Multiple states of turbulence at vanishing inertia

Abstract

Based on everyday experience fluid flows tend to be ordered and quiescent if inertial forces are low and held in check by viscosity. This intuition spectacularly fails in the case of complex macromolecular fluids like polymer melts, paints and biofluids. In such cases elastic fluid properties can drive turbulent motions at moderate and even vanishing Reynolds numbers. By studying viscoelastic flows in curved pipes we demonstrate that this low inertia phenomenology results from the competition of two hydrodynamic instabilities and respectively from the co-existence and interdependence of two distinct turbulent states. Unexpectedly the established categories of elastic and elasto-inertial turbulence (ET and EIT) fail to demarcate the actual turbulent states, fundamentally changing the perception of this phenomenon a century after its discovery.

Figures (3)

• Figure 1: Onset of multiple turbulent states and transformation from a center to a wall mode. (a) Normalized pressure fluctuation levels as a function of Reynolds number used to determine the onset. The fluctuation levels are calculated by $(\sigma_p-\sigma_{p}^{lam})/\sigma_{p}^{lam}$, where $\sigma_p$ is the standard deviation of pressure signals of each data point and $\sigma_{p}^{lam}$ is that of points representing laminar flow. Straight lines are fitted to different states for illustration. (b) Streamwise velocity fluctuations, $u^\prime/u^\prime_{\mathit{max}}$, as a function of distance from the pipe center, $r/(d/2)$, obtained from PIV. At $\mathit{Re} = 50, \mathit{Wi} = 50$, the peak of the fluctuations profile appears close to the pipe center. At $\mathit{Re} = 50, \mathit{Wi} = 75$, the peak has shifted towards the wall. (c) The onset of the center mode (black circles), the hoop stress mode as a primary instability (red circles), and the hoop stress as a secondary instability (open circles) as a function of curvature ratio, $d/R$. Inset depicts critical $\mathit{Wi}$ as a function of inverse curvature ratio, $R/d$. The red line is the Pakdel-McKinley criterion of two scales, written in the form of $\mathit{Wi}_c (1-\beta)^{1/2} = M_c[a/(R/d)+b]^{-1}$ (where $\beta$ denotes viscosity ratio between solvent and solution, and $M_c = 4.08$, $a = 1.83$ and $b = 0.01$ are fitting parameters), fitted to the data points of primary hoop stress mode (red solid circles). Each data point is obtained from 3 to 10 rehearsals of the experiment shown in (A). Error bars indicate standard deviation.
• Figure 2: Multiple turbulent states and the approach to the inertialess regime. (a) Onset of the center and hoop stress modes as a function of curvature ratio for 85% glycerol-200 ppm PAAM in 4 mm pipe ($\mathit{E} \approx 50$) and 80% glycerol-50 ppm PAAM in 1.6 mm pipe ($\mathit{E} \approx 80$). (b) Pressure fluctuation levels as a function of distance from the critical points of the center and hoop stress modes, respectively, in 1.6 mm pipes. Profiles of the hoop stress mode are shifted vertically by the amount of normalized pressure fluctuation at the onset. Solid lines are guides to the eye. (c) Streamwise velocity fluctuations, $u^\prime/u^\prime_{\mathit{max}}$, as a function of distance from the pipe center, $r/(d/2) = 0$, measured using PIV in 1.6 mm curved pipes. The pipe walls concave left. Data points in the vicinity of walls are omitted because of large uncertainties.
• Figure 3: Distinction between turbulent states. (a and b) Streamwise velocity fluctuations, $u^\prime/U$, close to the onsets of the center mode and wall mode, respectively. For the center mode, weak fluctuations (represented by yellow for positive $u^\prime$ regions and cyan for negative ones) are spatially organized in a chevron pattern, while for the wall mode, strong fluctuations (red and blue) emerge as streaks elongated in the streamwise direction. (c) Power spectra of the streamwise velocity component for turbulent states corresponding to the center and hoop stress modes. The velocity signals are sampled at locations close to the center line of the pipe.