Turbulent Wall Flows Research Articles

The modulation of particles on coherent structure of turbulent boundary layer in dilute liquid-solid two-phase flow with PIV

Liquid-solidtwo-phase wall turbulent flow is an important flow pattern in engineering field. Studying the modulation of particles on fluid plays a significant role in revealing the mechanism of momentum, heat and mass transfer, improving production efficiency and decreasing energy consumption. The statistical characteristics and coherent structure in the flat plate turbulent boundary layer in clean water condition and near-neutrally liquid-solid two-phase flow with 200 um polystyrene particles (dp+=2.29,dp+=duτv) measured by means of PIV technique were compared and analyzed. The variation rule of turbulent statistic characteristics such as mean velocity profile, turbulence intensity and Reynolds shear stress of fluid with polystyrene particles and clean water under the same free flow velocity was investigated. Based on the locally-average velocity structure function and condition sampling technique, the coherent structure was detected, and the two-dimensional spatial topologies such as the fluctuation velocity, Reynolds shear stress and spanwise vorticity in ejection and sweep events at different wall heights under two working conditions were extracted with phase-average method. It was found that in the buffer layer the flow velocity with particles was greater than that of clean water, while there was little difference between them in the logarithmic law region. Compared with the clean water, the streamwise turbulent intensity was decreased in particle-laden flow, whereas the normal turbulent intensity, the comprehensive turbulence intensity and the Reynolds shear stress were all increased, with the expense of a slightly increase of the friction drag in the fluid with particles by 3.57%. In the bursting of coherent structure, the addition of polystyrene particles suppressed the fluid streamwise fluctuation in a turbulent boundary layer, gave a rise to the normal fluctuation and Reynolds shear stress, while inhibited the streamwise development of the spanwise vortex and promoted its normal development. The addition of particles facilitated the fluctuation of the fluid microcluster, and as well as the transverse momentum transfer in the log law region of turbulent boundary layer. The particles could increase the probability of the coherent structure ejection events and had less influence on the sweep events. The number and intensity of the ejection event was the key to improve the flow normal turbulence intensity and even to improve the energy exchange near the wall. The presence of appropriate scale particles in wall turbulence could play an important role in strengthening the transfer of momentum, heat and mass, and promoting mixing and reaction in the engineering, only by slightly increasing the fraction resistance.

On mean flow universality of turbulent wall flows. I. High Reynolds number flow analysis

ABSTRACTThe universality and mathematical physical structure of wall-bounded turbulent flows is a topic of discussions over many decades. There is no agreement about questions like what is the physical mean flow structure, how universal is it, and how universal are theoretical concepts for local and global flow variations. These questions are addressed by using latest direct numerical simulation (DNS) data at moderate Reynolds numbers Re and experimental data up to extreme Re. The mean flow structure is explained by analytical models for three canonical wall-bounded turbulent flows (channel flow, pipe flow, and the zero-pressure gradient turbulent boundary layer). Thorough comparisons with DNS and experimental data provide support for the validity of models. Criteria for veritable physics derived from observations are suggested. It is shown that the models presented satisfy these criteria. A probabilistic interpretation of the mean flow structure shows that the physical constraints of equal entropies and equally likely mean velocity values in a region unaffected by boundary effects impose a universal log-law structure. The structure of wall-bounded turbulent flows is much more universal than previously expected. There is no discrepancy between local logarithmic velocity variations and global friction law and bulk velocity variations. Flow effects are limited to the minimum: the difference of having a bounded or unbounded domain, and the variation range of mean velocity values allowed by the geometry.

Numerical investigation of velocity ratio effect in combined wall and offset jet flows

In the present work, we propose to numerically study a combined turbulent wall and offset jet flow (this combination will be denoted WOJ). Several turbulence models were tested in this study: the standard k - ω, SST k - ω, standard k - e, RNG, and realizable k - e model. A parametric study was performed to determine the effect of offset ratio H and the velocity ratio r on the longitudinal and transverse positions of the merge point (MP), the combined point (CP), the upper vortex center (UVC) and the lower vortex center (LVC). Correlations that predict the position of these characteristic points of the WOJ flow as a function of H and r have been provided. Results show that any increase in the velocity ratio causes a displacement of the MP, CP, UVC and LVC further upstream along the longitudinal direction. Concerning the transverse positions, the increase of velocity ratio results in a deviation of the merge point (MP) and the lower vortex center (LVC) toward the strong jet (LWJ) whereas the transverse position of combined point (CP) and the upper vortex center (UVC) is almost independent of the velocity ratio.

An alternative floating element design for skin-friction measurement of turbulent wall flows

Indirect methods to estimate surface shear stress are commonly used to characterise rough-wall boundary-layer flows. The uncertainty is typically large and often insufficient to carry out quantitative analysis, especially for surface roughness where established scaling and similarity laws may not hold. It is, thus, preferable to rely instead on independent measurement techniques to accurately measure skin friction. The floating element was one of the first to be introduced, and still is the most popular for its features. Although its fundamental principle has remained unchanged, different arrangements have been suggested to overcome its inherent limitations. In this paper, we review some of these designs and further present an alternative that is able to correct for extraneous loads into the drag measurement. Its architecture is based on the parallel-shift linkage, and it features custom-built force transducers and a data acquisition system designed to achieve high signal-to-noise ratios. The smooth-wall boundary-layer flow is used as a benchmark to assess the accuracy of this balance. Values of skin-friction coefficient show an agreement with hot-wire anemometry to within 2% for Re_{theta } = 4times 10^3 up to 10^4. A rough surface of staggered distributed cubes with large relative height, delta /hsimeq 10, is also investigated. Results indicate the flow reaches the fully rough regime, at the measurement location, for the entire range of Reynolds number. Furthermore, the values of skin friction agree with existing estimations from alternative methods.Graphical abstractDrawings of the floating-element (FE) balance and skin-friction measurements for a smooth-wall boundary layer. On top: slice along the X-Y plane and top view (left) next to their corresponding pictures (right). Colors highlight distinctsubsystems, namely, the floating frame for drag measurement (yellow), the pitching moment mechanism (red) and the acquisitionsystem (blue). Bottom left: smooth-wall setup and calibration system. The FE is flush mounted with the wind tunnel floor and thepulley is attached to a linear traverse which allows setting its position at different wall-normal locations. During calibration, a lid isremoved to make way for the pulley to move into the test section. A wire is then strung over to suspend the weights. Bottom right: Skin friction over a smooth wall. The inset indicates the relative discrepancy between the FE values (blue) and those inferred fromhot-wire anemometry of the boundary-layer profile (red).

Turbulence investigation in the roughness sub-layer of a near wall flow

The turbulence behaviour along a wall roughened by pyramidal elements was analysed in the region extending from the apex of the roughness elements up to the external limit of the roughness sub-layer. The data used for the analysis were obtained by particle image velocimetry technique. The rough wall turbulent boundary layer flow is characterized by a relatively low Reynolds number. All the results on the rough wall were compared with data referring to the canonical flow on a smooth wall turbulent boundary layer. Mean values and turbulence quantities for the two flows collapse when approaching the external limit of the roughness sublayer. The quadrant analysis of the Reynolds shear stress, in the region near the surface, shows that the contribution of the sweep motions is about equivalent for the two flows (except for wall distances lower than 40 viscous units). The contribution of the ejection motions appears to be more important over the smooth wall than over the rough wall with increasing differences approaching the wall. The probability density functions of the streamwise fluctuating velocity field for the rough wall case appear to be positively skewed in the zone very close to the pyramid apex, in contrast with the behavior observed for the smooth wall case at corresponding distances from the wall. The integral and Taylor scales for the rough wall case appear to be strongly reduced by the presence of the roughness, while the Kolmogorov microscale shows higher values.

Експериментальна структура пристінної турбулентної течії при дії теплового випромінювання СО2-лазера

Експериментальна структура пристінної турбулентної течії при дії теплового випромінювання СО2-лазера

Local wall shear stress measurements with a thin plate submerged in the sublayer in wall turbulent flows

A local wall shear stress measurement technique has been developed using a thin plate, referred to as a sublayer plate which is attached to the wall in the sublayer of a near-wall turbulent flow. The pressure difference between the leading and trailing edges of the plate is correlated to the known wall shear stress obtained in the fully developed turbulent channel flow. The universal calibration curve can be well represented in dimensionless form, and the sensitivity of the proposed method is as high as that of the sublayer fence, even if the sublayer fence is enveloped by the linear sublayer. The results of additional experiments prove that the sublayer plate has fairly good angular resolution in detecting the direction of the local wall shear stress vector.

Prospectus: towards the development of high-fidelity models of wall turbulence at large Reynolds number.

Recent and on-going advances in mathematical methods and analysis techniques, coupled with the experimental and computational capacity to capture detailed flow structure at increasingly large Reynolds numbers, afford an unprecedented opportunity to develop realistic models of high Reynolds number turbulent wall-flow dynamics. A distinctive attribute of this new generation of models is their grounding in the Navier-Stokes equations. By adhering to this challenging constraint, high-fidelity models ultimately can be developed that not only predict flow properties at high Reynolds numbers, but that possess a mathematical structure that faithfully captures the underlying flow physics. These first-principles models are needed, for example, to reliably manipulate flow behaviours at extreme Reynolds numbers. This theme issue of Philosophical Transactions of the Royal Society A provides a selection of contributions from the community of researchers who are working towards the development of such models. Broadly speaking, the research topics represented herein report on dynamical structure, mechanisms and transport; scale interactions and self-similarity; model reductions that restrict nonlinear interactions; and modern asymptotic theories. In this prospectus, the challenges associated with modelling turbulent wall-flows at large Reynolds numbers are briefly outlined, and the connections between the contributing papers are highlighted.This article is part of the themed issue 'Toward the development of high-fidelity models of wall turbulence at large Reynolds number'.

A self-sustaining process model of inertial layer dynamics in high Reynolds number turbulent wall flows.

Field observations and laboratory experiments suggest that at high Reynolds numbers Re the outer region of turbulent boundary layers self-organizes into quasi-uniform momentum zones (UMZs) separated by internal shear layers termed 'vortical fissures' (VFs). Motivated by this emergent structure, a conceptual model is proposed with dynamical components that collectively have the potential to generate a self-sustaining interaction between a single VF and adjacent UMZs. A large-Re asymptotic analysis of the governing incompressible Navier-Stokes equation is performed to derive reduced equation sets for the streamwise-averaged and streamwise-fluctuating flow within the VF and UMZs. The simplified equations reveal the dominant physics within-and isolate possible coupling mechanisms among-these different regions of the flow.This article is part of the themed issue 'Toward the development of high-fidelity models of wall turbulence at large Reynolds number'.

Analysis of the coherent and turbulent stresses of a numerically simulated rough wall pipe

A turbulent rough wall flow in a pipe is simulated using direct numerical simulation (DNS) where the roughness elements consist of explicitly gridded three-dimensional sinusoids. Two groups of simulations were conducted where the roughness semi-amplitude h+ and the roughness wavelength λ+ are systematically varied. The triple decomposition is applied to the velocity to separate the coherent and turbulent components. The coherent or dispersive component arises due to the roughness and depends on the topological features of the surface. The turbulent stress on the other hand, scales with the friction Reynolds number. For the case with the largest roughness wavelength, large secondary flows are observed which are similar to that of duct flows. The occurrence of these large secondary flows is due to the spanwise heterogeneity of the roughness which has a spacing approximately equal to the boundary layer thickness δ.

Wall inclination effect in heat transfer characteristics of a combined wall and offset jet flow

In the present work, a numerical study of the heat transfer in an oblique combined turbulent wall and offset jet flow was performed. Different turbulence models were tested: the standard k−ω, SST k−ω, standard k−ɛ, RNG k−ɛ and realizable k−ɛ. A parametric study was also carried out to investigate the wall inclination β and the Reynolds number Re effect on the local Nusselt number Nu, local wall heat flux q, average Nusselt number Nu¯ and average wall heat flux q¯. Correlations witch link the local and the average Nusselt number and wall heat flux to the wall inclination β and the Reynolds number Re were also provided in this paper. Constant wall heat flux and constant wall temperature boundary conditions were considered in all computation. Analyzing the numerical results show that the heat transfer exchanged between the oblique wall and the whole flow is more intense for low values of wall inclinations β, for high values of Reynolds numbers Re and under constant wall heat flux boundary condition.

Manning’s formula and Strickler’s scaling explained by a co-spectral budget model

Manning’s empirical formula in conjunction with Strickler’s scaling is widely used to predict the bulk velocity$V$from the hydraulic radius$R_{h}$, the roughness size$r$and the slope of the energy grade line$S$in uniform channel and pipe flows at high bulk Reynolds numbers. Despite their importance in science and engineering, both Manning’s and Strickler’s formulations have waited for decades before finding a theoretical explanation. This was provided, for the first time, by Gioia & Bombardelli (Phys. Rev. Lett., vol. 88, 2002, 014501), labelled as GB02, using phenomenological arguments. Perhaps their most remarkable finding was the link between the Strickler and the Kolmogorov scaling exponents, the latter pertaining to velocity fluctuations in the inertial subrange of the turbulence spectrum and presumed to be universal. In this work, the GB02 analysis is first revisited, showing that GB02 employed severalad hocscaling assumptions for the turbulent kinetic energy dissipation rate and, although implicitly, for the mean velocity gradient adjacent to the roughness elements. The similarity constants arising from the GB02 scaling assumptions were presumed to be independent of$r/R_{h}$, which is inconsistent with well-known flow properties in the near-wall region of turbulent wall flows. Because of the dependence of these similarity constants on$r/R_{h}$, this existing theory requires the validity of the Strickler scaling to cancel the dependence of these constants on$r/R_{h}$so as to arrive at the Strickler scaling and Manning’s formula. Here, the GB02 approach is corroborated using a co-spectral budget (CSB) model for the wall shear stress formulated at the cross-over between the roughness sublayer and the log region. Assuming a simplified shape for the spectrum of the vertical velocity$w$, the proposed CSB model (subject to another simplifying assumption that production is balanced by pressure–velocity interaction) allows Manning’s formula to be derived. To substantiate this approach, numerical solutions to the CSB over the entire flow depth using different spectral shapes for$w$are carried out for a wide range of$r/R_{h}$. The results from this analysis support the simplifying hypotheses used to derive Manning’s equation. The derived equation provides a formulation for$n$that agrees with reported values in the literature over seven decades of$r$variations. While none of the investigated spectral shapes allows the recovery of the Strickler scaling, the numerical solutions of the CSB reproduce the Nikuradse data in the fully rough regime, thereby confirming that the Strickler scaling represents only an approximate fit for the friction factor for granular roughness.

Influence of Inertial Particles on Turbulence Characteristics in Outer and Near Wall Flow as Revealed With High Resolution Particle Image Velocimetry

A fully developed turbulent particle-gas flow in a rectangular horizontal channel 100 × 10 × 4000 mm3 is disclosed with high spatial resolution two-dimensional (2D) particle image velocimetry (PIV). The objective is to increase the knowledge of the mechanisms behind alterations in turbulent characteristics when adding two sets of relatively large solid spherical particles with mean diameters of 525 and 755 μm and particle size distributions of 450–600 and 710–800 μm, respectively. Reynolds numbers are 4000 and 5600 and relatively high volume fraction of 5.4 × 10−4 and 8.0 × 10−4 are tested. Both the near wall turbulent boundary layer flow and outer core flow are considered. Results show that the carrier phase turbulent intensities increase with the volume fraction of the inertial particles. The overall mean flow velocity is affected when adding the particles but only to a minor extent. Near the wall, averaged velocity decreases while fluctuating velocity components increase when particles are added to the flow. Quadrant analysis shows the importance of sweep near the wall and ejection events in the region defined by y+ > 20. In conclusion, high inertia particles can enhance turbulence even at relatively low particle Reynolds number <90. In the near bottom wall region, particles tend to be a source of instability reflected as enhancement in rms values of the normal velocity component.

Exact coherent structures at extreme Reynolds number

Exact coherent structures (ECS), unstable three-dimensional solutions of the Navier–Stokes equations, play a fundamental role in transitional and turbulent wall flows. Dempsey et al. (J. Fluid Mech., vol. 791, 2016, pp. 97–121) demonstrate that at large Reynolds number reduced equations can be derived that simplify the computation and facilitate mechanistic understanding of these solutions. Their analysis shows that ECS in plane Poiseuille flow can be sustained by a novel inner–outer interaction between oblique near-wall Tollmien–Schlichting waves and interior streamwise vortices.

Numerical study on evolution of subharmonic varicose low-speed streaks in turbulent channel flow

The evolution of two spanwise-aligned low-speed streaks in a wall turbulent flow, triggered by the instability of the subharmonic varicose (SV) mode, is studied by a direct numerical simulation (DNS) method in a small spatial-periodic channel. The results show that the SV low-speed streaks are self-sustained at the early stage, and then transform into subharmonic sinuous (SS) low-speed streaks. Initially, the streamwise vortex sheets are formed by shearing, and then evolve into zigzag vortex sheets due to the mutual induction. As the intensification of the SV low-speed streaks becomes prominent, the tilted streamwise vortex tubes and the V-like streamwise vortex tubes can be formed simultaneously by increasing \( + \frac{{\partial u}} {{\partial x}}\). When the SV low-speed streaks break down, new zigzag streamwise vortices will be generated, thus giving birth to the next sustaining cycle of the SV low-speed streaks. When the second breakdown happens, new secondary V-like streamwise vortices instead of zigzag streamwise vortices will be generated. Because of the sweep motion of the fluid induced by the secondary V-like streamwise vortices, each decayed low-speed streak can be divided into two parts, and each part combines with the part of another streak, finally leading to the formation of SS low-speed streaks.

Frequency–wavenumber mapping in turbulent shear flows

Spatial turbulence spectra for high-Reynolds-number shear flows are usually obtained by mapping experimental frequency spectra into wavenumber space using Taylor’s hypothesis, but this is known to be less than ideal. In this paper, we propose a cross-spectral approach that allows us to determine the entire wavenumber–frequency spectrum using two-point measurements. The method uses cross-spectral phase differences between two points – equivalent to wave velocities – to reconstruct the wavenumber–frequency plane, which can then be integrated to obtain the spatial spectrum. We verify the technique on a particle image velocimetry (PIV) data set of a turbulent boundary layer. To show the potential influence of the different mappings, the transfer functions that we obtained from our PIV data are applied to hot-wire data at approximately the same Reynolds number. Comparison of the newly proposed technique with the classic approach based on Taylor’s hypothesis shows that – as expected – Taylor’s hypothesis holds for larger wavenumbers (small spatial scales), but there are significant differences for smaller wavenumbers (large spatial scales). In the range of Reynolds number examined in this study, double-peaked spectra in the outer region of a turbulent wall flow are thought to be the result of using Taylor’s hypothesis. This is consistent with previous studies that focused on examining the limitations of Taylor’s hypothesis (del Álamo & Jiménez, J. Fluid Mech., vol. 640, 2009, pp. 5–26). The newly proposed mapping method provides a data-driven approach to map frequency spectra into wavenumber spectra from two-point measurements and will allow the experimental exploration of spatial spectra in high-Reynolds-number turbulent shear flows.

Spectral scaling in boundary layers and pipes at very high Reynolds numbers

One-dimensional energy spectra in flat plate zero pressure gradient boundary layers and pipe flows are examined over a wide range of Reynolds numbers ($2600\leqslant \mathit{Re}_{{\it\tau}}\leqslant 72\,500$). The spectra show excellent collapse with Kolmogorov scaling at high wavenumbers for both flows at all Reynolds numbers. The peaks associated with the large-scale motions (LSMs) and superstructures (SS) in boundary layers behave as they do in pipe flows, with some minor differences. The location of the outer spectral peak, associated with SS or very large-scale motions (VLSMs) in the turbulent wall region, displays only a weak dependence on Reynolds number, and it occurs at the same wall-normal distance where the variances establish a logarithmic behaviour and where the amplitude modulation coefficient has a zero value. The results suggest that with increasing Reynolds number the energy is largely confined to a thin wall layer that continues to diminish in physical extent. The outer-scaled wavelength of the outer spectral peak appears to decrease with increasing Reynolds number. However, there is still significant energy content in wavelengths associated with the SS and VLSMs. The location of the outer spectral peak appears to mark the start of a plateau that is consistent with a $k_{x}^{-1}$ slope in the spectrum and the logarithmic variation in the variances. This $k_{x}^{-1}$ region seems to occur when there is sufficient scale separation between the locations of the outer spectral peak and the outer edge of the log region. It does not require full similarity between outer and wall-normal scaling on the wavenumber. The extent of $k_{x}^{-1}$ region depends on the wavelength of the outer spectral peak (${\it\lambda}_{OSP}$), which appears to emerge as a new length scale for the log region. Finally, based on the observations from the spectra together with the statistics presented in Vallikivi et al. (J. Fluid Mech., 2015 (submitted)), five distinct wall-normal layers are identified in turbulent wall flows.

Reynolds-number-dependent turbulent inertia and onset of log region in pipe flows

AbstractA detailed analysis of the ‘turbulent inertia’ (TI) term (the wall-normal gradient of the Reynolds shear stress,$\mathrm{d} \langle -uv\rangle /\mathrm{d} y $), in the axial mean momentum equation is presented for turbulent pipe flows at friction Reynolds numbers$\delta ^{+} \approx 500$, 1000 and 2000 using direct numerical simulation. Two different decompositions for TI are employed to further understand the mean structure of wall turbulence. In the first, the TI term is decomposed into the sum of two velocity–vorticity correlations ($\langle v \omega _z \rangle + \langle - w \omega _y \rangle $) and their co-spectra, which we interpret as an advective transport (vorticity dispersion) contribution and a change-of-scale effect (associated with the mechanism of vorticity stretching and reorientation). In the second decomposition, TI is equivalently represented as the wall-normal gradient of the Reynolds shear stress co-spectra, which serves to clarify the accelerative or decelerative effects associated with turbulent motions at different scales. The results show that the inner-normalised position,$y_m^{+}$, where the TI profile crosses zero, as well as the beginning of the logarithmic region of the wall turbulent flows (where the viscous force is leading order) move outwards in unison with increasing Reynolds number as$y_m^{+} \sim \sqrt{\delta ^{+}}$because the eddies located close to$y_m^{+}$are influenced by large-scale accelerating motions of the type$\langle - w \omega _y \rangle $related to the change-of-scale effect (due to vorticity stretching). These large-scale motions of$O(\delta ^{+})$gain a spectrum of larger length scales with increasing$\delta ^{+}$and are related to the emergence of a secondary peak in the$-uv$co-spectra. With increasing Reynolds number, the influence of the$O(\delta ^{+})$motions promotes viscosity to act over increasingly longer times, thereby increasing the$y^{+}$extent over which the mean viscous force retains leading order. Furthermore, the TI decompositions show that the$\langle v \omega _z \rangle $motions (advective transport and/or dispersion of vorticity) are the dominant mechanism in and above the log region, whereas$\langle - w \omega _y \rangle $motions (vorticity stretching and/or reorientation) are most significant below the log region. The motions associated with$\langle - w \omega _y \rangle $predominantly underlie accelerations, whereas$\langle v \omega _z \rangle $primarily contribute to decelerations. Finally, a description of the structure of wall turbulence deduced from the present analysis and our physical interpretation is presented, and is shown to be consistent with previous flow visualisation studies.

The physics of energy transfer toward improved subgrid-scale models

Starting from physical insight on the energy transfer phenomena in wall turbulent flows, it is shown how modeling of subgrid stresses in large-eddy simulations can be improved. Each model should aim at reproducing the double feature of energy sink and source of the small scales of wall flows which become relevant when large filter lengths are considered. Here we propose one possible choice where the main ingredient is the coupling of the classical linear formulation of eddy viscosity with the nonlinear anisotropic features of the velocity increments tensor. This approach, which actually presents most of the features of the mixed models, captures the near-wall dynamics for very large filter lengths reproducing the small scales source physics responsible for backward energy transfer. A posteriori tests show excellent agreement with direct numerical simulation of turbulent channel flows even when very coarse grids are considered. The capability of the balance of the filtered second order structure function as a post-processing tool to evaluate the physics of any model is also shown.

The attached reverse and detached forward cascades in wall-turbulent flows

The present work describes the multidimensional behaviour of wall-bounded turbulence in the space of cross-scales (spanwise and wall-normal) and distances from the wall. This approach allows us to understand the cascade mechanisms by which scale-energy is transmitted scale-by-scale away from the wall, through the overlap layer, and into the bulk flow. Two distinct cascades are identified involving the attached and detached scales of motion, respectively. From the near-wall region, scale-energy is transferred towards the bulk, flowing through the attached scales of motion, while among the detached scales it converges towards small scales, ascending again to the channel centre. It is then argued that the attached scales of wall-bounded turbulence are involved in a reverse cascade process that starts from the wall and ends in the bulk flow. On the other hand, the detached scales belong to a direct forward cascade process towards dissipation. Hence, at a given distance from the wall the attached motion is fed by smaller attached scales located closer to the wall. In turn this attached motion is responsible for creating the scale-energy that sustains larger attached scales farther from the wall and smaller detached scales that are responsible for connecting the scale-energy at large scales to the dissipation at small scales through a forward cascade.