Supporting data for Matsuzawa, Zhu, et al., "Nonlinear Diffusion and Decay of a Blob of Turbulence Spreading Into a Quiescent Fluid", PNAS, (2026)

Open MIND · 2026-01-26

Paper: Nonlinear Diffusion and Decay of a Blob of Turbulence Spreading Into a Quiescent Fluid Authors: Takumi Matsuzawa, Minhui Zhu, Nigel Goldenfeld, William T. M. Irvine Description:This repository contains the data presented in Figures 1–5 of the paper, which examine the decay and propagation of turbulence generated by repeatedly colliding vortex rings. In the study, we also explored turbulence decay generated by oscillating a double grid. Each folder contains the data used to produce the main figures. For Figures 1C–F and 4C–F, we present time slices of ensemble-averaged fields, including energy and enstrophy. The original velocity field data is approximately 10 GB per sample. We performed 21, 10, and 10 replications for the blob, double oscillating grid, and single oscillating grid configurations, respectively. From these, Reynolds averaging was applied to extract the turbulent (fluctuating) components. Contents:Each folder contains HDF5 files with relevant data that made it into the plot. E.g. Figure 2B_tegral_lengthscale.h5 contains/blob/t_t0: t-t0/blob/ell: ell (Integral length scale) from the blob experiments/blob/ell_scaled: ell / Lbox (Scaled integral length scale)...

Supporting data for Matsuzawa, Zhu, et al., "Nonlinear Diffusion and Decay of a Blob of Turbulence Spreading Into a Quiescent Fluid", PNAS, (2026)

Zenodo (CERN European Organization for Nuclear Research) · 2026-01-26

Paper: Nonlinear Diffusion and Decay of a Blob of Turbulence Spreading Into a Quiescent Fluid Authors: Takumi Matsuzawa, Minhui Zhu, Nigel Goldenfeld, William T. M. Irvine Description:This repository contains the data presented in Figures 1–5 of the paper, which examine the decay and propagation of turbulence generated by repeatedly colliding vortex rings. In the study, we also explored turbulence decay generated by oscillating a double grid. Each folder contains the data used to produce the main figures. For Figures 1C–F and 4C–F, we present time slices of ensemble-averaged fields, including energy and enstrophy. The original velocity field data is approximately 10 GB per sample. We performed 21, 10, and 10 replications for the blob, double oscillating grid, and single oscillating grid configurations, respectively. From these, Reynolds averaging was applied to extract the turbulent (fluctuating) components. Contents:Each folder contains HDF5 files with relevant data that made it into the plot. E.g. Figure 2B_tegral_lengthscale.h5 contains/blob/t_t0: t-t0/blob/ell: ell (Integral length scale) from the blob experiments/blob/ell_scaled: ell / Lbox (Scaled integral length scale)...

Nonlinear diffusion and decay of a blob of turbulence spreading into a quiescent fluid

Proceedings of the National Academy of Sciences · 2026-02-12

Turbulence, left unforced, evolves under its own dynamics, invading surrounding quiescent fluid as it decays. A ubiquitous and familiar phenomenon, this fundamental aspect of turbulence has resisted the marriage of principled theory and experiment with no universal law yet capturing its evolution. Conventional flow chamber experiments have been hampered by boundary effects or strong mean flows that obscure the intrinsic dynamics of relaxation to quiescence. To circumvent these limitations, we create a spatially localized blob of turbulence using eight converging vortex generators focused at the center of a water tank, and observe its decay and expansion over decades in time using particle image velocimetry with logarithmic time sampling. The blob initially expands and decays until it reaches the walls of the tank and eventually transitions to a second regime of approximately spatially uniform decay. We interpret the turbulent dynamics as an interplay of nonlinear diffusion with associated steep fronts separating the turbulent and quiescent regions, and nonlinear decay, as described by the Kolmogorov-Barenblatt equation. We find direct evidence for this model within the expansion phase and decay phases of our turbulent blob and use it to account for the detailed behavior we observe. Our work provides a detailed spatially resolved narrative for the behavior of turbulence once the forcing is removed, and demonstrates unexpectedly that the turbulent cascade leaves an indelible footprint far into the decay process.

Beyond chaos: fluctuations, anomalies and spontaneous stochasticity in fluid turbulence

ArXiv.org · 2025-12-30

In this perspective, we consider the development of statistical hydrodynamics, focusing on the way in which the intrinsic stochasticity of turbulent phenomena was identified and is being explored. A major purpose of our discussion is to bring out the role of anomalies in turbulent phenomena, in ways that are not usually done, and to emphasize how the description of turbulent phenomena requires delicate considerations of asymptotic limits. The scope of our narrative includes selected historical aspects that are not usually emphasized, primarily due to G.I. Taylor, as well as discussions of certain aspects of the laminar-turbulent transition, the behaviour of turbulent drag at intermediate Reynolds numbers, and the statistics of fully-developed turbulence that exhibit spontaneous stochasticity.

Beyond chaos: fluctuations, anomalies and spontaneous stochasticity in fluid turbulence

arXiv (Cornell University) · 2025-12-30

In this perspective, we consider the development of statistical hydrodynamics, focusing on the way in which the intrinsic stochasticity of turbulent phenomena was identified and is being explored. A major purpose of our discussion is to bring out the role of anomalies in turbulent phenomena, in ways that are not usually done, and to emphasize how the description of turbulent phenomena requires delicate considerations of asymptotic limits. The scope of our narrative includes selected historical aspects that are not usually emphasized, primarily due to G.I. Taylor, as well as discussions of certain aspects of the laminar-turbulent transition, the behaviour of turbulent drag at intermediate Reynolds numbers, and the statistics of fully-developed turbulence that exhibit spontaneous stochasticity.

Interpreting convex integration results in hydrodynamics

European Mathematical Society Magazine · 2025-07-08

The aim of this article is to encourage debate of issues of the applications of modern methods of mathematical analysis in fluid dynamics. A recent surprising result derived by convex integration techniques shows non-uniqueness of weak solutions in initial value problems of the Navier–Stokes equations. The question of relevance of such a result to physical observed flows allows a variety of answers, some of which are discussed below.

Tricritical Directed Percolation Controls the Laminar-Turbulent Transition in Pipes with Body Forces

Physical Review Letters · 2025-08-05 · 2 citations

The laminar-turbulent transition in straight pipes is believed to occur through a continuous nonequilibrium phase transition in the directed percolation universality class. However, in curved pipes or in the presence of body forces it is possible to observe a discontinuous transition and other phenomenology which seem inconsistent with the emerging consensus. Here, we consider the perturbing effects of body forces and incorporate them into a minimal Landau theory of the transition. We calculate the phase diagram as a function of Reynolds number and body force strength, and show that above a threshold strength of the latter, there is a tricritical point which accounts for the observed discontinuity behavior, including spatially heterogeneous states. Our results are consistent with recent experiments in centrifugal pipes and direct numerical simulations of heated flows.

Self-consistent expansion and field-theoretic renormalization group for a singular nonlinear diffusion equation with anomalous scaling

Physical review. E · 2025-01-22

The method of self-consistent expansions is a powerful tool for handling strong coupling problems that might otherwise be beyond the reach of perturbation theory, providing surprisingly accurate approximations even at low order. First applied in its embryonic form to fully-developed turbulence, it has subsequently been successfully applied to a variety of problems that include polymer statistics, interface dynamics, and high-order perturbation theory for the anharmonic oscillator. Here, we show that the self-consistent expansion can be applied to singular perturbation problems arising in the theory of partial differential equations in conjunction with renormalization group methods. We demonstrate its application to Barenblatt's nonlinear diffusion equation for porous media filtration, where the long-time asymptotics exhibits anomalous dimensions that can be systematically calculated using the perturbative renormalization group. We find that even the first-order self-consistent expansion, when combined with the Callan-Symanzik equation, improves the approximation of the anomalous dimension obtained by the first-order perturbative renormalization group, especially in the strong coupling regime. We also develop a field-theoretic framework for deterministic partial differential equations to facilitate the application of self-consistent expansions to other dynamic systems and illustrate its application using the example of Barenblatt's equation. The scope of our results on the application of renormalization group and self-consistent expansions is limited to partial differential equations whose long-time asymptotics is controlled by incomplete similarity. However, our work suggests that these methods could be applied to a broader suite of singular perturbation problems such as boundary layer theory, multiple scales analysis, and matched asymptotic expansions, for which excellent approximations using renormalization group methods alone are already available.

Nonlinear Diffusion and Decay of an Expanding Turbulent Blob

ArXiv.org · 2025-05-28

Turbulence, left unforced, decays and invades the surrounding quiescent fluid. Though ubiquitous, this simple phenomenon has proven hard to capture within a simple and general framework. Experiments in conventional turbulent flow chambers are inevitably complicated by proximity to boundaries and mean flow, obscuring the fundamental aspects of the relaxation to the quiescent fluid state. Here, we circumvent these issues by creating a spatially-localized blob of turbulent fluid using eight converging vortex generators focused towards the center of a tank of water, and observe its decay and spread over decades in time, using particle image velocimetry with a logarithmic sampling rate. The blob initially expands and decays until it reaches the walls of the tank and eventually transitions to a second regime of approximately spatially uniform decay. We interpret these dynamics within the framework of a nonlinear diffusion equation, which predicts that the ideal quiescent-turbulent fluid boundary is sharp and propagates non-diffusively, driven by turbulent eddies while decaying with characteristic scaling laws. We find direct evidence for this model within the expansion phase of our turbulent blob and use it to account for the detailed behavior we observe, in contrast to earlier studies. Our work provides a detailed spatially-resolved narrative for the behavior of turbulence once the forcing is removed, and demonstrates unexpectedly that the turbulent cascade leaves an indelible footprint far into the decay process.

Power-laws in phylogenetic trees and the preferential coalescent

ArXiv.org · 2025-10-15

Phylogenetic trees capture evolutionary relationships among species and reflect the forces that shaped them. While many studies rely on branch length information, the topology of phylogenetic trees (particularly their degree of imbalance) offers a robust framework for inferring evolutionary dynamics when timing data is uncertain. Classical metrics, such as the Colless and Sackin indices, quantify tree imbalance and have been extensively used to characterize phylogenies. Empirical phylogenies typically show intermediate imbalance, falling between perfectly balanced and highly skewed trees. This regime is marked by a power-law relationship between subtree sizes and their cumulative sizes, governed by a characteristic exponent. Although a recent niche-size model replicates this scaling, its mathematical origin and the exponent's value remain unclear. We present a generative model inspired by Kingman's coalescent that incorporates niche-like dynamics through preferential node coalescence. This process maps to Smoluchowski's coagulation kinetics and is described by a generalized Smoluchowski equation. Our model produces imbalanced trees with power-law exponents matching empirical and numerical observations, revealing the mathematical basis of observed scaling laws and offering new tools to interpret tree imbalance in evolutionary contexts.