Applications of Lattice Boltzmann Methods
Dominik Bartuschat, Ulrich Rüde
New Delhi, IndiaJanuary 12, 2017
UGC-DAADLecture Series 2017
D. Bartuschat, M. Bauer, S. Bogner, C. Godenschwager, F. Schornbaum, U. Rüde
Chair for System Simulation, FAU Erlangen-Nürnberg
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Outline
● The waLBerla Simulation Framework● The Lattice Boltzmann Method and Complex Flows● Fluid-Particle Interactions● Charged Particles in Fluid Flow● Free Surface Flow
3
The waLBerla Simulation Framework
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
waLBerla
● Widely applicable Lattice Boltzmann framework● Suited for various flow applications● Large-scale, MPI-based parallelization● Dynamic application switches for heterogeneous architectures and
optimization
5
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
waLBerla Concepts
6
Block concept:● Domain partitioned into Cartesian grid of blocks● Blocks can be assigned to different processes● Blocks contain:
● cell data, e.g. fluid density, electric potential● global information e.g. location, MPI rank
Communication concept:● Simple communication mechanism on uniform grids, utilizing MPI● Ghost layers to exchange cell data with neighboring blocks
Sweep concept:● Sweeps are work steps of a time-loop, performed on block-parallel level● Example: MG sweep, contains sub-sweeps (restriction, prolongation, smoothing)
The Lattice Boltzmann Method
and flow in complex geometries
● Discrete lattice Boltzmann equation ● Describes probabilities fq that fluid molecules move with given velocities● Molecule collisions represented by collision operator Wq
fq(~xi +~cqdt, tn+dt)� fq(~xi , tn) = Wq (fq(~xi , tn))
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
The Lattice Boltzmann Method (LBM)
8
fq
● Domain discretized into cubic cells● Discrete velocities and associated
distribution functions per cell~cq
D3Q19 modelIllustration by Klaus Iglberger
˜
fq(~xi , tn) = fq(~xi , tn)�1
t�fq(~xi , tn)� f eqq (~xi , tn)
�
| {z }⌦q with single relaxation time
fq(~xi +~eq, tn+dt) = f̃q(~xi , tn)
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Stream-Collide
● Stream step:
9
The equation is solved in two steps:
● Collide step:
Fluid viscosity determined by , fluid velocity and density computable from nf t fq~u rf
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
●Sparse, but coherent geometry●Volume fraction 0.3%● Large number of small blocks●Multiple blocks per process● Load balancing required
Flow in Complex Geometries
10
Flow through Coronary Arteries
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Complex Geometry Initialization
11
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Domain Partitioning
● Domain partitioning of coronary tree dataset● Partitioning for aim of one block per process
● Excellent scaling on JUQUEEN with up to 1 trillion lattice cells*
12
JUQUEEN nodeboard512 processes, 485 blocks
JUQUEEN, full machine458 752 processes, 458 184 blocks
* C. Godenschwager et al. „A Framework for Hybrid Parallel Flow Simulations with a Trillion Cells in Complex Geometries“ (2013), doi:10.1145/2503210.2503273
mailto:[email protected]:[email protected]://dx.doi.org/10.1145/2503210.2503273http://dx.doi.org/10.1145/2503210.2503273
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Blood Flow and Perfusion● Flow simulation in arteries +
Myocardium as porous medium, modeled by LBM forcing term*● At aorta and endocardium: pressure boundary conditions● Blood flow visualization by particle tracing:
13
* Z. Guo, T. Zhao. „ Lattice Boltzmann model for incompressible flows through porous media“ (2002), doi:10.1103/PhysRevE.66.036304
mailto:[email protected]:[email protected]://dx.doi.org/10.1103/PhysRevE.66.036304http://dx.doi.org/10.1103/PhysRevE.66.036304
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Perfusion Results
14
Fluid-Particle Interaction with LBM
and tumbling spherocylinders in Stokes flow
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Fluid-Particle Interaction – 4-Way Coupling
● Particles mapped onto fixed lattice Boltzmann grid● Cells overlapped by object treated as moving boundary● Hydrodynamic forces on particle computed by momentum exchange method*
16
* N. Nguyen, A. Ladd. „Lubrication corrections for lattice-Boltzmann simulations of particle suspensions“ Phys. Rev. E (2002), doi:10.1103/PhysRevE.66.046708
Illustration by Jan Götz
mailto:[email protected]:[email protected]://dx.doi.org/10.1103/PhysRevE.66.046708http://dx.doi.org/10.1103/PhysRevE.66.046708
1/� =length
radius= 12
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Tumbling Spherocylinders in Stokes Flow
● Tumbling motion of elongated particles in Stokes flow● Four spherocylinders in periodic domain, aspect ratio
● LBM simulations with TRT operator and comparison against slender body formulation (examining influence of inertia, wall effects, and particle shape)*
17
* D. Bartuschat et al. „Two Computational Models for Simulating the Tumbling Motion of Elongated Particles in Fluids“ (2016), doi:10.1016/j.compfluid.2015.12.010
mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.compfluid.2015.12.010http://dx.doi.org/10.1016/j.compfluid.2015.12.010
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Two Tumbling Spherocylinders
● Flow Field around two spherocylinders,1/ε =12
18
50
100
150
200
250 ·10≠6
x[m
]
Á = 1/10 Á = 1/12 Á = 1/14
≠200
≠100
0
100
200·10≠6
u
x
[m/s
]
0 5 10 15 20 25
1.2
1.4
1.6
1.8
2
2.2·10≠3
t [s]
u
z
[m/s
]
● Motion dependent on aspect ratio 1/ε
➡ Convergence to preferred distance xmax due to inertia➡ Max. velocity uz for horizontal particle orientation,
min. uz for vertical orientation
Domain size: [576 dx]3
Fluid: Water (20°C)Time steps: 600000
dx=4.98µm, dt=4.55⋅10-5s, τ =6Particle density: 1492 kg/m3, radius = 4dxRuntime on LiMa: 16h, 768 cores
Charged Particles in Fluid Flow
for particle-laden electrokinetic flows
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Motivation
20
Interactions of large numbers of charged particles in fluid flows, influenced by external electric fields
© Kang and Li „Electrokinetic motion of particles and cells in microchannels“ Microfluidics and Nanofluidics
● Industrial applications:● Filtering particulates from exhaust gases● Charged particle deposition in cooling
systems of fuel cells
● Medical applications:● Optimization of Lab-on-a-Chip systems:● Sorting of different cells● Trapping cells and viruses
● Deposition of charged aerosol particles in respiratory tract
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Multi-Physics Simulation
21
Electro (quasi) statics
Fluiddynamics
Rigid body dynamics
hydrodynamic force
object movement
ion convectionforce on ions
electr
ostat
ic for
ce
charg
e den
sity
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Poisson Equation and Force on Particles
22
��F(~x) = re (~x)ee
~FC =�qparticle ·—F(~x)
● Electric potential described by Poisson equationwith particle‘s charge density on RHS:
● Discretized by finite volumes on lattice● Solved with cell-centered multigrid solver
implemented in waLBerla● Supersampling for computing overlap degree
to set RHS more accurately
● Electrostatic force on particle:
re
Illustration by Jan Götz
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
6-Way Coupling for Charged Particles
23
hydrodynam. force
object motion
Lubricationcorrection
electrostat. force
velocity BCs
object distance
LBM
correction force
charge distribution
Newtonian mechanicscollision response
treat BCsstream-collide step
Finite volumes
MGiterat.
treat BCsV-cycle
D. Bartuschat, U. Rüde. „Parallel Multiphysics Simulations of Charged Particles in Microfluidic Flows“, J. Comput. Sci. (2015), doi:10.1016/j.jocs.2015.02.006
mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.jocs.2015.02.006http://dx.doi.org/10.1016/j.jocs.2015.02.006
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Charged Particles Algorithm
24
foreach time step, do
// solve Poisson problem with particle charges:set RHS of Poisson equationwhile residual too high do
perform multigrid V-cycle to solve Poisson’s equation
// solve lattice Boltzmann equation// considering particle velocities:begin
set velocity BCs of particlesperform stream-collide step
// couple potential solver and LBM to pe:begin
apply hydrodynamic force to particlesapply electrostatic force to particlesperform lubrication correctionpe moves particles depending on forces
Charged Particles – Multigrid Solver
● Based on● Smoothing principle:
High-frequency error elimination by iterative solvers (e.g. GS)
● Coarse grid principle: Restriction to coarser grid transforms low-frequency error components to relative higher-frequency ones
● Smoothing on coarse grids● Prolongation of obtained correction
terms to finer grid
● Iterative method for efficient solution of sparse linear systems
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
26
● Applied recursively, V(νpre, νpost)-cycle
Multigrid
● All operations implemented as compact stencil operations
● Design goals:● Efficient and robust black-box solver● Handling complex boundary conditions on coarse levels● Naturally extensible to jumping coefficients
➡ Method of choice: Galerkin coarsening
● (FV) Stencils stored for each unknown● On finest level: quasi-constant stencils
● Averaging restriction, constant prolongation● Preserves D3Q7 stencil on coarse grids● Convergence rate deteriorates● Workaround for Poisson problem: Overrelaxing prolongation*
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
27
* Mohr, Wienands „Cell-centered Multigrid Revisited“, Comput. Vis. Sci. (2004)
P1 P2
P3 P4
Cell-Centered Multigrid - Implementation
Charged Particles – Validation
Radius: R = 600µmCharge: qe = 8000eSupersampling: factor 3
F(~r) = 14pee
qe|~r | if |
~r |� R
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
29
Validation of Electric Potential
Analytical solution for homogeneously charged particle in infinite domain:
0
50
100
150
200
250
0
0.1
0.2
0.3
·10�3
xL
F/V
Analytical solution
Numerical solution
Sphere surface
~rqe
R
x
Domain: [256 dx]3● Dirichlet BCs: exact solution● MG: 5 V(2,2)-cycles● Residual threshold 2⋅10-9
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
30
Determination of residual threshold:
Error hardly reduced after residual norm smaller than 10-9➡Residual threshold for simulations: 2⋅10-9
V(2,2) cyclesConv. rate 0.18
Validation of Electric Potential
0 1 2 3 4
5
10
�10
10
�9
10
�8
10
�7
10
�6
10
�5
Iteration
L 2n
orm
Error
Residual
Charged Particles – Results
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
32
Charged Particles in Fluid Flow
Channel: 2.56mm⇥7.68mm⇥1.92mm Particles: R = 80µm, qe =±40 000e Charged plates: F =±76.8Vdx = 10µm,t = 1.7,dt = 40µs Water (20 �C), Inflow 1 mm/s, Outflow 0 Pa else: no-slip & insulating BCs
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Charged Particles in Fluid Flow
33
•Computed on 144 cores (12 nodes) of RRZE - LiMa
•210 000 time steps
•15.7h runtime•643 unknowns per core
•6 MG levels
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
34
Weak scaling:● Costant size per core:
● 1283 cells● 9.4% moving obstacle cells
● Size doubled (in all dimensions)● Cell-centered MG
● V(3,3) with 7 levels● 6 to 116 CG coarse-grid iterations● Convergence rate: 0.07
● 2x4x2 cores per node
Experiments on SuperMUC:● 18 thin islands with 512 compute nodes, each:
● 16 cores (2 Xeon chips) @2.5 GHz● 32 GB DDR3 RAM
● Ranked #6 in TOP500 during experiments
Parallel Scaling Experiments
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
35
Weak Scaling for 240 Time Steps
32 768 cores7.1M particles
1 2 4 8 16 32 64 128
256
512
1024
2048
0
100
200
300
400
Number of nodes
Tota
lrun
times/s
LBMMapLubrHydrFpeMGSetRHSPtCmElectF
Overall parallel efficiency @2048 nodes: 83%
D. Bartuschat, U. Rüde. „Parallel Multiphysics Simulations of Charged Particles in Microfluidic Flows“ (2015), doi:10.1016/j.jocs.2015.02.006
mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.jocs.2015.02.006http://dx.doi.org/10.1016/j.jocs.2015.02.006
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
36
LBM 91 %MG - 1 V(3,3) 64 %
● Parallel efficiency @2048 nodes:
Weak Scaling for 240 Time Steps
➡ MG performance restricted by coarsest-grid solving
Meg
a la
ttice
site
upd
ates
per
sec
.
Meg
a flu
id la
ttice
site
upd
ates
per
sec
.
0 250 500 750 1000 1250 1500 1750 2000Number of nodes
0102030405060708090
103
MFL
UPS
(LB
M)
LBM Perform.20
40
60
80
100
120
103
MLU
PS (M
G)
MG Perform.
32 768 cores7.1M particles
D. Bartuschat, U. Rüde. „Parallel Multiphysics Simulations of Charged Particles in Microfluidic Flows“ (2015), doi:10.1016/j.jocs.2015.02.006
mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.jocs.2015.02.006http://dx.doi.org/10.1016/j.jocs.2015.02.006
Electrophoresis of Charged Particles
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Multi-Physics Simulation
38
Electro (quasi) statics
Fluiddynamics
Rigid body dynamics
hydrodynamic force
object movement
ion convectionforce on ions
electr
ostat
ic for
ce
charg
e den
sity
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Electrical Double Layer (EDL)
● Surfaces in contact with most liquids acquire surface charge● Surface charge balanced by oppositely charged ions from fluid
39
��y = eee Âi
zini•e� zi eykBT zi : valence of ions, e: elementary charge,
ni•: bulk ionic number concentration
● EDL potential described by Poisson-Boltzmann equation
x
yy
s
yd z
lD
solid
Shear plane
Stern plane
+
+
+
+
+
+
+
+
+
�
�
���
���
�
��
+
���
+�
�
�
+
+
+
�
�
�
�
diffuse layer
x
yy
s
yd z
lD
lD
Characteristic parameters: - potential EDL thickness z
Lattice Boltzmann equation with body force term
fq(~xi +~cqdt, t+dt)� fq(~xi , t) = Wq +dtFq(~fb)
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Electrostatic Force on Fluid in EDL
40
Electrostatic force on double layer charge~fb
= re
(y) ·~Eext
Electrical double layer potentialSymmetric Poisson-Boltzmann equation (PBE)
��y =�2z en•ee
sinh✓z eykB T
◆
Debye-Hückel approximation (DHA) for
��y =�k2 y, k = l�1D
|z |< 25mV
r
qy = z
R
lD
x
Radius: R = 120nmCharge: qs = −124eFluid: Water (20 �C)k R ⇡ 0.89
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
41
Domain: [128 dx]2 x 256 dx● SOR solver● Dirichlet BCs: exact solution● Residual reduction 2⋅10-7
Validation of Double Layer Potential
Analytical solution for sphere with uniform surface charge in infinite domain (Debye-Hückel approximation):
0 20 40
60
80 100 120
�10
�8
�6
�4
�2
0
·10�3
xL
y/V
Analytical
solution
Numerical
solution
y(~r) = z R|~r| e�k(|~r|�R) if |~r|� R
Channel: 1.28µm⇥2.56µm⇥1.28µm Particle: R = 120nm, qe = −124e z = −10mV, Ey =�4.7 · 10 6 V/mFluid: Water (20
�C), c• = 5µmol/l dx = 10nm,t = 6.5,dt = 0.2ns BCs: Periodic in axial direction,
else insulating & no-slip BCs
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Electrophoresis in Micro-Channel
Electrophoresis of charged particle along channel axis● Result after 30 000 time steps● Flow field, double layer potential, and ion charge distribution
42
U⇤EP = 0.21ms
Free Surface Flow
and effects of surface tension
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Free Surface Extension
44
●Volume of Fluid Approach*●Only liquid phase has to
be simulated**●Cells are classified as
either solid, liquid, gas or interface●LBM only performed in
fluid cells●Suitable for phases with
high density differences
*C. Hirt, B. Nichols. „Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries“ (1981), doi:10.1016/0021-9991(81)90145-5**C. Körner et al. „Lattice Boltzmann Model for Free Surface Flow for Modeling Foaming“ (2005), doi:10.1007/s10955-005-8879-8
mailto:[email protected]:[email protected]://dx.doi.org/10.1016/0021-9991(81)90145-5http://dx.doi.org/10.1016/0021-9991(81)90145-5http://dx.doi.org/10.1007/s10955-005-8879-8http://dx.doi.org/10.1007/s10955-005-8879-8
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Free Surface Extension
45
●Geometry Reconstruction:● computation of interface normals● compute normals at triple points● compute local curvature to account for
surface tension
●Surface Dynamics Simulation:● non-free-surface boundary treatment● free surface boundary treatment● LBM streaming step ( advection )● fill level updates ( mass advection )● LBM collision step● conversion of cell types
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Drop on Inclined Plane
46
kin. viscosity:surf. tension:drop diameter:drop resolution:τ:dt:
7.9‧10-6 m2/s 4.82‧10-2 N/m5 mm100 cells 0.5262.8‧10-6 s
SRT with Guo forcing term*
* Z. Guo et al. „ Discrete lattice effects on the forcing term in the lattice Boltzmann method“ (2002), doi: 10.1103/PhysRevE.65. 046308
mailto:[email protected]:[email protected]://dx.doi.org/10.1103/PhysRevE.65.%20046308http://dx.doi.org/10.1103/PhysRevE.65.%20046308
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Domain Setup
●Full free surface calculations are costly:● find interface cells● reconstruct interface normals and curvature● mass advection● cell conversions
47
●Restriction to L-shaped domain●No inclined geometry, use inclined gravity instead:
gravity
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Domain Setup
●Full free surface calculations are costly●Only fraction of complete domain covered with fluid/interface●Fluid covered region is moving➠ Dynamic load balancing required:
48
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Domain Setup
●Full free surface calculations are costly●Only fraction of complete domain covered with fluid/interface●Fluid covered region is moving➠ Dynamic load balancing required
49
●Simulation on 200 cores (on Emmy, RRZE)●Drop resolution of 100 cells➠ Runtime between 1 and 2 hours
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Boundary Setup
●No-slip boundary: solid wall, enforces zero velocity at boundary●Pressure boundary: pressure Dirichlet (set to capillary pressure)
50
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Varying Inclination Angle
51
15° 30° 45°
➠ Higher inclination causes drop to move further before being absorped
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Varying Contact Angle
52
50° 90° 110°
➠ Higher contact angle causes drop to move further before being absorped
Free Surface Flow with Particles
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Floating Objects
54
S. Bogner, U. Rüde. „ Simulation of floating bodies with lattice Boltzmann“ (2012), doi:10.1016/j.camwa.2012.09.012
mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.camwa.2012.09.012http://dx.doi.org/10.1016/j.camwa.2012.09.012
New Delhi, 12.01.2017 - Dominik Bartuschat - System Simulation Group - Applications of Lattice Boltzmann Methods(Contact: [email protected])
Rising Bubble
55
S. Bogner, U. Rüde. „ Simulation of floating bodies with lattice Boltzmann“ (2012), doi:10.1016/j.camwa.2012.09.012
mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.camwa.2012.09.012http://dx.doi.org/10.1016/j.camwa.2012.09.012
Thank you for your attention!