Size-dependent transient nature of localized turbulence in transitional channel flow

Localized turbulence is ubiquitous in subcritical transition to turbulence in wall-bounded shear flows. In canonical shear flows such as pipe and Couette flows, localized turbulence has been shown to be stochastically transient in nature, given that it can long live before suddenly reverting to laminar flow through a memoryless decaying process (Bottin & Chaté, 1998; Peixinho & Mullin, 2006; Hof et al., 2006; Eckhardt et al., 2007; Avila et al., 2010; Borrero-Echeverry et al., 2010; Avila et al., 2011; Shi et al., 2013). This transient nature is an essential ingredient (together with stochastic memoryless splitting) for establishing a relationship between the transition to turbulence in canonical shear flows and directed percolation phase transition (DP) (Hinrichsen, 2000; Barkley, 2016; Lemoult et al., 2016; Shih et al., 2016; Chantry et al., 2017; Klotz et al., 2022). These studies imply that both the transient nature and DP scenario may be universal to the transition of shear flows.

In transitional channel flow, the localized turbulence takes a form of banded structure tilted with respect to the streamwise direction (Tsukahara et al., 2005). The origin of this banded structure has been explained from the point of view of dynamical systems (Paranjape et al., 2020). In quasi-one dimensional (1D) channel flow realized in a narrow tilted computation domain, in which turbulent bands can only localize in the direction perpendicular to them, stochastic transient and splitting behaviors have also been reported, like in other canonical shear flows (Gomé et al., 2020, 2022). This scenario seems to suggest a (1+1)D (space + time) DP process close to the onset of sustained turbulence determined by the balance between the decay and splitting processes. However, the quasi-1D restriction does not allow the turbulent band to fully localize, leaving out the dynamics of upstream and downstream ends of the band.

In more realistic channel flows, where turbulence can be fully localized by developing a downstream end and an upstream end, the statistical characteristic of the globally sustained turbulence was proposed to be described by the (2+1)D DP model only in a limited range of transitional Reynolds number (Sano & Tamai, 2016; Shimizu & Manneville, 2019). Recent studies reported that a fully localized turbulent band becomes sustained already at low Reynolds numbers $Re_{\mathrm{cr}}\simeq 650\sim 660$ (Tao et al., 2018; Kanazawa, 2018; Paranjape, 2019; Mukund et al., 2021), far below the critical Reynolds numbers of $Re_{\mathrm{cr,DP}}\simeq 830$ (Sano & Tamai, 2016) and 905 (Shimizu & Manneville, 2019) for sustained turbulence in DP theory. This early sustained turbulence conflicts with DP that turbulence should be transient and the flow should eventually laminarize completely when the Reynolds number is below $Re_{\mathrm{cr,DP}}$. It also renders channel flow transition distinct from its counterparts in other shear flows, challenging transition via transient turbulence as a universal scenario for shear flows.

The self-sustaining of a turbulent band is mainly ascribed to the turbulence generation at its downstream end (Kanazawa, 2018; Paranjape, 2019; Shimizu & Manneville, 2019; Xiao & Song, 2020a, b; Liu et al., 2020; Mukund et al., 2021). Kanazawa (2018) proposed that a nonlinear relative periodic orbit is embedded at the downstream end of a turbulent band, generating velocity streaks and vortices periodically, while Xiao & Song (2020a) and Song & Xiao (2020) reported an inflectional instability associated with the local mean flow as the mechanism for turbulence generation there. However, a turbulent band immediately starts to decay if this mechanism is interrupted when the downstream end collides either with other turbulent bands (Shimizu & Manneville, 2019; Song & Xiao, 2020) or with channel side wall (Paranjape, 2019; Wu & Song, 2022). Free of disruption, a turbulent band even can be maintained down to $Re\simeq 500$, if the mechanism at the downstream end is maintained with some external body forces (Song & Xiao, 2020). The dynamics at this end is thus essential for the self-sustaining of turbulent bands (Mukund et al., 2021).

In this paper, we show that the dynamics at the downstream end of a turbulent band is in fact intrinsically transient by direct numerical simulation. Unlike the localized turbulence in pipe, Couette and quasi-1D channel flows, the transient behavior of the downstream end is featured with a size dependence.

It is computationally costly to track the dynamics of the entire localized bands in a large domain for a long time, given that a turbulent band may grow to hundreds of channel half heights ($h$) in length for $Re\gtrsim 660$ (Kanazawa, 2018; Tao et al., 2018; Paranjape, 2019; Shimizu & Manneville, 2019). Our simulations were performed in relatively smaller channels with periodic boundary conditions in the streamwise and spanwise directions, with a particular focus on the downstream end. The downstream end was tracked by using a frame of reference co-moving with the downstream end, and the band length is controlled by using a local artificial damping in a proper region of the moving frame (see figure 1 for an illustration). The role of this damping region is to damp out the turbulence that enters the region, for restricting the growth of the band. This also helps avoiding self-interaction of the turbulent band due to the periodic boundary condition. This technique, inspired by Kanazawa (2018), was used to isolate turbulent fronts in short periodic pipes for successfully investigating their local dynamics (Chen et al., 2022).

The Navier-Stokes equations in the frame of reference with a speed $\bm{c}=(c_{x},0,c_{z})$,

are solved, where $\bm{u}$ is the velocity with respect to the moving frame, $\mathrm{\bm{U}}$ is the parabolic base flow in the moving frame, $t$ is the time, $p$ is the pressure, and $c_{x}$ and $c_{z}$ are streamwise and spanwise speeds of the moving frame. The velocity is normalized by $3U_{b}/2$ and length by $h$, where $U_{b}$ is the bulk speed, correspondingly the Reynolds number is $Re=3U_{b}h/(2\nu)$ with $\nu$ the kinematic viscosity of the fluid. An artificial damping term $-\beta(z)(\bm{u}-\mathrm{\bm{U}})$ is added with

where $A$ is the nominal damping amplitude, $R$ the nominal half-width of the damping region and $B$ controls the decay of the damping strength at the boundary of the damping region ($z_{0}-R<z<z_{0}+R$). This damping is imposed only on the velocity deviation with respect to the parabolic base flow. In all simulations, we chose $B=0.5$, which gives a reasonably sharp drop of the damping at the boundary of the damping region, and $A=0.25$. The numerical code based on a high-order finite difference and Fourier spectral schemes used by Xiao & Song (2020a), Song & Xiao (2020) and Xiao & Song (2020b) is adapted for this study. The volume flux is kept constant during the simulations. The initial conditions are long-lived turbulent bands at close Reynolds numbers with the same band length, channel size and damping setting. In wall normal direction, 56 Chebyshev points are used for the finite difference discretization, and $h/\Delta x=5.2$ and $h/\Delta z=6.4$, which are comparable to the usual resolutions in the literature (Kanazawa, 2018; Tao et al., 2018; Xiao & Song, 2020a; Song & Xiao, 2020; Xiao & Song, 2020b). Time step size is $\Delta t=0.025$ for all the simulations. Higher wall-normal resolutions (72 and 96 points) and $\Delta t=0.015$ were also tested to make sure that the observation is not qualitatively affected.

With this technique, the position of the downstream end can be nearly fixed in the moving frame (see figure 1c for an example). Given the stable tilt angle of turbulent bands about the streamwise direction at low Reynolds numbers (around $42^{\circ}$ in the considered $Re$ regime) (Tao et al., 2018; Kanazawa, 2018; Paranjape, 2019; Xiao & Song, 2020a, b), the length of the band can be well controlled. This allows affordable simulations and an evaluation of the dependence of the band statistics on the controlled band length. With a similar idea, Taylor et al. (2016) controlled the total turbulent kinetic energy level by dynamically adjusting the Richardson number for a stratified flow. The difference is that their control affects the flow globally, whereas our damping technique only locally affects the turbulence in a small region at the upstream end of a turbulent band.

In such a setting, the downstream end can be tracked for a long time, and a stochastic transient behavior of the turbulence is observed, see figure 2 for $Re=655$ (close to the reported $Re_{\mathrm{cr}}$). The lengths of the bands are restricted to be around $L=69$. Figure 2(a) shows that the turbulence kinetic energy ($KE=\int_{V}(\bm{u}-\mathrm{\bm{U}})^{2}\mathrm{d}V/\int_{V}\mathrm{\bm{U}}^{2}\mathrm{d}V$ where $V$ denotes the whole flow domain) stays at a constant level before a sharp decrease toward zero, an indication of a sudden relaminarization of the flow. No significant difference for the band is visually seen in its lifetime before the sudden decay (see figure 2b–d). Nevertheless, noticeable difference can be observed in the decaying process, as shown in figure 2(e). Firstly, the streaks at the downstream end of the band appear to be tilted at much smaller angles with respect to the streamwise direction, a possible result of a reduced inflectional instability at the downstream end (Wu & Song, 2022). Secondly, the streaks away from the downstream end show less wiggling and fewer smaller structures, indicating weakened turbulent activities. The streak structures here cannot sustain themselves and are quickly dissipated. Once relaminarized, without a finite amplitude perturbation, the flow remains as laminar given the subcriticality of the channel flow at this low Reynolds number. See more details of the decay process in Appendix A.

The lifetime of a band can be measured by simply setting a threshold in $KE$ ($3\times 10^{-3}$ in this study), below which the turbulence is considered to have decayed. Thanks to the sharpness of the decrease, different thresholds would not cause large variation in the measured lifetime. Figure 2(a) shows that the lifetimes of the bands, starting from similar initial conditions, can differ by an order of magnitude (see the yellow and black curves), given the same $Re$ and band length. This indicates the stochasticity of the lifetime. Repeating the simulations with thirty different initial conditions at $Re=655$, we can define the survival rate $S$ as

where $N_{\mathrm{survived}}(t)$ denotes the number of runs in which turbulent band survived up to time $t$ and $N$ denotes the sample size. Plotting $S$ against time shows that the stochastic lifetimes seem to be exponentially distributed (see the black triangles in figure 3a). The exponential distribution of the lifetime of localized turbulence has been reported in pipe, Couette, and quasi-1D channel flow (Bottin & Chaté, 1998; Peixinho & Mullin, 2006; Hof et al., 2006; Avila et al., 2010; Borrero-Echeverry et al., 2010; Shi et al., 2013; Shimizu et al., 2019; Gomé et al., 2020, 2022), however, to our knowledge, such a characteristic is for the first time observed in channel flow without the quasi-1D restriction.

Statistics of the survival rate and lifetime were then performed at a few parameter groups $(Re,L)$, as shown in figure 3 where the results are grouped by the band length $L$. We considered $Re=650$, $655$ and $660$ for $L=69$, while smaller $L$’s in smaller channels for higher Reynolds numbers are considered given the rapidly increasing lifetime with $Re$ and so as the computation cost (see detailed domain configuration in the caption of figure 3). For most $(Re,L)$ groups, we performed approximately 30 runs with similar initial conditions (limited by the high computation costs), while 75 runs were performed at $(Re,L)=(670,38)$ considering the relatively low cost. The sample sizes are by no means large enough to give accurate statistical quantities, nevertheless, for all cases, we observed that the survival rate of turbulent bands $S$ exhibits an approximately exponential tail against time at sufficiently large times, an implication for an exponential distribution. For each $L$, the slope of the data in semi-logarithmic scale decreases as $Re$ increases, suggesting a larger mean lifetime.

Assuming an exponential distribution (suggested by figure 3), we estimated the mean lifetime (the expectation of the stochastic lifetime) of the turbulent bands using the method of Avila et al. (2010) and Gomé et al. (2020). Figure 4 shows the mean lifetime $\tau$ against $Re$ in groups of the band length. Clearly, the mean lifetime increases rapidly with $Re$ in this narrow $Re$ regime, and the data for $L=38$ and $54$ seem to suggest a nearly exponential growth. However, a distinct scaling cannot be concluded due to the limited number of $Re$ and the narrow $Re$ range. For a fixed $Re$, a sharp increase in $\tau$ with $L$ can be seen, given that $\tau$ increases by nearly ten thousand at $Re=655$, 660 and 665 when $L$ is increased by $15$. Clearly, the mean lifetime strongly depends on the band length.

Statistical studies at larger band lengths and higher Reynolds numbers are not affordable given the large length and time scales, nevertheless, we confirmed the transient behavior with a single simulation at a few parameter groups (see figure 5). We considered $L=84$ (realized in a channel of $L_{x}=L_{z}=100$) for $Re=655$, $660$ and $665$, and also considered $L=110$ (realized in a $(L_{x},L_{z})=(120,110)$ channel) for $Re=645$, $650$ and $660$. The two lengths are about 0.3 and 0.4 times of the equilibrium length of a turbulent band at $Re=660$ measured in a large domain (Kanazawa, 2018), respectively. For higher $Re$, we considered $Re=680$ and 685 in a channel of $L_{x}=L_{z}=60$ with $L=38$ for the turbulent bands, and observed large lifetimes (nearly $10^{5}$ time units at $Re=685$), see figure 5(b). The turbulent band also shows a transient characteristic in all these simulations. These observations give an expectation that the transient characteristic is likely to be retained beyond the band lengths and Reynolds numbers considered in the current work.

If a turbulent band is not restricted by the damping, it would grow in size before reaching the statistical equilibrium length (Kanazawa, 2018), and the average growth rate increases with $Re$ (Paranjape, 2019; Mukund et al., 2021). Even when the band length has reached the statistical equilibrium length, it fluctuates strongly due to the erratic decay of turbulence or band splitting at the upstream end (Kanazawa, 2018; Paranjape, 2019; Mukund et al., 2021). This differs from turbulent puffs in pipe flow which naturally have a nearly constant size (see e.g. Wygnanski et al. (1975); Eckhardt et al. (2007); Mullin (2011)), and therefore, the size is not relevant and the memoryless characteristic greatly eases the lifetime measurements. In channel flow, the ever change of the band length and the resulted vagueness of the transient characteristics, without any a priori expectation of the statistical distribution of the lifetimes, pose a challenge for lifetime measurements either in experiments or simulations.

Nevertheless, our results suggest that, as the length of a turbulent band grows, the mean lifetime would increase and the relaminarization of the bands would become increasingly rare in typical observation times in experiments and most numerical simulations ($\mathcal{O}(10^{3}\sim 10^{4})$), limited by the channel size and the computation cost. Shimizu & Manneville (2019) performed the longest simulations ($1.5\times 10^{5}$ time units), to our best knowledge, in a large domain of $(L_{x},L_{z})=(500,250)$ and reported that turbulent bands only become sustained above $Re\simeq 700$. This Reynolds number is higher than the critical Reynolds number of $650\sim 660$ for sustained turbulent bands proposed in other studies (Kanazawa, 2018; Tao et al., 2018; Paranjape, 2019; Kashyap et al., 2020b; Mukund et al., 2021), where the observation time was one order of magnitude shorter. The disagreement on the critical point may be partially attributed to whether or not the transient nature of turbulent bands was observed in different studies.

With a technique to control the length of a turbulent band, we identified the stochastic transient nature of the turbulence-generating dynamics at the downstream end at low Reynolds numbers, which was believed to sustain the entire turbulent band. This transient characteristic clarifies that localized turbulence in channel flow is similar to that in other shear flows and can also be described as a chaotic saddle in the perspective of dynamical systems. Our finding therefore supports that transition via transient localized turbulence is potentially a universal scenario in subcritical shear flows. Different from other flow systems, the peculiarity of channel flow is that, if the length of the band could be fixed, the lifetime would appear to be exponentially distributed and the decay process would be memoryless, and the mean lifetime would increase rapidly with the band length. However, when a turbulent band varies in length over time as in real channel flow, the size dependence translates into a time dependence. The statistical distribution of the lifetime cannot be directly inferred from our results and the lifetime is even difficult to measure. Nevertheless, the transient characteristics probably change with time and an increasing mean lifetime can be expected for a growing band.

Continuing efforts have been taken to prove that channel flow transition also belongs to the DP universality class (Sano & Tamai, 2016; Shimizu & Manneville, 2019; Manneville & Shimizu, 2020), thus unifying all canonical shear flows as a DP-type phase transition, which is however a difficult task given the complex dynamics exhibited by channel flow turbulence as presented here and elsewhere. Manneville & Shimizu (2020) constructed a probabilistic cellular automaton model for channel flow by taking an entire turbulent band as an excited site and observed the (1+1)D DP behavior under certain model parameters. Therefore, their study and Shimizu & Manneville (2019) seem to suggest a two-staged DP scenario, i.e. firstly a (1+1)D DP in the one sided regime and then a (2+1)D DP in the two sided regime. This model requires the same time-invariant probabilistic characteristics for all sites (i.e. all bands), as the classical DP model does also, which would equivalently require the statistics of the lifetime to be the same for bands with different lengths. The size dependence of the lifetime observed in our study seems to conflict with their model assumption. Mukund et al. (2021) reported that the splitting of a turbulent band at the upstream end also exhibits a size-dependent stochastic behavior and proposed that the splitting is a history-dependent instead of a memoryless process, which seems to conflict with the fundamental assumption of DP also. The finding of Mukund et al. (2021) and our observation question the simplification of an entire band as a percolation site with time-invariant probabilistic characteristics.

The far-field large-scale flow around a turbulent spot was found to decay algebraically (instead of exponentially) away from the localized turbulence, and very large domains are needed to accommodate it given this relatively slower decay (Kashyap et al., 2020a). Our domain sizes are not large enough in this respect. However, the far-field velocity seems to be of low amplitudes (see figure 7 in Appendix B), and therefore is not expected to qualitatively affect the turbulence generation and self-sustenance of the band, although may have some quantitative effects on the statistics of the lifetime. This is to be studied more quantitatively in the future. Besides, our current study covers small $Re$ and $L$ ranges so that cannot give the scaling of the mean lifetime with $Re$ and $L$ for bands under size control. Larger sample sizes and wider $Re$ and $L$ ranges need to be considered for future studies, and the mechanism underlying the size dependence of the lifetime remains to be theoretically clarified. The critical Reynolds number at the onset of globally sustained turbulence, above which the splitting outweighs the decay of turbulent bands, is also an important problem to be solved in the future. It would be too costly to purely rely on brute force DNS to resolve these problems, and more efficient algorithms, such as the extreme event algorithm (Gomé et al., 2022; Rolland, 2022) and the machine learning based method recently proposed for a simple Waleffe flow (Pershin et al., 2021), may be helpful. Simplified models may be necessary also when approaching the onset of globally sustained turbulence given the very large time and space scales involved (as in the case of pipe flow (Avila et al., 2011; Barkley, 2016; Mukund & Hof, 2018)). Our finding will be informative for designing such studies.

The authors acknowledge the financial support from the National Natural Science Foundation of China under grant number 11988102, 91852105, 91752113 and 92152106, as well as from Tianjin University under grant number 2018XRX-0027. The simulations were performed on Tianhe-2 at the National Supercomputer Centre in Guangzhou and Tianhe-1(A) at the National Supercomputer Centre in Tianjin.

The authors declare no conflicting interests.