Turbulent Cascades II

The decay of helicity and its impact on kinetic energy is analyzed at very large Reynolds numbers in homogeneous isotropic turbulence lacking mirror symmetry, using the Eddy-Damped Quasi-Normal Markovian closure. A theoretical time decay exponent for helicity is derived and assessed numerically. In addition to the initial well-known slowing down of non-linear transfers, it is further shown that helicity slightly accelerates the decay of kinetic energy in Batchelor turbulence, because it decreases the \(k^4\) back transfers, which is proved analytically using non-local expansions. Finally, unlike the kinetic energy spectrum in Batchelor turbulence, the permanence of large eddies is verified for the helical spectrum.

A spectral analysis of strictly axisymmetric turbulence is performed. We investigate in particular by direct numerical simulation the possible cascades of energy and helicity. Decaying and forced flows at moderate Reynolds numbers are considered. A dual cascade, in which energy is transferred to the large scales and helicity to the small ones, is first evidenced in helical flows. A similar scenario is then shown to hold in the absence of a net helicity: in this case, energy also cascades to the largest scales, and positively and negatively polarized helicity are transferred to the small ones.

There is a lack of high precision results for turbulence. Here we present a non-equilibrium thermodynamical approach to the turbulent cascade and show that the entropy generation \(\varDelta S_{tot}\) of the turbulent cascade fulfills in high precision the rigorous integral fluctuation theorem \(\langle e^{-\varDelta S_{tot}} \rangle _{u(\cdot )} = 1\). To achieve this result the turbulent cascade has to be taken as a stochastic process in scale, for which Markov property is given and for which an underlying Fokker-Planck equation in scale can be set up. For one exemplary data set we show that the integral fluctuation theorem is fulfilled with an accuracy better than \(10^{-3}\). Furthermore, we show that other basic turbulent features are well taking into account like the third order structure function or the skewness of the velocity increments.

The evolution of invariants of the velocity gradient tensor is examined to determine local topologies of flow within shear flow turbulence. In a temporal direct numerical simulation of a round jet, a large number of fluid pathlines were computed simultaneously, and values of invariants at locations along pathlines were stored. It turns out that trajectories in the invariant space, corresponding to fluid pathlines, are far more varied than those of the conditional mean field that has been determined before. Several trajectories have segments where the invariants have much larger values than that expected from their joint pdfs. Corresponding large changes are also observed in the space of the invariants of the strain rate tensor. Although less frequent, these large departures may have consequences for the evolution of turbulent flow fields.

In the group-theoretical model of stationary homogeneous turbulence (PRE 72, 016302, 2005), the renormalized form of the Navier-Stokes equations includes the turbulent viscosity, which appears not from averaging of the nonlinear term, but from the molecular viscosity term. The next raised question is as follows. In the immiscible gas-liquid turbulent flow, the motion equation is completed by the surface tension force, acting on the interface. When such a flow is averaged over some length-scale, there is no more interfaces. Then at the high Reynolds number, what is the renormalized form of governing equations in this flow? In the framework of approach of the aforementioned paper, the result is this: similar to the Smagorinsky viscosity, the “effective surface tension coefficient” appears in the invariant to scaling transformation form. Its expression is discussed in this paper.

Some recent developments on the physical mechanism of turbulence cascades are summarised. It is first shown that the energy cascade in statistically steady isotropic turbulence is local in scale, at least on average, and that temporal variations of the large-scale forcing are transferred to smaller scales as a ‘wave’ consistent with the classical Kolmogorov model. It is further shown that, when energy-containing structure are individually tracked in band-pass filtered velocity fields, they also behave classically. The correlation of their physical position with larger (or smaller) structures is highest towards the beginning (or end) of their lifetimes. The analysis is then extended to the structures of momentum flux in the logarithmic layer of turbulent channels. Small structures grow and shrink smoothly along their lifetimes, but larger ones change size mostly by splits and mergers involving structures of similar size. For the largest structures, splits predominate, although not overwhelmingly.

The aim of this paper is to identify the effects of the regenerating cycle phases on the Lagrangian acceleration statistics. Direct Numerical Simulation of fluid and inertial particles (\(St = 3.0\)) moving in a stationary homogeneous shear flow is performed and the autocorrelation functions of the norm and components of the Lagrangian acceleration vector are calculated. The energy balance between turbulent scales is first observed, and the range of scales, sensitive to growth and collapse phases, is identified. In link to this range, it was shown that the acceleration norm is correlated longer during a growth phase and shorter during a collapse phase. This effect is amplified when inertia of particles is increased. At the same time, it was shown that the acceleration vector components are invariant to the regenerating cycle.

We present results of high-resolution numerical simulations of compressible 2D turbulence forced at intermediate spatial scales with a solenoidal white-in-time external acceleration. A case with an isothermal equation of state, low energy injection rate, and turbulent Mach number \(M\approx 0.34\) without energy condensate is studied in detail. Analysis of energy spectra and fluxes shows that the classical dual-cascade picture familiar from the incompressible case is substantially modified by compressibility effects. While the small-scale direct enstrophy cascade remains largely intact, a large-scale energy flux loop forms with the direct acoustic energy cascade compensating for the inverse transfer of solenoidal kinetic energy. At small scales, the direct enstrophy and acoustic energy cascades are fully decoupled at small Mach numbers and hence the corresponding spectral energy slopes comply with theoretical predictions, as expected. At large scales, dispersion of acoustic waves on vortices softens the dilatational velocity spectrum, while the pseudo-sound component of the potential energy associated with coherent vortices steepens the potential energy spectrum.

The general exact solution of the Cauchy problem to the 3D Euler vortex equation for compressible flow in unbound space is obtained. This solution has singularity at finite time and coincides with the vortex solution of the 3D Hopf equation for particles motion by inertia. A closed description of the evolution of enstrophy and all higher moments for the corresponding vortex field is established, giving an exact solution to the problems of closure in the theory of turbulence. On the base of this solution the smooth solution of the Navier-Stokes 3D equation for viscous compressible medium is obtained taking into account the effective viscosity and representation for the pressure field, which follows from the integral entropy balance equation, not from the medium equation of state. The above provides a positive solution for the Clay Millennium Problem (www.​claymath.​org) just in the case of its generalization on the Navier-Stokes equation for the compressible medium, for which an absence of smooth solutions on finite time interval has been a priori assumed before.

The interface/turbulence interaction between two fluids in a turbulent environment has an important role in many technical processes, e.g. primary liquid atomization in combustion devices. Primary atomization has a significant role in spray formation and its characteristics. The resulting dynamics typically span 4–6 orders of magnitude in length scales, making detailed numerical simulations exceedingly expensive. This motivates the need for modeling approaches based on spatial filtering such as large-eddy simulation (LES). In this paper, a new approach based on One-Dimensional turbulence (ODT) is presented to describe the subgrid interface dynamics. ODT is a stochastic model simulating turbulent flow evolution along a notional one-dimensional line of sight by applying instantaneous maps that represent the effects of individual turbulent eddies on property fields. It provides affordable high resolution of interface creation and property gradients within each phase, which are key for capturing the local behavior as well as overall trends. ODT has previously been shown to reproduce the main features of an experimentally determined regime diagram for primary jet breakup. Here a new approach called VODT is presented which produces a size-conditioned as well as a total time rate of generation of droplets for given flow conditions at an interface. At the LES level, the total droplet generation from VODT is interpreted as a rate of mass conversion of LES-resolved liquid into unresolved droplets. Preliminary results of applying VODT to a cell with a planar-shear-layer are discussed at the end of the paper.

The energy transfer among the different scales of the turbulent structures is analysed by means of the generalised Kolmogorov equation (GKE). The equation is applied to a turbulent channel with the addition of a bump which creates a strong shear layer and separation bubble. The GKE can provide an intricate description of the energy scale-by-scale budget in both physical and separation space, through the identification of the regions of production and dissipation of energy. Conventional one-point statistics do not allow any analysis across scales. The GKE statistics show that the turbulent structures follow two paths: they are trapped by the recirculation bubble, deformed and dissipated or they are convected downstream by the shear layer and elongated in the streamwise direction. These paths correspond to the direct and inverse energy cascades, respectively. The main feature of this complex flow is that the energy dynamics depends, in a non-trivial way, on both the physical position and separation scales, and does not follow the classical energy path occurring in homogeneous isotropic turbulence.

We analyze the essence of nonlinear processes that underlie turbulence sustenance in spectrally stable shear flows. In these flows, the strong anisotropy of velocity shear-induced nonmodal growth phenomenon in spectral (k-)space, in turn, entails the anisotropy of nonlinear processes in this space. Consequently, the main novel nonlinear process is transverse, or angular redistribution of modes in Fourier space referred to as the nonlinear transverse cascade rather than a mere direct/inverse cascade. It is demonstrated that nonlinear coherent as well as turbulent states are sustained via a subtle interplay of the linear nonmodal growth (that has transient nature) and the nonlinear transverse cascade. This course of events exemplifies the well-known bypass scenario of subcritical turbulence in spectrally stable shear flows. In this proceedings paper, we present selected results of our simulations of hydrodynamic and MHD 2D plane shear flows to demonstrate the transverse cascade in action.

We review recent results concerning the idealized framework of incompressible homogeneous buoyancy-driven turbulence, shedding light on the mixing process occurring in variable density fluids subjected to accelerations. Self-similar analysis, results from numerical simulations and anisotropic spectral models establish the sensitivity of the late time dynamics to the distribution of energy at large scales, to the different properties of the mixing and to the resonances inside the mixing zone when a time-varying acceleration is applied. The isotropic and anisotropic part of turbulent spectra are also investigated. Different scenarii are proposed to explain how the turbulent scales within the inertial range are altered by buoyancy forces.

Experimental data measured with a 3d Shadow-Particle Tracking Velocimetry (S-PTV) setup in fully developed turbulence (\(\mathrm {Re_\lambda }=[175-225]\)) is presented. The underlying flow is of the von Kármán type and as other similar flows, its mean flow is bistable, the two states having the topology of a stagnation point with one contracting and two dilating directions. Tracer particle trajectories permit the investigation of the inhomogeneity and anisotropy of the smallest scales, namely acceleration statistics. The local variance and time-scale of acceleration components are shown to mimic the large scale properties of the flow, the time-scales being more anisotropic than the variances. We explain the hierarchy of time-scales by investigating the Lagrangian Taylor micro-scale which is related to acceleration and velocity variances, and discuss the very high Reynolds number regime.

We consider homogeneous turbulence submitted to the effect of an external rotation of the system. The presence of the Coriolis force results in anisotropic dynamics and structure of the flow, due to the presence of propagating inertial waves, and of a modified dynamics. The anisotropic structure of the flow is analyzed by maps of second- and third-order two-point correlation statistics in physical separation space, distinguishing between axial and perpendicular separation. Second-order statistics permit to assess the anisotropy of the flow which develops due to the presence of rotation. However, nonlinear dynamics has to be characterized by examining the third-order correlation term in the Kármán-Howarth-Monin equation, in which the non linear term appears as the divergence of the flux vector \(\mathbf {F}\). We show that maps of the components of \(\mathbf {F}\) permit to examine the detailed anisotropic interactions, and to discuss the results of our Direct Numerical Simulations to that from Kolmogorov theory, from wave turbulence theory and from experiments.

The study aims at furthering our understanding and quantifying the influence of coherent structures on small-scale turbulence and passive scalar mixing, in an annular jet configuration with large diameter ratios. This ’bluff-body’ geometry is close to that widely used in combustion for flame stabilization [1]. A passive contaminant is introduced in the flow, through a slight heating. We report the evolution along the jet axis of the following quantities: mean values of the longitudinal velocity and passive scalar (\(\bar{U}\) and \(\bar{\varTheta }\)), as well as the energy and scalar dissipation rates (\(\bar{\varepsilon }\) and \(\bar{\chi }\)). It is shown that these statistics:

Spacecraft observations show the presence, in the polar Solar Wind (SW) plasma, of outward propagating Alfvénic fluctuations with a very broadband spectrum. This is true only up to a certain frequency in the spectrum, after which a consistent amount of energy is present also in the spectra of inward propagating Alfvén and density fluctuations. A mechanism able to explain the production of inward propagating modes and density fluctuations is the parametric instability, in which Alfvén waves can decay and produce back-scattered waves and density perturbations. In this work, we show some recent results obtained with a \((2+1/2)\)D-MHD pseudo-spectral numerical code in which an attempt is made to reproduce an initial condition similar to that observed in the real SW. The evolution of the instability generates a nonlinear cascade that redistributes the energy towards the small scales in all directions. A striking feature of this evolution is the presence of a few localized, energy containing, coherent pressure-balanced structures.

The properties of helical turbulence in heterogeneous media are studied. It is shown that the amplification of large-scale eddy perturbations by initially homogeneous isotropic spiral turbulence is possible in an incompressible fluid with solid particles. The motion of solid particles provides non-zero divergence on a pulsating scale and thus provides non-zero values of Reynolds stresses in averaged equations. Eddy instability of helical turbulence against large-scale perturbations in an incompressible fluid with oscillating gas bubbles is found. It is shown that bubble oscillations provide an asymmetry of the Reynolds stresses in the averaged equations and the appearance of generation terms.

Evolution of droplets and turbulence in a small box which is ascending inside the maritime cumulus cloud has seamlessly been simulated for about 10 min from the view point of the microscopic dynamics. It is found that the kinetic energy spectrum obeys the Kolmogorov spectrum \(k^{-5/3}\) at low to moderate wavenumbers, while the spectra of the temperature and the water vapor mixing ratio are modified, close to \(k^{-1/3}\) at low wavenumbers and roll off more slowly than the exponential in the diffusive range. This modification of the spectra arises from the condensation-evaporation and the liquid water mass loading to the flow. It is also found that the spectra related to the cloud droplets consist of two contributions, one is from the spatially correlated part and the other is from the uncorrelated part which originates from the discreteness of droplets. The former dominates the spectrum at low to moderate wavenumbers and the latter at high wavenumbers. We argue the effects of the two contributions on the turbulence spectra.

A homogenous swarm of bubbles rising through a liquid generates anisotropic homogeneous random velocity fluctuations. The statistical properties of bubble-induced fluctuations differ from the classical shear-induced turbulence. The probability density functions are non Gaussian and show a succession of exponential evolutions. The power spectral densities exhibit a \(k^{-3}\) subrange for wavelengths around the bubble size. The understanding of these properties requires to consider that bubble-induced agitation involves two contributions of a different nature. The first one is not related to any flow instability and results from the anisotropic flow disturbances generated near the bubbles, principally in the vertical direction. The second one is the almost isotropic turbulence induced by the instability of the flow through a population of bubbles, which turns out to be the main cause of horizontal fluctuations. Even if the two contributions are coupled, only the second one deserves to be called bubble-induced turbulence.

We are studying dispersion of finite-size particles in a turbulent plane Couette flow by numerical simulations. The effect of particle non-sphericity was discussed (particles are neutrally buoyant and shape varies from oblate to prolate, aspect ratio is ranging from 0.5 to 2). Particle dispersion is analyzed also when inertia is considered for different particle densities for spherical particles (while keeping comparable Stokes number). This work yields evidences that the particle distribution in turbulent flow coherent structures is in general correlated to the cycle of regeneration of turbulence in Couette flow (the strongest correlation being for massless bubbles), and that the particle residence time in large scale vortices is equal to the characteristic time scale of the flow regeneration cycle.

We present a study of preferential concentration with Voronoï diagrams of finite size solid particles in a von Kármán flow. This flow is an interesting case of strongly inhomogeneous turbulence with high \(Re_{\lambda }\). We investigate preferential concentration of PMMA particles with density \(\rho _p=1400\) kg/m\(^3\) and diameter \(2.8<d_p/\eta <6.3\) for \(340<Re_{\lambda }<810\). We conclusively find that particles form clusters and voids. The geometry of these structures is therefore studied, and results compared with previous works in other flows.

The statistical features of turbulent flows depend on the locality properties of energy transfers among scales. The latter, in turn, may have consequences for the relative dispersion of passive particles. We consider a class of two-dimensional flows of geophysical interest, namely \(\alpha \)-turbulence models, possessing different locality properties. We numerically study relative dispersion in such flows using both fixed-time and fixed-scale indicators. The results are compared with predictions based on phenomenological arguments to explore the relation between the locality of the turbulent cascade and that of relative dispersion. We find that dispersion behaviors agree with expectations from local theories, for small enough values of the parameter \(\alpha \) (dynamics close to surface quasi geostrophy) and for sufficiently small initial pair separations. Non-local dispersion is instead observed for the largest \(\alpha \) considered (quasi-geostrophic model).

The effect of thermal inertia on the dynamics of particles with a thermal expansion coefficient larger than that of the fluid is investigated in Rayleigh-Bénard convection (RBC) using direct numerical simulations. A simple point-particles approach is used, where thermal expansion of both particles and fluid is included. These thermally responsive particles move towards the hot (bottom) or cold (top) plates, where they become lighter or heavier than the fluid to eventually escape this region of the flow. When the thermal response time of particles is large, this process is slow and particles spend more time at the walls than in the bulk. It is indeed shown that in this regime the number of particles at the plates is enhanced, compared to the uniform distribution found for tracer particles in RBC. A more complex point-particle approach, including non-linear effects on the drag forces, shows that non-linear thermal effects influence both the temperature and velocity statistics of the thermally responsive particles and cannot be ignored.

We investigate experimentally turbulence of surface gravity waves in the Coriolis facility in Grenoble by using both high sensitivity local probes and a time and space resolved stereoscopic reconstruction of the water surface. We show that the water deformation is made of the superposition of weakly nonlinear waves following the linear dispersion relation and of bound waves resulting from non resonant triadic interaction. Although the theory predicts a 4-wave resonant coupling supporting the presence of an inverse cascade of wave action, we do not observe such inverse cascade. We investigate 4-wave coupling by computing the tricoherence i.e. 4-wave correlations. We observed very weak values of the tricoherence at the frequencies excited on the linear dispersion relation that are consistent with the hypothesis of weak coupling underlying the weak turbulence theory.

Turbulence is a complex, multi-scale fluid process that can be strongly modified by the presence of multiple phases. In this work, we will discuss various aspects of the interaction between turbulence and interfaces with surface tension, as commonly encountered in liquid-gas flows. This study is based on a series of direct numerical simulations of homogeneous and isotropic turbulence in the presence of an initially flat interface that separates two fluids of equal densities and viscosities. This highly simplified flow configuration is selected as it isolates a critical aspect of turbulent liquid-gas flows and allows for deeper analysis. A second order numerical discretization that conserves mass, momentum, and kinetic energy is employed for all simulations. The scales of interface corrugation are presented, identifying the presence of a critical cutoff length scale below which surface tension suppresses interface deformation.

Advances to a dual-scale modeling approach [1] are presented to describe turbulent phase interface dynamics in a large-eddy-simulation-type spatial filtering context. Spatial filtering of the governing equations introduces several sub-filter terms that require modeling. Instead of developing individual closure-models for the terms associated with the interface, the dual-scale approach uses an exact closure by explicitly filtering a fully resolved realization of the phase interface. This resolved realization is maintained on a high-resolution over-set mesh. The advection equation for the phase interface on this DNS scale requires a model for the fully resolved interface advection velocity. This velocity is the sum of the filter scale LES velocity, available from the LES flow solver, and the sub-filter velocity fluctuation. The sub-filter velocity fluctuation is due to sub-filter turbulent eddies, reconstructed using a local fractal interpolation technique [2]. Results of the dual-scale model are compared to recent DNS of unit density and viscosity contrast interfaces in homogeneous isotropic turbulence without surface tension [3].

We present results of a particle image velocimetry (PIV) study conducted on forced plumes in a rotating system. The measurements were carried out for nine different background rotations and it was found that the plume precesses anticyclonically around the axis of background rotation. The data analysis has revealed that the precession rate increases linearly with the rate of background rotation.

Atmospheric boundary layer turbulence in stably stratified conditions is characterised by an intermittent, unsteady behaviour. The intermittency can result from localised flow acceleration due to non-turbulent motions, which can exhibit structures such as ramp-cliff convective patterns, waves or microfronts. Based on a timeseries clustering method, we characterise interactions between scales of motion in a dataset of near-surface stable boundary layer turbulence. Individual flow structures are investigated in two weak-wind flow regimes exhibiting distinct scale interaction properties. The signature of flow structures differs despite comparable wind and stability properties.

Low-rank multilevel approximation methods are often suited to attack high-dimensional problems successfully and they allow very compact representation of large data sets. Specifically, hierarchical tensor product decomposition methods, e.g., the Tree-Tucker format and the Tensor Train format emerge as a promising approach for application to data that are concerned with cascade-of-scales problems as, e.g., in turbulent fluid dynamics. Beyond multilinear mathematics, those tensor formats are also successfully applied in e.g., physics or chemistry, where they are used in many body problems and quantum states. Here, we focus on two particular objectives, that is, we aim at capturing self-similar structures that might be hidden in the data and we present the reconstruction capabilities of the Tensor Train decomposition method tested with 3D channel turbulence flow data.