Stationary Homogeneous Isotropic Turbulence Research Articles

Estimating the turbulent kinetic energy dissipation rate from one-dimensional velocity measurements in time

Abstract. The turbulent kinetic energy dissipation rate is one of the most important quantities characterizing turbulence. Experimental studies of a turbulent flow in terms of the energy dissipation rate often rely on one-dimensional measurements of the flow velocity fluctuations in time. In this work, we first use direct numerical simulation of stationary homogeneous isotropic turbulence at Taylor-scale Reynolds numbers 74≤Rλ≤321 to evaluate different methods for inferring the energy dissipation rate from one-dimensional velocity time records. We systematically investigate the influence of the finite turbulence intensity and the misalignment between the mean flow direction and the measurement probe, and we derive analytical expressions for the errors associated with these parameters. We further investigate how statistical averaging for different time windows affects the results as a function of Rλ. The results are then combined with Max Planck Variable Density Turbulence Tunnel hot-wire measurements at 147≤Rλ≤5864 to investigate flow conditions similar to those in the atmospheric boundary layer. Finally, practical guidelines for estimating the energy dissipation rate from one-dimensional atmospheric velocity records are given.

Vortex stretching and enstrophy production in stationary homogeneous isotropic turbulence

On average, the length of a material line element at any given time is longer than that at the initial time. Because a vortex line is frozen into the fluid at an infinite Reynolds number, it seems that the relation between the mean enstrophies (half the squared vorticity) at two time instances can be easily derived using a method similar to that for a material line element. However, this relation has not been analytically derived because a vortex line depends on the velocity even at the initial time. In this paper, we analytically show, using the conditional average, that the mean enstrophy normalized by the initial enstrophy at any given time is larger than one. We also analytically show that the mean enstrophy production is positive in stationary homogeneous isotropic turbulence for high Reynolds numbers.

A tutorial on reproducing a predefined autocovariance function through AR models: application to stationary homogeneous isotropic turbulence

Sequential methods for synthetic realisation of random processes have a number of advantages compared with spectral methods. In this article, the determination of optimal autoregressive (AR) models for reproducing a predefined target autocovariance function of a random process is addressed. To this end, a novel formulation of the problem is developed. This formulation is linear and generalises the well-known Yule-Walker (Y-W) equations and a recent approach based on restricted AR models (Krenk-Møller approach, K-M). Two main features characterise the introduced formulation: (i) flexibility in the choice for the autocovariance equations employed in the model determination, and (ii) flexibility in the definition of the AR model scheme. Both features were exploited by a genetic algorithm to obtain optimal AR models for the particular case of synthetic generation of homogeneous stationary isotropic turbulence time series. The obtained models improved those obtained with the Y-W and K-M approaches for the same model parsimony in terms of the global fitting of the target autocovariance function. Implications for the reproduced spectra are also discussed. The formulation for the multivariate case is also presented, highlighting the causes behind some computational bottlenecks.

Péclet-number dependence of small-scale anisotropy of passive scalar fluctuations under a uniform mean gradient in isotropic turbulence

We study passive scalar fluctuations convected by statistically stationary homogeneous isotropic turbulence under a uniform mean scalar gradient. In order to elucidate the parameter dependence of small-scale statistics of scalar fluctuations, we conduct direct numerical simulations of passive scalar turbulence with 59 different combinations of Reynolds number and Schmidt number. For all the cases, we compute time-average statistics of various quantities, which include the scalar derivative skewness and flatness, the ratio of parallel-to-perpendicular scalar-gradient variances, and the anisotropy parameter recently proposed (Hill, Phys. Rev. Fluids, vol. 2, 2017, 094601). Notably, the degree of small-scale anisotropy of passive scalar fluctuation is characterised by a universal function of the Peclet number , where is the root mean square velocity, the Taylor microscale of scalar fluctuation, the mass diffusivity. In the definition of the Peclet number, the use of , rather than the Taylor microscale of velocity fluctuation, is key to collapsing the data of different Reynolds and Schmidt numbers. When the Peclet number is low, large-scale anisotropic scalar structures emerge irrespective of the Reynolds number. These structures are elongated along the direction of the uniform mean scalar gradient, and their size is significantly larger than the integral length scale of velocity fluctuation.

Turbulence modulation in particle-laden stationary homogeneous isotropic turbulence using one-dimensional turbulence

Turbulence modulation in particle-laden stationary homogeneous isotropic turbulence is investigated using one-dimensional turbulence (ODT), a low-dimensional stochastic flow simulation model. For this purpose, ODT is extended in two ways. First, a forcing scheme that maintains statistical stationarity is introduced. Unlike direct numerical simulation (DNS) of forced turbulence, the ODT framework accommodates forcing that is not directly coupled to the momentum equation. For given forcing the ODT energy dissipation rate is therefore the same in particle-laden cases as in the corresponding single-phase reference case. Second, previously implemented one-way-coupled particle phenomenology is extended to two-way coupling using the general ODT methodology for flow modulation through interaction with any specified energy and momentum sources and sinks. As in a DNS comparison case for Re-lambda = 70, turbulence modulation is diagnosed primarily on the basis of the fluid-phase kinetic-energy spectrum. Because ODT involves subprocesses with straightforward physical interpretations, the ODT mechanisms of particle-induced turbulence modulation are clearly identified and they are plausibly relevant to particleladen Navier-Stokes turbulence. ODT results for the ratio of particle-phase and fluid-phase kinetic energies as a function of particle Stokes number and mass loading are reported for the purpose of testing these predictions in the future when these quantities are evaluated experimentally or using DNS.

Fully Correlated Stochastic Inter-Particle Collision Model for Euler\u2013Lagrange Gas\u2013Solid Flows

In Lagrangian stochastic collision models, a fictitious particle is generated to act as a collision partner, with a velocity correlated to the velocity of the real colliding particle. However, most often, the fluid velocity seen by this fictitious particles is not accounted for in the generation of the fictitious particle velocity, leading to a de-correlation between the fictitious particle velocity and the local fluid velocity, which, after collision, leads to an unrealistic de-correlation of the real particle velocity and the fluid velocity as seen by the particle. This de-correlation, in turn, causes a spurious decrease of the particle kinetic energy, even though the collisions are assumed perfectly elastic. In this paper, we propose a new model in which the generated fictitious particle velocity is correctly correlated to both the real particle velocity and the local fluid velocity at the particle, hence preventing the spurious loss of the total particle kinetic energy. The model is suitable for small inertial particles. Two algorithms for integrating the collision frequency are also compared to each other. The models are validated using large eddy simulation (LES) of mono-dispersed particle-laden stationary homogeneous isotropic turbulence. Simulations are conducted with spherical particles with different turbulent Stokes number, St_t = [0.75 - 5.8], and volume fractions, alpha _p = [0.014 - 0.044], and are compared to the results of the LES using a deterministic discrete particle simulation model.

Direct Numerical Simulation of Turbulent Combustion of Hydrogen—Air Mixtures of Various Compositions in a Two-Dimensional Approximation

A technique of two-dimensional direct numerical simulation of turbulent flame propagation in reacting gas mixtures under stationary homogeneous isotropic turbulence conditions is proposed. This technique is based on a detailed kinetic mechanism of combustion of a multicomponent mixture and uses no fitting parameters. It is applied to the calculation of turbulent combustion of a hydrogen-air mixture. A condition is proposed to compare the results of two-dimensional calculations (dependences of flame propagation velocity on turbulence intensity) with the data of actual three-dimensional experiments. The obtained agreement between the calculated and measured dependences confirmed the validity of the proposed condition. The effects of pressure on the flame propagation velocity were considered. The calculated concentrations of the active reaction centers-hydroxyl (OH) and H and O atoms-in turbulent flame are lower than those in laminar flame, which also agrees with experimental results.

The Renormalization-Group Method in the Problem on Calculation of the Spectral Energy Density of Fluid Turbulence

In order to find the shape of energy spectrum within the framework of the model of stationary homogeneous isotropic turbulence, the renormalization-group equations, which reflect the Markovian nature of the mechanism of energy transfer along the wavenumber spectrum, are used in addition to the dimensional considerations and the energy balance equation. For the spectrum, the formula depends on three parameters, namely, the wavenumber, which determines the upper boundary of the range of the turbulent energy production, the spectral flux through this boundary, and the fluid kinematic viscosity.

Streamlines in stationary homogeneous isotropic turbulence and fractal-generated turbulence

We compare streamline statistics in stationary homogeneous isotropic turbulence and in turbulence generated by a fractal square grid. We examine streamline segments characterised by the velocity difference and the distance l between extremum points. We find close agreement between the stationary homogeneous isotropic turbulence and the decay region of the fractal-generated turbulence as well as the production region of the fractal flow for small segments. The statistics of larger segments are very similar for the isotropic turbulence and the decay region, but differ for the production region. Specifically, we examine the first, second and third conditional mean . Noticeably, non-vanishing for are due to an asymmetry of positive and negative segments, i.e. those for which and , respectively. This asymmetry is not only kinematic, but is also due to dissipative effects and therefore contains cascade information.

A direct numerical simulation study of interface propagation in homogeneous turbulence

A 3D direct numerical simulation (DNS) study of the evolution of a self-propagating interface in forced constant-density statistically stationary homogeneous isotropic turbulence was performed by solving Navier–Stokes and level-set equations under a wide range of conditions that cover various (from 0.1 to 2.0) ratios of the interface speed $S_{L}$ to the r.m.s. turbulent velocity $U^{\prime }$ and various (50, 100 and 200) turbulent Reynolds numbers$\mathit{Re}$. By analysing computed data, the following issues were addressed: (i) dependence of the speed and thickness of the fully developed statistically planar mean front that envelops the interface on$U^{\prime }/S_{L}$and$\mathit{Re}$, (ii) dependence of the fully developed mean turbulent flux of a scalar$c$that characterizes the state of the fluid ($c=0$and 1 ahead and behind the interface respectively) on$U^{\prime }/S_{L}$and$\mathit{Re}$, (iii) evolution of the mean front speed, its thickness, and the mean scalar flux during the front development after embedding a planar interface into the forced turbulence and (iv) relation between canonical and conditioned moments of the velocity, velocity gradient and pressure gradient fields.

Stochastic vortex structure method for modeling particle clustering and collisions in homogeneous turbulence

Current particle dispersion models do not accurately predict the particle clustering that occurs in turbulent flow due to interaction of the particles with turbulent eddies. This clustering arises due to the effects of centrifugal forces which act to throw heavy particles out of the turbulent eddies, causing the particles to collect in high-concentration sheets lying between the eddies. The current paper proposes a stochastic vortex structure (SVS) model for simulation of particle clustering and collisions in turbulent flows. A new measure for particle drift relative to the fluid velocity is proposed that is related to the cross product of the fluid acceleration and velocity fields. Tests were conducted comparing the predictions of the SVS model with direct numerical simulation (DNS) and with three different stochastic Lagrangian methods in statistically stationary homogeneous isotropic turbulence with particles having Stokes numbers based on integral length scaling of order unity, assuming one-way fluid-particle coupling. The tests examined different turbulent flow features that are important for particle dispersion and clustering, as well as for prediction of the particle collision rate and collision distribution. The results indicate that the SVS model performs reasonably well for predicting particle concentration heterogeneity and collision rate, and that differences between the SVS and DNS results can be attributed to the fact that the SVS model neglects the small-scale velocity fluctuations within the turbulent flow.

Preferential concentration of poly-dispersed droplets in stationary isotropic turbulence

The preferential concentration of poly-dispersed water droplets with a range of Sauter mean diameters between 25 and 95 μm has been studied experimentally in stationary homogeneous isotropic turbulence with four different intensities, characterized by turbulent Reynolds numbers based on Taylor microscale, of Reλ = 107, 145, 185 and 213. The image processing method of recorded scattered light intensity images from droplets is described and its ability to identify droplets is assessed in terms of image quality. The influence of image processing parameters on measured characteristics of droplet clustering is evaluated. The radial distribution function (RDF) and 2D Voronoi analysis quantified the magnitude of preferential droplet concentration and the results from both methods agreed well. RDF showed that the characteristic length scale of resulting droplet clusters varies between 20 and 30 times the Kolmogorov length scale over all the experimental conditions. It was found that the preferential concentration is more appropriately described by a Stokes number, based on various representative diameters, namely the arithmetic mean diameter, D10, or the diameter, DN60 %, below which 60 % of the total droplet number in the spray is present, or the diameter, DV5 %, which carries 5 % of the total liquid volume in the spray. The magnitude of droplet preferential concentration was maximum when the proposed Stokes number was around unity for all experimental conditions. Little dependence of the magnitude of preferential concentration on turbulent Reynolds numbers was found, in contrast to the recent DNS findings (Tagawa et al. in J Fluid Mech 693:201–215, 2012).

Direct-numerical-simulation-based measurement of the mean impulse response of homogeneous isotropic turbulence

A technique for measuring the mean impulse response function of stationary homogeneous isotropic turbulence is proposed. Such a measurement is carried out here on the basis of direct numerical simulation (DNS). A zero-mean white-noise volume forcing is used to probe the turbulent flow, and the response function is obtained by accumulating the space-time correlation between the white forcing and the velocity field. This technique to measure the turbulent response in a DNS numerical experiment is a research tool in that field of spectral closures where the linear-response concept is invoked either by resorting to renormalized perturbations theories or by introducing the well-known fluctuation-dissipation relation (FDR). Although the results obtained in the present work are limited to relatively low values of the Reynolds number, a preliminary analysis is possible. Both the characteristic form and the time scaling properties of the response function are investigated in the universal subrange of dissipative wave numbers; a comparison with the response approximation given by the FDR is proposed through the independent DNS measurement of the correlation function. Very good agreement is found between the measured response and Kraichnan's description of random energy-range advection effects.

Acceleration-based classification and evolution of fluid flow structures in two-dimensional turbulence

Zero acceleration points (ZAPs) and flow structures around them are studied in a direct numerical simulation of two-dimensional energy-cascading stationary homogeneous isotropic turbulence with an extended k(-5/3) energy spectrum. A well-defined classification of ZAPs in terms of the acceleration gradient tensor's (∇a) invariants emerges naturally as a result of well-defined properties of and relations between these invariants at ZAPs. About half of all ZAPs are anti-ZAPs [with det(∇a)<0 ] and the number of vortical and straining ZAPs [with det(∇a)>0 ] is about the same. Vortical and straining ZAPs are swept by the local fluid velocity to a good statistical approximation whereas anti-ZAPs, which are present in every vortical and straining ZAP's creation and destruction events, are not. The average lifetime of ZAPs seems to scale with the time scale of the smallest eddies in the turbulence, though ZAPs (in particular vortical ones) are able to survive up to a few integral time scales. Our ZAP classification can also be applied to extended flow regions and it turns out that vortical and straining regions are mediated by regions containing anti-ZAPs. A discussion of the length scales and sizes characterizing these regions and the distances between ZAPs is also given.

The influence of the non-resolved scales of thermal radiation in large eddy simulation of turbulent flows: A fundamental study

The filtered radiative transfer equation has been solved without subgrid-scale model and compared to a-priori estimations from direct numerical simulation of statistically stationary homogeneous isotropic turbulence. The importance of the large eddy simulation non-resolved scales on the thermal radiation is analysed in this academic case. The influence of the mean and variance of temperature of the system on the subgrid-scale correlations is studied. It is shown that the turbulence–radiation interaction (TRI) is greater for low-temperature flows, where the radiative transfer is lower. Moreover, the strongest subgrid-scale correlations, which are the temperature self-correlation and the absorption coefficient-temperature correlation, have opposite effects, which suggests that it is better to neglect both correlations, instead of modelling only one. Therefore, the assumption of neglecting the TRI in LES leads to good predictions. Three-dimensional filtering effects, as well as the grid influence on non-local quantities, such as the radiation intensity, are also analysed. These effects have a local influence, but are negligible in the tested cases.

The dissipation rate coefficient of turbulence is not universal and depends on the internal stagnation point structure

The energy dissipation rate coefficient of statistically stationary homogeneous isotropic turbulence depends on the external force sustaining the turbulence irrespective of Reynolds number. This nonuniversality is established by proving that the Taylor length is proportional to the mean distance between stagnation points and thereby relating the energy dissipation rate coefficient to the stagnation point structure of the turbulence which is shown to depend on the structure of the large eddies. Confirmation of these relations is obtained at moderate Reynolds numbers by a series of direct numerical simulations where the large-scale forcing is systematically varied.

Dispersion of heavy particles in stably stratified turbulence

The dispersion of heavy inertial particles in statistically stationary stably stratified turbulence is studied by means of direct numerical simulations. The following issues have been addressed: What distinguishes dispersion in such stratified flows from dispersion processes in statistically stationary homogeneous isotropic turbulence? How is the dispersion process affected by the Stokes number of the inertial particles (0.1≲St=τp/τK≲10, with τp the particle response time and τK the Kolmogorov time)? What is the interplay between buoyancy and the Stokes number? And what is the effect, if any, of particle settling, nonlinear drag, and lift forces (particularly relevant for stratified turbulence with its vertical shear layers) on particle dispersion? The long-time dispersion in isotropic turbulence is found to be maximum around St=1, in agreement with the observation of preferential concentration for St≈1. In stably stratified turbulence such a maximum in the dispersion is only found for the horizontal direction. The horizontal and vertical dispersions in stably stratified turbulence show different behaviors due to the anisotropy of the flow, and in particular, vertical dispersion is strongly affected by the inertia of the particles. With increasing St the classical plateau found for vertical fluid particle dispersion becomes less pronounced and it even vanishes for Stokes numbers of O(10) and higher. Furthermore, the long-time vertical dispersion increases with increasing St. The effects of gravity, nonlinear drag, and lift forces have been considered in more detail. It turned out that the settling enhancement of inertial particles, as observed in isotropic turbulence, is suppressed by stratification and by nonlinear drag effects. Moreover, nonlinear drag only affects the dispersion in the vertical direction in stably stratified turbulence. Finally, it is found that lift forces can safely be neglected for dispersion studies under the current parameter settings.

Lagrangian stochastic models for turbulent relative dispersion based on particle pair rotation

The physical picture of a fluid particle pair as a couple of material points rotating around their centre of mass is proposed to model turbulent relative dispersion in the inertial range. This scheme is used to constrain the non-uniqueness problem associated to the Lagrangian models in the well-mixed class and the properties of the stochastic process derived are analysed with respect to some turbulent velocity characteristics. A simple illustrative Markov model is developed in stationary homogeneous isotropic turbulence and the particle separation statistics are compared with direct numerical simulation data. In spite of the simplicity of the model, a consistent comparison is observed in the inertial range, supporting the formulation proposed.

Reynolds-number dependence of line and surface stretching in turbulence: folding effects

The stretching rate, normalized by the reciprocal of the Kolmogorov time, of sufficiently extended material lines and surfaces in statistically stationary homogeneous isotropic turbulence depends on the Reynolds number, in contrast to the conventional picture that the statistics of material object deformation are determined solely by the Kolmogorov-scale eddies. This Reynolds-number dependence of the stretching rate of sufficiently extended material objects is numerically verified both in two- and three-dimensional turbulence, although the normalized stretching rate of infinitesimal material objects is confirmed to be independent of the Reynolds number. These numerical results can be understood from the following three facts. First, the exponentially rapid stretching brings about rapid multiple folding of finite-sized material objects, but no folding takes place for infinitesimal objects. Secondly, since the local degree of folding is positively correlated with the local stretching rate and it is non-uniformly distributed over finite-sized objects, the folding enhances the stretching rate of the finite-sized objects. Thirdly, the stretching of infinitesimal fractions of material objects is governed by the Kolmogorov-scale eddies, whereas the folding of a finite-sized material object is governed by all eddies smaller than the spatial extent of the objects. In other words, the time scale of stretching of infinitesimal fractions of material objects is proportional to the Kolmogorov time, whereas that of folding of sufficiently extended material objects can be as long as the turnover time of the largest eddies. The combination of the short time scale of stretching of infinitesimal fractions and the long time scale of folding of the whole object yields the Reynolds-number dependence. Movies are available with the online version of the paper.

Lagrangian conditional statistics, acceleration and local relative motion in numerically simulated isotropic turbulence

Lagrangian statistics of fluid-particle velocity and acceleration conditioned on fluctuations of dissipation, enstrophy and pseudo-dissipation representing different characteristics of local relative motion are extracted from a direct numerical simulation database of stationary (forced) homogeneous isotropic turbulence. The grid resolution in the simulations is up to 20483, and the Taylor-scale Reynolds number ranges from about 40 to 650, where characteristics of small-scale intermittency in the Eulerian flow field are well developed. A key joint statistic of the conditioning variables is the dissipation-enstrophy cross-correlation, which is asymmetric, but becomes less so at high Reynolds number. Conditional velocity autocorrelations are consistent with rapid changes in the velocity of fluid particles moving in regions of large velocity gradients. Examination of statistics conditioned upon enstrophy, especially in a local coordinate frame moving with the vorticity vector, and of the centripetal acceleration suggests the presence of vortex-trapping effects which persist for several Kolmogorov time scales. Further results on acceleration statistics and joint velocity-acceleration autocorrelations are also presented to help characterize in detail the properties of a joint stochastic process of velocity, acceleration and the pseudo-dissipation. Together with recent work on Eulerian conditional acceleration and Reynolds-number dependence of basic Lagrangian quantities, the present results are directly useful for the development of a new stochastic model formulated to account for intermittency and Reynolds-number effects as described in detail in a companion paper.