Turbulent transport of material particles: an experimental study of finite size effects

We present experimental Lagrangian statistics of finite sized, neutrally bouyant, particles transported in an isotropic turbulent flow. The particle's diameter is varied over turbulent inertial scales. Finite size effects are shown not to be trivially related to velocity intermittency. The global shape of the particle's acceleration probability density functions is not found to depend significantly on its size while the particle's acceleration variance decreases as it becomes larger in quantitative agreement with the classical k(-7/3) scaling for the spectrum of Eulerian pressure fluctuations in the carrier flow.     

Lagrangian view of time irreversibility of fluid turbulence

A turbulent flow is maintained by an external supply of kinetic energy, which is eventually dissipated into heat at steep velocity gradients. The scale at which energy is supplied greatly differs from the scale at which energy is dissipated, the more so as the turbulent intensity(the Reynolds number) is larger. The resulting energy flux over the range of scales, intermediate between energy injection and dissipation, acts as a source of time irreversibility. As it is now possible to follow accurately fluid particles in a turbulent flow field, both from laboratory experiments and from numerical simulations, a natural question arises: how do we detect time irreversibility from these Lagrangian data? Here we discuss recent results concerning this problem. For Lagrangian statistics involving more than one fluid particle, the distance between fluid particles introduces an intrinsic length scale into the problem. The evolution of quantities dependent on the relative motion between these fluid particles, including the kinetic energy in the relative motion, or the configuration of an initially isotropic structure can be related to the equal-time correlation functions of the velocity field, and is therefore sensitive to the energy flux through scales, hence to the irreversibility of the flow. In contrast, for singleparticle Lagrangian statistics, the most often studied velocity structure functions cannot distinguish the "arrow of time". Recent observations from experimental and numerical simulation data, however, show that the change of kinetic energy following the particle motion, is sensitive to time-reversal. We end the survey with a brief discussion of the implication of this line of work.     

On the Existence of Invariant Measure for Lagrangian Velocity in Compressible Environments

We study transport of a passive tracer particle in a time dependent turbulent flow in the medium with positive molecular diffusivity. We show that there exists then a probability measure equivalent to the underlying physical probability, corresponding to the Eulerian velocity field, under which the particle Lagrangian velocity observations are stationary. As an application we derive the existence of the Stokes drift and the effective diffusivity—the characteristics of the long time behavior of the particle motion.     

Lagrangian Velocity Fluctuations in Fully Developed Turbulence: Scaling, Intermittency, and Dynamics

New aspects of turbulence are uncovered if one considers the flow motion from the perspective of a fluid particle (known as the Lagrangian approach) rather than in terms of a velocity field (the Eulerian viewpoint). Using a new experimental technique, based on the scattering of ultrasound, we have obtained a direct measurement of particle velocities, resolved at all scales, in a fully turbulent flow. We find that the Lagrangian velocity autocorrelation function and the Lagrangian time spectrum are in agreement with the Kolmogorov K41 phenomenology. Intermittency corrections are observed and we give a measurement of the Lagrangian structure function exponents. They are more intermittent than the corresponding Eulerian exponents. We also propose a novel analysis of intermittency in turbulence: our measurement enables us to study it from a dynamical point of view. We thus analyze the Lagrangian velocity fluctuations in the framework of random walks. We find experimentally that the elementary steps in the walk have random uncorrelated directions but a magnitude that displays extremely long-range correlations in time. Theoretically, we study a Langevin equation that incorporates these features and we show that the resulting dynamics accounts for the observed one-point and two-point statistical properties of the Lagrangian velocity fluctuations. Our approach connects the intermittent statistical nature of turbulence to the dynamics of the flow.     

Relating Lagrangian passive scalar scaling exponents to Eulerian scaling exponents in turbulence

Intermittency is a basic feature of fully developed turbulence, for both velocity and passive scalars. Intermittency is classically characterized by Eulerian scaling exponent of structure functions. The same approach can be used in a Lagrangian framework to characterize the temporal intermittency of the velocity and passive scalar concentration of a an element of fluid advected by a turbulent intermittent field. Here we focus on Lagrangian passive scalar scaling exponents, and discuss their possible links with Eulerian passive scalar and mixed velocity-passive scalar structure functions. We provide different transformations between these scaling exponents, associated to different transformations linking space and time scales. We obtain four new explicit relations. Experimental data are needed to test these predictions for Lagrangian passive scalar scaling exponents.     

A PDF projection method: A pressure algorithm for stand-alone transported PDFs

In this paper, a new formulation of the projection approach is introduced for stand-alone probability density function (PDF) methods. The method is suitable for applications in low-Mach number transient turbulent reacting flows. The method is based on a fractional step method in which first the advection–diffusion–reaction equations are modelled and solved within a particle-based PDF method to predict an intermediate velocity field. Then the mean velocity field is projected onto a space where the continuity for the mean velocity is satisfied. In this approach, a Poisson equation is solved on the Eulerian grid to obtain the mean pressure field. Then the mean pressure is interpolated at the location of each stochastic Lagrangian particle. The formulation of the Poisson equation avoids the time derivatives of the density (due to convection) as well as second-order spatial derivatives. This in turn eliminates the major sources of instability in the presence of stochastic noise that are inherent in particle-based PDF methods. The convergence of the algorithm (in the non-turbulent case) is investigated first by the method of manufactured solutions. Then the algorithm is applied to a one-dimensional turbulent premixed flame in order to assess the accuracy and convergence of the method in the case of turbulent combustion. As a part of this work, we also apply the algorithm to a more realistic flow, namely a transient turbulent reacting jet, in order to assess the performance of the method.     

Remapping-free ALE-type kinetic method for flow computations

Based on the integral form of the fluid dynamic equations, a finite volume kinetic scheme with arbitrary control volume and mesh velocity is developed. Different from the earlier unified moving mesh gas-kinetic method [C.Q. Jin, K. Xu, An unified moving grid gas-kinetic method in Eulerian space for viscous flow computation, J. Comput. Phys. 222 (2007) 155–175], the coupling of the fluid equations and geometrical conservation laws has been removed in order to make the scheme applicable for any quadrilateral or unstructured mesh rather than parallelogram in 2D case. Since a purely Lagrangian method is always associated with mesh entangling, in order to avoid computational collapsing in multidimensional flow simulation, the mesh velocity is constructed by considering both fluid velocity (Lagrangian methodology) and diffusive velocity (Regenerating Eulerian mesh function). Therefore, we obtain a generalized Arbitrary-Lagrangian–Eulerian (ALE) method by properly designing a mesh velocity instead of re-generating a new mesh after distortion. As a result, the remapping step to interpolate flow variables from old mesh to new mesh is avoided. The current method provides a general framework, which can be considered as a remapping-free ALE-type method. Since there is great freedom in choosing mesh velocity, in order to improve the accuracy and robustness of the method, the adaptive moving mesh method [H.Z. Tang, T. Tang, Adaptive mesh methods for one-and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal. 41 (2003) 487–515] can be also used to construct a mesh velocity to concentrate mesh to regions with high flow gradients.     

Lagrangian dynamics and statistical geometric structure of turbulence

The local statistical and geometric structure of three-dimensional turbulent flow can be described by the properties of the velocity gradient tensor. A stochastic model is developed for the Lagrangian time evolution of this tensor, in which the exact nonlinear self-stretching term accounts for the development of well-known non-Gaussian statistics and geometric alignment trends. The nonlocal pressure and viscous effects are accounted for by a closure that models the material deformation history of fluid elements. The resulting stochastic system reproduces many statistical and geometric trends observed in numerical and experimental 3D turbulent flows, including anomalous relative scaling.     

Lagrangian velocity statistics in turbulent flows: effects of dissipation

We use the multifractal formalism to describe the effects of dissipation on Lagrangian velocity statistics in turbulent flows. We analyze high Reynolds number experiments and direct numerical simulation data. We show that this approach reproduces the shape evolution of velocity increment probability density functions from Gaussian to stretched exponentials as the time lag decreases from integral to dissipative time scales. A quantitative understanding of the departure from scaling exhibited by the magnitude cumulants, early in the inertial range, is obtained with a free parameter function D(h) which plays the role of the singularity spectrum in the asymptotic limit of infinite Reynolds number. We observe that numerical and experimental data are accurately described by a unique quadratic D(h) spectrum which is found to extend from h(min) approximately 0.18 to h(max) approximately 1.     

Two-dimensional Euler flows in slowly deforming domains

We consider the evolution of an incompressible two-dimensional perfect fluid as the boundary of its domain is deformed in a prescribed fashion. The flow is taken to be initially steady, and the boundary deformation is assumed to be slow compared to the fluid motion. The Eulerian flow is found to remain approximately steady throughout the evolution. At leading order, the velocity field depends instantaneously on the shape of the domain boundary, and it is determined by the steadiness and vorticity-preservation conditions. This is made explicit by reformulating the problem in terms of an area-preserving diffeomorphism g_Λ which transports the vorticity. The first-order correction to the velocity field is linear in the boundary velocity, and we show how it can be computed from the time derivative of g_Λ.The evolution of the Lagrangian position of fluid particles is also examined. Thanks to vorticity conservation, this position can be specified by an angle-like coordinate along vorticity contours. An evolution equation for this angle is derived, and the net change in angle resulting from a cyclic deformation of the domain boundary is calculated. This includes a geometric contribution which can be expressed as the integral of a certain curvature over the interior of the circuit that is traced by the parameters defining the deforming boundary.A perturbation approach using Lie series is developed for the computation of both the Eulerian flow and geometric angle for small deformations of the boundary. Explicit results are presented for the evolution of nearly axisymmetric flows in slightly deformed discs.     

Statistics of Lagrangian velocities in turbulent flows

We present a generalized Fokker-Planck equation for the joint position-velocity probability distribution of a single fluid particle in a turbulent flow. Based on a simple estimate, the diffusion term is related to the two-point two-time Eulerian acceleration-acceleration correlation. Dimensional analysis yields a velocity increment probability distribution with normal scaling v approximately t(1/2). However, the statistics need not be Gaussian.     

Identification of Regions with Inhomogeneous Particle Behavior in Dilute Gas‐solid Flows

The behavior of the particulate phase in a highly turbulent gas flow has been investigated in a vertical channel. Variations of the flow configuration (1. Flow past a cylinder, 2. flow past a wall‐mounted obstacle and 3. flow around a horizontally injected jet) have been subject to both experiments and numerical simulations. The velocity vector field of the solid phase has been measured by digital particle image velocimetry (DPIV). The measurements have been focused on particle‐obstacle collisions and crossflow in the vicinity of the jet nozzle using the lately developed twinpeak detection method. By application of this method regions of highly inhomogeneous particle behavior could be detected mainly upstream of the flow perturbation. Numerical results have been obtained by an Eulerian‐Lagrangian method on boundary‐fitted grids. Particle‐particle interactions as well as interphase exchange of momentum have been taken into account. The simulation results showed to be well in accordance with the experimental results.     

A multi-scale simulation method for high Reynolds number wall-bounded turbulent flows

We present an assessment and enhancement of the hybrid two-level large-eddy simulation method (A.G. Gungor and S. Menon, A new two-scale model for large eddy simulation of wall-bounded flows, Prog. Aerosp. Sci. 46 (2010), pp. 28–45), a multi-scale formulation for simulation of high Reynolds number wall-bounded turbulent flows. The assessment of the method is performed by examining role of static and dynamic blending functions used to perform hybridisation of two-level simulation (K. Kemenov and S. Menon, Explicit small-scale velocity simulation for high-Re turbulent flows, J. Comput. Phys. 220 (2006), pp. 290–311; K. Kemenov and S. Menon, Explicit small-scale velocity simulation for high-Re turbulent flows. Part 2: Non-homogeneous flows, J. Comput. Phys. 222 (2007), pp. 673–701) and large-eddy simulation methods. The sensitivity of first- and second-order turbulence statistics to the type of blending functions is investigated by simulating a fully developed turbulent flow in a channel at a friction Reynolds number Re_τ = 395 and comparing the results with those obtained using a direct numerical simulation. The first-order statistics do not show any significant differences for different blending functions, but the second-order statistics show some minor differences. The dynamic evaluation of the hybrid region and the blending function is necessary for non-equilibrium and complex flows where use of a static blending function can lead to inaccurate results. We propose two criteria for the dynamic evaluation; first evaluates extent of the hybrid region based on the subgrid turbulent kinetic energy and the second estimates the blending function based on a characteristic length scale. The computational efficiency of the method is enhanced by incorporating a hybrid programming paradigm where a standard domain decomposition by the message-passing-interface library is combined with the open multi-processing based parallelisation. A further enhancement of the method is achieved by incorporating a closure model for the unclosed hybrid terms in the governing equations, which appear due to hybridisation of two-level- and large-eddy-simulation methods. The model is based on an order of magnitude approximation and a preliminary assessment of the model shows improvement of turbulence statistics when used to simulate turbulent flow in a periodic channel. The assessment and improvements to the multi-scale method make it more suitable for simulation of practical wall-bounded turbulent flows at higher Reynolds number than a conventional large-eddy simulation. This is demonstrated by simulating two representative cases; turbulent flow at high Reynolds number in a periodic channel and flow over a bump placed on the lower surface of a channel, where a relatively coarser computational grid is found to be sufficient for reasonably accurate results.     

Eulerian–Lagrangian multiscale methods for solving scalar equations – Application to incompressible two-phase flows

The present article proposes a new hybrid Eulerian–Lagrangian numerical method, based on a volume particle meshing of the Eulerian grid, for solving transport equations. The approach, called Volume Of Fluid Sub-Mesh method (VOF-SM), has the advantage of being able to deal with interface tracking as well as advection–diffusion transport equations of scalar quantities. The Eulerian evolutions of a scalar field could be obtained on any orthogonal curvilinear grid thanks to the Lagrangian advection and a redistribution of particles on the Eulerian grid. The Eulerian concentrations result from the projection of the volume and scalar informations handled by the particles. The particle velocities are interpolated from the Eulerian velocity field. The VOF-SM method is validated on several scalar interface tracking and transport problems and is compared to existing schemes within the literature. It is finally coupled to a Navier–Stokes solver and applied to the simulation of two free-surface flows, i.e. the two-dimensional buckling of a viscous jet during the filling of a square mold and the three-dimensional dam-break flow in a tank.     

Lagrangian Dispersion in Gaussian Self-Similar Velocity Ensembles

We analyze the Lagrangian flow in a family of simple Gaussian scale-invariant velocity ensembles that exhibit both spatial roughness and temporal correlations. We argue that the behavior of the Lagrangian dispersion of pairs of fluid particles in such models is determined by the scale dependence of the ratio between the correlation time of velocity differences and the eddy turnover time. For a non-trivial scale dependence, the asymptotic regimes of the dispersion at small and large scales are described by the models with either rapidly decorrelating or frozen velocities. In contrast to the decorrelated case, known as the Kraichnan model and exhibiting Lagrangian flows with deterministic or stochastic trajectories, fast separating or trapped together, the frozen model is poorly understood. We examine the pair dispersion behavior in its simplest, one-dimensional version, reinforcing analytic arguments by numerical analysis. The collected information about the pair dispersion statistics in the limiting models allows to partially predict the extent of different phases of the Lagrangian flow in the model with time-correlated velocities.     

A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov–Poisson equations

The Vlasov–Poisson equations describe the evolution of a collisionless plasma, represented through a probability density function (PDF) that self-interacts via an electrostatic force. One of the main difficulties in numerically solving this system is the severe time-step restriction that arises from parts of the PDF associated with moderate-to-large velocities. The dominant approach in the plasma physics community for removing these time-step restrictions is the so-called particle-in-cell (PIC) method, which discretizes the distribution function into a set of macro-particles, while the electric field is represented on a mesh. Several alternatives to this approach exist, including fully Lagrangian, fully Eulerian, and so-called semi-Lagrangian methods. The focus of this work is the semi-Lagrangian approach, which begins with a grid-based Eulerian representation of both the PDF and the electric field, then evolves the PDF via Lagrangian dynamics, and finally projects this evolved field back onto the original Eulerian mesh. In particular, we develop in this work a method that discretizes the 1 + 1 Vlasov–Poisson system via a high-order discontinuous Galerkin (DG) method in phase space, and an operator split, semi-Lagrangian method in time. Second-order accuracy in time is relatively easy to achieve via Strang operator splitting. With additional work, using higher-order splitting and a higher-order method of characteristics, we also demonstrate how to push this scheme to fourth-order accuracy in time. We show how to resolve all of the Lagrangian dynamics in such a way that mass is exactly conserved, positivity is maintained, and high-order accuracy is achieved. The Poisson equation is solved to high-order via the smallest stencil local discontinuous Galerkin (LDG) approach. We test the proposed scheme on several standard test cases.     

On the application of the two-level large-eddy simulation method to turbulent free-shear and wake flows

We present application of the hybrid two-level large-eddy simulation (TLS-LES) method, a multi-scale simulation model, to turbulent free-shear and wake flows at moderately high Reynolds number. The TLS-LES method combines the scale-separation-based two-level simulation (TLS) model with the spatial-filtering-based conventional large-eddy simulation (LES) model in an additive manner using a normalised blending function. The additive blending can be performed in a static or a dynamic manner. We demonstrate that the method, which has been originally developed for wall-bounded flows, can be used to simulate flows in complex configurations without requiring any further adjustments to the model. In this study, three canonical flows are simulated, which are representative of free-shear and wake flows. These cases include a temporally evolving mixing layer, flow past a circular cylinder in a uniform flow and flow past a finite-span airfoil placed in a uniform flow at three different angle of attacks. We analyse the role of static and dynamic blending functions, large-scale grid resolution and the effect of small scales on the instantaneous flow features and turbulence statistics. The results obtained from these cases demonstrate robustness, accuracy and consistency of the multi-scale TLS-LES method and show that the method is suitable for investigation of turbulent flows that encompass features such as massive separation, reattachment, transition to turbulence and unsteady wake, which are challenging to model numerically.