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# Mean flow of turbulent boundary layers over porous substrates

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## Abstract

Mean-flow measurements of turbulent boundary layers over porous walls (permeable and rough) with varying pore size (s), permeability (K) and thickness (h) are presented across a wide range of friction Reynolds numbers (Reτ≈2000–18000) and permeability based Reynolds numbers (ReK≈1.5–60). The mean wall shear stress was determined using a floating element drag balance and the boundary layer profiles were acquired using hot-wire anemometry. Substrate permeability is shown to increase the magnitude of the mean velocity deficit. The use of a modified indicator function, assuming “universal” values for von Karman constant (κ=0.39) supports previous results where a strongly modified logarithmic region was observed. The indicator function was also used to estimate the zero-plane displacement (yd), the roughness function (ΔU+), and equivalent sandgrain roughness (ks). At high Reynolds numbers, the roughness function data collapses on to the Nikuradse's fully rough asymptote. However, at low roughness Reynolds numbers (ks+<100), we observe the flow to be transitionally rough, evolving with Nikuradse-type behavior. The equivalent sandgrain roughness ks for each substrate appears to include roughness and permeability contributions. These two contributions can be separated using data obtained from the same substrates with different thickness. This may allow us to model the porous wall as a combination of rough and permeable wall.

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Experimental researeli on flow over permeable surfaces is reported, in which flow over a permeable, rough surface is compared with flow over non-permeable, rough and smooth surfaces. The experiments were conducted in a wind tunnel using two permeable surfaces (Fig. 4) of relative thicknesses D/de = 2,73 and 9,67.Velocity profiles and their turbulent fluctuations (Fig. 2, Fig. 5 and Fig. 8) were measured with a constant-temperature hotfilm-anemometer. The universal velocity-defect law, eq. 2, was applied to obtain the shear velocity, u*.; and the law of the wall, eq. 1, was used to obtain the reference level (Fig. 3) and the roughness height, y', (Fig. 6); the impulse equation was used to check these results. The local resistance coefficient, e'f, versus the Reynolds number, Re, given in Fig. 7, may be considered as the resulting display of data. It is concluded that the boundary resistance of the tested permeable surface is higher than that of the non-permeable boundaries having identical roughness.
Article
In order to understand the effects of the wall permeability on turbulence near a porous wall, flow field measurements are carried out for turbulent flows in a channel with a porous bottom wall by a two-component particle image velocimetry (PIV) system. The porous media used are three kinds of foamed ceramics which have almost the same porosity (∼0.8) but different permeability. It is confirmed that the flow becomes more turbulent over the porous wall and tends to be turbulent even at the bulk Reynolds number of Reb=1300 in the most permeable wall case tested. Corresponding to laminar to turbulent transition, the magnitude of the slip velocity on the porous wall is found to increase drastically in a narrow range of the Reynolds number. To discuss the effects of the wall roughness and the wall permeability, detailed discussions are made of zero-plane displacement and equivalent wall roughness for porous media. The results clearly indicate that the turbulence is induced by not only the wall roughness but the wall permeability. The measurements have also revealed that as Reb or the wall permeability increases, the wall normal fluctuating velocity near the porous wall is enhanced due to the effects of the wall permeability. This leads to the increase of the turbulent shear stress resulting in higher friction factors of turbulence over porous walls.
Article
The main objectives of this study are to suggest a proper boundary condition at the interface between a permeable block and turbulent channel flow and to investigate the characteristics of turbulent channel flow with permeable walls. The boundary condition suggested is an extended version of that applied to laminar channel flow by Beavers & Joseph (1967) and describes the behaviour of slip velocities in the streamwise and spanwise directions at the interface between the permeable block and turbulent channel flow. With the proposed boundary condition, direct numerical simulations of turbulent channel flow that is bounded by the permeable wall are performed and significant skin-friction reductions at the permeable wall are obtained with modification of overall flow structures. The viscous sublayer thickness is decreased and the near-wall vortical structures are significantly weakened by the permeable wall. The permeable wall also reduces the turbulence intensities, Reynolds shear stress, and pressure and vorticity fluctuations throughout the channel except very near the wall. The increase of some turbulence quantities there is due to the slip-velocity fluctuations at the wall. The boundary condition proposed for the permeable wall is validated by comparing solutions with those obtained from a separate direct numerical simulation using both the Brinkman equation for the interior of a permeable block and the Navier–Stokes equation for the main channel bounded by a permeable block.
Article
Careful reassessment of new and pre-existing data shows that recorded scatter in the hot-wire-measured near-wall peak in viscous-scaled streamwise turbulence intensity is due in large part to the simultaneous competing effects of the Reynolds number and viscous-scaled wire length l+. An empirical expression is given to account for these effects. These competing factors can explain much of the disparity in existing literature, in particular explaining how previous studies have incorrectly concluded that the inner-scaled near-wall peak is independent of the Reynolds number. We also investigate the appearance of the so-called outer peak in the broadband streamwise intensity, found by some researchers to occur within the log region of high-Reynolds-number boundary layers. We show that the ‘outer peak’ is consistent with the attenuation of small scales due to large l+. For turbulent boundary layers, in the absence of spatial resolution problems, there is no outer peak up to the Reynolds numbers investigated here (Reτ = 18830). Beyond these Reynolds numbers – and for internal geometries – the existence of such peaks remains open to debate. Fully mapped energy spectra, obtained with a range of l+, are used to demonstrate this phenomenon. We also establish the basis for a ‘maximum flow frequency’, a minimum time scale that the full experimental system must be capable of resolving, in order to ensure that the energetic scales are not attenuated. It is shown that where this criterion is not met (in this instance due to insufficient anemometer/probe response), an outer peak can be reproduced in the streamwise intensity even in the absence of spatial resolution problems. It is also shown that attenuation due to wire length can erode the region of the streamwise energy spectra in which we would normally expect to see kx−1 scaling. In doing so, we are able to rationalize much of the disparity in pre-existing literature over the kx−1 region of self-similarity. Not surprisingly, the attenuated spectra also indicate that Kolmogorov-scaled spectra are subject to substantial errors due to wire spatial resolution issues. These errors persist to wavelengths far beyond those which we might otherwise assume from simple isotropic assumptions of small-scale motions. The effects of hot-wire length-to-diameter ratio (l/d) are also briefly investigated. For the moderate wire Reynolds numbers investigated here, reducing l/d from 200 to 100 has a detrimental effect on measured turbulent fluctuations at a wide range of energetic scales, affecting both the broadband intensity and the energy spectra.
Article
We review the experimental evidence on turbulent flows over rough walls. Two parameters are important: the roughness Reynolds number k + s , which mea-sures the effect of the roughness on the buffer layer, and the ratio of the boundary layer thickness to the roughness height, which determines whether a logarithmic layer survives. The behavior of transitionally rough surfaces with low k + s depends a lot on their geometry. Riblets and other drag-reducing cases belong to this regime. In flows with δ/k 50, the effect of the roughness extends across the boundary layer, and is also variable. There is little left of the original wall-flow dynamics in these flows, which can perhaps be better described as flows over obstacles. We also review the evidence for the phenomenon of d-roughness. The theoretical arguments are sound, but the experimental evidence is inconclusive. Finally, we discuss some ideas on how rough walls can be modeled without the detailed computation of the flow around the roughness elements themselves.
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