2.1. Experimental procedure

For specified $(\varOmega , \phi )$ , the ‘up-sweep’ (US) run consists of a ramp-up protocol where the rotation frequency is increased quasi-statically from $f = f_{\textit{min}} \sim 0$ to $f = f_{\textit{max}} = \max \{f_i, f_o\} = 50$ Hz. No separate ‘down-sweep’ (DS) run, starting from $f=f_{max }$ to reach $f_{\textit{min}}$ , was performed, but several experiments (with $\phi =0, 0.15$ and $0.3$ ) were repeated with a ‘up–down-sweep’ (UDS) protocol in which the DS run was followed by an US run. For the UDS run, a ramp-down protocol is followed by decreasing the rotation frequency from $f = f_{\textit{max}}$ to $f_{\textit{min}}$ after maintaining the rotation frequency constant at $f = f_{\textit{max}}$ over an extended period of $600$ s post the US run; the influence of this waiting time-window on the final TTV state is discussed in § 4.2. For all experiments, the ramping rate is taken to be

(2.2) \begin{equation} \left |\frac {\mathrm{d}{\textit{Re}}(\phi )}{\mathrm{d}\tau }\right | = \frac {0.05}{\mu _r(\phi )}, \quad \mbox{with} \quad {\textit{Re}}(\phi ) = \max \{{\textit{Re}}_i(\phi ), {\textit{Re}}_o(\phi )\} \quad \mbox{and} \quad \tau =\frac {t}{\tau _{vis}(\phi )}, \end{equation}

where ${\textit{Re}}_i(\phi ) = \rho r_i \omega _i \delta /\mu (\phi )$ and ${\textit{Re}}_o(\phi ) = \rho r_o \omega _o \delta /\mu (\phi )$ refer to the inner and outer Reynolds numbers, respectively, and $\tau _{vis}(\phi ) = \rho \delta ^2/\mu (\phi )$ is the viscous diffusion time. The additional factor, $\mu _r(\phi )$ , in the denominator of (2.2) takes care of the fact that the suspension experiments are carried out at a progressively slower ramping rate with increasing particle loadings (Alam & Ghosh Reference Alam and Ghosh2023; Ghosh & Alam Reference Ghosh and Alam2024). Note further that, during the US and DS runs, the rotation frequency is increased and decreased in steps by keeping it constant over a period of $10$ seconds to ensure the stability of various flow states. This $10$ s time-window corresponds to approximately $17\tau _{vis}$ for the particle-free ( $\phi =0$ ) Newtonian fluid, while the same stands for about $ 70 \tau _{vis}$ at $\phi = 0.3$ , thus allowing more time for the suspension to relax to a stable state with increasing $\phi$ .

Table 1.

Explored range of Reynolds and Stokes numbers at a rotation ratio of $\varOmega =\omega _o/\omega _i=-0.35$ ; ${\textit{Re}}_s^{cr}$ refers to the shear Reynolds number (3.1) at the onset of primary bifurcation during the US run (see figure 6).

Table 1. Long description

A table with 6 rows and 4 columns. The columns are labeled Particle volume fraction phi, Shear Reynolds number (3.1), Particle Reynolds number (2.3), and Stokes number (2.4). The table presents the explored range of Reynolds and Stokes numbers at a rotation ratio. Row 1: phi 0, Shear Reynolds number (201.1, 2093.0), Particle Reynolds number (0.128, 1.335), Stokes number Rep/18. Row 2: phi 0.05, Shear Reynolds number (203.5, 1786.3), Particle Reynolds number (0.127, 1.116), Stokes number Rep/18. Row 3: phi 0.10, Shear Reynolds number (177.1, 1440.3), Particle Reynolds number (0.113, 0.917), Stokes number Rep/18. Row 4: phi 0.15, Shear Reynolds number (169.0, 1157.3), Particle Reynolds number (0.108, 0.738), Stokes number Rep/18. Row 5: phi 0.20, Shear Reynolds number (163.9, 905.1), Particle Reynolds number (0.104, 0.578), Stokes number Rep/18. Row 6: phi 0.30, Shear Reynolds number (166.4, 513.7), Particle Reynolds number (0.102, 0.317), Stokes number Rep/18.

2.2. Non-Brownian suspensions of inertial particles

The particle Reynolds number is estimated from

(2.3) \begin{equation} {\textit{Re}}_p(\varOmega ,\phi ) = \frac { \rho \dot {\gamma }_{\textit{app}} d_p^2}{\mu (\phi ) } = {\textit{Re}}_i(\phi )\left |1 - \frac {\varOmega }{\eta }\right | \left (\frac {d_p}{\delta }\right )^2 , \end{equation}

with its maximum being ${\textit{Re}}_p^{\textit{max}}\equiv {\textit{Re}}_p(\phi =0)=(1.23, 1.335)$ at $\varOmega = (-0.25, -0.35)$ ; here $\dot {\gamma }_{\textit{app}}=|\omega _o r_o - \omega _i r_i|/\delta$ is the apparent shear rate across the Couette gap. The particle Stokes number is estimated from

(2.4) \begin{equation} \textit{St}_p(\varOmega ,\phi ) = \frac {m_p \dot {\gamma }_{\textit{app}}}{3 \pi \mu (\phi ) d_p} =\frac {{\textit{Re}}_p(\varOmega ,\phi )}{18}, \end{equation}

with its maximum being $\textit{St}_p^{\textit{max}}= (0.068, 0.074)$ at $\varOmega = (-0.25, -0.35)$ . The corresponding maximum Péclet number is

(2.5) \begin{equation} \textit{Pe}_p(\varOmega ) = \frac {3 \pi \mu (0) d_p^3 \dot {\gamma }_{\textit{app}}}{4k_B T} = (1.86, 2.03) \times 10^9, \end{equation}

where $k_B$ is the Boltzmann constant and $T = 22 \pm 0.5\, ^{\circ }\text{C}$ is the temperature.

Table 1 provides a summary of flow parameters explored in this study. It is clear that while the working fluid can be treated as a suspension of non-Brownian particles with $\textit{Pe}_p=O(10^9)$ , the suspension particles are inertial since ${\textit{Re}}_p = O(1)$ over the range of bulk Reynolds numbers, from its neutral stability threshold to its maximum explored value. Therefore, the effect of particle inertia on our reported results on pattern transitions cannot be ignored.

2.3. Image analysis

A small amount ( $\lt 0.1\,\%$ ) of anisotropic mica flakes is added to the suspension for flow visualisation, with the TC cell being illuminated using three $41$ W LED panels. While a Nikon (D750 DSLR) camera is used to capture continuous video images of surface flow patterns at a frame rate of $60$ s $^{-1}$ , the high-speed imaging at a frame rate of $200$ s $^{-1}$ utilising a Phantom v $9$ (DANTEC DYNAMICS) CMOS camera is done to capture high-frequency modes. Further, the torque exerted by the fluid, measured at the inner cylinder by the upper motor of the rheometer, is stored by the RheoCompass software on a desktop computer.

The captured images, along with the torque data, are then analysed to extract information on various state transitions leading towards turbulence. Unless mentioned explicitly, the frequency maps presented in §§ 3 and 4 have been constructed by analysing video images captured by the high-speed camera at 200 frames per second. Appendix A provides details about the methodology for the spectral analyses of video images, and the limitations of the image analysis to resolve high- and low-frequency modes are discussed in Appendix A.1.

3.1. Prechaotic states and the phase digram of patterns up to turbulence

The pattern transitions at $\varOmega = -0.35$ , up to the prechaotic states, are shown in figure 2(a–d) during US runs for (a,b) particle-free ( $\phi =0$ ) and (c,d) particle-laden ( $\phi =0.15$ ) suspensions. While figure 2(a,c) display the space- ${\textit{Re}}_s(\phi )$ diagrams that have been constructed by sequentially stacking a vertical strip of thickness 1 pixel along the height of the TC cell as the shear Reynolds number is increased during quasi-static ramping, figure 2(b,d) shows the corresponding frequency maps $\omega /\omega _i$ versus ${\textit{Re}}_s(\phi )$ . For the particle-free ( $\phi =0$ ) case in figure 2(a,b), it is clear that with increasing ${\textit{Re}}_s$ , the CCF first gives birth to the stationary TVF that subsequently bifurcates into time-dependent flow state WTV, characterised by a single temporal frequency, marked by the filled blue circles in figure 2(b). The overall transition sequence of primary and secondary bifurcations (CCF $\rightarrow$ TVF $\rightarrow$ WTV) in figure 2(a,b) look qualitatively similar to those at $\varOmega = -0.25$ (and also at $\varOmega =-0.5$ as reported by Singh et al. (Reference Singh, Ghosh and Alam2022)) and holds for dilute suspensions ( $\phi \leqslant 0.05$ ) over the present explored range of counter-rotation ratio $-0.35\leqslant \varOmega \leqslant 0$ . The discussions on tertiary and quaternary bifurcations from the WTV state is deferred to § 3.1.1.

Note that the number of vortices/rolls in each states of TVF/WTV/MWV is marked in figure 2(b,d) which indicate abrupt changes in the number of rolls at WTV $\rightarrow$ MWV transitions. The related issue of the transition between different number of roll states, i.e. ‘multiroll’ transtions (Benjamin Reference Benjamin1978a ,Reference Benjamin b ; Mullin et al. Reference Mullin, Heise and Pfister2017), was discussed for suspension TCF in our recent work (Singh et al. Reference Singh, Ghosh and Alam2022) and will not be pursued here. In present experiments, all captured images (see, for example, figures 2, 3 and 4 a,c) show only a part of the TC cell $z/\delta \in (0,7)$ since its top part $z/\delta \in (7,8.5)$ is fitted with a perspex lid/end cap that blocks images of a pair of rolls/vortices adjacent to the top end plate. Therefore, including the missing pair, the number of rolls in figure 2(a,c) should be increased by $2$ , i.e. ( $n+2$ )-rolls states, that cover the whole axial span of the TC cell $z/\delta \in (0,8.5)$ . This point must be noted for all displayed images that are discussed in the rest of this paper.

Figure 3.

RIB and SVF (or up-propagating/left-handed spiral) states at (a,b) ${\textit{Re}}_s=173.5$ and (c,d) ${\textit{Re}}_s=181.35$ for $\phi =0.15$ , with other parameters as in figure 2(c,d). The red up and down triangles in (b,d) refer to the rotational frequencies (and their integer multiples) of the outer and inner cylinders, respectively. The displayed time window on the right of (a,c) refers to a time period of $4 s\approx 12.6 \tau _{vis}(\phi )$ , where $\tau _{vis}(\phi )=\rho \delta ^2/\mu (\phi )$ is the viscous time scale.

Figure 4.

Same as figure 2 but at $\phi =0.3$ for (a,b) US and (c,d) DS runs. Markers/symbols in (b,d) carry the same meaning as in figure 2(b,d).

For a semi-dilute suspension with a particle volume fraction of $\phi = 0.15$ in figure 2(c,d), the RIB and SVF states are born from the CCF state as the primary instability mode; this is confirmed from the space- ${\textit{Re}}_s(\phi )$ diagram and the frequency map in figures 2(c) and 2(d), respectively, at $\varOmega =-0.35$ . The snapshots, time evolutions and the spectral contents of the RIB and SVF states can be ascertained from figures 3(a,b) and 3(c,d), respectively. Recall that both RIB and SVF represent ‘non-axisymmetric’ time-dependent states, and were originally discovered by Majji et al. (Reference Majji, Banerjee and Morris2018), later confirmed by Ramesh et al. (Reference Ramesh, Bharadwaj and Alam2019), in the context of suspension TCF with pure inner cylinder rotation ( $\varOmega =0$ ). Note that the RIB state results from the interaction between up- and down-propagating (left- and right-handed) spirals, and represents an axially stationary, azimuthally rotating wave; hence, the RIB is also referred to as the ‘rotating standing wave’ (RSW), having a chequerboard-type pattern (Chossat & Iooss Reference Chossat and Iooss1994; Majji et al. Reference Majji, Banerjee and Morris2018; Ramesh & Alam Reference Ramesh and Alam2020; Singh et al. Reference Singh, Ghosh and Alam2022) as in the right panel of figure 3(a). Therefore, the dominant peak at $\omega _{\textit{RIB}}/\omega _i \approx 0.12$ in figure 3(b) represents the azimuthal propagation frequency of the RIB.

Our recent work (Ghosh & Alam Reference Ghosh and Alam2024) showed that the dominant frequencies of RIB and SVF states appearing in the co-rotation regime ( $\varOmega \geqslant 0$ ) of suspension TCF can be vastly different. This difference persists in the counter-rotation regime too; for example, $\omega _{\textit{RIB}}/\omega _i \approx 0.12$ and $\omega _{\textit{SVF}}/\omega _i \approx 0.35$ in figure 3(b,d); note that these frequencies, in terms of the effective rotation rate $\omega _{sh}=|\omega _i-\omega _o/\eta |$ , are given by

(3.2) \begin{equation} \frac {\omega _{\textit{RIB}}}{\omega _{sh}} \approx 0.16 \quad \mbox{and} \quad \frac {\omega _{\textit{SVF}}}{\omega _{sh}} \approx 0.47. \end{equation}

The latter value in (3.2) agrees well with those reported (i) $\omega _{\textit{SVF}}/\omega _{sh}\approx 0.5$ by Majji et al. (Reference Majji, Banerjee and Morris2018) and (ii) $\omega _{\textit{SVF}}/\omega _{sh}\approx 0.45$ by Ramesh et al. (Reference Ramesh, Bharadwaj and Alam2019) for the case of pure inner-cylinder rotation ( $\varOmega =0=\omega _o$ ). In addition, we verified that the normalised frequencies of RIB and SVF states are nearly independent of ${\textit{Re}}_s(0) = {\textit{Re}}_s(\phi )\mu _r(\phi )$ (and, hence, not affected by fluid inertia) at $\varOmega =-0.35$ , a signature of ‘particle-induced’ spiral patterns (Ghosh & Alam Reference Ghosh and Alam2024). The latter finding is in contrast to ‘fluid-induced’ spirals (that appear in the strongly counter-rotation regime of TCF, Andereck et al. (Reference Andereck, Liu and Swinney1986)) whose propagation frequency depends on the fluid inertia and is known to decrease with increasing ${\textit{Re}}_s$ (Andereck et al. Reference Andereck, Liu and Swinney1986; Singh et al. Reference Singh, Ghosh and Alam2022).

When the particle loading is increased to $\phi = 0.3$ , only the SVF state shows up as the primary bifurcating state during both US and DS runs, see figure 4. Collectively, figures 2(c,d) and 4(a–d) indicate that, although the particle loading modifies the nature of the primary instability mode beyond a critical value of $\phi \approx 0.1$ , the subsequent pattern transitions from the RIB/SVF state, SVF/RIB $\leftrightarrow$ TVF $\leftrightarrow$ WTV, remain qualitatively similar to the particle-free ( $\phi =0$ ) case as in figure 2(a,b). We conclude that for small values of counter-rotation ratio $\varOmega \geqslant -0.35$ , the primary bifurcation from the CCF results in non-axisymmetric travelling waves (RIB/SVF), via a Hopf bifurcation, for semi-dilute suspensions ( $\phi \geqslant 0.1$ ), however, the first bifurcation of CCF $\to$ TVF, via a pitchfork bifurcation, holds for dilute suspensions ( $\phi \leqslant 0.05$ ).

3.1.1. Relative periodic orbits (MWV) and quasi-periodic states (DMWV)

The WTV states are, in fact, rotating waves (Gollub & Swinney Reference Gollub and Swinney1975; Andereck et al. Reference Andereck, Liu and Swinney1986; Chossat & Iooss Reference Chossat and Iooss1994) that rotate along the azimuthal direction with a finite speed ( $v_\theta = f_w\lambda _\theta$ , where $\lambda _\theta$ is the azimuthal wavelength of wavy vortices); they do not represent truly periodic solutions but are categorised as ‘relative equilibria’ (Krupa Reference Krupa1990; Cvitanović et al. Reference Cvitanović, Artuso, Mainieri, Tanner and Vattay2015; Feldmann et al. Reference Feldmann, Borrero-Echeverry, Burin, Avila and Avila2023). Referring to figures 2 and 4, with increasing Reynolds number, the WTV gives birth to MWV (Gorman, Swinney & Rand Reference Gorman, Swinney and Rand1981; Gorman & Swinney Reference Gorman and Swinney1982; Coughlin & Marcus Reference Coughlin and Marcus1992; Takeda Reference Takeda1999) that is characterised by a modulation (incommensurate) frequency ( $f_*$ , QP $_1$ ) of its dominant wavy vortex frequency $f_w$ (such that the frequency ratio $f_w/f_*$ is an irrational number). In dynamical systems theory (Cvitanović et al. Reference Cvitanović, Artuso, Mainieri, Tanner and Vattay2015), the MWV states are classified as ‘relative periodic orbits’ (Wulff et al. Reference Wulff, Lamb and Melbourne2001) since they represent periodic orbits when viewed from a reference frame moving with the speed of the rotating WTV mode. With further increase in ${\textit{Re}}_s$ , the MWV gives birth to a state, having a secondary modulation frequency ( $f_2$ , QP $_2$ ). The latter state with two incommensurate frequencies (QP $_1$ and QP $_2$ ; $f_*/f_2 \neq {\mathcal N}$ ) represents a truly ‘quasi-periodic’ state (Andereck et al. Reference Andereck, Liu and Swinney1986; Coughlin et al. Reference Coughlin, Markus, Tagg and Swinnney1991; Chossat & Iooss Reference Chossat and Iooss1994; Altmeyer et al. Reference Altmeyer, Do, Marques and Lopez2012; Cvitanović et al. Reference Cvitanović, Artuso, Mainieri, Tanner and Vattay2015; Feldmann et al. Reference Feldmann, Borrero-Echeverry, Burin, Avila and Avila2023).

Figure 5.

Snapshot and power spectra, depicting WTV $\to$ MWV $\to$ DMWV transition, with increasing ${\textit{Re}}_s$ over $467.66\leqslant {\textit{Re}}_s\leqslant 678.43$ for particle-free ( $\phi =0$ ) TCF at $\varOmega =-0.35$ . Markers/symbols in each spectra carry the same meaning as in figure 2(b,d).

The spectral signatures of the above transition process towards a quasi-periodic state (characterised by two incommensurate frequencies) can be ascertained from figure 5 that displays (i) the snapshots (upper panels) and (ii) the power spectra (lower panels) at different values of ${\textit{Re}}_s$ for the particle-free ( $\phi =0$ ) TCF; other parameters are as in figure 2(a,b). While figure 5(a) represents a WTV state, with its wavy vortex frequency ( $f_w$ ) being marked by a filled-blue circle, the remaining panels figure 5(b–f) contain one/two modulation frequencies of the underlying wavy mode $f_w$ . The primary and secondary modulation/incommensurate frequencies are marked by stars ( $f_*$ ) and filled-green triangles ( $f_2$ ), respectively, in figure 5(b–d). It is seen that the energy content of the $f_*$ -mode increases with increasing ${\textit{Re}}_s$ and can exceed that of $f_w$ , see figure 5(b,c,d). Clearly, each power spectrum in figure 5(b,c,d) represents a relative periodic orbit (Cvitanović et al. Reference Cvitanović, Artuso, Mainieri, Tanner and Vattay2015), with $f_*$ being the modulation frequency of the wavy vortex mode, and hence the resulting state is called MWV (Gorman & Swinney Reference Gorman and Swinney1982; Coughlin & Marcus Reference Coughlin and Marcus1992; Takeda Reference Takeda1999). A second modulation frequency $f_2$ appears in the PSD with further increasing ${\textit{Re}}_s$ , see figure 5(e, f), and the energy contents of $f_*$ - and $f_2$ -modes increase and decrease, respectively, in the same limit. These quasi-periodic states, having two incommensurate frequencies, will henceforth be referred to as ‘doubly’ modulated wavy vortex (DMWV) in the present paper.

The spectral data, such as those from figure 5(a–f), have been used to generate the frequency maps in figures 2(b) and 2(d), marking the phase-boundary between MWV and DMWV states. Note in figure 2 that the two-frequency quasi-periodic state persists over a large range of ${\textit{Re}}_s$ at $\phi =0$ (figure 2 b) and $\phi =0.15$ (figure 2 d). During a successive US and DS run at the highest particle loading of $\phi =0.3$ (figure 4 a,b), the quasi-periodic DMWV state was identified only during the DS run, see figure 4(d) at ${\textit{Re}}_s\sim 350$ .

3.1.2. Phase diagram of patterns up to turbulence

For shear Reynolds numbers beyond the MWV/DMWV states, the TCF admits chaotic wavy vortex (CWV), TWV and TTV states that are summarised in figure 6 as phase diagrams on (a) ( $\phi$ - ${\textit{Re}}_s(\phi )$ ) and (b) ( $\phi$ - ${\textit{Re}}_s(0)$ ) planes at $\varOmega = -0.35$ . It is seen that the RIB appears as the primary bifurcating state for $0.1 \leqslant \phi \leqslant 0.2$ during the US run only and ceases to exist during the DS run as confirmed at $\phi = 0.15$ , indicating a hysteretic nature of the primary instability. Overall, the primary bifurcation in suspension TCF leads to a non-axisymmetric propagating mode, CCF $\leftrightarrow$ RIB/SVF, if the particle loading is $\phi \geqslant 0.1$ and this bifurcation appears to be hysteretic. The secondary and tertiary bifurcating states are (i) a stationary TVF and (ii) a propagating WTV, respectively, with latter giving birth to a MWV state, characterised by a single incommensurate frequency, before quasi-periodic motion sets in.

Figure 6(a) illustrates that the overall effect of particle loading on various state transitions up to the TTV state is to decrease the onset values of ${\textit{Re}}_s(\phi )$ at which the respective bifurcations take place. This confirms the destabilising role of particles at the onset of both prechaotic (CCF $\to$ RIB/SVF $\to$ TVF $\to$ WTV $\to$ MWV $\to$ DMWV) and postchaotic (DMWV $\to$ CWV $\to$ TWV $\to$ TTV) flow states at $\varOmega = -0.35$ . The transition from the MWV/DMWV states to a chaotic state (CWV) is discussed in § 3.2 and that from CWV to turbulent states (TWV and TTV) in § 3.3.

Figure 6.

Phase diagrams of patterns on (a) $(\phi , {\textit{Re}}_s(\phi ))$ and (b) $(\phi , {\textit{Re}}_s(0))$ planes at $\varOmega = -0.35$ . The empty and filled markers represent the data for US and DS runs, respectively. The hatched regions (horizontal and vertical lines) present the parameter space where VSM is observed within the CWV (horizontal lines) and TWV (vertical lines with yellow shading) states. The cross-hatched orange-shaded region contains modes with two incommensurate frequencies, representing ‘quasi-periodic’ state of DMWV. The lower-order states are defined in figure 2; other acronyms refer to TWV and TTV states.

3.2. Transition to chaos and the VSM phenomenon

The development of a chaotic flow state at $\varOmega = -0.35$ , beyond MWV/DMWV, involves complex spatiotemporal dynamics as we demonstrate here. A new phenomenon of the splitting and merging of vortices in the background of a chaotic state is identified which occurs over a large range of ( ${\textit{Re}}_s, \phi$ ) as marked by the hatched regions in figure 6. The dynamical signatures of this new transition, the characteristics of the underlying chaos and the role of particles on them are discussed in §§ 3.2.1–3.2.3.

Figure 7.

Transition from quasi-periodic to chaotic states in particle-free ( $\phi =0$ ) TCF during US runs at $\varOmega =-0.35$ . Each subpanel consists of a space–time plot (top panel) and its power spectra (bottom panel), with parameter values of (a,b) ${\textit{Re}}_s = 653.63$ (DMWV, quasi-periodic), (c,d) ${\textit{Re}}_s = 728.0$ (VSM + CWV, chaotic) and (e, f) ${\textit{Re}}_s = 930.1$ (VSM + CWV, chaotic). The red down and up triangles in bottom panels mark the frequencies of the inner ( $f_{\boldsymbol{\nabla }}$ ) and outer ( $f_{\Delta }$ ) cylinders, respectively, and their harmonics. The filled-blue circle ( $f_w$ ) is the frequency of the wavy vortex mode, while the blue cross in panel $b$ denotes its harmonic; the orange star ( $f_*$ , i.e. QP $_1$ ), the filled-green triangle ( $f_2$ , i.e. QP $_2$ ) and the filled-magenta square ( $f_3$ , i.e. QP $_3$ ) in (b,d) denote primary, secondary and tertiary modulation frequencies, respectively, of the wavy mode $f_w$ .

Figure 8.

Snapshots depicting the transition among different numbers of vortices (rolls) during VSM events for a particle-free ( $\phi =0$ ) Newtonian fluid at ${\textit{Re}}_s = 728.0$ during the US run at $\varOmega = -0.35$ . The brighter and darker bands in each image represent ‘outflow’ (i.e. the radial flow is directed towards the outer cylinder) and ‘inflow’ (i.e. the radial flow is directed towards the inner cylinder) jets, respectively, and the centreline of each dark/bright band approximately marks the boundary between two counter-rotating wavy-Taylor vortices. For example, the band numbers ( $1, 3, 5$ ) in ( $a$ ) are accompanied by six counter-rotating vortices/rolls and hence ( $a$ ) represents a six-rolls state. A supplementary movie 1 is available at https://doi.org/10.1017/jfm.2026.12184.

3.2.1. VSM in particle-free TCF and the spectral characterisation of VSM + CWV state

For the particle-free ( $\phi =0$ ) TCF, the space–time plot for a DMWV state at ${\textit{Re}}_s = 653.6$ is shown in figure 7(a) and its power spectrum in figure 7(b) reveals the presence of three frequency peaks $f_w$ , $f_*$ (QP $_1$ ) and $f_2$ (QP $_2$ ), with the later two representing the primary and secondary, respectively, modulation/incommensurate frequencies of the wavy mode $f_w$ . Therefore, according to the definition in § 3.1.1, figure 7(a,b) represents a truly quasi-periodic state. Being in a quasi-periodic (QP $_1$ +QP $_2$ ) state implies that the flow is close a chaotic state that ideally requires the presence of three (QP $_1$ , QP $_2$ and QP $_3$ ) incommensurate frequencies (Ruelle & Takens Reference Ruelle and Takens1971), although the tertiary modulation frequency (i.e. a 3-tori solution (Altmeyer et al. Reference Altmeyer, Do, Marques and Lopez2012)) is likely unstable and, hence, difficult to realise in experiments.

With further increasing ${\textit{Re}}_s$ , we find the first evidence of chaos at ${\textit{Re}}_s = 728.0$ , see figure 7(c,d). Indeed, the power spectrum in figure 7(d) contains three incommensurate frequencies, marked by $f_*$ (QP $_1$ ), $f_2$ (QP $_2$ ) and $f_3$ (QP $_3$ ), in addition to the wavy vortex frequency at $f_w$ (marked by a filled-blue circle). The corresponding space–time plot in figure 7(c) reveals that this chaotic state is associated with repeated splitting and merging of vortices/rolls where the roll adjacent to the bottom boundary first splits and then the roll-merging takes place around the mid-height of the TC cell: we designate this state as ‘VSM + CWV’. Interestingly, the tail of the spectrum beyond $f_w$ in figure 7(d) decays gradually with increasing $\omega$ . The latter signature makes the PSD of ‘VSM + CWV’ different from the CWV state (Takeda Reference Takeda1999; Dutcher & Muller Reference Dutcher and Muller2009); the spectral characteristics of the pure CWV state are discussed in § 3.3.

The dynamics of VSM at ${\textit{Re}}_s = 728.0$ , involving transitions among different numbers of vortices over time, is depicted in figure 8(a–f) via snapshots. In each panel, the brighter and darker bands denote ‘outflow’ (i.e. the radial flow is directed towards the outer cylinder) and ‘inflow’ (i.e. the radial flow is directed towards the inner cylinder) jets, respectively, with each band representing a pair of counter-rotating vortices/rolls; for example, the band numbers ( $1, 3, 5$ ) in panel ( $a$ ) are accompanied by six counter-rotating vortices/rolls and, hence, figure 8(a) represents a six-rolls state. While the top panels in figure 8 illustrates the roll-splitting event at three instants, the bottom panels in figure 8 portrays the roll-merging event at three instants following the roll-splitting event. From figure 8(a–c), we find that during the roll-splitting event the roll adjacent to the bottom boundary of a six-rolls state (figure 8 a) splits in two successions to produce an ‘abnormal’ (Benjamin Reference Benjamin1978a , Reference Benjaminb ) seven-rolls state (figure 8 b) and a eight-rolls state (figure 8 c), respectively. A pair of adjacent vortices of the eight-rolls state, located at the mid-height of the TC cell, then merges together to form an interim seven-rolls state (figure 8 d) which eventually transitions back to a six-rolls state (figure 8 e, f) through another merging event. Note that the number of rolls remains unchanged before and after the VSM event (figure 8 a, f). The disappearance of rolls around the mid-height of the TC cell, following a roll-merging event, results in a ‘pitchfork’ shape in the space–time plot (see figure 7 c,e) that allows us to identify the VSM events exclusively from the space–time diagrams.

While the VSM event is triggered by the splitting of bottom boundary roll in figure 7(c,d), we also observed VSM driven by the splitting of top boundary roll that occurs randomly at ${\textit{Re}}_s \geqslant 825$ . For example, figure 7(e, f) depicts the dynamics of VSM at ${\textit{Re}}_s = 930.1$ that involves the splitting of top and bottom boundary rolls alternately. Note that the frequency of VSM events increases with increasing ${\textit{Re}}_s$ which is confirmed by comparing the space–time plots in figures 7(c) and 7(e). The corresponding power-spectrum in figure 7(f) is found to be devoid of any modulation frequency, with a broader, shallower peak of the underlying wavy mode compared with the chaotic state at ${\textit{Re}}_s = 728.0$ (figure 7 d). The increased frequency of VSM events at ${\textit{Re}}_s=930.1$ is likely responsible for the disappearance of quasi-periodic modulations of the underlying CWV in figure 7(f). The tails of both PSDs in figure 7(d, f) show a gradual decay with increasing frequency at $f\gt f_w$ : this is the main spectral characteristic feature of the VSM + CWV state.

Figure 9.

Same as figure 7, but for $\phi = 0.15$ , depicting the transition from MWV to DMWV to chaotic states. The parameter values are (a,b) ${\textit{Re}}_s(\phi ) = 449.5$ (MWV, with its modulation frequency at $f_*$ ), (c,d) ${\textit{Re}}_s(\phi ) = 584.1$ (quasi-periodic DMWV, with two incommensurate frequencies at $f_*$ (QP $_1$ ) and $f_2$ (QP $_2$ )) and (e, f) ${\textit{Re}}_s(\phi ) = 606.5$ (VSM + CWV, chaotic). Different symbols in (b,d, f) carry the same meaning as those in figure 7.

Figure 10.

VSM events for a suspension with $\phi =0.15$ at ${\textit{Re}}_s(\phi ) = 606.5$ and $\varOmega = -0.35$ . In (a), the numbers ( $1, 3, 5$ ) and ( $2,4$ ) refer to outflow (i.e. the radial flow is directed towards the outer cylinder) and inflow (i.e. the radial flow is directed towards the inner cylinder) jets, respectively. Other details are same as in fgure 8.

3.2.2. Role of particles on VSM and another variant of CWV: CWV $_B$

Recall from the phase diagrams in figure 6 that the VSM events within the CWV state persists in suspension TCF up to a particle loading of $\phi \leqslant 0.2$ at $\varOmega = -0.35$ . The transition sequence from the MWV state to VSM + CWV is depicted in figure 9 for a suspension with $\phi =0.15$ . Figure 9(a,b) at ${\textit{Re}}_s(\phi ) = 449.5$ represents a MWV state for which the modulation frequency, marked by an orange star ( $f_*$ , i.e. QP $_1$ ), is visible from its power spectra in figure 9(b); at a higher value of ${\textit{Re}}_s(\phi ) = 584.1$ in figure 9(c,d), we find a ‘two-frequency’ DMWV state for which the second incommensurate frequency is marked by the filled-green triangle ( $f_2$ , i.e. QP $_2$ ) in figure 9(d). The presence of primary ( $f_*$ ) and secondary ( $f_2$ ) modulation/incommensurate frequencies in figure 9(d) signals that the flow is close to a chaotic state.

The first appearance of VSM at $\phi = 0.15$ is found at ${\textit{Re}}_s(\phi ) \approx 606.5$ , see figure 9(e, f). Two VSM events can be identified from the ‘pitchfork shape’ regions present in the space–time plot in figure 9(e), while its power spectrum is shown in figure 9(f). The latter contains a broad shallow peak along with a gradually decaying tail at $f\gt f_w$ – these spectral features are identical to those of the particle-free case in figure 7( $f$ ), and therefore figure 9(e, f) indeed represents a chaotic state of VSM + CWV. The dynamics of VSM processes at $\phi = 0.15$ can be ascertained from figure 10 at ${\textit{Re}}_s(\phi ) = 606.5$ : the dynamical events, involving transitions among different numbers of vortices over time, are found to be similar to that at $\phi = 0$ (figure 8). With further increasing ${\textit{Re}}_s(\phi )$ , we find that the frequency of VSM increases, similar to the particle-free case, and the corresponding power spectra (not shown) is characterised by a broader and shallower peak against the noisier background with a decaying tail.

Figure 11.

Same as figure 9, but for $\phi = 0.2$ , depicting the transition from CWV $_B$ to VSM + CWV. The parameter values are (a,b) ${\textit{Re}}_s(\phi ) = 518.42$ (CWV $_B$ ), (c,d) ${\textit{Re}}_s(\phi ) = 563.2$ (CWV $_B$ ) and (e, f) ${\textit{Re}}_s(\phi ) = 633.15$ (VSM + CWV). Symbols, marking different frequency peaks in panels (b,d,f), carry the same meaning as those in figure 7.

Recall from figure 6 that the VSM occurs after the flow transitions to a chaotic state at $\phi =0.2$ and was not observed at $\phi = 0.3$ within the explored range of ${\textit{Re}}_s(\phi )$ (with $f_{\textit{max}} = 50$ Hz). To check if there are differences between two chaotic states with and without VSM events, we show the space–time plot and its power spectrum in figures 11(a,b), 11(c,d) and 11(e, f) that refer to ${\textit{Re}}_s(\phi ) = 518.42$ (without VSM), $563.2$ (without VSM) and $633.15$ (with VSM), respectively, at $\phi =0.2$ . The power spectra in figures 11(b,d) contain a weak, broad, low-frequency mode, marked by $B$ in each panel, whereas the power-spectra in figure 11(f) looks similar to that for the VSM + CWV state at $\phi \leqslant 0.15$ (figures 7 f and 9 f).

The appearance of a low-intensity broad peak seems to be a robust feature in present experiments of suspension TCF with $\phi \geqslant 0.2$ ; for example, it persists at a higher particle loading of $\phi =0.3$ too, see figures 12(a,b) and 12(c,d) that represent US and DS runs, respectively, at the same value of ${\textit{Re}}_s(\phi )\approx 450$ . The power spectra in figures 12(b) and 12(d) are characterised by the presence of a low-frequency broad mode $B$ . This spectral signature is similar to the chaotic vortices discovered by Gollub & Swinney (Reference Gollub and Swinney1975) and Fenstermacher et al. (Reference Fenstermacher, Swinney and Gollub1979) in the context of the particle-free TCF with pure inner-cylinder rotation ( $\varOmega =0$ ). Therefore, following Gollub & Swinney (Reference Gollub and Swinney1975) and Fenstermacher et al. (Reference Fenstermacher, Swinney and Gollub1979), we mark the latter state as ‘CWV $_B$ ’ which is spectrally distinct from the CWV state induced by the VSM (i.e. VSM + CWV). Note that the transition CWV $_B$ $\rightarrow$ VSM + CWV at $\phi = 0.2$ takes place with a gradual disappearance of the low-frequency $B$ mode over $518.0\leqslant {\textit{Re}}_s(\phi ) \lt 565.4$ beyond which the VSM emerges, leading to a ‘VSM + CWV’ state (such as in figure 11(f)).

The low-intensity broad peak ‘ $B$ ’ in the CWV $_B$ state suggests that the associated chaos is deterministic and is low-dimensional in nature (Brandstäter et al. Reference Brandstäter, Swift, Swinney, Wolf, Farmer, Jen and Crutchfield1983; Brandstäter & Swinney Reference Brandstäter and Swinney1987). While both CWV and CWV $_B$ can be classified as temporally chaotic states that maintain spatial coherence, the spectrum of the VSM + CWV state fits the description of spatiotemporal chaos (Cross & Hohenberg Reference Cross and Hohenberg1994; Feldmann et al. Reference Feldmann, Borrero-Echeverry, Burin, Avila and Avila2023).

Figure 12.

(a,c) Snapshots (left panel) and space–time diagrams (right panel) and (b,d) power spectra, depicting spectral characteristics of CWV $_B$ during (a,b) US and (c,d) DS runs at $\phi = 0.3$ and $\varOmega =-0.35$ ; other parameters are (a,b) ${\textit{Re}}_s(\phi ) = 450.5$ and (c,d) ${\textit{Re}}_s(\phi ) = 449.7$ . The displayed time-window on the right of (a,c) refers to a time-period of $1s \approx 7.1\tau _{vis}(\phi )$ . Symbols, marking different frequency peaks in (b,d), carry the same meaning as those in figure 7.

Figure 13.

(a) Locations of VSM events as functions of ${\textit{Re}}_s$ near the top plate (upper vertical lines) and lower-plate (lower lines) during an up-sweep-run at $\phi =0$ . (b) The probability density function (PDF) of the normalised interevent time (IET) $\Delta t^{\textit{VSM}} = \Delta t/t_{sh}$ between successive VSM events, where $t_{sh} = \tau _{vis}/{\textit{Re}}_s(\phi )$ is the shear time scale. In the main panel, $P$ and $S$ denote the primary and secondary peaks of the PDF, respectively; the inset in (b) is an enlarged view of the secondary peak S.

3.2.3. Characterisation of VSM events and the nature of VSM-induced chaos

As clarified in § 3.2.1, the splitting of vortices was found to be initiated near the top and bottom end plates, followed by the merging of vortices around centre of the TC cell; these VSM events have been identified from the continuous space–time diagram captured through the DSLR camera. The successive temporal locations (at different values of $t\to {\textit{Re}}_s$ ) of VSM events during an US run at $\phi =0$ are marked by the vertical lines in figure 13(a); the upper and lower lines, respectively, refer to VSM events originating from the top and bottom plates.

From the space–time diagrams such as in figure 13(a), we estimated the interevent time (IET; i.e. the time between two successive VSM events) $\Delta t$ . Note that the time window over which one VSM event takes place is very small compared with the IET, $\Delta t$ , between two consecutive VSM events. The PDF of the normalised IET,

(3.3) \begin{equation} \Delta t^{\textit{VSM}} = \frac {\Delta t}{t_{sh}}, \quad \mbox{with} \quad t_{sh} = \frac {\tau _{vis}}{{\textit{Re}}_s(\phi )} \end{equation}

being the shear time scale, is displayed in figure 13(b) for $\phi \in (0,0.2)$ . It is seen that the PDF of IET is skew-symmetric and multimodal that features a primary or dominant peak (P) at $\Delta t_P^{\textit{VSM}} \approx 4000$ for all $\phi$ during US and DS runs. Figure 13(b) also reveals the presence of a secondary peak (S, see the enlarged view in the inset) at $\Delta t_S^{\textit{VSM}}\in (20\,000, 30\,000)$ whose time scale is an order of magnitude larger than the primary one for all $\phi$ . The multimodal nature of the PDF suggests that the VSM is a multiscale phenomenon, with the most VSM events occurring around a primary (fast) time scale $\Delta t_P^{\textit{VSM}}$ , while fewer events take place over a secondary (slower) time scale $\Delta t_S^{\textit{VSM}}$ .

Looking at the effect of particle loading on IET in figure 13(b), we find that the intensity of $\Delta t_P^{\textit{VSM}}$ increases with increasing particle loading upto $\phi = 0.15$ , implying the occurrence of more VSM events over the primary time scale; however, there is a decline in the intensity of the P peak at $\phi = 0.2$ , indicating that a further higher particle loading beyond $\phi \gt 0.15$ reduces the number of VSM events. Interestingly, the location of the P peak ( $\Delta t_P$ ) is almost independent of the particle loading. On the other hand, the overall effect of particle loading on the secondary time scale $\Delta t_S^{\textit{VSM}}$ (see the inset of figure 13 b) is to decrease in its intensity (implying fewer events), with a shift towards higher values of $\Delta t_S$ . The latter findings imply that the secondary VSM events occurs over a longer time scale and becomes more infrequent with increasing particle loading $\phi$ .

Figure 14.

Return maps (3.4) of IET (3.3) at (a) $\phi = 0$ , (b) $0.15$ and (d) $0.2$ , respectively, during the US run; (c) return map of IET at $\phi = 0.15$ during the DS run.

The return maps for IETs are shown in figure 14(a–d) at (a) $\phi = 0$ , (b,d) $\phi = 0.15$ , and (c) $\phi = 0.2$ during (a,b,d) US and (c) DS runs. Note that each marker in figure 14(a–d) stands for a pair

(3.4) \begin{equation} \left (\Delta t_n^{\textit{VSM}}, \Delta t_{n+1}^{\textit{VSM}}\right )\!, \end{equation}

with $\Delta t_n^{\textit{VSM}}$ and $\Delta t_{n+1}^{\textit{VSM}}$ being the $n$ th and $(n+1)$ th IETs, respectively. While a single cluster in figure 14(c) portrays the existence of the primary time scale $\Delta t_P^{\textit{VSM}}$ (and the secondary peak S is absent during the DS run at $\phi = 0.15$ ), the multiple clusters in figure 14(a,b,d) reveal the dominance of $\Delta t_P^{\textit{VSM}}$ over $\Delta t_S^{\textit{VSM}}$ during the US run. It is also evident from figure 14(a–d) that the VSM events are temporally uncorrelated, implying that the occurrence of a VSM event is independent of the previous event, i.e. IETs follow a Markov process. Furthermore, the shape of the cluster around $\Delta t_P^{\textit{VSM}}$ in figure 14(a–d) and the corresponding PDF in figure 13(b) resembles that of a skewed Gaussian. Based on these findings, we conclude that the chaos induced by VSM (i.e. the VSM + CWV state) is stochastic rather than deterministic (Shaw Reference Shaw1984).

Figure 15.

(a) Space– ${\textit{Re}}_s$ diagram, (b) normalised frequency map and (c) variation of ${\textit{Nu}}_{\omega }$ with ${\textit{Re}}_s$ illustrating the transition to TTV following CWV during the US run at $(\phi , \varOmega ) = (0, -0.35)$ . While the filled black diamonds in (b) represent the dominant frequency ( $f_{\textit{TWV}}$ ) of the TWV state, the filled blue circles represent the frequency of the VSM + CWV state. The grey patches in (b,c) within the ‘VSM + TWV’ region represent VSM-events, while the intervening white patches denote pure TWV modes; see § 3.3 for a detailed characterisation of the VSM + TWV state. The red patch in each panel denotes the range of ${\textit{Re}}_s$ over which the frequency of the TWV mode gradually diminishes with increasing ${\textit{Re}}_s$ (see Appendix B). The dashed lines in each panel mark the approximate boundary between two successive states.

3.3. Postchaotic flow states and the transition to turbulence

The spectral development towards turbulence (Coles Reference Coles1965; Grossmann et al. Reference Grossmann, Lohse and Sun2016) is well documented for the case of pure inner cylinder rotation ( $\varOmega = 0$ ) in the context of particle-free Newtonian TCF (Gollub & Swinney Reference Gollub and Swinney1975; Andereck et al. Reference Andereck, Liu and Swinney1986; Takeda Reference Takeda1999; Dutcher & Muller Reference Dutcher and Muller2009) and suspension TCF (Dash et al. Reference Dash, Anantharaman and Poelma2020). At $\varOmega =0$ , the turbulent state, referred to as TTV (Andereck et al. Reference Andereck, Liu and Swinney1986), comprises Taylor vortices, having small-scale spatiotemporal features in the absence of any large-scale wavy motion.

For present experiments with particle-free Newtonian fluid, the development of turbulent states from the VSM + CWV state is illustrated in figure 15 during an US run. It is seen that the VSM + CWV state exists at ${\textit{Re}}_s \lt 1064$ beyond which the VSM events become irregular due to the emergence of TWV. The intermittent appearance of TWV over $1064 \leqslant {\textit{Re}}_s \lt 1132.2$ can be discerned from the space– ${\textit{Re}}_s$ diagram and ${\textit{Nu}}_{\omega }{-}{\textit{Re}}_s$ plot displayed in figures 15(a) and 15(c), respectively. The effective Nusselt number ${\textit{Nu}}_\omega$ is defined via (Dubrulle et al. Reference Dubrulle, Dauchot, Daviaud, Longaretti, Richard and Zahn2005; Eckhardt et al. Reference Eckhardt, Grossmann and Lohse2007):

(3.5) \begin{equation} \textit{Nu}_{\omega } = \frac {G}{G_{\textit{lam}}}, \end{equation}

where $G = \rho {\mathcal T}/(2\pi h \mu (\phi )^2)$ is the dimensionless torque, with $\mathcal T$ being the dimensional measured torque and $G_{\textit{lam}} = \eta {\textit{Re}}_s(\phi )/(1-\eta )^2$ is the dimensionless laminar torque for the flow between two infinitely long cylinders. Figure 15(c) indicates that ${\textit{Nu}}_\omega$ varies relatively smoothly with ${\textit{Re}}_s$ in the TWV state, while the same in the VSM + TWV state contains fluctuations. With increasing ${\textit{Re}}_s$ , the fluctuations in ${\textit{Nu}}_{\omega }$ finally disappear at ${\textit{Re}}_s \geqslant 1132.2$ and the pure TWV state is found at $1132.2 \leqslant {\textit{Re}}_s \leqslant 1230.1$ , see figure 15(c). Therefore, the fluctuations in ${\textit{Nu}}_{\omega }$ in figure 15(c) can be used as a metric to demarcate the boundary (i.e. threshold ${\textit{Re}}_s$ ) between VSM + TWV and TWV states.

Figure 16.

Snapshots (top left panels), space–time diagrams (top right panels) and power spectra (bottom panels) for different ${\textit{Re}}_s$ at $(\phi , \varOmega ) = (0, -0.35)$ depicting the transition to TTV from TWV during US run. The parameter values are (a,b) ${\textit{Re}}_s = 1168.15$ (TWV), (c,d) ${\textit{Re}}_s = 1199.15$ (TWV), (e, f) ${\textit{Re}}_s = 1230.15$ (approximate boundary of TWV $\to$ TTV) and (g,h) ${\textit{Re}}_s = 1354.1$ (TTV). The filled-black diamond in (b,d) marks the dominant frequency of the TWV state; the red down and up triangles in (b,d, f,h) mark the frequencies of the inner ( $f_{\boldsymbol{\nabla }}$ ) and outer ( $f_{\Delta }$ ) cylinders, respectively, and their harmonics. The displayed time window in each of (a,c,e,g) refers to $2s\approx 3.5\tau _{vis}(\phi )$ .

The characteristic features of TWV at ${\textit{Re}}_s = 1168.15$ are shown in figure 16(a,b) through a snapshot and a space–time plot (figure 16 a) and the corresponding power spectrum (figure 16 b). While the snapshot in figure 16(a) shows the emergence of small-scale flow structures, its power spectrum in figure 16(b) contains a sharp peak at $\omega _{\textit{TWV}}/\omega _i \approx 1.29$ (that represents the propagating wavy vortices in the background of turbulence) beyond which the spectra has a ‘flat’, broad tail. Note that the flat, noisy tail of TWV spectra in figure 16(b) distinguishes itself from the gradually ‘decaying’ noisy tail of the VSM + CWV spectra (at $f\gt f_w$ ) such as in figure 7(d, f).

While the power spectra of a pure TWV state, having a sharp peak at $f_{\textit{TWV}}$ , is shown in figure 16(b), its transition to the TTV state for the particle-free case can be ascertained from figure 16(d, f,h). It is seen that the intensity of $f_{\textit{TWV}}$ gradually decreases from figure 16(b) with increasing ${\textit{Re}}_s$ , resulting in a low-intensity peak in figure 16(d): this frequency peak becomes a ‘shallower and wider’ peak at ${\textit{Re}}_s=1230.15$ (see figure 16 f) that eventually gets buried inside the featureless, broad noisy spectrum at ${\textit{Re}}_s \approx 1294.6$ , signifying the emergence of the TTV state. The turbulent nature of TTV can be ascertained from increased small-scale structures in the snapshot in figure 16(g) at ${\textit{Re}}_s = 1354.1$ , and its power spectrum in figure 16(h) reveals its spectral characteristics: a flat, featureless, noisy tail for the TTV state. It is clear from figures 16(d) and 16(f) that the TWV $\rightarrow$ TTV transition occurs over a range of ${\textit{Re}}_s$ over which the height of $f_{\textit{TWV}}$ (i) gradually decreases, (ii) becomes wider (such that a single peak is not identifiable such as in figure 16(f)) and (iii) eventually gets buried with the noisy tail at ${\textit{Re}}_s \approx 1260$ : this is discussed further in Appendix B. The above range of ${\textit{Re}}_s \in (1200, 1260)$ is marked by a red patch in figure 15(a–c), and the TWV $\to$ TTV transition is identified with the absence of a ‘sharp’ peak at ${\textit{Re}}_s\approx 1230$ (figure 15 f), marked by a dashed line within the red patch.

Figure 17.

(a,d) Space– ${\textit{Re}}_s$ diagrams, (b,e) normalised frequency maps and (c, f) ${\textit{Nu}}_{\omega }$ versus ${\textit{Re}}_s(\phi )$ depicting the transition to TTV during (a–c) US and (d–f) DS runs at $(\phi , \varOmega ) = (0.15, -0.35)$ . The stars and squares in (b,e) refer to primary (QP $_1$ ) and secondary (QP $_2$ ) modulation (incommensurate) frequencies, respectively; refer to frequency peaks marked by $f_*$ and $f_2$ in figure 18(f,h). Other symbols refer to those as in figure 15; the dashed lines in each panel mark the approximate boundary between two successive states. The presence of the first modulation frequency ( $f_*$ ) is a key spectral characteristic of the pure CWV state that distinguishes it from the (modulation-free) TWV state.

The analogue of figure 15(a–c) for suspensions with $\phi = 0.15$ is shown in figure 17(a–c). While the VSM + CWV $\rightarrow$ CWV $\rightarrow$ TWV transition is evident from the space– ${\textit{Re}}_s(\phi )$ diagram (figure 17 a) and ‘ ${\textit{Nu}}_{\omega }$ versus ${\textit{Re}}_s(\phi )$ ’ plot (figure 17 c), the TWV $\rightarrow$ TTV transition can be ascertained by the absence of $f_{\textit{TWV}}$ in the frequency map in figure 17(b). However, during the DS run (see figure 17 d–f), we find the reappearance of VSM events at around ${\textit{Re}}_s(\phi )\approx 850$ , indicated by fluctuations in ${\textit{Nu}}_\omega$ in figure 17(f); this results in a transition sequence of VSM + CWV $\leftarrow$ CWV $\leftarrow$ VSM + TWV $\leftarrow$ TWV $\leftarrow$ TTV. A snapshot, its space–time map over 2 s and the corresponding power spectrum are displayed in figure 18 for (a,b) a pure TWV state ( ${\textit{Re}}_s=906.02$ ), (c,d) a VSM + TWV state ( ${\textit{Re}}_s=849.95$ ) and (e, f) a pure CWV state ( ${\textit{Re}}_s=821.89$ ): they represent the DS run in figure 17(d–f). The vortex merging event is clearly evident in the space-time map of figure 18(c). The signature of the vortex spliting and merging events on the PSD of TWV can be ascertained from a comparison between figures 18(b) and 18(d) – while the former contains at sharp peak at $f_{\textit{TWV}}$ (marked by a black diamond), figure 18(d) contains a shallow, broad peak, with a gradually ‘decaying’ tail at $f\gt f_{\textit{TWV}}$ . The latter is, in fact, a key spectral signature of the VSM + CWV state (namely, figures 7 d, f, 9 d, f and 11 f) and, therefore, the interactions of the TWV mode with VSM events lead to a state (VSM + TWV) whose spectral signatures resemble that of the VSM + CWV state.

The spectrum of a TWV state (figure 18 $b$ ) differs from that of a pure CWV state (figure 18 $f$ ) on one crucial aspect: the latter contains at least one modulation frequency at $f_*$ (QP $_1$ , marked by a star symbol in panel figure 18(f)). This distinctive spectral feature is further demonstrated in figure 18(h) that represents a CWV state ( ${\textit{Re}}_s=858.9$ ) during the US run of figure 17(a–c). The latter spectrum also contains a second modulation (incommensurate) frequency (at $f_2\gt f_w$ , marked by a green triangle) which was not identified in its DS run (figure 17 e). Possible reasons for not able to resolve all incommensurate frequencies of quasi-periodic/chaotic states could be due to (i) experimental protocols and/or (ii) limitations of image analysis as discussed briefly in Appendix A.1. The spectral distinctions of the pure CWV state, having high- ( $f_2\gt f_w$ ) and low-frequency ( $f_*\lt f_w$ ) modulations, and its evolution with increasing ${\textit{Re}}_s$ are characterised in detail in Appendix C.

Figure 18.

(a,c,e) Snapshots and space–time plots and (b,d, f) the power-spectra for (a,b) a pure TWV state at ${\textit{Re}}_s=906.02$ , (c,d) a ‘VSM + TWV’ state at ${\textit{Re}}_s(\phi )=849.95$ and (e, f) a pure CWV state at ${\textit{Re}}_s=821.89$ during the ‘DS’ run. (g,h) Same as (e, f) for a pure CWV state at ${\textit{Re}}_s=858.9$ during the ‘US’ run. Other parameters are $\phi =0.15$ and $\varOmega =-0.35$ as in figure 17. The displayed time window on the right of in each panel (a,c,e,g) refers to $2s\approx 6.3\tau _{vis}(\phi )$ . The filled-black diamonds in (b,d) mark the azimuthal propagation frequency of the TWV state; in (f,h), the star and filled-magenta squares represent the primary ( $f_*$ ) and secondary ( $f_2$ ) modulation/incommensurate frequencies, respectively. In (b,d, f,h), the filled-blue circles refer to the frequency $f_w$ of the wavy vortex mode; the red down and up triangles mark the frequencies of the inner ( $f_{\boldsymbol{\nabla }}$ ) and outer ( $f_{\Delta }$ ) cylinders, respectively, and their harmonics.

On the whole, irrespective of the particle loading $\phi \leqslant 0.2$ , we find that the power spectrum of TTV is flat and featureless with a broad noisy tail, and the final transition to TTV is characterised by the disappearance of the dominant frequency $f_{\textit{TWV}}$ of the TWV state at $\varOmega =-0.35$ . The latter spectral characteristic is the same as that for the TWV $\leftrightarrow$ TTV transition in particle-free Newtonian TCF (Takeda Reference Takeda1999; Dutcher & Muller Reference Dutcher and Muller2009) with pure inner-cylinder rotation ( $\varOmega =0$ ).

Returning to the phase diagram in figure 6, the parameter space on ( $\phi - {\textit{Re}}_s(\phi )$ ) and ( $\phi - {\textit{Re}}_s(0)$ ) planes where the VSM $+$ TWV state exists is marked by the yellow patch regions with vertical lines. The final transition to turbulence (TTV) occurred from (i) the TWV state at $\phi =(0, 0.1, 0.15, 0.2)$ and (ii) the VSM + TWV state at $\phi = 0.05$ . However, at $\phi = 0.3$ , the final state was found to be a chaotic state (CWV $_B$ as in figure 6) within the maximum explored range of ${\textit{Re}}_s(\phi )$ (with $f_{max } = 50$ Hz) in our setup, but such higher-order transitions leading to TTV may also occur at $\phi = 0.3$ if the experiments are carried out over an extended range of ${\textit{Re}}_s$ . In any case, the difference in transition scenarios during US and DS runs in figure 17 indicates that the postchaotic transition to TWV is hysteretic, whereas the onset of TTV appears to be non-hysteretic.

Figure 19.

(a) Variation of the normalised propagation frequency ( $\omega _{\textit{TWV}}/\omega _{sh}$ ) of TWV at different particle loading and (b) its scaling; here, $\omega _{sh} = |\omega _i - \omega _o/\eta |$ is the effective rotation rate. Other parameters are $\phi =0.15$ and $\varOmega =-0.35$ .

3.4. Frequency of TWV mode and its scaling

The spectral data on the propagation frequency of the TWV state, a precursor to the TTV state, have been extracted from the power spectrum plots, such as in figure 16(b,d), for various $\phi$ at $\varOmega =-0.35$ and are plotted as a function of the suspension shear Reynolds number ${\textit{Re}}_s(\phi )$ in figure 19(a); these data have been normalised by the effective rotation rate $\omega _{sh} = |\omega _i - \omega _o/\eta |$ . It is seen from figure 19(a) that $\omega _{\textit{TWV}}/\omega _{sh}$ , for each $\phi$ , decreases with increasing ${\textit{Re}}_s(\phi )$ ; note that the data for $\phi =0.05$ (marked by red triangles) follow closely those for $\phi =0$ (blue triangles), however, the data for $\phi \geqslant 0.1$ show noticeable variations. The overall decrease of $\omega _{\textit{TWV}}/\omega _{sh}$ with increasing ${\textit{Re}}_s(\phi )$ can be attributed to the decrease in the number of azimuthal waves (and eventually degenerating into TTV) as found previously by Fenstermacher et al. (Reference Fenstermacher, Swinney and Gollub1979) for the particle-free case. It is interesting to note that the US and DS frequency data for $\phi =0.15$ (marked by magenta coloured up and down triangles, respectively) overlap nicely with each other, but show a discontinuous jump at ${\textit{Re}}_s(\phi )\approx 875$ ; similar jumps are also observed for the data at $\phi =0.05$ (red triangles) and $0.2$ (cyan triangles) in figure 19(a). A likely reason for such jumps in $\omega _{\textit{TWV}}/\omega _{sh}$ is an abrupt change in the number of azimuthal waves in TWV (although it is difficult to quantify this from images).

Figure 19(b) demonstrates that $\omega _{\textit{TWV}}/\omega _{sh}$ at different $\phi$ can be collapsed to obtain a scaling relation of the form,

(3.6) \begin{equation} \frac {\omega _{\textit{TWV}}}{\omega _{sh}} = \beta _0 {\textit{Re}}_{s}(0)^{-\beta _1} \mu _r(\phi )^{\beta _2} = \beta _0 {\textit{Re}}_s(\phi )^{-\beta _1} \mu _r(\phi )^{\beta _2 - \beta _1}, \end{equation}

where $(\beta _0, \beta _1, \beta _2) \approx (720.11, 0.967, 0.502)$ and $\mu _r(\phi )=\mu (\phi )/\mu (0)$ is relative viscosity of the suspension. The scaling (3.6) holds over a range of ‘pure-fluid’ shear Reynolds number ${\textit{Re}}_s(0) \in (1100, 1950)$ for $\phi \leqslant 0.2$ . For the pure-fluid case, the decay of $\omega _{TWV}$ can be approximated as ${\textit{Re}}_s^{-1}$ , however, there is no prior data to verify this scaling.