Modeling the effects of small turbulent scales on the drag force for particles below and above the Kolmogorov scale

Consistently with observations from recent experiments and DNS, we focus on the effects of strong velocity increments at small spatial scales for the simulation of the drag force on particles in high Reynolds number flows. In this paper, we decompose the instantaneous particle acceleration in its systematic and residual parts. The first part is given by the steady-drag force obtained from the large-scale energy-containing motions, explicitly resolved by the simulation, while the second denotes the random contribution due to small unresolved turbulent scales. This is in contrast with standard drag models in which the turbulent microstructures advected by the large-scale eddies are deemed to be filtered by the particle inertia. In our paper, the residual term is introduced as the particle acceleration conditionally averaged on the instantaneous dissipation rate along the particle path. The latter is modeled from a log-normal stochastic process with locally defined parameters obtained from the resolved field. The residual term is supplemented by an orientation model which is given by a random walk on the unit sphere. We propose specific models for particles with diameter smaller and larger size than the Kolmogorov scale. In the case of the small particles, the model is assessed by comparison with direct numerical simulation (DNS). Results showed that by introducing this modeling, the particle acceleration statistics from DNS is predicted fairly well, in contrast with the standard LES approach. For the particles bigger than the Kolmogorov scale, we propose a fluctuating particle response time, based on an eddy viscosity estimated at the particle scale. This model gives stretched tails of the particle acceleration distribution and dependence of its variance consistent with experiments

Model for the dynamics of micro-bubbles in high-Reynolds-number flows

We propose a model for the acceleration of micro-bubbles (smaller than the dissipative scale of the flow) subjected to the drag and fluid inertia forces in a homogeneous and isotropic turbulent flow. This model, that depends on the Stokes number, Reynolds number and the density ratio, reproduces the evolution of the acceleration variance as well as the relative importance and alignment of the two forces as observed from direct numerical simulations (DNS). We also report that the bubble acceleration statistics conditioned on the local kinetic energy dissipation rate are invariant with the Stokes number and the dissipation rate. Based on this observation, we propose a stochastic model for the instantaneous bubble acceleration vector accounting for the small-scale intermittency of the turbulent flows. The norm of the bubble acceleration is obtained by modelling the dissipation rate along the bubble trajectory from a log-normal stochastic process, whereas its orientation is given by two coupled random walks on a unit sphere in order to model the evolution of the joint orientation of the drag and inertia forces acting on the bubble. Furthermore, the proposed stochastic model for the bubble acceleration is used in the context of large eddy simulations (LES) of turbulent flows laden with small bubbles. To account for the turbulent motion at scales smaller than the mesh resolution, we decompose the instantaneous bubble acceleration in its resolved and residual parts. The first part is given by the drag and fluid inertia forces computed from the resolved velocity field, and the second term refers to the random contribution of small unresolved turbulent scales and is estimated with the stochastic model proposed in the paper. Comparisons with DNS and standard LES, show that the proposed model improves significantly the statistics of the bubbly phase

Particle-laden flows forced by the disperse phase: Comparison between Lagrangian and Eulerian simulations

The goal of the present work is to assess the ability of Eulerian moment methods to reproduce the physics of two-way coupled particle-laden turbulent flow systems. Previous investigations have been focused on effects such as preferential concentration, and turbulence modulation, but in regimes in which turbulence is sustained by an imposed external forcing. We show that in such regimes, Eulerian methods need resolutions finer than nominal Kolmogorov scale in order to capture statistics of particle segregation, but gas and disperse phase velocity variances can be captured with resolutions comparable to the Kolmogorov length. The work is then extended to address the question whether Eulerian methods are suitable in scenarios in which the continuum field of interest (temperature or momentum) is itself primarily driven by particles. To this end we have extended our analysis to the problem of turbulence driven by heated particles (Zamansky et al. PoF 2014) and have assessed capabilities of Eulerian methods in capturing particle segregation, as well as statistics of the temperature and velocity fields. Separate investigations are developed for cases with and without buoyancy driven turbulence. For each case corresponding Lagrangian calculations are developed and convergence of statistics with respect to the number of particles is established. Then the statistically- converged Lagrangian and Eulerian results are compared. Results show that accurate capture of segregation by the Eulerian methods always requires resolutions much higher than the nominal Kolmogorov scale. In scenarios for which a continuum phase is forced by particles, results from Eulerian methods show some sensitivity of predicted continuum statistics to the mesh resolution. This sensitivity was found to be largest for the case of a temperature field forced by hot particles, but without presence of buoyancy. In this case a Eulerian method with nominal Kolmogorov resolution was found to be insufficient for capture of temperature statistics. When additional coupling between particles and continuum phase is introduced by including the buoyancy effects, this sensitivity is suppressed in the temperature field, but some sensitivity to the Eulerian mesh resolution were detected in the momentum fields

Turbulent thermal convection driven by heated inertial particles

The heating of particles in a dilute suspension, for instance by radiation, chemical reactions or radioactivity, leads to local temperature fluctuations in the fluid due to the non-uniformity of the disperse phase. In presence of a gravity field, the fluid is set in motion by the resulting buoyancy forces. When the particle density is different than the fluid, the fluid motion alters the spatial distribution of particles and possibly strengthens their concentration inhomogeneities. This in turn causes more intense local heating. Direct numerical simulations in the Boussinesq limit show this feedback loop. Various regimes are identified depending on the particle inertia. For very small particle inertia, the macroscopic behavior of the system is the result of many thermal plumes that are generated independently of each other. For significant particle inertia, clusters of particles are observed and their dynamics control the flow. The emergence of very intermittent turbulent fluctuations shows that the flow is influenced by the larger structures (turbulent convection) as well as by the small-scale dynamics that affect particle segregation and thus the flow forcing. Assuming thermal equilibrium between the particles and the fluid (i.e. infinitely fast thermal relaxation of the particle), we investigate the evolution of statistical observables with the change of main control parameters (namely the particle number density, the particle inertia and the domain size), and propose scaling argument for these trends. Concerning the energy density in the spectral space, it is observed that the turbulent energy and temperature spectra follow a power law, the exponent of which varies continuously with the Stokes number. Furthermore the study of the spectra of the temperature and momentum forcing (and thus of the concentration/temperature and velocity/temperature correlations) gives strong support to the proposed feedback loop mechanism. We then discuss the intermittency of the flow, and analyze the effect of relaxing some of the simplifying assumptions, thus assessing the relevance of the original studied configuration

Experimental study of bubble detection in liquid metal.

Bubble detection in liquid metal is an important issue for various technological applications. For instance, in the framework of Sodium Fast Reactors design, the presence of gas in the sodium flow of the primary and secondary loops is an issue of crucial importance for safety and reliability. Here, the two main gas measurement methods in sodium are ultrasonic testing and eddy-current testing; we investigate the second method in our study. In a first approach, we have performed experiments with liquid metal galinstan containing insulating spherical beads of millimeter size. The liquid metal is probed with an Eddy-current Flowmeter (ECFM) in order to detect the beads and characterize their diameter and position. Results show that the signal measured by the ECFM is correlated with the effect of these parameters. Finally, an analytical model is proposed and compared to the experimental results

Radiation induces turbulence in particle-laden fluids.

When a transparent fluid laden with solid particles is subject to radiative heating, non-uniformities in particle distribution result in local fluid temperature fluctuations. Under the influence of gravity, buoyancy induces vortical fluid motion which can lead to strong preferential concentration, enhancing the local heating and more non- uniformities in particle distribution. By employing direct numerical simulations this study shows that the described feedback loop can create and sustain turbulence. The velocity and length scale of the resulting turbulence is not known a priori, and is set by balance between viscous forces and buoyancy effects. When the particle response time is comparable to a viscous time scale, introduced in our analysis, the system exhibits intense fluctuations of turbulent kinetic energy and strong preferential concentration of particle

Drag modulation in turbulent boundary layers subject to different bubble injection strategies

The aim of this study is to investigate numerically the interaction between a dispersed phase composed of micro-bubbles and a turbulent boundary layer flow. We use the Euler–Lagrange approach based on Direct Numerical Simulation of the continuous phase flow equations and a Lagrangian tracking for the dispersed phase. The Synthetic Eddy Method (SEM) is used to generate the inlet boundary condition for the simulation of the turbulent boundary layer. Each bubble trajectory is calculated by integrating the force balance equation accounting for buoyancy, drag, added-mass, pressure gradient, and the lift forces. The numerical method accounts for the feedback effect of the dispersed bubbles on the carrying flow. Our approach is based on local volume average of the two-phase Navier–Stokes equations. Local and temporal variations of the bubble concentration and momentum source terms are accounted for in mass and momentum balance equations. To study the mechanisms implied in the modulation of the turbulent wall structures by the dispersed phase, we first consider simulations of the minimal flow unit laden with bubbles. We observe that the bubble effect in both mass and momentum equations plays a leading role in the modification of the flow structures in the near wall layer, which in return generates a significant increase of bubble volume fraction near the wall. Based on these findings, we discussed the influence of bubble injection methods on the modulation of the wall shear stress of a turbulent boundary layer on a flat plate. Even for a relatively small bubble volume fraction injected in the near wall region, we observed a modulation in the flow dynamics as well as a reduction of the skin friction

Fluid inertia effects on the motion of small spherical bubbles or solid spheres in turbulent flows

In this paper we study finite particle Reynolds number effects up to Re p=50 on the dynamics of small spherical bubbles and solid particles in an isotropic turbulent flow. We consider direct numerical simulations of light pointwise particles with various expressions of the drag force to account for finite Re p and the type of particle. Namely, we consider the Stokes drag law, the Schiller and Neumann relation and the Mei law. We show that an effective Stokes number, based on the mean value of the drag coefficient to account for the inertial effects involved in the drag law, gives a quasi-self-similar evolution of the variances of the bubble acceleration and of the forces exerted on the particle. This allows us to provide a satisfactory prediction of these quantities using Tchen's theory at finite particle Reynolds number. Based on these relations, we can specify the conditions under which the total inertial force (sum of the added mass and the Tchen contributions) is negligible compared with the drag force. Thus, for particles of very small dimensions, the fluid inertia force is negligible, provided the density ratio is of order 1 or larger. However, when the particle inertia becomes consequential, the threshold value of the density ratio increases significantly. Although this corresponds to the limit of the validity of the model, this draws attention to the fact that, for large Stokes numbers, the added mass and fluid inertia forces could play a more important role than is usually attributed to them

Magnetoconvection transient dynamics by numerical simulation

We investigate the transient and stationary buoyant motion of the Rayleigh-Bénard instability when the fluid layer is subjected to a vertical, steady magnetic field. For Rayleigh number, Ra, in the range 10^3-10^6, and Hartmann number, Ha, between 0 and 100, we performed three-dimensional direct numerical simulations. To predict the growth rate and the wavelength of the initial regime observed with the numerical simulations, we developed the linear stability analysis beyond marginal stability for this problem. We analyzed the pattern of the flow from linear to nonlinear regime. We observe the evolution of steady state patterns depending on Ra/Ha2 and Ha. In addition, in the nonlinear regime, the averaged kinetic energy is found to depend on Ra and to be independent of Ha in the studied range

Destabilization of a liquid metal by nonuniform Joule heating

We study the effect of an impressing AC magnetic field at the bottom of a liquid metal layer of thickness h. In this situation the fluid is set in motion by the buoyancy forces caused by internal heat sources. The heat sources, caused by the Joule effect induced by the AC field, present an exponentially decaying profile, with characteristic length δ. As the magnetic field is horizontal, the Lorentz force has no influence on the dynamics of the system since it contributes only to the magnetic pressure. We propose an analysis of both the transient and fully developed regimes using linear stability analysis (LSA) and direct numerical simulations (DNSs). The transient period is governed by the temporal evolution of the temperature field as well as the development of the convective instability, which can be concomitant and therefore requires adopting a transient LSA algorithm to track these two effects. The DNSs have been performed for various distributions of the heat sources and various total heat input. This corresponds to independently varying δ/h in the range 0.04≤δ/h≤0.45 and a Rayleigh number 1.1×104≤Ra≤1.2×105. We observe the relaxation of the temperature up to the steady conductive profile before the transition to the nonlinear regime when Ra is small, whereas for larger Ra, nonlinear effects appear during the relaxation of the temperature profile. The unsteadiness of the temperature field significantly alters the development of the instability because of a much smaller growth rate. Surprisingly, we observe that δ/h has only a limited influence on averaged quantities as well as on the patterns for both the linear and nonlinear regimes. This comes with the fact that the profiles present an apparent reflectional symmetry, despite the asymmetry of the governing equations