Subcritical transition to turbulence: what we can learn from the physics of glasses
In this note, we discuss possible analogies between the subcritical transition to turbulence in shear flows and the glass transition in supercooled liquids. We briefly review recent experimental and numerical results, as well as theoretical proposals, and compare the difficulties arising in assessing the divergence of the turbulence lifetime in subcritical shear flow with that encountered for the relaxation time in the study of the glass transition. In order to go beyond the purely methodological similarities, we further elaborate on this analogy and propose a simple model for the transition to turbulence, inspired by the Random Energy Model (a standard model for the glass transition), with the aim to possibly foster yet unexplored directions of research in subcritical shear flows.
pacs:47.27.Cn, 47.27.eb, 64.70.P-
Statistical physics has devoted a lot of effort to the study of fully developed turbulence, but much less to that of the transition to turbulence Frisch (1995), which occurs when the Reynolds number, the ratio of the advection time to the viscous time, is increased. The transition is commonly observed in flow regimes lacking linear instability and is referred to as globally subcritical Joseph (1976); Grossmann (2000); Dauchot and Manneville (1997).
The plane Couette flow, driven by two plates moving parallel to each other in opposite directions is linearly stable at all Reynolds number and, as such, is the epitome of globally subcritical transitions Romanov (1973). Other flows usually transit to turbulence before linear instability sets in. These include the circular Poiseuille flow (cPf) and the plane Poiseuille flow (pPf), which are driven by a pressure gradient respectively along a circular pipe or between two parallel plates, as well as the counter-rotating Taylor Couette flow (TCf), driven by two concentric cylinders rotating in opposite directions. In all these cases, the transition is particularly delicate to understand owing to its abrupt character. A complex spatio-temporal dynamics is observed, involving in particular the nucleation and the growth or decay of turbulent domains called ’puffs’ (pPf) or ’spots’ (pCf) –see, e.g., Darbyshire and Mullin (2006); Willis et al. (2008) for cPf, Tillmark and Alfredsson (1992); Daviaud et al. (1992); Dauchot and Daviaud (1995) for pCf, Carlson et al. (1982) for pPf, and Coles (1965); Coles and Van Atta (1967) for TCf.
A recent surge of interest has been motivated by the audacious proposal that shear flow turbulence could remain transient up to arbitrarily large Reynolds number, opening ways towards a better control of such turbulent regimes Hof et al. (2006). This proposal was motivated by experimental and numerical observations in cPf Hof et al. (2006) and pCf Schneider et al. (2010a) regarding the statistics of turbulent lifetimes, which contradicted those previously obtained in cPf Faisst and Eckhardt (2004); Peixinho and Mullin (2006) and pCf Bottin et al. (1998a); Bottin and Chaté (1998). These contradictory results, together with the experimental discovery of a spectacular long wavelength periodic organization of the laminar-turbulent coexistence in pCf and TCf Prigent et al. (2002, 2003), has motivated further experiments in TCf Borrero-Echeverry et al. (2010) and cPf Avila et al. (2011); Samanta et al. (2011), the development of various models Eckhardt et al. (2008); Lagha and Manneville (2007); Gibson et al. (2008); Manneville (2009a); Barkley (2011) and an impressive number of numerical studies Barkley and Tuckerman (2005); Willis and Kerswell (2007); Schneider and Eckhardt (2008); Duguet et al. (2008); Tuckerman et al. (2008); Duguet et al. (2009); Schneider et al. (2010b); Duguet et al. (2010); Moxey and Barkley (2010); Manneville and Rolland (2011); Avila et al. (2011); Philip and Manneville (2011); Manneville (2012). As a result, some comprehension of the mechanisms at play in the coexistence dynamics, as well as a better knowledge of the organization of phase space, involving many unstable solutions of the Navier-Stokes equation, has been gained.
Interestingly, the presence in phase space of many unstable solutions and the existence of finite, yet extremely large, relaxation times, are reminiscent of the physics of glasses (see, e.g., Struik (1978); Angelani et al. (2000); Berthier and Biroli (2011)). In particular, whether the structural relaxation times of a glass really diverges at a given finite temperature or remains very large but finite at any positive temperature is an important question –related to the existence of a genuine phase transition to an ideal glass state– that remains largely open Kivelson et al. (1996). However, the intense activity related to this specific issue has triggered along the way different (and perhaps even more interesting) questions, driving the field of glasses towards important conceptual progresses Berthier and Biroli (2011).
In this note, we explore the analogy between the subcritical transition to turbulence and the glass transition from several viewpoints. After a concise review of the major results on the transition to turbulence, we discuss the limitations of fitting procedures in assessing the divergence of the turbulence lifetime, drawing inspiration from similar discussions in the glass literature (Sec. II). Then, we briefly review the theoretical scenarios and models that have been proposed to describe the subcritical transition to turbulence, and we tentatively discuss the analogy with glasses at a conceptual level (Sec. III). The understanding of the glass transition has greatly benefited from the study of oversimplified models like the Random Energy Model Derrida (1980, 1981), which describes the statistical behavior of a system evolving in a random energy landscape. In this spirit, we try to transpose the Random Energy Model, keeping in mind both its strengths and weaknesses, to the modeling of the subcritical transition to turbulence, in order to possibly gain insight into the statistical mechanisms at play in this transition (Sec. IV). As a result, we obtain an estimate of the turbulence lifetime as a function of the Reynolds number close to the transition, an estimate which qualitatively agrees amazingly well with the observed phenomenology.
Clearly, this qualitative agreement does not in itself prove the analogy to be specifically deep, but it suggests that it deserves to be further explored. More generally, we hope, in the spirit of Pomeau’s seminal paper Pomeau (1986), that the analogy presented here could foster contributions from the statistical physics community to the old standing problem of the transition to turbulence, taking advantage of recently developed concepts in the statistical physics of disordered systems. Conversely, the development of techniques such as Particle Image Velocimetry, and the exponential increase of the numerical capacities could help in validating or invalidating the assumptions made on some properties of turbulence in following the present analogy.
Ii Turbulence lifetime
ii.1 A review of experimental and numerical observations
A standard characterization of the subcritical transition to turbulence is the determination of the average turbulence lifetime, following either a perturbation or a quench, as a function of Reynolds number. We thus start by briefly reviewing the experiments and direct numerical simulations reporting the increase of the turbulence lifetime when the Reynolds number is increased. To our knowledge, the first systematic measurement of turbulence lifetimes was conducted in the pCf Bottin et al. (1998a); Bottin and Chaté (1998). Two different kinds of experiments have been performed, differing by the way the initial condition is prepared. In what we shall call type-A experiments, the Reynolds number, , is set to its value of interest and the laminar flow is disturbed locally at the initial time. In type-B experiments, a turbulent flow is prepared at high , and quenched at the initial time down to the value of interest. In both cases, one monitors the evolution of the turbulent fraction , which characterizes the coexistence dynamics of laminar and turbulent domains (see Fig 1). For , fluctuates around some average value, which remains finite on experimental timescales. For , relaxes towards zero, without displaying any long transient regime. In between, for , exhibits a first rapid decay, followed by a long transient quasi-steady regime, before a large fluctuation sets it to zero. The lifetime of these transients are exponentially distributed and the average value was reported to diverge like .
The cPf was later investigated in various ways. In Peixinho and Mullin (2006); Darbyshire and Mullin (2006) a puff was generated inside a constant flow rate pipe flow by introducing a short duration perturbation. Then R was reduced and the subsequent evolution of the puff was monitored as it progressed downstream. The probability of observing a localized disturbed region of flow as a function of distance downstream is exponential and the time required for half of the initial states to decay, , was reported to diverge like , in agreement with the observations made in the pCf. Other protocols lead to the same conclusions Peixinho and Mullin (2006); Darbyshire and Mullin (2006).
However, these results were challenged later by another experimental study Hof et al. (2006). In a pressure driven flow through a very long pipe, the authors could record much longer dimensionless observation times. They could determine the probability to be turbulent after a time period given by the distance between the perturbation location and the outlet, as a function of flow rate. For short times, the data are within the error bars of Peixinho and Mullin (2006); Darbyshire and Mullin (2006) but for longer times they deviate from the divergent behavior reported above and are better represented by an exponential variation: , without singularity (here and in what follows, and denote generic fit parameters). Finally in a recent experimental study of turbulence in pipe flow spanning height orders of magnitude in time, drastically extending all previous investigations, it was claimed that the turbulent state remains transient, with a mean lifetime, which depends super-exponentially on the Reynolds number: Hof et al. (2008).
Intense numerical simulations of the cPf have also been conducted, but did not clarify the situation. In Faisst and Eckhardt (2004); Willis and Kerswell (2007) a diverging behavior of the turbulent lifetimes compatible with the experimental results of Peixinho and Mullin (2006); Darbyshire and Mullin (2006) is reported. Later in Schneider and Eckhardt (2008), the authors –one of which is common to Faisst and Eckhardt (2004)– conducted further simulations and reanalyzed older data, concluding to an exponential dependence such as the one reported in Hof et al. (2006). Altogether despite intense experimental and numerical effort, no definitive answer regarding the divergence or finiteness of turbulence lifetime could be obtained from the fit of data by phenomenological functional forms.
ii.2 Fitting procedures: lessons from glass physics
As stated in the introduction, this issue is not specific to the transition to turbulence. When a liquid is suddenly quenched below its crystallization temperature and if crystallization can be avoided, the liquid enters a state, called supercooled liquid, in which the relaxation time increases by several orders of magnitude over a limited range of temperature Struik (1978). A divergence at a finite temperature of the relaxation time would signal an ideal glass transition, and would thus be of high interest, at least at a conceptual level. Despite huge efforts made to measure the variations of the relaxation time over an experimental window of more than ten decades, no clear consensus has been obtained yet. More precisely, the available data are both consistent with fits including a divergence at a finite temperature , and with fits diverging only at Kivelson et al. (1996).
The same difficulty is also expected to appear in the context of turbulence. We illustrate this point on experimental data recently obtained in the case of the TCf Borrero-Echeverry et al. (2010), when only the external cylinder is rotating. The TCf is then, like the pCf, linearly stable for all R. Also, because the TCf is a closed flow, one can record very long times. In this experiment, the angular rotation of the external cylinders fixes the Reynolds number. The flow was perturbed by rapidly accelerating the inner cylinder in the direction opposite to the rotation of the outer cylinder and immediately stopping it. After a short regime of featureless turbulence, the flow exhibits long transients characterized by the coexistence of laminar and turbulent domains, before eventually relaxing towards the laminar flow. The distribution of lifetimes is again exponential, and the authors argue that the mean turbulent lifetime does not diverge and rather behaves in the transitional regime as a double exponential , as observed in the cPf Hof et al. (2008).
It is interesting to note that in the oldest experiments, the debate about the functional dependence of the average turbulent lifetime on the Reynolds number was concentrating on the choice between the two following forms:
whereas the most recent experiments, both in the case of the cPf Hof et al. (2008) and the TCf Borrero-Echeverry et al. (2010), have access to much longer experimental timescales and point at a double exponential behavior. This last functional form ensures a very fast increase of without singularity, and could give the impression that it solves the above debate. However as learnt from the physics of glasses, the debate has actually been shifted towards two alternative functional forms, namely:
As a matter of fact, Eq. (4), which has (to the best of our knowledge) not yet been proposed in the context of the transition to turbulence, is a very standard form called the Vogel-Fulcher-Tammann (VFT) law in the physics of glasses Kivelson et al. (1996).
Figure 2 displays the data obtained in TCf Borrero-Echeverry et al. (2010) –which are available online as supplementary material– together with possible fits by the four functional forms proposed above. Note that we have only performed global fits of the data, without trying to extract various regimes and crossovers as can be done in the case of glasses Kivelson et al. (1996). One clearly observes that indeed the relevant variable to describe the growth of the turbulent lifetimes is , as soon as really large times are considered. However, one also sees that apart from the simple exponential form Eq. (1), all other descriptions are not discriminable, so that there is no definitive way to rule out or confirm the existence of a singularity. One faces the same difficulty as in the physics of glasses: the lifetimes to be measured become very large, which makes it difficult to accumulate significant statistics. The experimental results are thereby confined to a finite range of Reynolds number or temperature, from which even with high-quality data sets, the divergence of a characteristic time cannot convincingly be determined from fits.
Before concluding this section, let us mention that the double-exponential form Eq. (3) has been justified on the basis of extreme value statistics Goldenfeld et al. (2010). However, as stated by the authors, the argument is only local, as it involves an expansion in around a given reference value. Hence no clear conclusion can be drawn from the theoretical argument of Goldenfeld et al. (2010) on the issue of the divergence of at a finite or infinite value of . Finally, let us emphasize that for now, we have left aside all issues related to finite size effects, which in turn can severely alter the functional dependence of time and length scales in transitional regimes.
Iii Theoretical scenarios and models
After discussing the empirical results, a natural question is to know how one can understand, from a more theoretical perspective, the globally subcritical transition to turbulence. This transition is by definition controlled by solutions of the Navier-Stokes equation, which do not branch continuously from the laminar flow solution when the Reynolds number is increased Kawahara et al. (2012). These solutions –of various kinds, stationary states, traveling waves, or more complex coherent structures– are unstable and form hyperbolic states, with stable and unstable manifolds. Early indications of the existence of these solutions were reported in pCf, both numerically Nagata (1990); Clever and Busse (1992); Cherhabili and Ehrenstein (1996) and experimentally Bottin et al. (1998b). More recently, they were also observed in the cPf de Lozar et al. (2012). The intricate network made of these manifolds and their connections then serves as a skeleton for the turbulent flow.
iii.1 Low dimensional models
In principle one would like to collect all such states, estimate their dynamical weight and calculate statistical averages from periodic orbit theory Cvitanovic and Eckhardt (1991). In practice, one must restrict the analysis to low-dimensional models Waleffe (1997); Dauchot and Vioujard (2000); Eckhardt and Mersmann (1999); Moehlis et al. (2004); Skufca et al. (2006); Vollmer et al. (2009) or to simulations Schmiegel and Eckhardt (1997); Itano and Toh (2001); Faisst and Eckhardt (2004); Skufca et al. (2006); Hof et al. (2006); Eckhardt et al. (2008); Schneider et al. (2007); Schneider and Eckhardt (2008), performed in the so-called minimal flow unit assumption Jimenez and Moin (1991). Doing so, it was shown that the regions of initial conditions for which long lifetimes exhibit strong fluctuations and a sensitive dependence on initial conditions were separated from the regions with short lifetimes and smooth variations by a border, the so-called ”edge of chaos” Itano and Toh (2001); Skufca et al. (2006); Vollmer et al. (2009). Later, some exact solutions with codimension-1 stable manifolds have been identified as edge states, that is, solutions that locally form the stability boundary between laminar and turbulent dynamics Duguet et al. (2009); Mellibovsky et al. (2009); Duguet et al. (2010); Schneider et al. (2010b). These important results contributed to make concrete the picture borrowed from dynamical system theory of a turbulent repellor, separated from the laminar state by a set of edge states connected through heteroclinic manifolds. In particular the existence of the above non trivial solutions has served to understand the exponential distribution of lifetimes in the transitional regime.
iii.2 Spatially extended models
Unfortunately the above picture does not bring a complete description of the subcritical transition to turbulence. As argued in Lagha and Manneville (2007); Manneville (2009a, b), the reason is that the dynamics being either projected on a small set of modes or limited to small computational domains with periodic boundary conditions, it cannot capture the genuinely spatiotemporal coexistence of laminar and turbulent states observed in open and unbounded flows. In particular, it can neither capture the long wavelength modulation of turbulent intensity, nor the regime of alternating laminar and turbulent stripes, first observed experimentally in pCf and TCf Prigent et al. (2002, 2003), and then reproduced numerically in pCf Barkley and Tuckerman (2005); Tuckerman et al. (2008); Duguet et al. (2010); Manneville and Rolland (2011); Rolland and Manneville (2011); Manneville (2012).
As a matter of fact it is known for long that, according to the scenario called spatiotemporal intermittency (STI Kaneko (1985)), transient chaotic states locally distributed in space, e.g. on a lattice, may evolve into a sustained turbulent global state due to spatial couplings Chaté and Manneville (1988); Chaté and Manneville (1994); Bohr et al. (2001); Grassberger and Schreiber (1991). Following this path, it was demonstrated that a simple 1D-model of cPf, composed of coupled maps, does indeed captures remarkably well the character of the turbulent pipe flow in the transitional regime and exhibits a critical transition towards sustained turbulence via spatiotemporal intermittency Barkley (2011). The transition is further believed to belong to the Directed Percolation class Chaté and Manneville (1988); Chaté and Manneville (1994); Grassberger (2006), as already suggested in Pomeau (1986) for pCf, and recently reconsidered in cPf Sipos and Goldenfeld (2011).
Finally, it was shown by means of fully resolved direct numerical simulations of the Navier Stokes equation, that there exists a crossover length scale of the order of times the cross-stream length below which the spatio-temporal processes at play in large-scale simulations and experiments are not captured Manneville (2009a). Since then, a number of numerical investigations of large aspect ratios cPf and pCf have reproduced the complex spatio-temporal coexistence of laminar and turbulent states, and identified the first hydrodynamics mechanisms at play Duguet et al. (2010); Moxey and Barkley (2010); Manneville (2012); Avila et al. (2011).
iii.3 Analogies and differences with glasses
We now wish to discuss from a theoretical perspective the analogies, as well as the differences, between flows close to the transition to turbulence (in short, transitional flows) and liquids close to the glass transition. To this aim, it is useful to first summarize the essential features of the subcritical transition to turbulence:
Subcriticality: While the laminar flow is stable against infinitesimal perturbations, finite amplitude perturbations may trigger an abrupt transition towards a disordered flow. Such a disordered flow can also be obtained by quenching fully turbulent flows.
Spatio-temporal intermittency: This disordered flow is made of turbulent domains, which move, grow, decay, split and merge leading to spatio-temporal intermittency, that is a coexistence dynamics in which active/turbulent regions may invade absorbing/laminar ones, where turbulence cannot emerge spontaneously.
Transients and Meta-stability: For large enough Reynolds number this disordered flow has long lifetimes, which are distributed exponentially. Whether the associated characteristic time diverges at a finite Reynolds number is still a matter of debate. For low Reynolds number, say , or small enough disturbances, the flow relaxes rapidly towards the laminar flow.
Unstable states: When increasing the Reynolds number a larger and larger number of unstable finite amplitude solutions appear in phase space. Some have been identified as edge states separating the others from the laminar state.
As mentioned in the introduction, some of these features are also shared, at a qualitative level, with glasses. For instance, the presence of long transient relaxing states is a key feature of glasses Struik (1978). Also, the existence of many unstable solutions is reminiscent of the energy landscape picture of glasses Sciortino (2005). Indeed, the slow relaxation in glasses has been argued to result from the wandering of the phase-space point representing the system in a complex energy landscape Bouchaud et al. (1998), mostly composed of many unstable fixed points Angelani et al. (2000); Broderix et al. (2000); Grigera et al. (2002) (though local minima also play an important role at low enough temperature). The most striking feature of the glass transition, the rapid increase of the relaxation time by several orders of magnitude over a moderate range of temperature, is also interpreted as a consequence of this complex dynamics in phase space. These results from glass theory suggest that the complex structure of phase space in transitional flows, with the presence of many unstable solutions, plays an important role in the properties of the subcritical transition to turbulence. To elaborate on this idea, we propose in the next section an extension of the simplest model of the glass transition, namely the Random Energy Model Derrida (1980, 1981), to the context of the transition to turbulence.
Other possible similarities between the transition to turbulence and the glass transition can be pointed out, considering now the real-space dynamics. For instance, one of the recurrent feature of glassy systems is the heterogeneities of the dynamics: slowly and rapidly relaxing regions coexist in real space, permanently evolving in a complex spatiotemporal organisation van Saarloos et al. (2011). This is reminiscent of the dynamics observed in subcritical transitional flows, where regions with different level of fluctuations coexist. And indeed, some of the one-dimensional models introduced to describe such dynamical heterogeneities in glasses, the so-called kinetically constrained model Chandler and Garrahan (2010); Ritort and Sollich (2003) exhibit spatiotemporal dynamics which are very similar to those observed in the one-dimensional models introduced to discuss the transition via spatiotemporal intermittency Chaté and Manneville (1994); Barkley (2011), especially when looking at spatio-temporal diagrams. For some of the KCM models, the critical point observed in the limit of zero temperature even belongs to the directed percolation class.
Let us emphasize that beyond the possible analogies discussed above, there are also many important differences between the glass transition and the transition to turbulence. A first difference is that supercooled liquids, in which the relaxation time strongly increases when lowering temperature close to the glass transition, are equilibrium systems, while transitional flows are intrinsically non-equilibrium systems. Indeed, the control parameter of the transition (the Reynolds number) may be thought of as a distance to equilibrium, which has to be increased to reach the turbulent state. Another difference is that the turbulent lifetime is the time before the flow falls into the absorbing laminar state, while the relaxation time in glasses is defined from the relaxation of density, or stress, correlations –no absorbing state is involved in this case.
A precise mapping between the glass transition and the transition to turbulence should thus not be expected, and the proposed analogy should not be considered in a strict sense. As we shall see now, there is for instance no direct mapping between say the Reynolds number and the temperature. The idea underlying the present work is rather to take advantage of the methodological and conceptual tools developed in the framework of the glass transition to shed some light on the subcritical transition to turbulence, keeping in mind the limitations of such an approach. Still, we shall see as a first illustration that it allows us to discuss in an original way the dependence of the turbulence lifetime on the Reynolds number.
Iv A Random Energy Model for transitional flows
Along the lines described in the last section, we now introduce a simple model that captures, as an essential ingredient, the wandering of the system on a complex landscape. This model is a variant of the Random Energy Model Derrida (1980, 1981), a toy model which has proved useful in the understanding of the glass transition, in spite of its oversimplified character. As a byproduct, our model yields interesting predictions for the dependence of the turbulent lifetime on the Reynolds number, as discussed below.
iv.1 Diffusion in the energy landscape
As a first step, it is necessary to statistically characterize the properties of the energy landscape, in particular the number of unstable solution at a given energy above the laminar state, as function of the Reynolds number and of the volume of the flow. Though numerical investigations of turbulent flows have not been able yet to characterize the number of unstable solutions as a function of the volume of the flow, the analogy with glasses suggests that this number of solutions may grow exponentially with the volume of the system. Characterizing the state of the flow by its turbulent energy per unit volume, (that is the excess kinetic energy with respect to the laminar flow), we assume that the number of unstable solutions at a given energy density and Reynolds number grows exponentially with the volume according to
thus defining an entropy density . At low Reynolds number, no unstable states exist, so that we assume that the entropy is equal to zero for all if is less than a characteristic value . For , we assume that on an interval , and otherwise, meaning that unstable states exist only in the energy interval .
Turning to dynamics, we assume on the basis of the experimental and numerical observations reported in section III that the turbulent flow spends most of its time close to unstable solutions, and that the evolution of the flow can be considered as a succession of jumps between different unstable solutions. If however the flow ends up in the laminar state, the evolution stops until an external perturbation is imposed. Taking into account the presence of the absorbing laminar state is obviously essential to determine the lifetime of the turbulent flow. This will be the focus of Sect. IV.2. Yet, in a first stage, it is interesting to consider the evolution of the turbulent flow in the absence of the absorbing laminar state, in order to make the analogy with glass models emerge more clearly.
As it is unlikely that a large amount of energy could be injected or dissipated within a short time period, one expects that the energy of successively visited unstable solutions are close to one another. At a coarse-grained level, it is then natural to assume that the energy evolves diffusively. In order to take into account the variation with of the number of unstable states, the evolution should also be biased toward values of the energy with a high entropy . More precisely, the bias should depend on the derivative of the entropy with respect to the energy (a constant entropy introduces no bias in the dynamics). Altogether, the simplest evolution equation for the energy incorporating the above ingredients is the following Langevin equation
where the prime denotes a derivative with respect to . To enforce the finite range of values , reflecting boundary conditions are assumed at and . The parameter is a proportionality coefficient to be determined later on, included for dimensional reasons. The term accounts for dissipative effects, and is a white noise describing the energy injection mechanism, satisfying
where is a diffusion coefficient in energy space. These are obviously strong simplifications: the dissipation rate could in principle depend on and the noise should rather be considered as colored and multiplicative in such non-equilibrium systems, but we wish to keep the model as simple as possible for the sake of illustration. The assumption of a constant dissipation rate is however justified in the limit where the width of the accessible energy range is small with respect to . Besides, considering that the noise is self-generated by the turbulent fluctuations, and thus results from the superposition of a number of contributions proportional to the volume , one expects the diffusion coefficient to scale as . Note that all parameters , and may depend on the Reynolds number .
The Fokker-Planck equation describing the evolution of the probability distribution then reads
The stationary solution is obtained as
Following standard statistical physics arguments, one expects the distribution to be proportional to the number of unstable states , which imposes . Introducing the parameter , the stationary distribution then reads
where we have emphasized the -dependence of the parameter , which in the present context describes the balance between the energy injection and the dissipation, as does the inverse temperature at equilibrium. If is large, the distribution is dominated by the energy which maximizes the argument of the exponential, namely . If the maximum of lies within the interval , the most probable energy is the solution of
Assuming the entropy to be a concave function of (see figure 3),
is a decreasing function of , and thus has its maximum at .
We now introduce the key element of the model, which we borrow from the Random Energy Model Derrida (1980, 1981). The specificity of the latter, which leads to a glass transition, is that the entropy has a finite slope at the minimum energy. By analogy, we thus assume that takes a finite value, denoted as , when .
From a statistical physics point of view, the presence of a finite slope of the entropy at the minimum energy is related to the presence of long-range interactions in the system. Indeed, one can check that for short-range interacting systems, the entropy has an infinite slope at the minimal energy Stillinger (1988). In the Random Energy Model, the fact that all energy levels are statistically independent implicitly results from long-range (mean-field type) interactions. Indeed, the Random Energy Model can be interpreted as the limit of mean-field spin-glass models where interactions involve spins (instead of spins for, e.g., the Ising model), when Derrida (1981).
In the context of the subcritical transition to turbulence, the presence of the pressure field and of large scale unstable solutions, such as the unstable longitudinal vortices, naturally induces such long-range correlations. Note also that long-range correlations are well-known to be present in the fully turbulent regime, as seen for instance by the presence of non-Gaussian fluctuations in the flow Bramwell et al. (1998); Portelli et al. (2003) The generic presence of long-range correlations in turbulent flows thus make plausible the assumption of a finite slope of the entropy at the minimal energy where unstable solutions exists. Clearly, this hypothesis would need to be checked in numerical simulations of realistic flows, which is however a complicated task. We thus presently take this assumption as a working hypothesis motivated by the analogy with glasses, and explore its consequences in the framework of subcritical turbulence modeling.
Coming back to the model, we see that if , Eq. (11) generally admits a solution . In contrast, if , Eq. (11) has no solution, and is maximum at . The probability distribution then concentrates on . Intuitively, one expects to be a decreasing function of (that is, the temperature characterizing the fluctuations increase with the Reynolds number). On the other hand, the total number of unstable states increases with the Reynolds number, and it is thus plausible that increases (or at least remains constant) with . This suggests the existence of a Reynolds number such that . In this case, the average energy is larger than for , while the dynamics in phase space concentrates on the states of minimal energy for .
As emphasized at the beginning of this section, these conclusions hold under the unphysical hypothesis that no laminar state is present. However, if the paths leading from the unstable states to the laminar one are rare enough, the flow is likely to visit a large number of unstable states, and should thus partially equilibrate, before ending up into the laminar state. It is then plausible that the equilibrium distribution given in Eq. (10) qualitatively describes this quasi-equilibrium regime. A natural assumption is that most of the paths leading to the laminar state are connected to unstable states close to , the so-called edge states in the context of turbulence. As for , the average energy remains close to , the flow should reach the laminar state in a reasonably short time. Conversely, for , the typical energy remains well above the threshold , and one expects that it takes a very large time to find the laminar state, as it implies excursions very far from the typical energy.
Hence, the Reynolds number appears as a transition (or crossover) value between a regime of short turbulent lifetime and a regime of large lifetime. Note also that the turbulent lifetime should essentially vanish below the Reynolds value where unstable states cease to exist.
iv.2 Determination of the turbulent lifetime
In this section, we now try to put the above arguments on a more quantitative basis. We define the turbulent lifetime as the mean time to reach the laminar state after a sudden quench from a higher Reynolds number value, where turbulence is established. This situation can be modeled using Eq. (6) for the stochastic dynamics of , with now an absorbing (instead of reflecting) boundary at to account for the presence of the laminar state. The initial condition at is chosen as , to model the quench from high energy turbulent states. Determining the turbulent lifetime then amounts to computing the mean first passage time at the absorbing boundary .
Such a calculation is however difficult for an arbitrary functional form of the entropy and we have to restrict the choice of to the linear form
over the interval . In this case, the mean first passage time can be computed from the solution of the associated Fokker-Planck equation Redner (2001), and one finds
with and , and where the function is given by
For large , the argument of the function in Eq. (13) is large as soon as , that is . The value of is then given, to a good approximation, by the asymptotic behavior of when , which reads
Hence, is given for by
which turns out to be independent of the volume , as intuitively expected in the large limit. In terms of Reynolds number, one thus has a power-law divergence for close to (),
However, for any finite volume this divergence is cut off when approaches , as soon as with some constant , and a crossover is observed to the exponential form obtained from Eq. (16)
Contrary to Eq. (17), expression (19) involves the volume . For , becomes infinite, and a true power-law divergence is observed for . For very large but finite , the divergence can be observed in practice only on a narrow range of Reynolds number, before becomes exceedingly large. On this narrow range, behaves linearly with . In contrast, if is not too large, the range of over which the divergence is observed broadens, and corrections to the linear behavior of with can become observable, possibly leading to a super-exponential behavior of as a function of . Though sub-exponential behavior cannot be discarded, one expects at least to increase with , which goes in favor of the super-exponential case.
The initial motivation of the analogy proposed in this paper was two-fold. First the intense debate that animated the transition to turbulence community regarding the possible divergence of the turbulent lifetimes at a finite Reynolds number was reminiscent of a similar situation encountered in the physics of glasses a few decades earlier. Second, the idea that the dynamics is controlled by unstable solutions away from the laminar state shared some similarity with the role played by the large number of saddles at the onset of the glass transition. The goal of the analogy presented here was to make these intuitions more precise.
We have shown that indeed, even with very good data, one cannot discriminate a singular dependence from a regular but very fast increase of the turbulent lifetimes, especially if one includes the possibility of a Vogel-Fulcher-Tammann like singularity. We have also seen that finite size effects may lead to a crossover, which cannot be resolved experimentally or numerically because of the extremely large timescales at play.
The model presented here was designed to be as simple as possible, taking inspiration from the Random Energy Model with the aim to illustrate the analogy between glasses and transitional flows. As such, it does not claim to be realistic in any way, and some of its main limitations are rather obvious: the spatial structure of the flow is not taken into account, and the key ingredient (the finite slope of the entropy at minimum energy) is taken as a working hypothesis, motivated by the analogy with glasses. It is however quite remarkable that such a simplified model yields a crossover between a power law and an exponential form, in qualitative agreement with the experimental results. Note also that this result on the turbulent lifetime is not a straightforward mapping from the Random Energy Model, since the latter is a purely static model, with no dynamics involved, and without any equivalent of the laminar state considered here.
These encouraging first results call for checks in direct numerical simulations of the hypotheses underlying the model. Counting the number of unstable solutions as a function of their energy density, that is accessing , would be a major step towards the characterization of the transition to turbulence. This is obviously a difficult task, but still far less ambitious than characterizing the stability properties of these solutions and describing the complex interplay of their stable and unstable manifold. This simplification is in essence the gain obtained when switching from a dynamical system point of view to a statistical physics one. A first step would be to investigate a similar approach in simpler non linear spatio-differential equations, where spatiotemporal intermittency has been studied, like the Kuramoto-Sivashinsky equation or the complex Ginzburg Landau one Kaneko (1985). Valuable insights could also be obtained by measuring in direct numerical simulations the dissipation rate as a function of the energy density, as well as characterizing the statistical properties of the turbulent energy fluctuations in the intermediate range of Reynolds numbers.
In the above section, we have considered as the volume of the system. However, in the spirit of real-space approaches, the relevant volume to be considered may rather be the volume of coherent regions of the flow, namely regions over which correlations extend. In a very large aspect ratio experiment, it is plausible (though not obvious) that far away regions in the system experience no interactions. As a result, the volume would acquire a more intrinsic nature: it would then be self-determined by the flow dynamics and not by the arbitrary size of the experiments.
Such a coherence volume cannot be accessed in the framework of models similar to the Random Energy Model, which is mean-field in nature. However, if the analogy with the physics of glasses proves to be fruitful, it would be of interest to consider its most recent developments (including in particular the Random First Order Transition scenario Stevenson and Wolynes (2005)), which precisely address the real-space description issue Berthier and Biroli (2011). Pomeau Pomeau (1986) suggested more than twenty-five years ago that the growth and death of the laminar and turbulent regions could obey a first-order nucleation-like dynamics (albeit of a peculiar type, given the fluctuating active property of the turbulent state and the absorbing character of the laminar state). Let us conclude with the somewhat naive suggestion that taking inspiration from the Random First Order Transition theory of glasses might be a way to extend the standard laminar-turbulent coexistence scenario to a situation where a large number of turbulent states (associated to local unstable solutions of the Navier-Stokes equation) coexist with the laminar state.
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