# A scale-based study of the Reynolds number scaling for the near-wall
streamwise turbulence intensity in wall turbulence

Cheng Cheng  Lin Fu<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

###### Abstract

Very recently, a defect model which depicts the growth tendency of the near-
wall peak of the streamwise turbulence intensity has been developed (Chen $\&$
Sreenivasan, J. Fluid Mech. (2021), vol.908, R3). Based on the finiteness of
the near-wall turbulence production, this model predicts that the magnitude of
the peak will approach a finite limit as the Reynolds number increases. In the
present study, we revisit the basic hypotheses of the model, such as the
balance between the turbulence production and the wall dissipation in the
region of peak production, the negligible effects of the logarithmic motions
on the wall dissipation, and the typical time-scale that the outer-layer flow
imposes on the inner layer. Our analyses show that some of them are not
consistent with the characteristics of the wall-bounded turbulence. Moreover,
based on the spectral stochastic estimation, we develop a framework to assess
the wall dissipation contributed by the energy-containing eddies populating
the logarithmic region, and uncover the linkage between its magnitude and the
local Reynolds number. Our results demonstrate that these multi-scale eddies
make a non-negligible contribution to the formation of the wall dissipation.
Based on these observations, we verify that the classical logarithmic model,
which suggests a logarithmic growth of the near-wall peak of the streamwise
turbulence intensity with regard to the friction Reynolds number, is more
physically consistent, and still holds even with the latest high-Reynolds-
number database.

###### keywords:

††journal: XXX

##  1 Introduction

In recent years, in terms of the wall-bounded turbulence, the Reynolds number
dependence of the peak of the streamwise turbulence intensity
$\overline{u^{\prime 2}_{p}}^{+}$ in the vicinity of the wall attracts lots of
attention (here, superscript $+$ denotes the normalization with wall units,
and overbar indicates the ensemble average). This phenomenon has not only been
verified by direct numerical simulations (DNS), for example, in [1], [2], [3],
[4], but also laboratory experiments, such as in [5], [6], [7], to name a few.
In general, at least for the wall-bounded flow with $Re_{\tau}\leq 35000$
($Re_{\tau}=hu_{\tau}/\nu$, where $h$ denotes the channel half-height, the
boundary layer thickness, or the pipe radius, $u_{\tau}$ the wall friction
velocity, and $\nu$ the kinematic viscosity), its establishment is evident
[8]. This observation suggests the failure of the wall scaling for turbulence
intensity, and indicates the scale interactions between the near-wall and the
outer-layer motions. Throughout this paper, the terms ‘eddy’ and ‘motion’ are
exchangeable.

Over the past decades, researchers have devoted themselves to understanding
and predicting the growth tendency of $\overline{u^{\prime 2}_{p}}^{+}$. The
most influential speculation is the logarithmic model (LM), which takes the
form of

$\overline{u^{\prime 2}_{p}}^{+}=A\ln(Re_{\tau})+B,$ (1)

where $A$ and $B$ are two constants. Marusic et al. [8] fitted the existing
DNS and experimental data, and estimated their values as $A=0.63$ and
$B=0.38$. They further pointed out that the logarithmic growth tendency to
$Re_{\tau}$ is consistent with the celebrated attached-eddy model (AEM) [9,
10]. The increase of $\overline{u^{\prime 2}_{p}}^{+}$ can be interpreted as
the outcome of the footprints of the energy-containing eddies populating the
logarithmic and outer layers on the near-wall region. It should be noted that,
if Eq. (1) is valid, it indicates that the magnitude of $\overline{u^{\prime
2}_{p}}^{+}$ is unbounded as the growth of $Re_{\tau}$.

Very recently, Chen and Sreenivasan [11] developed an alternative formula to
describe the growth tendency of $\overline{u^{\prime 2}_{p}}^{+}$, and
predicted that its magnitude is finite when $Re_{\tau}\to\infty$. Their
derivation can be briefly introduced as follows. The balance equation of the
streamwise turbulence kinetic energy $\frac{1}{2}\overline{u^{\prime 2}}^{+}$
can be expressed as

$0=P^{+}+D^{+}-\epsilon^{+}+\Pi_{x}^{+}+T_{t,x}^{+},$ (2)

where $P^{+}$ is the turbulence production term, and equals to
$-\overline{u^{\prime}v^{\prime}}^{+}\partial\bar{u}^{+}/\partial y^{+}$,
i.e., the product of the mean shear and the Reynolds shear stress,
$D^{+}=\frac{1}{2}\left(\partial^{2}/\partial y^{+2}\right)\overline{u^{\prime
2}}^{+}$ is the turbulence diffusion term,
$\epsilon^{+}=\overline{\left|\nabla u^{\prime}\right|^{2+}}$ is the
dissipation term, $\Pi_{x}^{+}=\overline{p^{{}^{\prime}+}\partial
u^{{}^{\prime}+}/\partial x^{+}}$ is the pressure-strain term, and
$T_{t,x}^{+}$ is the turbulence transport term. Here, $x$, $y$, and $z$ denote
the coordinates in the streamwise, wall-normal, and spanwise directions,
respectively. In the vicinity of the wall, the turbulence diffusion ${D}^{+}$
is thought to be balanced with the dissipation $\epsilon^{+}$. By the Taylor
expansion of $\overline{u^{\prime 2}}^{+}$, the order of the peak magnitude
can be evaluated as

$\overline{u^{\prime 2}_{p}}^{+}\sim
D_{w}^{+}y_{p}^{+2}=\epsilon_{w}^{+}y_{p}^{+2},$ (3)

where $y_{p}^{+}$ is the wall-normal position of the peak, which approximately
equals 15, and the subscript $w$ stands for the value at the wall. If
$y_{p}^{+}$ is independent on the Reynolds number, Eq. (3) demonstrates that
the peak value is determined by the wall dissipation. If $\epsilon_{w}^{+}$ is
finite when $Re_{\tau}\to\infty$, so does $\overline{u^{\prime 2}_{p}}^{+}$.
They further argued that the magnitude of $\epsilon_{w}^{+}$ is bounded, since
the maximum value of the turbulent production is bounded by $1/4$, and the
energy production balances with the dissipation at the location of the peak
production (denoted as $y_{pp}^{+}$, approximately equals 12). Hence, the
magnitude of $\overline{u^{\prime 2}_{p}}^{+}$ should also be bounded.

The scaling law of $\epsilon_{w}^{+}$ is then derived. Chen and Sreenivasan
[11] hypothesized that the defect dissipation, i.e,
$\epsilon_{d}^{+}=1/4-\epsilon_{w}^{+}$, results from the fact that a fraction
of energy is transferred to the outer region and not dissipated in the near-
wall region locally. The time-scale of this process is $\eta_{o}/u_{\tau}$,
which can also be considered as the superposition time-scale of outer motions
on the inner layer. Here, $\eta_{o}=\nu^{3/4}/\epsilon_{o}^{1/4}$ is the
Kolmogorov scale of the outer layer with the corresponding dissipation
$\epsilon_{o}$ as $u_{\tau}^{3}/h$. Further normalization with the wall
scaling shows that $\epsilon_{d}^{+}\sim Re_{\tau}^{-1/4}$, and therefore, the
wall dissipation follows that

$\epsilon_{w}^{+}=1/4-\beta/Re_{\tau}^{1/4},$ (4)

where $\beta$ is a constant. Substituting Eq. (4) into Eq. (3), one can
finally obtain the predition of $\overline{u^{\prime 2}_{p}}^{+}$, which is

$\overline{u^{\prime 2}_{p}}^{+}=\alpha(1/4-\beta/Re_{\tau}^{1/4}),$ (5)

where $\alpha$ is another constant. $\alpha$ and $\beta$ are fitted to be 46
and 0.42, respectively. Hereafter, we denote the defect models (4) and (5) as
DM. Monkewitz [12] developed a composite model for the profiles of
$\overline{u^{\prime 2}}^{+}$ based on DM. Chen and Sreenivasan [13] further
generalized the defect model to other near-wall physics, such as the
turbulence diffusion.

Considering the two models mentioned above, some fundamental questions can be
raised. First of all, which model is more physically consistent ? Second,
whether we can develop a methodology to validate the LM or DM ? The present
study mainly addresses these two questions.

##  2 Analyses and discussions on DM

Now, we pay attention to the arguments made by Chen and Sreenivasan [11]. The
first point is that the Taylor series expansion of $\overline{u^{\prime
2}}^{+}$, namely Eq. (3), is the result of the balance between the wall
diffusion and dissipation in the vicinity of the wall. Smits et al. [14]
observed that at $y^{+}=10$, the dominated mechanism of
$\frac{1}{2}\overline{u^{\prime 2}}^{+}$ is the turbulence production, and the
signs of ${D}^{+}$ and $\epsilon^{+}$ are inverse at this position in a
channel flow with $Re_{\tau}=5200$. Accordingly, the validity of the Taylor
expansion of $\overline{u^{\prime 2}}^{+}$ (Eq. (3)) at $y_{p}^{+}$ is
ambiguous. It is noted that the similar tendency can be observed in the up-to-
date DNS of channel flows at $Re_{\tau}=10050$ [15] and $Re_{\tau}=7987$ [16]
(see Fig. 1($a$)). However, Smits et al. [14] also found that the near-wall
profiles of $\overline{u^{\prime 2}}^{+}$ extracted from the existing DNS data
of different types of wall turbulence collapse well if scaled with the
streamwise wall-shear stress fluctuation
$\overline{\tau_{x}^{{}^{\prime}2}}^{+}$, which is identical to the wall
dissipation $\epsilon_{w}^{+}$. Hence, Eq. (3) is still a reasonable
proposition. On the other side, Pirozzoli et al. [4] reported that the value
of $y_{p}^{+}$ in DNS of the pipe flows with $Re_{\tau}$ ranging from
$180-6020$ slightly increases with $Re_{\tau}$, which may have non-negligible
effects on the magnitude of $\overline{u^{\prime 2}_{p}}^{+}$ as per Eq. (3).
As discussed by Chen and Sreenivasan [11], the existing data can not offer a
sufficient evidence that $y_{p}^{+}$ is a strong $Re_{\tau}$-dependent
variable. For example, for a turbulent channel flow with $Re_{\tau}=180$, the
peak value of $\overline{u^{\prime 2}_{p}}^{+}$ locates at $15.3$ [17]. The
latest DNS data with $Re_{\tau}=10050$ also shows that $y_{p}^{+}=15.4$ [15].
It can be seen that $y_{p}^{+}$ is insensitive to the Reynolds number.
Besides, the measurement of an atmospheric boundary layer with
$Re_{\tau}\approx 900000$ also shows that $y_{p}^{+}\approx 13.4$ with
uncertainty between 11.6 and 15.2 [18]. Schlatter and Örlü [19] summarized the
values of $y_{p}^{+}$ for different simulations of boundary layers and
channels. They claimed that there exists a fairly large spread of the data
with a variation of up to $10\%$ at a fixed $Re_{\tau}$ (see Fig. 4($b$) of
their paper). Hence, in the present study, we still treat it as a Reynolds-
number insensitive quantity.

Figure 1: Energy budget for $\frac{1}{2}\overline{u^{\prime 2}}^{+}$ in
channel flows at $Re_{\tau}=10050$ [15] and $Re_{\tau}=7987$ [16]: ($a$) near-
wall region; ($b$) logarithmic region. The black dashed line in ($a$) is
plotted to highlight the position $y^{+}_{pp}$. Figure 2: Profiles of ($a$)
$P^{+}/\epsilon^{+}$, and ($b$) $\epsilon^{+}/\epsilon_{w}^{+}$ from DNS of
channel flows. The red dashed lines in ($a$) and ($b$) are plotted to
highlight the position $y^{+}_{pp}$.

Another important point of DM is the defect model of $\epsilon_{w}^{+}$. It is
established based on the hypothesis that the energy production balances with
the dissipation at $y_{pp}^{+}$, so the wall dissipation is also bounded due
to the bounded production. However, Fig. 1($a$) shows that the energy balance
at $y_{pp}^{+}$ is relatively complicated, as all the terms are non-
negligible. Thus, the proposition of DM that $P^{+}$ balances $\epsilon^{+}$
at the location, where the production is maximum, is questionable. Moreover,
Fig. 2($a$) shows the ratio between $P^{+}$ and $\epsilon^{+}$ in the near-
wall region of turbulent channel flows calculated from DNS data with
$Re_{\tau}$ ranging from $O(10^{2})$ to $O(10^{4})$ [17, 20, 21, 1, 3, 15]. It
can be seen that $P^{+}/\epsilon^{+}$ is Reynolds-number independent at
$y_{pp}^{+}$, which can be expressed as

$\frac{P^{+}(y_{pp}^{+})}{\epsilon^{+}(y_{pp}^{+})}=\phi,$ (6)

where $\phi$ is a constant, and approximately equals 2.3. Eq. (6) indicates
that $\epsilon^{+}(y_{pp}^{+})$ is bounded on the premise of the finiteness of
$P^{+}(y_{pp}^{+})$, and the energy produced at $y_{pp}^{+}$ can not be
dissipated locally and completely even when $Re_{\tau}\to\infty$, which is not
consistent with the deduction of DM. As DM predicts that the near-wall
production will be dissipated locally and entirely when $Re_{\tau}$ is high
enough. To illustrate the relationship between $\epsilon_{w}^{+}$ and
$\epsilon^{+}$, Fig. 2($b$) shows the variations of
$\epsilon^{+}/\epsilon_{w}^{+}$ as functions of $y^{+}$. At $y_{pp}^{+}$,
clear Reynolds-number dependence can be observed, which can be expressed as

$\frac{\epsilon^{+}(y_{pp}^{+})}{\epsilon_{w}^{+}}=f(Re_{\tau}),$ (7)

where $f(Re_{\tau})$ is a function of $Re_{\tau}$. The substitution of Eq. (7)
into Eq. (6) yields

$\epsilon_{w}^{+}=\frac{P^{+}(y_{pp}^{+})}{\phi f(Re_{\tau})}.$ (8)

Eq. (8) suggests that the finiteness of $\epsilon_{w}^{+}$ is not only
dependent on the asymptotic behavior of $P^{+}(y_{pp}^{+})$, but also the
properties of the function $f(Re_{\tau})$, which can not be determined by the
existing limited database. Hence, the bounded $P^{+}(y_{pp}^{+})$ and
$\epsilon^{+}(y_{pp}^{+})$ do not directly imply the boundness of
$\epsilon_{w}^{+}$, thus $\overline{u^{\prime 2}_{p}}^{+}$.

The last important point of DM is that $\epsilon_{d}^{+}$ is manifested as the
consequence of the near-wall energy transferred to the outer layer and
dissipated nonlocally with the time-scale $\eta_{o}/u_{\tau}$ and with the
outer dissipation rate $\epsilon_{o}$ equaling $u_{\tau}^{3}/h$. This scenario
can be reinterpreted from the perspective of the energy-containing motions,
i.e., the deviation of the wall dissipation from 1/4 can be regarded as the
influences from large-scale motions (LSMs) populating the outer layer with the
characteristic scale $h$, and the typical time-scale of this process is
$\eta_{o}/u_{\tau}$. That is to say, the logarithmic region has negligible
effects on the magnitude of the wall dissipation. However, it is known to all
that there are multi-scale motions in the high-Reynolds number wall-bounded
turbulence [22, 23]. In addition to the near-wall motions with the
characteristic length scale $\nu/u_{\tau}$ and the LSMs with outer-flow
scaling $h$, there is still a crowd of energy-containing motions with typical
length scale $y$, namely their distance from the wall, populating the
logarithmic region. These $y-$scaling motions have been demonstrated to be
responsible for the formation of the classical log law of the streamwise mean
velocity [24], and become more and more in number with increasing Reynolds
number [23, 25]. Fig. 1($b$) shows the streamwise kinetic energy budget in the
logarithmic region of channel flows at $Re_{\tau}=10050$ [15] and
$Re_{\tau}=7987$ [16]. In the logarithmic region, $P^{+}$ and $\epsilon^{+}$
are not fully balanced with each other. Aside from $\epsilon^{+}$, the
pressure-strain term $\Pi_{x}^{+}$ also balances a fraction of the turbulence
production. It redistributes the streamwise kinetic energy to the spanwise and
wall-normal components, as the spectral scale properties of the wall-normal
and spanwise pressure-strain terms are in accordance with $\Pi_{x}^{+}$ and
only opposite in sign (the reader can refer to [26] and [27] for details).
Further to this, the induced wall-normal transport would stimulate the near-
wall eddies by scale interactions (see Fig. 3). Interestingly, this type of
interaction process has recently been observed by Doohan et al. [28] in their
numerical experiment, and named as ‘driving process’. The main results of this
process are reported to be the transient amplifications of the local small-
scale productions and their subsequent dissipation [28]. They also found that
the time-scale of the driving mechanism is determined by the self-sustaining
process of large-scale eddies. It can be envisioned that the time-scale is
proportional to $y/u_{\tau}$, as $y$ and $u_{\tau}$ are the characteristic
length scale and velocity scale of a $y-$scaling eddy populating the
logarithmic region, respectively. Our proposition is in line with the original
claim of Townsend [9], who pointed out that the dissipation length scale in
the constant-stress equilibrium layer must be proportional to the wall
distance. In fact, this property is the essential building block of the
celebrated attached-eddy model, because it signifies that there must exist
eddies populating the logarithmic region and extending to the wall [9]. The
study of Lozano-Durán and Jiménez [29] can also verify our assessment. They
found that the lifetimes of the wall-attached structures extending to the
logarithmic region are in direct proportion to their wall-normal heights.
Hence, the corresponding dissipation imposed on the near-wall flow by the
turbulence at the location $y$ can also be estimated as

$\int_{y}^{h}(y^{*})^{-1}u_{\tau}^{3}/y^{*}dy^{*}\sim u_{\tau}^{3}/y,$ (9)

whose variation at $Re_{\tau}=10050$ is shown in Fig. 1($b$). Here,
$(y^{*})^{-1}$ is the probability density of the $y^{*}-$scaling eddies [9,
30, 31]. It can be seen that it shares a similar order of magnitude with
$\Pi_{x}^{+}$, which verifies the above arguments indirectly. Accordingly, we
may conclude that the original DM is not consistent with the multi-scale
characteristics of high-Reynolds number wall turbulence.

Finally, it is worth mentioning that the physical interpretation of the
typical time-scale of outer-region additional dissipation in DM, i.e.,
$t_{a}=\eta_{o}/u_{\tau}$, is ambiguous, since $\eta_{o}$ is the Kolmogorov
scale in the outer region, whereas $u_{\tau}$ is the characteristic velocity
scale of the energy-containing eddies. Actually, the Kolmogorov velocity scale
in outer region is $u_{\eta}=(\nu\epsilon_{o})^{1/4}$, thus, the ratio between
$t_{a}$ and Kolmogorov time-scale $t_{\eta}$ can be estimated as

$\frac{t_{a}}{t_{\eta}}=\frac{u_{\eta}}{u_{\tau}}=(\frac{\nu}{u_{\tau}h})^{1/4}=Re_{\tau}^{-1/4}.$
(10)

Eq. (10) shows that $t_{a}$ will be far less than $t_{\eta}$ when
$Re_{\tau}\to\infty$, and becomes the smallest time-scale in the outer region.
This conclusion is counter-intuitive.

Figure 3: A schematic of the scale interactions between the energy-containing
eddies in the near-wall and the logarithmic regions. Each circle represents an
individual eddy.

##  3 LM versus DM

###  3.1 A scale-based model

According to above discussions, we can see that the defect model of wall
dissipation is questionable, and the eddies occupying the logarithmic region
also have impacts on the near-wall dissipation. Even if we insist that the
defect model of wall dissipation is valid, and adjust the time-scale to
$y/u_{\tau}$, the normalized defect dissipation $\epsilon_{d}$ at a given
wall-normal height $y$ can be assessed as

$\epsilon_{d}^{+}(y^{+})\sim(u_{\tau}^{3}/y)/(u_{\tau}^{4}/\nu)=(y^{+})^{-1}.$
(11)

Thus, beyond the near-wall region, the total defect dissipation
$\epsilon_{d,t}^{+}$ is given by

$\epsilon_{d,t}^{+}\sim\int_{y_{s}^{+}}^{h^{+}}(y^{+})^{-1}dy^{+}\sim\ln(Re_{\tau}),$
(12)

where $y_{s}^{+}$ denotes the starting location of the logarithmic layer,
which is classically believed to be $80\leq y_{s}^{+}\leq 100$ [22, 32]. It is
noted that the effects of LSMs are also included in Eq. (12). When
$Re_{\tau}\to\infty$, $\epsilon_{w}^{+}$ follows a new law as

$\epsilon_{w}^{+}=1/4-C_{1}\ln(Re_{\tau}),$ (13)

where $C_{1}$ is a constant. Eq. (13) indicates that when $Re_{\tau}$ is high
enough, $\epsilon_{w}^{+}$ will be negative, and, apparently, this is not
possible. Over the past two decades, several works also reported that the
lower bound of the logarithmic region is Reynolds-number-dependent, namely,
$y_{s}^{+}=A_{1}Re_{\tau}^{0.5}$, where $A_{1}$ is a constant estimated
between 1 and 3 [33, 34, 35, 24]. This is also in line with the mesolayer
scaling proposed by Afzal [36]. For this circumstance, the total defect
dissipation $\epsilon_{d,t}^{+}$ can be deduced to follow Eq. (12), and thus
$\epsilon_{w}^{+}$ also obeys the unphysical scaling Eq. (13).

Hence, it is instructive to abandon the defect description. Instead, the wall
dissipation should be regarded as the accumulated outcome of the energy-
containing motions, i.e.,

$\epsilon_{w}^{+}\sim\int_{y_{s}^{+}}^{h^{+}}(y^{+})^{-1}dy^{+}=C_{2}\ln(Re_{\tau})+C_{3},$
(14)

where $C_{2}$ and $C_{3}$ are two constants, and this model includes the
effects of the near-wall turbulence on the wall dissipation. The substitution
of Eq. (14) into Eq. (3) yields the LM, i.e., Eq. (1). This result highlights
that the LM is more physically consistent.

Upon the analyses above, it is transparent that the dispute between the two
models is whether the logarithmic region has remarkable effects on the
magnitude of the wall dissipation. DM regards the near-wall region as the
energy source, and the departure of the wall dissipation from the ultimate
state is ascribed to the energy transferred to the outer region and dissipated
locally. While LM treats the generation of the wall dissipation as the
additive effects resulting from the multi-scale motions. Accordingly, if we
can develop a methodology to examine the effects of the log-region flow on the
magnitude of wall dissipation, then we can further decide which model is
physically consistent. The following subsection is focused on this issue.

###  3.2 Validation by a numerical framework

In this section, we employ the spectral stochastic estimation, a numerical
approach, to verify our argument by analyzing the DNS database. The database
used in the present study includes the incompressible turbulent channel flows
at $Re_{\tau}=2003$ and $Re_{\tau}=4179$, which have been extensively
validated by previous studies [1, 37, 38, 39, 40]. These two simulations were
conducted in a computational domain of $8\pi h\times 3\pi h\times 2h$ and
$2\pi h\times\pi h\times 2h$ in the streamwise, spanwise, and wall-normal
directions, respectively. Forty raw snapshots are used for the ensemble
average in the present study for both cases. All these data are provided by
the Polytechnic University of Madrid. It is worth noting that the key results
reported below are not sensitive to the number of instantaneous flow fields
employed for accumulating statistics (see Appendix A).

According to the inner-outer interaction model [41], the large-scale motions
would exert the footprints on the near-wall region, i.e., the superposition
effects. Baars et al.[42] demonstrated that this component (denoted as
$u_{L}^{{}^{\prime}+}(y^{+})$) can be obtained by the spectral stochastic
estimation of the streamwise velocity fluctuation at the logarithmic region
$y_{o}^{+}$, namely,

$u_{L}^{{}^{\prime}+}\left(x^{+},y^{+},z^{+}\right)=F_{x}^{-1}\left\\{H_{L}\left(\lambda_{x}^{+},y^{+}\right)F_{x}\left[u_{o}^{{}^{\prime}+}\left(x^{+},y_{o}^{+},z^{+}\right)\right]\right\\},$
(15)

where $u_{o}^{{}^{\prime}+}$ denotes the streamwise velocity fluctuation at
$y_{o}^{+}$ in the logarithmic region, and, $F_{x}$ and $F_{x}^{-1}$ denote
FFT and inverse FFT in the streamwise direction, respectively. $H_{L}$ is the
transfer kernel, which evaluates the correlation between
$u^{{}^{\prime}+}(y^{+})$ and $u_{o}^{{}^{\prime}+}(y_{o}^{+})$ at a given
length scale $\lambda_{x}^{+}$, and can be calculated as

$H_{L}\left(\lambda_{x}^{+},y_{o}^{+}\right)=\frac{\left\langle\hat{u^{\prime}}\left(\lambda_{x}^{+},y^{+},z^{+}\right)\hat{u_{o}^{\prime}}^{*}\left(\lambda_{x}^{+},y_{o}^{+},z^{+}\right)\right\rangle}{\left\langle\hat{u_{o}^{\prime}}\left(\lambda_{x}^{+},y_{o}^{+},z^{+}\right)\hat{u_{o}^{\prime}}^{*}\left(\lambda_{x}^{+},y_{o}^{+},z^{+}\right)\right\rangle},$
(16)

where $<\cdot>$ represents the averaging in the temporal and spatially
homogeneous directions, $\hat{u^{\prime}}$ is the Fourier coefficient of
$u^{\prime}$, and $\hat{u^{\prime}}^{*}$ is the complex conjugate of
$\hat{u^{\prime}}$.

In this work, we mainly focus on the quantity $\epsilon_{w}^{+}$ generated by
the logarithmic eddies. Thus, the predicted near-wall position $y^{+}$ is
fixed at $y^{+}=0.3$, and the outer reference height $y_{o}^{+}$ varies from
$100$ (namely $y_{s}^{+}$) to $0.2h^{+}$, i.e., the upper boundary of
logarithmic region [22, 43]. Once $u_{L}^{{}^{\prime}+}$ is obtained, the
superposition component of $\epsilon_{w}^{+}$ can be calculated by definition
(i.e., $(\frac{\partial u_{L}^{{}^{\prime}+}}{\partial y^{+}})^{2}$ at the
wall), and denoted as $\epsilon_{w,L}^{+}(y_{o}^{+})$. According to the
hierarchical energy-containing eddies in high-Reynolds number wall turbulence
[23], $\epsilon_{w,L}^{+}(y_{o}^{+})$ represents the superposition contributed
from the wall-coherent motions with their heights larger than $y_{o}^{+}$.
Thus, the difference value
$\Delta\epsilon_{w}^{+}(x,y_{o}^{+},z)=\epsilon_{w}^{+}(x,y^{+},z)-\epsilon_{w,L}^{+}(x,y_{o}^{+},z)$
can be interpreted as the cumulation of the superposition from the wall
coherent eddies with their heights less than $y_{o}^{+}$ and the contribution
of the near-wall small-scale motions. Detached eddies cannot contribute to it,
because they cannot interact with the wall indeed. Considering that
$y_{o}^{+}$ can be interpreted as local $Re_{\tau}$, the increase of
$y_{o}^{+}$ corresponds to the enlargement of $Re_{\tau}$. In this way, the
Reynolds-number effects on the quantity of $\epsilon_{w}^{+}$ can be verified
directly.

Figure 4: Variation of the mean $\Delta\epsilon_{w}^{+}$ as a function of
$y_{o}^{+}$ for turbulent channel flows at ($a$) $Re_{\tau}=2003$ and ($b$)
$Re_{\tau}=4179$ with $y_{o}^{+}$ varying from $100$ to $0.2h^{+}$, and the
red lines denote the corresponding exponential fitting.

Fig. 4 shows the variation of the mean $\Delta\epsilon_{w}^{+}$ as functions
of $y_{o}^{+}$ for turbulent channel flows at $Re_{\tau}=2003$ and
$Re_{\tau}=4179$. It can be seen that $\Delta\epsilon_{w}^{+}$ increases with
$y_{o}^{+}$. It highlights the fact that the energy-containing eddies
populating the logarithmic region have made a non-negligible contribution to
the formation of the wall dissipation. This essential factor, however, has not
been taken into consideration by Chen and Sreenivasan [11]. Moreover, a clear
exponential relationship between the mean $\Delta\epsilon_{w}^{+}$ and
$y_{o}^{+}$ can be observed for both cases, which provides a direct evidence
for the logarithmic variation, i.e., Eq. (14). It is noted that the diversity
in the upper part of the logarithmic region of the case with $Re_{\tau}=4179$
is due to the relatively small computational domain size in this database.
Again, this analysis demonstrates that the LM is more physically consistent.

The relationship between the present results and the attached-eddy model
merits a discussion. The key consensus between them is that the eddies
populating the logarithmic region would exert non-negligible effects on the
near-wall turbulence. According to the attached-eddy hypothesis, the inactive
part of the attached eddies would penetrate into the near-wall region [9, 44,
45]. The typical time-scale $y/u_{\tau}$ discussed above can be considered as
the characteristic time-scale imposing on the near-wall flow by the footprints
of the attached eddies with the wall-normal height $y$. Our previous work has
demonstrated that the numerical framework described above can isolate the
footprints of the attached eddies populating the logarithmic region accurately
[46]. Hence, the results shown in Fig. 4 are consistent with the attached-eddy
hypothesis. Moreover, as the Reynolds number increases, the span of the
logarithmic region grows with the total number of the attached eddies becoming
larger simultaneously, which consequently brings about the growth of
$\epsilon_{w}^{+}$ and $\overline{u^{\prime 2}_{p}}^{+}$.

##  4 Scalings of $\epsilon_{w}^{+}$ and $\overline{u^{\prime 2}_{p}}^{+}$ in
turbulent channel and pipe flows

Now, we focus on the direct numerical validation of $\epsilon_{w}^{+}$ and
$\overline{u^{\prime 2}_{p}}^{+}$ scalings with DNS database. Here, we only
show the results of turbulent channel and pipe flows to highlight the
differences between DM and LM, as the DNS data of channel and pipe flows are
more abundant relatively, and cover a wider range of $Re_{\tau}$ [17, 47, 20,
48, 1, 49, 50, 51, 3, 16, 4, 15]. Fig. 5($a$) compares the defect model of Eq.
(4) fitted by Chen and Sreenivasan [11], the logarithmic growth of Eq. (14)
fitted by Smits et al. [14] ($\epsilon_{w}^{+}=0.08+0.0139\ln(Re_{\tau})$),
and the logarithmic growth fitted by the present study for $\epsilon_{w}^{+}$,
i.e.,

$\epsilon_{w}^{+}=0.0163\ln(Re_{\tau})+0.0603.$ (17)

The comprehensive DNS results of channel and pipe flows are included and
presented by symbols. It is noted that there is a slight difference between
the slope of the red lines in Fig. 5($a$) and Fig. 4. This can be ascribed to
the effects of LSMs in raw DNS data, which can not be captured by the spectral
stochastic estimation conducted in section 3.2. Fig. 5($b$) compares the
defect model of Eq. (5) fitted by Chen and Sreenivasan [11] and the
logarithmic growth of Eq. (1) fitted by the present study for
$\overline{u^{\prime 2}_{p}}^{+}$ using the DNS data with $Re_{\tau}\gtrsim
1000$, i.e.,

$\overline{u^{\prime
2}_{p}}^{+}=0.744\ln(Re_{\tau})+2.751=45.63\epsilon_{w}^{+}.$ (18)

The similar fitting reported by [14] is also included ($\overline{u^{\prime
2}_{p}}^{+}=46(0.08+0.0139\ln(Re_{\tau}))$). Both of the two logarithmic law
fittings are representative of the variational tendencies. According to Chen
and Sreenivasan [11], a more precise version of Eq. (3) is given by

${\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\overline{u^{\prime
2}_{p}}^{+}\approx(y_{p}^{+2}/4)\epsilon_{w}^{+}.}}$ (19)

Comparing the fitting result Eq. (18) with Eq. (19), we can obtain
$y_{p}^{+}\approx 13.5$. It is very close to the widely-accepted value
mentioned above.

Figure 5: Variations of ($a$) $\epsilon_{w}^{+}$ and ($b$)
$\overline{u^{\prime 2}_{p}}^{+}$ as functions of $Re_{\tau}$. The
comprehensive DNS results of the channel and pipe flows are presented by
symbols. The blue dashed lines, the solid lines, and the dash-dotted lines
denote the predictions from the logarithmic growth fitted by Smits et al.
[14], by the present study, and the defect model fitted by Chen and
Sreenivasan [11].

To further quantify the fitting errors obtained with LM and DM, we calculate
the root-mean-square deviation (RMSD) of the two models. It takes the form of

$\mathrm{RMSD}=\sqrt{\frac{\sum_{t=1}^{n}\left(\hat{\phi}_{t}-\phi_{t}\right)^{2}}{n}},$
(20)

where $n$ is the number of the DNS samples, and $\hat{\phi}_{t}$ and
$\phi_{t}$ denote the predicted value and the DNS value of the target physical
quantity at a given $Re_{\tau}$, respectively. Towards the databases shown in
Fig. 5, the RMSDs of the logarithmic law fitted by the present study, by Smits
et al. [14], and the defect power law proposed by Chen and Sreenivasan [11]
are 0.0026, 0.0049 and 0.0029 for $\epsilon_{w}^{+}$, respectively. The
counterparts of $\overline{u^{\prime 2}_{p}}^{+}$ are 0.117, 0.135, and 0.166
for the corresponding three model predictions. It can be seen that LM is not
inferior to DM in characterizing the variation tendencies of
$\epsilon_{w}^{+}$ and $\overline{u^{\prime 2}_{p}}^{+}$, as a whole.

At last, it should be acknowledged that the grid spacings of DNS can have
effects on the magnitudes of the near-wall physics discussed in the present
study. Yang et al.[52] have shown that for resolving rare and high-intensity
wall-shear stress events in wall-bounded turbulence, the desired grid spacing
of DNS should be finer than the standard grid resolution. To be specific, the
standard grid resolution can only resolve about $90\%$-$95\%$ rare events in a
turbulent channel flow with $Re_{\tau}=10000$. In this regard, the grid
spacings of the DNS conducted by Hoyas et al. [15] with $Re_{\tau}=10050$ and
Kaneda and Yamamoto [16] with $Re_{\tau}=7987$ are relatively coarse (for
details of the computational setups, the readers can refer to their papers).
Thus, the conclusions drawn at present may not be very rigorous by just
dissecting the data shown in Fig. 5, due to the lack of solid DNS data with
higher Reynolds number. However, we can still justify which model is more
consistent with the physical characteristics of the wall-bounded turbulence.
Scientifically speaking, the physical implications behind these formulas,
rather than the formulas themselves, are critical indeed, and they truly
enhance our understandings of the complicated turbulence. After all, there is
a range of functions that can match the current databases well from the point
of mathematics, even better than the defect-power model and the logarithmic
model discussed here with smaller RMSD, regardless of their physical
significance. In summary, based on the spectral stochastic estimation, our
study provides a numerical framework for investigating the physical nature of
the wall-bounded turbulence. Upon the analysis of the present study, it can be
concluded with caution that the classical logarithmic model is more physically
consistent.

##  5 Concluding remarks

The main conclusion of the present study is that the logarithmic models, which
depict the Reynolds number scalings of $\epsilon_{w}^{+}$ and
$\overline{u^{\prime 2}_{p}}^{+}$, are more physically consistent compared
with the defect model developed by Chen and Sreenivasan [11]. The analyses of
the basic hypotheses of DM suggest that some propositions are not consistent
with the characteristics of the wall turbulence. They are summarized as below.

($a$) The bounded energy production at $y_{pp}^{+}$ only indicates that the
dissipation at $y_{pp}^{+}$ is finite. It does not imply that the wall
dissipation is bounded.

($b$) The eddies populating the logarithmic region can exert non-trivial
effects on the wall dissipation magnitude, which have not been taken into
consideration by DM.

($c$) The rational time-scale with which the energy-containing eddy interacts
with the near-wall flow is $y/u_{\tau}$, rather than $\eta_{o}/u_{\tau}$.

Moreover, to verify our analyses, we develop a framework to quantify the wall
dissipation generated by the logarithmic eddies in turbulent channel flows at
$Re_{\tau}=2003$ and $Re_{\tau}=4179$. Our results show that the normalized
wall dissipation stemming from these eddies has a clear exponential
relationship with their wall-normal heights $y^{+}$. Based on the above
observations, we believe that the logarithmic model is more physically
consistent, and still holds even with the latest high-Reynolds-number
database.

## Acknowledgments

We are grateful to the authors cited in Fig. 1, Fig. 2, and Fig. 5 for making
their invaluable data available. We also thank Professor Jiménez for making
the DNS snapshots used in section 3.2 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).

## Appendix A. Statistic sensitivity to the number of instantaneous flow
fields

Figure 6: Mean $\Delta\epsilon_{w}^{+}$ variation as functions of $y_{o}^{+}$
for the cases with ($a$) $Re_{\tau}=2003$ and ($b$) $Re_{\tau}=4179$ with
different $N_{f}$.

The influences of the number of instantaneous flow fields ($N_{f}$) for
accumulating statistics are examined. Fig. 6 shows the effects of $N_{f}$ on
the statistic mean $\Delta\epsilon_{w}^{+}$ for the cases with
$Re_{\tau}=2003$ and $Re_{\tau}=4179$. Alteration of the statistical samples
mainly affects the intercepts of the logarithmic variations of the mean
$\Delta\epsilon_{w}^{+}$ with respect to $y_{o}^{+}$, but the logarithmic
tendencies are not changed.

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11institutetext: University of California, Santa Cruz1, University of
California, Davis2,
Columbus State University3
11email<EMAIL_ADDRESS>
# HealNet - Self-Supervised Acute Wound Heal-Stage Classification

Héctor Carrión1 Mohammad Jafari3 Hsin-Ya Yang2 Roslyn Rivkah Isseroff2 Marco
Rolandi1 Marcella Gomez1 Narges Norouzi1∗

###### Abstract

Identifying, tracking, and predicting wound heal-stage progression is a
fundamental task towards proper diagnosis, effective treatment, facilitating
healing, and reducing pain. Traditionally, a medical expert might observe a
wound to determine the current healing state and recommend treatment. However,
sourcing experts who can produce such a diagnosis solely from visual
indicators can be difficult, time-consuming and expensive. In addition,
lesions may take several weeks to undergo the healing process, demanding
resources to monitor and diagnose continually. Automating this task can be
challenging; datasets that follow wound progression from onset to maturation
are small, rare, and often collected without computer vision in mind. To
tackle these challenges, we introduce a self-supervised learning scheme
composed of (a) learning embeddings of wound’s temporal dynamics, (b)
clustering for automatic stage discovery, and (c) fine-tuned classification.
The proposed self-supervised and flexible learning framework is biologically
inspired and trained on a small dataset with zero human labeling. The HealNet
framework achieved high pre-text and downstream classification accuracy; when
evaluated on held-out test data, HealNet achieved 97.7% pre-text accuracy and
90.62% heal-stage classification accuracy.

###### Keywords:

Wound healing Self-supervised medical image processing Clustering.

##  1 Introduction and Related Work

Wound healing is a variable, painful, and often prolonged process that can
impact the quality of life for millions of people worldwide. Approximately 6.5
million patients present chronic wounds annually in the United States,
requiring a treatment cost of over $25 billion per year [1]. In 2014, acute
wounds resulted in 17.2 million hospital visits [2], while the odds of an
acute wound requiring further care significantly increases with age, diabetes,
and obesity [1]; thus a large portion of acute wounds will not resolve
uneventfully and burden healthcare resources. An accessible and automated
protocol to identify, track, and predict wound heal-stages could help tackle
these challenges while reducing costs.

Biologically routine wound healing mechanics are four precise and programmed
stages: hemostasis, inflammation, proliferation, and maturation [3]. Numerous
chemical reactions and interactions between cells occur within these phases
under and above the wound bed. However, the visual indicators that result from
or relate to each stage are medically under-specified. Therefore,
dermatologists are often limited to studying wound area changes or rely on
mice wound re-epithelialization and histological analysis (which requires
harvesting the wound-bed and thus sacrificing the mice) to understand wound-
stage progression [4]. Invasiveness, the cost of long-running experiments, and
the lack of qualified experts that can analyze image-based results contribute
to the lack of large-scale, complete wound-stage progression image datasets.

Figure 1: (Step 1) The self-supervised network is used to generate
16-dimensional feature vectors. We cluster these feature vectors using k-means
with $k=4$. (Step 2) The new pseudo-labels (based on clusters) are used to
assign each image a heal-stage class. The network is then re-tasked and fine-
tuned using pseudo-labels, predicting the wound stage.

Machine Learning (ML) for automated wound analysis has not been approached
from the heal-stage angle. Methods exist for wound detection and size
estimation [5], where obtaining ground-truth labels are relatively
straightforward. Other classification systems aim to identify different
characteristics of wounds or ulcers [6]. Acute wound linear regression models
incorporate location, age, size, and infection features to generate time-to-
heal estimates [7]. Recent lesion segmentation work has found significant
inter-and-intra annotator variance even among experts [8], suggesting that
precisely defining wound-stage visual indicators is not trivial.

Weakly-supervised and self-supervised learning approaches have shown the
ability to improve the performance of tasks with little or no annotated data
by optimizing a pre-text objective function that learns representations
without (or with few) explicit human labels. For example, learning color from
grayscale images (colorization) [9, 10], shuffling and predicting image tile
permutations (Jigsaw) [11], removing and re-generating image regions
(inpainting) [12, 13] rotation prediction (RotNet) [14] and classifying
augmented positive and negative pairs (contrastive learning) [15, 16]. In
modalities like video and audio, the inherent temporal structure of the data
can be exploited to create other proxy objective functions, for example
shuffling video frames and predicting temporal order [17]. Further work has
shown that these embeddings can be clustered and re-used as pseudo-labels to
improve representation quality for weakly-supervised and self-supervised
models [18].

Due to the nature of wound healing, capturing progression over time involves
learning the inherent temporal structure. Thus, a proxy task is created to
learn wound healing temporal dynamics. We study a small, unlabeled dataset
containing daily mouse wound observations from onset to maturation. We
leverage the dataset to generate wound image pairs in the temporal forward
(positive) and backward (negative) directions. This step increases our dataset
size by a factor of $15$. The wound pairs are then used to train a network to
determine the input pair’s temporal validity and thus encoding features
relevant to wound healing temporal dynamics. These embeddings are then
clustered into four groups (one per heal-stage) to generate pseudo-labels
without human intervention. Finally, a classifier is re-tasked and fine-tuned
on pseudo-labels to predict the heal-stage of input images (Fig. 1).

Figure 2: (left side) Due to the nature of wound healing, images capturing
progression naturally contain a temporal structure. We generate image pairs in
the forwards (positive) and backward (negative) temporal directions; this
serves as the input for our self-supervised temporal coherency training step.
(right side) Randomly sampled image input pairs.

##  2 Description of Data

The experiment for generating the image dataset used in this work is described
in [19], from which we generate circular wound-only crops. The images contain
wounds from eight mice (four young and four aged) imaged daily over sixteen
days. Each individual had a wound inflicted on the left side and another on
the right side. The resulting dataset is 256 images (8 mice $\times$ 2 wounds
$\times$ 16 days). Note that this dataset was not captured with computer
vision in mind; consequently, we need to account for challenges such as blur,
occlusion and illumination noise. To address this, data augmentation was used
in training.

Wound pairs have been created for the pre-text learning task in order to
capture the temporal healing process. We generated pairs in the temporal
forwards (positive) and backwards (negative) directions. Fig. 2 illustrates
the process of creating these pairs. We did not generate pairs containing
same-day same-wound images. Thus, the resulting dataset contains 3,840 image
pairs (16 wounds $\times$ 16 days $\times$ 15 pairs per day). In order to
allow our validation and test sets to capture any differing dynamics between
the two cohorts of young and aged mice, we randomly selected two young wounds
and two aged wounds from the dataset; a couple of unique aged and young wounds
is placed into the validation set and a second couple into the testing set.
After this split, the resulting image pair dataset contains 2,880 training
samples (75%), 480 validation, and testing samples (12.5% each). The single
image dataset then contains 192 training samples (75%), 32 validation, and
testing samples (12.5% each). We consistently keep the same validation and
test wounds held out throughout the training pipeline and between our pre-text
and downstream tasks. Finally, we cropped identical circles around the
centroid of each wound image to ensure no signals other than the wound itself
are contained in the training data.

Figure 3: Positive and negative wound pairs train the network. The image
encoder forms a Siamese tuple configuration with shared parameters; each
branch outputs a 16-dimensional feature vector. The feature vectors are
concatenated and passed through a fully-connected binary classifier,
outputting the input pair temporal validity.

##  3 Approach

The HealNet scheme involves a self-supervised learning step, followed by a
clustering step, and a second learning step for wound-stage classification.
Across all steps, the same image encoder neural network backbone is used and
the trained weights are kept. Given the limited size of our training dataset,
we chose DenseNet121 [20] as the HealNet backbone architecture. DenseNet is
fully convolutional and thus includes fewer parameters than other commonly
used feature extractors while showing strong performance metrics. The first
learning step revolves around self-supervised pre-text learning and thus
includes a concatenation layer and a fully-connected binary classifier as the
network head. The second learning step resembles a more traditional image
classifier, replacing the previous head with a single fully-connected layer
for heal-stage classification. Before training, DenseNet is initialized to
ImageNet [21] weights.

###  3.1 Temporal Coherency Self-supervised Training

To predict the temporal validity of our image pairs and thus encode temporal
dynamics, we define a tuple Siamese architecture [22] using two parallel
stacks of identical feature extraction networks. Each stack shares weights and
takes a single image from the pair as input while outputting a representation
encoding (of size 16). We concatenate both encodings and feed them into a
single fully-connected binary classification layer that determines whether the
input pair is negative or positive. The loss function is binary cross-entropy
and the optimizer is Adam [23] with default parameters. Fig. 3 summarizes the
temporal coherency training scheme.

###  3.2 Unsupervised Wound Stage Clustering

In order to identify and cluster wound healing stages, we use unsupervised
k-means clustering. We apply unsupervised k-means clustering to the
representations generated by our temporal coherency network to generate our
clusters. We set $k=4$ since medical literature suggests four main healing
stages: hemostasis, inflammation, proliferation, and maturation. We use these
cluster mappings to assign pseudo-labels to our dataset. Fig. 1 illustrates
the two steps of assigning pseudo-labels using k-mean clustering and
subsequently using these pseudo-labels for heal-stage classification [18]. The
question of how we specifically relate each cluster to each heal-stage is
covered in Section 4.

###  3.3 Wound Stage Classification

As shown in Fig. 1, the final step of the pipeline is to re-task and fine-tune
the pre-text network towards predicting the heal-stage of input images based
on cluster assigned pseudo-labels. The re-tasking is done by adding a fully-
connected layer with a softmax activation function and four output neurons
(one per healing stage). The fine-tuning is accomplished by further training
and updating the weights of the model using a low learning rate ($\eta=0.001$)
and the Adam [23] optimizer.

##  4 Results and Discussion

This section covers loss and accuracy scores for both learning steps, as well
as evaluates the quality of cluster-based stage discovery from a cluster-
member appearance perspective, from a cluster-member wound-age statistics
perspective and via agreement against human heal-stage annotators. Finally, we
will cover the metrics achieved by the model on held-out test data at each
learning step.

###  4.1 Temporal Coherency Self-supervised Evaluation

The self-supervised model is trained until convergence. Our results show the
network is not overfitting, achieves high accuracy on the validation set, and
can predict the temporal validity of the train, validation, and test sets. The
prediction accuracy for the temporal validity task is 97.2%, 95.0%, and 97.7%
on training, validation, and test sets, respectively, as compiled in Table 2.

###  4.2 Wound Healing Clustering Evaluation

We performed two-dimensional PCA in order to visualize the clusters along with
their centroids. We can observe in Fig. 4 that our clusters are mostly close
together, especially in the later stages. Given how wound progression is
slowly evolving and the time-step between observations is equal to 1 day, this
closeness and some overlap are expected. We can also observe how the
hemostasis cluster contains the highest spread. This is likely because this
cluster mostly contains day 0 images where the wound bed is uncovered and
artificially illuminated and day 1 images where the wound bed is no longer
artificially illuminated and covered with a protective plastic film, causing
some distance between the embeddings. Due to the noisy nature of the dataset,
some spread is also expected.

Figure 4: K-means ($k=4$) cluster visualization with centroids using
2-dimensional PCA.

To further evaluate the quality of the clusters, we looked into the images in
each cluster (for young and aged mice separately). The median wound age (in
days) and first and third quartile of wound age are calculated for each
cluster. As can be seen from Fig. 5, images in each of the clusters contain
visual cues for hemostasis (fresh and clearly defined wound-edge),
inflammation (swelling of the wound edge, wet and shiny appearance),
proliferation (dry and matte appearance with rough and uneven texture), and
maturation (complete or near-complete wound closure). In addition, there is a
clear temporal progression in the median wound day age that also
differentiates our clusters. It would be interesting to note that a differing
wound healing rate between young and aged mice is encoded and shown in the
median, first, and third quantiles of each cohort. Previous work on this
dataset [19] also found a similar divergence in terms of wound-size
progression between young and aged mice, aligning with these results. These
two aspects are used in relating each cluster to a matching heal-stage. To
validate these assignments, a blind survey of 5 human non-experts was
performed where we found a top-1 agreement of over 80% between the self-
supervised and human labels. Table 1 outlines detailed statistics on cluster
temporal distribution and Table 2 describes human annotator agreement per
datasplit.

Figure 5: (left) Cluster member examples split between young and aged mice and
(right) Temporal distribution box plots. Note: here we show the original non-
cropped images to verify the medical red rings (splints) have not biased the
cluster members.

We align the cluster groups with a heal-stage according to two metrics: visual
appearances and wound age distribution metrics.

Table 1: Cluster group wound-age (experimental days 0 - 15) statistics.

Age Group | Cluster | Q1 | Median(Q2) | Q3 | Average
---|---|---|---|---|---
Young | Cluster 1 | 0.0 | 0.0 | 1.0 | 0.5
Cluster 2 | 2.0 | 3.0 | 4.0 | 3.0
Cluster 3 | 6.0 | 7.0 | 8.0 | 7.1
Cluster 4 | 11.0 | 12.5 | 14.0 | 12.5
Aged | Cluster 1 | 0.0 | 1.0 | 1.0 | 1.0
Cluster 2 | 3.0 | 4.0 | 5.5 | 4.5
Cluster 3 | 8.0 | 9.0 | 11.0 | 9.6
Cluster 4 | 13.0 | 14.0 | 15.0 | 13.4

###  4.3 Wound Stage Classification

The wound stage classification model is trained until convergence. While this
task contains fewer training samples ($15\times$ less), the loss curves show
the network is not overfitting, achieves relatively high accuracy on the
validation set, and can predict the heal-stage of training, validation, and
test sets (Table 3). The accuracy for the wound stage classification task is
95.8%, 87.5%, and 90.6% on training, validation, and test sets, respectively.
Finally, we trained an identical architecture as a straight-forward CNN
Classifier (without the pre-text task) utilizing the collected human survey
labels in order to compare performance. Under these conditions the network
achieved a lower accuracy of 83.3%, 81.3%, and 78.1% on training, validation,
and test sets, respectively. Table 3 summarizes the loss and accuracy scores
described above.

Table 2: Human annotator vs. automatic pseudo-label agreement.

Data Split | Agreement (%)
---|---
Train | 80.7
Validation | 71.9
Test | 87.5
Overall | 80.5

Table 3: Performance metrics across both temporal coherency and wound stage
classification tasks for training, validation, and test sets.

Task | Train Acc. | Val. Acc. | Test Acc.
---|---|---|---
Supervised Classifier (baseline) | 83.3% | 81.3% | 78.1%
Temporal Coherency (pre-text) | 97.2% | 95.0% | 97.7%
Wound Stage Classification (downstream) | 95.8% | 87.5% | 90.6%

##  5 Conclusion

HealNet is a self-supervised learning scheme for automatic wound stage
discovery and classification. It forms a novel combination of pre-text
temporal feature extraction, k-means clustering, and pseudo-label re-tasking
and fine-tuning. The results demonstrate the effectiveness and performance of
the presented method with zero human-assisted data annotation. Considering the
cost-effectiveness of the approach compared to medical experts’ continuous
diagnosis, the developed pipeline is a promising tool for augmenting,
accelerating, and improving traditional human diagnosis. Given that there
exists a high chance for expert disagreement in the medical field, a self-
supervised approach could aid in decision-making or in compiling a true
‘golden-master’ list of visual indicators relating to the healing process, the
healing stages and the temporal patterns between them. The proposed HealNet
framework can also be used by the scientific community to discover obscure and
unseen patterns in their unlabeled data at low cost.

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