Numerical Simulation of

Granular Flow

Based on Micropolar Fluid Theory

Shin-Ichiro Serizawa/Tomoyuki Ito (serizawa.shinichiro@gmail.com)

Needs of Granular Flow Analysis

From Natural Phenomena to Industry

Prediction of Natural Phenomena

Landslide

Avalanche

Pyroclastic flow

Industry

Civil Engineering

Agriculture

Pharmacy

Electrophtography

Numerical Analysis Method

Distinct Element Method

Each Particle Interactions are presented by Cundall Model

Cundall, P.A., Strack, O.D.L., A Discrete Numerical Model for Granular Assemblies, Geotechnique, 29-1(1979), 47-65.

DEM Takes Huge Computing Cost.

Ct

Kt Kn

Cn

m

rj

ri

mj

mi

Cundall Model

Description of Granular Flow as the Constitutive Law model is required.

FEM/FDM and etc. can be applied to solve.

Reduction of Computational Cost

Hibler’s Rheology Model

Numerical Model of the Sea Drift Ices

Hibler, W. D., III. A Dynamic Thermodynamic Sea Ice Model, Journal of Physical Oceanography, 9-4 (1979), 815-846.

Viscosity coefficient h

Decided by Principle Strain Speed

If Large then Act as Plastic Flow

Else if then Act as Viscous flow

Mohr-Coulomb Yield Criterion

Internal Friction Angle f 1 2 1 2 sin f

max

1 2

P sinmin ,

fh h

Hibler’s Rheology Model

Governing Equations

Conservation Law

Momentum

Constitutive Equation

ij ij ij kk ij

1P 2

2 h

iji , j

d

dt

vf

k ,k

d0

dt

v

Problems of Hibler’s Rheology Model

The Rotations of Particles are not Considered.

The Grain Size is not Explicitly Described in Constitutive Equation.

Micoropolar Fluid Theory

Microstructure in Continuum

Micro Rotation w

Characteristic Length d

1

11

2

m32

m31

22

w3

d

12

21

3

v1

v2

Micoropolar Fluid Theory

Governing Equations

Conservation of Mass

Momentum

Angular Momentum

Constitutive Equation

Stress

Coupled Stress

iji , j ijk jk i

dI c

dt m

w

iji , j i

d

dt

vf

k ,k

d0

dt

v

ij k ,k ij i , j j ,i c i , j m h w h w h w h w

j ,i i , j

ij k ,k i

j ,i i , j

kj r ijk

v v2

v vP v 2

22h w h

Model Based on Micropolar Fluid Theory

Kanatani, K., A Micropolar Continuum Theory for the Flow of Granular Materials, International Journal of Engineering Science, 17-4, (1979), 419–432.

Mitarai, N. Hayakawa H. and Nakanishi, H., Collisional Granular Flow as a Micropolar Fluid, Phys. Rev. Lett. 88, (2002), 174301.

Extended Hibler Model

Viscosity

Decided by Equivalent Strain Speed

Micro Rotation Viscosity

Angular Viscosity

Pressure Equation

Pressure is Non Negative value

max

P sinmin ,h

fh

1

2ij ij1 2 ij ji 2

ij3 ij

e e e ed k k

2

g gg

1

3ij ij kk ije

r f ( d ) Const.h

2

c

1I

1d

0h h h

max

0

0 00

0

P P

1

3ij ij kk ijk

Smoothed Particle Hydrodynamics

Physics Quantities are expressed by Kernel Function Lucy, L. B., A numerical approach to the testing of the fission

hydrodynamics, Astron., J., 82-12 (1977), 1013- 1024.

Gingold R. A. Monaghan, J. J., Smoothed particle hydrodynamics: Theory and application to non spherical stars, Mon. Not, Roy. Astron. Soc., 181 (1977), 375-389.

Approximation by Kernel Functions

Example by SPH

Schematic of Sand Pile Formation

0

0.1

0.2

0.3

0.4

0.5

0 0.25 0.5 0.75 1

Hie

gh

t (m

)

x (m)

25 particles

50 particles

x

y

xmax

Result of Pseudo Viscous Fluid

Broken Dam

MaxvMinv

0.0 4.91 (/s) -4.91

0.0 4.91 (/s) -4.91

Result of Extended Hibler Model

Velosity:v

Angular Velosity:w Counter Clockwise No Rotation Clockwise

0.0 0.78 (m/s)

0.0 0.78 (m/s)

Velosity:v

Angular Velosity:w Counter Clockwise No Rotation Clockwise

Parameters in Constitutive Equations

Internal Friction Angle : f

Characteristic Length : d

Internal Friction Angle

Internal Friction Angle and Angle of Repose

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.2 0.4 0.6 0.8 1.0

70

60

45

30

15

y

x

15f 30f 45f 60f 75f

Influence of Parameters in Equations

Internal Friction Angle : f

Characteristic Length : d

Characteristic Length and Fluidity

Propagated Front Position

as Index of Fluidity

0.8

0.9

0.9

1.0

1.0

1x10-5

1x10-4

1x10-3

1x10-2

1x10-1

1x100

x max

d

Hibler Model Radius of Kernel Function:h

xmax

Conclusion

The constitutive model of granular flow based on micropolar fluid theory is presented.

Part of granular matter flow with parallel and rotational motion and the other part do not flow.

The proposed model can reproduce sand pile unlike fluid.

Angle of repose depends on internal friction angle of granular matter.

The size of granular matter has an influence on fluidity.

The result is presented in The 63rd Japan National Congress of Theoretical and Applied Mechanics.

Tokyo Institute of Technology

2014.9.26