2.1. Suspension of neutrally buoyant spheres

A neutrally buoyant suspension is prepared with polymethyl methacrylate (PMMA, with refractive index $n_p=1.4873$; Sigma Aldrich) micro-spheres having mean diameter $d_p=(50\pm 10)\,\mathrm {\mu }$m and material density $\rho _p\approx 1190\,{\rm kg}\,{\rm m}^{-3}$ that are of slightly larger diameter and density than those used in previous works (Ramesh et al. Reference Ramesh, Bharadwaj and Alam2019; Ramesh & Alam Reference Ramesh and Alam2020). The solvent is a ternary mixture of water, glycerine and ammonium thiocyanate NH$_4$SCN ($21.89\,\%$, $44.34\,\%$ and $33.77\,\%$ by weight, respectively) having density $\rho \approx 1190.47\,{\rm kg}\,{\rm m}^{-3}$ and refractive index $n = 1.4842$.

The effective shear viscosity of the suspension, $\mu (\phi )$, was measured using a parallel-plate module in an Anton Paar MCR-302 rheometer (Ramesh et al. Reference Ramesh, Bharadwaj and Alam2019) over a range of particle volume fraction $\phi \leq 0.3$. The measured viscosity agrees closely with the Krieger–Dougherty relation (Krieger & Dougherty Reference Krieger and Dougherty1959; Guazzelli & Pouliquen Reference Guazzelli and Pouliquen2018)

(2.1)\begin{equation} \mu(\phi) = \mu_l \left(1- \frac{\phi}{\phi_m}\right)^{{-}2} = \mu(0)\,\mu_r(\phi), \end{equation}

where $\mu (0)\equiv \mu _l \approx 7.90\,{\rm mPa}\,{\rm s}$ (at $T = 22\,^{\circ }{\rm C}$) is the shear viscosity of the solvent, $\phi _m=0.585$ is the maximum volume fraction of particles close to jamming, and

(2.2)\begin{equation} \mu_r(\phi)= \frac{\mu(\phi)}{\mu(0)} \geq 1 \end{equation}

is the relative viscosity of suspension or the Krieger–Dougherty factor.

2.2. Twin-drive MCR-702 rheometer and the experimental procedure

Experiments are carried out in a TC cell mounted on a twin-drive MCR-702 rheometer (Anton Paar GmbH, Austria), which is equipped with two motors. While the outer acrylic cylinder (with radius $r_o=17.95$ mm) mounted on a custom-built shaft is connected to the lower motor, the inner cylinder (a cylindrical bob of radius $r_i=16$ mm, made of black anodized aluminium, identical to that in Ramesh et al. Reference Ramesh, Bharadwaj and Alam2019) is driven by the upper motor. A three-dimensional-printed circular acrylic lid is fitted tightly at the top of the outer cylinder, leaving an axial clearance of approximately $0.5$ mm at the top and bottom of the inner cylinder, and therefore both axial boundaries rotate with the outer cylinder. The annular gap between the two cylinders is $\delta =r_o-r_i=1.95$ mm, and the length of the inner cylinder is $h=16.5$ mm, resulting in $\eta =r_i/r_o=0.891$, $\kappa =\delta /r_i=0.122$ and $\varGamma =h/\delta =8.46$. These geometrical parameters can be contrasted with $(\eta,\kappa,\varGamma )_{ALS}=(0.883, 0.132, 30)$ of Andereck et al. (Reference Andereck, Liu and Swinney1986), to which we shall compare our phase diagram of patterns in the counter-rotating regime in § 3.3. Note that the ratio between the Couette gap and the particle diameter is $\delta /d_p\approx 39$, which is comparable to the $37.5$ of Ramesh et al. (Reference Ramesh, Bharadwaj and Alam2019).

2.3. Spectral analyses of scattered light intensity

To extract quantitative data like the frequencies and wavelengths of the patterns (Andereck et al. Reference Andereck, Liu and Swinney1986; Dutcher & Muller Reference Dutcher and Muller2009), the images are analysed using a one-dimensional fast Fourier transform (FFT) of the intensity profile of the scattered light as captured in video images. More specifically, a time series of the mean intensity $I(t;z_k)=\{I(t_j; z_k),\, j=1,2,\ldots, N\}$ – with each $I(t_j; z_k)$ being calculated over a square box of $30\times 30$ pixels around the axial location $z=z_k$ – is constructed from $N$ successive snapshots; note that $t_j=t_0+j\,\Delta t$, where $\Delta t$ is the time delay between two successive snapshots. The temporal power spectral density (PSD) of the resulting time series involves calculating the complex modulus

(2.3)\begin{equation} E\left(\frac{\omega}{\omega_i}; z_k\right) = |\hat{I}(\omega; z_k)|^2 = \left|\mbox{FFT}_{\omega}\{I(t; z_k)\}\right|^2 \end{equation}

as a function of frequency $\omega$. While presenting data on the frequency content of the scattered light, we will report the mean PSD

(2.4)\begin{equation} E_{av} = \frac{1}{2}\left(E\left(\frac{\omega}{\omega_i}; \frac{z_1}{\delta}=3.2\right) + E\left(\frac{\omega}{\omega_i}; \frac{z_2}{\delta}=5.4\right) \right), \end{equation}

which is defined as the average of the PSD values at two axial locations, $z_k/\delta =3.2$ and $5.4$ (i.e. above and below the mid-height $z/\delta \approx 4.23$ of the TC cell) for $N=300$ snapshots that cover a time window of $5$ s. The results do not change if the averaging (2.4) is done over three or more axial locations.

Similarly, considering the instantaneous light intensity $I(z(t))\equiv I(z;t)$ along an axial strip (of height $z=h_1\leq h$ along the length of the cylinders) of thickness $1$ pixel from each snapshot, and stacking such profiles from $N=300$ successive snapshots, yielded an intensity profile $I(z) = \{ I(z(t_0+j\,\Delta t)), \, j=1,2,\ldots,N\}$ over an extended spatial coordinate $z=(0,Nh_1)$. The spatial PSD of this composite signal $I(z)$,

(2.5)\begin{equation} E\left(k=\delta/\lambda\right) = \left|\mbox{FFT}_{k} \{I(z)\}\right|^2, \end{equation}

is calculated as a function of the normalized wavenumber $k=\delta /\lambda$ (i.e. the spatial frequency). Since we will be analysing cropped images (see figure 4 and the discussion in § 4.1.2) having axial length $z/\delta =h_1/\delta \approx 6.83$, the spatial FFT plots will contain a peak at $\delta /\lambda =1/6.83 \approx 0.146$ as well as at its integer multiples (see figures 9 and 17).

While (2.4) provides the frequency content of the signal that helps to identify the propagating or stationary nature of the underlying patterns, (2.5) yields the axial wavelength ($\lambda /\delta$) of patterns and its (long- and short-wave) modulations.

3.1. Shear Reynolds number and the rotation ratio

The experimental results are characterized in terms of the shear Reynolds number (Dubrulle et al. Reference Dubrulle, Dauchot, Daviaud, Longaretti, Richard and Zahn2005),

(3.1)\begin{equation} \mbox{Re}_{s}(\phi,\varOmega;\eta) \stackrel{def}{=} \frac{\rho |U_s|\delta}{\mu(\phi)} \equiv \frac{2}{1+\eta}\,|(1-\varOmega)|\,\mbox{Re}_i(\phi) \equiv \frac{2\eta}{1+\eta}\,\frac{|(1-\varOmega)|}{\varOmega}\,\mbox{Re}_o(\phi), \end{equation}

which is based on the shear velocity $U_s=(\omega _i -\omega _o) r_g^2/r_{av}$ and the suspension viscosity $\mu (\phi )$, with $r_{av}=(r_i+r_o)/2$ and $r_g=\sqrt {r_i r_o}$ referring to arithmetic and geometric mean radii, respectively. Note that $\mbox {Re}_s$ can be interpreted as the ratio between a viscous time scale ($\rho \delta ^2/\mu (\phi )$) and a shear time scale ($\delta /U_s$). The latter two expressions in (3.1) tie $\mbox {Re}_s$ to both inner and outer Reynolds numbers ($\mbox {Re}_i, \mbox {Re}_o$) in terms of the radius ratio $\eta =r_i/r_o$ and the rotation ratio

(3.2)\begin{equation} \varOmega \stackrel{def}{=} \frac{\omega_o}{\omega_i} \equiv \eta\,\frac{\mbox{Re}_o}{\mbox{Re}_i}. \end{equation}

Note that the ‘$\varOmega =\mbox {constant}$’ lines in figure 1(b) are drawn by using the present value of $\eta =0.891$ in (3.2). Equation (3.1) is directly related to Taylor number,

(3.3)\begin{equation} {Ta}(\phi) \stackrel{def}{=} \left(\frac{r_{av}}{r_g}\right)^4\delta^2 r_{av}^2 (\omega_i-\omega_o)^2\left(\frac{\rho}{\mu(\phi)}\right)^2 \equiv \frac{(1+\eta)^8}{256\eta^4}\, \mbox{Re}_s^2 , \end{equation}

via a geometric factor $(1+\eta )^8/256\eta ^4\equiv \sigma ^2$, where $\sigma$ is called the geometrical Prandtl number (Eckhardt, Grossmann & Lohse Reference Eckhardt, Grossmann and Lohse2007). Another control parameter (often used in astrophysical contexts for TCF) is the rotation number (Dubrulle et al. Reference Dubrulle, Dauchot, Daviaud, Longaretti, Richard and Zahn2005)

(3.4)\begin{equation} R_\varOmega \stackrel{def}{=} \frac{\omega_{rf}\delta}{U_s} = (1-\eta)\,\frac{\mbox{Re}_i + \mbox{Re}_o}{\mbox{Re}_i - \eta\,\mbox{Re}_o} = (1-\eta)\,\frac{1+\varOmega/\eta}{1-\varOmega}, \end{equation}

which can be used instead of $\varOmega$; note that $\omega _{rf}=(\omega _i r_i + \omega _o r_o)/r_{av}$ is the reference angular velocity. It is clear that $R_\varOmega$ is the ratio between a shear time scale ($\delta /U_s$) and a rotation time scale ($1/\omega _{rf}$). Collectively, the description of the counter-rotating TCF in terms of $(\mbox {Re}_s, R_\varOmega )$, instead of ($\mbox {Re}_i, \mbox {Re}_o$), helps to isolate the shearing and centrifugal effects as noted by Dubrulle et al. (Reference Dubrulle, Dauchot, Daviaud, Longaretti, Richard and Zahn2005).

Since we conducted experiments at fixed values of $\varOmega$, the combination ($\mbox {Re}_s(\phi ), \varOmega$) is a natural choice to characterize our results for given $\phi$, instead of presenting results in terms of $(\mbox {Re}_i, \mbox {Re}_o$) or ($\mbox {Re}_s, R_\varOmega$). For specified values of $(\phi, \varOmega )$, three to eight sets (up-sweep and down-sweep) of experiments were performed using the ramping protocols described in § 2.2.1.

For suspension TCF, most experiments have been conducted for particle volume fractions $\phi =0.05, 0.1, 0.2$ by specifying the rotation ratio $\varOmega < 0$. Given the maximum frequency $f_{max}=20$ Hz of two motors, the maximum value of $\mbox {Re}_s(\phi,\varOmega )$ depends on ($\phi, \varOmega$); for example, $\mbox {Re}_s^{max}(\phi =0.2,\varOmega )\approx (298, 541)$ and $\mbox {Re}_s^{max}(\phi =0,\varOmega ) \approx (689$, 1252) at $\varOmega =(-0.1, -1)$. The corresponding range of inner and outer Reynolds numbers ($\mbox {Re}_i^{max}, \mbox {Re}_o^{max}$) can be estimated from (3.1). Defining the apparent shear rate as $\dot \gamma _{app}=|\omega _i r_i-\omega _o r_o|/\delta$, the upper bound on the particle Reynolds number is

(3.5)\begin{equation} \mbox{Re}_p^{max}= \frac{\rho\dot\gamma_{app} d_p^2}{\mu_l} = \mbox{Re}_i \left(\frac{d_p}{\delta}\right)^2 \left|1- \frac{\varOmega}{\eta}\right| \approx (0.15, 1.5) \end{equation}

at $\mbox {Re}_i=(10^2, 10^3)$ for $\varOmega =-1$. The corresponding Péclet number is

(3.6)\begin{equation} {Pe}= \frac{3{\rm \pi}\mu_ld_p^3\dot\gamma_{app}}{4k_BT} = \left(\frac{\rho\dot\gamma_{app} d_p^2}{\mu_l}\right) \left(\frac{3{\rm \pi}\mu_l^2 d_p}{4k_BT\rho} \right) \approx 1.517\times 10^9\mbox{Re}_p, \end{equation}

where $k_B=1.38065\times 10^{-23}\,{\rm J}\,{\rm K}^{-1}$ is the Boltzmann constant. Therefore, our suspension consists of non-Brownian particles ($ {Pe}=O(10^9)$), but the particle inertia ($\mbox {Re}_p=O(1)$) is likely to have some effects on the observed patterns, especially those that appear at higher values of $\mbox {Re}_s(\phi )$.

3.2. Torque at the inner cylinder and the pseudo-Nusselt number

Let ${\mathcal {T}}$ be the measured torque (in N m) at the inner cylinder, which is made dimensionless via (Wendt Reference Wendt1933; Taylor Reference Taylor1936; Donnelly & Schwarz Reference Donnelly and Schwarz1965; Dubrulle & Hersant Reference Dubrulle and Hersant2002; Dubrulle et al. Reference Dubrulle, Dauchot, Daviaud, Longaretti, Richard and Zahn2005; Eckhardt et al. Reference Eckhardt, Grossmann and Lohse2007)

(3.7)\begin{equation} G= \frac{\mathcal{T}\rho}{2{\rm \pi} h\,\mu(\phi)^2} , \end{equation}

where we have used the suspension viscosity $\mu (\phi )$ from (2.1) instead of the fluid viscosity $\mu (0)=\mu _l$. For the case of infinite aspect ratio TC geometry (i.e. without axial end-plates), the laminar value of torque (3.7) in the CCF state is given by (Dubrulle & Hersant Reference Dubrulle and Hersant2002; Eckhardt et al. Reference Eckhardt, Grossmann and Lohse2007)

(3.8)\begin{align} G_{lam}&= \frac{\eta}{(1-\eta)^2}\,\mbox{Re}_s(\phi) \end{align}

(3.9)\begin{align} &\equiv \frac{2\eta}{(1-\eta)^2(1+\eta)}\,|1-\varOmega|\,\mbox{Re}_i(\phi) \end{align}

(3.10)\begin{align} &\equiv \frac{2\eta^2}{(1-\eta)^2(1+\eta)}\,|\varOmega^{{-}1}-1|\,\mbox{Re}_o(\phi). \end{align}

Note that the above dimensionless quantity $G_{lam}$ is also defined in terms of $\mu (\phi )$. A convenient way to present the torque data is given in terms of the pseudo/omega Nusselt number (Dubrulle & Hersant Reference Dubrulle and Hersant2002; Eckhardt et al. Reference Eckhardt, Grossmann and Lohse2007)

(3.11)\begin{equation} \mbox{{Nu}}_\omega = \frac{J^\omega}{J^\omega_{lam}} \equiv \frac{G}{G_{lam}}, \end{equation}

where

(3.12a,b)\begin{equation} J^\omega=(\mu(\phi)/\rho)^2 G \quad \text{and} \quad J^{\omega}_{lam}=(\mu(\phi)/\rho)^2 G_{lam} \end{equation}

are the flux of angular velocity from the inner to the outer cylinder and its laminar value, respectively. We will follow the above definitions (3.11)–(3.12a,b) to present the torque data for suspension TCF in §§ 4 and 5.

3.3. Validation: comparison with previous works

3.3.1. Particle-free TCF under counter-rotation

For the particle-free ($\phi =0$) TCF, figure 2(a) displays the up-sweep data on critical shear Reynolds numbers $\mbox {Re}_s^{c1}\ {\rm and}\ \mbox {Re}_s^{c2}$ for primary (${\rm CCF}\to {\rm TVF}$ or ${\rm CCF}\to {\rm SVF}$) and secondary (${\rm TVF}\to {\rm WTV}$ or ${\rm SVF}\to {\rm ISV}$) transitions, respectively. For a comparison, this figure also includes the data of Andereck et al. (Reference Andereck, Liu and Swinney1986) on $\mbox {Re}_s^{c1}\ {\rm and}\ \mbox {Re}_s^{c2}$ (marked by square and diamond symbols) that have been extracted from their figure 2 (see figure 1b). Our estimates of $\mbox {Re}_s^{c1}$ are slightly larger but in good agreement with those of Andereck et al. (Reference Andereck, Liu and Swinney1986) at $\varOmega \geq -0.5$, but a large quantitative difference is found at $\varOmega = -1$, for which the primary bifurcation yields an oscillatory state (SVF). In general, the onset of travelling vortices is delayed significantly (e.g. $\mbox {Re}_s^{c2}>\mbox {Re}_s^{c2}|_{ALS}$) for all $\varOmega$ in the present set-up.

Figure 2.

(a) Critical shear Reynolds number (see (3.1)) for primary (blue line) and secondary (magenta line) transitions in particle-free ($\phi =0$) TCF at $\varOmega < 0$; see text for details. While the red squares and black diamonds denote the (up-sweep) data of Andereck et al. (Reference Andereck, Liu and Swinney1986) for primary and secondary transitions, respectively, the blue and magenta triangles refer to the corresponding data of the present experiments; two arrows mark the data of Ramesh et al. (Reference Ramesh, Bharadwaj and Alam2019) for $\varOmega =0$. (b) Phase diagram of patterns in the $(\mbox {Re}_s(\phi ), \phi )$-plane at $\varOmega \equiv \omega _o/\omega _i=-10^{-3}\approx 0$ and its comparison with Ramesh et al. (Reference Ramesh, Bharadwaj and Alam2019) for $\varOmega =0$, marked by circles and black lines. For all cases, the ramping rate is ${\rm d}\mbox {Re}(\phi )/{\rm d}\tau _d=0.05$.

Note that the delayed transition to oscillatory states in a small aspect ratio TC cell ($\varGamma =11$) was confirmed by Ramesh et al. (Reference Ramesh, Bharadwaj and Alam2019) at $\varOmega =0$ (see their table 2); their data on the first two critical Reynolds numbers,

(3.13)\begin{equation} (\mbox{Re}_s^{c1}, \mbox{Re}_s^{c2}) =\frac{2}{(1+\eta)}\,(\mbox{Re}_i^{c1},\mbox{Re}_i^{c2}) \approx (128, 180), \end{equation}

marked by two arrows on the right of figure 2(a), are in relatively better agreement with the present data than with Andereck et al. (Reference Andereck, Liu and Swinney1986). Although the radius ratio and curvature $(\eta,\kappa )=(0.891, 0.122)$ of our TC cell are close to $(\eta,\kappa )=(0.883, 0.132)$ of Andereck et al. (Reference Andereck, Liu and Swinney1986), the present aspect ratio ($\varGamma =8.46$ versus $\varGamma _{ALS}=30$) is approximately $3.55$ times less than that in the latter work. This axial constraint is known to make the Ekman vortices (Coles Reference Coles1965) stronger in a small-$\varGamma$ TC cell, which imparts an overall stabilizing effect (Cole Reference Cole1976; Benjamin Reference Benjamin1978b; Lorenzen, Pfister & Mullin Reference Lorenzen, Pfister and Mullin1983; Mullin, Heise & Pfister Reference Mullin, Heise and Pfister2017; Gopan & Alam Reference Gopan and Alam2020; Aghor & Alam Reference Aghor and Alam2021), thereby delaying the onset of oscillatory modes (WTV/SVF).

Notwithstanding the quantitative differences, we conclude that the present set-up identifies the onset of both primary and secondary states in the counter-rotation regime of the particle-free TCF of an incompressible Newtonian fluid.

5.1. ISV and their spectral characteristics

To distinguish the transition sequence depicted in figure 14(a) and the spectral evolution of the underlying modes, we display in figures 15(a–f) a series of (i) images (upper left), (ii) the corresponding space–time maps (upper right) and their (iii) temporal (lower left) and (iv) spatial (lower right) spectra with increasing shear Reynolds number $\mbox {Re}_s=515$, $577.2$, $598$, $600$, $674.3$ and $803$; the particle volume fraction is set to $\phi =0.1$ for all cases. By drawing an analogy with the movement of the right-handed (left-handed) screw/helix as positive (negative), we will denote the right- and left-handed spiral/helical vortices by SVF$^{+}$ and SVF$^{-}$, respectively. The backward-facing inclined bands in the upper right image of figure 15(a) confirm the existence of a right-handed spiral (SVF$^+$) at $\mbox {Re}_s=515$; the rotation/propagation frequency of this spiral is ${\omega /\omega _i\approx 0.86}$ (see the blue circle in the left lower plot in figure 15a), with its harmonic being located at ${\omega _h/\omega _i\approx 2\omega /\omega _i}$ (denoted by a grey cross). The spatial power spectrum of a vertical strip of width 1 pixel (marked by the red dashed line), shown in the lower right plot of figure 15(a), contains a peak at $\delta /\lambda \approx 1$ (marked by a filled green circle), indicating that the spacing of spirals scales approximately with annular gap $\delta$.

Figure 15.

(a–f) Snapshots of spiral vortices (SVF/ISV/NIS, top left) and their time evolution over $2.5$ s (top right), and the power spectra of scattered light intensity $I(z,t)$ (bottom): mean temporal FFT (bottom left) averaged over two axial locations at $z/\delta = 3.2$ and $5.4$ (marked by black squares in each image), and spatial FFT (bottom right) along a vertical strip of width $1$ pixel, marked by the red dashed line in each image. The rotation ratio is $\varOmega =-1.0$, and the particle volume fraction is $\phi =0.1$: (a) $\mbox {Re}_s=515$ (SVF), (b) $\mbox {Re}_s=577.2$ (ISV), (c) $\mbox {Re}_s=598$ (ISV), (d) $\mbox {Re}_s=600$ (ISV), (e) $\mbox {Re}_s=674.3$ (NIS), and (f) $\mbox {Re}_s=803$ (NIS). The bright horizontal patch in each image and space–time map in (a–d) refers to mica-flakes getting accumulated around the mid-height of the TC cell; see Appendix B for details. The open red circles in the spatial spectra in (a–f) refer to the axial periodicity of the cropped images ($\delta /z=1/6.83\approx 0.146$) and its harmonics; while the green circles (at $\delta /\lambda \approx 1$ in a–c) refer to the spacing between spirals, the additional spatial peaks (denoted by blue stars) appear only in the ISV state (at $\delta /\lambda \sim 0.42$ in b,c, and at $\sim 0.48$ in d). All data refer to up-sweep runs at ramping rate ${\rm d}\mbox {Re}(\phi )/{\rm d}\tau _d=0.05$.

Figure 15(b) is an analogue of figure 15(a) but at a higher value of $\mbox {Re}_s=577.2$; the corresponding space–time map confirms the existence of two separate spirals (left- and right-handed) that originate from the bottom and top of the TC cell. These two spirals SVF$^+$ and SVF$^-$ meet and terminate around the mid-height ($z/\delta \approx 4.2$) of the TC cell – we call this state interpenetrating spiral vortices (ISV). Note that the temporal and spatial power spectra of the ISV in figure 15(b) contain spectral features similar to the case of a single SVF at $\mbox {Re}_s=515$ as if the two spirals are weakly interacting or are about to interact with each other. Such a weakly interacting temporal spectrum along with coexisting left- and right-handed spirals are taken to be the signatures for the onset of ISV from the SVF state at $\mbox {Re}_s=\mbox {Re}_s^{c2}>\mbox {Re}_s^{c1}$. Note further that locating the ${\rm SVF}\leftrightarrow {\rm ISV}$ transition solely based on the frequency data can be problematic, but another metric, in terms of the torque data (pseudo-Nusselt number), to identify the onset of ISV will be discussed in § 5.3.

With further increase of $\mbox {Re}_s>\mbox {Re}_s^{c2}$, the left- and right-handed spirals indeed start penetrating into each other along the length of the cylinders, with the emergence of diamond-shaped structures in the space–time maps, such as in figures 15(c,d) that correspond to $\mbox {Re}_s=598$ and $600$, respectively. The interaction of SVF$^+$ and SVF$^-$ has a clear signature in its temporal power spectrum too: (i) the strengths (heights) of the peak and its harmonic decrease with increasing $\mbox {Re}_s$, and (ii) the appearance of a low-frequency peak, marked by an orange star in the lower left plots of figures 15(c,d). In fact, the low-frequency peak at $\mbox {Re}_s=598$ (figure 15c) is broadened at $\mbox {Re}_s=600$ (figure 15d), presumably due to nonlinear interactions at higher $\mbox {Re}_s$ – a comparison between the respective space–time maps tends to support the latter claim. On the other hand, the spatial power spectra in figures 15(b–d) contain dominant peaks (denoted by blue stars) that are located at $\delta /\lambda \approx 0.42$ in figures 15(b,c), but at $\delta /\lambda \approx 0.48$ in figure 15(d); these additional peaks do indicate the emergence of some spatial length scale in the ISV state, but we are unable to identify this length scale. In any case, the above temporal spectral characteristics of ISV mirror closely those found in figures 7(c,d) of Andereck et al. (Reference Andereck, Liu and Swinney1986) in the context of particle-free TCF, although they have not mentioned explicitly how they identified the transition between SVF and ISV states.

5.2. Non-propagating interpenetrating spirals

In the nonlinear regime of two interacting spirals, the diamond-shaped structures are found all over the surface of the outer cylinder, as confirmed from the space–time maps in figures 15(e, f) that refer to $\mbox {Re}_s=674.3$ and $803$, respectively. The frequency peaks disappear completely from the respective temporal power spectra, which are broad and featureless with a noisy background – also, there seems to be a power-law decay of the spectra at large enough $\omega$. In addition, the spatial spectra in figures 15(e, f) do not contain any structural information, confirming the non-periodic nature of these patterns. Therefore, with increasing $\mbox {Re}_s>\mbox {Re}_s^{c2}$, the ISV state degenerates into a non-propagating, non-periodic pattern at $\mbox {Re}_s\geq \mbox {Re}_s^{c3}(\phi )$ – this tertiary state is called non-propagating interpenetrating spirals (NIS).

Comparing our spectra for the NIS state (see the lower left plots in figures 15e, f) with figure 8(c) of Andereck et al. (Reference Andereck, Liu and Swinney1986), we find that the NIS possess spectral characteristics similar to those of the so-called wavy interpenetrating spirals (WIS). Interestingly, although the waves are not clear in our image in figure 15(e), they are visible in figure 15(f). Further, it has been noted by Andereck et al. (Reference Andereck, Liu and Swinney1986) that the visual appearance of WIS is time-dependent in the sense that the waves appear and disappear intermittently in time, and the spectra of WIS were always found to be devoid of frequency peaks with a noisy background – both spectral features are similar to those found for the NIS state in the present experiments. Therefore, in the following, we will use the acronym NIS or WIS to refer to the same non-propagating (and aperiodic) tertiary state.

The above route of the spectral evolution of spiral vortices with increasing $\mbox {Re}_s$ persists at higher values of $\phi$ too, an example of which is shown in figure 16 for $\phi =0.2$. Note that for the range $\mbox {Re}_s(\phi )< 550$ studied (with $f_{max}=20$ Hz), we did not find the tertiary state (NIS) at $\phi =0.2$. Since the ISV can be considered as a quasi-periodic state, collectively figures 15 and 16 confirm that the transition scenario ${\rm CCF}\to {\rm SVF}\to {\rm ISV}\to {\rm NIS}$ towards an aperiodic state (NIS) follows the well-known quasi-periodic route of transition to chaos of Ruelle & Takens (Reference Ruelle and Takens1971) that occurs via a finite number of Hopf bifurcations. Similar spectral evolution (but with different types of patterns) leading to elastic turbulence seems to hold in the TCF of viscoelastic fluids too (Dutcher & Muller Reference Dutcher and Muller2013) – the present suspension TCF can be explored by considering such non-Newtonian solvents in future.

Figure 16.

Evolution of spirals (upper images, space–time map over $2.5$ s) and their power spectra (lower plots) with increasing shear Reynolds number: (a) $\mbox {Re}_s=400$ (SVF), (b) $\mbox {Re}_s=475$ (ISV, near onset), and (c) $\mbox {Re}_s=530$ (ISV). Other parameters are $\phi =0.2$ and $\varOmega =-1$. These images were constructed from the up-sweep data of a UD run at ramping rate ${\rm d}\mbox {Re}(\phi )/{\rm d}\tau _d=0.025$ with a different LED light facility (see the last paragraph in § 2.2.2).

5.3. Signatures of ${\rm CCF}\leftrightarrow {\rm SVF}\leftrightarrow {\rm ISV}\leftrightarrow {\rm NIS}$ transitions in the torque data

To identify the signatures of transitions among CCF, SVF, ISV and NIS states on the torque data, we plot the variations of the pseudo-Nusselt number $\mbox {Nu}_{\omega }$ (see (3.11)) with $\mbox {Re}_s$ in the upper plots of figures 17(a) and 17(b), respectively, for up-sweep ($\mbox {Re}_s\uparrow$) and down-sweep ($\mbox {Re}_s\downarrow$) runs; the particle volume fraction is $\phi =0.1$, with other parameters as in figure 14. The space–$\mbox {Re}_s(\phi )$ map and the related evolution of the frequency peaks are displayed in the middle and lower images of figure 17, respectively. The analogous plots for the particle-free ($\phi =0$) case are displayed in figures 17(c,d). Irrespective of the value of $\phi$, it is clear from figures 17(a–d) that there is a change in the slope of the $\mbox {Nu}_\omega$–$\mbox {Re}_s$ curve at ${\rm CCF}\leftrightarrow {\rm SVF}$ transition that agrees closely with the appearance of propagation frequencies of spiral vortices in the respective power spectrum as confirmed in the lower plots of figures 17(a–d).

Figure 17.

Effective Nusselt number $\mbox {Nu}_\omega$ (top), the space–$\mbox {Re}_s(\phi )$ map (middle) and the normalized frequency $\omega /\omega _i$ (bottom) for (a,c) up-sweep and (b,d) down-sweep runs. Parameter values are (a,b) $\phi =0.1$ and (c,d) $\phi =0$, with $\varOmega =-1.0$; the ramping rate is ${\rm d}\mbox {Re}(\phi )/{\rm d}\tau _d=0.05$. While the blue circles and grey crosses denote the fundamental SVF mode ($\omega _1=\omega _{sp}$) and its harmonic, respectively, the orange stars and green triangles denote two incommensurate frequencies ($\omega _2$ and $\omega _3$) that appear in the ISV region.

The onset of the secondary transition, ${\rm SVF}\to {\rm ISV}$, is accompanied by the appearance of incommensurate low-frequency peaks (marked by orange stars and green triangles in the frequency maps of figure 17) along with a diminishing amplitude of the fundamental SVF mode (blue circles). Note, however, that there are ranges of $\mbox {Re}_s(\phi )$ in the ISV region (see the lower plot in figure 17a) over which the incommensurate frequencies are not seen. On the other hand, the tertiary transition, ${\rm SVF}\to {\rm ISV}$, has two visible signatures in the torque data: (i) $\mbox {Nu}_\omega$ starts fluctuating around some mean value that increases with increasing $\mbox {Re}_s$, and/or (ii) there is a discontinuous jump in $\mbox {Nu}_\omega$. Note that the magnitude of fluctuations in $\mbox {Nu}_\omega$ seems to decrease with increased particle loading that can be ascertained from a visual comparison of the torque data in the upper plots of figures 17(a) and 17(c). The latter observation possibly implies some dampening effect of particles on interpenetrating spirals as they pierce through each other along the length of the cylinders.

Referring to the space–time maps in figures 15(c,d), we infer that the underlying dynamical process for the interaction between left- and right-handed spirals, which gives birth to the ISV state, is asymmetric near the onset at $\mbox {Re}_s = \mbox {Re}_s^{c2}$ – this asymmetric interaction is likely to be responsible for observed fluctuations in the measured torque. The torque fluctuations eventually saturate and persist at large enough $\mbox {Re}_s$ when two spirals completely penetrate into each other in the axial direction. That the fluctuations in $\mbox {Nu}_\omega$ indeed become saturated can be ascertained from the upper plots in figure 17(a–d) near the onset of the tertiary transition ${\rm ISV}\to {\rm NIS}$ at $\mbox {Re}_s=\mbox {Re}_s^{c3}\gg \mbox {Re}_s^{c2}$. The tertiary state (NIS/WIS) is of course non-periodic and is devoid of frequency peaks in its power spectrum, as confirmed from the frequency maps in figure 17.

Based on the analyses of the temporal spectrum (§ 5.1) and the torque data (§ 5.3), we can categorize interpenetrating spirals into two subclasses: (i) weakly interacting ISV (figure 15b), and (ii) strongly/nonlinearly interacting ISV (figures 15c,d and 16b,c); while the former are characterized by a single dominant frequency (i.e. the SVF mode), the onset of the nonlinearly interacting ISV is characterized by torque fluctuations and/or two incommensurate frequencies. Returning to the phase diagram in figure 14(a), we note that the phase boundaries across ${\rm CCC}\leftrightarrow {\rm SVF}$, ${\rm SVF}\leftrightarrow {\rm ISV}$ and ${\rm ISV}\leftrightarrow {\rm NIS}$ transitions were drawn based on (i) the frequency analyses of the video images, as well as (ii) the analyses of the torque data, with the latter being a reliable metric to identify the onset of the ISV state.

5.4. Speed of spiral vortices and its scaling and comparison with Andereck et al. ( Reference Andereck, Liu and Swinney1986)

The propagation speed of spirals, $v_{sp}$, can be estimated from (i) its rotation frequency $\omega _{sp}$ and (ii) the axial wavelength $\lambda _z$ of spirals:

(5.2)\begin{equation} \frac{v_{sp}}{\omega_r \delta}= \frac{\omega_{sp}\lambda_z}{\omega_r \delta} \equiv \frac{\omega_{sp}}{\omega_r}\,\frac{\lambda_z}{\delta} \approx \frac{\omega_{sp}}{\omega_r}, \end{equation}

with the latter approximation $\lambda _z\approx \delta$ being verified from the spatial FFT analysis (see figures 15a,b); here, $\omega _r=|\omega _i-\omega _o|$ is the relative angular speed between the two cylinders. The propagation frequency $\omega _{sp}$ has been determined from the temporal FFT analysis, resulting in frequency maps such as those in the bottom plots of figures 17(a–d).

The variations of the normalized rotation speed $\omega _{sp}/\omega _r$ of spirals as a function of the shear Reynolds number $\mbox {Re}(\phi )$ are displayed in figure 18(a) for different values of $\phi$. It is seen that $\omega _{sp}/\omega _r\approx 0.5$ at the onset of spiral vortices (i.e. at $\mbox {Re}_s= \mbox {Re}_s^{c1}$) and then $\omega _{sp}/\omega _r$ decreases with increasing $\mbox {Re}_s(\phi )$ at any $\phi$. The terminal data for each $\phi$ in figure 18(a) represent the onset of an oscillatory ISV state (see figure 17). Figure 18(b) indicates that our wave speed data can be collapsed via the following scaling relations:

(5.3)\begin{align} \frac{\omega_{sp}}{\omega_r} &= {\alpha_0}\,{\mbox{Re}_s(\phi)}^{-\alpha_1}\,\mu_r^{-\alpha_2} \end{align}

(5.4)\begin{align} &= {\alpha_0}\,{\mbox{Re}_s(0)}^{-\alpha_1}\,\mu_r^{(\alpha_1-\alpha_2)}, \end{align}

with $(\alpha _0, \alpha _1, \alpha _2) \approx (65.28, 0.78, 0.36)$; here, $\mbox {Re}_s(0)=\mbox {Re}_s(\phi )\,\mu _r$ is the shear Reynolds number based on the viscosity of the solvent ($\mu (0)=\mu _l$), and the relative viscosity $\mu _r(\phi )$ is defined in (2.2). We need to point out that our data in figure 18(b) span a small range $\mbox {Re}_s(\phi )\in (300,600)$ over which the above scaling laws are deemed to hold for $\phi \leq 0.2$.

Figure 18.

Variations of (a) the normalized propagation frequency ($\omega _{sp}/\omega _r$) of spirals with $\mbox {Re}_s(\phi )$, and (b) its scaling; $\mu _r(\phi )$ is the relative viscosity of suspension, $\omega _r=|\omega _i-\omega _o|$ is the relative angular speed of the two cylinders, and $\varOmega =-1.0$.

For the particle-free ($\phi =0$) case, (5.4) boils down to $\omega _{sp}/\omega _r\propto \mbox {Re}_s^{-0.78}$, decreasing axial speed of spirals with increasing $\mbox {Re}_s$. Note that we have determined the speed of laminar spirals at a fixed rotation ratio $\varOmega =-1$ for which there are no experimental data to make a direct comparison. In this regard, the earliest work to measure the speed of spirals is Coles (Reference Coles1962), who investigated the regime of ‘spiral turbulence’ (Coles Reference Coles1965; Van Atta Reference Van Atta1966). These authors found that the laminar and turbulent regions of the flow formed an alternating pattern of helical stripes (spirals) that propagate with approximately the mean angular speed of the two cylinders, i.e. $\omega _{sp}/\omega _r\approx 0.5$. Detailed quantitative measurements were carried out by Andereck et al. (Reference Andereck, Liu and Swinney1986), who reported a nearly constant value (see their figure 12)

(5.5)\begin{equation} \frac{\omega_{sp}}{\omega_o} \approx 0.6 \end{equation}

for turbulent spirals over a range of inner-cylinder Reynolds numbers $\mbox {Re}_i\in (700, 1600)$ at an outer-cylinder Reynolds number $\mbox {Re}_o=-3500$; note that the rotation ratio was also varied in the range $\varOmega =\eta (\mbox {Re}_o/\mbox {Re}_i)\in -(4.415,1.93)$. Using $\omega _o=\omega _r/(1/\varOmega -1)$ and $\mbox {Re}_s=2\,|1-\varOmega |\,\mbox {Re}_i/(1+\eta )$, their data (5.5) lead to the following estimates:

(5.6)\begin{equation} \frac{\omega_{sp}}{\omega_r} = \left\{ \begin{array}{@{}ll} 0.489 & \mbox{at}\ \mbox{Re}_s=4026 \ (\mbox{Re}_i=700,\ \mbox{Re}_o={-}3500,\ \varOmega={-}4.415),\\ 0.432 & \mbox{at}\ \mbox{Re}_s=4556 \ (\mbox{Re}_i=1200,\ \mbox{Re}_o={-}3500,\ \varOmega={-}2.575),\\ 0.395 & \mbox{at}\ \mbox{Re}_s=4981 \ (\mbox{Re}_i=1600,\ \mbox{Re}_o={-}3500,\ \varOmega={-}1.931). \end{array} \right. \end{equation}

The numerical values of $\omega _{sp}/\omega _r$ and its decreasing trend with increasing $\mbox {Re}_s$ agree with our findings in figure 18, although their data can be fitted via $\omega _{sp}/\omega _r\sim \mbox {Re}_s^{-1}$, which differs from the present exponent $\alpha _1\approx 0.78$ in (5.4). Furthermore, the above comparative analysis indicates that the laminar and turbulent spirals propagate at similar speeds.