Abstract
The mechanisms and universality class underlying the remarkable phenomena at the transition to turbulence remain a puzzle 130 years after their discovery^{1}. Near the onset to turbulence in pipes^{1}, plane Poiseuille flow^{2} and Taylor–Couette flow^{3}, transient turbulent regions decay either directly^{4} or through splitting^{5,6,7,8}, with characteristic timescales that exhibit a superexponential dependence on Reynolds number^{9,10}. The statistical behaviour is thought to be related to directed percolation (DP; refs 6,11,12,13). Attempts to understand transitional turbulence dynamically invoke periodic orbits and streamwise vortices^{14,15,16,17,18,19}, the dynamics of longlived chaotic transients^{20}, and model equations based on analogies to excitable media^{21}. Here we report direct numerical simulations of transitional pipe flow, showing that a zonal flow emerges at large scales, activated by anisotropic turbulent fluctuations; in turn, the zonal flow suppresses the smallscale turbulence leading to stochastic predator–prey dynamics. We show that this ecological model of transitional turbulence, which is asymptotically equivalent to DP at the transition^{22}, reproduces the lifetime statistics and phenomenology of pipe flow experiments. Our work demonstrates that a fluid on the edge of turbulence exhibits the same transitional scaling behaviour as a predator–prey ecosystem on the edge of extinction, and establishes a precise connection with the DP universality class.
Main
Turbulent fluids are ubiquitous in nature, arising for sufficiently large characteristic speeds U, depending on the kinematic viscosity ν and a characteristic system scale, such as the diameter of a pipe D. Turbulent flows are complex, stochastic, and unpredictable in detail, but transition at lower velocities to a laminar flow, which is simple, deterministic and predictable. This transition is controlled by the dimensionless parameter known as the Reynolds number, which in the pipe geometry of interest here is given by Re ≡ UD/ν, and occurs in the range 1,700 ≲ Re ≲ 2,300. The laminarturbulence transition has presented a challenge to experiment and theory since Osborne Reynolds’ original observation of intermittent ‘flashes’ of turbulence^{1}.
To explore this transitional regime, we have performed direct numerical simulations of the Navier–Stokes equations in a pipe of length L = 10D, using the opensource code ‘Open Pipe Flow’^{23}, as described in Supplementary Methods. The Reynolds number at which transitional turbulence occurs is higher for short pipes^{23}, and the simulations reported here for L = 10D were performed at a nominal value Re = 2,600, which we estimate to be equivalent to Re ≲ 2,200 in long pipe data^{7} based on estimates of when puff decay transitions to puff splitting. We confirmed that our results did not qualitatively change for a longer pipe with L = 20D. We denote the timedependent velocity deviation from the Hagen–Poiseuille flow by u = (u_{z}, u_{θ}, u_{r}). Because we were interested in transitional behaviour, we looked for a decomposition^{2,6,24,25} of largescale modes that would indicate some form of collective behaviour, as well as smallscale modes that would be representative of turbulent dynamics. In particular, we report here the behaviour of the velocity field , where the bar denotes average over z and θ, and . We refer to this as the zonal flow. In Fourier space, the zonal flow is given by , where k is the axial wavenumber and m is the azimuthal wavenumber, r is the real space radial coordinate and the tilde denotes Fourier transform in the θ and z directions only. Turbulence was represented by shortwavelength modes, whose energy is .
Shown in Fig. 1a is a time series for the energy of the zonal flow, compared with the energy E_{T}(t) of the turbulent energy. The curves show clear persistent oscillatory behaviour, modulated by longwavelength stochasticity, as shown in the phase portrait of Fig. 1b. In Fig. 1c, we have calculated the phase shift between the turbulence and zonal flows, with the result that the turbulent energy leads the zonal flow energy by ∼π/2. This suggests that these oscillations can be interpreted as a timeseries resulting from activator–inhibitor dynamics, such as occurs in a predator–prey ecosystem. Predator–prey ecosystems are characterized by the following behaviour: the ‘prey’ mode activates the ‘predator’ mode, which then grows in abundance. At the same time, the growing predator mode begins to inhibit the prey mode. The inhibition of the prey mode starves the predator mode, and it too becomes inhibited. The inhibition of the predator mode allows the prey mode to reactivate, and the population cycle begins again.
The flow configuration for the predator mode is shown in Fig. 1d, and consists of a series of azimuthally symmetric modes with direction reversals as a function of radius r. Such banded shear flows are known as zonal flows and are of special significance in plasma physics, astrophysical and geophysical flows, owing to their role in regulating turbulence^{26}. The purely azimuthal component of the zonal flow, denoted by , is spatially uniform in z, and the lack of a radial component means that it is not driven by pressure gradients. Thus, it can exist only as a result of nonlinear interactions with turbulent modes. In this sense, it is a collective mode, one with special significance for transitional turbulence.
The simplest way for such an azimuthal shear flow to couple to turbulent fluctuations is through the Reynolds stress τ: however, a uniform Reynolds stress cannot drive a shear flow, so the first symmetryallowed possibility is the radial gradient of the Reynolds stress^{26}, as expressed in the Reynolds momentum equation. Thus, to probe the dynamics that govern the emergence of the zonal flow, we have calculated the timeaveraged radial gradient of the instantaneous Reynolds stress, τ ≡ u_{θ}′u_{r}′, where , and show in Fig. 1f the 4.5timeunitrunningmean time series of and the radial gradient ∂_{r}τ. Both quantities have been averaged over 0 ≤ z ≤ L, 0 ≤ θ ≤ 2π and R_{0} ≤ r < R, where R = D/2, R_{0} = 0.641R, and the resulting time series are clearly highly correlated. In general, it is the case that zonal flows are driven by statistical anisotropy in turbulence, but are themselves an isotropizing influence on the turbulence through their coupling to the Reynolds stress^{27,28,29}. The fact that turbulence anisotropy activates the zonal flow, and that zonal flow inhibits the turbulence, is responsible for the predator–prey oscillations observed in the numerical simulations.
These numerical results suggest that the largescale zonal flow and the smallscale turbulence are necessary, and perhaps even sufficient components of an effective coarsegrained description of transitional turbulence in the spirit of Landau theory. Following the usual logic of the modern theory of phase transitions^{30}, we construct the effective theory from symmetry principles alone, as there are no small parameters with which to perform a systematic derivation. If correct, this effective predator–prey theory should undergo spatiotemporal fluctuations whose functional form matches the observations for the lifetime and splitting time of turbulent puffs in a pipe.
The simplest system that corresponds to our direct numerical simulations of the Navier–Stokes equations has three trophic levels: nutrient (E), Prey (B) and Predator (A), which correspond in the fluid system to laminar flow, turbulence and zonal flow respectively. The interactions between individual representatives of these levels are given by the following reactions:
where d_{A} and d_{B} are the death rates of A and B, p is the predation rate, b is the prey birth rate due to consumption of nutrient, 〈ij〉 denotes hopping to nearest neighbour sites, and D_{A} and D_{B} are the nearestneighbour hopping rate for predator and prey respectively, assumed for simplicity here to be the same value D_{AB} for predator and prey. The ‘mutation’ term (B → A) is symmetryallowed and has the interesting consequence that the diagram of the predator–prey model matches that of pipe transitional turbulence (Supplementary Methods).
We simulated this predator–prey model, using methods described in Supplementary Methods, in a thin twodimensional strip on a 401 × 11 lattice. The control parameter is the prey birth rate b. When b is small enough, the population is metastable, and cannot sustain itself, decaying with a finite lifetime τ^{d}(b). As b increases, the lifetime of the population increases rapidly: in particular the prey lifetime increases rapidly with b. At large enough values of b, the decay of the initial population is not observed, but instead the initially localized population splits after a time τ^{s}(b), spreading outwards and spontaneously splitting into multiple clusters, as shown in the space–time plot of clusters of prey of Fig. 2a.
To quantify these observations, we have measured both the lifetime of population clusters in the metastable region and their splitting time using a procedure directly following that of the turbulence experiments and simulations^{7}, and described in Supplementary Methods. We comment that both timescales involve implicitly measurements of quantities that exceed a given threshold, and thus it is natural that the results are found to conform to extreme value statistics^{12,31}.
In Fig. 2a we show the phenomenology of the dynamics of initial clusters of prey, corresponding to the predator–prey analogue for the experiments in pipe flow which followed the dynamics of an initial puff of turbulence injected into the flow^{4}. Depending on the prey birth rate, the cluster decays either homogeneously or by splitting, precisely mimicking the behaviour of turbulent puffs as a function of Reynolds number. The extraction from data of decay times is described in Supplementary Methods. In Fig. 2b is shown the semilog plot of lifetime for both decay and splitting as a function of prey birth rate, the upward curvature indicative of superexponential behaviour. The inset to Fig. 2b shows a double exponential plot of puff lifetime and splitting time versus prey birth rate, the straight line being the fit to the functional form indicated in the caption. These figures indicate a remarkable similarity to the corresponding plots obtained for transitional pipe turbulence in both experiments^{4} and direct numerical simulations^{7}, and demonstrate conclusively that experimental observations are well captured by an effective twofluid model of pipe flow turbulence with predator–prey interactions between the zonal flow and the small scale turbulence.
Our simulations show that the predator–prey model expressed by equation (1) exhibits a rich phase diagram that captures the main features observed in transitional turbulence in pipes. We can understand the qualitative features of the phase diagram from linear stability analysis of the mean field solution of the predator–prey equations^{22}. Near the transition, the solutions are linearly stable, all eigenvalues are real and there are no spatialtemporal oscillations. But for higher values of b, the eigenvalues develop an imaginary part, a necessary condition for the breakdown of spatially homogeneous prey domains into periodic travelling wave states^{32}. The phase diagram is sketched in Fig. 3, along with the corresponding phase diagram for transitional pipe turbulence as determined by experiment. The phenomenology of the predator–prey system mirrors that of turbulent pipe flow.
To determine the universality class of the nonequilibrium phase transition from laminar to turbulent flow, we use the twofluid predator–prey mode in equation (1). Near the transition to prey extinction, the prey population is very small and no predator can survive, and thus equation (1) simplifies to
These equations are exactly those of the reactiondiffusion model for directed percolation^{33}. The argument here is heuristic but the result is correct and can be obtained systematically from statistical field theory techniques, as described in Supplementary Methods.
The observation of the emergence of a zonal flow, excited by the developing turbulent degrees of freedom and the demonstration of its role in determining the phenomenology of transitional pipe turbulence has an interesting consequence: the zonal flow can be assisted by rotating the pipe, and this should catalyse the transition to turbulence, causing it to occur at lower Re. Indeed experiments on axiallyrotating pipes^{34} are consistent with this prediction.
Our work underscores not only the potential importance of zonal flows in other transitional turbulence situations^{9,10}, but also shows the utility of coarsegrained effective models for nonequilibrium phase transitions, even to states as perplexing as fluid turbulence.
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Acknowledgements
We gratefully acknowledge helpful discussions with Y. Duguet and Z. Goldenfeld. We especially thank A. Willis for permission to use his code ‘Open Pipe Flow’^{23}. This work was partially supported by the National Science Foundation through grant NSFDMR1044901.
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H.Y.S. and N.G. designed the project. Computer simulations of pipe turbulence were performed by T.L.H. Computer simulations of stochastic predator–prey dynamics were performed by H.Y.S. All authors contributed to the interpretation of the data and the writing of the paper.
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Shih, HY., Hsieh, TL. & Goldenfeld, N. Ecological collapse and the emergence of travelling waves at the onset of shear turbulence. Nature Phys 12, 245–248 (2016). https://doi.org/10.1038/nphys3548
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