Structures of elastoinertial turbulence in pipe flow

TL;DR

This study investigates elastoinertial turbulence (EIT) in axisymmetric pipe flow by developing VESPOD, a viscoelastic energy-based variant of spectral POD that simultaneously decomposes velocity and polymer-stress fields. The method reveals that EIT dynamics are dominated by three traveling-wave families whose higher-wavenumber structures are simple harmonics of the fundamental, with polymeric stress forming thin sheets at critical layers that nest across waves. These findings, analyzed across domain lengths and parameter variations, show a robust three-family organization at moderate Re and Wi, while non-harmonic modes emerge at higher control parameters, offering a unified energy-based picture of EIT that links pipe and channel turbulence while highlighting modality differences. The results have implications for understanding drag reduction limits and guiding models of viscoelastic turbulence in complex geometries.

Abstract

Elastoinertial turbulence (EIT) is a self-sustaining chaotic state resulting from the interplay between inertia and elasticity in the flow of dilute polymeric solutions, and its emergence is believed to limit the achievable drag reduction in turbulence flow using polymer additives. In the present study, we introduce a viscoelastic variant of spectral proper orthogonal decomposition (VESPOD) that decomposes velocity and polymeric stress fields of EIT together into well-defined orthogonal oscillating modes such that the decomposition is optimal in the terms of the total mechanical energy of the flow. Using this technique, we investigate the dominant coherently evolving structures underlying the dynamics of EIT in axisymmetric pipe flow. By analyzing distinct peaks in the leading eigenvalue of the VESPOD eigenvalue spectrum, we find that the dynamics of EIT in pipe flow is dominated by three distinct families of traveling waves, where the higher wavenumber structures of each family are simple harmonics of their respective fundamental waves. The radial velocity fields of the traveling waves are characterized by the formation of large-scale structures spanning the pipe radial direction. However, the polymeric stress fields corresponding to them are characterized by the formation of thin inclined sheets of high stress fluctuations at the critical layers of the respective waves, i.e.~ the locations where the wave speed of the VESPOD mode matches the mean streamwise velocity. Additionally, these sheets exhibit nested structures, where the polymeric sheets of faster waves are confined by those of the immediately slower waves.

Figures (12)

• Figure 1: Instantaneous ($a$) radial velocity, ($b$) streamwise velocity, and ($c$) $zz$-component of stretch tensor of elastoinertial turbulence (EIT) in axisymmetric pipe flow at $\Rey=3000$, $\textit{Wi}=35$, and $L=5$. Means (averaged over $z$ and $t$) of ($d$) velocity components and ($e$) stretch components. Dotted lines ($d,e$) represent the laminar profiles at the same parameter.
• Figure 2: ($a$) Leading eigenvalue of the VESPOD energy spectrum of the structures having wavenumber $\kappa=1$ underlying EIT at $\Rey=3000$, $\textit{Wi}=35$, and $L=5$ (Inset: spectrum in log-log scale). ($b$) Leading eigenvalues of the VESPOD energy spectra for structures having different wavenumbers. Red symbols represent the peaks in the leading eigenvalues (Circle $\bullet$, star $\star$, and triangle $\blacktriangle$ are first, second, and third peaks, respectively).
• Figure 3: VESPOD mode structures of ($a-c$) radial velocity ($u_r'$), ($d-f$) streamwise velocity ($u_z'$), and ($g-i$) $zz$-component of stretch tensor ($T_{zz}'$) corresponding to the peaks in the leading eigenvalue of VESPOD energy spectrum of $\kappa=1$. Locations of critical layers have been denoted with white dotted lines ($g-i$). Other parameters are $\Rey=3000$, $\textit{Wi}=35$, and $L=5$.
• Figure 4: Wave speeds of the VESPOD mode structures corresponding to the peaks in the energy spectra: ($a$) Wave speed vs wave number and ($b$) Wave speed vs peak index. Parameter values are $\Rey=3000$, $\textit{Wi}=35$, and $L=5$.
• Figure 5: VESPOD mode structures of $u_r'$ corresponding to different peaks in the leading eigenvalue of energy spectrum of wavenumber ($a-c$) $\kappa=2$, ($d-f$) $\kappa=3$, ($g-i$) $\kappa=4$, and ($j-l$) $\kappa=5$. Other parameters are $\Rey=3000$, $\textit{Wi}=35$, and $L=5$.