The interplay of inertia and elasticity in polymeric flows

TL;DR

This study uses direct numerical simulations of the Oldroyd-B model to analyze polymeric turbulence across wide ranges of inertia $Re$ (via $Re_\lambda$) and elasticity $De$. It reveals a non-unique, multi-regime energy spectrum arising from competing flux contributions that slow the fluid nonlinear cascade, with Newtonian and polymeric scaling coexisting and transitioning as $Re_\lambda$ and $De$ vary. The work links real-space statistics (structure functions and kurtosis) to Fourier-space scaling, showing distinct polymer spectra, altered intermittency, and polymer stretching that correlates with extensional regions more strongly than with rotation. Overall, the paper bridges elastic turbulence and elastic-inertial turbulence within homogeneous, isotropic flows, offering insights into energy transfer, dissipation, and the role of polymer elasticity in turbulent dynamics.

Abstract

Addition of polymers modifies a turbulent flow in a manner that depends non-trivially on the interplay of fluid inertia, quantified by the Reynolds number $Re$, and the elasticity of the dissolved polymers, given by the Deborah number $De$. We use direct numerical simulations to study polymeric flows at different $Re$ and $De$ numbers, and uncover various features of their dynamics. Polymeric flows exhibit a multiscaling energy spectrum that is a function of $Re$ and $De$, owing to different dominant contributions to the total energy flux across scales. This behaviour is also manifested in the real space scaling of structure functions. We also shed light on how the addition of polymers results in slowing down the fluid non-linear cascade resulting in a depleted flux, as velocity fluctuations with less energy persist for longer times in polymeric flows. These velocity fluctuations exhibit intermittent, large deviations similar to that in a Newtonian flow at large $Re$, but differ more and more as $Re$ becomes smaller. This observation is further supported by the statistics of fluid energy dissipation in polymeric flows, whose distributions collapse on to the Newtonian at large $Re$, but increasingly differ from it as $Re$ decreases. We also show that polymer dissipation is significantly less intermittent compared to fluid dissipation, and even less so when elasticity becomes large. Polymers, on an average, dissipate more energy when they are stretched more, which happens in extensional regions of the flow. However, owing to vortex stretching, regions with large rotation rates also correlate with large polymer extensions, albeit to a relatively less degree than extensional regions.

Figures (16)

• Figure 1: Representative snapshots of normalized fluid energy dissipation rates $\epsilon_{{\rm f}}$ at (a) large and (b) small $Re$ for $De \approx 1$. Large $Re_\lambda$ polymeric flows exhibit a wide range of flow structures, while the small $Re_\lambda$ flows have only large scale structures.
• Figure 2: The non-unique scaling of the fluid energy spectrum $E(k)$ in polymeric flows at different (a) $De$ and (b) $Re$ numbers. The spectra have been shifted vertically for visual clarity by factors of powers of 10. Three distinct scaling regimes (in different shades) have been shown in dashed ($k^{-5/3}$, Newtonian), dash-dotted ($k^{-2.3}$, polymeric) and dotted ($k^{-\gamma}$; $\gamma \geq 4$, smooth) lines. All three regimes coexist when De $\approx 1$. (a) The smallest $De$ has a close to Newtonian behaviour as the elastic effects are minimal. (b) The triple scaling at largest $Re$ gives way to a unique, smooth $k^{-4}$ regime at the smallest $Re_\lambda$.
• Figure 3: Normalized flux contributions at (a) Re$_\lambda \approx 450$, (b) $240$, and (c) $40$ for flows with De $\approx 1$. The polymeric contribution $\mathcal{P}$ is split into flux $\Pi_{\rm p}$ and dissipation $\mathcal{D}_{\rm p}$ contributions as $\mathcal{P} = \Pi_{\rm p} + \mathcal{D}_{\rm p}$ as in Marco23. (a) At large $Re_\lambda$, three distinct regimes are determined by different dominant contributions: large scales are dominated by the fluid non-linear flux $\Pi_{\rm f}$, intermediate scales by the polymer flux $\Pi_{\rm p}$, and small scales by the polymer dissipation $\mathcal{D}_{\rm p}$. (b) At moderate $Re_\lambda$, the fluid-nonlinearity $\Pi_{{\rm f}}$ is weakened and comprises two distinct regimes dominated by $\Pi_{\rm p}$ and $\mathcal{D}_{\rm p}$. (c) At extremely small $Re_\lambda$, only the dissipative terms $\mathcal{D}_{\rm f}$, $\mathcal{D}_{\rm p}$ remain important. All terms are normlised by $\epsilon_{\rm t}$.
• Figure 4: The scaling behaviour of flux contributions obtained by replotting \ref{['fig:Flux']} on a log-log scale. The approximate scaling forms are shown as dash-dotted lines. The fluid flux $\Pi_{{\rm f}}$ decays as a power-law in the presence of polymers at large to moderate $Re_\lambda$, implying a slow and weak fluid non-linear cascade.
• Figure 5: The plots of various timescales discussed in section \ref{['sect:Time']} for (a) Re$_\lambda$ = 450 and (b) 240, at $De = 1$. We show the compensated plots for the eddy turnover time (triangles) and the time obtained from the nonlinear (squares) and polymer fluxes.