In this work, we perform numerical simulations of forced two-phase isotropic turbulence to study the stationary states of a two-phase mixture. We first formulate three different approaches to force a two-phase turbulent flow that maintains a constant (a) mixture kinetic energy, (b) mixture kinetic energy + surface energy, and (c) individual phase kinetic energy, and study their effect on the final stationary state of the system. We show that forcing the two-phase system eliminates the arbitrariness associated with the initialization of the second phase in two-phase turbulence simulations. Next, we study the global statistics of the two-phase mixture in the stationary state for varying density ratios ($10^{-3}$ to $10^3$) and viscosity ratios ($10^{-3}$ to $10^3$), void fraction (dilute to dense regimes), and Weber numbers (breakup and non-breakup regimes). We utilize a spatial filtering approach, that is applicable for the general case of incompressible/compressible two-phase flows, to compute the energy spectra of each of the phases and study the turbulent kinetic energy distribution across scales. We find that the interface behaves as a soft wall, inhibiting the accommodation of eddies inside the dispersed phase. Finally, we show that bubbles and drops exhibit different turbulent characteristics for the same global conditions of the carrier phase, with bubbles exhibiting higher turbulent intensities inside them and droplets behaving ballistically, similar to solid particles. This study acts as a foundational work on stationary two-phase turbulence, which can be used for further analysis and the development of subgrid-scale models for larger-scale simulations of two-phase turbulent flows.
We provide a consistent theory of turbulence in the presence of shear and rotation. Starting from a quasi-linear equation for the fluctuating fields, we derive turbulence amplitude and turbulent transport coefficients, taking into account the effect of shear and rotation on turbulence. We show that the shear reduces both turbulence amplitude and transport, more strongly in the direction parallel to the shear than in the perpendicular one, effectively inducing an anisotropic turbulence. The rotation further reduces turbulence amplitude and transport. The interaction between the shear and the ro…
We report on simulations of two-dimensional turbulence in the inverse energy cascade regime. Focusing on the statistics of Lagrangian tracer particles, scaling behavior of the probability density functions of velocity fluctuations is investigated. The results are compared to the three-dimensional case. In particular an analysis in terms of compensated cumulants reveals the transition from a strong non-Gaussian behavior with large tails to Gaussianity. The reported computation of correlation functions for the acceleration components sheds light on the underlying dynamics of the tracer particles…
By extracting unstable invariant solutions directly from body-forced three-dimensional turbulence, we study the dynamical processes at play when the forcing is large scale and either unidirectional in the momentum or the vorticity equations. In the former case, the dynamical processes familiar from recent work on linearly-stable shear flows - variously called the Self-Sustaining Process (Waleffe 1997) or Vortex-Wave Interaction (Hall & Smith 1991; Hall & Sherwin 2010) - are important even when the base flow is linearly unstable. In the latter case, where the forcing drives Taylor-Green vortice…
Forced turbulence combined with the effect of rotation and shear flow is studied. In a previous paper [Leprovost and Kim, PRE in press (2008)], we considered the case where the shear and the rotation are perpendicular. Here, we consider the complementary case of parallel rotation and shear, elucidating how rotation and flow shear influence the generation of shear flow (e.g. the direction of energy cascade), turbulence level, transport of particles and momentum. In contrast with the case where rotation and shear are perpendicular, we found that rotation affects turbulence amplitude only for ver…
We show, by direct numerical simulations, that heavy inertial particles (characterized by Stokes number $\St$) in inhomogeneously forced statistically stationary isothermal turbulent flows cluster at the minima of mean-square turbulent velocity. Two turbulent transport processes, turbophoresis and turbulent diffusion together determine the spatial distribution of the particles. If the turbulent diffusivity is assumed to scale with turbulent root mean square velocity, as is the case for homogeneous turbulence, the turbophoretic coefficient can be calculated. Indeed, for the above assumption, th…
The weak version of universality in turbulence refers to the independence of the scaling exponents of the $n$th order strcuture functions from the statistics of the forcing. The strong version includes universality of the coefficients of the structure functions in the isotropic sector, once normalized by the mean energy flux. We demonstrate that shell models of turbulence exhibit strong universality for both forced and decaying turbulence. The exponents {\em and} the normalized coefficients are time independent in decaying turbulence, forcing independent in forced turbulence, and equal for dec…
We present a spatio-temporal analysis of a statistically stationary rotating turbulence experiment, aiming to extract a signature of inertial waves, and to determine the scales and frequencies at which they can be detected. The analysis uses two-point spatial correlations of the temporal Fourier transform of velocity fields obtained from time-resolved stereoscopic particle image velocimetry measurements in the rotating frame. We quantify the degree of anisotropy of turbulence as a function of frequency and spatial scale. We show that this space-time-dependent anisotropy is well described by th…
We consider turbulence in a stratified 'Kolmogorov' flow, driven by horizontal shear in the form of sinusoidal body forcing in the presence of an imposed background linear stable stratification in the third direction. This flow configuration allows the controlled investigation of the formation of coherent structures, which here organise the flow into horizontal layers by inclining the background shear as the strength of the stratifica- tion is increased. By numerically converging exact steady states from direct numerical simulations of chaotic flow, we show, for the first time, a robust connec…
We demonstrate an unexpected connection between isotropic turbulence and wall-bounded shear flows. We perform direct numerical simulations of isotropic turbulence forced at large scales at moderate Reynolds numbers and observe sudden transitions from a chaotic dynamics to a spatially simple flow, analogous to the laminar state in wall bounded shear flows. We find that the survival probabilities of turbulence are exponential and the typical lifetimes increase superexponentially with the Reynolds number. Our results suggest that both isotropic turbulence and wall-bounded shear flows qualitativel…
In this paper, we investigate the statistical features of the fully developed, forced, rapidly rotating, {turbulent} system using numerical simulations, and model {the} energy {spectrum} that {fits} well with the numerical data. Among the wavenumbers ($k$) larger than the Kolmogorov dissipation wavenumber, the energy is distributed such that the suitably non-dimensionized energy spectrum is ${\bar E}({\bar k})\approx \exp(-0.05{\bar k})$, where overbar denotes appropriate non-dimensionalization. {For the wavenumbers smaller than that of forcing, the energy in a horizontal plane is much more th…
We report an experimental investigation on the influence of an external magnetic field on forced 3D turbulence of liquid gallium in a closed vessel. We observe an exponential damping of the turbulent velocity fluctuations as a function of the interaction parameter N (ratio of Lorentz force over inertial terms of the Navier-Stokes equation). The flow structures develop some anisotropy but do not become bidimensional. From a dynamical viewpoint, the damping first occurs homogeneously over the whole spectrum of frequencies. For larger values of N, a very strong additional damping occurs at the hi…
In this Part 2 we study further experimental properties of two-layer exchange flows in a stratified inclined duct (SID), which are turbulent, strongly-stratified, shear-driven, and continuously-forced. We analyse the same state-of-the-art data sets using the same core shear layer methodology as in Part 1, but we focus here on turbulent energetics and mixing statistics. The detailed analysis of kinetic and scalar energy budgets reveals the specificity and scalings of SID turbulence, while energy spectra provide insight into the current strengths and limitations of our experimental data. The ani…
We inquire about the properties of 2d Navier-Stokes turbulence simultaneously forced at small and large scales. The background motivation comes by observational results on atmospheric turbulence. We show that the velocity field is amenable to the sum of two auxiliary velocity fields forced at large and small scale and exhibiting a direct-enstrophy and an inverse-energy cascade, respectively. Remarkably, the two auxiliary fields reconcile universal properties of fluxes with positive statistical correlation in the inertial range.
A simple scheme of forcing turbulence away from decay was introduced by Lundgren some time ago, the `linear forcing', which amounts to a force term linear in the velocity field with a constant coefficient. The evolution of linearly forced turbulence towards a stationary final state, as indicated by direct numerical simulations (DNS), is examined from a theoretical point of view based on symmetry arguments. In order to follow closely the DNS the flow is assumed to live in a cubic domain with periodic boundary conditions. The simplicity of the linear forcing scheme allows one to re-write the pro…
In this paper, we present an extensive study of linearly forced isotropic turbulence. By using an analytical method, we identified two parametric choices that are new to our knowledge. We proved that the underlying nonlinear dynamical system for linearly forced isotropic turbulence is the general case of a cubic Lienard equation with linear damping (Dumortier and Rousseau 1990).
We characterize the statistical and geometrical properties of the cyclone-anticyclone asymmetry in a statistically-steady forced rotating turbulence experiment. Turbulence is generated by a set of vertical flaps which continuously inject velocity fluctuations towards the center of a tank mounted on a rotating platform. We first characterize the cyclone-anticyclone asymmetry from conventional single-point vorticity statistics. We propose a phenomenological model to explain the emergence of the asymmetry in the experiment, from which we predict scaling laws for the root-mean-square velocity in g…
In this letter, we present an extensive study of the linearly forced isotropic turbulence. By using analytical method, we identify two parametric choices, of which they seem to be new as far as our knowledge goes. We prove that the underlying nonlinear dynamical system for linearly forced isotropic turbulence is the general case of a cubic Lienard equation with linear damping. We also discuss a Fokker-Planck approach to this new dynamical system,which is bistable and exhibits two asymmetric and asymptotically stable stationary probability densities.
A dynamical model is proposed for isotropic turbulence driven by steady forcing that yields a viscosity independent dynamics for the small-scale (inertial) regime. This reproduces the Kolmogorov spectrum for the two-point velocity correlation function in the fully developed (stationary) stage, while predicting intermittency corrections for higher order moments. The model also yields a transient stage with a power-law time evolution. The crossover time to fully developed turbulence scales with the turbulent system size as $\sim L^{11/3}$. The physical origin of the transient behavior is explain…
We study experimentally the dynamics and statistics of capillary waves forced by random steep gravity waves mechanically generated in laboratory. Capillary waves are produced here by gravity waves from nonlinear wave interactions. Using a spatio-temporal measurement of the free-surface, we characterize statistically the random regimes of capillary waves in the spatial and temporal Fourier spaces. For a significant wave steepness ($0.2-0.3$), power-law spectra are observed both in space and time, defining a turbulent regime of capillary waves transferring energy from large scale to small scale.…
We study the experimental properties of exchange flows in a stratified inclined duct (SID), which are simultaneously turbulent, strongly stratified by a mean vertical density gradient, driven by a mean vertical shear, and continuously forced by gravity. We focus on the core shear layer away from the duct walls, where these flows are excellent experimentally-realisable approximations of canonical hyperbolic-tangent stratified shear layers, whose forcing allows mean and turbulent properties to reach quasi steady states. We analyse state-of-the-art data sets of the time-resolved density and veloc…
Classically, large-scale forced turbulence is characterized by a transfer of energy from large to small scales via nonlinear interactions. We have investigated the changes in this energy transfer process in broad-band forced turbulence where an additional perturbation of flow at smaller scales is introduced. The modulation of the energy dynamics via the introduction of forcing at smaller scales occurs not only in the forced region but also in a broad range of length-scales outside the forced bands due to non-local triad interactions. Broad-band forcing changes the energy distribution and energ…
Conflict between formation of a cyclonic vortex and isotropization in forced homogeneous rotating turbulence is numerically investigated. It is well known that a large rotation rate of the system induces columnar vortices to result in quasi-two-dimensional (Q2D) flow, while a small rotation rate allows turbulence to be three-dimensional (3D). It is found that the transition from the Q2D turbulent flow to the 3D turbulent flow and the reverse transition occur at different values of the rotation rates. At the intermediate rotation rates, bistability of these two statistically steady states is ob…
A model for the Reynolds number dependence of the dimensionless dissipation rate $C_{\varepsilon}$ was derived from the dimensionless Kármán-Howarth equation, resulting in $C_{\varepsilon}=C_{\varepsilon, \infty} + C/R_L + O(1/R_L^2)$, where $R_L$ is the integral scale Reynolds number. The coefficients $C$ and $C_{\varepsilon,\infty}$ arise from asymptotic expansions of the dimensionless second- and third-order structure functions. This theoretical work was supplemented by direct numerical simulations (DNSs) of forced isotropic turbulence for integral scale Reynolds numbers up to $R_L=5875$ ($…
Stratified flows forced by internal waves similar to those obtained in the Coriolis platform (LEGI, Grenoble, France) \cite{Savaro2020} are studied by pseudospectral triply-periodic simulations. The experimental forcing mechanism consisting in large oscillating vertical panels is mimicked by a penalization method. The analysis of temporal and spatiotemporal spectra reveals that the flow for the strongest forcing in the experiments is composed of two superposed large and quasi-steady horizontal vortices, of internal waves in box modes and of much weaker waves outside the modes. Spatial spectra …
From the energy balance in wavenumber space expressed by the Lin equation, we derive a new form for the local Karman-Howarth equation for forced isotropic turbulence in real space. This equation is then cast into a dimensionless form, from which a combined analytical and numerical study leads us to deduce a new model for the scale-independent nondimensional dissipation rate $\Ceps$, which takes the form $\Ceps = \Cinf + C_L/R_L$, where the asymptotic value $\Cinf$ can be evaluated from the third-order structure function. This is found to fit the numerical data with $\Cinf = 0.47 \pm 0.01$ and …
Developing data-driven subgrid-scale (SGS) models for large eddy simulations (LES) has received substantial attention recently. Despite some success, particularly in a priori (offline) tests, challenges have been identified that include numerical instabilities in a posteriori (online) tests and generalization (i.e., extrapolation) of trained data-driven SGS models, for example to higher Reynolds numbers. Here, using the stochastically forced Burgers turbulence as the test-bed, we show that deep neural networks trained using properly pre-conditioned (augmented) data yield stable and accurate a …
The normalized non-dimensional von Kármán-Howarth equation for isotropic homogeneous decaying and forced steady turbulence is integrated to obtain expressions for the dissipation rate coefficient $C_ε=(L ε)/< u^2 >^{3/2}$, where $L$ denotes the longitudinal integral length scale, $ε$ the mean dissipation rate and $< u^2 >$ the mean variance of the longitudinal velocity fluctuations. For decaying turbulence the final exact expressions for $C_ε$ for the low and high Reynolds number limit depend on the decay exponent $n$, which is known to depend on the initial velocity structure at the turbulenc…
Drift of interfaces in forced stably-stratified turbulence and the role of vertically-sheared helical structures
Department of Physics & INFN, University of Rome “Tor Vergata”（罗马第二大学物理系及意大利国家核物理研究所）；INFN Sezione di Lecce（意大利国家核物理研究所莱切分部）；CNR NANOTEC（意大利国家研究委员会纳米技术研究所）
We study the evolution of kinetic and magnetic energy spectra in magnetohydrodynamic flows in the presence of strong cross helicity. For forced turbulence, we find weak inverse transfer of kinetic energy toward the smallest wavenumber. This is plausibly explained by the finiteness of scale separation between the injection wavenumber and the smallest wavenumber of the domain, which here is a factor of 15. In the decaying case, there is a slight increase at the smallest wavenumber, which is probably explained by the dominance of kinetic energy over magnetic energy at the smallest wavenumbers. Wi…
We study experimentally temperature fluctuations in stably stratified forced turbulence in air flow. In the experiments with an imposed vertical temperature gradient, the turbulence is produced by two oscillating grids located nearby the side walls of the chamber. Particle Image Velocimetry is used to determine the turbulent and mean velocity fields, and a specially designed temperature probe with sensitive thermocouples is employed to measure the temperature field. We found that the ratio $\big[(\ell_x \nabla_x \bar{T})^2 + (\ell_y \nabla_y \bar{T})^2 + (\ell_z \nabla_z \bar{T})^2\big] / <θ^2…
The decay of kinetic helicity is studied in numerical models of forced turbulence using either an externally imposed forcing function as an inhomogeneous term in the equations or, alternatively, a term linear in the velocity giving rise to a linear instability. The externally imposed forcing function injects energy at the largest scales, giving rise to a turbulent inertial range with nearly constant energy flux while for linearly forced turbulence the spectral energy is maximum near the dissipation wavenumber. Kinetic helicity is injected once a statistically steady state is reached, but it is…
We demonstrate an inverse energy cascade in a minimal model of forced 2D quantum vortex turbulence. We simulate the Gross-Pitaevskii equation for a moving superfluid subject to forcing by a stationary grid of obstacle potentials, and damping by a stationary thermal cloud. The forcing injects large amounts of vortex energy into the system at the scale of a few healing lengths. A regime of forcing and damping is identified where vortex energy is efficiently transported to large length scales via an inverse energy cascade associated with the growth of clusters of same-circulation vortices, a Kolm…
The pseudospectral method, in conjunction with a new technique for obtaining scaling exponents $ζ_n$ from the structure functions $S_n(r)$, is presented as an alternative to the extended self-similarity (ESS) method and the use of generalized structure functions. We propose plotting the ratio $|S_n(r)/S_3(r)|$ against the separation $r$ in accordance with a standard technique for analysing experimental data. This method differs from the ESS technique, which plots $S_n(r)$ against $S_3(r)$, with the assumption $S_3(r) \sim r$. Using our method for the particular case of $S_2(r)$ we obtain the n…
Turbulent dynamo theories have faced difficulties in obtaining evolution of large-scale magnetic fields on short dynamical time-scales due to the constraint imposed by magnetic helicity balance. This has critical implications for understanding the large-scale magnetic field evolution in astrophysical systems like the Sun, stars and galaxies. Direct numerical simulations (DNS) in the past with isotropically forced helical turbulence have shown that large-scale dynamo saturation time-scales are dependent on the magnetic Reynolds number (Rm). In this work, we have carried out periodic box DNS of …
A calculational approach in fluid turbulence is presented. Use is made of the attracting nature of the fluid-dynamic dynamical system. An approach is offered that effectively propagates the statistics in time. Loss of sensitivity to an initial probability density functional and generation of stationary statistical effects is speculated.
We have performed interface-resolved direct numerical simulations of forced homogeneous-isotropic turbulence in a dilute suspension of spherical particles in the Reynolds number range Re-lambda=115-140. The solid-fluid density ratio was set to 1.5, gravity was set to zero, and two particle diameters were investigated corresponding to approximately 5 and 11 Kolmogorov lengths. Note that these particle sizes are clearly outside the range of validity of the point-particle approximation, as has been shown by Homann & Bec (2010). At the present parameter points the global effect of the particles up…
Numerical simulations are made for forced turbulence at a sequence of increasing values of Reynolds number, R, keeping fixed a strongly stable, volume-mean density stratification. At smaller values of R, the turbulent velocity is mainly horizontal, and the momentum balance is approximately cyclostrophic and hydrostatic. This is a regime dominated by so-called pancake vortices, with only a weak excitation of internal gravity waves and large values of the local Richardson number, Ri, everywhere. At higher values of R there are successive transitions to (a) overturning motions with local reversal…
We carry out a detailed study of dynamic multiscaling in the turbulent nonequilibrium, but statistically steady, state of the stochastically forced one-dimensional Burgers equation. We introduce the concept of $\textit{interval collapse times}$ $τ_{\rm col}$, the time taken for an interval of length $\ell$, demarcated by a pair of Lagrangian tracers, to collapse at a shock. By calculating the dynamic scaling exponent of the order-$p$ moment of $τ_{\rm col}$, we show that (a) there is $\textit{not one but an infinity of characteristic time scales}$ and (b) the probability distribution function …
We consider the case of finite-size spherical particles which are settling under gravity in a homogeneous turbulent background flow. Turbulence is forced with the aid of the random forcing method of Eswaran and Pope [Comput. Fluids, 16(3):257-278, 1988], while the solid particles are represented with an immersed-boundary method. The forcing scheme is used to generate isotropic turbulence in vertically elongated boxes in order to warrant better decorrelation of the Lagrangian signals in the direction of gravity. Since only a limited number of Fourier modes are forced, it is possible to evaluate…
A study of the relationship between Lagrangian statistics and flow topology in fluid turbulence is presented. The topology is characterized using the Weiss criterion that provides a simplified tool to partition the flow into topologically different regions: elliptic (vortex dominated), hyperbolic (deformation dominated), and intermediate (turbulent background). The flow corresponds to forced two-dimensional Navier-Stokes turbulence in doubly periodic and circular bounded domains with non-slip boundary conditions. In the double periodic domain, the probability density function (pdf) of the Weis…
In this study, we performed highly resolved large-eddy simulations (LES) to investigate the influence of variable properties on the forced turbulent convection in a channel. The constant heat flux boundary condition permits wall temperature fluctuations and thus induces variations of fluid properties. Since the effect of viscosity on the flow exhibits $Re_τ^{-1}$ scaling, we only considered $Re_τ= 180$ in the present study. Compared to the flow with constant properties, results indicate that the variable properties have trivial effects on the mean velocity and temperature profiles, Reynolds sh…
In present study, we discuss results of applicability of discrete filters for large eddy simulation (LES) method of forced compressible magnetohydrodynamic (MHD) turbulent flows with the scale-similarity model. Influences and effects of discrete filter shapes on the scale-similarity model are examined in physical space using a finite-difference numerical schemes. We restrict ourselves to the Gaussian filter and the top-hat filter. Representations of this subgrid-scale model which correspond to various 3- and 5-point approximations of both Gaussian and top-hat filters for different values of pa…
Aims: We determine the components of the $Λ$-effect tensor that quantifies the contributions to the turbulent momentum transport even for uniform rotation. Methods: Three-dimensional numerical simulations are used to study turbulent transport in triply periodic cubes under the influence of rotation and anisotropic forcing. Comparison is made with analytical results obtained via the so-called minimal tau-approximation. Results: In the case where the turbulence intensity in the vertical direction dominates, the vertical stress is always negative. This situation is expected to occur in stellar co…
In order to better understand deviations from equilibrium in turbulent flows, it is meaningful to characterize the dynamics rather than the statistics of turbulence. To this end, the Lyapunov theory provides a useful description of turbulence through the study of the perturbation dynamics. In this work, the Lyapunov spectrum of forced homogeneous isotropic turbulent flows is computed. Using the Lyapunov exponents of a flow at different Reynolds numbers, the scaling of the dimension of the chaotic attractor for a three-dimensional homogeneous isotropic flow (HIT) is obtained for the first time …
Large scale features of a randomly isotropically forced incompressible and unbounded rotating fluid are examined in perturbation theory. At first order in both the random force amplitude and the angular velocity we find two types of modifications to the fluid equation of motion. The first correction transforms the molecular shear viscosity into a (rotation independent) effective viscosity. The second perturbative correction leads to a new large scale non-dissipative force proportional to the fluid angular velocity in the slow rotation regime. This effective force does no net work and alters th…
We present experimental evidence for a double cascade of kinetic energy in a statistically stationary rotating turbulence experiment. Turbulence is generated by a set of vertical flaps which continuously injects velocity fluctuations towards the center of a rotating water tank. The energy transfers are evaluated from two-point third-order three-component velocity structure functions, which we measure using stereoscopic particle image velocimetry in the rotating frame. Without global rotation, the energy is transferred from large to small scales, as in classical three-dimensional turbulence. Fo…
