# Large-scale motions and self-similar structures in compressible turbulent
channel flows

Cheng Cheng  Department of Mechanical and Aerospace Engineering, The Hong Kong
University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong  Lin
Fu  Corresponding author. Email<EMAIL_ADDRESS>Department of Mechanical and
Aerospace Engineering, The Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong  Department of Mathematics, The Hong Kong
University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
HKUST Shenzhen-Hong Kong Collaborative Innovation Research Institute, Futian,
Shenzhen, China

(August 27, 2024)

###### Abstract

In this work, we study the scale characteristics of the log- and outer-region
motions and structures in subsonic and supersonic turbulence. To this end, a
series of direct numerical simulations of the compressible turbulent channel
flow at medium Reynolds numbers are performed. Based on this database, the
streamwise and spanwise length scales of the outer-region motions are
investigated by the two-point correlations and the one-dimensional spectra.
The energy distribution among the multi-scale structures in the outer region
is found to be dominated by the semilocal friction-Reynolds-number effects
rather than the Mach-number effects. This conclusion not only holds for the
velocity fluctuations but also the fluctuations of the thermodynamic
variables. Besides, the streamwise and spanwise length scales of the outer
motions do not alter significantly when the flow passes the sound barrier as
reported by a previous experimental study (Bross et al., J. Fluid Mech., vol.
911, 2021, A2). On the other hand, the self-similar structures populating the
logarithmic region are investigated by adopting a linear coherence spectrum.
The streamwise/wall-normal aspect ratio of the self-similar wall-attached
structures of the streamwise velocity and temperature fluctuations is
approximately 15.5, and the counterpart of density and pressure fluctuations
is 1.8. The present study confirms the existence of self-similar structures in
compressible wall turbulence and assesses their geometrical characteristics.

##  I Introduction

Since the discovery of the streaks in the near-wall region of the turbulent
boundary layers by Kline  _et al._ [1], the coherent structures, as well as
the energy-containing motions (eddies) in wall-bounded turbulence have been
extensively investigated over the past decades. Now, it is understood that in
addition to the streaks and the vortical structures residing in the near-wall
region [1, 2], there are also large-scale motions (LSMs), very-large-scale
motions (VLSMs), and self-similar motions populating the logarithmic and outer
regions. These three typical energy-containing motions have been unmasked in
high-Reynolds-number pipes [3, 4], channels [5, 6], and zero-pressure-gradient
turbulent boundary layers [7, 8]. Besides, they have also been identified in
compressible wall-bounded turbulence [9, 10, 11]. Further studies have also
revealed that these energy-containing motions not only exert pronounced
influences on the near-wall turbulent intensities [12, 13], but also the
unsteadiness of the separated bubble downstream of the shockwave/boundary-
layer interactions [14, 15]. Thus, the investigation of their physical
characteristics has remarkable significance for drag reduction and flow-
separation control of the high-speed vehicles. It should be noted that in
internal flows, the long streaky low-streamwise-momentum motions are often
referred to as ‘VLSMs’, whereas in external flows they are more commonly
referred to as ‘superstructures’ [16]. In the present study, we do not
differentiate between the two terminologies and denote these motions in
internal/external flows as VLSMs collectively.

LSMs have a streamwise scale of approximately $1-2\delta$ ($\delta$ indicates
the boundary thickness), and VLSMs are generally believed to extend several
$\delta$ long along the streamwise direction (the exact length is disputed)
and meander in spanwise direction [16]. The investigations of the
incompressible wall-bounded turbulence have provided the majority of the
knowledge concerning these motions. It is speculated that the coherent-
structure organizations of compressible wall turbulence in the logarithmic and
outer regions would resemble those of incompressible flows. However, this
hypothesis is still controversial.

As early as 1989, Smits  _et al._ [17] compared the turbulent structures of
subsonic ($Ma=0.1$, here $Ma$ denotes the Mach number) and supersonic
($Ma=2.9$) boundary layers, and found that their spanwise scales are almost
identical, but the streamwise scales in the supersonic flow are about half the
size of those in subsonic flow. Their analyses are based on the space-time
correlations of the mass-flux fluctuations measured from the hot-wire (HW)
probes. Smits and Dussauge [18] also summarized the experimental measurements
(hot-wire measurements dominate) and came to the conclusion that the
streamwise length scales of outer motions decrease obviously with increasing
Mach number, whereas their spanwise scales have no statistical changes with
the Reynolds and Mach numbers. However, Spina  _et al._ [19] reported that the
average spanwise extent of the largest eddies in a Mach 3 turbulent boundary
layer is similar to that of subsonic turbulent boundary layers by adopting
hot-wire and flow visualizations, whereas their average streamwise scales are
about twice those of low Reynolds number, subsonic turbulent boundary layers.

Nevertheless, these propositions are not totally in accordance with the
measurement via particle image velocimetry (PIV). Ganapathisubramani  _et al._
[9] used planar PIV to investigate the scale characteristics of the energy-
containing motions in a Mach 2 turbulent boundary layer. By employing the two-
point correlation, they discovered that the streamwise length scale of the
streamwise velocity fluctuations at the outer region is nearly four-time long
as that in incompressible cases, and their spanwise length scale is slightly
larger than that of the subsonic flow. However, Williams  _et al._ [20] also
investigated a hypersonic turbulent boundary layer at Mach 7.5 by PIV, and
their conclusion is different, e.g., the streamwise correlation lengths are
less sensitive to the compressibility. Very recently, Bross  _et al._ [11]
examined a sequence of compressible turbulent boundary layers varying from
subsonic to supersonic, by using planar 2-D PIV and stereo-PIV, and they
observed that there is a scale expansion in both streamwise and spanwise
directions when flow passes the sound barrier. It’s interesting to note that,
rather than the absolute value of the Mach number, whether this phenomenon
takes place relies on whether the flow speed is sub- or supersonic.

Direct numerical simulation (DNS) is another tool to tackle this question.
However, the findings from DNS diverge significantly from those of the
aforementioned experimental studies. Ringuette  _et al._ [21] performed a DNS
study of a Mach 3 turbulent boundary layer and found that VLSMs in the
compressible flow are similar to those in incompressible cases, and their
length scale is mostly shaped by the Reynolds-number effects. Pirozzoli and
Bernardini [22] analyzed the structural organization of a Mach 2 turbulent
boundary layer and reported that their streamwise length scales are identical
to the incompressible cases, whereas the spanwise length scales are slightly
larger. Later, they also observed that the spanwise length scales of the
motions in the outer region of the Mach 1.5 turbulent channel flows are close
to the incompressible flows [10]. The scale invariance of LSMs in supersonic
wall turbulence has been drawn in a series of works of their group and
collaborators [22, 23, 10, 24]. Recently, similar results are also provided by
Yao and Hussain [25] and Huang  _et al._ [26] after examining their own DNS
database of compressible channel flows and supersonic/hypersonic turbulent
boundary layers with different thermal boundary conditions. The latter study
also reveals that the wall cooling condition plays a minor role in shaping the
length scales of LSMs and VLSMs. Table 1 summarizes the experimental and
numerical studies on the length scales of the outer-region motions in
compressible wall-bounded turbulence. Reviewing the works of predecessors, it
can be found that the scale characteristics of the motions in the outer region
of the compressible wall turbulence are ambiguous from the experimental and
numerical sides. Most of the DNS studies hold the view that their length
scales are not sensitive to compressibility, whereas for the experimental
studies, it has been a question of different opinions.

Reference | Year | Flow | Ma | Method | $\lambda_{x}$ | $\lambda_{z}$ | Note
---|---|---|---|---|---|---|---
Smits  _et al._ [17] | 1989 | TBL | 0.1-2.9 | HW | $\searrow$ | $\rightarrow$ | None
Spina  _et al._ [19] | 1994 | TBL | 3 | HW | $\nearrow$ | $\rightarrow$ | None
Smits and Dussauge [18] | 2006 | TBL | 0.035-11 | HW | $\searrow$ | $\rightarrow$ | HW dominant
Ganapathisubramani  _et al._ [9] | 2006 | TBL | 2.0 | PIV | $\nearrow$ | $\nearrow$ | None
Williams  _et al._ [20] | 2018 | TBL | 7.5 | PIV | $\rightarrow$ | None | None
Bross  _et al._ [11] | 2021 | TBL | 0.3-3 | PIV | $\nearrow$ | $\nearrow$ | Transonic expansion
Pirozzoli and Bernardini [22] | 2011 | TBL | 2 | DNS | $\rightarrow$ | $\nearrow$ | None
Pirozzoli [23] | 2012 | TBL/CP | 0.3-4 | DNS | None | $\rightarrow$ | None
Modesti and Pirozzoli [10] | 2016 | CH | 0.1-3 | DNS | None | $\rightarrow$ | None
Yao and Hussain [25] | 2020 | CH | 0-1.5 | DNS | $\rightarrow$ | $\rightarrow$ | None
Huang  _et al._ [26] | 2022 | TBL | 2.5-10.9 | DNS | $\rightarrow$ | $\rightarrow$ | None
Cogo  _et al._ [24] | 2022 | TBL | 2-5.86 | DNS | None | $\rightarrow$ | None
Table 1: Summary of the experimental and numerical studies on the length
scales of the outer-region motions in compressible wall-bounded turbulence.
Here, TBL, CP, and CH denote turbulent boundary layer, Couette-Poiseuille
flow, and channel flow, respectively. $\lambda_{x}$ and $\lambda_{z}$ denote
the streamwise and spanwise length scales, respectively. The symbols
$\nearrow$ and $\searrow$ denote larger and smaller than the
incompressible/subsonic counterparts, respectively, whereas $\rightarrow$
denotes no change.

On the other hand, it is documented that there are self-similar motions
populating the logarithmic region of the wall turbulence [27, 28, 29, 30].
According to the well-known attached-eddy model, the logarithmic region is
occupied by an array of self-similar energy-containing motions (or eddies)
with their roots attached to the near-wall region [31, 32]. Hence, examining
the geometrical features of these self-similar structures is beneficial for
both developing the attached-eddy hypothesis and advancing the knowledge of
turbulent dynamics. In this respect, nearly all the works concentrate on
incompressible wall-bounded turbulence. For example, Baars  _et al._ [27]
identified the self-similar structures of the streamwise velocity fluctuations
in incompressible boundary layers by a linear coherence spectrum and found the
streamwise/wall-normal aspect ratio ($AR$) of them is roughly 14. This scale
characteristic is then employed to construct a filter for isolating the
spectral energy results from the wall-attached eddies [33]. Whether it still
holds in compressible turbulence? Whether the thermodynamic variables bear the
self-similar characteristics as the streamwise velocity fluctuations? If yes,
what is their $AR$? None of these issues have been reported to be thoroughly
investigated. Very recently, Yu  _et al._ [34] employed the proper orthogonal
decomposition to identify the self-similar structures of the streamwise
velocity and the temperature fluctuations in a compressible channel flow, but
their geometrical characteristics have not been clarified.

The work of the present study is divided into two parts. In the first part, we
conduct a series of DNSs of the subsonic and the supersonic turbulent channel
flows at medium Reynolds numbers (the maximal friction Reynolds number is
1150). Then, in conjunction with the open-source incompressible data, the
Mach-number effects on the scale characteristics of the outer-region motions
are dissected, together with the scale characteristics of the thermodynamic
variables, such as $p^{\prime}$ (pressure fluctuation), $T^{\prime}$
(temperature fluctuation), and $\rho^{\prime}$ (density fluctuation).
Particular attention is paid to the scale expansion when flow passes the sound
barrier reported by a recent study [11]. In the second part, we elaborate on
the geometrical characteristics of the self-similar structures populating the
logarithmic region. We generalize the work of Baars  _et al._ [27] to the
compressible flows by leveraging the database set up in the present study. We
not only compare the geometrical properties of the self-similar wall-attached
velocity fluctuation with the incompressible counterparts but also unravel
those of the thermodynamic variables. The present work may help develop an
advanced scale-resolving method for predicting the compressible wall-bounded
flows [35, 36].

The focuses of the present study are summarized as follows.

(1) The scale characteristics of the outer-region motions in subsonic and
supersonic wall turbulence, not only for the velocity fluctuations, but also
the fluctuations of the thermodynamic variables. The latter has rarely been
reported before. Particular attention is paid to the scale expansion when flow
passes the sound barrier reported by a recent study [11].

(2) The possible factors that may account for the differences between the
experimental and numerical studies, such as the variable for study, the method
to normalize the spectrum, and the wall-normal position of the detected plane.

(3) The scale characteristics of the self-similar structures in the
logarithmic region, not only for the streamwise velocity fluctuations, but
also the thermodynamic variables.

The paper is organized as follows: In Sec. II, the DNS setup and numerical
validation are provided in detail. The scale characteristics of the outer-
region motions in compressible channel flow are analyzed by the two-point
correlation and the one-dimensional spectrum in Sec. III, concurrently, the
geometrical characteristics of the self-similar structures in the logarithmic
region are also investigated in this section. Conclusions are given in Sec.
IV.

##  II NUMERICAL setup and validation

DNSs of compressible turbulent channel flows have been conducted with a
finite-difference code, by solving the three-dimensional unsteady compressible
Navier–Stokes equations. The convective terms are discretized with a seventh-
order upwind-biased scheme, and the viscous terms are evaluated with an
eighth-order central difference scheme. Time advancement is performed using
the third-order strong-stability-preserving (SSP) Runge–Kutta method [37]. A
constant molecular Prandtl number $Pr$ of 0.72 and a specific heat ratio
$\gamma$ of 1.4 are employed. The dependence of dynamical viscosity $\mu$ on
temperature $T$ is given by Sutherland’s law, i.e.,

$\mu=\mu_{0}\frac{T_{0}+S}{T+S}\left(\frac{T}{T_{0}}\right)^{3/2},$ (1)

where $S=110.4K$ and $T_{0}=273.1K$.

All the DNSs are carried out in a rectangular box with its sizes along
streamwise ($x$), spanwise ($z$), and wall-normal ($y$) directions denoted as
$L_{x}$, $L_{z}$, and $L_{y}$, respectively. In the streamwise and spanwise
directions, the mesh is uniformly spaced, whereas, in the wall-normal
direction, the mesh is hyperbolically clustered towards the walls. The
isothermal no-slip conditions are imposed at the top and bottom walls, and the
periodic boundary condition is imposed in the wall-parallel directions, i.e.,
$x$ and $z$ directions. All simulations begin with a parabolic velocity
profile with random perturbations superimposed and uniform temperature and
density values. A body force is imposed in the streamwise direction to
maintain a constant mass flow rate, and a corresponding source term is also
added in the energy equation [38].

In the present study, we carry out two simulations at a bulk Mach number
$M_{b}=U_{b}/C_{w}=0.8$ ($U_{b}$ is the bulk velocity, and $C_{w}$ is the
speed of sound at wall temperature) and $Re_{b}=\rho_{b}U_{b}h/\mu_{w}=7667$
and $17000$ ($\rho_{b}$ denotes the bulk density, $h$ the channel half-height,
and $\mu_{w}$ the dynamic viscosity at the wall). Two other DNSs at a bulk
Mach number $M_{b}=1.5$ and $Re_{b}=9400$ and $20020$ are also conducted. For
all cases, the computational domain has the same dimensions, namely,
$L_{x}\times L_{z}\times L_{y}=4\pi h\times 2\pi h\times 2h$. Previous studies
have verified that these setups of dimensions can capture most of the large-
scale motions in the outer region of the boundary layer [39, 40]. Details of
the parameter settings are listed in Table 2. Two incompressible cases at
$Re_{\tau}=547$ and $Re_{\tau}=934$ by Del Álamo and Jiménez [5] and Del Álamo
_et al._ [41] are also employed for comparison. Details of the parameter
settings are listed in Table 3.

Case | $M_{b}$ | $Re_{b}$ | $Re_{\tau}$ | $Re_{\tau}^{*}$ | $\Delta x^{+}$ | $\Delta z^{+}$ | $\Delta y_{min}^{+}$ | $\Delta y_{max}^{+}$ | $Tu_{\tau}/h$
---|---|---|---|---|---|---|---|---|---
Ma08Re8K | 0.8 | 7667 | 436 | 382 | 10.8 | 6.9 | 0.44 | 5.4 | 49.4
Ma08Re17K | 0.8 | 17000 | 882 | 778 | 10.8 | 6.5 | 0.63 | 6.4 | 15.3
Ma15Re9K | 1.5 | 9400 | 594 | 395 | 7.3 | 3.7 | 0.5 | 5.9 | 30.2
Ma15Re20K | 1.5 | 20020 | 1150 | 780 | 9.3 | 4.7 | 0.49 | 6.9 | 9.1
Table 2: Parameter settings of the compressible DNS database. Here, $M_{b}$ denotes the bulk Mach number. $Re_{b}$, $Re_{\tau}$ and $Re_{\tau}^{*}$ denote the bulk Reynolds number, friction Reynolds number, and semilocal friction Reynolds number, respectively. $\Delta x^{+}$ and $\Delta z^{+}$ denote the streamwise and spanwise grid resolutions in viscous units, respectively. $\Delta y_{min}^{+}$ and $\Delta y_{max}^{+}$ denote the finest and the coarsest resolution in the wall-normal direction, respectively. $Tu_{\tau}/h$ indicates the total eddy turnover time used to accumulate statistics. Case | $Re_{\tau}$ | $Re_{\tau}^{*}$ | $L_{x}(h)$ | $L_{y}(h)$ | $L_{z}(h)$ | $\Delta x^{+}$ | $\Delta z^{+}$ | $\Delta y_{min}^{+}$ | $\Delta y_{max}^{+}$ | $Tu_{\tau}/h$
---|---|---|---|---|---|---|---|---|---|---
Ma00Re10K | 547 | 547 | $8\pi$ | 2 | $4\pi$ | 13.4 | 6.8 | 0.04 | 6.7 | 22
Ma00Re18K | 934 | 934 | $8\pi$ | 2 | $3\pi$ | 11.5 | 5.7 | 0.03 | 7.6 | 12
Table 3: Parameter settings of the incompressible DNS database.

Both the Reynolds (denoted as $\bar{\phi}$) and the Favre averaged (denoted as
$\widetilde{\phi}=\overline{\rho\phi}/\overline{\rho}$) statistics are used in
the present study. The corresponding fluctuating components are represented as
$\phi^{\prime}$ and $\phi^{\prime\prime}$, respectively. Hereafter, we use the
superscript $+$ to represent the normalization with the friction velocity
(denoted as $u_{\tau}$, $u_{\tau}=\sqrt{\tau_{w}/\rho_{w}}$, $\tau_{w}$ is the
mean wall-shear stress), and the viscous length scale (denoted as
$\delta_{\nu}$, $\delta_{\nu}=\nu_{w}/u_{\tau}$, $\nu_{w}=\mu_{w}/\rho_{w}$).
We also use the superscript $*$ to represent the normalization with the
semilocal wall units, i.e., $u_{\tau}^{*}=\sqrt{\tau_{w}/\overline{\rho}}$ and
$\delta_{\nu}^{*}=\overline{\nu(y)}/u_{\tau}^{*}$. Hence, the relationship
between the semilocal friction Reynolds number and the friction Reynolds
number is
$Re_{\tau}^{*}=Re_{\tau}\sqrt{\left(\overline{\rho_{c}}/\bar{\rho}_{w}\right)}/\left(\overline{\mu_{c}}/\bar{\mu}_{w}\right)$.
The subscript $c$ refers to the quantities evaluated at the channel center. It
is noted that the cases Ma08Re8K and Ma08Re17K share similar $Re_{\tau}^{*}$
with the cases Ma15Re9K and Ma15Re20K, respectively.

Figure 1: Profiles of mean streamwise velocity, mean temperature, and mean
density for the cases Ma08Re8K ($a$) and Ma08Re17K ($c$); profiles of the
Reynolds stress for the cases Ma08Re8K ($b$) and Ma08Re17K ($d$). The profiles
of $T/T_{w}$ are shifted upward by $\Delta(T/T_{w})=0.2$ for better
comparison.

Figure 2: Profiles of mean streamwise velocity, mean temperature, and mean
density for the cases Ma15Re9K ($a$) and Ma15Re21K ($c$); profiles of the
root-mean-square of the velocity fluctuation for the cases Ma15Re9K ($b$) and
Ma15Re21K ($d$). The profiles of $T/T_{w}$ are shifted upward by
$\Delta(T/T_{w})=0.2$ for better comparison.

To verify the compressible dataset of present study, Fig. 1 compares the DNS
results of Ma08Re8K and Ma08Re17K with the flow statistics of Yao and Hussain
[25] at identical $Ma_{b}$ and $Re_{b}$, respectively. Both the mean
quantities and the Reynolds stress $\tau_{ij}=\bar{\rho}R_{ij}$ with
$R_{ij}=\widetilde{u_{i}^{\prime\prime}u_{j}^{\prime\prime}}=\widetilde{u_{i}u_{j}}-\widetilde{u_{i}}\widetilde{u_{j}}$
($v$ and $w$ denote the wall-normal and spanwise velocity, respectively) are
compared. Fig. 2 compares the mean quantities and the root-mean-square
(r.m.s.) of the velocity fluctuations of Ma15Re9K and Ma15Re20K with the
results reported by Modesti and Pirozzoli [10] at $M_{b}=1.5$, $Re_{b}=7667$
and $M_{b}=1.5$, $Re_{b}=17000$, respectively. All the profiles of the
concerned quantities agree reasonably with the previous studies. The minor
differences shown in Fig. 2 are ascribed to the distinctions in $Re_{b}$. All
these confirm the accuracy of the present database.

Figure 3: Profiles of mean local Mach number for all compressible cases. The
red dashed line indicates $y=0.2h$, i.e., the wall-normal position for
investigation in the present study.

Fig. 3 plots the variations of the mean local Mach number ($M_{l}$) as
functions of the wall-normal height $y/h$. For the cases at $M_{b}=0.8$,
$M_{l}\approx 0.69$ at $y=0.2h$, whereas for the cases at $M_{b}=1.5$,
$M_{l}\approx 1.16$ at the same wall-normal height. Considering that $y=0.2h$
is the upper bound of the logarithmic region [42], dissecting the scale
characteristics at this wall-normal height can shed light on the differences
between the large-scale motions in the subsonic and supersonic wall-bounded
turbulence. The results presented below are mainly focused on this wall-normal
position. We also show that the conclusions drawn below are not changed with
varying wall-normal heights within the outer region (see appendix A).

##  III Results and discussions

###  III.1 Variable for characterizing outer-layer motions

Before processing our analysis, it is crucial to clarify which physical
variable is appropriate to characterize the scales of the outer-layer motions.
Several variables have been employed by previous studies to scrutinize the
scale characteristics of compressible wall-bounded turbulence. For example,
Smits  _et al._ [17] adopted the correlated signals of the mass-flux
fluctuations $(\rho u)^{\prime}$ to depict the spatial organizations of the
coherent structures in the compressible boundary layers. Ganapathisubramani
_et al._ [9] used the two-point correlation of $u^{\prime}$ to measure the
length scale of VLSMs in a turbulent boundary layer of Mach 2. Moreover,
$\sqrt{\rho}u^{{}^{\prime\prime}}$ has also been visualized to display the
formation of the energy-containing motions vividly [25, 43, 26].

Figure 4: Streamwise ($a$) and spanwise ($b$) two-point correlations of
$u^{\prime}$, $\sqrt{\rho}u^{{}^{\prime\prime}}$, and $(\rho u)^{\prime}$ for
the case Ma15Re20K at $y/h=0.2$.

In this study, the behaviors of these quantities are compared by utilizing the
streamwise and spanwise two-point correlations based on the case Ma15Re20K.
For the streamwise two-point correlation of variable $\psi$, it takes the form
of

$R_{x}\left(r_{x},y\right)=\frac{\left\langle\psi(x,y,z)\psi\left(x+r_{x},y,z\right)\right\rangle}{\left\langle\psi^{2}\right\rangle},$
(2)

where $r_{x}$ is the separation distance along the streamwise direction, and
$<\cdot>$ represents the averaging in the temporal and spatially homogeneous
directions. The spanwise two-point correlation $R_{z}$ can be defined in a
similar way. Fig. 4($a$) and ($b$) show the variations of the streamwise and
spanwise two-point correlations of the above-mentioned variables at $y=0.2h$,
respectively. We only show the profiles at $r_{x}>0$ and $r_{z}>0$ due to the
symmetries of the two correlations with respect to $r_{x}=0$ and $r_{z}=0$,
respectively. It can be seen that the profiles of $R_{x}$ and $R_{z}$ of all
the quantities overlap with each other well. We have checked that the same
conclusion applies to other velocity fluctuation components, and the
assessment is not shown here for brevity. It indicates that these physical
quantities share identical length-scale characteristics statistically, at
least when $M_{b}\leq 1.5$. It also agrees with the observation of Ringuette
_et al._ [21], who reported that the correlations of $(\rho u)^{\prime}$ and
$u^{\prime}$ are very similar in a turbulent boundary layer at Mach 3. Whether
it holds at higher Mach numbers remains to be checked. Thus, the results
presented below mainly focus on the variables weighted by $\sqrt{\rho}$.

###  III.2 Large-scale motions in subsonic and supersonic turbulent channel
flows

####  III.2.1 Density-weighted velocity fluctuations

Figure 5: Streamwise ($a$,$c$,$e$) and spanwise ($b$,$d$,$f$) premultiplied
one-dimensional spectra of $\sqrt{\rho}u^{{}^{\prime\prime}}$ ($a$,$b$),
$\sqrt{\rho}v^{{}^{\prime\prime}}$ ($c$,$d$), and
$\sqrt{\rho}w^{{}^{\prime\prime}}$ ($e$,$f$) for all cases at $y/h=0.2$.

In this section, we dedicate to comparing the scale characteristics of the
large-scale motions in subsonic and supersonic channel flows. Fig. 5($a$) and
($b$) show the premultiplied streamwise and spanwise spectra of the density-
weighted streamwise velocity fluctuations ($\sqrt{\rho}u^{{}^{\prime\prime}}$)
at $y=0.2h$. The area under the spectral curve represents the intensity of the
density-weighted Reynolds stress
$\overline{(\sqrt{\rho}u^{{}^{\prime\prime}})^{2}}$. For the streamwise
spectra, all the spectra peak at $\lambda_{x}\approx 2h$, and no observable
differences in the streamwise wavelengths of the most energetic scales
($\lambda_{x,me}$) between the incompressible, subsonic, and supersonic cases
can be found. The spectral energy of the peaks is determined by the magnitude
of $Re_{\tau}^{*}$ instead. For the spanwise spectra, all the cases peak at
$\lambda_{z}\approx 1h$, the typical width of VLSMs [42], and the distinctions
between the $\lambda_{z,me}$ of all cases are still inconspicuous. Similar
conclusions about the length scales of $\sqrt{\rho}u^{{}^{\prime\prime}}$ in
the outer region have also been drawn by [25].

Very recently, Bross  _et al._ [11] conducted an experimental study and
observed that the $\lambda_{x,me}$ in the outer region of the supersonic
turbulent boundary layers are slightly larger than those of the subsonic
cases, whereas the $\lambda_{z,me}$ of the supersonic cases have a distinct
increase compared with the subsonic cases. In compressible turbulent channel
flows, at least within the Reynolds and Mach numbers under scrutinizing by the
present study, this phenomenon is not that remarkable. The possible
explanations are given in section III.2.3. As a side note, Bross  _et al._
[11] normalized the spectra of the streamwise velocity fluctuations in a
different way, i.e.,
$(\overline{\rho}/\rho_{w})k_{x}E_{u^{\prime}u^{\prime}}/u_{\tau}^{2}$ and
$(\overline{\rho}/\rho_{w})k_{z}E_{u^{\prime}u^{\prime}}/u_{\tau}^{2}$. We
have verified that the conclusions reached above are unaffected by normalizing
the spectra in this manner (see section III.2.3).

The premultiplied streamwise and spanwise spectra of
$\sqrt{\rho}v^{{}^{\prime\prime}}$ and $\sqrt{\rho}w^{{}^{\prime\prime}}$ are
also shown in Fig. 5($c$-$d$) and ($e$-$f$), respectively. It can be seen that
no clear Mach-number dependence of the wavelengths of the spectral peaks can
be identified for all cases, regardless of the spectral direction. Moreover,
the spectra of the cases at larger $Re_{\tau}^{*}$ are more energetic in small
length scales, and the cases at similar $Re_{\tau}^{*}$ collapse well.
Although less obvious, this phenomenon can also be seen in the premultiplied
streamwise and spanwise spectra of $\sqrt{\rho}u^{{}^{\prime\prime}}$. These
observations highlight the fact that the semilocal Reynolds-number effects
dominate the energy distribution among the multi-scale eddies in the outer
region, regardless of whether the flow passes the sound barrier or not, and
whether the flow is compressible or not. They are also consistent with some
previous studies [44, 45, 43, 46, 47], which emphasize that the semilocal
Reynolds number $Re_{\tau}^{*}$ is a key similarity parameter for the
compressible wall-bounded turbulence. In summary, the present study does not
observe that there is a sudden increase in length scales of the out-layer
motions when the flow transitions from subsonic to supersonic state.

####  III.2.2 Thermodynamic variables

Figure 6: Streamwise ($a$,$c$,$e$) and spanwise ($b$,$d$,$f$) premultiplied
one-dimensional spectra of $T^{\prime}$ ($a$,$b$), $p^{\prime}$ ($c$,$d$), and
$\rho^{\prime}$ ($e$,$f$) for all compressible cases at $y/h=0.2$. All the
spectra are normalized by their maximum values.

In this section, we turn to report the scale characteristics of the
thermodynamic variables, i.e., $T^{\prime}$, $p^{\prime}$, and $\rho^{\prime}$
at outer region $y=0.2h$. Fig. 6($a$-$b$), ($c$-$d$), and ($e$-$f$) show the
premultiplied streamwise and spanwise spectra of $T^{\prime}$, $p^{\prime}$,
and $\rho^{\prime}$ of the compressible cases, respectively. All the spectra
are normalized by their maximum values for comparison. It is apparent that the
cases with similar $Re_{\tau}^{*}$ share akin scale-dependent energy
distributions for all three thermodynamic variables. This observation supports
the viewpoint put forward above that $Re_{\tau}^{*}$ is the key parameter in
shaping the spectra. Moreover, no clear Mach-number effects on the scale
characteristics can be identified between the cases with similar
$Re_{\tau}^{*}$. This demonstrates once again that there is no sudden increase
in length scales of the outer-region motions when the flow becomes supersonic
from a subsonic state, as the scale characteristics of the thermodynamic
variables are highly linked with those of LSMs and VLSMs [22].

####  III.2.3 Discussions

The reason why the DNS results differ from the experimental results merits a
discussion. Ganapathisubramani  _et al._ [9] and Bross  _et al._ [11]
conjectured that analysis of mass-flux $(\rho u)^{\prime}$ versus $u^{\prime}$
may result in distinct conclusions about the length scales of the outer
motions. The origin of this discrepancy is ascribed to the different velocity
measurement tools in the experiments, i.e., the hot wire versus the PIV.
However, Fig. 4 above demonstrates that the discrepancies of the mean length
scale between these two variables at $y=0.2h$ are minor, at least, within the
Reynolds and Mach numbers under consideration. Fig. 7 compares the streamwise
and spanwise spectra of the streamwise velocity fluctuations for Ma15Re9K and
Ma15Re20K with different density-weighting approaches and without density
weighting. It is noted that
$(\overline{\rho}/\rho_{w})k_{x}E_{u^{\prime}u^{\prime}}/u_{\tau}^{2}$ and
$(\overline{\rho}/\rho_{w})k_{z}E_{u^{\prime}u^{\prime}}/u_{\tau}^{2}$ are
commonly adopted by experimental studies to exhibit the scale characteristics
of the velocity signals. It can be seen that the density-weighted spectra
overlap with each other. It indicates that the different weighting methods do
not alter the energy distributions among scales within the Reynolds and Mach
numbers under consideration. In addition, though the spectra without density
weighting have unique forms, the length scales of their peaks are consistent
with the density-weighted ones, that is, $\lambda_{x,e}\approx 2h$ and
$\lambda_{z,e}\approx 1h$. In other words, for the spectra of the current
examples under examination, the consequences of weighting with density on the
values of $\lambda_{x,e}$ and $\lambda_{z,e}$ are negligible.

Figure 7: Streamwise ($a$) and spanwise ($b$) premultiplied spectra of the
streamwise velocity fluctuations for Ma15Re9K and Ma15Re20K at $y=0.2h$ with
different density-weighted approaches and without density weighting.

Another possible reason is that the configuration of the wall turbulence here
is different from that of the work of experimentalists, who usually
investigate the length scales of outer motions in turbulent boundary layers,
instead of channel flow [17, 9, 48, 11]. Some previous studies on
incompressible wall turbulence have reported that the average streamwise
length scale of VLSMs in internal flow (e.g., pipe and channel flow) is larger
than that of the turbulent boundary layer [49, 50, 8]. Besides, the friction
Reynolds numbers of the cases in the present study are $O$($10^{2}$) to
$O$($10^{3}$), whereas for the works of experimentalists, they are usually
$O$($10^{3}$) to $O$($10^{4}$) [9, 11]. Hence, the observations reported in
the present study remained to be verified in high-Reynolds-number wall
turbulence. We also notice that, in the experiments conducted by Bross  _et
al._ [11], the walls are hot (the wall temperature is higher than that of free
stream), whereas for the turbulent channel flows in the present study, the
walls are cool (see Fig. 1 and 2). This might be another non-negligible
factor.

At last, the wall-normal location of the measurement plane deserves a
discussion. When examining the spanwise length scales of the outer motions,
Bross  _et al._ [11] set the measurement plane at approximately $y/\delta=0.1$
for the subsonic cases, and at $y/\delta=0.2$ for the supersonic cases. They
found that the average width of the outer motions is larger for the supersonic
cases. In Fig. 8, we repeat this process, and compare the spanwise spectra of
$\sqrt{\rho}u^{{}^{\prime\prime}}$ for different cases. A similar phenomenon
can also be observed. However, as the spanwise length scales of the energy-
containing motions populating the logarithmic and outer regions scale in the
outer unit and increase with their wall-normal heights [41, 51, 52], the
discrepancies of $\lambda_{z,e}$ for the supersonic and the subsonic cases
shown in Fig. 8 here and Fig. 15 of [11] are ascribed to the wall-normal
growth of the motions, rather than the Mach-number effects. Thus, to preclude
the scale-growth effects and uncover the Mach-number effects on length scales
of LSMs and VLSMs, the wall-normal position of the measurement planes for the
supersonic and the subsonic cases should be identical and located in the
logarithmic or outer regions.

Figure 8: Spanwise premultiplied spectra of $\sqrt{\rho}u^{{}^{\prime\prime}}$
at $y=0.1h$ for the subsonic cases, and at $y=0.2h$ for the supersonic cases.

###  III.3 Self-similar structures in compressible turbulent channel flows

In this section, we pay attention to the self-similar structures in
compressible turbulent channel flows. To inspect the structures that are
highly correlated with the near-wall flows, the linear coherence spectrum
(LCS) of the variable $\psi$ is adopted, which is given by

$\gamma^{2}\left(y_{o},y;\lambda_{x}\right)\equiv\frac{\left|\left\langle\widehat{\psi}\left(y_{o};\lambda_{x}\right)\widehat{\psi}^{*}\left(y;\lambda_{x}\right)\right\rangle\right|^{2}}{\left\langle\left|\widehat{\psi}\left(y_{o};\lambda_{x}\right)\right|^{2}\right\rangle\left\langle\left|\widehat{\psi}\left(y;\lambda_{x}\right)\right|^{2}\right\rangle},$
(3)

where $\psi(y_{o})$ denotes the variable $\psi$ at the location $y_{o}$ in the
logarithmic region, $\psi(y)$ denotes the variable $\psi$ at the location $y$
in the near-wall region, $\widehat{\psi}$ is the Fourier coefficient of
$\psi$, $\widehat{\psi}^{*}$ is the complex conjugate of $\widehat{\psi}$, and
$\mid\cdot\mid$ is the modulus. $\gamma^{2}$ evaluates the square of the
scale-specific correlation between $\psi(y_{o})$ and $\psi(y)$ with
$0\leq\gamma^{2}\leq 1$ [53]. LCS has been adopted to study the self-similar
wall-attached structures in incompressible boundary layers [27, 33], and the
coherence between the temperature and the velocity fields in compressible wall
turbulence [54]. To the authors’ knowledge, no study on the geometrical
characteristics of the self-similar structures in compressible turbulent
channel flows by employing LCS has been reported, particularly the structures
of the thermodynamic variables. In the present study, we fill this gap via
analyzing the cases Ma08Re17K and Ma15Re20K, as their Reynolds numbers are
relatively higher.

Figure 9: Variations of $\gamma^{2}$ as functions of $\lambda_{x}/y_{o}$ at
two selected wall-normal heights, i.e., $y_{o}=0.1h$ and $y_{o}=0.15h$, for
$\psi=\sqrt{\rho}u^{{}^{\prime\prime}}$ ($a$), $T^{\prime}$ ($b$),
$p^{\prime}$ ($c$), and $\rho^{\prime}$ ($d$), respectively. The dashed lines
in the subfigures denote the logarithmic growth Eq. (4).

In the present study, the near-wall region is fixed at $y^{+}\approx 0.6$, and
the outer reference height $y_{o}$ is located in the logarithmic region. We
have checked that as long as $y^{+}\leq 15$, the results presented below are
insensitive to the choice of $y^{+}$. Fig. 9($a$-$d$) show the variations of
$\gamma^{2}$ as functions of $\lambda_{x}/y_{o}$ at two selected wall-normal
heights, i.e., $y_{o}=0.1$ and $y_{o}=0.15$, for
$\psi=\sqrt{\rho}u^{{}^{\prime\prime}}$, $T^{\prime}$, $p^{\prime}$, and
$\rho^{\prime}$, respectively. For $\psi=\sqrt{\rho}u^{{}^{\prime\prime}}$ and
$T^{\prime}$, $\gamma^{2}$ grows linearly with $\ln(\lambda_{x}/y_{o})$ for
$\lambda_{x}/y_{o}\geq 20$, whereas for $\psi=p^{\prime}$ and $\rho^{\prime}$,
the similar linear relations can be recognized for $2\leq\lambda_{x}/y_{o}\leq
10$. Moreover, for the wall turbulence at different wall-normal heights and
Mach numbers, the $\gamma^{2}$ profiles of a target variable $\psi$ roughly
coincide with each other in these regions. It indicates that there are self-
similar structures of $\psi$ populating the logarithmic region, whose scale
characteristics are not sensitive to the Mach numbers. This scenario is
consistent with the celebrated attached-eddy model [31, 32], which
hypothesizes that the energy-containing motions in the logarithmic region are
self-similar and can permeate into the near-wall region.

$\psi$ | $C_{1}$ | $C_{2}$ | $AR$
---|---|---|---
$\sqrt{\rho}u^{{}^{\prime\prime}}$ | 0.194 | -0.531 | 15.44
$T^{\prime}$ | 0.20 | -0.552 | 15.79
$p^{\prime}$ | 0.339 | -0.190 | 1.75
$\rho^{\prime}$ | 0.177 | -0.107 | 1.83
Table 4: The fitted constants $C_{1}$, $C_{2}$ in Eq. (4), and the
corresponding $AR$ assessed by Eq. (5) for different $\psi$.

The magnitude of $\gamma^{2}$ within the self-similar region takes the form of

$\gamma^{2}=C_{1}\ln\left(\frac{\lambda_{x}}{y_{o}}\right)+C_{2},$ (4)

where $C_{1}$ and $C_{2}$ are two constants and can be estimated by fitting
the $\gamma^{2}-$ spectra over this range. Baars  _et al._ [27] proposed that
the streamwise/wall-normal aspect ratio of the self-similar structures can be
assessed by

$\left.AR\equiv\frac{\lambda_{x}}{y_{o}}\right|_{\gamma^{2}=0}=\exp\left(\frac{-C_{2}}{C_{1}}\right),$
(5)

according to the hierarchical structure of the self-similar attached eddies
[32]. The values of $C_{1}$, $C_{2}$, and $AR$ for distinct $\psi$ are listed
in Table 4. It can be seen that the $AR$ of $\sqrt{\rho}u^{{}^{\prime\prime}}$
and $T^{\prime}$ are very close to each other and appropriately equal to 15.5.
This result is in accordance with the assessment conducted by Baars  _et al._
[27], who found the $AR$ of the self-similar structures of the streamwise
velocity fluctuations in incompressible turbulent boundary layers is 14.
Besides, it is not unexpected that the $AR$ of $T^{\prime}$ is closely
approximate to that of $\sqrt{\rho}u^{{}^{\prime\prime}}$, as a number of
studies show that the strong Reynolds analogy (SRA) is valid in compressible
wall turbulence [55, 38, 56, 35]. For $p^{\prime}$ and $\rho^{\prime}$, the
$AR$ is approximately equal to 1.8, and significantly smaller than those of
$\sqrt{\rho}u^{{}^{\prime\prime}}$ and $T^{\prime}$. This observation is
reminiscent of the property of $p^{\prime}$ in incompressible wall turbulence,
that is, $p^{\prime}$ is more energetic at small scales ($\lambda_{x}\leq
C_{3}h$, $C_{3}$ is a constant) [57]. Baars  _et al._ [27] and Baars and
Marusic [33] also noticed that the $\gamma^{2}-$spectra of $u^{\prime}$ in
incompressible turbulent boundary layers tend to be constant at an outer-
scaling limit of $\lambda_{x}/h\approx 10$. It can not be observed by the
present study for $\psi=\sqrt{\rho}u^{{}^{\prime\prime}}$ and $T^{\prime}$ due
to the limited size of the computational domain. However, for
$\psi=p^{\prime}$ and $\rho^{\prime}$, the $\gamma^{2}-$spectra shown in Fig.
9($c$) and ($d$) departure from the power-law relation Eq. (4) for
$\lambda_{x}/y_{0}\geq 10$, which results from the very large-scale structures
of these two variables. Again, this shows that the scale characteristics of
$p^{\prime}$ and $\rho^{\prime}$ are completely different from those of
$\sqrt{\rho}u^{{}^{\prime\prime}}$ and $T^{\prime}$ in compressible wall
turbulence.

Finally, we want to point out that the geometrical characteristics of the
self-similar structures reported in the present study may aid in developing
physics -based data-driven filters in compressible wall turbulence study, just
as the work of [33] in incompressible flows. Besides, the LCS employed here
can also be used to shed light on the effects of the wall thermal boundary
conditions on the sizes of the self-similar structures in the compressible
flows. For example, Huang  _et al._ [26] observed that strong wall cooling can
lead to a remarkable reduction in the scale separation between the small and
large eddies. The LCS can be developed as an effective tool to quantify this
variation.

##  IV Conclusions

In the present study, we conduct a series of DNSs of the subsonic and
supersonic turbulent channel flows at moderate Reynolds numbers. By employing
the DNS database, we compare the scale characteristics of the outer motions in
subsonic and supersonic turbulence and dissect the geometrical characteristics
of the self-similar structures for the streamwise velocity fluctuations and
the fluctuations of the thermodynamic variables. The conclusions are
summarized below.

($a$) The semilocal friction Reynolds number rather than the Mach number
determines the energy distribution among the multi-scale structures in the
outer region, not only for the velocity fluctuations but also for the
fluctuations of the thermodynamic variables. In addition, contrary to what a
prior experimental study claimed, we have not found that the streamwise and
spanwise length scales of the outer motions alter significantly when flow
passes the sound barrier.

($b$) The streamwise/wall-normal aspect ratio of the self-similar wall-
attached structures of $\sqrt{\rho}u^{{}^{\prime\prime}}$ and $T^{\prime}$ is
$15.5$, which is similar to that of the self-similar structures in
incompressible wall turbulence. The counterpart of $p^{\prime}$ and
$\rho^{\prime}$ is $1.8$, which has not been reported before. The very large-
scale structures of $p^{\prime}$ and $\rho^{\prime}$ are those with
$\lambda_{x}/y_{0}\geq 10$. These scale characteristics are found to be not
sensitive to the Mach and Reynolds numbers.

It is noted that the conclusions drawn here are only verified by the database
of the compressible channel flows. Whether they hold in wall turbulence with
different configurations requires further research. Besides, the wall
temperature is deemed to impose non-negligible effects on the energy-
containing motions in wall-bounded turbulence. Hence, it is of great
significance to investigate them in turbulent boundary layers with different
wall temperatures.

###### Acknowledgements.

We would like to thank Professor Jiménez for making the incompressible DNS
data available. L.F. acknowledges the fund from the Research Grants Council
(RGC) of the Government of Hong Kong Special Administrative Region (HKSAR)
with RGC/ECS Project (No. 26200222), the fund from Guangdong Basic and Applied
Basic Research Foundation (No. 2022A1515011779), and the fund from the Project
of Hetao Shenzhen-Hong Kong Science and Technology Innovation Cooperation Zone
(No. HZQB-KCZYB-2020083).

##  V APPENDIX A: Scale characteristics at other wall-normal position

Fig. 10 shows the streamwise and spanwise two-point correlations and spectra
of $\sqrt{\rho}u^{{}^{\prime\prime}}$ at $y=0.3h$ for all cases. For the two-
point correlations, all the profiles nearly coincide (except the $R_{x}$ of
Ma0018K). For the spectra, the $\lambda_{x,me}$ of all cases is approximately
equal to 2.5, whereas for $\lambda_{z,me}$, that is 1.2. Again, all these
suggest that there is no recognizable scale expansion of the outer-region
motions in both streamwise and spanwise directions when flow passes the sound
barrier.

Figure 10: Streamwise ($a$) and spanwise ($b$) two-point correlations of
$\sqrt{\rho}u^{{}^{\prime\prime}}$ for all cases at $y/h=0.3$; streamwise
($c$) and spanwise ($d$) premultiplied one-dimensional spectra of
$\sqrt{\rho}u^{{}^{\prime\prime}}$ for all cases at $y/h=0.3$.

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# CLUE: Towards Discovering Locked Cryptocurrencies in Ethereum

Xiaoqi Li  Department of Computing, The Hong Kong Polytechnic UniversityHong
Kong SARChina<EMAIL_ADDRESS>,  Ting Chen  Center for Cybersecurity,
University of Electronic Science and Technology of ChinaChengduChina
<EMAIL_ADDRESS>,  Xiapu Luo  Department of Computing, The Hong Kong
Polytechnic UniversityHong Kong SARChina<EMAIL_ADDRESS>and  Chenxu
Wang  School of Software Engineering, Xi’an Jiaotong UniversityXi’anChina
<EMAIL_ADDRESS>
(2021)

###### Abstract.

As the most popular blockchain that supports smart contracts, there are
already more than 296 thousand kinds of cryptocurrencies built on Ethereum.
However, not all cryptocurrencies can be controlled by users. For example,
some money is permanently locked in wallets’ accounts due to attacks. In this
paper, we conduct the first systematic investigation on locked
cryptocurrencies in Ethereum. In particular, we define three categories of
accounts with locked cryptocurrencies and develop a novel tool named Clue to
discover them. Results show that there are more than 216 million dollars value
of cryptocurrencies locked in Ethereum. We also analyze the reasons (i.e.,
attacks/behaviors) why cryptocurrencies are locked. Because the locked
cryptocurrencies can never be controlled by users, avoid interacting with the
accounts discovered by Clue and repeating the same mistakes again can help
users to save money.

Distributed System Security, Ethereum, Cryptocurrency

††journalyear: 2021††conference: the 36th ACM/SIGAPP Symposium on Applied
Computing; March, 2021; Gwangju, Korea††article: 4††price: 15.00††isbn:
978-1-4503-XXXX-X/18/06††ccs: Security and privacy Distributed systems
security††ccs: Software and its engineering Software usability††ccs: Security
and privacy Software security engineering

##  1\. Introduction

As the most popular blockchain that supports smart contracts, there are many
kinds of contract-based cryptocurrencies built in Ethereum. Apart from ETH,
which is the native cryptocurrency of Ethereum, more than 296 thousand
cryptocurrency contracts are deployed in Ethereum . These cryptocurrencies
have high market capitalization. For example, the ETH has a total value of
about 20 billion dollars (1, ), and USDT has a total value of more than four
billion dollars (2, ). Note that all the cryptocurrencies’ prices in this
paper are based on statistics in September, 2020 .

However, not all cryptocurrencies can be controlled by users. Actually, much
value of cryptocurrency is permanently locked in some accounts. For example,
the attacker escalated his privilege and destructed Parity’s multi-sig library
contract in 2017 (li2020glaser, ), which locked all the ETH stored in Parity
wallet accounts. Through our analysis, there are 203 wallet accounts with more
than 515,035 locked ETH, which is worth more than 192 million dollars. Many
users still sent cryptocurrencies to the attacked wallet accounts, leading
more money permanently lost. If the accounts with locked cryptocurrencies can
be detected and alerted in time, users can reduce their economic losses.

Unfortunately, there still lacks systematic research on the locked
cryptocurrencies in Ethereum. To fill this gap, we propose and develop a novel
tool named Clue (disCovering Locked cryptocUrrency in Ethereum), which can
discover three categories of accounts with more than 216 million dollars value
of locked cryptocurrencies. In particular, we discover two categories of
contract accounts with locked cryptocurrencies due to contract destruction or
attacks, and one category of EOAs (Externally Owned Accounts) with locked
cryptocurrencies due to users’ unreasonable behaviors. Note that calling to
accounts with locked cryptocurrencies not only wastes system computation
resources, but also wastes users’ money.

The main contributions of this paper are as follows:

(1) To the best of our knowledge, we conduct the first research that
systematically analyzes locked cryptocurrencies in Ethereum. We propose and
define three categories of accounts with locked cryptocurrencies, i.e., one
kind of EOAs and two kinds of smart contract accounts.

(2) We implement a tool named Clue to detect each category of accounts with
locked cryptocurrencies. For smart contract accounts, we analyze their account
states in StateDB and analyze their historical transactions, to discover
destructed contracts. Leveraging symbolic execution, we analyze the runtime
bytecodes of smart contracts to discover attacked Parity wallet contracts. For
EOAs, we mainly use account state analysis and transaction analysis to detect
contract-creation failure EOAs.

(3) We analyze the attacks/behaviors related to the discovered locked
cryptocurrencies, which can explain why they are locked and help users to save
money. We also conduct experiments to evaluate its quantity and accuracy. A
total of 216,186,551.12$ value of cryptocurrencies are discovered by Clue, and
all of the discovered cryptocurrencies are permanently locked in Ethereum.

##  2\. Background and Related Work

Ethereum: Ethereum is the most popular blockchain system that supports smart
contracts (5, ). There are two kinds of accounts in Ethereum, i.e., EOA and
contract account (li2020stan, ). EOA is controlled by user through its private
key, which does not store any code. Contract account is created by EOA or
another contract, which stores the runtime bytecodes of the contract. Smart
contract is a program deployed and executed in blockchain
(chen2018understanding, ). Every node in Ethereum runs an EVM (Ethereum
Virtual Machine) and the runtime bytecodes are executed in the EVM. When a
user calls the smart contract, he/she needs to send transaction with gas to
the address of the target contract (chen2018towards, ). Every operation of
contract runtime bytecodes consumes specific amount of gas when they are
executed in the EVM (chen2017under, ). Developers/users can also destruct the
deployed smart contract through executing SELFDESTRUCT operation (chen2020, ).

Cryptocurrency: Cryptocurrency is digital assets based on blockchain
techniques (6, ). There are two categories of cryptocurrencies in Ethereum,
i.e., ETH and CBC (Contract-Based Cryptocurrency) (6, ). ETH is the native
cryptocurrency of Ethereum. Apart from ETH, there are many other kinds of
cryptocurrencies based on contracts, and ERC20 is the most popular standard of
CBC (li2020stan, ). All the CBC analyzed in this paper are compliant with
ERC20. Both EOA and contract account can hold cryptocurrency. EOA can transfer
out ETH by initiating transactions from it, and contract can transfer out ETH
by executing specific operations (e.g., CALL, SELFDESTRUCT). Note that
accounts can only transfer out their CBC by calling the corresponding
ERC20-based smart contract. The ERC20 standard provides some basic functions
and events that must be implemented of CBC in Ethereum. If the user $U_{a}$
wants to transfer out CBC, $U_{a}$ can call transfer(). Furthermore, the user
$U_{a}$ can authorize another account $U_{b}$ to transfer out CBC through
calling transferFrom(). Before $U_{b}$ transfers out $U_{a}$’s CBC, $U_{a}$
must authorize the account $U_{b}$ through calling approve().

Account State: Every account in Ethereum has four state fields stored in
StateDB (State DataBase) (li2020glaser, ). For each account $a$, we mainly
analyze three fields. Code $\sigma{[a]}_{c}$ stores the smart contract’s
runtime bytecodes, which is empty if $a$ is an EOA. Balance $\sigma{[a]}_{b}$
stores the ETH balance value (in Wei) of the account. Nonce $\sigma{[a]}_{n}$
stores the number of transactions sent from EOA, or the number of contracts
created by contract account.

Related Work: Chen et al. (17, ) detected Ponzi schemes, which are classic
frauds and might cheat users’ ETH. They built a classification model to detect
latent Ponzi schemes by using data mining and machine learning methods. Cheng
et al. (18, ) analyzed the attack that steals cryptocurrencies exploiting
unprotected JSON-RPC endpoints. They designed and implemented a honeypot that
could capture real attacks in the wild. Ji et al. (ji2020deposafe, )
implemented a tool named Deposafe to detect and exploit the fake deposit
vulnerability in ERC-20 tokens. However, all of the above work analyzed the
cryptocurrencies illegally possessed by criminals, and they did not analyze
locked cryptocurrencies that does not belong to anyone. (19, ) measured the
network properties and structures of ERC20 smart contracts, and (6, ) analyzed
inconsistent behaviors in ERC20 smart contracts. However, they focused on
analyzing the smart contracts’ implementations and invocations, whose purposes
differ from ours. There are some other work analyzed cryptocurrencies in
Ethereum (frowis2019detecting, ) or other blockchains (howell2020initial, ).

Figure 1. Overview of Clue’s architecture. There are mainly two modules in
Clue: detection of locked cryptocurrencies in contract and detection of locked
cryptocurrencies in EOA.

##  3\. Locked Cryptocurrencies

Destructed Contract: In Ethereum, the smart contract can be destructed and
transfer out all its stored ETH through executing the SELFDESTRUCT operation.
After destruction, the smart contract account will be deleted from StateDB.
However, some users may not know in time of the smart contract’s destruction
and still send ETH/CBC to it, which leads to the sent ETH/CBC be locked. After
sending ETH to the destructed smart contract account, the contract account
with the same address before destruction will be created again in the StateDB.
For the CBC held by the destructed contract account, most of it will also be
permanently locked. Because the destructed contract account stores no runtime
bytecodes, it cannot send out transaction. Therefore, the destructed contract
account cannot transfer out its CBC through calling transfer() function, or
authorize another account to transfer out its CBC through calling the
transferFrom() function. Above all, all the ETH and most of the CBC held by
the destructed contract accounts are permanently locked in Ethereum. For
example, one smart contract named Insightsnetworkcontributions (Address:
0x97eC9BFb…) is discovered by Clue as destructed contract with locked
cryptocurrencies. It has been transferred more than 208 ETH after its
destruction, which is worth more than 79 thousand dollars.

Attacked Parity Contract: In 2017, the attacker escalated his privilege and
destructed the multi-sig library of Parity wallets, leading to all the ETH and
most of the CBC held by wallet contracts that depend on the library locked
permanently (li2020glaser, ). The attacker destructed the wallets’ library in
the following process. First, the attacker called the library’s functions
initWallet() and initMultiowned() through the fallback function, to escalate
his/her privilege. Second, the attacker destructed the library contract
through calling function kill(). After the library’s destruction, all the
wallet contracts can no longer call the library and executing its functions.
Therefore, all the ETH stored in the attacked Parity wallet contracts is
permanently locked. Furthermore, all the CBC held by the attacked wallet
contracts is also locked. This is because the wallet contract cannot call the
ERC20 contracts. For example, one attacked Parity wallet contract (Address:
0x0da3cB30…) discovered by Clue stores 2,576.35ETH. After the attack, there
were 17.88ETH transferred to this wallet contract. Furthermore, there is also
2.09$ value of CBC locked in it. If these wallet accounts with locked
cryptocurrencies can be detected and alerted in time, the users might no
longer transfer cryptocurrencies to it, which can help users to save money.

Contract-creation Failure EOA: When the user deploys a smart contract in
Ethereum, he/she will still receive one fake contract address if the contract-
creation fails. Indeed, the received contract address does not exist in
StateDB just after the contract-creation failure. However, some users might
wrongly ignore the failure message and still transfer cryptocurrencies to the
fake contract address, leading to cryptocurrencies locked permanently. Because
the address with locked cryptocurrencies never stores code, we classify it as
EOA. For Example, one EOA (Address: 0x5488b0a0…) discovered by Clue locks 19
ETH in value of ~7,088.71 dollars and some CBC in value about seven dollars.
After the user encountered an out-of-gas error during contract-creation, he
still called the fake contract address three times.

##  4\. Clue

The overview of Clue’s architecture is shown in Figure 1, which mainly
consists of two modules: (1) Locked cryptocurrencies in contracts. For
destructed contract, we debug accounts’ historical transactions and detect
destructed contracts through transaction trace analysis and balance analysis.
For attacked Parity contract, we statically analyze contracts’ runtime
bytecodes and detect wallet contracts through symbolic execution. (2) Locked
cryptocurrencies in EOAs. We export sensitive EOAs from StateDB and detect
contract-creation failure accounts with locked cryptocurrencies through
transaction and balance analysis.

Detection of Destructed Contracts: The detection of destructed contracts with
locked cryptocurrencies is divided into four steps. First, for accounts stored
in the StateDB, we debug their historical external transactions through Geth
API debug.traceTransaction(). From the execution trace of the external
transaction, we analyze whether it ever executed the SELFDESTRUCT operation,
which is used for destructing the contract account. Second, for the external
transaction that executed SELFDESTRUCT, we leverage Ethereum RPC API to get
the detailed information of the transaction. Because the execution of
SELFDESTRUCT will produce internal transaction, we get the detailed
information of the internal transaction according to the hash of external
transaction. Third, leveraging the transaction’s execution trace and detailed
information, we analyze the specific address of the destructed contract
account. If the type field of one internal transaction is “suicide”, we can
conclude that it is used for destructing the contract account. Then we export
the sender address of the internal transaction, which is the address of
destructed contract. Fourth, we analyze ETH/CBC balance of the destructed
contract through Ethereum RPC-APIs. At last, the destructed contracts with
locked cryptocurrencies can be discovered.

Detection of Attacked Parity Contracts: The detection of attacked Parity
contracts with locked cryptocurrencies is divided into four steps. First, for
contract accounts stored in the StateDB, we export their runtime bytecodes
from the code field $\sigma{[a]}_{c}$. Second, we statically analyze the
bytecodes. In particular, we use Disasm (10, ) to disassemble the runtime
bytecodes and detect hardcoded Parity library pattern. Third, we leverage
symbolic execution techniques to analyze the runtime bytecodes with the
hardcode pattern. We use Oyente (luu2016making, ) as the symbolic execution
engine. During the symbolic execution process of runtime bytecodes, we monitor
the external call related operations. If we encounter external call
operation’s execution, we analyze its second operand $P_{a}$, which is used
for the target address of the external call. If $P_{a}$ is a real value and
equals with the hardcoded Parity library’s address, we can conclude that the
corresponding analyzed contract account is an attacked Parity contract.
Furthermore, the attacked Parity contract cannot call ERC20 contracts to
transfer out its CBC. This is because $P_{a}$ does not equal with ERC20
contracts’ addresses or associated with transaction’s input data. Fourth, we
analyze the ETH/CBC balances for the detected contracts in the third step.

Detection of Contract-creation Failure EOAs: We leverage account state
analysis and transaction analysis to detect contract-creation failure EOAs
with locked cryptocurrencies, which is divided into three steps. First, we
traverse the StateDB and filter out sensitive EOAs. The sensitive EOAs have
the following state features: nonce $\sigma{[a]}_{n}$ is zero, and code
$\sigma{[a]}_{c}$ is empty. The sensitive EOAs with these features never send
out any transaction. $\sigma{[a]}_{c}$ field is empty indicates that the
account $a$ is an EOA. For an EOA, its $\sigma{[a]}_{n}$ field stores the
number of transactions sent from it. Second, leveraging Ethereum RPC-API, we
fetch and analyze sensitive EOA’s oldest transaction, to verify that it
encountered an error and returned a smart contract address. As described in
Section 3, the contract-creation transaction will also return a fake contract
address when it fails with errors. Third, we analyze ETH/CBC balance of the
detected EOAs in the second step through Ethereum RPC-APIs. At last, contract-
creation failure accounts with locked cryptocurrencies can be discovered.

##  5\. Evaluation

We carry out experiments to answer the following research questions: RQ1
(Quantity): How much value of locked cryptocurrencies can be detected by Clue?
RQ2 (Accuracy): To what extent can Clue accurately discover locked
cryptocurrencies?

Table 1. Statistics of locked cryptocurrencies and related accounts detected through Clue. (❍: discovered candidate accounts. ⚫: accounts with locked cryptocurrencies.) Category | Discovered account | Locked ETH | Locked CBC
---|---|---|---
Destructed contract | 5,916,076❍ — 173⚫ | 123,841.02$ | 25,036,305.09$
Attacked Parity contract | 658❍ — 203⚫ | 190,060,328.19$ | 950,380.79$
Contract-creation failure EOA | 3,720❍ — 191⚫ | 15,640.76$ | 55.27$
Total | 5,920,454❍ — 567⚫ | 190,199,809.97$ | 25,986,741.15$

RQ1 Quantity: We evaluate the quantity of locked cryptocurrencies detected by
Clue, whose statistics are shown in Table 1. Applying Clue to all Ethereum
StateDB data, we totally discover 216,186,551.12$ value of locked
cryptocurrencies. The related accounts’ addresses for each category and
analyzed transaction data are published
onhttps://figshare.com/articles/dataset/11605296. For the destructed
contracts, many of them were created due to DoS attacks (chen2017adaptive, ).
The attacker created large amount of smart contracts and destructed them
through SELFDESTRUCT operation. Most of these destructed contracts are not
called any more by normal users. Therefore, most of the destructed contracts
do not lock any cryptocurrency. For the attacked Parity contracts, we totally
discover 658 related accounts, while Etherscan only tags 153 of them. For
contract-creation failure EOAs, their locked cryptocurrencies’ value is small,
because users might stop calling these accounts after they realize the
contract-creation failure. The locked CBC of destructed contracts does not be
transferred out during contracts’ destruction, which leads to more locked CBC
than ETH. Furthermore, all these detected accounts might lock more
cryptocurrencies with Ethereum’s running, and we also plan to measure locked
cryptocurrencies’ time accumulation in our future work. Answer to RQ1: For the
proposed three kinds of Ethereum accounts, we totally discover 216,186,551.12$
value of cryptocurrencies locked in them.

RQ2 Accuracy: For destructed contracts, we check all the 173 discovered
accounts through Etherscan. All of them have been tagged “Self-Destruct”, and
they all have more than zero value of ETH/CBC. Furthermore, there is no
ETH/CBC transferred out after their destruction. Similarly, all the 203
attacked Parity wallets have more than zero value of ETH/CBC, and their
ETH/CBC never be transferred out after the Parity attack (Transaction hash:
0x47f7cff7…). In addition, we decompile these contracts leveraging Panoramix
(16, ), and they all call the attacked Parity wallets’ library. For the
contract-creation failure accounts, we check all the 191 discovered accounts
that lock cryptocurrencies through Etherscan. All of these accounts
encountered errors during contract-creation, and they all have more than zero
value of ETH/CBC. Also, their ETH/CBC is never transferred out. Answer to RQ2:
100% of the 567 accounts discovered by Clue store cryptocurrencies, and all of
these cryptocurrencies are locked permanently.

##  6\. Discussion and Conclusion

Discussion: We propose and detect three categories of accounts with locked
cryptocurrencies in Ethereum, while there might also exist other categories.
In our future work, we plan to analyze more categories of accounts with locked
cryptocurrencies. We run Clue on all the Ethereum accounts’ data. Although the
number of discovered accounts with locked cryptocurrencies is small (i.e.,
567), the value of locked cryptocurrencies is great. To the best of our
knowledge, there is still no research of how many accounts with locked
cryptocurrencies exist in Ethereum, and our work fills this gap. In other
blockchain systems (e.g., Bitcoin, EOS), there might also exist locked
cryptocurrencies. We plan to detect more cryptocurrencies in other blockchain
systems in future work.

Conclusion: In this paper, we analyzed cryptocurrencies locked permanently in
Ethereum. We defined three categories of accounts with locked cryptocurrencies
and implemented a tool named Clue, which discovered more than 216 million
dollars value of locked cryptocurrencies. We also analyzed why these
cryptocurrencies are locked, which can help users/developers to avoid losing
money.

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