Study of the instability of the Poiseuille flow using a thermodynamic formalism

Significance

The stability of Poiseuille flow is among the most classical problems in fluid mechanics going back to the pioneering work of Reynolds. Despite a huge amount of effort, particularly recent advances in experimental studies, we still lack a suitable theoretical framework. In this paper, we develop a thermodynamic formalism for studying this problem. We show that this formalism allows us to identify a unique critical Reynolds number at which the laminar flow loses stability. We also study the free energy, action, and other thermodynamic relations for the macroscopic quantities associated with the flow. The formalism developed here should be applicable to a large class of problems exhibiting subcritical instabilities.

Abstract

The stability of the plane Poiseuille flow is analyzed using a thermodynamic formalism by considering the deterministic Navier–Stokes equation with Gaussian random initial data. A unique critical Reynolds number, $R{e}_c\approx \mathrm{2,332}$, at which the probability of observing puffs in the solution changes from 0 to 1, is numerically demonstrated to exist in the thermodynamic limit and is found to be independent of the noise amplitude. Using the puff density as the macrostate variable, the free energy of such a system is computed and analyzed. The puff density approaches zero as the critical Reynolds number is approached from above, signaling a continuous transition despite the fact that the bifurcation is subcritical for a finite-sized system. An action function is found for the probability of observing puffs in a small subregion of the flow, and this action function depends only on the Reynolds number. The strategy used here should be applicable to a wide range of other problems exhibiting subcritical instabilities.

The instability of shear flows, of which the Poiseuille flow is a canonical example, is among the most classical and most challenging problems in fluid mechanics, and a huge amount of effort has been devoted to it (1–13). The most definitive advance has been the recent experimental work by Avila et al. (9): By measuring the puff decaying and splitting times, they obtained an estimate for the critical Reynolds number, at around 2,040, for the 3D pipe flow. On the theoretical side, although the normal mode analysis in the linear stability theory of Poiseuille flow is regarded as being among the most important chapters of theoretical fluid dynamics and is associated with the names of Orr–Sommerfeld, Heisenberg, C. C. Lin, etc., it is generally accepted that normal mode analysis is insufficient in describing the instability of Poiseuille flow. If anything, Poiseuille flow should be regarded as an example of subcritical instability, perhaps the most well-known one.

Currently, we still lack theoretical tools for tackling such instabilities. One interesting proposal has been to draw an analogy with phase transitions (13–18): The bifurcation diagram for subcritical instabilities resembles the phase diagram for first-order phase transitions. However, to develop some quantitative tools out of this analogy, one also faces some serious issues. The first is that shear flow is well known to be a nonconservative and nonequilibrium system. More importantly, what we are interested in is the relative stability, i.e., the basins of attraction for laminar and turbulent states. This is opposite to the ergodic hypothesis that is the cornerstone of statistical mechanics. In fact, at least for finite-size systems undergoing subcritical instabilities, the complication arises precisely because the system is not ergodic. This casts serious doubt on the analogy with phase transitions in statistical mechanics.

While philosophical issues exist, we propose a different scenario for thinking about the laminar–turbulent transition from the viewpoint of thermodynamics. We view it as an initial value problem with the initial value being the laminar flow plus noise. This will be our ensemble. We then study the long-time behavior of flow with initial conditions generated from this ensemble. The statistical steady states thus encode the information we are interested in. For example, we can extract various free energies and thermodynamic relations. In particular, in the thermodynamic limit, one can define a unique critical Reynolds number for Poiseuille flow to become unstable, and one can characterize this transition using concepts similar to those of traditional statistical physics.

In this paper, we carry out this program for the 2D shear flow. In this case, the role of the puff states is played by traveling waves. We define and analyze the free energy, action, and thermodynamic relations. We show that in the thermodynamic limit, there is a well-defined critical Reynolds number, $R{e}_c\approx \mathrm{2,332}$: Below $R{e}_c$, traveling waves are unstable; above $R{e}_c$, the Poiseuille flow becomes unstable. As it turns out, this transition is more like a continuous phase transition, rather than a first-order transition.

Before dwelling on the channel flow problem, let us first use a concrete example to discuss the statistical–mechanical framework for systems with deterministic dynamics and random initial conditions.

Zero-Temperature Kinetic Ising Model

Model Setup.

We introduce our formalism using a prototypical example, which can be regarded as a zero-temperature kinetic Ising model. Consider a 1D, regular, and periodic lattice of size L. On each lattice site, there is a spin that can take values $+1$ (spin up) or $-1$ (spin down). A microstate of the system is $x=\left(x_1,\dots ,x_n\right)\in {\left\{-1,+1\right\}}^L$. In analogy with the canonical ensemble in classical statistical mechanics, we consider an ensemble of initial conditions given by the distribution ${\rho }_L\left(x;p\right)=p^{{\displaystyle {\sum }_{i=1}^L{\delta }_{+1,x_i}}}{\left(1-p\right)}^{{\displaystyle {\sum }_{i=1}^L{\delta }_{-1,x_i}}}$. That is, each spin is independent, pointing up with probability p and down with probability $1-p$. Next, we introduce deterministic dynamics on the microstates. Associate with each microstate x an energy $E\left(x\right)=-{\displaystyle {\sum }_{i=1}^L{x}_i{x}_{i+1}}-h{\displaystyle {\sum }_{i=1}^L{x}_i}$. The first term promotes spin alignment among nearest neighbors whereas the second term promotes spin alignment with an external field of strength $h\in \mathrm{ℝ}$. Proceeding sequentially through the lattice from site 1 to L, we flip each spin ($x_i\mapsto -x_i$) if doing so decreases the energy E. Each complete iteration through the lattice is denoted by $T_h$, which maps a microstate x onto a new microstate $x^{\prime }=T_h\left(x\right)$. It is not hard to see that the dynamics quickly drives the microstate x to an attractor $T_h^{\infty }\left(x\right)$, depending on both the initial x and the value of h.

Now, consider macrostates z defined by $z=c_L\left(x\right)=L^{-1}{\displaystyle {\sum }_{i=1}^L{x}_i}$ (magnetization). The dynamics and the initial distribution then induce a limiting distribution over the macrostates given by ${\widehat{\rho }}_L\left(z;p,h\right)={\displaystyle {\sum }_{c_L\left(T_h^{\infty }\left(x\right)\right)=z}{\rho }_L\left(x;p\right)}$. The free energy is defined analogously to that in classical statistical mechanics,

$$F\left(z;p,h\right)=\underset{L\to \infty }{\lim }-L^{-1}\ln {\widehat{\rho }}_L\left(z;p,h\right).$$ [1]

The free energy allows us to identify phase transitions. Notice that Eq. 1 implies ${\widehat{\rho }}_L\approx e^{-LF}$ so that the most probable values of z must coincide with the minima of F in the thermodynamic limit. When the minimum is a singleton, we denote it as $z*\left(p,h\right)$. If $z*$ is not an analytic function of $\left(p,h\right)$, we say that there is a phase transition. Furthermore, if $z*$ is discontinuous, there is a first-order phase transition. Otherwise, we have a continuous transition.

Free Energy.

For this model, the free energy can be calculated exactly. When $h>0$, all initial configurations converge to $\left(+1,\dots ,+1\right)$, except for $x=\left(\pm 1,-1,\dots ,-1\right)$, which occurs with probability ${\left(1-p\right)}^{L-1}$ and converges to $\left(-1,\dots ,-1\right)$. Hence, the free energy is

$$F\left(z;p,h\right)=\{\begin{array}{ll}-\ln \left(1-p\right)\hfill & z=-1,\hfill \\ 0\hfill & z=+1,\hfill \\ \mathrm{\infty }\hfill & \mathrm{otherwise}\text{.}\hfill \end{array}$$ [2]

Similarly, when $h<0$, we have

$$F\left(z;p,h\right)=\{\begin{array}{ll}0\hfill & z=-1,\hfill \\ -\ln p\hfill & z=+1,\hfill \\ \mathrm{\infty }\hfill & \mathrm{otherwise}\text{.}\hfill \end{array}$$ [3]

The case $h=0$ is more complicated. See Supporting Information for an exact derivation. The result is

$$F\left(z;p,0\right)=z{\lambda }_m\left(z,p\right)-\ln k\left({\lambda }_m\left(z,p\right);p\right),$$ [4]

where ${\lambda }_m\left(z,p\right)$ is implicitly defined by $zk\left({\lambda }_m\left(z,p\right);p\right)=\left[\partial k\left({\lambda }_m\left(z,p\right);p\right)\right]/\partial \lambda$ and $k\left(\lambda ;p\right)$ is the largest eigenvalue of the augmented transition matrix $\tilde{M}\left(\lambda ;p\right)$ in Eq. S8. In Fig. 1, we plot the free energy, which has a unique minimum at $z*\left(p,0\right)=1-2p-\left[2\left(1-2p\right)/\left(1-p+p^2\right)\right].$ To summarize,

$$z^{*}\left(p,h\right)=\{\begin{array}{ll}+1\hfill & h>0,\hfill \\ 1-2p-\frac{2\left(1-2p\right)}{1-p+p^2}\hfill & h=0,\hfill \\ \mathrm{-}1\hfill & h<0.\hfill \end{array}$$ [5]

We see that $z*$ is discontinuous at $h=0$, i.e., there is a first-order phase transition.

Fig. 1.

Free energy vs. net magnetization at $h=0$, for various values of p. The free energy minimum $z*$ increases with increasing p.

Just like traditional statistical mechanics, the free energy here plays the role of a large deviation rate function (19). The minimum of the free energy is the most probable value of the macrostate in the thermodynamic limit. For all other values, the free energy gives the asymptotic rate of exponential decay, as $L\to \infty$, of the probability to observe them. Here, the probability of observing a state corresponds to the probability for the initial condition to fall in the basin of attraction of that state. Intuitively, given some initial distribution, the free-energy landscape highlights the relative strengths (i.e., size of the basins of attraction) of possible attractors that can be reached under the specified deterministic dynamics.

For this particular example, a well-defined thermodynamic limit is shown to exist by explicit calculation. At this point, there are no general theorems that guarantee the existence of such a limit. A natural conjecture is that the thermodynamic limit should exist for systems with local interactions and initial data with short-range correlations.

Two-Dimensional Channel Flow

Governing Equations and Basic Phenomena.

Consider the 2D Poiseuille flow between two infinite parallel plates $y=\pm 1$, governed by the 2D incompressible Navier–Stokes equations

$$\nabla \cdot \mathbf{u}=0,\hspace{1em}{\partial }_t\mathbf{u}+\mathbf{u}\cdot \nabla \mathbf{u}=-\nabla p+R{e}^{-1}{\nabla }^2\mathbf{u}+\mathbf{f},$$ [6]

where $Re$ is the Reynolds number. The flow is driven by a constant average pressure gradient $\mathbf{f}=\left(f_x,f_y\right)=\left(2/Re,0\right)$. The flow possesses a laminar state characterized by a parabolic velocity profile $u_P=1-y^2$ (20–24). Although linear stability theory gives a critical Reynolds number $R{e}_{*}=\mathrm{5,772}$ above which the laminar solution becomes linearly unstable, there are also stable traveling solutions at subcritical Reynolds numbers $Re<R{e}_{*}$ (21). A classical question that arises is: Which one of the stable solutions survives under perturbations?

Thermodynamic Formalism with Random Initial Condition.

We now study the channel flow using randomized initial conditions. Periodic boundary conditions in the streamwise direction and nonslip conditions at walls are assumed. The random initial velocity field is given by a superposition of the laminar Poiseuille flow and a Gaussian white noise: ${\mathbf{u}}_0={\mathbf{u}}_P+\mathit{ϵ}{\mathbf{u}}_G$. The field ${\mathbf{u}}_G$ is numerically constructed by white noise projected to the subspace satisfying the incompressibility condition and the nonslip boundary conditions at walls. This forms the ensemble of initial conditions with distribution ${\rho }_L\left({\mathbf{u}}_0;\mathit{ϵ}\right)$. For time integration, we use a second-order backward differentiation scheme for linear terms, and a second-order explicit scheme for nonlinear terms. A fast spectral scheme (25) is used to solve the 2D Stokes equations resulting from the time discretization. We find it sufficient to use 33 Legendre modes in the y direction and 256 Fourier modes in the x direction per $L=100$ in plate length. Most of the simulation results reported are for $L=\mathrm{1,600}$ with total integration times about 10,000 time units. This is enough for providing convergent quantities in the thermodynamic limit and for the integral quantities, such as mass flux and mean energy, to become steady. For each pair of parameters $\left(Re,\mathit{ϵ}\right)$, about 200–1,000 initial conditions were performed. Typically, 1,000 initial conditions were used for PDF curves.

Puff Solutions.

Fig. 2 shows a profile for the local average energy $q\left(x\right)=\left(1/2\right){\displaystyle {\int }_{-1}^1\left(1/2\right)\left(u^2+v^2\right)dy}$ in a realization at $Re=\mathrm{2,800}$ and $\mathit{ϵ}=0.18$. We observe a stable train of localized wave packets, similar to that reported by Jimenez (24) in an $L=8\pi$ channel flow for $Re<\mathrm{5,000}$. A localized wave packet (puff) has a characteristic length scale of 20 and resembles those in the 3D pipe flow (26, 27).

Fig. 2.

Profile of local average energy q in a $L=\mathrm{1,600}$ channel at $Re=\mathrm{2,800}$ and $\mathit{ϵ}=0.18$. (A) Global view. Localized packets (puffs) are observed. (B) A magnified view of individual puffs.

It is known that in 2D channel flow, there exist secondary flows in the form of traveling waves (20, 21). The 2D secondary flows were discovered in the range $Re\ge \mathrm{2,750}$ for the length $L\ge 6\pi$ (22). We found that the critical Reynolds numbers for the one-puff solution to be sustainable are $R{e}_{p,L}=\mathrm{2,341.8},\hspace{0.17em}\mathrm{2,336.9},\hspace{0.17em}\mathrm{2,334.4},\hspace{0.17em}\mathrm{and}\hspace{0.17em}\mathrm{2,333.3}$ for $L=400,\hspace{0.17em}800,\hspace{0.17em}\mathrm{1,600},\hspace{0.17em}\mathrm{and}\hspace{0.17em}\mathrm{3,200}$, respectively. These critical Reynolds numbers were estimated by tracking the one-puff solution continuously while decreasing $Re$. Immediately above these Reynolds numbers, the one-puff solution is linearly stable. From these critical Reynolds numbers at finite L, we obtain an estimate of the critical Reynolds number at infinite L, $R{e}_p\approx \mathrm{2,332}$. In fact, we also observe the relation

$$R{e}_{p,L}-R{e}_p\approx L^{-1}.$$ [7]

Puff Density and Free Energy.

Next we study the statistics of a macrostate variable, the puff density $z=n/L$, where n denotes the number of puffs. The initial distribution ${\rho }_L\left({\mathbf{u}}_0;\mathit{ϵ}\right)$ constructed above induces a macrostate distribution ${\widehat{\rho }}_L\left(z;Re,\mathit{ϵ}\right)$ on the puff density at steady state. According to the framework introduced earlier, a free energy can be defined,

$$F\left(z;Re,\mathit{ϵ}\right)=\underset{L\to \infty }{\lim }-L^{-1}\ln {\widehat{\rho }}_L\left(z;Re,\mathit{ϵ}\right).$$ [8]

The existence of the thermodynamic limit in Eq. 8 is supported by numerical results. Fig. 3A depicts $-L^{-1}\ln {\widehat{\rho }}_L\left(z;Re,\mathit{ϵ}\right)$ at $Re=\mathrm{2,800}$ and $\mathit{ϵ}=0.18$, for $L=400,\hspace{0.17em}800,\hspace{0.17em}\mathrm{and}\hspace{0.17em}\mathrm{1,600}$. After shifting along the y direction by an L-dependent amount, all three functions collapse onto a single profile shown in the solid line in the figure. This indicates that good convergence is achieved at $L=\mathrm{1,600}$. Note that the free energy F is approximately quadratic for small deviations from the most probable value $z*$. In Fig. 3B, the free energy is plotted for $Re=\mathrm{2,600},\hspace{0.17em}\mathrm{2,700}$. We see that $z*$ increases with increasing Reynolds number. Fig. 3C shows $z*$ as a function of $Re$. The symbols represent the data points obtained from direct simulations, and the solid line fit is obtained from later discussions on the statistics of the interpuff distance (Eq. 11). The dependence of $z*$ on $Re$ gives a natural thermodynamic definition of a critical Reynolds number $R{e}_c$. Above this point, the most probable puff density $z*$ is nonzero, whereas below this point, it is zero. In the following, we will study the spatial structure of the puff solutions to obtain an accurate relation between $z*$ and $Re$ and clarify the relationship between $R{e}_c$ and $R{e}_p$. In particular, we will demonstrate that $R{e}_c=R{e}_p$, and that $z*$ is continuous but not smooth at $Re=R{e}_c$, implying a continuous phase transition.

Fig. 3.

Free energy and its minimum. (A) Symbols denote normalized logarithm of the PDFs of puff density for $Re=\mathrm{2,800}$, $\mathit{ϵ}=0.18$, and $L=400,\hspace{0.17em}800,\hspace{0.17em}\mathrm{and}\hspace{0.17em}\mathrm{1,600}$. The solid line represents a quadratic curve derived from a Gaussian distribution. (B) Free energy for $\mathit{ϵ}=0.18$ and $Re=\mathrm{2,600},\hspace{0.17em}\mathrm{2,700}$, suggesting that the minimum $z*$ increases as $Re$ increases. (C) Symbols denote free energy minimum $z*$ (which can be thought of as the order parameter) versus $Re$ calculated from direct simulations. The solid line is a power-law fit obtained from Eq. 11, showing a continuous transition.

Statistics of Interpuff Distance.

To elucidate the spatial structure of puff solutions, we calculate the complementary cumulative distribution function (CCDF) $G\left(\lambda \right)$ of interpuff distance. This is the probability that the distance between two neighboring puffs is greater than λ. As shown in Fig. 4A, for the noise amplitude $\mathit{ϵ}=0.18$, $G\left(\lambda \right)$ fall on the same exponential form,

$$G\left(\lambda \right)=\exp \left[-\alpha \left(\lambda -{\lambda }_0\right)\right],$$ [9]

for a variety of Reynolds numbers. This is similar to the exponential form of CCDFs for interpuff distance or turbulence length scale in 3D pipe flow (26, 27). The ${\lambda }_0$ account for the fact that two puffs cannot be arbitrarily close: Each puff produces a perturbation to the background flow. If two puffs are too close, the combined perturbation is large and both puffs become unstable. Hence, there is a minimum puff separation length $l_0\approx {\lambda }_0$. Note that the minimum separation length $l_0$ is not exactly equal to ${\lambda }_0$ since, when λ is close to $l_0$, the puff interaction becomes complicated and the distance is no longer exponentially distributed (Fig. 4A, small λ). However, $l_0$ and ${\lambda }_0$ are similar in magnitude. When viewed at large length scales ($\gg {\lambda }_0$), the exponential form holds. Eq. 9 determines a characteristic length scale $1/\alpha$, and a characteristic density α for each Reynolds number. In the thermodynamic limit, the average density α coincides with the minimum of the free energy $z*$.

Fig. 4.

Statistics of interpuff distance. (A) Normalized logarithms of the CCDFs of interpuff distance at various Reynolds numbers for $\mathit{ϵ}=0.18$. Solid line is least square fit of the exponential form. Deviations from exponential distribution are observed for small λ due to puff interactions. Deviations at large λ are due to finite size effects. (B) The exponent α for CCDF of puff distance versus Reynolds numbers, for $\mathit{ϵ}=0.12,\hspace{0.17em}0.14,\hspace{0.17em}0.16,\hspace{0.17em}\mathrm{and}\hspace{0.17em}0.18$. Lines are least square estimations of the power-law fit. These highlight the universality of the power-law behavior. (C) The logarithm of the exponent. The line represents a uniform approximation.

Define the normalized Reynolds number $R{e}_n=\left(Re-R{e}_c\right)/R{e}_c$. In Fig. 4B, we plot the curves of the α (=$z*$) against $R{e}_n$ for noise amplitudes $\mathit{ϵ}=0.12,\hspace{0.17em}0.14,\hspace{0.17em}0.16,\hspace{0.17em}\mathrm{and}\hspace{0.17em}0.18$. Power-law scaling relations between α and $R{e}_n$ are observed,

$$\alpha \approx R{e}_n^{\gamma }.$$ [10]

This is different from the superexponential relation observed in pipe flow (27). The power-law scaling implies the divergence of interpuff distance at the critical Reynolds number $R{e}_c$. This suggests that the volume of the basin of attraction for puff solutions vanishes at $R{e}_c$.

Action Function of the Transition.

By applying a least square fit, we found that the scaling exponent γ depends on ε in the form $\gamma =C_{\mathit{ϵ}}{\mathit{ϵ}}^{-2}$, where $C_{\mathit{ϵ}}=0.075$. The form of γ can be attributed to the fact that Gaussian noise of variance of $\sim {\mathit{ϵ}}^2$ is added to the initial field. A uniform 2D approximation is constructed by an exponential fit

$$z^{*}\left(\mathit{ϵ},R{e}_n\right)=\alpha \left(\mathit{ϵ},R{e}_n\right)=C_{\alpha }\exp \left[C_{\mathit{ϵ}}{\mathit{ϵ}}^{-2}\log \left(C_{Re}R{e}_n\right)\right],$$ [11]

where $C_{\alpha }=0.45$ and $C_{Re}=1.15$ (Fig. 4C, solid line). The value of α determines the puff density in a long channel: The probability of finding a puff in a relatively short subregion of the channel with length l ($l\gg {\lambda }_0$) is about $P\left(l\right)=\alpha l$. One can express $P\left(l\right)$ in a more familiar form,

$$P\left(l\right)\approx C_{\alpha }l\exp \left[-{\mathit{ϵ}}^{-2}S\left(Re\right)\right],$$ [12]

where the action function is given by

$$S\left(Re\right)=-C_{\mathit{ϵ}}\log \left[C_{Re}\left(Re-R{e}_c\right)/R{e}_c\right].$$ [13]

The action function $S\left(Re\right)$ represents the cost of inducing a transition from the laminar base flow to a local puff. It is a decreasing function of $Re$, which indicates that as $Re$ becomes smaller, it is harder for the Gaussian noise to trigger a transition. $S\left(Re\right)$ does not depend on the channel length L or the noise amplitude ε. Thus, it is an intrinsic property of the 2D channel flow in the thermodynamic limit.

Relationship Between Re_c and Re_p.

We have defined two critical Reynolds numbers. $R{e}_p$ is defined as the point where a single-puff solution in the infinite channel becomes sustainable. Hence, at $Re<R{e}_p$, the puff density is 0. $R{e}_c$ is defined as the point where the puff density $z*$ becomes nonzero. Clearly, $R{e}_p\le R{e}_c$. We argue that they are equal: In an infinite channel, some puffs always form due to randomness in the initial condition. If $Re>R{e}_p$, puffs sufficiently far apart can be sustained. This gives rise to a nonzero puff density $z*>0$ depending on both the noise level and the minimum interpuff distance. In particular, this implies that $R{e}_p\ge R{e}_c$ and, hence, $R{e}_p=R{e}_c$. The argument here relies on the fact that puffs sufficiently far apart do not interact and can be sustained indefinitely. This need not be true in 3D problems. The fact that $R{e}_p=R{e}_c$ in 2D is further supported by the scaling relations we obtain above. This is a nontrivial statement since $R{e}_p$ is much easier to obtain than $R{e}_c$ due to the difficulties in computing $z*$ accurately near $R{e}_c$. In fact, this is among the most significant issues in 3D pipe flows.

Thermodynamic Relations.

In thermodynamics, a class of important relations exists among macroscopic observables. These are known as thermodynamic relations (e.g., the ideal gas equation of state). We show that such relations also exist for the channel flow by calculating the skin friction coefficient $C_f=-{\langle {\partial }_{x}p\rangle }_{xyt}h/\left({\rho }_0{U}^2/2\right)$ (22) at $L=400,\hspace{0.17em}800,\hspace{0.17em}\mathrm{and}\hspace{0.17em}\mathrm{1,600}$. Here, $h=1$ is the half-channel width, ${\rho }_0=1$ is the fluid density, and U is the mean streamwise velocity; ${\langle \rangle }_{xyt}$ denotes space–time average.

We found that a normalized skin friction coefficient can be constructed by setting $C_{fn}=\left(C_f/C_{f0}-1.0\right)/\alpha$, where $C_{f0}=9/Re$ is the skin friction coefficient for laminar flow (22). The normalized skin friction coefficient represents the amount of additional drag (on top of the laminar flow) introduced by each puff. As the pipe length L increases, $C_{fn}$ converges to a power law with respect to $R{e}_n$ (Fig. 5). The power-law fit is of the form $C_{fn}=C_{fn0}+A_{C}R{e}_n^{B_C}$, and is indicated by the solid line in Fig. 5 ($C_{fn0}=3.5,\hspace{0.17em}{A}_C=17,\hspace{0.17em}{B}_C=0.73$). $C_{fn0}$ is the limiting value of $C_{fn}$ as $R{e}_n\to 0$. We expect the power-law form to hold for smaller $R{e}_n$ than those considered here, provided that L is large enough. From the scaling laws of $C_{fn}$, we obtain the dependence of the unnormalized skin friction coefficient $C_f$ on the Reynolds number. The results are shown in Fig. 6. Again, a continuous transition is observed.

Fig. 5.

Normalized skin friction coefficient versus the normalized Reynolds number for the one-puff solution at various channel lengths. Solid line is least square estimation of power-law fit. We see that $C_{fn}$ converges to a power-law form.

Fig. 6.

Skin friction coefficients. (A) Symbols denote numerical results of $C_f-C_{f0}$. Lines are fitted power-law. (B) Skin friction coefficient $C_f$ as a function of $Re$ obtained from the fitted coefficients. We observe a continuous transition, as in the case for the puff density.

With the uncovered power-law forms, we can now relate the skin friction coefficient to the mean puff density $z*$. From the above, we have $C_f=C_{f0}\left(1+C_{fn}{z}^{*}\right)$ and $C_{fn}=C_{fn0}+A_{C}R{e}_n^{B_C}$. Therefore,

$$C_f=\frac{9}{R{e}_c\left[1+R{e}_n\left(z^{*}\right)\right]}\left(1+\left[C_{fn0}+A_{C}R{e}_n{\left(z*\right)}^{B_C}\right]z^{*}\right),$$ [14]

where $R{e}_n\left(z*\right)$ is defined implicitly by Eq. 11. This is a thermodynamic relation relating the macrostate $z*$ and $C_f$. In Fig. 7, we plot these curves for various noise levels. This is somewhat analogous to the p-V isotherms for ideal gas. Relations of this type exist for other macroscopic quantities, such as mass flux and average energy dissipation.

Fig. 7.

Thermodynamic relation (14) relating the skin friction coefficient and the puff density for $\mathit{ϵ}=0.12,\hspace{0.17em}0.14,\hspace{0.17em}0.16,\hspace{0.17em}\mathrm{and}\hspace{0.17em}0.18$. This is, in some sense, analogous to the p-V isotherms in the ideal gas equation of state.

Discussion

Essentially, the picture of the laminar–puff transition is as follows: For $Re<R{e}_c=R{e}_p$, there are no sustainable puff solutions and the flow remains laminar under arbitrary perturbations. In this case, the puff density is 0 with probability 1 for all ε. For $Re>R{e}_c$, the probability of finding a puff in a subregion of length l (${\lambda }_0\ll l\ll L$) is $\alpha l$ where $\alpha \approx \exp \left[-{\mathit{ϵ}}^{-2}S\left(Re\right)\right]$. Once formed, a puff is stable. Its structure remains intact and does not interact with other puffs sufficiently far away. In the thermodynamic limit, only solutions with puffs survive, and the laminar flow loses the competition. For a small initial noise ε and a large spatial scale ($\gg {\lambda }_0$), we can view the emergence of puffs as a spatial Poisson process with rate α. In particular, as $L\to \infty$, the most probable puff density is simply the mean of this Poisson process, α. Hence (compare Eq. 5),

$$z^{*}\left(Re,\mathit{ϵ}\right)=\{\begin{array}{ll}0\hfill & Re<R{e}_c,\hfill \\ C_{\alpha }\exp \left[-{\mathit{ϵ}}^{-2}S\left(Re\right)\right]\hfill & Re>R{e}_c,\hfill \end{array}$$ [15]

where $S\left(Re\right)$ is the action defined in Eq. 13. Since $S\left(Re\right)$ diverges as $Re\downarrow R{e}_c$, we have ${\lim }_{Re\downarrow R{e}_c}{z}^{*}\left(Re,\mathit{ϵ}\right)=0$, and, thus, this is a continuous transition (Fig. 3C). The order of the transition increases with decreasing ε.

We now discuss the significance of $R{e}_c$ and $R{e}_p$ in relation with known results in the literature. In the classical, dynamical systems treatment, one usually considers a finite channel of length L. In this case, 2D traveling wave solutions arise from a subcritical, saddle-node bifurcation at $Re=R{e}_{p,L}$ that depends on the channel length. For example, for $L=2\pi /1.3175$, $R{e}_{p,L}\approx \mathrm{2,939}$ (28), and for $L=2\pi /0.15$, $R{e}_{p,L}\approx \mathrm{2,750}$ (22). In the thermodynamic framework, we are interested in the limit $L\to \infty$, which allows us to define an $R{e}_c$ that is analogous to a phase transition point in statistical mechanics and thermodynamics. As we argued above, in the strictly 2D flow we considered here, $R{e}_c=R{e}_p={\lim }_{L\to \infty }R{e}_{p,L}$. However, this might not hold for 3D problems. For example, in 3D plane Poiseuille flow, 3D traveling wave solutions arise from secondary bifurcations from the stable 2D branch (29, 30). Furthermore, 3D traveling wave solutions have been found at much lower Reynolds numbers (31–33). In 3D pipe flow, the instability of puff structures makes even defining $R{e}_p$ problematic. Nevertheless, one can always define $R{e}_c$ for appropriately chosen macrostates, with no explicit reference to the underlying dynamical system. In other words, the thermodynamic treatment presented here is especially useful for dynamical systems with complex bifurcation diagrams.

One also notices an interesting feature of the 2D laminar–turbulence transition. From the dynamical systems viewpoint, we have a subcritical bifurcation for a finite system. That is, for $Re<R{e}_c$, the laminar flow is the only linearly stable solution. For $Re>R{e}_c$, there are coexisting stable solutions (laminar and puff), each with a distinct domain of attraction that together form a partition of the phase space. However, at $Re=R{e}_c^{+}$, the volume of the domain of attraction of puff solutions vanishes. This results in the divergence of $S\left(Re\right)$, and hence a continuous phase transition, despite having a subcritical bifurcation in the underlying dynamical system. Naively, one might have associated a subcritical bifurcation with a first-order phase transition because of the presence of jumps in both cases. The lesson that we learned here is that there are subtle differences between subcritical bifurcations and first-order phase transitions.

Lastly, in this study, we have focused on random initial conditions in the form of (projected) additive white noise. A natural question is what changes if one were to consider other initial ensembles, such as multiplicative or colored noise. For the problem considered here, things are expected to remain qualitatively the same. The reason is as follows: As discussed before, $R{e}_c=R{e}_p$. However, $R{e}_p$ is a property of the flow and is independent of the distribution of the initial noise. Hence, as long as the initial noise has positive probability of driving the flow into the basins of attraction of puff solutions, one would obtain the same transition point $R{e}_c$. However, at fixed $Re>R{e}_c$, the puff density distribution will quantitatively depend on the details of the initial noise. For example, we found that adding a small spatial correlation to the noise can lead to higher puff densities at fixed $Re>R{e}_c$. It will be interesting to see if equivalence-of-ensemble results can be proved for suitable initial ensembles.

Conclusion

This study allows us to draw three main conclusions:

• i)Regarding the laminar–turbulent transition in the 2D channel flow, we have found that as soon as the puff solutions become linearly stable, they are the only ones that survive at the thermodynamic limit. A unique critical Reynolds number is found for this transition: $R{e}_c\approx \mathrm{2,332}$, and the transition is continuous. Other macroscopic quantities, such as skin friction coefficient, can also be studied, and they satisfy certain thermodynamic relations.
• ii)We have developed a thermodynamic formalism for studying deterministic dynamics with random initial data. On a first look, this is very far from traditional statistical mechanics, which assumes ergodicity. Nevertheless, a thermodynamic formalism can still be developed by quantifying the probability of the domains of attraction for different macrostates. We believe this formalism can be useful for studying a wide range of examples.
• iii)We noted some differences between bifurcations in dynamical systems theory and phase transitions in statistical mechanics. Naively, one would have expected that a subcritical bifurcation should correspond to a first-order phase transition. However, the act of taking the thermodynamic limit can give unexpected results. In fact, the laminar–turbulent transition for 2D channel flows provides an example of a continuous transition in the thermodynamic limit and yet a subcritical bifurcation for finite-size systems.

A natural question is the significance of the critical Reynolds number $R{e}_{*}=\mathrm{5,772}$ derived from linear stability analysis (21). This is the point at which the laminar flow loses linear stability, so that arbitrary perturbation will drive it to turbulence. In our framework, this critical Reynolds number corresponds to the point where the cost S of inducing a transition vanishes. Therefore, the various regimes of the channel flow can be effectively characterized by the action function $S\left(Re\right)$: For $Re\le R{e}_c$, there are no linearly stable puff solutions, so $S\left(Re\right)=+\infty$. For $Re\ge R{e}_{*}$, laminar solutions become unstable thus $S\left(Re\right)=0$. Between $R{e}_c$ and $R{e}_{*}$, the action S is finite and positive.

Another question is how much of this is true in the laminar–turbulent transitions in 3D plane Poiseuille flow and 3D pipe flow. In these cases, the bifurcation diagrams and the dynamical behaviors of traveling wave solutions are much more complex, and a thermodynamic treatment might prove useful. We hope that this work provides such a framework to think about 3D laminar–turbulent transitions.

Free Energy for the Kinetic Ising Model

Here we give a detailed derivation of the free energy for the kinetic Ising model introduced in the main text. Recall that the free energy is defined as

$$F\left(z;p,h\right)=\underset{L\to \infty }{\lim }-L^{-1}\ln {\widehat{\rho }}_L\left(z;p,h\right),$$ [S1]

where ${\widehat{\rho }}_L$ is the limiting distribution of the net magnetization under random initial spin distribution and deterministic Ising dynamics. When $h\ne 0$, the attractors and their domains of attraction are simple to characterize and the expressions for the free energy are easily derived, as discussed Zero-Temperature Kinetic Ising Model. For completeness, we reproduce them here:

$$F\left(z;p,h\right)=\{\begin{array}{ll}-\ln \left(1-p\right)\hfill & z=-1,\hfill \\ 0\hfill & z=+1,\hfill \\ \mathrm{\infty }\hfill & \mathrm{otherwise,}\hfill \end{array}$$ [S2]

for $h>0$ and

$$F\left(z;p,h\right)=\{\begin{array}{ll}0\hfill & z=-1,\hfill \\ -\ln p\hfill & z=+1,\hfill \\ \mathrm{\infty }\hfill & \mathrm{otherwise,}\hfill \end{array}$$ [S3]

for $h<0$.

We now discuss the case $h=0$, where the energy function has the form

$$E\left(x\right)=-{\displaystyle \sum _{i=1}^L{x}_i{x}_{i+1}}.$$ [S4]

Hence, a spin will be flipped if and only if it is surrounded by two neighboring spins both pointing in the opposite direction. For example, $\left(\dots ,-1,+1,-1,\dots \right)\to \left(\dots ,-1,-1,-1,\dots \right)$. Starting from an initial condition x, the dynamics will drive the system into a configuration consisting of blocks of positive and negative spins, each of size at least 2. Notice also that only one iteration through the lattice is required to converge to an attractor $y=T_h^{\infty }\left(x\right)=T_h\left(x\right)$.

With the above observation on the dynamics, we can determine ${\widehat{\rho }}_L$. The trick is to view space as “time.” That is, we treat the final spin configuration y as a sample path of discrete Markov chain. Obviously, since the spin interaction concerns both left and right neighbors, the chain cannot be Markov if its state consists of single-site spins. Instead, we consider pairwise spins, such that the state space of the Markov chain consists of the states $\left\{\left(+1,+1\right),\left(+1,-1\right),\left(-1,+1\right),\left(-1,-1\right)\right\}$. Let us emphasize that this is not simply translating the path $\left\{y_1,y_2,\dots ,y_L\right\}$ into $\left\{\left(y_1,y_2\right),\left(y_2,y_3\right),\dots ,\left(y_{L-1},y_L\right)\right\}$, since doing so will break the Markov property. Instead, the chain we consider is $\left\{\left(y_1,x_2\right),\left(y_2,x_3\right),\dots ,\left(y_{L-1},x_L\right)\right\}$. The first spin in each pair is the final configuration, whereas the second spin is the initial condition. The transition probability matrix for this Markov chain is

$$M\left(p\right)=\left(\begin{array}{cccc}p& 1-p& 0& 0\\ p& 0& 1-p& 0\\ 0& 0& 1-p& p\\ p& 0& 1-p& 0\end{array}\right),$$ [S5]

which takes into account the flipping dynamics described above.

The net magnetization z is given by

$$z=z\left(y\right)=\frac{1}{L}{\displaystyle \sum _{i=1}^L{y}_i}.$$ [S6]

To calculate the asymptotic distribution of z and hence the free energy, we appeal to large deviation theory. According to the Gärtner–Ellis theorem (34), the free energy is given by

$$F\left(z;p,0\right)=\underset{\lambda \in \mathrm{ℝ}}{\sup }\left\{\lambda z-\ln k\left(\lambda ;p\right)\right\},$$ [S7]

where $k\left(\lambda ;p\right)$ is the largest eigenvalue of the augmented transition matrix

$$\tilde{M}\left(\lambda ;p\right)=\left(\begin{array}{cccc}{e}^{\lambda }p& e^{\lambda }\left(1-p\right)& 0& 0\\ e^{\lambda }p& 0& e^{-\lambda }\left(1-p\right)& 0\\ 0& 0& e^{-\lambda }\left(1-p\right)& e^{-\lambda }p\\ e^{\lambda }p& 0& e^{-\lambda }\left(1-p\right)& 0\end{array}\right).$$ [S8]

It can be shown by differentiation that the RHS of Eq. S7 has a unique stationary maximum for all $0<z,p<1$, and we call the maximizing argument ${\lambda }_m\left(z,p\right)$, implicitly defined by

$$zk\left({\lambda }_m\left(z,p\right);p\right)=\frac{\partial k\left({\lambda }_m\left(z,p\right);p\right)}{\partial \lambda }.$$ [S9]

Its exact expression is available, but it is cumbersome to write down, so we omit it here. The free energy is hence

$$F\left(z;p,0\right)=z{\lambda }_m\left(z,p\right)-\ln k\left({\lambda }_m\left(z,p\right);p\right).$$ [S10]

This has been plotted in Fig. 1.

In particular, the minimum of the free energy $z^{\mathrm{\star }}$ can be found by direct differentiation of Eq. S10, and is given by

$$z^{\mathrm{\star }}\left(p,0\right)=1-2p-\frac{2\left(1-2p\right)}{1-p+p^2}.$$ [S11]

Alternatively, one can also find this expression by taking expectation with respect to the invariant distribution of the Markov chain.

Therefore, the dependence of the magnetization on thermodynamic parameters is given by

$$z^{\mathrm{\star }}\left(p,h\right)=\{\begin{array}{ll}+1\hfill & h>0,\hfill \\ 1-2p-\frac{2\left(1-2p\right)}{1-p+p^2}\hfill & h=0,\hfill \\ \mathrm{-}1\hfill & h<0.\hfill \end{array}$$ [S12]