Numerical calculations of Helium-II hydrodynamics show that a dense tangle of superfluid vortices induces in an initially stationary normal fluid a highly dissipative, complex, vortical flow pattern ("turbulence") with a -2.2 energy spectrum scaling exponent and fluctuations Reynolds number of order unity. In this normal fluid flow the effects of mutual friction excitation from the superfluid vortices and those of viscous stresses are of the same order. The results suggest that in previous experiments the dynamics of decaying, high Reynolds number, quantum turbulent flows could only weakly be affected by the quantized vortices. As a consequence, their energy spectra would be (to a very good approximation) the classical, Navier-Stokes type, energy spectra of the normal fluid component.
This movie illustrates the recent numerical and experimental discovery of a new state of turbulence in dilute polymer solutions, Elasto-Inertial Turbulence. Elasto-inertial turbulence is characterized by a chaotic flow state in which both inertial and elastic effects, whose relative contributions vary with the Reynolds number, control the flow dynamics.
Using time-resolved measurements of the orientation of rigid inertial fibers in a turbulence-tank, we investigate the autocorrelation of their tumbling rate. The correlation time ($τ_d$) is well predicted by Kolmogorov inertial-range scaling based on the fiber length (L) when the fiber inertia can be neglected. For inertial fibers, we propose a simple model considering fiber inertia (measured by a tumbling Stokes number) and a viscous torque which accurately predicts both the correlation time and the variance of the tumbling rate.
This article accompanies the submission of a fluid dynamics video (entry 102248) of inertial cloud droplet clustering in isotropic turbulence for the Gallery of Fluid Motion of the 66th Annual Meeting of the American Physical Society, Division of Fluid Dynamics.
We study the rotational dynamics of {\it inertial} disks and rods in three-dimensional, homogeneous isotropic turbulence. In particular, we show how the alignment and the decorrelation time-scales of such spheroids depend, critically, on both the level of inertia and the aspect ratio of these particles. These results illustrate the effect of inertia---which leads to a preferential sampling of the local flow geometry---on the statistics of both disks and rods in a turbulent flow. Our results are important for a variety of natural and industrial settings where the turbulent transport of asymmetr…
Two dimensional channel flow simulations of FENE-P fluid in the elasto-inertial turbulence regime reveal distinct regimes ranging from chaos to a steady travelling wave which takes the form of an arrowhead structure. This coherent structure provides new insights in the polymer/flow interactions driving EIT, which are observed in a set of controlled numerical experiments and the study of transfer between elastic and turbulent kinetic energy.
We investigate numerically the dynamics and statistics of inertial particles transported by stratified turbulence, in the case of particle density intermediate in the average density profile of the fluid. In these conditions, particles tend to form a thin layer around the corresponding fluid isopycnal. The thickness of the resulting layer is determined by a balance between buoyancy (which attracts the particle to the isopycnal) and inertia (which prevents them from following it exactly). By means of extensive numerical simulations, we explore the parameter space of the system and we find that …
Elasto-inertial turbulence is a new state of turbulence found in flows with polymer additives . The dynamics of turbulence generated and controlled by such additives is investigated from the perspective of the coupling between polymer dynamics and flow structures. Direct numerical simulations of channel flow with Reynolds numbers ranging from 1000 to 6000 (based on the bulk and the channel height) are used to study the formation and dynamics of elastic instabilities and their effects on the flow. The resulting mechanism of interactions between polymer dynamics and the flow helps resolve a long…
Dynamic properties of elasto-inertial turbulence (EIT) are studied in a Taylor-Couette geometry. EIT is a chaotic flow state that develops upon both non-negligible inertia and viscoelasticity. A combination of direct flow visualisation and torque measurement allows to verify the earlier onset of EIT compared to purely inertial instabilities (and inertial turbulence). The scaling of the pseudo-Nusselt number with inertia and elasticity is discussed here for the first time. Variations in the friction coefficient, temporal frequency spectra, and spatial power density spectra highlight that EIT un…
Rotating turbulence is commonly known for being dominated by geostrophic vortices that are invariant along the rotation axis and undergo inverse cascade. Yet, it has recently been shown to sustain fully three-dimensional states with a downscale energy cascade. In this letter, we investigate the statistical properties of three-dimensional rotating turbulence by the means of direct numerical simulations in a triply periodic box where geostrophic vortices are specifically damped. The resulting turbulent flow is an inertial wave turbulence that verifies the Zakharov-Kolmogorov spectrum derived ana…
Turbulence generally arises in shear flows if velocities and hence inertial forces are sufficiently large. In striking contrast, viscoelastic fluids can exhibit disordered motion even at vanishing inertia. Intermediate between these cases, a novel state of chaotic motion, `elasto-inertial turbulence' (EIT), has been observed in a narrow Reynolds number interval. We here determine the origin of EIT in experiments and show that characteristic EIT structures can be detected across an unexpectedly wide range of parameters. Close to onset a pattern of chevron shaped streaks emerges in excellent agr…
In this paper, we present an experimental investigation of the turbulent saturation of the flow driven by parametric resonance of inertial waves in a rotating fluid. In our set-up, a half-meter wide ellipsoid filled with water is brought to solid body rotation, and then undergoes sustained harmonic modulation of its rotation rate. This triggers the exponential growth of a pair of inertial waves via a mechanism called the libration-driven elliptical instability. Once the saturation of this instability is reached, we observe a turbulent state for which energy is injected into the resonant inerti…
In this two--part study, we present the development and analysis of a stochastic theory for characterizing the relative positions of monodisperse, low-inertia particle pairs that are settling rapidly in homogeneous isotropic turbulence. In the limits of small Stokes number and Froude number such that $Fr \ll St_η\ll 1$, closures are developed for the drift and diffusion fluxes in the probability density function (PDF) equation for the pair relative positions. The theory focuses on the relative motion of particle pairs in the dissipation regime of turbulence, i.e., for pair separations smaller …
In this paper we investigate both theoretically and numerically the forward in time (FIT) and backward in time (BIT) dispersion of fluid and inertial particle pairs in isotropic turbulence. Fluid particles are known to separate faster BIT than FIT in three-dimensional turbulence, and we find that inertial particles do the same. However, we find that the irreversibility in the inertial particle dispersion is in general much stronger than that for fluid particles. For example, the ratio of the BIT to FIT mean-square separation can be up to an order of magnitude larger for inertial particles than…
Turbulence is ubiquitous in nature yet even for the case of ordinary Newtonian fluids like water our understanding of this phenomenon is limited. Many liquids of practical importance however are more complicated (e.g. blood, polymer melts or paints), they exhibit elastic as well as viscous characteristics and the relation between stress and strain is nonlinear. We here demonstrate for a model system of such complex fluids that at high shear rates turbulence is not simply modified as previously believed but it is suppressed and replaced by a new type of disordered motion, elasto-inertial turbul…
Recently, the nature of viscoelastic drag-reducing turbulence (DRT), especially maximum drag reduction (MDR) state, has become a focus of controversy. It has long been regarded as polymers-modulated inertial turbulence (IT), but is challenged by the newly proposed concept of elasto-inertial turbulence (EIT). This study is to re-picture DRT in parallel plane channels by introducing dynamics of EIT based on statistical and budget analysis for a series of flow regimes from the onset of DR to EIT. Energy conversion between velocity fluctuations and polymers as well as polymeric pressure redistribu…
We report on laboratory experiments of wave-driven rotating turbulence. A set of wavemakers produces inertial-wave beams that interact nonlinearly in the central region of a water tank mounted on a rotating platform. The forcing thus injects energy into inertial waves only. For moderate forcing amplitude, part of the energy of the forced inertial waves is transferred to subharmonic waves, through a standard triadic resonance instability. This first step is broadly in line with the theory of weak turbulence. Surprisingly however, stronger forcing does not lead to an inertial-wave turbulence reg…
We study the transition from fluid at rest to turbulence in a rotating water cylinder. We show that the energy, injected at a given height, is transported by inertial wave packets through the fluid volume. These waves propagate at velocities consistent with those calculated from linearized theory, even when they possess large amplitudes. A clear "front" in the temporal evolution of the energy power spectrum is detected, defining a time scale for energy transport at the linear wave speed in the system. Nonlinear energy transfer between modes is governed by a different time scale that can be muc…
The goal of the present study is: (i) to demonstrate the two-dimensional nature of the elasto-inertial instability in elasto-inertial turbulence (EIT), (ii) to identify the role of the bi-dimensional instability in three-dimensional EIT flows and (iii) to establish the role of the small elastic scales in the mechanism of self-sustained EIT. Direct numerical simulations of FENE-P fluid flows are performed in two- and three-dimensional channels. The Reynolds number is set to $\mathrm{Re}_τ= 85$ which is sub-critical for 2D flows but beyond transition for 3D ones. The polymer properties correspon…
The effect of turbulence on the mass and heat transfer between small heavy inertial particles (HIP) and an embedding fluid is studied. Two effects are identified. The first effect is due to the relative velocity between the fluid and the particles, and a model for the relative velocity is presented. The second effect is due to the clustering of particles, where the mass transfer rate is inhibited due to the rapid depletion of the consumed species inside the dense particle clusters. This last effect is relevant for large Damkohler numbers and it may totally control the mass transfer rate for Da…
Two-way coupled direct numerical simulations are used to investigate the effects of inertial particles on self-sustained, turbulent coherent structures (i.e. the so-called the regeneration cycle) in plane Couette flow at low Reynolds number just above the onset of transition. Tests show two limiting behaviors with increasing particle inertia, similar to the results from the linear stability analysis of saffman, 1962: low-inertia particles trigger the laminar-to-turbulent instability whereas high-inertia particles tend to stabilize turbulence due to the extra dissipation induced by particle-flu…
The fate of small particles in turbulent flows depends strongly on the surrounding fluid's velocity gradient properties such as rotation and strain-rates. For non-inertial (fluid) particles, the Restricted Euler model provides a simple, low-dimensional dynamical system representation of Lagrangian evolution of velocity gradients in fluid turbulence, at least for short times. Here we derive a new restricted Euler dynamical system for the velocity gradient evolution of inertial particles such as solid particles in a gas or droplets and bubbles in turbulent liquid flows. The model is derived in t…
Previous studies on nonspherical particle-fluid interaction were mostly confined to elongated fiber-like particles, which were observed to induce turbulence drag reduction. However, with the presence of tiny disk-like particles how wall turbulence is modulated and whether drag reduction occurs are still unknown. Motivated by those open questions, we performed two-way coupled direct numerical simulations of inertialess spheroids in turbulent channel flow by an Eulerian-Lagrangian approach. The additional stress accounts for the feedback from inertialess spheroids on the fluid phase. The results…
We use a coarse-graining approach to prove that inter-scale transfer of kinetic energy in compressible turbulence is dominated by local interactions. Locality here means that interactions between disparate scales decay at least as fast as a power-law function of the scale-disparity ratio. In particular, our results preclude transfer of kinetic energy from large-scales directly to dissipation scales, such as into shocks, in the limit of high Reynolds number turbulence as is commonly believed. The results hold in broad generality, at any Mach number, for any equation of state, and without the re…
In our recent works we proposed a theory of turbulence in inertial gas flow via the mean field effect of an intermolecular potential. We found that, in inertial flow, turbulence indeed spontaneously develops from a laminar initial condition, just as observed in nature and experiments. However, we also found that density and temperature in our inertial flow model behave unrealistically. The goal of the current work is to demonstrate technical possibility of modeling compressible, turbulent flow at low Mach number where both density and temperature behave in a more realistic fashion. Here we foc…
Ice crystals settling through a turbulent cloud are rotated by turbulent velocity gradients. In the same way, turbulence affects the orientation of aggregates of organic matter settling in the ocean. In fact most solid particles encountered in Nature are not spherical, and their orientation affects their settling speed, as well as collision rates between particles. Therefore it is important to understand the distribution of orientations of non-spherical particles settling in turbulence. Here we study the angular dynamics of small prolate spheroids settling in homogeneous isotropic turbulence. …
In this paper we present a novel hydrodynamic experiment using liquid $^4$He. The flow is forced inertially by a canonical oscillating grid using either its normal (He~I) or superfluid (He~II) phase, generating a statistically stationary turbulence. We characterise the turbulent properties of the flow using 2D Lagrangian Particle tracking on hollow glass micro-spheres. As expected for tracer particles, the Voronoï tessellation on particle positions does not show a significant departure from a random Poisson process neither in He~I nor He~II phase. Particles' positions are tracked with high tem…
In this paper, using Pao's conjecture [Y.-H. Pao, Phys. Fluids 11, 1371 (1968)], we derive expressions for the spectra and fluxes of kinetic energy and enstrophy for two-dimensional (2D) forced turbulence that extend beyond the inertial range. In these expressions, the fluxes and the spectra contain additional factors of the exponential form. To validate these model predictions, we perform numerical simulations of 2D turbulence with an external force applied at $k=k_f$ in the intermediate range. The numerical results match with the model predictions, except for the energy and enstrophy fluxes …
I shortly describe the main results on elastically driven instabilities and elastic turbulence in viscoelastic inertia-less flows with curved streamlines. Then I describe a theory of elastic turbulence and prediction of elastic waves at Re<<1 and Wi>>1, which speed depends on the elastic stress similarly to the Alfven waves in magneto-hydrodynamics and in in a contrast to all other fluid flows with wave speed depending on medium elasticity. Since the established and testified mechanism of elastic instability of viscoelastic flows with curvilinear streamlines becomes ineffective at zero curvatu…
Suspensions of active agents with nematic interactions exhibit complex spatio-temporal dynamics such as mesoscale turbulence. Since the Reynolds number of microscopic flows is very small on the scale of individual agents, inertial effects are typically excluded in continuum theories of active nematic turbulence. Whether active stresses can collectively excite inertial flows is currently unclear. To address this question, we investigate a two-dimensional continuum theory for active nematic turbulence. In particular, we compare mesoscale turbulence with and without the effects of advective inert…
An analytic model for steady state turbulence is employed to obtain the inertial range power spectrum of compressible turbulence. We assume that for homogeneous turbulence, the timescales controlling the energy injected at a given wavenumber from all smaller wave-numbers, are equal for each spatial component. However, the longitudinal component energy is diverted into compression, so the rate controlling the energy that is transferred to all larger wave-numbers by the turbulent viscosity is reduced. The resulting inertial range is a power law with index -2. Indeed such power spectra were obser…
Using Direct Numerical Simulations (DNS), we examine the effects of Taylor Reynolds number, $R_λ$, and Froude number, $Fr$, on the motion of settling, bidisperse inertial particles in isotropic turbulence. Particle accelerations play a key role in the relative motion of bidisperse particles, and we find that reducing $Fr$ leads to an enhancement of the accelerations, but a suppression of their intermittency. For Stokes numbers $St>1$, the effect of $R_λ$ on the accelerations is enhanced by gravity, since settling causes the particle accelerations to be affected by a larger range of flow scales…
Effects of quasi-point vortices on the inertial range of scales in homogeneous two-dimensional turbulence (classic and quantum) have been studied using the notion of distributed chaos. Results of direct numerical simulations of decaying turbulence, turbulence with small-scale forcing, and turbulent thermal convection on a sphere as well as results of the Global Atmospheric Sampling Program (GASP) measurements in the tropospheric and stratospheric turbulence over mountainous terrain (the small-scale forcing) have been used for this purpose. The superfluid and Bose-Einstein two-dimensional turbu…
We study the transition from laminar flow to fully developed turbulence for an inertially-driven von Karman flow between two counter-rotating large impellers fitted with curved blades over a wide range of Reynolds number (100 - 1 000 000). The transition is driven by the destabilisation of the azimuthal shear-layer, i.e., Kelvin-Helmholtz instability which exhibits travelling/drifting waves, modulated travelling waves and chaos below the emergence of a turbulent spectrum. A local quantity -the energy of the velocity fluctuations at a given point- and a global quantity -the applied torque- are …
Turbophoresis leading to preferential concentration of inertial particles in regions of low turbulent diffusivity is a unique feature of inhomogeneous turbulent flows, such as free shear flows or wall-bounded flows. In this work, the theory for clustering of weakly inertial particles in homogeneous turbulence is extended to the inhomogeneous case of a turbulent channel flow. The inhomogeneity contributes to the cluster formation in addition to clustering in homogeneous turbulence. A space-dependent rate for the creation of inhomogeneous particle concentration is derived in terms of local stati…
We study experimentally the spatial distribution, settling, and interaction of sub-Kolmogorov inertial particles with homogeneous turbulence. Utilizing a zero-mean-flow air turbulence chamber, we drop size-selected solid particles and study their dynamics with particle imaging and tracking velocimetry at multiple resolutions. The carrier flow is simultaneously measured by particle image velocimetry of suspended tracers, allowing the characterization of the interplay between both the dispersed and continuous phases. The turbulence Reynolds number based on the Taylor microscale ranges from $Re_λ…
It is well known that the reversibility of Stokes flow makes it difficult for small microorganisms to swim. Inertial effects break this reversibility, allowing new mechanisms of propulsion and feeding. Therefore it is important to understand the effects of unsteady and fluid inertia on the dynamics of microorganisms in flow. In this work, we show how to translate known inertial effects for non-motile organisms to motile ones, from passive to active particles. The method relies on a principle used earlier by Legendre and Magnaudet (1997) to deduce inertial corrections to the lift force on a bub…
The purpose of the present work is to trace parallels between the known inertia forces in fluid dynamics with the inertia forces in electromagnetism that are known to induce resistance forces on masses both due to acceleration and at constant velocity. It is shown that the force exerted on a particle by an ideal fluid produces two effects: i) resistance to acceleration and, ii) an increase of mass with velocity. These resistance forces arise due to the fluid dragged by the particle, where the bare mass of the particle at rest changes when in motion ("dressed" particle). It is demonstrated that…
Direct numerical simulations of homogeneous decaying turbulence with mild background rotation show the existence of a systematic and significant imbalance between the non-linear energy cascade to small scales and its dissipation. By starting the decay from a statistically stationary and fully developed rotating turbulence state, where the dissipation and the energy flux are approximately equal, the data shows a growing imbalance between the two until a maximum is reached when the dissipation is about twice the energy flux. This dichotomy of behaviours during decay is reminiscent of the non-equ…
Direct numerical simulations of the forced Navier-Stokes equations were performed, in which each shell-averaged quantity evolved from a value appropriate to an initial Gaussian state, to fluctuate about a mean value. Once the transient had passed, mean values (and their associated statistics) were obtained by sampling the evolved time-series at intervals of the order of an eddy-turnover time. This was repeated for a range of Taylor-Reynolds numbers from 10.6 to 335.2. With increasing Reynolds number, our results for energy spectra, transfer spectra and inertial flux supported the Kolmogorov-Ob…
Multiscale statistical analyses of inertial particle distributions are presented to investigate the statistical signature of clustering and void regions in particle-laden incompressible isotropic turbulence. Three-dimensional direct numerical simulations of homogeneous isotropic turbulence at high Reynolds number ($Re_λ\gtrsim 200$) with up to $10^9$ inertial particles are performed for Stokes numbers ranging from $0.05$ to $5.0$. Orthogonal wavelet analysis is then applied to the computed particle number density fields. Scale-dependent skewness and flatness values of the particle number densi…
Turbulent flows preferentially concentrate inertial particles depending on their stopping time or Stokes number, which can lead to significant spatial variations in the particle concentration. Cascade models are one way to describe this process in statistical terms. Here, we use a direct numerical simulation (DNS) dataset of homogeneous, isotropic turbulence to determine probability distribution functions (PDFs) for cascade multipliers, which determine the ratio by which a property is partitioned into sub-volumes as an eddy is envisioned to decay into smaller eddies. We present a technique for…
Inertial particle data from three-dimensional direct numerical simulations of particle-laden homogeneous isotropic turbulence at high Reynolds number are analyzed using Voronoi tessellation of the particle positions, considering different Stokes numbers. A finite-time measure to quantify the divergence of the particle velocity by determining the volume change rate of the Voronoi cells is proposed. For inertial particles the probability distribution function (PDF) of the divergence deviates from that for fluid particles. Joint PDFs of the divergence and the Voronoi volume illustrate that the di…
In particle-laden turbulent flows the turbulence in carrier fluid phase gets affected by the dispersed particle phase for volume fraction above $10^{-4}$ and hence reverse coupling or two-way coupling becomes relevant in that volume fraction regime. In a recent study by Muramulla $et.al.^1$, a discontinuous decrease of turbulence intensity is observed in a vertical particle-laden turbulent channel-flow for a critical volume fraction O($10^{-3}$). The collapse of turbulent intensity is found out to be a result of catastrophic reduction of turbulent energy production rate. Mechanistically, parti…
Fluids with internal microstructure like dense suspensions, biological and polymer added fluids, are commonly found in the turbulent regime in many applications. Their flow is extremely difficult to be studied as microstructure complexity and Reynolds number increase, even nowadays. This bottleneck is novelty and efficiently treated here by the micropolar theory. Our findings support that when denser microstructure occurs turbulence is intensified by the chaotic rotation of internal fluid's elements. Unlike in Newtonian fluids, shear stress is now decreased, and viscous and microstructure inte…
For low-Reynolds number shear-flows of neutrally-buoyant suspensions, the shear stress is often modeled using an effective viscosity that depends only on the solid fraction. As the Reynolds number ($Re$) is increased and inertia becomes important, the effective viscosity also depends on the Reynolds number itself. The current experiments measure the torque for flows of neutrally-buoyant particles in a coaxial-cylinder rheometer for solid fractions, $ϕ$, from 10\% to 50\% and Reynolds numbers based on particle diameter from 2 to 1,000. For experiments for Reynolds of O(10) and solid fractions l…
Particles with density different from that of the advecting turbulent fluid cluster due to the different response of light/heavy particles to turbulent fluctuations. This study focuses on the quantitative characterization of the segregation of dilute poly-disperse inertial particles evolving in turbulent flow, as obtained from Direct Numerical Simulation of the Navier-Stokes equations. We introduce an indicator of segregation amongst particles of different inertia and/or size, from which a length scale r_{seg} quantifying the degree of segregation of two different types of particles is deduced…
