2.1. Numerical methods of DNS

The governing equations for the incompressible turbulent channel flow include the continuity equation

(2.1) \begin{equation} \frac { \partial u_i}{ \partial x_i} = 0, \end{equation}

and the momentum equation

(2.2) \begin{equation} \frac { \partial u_i}{ \partial t} +\frac {\partial u_i u_{\!j}}{\partial x_{\!j}}= -\frac {1}{\rho }\frac { \partial p}{ \partial x_i} + \nu \frac {\partial ^2 u_i}{\partial x_{\!j} \partial x_{\!j}}. \end{equation}

In (2.1) and (2.2), $p$ denotes the pressure, $\rho$ is the density and $\nu$ is the kinematic viscosity. The velocity $u_i$ comprises three velocity components $u_1=u$ , $u_2=v$ and $u_3=w$ , corresponding to the velocity components in the streamwise ( $x$ ), vertical ( $y$ ) and spanwise ( $z$ ) directions, respectively.

Equations (2.1) and (2.2) are solved using a hybrid finite-differential/spectral method. The velocity is expanded in Fourier series in the periodic streamwise ( $x$ ) and spanwise ( $z$ ) directions as

(2.3) \begin{equation} u_i(x,y,z,t) = \sum _{k_x} \sum _{k_z} \widehat {u}_i(k_x,y,k_z,t) \exp {[i(k_x x + k_z z)]}, \end{equation}

where $k_x$ and $k_z$ denote the streamwise and spanwise wavenumbers, respectively, and $\widehat {u}_i$ represents the Fourier coefficient of $u_i$ . In the vertical ( $y$ ) direction, a staggered grid is employed. The streamwise velocity ( $u$ ), spanwise velocity ( $w$ ) and pressure ( $p$ ) are defined at cell centres, while the vertical velocity ( $v$ ) is located at cell faces. A sixth-order compact finite-differential scheme is used for spatial discretisations in the $y$ -direction. No-slip boundary conditions are imposed at the two walls located at $y = \pm h$ . Time integration is advanced using a third-order Runge–Kutta (RK3) scheme.

Within each RK3 substep, a fractional-step method is employed to enforce the divergence-free condition given by (2.1). In the spectral space, each substep of the RK3 scheme consists of the following three stages:

(2.4) \begin{equation} \big(\Delta t^{-1} - \beta _l L\big)\widehat {u}_i^* = \Delta t^{-1}\widehat {u}_i^{l-1} - \gamma _l \widehat {C}\big(u_i^{l-1}\big) - \eta _l \widehat {C}\big(u_i^{l-2}\big) + \beta _l L\big(\widehat {u}_i^{l-1}\big), \end{equation}

(2.5) \begin{equation} \Delta t \delta _l \frac {1}{\rho } \textit{DG} \widehat {p}^l = \frac {\partial \widehat {u}_i^*}{\partial x_i} \end{equation}

and

(2.6) \begin{equation} \widehat {u}_i^l = \widehat {u}_i^* - \Delta t \delta _l \frac {\partial \widehat {p}^l}{\rho \partial x_i}. \end{equation}

Here, $l = 1, 2$ and $3$ denotes the substep index of the RK3 scheme. The coefficients are given by

(2.7) \begin{align} & \beta _1 = \frac {4}{15},\quad \beta _2 = \frac {1}{15},\quad \beta _3 = \frac {1}{6}, \end{align}

(2.8) \begin{align} & \gamma _1 = \frac {8}{15},\quad \gamma _2 = \frac {5}{12},\quad \gamma _3 = \frac {3}{4}, \end{align}

(2.9) \begin{align} &\quad \,\, \eta _1 = 0,\quad \eta _2 = -\frac {17}{60},\quad \eta _3 = -\frac {5}{12}, \end{align}

(2.10) \begin{align} & \delta _1 = \frac {8}{15},\quad \delta _2 = \frac {2}{15},\quad \delta _3 = \frac {1}{3}. \end{align}

For $l=1$ and $3$ , the velocities $u_i^0$ and $u_i^3$ represent the flow fields at the $n{\mathrm{th}}$ and $(n+1){\mathrm{th}}$ time steps, respectively, separated by a time interval $\Delta t$ . Here, $D$ and $G$ represent the divergent and gradient operators, respectively. The term $L(\widehat {u}_i)$ denotes the Fourier coefficient of the viscous term $\partial ^2 u_i / (\partial x_{\!j} \partial x_{\!j})$ in (2.2). The term $\widehat {C}(u_i)$ represents the Fourier coefficient of the nonlinear convection term $\partial (u_i u_{\!j})/\partial x_{\!j}$ in (2.2), which is expressed as

(2.11) \begin{equation} \widehat {C}(u_i)(k_x,y,k_z,t) = i k_x \widehat {u_iu_{\!j}} \delta _{1j} + \frac {\partial \widehat {u_iu_{\!j}}}{\partial y} \delta _{2j} + i k_z \widehat {u_iu_{\!j}}\delta _{3j}. \end{equation}

For given wavenumbers $k_{x0}$ and $k_{z0}$ , the Fourier coefficient $\widehat {u_i u_{\!j}}$ is equal to

(2.12) \begin{equation} \widehat {u_i u_{\!j}}(k_{x0},y,k_{z0}) = \sum _{k_x, k_z} \widehat {u}_i (k_x,y, k_z) \widehat {u}_{\!j} (k_{x0}-k_x,y, k_{z0}-k_z), \end{equation}

which explicitly accounts for the triadic nonlinear interactions among turbulent motions across different length scales. The $3/2$ rule is employed to remove the aliasing errors in the nonlinear terms (Kravchenko & Moin Reference Kravchenko and Moin1997). Our numerical tool has been validated in our previous studies, such as opposition control of turbulent channel flows (Deng & Xu Reference Deng and Xu2012; Deng et al. Reference Deng, Huang and Xu2016), coherent structures in rotating channel flows (Yang & Wang Reference Yang and Wang2018; Yang et al. Reference Yang, Deng, Wang and Shen2020) and open channel flows driven by surface shear stresses (Deng et al. Reference Deng, Yang, Xuan and Shen2019, Reference Deng, Yang, Xuan and Shen2020; Deng et al. Reference Deng, Yang and Shen2022).

2.2. Set-up of numerical experiments

According to Hwang (Reference Hwang2015) and Marusic & Monty (Reference Marusic and Monty2019), energetic attached eddies are composed of self-similar long streaks and short vortical structures, with their vertical height growing with their length scales. Crucially, long streaks and short vortices sharing the same spanwise length scale form a self-sustaining unit (Hwang Reference Hwang2015; Cossu & Hwang Reference Cossu and Hwang2017), the smallest of which corresponds to the near-wall cycle. Therefore, to reveal the dependence of near-wall turbulent motions on other wider ones, we designed numerical experiments to suppress the spanwise-wide turbulent motions. Specifically, in fully developed turbulent channel flows, all three velocity components with their spanwise wavelength satisfying $\lambda _z = 2\pi / |k_z| \gt \lambda _{zf}$ are set to zero after a prescribed time $t_0$ . This high-pass spectral filter is defined as

(2.13) \begin{equation} \widehat {u}_i^l(k_x, y, k_z, t \geqslant t_0) = 0 \quad \text{for} \quad |k_z| \lt k_{zf} = \frac {2\pi }{\lambda _{zf}} \quad \text{and} \quad k_x^2 + k_z^2 \neq 0, \end{equation}

where $k_{zf}$ denotes the cutoff spanwise wavenumber corresponding to the cutoff spanwise wavelength $\lambda _{zf}$ . Here, the high-pass filter is not applied in the streamwise wavenumbers. This is because, among numerous established numerical experiments (Hamilton et al. Reference Hamilton, Kim and Waleffe1995; Jiménez & Pinelli Reference Jiménez and Pinelli1999; Hwang Reference Hwang2013; Bae et al. Reference Bae, Lozano-Durán and McKeon2021; Jiménez Reference Jiménez2022), those suppressing long structures (Hamilton et al. Reference Hamilton, Kim and Waleffe1995; Jiménez & Pinelli Reference Jiménez and Pinelli1999; Jiménez Reference Jiménez2022) were typically employed to investigate the dependence of short vortical structures on long streaks, a topic that lies beyond the scope of this study.

The filter defined in (2.13) is applied after the prediction step (2.4) and prior to the projection step (2.5) within each RK3 substep. This sequence ensures mass conservation. For the mean flow and turbulent motions at retained wavelengths, the momentum is strictly conserved, as they are obtained by solving the momentum equation in the spectral space. For turbulent motions suppressed at the targeted wavenumbers, i.e. $|k_z|\lt k_{zf}$ in this study, the momentum is removed at every step. Such methodology, which directly removes the momentum at specified wavenumbers, has been used in numerous established numerical experiments to investigate the sustaining mechanism of wall-bounded turbulence (Hamilton et al. Reference Hamilton, Kim and Waleffe1995; Jiménez & Pinelli Reference Jiménez and Pinelli1999; Hwang Reference Hwang2013; Bae et al. Reference Bae, Lozano-Durán and McKeon2021; Jiménez Reference Jiménez2022). For instance, in the numerical experiments of Hwang (Reference Hwang2013), the right-hand side of the prediction step (2.4) was set to zero for the specified wavenumbers to remove the momentum at these length scales, which yields an effect equivalent to our filter. We have successfully replicated the numerical experiments of Hwang (Reference Hwang2013) and Jiménez (Reference Jiménez2022) in Appendix A. The close agreement between our results and theirs validates the effectiveness and robustness of the current approach in suppressing turbulent motions within the targeted wavenumber range.

Using the spanwise filter in (2.13), the internal nonlinear interactions for turbulent motions with $\lambda _z \leqslant \lambda _{zf}$ are retained, while all contributions from wider turbulent motions with $\lambda _z \gt \lambda _{zf}$ through triadic nonlinear interactions (2.11)–(2.12), i.e. the ‘scattering’ mechanism (Tobias & Marston Reference Tobias and Marston2017; Hernández et al. Reference Hernández, Yang and Hwang2022b ), are eliminated. In particular, when $\lambda _{zf}^+ \gt 110$ , nonlinear interactions among canonical near-wall long streaks and short streamwise vortices in the self-sustaining cycle (Hamilton et al. Reference Hamilton, Kim and Waleffe1995; Waleffe Reference Waleffe1997; Jiménez & Pinelli Reference Jiménez and Pinelli1999; Panton Reference Panton2001; Schoppa & Hussain Reference Schoppa and Hussain2002) and the self-bursting process (Jiménez Reference Jiménez2022) are preserved. By changing the values of $\lambda _{zf}$ and monitoring the flow state, one can determine the threshold value of $\lambda _{zf}$ , i.e. the minimal value $\lambda _{zf}^{\textit{min}}$ to maintain near-wall turbulence, regardless of whether the maintenance of near-wall coherent structures is dominated by the canonical self-sustaining cycle or the recently proposed self-bursting process.

Since this study focuses on the near-wall region, our analyses are primarily based on numerical experiments conducted at a relatively low bulk Reynolds number $\textit{Re}_m = U_m h / \nu = 2800$ , where $U_m$ denotes the bulk mean velocity. In the undamped (UnD) case, i.e. without application of the spectral filter (2.13), the corresponding friction Reynolds number is $\textit{Re}_\tau = u_\tau h / \nu \approx 180$ . To achieve a precise estimation of the spanwise length scales critical for near-wall turbulence, full-scale DNS is performed in a computational domain size of $L_x \times L_y \times L_z = 4\pi h \times 2h \times 2\pi h$ . This domain size is significantly larger than a minimal flow unit (Jiménez & Moin Reference Jiménez and Moin1991; Jiménez & Pinelli Reference Jiménez and Pinelli1999), thereby enabling a denser distribution of the discrete spanwise wavelengths in the spectral space and allowing for finer resolution of the critical spanwise length scales. A grid of $192 \times 128 \times 192$ is employed, corresponding to the grid spacings of $\Delta x^+ \approx 11.8$ , $\Delta z^+ \approx 5.9$ and $\Delta y^+_{\textit{min}} \approx 0.13$ . Hereinafter, the superscript ‘+’ represents the normalisation using the friction velocity $u_\tau$ in the case UnD. In this study, several values of the cutoff spanwise wavelength $\lambda _{zf}$ are considered to determine $\lambda _{zf}^{\textit{min}}$ , as listed in table 1. Additional numerical experiments at a higher Reynolds number $\textit{Re}_m = 11\,000$ (corresponding to $\textit{Re}_\tau \approx 590$ ) are also conducted to discuss the Reynolds number dependence of the critical cutoff wavelength $\lambda _{zf}^{\textit{min}}$ for sustaining near-wall turbulence in § 6.

Table 1.

Parameters of numerical experiments at a fixed bulk Reynolds number $\textit{Re}_m = 2800$ . Here, $\lambda _{zf}$ denotes the cutoff spanwise wavelength in the high-pass spectral filter (2.13) and $\textit{Re}_\tau$ is the friction Reynolds number in different cases. The superscript ‘+’ denotes the normalisation using the kinematic viscosity $\nu$ and the friction velocity $u_\tau$ in the undamped (UnD) case.

Table 1. Long description

The table presents parameters of numerical experiments at a fixed bulk Reynolds number. It includes seven rows and eleven columns. The columns are labeled as follows: Case, Re_m, L_x x L_y x L_z, Delta x+, Delta y_min+, Delta z+, lambda_z_c+, lambda_z_c/h, Re_t, and Flow state. The rows are labeled with different cases: UnD, DZ01, DZ02, DZ03, DZ04, and DZ05. Each row provides specific values for each parameter. For example, Row 1: UnD, 2800, 4 pi h x 2 h x 2 pi h, 11.8, 0.13, 5.9, 565, 3.14, 178.9, Fully turbulent. Row 2: DZ01, 2800, 4 pi h _ 2 h _ 2 pi h, 11.8, 0.13, 5.9, 376, 2.09, 177.6, Turbulent. Row 3: DZ02, 2800, 4 pi h _ 2 h _ 2 pi h, 11.8, 0.13, 5.9, 282, 1.57, 176.2, Turbulent. Row 4: DZ03, 2800, 4 pi h x 2 h x 2 pi h, 11.8, 0.13, 5.9, 226, 1.26, 155.2, Transitional. Row 5: DZ04, 2800, 4 pi h x 2 h x 2 pi h, 11.8, 0.13, 5.9, 188, 1.05, 66.4, Relaminarising.

4.1. Mean velocity and Reynolds stresses

Figure 3.

Variations of the profiles of (a) the mean velocity $\langle u\rangle ^+$ and (b) the Reynolds shear stress $-\langle u' v' \rangle ^+$ with the spanwise cutoff wavelength $\lambda _{zf}^+$ at the bulk Reynolds number $\textit{Re}_m=2800$ . In the cases DZ01–DZ04, $\lambda _{zf}^+$ varies from 565 to 226. The results in a minimal channel with its spanwise domain size limited to $L_z^+=188$ are also shown (the case Mini). All profiles are normalised by the friction velocity $u_\tau$ in the case UnD.

Figure 3. Long description

Two line graphs depict variations of profiles of mean velocity and Reynolds shear stress with spanwise cutoff wavelength. Panel A: The line graph shows the mean velocity profiles. The x-axis is labeled y+ and ranges from 10^0 to 10^2. The y-axis is labeled <u>+ and ranges from 0 to 20. The graph includes multiple data series represented by different line styles and colors: solid black line for UnD, dashed magenta line for DZ01, dotted blue line for DZ02, dashed blue line for DZ03, dashed orange line for DZ04, and solid green line for Mini. Panel B: The line graph shows the Reynolds shear stress profiles. The x-axis is labeled y+ and ranges from 10^0 to 10^2. The y-axis is labeled −<u’v’>+ and ranges from 0 to 0.9. The graph includes the same data series as Panel A. The profiles are normalized by the friction velocity in the case UnD.

Figure 3 shows the variations of the profiles of the mean velocity $\langle u\rangle ^+$ and the Reynolds shear stress $-\langle u'v' \rangle ^+$ with the cutoff wavelength $\lambda _{zf}^+$ at the bulk Reynolds number $\textit{Re}_m = 2800$ . For consistent comparison, all statistics are normalised using the friction velocity $u_\tau$ obtained in the case UnD. As shown in figure 3(a), the mean velocity $\langle u\rangle ^+$ exhibits a slight increase in the region $y^+ \lt 80$ as $\lambda _{zf}^+$ is reduced (more spanwise length scales are suppressed) in cases DZ01–DZ03. In the case DZ04, $\langle u \rangle ^+$ is only enhanced for $20\lt y^+\lt 80$ . The reduction in $\langle u \rangle ^+$ below $y^+=20$ in the case DZ04 corresponds to the significant reduction in wall shear stress in figure 1. For $y^+ \gt 80$ , the mean velocity $\langle u \rangle ^+$ in the cases DZ01–DZ04 is lower than that in the case UnD, consistent with the imposed constraint of the fixed flow rate.

This behaviour can be linked to the variation of the Reynolds shear stress through the Reynolds-averaged momentum equation normalised by $u_{\tau }$ in the case UnD, which is written as

(4.1) \begin{equation} \frac {1}{\textit{Re}_{\tau 0}}\frac {\mathrm{d}\langle u\rangle ^+}{\mathrm{d}(y/h)} - \langle u'v' \rangle ^+ = \left (\frac {\textit{Re}_\tau }{\textit{Re}_{\tau 0}}\right )^2 \left (1 - \frac {y}{h}\right )\!. \end{equation}

Here, $\textit{Re}_\tau$ denotes the friction Reynolds number in each numerical experiment and $\textit{Re}_{\tau 0}$ that in the case UnD. According to table 1, the ratio $(\textit{Re}_\tau / \textit{Re}_{\tau 0})^2$ is slightly smaller than $1$ for the cases DZ02 and DZ03. Consequently, the slight increase in $\langle u\rangle ^+$ observed within $y^+ \lt 80$ for these two cases can be linked primarily to the slight reduction in the Reynolds shear stress $-\langle u'v' \rangle ^+$ , as evident in figure 3(b). For the case DZ04, $(\textit{Re}_\tau / \textit{Re}_{\tau 0})^2$ is approximately 0.7, and thus the Reynolds shear stress $-\langle u'v' \rangle ^+$ is significantly reduced. Given that the near-wall Reynolds shear stress is predominantly generated by coherent structures, the variation in $-\langle u'v' \rangle ^+$ suggests, at first glance, that the high-pass filter (2.13) only significantly alters near-wall coherent structures in the case DZ04.

Figure 4.

Variations of the profiles of the three components of the Reynolds normal stresses with the spanwise cutoff wavelength $\lambda _{zf}^+$ at the bulk Reynolds number $\textit{Re}_m=2800$ : (a) $\langle u^\prime u^\prime \rangle ^+$ ; (b) $\langle v^\prime v^\prime \rangle ^+$ and (c) $\langle w^\prime w^\prime \rangle ^+$ . In the cases DZ01–DZ04, $\lambda _{zf}^+$ varies from 565 to 226. The results in a minimal channel with its spanwise domain size limited to $L_z^+=188$ are also shown (the case Mini). All profiles are normalised by the friction velocity $u_\tau$ in the case UnD.

However, the three components of the Reynolds normal stresses in the cases DZ01–DZ04 all exhibit pronounced deviations from those in the case UnD in figure 4, even though the modifications of the mean velocity and Reynolds shear stress by the filter are relatively weak in cases DZ01–DZ03. As shown in figure 4(a), filtering out spanwise-wide motions leads to a moderate increase in the near-wall peak of the streamwise normal stress $\langle u'u' \rangle ^+$ in the cases DZ01–DZ03. Conversely, both the vertical ( $\langle v'v' \rangle ^+$ ) and spanwise ( $\langle w'w' \rangle ^+$ ) components are substantially reduced with decreasing $\lambda _{zf}^+$ (figure 4 b, c). This opposing behaviour suggests that the weak change in the Reynolds shear stress $-\langle u'v' \rangle ^+$ in the cases DZ01–DZ03 results from a compensating effect between the enhanced $u^\prime$ and the diminished $v^\prime$ . When the cutoff wavelength decreases further from $\lambda _{zf}^+=282$ (case DZ03) to $\lambda _{zf}^+=226$ (case DZ04), the peak values of $\langle u'u' \rangle ^+$ , $\langle v'v' \rangle ^+$ and $\langle w'w' \rangle ^+$ are all reduced compared with those in case DZ03.

It is well established that, in the near-wall region, $\langle u'u' \rangle ^+$ is predominantly associated with long streaks (Kim et al. Reference Kim, Moin and Moser1987; Jiménez Reference Jiménez1998; Panton Reference Panton2001; Jiménez et al. Reference Jiménez, Del Álamo and Flores2004; Marusic et al. Reference Marusic, Mathis and Hutchins2010; Hwang Reference Hwang2015; Mizuno Reference Mizuno2016; Cassinelli et al. Reference Cassinelli, de Giovanetti and Hwang2017; Jiménez Reference Jiménez2018), whereas $\langle v'v' \rangle ^+$ and $\langle w'w' \rangle ^+$ are primarily generated by relatively short streamwise vortical structures (Robinson Reference Robinson1991; Jeong et al. Reference Jeong, Hussain, Schoppa and Kim1997; Schoppa & Hussain Reference Schoppa and Hussain2002; Jiménez et al. Reference Jiménez, Del Álamo and Flores2004; Hwang Reference Hwang2015; Cassinelli et al. Reference Cassinelli, de Giovanetti and Hwang2017). Therefore, it can be deduced that the removal of spanwise-wide turbulent motions mildly amplifies near-wall long streaks while strongly attenuating short streamwise vortices. The contrasting response of long streaks and short vortical structures is verified in further detail in the next section.

4.2. Near-wall turbulent motions at different length scales

Figure 5.

Comparisons of the premultiplied streamwise and spanwise energy spectra of the streamwise velocity fluctuations: (a) $k_x E_{u'u'}^+(\lambda _x^+, y^+)$ with contours 0.05 (thick lines) and 1.2 (thin lines) and (b) $k_z E_{u'u'}^+(\lambda _z^+, y^+)$ with contours 0.35 (thick lines) and 4.5 (thin lines), among the cases UnD and DZ01–DZ04. Here: ——, black, UnD; - $\boldsymbol{\cdot}$ - $\boldsymbol{\cdot}$ -, red, DZ01; $\ldots \!..$ , blue, DZ02; - - - -, steel blue, DZ03; - $\boldsymbol{\cdot }\boldsymbol{\cdot}$ - $\boldsymbol{\cdot }\boldsymbol{\cdot}$ , orange, DZ04. The superscript ‘ $+$ ’ represents the normalisation using $u_\tau$ in the case UnD.

Figure 5. Long description

Two contour plots compare premultiplied streamwise and spanwise energy spectra of streamwise velocity fluctuations among different cases. Panel A: The contour plot shows the premultiplied streamwise energy spectra with contours at 0.05 (thick lines) and 1.2 (thin lines). The x-axis is labeled λx+ and ranges from 10^2 to 10^3. The y-axis is labeled y+ and ranges from 10^0 to 10^2. The legend indicates different cases: black solid line for UnD, red dashed line for DZ01, blue dashed line for DZ02, steeblue dashed-dotted line for DZ03, and orange dashed line for DZ04. Panel B: The contour plot shows the premultiplied spanwise energy spectra with contours at 0.35 (thick lines) and 4.5 (thin lines). The x-axis is labeled λz+ and ranges from 10^2 to 10^3. The y-axis is labeled y+ and ranges from 10^0 to 10^2. The legend is the same as in Panel A. The superscript represents normalization using the case UnD.

To elucidate the impacts of the absence of spanwise-wide turbulent motions on long streaks and short vortical structures in the near-wall region, we compare the energy spectra, Reynolds stresses associated with near-wall structures and instantaneous flow fields in the numerical experiments DZ01–DZ04 with those in the case UnD in this section.

Figure 5 shows the premultiplied streamwise and spanwise energy spectra of the streamwise velocity fluctuations, denoted respectively as $k_x E_{u'u'}^+$ and $k_z E_{u'u'}^+$ , for the cases UnD and DZ01–DZ04. In the case UnD, the spectra peak is located at approximately $\lambda _x^+ =1000$ and $\lambda _z^+ = 100$ , corresponding to the canonical streamwise and spanwise length scales of near-wall streaks (Jiménez Reference Jiménez1998; Jiménez et al. Reference Jiménez, Del Álamo and Flores2004; Mizuno Reference Mizuno2016; Cassinelli et al. Reference Cassinelli, de Giovanetti and Hwang2017; Lee & Moser Reference Lee and Moser2019). The peak at $\lambda _z^+ = 100$ remains clearly present, indicating that the average spanwise spacing of near-wall streaks is almost unaffected in the numerical experiments. Due to the application of the high-pass filter, energy accumulates near the cutoff spanwise wavelength in the premultiplied spanwise spectra $k_z E_{u'u'}^+$ . This phenomenon is analogous to the energy accumulation observed near the cutoff streamwise wavenumber in the numerical experiments of Jiménez (Reference Jiménez2022), where long streaks were damped. In contrast, the streamwise spectra $k_x E_{u'u'}^+$ are markedly altered in the absence of turbulent motions wider than $\lambda _{zf}^+$ . As $\lambda _{zf}^+$ decreases, the spectral peak shifts towards larger streamwise wavelengths. Notably, for $\lambda _{zf}^+ = 282$ and $226$ , the peak extends towards $\lambda _x^+ \to \infty$ , suggesting a strong amplification of long streaks (also shown later in figure 8). Consequently, compared with the case UnD, streamwise velocity fluctuations with $\lambda _x^+ \lt 1000$ are suppressed, while those with $\lambda _x^+ \gtrsim 1000$ are enhanced. This shift indicates that the observed increase in $\langle u'u' \rangle ^+$ in the filtered cases (figure 4 a) arises primarily from the amplification of long streaks ( $\lambda _x^+ \gtrsim 1000$ ). Moreover, the trend that the peak of $k_x E_{u'u'}^+$ shifts towards $\lambda _x^+ \to \infty$ as $ \lambda _{zf}^+$ decreases is analogous to the behaviour observed in the linear transient growth analyses, which exhibit the largest growth rates of streaks when $\lambda _x^+ \to \infty$ (Butler & Farrell Reference Butler and Farrell1993; del Álamo & Jiménez Reference del Álamo and Jiménez2006; Pujals et al. Reference Pujals, García-Villalba, Cossu and Depardon2009; Deng et al. Reference Deng, Xu, Huang and Cui2013; Deng et al. Reference Deng, Yang and Shen2025). This similarity suggests that turbulent structures wider than $\lambda _{zf}^+ = 200$ also play an active role in the nonlinear processes responsible for shaping the characteristic streamwise length scales of long streaks.

Figure 6.

Comparisons of the premultiplied streamwise energy spectra of (a) $k_x E_{v'v'}^+(\lambda _x^+, y^+)$ with contours 0.0065 (thick lines) and 0.065 (thin lines), (b) $k_x E_{w'w'}^+(\lambda _x^+, y^+)$ with contours 0.012 (thick lines) and 0.12 (thin lines) and (c) $k_x E_{-u'v'}^+(\lambda _x^+, y^+)$ with contours 0.0004 (thick lines) and 0.06 (thin lines), among the cases UnD and DZ01–DZ04. Here: ——, black, UnD; - $\boldsymbol{\cdot}$ - $\boldsymbol{\cdot}$ -, red, DZ01; $\ldots \!..$ , blue, DZ02; - - - -, steel blue, DZ03; - $\boldsymbol{\cdot }\boldsymbol{\cdot}$ - $\boldsymbol{\cdot }\boldsymbol{\cdot}$ , orange, DZ04. The superscript ‘ $+$ ’ represents the normalisation using $u_\tau$ in the case UnD.

Figure 7.

Variations of the profiles of the Reynolds stresses contributed by (a i–d i) long streaks $\langle u_i^{L} u_{\!j}^{L} \rangle ^+$ and (a ii–d ii) short vortical structures $\langle u_i^{S} u_{\!j}^{S} \rangle ^+$ with the cutoff wavelength $\lambda _{zf}^+$ in the cases DZ01–DZ04. All profiles are normalised by the friction velocity $u_\tau$ in the case UnD.

Figure 7. Long description

The image contains eight line graphs arranged in two rows and four columns. Each graph shows the variations of Reynolds stress profiles. The x-axis of each graph is labeled y+ on a logarithmic scale, and the y-axis represents different normalized Reynolds stress components. Panel (a1) to (d1) in the first row show the profiles for long streaks, while panel (a2) to (d2) in the second row show the profiles for short vortical structures. Each graph includes multiple lines representing different cases (UnD, DZ01, DZ02, DZ03, DZ04) with distinct styles (solid, dashed, dotted). The profiles are normalized by the friction velocity in the case UnD. The graphs illustrate how the Reynolds stress contributions vary with the cutoff wavelength for different cases.

Figure 8.

Instantaneous flow fields in the cases UnD and DZ01–DZ04. The isosurface shows the Q-criterion with $Q=2$ . The contours display the streamwise velocity fluctuations $u^\prime$ on the horizontal plane located at $y^+=15$ .

Similar to the streamwise velocity fluctuations, the peaks of spanwise energy spectra of the vertical ( $v'$ ) and spanwise ( $w'$ ) velocity fluctuations are weakly affected by the suppression of spanwise-wide turbulent motions (not shown here). Therefore, our analysis of these two components focuses primarily on their streamwise energy spectra. Figure 6 presents the premultiplied streamwise energy spectra of the vertical and spanwise velocity fluctuations as well as the co-spectra of $u^\prime v^\prime$ , i.e. $k_xE_{v'v'}^+(\lambda _x^+, y^+)$ , $k_x E_{w'w'}^+(\lambda _x^+, y^+)$ and $k_xE_{-u'v'}^+(\lambda _x^+, y^+)$ , respectively. In contrast to the pronounced shift observed in the peak of $k_x E_{u'u'}^+$ , the peaks of $k_x E_{v'v'}^+$ and $k_x E_{w'w'}^+$ remain localised at $\lambda _x^+ = 200$ – $400$ , consistent with the characteristic streamwise length scales of quasi-streamwise vortices (Jeong et al. Reference Jeong, Hussain, Schoppa and Kim1997; Schoppa & Hussain Reference Schoppa and Hussain2002; Jiménez et al. Reference Jiménez, Del Álamo and Flores2004; Cassinelli et al. Reference Cassinelli, de Giovanetti and Hwang2017), even when spanwise-wide motions are artificially suppressed in the cases DZ01–DZ04. Around these peaks, the intensities of both $k_x E_{v'v'}^+$ and $k_x E_{w'w'}^+$ gradually decrease as $\lambda _{zf}^+$ is reduced in the cases DZ01–DZ04. Therefore, the attenuation of $\langle v'v' \rangle ^+$ and $\langle w'w' \rangle ^+$ (figure 4 b, c) is possibly attributed to the attenuation of near-wall vortices (also shown in figures 7 and 8).

The enhancement of the long streamwise velocity fluctuations, coupled with the attenuation of the relatively short streamwise and vertical velocity fluctuations, also induces corresponding modifications in their streamwise cospectra. Figure 6(c) compares the premultiplied streamwise co-spectrum $k_x E_{-u'v'}^+(\lambda _x^+, y^+)$ among different numerical experiments. As shown, the peak of $k_x E_{-u'v'}^+$ shifts progressively towards $\lambda _x^+ \to \infty$ as $\lambda _{zf}^+$ is reduced, a trend analogous to that observed in $k_x E_{u'u'}^+$ (figure 5 a). As such, the enhancement of $E_{-u'v'}^+$ at the large streamwise wavelength compensates for its reduction in the relatively short streamwise wavelength, thereby yielding only a minor net change in $-\langle u'v' \rangle$ in the cases DZ01–DZ03 (figure 3 b). For the case DZ04, $E_{-u'v'}^+$ in the near-wall region is reduced at both short and long streamwise length scales (also shown later in figure 7 d i, d ii). Consequently, this leads to a notable decrease in the total Reynolds shear stress $-\langle u'v' \rangle$ for the case DZ04 (figure 3 b).

The analysis of the spectra presented here reveals that, in the absence of spanwise-wide turbulent motions, the long streamwise velocity fluctuations are amplified, whereas the three components of short velocity fluctuations are consistently attenuated. To quantify the opposite impacts of spanwise-wide turbulent motions on the near-wall long and short structures, we decompose the velocity fluctuations into the spanwise-wide part, the near-wall long part and the short part as follows:

(4.2) \begin{equation} u_i^\prime = u_i^{W}+u_i^{L} +u_i^{S}. \end{equation}

According to the threshold value of the spanwise cutoff wavelength required for maintaining near-wall turbulence (3.1), turbulent motions with $\lambda _z^+ \gt 200$ are considered as the spanwise-wide part $u_i^{W}$ . Among the left turbulent motions with $\lambda _z^+\leqslant 200$ , turbulent motions of $\lambda _x^+ \gtrsim 1000$ are denoted as $u_i^L$ and those of $\lambda _x^+\lt 1000$ as $u_i^S$ .

Figure 7 displays the variations of the profiles of the Reynolds stresses associated with the long and short turbulent motions with the cutoff wavelength $\lambda _{zf}^+$ . It can be observed that long turbulent motions are dominated by the streamwise velocity fluctuations, consistent with the features of long streaks (Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967; Smith & Metzler Reference Smith and Metzler1983; Kim et al. Reference Kim, Moin and Moser1987; Robinson Reference Robinson1991; Jeong et al. Reference Jeong, Hussain, Schoppa and Kim1997; Panton Reference Panton2001; Hwang & Cossu Reference Hwang and Cossu2010; Hwang Reference Hwang2015; Jiménez Reference Jiménez2018). The variations of $\langle u_i^{L} u_{\!j}^{L} \rangle ^+$ with $\lambda _{zf}^+$ indicate that long streaks are more intensified in the absence of spanwise-wide turbulent motions. In particular, $\langle u^{L} u^{L} \rangle ^+$ and $-\langle u^{L} v^{L} \rangle ^+$ increase with reducing $\lambda _{zf}^+$ for the cases DZ01–DZ03 (figure 7 a i, d i), and the magnitudes of $\langle v^{L} v^{L} \rangle ^+$ and $\langle w^{L} w^{L} \rangle ^+$ are gradually suppressed. For the short component, the three components of velocity fluctuations are all pronounced (figure 7 a ii, b ii, c ii), which agrees with the characteristics of short vortical structures (Kim et al. Reference Kim, Moin and Moser1987; Robinson Reference Robinson1991; Jeong et al. Reference Jeong, Hussain, Schoppa and Kim1997; Adrian Reference Adrian2007; Hwang Reference Hwang2015; Jiménez Reference Jiménez2018). The four components of the Reynolds stresses $\langle u_i^{S} u_{\!j}^{S} \rangle ^+$ all decrease as $\lambda _{zf}^+$ decreases, suggesting the suppression of short vortical structures. For the transitional case DZ04, accompanied by a marked suppression of short vortical structures as well as long vertical and spanwise velocity fluctuations, the near-wall peak of $\langle u^{L} u^{L} \rangle ^+$ ceases to increase (figure 7 a i).

The opposite change of long streaks and short vortical structures is also corroborated by the instantaneous flow fields displayed in figure 8. As shown, streamwise vortices are markedly weakened as $\lambda _{zf}^+$ decreases, while the near-wall low- and high-speed streaks become more coherent, elongated and intensified.

In summary, the absence of spanwise-wide turbulent motions leads to the amplification of near-wall long streaks and the suppression of short vortical structures. According to the near-wall self-sustaining cycle (Hamilton et al. Reference Hamilton, Kim and Waleffe1995; Waleffe Reference Waleffe1997; Jiménez & Pinelli Reference Jiménez and Pinelli1999; Panton Reference Panton2001; Schoppa & Hussain Reference Schoppa and Hussain2002; Jiménez Reference Jiménez2018; Lozano-Durán et al. Reference Lozano-Durán, Constantinou, Nikolaidis and Karp2021), the intensities of these long streaks and short vortices are mutually dependent. The contrasting responses observed in the cases DZ01–DZ03 from full-scale DNS suggest that spanwise-wide turbulent motions may either facilitate the nonlinear energy transfer from long streaks to short vortical structures within the self-sustaining cycle, or alternatively exert opposing influences on other energy sources feeding these two types of near-wall coherent structures. Furthermore, although the intensities of near-wall long streaks increase monotonically with decreasing $\lambda _{zf}^+$ in the cases DZ01–DZ03, this enhancement ceases when short vortices are strongly suppressed in the case DZ04 and the flow ultimately relaminarises when $\lambda _{zf}^+ =188$ in the case DZ05. This implies that spanwise-wide turbulent motions are essential for sustaining near-wall turbulence, primarily by maintaining short vortical structures, thereby underscoring the central role of short vortical structures in near-wall dynamics, as emphasised in both the classical self-sustaining cycle (Hamilton et al. Reference Hamilton, Kim and Waleffe1995; Waleffe Reference Waleffe1997; Jiménez & Pinelli Reference Jiménez and Pinelli1999; Panton Reference Panton2001; Schoppa & Hussain Reference Schoppa and Hussain2002; Jiménez Reference Jiménez2018; Lozano-Durán et al. Reference Lozano-Durán, Constantinou, Nikolaidis and Karp2021) and the recently proposed self-bursting process (Jiménez Reference Jiménez2022). The dynamical mechanisms underlying the opposite variation trends in long streaks and short vortical structures with the cutoff wavelength, as well as the eventual relaminarisation, are analysed in detail in § 5.

5.1. Budgets of $\langle u_i^Lu_{\!j}^L\rangle$ and $\langle u_i^Su_{\!j}^S\rangle$

Applying the spectral filter based on the length scales in the definition of $u_i^L$ to the governing equation of the velocity fluctuations $u_i'$ , we obtain the governing equation of $u_i^L$ as

(5.1) \begin{equation} \frac {\partial u_i^L}{\partial t}+ \langle u_k \rangle \frac {\partial u_i^L}{\partial x_k}+ u_k^L\frac {\partial \langle u_i\rangle }{\partial x_k} + \frac {\partial (u_i' u_k')^L}{\partial x_k}=-\frac {1}{\rho }\frac {\partial p^L}{\partial x_i} + \nu \frac {\partial ^2 u_i^L}{\partial x_k x_k}. \end{equation}

Similarly, the governing equation for $u_i^S$ is given by

(5.2) \begin{equation} \frac {\partial u_i^S}{\partial t}+ \langle u_k \rangle \frac {\partial u_i^S}{\partial x_k}+ u_k^S\frac {\partial \langle u_i\rangle }{\partial x_k} + \frac {\partial (u_i' u_k')^S}{\partial x_k}=-\frac {1}{\rho }\frac {\partial p^S}{\partial x_i} + \nu \frac {\partial ^2 u_i^S}{\partial x_k x_k}. \end{equation}

The term $(u_i' u_k')^L$ in (5.1) represents the long part of $u_i' u_k'$ , while its counterpart $(u_i' u_k')^S$ in (5.2) is the short part of $u_i' u_k'$ . Based on the velocity decomposition in (4.2), these two nonlinear terms can be decomposed into

(5.3) \begin{equation} (u_i' u_k')^L = \underbrace {\big(u_i^Su_k^S\big)^L}_{{\mathcal{S}_i^L}}+ \underbrace {\big(u_i^Lu_k^L\big)^L}_{{\mathcal{L}_i^L}}+\underbrace {\big(u_i^Su_k^L+u_i^Lu_k^S\big)^L}_{{\mathcal{I}_i^L}}+\underbrace {\big[u_i^Wu_k^W+u_i^Wu_k^S+u_i^Su_k^W+u_i^Wu_k^L+u_i^Lu_k^W\big]^L}_{{\mathcal{W}_i^L}}, \end{equation}

and

(5.4) \begin{equation} (u_i' u_k')^S = \underbrace {\big(u_i^Su_k^S\big)^S}_{{\mathcal{S}_i^S}}+ \underbrace {\big(u_i^Lu_k^L\big)^S}_{{\mathcal{L}_i^S}}+\underbrace {\big(u_i^Su_k^L+u_i^Lu_k^S\big)^S}_{{\mathcal{I}_i^S}}+\underbrace {\big[u_i^Wu_k^W+u_i^Wu_k^S+u_i^Su_k^W+u_i^Wu_k^L+u_i^Lu_k^W\big]^S}_{{\mathcal{W}_i^S}}, \end{equation}

respectively. Regarding the notation, the superscript ‘L’ (or ‘S’) and subscript ‘i’ indicate that the terms $\mathcal{S}_i^L, \mathcal{L}_i^L,\mathcal{I}_i^L, \mathcal{W}_i^L$ (and their counterparts in (5.4)) appear in the transport equation for $u_i^{L}$ (or $u_i^S$ ). Physically, the term $\mathcal{S}$ represents the self-interaction of near-wall short structures, $\mathcal{L}$ denotes the self-interaction of near-wall long structures, $\mathcal{I}$ corresponds to the nonlinear interaction between near-wall long and short structures, and $\mathcal{W}$ represents the nonlinear interaction induced by spanwise-wide turbulent motions. Their physical meanings are listed in table 2.

Table 2.

Physical meanings of the nonlinear transport terms in (5.3), (5.4), (5.5) and (5.16).

Figure 1. Long description

Two line graphs depict the time histories of wall shear stress and turbulent kinetic energy in numerical experiments of turbulent channel flows. Panel A: The line graph shows the horizontally averaged wall shear stress. The x-axis represents the dimensionless time ((t − t0)Um/h) ranging from 0 to 300, and the y-axis represents the normalized wall shear stress (τw/ρUm^2) ranging from 0 to 5 times 10^−3. Different lines represent various cases with different damping parameters. Panel B: The line graph shows the bulk-averaged turbulent kinetic energy. The x-axis represents the dimensionless time ((t − t0)Um/h) ranging from 0 to 300, and the y-axis represents the normalized turbulent kinetic energy (E+) ranging from 0 to 3. Different lines represent various cases with different damping parameters. The legend indicates the specific parameters for each case.

The transport equation of the Reynolds stresses associated with long streaks, $\langle u_i^{L} u_{\!j}^{L} \rangle$ , is then derived by multiplying the equation for $u_i^L$ (5.1) by $u_{\!j}^L$ , adding it to the corresponding product of $u_i^L$ with the equation for $u_{\!j}^L$ , and averaging the result over the time and the horizontal plane, which yields

(5.5) \begin{equation} \frac { \partial \big\langle u_i^Lu_{\!j}^L \big\rangle }{ \partial t} = P_{ij}^{L} + \varGamma _{ij}^{L} + G_{ij}^{L} + \varepsilon _{ij}^{L} + D_{ij}^{L} + \underbrace {\textit{TS}_{ij}^{L}+ \textit{TL}_{ij}^{L}+ \textit{TI}_{ij}^{L}+\textit{TW}_{ij}^{L}}_{T_{ij}^L}. \end{equation}

Similar to the transport equation of the total Reynolds shear stress (Domaradzki et al. Reference Domaradzki, Liu, Härtel and Kleiser1994; Mizuno Reference Mizuno2016; Cho et al. Reference Cho, Hwang and Choi2018; Lee & Moser Reference Lee and Moser2019; Yang et al. Reference Yang, Deng, Wang and Shen2020), the terms on the right-hand side of (5.5) are the production term $P_{ij}^{L}$ , the pressure-redistribution term $\varGamma _{ij}^{L}$ , the pressure-diffusion term $G_{ij}^{L}$ , the dissipation term $ \varepsilon _{ij}^{L}$ , the viscous diffusion term $D_{ij}^{L}$ , the nonlinear turbulent transport term $\textit{TS}_{ij}^{L}$ associated with the self-interaction term of near-wall short structures $\mathcal{S}_i^L$ in (5.3), the nonlinear turbulent transport term $\textit{TL}_{ij}^{L}$ associated with the self-interaction term of near-wall long streaks $\mathcal{L}_i^L$ in (5.3), the nonlinear turbulent transport term $\textit{TI}_{ij}^{L}$ associated with the interaction term between near-wall long and short motions $\mathcal{I}_{i}^L$ in (5.3), and the nonlinear turbulent transport term $\textit{TW}_{ij}^{L}$ associated with the interaction term involving spanwise-wide turbulent motions $\mathcal{W}_i^L$ in (5.3), which are defined as

(5.6) \begin{equation} P_{ij}^{L} = - \big\langle u_{\!j}^{L} u_k^{L} \big\rangle \frac { \partial \langle u_i \rangle }{ \partial x_k} - \langle u_i^{L} u_k^{L} \rangle \frac { \partial \langle u_{\!j} \rangle }{ \partial x_k}, \end{equation}

(5.7) \begin{equation} \varGamma _{ij}^{L} = \frac { 1}{ \rho } \left \langle p^{L} \frac { \partial u_{\!j}^{L} }{ \partial x_i} \right \rangle + \frac { 1}{ \rho } \left \langle p^{L} \frac { \partial u_i^{L} }{ \partial x_{\!j}} \right \rangle \!, \end{equation}

(5.8) \begin{equation} G_{ij}^{L} = - \frac { 1}{ \rho } \left \langle \frac { \partial p^{L}u_{\!j}^{L} }{ \partial x_i}\right \rangle -\frac { 1}{ \rho } \left \langle \frac { \partial p^{L}u_i^{L} }{ \partial x_{\!j}} \right \rangle \!, \end{equation}

(5.9) \begin{equation} \varepsilon _{ij}^{L} = - 2 \nu \left \langle \frac { \partial u_i^{L}}{ \partial x_k} \frac { \partial u_{\!j}^{L}}{ \partial x_k} \right \rangle \!, \end{equation}

(5.10) \begin{equation} D_{ij}^{L} = \nu \frac { \partial ^2 \big\langle u_i^{L}u_{\!j}^{L} \big\rangle }{ \partial x_k \partial x_k}, \end{equation}

(5.11) \begin{equation} \textit{TS}_{ij}^{L} = - \left \langle u_{\!j}^{L} \frac { \partial \mathcal{S}_i^{L} }{ \partial x_k} \right \rangle - \left \langle u_i^{L} \frac { \partial \mathcal{S}_{\!j}^{L} }{ \partial x_k} \right \rangle \!, \end{equation}

(5.12) \begin{equation} \textit{TL}_{ij}^{L} = - \left \langle u_{\!j}^{L} \frac { \partial \mathcal{L}_i^{L} }{ \partial x_k} \right \rangle - \left \langle u_i^{L} \frac { \partial \mathcal{L}_{\!j}^{L} }{ \partial x_k} \right \rangle \!, \end{equation}

(5.13) \begin{equation} \textit{TI}_{ij}^{L} = - \left \langle u_{\!j}^{L} \frac { \partial \mathcal{I}_i^{L} }{ \partial x_k} \right \rangle - \left \langle u_i^{L} \frac { \partial \mathcal{I}_{\!j}^{L} }{ \partial x_k} \right \rangle \end{equation}

and

(5.14) \begin{equation} \textit{TW}_{ij}^{L} = - \left \langle u_{\!j}^{L} \frac { \partial \mathcal{W}_i^{L} }{ \partial x_k} \right \rangle - \left \langle u_i^{L} \frac { \partial \mathcal{W}_{\!j}^{L} }{ \partial x_k} \right \rangle , \end{equation}

respectively. The sum of $\textit{TS}_{ij}^{L}$ , $\textit{TL}_{ij}^{L}$ , $\textit{TI}_{ij}^{L}$ and $\textit{TW}_{ij}^{L}$ is the total turbulent transport term, i.e.

(5.15) \begin{equation} T_{ij}^{L} =\textit{TS}_{ij}^L+\textit{TL}_{ij}^L+\textit{TI}_{ij}^L+\textit{TW}_{ij}^L= - \left \langle u_{\!j}^{L} \frac { \partial ( u_i^\prime u_k^\prime )^{L} }{ \partial x_k} \right \rangle - \left \langle u_i^{L} \frac { \partial ( u_{\!j}^\prime u_k^\prime )^{L} }{ \partial x_k} \right \rangle \!. \end{equation}

It should be noted that these turbulent transport terms contain both the energy transfer among different length scales in the $x$ - and $z$ -directions, and the energy transport in the $y$ -direction.

The governing equation for the Reynolds stresses associated with short vortical structures, $\langle u_i^S u_{\!j}^S \rangle$ , follows a pattern similar to that of $\langle u_i^L u_{\!j}^L \rangle$ (5.5) as

(5.16) \begin{equation} \frac { \partial \big\langle u_i^Su_{\!j}^S \big\rangle }{ \partial t} = P_{ij}^{S} + \varGamma _{ij}^{S} + G_{ij}^{S} + \varepsilon _{ij}^{S} + D_{ij}^{S} + \underbrace {\textit{TS}_{ij}^{S}+ \textit{TL}_{ij}^{S}+ \textit{TI}_{ij}^{S}+\textit{TW}_{ij}^{S}}_{T_{ij}^S}. \end{equation}

These terms on the right-hand side of (5.16) can be expressed by simply replacing the superscript ‘L’ with ‘S’ in their counterparts in (5.6)–(5.15). The physical meanings of these nonlinear transport terms are listed in table 2.

Figure 9.

Profiles of the budget terms of $\langle u_i^Lu_{\!j}^L \rangle$ in the case UnD. These terms are normalised by $u_\tau ^4/\nu$ .

Figure 9. Long description

The image contains four line graphs labeled Panel A, Panel B, Panel C, and Panel D, each depicting different budget terms of turbulent motions in the near-wall region. Panel A shows the budget of uL+uL, with the y-axis labeled Budget of uL+uL and the x-axis labeled y+. Various lines represent different budget terms, including PL+ij, GL+ij, and others, each with distinct markers and colors. Panel B shows the budget of vL+vL, with the y-axis labeled Budget of vL+vL and the x-axis labeled y+. Similarly, different lines represent various budget terms. Panel C shows the budget of wL+wL, with the y-axis labeled Budget of wL+wL and the x-axis labeled y+. Panel D shows the budget of −uL+vL, with the y-axis labeled Budget of −uL+vL and the x-axis labeled y+. Each panel displays multiple lines with different markers and colors, indicating different budget terms and their variations across the y+ axis.

Figure 9 presents the budget terms of the Reynolds stresses $\langle u_i^L u_{\!j}^L \rangle$ for the case UnD at $\textit{Re}_m = 2800$ . As shown in figure 9(a), $\langle u^L u^L \rangle$ in the buffer layer gains energy predominantly through the production term $P_{11}^L = -2 \langle u^L v^L \rangle \, \partial \langle u \rangle / \partial y$ . The dominant sinks for $\langle u^L u^L \rangle$ in the near-wall region are the turbulent transport terms $\textit{TS}_{11}^L$ and $\textit{TW}_{11}^L$ , whereas the pressure-redistribution term $\varGamma _{11}^L$ plays an ignorable role in the overall balance. In contrast to the strong production of $\langle u^L u^L \rangle$ , the energy sources for $\langle v^L v^L \rangle$ and $\langle w^L w^L \rangle$ are considerably weaker, consistent with the relatively small amplitudes of $v^L$ and $w^L$ compared with the dominant streamwise velocity fluctuations $u^L$ (figure 7). As illustrated in figure 9(c), the long spanwise velocity fluctuations $w^L$ draw energy primarily from the nonlinear turbulent transport terms $\textit{TS}_{33}^L$ and $\textit{TW}_{33}^L$ , which are balanced by the viscous dissipation term and the negative pressure-redistribution term. For $\langle v^L v^L \rangle$ , the pressure-redistribution term $\varGamma _{22}^L$ acts as the main energy source in near-wall region (figure 9 b), which is largely balanced by the viscous dissipation term. Given the comparable magnitudes of the negative $\varGamma _{33}^L$ and the positive term $\varGamma _{22}^L$ , $\langle v^L v^L \rangle$ gains energy primarily from $\langle w^L w^L \rangle$ via the pressure-redistribution term. For the Reynolds shear stress $-\langle u^L v^L \rangle$ , the primary source is the production term $P_{12}^L=\langle v^L v^L\rangle \, \partial \langle u \rangle / \partial y$ (figure 9 d), which is primarily balanced by the turbulent transport term $\textit{TW}_{12}^L$ .

The budgets of $\langle u_i^L u_{\!j}^L\rangle$ can reveal the energy transfer process of near-wall long structures. The long spanwise velocity fluctuation $w^L$ extracts energy through the turbulent transport term associated with the self-interaction of short vortical structures, $\textit{TS}_{33}^L$ , and the term associated with spanwise-wide turbulent motions, $\textit{TW}_{33}^L$ . It should be noted that $\textit{TW}_{33}^L$ is mainly contributed by nonlinear interactions between spanwise-wide turbulent motions and near-wall short structures (see Appendix C). The energy of $w^L$ is subsequently transferred to the long vertical velocity $v^L$ through the pressure-redistribution term $\varGamma _{22}^L$ , which eventually contributes to the production of $u^L$ . The nonlinear transport terms $\textit{TW}_{ij}^L$ associated with spanwise-wide turbulent motions have dual roles, working as energy sinks for $\langle u^Lu^L\rangle$ and $-\langle u^Lv^L\rangle$ but as an energy source for $\langle w^Lw^L\rangle$ . The energy transfer process of near-wall long structures indicates that they are highly related to short vortical structures and spanwise-wide turbulent motions through the turbulent transport terms $\textit{TS}_{33}^L$ and $\textit{TW}_{33}^L$ (which is displayed in the blue box in figure 11).

Figure 10.

Profiles of the budget terms of $\langle u_i^S u_{\!j}^S\rangle$ in the case UnD. These terms are normalised by $u_\tau ^4/\nu$ .

Regarding short vortical structures, figure 10 shows the profiles of the budget terms of $\langle u_i^S u_{\!j}^S \rangle$ in the case UnD. As shown in figure 10(a), in the buffer layer, $\langle u^Su^S\rangle$ mainly obtains energy from the mean shear through the production term $P_{11}^{S}$ associated with the Reynolds shear stress $-\langle u^S v^S \rangle$ , which is mainly balanced by the viscous dissipation $\varepsilon _{11}^S$ and secondarily by the negative pressure-redistribution term $\varGamma _{11}^S$ . Corresponding to the negative $\varGamma _{11}^S$ , part of $\langle u^Su^S\rangle$ is redistributed to $\langle v^S v^S \rangle$ and $\langle w^S w^S\rangle$ via the positive $\varGamma _{22}^S$ and $\varGamma _{33}^S$ (figure 10 b, c), respectively, and then dissipated by the viscosity. In addition to the pressure-redistribution term, $\langle v^Sv^S\rangle$ also obtains considerable energy from the turbulent transport term associated with spanwise-wide turbulent motions $\textit{TW}_{22}^S$ . For the Reynolds shear stress contributed by short vortical structures $-\langle u^Sv^S\rangle$ , it is produced by $P_{12}^S$ involved in the mean shear and $\langle v^S v^S \rangle$ (figure 10 d).

From the energy budgets, we can find the energy transfer process of short vortical structures. The energy of $u^S$ is drawn from the mean velocity by $v^S$ through the production of $-\langle u^Sv^S\rangle$ . In turn, $v^S$ and $w^S$ receive energy from $u^S$ via the pressure-redistribution terms $\varGamma _{22}^S$ and $\varGamma _{33}^S$ . In addition to $\varGamma _{22}^S$ , $v^S$ also obtains energy from the turbulent transport term induced by spanwise-wide motions, $\textit{TW}_{22}^S$ .

Figure 11.

Sketch of the energy transfer processes of $\langle u_i^Su_{\!j}^S \rangle$ and $\langle u_i^Lu_{\!j}^L \rangle$ . Dashed lines refer to the energy transfer between long streaks and short vortical structures via the nonlinear interaction terms $\textit{TI}_{ij}^{S}$ and $\textit{TS}_{ij}^{L}$ . Here, the term $\textit{TW}_{33}^L$ is mainly contributed by nonlinear interactions between spanwise-wide turbulent motions and short vortical structures (see Appendix C).

Figure 11. Long description

A diagram of the energy transfer processes between long streaks and short vortical structures in near-wall turbulence. The diagram is divided into two main sections: long streaks on the left and short vortical structures on the right. The long streaks section includes components such as the Reynolds shear stress, the production term, and the nonlinear transport terms. The short vortical structures section includes components such as the turbulent kinetic energy, the production term, and the nonlinear transport terms. Arrows indicate the direction of energy transfer between these components. Dashed lines represent the energy transfer between long streaks and short vortical structures via the nonlinear interaction terms. The term is mainly contributed by nonlinear interactions between spanwise-wide turbulent motions and short vortical structures.

The energy cycles of near-wall turbulent motions described earlier are summarised in figure 11. It can be found that the energy cycle of short vortical structures is maintained via the energy drawn from the mean shear by $v^S$ , similar to those of the total Reynolds stresses (Kim et al. Reference Kim, Moin and Moser1987). However, the energy cycle of long structures is driven by $w^L$ that obtains energy from short vortical structures and spanwise-wide turbulent motions, and then transfers energy to $v^L$ to generate $u^L$ , somehow supporting the recent perspective of Jiménez (Reference Jiménez2022) that long streaks are bypass products of short vortical structures. In addition, the nonlinear interactions associated with spanwise-wide turbulent motions play distinct roles in the energy transfer processes of short vortical structures and long streaks. For short vortical structures, they mainly act as an extra energy source for $v^S$ to enhance the energy cycle of short vortical structures (in the pink box). For long structures, they work as an important energy source for $w^L$ that drives the energy cycle of long streaks (in the blue box), but work as sinks for $\langle u^Lu^L\rangle$ and $-\langle u^Lv^L\rangle$ . The robustness of the budgets and the underlying energy cycles has been examined in Appendix C by changing the wavelength used to define near-wall turbulent motions in the decomposition (4.2).

5.2. Dynamical mechanisms underlying the responses of near-wall turbulence

In this section, we elucidate the distinct responses of near-wall long structures and short vortical structures to the suppression of spanwise-wide turbulent motions in cases DZ01–DZ04, as well as the mechanism driving relaminarisation in case DZ05, based on the energy cycles summarised in figure 11. These physical interpretations are confirmed by the temporal evolutions of near-wall Reynolds stresses and the variations of the dominant budget terms in the quasi-steady state with the cutoff wavelength. The temporal evolutions of the integration of the horizontally averaged Reynolds stresses below $y^+\approx 70$ for near-wall short vortical structures, denoted as $\overline {\langle u_i^{S} u_{\!j}^{S} \rangle }^+_{xz}$ , and long streaks, denoted as $\overline {\langle u_i^{L} u_{\!j}^{L} \rangle }^+_{xz}$ , are illustrated in figure 12. Here, the overline represents the integration over $0\leqslant y^+ \leqslant 70$ , and ‘ $\langle \rangle _{xz}$ ’ denotes the averaging in the horizontal plane. The responses of the dominant budget terms for $\langle u_i^Su_{\!j}^S \rangle$ and $\langle u_i^Lu_{\!j}^L \rangle$ to the filter in the quasi-steady state are presented in figures 13 and 14, respectively.

Figure 12.

Time histories of (a i–d i) $\overline {\langle u_i^{S} u_{\!j}^{S} \rangle }^+_{xz}$ associated with short vortical structures and (a ii–d ii) $\overline {\langle u_i^{L} u_{\!j}^{L} \rangle }^+_{xz}$ associated with long streaks in the cases DZ03–DZ05. Here, the overline represents the integration over $0\leqslant y^+ \leqslant 70$ and ‘ $\langle \rangle _{xz}$ ’ denotes the averaging in the horizontal plane. All the simulations are initialised from a fully developed turbulent channel flow field at the same instant $t=t_0$ . The vertical dashed line indicates the time when the corresponding variable decays to zero.

For short vortical structures, as previously demonstrated in figure 11 (pink box), the turbulent transport term associated with spanwise-wide turbulent motions, $\textit{TW}_{22}^S$ , acts as an additional energy source for $\langle v^Sv^S \rangle$ , driving the energy cycle of these structures. It can be deduced that the suppression of spanwise-wide turbulent motions leads to a reduction in $\textit{TW}_{22}^S$ . The attenuation of $\textit{TW}_{22}^S$ can disrupt the budget balance of $\langle v^Sv^S \rangle$ , resulting in a negative temporal derivative (i.e. $\partial \langle v^S v^S \rangle / \partial t \lt 0$ ) and a consequent decay in $\langle v^S v^S \rangle$ . The reduction in $\langle v^S v^S \rangle$ can weaken the production term $P_{12}^S$ , thereby diminishing $-\langle u^S v^S \rangle$ . The weakened $-\langle u^S v^S \rangle$ can subsequently suppress the production term $P_{11}^S$ , thereby reducing the magnitude of $u^S$ . Accompanying the reduction in $u^S$ , the energy transferred from $u^S$ to $v^S$ and $w^S$ through the pressure-redistribution term should also be reduced. As the cutoff wavelength decreases, the reduction in $\textit{TW}_{22}^S$ can become more pronounced, leading to a more obvious suppression of near-wall short vortical structures.

The previous deduction is corroborated by the temporal evolutions of $\overline {\langle u_i^Su_{\!j}^S\rangle }_{xz}(t)$ shown in figure 12(a i–d i), as well as the variations of the budget terms associated with short vortical structures obtained in the quasi-steady states in figure 13. As depicted in figure 12(b i), the integrated Reynolds stress $\overline {\langle v^Sv^S\rangle }_{xz}$ exhibits a marked reduction immediately following the application of the high-pass filter. As the cutoff wavelength decreases in cases DZ03–DZ05, the decay rates accelerate, leading to the complete suppression in case DZ05 at $t-t_0\approx 20 h/U_m$ . This accelerated reduction in $\overline {\langle v^Sv^S\rangle }_{xz}$ aligns with the observation that the attenuation of $\textit{TW}_{22}^S$ becomes more pronounced as the cutoff wavelength decreases (figure 13 b). Accompanying the monotonic reduction in $\overline {\langle v^Sv^S\rangle }_{xz}$ , the quantities $\overline {\langle u^Su^S\rangle }_{xz}$ , $\overline {\langle w^Sw^S\rangle }_{xz}$ and $\overline {-\langle u^Sv^S\rangle }_{xz}$ are also gradually suppressed (figure 12 a i, c i, d i). The monotonic suppression of $\overline {\langle u^Su^S\rangle }_{xz}$ , $\overline {\langle w^Sw^S\rangle }_{xz}$ and $\overline {-\langle u^Sv^S\rangle }_{xz}$ is attributed to the reduction in the energy production terms $P_{11}^S$ and $P_{12}^S$ , as well as the pressure-redistribution terms $\varGamma _{22}^S$ and $\varGamma _{33}^S$ (figure 13). Notably, in case DZ05, coinciding with the vanishing of $\overline {\langle v^Sv^S\rangle }_{xz}$ at $t-t_0\approx 20 h/U_m$ , the terms $\overline {\langle u^Su^S\rangle }_{xz}$ , $\overline {\langle w^Sw^S\rangle }_{xz}$ and $\overline {-\langle u^Sv^S\rangle }_{xz}$ also decay to zero at $t-t_0\approx 30 h/U_m$ (figure 12 a i, c i, d i). These decay sequences further confirm that $v^S$ plays a critical role in driving the energy cycle of near-wall short vortices and its magnitude is significantly impacted by the energy provided by the nonlinear term associated with spanwise-wide turbulent motions $\textit{TW}_{22}^S$ .

Figure 13.

Variations of the dominant budget terms of $\langle u_i^Su_{\!j}^S \rangle$ with the cutoff wavelength $\lambda _{zf}^+$ in the numerical experiments. Here: ——, UnD; - $\boldsymbol{\cdot}$ - $\boldsymbol{\cdot}$ -, DZ01; $\ldots \!..$ , DZ02; - - - -, DZ03; - $\boldsymbol{\cdot }\boldsymbol{\cdot}$ - $\boldsymbol{\cdot }\boldsymbol{\cdot}$ , DZ04. Notably, negative $\textit{TW}_{11}^{S+}$ and $\textit{TW}_{22}^{S+}$ are shown.

Figure 13. Long description

The image contains four line graphs labeled (a), (b), (c), and (d), each depicting variations of dominant budget terms with the cutoff wavelength in numerical experiments. Panel A: A line graph shows the variations of P11+ and −TW11+ with y+. The x-axis is labeled y+ on a logarithmic scale ranging from 10^0 to 10^2. The y-axis ranges from −0.05 to 0.15. Different line styles represent various datasets: solid, dashed, dotted, and dash-dotted lines. Panel B: A line graph shows the variations of Γ22+ and −TW22+ with y+. The x-axis is labeled y+ on a logarithmic scale ranging from 10^0 to 10^2. The y-axis ranges from −0.01 to 0.01. Different line styles represent various datasets: solid, dashed, dotted, and dash-dotted lines. Panel C: A line graph shows the variations of Γ33+ and TW33+ with y+. The x-axis is labeled y+ on a logarithmic scale ranging from 10^0 to 10^2. The y-axis ranges from −0.01 to 0.03. Different line styles represent various datasets: solid, dashed, dotted, and dash-dotted lines. Panel D: A line graph shows the variations of P12+ and TW12+ with y+. The x-axis is labeled y+ on a logarithmic scale ranging from 10^0 to 10^2. The y-axis ranges from −0.01 to 0.04. Different line styles represent various datasets: solid, dashed, dotted, and dash-dotted lines.

Figure 14.

Variations of the dominant budget terms of $\langle u_i^Lu_{\!j}^L \rangle$ with the cutoff wavelength $\lambda _{zf}^+$ in the numerical experiments. Here: ——, UnD; - $\boldsymbol{\cdot}$ - $\boldsymbol{\cdot}$ -, DZ01; $\ldots \!..$ , DZ02; - - - -, DZ03; - $\boldsymbol{\cdot }\boldsymbol{\cdot}$ - $\boldsymbol{\cdot }\boldsymbol{\cdot}$ , DZ04. Notably, negative $\textit{TW}_{33}^{L+}$ is shown here.

Regarding the Reynolds stresses associated with long streaks, as the variation tendency of $\langle u^Lu^L \rangle$ is different from those of $\langle v^Lv^L \rangle$ and $\langle w^Lw^L \rangle$ , we first elucidate the reduction in $\langle v^Lv^L \rangle$ and $\langle w^Lw^L \rangle$ . As previously shown in the blue box of figure 11, the component $w^L$ primarily derives energy from the nonlinear interaction term associated with short vortical structures, $\textit{TS}_{33}^L$ , and that associated with spanwise-wide turbulent motions, $\textit{TW}_{33}^L$ . The removal of spanwise-wide turbulent motions should result in a reduction of $\textit{TW}_{33}^L$ . Concurrently, the suppression of short vortical structures by the filter diminishes $\textit{TS}_{33}^L$ (self-interaction of short vortical structures) and $\textit{TW}_{33}^L$ (dominated by nonlinear interaction between spanwise-wide turbulent motions and short vortical structures). Due to the attenuation of these two dominant source terms, $\langle w^L w^L \rangle$ should exhibit a decrease after the application of the high-pass filter. Meanwhile, $v^L$ acquires energy mainly from $w^L$ via the pressure-redistribution term $\varGamma _{22}^L$ . Consequently, it should decay with $w^L$ . As the cutoff wavelength decreases, the suppression of $\textit{TS}_{33}^L$ and $\textit{TW}_{33}^L$ should become more pronounced, and thus the reduction rates of $v^L$ and $w^L$ increase.

The previous deduction of the mechanism responsible for the reduction in $v^L$ and $w^L$ can be corroborated by the decay of $v^L$ and $w^L$ in cases DZ03–DZ05 over time following the application of the high-pass filter (figure 12 b ii, c ii), as well as the monotonic reduction in the energy source terms for $w^L$ ( $\textit{TS}_{33}^L$ and $\textit{TW}_{33}^L$ ) and $v^L$ (the pressure-redistribution term $\varGamma _{22}^L$ ) with decreasing $\lambda _{zf}^+$ (figure 14 b, c). Notably, in case DZ05, once short vortical structures are fully suppressed at $t-t_0\approx 30 h/U_m$ (figure 12 a i–d i), the energy transferred to $w^L$ through the nonlinear transport terms $\textit{TS}_{33}^L$ (self-interaction of short vortical structures) and $\textit{TW}_{33}^L$ (dominated by nonlinear interaction between spanwise-wide turbulent motions and short vortical structures) also decays to zero. This results in the subsequent decay of both $w^L$ and $v^L$ , which vanish at $t-t_0 \approx 40 h/U_m$ (figure 12 b ii, c ii). Such temporal evolution also confirms the importance of short vortical structures for sustaining $v^L$ and $w^L$ .

Distinct from the temporal evolutions of $\overline {\langle v^Lv^L\rangle }_{xz}$ and $\overline {\langle w^Lw^L\rangle }_{xz}$ , $\overline {\langle u^Lu^L\rangle }_{xz}$ and $\overline {-\langle u^Lv^L\rangle }_{xz}$ (figure 12 a ii, d ii) in the cases DZ03–DZ05 all grow with time just after the application of the high-pass filter. For the cases DZ04 and DZ05, they decay after the growth. In particular, in the case DZ05, they decay to zero after $v^L$ and $w^L$ are fully suppressed.

The temporal evolution of $\overline {-\langle u^Lv^L\rangle }_{xz}$ at various $\lambda _{zf}^+$ can be elucidated by examining the variations of its dominant energy sink associated with spanwise-wide turbulent motions, $\textit{TW}_{12}^L$ , and its major energy source, $P_{12}^L$ . Due to the removal of spanwise-wide turbulent motions, the magnitude of the energy sink $\textit{TW}_{12}^L$ is abruptly suppressed. In contrast, since $\overline {\langle v^Lv^L\rangle }_{xz}$ decays gradually over time (figure 12 b ii), the energy source term $P_{12}^L$ evolves more slowly than $\textit{TW}_{12}^L$ . Consequently, shortly after the application of the high-pass filter, $-\partial \langle u^L v^L\rangle /\partial t$ becomes positive, resulting in the initial growth of $\overline {-\langle u^Lv^L\rangle }_{xz}$ . The initial growth of $\overline {-\langle u^Lv^L\rangle }_{xz}$ enhances the primary energy source of $\overline {\langle u^Lu^L\rangle }_{xz}$ , namely the production term $P_{11}^L$ . Concurrently, the removal of spanwise-wide turbulent motions suppresses the energy sink $\textit{TW}_{11}^L$ . Consequently, $\overline {\langle u^Lu^L\rangle }_{xz}$ also exhibits the initial growth. Thus, the initial increase in both $\overline {-\langle u^Lv^L\rangle }_{xz}$ and $\overline {\langle u^Lu^L\rangle }_{xz}$ is attributed to the attenuation of the nonlinear turbulent transport terms associated with spanwise-wide motions ( $\textit{TW}_{11}^L$ and $\textit{TW}_{12}^L$ ), which is validated by the monotonic decrease of their magnitudes with the decrease of $\lambda _{zf}^+$ (figure 14 a, d).

As time progresses, in cases DZ04 and DZ05 where $\lambda _{zf}^+$ is sufficiently small, the suppression of $\overline {\langle v^Lv^L\rangle }_{xz}$ becomes pronounced (figure 12 b ii). This leads to an obvious decrease in the production term $P_{12}^L$ that eventually counteracts the reduction in the magnitude of the energy sink $\textit{TW}_{12}^L$ . Consequently, $\overline {-\langle u^Lv^L\rangle }_{xz}$ begins to decay following the decay of $v^L$ . Following this trend, the production term $P_{11}^L$ diminishes, reducing the growth rate of $\overline {\langle u^Lu^L\rangle }_{xz}$ and arresting its growth in case DZ04. The corresponding reduction in $P_{11}^L$ and $P_{12}^L$ in case DZ04 is illustrated in figures 14(a) and 14(d).

It is worthwhile noting that as the energy cycle of long streaks is driven by $w^L$ and $v^L$ , once $w^L$ and $v^L$ are fully suppressed at $t-t_0 \approx 40 h/U_m$ in case DZ05 following full suppression of short vortical structures, $\overline {-\langle u^L v^L\rangle }_{xz}$ and $\overline {\langle u^L u^L\rangle }_{xz}$ also decay to zero after $t-t_0 \approx 40 h/U_m$ .

In summary, the distinct responses of long streaks and short vortical structures in cases DZ01–DZ03 are primarily attributed to the distinct roles played by nonlinear terms associated with spanwise-wide turbulent motions for $u^L$ and $v^S$ . The arrested growth of $u^L$ in case DZ04 is associated with the suppression of short vortical structures, which provide energy to $w^L$ along with spanwise-wide motions via nonlinear terms to drive the energy cycle of long streaks. The observed temporal sequence (that $v^S$ decays to zero first, followed by $u^S$ and $w^S$ , then $v^L$ and $w^L$ , and finally $u^L$ ) further confirms that the relaminarisation in case DZ05 is attributed to the full suppression of short vortical structures, which results from the reduction in the extra energy transferred to $v^S$ via nonlinear transport term associated with spanwise-wide turbulent motions.