Particles in Fluids. Sedimentation Fluidized beds Size segregation under shear Pneumatic transport Filtering Saltation Rheology of suspensions

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1 Particles in Fluids Sedimentation Fluidized beds Size segregation under shear Pneumatic transport Filtering Saltation Rheology of suspensions Sandstorm

2 Fluidized Bed

3 Equation of motion v of v fluid v vx vx vx x y z x y z and px v y vy vy are velocity and vpressure field of the fluid, x vy vz and μ its density and dynamic x viscosity. y z v z vz vz v v v v (x) ( ) Incompressible Navier-Stokes x y z x y equation: z v 1 v( v) p v t ( v ) 0 const t v 0

4 Initial and Boundary Conditions of NS eqs. v 1 2 ( v ) v p v, v 0 t viscosity v( x, t0) V0( x), p( x, t0) P0( x) v(, t) v ( t), p(, t) p ( t) 0 0 v( x, t) velocity field, p( x, t) pressure field Reynolds number Re Re Vh V is characteristc velocity h is characteristic length μ is dynamic viscosity Re << 1 is the Stokes limit (laminar flow) Re >> 1 is turbulent limit (Euler equation)

5 Solvers for NS equation CFD = Computational Fluid Dynamics Classical fluid solvers Penalty method with MAC Finite Volume Method (FLUENT) Turbulent case: k-ε model or spectral method Solving the differential equations numerically. Discrete fluid solvers Lattice Gas Automata (LGA) Lattice Boltzmann Method (LBM) Dissipative Particle Dynamics (DPD) Smooth Particle Hydrodynamics (SPH) Stochastic Rotation Dynamics (SRD) Direct Simulation Monte Carlo (DSMC)

6 One particle in fluid e.g. pull sphere through fluid no-slip condition: v v particle v particle Γ fluid moving boundary condition create shear in fluid : exchange momentum drag force (Bernoulli s principle) Drag force F D da stress tensor v v i j ij xj x i ij p η = μ is static viscosity

7 v Homogeneous flow R Re << 1 Stokes law: F D = 6π η R v (exact for Re = 0) R is particle radius, v is relative velocity Re >> 1 Newton s law: F D = 0.22π R 2 v 2 general drag law: C D is the drag coefficient 2 F D C D 8 Re 2 Drag coefficient C D Re Reynolds number Re = Dv/μ

Inhomogeneous flow In velocity or pressure gradients: Lift forces are perpendicular to the direction of the external flow, important for wings of airplanes.

8 Inhomogeneous flow In velocity or pressure gradients: Lift forces are perpendicular to the direction of the external flow, important for wings of airplanes. lift force: C L is lift coefficient when particle rotates: Magnus effect important for soccer Many particles in fluids The fluid velocity field follows the incompressible Navier Stokes equations. Many industrial processes involve the transport of solid particles suspended in a fluid. The particles can be sand, colloids, polymers, etc. The particles are dragged by the fluid with a force: 2 F D C D 8 Re 2 simulating particles moving in a sheared fluid

9 Stokes limit hydrodynamic interaction between the particles v i ji M ij ( ri rj ) v mobilitymatrix for Re = 0 mobility matrix exact Stokesian Dynamics (Brady and Bossis) invert a full matrix only a few thousand particles j Numerical techniques 1 2 Calculate stress tensor directly by evaluating the gradients of the velocity field through interpolation on the numerical grid, e.g. using Chebychev polynomials (Kalthoff et al.). Method of Fogelson and Peskin: Advect markers that were placed in the particle and then put springs between their new an their old position. These springs then pull the particle.

10 Sedimentation Sedimentation ist the descent of particles in a fluid due to the action of gravity. The interaction between the particles and the fluid is given by the condition that the velocity of the fluid on the entire surface of each particle is equal to the velocity of this particle. Measure settling velocity, i.e. velocity of the upper front. If particles are of different species then one has several fronts. Open question: size dependence of the density fluctuations. Sedimentation Glass beads descending in silicon oil using penalty method with MAC grid comparing experiment and simulation

11 Sedimentation velocity settling velocity v S = v 0 (1-Φ) Φ = volume fraction of particles Sedimentation of platelets Oblate ellipsoids descend in a fluid under the action of gravity. This has applications in biology (blood), industry (paint) and geology (clay).

Shear flow using LBM (= Lattice Bolzmann Method) A.J.C. Ladd, J.

1994 Simulating clay Assume spherical particles Use molecular dynamics

particle-particle interaction: - screened Coulomb potential (ions /

12 Shear flow using LBM (= Lattice Bolzmann Method) A.J.C. Ladd, J.Fluid Mech Simulating clay Assume spherical particles Use molecular dynamics for the simulation DLVO potentials describe the dominant particle-particle interaction: - screened Coulomb potential (ions / counterions), repulsive - Van-der-Waals-attraction for short distances Hertz force for overlapping particles Lubrication force ~ Al 2 O 3 interaction potentials

13 Shear flow Shear viscosity

14 Viscosity at small v s Sheared Blood

15 Porous Media Calculation of fluid motion We create a two-dimensional realization of a porous medium by placing randomly disks that do not overlap (RSA). We solve the incompressible Navier-Stokes equation with a commercial discrete volume solver on an adaptive triangulated mesh FLUENT

16 Flow through porous medium Flow through a porous medium important for oil recovery, filtration and fluidized beds Color represents the absolute value of the fluid velocity.

$Kozeny Carman: 3 2 0 ( 1 ) where: 2 K0 h /12 Darcy ε is the porosity = void fraction is the reference value for an empty$

17 Flow through porous medium K K Macroscopic permeability p K p Flux depends on pressure drop as: We verify the law of Kozeny Carman: ( 1 ) where: 2 K0 h /12 Darcy ε is the porosity = void fraction is the reference value for an empty channel.

δ y u x inertial impaction u direct interception D diffusion =

18 Filtration Massive tracer particles of diameter d p, velocity u p and density p are released. δ y u x inertial impaction u direct interception D diffusion = capture efficiency Stokes number: St : = D and St 0: = 0 Trajectories of particles St = St =

19 Trajectories porosity = 0.7, Stokes number St = 0.1 colours code the absolute value of the velocity add the action of gravity Dependence on gravity

Capture efficiency log 10 (δ/d) 0 1 2 3 with gravity 1.0 log 10 (δ/d) 0 1 ε=0.85 ε=0.90 ε=0.95 without gravity 0.

20 Capture efficiency log 10 (δ/d) with gravity 1.0 log 10 (δ/d) 0 1 ε=0.85 ε=0.90 ε=0.95 without gravity log 10 (St St c ) log 10 [St/(ε ε c )] Without gravity we find a critical Stokes number St c = ( St St ) ~ c with 0.5 Non-captured particles is fraction of non-captured particles. typical decay: ( x) e is the penetration depth for 1 it is easy to show that: = D/4(1-) x / we find more generally: = D 2 /4(1-) ε = 0.7

21 Aeolian Sand transport Transport by Saltation

With more energy they impact again against the surface and eject a splash of new

22 even on a wet surface The Mechanism of Saltation Grains are drawn from the ground and accelerated by the wind. With more energy they impact again against the surface and eject a splash of new particles. In this way more and more grains saltate until saturation is reached due to momentum conservation. h Wind Ralph A. Bagnold h

23 Dependence on grain size typical grain diameters for saltation on earth: m wind tunnel measurements Bagnold and Chepil Wind Channel in Aarhus

24 Measuring the wind velocity in channel Schematic saltating trajectory mobile wall at top u x (y) y > > x > u p θ = ejection angle u p = particle velocity u x (y) = wind velocity profile

25 The turbulent air flow logarithmic velocity profile of the horizontal component of the velocity as function of height y: u x u z ln z0 * z 0.4 is the von Kármán constant z 0 is the roughness length viscosity of air: η = kg m -1 s -1 density of air: = kg m -3 Solve it with k- model using FLUENT. a commercial finite volume solver on an adaptive triangulated mesh Types of transient behaviour force on particle: 0.08 du dt p F D 1 u u g p p 0.06 u * < u t u * > u t y(m) x(m) threshold velocity u t 0.35

26 Steady state saturated flux q s Saturated flux Bagnold (1941): 3 q s u * 1.0 simulation Eq. (7) Lettau Lettau Bagnold Lettau and Lettau: (1978) fit of solid line: q s (kg m 1 s 1 ) q s u * u t q s a 2 ( u* ut ) u * (m/s)

27 Wind velocity profile difference between disturbed and undisturbed velocity profile u * = 0.51 y(m) y(m) q=0.010 q=0.015 q=0.020 q=0.030 q= [u x (0) u x (q)]/q collapse when normalizing with flux q u x (0) u x (q) (m/s) Height of saltation layer y max height of maximum loss of velocity linear increase with u * y max (m) 0.05 u t 0.35 becomes zero at: u t = u * (m/s)

28 Splash in 3d BobFest, Duke University, Oct.12-13, 2013 θ = 0.15 in 3d

29 Role of collisions: saltons Saltons jumping on the soft bed θ= 0.90 e = 0.7 BobFest, Duke University, Oct.12-13, 2013 Contribution to the flux

Planet Mars Parameters on Mars g d air Earth 9.81m grain 2 s 1.225 kg 2650 kg 250 m u *t 0.2 m/s m 1.8 10 kg/sm 3 m 3 g d air Mars 3.71m grain 2 s 0.02 kg 3200 kg 600 m u *t 2.

30 Planet Mars Parameters on Mars g d air Earth 9.81m grain 2 s kg 2650 kg 250 m u *t 0.2 m/s m kg/sm 3 m 3 g d air Mars 3.71m grain 2 s 0.02 kg 3200 kg 600 m u *t 2.0 m/s m [Greeley and Iversen (1985)] kg/sm 3 m 3 (Viscosity of CO 2 at C) - u * on Earth is 0.4 m/s and on Mars, Pathfinder Mission 1997 found u * close to threshold. Further, it has been found that the angle of the slip face of martian dunes is the same as of terrestrial dunes.

31 Saltation on Mars White (1979) Greenley et al. (1996) fl 3 u t u t qs C u* 1 1 g u* u* C = 18 for Earth C = 2.9 for Mars 2 C = 19 d / l v l v = (μ 2 /g) 1/3 Saltation on Mars Q s q q s s0 q d u u u g 2 fl s0 2 3 t * t

32 Saltation on Mars Length L salt and height H salt of saltation trajectory L t u u salt v * t H t u u t salt v * t v / g Dune velocities

33 Particle mixing in turbulent channel flow Consider a channel through which particles are dragged by a turbulent flow. Do not calculate the whole fluid field, but rather use a stochastic approach to model the influence of the turbulent velocity field on the particle movement. The particle density is constant throughout the system. Concentrate on a small region in the center of the channel, which means we ignore the effects of the walls. Study the mixing of two types of spherical particles. Particle mixing in turbulent channel flow Fluid velocity inside the channel: u( t) u u ( t) t mean fluid velocity intrinsic fluctuations Use the empirical drag law to couple the particle movement to the fluid velocity u (t)

Acceleration of tracer particles in fully developed turbulence Measured trajectory of a tracer particle in a turbulent water flow at Reynolds number Re=63 000.

34 Acceleration of tracer particles in fully developed turbulence Measured trajectory of a tracer particle in a turbulent water flow at Reynolds number Re= Experimental measurements of the probability density function (pdf) of the acceleration of tracer particles show a clearly non Gaussian behavior. La Porta et al., Nature 409 (2001), 1017 N. Mordant et al., Physica D 193 (2004), Intrinsic velocity fluctuations We study the influence of the intrinsic velocity fluctuations on the mixing. The mean fluid velocity is kept constant: u const. The fluctuating part is determined by using the stochastic model introduced by A. M. Reynolds (Phys. Rev. Lett. 91 (2003), ) that reproduces well the experimentally observed distributions and autocorrelations of velocities and accelerations of tracer particles in fully developed turbulence. (t) The model calculates a time series for the velocity of a tracer particle. For every real particle we generate a tracer particle and evolve it in time. The velocity of the tracer particle then gives us a stream line and the real particle is dragged into the direction of this line. u t

35 Particle mixing in turbulent channel flow Particle mixing in turbulent channel flow ( y) 1 1( y) 2( y) At different positions x s along the channel we make a vertical slice and measure the relative particle densities of every type of particles (see inset). From the differences of these densities we calculate the density differences μ(y).

Particle transport by water under water dunes in front of San Francisco bay Particle transport by water - Transport Mechanisms: - 1. Creep rolling and sliding of grains on the soil - 2.

36 Particle transport by water under water dunes in front of San Francisco bay Particle transport by water - Transport Mechanisms: - 1. Creep rolling and sliding of grains on the soil - 2. Saltation hops of grains near the soil - 3. Sheet Flow completely mobile sand bed, grains moving in granular sheets - 4. Suspension turbulent lift forces overcome gravity, particles can travel very long distances

37 Particle transport by water Particle transport by water

38 Particle transport by water

Computational Physics IfB, ETH Zürich Switzerland. International School of Physics Enrico Fermi. Physics and Biology

Particles in Fluids Hans J. Herrmann Computational Physics IfB, ETH Zürich Switzerland International School of Physics Enrico Fermi Complex Materials in Physics and Biology Varenna, July 29-July 3, 2010