We adress the problem of interactions between the longitudinal velocity increment and the energy dissipation rate in fully developed turbulence. The coupling between these two quantities is experimentally investigated by the theory of stochastic Markovian processes. The so-called Markov analysis allows for a precise characterization of the joint statistical properties of velocity increment and energy dissipation. In particular, it is possible to determine the differential equation that governs the evolution along scales of the joint probability density of these two quantities. The properties of this equation provide interesting new insights into the coupling between energy dissipation and velocity incrementas leading to small scale intermittency.
We discuss the possibility of dual local and non-local cascades in a 3D turbulent Beltrami flow, with inverse energy cascade and direct helicity cascade, by analogy with 2D turbulence. We discuss the corresponding energy spectrum in both local and non-local case. Comparison with a high Reynolds number turbulent von Karman flow is provided and discussed.
We address the problem of differences between longitudinal and transverse velocity increments in isotropic small scale turbulence. The relationship of these two quantities is analyzed experimentally by means of stochastic Markovian processes leading to a phenomenological Fokker- Planck equation from which a generalization of the Karman equation is derived. From these results, a simple relationship between longitudinal and transverse structure functions is found which explains the difference in the scaling properties of these two structure functions.
Features of the turbulent cascade are investigated for various datasets from three different turbulent flows. The analysis is focused on the question as to whether developed turbulent flows show universal small scale features. To answer this question, 2-point statistics and joint multi-scale statistics of longitudinal velocity increments are analysed. Evidence of the Markov property for the turbulent cascade is shown, which corresponds to a 3-point closure that reduces the joint multi-scale statistics to simple conditional probability density functions (cPDF). The cPDF are described by the Fok…
Interscale energy transfers at the vicinity of the turbulent/non-turbulent interface are from small to large scales in directions close to the interface's tangent plane where motions are predominantly stretching, but from large to small scales in the other directions where motions are predominantly compressive and significantly correlated with square angular momentum. An important role in this predominance is played by the extreme compressive motions which can be significantly more likely than extreme stretching motions even where motions are on average stretching. The most intense interscale …
We discuss the phenomenology of the split energy cascade in a three-dimensional thin fluid layer by mean of high resolution numerical simulations of the Navier-Stokes equations. We observe the presence of both an inverse energy cascade at large scales, as predicted for two-dimensional turbu- lence, and of a direct energy cascade at small scales, as in three-dimensional turbulence. The inverse energy cascade is associated with a direct cascade of enstrophy in the intermediate range of scales. Notably, we find that the inverse cascade of energy in this system is not a pure 2D phenomenon, as the …
We investigate the transition from forward to inverse energy cascade in turbulent flows in thin layers, varying the functional form of the forcing and the thickness of the layer. We show that, as the forcing function becomes more three-dimensional, the inverse cascade is suppressed and the critical height hc, where the transition occurs, is decreased. We study the dependence of this critical height on a parameter r which measures the dimensionality of the forcing and thus construct a phase space diagram in the parameter space r-h. We discuss the effect of Reynolds number and domain size.
By tracking small particles in the bulk of an intensely turbulent flow, we show that even a very small concentration of long-chain polymers disrupts the usual turbulent energy cascade. The polymers affect scales much larger than their physical size, from the dissipation range to the inertial range. The effect depends strongly on the polymer concentration. While the dissipative-scale statistics change continuously as the polymer concentration is increased, the inertial-range energy transfer rate is only altered by the polymer additives when the concentration is above a threshold (approximately …
Using direct numerical simulations we demonstrate that magnetic helicity exhibits a bidirectional turbulent cascade at high but finite magnetic Reynolds numbers. Despite the injection of positive magnetic helicity in the flow, we observe that magnetic helicity of opposite signs is generated between large and small scales. We explain these observations by carrying out an analysis of the magnetohydrodynamic equations reduced to triad interactions using the Fourier helical decomposition. Within this framework, the direct cascade of positive magnetic helicity arises through triad interactions that…
In incompressible and periodic statistically stationary turbulence, exchanges of turbulent energy across scales and space are characterised by very intense and intermittent spatio-temporal fluctuations around zero of the time-derivative term, the spatial turbulent transport of fluctuating energy, and the pressure-velocity term. These fluctuations are correlated with each other and with the intense intermittent fluctuations of the interscale energy transfer rate. These correlations are caused by the sweeping effect, the link between non-linearity and non-locality, and also relate to geometrical…
Conditional statistics of homogeneous isotropic turbulent flow is investigated by means of high-Reynolds number direct numerical simulations performed with $2048^3$ collocation points. Eulerian as well as Lagrangian velocity increment statistics under several conditions are analyzed and compared. In agreement with experimental data longitudinal probability density functions $P(δ^{||}_l u|ε_l)$ conditioned on a scale-averaged energy dissipation rate are close to Gaussian distributions over all scales within the inertial range of scales. Also transverse increments conditioned on either the dissi…
In three dimensional turbulence there is on average a cascade of kinetic energy from the largest to the smallest scales of the flow. While the dominant idea is that the cascade occurs through the physical process of vortex stretching, evidence for this is debated. In the framework of the Karman-Howarth equation for the two point turbulent kinetic energy, we derive a new result for the average flux of kinetic energy between two points in the flow that reveals the role of vortex stretching. However, the result shows that vortex stretching is in fact not the main contributor to the average energy…
To the naked eye, turbulent flows exhibit whirls of many different sizes. To each size, or scale, corresponds a fraction of the total energy resulting from a cascade in five dimensions: scale, time and three-dimensional space. Understanding this process is critical to modeling strategies of geophysical and industrial flows. By tracking the flow regions containing energy in different scales, we have evidenced the statistical predominance of a cross-scale link whereby fluid lumps of energy at scale $Δ$ appear within lumps of scale $2Δ$ and die within those of scale $Δ/2$. Our approach uncovers t…
We present a statistical analysis of the two-point vorticity probability density of the vorticity field generated in the inverse cascade of stationary two-dimensional turbulence.
The transfer of kinetic energy from large to small scales is a hallmark of turbulent flows. Yet, a precise mechanistic description of this transfer, which is expected to occur via an energy cascade, is still missing. Several conceptually simple configurations with vortex tubes have been proposed as a testing ground to understand the energy cascade. Here, we focus on incompressible flows and compare the energy transfer occurring in a statistically steady homogeneous isotropic turbulent (HIT) flow with the generation of fine-scale motions in configurations involving vortex tubes. We start by fil…
The effect of domain anisotropy on the inverse cascade occurring within the geostrophic turbulence regime of rapidly rotating Rayleigh-Bénard convection (RRBC) is investigated. In periodic domains with square cross-section in the horizontal a domain-filling dipole state is present. For rectangular periodic domains a Kolmogorov-like flow consisting of a periodic array of alternating unidirectional jets with embedded vortices is observed, together with an underlying weak meandering transverse jet. Similar transitions occurring in weakly dissipative two-dimensional flows driven by externally impo…
The current work presents an experimental investigation of the dynamic interactions between flow scales caused by repeated actions of the nonlinear term of the Navier-Stokes equation. Injecting a narrow band oscillation, representing a single Fourier mode, into a round jet flow allows the measurement of the downstream generation and development of higher harmonic spectral components and to measure when these components are eventually absorbed into fully developed turbulence. Furthermore, the dynamic evolution of the measured power spectra observed corresponds well to the measured cascaded dela…
We study the influence of the energy spectrum on the extent of the cascade range of a passive tracer in turbulent flows. The interesting cases are when there are two different spectra over the potential range of the tracer cascade (in 2D when the tracer forcing is in the inverse energy cascade range, and in 3D when the Schmidt number Sc is large). The extent of the tracer cascade range is then limited by the width of the range for the shallower of the two energy spectra. Nevertheless, we show that in dimension $d=2,3$ the tracer cascade range extends (up to a logarithm) to $κ_{d\text{D}}^{p}$,…
To explain the observed decay of superfluid turbulence at very low temperature, it has been proposed that a cascade of Kelvin waves (analogous to the classical Kolmogorov cascade) transfers kinetic energy to length scales which are small enough that sound can be radiated away. We report results of numerical simulations of the interaction of quantized vortex filaments. We observe the development of the Kelvin-waves cascade, and compute the statistics of the curvature, the amplitude spectrum (which we compare with competing theories) and the fractal dimension.
The conventional approach to the turbulent energy cascade, based on Richardson-Kolmogorov phenomenology, ignores the topology of emerging vortices, which is related to the helicity of the turbulent flow. It is generally believed that helicity can play a significant role in turbulent systems, e.g., supporting the generation of large-scale magnetic fields, but its impact on the energy cascade to small scales has never been observed. We suggest for the first time a generalized phenomenology for isotropic turbulence with an arbitrary spectral distribution of the helicity. We discuss various scenar…
This study presents a numerical simulation of a 3D viscous flow in the VKI-Genoa cascade that takes into account the laminar-turbulent transition. The numerical simulation is performed using the Reynolds-averaged Navier-Stokes equations and the two-equation k-omega SST turbulence model. The algebraic Production Term Modification model is used for modeling the laminar-turbulent transition. Computations of both fully turbulent and transitional flows are carried out. The contours of the Mach number, the turbulence kinetic energy, the entropy function, as well as limiting streamlines are presented…
The paper is devoted to the investigation of the turbulent water boundary layer in the jet mixing flows using high-speed infrared (IR) thermography. Two turbulent mixing processes were studied: a submerged water jet impinging on a flat surface and two intersecting jets in a round disc-shaped vessel. An infrared camera (FLIR Systems SC7700) was focused on the window transparent for IR radiation; it provided high-speed recordings of heat fluxes from a thin water layer close to the window. Temperature versus time curves at different points of water boundary layer near the wall surface were acquir…
Helicity, as one of only two inviscid invariants in three-dimensional turbulence, plays an important role in the generation and evolution of turbulence. From the traditional viewpoint, there exists only one channel of helicity cascade similar to that of kinetic energy cascade. Through theoretical analysis, we find that there are two channels in helicity cascade process. The first channel mainly originates from vortex twisting process, and the second channel mainly originates from vortex stretching process. By analysing the data of direct numerical simulations of typical turbulent flows, we fin…
Passive scalar cascade in the intermediate layer of turbulent channel flow for $Pr\leq 1$
Univ. Lille（里尔大学）；CNRS（法国国家科学研究中心）；ONERA（法国航空航天研究院）；Arts et Metiers Institute of Technology（巴黎高科国立高等工艺学院）；Centrale Lille（里尔中央理工学院）；Graduate School of Engineering Science, University of Osaka（大阪大学工学研究科）
Within the framework of random multiplicative energy cascade models of fully developed turbulence, expressions for two-point correlators and cumulants are derived, taking into account a proper conversion from an ultrametric to an Euclidean two-point distance. The comparison with two-point statistics of the surrogate energy dissipation, extracted from various wind tunnel and atmospheric boundary layer records, allows an accurate deduction of multiscaling exponents and cumulants. These exponents serve as the input for parametric estimates of the probabilistic cascade generator.
The problem of the occurrence of an energy cascade for Alfvénic turbulence in solar wind plasmas was hystorically addressed by using phenomenological arguments based to the weakness of nonlinear interactions and the anisotropy of the cascade in wave vectors space. Here, this paradox is reviewed through the formal derivation of a Yaglom relation from anisotropic Magnetohydrodynamic equation. The Yaglom relation involves a third-order moment calculated from velocity and magnetic fields and involving both Elsässer vector fields, and is particularly useful to be used as far as spacecraft observati…
The well known mean-field model of a turbulence driven dynamo is reviewed in relation to Laboratory experiments in which a turbulent cascade is created by a pair of large rotors. It is argued that in such experiments the alpha-effect, driving a dynamo field, will be much less than the dissipative beta-effect. Consequently a mean field dynamo cannot be sustained. This conclusion is supported by recent measurements of the alpha and beta effects in the Madison Dynamo Experiment.
We present first results of an implicit large eddy simulation of the MTU T161 low pressure turbine at a Reynolds number of 90,000 and Mach number of 0.6, both based on isentropic exit conditions, using a high order discontinuous Galerkin method. The aim is to validate the numerical setup with respect to available experimental data. We discuss the steps taken to create realistic inflow boundary conditions in terms of end wall boundary layer thickness and free stream turbulence intensity. This is achieved by tailoring the input distribution of Reynolds stresses and turbulent length scale to a Fo…
Cascades of turbulent kinetic energy and multicomponent scalars in a momentum-scalar coupling turbulence driven by multiscale forces under homogeneous and isotropic hypotheses
The interaction between small- and large-scale structures, and the coexisting bottom-up and top-down processes are studied in a turbulent plane Couette flow, where space-filling longitudinal rolls appear at relatively low values of the Reynolds number $Re$. A DNS database at $Re_τ=101$ is built to replicate the highest $Re$ considered in a recent experimental work by Kawata and Alfredsson (Phys. Rev. Lett., vol.120, 2018, 244501). Our study is based on the exact budget equations for the second-order structure function tensor $\langle δu_i δu_j \rangle$, i.e. the Anisotropic Generalized Kolmogo…
The circulation around any closed loop is a Lagrangian invariant for classical, smooth solutions of the incompressible Euler equations in any number of space dimensions. However, singular solutions relevant to turbulent flows need not preserve the classical integrals of motion. Here we generalize the Kelvin theorem on conservation of circulations to distributional solutions of Euler and give necessary conditions for the anomalous dissipation of circulations. We discuss the important role of Kelvin's theorem in turbulent vortex-stretching dynamics and conjecture a version of the theorem which m…
Exploring 2D turbulent properties in anisotropic and disordered Fourier space: Insights into inverse cascades and universal superdiffusion from randomly sampled triadic interaction
National Research Council, Institute of Atmospheric Pollution Research（国家研究委员会，大气污染研究所）；University of Calabria（卡拉布里亚大学）；Dipartimento di Fisica, University of Calabria（卡拉布里亚大学物理系）
The Log-Poisson phenomenological description of the turbulent energy cascade is evoked to discuss high-order statistics of velocity derivatives and the mapping between their probability distribution functions at different Reynolds numbers. The striking confirmation of theoretical predictions suggests that numerical solutions of the flow, obtained at low/moderate Reynolds numbers can play an important quantitative role in the analysis of experimental high Reynolds number phenomena, where small scales fluctuations are in general inaccessible from direct numerical simulations.
In this paper, we introduce a new way to estimate the scaling parameter of a self-similar process by considering the maximum probability density function (pdf) of tis increments. We prove this for $H$-self-similar processes in general and experimentally investigate it for turbulent velocity and temperature increments. We consider turbulent velocity database from an experimental homogeneous and nearly isotropic turbulent channel flow, and temperature data set obtained near the sidewall of a Rayleigh-Bénard convection cell, where the turbulent flow is driven by buoyancy. For the former database,…
The exact solution of the Cauchy problem for a Fokker-Planck equation used by R. Friedrich and J. Peinke for the description of a turbulent cascade, considered as a stochastic process of Markovian type, is obtained in the frame of M. Suzuki approach.
While in classical turbulence helicity depletes nonlinearity and can alter the evolution of turbulent flows, in quantum turbulence its role is not fully understood. We present numerical simulations of the free decay of a helical quantum turbulent flow using the Gross-Pitaevskii equation at high spatial resolution. The evolution has remarkable similarities with classical flows, which go as far as displaying a dual transfer of incompressible kinetic energy and helicity to small scales. Spatio-temporal analysis indicates that both quantities are dissipated at small scales through non-linear excit…
Here we show a procedure to generate an analytical structure producing a cascade that scales as the energy spectrum in isotropic homogeneous turbulence. We obtain a function that unveils a non-self-similar fractal at the origin of the cascade. It reveals that the backbone underlying $-5/3$ cascades is formed by deterministic nested polynomials with parameters tuned in a Hopf bifurcation critical point. The cascade scaling is exactly obtainable (not by numerical simulations) from deterministic low dimensional nonlinear dynamics. Consequently, it should not be exclusive for fluids but also prese…
Breaking waves entrain gas beneath the surface. The wave-breaking process energizes turbulent fluctuations that break bubbles in quick succession to generate a wide range of bubble sizes. Understanding this generation mechanism paves the way towards the development of predictive models for large-scale maritime and climate simulations. Garrett et al. (2000) suggested that super-Hinze-scale turbulent breakup transfers entrained gas from large to small bubble sizes in the manner of a cascade. We provide a theoretical basis for this bubble-mass cascade by appealing to how energy is transferred fro…
Some multidimensional generalizations of the Fokker-Planck equation used by R. Friedrich and J. Peinke for the description of a turbulent cascade as a stochastic process of Markovian type, are considered. The exact solutions of the Cauchy problems for these equations are found with the operator methods.
We study statistical properties of turbulent inverse cascades in a class of nonlinear models describing a scalar field transported by a two-dimensional incompressible flow. The class is characterized by a linear relation between the transported field and the velocity, and include several cases of physical interest, such as Navier-Stokes, surface quasi-geostrophic and Charney-Hasegawa-Mima equations. We find that some statistical properties of the inverse turbulent cascades in such systems are conformal invariant. In particular, the zero-isolines of the scalar field are statistically equivalent…
Experiments and numerical simulations of turbulent $^4$He and $^3$He-B have established that, at hydrodynamic length scales larger than the average distance between quantum vortices, the energy spectrum obeys the same 5/3 Kolmogorov law which is observed in the homogeneous isotropic turbulence of ordinary fluids. The importance of the 5/3 law is that it points to the existence of a Richardson energy cascade from large eddies to small eddies. However, there is also evidence of quantum turbulent regimes without Kolmogorov scaling. This raises the important questions of why, in such regimes, the …
This note studies the mechanism of turbulent energy cascade through an opportune bifurcations analysis of the Navier--Stokes equations, and furnishes explanations on the more significant characteristics of the turbulence. A statistical bifurcations property of the Navier--Stokes equations in fully developed turbulence is proposed, and a spatial representation of the bifurcations is presented, which is based on a proper definition of the fixed points of the velocity field. The analysis first shows that the local deformation can be much more rapid than the fluid state variables, then explains th…
The possibility of subcritical behaviour in the geostrophic turbulence regime of rapidly rotating thermally driven convection is explored. In this regime a non-local inverse energy transfer may compete with the more traditional and local direct cascade. We show that, even for control parameters for which no inverse cascade has previously been observed, a subcritical transition towards a large-scale vortex state can occur when the system is initialized with a vortex dipole of finite amplitude. This new example of bistability in a turbulent flow, which may not be specific to rotating convection,…
We observe electro-hydrodynamically driven turbulent flows at low Reynolds numbers in a two-fluid emulsion consisting of micron-scale droplets. In the presence of electric fields, the droplets produce interacting hydrodynamic flows which result in a dynamical organization at a spatial scale much larger than the size of the individual droplets. We characterize the dynamics associated with these structures by both video imaging and a simultaneous, in situ, measurement of the time variation of the bulk Reynolds stress with a rheometer. The results display scale invariance in the energy spectra in…
In nature turbulent flows exist that are neither simply 2D nor 3D but boundary conditions, such as varying stratification, force them towards the one or the other. Here, we report the first evidence of the co-existence of 2D and 3D turbulence in an experimental flow driven by Faraday waves in water. We find that an inverse energy cascade at the fluid surface and a direct energy cascade in the 3D bulk flow underneath exist simultaneously. We base our analysis on temporally and spatially well-resolved velocity fields measured at horizontal and vertical planes. The findings suggest that the stron…
As a minimal mathematical model generating cascade analogous to that of the Navier-Stokes turbulence in the inertial range, we propose a one-dimensional partial-differential-equation model that conserves the integral of the squared vorticity analogue (enstrophy) in the inviscid case. With a large-scale forcing and small viscosity, we find numerically that the model exhibits the enstrophy cascade, the broad energy spectrum with a sizable correction to the dimensional-analysis prediction, peculiar intermittency and self-similarity in the dynamical system structure.
Turbulence is an intriguing non-equilibrium state, which originates from fluid mechanics and has far-reaching consequences in the description of climate physics, the characterization of quantum hydrodynamics, and the understanding of cosmic evolution. The concept of turbulent cascade describing the energy redistribution across different length scales offers one profound route to reconcile fundamental conservative forces with observational energy non-conservation of accelerating expansion of the universe bypassing the cosmological constant. Here, we observe a dimension crossing turbulent energy…
We present a user-friendly open-source MATLAB\textsuperscript{\textregistered} package developed by the research group Turbulence, Wind energy and Stochastics (TWiSt) at the Carl von Ossietzky University of Oldenburg. Firstly, this package helps the user to perform a very basic statistical analysis of a given turbulent data set which we believe to be useful to the entire turbulence community. It can be used to estimate the statistical quantities of turbulence such as the spectrum density, turbulent intensity, integral length scale, Taylor microscale, Kolmogorov scale and dissipation rate. Diff…
Inverse cascades of kinetic energy and thermal variance in the subset of vertically homogeneous modes in spectral space are found to cause a slow aggregation to a pair of convective supergranules that eventually fill the whole horizontally extended, three-dimensional, turbulent Rayleigh-Bénard convection layer when a heat flux is prescribed at the top and bottom. An additional weak rotation of the layer around the vertical axis stops this aggregation at a scale that is smaller than the lateral domain extension and ceases the inverse cascade for the thermal variance. The inverse cascade for the…
