On Chinese space station: pioneering space experiments unraveling the hydrodynamic instability of annular thermocapillary convection

Abstract

On the Chinese space station, a space scientific experiment, named “The microgravity research on oscillation characteristics and transition issues of annular flow”, has completed its research plan for the first stage. The payload, supporting multiple annular fluid models with a heating, rotating or lifting central column, is developed for conducting a series of experiments in the fluid physics rack. This paper presents the space experiment with a fixed inner column and focuses on the volume ratio effect on the critical instability of thermocapillary convection. To observe the transition from steady flow to oscillatory flow, an infrared camera is adopted to record the temperature pattern of the annular flow. The instability critical point, critical frequency and instability mode are analyzed by the dynamic mode decomposition (DMD). The space experiment shows that the volume ratio(Vr) has a significant influence on the mode of thermocapillary instability by the surface configuration. The critical curves are divided into two branches according to critical frequencies, which are low-frequency modes (f = 0.056 ~ 0.072 Hz, Vr = 0.625 ~ 1.065) and high-frequency modes (f = 0.301 ~ 0.419 Hz, Vr = 0.525 ~ 0.575), respectively. The low-frequency modes include various azimuthal wave modes, i.e. regular wave (m = 3, m = 4) and irregular wave (m = 4 ~ 5). The high-frequency mode is the combination of radial waves and azimuthal waves. New coupled patterns, such as one spiral wave with m = 1 azimuthal mode, two spiral waves with m = 2 azimuthal mode and annular radial wave, etc., are discovered. Different modes are growing, coupling and competing during the onset of oscillatory instability. As volume ratio is changing, the mode transition is found as a continuous process of competition, rather than an abrupt change with clear volume ratio boundary.

Introduction

Thermocapillary convection in annular pools is a hydrodynamic model for the crystal growth by Czochralski method. When a radial temperature gradient is established via heating or cooling the central column, thermocapillary convection arises from the surface tension gradient^1,2. On the one hand, the dynamics of thermocapillary convection can well explain the generation of defects or segregation during crystal growth, which is helpful to improve the crystal quality. On the other hand, annular flow has periodic boundaries in the azimuthal direction, which provides an ideal model for studying flow instability issues, such as hydrothermal waves and surface waves^3,4. Therefore, the hydrodynamic instability, pattern formation, mode evolution, and transition paths to chaos in annular flow have attracted much attention from the scientific community^5,6,7.

The critical stability specifically refers to the transition from axisymmetric time-independent flow to oscillatory flow when the temperature difference exceeds a certain critical value^8,9. The mechanism of the oscillation is ascribed to the hydrothermal wave, which is proposed by Smith and Davis¹⁰ when they studied the thermocapillary instability in an infinitely horizontal layer. Garnier and Chiffaudel¹¹ found that thermocapillary waves in an annular pool propagate spirally from the inside cold column to the outside. The formation of spiral waves is well explained by the hydrothermal wave theory in which the wave direction always maintains a certain angle with the temperature gradient. Garnier et al.¹² found that the oscillatory mode appears as “flower wave” when a temperature gradient is applied reversely. Yu et al.¹³ considered the solute capillary effect on the thermocapillary convection in a shallow annular pool. The bidirectional waves and multi-frequency oscillations were observed, indicating that multiple basic modes exist in the complex thermo-solute-capillary systems.

In the earlier study of microgravity fluid dynamics, two space experiments were proposed to study the thermocapillary instability in annulus flow. Schwabe et al.^14,15,16 completed space experiments in the FOTON-12 satellite, known as MAGIA (Marangoni Grown Instabilities in an Annulus). They studied the thermocapillary waves in the central cooled annular pools with different aspect ratios. Various unsteady modes are observed, such as unstable traveling or standing waves, two opposite traveling waves from source point, and local traveling waves. Kamatani et al.^17,18 carried out a series of space experiments from 1992 to 1995 to study thermocapillary convection with different sizes of heating columns. A two-lobed rotating pattern (azimuthal traveling wave with m = 2) and two-/three-lobed pulsating patterns (standing waves with m = 2 or 3) were reported in their experiments. However, on the International Space Station, more research has focused on thermocapillary convection in liquid bridges. For example, the MEIS-1, MEIS-2, and MEIS-4 experiment series conducted by the JAXA also validated the theoretical models of hydrothermal waves (HTW).

For simplicity most of space experiment focus on the flat or cylindrical surface, which means the volume ratio Vr = 1. Hu et al.¹⁹ put forward to researching the volume ratio effect and the transition process in china’s space experiment. The volume ratio effect is a key influencing factor in microgravity flow, which changes the interfacial curvature and the instability mode²⁰. –²¹ In the SJ-10 scientific satellite, Kang and Duan et al. studied^22,23 the thermocapillary convection in the inner-heated annular cavity, and conducted 23 sets of experiments with different volume ratios. Regular waves are found in large volume ratio, where the azimuthal wave number decreases from m = 4 to m = 3 as the volume ratio decreases. The oscillatory fluctuation enters an irregular wave state in small volume ratio. The experiment gives that the critical curve with volume ratio has two branches, which are consistent with the volume ratio effect put forward by Hu et al.²⁴ in the earlier research on liquid bridges. Moreover, to confirm Hu’s theory, Kang and Duan²⁵ also conducted the research on thermocapillary flow in liquid bridges with a variety of volume ratios and aspect ratios in the Tiangong-2 laboratory. All these studies have identified the high-frequency mode at small Vr and the low-frequency mode at large Vr, corresponding to a two-branch marginal stability curve. However, the composition of these modes and their competitive interactions during instability require more in-depth investigation.

Based on experiments conducted on the SJ-10 satellite, our team designed and implemented a space experiment on annular flow to systematically investigate oscillatory modes under different volume ratios, with additional consideration given to central column rotation. This project, named “Research on Oscillation Characteristics and Transition of Annular Flow under Microgravity,” aims to establish a hydrodynamic model for the Czochralski process that incorporates central column heating, rotating, and pulling. The coupling between forced convection and Marangoni flow is predicted to exhibit rich mode competition phenomena, with quasi-steady flow having been previously observed in ground-based experiments²⁶. Concurrently, another experiment named “Study of Supercritical Flow” is conducted using an extravehicular payload mounted on the Tianzhou-6 cargo spacecraft, designed to investigate supercritical bifurcations and transitions in annular thermocapillary instability at high Marangoni numbers²⁷. These experiments will collect precious data about the new models of annular flow and their supercritical transition process.

Since China’s Manned Space Station laboratory provides extensive opportunities for long-term, crew-attended space experiments, we conducted experiments to investigate the volume ratio effect on wave modes. The annular flow payload was installed in the fluid physics rack, which is capable of transmitting tens to hundreds of gigabytes of experimental data²⁸. The data-driven approach enables analysis of the mode components of both high-frequency and low-frequency modes, as well as their competitive interactions. By comparing mode component variations across different volume ratios, we hope to gain deeper insights into how interfacial effects influence critical mode selection. Understanding these critical instabilities establishes the foundation for studying more complex coupled flows involving rotation and pulling effects.

This paper systematically investigates the critical instabilities of the annular thermocapillary convection in China’s Manned Space Station. Through dynamic mode decomposition, the image data sequences are analyzed to capture the oscillatory mode. This study focuses on the critical wave modes and the mode evolution. By comparing the critical onset, frequency and spatial-temporal characteristics of the wave modes, this study aims to investigate the selection of wave patterns by the changing of volume ratio. A better understanding of the volume ratio effect on the mode instability and transition is obtained.

Results

Critical conditions for oscillatory instability

When the heating column is fixed, the critical conditions for the transition from axisymmetric flow to oscillatory flow are presented in Table  1. It shows 12 typical space experiments with volume ratios ranging from 0.525 to 1.060. In each case, the inner column is heated linearly for 90 min, and an infrared camera, a CCD and thermocouples are used to observe the transition of flow state. Each experimental case generates 15 GB of data, including infrared image data (14.2GB, about 23240 images).

The results show that the volume ratio has a significant influence on the critical temperature difference and wave mode. The maximum critical condition (case10: Vr = 0.625) is approximately three times that of the minimum critical condition (case12: Vr = 0.525). The influence of volume ratio on the dimensionless critical conditions will be discussed in “Dimensionless critical condition and frequency”. The critical conditions depend on the volume ratio. Consequently, the fluid temperature at the onset changes, which has a large influence on the viscosity of silicone oil. The viscosity conforms to the following expression:

$$\:\nu\:={\nu\:}_{25}exp\left(5.892\frac{25-T}{273.15+T}\right).$$

(1)

As the kinematic viscosity decreases with rising temperature, the Prandtl numbers ($\:Pr=\nu\:/\kappa\:$) at the onset are slightly different, as shown in Table 1.

Table  1 also presents the influence of volume ratio on the critical frequency. According to the frequency, the critical modes are divided into two categories: (1) low-frequency mode (f = 0.056 ~ 0.072 Hz, Vr = 0.625 ~ 1.065); (2) high-frequency mode (f = 0.301 ~ 0.419 Hz, Vr = 0.525 ~ 0.575). Figure  1 (a) depicts the critical temperature differences for volume ratios ranging from 0.525 to 1.06. Correspondingly, the critical frequency versus the volume ratios is plotted in Fig.  1 (b). The critical curves can be divided into two discontinuous branches, corresponding to two instability mechanisms: high-frequency mode (small volume ratio: Vr < 0.6) and low-frequency mode (large volume ratio: Vr > 0.6). In high-frequency mode, the critical temperature difference is small (~ 10℃) and the critical frequency is 0.3 ~ 0.4 Hz. In low-frequency mode, the critical temperature difference shows a decrease from 32℃ to 12℃, a 62.5% reduction. Meanwhile, the decrease of critical frequency is relatively slow, which decreases from 0.072 Hz to 0.056 Hz, a 22.2% reduction.

Compared the space experiment with the numerical simulation works in Ref.³, the right branch of critical curve is in qualitative agreement. The simulation investigates 8 thermocapillary convection cases with different ratios within 0.809 ≤ Vr ≤ 1.173 under microgravity. It found that the critical frequency decreases monotonically from fc = 0.0464 Hz to 0.0339 Hz, and the critical temperature difference similarly exhibits a monotonic decrease from ΔTc = 12.92 °C to 6.05 °C .Two azimuthal wave modes with m = 4 and m = 3 are observed and the wave number decreases with increasing volume ratio. Furthermore, oscillations commence as standing waves at the critical condition, and these standing waves transition into traveling waves at a slightly supercritical condition. However, the space experiment captured plenty of competition processes near the critical condition, which are not expected by numerical simulations. Especially, the high-frequency mode occurring on the highly deformed surface under small Vr conditions is not predicted by the numerical simulations.

Table 1 The critical condition of space experiment cases.

The low-frequency oscillation and high-frequency oscillation correspond to two types of hydrothermal waves. The low-frequency modes are further subdivided into azimuthal waves with m = 3 (Vr > 0.90), m = 4 (0.75 < Vr < 0.90), and irregular waves (0.60 < Vr < 0.75). Experimental observations show that as the wave number increases with decreasing volume ratio, the m = 5 mode emerges at small Vr. Consequently, the competitive process between m = 4 and m = 5 is found in the irregular waves at small volume ratios. The wave pattern change of high-frequency mode does not lead to a sudden change in frequency or critical condition. When Vr < 0.60, the azimuthal wave transitions to the coupled mode of radial wave and azimuthal wave, which brings about abrupt jumps in both the critical temperature difference and frequency. The coupled wave in radial and azimuthal directions is a new phenomenon discovered in this space experiment. The wave modes of annular thermocapillary convection will be discussed in detail in “Volume ratio effect on wave patterns”.

Fig. 1

The volume ratio effect on critical condition. (a) The critical temperature difference, (b) the critical frequency.

Volume ratio effect on wave patterns

According to the wave patterns and dynamics, the low-frequency mode and the high-frequency mode are composed of different wave types. The low-frequency mode is wave propagating in azimuthal direction, including: (A) m = 3 azimuthal wave, (B) m = 4 azimuthal wave and (C) irregular wave. While the high-frequency mode is (D) coupled radial and azimuthal waves, in which the waves propagate in the radial and azimuthal direction coexist.

A. m = 3 azimuthal wave

The azimuthal wave with m = 3 emerges in the experiments with a relatively large volume ratio(Vr ranges from 0.90 to 1.06). As shown in Fig. 2, the typical image sequence of the azimuthal wave with m = 3 is presented (at Vr = 1.025 ). The critical temperature difference at which the wave pattern first appears is ΔT = 11.461℃ (at t = 1725 s). After a growing process, the wave develops to a stable oscillatory state as shown in Fig. 2, at ΔT = 13.879℃ (t = 2102s ~ 2120.5 s). The oscillation mode is a counterclockwise rotating structure with three petals marked as 1, 2, and 3 respectively. One period of the m = 3 travelling wave corresponds to the petals rotating by 120°. One period is completed when the petal rotates 120°. Figure 2 shows one period of oscillation, in which Fig. 2(f) returns to the initial state of Fig. 2(a). Therefore, the critical mode when Vr = 1.025 is a counterclockwise travelling wave with m = 3, and its period is 18.5s.

Fig. 2

m = 3 azimuthal wave when Vr = 1.025. (t₀ = 2102s, Period T = 18.5s, ΔT = 13.879℃).

B. m = 4 azimuthal wave

The wave mode of m = 4 predominates with a volume ratio of 0.750 ~ 0.900. Figure 3 shows the typical image sequence of the azimuthal wave with m = 4 (at Vr = 0.875). The critical temperature difference at which the instability occurs in this condition is ΔT = 19.206℃ (t = 3215s). While Fig. 3 gives a stable oscillation when the temperature difference is ΔT = 22.457℃. Four petal-shaped structures, marked as 1, 2, 3, and 4 respectively, are rotating clockwise. When the petal-shaped structure rotates 90°, it goes through an oscillation period, which is 32.5s. Compared with the wave configuration of m = 3 (Fig. 2), the maximum points of petal are more concentrated to the central column and the peak-to-peak values are larger. The wave intensity and configuration fluctuate slightly during the period. It implies that the wave is not an ideal travelling wave which may couple with standing wave or other wave modes.

Fig. 3

m = 4 azimuthal wave when Vr = 0.875. (t₀ = 3555s, Period T = 16s, ΔT=$\:22.457$℃).

C. Irregular wave

Irregular waves emerge at a small volume ratio within the range of Vr = 0.600 ~ 0.750. Figure 4 shows one oscillation period of the Vr = 0.725 case, where the dynamics of petals exhibit complex characteristics. The alternately pulsating petals and rotating petals coexistin this wave pattern. The upper part of Fig. 4 is a clockwise travelling wave, and the lower part is an alternately appearing standing wave. The blue arrow in the figure indicates the moving direction of the petal, and the red label marks newly generated petals due to the standing wave. It is found that the number of petals changes at different snapshots. For instance, the petal number m = 4 in Fig. 4(a), (d), and (f), whereas the petal number m = 5 in Fig. 4(b), (c), and (e). This indicates that there is a strong coupling between the wave number m = 4 mode and the wave number m = 5 mode.

Fig. 4

m = 4 azimuthal wave when Vr = 0.725. (t₀ = 4229s, Period T = 16.5s,ΔT = 25.418℃).

D. Coupled radial and azimuthal waves

When the volume ratio is reduced below 0.6, the critical instability transitions to a high-frequency mode. The high-frequency mode is a new waveformwhose pattern is coupled with the azimuthal wave and the radial wave. The waveform can be divided into two regions of fluctuation: the azimuthal wave occupies the inner region, while the radial wave is generated at the outer region. Figures 5 and 6 show the coupled radial and azimuthal waves in the case of Vr = 0.575 and 0.525. The boundary of two regions is marked by a dashed circle in Fig. 5(a) and Fig. 6(a). The diameter of the azimuthal wave decreases from 25.9 mm to 20.8 mm as the volume ratio decreases from Vr = 0.575 to Vr = 0.525. The critical frequency of high-frequency mode is more sensitive to the volume ratio than the low-frequency mode. This sensitivity of the high-frequency mode to volume ratio can be explained by the significant change in the high-frequency mode’s waveform. The decrease of the scale of azimuthal wave region increases the critical frequency. As shown in Fig.  1(b), there is a significant increaseof the critical frequency when the volume ratio decreases.

The inner region for azimuthal wave can be further separated into two annular layers: the inside layer and the outside layer, as shown in Fig. 5(b). Each layer contains petal-like structures such as the cold petals and hot petals. The change in azimuthal wave scale leads to different wave number selections in the azimuthal wave region. The shrinkage of the azimuthal region increases the ratio of circumference to width, which increases the capacity for azimuthal waves. Thus, the mode of Vr = 0.575 is m = 1 standing wave, in which the petals are shifting in Fig. 5, and the mode of Vr = 0.525 is m = 2 travelling wave, in which the petals are rotating in Fig. 6.

The waveform selection of our radial wave is related to the wave mode of the azimuthal wave. When the volume ratio Vr = 0.525, the azimuthal m = 1 wave generates a concentric travelling wave. The concentric travelling wave prefers to couple with a standing wave, because the standing wave fluctuates synchronously in the azimuthal direction. Different from them = 1 wave at Vr = 0.575, the m = 2 travelling wave is in the form of two rotating petals, which generate two spiral waves propagating outwards. Therefore, we infer that the outer radial waveform is determined by the inner azimuthal waveform, since the oscillation of the azimuthal wave provides the disturbance source for the radial wave in the outer region.

Fig. 5

Coupled radial and azimuthal waves when Vr = 0.575. (t₀ = 2496s, Period T = 3.25s,ΔT = 14.678℃).

Fig. 6

Coupled radial and azimuthal waves when Vr = 0.525. (t₀ = 2247s, Period T = 2.25s, ΔT = 10.911 ℃).

Discussion

Dimensionless critical condition and frequency

Figure 2(a) shows a jagged distribution of the critical temperature differences versus the volume ratio. This is related to the fluctuation of ambient temperature, because two experiments are conducted within one day and the second experiment is conducted in a higher ambient temperature. According to the properties of silicone oil, the high temperature will lead to a low viscosity. To eliminate the environmentally induced fluctuations of critical conditions, dimensionless analysis of the critical conditions of instability is required.

Since the air convection in microgravity is suppressed, we simplify the liquid interface as an adiabatic boundary. Thus, the influence of the Biot number is not considered. Thermocapillary convection is characterized by the Marangoni number and the Prandtl number:

$$\:Ma=\frac{\left|{{\upsigma\:}}_{\text{T}}\right|D\varDelta\:\text{T}}{\mu\:\kappa\:},$$

(2)

$$\:Pr=\frac{\nu\:}{\kappa\:}.$$

(3)

The Reynolds number can be calculated using the Marangoni number and the Prandtl number, that is, it can be expressed as

$$\:Re=Ma/Pr.$$

(4)

In Table  1, the Prandtl (Pr) and Reynolds (Re) numbers under critical instability conditions are calculated. Figure 7 illustrates the critical Reynolds number as a function of the volume ratio. The results reveal that the critical condition given by the Reynolds number can yield a smoother critical curve.

Fig. 7

The critical Re number vs. volume ratio.

By fitting the critical condition of high-frequency mode, the scaling law between the critical Re and volume ratio satisfies Ma ~ Re ~ Vr^− 3. With the decrease of volume ratio leads to a concave surface, the equivalent depth H also decreases, with H ~ Vr^− 3. We proposethat the increase of Reynolds number as the volume ratio decreases is related to the equivalent depth, because the flow with a thin layer is more stable due to the viscous dissipation. The stability of thermocapillary convection is primarily influenced by the coupling effects of surface tension, thermal diffusion, and viscosity. Surface tension is the driving force, while viscosity and thermal diffusion play a dissipative role. We use the equivalent depth H to define the characteristic scales of these effects, shown in Table 2.

Table 2 Characteristic scales of thermocapillary instability.

The effect of surface tension is mainly related to the length in the radial direction and the characteristic acceleration is

$$\:{\text{a}}_{{\upsigma\:}}=\frac{\left|{-{\upsigma\:}}_{\text{T}}\right|\cdot\:\varDelta\:\text{T}}{{\uprho\:}{\text{D}}^{2}}.$$

(5)

Therefore, the characteristic time scale of surface tension is:

$$\:{\tau\:}_{\sigma\:}^{2}=\frac{D}{{a}_{\sigma\:}}=\frac{\rho\:{D}^{3}}{\left|{-{\upsigma\:}}_{\text{T}}\right|\cdot\:\varDelta\:\text{T}}.$$

(6)

Since the annular thermocapillary flow is axisymmetric, the influence of viscosity is related to the scale of radial cross-section, $\:HD$. Therefore, the characteristic time of viscous action is

$$\:{\tau\:}_{vis}=\frac{HD}{v}.$$

(7)

The height of annular flow H is chosen when analyzing the heat diffusion, so the time scale of thermal diffusion is

$$\:{\tau\:}_{th}=\frac{{H}^{2}}{\kappa\:}.$$

(8)

Through these characteristic times, a new Marangoni number is derived as:

$$\:\stackrel{\sim}{Ma}=\frac{{\tau\:}_{vis}\cdot\:{\tau\:}_{th}}{{\tau\:}_{\sigma\:}^{2}}=\frac{\left|{-{\upsigma\:}}_{\text{T}}\right|\cdot\:\varDelta\:\text{T}\cdot\:{D}}{\rho\:v\kappa\:}{\cdot\:\left(\frac{H}{D}\right)}^{3}\sim Ma{\cdot\:Vr}^{3}\sim constant.$$

(9)

which is an approximately constant critical condition.

The critical frequency is also a key parameter, which decreases as the volume ratio increases. The traditional definition of the dimensionless frequency is $\:\stackrel{\sim}{{f}_{\text{c}}}=\frac{{f}_{c}{D}^{2}}{v}$. Figure 8 shows the dimensionless frequency and the fitting relation is $\:\stackrel{\sim}{{f}_{\text{c}}}$ $\:\sim{Vr}^{-0.96}$. The increase of volume ratio, on the one hand, increases the spatial scale of flow, and, on the other hand, decreases the average fluid viscosity due to the low critical temperature difference. To consider both the influence of flow scale and viscosity, we have selected the time scale of viscosity $\:{\tau\:}_{\text{v}\text{i}\text{s}}=HD/v$ to define the dimensionless frequency as

$$\:\stackrel{\sim}{{F}_{c}}={f}_{c}{\cdot\:\tau\:}_{vis}=\frac{{f}_{c}HD}{v}\sim constant.$$

(10)

Figure 8 shows that the dimensionless frequency $\:\stackrel{\sim}{{F}_{\text{c}}}$ is close to the constant 6.43. This resultimplies that the thermocapillary oscillation is strongly correlated with the viscous dissipation effect. In conclusion, the dimensionless critical condition and critical frequency are close to a constant, which demonstrates that the characteristic scales in Table 2 can explain the instability mechanism of the high-frequency mode.

Fig. 8

Dimensionless critical frequency vs. volume ratio.

The dimensionless frequency of the low-frequency mode can be extended to the high-frequency mode. The abrupt increase of frequency can be explained by the waveform change, which is caused by the mode transition at Vr = 0.6. In the low-frequency mode, the azimuthal wave, such as the m = 3 wave, the m = 4 wave and the irregular wave, occupies the whole region of the pool. As the mode transitions to the coupled wave, the diameter of the azimuthal wave region suddenly decreases. The diameter of azimuthal wave $\:{D}_{\text{a}\text{z}}$ is 25.9 mm at Vr = 0.575. To estimate the mean height of azimuthal wave region, the meniscus is solved by Young-Laplace equation. By taking the average of the surface height where r < $\:{\text{D}}_{\text{a}\text{z}}$/2, the mean height of azimuthal wave region is $\:{H}_{\text{a}\text{z}}$ = 5.06 mm. According to the dimensionless frequency F_c=6.43, we estimate the frequency as $\:{f}_{\text{c}}=\nu\:{F}_{\text{c}}/\left({H}_{\text{a}\text{z}}{D}_{\text{a}\text{z}}\right)$ = 0.26 Hz, which is in agreement with the experimental frequency of 0.301 Hz. Therefore, the dimensionless frequency $\:\stackrel{\sim}{{F}_{\text{c}}}$ in Eq. (10) can characterize well the oscillations of both low-frequency and high-frequency modes.

DMD analysis of the oscillation process

By selectingan appropriate time window, we sample the time-series of infrared images covering the whole process from stability to oscillation. The dynamic mode decomposition (DMD) technique is applied to learn the mode components and their evolution during the onset process. Interestingly, the DMD technique obtains the high-order modes that cannot be directly observed with the naked eye. The mode evolution of four typical wave patterns, ordered by decreasing volume ratio, are discussed as follows.

A. m = 3 azimuthal wave

The volume ratio condition for the m = 3 azimuthal wave ranges from 0.900 to 1.060. Figure 2 shows the critical waves for Vr = 1.025.We conduct DMD analysis to investigatehow the wave mode evolvesfrom the critical instability to the m = 3 travelling wave. The infrared image sequencefrom t = 1625 ~ 2125s is chosen, which covers the whole growing process of instability. Figure 9 shows the real part and phases of the first five DMD modes, labeled as mode-1, -2, -3, -4, and − 5. The real part of the DMD modes, given in Fig. 9 (a1)-(e1), shows the wave patterns. The mode phases, whose gradient represents the wave direction and velocity, are shown in Fig. 9 (a2)-(e2). Thus, the phase of travelling waves exhibits a gradual gradient, while standing waves shows abrupt changes. Table 3 provides the mode characteristics, such as the wavenumber m, the oscillation frequency f, and the growth rate σ.

The DMD analysis reveals that the growing process involves interaction between multiple waves, including the m = 3 wave (mode-1 and − 2), m = 4 wave (mode − 5) and m = 6 wave(mode-3 and − 4). Though mode-1 and mode-2 have the same wave number m = 3, their propagating dynamics are different. It can be seen from the phase diagram in Fig. 9 (a2) and (b2) that the mode-1 is a travelling wave while the mode-2 is a coupled mode of travelling wave and standing wave. As shown in Table 3,the mode-3 and mode-4 are the secondary harmonics, which have double wave numbers and frequencies of mode-1 and mode-2. The mode-3 is a m = 6 azimuthal travelling wave with curved petals, while mode − 4 is a m = 6 standing wave with axisymmetric petals. Interestingly, though the m = 3 waves and their harmonics predominate, a m = 4 wave, which is mode-5 shown in Fig. 9 (e1)-(e2), is discovered during the critical instability. The wavenumbers of m = 3 and m = 4 are incommensurable, so they are two basic modes: (1) m = 3 (mode-1 and mode-2) and (2) m = 4 (mode-5).

Fig. 9

Dynamic mode decomposition (Vr = 1.025, t = 1625–2125 s).

Table 3  The DMD modes during the critical instability in the Vr = 1.025 case.

Since the evolution of basic modes determines the instability process, the contribution of basic modes to instability is studied. Figure 10 (a), (b) and (c) respectively show the spatial-temporal evolution diagrams of mode-1, mode-2 and mode-5. The spatial-temporal evolution of mode-1 in Fig. 10 (a) presents a striped structure, which indicates a clockwise travelling wave. The spatial-temporal evolution of mode-2 in Fig. 10 (b) presents a speckled structure, which indicates a standing wave. The competition between mode-1 and mode-2 drives the critical oscillation to evolve from a standing wave to a travelling wave. Figure 11 (b) shows the reconstructed spatial-temporal evolution obtained by the superposition of mode-1 and mode-2. The reconstructed spatial-temporal evolution is consistent with the experimental result (Fig. 11 (c)), where the speckled structure appears first and evolves into the striped structure.

The critical oscillation process also includes the competitive process between modes form = 3 (mode-1 and mode-2) and modes form = 4 (mode-5). From the spatial-temporal evolution in Fig. 10 (c), the pattern is dominated by speckles and is an m = 4 standing wave. To study the influence of the m = 4 mode, we compare the spatial-temporal evolution of the m = 3 mode and the superposition of m = 3 and 4 in Fig. 11 (b, c). It is found that the superposition of the m = 4 mode on the m = 3 mode has a negligible effect on the spatial-temporal evolution. Therefore, the critical oscillation process is dominated by the competition between mode-1 and mode-2, where the m = 3 mode transitions from a standing wave to a travelling wave.

The nonlinear competition between basic modes determines the critical instability process. Figure 11 (a) shows the amplitude evolution of basic modes during the linear stability and nonlinear saturation. The amplitude of mode-1 increases monotonically before saturation. The amplitude of mode-2 first increases, and then decreases after the saturation of mode-1. It means that the travelling wave mode-1 has an advantage in its competition with the standing wave mode-2. Thus, the transition from standing wave to travelling wave can be explained by the amplitude evolution of the modes. In addition, the amplitude of mode-5 is much weaker than that of mode-1 and mode-2. It grows when 100s < t < 200s, and decays after t > 200s. The timing of mode-5 decay exactly coincides with the growth of the m = 3 mode at t > 200s. The decay of mode-5 can be attributed to the competition from the m = 3 modes (mode-1 and mode-2).

Fig. 10

The azimuthal spatial-temporal evolution of DMD modes. (a) The azimuthal spatial-temporal evolution of mode-1, (b) the azimuthal spatial-temporal evolution of mode-2, (c) the azimuthal spatial-temporal evolution of mode-5.

Fig. 11

Comparison of mode reconstruction and experiment. (a) The amplitude evolution of basic modes, (b) the superposition of modes-1,-2, (c) the superposition of modes-1,-2,-5, (d) the experimental spatial-temporal evolution.

B. m = 4 azimuthal wave

The range for the m = 4 azimuthal wave is Vr = 0.75 ~ 0.90, and the case of Vr = 0.875 is in the adjacent region between the m = 3 wave and the m = 4 wave. The DMD mode of its critical instability is shown in Figs. 3, 4, 5 and 6. The basic modes are the m = 4 wave and the m = 3 wave. Mode-1 is the m = 4 azimuthal travelling wave with a frequency of 0.062 Hz, and Mode-2 is the m = 4 azimuthal standing wave with a frequency of 0.061 Hz. Another basic mode is the mode-4, which is an m = 3 standing wave. Mode-3 and Mode-5 are the m = 8 harmonics, whose frequencies are close to the double frequency of the basic mode. The mode-3 mode is a m = 8 travelling wave. Although mode-5 still exhibits an m = 8 azimuthal wave pattern, its azimuthal wave character has been greatly attenuated. The pattern of mode-5 presents a blurred petal-like structure in Fig. 12 (d).

Fig. 12

Dynamic mode decomposition(Vr = 0.875, t = 3075–3575 s ).

Table 4 The DMD modes during the critical instability in the Vr = 0.875 case.

Figure 13 compares the amplitude evolution and the spatial-temporal evolution of three basic modes. The onset is the wave disturbance from a standing wave to a travelling wave. The spatial-temporal evolution from the experiment can be found in Fig. 13 (d). The basic modes of the Vr = 0.875 case [m = 4 (mode-1 and mode-2) and m = 3 (mode-4)] are given in Fig. 13 (a). The superposition of mode-1 and mode-2 is given in Fig. 13 (b), which reflects the evolution from standing wave to travelling wave. The spatial-temporal evolution of mode-1 and mode-2 fits the experimental result well, and the influence of m = 3 mode-4 can be neglected in Fig. 13 (c). Although the amplitude of mode-4 is small, it has an increasing tendency which forecasts the instability of m = 3. Therefore, though the critical instability of m = 3 is dominated in the region of Vr = 0.75 ~ 0.90, the high-order modes of the onset are influenced by the unstable mode in its neighboring region (Tables 4, 5 and 6).

Fig. 13

Comparison of mode reconstruction and experiment. (Vr = 0.875). (a) The amplitude evolution of basic modes, (b) the superposition of modes-1,-2, (c) the superposition of modes-1,-2,-4, (d) the experimental spatial-temporal evolution.

C. Irregular wave

The irregular wave happens in the Vr = 0.600 ~ 0.750 range, where the wave pattern is no longer a standard travelling or standing wave. The DMD analysis shows that the irregular wave is dominated by a few modes. As shown in Fig. 14, the irregular wave modeconsists mainly of the first three DMD modes.Mode-4 and mode-5 exhibit no obvious wave patterns. The basic modes are the m = 5 modes (mode-1) and the m = 4 modes(mode-2 and mode-3). The phase diagrams show that these modes exhibit strange characteristics of both travelling waves and standing waves. Different from the previous critical mode, there are two positive growth modes of m = 5 (mode-1) and m = 4 (mode-2), which have growth rates of 3.57 × 10^− 3 and 4.90 × 10^− 3. This indicates the presence of two linearly unstable modes that compete with each other in the critical state. The frequencies of mode-1, -2, and − 3 are 0.061 Hz, 0.068 Hz, and 0.065 Hz, respectively. The interaction of different frequency modes results in non-periodical oscillation. Therefore, the irregular wave contains several modes whose waveforms are coupled with local travelling waves and local standing waves.

Fig. 14

Dynamic mode decomposition(Vr = 0.725, t = 3750–4250 s ).

Table 5 The DMD modes during the critical instability in the Vr = 0.725 case.

Figure 15 (a) shows the competition process of the three basic modes. The linear growing state of basic modes begins at t = 250s. Mode-1, mode-2 and mode-3 are increasing simultaneously, and mode-1 is the fastest growing mode. After t > 350s, mode-2 grows rapidly and mode-3 is suppressed. When t = 350s, the amplitudes of mode-1 and mode-2 are almost equal. By comparing with the spatial-temporal evolution in Fig. 15 (b–c), the mode-3 cannot be neglected, which leads to the local travelling wave at the very beginning of the onset. During the initial stage (250 s < t < 350 s), with the simultaneous growth of mode-1, -2 and − 3, the azimuthal wave mode is locally counter-propagating travelling waves with obvious sources and sinks. When t > 350s, the local travelling wave evolves into a mixture of local standing waves and local travelling waves. Therefore, the irregular wave is a complex combination of standing and travelling waves.

Fig. 15

Comparison of mode reconstruction and experiment. (Vr = 0.725, t = 3750–4250 s ). (a) The amplitude evolution of basic modes, (b) the superposition of modes-1,-2, (c) the superposition of modes-1,-2,-3, (d) the experimental spatial-temporal evolution.

D. Coupled radial and azimuthal waves

In the former section A, B and C where Vr > 0.6, the waves are low-frequency waves, while when Vr < 0.6 the waves are high-frequency waves. The high-frequency wave pattern is a complex couple of radial wave at the outer region and azimuthal wave at the inner region. The DMD mode analysis of Vr = 0.575 is shown in Fig. 16. The basic modes are mode-1, mode-2 and mode-3, while mode-4 is a harmonic with doubled frequency. Mode-1, mode-2, and mode-3 are dominated by a two-layer standing wave inside, while outside, they are dominated by radial travelling waves. The mode-4 is an m = 2 azimuthal wave with only one layer and the radial wave is a spiral wave, which covers a large area and overlaps with the second layer of azimuthal waves in mode-1,-2 and-3. Thus, radial waves and standing waves are not separated but overlapped. The phase lag observed between the layers implies that a radial wave component exists in the azimuthal wave area.

Fig. 16

Dynamic mode decomposition(Vr = 0.575, t = 1067–1333 s ).

The DMD modes of the Vr = 0.525 case are shown in Fig. 17. Mode-1 is m = 2 waves coupled with two spiral waves. The inner structure is two rotating hot lobes, and the outer structure is two spiral waves rotating synchronously. The radial wave region of mode-2 exhibits an eccentric annular wave propagating outside. This wave is related to the “source” and “sink” structures at the inner azimuthal wave region. The disturbance is generated from the “source”, then propagates separately and finally converges to the “sink”. As a result, the radial wave is excited with a phase difference in azimuthal direction, which leads to the eccentric circular fluctuation in the radial wave region. In Mode-3, we only observed one C-shaped hot lobe in the second-layer azimuthal wave, and only one spiral wave structure exists in the outer radial wave region. Mode-4, a secondary harmonic, has significant spiral wave characteristics. These modes are asymmetric in the circumferential direction, which may be related to the asymmetry of the liquid surface in the extremely small volume ratio. Although both the m = 1 and m = 2 waves are found, the dominant wave instability is the m = 2 azimuthal travelling wave coupled with two spiral waves.

Fig. 17

Dynamic mode decomposition (Vr = 0.525, t = 933–1200 s ).

Table 6 The DMD modes during the critical instability in the Vr = 0.725 case.

Mode competitions

According to the linear stability theory, the critical mode is the one that first has a non-negative growth rate during the heating-up process. The critical mode is unique except at the intersection point of the two critical curves. Since the unique mode will grow exponentially and overcome other competing modes, a sudden transition in the critical oscillatory state of the annular flow as the volume ratio changes is expected. Thus, the critical oscillation selects one mode from m = 3, m = 4 or other modes. However, space experiments have shown that multiple modes competed during the onset process. As the volume ratio decreases, a transition from m = 3 to m = 4 is continuous, so that in the adjacent areas around Vr = 0.9, m = 3 and m = 4 modes coexist and compete. For a similar mechanism, the competition of low-frequency mode and high-frequency mode is also found in their adjacent region.

As the wavenumber increases with the decrease of the volume ratio, the critical instability transitions from the m = 3 wave to the m = 4 wave at Vr = 0.9. A strong coupling between the modes of m = 3 and m = 4 is observed at Vr = 0.925. Since the wave is dominated by the m = 3 mode, the DMD mode components, as shown in Fig. 18, are similar to those in Fig. 9 (at Vr = 1.025). However, the m = 4 mode becomes the 4th mode, whose amplitude has exceeded that of the m = 6 standing wave mode. Figure 19 (a) shows the amplitude evolution of the basic modes (mode-1, mode-2 and mode-4), and the spatial-temporal evolutions reconstructed by mode-1/ -2 combined and modes-1/ -2/ -3 are shown in Fig. 19 (b-c).

Figure 19 (b) shows that mode-1 and mode-2 characterize the transition from standing wave to travelling wave, in which the speckled structure transforms into the striped structure. The nodes of a standing wave are slowly rotating with an angular velocity of 0.0525°/s. The rotation velocity can be explained by the frequency difference $\:\varDelta\:f$ between mode-1 and mode-2. As $\:\varDelta\:f=0.001\text{H}\text{z}$is given in Table 7, the drift velocity is theoretically calculated by $\:\frac{\partial\:\phi\:}{\partial\:t}=\frac{360\cdot\:\varDelta\:f}{2\cdot\:m}=0.06^\circ\:/s$.Therefore, mode-1 and mode-2 can reflect both the transformation from standing wave to travelling wave and the drift phenomenon.

Fig. 18

Dynamic mode decomposition (Vr = 0.925, t = 2625–3375 s).

Table 7 The DMD modes during the critical instability in the Vr = 0.725 case.

In Fig. 19 (d), a long-period modulation of spatial-temporal evolution is found in the initial stage of instability (200 ~ 400s). The m = 3 mode (Fig. 19 (b)) cannot reconstruct this long-period structure. The modulation period contains 8 oscillatory periods indicating that there exist two competing modes, with frequency difference of about f/8 = 0.0078 Hz. The frequency of mode-4 is 0.0545 Hz, which differs from the m = 3 mean frequency by 0.0081 Hz. Thus, it can be inferred that the beat frequency phenomenon of the m = 3 mode and the m = 4 mode results in this long-period modulation structure. By superimposing the m = 4 (mode-4) with the m = 3 mode (mode-1 and mode-2), the spatial-temporal evolution in Fig. 19 (c) can reconstruct the long-period instability well. Therefore, when Vr = 0.925, the m = 4 mode is strong enough to affect the critical m = 3 mode and forms a long-period modulated instability process.

Fig. 19

Comparison of mode reconstruction and experiment. (a) The amplitude evolution of basic modes, (b) the superposition of modes-1,-2, (c) the superposition of modes-1,-2,-4, (d) the experimental spatial-temporal evolution (Vr = 0.925, t = 2625–3375 s).

There are two trends of mode transition in the case of Vr = 0.775. The DMD mode of critical instability at Vr = 0.775 is shown in Fig. 20. (1) When the volume ratio is reduced to Vr = 0.775, a wave component of m = 5 appears, which conforms to the trend that the wave number increases as the volume ratio decreases. The instability is still dominated by the azimuthal wave of m = 4 (mode-1 and mode-2), but basic modes of m = 5 (mode-5) and m = 3 (mode-6) appear. The appearance of the azimuthal wave m = 5 conforms to the changing trend of wave number increasing with the decrease of volume ratio. (2) The m = 8 azimuthal standing wave, which is the secondary harmonic, is weakened and forms a travelling wave in the radial direction. Compared with the modes of Vr = 1.025, 0.925, 0.875,and 0.775, as shown in Fig. 9 (d), 12 (e), 18 (e), and 20 (d), the standing harmonics gradually decrease their azimuthal wave intensity and tend to develop into a radial wave mode. In the Fig. 20 (d2), we clearly observe the phase gradient along the radial direction, indicating that the mode exhibits radial propagation characteristics (Table 8).

Fig. 20

Dynamic mode decomposition(Vr = 0.775, t = 3600 ~ 3450s).

Table 8 The DMD modes during the critical instability in the Vr = 0.775 case.

Method

Space experimental payload

The “Research on Oscillation Characteristics and Transition of Annular Flow under Microgravity” is a space experiment project carried out in the fluid physics rack on China’s Manned Space Station. The fluid physics rack is a fluid experimental platform that supports comprehensive measurement, including fluid dynamics and complex fluid research. It provides power supply, communication, experimental control, and data management for the payload (experiment cell) and offers 15 observation methods for scientific research. The Fluid Physics Experiment Rack is equipped with a magnetic levitation vibration isolation system featuring three operational modes:: locked mode, free-floating mode, and excitation mode. (1) In locked mode, mechanical clamps secure rigid connections between the experimental platform and the rack; (2) In free-floating mode, the experimental platform is released, enabling stable levitation control via multiple magnetic yokes to substantially minimize residual acceleration interference on experiments; (3) In excitation mode, stimuli within specific frequency is applied by the magnetic yokes while the system is free-floating, facilitating the study of g-jitter effects.

We have developed a payload, the annular flow experiment cell, to conduct research on the oscillation characteristics and transition of annular flow under microgravity. The three-dimensional size of the payload is 320 mm×200 mm×220 mm, in accordance with the standards of the fluid physics rack, as shown in Fig. 21 (a). It includes three electrical interfaces for power supply, communication, and scientific data transmission. An infrared window is set at the topside of the payload, through which the infrared camera of fluid physics rack observes the annular flow, as shown in Fig. 21 (b). As the thermocapillary flow is driven by a surface temperature gradient, the image of surface temperature is our main scientific data.

Fig. 21

The payload and scientific experimental model. (a) Space experiment payload, (b) schematic of the payload, (c) temperature difference of the liquid pool, (d) deformed liquid surface.

The schematic of the payload is shown in Fig. 22 (b). The annular liquid pool is mounted on a platform via 6 support legs. Beneath the platform, a rotation motor and lifting stage jointly control the central heating column, enabling both circumferential rotation and vertical lifting motion. Temperature difference is established, as shown in Fig. 22 (c), by the heater in the central column and the 6 refrigeration chips attached on the pool wall. The experiment utilizes a PI motor to drive the liquid cylinder for precise working fluid injection, achieving accurate control of volume ratios for different working fluids. The injection process is monitored by a CCD camera positioned at the upper side with corresponding LED illumination. The deformed liquid surface record by the camera is shown in Fig. 22 (d). Thermal capillary instability phenomena during heating are scientifically observed via an infrared camera provided by the Fluid Physic Rack.

The height of the pool H is 12 mm, the diameter D is 40 mm, and the diameter of the heating column d is 10 mm. The pool wall and the center column is made of copper with high thermal conductivity. The bottom plate is made of polysulfone plastic base plate, with its poor thermal conductivity. The free surface is opened to the environment, which the heat transfer can be simplified as Newton’s cooling law. Two thermocouples are fixed at the cooling pool and heating column, and the temperatures are directly recorded by the fluid physics rack through electrical interfaces. The fluid physics rack performs PID control on the cooling and heating power to maintain the temperature difference. The working fluid is 2 cSt silicone oil, whose physical property parameters are shown in Table 9.

Table 9 Physics properties of KF96 2cSt silicon oil at 25℃.

On board the Tianzhou-6 cargo spacecraft, the payload (annular flow experiment cell) was launched on May 10, 2023. On May 20, 2023, the Shenzhou-16 crew installed the payload to the fluid physics rack. Subsequently, we commenced the space experiments by remote commands. On November 4, 2023, the project completed its first stage of on-orbit experiments. Before the experiment, we measured the RMS of residual acceleration under the fluid rack’s locked mode. The g-jitter ranges from 10^− 5 to 10^− 3 m/s² within the 0.01 ~ 10 Hz frequency band, which covers the dominant frequency of thermocapillary oscillations ranging from 0.05 ~ 0.5 Hz. To analyze the impact of g-jitter on interface morphology and Marangoni convection, we introduced both static and dynamic Bond numbers.

(1) Static Bond Number, which is ratio of residual gravity to surface tension:

$$\:\text{B}{\text{o}}_{\text{s}\text{t}\text{a}\text{t}\text{i}\text{c}}=\frac{{\uprho\:}{\text{g}}_{\text{r}\text{e}\text{s}}{\text{L}}^{2}}{{\upsigma\:}}=0.076$$

(2) Dynamic Bond Number, which is ratio of buoyancy force to thermocapillary force:

$$\:\text{B}{\text{o}}_{\text{dynamic}}\text{=}\frac{{\rho \beta}{\Delta}\text{T}{\text{g}}_{\text{res}}{\text{D}}^{\text{2}}}{{{\sigma}}_{\text{T}}{\Delta}\text{T}\text{/}\text{L}}\text{=}\frac{{\rho \beta}{\text{g}}_{\text{res}}{\text{D}}^{\text{3}}}{{{\sigma}}_{\text{T}}}{=0.003}$$

Since the $\:\text{B}{\text{o}}_{\text{s}\text{t}\text{a}\text{t}\text{i}\text{c}}\ll\:1$ and $\:\text{B}{\text{o}}_{\text{d}\text{y}\text{n}\text{a}\text{m}\text{i}\text{c}}\ll\:1$, the influence of g-jitter can be neglected. We maintained the vibration isolation system in locked mode throughout the microgravity experiment. A total of 96 sets of experimental cases were accomplished, including three typical models: fixed column, rotating column, and lifting column. In the first stage, the volume ratio covers the range of 0.45 to 1.06 and the temperature difference reached 30 °C.

Data analyse method

We have obtained a substantial amount of space experimental data on oscillatory flow in annular thermocapillary flow. Annular flow transitions from axisymmetric steady flow to oscillatory flow driven by the thermocapillary effect with the increase of temperature difference. This transition, a form of Hopf bifurcation, is a remarkable characteristic of thermocapillary convection. Oscillatory flow behaves like a wave in space, and the amplitude grows or decays during the instability process. The formation and evolution of instability modes in axisymmetric models area key scientific issue. Generally, one or a few modes first lose their stability in the critical state. Moreover, in the supercritical state, when the temperature difference is increased to a higher value, the critical modes will excite harmonics or undergo transformation due to non-linearity. Dynamic Mode Decomposition (DMD) is a good tool for learning the intrinsic oscillation modes from the experimental data. By modal decomposition, we can analyze the dynamics of complex oscillatory flow in thermocapillary flow.

Dynamic Mode Decomposition (DMD) can extract modes and eigenvalues from the data of a complex dynamic system. The eigenvalues contain the growth rate and frequency, which are the critical characteristics for flow stability analysis. The basic principle of DMD is as follows: (1) Assume the unknown physical system, $\:{x}_{k+1}=f\left({x}_{k}\right)$, can be denoted as a linear system, $\:{x}_{k+1}=\varvec{A}{x}_{k}$, where $\:{x}_{k}$ represents the observation vector at k time step; (2) Estimate matrix A from the time series$\:{x}_{k},k=\text{1,2}...,T$. The dynamic modes are then obtained as the eigenvectors of matrix A, while the corresponding eigenvalues determine the growth rates and frequencies of these modes.

If we define $\:{\varvec{X}}_{1}^{T}=\left[{x}_{1},{\text{x}}_{2},{x}_{3},...,{x}_{T}\right]$, the matrix A can be estimated directly by $\:{\mathbf{A}=\varvec{X}}_{2}^{\text{T}}{\left({\varvec{X}}_{1}^{\text{T}-1}\right)}^{\dagger}$directly, since $\:{\varvec{X}}_{2}^{\text{T}}=\varvec{A}{\varvec{X}}_{1}^{\text{T}-1}$. Since the dimension of matrix A is excessively large, a dimension reduction technique is used to by analyzing DMD matrix, which is a low-rank approximation of matrix A. The steps of DMD are as follows: (1) Use Singular Value Decomposition (SVD) to decompose the matrix $\:{\mathbf{X}}_{1}^{\text{T}-1}=\mathbf{U}\varvec{\Sigma\:}{\mathbf{V}}^{\mathbf{*}}$. (2) Define the DMD matrix as $\:{\varvec{F}}_{\text{d}\text{m}\text{d}}={\varvec{U}}^{\varvec{*}}{\varvec{X}}_{2}^{\text{T}}\varvec{V}{\varvec{\varSigma\:}}^{-1}$. (3) Perform eigenvalue decomposition on the DMD matrix to obtain the eigenvaluesmatrix $\:\text{D}$ and eigenvectors $\:\text{Y}$. (4) Since the DMD matrix $\:{\text{F}}_{\text{d}\text{m}\text{d}}$ is the adjoint matrix of matrix $\:\varvec{A}$, the eigenvalues of these two matrices are equal. The relationship between the eigenvectors $\:\varvec{\varPhi\:}$ of $\:\varvec{A}$ and the eigenvectors $\:\varvec{Y}$ of $\:{\varvec{F}}_{\text{d}\text{m}\text{d}}$ is $\:\varvec{\varPhi\:}=\varvec{U}\varvec{Y}$.

The DMD method is employed to identify, extract, and reconstruct the thermocapillary modes from infrared image data. Figure 22 shows the data analysis process. First, DMD is applied to decompose the infrared image sequence within a specific time window, yielding the dominant dynamic modes and their corresponding eigenvalues. Subsequently, the change in mode amplitude over time is calculated. Finally, specific modes are selected to reconstruct the fluctuation waveform or spatial-temporal graph, aiming to quantify the contribution of each mode to the oscillatory instability. By comparing the reconstruction of the spatial-temporal evolution with the classical spatial-temporal analysis, we can validate whether the DMD modes can accurately characterize the actual fluctuation. As depicted in Fig. 22, the reconstruction of the spatial-temporal evolution using only 3 modes closely matches the azimuthal spatial-temporal evolution. Additionally, classical spatial-temporal analysis intuitively reveals the evolution of azimuthal waves and quickly pinpoints the timing of instability events.

Fig. 22

The data processing workflow for mode decomposition and reconstruction of spatial-temporal evolution.

Criterion for critical oscillation

The critical instability of annular thermocapillary convection refers to the transition from a steady axisymmetric flow to a periodic oscillatory flow during the linear heating process. Figure 23 (a) shows the infrared image at a temperature difference of 11.055 °C (t = 1975 ~ 2050s) in the space experimental case of Vr = 1.06. The petal-shaped waves with wave number m = 3 can be clearly observed. By extracting the spatial-temporal evolution of the azimuthal temperature from t = 2000s to 2500 s, as shown in Fig. 23 (b), the evolution of azimuthal waves within a time window of 500s is given. The critical oscillation occurs between t = 2150–2225 s, corresponding to a critical temperature difference of 12.375 °C. The azimuthal waveform can be obtained from the pattern change of the evolution diagram. Spots and stripes, representing standing waves and travelling waves respectively, are two typical patterns observed in such evolutions. In Fig. 23 (b), the critical oscillation is initially a standing wave and then transitions to a travelling wave when t > 2300 s. Therefore, in the case of Vr = 1.06, the instability manifests as an m = 3 azimuthal wave which transitions from standing wave to travelling wave.

Fig. 23

The infrared image and the azimiuthal spatial-temporal evolution. (a) The infrared image of m = 3, (b) the azimiuthal spatial-temporal evolution.

The critical temperature difference is a key parameter for studying the instability of thermocapillary convection. It refers to the temperature difference at which surface fluctuation first appears during the linear heating process. We sample the infrared image sequence using a sliding time window and then use DMD to analyze the oscillation mode. When the time window slides to the unstable region, an oscillating mode with positive growth is obtained in the DMD mode. Figure 24 shows the first modes of DMD analysis with a sliding time window of 75 s in the case of Vr = 1.060. In Fig. 24 (a), the mode at t = 1975 ~ 2050 s is random noise without oscillation. In Fig. 24 (b), the fluctuation structure with m = 3 begins to appear at t = 2075 ~ 2150 s. According to linear stability theory, the critical mode is defined as an oscillating mode with positive growth that appears when the criteria $\:f\ne\:0$ and $\:{\upsigma\:}\ge\:0$ is satisfied. However, the oscillating mode in Fig. 24(b) has a 0 Hz frequency and a negative growth rate, so it does not meet the critical oscillation condition. Obvious oscillation first appears in Fig. 24(c) with a frequency of f = 0.056 Hz and a growth rate of 5 × 10⁻³, which satisfies the onset criterion of instability. The time window of t=2150 ~ 2225 s corresponds to the temperature difference range (12.097~12.652℃) due to the linear heating process. Therefore, we define the criterion of critical instability as $\:f\ne\:0$ and $\:{\upsigma\:}\ge\:0$for quantitative analysisof the critical conditions.

Fig. 24

The critical instability process(Vr = 1.060). (a) T  = 11.055℃ (t = 1975 ~ 2050 s), (b) T = 11.790℃ (t = 2075–2150 s), (c ) 12.375 (t = 2150–2225 s).

We primarily employs an infrared camera to measure the surface morphology of the liquid pool, complemented by thermocouples monitoring temperatures at both the hot and cold ends. The cooled-type infrared camera, operating in the 2.0–4.8 μm spectral range, exhibited a temperature measurement noise level of 0.01 °C (3σ) in the space experiment. The uncertainty in the critical temperature difference for the onset of oscillations, as detailed in Table 1, ranges from ± 0.247 °C to 0.289 °C. This uncertainty primarily stems from the 75-second sliding window employed for oscillation detection. The continuous heating of the hot end within this temporal window introduces uncertainty in pinpointing the exact critical point. For example, the time window of t = 2150–2225 s in Fig. 24(c) corresponds to the temperature difference range (12.097 ~ 12.652℃) due to the linear heating process. It is noteworthy that this uncertainty is significantly greater than the error of thermocouples used for hot and cold end temperature monitoring, which achieved an accuracy of 0.02 °C after water-bath calibration.

The Dynamic Mode Decomposition (DMD) method is employed for modal analysis, which offers a distinct advantage in identifying weak oscillatory modes during the critical process. It enables the capture of modal characteristics at a very early stage of critical oscillation. We evaluated the errors associated with the DMD decomposition. First, the frequency error of DMD was analyzed. The fundamental approach involved performing DMD decompositions on data samples of different time windows and then computing variation of dominant frequency. Taking the case of Vr = 1.06 for example, the frequency deviation across different time windows (62.5 ~ 125s) is 0.0009 Hz, corresponding to a relative error of 1.7%. Furthermore, it validates that our chosen window length of 75 s, which encompasses four oscillation cycles, is sufficient for an accurate frequency analysis of the oscillatory modes. Subsequently, to investigate the influence of image noise on the accuracy of Dynamic Mode Decomposition (DMD), we generated infrared image data in which white noise with varying Signal-to-Noise Ratios (SNRs) was introduced. By applying the DMD technique to these datasets with different SNRs, we demonstrate that even when the Signal-to-Noise Ratio (SNR) decreases to 0.2, meaning the noise amplitude is five times that of the signal, the DMD method can still accurately identify the oscillatory modes. Based on this finding, it is inferred that the minimum detectable peak-to-peak amplitude of an oscillatory mode is 0.002 °C, as the noise of infrared camera is 0.01 °C .