We investigate the effects of turbulent fluctuations on the Lagrangian statistics of absorption of a scalar field by tracer particles, as a model for nutrient uptake by suspended non-motile microorganisms. By means of extensive direct numerical simulations of an Eulerian-Lagrangian model we quantify, in terms of the Sherwood number, the increase of the scalar uptake induced by turbulence and its dependence on the Peclet and Reynolds numbers. Numerical results are compared with classical predictions for a stationary shear flow extended here to take into account the presence of a restoring scalar flux. We find that mean field predictions agree with numerical simulations at low Peclet numbers but are unable to describe the large fluctuations of local scalar uptake observed for large Peclet numbers. We also study the role of velocity fluctuations in the local uptake by looking at the temporal correlation between local shear and uptake rate and we find that the latter follows fluid velocity fluctuations with a delay given by Kolmogorov time scale. The relevance of our results for aquatic microorganisms is also discussed.
We study the effect of a turbulent flow of liquid sodium generated in the von Kármán geometry, on the localized field of a magnet placed close to the frontier of the flow. We observe that the field can be transported by the flow on distances larger than its integral length scale. In the most turbulent configurations, the mean value of the field advected at large distance vanishes. However, the rms value of the fluctuations increases linearly with the magnetic Reynolds number. The advected field is strongly intermittent.
The rotational dynamics of anisotropic particles advected in a turbulent fluid flow are important in many industrial and natural setting. Particle rotations are controlled by small scale properties of turbulence that are nearly universal, and so provide a rich system where experiments can be directly compared with theory and simulations. Here we report the first three-dimensional experimental measurements of the orientation dynamics of rod-like particles as they are advected in a turbulent fluid flow. We also present numerical simulations that show good agreement with the experiments and allow…
A phenomenological model for the dissipation of scalar fluctuations due to the straining by the fluid motion is proposed in this letter. An explicit equation is obtained for the time evolution of the probability distribution function of a coarse-grained scalar concentration. The model relies on a self-convolution process. We first present this model in the Batchelor regime and then extend empirically our result to the turbulent case. The inclusion of this model in more general transport equations, including spatial gradients, is discussed in relation with 2D turbulence and stratified flows.
We study the melting dynamics of large ice balls in a turbulent von Karman flow at very high Reynolds number. Using an optical shadowgraphy setup, we record the time evolution of particle sizes. We study the heat transfer as a function of the particle scale Reynolds number for three cases: fixed ice balls melting in a region of strong turbulence with zero mean flow, fixed ice balls melting under the action of a strong mean flow with lower fluctuations, and ice balls freely advected in the whole flow. For the fixed particles cases, heat transfer is observed to be much stronger than in laminar f…
The motion of a large, neutrally buoyant, particle, freely advected by a turbulent flow is determined experimentally. We demonstrate that both the translational and angular accelerations exhibit very wide probability distributions, a manifestation of intermittency. The orientation of the angular velocity with respect to the trajectory, as well as the translational acceleration conditioned on the spinning velocity provide evidence of a lift force acting on the particle.
Advection-diffusion problems of magnetic field and tracer field are analyzed using the field theoretic perturbative renormalization group. Both advected fields are considered to be passive, i.e., without any influence on the turbulent environment, and advecting velocity field is generated by compressible version of stochastic Navier-Stokes equation. The model is considered in the vicinity of space dimension $d=4$ and is a continuation of previous work [N.V. Antonov et al., Phys. Rev. E 95, 033120 (2017)]. The perturbation theory near the special dimension $d=4$ is constructed within a double e…
The rate of collision and the relative velocities of the colliding particles in turbulent flows is a crucial part of several natural phenomena, e.g., rain formation in warm clouds and planetesimal formation in a protoplanetary disks. The particles are often modeled as passive, but heavy and inertial. Within this model, large relative velocities emerge due to formation of singularities (caustics) of in the gradient matrix of the velocities of the particles. Using extensive direct numerical simulations of heavy particles in both two (direct and inverse cascade) and three dimensional turbulent fl…
We study the 6-dimensional dynamics -- position and orientation -- of a large sphere advected by a turbulent flow. The movement of the sphere is recorded with 2 high-speed cameras. Its orientation is tracked using a novel, efficient algorithm; it is based on the identification of possible orientation `candidates' at each time step, with the dynamics later obtained from maximization of a likelihood function. Analysis of the resulting linear and angular velocities and accelerations reveal a surprising intermittency for an object whose size lies in the integral range, close to the integral scale …
Heavy inertial particles transported by a turbulent flow are shown to concentrate in the regions where an advected passive scalar, such as temperature, displays very strong front-like discontinuities. This novel effect is responsible for extremely high levels of fluctuations for the passive field sampled by the particles that impacts the heat fluxes exchanged between the particles and the surrounding fluid. Instantaneous and averaged heat fluxes are shown to follow strongly intermittent statistics and anomalous scaling laws.
This article investigates the formation of spanwise vorticity in the velocity streaks of the oblique laminar- turbulent bands of plane Couette flow (PCF) by mean of Direct Numerical Simulations (DNS). The spanwise vorticity is created by a roll-up type development of the streamwise-wall normal shear layer of the velocity streaks. It is advected by the large scale flow along the bands. We propose a criterion on spanwise vorticity which detects these events in order to perform systematic measurements. Beside of the streamwise and spanwise correlation lengths of the rolls, their advection velocit…
Premixed turbulent flames, encountered in power generation and propulsion engines, are an archetype of a randomly advected, self-propagating surface. While such a flame is known to exhibit large-scale intermittent flapping, the possible intermittency of its small-scale fluctuations has been largely disregarded. Here, we experimentally reveal the inner intermittency of a premixed turbulent V-flame, while clearly distinguishing this small-scale feature from large-scale outer intermittency. From temporal measurements of the fluctuations of the flame, we find a frequency spectrum that has a power-…
We investigate the dynamics of very large particles freely advected in a turbulent von Karman flow. Contrary to other experiments for which the particle dynamics is generally studied near the geometrical center of the flow, we track the particles in the whole experiment volume. We observe a strong influence of the mean structure of the flow that generates an unexpected large-scale sampling effect for the larger particles studied; contrary to neutrally buoyant particles of smaller yet finite sizes that exhibit no preferential concentration in homogeneous and isotropic turbulence (Fiabane et al.…
This study revisits the problem of advective transfer and spectra of a diffusive scalar field in large-scale incompressible flows in the presence of a (large-scale) source. By ``large-scale'' it is meant that the spectral support of the flows is confined to the wave-number region $k<k_d$, where $k_d$ is relatively small compared with the diffusion wave number $k_κ$. Such flows mediate couplings between neighbouring wave numbers within $k_d$ of each other only. It is found that the spectral rate of transport (flux) of scalar variance across a high wave number $k>k_d$ is bounded from above by $U…
Spatial regions that do not mix effectively with their surroundings and thus contribute less to the heat transport in fully turbulent three-dimensional Rayleigh-Bénard flows are identified by Lagrangian trajectories that stay together for a longer time. These trajectories probe Lagrangian coherent sets (CS) which we investigate here in direct numerical simulations in convection cells with square cross section of aspect ratio $Γ= 16$, Rayleigh number $Ra = 10^{5}$, and Prandtl numbers $Pr = 0.1, 0.7$ and $7$. The analysis is based on $N=524,288$ Lagrangian tracer particles which are advected in…
We consider the model of a transverse vector (e.g. magnetic) field with the most general form of the nonlinearity, known as the ${\cal A}$ model, passively advected by a strongly compressible turbulent flow, governed by the randomly stirred Navier-Stokes equation. The full stochastic problem is equivalent to a certain renormalizable field theoretic model with an infrared attractive fixed point. Thus, the scaling behaviour for the large-scale, long-distance behaviour is established. However, the question whether the parameter ${\cal A}$ tends to a certain fixed-point value of the renormalizatio…
The statistics of polymers advected by a turbulent flow are investigated. To limit the polymer lengths above to coil-stretch transition, a FENE-P type relaxation law is used. The turbulence is modeled by a random strain, delta-correlated in time and with Gaussian statistics. The distribution of polymer lengths for both the coiled and stretched states are derived, from which are obtained analytical expressions for the moments of the distribution. The polymer stress on the fluid decreases linearly with the inverse of the Deborah number. The degradation (breaking) of the polymers is also discusse…
We obtain the probability distribution functions (PDFs) of the time that a Lagrangian tracer or a heavy inertial particle spends in vortical or strain-dominated regions of a turbulent flow, by carrying out direct numerical simulation (DNS) of such particles advected by statistically steady, homogeneous and isotropic turbulence in the forced, three-dimensional, incompressible Navier-Stokes equation. We use the two invariants, $Q$ and $R$, of the velocity-gradient tensor to distinguish between vortical and strain-dominated regions of the flow and partition the $Q-R$ plane into four different reg…
We investigate experimentally the statistical properties of active and passive scalar fields in turbulent Rayleigh-Bénard convection in water, at $Ra\sim10^{10}$. Both the local concentration of fluorescence dye and the local temperature are measured near the sidewall of a rectangular cell. It is found that, although they are advected by the same turbulent flow, the two scalars distribute differently. This difference is twofold, i.e. both the quantities themselves and their small-scale increments have different distributions. Our results show that there is a certain buoyant scale based on time…
We develop an adversarial-reinforcement learning scheme for microswimmers in statistically homogeneous and isotropic turbulent fluid flows, in both two (2D) and three dimensions (3D). We show that this scheme allows microswimmers to find non-trivial paths, which enable them to reach a target on average in less time than a naive microswimmer, which tries, at any instant of time and at a given position in space, to swim in the direction of the target. We use pseudospectral direct numerical simulations (DNSs) of the 2D and 3D (incompressible) Navier-Stokes equations to obtain the turbulent flows.…
Large eddy simulations are performed to study tonal noise generation by a NACA0012 airfoil at an angle of attack ${α= 3}$ deg. and freestream Mach number of ${M_{\infty} = 0.3}$. Different Reynolds numbers are analyzed spanning ${0.5 \times 10^5 \le Re \le 4 \times 10^5}$. Results show that the flow patterns responsible for noise generation appear from different laminar separation bubbles, including one observed over the airfoil suction side and another near the trailing edge, on the pressure side. For lower Reynolds numbers, intermittent vortex dynamics on the suction side results in either c…
We present an investigation of the statistics of velocity gradient related quantities, in particluar energy dissipation rate and enstrophy, along the trajectories of fluid tracers and of heavy/light particles advected by a homogeneous and isotropic turbulent flow. The Refined Similarity Hypothesis (RSH) proposed by Kolmogorov and Oboukhov in 1962 is rephrased in the Lagrangian context and then tested along the particle trajectories. The study is performed on state-of-the-art numerical data resulting from numerical simulations up to Re~400 with 2048^3 collocation points. When particles have sma…
Direct numerical simulations (DNS) of particle-laden turbulent flow in straight, mildly curved and strongly bent pipes are performed in which the solid phase is modelled as small heavy spherical particles. A total of seven populations of dilute particles with different Stokes numbers, one-way coupled with their carrier phase, are simulated. The objective is to examine the effect of the curvature on micro-particle transport and accumulation. It is shown that even a slight non-zero curvature in the flow configuration strongly impact the particle concentration map such that the concentration of i…
We find actual evidence, relying upon vorticity time series taken in a high Reynolds number atmospheric experiment, that to a very good approximation the surface boundary layer flow may be described, in a statistical sense and under certain regimes, as an advected ensemble of homogeneous turbulent systems, characterized by a lognormal distribution of fluctuating intensities. Our analysis suggests that usual direct numerical simulations of homogeneous and isotropic turbulence, performed at moderate Reynolds numbers, may play an important role in the study of turbulent boundary layer flows, if s…
We explore the transport mechanisms of heat in two- and three-dimensional turbulent convection flows by means of the long-term evolution of Lagrangian coherent sets. They are obtained from the spectral clustering of trajectories of massless fluid tracers that are advected in the flow. Coherent sets result from trajectories that stay closely together under the dynamics of the turbulent flow. For longer times, they are always destroyed by the intrinsic turbulent dispersion of material transport. Here, this constraint is overcome by the application of evolutionary clustering algorithms that add a…
A theory of clustering of inertial particles advected by a turbulent velocity field caused by an instability of their spatial distribution is suggested. The reason for the clustering instability is a combined effect of the particles inertia and a finite correlation time of the velocity field. The crucial parameter for the clustering instability is a size of the particles. The critical size is estimated for a strong clustering (with a finite fraction of particles in clusters) associated with the growth of the mean absolute value of the particles number density and for a weak clustering associat…
We present numerical studies of stretching of Hookean dumbbells in a turbulent Navier-Stokes flow with a linear mean profile, <u_x>=Sy. In addition to the turbulence features beyond the viscous Kolmogorov scale η, the dynamics at the equilibrium extension of the dumbbells significantly below eta is well resolved. The variation of the constant shear rate S causes a change of the turbulent velocity fluctuations on all scales and thus of the intensity of local stretching rate of the advecting flow. The latter is measured by the maximum Lyapunov exponent lambda_1 which is found to increase as λ_1 …
We introduce a new method to measure Lagrangian vorticity and the rotational dynamics of anisotropic particles in a turbulent fluid flow. We use 3D printing technology to fabricate crosses (two perpendicular rods) and jacks (three mutually perpendicular rods). Time-resolved measurements of their orientation and solid-body rotation rate are obtained from stereoscopic video images of their motion in a turbulent flow between oscillating grids with $R_λ$=$91$. The advected particles have a largest dimension of 6 times the Kolmogorov length, making them a good approximation to anisotropic tracer pa…
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）
We study the spreading of viruses, such as SARS-CoV-2, by airborne aerosols, via a new first-passage-time problem for Lagrangian tracers that are advected by a turbulent flow: By direct numerical simulations of the three-dimensional (3D) incompressible, Navier-Stokes equation, we obtain the time $t_R$ at which a tracer, initially at the origin of a sphere of radius $R$, crosses the surface of the sphere \textit{for the first time}. We obtain the probability distribution function $\mathcal{P}(R,t_R)$ and show that it displays two qualitatively different behaviors: (a) for $R \ll L_{\rm I}$, $\m…
As a test bed for the growth of protoplanetary bodies in a turbulent circumstellar disk we examine the fate of a boulder using direct numerical simulations of particle seeded gas flowing around it. We provide an accurate description of the flow by imposing no-slip and non-penetrating boundary conditions on the boulder surface using the immersed boundary method pioneered by Peskin (2002). Advected by the turbulent disk flow, the dust grains collide with the boulder and we compute the probability density function (PDF) of the normal component of the collisional velocity. Through this examination…
Flow between concentric spheres of radius ratio $η= r_\mathrm{i}/r_\mathrm{o} = 0.35$ is studied in a 3 m outer diameter experiment. We have measured the torques required to maintain constant boundary speeds as well as localized wall shear stress, velocity, and pressure. At low Ekman number $E = 2.1\times10^{-7}$ and modest Rossby number $0.07 < Ro < 3.4$, the resulting flow is highly turbulent, with a Reynolds number ($Re=Ro/E$) exceeding fifteen million. Several turbulent flow regimes are evident as $Ro$ is varied for fixed $E$. We focus our attention on one flow transition in particular, be…
We have developed novel instrumentation for making Lagrangian measurements of temperature in diverse fluid flows. A small neutrally buoyant capsule is equipped with on-board electronics which measure temperature and transmit the data via a wireless radio frequency link to a desktop computer. The device has 80 dB dynamic range, resolving milli-Kelvin changes in temperature with up to 100 ms sampling time. The capabilities of these "smart particles" are demonstrated in turbulent thermal convection in water. We measure temperature variations as the particle is advected by the convective motion, a…
We show that the decay of a passive scalar $θ$ advected by a random incompressible flow with zero correlation time in Batchelor limit can be mapped exactly to a certain quantum-mechanical system with a finite number of degrees of freedom. The Schroedinger equation is derived and its solution is analyzed for the case when at the beginning the scalar has Gaussian statistics with correlation function of the form $e^{-|x-y|^2}$. Any equal-time correlation function of the scalar can be expressed via the solution to the Schroedinger equation in a closed algebraic form. We find that the scalar is int…
Identification and extraction of vortical structures and of waves in a disorganised flow is a mayor challenge in the study of turbulence. We present a study of the spatio-temporal behavior of turbulent flows in the presence of different restitutive forces. We show how to compute and analyse the spatio-temporal spectrum from data stemming from numerical simulations and from laboratory experiments. Four cases are considered: homogeneous and isotropic turbulence, rotating turbulence, stratified turbulence, and water wave turbulence. For homogeneous and isotropic turbulence, the spectrum allows id…
We provide closed form solutions for an equation which describes the transport of turbulent kinetic energy in the framework of a turbulence model with a single equation.
An exact relation is derived between scalar dissipation due to molecular diffusivity and the randomness of stochastic Lagrangian trajectories for flows without bounding walls. This "Lagrangian fluctuation-dissipation relation" equates the scalar dissipation for either passive or active scalars to the variance of scalar inputs associated to initial scalar values and internal scalar sources, as those are sampled backward in time by the stochastic Lagrangian trajectories. As an important application, we reconsider the phenomenon of "Lagrangian spontaneous stochasticity" or persistent non-determin…
The intermittency of a passive scalar advected by three-dimensional Navier-Stokes turbulence at a Taylor-scale Reynolds number of $650$ is studied using direct numerical simulations on a $4096^3$ grid; the Schmidt number is unity. By measuring scalar increment moments of high orders, while ensuring statistical convergence, we provide unambiguous evidence that the scaling exponents saturate to $1.2$ for moment order beyond about $12$, indicating that scalar intermittency is dominated by the most singular shock-like cliffs in the scalar field. We show that the fractal dimension of the spatial su…
Hydroacoustic Absorption and Amplification by Turbulence
Ningbo University（宁波大学）；Zhejiang Provincial Engineering Research Center for the Safety of Pressure Vessel and Pipeline, Ningbo University（宁波大学压力容器与管道安全浙江省工程研究中心）；Key Laboratory of Impact and Safety Engineering (Ningbo University), Ministry of Education（教育部冲击与安全工程重点实验室（宁波大学））；Ningbo Jiangbei District People's Government（宁波市江北区人民政府）
A Lagrangian fluctuation-dissipation relation has been derived in a previous work to describe the dissipation rate of advected scalars, both passive and active, in wall-bounded flows. We apply this relation here to develop a Lagrangian description of thermal dissipation in turbulent Rayleigh-Bénard convection in a right-cylindrical cell of arbitrary cross-section, with either imposed temperature difference or imposed heat-flux at the top and bottom walls. We obtain an exact relation between the steady-state thermal dissipation rate and the time for passive tracer particles released at the top …
University of Warwick（华威大学）；Indian Institute of Science（印度科学学院）；Institute For Plasma Research（等离子体研究所）
Inertial particles advected by a background flow can show complex structures. We consider inertial particles in a 2D Taylor-Green (TG) flow and characterize particle dynamics as a function of the particle's Stokes number using dynamic mode decomposition (DMD) method from particle image velocimetry (PIV) like-data. We observe the formation of caustic structures and analyze them using DMD to (a) determine the Stokes number of the particles, and (b) estimate the particle Stokes number composition. Our analysis in this idealized flow will provide useful insight to analyze inertial particles in mor…
We consider the effects of thermal noise on the Batchelor-Kraichnan theory of high Schmidt-number mixing in the viscous-dissipation range of turbulent flows. Using fluctuating hydrodynamics for a binary fluid mixture at low Mach numbers, we justify linearization around the deterministic Navier-Stokes solution in the dissipation range. For the latter solution we adopt the standard Kraichnan model and derive asymptotic high-Schmidt limiting equations for the concentration field, in which the thermal velocity fluctuations are exactly represented by a Gaussian random velocity which is white in tim…
We consider the two-dimensional (2D) flow in a flat free-slip surface that bounds a three-dimensional (3D) volume in which the flow is turbulent. The equations of motion for the two-dimensional flow in the surface are neither compressible nor incompressible but strongly influenced by the 3D flow underneath the surface. The velocity correlation functions in the 2D surface and in the 3D volume scale with the same exponents. In the viscous subrange the amplitudes are the same, but in the inertial subrange the 2D one is reduced to 2/3 of the 3D amplitude. The surface flow is more strongly intermit…
The effects of a large-scale shear on the energy spectrum of a passively advected scalar field are investigated. The shear is superimposed on a turbulent isotropic flow, yielding an Obukhov-Corrsin $k^{-5/3}$ scalar spectrum at small scales. Shear effects appear at large scales, where a different, anisotropic behavior is observed. The scalar spectrum is shown to behave as $k^{-4/3}$ for a shear fixed in intensity and direction. For other types of shear characteristics, the slope is generally intermediate between the -5/3 Obukhov-Corrsin's and the -1 Batchelor's values. The physical mechanisms …
