3. Results and discussion

Figure 1(a) shows the typical snapshots of flow structures in vibroconvection with different dimensionless frequencies $\omega =10^5$, $10^6$ and $10^7$ at fixed dimensionless amplitude ${\textit {a}}=0.01$ and fixed Prandtl number ${\textit {Pr}} = 4.38$. It is seen that the shaking by external vibration strongly destabilizes the conductive state and generates large distortion of the temperature field in bulk regions by creating an artificial gravity (Beysens et al. Reference Beysens, Chatain, Evesque and Garrabos2005; Beysens Reference Beysens2006). With increasing $\omega$, it is vibration-induced artificial gravity that becomes strong enough to destabilize the TBL and trigger abundant thermal plumes. Those plumes are transported into bulk regions and self-organized into columnar structures. This indicates that the feature of the main structures responsible for heat transport in microgravity vibroconvection are different from those in the gravity-induced RB convection. The vertical columnar structure in the bulk is very similar to the columnar pattern observed in experiments on the interface between two miscible liquids under vibration in microgravity (Gaponenko et al. Reference Gaponenko, Torregrosa, Yasnou, Mialdun and Shevtsova2015).

Figure 1.

Flow structure in microgravity vibroconvection. (a) Instantaneous 3-D flow structure visualized by the volume rendering of the instantaneous temperature field (see supplementary movies available at https://doi.org/10.1017/jfm.2024.368) under different vibration frequencies $\omega = 10^5$ (a i), $10^6$ (a ii) and $10^7$ (a iii) at fixed amplitude ${\textit {a}} = 0.01$ and Prandtl number ${\textit {Pr}} = 4.38$. The subpanels below show the corresponding temperature contours extracted on the horizontal slice at the edge of TBL. (b) Power spectrum of fluctuating temperature in bulk zones. (c) The variation of the characteristic wavenumber $k_m$ as functions of $\omega$. (d) Percentage of BL (solid symbols) and bulk (hollow symbols) to the global averaged thermal dissipation rate, as functions of vibration frequency.

To quantitatively analyse the feature of columnar structures, we extract the instantaneous temperature field in bulk zones and calculate the power spectrum $P(k)$ of temperature fluctuations by applying the Fourier transform in the vibrational direction as shown in figure 1(b). It is found that there exists a characteristic wavenumber $k_{m}$, at which the wavenumber distribution function $P(k)$ is maximal. Indeed, $k_{m}$ characterizes the number of columnar structures in vibroconvection. We then plot the variation of $k_{m}$ as functions of $\omega$ in figure 1(c). It is shown that $k_m$ monotonically increases with increasing $\omega$, indicating that more columnar structures are formed under a stronger vibrational driving force. Such a kind of relation for the columnar structures was experimentally established while vibrating two miscible liquids (see figure 5 in the work of Gaponenko et al. (Reference Gaponenko, Torregrosa, Yasnou, Mialdun and Shevtsova2015)). This is consistent with the fact that larger heat transport enhancement occurs at larger $\omega$. Further, to examine the role of the TBL in vibroconvective heat transport processes, we decompose the globally averaged thermal dissipation rate $\epsilon _T = \kappa \lvert \boldsymbol {\nabla } T \rvert ^2$ into their boundary layer (BL) and bulk contributions, and then plot the variation of relative contributions as functions of $\omega$ in figure 1(d), as suggested by the Grossmann–Lohse theory (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009; Stevens et al. Reference Stevens, van der Poel, Grossmann and Lohse2013). It is seen in figure 1(d) that the BL contribution of $\epsilon _T$ is much larger than the bulk one, suggesting the BL-dominant thermal dissipation. This reveals that the dynamics of BLs plays a crucial role on the underlying mechanism of heat transport in vibroconvective turbulence.

Next, we address the question of how the global heat transport depends on the control parameters of vibroconvection. First, we examine the dependence of heat transport on the vibration frequency. Figure 2(a,d) shows the measured ${\textit {Nu}}$ as functions of frequency $\omega$ in a log–log plot for different amplitudes ${\textit {a}}$ in 3-D and 2-D cases. Here, the ${\textit {Nu}}$ number, as the non-dimensional ratio of the measured heat flux to the conductive one, is calculated by ${\textit {Nu}} = \langle w T - \kappa \partial _z T\rangle /(\kappa \varDelta /H)$, where $w$ is the vertical velocity and $\langle {\cdot } \rangle$ denotes the time and space averaging. It is observed that the ${\textit {Nu}}$–$\omega$ scaling relation is not unique for a specific amplitude, namely, there seems to be a transition from ${\textit {Nu}} \sim \omega$ to ${\textit {Nu}} \sim \omega ^{2/3}$ in both 3-D and 2-D cases, as shown by the dashed lines or in figure 5 in Appendix B. Note that the precise values of scaling exponents are obtained from the physical model we proposed below, not adjusted from the fitting with the numerical data.

Figure 2.

Heat transport scaling in vibroconvective turbulence. (a–c) The measured Nusselt number ${\textit {Nu}}$ as functions of vibration frequency $\omega$, vibrational Rayleigh number ${\textit {Ra}}_{vib}$, oscillational Rayleigh number ${\textit {Ra}}_{os}$ for 3-D cases. (d–f) The measured Nusselt number ${\textit {Nu}}$ as functions of vibration frequency $\omega$, vibrational Rayleigh number ${\textit {Ra}}_{vib}$, oscillational Rayleigh number ${\textit {Ra}}_{os}$ for 2-D cases. The dashed lines in the panels are ${\textit {Nu}} \sim \omega$ (lower), ${\textit {Nu}} \sim \omega ^{2/3}$ (upper) in (a,d), ${\textit {Nu}} \sim {\textit {Ra}}_{vib}^{1/2}$ in (b,e), ${\textit {Nu}} \sim {\textit {Ra}}_{os}^{1/3}$ in (c,f). Those precise scaling relations are theoretically deduced by our proposed physical model in the paper. Note that the vibration amplitude range in the 3-D cases is from ${\textit {a}}=10^{-3}$ to ${\textit {a}}=10^{-1}$ and in the 2-D cases is from ${\textit {a}}=10^{-3}$ to ${\textit {a}}=3 \times 10^{-1}$.

Further, we examine the dependency of heat transport on the two important analogous Rayleigh numbers in vibroconvective turbulence, which are the vibrational Rayleigh number ${\textit {Ra}}_{vib}$ and oscillational Rayleigh number ${\textit {Ra}}_{os}$. The first one is the vibrational Rayleigh number ${\textit {Ra}}_{vib} = (\alpha A \varOmega \Delta H)^2/(2\nu \kappa )$, i.e. ${\textit {Ra}}_{vib} = \tfrac {1}{2}{\textit {a}}^2\omega ^2 {\textit {Pr}}$, which is obtained from applying the averaged approach on vibroconvective equations in the limit of small amplitudes and high frequencies, and quantifies the intensity of the external vibrational source. Figure 2(b,e) depict, respectively, the measured ${\textit {Nu}}$ as functions of ${\textit {Ra}}_{vib}$ in a log–log plot at different amplitudes for 3-D and 2-D cases. We find that at small ${\textit {Ra}}_{vib}$, numerical data almost collapse together on the same scaling law, i.e. ${\textit {Nu}} \sim {\textit {Ra}}_{vib}^{1/2}$, as shown by the dashed lines. However, at large ${\textit {Ra}}_{vib}$, a significant departure from this scaling behaviour is observed for large amplitudes. The other is the oscillational Rayleigh number ${\textit {Ra}}_{os} = \alpha A \varOmega ^2 \Delta H^3/(\nu \kappa )$, i.e. ${\textit {Ra}}_{os} = {\textit {a}} \omega ^2 {\textit {Pr}}$, which is analogous to Rayleigh number in RB convection but replacing the gravitation by the vibration-induced acceleration. Figure 2(c,f) shows the variation of ${\textit {Nu}}$ as functions of ${\textit {Ra}}_{os}$ for various amplitudes in 3-D and 2-D cases. We find that at large ${\textit {Ra}}_{os}$, numerical data almost collapse onto the same scaling relation ${\textit {Nu}} \sim {\textit {Ra}}_{os}^{1/3}$ as shown by the dashed line, but at small ${\textit {Ra}}_{os}$, numerical data points deviate considerably from this scaling for small amplitudes. Both scaling relations ${\textit {Nu}} \sim {\textit {Ra}}_{vib}^{1/2}$ for small ${\textit {Ra}}_{vib}$ and ${\textit {Nu}} \sim {\textit {Ra}}_{os}^{1/3}$ for large ${\textit {Ra}}_{os}$ can be further confirmed from the compensated plots in figure 6 in Appendix C. From above, using solely the common control parameters like $\omega$, ${\textit {Ra}}_{vib}$ or ${\textit {Ra}}_{os}$, unifying the heat transport scaling in vibroconvective turbulence cannot be achieved.

Now, there are two important questions remaining to be answered in vibroconvective turbulence: one is why there exists two different heat transport scaling laws, i.e. ${\textit {Nu}} \sim {\textit {Ra}}_{vib}^{1/2}$ and ${\textit {Nu}} \sim {\textit {Ra}}_{os}^{1/3}$; the other is whether a unified constitutive law emerges in vibroconvective turbulence. Hereafter, we propose a physical model to address the first question. From the analysis above, we know that the BL-contribution to the global thermal dissipation rate is dominant, implying that the BL dynamics plays a crucial role in the heat transport mechanism. In vibroconvective turbulence, there are two types of BL: the thermal boundary layer (TBL) with the thickness of $\delta _{th}$, which is estimated by $\delta _{th}\approx H/(2{\textit {Nu}})$; the other is the oscillating boundary layer (OBL) induced by the external vibration. The modulation depth of OBL referring to $\delta _{os}$ is defined as the depth, at which the delaying rate of the intensity of vibration-induced shear effect is equal to $99\,\%$. Considering the intensity of vibrational modulation falling off exponentially from the surface, one easily obtains $\delta _{os} = -\ln (1-0.99) \delta _S \approx 4.605 \delta _S$ where $\delta _S = \sqrt {2\nu /\varOmega }$ is the Stokes layer thickness. First, when $\delta _{th} > \delta _{os}$ as sketched in figure 3(ai), by taking into account the balance between the convective and conductive transports within TBL, the dimensional analysis of the governing equation of the temperature field gives rise to $w {\varDelta }/{\delta _{th}} \sim \kappa {\varDelta }/{\delta ^2_{th}}$. And, in the momentum equation, the balance between the vibration-induced buoyancy and the viscous dissipation leads to $\alpha A\varOmega ^2 \varDelta \sim \nu {u}/{\delta ^2_{os}}$ with $u$ the horizontal velocity. Using both above relations, assuming that the magnitude of velocity components $u$ and $w$ follows a similar scaling behaviour, i.e. the ratio $w/u$ is approximately constant, together with $\delta _{os} \sim \sqrt {\nu /\varOmega }$ and $\delta _{th} \sim H/{\textit {Nu}}$, one obtains the scaling relation between ${\textit {Nu}}$ and ${\textit {Ra}}_{vib}$,

(3.1)\begin{equation} {\textit{Nu}} \sim {\textit{Ra}}_{vib}^{1/2} {\textit{Pr}}^{1/2}. \end{equation}

The scaling relation in (3.1) shows that vibroconvective heat transport is independent of viscosity $\nu$, but depends on thermal diffusion coefficient $\kappa$. This implies that the dynamics of TBL is dominant to heat transport in cases of $\delta _{th} > \delta _{os}$.

Figure 3.

Unified constitutive law in vibroconvective turbulence. (a) Sketch of the (a i) TBL-dominant regime and the (a ii) OBL-dominant regime. (b,d) The TBL thickness $\delta _{th}$ (symbols) and the critical modulation depth $\delta _{os}$ (solid line) of OBL as a function of the oscillational Reynolds number ${\textit {Re}}_{os}$ for 3-D and 2-D cases. The insets in (b,d) show the ratio $\delta _{os}/\delta _{th}$ as a function of the oscillational Reynolds number ${\textit {Re}}_{os}$. The solid dashed line indicates $\delta _{os}/\delta _{th} = 1$. (d,e) The unified scaling law exhibited between $a {\textit {Nu}}$ and the oscillational Reynolds number ${\textit {Re}}_{os}$ for 3-D and 2-D cases. The emergence of a universal constitutive law of vibroconvective turbulence is clearly observed. Here, the TBL-dominated regime is coloured light purple and OBL-dominated regime light cyan in (b–e). The coloured symbols have the same meaning as those in figure 2.

When $\delta _{th} < \delta _{os}$ as sketched in figure 3(a ii), the balance between the vibration-induced buoyancy and the viscous dissipation within TBL allows one to rewrite the momentum equation using dimensional analysis: $\alpha A\varOmega ^2 \varDelta \sim \nu {u}/{\delta ^2_{th}}$. Combining the above equation and $w {\varDelta }/{\delta _{th}} \sim \kappa {\varDelta }/{\delta ^2_{th}}$ for the temperature equation, together with the above assumption that the ratio $w/u$ is approximately constant, $\delta _{os} \sim \sqrt {\nu /\varOmega }$ and $\delta _{th} \sim H/{\textit {Nu}}$, one deduces the scaling relation between ${\textit {Nu}}$ and ${\textit {Ra}}_{os}$,

(3.2)\begin{equation} {\textit{Nu}} \sim {\textit{Ra}}_{os}^{1/3}. \end{equation}

The heat transport scaling in (3.2) is similar to that of RB convection in the classical regime through replacing the gravitation by vibration-induced acceleration. Both heat transport scalings predicted in (3.1) and (3.2) agree well with numerical results shown in figure 2. The competition between TBL and OBL results in the two different heat transport scaling relations, namely, ${\textit {Nu}} \sim {\textit {Ra}}_{vib}^{1/2}$ and ${\textit {Nu}} \sim {\textit {Ra}}_{os}^{1/3}$.

Furthermore, we address the second question of whether the universal constitutive law of vibroconvective turbulence emerges. First, to quantify the dynamics of OBL, we define the oscillational Reynolds number ${\textit {Re}}_{os} = \alpha \Delta A \varOmega \delta _{os}/\nu$, which is related to the vibrational velocity with the Boussinesq parameter $\alpha \Delta A \varOmega$ and the modulation depth $\delta _{os}$ and obeys the relation ${\textit {Re}}_{os} = 4.605 {\textit {a}} (2\omega )^{1/2}$. Second, we study the dependency of ${\textit {Nu}}$ on ${\textit {Re}}_{os}$. It is intriguing to find that both ${\textit {Nu}} \sim {\textit {Ra}}_{vib}^{1/2}$ and ${\textit {Nu}} \sim {\textit {Ra}}_{os}^{1/3}$ scaling laws can be rewritten as ${\textit {Nu}} \sim {\textit {a}}^{-1} {\textit {Re}}_{os}^\beta$ with $\beta =2$ for the TBL-dominant heat transport regime ($\delta _{th}>\delta _{os}$), and $\beta =4/3$ for the OBL-dominant heat transport regime ($\delta _{th}<\delta _{os}$). Therefore, we conclude that due to the competition between the dynamics TBL and OBL on heat transport, the underlying mechanism of heat transport in vibroconvective turbulence can be categorized into the two following regimes.

1. (i) The TBL-dominant regime ($\delta _{th} > \delta _{os}$): the OBL is submerged into TBL. Thermal plumes facilitated by vibration-induced strong shear detach from OBL and move into TBL. The plume dynamics is then mainly dominant by the molecular diffusion between OBL and TBL. Those plumes thermally diffuse and then self-organize into columnar structures in bulk zones, which transport heat from the bottom hot plate to the top cold one. The heat transport scaling exhibits the scaling ${\textit {Nu}} \sim {\textit {a}}^{-1} {\textit {Re}}_{os}^{2}$.

2. (ii) The OBL-dominant regime ($\delta _{os} > \delta _{th}$): the TBL is nested into OBL. The OBL dominates the dynamics of thermal plumes ejected from TBL by vibration-induced strong shear. Between OBL and TBL, the shear effect mixes those plumes and sweep away some of them (Scagliarini, Gylfason & Toschi Reference Scagliarini, Gylfason and Toschi2014; Blass et al. Reference Blass, Zhu, Verzicco, Lohse and Stevens2020; Jin et al. Reference Jin, Wu, Zhang, Liu and Zhou2022). The remaining plumes then move into bulk zones and self-organize into columnar structures. In this regime, due to the plume-sweeping mechanism between OBL and TBL, the heat transport is depleted and obeys the scaling with a smaller scaling relation exponent ${\textit {Nu}} \sim {\textit {a}}^{-1} {\textit {Re}}_{os}^{4/3}$.

Finally, we use the simulated data to confirm the theoretically deduced unified constitutive law. First, we plot in figure 3(b,d) the variation of both ${\textit {a}} \delta _{th}$ and ${\textit {a}} \delta _{os}$ as functions of ${\textit {Re}}_{os}$. It is shown that for all fixed amplitudes, the value of both $\delta _{th}$ and $\delta _{os}$ monotonically decreases as increasing ${\textit {Re}}_{os}$, and $\delta _{th}$ decreases faster than $\delta _{os}$. The intersection point between the curves of ${\textit {a}} \delta _{th}$ and ${\textit {a}} \delta _{os}$ divides the plane into two regions, which corresponds to the TBL-dominant regime in the left-hand side ($\delta _{th} > \delta _{os}$) and OBL-dominant regime in the right-hand side ($\delta _{th} > \delta _{os}$). As depicted in the inset of figure 3(b,d), the dividing line between TBL-dominant and OBL-dominant regimes is nearly at the position of $\delta _{os}/\delta _{th} = 1$. This confirms that the underlying mechanism of vibroconvective heat transport is attributed to the competition between the dynamics of TBL and OBL. Second, we plot the calculated ${\textit {a}} {\textit {Nu}}$ as functions of ${\textit {Re}}_{os}$ as shown in figure 3(c,e). It is expected that all numerical data collapse together onto the derived universal constitutive law. Evidently, the numerical data and theoretical model show an excellent agreement. This confirms the emergence of the universal constitutive law of vibroconvective turbulence in microgravity. The evidence of this unified heat transport heat scaling is also shown from the compensated plots in figure 7 in Appendix D.