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

Size: px

Start display at page:

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

Samuel Hamilton
5 years ago
Views:

International School of Physics Enrico Fermi Complex Materials

1 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 Dept. de Física Univ. Fed. do Ceará Fortaleza, Brasil

2 Particles in Fluids Sedimentation Fluidized beds Size segregation g under shear Hydraulic transport Filtering Saltation ti Rheology of suspensions

3 Equation of motion of fluid v ( ) p () x y p, (x)and are velocity and pressure field of the fluid, and μ its density and dynamic viscosity. Incompressible Navier-Stokes equation: v 1 v( v) p v tt ( v ) 0 const tt vv 0

4 Initial and Boundary Conditions of NS eqs. v 1 ( v ) v p 2 v, v t viscosity v( x, t ) V ( x), p( x, t ) P ( x) v(, t) v ( t), p(, t) p ( t) v ( x, t ) p ( xt, ) velocity field, pressure field

5 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)

6 Solvers for NS equation Penalty method with MAC Finite Volume Method (FLUENT) Turbulent case: k-ε model or spectral method Lattice Boltzmann Discrete methods: DPD, SPH, SRD, LG CFD = Computational Fluid Dynamics

7 v Navier Stokes equation k1 v t k Apply on both sides : v v p v v v 2 ( ) k1 k k k k1 k 2 2 p k 1 v k v k v k t Insert incompressibility condition: v v 0 k 1 k

8 Navier Stokes equation 2 p ( v ) v 1 k k k Poisson equation determine pressure p k+1 To solve it one needs boundary conditions for the pressure which one obtains projecting the NS equation on the boundary. This must be done numerically.

9 Operator splitting Introduce auxiliary variable field v * v v v v * * k 1 k 2 pk 1 vk ( vk ) vk t and split in two equations: * v v k t v k 1 t 2 vk ( vk ) vk v * p k 1 v *

10 Operator splitting Applying on one obtains 2 v v * k1 t p k1 p v t * k1 Projecting on the normal n to the boundary one obtains: p k1 n 1 * t n p n v v k1 k 1

11 Spatial discretization MAC = Marker and Cell is a staggered lattice: x h is the lattice spacing y Place components of velocity on centers of edges and pressures in the centers of the cells.

12 Spatial discretization 1 p p p 1 xi, 12, j h i, j i, j x 2 1 p i, j p i, j pi, j pi, j pi, j pi, j h y

13 Spatial discretization 1 v v v v v h * * * * * i, j x, i12, j x, i12, j y, i, j12 y, i, j12 x 2 p k1 v t * y Poisson equation for the pressure p k+1 is solved on the centers of the cells ().

14 Spatial discretization v v t p v v v 2 ( ) k1 k k1 k k k The equations for the velocity components are solved on the edges. v v v x v v y v x x x y x

15 Spatial discretization x x v x 1 x x v v v v h 1 i, j h i, j i, j x i, j x 2 y x y v 1 y y y y v v v v v y 1 4 i, j i, j i1, j i1, j i, j v v x x h i j 1 i j h,, 2 2

16 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)

17 One particle in fluid e.g. gp pull sphere through fluid no-slip condition: v v particle v particle Γ fluid moving boundary condition create shear in fluid : exchange momentum

18 Drag force drag force F D da (Bernoulli s principle) v v p i j ij ij x x j i stress tensor j η = μ is static viscosity

19 Homogeneous flow v R Re << 1 Stokes law: F D = 6π η R v D (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: Re 2 8 C D is the drag coefficient D 2 F C D D

20 Drag coefficient C D

21 Inhomogeneous flow In velocity or pressure gradients: Lift forces are perpendicular to the direction of the external flow, important tf for wings of airplanes. lift force: C L is lift coefficient when particle rotates: Magnus effect important for soccer

22 Many particles in fluids The fluid velocity field follows the incompressible Navier Stokes equations. Many industrial i 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 8 C D Re 2 simulating particles moving in a sheared fluid

23 Stokes limit hydrodynamic interaction between the particles v i M ( r r ) v ij i j j i mobility matrix for Re = 0 mobility matrix exact Stokesian Dynamics (Brady and Bossis) invert a full matrix only a few thousand particles j

24 Calculation of fluid motion v (x) px () and are velocity and pressure field of the fluid, and its density and dynamic viscosity. We solve the incompressible Navier-Stokes equation: v 1 v( v) p v tt FLUENT a commercial discrete volume solver on an adaptive triangulated mesh BETTER: USE REGULAR GRID v 0

25 Numerical techniques 1 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.). 2 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.

Sedimentation Sedimentation ist the descent of particles in a fluid due to the

The interaction between the particles and the fluid is given by the condition

to the velocity of this particle. Measure settling velocity, i.e. velocity of the upper front.

26 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. ti

27 Sedimentation Glass beads descendingdi in silicon oil using penalty method with MAC grid comparing experiment and simulation Thesis of Kai Hoefler

28 Sedimentation velocity Martin Hecht, 2006

29 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). Thesis of Frank Fonseca

30 Draft, Kiss and Tumble draft kiss tumble

31 Draft, Kiss and Tumble vertical velocity horizontal velocity

32 Shear flow using LBM (= Lattice Bolzmann Method) A.J.C. Ladd, J.Fluid Mech. 1994

33 Velocity distributions under shear using LBM technique of T. Ladd Jens Harting 2006 exponential tails

34 Scaling 2 and z z z z z Pv ( ) dv 1 vpv ( ) dv 1 Jens Harting 2006

35 Scaling Dependence on viscosity ν and volume fraction v RMS z v 2 z Jens Harting 2006

36 Dependence on shear velocity Shear under gravity Jens Harting 2006

37 Dependence on shear velocity Jens Harting 2006

38 Simulating clay Assume spherical particles Use molecular dynamics for the simulation DLVO potentials describe the dominant particle-particle interaction: - screened ed Coulomb o potentialt (ions / counterions), repulsive - Van-der-Waals-attraction for short distances Hertz force for overlapping particles Lubrication force ~ Al 2 O 3 interaction potentials Martin Hecht, 2006

39 Simulation of the fluid Stochastic Rotation Dynamics (SRD) Martin Hecht, 2006 introduction of representative fluid particles collective interaction by rotation of local particle velocities very simple dynamics, but recoveres hydrodynamics d correctly Brownian motion is intrinsic

40 Shear flow Martin Hecht, 2006

41 Shear viscosity Martin Hecht, 2006

42 Viscosity at small v s Martin Hecht, 2006

43 Density correlation Martin Hecht, 2006

44 Future Perspectives Consider particles of more complex shapes. Consider deformable particles. Consider particle rotations (lift forces). Consider polydisperse size distributions. Consider turbulent flow, i.e. large Re. Consider slippage at walls. Consider non-newtonian fluids.

45 Filtration José Soares Andrade Jr. Ascanio Dias Araújo Departamento de Física UFC, Fortaleza, Brazil

46 Flow through a porous medium important for oil recovery, filtration and fluidized beds two-dimensional realization by placing randomly disks that t do not overlap (RSA)

47 Distribution of channel widths 2 normalized channel width: l * = l/d ) P(l * ε=0.6 ε=0.7 ε=0.8 ε=0.9 D = diameter of disks l *

48 Data collapse collapse for different porosities by normalizing with: l D 2 3(1 ) 1 P(l)<l> ε=0.6 ε=0.7 ε=0.8 ε=0.9 crossover at: l D giving i a porosity: l/<l> cross 1 /

49 Macroscopic permeability We verify the law of Kozeny Carman: K K (1 ) where: K h 2 0 h /12

50 Distribution of velocities is absolute value of the local velocity. V is the (uniform) velocity of the fluid at the entrance. P(v/V) )/ε lo og θ)ε 2 10 [P(l * cosθ ] log 10 [(l * cosθ)/ε 2 ] scaling with 2 solid line: Gaussian fit ε=0.6 ε=0.7 ε=0.8 ε= (v/v)ε 2

51 Distribution of fluxes best fit (dashed red line): P(( ) e / with independent on porosity 0 log 10 P(φ) 2 ε=0.6 ε=0.7 ε=0.8 ε= K/K ε 3 /(1 ε) log 10 φ 10 full black line is the convolution: P P v P l cos vl cos dvd l cos

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

53 Simple models u 1 Periodic array of disks Marshall (1993) z y x D h Parallel pipes x x+δx

54 Capture efficiency log (δ/d) with gravity 1.0 log 10 (δ/d) 0 1 without gravity ε=0.85 ε=0.90 ε= log 10 (St St c ) Without gravity we find a critical Stokes number St c = ( St St c ) ~ c with log 10 [St/(ε ε c )]

55 Dependence on gravity

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

57 Trajectories of particles St = St =

58 Non-captured particles is fraction of non-captured particles. typical decay: ( x ) e x / is the penetration depth for 1 it is easy to show that: we find more generally: = D/4(1-) = D 2 /4(1-)

59 Scaling of the penetration depth solid line: (1-)/D = St -1 with = 0.58 that means =1 for St c = 0

60 Scaling of 1.0 φ φ(l) ε=0.6 ε=0.7 ε=0.8 scaling variable St(1-)x/L L = length 1 St(1 ε) ) St(1 ε)x/l of filter

61 Future Perspectives Consider interactions between particles. Consider feedback of particles on fluid. Consider particles with finite radius. Consider thermal diffusion of particles. Consider turbulent flows, i.e. large Re. Consider three dimensional simulations.

62 Non-Newtonian Newtonian flow Apiano Morais, Hansjoerg Seybold, Jose Soares Andrade n1 Cross fluid: K, 1 2 Bingham fluid: 0 for K B 0 0 for 0 0

63 Non-Newtonian Newtonian flow Darcy s law: u 0 kd p L k D is global permeability for Cross fluid define hydraulic conductivity: p k u0k L 11 n

64 Non-Newtonian Newtonian flow Cross fluid porosity ε =

65 Non-Newtonian Newtonian flow k 1 ( 1) 75 n n n 1 2(1 n) n ( n1) n 0 K ( 1) n 12 n k d e n 25 3n 1 3 k 1 = k D (n=1) Re' k u 1 n 2n Kd p data collapse Cross fluid

66 Non-Newtonian Newtonian flow Cross fluid d e = 1.58 for ε = 0.5 and d e = 0.35 for ε = 0.7

67 Non-Newtonian Newtonian flow Bingham fluid Re B u0d p K B k K u L p B B 0 k 1 is lowest value of k B at small Re.

68 Future Perspectives Consider shear thinning rheologies. Consider particles in fluid (filtration). Consider slippage at the walls. Consider hardening in time (glue). Consider turbulent flows, i.e. large Re. Consider anisotropic rheologies.

69 Northeast Brazil Lençóis Maranhenses

70 Quicksand

71 Quicksand

72 Quicksand

73 Quicksand Dirk

74 Quicksand

75 Quicksand Buda

76 Quicksand Buda + Dirk

77 Quicksand

78 Quicksand

79 Quicksand

80 Q Quicksand 2d simulation Use Contact Dynamics to produce cohesive packing p g and then p push into it an intruder disk with force Fd. Dirk Kadau

81 Quicksand Dirk Kadau

82 Quicksand Dirk Kadau F c = cohesive force

83 Penetration tests penetration depth vary density 1 low density smaller 0.8 force Dirk Kadau w/w T (diff pos) (diff configs) 0.4 (diff configs) 0.35 (diff configs) F d /F c

84 Penetration tests penetration ti depth 1 scaling by square of 0.8 volume fraction data collapse w/w T Dirk Kadau F d /F c /(vol frac) 2

85 Penetration tests after initial phase: constant velocity force-velocity relation: - quadratic - shifted by force threshold ν = Dirk Kadau

86 Penetration tests v (F F ) drive thresh θ = 0.50 (ν=0.432) θ = 0.49 (ν = 0.4) θ = 0.50 (ν = 0.35) exponent θ = ½ v100 relate to pinning/depinnig transition ν= ν= 0.4 ν= Dirk Kadau 1 (F drive - F thresh ) / F c

87 Shear strength Dirk Kadau

88 Outlook Shear thickening Wall interactions (microfluidics) Polymeric fluids Phase transformations (cooling, drying) Viscoelasticity Frequency dependence d Polydisperse systems

89 Aeolian Sand transport

90 Transport by Saltation

91 even on a wet surface

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

92 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

93 Dependence on grain size typical grain diameters for saltation on earth: m wind tunnel measurements Bagnold and Chepil

94 Schematic saltating trajectory mobile wall at top u x (y) y x M.P. Almeida u p > > > θ = ejection angle u p = particle velocity u x (y) = wind velocity profile

95 The turbulent air flow logarithmic velocity profile of the horizontal component of the velocity as function of height y: u x y u* ln z z0 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

96 Types of transient behaviour force on particle: du dt p F D 1 u u g p p u * < u t u * > u t y(m m) x(m) threshold h velocity u t

97 Steady state saturated flux q s

98 Saturated flux Bagnold (1941): 1.0 q s u * 3 simulation Eq. (7) Lettau Lettau Bagnold Lettau and Lettau: (1978) q 1 s 1 s (kg m ) q s q fit of solid line: s a 2 ( u* ut ) u u * t u * (m/s) Physical Review Letters, January 2006

99 Wind velocity profile difference between disturbed and undisturbed velocity profile u * = 0.51 y(m 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)

100 Wind velocity profile

101 Height of saltation layer y max height of maximum loss of velocity linear y max (m m) increase with u * u t becomes zero at: u t = u * (m/s)

102 Planet Mars Eric Parteli M. Almeida, E. Parteli, J.S. Andrade, HJH PNAS 105, 6222 (2008)

103 Parameters on Mars Earth g 9.81m air grain s kg 2650 kg d 250 m u *t 0.2 m/s m 3 m kg/sm 3 Mars g 3.71m air grain s 0.02 kg kg d 600 m u *t 2.0 m/s m 3 m [Greeley and Iversen (1985)] kg/sm (Viscosity of CO 2 at C) 3 - 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.

104 Saltation on Mars White et al. (1979), Greenley et al. (1996) fl 3 u t u t qs C u* 1 1 g u* u* C = 18 for Earth C =29forMars 2.9 C = 19 d / l v l v = (ν 2 /g) 1/3 2

105 Saltation on Mars Q s q s q s0 q d u u u g 2 fl s t * t

106 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 2 2 3

107 Saltation on Mars H L salt gd H salt l v gd ll l v = (ν 2 /g) 1/3 v

108 Saltation on Mars Impact angle and height to length ratio of saltation trajectory as function of u * L salt H u H salt *

109 Dune velocities

110 Particle transport by water under water dunes in front of San Francisco bay

111 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

112 Erosion and Deposition - Erosion E is a function of the surface drag. It is proportional to the ratio between particles N susceptible to be entrained and erosion time T which is the time needed to dislodge a particle from the bed. E=N/T. - Deposition D scales with the sand flux q and the proportionality defines the deposition length L=q/D - In contrast tto wind driven sand motion the deposition length appears to be the relevant length scale and not the drag length d ρ grain / ρ fluid. - Topography changes due to sand transport: - Equilibrium of erosion E, deposition D h (1 ) D t E

113 Reduction of the shear stress due to the grains b - Moving sand grains substract momentum from the fluid leading to a grain shear stress g - is modeled dldby a function of fqq and - this relationship differs strongly from that t one known for wind-driven di sand transport g g Nonlinearity, nonequilibrium and complexity, Tepoztlán, Nov.27-Dec.3, 2005

114 Comparison to experiment Comparison to data: Data exist for the inverse saturation length as function of the shear stress without sand transport. - Most experimental data are scattered between the red and the green curve. - Our model lin linear approximation (blue curve) compares well to the data - Data exist for the inverse Thomas Pähtz E D q eq L s q Ls ( 0.039) d

115 Future Perspectives Consider interactions between grains. Consider distribution of ejection angles. Consider finite radius of grains. Consider polydisperse size distributions. Consider the temporal fluctuations inherent to turbulent flow. Consider three dimensional simulations.

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

Particles in Fluids. Sedimentation Fluidized beds Size segregation under shear Pneumatic transport Filtering Saltation Rheology of suspensions Particles in Fluids Sedimentation Fluidized beds Size segregation under shear Pneumatic transport Filtering Saltation Rheology of suspensions Sandstorm Fluidized Bed Equation of motion v of v fluid v vx