2. Computational set-up

The current work employs bio-inspired surfaces of biofouling type that are commonly found on the hulls of naval vessels (see for example Schultz Reference Schultz2007). Specifically, four topographies are considered, one in a staggered arrangement (BS39) comprising only barnacle-type organisms and three in random arrangements comprising (i) only barnacle-type (BR39) organisms, (ii) only tubeworm-type (TR39) organisms and (iii) a mixture (MR39) of barnacle- (20 %) and tubeworm-type (80 %) organisms (see figure 1). The barnacle-type organisms are modelled via truncated cones, following the work of Womack et al. (Reference Womack, Volino, Meneveau and Schultz2022), while the tubeworm-type organisms are designed to mimic scans of actual organisms found on ship hulls (see Kaminaris & Balaras Reference Kaminaris and Balaras2024, for details). All topographies have the same surface coverage, $\lambda _p\approx 40\,\%$ , and frontal solidity, $13.11\,\%\lt \lambda _f\lt 15.9\,\%$ while they all have skewness, $Sk\gt 1$ , which is characteristic of biofouling surfaces. The main difference between the barnacle-type (BS39, BR39) and the mixed- (MR39) or tubeworm-type (TR39) topographies is the mean height, which is $\overline {h}/\delta _{\textit{LE}} \approx 0.045$ for the former and $\overline {h}/\delta _{\textit{LE}} \approx 0.025$ for the latter ( $\delta _{\textit{LE}}$ is the boundary-layer thickness at the leading edge of the roughness). The key surface statistics for all topographies are listed in table 1. Furthermore, all flows are fully rough, $k_s^+\gt$ 100. A smooth-wall flat plate is also analysed as a reference case.

Table 1.

Surface characteristics for the biofouling-type topographies considered in this work.

The computational domain is shown in figure 2(b), where the roughness patch is positioned $15\delta _{\textit{LE}}$ downstream from the inflow plane and extends $60\delta _{\textit{LE}}$ , followed by another smooth-wall portion. The overall dimensions of the computational domain are $100\delta _{\textit{LE}} \times 14\delta _{\textit{LE}} \times 10\delta _{\textit{LE}}$ in the streamwise, spanwise and wall-normal directions, respectively. A precursor simulation is performed to extract velocity boundary conditions for the inflow plane: a laminar boundary layer is tripped by a dimple topography and is sampled at the streamwise distance where the desired Reynolds number is matched (see figure 2 a). A structured Cartesian grid is used, and the boundary conditions on the roughness surface are imposed via an immersed boundary method (Balaras Reference Balaras2004). The grid within the roughness area is approximately uniform in all directions placing $40 \times 40 \times 70$ points in the streamwise, spanwise and wall-normal directions, respectively, for each barnacle organism. For the case of tubeworm organisms, their maximum dimension is resolved with approximately $140$ grid points. This translates into a grid resolution of $\Delta$ x $^{+}=10.5$ , $\Delta$ z $^{+}=9$ and $0.9 \leqslant \Delta$ y $^{+} \leqslant 20$ (based on the friction velocity, $u_\tau$ , on the smooth part of the plate), while the resolution in terms of Kolmogorov units is $\Delta y/\eta \sim 1.9$ right above the roughness crest of the barnacle-type surfaces, which is the location where $\eta$ becomes the smallest, and everywhere in the domain less than 6. For details on the grid resolution, the boundary conditions, as well as the numerical methods employed in the current study see Kaminaris et al. (Reference Kaminaris, Balaras, Schultz and Volino2023). In the latter, a detailed validation against the experimental work of Womack et al. (Reference Womack, Volino, Meneveau and Schultz2022) is also provided. It is noted that the friction Reynolds number range obtained between the leading edge and trailing edge of the roughness patch for the rough-wall topographies is $1700 \leqslant {{\textit{Re}}_{\tau ,{\textit{rough}}}} \leqslant 3150$ , and $750 \leqslant {{\textit{Re}}_{\tau ,{\textit{smooth}}}} \leqslant 1300$ for the smooth-wall case.

Figure 2.

Schematic of the computational domain. (a) Precursor simulation; (b) production runs.

Figure 3.

(a) Streamwise evolution of the boundary-layer thickness, $\delta /\delta _{\textit{LE}}$ ; (b) comparison of the streamwise evolution of the boundary-layer thickness, $\delta /\delta _{\textit{LE}}$ in the case of the BS39 case between the two different inlet boundary conditions used (see legend for details); (c) mean streamwise velocity profiles in inner coordinates at $x/\delta _{\textit{LE}}=65$ ; (d) mean streamwise velocity profiles in defect law at ${\textit{Re}}_\tau \approx$ 2100; (e) streamwise velocity root-mean-square profiles in outer coordinates at ${\textit{Re}}_\tau \approx$ 2100; ( f) streamwise velocity profiles in inner coordinates separated by a streamwise distance. For clarity symbols in (a)–(b) correspond to every hundredth streamwise point, while (c)–(e) correspond to every fifth wall-normal point. Smooth wall; BS39; BR39; MR39; TR39.

3. Superstructure evolution

First, in order to establish the impact each rough-wall case has on the mean flow, the boundary-layer evolution is presented in figure 3(a), by means of the boundary-layer thickness, $\delta /\delta _{\textit{LE}}$ over all the topographical cases studied herein. It can be clearly seen that the presence of roughness significantly impacts the boundary-layer growth, while the topographies with higher mean roughness heights, $\overline {h}$ (corresponding to higher barnacle-type concentrations) exhibit the highest growth rate. The surface cases BS39 and BR39, which all comprise approximately same height roughness elements, show minimal differences despite their different distributions. The hump formed in the boundary-layer thickness at the leading edge of the roughness is the result of the significant blockage ratios, $\overline {h}/\delta _{\textit{LE}}$ , which imposes pressure drag by the roughness and redirects the flow to overcome the roughness elements. It is noted that there is a small difference of ${\approx}7\,\%$ in the Reynolds number at the inlet of the computational domain between the smooth, BS39, BR39 and the MR39, TR39 cases, which, however, was not found to result in any difference in the boundary-layer evolution over the rough-wall patch according to a numerical experiment performed in the BS39 case (please see figure 3 b). Therefore, the inlet boundary condition is treated as same for all cases. To further quantify the roughness impact on the mean flow the streamwise velocity profile in inner coordinates is provided in figure 3(c). Through the latter, it can be seen that the roughness function, $\Delta U^+$ , which encapsulates the momentum deficit due to the presence of the roughness, also increases as the mean roughness height increases, similarly to the observations made regarding the boundary-layer growth. On the other hand, despite the significant impact the roughness has in the near-wall region, the outer part of the flow seems to behave similarly and independent of the topographical terrain underneath, as suggested by the mean streamwise velocity and streamwise velocity root-mean-square profiles in figure 3(d,e), plotted using the velocity scale introduced by Zagarola & Smits (Reference Zagarola and Smits1998) and further validated in boundary-layer configurations by Connelly, Schultz & Flack (Reference Connelly, Schultz and Flack2006). It is noted that the small differences between the rough- and smooth-wall data in figure 3(e) are expected and due to the difference in ${\textit{Re}}_\tau$ between the cases, given that the maximum smooth-wall friction Reynolds number is ${\textit{Re}}_\tau \approx 1\,300$ while the rough-wall ${\textit{Re}}_\tau \approx 2\,100$ ( ${\textit{Re}}_\tau =u_\tau \delta /\nu$ , where $u_\tau$ and $\delta$ is the local friction velocity and boundary-layer thickness), while $\nu$ is the kinematic viscosity. Note that the local friction velocity is computed via direct force integration on the surface (see Kaminaris et al. Reference Kaminaris, Balaras, Schultz and Volino2023 for details). The strong dependency of the Reynolds stresses on the Reynolds number is highlighted in great detail in Squire et al. (Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016b ) and Sillero et al. (Reference Sillero, Jiménez and Moser2013). Finally, the equilibrium state of the boundary layer is demonstrated via figure 3( f), where multiple streamwise velocity profiles are plotted along the boundary-layer evolution. The latter should not be taken for granted, since the boundary-layer flows of the present work initially develop over a smooth-wall region before entering the rough-wall patch and thus they carry some transition effects. Immediately downstream of the leading edge of the roughness the boundary layers have not yet reached an equilibrium state, which is eventually achieved after some development region. Figure 3( f) indicates that the flow herein starts following the canonical velocity profile again at $\approx$ 10 $\delta _{\textit{LE}}$ downstream of the leading edge of the roughness. The latter was also found to be largely the same for all the surfaces considered. Note the symbol $\textit {d}$ used in figure 3(c–f) corresponds to the zero-plane displacement, namely the virtual wall origin, that is, the distance which the wall has to be shifted to yield the same slope of the rough-wall logarithmic layer as the smooth-wall one, given the same constants $\kappa$ and $B$ .

Figure 4.

Top view of the isosurfaces of the streamwise velocity fluctuations, $u^\prime$ , in the cases of (a) smooth wall; (b) BS39; (c) BR39; (d) MR39; (e) TR39. Positive isosurface $u^\prime /U_e=0.13$ shown in red and negative isosurface $u^\prime /U_e=-0.13$ shown in blue. Green lines indicate the lengths of two typical structures at the upstream and downstream part of the BR39 topography.

Figure 5.

Side view of the contours of streamwise velocity fluctuations, $u^\prime /U_e$ , in the case of the rough wall with the random BR39 case.

Figure 6.

Streamwise evolution of the boundary-layer sublayers for BR39 case. Outer layer represented by lighter blue colour and its centreline by ; Logarithmic layer represented by darker blue colour and its centreline by .

Figure 4 presents a top view of iso-surfaces of positive and negative instantaneous streamwise velocity fluctuations, $u^\prime (\underline {x},t) = u(\underline {x},t) - \overline {u}(\underline {x})$ (where $\underline {x}=x\hat {i}+y\hat {j}+z\hat {k}$ ), highlighting the superstructures. Qualitatively, the superstructures become stronger and wider as they grow within the rough-wall patches compared with the smooth-wall ones, despite the fact that, in all simulations, the same boundary layer enters the domain of interest. Furthermore, the known tendency of the smooth-wall superstructures towards streamwise misalignment (also known as meandering) is also observed in the rough-wall topographies. In regard to the superstructures formed over the topographies with lower mean heights, $\overline {h}$ , (i.e. topographies with lower barnacle-type concentrations) shown in figure 4(d,e), it can be seen that they form a more slender shape compared with the increased mean heights ones (i.e. increased barnacle-type concentrations) of figure 4(b,c). The significant growth the superstructures exhibit is highlighted in figure 4(c), where both an upstream small superstructure of approximately $3\delta _{\textit{LE}}$ length and a downstream one of approximately $25\delta _{\textit{LE}}$ are indicated. In line with previous works (see for example Kevin et al. Reference Kevin, Monty and Hutchins2019a , Reference Kevin, Monty and Hutchins2019b ) we see that, also for the case of rough walls, these structures exhibit a streamwise inclination, which results in superstructures climbing to neighbouring ones of opposite sign as they evolve downstream (figure 5). This inclination is present throughout the boundary-layer evolution and qualitatively at almost constant angles of around $8^{\circ }$ .

Figure 7.

Streamwise two-point correlations of the streamwise velocity fluctuations, $R^{x}_{u^\prime u^\prime }$ , at $x/\delta _{\textit{LE}}=62.5$ and various wall-normal locations. Left side: smooth wall, ${\textit{Re}}_\tau =1150$ ; right side: BR39 case, ${\textit{Re}}_\tau =2900$ ; (a,b) at the centreline of viscous/roughness sublayer ( $y/\delta _{\textit{smooth}}=0.005$ , $y/\delta _{\textit{BR}39}=0.097$ ); (c,d) at the centreline of the logarithmic layer ( $y/\delta _{\textit{smooth}}=0.083$ , $y/\delta _{\textit{BR}39}=0.15$ ); (e, f) at the centreline of the outer layer ( $y/\delta _{\textit{smooth}}=0.57$ , $y/\delta _{\textit{BR}39}=0.59$ ); $\bullet$ experiments by Hutchins & Marusic (Reference Hutchins and Marusic2007) at ${\textit{Re}}_\tau =1120$ ; dashed lines correspond to the correlation cutoff of $R^{x}_{u^\prime u^\prime }=0.1$ .

In order to estimate the superstructure length, the time- and spanwise-averaged two-point correlations of the streamwise velocity fluctuations with respect to the streamwise lag are computed as

(3.1) \begin{equation} R^{x}_{u^\prime u^\prime }(x,y,\Delta x)= \left\langle \sum _{n=1}^{N_{\!x}}\left[u^\prime (x,y,z,t) u^\prime (x+\Delta x,y,z,t)\right]/\sum _{n=1}^{N_{\!x}}\big[u^{\prime 2}(x,y,z,t)\big]\right\rangle _{z,t}, \end{equation}

and presented in figure 7 for three different wall-normal locations: (i) the viscous sublayer/roughness sublayer (here referring to the wall-normal distance between the roughness crest an the onset of the logarithmic layer), (ii) the logarithmic layer and the (iii) outer layer. The two-point correlation functions are widely accepted as a rigorous way of estimating a signal correlation length and here are computed in a window of $10\delta$ -length, centred at $x/\delta _{\textit{LE}}=62.5$ . It should be noted that, despite the superstructure extent across the whole boundary-layer height, as was shown in figure 5, it is of interest to investigate their length-scale dependency on the wall-normal distance relative to the location of the three major sublayers. Thus, the normalised two-point correlations are computed at the centreline of each sublayer, which is tracked in space following the boundary-layer growth, as shown in figure 6. The sublayer limits are estimated based on the inner-scaled streamwise velocity profile (see figure 3 c). Specifically, the start of the outer layer is considered at $y/\delta =0.145$ for the smooth wall, at $y/\delta =0.19$ for the BS39, at $y/\delta =0.195$ for the BR39, at $y/\delta =0.148$ for the MR39 and at $y/\delta =0.14$ for the TR39 topography. In the context of the current paper, the roughness sublayer refers to the thin layer immediately above the roughness peak and the start of the logarithmic layer, as defined by the mean streamwise velocity profile in inner coordinates. The correlation length, $\mathscr{L}_{\!x}$ , is defined as the distance between the first two symmetric crossings of the $R^{x}_{u^\prime u^\prime }$ distribution at the $0.1$ cutoff ordinate. By looking at figure 7(a,c,e) one can see that the streamwise extent of the smooth-wall superstructures increases as one moves away from the wall and becomes maximum in the logarithmic layer, beyond which it decreases again. The latter also verifies the observations made in Hutchins & Marusic (Reference Hutchins and Marusic2007), whose two-point correlations are superposed in figure 7(c,e) and shown in black circles, indicating an excellent agreement between the two studies. On the other hand, the streamwise extent of the superstructures, in the case of the BR39 arrangement, grows linearly with the wall-normal distance, as shown in figure 7(b,d, f). The same trends were obtained for the rest of the rough-wall cases, which are not shown here for the sake of brevity. Interestingly, the streamwise extent of the superstructures in the case of the rough walls is found to be reduced compared with the corresponding one of the smooth wall. However, it is noted that, in the present figures, the correlation lags are scaled by the local boundary-layer thickness, $\delta$ , extracted from the middle of the correlation window, which in the case of the rough walls is significantly increased. The rough-wall boundary-layer evolution is not universal, but rather dependent on each specific topography studied, and as will be shown in the next section, different topographies result in different correlation lengths, $\mathscr{L}_{\!x}/\delta$ , even for the same incoming ${\textit{Re}}$ boundary layer.

Figure 8.

Spanwise two-point correlations of the streamwise velocity fluctuations, $R^{z}_{u^\prime u^\prime }$ , at $x/\delta _{\textit{LE}}=62.5$ and various wall-normal locations. Left side: smooth wall, ${\textit{Re}}_\tau =1150$ ; right side: BR39 case ${\textit{Re}}_\tau =2900$ ; (a,c) at the centreline of viscous/roughness sublayer ( $y/\delta _{\textit{smooth}}=0.005$ , $y/\delta _{\textit{BR}39}=0.097$ ); (b,d) at the centreline of the logarithmic layer ( $y/\delta _{\textit{smooth}}=0.083$ , $y/\delta _{\textit{BR}39}=0.15$ ); (e, f) at the centreline of the outer layer ( $y/\delta _{\textit{smooth}}=0.57$ , $y/\delta _{\textit{BR}39}=0.59$ ); $\bullet$ experiments by Hutchins & Marusic (Reference Hutchins and Marusic2007) at ${\textit{Re}}_\tau =1120$ ; dashed lines correspond to the correlation cutoff of $R^{z}_{u^\prime u^\prime }=0.1$ .

The spanwise extent (i.e. width) of the superstructures can be also computed from

(3.2) \begin{equation} {R^z}_{u^\prime u^\prime }(x,y,\Delta z)= \left\langle \sum _{n=1}^{N_z}\left[u^\prime (x,y,z,t) u^\prime (x,y,z+\Delta z,t)\right]/\sum _{n=1}^{N_z}\big[u^{\prime 2}(x,y,z,t)\big]\right\rangle _{t}, \end{equation}

for the same streamwise and wall-normal locations and it is shown in figure 8 for both the smooth wall and the BR39 topography. Here, the whole spanwise extent of the computational box is used as a sampling window. From figure 8 we can infer that, contrary to the streamwise extent dependency on the wall-normal distance, in this case both the smooth wall and the BR39 topography exhibit larger widths as the wall-normal distance increases, reaching the boundary-layer edge. Figure 8(c,e) further validates the current findings regarding the smooth-wall superstructure shape when compared with the respective ones of Hutchins & Marusic (Reference Hutchins and Marusic2007). It is noted that small differences in the latter in the strength of the anti-correlated regions could be attributed to the fact that the autocorrelation computed from the DNS data contains a much larger sample of superstructure entities in the spanwise direction compared with Hutchins & Marusic (Reference Hutchins and Marusic2007). Figure 8 indicates shorter structures than the ones observed in figure 4. The latter is the result of the fact that the autocorrelation functions are averaged in time and thus they encapsulate the length of all flow structures not just of the largest/er ones that can be see in some instantaneous snapshots. Alternative approaches such as a curve fitting in the $u^\prime$ isosurfaces can easily bias the length estimates as they are entirely based on a given isosurface threshold, as well as on the order of the polynomial chosen to represent the curve, the selection of which is subjective. In addition, the $u^\prime$ isosurfaces often branch to other isosurfaces, thus making the selection of the superstructure direction ambiguous. On the other hand, the autocorrelation functions provide a length estimate based on a more mathematically rigorous, normalised coefficient.

Figure 9.

(a) Window size for the computation of $R^{x}_{u^\prime u^\prime }$ ; (b) evolution of the superstructure length, $\mathscr{L}_{\!x}$ (computed in the logarithmic layer for BR39 case), for different window sizes. Window lengths shown in legend. The reference boundary-layer thickness at the centre of the domain is equal to $\overline {\delta }_{\textit{BR}39}\approx 1.8\delta _{\textit{LE}}$ .

Although, a single location measurement reveals a lot of information about the superstructure size, it does not provide any information about the growth of these structures along and across the boundary layer. Thus, their growth is captured herein by employing a windowing technique at various streamwise and wall-normal locations. Besides the streamwise and spanwise correlations we also compute the temporal correlation

(3.3) \begin{equation} {R}^t_{u^\prime u^\prime }(\underline {x}, \Delta t)= \sum _{n=1}^{N_t}\left[u^\prime (\underline {x},t) u^\prime (\underline {x},t+\Delta t)\right]/\sum _{n=1}^{N_t}\big[u^{\prime 2}(\underline {x},t)\big]. \end{equation}

The size of the sampling window at different streamwise locations is well defined for the case of the spanwise, $R^{z}_{u^\prime u^\prime }$ , and the temporal, $R^{t}_{u^\prime u^\prime }$ , correlations and coincides with the size of the computational domain in the periodic spanwise direction and the length of the time-series velocity signal, respectively. For the case of streamwise correlations, $R^{x}_{u^\prime u^\prime }$ , the sampling-window selection is not trivial because the correlation direction coincides with the direction in which the superstructures grow: if one selects the full streamwise extent of the domain, then the resulting streamwise correlations will be independent of the streamwise location, while if it is too narrow it will produce non-physical length scales. We conducted a sampling-window sensitivity analysis, where multiple windows of various lengths defined as an integer multiple of the local boundary-layer thickness, $\delta$ , are employed as shown in figure 9(a) relative to the flow scales in the case of the BR39 topography. The resulting length scales of the $R^{x}_{u^\prime u^\prime }$ correlations (all with same cutoff of $0.1$ ) at various streamwise locations for the BR39 topography are plotted in figure 9(b). It can be seen that no significant effect is detected for window lengths greater than $6\delta$ . In the case of the smooth wall the $\mathscr{L}_{\!x}/\delta _{\textit{LE}}$ convergence is observed in windows of size $8\delta$ and greater. Here, to ensure that even the largest instantaneous structures are captured, the window with extent equal to $10\delta$ is chosen everywhere in the present work.

Figure 10.

Contours of two-point correlations at the logarithmic layer with (a) spanwise lag, $R^{z}_{u^\prime u^\prime }$ ; (b), (d) streamwise lag, $R^{x}_{u^\prime u^\prime }$ ; (c), (e) temporal lag, $R^{t}_{u^\prime u^\prime }$ ; all in the case of the rough wall for the $BR39$ case; (d), (e) correspond to the streamwise location of $x/\delta _{\textit{LE}}=36$ , while (b), (c) correspond to $x/\delta {\textit{LE}}=59$ .

All two-point correlations were computed at ten streamwise locations separated by $3.4\delta _{\textit{LE}}$ , neglecting the leading and trailing parts of a roughness patch in order to eliminate any possible effects arising from the rough-to-smooth-wall transitions. Three wall-normal locations are considered as in figure 7. The streamwise evolution of the superstructure width can be seen in figure 10(a), where contours of $R^{z}_{u^\prime u^\prime }$ are shown at the logarithmic layer. The solid dark-blue region, which represents the spanwise correlation length or equivalently the width of the characteristic superstructure, is followed by a thin white region that corresponds to the correlation zero crossing. It is evident that this region thickens as one moves downstream, indicating that the width of superstructures has an outer scaling growth which is in agreement with the qualitative observations in figure 4. Figure 10(a) indicates that the width growth of a characteristic superstructure is closely followed throughout the domain by one of different sign, shown in dark orange, representing the anti-correlated regions. Furthermore, the spanwise alternation of $R^{z}_{u^\prime u^\prime }$ implicitly confirms the spanwise alternation detected in the signs of the superstructures themselves. In addition, the spanwise variation of the superstructure length can be seen in figure 10(b,d), which is centred at two distinct streamwise locations at $x/\delta _{\textit{LE}}=36$ and $x/\delta _{\textit{LE}}=59$ , respectively. The amplitude of the dark-blue region corresponding to the superstructure correlation length is significantly enhanced at the downstream location, providing further support for the outer scaling nature of superstructures. Finally, the superstructure characteristic time-scale variation with respect to the spanwise distance can be also seen in figure 10(c,e) for the same streamwise locations. Again, as in the case of the correlation lengths, the amplitude increases as the boundary layer grows, which suggests that larger-scale superstructures exhibit larger time scales. Very similar behaviour was also obtained for the rest of the topographical arrangements.

Figure 11.

Streamwise evolution of: (a) length, $\mathscr{L}_{\!x}/\delta _{\textit{LE}}$ ; (b) width, $\mathscr{L}_z/\delta _{\textit{LE}}$ ; (c) time scale, $\unicode{x1D4C9}\delta _{\textit{LE}}/U_e$ , of superstructures at the centreline of the logarithmic layer scaled by the smooth-wall leading-edge boundary-layer thickness. smooth wall; BS39; BR39; MR39; TR39.

Figure 12.

Streamwise evolution of: (a) length, $\mathscr{L}_{\!x}/\delta$ ; (b) width, $\mathscr{L}_z/\delta$ of superstructures at the centreline of the logarithmic layer scaled by the local boundary-layer thickness. smooth wall; BS39; BR39; MR39; TR39.

In order to more comprehensively capture the growth of superstructures within the boundary-layer, their spatio-temporal evolution is presented in figure 11 by performing a spanwise averaging over the full spanwise extent of the planar correlation maps shown in figure 10(b,c,d,e). Specifically, figure 11(a) suggests that the superstructure length grows faster over the rough-wall cases compared with the smooth wall, although no significant difference is observed between BS39 and BR39 indicating that the roughness distribution has minimal effect not just on the mean evolution of the boundary layer but also on its dynamics. Note that the latter observation does not necessarily imply that the roughness distribution has minimal impact on the flow in general, as it has been shown that flow phenomena such as the mean secondary motions maintain a strong dependency among others on the roughness distribution (see for example Barros & Christensen Reference Barros and Christensen2014; Kaminaris et al. Reference Kaminaris, Balaras, Schultz and Volino2023). The existence of increased superstructure lengths over the smooth-wall case compared with the rough-wall ones is probably a product of a breaking mechanism that takes place at the smooth-to-rough-wall transition, at which the coherence of incoming superstructures from the upstream smooth-wall portion of the computational domain is interrupted by the roughness elements. In the region downstream of the roughness leading edge the rough-wall superstructures reorganise themselves and eventually their streamwise coherence will surpass the respective one of the smooth-wall superstructures. The very good collapse of the correlation lengths from all the streamwise locations into a line strongly suggests that the outer scaling growth of superstructures follows a linear behaviour, at least up to ${\textit{Re}}_\tau \leqslant 3000$ . The same linear behaviour is also obtained for the widths of superstructures, which, similarly to the lengths, exhibit a higher growth rate in the cases of rough walls with minimal differences between BS39 and BR39 (barnacle-type only organisms). Similar arguments can be made for the time-scale evolution as well. More strikingly, the length scales when scaled by the local boundary-layer thickness (extracted at the middle of each sampling window) result in a constant ratio, $\mathscr{L}_{\!x}/\delta =c_1$ and $\mathscr{L}_z/\delta =c_2$ , as shown in figure 12. This is found to be true for both the smooth-wall case and all the rough-wall cases, as well as across all the three major sublayers (here, only logarithmic-sublayer evolution is shown), and thus this non-dimensional length scale is believed to constitute a flow invariant. Strong support for the latter argument is implicitly provided by Hutchins & Marusic (Reference Hutchins and Marusic2007), where two-point correlations computed in various ${\textit{Re}}_\tau$ regimes collapsed into a single curve when scaled by the appropriate boundary-layer thickness. The latter is equivalent here to performing two-point correlations over different streamwise stations downstream in nominally the same boundary layer in which the ${\textit{Re}}_\tau$ increases. Thus, scaling arguments could be made for flows of significantly higher ${\textit{Re}}$ . It should be stated that alternation of the correlation cutoff did not break the scaling at all, instead resulting in different $c_1$ and $c_2$ constants. Besides the correlation cutoff, the absolute values of the $c_1$ and $c_2$ constants are found to also depend on the underneath topography studied, while dependency on the characteristic ${\textit{Re}}$ of the flow remains to be investigated. Although figure 12 indicates overall constant $\mathscr{L}_{\!x}/\delta$ and $\mathscr{L}_z/\delta$ ratios for all the topographies, some cases show more deviation compared with the rest. The small discrepancies found in some of the case studies, i.e. TR39 and BS39, are believed to exist because of two main reasons. First, in the case of the streamwise length-scale ratios the small deviations are presumed to be the result of the fact that, although the streamwise autocorrelations are performed in windows, the centres of which follow closely the sublayer growth rate, the autocorrelations per se are performed in the horizontal direction at each window. Thus, there might be a mismatch between the angle of the inclined superstructures and the horizontal direction of the correlations even within the window extent, which might also be to some degree case dependent. Second, there is an underlying assumption that the part of superstructures at each sublayer follows the sublayer growth, which might not be strictly true. Interestingly, the fact that the numerical experiments span a distance of ${\approx}60\delta _{\textit{LE}}$ , at which the ${\textit{Re}}_\tau$ is almost doubled (between the leading and trailing edges of the roughness), as well as that the surfaces under investigation differ significantly and still manage to sustain structures that grow with the boundary layer in a proportional fashion, provides strong support for the universal existence of such an invariants, $\mathscr{L}_{\!x}/\delta$ and $\mathscr{L}_z/\delta$ . Finally, it is noted that the snapshots used to build the latter scale evolutions spanned a total time of $T\approx 34\overline {\delta }_{\textit{BR}39}/U_e$ (where $\overline {\delta }_{\textit{BR}39}\approx 1.8\delta _{\textit{LE}}$ ), which in fact after performing spanwise averaging is equivalent to $T\approx 950\overline {\delta }_{\textit{BR}39}/U_e$ . Attention should be drawn to the fact that expanding in the spanwise direction does not necessarily yield independent flow samples, however, in the present case the spanwise autocorrelation functions of figure 11(b) act as a guide to inform us about the number of such independent samples and specifically they indicate that $\approx 28$ (i.e. $L_{z,\textit{domain}}/\mathscr{L}_{z,BR39}\approx 14/0.5\approx 28$ ) distinct superstructures can be found across the spanwise extent of the boundary layer.

Figure 13.

Isosurfaces of the time-averaged volumetric autocorrelations, $R^{\underline {\unicode{x1D4CD}}}_{u^\prime u^\prime }(x, y, \Delta \underline {x})$ , at three different streamwise stations in the case of (a), (c) the smooth wall and (b), (d) the BR39 case. (a), (b) Correspond to a 3-D view and (c), (d) to a streamwise/wall-parallel view of the same structures. Positive (dark-purple) isosurfaces are visualised with a threshold of $0.05$ and negative (dark-orange) isosurfaces with $-0.05$ .

The analysis made so far provides an estimation of the superstructures size, as well as an insight into their streamwise evolution along the boundary layer. However, it does not provide any information regarding their 3-D silhouette. To address that, volumetric two-point correlations

(3.4) \begin{equation} R^{\underline {\unicode{x1D4CD}}}_{u^\prime u^\prime }(x, y, \Delta \underline {x})= \left\langle \sum _{n=1}^{N_{\textit{xyz}}}\left[u^\prime (\underline {x},t) u^\prime (\underline {x}+\Delta \underline {x},t)\right]/\sum _{n=1}^{N_{\textit{xyz}}}\big[u^{\prime 2}(\underline {x},t)\big]\right\rangle _{t}, \end{equation}

are computed at three distinct regions of the boundary layer each of $10\delta \times 14\delta _{\textit{LE}} \times \delta$ size in the streamwise, spanwise and wall-normal directions, respectively, centred at the centroid at $x/\delta _{\textit{LE}}=32$ , $x/\delta _{\textit{LE}}=46$ and $x/\delta _{\textit{LE}}=59$ with respect to the streamwise coordinate and at $y/\delta =0.5$ with respect to the wall normal. It is pointed out that, in the BR39 case, the volumetric region utilised in the two-point correlations is of $10\delta \times 14\delta _{\textit{LE}} \times 0.9\delta$ size and thus centred at $y/\delta =0.55$ , because the wall-normal distance up to the nominal roughness crest is discarded. In this way a characteristic 3-D form of the superstructure evolution is obtained along the boundary layer and presented in figure 13(a,b). In the latter the positively correlated region depicting the characteristic superstructure of a given sign is shown in dark purple and the negatively correlated ones in dark orange; all scaled by the fixed smooth-wall incoming boundary-layer thickness $\delta _{\textit{LE}}$ at the leading edge of the roughness. It is evident that as the superstructures move downstream they become longer and wider, as was also visually observed in figure 4. Furthermore, through the side view of figure 13(c,d) one can easily see that the superstructures are inclined with respect to the mean-flow direction, as well as that their shape changes as the boundary layer evolves, by stretching from slightly above the wall all the way to the boundary-layer edge. The latter is even more evident in the BR39 topography (figure 13 d), where upstream superstructures that form an oval-like shape are converted into more elongated ones with pronounced ‘tails’ at the further downstream locations. Visual inspection of figure 5 reveals similar behaviour along the boundary layer, as expected. It should be noted that regarding the smooth-wall $R^{\underline {\unicode{x1D4CD}}}_{u^\prime u^\prime }$ , Sillero et al. (Reference Sillero, Jiménez and Moser2014) and Kevin et al. (Reference Kevin, Monty and Hutchins2019b ) depicted structures of a very similar shape. However, they found that the correlations were not symmetric with respect to the streamwise lag, but rather enhanced at the negative-lag region. The pronounced ‘tails’ in the isosurfaces of $R^{\underline {\unicode{x1D4CD}}}_{u^\prime u^\prime }$ at the negative-lag regions are also well captured here, as mentioned above. However, it should be noted that, since volumetric correlations are used in the present work, the structures throughout the boundary layer are symmetric with respect to their local streamwise lag. On the other hand, when 2-D autocorrelations are performed at various wall-normal locations with respect to a fixed reference height (not shown here), i.e. $y/\delta \approx$ 0.5, the resulting structures, although maintaining non-equal upstream and downstream correlation parts, were found to be more symmetric compared with the ones in Sillero et al. (Reference Sillero, Jiménez and Moser2014) and Kevin et al. (Reference Kevin, Monty and Hutchins2019b ). This is probably due to the ${\textit{Re}}_\tau$ differences between the present and Sillero et al. (Reference Sillero, Jiménez and Moser2014) and Kevin et al. (Reference Kevin, Monty and Hutchins2019b ) studies. The friction ${\textit{Re}}$ ranges for the smooth wall and the BR39 topography shown in figure 13(a–d) are $850\leqslant {\textit{Re}}_\tau \leqslant 1100$ and $1900 \leqslant {\textit{Re}}_\tau \leqslant 2800$ , respectively. The smooth-wall boundary layer, despite clearly depicting a logarithmic layer, likely holds little separation between the near-wall and the outer-layer structures at ${\textit{Re}}_\tau \approx 1000$ and thus an increased inclination of $\overline {R}^{t}_{u^\prime u^\prime }$ towards the negative-lag region is expected at further downstream locations. The latter is in agreement with the arguments made in Hutchins & Marusic (Reference Hutchins and Marusic2007) for smooth-wall boundary layers at a similar ${\textit{Re}}_\tau$ , with which the current smooth-wall DNS data showed very good agreement, as was previously shown in figures 7 and 8(a,c,e). Finally, a 2-D comparison of the two-point correlations between the smooth wall and the BR39 topography centred at $x/\delta _{\textit{LE}}=59$ and $\Delta z=0$ is shown figure 14 in terms of the local streamwise and wall-normal lag. It can be seen that the BR39 topography forms bulkier structures with steeper inclination angles compared with the smooth-wall ones, while in both cases anti-correlated regions are detected upstream and downstream of the main correlated region that are found to be stronger in the rough-wall case. Similar arguments where made in Kevin et al. (Reference Kevin, Monty and Hutchins2019b ) over a converging–diverging riblet surface. Volino et al. (Reference Volino, Schultz and Flack2007) had also detected structures with stronger inclination angles under the presence of a rough wall. It is noted, however, that the smooth- and rough-wall comparison herein is made on the basis of the same streamwise coordinate given the presence of the same incoming boundary layer, $\delta _{\textit{LE}}$ , and thus these differences may not hold when compared at the same ${\textit{Re}}_\tau$ . Moreover, the positively correlated smooth-wall $R^{\underline {\unicode{x1D4CD}}}_{u^\prime u^\prime }$ structure seems to be slightly longer, for a given isoline cutoff, as was also shown above from the 1-D correlations of figure 7. It should be noted, however, that the 1-D correlation functions do not directly correspond to the length scales captured by the volumetric correlation functions even in the case when the correlations are performed along a single direction, since in the latter a volume block is correlated and not just a 1-D signal, that is, structures located at different wall-normal locations.

Figure 14.

Side-view contours of the time-averaged volumetric autocorrelations, $R^{\underline {\unicode{x1D4CD}}}_{u^\prime u^\prime }(x, y, \Delta \underline {x})$ for the BR39 topography extracted at $\Delta z=0$ , centred at $x/\delta _{\textit{LE}}=59$ and $y/\delta =0.55$ and superposed by the respective smooth-wall isolines centred at $y/\delta =0.5$ and at the same streamwise coordinate. Positive smooth-wall isolines $(0.01$ , $0.02$ , $0.04$ , $0.08$ , $0.16$ , $0.32)$ are shown in white and negative isolines ( $-0.004$ , $-0.0045$ , $-0.005$ , $-0.0055$ ) in black; Note that, for the smooth-wall autocorrelation functions, $R^{\underline {\unicode{x1D4CD}}}_{u^\prime u^\prime }$ , the whole boundary-layer height is used, while for the BR39 topography only the boundary-layer part above the nominal roughness crest.

4. Evidence of counter-rotating roll modes

An increased number of studies investigating the characteristics of superstructures in turbulent boundary layers have reported evidence of instantaneous counter-rotating vortices that flank the superstructures and exhibit streamwise coherence (see for example Marusic et al. Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021; Kevin et al. Reference Kevin, Monty and Hutchins2019a , Reference Kevin, Monty and Hutchins2019b ). In this work we investigate the behaviour of such large-scale flow motions using modal analysis by means of POD. One could alternatively employ a low-pass filter (e.g. Gaussian) on the instantaneous velocity field, instead of performing a POD. The filter, however, would require a kernel size to be chosen that even in the case that it would be associated with the autocorrelation length scales, it could in fact impose a specific length scale on the reconstructed flow field. On the other hand, when POD is employed the reconstructed flow field is based on the number of leading modes selected, which are freely elected to represent the most energetic structures, given enough of a time sample. We employ POD through the ‘method of snapshots’ which is ideal for cases where the number of spatial points is significantly increased compared with the number of instances (i.e. tall/skinny matrices), as in the current work. The method can be summarised in five main steps: (i) formation of the snapshot matrix, $Y$ , for the velocity fluctuations as (n,m), where n stands for the total number of points of the two/three-dimensional field multiplied by the number of velocity components and m for the number of instances; (ii) computation of the temporal correlation matrix, $C_t=Y^TY$ , which will be of compact size (m,m); (iii) computation of the eigenvectors (or temporal coefficients), $\varPsi$ , and eigenvalues, $\lambda$ , from $C_t\lambda =\lambda \varPsi$ ; (iv) projection of the temporal eigenvectors onto the velocity field through the singular value decomposition definition to obtain the spatial eigenvectors, $\varPhi$ ; (v) construction of a reduced-order model (ROM) based on a lower number of modes, $\hat {u}^\prime =\sum _{n=1}^{N} \alpha _n(t)\varPhi _n(\underline {x})$ . The ROM reconstruction for the whole 3-D field has been achieved by parallelising the POD algorithm using a classical domain decomposition along the streamwise direction by employing MPI library calls, which allowed for inter-core communication in order for the matrix $C_t$ to be informed from the flow states along the whole extent of the boundary layer.

Here, we reconstruct the cross-plane velocity fluctuations by using only the first dominant mode, since the scope of the current study is not to create a ROM that could substitute the need for performing experiments/simulations, but rather to explore the underlying flow patterns in a lower-order flow field. The POD helps in identifying the latent large-energy-carrying structures, and it thus can be used as a ‘filtering’ operator to clean up the rather noisy instantaneous velocity field. Figure 15(a,b) shows the streamwise velocity fluctuations superposed by the magnitude-scaled cross-plane ones in vector format at a given instant and streamwise location in the case of the smooth wall and the BR39 topography, respectively. In figure 15(c,d) the ROM fields are shown for the same instant and streamwise location as in the figures above (figure 15 a,b). One can thus readily detect that, in the ROM representations, well-shaped counter-rotating motions are formed, that are otherwise not visible in the original field, encharged with the cross-plane momentum redistribution. These counter-rotating motions are also found to flank the high- and low-speed superstructures; this behaviour holds true across the whole span of the boundary layer. It is also found that these vortical motions occur in fixed spanwise and wall-normal locations as time evolves, with the only major parameter being altered being their rotation sign, which changes to accommodate the superstructure passage through the cross-plane. The latter can be seen in figure 15(e, f) which shows the ROM of velocity fluctuations at a later instant. Hence, the observation of roll modes that flank the different sign superstructures is confirmed in both the smooth and rough walls. If one also factors in the excellent agreement observed in the outer-layer similarity (see figure 3 d,e) it could be inferred that the way momentum organises itself in high Reynolds number wall-bounded flows might be independent of the roughness terrain underneath. The ROM in the rest of rough-wall cases, not shown here for brevity, follows the same behaviour, as well. It is also noted that the conjugate pair of counter-rotating vortices can be found at different wall-normal locations in spite of the fact they may flank the same superstructure. In fact, when the cross-plane vector fluctuations are not scaled by magnitude, one can easily observe that each roll mode exhibits a strength asymmetry compared with each conjugate pair and even across to its own reference core (not shown here). The latter is also in agreement with the observations made in Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021) and Kevin et al. (Reference Kevin, Monty and Hutchins2019a , Reference Kevin, Monty and Hutchins2019b ). The instantaneous nature of the ROM field discussed here can be better understood through the visualisations provided in the accompanying movies. Furthermore, to enhance the validity of our observations a time sample and frequency sensitivity analysis is also performed for the BR39 topography that can be found in Appendix

Figure 15.

Cross-plane of streamwise velocity fluctuations scaled in local units at $x/\delta _{\textit{LE}}=65$ . Top: original streamwise velocity fluctuations, $u^\prime$ , for (a) the smooth wall ( ${\textit{Re}}_\tau =1200$ ); (b) BR39 case ( ${\textit{Re}}_\tau =3000$ ). Middle: ROM of streamwise velocity fluctuations, $\hat {u}^\prime$ , (only first most dominant mode) for (c) smooth wall; (d) BR39. Bottom: same as above at a later time instant when rotation sign changes.

In order to better understand the streamwise footprint of the counter rotating motions ROMs were constructed for the 3-D velocity fluctuation field of the whole extent of the boundary layer at a downscaled domain of a factor of four (by skipping one grid point in the streamwise and spanwise directions) to allow for a more computationally efficient analysis to be performed. A top view of the isosurfaces of the velocity fluctuations, $\hat {u}^\prime$ superposed by the ROM $\hat {Q}$ -criterion coloured by the streamwise fluctuating vorticity, $\hat {\omega }'_{\!x}$ can be seen in figure 16(a). It is evident that the roll modes detected in figure 15(c–f) preserve a significant streamwise coherence by closely following the superstructure evolution, as expected based on the 2-D superstructure representations of figure 15(c–f). It is noted that, besides the roll modes, the ROM superstructures themselves exhibit an increased streamwise coherence compared with the ones found in the original full-order flow field, as expected. Similar findings reported in Dong et al. (Reference Dong, Pan, Wang and Yuan2023), who, through performing POD over a smooth-wall boundary, reconstructed a lower-order superstructure field with increased coherence and reduced noise. To better understand the interaction between the superstructures and the counter-rotating vortices multiple magnified regions along the boundary layer are provided in figure 16(a). Specifically, it can be seen that clockwise vortices are formed between the low- to high-speed superstructures (by marching from the negative to the positive spanwise coordinates), transferring high momentum fluid from the upper portion of the boundary layer lower to the wall, as was also shown from the 2-D cross-sections of figure 15. Figure 16(b) isolates the isosurfaces of only the roll modes allowing for a cleaner depiction of their layout, which spans several boundary-layer thicknesses in the order of $10{-}15\overline {\delta }_{\textit{BR}39}$ . Finally, it is emphasised that these computations evolving whole boundary-layer volumetric DNS datasets are of high computational cost, with each POD run lasting for more than a week on two high-memory (3 TB) nodes processing over 1.7 TB of instantaneous data.

Figure 16.

Reduced-order model of the flow accounting only for the highest energy mode; isosurfaces of high-speed superstructures at $\hat {u}^\prime /U_e=0.002$ shown in red and low-speed superstructures at $\hat {u}^\prime /U_e=-0.002$ shown in blue; isosurfaces of $\hat {Q}=1.7\times 10^{-5}$ coloured by the streamwise vorticity fluctuations, $\hat {\omega }'_{x}$ ; all in the case of the rough wall with the random BR39 case.