Grain Sedimentation withSPH-DEM and its Validation
Martin Robinson1, Marco Ramaioli2 and StefanLuding3
1OCCAM, Mathematical Institute, University of Oxford, United Kingdom2Nestle Research Center, Lausanne, Switzerland
3Multiscale Mechanics, University of Twente, Enschede, Netherlands
http://people.maths.ox.ac.uk/robinsonm/
http://www.pardem.eu/
Introduction
• Fluid-particle systems are ubiquitous in nature and industry
• Goal: development of meshfree fluid-particle simulation method
– Smoothed Particle Hydrodynamics and Discrete Element Method (SPH-DEM)
– Purely particle-based method results in no mesh-based issues such asmesh deformation or topology changes
– Method is well suited for applications involving a free surface, including(but not limited to) debris flows, avalanches, sediment transport orerosion in rivers and beaches and liquid-powder dispersion and mixingin the food processing industry
• Applied to:
– Validation test cases using single/multi particle sedimentation
– Dispersion of granular bed by liquid jet (see poster PG2013 0296)
0s 2s 7s Time
SPH-DEM implementation of aDiscrete Particle Method
Smoothed Particle Hydrodynamics (SPH)
• Smoothed Particle Hydrodynamics is a Lagrangianmethod for modelling fluids
• Fluid eqns. discretised over a disordered set ofinterpolation points that move with the fluid ve-locity
• Lack of mesh simplifies problems involving com-plex geometries, free-surfaces and multi-phase in-terfaces
Discrete Elment Model (DEM)
• DEM represents solid particles as discrete ele-ments (spheres) that interact via contact forces
• Use linear dash-pot contact model for simulationsshown here
• This study mainly concerned with sedimenta-tion and testing fluid-particle coupling. Particle-particle contacts have minimal effect.
Fluid-particle Coupling
• Fluid equations based on locally averaged Navier-Stokes equations by Anderson and Jackson (1967)
• Smooth porosity field calculated from DEM parti-cle positions using standard SPH smoothing kernel
• Coupling SPH with DEM leads to a purelyparticle-based method for fluid-particle simulation.
• Retains flexibility of a meshless method
• Advantages for code development and paralleliza-tion
Calculation of porosity field. Solid DEM particles are filled black circles,SPH particles are shown as circles shaded by the value of the interpolationkernel W (h). The smooth porosity field ε (shown right) is calculated byconvolution of SPH interpolation kernels with DEM particle volumes (i.e.εa = 1 −
∑jWaj(hc)Vj)
3D Sedimentation Test Cases
Test suite of three sedimentation cases with known analytical solutions tocompare with simulation results. Tests are ordered by increasing challenge.
1. Single Particle Sedimentation (SPS)
• Single solid particle falling in a fluid column
• Periodic boundary conditions in x and y directions. No-slip solidboundary at lower z boundary
• Simple case, tests force balance between buoyancy and drag
• Given Stokes drag, can find analytic expression for solid particle veloc-ity
2. Sedimentation of a constant porosity block (CPB)
• Rigid porous block falling in the same water column
• Porous block constructed of a regular array of DEM particles
• Constant (translating) porosity and velocity field
• Given drag term (we use Di Felice drag, valid for higher Re and packedspheres), we calculate expected terminal velocity versus porosity ofblock
x
y
z x
y
x
y
z
(left) SPS. w = 4× 10−3 m and h = 6× 10−3 m (right) CPB and RTI
3. Rayleigh Taylor Instability (RTI)
• Same initial conditions as CPB, but now DEM particles free to moverelative to each other
• Similar to classical Rayleigh Taylor Instability, with some differences:Smooth transition of effective density from more dense suspension tolighter fluid, flow of fluid through suspension
• Inhomogeneous and dynamic porosity and velocity field
• Linearising around the initial unstable condition, we calculate lowerand upper bounds on the exponential growth rate of the instability
Solid and Fluid Parameters
Property ValueDensity 2500 kg/m3
Diameter 1 × 10−4mSpring Stiffness 1 × 10−4kg/s2
Damping 0 kg/s
Table 1: DEM parameters. Contact law used islinear spring dash-pot
Property Air Water Glycerol-waterDensity (kg/m3) 1.18 1000 1150Viscosity (Pa · s) 1.86 × 10−5 8.9 × 10−4 8.9 × 10−3
Rep 3.19 0.85 0.011Ar 83.89 18.57 0.192
Table 2: Fluid parameters. Dimensionless numbersshown for single particle falling in fluid
Single Particle Sedimentation (SPS)
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10
u z/ut
t/td
Water One-wayWater Two-way
Water-Glycerol One-wayWater-Glycerol Two-way
Air One-wayAir Two-wayStokes Law
-4-2024
0 1 2 3 4 5
%Error
t/td
-6
-5
-4
-3
-2
-1
0
1
2
3
1 2 3 4 5 6
%E
rror
inTe
rmin
alV
eloc
ity
h/d
• SPH-DEM simulations reproduced the analytical solutions very well, withless than 1% error over a wide range of Particle Reynolds Numbers0.011 ≤ Rep ≤ 9 and fluid resolutions.
• Necessary condition: the fluid resolution h is sufficiently coarse comparedto particle diameter (h > 2d) to reduce spurious fluctuations in porosity
Sedimentation of a constantporosity block (CPB)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0.5 0.6 0.7 0.8 0.9 1
u/u t
ε
SPH-DEM WaterSPH-DEM Water-Glycerol
Expected
• SPH-DEM reproduced the expected terminal velocity of the block within5% over prediction, over a range of porosities 0.6 < ε < 1.0 and ParticleReynolds Numbers 0.002 ≤ Rep ≤ 0.85
• Over prediction of the terminal velocity is due to smoothing of the poros-ity field near the edges of the block and reduces with a finer fluid reso-lution
• Some fluctuations in velocity of the SPH particles near the edges of theblock due to high porosity gradients. Adding a small amount of artificialviscosity was sufficient to damp these fluctuations and prevent them fromaffecting the terminal velocity of the block.
Rayleigh Taylor Instability (RTI)
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
minimum
particleheight(m)
time (s)
SPH-DEM (αart=0.1; ε=0.8)SPH-DEM (αart=0.0; ε=0.8)Two-fluid Model (ε=0.8)Two-fluid Model (ε=0.93)
• Results compare the motion of lowest DEM particle to the predictedgrowth of the interface in a two-fluid model of RT (Chandrasekhar,1961). Two-fluid model used to obtain upper and lower bounds
• SPH-DEM successfully reproduced the instability and its growth rate forboth water and water-glycerol
• Addition of artificial viscosity (αart = 0.1) erroneously reduces thegrowth rate
• For this test case the addition of artificial viscosity was not necessary forstability; due to the relatively high porosity ε = 0.8 and lower porositygradients at the interface between the suspension and clear fluid
Outlook
• Applications
– Injection of water jet into a granular bed (food processing industry)
– To predict the shape of the front correctly, one has to consider thefree surface and the absence of dissipation on the air side, both in theSPH-DEM model
• New physics
– The choice of appropriate drag laws (e.g. for polydisperse flows)
– inclusion of the added mass and lift forces
– More realistic DEM particle contact forces and
– The inclusion of contact friction and lubrication forces
– The inclusion of surface tension effects.
• Improvements
– Primary concern is SPH velocity fluctuations near high porosity gradi-ents
– Can be suppressed with addition of artificial viscosity, but goal is toremove this numerical instability from the method.
References
Martin Robinson, Marco Ramaioli, and Stefan Luding. Fluid-particle flow and validation using two-way-coupled mesoscale sph-dem. Submitted, preprint available at http:
// arxiv. org/ abs/ 1301. 0752 , 2013a.
Martin Robinson and Marco Ramaioli. Mesoscale fluid-particle interaction using two-way coupled SPH and the discrete element method. In 6th SPHERIC Workshop, pages 72–78,May 2011.
Mohammadreza Ebrahimi, Prashant Gupta, Martin Robinson, Martin Crapper, Jin Sun, Marco Ramaioli, Stefan Luding, and Jin Y. Ooi. Comparison of coupled DEM-CFD andSPH-DEM methods in single and multiple particle sedimentation test cases. In Proc. of III International Conference on Particle-based Methods, 2013.
Martin Robinson, Marco Ramaioli, and Stefan Luding. Grain Sedimentation with SPH-DEM and its Validation. In Proc. of Powders Grains, 2013b.
Martin Robinson, Stefan Luding, and Marco Ramaioli. SPH-DEM simulations of grain dispersion by liquid injection. In Proc. of Powders Grains, 2013c.
Martin Robinson, Stefan Luding, and Marco Ramaioli. Simulations of grain dispersion by liquid injection using SPH-DEM. In preparation, 2013d.
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