3.1. Roughness effect on turbulence statistics

In flows over spanwise-periodic roughness strips, the time-mean velocity field is intrinsically heterogeneous in the wall-parallel plane due to the presence of SCs, wake sheltering and pressure-drag footprints imposed by the roughness geometry. Under such conditions, a classical Reynolds decomposition is insufficient, as it does not distinguish turbulence-driven transport from stationary, geometry-induced momentum redistribution. To separate these effects, we employ a triple decomposition within a double-averaging framework, in which temporal averaging is followed by spatial averaging over the wall-parallel plane.

The instantaneous velocity field is decomposed as

(3.1) \begin{equation} u_i(x,y,z,t)=\langle \bar {u}_i \rangle + \tilde {u}_i(x,y,z) + u_i^{\prime}(x,y,z,t), \end{equation}

where an overbar denotes temporal averaging, angle brackets denote spatial averaging in the wall-parallel directions and $\langle \bar {u}_i \rangle$ is the double-averaged base flow, which depends only on the wall-normal coordinate. The term $\tilde {u}_i=\bar {u}_i-\langle \bar {u}_i \rangle$ is the form-induced component associated with the spatial organisation of the time-mean flow imposed by the roughness geometry. The fluctuation $u_i^{\prime}$ denotes the turbulent component and satisfies $\overline {u_i^{\prime}}=0$ at each spatial location.

The quantity relevant to the following analysis is the double-averaged fluctuation stress, excluding the viscous contribution,

(3.2) \begin{equation} \tau _{\textit{ij}}^{\textit{DA}} \equiv \left \langle \left (\bar {u}_i-\langle \bar {u}_i\rangle \right )\left (\bar {u}_j-\langle \bar {u}_j\rangle \right )\right \rangle = \langle \tilde {u}_i \tilde {u}_j \rangle + \left \langle \overline {u_i^{\prime} u_{\!j}^{\prime}} \right \rangle , \end{equation}

where the first term, $\langle \tilde {u}_i \tilde {u}_j \rangle$ , is the dispersive stress arising from spatial correlations of the time-mean heterogeneous flow and the second term, $\langle \overline {u_i^{\prime} u_{\!j}^{\prime}} \rangle$ , is the double-averaged turbulent Reynolds stress. Their sum defines the total double-averaged fluctuation stress discussed later. Accordingly, the quantity plotted in figures 5 and 6 is not the full second moment $\langle \overline {u_i u_{\!j}} \rangle$ , but the fluctuation stress $\tau _{\textit{ij}}^{\textit{DA}}$ defined in (3.2). Unless otherwise stated, the velocity components appearing in the previous decomposition are normalised by the friction velocity $u_{\tau }$ , the second-order stress quantities are normalised by $u_{\tau }^{2}$ and the wall-normal coordinate is expressed in inner units relative to the roughness crest as $(y-y_0)^{+}$ .

In the double-averaged streamwise momentum equation, the wall-normal gradient of the relevant dispersive-stress component appears alongside roughness-induced drag contributions obtained from surface integrals, namely pressure drag and viscous drag per unit fluid volume. Together with the turbulent stress divergence and the viscous mean-shear term, these contributions close the momentum balance. In the kinetic-energy budget, the spatial heterogeneity of the time-mean flow gives rise to wake production and dispersive-transport terms that mediate energy transfer from the base flow to turbulence at the scale of the roughness elements. These processes cannot be absorbed into classical Reynolds stresses and must therefore be treated explicitly. This formalism follows the double-averaged Navier–Stokes framework developed for rough beds, which establishes the relevant averaging theorems, and clarifies the physical interpretation of dispersive stresses and drag terms (Nikora et al. Reference Nikora, McEwan, McLean, Coleman, Pokrajac and Walters2007). The importance of form-induced fluxes and wake production in second-order budgets was already highlighted in canopy-flow studies employing time-then-space averaging (Raupach & Shaw Reference Raupach and Shaw1982). More recently, Ma et al. (Reference Ma, Xu, Sung, Tian and Huang2025) further demonstrated through DNS of rough-wall channels that compatible triple-decomposition and double-averaging analyses are essential for disentangling geometry-locked transport from genuinely turbulent inter-scale transfers over distributed roughness. Taken together, this framework provides the basis for distinguishing stationary geometry-locked transport from turbulent transport in the rough-wall flows considered here. In the following, we first examine the wall-normal distributions of the double-averaged fluctuation stresses for all cases. When a further scale separation of the turbulent term is required, it is introduced later together with the filtering procedure used to define the corresponding large- and small-scale fields.

Figure 5.

Wall-normal profiles of the double-averaged fluctuation stresses $\tau _{\textit{ij}}^{\textit{DA}}$ for the smooth-wall case, the homogeneous roughness reference case R17 homogeneous and the spanwise-periodic roughness-strip configurations: (a–d) $\tau _{11}^{\textit{DA}}$ , $\tau _{22}^{\textit{DA}}$ , $\tau _{33}^{\textit{DA}}$ and $-\tau _{12}^{\textit{DA}}$ , respectively. Unless otherwise stated, the stresses are normalised by $u_{\tau }^{2}$ , and the wall-normal coordinate is expressed in inner units relative to the roughness crest as $(y-y_0)^{+}$ . Panels (b, d) show the spanwise-averaged profiles for the smooth wall (black), the homogeneous roughness reference case R17 homogeneous (black dashed line), and the spanwise-heterogeneous rough-wall cases R13, R17 and R20.

Figure 6.

Spanwise–wall-normal ( $y$ – $z$ ) contours, and corresponding wall-normal profiles of the dispersive and turbulent contributions to the double-averaged fluctuation stresses for the Smooth, R13, R17 and R20 cases: (a, c) turbulent contributions; (b, d) corresponding dispersive contributions. In each panel, the $y$ – $z$ contours are shown with the wall-normal coordinate expressed in inner units as $(y-y_0)^{+}$ ; panels (b, d) show the spanwise-averaged profiles for the Smooth case, the additional homogeneous roughness reference case R17 homogeneous (black dashed line), and the spanwise-heterogeneous rough-wall cases R13, R17 and R20.

Figure 5 presents the wall-normal profiles of $\tau _{\textit{ij}}^{\textit{DA}}$ for the smooth-wall case, the homogeneous roughness reference case R17 homogeneous and the spanwise-periodic roughness-strip configurations with different element heights. The wall-normal coordinate is defined relative to the roughness crest as $(y-y_0)^{+}$ , with $y_0$ denoting the roughness-element crest, to facilitate a direct comparison of the flow statistics above and below the roughness elements.

Compared with the smooth wall, both the homogeneous and spanwise-heterogeneous rough-wall cases exhibit pronounced modifications of the stress distributions. However, the responses differ substantially between the homogeneous reference case and the strip-roughness configurations, indicating that roughness height alone does not fully determine the stress amplification. Among all components, the streamwise normal stress $\tau _{11}^{\textit{DA}}$ shows the most prominent enhancement. Its peak magnitude increases markedly in all rough-wall cases, but the increase is far greater for the spanwise-heterogeneous strip cases than for the homogeneous roughness reference. This contrast indicates that the strong amplification of $\tau _{11}^{\textit{DA}}$ is not simply a generic consequence of wall roughening, but is substantially reinforced by the spanwise heterogeneity of the roughness arrangement.

In contrast, the wall-normal and spanwise normal stress components, $\tau _{22}^{\textit{DA}}$ and $\tau _{33}^{\textit{DA}}$ , exhibit more moderate changes. Although their magnitudes are elevated relative to the smooth-wall case, their overall wall-normal distributions remain broadly similar across the rough-wall configurations. The homogeneous roughness case follows the same general trend, but its magnitude remains systematically below those of the strip-roughness cases over much of the wall-normal range. The shear stress component $-\tau _{12}^{\textit{DA}}$ is likewise enhanced by roughness, with the spanwise-heterogeneous cases displaying a stronger amplification and a clearer upward displacement of the peak than the homogeneous reference case.

These results demonstrate that the influence of roughness on the double-averaged fluctuation Reynolds stresses is inherently anisotropic, with the streamwise component providing the dominant contribution to the overall stress response. More importantly, the comparison with R17 homogeneous shows that the pronounced enhancement observed in the strip-roughness cases cannot be attributed to roughness magnitude alone. Rather, it reflects an additional contribution associated with the spanwise organisation imposed by the heterogeneous roughness layout. To clarify whether this enhanced stress response is predominantly turbulent or form-induced in origin, the individual turbulent and dispersive contributions are examined in figure 6.

Figure 6 presents the spatial distributions and wall-normal profiles of the turbulent and form-induced contributions to the double-averaged fluctuation Reynolds stresses. The results show that the pronounced enhancement of $\tau _{\textit{ij}}^{\textit{DA}}$ observed over spanwise roughness strips is predominantly attributable to the form-induced stress component. This contribution exhibits spatially organised structures whose characteristic scales are closely tied to the roughness geometry, reflecting a persistent, geometry-locked organisation of the time-mean flow. In contrast, the corresponding variations in the genuinely turbulent stress component remain comparatively modest across all cases.

The inclusion of the homogeneous roughness reference case provides an important additional point of comparison. Although R17 homogeneous also produces a non-zero form-induced contribution, indicating that homogeneous wall roughening can generate a certain degree of stationary spatial organisation in the time-mean flow, its magnitude is substantially weaker than that observed in the spanwise-heterogeneous strip cases. This difference is especially evident in $\langle \tilde {u}\tilde {u} \rangle ^{+}$ and $-\langle \tilde {u}\tilde {v} \rangle ^{+}$ , for which the heterogeneous roughness cases show much stronger near-crest intensities and a more pronounced wall-normal extent. The comparison therefore demonstrates that the large form-induced stress enhancement reported here is not a universal consequence of roughness alone, but is significantly amplified by the spanwise heterogeneity of the roughness arrangement.

This contrast indicates that the roughness strips do not modify the overall stress level through a uniform amplification of small-scale turbulence. Instead, their primary effect is to introduce stable spanwise heterogeneity that reorganises the time-mean flow and its associated momentum-transport pathways. In this sense, the dominant contribution of the form-induced stress highlights the role of stationary, roughness-imposed flow organisation rather than enhanced turbulent agitation. The comparison with R17 homogeneous reinforces this interpretation by showing that, while homogeneous roughness can produce a limited form-induced component, the much stronger geometry-locked transport arises specifically from the stripwise heterogeneity.

Notwithstanding its statistical dominance, the dynamical origin of the form-induced stress warrants closer examination, particularly with regards to how these geometry-locked structures are sustained in the wall-normal direction and how far they extend into the outer region. From this perspective, it is instructive to analyse the flow-organisation mechanisms underpinning these stresses by jointly considering the characteristics of the secondary currents and the associated velocity-correlation footprints. Such an approach enables a more detailed assessment of how roughness-strip-induced secondary currents modulate wall-normal momentum transport and influence the wall-normal positioning of VLSMs, thereby clarifying the distinct roles of homogeneous roughness and spanwise-heterogeneous roughness in the modulation of wall-bounded turbulence.

3.2. Roughness-induced spanwise organisation of the flow

This section examines the SCs induced by spanwise roughness strips and the resulting spanwise organisation of the flow. The analysis focuses first on the modification of wall-normal and spanwise pathways of high- and low-momentum fluid in the presence of roughness. It then characterises the spatial distribution and intensity of the associated SCs, with emphasis on their spanwise coherence and wall-normal extent.

Figure 7 presents contours of the mean streamwise velocity, overlaid with the wall-parallel velocity vectors, illustrating the spanwise arrangement of HMPs and LMPs. The spanwise locations of the HMPs and LMPs are explicitly indicated in the figure. From a statistical perspective, their distribution exhibits a clear spanwise-alternating pattern that is consistent with previous observations in turbulent flows over spanwise-heterogeneous roughness. In particular, persistent SCs reorganise the time-mean momentum field into geometry-locked high- and low-momentum pathways, such that high-momentum regions preferentially form above the roughness strips, whereas low-momentum pathways are established within the inter-strip gaps (Barros & Christensen Reference Barros and Christensen2014; Willingham et al. Reference Willingham, Anderson, Christensen and Barros2014; Anderson et al. Reference Anderson, Barros, Christensen and Awasthi2015).

Figure 7.

Contours of mean streamwise velocity $\overline {U}/U_b$ for (a) the reference smooth case, and for (b) R13, (c) R17 and (d) R20. The blue dashed line indicates the location of the low-momentum pathway (LMP), while the red dashed line indicates the location of the high-momentum pathway (HMP).

This organisation indicates that the roughness strips induce a systematic redistribution of streamwise momentum through secondary motions. Specifically, momentum is transported upward from the regions above the roughness elements towards higher wall-normal locations and subsequently conveyed downward into the inter-strip gaps. As a result, alternating regions of high and low streamwise momentum are established in the spanwise direction, forming a persistent large-scale organisation of the mean flow that is locked to the roughness geometry. The wall-normal reach of the high-momentum pathways further indicates that the influence of the roughness-induced secondary motions is not confined to the near-wall region, but extends to elevated wall-normal locations, motivating a quantitative characterisation of the intensity and spatial extent of these secondary motions.

To characterise the directionality of the secondary vortical motions, we employ the signed swirling strength, which has been widely used in previous studies of secondary motions over spanwise-heterogeneous surfaces (Vanderwel & Ganapathisubramani Reference Vanderwel and Ganapathisubramani2015; Vanderwel et al. Reference Vanderwel, Stroh, Kriegseis, Frohnapfel and Ganapathisubramani2019; Wangsawijaya et al. Reference Wangsawijaya, Baidya, Chung, Marusic and Hutchins2020). At each instantaneous snapshot, it is defined from the instantaneous cross-plane velocity field as

(3.3) \begin{equation} \varLambda _{ci}^{*}(x,y,z,t) = \varLambda _{ci}(x,y,z,t)\, \frac {\varOmega _x(x,y,z,t)}{\left |\varOmega _x(x,y,z,t)\right |}, \end{equation}

which is equivalent to applying the sign function $\mathrm{sgn}(\varOmega _x)$ to assign the sense of rotation according to the local instantaneous streamwise vorticity. Here, $\varLambda _{ci}$ is the imaginary part of the complex eigenvalue of the instantaneous local in-plane velocity-gradient tensor,

(3.4) \begin{equation} \begin{bmatrix} \partial _y v(x,y,z,t) & \partial _z v(x,y,z,t)\\ \partial _y w(x,y,z,t) & \partial _z w(x,y,z,t) \end{bmatrix}, \end{equation}

where $\varOmega _x(x,y,z,t)=\partial _y w-\partial _z v$ is the instantaneous streamwise vorticity. Positive and negative values of $\varLambda _{ci}^{*}$ therefore represent clockwise and counter-clockwise vortical motions, respectively, within the $y$ – $z$ plane. It should be noted that $\varLambda _{ci}^{*}$ is first evaluated from each instantaneous velocity field and the quantity visualised is the ensemble average of these instantaneous signed swirling-strength fields, rather than the signed swirling strength computed from the mean secondary-flow field. For the wall-normal coordinate, a crest-zero convention is adopted to align the reference with the roughness geometry, $y^{+}={\textit{Re}}_{\tau }(y-y_0)$ . This convention facilitates direct comparison across different roughness cases by fixing the zero of $y^{+}$ at the roughness crests. Fluid points lying within the solid roughness are excluded from all integrations through a geometric mask applied to the computational domain.

Figure 8 shows the spatial distribution of large-scale SCs, characterised by the mean swirl strength $\varLambda _{ci}^{*}$ weighted by the sign of the streamwise vorticity, for the smooth-wall case and the roughness-strip configurations. The results indicate that the intensity of the secondary motions attains its maximum in the vicinity of the roughness-strip crests. In the wall-normal direction, a pronounced near-wall double-peak distribution is observed, with the two local maxima located at wall-normal heights corresponding to the roughness crest and to a position below the roughness trough. This wall-normal arrangement is consistent with the contour plots, which reveal two pairs of counter-rotating secondary cells with opposite sense of rotation.

This spatial organisation provides an explanation for the elevated streamwise form-induced component previously observed within the inter-strip gaps. Specifically, the combined action of the secondary motions originating near the roughness crests and those associated with the lower wall-normal region leads to enhanced wall-normal and spanwise transport of streamwise momentum into the gap regions. As the roughness height increases, the secondary-current signature becomes more distinct, as reflected by the clearer HMP/LMP organisation in the mean streamwise velocity field together with the stronger dispersive contributions in figure 6.

Figure 8.

Large-scale SCs characterised by the mean swirl strength $\varLambda _{ci}^{*}$ weighted by the sign of the streamwise vorticity $\varOmega _x$ . Red and blue colours denote counter-clockwise (CCW) and clockwise (CW) SCs, respectively: (a) the smooth-wall case; (b–d) cases R13, R17 and R20, respectively.

3.3. Impact of SCs on VLSMs

The previous results characterise the momentum redistribution associated with the roughness-induced secondary motions. To examine how this redistribution relates to the organisation of VLSMs, it is necessary to first identify the large-scale turbulent structures in a systematic manner. To this end, we employ the two-stage filtering methodology proposed by Hwang et al. (Reference Hwang, Lee, Sung and Zaki2016) to extract the characteristic large-scale structures from the instantaneous streamwise velocity field. In the first step, a two-dimensional Gaussian filter is applied to each $y$ – $z$ plane of the instantaneous fluctuation field $u'(x,y,z)$ . The standard deviation of the Gaussian kernel is allowed to vary with wall-normal distance so as to reflect the growth of the spanwise length scale of the motions with $y$ . For each tested case, this wall-normal dependence is represented by the cubic relation

(3.5) \begin{equation} \sigma _n/\delta = a_n\left (y/\delta \right )^3 + b_n\left (y/\delta \right )^2 + c_n\left (y/\delta \right ) + d_n, \end{equation}

where the coefficients $(a_n,b_n,c_n,d_n)$ depend on the Reynolds number and roughness condition of case $n$ .

The coefficients are determined a posteriori from the DNS database. At each wall-normal location $y$ , we compute the spanwise two-point correlation of the streamwise velocity fluctuations,

(3.6) \begin{equation} R_{uu}(y,\Delta z) = \frac {\left \langle u'(x,y,z)\,u'(x,y,z+\Delta z)\right \rangle _{x,t}} {\left \langle u'^2(x,y,z)\right \rangle _{x,t}}, \end{equation}

and extract a characteristic spanwise length scale $l_z(y)$ defined as the separation distance $\Delta z$ at which the correlation decays to approximately one half of its zero-lag value, i.e. $R_{uu}(y,l_z(y))\approx 0.5$ .

This procedure provides a smoothly varying estimate of the local spanwise streak spacing, from which the wall-normal dependence of the Gaussian filter width is subsequently obtained. The Gaussian width $\sigma _n(y)$ is then chosen such that the half-width of the kernel corresponds to this spanwise scale, which prevents adjacent positive and negative fluctuations from cancelling each other and yields a smoothly varying filtered field in the spanwise direction. The discrete values of $\sigma _n(y)/\delta$ obtained in this way are finally fitted by least squares to the cubic form above, producing the coefficients $(a_n,b_n,c_n,d_n)$ that are used in the filtering procedure. The numerical values of these coefficients for all cases considered are reported in table 2.

Table 2.

Coefficients $a_n$ , $b_n$ , $c_n$ and $d_n$ defining the wall-normal variation of the Gaussian filter width $\sigma$ in (3.5) for different cases.

To extract the characteristic large-scale motions from the instantaneous streamwise velocity field, we adopt the two-stage filtering methodology proposed by Hwang et al. (Reference Hwang, Lee, Sung and Zaki2016). In the first step, a two-dimensional Gaussian filter with a wall-normal-dependent width is applied to each $y$ – $z$ plane of the instantaneous velocity field $u'(x,y,z)$ , producing a smoothly varying field in the spanwise direction. In the second step, a long-wavelength-pass filter is applied along the streamwise direction to retain only motions with streamwise wavelengths $\lambda _x\geqslant h$ . This sequential filtering isolates the large-scale and very-large-scale structures, while avoiding the discontinuities that would arise if the streamwise filter were applied directly to the unfiltered field. Because the combined filters attenuate the amplitude of the velocity fluctuations, the streamwise velocity threshold used for structure identification must be corrected. The corrected threshold is defined as $u_{\textit{th}}^{\prime}(y)=f(y)u_{\textit{th,raw}}^{\prime}$ , with $f(y)=\langle u'(x)\,\hat {u'}(x)\rangle /\langle u'(x)\,u'(x)\rangle$ , where $\hat {u'}(x,y,z)$ is the filtered velocity. The factor $f(y)$ accounts for the local weakening of the filtered signal and ensures that the streak cores are consistently detected across the wall-normal domain. Local extrema of $\hat {u'}$ along the spanwise direction are then identified at each $(x,y)$ by locating the positions where $\partial \hat {u'}/\partial z=0$ . Each extremum is classified according to its filtered velocity level relative to the corrected threshold,

(3.7) \begin{equation} CP(x,y,z)= \begin{cases} +1 & \text{if } \partial \hat {u'}/\partial z=0 \text{ and } \hat {u'}\gt u_{\textit{th}}^{\prime}(y),\\ -1 & \text{if } \partial \hat {u'}/\partial z=0 \text{ and } \hat {u'}\lt -u_{\textit{th}}^{\prime}(y),\\ 0 & \text{otherwise}, \end{cases} \end{equation}

with ${\textit{CP}}=+1$ indicating high-speed streak cores and ${\textit{CP}}=-1$ marking low-speed ones. Neighbouring extrema that are streamwise-connected are grouped into a unique object, which defines an individual elongated streak-like structure. For each detected structure, its streamwise length $L_{\textit{CP}}$ is measured from the most upstream to the most downstream ${\textit{CP}}$ location and the spanwise boundaries are adjusted to the positions where $\partial \hat {u'}/\partial z$ changes sign. The organised motions are classified based on their normalised streamwise length. In the present study, a structure is identified as an LSM if $1 \leqslant L_{\textit{CP}}/h \lt 3$ and as a VLSM if $L_{\textit{CP}}/h \geqslant 3$ . This length-based criterion is applied independently to high-speed and low-speed structures, yielding four conditional classes: negative LSMs (nLSMs), negative VLSMs (nVLSMs), positive LSMs (pLSMs) and positive VLSMs (pVLSMs).

Figure 9 presents the ensemble-averaged number of conditional events, $\langle N_{\textit{CP}} \rangle$ , as a function of the normalised streamwise length $L_{\textit{CP}}/L_x$ for nVLSMs and pVLSMs. The structures are classified as VLSMs according to the criterion $L_{\textit{CP}}/h \geqslant 3$ , while the abscissa is presented in the normalised form $L_{\textit{CP}}/L_x$ to facilitate comparison of the streamwise extent within the finite computational domain. Results are shown for the smooth-wall configuration, and for the rough-wall cases R13, R17 and R20. The shaded regions highlight the range of large-scale motions over which roughness effects are most pronounced. In evaluating the streamwise length $L_{\textit{CP}}/L_x$ , only fluid points above the roughness elements are considered; contributions from the inter-strip gap regions are excluded to focus on the large-scale organisation associated with the roughness-induced flow modulation.

Compared with the smooth-wall case, the occurrence of long streamwise-coherent motions is generally reduced over the rough-wall cases. However, the response is distinctly sign-dependent. For positive-velocity structures, the conditional event count decreases systematically with increasing roughness height over most of the resolved streamwise-length range. By contrast, for negative-velocity structures, the conditional event count increases monotonically from R13 to R20 among the rough-wall cases, indicating that roughness progressively promotes the relative persistence of negative long streamwise-coherent motions. These results suggest that spanwise-heterogeneous roughness does not simply amplify or suppress VLSMs in a uniform manner; rather, it induces a sign-asymmetric reorganisation of the large-scale streamwise structures, consistent with the secondary-current-mediated modulation emphasised.

Figure 9.

Ensemble-averaged number of conditional events, $\langle N_{\textit{CP}} \rangle$ , as a function of the normalised streamwise length $L_{\textit{CP}}/L_x$ for nVLSMs and pVLSMs. VLSMs are classified according to the criterion $L_{\textit{CP}}/h \geqslant 3$ , while the abscissa is plotted as $L_{\textit{CP}}/L_x$ . Results are shown for the smooth-wall configuration, and for the rough-wall cases R13, R17 and R20. The shaded regions highlight the range of large-scale motions over which roughness effects are most pronounced. In evaluating the streamwise length $L_{\textit{CP}}/L_x$ , only fluid points above the roughness elements are considered; contributions from the inter-strip gap regions are excluded to focus on the large-scale organisation associated with the roughness-induced flow modulation.

Figure 10.

Wall-normal distribution of VLSM structure count density as a function of the streamwise length scale $L_{\textit{CP}}/h$ : (a–d) density of positive- $u$ VLSMs for the Smooth, R13, R17 and R20 cases, respectively; (e–h) corresponding density of negative- $u$ VLSMs in the same order. Coloured circles represent the VLSM count density $\langle N_{\textit{CP}}\rangle ^{+/-}$ , defined as the number of detected structures per frame per unit effective area within each $(L_{\textit{CP}},y)$ bin. Both the colour intensity and the marker size encode the local count density, with darker colours and larger symbols indicating higher densities. Light-coloured or empty circles correspond to bins with negligible or zero detected structures.

Further insight into the roughness-modified organisation of VLSMs is provided by the wall-normal distribution of the VLSM structure count density as a function of $L_{\textit{CP}}$ in figure 10. In the present low- ${\textit{Re}}_{\tau }$ dataset, the smooth-wall case itself exhibits a discernible outer-layer preference in the occurrence of VLSM structures, consistent with the established view that large-scale motions preferentially reside away from the immediate near-wall region. Relative to this baseline distribution, the rough-wall cases exhibit an additional and more clearly defined concentration of the roughness-enhanced VLSM population at wall-normal positions close to the roughness crests.

Figure 11.

Instantaneous contours of the streamwise velocity fluctuation $u-\langle \bar {u}\rangle$ on an $x$ – $z$ plane at $y/h=0.3$ for (a) the smooth-wall case and the rough-wall cases: (b) R13; (c) R17 and (d) R20.

This near-crest concentration persists over the range of streamwise lengths for which VLSMs are enhanced in the rough-wall cases. The role of roughness is therefore better understood not as generating wall-normal preference in itself, but as superimposing a roughness-conditioned reorganisation on the underlying outer-layer tendency, such that part of the VLSM population becomes preferentially localised near the roughness crests through the action of roughness-induced secondary currents.

Because the present simulations are performed at relatively low friction Reynolds numbers, for which the scale separation between near-wall turbulence and VLSMs remains limited, the following interpretation is restricted to the present low- ${\textit{Re}}_{\tau }$ open-channel DNS. Within this parameter range, the main result of figure 10 is that, relative to the present smooth-wall reference, spanwise-heterogeneous roughness produces a stronger and more explicit roughness-conditioned wall-normal localisation of VLSM occurrence.

Moreover, when combined with the conditional statistics in the spanwise direction, these wall-normal distributions reveal a clear spatial segregation of VLSMs. nVLSMs are preferentially concentrated above the roughness strips, whereas pVLSMs are more frequently found in the inter-strip regions. This spatial segregation is also visible in the instantaneous flow fields shown in figure 11, where the large-scale streaky structures at $y/h = 0.3$ exhibit an apparent spanwise organisation relative to the underlying roughness geometry. Figure 11 therefore serves as a qualitative illustration of the instantaneous organisation, while the corresponding statistical support is provided by figures 10, 12 and 13.

Taken together, these results indicate that spanwise-heterogeneous roughness fundamentally alters the balance between suppression and reorganisation of large-scale coherent motions. While the formation of LSMs is weakened, the VLSM population is reorganised in a roughness-conditioned and sign-dependent manner governed by secondary currents, highlighting a more intricate interaction between roughness-induced secondary motions and large-scale turbulence than previously assumed.

Figure 12.

Contours of the conditional two-point correlation $R(u_I^{\prime},u')$ in the $y$ – $z$ plane for negative and positive very-large-scale motions (nVLSMs and pVLSMs). Results are shown for the smooth-wall case, and for the rough-wall cases R13, R17 and R20.

Figure 13.

Contours of the conditional two-point correlation $R(u_I^{\prime},w')$ in the $y$ – $z$ plane for nVLSMs and pVLSMs over the smooth wall and rough-wall cases (R13, R17 and R20). Black solid lines denote positive iso-contours of $R(u_I^{\prime},v')$ , while black dashed lines indicate negative iso-contours, illustrating the coupled spatial organisation of wall-normal and spanwise velocity correlations associated with large-scale structures.

To characterise the spatial organisation of long streamwise velocity fluctuations, the conditional two-point correlation is employed. Long streamwise structures are first identified based on the area indicator function $I$ , and the conditional correlation between the streamwise velocity fluctuation associated with a given event, $u_I^{\prime}$ , and the fluctuating velocity field, $u_{\!j}^{\prime}$ , is defined as

(3.8) \begin{equation} R\big[u_I^{\prime}, u_{\!j}^{\prime}\big]\big(r_x, y, r_z; y_{\textit{ref}}\big) = \frac { \left \langle u_I^{\prime}\big(x,z; y_{\textit{ref}}\big)\, u_{\!j}^{\prime}\big(x+r_x, y, z+r_z\big) \right \rangle }{ \sigma _{u'}\big(y_{\textit{ref}}\big)\, \sigma _j(y) }, \end{equation}

where the subscript $j$ denotes the velocity components $(u',\,v',\,w')$ , $\sigma$ is the corresponding standard deviation and $\langle \boldsymbol{\cdot }\rangle$ represents ensemble averaging.

In the present study, the wall-normal reference location is fixed at $y_{\textit{ref}}/h = 0.3$ for all cases. In the conditional-correlation figures for the rough-wall configurations, however, the spanwise reference position is chosen according to the most probable spanwise occurrence of the conditioned VLSM type. Consistent with the statistical distributions discussed previously, nVLSMs are preferentially concentrated above the roughness strips, whereas pVLSMs occur more frequently in the inter-strip regions. Accordingly, the conditional correlations are evaluated at the spanwise position where the corresponding structure has the highest occurrence probability. In these figures, $z/h = 0$ therefore denotes the reference spanwise position of the conditioned event, rather than a universal geometric midpoint of the computational domain.

Based on the above-mentioned definition, three conditional correlations are examined in the present study. The streamwise auto-correlation $R[u^{\prime}_I,u']$ characterises the spatial coherence and streamwise extent of the long- $u'$ structures and their statistical connection with the surrounding streamwise velocity fluctuations. The conditional cross-correlation $R[u^{\prime}_I,v']$ reflects the wall-normal motions associated with the identified events, and therefore provides insight into the ejection- and sweep-type dynamics linked to the long streamwise structures. In addition, the conditional cross-correlation $R[u^{\prime}_I,w']$ quantifies the spanwise velocity fluctuations induced by the long- $u'$ motions and is particularly useful for revealing the presence of large-scale circulations and converging or diverging spanwise motions in the vicinity of the wall.

Figure 12 presents contours of the conditional two-point correlation $R(u_I^{\prime},u')$ in the $y$ – $z$ plane for nVLSMs and pVLSMs, for both the smooth-wall case and the rough-wall configurations. This diagnostic quantifies the spatial footprint of large-scale streamwise velocity fluctuations conditioned on VLSM events. Across all cases, the conditional correlation fields reveal that, although an outer-layer occurrence tendency remains evident in the VLSM statistics, the peak values of $R(u_I^{\prime},u')$ are consistently found much closer to the wall. This near-wall localisation of the correlation maximum persists for both nVLSMs and pVLSMs, and becomes increasingly pronounced with increasing roughness height. Notably, the wall-normal position of the correlation peak closely coincides with the near-wall maximum of the form-induced streamwise stress component $\langle \tilde {u}\tilde {u} \rangle ^{+}$ identified in the preceding analysis, which originates from momentum redistribution driven by roughness-induced secondary motions.

This correspondence demonstrates a statistical association between the near-wall momentum redistribution induced by SCs and the spatial footprint of VLSMs. Although VLSMs are preferentially observed in the outer region, their conditional correlation fields indicate that the streamwise velocity component remains strongly correlated with near-wall regions subject to SC-driven momentum reorganisation. Consequently, the influence of the secondary motions manifests as a persistent near-wall signature in the large-scale streamwise velocity field, which maintains correlation with VLSM events occurring farther from the wall.

Figure 13 presents the conditional two-point correlation $R(u_I^{\prime},w')$ in the $y$ – $z$ plane for nVLSMs and pVLSMs, with iso-contours of $R(u_I^{\prime},v')$ superimposed using black solid and dashed lines to denote positive and negative values, respectively. This representation highlights the coupled spatial organisation of spanwise and wall-normal velocity correlations associated with large-scale streamwise velocity events.

Compared with the smooth-wall case, the rough-wall configurations exhibit a substantially more intricate spatial organisation of the conditional correlation fields. In particular, an alternating wall-normal arrangement of regions associated with dispersive and converging motions emerges, reflecting the modulation of the spanwise velocity component imposed by the roughness strips. This organisation is markedly weaker over the smooth wall, indicating a roughness-induced restructuring of the velocity-correlation field. The locations of the dominant correlation peaks further underscore a close statistical association between near-wall motions and the spatial organisation of VLSMs. Although the large-scale motions are identified away from the wall, the conditional correlations reveal that their spanwise and wall-normal velocity signatures remain strongly linked to near-wall regions influenced by roughness-induced secondary motions. The combined examination of $R(u_I^{\prime},u')$ and $R(u_I^{\prime},w')$ elucidates the flow-field environment in which VLSMs are embedded, demonstrating that their spatial organisation is closely connected to near-wall motions modulated by secondary-current systems under spanwise-heterogeneous roughness.

Figure 14.

Instantaneous streamwise velocity fluctuations at selected wall-normal locations (left, $y/h=0.15$ , 0.3, 0.5 and 0.8 for case R20), the corresponding roughness configuration (centre), and schematic conditional footprint patterns (right) illustrating converging and dispersive transverse motions associated with pVLSMs and nVLSMs. The schematic footprints demonstrate distinct spanwise phase relationships between converging and dispersive motions relative to the roughness geometry, highlighting a dominant organisational mode characterised by convergence over the roughness strips and dispersion over the inter-strip gaps, as well as a secondary mode with a shifted phase relationship.

Figure 14 combines instantaneous streamwise velocity fluctuations at selected wall-normal locations with the roughness configuration and schematic conditional footprint patterns, providing a composite representation of the spatial organisation of large-scale motions over spanwise roughness strips. The left panels show instantaneous streamwise velocity fluctuations at multiple wall-normal heights ( $y/h=0.15$ , 0.3, 0.5 and 0.8) for case R20, the central panel illustrates the corresponding roughness geometry, and the right panels present schematic representations of converging and dispersive transverse motions associated with pVLSMs and nVLSMs under different spanwise phase relationships.

By combining the instantaneous velocity fields with the footprint structures revealed by the conditional velocity correlations, a coherent mechanistic picture of roughness-induced large-scale organisation emerges. Spanwise-heterogeneous roughness introduces a persistent geometric constraint on the near-wall flow, leading to a phase-locked spatial organisation of large-scale coherent motions through systematic cross-strip momentum redistribution. This organisation manifests as alternating converging and dispersive transverse motions with a fixed spanwise phase relative to the roughness geometry.

In particular, the conditional correlation patterns indicate paired converging and dispersive motions whose spanwise locations are aligned with regions occupied by pVLSMs and nVLSMs. Dispersive motions are preferentially associated with the inter-strip gaps, whereas converging motions predominantly occur above the roughness strips. This phase correspondence is consistent with a dominant organisational mode characterised by convergence over the strips and dispersion over the gaps. Through the associated spanwise-velocity correlation footprints, this mode supports the interpretation of a spanwise-segregated arrangement in which nVLSMs preferentially occur above the roughness strips, whereas pVLSMs are more frequently found in the inter-strip regions, with a periodicity consistent with that of the roughness pattern.

Within a limited near-wall region, a secondary organisational mode is also observed, characterised by an opposite phase relationship but markedly weaker intensity. The coexistence of these modes indicates that the roughness-induced organisation is not monolithic, but instead involves multiple competing arrangements whose relative prominence varies with wall-normal position. With increasing wall-normal distance (e.g. $y/h \approx 0.5$ –0.8), the direct geometric constraint imposed by the roughness progressively weakens. Correspondingly, the strip-periodic signatures and phase locking evident in the conditional correlation fields decay substantially. At these heights, the spanwise localisation of pVLSMs and nVLSMs is no longer tightly controlled by the roughness-strip positions, and the large-scale organisation gradually approaches that characteristic of smooth-wall flows, exhibiting weaker cross-strip modulation, reduced coherence and a more spatially diffuse structure.

3.4. Inter-scale energy transfer

The preceding results show that spanwise-heterogeneous roughness generates pronounced SCs, which reorganise the large-scale streamwise velocity field, and modulate the occurrence and organisation of VLSMs with large characteristic streamwise lengths $L_{\textit{CP}}$ . At the same time, the rough-wall cases exhibit an overall increase in wall shear stress, consistent with previous studies on roughness-induced drag enhancement. While the present results reveal a sign-dependent reorganisation of high- $L_{\textit{CP}}$ VLSM structures under rough-wall conditions, Jing et al. (Reference Jing, Duan, Li, Cao and Zhu2025) reported that the contribution of VLSMs to wall shear stress is significantly reduced in such flows, suggesting a decoupling between VLSM coherence and near-wall momentum transfer. This apparent discrepancy indicates that the SC-induced reorganisation of VLSMs does not directly translate into enhanced near-wall stress contributions, but instead reflects a redistribution of energy and momentum across scales and wall-normal locations. To elucidate this mechanism, the following section examines the associated energy-transfer processes, with particular emphasis on how inter-scale redistribution governs the role of VLSMs in near-wall turbulence modulation.

Figure 15 presents the premultiplied one-dimensional streamwise wavenumber spectra of the streamwise velocity and the Reynolds shear stress at representative spanwise locations relative to the roughness strips. As demonstrated earlier, spanwise-heterogeneous roughness induces converging and dispersive transverse motions with a fixed spanwise phase in the near-wall region, whose spatial distribution is closely aligned with the roughness-strip geometry. In regions dominated by converging motions, streamwise velocity fluctuations are more readily accumulated and sustained over larger streamwise length scales. This behaviour is reflected in the pronounced enhancement of the premultiplied streamwise velocity spectra $k_x F_{uu}(k_x)/u_{\tau }^2$ within the range $\lambda _x/h = \mathcal{O}(1)$ above the roughness strips, where $u_{\tau }$ is the friction velocity and serves as the characteristic near-wall velocity scale.

At the same spanwise locations, the presence of converging transverse motions promotes cross-strip momentum accumulation and exchange with the mean flow. As a consequence, the Reynolds shear stress spectra $|k_x C_{uv}(k_x)|/u_{\tau }^2$ exhibit a concurrent increase at comparable streamwise wavelengths. This indicates that the large-scale streamwise structures supported in these regions are more effective in contributing to momentum transport. In contrast, the inter-strip gap regions are characterised predominantly by dispersive transverse motions. These motions tend to redistribute momentum laterally and weaken the streamwise coherence of velocity fluctuations, thereby limiting the contribution of large-scale structures to both $k_x F_{uu}(k_x)/u_{\tau }^2$ and $|k_x C_{uv}(k_x)|/u_{\tau }^2$ . Accordingly, the spectral energy and stress levels at $\lambda _x/h=\mathcal{O}(1)$ are reduced over the gaps relative to those observed above the roughness strips.

Overall, the spanwise heterogeneity observed in the premultiplied spectra does not simply reflect differences in local turbulence intensity. Instead, it is closely associated with the roughness-induced organisation of converging and dispersive transverse motions, highlighting a large-scale transport mechanism modulated by SCs that governs the spatial distribution of spectral energy and momentum flux. To examine the inter-scale energy-transfer processes underlying the observed spectral organisation, the analysis is formulated in terms of the spectral turbulent kinetic energy transport equation, which provides a suitable framework for assessing energy redistribution under roughness-induced SCs.

Figure 15.

Premultiplied one-dimensional streamwise wavenumber spectra of (a–c) the streamwise velocity variance $k_x F_{uu}(k_x)/u_{\tau }^2$ and (d–f) the Reynolds shear stress $|k_x C_{uv}(k_x)|/u_{\tau }^2$ at three representative spanwise locations relative to the roughness strips. Results are shown for the smooth-wall case and three spanwise-heterogeneous roughness cases (R13, R17 and R20). The abscissa denotes the streamwise wavelength normalised by the channel height, $\lambda _x/h$ . Panels (a, d), (b, e) and (c, f) correspond to $z/h=2.4$ (over gap), $z/h=3.0$ (over stripes) and $z/h=3.6$ (over gap), respectively.

Figure 15. Long description

The image contains six line graphs depicting the premultiplied one-dimensional streamwise wavenumber spectra of streamwise velocity variance and Reynolds shear stress at three representative spanwise locations relative to the roughness strips. The graphs are organized into two rows and three columns, labeled as panels (a) to (f). Panel A, Panel B, and Panel C in the top row show the streamwise velocity variance, while Panel D, Panel E, and Panel F in the bottom row show the Reynolds shear stress. Each panel compares the smooth-wall case with three spanwise-heterogeneous roughness cases (R13, R17, and R20). The x-axis in all panels represents the streamwise wavelength normalized by the channel height, denoted as λx/h. The y-axis in panels (a), (b), and (c) represents kxFuu/(uτ)^2, while in panels (d), (e), and (f) it represents |kxCuv|/(uτ)^2. The legend indicates the different cases: Smooth (black), R13 (red), R17 (blue), and R20 (green). Panels (a) and (d) correspond to the location over the gap, panels (b) and (e) correspond to the location over the stripes, and panels (c) and (f) correspond to the location over the gap. Each graph shows how the spectra vary across different spanwise locations and roughness conditions, highlighting the energetic and coherent structures in the outer layer of wall turbulence.

In the present study, we employ the spectral energy-transfer framework of Cho, Hwang & Choi (Reference Cho, Hwang and Choi2018) to investigate how the VLSMs, induced by the uniformly distributed roughness strips at the open-channel bed, modify the inter-scale energy exchange processes in turbulent open-channel flow. Understanding this modulation is essential, as the VLSMs introduce pronounced large-scale coherence and modify the transfer pathways through which turbulent kinetic energy (TKE) is redistributed across scales. To diagnose such mechanisms, we analyse the spectral form of the TKE transport equation, for which a decomposition in the spanwise direction is particularly appropriate. As emphasised by Hwang (Reference Hwang2015), the characteristic size of the energy-containing motions in wall turbulence is most clearly organised by the spanwise length scale; consequently, a Fourier-mode decomposition in the spanwise direction provides a natural and physically meaningful basis for isolating structures of different scales. With this decomposition, the fluctuation field is expressed as

(3.9) \begin{equation} u_i^{\prime}(t,x,y,z)=\int _{-\infty }^{\infty }\hat {u_i}^{\prime}(t,x,y,k_z)\,e^{ik_z z}\,\mathrm{d}k_z, \end{equation}

where $\hat {\phantom {u}}$ denotes the Fourier-transformed coefficient and $k_z$ is the spanwise wavenumber. Substituting (3.9) into the fluctuation momentum equations and multiplying by $\hat {u_i^{\prime}}^{*}(k_z)$ , where the superscript $*$ denotes the complex conjugate, followed by averaging in time and the streamwise direction, yields the spectral turbulent kinetic energy budget,

(3.10) \begin{equation} \begin{aligned} \overline {\frac {\partial \hat {e}(k_z)}{\partial t}} &= \underbrace { \left \langle \textit{\textit{Re}}\left \{ - \overline {\hat {u'}^{*}(k_z)\,\hat {v'}(k_z)}\, \frac {\mathrm{d}U}{\mathrm{d}y} \right \} \right \rangle _x }_{\hat {P}(y,k_z)} + \underbrace { \left \langle - \nu \overline {\frac {\partial \hat {u_i^{\prime}}^{*}(k_z)}{\partial x_j} \frac {\partial \hat {u_i^{\prime}}(k_z)}{\partial x_j}} \right \rangle _x }_{\hat {\varepsilon }(y,k_z)} \\[8pt] &\quad + \underbrace { \left \langle \textit{\textit{Re}}\left \{ - \overline { \hat {u_i^{\prime}}^{*}(k_z)\, \frac {\partial }{\partial x_j} \left ( \widehat {u_i^{\prime} u_{\!j}^{\prime}}(k_z) \right ) } \right \} \right \rangle _x }_{\hat {T}_{\textit{turb}}(y,k_z)} + \underbrace { \left \langle \textit{\textit{Re}}\left \{ \frac {\mathrm{d}}{\mathrm{d}y} \left ( - \overline {\frac {\hat {p'}(k_z)\,\hat {v'}^{*}(k_z)}{\rho }} \right ) \right \} \right \rangle _x }_{\hat {T}_{p}(y,k_z)} \\[8pt] &\quad + \underbrace { \left \langle \nu \frac {\mathrm{d}^2 \overline {\hat {e}(k_z)}}{\mathrm{d}y^2} \right \rangle _x }_{\hat {T}_{\nu }(y,k_z)} . \end{aligned} \end{equation}

Here, $\hat {P}(y,k_z)$ is the mean-shear production, $\hat {\varepsilon }(y,k_z)$ and $\hat {T}_{\nu }(y,k_z)$ denote viscous dissipation and transport, and $\hat {T}_{p}(y,k_z)$ is the pressure-transport term. The turbulent transport term $\hat {T}_{\textit{turb}}(y,k_z)$ is the sole contributor to genuine inter-scale energy transfer and, therefore, forms the core of the present analysis.

The premultiplied spectral budgets of turbulent kinetic energy reveal a clear separation between the spatial characteristics of production and dissipation in the presence of spanwise-heterogeneous roughness. As shown in figure 16(a, c), turbulent kinetic energy production above the roughness crests exhibits a pronounced intermittency in the $(y^{+},\lambda _z^{+})$ space. This intermittency reflects the strong dependence of production on locally enhanced mean shear and its modulation by roughness-induced SCs, which organise large-scale coherent motions in a spatially non-uniform manner. In contrast, the dissipation term remains comparatively continuous across both wall-normal and spanwise scales, indicating that near-wall small-scale motions are persistently sustained rather than being directly tied to localised production events.

Figure 16.

Premultiplied spectral budgets of turbulent kinetic energy for the four cases (Smooth, R13, R17 and R20) in the $(y^{+},\lambda _z^{+})$ space: (a) production; (b) turbulent transport; (c) viscous transport and (d) viscous dissipation. The grey dashed line marks the roughness crest location $y_0$ . The solid black line denotes $\lambda _z = 5y$ , corresponding to the characteristic spanwise scale of energy-containing motions and marking the ridge of turbulent kinetic energy production associated with attached eddies. The dashed black line indicates $\lambda _z = 5\eta$ , where $\eta$ is the Kolmogorov length scale. This line identifies the dominant scale of viscous transport and dissipation.

The transport terms further highlight this imbalance. Consistent with the interpretation proposed by Hinze (Reference Hinze1967), SCs develop in regions characterised by a local imbalance between turbulent production and dissipation, typically near spanwise transitions in surface conditions. Turbulent and viscous transport, shown in figures 16(b) and 16(c), redistribute energy away from regions of intermittent production and sustain near-wall turbulence through spanwise and wall-normal transport pathways associated with secondary motions. As a result, production and dissipation become increasingly decoupled in space, demonstrating that the classical local balance between these two terms no longer holds under spanwise-heterogeneous roughness. Instead, a transport-dominated mechanism emerges in which SCs reorganise the near-wall energy budget by mediating the redistribution of energy across both spatial locations and scales.

Figure 17.

Premultiplied spectral budgets of turbulent kinetic energy for the four cases (Smooth, R13, R17 and R20) in outer scaling, shown in the $(y/h,\lambda _z/h)$ space: (a) production; (b) turbulent transport; (c) viscous transport and (d) viscous dissipation. The grey dashed line denotes the roughness-crest location $y_0/h$ .

While figure 16 in inner scaling highlights the near-wall spectral organisation, and the local imbalance among production, transport and dissipation, such a representation is less suited to assessing the wall-normal extent of the roughness effect from the perspective of outer-layer large-scale turbulence. To complement this view, the same premultiplied spectral budgets are plotted in outer scaling in figure 17, with both wall-normal distance and spanwise wavelength normalised by $h$ . In this form, the distinction between the smooth- and rough-wall cases remains clear over an $O(h)$ wall-normal range. In particular, the rough-wall cases retain substantially stronger and more spatially organised production above the roughness crests, indicating that the influence of roughness-induced secondary currents is not confined to the immediate vicinity of the roughness elements, but extends into the outer region through a large-scale reorganisation of the mean-shear-dependent energy input. The turbulent-transport term exhibits an even more pronounced roughness dependence, with stronger negative contributions persisting over a broad wall-normal range, which indicates that the energy introduced at these scales is actively redistributed rather than locally balanced. In contrast, the viscous-transport term remains restricted to a thin near-wall layer when expressed in outer units, showing that the dominant roughness effect at large scales is not an outward extension of viscous activity. The dissipation field, although still concentrated towards smaller scales, is likewise enhanced in the rough-wall cases over a wider wall-normal extent, suggesting that the redistributed energy is ultimately transferred towards smaller-scale motions. Taken together, the outer-scaled representation reinforces the interpretation that spanwise-heterogeneous roughness introduces an energetically significant outer-layer pathway, through which secondary currents reorganise the balance among production, transport and dissipation across both space and scale.

To further assess how SC-dominated motions contribute to this transport process, it is necessary to identify the characteristic spanwise scales at which these motions are active. This is achieved by first examining the premultiplied spanwise energy spectra of the streamwise velocity fluctuations, which provide a quantitative measure of the dominant spanwise wavelengths associated with roughness-induced large-scale organisation. Based on this, the subsequent analysis focuses on the premultiplied spectral budgets of turbulent kinetic energy in the $(y^{+},\lambda _z^{+})$ space, as shown in figure 16, where the scale-dependent roles of production, transport and dissipation can be evaluated in a unified framework.

Figure 18 presents the premultiplied spanwise energy spectra of the streamwise velocity fluctuations as functions of $y^{+}$ and $\lambda _z/h$ . To further clarify the dominant spanwise scales in different wall-normal regions, we additionally examine the corresponding one-dimensional premultiplied spectra averaged over two wall-normal intervals, namely the near-wall region $y/h \in [0,0.3]$ and the outer region $y/h \in [0.3,1.0]$ , as shown in figures 18(e) and 18(f). This analysis reveals two distinct characteristic scales. The smooth-wall case already exhibits an outer-region peak at $\lambda _z/h = O(1)$ , indicating that an outer-scale spanwise organisation is present even in the absence of roughness-induced SCs. However, in the rough-wall cases, this outer-scale peak becomes substantially stronger, showing that spanwise-heterogeneous roughness reinforces and geometrically locks this outer-scale organisation. We therefore interpret the strengthened outer-region peak at $\lambda _z/h = O(1)$ as the outer-scale organisational footprint of the roughness-induced SCs. By contrast, the spectra in the near-wall region above the roughness crests exhibit a separate peak at a smaller spanwise wavelength, $\lambda _z/h \approx 0.3$ . This smaller peak is not interpreted as the unique spectral signature of the SCs, but rather as a roughness-locked near-wall active scale. It should also be noted that the spectra shown in figure 18 are spanwise-averaged. Such averaging smooths local near-bed signatures and weakens the corresponding spectral peak close to the bed.

Figure 18.

Premultiplied spanwise energy spectra of the streamwise velocity fluctuations in the $(y^{+},\lambda _z/h)$ plane: (a)–(d) smooth, R13, R17 and R20 cases, respectively. For the rough-wall cases, two distinct spectral features are identified: a smaller near-wall peak located slightly above the roughness crests and an outer-region peak at $\lambda _z/h = O(1)$ . The red markers in panels (b)–(d) denote the roughness-locked near-wall active scale, whereas the blue markers denote the outer-scale organisational footprint associated with the roughness-induced secondary-current organisation. (e) and (f) Corresponding one-dimensional premultiplied spanwise spectra averaged over $y/h \in [0,0.3]$ and $y/h \in [0.3,1.0]$ , respectively.

The distinction between the strengthened outer-scale footprint and the smaller near-wall active scale is essential for interpreting the nonlinear transport process, because the two scales may share negative net turbulent transport while exhibiting different internal transfer characteristics. Based on this, the following analysis performs separate triadic decompositions at these two scales. The decomposition at $\lambda _z/h = O(1)$ is introduced to isolate the transport mechanism associated with the outer-scale SC organisation itself, whereas that at $\lambda _z/h \approx 0.3$ is used to identify the energetically most active near-wall redistribution pathway above the roughness crests. This two-scale decomposition is necessary because a single target scale would conflate two physically distinct processes: the roughness-enhanced outer-scale organisation and the roughness-locked near-wall redistribution, through which energy received from larger scales is subsequently transferred to smaller scales. By separating these two target scales, the triadic analysis can more precisely assess how roughness-induced SCs reorganise the cross-scale energy pathway from outer-scale organisation to near-wall redistribution.

Following Cho et al. (Reference Cho, Hwang and Choi2018), the reference lines at $l/k_{z,0}=1$ and $m/k_{z,0}=1$ distinguish interactions involving eddies larger than the target scale from those involving smaller eddies. At a given wall-normal location $y^{+}$ and target spanwise wavenumber $k_{z,0}$ , corresponding to the spanwise wavelength $\lambda _{z,0}=2\pi /k_{z,0}$ , the turbulent transport spectrum can be written as

(3.11) \begin{equation} \hat {T}_{\textit{turb}}(y,k_{z,0}) = \left \langle \textit{\textit{Re}} \left \{ -\overline {\hat {u_i^{\prime}}^{*}(y,k_{z,0}) \frac {\partial }{\partial x_j} \sum _{l+m=k_{z,0}} \hat {u_i^{\prime}}(y,l)\hat {u_{\!j}^{\prime}}(y,m)} \right \} \right \rangle _x , \end{equation}

where $l$ and $m$ denote the spanwise wavenumbers participating in the triadic interaction, the overbar $\overline {(\boldsymbol{\cdot })}$ denotes time averaging, and $\langle \boldsymbol{\cdot }\rangle _x$ denotes averaging in the streamwise direction. Equation (3.11) shows that the turbulent transport at the scale $k_{z,0}$ arises exclusively from nonlinear triadic interactions among Fourier modes whose spanwise wavenumbers satisfy the resonance condition $l+m=k_{z,0}$ .

Figure 19.

Schematic classification of triadic interactions in the $(l/k_{z,0},\, m/k_{z,0})$ plane based on the spanwise wavelengths of the interacting eddies relative to the target Fourier mode $\lambda _{z,0}=2\pi /k_{z,0}$ . The dashed lines at $l/k_{z,0}=1$ and $m/k_{z,0}=1$ separate interactions involving eddies larger than $\lambda _{z,0}$ from those involving smaller-scale eddies.

Because the Fourier transform is applied to real-valued velocity fields, the Fourier coefficients at negative wavenumbers correspond to the complex conjugates of those at positive wavenumbers. Accordingly, for positive $k_{z,0}$ , the convolution sum in (3.11) can be rearranged as

(3.12) \begin{equation} \sum _{l+m=k_{z,0}} \!\hat {u_i^{\prime}}(l)\hat {u_{\!j}^{\prime}}(m) = \!\sum _{\substack {l+m=k_{z,0}\\ l,m\geqslant 0}} \!\hat {u_i^{\prime}}(l)\hat {u_{\!j}^{\prime}}(m) + \!\sum _{\substack {-l+m=k_{z,0}\\ l,m\geqslant 0}} \!\hat {u_i^{\prime}}(-l)\hat {u_{\!j}^{\prime}}(m) +\! \sum _{\substack {l-m=k_{z,0}\\ l,m\geqslant 0}} \!\hat {u_i^{\prime}}(l)\hat {u_{\!j}^{\prime}}(-m), \end{equation}

indicating that all contributing triads can be represented within the first quadrant of the $l$ – $m$ plane ( $l,m\gt 0$ ). Moreover, (3.11) implies the symmetry relation

(3.13) \begin{equation} \hat {T}_{\textit{turb}}\big(y,-k_{z,0}\big) = \hat {T}_{\textit{turb}}^{*}\big(y,k_{z,0}\big) = \hat {T}_{\textit{turb}}\big(y,k_{z,0}\big), \end{equation}

so that negative values of $k_{z,0}$ need not be considered separately.

Within this framework, each triadic interaction is classified according to the spanwise wavelengths of the interacting eddies relative to that of the target Fourier mode $\lambda _{z,0}$ . As illustrated schematically in figure 19, the axes $l/k_{z,0}$ and $m/k_{z,0}$ distinguish interactions involving eddies larger than $\lambda _{z,0}$ , smaller than $\lambda _{z,0}$ or mixed-scale interactions between the two. The four regions of the $l$ – $m$ plane therefore correspond to interactions between two larger-scale eddies, two smaller-scale eddies or combinations of larger- and smaller-scale motions relative to the target scale.

Figure 20.

Triadic interaction maps in the $(l/k_{z,0},\,m/k_{z,0})$ plane for the target scale corresponding to the outer-scale organisational footprint identified in figure 18: (a) smooth-wall case for reference, and (b–d) cases R13, R17 and R20, respectively. Warm colours denote net energy influx to the selected target mode, while cool colours denote net energy removal from it. The horizontal and vertical dashed lines at $l/k_{z,0}=1$ and $m/k_{z,0}=1$ separate interactions involving eddies larger than the target scale from those involving smaller eddies, following the classification of Cho et al. (Reference Cho, Hwang and Choi2018). The dotted lines indicate the admissible wavenumber-matching branches for the triadic interactions.

Figure 21.

Triadic interaction maps in the $(l/k_{z,0},\,m/k_{z,0})$ plane. Coloured markers denote triads satisfying the wavenumber-matching conditions, with red and blue symbols representing different interaction channels as defined in the analysis. The dotted diagonal lines indicate $(l=m)$ and its shifted counterparts, while the horizontal and vertical dashed lines mark the reference wavenumber $k_{z,0}$ : (a) the smooth-wall case, and (b–d) cases R13, R17 and R20, respectively.

To clarify the scale-dependent origin of the nonlinear turbulent transport, we separately examine the triadic interactions at the two characteristic spanwise scales identified from figure 18. We begin with the larger scale, $\lambda _z/h = O(1)$ , which corresponds to the strengthened outer-scale organisational footprint, and then consider the smaller roughness-locked near-wall active scale at $\lambda _z/h \approx 0.3$ .

The triadic interaction maps at the outer-scale organisational footprint are shown in figure 20. At this target scale, the rough-wall cases exhibit a transport pattern that remains broadly consistent with the classical picture of negative turbulent transport. In particular, the contributions associated with triads involving eddies larger than the selected target scale are predominantly negative, indicating that larger-scale motions participate in the net energy removal from the target mode. This behaviour is characteristic of a classically negatively transported scale, for which the selected mode is embedded in a forward redistribution process and loses energy through nonlinear interactions with larger energy-containing motions. In this sense, the scale $\lambda _z/h = O(1)$ captures the dominant outer spectral signature of the roughness-induced spanwise organisation and behaves as the primary organisational scale of the secondary-current system in spectral space.

A qualitatively different transfer pattern is observed when the target scale is shifted to the smaller roughness-locked near-wall active scale $\lambda _z/h \approx 0.3$ , as shown in figure 21. At this scale, the dominant contribution from larger-scale motions becomes predominantly positive in the rough-wall cases, indicating that the selected target mode receives net energy input from eddies larger than itself. However, the total turbulent transport at this scale remains negative. This combination of positive large-scale input and negative net transport shows that the injected energy is not accumulated at the target scale. Instead, $\lambda _z/h \approx 0.3$ is characterised by a through-flow type of transport behaviour: it acts as a near-wall relay scale that receives energy from larger-scale motions and rapidly redirects it towards smaller scales. In this sense, the energetic role of the $\lambda _z/h \approx 0.3$ mode is not to store energy, but to mediate the most active stage of near-wall roughness-induced redistribution above the roughness crests.

Taken together, figures 20 and 21 show that roughness reorganises the large-to-small-scale energy pathway through the coupled action of two dynamically distinct but physically connected spanwise scales. At $\lambda _z/h = O(1)$ , corresponding to the strengthened outer-scale organisational footprint, the selected target mode behaves as a classically negatively transported organisational scale: larger-scale motions make a dominant contribution to the net energy removal from this mode, consistent with the standard integral-scale picture of negative turbulent transport. In contrast, at the smaller roughness-locked near-wall active scale, $\lambda _z/h \approx 0.3$ , the target mode exhibits a different transport character. Although the net turbulent transport remains negative, the dominant contribution from larger-scale motions is positive, indicating that this scale first receives energy from larger scales and then redirects it towards smaller motions. The two scales therefore play complementary roles in the roughness-induced transport process: the outer $O(1)$ scale captures the dominant spanwise organisation associated with the secondary-current system, whereas the smaller near-wall scale marks the location at which roughness-induced redistribution is most energetically active.

This distinction is essential because both target scales are associated with negative net turbulent transport, yet they represent different internal pathways of energy transfer. The outer-scale footprint behaves as a negatively transported organisational scale embedded in the classical forward redistribution process, whereas the smaller near-wall active scale behaves as a relay scale that mediates the onward transfer of energy after receiving input from larger-scale motions. The energetic role of roughness-induced SCs should therefore be understood not through a single preferred wavelength, but through a scale-dependent coupling between outer-scale organisation and near-wall redistribution.

3.5. Kinematic organisation and energy-redistribution pathways

Spanwise roughness strips induce streamwise-aligned secondary currents (SCs) that penetrate from the near-wall region into the outer layer, forming persistent and geometry-locked circulation patterns. This behaviour is consistent with previous DNS studies showing that spanwise heterogeneity in wall conditions gives rise to secondary roll cells whose structure and strength are governed by the roughness elevation contrast and the associated drag distribution (Anderson et al. Reference Anderson, Barros, Christensen and Awasthi2015; Stroh et al. Reference Stroh, Schäfer, Frohnapfel and Forooghi2020). The present results extend these observations by quantifying the dominant contribution of form-induced stresses to momentum redistribution and by identifying the spectral signatures through which these geometry-imposed secondary motions reorganise the flow.

The resulting flow heterogeneity is primarily reflected in the dominance of form-induced stresses in the total Reynolds stress budget. Rather than a uniform amplification of turbulence intensity, momentum transport is reorganised through large-scale, geometry-locked SCs. Similar behaviour has been reported by Medjnoun et al. (Reference Medjnoun, Vanderwel and Ganapathisubramani2018), who showed that spatial variations in wall shear generate organised dispersive stresses with limited impact on local turbulence production. More recently, Ma et al. (Reference Ma, Xu, Sung, Tian and Huang2025) demonstrated that form-induced structures in rough-wall channel flows govern inter-scale energy redistribution, effectively decoupling large-scale organisation from enhanced small-scale turbulence. The present results further indicate that the spatial structure of Reynolds stress is largely controlled by the persistence and geometry of the SCs, rather than by a simple increase in background turbulent activity.

These SCs organise the mean flow into alternating high-momentum pathways (HMPs) and low-momentum pathways (LMPs) aligned with the roughness geometry and extending from the near-wall region into the outer layer. Similar streamwise-aligned roll cells and large-scale spatial organisation have been reported in rough-wall flows with comparable heterogeneity scales (Wangsawijaya et al. Reference Wangsawijaya, Baidya, Chung, Marusic and Hutchins2020; Zampiron et al. Reference Zampiron, Cameron and Nikora2020). Within this organised mean-flow framework, large-scale motions exhibit a differentiated response: while LSMs are suppressed, VLSMs persist and become preferentially localised. In particular, the present results reveal a clear spatial segregation of positive- and negative-velocity VLSMs across the roughness pattern, with nVLSMs preferentially concentrated above the roughness strips and pVLSMs more frequently found in the inter-strip regions.

Although VLSMs are identified in the outer region, the conditional correlation analysis indicates that, within the present friction-Reynolds-number range and computational-domain size, their strongest statistical footprint remains located near the wall and is collocated with regions of enhanced form-induced stress. This observation suggests that the formation and organisation of VLSMs remain closely connected to near-wall turbulent vortical motions, whose intensity sustains persistent form-induced momentum transport. In this sense, the VLSMs observed in the present simulations should not be interpreted as dynamically isolated outer-layer structures, but rather as motions that remain strongly influenced by roughness-induced near-wall turbulence heterogeneity under the present flow conditions. Jing et al. (Reference Jing, Duan, Li, Cao and Zhu2025) reported that the contribution of VLSMs to wall shear stress is reduced in rough-wall flows, potentially owing to the interference of strong secondary motions. Building on this observation, the present results suggest that, at the Reynolds number and domain size considered here, VLSMs remain dynamically coupled to near-wall turbulent vortical activity, which mediates the redistribution of VLSM-related energy towards form-induced stresses and smaller-scale motions. Owing to computational limitations, however, whether the same coupling persists at significantly higher friction Reynolds numbers and in substantially longer computational domains remains to be examined in future numerical or experimental studies.

The spectral analyses refine this picture by showing that the roughness-induced transport pathway cannot be described by a single spanwise scale. Instead, two dynamically distinct but physically connected characteristic scales emerge. The first is an outer-scale organisational footprint at $\lambda _z/h = O(1)$ , which is also present in the smooth-wall flow but becomes substantially stronger in the rough-wall cases. This indicates that spanwise-heterogeneous roughness does not generate the outer-scale organisation from scratch, but rather reinforces and geometrically locks a pre-existing outer-scale spanwise organisation through the action of the SCs. The second is a smaller roughness-locked near-wall active scale at $\lambda _z/h \approx 0.3$ , located slightly above the roughness crests, where the negative turbulent transport is strongest.

These two scales occupy different positions in the multi-scale transfer pathway linking the outer large-scale motions to the near-wall region. The larger $\lambda _z/h = O(1)$ scale corresponds to the dominant spanwise organisation of the SC system and represents the scale at which the roughness-enhanced secondary-current structure is most clearly expressed in spectral space. In contrast, the smaller $\lambda _z/h \approx 0.3$ scale corresponds to the near-wall redistribution stage, where energy transferred from larger scales is most actively processed and redirected towards smaller motions. In this sense, the two scales should not be viewed as competing definitions of the SC signature, but rather as two connected levels of the same roughness-induced transport pathway: the $O(1)$ scale reflects outer-scale organisation, whereas the $0.3$ scale reflects near-wall redistribution.

Figure 22.

Schematic of the kinematic organisation and energy-redistribution pathways associated with roughness-induced SCs, and their interaction with positive- and negative-velocity very-large-scale motions (pVLSMs and nVLSMs) over spanwise-heterogeneous roughness.

Triadic interaction analysis further clarifies the distinct energetic roles of these two scales. At the outer-scale organisational footprint, the transport pattern remains broadly consistent with the classical picture of negative turbulent transport: larger-scale motions contribute predominantly to the net energy removal from the selected target mode. This indicates that the outer-scale footprint behaves as a classically negatively transported organisational scale associated with the large-scale spanwise structure of the SC system. At the smaller roughness-locked near-wall active scale, however, the dominant contribution from larger-scale motions becomes positive, even though the net turbulent transport at that scale remains negative. This means that the near-wall target mode first receives energy from larger-scale motions and then rapidly redirects that energy towards smaller-scale turbulence. The smaller near-wall scale therefore acts as a receiving-and-redistributing relay scale, rather than as a simple energy reservoir.

Viewed together, these results suggest a scale-ordered pathway in which very-large-scale motions and other outer large-scale structures provide the upstream energy source, the $O(1)$ spanwise scale organises that energy within the roughness-induced secondary-current system and the smaller $\lambda _z/h \approx 0.3$ scale mediates the most active near-wall redistribution towards smaller-scale turbulence and form-induced stresses. The energetic role of roughness-induced SCs is therefore not exhausted by identifying a single preferred wavelength or by viewing them as a single-scale extractor. Instead, the SCs reorganise the large-to-small-scale energy pathway through the coupled action of a strengthened outer-scale organisational footprint and a smaller near-wall active redistribution scale. The former captures the dominant spanwise organisation of the SC system, whereas the latter marks the location at which roughness-induced redistribution is most energetically active.

Figure 22 synthesises these findings by illustrating the coupled kinematic and energetic roles of SCs. From a kinematic perspective, roughness-induced SCs generate alternating converging and dispersive transverse motions that spatially segregate positive- and negative-velocity VLSMs, with nVLSMs preferentially concentrated above the roughness strips and pVLSMs more frequently found in the inter-strip regions. From an energetic perspective, the outer-scale organisational footprint of the SC system links the roughness-imposed spanwise structure to the broader outer large-scale motions, while the smaller near-wall active scale mediates the onward redistribution of energy received from larger motions towards smaller-scale turbulence and form-induced stresses. Together, these perspectives highlight the central role of roughness-induced SCs in controlling both the kinematic organisation and the energy-redistribution pathways of VLSMs in spanwise-heterogeneous flows.