Mecánica de Fluidos - Merle Potter y David Wiggert - Tercera Edicion
Smoothed Particle Hydrodynamics (SPH): corrections and … · 2013-11-19 · For further reading...
Transcript of Smoothed Particle Hydrodynamics (SPH): corrections and … · 2013-11-19 · For further reading...
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Smoothed Particle Hydrodynamics (SPH):corrections and applications
Q.Z. Hou
April 7 2010
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Outline
1 Motivation
2 SPH and corrective schemesSmoothed particle hydrodynamicsCorrective schemes
3 Test problemsWater hammerHeat conductionStress wave propagationShock tube
4 Summary and future workBibliography
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Outline
1 Motivation
2 SPH and corrective schemesSmoothed particle hydrodynamicsCorrective schemes
3 Test problemsWater hammerHeat conductionStress wave propagationShock tube
4 Summary and future workBibliography
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Physical problem
Laboratory set-up
Bozkus and Wiggert (1997)
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Practical problem
Industrial pipelines
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Oil pipeline
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Steam pipeline
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Difficulties in traditional CFD methods
Two fundamental frameworks:Eulerian description: mesh fixed in space, objects moveacross it; calculation of mass, momentum and energydistributionLagrangian description: mesh attached to the material,cells deform; no mass flux, flux of momentum and energyat cell boundaries
Difficulties:Eulerian grid: irregular/complex geometry, locations ofinhomogeneities, free surfaces, moving interfaces,deformable boundariesLagrangian grid: mesh generation, large deformationboth: explosion, high-velocity impact, discrete particles
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Outline
1 Motivation
2 SPH and corrective schemesSmoothed particle hydrodynamicsCorrective schemes
3 Test problemsWater hammerHeat conductionStress wave propagationShock tube
4 Summary and future workBibliography
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What is SPH?
SPH: smoothed particle hydrodynamics
1. movies3D simulation of dam breakpouring water in a glass (like commercial advertisement forbeer).Hollywood, Pixar Animation Studio, Industry Light andMagic, fluid simulations in Ice Age and 2012
2. key ideafollow particles’ movement, information attached to them isused as weighted average in numerical approximation
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Fundamentals of SPH
Integral representation
f (x) =
∫Ω
f (ξ)δ(x − ξ,h) dV
Function reconstruction
f (x) ≈∫
Ωf (ξ)W (x− ξ,h) dV
Representation of the derivative
∇f (x) ≈ −∫
Ωf (ξ)∇W (x − ξ,h)dV
with smoothing function W (x − ξ,h)and smoothing length h
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Smoothing function
Requirements:
unity:∫
Ω W (x − ξ,h) dξ = 1;compact support: W (x − ξ,h) = 0, |x − ξ| > κh;positivity: W (x − ξ,h) ≥ 0, |x − ξ| ≤ κh;symmetry: W (x − ξ,h) = W (ξ − x ,h);δ function consistency:
limh→0
W (x − ξ,h) = δ(x − ξ);
monotonically decreasing with respect to |x − ξ|smooth
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Cubic spline – smoothing function W
Cubic spline and its derivatives
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Discretisation (particle approximation)
Ω is divided into N particles, with volume ∆Vj , (j = 1, · · · ,N)
f (x) ≈∫
Ωf (ξ)W (x − ξ,h) dV
⇒ f (x i) ≈∑
f (ξj)W (x i − ξj ,h)∆Vj
movement of particlesgive each particle a fixed mass mj and replace the volume ∆Vjby
∆Vj =mj
ρj
position evolutionxn+1
i = xni + ∆t vn
i
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Discretisation for derivative
∇f (x) ≈ −∫
Ωf (ξ)∇W (x − ξ,h)dV
⇒ ∇f (x i) ≈ −∑
f (ξj)∇W (x i − ξj ,h)∆Vj
introduce the following notation
Wij := W (x i − ξj ,h),
Wij,α :=(∂Wij∂xα
)i, where α indicates a spatial direction
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Particle distribution
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1D example: f (x) = x , x ∈ [0, 1],N = 5, κh = 12
f1 ≈∑5
j=1 fjW1j∆Vj = f1W11∆V4 + f2W12∆V2 = 16 f2 = 1
24 6= 0
function reconstruction derivative approximation
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Corrective schemes (1D)
Expanding Taylor series for f (ξ) around point x yields
f (ξ) ≈ f (x) +∂f (x)
∂x(ξ − x)
Multiplying with W and integrating over Ω∫Ω
f (ξ)WdV ≈ f (x)
∫Ω
WdV +∂f (x)
∂x
∫Ω
(ξ − x)WdV
∫Ω
f (ξ)WdV ≈ f (x)
∫Ω
WdV
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Corrective scheme for derivative
Taylor series expansion
f (ξ) ≈ f (x) +∂f (x)
∂x(ξ − x)
Multiplying with ∂W∂ξ and integrating over Ω∫
Ωf (ξ)
∂W∂ξ
dV ≈ f (x)
∫Ω
∂W∂ξ
dV +∂f (x)
∂x
∫Ω
(ξ − x)∂W∂ξ
dV
CSPM: corrective smoothed particle method
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Corrective scheme in discretized form
function approximation∫Ω
f (ξ)WdV ≈ f (x)
∫Ω
WdV
⇒ f (x) ≈∫
Ω f (ξ)WdV∫Ω WdV
⇒ f (xi) ≈∑
f (xj)Wij∆Vj∑Wij∆Vj
derivative approximation∫Ω
f (ξ)∂W∂ξ
dV ≈ f (x)
∫Ω
∂W∂ξ
dV +∂f (x)
∂x
∫Ω
(ξ − x)∂W∂ξ
dV
⇒(∂f (x)
∂x
)i≈∑
[f (xj)− f (xi)]Wij,x ∆Vj∑(xj − xi)Wij,x ∆Vj
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Same 1D example
function reconstruction derivative approximation
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Same 1D example: f = (f1, f2, f3, f4, f5)T
f ≈ Af , ∂f/∂x ≈ Bf
Interpolation matrix A2/3 1/61/6 2/3 1/6
1/6 2/3 1/61/6 2/3 1/6
1/6 2/3
,
4/5 1/51/6 2/3 1/6
1/6 2/3 1/61/6 2/3 1/6
1/5 4/5
Difference matrix B
1∆x
0 1
2−1
2 0 12
−12 0 1
2−1
2 0 12
−12 0
,1
∆x
−1 1−1
2 0 12
−12 0 1
2−1
2 0 12
−1 1
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2D: f (x , y) = sin πx sin πy , (x , y) ∈ [0,1]× [0,1]
Reconstruction of function f and particle approximation of thefirst derivative fx
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Outline
1 Motivation
2 SPH and corrective schemesSmoothed particle hydrodynamicsCorrective schemes
3 Test problemsWater hammerHeat conductionStress wave propagationShock tube
4 Summary and future workBibliography
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Water hammer
Typical reservoir-pipe-valve (RPV) system
Governing equations
∂v∂t
=1ρ
∂p∂x
+∂φ
∂x, (momentum equation)
∂p∂t
= K∂v∂x
(continuity equation)
linear artificial viscosity φ =
Cv,x , v,x < 00, v,x > 0
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Solving two first order equations by CSPM
CSPM:(∂f∂x
)i≈
∑[f (xj )−f (xi )]Wij,x ∆Vj∑
(xj−xi )Wij,x ∆Vjfor f = v and f = p
IC: v0i = v0; p0
i = p0
Algorithm for single time stepstep 1: apply BC vn+1
N = 0step 2: calculate pressure
pn+1i = pn
i + ∆t[K(∂v∂x
)i
]step 3: apply BC pn+1
1 = p0step 4: calculate velocity
vn+1i = vn
i + ∆t[
1ρ
(∂p∂x
)i
+
(∂φ
∂x
)i
]
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Results
instantaneous valve closing linear valve closing
Pressure history at valve
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Heat conduction in a bar
Temperature distribution at different times, left - energy balance(two first order eqs.), right - heat conduction (one second order
eq.)
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Wave propagation in an elastic bar
Snapshot at t = 10 µs
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Shock tube (moving particles)
Shock tube problem
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Outline
1 Motivation
2 SPH and corrective schemesSmoothed particle hydrodynamicsCorrective schemes
3 Test problemsWater hammerHeat conductionStress wave propagationShock tube
4 Summary and future workBibliography
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Summary
difficulties in traditional methodsfundamentals of SPHcorrections to SPH1D test problems of CSPM
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Future work
new corrective schemesirregular particlesmoving particles2D / 3D problems
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For further reading
Bozkus Z and Wiggert D.C (1997)Liquid slug motion in a voided line. Journal of Fluids andStructures, 11, 947-963
Liu G.R and Liu M.B (2003)Smoothed Particle Hydrodynamics: A Meshfree ParticleMethod.
Chen J.K., Beraun J.E., Carney T.C (1999)A corrective smoothed particle method for boundary valueproblems in heat conduction. International Journal forNumerical Methods in Engineering, 46: 231-252
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For further reading
Bozkus Z and Wiggert D.C (1997)Liquid slug motion in a voided line. Journal of Fluids andStructures, 11, 947-963
Liu G.R and Liu M.B (2003)Smoothed Particle Hydrodynamics: A Meshfree ParticleMethod.
Chen J.K., Beraun J.E., Carney T.C (1999)A corrective smoothed particle method for boundary valueproblems in heat conduction. International Journal forNumerical Methods in Engineering, 46: 231-252
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For further reading
Bozkus Z and Wiggert D.C (1997)Liquid slug motion in a voided line. Journal of Fluids andStructures, 11, 947-963
Liu G.R and Liu M.B (2003)Smoothed Particle Hydrodynamics: A Meshfree ParticleMethod.
Chen J.K., Beraun J.E., Carney T.C (1999)A corrective smoothed particle method for boundary valueproblems in heat conduction. International Journal forNumerical Methods in Engineering, 46: 231-252