numerical simulation of liquid sloshing and fluid-structure interaction (fsi) inside closed volumes

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

Certification

Conceptual Design

Contra Expertise

Outsourcing

FEA/CFD Analyses

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

Nonlinear wave evolution – Results

Animation

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

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