SPH simulation within the OpenFOAM framework · OutlineIntroductionMotivationSmoothed Particle HydrodynamicsNumerical ExamplesReferences SPH simulation within the OpenFOAM framework

Outline Introduction Motivation Smoothed Particle Hydrodynamics Numerical Examples References

SPH simulation within the OpenFOAMframework

Pankaj Kumar, Qing Yang, Van Jones, and Leigh McCue

Aerospace and Ocean Engineering

Virginia Polytechnic Institute and State University

June 14, 2011


Introduction

Motivation

Smoothed Particle Hydrodynamics

Numerical Examples


Outline

Introduction

Motivation


Numerical Examples


Grid Based Numerical Schemes

Reference [2]

Reference [1]


Outline

Introduction

Motivation


Numerical Examples


Grid Based Numerical Methods


Smoothed Particle Hydrodynamics (SPH)


Outline

Introduction

Motivation


Numerical Examples


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
































Integral Representation / Kernel Approximation(Liu and Liu)

f (x) =

∫Ω

f (x′)δ(x− x′)dx′

Ω is the volume of the domain and prime denotes neighboringvariables.δ(x− x′) is the Dirac delta function:

δ(x− x′) =

1 if x = x′

0 if x 6= x′

Replacing the dirac delta function with a smoothing function, in SPHconvention called kernel function, W(x− x′, h)

f (x) =

∫Ω

f (x′)W(x−x′, h)dx′ ∇f (x) =

∫Ω

[∇f (x′)

]W(x−x′, h)dx′



f (x) =

∫Ω



δ(x− x′) =

1 if x = x′

0 if x 6= x′


f (x) =

∫Ω

f (x′)W(x−x′, h)dx′ ∇f (x) =

∫Ω

[∇f (x′)




f (x) =

∫Ω



δ(x− x′) =

1 if x = x′

0 if x 6= x′


f (x) =

∫Ω

f (x′)W(x−x′, h)dx′ ∇f (x) =

∫Ω

[∇f (x′)



Kernel Function(Liu and Liu)

1. Even function2. Normalization condition / unity condition:∫

ΩW(x− x′, h)dx′ = 1

3. Delta function property:

limh→0

W(x− x′, h) = δ(x− x′)

4. W(x− x′, h) ≥ 0, monotonically decreasing5. Compact condition:

W(x− x′, h) = 0 when |x− x′| > kh




ΩW(x− x′, h)dx′ = 1


limh→0

W(x− x′, h) = δ(x− x′)


W(x− x′, h) = 0 when |x− x′| > kh




ΩW(x− x′, h)dx′ = 1


limh→0

W(x− x′, h) = δ(x− x′)


W(x− x′, h) = 0 when |x− x′| > kh




ΩW(x− x′, h)dx′ = 1


limh→0

W(x− x′, h) = δ(x− x′)


W(x− x′, h) = 0 when |x− x′| > kh




ΩW(x− x′, h)dx′ = 1


limh→0

W(x− x′, h) = δ(x− x′)


W(x− x′, h) = 0 when |x− x′| > kh


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

ρjf (xj)W(xi − xj, h)

∇ · f (xi) ≈ −N∑

j=1

mj

ρjf (xj)∇W(xi − xj, h)


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

ρjf (xj)W(xi − xj, h)

∇ · f (xi) ≈ −N∑

j=1

mj

ρjf (xj)∇W(xi − xj, h)


Governing Equations of Fluid Dynamics(Liu and Liu)

Conservation of Mass

DρDt

= −ρ∇ ·~v

SPH formulation of Conservation of Mass

Dρi

Dt= ρi

N∑j=1

mj

ρjvβij∂Wij

∂xβi



Conservation of Mass

DρDt

= −ρ∇ ·~v

SPH formulation of Conservation of Mass

Dρi

Dt= ρi

N∑j=1

mj

ρjvβij∂Wij

∂xβi



Conservation of Momentum

Dvα

Dt= −1

ρ

∂σαβ

∂xβ

SPH formulation of Conservation of Momentum

DvαiDt

=

N∑j=1

mjσαβi + σαβj

ρiρj

∂Wij

∂xβi



Conservation of Momentum

Dvα

Dt= −1

ρ

∂σαβ

∂xβ

SPH formulation of Conservation of Momentum

DvαiDt

=

N∑j=1

mjσαβi + σαβj

ρiρj

∂Wij

∂xβi


Equation of State(Liu and Liu)

• Tait equation of state for water (artificial compressibility):

P = B[(

ρ

ρ0

)γ

− 1]

P is the gauge pressure, B = 3.047 kilobars and γ = 7.15 for

water based on experimental data. Since ∆t = CFLp/√

Bρ

, a

numerical speed of sound Cs0 chosen to limit density variationwithin 1% and B =

Cs20ρ0γ

• Ideal gas law for air:P = ρRT

P is the absolute pressure, R is the individual gas constant, for airR = 286.9(J/kgK), T is the Kelvin temperature.




P = B[(

ρ

ρ0

)γ

− 1]



Bρ

, a


Cs20ρ0γ






P = B[(

ρ

ρ0

)γ

− 1]



Bρ

, a


Cs20ρ0γ




Boundary ConditionMonaghan-type Boundary Condition (Monaghan): The force per unit massacting from a boundary particle to a fluid particle within its neighboringdomain is:

~f = ~nR(η)P(ξ)

~n: Normal of the boundary particle; η: Normal distance from the fluidparticle to the boundary particle; ξ: the distance between the fluid particleand the boundary particle projected on the tangent of the boundary particle.

R(η) = A1− q√

q

q = η2∆p , A = 1

h

(0.01ca

2 + β caVab nb). ∆p: the boundary particle spacing,

c is the numerical speed of sound of particles, and:

β =

0 if Vab > 01 if Vab < 0

P(ξ) =12

(1 + cos

(πξ

∆p

))


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


Numerical Examples


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, γ = 7.0, B = 1

γρC2s


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


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, γX = 7.0, BX = 17.4ρgH, CY = 155

√2gH,

γY = 7.0, BY = BX ,


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


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


Dam Collapse: Multi PhaseTime Evolution of Water-Front Toe

0 0.5 1 1.5 2 2.52

2.5

3

3.5

4

4.5

5

5.5

x front

/H

t(g/H)0.5

SPHRAS


Conclusion

• SPH solver is developed into the OpenFOAM framework.

• Direct comparison of results for meshfree and grid basedmethods are possible into the OpenFOAM framework.

• It is possible to use SPH and RANS together to capitalize on thebest of both using OpenFOAM framework.


Future Work

• To develop a utility to generate a temporary boundary particlesalong the fluid surface to bring the system into equilibrium attime t = 0.

• To develop a utility to generate ghost particles for SPHsimulation.

• To develop a solver utilizing the benefits of SPH and RANStogether.


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, 191:448–475, 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, 110:399–406, 1992.

http://www.arl.hpc.mil/Publications/eLink_Fall05/cover.html

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



SPH simulation within the OpenFOAM framework · OutlineIntroductionMotivationSmoothed Particle HydrodynamicsNumerical ExamplesReferences SPH simulation within the OpenFOAM framework

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