Open Statistical Physics Open University,10/03/2010 A kinetic theory for the transport of small particles in turbulent flows Michael W Reeks School of Mechanical & Systems Engineering, Newcastle University

Open Statistical Physics Open University,10/03/2010 Environmental /industrial processes Mixing & combustion pollutant dispersion fouling / deposition clean up radioactive releases slurry /pneumatic conveying aerosol formation

Open Statistical Physics Open University,10/03/2010 Modelling Particle Flows Particle Tracking (Lagrangian) –track particles through a random flow by solving particle equation of motion Two-Fluid Model (Eulerian) –Continuum equations for continuous (carrier flow) and dispersed phase (particles) –constitutive relations /closure approximations –boundary conditions

Open Statistical Physics Open University,10/03/2010 Objectives application of formal closure methods in dilute flows to –derive continuum equations /constitutive relations for the particle phase compare with traditional heuristic approach criteria for their validity –incorporate the influence of turbulent structures on particle motion into continuum equations One particle dispersion Two particle (pair) dispersion Drift in inhomogeneous turbulence

Open Statistical Physics Open University,10/03/2010 particle motion in plain vortex and straining flow

Open Statistical Physics Open University,10/03/2010

Caustics – Mehlig & Wilkinson

Open Statistical Physics Open University,10/03/2010 Settling in homogeneous turbulence Maxey 1988, Maxey & Wang 1992

Open Statistical Physics Open University,10/03/2010 –Begin with particle equation of motion e.g. for gas-solid flows –Separate particle velocities & aerodynamic forces into mean & fluctuating components –Average over all realisations of the flow Reynolds stresses Inter-phase momentum transfer Mass X accel Two-Fluid Model mean and fluctuating carrier flow velocities

Open Statistical Physics Open University,10/03/2010 Kinetic/ PDF Approach –analogous to the kinetic theory of gases –uses an equation analogous to the Maxwell- Boltzmann Equation to derive the two-fluid equations for a dispersed flow mass-momentum and energy equations (c.f. RANS for continuous phase) constitutive relations boundary conditions

Open Statistical Physics Open University,10/03/2010 Modelling of Particle Flows 11 Two PDF approaches Apply closure to transport equations for particle phase space density u p is the fluid velocity seen by the particle at x (Simonin approach)

Open Statistical Physics Open University,10/03/2010 Closure approximations for Reeks, Zaichek, Swailes, Minier

Open Statistical Physics Open University,10/03/2010 Modelling of Particle Flows 13 Body force Mass x acceleration Turbulent stress Momentum Equation - PDF Approach Equation of state

Open Statistical Physics Open University,10/03/2010 Momentum equation as a diffusion equation

Open Statistical Physics Open University,10/03/ Particle Reynolds stress transport eqns - Reynolds stresses depend on shearing of both phases - Requires closure for Reynolds stress flux

Open Statistical Physics Open University,10/03/ Chapman Enskog Approximation

Open Statistical Physics Open University,10/03/2010 Application- PDF solutions Transport and deposition in turbulent boundary layers Deposition velocity particle relaxation time Velocity distribution at a wall for particles settling under gravity in a turbulent flow with particla absorption at the wall

Open Statistical Physics Open University,10/03/2010 Divergence of the particle velocity field along a particle trajectory measures the change in particle concentration zero for particles which follow an incompressible flow non zero for particles with inertia particle streamlines

Open Statistical Physics Open University,10/03/ Application to kinetic approach

Open Statistical Physics Open University,10/03/2010 Pair dispersion and segregation Two colliding spheres radii r 1, r 2 r1r1 r2r2 Collision sphere r g (r)

Open Statistical Physics Open University,10/03/2010 Kinetic Equation for P(w,r,t)and moment equations momentum w = relative velocity between identical particle pairs, distance r apart Δu(r) = relative velocity between 2 fluid pts, distance r apart Structure functions Net turbulent Force mass convection β = St -1, St=Stokes number Probability density(Pdf) mass

Open Statistical Physics Open University,10/03/2010 Kinetic Equation predictions Zaichik and Alipchenkov, Phys Fluids 2003

Open Statistical Physics Open University,10/03/2010 Summary / Conclusions Particle transport and segregation in a turbulent flow –Kinetic / pdf approach (single particle transport) Treatment of the dispersed particle phase as a fluid –Continuum equations –Constitutive relations –Boundary conditions (perfectly / partially absorbing) –Kinetic approach for particle pair transport radial distribution function Role of compressibility in the formulation of a kinetic equation –Net relative drift velocity between particle pairs – enhancement local concentration of neighbouring particles

Open Statistical Physics Open University,10/03/2010 Moments of particle number density St=0.05St=0.5 Particle number density is spatially strongly intermittent Sudden peaks indicate singularities in particle velocity field

Open Statistical Physics Open University,10/03/2010 Influence of turbulent structures x,t X p,s Consider the instantaneous concentration  (x,t)derived from an initial concentration  (x,0) and a particle velocity field v p (x,t). The conservation of mass equation is

Open Statistical Physics Open University,10/03/2010 Dispersion in a random compressible flow Drift velocity(Maxey) Diffusion tensor D (Taylor’s theory) Net particle flux

Open Statistical Physics Open University,10/03/2010 Segregation of inertial particles in turbulent flows M. W. Reeks, R. IJzermans, E. Meneguz, Y.Ammar Newcastle University, UK M. Picciotto, A. Soldati University of Udine, It ‘ Fractals, singularities, intermittency, and random uncorrelated motion ’

Open Statistical Physics Open University,10/03/2010 De-mixing of particles Particles suspended in a turbulent flow do not mix but segregate –depends upon the particles inertial response to: structure and persistence of the turbulence Important in mixing and particle collision processes –growth of PM10 and cloud droplets in the atmosphere the onset of rain. Presentation is about quantifying segregation – analysing statistics and morphology of the segregation using a Full Lagrangian Method (FLM) –use of compressibility to reveal » fractal nature » intermittency » random uncorrelated motion

Open Statistical Physics Open University,10/03/2010 particle motion in a vortex and straining flow Stokes number St = τ p /τ f ~1

Open Statistical Physics Open University,10/03/2010 Segregation in isotropic turbulence

Open Statistical Physics Open University,10/03/2010 Segregation - dependence on Stokes number St=τ p /τ f

Open Statistical Physics Open University,10/03/2010 Segregation in counter-rotating vortices Flow pattern translated randomly in space with finite life- time

Open Statistical Physics Open University,10/03/2010 Caustics – Mehlig & Wilkinson

Open Statistical Physics Open University,10/03/2010 Compressibility of a particle flow Falkovich, Elperin,Wilkinson, Reeks zero for particles which follow an incompressible flow non zero for particles with inertia measures the change in particle concentration Divergence of the particle velocity field along a particle trajectory particle streamlines Compressibility (rate of compression of elemental particle volume along particle trajectory)

Open Statistical Physics Open University,10/03/2010 Measurement of the compressibility Deformation of elemental volume Compression - fractional change in elemental volume of particles along a particle trajectory can be obtained directly from solving the particle eqns. of motion  Avoids calculating the compressibility via the particle velocity field  Can determine the statistics of ln J(t) easily.  The process is strongly non-Gaussian – highly intermittent - x p (t),v p (t),J ij (t),J(t)) - Fully Lagrangian Method

Open Statistical Physics Open University,10/03/2010 Particle trajectories in a periodic array of vortices

Open Statistical Physics Open University,10/03/2010 Deformation Tensor J

Open Statistical Physics Open University,10/03/2010 Singularities in particle concentration

Open Statistical Physics Open University,10/03/2010 Compressibility Simple 2-D flow field of counter rotating vortices KS random Fourier modes: distribution of scales, turbulence energy spectrum

Open Statistical Physics Open University,10/03/2010 Moments of particle number density Along particle trajectory: particle number density n related to J by : Particle averaged value of is related to spatially averaged value: Trivial limits: (equivalent to counting particles) Any space-averaged moment is readily determined, if J is known for all particles in the sub-domain

Open Statistical Physics Open University,10/03/2010 Moments of particle number density St=0.05St=0.5 Particle number density is spatially strongly intermittent Sudden peaks indicate singularities in particle velocity field

Open Statistical Physics Open University,10/03/2010 Comparison with analytical estimate If St is sufficiently small: For first time, numerical support for theory of Balkovsky et al (2001, PRL): “  is convex function of  ”.

Open Statistical Physics Open University,10/03/2010 Random uncorrelated motion Quasi Brownian Motion - Simonin et al Decorrelated velocities - Collins Crossing trajectories - Wilkinson RUM - Ijzermans et al. Free flight to the wall - Friedlander (1958) Sling shot effect - Falkovich Falkovich and Pumir (2006)

Open Statistical Physics Open University,10/03/2010 Radial distribution function (RDF) g(r) r g (r)

Open Statistical Physics Open University,10/03/2010 DNS: details of the code Statistically stationary HIT Pseudo-spectral code Grid 128x128x128 Re =65 Forcing is applied at the lowest wavenumbers NSE for an incompressible viscous turbulent flow : In a DNS of HIT, the solution domain is in a cube of size L, and : inertial particles are random distributed at t=0 in a box of L=2 Interpolation of the velocity the particle position with a 6th order Lagrangian polynomial Trajectories and equations calculated by RK4 method Initial conditions so that volume is initially a cube

Open Statistical Physics Open University,10/03/2010 Averaged value of compressibility vs time Elena Meneguz 46 Qualitatively the same trend with respect to KS We expect a different threshold value WHAT CAUSES THE POSITIVE VALUES???

Open Statistical Physics Open University,10/03/2010 Moments of particle concentration intermittency due to the presence of singularities in the pvf

Open Statistical Physics Open University,10/03/2010 Turbulent Agglomeration Two colliding spheres radii r 1, r 2 r1r1 r2r2 test particle Saffman & Turner model Agglomeration in DNS turbulence L-P Wang et al. - examined S&T model Frozen field versus time evolving flow field Absorbing versus reflection Brunk et al. - used linear shear model to assess influence of persistence of strain rate, boundary conditions, rotation Collision sphere

Open Statistical Physics Open University,10/03/2010 Agglomeration of inertial particles Sundarim & Collins(1997), Reade & Collins (2000): measurement of rdfs and impact velocities as a function of Stokes number St Net relative velocity between colliding spheres along their line of centres RDF at r c

Open Statistical Physics Open University,10/03/2010 Inertial collisions (RUM) Ratio of the RMS of the relative velocity of colliding particles over the corresponding RMS of the relative fluid velocity; collision radius r c/ η k =0.1 r1r1 r2r2 r c =r 1 +r 2 particle Stokes number St

Open Statistical Physics Open University,10/03/2010 Probabalistic Methods

Open Statistical Physics Open University,10/03/2010 Kinetic Equation and its Moment equations Zaichik, Reeks,Swailes, Minier) momentum w = relative velocity between identical particle pairs, distance r apart Δu(r) = relative velocity between 2 fluid pts, distance r apart Structure functions Net turbulent Force (diffusive) mass convection β = St -1 Probability density(Pdf) mass

Open Statistical Physics Open University,10/03/2010 Kinetic Equation predictions Zaichik and Alipchenkov, Phys Fluids 2003

Open Statistical Physics Open University,10/03/2010 Dispersion and Drift in compressible flows ( Elperin & Kleorin, Reeks, Koch & Collins, Reeks ) w(r,t) the relative velocity between particle pairs a distance r apart at time t Particles transported by their own velocity field w(r,t) Conservation of mass (continuity) Random variable

Open Statistical Physics Open University,10/03/2010 Summary / Conclusions Overview –Transport, segregation, agglomeration dependence on Stokes number –Use of particle compressibility d/dt(lnJ) –Singularities, intermittency, fractals, random uncorrelated motion –Measurement) and modeling of agglomeration (RDF and de-correlated velocities PDF (kinetic) approach, diffusion / drift in a random compressible flow field –New PDF approaches – statistics of acceleration points (sweep/stick mechanisms)(Coleman & Vassilicos)