11institutetext:  Cluster of Excellence SimTech, University of Stuttgart,
Pfaffenwaldring 5a, 70569 Stuttgart, Germany, 11email<EMAIL_ADDRESS>stuttgart.de 22institutetext:  Max Planck Institute for Intelligent Systems,
Heisenbergstraße 3, 70569 Stuttgart, Germany 33institutetext:  Institute of
Aerospace Thermodynamics (ITLR), Pfaffenwaldring 31, 70569 Stuttgart, Germany

# HLRS2021

xu.chu  Xu Chu  11 Wenkang Wang  22 Bernhard Weigand  33112233

# An investigation of information flux between turbulent boundary layer and
porous medium

xu.chu  Xu Chu  11 Wenkang Wang  22 Bernhard Weigand  33112233

###### Abstract

The interaction between boundary layer turbulence and a porous layer is the
cornerstone to the interface engineering. In this study, the spatial resolved
transfer entropy is used to assess the asymmetry of the causal interaction
next to a permeable wall. The analysis was based on pore-resolved direct
numerical simulation of turbulent channel flow over a cylinder array. The
spatial map of transfer entropy reveals the information flux between the
porous medium and arbitrary nearby position.

##  1 Introduction

Turbulent flow over permeable interfaces is ubiquitous in nature and
engineering applications, from sediment transport to the transpiration cooling
in gas turbines. Porous media are generally featured with complex topology in
space (Chu et al, 2018). A wide range of properties has been identified to
have a significant effect on the mass and momentum transport across the porous
medium-free flow interface. However, understanding and optimizing macro/micro-
scale characteristics of porous media remains a challenge. This is due to both
the difficulty of experimental measurements at the pore scale Terzis et al
(2019) as well as the high costs of numerical simulations to resolve the full
spectrum of scales.

Early studies of turbulent flow over permeable beds have addressed the effect
of varying permeability on surface flows. The turbulent surface flow is
similar to that of a canonical boundary layer when the interfacial
permeability is low, as the near-wall structures are less disrupted. However,
as the permeability increases, large-scale vortical structures emerge in the
surface flow. This is attributed to Kelvin-Helmholtz (KH) type instabilities
from inflection points of the mean velocity profile. More recently, the
application of porous media for drag reduction inspired a series of in-depth
studies on anisotropic permeability. Direct Numerical Simulation (DNS)
exhibits an edge of observing and analyzing turbulent physics in a confined
small space, not only for the canonical cases as channel flows and pipe flows
Chu et al (2016b, 2020a); McEligot et al (2018, 2020); Pandey et al (2017b).
Suga et al. Suga et al (2020) investigated the influence of the anisotropic
permeability tensor of porous media at a higher permeable regime. It was found
that streamwise and spanwise permeabilities enhance turbulence whilst vertical
permeability itself does not. In particular, the enhancement of turbulence is
remarkable over porous walls with streamwise permeability, as it allows the
development of streamwise large-scale perturbations induced by the
Kelvin–Helmholtz instability.

The concept of transfer entropy was proposed Schreiber (2000) as a tool to
evaluate the directional information transfer between a source signal and a
target signal. This definition of transfer entropy measures the information
contained in the source about the next state of the target that was not
already contained in the target’s past. This allows one to differentiate the
direction of the information flux, and can therefore be used to quantify
causal interactions. Recently, Lozano-Durán Lozano-Durán et al (2020, 2021);
Lozano-Durán and Arranz (2022) highlighted the importance of causal inference
in fluid mechanics and proposed leveraging information-theoretic metrics to
explore causality in turbulent flows. Wang et al. Wang et al (2021a) used
transfer entropy as a marker to evaluate the causal interaction between
turbulent channel flow and porous media consisting of circular cylinders. The
POD time coefficients of the leading order modes were extracted as the
representative signals for surface and sub-surface flow.

The explorations above illustrate the potential strength of information-
theoretic tools in revealing the surface/sub-surface coupling dynamics.
However, there are still numerous challenges and unanswered questions. In the
current work, we will compute spatial resolved transfer entropy between
turbulent fluctuations in channel flows over porous media.

##  2 Numerical method

In our DNS, three-dimensional incompressible Navier$-$Stokes equations are
solved in a non-dimensional form without models,

$\frac{\partial u_{j}}{\partial x_{j}}=0$ (1) $\frac{\partial u_{i}}{\partial
t}+\frac{\partial u_{i}u_{j}}{\partial x_{j}}=-\frac{\partial p}{\partial
x_{i}}+\frac{1}{Re_{D}}\frac{\partial^{2}u_{j}}{\partial x_{i}\partial
x_{j}}+\Pi\delta_{i1}$ (2)

where $\Pi$ is a constant pressure gradient in the mean-flow direction. The
governing equations are non-dimensionalized by normalizing lengths by the
half-width of the whole simulation domain $H$(figure 1a), velocities by the
averaged bulk velocity $U_{b}$ of the free flow region ($y/H=[0,1]$), such
that time is non-dimensionalized by $H/U_{b}$. The spectral/$hp$ element
solver Nektar++ (Cantwell et al, 2011; Chu et al, 2019, 2020b, 2021b, 2021a;
Pandey et al, 2020; Wang et al, 2021b) is used to discretize the numerical
domain containing complex geometrical structures. The solver allows for
arbitrary-order spectral/$hp$ discretizations with hybrid-shaped elements. The
time-stepping is performed with a second-order mixed implicit-explicit (IMEX)
scheme. The time step is fixed to $\Delta t/(H/U_{b})=5\times 10^{-4}$.

|
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Figure 1: (a) Configuration of the computational domain (C05). The blue and
red isosurfaces show the wall-normal fluctuation $v^{\prime}/u^{p}_{\tau}$ at
level $-0.4$ and $0.4$, respectively. Panels (b-d) show the profiles of
streamwise $u^{\prime}$, wall-normal $v^{\prime}$, and spanwise $w^{\prime}$
fluctuation intensity, respectively.

Hereafter, the velocity components in the streamwise $x$, wall-normal $y$, and
spanwise $z$ directions are denoted as $u$, $v$, and $w$, respectively. The
domain size ($L_{x}/H\times L_{y}/H\times L_{z}/H$) is $10\times 2\times
0.8\pi$ in all cases. The lower half ($y/H=[-1,0]$) contains the porous media
and the upper half ($y/H=[0,1]$) is the free channel flow. The porous layer
consists of 50 cylinder elements along the streamwise direction and 5 rows in
the wall-normal direction as illustrated in figure 1. The distance between two
nearby cylinders is fixed at $D/H=0.2$. No-slip boundary conditions are
applied to the cylinders, the upper wall, and the lower wall. Periodic
boundary conditions are used in streamwise and spanwise directions. The
geometry is discretized using quadrilateral elements on the $x-y$ plane with
local refinement near the interface. Lagrange polynomials (polynomial order
$P=5-9$) are used within each element on the $x-y$ plane. The spanwise
direction is extended with a Fourier spectral method.

case | $\varphi$ | $Re_{\tau}^{\mathrm{p}}$ |  $\Delta x^{\mathrm{p}+}$/$\Delta y^{\mathrm{p}+}$/$\Delta z^{\mathrm{p}+}$ | $Re_{\tau}^{\mathrm{s}}$ |  $\Delta x^{\mathrm{s}+}$/$\Delta y^{\mathrm{s}+}$/$\Delta z^{\mathrm{s}+}$ |  $\sqrt{K_{xx}}^{\mathrm{p}+}$,$\sqrt{K_{yy}}^{\mathrm{p}+}$ | $\sqrt{K_{zz}}^{\mathrm{p}+}$ | $C_{xx}^{\mathrm{p+}},C_{yy}^{\mathrm{p+}}$ | $r^{\mathrm{p+}}$
---|---|---|---|---|---|---|---|---|---
C05 | 0.5 | 336 | 2.2/0.38/5.3 | 180 | 6.3/0.43/4.5 | 4.55 | 8.86 | 0.37 | 42
C06 | 0.6 | 464 | 3.5/0.50/6.8 | 190 | 5.5/0.47/5.4 | 9.34 | 15.23 | 0.58 | 48
C07 | 0.7 | 625 | 4.2/0.55/8.4 | 160 | 5.2/0.51/5.1 | 20.65 | 30.98 | 2.99 | 52
C08 | 0.8 | 793 | 3.1/0.51/5.0 | 170 | 4.1/0.40/2.8 | 36.83 | 53.99 | 13.58 | 52
Table 1: Simulation parameters. The porosity of the porous medium region is
$\varphi$. The friction Reynolds numbers are $Re_{\tau}^{\mathrm{p}}$ and
$Re_{\tau}^{\mathrm{s}}$ for the porous and impermeable top walls
respectively. $\Delta x^{\mathrm{p}+}$/$\Delta y^{\mathrm{p}+}$/$\Delta
z^{\mathrm{p}+}$ and $\Delta x^{\mathrm{s}+}$/$\Delta y^{\mathrm{s}+}$/$\Delta
z^{\mathrm{s}+}$ are the respective grid spacings in wall units of porous wall
and top wall, respectively. $\sqrt{K_{\alpha\alpha}}^{\mathrm{p}+}$ and
$C_{\alpha\alpha}^{\mathrm{p+}}$ are the diagonal components of the
permeability tensor and Forchheimer coefficient, respectively, in the
direction of $\alpha$ ($\alpha\in\\{x,y,z\\}$). $r^{\mathrm{p+}}$ is the
radius of the cylinders. All the variables are normalized using
$u_{\tau}^{\mathrm{p}}$ and $\nu$.

Four DNS cases are investigated with varying porosity
$\varphi=0.5,0.6,0.7,0.8$, which is defined as the ratio of the void volume to
the total volume of the porous structure. The parameters of the simulated
cases are listed in table 1, where the cases are named after their respective
porosity. The superscripts $(\cdot)^{\mathrm{p}}$ and $(\cdot)^{\mathrm{s}}$
represent variables of the permeable wall and the top smooth wall sides,
respectively. Variables with superscript + are scaled by the friction
velocities $u_{\tau}$ of their respective side and the viscosity $\nu$. Note
that the distance between cylinders is fixed, and the porosity is changed by
varying the radius of the cylinders. The normalized cylinder radius is in the
range $r^{\mathrm{p}+}=42$–$52$ for all the cases tested (see table 1) such
that the effect of surface roughness is assumed to be on a similar level. The
Reynolds number of the top wall boundary layer is set to be
$Re_{\tau}^{\mathrm{s}}=\delta^{\mathrm{s}}u_{\tau}^{\mathrm{s}}/\nu\approx
180$ for all cases ($\delta$ is the distance between the position of maximum
streamwise velocity and the wall). In this manner, changes in the top wall
boundary layer are minimized. On the top smooth wall side, the streamwise cell
size ranges from $4.1\leq\Delta x^{\mathrm{s}+}\leq 6.3$ and the spanwise cell
size is below $\Delta z^{\mathrm{s}+}=5.4$. On the porous media side, $\Delta
z^{\mathrm{p}+}$ is below 8.4, whereas $\Delta x^{\mathrm{p}+}$ and $\Delta
y^{\mathrm{p}+}$ are enhanced by polynomial refinement of a local mesh
(Cantwell et al, 2011). The total number of grid points ranges from 88$\times
10^{6}$ (C04) to 595$\times 10^{6}$ (C08). Each cylinder in the porous domain
is resolved with 80 to 120 grids along the perimeter. By performing parameter
tests on porous media and fitting the Darcy-Forchheimer equation, the
permeability tensor $K$ and Forchheimer coefficient $C$ may be obtained.

The velocity field $\bm{u}(\bm{x},t)=(u,v,w)$ can be further decomposed into
the time-averaged velocity $\overline{\bm{u}}=\int{\bm{u}}\mathrm{d}t/T$ and
turbulent fluctuation
$\bm{u}^{\prime}=(u^{\prime},v^{\prime},w^{\prime})=\bm{u}-\overline{\bm{u}}$.
Figure 1(a) shows a snapshot of wall normal fluctuation $v\prime$. The
intensity profiles of $u^{\prime}$, $v^{\prime}$, and $w^{\prime}$ are shown
in Fig.1(b-d), respectively. The turbulent intensity grows with porosity for
both the channel and porous medium region. In the current study, we focus on
the up and down-welling motions at the upper surface of the porous medium,
which is represented by the fluctuation $v^{\prime}$.

##  3 Results and analysis

###  3.1 Transfer entropy

In the current work, the transfer entropy (Schreiber, 2000; Lozano-Durán et
al, 2021; Lozano-Durán and Arranz, 2022; Williams and Lozano-Duran, 2022;
Arranz and Lozano-Duran, 2022) is used to evaluate the direction of coupling,
i.e., the cause-effect relationship, between two time-series. The transfer
entropy originates from the framework of information theory, and is a specific
version of the mutual information for conditional probability. It represents
the dependence of the future state of process $X$ on the process $Y$, which is
measured as the decrease of uncertainty (entropy) of $X$ by knowing the past
state of $Y$. The transfer entropy from $X$ and $Y$ can be defined as:

$T_{Y\rightarrow X}(\Delta t)=H(X_{t}|X_{t-1})-H(X_{t}|{X_{t-1},Y_{t-\Delta
t}})$ (3)

where $\Delta t$ is the time-lag to evaluate causality, and $H(A|B)$ is the
conditional Shannon entropy of the variable $A$ given $B$, which is defined as

$H(A|B)=-E\left[\mathrm{log}(p(A|B)\right]=-E\left[\mathrm{log}\left(\frac{p(A,B)}{p(B)}\right)\right]=E[\mathrm{log}_{2}(p(B))]-E[\mathrm{log}_{2}(p(A,B))]$
(4)

where $p(\cdot)$ is the probability density function, and $E[\cdot]$ denotes
the expectation value. $H(A|B)$ denotes the chunk of information in $A$ that
has nothing to do with $B$.

In order to quantify the strength of causality, normalization is necessary to
scale the magnitude of causality within $[0,1]$ and eliminate small values
caused by statistical errors. The normalized transfer entropy can be defined
as:

$\tilde{T}_{Y\rightarrow X}=\frac{T_{Y\rightarrow X}-E[T_{Y^{s}\rightarrow
X}]}{H(X_{t}|X_{t-1})}$ (5)

where $E[T_{Y^{s}\rightarrow X}]$ is an estimation of the statistical bias.
$Y^{s}$ is the surrogate variable of $Y$, which is acquired by randomly
permuting $Y$ in time to break its causal links with $X$. The conditional
entropy $H(X_{t}|X_{t-1})$ is the intrinsic uncertainty of $X$ knowing its
history. Eq.5 thus represents the fraction of information in the target $X$
not explained by its past that is explained by $Y$ in conjunction with that
past.

#### Local transfer entropy

The physical interpretation of the quantity of transfer entropy can be hard to
grasp and sometimes the outcome is confused with the more-widely used
correlation method. In fact, the two methods are indeed closely related. In
the following, the results from both analysing tools will be presented and
compared. In the current section, we will present a simple example to
illustrate the strength of transfer entropy and explore the connection between
the information transfer and the flow motion.

The definition of transfer entropy quantifies the statistical coherence
between time-evolving systems in a global and averaged manner. Using Eq.4, we
rewrite Eq.3 in the following form:

$T_{Y\rightarrow X}(\Delta
t)=-E\left[\frac{\mathrm{log}_{2}(p(X_{t}|X_{t-1}))}{\mathrm{log}_{2}(p(X_{t}|X_{t-1},y_{t-\Delta
t}))}\right]$ (6)

We define the local transfer entropy $\zeta$ as

$\zeta_{Y\rightarrow X}(t,\Delta t)=\log_{2}\frac{p\left(x_{t}\mid
x_{t-1},y_{t-\Delta t}\right)}{p\left(x_{t}\mid x_{t-1}\right)}$ (7)

where $x_{t}$ and $y_{t}$ are instantaneous measurements at time $t$ of
processes $X$ and $Y$, respectively. The transfer entropy in Eq.3 is
essentially the expectation value of the local transfer entropy, that is:

$T_{Y\rightarrow X}=E[\zeta_{Y\rightarrow X}(t,\Delta t)]$ (8)

To explain the idea of (local) transfer entropy more explicitly, figure 2(a)
shows the excerpts of wall-normal fluctuation $v_{1}^{\prime}(t)$ and
$v_{2}^{\prime}(t)$ of case C05, which are extracted from the middle of the
first-layer cylinder gaps ($y=-r_{c}$) and crest position ($y=0$),
respectively. The two signals $v_{1}^{\prime}$ and $v_{2}^{\prime}$ are
acquired from the same streamwise position. Despite their clear differences in
scale content, the two signals appear to be connected, with the positive and
negative peaks appearing to be coordinated. This feature is also evidenced by
the cross-correlation profile between $v_{1}^{\prime}$ and $v_{2}^{\prime}$
(see Fig.2b), which is defined as

$R_{vv}(\Delta t)=\overline{v_{1}^{\prime}(t)v_{2}^{\prime}(t+\Delta
t)}/(\sigma_{v_{1}^{\prime}}\sigma_{v_{2}^{\prime}}),$ (9)

where $\Delta t$ is the shifted time interval between the two signals. The
profile shows a positive peak value of 0.48 at $\Delta tU_{b}/H\approx 0.02$,
revealing that the two signals are largely correlated. However, a high
correlation value does not always reflect information flow or a strong causal
relationship because $R$ contains no discrimination about direction. As a
comparison, figure 2(c) shows the transfer entropy
$\tilde{T}_{v_{2}^{\prime}\rightarrow v_{1}^{\prime}}$ and
$\tilde{T}_{v_{1}^{\prime}\rightarrow v_{2}^{\prime}}$ as functions of $\Delta
t$. First of all the time delay for transfer entropy can only be positive due
to the constraint that the ‘cause’ always happens before the ‘effect’.
Secondly, there is a huge disparity between the magnitude of
$\tilde{T}_{v_{2}^{\prime}\rightarrow v_{1}^{\prime}}$ and
$\tilde{T}_{v_{1}^{\prime}\rightarrow v_{2}^{\prime}}$, which reveals the
strong dependence of information flux on the coupling direction. For the
current signal set, the transfer entropy in both directions reaches a maximum
at $\Delta tU_{b}/H\approx 0.15$.

Figure 2: Comparison between correlation and transfer entropy. (a) Excerpts of
instantaneous vertical fluctuations $v_{1}^{\prime}$ and $v_{2}^{\prime}$ at
$y=-r_{c}$ (solid line) and $y=0$, respectively; (b) profile of cross-
correlation between $v_{1}^{\prime}$ and $v_{2}^{\prime}$. (c) Transfer
entropy $\tilde{T}_{v_{2}^{\prime}\rightarrow v_{1}^{\prime}}$ (solid line)
and $\tilde{T}_{v_{1}^{\prime}\rightarrow v_{2}^{\prime}}$ (dashed line) as a
function of time delay $\Delta t$. Figure 3: Interpretation of transfer
entropy from the scope of local events. (a) local transfer entropy
$\zeta_{v_{2}^{\prime}\rightarrow v_{1}^{\prime}}$ (solid line) and
$\zeta_{v_{1}^{\prime}\rightarrow v_{2}^{\prime}}$ (dashed line) corresponding
to the time series shown in Fig.2(a). The colour in panels (b) and (d) show
the accumulative maps of $\zeta_{v_{2}^{\prime}\rightarrow v_{1}^{\prime}}$
and $\zeta_{v_{1}^{\prime}\rightarrow v_{2}^{\prime}}$ in the space of
$(v_{1}^{\prime},v_{2}^{\prime})$, respectively. The isolines are the joint
probability density function of $p(v_{1}^{\prime},v_{2}^{\prime})$ with a time
delay $\Delta t$ imposed on the source signal. The levels are from 1$\times
10^{-3}$ to 3$\times 10^{-3}$ with a step of 5$\times 10^{-4}$.

To further explain the concept of transfer entropy, Fig. 3(a) shows the local
transfer entropy corresponding to the time-series in Fig.2(a). The time delay
$\Delta t$ is fixed at $\Delta tU_{b}/H=0.15$, corresponding to the maximum
value of transfer entropy. There are several things to be noted. First, the
$\zeta$ of the two transfer directions are totally different. The magnitude of
$\zeta_{v_{2}^{\prime}\rightarrow v_{1}^{\prime}}$ is generally larger than
that of $\zeta_{v_{1}^{\prime}\rightarrow v_{2}^{\prime}}$, which explains the
disparity between the two profiles of $\tilde{T}$ in Fig.2(c). Secondly, both
position and negative values exist for local transfer entropy. Unlike the
global transfer entropy that is positive by definition Schreiber (2000), the
value of local transfer entropy is not constrained. A positive $\zeta(t,\Delta
t)$ indicate an informative source for a speciﬁc event set
$\\{x_{t},x_{t-1},y_{t-\Delta t}\\}$, while a negative $\zeta(t,\Delta t)$
occurs when the probability of observing the actual next state of the
destination given the value of the source $p\left(x_{t}\mid
x_{t-1},y_{t-\Delta t}\right)$, is lower than the probability knowing only the
history of the destination $p\left(x_{t}\mid x_{t-1}\right)$. In this case,
the source element is actually misleading regarding the destination’s state
transition.

In the brief excerpt of $\zeta_{v_{2}^{\prime}\rightarrow v_{1}^{\prime}}$
(Fig. 3a), we observed several major positive peaks, when both
$v_{1}^{\prime}$ and $v_{2}^{\prime}$ are of strong magnitude. However, this
can not be taken as an evidence that positive local transfer entropy captures
the shifted peaks between $v_{1}^{\prime}$ and $v_{2}^{\prime}$, as there are
also locations where shifted peaks do not generate strong positive $\zeta$. To
understand the true event contributing to positive $\zeta$, we computed the
accumulative map of $\zeta$ in the space of $(v_{1}^{\prime},v_{2}^{\prime})$
(Fig.3b,c). In addition, the joint probability density function (JPDF)
$p(v_{1}^{\prime},v_{2}^{\prime})$ is superimposed on the accumulative map as
dashed isolines for comparison.

It is shown in Fig.3(b) that $\zeta_{v_{2}^{\prime}\rightarrow
v_{1}^{\prime}}$ is mainly positive in the second (i.e.,
$(v_{1}^{\prime}(t)<0,v_{2}^{\prime}(t-\Delta t)>0)$) and fourth quadrants
(i.e., $(v_{1}^{\prime}(t)>0,v_{2}^{\prime}(t-\Delta t)<0)$), and negative in
the remaining areas. Moreover, the events with $v_{1}^{\prime}$ and
$v_{2}^{\prime}$ of a small magnitude are mostly ‘misleading’, although they
are of a larger number (see the JPDF indicated by dashed isolines). We focus
on the major positive peak in the 4th quadrant as it represents dominant
scenario of the predictive event sets. The peak locates at
$(v_{1}^{\prime}(t)/\sigma_{v_{1}^{\prime}},v_{2}^{\prime}(t-\Delta
t)/\sigma_{v_{2}^{\prime}})=(0.5,-1.5)$. This is consistent with the positive
$\zeta_{v_{2}^{\prime}\rightarrow v_{1}^{\prime}}$ peaks (indicated by arrows)
in Fig.2(a), where a large negative $v_{2}^{\prime}$ happens first, and then a
minor positive $v_{1}^{\prime}$ is found after a time delay of around $\Delta
tU_{b}/H=0.15$. Such a combination of source and target events is essentially
different from the events contributing to the correlation peaks in Fig.2(b),
i.e., those slightly shifted peaks. During the computation of transfer
entropy, the information carried by the immediate past of the destination is
subtracted from the predictive information. This may explain why the shifted
peaks are not identified as informative events by transfer entropy, as the
information of about the peak could be already encoded in the history of the
destination signal. The accumulative map of $\zeta_{v_{1}^{\prime}\rightarrow
v_{2}^{\prime}}$ (Fig.3c) shares a similar pattern with that of
$\zeta_{v_{2}^{\prime}\rightarrow v_{1}^{\prime}}$ in Fig.3(b), but with a
much smaller magnitude. This is consistent with the previous observation of
Fig.3(a).

###  3.2 Inter-layer correlation and transfer entropy

Figure 4:  (a1-d1) Map of inter-layer correlation coefficient of $v^{\prime}$
fluctuations. Panels (a1-d1) represent cases C05-C08, respectively. The color
and isolines show the logarithm of the correlation coefficient, with levels
from -1.5 to -0.01 with a step of 0.02. (a2-d2) Map of transfer entropy
between different $y$ positions. Panels (a2-d2) represent cases C05-C08,
respectively. The color and isolines show the logarithm of the transfer
entropy, with levels from -6 to -2 with a step of -1.

#### Correlation map

To illustrate the statistical connection between wall-normal positions, we
calculate the map of maximum correlation coefficient for the $v$ fluctuations
at different $y$ positions, which is defined as

$R^{\mathrm{max}}_{vv}(y_{1},y_{2})=\mathop{\mathrm{max}}\limits_{\Delta
t}\left\\{\frac{\overline{v^{\prime}(t;y_{2})v^{\prime}(t+\Delta
t;y_{1})}}{\sigma_{v^{\prime}}(y_{1})\sigma_{v^{\prime}}(y_{2})}\right\\}$
(10)

where $\mathop{\mathrm{max}}\limits_{\Delta t}$ denotes the maximum regarding
the variable of $\Delta t$. Figure 4(a1-d1) show the contours of
$R^{\mathrm{max}}_{vv}(y_{1},y_{2})$ for all the cases. By definition, the
correlation maps are exactly symmetrical about the diagonal line
$y_{1}=y_{2}$, since the correlation doesn’t differentiate the order of the
two input signals.

For lower porosity cases C05 and C06 (Figure 4a1,b1), the correlation map
shows a ‘neck’ around $(y_{1},y_{2})=(0,0)$ which separates the contour into
two part, i.e., the part with both $y_{1}$ and $y_{2}$ above the interface,
and the one involving porous medium flow ($y_{1}<0$ or $y_{2}<0$). Without
losing generality, we can consider $y_{2}$ (the vertical axis) as the position
of the reference $v^{\prime}$, and its correlated range is shown on $y_{1}$
axis (the horizontal axis). The fluctuation $v^{\prime}$ at the interface
($y_{2}=0$) has a quite strong connection with the flow below the interface
($y_{1}<0$), but a limited correlated range in the channel region ($y_{1}>0$).
This is probably due to the scale separation between eddies of the near-wall
region and the outer layers. As $y_{2}$ increases ($y_{2}\geq 0$), the
correlated range of $y_{1}$ increases linearly with $y_{2}$, which is
associated with the growth of the scale of coherent structures. The correlated
$y_{1}$ range increases more drastically when $y_{2}$ descends into the porous
media area ($y_{2}<0$). In fact, the correlation coefficient between
surface/subsurface flow can even exceed that of two points in the channel.
This is a direct evidence that the motions in the transitional layer of the
porous medium are interacting with the flow structures in the boundary layer.
As porosity grows (C07 and C08), the intensity of the correlation map
increases, and the ‘neck’ between the channel region and the porous media
region becomes ambiguous. However, the outline of the correlation map remains
similar.

The correlation map provides a point of view into the connection of signals at
different layers. However, as explained earlier, this method can not be used
to infer the causal relationship. In the next section, the transfer entropy is
used to clarify this problem.

#### Transfer entropy map

In this section, the same set of data as in Sec.3.2 will be used to compute
transfer entropy, and the result is shown in Figure 4(a2-d2). Since the input
signals for transfer entropy are assigned as target or source, the $y$
positions of the target and source $v^{\prime}$ signals are defined as
$y_{\mathrm{target}}$ and $y_{\mathrm{source}}$, respectively. For each set of
$(y_{\mathrm{target}},y_{\mathrm{source}})$, the delay time $\Delta t$ is
selected so that the maximum value of transfer entropy is reached.

In contrast to the correlation map, the transfer entropy map is clearly
asymmetrical, especially near and below the crest position. When the
destination $y_{\mathrm{target}}$ is located in the channel region, the
transfer entropy decays quickly as $y_{\mathrm{target}}$ moves away from
$y_{\mathrm{target}}$. In contrast, for a target signal below the interface
($y_{\mathrm{target}}<0$), the information flux from the channel flow decays
much slower as the rising of $y_{\mathrm{source}}$, suggesting that the
fluctuations below the interface is actively subject to the influence of
channel flow. The magnitude of the transfer entropy from porous medium to
channel flow is almost negligible for the lowest porosity case C05. As the
porosity increases, the information flux from below the interface to above
(bottom-up effect) rises accordingly, but the top-down coupling always
dominates for the cases tested.

###  3.3 Spatial-resolved transfer entropy

In the previous section, we explored the transfer entropy in $y$ direction,
which elucidated the causal relationship between the channel and porous medium
flow. Note that only one streamwise location is selected which is right at the
middle of two nearby cylinders. In this section, the influence of the
streamwise position on the transfer entropy will also be investigated.
However, instead of enumerating all possible combinations of spatial locations
in sec3.2, we specify $v^{\prime}$ at the throat between the top-layer
cylinders ($y=-r_{c}$) as the representative signal of the porous medium, and
explore the information flux between it and nearby spatial points.

Figure 5:  Map of correlation coefficient $R_{vv}(\Delta x,y)$ in $x-y$ plane.
Panels (a-d) represent cases C05-C08, respectively. The reference point is
fixed at the middle of cylinders, i.e., $(\Delta x,y)=(0,-r_{c})$. The color
and the isolines show the logarithm of the correlation coefficient, with
levels from -2 to -0.02 with a step of 0.02.

As in the previous sections, we first show the two dimensional correlation map
as a comparison with the transfer entropy. With the reference location
selected at $(x_{\mathrm{ref}},y_{\mathrm{ref}})$=($x_{c},-r_{c}$), the
definition of the correlation map is a little bit different from that in
Eq.10, i.e.,

$R_{vv}(\Delta x,y)=\mathop{\mathrm{max}}\limits_{\Delta
t}\left\\{\frac{\overline{v^{\prime}(t;x_{\mathrm{ref}},y_{\mathrm{ref}})v^{\prime}(t+\Delta
t;x_{\mathrm{ref}},y)}}{\sigma_{v^{\prime}}(x_{\mathrm{ref}},y_{\mathrm{ref}})\sigma_{v^{\prime}}(x_{\mathrm{ref}}+\Delta
x,y)}\right\\}$ (11)

The results are shown in Fig.5. There is a strong correlation between the
$v^{\prime}$ of nearby upstream and downstream pore units for all cases
tested. As discussed by Wang et al (2021a), this strong correlation can be
attributed to the interaction of fluids in adjacent pores under the constrain
of continuity. A strong up-welling motion at one pore could induce down-
welling motions of nearby pore units. In addition, Kim et al. Kim et al (2020)
show the up-welling and down-welling motions at the permeable surface is under
the modulation of large scale structures from the channel flow. Particularly,
up-welling motions happen during the passage of a low-speed structure, while
down-welling motions usually correspond to high-speed motions. This is also
supported by the large scale statistical structure in the channel region,
which encapsulate the correlated pore unit below (Fig.5).

##  4 Computational performance

The supercomputing systems used for the DNS was HAWK located at the High
Performance Computer Center Stuttgart (HLRS). The new flagship machine HAWK,
based on the Hewlett Packard Enterprise platform running AMD EPYC 7742
processor code named Rome, has a theoretical peak performance of 26 petaFLOPs,
and consists of a 5,632-node cluster. One AMD EPYC 7742 CPU consists of 64
cores, which leads to 128 cores on a single node sharing 256 GB memory on
board. This means that a total of 720,896 cores are available on the HAWK
system.

Figure 6: Scaling behavior of the solver on ‘HAWK’, wall clock time per time
step on the left side. On the right side the efficiency for different numbers
of MPI ranks.

The scalability of the high-order solver was tested on HAWK from 8,192 to
65,536 CPU cores. With sufficient computational resource, the wall clock time
per time step could reach 0.2-0.4 second, which is optimal to perform direct
numerical simulations for turbulent flows. A minimum efficiency of $70\%$ can
be achieved with the maximal number of MPI ranks. Compared to the scalability
test of the low-order FVM code OpenFOAM Evrim et al (2020); Chu et al (2016b,
a); Pandey et al (2017a, 2018b, 2018a); Yang et al (2020) on the previous HPC
machines in HLRS, the current high-order solver shows an excellent scalability
on the state of art HPC cluster HAWK.

##  5 Conclusions

The causal relationship between surface and sub-surface flow is investigated
by using interface-resolved DNS data. Using turbulent fluctuations as the
source and target signal, we calculated the transfer entropy between the top-
layer pores and a wide range of positions in the channel. The maps of the
spatial distribution of transfer entropy illustrate that the information flux
from the porous medium to the free flow is limited to the near-wall region,
even for the highest porosity case. On the other hand, the causal interaction
in the ‘top-down’ direction is significantly stronger, and there is still
observable causality between a higher layer of the channel flow (up to
$y/H=0.2$) and flow motions at top layer pores.

##  6 Acknowledgement

The study has been financially supported by SimTech (EXC 2075/1–390740016) and
SFB-1313 (Project No.327154368) from Deutsche Forschungsgemeinschaft (DFG).
The authors gratefully appreciate the access to the high performance computing
facility _Hawk_ at HLRS, Stuttgart of Germany.

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