3.1. Effect of system size

To investigate the impact of system size on flow patterns and spectral properties, we varied the system sizes while maintaining the forcing and fluid layer depth (i.e. bottom friction) unchanged. Figure 1 shows the instantaneous vorticity fields for five different cases of $L_B/l_f$. Note that for the case of the largest $L_B/l_f=29.9$, the size of the field of view is $L/l_f=22.2$, smaller than $L_B/l_f$. Our results reveal that the flow patterns differ significantly for different values of $L_B$. Specifically, for large $L_B/l_f$ ($=29.9, 22.2$), the large-scale streamers identified by the co-directional velocity vectors, connecting the neighbouring vortices, can be observed, whereas for small $L_B/l_f$ ($=13.2, 10.6$), a coherent vortex with a size close to $L_B$ dominates the flow field. The mechanism of the formation of a system-sized coherent vortex is that the streamers, produced by the same-sign neighbouring vortices in the initial stage of the flow evolution, are guided by the side boundaries to evolve into a large-scale vortex in a finite domain (Xia & Shats Reference Xia and Shats2012). These observations are consistent with the previous experiments by Fang & Ouellette (Reference Fang and Ouellette2021). According to the arguments given in previous studies (Paret & Tabeling Reference Paret and Tabeling1998; Chertkov et al. Reference Chertkov, Connaughton, Kolokolov and Lebedev2007), the presence of a system-size vortex in the cases $L_B/l_f =13.2$ and 10.6 here indicates the occurrence of spectral condensation.

Figure 1.

The instantaneous vorticity fields $\omega$ overlaid with the down-sampled velocity vectors under the same forcing and bottom damping in the cases of different system sizes: (a) $L_B/l_f = 29.9$, (b) $L_B/l_f = 22.2$, (c) $L_B/l_f = 16.8$, (d) $L_B/l_f = 13.2$, (e) $L_B/l_f = 10.6$.

We plot the time evolution of the kinetic energy in figure 2(a). The damping rate $\gamma$ ($=\alpha l_f^2/\nu$) was estimated by fitting the exponential decay of kinetic energy after switching off the forcing (Xia et al. Reference Xia, Shats and Falkovich2009; Fang & Ouellette Reference Fang and Ouellette2017), i.e. $E(t)\propto {\rm e}^{-2\alpha t}$, where $E(t)= \sum [u_1(t)^2+u_2(t)^2]/2N^2$ is the kinetic energy at time $t$ (with $N^2$ the number of velocity vectors in a snapshot). As shown in table 1, $\gamma$ exhibits a moderate decreasing trend with the increase of $L_B$. This could be attributed to the different flow patterns or the additional dissipation caused by the solid boundaries which may be more severe for small domain sizes. Figure 2(b) shows the fraction of kinetic energy in the mean flow field $E_m/E_0$ as a function of $L_B/l_f$, where $E_0 = u_{rms}^2/2$ (see figure 2a) and $E_m=\sum (\bar u_1^2+\bar u_2^2)/2N^2$ (with $\bar u_1$ and $\bar u_2$ the time-averaged velocity fields in the steady state). Our results indicate that the flow field becomes more stable as $L_B$ decreases, which is consistent with previous studies (Xia et al. Reference Xia, Shats and Falkovich2009; Fang & Ouellette Reference Fang and Ouellette2021). Figure 2(c) plots the friction-dominated scale $L_\alpha$ ($=u_{rms}/\alpha$) as a function of $L_B/l_f$. It is seen that for the cases $L_B/l_f$ =13.2 and 10.6, $L_\alpha /L_B\gtrsim 1.0$ is satisfied, which was previously considered to indicate spectral condensation, and seems to agree with the observations of the flow patterns shown in figure 1.

Figure 2.

(a) The time evolution of the kinetic energy $E(t)$. The red dashed line denotes the linear growth stage after the forcing is switched on, which can be used to estimate the energy injection rate $\epsilon _{i}={\rm d}E/{\rm d}t$, and the blue dotted curve denotes the exponential decay after the forcing is switched off, i.e. $E\propto {\rm e}^{-2\alpha t}$, from which $\alpha$ can be estimated. (b) The proportion of kinetic energy in the mean flow field $E_m/E_0$. (c) The scale ratio $L_\alpha /L_B$ as a function of the system size $L_B/l_f$.

Table 1.

Physical parameters of the thin-layer turbulent flows for the different system sizes under the same forcing and bottom friction conditions.

In this system, the energy $\epsilon_{i}$ injected at the forcing scale is dissipated by two mechanisms due to the large-scale friction $\epsilon _{\alpha }$ and viscosity $\epsilon _{\nu }$, respectively, i.e. $\epsilon _{i} = \epsilon _{\alpha }+\epsilon _{\nu }$, where $\epsilon _{\alpha }=\alpha u_{rms}^2$ and $\epsilon _{\nu }=\nu \omega _{rms}^2$. The energy injection rate $\epsilon _{i}$ can be estimated by the linear growth stage of the kinetic energy after switching on the forcing (Xia et al. Reference Xia, Shats and Falkovich2009), as shown in figure 2(a). Table 1 shows the energy budget of the turbulent flows for different cases of $L_B/l_f$. It is seen that more energy is dissipated by friction than by viscosity, i.e. $\epsilon _{\alpha }>\epsilon _{\nu }$, which facilitates the process of an inverse energy cascade. Moreover, the energy injection rate $\epsilon _{i}$ is larger than $\epsilon _{\alpha }+\epsilon _{\nu }$, which could be due to the additional dissipation by the boundaries $\epsilon _{bound}$ ($\approx \epsilon _{i}-\epsilon _{\alpha }-\epsilon _{\nu }$), although it is relatively weak and much smaller than $\epsilon_\alpha$. However, $\epsilon _{bound}$ does not seem to have a clear relationship with $L_B/l_f$.

In addition to the qualitative observations of spectral condensation from the regular vortex pattern, its more quantitative manifestation is the deviation of the spectrum from the classical $k^{-5/3}$ scaling in the inverse cascade range. The energy spectra for different system sizes $L_B/l_f$ are plotted in figure 3. It is surprising that the spectra show a similar pattern in all cases, i.e. a close to $k^{-3}$ scaling below the forcing wavenumber. In previous studies (Sommeria Reference Sommeria1986; Chertkov et al. Reference Chertkov, Connaughton, Kolokolov and Lebedev2007; Xia et al. Reference Xia, Punzmann, Falkovich and Shats2008), the $k^{-3}$ behaviour in the spectrum of the inverse cascade was taken as evidence for the occurrence of the spectral condensation. In this sense, the information conveyed by the flow patterns shown in figure 1, the scale ratio $L_\alpha /L_B$ in figure 2(c) and the energy spectra in figure 3 appear to be different. Specifically, $L_\alpha /L_B$ could predict the occurrence of the system-size vortex, but is not able to capture the spectral properties.

Figure 3.

The energy spectra for different system sizes $L_B/l_f$.

Figure 4(a) illustrates the temporal evolution of the energy spectrum for the case $L_B/l_f = 16.8$ ($L_\alpha /L_B=0.87$). In the initial stages, the spectrum exhibits a $k^{-5/3}$ scaling, followed by a transition to a steeper $k^{-3}$ scaling. In current experiments, the inertial range of inverse cascade in the energy spectrum is not wider than half a decade, which seems to be the norm in all previous experiments (Sommeria Reference Sommeria1986; Paret & Tabeling Reference Paret and Tabeling1997; Xia et al. Reference Xia, Shats and Falkovich2009). Chertkov et al. (Reference Chertkov, Connaughton, Kolokolov and Lebedev2007) suggested that the $k^{-3}$ spectrum results from the presence of coherent structures and does not represent the turbulent velocity fluctuations involved in the cascade. It is noteworthy that the time evolution of the spectrum shown in figure 4(a) is consistent with the numerical findings of Chertkov et al. (Reference Chertkov, Connaughton, Kolokolov and Lebedev2007), despite the absence of a system-size vortex in our study with large $L_B/l_f$ (see figures 1a–c). Furthermore, when the mean flow is subtracted, the spectrum of the velocity fluctuations, as depicted in figure 4(b), recovers a $k^{-5/3}$ scaling, in good agreement with previous studies (Chertkov et al. Reference Chertkov, Connaughton, Kolokolov and Lebedev2007; Xia et al. Reference Xia, Punzmann, Falkovich and Shats2008, Reference Xia, Shats and Falkovich2009).

Figure 4.

(a) The time evolution of the energy spectrum for the case $L_B/l_f = 16.8$. (b) Comparison of energy spectra for the whole velocity field with those that contain only the fluctuation part for the two cases $L_B/l_f = 10.6$ and $L_B/l_f = 22.2$.

3.2. Effect of damping rate

In this subsection, we investigate the effect of the damping rate $\gamma$ on flow patterns and spectral properties for large system sizes ($L_B/l_f>25$, $L_\alpha /L_B<1.0$). It is noted that due to $L_\alpha /L_B<1.0$, we refer to the configuration considered in this subsection as unbounded turbulence. Figures 5(a,d) show instantaneous snapshots of the vorticity field at moderate and low damping rates under the same forcing conditions and system size ($L_B/l_f=38.4$). Notably, the flow patterns differ significantly between the two damping rates. Specifically, for $\gamma = 5.7$, the intermediate-scale turbulent vortices coexist with the large-scale streamers, accompanied by several small vortices. In the case $\gamma = 8.8$, however, the length scales of the turbulent structures are smaller, and the vortices are stronger, indicated by the higher vorticity intensity. These observations are reasonable because the reduced bottom friction facilitates energy transfer to larger scales when the system size is sufficiently large.

Figure 5.

The instantaneous snapshots of (a,d) the vorticity field and (b,e) energy-flux fields (filtering kernel size $r = 1.5l_f$) overlaid with the down-sampled velocity vectors for $L_B/l_f=38.4$ and $Re_f \approx 75$ at (a,b) low damping rate ($\gamma =5.7$) and (d,e) moderate damping rate ($\gamma =8.8$). Note that the flow fields in (a,b) are at the same instant shown with different contour maps, as in (d,e). (c) The energy-flux field that is observed 4.3 s subsequent to that in (b) or (a). The dashed-line rectangles in (b,c) show that the small-scale vortices are ‘swept out’ by the subsequent large-scale motions indicated by the bold arrow. ( f) Probability density functions (PDFs) of energy flux $\varPi ^{(1.5l_f)}$ for two damping rates; the inset plots the corresponding PDFs of normalized energy flux.

We implement the filter-space technique (Rivera et al. Reference Rivera, Daniel, Chen and Ecke2003; Chen et al. Reference Chen, Ecke, Eyink, Rivera, Wan and Xiao2006; Liao & Ouellette Reference Liao and Ouellette2013) to calculate the scale-to-scale energy transfer flux $\varPi ^{(r)}$ in physical space based on the filter velocity field with kernel width $r$. The energy flux $\varPi ^{(r)}$ is defined as $\varPi ^{(r)}=-\tau _{ij}^{(r)}s_{ij}^{(r)}$, where $\tau _{ij}^{(r)}=(u_iu_j)^{(r)}-u_i^{(r)}u_j^{(r)}$ and $s_{ij}^{(r)}=(\partial _{x_j}u_i^{(r)}+\partial _{x_i}u_j^{(r)})/2$ are the rate of strain and stress tensor, respectively. Likewise, the enstrophy transfer fluxes $Z^{(r)}=-\partial _{x_i}\omega ^{(r)}((u_i\omega )^{(r)}-u_i^{(r)}\omega ^{(r)})$ are calculated using the filter vorticity field. As with the previous definition, $\varPi ^{(r)}>0$ ($Z^{(r)}>0$) indicates energy (enstrophy) flux to smaller length scales, while $\varPi ^{(r)}<0$ ($Z^{(r)}<0$) denotes energy (enstrophy) flux towards larger scales. Figures 5(b,e) show the instantaneous snapshots of the energy transfer fluxes with $r = 1.5l_f$, i.e. $\varPi ^{(1.5l_f)}$, and figure 5( f) shows the probability density functions (PDFs) of $\varPi ^{(1.5l_f)}$ for two damping rates. It can be seen that the alternating lobes of intense spectral energy flux are associated with strong vortices in the flow field (especially in figure 5e), which is consistent with previous studies (Xiao et al. Reference Xiao, Wan, Chen and Eyink2009; Liao & Ouellette Reference Liao and Ouellette2013). Moreover, one sees that the scale-to-scale energy transfers at $r = 1.5l_f$ for $\gamma = 8.8$ are much stronger than for $\gamma = 5.7$. This may seem counter-intuitive, as lower bottom friction would reasonably lead to a stronger inverse energy transfer. As shown in figures 5(b,c), the relatively small-scale vortices (inside the dashed rectangles) seem to be ‘eliminated’ by the subsequent large-scale turbulent structures (or called streamers, indicated by the green arrow), resulting in the weakening energy transfers at these scales. This aspect is illustrated more clearly in figure 6(a).

Figure 6.

(a) Ensemble-averaged scale-to-scale energy and enstrophy (inset) transfer fluxes. (b) Energy spectra for different values of $\gamma$. Note that for all cases here, $Re_f= 75\pm 5$, $L_B/l_f=38.4$, and $L_\alpha /L_B$ is within the range 0.19–0.43.

Figure 6(a) plots the spatio-temporally averaged energy transfer fluxes $\langle {\varPi }^{(r)}\rangle$ as functions of $r/l_f$ for different damping rates $\gamma$. For all cases, $\langle {\varPi }^{(r)}\rangle \lesssim 0$ holds true, even at small scales. This indicates the dominant effect of the inverse energy transfers, which is also characterized by the negative-skewed PDFs shown in figure 5( f). It is seen that the inverse energy flux for four larger $\gamma$ cases reaches a maximum at approximately the forcing scale ($r/l_f\approx 1.2$), consistent with previous studies (Rivera et al. Reference Rivera, Daniel, Chen and Ecke2003; Liao & Ouellette Reference Liao and Ouellette2013). However, for $\gamma = 5.7$, the peak value of the inverse energy flux occurs at $r/l_f\approx 4$ and is the smallest among the five cases. In addition, for $r/l_f<3$, the inverse energy flux decreases with decreasing damping rate, while an opposite trend is observed for $r/l_f>3$. Figure 6(a) suggests that although turbulent structures can grow larger at smaller damping rates (as evidenced by the large-scale streamers shown in figures 5a–c), the existence of these large-scale coherent structures suppresses the energy flux in the intermediate scales. Moreover, this suppression effect also leads to the reduced enstrophy flux below the forcing scale, as shown in the inset of figure 6(a).

Figure 6(b) illustrates the effect of damping rate $\gamma$ on the spectral properties of the quasi-2-D unbounded turbulence. Notably, the fine scales ($k\gtrsim 1000$ m$^{-1}$) dominated by viscous dissipation are nearly unaffected by the large-scale friction, which could be expected. Conversely, the kinetic energy around the forcing wavenumber is diminished as $\gamma$ decreases. This observation is consistent with the behaviour of enstrophy and energy fluxes with respect to $\gamma$, as illustrated in figure 6(a). Furthermore, we found that with the decrease of $\gamma$, the spectrum in the enstrophy inertial range becomes flatter (as a result of reduced kinetic energy at intermediate scales) and approaches the theoretical prediction of $k^{-3}$ (Kraichnan Reference Kraichnan1967; Boffetta & Musacchio Reference Boffetta and Musacchio2010). This trend aligns with previous experiments (Boffetta et al. Reference Boffetta, Cenedese, Espa and Musacchio2005). The steepening of the energy spectrum in the enstrophy inertial range has been observed commonly in previous studies, and potential causes include the presence of viscous dissipation (Tran & Bowman Reference Tran and Bowman2004), the existence of large-scale long-lived vortices (Kramer et al. Reference Kramer, Keetels, Clercx and van Heijst2011), and linear damping (Boffetta et al. Reference Boffetta, Cenedese, Espa and Musacchio2005). However, our experimental results indicate that it is the suppression of small-scale turbulence by large-scale coherent structures that leads to the spectrum in the forward enstrophy cascade approaching $k^{-3}$.

The spectral properties in the inverse energy cascade are now being analysed. With a large damping rate ($\gamma =13.9$), the energy spectrum is observed to be flatter than $k^{-5/3}$ despite the presence of strong inverse energy transfers (as seen in figure 6a). This deviation indicates that the energy flux is strongly influenced by the bottom damping and undergoes significant changes within the energy cascade range. As $\gamma$ decreases, the energy spectrum gradually recovers the classical $k^{-5/3}$ scaling. Further reduction in the damping rate to $\gamma =5.7$ leads to a steeper spectrum close to $k^{-3}$, which is usually taken as evidence for the occurrence of spectral condensation. It is worth noting that the ratio $L_\alpha /L_B$ for all cases shown in figure 6 is in the range 0.19–0.43, which is notably smaller than 1.0.

To further investigate the scaling behaviour of the spectrum in the inverse cascade range, the scaling exponent $\zeta _{IC}$ is determined by least squares fitting for different cases of damping rate, as depicted in figure 7. The inset of figure 7(a) plots $\zeta _{IC}$ as functions of $L_\alpha /L_B$ for various $L_B/l_f$. Interestingly, the transition of the spectrum from the classical $k^{-5/3}$ scaling to the $k^{-3}$ scaling occurs at different values of $L_\alpha /L_B$, all of which are smaller than 1.0. Furthermore, it has been demonstrated in § 3.1 that the system size $L_B/l_f$ has little impact on the spectral properties. These findings challenge the existing understanding of spectral condensation, which says that condensation would not be expected for $L_\alpha /L_B < 1.0$ (Danilov & Gurarie Reference Danilov and Gurarie2000; Boffetta & Musacchio Reference Boffetta and Musacchio2010; Alexakis & Biferale Reference Alexakis and Biferale2018). Therefore, $L_\alpha /L_B$ may not be a suitable parameter for predicting the occurrence of spectral condensation.

Figure 7.

The exponent $\zeta _{IC}$ of the spectrum in the inverse cascade range as functions of two dimensionless parameters: (a) $\gamma$ and (b) $\lambda /l_f$. The data points in the present experiments (solid circles) are coloured by $Re_f$. The data from previous thin-layer turbulence experiments are also shown as hollow symbols for comparison, and their key parameters are listed in table 2. The inset in (a) shows $\zeta _{IC}$ as a function of $L_\alpha /L_B$ with different $L_B/l_f$: 38.4 (circles), 29.8 (left triangles), 27.7 (upper triangles) and 26.0 (hexagrams), and the thin dashed lines indicate that there is no universal value of $L_\alpha /L_B$ for the spectral transition. The inset in (b) shows $\lambda /l_f$ as a function of $\gamma$; the data shown by asterisks are from the soap-film experiments (Rivera et al. Reference Rivera, Vorobieff and Ecke1998; Rivera & Wu Reference Rivera and Wu2002; Zhou et al. Reference Zhou, Fang, Ouellette and Xu2020); the grey dash-dotted line denotes $\lambda /l_f\propto \gamma ^{-1.2}$.

Table 2.

Parameters of thin-layer quasi-2-D turbulence. The sources are (a) Paret & Tabeling (Reference Paret and Tabeling1997), (b) Xia et al. (Reference Xia, Punzmann, Falkovich and Shats2008, Reference Xia, Francois, Punzmann, Byrne and Shats2016), (c) Rivera & Ecke (Reference Rivera and Ecke2016), and (d) present work (only the cases $L_\alpha /L_B<1.0$ are shown). The last row denotes whether a steepened spectrum in the inverse cascade range is observed in the corresponding parameter space.

As shown in figure 7(a), we observe a transition in the scaling exponent $\zeta _{IC}$ from $-$5/3 to $-$3 at $\gamma \approx 6.8$. This value of $\gamma$ agrees to a certain degree with the numerical studies of Doludenko et al. (Reference Doludenko, Fortova, Kolokolov and Lebedev2022), in which $\gamma \approx 10$ is considered as a critical value for distinguishing the chaotic state and the coherent vortex state in 2-D turbulence. Furthermore, the results from the studies by Xia et al. (Reference Xia, Punzmann, Falkovich and Shats2008, Reference Xia, Francois, Punzmann, Byrne and Shats2016), using a similar forcing scale ($l_f = 10$ mm) in their experiments, demonstrate consistency with the observed trends in our data. However, the data from Paret & Tabeling (Reference Paret and Tabeling1997) ($l_f=15$ mm) and Rivera & Ecke (Reference Rivera and Ecke2016) ($l_f=18$ mm) appear to be inconsistent with our findings. This inconsistency suggests that $\gamma = \alpha l_f^2/\nu$, which incorporates the effect of $l_f$ (or $l_f^2$), may not be the best quantity for describing spectral properties, as the forcing scale itself does not play a crucial role in the energy flux when $L_B/l_f \gg 1.0$ (Paret & Tabeling Reference Paret and Tabeling1998; Xia et al. Reference Xia, Punzmann, Falkovich and Shats2008; Fang & Ouellette Reference Fang and Ouellette2021). Nevertheless, the relative strength of bottom friction to viscous dissipation, represented by the ratio $\alpha /\nu$, was found to have a crucial impact on the spectral properties.

We have seen a reduction in vorticity intensity $\varOmega$ as the damping rate $\gamma$ decreases, as shown in figures 5(a,d). Consequently, we can expect an increase in the Taylor microscale ($\lambda =\sqrt {E_0/\varOmega }=u_{rms}/\omega _{rms}$) with decreasing $\gamma$, which is demonstrated in the inset of figure 7(b). Specifically, the dimensionless Taylor microscale $\lambda /l_f$ presents a monotonic dependence on the damping rate, with a scaling $\lambda /l_f\propto \gamma ^{-1.2}$. The reason for this scaling is currently unclear and requires further research in the future. To shed some light on the transition in $\zeta _{IC}$, we plot it as a function of the ratio $\lambda /l_f$ in figure 7(b). Interestingly, we find that data from this work and from previous experimental studies, following a consistent trend, undergo a transition from $-5/3$ to $-3$ at $\lambda /l_f\approx 0.63$, and saturate when $\lambda /l_f\approx 1.0$. It may not be surprising that the Taylor microscale is able to capture certain spectral properties, as both $E_0$ and $\varOmega$ are integrals of the energy spectrum. However, this does not explain why the transition occurs specifically at $\lambda /l_f\approx 0.63$. Recalling that the suppression of intermediate scales (at approximately the forcing scale) by the large-scale structures leads to the variation of energy flux and spectral properties (see figures 5 and 6), it is natural to ask when this suppression comes into play. One possibility is that the suppression effect becomes significant when the large-scale streamers are stronger than the force-fed vortices, resulting in the latter being submerged or swept away by the former. As a result, the kinetic energy and energy flux at intermediate scales are remarkably diminished. In the current configuration of unbounded turbulence, the coherent structures manifest as large-scale streamers, and their strength can be characterized by $u_{rms}$, while the strength of forcing-scale vortices is represented by $\omega _{rms}$ (which is manifested by the enstrophy spectrum peaking at the forcing scale). Therefore, the condition for the dominant role of this suppression effect could be expressed as $u_{rms}>\omega _{rms}\times l_f/2$ (where $l_f/2$ is the radius of the forcing-scale vortices), i.e. $\lambda /l_f>0.5$. This argument agrees well with the observation in figure 7(b).

Additionally, it is noteworthy that the above argument is also applicable to 2-D soap-film experiments, in which the damping rate is relatively large. In previous studies of soap-film turbulence, the dimensionless Taylor microscale $\lambda /l_f$ did not exceed 0.4 (see the inset of figure 7b) (Rivera et al. Reference Rivera, Vorobieff and Ecke1998; Rivera & Wu Reference Rivera and Wu2002), notably smaller than the critical value 0.63. Correspondingly, the energy spectrum in the inverse cascade of soap-film turbulence is relatively flatter than $k^{-5/3}$, or there is no net inverse energy transfer (Rivera & Wu Reference Rivera and Wu2002; Cerbus & Goldburg Reference Cerbus and Goldburg2013; Zhou et al. Reference Zhou, Fang, Ouellette and Xu2020). Therefore, spectral condensation is not expected in these experiments. From this perspective, the dimensionless Taylor microscale $\lambda /l_f$ appears to be a valuable diagnostic parameter for characterizing the extent of inverse energy cascade and indicating the occurrence of spectral condensation in (quasi-)2-D turbulence experiments. We stress, however, that there is no causality here between the Taylor microscale and spectral condensation, and it is only their correlation (see figure 7b) that allows us to use the former as a diagnostic indicator of the latter.