Why do only some riblets promote spanwise rollers?

Abstract

Linear-stability modelling suggests that all sufficiently large riblets promote maximally growing spanwise rollers (García-Mayoral & Jiménez 2011 J. Fluid Mech. vol. 678, 317–347), yet direct numerical simulations (DNS) have shown that this is not the case (Endrikat et al. 2021 J. Fluid Mech. vol. 913, A37) some riblet shapes do not form spanwise rollers at all. Thus, the drag-reduction breakdown across all riblet shapes cannot be solely attributed to maximally growing spanwise rollers, prompting a reappraisal of the modelling. In this paper, comparing DNS data with riblet-resolving linear-stability predictions shows that the spanwise rollers are actually marginal modes, not maximally growing instabilities. This riblet-resolved linear analysis also predicts that not all riblet shapes promote spanwise rollers, in agreement with DNS, and unlike earlier linear-stability modelling, which relied on a one-dimensional (1-D) mean flow and on an over-simplified effective wall-admittance boundary condition. These riblet-resolved calculations further inform how to capture the effect of the riblet shape in a 1D model. Once captured, predictions with an effective boundary condition match riblet-resolved results, but still do not indicate what features of the riblet geometry promote the roller instability. Thus, the wall admittance is measured near the riblet crests, in both the riblet-resolved linear analysis and DNS, to show that the in-groove dynamics is dominated by a balance between the overlying pressure and unsteady inertia, and not viscous diffusion, as previously assumed. This pressure–unsteady-inertia balance sets the linear scaling of the wall admittance with riblet size, as observed in DNS, and is a key factor in setting the streamwise wavelength of the spanwise rollers. Furthermore, modelling this pressure–unsteady-inertia balance in the wall admittance reveals the role of riblet slenderness in promoting spanwise rollers, which provides the missing link in previous correlations between the riblet geometry and the presence or lack of rollers.

1. Introduction

1.1. Background

Interest in spanwise-coherent Kelvin–Helmholtz-like rollers stems from their formation over a range of non-smooth surfaces, examples including riblets (García-Mayoral & Jiménez 2011b ; Endrikat et al. 2021a , b ), permeable substrates (Jiménez et al. 2001; Breugem, Boersma & Uittenbogaard 2006; Gómez-de-Segura et al. 2018b ; Suga et al. 2018; Gómez-de-Segura & García-Mayoral 2019; Motoki et al. 2022; Habibi Khorasani, Luhar & Bagheri 2024) and plant canopies (Raupach, Finnigan & Brunei 1996; Finnigan 2000; Nepf et al. 2007; Ghisalberti 2009; Nepf 2012; Sharma & García-Mayoral 2020). Understanding how and why spanwise rollers form motivates this work, given that permeable substrates and engineered rough surfaces, which are susceptible to rollers, hold the potential to reduce drag, enhance heat transfer and suppress noise (García-Mayoral & Jiménez 2011b ; Gómez-de-Segura et al. 2018b ; Endrikat et al. 2021a ; Kuwata 2022; Rouhi et al. 2022; Hartog et al. 2024). For these surface textures, spanwise rollers are generated by a velocity difference across the interface between the fluid region and the effectively porous fluid–solid region, where this velocity difference feeds perturbation energy production (Jiménez et al. 2001; García-Mayoral & Jiménez 2011b ). The finer details of the surface texture tend to influence the appearance and intensity of the spanwise rollers, not only through the mean shear at the interface, but also through the effective wall-normal permeability of the surface texture (Endrikat et al. 2021a ). In addition, the wall-normal permeability experienced by turbulent structures tends to depend on their characteristic lengths in the streamwise and spanwise directions (Gómez-de-Segura et al. 2018b ; Chavarin et al. 2021; Endrikat et al. 2021a ; Hao & García-Mayoral 2025). This turbulent-scale dependence can then lead to rollers which are either (i) flow structures which extend no further than the buffer layer ( $z^+ \lesssim 50$ ), and which are viscous-scaled through their localisation about the local maximum of $-\partial _{zz} U^+$ (García-Mayoral & Jiménez 2011b , 2012), where $z$ is the wall-normal height, $U$ the mean streamwise velocity and where the superscript $()^+$ denotes normalisation by the friction velocity $u_\tau$ and kinematic viscosity $\nu$ , or (ii) flow structures which extend the full height of the domain $z \lesssim \delta$ , where $\delta$ is the domain height (Jiménez et al. 2001; Kuwata 2022; Motoki et al. 2022). Near-wall/buffer-layer rollers are often observed over streamwise-aligned riblets, permeable substrates (Gómez-de-Segura et al. 2018b ; Habibi Khorasani et al. 2024) and dense filament canopies (Sharma & García-Mayoral 2020), and are the focus of this work, although with scope limited only to riblets.

For riblets, linear-stability modelling predicts that rollers form only once the effective wall admittance (the ratio of wall-normal velocity and pressure fluctuations) provided by the riblets exceeds some threshold. This effective wall admittance can be estimated from a Stokes-flow problem within the riblet grooves (García-Mayoral & Jiménez 2011b ), and was further shown to be related to the square-root of groove area $\ell _g^+$ for conventional riblets (García-Mayoral & Jiménez 2011b ). For riblets, $\ell _g^+$ has also been shown to reduce the scatter in the drag-reduction curves measured across different shapes, including the point of maximum drag reduction at $\ell _g^+ \approx 11$ (García-Mayoral & Jiménez 2011a , b ). This relationship between linearly unstable spanwise rollers and the effective wall admittance of the riblets thereby appeared to both explain the improved collapse of the drag-reduction curves and identify the point of maximum drag reduction. However, this roller–wall-admittance relationship was not supported by the direct numerical simulations (DNS) of Endrikat et al. (2021a ), who showed that at matched $\ell _g^+$ , some riblet shapes formed spanwise-coherent rollers, but others did not. Endrikat et al. (2021a ) further showed that spanwise rollers were not the only contribution to the drag-reduction breakdown for riblets, as previously thought. Thus, a complete description of the drag breakdown remains an open question (Modesti et al. 2021; Chan et al. 2023; Viggiano et al. 2024), such that the optimal riblet size of $\ell _g^+ \approx 11$ can only be identified empirically.

1.2. Outline

This paper reconciles previous inconsistencies between linear predictions and DNS observations of spanwise rollers. This is achieved by improving our dynamical understanding of spanwise rollers based on two hypotheses. First, that the rollers are actually marginal instabilities. Second, that the emergence of rollers is driven by a balance between the unsteady term and pressure gradient within the riblet grooves, which depends on riblet slenderness. This is contrary to the existing assumption that the dynamics within the groove were dictated by a balance between the pressure gradient and viscous term. These hypotheses are verified by performing two-dimensional (2-D) riblet-resolved linear-stability analysis and further supported by new analysis of an existing DNS database (Endrikat et al. 2021a , b ; Wong et al. 2024). A simpler, empirical one-dimensional (1-D) model is also developed, based on these hypotheses and the 2-D analysis, to rapidly predict many of the key characteristics of the spanwise rollers. The empirical boundary condition provides useful insights for riblet sizes up to and slightly beyond their design point (of maximum drag reduction), although the 1-D model eventually breaks down for larger riblets (earlier than its 2-D counterpart). The formal homogenisation procedure required to obtain a more rigorous 1-D boundary condition is, however, shown to be more expensive than performing the 2-D linear-stability analysis in its stead, such that the 2-D analysis forms a valuable design tool for a more refined riblet-shape optimisation.

First, § 2 presents further evidence that four of the six riblet shapes tested in Endrikat et al. (2021a ) (see table 1) do not promote strong spanwise rollers, quantified by a near-zero roller drag penalty. However, a range of instantaneous flow visualisations (Chu & Karniadakis 1993; Goldstein, Handler & Sirovich 1995; Endrikat et al. 2021a , b ; Rouhi et al. 2022) suggest that weak rollers persist, at least for some of these riblet shapes. Section 2 thereby establishes a more sensitive gauge of roller intensity across riblet shapes and sizes, from new analysis of minimal-channel riblet DNS data (Endrikat et al. 2021a ; Modesti et al. 2021; Wong et al. 2024), as generating new DNS data was not required. Details for almost all cases in the DNS dataset, and riblet geometric parameters, are tabulated in Wong et al. (2024); details for the remainder of the DNS cases (for the largest $\ell _g^+$ riblets) can be found in Endrikat et al. (2021a ). Overall, this more sensitive measure of roller intensity, based on the pressure fluctuation $\langle p'p'\rangle$ profiles, enables a more detailed appraisal of the linear-stability analysis in § 3, even for those riblet shapes which have roller drag penalties lying within numerical tolerances (Endrikat et al. 2021a ).

Table 1. Six of the riblet shapes considered in past minimal-channel DNS (Endrikat et al. 2021a ; Modesti et al. 2021; Wong et al. 2024), their corresponding geometric parameters (spanwise spacing $s$ , height $k$ , tip-angle $\alpha$ , base thickness $t_r$ , square-root of groove area $\ell _g$ ) and the viscous-scaled square-root of groove areas at which past DNS were conducted. Whether a drag penalty was attributed to spanwise rollers forming over the riblets (Endrikat et al. 2021a , figure 11 b) is also listed for each shape.

Section 3 also reconsiders the wall-admittance threshold for spanwise rollers. The DNS measurements of Endrikat et al. (2021a , figure 13b) showed that crest-measured wall admittances for riblets are $\hat {w}^+/\hat {p}^+\approx 0.02$ to $\approx 0.03$ when rollers begin to appear ( $\ell _g^+ \approx 10$ to 15), where $w$ is the wall-normal velocity, $\hat {w}$ its Fourier transform, $p$ is the pressure, and where $\hat {w}^+/\hat {p}^+$ was integrated across all wavelengths $\lambda _y^+ \gtrsim 250$ and $65 \lesssim \lambda _x^+ \lesssim 290$ . Note $x$ is the streamwise, $y$ the spanwise and $z$ the wall-normal coordinate. However, at $\ell _g^+ \approx 10$ to 15, the wall admittance of the model boundary condition of García-Mayoral & Jiménez (2011b ) was 5–10 times higher than DNS measurements (Endrikat et al. 2021a ), i.e. $\hat {w}^+/\hat {p}^+\approx 0.1$ to $\approx 0.3$ , integrated across the same $\lambda _x^+$ and $\lambda _y^+$ ranges. Indeed, these $\hat {w}^+/\hat {p}^+\gtrsim 0.1$ model-predicted roller modes shared many of the characteristics of the maximally growing Kelvin–Helmholtz-like rollers which form in the free-shear-layer limit (García-Mayoral & Jiménez 2011b , comparing figures 19c and 19e), and notably, had streamwise wavelengths of $\lambda _x^+ \approx 60$ , rather than $\lambda _x^+ \approx 150$ as measured for rollers over riblets in DNS (García-Mayoral & Jiménez 2011b ; Endrikat et al. 2021a ). Separately, the 2-D resolvent analysis of Chavarin & Luhar (2020) found spanwise roller modes with high relative gains at a streamwise wavelength ( $\lambda _x^+ \approx 130$ ) similar to DNS for $\ell _g^+ \approx 13$ blade riblets, by essentially assuming zero growth (marginal) modes. Marginal modes are zero-growth-rate modes for a given $\ell _g^+$ , $\lambda _x^+$ combination, where a change in either $\ell _g^+$ or $\lambda _x^+$ results in the growth rate of the mode switching sign (unlike neutral modes, across which the growth rate does not switch sign), following the definition in Drazin & Reid (2004). In summary, these results suggest that rollers over riblets are not maximally growing instabilities. Comparison between riblet-resolved linear-stability predictions and DNS data in § 3 confirms that rollers over riblets are not maximally growing. Instead, they are marginally stable structures emerging at lower wall-admittance values, a conclusion which is further verified by the linear-stability analysis accurately predicting key characteristics (streamwise wavelengths and wave speeds) of the spanwise rollers.

After further showing that only some riblet shapes promote these marginal rollers, the question then turns to explaining why. In Endrikat et al. (2021a ), the sustenance of spanwise rollers was well correlated with high momentum absorption near the riblet crests, and not the effective riblet wall admittance alone. This conclusion is supported in § 4, as riblet-resolved linear-stability predictions of rollers prove sensitive to some of the details of the mean-velocity profile within the grooves, where the mean profile encapsulates the momentum absorption near the crest plane. The effective wall-admittance boundary condition is also reconsidered, in an effort to develop an improved dynamical model to predict roller formation. Previously, the wall admittance was assumed proportional to riblet size and streamwise scale, as $\hat {w}^+/\hat {p}^+ \propto {\ell _g^+}^3{\lambda _x^+}^{-2}$ (García-Mayoral & Jiménez 2011b ), where $\lambda _x$ is the streamwise wavelength; separately, for permeable substrates, a streamwise-scale-independent model boundary condition $\hat {w}^+/\hat {p}^+ \propto \mathrm{constant}$ has been assumed (Jiménez et al. 2001; Motoki et al. 2022). However, § 4 shows that the wall admittance provided by riblets does not vary as $\hat {w}^+/\hat {p}^+ \propto {\lambda _x^+}^{-2}$ , as the dominant balance near the riblet crests is not between viscous diffusion and overlying pressure. Assuming the overlying pressure is instead balanced by unsteady in-groove inertia near the crests both explains the trends in the DNS measurements (Endrikat et al. 2021a ) and corrects the streamwise-scale dependence of the wall-admittance model to $\hat {w}^+/\hat {p}^+ \propto {\lambda _x^+}^{-1}$ .

Section 5 tests this understanding of the improved wall-admittance boundary condition with 1-D linear-stability analysis. This linear analysis provides reasonable predictions of the streamwise wavelengths and wave speeds of rollers at their onset, but eventually suggests large ( $\ell _g^+ \gtrsim 25$ ) $90^\circ$ triangular and asymmetric triangular riblets would sustain rollers, unlike the riblet-resolved analysis and DNS. Accurately modelling not only the wall admittance but also the streamwise slip is required for larger riblets. Finally, conclusions are provided in § 6.

2. Identifying rollers over riblets from DNS

The aim of this section is to develop a more sensitive measure of the intensity of spanwise rollers forming over riblets, so as to provide a more detailed comparison between DNS and linear-stability analysis. Endrikat et al. (2021a ) identified strong rollers for blade and 30 $^\circ$ triangular riblets by measuring a roller drag penalty $0.1 \lesssim \Delta U^+_{\textit{KH}} \lesssim 0.5$ . The roller drag penalty $\Delta U^+_{\textit{KH}}$ is a measure of the change in drag produced by a difference in Reynolds stresses between smooth and riblet surfaces for large spanwise-wavelength Fourier modes. This measure was calculated following the procedure in MacDonald et al. (2016), similar to García-Mayoral & Jiménez (2011b ), except (i) integrated in spectral space across $65 \lesssim \lambda _x^+ \lesssim 290$ , $250 \lesssim \lambda _y^+ \lesssim \infty$ and for $0 \leqslant z^+ \lesssim 100$ and (ii) neglecting any $z^+$ where the $ \lambda _x^+$ , $ \lambda _y^+$ integrated Reynolds stresses over the riblet are negative. Weak rollers, with $\Delta U^+_{\textit{KH}} \approx 0$ , were still suggested by visualisations of the instantaneous wall-shear stress for the remaining riblet shapes. Similar visualisations are provided in figure 1. Following Endrikat et al. (2021a ) and Rouhi et al. (2022), the (instantaneous) presence of rollers over riblets can be inferred from regions of high negative wall-shear stress or wall pressure which are approximately spanwise uniform. Strong rollers form over 30 $^\circ$ triangular riblets, as indicated by spanwise-coherent regions of negative wall-shear stress and (phase-shifted) spanwise-coherent regions of wall pressure. Weaker rollers are suggested for blade , $30^\circ$ trapezoidal and $60^\circ$ triangular riblets, with both reduced spanwise coherence, and fewer rollers overall. Rollers become almost non-existent for 90 $^\circ$ triangular and asymmetric triangular riblets, with almost no negative wall-shear stress events (for this threshold), although the wall pressure may still indicate the presence of one or two weak rollers. Overall, while an accurate measure of the drag penalty of these rollers may still be $\Delta U^+_{\textit{KH}} \approx 0$ , $\Delta U^+_{\textit{KH}}$ is sensitive to both the choice of turbulence origin $\ell _T^+$ and ad hoc cutoffs in spectral space. Thus, an alternative gauge of roller intensity is sought.

Spectral signatures of the rollers, which could quantify their intensity, are also present in almost all near-wall measurements of fluctuation products, e.g. $\hat {u}'\hat {u}'$ , $\hat {v}'\hat {v}'$ , $\hat {w}'\hat {w}'$ , $\hat {u}'\hat {w}'$ , $\hat {p}'\hat {p}'$ (García-Mayoral & Jiménez 2011b ; Endrikat et al. 2021a , b ), where $u$ is the streamwise velocity, $v$ the spanwise velocity and where the superscript () $'$ denotes a fluctuation. Of these, the spectra of $\hat {p}'\hat {p}'$ seem to be the most uniform in the wall-normal direction (MacDonald et al. 2017), minimising the issue of an ambiguous $\ell _T^+$ . Equally, flow visualisations have shown $p'$ maintains spanwise coherence even well below the interface of a porous substrate (Habibi Khorasani et al. 2024). Thus, the magnitude of the time- and spanwise-averaged $\langle p'p' \rangle$ profiles are considered figure 2 for each riblet shape, intrinsically averaged below the riblet crests. From figure 2, it is readily apparent that rollers are promoted by some riblet shapes, e.g. 30 $^\circ$ triangular riblets especially (figure 2 c), given the increase in $\langle p'p' \rangle$ relative to its smooth-wall counterpart.

Figure 1. Plan view of spanwise rollers via the instantaneous wall-shear stress (black) and the excess instantaneous wall pressure (grey) for $\ell _g^+ \approx 12$ riblets of various shapes. The threshold for the wall-shear stress is $\tau ^+ = -0.2$ , as in Endrikat et al. (2021b ), while the threshold for the excess instantaneous wall pressure is $p^+ - \langle p^+\rangle _{x_r,y_r} = -2$ ; here $\langle \boldsymbol{\cdot }\rangle _{x_r,y_r}$ indicates averaging across all points on the riblet surface.

Based on the results of figure 2, the intensity of the spanwise rollers is quantified by $\mathrm{max}(\langle p'p'\rangle ^+)/\mathrm{max}(\langle p_{sm}'p_{sm}'\rangle ^+)$ , taking the maxima across all $z^+$ , and where $p_{sm}'$ is the pressure fluctuation over a smooth wall. These intensities are collected for each of the riblet shapes and plotted as a function of $\ell _g^+$ in figure 3(a). If the threshold for sustaining rollers were at some fixed $\ell _g^+$ , all these intensities would increase in concert, regardless of riblet shape, say, at $\ell _g^+ \approx 11$ . However, as shown in figure 3(a), the roller intensity increases for the $30^\circ$ triangular riblets at $\ell _g^+ \approx 11$ , but is delayed slightly for some of the other riblet geometries, e.g. the blade riblets at $\ell _g^+ \approx 13$ , consistent with the prediction for similar blade riblets in the 2-D resolvent analysis of Chavarin & Luhar (2020). Equally, the intensities of rollers varies greatly with riblet shape (figure 3 a). The highest intensities are attained with $30^\circ$ triangular riblets, with $\mathrm{max}(\langle p'p'\rangle ^+)/\mathrm{max}(\langle p_{sm}'p_{sm}'\rangle ^+)$ reaching $\approx 3$ , followed by blades , $30^\circ$ trapezoidal and $60^\circ$ triangular riblets with increasingly reduced intensities. For blade and $30^\circ$ trapezoidal riblets, the roller intensities peak at $\ell _g^+ \approx 20$ to $25$ . For even larger riblets, $\ell _g^+ \gtrsim 40$ , this measure of the roller intensity then drops, with $\mathrm{max}(\langle p'p'\rangle ^+)/\mathrm{max}(\langle p_{sm}'p_{sm}'\rangle ^+)$ tending back toward unity, and hinting at a weakening of the roller mode as the riblet spacing increases. These minimal-channel DNS results are consistent with the recent experiments of Abu Rowin et al. (2025), who also showed that rollers weaken for slender triangular riblets at $\ell _g^+ \gtrsim 40$ .

Figure 2. Time- and plane-averaged $\langle p'p' \rangle ^+$ profiles for riblets (solid coloured) and a smooth wall (dashed black), having reprocessed the DNS datasets of Endrikat et al. (2021a ), Modesti et al. (2021) and Wong et al. (2024), and intrinsically averaging below the crests. The magnitude of $\langle p'p' \rangle ^+$ relative to that over a smooth wall gives an indication of the presence (or lack) of spanwise rollers. The diverging blue-grey-red colour scheme indicates the riblet size (blue smallest $\ell _g^+$ , red largest), where matched colours between panels represent approximately matched $\ell _g^+$ riblets.

When rollers form, not only does the intensity of $\langle p'p'\rangle ^+$ increase (figure 3 a), but the peak pressure of the 2-D cospectra of $\hat {p}'\hat {p}'$ also shifts from $\lambda _x^+ \approx 200$ to $\lambda _x^+ \approx 150$ . This was shown for blade riblets in García-Mayoral & Jiménez (2011b ) and for $30^\circ$ triangular riblets in Endrikat et al. (2021b ). Thus, slices of the 2-D cospectra of $\hat {p}'\hat {p}'$ are shown in Appendix A, figure 10 at $z^+ + \ell _T^+\approx 10$ for all riblet sizes and shapes for which the roller intensity was measured in figure 3(a). From these cospectra, the streamwise wavelengths of the peak pressure are collected in figure 3(b). Between $\ell _g^+ \approx 11$ to $\ell _g^+ \approx 13$ , the shift in peak pressure from $\lambda _x^+ \approx 200$ to $\lambda _x^+ \approx 150$ is evident for $30^\circ$ triangular , blade and $30^\circ$ trapezoidal riblets, indicating the appearance of spanwise rollers. However, for the remaining three riblets – $60^\circ$ triangular , $90^\circ$ triangular and asymmetric triangular riblets – there is no clear shift in wavelength. Given the slightly higher roller intensities for $60^\circ$ triangular riblets, very weak rollers may persist for this riblet shape, while $90^\circ$ triangular and asymmetric triangular riblets are essentially ruled out of having rollers of any relevant intensity.

Figure 3. Identifying which riblets have rollers and gauging their intensity through (a) the ratio of the maximum of the $x$ - $y$ - $t$ averaged pressure fluctuations $\langle p'p' \rangle ^+$ , relative to that over a smooth wall, cf. figure 2, and (b) the streamwise wavelength of the local maximum in the pressure spectra (integrated across all $\lambda _y^+$ and all $z^+ + \ell _T^+ \lesssim 10$ ); see Appendix A for more. Thin solid black lines indicate the equivalent smooth-wall values, $\mathrm{max}(\langle p'p'\rangle ^+)/\mathrm{max}(\langle p_{sm}'p_{sm}'\rangle ^+) = 1$ and $\lambda _x^+ \approx 200$ , respectively. Although the ratio $\mathrm{max}(\langle p'p'\rangle ^+)/\mathrm{max}(\langle p_{sm}'p_{sm}'\rangle ^+)$ should be approximately independent of channel size at matched $\mathit{Re}_\tau$ , where $\mathit{Re}_\tau$ is the friction Reynolds number, the magnitude of $\langle p'p' \rangle$ in minimal-channel DNS is about 20 % larger near the wall than its full channel equivalent; for further discussion, see MacDonald et al. (2017, § 3.2).

3. Riblet-resolved linear-stability analysis

The question of whether spanwise rollers are actually a maximally growing instability was raised in § 1.2. Rollers were believed to be a maximally growing instability from the results of linear-stability modelling with an effective wall-admittance boundary condition (García-Mayoral & Jiménez 2011b ), and specifically a boundary condition which was later shown to incorrectly scale with riblet size (Endrikat et al. 2021a ), i.e. $\hat {w}^+/\hat {p}^+\propto {\ell _g^+}$ and not $\propto {\ell _g^+}^3$ as previously assumed. Thus, to avoid any uncertainties introduced by an effective wall-admittance boundary condition, the riblets are first resolved directly as a no-slip, impermeable surface in this section. The results of the riblet-resolved linear-stability analysis are then compared with DNS to show that rollers are actually near marginally stable and not maximally growing as previously believed. This linear-stability analysis then also provides a means of predicting which riblet shapes are likely to promote spanwise rollers.

3.1. Methods

Linear perturbations in the velocity and pressure are expressed as $\{\boldsymbol{u}',p'\}(x,y,z,t) = \{\boldsymbol{\hat {u}},\hat {p}\}(y,z)\exp [\mathrm{i} (\kappa _x x + \kappa _y y - \omega t)]$ , with streamwise wavenumber $\kappa _x$ , spanwise wavenumber $\kappa _y$ and frequency $\omega$ ; again, $z$ denotes the wall-normal coordinate, and $t$ time. Focus is placed on perturbations with small spanwise wavenumber $\kappa _y$ , i.e. large spanwise wavelength $\lambda _y = 2\unicode{x03C0} /\kappa _y$ , as spanwise-coherent rollers typically have $\lambda _y^+ \gtrsim 120$ (García-Mayoral & Jiménez 2011b ; Endrikat et al. 2021a ), and as suggested by the higher magnitudes of the pressure cospectra at $\lambda _y^+ \gtrsim 100$ in Appendix A. For simplicity, $\kappa _y=0$ is considered henceforth, noting that of the other large wavelength modes tested ( $\lambda _y^+ \approx 125$ and $\lambda _y^+ \approx 250$ ), all had higher decay rates than the spanwise-infinite ( $\kappa _y=0$ ) modes (Appendix B).

The perturbations satisfy the linearised Navier–Stokes equations

(3.1) \begin{align} \big[-\mathrm{i} \omega ^+ + U^+ \mathrm{i} \kappa _x^+ + {\kappa _x^+}^2 - \partial _{yy} - \partial _{zz}\big] \hat {u}^+ + \hat {v}^+ \partial _{y} U^+ + \hat {w}^+ \partial _{z} U^+ &= -\mathrm{i} \kappa _x^+ \hat {p}^+ , \end{align}

(3.2) \begin{align} \big[-\mathrm{i} \omega ^+ + U^+ \mathrm{i} \kappa _x^+ + {\kappa _x^+}^2 - \partial _{yy} - \partial _{zz} \big] \hat {v}^+ &= -\partial _{y} \hat {p}^+, \end{align}

(3.3) \begin{align} \big[-\mathrm{i} \omega ^+ + U^+ \mathrm{i} \kappa _x^+ + {\kappa _x^+}^2 - \partial _{yy} - \partial _{zz}\big] \hat {w}^+ &= -\partial _{z} \hat {p}^+ , \end{align}

(3.4) \begin{align} \mathrm{i} \kappa _x^+ \hat {u}^+ + \partial _{y} \hat {v}^+ + \partial _{z} \hat {w}^+ &= 0, \end{align}

here expressed in viscous units and having linearised about a one-component, two-dimensional mean flow $U(y,z)$ . The mean flow satisfies

(3.5) \begin{equation} \left(1+\nu _e^+\right) \partial _{yy} U^+ + \partial _{z} \left[\left(1+\nu _e^+\right) \partial _{z} U^+\right] = 1/\mathit{{Re}}_\tau , \end{equation}

where $\mathit{{Re}}_\tau = u_\tau \delta /\nu$ is the friction Reynolds number based on a domain height $\delta$ , and where the eddy viscosity $\nu _e^+$ captures the influence of the Reynolds stresses ( $\overline {u'w'}$ ) on the mean flow. Note that eddy viscosity is applied only when obtaining the mean flow (3.5), i.e. eddy viscosity is not included in (3.1)–(3.3) when solving for the perturbations. Eddy viscosity is not overly important for accurately predicting spanwise rollers over riblets, given their near-wall location (Gómez-de-Segura 2019), and including eddy viscosity in (3.1)–(3.3) leads to only a small reduction in growth rate (figure 4 d, comparing red solid and red dashed curves).

Boundary conditions in the spanwise direction are periodic (with period corresponding to riblet spacing $s^+$ ), and along the riblet wall are no-slip, on both the mean flow ( $U^+=0$ ) and the perturbations ( $\boldsymbol{\hat {u}}=0$ ). Symmetry conditions at the channel centreline ( $\partial _{z} U^+ = 0$ , $\partial _{z} \hat {u}^+=\partial _{z} \hat {v}^+=w^+=0$ ) are placed at $z^+ = \mathit{{Re}}_\tau -\ell _T^+$ , taking the riblet crests at $z^+=0$ , to ensure a consistent domain height for both riblet and smooth-wall calculations, and where $\ell _T^+$ is the turbulence origin. How $\ell _T^+$ is selected for each riblet size and shape is discussed in Appendix C. A Cess profile for the eddy viscosity $\nu _{e,S}^+(z^+) = 0.5(1 + (\kappa \mathit{{Re}}_\tau (1-z'^2)(1+2z'^2)(1-\exp (-(1-|z'|)\mathit{{Re}}_\tau /A))/3)^2 )^{1/2} - 0.5$ is assumed (Reynolds & Tiederman 1967), where $z' = (z^+-\mathit{{Re}}_\tau )/\mathit{{Re}}_\tau$ . The constants selected for the Cess profile (von Kármán constant $\kappa = 0.46$ and van Driest constant $A=30.7$ ) minimise the sum-squared difference relative to the full-channel smooth-wall DNS mean profile of Endrikat et al. (2021a ) at $\mathit{{Re}}_\tau = 395$ , following Moarref & Jovanović (2012), cf. $\kappa = 0.45$ , $A = 29.4$ at $\mathit{{Re}}_\tau = 547$ . The eddy-viscosity profile for riblets $\nu _{e,R}$ is unchanged, except for a wall-normal shift by $\ell _T^+$ , i.e. $\nu _{e,R}^+(z^+)=\nu _{e,S}^+(z^++\ell _T^+)$ , and where $\nu _{e,R}^+(z^+)=0$ for all $z^+\leqslant -\ell _T^+$ .

Solutions to the eigenvalue problem (3.1)–(3.4) and the mean flow (3.5) are obtained in FreeFem++ (Hecht 2012), having defined the riblet shape and grid sizes in Gmsh (Geuzaine & Remacle 2009). The meshes are unstructured, to maintain grid sizes $\varDelta _y^+ \approx \varDelta _z^+ \approx 0.1$ near the riblets, and $\varDelta _y^+ \approx \varDelta _z^+ \approx 2$ in the far-field. Based on the typical spacing of a roller-promoting riblet ( $s^+ \approx 20$ ), these grid sizes correspond to $\approx$ 200 points per riblet spacing near the crests, and, e.g. $\approx$ 40 points across the tips of the blade riblets. The finite elements are triangular, second-order in velocity and first-order in pressure. The eigenvalue subroutine uses a shifted-inverse method, with a typical initial guess of a wave speed of $c^+ \approx 4$ (and zero growth rate), and with the nearest four eigenvalues requested to a relative tolerance of $10^{-12}$ . The grid-resolution error was also assessed for trapezoidal riblets in the range $70 \lesssim \lambda _x^+ \lesssim 400$ (figure 4 d, comparing red solid and black dashed curves) as relevant to spanwise rollers. The errors in the growth rate were below $0.1$ % relative to halved grid sizes, $\Delta _y^+ \approx \Delta _z^+ \approx 0.05$ near the riblets and $\varDelta _y^+ \approx \varDelta _z^+ \approx 1$ in the far-field; for reference, this required $\approx 1.14$ M elements, compared with $\approx 0.28$ M elements for the baseline resolution used henceforth.

3.2. Predicting which riblets have rollers

As shown in figure 4, linear-stability analysis in which the detailed riblet shapes are resolved, and not modelled with an effective boundary condition, yields predictions consistent with DNS (García-Mayoral & Jiménez 2011b ; Endrikat et al. 2021a , b ). This includes predicted wavelengths near $\lambda _x^+ \approx 150$ (figure 4 b), predicted wave speeds near $c^+ \approx 6$ (figure 4 c), and the roller mode first appearing at $\ell _g^+ \approx 10$ upon increasing $\ell _g^+$ (figure 4 a), all consistent with DNS of rollers over riblets.

Figure 4. Predictions of the riblet-resolved linear-stability analysis for riblets with sizes $\ell _g^+ \lesssim 30$ . Positive growth rates indicate growing rollers, which are not achieved by all riblet shapes. (a) Growth rate of the spanwise-infinite mode, at the local maximum in growth rate (only if present). (b) Corresponding streamwise wavelength, with the shaded region indicating typical values for rollers from DNS (García-Mayoral & Jiménez 2011b ; Endrikat et al. 2021a ). (c) Corresponding wave speed, with the shaded region again indicating typical values for rollers from DNS. In (a–c), solid lines use DNS-interpolated $\ell _T^+$ values for the Cess profile, and dashed lines use a priori viscous vortex model $\ell _T^+$ values. See Appendix C for more. (d) Assessment of the sensitivity of the growth rate predictions to the numerical resolution and to modelling assumptions. Only the choice of the turbulence origin $\ell _T^+$ proves greatly important.

The 2-D linear-stability analysis also correctly predicts that rollers should be most intense for $30^\circ$ triangular riblets, as their roller growth rates are the highest across the riblet shapes tested (figure 4 a). In addition, rollers would be next-most intense for blade , $30^\circ$ trapezoidal and then 60 $^\circ$ triangular riblets, respectively, consistent with the DNS ordering of intensities (figure 3 a). However, the 2-D linear-stability analysis differs slightly from the DNS in terms of when rollers first become marginal. For example, from DNS measurements (figure 3), rollers are expected at $\ell _g^+ \approx 13$ for both blade and $30^\circ$ trapezoidal riblets, especially from the shift in wavelength of the peak pressure in figure 3(b). However, marginal rollers are predicted in the linear-stability analysis past $\ell _g^+ \approx 14$ and $\ell _g^+ \approx 17$ , respectively, for these riblet shapes. Only for $30^\circ$ triangular riblets are rollers predicted at $\ell _g^+ \approx 11$ , consistent with when rollers are expected from DNS (figure 3).

While the linear-stability predictions in figure 4 (solid lines) overall agree with DNS observations, the stability results themselves still implicitly depend on the DNS through the DNS-interpolated model for the eddy viscosity origin, $\ell _T^+$ (discussed in Appendix C). Thus, additional linear-stability predictions are provided in figure 4 based on a priori $\ell _T^+$ values (dashed lines). The a priori $\ell _T^+$ values are taken from the viscous vortex model of Wong et al. (2024) and are accurate only up to $\ell _g^+ \approx 10$ , with the error in the extrapolation for larger $\ell _g^+$ shown in Appendix C. The now completely a priori 2-D linear-stability predictions (dashed lines) overall remain consistent with the DNS observations of spanwise rollers over riblets. The a priori predictions correctly identify which riblets promote rollers and which do not, and successfully sort riblet geometries by roller intensity. However, the a priori predictions do tend to somewhat overpredict growth rates, as the shear layer is up to $\approx 4$ viscous units thicker with the extrapolated a priori $\ell _T^+$ values than with the DNS-interpolated $\ell _T^+$ values. There is also some degree of sensitivity in the roller wave speed, and sometimes also roller wavelength, with small changes in $\ell _T^+$ (1–2 viscous units). Overall, further work is required to improve predictions of $\ell _T^+$ for larger riblets. Nevertheless, these results demonstrate that useful a priori linear-stability predictions can still be made for other riblet geometries (without further model improvements), for riblets up to and slightly past their design point of maximum drag reduction ( $\ell _g^+\approx 10$ ), and without running any DNS. This does not alter the main message though, i.e. that an accounting of the shift in turbulence $\ell _T^+$ is necessary to correctly predict the occurrence of rollers (figure 4 d, comparing solid and dotted lines). Notably, taking $\ell _T^+ = 0$ (crest-origin) regardless of riblet size and shape would result in all riblet shapes tested promoting spanwise rollers for all $\ell _g^+$ past the onset of the instability (not shown). In reality, for large riblets, the turbulent flow descends into the riblet grooves (i.e. $\ell _T^+ \gt 0$ ), leading to an increased lack of mean shear and causing the rollers to decay at larger $\ell _g^+$ .

The results in figure 4(a) also provide a means to predict the intensity of rollers over these and similarly shaped riblets. For example, weak rollers were suggested by flow-reversal within the grooves of $\ell _g^+ \approx 19$ , $\alpha \approx 53^\circ$ tip-angle triangular riblets, from the DNS of Chu & Karniadakis (1993, figure 30). Equally, weak rollers would be predicted in the linear-stability analysis, as the similarly sized $\alpha \approx 60^\circ$ tip-angle triangular riblets have near-marginal rollers according to figure 4(a), and as more slender riblets tend to increase the likelihood of rollers (comparing green through purple through yellow curves, in order of decreasing $\alpha$ , or increasing slenderness).

Furthermore, the 2-D linear-stability analysis also predicts that not all riblets promote rollers, unlike the 1-D analysis of García-Mayoral & Jiménez (2011b ). Figure 4(a) shows that rollers would almost never form for $90^\circ$ triangular and asymmetric triangular riblets, in agreement with the lack of rollers observed for those riblet shapes in figures 1–3, and in agreement with DNS (Endrikat et al. 2021a , b ). Note that the maximum growth rate for a given riblet size is plotted only in figure 4(a) when a local maximum in growth rate is present, such that results are not plotted past $\ell _g^+ \approx 24$ for $90^\circ$ triangular riblets.

These 2-D linear-stability results further demonstrate that spanwise rollers are not a maximally growing instability, as the growth rates remain near-zero (marginal) in figure 4(a) for riblets with sizes $10 \lesssim \ell _g^+ \lesssim 20$ . Note that the growth rates from the 2-D linear-stability analysis are $\mathrm{Im}(\omega ^+) \lesssim 0.04$ , across a relatively wide range of $\ell _g^+$ , and that these growth rates are significantly smaller than those measured for maximally growing rollers, $\mathrm{Im}(\omega ^+) \approx 0.2$ (García-Mayoral & Jiménez 2011b ). That rollers are marginal modes thereby explains the agreement between the current linear-stability predictions and those of the resolvent analysis of blade riblets in Chavarin & Luhar (2020); additional comparisons between linear-stability and resolvent analysis are provided in Appendix D. Equally, that rollers are marginal explains why García-Mayoral & Jiménez (2011b ), who assumed maximally growing rollers, underpredicted wavelengths of $\lambda _x^+ \approx 60$ for spanwise rollers. As shown in figure 4(b), marginal rollers have wavelengths $100 \lesssim \lambda _x^+ \lesssim 200$ , consistent with DNS. This realisation further motivates the reconsideration of a 1-D linear-stability model in § 5.

The 2-D linear-stability analysis also suggests rollers weaken and eventually vanish for all riblet shapes at larger $\ell _g^+$ (only computed up to $\ell _g^+ \approx 30$ in the linear analysis). However, some discrepancies between the linear-stability analysis and the DNS are again observed. For example, the 2-D analysis predicts that rollers would decay by $\ell _g^+ \approx 25$ for $30^\circ$ trapezoidal riblets, whereas from the DNS analysis, weak rollers appear to persist until $\ell _g^+ \approx 40$ , with $\mathrm{max}(\langle p'p'\rangle ^+)/\mathrm{max}(\langle p_{sm}'p_{sm}'\rangle ^+)$ remaining slightly above unity in figure 3(a), although slowly decreasing with increasing $\ell _g^+$ . As an essentially inviscid instability (Jiménez et al. 2001; Drazin & Reid 2004; García-Mayoral & Jiménez 2011b ), the roller mode is very sensitive to the magnitude of $\partial _{zz} U$ near the crests, which sets the size and intensity of the shear layer. With increasing riblet size, turbulence descends further within the riblet grooves, so $\ell _T^+$ increases and $\partial _{zz} U$ reduces. Perturbation energy production via $ \mathrm{i}\kappa _x^+ \hat {w}^+ \partial _{zz} U^+$ thereby falls with increasing $\ell _g^+$ , and so, based on the 2-D linear model with $\ell _T^+$ -shifted Cess, it is only $\ell _g^+ \lesssim 30$ riblets that sustain marginal rollers. Thus, the weakening of the roller mode can be explained within the simplified framework of the linear-stability analysis. Note that rollers weaken at larger $\ell _g^+$ in spite of the increases to the wall admittance with riblet size, recalling that $\hat {w}^+/\hat {p}^+$ scales approximately linearly with $\ell _g^+$ (Endrikat et al. 2021a , figure 13 b). In addition, the validity of the linear analysis (and this explanation) is supported by its predictions remaining generally consistent with both DNS (Endrikat et al. 2021a ) at the same $\mathit{{Re}}_\tau$ and recent experiments (Abu Rowin et al. 2025) at much higher $\mathit{{Re}}_\tau$ . Note that there is no immediate reason why the model predictions (and explanation) should be invalidated at higher $\mathit{{Re}}_\tau$ , given the insensitivity of the near-wall mean-velocity curvature to $\mathit{{Re}}_\tau$ , as discussed in García-Mayoral & Jiménez (2012).

Linear predictions are not provided in figure 4 for $\ell _g^+ \gt 30$ , as the Cess approximation of the mean flow is an increasingly poor representation of the DNS mean flows for increasingly large riblets, and as the roller dynamics may become increasingly nonlinear. However, riblets are typically manufactured to ensure $\ell _g^+ \approx 10$ at their design point, as assumed in, for example, Mele et al. (2020) and Mele, Saetta & Tognaccini (2023), such that the eventual breakdown of the Cess approximation at larger $\ell _g^+$ barely impacts the usefulness of the 2-D predictions for riblet design.

Further improvements to the mean-flow approximation are not considered here, although the importance of various modelling choices is briefly considered for not-too-large riblets ( $\ell _g^+ \approx 15$ ) in figure 4(d). As shown, the constants employed in the Cess eddy-viscosity profile, the inclusion of eddy viscosity when solving for the perturbations, and the numerical resolution, all do not greatly alter the growth rates predicted for the roller instability. The only modelling choice which proves relevant is the choice of turbulence origin $\ell _T^+$ . Slight differences in the location of the turbulence origin (DNS-fitted or DNS-matched) provide commensurately small differences in the predicted growth rate, and so do not compromise growth predictions. Note that the $\ell _T^+$ values here are based on quadratic fits to DNS data and remain within $\approx \pm 30\,\%$ of DNS-matched values of $\ell _T^+$ , where the DNS-matched values satisfy $\Delta U^+ = -(\ell _U^+ - \ell _T^+)$ at each DNS data point (discussed further in Appendix C). Further evidence that the fitted $\ell _T^+$ values are satisfactory is provided in Appendix E, figures 15 and 16. Predictions from riblet-resolved 2-D mean flows (with smooth-wall Cess eddy-viscosity profiles shifted by fitted- $\ell _T^+$ ) are shown to be consistent with predictions from DNS 2-D mean flows (Appendix E), for all riblet geometries tested. However, an unrealistic estimate for the turbulence origin, e.g. assuming the turbulence origin is at the riblet crests ( $\ell _T^+ = 0$ ), noticeably alters the predicted growth rate, to the extent that growing or decaying rollers may be misidentified (figure 4 d). Thus, care is warranted for very large riblets, when either selecting the virtual origin, or otherwise obtaining an estimate of the mean flow.

3.3. Further evidence that the spanwise rollers are a marginal instability

To further demonstrate that the rollers in DNS are a marginal instability, the wall admittances measured in the 2-D linear-stability analysis are compared with the wall admittances measured in DNS for Fourier modes representative of spanwise rollers ( $\lambda _y^+ \rightarrow \infty$ ), for blade and trapezoidal riblets, up to $\ell _g^+ \approx 20$ . These wall-admittance measurements are performed only to further support the finding that rollers are marginal modes, rather than as a separate means of identifying spanwise rollers, as used previously (Endrikat et al. 2021a ).

Figure 5. Comparing the wall admittance measured in the DNS with that predicted by the 2-D linear-stability analysis (measured at $z^+ = -\ell _U^+$ , zeroing $\hat {w}^+$ and $\hat {p}^+$ within the riblets). Coloured contours are probability density functions from the DNS for spanwise modes with wavelengths $\lambda _y^+ \rightarrow \infty$ . Solid lines are 2-D linear-stability analysis ( $\kappa _y^+ = 0$ mode; black, decaying; cyan, growing). Dot-dashed black lines are from the 1-D model, (4.2).

The wall admittance $\hat {w}^+/\hat {p}^+$ is measured in the 2-D linear-stability analysis by superficially spanwise averaging $\hat {w}^+$ and intrinsically spanwise averaging $\hat {p}^+$ at $z^+ = -\ell _U^+$ , and calculating the resulting amplitude and phase of $\hat {w}^+/\hat {p}^+$ as a function of $\lambda _x^+$ . For the corresponding DNS measurements, the same process is applied for each instantaneous snapshot, of which there are $\approx 150$ to $700$ snapshots depending on the case in question. From these instantaneous measurements of $\hat {w}^+/\hat {p}^+$ , probability density functions of the amplitude and phase are constructed, the former with 30 logarithmically spaced bins placed between amplitudes $10^{-3}$ and $1$ , and the latter with 30 linearly spaced bins between $-\unicode{x03C0}$ and $\unicode{x03C0}$ (then translated by $2\unicode{x03C0}$ and converted to degrees when plotted), following Endrikat et al. (2021b ). To account for the logarithmically spaced bins, the probability densities for the amplitude of $\hat {w}^+/\hat {p}^+$ are premultiplied by the bin centres. Finally, note that zeroing within the roughness has not previously been observed to corrupt spectral measurements for spanwise wavelengths larger than the roughness (riblet) spacing (Sharma & García-Mayoral 2020).

As shown in figure 5, the wall-admittance amplitudes $|\hat {w}^+/\hat {p}^+|$ and phases $\angle (\hat {w}^+/\hat {p}^+)$ predicted by the 2-D linear-stability analysis are in reasonable agreement with DNS measurements, for riblets with $\ell _g^+ \gtrsim 15$ and at streamwise wavelengths $100 \lesssim \lambda _x^+ \lesssim 300$ . Specifically, when growing rollers are measured in the 2-D linear-stability analysis, as indicated by the cyan portions of the solid lines in figure 5, the wall-admittance amplitudes $|\hat {w}^+/\hat {p}^+|$ and phases $\angle (\hat {w}^+/\hat {p}^+)$ are near the medians of the DNS probability density functions (ridges of the coloured contours). Thus, the spanwise rollers in the DNS resemble near-marginal modes for these blade and $30^\circ$ trapezoidal riblets with $\ell _g^+ \gtrsim 15$ and across $100 \lesssim \lambda _x^+ \lesssim 300$ . For smaller riblets ( $\ell _g^+ \lesssim 13$ ) and at smaller wavelengths $\lambda _x^+ \lesssim 100$ , the linear-stability predictions and DNS results differ, although rollers are rarely, if ever, present in these ranges in the DNS (García-Mayoral & Jiménez 2011b ; Endrikat et al. 2021a ).

Finally, figure 5 shows that the wall-admittance amplitudes $|\hat {w}^+/\hat {p}^+|$ measured at the riblet-shape-dependent mean origin $z^+ = -\ell _U^+$ are still typically order $0.1$ for roller-promoting riblets. These mean-origin admittance measurements have similar magnitudes to the crest-admittance measurements of Endrikat et al. (2021a , figure 12), and remain orders of magnitude below the wall admittances required for riblets to emulate a free-shear layer, as previously assumed (García-Mayoral & Jiménez 2011b ). Thus, riblets appear unable to support a maximally growing Kelvin–Helmholtz mode, and at best, support its marginal precursor. However, realising that rollers are marginal is not alone sufficient for the 1-D linear-stability model of García-Mayoral & Jiménez (2011b ) to then correctly predict rollers, or a lack thereof, for a given riblet shape. Accurate predictions for rollers also require improved approximations of both the effective wall-admittance boundary condition and the mean-velocity profile, as considered in § 4.

4. Towards a 1-D linear-stability model: understanding the flow dynamics within the riblet groove

Having shown that spanwise rollers can be well predicted with 2-D linear-stability analysis (§ 3), the question then becomes whether it is necessary to resolve the details of the riblet shapes, e.g. is an effective boundary condition adequate? Various approaches to obtain effective boundary conditions for riblets, or other rough or permeable surfaces, have been proposed (Luchini, Manzo & Pozzi 1991; García-Mayoral & Jiménez 2011b ; Luchini 2013; Lācis & Bagheri 2017; Gómez-de-Segura et al. 2018a ; Bottaro 2019; Bottaro & Naqvi 2020; Lācis et al. 2020; Naqvi & Bottaro 2021; Habibi Khorasani et al. 2022). These effective boundary conditions typically relate the velocity components to their gradients (or the pressure) along some plane, e.g. at the riblet crests. For example, the effective boundary conditions relevant to spanwise-infinite modes over riblets could be expressed as $\hat {u}^+ = \hat {C}_{uu} \partial _z \hat {u}^+ + \hat {C}_{up} \hat {p}^+$ and $\hat {w}^+ = \hat {C}_{wu} \partial _z \hat {u}^+ + \hat {C}_{wp} \hat {p}^+$ , here expressed in Fourier space, as in Gómez-de-Segura et al. (2018a ), Gómez-de-Segura & García-Mayoral (2020) and Hao & García-Mayoral (2025). The challenge is then to obtain the coefficients $\hat {C}_{uu}$ , $\hat {C}_{up}$ , $\hat {C}_{wu}$ and $\hat {C}_{wp}$ , which in this case depend not only on the riblet size and shape, but also on $\lambda _x^+$ , $\mathrm{{Re}}(\omega ^+)$ and $\mathrm{Im}(\omega ^+)$ . Accurately obtaining these coefficients often relies on texture-resolved calculations in a representative volume element (Bottaro 2019), such as in the 2-D linear-stability analysis in § 3. Moreover, these coefficients provide little perspective with which to explain why only some riblets promote rollers, as it becomes difficult to inspect the relationships between the coefficients, the riblet shapes and the growth of the roller instability. However, given that the 2-D linear-stability analysis has already been performed (i.e. the $\omega ^+$ of interest at which $\hat {C}_{uu}$ , $\hat {C}_{up}$ , $\hat {C}_{wu}$ and $\hat {C}_{wp}$ should be evaluated are already known, from § 3), an assessment of the accuracy of effective boundary conditions can be quickly made, as shown in figure 6. Even with perfect information (i.e. taking from the riblet-resolved 2-D-analysis measurements of the spanwise-averaged mean flow and spanwise-averaged perturbations) the use of a 1-D effective boundary condition placed at the riblet crests $z^+=0$ yields predictions with some error (comparing square markers to solid line). This error almost entirely vanishes once the 1-D effective boundary condition is placed $\approx 6$ viscous units above the riblet crests (circle markers), indicating that both spanwise variations in the mean flow and spanwise variations in the perturbations, which lead to cross-terms upon spanwise averaging, are relevant near the riblet crests ( $z^+ \lesssim 6$ ). These spanwise variations in the riblet-resolved analysis must be captured, at least in an aggregate sense, for the 1-D analysis to be able to replicate the 2-D results. The importance of spanwise variations in the mean is further considered in § 4.2.

Figure 6. Testing effective boundary conditions for riblets, here, for $\ell _g^+ \approx 12$ trapezoidal riblets. (a) Growth rates. (b) Wave speeds. Solid lines are from the 2-D linear-stability analysis, identical to figures 8(g) and 8( j), respectively. Symbols are from 1-D linear-stability analysis, with more general wall-admittance boundary conditions of $\hat {u}^+ = \hat {C}_{uu} \partial _z \hat {u}^+ + \hat {C}_{up} \hat {p}^+$ and $\hat {w}^+ = \hat {C}_{wu} \partial _z \hat {u}^+ + \hat {C}_{wp} \hat {p}^+$ , where $\hat {C}_{uu}$ , $\hat {C}_{up}$ , $\hat {C}_{wu}$ and $\hat {C}_{wp}$ were calculated for each $\lambda _x^+$ based on the growth rates and wave speeds of the leading eigenvalues from the 2-D analysis.

As calculating the coefficients $\hat {C}_{uu}$ , $\hat {C}_{up}$ , $\hat {C}_{wu}$ and $\hat {C}_{wp}$ for an effective boundary condition is no more informative than 2-D linear-stability analysis, the choice is instead made to develop a model boundary condition for the 1-D analysis in § 4.1. As shown in García-Mayoral & Jiménez (2011b ), the relationship between $\hat {w}^+$ and $\hat {p}^+$ dictated the intensity of the roller instability, while Hao & García-Mayoral (2025) further highlighted the importance of the overlying pressure fluctuations in driving wall-normal transpiration. Thus, obtaining a model for the $\hat {C}_{wp}$ coefficient in particular is pursued. Furthermore, from the results presented in figure 5, such a boundary condition should relate the wall-normal velocity and pressure $\hat {w}^+/\hat {p}^+$ as a function of streamwise wavelength. For example, if $\hat {w}^+/\hat {p}^+$ varies as a decreasing function of $\lambda _x^+$ , this physically represents coherent structures with large streamwise length scales struggling to penetrate the in-groove flow, and thereby experiencing a low effective wall admittance, while structures with small streamwise lengths scales experience higher wall admittances as they easily enter the riblet grooves. Such scale-dependent wall admittances have been proposed to model a range of other permeable and compliant surfaces (Brooke Benjamin 1960; Landahl 1962; Carpenter & Garrad 1986; Gómez-de-Segura et al. 2018b ; Gómez-de-Segura 2019; Gómez-de-Segura & García-Mayoral 2020; Hao & García-Mayoral 2025), although scale-independent wall admittances have also been considered (Jiménez et al. 2001; Bottaro 2019; Motoki et al. 2022). Section 4.1 is thereby devoted to developing a model for the effective wall-admittance boundary condition valid for streamwise-aligned riblets, including the dependence of $\hat {w}^+/\hat {p}^+$ on the streamwise length scale $\lambda _x^+$ , and on the riblet size and shape. Section 4.2 then further considers the sensitivity of marginal rollers to approximations of the mean flow, while the riblet shapes are still resolved, i.e. without simultaneously introducing uncertainty in both the approximation of the mean flow, and the effective wall-admittance boundary condition.

4.1. Relating the wall admittance to the streamwise wavelength

As a first step in developing an expression for $\hat {w}^+/\hat {p}^+$ , the median amplitude and phase of the probability density functions for $\ell _g^+ \approx 16$ blade riblets are plotted in figure 7 as a function of wall-normal height within the riblet groove. Note that the median amplitude (or phase) tracks the crest of the corresponding probability density function for a given wall-normal height, and that the probability density functions at $z^+ = -\ell _U^+ \approx -2.9$ were depicted in figure 5(q) for the amplitude, and figure 5(s) for the phase, for these $\ell _g^+ \approx 16$ blade riblets. Figure 7 then serves to test the previous assumption that rollers experienced an effective wall admittance due to streamwise variations in the mass flux within the riblet grooves, driven by the overlying pressure and balanced by viscous diffusion (García-Mayoral & Jiménez 2011b ). In relation to figure 7, these assumptions imply the admittance amplitude $|\hat {w}^+/\hat {p}^+|$ should vary as ${\lambda _x^+}^{-2}$ and the admittance phase be constant at $\angle (\hat {w}^+/\hat {p}^+) = 180^\circ$ . To see this, note that the streamwise momentum equation with a dominant balance between pressure and viscous diffusion simplifies to ${\nabla} _{yz}^2 \hat {u}^+ = \mathrm{i} \kappa _x^+ \hat {p}^+$ within the groove. The in-groove velocity can be expressed as $\hat {u}^+ = -\mathrm{i}\kappa _x^+ \hat {p}^+ f^+(y^+,z^+)$ , as shown in García-Mayoral & Jiménez (2011b ), by assuming the pressure is uniform within the groove cross-section and by defining an auxiliary function $f^+$ which satisfies ${\nabla} _{yz}^2 f^+ = -1$ (and has the same boundary conditions as $\hat {u}^+$ ). Substituting continuity $-\mathrm{i} \kappa _x^+ \hat {u}^+ = \partial _z \hat {w}^+$ into the in-groove velocity expression, to relate $\partial _z\hat {w}^+$ to $\hat {p}^+$ , yields $\partial _z \hat {w}^+ = -{\kappa _x^+}^2 \hat {p}^+ f^+(y^+,z^+)$ . Finally, integrating across the groove cross-section (from $z^+\leqslant 0$ to the valley) gives $\langle \hat {w}^+\rangle |_{z^+} = -({\kappa _x^+}^2\, \hat {p}^+|_{z^+}/s^+) \iint _{A^+_{g,z}} f^+ \,\mathrm{d}y^+\mathrm{d}z^+$ , i.e. equation (6.9) of García-Mayoral & Jiménez (2011b ), except with $A_{g,z}^+$ , the groove area below the height $z^+$ , in the place of $A_{g}^+$ , the groove area below the crests (at $z^+ = 0$ ) and assuming no-slip, impermeable riblet walls. The integral $\iint _{A^+_{g,z}} f^+ \,\mathrm{d}y^+\mathrm{d}z^+$ is real, and depends just on the riblet geometry (García-Mayoral & Jiménez 2011b ), so the real prefactor ${\kappa _x^+}^2/s^+ = 4\unicode{x03C0} ^2/({\lambda _x^+}^2s^+)$ dictates the scale-by-scale relationship between $\langle \hat {w}^+\rangle$ and $\hat{p}^+$ which for this pressure–viscous balance is proportional to ${\lambda _x^+}^{-2}$ with a $180^\circ$ phase difference.

Figure 7. Median magnitude and phase of the wall admittance as a function of wall-normal height within the groove of blade riblets, to infer the dominant balances in the governing equations. An admittance magnitude varying as ${\lambda _x^+}^{-1}$ and a 270 $^\circ$ phase correspond to a balance dominated by unsteady inertia and overlying pressure. An admittance magnitude varying as ${\lambda _x^+}^{-2}$ and a 180 $^\circ$ phase correspond to a balance dominated by viscous diffusion and overlying pressure.

As shown in figure 7, however, such a pressure–viscous-diffusion balance is only satisfied deep within the riblet grooves, within half a viscous unit of the groove floor ( $z^+ \lesssim -12.2$ , maroon curve, with the groove floor at $z^+ \approx -12.5$ ), and implying a very thin viscous sublayer for these long spanwise-wavelength $\lambda _y^+ \rightarrow \infty$ modes. Thus, beyond one viscous unit from the valley floor, the DNS measured amplitudes and phases no longer satisfy the viscous-dominated wall-admittance boundary condition assumed by García-Mayoral & Jiménez (2011b ) for these $\ell _g^+ \approx 16$ riblets. Although much smaller riblets $\ell _g^+ \lesssim 10$ may have viscous-dominated in-groove flow, as suggested by the slightly steeper admittance trends in figure 5, these smaller riblets are equally unlikely to ever sustain rollers. Thus, a new wall-admittance boundary condition is developed, valid for riblets of sizes $10 \lesssim \ell _g^+ \lesssim 20$ .

One approach that provides wall-admittance trends consistent with those in figure 7 is to assume the overlying pressure is instead balanced by unsteady inertia within the groove. Under such a pressure–unsteady-inertia balance, the streamwise momentum equation simplifies to $-\mathrm{i} \kappa _x^+ c^+ \hat {u}^+ = -\mathrm{i} \kappa _x^+ \hat {p}^+$ within the groove, and where unsteady inertia refers specifically to $\partial _t \hat {u}^+$ in this paper. Linearised mean-flow advection terms $U^+\mathrm{i}\kappa _x^+ \{\hat {u}^+,\hat {v}^+,\hat {w}^+\}$ , $\hat {v}^+\partial _yU^+$ and $\hat {w}^+\partial _zU^+$ are also neglected below $z^+ = -\ell _U^+$ , as the mean velocity and its gradients drop noticeably within the riblet grooves. From DNS measurements for $\ell _g^+ \lesssim 20$ riblets, the superficially averaged mean velocity at the mean origin is at least half the crest mean velocity, i.e. $\langle U^+ \rangle |_{z^+ = -\ell _U^+}\lesssim 0.5\langle U^+ \rangle |_{z^+ = 0}$ . Only blade riblets have noticeably higher mean velocities at the mean origin, of up to $\approx 67$ % of the crest velocity. Substituting continuity $-\mathrm{i} \kappa _x^+ \hat {u}^+ = \partial _z \hat {w}^+$ into the streamwise momentum equation to eliminate $\hat {u}^+$ provides $\partial _z \hat {w}^+ = -\mathrm{i} \kappa _x^+ \hat {p}^+ / c^+$ . Applying the same technique as previously and integrating over the groove area yields $\langle \hat {w}^+\rangle |_{z^+} = -[\mathrm{i}\kappa _x^+\, \hat {p}^+|_{z^+}/(c^+s^+) ]\iint _{A^+_{g,z}} \mathrm{d}y^+\mathrm{d}z^+$ for any $z^+\leqslant 0$ , under the same assumption of a pressure uniformly equal to $\hat {p}^+|_{z^+}$ over the groove cross-section and with impermeable riblet walls. Evaluated at the virtual origin for the mean flow, this yields a wall-admittance boundary condition of

(4.1) \begin{equation} \frac {\langle \hat {w}^+\rangle }{\hat {p}^+}\Bigg |_{z^+ = -\ell _U^+} = \frac {-\mathrm{i} \kappa _x^+}{c^+} \frac {A_{g,-\ell _U}^+}{s^+}, \end{equation}

where $A_{g,-\ell _U}^+$ is the groove area below $z^+ = -\ell _U^+$ and where $z^+ = 0$ is at the riblet crests. Expressing the effective boundary condition as (4.1) showcases the relationship between the wall admittance and the riblet geometry, via $A_{g,-\ell _U}^+/s^+$ , which may not have been readily apparent from brute-force calculations of the wall-admittance coefficient $\hat {C}_{wp}$ for different riblet sizes. The effective boundary condition (4.1) is composed of $-\mathrm{i}\kappa _x^+/c^+ = -2\unicode{x03C0} \mathrm{i}/(\lambda _x^+c^+)$ multiplied by the real, purely geometric factor $A_{g,-\ell _U}^+/s^+$ , and further highlights that having assumed a balance between pressure and unsteady inertia within the groove provides a wall admittance proportional to ${\lambda _x^+}^{-1}$ and a constant $270^\circ$ phase difference when $c^+$ is real (i.e. the mode is marginal), consistent with the trends in DNS shown in figure 7. Specifically, the wall-admittance amplitudes measured in DNS vary as ${\lambda _x^+}^{-1}$ at wall-normal heights near the mean origin ( $z^+ = -\ell _U^+$ ), while the wall-admittance phases are slightly lower than $270^\circ$ , but still remain reasonably constant across $\lambda _x^+$ . Only much deeper within the riblet grooves, between $-12 \lesssim z^+ \lesssim -6$ , are both unsteady inertia and viscous diffusion equally relevant in balancing the overlying pressure. Similarly, the 2-D linear-stability analysis predicted trends of ${\lambda _x^+}^{-1}$ for the admittance and $\approx 270^\circ$ for the phase at $z^+ = -\ell _U^+$ , when rollers were near marginal, recalling figure 5. Thus, in the 1-D analysis, a pressure–unsteady-inertia balance will be encoded in the effective wall-admittance boundary condition at $z^+ = -\ell _U^+$ , rather than a pressure–viscous-diffusion balance.

To further reinforce that a pressure–unsteady-inertia balance is dominant near $z^+ \approx -\ell _U^+$ , the model boundary condition (4.1) is compared with 2-D linear-stability analysis and DNS measurements in figure 5, taking a fixed (real) wave speed of $c^+ = 6$ ; see (4.2). When rollers are near marginal, the wall-admittance amplitudes given by (4.2), dot-dashed lines, agree well with the wall-admittance amplitudes measured from the 2-D linear-stability analysis, solid black/cyan lines, and from DNS, coloured contours. This agreement is observed in both the matched ${\lambda _x^+}^{-1}$ trends with streamwise wavelength, and the approximate $A_{g,-\ell _U}^+/s^+$ scaling of the wall-admittance amplitude with riblet size. Thus, the 1-D boundary condition (4.2), which captures the importance of unsteady inertia within the riblet grooves, reasonably represents the admittance amplitude provided by grid-resolved riblets. However, the admittance phases measured in the 2-D linear-stability analysis and DNS at $z^+ = -\ell _U^+$ are slightly lower than the predicted phase of 270 $^\circ$ from (4.1). Thus, (4.1) is henceforth approximated as

(4.2) \begin{equation} \frac {\hat {w}^+}{\hat {p}^+}\Bigg |_{z^+ = -\ell _U^+} = \frac {\kappa _x^+}{c_{\textit{ref}}^+} \frac {A_{g,-\ell _U}^+}{s^+}\exp (\mathrm{i} \phi _{\textit{ref}}), \end{equation}

taking $c_{\textit{ref}}^+ = 6$ as a typical reference wave speed and $\phi _{\textit{ref}} = 250^\circ$ as a typical phase for near-marginal spanwise rollers, and having verified that taking $\phi _{\textit{ref}} = 250^\circ$ , consistent with DNS (figure 5), yields slightly more accurate predictions than $\phi _{\textit{ref}} = 270^\circ$ . Equation (4.2), in addition to a no-slip condition $\hat {u}^+=0$ , then form the effective boundary conditions to be applied in the 1-D analysis.

4.2. Approximations to the mean-velocity profile: are spanwise variations important?

There is the potential for a greater degree of error when performing 1-D rather than 2-D analysis, as both the effective boundary condition and mean-velocity profile are approximated. As was shown in figure 6, however, these errors vanish when both (i) rigorous boundary conditions relating both $\hat {u}^+$ and $\hat {w}^+$ to both $\partial _z \hat {u}^+$ and $\hat {p}^+$ are obtained (instead of modelling these relationships) and (ii) the effective boundary conditions are placed above the riblet crests, in a region where the mean flow is spanwise uniform (and using this spanwise-uniform mean flow in the 1-D analysis). Error (ii) specifically stems from the absence of cross-terms in the 1-D analysis. Cross-terms of the form, for example, $\langle (U^+ - \langle U^+\rangle _y)\mathrm{i}\kappa _x^+(\hat {u}^+ - \langle \hat {u}^+\rangle _y)\rangle _y$ , result from taking the 2-D advection term $U^+(y^+,z^+)\mathrm{i}\kappa _x^+ \hat {u}^+(y^+,z^+)$ , subtracting the 1-D advection term $\langle U^+\rangle _y\mathrm{i}\kappa _x^+\langle \hat {u}^+\rangle _y$ , and spanwise averaging. Note that cross-terms also immediately vanish if the 2-D mean flow happens to be spanwise uniform, such that error (ii) is non-zero only below a certain wall-normal height.

Figure 8. Assessing the importance of spanwise variations in the mean flow. Growth rates and wave speeds predicted for rollers over blade and trapezoidal riblets obtained with spanwise-uniform mean flows (middle column), remain similar to those with spanwise-varying mean flows (left column; as in figure 4). Zeroing the mean flow below the mean origin $z^+ = -\ell _U^+$ (right column) is also relatively unimportant when it comes to predicting the roller growth rates, but tends to underpredict wave speeds. Note that the superficial averages of the mean flows above $z^+ = -\ell _U^+$ (right column) are identical to those used in the 1-D linear-stability analysis (figure 9).

The importance of the cross-terms, and thereby the spanwise variations in the mean, are here considered by modifying the mean flow, while applying impermeable, no-slip boundary conditions on the perturbations along the riblet surface. This avoids simultaneously introducing uncertainties in the modelling approximations for both the mean flow and the perturbations. Specifically, the cross-terms are eliminated from the 2-D analysis (figure 8) by replacing the spanwise-varying mean flow $U^+(y^+,z^+)$ with its spanwise average $\langle U^+\rangle _y (z^+)$ . Here, the intrinsic average is used below the crests (as defined in Appendix E), to ensure a smooth wall-normal variation in the 1-D superficially averaged mean-velocity profile even if there is a sudden change in the fluid volume fraction, e.g. as for blade riblets. As shown in figure 8, the growth-rate and wave-speed predictions based on both spanwise-varying mean flows (left column, with cross-terms), and intrinsically averaged spanwise-uniform mean flows (middle column, without cross-terms) are similar, for the riblets of sizes $\ell _g^+ \lesssim 20$ tested. Thus, it is unlikely that the cross-terms, that are by default absent in the 1-D analysis, will greatly impact the accuracy of the 1-D predictions, based on this 2-D analysis with and without cross-terms. Equally, spanwise variations in the mean flow are a detail which need not be retained in the 1-D analysis, in any sense, to predict rollers reasonably well for riblets of sizes $\ell _g^+ \lesssim 20$ .

The importance of the mean profile below the mean-flow origin $\ell _U^+$ is then considered. Although capturing the full extent of the shear layer (up to $z^+ = -\ell _U^+$ ) is important when predicting rollers (Sharma & García-Mayoral 2020), the details of the mean flow below $z^+ = -\ell _U^+$ may not be relevant. As shown in figure 8, zeroing the mean velocity below $z^+ = -\ell _U^+$ yields growth-rate predictions (right column) which remain similar to those with unmodified mean flows (left column). However, zeroing below $z^+ =-\ell _U^+$ reduces the wave speeds predicted for larger riblets, indicating that rollers are sensitive to the mean-velocity profile within the riblet grooves. Although not shown here, linearly extrapolating the spanwise-averaged velocity profile below the crests of the riblets, such that $\langle U^+\rangle _e (z^+) = 0$ at $z^+ = -\ell _U^+$ , also led to an underprediction of wave speeds. Thus, although spanwise variations in the mean profile are unimportant to the growth of the spanwise roller instability (for $\ell _g^+ \lesssim 20$ ), the wave speed can be inaccurately predicted when the in-groove mean flow is not well approximated, e.g. by zeroing the mean velocity at or below $z^+ = -\ell _U^+$ .

Based on the similarities in the growth rates shown in figure 8, the superficial averages of the riblet-resolved 2-D mean profiles will be employed in the 1-D linear-stability analysis in § 5. The 1-D analysis then requires a single riblet-resolved 2-D calculation per riblet size, with an $\ell _T^+$ -shifted smooth-wall Cess eddy-viscosity profile, to obtain the superficially averaged mean flow. Note that the growth rates predicted from 1-D mean flows, calculated with a boundary condition of $U^+=0$ at $z^+ = -\ell _U^+$ , and still with $\ell _T^+$ -shifted Cess profiles, were not consistent with predictions from 2-D calculations, as shown in Appendix E, figure 15. However, to demonstrate that the 1-D analysis can still provide useful standalone predictions, i.e. without any assisting 2-D calculations, results from 1-D linear-stability analysis with 1-D mean flows ( $U^+=0$ at $z^+ = -\ell _U^+$ ) are also included.

5. The 1-D linear predictions based on a dynamical model of the in-groove flow

As posed at the start of § 4, the question of whether the riblets can be adequately replaced by an effective boundary condition is now considered. This involves the use of the model boundary condition (4.2), as a means to relate the riblet geometry to the wall admittance, and thereby explain why only some riblets promote spanwise rollers.

The 1-D analysis is performed in MATLAB, solving a primitive variable formulation of the linearised Navier–Stokes equations, based on Luhar, Sharma & McKeon (2014) and Luhar, Sharma & McKeon (2015). The no-slip ( $\hat {u}^+ = \hat {v}^+=0$ ) and wall-admittance (4.2) boundary conditions are placed at $z^+ = -\ell _U^+$ , and implemented with row replacement. Symmetry conditions at the centreline follow from appropriate modifications to the spectral differentiation matrices (Weideman & Reddy 2001), with interest in symmetric streamwise velocity perturbations. The lower half of the domain is then discretised with $N_c = 160$ Chebyshev nodes (for $\mathit{{Re}}_\tau = 395$ ), in line with the $N_c \geqslant 100$ Chebyshev nodes (half-domain) of Luhar et al. (2015), and the $N_s = 200$ splines (full domain) of Jiménez et al. (2001), both at $\mathit{{Re}}_\tau \approx 2003$ . The eigenvalue problem is solved in generalised form using the function eigs with tolerance $10^{-12}$ , and targeting the leading four eigenvalues near $c^+ \approx 6$ . Eigenvalues which prove sensitive to grid resolution with tolerance $10^{-3}$ (comparing $N_c = 160$ to $N_c = 170$ ) are omitted, although spurious modes are typically far from the eigenmode of interest.

5.1. Predictions of the 1-D linear-stability analysis

The results of the 1-D and 2-D analyses are compared in figure 9. Overall, the 1-D analyses (figure 9, middle and right columns) provide predictions which are reasonably consistent with those of both the 2-D linear-stability analysis (figure 9, left column) and the DNS, in terms of which riblet shapes promote rollers. Specifically, both 1-D analyses predict that $30^\circ$ triangular riblets should form rollers at the smallest $\ell _g^+$ values and should promote the most intense rollers, followed then by blade , $60^\circ$ triangular and $30^\circ$ trapezoidal riblets, respectively. This ordering of roller intensities is similar to that observed in both the 2-D linear-stability analysis and DNS, and follows from the scaling of the wall-admittance boundary condition (4.2) with riblet size and shape, and streamwise scale. The ${\lambda _x^+}^{-1}$ variation with streamwise scale penalises the wall admittance less aggressively at larger streamwise wavelengths (than a ${\lambda _x^+}^{-2}$ scale dependence), in line with a balance between unsteady inertia and pressure within the riblet grooves. This allows for larger wall admittances at the streamwise wavelengths $90 \lesssim \lambda _x^+ \lesssim 140$ , figure 9(d), at which rollers are predicted in the 1-D analysis. Equally, the $A_{g,-\ell _U}^+/s^+$ scaling of the wall-admittance boundary condition (4.2) captures enough of a measure of the riblet size and shape, particularly the riblet slenderness (per $\ell _g$ ), i.e. $A_{g,-\ell _U}/(\ell _g s)$ , so as to provide reasonable predictions in the 1-D analysis (further evidence is provided in Appendix F by comparing the leading eigenmodes from the 2-D and 1-D analyses). The riblet slenderness $A_{g,-\ell _U}/(\ell _g s)$ is expressed per $\ell _g$ to enable comparison at roller-onset sizes of matched $\ell _g^+ \approx 10$ (García-Mayoral & Jiménez 2011b ), and so provides a rough guide as to which riblet shapes are likely to have the most intense rollers. These slenderness values $A_{g,-\ell _U}/(\ell _g s)$ are, , $0.77$ ; , 0.49; , 0.42; , 0.38; , 0.28; , 0.26. Thus, the ordering of roller intensity from the 2-D linear-stability analysis and DNS is (approximately) reproduced, just from this $A_{g,-\ell _U}/(\ell _g s)$ measure of riblet slenderness, as informed by the development and modelling of the wall-admittance boundary condition in § 4.1.

6. Conclusions

This paper has highlighted the importance of riblet-resolved calculations for accurately predicting spanwise roller instabilities forming over riblets, and more generally, of the importance of texture-resolved calculations to inform modelling choices for non-smooth surfaces. Riblet-resolved calculations proved important both in establishing when and why effective boundary conditions break down, and in identifying the key features of the mean flow within the riblet grooves. Furthermore, physical insights into the flow dynamics in a linearised setting proved consistent with new insights from DNS analysis, and reasonable predictions for which riblet sizes and shapes promote rollers were obtained with both the a priori and a posteriori 2-D models.

Figure 9. Comparing the 2-D linear-stability analysis which resolves the riblets (left column) to the 1-D analysis with a wall-admittance model capturing the in-groove physics (middle and right columns). The 1-D wall-admittance boundary conditions are $\hat {w}^+/\hat {p}^+ = (A_{g,-\ell _U}^+/s^+)(\kappa _x^+/6)\exp (250\unicode{x03C0} \mathrm{i}/180)$ and $\hat {u}^+=0$ at $z^+ = -\ell _U^+$ . The 1-D analysis is performed both with superficially averaged mean-velocity profiles obtained from riblet-resolved 2-D calculations for each riblet size (middle column) and with mean-velocity profiles obtained over no-slip walls placed at $z^+ = -\ell _U^+$ (right column), so as to provide a set of standalone 1-D predictions. Overall, the 1-D analysis reasonably captures the prevalence or lack of rollers based on the roller growth rates (first row), as well as the roller wavelengths (second row) and wave speeds (third row). However, the 1-D analysis tends to predict growing rollers at smaller $\ell _g^+$ than the 2-D analysis, and tends to underpredict the wave speeds when using 1-D mean flows over a no-slip wall (right column), as the 2-D superficially averaged mean flows retain a slip velocity at $\ell _U^+$ .

Through the use of riblet-resolved 2-D linear-stability analysis, in concert with DNS and 1-D linear-stability analysis, previous theories regarding spanwise rollers (García-Mayoral & Jiménez 2011b ) were reappraised. First, riblet grooves did not provide sufficient wall admittance to emulate a free-shear layer. Thus, maximally growing free-shear-layer instabilities are not promoted by riblets, with the rollers over riblets therefore not strictly Kelvin–Helmholtz rollers. The rollers are sensitive to (i) the finite wall admittance of the riblets, (ii) the mean shear in the mixing layer (which depends critically on the choice of turbulence origin $\ell _T^+$ ) and (iii) the details of the mean-velocity profile within the riblet grooves, the latter two related to the momentum absorption near the riblet tips (Endrikat et al. 2021a ).

The spanwise rollers, being far from the maximally growing limit, were further shown to be near-marginally-stable modes. This finding suggests that the spanwise rollers forming over some other non-smooth surfaces may also be marginal modes, especially whenever rollers form with similar streamwise wavelengths $\lambda _x^+ \approx 150$ to those over riblets, e.g. as observed for some permeable substrates (Gómez-de-Segura & García-Mayoral 2019; Chavarin et al. 2021). The characteristics of the marginal rollers over riblets, as predicted by the 2-D linear-stability analysis herein, also matched DNS observations (García-Mayoral & Jiménez 2011b , 2012; Endrikat et al. 2021a ) and 2-D resolvent predictions (Chavarin & Luhar 2020), having streamwise wavelengths of $\lambda _x^+ \approx 150$ and wave speeds $c^+ \approx 5$ . This resolved previous discrepancies in the predictions of $\lambda _x^+ \approx 60$ for the streamwise wavelength of assumed-maximally growing rollers (García-Mayoral & Jiménez 2011b ; Gómez-de-Segura et al. 2018b ), compared with the wavelength $\lambda _x^+ \approx 150$ of marginal rollers shown herein and in resolvent analyses (Chavarin & Luhar 2020; Chavarin et al. 2021). Furthermore, the 2-D linear-stability analysis correctly identified which riblet shapes promote (marginal) spanwise rollers, and which did not, in agreement with DNS observations. With a model for the turbulence origin $\ell _T^+$ , this formed a predictive tool able to determine which riblets promote rollers. It still remains to determine the drag contribution attributed to the additional Reynolds stresses carried by the rollers, which has yet to be definitively measured (Endrikat et al. 2021a ; Viggiano et al. 2024), and which ultimately requires an improved means of measuring or modelling the turbulence origin $\ell _T^+$ . In addition, it remains to clarify the mechanism(s) responsible for the complete drag-reduction breakdown for all riblet shapes (Modesti et al. 2021; Endrikat et al. 2022; Chan et al. 2023).

With the overall success of the 2-D linear-stability analysis, the viability of accurate 1-D linear predictions was reconsidered. This required a 1-D effective boundary condition to represent the riblets, with the choice made to model the boundary condition with a wall admittance, toward identifying which geometrical features of a given riblet shape promote rollers. As shown herein, the wall-admittance measurements from both DNS and 2-D linear-stability analysis were at odds with the conventional theory of viscous-dominated in-groove physics, which informed previous 1-D modelling (García-Mayoral & Jiménez 2011b ). Instead, for roller-promoting riblets, the overlying pressure, which drives the roller instability, was shown to be balanced by unsteady inertia within the groove. Only within half a viscous unit of the groove floor was viscous-dominated flow physics recovered. This pressure–unsteady-inertia balance revealed the importance of riblet slenderness in controlling the intensity of spanwise rollers, via the effective wall admittance. This pressure–unsteady-inertia balance also explained, for example, why rollers are promoted by slender 30 $^\circ$ triangular riblets and not by less-slender $90^\circ$ triangular riblets.

Once applied in the 1-D analysis, this wall-admittance boundary condition provided reasonable predictions as to which riblet shapes promote rollers, for riblets with grooves sizes $10 \lesssim \ell _g^+ \lesssim 20$ . This 1-D boundary condition appropriately captured both the scaling of the wall admittance with riblet size and streamwise scale, which was further reflected in the 1-D analysis reasonably predicting the streamwise wavelength for maximum growth ( $\lambda _x^+ \approx 100$ ). The 1-D analysis also indicated that intense rollers would form for the most slender riblets, 30 $^\circ$ triangular and blade riblets especially, as in DNS. However, even though this 1-D effective boundary condition captured the in-groove physics, the 1-D analysis was unable to conclusively identify whether certain riblet shapes never promote spanwise rollers at very large riblet sizes $\ell _g^+ \gtrsim 25$ . For these very large riblets, the effective wall-admittance boundary condition broke down, with the accuracy of the mean-flow approximation a secondary (but still relevant) concern.

Beyond this, the model reduction, from DNS to 2-D to 1-D linear-stability analysis, still served to provide a generalisable and predictive explanation for rollers, i.e. a relationship between a purely geometric parameter, the riblet slenderness, $A_{g,-\ell _U}/(\ell _g s)$ , and the dynamics (intensity) of the spanwise rollers. Identifying the correct physical balance within the riblet grooves directly indicated the importance of riblet slenderness, without having to rely on any correlations in DNS. Riblet slenderness being a shape-dependent quantity also explained why only some riblets (at matched size) promote rollers, as (i) groove area is only size-dependent, and (ii) marginal rollers are more sensitive to changes in the effective wall admittance than maximally growing rollers. These findings stand in stark contrast to the previous theory, which suggested that all sufficiently large riblets should form maximally growing rollers (García-Mayoral & Jiménez 2011b ).

Given that a 1-D wall-admittance boundary condition can predict marginal spanwise rollers for riblet sizes of practical relevance, similar analyses may prove insightful for other surfaces, e.g. for high-admittance permeable substrates (Zampogna & Bottaro 2016; Lācis & Bagheri 2017; Habibi Khorasani et al. 2024), which, although drag increasing (Chu et al. 2019; Shahzad, Hickel & Modesti 2023; Hartog et al. 2024), may serve to mitigate noise or enhance heat transfer. This work also cautions results derived solely from modelled boundary conditions without accompanying texture-resolved calculations, e.g. for permeable substrates modelled with a scale-independent wall admittance (Jiménez et al. 2001; Motoki et al. 2022), as although linear-stability analysis may be consistent with DNS using the same modelled boundary conditions, the observed modes may not be realisable from any manufactured surface. Thus, texture-resolved calculations should be leveraged, if a parametric sweep would prove affordable, e.g. to analyse the permeable substrates directly. Otherwise, a smaller number of texture-resolved calculations should be performed and interrogated, so as to develop an effective model boundary condition able to relate the substrate properties to the dynamics of the overlying rollers.