Isotropic Turbulent Velocity Research Articles

Characteristics of clustered particle relative velocity in homogeneous and isotropic turbulence

Particle collisions are mainly governed by the preferential concentration of inertia particles and the formation of fold caustics. By fold caustics, we mean that relative velocity does not go smoothly to zero when the particle separations decrease due to inertia. Despite the importance of the second-order relative velocity structure function, there has been relatively little experimental research on the formation of caustics due to the high accuracy requirements for the particle relative velocity measurements. In the dissipation range, an obvious departure between the second-order structure function of particles normalized by the square of the Kolmogorov velocity and the Kolmogorov turbulent scaling of r2 was observed for all four experimental conditions. In the inertial range, the second-order structure function normalized by the square of the Kolmogorov velocity was consistent with the scale form of r2/3 in all cases. The conditional second-order relative velocity structure function of particles in clusters differs from that of the arithmetically averaged particle statistics in both the dissipation range and the inertial subrange. In the dissipation range, caustics are present for all categories of particles. In the inertial range, the scaling of the second-order velocity structure function is almost identical for all categorized particle types.

Transition of fluctuations from Gaussian state to turbulent state.

Variation of the statistical properties of an incompressible velocity, passive vector and passive scalar in isotropic turbulence was studied using direct numerical simulation. The structure functions of the gradients, and the moments of the dissipation rates, began to increase at about from the Gaussian state and grew rapidly at in the turbulent state. A contour map of the probability density functions (PDFs) indicated that PDF expansion of the gradients of the passive vector and passive scalar begins at around , whereas that of the longitudinal velocity gradient PDF is more gradual. The left tails of the dissipation rate PDF were found to follow a power law with an exponent of 3/2 for the incompressible velocity and passive vector dissipation rates, and 1/2 for the scalar dissipation rate and the enstrophy; they remained constant for all Reynolds numbers, indicating the universality of the left tail. The analytical PDFs of the dissipation rates and enstrophy of the Gaussian state were obtained and found to be the Gamma distribution. It was shown that the number of terms contributing to the dissipation rates and the enstrophy determines the decay rates of the two PDFs for low to moderate amplitudes.This article is part of the theme issue ‘Scaling the turbulence edifice (part 1)’.

Another look at rotational sampling of atmospheric turbulence with focus on the transference of energy from different frequency intervals of the Eulerian spectrum to the rotational spectrum

Rotational sampling is of paramount importance in the loading of wind turbine blades due to wind turbulence We revisit the classical model of rotational sampling of the longitudinal component of a statistically stationary, homogeneous, isotropic turbulent velocity field experienced by a point rotating in a plane perpendicular to the mean velocity. As is well known, this is the simplest frame work, since it allows to determine the non-dimensional rotational spectrum in terms of the non-dimensional frequency and only two more non dimensional parameters, the non dimensional radial position of the rotating point and the non dimensional rotational speed. We systematize the parametric analysis of the rotational spectrum proposed independently by Connel in 1981 and Kristensen et al. in 1982, in terms of the aforementioned two non dimensional parameters. We propose a quantification of the transference of energy from different frequency intervals of the Eulerian spectrum to the rotational spectrum. We verify that regions of the Eulerian spectrum corresponding to different frequency intervals are transformed into contributions to the rotational spectrum that expand to the whole frequency range without preserving the variance. Certain frequency intervals of the Eulerian spectrum can lead to negative contributions to the rotational spectrum along certain frequency ranges.

Triple Hill’s Vortex Synthetic Eddy Method

The generation of initial or inflow synthetic turbulent velocity or scalar fields reproducing statistical characteristics of realistic turbulence is still a challenge. The synthetic eddy method, previously introduced in the context of inflow conditions for large eddy simulations, is based on the assumption that turbulence can be regarded as a superposition of coherent structures. In this paper, a new type of synthetic eddy method is proposed, where the fundamental eddy is constructed by superposing three Hill's vortices, with their axes orthogonal to each other. A distribution of Hill's vortices is used to synthesize an anisotropic turbulent velocity field that satisfies the incompressibility condition and match a given Reynolds stress tensor. The amplitudes of the three vortices that form the fundamental eddy are calculated from known Reynolds stress profiles through a transformation from the physical reference frame to the principal-axis reference frame. In this way, divergence-free anisotropic turbulent velocity fields are obtained that can reproduce a given Reynolds stress tensor. The model was tested on both isotropic and anisotropic turbulent velocity fields, in the framework of grid turbulence decay and turbulent channel flow, respectively. The transition from artificial to realistic turbulence in the proximity to the inflow boundary was found to be small in all test cases that were considered.

Modelling of the turbulent burning velocity based on Lagrangian statistics of propagating surfaces

We propose a predictive model of the turbulent burning velocity in homogeneous isotropic turbulence (HIT) based on Lagrangian statistics of propagating surfaces. The propagating surfaces with a constant displacement speed are initially arranged on a plane, and they evolve in non-reacting HIT, behaving like the propagation of a planar premixed flame front. The universal constants in the model of characterize the enhancement of area growth of premixed flames by turbulence, and they are determined by Lagrangian statistics of propagating surfaces. The flame area is then modelled by the area of the propagating surfaces at a truncation time. This truncation time signals the statistical stationary state of the evolutionary geometry of the propagating surfaces, and it is modelled by an explicit expression using limiting conditions of very weak and strong turbulence. Another parameter in the model of characterizes the effect of fuel chemistry on , and it is pre-determined by the very few available data points of from experiments or direct numerical simulation (DNS) in weak turbulence. The proposed model is validated using three DNS series of turbulent premixed flames with various fuels. The model prediction of generally agrees well with DNS in a wide range of premixed combustion regimes, and it captures the basic trends of in terms of the turbulence intensity, including the linear growth in weak turbulence and the ‘bending effect’ in strong turbulence.

Turbulent wind field representation and conditional mean-field simulation.

The covariance structure of a homogeneous isotropic turbulent wind velocity field is derived in terms of modified Bessel functions for an extended form of the Kàrmàn velocity spectrum, including explicit expressions for the transverse coherence functions. A concept of transformed isotropic turbulence is introduced to account for differences in the axial, transverse and vertical fluctuating wind velocities and length scales in natural wind. A special form of the auto-regressive simulation format is developed for convected turbulence with exponentially increasing intervals to the regression planes. In each step, the wind velocity field in a transverse plane is represented by a conditional mean field and a stochastic contribution determined explicitly by the time-space covariances. Simulation results are presented for a square area of dimension less than the integral length scale, representative of buildings and wind turbines, and a horizontal line of length six times the length scale, representative of a long-span bridge. The simulations demonstrate high accuracy of simulated spectral densities, covariance functions and transverse coherence functions. The simulated results do not show visible dependence on the specific points used for the simulated records. The efficiency and the free simulation point configuration suggest high competitiveness compared to fast Fourier transform-based spectral methods.

Effects of Reynolds number and Stokes number on particle-pair relative velocity in isotropic turbulence: a systematic experimental study

The effects of Reynolds number ($R_{\unicode[STIX]{x1D706}}$) and Stokes number ($St$) on particle-pair relative velocity (RV) are investigated systematically using a recently developed planar four-frame particle tracking technique in a novel homogeneous and isotropic turbulence chamber. We compare the measured results with direct numerical simulation (DNS), verifying whether the conclusions of the DNS for simplified conditions and limited $R_{\unicode[STIX]{x1D706}}$ are still valid in reality. Two experiments are performed: varying $R_{\unicode[STIX]{x1D706}}$ between 246 and 357 at six $St$ values, and varying $St$ between 0.02 and 4.63 at five $R_{\unicode[STIX]{x1D706}}$ values. The measured mean inward particle-pair RV $\langle w_{r}^{-}\rangle$ as a function of separation distance $r$ is compared with the DNS under closely matched conditions. At all experimental conditions, an excellent agreement is achieved, except when the particle separation distance $r\lesssim 10\unicode[STIX]{x1D702}$ ($\unicode[STIX]{x1D702}$ is the Kolmogorov length scale), where the experimental $\langle w_{r}^{-}\rangle$ is consistently higher, possibly due to particle polydispersity and finite laser thickness in the experiments (Dou et al., arXiv:1712.07506, 2017). At any fixed $St,\langle w_{r}^{-}\rangle$ is essentially independent of $R_{\unicode[STIX]{x1D706}}$, echoing the DNS finding of Ireland et al. (J. Fluid Mech., vol. 796, 2016, pp. 617–658). At any fixed $R_{\unicode[STIX]{x1D706}}$, $\langle w_{r}^{-}\rangle$ increases with $St$ at small $r$, showing dominance of the path-history effect in the dissipation range when $St\gtrsim O(1)$, but decreases with $St$ at large $r$, indicating dominance of inertial filtering. We further compare the $\langle w_{r}^{-}\rangle$ and RV variance $\langle w_{r}^{2}\rangle$ from experiments with DNS and theoretical predictions by Pan & Padoan (J. Fluid Mech., vol. 661, 2010, pp. 73–107). For $St\lesssim 1$, experimental $\langle w_{r}^{-}\rangle$ and $\langle w_{r}^{2}\rangle$ match these values well at $r\gtrsim 10\unicode[STIX]{x1D702}$, but they are higher than both DNS and theory at $r\lesssim 10\unicode[STIX]{x1D702}$. For $St\gtrsim 1$, $\langle w_{r}^{-}\rangle$ from all three match well, except for $r\lesssim 10\unicode[STIX]{x1D702}$, for which experimental values are higher, while $\langle w_{r}^{2}\rangle$ from experiment and DNS are much higher than theoretical predictions. We discuss potential causes of these discrepancies. What this study shows is the first experimental validation of $R_{\unicode[STIX]{x1D706}}$ and $St$ effect on inertial particle-pair $\langle w_{r}^{-}\rangle$ in homogeneous and isotropic turbulence.

Renormalized cumulants and velocity derivative skewness in Kolmogorov turbulence

We apply a renormalized perturbative scheme to the Navier–Stokes equation for an incompressible isotropic turbulent velocity field. This allows us to obtain the renormalized expressions for second- and third-order cumulants of the velocity derivative directly from the corresponding Feynman diagrams. The resulting expressions are integrated numerically by excluding and including the dissipation range assuming Kolmogorov and Pao’s phenomenological expressions for the energy spectrum. The ensuing values for skewness are found to be S = −0.647 (when the dissipation range is excluded) and (when the dissipation is included). These estimated values are compared with various experimental, numerical and theoretical results.

Clustering of vertically constrained passive particles in homogeneous isotropic turbulence.

We analyze the dynamics of small particles vertically confined, by means of a linear restoring force, to move within a horizontal fluid slab in a three-dimensional (3D) homogeneous isotropic turbulent velocity field. The model that we introduce and study is possibly the simplest description for the dynamics of small aquatic organisms that, due to swimming, active regulation of their buoyancy, or any other mechanism, maintain themselves in a shallow horizontal layer below the free surface of oceans or lakes. By varying the strength of the restoring force, we are able to control the thickness of the fluid slab in which the particles can move. This allows us to analyze the statistical features of the system over a wide range of conditions going from a fully 3D incompressible flow (corresponding to the case of no confinement) to the extremely confined case corresponding to a two-dimensional slice. The background 3D turbulent velocity field is evolved by means of fully resolved direct numerical simulations. Whenever some level of vertical confinement is present, the particle trajectories deviate from that of fluid tracers and the particles experience an effectively compressible velocity field. Here, we have quantified the compressibility, the preferential concentration of the particles, and the correlation dimension by changing the strength of the restoring force. The main result is that there exists a particular value of the force constant, corresponding to a mean slab depth approximately equal to a few times the Kolmogorov length scale η, that maximizes the clustering of the particles.

Random vectorial fields representing the local structure of turbulence

We propose a method to build up a random homogeneous, isotropic and incompressible turbulent velocity field that mimics turbulence in the inertial range. The underlying Gaussian field is given by a modified Biot-Savart law. The long range correlated nature of turbulence is then incorporated heuristically using a non linear transformation inspired by the recent fluid deformation imposed by the Euler equations. The resulting velocity field shows a non vanishing mean energy transfer towards the small scales and realistic alignment properties of vorticity with the eigenframe of the deformation rate.

Deriving fluid-particle correlation closures for Eulerian two-fluid models through use of Langevin equations

The correlation between the fluctuating particle and gas velocity in isotropic turbulence is studied with a set of stochastic differential equations taking into account both particle–particle collisions and the particle feedback on the turbulence. The principal aim of this work is to use the Langevin equations to formulate closures for two-fluid gas–particle flow models. Using Itô calculus we derived solutions for the turbulent kinetic energy of the particle phase and the particle-gas velocity correlations. If particle–particle collisions and particle feedback on the turbulence are neglected the new relations approach the ones derived by Tchen and Hinze but if these effects are included additional terms in the relations appear. In this study we only use a very simple model for the particle–particle collisions. The new relation and the classical relation of Tchen and Hinze for the particle turbulent kinetic energy as well as a relation based on the kinetic theory of granular flows have been implemented in a two-fluid model for turbulent gas–particle flow in a channel in order to make comparison for different particle Stokes numbers. Results show that while the two-fluid model using Hinze’s relations only gives good results for small Stokes numbers, the new relation yields significant improvements for a large range of Stokes numbers.

Scaling of Conditional Lagrangian Time Correlation Functions of Velocity and Pressure Gradient Magnitudes in Isotropic Turbulence

We study Lagrangian statistics of the magnitudes of velocity and pressure gradients in isotropic turbulence by quantifying their correlation functions and their characteristic time scales. In a recent work (Yu and Meneveau, Phys Rev Lett 104:084502, 2010), it has been found that the Lagrangian time-correlations of the velocity and pressure gradient tensor and vector elements scale with the locally-defined Kolmogorov time scale, evaluated from the locally-averaged dissipation-rate (ϵ r ) and viscosity (ν) according to $\tau_{K,r}=\sqrt{\nu/\epsilon_r}$ . In this work, we study the Lagrangian time-correlations of the absolute values of velocity and pressure gradients. It has long been known that such correlations display longer memories into the inertial-range as well as possible intermittency effects. We explore the appropriate temporal scales with the aim to achieve collapse of the correlation functions. The data used in this study are sampled from the web-services accessible public turbulence database ( http://turbulence.pha.jhu.edu ). The database archives a 10244 (space+time) pseudo-spectral direct numerical simulation of forced isotropic turbulence with Taylor-scale Reynolds number Re λ = 433, and supports spatial differentiation and spatial/temporal interpolation inside the database. The analysis shows that the temporal auto-correlations of the absolute values extend deep into the inertial range where they are determined not by the local Kolmogorov time-scale but by the local eddy-turnover time scale defined as $\tau_{e,r}= r^{2/3}\epsilon_r^{-1/3}$ . However, considerable scatter remains and appears to be reduced only after a further (intermittency) correction factor of the form of (r/L) χ is introduced, where L is the turbulence integral scale. The exponent χ varies for different variables. The collapse of the correlation functions for absolute values is, however, less satisfactory than the collapse observed for the more rapidly decaying strain-rate tensor element correlation functions in the viscous range.

Lagrangian Refined Kolmogorov Similarity Hypothesis for Gradient Time Evolution and Correlation in Turbulent Flows

We study time evolution of velocity and pressure gradients in isotropic turbulence by quantifying their autocorrelation functions and decorrelation time scales. The Lagrangian analysis uses data in a public database generated by direct numerical simulation at a Reynolds number Re{lambda} approximately 433. It is confirmed that when averaging over the entire domain, correlation functions decay on time scales on the order of the global Kolmogorov turnover time scale. However, when performing the analysis in different subregions of the flow, turbulence intermittency leads to large spatial variability in the decay time scales. Remarkably, excellent collapse of the autocorrelation functions is recovered when using a locally defined Kolmogorov time scale. This provides new evidence for the validity of Kolmogorov's refined similarity hypothesis, but from a Lagrangian viewpoint that provides a natural frame to describe the dynamics of turbulence.

Transporting passive additives in a turbulent medium under convective mixing

The vertical transport of passive additives in a medium in the presence of both turbulent and convective mixing is studied. It is shown that, despite the fact that turbulent and convective fluxes are uncorrelated, no additive mechanism of mixing exists. On the basis of a theoretical model and a comparison with model calculations, it is shown that, in the presence of both turbulent and convective mixing in a medium, the effect of the interference of turbulent and convective transport mechanisms takes place. This effect is manifested in the following way: the presence of convection decreases the coefficient of turbulent transport and the presence of turbulence decreases the coefficient of convective transport; in this case, this effect depends on the parameter of the inhomogeneity (extension) of convective fluxes. The relative contribution of each of the mechanisms to the total transport is estimated. It is shown that the interference of turbulence and convection results in an anisotropy of the coefficient of turbulent transport in an isotropic turbulent velocity field. The effect of the coefficient of molecular diffusion is assumed to be negligibly small, and its effect is not taken into consideration when analyzing model results. The interference of the turbulent and convective mechanisms of transport is explained by the inclusion of an additional mechanism that decreases the distance of linear particle propagation.

A Paradox concerning the Refined Similarity Hypothesis of Kolmogorov for Isotropic Turbulence

It is proved that the famous third-order structure function of the velocity in homogeneous isotropic turbulence derived by Kolmogorov implies the statistical interdependence of the difference and sum of the velocities at two points separated by a distance r. However, this is inconsistent with Kolmogorov's hypothesis regarding the difference between such velocities, made in 1962, in the sense that it implies that the statistics of the non-dimensional prefactor of the velocity difference are not universal but, instead, depend on the velocity sum, at least. The only resolution of this apparent paradox that can maintain the spirit of Kolmogorov's original hypothesis, which has been so important in the study of intermittency in turbulence, is given.

Effect of Schmidt number on the velocity–scalar cospectrum in isotropic turbulence with a mean scalar gradient

We consider transport of a passive scalar by an isotropic turbulent velocity field in the presence of a mean scalar gradient. The velocity–scalar cospectrum measures the distribution of the mean scalar flux across scales. An inequality is shown to bound the magnitude of the cospectrum in terms of the shell-summed energy and scalar spectra. At high Schmidt number, this bound limits the possible contribution of the sub-Kolmogorov scales to the scalar flux. At low Schmidt number, we derive an asymptotic result for the cospectrum in the inertial–diffusive range, with a power law is observed in the inertial–diffusive range. Results are reported for the cospectrum from a direct numerical simulation at a Taylor Reynolds number of 265, and a comparison is made at Schmidt number order unity between theory, simulation and experiment.

Statistical treatment of scalar transfer in isotropic turbulence

It has recently been known that local high-gradient regions of an advected scalar, such as temperature or mass contaminant, in a turbulent state of fluid form thin sheets, randomly oriented and moving around with turbulent motion. Here is presented a joint multifractal model for velocity and scalar dissipations in isotropic turbulence which can predict the statistical distributions of the worms, vorticity-concentrated regions, as well as the above-described scalar-gradient sheets. This model allows us to derive turbulent diffusion coefficient in isotropic turbulence scaled by Reynolds number and Prandtl number, which predominates far over molecular diffusion coefficient.

A closure method for random advection of a passive scalar

A novel functional method is applied to calculate the statistics of a passive scalar in an isotropic turbulent velocity field. The method yields asymptotic series expansions for small velocity correlation time from which approximate closure equations are derived. The closure method admits a diagram expansion, and is implemented as a Mathematica program. Padé summation of the asymptotic series yields accurate values for the effective diffusivity and gives formulas expressing the Lagrangian correlation of the velocity in terms of the Eulerian correlation. The approximations compare very favorably with numerical simulations of advection by a Gaussian velocity field.

Planar isotropy of passive scalar turbulent mixing with a mean perpendicular gradient.

A recently proposed evolution equation [Vaienti et al., Physica D 85, 405 (1994)] for the probability density functions (PDF's) of turbulent passive scalar increments obtained under the assumptions of fully three-dimensional homogeneity and isotropy is submitted to validation using direct numerical simulation (DNS) results of the mixing of a passive scalar with a nonzero mean gradient by a homogeneous and isotropic turbulent velocity field. It is shown that this approach leads to a quantitatively correct balance between the different terms of the equation, in a plane perpendicular to the mean gradient, at small scales and at large Péclet number. A weaker assumption of homogeneity and isotropy restricted to the plane normal to the mean gradient is then considered to derive an equation describing the evolution of the PDF's as a function of the spatial scale and the scalar increments. A very good agreement between the theory and the DNS data is obtained at all scales. As a particular case of the theory, we derive a generalized form for the well-known Yaglom equation (the isotropic relation between the second-order moments for temperature increments and the third-order velocity-temperature mixed moments). This approach allows us to determine quantitatively how the integral scale properties influence the properties of mixing throughout the whole range of scales. In the simple configuration considered here, the PDF's of the scalar increments perpendicular to the mean gradient can be theoretically described once the sources of inhomogeneity and anisotropy at large scales are correctly taken into account.

Large-eddy simulation of passive scalar diffusion in isotropic turbulence

A spectral large-eddy simulation of the velocity and passive temperature fields in three-dimensional isotropic turbulence is developed at high Reynolds numbers and a Prandtl number of the order of 1. The temperature spectrum in the large scales is shown to develop a range of the form ET(k)≈0.1η[〈u2〉/ε]k−1, where ε and η are the kinetic energy and temperature variance dissipation rates, respectively. This range, which is a result of shearing by large scale velocity gradients, is followed by a k−5/3 inertial-convective range at higher wavenumbers. The temperature variance decays much faster in time than the kinetic energy. Finally, the spectral eddy conductivity rises logarithmically with k toward small wavenumbers, contrary to the eddy viscosity that displays a plateau.