders appear to be ‘extend’ vertically towards the freestream.

###### keywords:

mixing and dispersion, wakes, turbulent boundary layers, urban fluid dynamics,
mixing enhancement, turbulence control

††articletype: RESEARCH ARTICLE

Impact Statement

Canopy flow covers extensive phenomena encountered in both nature and
engineering applications. The characteristics of canopies (also referred to as
porous obstructions) is known to affect the mixing, momentum, and scalar (e.g.
mass and heat) exchange processes in the flow. Examples include flow around
urban areas, wind/tidal farms, patches of roughness in ship hulls, in-line
heat-exchangers among various others. In almost all these applications, the
patch of obstacles/roughness are typically characterised by its geometric
properties such as solidity (frontal and plan) or porosity and it is important
to understand the flow/scalar characteristics as well as transport processes
for varied geometric parameters. For instance, an urban canopy with given
distribution of buildings and a pollutant source at ground level could result
in different wake/dispersion characteristics depending on the frontal or plan
solidity of distribution. This study attempts to carry out simplified
experiments that will enable us to understand the intricacies of scalar-
momentum transport around such canopies and allow us to build useful models
for engineering design and/or prediction for engineering and environmental
flows.

##  1 Introduction

Canopy flow refers to a scenario where a boundary layer develops over and
around a porous surface/obstruction. Examples of such phenomenon are abound in
both nature and engineering applications, including but not limited to: flows
past aquatic and terrestrial vegetations, atmospheric boundary layers
developing over forests, clusters of tall buildings in cities, and wind/tidal
farms, which consist of arrays of wind/tidal turbines. These obstructions are
known to affect the scalar, momentum, and energy transport process in the flow
within the canopy and downstream of the canopies. Aquatic vegetations, for
example, provide shielding (Kemp et al., 2000), affect the sediment formation
and erosion (De Langre, 2008; Nepf, 2012a, b; Tinoco and Coco, 2016), and also
carbon sequestration in coastal ecosystems (Lei and Nepf, 2021). Urban and
vegetation canopies induce coherent, multi-scale turbulent motions in the
atmospheric boundary layer (Katul et al., 1997; Kanda, 2006; Huang et al.,
2009a, b, 2011; Li and Bou-Zeid, 2011), which governs flow mixing, momentum,
heat and mass exchange within the layer (Raupach and Thom, 1981; Finnigan,
2000; Poggi et al., 2004), including pollutants and other chemical compounds
(Poggi et al., 2006; Aristodemou et al., 2018). While the configuration and
tuning of turbines in wind/tidal farms are vital in maximising power output
(Vennell, 2011; Myers and Bahaj, 2012), they may also alter the meteorological
pattern and ecosystem of the installation sites. For example, installation of
wind turbines has been shown to affect the heat distribution and humidity of
the surrounding air (Baidya Roy et al., 2004; Rajewski et al., 2016), while
wakes generated by tidal turbines can potentially modify sedimentation process
and bacterial population in estuarine waters (Ahmadian et al., 2012).
Considering its extensive industrial applications and environmental impacts,
understanding the transport mechanism of the flow surrounding porous
obstructions is, therefore, essential.

Previous studies on porous obstructions (such as examples given above)
typically involved some simplifications from real-life conditions. The
obstruction may be simplified as an array of cylinders mounted on an isolated
patch with an incoming flow upstream of the cylinders (illustrated in figure
1). Numerical simulations and experiments have been conducted on the flow
surrounding such cylindrical array (Ball et al., 1996; Nepf et al., 1997;
Nepf, 1999; White and Nepf, 2003; Tanino and Nepf, 2008; Nicolle and Eames,
2011; Rominger and Nepf, 2011; Chen et al., 2012; Zong and Nepf, 2012; Chang
and Constantinescu, 2015; Chang et al., 2017), where the cylinders are as tall
as, or protruding from, a free surface, as commonly observed in emergent
aquatic vegetations. Here, the flow is dominated by Kármán vortex street in
wall-normal axis trailing behind the cylinders and thus can be considered two-
dimensional (Taddei et al., 2016; Nicolai et al., 2020). A more general
approach is to fully submerge the cylinders in the flow such that $h\ll\delta$
(figure 1b). Studies have been conducted for cylinders attached to a bottom
wall (as in figure 1b, see also Chen et al., 2013; Taddei et al., 2016;
Nicolai et al., 2020) and/or suspended from a free surface (Plew, 2011; Tseung
et al., 2016; Zhou and Venayagamoorthy, 2019). In either cases, the patch is
considered as 3-D, as flow mixing occurs on the sides of, downstream of, and
above the body. Taddei et al. (2016); Zhou and Venayagamoorthy (2019); Nicolai
et al. (2020) have identified three mechanisms in which the flow may escape
(‘bleed’) from a fully submerged porous obstruction: (i) the lateral bleeding
from the interior of the patch to the sides (spanwise direction, see figure
1a), (ii) the trailing edge bleeding, which is responsible in the formation of
wake downstream of the body (figure 1a), and (iii) the vertical bleeding, in
which the flow escapes from the interior to the top surface of the patch and
towards the freestream (figure 1b).

Figure 1: Illustrations of (a) lateral and trailing edge bleeding, and (b)
vertical bleeding of cylindrical arrays on a circular patch. $U_{bl}$ is the
incoming boundary-layer velocity, whose thickness is $\delta_{bl}$, and $h$ is
the height of the cylinders.

It has been reported that the solidity of a porous obstruction is a critical
parameter in determining drag (Nepf, 1999; Taddei et al., 2016; Chang et al.,
2017; Zhou and Venayagamoorthy, 2019), the structure of wake formed by the
obstruction (Ball et al., 1996; Nicolle and Eames, 2011; Zong and Nepf, 2012;
Chen et al., 2012; Chang and Constantinescu, 2015; Chang et al., 2017; Zhou
and Venayagamoorthy, 2019; Nicolai et al., 2020), scalar dispersion (Nepf et
al., 1997; White and Nepf, 2003; Tanino and Nepf, 2008), and sediment
disposition (Chen et al., 2012; Chang and Constantinescu, 2015). For the
cylindrical array shown in figure 1, the solidity is measured in terms of the
planar area covered by cylinders per total surface, $\phi\equiv
N_{c}(d/D)^{2}$ (also commonly referred to as the solid volume fraction),
where $N_{c}$ is the number of cylinders in a circular patch, $D$ is the
diameter of the circular patch, and $d$ is the diameter of the cylinder (see
figure 2a,d for graphical descriptions of $D$ and $d$). It has been observed
by Nicolle and Eames (2011); Taddei et al. (2016); Zhou and Venayagamoorthy
(2019) that total drag of the patch $C_{D}$ increases with the increasing
$\phi$ then drops for a solid obstruction ($\phi=1$, figure 2d). Higher patch
solidity also yields shorter and taller wake (in streamwise and wall-normal
direction, respectively, see Nicolai et al., 2020) in the trailing edge, and
increases both lateral and vertical bleeding (Zhou and Venayagamoorthy, 2019,
see also figure 1). Further, it has been reported in Nicolle and Eames (2011)
that at low $\phi$ ($\phi<0.05$) each cylinder in an array behaves as a single
obstruction, and as the solidity increases to $\phi>0.15$ the cylindrical
array behaves closer to a solid obstruction whose size is equal to that of the
patch ($\phi=1$, figure 2d). Within the range of moderate solidity
($0.05\leq\phi\leq 0.15$), suppression of vortex street formation downstream
of 2-D porous patches has been observed. In its place is a steady wake
(constant velocity) region, which is a consequence of flow bleeding. Zong and
Nepf (2012) and Chen et al. (2012) showed that the streamwise length of this
steady wake region scales on $C_{D}aD$, where $a$ is the frontal area per unit
volume $a\equiv N_{c}d/(0.25\pi D^{2})$. Similarly, scaling by $C_{D}ab$ and
$C_{D}ah$ have also been observed in Rominger and Nepf (2011) for a square
patch and Chen et al. (2013) for a fully submerged patch, respectively, where
$b$ is the half-width of the patch.

Although the characteristics of the wake and drag of a porous obstruction
shown in figure 1 have been discussed in details for both 2-D and 3-D patches,
an element that is still missing from these analyses is the intricacy between
bleeding and scalar transport in 3-D porous patches. So far, discussions about
scalar transport are limited to lateral and longitudinal dispersion in 2-D
patches (White and Nepf, 2003; Tanino and Nepf, 2008), sediment deposition
(Chen et al., 2012; Tinoco and Coco, 2016), qualitative flow visualisation
using dye injection (Ball et al., 1996; Rominger and Nepf, 2011; Zong and
Nepf, 2012), and non-simultaneous velocity and scalar measurements due to
limitations of the experimental facilities (Nepf, 1999; Tanino and Nepf,
2008). Here, a 3-D porous patch serves as a more realistic model of urban
landscapes, vegetation canopies, and wind farms, which are typically subsumed
within the atmospheric boundary layer. Inclusion of scalar measurements (in
addition to the velocity components) has many potential benefits, for example:
in air pollution studies, where one might consider cluster of densely
populated urban landscape as a porous obstruction and the pollutant as scalar,
or in studies about sediment disposition in ocean beds.

In this study we aim to address the following question: given a point source
of scalar located at ground level just upstream of 3-D porous patches, what
are the flow and scalar characteristics in the wake and above the patches for
various solidities? To answer this question, we conduct a parametric study of
turbulent boundary layers impinging on an array of cylinders mounted on a
circular patch (see figure 1). A point source upstream of the patch is created
by releasing a passive fluorescent dye. The solidity $\phi$ is varied by
systematically increasing the number of cylinders in the patch, such that
$0.098\leq\phi\leq 1$. This covers both medium and high solidity range
described in Nicolle and Eames (2011). The axis system $(x,y,z)$ corresponds
to the streamwise, wall-normal, and spanwise direction, respectively.
Simultaneous planar PIV-PLIF measurements in $x$–$y$ plane provide the
instantaneous streamwise–wall-normal velocity components $u$ and $v$,
respectively, and scalar concentration field $c$, which can be decomposed
further into their time-average and fluctuation about the mean

$u(x,y,t)=U(x,y)+u^{\prime}(x,y,t);\quad v=V+v^{\prime};\quad c=C+c^{\prime}$
(1)

It should be noted that due to the choice of the measurement plane, discussion
about flow bleeding in the current study is limited only to the trailing edge
and vertical bleeding (figure 1).

##  2 Experimental setup

###  2.1 Test cases

Experiments are carried out in the closed loop water flume at the University
of Southampton. The test section has the size of 6250 mm $\times$ 1200 mm
(length $\times$ width). In all measurements, the water level is kept constant
at $600$ mm and the freestream velocity is set to $U_{\infty}\approx 0.45$
ms-1.

A circular patch with a diameter ($D$) of 100 mm is mounted on the floor of
the tunnel at 5500 mm downstream of the test section entrance, which
corresponds to the patch Reynolds number $Re_{D}\equiv U_{\infty}D/\nu\approx
4.5\times 10^{4}$, where $\nu=1.0035\times 10^{-6}$ m2s-1 is the kinematic
viscosity of water at $20^{\circ}$C. The patch constitutes an array of rigid
cylinders, each has the diameter of $d=5$ mm and height $h=30$ mm (figures 1b
and 2a,d). The cylinders are arranged along concentric, evenly-spaced circles
whose axis is at the centre of the circular patch, such that the distance
between two consecutive cylinders is constant within the patch (see figures
2a–c). The solidity of the patch is parametrically varied by gradually
increasing the number of cylinders $N_{c}$ while keeping $D$ and $d$ constant.
Table 1 and figures 2(a–c) show three different circular patches: C39, C64,
C133, which correspond to $N_{c}$ = 39, 64, 133, which cover a range of patch
density of $0.1\lesssim\phi\lesssim 0.33$ ($2.5\lesssim aD\lesssim 8.5$). A
solid obstruction case, i.e. a cylinder covering the entire circular patch
(such that $N_{c}=1$ and $\phi=1$), is included in addition to the porous
cases as case ‘CS’ in table 1 and figure 2(d). The solid cylinder in case CS
has an equal height $h$ to the cylinder arrays in other test cases.

###  2.2 Validation with smooth-wall turbulent boundary layer

A smooth-wall turbulent boundary layer (‘TBL’) case is first tested to
validate PIV measurements in the present study. The measurement is conducted
in the same test section as other test cases by removing the circular patch.
This flow has the Reynolds number (based on momentum thickness $\theta$) of
$Re_{\theta}\equiv\theta U_{\infty}/\nu\approx 3000$ and friction Reynolds
number $Re_{\tau}\equiv\delta_{bl}U_{\tau bl}/\nu\approx 1500$, where
$\delta_{bl}=82\pm 3\%$ mm is the boundary-layer thickness, $U_{\tau
bl}\equiv\sqrt{\tau_{0}/\rho}=0.019\pm 2\%$ ms-1 is the friction velocity for
this case, $\tau_{0}$ is the wall shear stress, and $\rho=998.12$ kg m-3 is
the density of water at 20∘C. Figure 2(e) shows the profiles of mean
streamwise velocity $U^{+}\equiv U/U_{\tau bl}$ and the variance of velocity
fluctuations
$\overline{u^{\prime}u^{\prime}}^{+}\equiv\overline{u^{\prime}u^{\prime}}/U_{\tau
s}^{2}$ as functions of wall-normal location $y^{+}\equiv yU_{\tau s}/\nu$,
where subscript ‘+’ denotes viscous scaling and the viscous lengthscale
$\nu/U_{\tau s}=0.053$ mm. The logarithmic region of the mean velocity
collapses to the line $1/\kappa\log y^{+}+A$, where $\kappa=0.41$ is the von
Kármán constant and the $A=5.0$ is the log-law shift (Schlichting and Gersten,
2017). The variance $\overline{u^{\prime}u^{\prime}}$ is in good agreement
with that obtained from direct numerical simulation (DNS) of Sillero et al.
(2014) at $Re_{\theta}=4000$. Note that the DNS results presented in figure
2(e) are filtered according to the spatial resolution of the PIV measurements
(Lee et al., 2016), $\Updelta x^{+}\times\Updelta y^{+}=15\times 15$ in
streamwise and wall-normal directions, respectively.

Cases | $N_{c}$ | $d/D$ | $h/D$ | $\phi$ | $aD$ | Colour | PIV | PLIF | FOV 1 | FOV 2 | FOV 3
---|---|---|---|---|---|---|---|---|---|---|---
C39 | 39 | 0.05 | 0.3 | 0.098 | 2.483 | $\bullet$ | ✓ | ✓ | ✓ | ✓ | ✓
C64 | 64 | 0.05 | 0.3 | 0.160 | 4.074 | $\bullet$ | ✓ | ✓ | ✓ | ✓ | ✓
C133 | 133 | 0.05 | 0.3 | 0.333 | 8.467 | $\bullet$ | ✓ | ✓ | ✓ | ✓ | ✓
CS | 1 | 1 | 0.3 | 1 | 1.273 | $\bullet$ | ✓ | ✓ | ✓ | ✓ | ✗
TBL | – | – | – | – | – | – | ✓ | ✗ | ✓ | ✓ | ✗
Table 1: List of cylinder array test cases. $N_{c}$ is the number of cylinders
in a patch, $d$ and $D$ are the diameters of the cylinders and the patch,
respectively, $h$ is the height of the cylinders, $\phi$ is patch solidity,
and $a$ is the frontal area per unit volume. ‘PIV’ and ‘PLIF’ columns show the
availability of PIV and PLIF images of each test case. The last three columns
show the availability of those images in each FOV.

Figure 2: Illustration of cylinder array test cases: (a) C39, (b) C64, (c)
C133, and (d) the solid case (CS) in $x$–$z$ plane (top view), $d$ is cylinder
diameter and $D$ is the circular patch diameter. (e) Mean streamwise velocity
$U^{+}$ ($\circ$) and variance of velocity fluctuation
$\overline{u^{\prime}u^{\prime}}^{+}$ ($\bullet$) of the smooth-wall turbulent
boundary-layer (TBL) case as functions of wall-normal location $y^{+}$ at
$Re_{\theta}\approx 3000$, where $\theta$ is the momentum thickness. Dashed
line ( ) is $1/(0.41)\log y^{+}+5.0$, solid line ( ) is the DNS results of
Sillero et al. (2014) at $Re_{\theta}=4000$, filtered according to the spatial
resolution of the PIV measurements (Lee et al., 2016). Data are downsampled
for clarity.

###  2.3 Particle image velocimetry (PIV) and planar laser-induced
fluorescence (PLIF)

Figure 3: Contours of instantaneous (a) streamwise velocity $u/U_{h/2}$ and
(b) scalar concentration $c/C_{s}$ (in logarithmic scale) of case C39,
obtained from (a) PIV and (b) PLIF measurements, respectively, where $U_{h/2}$
is the incoming velocity at half-height of the cylinders. Dashed lines show
the extent of FOV 1, 2 and 3. Black rectangles illustrate the cylinder array
of height $h$ mounted on a circular patch. The origin $x=0$ is at the centre
of the patch and $y=0$ is on the floor of the tunnel. Measurement data are not
available in the hatched regions (between the cylinders). In (b), blue arrow
shows the location of the scalar point source.

For simultaneous planar PIV and PLIF measurements, the measurement domain is
located in the streamwise–wall-normal ($x$–$y$) plane, slicing the centerline
of the circular patch (figure 3) and illuminated by Nd:YAG 100 mJ pulsed
laser. The measurement domain is created by combining 3 successive experiments
in field-of-view (FOV) 1, FOV 2, and FOV 3, which extends downstream of the
circular patch. Dashed lines in figure 3 illustrate these FOVs and the
overlaps between each FOV. It should be noted that although the PIV and PLIF
measurements are conducted simultaneously in each FOV, these measurements are
not conducted simultanenously between FOV 1, 2 and 3. PIV and PLIF images are
recorded first in FOV 1, then the cameras and optics are traversed downstream
to form FOV 2, then FOV 3. Measurements are conducted in all three FOVs for
the cylindrical array cases (C39, C64, and C133), and only the first two FOVs
for the baseline cases (CS and TBL, see table 1).

In each FOV, PIV images are recorded by a pair of 4 MP CMOS cameras, equipped
with wavelength filters to filter out the PLIF signal. As many as 3000 image
pairs are acquired for each case at the rate of 4 Hz, which corresponds to the
boundary-layer turnover rate of $TU_{\infty}/\delta_{bl}=1.4$. The timing
between a pair of PIV images is $\sim$1 ms ($t^{+}\equiv\Updelta tU_{\tau
bl}^{2}/\nu\approx 0.32$). Image cross-correlation and post-processing of the
resulting velocity vectors are conducted using a commercial software (DaVis,
LaVision GmbH), with the final window size of $24\times 24$ pixels (50%
overlap) and image pixel size of 15 pixels/mm, which corresponds to the
spatial resolution of $\Updelta x^{+}\times\Updelta y^{+}=15\times 15$ for
case TBL. Each FOV has the size of 253 mm $\times 126$ mm ($x\times y$), which
approximately extends 2.5 times the patch diameter (2.5$D$) in streamwise and
4$h$ in height. When the three FOVs are stitched together, the total length in
$x$ is 689 mm ($6.89D$).

For PLIF measurements, a point source is created upstream of the circular
patch by injecting Rhodamine 6G dye solution (Schmidt number $Sc\approx 2500$)
to the flow at the rate of 30 cc/min. The dye is supplied by gravity through a
thin tube (3 mm diameter) embedded at the floor of the tunnel at 45∘ angle.
The tube is located at 125 mm upstream from the centre the patch ($x/D=1.25$,
see figure 3b). PLIF images are acquired using a 5.5 MP 16-bit sCMOS camera
equipped with a wavelength filter to block all light except the PLIF signal
(Vanderwel and Tavoularis, 2014). To reduce the uncertainty of the scalar
concentration measurements, the laser power is recorded at each pulse by an
in-line power energy monitor. To maximise the signal-to-noise ratio of the
scalar concentration, the concentration of the dye solution at point source
$C_{s}$ is varied based on the location of the FOV relative to the point
source. For FOV 1 (closest to the point source), $C_{s}=0.3$ mg/L, while for
FOV 2 and FOV 3, $C_{s}=10$ mg/L. The measured scalar concentration presented
in this study is normalised by $C_{s}$.

##  3 Results and discussions

Figure 4: Contours of time-averaged streamwise velocity $U/U_{\infty}$ of
case: (a) C39, (b) C64, (c) C133, and (d) CS. Lines are the time-averaged
spanwise vorticity $\Omega_{y}D/U_{h/2}$ with the following contour levels:
$-8,-7,...,-2$ ( ) and $0.4,0.8,...,1.6$ ( ).

Figure 5: Contours of time-averaged scalar concentration $C/C_{s}$ (in
logarithmic scale) of case: (a) C39, (b) C64, (c) C133, and (d) CS. Lines are
the time-averaged wall-normal velocity $V/U_{h/2}$ with the following contour
levels: $-0.07,-0.06,-0.05$ ( ) and $0.075,0.1,...,0.2$ ( ). Measurement data
are not available in the hatched regions.

###  3.1 Trailing edge bleeding, the extent of the wake and scalar
concentration

Figures 5(a–d) show the contours of time-averaged streamwise velocity $U$ for
all porous test cases and the solid case (CS) listed in table 1. The contours
are normalised by $U_{h/2}$, which is defined as the incoming, undisturbed
streamwise velocity upstream of the patch ($x/D=-1.3$) at the half-height of
the cylinder, $y/h=0.5$. The figures show the formation of wake for all porous
and solid cases in the form of velocity deficit downstream of the patches. The
magnitude of the deficit is significant, which signals the presence of patch-
sized flow phenomena (Taddei et al., 2016) as opposed to those of individual
cylinders observed in patches with low solidity (Chen et al., 2012; Chang and
Constantinescu, 2015; Taddei et al., 2016; Nicolai et al., 2020). In figures
5(b–c), blue-shaded contours show $U<0$, which correspond to the recirculation
of the flow downstream of the patch. The strength of the recirculating flow
appears to be affected by the solidity of the patch: recirculation is not
apparent in case C39 (case with the lowest solidity, figure 5a), a weak
recirculation then appears in case C64 (figure 5b), which grows stronger in
case C133 (figure 5c). It is also noted that the recirculation bubbles are
shifted downstream as solidity decreases. In case C64 (figure 5b) the bubble
is observed at $1\lesssim x/D\lesssim 2$, while in case C133 (figure 5c) and
the solid case CS (figure 5d), the bubbles form immediately at the trailing
edge of the patches ($x/D=0.5$). Similar phenomenon has been observed in 2-D
patches (Chang and Constantinescu, 2015), 3-D patches (Taddei et al., 2016;
Zhou and Venayagamoorthy, 2019), and porous plates (Castro, 1971). Contours of
positive (upward) wall-normal velocity $V$ above the patches are shown in
solid lines in figure 5. For the porous cases in figures 5(a–c), vertical
bleeding is apparent above the patches, characterised by the contours of $V>0$
at $-0.5\leq x/D\leq 0.5$ and $y/h\geq 1$. This will be left for further
discussion in §3.2.

Superimposed with the contours of $U$ in figures 5(a–d) are the contours of
time-averaged spanwise vorticity $\Omega_{y}\equiv\partial V/\partial
x-\partial U/\partial y$ normalised by $U_{h/2}/D$. Black solid lines are the
contours of $\Omega_{y}<0$ (clockwise) and the blue lines $\Omega_{y}>0$
(counter-clockwise). Negative $\Omega_{y}$ show the development of shear
layers forming above and downstream of the patches, which appear to be
stronger (higher in magnitude) as solidity increases. Similar to the
observation in Taddei et al. (2016), here we also observe the elevation of the
shear layer further upward from the patches as solidity increases (compare,
for example, figure 5c to figures 5a,b). The lift-off is attributed to
vertical bleeding from the interior of the patches, which is noticeably absent
in the solid case CS (figure 5d) where there is no vertical bleeding.

Figures 5(a–d) show the contours of time-averaged scalar concentration $C$
(normalised by the concentration at the source $C_{s}$ and presented in
logarithmic scale) for all porous test cases and the solid case (CS) listed in
table 1. Dark-shaded contours in figures 5(a–c) illustrate how the scalar
permeates through and downstream of the patches. With less obstructions to
block the flow, high scalar concentration (purple-shaded) is observed
downstream of the C39 and C64 patch in figures 5(a,b), respectively. As
blockage ($\phi$) increases, TE bleeding decreases and consequently, the
downstream concentration decays (see C133 patch, figure 5c). Instead, the
scalar escapes the patch by means of vertical bleeding, as shown by higher
scalar concentration above the patch ($y/h\geq 1$) in figure 5(c). As $\phi$
reaches 1 (solid case), TE and vertical bleeding are non-existent, as
evidenced by $C\approx 0$ above the patch and low magnitude of $C$ downstream
of the patch (figure 5d). Here, the observed concentration is mainly the
result of flow mixing in the wake instead of the scalar permeating through the
patch.

Figure 6: Contours of time-averaged (a) defect velocity $\Updelta
U/U_{h/2}=0.3$, (b) wall-normal velocity $V/U_{h/2}=0.1$ at $x/D\leq 0.5$, and
(c) scalar concentration $C/C_{s}=3\times 10^{-4}$. Colour scheme for each
case (C39, C64, C133, and CS) is described in table 1. The horizontal and
vertical extent of the wake are defined by $L_{w}$ and $H_{w}$, respectively,
while $L_{c}$ and $H_{c}$ are the horizontal and vertical extent of the scalar
dispersion. Hatched regions illustrate the location of the patch.

Figure 7: (a) Normalised length of wake $L_{w}/D$ ($\bullet$) and scalar
dispersion $L_{c}/D$ ($\blacklozenge$) (as illustrated in figure 7) of all
test cases as a function of solidity $\phi$, (b) height of wake $H_{w}/D$
($\bullet$) and scalar dispersion $H_{c}/D$ ($\blacklozenge$), and (c)
integrated velocity bleeding $U_{bleed}/U_{h/2}$ at the trailing edge $x=D/2$.
In (a,b), hatched regions illustrate the sensitivity to the threshold level
defining the length/height ($\pm 20\%$). In (c), $\circ$: same quantities
measured in Nicolai et al. (2020) at $Re_{\theta}\approx 60000$. (d)
$1-(U_{bleed}/U_{h/2})$ as a function of $x$ downstream of the patch,  : the
reference smooth-wall case TBL. (e) Integrated scalar concentration
$C_{te}/C_{s}$ as a function of $x$. Colour scheme for each case (C39, C64,
C133, and CS) is described in table 1.

Figures 5 and 5 show the dependency of wake and scalar concentration
downstream of the porous bodies as a result of TE bleeding. We attempt to
measure the extent of the wake by utilising a similar method in Nicolai et al.
(2020). First, we define velocity defect as $\Updelta U=U_{bl}-U$, where
$U_{bl}(y)$ is the incoming boundary layer velocity profile upstream of the
patch (at $x/D=-1.3$), and arbitrarily choose a threshold for the magnitude of
$\Updelta U$ as a wake definition. Figure 7(a) shows the contours of $\Updelta
U/U_{h/2}$ for all porous cases and the solid case (table 1) at the chosen
threshold of 0.3. With this threshold, we are able to define the length and
height of the wake, $L_{w}$ and $H_{w}$, respectively (figure 7a). $L_{w}$ is
defined as the length of wake measured from the centreline of the patch
($x=0$) to the $x$-maxima of the chosen contour level, while $H_{w}$ is the
height of wake measured from the floor of the tunnel ($y=0$) to the $y$-maxima
of the same contour level. Figures 7(a,b) show the normalised length and
height of the wake, $L_{w}/D$ and $H_{w}/h$, respectively, as a function of
solidity $\phi$. The figures show that $L_{w}$ decreases with increasing
$\phi$, but $H_{w}$ increases (and possibly reaches a maxima at certain value
of $\phi$) and then decreases as $\phi\rightarrow 1$, where the obstacle is no
longer porous and there is no flow escaping from the top of the patch. The
wake ‘blow-up’ (i.e. taller wake) in higher $\phi$ has also been recorded in
Taddei et al. (2016); Nicolai et al. (2020). These tendencies of $L_{w}$ and
$H_{w}$ are insensitive to the change in threshold, as illustrated by the
hatched regions in figures 7(a,b), which correspond to the same quantities
measured when the threshold for $\Updelta U/U_{h/2}$ is varied by $0.3\pm
20\%$. As a comparison, $L_{w}$ and $H_{w}$ measured in Nicolai et al. (2020)
are shown in white-filled circles in figures 7(a,b). The study involved
similar arrays of cylindrical patches of various $\phi$ ($0.018\leq\phi\leq
1$) that impinged a developing turbulent boundary layer at $Re_{\theta}\approx
60000$ (20 times higher than that of the current study). It should be noted
that the measured $L_{w}$ and $H_{w}$ are greater in magnitude as the wake is
defined at a lower threshold of $\Updelta U/U_{h/2}=0.2$ instead of 0.3 due to
the increasing streamwise length of the FOV ($8D$ instead of $6.89D$). It
should also be noted that the values of $H_{w}$ (white-filled circles in
figure 7b) are scaled by a factor of $3.58h/\delta_{bl}$, where 3.58 is the
ratio of incoming boundary layer thickness to cylinder height $\delta_{bl}/h$
in Nicolai et al. (2020). However, the differences between studies do not
change the observed trend in $L_{w}$ and $H_{w}$ as a function of $\phi$.

Similarly, a threshold can also be applied to the scalar concentration field
$C$ to measure the extent of scalar dispersion downstream of the patch. Figure
7(b) shows the contours of $C/C_{s}$ for all porous cases and the solid case
(table 1) at the chosen threshold of $3\times 10^{-4}$. The length and height
of scalar dispersion $L_{c}$ and $H_{c}$, respectively, are measured from the
centreline of the patch $x=0$ to the $x$-maxima of the chosen contour level
and from the the floor of the tunnel $y=0$ to the $y$-maxima, respectively.
Filled diamonds in figures 7(a,b) show the measured $L_{c}$ and $H_{c}$ as a
function of $\phi$. Similar to those of the wake extent, $L_{c}$ and $H_{c}$
decreases and increases, respectively, with increasing $\phi$. It should be
noted that although scalar concentration data are available for case CS, the
magnitude of $C$ downstream of the patch is lower than that of the chosen
threshold (see figure 5d) and thus we are unable to measure $L_{c}$ and
$H_{c}$ of this case.

We further quantify the velocity of TE bleeding $U_{bleed}$. $U_{bleed}$ is
defined as $U$ downstream of the patch ($x\geq D/2$) integrated over $y$. This
definition can be extended further to the scalar concentration at the trailing
edge $C_{te}$, such that

$\left.\begin{array}[]{r
l}U_{bleed}(x)&=\frac{1}{h}\int_{0}^{h}U(x,y)\;\mathrm{d}y\\\
C_{te}(x)&=\frac{1}{h}\int_{0}^{h}C(x,y)\;\mathrm{d}y\end{array}\right\\}x\geq
D/2$ (2)

Figure 7(c) shows the magnitude of $U_{bleed}$ at the trailing edge, $x=D/2$,
as a function of $\phi$. White-filled circles are $U_{bleed}$ measured by
Nicolai et al. (2020) for similar cylindrical arrangements at
$Re_{\theta}\approx 60000$. Despite the difference in the order of $Re$, both
Nicolai et al. (2020) and the present study show that $U_{bleed}$ decreases
with $\phi$ at the trailing edge. Figures 7(d) and 7(e) further show the
defect $1-U_{bleed}/U_{h/2}$ and $C_{te}/C_{s}$ as functions of $x$ downstream
of the patch ($x\geq D/2$). Both figures show the decay of wake and scalar
concentration with $x$, with (generally) faster decay as $\phi$ increases in
the porous cases. In particular, $U_{bleed}$ in figure 7(d) shows that the
solid case CS recovers the fastest to the upstream condition (i.e., the
incoming flow, dashed line is the smooth-wall case TBL), followed by the
highest solidity case C133. We note that the decay of the wake for the porous
and solid cases (figure 7d) and the decay of $C$ for the porous cases (figure
7e) seem to follow a certain power law, possibly similar to that of Wygnanski
et al. (1986) for various solid and porous wake generators, which is left for
future works. The exception to this is the scalar concentration downstream of
case CS (black solid line in figure 7e), which is approximately constant in
$x$ and much lower in magnitude compared to the porous cases.

Figure 8: Contours of scalar concentration $\log_{10}(C/C_{s})$ for cases: (a)
C39, (b) C64, (c) C133, and (d) CS. Blue solid lines ( ) are negative
streamwise velocity, in (b): $U/U_{h/2}=-0.06,-0.04,-0.02$ and in (c,d):
$-0.4,-0.36,...-0.04$.

A closer look into figure 7(e) reveals a curious observation: the integrated
scalar concentration of case C64 peaks at $x/D\approx 1$, and the magnitude of
this peak is much higher than that of cases with lower and higher $\phi$ (C39
and C133). Figures 8(a–d) show the coloured contours of $C$ at the trailing
edge for all porous and solid cases, with blue contour lines superimposed in
the figures showing $U<0$, which corresponds to flow recirculation. For case
C64, figure 8(b) shows a high scalar concentration (purple-shaded region) at
the trailing edge and up to $x/D\approx 1$, which is responsible for the spike
in figure 7(e), immediately followed by the emergence of a recirculation
bubble. In other words, the scalar is trapped between the porous patch
upstream and the recirculation bubble that prevents the scalar from convecting
downstream. By contrast, case C39 (figure 8a) that has lower solidity shows no
recirculation bubble, and thus the scalar is freely transported downstream of
the patch. Case C133 (figure 8c), on other hand, shows the recirculation
bubble being shifted to the trailing edge. The scalar in this case is largely
prevented from escaping towards the trailing edge (this explains the decrease
in $L_{w}$ and $L_{c}$ as solidity increases, see figure 7a). Conservation
laws dictate two possible outlets left for the flow: lateral and vertical
bleeding. The higher magnitude of $C$ above the patch, upstream of the bubble
($y/h\geq 1$, $0\leq x/D\leq 0.5$) confirms increasing vertical bleeding in
higher solidity, while the increase of lateral bleeding with $\phi$ has been
confirmed in Zhou and Venayagamoorthy (2019). In some sense, the flow lingers
downstream of the porous obstruction (characterised by high magnitude of $C$
in this region in case C64), it is as if the obstruction is extended in the
streamwise direction. The length of this extension seems to be inversely
proportional to $\phi$, as it is longer than the FOV for case C39 (or perhaps
infinitely long), approximately equal to $1.25D$ for case C64 (figure 8b), and
0 for case C133 (figure 8c). Possibly, the extension length is akin to the
length of the steady wake region observed in 2-D patches (Ball et al., 1996;
Zong and Nepf, 2012; Chen et al., 2012). When the obstruction is impermeable
(figure 8d), the mechanism differs. It acts as a solid body, shedding a pair
of shear layers at the trailing edge (see contours of positive and negative
$\Omega_{y}$ and $V$ at $0.5\leq x/D\leq 1.5$ in figures 5d and 5d,
respectively).

###  3.2 Vertical bleeding and the extent of scalar concentration

Contours of $V>0$ above the patches are shown in solid lines in figure 5. The
contours reveal a very different pattern of flow above the patches for porous
cases (figure 5a–c) compared to the solid case CS (figure 5d). In case CS,
flow separation is characterised by high magnitude of $V$ observed in the
leading edge of the patch ($x/D=-0.5$). Compared to the solid case, the
magnitude of $V$ in the leading edge is lower for all porous cases (figure
5a–c), as the flow is able to penetrate the patches instead of being diverted
upward (see Taddei et al., 2016 for similar 3-D patches and Zhou and
Venayagamoorthy, 2019 for suspended patches, note that the sign of $V$ is
inverted in the latter). What is only observed in the porous patches, however,
is that the presence of $V>0$ region across the top surface of the patches
($-0.5\leq x/D\leq 0.5$ and $y/h\geq 1$) due to vertical bleeding. The
magnitude and extent of this region appear to be dependent upon solidity. As
as comparison, the contours of $V/U_{h/2}$ for the porous cases in figure 7(b)
show that for the same magnitude ($V/U_{h/2}=0.1$), $V$ is extended upward (in
$y$) as $\phi$ increases.

Figure 9: (a) Integrated vertical bleeding $V_{bleed}/U_{h/2}$ above the patch
($y/h>1$) as a function of $y$.  : the reference smooth-wall case TBL. Hatched
region shows $V_{bleed}\pm 1\%$ of $U_{h/2}$. (b) Integrated scalar
concentration above the patch $C_{v}/C_{s}$. Colour scheme for each case (C39,
C64, C133, and CS) is described in table 1.

Similar to the analysis in §3.1, we quantify the magnitude of vertical
bleeding as the integrated $V$ above the patch ($-D/2\leq x\leq D/2$ and
$y\geq h$), and this can be further extended to include the scalar
concentration,

$\left.\begin{array}[]{r
l}V_{bleed}(y)&=\frac{1}{D}\int_{-D/2}^{D/2}U(x,y)\;\mathrm{d}x\\\
C_{v}(y)&=\frac{1}{D}\int_{-D/2}^{D/2}C(x,y)\;\mathrm{d}x\end{array}\right\\}y\geq
h$ (3)

Figures 9(a,b) show the integrated vertical bleeding $V_{bleed}$ and scalar
concentration $C_{v}$ as functions of $y$ above the patch. Both figures show
higher magnitude of $V_{bleed}$ and $C_{v}$ closer to the patch, which decay
further towards the freestream. Dependency on $\phi$ is apparent on both.
$V_{bleed}$ in figure 9(a) generally increases with $\phi$, with a
significantly smaller magnitude (within $\pm 1\%$ of the incoming flow
$U_{h/2}$, see hatched region in figure 9a) for CS and TBL (black solid and
dashed lines in figure 9a, respectively). For case CS, small $V_{bleed}$ is
the result of averaging between high magnitude of $V>0$ in the leading edge
and $V<0$ above the patch (see contour lines in figure 5d), while for case TBL
($V=0$) this can be attributed to the error of the measurements. The
integrated scalar concentration in figure 9(b) is 0 for the solid case CS
(since there is no vertical bleeding, see black solid line in figure 9b),
while for the porous cases it increases with $\phi$ and decays to 0 further
from the patch.

Figure 10: Contours of scalar concentration $\log_{10}(C/C_{s})$ for cases:
(a) C39, (b) C64, (c) C133, and (d) CS. Black solid lines ( ) are the
normalised vorticity in $y$ axis $\Omega_{y}D/U_{h/2}=-8,-7,...,-4$.

In addition to increasing vertical bleeding with $\phi$, we also observed that
the maxima of $C_{v}$ in figure 9(b) occurs at the top surface of the patch
$y/h=1$ for all porous cases, but the maxima of $V_{bleed}$ in figure 9(a)
occurs further from the patch (for example, it occurs at $y/h\approx 1.5$ for
case C133 in figure 9a). Figures 10(a–c) offer a closer look of the flow above
the porous patches. The coloured contours are the scalar concentration
$C/C_{s}$ in logarithmic scale and the solid black lines are the contours of
$\Omega_{y}<0$, representing the shear layer forming above the patches. In
case C39 (figure 10a) the shear layer shields a region of (relatively) high
scalar concentration (orange-shaded contours in figure 10a) below the layer.
As solidity increases in case C64 (figure 10b), higher $C$ is observed
downstream of the patch (purple-shaded coloured contours), which coincides
with elevation of the shear layer compared to that of C39 in figure 10(a). In
the case with the highest solidity (C133 in figure 10c), recirculation occurs
on the trailing edge (figure 8c) followed by higher $C$ above the patch
compared to that downstream of the patch (figure 10c). Here the shear layer is
elevated further upward, shielding the observed high $C$ above the patch
($0.25\lesssim x/D\lesssim 0.5$). Similar to the analogy of ‘artificial’
streamwise extension in the porous obstructions discussed in §3.1, here there
seems to be an artificial _upward_ extension of the obstruction as $\phi$
increases. This perhaps explains the phenomenon observed in figure 9, that the
maxima of vertical bleeding occurs further from the patches but the maxima of
scalar concentration occurs exactly at the top surface of the patches.

##  4 Conclusions and Recommendations

We conduct planar PIV-PLIF measurements of the flow surrounding 3-D, circular
porous patches of various solidity $\phi$ ($0.098\leq\phi\leq 1$) to study the
characteristics of flow in the wake of and above the patches. Porosity is
generated by mounting rigid cylinders of height $h$ and diameter $d$ on the
patch (whose diameter is $D$) and the variation in $\phi$ is achieved by
systematically increasing the number of cylinders in a patch. The patches are
fully submerged within the incoming flow, with a fluorescent dye injected from
the floor of the tunnel, creating a point source upstream of the patches.
Simultaneous PIV-PLIF measurements in $x$–$y$ plane provide streamise and
wall-normal velocity components, as well as the concentration of the
fluorescent tracer in the FOV.

Examination of the time-averaged velocity and scalar fields suggests that the
trailing edge (longitudinal) and vertical flow bleeding from the obstructions
decreases and increases with increasing $\phi$, respectively, as previously
reported in Taddei et al. (2016); Zhou and Venayagamoorthy (2019); Nicolai et
al. (2020). In addition to that, we observe that $\phi$ affects the scalar
concentration in the wake and above the patch. In particular, present results
suggest that $\phi$ determines

  1. (i)

the horizontal extent of the wake ($L_{w}$) and scalar concentration ($L_{c}$)
downstream of the porous patches, with $L_{w}$ and $L_{c}$ decreasing with
$\phi$ (figure 7a). The decrease is attributed to the shift of recirculation
bubbles towards the trailing edge of the patches as $\phi$ increases. High
scalar concentration is observed downstream of the patch as the flow is
trapped between patches and recirculation bubble, acting as an elongation of
the obstructions in streamwise direction.

  2. (ii)

the vertical extent of the wake ($H_{w}$) and scalar concentration ($H_{c}$)
downstream of the porous patches, with $H_{w}$ and $H_{c}$ increasing with
$\phi$ (figure 7b). As $\phi$ increases further, recirculation bubble occurs
at the trailing edge of the patch and the flow is trapped in the interior of
the patch. The flow largely escapes by means of vertical (and lateral)
bleeding. High concentration of scalar, elevation of shear layer and $V$ above
the patch suggest that the flow is trapped between the top surface of the
patch, from which the flow bleeds, and the shear layer. Here we observe an
artificial extension of the obstacle further upward (i.e. the patch is
‘taller’ from the perspective of the flow) as $\phi$ increases.

As a final remark, the mean scalar/flow fields understanding both momentum
_and_ scalar transport of the flow surrounding 3-D porous obstructions and
building useful models for engineering design, prediction for meteorological
or oceanographic flows, etc. The results presented in this study are limited
to the time-averaged velocity and scalar components. However, we note that
intermittency (i.e. discrepancies between the time-averaged and instantaneous
flow fields) has been reported in Cassiani et al. (2008) and Taddei et al.
(2016). In the future, analysis about the fluctuating components
($u^{\prime}$, $v^{\prime}$, and $c^{\prime}$) should be included to obtain
the full picture of the flow characteristics. The decay of wake and scalar
concentration in the trailing edge, shown in figures 7(d,e), indicate a
possible wake model for both porous and solid cases, which should also be
investigated in the future.

#### Funding Statement

The authors gratefully acknowledge the financial support from EPSRC (Grant
Nos: EP/P021476/1 and EP/S013296/1).

#### Declaration of Interests

The authors declare no conflict of interest.

#### Author Contributions

CN designed, performed all measurements, and post-processed the data. DDW
post-processed, analysed the data, and wrote the manuscript, with the
supervision from BG.

#### Data Availability Statement

The data that support the findings of this study will be made available upon
publication.

#### Ethical Standards

The research meets all ethical guidelines, including adherence to the legal
requirements of the study country.

#### Supplementary Material

Not available.

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# Recent progress on the nonlocal power corrections to the inclusive penguin
decays $\bar{B}\to X_{s}\gamma$ and $\bar{B}\to X_{s}\ell\ell$ 111MITP-23-008.
DESY-23-29. Based on talks given by T.H. at the LF(U)V workshop, Zurich,
4.-6.7.2022 and at the LFC22 workshop, Trento, 29.8.-2.9.2022.

Michael Benzke
Institute for Theoretical Physics, University Hamburg
Luruper Chaussee 149, D-26761 Hamburg, Germany

Tobias Hurth
PRISMA+ Cluster of Excellence and Institute for Physics (THEP)
Johannes Gutenberg University, D-55099 Mainz, Germany

###### Abstract

We report on recent progress in the field of nonlocal (so-called resolved)
contributions to the inclusive penguin decays which presently belong to the
largest uncertainties in these inclusive decay modes. There is still a very
large scale and a large charm mass dependence in the present leading order
results which can in the future be decreased by including the $\alpha_{s}$
corrections.

##  1 Introduction

The inclusive decay modes $\bar{B}\to X_{s,d}\gamma$ and $\bar{B}\to
X_{s,d}\ell^{+}\ell^{-}$ are well known for being theoretically very clean
modes of the indirect search for new physics via flavour observables and
golden modes of the Belle-II experiment (for reviews see Refs. [1, 2, 3]). The
Belle-II experiment at KEK will accumulate two orders of magnitude larger data
samples than the $B$ factories [4]. This will lead to a very high experimental
precision in the penguin modes which has to be matched by the accuracy of
theoretical predictions.

Within the heavy mass expansion (HME) these inclusive so-called penguin modes
are domi- nated by the partonic contributions which can be calculated
perturbatively and sub-leading contributions start at the quadratic level,
$(\Lambda/m_{b})^{2}$ only. However, it is well known that this operator
product expansion breaks down in these inclusive modes if one considers
operators beyond the leading ones. This breakdown manifests in nonlocal power
corrections, also called resolved contributions. They are characterised by
containing subprocesses in which the photon couples to light partons instead
of connecting directly to the effective weak-interaction vertex [7].

These resolved contributions can be systematically calculated using soft-
collinear effective theory (SCET). In case of the inclusive $\bar{B}\to
X_{s}\gamma$ decay all resolved contributions to $O(1/m_{b})$ have been
calculated some time ago [5, 6, 7]. Also the analogous contributions to the
inclusive $\bar{B}\to X_{s,d}\ell^{+}\ell^{-}$ decays have been analysed to
$O(1/m_{b})$ [8, 9]. In both cases an additional uncertainty of $4-5\%$ was
found which represents the largest uncertainty in the prediction of the decay
rate of $\bar{B}\to X_{s}\gamma$ [10] and of the low-$q^{2}$ observables of
$\bar{B}\to X_{s,d}\ell^{+}\ell^{-}$ [11, 12].

In both penguin decays there are four resolved contributions at $O(1/m_{b})$,
namely from the interference terms ${\cal O}_{7\gamma}-{\cal O}_{8g}$, ${\cal
O}_{8g}-{\cal O}_{8g}$, and ${\cal O}^{c}_{1}-{\cal O}_{7\gamma}$, but also
from ${\cal O}^{u}_{1}-{\cal O}_{7\gamma}$. 222For the definition of the
operators the reader is guided to Ref. [9] The latter is CKM suppressed, but
was shown to vanish [5]. The ${\cal O}^{c}_{1}-{\cal O}_{7\gamma}$ piece is
the largest contribution in both penguin decays. As was already noted in Ref.
[8, 9], there are subleading contributions due to the interference of ${\cal
O}_{9,10}$ and ${\cal O}^{c}_{1}$ at order $1/m_{b}^{2}$ which are numerically
relevant due to the large ratio $C_{9,10}/C_{7\gamma}$ and which have be to
added in the future.

Recently, a new theoretical input [17, 18] allowed to reduce the impact of the
${\cal O}^{c}_{1}-{\cal O}_{7\gamma}$ contribution. However, in a more recent
analysis of the ${\cal O}^{c}_{1}-{\cal O}_{7\gamma}$ resolved contribution a
smaller reduction was found and additional uncertainties were identified [19].
We discuss the reasons for these discrepancies between the two analyses in
Refs. [18, 19] below.

In particular, a large scale dependence and also a large charm mass dependence
were identified in the lowest order result of this resolved contribution,
which calls for a systematic calculation of $\alpha_{s}$ corrections and
renormalisation group (RG) summation [19]. For this task a factorisation
formula for the subleading resolved corrections is needed which is valid to
all orders in the strong coupling constant $\alpha_{s}$. Here another new
input was given in Ref. [13] where a previous failure of factorisation in
specific resolved contributions was healed by using new refactorisation
techniques [14, 15, 16].

In this status report we focus on these two issues, namely the factorisation
theorem for resolved contributions and the estimate of the ${\cal
O}^{c}_{1}-{\cal O}_{7\gamma}$ contribution.

##  2 General properties of resolved contributions

The resolved contributions in the penguin decay $\bar{B}\to
X_{s,d}\ell^{+}\ell^{-}$ were calculated in the presence of a cut in the
hadronic mass $M_{X}$ which is needed also at the Belle-II experiment in order
to suppress huge background from double semi-leptonic decays. But it was shown
[8, 9] that the resolved contributions stay nonlocal when the hadronic cut is
released. Therefore they represent an irreducible uncertainty. In addition it
was shown that the support properties of the shape function imply that the
resolved contributions (besides the ${\cal O}_{8g}-{\cal O}_{8g}$ one) are
almost cut-independent. The analogous statements for the resolved contribution
in the penguin decay $\bar{B}\to X_{s,d}\gamma$ are also valid when the photon
energy cut is moved out of the endpoint region.

One finds a factorisation formula for the various contributions to the
inclusive penguin decays [5]: 333The symbol $\otimes$ denotes the convolution
of the soft and jet functions.

$\displaystyle d\Gamma(\bar{B}\to
X_{s}\,\gamma,\ell^{+}\ell^{-})=\sum_{n=0}^{\infty}\,\frac{1}{m_{b}^{n}}\,\sum_{i}\,H_{i}^{(n)}J_{i}^{(n)}\otimes
S_{i}^{(n)}$
$\displaystyle\qquad+\sum_{n=1}^{\infty}\,\frac{1}{m_{b}^{n}}\,\bigg{[}\sum_{i}\,H_{i}^{(n)}J_{i}^{(n)}\otimes
S_{i}^{(n)}\otimes\bar{J}_{i}^{(n)}+\sum_{i}\,H_{i}^{(n)}J_{i}^{(n)}\otimes
S_{i}^{(n)}\otimes\bar{J}_{i}^{(n)}\otimes\bar{J}_{i}^{(n)}\bigg{]}\,.$ (2.1)

The first line describes the so-called direct contributions, while the second
line contains the resolved contributions. The latter appear first only at the
order $1/m_{b}$ in the heavy-quark expansion. Here hard functions
$H_{i}^{(n)}$ describe physics at the high scale $m_{b}$. $J_{i}^{(n)}$ are
the so-called jet functions which represent the physics of the hadronic final
state $X_{s}$ at the intermediate hard-collinear scale
$\sqrt{m_{b}\Lambda_{\rm QCD}}$. The soft functions $S_{i}^{(n)}$, the so-
called shape functions, parametrise the hadronic physics at the scale
$\Lambda_{\text{QCD}}$. Within the resolved contributions we have new jet
functions $\bar{J}_{i}^{(n)}$ due to the coupling of virtual photons with
virtualities of order $\sqrt{m_{b}\Lambda_{\text{QCD}}}$ to light partons
instead of the weak vertex directly.

However, the specific resolved ${O}_{8g}-{O}_{8g}$ contribution does not
factorise because the convolution integral is UV divergent. The authors of
Ref. [5] claimed that there is an essential difference between divergent
convolution integrals in power-suppressed contributions of exclusive $B$
decays and the divergent convolution integral in the present case, while the
former were of IR origin, the latter divergence were of UV nature.
Nevertheless, using a hard cut-off in the resolved contribution, the sum of
direct and resolved ${O}_{8g}-{O}_{8g}$ contributions was shown to be scale
and scheme independent at the lowest order. But the failure of factorisation
did not allow for a consistent resummation of large logarithms. In a recent
paper, the divergences in the resolved and in the direct contributions were
identified as endpoint divergences. It was shown that also the divergence in
the direct contribution can be traced back to a divergent convolution integral
[13]. Recently new techniques [14, 15, 16] were presented in specific collider
applications, which allow for an operator-level reshuffling of terms within
the factorisation formula so that all endpoint divergences cancel out. This
idea of refactorisation was now implemented in this flavour example of the
resolved contributions which includes nonperturbative soft functions, the
subleading shape functions, not present in collider applications [13]. A
renormalised factorisation theorem on the operator level for these resolved
contributions was derived to all orders in the strong coupling constant. This
new result establishes the validity of the general factorisation theorem,
given in Eq. 2, - also for the ${O}_{8g}-{O}_{8g}$ contributions. This theorem
now allows for higher-order calculations of the resolved contributions and
consistent summation of large logarithms [13].

##  3 Calculation of the resolved ${\cal O}^{c}_{1}-{\cal O}_{7\gamma}$
contribution

###  3.1 General stragegy

Following the analysis in Refs. [10], the SM prediction for the branching
ratio of ${\mathcal{B}}_{s\gamma}$ with a certain cut $E_{0}$ in the photon
energy spectrum is based on the formula

${\mathcal{B}}(\bar{B}\to
X_{s}\gamma)_{E_{\gamma}>E_{0}}={\mathcal{B}}(\bar{B}\to
X_{c}\ell\bar{\nu})\left|\frac{V^{*}_{ts}V_{tb}}{V_{cb}}\right|^{2}\frac{6\alpha_{\mathrm{e}m}}{\pi\;C}\left[P(E_{0})+N(E_{0})\right],$
(3.2)

where the so-called semi-leptonic phase-space factor $C$ is determined using
the Heavy Quark Effective Theory (HQET) methods [21].

It is important to emphasise here that the perturbative contribution
$P(E_{0})$ is calculated using the local operators of the electroweak
effective Hamiltonian, while the resolved contributions in the nonperturbative
contribution $N(E_{0})$ are calculated using SCET. Thus, scale choices and
input parameters are in principle independent of each other in both
contributions. One should keep this in mind when the uncertainty due to the
resolved contributions is calculated relative to the perturbative decay rate.
As in the original analysis in Ref. [5], the perturbative decay rate at
leading order accuracy is used in the following. Moreover, the hard scale is
chosen in this perturbative contribution. 444The perturbative rate at higher
orders is often calculated at a scale slightly smaller than the hard scale for
other reasons, namely for the stabilisation of the charm mass renormalisation
dependence (see i.e. Ref. [10]) Because no $\alpha_{s}$ corrections or any RG
improvements are considered in the calculation of the resolved power
corrections, the scale choice is ambiguous. One first fixes the Wilson
coefficients in the resolved contribution at the hard scale, but then one
varies the scale of the Wilson coefficients in the resolved contributions
between the hard and the hard-collinear scale to make the scale dependence of
the resolved contributions manifest.

In the following the focus will be on the most important contribution due to
the interference of ${\cal O}^{c}_{1}-{\cal O}_{7\gamma}$. Using the original
notation of Ref. [5] one can write this resolved contribution normalised to
the perturbative leading order result as

${\cal F}_{\rm b\to
s\gamma}^{17}=\frac{C_{1}(\mu)\,C_{7\gamma}(\mu)}{(C_{7\gamma}(\mu_{\rm\mbox{{\tiny
OPE}}}))^{2}}\,\frac{\Lambda_{17}(m_{c}^{2}/m_{b},\mu)}{m_{b}}\,,$ (3.3)

where $\mu_{\rm\mbox{{\tiny OPE}}}$ denotes the perturbative scale, $\mu$ the
scale within the resolved contribution. At subleading power one finds [5]:

$\Lambda_{17}\Big{(}\frac{m_{c}^{2}}{m_{b}},\mu\Big{)}=e_{c}\,\mbox{Re}\int_{-\infty}^{\infty}\frac{d\omega_{1}}{\omega_{1}}\left[1-F\\!\left(\frac{m_{c}^{2}-i\varepsilon}{m_{b}\,\omega_{1}}\right)+\frac{m_{b}\,\omega_{1}}{12m_{c}^{2}}\right]h_{17}(\omega_{1},\mu)\,,$
(3.4)

with the penguin function $F(x)=4\,x\,{\rm arctan}^{2}(1/\sqrt{4x-1})$. The
shape function $h_{17}$ is given by the following HQET matrix element:

$h_{17}(\omega_{1},\mu)=\int\frac{dr}{2\pi}\,e^{-i\omega_{1}r}\frac{\,\langle
B\\!\mid\,\bar{h}(0){\overline{n}}\\!\\!\\!/i\gamma_{\alpha}^{\perp}\overline{n}_{\beta}gG^{\alpha\beta}(r\overline{n})h(0)\,\mid\\!B\rangle\,}{2M_{B}}\,,$
(3.5)

where $n$ and $\overline{n}$ are the light-cone vectors and $h$ and $G$ are
the heavy quark and gluon field, respectively. Soft Wilson lines are
suppressed in the notation. The variable $\omega_{1}$ corresponds to the soft
gluon momentum.

The general strategy to estimate the convolution integral of the perturbative
jet functions and the nonperturbative shape function $h_{17}$ is the
derivation of general properties of the shape functions. One shows for example
PT invariance in this case which implies that the matrix element is real.
Moreover, one finds moments of this HQET matrix element: While the zero-moment
was already known in the first analyses of the resolved contributions, the
second moment was recently derived using HQET techniques; moreover rough
dimensional estimates of the higher order moments were proposed [17, 18]. One
can also naturally assume that the support properties and the values of the
soft shape function are within the hadronic range.

Besides these general properties and the estimates on the moments nothing
further is known about the structure of the subleading shape functions.
Therefore, the two new analyses of the resolved contributions [18, 19] follow
here exactly the same strategy; they use a complete set of basis functions,
namely the Hermite functions in order to make a systematic analysis of all
possible model functions fulfilling the known properties of the shape
function. This systematic approach to the shape functions was already used in
several previous analyses [22, 23, 24]. Obviously this systematic approach
allows to avoid any prejudice regarding the unknown functional form of the
shape functions and, thus, leads to a valid estimate of the resolved
contribution. Any additional assumption calls for a clear justification.

###  3.2 Numerical results

In both new analyses [18, 19] the maximum value of the convolution integral
between jet and shape functions was found for Hermite polynomials of degree
six. Higher degree polynomials do not lead to larger values. Both analyses
found a significant reduction in the values of the resolved contributions due
to the new input of the second moment of the shape function.

The final result for the resolved ${\cal O}^{c}_{1}-{\cal O}_{7\gamma}$
contribution in Ref.[19] is:

${\cal F}_{\rm b\to s\gamma}^{17}\in[-0.4\%,\,4.7\%]\,,$ (3.6)

which represents a large reduction compared to the original estimate in
Ref.[5]. Some comments are in order:

  * •

In the present result no $\alpha_{s}$ corrections are included and no RG
improvements are done. Thus, this implies a large scale dependence in our
results. In the leading order result the only scale is in the Wilson
coefficients representing the hard function. Varying the LO Wilson
coefficients $C_{1}(\mu)\,C_{7\gamma}(\mu)$ in the resolved contribution from
the hard scale to the hard-collinear scale increases the result by more than
$40\%$. This represents an additional uncertainty of the result not included
in Eq.3.6.

  * •

The result in Eq. 3.6 includes a large kinematical $1/m_{b}^{2}$ contribution
from the ${\cal O}^{c}_{1}-{\cal O}_{7\gamma}$ interference. One can show by
inspection of all resolved $1/m_{b}^{2}$ contributions that this kinematical
$1/m_{b}^{2}$ term is the only one with the same shape function of the order
$1/m_{b}$ as in the $1/m_{b}^{1}$ term. All other resolved $1/m_{b}^{2}$
contributions in the interference of ${\cal O}^{c}_{1}-{\cal O}_{7\gamma}$
include shape functions of the order $1/m_{b}^{2}$. Those terms are not
calculated yet. Therefore, the signifi- cantly large $1/m_{b}^{2}$ term due to
kinematical factors in the ${\cal O}^{c}_{1}-{\cal O}_{7\gamma}$ term was
included in the result 3.6 as conservative estimate of those higher order
resolved contributions. This $1/m_{b}^{2}$ term was already included in the
original analysis in Ref. [5]. There it was also shown that other
$1/m_{b}^{2}$ contributions due to the interference of ${\cal O}^{c}_{1}-{\cal
O}_{8g}$ and ${\cal O}^{c}_{1}-{\cal O}_{1}$ are numerically negligible.

The authors of the new analysis in Ref. [18] find a much larger reduction and
end up with 555We translated this result to our scale fixing. The authors of
Ref. [18] find ${\cal F}_{\rm b\to s\gamma}^{17}\in[-0.3\%,\,1.6\%]$ in their
paper using the hard-collinear scale in the resolved and the perturbative
contribution instead of the hard scale.

${\cal F}_{\rm b\to s\gamma}^{17}\in[-0.4\%,\,1.9\%]\,,$ (3.7)

There are two main reasons for this difference with the result of Ref. [5],
given in Eq. 3.6:

  * •

The charm mass dependency originates from the anti-hard-collinear jet function
$\bar{J}_{i}^{(n)}$ represented by the charm loop with a soft gluon emission.
Therefore it is appropriate to use the running charm mass at the hard-
collinear scale ${m}_{c}^{\rm MS}(\mu_{\rm hc})$. The charm mass ambiguity of
the charm mass was made manifest by the variation of the hard-collinear scale
$\mu_{\rm hc}\sim\sqrt{m_{b}\,\Lambda_{\rm QCD}}$ from $1.3\,{\rm GeV}$ to
$1.7\,{\rm GeV}$ within the recent analysis in Ref. [5]. Using the present PDG
value of the charm mass being ${m}_{c}^{\rm MS}(m_{c})=(1.27\pm
0.02)\,\text{GeV}$ and using three-loop running with $\alpha_{s}(m_{c})=0.395$
and $\alpha_{s}(m_{Z})=0.1185$ down to the hard-collinear scale, one finds
$m_{c}^{\rm MS}(1.5\,{\rm GeV})=1.19\,{\rm GeV}$ as central value at
$1.5\,{\rm GeV}$. The change of the hard-collinear scale indicated above then
leads to [5]

$1.14\,{\rm GeV}\leq m_{c}\leq 1.26\,{\rm GeV}.$ (3.8)

The parametric errors of ${m}_{c}^{\rm MS}(m_{c})$ and $\alpha_{s}$ are
neglected in view of the larger uncertainty due to the change of the hard-
collinear scale $\mu_{\rm hc}$.
In the recent analysis in Ref. [18], two-loop running leads to the central
value $m_{c}^{\rm MS}(1.5\,{\rm GeV})=(1.20+0.03)\,{\rm GeV}$. The parametric
uncertainties, but no change of the hard-collinear scale is taken into
account. One then finds the following variation of the charm mass

$1.17\,{\rm GeV}\leq m_{c}\leq 1.23\,{\rm GeV},$ (3.9)

which was used in the analysis in Ref. [18], but it is unnaturally small and,
thus, represents an underestimation of the charm mass dependence.

  * •

The second reason is that the authors of Ref. [18] did not include any
estimate for the $1/m_{b}^{2}$ corrections. In view of the fact that the only
resolved $1/m_{b}^{2}$ term with the same shape function as in the
$1/m_{b}^{1}$ term is very large, this may lead to a further underestimation
of the overall uncertainty.

Finally, it is important to note that the local Voloshin term is subtracted
from the resolved contribution ${\cal F}_{\rm b\to s\gamma}^{17}$. This has
been done in all analyses of this specific resolved contribution to the
$\bar{B}\to X_{s}\gamma$ decay rate. Therefore this nonperturbative
contribution has still to be added to the decay rate. It is given by
$\Lambda_{17}^{\rm Voloshin}=(-1)(m_{b}\lambda_{2})/(9m_{c}^{2})$ and
corresponds to

${\cal F}_{\rm b\to s\gamma}^{\rm
Voloshin}=-\frac{C_{1}(\mu)\,C_{7\gamma}(\mu)\,\lambda_{2}}{(C_{7\gamma}(\mu_{\rm\mbox{{\tiny
OPE}}}))^{2}\,9\,m_{c}^{2}}=+3.3\%\,,$ (3.10)

If one neglects the shape function effects and treats the charm quark mass as
heavy one can derive the local Voloshin term from the resolved contribution
${\cal O}^{c}_{1}-{\cal O}_{7\gamma}$ (see section 3.2 of Ref. [9] for more
details). But the local Voloshin term, derived in Refs.[25, 26, 27, 28], does
not account for the complete resolved contribution as one can read off from
the additional contribution given in Eq. 3.6.

We close this progress report by another recent improvement of estimates of
the resolved contribution. Based on Ref. [32] an estimation of the resolved
contribution to the ${\cal F}^{78}$ was offered in Ref. [5] using experimental
data on $\Delta_{0-}$ of the isospin asymmetry of inclusive neutral and
charged $B\to X_{s}\gamma$ decay [29, 30]. In the recent analysis [18], the
authors derived new bounds by taking into account a new Belle measurement of
$\Delta_{0-}$ [31], which leads to the experimental determination of ${\cal
F}^{78}$ being the same order of magnitude as the determination using the
vacuum insertion approximation (VIA) [5] but includes the prospect to be
improved by more precise data.

## Acknowledgements

TH thanks the organisers of the LF(U)V workshop in Zurich and of the LFC22
workshop in Trento for their generous hospitality and the excellent
organisation of the workshops. TH is supported by the Cluster of Excellence
“Precision Physics, Fundamental Interactions, and Structure of Matter”
(PRISMA+ EXC 2118/1) funded by the German Research Foundation (DFG) within the
German Excellence Strategy (Project ID 39083149).

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