Interscale energy transfer in turbulent channels

TL;DR

This paper addresses how energy transfers between lengthscales occur in wall-bounded turbulence by mapping interscale transfers in a periodic channel through triadic interactions that satisfy $\bm{k}_c + \bm{k}_d = \bm{k}_r$. It introduces a framework that splits the nine velocity-derivative contributions of $\partial_i(u_i u_j)$, computes time-averaged energy-transfer metrics $\mathcal{M}(\hat{u}_j(\bm{k}_r), \hat{n}_{ij}(\bm{k}_c, \bm{k}_d))$, and presents log-wavelength donor/catalyst maps across wall-normal positions for $\mathrm{Re}_\tau \approx 180$. The study uses spectral binning to reduce data size, scales maps across recipient modes and heights, and validates the approach with reduced-order simulations that retain only the identified significant interactions. It finds that turbulence can be sustained when about 30% or more of the nonlinear interactions are kept, while retaining fewer interactions leads to near-wall deviations and potential energy imbalances, highlighting the potential for targeted reduced-order modeling and flow-control strategies. The framework offers a path toward efficient turbulence modeling by pinpointing the most influential interscale exchanges and guiding the design of reduced nonlinear dynamics that preserve essential cascade mechanisms.

Abstract

We investigate the energy cascade in wall-bounded turbulence by analysing the interscale transfer between streamwise and spanwise length scales in periodic channels. This transfer originates from the nonlinear interactions in the advective term of the Navier-Stokes equations, which satisfy the classical triadic compatibility relations. Each triadic interaction is examined individually, and its corresponding nonlinear momentum and energy transfer are mapped to assess its relative importance in sustaining turbulence. Motivated by the anisotropy of the flow, we interpret each contribution $\partial_i(u_i u_j)$ to the advection term as carrying distinct physical information, and therefore analyse them separately. Time-averaged maps of the energy transfer across all length scales and wall-normal positions for a channel flow at $Re_τ\approx 180$ are used to explore the mechanisms underlying the cascade process. As a proof of concept, reduced-order simulations are performed by retaining only the interactions identified as responsible for significant energy transfer based on our framework. Turbulent dynamics are successfully reproduced when 30% or more of the total interactions are included, while noticeable deviations emerge in the near-wall region when this proportion is further reduced.

Figures (10)

• Figure 1: Sketch of interscale interactions, adapted from de_salis_young_inter-scale_2024. Interaction I involves a recipient mode $\bm{k}_{3}$ receiving energy from an interaction between mode $\bm{k}_1$ and $\bm{k}_2$; $\bm{k}_{3}$in turn interacts with $\bm{k}_{6}$ in interaction III to transfer energy into $\bm{k}_{7}$, and so on to viscosity.
• Figure 2: Maps of inter-scale energy transfer in (a) donor and (b) catalyst wavenumbers to recipient lengthscale $\lambda_{t,x}^+ = 188, \lambda_{t,z}^+ = 94$ at $y^+ = 15$ from the $uu$ and $wu$ contributions to the advection term for an $Re_\tau = 180$ channel flow. The colour scale in wall units is from -0.1 (blue) to 0.1 (red). The black dashed lines indicate the recipient wavenumber $\bm{k}_r$.
• Figure 4: Log-$\lambda$ maps of inter-scale energy transfer in (a) donor and (b) catalyst modes to recipient lengthscale $\lambda_{t,x}^+ = 188, \lambda_{t,z}^+ = 94$ at $y^+ = 15$ from the $uu$ and $wu$ contributions to the advection term for an $Re_\tau = 180$ channel flow. The colour scale in wall units is from $-2\times10^{-5}$ (blue) to $2\times10^{-5}$ (red), scales based on methods detailed in § \ref{['sec:scaling']}. The black dashed lines indicate the recipient wavenumber $\bm{k}_r$. Circled regions in black in the corresponding maps of (a) and (b) represent the same interactions.
• Figure 5: Spectral binning procedure illustrated in log–$\lambda$ space. Black points indicate the computed modes, and grey regions represent the selected bins. Red circles mark the geometric centres where the averaged data are stored, while the blue cross denotes the mode closest to each geometric centre.
• Figure 6: (a) Total nonlinear energy transferred ($\hat{N}$) into each recipient wavelength at $y^+ = 15$. Coloured contours from blue to red are from -0.05 to 0.05, with increments of 0.005, in viscous units. (b) Total nonlinear energy transferred into each recipient wavelength integrated in $y$ across the channel. Solid and dashed contours are results for the present channel, and coloured ones data from Ding2025. Contour levels are from -1.4 to 1.4 with increments of 0.2.