Directed percolation describes lifetime and growth of turbulent puffs and slugs

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Directed percolation describes lifetime and growth of turbulent puffs and slugs

Maksim Sipos and Nigel Goldenfeld

Phys. Rev. E 84, 035304(R) – Published 29 September 2011

Abstract

We show that directed percolation (DP) simulations in a pipe geometry in $3 + 1$ dimensions capture the observed complex phenomenology of the transition to turbulence. At low Reynolds numbers (Re), turbulent puffs form and spontaneously relaminarize. At high Re, turbulent slugs expand uniformly into the laminar regions. In a spatiotemporally intermittent state between these two regimes of Re, puffs split and turbulent regions exhibit laminar patches. DP also captures some of the quantitative features of the transition, with a superexponentially diverging characteristic lifetime below the transition. Above the percolation threshold, active (turbulent) clusters expand into the inactive (laminar) phase with a well-defined velocity whose scaling with control parameter (Reynolds number or percolation probability) is consistent with experimental results. Our results provide strong evidence in favor of a conjecture of Pomeau.

Received 30 January 2011

DOI:https://doi.org/10.1103/PhysRevE.84.035304

Authors & Affiliations

Maksim Sipos and Nigel Goldenfeld

Department of Physics, Loomis Laboratory of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801-3080, USA

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Vol. 84, Iss. 3 — September 2011

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Figure 1
Comparison of the phenomenology of transitional turbulence as a function of Re with that of DP in $3 + 1$ dimensions as a function of $p$ , both in a pipe geometry.Reuse & Permissions
Figure 2
(a) Illustration of the decay of an active cluster in subcritical directed percolation in $3 + 1$ dimensions in pipe geometry. In this figure the length of the pipe extends horizontally, and the active sites in percolation are displayed with green cubes. Inactive sites are not shown. (b) Different front shapes in the growing front bond DP model. Rough fronts occur for small $p - p_{c}$ whereas smoother fronts occur for large $p - p_{c}$ . (c) Time evolution of an initially active 180 contiguous sites percolating downward simulated via bond percolation for $p = 0.61 < p_{c}$ . The active state (in black) fully decays into the absorbing state (white) after about 250 steps.Reuse & Permissions
Figure 3
(a) Superexponential scaling of the characteristic lifetime $τ$ . The line indicates the fit to (2). Error bars indicate 95% confidence intervals from a Kolmogorov-Smirnov test (for $p > 0.62$ ) and 90% confidence intervals from a $χ^{2}$ test (for $p \leq 0.62$ ). In the inset, $τ_{0} = 0.017$ . (b) Numerical data for survival probabilities $P (p, t)$ (points) and a fit to (1) with $τ$ given by (2) (solid lines). Data shown for $t = 300$ (red crosses), 1000 (green dots), 6000 (blue pluses) and 80 000 (cyan triangles). The value of $p_{c}$ observed corresponds to that of ( $1 + 1$ )-dimensional bond percolation. (c) Measured survival probabilities as functions of $t$ for four different values of percolation probability $p$ . Blue solid line indicates measured data, whereas red dashed line indicates a fit to the exponential distribution. Deviations from exponential distribution for small $t$ are due to nonzero $t_{0}$ .Reuse & Permissions
Figure 4
Measured values of front propagation velocity $G$ (red crosses) compared to theoretical prediction for ( $3 + 1$ )-dimensional and ( $1 + 1$ )-dimensional DP. Green vertical line indicates the value of $p$ for which $ξ_{⊥}$ exceeds $1.2 R$ . We call this value $p_{R}$ . (a) Plot indicating the power law crossover of $G (p)$ . Inset shows that $G / {(p - p_{c})}^{0.637}$ is roughly constant for $p < p_{R}$ and similarly $G / {(p - p_{c})}^{0.524}$ for $p > p_{R}$ . (b) Both regimes of DP have the same critical point $p_{c}$ (within the error of our measurements). The linear fits to $G^{1 / γ}$ are shown in solid lines. Extrapolations are indicated with dashed lines, and they cross at the same $p_{c}$ . This value of $p_{c}$ corresponds to that of ( $3 + 1$ )-dimensional bond percolation.Reuse & Permissions

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