## Abstract

The interaction of the overlying turbulent flow with riblets, and its impact on their drag reduction properties are analysed. In the so-called viscous regime of vanishing riblet spacing, the drag reduction is proportional to the riblet size, but for larger riblets the proportionality breaks down, and the drag reduction eventually becomes an increase. It is found that the groove cross section *A*^{+}_{g} is a better characterization of this breakdown than the riblet spacing, with an optimum . It is also found that the breakdown is not associated with the lodging of quasi-streamwise vortices inside the riblet grooves, or with the inapplicability of the Stokes hypothesis to the flow along the grooves, but with the appearance of quasi-two-dimensional spanwise vortices below *y*^{+}≈30, with typical streamwise wavelengths . They are connected with a Kelvin–Helmholtz-like instability of the mean velocity profile, also found in flows over plant canopies and other surfaces with transpiration. A simplified stability model for the ribbed surface approximately accounts for the scaling of the viscous breakdown with *A*^{+}_{g}.

## 1. Introduction

Riblets are small surface protrusions aligned with the direction of flow, which confer an anisotropic roughness to a surface. They are one of the few techniques that have been successfully applied to the reduction of the skin friction in turbulent boundary layers, both in the laboratory and in full aerodynamic configurations.

Riblets of very different geometries have been tested in wind tunnels, demonstrating drag reductions of the order of 10 per cent over flat plates. Walsh & Lindemann [1] tested several shapes, including triangular, notched-peak, sinusoidal and U-shaped riblets, obtaining maximum drag reductions of 7–8% for riblet spacings of approximately 15 wall units. A fairly broad early review was that of Walsh [2], and more recent ones are those of Choi [3], who emphasizes the work of the ERCOFTAC drag reduction group, and Bushnell [4], which is oriented towards drag-reduction techniques for aircraft. In his review on turbulent flows over rough walls, Jiménez [5] viewed drag reduction by riblets as a transitional roughness effect.

Riblet experiments have also been conducted in oil channels, which allow larger riblet dimensions and better control of the geometry, although typically at lower Reynolds numbers and shorter development lengths than wind tunnels. For example, Bechert *et al*. [6] conducted extensive tests on blade-shaped and trapezoidal-groove riblets, and proposed the latter as a compromise between optimum performance and practical fabrication and maintenance.

A recurrent theme of riblet research has been the motivation to study biological surfaces, which are often geometrically complex. Bruse *et al*. [7] conducted oil channel tests of shark-skin replicas, hairy surfaces and riblets with adjustable geometry, and Bechert *et al*. [8] reviewed the drag-reduction properties of biological surfaces and their replicas. More recently, Itoh *et al*. [9] tested the flow over seal fur, obtaining drag reductions of 12 per cent.

Off-design conditions have also been considered. We will discuss later the consequences of yaw and tip erosion, but we should mention here the effect of adverse pressure gradients, first reviewed by Walsh [2]. Although that condition is probably not the deciding factor in many practical applications, because the skin friction of adverse pressure gradient boundary layers tends to be low, pressure gradients of either sign are present over large areas of most practical configurations, and their effect on riblets remains uncertain. Walsh [2] found that riblet performance improved under adverse pressure gradients, but mentioned contradictory results by other authors, which he justified by the difficulty of using drag balances under those conditions. Coustols & Savill [10] summarized the results of several previous investigations, and concluded that the pressure gradients typically found over aircraft wings, whether adverse or favourable, had relatively little effect on the performance of riblets, while Debisschop & Nieuwstadt [11] tested riblets in a wind tunnel with dimensionless adverse pressure gradients an order of magnitude larger than those discussed in Walsh [2], and found that the maximum drag reduction of triangular riblets could increase roughly from 7 to 13 per cent.

Riblets have been used successfully to reduce the overall drag of aerofoils [12] and aircraft [13] with optimum riblet spacings of the order of 30–70 μm. Szodruch [14] reports on the flight tests of a commercial aeroplane (Airbus 320) with riblets over 70 per cent of its surface, and estimates an overall 2 per cent drag reduction, based on the fuel savings obtained. A summary of those tests, including maintenance and durability issues, can be found in Robert [15]. The discrepancy between the optimum laboratory performance and full configurations is probably to be expected from any method based on the reduction of skin friction. Not all the drag of an aircraft is friction [16], and much of the latter is distributed over three-dimensional or geometrically complex areas where drag control is difficult to optimize. Note that those limitations might not apply to configurations that are very different from commercial aircraft, such as gliders or other high-performance vehicles.

There is anecdotal evidence of the successful use of riblets in applications other than aircraft, particularly in sporting events in which cost and maintenance considerations are less important than in commercial aviation. The hulls of the USA challengers in the America’s Cup 1987 and 2010 sailing competitions were fitted with riblets, which had been banned by the regulations in intervening years. Both challenges succeeded, although it is impossible to determine whether riblets had any real role. Riblets were also used in the 1984 Olympic rowing events, but they were subsequently forbidden in official racing, together with all other devices that ‘modify the properties of the boundary layer’.

The physical mechanism of the drag reduction by riblets has been investigated in detail, although some aspects remain controversial. In particular, mean and local velocity profiles and turbulent statistics within and above the riblet grooves have been reported for experiments in wind tunnels [17]–[19], water channels [20] and numerical experiments [21]–[26].

Walsh & Lindemann [1] showed that the Reynolds number dependence of the effect of riblets on the skin friction could be expressed in large part in terms of the riblet dimensions expressed in wall units, *L*^{+}=*Lu*_{τ}/*ν*, where *ν* is the kinematic viscosity and *u*_{τ}=*τ*^{1/2} is the friction velocity defined in terms of the skin friction *τ*, where we have assumed unit fluid density for simplicity. The same will be done throughout the paper. A popular measure of the riblet size *L* is the groove spacing, *s*, but other dimensions, such as the depth *h*, have also been used. We will propose below an alternative that collapses the experiments better, and that has some theoretical support. Figure 1 shows a typical curve of drag reduction as a function of the spacing, in which different drag regimes can be defined according to how the drag depends on *s*^{+}. In the viscous regime, formally *s*^{+}≪1 but in practice –15, the contribution of the nonlinear terms to the flow within and in the immediate neighbourhood of the riblet grooves is negligible and, if *τ*_{0} is the skin friction for the smooth wall, the drag reduction *DR*=−Δ*τ*/*τ*_{0} is proportional to *s*^{+}. The viscous regime breaks down near *s*^{+}=*s*^{+}_{opt}, the optimum spacing for which drag reduction is maximum, and, eventually, the reduction becomes a drag increase, adopting a typical *k*-roughness behaviour [5]. The parameters that determine the optimum performance of a given riblet are its optimum size and the slope of the drag curve in the viscous regime. Both depend on the geometry, but the qualitative drag curve is always as just described.

This paper is organized as follows. Section 2 considers the suitability of the parameters traditionally used to scale the drag-reduction curves, and proposes an alternative. Section 3 reviews the viscous regime of drag reduction, including the effects of yaw and tip rounding, and §4 centres on the range of optimum drag reduction, with particular emphasis on the reasons for the breakdown of the viscous regime, and on the linear stability of the flow. The conclusions are then summarized.

## 2. Scaling of the drag-reduction curves

We saw above that the drag reduction by riblets is a function of the riblet size expressed in wall units, and that it has become common to characterize that size by the spacing *s*^{+}. We now enquire whether some other choice of riblet dimensions describes experimental evidence better. Considering a generic length *L*, the drag behaviour in the viscous regime is characterized by the slope
2.1so that *DR*≈*m*_{L} *L*^{+}. We will see in §3 that this slope can usually be computed theoretically, but a successful parameter should also predict the location *L*^{+}_{opt} of the breakdown of the linear behaviour, and collapse as much as possible the experimental drag curves for . An approximation that has often been used is that *s*^{+}_{opt}≈15, but this quantity is a function of the riblet geometry. As an example, figure 2 shows the optimum spacing against the ratio of the groove cross section to the square of the spacing, *A*_{g}/*s*^{2}, for several riblet shapes. For a given spacing, higher values of *A*_{g}/*s*^{2} imply deeper grooves, and, although *s*^{+}_{opt} is always in the range 10–20, it is clear from the figure that deeper grooves break down earlier, and that their maximum drag reduction is achieved for narrower riblets.

Using experimental results from Bechert *et al*. [6] for triangular, trapezoidal, blade and scalloped riblets, García-Mayoral & Jiménez [26] tested alternative scalings to find whether it was possible to express drag reduction in terms of a geometric parameter that captured both the influence of the riblet spacing and shape. The best results were achieved for the square root of the groove cross section, . The optimum values of *s*^{+} or *h*^{+} have scatters of the order of 40 per cent, while the optimum ℓ^{+}_{g} only varies by approximately 10 per cent around ℓ^{+}_{g,opt}≃10.7±1.0, for all the geometries reviewed.

Figure 3 compares experimental drag curves for a wide variety of riblet geometries, reduced with the appropriate viscous slope to compensate for the differences in viscous performance. The figure shows that there is good collapse of the data with ℓ^{+}_{g}, at least for . For the different geometries portrayed, is roughly 83 per cent of the value that would result from extrapolating the linear viscous regime up to ℓ^{+}_{g, opt}. The approximation
2.2using a fixed ℓ^{+}_{g, opt}=10.7, is quite accurate for conventional riblets. That is tested in figure 4a, which compares experimental values of *m*_{ℓ} and . Even for riblets with depth-to-width ratios as low as 0.2, the error of the approximation (2.2) is below 20 per cent.

However, it should be stressed that, although ℓ^{+}_{g} collapses the drag curves better than *s*^{+} or *h*^{+} for conventional geometries, there is no reason why it should do the same for unconventional configurations, such as the fibre riblets in Bruse *et al*. [7], the seal fur surface proposed by Itoh *et al*. [9] or the T-shaped riblets mentioned in Walsh [2]. Taking the latter as examples, the grooves become increasingly isolated from the overlying flow as the wall-parallel segments of the T-fences close into each other, but the groove area (or the groove spacing) changes little. For the limit of fully sealed grooves, the geometry would behave as a flat surface, and modifying ℓ^{+}_{g} should have no performance impact. The scaling with ℓ^{+}_{g} can only be considered as an empirical curve fit that works better than others for the experiments on existing geometries, but which should not be used uncritically for configurations that are very different from conventional ones. On the other hand, some theoretical justification for its use in conventional riblets is provided in §4.

## 3. The viscous regime

We now discuss the concept of protrusion height, which is generally considered to describe well the behaviour of riblets in the viscous regime. From here on, we will denote the streamwise, wall-normal and spanwise coordinates by *x*, *y* and *z*, respectively, and the corresponding velocity components by *u*, *v* and *w*.

There is a thin near-wall region in turbulent flows over smooth walls where viscous effects are dominant, nonlinear inertial effects can be neglected, and the mean velocity profile is linear. Its thickness is 5–10 wall units [27]. From the point of view of a small protrusion in this layer, the outer flow can be represented as a time-dependent, but otherwise uniform shear. Riblets destroy this uniformity near the wall but, if *s*^{+} is small enough, the flow still behaves as a uniform shear for *y*≫*h*. Because the equations of motion are locally linear, the riblets are uniform in the streamwise direction, and the shear varies only slowly with *x* when compared with riblet dimensions, the problem reduces to two uncoupled two-dimensional sub-problems in the *z*–*y* cross plane. The first one is the longitudinal flow of *u*, driven at *y*^{+}≫1 by a streamwise shear
3.1and the second one is the transverse flow of *v* and *w*, driven by
3.2Far from the wall, the effect of the riblets reduces to the virtual origins Δ_{u} and Δ_{w} [28]. Bechert & Bartenwerfer [29] had suggested that Δ_{u} was the explanation of the drag reduction, essentially because it moved turbulence away from the wall, but Luchini *et al*. [30] noted that the important quantity was the offset Δ*h*=Δ_{w}−Δ_{u}, the ‘protrusion height’, which is independent of any arbitrary reference wall position. Intuitively, if the cross-flow has a higher virtual origin than the longitudinal one (Δ*h*>0), the spanwise flow induced by the overlying streamwise vortices is impeded more severely than over a smooth wall. The streamwise vortices are displaced away from the wall, and the turbulent mixing of streamwise momentum is reduced. Since this mixing is responsible for the high local shear near the wall [31], its reduction results in a lower skin friction. The numerical calculation of Δ*h* only involves the two stationary two-dimensional Stokes problems for Δ_{u} and Δ_{w}, which are computationally much less intensive than the three-dimensional, time-dependent, turbulent flow over a ribbed wall.

The relation between protrusion height and drag is modified by the effect of the rest of the boundary layer velocity profile. The classical theory of wall turbulence is that surface manipulations only modify the intercept of the logarithmic velocity profile, while both the Kármán constant, *κ*≈0.4, and the wake function are unaffected [32]. The free-stream velocity, *U*_{δ}, can then be expressed as
3.3where *δ* is a suitable flow thickness, *δ*^{+} is the friction Reynolds number and *B* includes both the near-wall intercept and the contribution from the ‘wake’ component. The effect of a given ribbed surface would be to change *B*, which is equivalent to the ‘roughness function’ used to characterize rough surfaces [5]. For constant *U*_{δ} and small relative variations of the friction, it follows from equation (3.3) that
3.4where the first term in the denominator is due to the change of *u*_{τ} in *U*^{+}_{δ}, and the second one comes from the corresponding change in *δ*^{+}. Comparison between riblets at different Reynolds numbers should be done in terms of Δ*B*, not of Δ*τ*/*τ*_{0}, and the same is true when reducing experimental data to practical applications. The classical theory of wall turbulence suggests that, if the riblet size is much smaller than *δ*, which holds easily in the drag-reducing range, the effect of riblets should be confined to the near-wall layer, and that any change in *B* should only depend on the geometry of the riblets scaled in wall units. Note that equation (3.4) predicts that riblets lose effectiveness at high *δ*^{+}, such as in practical aerodynamic configurations, but only through the logarithmically slow decrease in *c*_{f0}.

Because of the linearity of the viscous regime, the change Δ*B* should be proportional to the protrusion height Δ*h*^{+},
3.5with a universal coefficient *μ*_{0}. That was tested by Jiménez [33], who performed direct simulations of channels at very low *δ*^{+}, in which the offset of the boundary conditions was modelled independently of the presence of actual riblets. He obtained *μ*_{0}≈0.66. A different argument by Bechert *et al*. [6], based on the uniform translation of the velocity profile, suggests *μ*_{0}≈0.785. Equation (3.5) is tested against experiments in figure 4b, but the experimental scatter is too large to distinguish between the two coefficients, or even to decide on the applicability of equation (3.5). At least some of the scatter is due to experimental artefacts. Wind tunnel and oil channel experiments are not strictly comparable, mostly because of the very different development lengths of the two set-ups, but also because of the different levels of geometric control, not always in the expected direction. For example, the open squares in figure 4 are blade riblets from Bechert *et al*. [6]. Those with crosses were mounted on a different base than those without them, and they agree better with the theory. Bechert noted the discrepancy, and repeated a few experiments after sealing the riblet base, increasing the drag reduction by about one-fifth. Similar caveats apply to the other riblets in the figure, including those that appear to agree with the theory.

However, if we believe the theoretical predictions in spite of the experimental ambiguities, equations (3.5) and (3.4) can be combined in a formula for the viscous drag-reduction slope *m*_{ℓ} that only depends on Stokes calculations,
3.6The results could then be used in the equivalent of figures 4a or 3b to predict the riblet performance, at least up to the limit of optimal drag reduction.

### (a) The effect of yaw

The tools discussed above can be used to make useful predictions of the behaviour of riblets in off-design situations. A simple case is the effect of a misalignment angle *θ* between the riblets and the flow, which was first measured experimentally by Walsh [2]. He found that its effect on drag reduction was negligible up to *θ*=15^{°}, which has been confirmed several times since then. The experiments also find that the drag reduction vanishes at *θ*=25–35^{°}, and that the maximum drag increase occurs for *θ*=90^{°}. A list of relevant references can be found in the paper by Koeltzsch *et al*. [34].

Consider now the viscous regime. Since the problem is linear, the longitudinal and transverse velocity with respect to the free stream can be projected into the frame of reference of the riblets, with the result that the offsets with respect to the flow are linear combinations of the offsets of perfectly aligned riblets, and that the protrusion height decays with the yaw as [35]. That agrees qualitatively with the experiments mentioned above.

On the other hand, that simple dependence does not extend away from the viscous regime. Hage *et al*. [35] reported on the effect of yaw on riblets near the optimum spacing, and found a typically larger geometry-dependent degradation than in the viscous case, increasing strongly as *s*^{+} exceeds the optimum value. Since the viscous breakdown is an indication of the effect of nonlinearity on the riblets, it is not surprising that the linear predictions do not apply in that limit.

### (b) The impact of tip rounding

Another effect that can be analysed using the results of the previous two sections is riblet erosion, which is a major concern for industrial applications. Walsh [37] measured drag reductions for triangular riblets with rounded peaks, finding a performance loss of up to 40 per cent for a tip radius *R*≈0.08*s*. He found no significant performance degradation from the rounding of the groove bottoms.

We can infer from the discussion in §2 that the details of the tip geometry should not affect significantly the viscous breakdown, because they barely modify the groove cross section. That is supported by the experiments of Bechert *et al*. [6], who tested blade and scalloped geometries in which only the tip thickness changed. The maximum drag reductions changed with the tip thickness, but not the optimum *s*^{+}, suggesting that the differences in performance were due to the changes in the slope of the drag-reduction curve in the viscous regime.

Viscous results for different geometries with tip rounding are presented in figure 5. The triangular riblets show a dramatic performance decrease, in agreement with Walsh [37], but the protrusion heights for flat-top blades, which perform worse for *R*=0 than the sharp triangles, change little with tip rounding, and even increase slightly with growing radii. It is common knowledge [6,30] that sharper riblets have higher protrusion heights. Tip rounding reduces the sharpness of those configurations, and degrades their performance, but the rounding of the tips of initially blunt blades in a sense sharpens them, thus improving their performance. For practical applications in which erosion is an issue, it is probably preferable to use riblets that do not depend initially too much on the sharpness of their tips.

## 4. The breakdown of the viscous regime

In this section, we discuss the physical mechanism leading to the breakdown of the viscous regime. This is a key issue in riblet design, because we have seen that the optimum drag reduction is proportional to the riblet size at breakdown, and using larger riblets would lead to better optimum performance.

The theories proposed in the literature fall into two broad groups. The first one is that the effect of the riblets on the cross-flow loses effectiveness once they move beyond the Stokes regime. For example, Goldstein & Tuan [24] suggested that the deterioration is due to the generation of secondary streamwise vorticity over the riblets, because the unsteady cross-flow separates and sheds small-scale vortices that create extra dissipation. However, it is known that spanwise wall oscillations, which also presumably introduce unsteady streamwise vorticity, can decrease drag [38], and that spanwise boundary conditions that inhibit the creation of secondary wall vorticity can increase it [39], suggesting that the presence of extra vorticity near the wall need not be detrimental for drag reduction.

The second group of theories assumes that the observed optimum wavelength, *s*^{+}≈10–20, is related to the scale of the turbulent structures in the wall region, such as in the observations by Choi *et al*. [21], Suzuki & Kasagi [20] and Lee & Lee [19], that the increase in drag coincides with the lodging of the quasi-streamwise vortices within the riblet grooves. However, all those observations are for *s*^{+}=30–40, well past the optimum size.

García-Mayoral & Jiménez [26] recently documented a different scenario in a series of direct numerical simulation experiments with riblets spanning the full range of sizes from drag reduction to drag increase. They observed, for riblet sizes around the optimum, the formation of near-wall spanwise vortex rollers whose intensity grows rapidly with the riblet size. Those structures can be seen in the two-dimensional spectra of the flow variables, shown in figure 6a for the Reynolds shear stress *τ*=−〈*uv*〉, and their dimensions for different riblet sizes remain essentially constant in wall units. They have streamwise wavelengths , and only exist below *y*^{+}≈30 when *y*^{+} is referred to the plane of the riblet tips.

They are long in the spanwise direction, extending from to the full channel width. An example can be seen in figure 6b, which shows instantaneous spanwise-averaged streamlines in the *x*–*y* plane. The rollers are centred at *y*^{+}≈10–15, and penetrate slightly into the riblet grooves. The streamwise separation between rollers of the same sign is , in agreement with the spectra. Note that, since this figure is a spanwise average over a box with , the aspect ratio of the surviving rollers is at least 10 with respect to *x* and 30 with respect to *y*, implying a quasi-two-dimensional phenomenon in an *x*–*y* plane.

Although, to our knowledge, those structures have not been reported before, they can be seen, in retrospect, in some of the visualizations of Goldstein *et al*. [23] and Chu & Karniadakis [22], and it is interesting that, even in conditions very different from ours, their streamwise wavelengths are also in the range –200.

The new structures account for most of the degradation of riblet performance with size. Consider two channels with identical half-width, *δ*, and centreline velocity *U*_{δ}, one of them with riblets (R), and the other without (S). Define an approximate wall friction, *u*^{2}_{*}=−(∂_{x}*P*)*δ*, as the extrapolation of the total stress, *τ*(*y*)+*ν*∂_{y}*U*, to *y*=0, which is the plane of the riblet tips. Note that this is not the skin friction that should be used in the true friction coefficient of the ribbed channels, because it neglects the effect of the streamwise pressure gradient over the cross section of the grooves, but it has the same qualitative behaviour as the real one, and can be used for the present argument. That can be seen by comparing the dashed line in figure 7, which is the friction coefficient computed in this way, with the filled circles in figure 3b, which are the true friction coefficients for the same cases. A complete treatment of the following discussion can be found in García-Mayoral & Jiménez [26]. Integrating the mean momentum equation for the two channels and defining the approximate friction coefficient as , it can be shown by integrating the momentum equation between the planes of the riblet tips that the drag change owing to the riblets is
4.1where quantities with asterisks are normalized with their own *u*_{*}, and subscripts refer to the channel type. The two terms on the right-hand side are plotted independently in figure 7. The first one is the slip velocity at the plane of the riblet tips, owing to the presence of the grooves, and always reduces skin friction. That is essentially the mechanism of drag reduction in the viscous regime, and it is interesting that it remains proportional to the riblet size across the figure, even after the drag starts to increase. The slip velocity in figure 7 follows almost exactly the predictions of the longitudinal Stokes problem (3.1), even for the larger riblets, showing that the deterioration of the drag is not due to the breakdown of the viscous hypothesis within the groove. It turns out that the velocities within the groove are small enough that their Reynolds numbers remain small.

The deterioration is due to the extra Reynolds stress in the second term of equation (4.1), whose integrand is small everywhere except near *y*=0, because the two stresses have been scaled to approximately coincide far from the wall. Moreover, the figure includes, in open symbols, the integral of the full Reynolds stress, and, in filled ones, the result of considering only the cospectrum in the spectral region of the new structures, , and . It is clear that the stresses responsible for the drag degradation are those of the new spanwise structures.

The formation of structures perpendicular, rather than parallel, to the riblets may seem surprising, but it is not completely unexpected. Similar spanwise rollers have been reported over vegetable canopies [40,41], and over permeable [42] and porous walls [43]. The length scale of the structures varies depending on the particular problem, but, although few quantitative analyses exist in the literature [42,44], the phenomenon has always been attributed to a Kelvin–Helmholtz-like instability. In essence, the mean profile of a boundary layer almost has an inflection point at the wall, and the reason that it remains inviscidly stable is that the impermeability condition, *v*=0, precludes the antisymmetric unstable eigenfunctions characteristic of Kelvin–Helmholtz. Once any modification of the wall allows local transpiration, the inflection-point instability reappears.

García-Mayoral & Jiménez [26] adapted that general idea to ribbed surfaces. The conceptual model is that the longitudinal Stokes flow along the grooves is driven by the pressure variation of the overlying turbulent flow, and that the resulting longitudinal variations of the velocity within the grooves create a wall-normal transpiration that acts as a boundary condition for an inviscid Rayleigh equation for linearized perturbations around the mean velocity profile in *y*>0. The boundary condition has the form
4.2where the two signs of the last term apply, respectively, to the upper and lower walls and *U*^{′} is the gradient of the mean velocity at the wall. The new parameter *L*_{w} can be interpreted as a length scale for the groove cross section, and is defined as
4.3where *f* is the velocity owing to a normalized pressure gradient along the groove, ∂_{x}*P*=−*ν*, and satisfies ∇^{2}*f*=−1. It has dimensions of length squared and depends only on the groove geometry. The flow is unstable for all *L*_{w}>0, but figure 8a shows that the instability only becomes significant for , essentially independently of the Reynolds number of the flow above the wall.

The values of *L*_{w} for several conventional riblet shapes are compiled in figure 8b, which shows that *L*_{w}≈0.35 ℓ_{g} for groove aspect ratios larger than *A*_{g}/*s*^{2}≈0.4. Together with the stability threshold just mentioned, this result suggests that the flow becomes unstable above ℓ^{+}_{g}≈11, giving some theoretical support to the scaling of the drag curves found empirically in §2.

## 5. Conclusions

We have reviewed the regimes for drag reduction in ribbed surfaces, with particular emphasis on the practical information that can be extracted from the viscous regime, and on the conditions under which that regime breaks down. We have shown that the existing experiments for the location of the breakdown collapse better with a new length scale, , based on the groove area, than with more classical choices such as the riblet spacing or depth. The best estimate for optimum drag reduction is ℓ^{+}_{g}≃10.7, which, together with the drag slope in the viscous limit, can be used to predict riblet performance within 20 per cent. We have shown that this probably predicts the effect of tip erosion, which, somewhat surprisingly, is not always deleterious, but that the effect of yaw for optimized riblets is only qualitatively predicted by the viscous regime.

Using direct simulations of ribbed channels spanning the range from viscous drag reduction to drag increase, we have shown that the degradation for large riblets of the linear regime of drag reduction is not connected with the breakdown of the Stokes behaviour of the longitudinal velocity along the riblet grooves. Even when the drag is already increasing, the slip velocity at the plane of the riblet tips remains proportional to the riblet size. The extra drag comes from a system of spanwise vortices below *y*^{+}≈30, with dimensions that scale in wall units, independently of the riblet size. We have connected those rollers to a Kelvin–Helmholtz-like instability common to other systems with surface transpiration, such as canopies, and permeable and porous surfaces, and we have described a model for ribbed surfaces that provides some theoretical justification for the experimental scaling of the breakdown with the groove area.

## Acknowledgements

This work was supported in part by the CICYT grant TRA2009-11498, and by the sixth framework AVERT programme of the European Commission, AST5-CT-2006-030914. R.G.-M. was supported by an FPI fellowship from the Spanish Ministry of Education and Science.

## Footnotes

One contribution of 15 to a Theme Issue ‘Flow-control approaches to drag reduction in aerodynamics: progress and prospects’.

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