Settling of Small Particles in Homogeneous Turbulence: Settling Velocity Enhancement by Two-Way Coupling T. Bosse, L. Kleiser (ETHZ), E. Meiburg (UCSB) Motivation One-way coupled flows - particle-laden mixing layer Two-way coupled flows - particles settling in homogeneous turbulence - dynamics of a suspension drop Summary

Motivation Particle-air interaction influences: Growth / Amplification Front velocity Deposition Runout length

Motivation Turbidity current. Turbidity current: Sediment flow down the continental slope Repeated turbidity currents in the same region can lead to the formation of hydrocarbon reservoirs Effective settling rate determines properties of sediment layer: - particle layer thickness distribution - particle size distribution Other applications: water/air quality, dust storms, cloud dynamics, medical devices, spray combustion, industrial processes...

Dilute flows Volume fraction of particles of O(10 -3 ): particle radius « particle separation particle radius « characteristic length scale of flow coupling of fluid and particle motion primarily through momentum exchange, not through volumetric effects effects of particles on fluid continuity equation negligible

Very dilute flows: One-way coupling Small mass fraction of heavy particles (dusty gas, dilute spray): particles move independently of each other particles have negligible effect on the fluid motion can first solve for fluid motion, afterwards for particle dynamics Particle dynamics is governed by balance of : particle inertia viscous drag force gravity added mass, lift forces, pressure gradients in the fluid, and Basset history term can be neglected

Three physically relevant time scales: aerodynamic response time of particle  a characteristic time of flow field  f particle settling time  s Very dilute flows: One-way coupling (cont’d) Two dimensionless parameters govern particle motion:

Very dilute flows: One-way coupling (cont’d) Solve ODE for each particle (Maxey and Riley 1983): Stokes drag gravity Continuity and momentum equation for single-phase fluid:

Example: Particle laden mixing layer Martin and Meiburg (1994) small particle inertia, weak gravitational effects: particles follow fluid motion no local accumulation of particles clear and particle laden fluid mix through entrainment

Particle laden mixing layer (cont’d) Martin and Meiburg (1994) intermediate particle inertia, weak gravitational effects: particles are ejected from vortex centers optimal ejection of particles with intermediate Stokes number (Crowe et al.) local accumulation of particles in bands midway between vortex\centers

Particle laden mixing layer (cont’d) Martin and Meiburg (1994) intermediate particle inertia, strong gravitational effects: sedimenting particles are ejected by vortices organization of the particle concentration field into sedimenting bands

 fluid velocity  particle velocity Particles settling in homog. turbulence: One-way coupling Maxey (1987), Wang and Maxey (1993): simulation - analyze cellular flows and isotropic turbulence under one-way coupling - particles accumulate in regions of low vorticity and high strain - increase in mean settling velocity as compared to Stokes velocity because of ‘preferential sweeping’ towards regions of downward fluid velocity g inertial biaspreferential sweeping +

Particles settling in homog. turbulence: Two-way coupling Aliseida et al. (2002), Yang and Shy (2005): - wind tunnel/closed container experiments, spray droplets/solid particles - fluid is accelerated downwards in regions of high particle concentration, which leads to enhanced settling - large discrepancy between the two studies w.r.t. magnitude of this effect g inertial biaspreferential sweeping + g collective particle drag +

Dilute, two-way coupled flows Suspended particles occupy small volume fraction, but have O(1) mass fraction, strong particle inertia: each particle locally exerts force on the fluid (equal and opposite to the fluid force acting on the particle) volume coupling can still be neglected Suspension dynamics can be described by: incompressible continuity equation Navier-Stokes equation plus additional force term set of ODE’s for each particle’s location, velocity

Dilute, two-way coupled flows: Governing equations inverse drag force Dimensionless parameters: Scaling with Taylor microscale and rms-velocity u’:

Dilute, two-way coupled flows (cont’d) Dimensionless parameters: As will be seen, results suggest that it is preferable to scale the particle equation with Kolmogorov scales:

Simulation approach Fluid equations: Fourier pseudospectral method, RK/CN time stepping Turbulence forcing procedure according to Eswaran & Pope (1988) Particles: Lagrangian tracking Coupling terms: Trilinear interpolation between particle and grid point locations Steps: 1.Fluid only: Run simulation until statistically stationary 2.Add particles with random spatial distribution, Stokes setting velocity 3.Run with one-way coupling until statistically stationary 4.Turn on two-way coupling 5.Run until statistically stationary

Simulation approach: Related work For dilute flows with many particles, several variations of force coupling: Lagrangian-based Navier-Stokes approaches (Elghobashi et al., Eaton et al., Walther and Koumoutsakos, Lohse et al., etc….) Stokeslet-based simulations (Nitsche and Batchelor ’97, Machu et al. ’01) Multipole expansions (Maxey and Patel ’01) For O(10-100) particles: DNS (Joseph, Glowinski et al.) Force coupling method (Maxey and Dent ’98) For dense particle loading: Two-fluid simulations (Drew ’83, Crowe et al. ’96) closure assumptions needed

Settling velocity enhancement most pronounced for Results: One-way coupling Validation against Wang & Maxey (1993): WM,

Results: One-way coupling (cont’d) Temporal evolution of particle concentration distribution: Large particle-free regions emerge Regions of high particle concentration grow Regions of moderate particle concentration decrease Good agreement with Wang & Maxey (1993)

Correlation between particle volume fraction and vorticity magnitude: Results: Two-way coupling

Results: Two-way coupling (cont’d) Settling velocity enhancement: Two-way coupling effects increase with particle volume fraction Increase in settling velocity noticeable above volume fraction O(10 -5 )

Results: Two-way coupling (cont’d) Particle concentration distribution: Small particle volume fractions: probability functions not affected by two-way coupling Larger particle volume fractions: fewer particle-free regions

Results: Two-way coupling (cont’d) Enhancement of particle settling velocity: Enhancement due to two-way coupling above volume fractions O(10 -6 ) Above volume fractions O(5 x ), turbulence properties are modified, so that the settling velocity enhancement increases less than linearly

Results: Two-way coupling (cont’d) If turbulence properties are kept constant by adjusting forcing: Nearly linear increase in settling velocity with volume fraction : Re  adjusts itself _____ : Re kept constant

Results: Two-way coupling (cont’d) Mechanism of settling velocity enhancement: Downward fluid velocity increases in regions of high particle concentration Increased downward fluid velocity enhances particle settling velocity Vertical fluid velocity as function of particle volume fraction: Settling velocity enhancement as function of particle volume fraction:

Results: Comparison with experiments simulations underpredict two-way coupling effects measured by Aliseida et al. (‘02) simulations overpredict two-way coupling effects measured by Yang and Shy (’05) experiment, one-way two-way, Comparison with Aliseda et al. (2002) Comparison with Yang & Shy (2005) two-way,  one-way,  experiment, ( )( )

Results: Comparison with experiments Potential reasons for discrepancies: experiments: particle size distribution, simulation: monodisperse particles simulations: - match turbulence Re number, but other turbulence parameters may be somewhat different - low order interpolation may cause some errors, but a few per cent at most experiments: - particles may induce mean downward fluid motion in the windtunnel test section

One-way coupling: –Successful validation against Wang & Maxey, JFM (1993) –Strongest particle-fluid interaction for Stokes numbers around unity –Large inhomogeneities in particle distribution, correlation between vorticity and particle concentration Two-way coupling –Particle settling velocity enhancement found for –Monotonic increase of with particle volume fraction, relation roughly linear if microscale Reynolds number kept constant –Turbulence modification sets in for : particles have dissipative effect on turbulent carrier fluid –Collective particle drag responsible for additional settling velocity enhancement compared to one-way coupling Comparison with experiment –Still significant differences between numerical and experimental results –Further research necessary Summary

Numerical simulation of a suspension drop Mitts (1996) Bosse (2002)

Re d = 0.01 Numerical simulation of a suspension drop

Re d = 1 Numerical simulation of a suspension drop

Re d = 300 Numerical simulation of a suspension drop

Challenges in the simulation of particle laden flows: different parameter ranges dominated by different physical mechanisms large variety of numerical approaches (Lagrangian, Eulerian, two-fluid, statistical, hybrid….) two-way coupling between fluid and particles: momentum, volume, thermal, chemical… interaction between suspension and bed: particle deposition, erosion, sorting… Summary