Scaling and dynamics of turbulence over sparseA canopies

Turbulent flows within and over sparse canopies are investigated using direct numerical simulations. We focus on the effect of the canopy on the background turbulence, the part of the flow that remains once the element-induced flow is filtered out. In channel flows, the distribution of the total stress is linear with height. Over smooth walls, the total stress is only the `fluid stress' $\tau_f$, the sum of the viscous and the Reynolds shear stresses. In canopies, in turn, there is an additional contribution from the canopy drag, which can dominate within. We find that, for sparse canopies, the ratio of the viscous and the Reynolds shear stresses in $\tau_f$ at each height is similar to that over smooth-walls, even within the canopy. From this, a height-dependent scaling based on $\tau_f$ is proposed. Using this scaling, the background turbulence within the canopy shows similarities with turbulence over smooth walls. This suggests that the background turbulence scales with $\tau_f$, rather than with the conventional scaling based on the total stress. This effect is essentially captured when the canopy is substituted by a drag force that acts on the mean velocity profile alone, aiming to produce the correct $\tau_f$, without the discrete presence of the canopy elements acting directly on the fluctuations. The proposed mean-only forcing is shown to produce better estimates for the turbulent fluctuations compared to a conventional, homogeneous-drag model. The present results thus suggest that a sparse canopy acts on the background turbulence primarily through the change it induces on the mean velocity profile, which in turn sets the scale for turbulence, rather than through a direct interaction of the canopy elements with the fluctuations. The effect of the element-induced flow, however, requires the representation of the individual canopy elements.Cambridge Commonwealth, European and International Trust PRACE DECI-15 European Research Counci

Spectral Analysis of the Slip-Length Model for Turbulence over Textured Superhydrophobic Surfaces.

We assess the applicability of slip-length models to represent textured superhydrophobic surfaces. From the results of direct numerical simulations, and by considering the slip length from a spectral perspective, we discriminate between the apparent boundary conditions experienced by different lengthscales in the overlying turbulent flow. In particular, we focus on the slip lengths experienced by lengthscales relevant to the near wall turbulent dynamics. Our results indicate that the apparent failure of homogeneous slip-length models is not the direct effect of the texture size becoming comparable to the size of eddies in the flow. The texture-induced signal scatters to the entire wavenumber space, affecting the perceived slip length across all lengthscales, even those much larger than the texture. We propose that the failure is caused by the intensity of the texture-induced flow, rather than its wavelength, becoming comparable to the background turbulence

Turbulent flows over dense filament canopies

Turbulent flows over dense canopies of rigid filaments of small size are investigated for different element heights and spacings using DNS. The flow can be decomposed into the element-coherent, dispersive flow, the Kelvin--Helmholtz-like rollers typically reported over dense canopies, and the background, incoherent turbulence. The canopies studied have spacings $s^+ = 3$--$50$, which essentially preclude the background turbulence from penetrating within. The dispersive velocity fluctuations are also mainly determined by the spacing, and are small deep within the canopy, where the footprint of the Kelvin--Helmholtz-like rollers dominates. Their typical streamwise wavelength is determined by the mixing length, which is essentially the sum of its height above and below the canopy tips. For the present dense canopies, the former remains roughly the same in wall-units, and the latter, which scales with the drag length, depends linearly on the spacing. This is the result of the drag being essentially viscous and governed by the planar layout of the canopy. In shallow canopies, the proximity of the canopy floor inhibits the formation of Kelvin--Helmholtz-like rollers, with essentially no signature for height-to-spacing ratios $h/s \approx 1$, and no further inhibition beyond $h/s \approx 6$. Very small spacings also inhibit the rollers, due to their obstruction by the canopy elements. The obstruction decreases with increasing spacing and the signature of the instability intensifies, even if for canopies sparser than those studied here the instability eventually breaks down. Simple models based on linear stability can capture some of the above effects.Cambridge Commonwealth, European and International Trust EPSRC Tier-2 grant EP/P020259/

Analysis of anisotropically permeable surfaces for turbulent drag reduction

The present work proposes the use of anisotropically permeable substrates as a means to reduce turbulent skin friction. We conduct an a priori analysis to assess the potential of these surfaces, based on the effect of small-scale surface manipulations on near-wall turbulence. The analysis, valid for small permeability, predicts a monotonic decrease in friction as the streamwise permeability increases. Empirical results suggest that the drag-reducing mechanism is however bound to fail beyond a certain permeability. We investigate the development of Kelvin-Helmholtz-like rollers at the surface as a potential mechanism for this failure. These rollers, which are a typical feature of turbulent flows over permeable walls, are known to increase drag, and their appearance to limit the drag-reducing effect. We propose a model, based on linear stability analysis, which predicts the onset of these rollers for sufficiently large permeability, and allows us to bound the maximum drag reduction that these surfaces can achieve

Imposing virtual origins on the velocity components in direct numerical simulations

The relative wall-normal displacement of the origin perceived by different components of near-wall turbulence is known to produce a change in drag. This effect is produced for instance by drag-reducing surfaces of small texture size like riblets and superhydrophobic surfaces. To facilitate the research on how these displacements alter near-wall turbulence, this paper studies different strategies to model such displacement effect through manipulated boundary conditions. Previous research has considered the effect of offsetting the virtual origins perceived by the tangential components of the velocity from the reference, boundary plane, where the wall-normal velocity was set to zero. These virtual origins are typically characterised by slip-length coefficients in Robin, slip-like boundary conditions. In this paper, we extend this idea and explore several techniques to define and implement virtual origins for all three velocity components on direct numerical simulations (DNSs) of channel flows, with special emphasis on the wall-normal velocity. The aim of this work is to provide a suitable foundation to extend the existing understanding on how these virtual origins affect the near-wall turbulence, and ultimately aid in the formulation of simplified models that capture the effect of complex surfaces on the overlying flow and on drag, without the need to resolve fully the turbulence and the surface texture. From the techniques tested, Robin boundary conditions for all three velocities are found to be the most satisfactory method to impose virtual origins, relating the velocity components to their respective wall-normal gradients linearly. Our results suggest that the effect of virtual origins on the flow, and hence the change in drag that they produce, can be reduced to an offset between the virtual origin perceived by the mean flow and that perceived by the overlying turbulence, and that turbulence remains otherwise smooth-wall-like, as proposed by Luchini (1996). The origin for turbulence, however, would not be set by the spanwise virtual origin alone, but by a combination of the spanwise and wall-normal origins. These observations suggest the need for an extension of Luchini’s virtual-origin theory to predict the change in drag, accounting for the wall-normal transpiration when its effect is not negligible

Turbulent drag reduction by anisotropic permeable substrates-analysis and direct numerical simulations

We explore the ability of anisotropic permeable substrates to reduce turbulent skin-friction, studying the influence that these substrates have on the overlying turbulence. For this, we perform DNSs of channel flows bounded by permeable substrates. The results confirm theoretical predictions, and the resulting drag curves are similar to those of riblets. For small permeabilities, the drag reduction is proportional to the difference between the streamwise and spanwise permeabilities. This linear regime breaks down for a critical value of the wall-normal permeability, beyond which the performance begins to degrade. We observe that the degradation is associated with the appearance of spanwise-coherent structures, attributed to a Kelvin-Helmholtz-like instability of the mean flow. This feature is common to a variety of obstructed flows, and linear stability analysis can be used to predict it. For large permeabilities, these structures become prevalent in the flow, outweighing the drag-reducing effect of slip and eventually leading to an increase of drag. For the substrate configurations considered, the largest drag reduction observed is $\approx 20-25\%$ at a friction Reynolds number $\delta^+ = 180$

Modulation of near-wall turbulence in the transitionally rough regime

Direct numerical simulations of turbulent channels with rough walls are conducted in the transitionally rough regime. The effect that roughness produces on the overlying turbulence is studied using a modified triple decomposition of the flow. This decomposition separates the roughness-induced contribution from the background turbulence, with the latter essentially free of any texture footprint. For small roughness, the background turbulence is not significantly altered, but merely displaced closer to the roughness crests, with the change in drag being proportional to this displacement. As the roughness size increases, the background turbulence begins to be modified, notably by the increase of energy for short, wide wavelengths, which is consistent with the appearance of a shear-flow instability of the mean flow. A laminar model is presented to estimate the roughness-coherent contribution, as well as the displacement height and the velocity at the roughness crests. Based on the effects observed in the background turbulence, the roughness function is decomposed into different terms to analyse different contributions to the change in drag, laying the foundations for a predictive model

Turbulent Drag Reduction Using Anisotropic Permeable Substrates.

The behaviour of turbulent flow over anisotropic permeable substrates is studied using linear stability analysis and direct numerical simulations (DNS). The flow within the permeable substrate is modelled using the Brinkman equation, which is solved analytically to obtain the boundary conditions at the substrate-channel interface for both the DNS and the stability analysis. The DNS results show that the drag-reducing effect of the permeable substrate, caused by preferential streamwise slip, can be offset by the wall-normal permeability of the substrate. The latter is associated with the presence of large spanwise structures, typically associated to a Kelvin-Helmholtz-like instability. Linear stability analysis is used as a predictive tool to capture the onset of these drag-increasing Kelvin-Helmholtz rollers. It is shown that the appearance of these rollers is essentially driven by the wall-normal permeability Ky+ . When realistic permeable substrates are considered, the transpiration at the substrate-channel interface is wavelength-dependent. For substrates with low Ky+ , the wavelength-dependent transpiration inhibits the formation of large spanwise structures at the characteristic scales of the Kelvin-Helmholtz-like instability, thereby reducing the negative impact of wall-normal permeability

Geometry-induced fluctuations in the transitionally rough regime

Direct numerical simulations of turbulent flows over rough surfaces are conducted to investigate the physics of the transitionally rough regime. Different roughness sizes are analysed within the transitional regime, while keeping the shape of the surface geometry constant. To study the effect of roughness on the flow field, a novel decomposition is used to divide the velocity into two components: a turbulent, geometry-independent contribution, and a geometry-induced contribution, whose intensity is modulated by the overlying turbulence. In the onset of the transitionally rough regime, the turbulent component remains essentially unmodified, and it is anticipated that all the roughness effects can be attributed entirely to the geometry-induced fluctuations. As the roughness size increases further, the turbulent component is also modified, and the fluid-surface interaction becomes more complex.This work was supported by the European Research Council through the II Multiflow Summer Workshop, and by the British Engineering and Physical Sciences Research Council through grant number EP/M506485/1