# Small-scale anisotropy and ramp-cliff structures in turbulent shear flows

Xuechun Gong, Ping-Fan Yang, Haitao Xu, Alain Pumir

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Xuechun Gong, Ping-Fan Yang, Haitao Xu, Alain Pumir. Small-scale anisotropy and ramp-cliff structures in turbulent shear flows. Journal of Fluid Mechanics, 2025, 1009, pp.R6. 10.1017/jfm.2025.176. hal-05375168

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# Small-Scale Anisotropy and Ramp-Cliff Structures in Turbulent Shear Flows

Xuechun Gong¹, Ping-Fan Yang²,³†, Haitao Xu¹ and Alain Pumir⁴

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¹Center for Combustion Energy and School of Aerospace Engineering, Tsinghua University, Beijing

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100084, PR China

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²School of Aeronautics and Institute of Extreme Mechanics, Northwestern Polytechnical University, Xi’an

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710072, PR China

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³National Key Laboratory of Aircraft Configuration Design, Xi’an 710072, PR China

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⁴Ecole Normale Supérieure de Lyon, CNRS, Laboratoire de Physique, F-69342, Lyon, France

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(Received xx; revised xx; accepted xx)

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Experimental and numerical observations in turbulent shear flows point to the persistence of the anisotropy imprinted by the large scale velocity gradient down to the smallest scales of turbulence. This is reminiscent of the strong anisotropy induced by a mean passive scalar gradient, which manifests itself by the “ramp-cliff” structures. In the shear flow problem, the anisotropy can be characterized by the odd-order moments of $\partial_y u$, where $u$ is the fluctuating streamwise velocity component, and $y$ is the direction of mean shear. Here, we extend the approach proposed by Buaria et. al. (Phys. Rev. Lett., 126, 034504, 2021) for the passive scalar fields, and postulate that fronts of width $\delta \sim \eta Re_\lambda^{1/4}$ explain the observed small-scale anisotropy for shear flows. This model is supported by the collapse of the positive tails of the probability density functions (PDFs) of $(\partial_y u)/(u'/\delta)$ in turbulent homogeneous shear flows (THSF) when the PDFs are normalized by $\delta/L$, where $u'$ is the root-mean-square of $u$ and $L$ is the integral length scale. The predictions of this model for the odd-order moments of $\partial_y u$ in THSF agree well with direct numerical simulation (DNS) and experimental results. Moreover, the extension of our analysis to the log-layer of turbulent channel flows (TCF) leads to the prediction that the odd-order moments of order $p(p > 1)$ of $\partial_y u$ have power-law dependences on the wall distance $y^+$: $\langle (\partial_y u)^p \rangle / \langle (\partial_y u)^2 \rangle^{p/2} \sim (y^+)^{(p-5)/8}$, which is consistent with DNS results.

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**Key words:**

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## 1. Introduction

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The overwhelming majority of turbulent flows, relevant to natural or technological processes, is anisotropic due to the specificity of the motion at large scales, which may include a shear or a straining motion. In his seminal work, Kolmogorov (1941) postulated that at

<footnote>† Email address for correspondence: yangpingfan@nwpu.edu.cn</footnote>
very high Reynolds numbers and far from boundaries, the statistical properties of turbulent motion at sufficiently small scales are isotropic, insensitive to the anisotropy imposed at the largest scales. This fundamental hypothesis of the K41 theory plays a crucial role for many subsequent turbulence theories and models. In practice, however, Reynolds numbers are large, but finite, leaving open the quantitative question of to what extent small scales are influenced by the flow properties at large-scale. Specifically, in turbulent shear flows, small-scale anisotropy can be characterized by the odd-order moments of $∂_y u$, where $u$ is the fluctuating streamwise velocity component and $y$ is the direction of mean shear. The analysis of Corrsin (1958) and Lumley (1967) leads to the conclusion that as the Reynolds number increases, the skewness of $∂_y u$ decays as $Re_λ^{-1}$, where $Re_λ = u'λ/v$ is the Reynolds number based on the Taylor microscale $λ = u'/(\langle(∂_x u)^2\rangle^{1/2})$, $u'$ is the root-mean-square of the streamwise velocity $u$, and $v$ is the kinematic viscosity. The prediction, which also applies to the decay of anisotropy in passive scalar fluctuations in turbulent flow fields, was first challenged by experimental measurements of scalar statistics under the existence of a mean scalar gradient $G$ (Tong & Warhaft 1994; Warhaft 2000; Shraiman & Siggia 2000), along with numerical studies of simplified models (Holzer & Siggia 1994), and direct numerical simulations (DNS) (Pumir 1994). These results together with more recent DNS results at much higher $Re_λ$ (Buaria et al. 2021), clearly establish that the skewness of the fluctuating scalar gradient $∂_l θ = G · ∇θ/|G|$ is $O(1)$ and persists as $Re_λ$ increases, i.e., $S_{∂_l θ} \sim Re_λ^0$, casting doubt on the classical prediction of $Re_λ^{-1}$. This anomalous behavior, which shows a clear violation of the anisotropy of the scalar field at the smallest scales in the flow, can be related to the presence of large-scale “ramp-cliff” structures observed long ago (see e.g., Mestayer et al. 1976; Gibson et al. 1977; Sreenivasan et al. 1979; Mestayer 1982). These structures feature a wide “ramp” region where the physical quantity changes gradually, followed by a “cliff” region where changes are sharp. The anisotropy originates from a strong alignment of the scalar gradients in the cliff with the mean scalar gradient. Recently, Sreenivasan (2019) proposed a method to estimate the order of magnitude of $S_{∂_l θ}$ based on ramp-cliff structures in passive scalar fields. Based on this idea, Buaria et al. (2021) developed a model providing quantitative predictions for the $Re_λ$ scaling of the odd-order moments of $∂_l θ$, which agree well with numerical results.

The study of small-scale anisotropy in scalar fields with a mean gradient has raised the question of whether similar behaviors occur in the turbulent velocity field with a mean shear $S = d\langle U \rangle/dy$ (Pumir & Shraiman 1995; Casciola et al. 2003; Smits et al. 2011), where $\langle U \rangle$ is the mean velocity taken here to be varying only in the $y$-direction. For turbulent homogeneous shear flows (THSF) with a constant mean shear, experimental (Garg & Warhaft 1998; Shen & Warhaft 2000; Ferchichi & Tavoularis 2000; Warhaft & Shen 2001) and DNS results (Rogers 1986; Pumir 1996) have demonstrated that the skewness of $∂_y u$, $S_3 = \langle (∂_y u)^3 \rangle / \langle (∂_y u)^2 \rangle^{3/2}$, exhibits a scaling of $Re_λ^{-1/2}$, which also differs from the classical prediction of $Re_λ^{-1}$, although less strongly than for the passive scalar. Flow visualizations demonstrate the presence of large-scale “ramp-cliff” structures in THSF (Pumir & Shraiman 1995). This suggests to apply the models proposed for the scalar (Buaria et al. 2021) also to the velocity fields in THSF. As we show here, this approach leads to quantitative predictions for small-scale anisotropy in velocity statistics, particularly the scaling of odd-order moments of $∂_y u$ with $Re_λ$.

Furthermore, ramp-cliff structures also have close connections with the existence of uniform momentum zones observed in turbulent boundary layers (Meinhart & Adrian 1995). Additionally, the suggestive evidence for “ramp-cliff” structures in instantaneous streamwise velocity profiles of fully-developed turbulent channel flows (TCF), especially in the inner region, are provided by Pumir et al. (2016). Meanwhile, a significant decay of $S_3$ with respect to the wall distance in the log-layer region has also been observed by Vreman & Kuerten
82 (2014). Based on the similarities between THSF and TCF for both large (Sekimoto et al. 2016; Dong et al. 2017) and small scales (Pumir et al. 2016; Pumir 2017), we extend the discussions of small-scale anisotropy to TCF. This allows us to predict that the odd-order moments of $\partial_y u$ depend on the distance $y$ to the wall as a power-law function.
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86 The rest of the paper is organized as follows: In Section 2, we briefly introduce the experimental and DNS data used in this work. Then in Section 3, we review the model proposed by Buaria et al. (2021) for scalar fields and extend it to THSF to characterize the small-scale anisotropic properties of the odd-order moments of $\partial_y u$, and derive the scaling of the characteristic length scale of the cliff, $\delta$, in THSF. In Section 4, we generalize the discussions to the log-layer region of TCF and obtain how anisotropy decays as the distance from the wall increases. Finally, we summarize and conclude our findings in Section 5.
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## 2. Data used

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The THSF data analysed in this work include both DNS and experimental data. Numerically, the Navier-Stokes equations are solved by DNS in a periodic domain by using pseudo-spectral method developed by Rogallo (1981) to efficiently account for the shear. The code is described in Pumir & Shraiman (1995) and Pumir (1996). Physically, turbulence is maintained by the interaction between the mean shear and the velocity fluctuations, without any extra forcing. We chose a domain of size $4\pi \times 2\pi \times 2\pi$, following the work of Sekimoto et al. (2016), who showed that the corresponding THSF provide a good model for shear driven turbulence. Correspondingly, the system is discretized with $2M \times M \times M$ Fourier modes. To the simulations with $M = 200$ and $M = 320$, discussed in Pumir (2017) and Yang et al. (2021), we added here a new simulation with $M = 512$. The corresponding Reynolds numbers, defined by $Re_\lambda = \langle u^2 \rangle / [\nu (\partial_x u)^2]^{1/2}$, are $Re_\lambda = 120, 145$ and $230$, respectively. Defining the Kolmogorov length as $\eta = (\nu^3/\epsilon)^{1/4}$, where $\nu$ is viscosity and $\epsilon$ is the kinetic energy dissipation per unit mass, and $k_{\max}$ the largest mode faithfully represented in the simulation, the product $k_{\max}\eta$ was $\gtrsim 1.3$. The total durations of the runs, $T_{\text{tot}}$, were equal to $720 \text{ S}^{-1}$ ($M = 200$), $432 \text{ S}^{-1}$ ($M = 320$) and $190 \text{ S}^{-1}$ ($M = 512$).
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For TCF, we used the DNS data from the Johns Hopkins Turbulence Databases (JHTDB). In this work, we mainly use the dataset with the Reynolds number $Re_\tau \equiv u_\tau h/\nu = 5200$ (Lee & Moser 2015), where $u_\tau \equiv \sqrt{\tau_w/\rho}$ is the friction velocity, $h$ is the half height of the channel, $\tau_w$ is the wall shear stress and $\rho$ is the density of the fluid. The DNS of TCF are conducted within a rectangular domain of size $8\pi$, $2$ and $3\pi$ in the streamwise ($x$), wall-normal ($y$), and spanwise ($z$) directions, respectively. We downloaded data from the instantaneous velocity fields at $t = 0, 1, \dots, 8$ and $10$, total $10$ time instants. Pumir et al. (2016) pointed out that at $t = 9$, the skewness of $\partial_y u$ at the center of the channel does not vanish, which is inconsistent with the requirement of symmetry. For this reason, we have not included the data at $t = 9$ in our analysis. From each velocity field, we selected $27$ different wall-parallel ($x-z$) planes in both the upper and lower parts of the channel and downloaded $N_x \times N_z = 3200 \times 1200$ data points from each plane, resulting in $3200 \times 1200 \times 10 \times 2 \sim 7 \times 10^7$ samples at each wall distance. For comparison, we also considered the lower Reynolds dataset with $Re_\tau = 1000$ (Graham et al. 2016). Similar to Pumir et al. (2016), we chose $10$ snapshots at $t = 2.5 + t \times n, n = 0, \dots, 9$, and from each snapshot we selected $47$ $x-z$ planes from both sides of the channel and downloaded approximately $\sim 2 \times 10^7$ at each wall distance.
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The experimental results for $Re_\lambda = 158$ and $Re_\lambda = 811$ are from the hot-wire measurement in homogeneous shear flows in a wind tunnel by Shen & Warhaft (2000).
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<figure><img src="image_2.png" /><figcaption>Figure 1: (a) Summarized experimental and numerical results for the skewness of $\partial_y u$ in THSF. The red solid line and the black dashed line indicate the $Re_\lambda^{-1}$ and $Re_\lambda^{-1/2}$ scaling, respectively. (b) A cartoon of the ramp-cliff structures in the instantaneous streamwise velocity profile for THSF.</figcaption></figure>

### 3. Ramp-cliff structures of velocity in THSF

The local isotropy assumption of the K41 theory states that when the Reynolds number $Re_\lambda$ is sufficiently high, the influence of large-scale anisotropy on small scales is negligible. One of the consequences of local isotropy is that the odd-order moments of $\partial_y u$ should be approximately zero. The local isotropy hypothesis is typically investigated in idealized turbulent flows, such as THSF, which break isotropy but retain homogeneity. The lowest nonzero odd-order moment, the skewness $S_3$ of the THSF data from this work are shown in figure 1(a), together with numerical (Rogers 1986; Pumir 1996, 2017) and experimental (Tavoularis & Corrsin 1981; Garg & Warhaft 1998; Ferchichi & Tavoularis 2000; Shen & Warhaft 2000) data. Here, we consider the possibility of a relation between the scaling of the skewness as $S_3 \sim Re_\lambda^{-1/2}$ (Shen & Warhaft 2000), and the ramp-cliff structures observed in the velocity field of THSF from DNS (Pumir & Shraiman 1995), similar to what had been studied in the case of a passive scalar (Buaria et al. 2021). For this reason, we briefly review the model proposed by Buaria et al. (2021) for passive scalar fields and then extend it to the turbulent velocity field in THSF.

Buaria et al. (2021) estimated that the odd-order moments of the partial derivative of $\theta$ in the direction parallel to the imposed scalar mean gradient are dominated by contributions from the “cliff” sides of the ramp-cliff structures, where the scalar jumps by $\sim \theta' \equiv \langle \theta^2 \rangle^{1/2}$ over a very short scale $\delta_\theta$ and thus occupies a fraction of $\delta_\theta/L$. This leads to:

$$ \langle (\partial_t \theta)^p \rangle \sim \left( \frac{\theta'}{\delta_\theta} \right)^p \frac{\delta_\theta}{L}. \qquad (3.1) $$

Quantitative predictions then rely on the correct estimation of $\delta_\theta$. For passive scalars, $\delta_\theta$ is controlled by the competition between diffusion and turbulent advection. The extension of these ideas to the THSF problem requires some care: judging from the difference in the power laws for $S_{\partial_t \theta} \sim Re_\lambda^0$ and $S_3 \sim Re_\lambda^{-1/2}$, very different length scales are involved for the scalar and for the velocity fields. Therefore, our next step is to determine this length scale.

As sketched in figure 1(b), the size of the entire ramp-cliff structure in the velocity field
<figure><img src="image_3.png" /><figcaption>Figure 2: PDFs of the velocity derivative $\partial_y u$ from DNS ($Re_\lambda$ = 120, $Re_\lambda$ = 144 and $Re_\lambda$ = 230) and experiments ($Re_\lambda$ = 158 and $Re_\lambda$ = 811, see Shen & Warhaft (2000)). (a) $\partial_y u$ normalized by its root-of-mean values. (b) $\partial_y u$ normalized by $u'/\delta$ ($z \equiv \partial_y u/((\partial_y u)^2)^{1/2}Re_\lambda^{1/4}$). The PDFs are normalized by $\delta/L$ according to Eq. (3.5).</figcaption></figure>

is on the order of the integral length scale $L$, while the cliff occupies a much smaller scale $\delta$, within which significant variations in streamwise velocity occur with a typical magnitude of $u'$. Therefore, the typical value of $\partial_y u$ in the cliff region is $u'/\delta$, and the fraction of the cliff region is proportional to $\delta/L$; while in the ramp region these are $u'/L$ and $1 - \delta/L$, respectively. Since $\delta \ll L$, thus for odd orders $p = 3, 5, \dots$, the contribution to the moments $\langle (\partial_y u)^p \rangle$ from the cliff regions can be estimated as

$$ \langle (\partial_y u)^p \rangle \sim \left( \frac{u'}{\delta} \right)^p \frac{\delta}{L}, \qquad (3.2) $$

while the contribution from the ramp regions is negligible in comparison. Letting $p = 3$ and normalizing Eq. (3.2) by $\langle (\partial_y u)^2 \rangle^{3/2}$, we obtain a prediction for $S_3$

$$ S_3 \equiv \frac{\langle (\partial_y u)^3 \rangle}{\langle (\partial_y u)^2 \rangle^{3/2}} \sim \left(\frac{u'}{\delta}\right)^3 \frac{\delta}{L} \left(\frac{\epsilon}{\nu}\right)^{3/2} \sim \frac{\nu^{3/2}}{\delta^2 \epsilon^{1/2}} \sim \left(\frac{\eta}{\delta}\right)^2 \qquad (3.3) $$

In the steps above, we used the relation between the turbulent dissipation rate and the velocity derivatives, $\epsilon \sim \nu \langle (\partial_y u)^2 \rangle$; the so-called zeroth law of turbulence, $u'^3/L \sim \epsilon$; and the definition of the Kolmogorov scale $\eta = (\nu^3/\epsilon)^{1/4}$, respectively. The empirical relation $S_3 \sim Re_\lambda^{-1/2}$, see figure 1(a), thus leads to the conclusion that the characteristic cliff size behaves as:

$$ \delta \sim Re_\lambda^{1/4} \eta. \qquad (3.4) $$

We stress that the scaling of the length scale $\delta$ in the cliff region has been obtained solely from the scaling of the lowest non-trivial odd-order moment, namely the skewness $S_3$. Whether the length scale $\delta$ defined by Eq. (3.4) correctly provides an accurate description of higher-order moments such as $S_5$ and $S_7$, requires further testing. To this end, we focus on the large positive events of $\partial_y u$, and more specifically on the positive tails of the probability density functions (PDF) of $\partial_y u$, since the cliff regions dominate the contribution to the large positive events of $\partial_y u$. Figure 2(a) shows the PDFs of $\partial_y u$ normalized by the root-mean-square value $\langle (\partial_y u)^2 \rangle^{1/2}$. These PDFs are collected from various sources, including two
experimental cases with $Re_λ = 158$ and $Re_λ = 811$ from Shen & Warhaft (2000), as well as three DNS cases with $Re_λ = 120$, $Re_λ = 145$ (Pumir 2017) and $Re_λ = 230$. We observe that as the Reynolds number increases, the positive tails extend further, albeit not as rapidly as the negative ones, which is consistent with the decrease in the skewness $S_3$ when $Re_λ$ increases, as shown by figure 1(a).

On the other hand, the positive tails of the PDFs should collapse if we choose the assumed characteristic scale of the velocity gradient in these regions, $u'/δ$, to normalize the $∂_y u$ and use $δ/L$ to normalize the values of the PDFs. Noticing that $u'^3/L \sim ε$, $L/η \sim Re_λ^{3/2}$ and $ε \sim v\langle(∂_y u)^2\rangle$, one obtains from Eq. (3.4):

$$u'/δ \sim \langle(∂_y u)^2\rangle^{1/2} Re_λ^{1/4}, \quad δ/L \sim Re_λ^{-5/4}, \qquad (3.5)$$

which allows us to determine the normalization factors from the known data and thus provides a convenient way to transfer the normalization in figure 2(a) to the normalizations by $u'/δ$ and $δ/L$. Figure 2(b) shows the PDFs of the rescaled velocity derivative $z \equiv ∂_y u / ((∂_y u)^2)^{1/2} Re_λ^{1/4}$, with the values of PDFs, $P(z)$, normalized by $Re_λ^{-5/4}$. Figure 2(b) suggests a generally good collapse of the positive tails for different $Re_λ$ from both DNS and experiments, aside for the experimental data at $Re_λ = 158$, which may be attributed to the limited number of statistics in this case compared to the other datasets. The collapse of the positive tails of the rescaled PDFs can be quantitatively evaluated by taking the ratios of the PDF curves at different Reynolds numbers. The results (not shown) indicate that these ratios indeed exhibit a plateau near 1 in the region of $z \ge O(1)$, corresponding to the large positive derivatives. The collapse supports that the large positive events of $∂_y u$ indeed scale as $u'/δ$ with a probability proportional to $δ/L$.

Using the expression for $δ$, Eq. (3.4), and substituting into Eq. (3.2), one obtains the scaling of the normalized $p$-th odd-order moments of the streamwise velocity derivative $∂_y u$, $S_p$ as a function of $Re_λ$:

$$S_p \equiv \frac{\langle (\partial_y u)^p \rangle}{\langle (\partial_y u)^2 \rangle^{p/2}} \sim Re_\lambda^{(p-5)/4}. \qquad (3.6)$$

Specifically, the cases for $p = 5, 7$ correspond to $S_5 \equiv \langle(∂_y u)^5\rangle/\langle(∂_y u)^2\rangle^{5/2}$ and $S_7 \equiv \langle(∂_y u)^7\rangle/\langle(∂_y u)^2\rangle^{7/2}$, respectively. The prediction from Eq. (3.6) for $S_5$ is $S_5 \sim Re_λ^0$, which agrees very well with the experimental result of $\sim Re_λ^{-0.02}$ in Shen & Warhaft (2000). The prediction for $S_7 \sim Re_λ^{0.5}$, is not inconsistent with the data of Shen & Warhaft (2000) and Warhaft & Shen (2001), which reported $S_7 \sim Re_λ^{0.63}$, considering the scatter of the data (see e.g. Fig.11 of Shen & Warhaft (2000)). Notably, our model predicts that the normalized odd-order moments $S_p$ with $p \ge 5$ do not decay towards zero as $Re_λ \to \infty$, which clearly are not in favour of the assumption of return to isotropy.

We notice that in the analysis above, we have assumed that the exponent of the scaling of $S_3$ as a function of $Re_λ$ is $-1/2$, which slightly differs from the scaling of $S_3 \sim Re_λ^{-0.52}$ proposed by Shen & Warhaft (2000). A straightforward analysis shows that our results are robust under this small changes in the exponents.

The above results substantiate the close relation between the odd-order moments of $∂_y u$ and the ramp-cliff structures in THSF. They provide the motivation for exploring further their relations in more complex shear flows such as TCF.

## 4. The anisotropy of velocity gradients in TCF

One of the conditions for the turbulent flow to restore isotropy in K41 theory is that the Reynolds number should be sufficiently high, as explained in Section 3. The other requirement
for local isotropy at small scales is that the flow is far enough from any boundary. Therefore, in wall bounded shear turbulence, with the presence of a wall, an important question is the dependence of small-scale anisotropy on the distance to the wall (i.e. the wall distance $y$). Specifically, we ask whether the odd-order moments of $\partial_y u$ diminish as $y$ increases in TCF. In near-wall turbulence, the dimensionless wall distance $y^+ \equiv y/\delta_v = u_\tau y/\nu$ is the physical distance to the wall measured in units of the thickness of the viscous sublayer $\delta_v \equiv \nu/u_\tau$ (see e.g. Pope 2000). Consequently, $y^+$ can also be regarded as an effective local Reynolds number representing the scale separation between the local integral scale and the viscous scale at that particular position. For this reason, we expect that in the log-layer region where $y^+ \gg 1$ and $y/h \ll 1$, the behavior of the odd-order moments of $\partial_y u$ should show similarities to the case of THSF, which may also not support the return to isotropy assumption (Pumir & Shraiman 1995).

It has been observed that normalized third-order moment $S_3$ decreases as $y^+$ increases in the log-layer region (Vreman & Kuerten 2014; Pumir et al. 2016). However, a quantitative description of the $y^+$ dependence of $S_3$, or of the higher-moments such as $S_5$ and $S_7$, is not available. Pumir et al. (2016) also observed large-scale ramp-cliff structures in the instantaneous streamwise velocity field in TCF. This is also consistent with the existence of uniform streamwise momentun regions in boundary layers (Meinhart & Adrian 1995). These observations encourage us to generalize the discussions from the last section on THSF to the odd-order moments $S_p$ ($p > 1$) in TCF. This is particularly relevant for predicting the $y^+$ scaling of $S_p$ in the log-layer region.

As before, we assume that the cliff regions dominate the contribution to the odd-order moments of $\partial_y u$ and generalize Eq. (3.2) to the log-layer of TCF. To estimate the typical length scale $\delta$ of the cliff regions and its $y^+$ dependence, we assume that $\delta$ still satisfies Eq. (3.4) locally, i.e. $\delta \sim Re_\lambda^{1/4} \eta$ at any given $y^+$ by making the analogy that the ramp-cliff structures in TCF are similar to those in THSF at the same local Reynolds number. Next, we need to estimate the $y^+$ dependence of $Re_\lambda$ and $\eta$ in the log-layer of TCF. From the definition of $Re_\lambda$ and noticing that in the log-layer, the fluctuation velocity $u'$ is nearly constant (Pope 2000) and the balance between local production and dissipation leads to $\epsilon \sim \langle (\partial_x u)^2 \rangle \sim (y^+)^{-1}$ (Pumir et al. 2016; Li et al. 2024), we have:

$$Re_\lambda \equiv \frac{u'^2}{\nu \langle (\partial_x u)^2 \rangle^{1/2}} \sim \frac{1}{\nu \langle (\partial_y u)^2 \rangle^{1/2}} \sim (y^+)^{1/2}. \quad (4.1)$$

Meanwhile, since $\epsilon \sim (y^+)^{-1}$, the Kolmogorov length $\eta$ scales as

$$\eta = \nu^{3/4} \epsilon^{-1/4} \sim (y^+)^{1/4}, \quad (4.2)$$

Then from Eqs. (4.1) and (4.2) we obtain:

$$\delta \sim Re_\lambda^{1/4} \eta \sim (y^+)^{3/8}. \quad (4.3)$$

In Section 3 (fig. 2(b)) we observed that normalizing $\partial_y u$ and its PDF values respectively by $u'/\delta$ and $\delta/L$ can collapse the positive tails of the PDFs of $\partial_y u$ in THSF. Figure 3(a) generalizes this approach to the PDFs of $\partial_y u$ at different positions in the log-layer of TCF. Namely, fig. 3(a) shows the PDFs of $\partial_y u$ normalized according to Eq. (3.5) with the local $\langle (\partial_y u)^2 \rangle^{1/2}$ and $Re_\lambda$ determined from their definitions at four different positions $y^+ = 128, 256, 512$ and $1024$ in the log-layer of the TCF at $Re_\tau = 5200$. The positive tails of the PDFs at different $y^+$ collapse very well, supporting our assumption that the large positive events of $\partial_y u$ in the log-layer of TCF are indeed dominated by cliff structures, with a typical scale of $u'/\delta$ and probability $\delta/L$, similar to the case in THSF. Based on this observation,
<figure><img src="image_4.png" /><figcaption>Figure 3: (a) The PDFs of $\partial_y u$ scaled by $u'/\delta$ and $\delta/L$ using the expression of Eq. (3.5) at $y^+ = 128, 256, 512$ and $1024$ in the log-layer of the TCF with the Reynolds number $Re_\tau = 5200$. (b)-(d) The variations of normalized odd-order moments $S_3, S_5$ and $S_7$ of $\partial_y u$ in TCF at two Reynolds numbers $Re_\tau = 1000$ and $Re_\tau = 5200$. The green solid lines illustrate predictions by Eq. (4.5). The curves in the insets of (b) and (d) are compensated by corresponding $y^+$-scaling expressed in Eqs (4.6) in the log-layer region. Vertical dashed lines in (b)-(d) mark the upper bounds of the log-layer region at the two Reynolds numbers.</figcaption></figure>

267 we can further discuss the $y^+$ dependence of the normalized odd-order moments of $\partial_y u$, $S_p$, within the log-layer region.

268 In addition to the scaling of $Re_\lambda$, $\eta$ and $\delta$ shown in Eqs. (4.1) - (4.3), to predict the scaling of $S_p$ using Eq. (3.2), we still need to determine the $y^+$ dependence of the integral length scale $L$. We note that near the wall, $L$ is controlled by the largest eddy allowed by the distance to the wall, i.e.,

273 $$L \sim y^+. \quad (4.4)$$

274 Combining these results from Eq. (4.1) to Eq. (4.4), we obtain:

275 $$S_p \equiv \frac{\langle (\partial_y u)^p \rangle}{\langle (\partial_y u)^2 \rangle^{p/2}} \sim (y^+)^{(p-5)/8}. \quad (4.5)$$
For the specific values of $p = 3, 5$ and $7$, the predictions are:

$$S_3 \sim (y^+)^{-1/4}, \quad S_5 \sim (y^+)^0, \quad S_7 \sim (y^+)^{1/4}. \qquad (4.6)$$

Figures 3(b)-3(d) present the y-dependence of $S_3$, $S_5$ and $S_7$ in TCF at two different Reynolds numbers $Re_τ = 5200$ and $Re_τ = 1000$. For a better comparison between the results from these two data sets and our predictions, insets in figures 3(b) and 3(d) show these odd-order moments compensated by the corresponding $y^+$-scaling given by Eqs. (4.6). It can be seen that the results for the skewness $S_3$ is consistent with a power-law decay in the log-layer region as $y^+$ increases until the center region ($y/h \ge 0.3$) is reached, where the skewness hits a plateau, before going to zero at the center of the channel due to symmetry requirement. We notice that for $Re_τ = 1000$, the log-layer is very narrow. By performing a least-squares fit on the $Re_τ = 5200$ data over the range $60 < y^+ < 1024$, we obtain a scaling of $S_3 \sim (y^+)^{-0.22}$, which is consistent with the prediction of $(y^+)^{-1/4}$ by Eq. (4.6). We also observe that the normalized fifth-order moment $S_5$ is nearly constant in figure 3(c), and the seventh-order moment shows an approximate scaling of $S_7 \sim (y^+)^{1/4}$ in the log-layer region in figure 3(d), consistent with the predictions by Eq. (4.6). In summary, Eq. (4.5), derived by stressing the role of the ramp-cliff structures, agrees with DNS results at least for the odd-order moments $S_ρ$ with $p \le 7$. These results suggest that in the log-layer of TCF, the large positive events of $\partial_y u$ are also predominantly contributed by cliff structures, extending our understanding of small-scale anisotropy from the simple THSF to the more complex case with the presence of a wall.

## 5. Conclusion

In conclusion, we propose that in turbulent shear flows, the ramp-cliff structures induced by the large-scale mean shear provide the dominant contribution to the small-scale anisotropy, which can be captured quantitatively by generalizing the model proposed by Buaria et al. (2021) to the velocity fields in turbulent shear flows and specifying the typical length scale of the cliff structures as $\delta \sim Re_λ^{1/4}\eta$. When normalized accordingly, the positive tails of the PDFs of $\partial_y u$ collapse well, which provides support to our model and confirms that the large positive events of $\partial_y u$ scale as $u'/\delta$ with a probability proportional to $\delta/L$. The model then predicts the Reynolds number dependence of the odd-order moments of $\partial_y u$, which is in good agreement with the available experimental and numerical results. Specifically, the model explains the observation that the normalized odd-order moments $S_ρ$ with $p > 3$ do not decay to zero as the Reynolds number increases, which contradicts the assumption of return to isotropy. Furthermore, we extend the discussion to the log-layer of TCF, where we identified a power-law scaling for the odd-order moments of $\partial_y u$ versus the wall distance $y^+: \langle (\partial_y u)^p \rangle / \langle (\partial_y u)^2 \rangle^{p/2} \sim (y^+)^{(p-5)/8}$, and compared it with high Reynolds number DNS data. In summary, this study not only substantiates the existence of ramp-cliff structures in turbulent shear flows but also elucidates the typical length scale of the cliff region, where the effects of small-scale anisotropy are most pronounced. A challenging question for further study is to provide a theoretical explanation for the scaling of $\delta$. Furthermore, it would be interesting to extend the ideas presented in this work to other free shear flows, such as wakes or jets, for which the limited data available to us suggest that the ramp-cliff structures also exist in the velocity fields of the free shear flows.

**Acknowledgements.** We are grateful to X. Shen and Z. Warhaft for making the original data from Shen & Warhaft (2000) available for us to show in figure 2. XG, PFY and HX acknowledge support from the National Natural Science Foundation of China (NSFC) under grants 12202452, 11672157, 91852104, and 11988102. Alain Pumir was supported by the French Agence Nationale de la Recherche under Contract
323 No. ANR-20-CE30-0035 (project TILT), and acknowledge the Max-Planck Institut for Dynamics and Self-

324 Organisation.

325 **Declaration of interests.** The authors report no conflict of interest.

326 **Author ORCIDs.**

327 Xuechun Gong https://orcid.org/0009-0003-5499-0572;

328 Ping-fan Yang https://orcid.org/0000-0003-1949-500X;

329 Haitao Xu https://orcid.org/0000-0002-2863-7658;

330 Alain Pumir https://orcid.org/0000-0001-9946-7353.

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