Spectral Universality of Elastoinertial Turbulence

Dissolving small amounts of polymer into a Newtonian fluid can dramatically change the dynamics of transitional and turbulent flows. We investigate the spatiotemporal dynamics of a submerged jet of dilute polymer solution entering a quiescent bath of Newtonian fluid. High-speed digital Schlieren imaging is used to quantify the evolution of Lagrangian features in the jet revealing a rich sequence of transitional and turbulent states. At high levels of viscoelasticity, we identify a new distinct transitional pathway to elastoinertial turbulence (EIT) that does not feature the conventional turbulent bursts and instead proceeds via a shear-layer instability that produces elongated filaments of polymer due to the nonlinear effects of viscoelasticity. Even though the pathways to the EIT state can be different, and within EIT the spatial details of the turbulent structures vary systematically with polymer microstructure and concentration, there is a universality in the power-law spectral decay of EIT with frequency, $f^{-3}$, independent of fluid rheology and flow parameters.

Figure 1

Flow transitions in a submerged jet. (a)–(e) Laminar (green circle), turbulent bursts (yellow circle), NT (red circle), EIF (blue circle), and EIT (violet circle) state. (f) State diagram for $L_{\max }=91.4$, i.e., PEO with $M_w=8\times {10}^6\mathrm{g}/\mathrm{mol}$. To represent the Newtonian jet on this state diagram, we use a linear ordinate for $\mathrm{El}\le {10}^{-3}$.

Figure 2

(a) From left to right: snapshots of a Newtonian (water) jet and PEO solution jets with $M_w=2\times {10}^6\mathrm{g}/\mathrm{mol}$, $c=100\mathrm{ppm}$, $M_w=4\times {10}^6\mathrm{g}/\mathrm{mol}$, $c=50\mathrm{ppm}$, and $M_w=8\times {10}^6\mathrm{g}/\mathrm{mol}$, $c=12.5\mathrm{ppm}$, respectively. (b) Kymographs of the concentration fluctuations at the Eulerian cross section at a distance $120R$ from the nozzle, shown with a white rectangle. (c) Evolution of the scale of segregation $S$ with polymer extensibility $L_{\max }^2$. The error bars from repeated measurements of $S$ are smaller than the symbol size.

Figure 3

(a) Frequency spectra of the fluctuations in the local concentration in turbulent polymer jets at $\mathrm{Re}=400$ and for a vertical distance $120R$ from the nozzle at the centerline of the jet. The abscissa and ordinate are nondimensionalized with the Zimm relaxation time, ${\lambda }_{\text{Zimm}}$, and the square of the mean local concentration, $⟨c{⟩}^2$, respectively. The dashed line denotes ${\lambda }_{\text{Zimm}}f=1$. The corresponding Newtonian jet is stable at this condition. (b) Temporal power spectral density for concentration fluctuations at $\mathrm{Re}=800$ for the Newtonian jet and polymer solutions with different molecular weights and concentrations. The abscissa is nondimensionalized with the strain rate at the exit of the nozzle, $U/R$. The dashed line denotes the start of the inertial range for Newtonian turbulence. The data acquisition point is shown with a hollow circle in the image shown on the right. The insets in (b) compare the raw time signal of concentration fluctuations for the Newtonian jet and for a jet of PEO solution ($M_w=8\times {10}^6\mathrm{g}/\mathrm{mol}$, $c=12.5\mathrm{ppm}$).