Lagrangian Correlation Time Research Articles

Particle transport in a turbulent pipe flow: direct numerical simulations, phenomenological modelling and physical mechanisms

In particle-laden turbulent wall flows, transport of particles towards solid walls is phenomenologically thought to be governed by the wall-normal turbulence intensity supporting the underlying particle–eddy interactions that are usually modelled by a combination of turbophoresis and turbulent diffusion. We estimate the turbophoretic and turbulent diffusive coefficients as a function of wall-normal coordinate directly from a generated direct numerical simulation (DNS) database of low volume fraction point particles in a turbulent pipe flow. These coefficients are then used in an advection–diffusion equation to estimate the particle concentration as a function of wall-normal distance and time, with favourable comparison against DNS for smaller Stokes number ( $St^+$ ) particles suggesting a limitation of the common gradient diffusion hypothesis for larger $St^+$ particles. Using DNS we explore the non-trivial effects of $St^+$ , pipe wall condition (particle absorbing or elastic) as well as the influence of drag and lift force on the velocity and particle statistics giving rise to different particle concentrations. We then appraise various Eulerian-based models of turbophoretic and turbulent diffusive coefficients and, finally, use physical insights from Lagrangian correlation times, conditional quadrant analysis and flow topology to shed further light on the particle transport as a function of various parameters and the limits of gradient diffusion hypothesis.

Particle motion within the viscous sublayer of a turbulent shear flow

The motion of isolated heavy particles on the bottom wall of a turbulent shear flow is investigated experimentally (using high-frequency particle tracking) and correlated to that of the fluid (from high-resolution LDV measurements). Numerous particle diameters, particle densities, and flow velocities are considered, where the particles remain within the viscous sublayer (particle Reynolds number lower than 10). It is shown that fluid flow fluctuations within the viscous sublayer (varying Couette flow) induce particle velocity fluctuations of the same magnitude, slightly smaller. The mean particle velocity, standard deviation, and Lagrangian correlation time are found to scale with the viscous scales. Particles roll and slide without takeoff, unlike what happens in viscous laminar flow where the lift force may overcome the immersed particle weight. Arguments are given for this difference. Probability density functions for the streamwise and spanwise velocity are found to follow Gamma and Gaussian laws, respectively, which also scale with the viscous scales.

Numerical computation of the transport matrix in toroidal plasma with a stochastic magnetic field

A new numerical method, based on integrating along the full orbit of guiding centers, to compute the transport matrix is realized. The method is successfully applied to compute the phase-space diffusion tensor of passing electrons in a tokamak with a stochastic magnetic field. The new method also computes the Lagrangian correlation function, which can be used to evaluate the Lagrangian correlation time and the turbulence correlation length. For the case of the stochastic magnetic field, we find that the order of magnitude of the parallel correlation length can be estimated by qR0, as expected previously.

Reynolds and Prandtl number scaling of viscous heating in isotropic turbulence

Viscous heating is investigated using high-resolution direct numerical simulations. Scaling relations are derived and verified for different values of the Reynolds and Prandtl numbers. The scaling of the heat fluctuations is shown to depend on Lagrangian correlation times and on the scaling of dissipation-rate fluctuations. The convergence of the temperature spectrum to asymptotic scaling is observed to be slow, due to the broadband character of the temperature production spectrum and the slow convergence of the dissipation-rate spectrum to its asymptotic form.

Correlations between the instantaneous velocity gradient and the evolution of scale-to-scale fluxes in two-dimensional flow.

Using high-resolution particle tracking velocimetry and filter-space techniques, we study the links between the scale-to-scale transfer of energy and enstrophy and instantaneously rotational and straining regions as determined by the classic Okubo-Weiss parameter in a quasi-two-dimensional laboratory flow. Although the Okubo-Weiss parameter has shortcomings, we find that, when suitably conditioned, it is surprisingly a good predictor for the future evolution of the spectral fluxes. By studying Lagrangian correlation functions, we explain our findings by showing that both the spectral fluxes and the Okubo-Weiss parameter are independently correlated for long times along fluid-element trajectories and thus that any predictive capacity of the Okubo-Weiss parameter arises because it is coupled to fluid advection. Our results suggest potential strategies for forecasting in complex flows by looking for quantities with long Lagrangian correlation times.

Dependence of turbulent advection on the Lagrangian correlation time.

In turbulent scalar mixing, starting from random initial conditions, the root-mean-square advection term rapidly drops as the flow and the scalar field organize. We show first analytically, for the simplified case of a blob in shear flow with a finite correlation time, how the advection term is reduced compared to a randomly aligned scalar structure. This picture is then generalized to turbulent mixing. These examples show that the rapid depletion of advection depends on the lifetime of turbulent structures, compared to the local straining time scale. A turbulence closure is used to show that the Lagrangian correlation time indeed determines the deviation from Gaussian behavior. In particular it is shown that in the inertial range the depletion mechanism is self-similar, since a constant ratio is observed between the advection spectrum and its Gaussian equivalent. Finally, direct numerical simulation shows that in the limit of an infinite correlation time of the turbulent eddies, corresponding to a frozen velocity field, the mean-square advection tends to a zero fraction of its Gaussian estimate.

TURBULENCE-INDUCED RELATIVE VELOCITY OF DUST PARTICLES. III. THE PROBABILITY DISTRIBUTION

Motivated by its important role in the collisional growth of dust particles in protoplanetary disks, we investigate the probability distribution function (PDF) of the relative velocity of inertial particles suspended in turbulent flows. Using the simulation from our previous work, we compute the relative velocity PDF as a function of the friction timescales, tau_p1 and tau_p2, of two particles of arbitrary sizes. The friction time of particles included in the simulation ranges from 0.1 tau_eta to 54T_L, with tau_eta and T_L the Kolmogorov time and the Lagrangian correlation time of the flow, respectively. The relative velocity PDF is generically non-Gaussian, exhibiting fat tails. For a fixed value of tau_p1, the PDF is the fattest for equal-size particles (tau_p2~tau_p1), and becomes thinner at both tau_p2<tau_ p1 and tau_p2>tau_p1. Defining f as the friction time ratio of the smaller particle to the larger one, we find that, at a given f in 1/2<f<1, the PDF fatness first increases with the friction time, tau_ph, of the larger particle, peaks at tau_ph~tau_eta, and then decreases as tau_ph increases further. For 0<f<1/4, the PDF shape becomes continuously thinner with increasing tau_ph. The PDF is nearly Gaussian only if tau_ph is sufficiently large (>>T_L). These features are successfully explained by the Pan & Padoan model. Using our simulation data and some simplifying assumptions, we estimated the fractions of collisions resulting in sticking, bouncing, and fragmentation as a function of the dust size in protoplanerary disks, and argued that accounting for non-Gaussianity of the collision velocity may help further alleviate the bouncing barrier problem.

TURBULENCE-INDUCED RELATIVE VELOCITY OF DUST PARTICLES. II. THE BIDISPERSE CASE

We extend our earlier work on turbulence-induced relative velocity between equal-size particles (Pan and Padoan, Paper I) to particles of arbitrarily different sizes. The Pan and Padoan (PP10) model shows that the relative velocity between different particles has two contributions, named the generalized shear and acceleration terms, respectively. The generalized shear term represents the particles' memory of the spatial flow velocity difference across the particle distance in the past, while the acceleration term is associated with the temporal flow velocity difference on individual particle trajectories. Using the simulation of Paper I, we compute the root-mean-square relative velocity, <w^2>^1/2, as a function of the friction times, tau_p1 and tau_p2, of the two particles, and show that the PP10 prediction is in satisfactory agreement with the data, confirming its physical picture. For a given tau_p1 below the Lagrangian correlation time of the flow, T_L, <w^2>^1/2 as a function of tau_p2 shows a dip at tau_p2~tau_p1, indicating tighter velocity correlation between similar particles. Defining a ratio f=tau_pl/tau_ph, with tau_pl and tau_ph the friction times of the smaller and larger particles, we find that <w^2>^1/2 increases with decreasing f due to the generalized acceleration contribution, which dominates at f<1/4. At a fixed f, our model predicts that <w^2>^1/2 scales as tau_ph^1/2 for tau_p,h in the inertial range of the flow, stays roughly constant for T_L <tau_ph < T_L/f, and finally decreases as tau_ph^-1/2 for tau_ph>>T_L/f. The acceleration term is independent of the particle distance, r, and thus reduces the r-dependence of <w^2>^1/2 in the bidisperse case.

On the anisotropy of the turbulent passive scalar in the presence of a mean scalar gradient

AbstractWe investigate the origin of the scalar gradient skewness in isotropic turbulence on which a mean scalar gradient is imposed. The problem of the advection of an anisotropic scalar field is reformulated in terms of the advection of an isotropic vector field. For this field, triadic closure equations are derived. It is shown how the scaling of the scalar gradient skewness depends on the choice of the time scale used for the Lagrangian decorrelation of the vector field. The persistent anisotropy in the small scales for the third-order statistics is shown to be perfectly compatible with Corrsin–Obukhov scaling for second-order quantities, since second- and third-order scalar quantities are governed by a different triad correlation time scale. Whereas the inertial range dynamics of second-order scalar quantities is governed by the Lagrangian velocity correlation time, the third-order quantities remain correlated over a time related to the large-scale dynamics of the scalar field. It is argued that this time is determined by the average time it takes for a fluid particle to travel between ramp-cliff scalar structures.

Time scales of turbulent relative dispersion

Tracers in a turbulent flow separate according to the celebrated t3/2 Richardson-Obukhov law, which is usually explained by a scale-dependent effective diffusivity. Here, supported by state-of-the-art numerics, we revisit this argument. The Lagrangian correlation time of velocity differences increases too quickly for validating this approach, but acceleration differences decorrelate on dissipative time scales. Phenomenological arguments are used to relate the behavior of separations to that of a "local energy dissipation," defined as the average ratio between the cube of the longitudinal velocity difference and the distance between the two tracers. This quantity is shown to stabilize on short time scales and this results in an asymptotic diffusion ∝t1/2 of velocity differences. The time of convergence to this regime is shown to be that of deviations from Batchelor's initial ballistic regime, given by a scale-dependent energy dissipation time rather than the usual turnover time. It is finally demonstrated that the fluid flow intermittency should not affect this long-time behavior of the relative motion.

Lagrangian time correlations of vorticity alignments in isotropic turbulence: Observations and model predictions

Motivated by results from recent particle tracking experiments in turbulence Xu et al. [Nature Phys. 7, 709 (2011)], we study the Lagrangian time correlations of vorticity alignments with the three eigenvectors of the deformation-rate tensor. We use data from direct numerical simulations (DNS), and explore the predictions of a Lagrangian model for the velocity gradient tensor. We find that the initial increase of correlation of vorticity direction with the most extensive eigen-direction observed by Xu et al. is reproduced accurately using the Lagrangian model, as well as the evolution of correlation with the other two eigendirections. Conversely, time correlations of vorticity direction with the eigen-frame of the pressure Hessian tensor show differences with the model.

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.

Nuclear magnetic resonance: A new tool for the validation of multiphase multidimensional CFD codes

Turbulent transport models and data in bubbly flows are briefly reviewed since they play an important role in the modeling of boiling flows in forced convection. Shortcomings of earlier measurements of the eddy diffusivity by NMR (Lemonnier and Leblond, 2007b) are analyzed and a new procedure is presented which is now consistent with the procedure of Gatenby and Gore (1994) developed for single-phase turbulent flow characterization. The newly estimated eddy diffusivity agrees now with that previously obtained by Serizawa et al. (1975b) with a thermal method and that of the model of Sato and Sadatomi (1981). This procedure also provides the liquid velocity fluctuation RMS and the Lagrangian correlation time of velocity fluctuations. In addition, the same NMR technique provides also the area-averaged liquid velocity and void fraction. Bubbly flow data up to transition to slug flow are provided which also agree with existing drift-flux models (Ishii and Hibiki, 2006). It is finally discussed how the NMR method can be extended to local measurements and may provide a fully non-intrusive diagnostic in two-phase flows and which is not limited to bubbly flow.

Subgrid-scale contributions to Lagrangian time correlations in isotropic turbulence

The application of large-eddy simulation (LES) to particle-laden turbulence raises such a fundamental question as whether the LES with a subgrid scale (SGS) model can correctly predict Lagrangian time correlations (LTCs). Most of the currently existing SGS models are constructed based on the energy budget equations. Therefore, they are able to correctly predict energy spectra, but they may not ensure the correct prediction on the LTCs. Previous researches investigated the effect of the SGS modeling on the Eulerian time correlations. This paper is devoted to study the LTCs in LES. A direct numerical simulation (DNS) and the LES with a spectral eddy viscosity model are performed for isotropic turbulence and the LTCs are calculated using the passive vector method. Both a priori and a posteriori tests are carried out. It is observed that the subgrid-scale contributions to the LTCs cannot be simply ignored and the LES overpredicts the LTCs than the DNS. It is concluded from the straining hypothesis that an accurate prediction of enstrophy spectra is most critical to the prediction of the LTCs.

Stretching in a model of a turbulent flow

Using a multi-scaled, chaotic flow known as the KS model of turbulence [J.C.H. Fung, J.C.R. Hunt, A. Malik, R.J. Perkins, Kinematic simulation of homogeneous turbulence by unsteady random fourier modes, J. Fluid Mech. 236 (1992) 281–318], we investigate the dependence of Lyapunov exponents on various characteristics of the flow. We show that the KS model yields a power law relation between the Reynolds number and the maximum Lyapunov exponent, which is similar to that for a turbulent flow with the same energy spectrum. Our results show that the Lyapunov exponents are sensitive to the advection of small eddies by large eddies, which can be explained by considering the Lagrangian correlation time of the smallest scales. We also relate the number of stagnation points within a flow to the maximum Lyapunov exponent, and suggest a linear dependence between the two characteristics.

Stochastic Models for the Kinematics of Moisture Transport and Condensation in Homogeneous Turbulent Flows

Abstract The transport of a condensing passive scalar is studied as a prototype model for the kinematics of moisture transport on isentropic surfaces. Condensation occurs whenever the scalar concentration exceeds a specified local saturation value. Since condensation rates are strongly nonlinear functions of moisture content, the mean moisture flux is generally not diffusive. To relate the mean moisture content, mean condensation rate, and mean moisture flux to statistics of the advecting velocity field, a one-dimensional stochastic model is developed in which the Lagrangian velocities of air parcels are independent Ornstein–Uhlenbeck (Gaussian colored noise) processes. The mean moisture evolution equation for the stochastic model is derived in the Brownian and ballistic limits of small and large Lagrangian velocity correlation time. The evolution equation involves expressions for the mean moisture flux and mean condensation rate that are nonlocal but remarkably simple. In a series of simulations of homogeneous two-dimensional turbulence, the dependence of mean moisture flux and mean condensation rate on mean saturation deficit is shown to be reproducible by the one-dimensional stochastic model, provided eddy length and time scales are taken as given. For nonzero Lagrangian velocity correlation times, condensation reduces the mean moisture flux for a given mean moisture gradient compared with the mean flux of a noncondensing scalar.

Large-eddy Simulation of the Dispersion of Solid Particles in a Turbulent Boundary Layer

A large-eddy simulation (LES) with the dynamic Smagorinsky-Germano subgrid-scale (SGS) model is used to study the dispersion of solid particles in a turbu- lentboundarylayer.SolidparticlesaretrackedinaLagrangianway.Theinstantaneous velocity of the surroundingfluid is considered to have a large-scale part (directly com- puted by the LES) and a small-scale part. The SGS velocity of the surrounding fluid is given by a three-dimensional Langevin model written in terms of SGS statistics at a mesh level. An appropriate Lagrangian correlation time scale is considered in order to include the influences of gravity and inertia of the solid particle. Inter-particle colli- sions and the influence of particles on the mean flow are also taken into account. The results of the LES are compared with the wind-tunnel experiments of Nalpanis et al. (1993 J Fluid Mech251: 661-685) and of Taniere et al. (1997 Exp in Fluids23:463-471) on sand particles in saltation and in modified saltation, respectively.

A DISCUSSION ON ADVECTION DISPERSION IN LAMINAR PIPE FLOW

In solute transport through porous media in column experiment, increasing of apparent dispersion coefficient D as distance can be observed. This kind of phenomena has been thought by heterogeneous permeability. But Taylor theoretically explained by Lagrangian correlation function that constant coefficient D requires a time while solute particles go through various velocities. This concept can be used in porous media also.As a phenomenon to simplify porous media flow, solute transport in laminar pipe flow is studied. Solute transport is simulated by stochastic-Eulerian approach and compared with previous theories from Lagrange or Euler view points. Results show good fitting to the Lagrangian correlation theory and time scale t0, in which D changes. Furthermore, they indicate that D is underestimated by using 1-D Eulerian theory in time scale same as t0.

Dispersion of passive tracers in the direct enstrophy cascade: Experimental observations

The paper reports a statistical analysis of the separation of pairs in the enstrophy cascade, in a two-dimensional flow. The flow is produced experimentally, using electromagnetic forcing. Two regimes of separation are found. At early times (i.e., within two integral times) the separation process is exponential, following Lin’s law [J. T. Lin, J. Atmos. Sci. 29, 394 (1972)]. In this range of time, the probability density functions (PDFs) of separations are self-similar in time, developing stretched exponential tails, and one may define a Lagrangian correlation time. Above two integral times, an algebraic regime takes place, with exponential tails for the PDFs of the separations, and self-similar Lagrangian correlation functions. The present work thus confirms the existence of two regimes, by analyzing a number of statistical quantities including the Lagrangian PDF and the temporal Lagrangian correlation functions which so far, were left undetermined for the Batchelor regime.

Particle transport in a random velocity field with Lagrangian statistics.

The transport properties of a random velocity field with Kolmogorov spectrum and time correlations defined along Lagrangian trajectories are analyzed. The analysis is carried out in the limit of short correlation times, as a perturbation theory in the ratio, scale by scale, of the eddy decay and turnover time. Various quantities such as the Batchelor constant and the dimensionless constants entering the expression for particle relative and self-diffusion are given in terms of this ratio and of the Kolmogorov constant. Particular attention is paid to particles with finite inertia. The self-diffusion properties of a particle with Stokes time longer than the Kolmogorov time are determined, verifying on an analytical example the dimensional results of Olla [Phys. Fluids 14, 4266 (2002)]. Expressions for the fluid velocity Lagrangian correlations and correlation times along a solid particle trajectory are provided in several parameter regimes, including the infinite Stokes time limit corresponding to Eulerian correlations. The concentration fluctuation spectrum and the nonergodic properties of a suspension of heavy particles in a turbulent flow, in the same regime, are analyzed. The concentration spectrum is predicted to obey, above the scale of eddies with lifetime equal to the Stokes time, a power law with universal -4/3 exponent, and to be otherwise independent of the nature of the turbulent flow. A preference of the solid particle to lie in less energetic regions of the flow is observed.