2. Data used

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 the pseudo-spectral method developed by Rogallo (Reference Rogallo1981) to efficiently account for the shear. The code is described in Pumir & Shraiman (Reference Pumir and Shraiman1995) and Pumir (Reference Pumir1996). 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 $4 \unicode{x03C0} \times 2 \unicode{x03C0} \times 2 \unicode{x03C0}$ , following the work of Sekimoto et al. (Reference Sekimoto, Dong and Jiménez2016), who showed that the corresponding THSF provide a good model for shear driven turbulence. Correspondingly, the system is discretised with $2 M \times M \times M$ Fourier modes. To the simulations with $ M = 100$ and $M = 160$ , discussed in Pumir (Reference Pumir2017) and Yang et al. (Reference Yang, Pumir and Xu2021), we added here a new simulation with $M = 256$ . The corresponding Reynolds numbers, defined by $Re_\lambda = \langle u^2 \rangle /[ \nu \langle (\partial _x u)^2 \rangle ^{1/2}]$ , are $Re_\lambda = 120$ , $145$ and $230$ , respectively. Defining the Kolmogorov length as $\eta = (\nu ^3 /\varepsilon )^{1/4}$ , where $\nu$ is viscosity and $\varepsilon$ is the kinetic energy dissipation rate 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, $T_{tot}$ , were equal to $720 \, S^{-1}$ ( $M = 100$ ), $432 \, S^{-1}$ ( $M = 160$ ) and $190 \, \textrm S^{-1}$ ( $M=256$ ).

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 Reference Lee and Moser2015), 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 $8 \unicode{x03C0}$ , $2$ and $3 \unicode{x03C0}$ in the streamwise ( $x$ ), wall-normal ( $y$ ) and spanwise ( $z$ ) directions, respectively. We downloaded data from the instantaneous velocity fields at $t = 0$ , $1$ , $\ldots$ , $8$ and $10$ , a total of 10 time instants. Pumir et al. (Reference Pumir, Xu and Siggia2016) pointed out that at $t = 9$ , the skewness of $\partial _y u$ at the centre 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. Reference Graham2016). Similar to Pumir et al. (Reference Pumir, Xu and Siggia2016), 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 $\sim 2\times 10^7$ at each wall distance.

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 (Reference Shen and Warhaft2000).

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 idealised turbulent flows, such as THSF, which break isotropy but retain homogeneity. The lowest non-zero odd-order moment, the skewness $S_3$ of the THSF data from this work is shown in figure 1(a), together with numerical (Rogers Reference Rogers1986; Pumir Reference Pumir1996, Reference Pumir2017) and experimental (Tavoularis & Corrsin Reference Tavoularis and Corrsin1981; Garg & Warhaft Reference Garg and Warhaft1998; Ferchichi & Tavoularis Reference Ferchichi and Tavoularis2000; Shen & Warhaft Reference Shen and Warhaft2000) 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 Reference Shen and Warhaft2000), and the ramp–cliff structures observed in the velocity field of THSF from DNS (Pumir & Shraiman Reference Pumir and Shraiman1995), similar to what had been studied in the case of a passive scalar (Buaria et al. Reference Buaria, Clay, Sreenivasan and Yeung2021). For this reason, we briefly review the model proposed by Buaria et al. (Reference Buaria, Clay, Sreenivasan and Yeung2021) for passive scalar fields and then extend it to the THSF.

Figure 1.

(a) Summarised 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 sketch of the ramp–cliff structures in the instantaneous streamwise velocity profile for THSF, which is the sum of the mean velocity, $\langle U \rangle = Sy$ (dotted black line) and the fluctuation, u(y) (solid red line), where S is the mean shear rate and y is the coordinate in the direction of the mean shear. The lower part shows the zoom-in of the instantaneous fluctuation velocity in the green region.

Buaria et al. (Reference Buaria, Clay, Sreenivasan and Yeung2021) 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

(3.1) \begin{equation} \langle (\partial _{l} \theta )^p \rangle \sim \left ( \frac {\theta ^{\prime }}{\delta _{\theta }} \right )^p \frac {\delta _{\theta }}{L} \, . \end{equation}

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 $S_{\partial _l \theta } \sim Re_\lambda ^0$ 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 is of 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 $u^{\prime }$ . Therefore, the typical value of $\partial _y u$ in the cliff region is $u^{\prime }/\delta$ , and the fraction of the cliff region is proportional to $\delta / L$ ; while in the ramp region these are $u^{\prime }/L$ and $1- (\delta /L)$ , respectively. Since $\delta \ll L$ , for odd orders $p = 3$ , $5$ , $\ldots$ , the contribution to the moments $\langle (\partial _y u)^p \rangle$ from the cliff regions can be estimated as

(3.2) \begin{equation} \langle (\partial _y u)^p \rangle \sim \left( \frac {u^{\prime }}{\delta } \right)^p \frac {\delta }{L} , \end{equation}

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

(3.3) \begin{equation} S_3 \equiv \frac {\langle ( \partial _y u )^3 \rangle }{ \langle (\partial _y u)^2 \rangle ^{3/2} } \sim \left( \frac {u^{\prime }}{\delta } \right)^3 \frac {\delta }{L} \bigg / \left( \frac {\varepsilon }{\nu } \right)^{3/2} \sim \frac {\nu ^{3/2}}{\delta ^2 \varepsilon ^{1/2}} \sim \left (\frac {\eta }{\delta }\right )^2. \end{equation}

In the steps earlier, we used the relation between the turbulent dissipation rate and the velocity derivatives, $\varepsilon \sim \nu \langle (\partial _y u )^2\rangle$ ; the so-called zeroth law of turbulence, $u^{\prime 3}/L \sim \varepsilon$ ; and the definition of the Kolmogorov scale $\eta \equiv (\nu ^3/\varepsilon )^{1/4}$ , 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

(3.4) \begin{equation} \delta \sim Re_{\lambda }^{1/4}\eta . \end{equation}

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 (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 $\partial _y u$ normalised 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 $Re_{\lambda }= 158$ and $Re_{\lambda }= 811$ from Shen & Warhaft (Reference Shen and Warhaft2000), as well as three DNS cases with $Re_{\lambda }= 120$ , $Re_{\lambda }= 145$ (Pumir Reference Pumir2017) and $Re_{\lambda }= 230$ . 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 $Re_\lambda$ increases, as shown by figure 1(a).

Figure 2.

The 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 Reference Shen and Warhaft2000). Panel (a) shows $\partial _y u$ normalised by its root-of-mean values. Panel (b) shows $\partial _y u$ normalised by $u^{\prime }/\delta$ ( $z \equiv \partial _y u/(\langle (\partial _y u )^2\rangle ^{1/2} Re_{\lambda }^{1/4})$ ). The PDFs are normalised by $\delta /L$ according to (3.5).

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, $u^{\prime }/\delta$ , to normalise the $\partial _y u$ and use $\delta /L$ to normalise the values of the PDFs. Noting that $u^{\prime 3}/L \sim \varepsilon$ , $L/\eta \sim Re_{\lambda }^{3/2}$ and $\varepsilon \sim \nu \langle (\partial _y u )^2\rangle$ , one obtains from (3.4)

(3.5) \begin{align} u^{\prime }/\delta \sim \langle (\partial _y u )^2\rangle ^{1/2} Re_{\lambda }^{1/4},\,\,\,\,\,\,\delta /L \sim Re_{\lambda }^{-5/4} , \end{align}

which allows us to determine the normalisation factors from the known data and thus provides a convenient way to transfer the normalisation in figure 2(a) to the normalisations by $u^{\prime }/\delta$ and $\delta /L$ . Figure 2(b) shows the PDFs of the rescaled velocity derivative $z \equiv \partial _y u/(\langle (\partial _y u )^2\rangle ^{1/2} Re_{\lambda }^{1/4})$ , with the values of PDFs, $P(z)$ , normalised by $Re_{\lambda }^{-5/4}$ . Figure 2(b) suggests a generally good collapse of the positive tails for different $Re_{\lambda }$ from both DNS and experiments, aside for the experimental data at $Re_\lambda = 158$ , which may be attributed to the limited number of statistics in this case compared with 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 ratio analysis (not shown) was performed for the DNS cases at two Reynolds numbers and indicated that these ratios indeed exhibit a plateau near one in the region of $z \gtrsim O(1)$ , corresponding to the large positive derivatives. The collapse supports that the large positive events of $\partial _y u$ indeed scale as $u^{\prime }/\delta$ with a probability proportional to $\delta /L$ .

Using the expression for $\delta$ , (3.4), and substituting into (3.2), one obtains the scaling of the normalised $p$ th odd-order moments of the streamwise velocity derivative $\partial _y u$ , $S_p$ as a function of $Re_{\lambda }$ :

(3.6) \begin{equation} S_p \equiv \frac {\langle (\partial _y u)^p \rangle }{\langle (\partial _y u)^2 \rangle ^{p/2}} \sim Re_{\lambda }^{(p-5)/4}. \end{equation}

Specifically, the cases for $p = 5,7$ correspond to $S_{5} \equiv \langle (\partial _y u)^5 \rangle /\langle (\partial _y u)^2 \rangle ^{5/2}$ and $S_{7} \equiv \langle (\partial _y u)^7 \rangle /\langle (\partial _y u)^2 \rangle ^{7/2}$ , respectively. The prediction from (3.6) for $S_{5}$ is $S_{5} \sim Re_{\lambda }^0$ , which agrees very well with the experimental result of $\sim Re_{\lambda }^{-0.02}$ in Shen & Warhaft (Reference Shen and Warhaft2000). The prediction for $S_{7} \sim Re_\lambda ^{0.5}$ is not inconsistent with the data of Shen & Warhaft (Reference Shen and Warhaft2000) and Warhaft & Shen (Reference Warhaft and Shen2001), which reported $S_{7} \sim Re_\lambda ^{0.63}$ , considering the scatter of the data (see e.g. figure 11 of Shen & Warhaft Reference Shen and Warhaft2000). Notably, our model predicts that the normalised odd-order moments $S_{p}$ with $p \geqslant 5$ do not decay towards zero as $Re_{\lambda } \to \infty$ , which clearly are not in favour of the assumption of return to isotropy.

We note that in the analysis earlier, we have assumed that the exponent of the scaling of $S_3$ as a function of $Re_\lambda$ is $-1/2$ , which differs slightly from the scaling of $S_3 \sim Re_\lambda ^{-0.52}$ proposed by Shen & Warhaft (Reference Shen and Warhaft2000). A straightforward analysis shows that the collapse of the normalised PDFs is robust under small changes in the exponents.

The earlier results substantiate the close relation between the odd-order moments of $\partial _y u$ 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 § 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 $y^+ \equiv y/\delta _{\nu } = u_{\tau }y/\nu$ is the physical distance to the wall measured in units of the thickness of the viscous sublayer $\delta _{\nu } \equiv \nu /u_{\tau }$ (see e.g. Pope Reference Pope2000). 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 behaviour 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 Reference Pumir and Shraiman1995).

It has been observed that normalised third-order moment $S_{3}$ decreases as $y^+$ increases in the log-layer region (Vreman & Kuerten Reference Vreman and Kuerten2014; Pumir et al. Reference Pumir, Xu and Siggia2016). 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. (Reference Pumir, Xu and Siggia2016) 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 momentum regions in boundary layers (Meinhart & Adrian Reference Meinhart and Adrian1995). These observations encourage us to generalise the discussions from the last section on THSF to the odd-order moments $S_p$ ( $p\gt 1$ ) 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 generalise (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 (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 $u^{\prime }$ is nearly constant (Pope Reference Pope2000) and the balance between local production and dissipation leads to $\varepsilon \sim \nu \langle (\partial _x u)^2\rangle \propto (y^+)^{-1}$ (Pumir et al. Reference Pumir, Xu and Siggia2016; Li et al. Reference Li, Hu and Fang2024), we have

(4.1) \begin{equation} Re_{\lambda } \equiv \frac {{u^{\prime }}^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}. \end{equation}

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

(4.2) \begin{equation} \eta \equiv \nu ^{3/4}\varepsilon ^{-1/4} \sim (y^{+})^{1/4}. \end{equation}

Then from (4.1) and (4.2) we obtain

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

In § 3 (figure 2 b) we observed that normalising $\partial _y u$ and its PDF values respectively by $u^{\prime }/\delta$ and $\delta /L$ can collapse the positive tails of the PDFs of $\partial _y u$ in THSF. Figure 3(a) generalises this approach to the PDFs of $\partial _y u$ at different positions in the log-layer of TCF. Namely, figure 3(a) shows the PDFs of $\partial _y u$ normalised according to (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 $u^{\prime }/\delta$ and probability $\delta /L$ , similar to the case in THSF. Based on this observation, we can further discuss the $y^+$ dependence of the normalised odd-order moments of $\partial _y u$ , $S_p$ , within the log-layer region.

Figure 3.

(a) The PDFs of $\partial _y u$ scaled by $u^{\prime }/\delta$ and $\delta /L$ using the expression of (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 normalised odd-order moments $S_3$ , $S_5$ and $S_7$ of $\partial _y u$ in TCF at two Reynolds numbers, $Re_{\tau } = 1000$ (blue plus symbols) and $Re_{\tau } = 5200$ (red cross symbols). The green solid lines illustrate predictions by (4.5). The curves in the insets of (b) and (d) are compensated by corresponding $y^+$ -scaling expressed in (4.6) in the log-layer region. Vertical dash-dotted lines in (b–d) mark the upper bounds of the log-layer region at the two Reynolds numbers.

In addition to the scaling of $Re_\lambda$ , $\eta$ and $\delta$ shown in (4.1)–(4.3), to predict the scaling of $S_p$ using (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.

(4.4) \begin{equation} L \sim y^+. \end{equation}

Combining these results from (4.1) to (4.4), we obtain

(4.5) \begin{equation} S_p \equiv \frac {\langle (\partial _y u)^p \rangle }{\langle (\partial _y u)^2 \rangle ^{p/2}} \sim (y^{+})^{(p-5)/8}. \end{equation}

For the specific values of $p = 3,5$ and $7$ , the predictions are

(4.6) \begin{align} S_{3} \sim (y^{+})^{-1/4}, \quad S_{5} \sim (y^{+})^{0}, \quad S_{7} \sim (y^{+})^{1/4}. \end{align}

Figures 3(b) and 3(d) present the $y$ -dependence of $S_3$ , $S_5$ and $S_7$ in TCF at two different Reynolds numbers. $Re_\tau = 5200$ and $Re_\tau = 1000$ . 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 (4.6). It can be seen that the results for the skewness $S_3$ are consistent with a power-law decay in the log-layer region as $y^+$ increases until the centre region ( $y/h \geqslant 0.3$ ) is reached, where the skewness hits a plateau, before going to zero at the centre of the channel due to symmetry requirement. We notice that for $Re_\tau = 1000$ , the log-layer is very narrow. By performing a least-squares fit on the $Re_\tau = 5200$ data over the range $60 \lt y^+ \lt 1024$ , we obtain a scaling of $S_{3} \sim (y^{+})^{-0.22}$ , which is consistent with the prediction of $(y^{+})^{-1/4}$ by (4.6). We also observe that the normalised 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 (4.6). In summary, (4.5), derived by stressing the role of the ramp–cliff structures, agrees with DNS results at least for the odd-order moments $S_p$ with $p \leqslant 7$ . 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.