The transport equations for velocity variances are investigated using data from DNS of incompressible channel flows at $Re_\tau$ up to 5200. Each term in the transport equation has been spectrally decomposed to expose the contribution of turbulence at different length scales to the processes governing the flow of energy in the wall-normal direction, in scale and among components. 
The outer-layer turbulence is dominated by very large-scale streamwise elongated modes. Away from the wall, production occurs primarily in these large-scale streamwise-elongated modes in the streamwise velocity, but dissipation occurs nearly isotropically in both velocity components and scale. For this to happen, the energy is transferred from the streamwise elongated modes to modes with a range of orientations through non-linear interactions, and then transferred to other velocity components. This allows energy to be transferred more-or-less isotropically from these large scales to the small scales at which dissipation occurs. The VLSMs also transfer energy to the wall-region resulting in a modulation of the autonomous near-wall dynamics. 
The near-wall energy flows are consistent with the well-known autonomous near-wall dynamics. Through the overlap region between outer and inner layer turbulence, there is a self-similar structure to the energy flows. The VLSM production occurs at spanwise scales that grow with $y$. There is transport of energy away from the wall over a range of scales that grows with $y$. And, there is transfer of energy to small dissipative scales which grow like $y^{1/4}$. Finally, the small-scale near-wall processes characterised by wavelengths less than 1000 wall units are largely Reynolds number independent, while the larger-scale outer layer process are strongly Reynolds number dependent. The interaction between them appears to be relatively simple.

Spectral analysis of the budget equation in turbulent channel flows at high Re

The transport equations for the variances of the velocity components are investigated using data from direct numerical simulations of incompressible channel flows at friction Reynolds number ( ) up to . Each term in the transport equation has been spectrally decomposed to expose the contribution of turbulence at different length scales to the processes governing the flow of energy in the wall-normal direction, in scale and among components. The outer-layer turbulence is dominated by very large-scale streamwise elongated modes, which are consistent with the very large-scale motions (VLSM) that have been observed by many others. The presence of these VLSMs drives many of the characteristics of the turbulent energy flows. Away from the wall, production occurs primarily in these large-scale streamwise-elongated modes in the streamwise velocity, but dissipation occurs nearly isotropically in both velocity components and scale. For this to happen, the energy is transferred from the streamwise-elongated modes to modes with a range of orientations through nonlinear interactions, and then transferred to other velocity components. This allows energy to be transferred more-or-less isotropically from these large scales to the small scales at which dissipation occurs. The VLSMs also transfer energy to the wall region, resulting in a modulation of the autonomous near-wall dynamics and the observed Reynolds number dependence of the near-wall velocity variances. The near-wall energy flows are more complex, but are consistent with the well-known autonomous near-wall dynamics that gives rise to streaks and streamwise vortices. Through the overlap region between outer- and inner-layer turbulence, there is a self-similar structure to the energy flows. The VLSM production occurs at spanwise scales that grow with . There is transport of energy away from the wall over a range of scales that grows with . Moreover, there is transfer of energy to small dissipative scales which grows like , as expected from Kolmogorov scaling. Finally, the small-scale near-wall processes characterised by wavelengths less than 1000 wall units are largely Reynolds number independent, while the larger-scale outer-layer processes are strongly Reynolds number dependent. The interaction between them appears to be relatively simple.

Spectral analysis of the budget equation in turbulent channel flows at high Reynolds number

Wall turbulence is a ubiquitous phenomenon in nature and engineering applications, yet predicting such turbulence is difficult due to its complexity. High-Reynolds-number turbulence arises in most practical flows, and is particularly complicated because of its wide range of scales. Although the attached-eddy hypothesis postulated by Townsend can be used to predict turbulence intensities and serves as a unified theory for the asymptotic behaviours of turbulence, the presence of coherent structures that contribute to the logarithmic behaviours has not been observed in instantaneous flow fields. Here, we demonstrate the logarithmic region of the turbulence intensity by identifying wall-attached structures of the velocity fluctuations ( structures. These findings suggest that the identified structures are prime candidates for Townsend’s attached-eddy hypothesis and that they can serve as cornerstones for understanding the multiscale phenomena of high-Reynolds-number boundary layers.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2243925528E22AE6C4125288B89953DE/S0022112018007279a.pdf/div-class-title-wall-attached-structures-of-velocity-fluctuations-in-a-turbulent-boundary-layer-div.pdf

Wall-attached structures of velocity fluctuations in a turbulent boundary layer

The spectra and correlations of the velocity fluctuations in turbulent channels, especially above the buffer layer, are analysed using new direct numerical simulations with friction Reynolds numbers up to Re at very large ones.

Scaling of the energy spectra of turbulent channels

Trends in turbomachinery turbulence treatments

Here, we report the measurements of two-dimensional (2-D) spectra of the streamwise velocity (u) in a high-Reynolds-number turbulent boundary layer. A novel experiment employing multiple hot-wire probes was carried out at friction Reynolds numbers ranging from 2400 to 26 000. Taylor's frozen turbulence hypothesis is used to convert temporal-spanwise information into a 2-D spatial spectrum which shows the contribution of streamwise (λx) and spanwise (λy) length scales to the streamwise variance at a given wall height (z). At low Reynolds numbers, the shape of the 2-D spectra at a constant energy level shows λy/z ∼ (λx/z)1/2 behaviour at larger scales, which is in agreement with the existing literature at a matched Reynolds number obtained from direct numerical simulations. However, at high Reynolds numbers, it is observed that the square-root relationship tends towards a linear relationship (λy ∼ λx), as required for self-similarity and predicted by the attached eddy hypothesis.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/548E2EFD8F94B4ED6EBE200A3B5E5C18/S0022112017003597a.pdf/div-class-title-two-dimensional-energy-spectra-in-high-reynolds-number-turbulent-boundary-layers-div.pdf

Two-dimensional energy spectra in high-Reynolds-number turbulent boundary layers

Simulations and experiments at low Reynolds numbers have suggested that skin-friction drag generated by turbulent fluid flow over a surface can be decreased by oscillatory motion in the surface, with the amount of drag reduction predicted to decline with increasing Reynolds number. Here, we report direct measurements of substantial drag reduction achieved by using spanwise surface oscillations at high friction Reynolds numbers ([Formula: see text]) up to 12,800. The drag reduction occurs via two distinct physical pathways. The first pathway, as studied previously, involves actuating the surface at frequencies comparable to those of the small-scale eddies that dominate turbulence near the surface. We show that this strategy leads to drag reduction levels up to 25% at [Formula: see text] = 6,000, but with a power cost that exceeds any drag-reduction savings. The second pathway is new, and it involves actuation at frequencies comparable to those of the large-scale eddies farther from the surface. This alternate pathway produces drag reduction of 13% at [Formula: see text] = 12,800. It requires significantly less power and the drag reduction grows with Reynolds number, thereby opening up potential new avenues for reducing fuel consumption by transport vehicles and increasing power generation by wind turbines.

/pdf/an-energy-efficient-pathway-to-turbulent-drag-reduction-54dbtdpvvp.pdf

An energy-efficient pathway to turbulent drag reduction.

In this paper, we present the two-dimensional (2-D) energy cross-spectrum of the streamwise velocity (, where this self-similar trend is obscured by coexisting scales. The present analysis reaffirms that a self-similar structure, conforming to Townsend’s attached eddy hypothesis, is ingrained in the flow.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/C1E07AB4D7C7FE57B89A32C1D302325E/S0022112020001391a.pdf/div-class-title-two-dimensional-cross-spectrum-of-the-streamwise-velocity-in-turbulent-boundary-layers-div.pdf

Two-dimensional cross-spectrum of the streamwise velocity in turbulent boundary layers

Two-dimensional (2D) spectra of the streamwise velocity component, measured at friction Reynolds numbers ranging from 2400 to 26000, are used to refine a model for the logarithmic region of turbulent boundary layers. Here we focus on the attached eddy model (AEM). The conventional AEM assumes the boundary layer to be populated with hierarchies of self-similar wall-attached (Type A) eddies alone. While Type A eddies represent the dominant energetic large-scale motions at high Reynolds numbers, the scales that are not represented by such eddies are observed to carry a significant proportion of the total kinetic energy. Therefore, in the present study, we propose an extended AEM that incorporates two additional representative eddies. These eddies, named Type CA and Type SS, represent the self-similar but wall-incoherent low-Reynolds-number features and the non-self-similar wall-coherent superstructures, respectively. The extended AEM is shown to better predict a greater range of energetic length scales and capture the low- A nd high-Reynolds-number scaling trends in the 2D spectra of all three velocity components. A discussion on spectral self-similarity and the associated k-1 scaling law is also presented.

Dileep Chandran

Papers

Two-dimensional energy spectra in high-Reynolds-number turbulent boundary layers

An energy-efficient pathway to turbulent drag reduction.

Two-dimensional cross-spectrum of the streamwise velocity in turbulent boundary layers

Spectral-scaling-based extension to the attached eddy model of wall turbulence

Periodicity of large-scale coherence in turbulent boundary layers