Inertial Turbulence Research Articles

Scaling Transition of Active Turbulence from Two to Three Dimensions.

Turbulent flows are observed in low-Reynolds active fluids, which display similar phenomenology to the classical inertial turbulence but are of a different nature. Understanding the dependence of this new type of turbulence on dimensionality is a fundamental challenge in non-equilibrium physics. Real-space structures and kinetic energy spectra of bacterial turbulence are experimentally measured from two to three dimensions. The turbulence shows three regimes separated by two critical confinement heights, resulting from the competition of bacterial length, vortex size and confinement height. Meanwhile, the kinetic energy spectra display distinct universal scaling laws in quasi-2D and 3D regimes, independent of bacterial activity, length, and confinement height, whereas scaling exponents transition in two steps around the critical heights. The scaling behaviors are well captured by the hydrodynamic model wedevelop, which employs image systems to represent the effects of confining boundaries. The study suggests a framework for investigating the effect of dimensionality on non-equilibrium self-organizedsystems.

Harnessing elastic instabilities for enhanced mixing and reaction kinetics in porous media

Turbulent flows have been used for millennia to mix solutes; a familiar example is stirring cream into coffee. However, many energy, environmental, and industrial processes rely on the mixing of solutes in porous media where confinement suppresses inertial turbulence. As a result, mixing is drastically hindered, requiring fluid to permeate long distances for appreciable mixing and introducing additional steps to drive mixing that can be expensive and environmentally harmful. Here, we demonstrate that this limitation can be overcome just by adding dilute amounts of flexible polymers to the fluid. Flow-driven stretching of the polymers generates an elastic instability, driving turbulent-like chaotic flow fluctuations, despite the pore-scale confinement that prohibits typical inertial turbulence. Using in situ imaging, we show that these fluctuations stretch and fold the fluid within the pores along thin layers ("lamellae") characterized by sharp solute concentration gradients, driving mixing by diffusion in the pores. This process results in a [Formula: see text] reduction in the required mixing length, a [Formula: see text] increase in solute transverse dispersivity, and can be harnessed to increase the rate at which chemical compounds react by [Formula: see text]-enhancements that we rationalize using turbulence-inspired modeling of the underlying transport processes. Our work thereby establishes a simple, robust, versatile, and predictive way to mix solutes in porous media, with potential applications ranging from large-scale chemical production to environmental remediation.

Influence of atmospheric wind on the propagation of radio waves

Background. It is necessary to study the influence of the physical characteristics of the atmosphere, in particular, wind on atmospheric turbulence and, consequently, on the characteristics of the radio signal it is shown. Aim. The dependence of the time-spectral function of the radio signal energy flux on the wind speed in the troposphere in the antenna plane is found. Methods. A method of transition from the Cartesian coordinate system in the antenna plane to the polar coordinate system of wave numbers has been developed. Based on this method the relationship between the Fourier spectral function of the correlation moment and the representation of the Bessel function is found. For the Fourier spectral function of the correlation moment, the previously obtained solution of the differential equation for the fluctuations of the eikonal amplitude of electromagnetic wave in a turbulent atmosphere at the front of the electromagnetic wave at the coordinate of the receiving antenna is used. Using the inverse Fourier transform the relationship between the time-spectral function of the radio signal energy flux and the time-correlation function of this flux is found. Results. Based on the study of the time-correlation function of the radio signal energy flow its relationship with the two-point correlation moment characterizing the fluctuations of the eikonal amplitude of the radio signal is found. To analyze the effect of wind, a turbulence model was used, reflecting the inertial turbulence interval, in which the energy flow from larger turbulent vortices to smaller vortices is determined by the viscous dissipation of the smallest vortices. Conclusion. Numerical calculations have shown that such a wind in the antenna plane blows away turbulent vortices in this plane, bettering the quality of the received radio signal.

Nonequilibrium hyperuniform states in active turbulence

We demonstrate that the complex spatiotemporal structure in active fluids can feature characteristics of hyperuniformity. Using a hydrodynamic model, we show that the transition from hyperuniformity to nonhyperuniformity and antihyperuniformity depends on the strength of active forcing and can be related to features of active turbulence without and with scaling characteristics of inertial turbulence. Combined with identified signatures of Levy walks and nonuniversal diffusion in these systems, this allows for a biological interpretation and the speculation of nonequilibrium hyperuniform states in active fluids as optimal states with respect to robustness and strategies of evasion and foraging.

Pattern selection and the route to turbulence in incompressible polar active fluids

Active fluids, such as suspensions of microswimmers, are well known to self-organize into complex spatio-temporal flow patterns. An intriguing example is mesoscale turbulence, a state of dynamic vortex structures exhibiting a characteristic length scale. Here, we employ a minimal model for the effective microswimmer velocity field to explore how the turbulent state develops from regular, stationary vortex patterns when the strength of activity resp. related parameters such as nonlinear advection or polar alignment strength—is increased. First, we demonstrate analytically that the system, without any spatial constraints, develops a stationary square vortex lattice in the absence of nonlinear advection. Subsequently, we perform an extended stability analysis of this nonuniform ‘ground state’ and uncover a linear instability, which follows from the mutual excitement and simultaneous growth of multiple perturbative modes. This extended analysis is based on linearization around an approximation of the analytical vortex lattice solution and allows us to calculate a critical advection or alignment strength, above which the square vortex lattice becomes unstable. Above these critical values, the vortex lattice develops into mesoscale turbulence in numerical simulations. Utilizing the numerical approach, we uncover an extended region of hysteresis where both patterns are possible depending on the initial condition. Here, we find that turbulence persists below the instability of the vortex lattice. We further determine the stability of square vortex patterns as a function of their wavenumber and represent the results analogous to the well-known Busse balloons known from classical pattern-forming systems such as Rayleigh–Bénard convection experiments and corresponding models such as the Swift–Hohenberg equation. Here, the region of stable periodic patterns shrinks and eventually disappears with increasing activity parameters. Our results show that the strength of activity plays a similar role for active turbulence as the Reynolds number does in driven flow exhibiting inertial turbulence.

Universal relative scaling of longitudinal structure functions in shear-dominated turbulence

Shear significantly influences turbulence in the energy-containing range of shear-dominated flows, and the longitudinal structure functions do not have a universal form as they do in homogeneous isotropic turbulence. Despite this, the relative scaling of structure functions exhibits universal sub-Gaussian behaviour in shear-dominated flows, in particular for turbulent boundary layers, channels and Taylor–Couette flows. Our investigation of a turbulent vertical buoyancy layer at $Pr = 0.71$ using direct numerical simulation shows this universality even in moderate-Reynolds-number buoyancy-driven but shear-dominated boundary layers. It is demonstrated that the universality is related to the energy density of the eddies, which attains a hierarchical equilibrium in the energy-containing range of shear-dominated turbulence. We conjecture that the universal sub-Gaussian behaviour of the energy density of the energy-containing range, which was considered to be non-trivial in prior studies, is related to the universal anomalous scaling exponents of the inertial subrange turbulence. Based on this conjecture, we propose a hypothesis that relates large-scale eddies and the intermittent dissipation field in shear-dominated turbulence, highlighting a relationship between large and small scales. A phenomenological model is also developed to predict the scaling, which is verified using data from a turbulent boundary layer, half-channel and vertical buoyancy layer at friction Reynolds numbers spanning four orders of magnitude. Excellent agreement is observed.

Turbulence Unsteadiness Drives Extreme Clustering.

We show that the unsteadiness of turbulence has a drastic effect on turbulence parameters and in particle cluster formation. To this end we use direct numerical simulations of particle laden flows with a steady forcing that generates an unsteady large-scale flow. Particle clustering correlates with the instantaneous Taylor-based flow Reynolds number, and anticorrelates with its instantaneous turbulent energy dissipation constant. A dimensional argument for these correlations is presented. In natural flows, unsteadiness can result in extreme particle clustering, which is stronger than the clustering expected from averaged inertial turbulence effects.

The influence of turbulence and inertia in radial fracture flow

Industrial applications of flow through fractures such as geothermal energy or hydraulic stimulation involve forcing large flow rates through small fractures, thereby inducing inertial fluid behaviours and turbulence. The most common fracture flow model, Poiseuille flow (the cubic law), is incapable of capturing these phenomena and thus the impact of inertial and turbulent forces in fracture flow has remained relatively unexplored. The GG22 flow model is a newly derived fracture flow model that is capable of capturing inertial, transient and turbulent forces. In this article, we apply the GG22 flow model to hydraulic stimulation of radial fractures for the first time to determine how these phenomena manifest. We show that inertia and turbulence only manifest near the wellbore (within 30 radii) and lead to changes in fracture shape and injection pressure but have little impact on tip behaviour. Turbulence increases wellbore pressure and aperture while inertia decreases wellbore pressure and aperture. The majority of the pressure loss along the fracture occurs near the wellbore and is captured by turbulence where entrance correction factors would otherwise be needed. Using water, turbulence is the dominant mechanism that causes departures from Poiseuille flow at high $Re$ . The solution departs immediately upon the manifestation of turbulence ( $Re\geq 2\times 10^3$ ), while inertial effects manifest at higher flow rates ( $Re\geq 2\times 10^4$ ). Using slickwater, the opposite trend is observed: inertial effects manifest first ( $Re\geq 5\times 10^3$ ), while turbulent effects are delayed ( $Re\geq 10^4$ ). In both cases, the threshold for departures from the Poiseuille flow solution are low and the differences are large.

Persistence in active turbulence.

Active fluids such as bacterial swarms, self-propelled colloids, and cell tissues can all display complex spatiotemporal vortices that are reminiscent of inertial turbulence. This emergent behavior, despite the overdamped nature of these systems, is the hallmark of active turbulence. In this Letter, using a generalized hydrodynamic model, we present a study of the persistence problem in active turbulence. We report that the persistence time of passive tracers inside the coherent vortices follows a Weibull probability density whose shape and scale are decided by the strength of activity-contrary to inertial turbulence that displays power-law statistics in this region. In the turbulent background, the persistence time is exponentially distributed that is remindful of inertial turbulence. Finally we show that the driver of persistence inside the coherent vortices is the temporal decorrelation of the topological field, whereas it is the vortex turnover time in the turbulent background.

Evaluating the performance of turbulence closure models for tidal stream resource characterization

Harvesting tidal stream energy from the ocean for electricity generation has been considered as an energy resource alternative to fossil fuels for mitigating the negative impact of climate change and enhancing energy security and coastal resilience. Numerical models have been used extensively to characterize and assess tidal resources at potential tidal energy development sites. Turbulence plays an important role in site selection and tidal turbine farm deployment. However, most of the numerical modeling studies for tidal energy resource characterization do not include turbulence characteristics because of the limitation of Reynolds averaged Navier–Stokes coastal ocean model in resolving the inertial sub-range turbulence scales. However, studies also demonstrated that turbulence properties, such as turbulence intensity and turbulence kinetic energy, simulated by the coastal ocean models based on turbulence closures can be useful in assisting tidal resource characterization at tidal energy sites. In this study, we evaluated four General Ocean Turbulence Model (GOTM) turbulence closure models implemented in a tidal hydrodynamic model –Finite Volume Community Ocean Model (FVCOM) to characterize the tidal energy resource in the Western Passage, Maine, USA – a top ranked tidal energy site. The four turbulence closure models used in this analysis are k–kl (Mellor–Yamada Level 2.5), k-ε, k–ω, and a generic model. Model sensitivities showed that the MY2.5 model performed the best in comparison with the field measurements. In particular, the simulated time series of turbulence intensity and kinetic energy matched the observed data very well in the Western Passage. Detailed analysis was conducted to characterize turbulence properties on the horizontal plane and at selective cross-sections. This study demonstrated that turbulence properties simulated by a coastal ocean model, along with tidal hydrodynamic properties, can be very informative for tidal energy resource characterization and project site selection, as recommended by the International Electrotechnical Commission (IEC) technical specifications (IEC TS 62600-201).

Maximum drag reduction state of viscoelastic turbulent channel flow: marginal inertial turbulence or elasto-inertial turbulence

The essence of the maximum drag reduction (MDR) state of viscoelastic drag-reducing turbulence (DRT) is still under debate, which mainly holds two different types of views: the marginal state of inertial turbulence (IT) and elasto-inertial turbulence (EIT). To further promote its understanding, this paper conducts a large number of direct numerical simulations of DRT at a modest Reynolds numberRewith$Re = 6000$for the FENE-P model that covers a wide range of flow states and focuses on the problem of how nonlinear extension affects the nature of MDR by varying the maximum extension length$L$of polymers. It demonstrates that the essence of the MDR state can be both IT and EIT, where$L$is somehow an important parameter in determining the dominant dynamics. Moreover, there exists a critical$L_c$under which the minimum flow drag can be achieved in the MDR state even exceeding the suggested MDR limit. Systematic analyses on the statistical properties, energy spectrum, characteristic structures and underly dynamics show that the dominant dynamics of the MDR state gradually shift from IT-related to EIT-related dynamics with an increase of$L$. The above effects can be explained by the effective elasticity introduced by different$L$at a fixed Weissenberg number (Wi) as well as the excitation of pure EIT. It indicates that larger$L$introduces more effective elasticity and is favourable to EIT excitation. Therefore, we argue that the MDR state is still dominated by IT-related dynamics for the case of small$L$, but replaced by EIT-related dynamics at high$L$. The obtained results can harmonize the seemingly controversial viewpoints on the dominant dynamics of the MDR state and also provide some ideas for breaking through the MDR limit, such as searching for a polymer solution with a proper molecular length and concentration.

Friction dynamics of elasto-inertial turbulence in Taylor-Couette flow of viscoelastic fluids.

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 visualization and torque measurement allows to verify the earlier onset of EIT compared with 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 undergoes an intermediate behaviour before transitioning to its fully developed chaotic state that requires both high inertia and elasticity. During this transition, the contribution of secondary flows to the overall friction dynamics is limited. This is expected to be of great interest in the aim of achieving efficiency mixing at low drag and low but finite Reynolds number. This article is part of the theme issue "Taylor-Couette and related flows on the centennial of Taylor's seminal Philosophical transactions paper (Part 2)".

Intermittency, fluctuations and maximal chaos in an emergent universal state of active turbulence

The phenomenon of active turbulence, a complex organization of matter driven at the scale of its constituent agents, is puzzling. Specifically, the lack of scale-separation in low-Reynolds-number active flows breaks away from the familiar notions of the energy cascade and approximate scale-invariance of inertial turbulence. Here, using a generalized hydrodynamic model developed for bacterial turbulence, we provide compelling analytical and numerical evidence that, beyond a critical drive, active turbulence indeed attains universality akin to inertial turbulence. In this asymptotic state, the energy spectrum scales as k−3/2, reminiscent of some classes of inertial turbulence. The flow also exhibits spatio-temporal intermittency beyond the transition, as seen from non-Gaussian fluctuations in velocity differences. With these tell-tale fingerprints, active turbulence is placed closer in phenomenology to inertial turbulence than previously held. We show, however, that as a consequence of a finite range of scales, the degree of chaoticity and hence mixing efficiency saturates to a maximum in the asymptotic regime, unlike unbounded chaos in inertial turbulence. We conclude that active turbulence, depending on the level of drive, can switch between fundamentally distinct non-universal and universal states. Active fluids exhibit regimes with a complex spatio-temporal structure reminiscent of inertial turbulence. Now, inertial and active turbulence are theoretically shown to be closely related indeed.

Dynamic mode structure of active turbulence

The collective motion of dense suspensions of swimming bacteria is typical of a broad class of active materials, which serve an array of important biological and ecological functions. This work combines microfluidic experiments with modal analysis, typically reserved for inertial turbulence, to quantify the active turbulence of bacterial suspensions. Our results unveil the underlying constituent flow structures responsible for the interactions of chaotic bacterial motion with solid boundaries and external flows, and establish an analysis framework to facilitate new experimental and modeling approaches in active matter systems.

Experimental insights into elasto-inertial transitions in Taylor-Couette flows.

Since the seminal work of Taylor in 1923, Taylor-Couette (TC) flow has served as a paradigm to study hydrodynamic instabilities and bifurcation phenomena. Transitions of Newtonian TC flows to inertial turbulence have been extensively studied and are well understood, while in the past few years, there has been an increasing interest in TC flows of complex, viscoelastic fluids. The transitions to elastic turbulence (ET) or elasto-inertial turbulence (EIT) have revealed fascinating dynamics and flow states; depending on the rheological properties of the fluids, a broad spectrum of transitions has been reported, including rotating standing waves, flame patterns (FP), and diwhirls (DW). The nature of these transitions and the relationship between ET and EIT are not fully understood. In this review, we discuss experimental efforts on TC flows of viscoelastic fluids. We outline the experimental methods employed and the non-dimensional parameters of interest, followed by an overview of inertia, elasticity and elasto-inertia-driven transitions to turbulence and their modulation through shear thinning or particle suspensions. The published experimental data are collated, and a map of flow transitions to EIT as a function of the key fluid parameters is provided, alongside perspectives for the future work. This article is part of the theme issue 'Taylor-Couette and related flows on the centennial of Taylor's seminal Philosophical Transactions paper (part 1)'.

Coexistence of Active and Hydrodynamic Turbulence in Two-Dimensional Active Nematics.

In active nematic liquid crystals, activity is able to drive chaotic spatiotemporal flows referred to as active turbulence. Active turbulence has been characterized through theoretical and experimental work as a low Reynolds number phenomenon. We show that, in two dimensions, the active forcing alone is able to trigger hydrodynamic turbulence leading to the coexistence of active and inertial turbulence. This type of flow develops for sufficiently active and extensile flow-aligning nematics. We observe that the combined effect of an extensile nematic and large values of the flow-aligning parameter leads to a broadening of the elastic energy spectrum that promotes a growth of kinetic energy able to trigger an inverse energy cascade.

Enhanced Diapycnal Mixing in the Deep Ocean Around the Island of Taiwan

AbstractThe deep ocean to the east and south of Taiwan Island‒including the Okinawa Trough, Ryukyu Arc, Philippine Sea, Luzon Strait, and the northeast South China Sea‒is one of the most prominent mixing hotspots of the global ocean. The high‐resolution seismic reflection technique can provide detailed views of the water's finestructure and near‐transient turbulent mixing therein. Using legacy data from two seismic cruises, we present the finestructure and quantify turbulent mixing within the thermocline and sub‐thermocline (100–1,000 m depths) to the east and south offshore Taiwan. We derive dissipation rate ε and diapycnal diffusivity Kρ by analyzing the inertial convective turbulence regime of horizontal slope spectra from the seismically auto‐tracked wave‐fields. Dissipation and diffusivities are locally enhanced close to the topography exceeding 10−6 W⋅kg−1 and 10−2 m2⋅s−1, and gradually decrease to 10−9 W⋅kg−1 and 10−4‒10−5 m2⋅s−1 beyond tens of kilometers. The mean diffusivity distribution pattern suggests that a large fraction of the energy dissipates close to the topography and a significant fraction is radiated into the open ocean. Both the tides and west boundary current drive the significantly enhanced mixing close to the topography of ridges, isolated islands and continental slopes. Kuroshio and eddy activities are responsible for the local anomalously enhanced mixing patches far away from the topography. Comparisons among seismic results, numerical simulations and eddy analyses suggest that the energy cascades downward from large‐scale motions to small‐scale turbulence via multiple mechanisms of tide‐topography interaction, flow‐topography interaction, and frontal instability around the island of Taiwan.

Repicturing viscoelastic drag-reducing turbulence by introducing dynamics of elasto-inertial turbulence

Recently, the nature of viscoelastic drag-reducing turbulence (DRT), especially the maximum drag reduction (MDR) state, has become a focus of controversy. It has long been regarded as polymer-modulated inertial turbulence (IT), but is challenged by the newly proposed concept of elasto-inertial turbulence (EIT). This study is to repicture DRT in parallel plane channels by introducing dynamics of EIT through statistical, structural and budget analysis for a series of flow regimes from the onset of drag reduction to EIT. Some underlying mechanistic links between DRT and EIT are revealed. Energy conversion between velocity fluctuations and polymers as well as pressure redistribution effects are of particular concern, based on which a new energy self-sustaining process (SSP) of DRT is repictured. The numerical results indicate that at low Reynolds number ( $Re$ ), weak IT flow is replaced by a laminar regime before the barrier of EIT dynamics is established with the increase of elasticity, whereas, at moderate $Re$ , EIT-related SSP can get involved and survive from being relaminarized. This further explains the reason why relaminarization phenomenon is observed for low $Re$ while the flow directly enters MDR and EIT at moderate $Re$ . Moreover, with the proposed energy picture, the newly discovered phenomenon that streamwise velocity fluctuations lag behind those in the wall-normal direction can be well explained. The repictured SSP certainly justifies the conjecture that IT nature is gradually replaced by that of EIT in DRT with the increase of elasticity.

Characteristics of intrinsic non-stationarity and its effect on eddy-covariance measurements of CO<sub>2</sub> fluxes

Abstract. Stationarity is a critical assumption in the eddy-covariance method that is widely used to calculate turbulent fluxes. Many methods have been proposed to diagnose non-stationarity attributed to external non-turbulent flows. In this paper, we focus on intrinsic non-stationarity (IN) attributed to turbulence randomness. The detrended fluctuation analysis is used to quantify IN of CO2 turbulent fluxes in the downtown of Beijing. Results show that the IN is common in CO2 turbulent fluxes and is a small-scale phenomenon related to the inertial sub-range turbulence. The small-scale IN of CO2 turbulent fluxes can be simulated by the Ornstein–Uhlenbeck (OU) process as a first approximation. Based on the simulation results, we find that the flux-averaging time should be greater than 27 s to avoid the effects of IN. Besides, the non-stationarity diagnosis methods that do not take into account IN would possibly make a wrong diagnosis with some parameters.

Experimental Investigation on the Hydraulic Characteristics of the Two-Phase Flow in the Intersecting Rock Fractures

An appropriate understanding of the hydraulic characteristics of the two-phase flow in the rock fracture network is important in many engineering applications. To investigate the two-phase flow in the fracture network, a study on the two-phase flow characteristics in the intersecting fractures is necessary. In order to describe the two-phase flow in the intersecting fractures quantitatively, in this study, a gas-water two-phase flow experiment was conducted in a smooth 3D model with intersecting fractures. The results in this specific 3D model show that the flow structures in the intersecting fractures were similar to those of the stratified wavy flow in pipes. The nonlinearity induced by inertial force and turbulence in the intersecting fractures cannot be neglected in the two-phase flow, and the Martinelli-Lockhart model is effective for the two-phase flow in intersecting fractures. Delhaye’s model can be adapted for the cases in this experiment. The turbulence of the flow can be indicated by the values of C in Delhaye’s model, but resetting the appropriate range of the values of C is necessary.