Numerical simulation of liquid sloshing and fluid-structure interaction (FSI) inside closed volumes
Ing. R. Euser
• Introduction – About Femto Engineering – Purpose of presentation
• Liquid sloshing – Oil and gas applications – Numerical solution (SPH) – Simulation examples
• Fluid-Structure interaction (FSI) – Oil and gas applications – Numerical solutions (SPH-FEM and SPH2) – Simulation examples
Contents
Introduction
About Femto – Business proposal
Consultancy
Product Optimization
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Conceptual Design
Contra Expertise
Outsourcing
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CAE Software
Sales Benelux CAE Software
Support
Customization
Software Development
Training
About Femto – Competences
Linear FEM Nonlinear FEM
Conceptual Design (3D-CAD)
SPH and SPH-FEM coupling
About Femto – Work field
Offshore & Maritime
High Tech & Aerospace
Packaging & Ind. Design
Machinery & Process Equipment
Industries
Machinery & Process Equipment
About Femto – Key points
• Specialists in CAE: > 15 years active
• NAFEMS membership since 2001
• Offices in Delft & Münster (Germany)
• Organization of ~ 17 employees
• Software development & customization
• FEM Partner for Siemens PLM Software
To inform you about our numerical solutions for liquid sloshing and liquid impact problems in the oil and gas industry.
Purpose of presentation
Liquid sloshing
Oil and gas applications
Transport tanks LNG carriers
Storage tanks Liquid dampers
• To model liquids and gases we use Smoothed Particle Hydrodynamics (SPH).
• SPH is a numerical discretization method that uses discrete elements (particles) to simulate the behavior of fluids.
• The SPH method uses the following formulation:
Numerical solution - Overview
j
ijj
j
j
i hrWxfm
xf ,
Var. Description
i Particle index
j Neighbor index
m Particle mass
ρ Density
W Smoothing kernel
r Neighbor distance
h Smoothing length
The accuracy of the SPH method highly depends on the type of smoothing kernel that is being used. • In our solution we use a B-Spline kernel function:
• Smoothing length is updated by:
Numerical solution – Smoothing kernel
iii h
dt
dhv
3
1
hr
hrhh
r
hrh
r
h
r
hhrW
20
226
1
02
1
3
2
2
3,
3
32
3
Var. Description
v Particle velocity
t Time
Fluid behavior is described using the Navier-Stokes equations. Approximating the laws of conservation for the Navier-Stokes equations with SPH gives:
• Conservation of mass:
• Conservation of momentum:
• Conservation of energy:
Numerical solution – Navier-Stokes equations
j i
ij
ij
j
j
ii
x
Wv
m
Dt
D
j i
ij
ji
ji
ji
x
Wm
Dt
Dv
222
1
i
i
i
j i
ij
ij
ji
ji
ji
x
Wv
ppm
Dt
De
Var. Description
α, β Coordinate directions
x Particle position
σ Stress tensor
e Internal energy
p Pressure
μ Dynamic viscosity
ε Shear strain rate
When neglecting the viscous terms of the Navier-Stokes equations we get the Euler equations. The SPH approximations for the Euler equations are:
• Conservation of mass:
• Conservation of momentum:
• Conservation of energy:
Numerical solution – Conservation equations
j i
ij
ij
j
j
ii
x
Wv
m
Dt
D
j i
ij
ji
ji
ji
x
Wppm
Dt
Dv
j i
ij
ij
ji
ji
ji
x
Wv
ppm
Dt
De
2
1
• Incompressible fluids: (Tait equation)
• Compressible fluids: (Tammann equation)
1
0
2
0
cp
Depending on the type of fluid to be simulated, the pressure is calculated explicitly using a suitable equation of state:
Numerical solution – Equation of state
ep 1
Var. Description
ρ0 Reference density
c Speed of sound
γ Polytropic constant
e Specific internal energy
The diagram on the left shows a general overview of
the explicit solver structure of our SPH solution.
The next few slides show an example of how to solve an inviscid incompressible fluid
flow.
Numerical solution – Solver structure
Search for neighbors
Calculate smoothing kernel
Calculate rate of density change
Calculate pressure
Calculate rate of momentum change
Update particles
Calculate time step
Numerical solution – Solving example
Search for neighbors
Calculate smoothing kernel
Calculate rate of density change
Calculate pressure
Calculate rate of momentum change
Update particles
Calculate time step
Load case: A rectangular 2D tank with a volume of water in it, subject to gravity.
The colors of the particles show their velocities.
g
Numerical solution – Solving example
Search for neighbors
Calculate smoothing kernel
Calculate rate of density change
Calculate pressure
Calculate rate of momentum change
Update particles
Calculate time step
Depending on the type of problem being solved, a suitable Nearest
Neighbor Search (NNS) algorithm can be used. In most cases a so-called Tree
Search algorithm is used.
Numerical solution – Solving example
Search for neighbors
Calculate smoothing kernel
Calculate rate of density change
Calculate pressure
Calculate rate of momentum change
Update particles
Calculate time step
hr
hrhh
r
hrh
r
h
r
hhrW
20
226
1
02
1
3
2
2
3,
3
32
3
Numerical solution – Solving example
Search for neighbors
Calculate smoothing kernel
Calculate rate of density change
Calculate pressure
Calculate rate of momentum change
Update particles
Calculate time step
j i
ij
ij
j
j
ii
x
Wv
m
Dt
D
Numerical solution – Solving example
Search for neighbors
Calculate smoothing kernel
Calculate rate of density change
Calculate pressure
Calculate rate of momentum change
Update particles
Calculate time step
1
0,
2
0,
i
iii
i
cp
Numerical solution – Solving example
Search for neighbors
Calculate smoothing kernel
Calculate rate of density change
Calculate pressure
Calculate rate of momentum change
Update particles
Calculate time step
g
x
Wppm
Dt
Dv
j i
ij
ji
ji
ji
Numerical solution – Solving example
Search for neighbors
Calculate smoothing kernel
Calculate rate of density change
Calculate pressure
Calculate rate of momentum change
Update particles
Calculate time step
tDt
Dvvv
ti
titi
0
01
,
,,
tvxxtititi
0,01 ,,• Position:
• Velocity:
• Density: tDt
D ti
titi 0
01
,
,,
Numerical solution – Solving example
Search for neighbors
Calculate smoothing kernel
Calculate rate of density change
Calculate pressure
Calculate rate of momentum change
Update particles
Calculate time step
i
ii
c
ht min
The following general condition is required for our SPH solution:
• In order to prove the numerical accuracy of our SPH solution in the area of liquid sloshing, we performed two (water based) hydrodynamics benchmarks:
– Nonlinear wave evolution (2D)
– Tank sloshing (3D)
• Both benchmarks are based on real experiments.
Numerical solution – Verification
Nonlinear wave evolution – Load case
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0
0.5
1
1.5
2
Gate velocity
t [s]
v [m
/s]
Nonlinear wave evolution – Model
Tank sloshing – Load case
Tank sloshing – Model
• Number of particles: 108,540
• Height probes: Vertical rows of particles
• Pressure sensors: Finite elements of container
Tank sloshing – Results
0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
Water height @ H2Average error = 14.59 %
Radioss SPH
Experiment
t [s]
h [m
]
0 1 2 3 4 5 6
-1000
1000
3000
5000
7000
9000
Pressure @ P2Average error = 5.49 %
Radioss SPH
Experiment
t [s]
h [m
]
Animation
Simulation examples
Transport tank sloshing Lifeboat water entry
Fluid-structure interaction (FSI)
Oil and gas applications
Shut-off valves
Pumping systems Tank baffles
Chemical reactors
• To model the fluid-structure interaction between SPH based fluids and solid structures, we use two different coupling schemes: – SPH-FEM coupling scheme: A coupling
between SPH particles (for the fluid) and finite elements (for the solid structure).
– SPH2 coupling scheme: A fully SPH based coupling in which both the fluid and the solid structure are modeled with SPH.
• The type of coupling scheme to be used depends on the type of problem.
Numerical solution – Overview
The SPH-FEM coupling scheme has the following properties: • Master surfaces (solid surfaces) and slave
nodes (SPH fluid particles). • The interface gap determines if there’s any
contact between the SPH particle and the solid surface.
• Normal force computation depends on both the penetration distance and the rate of penetration:
• Tangential force computation depends on both the normal force and the tangential velocity:
Numerical solution – SPH-FEM scheme
t
pmkCpkF psssn
2
Var. Description
Fn Normal force
ks Interface spring stiffness
p Penetration
Cs Viscous damping coefficient on interface stiffness
mp Particle mass
k0 Initial interface spring stiffness
G Interface gap
Ft Tangential force
μ (Viscous) friction coefficient
Fa Adhesion force
Cf Viscous damping coefficient on interface friction
vt Tangential velocity vector
pG
Gkks 0
ant FFF ,min
pstfa mkCF 2v
• As for the SPH based coupling scheme, the coupling condition is automatically defined by the smoothing kernel.
• In order to model solids with SPH the entire stress tensor has to be taken into account, which leads to the following equation for the change of momentum:
• The stress tensor may be split into a normal component and a deviatoric component:
Numerical solution – SPH2 scheme
Var. Description
δ Dirac Delta function
τ Deviatoric stress tensor
j i
ij
ji
ji
ji
x
Wm
Dt
Dv
p
• In order to prove the numerical accuracy of the two coupling schemes and to show their field of application, we performed two different benchmarks:
– Dam break (2D)
– Water column impact (2D)
• Again, both benchmarks are based on real experiments.
Numerical solution – Verification
Dam break – Load case
Var. Value
H 0.14 m
L 0.079 m
A 0.1 m
s 5.0 mm
Rubber
ρ 1,100 kg/m3
E 6.0 MPa
ν 0.4 H
L
A
Elastic gate
s
Water
Dam break – Model
• 2D SPH2 model
• Particle count: 14,632
• Particle size: 1.0 mm
Dam break – Results
Animation
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0 0.1 0.2 0.3 0.4
De
fle
ctio
n [
m]
Deflection Average error: 2.63 %
Sim-X
Exp-X
Sim-Y
Exp-Y
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 0.1 0.2 0.3 0.4
Hei
ght
[m]
Time [s]
Water height Average error: 5.30 %
Sim
Exp
Water column impact – Load case
Var. Value
A 0.1 m
B 0.1 m
L 1.0 m
h 10.0 mm
s 15.0 mm
u 1.0 m/s
Rubber
ρ 1,100 kg/m3
E 12.0 MPa
ν 0.4
B
L
A
s
Water
h
u
Water column impact – Model
2D SPH2 model
Particle count 6,500
Particle size 2.0 mm
The purpose of this load case is to compare the SPH-FEM coupling scheme against the SPH2 coupling scheme, so we created two models:
2D SPH-FEM model
Particle count 2,500
Particle size 2.0 mm
Element count 4,000
Element size 2.0 mm
Water column impact – Results
Animation
SPH2
SPH-FEM
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 0.05 0.1 0.15
Dis
pla
cem
ent
[m]
t [s]
Displacement at center
SPH-SPH
SPH-FEM
Simulation examples
Oil tank leakage Container ship launch
• Liquid sloshing: – Applications: Flow problems deal with free-
surface flows and therefore with (nonlinear) sloshing motions.
– Numerical solution: Smoothed Particle Hydrodynamics (SPH)
• Fluid-structure interaction (FSI): – Applications: Free-surface flows that impact
solid structures and fluid flows that interact with soft (hyper-elastic) materials.
– Numerical solutions: SPH-FEM and SPH2 fluid-structure coupling scheme
Summary
Molslaan 111 2611 rk delft T +31 (0) 15 285 05 80 F +31 (0) 15 285 05 81