Routes to turbulent Taylor vortices: Vortex splitting and merging in chaotic state and the role of particles

A novel experimental finding of vortex splitting and merging (VSM) route to turbulence is reported in the counter-rotation regime of the Taylor–Couette flow of a Newtonian fluid, and the role of neutrally buoyant inertial particles on this transition pathway is analysed up to a particle volume fraction of

over a range of rotation ratios

negative 0.35 less than or slanted equals upper Omega equals omega Subscript o Baseline divided by omega Subscript i Baseline less than 0 − 0.35 ⩽ Ω = ω o / ω i < 0 $-0.35\leqslant \varOmega =\omega _o/\omega _i\lt 0$

(where

omega Subscript i ω i $\omega _i$

and

omega Subscript o ω o $\omega _o$

are the angular speeds of the inner and outer cylinders, respectively). The onset of VSM is found to coincide with the transition of a quasi-periodic state (‘doubly’ modulated wavy vortex [DMWV], characterised by two incommensurate frequencies) to a chaotic state (chaotic wavy vortex [CWV], in the background of VSM) at

phi less than or slanted equals 0.15 ϕ ⩽ 0.15 $\phi \leqslant 0.15$

, giving rise to a new state of ‘VSM + CWV’ in which the VSM lives within a distinct CWV state. With further increasing particle loading, however, the VSM appears via MWV

right arrow → $\to$

DMWV

right arrow → $\to$

CWV

Subscript normal upper B Baseline right arrow B → $_{\mathrm{B}}\to$

VSM + CWV

right arrow → $\to$

CWV at

phi equals 0.2 ϕ = 0.2 $\phi = 0.2$

and is found to be absent at

phi equals 0.3 ϕ = 0.3 $\phi = 0.3$

; another variant (CWV

Subscript normal upper B B $_{\mathrm{B}}$

) of CWV, characterised by an additional broad low-frequency peak, bifurcates from the quasi-periodic (DMWV) state at

phi greater than or slanted equals 0.2 ϕ ⩾ 0.2 $\phi \geqslant 0.2$

. The VSM events are shown to be temporally uncorrelated, with a multimodal skewed probability density function for the interevent time interval, implying that the VSM-induced chaos (VSM + CWV) is stochastic. The final transition to turbulent Taylor vortex (TTV) is found to occur via a propagating state of turbulent wavy vortices (TWV). The power spectrum of TTV is flat and featureless with a broad noisy tail, whereas the flat spectrum of TWV contains a sharp peak that represents a large-scale wavy vortex mode, whose disappearance marks the onset of TTV. It is shown that the propagation frequency of the TWV mode decreases with increasing shear Reynolds number (

italic Re Subscript s Baseline left parenthesis phi right parenthesis Re s ( ϕ ) ${\textit{Re}}_s(\phi )$

) and follows a power-law scaling in terms of

italic Re Subscript s Baseline left parenthesis phi right parenthesis Re s ( ϕ ) ${\textit{Re}}_s(\phi )$

and the relative viscosity of the suspension. Our results establish that the Ruelle–Takens route to chaos (Ruelle & Takens, Commun. Math. Phys. , 1971, vol. 20, pp. 167–192) holds in counter-rotating suspension Taylor–Couette flow over a range of particle loading

phi less than or slanted equals 0.3 ϕ ⩽ 0.3 $\phi \leqslant 0.3$

at

negative 0.35 less than or slanted equals upper Omega less than 0 − 0.35 ⩽ Ω < 0 $-0.35\leqslant \varOmega \lt 0$

.