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 Reference García-Mayoral and Jiménez2011b ; Endrikat et al. Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a ), 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 Reference Gómez-de-Segura2019), 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 Reference Reynolds and Tiederman1967), 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. (Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a ) at $\mathit{{Re}}_\tau = 395$ , following Moarref & Jovanović (Reference Moarref and Jovanović2012), 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 Reference Hecht2012), having defined the riblet shape and grid sizes in Gmsh (Geuzaine & Remacle Reference Geuzaine and Remacle2009). 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 Reference García-Mayoral and Jiménez2011b ; Endrikat et al. Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a , Reference Endrikat, Modesti, MacDonald, García-Mayoral, Hutchins and Chungb ). 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 Reference García-Mayoral and Jiménez2011b ; Endrikat et al. Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a ). (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. (Reference Wong, Camobreco, García-Mayoral, Hutchins and Chung2024) 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 (Reference Chu and Karniadakis1993, 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 (Reference García-Mayoral and Jiménez2011b ). 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. Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a , Reference Endrikat, Modesti, MacDonald, García-Mayoral, Hutchins and Chungb ). 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 Reference García-Mayoral and Jiménez2011b ). 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 (Reference Chavarin and Luhar2020); additional comparisons between linear-stability and resolvent analysis are provided in Appendix D. Equally, that rollers are marginal explains why García-Mayoral & Jiménez (Reference García-Mayoral and Jiménez2011b ), 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. Reference Jiménez, Uhlmann, Pinelli and Kawahara2001; Drazin & Reid Reference Drazin and Reid2004; García-Mayoral & Jiménez Reference García-Mayoral and Jiménez2011b ), 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. Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a , 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. Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a ) at the same $\mathit{{Re}}_\tau$ and recent experiments (Abu Rowin et al. Reference Abu Rowin, Deshpande, Wang, Kozul, Chung, Sandberg and Hutchins2025) 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 (Reference García-Mayoral and Jiménez2012).

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. (Reference Mele, Tognaccini, Catalano and de Rosa2020) and Mele, Saetta & Tognaccini (Reference Mele, Saetta and Tognaccini2023), 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. Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a ).

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. (Reference Endrikat, Modesti, MacDonald, García-Mayoral, Hutchins and Chung2021b ). 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 Reference Sharma and García-Mayoral2020).

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 Reference García-Mayoral and Jiménez2011b ; Endrikat et al. Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a ).

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. (Reference Endrikat, Modesti, García-Mayoral, Hutchins and Chung2021a , 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 Reference García-Mayoral and Jiménez2011b ). 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 (Reference García-Mayoral and Jiménez2011b ) 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.

4.1. 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 Reference García-Mayoral and Jiménez2011b ). 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 (Reference García-Mayoral and Jiménez2011b ), 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 (Reference García-Mayoral and Jiménez2011b ), 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 Reference García-Mayoral and Jiménez2011b ), 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 (Reference García-Mayoral and Jiménez2011b ) 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 Reference Sharma and García-Mayoral2020), 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.

5.1. 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 Reference García-Mayoral and Jiménez2011b ), 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.