Experimental investigation of turbulent transport of material particles

We report measurements of Lagrangian velocity and acceleration statistics of particles transported in a turbulent flow obtained with an acoustic Doppler velocimelly technique. We consider a homogeneous isotropic grid turbulence generated in a wind tunnel. As a first step we study isolated particles dynamics with a particular focus on the influence of particles finite size on their response to the turbulence forcing. As particles we use neutrally buoyant soap bubbles inflated with helium. The size of the particles can be adjusted from 1.5 mm to 6 mm, corresponding to inertial range scales. We show that the response time of the particles to the turbulence forcing increases with their s i ze . We analyze our data in the frame of two times stochastic models, and show that the cut-of! time scale in the Lagrangian energy spectrum of the particles dynamics has a dependence on their diameter consistent with a low­ pa ss .filtering of the turbulent cascade by the particles finite size. des bulles de savons gonflees a /'helium de sorte a etre rendues iso-densite dans !'air. Leur faille, ajustable entre 1.5 mm et 6 mm, correspond a des echelles inertielles de Ia turbulence. Nous montrons que le temps de reponse des particules augmente avec leur faille et analysons ces resultats a Ia lumiere de modeles stochastiques a deux temps. Nous montrons ainsi' que le spectre Lagrangien d'energie des particules presente bien une coupure a petite echelle liee a leur tail/e.

the influence of particles finite size on their response to the turbulence forcing. As particles we use neutrally buoyant soap bubbles inflated with helium. The size of the particles can be adjusted from 1.5 mm to 6 mm, corresponding to inertial range scales. We show that the response time of the particles to the turbulence forcing increases with their size. We analyze our data in the frame of two times stochastic models, and show that the cut-of! time scale in the Lagrangian energy spectrum of the particles dynamics has a dependence on their diameter consistent with a low pass .filtering of the turbulent cascade by the particles finite size.

Introduction
Particle laden turbulent flows play an important role in various situations such as industrial processes or atmospheric dispersion of pollutan ts for instance. When the particles are neutrally buoyant and small (typically comparable in size with the dissipation scale of the surround ing turbulence) they behave as tracers for fluid particles. However, in many practical situations, the particles are heavier and/or larger, their dynamics is then af f ected by inert ial effects and it deviates from fluid particles dynamics, (Maxey et al (1983), Aliseda et al. (2002), Ayyalasomayajula et al. (2006)). The precise role of size and density of the particles in the modification of their dynamics with respect to fluid tracers, remains largely an open question.
Here, we report measurements of Lagrangian velocity and acceleration stausttcs of material particles transported in a grid generated windtunnel turbulent flow, with a Reynolds number (based on Taylor microscale) of R:�.-200. The dissipation scale T} is 200 J. lm and the energy injection scale L is 2.5 em. As a first step, we only explore particles finite size effects. To decouple the role of size and density of the particles, we consider neutrally buoyant particles, which are soap bubbles inflated with helium and which diameter can be adjusted from 1 .5 mm to 6 mm which corresponds to inertial range scales. The Lagrangian measurements are obtained with an acoustic Doppler velocimetry technique (figure I a): from the instantaneous Doppler frequency shift of acoustic waves scattered by a particle in a turbulent flow, we measure the velocity of the particle (Poulain et al. (2004)). The instantaneous fr equency is determined with a parametric maximum oflikelyhood algorithm (Mordant et al. (2001)). The particles can be tracked over a period covering several dissipation time scales, corresponding to a significant fraction of the integral time scale of the flow.

Experimental Approach
The Lagrangian measurements are obtained with an acoustic Doppler velocimetry teclmique (figure I a): from the instantaneous Doppler frequency shift of acoustic waves scattered by a particle in a turbulent flow, we measure the velocity of the particle (Poulain 2004). The instantaneous frequency is determined with a parametric maximum of likelyhood algorithm derived by Mordant et al. (2004). The particles can be tracked over a period covering several dissipation time scales, corresponding to a significant fraction of the integral time scale of the flow.

Experiemental Results
In order to investigate the influence of particle size on its Lagrangian dynamics, we first consider how the Lagrangian velocity autocorrelation function is affected when we change the bubbles diameter. Note that we only show a relatively short time lags range, for which we have enough Lagrangian trajectories to ensure a good statistical convergence.
From the curvature at r=O we can estimate an equivalent Lagrangian Taylor time scale r;_(D) associated to the Lagrangian dynamics of a particle of diameter D. Figure 2a shows a clear dependence of r;. on particle size. We note that as the particle size decreases, r; appears to approach an asymptotic value (which we can estimate here around 25 ms) which corresponds to the intrinsic Lagrangian microscale of the turbulent flow, as smaller particles approach fluid tracers. The increase of the microscale of the Lagrangian dynamics of the particles as their size increases suggests a longer response time of larger particles to the turbulence forcing. This is consistent with the intuitive phenomenology, that large parti cles do not feel velocity gradients at sea les smaller than their size , and therefore, they must filter in some way the turbulent energy cascade at some small scale related to their size. To test fu rther this scenario, we analyse the Lagrangian velocity correlation function in the frame of two times stochastic model given by

Saw ford et a!. (I 991).
In this description, the autocorrelation function is given by a double exponential law : (1) where r0 is a small time scal e chamcterizing the cut-off of the particles Lagrangian energy spectrum. For fluid particle tracers, for instance, ro is directly related to the viscous dissipation time r,1. For particles with finite diameter D in the inertial range, in the scenario described above where the fluid turbulent energy is low-pass filtered by the particle at a scale corresponding to its diameter D, the corresponding cut-off time scale can be estimated in the framework of K41 phenomenology as 'tv � e113 D713, where <: is the energy dissipation rate.

Conclusion
In the work presented above, the finite size eff ects of the particles, in a homogeneous isotropic turbulent flow have been studied. We have fo und that the Lagrangian response time of the material particles increases as their diameter increases, which is due to the fact that particles are insensible to the velocity gradients at scales smaller than their diameter. As a result of this phenomenon we have observed filtering in the Lagrangian energy spectrum. Lagrangian rnicroscale time, r;, have been determined by fitting parabolas to Lagrangian velocity autocorrelation. It has been found that r;, decreases as the diameter of the particle decreases and it tends to reach an asymptotic value. Later on, a two times stochastic model is used to determine small time scale, r0. The realation derived from K4l phenomenology, r0 -D 21 3 appears to work in a good agreement when we plot ro I D213 as a function of particle's diameter. Other diagnosis not discussed here, based for instance on measurements of the acceleration variance of the particles as a function of their diameter also confirm this idea. Further investigation will explore the role of particles density.