Vortex merging and splitting: A route to elastoinertial turbulence in Taylor-Couette flow

We report experimental evidence of a new merge-split transition (MST) to elastoinertial turbulence (EIT) in Taylor-Couette flows of viscoelastic polymer solutions, caused by merging and splitting of base Taylor vortices when crossed by elastic axial waves (rotating standing waves, RSW). These vortex merging and splitting events are not due to transient behavior, finite aspect ratio, or shear-thinning behavior. They are random in nature and increase in frequency with Re; when superimposed on a RSW flow state they cause abrupt changes in the axial spatial wavelength, leading to the transition from a RSW to the EIT state. We thus identify MST as an inertial feature solely triggered by elasticity and independent of any shear-thinning behavior.

Figure 1

Schematic of the experimental setup showing the Taylor-Couette cell and the visualization arrangement.

Figure 2

Oscillatory shear rheological characterization of the working fluid ($1\%$ strain). The elastic timescale is obtained by removing the contribution of the solvent viscosity ${\mu }_s$ to $G^{″}$ curves as $\widetilde{G^{″}}=G^{″}-{\mu }_s\times \omega$ and finding the crossover between G' and $\widetilde{G^{″}}$ (vertical dashed line).

Figure 3

Reynolds-space and frequency-space diagrams of the ramp-up experiment. (a) $z$ root mean square $i^{*}$ of the (b) Reynolds-space flow map intensity, such that $i^{*}\left(\text{Re}\right)={〈I(\mathrm{z},\text{Re})〉}_{\mathrm{rms},\mathrm{z}}$ with $I$ the gray-scale intensity of flow map 3 (b). (c) Frequency map where the frequency axis is scaled by the maximum inner cylinder frequency reached at the end of the experiment, $f_{\max }=2\pi /{\mathrm{\Omega }}_{\max }=9.45$ Hz. Vertical black dotted lines denote transitions from CF to TVF and to RSW states. The more gradual transition from RSW to EIT is also illustrated by a vertical gray dotted line. The cylinder frequency increase with Re is illustrated with the black dashed line. Horizontal dashed lines matching RSW state frequency ridges are plotted at $f=k\times f_e$, $k=[\frac{1}{3},\frac{2}{3},1]$.

Figure 4

Steady-state recordings at constant Re of successive RSW and EIT states. (a)–(d) Time-space diagrams over 50-s time spans. (e)–(h) Detailed section of the same flow maps on the first 5 s and in the central region (dashed rectangles). Merging and splitting of Taylor vortices in the RSW flow state are indicated by full and dotted line ellipses, respectively. Note that the EIT flow state is essentially composed of a high number of indistinguishable merging and splitting along with axial oscillations. (i) and (j) are detailed closeups on splitting [(i) S2, $t=18\text{s}$] and merging [(j) S3, $t=38\text{s}$] events (RSW), where dashed lines are a guide to the eye showing the base Taylor vortices.

Figure 5

Role of RSW waves in the merging and splitting of Taylor vortices, illustrated in the unwrapped ($\theta$,z) plane. Only a central vertical line of this plane is sampled to produce flow maps shown in Figs. 3 and 4, as explained in Fig. 1.

Figure 6

Evolution with Re of the frequency spectra of flow states obtained from steady-state experiments (a) and extracted from the frequency map in Fig. 3 (b). Frequencies ($x$ axis) are scaled by the elastic frequency $f_e$ on (a) and by the cylinder's rotation frequency $\mathrm{\Omega }/2\pi$ on (b). In (b), vertical black and gray dotted lines denote this latter frequency and its first harmonic, respectively. Vertical mixed blue lines are plotted at $f=\frac{1}{3}{f}_e$. In (a) vertical dotted and mixed lines denote the $f=f_e$ and $f=\frac{1}{3}{f}_e$ peaks, respectively.

Figure 7

(a) Instantaneous and (b) time-averaged spatial spectra (Fourier transform on intensity profiles in the axial direction) of steady-state experiments as a function of the normalized wavelength $\lambda /d$ (a) and normalized wave number $\tilde{k}=d/\lambda$ (b).

Figure 8

Axial spatial wavelength $\lambda /d$, expressed as a function of time for steady-state experiments (a), and as a function of Re for the ramp-up experiment (d). (b) Shows the evolution of normalized standard deviation $S$ of $\lambda /d$ on the full duration of each steady-state experiment as a function of Re. (c) Evidences the $S\sim {\text{Re}}^2\sim N_1$ scaling by plotting $S/{\text{Re}}^2$ as a function of Re.

Figure 9

Spatiotemporal spectra obtained by two-dimensional FFT in RSW (a) and EIT (b) steady-state experiments. Spatially averaged spectra are projected in planes a3 and b3. Temporally averaged spectra are projected in planes a2 and b2. Projections are indicated by dashed arrows.