2. Experimental set-up

A dielectric fluid is confined between two cylindrical electrodes maintained at different temperatures. The inner cylinder, whose outer surface radius is $R_1 = 5$ mm, is made of aluminum with a black nickel oxide coating on the outer surface. The inner surface of the outer cylinder has radius $R_2 = 10$ mm. It is made of boron silicate glass, and the inner surface is coated with a transparent conducting oxide, so that the gap is optically free for visualisation techniques through the outer cylindrical surface. Both electrodes have height $H = 100$ mm, which gives aspect ratio $\varGamma = H / (R_2 - R_1) = 20$. The top and bottom lids are made of acrylic glass so the axial direction is optically free. The inner electrode is maintained at temperature $T_1$, and the outer electrode is maintained at temperature $T_2$, so a temperature difference $\Delta T = T_1 - T_2$ between the two surfaces is achieved. In addition, a high alternative electric field is applied with frequency 200 Hz within the dielectric fluid by connecting the inner electrode to the phase of a high-voltage amplifier and by grounding the outer electrode.

As working fluids, silicone oils B3 and B5 are used due to their satisfying dielectric properties (see table 1). In particular, their dielectric constant varies slowly with the applied frequency of the electric field, and internal heating can be neglected since the dissipation factor is low (Yoshikawa et al. Reference Yoshikawa, Kang, Mutabazi, Zaussinger, Haun and Egbers2020). Another advantage is that the silicone oils have a refractive index close to that of acrylic glass, so the reflections and refraction between the transparent materials are limited. Additionally, it was shown that these silicone oils can be used together with a suspension of hollow glass spheres (HGS) to perform particle image velocimetry (PIV) measurements (Adrian & Westerweel Reference Adrian and Westerweel2011). The hollow glass spheres have average radius $5\ \mathrm {\mu }{\rm m}$ and density $1100\ {\rm kg}\ {\rm m}^{-3}$. They are slightly denser than the silicone oils, so the Earth gravity and the centrifugal acceleration occurring during the acceleration phase of the rocket should lead to the sedimentation of the particles. However, it was shown by Seelig et al. (Reference Seelig, Meyer, Gerstner, Meier, Jongmanns, Baumann, Heuveline and Egbers2019) that the sedimentation velocity of the Potter hollow glass spheres due to the Earth gravity in silicone oil AK5 (with the same properties as B5) was of the order of $10^{-6} {\rm m}\ {\rm s}^{-1}$. During the flight, the gap filling pumps in the mixing loop (see figure 2) are activated just at the beginning of the $\mathrm {\mu }{\rm g}$ phase, which ensures the homogeneous distribution of particles inside the system. Once the system is under weightlessness, no external forces act on the density mismatch between the particles and the fluid. However, the application of the electric field induces the DEP force on the particles, which can result in an electric sedimentation. Indeed, the relative permittivity of the hollow glass spheres is 4.6, which is different from that of the silicone oils. The particles have a larger permittivity than the fluid, and will then tend to be attracted to areas with larger strength in the electric field, i.e. to the inner cylinder. That mechanism is known in biological science since it can be used to separate cells by their permittivity (Pohl & Kaler Reference Pohl and Kaler1979; Zhu, Tzeng & Xuan Reference Zhu, Tzeng and Xuan2010), and one can derive the electric sedimentation of the particle from the DEP force applied to them. Seelig et al. (Reference Seelig, Meyer, Gerstner, Meier, Jongmanns, Baumann, Heuveline and Egbers2019) have estimated that for temperature difference 7 K and peak voltage 7 kV, the electric sedimentation velocity measured at the inner cylinder is $6.3\times 10^{-6}\ {\rm mm}\ {\rm s}^{-1}$. Considering the relatively short duration of the $\mathrm {\mu }{\rm g}$ condition (6 min) and the thermo-electric convective fluid velocity being of the order of $0.1\unicode{x2013}1\ {\rm mm}\ {\rm s}^{-1}$ (see Meier et al. Reference Meier, Jongmanns, Meyer, Seelig, Egbers and Mutabazi2018; Meyer et al. Reference Meyer, Meier, Jongmanns, Seelig, Egbers and Mutabazi2019; Szabo et al. Reference Szabo, Meier, Meyer, Barry, Motuz, Mutabazi and Egbers2021), it is assumed that the sedimentation of the particles can be neglected.

Figure 2.

Schematic drawing of the fluid loops for the temperature control and gap mixing.

Two independent fluid loops are used to maintain the temperature difference $\Delta T$ (see figure 2). The heating fluid flows through the inner cylinder, whereas the cooling fluid flows through a cylindrical tank situated around the outer cylinder. The cooling tank is made of boron silicate glass and is coated with an anti-reflective surface for the use of light-based measurement techniques. The loops have flows of heating and cooling fluids in opposite directions in order to keep the temperature difference constant along the axial direction. The temperature is measured by a set of thermosensors positioned at the inlet and outlet of the experiment cell. Each of the fluid loops is composed of a small fluid reservoir, a pump and a Peltier element that allows for the temperature at the inlet of the cell to be controlled. The chosen fluid in these loops is silicone oil in order to avoid flash-overs that can occur if a leakage allows the cooling or heating fluid to enter inside the gap where the strong electric field is applied.

Additionally to the temperature control, a volumetric flux sensor was placed in the heating loop of the system. Coupling this measurement with those of the temperature at the inlet and at the outlet of the inner cylinder would lead to the rate of heat transfer at the inner cylindrical surface per unit area. The determination of the Nusselt number could then be done by scaling that heat flux with that of the purely conductive one. Unfortunately, an important uncertainty in the temperature measurement combined with a slight increase of the loops’ temperature during the experimental run attributed to a slight drop of pumping efficiency that did not allow the estimation of the Nusselt number to be reliable.

To perform the PIV techniques, the gap is illuminated with a continuous blue (450 nm) laser light sheet in a radial axial plane (see figure 3a). A camera records the motion of the particles inside that plane at 10 frames per second. The radial and axial components of the velocity field $\boldsymbol {u} = V_r\boldsymbol {e}_r + V_z\boldsymbol {e}_z$ can be measured in the illuminated $(r , z)$ plane. Two mirrors are used in order to achieve a compact system, suitable for fitting in the sounding rocket. Simultaneously, the shadowgraph technique is used to visualise density variations along the azimuthal direction. The gap is lightened along the axial direction with a telecentric homogeneous light illumination of green (520 nm) colour (see figure 3a). Passing through the height of the gap, the light beams are refracted due to temperature variations. The local brightness then changes compared to the original light intensity and is measured at the other side of the cell with a camera equipped with a telecentric objective. The camera is recording the fluid at 5 frames per second. The obtained two-dimensional shadowgraph results from the three-dimensional temperature profile and is used to identify breaking of intensity axial symmetry within the gap. Due to reflections of the light on the hollow glass spheres or on other surfaces, the laser light sheet might be visible on the shadowgraphs, despite the telecentric objective that discriminates most of the rays incoming with a certain incident angle. To limit that cross-contamination of the two visualisation techniques, the illumination by the telecentric system is chosen to be sufficiently large so that the laser light sheet is not visible in the shadowgraphs. In counterpart, a blue filter is equipped on the PIV cameras so only the motion of particles occurring in the laser plane is visible.

Figure 3.

Mechanical drawings of (a) an experimental cell with the optics for the measurement techniques, and (b) the TEKUS module equipped with four experimental set-ups as shown in (a) (illustrations courtesy of Airbus Defense and Space).

The complete experiment consists of four identical cylindrical annuli. Each of them has its own fluid loops for control of the temperature differences, and is connected to separated high-voltage amplifiers (see figure 3b). The experimental parameters were chosen so the electric Rayleigh numbers, as defined in (1.4), are higher than the theoretical threshold for the onset of thermo-electric convection, but relatively close to it. In that manner, the growth of the instability is expected to be small, yet positive, for all phases of applied voltage. An overview of the experiment parameters is given in table 2, where the fluid, the applied temperature difference and the applied peak voltage are indicated for each of the cells in their dimensional and dimensionless forms. For each parameter, the duration of the phase of the applied electric field is specified. The value of the electric gravity measured at the mid-gap, $g_e(\bar {R})$, and that of $L$ are also given. In the cylindrical annulus with radius ratio $\eta = 0.5$, the curvature of the electrodes is the main source of non-homogeneity of the electric field, leading to an electric gravity that is strongly dependent on the applied electric potential, and weakly coupled with the temperature difference between the two surfaces.

Table 2.

Experimental parameters in their dimensional and dimensionless forms used during the TEXUS-57 flight, together with the duration $\Delta t$ of the applied force during the $\mathrm {\mu }{\rm g}$ phase, the intensity of the electric gravity at the mid-gap, and the corresponding value of the electric Rayleigh number.

Before the application of the high voltage, a certain period of time is dedicated to the stabilisation of the conductive state of the flow. During that period, the pump of the filling loop was activated to homogenise the temperature profile within the gap, as well as to distribute uniformly the concentration of particles. After that mixing phase, approximately 30 s is needed for the fluid flow to dissipate and for its temperature profile to establish. Considering the low value of the thermal diffusivity of silicone oils, the temperature is not expected to reach a steady state, but it was shown by Meyer et al. (Reference Meyer, Meier, Jongmanns, Seelig, Egbers and Mutabazi2019) that the growth rate of perturbation is increased when the initial conditions at which the electric field is applied are closer to the conductive state. The first set of high-voltage application lasts for 200 s. The next phase of applied high voltage is performed with higher values of the electric tension. Since it is expected that the growth rate of convective flow gets higher, the duration of the second phase is 80 s shorter than the previous one. With this experimental set-up and the chosen parameters, over the 360 s of available $\mathrm {\mu }{\rm g}$ conditions, 320 s are used effectively to investigate the dynamic of the dielectric fluids for 8 different values of the electric Rayleigh number.

3. Linear stability theory

Before the results section, the linear stability theory is presented to identify some basic features of the thermo-electric instability and to introduce some predictions from that analysis. The method for the LSA of the problem in a cylindrical annulus has been performed numerous times in different configurations, i.e. in $\mathrm {\mu }{\rm g}$ conditions (see Malik et al. Reference Malik, Yoshikawa, Crumeyrolle and Mutabazi2012; Yoshikawa et al. Reference Yoshikawa, Crumeyrolle and Mutabazi2013), under Earth gravity conditions (see Meyer et al. Reference Meyer, Jongmanns, Meier, Egbers and Mutabazi2017, Reference Meyer, Crumeyrolle, Mutabazi, Meier, Jongmanns, Renoult, Seelig and Egbers2018), or with the rotation of one or both cylinders (see Yoshikawa et al. Reference Yoshikawa, Meyer, Crumeyrolle and Mutabazi2015; Kang et al. Reference Kang, Meyer, Yoshikawa and Mutabazi2019). We now introduce briefly the main information about linear stability theory, and we refer to the above-mentioned literature for the exhaustive description of the method, and for their tests of validity.

The velocity field $\boldsymbol {v} = u\boldsymbol {e}_r + v\boldsymbol {e}_{\varphi } + w\boldsymbol {e}_z$ , the temperature deviation from the reference temperature $\theta$, the generalised pressure ${\rm \pi}$, and the electric potential $\phi$ satisfy the continuity equation, the momentum equation, the energy equation and the Gauss law, written as

(3.1a)$$\begin{gather} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{v} = 0 , \end{gather}$$

(3.1b)$$\begin{gather}\frac{\partial \boldsymbol{v}}{\partial t} + \boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{v} ={-} \boldsymbol{\nabla} {\rm \pi}+ \nu\,\varDelta \boldsymbol{v} - \alpha \theta \boldsymbol{g}_e , \end{gather}$$

(3.1c)$$\begin{gather}\frac{\partial \theta}{\partial t} + \boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} \theta = \kappa\,\varDelta \theta , \end{gather}$$

(3.1d)$$\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} \left( \varepsilon\,\boldsymbol{\nabla} \phi \right) = 0 . \end{gather}$$

No-slip boundary conditions are adopted at both cylindrical surfaces, and the temperatures of the inner and outer cylinders are set to $T_1$ and $T_2$, respectively. Neglecting the periodic nature of the electric tension, the root-mean-square electric potential $V_0 = \sqrt {2}\,V_p / 2$ is applied at the inner cylinder while the outer cylinder is grounded ($\phi = 0$). The cylindrical annulus is considered to be of infinite length. The set of equations (3.1) and the boundary conditions are non-dimensionalised by scaling the lengths, time, temperature and electric potential by the gap size $d$, the characteristic time of viscous dissipation $d^2 / \nu$, the temperature difference between the two cylinders $\Delta T$, and the effective electric potential $V_0$, respectively. The stability of the flow is studied in the vicinity of the base flow, which is defined as the stationary, axisymmetric and axially invariant solution of the problem. The dimensionless base state is given by

(3.2a–c)\begin{equation} \boldsymbol{v}_b = \boldsymbol{0}, \quad \theta_b = \frac{\log \left[r \left(1 - \eta\right) \right]}{\log \eta}, \quad \phi_b = \frac{\log \left(1 - \gamma_e \theta_b \right)}{\log \left( 1 - \gamma_e \right)} . \end{equation}

An infinitesimal perturbation $( \tilde {u},\tilde {v},\tilde {w},\tilde {{\rm \pi} },\tilde {\theta },\tilde {\phi } )^{\rm T}$ is added to the base state, where the exponent ${\rm T}$ refers to the transpose. The perturbation is then developed into normal modes $\exp [s t + {\rm i} (n \varphi + kz ) ]$ with a complex amplitude $( \hat {u},\hat {v},\hat {w},\hat {{\rm \pi} },\hat {\theta },\hat {\phi } )^{\rm T}$. The parameter $s$ can be complex: $s = \sigma ^* - {\rm i} \omega ^*$, where $\sigma ^*$ is the growth rate of perturbations, and $\omega ^*$ is the propagation frequency. Here, $\sigma ^*$ and $\omega ^*$ are dimensionless, but their dimensional forms $\sigma = \sigma ^*/\tau _{\nu }$ and $\omega = \omega ^*/\tau _{\nu }$ are used later for a direct connection to the experiment. The orientation and size of the mode is determined by the azimuthal mode number $n$ and the axial wavenumber $k$. The dimensionless set of equations for the amplitude of perturbations reads

(3.3a)$$\begin{gather} 0 = \left( \frac{{\rm d}}{{\rm d} r} + \frac{1}{r} \right) \hat{u} + \frac{{\rm i}n}{r}\,\hat{v} + {\rm i}k\hat{w} , \end{gather}$$

(3.3b)$$\begin{gather}s\hat{u} = \varDelta \hat{u} - \frac{{\rm d} \hat{\rm \pi}}{{\rm d} r} - \frac{\hat{u}}{r^{2}} - \frac{2 {\rm i} n}{r^{2}}\,\hat{v} - \frac{\gamma_e V_E^2}{Pr} \left( g_{e,b} \hat{\theta} + \hat{g}_{e,r} \theta_b \right) , \end{gather}$$

(3.3c)$$\begin{gather}s \hat{v} = \varDelta \hat{v} - \frac{{\rm i} n}{r}\,\hat{\rm \pi} - \frac{\hat{v}}{r^2} + \frac{2 {\rm i} n}{r^2} \hat{u} - \frac{\gamma_e V_E^2}{Pr}\,\theta_b \hat{g}_{e,\varphi} , \end{gather}$$

(3.3d)$$\begin{gather}s \hat{w} = \varDelta \hat{w} - {\rm i} k \hat{\rm \pi} - \frac{\gamma_e V_E^2}{Pr}\,\theta_b \hat{g}_{e,z} , \end{gather}$$

(3.3e)$$\begin{gather}s \hat{\theta} = \frac{1}{Pr}\,\varDelta \hat{\theta} - \frac{{\rm d} \theta_b}{{\rm d} r}\,\hat{u} , \end{gather}$$

(3.3f)$$\begin{gather}0 = \left(1 - \gamma_e \theta_b \right) \varDelta \hat{\phi} - \gamma_e\,\frac{{\rm d} \theta_b}{{\rm d} r}\, \frac{{\rm d} \hat{\phi}}{{\rm d} r} - \gamma_e\,\frac{{\rm d} \phi_b}{{\rm d} r} \left( \frac{{\rm d}}{{\rm d} r} + \frac{1}{r} \right) \hat{\theta} - \gamma_e\,\frac{{\rm d}^2 \phi_b}{{\rm d} r^2}\,\hat{\theta} , \end{gather}$$

where the Laplacian operator $\varDelta$ reads

(3.4)\begin{equation} \varDelta = \frac{1}{r}\,\frac{{\rm d}}{{\rm d} r} \left( r\,\frac{{\rm d}}{{\rm d} r} \right) - \frac{n^2}{r^2} - k^2 . \end{equation}

The dimensionless electric potential $V_E = V_p / \sqrt {2}\,V_{f}$ and the thermo-electric parameter $\gamma _e = e\,\Delta T$ have been introduced, where $V_f = ( \rho _{ref}\,\kappa \nu / \varepsilon _{ref} )^{1/2}$ is the characteristic voltage for a given fluid. The base electric gravity is given by $g_{e,b}= \boldsymbol {\nabla } ( \boldsymbol {\nabla } \phi _b )^2$. The amplitude of all perturbations vanishes at the cylindrical surfaces $\hat {u} = \hat {v} = \hat {w} = \hat {\theta } = \hat {\phi } = 0$. The momentum equations (3.3b)–(3.3d) include the perturbation of the electric gravity along the three spatial components:

(3.5a–c)\begin{equation} \hat{g}_{e,r} = \frac{{\rm d} \phi_b}{{\rm d} r}\,\frac{{\rm d}^2 \hat{\phi}}{{\rm d} r^2} + \frac{{\rm d}^2 \hat{\phi}_b}{{\rm d}r^2}\,\frac{{\rm d} \hat{\phi}}{d r}, \quad \hat{g}_{e,\varphi} = \frac{{\rm i}n}{r}\,\frac{{\rm d} \phi_b}{{\rm d} r}\, \frac{{\rm d}\hat{\phi}}{{\rm d} r}, \quad \hat{g}_{e,z} = {\rm i}k\,\frac{{\rm d} \phi_b}{{\rm d} r}\, \frac{{\rm d} \hat{\phi}}{{\rm d}r} . \end{equation}

The system of (3.3) is solved using a spectral collocation method as described in Yoshikawa et al. (Reference Yoshikawa, Crumeyrolle and Mutabazi2013), with the order of the Chebyshev polynomial as 30 to ensure the convergence of calculations.

The growth rate is determined for fixed values of the control parameters $(\eta, Pr , \zeta, \gamma _e , V_E )$, and for various values of $k$ and $n$. The global maximum of the growth rate determines the most critical modes for the given set of parameters, and the state for which the maximum value coincides with $\sigma ^* = 0$ corresponds to the critical state $(V_{E,c}, k_c, n_c, \omega ^*_c)$. Figure 4 shows the evolution of the growth rate in dimensional form as a function of the axial wavenumber $k$ for various control parameters. For $\gamma _e = 0.0053$, the critical state is obtained for $V_{E,c} = 673$, $k_c = 1.528$ and $n_c = 4$. The growth rate of the most critical state is also shown for the different control parameters applied to the silicone oil B5 during the TEXUS-57 flight. The growth rate increases with increasing $V_E$ and $\gamma _e$, i.e. with increasing $L$. The number of modes in the azimuthal direction increases with $L$, from $n=4$ at the critical state, to $n=6$ for the highest value of $L$ applied to the liquid. All the determined modes are stationary ($\omega ^* = 0$) and helical ($n \neq 0$ and $k \neq 0$). In Yoshikawa et al. (Reference Yoshikawa, Crumeyrolle and Mutabazi2013), it is demonstrated that the critical state of the flow is independent of the Prandtl number. Another important result is that when the radius ratio is not too large, the critical values of $L_c$, $k_c$ and $n_c$ are independent of the temperature difference between the two cylinders. In the special case $\eta = 0.5$, the critical Rayleigh number is $L=1498$.

Figure 4.

Dimensional growth rate $\sigma$ of the most critical mode as a function of the axial wave number for silicone oil B5 and for various control parameters.

4. Experimental results

As seen in table 2, the electric Rayleigh number takes values from $L = 2774$ to $L = 9227$ in the present experiment. It corresponds to 1.85 and 6.17 times the critical value $L_c = 1498$ determined by Yoshikawa et al. (Reference Yoshikawa, Crumeyrolle and Mutabazi2013). For silicone oil B5 in the cylindrical annulus with the same dimension as for the present experiment, the minimum value of $L$ for which an instability has been observed during the weightless environment of a parabolic flight was $L = 3995$. Under the Earth gravity, Seelig et al. (Reference Seelig, Meyer, Gerstner, Meier, Jongmanns, Baumann, Heuveline and Egbers2019) conducted an experiment for a vertical cylindrical annulus with the same dimensions and with the same dielectric liquid (silicone oil AK5). For temperature differences 5 and 10 K, thermo-electric instabilities were detected for $V_E \approx 1020$ at the lowest. During the $\mathrm {\mu }{\rm g}$ condition of the TEXUS flight, the PIV and the shadowgraph techniques highlighted the growth of a thermo-electric instability for all investigated values of $L$.

The results from the PIV techniques are shown in figure 5 for all investigated values of $L$ at the end of their respective phase of active high voltage. The plots indicate the velocity field and the isosurface of the azimuthal component of the vorticity $\omega _{\varphi } = \partial V_z / \partial r - \partial V_r / \partial z$ measured in the $(r,z)$ plane where the laser light sheet is illuminating the particles. The velocity field clearly indicates the occurrence of thermo-electric convection for all values of $L$. For $L = 2774$, counter-rotating pairs of vortices are observed only in the vicinity of the lids. Between $z = 10$ mm and $z = 70$ mm, the flow is mainly radially outwards, indicating a sheet-like convective flow, therefore $V_z$ and $\omega _{\varphi }$ are small in that area. For higher values of $L$, the counter-rotating vortices are more pronounced and extend along the whole height of the gap. For all values of $L$, the flow intensity seems to be increasing continuously with the increase of $L$. Numerical simulations were performed by Kang & Mutabazi (Reference Kang and Mutabazi2021) for $L = 6246$ in the case of weightlessness. They predicted helical modes with 5–6 pairs of vortices along the axial direction, and amplitude of the azimuthal vorticity approximately $\omega _{\varphi } = 0.2\ {\rm s}^{-1}$. Their results are in good qualitative agreement with the present experimental ones. The number of modes along the axial direction seems higher in the experiments, which could explain why the numerical result slightly underestimates the amplitude of the $\omega _{\varphi }$.

Figure 5.

Velocity field and surface of isovalues of the azimuthal component of the vorticity at the end of the phase of high-voltage application for (a) $L = 2774$, (b) $L = 2933$, (c) $L = 3995$, (d) $L = 4583$, (e) $L = 5586$, ( f) $L = 5905$, (g) $L = 8044$ and (h) $L = 9227$.

The shadowgraphs corresponding to the different values of $L$ are shown in figure 6. To obtain those shadowgraphs, the grey level of a given image is normalised with that of a reference image. That reference corresponds to the case when no temperature difference and no electric tension are applied. The value of the local light intensity $I$ is $I = 1$ when the local grey level is equal to that of the reference case. For all values of $L$, the light intensity profiles overcome a break of symmetry along the azimuthal direction, indicating the occurrence of the thermo-electric instability. For values up to $L = 3995$ of the electric Rayleigh number, the light intensity is slightly larger around the mid-gap. This higher intensity has a certain extension along the azimuthal direction where the non-homogeneity of the light intensity is slightly visible. For larger values of $L$, the non-axisymmetric behaviour of the light intensity profile is clearly visible. The light intensity measured by shadowgraphy highlights the formation of hot and cold jets at different azimuthal positions that, combined with the previous results of PIV along the axial direction, indicates that the thermo-electric instability tends to have a helical structure, as predicted by Malik et al. (Reference Malik, Yoshikawa, Crumeyrolle and Mutabazi2012). For all parameters, the pattern observed in the shadowgraph is stationary. At least for a given fluid, the light intensity variations increase with increasing $L$; however, it seems that the silicone oil B3 is more sensitive to the shadowgraph technique than the silicone oil B5, so that the light intensities seem globally higher for B3 at equivalent values of $L$. Small differences in refractive index, or its variation rate with respect to the density, might explain such an observation. Although the shadowgraphs indicate the break of symmetry of the flow along the azimuthal direction, the technique does not allow a clear determination of the mode number. The main difficulty comes from the fact that the measurement is performed along a large fluid distance, which prevents the proper tracking of the jets along the axial direction.

Figure 6.

Shadowgraphs at the end of the phase of high-voltage application for (a) $L = 2774$, (b) $L = 2933$, (c) $L = 3995$, (d) $L = 4583$, (e) $L = 5586$, ( f) $L = 5905$, (g) $L = 8044$ and (h) $L = 9227$.

The time evolution of the velocity field and of the light intensity profiles can be analysed through the use of space–time diagrams. The radial velocity is measured at the mid-gap along the axial direction, and the light intensity is measured at the mid-gap along the azimuthal direction. The time variations of those measurements are shown in figure 7 for two values of $L$. The temporal evolution of the radial velocity highlights the growth of the convective pattern starting from the lids of the cylindrical annulus for electric Rayleigh numbers up to $L = 3995$ (see figures 7a,c). This perturbation then brings about the convection at the mid-height of the cylinders. This mechanism seems to be in good agreement with the numerical simulation conducted by Gerstner (Reference Gerstner2020), who described the growth of toroidal vortices close to the boundaries. The growth of these axisymmetric modes then propagates progressively towards the mid-height of the cell, at which point the modes become non-axisymmetric. For larger values of $L$, the thermo-electric convection grows from the lids as well as from the bulk of the cells. The space–time diagrams for both the radial velocity and the light intensity show that the general amplitude of the perturbations reaches saturation, and that the instability is stationary. However, the modes seem to be still slowly evolving at the end of the phase of applied high voltage for some values of $L$. For instance, for $L = 9227$ (see figures 7b,d) the space–time diagram of $V_r$ indicates that some outward jets (positive radial velocity) are being split in two whilst some inward jets (negative radial velocity) are merging. The space–time diagram for the light intensity at the same value of $L$ is also indicating a certain modification of the shape of the modes. This behaviour is observed for different values of $L$ and seems to indicate that the three-dimensional convective cells are taking a long time to adapt to their available space and to reach a steady state, or that they might not be able to do so.

Figure 7.

Space–time diagrams of (a,b) the radial component of the velocity along the axial direction and (c,d) the light intensity along the azimuthal direction. Both quantities are measured at the mid-gap for (a,c) $L = 2933$ and (b,d) $L = 9227$. The origin of the time is taken at the instant when the high voltage is applied.

In an analogous problem, Odenbach (Reference Odenbach1995) used the amplitude of the perturbation of temperature to determine the critical value of the magnetic Rayleigh number. For his investigation, a magnetic field was applied to a ferrofluid during the $\mathrm {\mu }{\rm g}$ phase of a sounding rocket flight. The extrapolation method used to evaluate the critical control parameter relies on the dynamic of the formation of coherent structures in media. The nonlinear Ginzburg–Landau equation describes weakly nonlinear spatio-temporal phenomena in the vicinity of the primary bifurcation. For a stationary and periodic instability, as is the case here, the equation can be written as (Cross & Hohenberg Reference Cross and Hohenberg1993; Manneville Reference Manneville2005)

(4.1)\begin{equation} \tau_0\,\frac{\partial A}{\partial t} = \delta A + \xi_0^2\,\frac{\partial^2 A}{\partial z^2} - l\,|A|^2\,A , \end{equation}

where $A$ is the amplitude of a perturbation mode, and $\tau _0$ and $\xi _0$ are the characteristic time and the coherent length of a perturbation, respectively. These quantities that correspond to linear mechanisms are scales for the variation of a perturbation in time and in space. The Landau constant $l$ originates from a nonlinear mechanism that leads to the saturation of the mode amplitude when it is positive (super-critical bifurcation). The reduced control parameter $\delta = ( L - L_c ) / L_c$ indicates the proximity to the critical threshold and should not be too high, so that (4.1) is valid.

During the TEXUS-57 flight, all the studied control parameters are above the critical values, and an unstable regime was observed systematically. Indeed, it was a choice not to investigate parameters that would allegedly result in a purely conductive state. The distinction between the conductive base state when $L < L_c$ and the unstable regime when $L > L_c$ was, however, determined experimentally by Seelig et al. (Reference Seelig, Meyer, Gerstner, Meier, Jongmanns, Baumann, Heuveline and Egbers2019) under the influence of a vertical gravity field, and it was also investigated through numerical simulations and LSA when considering $\mathrm {\mu }{\rm g}$ conditions. We therefore assume the supercriticality of the bifurcation and admit that conveniently, (4.1) describes the dynamics of the amplitude of the modes of instability.

The norm of the radial component of the velocity at a given radial position is averaged over the height of the cavity and used to evaluate the amplitude of the modes, so that

(4.2)\begin{equation} A = \frac{1}{H}\int_0^H |V_r| \,{\rm d}z. \end{equation}

By making this choice, the variation of the amplitude close to a solid boundary is hidden and the second term on the right-hand side of (4.1) should be cancelled. Indeed, the mode structure is three-dimensional, so it is not relevant to evaluate the local amplitude envelope at a single azimuthal angle, since it was shown that the modes also have a certain periodicity in the azimuthal direction. The amplitude $A$ measured at the mig-gap is plotted over time in figure 8 for various values of the electric Rayleigh number during the first 120 s of the phase of applied voltage. For each of the values of $L$, the amplitude of the modes initially increases exponentially with a constant growth. After a sufficiently long time, the amplitude saturates to a constant value $A_s$, which depends on $L$. For sufficiently large values of $L$, at the transition from constant growth to saturation, the amplitude overshoots its saturation value and may start to oscillate around the value $A_s$. One can see that the higher the electric Rayleigh number, the higher the growth rate of the mode during the linear growth of the perturbation amplitude.

Figure 8.

Evolution of the amplitude of perturbation $A$ at the mig-gap as defined in (4.2) as a function of time for different values of $L$. The initial time is synchronised with the beginning of the high-voltage application phase.

When a perturbation mode saturates, its amplitude becomes stationary and (4.1) reduces to $|A|^2 = \delta / l$, which indicates a proportionality between the square of the amplitude and the electric Rayleigh number. That mechanism is observed in numerous problems of pattern formation in fluid mechanics (Odenbach Reference Odenbach1995; Le Gal et al. Reference Le Gal, Harlander, Borcia, Le Dizès, Chen and Favier2021). The evolution of $A_{s}^2$ with $L$ is shown in figure 9. For two values of the electric Rayleigh number, i.e. $L = 5586$ and $L = 8044$, the value of $A$ did not converge to a constant so that $A_s$ could not be extracted in those cases. For all other values of $L$, the time evolution of the amplitude $A$ is evaluated at a fixed radius corresponding to the mid-gap $\bar {R}$, as well as at the neighbouring radial positions $R \pm {\rm d}r$, with ${\rm d}r = 0.2$ mm. The last 10 s of the phases of applied electric field are used for each radius to generate reliable statistics data, from which the average provides the value $A_s$, and the standard deviation gives the error bar that takes into account uncertainties in the velocity and in the positioning of the mid-gap. A linear fit between the square of the amplitude $A_s$ of the modes and the electric Rayleigh number is given, and its extrapolation at $A_{s}^2 = 0$ provides the critical value $L_c = 1628$ for the electric Rayleigh number. This value is to be compared with the value of $L_c = 1498$ derived from LSA in Yoshikawa et al. (Reference Yoshikawa, Crumeyrolle and Mutabazi2013) and from numerical simulations in Travnikov, Crumeyrolle & Mutabazi (Reference Travnikov, Crumeyrolle and Mutabazi2015) for any values of the Prandtl number and for a radius ratio $\eta = R_1/R_2 = 0.5$. The error in the prediction made by the linear interpolation is approximately $L_c \pm 220$, which corresponds to a 15 % accuracy in the determination of the critical parameter. Using the critical value determined above, i.e. $L_c = 1628$, the calculation of the slope of the linear fit with respect to the parameter $\delta$ provides an experimental value for the Landau constant $l = ( \partial |A_s|^2 / \partial \delta )^{-1} = 97.22\ {\rm s}^2\ {\rm mm}^{-2}$.

Figure 9.

Evolution of the square of the saturated amplitude as a function of the electric Rayleigh number.

Another method to derive the critical value of $L$ is to study the growth of the perturbation amplitude $A$ (see Yoshikawa & Wesfreid Reference Yoshikawa and Wesfreid2011). For small amplitudes far from the solid boundaries, (4.1) implies an exponential evolution of $A$ with time. Writing the perturbation amplitude as $A(r,\varphi,t) = A_{0}(r,\varphi )\exp (\sigma t)$, where $\sigma$ is the growth rate of the perturbation, (4.1) leads to $\sigma = \delta /\tau _0$. That latest result indicates that when the perturbation starts to grow, the growth rate of $A$ is proportional to the electric Rayleigh number. To determine the growth rate $\sigma$, a certain interval of time is chosen so that the evolution of $A$ satisfies an exponential growth. The slope of the curve $\log (A) = f(t)$ within the selected interval corresponds to the growth rate $\sigma$. The growth rate is determined through the evolution of the amplitude measured at various radial positions in the interval $r \in [6.5, 8.5]$ mm, where $A$ is evaluated with sufficient accuracy. The interval choice for the estimation of the linear growth is also varied slightly so the resulting error is taken into account.

The variations of $\sigma$ as functions of the electric Rayleigh number are shown in figure 10. The growth rate is zero at the onset of thermo-electric convection, i.e. at $L = L_c$, and increases linearly with $L$ for relatively small values of $L$. A linear fit is applied as a function of $L$ for $L < 6500$, so that the nonlinear term in (4.1) can be neglected. Using that fit to determine the critical value of the electric Rayleigh number, the evolution of $\sigma$ gave the value $L_c = 1695 \pm 400$. However, the two methods gave similar values of the critical parameter, and the difficulty of fitting efficiently an exponential growth to the evolution of $A$ brings about a critical electric Rayleigh number with approximately 25 % accuracy. Using the determined value $L_c = 1695$, the evaluation of the slope of $\sigma$ as a function of $L$ provides an experimental value for the coherent time $\tau _0 = ( \partial \sigma / \partial \delta )^{-1} = 13.37$ s. Figure 10 also shows the evolution of the growth rate obtained from LSA for silicone oils B3 and B5 by fixing all parameters to those of the critical state, except for $V_E$, whose variation induces the variation of $L$. The two curves are linear and cross the line $\sigma = 0\ {\rm s}^{-1}$ at $L = 1498$. The value of $\sigma$ for $L > 1498$ depends slightly on the fluid viscosity, and its dependence on $\gamma _e$ can be neglected. There is a qualitative agreement between the results from LSA and the experimental ones, and the characteristic times $\tau _0$ derived from LSA are $\tau _0 = 16.2$ s for B5, and $\tau _0 = 16.7$ s for B3. These values are in relatively good agreement with the experimental results.

Figure 10.

Evolution of the growth rates of $A$ as functions of the electric Rayleigh number. The results from LSA are also shown for the two different fluids with $k = k_c = 1.528$ and $n = n_c = 4$.

Multiplying (4.1) with the amplitude and neglecting the end effect by suppressing the spatial derivative of the amplitude, we obtain the Stuart–Landau equation (Stuart Reference Stuart1960):

(4.3)\begin{equation} \frac{\partial |A|^2}{\partial t} = \frac{2 \delta}{\tau_0}\,|A|^2 - \frac{2 l}{\tau_0}\,|A|^4. \end{equation}

This equation admits a solution written in the form (Landau & Lifshitz Reference Landau and Lifshitz2013; Drazin & Reid Reference Drazin and Reid2004)

(4.4)\begin{equation} |A(t)|^{{-}2} = \frac{l}{\delta} + \left( |A_0|^{{-}2} - \frac{l}{\delta} \right) \exp \left( - 2 \delta\,\frac{t}{\tau_0} \right) , \end{equation}

where $A_0 = A(t=0)$ is the amplitude at the initial time. The parameter $\delta$ is based on the critical electric Rayleigh number extracted from the saturated amplitude evaluation, i.e. $L_c = 1628$. Using the values of the Landau constant $l$ and the characteristic time $\tau _0$ determined experimentally, we can plot the evolution of the amplitude and compare it with the experimental one. The comparison is shown in figure 11 for various values of $L$. For each of the curves, the initial value of the modes’ amplitude $A_0$ is selected so that the analytically determined curve fits the experimental one the best. The exponential growth in time of the amplitude derived from (4.4) is in quantitative good agreement with the experimental observation. The agreement is still valid for $L = 9227$, for which the growth rate $\sigma$ was found to have an offset compared to the linear extrapolation derived in figure 10. For some given values of $L$, the saturated amplitude $A_s$ determined experimentally might differ from the asymptotic values $A(t \rightarrow \infty ) = \sqrt {\delta /l}$, but the error exhibited a maximum of 15 %. It is found experimentally that for sufficiently large values of $L$, the amplitude overshoots its saturated value and starts to oscillate around the equilibrium state. The solution of the Stuart–Landau equation does not include that mechanism, and higher-order terms would be required in (4.3). Overall, the good agreement justifies the use of the Stuart–Landau equation to characterise the amplitude evolution, and to validate the use of a constant growth rate for the determination of the experimental data shown in figure 10.

Figure 11.

Comparison of the amplitude evolution measured experimentally and calculated analytically from (4.4).