Defect-mediated turbulence in bubbly Taylor-Couette flow

We investigate experimentally the defect-mediated turbulence (DMT) which is induced by bubbles injection in a Taylor-Couette flow when the inner cylinder is rotating while the outer cylinder is fixed. Bubbles of 1.2 mm in diameter are injected at the bottom of a Taylor-Couette device of radii ratio equal to 0.91. The tangential Reynolds number range is $[2200,19300]$ and the air injection rate varies up to 800 ml/min. For these conditions of the experiments, bubbles are trapped in the gap by the Taylor vortices and arranged as patterns (toroidal, wavy toroidal, spirals, and wavy spirals). Visualizations of the bubble patterns were carried out. When decreasing the Reynolds number or increasing the air injection rate, spiral and toroidal patterns can coexist in a composite flow. Defects occur in the bubble's patterns (merging or splitting of the Taylor vortex pairs). By analyzing the space-time diagram of bubbles patterns and their complex demodulation, we highlight different regimes and transitions in the DMT of the bubbly Taylor-Couette flow. The control parameter of the transitions is the air volumetric fraction, which evolves as the ratio between the axial injection Reynolds number and the tangential Reynolds number. By increasing the air volumetric fraction, the defects in the DMT flows are classified as three flow regimes: (i) structured composite flow where the defects are periodic in space and time, (ii) intermittency defects chaos where the defects zones alternate randomly with the patterns in time and space, and (iii) developed defects chaos with a large defects density. The statistical properties of these three regimes of the DMT are analyzed in the framework of the complex Ginzburg-Landau equation.

Figure 1

Conceptual sketch of the bubbly Taylor-Couette flow: (a) Taylor-Couette system; (b) view of preferential localization of the bubbles in the gap (bubbles are represented as full circles).

Figure 2

Scheme of the IRENav Taylor-Couette facility.

Figure 3

Map of the bubbles localization in the gap as a function of parameters $C$ and $H_{\text{new}}$ Fokoua et al. [9].

Figure 4

Visualization of the bubble patterns in the $65\%$ glycerol mixture for $\text{Re}=2648$; ${\text{Re}}_g=0.16$; $Q_g=160\mathrm{ml}\text{/}\min ;$ $\alpha =0.0088\%$: (a) instantaneous image, (b) space-time diagram, (c) first filtered diagram, and (d) homogenized diagram with defects (circle).

Figure 5

Examples of space-time diagrams sequences that evidence different bubbly basic patterns. Examples are shown for the mixture of $40\%$ glycerol: (a) toroidal pattern T ($\text{Re}=8156$; $Q_g=32\text{ml/min}$; ${\text{Re}}_g=0.13$; $\alpha (\%)=0.0024$); (b) wavy Toroidal pattern WT ($\text{Re}=17000$; $Q_g=40\mathrm{ml}\text{/}\min$; ${\text{Re}}_g=0.19$; $\alpha (\%)=0.0016$); (c) spiral pattern S ($\text{Re}=14500$; $Q_g=60.8\mathrm{ml}\text{/}\min$; ${\text{Re}}_g=0.28$; $\alpha (\%)=0.0027$). The axial phase velocity determined by the slope of the stripes is $V_{\text{phase}}=0.071\text{m/s}$; $V_{\text{phase}}/V_b=0.72$; (d) wavy spiral pattern WS ($\text{Re}=11361$; $Q_g=80\mathrm{ml}\text{/}\min$; ${\text{Re}}_g=0.36$; $\alpha (\%)=0.0045$). The axial phase velocity is $V_{\text{phase}}=0.039\text{m/s}$; $V_{\text{phase}}/V_b=0.41$.

Figure 6

Examples of space-time diagrams that evidence the different bubbly composite patterns regimes: (a) Structured composite patterns regime $\text{Re}=8339$; $Q_g=56\text{ml/min}$; $\alpha (\%)=0.0042$; ${\text{Re}}_g=0.24$; $40\%$ of glycerol, highlights periodical punctual defects in circles. (b) Structured composite patterns regime $\text{Re}=2478$; $Q_g=51\mathrm{ml}\text{/}\min$; $\alpha (\%)=0.0031$; ${\text{Re}}_g=0.054$; $65\%$ of glycerol, highlights periodical punctual defects in circles. (c) Intermittency defect chaos regime, $\text{Re}=3714$; $Q_g=160\mathrm{ml}\text{/}\min$; $\alpha (\%)=0.0065$; ${\text{Re}}_g=0.17$; $65\%$ of glycerol, highlights burst of defect in ellipse. (d) Developed defect chaos regime, $\text{Re}=3669$; $Q_g=800\mathrm{ml}\text{/}\min$; $\alpha (\%)=0.038$; ${\text{Re}}_g=0.98$; $65\%$ of glycerol, no basic patterns identified beyond one rotation period.

Figure 7

Map of the bubbly patterns regimes in the ($\text{Re},{\text{Re}}_g)$ plane. Isocontours of characteristic values of the parameter $\alpha$ are also plotted with lines and colors.

Figure 8

Bubbles jumping on a zoom of the space-time diagrams of the bubbles brightness: (a) before applying filtering and (b) after applying filtering and homogenization procedures.

Figure 9

Sketch that summarizes the transitions from the unique pattern regime to the developed defect chaos regime in the BTCF (for $2200<\text{Re}<5600$ in the explored region of the map of Fig. 7 for the $65\%$ glycerol mixture). ${\alpha }_{\text{def}}$ is the critical value of $\alpha$ beyond which the first defect occurs. This value will be characterized in the part dedicated to the statistical analysis of the defects. ${\alpha }_U=0.07\%$ is the lower value of $\alpha$ of the unexplored yet zone.

Figure 10

Power spectrum versus frequency for: the mixutre of 65% glycerol (a) structural composite pattern $\left[\alpha \right(\%)=0.001]$, (b) intermittency defect chaos $\left[\alpha \right(\%)=0.006]$, and (c) developed defect chaos $\left[\alpha \right(\%)=0.067]$. The dashed and dotted lines are, respectively, the background noise and zero energy.

Figure 11

Evolution of the characteristic frequencies normalized by the inner cylinder angular velocity as a function of $\alpha$ (red, principal frequency; blue, secondary frequencies).

Figure 12

Evolution of the normalized axial wavelength as a function of Re number.

Figure 13

Evolution of the normalized axial wavelength as a function of $\alpha$ control parameter.

Figure 14

(a) Time correlation and (b) length correlation and their best exponential fit for $\text{Re}=3530$; ${\text{Re}}_g=0.32$; $Q_g=30\mathrm{ml}\text{/}\min$; $\alpha =0.0013\%$, $65\%$ glycerol mixture.

Figure 15

Dimensional correlation time as a function of the control parameter $\alpha$ for mixtures of $65\%$ and $40\%$ glycerol.

Figure 16

Dimensionless axial correlation length as a function of control parameter $\alpha$ for $65\%$ and $40\%$ mixture.

Figure 17

Evolution of the axial diffusion velocity scaled by bubbles velocity $V_b$ with the parameter $\alpha (\%).$

Figure 18

Complex demodulation for ($\text{Re}=2648$; $Q_g=160\mathrm{ml}\text{/}\min$; $\alpha =0.0087\%$; ${\text{Re}}_g=0.16$; $65\%$ of glycerol mixture: (a) space-time diagram of analytical prolongment, (b) space-time diagram of phase, (c) module, (d-1) axial profile of the module near a defect ($t=5.1T_i$; $z=11d$), (d-2) axial profile of the module near a hole ($t=18.1T_i$; $z=9d$), (e) cartography of binarized module (defects in white) superimposed with the cartography of phase.

Figure 19

Evolution of the average number of defects according to the parameter $\alpha$ for mixtures $65\%$ and $40\%$ of glycerol, the solid line represents the law of adjustment proposed by Egolf and Greenside [31] [Eq. (19)] for the mixture of $65\%$ glycerol.

Figure 20

Normalized averaged lifetime of the defects according to $\alpha$ for mixtures of $65\%$ and $40\%$ glycerol. Error bars represent standard deviations.

Figure 21

Normalized time separation between two consecutive defects according to $\alpha$ for mixtures of $65\%$ and $40\%$ glycerol. Error bars represent standard deviations.

Figure 22

Inverse of the averaged separation time in function of the mean number of defects.

Figure 23

Histogram of the separation time between two consecutive defects fitted by exponential law for $\text{Re}=2867;Q_g=99.2\mathrm{ml}\text{/}\min ;\alpha (\%)=0.0054$; mixture $65\%$ of glycerol.

Figure 24

Covariance of the consecutive separation times between defects as a function of $\alpha (\%)$ for the mixture of $65\%$ glycerol.

Figure 25

Ratio of the correlation length to the averaged axial defect separation length according to $\alpha$ for the mixture of $65\%$ glycerol.