We use the mean-bacterial-velocity model to investigate the \textit{irreversibility} of two-dimensional (2D) \textit{bacterial turbulence} and to compare it with its 2D fluid-turbulence counterpart. We carry out extensive direct numerical simulations of Lagrangian tracer particles that are advected by the velocity field in this model. Our work uncovers an important, qualitative way in which irreversibility in bacterial turbulence is different from its fluid-turbulence counterpart: For large positive (or large but negative) values of the \textit{friction} (or \textit{activity}) parameter, the probability distribution functions of energy increments, along tracer trajectories, or the power are \textit{positively} skewed; so irreversibility in bacterial turbulence can lead, on average, to \textit{particles gaining energy faster than they lose it}, which is the exact opposite of what is observed for tracers in 2D fluid turbulence.
Bacterial cellulose is an important class of biomaterials which can be grown in well-controlled laboratory and industrial conditions. The cellulose structure is affected by several biological, chemical and environmental factors, including hydrodynamic flows in bacterial suspensions. In this work, we explore the possibility of using well controlled turbulent flow to control the bacterial cellulose production. Turbulent flows leading to random motion of fluid elements may affect the structure of the extracellular polymeric matrix produced by bacteria. Here we show that two-dimensional turbulence…
We experimentally study density fluctuations and energy spectra of bulk \textit{E. coli} suspensions of different concentrations. Our results verify the predicted scaling law of giant number fluctuations in three-dimensional (3D) wet active fluids. We find that such a scaling behavior persists at small scales even in low concentration suspensions well below the transition concentration to active turbulence. Our experiments also support the prediction on the energy spectra of dilute pusher swimmers and illustrate the spectral properties of the active turbulence of dense bacterial suspensions in…
Understanding the role of frictional drag in diffusive transport is an important problem in the field of active turbulence. Using a continuum model that applies well to bacterial suspensions, we investigate the role of Ekmann friction on the transport of passive (Lagrangian) tracers that go with the local flow. We find that the crossover from ballistic to diffusive regime happens at a time scale $τ_c$ that attains a minimum at zero friction, meaning that both injection and dissipation of energy delay the relaxation of tracers. We explain this by proposing that $τ_c \sim 2 \ell^*/u_{\text{rms}}…
We experimentally investigate the effects of polymer additives on the collective dynamics of swarming Serratia marcescens in quasi two-dimensional (2D) liquid films. We find that even minute amounts of polymers (< 20 ppm) can significantly enhance swimming speed and promote largescale coherent structures. Velocity statistics show that polymers suppress large velocity fluctuation, transforming the velocity distributions from super-Gaussian to Gaussian. Spatial and temporal correlation functions suggest that polymers increase both the size and lifetime of flow structures. The energy spectra show…
In this work, single particle dispersion was analyzed for a bacterial turbulence by retrieving the virtual Lagrangian trajectory via numerical integration of the Lagrangian equation. High-order displacement functions were calculated for cases with and without mean velocity effect. Two-regime power-law behavior for short and long time evolutions were identified experimentally, which were separated by the Lagrangian integral time. For the case with the mean velocity effect, the experimental Hurst numbers were determined to be $0.94$ and $0.97$ for short and long times evolutions, respectively. F…
Turbulence is ubiquitous, from oceanic currents to small-scale biological and quantum systems. Self-sustained turbulent motion in microbial suspensions presents an intriguing example of collective dynamical behavior amongst the simplest forms of life, and is important for fluid mixing and molecular transport on the microscale. The mathematical characterization of turbulence phenomena in active non-equilibrium fluids proves even more difficult than for conventional liquids or gases. It is not known which features of turbulent phases in living matter are universal or system-specific, or which ge…
The active interaction between the bacteria and fluid generates turbulent structures even at zero Reynolds number. Velocity of such a flow obtained experimentally has been quantitatively investigated based on streamline segment analysis. There is a clear transition at about $16$ times of the organism body length separating two different scale regimes, which may be attributed to the different influence of the viscous effect. Surprisingly the scaling extracted from the streamline segment indicates the existence of scale similarity even at the zero Reynolds number limit. Moreover the multifractal…
In this paper, an experimental velocity database of a bacterial collective motion , e.g., \textit{B. subtilis}, in turbulent phase with volume filling fraction $84\%$ provided by Professor Goldstein at the Cambridge University UK, was analyzed to emphasize the scaling behavior of this active turbulence system. This was accomplished by performing a Hilbert-based methodology analysis to retrieve the scaling property without the $β-$limitation. A dual-power-law behavior separated by the viscosity scale $\ell_ν$ was observed for the $q$th-order Hilbert moment $\mathcal{L}_q(k)$. This dual-power-la…
The collective motion of microswimmers in suspensions induce patterns of vortices on scales that are much larger than the characteristic size of a microswimmer, attaining a state called bacterial turbulence. Hydrodynamic turbulence acts on even larger scales and is dominated by inertial transport of energy. Using an established modification of the Navier-Stokes equation that accounts for the small scale forcing of hydrodynamic flow by microswimmers, we study the properties of a dense supensions of microswimmers in two dimensions, where the conservation of enstrophy can drive an inverse cascade…
Bacterial swarms display intriguing dynamical states like active turbulence. Using a hydrodynamic model we now show that such dense active suspensions manifest super-diffusion, via Lévy walks, which masquerades as a crossover from ballistic to diffusive scaling in measurements of mean-squared-displacements, and is tied to the emergence of hitherto undetected oscillatory streaks in the flow. Thus, while laying the theoretical framework of an emergent advantageous strategy in the collective behaviour of microorganisms, our study underlines the essential differences between active and inertial tu…
Dense active matter, from bacterial suspensions and microtubule bundles driven by motor proteins to cellular monolayers and synthetic Janus particles, is characterised by mesoscale turbulence, the emergence of chaotic flow structures. By immersing an ordered array of symmetric rotors in an active fluid, we introduce a microfluidic system that exploits spontaneous symmetry breaking in mesoscale turbulence to generate work. The lattice of rotors self-organises into a spin-state where neighbouring discs continuously rotate in permanent alternating directions due to combined hydrodynamic and elast…
From the smallest scales of quantum systems to the largest scales of intergalactic medium, turbulence is ubiquitous in nature. Often dubbed as the last unsolved problem of classical physics, it remains a time tested paradigm of dynamics far from equilibrium. The phenomenon even transcends to self-propelled fluids such as dense bacterial suspensions that can display turbulence at mesoscale even though the constituent particles move at Reynolds number below unity. It is intensely debated whether such fluids possess an effective temperature and obey fluctuation-dissipation relations (FDR) as they…
We investigate the predictability aspects of rotating turbulent flows through extensive numerical simulations of a shell model of rotating turbulence. In particular, we measure the large-scale predictability time and find that it increases with rotation rate to satisfy a power law in Rossby number with a scaling exponent of $-2/3$. Intriguingly, we find that before entering the algebraic growth stage, the error dynamics freezes for a time period determined by the finite Rossby number. We further analyse the scale dependence of the predictability time and observe that it tends to become scale i…
Hydrodynamics of Dense Active Fluids: Turbulence-Like States and the Role of Advected Activity
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（国际理论科学中心，塔塔基础研究所）；Department of Mechanical Engineering, Indian Institute of Technology Kanpur（坎普尔印度理工学院机械工程系）
Flagellated bacteria on nutrient-rich substrates can differentiate into a swarming state and move in dense swarms across surfaces. A recent experiment measured the flow in the fluid around an Escherichia coli swarm (Wu, Hosu and Berg, 2011 Proc. Natl. Acad. Sci. USA 108 4147). A systematic chiral flow was observed in the clockwise direction (when viewed from above) ahead of the swarm with flow speeds of about $10~μ$m/s, about 3 times greater than the radial velocity at the edge of the swarm. The working hypothesis is that this flow is due to the action of cells stalled at the edge of a colony …
We build on recent developments in the study of fluid turbulence [Gibbon \textit{et al.} Nonlinearity 27, 2605 (2014)] to define suitably scaled, order-$m$ moments, $D_m^{\pm}$, of $ω^\pm= ω\pm j$, where $ω$ and $j$ are, respectively, the vorticity and current density in three-dimensional magnetohydrodynamics (MHD). We show by mathematical analysis, for unit magnetic Prandtl number $P_M$, how these moments can be used to identify three possible regimes for solutions of the MHD equations; these regimes are specified by inequalities for $D_m^{\pm}$ and $D_1^{\pm}$. We then compare our mathematic…
We show that a shell-model version of the three-dimensional Hall-magnetohydrodynamic (3D Hall-MHD) equations provides a natural theoretical model for investigating the multiscaling behaviors of velocity and magnetic structure functions. We carry out extensive numerical studies of this shell model, obtain the scaling exponents for its structure functions, in both the low-$k$ and high-$k$ power-law ranges of 3D Hall-MHD, and find that the extended-self-similarity (ESS) procedure is helpful in extracting the multiscaling nature of structure functions in the high-$k$ regime, which otherwise appear…
The role of supersonic turbulence in structuring the interstellar medium (ISM) remains an unsettled question. Here, this problem is investigated using a newexact law of compressible isothermal hydrodynamic turbulence, which involves two-point correlations in physical space. The new law is shown to have a compact expression that contains a single flux term reminiscent of the incompressible case and a source term with a simple expression whose sign is given by the divergence of the velocity. The law is then used to investigate the properties of such a turbulence at integral Mach number $4$ produ…
Studying the atmospheric Planetary Boundary Layer (PBL) is crucial to understand the climate of a planet. The meteorological measurements by the instruments onboard InSight at a latitude of 4.5$^{\circ}$N make a uniquely rich dataset to study the active turbulent dynamics of the daytime PBL on Mars. Here we use the high-sensitivity continuous pressure, wind, temperature measurements in the first 400 sols of InSight operations (from northern late winter to midsummer) to analyze wind gusts, convective cells and vortices in Mars' daytime PBL. We compare InSight measurements to turbulence-resolvin…
The coupled dynamics of quantum turbulence (QT) and normal-fluid turbulence (NFT) have been a central challenge in quantum hydrodynamics, since it is expected to cause the unsolved T2 state of QT. We numerically studied the coupled dynamics of the two turbulences in thermal counterflow. NFT is driven by external forces to control its turbulent intensity, and the fast multipole method accelerates the calculation of QT. We show that NFT enhances QT via mutual friction. The vortex line density $L$ of the QT satisfies the statistical law $L^{1/2} \approx γV_{ns}$ with the counterflow velocity $V_{…
Turbulent fluid flows exhibit a complex small-scale structure with frequently occurring extreme velocity gradients. Particles probing such swirling and straining regions respond with an intricate shape-dependent orientational dynamics, which sensitively depends on the particle history. Here, we systematically develop a reduced-order model for the small-scale dynamics of turbulence, which captures the velocity gradient statistics along particle paths. An analysis of the resulting stochastic dynamical system allows pinpointing the emergence of non-Gaussian statistics and non-trivial temporal cor…
Active matter, composed of self-propelled entities, forms a wide class of out-of-equilibrium systems that display striking collective behaviors among which the so-called active turbulence where spatially and time disordered flow patterns spontaneously arise in a variety of {active systems}. De facto, the active turbulence naming suggests a connection with a second seminal class of out-of-equilibrium systems, fluid turbulence, and yet of very different nature with energy injected at global system scale rather than at the elementary scale of single constituents. Indeed the existence of a possibl…
We study the effect of different forcing functions and of the local gradient Richardson number Rig on the vertical mixing of Lagrangian tracers in stably stratified turbulence under the Boussinesq approximation, and present a wave and continuous-time random walk model for single- and two- particle vertical dispersion. The model consists of a random superposition of linear waves with their amplitude based on the observed Lagrangian spectrum of vertical velocity, and a random walk process to capture overturning that depends on the statistics of Rig among other Eulerian quantities. The model is i…
