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Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
SPH simulation within the OpenFOAM
framework
Pankaj Kumar, Qing Yang, Van Jones, and Leigh McCue
Aerospace and Ocean Engineering
Virginia Polytechnic Institute and State University
June 14, 2011
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Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
Introduction
Motivation
Smoothed Particle Hydrodynamics
Numerical Examples
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Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
Outline
Introduction
Motivation
Smoothed Particle Hydrodynamics
Numerical Examples
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Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
Grid Based Numerical Schemes
Reference [2]
Reference [1]
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Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
Outline
Introduction
Motivation
Smoothed Particle Hydrodynamics
Numerical Examples
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Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
Grid Based Numerical Methods
O li I d i M i i S h d P i l H d d i N i l E l R f
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Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
Smoothed Particle Hydrodynamics (SPH)
O tli I t d ti M ti ti S th d P ti l H d d i N i l E l R f
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Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
Outline
Introduction
Motivation
Smoothed Particle Hydrodynamics
Numerical Examples
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
Smoothed Particle Hydrodynamics (SPH)
(Liu and Liu)
A meshfree and Lagrangian particle method Particles can be uniformly / arbitrarily distributed
Adaptive in nature
Two key steps in SPH formulation:
1. Integral representation / Kernel approximation
2. Particle approximation
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
Smoothed Particle Hydrodynamics (SPH)
(Liu and Liu)
A meshfree and Lagrangian particle method Particles can be uniformly / arbitrarily distributed
Adaptive in nature
Two key steps in SPH formulation:
1. Integral representation / Kernel approximation
2. Particle approximation
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
Smoothed Particle Hydrodynamics (SPH)
(Liu and Liu)
A meshfree and Lagrangian particle method Particles can be uniformly / arbitrarily distributed
Adaptive in nature
Two key steps in SPH formulation:
1. Integral representation / Kernel approximation
2. Particle approximation
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
Smoothed Particle Hydrodynamics (SPH)
(Liu and Liu)
A meshfree and Lagrangian particle method Particles can be uniformly / arbitrarily distributed
Adaptive in nature
Two key steps in SPH formulation:
1. Integral representation / Kernel approximation
2. Particle approximation
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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y y p
Smoothed Particle Hydrodynamics (SPH)
(Liu and Liu)
A meshfree and Lagrangian particle method Particles can be uniformly / arbitrarily distributed
Adaptive in nature
Two key steps in SPH formulation:
1. Integral representation / Kernel approximation
2. Particle approximation
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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y y p
Smoothed Particle Hydrodynamics (SPH)
(Liu and Liu)
A meshfree and Lagrangian particle method Particles can be uniformly / arbitrarily distributed
Adaptive in nature
Two key steps in SPH formulation:
1. Integral representation / Kernel approximation
2. Particle approximation
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Integral Representation / Kernel Approximation
(Liu and Liu)
f(x) =
f(x)(x x)dx
is the volume of the domain and prime denotes neighboringvariables.
(x x) is the Dirac delta function:
(x x) =
1 ifx = x
0 ifx = x
Replacing the dirac delta function with a smoothing function, in SPH
convention called kernel function, W(x x, h)
f(x) =
f(x)W(xx, h)dx f(x) = f(x)
W(xx, h)dx
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Integral Representation / Kernel Approximation
(Liu and Liu)
f(x) =
f(x)(x x)dx
is the volume of the domain and prime denotes neighboringvariables.
(x x) is the Dirac delta function:
(x x) =
1 ifx = x
0 ifx = x
Replacing the dirac delta function with a smoothing function, in SPH
convention called kernel function, W(x x, h)
f(x) =
f(x)W(xx, h)dx f(x) = f(x)
W(xx, h)dx
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Integral Representation / Kernel Approximation
(Liu and Liu)
f(x) =
f(x)(x x)dx
is the volume of the domain and prime denotes neighboringvariables.
(x x) is the Dirac delta function:
(x x) =
1 ifx = x
0 ifx = x
Replacing the dirac delta function with a smoothing function, in SPHconvention called kernel function, W(x x, h)
f(x) =
f(x)W(xx, h)dx f(x) = f(x)
W(xx, h)dx
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Kernel Function
(Liu and Liu)
1. Even function
2. Normalization condition / unity condition:
W(x x, h)dx = 1
3. Delta function property:
limh0
W(x x, h) = (x x)
4. W(x x, h) 0, monotonically decreasing5. Compact condition:
W(x x
, h) = 0 when |x x
| > kh
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Kernel Function
(Liu and Liu)
1. Even function
2. Normalization condition / unity condition:
W(x x, h)dx = 1
3. Delta function property:
limh0
W(x x, h) = (x x)
4. W(x x, h) 0, monotonically decreasing5. Compact condition:
W(x x
, h) = 0 when |x x
| > kh
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Kernel Function
(Liu and Liu)
1. Even function
2. Normalization condition / unity condition:
W(x x, h)dx = 1
3. Delta function property:
limh0
W(x x, h) = (x x)
4. W(x x, h) 0, monotonically decreasing5. Compact condition:
W(x x
, h) = 0 when |x x
| > kh
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Kernel Function
(Liu and Liu)
1. Even function
2. Normalization condition / unity condition:
W(x x, h)dx = 1
3. Delta function property:
limh0
W(x x, h) = (x x)
4. W(x x, h) 0, monotonically decreasing5. Compact condition:
W(x
x
,h) = 0 when |
x
x
| >kh
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Kernel Function
(Liu and Liu)
1. Even function2. Normalization condition / unity condition:
W(x x, h)dx = 1
3. Delta function property:
limh0
W(x x, h) = (x x)
4. W(x x, h) 0, monotonically decreasing5. Compact condition:
W(x
x
,h) =
0 when|x
x
| >kh
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Particle Approximation
(Liu and Liu)
To approximate the continuous integral representations as discretizedform of summation over all the particles in the support domain.
f(xi) N
j=1
mj
j f(xj)W(xi xj, h)
f(xi) N
j=1
mj
jf(xj)W(xi xj, h)
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Particle Approximation
(Liu and Liu)
To approximate the continuous integral representations as discretizedform of summation over all the particles in the support domain.
f(xi) N
j=1
mj
j f(xj)W(xi xj, h)
f(xi) N
j=1
mj
jf(xj)W(xi xj, h)
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Governing Equations of Fluid Dynamics
(Liu and Liu)
Conservation of Mass
D
Dt= v
SPH formulation of Conservation of Mass
DiDt
= i
Nj=1
mjj
vij Wijxi
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Governing Equations of Fluid Dynamics
(Liu and Liu)
Conservation of Mass
D
Dt= v
SPH formulation of Conservation of Mass
DiDt
= i
Nj=1
mjj
vij Wijxi
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Governing Equations of Fluid Dynamics
(Liu and Liu)
Conservation of Momentum
Dv
Dt
=
1
x
SPH formulation of Conservation of Momentum
DviDt
=N
j=1
mj
i +
j
ijWij
xi
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Governing Equations of Fluid Dynamics
(Liu and Liu)
Conservation of Momentum
Dv
Dt
=
1
x
SPH formulation of Conservation of Momentum
DviDt
=N
j=1
mj
i +
j
ijWij
xi
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Equation of State
(Liu and Liu)
Tait equation of state for water (artificial compressibility):
P = B
0
1
P is the gauge pressure, B = 3.047 kilobars and = 7.15 forwater based on experimental data. Since t= CFLp/
B
, a
numerical speed of sound Cs0 chosen to limit density variation
within 1% and B =Cs200
Ideal gas law for air:P = RT
P is the absolute pressure, R is the individual gas constant, for air
R = 286.9(J/kgK), T is the Kelvin temperature.
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Equation of State
(Liu and Liu)
Tait equation of state for water (artificial compressibility):
P = B
0
1
P is the gauge pressure, B = 3.047 kilobars and = 7.15 forwater based on experimental data. Since t= CFLp/
B
, a
numerical speed of sound Cs0 chosen to limit density variation
within 1% and B =Cs200
Ideal gas law for air:P = RT
P is the absolute pressure, R is the individual gas constant, for air
R = 286.9(J/kgK), T is the Kelvin temperature.
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Equation of State
(Liu and Liu)
Tait equation of state for water (artificial compressibility):
P = B
0
1
P is the gauge pressure, B = 3.047 kilobars and = 7.15 forwater based on experimental data. Since t= CFLp/
B
, a
numerical speed of sound Cs0 chosen to limit density variation
within 1% and B =Cs200
Ideal gas law for air:P = RT
P is the absolute pressure, R is the individual gas constant, for air
R = 286.9(J/kgK), T is the Kelvin temperature.
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Boundary ConditionMonaghan-type Boundary Condition (Monaghan): The force per unit mass
acting from a boundary particle to a fluid particle within its neighboring
domain is: f = nR()P()
n: Normal of the boundary particle; : Normal distance from the fluidparticle to the boundary particle; : the distance between the fluid particleand the boundary particle projected on the tangent of the boundary particle.
R() = A1 q
q
q = 2p
, A = 1h
0.01ca
2 + caVab nb
. p: the boundary particle spacing,c is the numerical speed of sound of particles, and:
=
0 ifVab > 01 ifVab < 0
P() =1
21 + cos
p
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Particle Distribution
OpenFOAM utility blockMesh is used to generate the mesh.
Cell centers are used as the fluid particles.
OpenFOAM utility setFields is used to find different fluids incase of multiphase flow.
A utility monaghanBoundary is used to generate Monaghanboundary points.
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
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Particle Distribution
OpenFOAM utility blockMesh is used to generate the mesh.
Cell centers are used as the fluid particles.
OpenFOAM utility setFields is used to find different fluids incase of multiphase flow.
A utility monaghanBoundary is used to generate Monaghanboundary points.
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
P i l Di ib i
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Particle Distribution
OpenFOAM utility blockMesh is used to generate the mesh.
Cell centers are used as the fluid particles.
OpenFOAM utility setFields is used to find different fluids incase of multiphase flow.
A utility monaghanBoundary is used to generate Monaghanboundary points.
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
P i l Di ib i
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Particle Distribution
OpenFOAM utility blockMesh is used to generate the mesh.
Cell centers are used as the fluid particles.
OpenFOAM utility setFields is used to find different fluids incase of multiphase flow.
A utility monaghanBoundary is used to generate Monaghanboundary points.
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
O tli
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Outline
Introduction
Motivation
Smoothed Particle Hydrodynamics
Numerical Examples
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
D C ll ith F S f
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Dam Collapse with Free Surface
Dam-break problem and impact against a rigid obstetrical and walls.
Geometrical parameters:
H= 0.1m, L/H= 1, D/H= 1.5, d/H= 2, hb/H= 0.4, lb/H= 0.2,b/H= 1.4, Cs = 10
2gH, = 7.0, B = 1
C2s
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
Dam Collapse ith Free S rface
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Dam Collapse with Free Surface
t(g/H) = 0.0
t(g/H) = 2.97
t(g/H) = 1.98
t(g/H) = 3.96
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
Dam Collapse: Multi Phase
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Dam Collapse: Multi Phase
(Colagrossi and Landrini)
Dam-break problem and impact against a rigid walls.
Geometrical parameters:H= 0.6m, L/H= 2, D/H= 3, d/H= 5.366, Nx = 391, Ny = 2885,CX = 10.9
2gH, X = 7.0, BX = 17.4gH, CY = 155
2gH,
Y = 7.0, BY = BX,
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
Dam Collapse: Multi Phase using SPH
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Dam Collapse: Multi Phase using SPH
t(g/H) = 0.00
t(g/H) = 1.62
t(g/H) = 0.81
t(g/H) = 2.63
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
Dam Collapse: Multi Phase using RANS
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Dam Collapse: Multi Phase using RANS
t(g/H) = 0.00
t(g/H) = 1.62
t(g/H) = 0.81
t(g/H) = 2.63
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
Dam Collapse: Multi Phase
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Dam Collapse: Multi Phase
Time Evolution of Water-Front Toe
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
Conclusion
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Conclusion
SPH solver is developed into the OpenFOAM framework.
Direct comparison of results for meshfree and grid based
methods are possible into the OpenFOAM framework.
It is possible to use SPH and RANS together to capitalize on thebest of both using OpenFOAM framework.
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
Future Work
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Future Work
To develop a utility to generate a temporary boundary particlesalong the fluid surface to bring the system into equilibrium at
time t= 0.
To develop a utility to generate ghost particles for SPHsimulation.
To develop a solver utilizing the benefits of SPH and RANS
together.
Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References
References
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References
[1] URL http://www.arl.hpc.mil/Publications/eLink_Fall05/cover.html.
[2] URL http://dspace.mit.edu/bitstream/handle/1721.1/36385/16-21Spring-2003/OcwWeb/
Aeronautics-and-Astronautics/
16-21Techniques-of-Structural-Analysis-and-DesignSpring2003/CourseHome/index.htm.
[3] Andrea Colagrossi and Maurizio Landrini. Numerical simulation of interfacial flows by smoothed particle hydrodynamics.
Journal of Computational Physics, 191:448475, 2003.
[4] G. R. Liu and M. B. Liu. Smoothed particle hydrodynamics: a meshfree particle method. World Scientific, 2003.
[5] J. J. Monaghan. Simulating free surface flows with sph. Journal of Computational Physics, 110:399406, 1992.
http://www.arl.hpc.mil/Publications/eLink_Fall05/cover.htmlhttp://www.arl.hpc.mil/Publications/eLink_Fall05/cover.htmlhttp://dspace.mit.edu/bitstream/handle/1721.1/36385/16-21Spring-2003/OcwWeb/Aeronautics-and-Astronautics/16-21Techniques-of-Structural-Analysis-and-DesignSpring2003/CourseHome/index.htmhttp://dspace.mit.edu/bitstream/handle/1721.1/36385/16-21Spring-2003/OcwWeb/Aeronautics-and-Astronautics/16-21Techniques-of-Structural-Analysis-and-DesignSpring2003/CourseHome/index.htmhttp://dspace.mit.edu/bitstream/handle/1721.1/36385/16-21Spring-2003/OcwWeb/Aeronautics-and-Astronautics/16-21Techniques-of-Structural-Analysis-and-DesignSpring2003/CourseHome/index.htmhttp://dspace.mit.edu/bitstream/handle/1721.1/36385/16-21Spring-2003/OcwWeb/Aeronautics-and-Astronautics/16-21Techniques-of-Structural-Analysis-and-DesignSpring2003/CourseHome/index.htmhttp://dspace.mit.edu/bitstream/handle/1721.1/36385/16-21Spring-2003/OcwWeb/Aeronautics-and-Astronautics/16-21Techniques-of-Structural-Analysis-and-DesignSpring2003/CourseHome/index.htmhttp://dspace.mit.edu/bitstream/handle/1721.1/36385/16-21Spring-2003/OcwWeb/Aeronautics-and-Astronautics/16-21Techniques-of-Structural-Analysis-and-DesignSpring2003/CourseHome/index.htmhttp://dspace.mit.edu/bitstream/handle/1721.1/36385/16-21Spring-2003/OcwWeb/Aeronautics-and-Astronautics/16-21Techniques-of-Structural-Analysis-and-DesignSpring2003/CourseHome/index.htmhttp://www.arl.hpc.mil/Publications/eLink_Fall05/cover.htmlRecommended