Defect-mediated turbulence in ribbons of viscoelastic Taylor-Couette flow

Transition to defect-mediated turbulence in the ribbon patterns observed in a viscoelastic Taylor-Couette flow is investigated when the rotation rate of the inner cylinder is increased while the outer cylinder is fixed. In four polymer solutions with different values of the elasticity number, the defects appear just above the onset of the ribbon pattern and trigger the appearance of disordered oscillations when the rotation rate is increased. The flow structure around the defects is determined and the statistical properties of these defects are analyzed in the framework of the complex Ginzburg-Landau equation.

Figure 1

Master curve of the variation of the dynamic viscosity of polymer solutions of PEO in $95\%$ water–$5\%$ isopropyl alcohol with the dimensionless shear rate ${\lambda }_C\dot{\gamma }$ for three working concentrations.

Figure 2

Transition sequences from circular Couette flow (CCF) to the states of disordered oscillations in viscoelastic Taylor-Couette flow with concentrations $c\in \{500,600,700\}$: ${\text{Ta}}_c,{\text{Ta}}_d,{\text{Ta}}_i$, and ${\text{Ta}}_t$ are the threshold values of the ribbon pattern, of the defects mediated turbulence, of the spatio-temporal intermittency and of the inertio-elastic turbulence [16], respectively. ${\text{Ta}}_u$ is the lower boundary of the unexplored yet zone.

Figure 3

Space-time diagrams and frequency power spectra of pattern states for $E=0.011$: (a, d) ribbons without defects for $\text{Ta}=49.05$; (b, e) ribbons with defects for $\text{Ta}=50.1$; (c, f) spatio-temporal intermittency regime for $\text{Ta}=54.82$. The inset of Fig. 3 represents a zoom of power spectrum in the neighborhood of fundamental frequency.

Figure 4

Demodulation of the ribbon state observed at $\text{Ta}=44.5$ and $E=0.046$: (b) ribbons with defects the black and white dashed squares are the positions of right and left defects respectively; (a, d) phase and amplitude of the left spiral; (c, f) phase and amplitude of the right spiral; (e) spatial profiles of the amplitudes $\left|A\right|$ and $\left|B\right|$ of the right and left spirals in the vicinity of the core of defects. Black zones in amplitudes correspond to the neighborhood of the defects.

Figure 5

Spatiotemporal diagrams of the radial velocity component for ribbon state without defects (a) for $\text{Ta}=41.9$ and with defects (d) for $\text{Ta}=46$. Spatiotemporal diagrams of the the right (b, e) and left (c, f) spirals for ribbons without defects and with defects. The velocities are scaled by $d/\lambda$. The data correspond to a working solution with $E=0.137$.

Figure 6

Space-time diagram of: (a) ribbon pattern with defects, (b) amplitude of the pattern, (c) digitized amplitude for $E=0.020$ and $\text{Ta}=46.2$.

Figure 7

Variation with $\text{Ta}$: (a) of the mean number of defects for three values of the elasticity number: $E=0.011$ (stars), $E=0.020$ (squares), and $E=0.046$ (circles): the solid lines are the law (3) proposed by Ref. [37]; (b) of the correlation time for $E=0.020$, (c) of the correlation length for $E=0.020$. The insets of panels (b) and (c) are examples of the temporal and spatial correlation functions, respectively. The secondary peaks in the spatial correlation function come from the subharmonic mode.

Figure 8

(a) Digitized amplitude for $E=0.020$ and $\text{Ta}=46.2$ [extracted from Fig. 6]. The variation with $\text{Ta}$ of (b) the mean lifetime of defects, (c) the laminar time between two consecutive defects, for three values of $E$: $E=0.011$ (stars), $E=0.020$ (squares), and $E=0.046$ (circles). The solid lines are the fits from the formula (4). The error bars represent the standard deviations.

Figure 9

Histogram of the laminar time between two consecutive defects in ribbons for $\text{Ta}=47.4$ and $E=0.046$

Figure 10

Variation of the characteristic laminar time with the corrected mean defect number for the three values of $E$: $E=0.011$ (stars), $E=0.020$ (squares), and $E=0.046$ (circles)

Figure 11

Variation with $\tilde{\epsilon }$ of the ratio of the mean lifetime of defects to the mean laminar time of defects for thee three values of the elasticity number, $E=0.011$ (stars), $E=0.020$ (squares), and $E=0.046$ (circles). The error bars represent the standard deviations