2. Convection velocity of the SRNS fluctuations

2.1. The $r{-}u$ convection velocity difference in turbulent channel flows

The present results are obtained from direct numerical simulations (DNSs) of turbulent channel flow at a friction Reynolds number $R{e_\tau } \equiv {u_\tau }h/\nu = 550$ , where $u_\tau$ is the friction velocity, $h$ is the channel half-height and $\nu$ is the kinematic viscosity. The bulk Reynolds number is $Re\equiv U_bh/\nu =10\,000$ , where $U_b$ is the bulk velocity. Simulations were performed using a pseudo-spectral method with the $3/2$ dealiasing rule, implementing periodic boundary conditions in the streamwise and spanwise directions with no-slip conditions at the walls. The computational domain is $8\pi h \times 2h \times 3\pi h$ in the streamwise ( $x$ ), wall-normal ( $y$ ) and spanwise ( $z$ ) directions, discretised with $1536 \times 256 \times 1152$ grid points. The wall-normal grids are located on the Chebyshev collocation points, and the minimum and maximum wall-normal resolutions are $\min (\Delta _y^ + ) = 0.04$ near the wall and $\max (\Delta _y^ + ) = 6.67$ at the centre of the channel. Data are analysed at six wall-normal locations spanning key turbulent regimes: $y^+=5$ (viscous sublayer), $y^+=12$ (buffer layer), $y^+=44$ and $92$ (logarithmic region), $y^+=270$ (outer layer) and $y^+=550$ (channel centreline). The instantaneous flow fields are stored in the statistically stationary state. This validated dataset has been used in prior studies (Wu & He Reference Wu and He2020, Reference Wu and He2021b ; Zhang et al. Reference Zhang, Ning, Wu and He2025).

Consider the streamwise velocity fluctuations $u(x,z,t;y)$ at a given wall-normal location $y$ and time $t$ . If Taylor’s frozen-flow hypothesis (Taylor Reference Taylor1938) is satisfied, the streamwise velocity fluctuations $u$ are governed by

(2.1) \begin{equation} {\partial _t}u + {U_{\textit{cu}}}{\partial _x}u = 0, \end{equation}

where $U_{\textit{cu}}$ represents the convection velocity of $u$ . Convection velocity is employed to characterise the propagation of turbulent eddies and is a fundamental characteristic of space–time statistics. In reality, however, turbulent eddies undergo distortion while propagating downstream, deviating from frozen flow (Zhao & He Reference Zhao and He2009; Wilczek, Stevens & Meneveau Reference Wilczek, Stevens and Meneveau2015; Wu & He Reference Wu and He2023). Defining this distortion as $\sigma _u$ , the streamwise velocity fluctuations $u$ evolve according to

(2.2) \begin{equation} {\partial _t}u + {U_{\textit{cu}}}{\partial _x}u = {\sigma _u}, \end{equation}

where the convection velocity $U_{\textit{cu}}$ is determined by minimising the square of the distortion (Del Á lamo & Jiménez Reference Del Álamo and Jiménez2009; Wu & He Reference Wu and He2021a ), as formulated by

(2.3) \begin{equation} {U_{\textit{cu}}} \equiv \mathop {\arg }\limits _c \min \big\langle {({\partial _t}u + c{\partial _x}u)^2}\big\rangle = - {{\langle {\partial _t}u{\partial _x}u\rangle } \over {\big\langle {{({\partial _x}u)}^2}\big\rangle }}. \end{equation}

In the present work, $\left \langle {} \right \rangle$ denotes the ensemble average, which is performed in time and in the streamwise and spanwise directions due to statistically stationarity and homogeneity. Combining (2.2) and (2.3), we can obtain

(2.4) \begin{equation} \langle {\sigma _u}{\partial _x}u\rangle = 0. \end{equation}

Similarly, for the SRNS fluctuations $r \equiv {u^2} - \langle {u^2}\rangle$ , the convection velocity can be written as follows:

(2.5) \begin{equation} {U_{\textit{cr}}} \equiv \mathop {\arg }\limits _c \min \big\langle {({\partial _t}r + c{\partial _x}r)^2} \big\rangle = - {{\langle {\partial _t}r{\partial _x}r\rangle } \over {\big\langle {{({\partial _x}r)}^2}\big\rangle }} = - {{\big\langle {u^2}{\partial _t}u{\partial _x}u\big\rangle } \over {\big\langle {u^2}{{({\partial _x}u)}^2}\big\rangle }}. \end{equation}

The convection velocities $U_{\textit{cu}}$ and $U_{\textit{cr}}$ at all investigated locations are shown in figure 1(a), and the difference in convection velocity $U_{\textit{cr}}-U_{\textit{cu}}$ is shown in figure 1(b). In the near-wall region, $U_{\textit{cr}}$ exceeds $U_{\textit{cu}}$ , whereas within the logarithmic region and outer layer, $U_{\textit{cr}}$ is smaller than $U_{\textit{cu}}$ . These observations agree with the findings of Zhang et al. (Reference Zhang, Ning, Wu and He2025) obtained via wavenumber-dependent convection velocities. The difference in convection velocity is of the same order as the friction velocity. This non-negligible difference can provide some insights into the difference in the convection velocity between the rapid and slow components of the large-scale wall pressure fluctuations. The sources of large-scale wall pressure fluctuations are traced to the outer layer of the channel (Panton et al. Reference Panton, Lee and Moser2017; Anantharamu & Mahesh Reference Anantharamu and Mahesh2020), where the source term for the rapid component is the Reynolds stress fluctuations, while that for the slow component is the wall-normal velocity fluctuations. Moreover, the convection velocity of wall-normal velocity fluctuations is close to that of the streamwise velocity fluctuations investigated in this work (Kim & Hussain Reference Kim and Hussain1993; Geng et al. Reference Geng, He, Wang, Xu, Lozano-Durán and Wallace2015). Therefore, the $r{-}u$ convection velocity difference in the outer layer could potentially serve as an explanation for the observation that the convection velocity of the rapid components of the large-scale wall pressure fluctuations is larger than that of the slow components (Yang & Yang Reference Yang and Yang2022).

Zhang et al. (Reference Zhang, Ning, Wu and He2025) revealed that the observed difference in convection velocity stems from modulation effects imposed by streamwise velocity fluctuations on SRNS fluctuations, specifically altering their conditional convection velocity and amplitude. Next, we derive an exact expression for this convection velocity difference to explicitly demonstrate the modulation mechanism.

Figure 1.

(a) Convection velocities $U_{\textit{cu}}$ and $U_{\textit{cr}}$ at all investigated locations. (b) Convection velocity difference $U_{\textit{cr}}-U_{\textit{cu}}$ . The red solid line with squares represents the DNS results, the blue dashed line with deltas represents the results from (2.17) and the green dash-dotted line with gradient markers represents the results from (2.17), in which the fourth-order stress–velocity cross-spectra are obtained from the CS model. The results are normalised with wall units.

2.2. Exact expression for the $r{-}u$ convection velocity difference

Fiscaletti, de Kat & Ganapathisubramani (Reference Fiscaletti, de Kat and Ganapathisubramani2018) established distinct convection velocities for Reynolds shear stress fluctuations under positive and negative streamwise velocity fluctuation conditions, indicating that convection velocity associated with positive $u$ is larger than that associated with negative $u$ . Zhang et al. (Reference Zhang, Ning, Wu and He2025) demonstrated that streamwise velocity fluctuations $u$ exert similar convection velocity modulation effects on SRNS fluctuations $r$ . In this paper, to quantitatively investigate the modulation effect of $u$ on the convection dynamics of $r$ , we introduce the $u$ -conditioned convection velocity of $r$

(2.6) \begin{equation} \langle {U_{\textit{cr}}}|u\rangle \equiv \mathop {\arg }\limits _c \min \big\langle {({\partial _t}r + c{\partial _x}r)^2}|u \big\rangle = - {{\langle {\partial _t}r{\partial _x}r|u\rangle } \over {\big\langle {{({\partial _x}r)}^2}|u \big\rangle }}, \end{equation}

and we can then derive

(2.7) \begin{equation} \langle {U_{\textit{cr}}}|u\rangle = - {{\langle {\partial _t}u{\partial _x}u|u\rangle } \over {\big\langle {{({\partial _x}u)}^2}|u \big\rangle }} = {U_{\textit{cu}}} - {{\langle {\sigma _u}{\partial _x}u|u\rangle } \over {\big\langle {{({\partial _x}u)}^2}|u \big\rangle }}. \end{equation}

The $u$ -conditioned convection velocity of $r$ does not measure the overall convection velocity of $r$ , but rather the convection velocity of $r$ carried by specific events defined by the condition on $u$ . The difference between $U_{\textit{cr}}$ and $U_{\textit{cu}}$ is associated with the dependence of $\langle {U_{\textit{cr}}}|u\rangle$ on $u$ . More generally, for a variable $g(u)$ , where $g$ is a differentiable function of $u$ , the $u$ -conditioned convection velocity can be written as

(2.8) \begin{equation} \langle {U_{\textit{cg}}}|u\rangle = - {{\langle {\partial _t}g{\partial _x}g|u\rangle } \over {\big\langle {{({\partial _x}g)}^2}|u \big\rangle }} = - {{{{({\textrm {d}}g/{\textrm {d}}u)}^2}\langle {\partial _t}u{\partial _x}u|u\rangle } \over {{{({\textrm {d}}g/{\textrm {d}}u)}^2} \big\langle {{({\partial _x}u)}^2}|u \big\rangle }} = - {{\langle {\partial _t}u{\partial _x}u|u\rangle } \over {\big\langle {{({\partial _x}u)}^2}|u \big\rangle }}. \end{equation}

It can be seen that the $u$ -conditioned convection velocities are the same for different $g(u)$ . In contrast, the overall convection velocities of $g(u)$ may not be the same because they are influenced by ${\textrm {d}}g(u)/{\textrm {d}}u$ . Since conditioning on $u$ removes ${\textrm {d}}g(u)/{\textrm {d}}u$ , the conditional convection velocity can be regarded as a universal convection velocity, independent of the specific turbulent details inherent to $g(u)$ .

Numerically, $\langle {U_{\textit{cr}}}|u\rangle$ is calculated as follows:

(2.9) \begin{equation} \langle {U_{\textit{cr}}}|u\rangle = - {{\sum \limits _{u' \in {\textrm {bin}}(u)} {{\partial _t}u'{\partial _x}u'} } \over {\sum \limits _{u' \in {\textrm {bin}}(u)} {{{({\partial _x}u')}^2}} }}, \end{equation}

where ${\textrm {bin}}(u) = [u - 0.02\sqrt {\langle {u^2}\rangle } ,u + 0.02\sqrt {\langle {u^2}\rangle } ]$ . The DNS results of differences between $\langle {U_{\textit{cr}}}|u\rangle$ and $U_{\textit{cu}}$ at $y^+=5$ , $92$ and $550$ are shown in figure 2. As seen, $\langle {U_{\textit{cr}}}|u\rangle - {U_{\textit{cu}}}$ can be represented using linear approximation

(2.10) \begin{equation} \langle {U_{\textit{cr}}}|u\rangle - {U_{\textit{cu}}} = \gamma (u - b), \end{equation}

where $\gamma$ quantifies the intensity of the dependence of the conditional convection velocity $\langle {U_{\textit{cr}}}|u\rangle$ on streamwise velocity fluctuations $u$ . The parameter $b$ is related to the intercept of the linear approximation. As evidenced in figure 2, $\gamma \gt 0$ , indicating that the conditional convection velocity increases progressively with increasing streamwise velocity fluctuations $u$ .

Figure 2.

Difference between $\langle {U_{\textit{cr}}}|u\rangle$ and $U_{\textit{cu}}$ . Red solid lines represent the DNS results from (2.7), and blue dashed lines represent the linear approximation based on (2.10) with (2.14) and (2.15). Panels show (a) $y^+=5$ , (b) $y^+=92$ , (c) $y^+=550$ .

Substituting (2.7) into (2.10) yields

(2.11) \begin{equation} \langle {\sigma _u}{\partial _x}u|u\rangle = - \gamma (u - b) \big\langle {({\partial _x}u)^2}|u \big\rangle . \end{equation}

The quantity $\langle {\sigma _u}{\partial _x}u\rangle$ can be calculated using $\langle {\sigma _u}{\partial _x}u|u\rangle$ as follows:

(2.12) \begin{equation} \langle {\sigma _u}{\partial _x}u\rangle = \int {P(u)\langle {\sigma _u}{\partial _x}u|u\rangle {\textrm {d}}u} , \end{equation}

where $P(u)$ is the probability density function of $u$ . Substituting (2.11) into (2.12), we obtain

(2.13) \begin{equation} \langle {\sigma _u}{\partial _x}u\rangle = - \gamma \big\langle u{({\partial _x}u)^2}\big\rangle + \gamma b \big\langle {({\partial _x}u)^2}\big\rangle . \end{equation}

By combining (2.4) and (2.13), $b$ can be determined by

(2.14) \begin{equation} b = {{\big\langle u{{({\partial _x}u)}^2}\big\rangle } \over {\big\langle {{({\partial _x}u)}^2}\big\rangle }} = {1 \over 2}{{\langle {\partial _x}r{\partial _x}u\rangle } \over {\big\langle {{({\partial _x}u)}^2}\big\rangle }}. \end{equation}

Multiplying both sides of (2.11) by $u^2$ and taking the ensemble average with respect to $u$ yields $\gamma$

(2.15) \begin{equation} \gamma = - {{\big\langle {u^2}{\sigma _u}{\partial _x}u\big\rangle } \over {\big\langle {u^2}(u - b){{({\partial _x}u)}^2}\big\rangle }}. \end{equation}

The results of $\langle {U_{\textit{cr}}}|u\rangle - {U_{\textit{cu}}}$ obtained from the linear approximation (2.10) with (2.14) and (2.15) are shown in figure 2, and it can be seen that the linear approximation with appropriate parameters can correctly characterise the dependence of $\langle {U_{\textit{cr}}}|u\rangle$ on $u$ .

It is shown that the $r{-}u$ convection velocity difference is associated with the dependence of $\langle {U_{\textit{cr}}}|u\rangle$ on $u$ , and this dependence can be characterised by $\gamma$ . Next, we will derive an exact expression for ${U_{\textit{cr}}} - {U_{\textit{cu}}}$ that involves $\gamma$ . By substituting (2.2) into (2.5), we can obtain

(2.16) \begin{equation} {U_{\textit{cr}}} - {U_{\textit{cu}}} = - {{\big\langle {u^2}{\sigma _u}{\partial _x}u\big\rangle } \over {\big\langle {u^2}{{({\partial _x}u)}^2}\big\rangle }}. \end{equation}

Combining (2.14), (2.15) and (2.16), we obtain

(2.17) \begin{equation} {U_{\textit{cr}}} - {U_{\textit{cu}}} = \gamma \left ( {{{\langle {\partial _x}{r_{(4)}}{\partial _x}u\rangle } \over {\big\langle {{({\partial _x}r)}^2}\big\rangle }} - {1 \over 2}{{\langle {\partial _x}r{\partial _x}u\rangle } \over {\big\langle {{({\partial _x}u)}^2}\big\rangle }}} \right )\!. \end{equation}

Here, ${r_{(n)}} \equiv |u{|^n} - \langle |u{|^n}\rangle$ is referred to as the $n$ th-order stress. Specifically, when $n=2$ , $r_{(2)}$ is the SRNS fluctuation $r$ . Moreover, $\langle {\partial _x}{r_{(4)}}{\partial _x}u\rangle = \int \!\!\!\int {k_x^2{\mathop {\textrm{Re}}\nolimits } \langle {{\tilde r}_{(4)}}{{\tilde u}^ * }\rangle {\textrm {d}}{k_x}{\textrm {d}}\omega }$ is related to the real parts of the fourth-order stress–velocity cross-spectra $\langle {\tilde r_{(4)}}{\tilde u^ * }\rangle$ , and $\langle {\partial _x}r{\partial _x}u\rangle = \int \!\!\!\int {k_x^2{\mathop {\textrm{Re}}\nolimits } \langle \tilde r{{\tilde u}^ * }\rangle {\textrm {d}}{k_x}{\textrm {d}}\omega }$ is related to the real parts of the second-order stress–velocity cross-spectra $\langle \tilde r{\tilde u^ * }\rangle$ . In this paper, $k_x$ denotes the streamwise wavenumber, $\omega$ represents the frequency and $\boldsymbol{\tilde \cdot}$ denotes the space–time Fourier mode. For example, $\tilde u({k_x},\omega )$ refers to the space–time Fourier mode of streamwise velocity fluctuations. The spanwise direction is used for the ensemble average.

It is clear from (2.17) that the difference between $U_{\textit{cr}}$ and $U_{\textit{cu}}$ originates from the combined effects of the convection velocity modulation effect and stress–velocity cross-spectra. The $r{-}u$ convection velocity difference from (2.17) is shown in figure 1(b). The results obtained from the expression matched the DNS results. Consequently, the second- and fourth-order stress–velocity cross-spectra prove vital in accounting for the $r{-}u$ convection velocity difference.

3. Coherence similarity model for higher-order stress–velocity cross-spectra

Recent studies by Jacobi et al. (Reference Jacobi, Chung, Duvvuri and McKeon2021) and Cui & Jacobi (Reference Cui and Jacobi2023) demonstrated that second-order stress–velocity cross-spectra characterise the amplitude modulation of small-scale fluctuations by large-scale streamwise velocity fluctuations, where $u^2$ is used to represent the envelope of small-scale velocities. In this section, we establish connections between higher-order stress–velocity cross-spectra and their second-order counterparts.

According to the random sweeping hypothesis, the streamwise velocity fluctuation $u$ is independent of the small-scale velocity intensity $({\partial _x}u)^2$ (Praskovsky et al. Reference Praskovsky, Gledzer, Karyakin and Zhou1993; Katul et al. Reference Katul, Banerjee, Cava, Germano and Porporato2016; Huang & Katul Reference Huang and Katul2022), then

(3.1) \begin{equation} \langle {\partial _x}{r_{(n)}}{\partial _x}u\rangle = n \big\langle |u{|^{n - 2}}u \big\rangle \big\langle {({\partial _x}u)^2}\big\rangle . \end{equation}

Equation (3.1) can be rewritten as

(3.2) \begin{equation} \int \!\!\!\int {k_x^2{\mathop {\textrm{Re}}\nolimits } \langle {{\tilde r}_{(n)}}{{\tilde u}^ * }\rangle {\textrm {d}}{k_x}{\textrm {d}}\omega } = n \big\langle |u{|^{n - 2}}u \big\rangle \int \!\!\!\int {k_x^2 \big\langle |\tilde u{|^2}\big\rangle {\textrm {d}}{k_x}{\textrm {d}}\omega } . \end{equation}

Assuming that the integrands on both sides of (3.2) are equal, we have

(3.3) \begin{equation} {\mathop {\textrm{Re}}\nolimits } \langle {\tilde r_{(n)}}{\tilde u^ * }\rangle = n \big\langle |u{|^{n - 2}}u \big\rangle \big\langle |\tilde u{|^2}\big\rangle . \end{equation}

However, in wall turbulence, the velocity fluctuation $u$ and the small-scale velocity intensity $({\partial _x}u)^2$ are not statistically independent. This non-independence implies that the random sweeping hypothesis is invalid in our scenario. Therefore, a direct application of (3.3) would lead to substantial inaccuracies. For example, according to (3.3), we have ${\mathop {\textrm{Re}}\nolimits } \langle \tilde r{\tilde u^ * }\rangle = 0$ . In reality, ${\mathop {\textrm{Re}}\nolimits } \langle \tilde r{\tilde u^ * }\rangle$ is not only non-zero, but serves to characterise the amplitude modulation. To tackle this issue, we introduce a correction term on the right-hand side of (3.3) to account for the contribution arising from the statistical correlation between $u$ and $({\partial _x}u)^2$ , as follows:

(3.4) \begin{equation} {\mathop {\textrm{Re}}\nolimits } \langle {\tilde r_{(n)}}{\tilde u^ * }\rangle = n \big\langle |u{|^{n - 2}}u \big\rangle \big\langle |\tilde u{|^2}\big\rangle + {\mathop {\textrm{Re}}\nolimits } \big\langle \tilde r_{(n)}^c{\tilde u^ * }\big\rangle , \end{equation}

where $r_{(n)}^c$ represents the additional component in $r_{(n)}$ that is coherent with $u$ . When $n=2$ , we have ${\mathop {\textrm{Re}}\nolimits } \langle \tilde r{\tilde u^ * }\rangle = {\mathop {\textrm{Re}}\nolimits } \langle \tilde r_{(2)}^c{\tilde u^ * }\rangle$ .

Note that $\langle |u{|^n}\rangle$ varies at different wall-normal locations within the channel. We hypothesise that the turnover of turbulent structures transports $\langle |u{|^n}\rangle$ from other locations to a given position, resulting in the observed $r_{(n)}^c$ at that position. Let $\varDelta \langle |u{|^n}\rangle$ represent the characteristic difference in $\langle |u{|^n}\rangle$ between this position and other locations in the flow field such that the additional coherent stress $r_{(n)}^c$ is related to the characteristic stress difference $\varDelta \langle |u{|^n}\rangle$ . Therefore, we propose the following CS model for $r_{(n)}^c$ :

(3.5) \begin{equation} {{r_{(n)}^c} \over {\varDelta \langle |u{|^n}\rangle }} = {{r_{(2)}^c} \over {\varDelta \langle {u^2}\rangle }}, \end{equation}

which implies that the ratios of $r_{(n)}^c$ to $\varDelta \langle |u{|^n}\rangle$ at different orders are similar, all equalling the corresponding ratio at the second order. For a specific wall-normal location $y$ , the denominators $\varDelta \langle |u{|^n}\rangle$ and $\varDelta \langle {u^2}\rangle$ of (3.5) denote the spatial variations over some physical distances. These physical distances are intricately related to the size of the turbulent structures, which vary with $(x,z,t)$ . Therefore, $\varDelta \langle |u{|^n}\rangle$ and $\varDelta \langle {u^2}\rangle$ also change at various $(x,z,t)$ . In the CS model, however, it is the ratio $\varDelta \langle |u{|^n}\rangle /\varDelta \langle {u^2}\rangle$ that matters. Let $\langle |u{|^n}\rangle = {C_n}{\langle {u^2}\rangle ^{n/2}}$ ; then,

(3.6) \begin{equation} \varDelta \langle |u{|^n}\rangle \approx {n \over 2}{C_n}{\langle {u^2}\rangle ^{n/2 - 1}}\varDelta \langle {u^2}\rangle = {n \over 2}{{\langle |u{|^n}\rangle } \over {\langle {u^2}\rangle }}\varDelta \langle {u^2}\rangle . \end{equation}

Substituting (3.6) into (3.5) yields

(3.7) \begin{equation} r_{(n)}^c = {n \over 2}{{\langle |u{|^n}\rangle } \over {\langle {u^2}\rangle }}r_{(2)}^c. \end{equation}

Taking the space–time Fourier transform of (3.7), we have

(3.8) \begin{equation} \tilde r_{(n)}^c = {n \over 2}{{\langle |u{|^n}\rangle } \over {\langle {u^2}\rangle }}\tilde r_{(2)}^c. \end{equation}

It follows that

(3.9) \begin{equation} {\mathop {\textrm{Re}}\nolimits } \big\langle \tilde r_{(n)}^c{\tilde u^ * }\big\rangle = {n \over 2}{{\langle |u{|^n}\rangle } \over {\langle {u^2}\rangle }}{\mathop {\textrm{Re}}\nolimits } \big\langle \tilde r_{(2)}^c{\tilde u^ * }\big\rangle = {n \over 2}{{\langle |u{|^n}\rangle } \over {\langle {u^2}\rangle }}{\mathop {\textrm{Re}}\nolimits } \langle \tilde r{\tilde u^ * }\rangle . \end{equation}

From (3.4) and (3.9), we obtain the higher-order stress–velocity cross-spectra from the CS model

(3.10) \begin{equation} {\mathop {\textrm{Re}}\nolimits } \langle {\tilde r_{(n)}}{\tilde u^ * }\rangle = n \big\langle |u{|^{n - 2}}u\big\rangle \big\langle |\tilde u{|^2}\big\rangle + {n \over 2}{{\langle |u{|^n}\rangle } \over {\langle {u^2}\rangle }}{\mathop {\textrm{Re}}\nolimits } \langle \tilde r{\tilde u^ * }\rangle , \end{equation}

where the first term on the right-hand side is related to the space–time spectra $\langle |\tilde u{|^2}\rangle$ of $u$ , and the second term on the right-hand side is related to the second-order stress–velocity cross-spectra ${\mathop {\textrm{Re}}\nolimits } \langle \tilde r{\tilde u^ * }\rangle$ . If only the first term is present, the higher-order cross-spectra would exhibit similarity with the velocity spectra, whereas if only the second term exists, the higher-order cross-spectra would demonstrate similarity with the second-order cross-spectra. Although the second term is dominant, the first term is non-negligible in both the viscous sublayer and the outer layer, thereby giving rise to approximate similarity between the higher-order and second-order cross-spectra. Multiplying both sides of (3.10) by $k_x^2$ and integrating over $k_x$ and $\omega$ yields

(3.11) \begin{equation} \langle {\partial _x}{r_{(n)}}{\partial _x}u\rangle = \underbrace {n\big\langle |u{|^{n - 2}}u\big\rangle \big\langle {{({\partial _x}u)}^2}\big\rangle }_{{\textrm {Term}}\,{\textrm {I}}} + \underbrace {{n \over 2}{{\langle |u{|^n}\rangle } \over {\langle {u^2}\rangle }}\langle {\partial _x}r{\partial _x}u\rangle }_{{\textrm {Term}}\,{\textrm {II}}}, \end{equation}

where term I and term II correspond to the integrals of $\langle |\tilde u{|^2}\rangle$ and ${\mathop {\textrm{Re}}\nolimits } \langle \tilde r{\tilde u^ * }\rangle$ , respectively.

Figure 3.

The DNS results of $\langle {\partial _x}{r_{(n)}}{\partial _x}u\rangle /[n{\langle {u^2}\rangle ^{(n - 1)/2}}\langle {({\partial _x}u)^2}\rangle ]$ at all investigated locations, with (a) $n=2$ , (b) $n=3$ and (c) $n=4$ . In panels (b, c), the DNS results are compared with the results of term I, term II and the sum of the two terms in the CS model.

The results of $\langle {\partial _x}r{\partial _x}u\rangle$ at all investigated locations are shown in figure 3(a). It is observed that $\langle {\partial _x}r{\partial _x}u\rangle \gt 0$ in the near-wall region, whereas the logarithmic region and outer layer are $\langle {\partial _x}r{\partial _x}u\rangle \lt 0$ . In figures 3(b) and 3(c), comparisons of $\langle {\partial _x}{r_{(n)}}{\partial _x}u\rangle$ between the DNSs and the CS model at orders $n = 3$ and 4 demonstrate excellent agreement. Although not shown in this paper, the results of $\langle {\partial _x}{r_{(n)}}{\partial _x}u\rangle$ from the CS model at $n = 5$ and 6 also align with the DNS results. Crucially, term II plays a dominant role, whereas the contribution from term I is negligible in the buffer layer and logarithmic region. In the viscous sublayer and the outer layer, although term I cannot be ignored, its contribution is significantly smaller than that of term II. Therefore, without term II, the comparison of $\langle {\partial _x}{r_{(n)}}{\partial _x}u\rangle$ between the DNSs and the model exhibits a substantial discrepancy. By substituting $\langle {\partial _x}{r_{(4)}}{\partial _x}u\rangle$ from the CS model into (2.17), the model results of ${U_{\textit{cr}}} - {U_{\textit{cu}}}$ can be obtained. The model results in figure 1(b) reproduce the DNS profile. Substituting the dominant term (term II) into (2.17) leads to

(3.12) \begin{equation} {U_{\textit{cr}}} - {U_{\textit{cu}}} \approx \gamma {1 \over 2}\left ( {{{4\langle {u^4}\rangle \big\langle {{({\partial _x}u)}^2}\big\rangle } \over {\langle {u^2}\rangle \big\langle {{({\partial _x}r)}^2}\big\rangle }} - 1} \right ){{\langle {\partial _x}r{\partial _x}u\rangle } \over {\big\langle {{({\partial _x}u)}^2}\big\rangle }}. \end{equation}

Numerical calculations confirm that $4\langle {u^4}\rangle \langle {({\partial _x}u)^2}\rangle /(\langle {u^2}\rangle \langle {({\partial _x}r)^2}\rangle ) \gt 1$ at all wall-normal locations in the channel, establishing that the sign of ${U_{\textit{cr}}} - {U_{\textit{cu}}}$ is determined by the sign of $\langle {\partial _x}r{\partial _x}u\rangle$ . This explains the distinct behaviours observed: in the near-wall region, ${U_{\textit{cr}}} \gt {U_{\textit{cu}}}$ ; in the logarithmic region and outer layer, ${U_{\textit{cr}}} \lt {U_{\textit{cu}}}$ .

The contours of ${\mathop {\textrm{Re}}\nolimits } \langle {\tilde r_{(4)}}{\tilde u^ * }\rangle$ from the CS model and the DNSs are compared in figure 4. The model is in good agreement with the DNS. Additional validations at orders $n=3$ , 5 and 6 (not shown) confirm consistent accuracy. At $y^+=5$ , ${\mathop {\textrm{Re}}\nolimits } \langle {\tilde r_{(4)}}{\tilde u^ * }\rangle$ remains positive throughout the wavenumber–frequency domain, displaying a single-lobe structure. At other wall-normal locations, ${\mathop {\textrm{Re}}\nolimits } \langle {\tilde r_{(4)}}{\tilde u^ * }\rangle$ adopts a triple-lobe structure: the values in the first and third lobes are negative, whereas the value in the second lobe remains positive. The area of this second positive lobe systematically decreases with increasing wall distance. This finding is consistent with the observed sign transition of $\langle {\partial _x}{r_{(4)}}{\partial _x}u\rangle$ from positive to negative away from the wall. The result of the auto-spectra $\langle |\tilde u{|^2}\rangle$ of velocity fluctuations multiplied by $4\langle {u^3}\rangle$ is plotted in the middle column of figure 4. Note that the auto-spectra $\langle |\tilde u{|^2}\rangle$ always exhibit a single lobe, while the triple-lobe structure of ${\mathop {\textrm{Re}}\nolimits } \langle {{\tilde r}_{(4)}}{{\tilde u}^ * }\rangle$ implies that a complex space–time coupling occurs between streamwise velocity fluctuations $u$ and fourth-order stress $r_{(4)}$ . Since the dominant term II is related to ${\mathop {\textrm{Re}}\nolimits } \langle \tilde r{\tilde u^ * }\rangle$ , the features observed in ${\mathop {\textrm{Re}}\nolimits } \langle {\tilde r_{(4)}}{\tilde u^ * }\rangle$ fundamentally originate from ${\mathop {\textrm{Re}}\nolimits } \langle \tilde r{\tilde u^ * }\rangle$ , as confirmed by the similarity between the contours of ${\mathop {\textrm{Re}}\nolimits } \langle {\tilde r_{(4)}}{\tilde u^ * }\rangle$ and ${\mathop {\textrm{Re}}\nolimits } \langle \tilde r{\tilde u^ * }\rangle$ (see the right column of figure 4).

Figure 4.

Comparison of ${\mathop {\textrm{Re}}\nolimits } \langle {\tilde r_{(4)}}{\tilde u^ * }\rangle$ obtained from the DNS and the CS model at $y^+=5$ (a–c), $y^+=12$ (d–f), $y^+=44$ (g–i), $y^+=92$ (j–l), $y^+=270$ (m–o) and $y^+=550$ (p–r). Left: contours of ${\mathop {\textrm{Re}}\nolimits } \langle {\tilde r_{(4)}}{\tilde u^ * }\rangle$ obtained from the DNSs (coloured shades with solid lines) and the CS model (dashed lines). Middle: contours of $4\langle {u^3}\rangle \langle |\tilde u{|^2}\rangle$ . Right: contours of $2\langle {u^4}\rangle {\mathop {\textrm{Re}}\nolimits } \langle \tilde r{{\tilde u}^ * }\rangle /\langle {u^2}\rangle$ obtained from the DNSs (coloured shades with solid lines) and (3.21) (dashed lines).

Consider the governing equation for the streamwise velocity fluctuations

(3.13) \begin{equation} {\partial _t}u + U{\partial _x}u = - v{\partial _y}U - {\rho ^{ - 1}}{\partial _x}p + \nu {{\nabla} ^2}u - {\partial _x}r - {\partial _y}(uv - \langle uv\rangle ) - {\partial _z}(uw). \end{equation}

Here, $U$ is the mean velocity, $v$ and $w$ are the velocity fluctuations in the wall-normal and spanwise directions, respectively, $p$ is the pressure fluctuation and $\rho$ is the density. Also, $v{\partial _y}U$ is the mean shear term, ${\rho ^{ - 1}}{\partial _x}p$ is the pressure gradient term, $\nu {{\nabla} ^2}u$ is the viscous term and ${\partial _x}r$ , ${\partial _y}(uv - \langle uv\rangle )$ and ${\partial _z}(uw)$ correspond to different components of nonlinear terms. We can rewrite (3.13) as

(3.14) \begin{equation} {\partial _t}u + {U_{\textit{cu}}}{\partial _x}u = ({U_{\textit{cu}}} - U){\partial _x}u - {\partial _x}r + f, \end{equation}

where $f = - v{\partial _y}U - {\rho ^{ - 1}}{\partial _x}p + \nu {{\nabla} ^2}u - {\partial _y}(uv - \langle uv\rangle ) - {\partial _z}(uw)$ . By comparing (3.14) and (2.2), we obtain

(3.15) \begin{equation} {\sigma _u} = ({U_{\textit{cu}}} - U){\partial _x}u - {\partial _x}r + f. \end{equation}

Substituting (3.15) into (2.4) yields

(3.16) \begin{equation} {U_{\textit{cu}}} - U = {{\langle {\partial _x}r{\partial _x}u\rangle } \over {\big\langle {{({\partial _x}u)}^2}\big\rangle }} - {{\langle f{\partial _x}u\rangle } \over {\big\langle {{({\partial _x}u)}^2}\big\rangle }}. \end{equation}

If $f$ is neglected, (3.14) simplifies to only account for the advection by the mean velocity field and the nonlinear term involving $r$ , resulting in

(3.17) \begin{equation} {U_{\textit{cu}}} - U \approx {{\langle {\partial _x}r{\partial _x}u\rangle } \over {\big\langle {{({\partial _x}u)}^2}\big\rangle }}. \end{equation}

The sign of ${U_{\textit{cu}}} - U$ is the same as that of $\langle {\partial _x}r{\partial _x}u\rangle$ . This indicates that the second-order stress–velocity cross-spectrum, which characterises amplitude modulation, is associated with the difference between $U$ and $U_{\textit{cu}}$ . This finding aligns with the conclusion of Cui & Jacobi (Reference Cui and Jacobi2025), which reported that amplitude modulation behaviour is associated with the difference between the mean velocity and the convection velocity of streamwise velocity fluctuations. Both (3.12) and (3.17) demonstrate a close connection between amplitude modulation and some specific velocity differences in turbulence.

Taking the temporal Fourier transform of (3.14), we obtain

(3.18) \begin{equation} - {\textrm {i}}\big(\omega - {k_x}{U_{\textit{cu}}}\big)\tilde u = {\textrm {i}}{k_x}({U_{\textit{cu}}} - U)\tilde u - {\textrm {i}}{k_x}\tilde r + \tilde f. \end{equation}

Substituting (3.16) into (3.18), we obtain

(3.19) \begin{equation} - {\textrm {i}}\big(\omega - {k_x}{U_{\textit{cu}}}\big)\tilde u = {\textrm {i}}{k_x}\left ( {{{\langle {\partial _x}r{\partial _x}u\rangle } \over {\big\langle {{({\partial _x}u)}^2}\big\rangle }} - {{\langle f{\partial _x}u\rangle } \over {\big\langle {{({\partial _x}u)}^2}\big\rangle }}} \right )\tilde u - {\textrm {i}}{k_x}\tilde r + \tilde f. \end{equation}

Neglecting $f$ , we can derive

(3.20) \begin{equation} \tilde r \approx \left [ {{\omega \over {{k_x}}} - {U_{\textit{cu}}} + {{\langle {\partial _x}r{\partial _x}u\rangle } \over {\big\langle {{({\partial _x}u)}^2}\big\rangle }}} \right ]\tilde u \end{equation},

and

(3.21) \begin{equation} \langle \tilde r{\tilde u^ * }\rangle \approx \left [ {{\omega \over {{k_x}}} - {U_{\textit{cu}}} + {{\langle {\partial _x}r{\partial _x}u\rangle } \over {\big\langle {{({\partial _x}u)}^2}\big\rangle }}} \right ]\big\langle |\tilde u{|^2}\big\rangle . \end{equation}

The dashed lines in the right column of figure 4 plot the contours of $2\langle {u^4}\rangle {\textrm{Re}}\langle \tilde r{\tilde u^*}\rangle /\langle {u^2}\rangle$ , where ${\textrm{Re}}\langle \tilde r{\tilde u^*}\rangle$ is obtained from (3.21). It is observed that the results from (3.21) are close to the DNS results for large $k_x$ . However, (3.21) yields a dual-lobe structure: ${\textrm{Re}}\langle \tilde r{\tilde u^*}\rangle \gt 0$ in the wavenumber–frequency region $\omega \gt {k_x}({U_{\textit{cu}}} - \langle {\partial _x}r{\partial _x}u\rangle /\langle {({\partial _x}u)^2}\rangle )$ , while ${\textrm{Re}}\langle \tilde r{\tilde u^*}\rangle \lt 0$ in the wavenumber–frequency region $\omega \lt {k_x}({U_{\textit{cu}}} - \langle {\partial _x}r{\partial _x}u\rangle /\langle {({\partial _x}u)^2}\rangle )$ . Consequently, the transition from single-lobe to triple-lobe patterns should originate from the terms in $f$ that are neglected in the simplified equation.