Numerical and analytical studies of decaying, two-dimensional (2D) Navier-Stokes (NS) turbulence at high Reynolds numbers are reported. The effort is to determine computable distinctions between two different formulations of maximum entropy predictions for the decayed, late-time state. Both formulations define an entropy through a somewhat ad hoc discretization of vorticity to the "particles" of which statistical mechanical methods are employed to define an entropy, before passing to a mean-field limit. In one case, the particles are delta-function parallel "line" vortices ("points" in two dimensions), and in the other, they are finite-area, mutually-exclusive convected "patches" of vorticity which in the limit of zero area become "points." We use time-dependent, spectral-method direct numerical simulation of the Navier-Stokes equations to see if initial conditions which should relax to different late-time states under the two formulations actually do so.
We conduct a careful analysis of the data provided by Krogstad & Davidson (2011) and show that their data do not support their conclusions. According to their published data, their decaying approximately homogeneous isotropic turbulent flows are, invariably, clearly different from Saffman turbulence; and very clearly marked differences exist between the far downstream turbulence behaviors generated by their conventional grid and by their multiscale cross grids.
We present a numerical investigation of two-dimensional decaying turbulence in the Lagrangian framework. Focusing on single particle statistics, we investigate Lagrangian trajectories in a freely evolving turbulent velocity field. The dynamical evolution of the tracer particles is strongly dominated by the emergence and evolution of coherent structures. For a statistical analysis we focus on the Lagrangian acceleration as a central quantity. For more geometrical aspects we investigate the curvature along the trajectories. We find strong signatures for self-similar universal behavior.
A hallmark of fluid turbulence theory is the universal power law scaling of the velocity difference statistics between two points in space in the inertial range between the large energy injection scale and the small energy dissipation scale. Even at the highest Reynolds numbers available, laboratory and natural flows such universal power laws have not been convincingly demonstrated. Here we show for the decaying active grid turbulence of the Max Planck Variable Density Turbulence Tunnel that the velocity difference statistics at high Reynolds numbers do not exhibit a power law, but have a univ…
The way the increment statistics of turbulent velocity fluctuations scale with the increment size is a centerpiece of turbulence theories. We report data on decaying turbulence in the Max Planck Variable Density Turbulence Tunnel (VDTT), which show an approach of the inertial range statistics toward a nontrivial shape at small scales. By correcting for the contributions of energy decay to the large-scale statistics with a model, we find the scaling exponent of the second-order velocity increment statistics to be independent of the Reynolds number and equal to $0.693\pm0.003$ for $2000\lesssim …
The present paper reports on our effort to characterize vortical interactions in complex fluid flows through the use of network analysis. In particular, we examine the vortex interactions in two-dimensional decaying isotropic turbulence and find that the vortical interaction network can be characterized by a weighted scale-free network. It is found that the turbulent flow network retains its scale-free behavior until the characteristic value of circulation reaches a critical value. Furthermore, we show that the two-dimensional turbulence network is resilient against random perturbations but ca…
Direct numerical simulations of homogeneous decaying turbulence with mild background rotation show the existence of a systematic and significant imbalance between the non-linear energy cascade to small scales and its dissipation. By starting the decay from a statistically stationary and fully developed rotating turbulence state, where the dissipation and the energy flux are approximately equal, the data shows a growing imbalance between the two until a maximum is reached when the dissipation is about twice the energy flux. This dichotomy of behaviours during decay is reminiscent of the non-equ…
The scaling of the longitudinal velocity structure functions, $S_q(r) = < | δu (r) |^q > \sim r^{ζ_q}$, is analyzed up to order $q=8$ in a decaying rotating turbulence experiment from a large Particle Image Velocimetry (PIV) dataset. The exponent of the second-order structure function, $ζ_2$, increases throughout the self-similar decay regime, up to the Ekman time scale. The normalized higher-order exponents, $ζ_q / ζ_2$, are close to those of the intermittent non-rotating case at small times, but show a marked departure at larger times, on a time scale $Ω^{-1}$ ($Ω$ is the rotation rate), alt…
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 use two related non-stationarity functions as measures of the degree of scale-by-scale non-equilibrium in homogeneous isotropic turbulence. The values of these functions indicate significant non-equilibrium at the upper end of the inertial range. Wind tunnel data confirm Lundgren's (2002, 2003) prediction that the two-point separation $r$ where the second and third order structure functions are closest to their Kolmogorov scalings is proportional to the Taylor length scale $λ$, and that both structure functions increasingly distance themselves from their Kolmogorov equilibrium form as $r$ i…
Numerical and analytical studies of "final states" of two-dimensional (2D) decaying turbulence are reported. The first part of this work is trying to give a definition for final states of 2D decaying turbulence. Although the functional relation of $ω-ψ$ is frequently used as the characterization of those "final states," it is just a sufficient but not necessary condition so it is not proper to be used as the definition. It is found the way through the value of the effective area S covered by the scatter $ω-ψ$ plot, which is initially suggested by Read, is more general, and more suitable for th…
From DNS of turbulence decaying from specified initial conditions for the range of initial Taylor-Reynolds numbers 2.58 < Rλ(0) < 358.6, it was found that the shape of the iconic curve of dimensionless dissipation versus Reynolds number depended strongly on the choice of measurement time. For our preferred time, a composite based on peak values in the dissipation and inertial transfer curves, the result was virtually identical to the forced, stationary case. In the course of studying onset criteria, we found that the exponent for the power-law decay of the energy decreased with increasing Reyn…
We construct a discrete shell-model for two-dimensional turbulence that takes into account local and nonlocal interactions between velocity modes in Fourier space. In real space, its continuous limit is described by the one-dimensional Burgers equation. We find a novel approximate scaling solution of such an equation and show that it well describes the main characteristics of the energy spectrum in fully developed, freely decaying two-dimensional turbulence.
We investigate wind tunnel turbulence generated by both conventional and multi-scale grids. Measurements were made in a tunnel which has a large test-section, so that possible side wall effects are very small and the length assures that the turbulence has time to settle down to a homogeneous shear-free state. The conventional and multi-scale grids were all designed to produce turbulence with the same integral scale, so that a direct comparison could be made between the different flows. Our primary finding is that the behavior of the turbulence behind our multi-scale grids is virtually identica…
The previously reported non-equilibrium dissipation law is investigated in turbulent flows generated by various regular and fractal square grids. The flows are documented in terms of various turbulent profiles which reveal their differences. In spite of significant inhomogeneity and anisotropy differences, the new non-equilibrium dissipation law is observed in all these flows. Various transverse and longitudinal integral scales are measured and used to define the dissipation coefficient $C_{\varepsilon}$. It is found that the new non-equilibrium dissipation law is not an artefact of a particul…
Competition between two main attractors of the distributed chaos, one associated with translational symmetry (homogeneity) and another associated with rotational symmetry (isotropy), has been studied in freely decaying turbulence. It is shown that, unlike the case of statistically stationary homogeneous isotropic turbulence, the attractor associated with rotational symmetry (and controlled by Loitsyanskii integral) can dominate turbulent local dynamics in an intermediate stage of the decay, because the attractor associated with translational symmetry (and controlled by Birkhoff-Saffman integra…
In this paper we investigate the properties of rapidly rotating decaying turbulence using numerical simulations and phenomenological modelling. We find that as the turbulent flow evolves in time, the Rossby number decreases to $\sim 10^{-3}$, and the flow becomes quasi-two-dimensional with strong coherent columnar structures arising due to the inverse cascade of energy. We establish that a major fraction of energy is confined in Fourier modes $(\pm1,0,0)$ and $(0,\pm1,0)$ that correspond to the largest columnar structure in the flow. For wavenumbers ($k$) greater than the enstrophy dissipation…
We present results from a systematic numerical study of decaying turbulence in a dilute polymer solution by using a shell-model version of the FENE-P equations. Our study leads to an appealing definition of drag reduction for the case of decaying turbulence. We exhibit several new results, such as the potential-energy spectrum of the polymer, hitherto unobserved features in the temporal evolution of the kinetic-energy spectrum, and characterize intermittency in such systems. We compare our results with the GOY shell model for fluid turbulence.
Decaying turbulence is studied numerically using as initial condition a random flow whose shell-integrated energy spectrum increases with wavenumber k like k^q. Alternatively, initial conditions are generated from a driven turbulence simulation by simply stopping the driving. It is known that the dependence of the decaying energy spectrum on wavenumber, time, and viscosity can be collapsed onto a unique scaling function that depends only on two parameters. This is confirmed using three-dimensional simulations and the dependence of the scaling function on its two arguments is determined.
In the study by Sadeghi & Oberlack [JFM 899, A10 (2020)] it is claimed that new scaling laws are derived for the case of passive scalar dynamics under the influence of a constant mean gradient in decaying homogeneous isotropic turbulence. However, these scaling laws are not new and have already been derived and discussed in Bahri (2016). No novel analytical achievements are made by Sadeghi & Oberlack, as the title of their study misleadingly wants to suggest. In fact, the already established self-similar scaling laws obtained by Bahri through simple dimensional analysis are already more genera…
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…
We present a kinetic theory of two-dimensional decaying turbulence in the context of two-body and three-body vortex merging processes. By introducing the equations of motion for two or three vortices in the effective noise due to all the other vortices, we demonstrate analytically that a two-body mechanism becomes inefficient at low vortex density $n\ll 1$. When the more efficient three-body vortex mergings are considered {(involving vortices of different signs)}, we show that $n\sim t^{-ξ}$, with $ξ=1$. We generalize this argument to three-dimensional geostrophic turbulence, finding $ξ=5/4$, …
The breakup of a spherical droplet in a decaying homogeneous isotropic turbulence is studied by solving the Cahn-Hilliard-Navier-Stokes equations, using the discrete unified gas kinetic scheme combined with the free-energy-based phase-field model. We focus on the combined effects of turbulence and surface tension on the breakup process by assuming that the two fluid phases have the same density and same viscosity. The key physical parameters of the system include the volume fraction (6.54%), the initial Weber number (21.7), and the initial Taylor microscale Reynolds number (58). Three distinct…
We investigate non-equilibrium turbulence where the non-dimensionalised dissipation coefficient $C_{\varepsilon}$ scales as $C_{\varepsilon} \sim Re_{M}^{m}/Re_{\ell}^{n}$ with $m\approx 1 \approx n$ ($Re_M$ and $Re_{\ell}$ are global/inlet and local Reynolds numbers respectively) by measuring the downstream evolution of the scale-by-scale energy transfer, dissipation, advection, production and transport in the lee of a square-mesh grid and compare with a region of equilibrium turbulence (i.e. where $C_{\varepsilon}\approx \mathrm{constant}$). These are the main terms of the inhomogeneous, ani…
The decay of homogeneous isotropic turbulence in a variable viscosity fluid with a viscosity ratio up to 15 is analyzed by means of highly resolved direct numerical simulations (DNS) at low Reynolds numbers. The question addressed by the present work is how quantities such as the kinetic energy and the associated dissipation rate, as well as the inter-scale transport mechanism of turbulence are changed by local fluctuations of the viscosity. From the one-point budget equation of the turbulent kinetic energy, it is shown that the mean dissipation rate is nearly unchanged by variable viscosity e…
This paper explores the application of SPH to a Direct Numerical Simulation (DNS) of decaying turbulence in a two-dimensional no-slip wall-bounded domain. In this bounded domain, the inverse energy cascade, and a net torque exerted by the boundary, result in a spontaneous spin up of the fluid, leading to a typical end state of a large monopole vortex that fills the domain. The SPH simulations were compared against published results using a high accuracy pseudo-spectral code. Ensemble averages of the kinetic energy, enstrophy and average vortex wavenumber compared well against the pseudo-spectr…
The existence of drag reduction by polymer additives, well established for wall-bounded turbulent flows, is controversial in homogeneous, isotropic turbulence. To settle this controversy we carry out a high-resolution direct numerical simulation (DNS) of decaying, homogeneous, isotropic turbulence with polymer additives. Our study reveals clear manifestations of drag-reduction-type phenomena: On the addition of polymers to the turbulent fluid we obtain a reduction in the energy dissipation rate, a significant modification of the fluid energy spectrum especially in the deep-dissipation range, a…
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 present an experimental investigation of intense turbulence generated by a class of low-blockage space-filling fractal square grids. We confirm the existence of a protacted production region followed by a decaying region, as first reported by Hurst & Vassilicos (Physics of Fluids, 2007). We show that the centerline streamwise variation of most of the statistical properties of the turbulent flow can be scaled by a wake interaction length-scale $x_*$. We also confirm the finding of Seoud and Vassilicos (Physics of Fluids, 2007) that the ratio of the integral length-scale $L_u$ to the Taylor m…
In a two-dimensional (2D) classical fluid, a large-scale flow structure emerges out of turbulence, which is known as the inverse energy cascade where energy flows from small to large length scales. An interesting question is whether this phenomenon can occur in a superfluid, which is inviscid and irrotational by nature. Atomic Bose-Einstein condensates (BECs) of highly oblate geometry provide an experimental venue for studying 2D superfluid turbulence, but their full investigation has been hindered due to a lack of the circulation sign information of individual quantum vortices in a turbulent …
In linearly stable shear flows at moderate Re, turbulence spontaneously decays despite the existence of a codimension-one manifold, termed the edge of chaos, which separates decaying perturbations from those triggering turbulence. We statistically analyse the decay in plane Couette flow, quantify the breaking of self-sustaining feedback loops and demonstrate the existence of a whole continuum of possible decay paths. Drawing parallels with low-dimensional models and monitoring the location of the edge relative to decaying trajectories we provide evidence, that the edge of chaos separates state…
The free decay of MHD turbulence at large Reynolds numbers is studied numerically using a shell model. We study the statistical properties based on representative sample of realisations (128 realisations for each type of initial conditions) over the period of $10^5$ large-scale turnover times. The performed simulations show that the force-free non-helical MHD turbulence can demonstrate two different scenarios of evolution in spite of similar initial conditions. Within the first scenario, the cross-helicity accumulation is so fast that the energy cascade vanishes before significant magnetic ene…
We study the scaling properties of two-dimensional turbulence using dimensional analysis. In particular, we consider the energy spectrum both at large and small scales and in the "inertial ranges" for the cases of freely decaying and forced turbulence. We also investigate the properties of an "energy condensate" at large scales in spatially finite systems. Finally, an analysis of a possible inverse cascade in freely decaying turbulence is presented.
As the representative of the shock-capturing scheme, the Roe scheme fails to LES because important turbulent characteristics cannot be reproduced such as the famous k-5/3 spectral law owing to large numerical dissipation. In this paper, the Roe scheme is divided into five parts: , , , , and , which means basic upwind dissipation, pressure-difference-driven and velocity-difference-driven modification of the interface fluxes and pressure, respectively. Then, the role of each part on LES is investigated by homogeneous decaying turbulence. The results show that the parts , , and have little effect…
We show that in decaying hydromagnetic turbulence with initial kinetic helicity, a weak magnetic field eventually becomes fully helical. The sign of magnetic helicity is opposite to that of the kinetic helicity - regardless of whether or not the initial magnetic field was helical. The magnetic field undergoes inverse cascading with the magnetic energy decaying approximately like t^{-1/2}. This is even slower than in the fully helical case, where it decays like t^{-2/3}. In this parameter range, the product of magnetic energy and correlation length raised to a certain power slightly larger than…
Mass-Transfer Control With Microbubbles in Highly Turbulent Decaying Flows
George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology（乔治·W·伍德拉夫机械工程学院，佐治亚理工学院）；Renewable Bioproducts Institute, Georgia Institute of Technology（可再生生物制品研究所，佐治亚理工学院）
Numerical schemes used for the integration of complex flow simulations should provide accurate solutions for the long time integrations these flows require. To this end, the performance of various high-order accurate numerical schemes is investigated for direct numerical simulations (DNS) of homogeneous isotropic two-dimensional decaying turbulent flows. The numerical accuracy of compact difference, explicit central difference, Arakawa, and dispersion-relation-preserving schemes are analyzed and compared with the Fourier-Galerkin pseudospectral scheme. In addition, several explicit Runge-Kutta…
Statistical characteristics of freely decaying two-dimensional hydrodynamic turbulence at high Reynolds numbers are numerically studied. In particular, numerical experiments (with resolution up to $8192\times 8192$) provide a Kraichnan-type turbulence spectrum $E_k\sim k^{-3}$. By means of spatial filtration, it is found that the main contribution to the spectrum comes from the sharp vorticity gradients in the form of quasi-shocks. Such quasi-singularities are responsible for a strong angular dependence of the spectrum owing to well-localized (in terms of the angle) jets with minor and/or larg…
Despite the prominence of Onsager's point-vortex model as a statistical description of 2D classical turbulence, a first-principles development of the model for a realistic superfluid has remained an open problem. Here we develop a mapping of a system of quantum vortices described by the homogeneous 2D Gross-Pitaevskii equation (GPE) to the point-vortex model, enabling Monte-Carlo sampling of the vortex microcanonical ensemble. We use this approach to survey the full range of vortex states in a 2D superfluid, from the vortex-dipole gas at positive temperature to negative-temperature states exhi…
The von Karman-Howarth equation implies an infinity of invariants corresponding to an infinity of different asymptotic behaviours of the double and triple velocity correlation functions at infinite separations. Given an asymptotic behaviour at infinity for which the Birkhoff-Saffman invariant is not infinite, there are either none, or only one or only two finite invariants. If there are two, one of them is the Loitsyansky invariant and the decay of large eddies cannot be self-similar. We examine the consequences of this infinity of invariants on a particular family of exact solutions of the vo…
We study the decaying Burgers dynamics in $d$ dimensions for random Gaussian initial conditions. We focus on power-law initial energy spectra, such that the system shows a self-similar evolution. This is the case of interest for the "adhesion model" in cosmology and a standard framework for "decaying Burgers turbulence". We briefly describe how the system can be studied through perturbative expansions at early time or large scale (quasi-linear regime). Next, we develop a saddle-point method, based on spherical instantons, that allows to obtain the asymptotic probability distributions $\cP(η_r)…
Magnetohydrodynamic turbulence affects both terrestrial and astrophysical plasmas. The properties of magnetized turbulence must be better understood to more accurately characterize these systems. This work presents ideal MHD simulations of the compressible Taylor-Green vortex under a range of initial sub-sonic Mach numbers and magnetic field strengths. We find that regardless of the initial field strength, the magnetic energy becomes dominant over the kinetic energy on all scales after at most several dynamical times. The spectral indices of the kinetic and magnetic energy spectra become shall…
We present results from an ensemble of 50 runs of two-dimensional hydrodynamic turbulence with spatial resolution of 2048^2 grid points, and from an ensemble of 10 runs with 4096^2 grid points. All runs in each ensemble have random initial conditions with same initial integral scale, energy, enstrophy, and Reynolds number. When both ensemble- and time-averaged, inverse energy cascade behavior is observed, even in the absence of external mechanical forcing: the energy spectrum at scales larger than the characteristic scale of the flow follows a k^(-5/3) law, with negative flux, together with a …
The main objective of this work is to perform a detailed comparison of the lattice Boltzmann equation (LBE) and the recently developed discrete unified gas-kinetic scheme (DUGKS) methods for direct numerical simulation (DNS) of the decaying homogeneous isotropic turbulence (DHIT) in a periodic box. The flow fields and key statistical quantities computed by both methods are compared with those from pseudo-spectral (PS) method. The results show that the LBE and DUGKS have almost the same accuracy when the flow field is well-resolved, and that the LBE is less dissipative and is slightly more effi…
