pankaj kumar slides

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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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Introduction

Motivation

Smoothed Particle Hydrodynamics

Numerical Examples


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Outline

Introduction

Motivation


Numerical Examples


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Grid Based Numerical Schemes

Reference [2]

Reference [1]


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Outline

Introduction

Motivation


Numerical Examples


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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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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


Numerical Examples



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(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



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(Liu and Liu)


Adaptive in nature






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(Liu and Liu)


Adaptive in nature






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(Liu and Liu)


Adaptive in nature






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y y p


(Liu and Liu)


Adaptive in nature






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y y p


(Liu and Liu)


Adaptive in nature






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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



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(Liu and Liu)

f(x) =

f(x)(x x)dx



(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) =


W(xx, h)dx



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(Liu and Liu)

f(x) =

f(x)(x x)dx



(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) =


W(xx, h)dx



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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



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Kernel Function

(Liu and Liu)

1. Even function


W(x x, h)dx = 1


limh0

W(x x, h) = (x x)


W(x x

, h) = 0 when |x x

| > kh



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Kernel Function

(Liu and Liu)

1. Even function


W(x x, h)dx = 1


limh0

W(x x, h) = (x x)


W(x x

, h) = 0 when |x x

| > kh



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Kernel Function

(Liu and Liu)

1. Even function


W(x x, h)dx = 1


limh0

W(x x, h) = (x x)


W(x

x

,h) = 0 when |

x

x

| >kh



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Kernel Function

(Liu and Liu)

1. Even function2. Normalization condition / unity condition:

W(x x, h)dx = 1


limh0

W(x x, h) = (x x)


W(x

x

,h) =

0 when|x

x

| >kh



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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)



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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)



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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



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(Liu and Liu)

Conservation of Mass

D

Dt= v

SPH formulation of Conservation of Mass

DiDt

= i

Nj=1

mjj

vij Wijxi



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(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



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(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



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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.



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Equation of State

(Liu and Liu)


P = B

0

1


B

, a








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Equation of State

(Liu and Liu)


P = B

0

1


B

, a








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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



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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.



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P i l Di ib i


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P i l Di ib i


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O tli


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Outline

Introduction

Motivation


Numerical Examples


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


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


Dam Collapse: Multi Phase


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(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,


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


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




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Time Evolution of Water-Front Toe


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.


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.


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.html

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