Stationary Homogeneous Turbulence Research Articles

Turbulent energy transfers in physical and scale spaces at dissipative length scales

In statistically stationary homogeneous incompressible turbulence, the average energy transfer rate balance which exists at diffusion/dissipation-dominated length scales does not reflect what actually happens locally in space and time. We use a highly resolved direct numerical simulation of forced periodic turbulence to shed some light on the actual fluctuating dynamics which occur at these very small scales and which are rubbed off by averaging. Even though the viscous diffusion in physical space averages to zero and fluctuates less intensely than all other terms (except the energy input rate) in the local (in space–time) two-point energy balance, it fundamentally cannot be neglected. The local unsteadiness and the interspace turbulence transport terms cannot be ignored either in the interscale energy dynamics in spite of the fact that they also average to zero.

Implications of inertial subrange scaling for stably stratified mixing

We investigate the effects of the turbulent dynamic range on active scalar mixing in stably stratified turbulence by adapting the theoretical passive scalar modelling arguments of Beguier, Dekeyser & Launder (1978) (Phys. Fluids, vol. 21 (3), pp. 307–310) and demonstrating their usefulness through consideration of the results of direct numerical simulations of statistically stationary homogeneous stratified and sheared turbulence. By analysis of inertial and inertial–convective subrange scalings, we show that the relationship between the active scalar and turbulence time scales is predicted by the ratio of the Kolmogorov and Obukhov–Corrsin constants, provided mean flow parameters permit the two subrange scalings to be appropriate approximations. We use the resulting relationship between time scales to parameterise an appropriate turbulent mixing coefficient $\varGamma \equiv \chi /\epsilon$ , defined here as the ratio of available potential energy ( $E_p$ ) and turbulent kinetic energy ( $E_k$ ) dissipation rates. With the analysis presented here, we show that $\varGamma$ can be estimated by $E_p,E_k$ and a universal constant provided an appropriate Reynolds number is sufficiently high. This large Reynolds number regime appears here to occur at $ {{Re_b}} \equiv \epsilon / \nu N^{2} \gtrapprox 300$ where $\nu$ is the kinematic viscosity and $N$ is the characteristic buoyancy frequency. We propose a model framework for irreversible diapycnal mixing with robust theoretical parametrisation and asymptotic behaviour in this high- $ {{Re_b}}$ limit.

Scalings of scale-by-scale turbulence energy in non-homogeneous turbulence

A theory of non-homogeneous turbulence is developed and applied to boundary-free shear flows. The theory introduces assumptions of inner and outer similarity for the non-homogeneity of two-point statistics, and predicts power-law scalings of second-order structure functions that have some similarities with but also some differences from Kolmogorov scalings. These scalings arise as a consequence of these assumptions, of the general interscale and interspace energy balance, and of an inner–outer equivalence hypothesis for turbulence dissipation. They reduce to the usual Kolmogorov scalings in stationary homogeneous turbulence. Comparisons with structure function data from three qualitatively different turbulent wakes provide support for the theory's predictions but also raise new questions for future research.

Effect of compressibility on the local flow topology in homogeneous shear turbulence

The local flow topology based on the invariants of the velocity gradient tensor in stationary compressible homogeneous shear turbulence (HST) is studied by numerical simulations. In the compressible homogeneous shear turbulence, local compressibility decreases the flow volume fraction occupied by the focal, eddy, and shear flow structures both in compression regions and in strong expansion regions. The joint probability density function (pdf) of the second and third invariants of the deviatoric velocity gradient tensor exhibits a similar teardrop shape as for the homogeneous isotropic turbulence (HIT), and the tail of the joint pdf alongside the right branch of the null-discriminant curve is elongated as the turbulent Mach number increases. When conditioned on dilatation, the statistical preference for points in the fourth quadrants of the joint pdf is enhanced significantly by the compression motion. It is found that the shape of the joint pdf shows a good similarity between HST and HIT in strong compression regions, which is dependent on the root mean square dilatation, rather than the turbulent Mach number. In strong expansion regions, the shape of the joint pdf in HST has a long tail in the third quadrant, which is related to sheetlike expansion structures and does not exist in HIT. After the Helmholtz decomposition, the properties of local flow topology associated with the solenoidal component of the velocity field are found to be very similar to those in incompressible turbulence and are insensitive to the change in local dilatation and turbulent Mach number.

Effects of bulk viscosity on compressible homogeneous turbulence

The bulk viscosity appearing in the Navier-Stokes equations is generally assumed to be zero for dilute monatomic gases or incompressible fluids. With the growing interest in compressible flows, it is necessary to have a more clear understanding of the role of bulk viscosity and its effects on the properties of flow fields. In the present study, the effects of bulk viscosity on statistically stationary compressible homogeneous turbulence are studied by numerical simulations at the turbulent Mach number range 0.1–0.6. The results of compressible isotropic turbulence and compressible shear turbulence are compared and discussed for three bulk to shear viscosity ratios, i.e., μb/μ=0, 10, and 30. The compensated spectrum of the velocity component shows that a better Mt4 scaling is obtained with a higher bulk to shear viscosity ratio. This observation is confirmed by the one-point statistics of normalized compressible kinetic energy, Kc/Ks, which retains the Mt4 scaling that is predicted by pseudosound theory at turbulent Mach numbers up to 0.6 when μb/μ increases to 30. The analysis of small scale properties reveals that the compressibility of the flow is significantly reduced when bulk viscosity is involved, and the flow fields approach the nearly incompressible state, which is the reason that the Mt4 scaling is observed at a much higher turbulent Mach number compared to the previous studies on the compressible homogeneous turbulence without the bulk viscosity.

Time resolved measurements of droplet preferential concentration in homogeneous isotropic turbulence without mean flow

An experimental investigation of the time-dependent spatial distribution of droplet concentration in a “box” of stationary homogeneous and isotropic turbulence without mean flow was performed for polydispersed droplet clouds with a wide range of mean droplet diameters and droplet size distributions, characterized by a representative Stokes number, based on the droplet arithmetic diameter and the Kolmogorov time scale of the flow, varying between 0.2 and 6, and for turbulent Reynolds numbers of 98, 127, 147, and 235. A novel morphological tracking scheme was used to provide temporal tracking of droplet clusters according to cluster morphological features. The tracking scheme was based on Voronoï space tessellation, multivariate kernel density estimation, and mean shift feature space tracking. The developed tracking scheme was used to establish the relationship between the morphological features of droplet clusters, i.e., number of clusters per unit area and cluster length scales, and the velocity of droplet clusters. The time scale of the droplet clusters increased with both the turbulent Reynolds number and the Stokes number, but its dependence was stronger on the turbulence level. In addition, the effect of filling factors of turbulent flows by turbulent structures on droplet clustering was discussed, which led to suggestions of potential flow control methods to mitigate droplet preferential concentration.

Effect of compressibility on small scale statistics in homogeneous shear turbulence

The effect of compressibility on the small scale properties of stationary homogeneous shear turbulence (HST) is studied in a rectangular domain of size 4π × 2π × 2π by numerical simulations at four turbulent Mach numbers of Mt = 0.14, 0.2, 0.4, and 0.6. Compared to the compressible homogeneous isotropic turbulence (HIT), the small scale properties of HST are more sensitive to Mach number change and the velocity derivatives show a clear deviation from isotropy. At Mt = 0.14, the preferred eigenvalue ratio of the strain rate tensor is very close to −4:1:3 reported in incompressible turbulence. As Mt increases, the conditional probability density functions of the normalized eigenvalues become more dependent on the dilatation, and its ratio tends to −1:0:0 in the strong compression regions at Mt = 0.6, indicating the sheet-like structures of localized shock waves. The alignments between vorticity and eigenvectors are similar to the observations in HIT. After the Helmholtz decomposition, it is found that the compressible vortex stretching term is the primary cause for the enhancement of enstrophy production at high turbulent Mach numbers.

Effects of helicity on dissipation in homogeneous box turbulence

The dimensionless dissipation coefficient $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D700}L/U^{3}$, where $\unicode[STIX]{x1D700}$ is the dissipation rate, $U$ the root-mean-square velocity and $L$ the integral length scale, is an important characteristic of statistically stationary homogeneous turbulence. In studies of $\unicode[STIX]{x1D6FD}$, the external force is typically isotropic and large scale, and its helicity $H_{f}$ either zero or not measured. Here, we study the dependence of $\unicode[STIX]{x1D6FD}$ on $H_{f}$ and find that it decreases $\unicode[STIX]{x1D6FD}$ by up to 10 % for both isotropic forces and shear flows. The numerical finding is supported by static and dynamical upper bound theory. Both show a relative reduction similar to the numerical results. That is, the qualitative and quantitative dependence of $\unicode[STIX]{x1D6FD}$ on the helicity of the force is well captured by upper bound theory. Consequences for the value of the Kolmogorov constant and theoretical aspects of turbulence control and modelling are discussed in connection with the properties of the external force. In particular, the eddy viscosity in large-eddy simulations of homogeneous turbulence should be decreased by at least 10 % in the case of strongly helical forcing.

Spectra and Mach number scaling in compressible homogeneous shear turbulence

The effects of Mach number on the spectra and statistics of stationary compressible homogeneous shear turbulence (HST) are studied using numerical simulations in a rectangular domain (Lx = 4π, Ly = Lz = 2π) at turbulent Mach numbers from 0.05 to 0.66 and Taylor Reynolds numbers from 40 to 100. Long-term simulation results show that a statistically stationary state is obtained and the flow meets the strong acoustic equilibrium assumption at Mt ≈ 0.4. The analysis of spectral properties indicates that velocity and pressure tend toward a Mach number scaling of Mt2 suggested by acoustic dynamics at Mt ≳ 0.3. As for one-point statistics, it is found that a Mt4 scaling predicted by pseudo-sound theory holds for normalized compressible kinetic energy, Kc/Ks, at the small turbulent Mach number Mt ≲ 0.1. Kc/Ks approaches a Mt2 scaling at relatively higher turbulent Mach numbers, which is consistent with the spectral results. The normalized compressible dissipation rate, ϵc/ϵs, is nearly independent of Taylor Reynolds number and exhibits the same Mt4 scaling at small turbulent Mach numbers. The root mean square values of pressure, density, and temperature of compressible HST show good agreement with the Mt2 scaling, with the coefficient approximately doubled as compared with the compressible isotropic turbulence.

Limitations of the clump-correlation theories of shear-induced turbulence suppression

The clump theory, primarily constructed by Dupree [Phys. Fluids 15, 334 (1972)] based on the moment approach and then generalized to the correlation theory [Y. Z. Zhang and S. M. Mahajan, Phys. Fluids B 5, 2000 (1993)], has long served as a basis for constructing theories of turbulence suppression by shear flow. In order to reveal the “intrinsic approximation” invoked in the clump-correlation theory, we examine a model based on two dimensional magnetized drift waves. After a rigorous derivation of the exact response function—a key to average the Green function of the system—we show that the Dupree, Zhang-Mahajan approach is recovered as the lowest order approximation in a small dimensionless parameter ϒ which is a triple product of the correlation time, wave number, and fluctuating drift velocity. The clump-correlation theory, thus, constitutes the Gaussian and lowest order non-Markovian process for a homogeneous stationary turbulence. We also provide, especially for the tokamak community, a readily usable formula to evaluate the effectiveness of shear-flow suppression; this formula pertains regardless of the specific model of correlation time.

Coherent structures in statistically stationary homogeneous shear turbulence

The three-dimensional vortex clusters, and the structures based on the quadrant classification of the intense tangential Reynolds stress (Qs), are studied in direct numerical simulations of statistically stationary homogeneous shear turbulence (HST) at Taylor microscale Reynolds number $Re_{\unicode[STIX]{x1D706}}\approx 50{-}250$, with emphasis on comparisons with turbulent channels (CHs). The Qs and vortex clusters in HST are found to be versions of the corresponding detached (in the sense of del Álamo et al. (J. Fluid Mech., vol. 561 (2006), pp. 329–358)) structures in CHs, although statistically symmetrised with respect to the substitution of sweeps by ejections and vice versa. In turn, these are more symmetric versions of the corresponding attached Qs and clusters. In both flows, only co-gradient sweeps and ejections larger than the local Corrsin scale are found to couple with the shear. They are oriented anisotropically, and are responsible for carrying most of the total Reynolds stress. Most large eddies in CHs are attached to the wall, but it is shown that this is probably a geometric consequence of their size, rather than the reason for their dynamical significance. Most small Q structures associated with different quadrants are far from each other in comparison to their size, but those that are close to each other tend to form quasi-streamwise trains of groups of a sweep and an ejection paired side by side in the spanwise direction, with a vortex cluster in between, generalising to three dimensions the corresponding arrangement of attached eddies in CHs. These pairs are organised around an inclined large-scale conditional vortex ‘roller’, and it is shown that the composite structure tends to be located at the interface between high- and low-velocity streaks, as well as in strong ‘co-gradient’ shear layers that separate streaks of either sign in which velocity is more uniform. It is further found that the conditional rollers are terminated by ‘hooks’ reminiscent of hairpins, both upright and inverted. The inverted hook weakens as the structures approach the wall, while the upright one changes little. At the same time, the inclination of the roller with respect to the mean velocity decreases from $45^{\circ }$ in HST to quasi-streamwise for wall-attached eddies. Many of these observations are generalised to intense Reynolds stresses formed with different pairs of velocity components, and it is shown that most properties of the small structures can be traced to their definitions, rather than to their dynamics. It is concluded that the larger Reynolds-stress structures are associated with shear turbulence, rather than with the presence of a wall, while the smaller ones are generic to turbulence in general, whether sheared or not.

Transport equation for plasmas in a stationary-homogeneous turbulence

For a plasma in a stationary homogeneous turbulence, the Fokker-Planck equation is derived from the nonlinear Vlasov equation by introducing the entropy principle. The ensemble average in evaluating the kinetic diffusion tensor, whose symmetry has been proved, can be computed in a straightforward way when the fluctuating particle trajectories are provided. As an application, it has been shown that a mean parallel electric filed can drive a particle flux through the Stokes-Einstein relation, independent of the details of the fluctuations.

Transport, mixing and agglomeration of particles in turbulent flows

This paper describes methods and approaches that have been used to simulate and model the transport, mixing and agglomeration of small particles in a flowing turbulent gas. The transported particles because of their inertia are assumed not to follow the motion of the large scales of the turbulence and or the motion of the small dissipating scales of the turbulence. We show how both these behaviours can be represented by a PDF approach analogous to that used in Classical Kinetic Theory. For large scale dispersion the focus is on transport in simple generic flows like statistically stationary homogeneous and isotropic turbulence and simple shear flows. Special consideration is given to the transport and deposition of particles in turbulent boundary layers. For small scale transport the focus is on how the the small scales of turbulence together with the particle inertial response enhances collision processes like particle agglomeration. In this case the importance of segregation and the formation of caustics, singularities and random uncorrelated motion is highlighted and discussed.

Application of Millionschikov's hypothesis to homogeneous and isotropic turbulence possessing helicity

In this manuscript we construct a deductive theory for the stationary homogeneous turbulence, which is invariant under translations and rotations but lacks reflexional symmetry, considering the appropriate forms for the second- and third-order correlation functions. Appropriate invariant forms are considered for some second and third order correlation tensors, formed from the fluctuating components of velocity, acceleration, components of external force and pressures. In the light of the formulation of Millionschikov’s hypothesis we would point out further the treatment of first author [4] on the final period decay of isotropic turbulence using the points-merging technique employed here. We include also some enlightments of the process of future work that could be undertaken in this field of research.

Analytical Expressions for Doppler Spectra of Scatter from Hydrometeors Observed with a Vertically Directed Radar Beam

Abstract A generalized expression is derived for the correlation function of signals backscattered from hydrometeors observed with a vertically pointed radar beam in which particle size distribution, turbulence, and mean wind are not homogeneous. This study extends the previous work of Fang and Doviak by including the effects of particle size distribution on the measured Doppler spectrum. It shows the measured Doppler spectrum to be the volumetric mean of the weighted convolution of the normalized Doppler spectra associated with turbulence, mean wind, particle oscillation/wobble, and terminal velocity. Without particle oscillation/wobble, mean wind, and turbulence, the Doppler spectrum is the mirror image of the terminal velocity spectrum under the condition that the second-order effect of finite beamwidth can be ignored. This generalized Doppler spectrum reduces further to a previously derived expression if the particle size distribution, or equivalently reflectivity, is uniform. Provided there is a unique relationship between the particle’s terminal velocity and its effective diameter, the derived equations can be applied to scatterers consisting of ice particles as well as water droplets. This study derives the analytical expression for the Doppler spectrum of mean wind and also shows that if stationary homogeneous turbulence is the only contributor to spectral broadening, then the average of a large number of radar-measured Doppler spectra will be equal to the velocity probability density function of turbulence independent of the angular, range, reflectivity, and weighting functions.

Direct numerical simulation of stationary homogeneous stratified sheared turbulence

AbstractUsing direct numerical simulation, we investigate stationary and homogeneous shear-driven turbulence in various stratifications, ranging from neutral to very stable. To attain and maintain a stationary flow, we throttle the mean shear so that the net production stays constant for all times. This results in a flow that is characterized solely by its mean shear and its mean buoyancy gradient, independent of initial conditions. The method of throttling is validated by comparison with experimental spectra in the case of neutral stratification. With increasing stratification comes the emergence of vertically sheared large-scale horizontal motions that preclude a straightforward interpretation of flow statistics. However, once these motions are excluded, simply by subtracting the horizontal average, the underlying flow appears amenable to the standard methods of turbulence analysis. It is shown that a direct acknowledgement of the confining influence of the periodic simulation box can lead to a meaningful physical interpretation of the large scales. Once an appropriate confinement scale is identified, many features, including horizontal spectra, flux–gradient relationships and length scales, of stratified sheared turbulence can be readily understood, both qualitatively and quantitatively, in terms of Monin–Obukhov similarity theory. Finally, the similarity-theory framework is used to interpret the scaling of the vertical diapycnal diffusivity in stratified turbulence.

New sub-grid stochastic acceleration model in LES of high-Reynolds-number flows

The experimental observations of intermittent dynamics of Lagrangian acceleration in a "free" high-Reynolds-number turbulence are shown to be consistent with the Kolmogorov-Oboukhov theory. In line with Kolmogorov-Oboukhov's predictions, a new sub-grid scale (SGS) model is proposed and is combined with the Smagorinsky model. The new SGS model is focused on simulation of the non-resolved total acceleration vector by two stochastic processes: one for its norm, another for its direction. The norm is simulated by stochastic equation, which was derived from the log-normal stochastic process for turbulent kinetic energy dissipation rate, with the Reynolds number, as the parameter. The direction of the acceleration vector is suggested to be governed by random walk process, with correlation on the Kolmogorov's timescale. In the framework of this model, a surrogate unfiltered velocity field is emulated by computation of the instantaneous model-equation. The coarse-grid computation of a high-Reynolds-number stationary homogeneous turbulence reproduced qualitatively the main intermittency effects, which were observed in experiment of ENS in Lyon. Contrary to the standard LES with the Smagorinsky eddy-viscosity model, the proposed model provided: (i) non-Gaussianity in the acceleration distribution with stretched tails; (ii) rapid decorrelation of acceleration vector components; (iii) "long memory" in correlation of its norm. The turbulent energy spectra of stationary and decaying homogeneous turbulence are also better predicted by the proposed model.

On the Velocity Gradient in Stably Stratified Sheared Flows. Part 1: Asymptotic Analysis and Applications

We give a new derivation of the familiar linear relation for the dimensionless velocity gradient in the stably stratified surface layer and provide physical and empirical grounds for its universal applicability in stationary homogeneous turbulence over the whole range of static stabilities from Ri = 0 to very large Ri. Combining this relation with the budget equation for the turbulent kinetic energy we obtain the “equilibrium formulation” of the turbulent dissipation length scale, and recommend it for use in turbulence closure models.

Spectral properties of helical turbulence

The spectral properties of homogeneous stationary turbulence excited by a force that introduces considerable helicity, along with energy, into the flow are considered. It is shown that helicity is transferred on the inertial range as a passive admixture and its spectrum obeys the “−5/3” law much more accurately than the velocity fluctuation spectrum. The helicity is dissipated on the same scales as the energy, though the helicity transfer dynamics on the inertial range are different on the large and small scales. Numerical experiments are performed on the basis of a cascade model developed for describing helical turbulence.

Radiation statistics in homogeneous isotropic turbulence

An analysis of turbulence–radiation interaction (TRI) in statistically stationary (forced) homogeneous and isotropic turbulence is presented. A direct numerical simulation code was used to generate instantaneous turbulent scalar fields, and the radiative transfer equation (RTE) was solved to provide statistical data relevant in TRI. The radiation intensity is non-Gaussian and is not spatially correlated with any of the other turbulence or radiation quantities. Its power spectrum exhibits a power-law region with a slope steeper than the classical −5/3 law. The moments of the radiation intensity, Planck-mean and incident-mean absorption coefficients, and emission and absorption TRI correlations are calculated. The influence of the optical thickness of the medium, mean and variance of the temperature and variance of the molar fraction of the absorbing species is studied. Predictions obtained from the time-averaged RTE are also included. It was found that while turbulence yields an increase of the mean blackbody radiation intensity, it causes a decrease of the time-averaged Planck-mean absorption coefficient. The absorption coefficient self-correlation is small in comparison with the temperature self-correlation, and the role of TRI in radiative emission is more important than in radiative absorption. The absorption coefficient–radiation intensity correlation is small, which supports the optically thin fluctuation approximation, and justifies the good predictions often achieved using the time-averaged RTE.