MASTER'S THESIS - diva-portal.org

MASTER'S THESIS

Implementation of a Computational FluidDynamics Code for Propellant Sloshing

Analysis

Tiago Rebelo2013

Master of Science (120 credits)Space Engineering - Space Master

Luleå University of TechnologyDepartment of Computer Science, Electrical and Space Engineering

CRANFIELD UNIVERSITY

TIAGO ALEXANDRE RAMOS REBELO

IMPLEMENTATION OF A COMPUTATIONAL FLUID DYNAMICS

CODE FOR PROPELLANT SLOSHING ANALYSIS

SCHOOL OF ENGINEERING

MSc in Astronautics and Space Engineering

(SpaceMaster)

MSc Thesis

Academic Year: 2012 - 2013

CRANFIELD UNIVERSITY

SCHOOL OF ENGINEERING

MSc in Astronautics and Space Engineering

(SpaceMaster)

MSc Thesis

Academic Year 2012 - 2013

TIAGO ALEXANDRE RAMOS REBELO

Implementation of a Computational Fluid Dynamics Code for

Propellant Sloshing Analysis

Supervisors: Ph.D. Jennifer Kingston

M.Sc. Manuel Hahn

August 2013

This thesis is submitted in partial fulfilment (45%) of the requirements for the degree of

Master of Science in Astronautics and Space Engineering

© Cranfield University 2013. All rights reserved. No part of this publication may be reproduced without the

written permission of the copyright owner.

Implementation of a Computational FluidDynamics Code for Propellant Sloshing Analysis

MSc Thesis

Tiago Rebelo

Supported by:

Supervisors:

M.Sc. Manuel Hahn - EADS Astrium Satellites

Ph.D. Jennifer Kingston - Cranfield University

Ph.D. Johnny Ejemalm - Lulea University of Technology

August 2013

i

This M.Sc. thesis is dedicated to those whose work, sweat and tears

allowed me to reach this point...

...to my beloved Parents

...to my inspiring Grandparents

iii

“Se todo em cada coisa.

Poe quanto es

No mınimo que fazes.”

- Fernando Pessoa

“Be everything in each thing.

Put all of yourself

Into the slightest thing you do.”

- Fernando Pessoa

Abstract

Liquid propellant sloshing inside spacecraft tanks is of crucial importance to the

dynamics of the space vehicle. The interaction of the disturbance forces and torques,

caused by the moving fuel, with the solid body and the control system, might lead

to an increase in the AOCS actuators commands, which can degrade the vehicle’s

pointing performances and, in critical cases, generate unstable attitude and orbit

control. Thus, it is of major importance to accurately predict the behaviour of liquid

propellants sloshing inside spacecraft tanks.

This M.Sc. thesis is focused on this topic, being its major objective the implementa-

tion of a CFD software in an existing EADS Astrium simulation environment. The

integrated simulation environment is used to assess the influence of liquid propellant

sloshing for specific satellite missions.

From a defined set of requirements an open source CFD software based on FEM is

chosen - Elmer. The software is integrated and the final simulation environment is

evaluated for sloshing purposes using three different sloshing test cases.

The first two test cases deal with rectangular and cylindrical laterally excited tanks

where comparators are available - the results of the tests are validated against nu-

merical and experimental results.

The final test case is defined to reduce the gap between the simple test cases per-

formed to validate the software and the real sloshing problems faced in space vehicles.

A typical liquid propellant tank is selected and real mission conditions are simulated.

The liquid sloshing inside the laterally excited tank is deeply studied, being fully

characterized. The simulation environment is validated for the implemented liquid

sloshing problems.

vi

Acknowledgements

To start with, I would like to express my deepest gratitude to my supervisor at EADS

Astrium, Manuel Hahn. I am heartily thankful for the given opportunity, the guidance, the

encouragement and the constant support. This gratitude is extended to the AOCS/GNC

& Flight Dynamics department of Astrium Satellites, Friedrichshafen, Germany.

Special gratitude goes to my supervisor at Cranfield University, Jennifer Kingston. Her

support and help during the development of this work, but also during my stay in Cranfield,

are not forgotten.

At the Lulea University of Technology my gratitude goes to Victoria Barabash, for her

support in the many different challenges experienced during these 2 years. Also, for his

supervision during the development of this thesis, my gratitude to Johnny Ejemalm.

To Prof. Wolfgang A. Wall from the Institute for Computational Mechanics of the Technical

University of Munich for allowing me to develop my work at his institute, my gratitude.

Special acknowledgement goes to ESA’s directorate of Human Spaceflight and Operations,

for providing a real liquid propellant sloshing problem that brought challenge and value to

this thesis.

For his support and very useful inputs in all matters related with Elmer, my gratitude goes

to D.Sc. Peter Raback from the CSC - IT Center for Science, Finland.

My gratitude to all the entities that financially supported my M.Sc. studies, namely: ESA

Human Spaceflight and Operations directorate, through a study Scholarship; Erasmus

and Erasmus Mundus grants from Lulea University of Technology and the SpaceMaster

consortium; and last but not least the very important support of EADS Astrium during

my internships.

A special thanks goes to Anna Guerman, for giving me the opportunity to learn from her.

Without her I would never have found the beauties of space nor integrated this Master’s

programme.

For those who joined me in this incredible SpaceMaster journey, my deepest gratitude - it

would not have been the same without them.

Without any disregard to all the amazing people I met during these years abroad, my

special gratitude goes to Mauro Aja Prado, Ishan Basyal and Dries Agten, for their true

friendship.

To my family, for their unconditional love and support throughout my life, my deepest love

and gratitude. Special thanks to my parents, Joao and Maria, for providing the conditions

that allowed me to develop and aim higher; and to my sister Mara, for her support and

belief at all moments.

Finally, I want to thank Rita for her love throughout our common life. She gave me the

courage and support to take this programme to its end. Without her I would never have

made it, my unconditional love and gratitude goes to her.

viii

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Sloshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Lateral Sloshing . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Introduction to Damping . . . . . . . . . . . . . . . . . . . . . 11

2.1.3 Introduction to Non-linear Effects in Slosh . . . . . . . . . . . 12

2.1.4 Introduction to Micro-gravity Effects - Surface Tension . . . . 13

2.1.5 Other Types of Sloshing . . . . . . . . . . . . . . . . . . . . . 14

2.2 Computational Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Fluid Governing Equations . . . . . . . . . . . . . . . . . . . . 15

2.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.3 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.4 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.5 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.6 Solution Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

ix

Contents

3 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1 Functional Requirements . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 System Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 CFD Software Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.1 Selection Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.1.1 Available Codes . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.2 Satisfactory Codes . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.3 Top 3 Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.4 Final Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2.1 Available Codes . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2.2 Satisfactory Codes . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2.3 Top 3 Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2.4 Final Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Elmer - Open Source Finite Element Software . . . . . . . . . . . . 38

5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.2 Models / Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.3 Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.3.1 Graphical User Interface . . . . . . . . . . . . . . . . . . . . . 44

5.3.2 Command Line . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.4 Pre- and Post- Processing . . . . . . . . . . . . . . . . . . . . . . . . 47

5.4.1 Pre-Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.4.2 Post-Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6 Simulation Environment Setup . . . . . . . . . . . . . . . . . . . . . . 49

6.1 Simulation Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.2 Pre-Processing Methods . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.3 Post-Processing Methods . . . . . . . . . . . . . . . . . . . . . . . . . 51

7 Test case 1: Rectangular Tank . . . . . . . . . . . . . . . . . . . . . . 53

7.1 Test A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

7.1.1 Test Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 54

7.1.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.1.3 Results & Evaluation . . . . . . . . . . . . . . . . . . . . . . . 55

7.2 Test B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.2.1 Test Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.2.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 61


x

Contents

7.3 Test C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7.3.1 Test Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7.3.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 70


8 Test case 2: Cylindrical Tank . . . . . . . . . . . . . . . . . . . . . . . 78

8.1 Test Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

8.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

8.3 Results & Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

9 Test case 3: ESA Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

9.1 Test A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

9.1.1 Test Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 85

9.1.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 86


9.2 Test B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

9.2.1 Test Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 94

9.2.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 95


9.3 Test C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

9.3.1 Test Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 98

9.3.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 99


10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

11 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

A Test case 1 - Results: Test A . . . . . . . . . . . . . . . . . . . . . . . 115

B Test case 1 - Results: Test C . . . . . . . . . . . . . . . . . . . . . . . 122

C Test case 2 - Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

D Test case 3 - Results: Test A . . . . . . . . . . . . . . . . . . . . . . . 135

E Test case 3 - Results: Test B . . . . . . . . . . . . . . . . . . . . . . . 157

F Test case 3 - Results: Test C . . . . . . . . . . . . . . . . . . . . . . . 160

xi

List of Figures

2.1 Slosh wave shapes - first 2 antisymmetric x-modes for a rectangular

tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Slosh wave shapes - first 2 symmetric x-modes for a rectangular tank 8

2.3 Computational solution procedure process . . . . . . . . . . . . . . . 20

5.1 ElmerGUI main window . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.2 ElmerPost main window & graphics window . . . . . . . . . . . . . . 48

6.1 Software installation diagram . . . . . . . . . . . . . . . . . . . . . . 50

6.2 Simulation flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.3 Complete software installation diagram . . . . . . . . . . . . . . . . . 52

7.1 Rectangular tank - test A a): pressure at t = 0s . . . . . . . . . . . . 56

7.2 Rectangular sloshing tank - test A a): free surface shape evolution . . 57

7.3 Rectangular tank - test A a): CoG plots . . . . . . . . . . . . . . . . 58

7.4 Rectangular tank - test A a): sloshing amplitude plot . . . . . . . . . 59

7.5 Rectangular tank - test B a): pressure at t = 0s . . . . . . . . . . . . 63

7.6 Rectangular tank - test B b): pressure at t = 0s . . . . . . . . . . . . 63

7.7 Rectangular tank - test B a): CoG x-coord. Vs time . . . . . . . . . . 64

7.8 Rectangular tank - test B a): maximum wave amplitude (t = 3.48s) . 64

7.9 Rectangular tank - test B b): CoG plots . . . . . . . . . . . . . . . . 65

7.10 Rectangular tank - test B b): CoG x-coordinate Vs time . . . . . . . 65

7.11 Rectangular tank - test B b): sloshing amplitude plot . . . . . . . . . 66

7.12 Rectangular tank - test B b): PSD plot 1 . . . . . . . . . . . . . . . . 67

7.13 Rectangular tank - test B b): PSD plot 2 . . . . . . . . . . . . . . . . 67

7.14 Rectangular tank - test C: pressure at t = 0s . . . . . . . . . . . . . . 73

7.15 Rect. tank - test C - h = 0.050m longer dir.: free surface shape at

t = 0s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

7.16 Rect. tank - test C - h = 0.050m shorter dir.: free surface shape at

t = 0s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

7.17 Rect. tank - test C - h = 0.050m longer dir.: CoG x-coord. Vs time . 75

7.18 Rect. tank - test C - h = 0.050m shorter dir.: CoG x-coord. Vs time 75

7.19 Rect. tank - test C - h = 0.050m longer dir.: max. wave amplitude . 75

xii

List of Figures

7.20 Rect. tank - test C - h = 0.050m shorter dir.: max. wave amplitude . 76

8.1 Cylindrical tank - test: pressure at t = 0s . . . . . . . . . . . . . . . . 81

8.2 Cylindrical tank test - h = 0.050m: free surface shape at t = 0s . . . 81

8.3 Cylindrical tank test - h = 0.050m: CoG x-coordinate Vs time . . . . 82

8.4 Cylindrical tank test - h = 0.050m: max. wave amplitude . . . . . . . 82

9.1 ESA tank test A - MON-3: pressure at t = 0s . . . . . . . . . . . . . 89

9.2 ESA tank test A - MMH: pressure at t = 0s . . . . . . . . . . . . . . 89

9.3 ESA tank test A - MON-3 (50% fill ratio & fext = 0.70 Hz): CoG plots 90

9.4 ESA tank test A - MON-3 (50% fill ratio & fext = 0.70 Hz): sloshing

amplitude plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

9.5 ESA tank test A - MON-3 (50% fill ratio & fext = 0.70 Hz): PSD

plot 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92


plot 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92


plot 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93


plot 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

9.9 ESA tank test B - MON-3 or MMH: free surface shape at t = 0s . . . 96

9.10 ESA tank test B - MON-3: CoG x-coordinate Vs time . . . . . . . . 96

9.11 ESA tank test B - MON-3: max. wave amplitude . . . . . . . . . . . 97

9.12 ESA tank test B - MON-3: sloshing amplitude plot . . . . . . . . . . 97

9.13 ESA tank test C - MON-3 (60s simulation): CoG x-coord. Vs time . 100

9.14 ESA tank test C - MON-3 (60s simulation): wave amplitude at t = 30s100

9.15 ESA tank test C - MON-3 (60s simulation): wave amplitude at t = 45.5s101

9.16 ESA tank test C - MON-3 (60s simulation): sloshing amplitude plot . 101

9.17 ESA tank test C - MON-3 (60s simulation): CoG z-coord. Vs time . 102

9.18 ESA tank test C - MON-3 (20s simulation): wave amplitude at t = 0.54s103

9.19 ESA tank test C - MON-3 (20s simulation): wave amplitude at t =

19.60s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

9.20 ESA tank test C - MON-3 (20s simulation): CoG x-coord. Vs time . 104

9.21 ESA tank test C - MON-3 (20s simulation): CoG z-coord. Vs time . 105

A.1 Rectangular tank - test A b): pressure at t = 0s . . . . . . . . . . . . 116

A.2 Rectangular sloshing tank - test A b): free surface shape evolution . . 116

A.3 Rectangular tank - test A b): CoG plots . . . . . . . . . . . . . . . . 117

A.4 Rectangular tank - test A b): sloshing amplitude plot . . . . . . . . . 117

A.5 Rectangular tank - test A c): pressure at t = 0s . . . . . . . . . . . . 118

xiii

List of Figures

A.6 Rectangular sloshing tank - test A c): free surface shape evolution . . 118

A.7 Rectangular tank - test A c): CoG plots . . . . . . . . . . . . . . . . 119

A.8 Rectangular tank - test A c): sloshing amplitude plot . . . . . . . . . 119

A.9 Rectangular tank - test A d): pressure at t = 0s . . . . . . . . . . . . 120

A.10 Rectangular sloshing tank - test A d): free surface shape evolution . . 120

A.11 Rectangular tank - test A d): CoG plots . . . . . . . . . . . . . . . . 121

A.12 Rectangular tank - test A d): sloshing amplitude plot . . . . . . . . . 121

B.1 Rect. tank - test C - h = 0.100m longer dir.: free surface shape at

t = 0s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

B.2 Rect. tank - test C - h = 0.100m longer dir.: CoG x-coord. Vs time . 123

B.3 Rect. tank - test C - h = 0.100m longer dir.: max. wave amplitude . 123


t = 0s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124




t = 0s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125




t = 0s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126



B.13 Rect. tank - test C - h = 0.100m shorter dir.: free surface shape at

t = 0s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

B.14 Rect. tank - test C - h = 0.100m shorter dir.: CoG x-coord. Vs time 127

B.15 Rect. tank - test C - h = 0.100m shorter dir.: max. wave amplitude . 127


t = 0s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128




t = 0s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129




t = 0s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130


xiv

List of Figures


C.1 Cylindrical tank test - h = 0.100m: free surface shape at t = 0s . . . 132

C.2 Cylindrical tank test - h = 0.100m: CoG x-coordinate Vs time . . . . 132

C.3 Cylindrical tank test - h = 0.100m: max. wave amplitude . . . . . . . 132







D.1 ESA tank test A - MON-3 (25% fill ratio & fext = 0.70 Hz): CoG plots136


D.3 ESA tank test A - MON-3 (25% fill ratio & fext = 0.70 Hz): sloshing

amplitude plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137


amplitude plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

D.5 ESA tank test A - MON-3 (25% fill ratio & fext = 0.70 Hz): PSD

plot 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138


plot 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138


plot 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139


plot 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139



amplitude plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140




amplitude plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142


amplitude plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142


plot 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143


plot 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

xv

List of Figures


plot 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144


plot 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

D.19 ESA tank test A - MMH (25% fill ratio & fext = 0.70 Hz): CoG plots 145


D.21 ESA tank test A- MMH (25% fill ratio & fext = 0.70 Hz): sloshing

amplitude plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

D.22 ESA tank test A - MMH (25% fill ratio & fext = 1.50 Hz): sloshing

amplitude plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

D.23 ESA tank test A - MMH (25% fill ratio & fext = 0.70 Hz): PSD plot 1147







amplitude plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150


amplitude plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150








amplitude plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154


amplitude plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154





E.1 ESA tank test B - MMH (50% fill ratio): CoG x-coordinate Vs time . 158

E.2 ESA tank test B - MMH (50% fill ratio): max. wave amplitude . . . 158

E.3 ESA tank test B - MMH (50% fill ratio): sloshing amplitude plot . . 159

F.1 ESA tank test C - MON-3: PSD plot . . . . . . . . . . . . . . . . . . 161

xvi

List of Figures

F.2 ESA tank test C - MON-3 (20s simulation): sloshing amplitude plot . 161

F.3 ESA tank test C - MMH: PSD plot . . . . . . . . . . . . . . . . . . . 162

F.4 ESA tank test C - MMH (60s simulation): CoG x-coord. Vs time . . 162

F.5 ESA tank test C - MMH (60s simulation): wave amplitude at t = 30s 163

F.6 ESA tank test C - MMH (60s simulation): wave amplitude at t = 45.46s163

F.7 ESA tank test C - MMH (60s simulation): CoG z-coord. Vs time . . 164



F.10 ESA tank test C - MMH (20s simulation): CoG x-coord. Vs time . . 165

F.11 ESA tank test C - MMH (20s simulation): CoG z-coord. Vs time . . 166

F.12 ESA tank test C - MMH (60s simulation): sloshing amplitude plot . . 166

F.13 ESA tank test C - MMH (20s simulation): sloshing amplitude plot . . 166

xvii

List of Tables

4.1 Second phase: satisfactory codes - codes and characteristics. . . . . . 34

4.2 Third phase - codes and evaluated characteristics. . . . . . . . . . . . 35

7.1 Results - rectangular tank: test A. . . . . . . . . . . . . . . . . . . . 59

7.2 Comparison of results - rectangular tank: test A. . . . . . . . . . . . 60

7.3 Results - rectangular tank: test B. . . . . . . . . . . . . . . . . . . . . 68

7.4 Comparison of results - rectangular tank: test B. . . . . . . . . . . . 69

7.5 Results - rectangular tank: test C. . . . . . . . . . . . . . . . . . . . . 76

7.6 Comparison of results - rectangular tank: test C - 1 . . . . . . . . . . 76

7.7 Comparison of results - rectangular tank: test C - 2 . . . . . . . . . . 77

8.1 Results - cylindrical tank test. . . . . . . . . . . . . . . . . . . . . . . 83

8.2 Comparison of results - cylindrical tank test. . . . . . . . . . . . . . . 83

9.1 Results - ESA tank: test B. . . . . . . . . . . . . . . . . . . . . . . . 97

xviii

List of Abbreviations

AMG Algebraic Multigrid

AOCS Attitude and Orbit Control System

BDF Backward Differences Formulae

BEM Boundary Element Method

BiCGStab Biconjugate Gradient Stabilized

CFD Computational Fluid Dynamics

CG Conjugate Gradient

CGS Conjugate Gradient Squared

CoG Center of Gravity

ESA European Space Agency

FDM Finite Difference Method

FEM Finite Element Method

FVM Finite Volume Method

GCR Generalized Conjugate Residual

GMG Geometric Multigrid

GMRES Generalized Minimal Residual

GPL General Public License

GUI Graphical User Interface

HSO Human Spaceflight and Operations

ILU Incomplete LU

xix

List of Abbreviations

LAPACK Linear Algebra Package

LGPL Lesser General Public License

NTP Normal Temperature and Pressure

OS Operating System

PDE Partial Differential Equation

PSD Power Spectral Density

Sif Solver Input file

SUPG Streamline-Upwind Petrov-Galerkin

TFQMR Transpose-Free Quasi-Minimal Residual

UMFPACK Unsymmetric Multifrontal Sparse LU Factorization Package

xx

Chapter 1

Introduction

“Ever since there have been people, there have been explorers,

looking in places where others had not been before. Not everyone

does it, but we are part of a species where some members of the

species do, to the benefit of us all.”

- Neil deGrasse Tyson

Since the beginning of times humans have looked into the sky and wondered

at its beauties. For centuries we dreamt about leaving the Earth and going

further, beyond the sky, to achieve space flight. Due to the perseverance

and effort of some, this dream became a reality when the first artificial

satellite - Sputnik I - was launched in 1957 - at that point, a new era

began, the space age just started...

In less than 60 years of space exploration we landed humans on the Moon;

generated conditions to have humans orbiting the Earth on a permanent

basis; alighted several spacecraft in close planetary bodies (Venus, Mars

and Jupiter); studied extraterrestrial bodies; launched thousands of satell-

ites with numerous purposes to orbit the Earth; and are now about to break

another important barrier by flying a spacecraft into outer space - all these

important advances not only contributed to the scientific and technological

development of our society, but also made life on Earth much easier.

To achieve these breakthroughs, many questions had to be addressed and

many studies to be performed. For years, thousands of minds around the

globe worked and are still working to increase the capabilities of modern

space systems.

1

1. Introduction

The complexity level, now reached, allows us to deeply address some ques-

tions which have long been made and yet not fully answered, some of these

are the focus of this work:

• How are the dynamics of a space vehicle affected by the behaviour of

the liquid propellants inside its tanks?

• How accurately can the behaviour of the liquid propellants and their

interaction with the spacecraft be predicted?

Thanks to recent advances in science, engineering and technology, it is

now possible to develop deeper and further studies on this important topic

- liquid propellant sloshing inside spacecraft tanks. Even though, this has

been identified long ago as being of significant and sometimes even critical

influence on the dynamics of a spacecraft, it has not yet been completely

studied, mainly due to the difficulty that is predicting the liquids behaviour

inside the tanks.

When not carefully accounted for, the interaction of the disturbance forces

and torques caused by the moving fuel with the solid body and the control

system through the feedback loop can lead to an increase in the Atti-

tude and Orbit Control System (AOCS) actuators commands, which can

degrade the satellite pointing performances and in some critical cases ge-

nerate unstable attitude and orbit control.

This means that it is of major importance to accurately predict the be-

haviour of the liquid propellants inside the spacecraft tanks. This M.Sc.

thesis is focused on this topic, being its ultimate objective the “Imple-

mentation of a Computational Fluid Dynamics (CFD) Code for Propellant

Sloshing Analysis”.

1.1 Aim

The aim of this project is to implement in the existing EADS Astrium

simulation environment a CFD Code that shall be used to assess the in-

fluence of liquid propellant sloshing in specific satellite missions. Selected

propellant sloshing examples, defined by EADS Astrium and the European

Space Agency’s (ESA) directorate of Human Spaceflight and Operations

(HSO) (which also supported this project through a scholarship), shall be

incorporated, analysed and finally evaluated using the newly implemented

CFD code.

2

1. Introduction

The results of this work are planned to be used in the future by EADS

Astrium. The implemented CFD code is intended to become the favourite

sloshing analysis tool for the AOCS/GNC & Flight Dynamics Department

of Astrium Satellites (Friedrichshafen, Germany) where this work is being

developed under the supervision of M.Sc. Manuel Hahn.

1.2 Objectives

The following milestones were defined for this thesis project. Together they

define the general objectives that shall be accomplished during the project.

• Perform an extensive literature research to select the most suitable

CFD code based on defined requirements;

• Implement the selected CFD code in the existing Astrium’s pre- and

post- processing environment;

• Implement, analyse and evaluate selected propellant sloshing exam-

ples;

• Validate the simulation environment for the selected examples.

1.3 Outline

• Chapter 1 introduces the topic of this M.Sc. thesis, presents its aim

and the general objectives expected to be achieved during its devel-

opment.

• Chapter 2 presents a detailed but not exhaustive literature review

about the sloshing and CFD topics.

• Chapter 3 briefly introduces the requirements for the project.

• Chapter 4 presents the deep state-of-the-art investigation developed

to choose the most suitable CFD software for the purposes of this

project.

• Chapter 5 gives a brief introduction to the chosen CFD software -

Elmer.

3

1. Introduction

• Chapter 6 presents the setup of the complete simulation environment.

• Chapter 7 introduces, defines, explains the implementation, presents

the results and evaluates the tests performed for the rectangular tank

test case.

• Chapter 8 follows the same path of Chapter 7 and introduces the

cylindrical tank test case.

• Chapter 9 similarly to chapters 7 and 8 presents the cylindrical tank

with hemispherical domes (by ESA) test case.

• Chapter 10 presents the final conclusions drawn from this work.

• Chapter 11 attempts to explore future research lines and define work

that could be further developed.

4

Chapter 2

Literature Review

“The learning and knowledge that we have, is, at the most, but

little compared with that of which we are ignorant.”

- Plato

“He who receives ideas from me, receives instruction himself

without lessening mine; as he who lights his taper at mine

receives light without darkening me.”

- Thomas Jefferson

Before getting immersed in the project implementation, it is crucial to have

a general understanding of the relevant topics addressed in this Master’s

thesis. This chapter presents an explanation of the basic concepts related

with liquid slosh and CFD. It shall help the reader to get into the topic

without deeply getting into the overwhelming complexity of the concepts.

The subsequent sections provide an introduction to the liquid sloshing con-

cept - section 2.1, followed by a general overview of the theory behind CFD

- section 2.2.

2.1 Sloshing

As mentioned in chapter 1, liquid propellant sloshing in spacecraft tanks

can be of critical influence to the dynamics of the system, as well as to the

AOCS. The sloshing forces and torques imposed by the liquid motion in

the tank, together with the resulting shifts in the liquid’s center of gravity

(CoG), need to be carefully analysed.

5

2. Literature Review

In the following subsections a general introduction to the sloshing phe-

nomenon is presented. A simple analytical overview of lateral sloshing in

geometrically simple tanks containing ideal liquids in linear regime (small

wave amplitudes) is fully described. More complex sloshing problems -

which include also damping, non-linearity and micro-gravity effects, as well

as the derivation of the equations can be found in the literature, see [1]

and [2]. Even if not deeply described, the concepts of damping, non-linear

effects and micro-gravity effects are still briefly introduced later. However,

before getting deeper in the topic it is important to define the concept of

liquid sloshing inside rigid containers:

- “Any motion of a free liquid surface caused by any disturbance to a rigid

container partially filled with liquid.” [2]

2.1.1 Lateral Sloshing

Lateral sloshing is the simplest way of liquid sloshing inside containers. It

is defined as the formation of a standing wave on the surface of a liquid

when a tank partially filled is laterally excited. Under simplified conditions

the behaviour of the liquid can be defined by a set of equations, which

incorporate a set of liquid sloshing parameters.

The natural frequencies of the liquid, the velocity potential and the forces

and torques generated by its motion can be analytically obtained for sim-

ple tank geometries subjected to small external excitations under accel-

erated environments. Using as a basis the classical potential flow theory,

which involves treating the fluid as incompressible and inviscid, and solving

Laplace’s equation that satisfies the boundary conditions, these parameters

can be found.

In 1952 Graham and Rodriguez [3] introduced for the first time the 3- -

dimensional free surface natural frequencies of a liquid sloshing inside a

rectangular container. Later, in 1955 Housner [4] derived the analytical

solution for the first antisymmetric sloshing frequency of liquids sloshing

inside rectangular and cylindrical tanks. In 1966 Abramson [5] (republished

in 2000 by Dodge [1]) completely derived these parameters for several types

of tanks, having even introduced damping, non-linearity and micro-gravity

effects in its derivations.

6


In the following subsections, important sloshing parameters of ideal liquids

sloshing inside simple rigid containers are presented. The fluid is always

considered incompressible and the lateral excitation is considered much

smaller than the vertical acceleration acting on the tank, and therefore

negligible. For more details on the derivation of the equations please refer

to the above mentioned references.

Before presenting the above mentioned liquid sloshing parameters and re-

spective equations for both, rectangular and cylindrical tanks, let us first

introduce the concept of sloshing modes.

2.1.1.1 Sloshing Modes

The definition of sloshing modes results from the multiple configurations

to which the liquid’s surface may evolve when it sloshes inside a container.

Normally, this modes are defined by:

• n - the antisymmetric mode number;

• m - the symmetric mode number.

Therefore, in general, there are two main types of lateral sloshing: the

antisymmetric and the symmetric. Their modes are defined by the number

of wave peaks formed at the liquid’s surface, for example:

• for n = 1 there is a positive peak at one wall and a negative one at

the other;

• for m = 1 a positive peak occurs in the middle of the tank and two

negative ones appear in the walls.

The number of wave peaks increases together with the mode numbers n

and m, respectively for the antisymmetric and symmetric sloshing types. [1]

Antisymmetric Modes

The antisymmetric sloshing modes represent the most severe cases of liquid

slosh that can develop in spacecraft tanks. The slosh wave shapes for the

first two x-modes of a rectangular tank are presented in figure 2.1.

7


Figure 2.1: Slosh wave shapes - first 2 antisymmetric x-modes for a rectangulartank.

It is possible to visualize in figure 2.1 that the CoG shifts when the liquid

moves to provoke forces and torques that act on the tank shell. One im-

portant evidence is that the higher the sloshing mode number, the higher

the corresponding natural frequency and the smaller the CoG shift. Thus,

the smaller the generated disturbances and less significant the importance

of the higher sloshing modes.

Symmetric Modes

The symmetric sloshing modes are of less significance regarding the pro-

pellant sloshing disturbing effects in spacecraft tanks. Figure 2.2 presents

the slosh wave shapes for the first two symmetric x-modes in a rectangular

tank.

Figure 2.2: Slosh wave shapes - first 2 symmetric x-modes for a rectangular tank.

8


As it can be seen in figure 2.2, there is no lateral CoG shift in the liquid.

This makes the lateral forces and torques acting on the tank shell non-

-existent. As a side note, the frequencies of the symmetric modes are always

higher than those of the corresponding antisymmetric modes.

2.1.1.2 Rectangular Tank

Starting from the classic potential flow theory together with some necessary

assumptions and defined boundary conditions for this specific problem, the

natural frequencies of liquid’s sloshing inside rectangular containers can be

analytically derived. These depend on the height of the liquid inside the

tank, the tank shape and the vertical acceleration.

Antisymmetric modes only - 2D Natural Frequencies [5]

ω2n = π(2n− 1)

(ga

)tanh

[π(2n− 1)

(h

a

)](2.1)

Where:

ω: is the natural frequency

n: is the mode number

g: is the gravitational acceleration

a: is the width of the tank (in the x-direction)

h: is the height of the liquid inside the tank

Symmetric modes only - 2D Natural Frequencies [5]

ω2m = 2mπ

(ga

)tanh

[2mπ

(h

a

)](2.2)

Where:

m: is the mode number

9


Equations (2.1) and (2.2) present the natural frequencies for the anti-

symmetric and symmetric sloshing modes when the translational oscillation

of the tank occurs along the x-direction. If this oscillation occurs along the

y-direction, the equations are the same but, the width a is replaced by the

breadth b.

3D Free Surface Natural Frequencies [3]

ω2mn = gKmn tanh (Kmnh) (2.3)

Where:

Kmn = π√

(2m)2

a2 + (2n)2

b2

Equation (2.3) gives the natural frequencies of the modes which vary in

both x and y directions. The resulting wave shapes for this mode are a

combination of the 2D x- and y- mode shapes.

First Antisymmetric Sloshing Frequency - Liquid Water [4]

It is important to note that this equation was developed for inviscid fluids.

However, because in its development Housner applied simpler methods in

the resolution of the Partial Differential Equations (PDEs), the results

that can be obtained using this equation are slightly different from those

obtained by Abramson.

Recent publications [6], have used this formula to obtain the first antisym-

metric natural sloshing frequency of liquid water - according to them, the

results are slightly more accurate than those given by Abramson.

Therefore, for liquid water sloshing inside a rectangular tank vertically

accelerated and laterally excited, the first sloshing frequency is given by

equation (2.4):

ω21 = 2

√5

2

( gL

)tanh

(2

√5

2

(h

L

))(2.4)

Where:

L: is the length of the rectangular tank along the direction of excita-

tion

10


2.1.1.3 Cylindrical Tank

The equations previously presented for the rectangular tank case can also

be found in the literature for the cylindrical tank geometry [1]. However,

due to their increased complexity, they are not introduced here. Only the

first sloshing natural frequency for liquid water is presented below.

First Antisymmetric Sloshing Frequency - Liquid Water [4]

For liquid water inside a cylindrical tank vertically accelerated and laterally

excited, the first sloshing frequency is given by equation (2.5):

ω21 = 2

√27

8

( gD

)tanh

(2

√27

8

(h

D

))(2.5)

Where:

D: is the diameter of the cylindrical tank

2.1.2 Introduction to Damping

The analytic equations previously presented (subsection 2.1.1) for laterally

excited simple tanks do not consider the viscosity of the fluids. Meaning

that, damping effects are neglected. Thus, it is being considered that

the oscillation of the sloshing wave will continue over time even when the

external excitation is stopped. This is not representative of the real world,

where such thing does not happen.

For a viscous (non-ideal) fluid, damping will exist. Thus, once the external

excitation is stopped, the sloshing wave decreases in amplitude and will

also eventually stop.

Damping shall then, optimally, be considered, when performing sloshing

analysis. Nevertheless, because its consideration considerably increases the

complexity of the analytical equations previously exposed, CFD tools are

normally used to accurately replicate the effects of damping in sloshing.

As a side note, the main parameters that mostly influence damping are:

the viscosity of the sloshing liquid, the fill level, the shape and the tank

11


shell. Meaning that, to increase damping and reduce sloshing, not only

the properties of the liquid matter but also the tank geometry and its

properties. [1]

2.1.3 Introduction to Non-linear Effects in Slosh

It was already stated that the lateral sloshing examples analytically derived

and presented in subsection 2.1.1 include several simplifications, which

made the problem possible to be analytically solved. Among these are

the non-linear effects, which are also not considered. However, non-linear

effects always exist in liquid sloshing and therefore a brief introduction to

these shall be given.

For small sloshing amplitude waves, thus small external perturbations, non-

-linear effects are normally neglected - their effect is almost non-existent.

However, for large wave amplitudes and different forms of sloshing (such

as rotary sloshing), non-linear effects are crucial and dominate the sloshing

response. Thus, they shall also be modelled in CFD in order to accurately

predict the real behaviour of the liquid sloshing. This topic consists itself

in a complex field of studies, but a brief introduction can be found in [1, 2].

To ease the understanding, a simple manner of explaining the importance of

the non-linear effects in a flow is by means of the non-dimensional Reynolds

number (Re) - equation (2.6).

This very useful number gives a measure of the ratio of the inertial forces

by the viscous forces. Consequently, it quantifies the relative importance

of these types of forces for given flow conditions.

Re =ρvL

µ(2.6)

Where:

v: is the mean velocity of the object relative to the fluid

ρ: is the density of the fluid

L: is the characteristic length

µ: is the kinematic viscosity of the fluid

12


It is known that an increase in the Reynolds number leads to an increase in

the predominance of the non-linear effects in the fluid. Hence, looking at

equation (2.6) it is possible to see that the Reynolds number increases when

the mean velocity of the fluid increases or the viscosity decreases (keeping

the density constant). So, to avoid non-linear sloshing effects, the velocity

of the fluid shall be kept relatively small and the viscosity relatively high.

2.1.4 Introduction to Micro-gravity Effects - Surface Ten-

sion

Even though one of the simplifications assumed in the lateral sloshing ex-

amples of subsection 2.1.1 was that the liquid inside the spacecraft tank

was under accelerated conditions, the hypothesis of the liquid motion tak-

ing place in a micro-gravity environment exists and is very common in

space missions. Thus, it shall be briefly introduced.

In a micro-gravity environment, where the body forces become so small,

other small forces take place and are dominant in the behaviour of the

fluid. The most important of these are the surface tension forces at the

free surface of the liquid.

Surface tension (or capillary forces) is a complete field of studies by itself.

Thus, it would be totally out of the scope for this project to deeply intro-

duce this topic. Therefore, in this thesis development, it was considered

that the surface tension effects were not accountable for the defined test

cases. Thus, it was only important to understand what affects the predom-

inance of surface tension effects and at which point they become dominant

in the behaviour of the fluid. More details about this topic can also be

found in [1, 2].

To measure the predominance of the surface tension effects the Bond (or

Eotvos) number - equation (2.7), is used. This dimensionless number is

normally used as the most common comparison ratio for gravity and surface

tension forces.

Bo =ρaL2

γ(2.7)

Where:

ρ: is the density of the fluid

13


a: is the acceleration associated with the body force

L: is the characteristic length

γ: is the surface tension of the interface

Looking at equation (2.7) it is possible to understand that a high Bond

number represents a system relatively unaffected by the surface tension eff-

ects. Oppositely, a low number (typically less than one) indicates that the

surface tension effects dominate the fluid behaviour. Low Bond numbers

normally occur for fluids under micro-gravity conditions.

2.1.5 Other Types of Sloshing

Besides the lateral sloshing type here presented for 2 geometrically simple

tanks (subsection 2.1.1), there are several other types of sloshing. The most

interesting one is the rotary sloshing type which introduces predominant

non-linear effects in the fluid dynamics.

The study of this or other types of sloshing is out of the scope of this work.

Nevertheless, more information can be found in the literature, see [1] or [2].

2.2 Computational Fluid Dynamics

CFD is considered by many, the new, most interesting, branch of fluid dy-

namics. It acts together with the classical branches of pure experiment and

pure theory which are then supported and complemented by the cost-

-effective CFD tools. The role of CFD in engineering predictions has be-

come so strong that it has taken a permanent place in all the aspects of

fluid dynamics, from basic research to engineering design. [7]

CFD integrates disciplines such as fluid mechanics, mathematics and com-

puter science. The dynamics of a fluid can be characterized by mathemat-

ical equations (often called governing equations), which can then be solved

using numerical methods in sophisticated digital computers, by means of

computer programs or software packages. [8]

Since CFD became a trusted tool, the way in which engineering analy-

ses is performed has totally changed. The use of CFD substantially re-

duces lead times and costs in designs and productions compared to the

14


use of experimental approaches. Besides, it also offers the possibility to

solve complicated flow problems which could never be solved by analytical

means.

Nevertheless, one shall not blindly trust CFD results - they are only as

valid as the physical models incorporated in the governing equations and

boundary conditions. Hence, they are subject to various error sources

which can severely influence the accuracy of the results.

Numerical results shall always be thoroughly examined before believed to

be correct. Wonderful bright color pictures may provide a sense of reality

which might lead to mistaken conclusions. Therefore, at least, a basic

understanding of the theory behind CFD is needed to critically judge all

the results before trusting them.

2.2.1 Fluid Governing Equations

A fluid can be described by means of a set of mathematical equations

which represent its physical behaviour - the fluid governing equations. The

fundamental principles on which they are based are:

• Mass conservation - gives the continuity equation;

• Momentum conservation - gives the momentum equations (also known

as Navier-Stokes equations);

• Energy Conservation - gives the energy equation.

Various flow physics are governed by these fundamental principles which

might need to be applied together with some other modelling equations,

such as the turbulence ones.

The governing equations for a fluid flow general case in which the fluid is

considered non-turbulent, unsteady, 3-Dimensional, viscous, incompress-

ible and isothermal are presented here for reference - further in the devel-

opment of this work this is how the fluid is treated. More details, as well

as the derivation of these equations can be found in the literature, see for

example [7, 8, 9, 10] or [11] for a deeper explanation.

15


• Continuity equation:

∇ · (ρ~V ) = 0 (2.8)

• Momentum equations (or Navier-Stokes equations):

∇ · (ρu~V ) = −∂p∂x

+∂τxx∂x

+∂τyx∂y

+∂τzx∂z

+ ρfx (2.9)

∇ · (ρv~V ) = −∂p∂y

+∂τxy∂x

+∂τyy∂y

+∂τzy∂z

+ ρfy (2.10)

∇ · (ρw~V ) = −∂p∂z

+∂τxz∂x

+∂τyz∂y

+∂τzz∂z

+ ρfz (2.11)

Where:

~V = u~i+ v~j + w~k: is the vector velocity field

u = u(x, y, z, t): is the velocity component in the x-direction at time

t (unsteady flow)

v = v(x, y, z, t): is the velocity component in the y-direction at time t

(unsteady flow)

w = w(x, y, z, t): is the velocity component in the z-direction at time

t (unsteady flow)

ρ: is the density

τxx = λ∇ · ~V + 2µ∂u∂x

: is the the shear stress xx component

τyy = λ∇ · ~V + 2µ∂v∂y

: is the the shear stress yy component

τzz = λ∇ · ~V + 2µ∂w∂z

: is the the shear stress zz component

τxy = τyx = µ(

∂v∂x

+ ∂u∂y

): is the the shear stress xy or yx component

τxz = τzx = µ(∂u∂z

+ ∂w∂x

): is the the shear stress xz or zx component

τyz = τzy = µ(

∂w∂y

+ ∂v∂z

): is the the shear stress yz or zy component

µ: is the molecular viscosity

λ: is the bulk viscosity

16


p: is the pressure

fx, fy, fz: are the body forces per unit of mass acting on the fluid

element in x, y and z directions respectively and ρ~f · ~V = ρ(ufx +

vfy + wfz)

The above system of equations contains 4 equations in terms of 4 unknown

flow-field variables: the velocity field and the pressure.

2.2.2 Boundary Conditions

The mathematical equations previously presented govern the flow of a fluid.

Nevertheless, boundary conditions (and sometimes initial conditions) are

required to dictate the particular solutions to be obtained from the govern-

ing equations. There are 2 types of boundary conditions that, normally,

are defined for a fluid. Without getting into deep details (see [8] for more

information) these are:

• Dirichlet boundary conditions - give the boundary conditions re-

lated with velocity and pressure. The velocity components and/or the

pressure are defined at the boundary.

• Neuman boundary conditions - give the boundary conditions re-

lated with accelerations and forces. The acceleration components

and/or the force are defined at the boundary.

2.2.3 Meshing

A mesh is a discrete representation of the geometry of a problem which is

intended to be solved using CFD. The mesh designates the cells or elements

on which the flow is to be solved. Below an introduction to the different

types of mesh grids, as well as to the different mesh elements is presented.

2.2.3.1 Mesh Grids

In a simple manner, there are two types of meshes which are character-

ized by the connectivity of their points. Structured meshes (or grids) have

a regular connectivity, which means that each point of the mesh has the

17


same number of neighbours. The unstructured ones instead, have an irreg-

ular connectivity where each point can have a different number of neigh-

bours. [10]

Below, an introduction to these two types of grids is provided, the most

common types of mesh elements − tetrahedral and hexahedral, are also

discussed.

Structured Grid

In this type of grid, the points of an elemental cell can be easily addressed

by two indices (i, j) in two dimensions or three indices (i, j, k) in three

dimensions. The connectivity is very simple seeing that cells adjacent to

a given element face are identified by the indices and the cell edges form

continuous mesh lines that begin and end on opposite elemental faces. In

2D, the central cell is connected by four neighbouring cells. In 3D, the

central cell is connected by six neighbouring cells. This type of grids have

the advantage of allowing easy data management and connectivity, which

occurs in a regular fashion, making programming easy. [8]

Nevertheless, the disadvantage of adopting such a mesh, particularly for

more complex geometries is the increase in grid non-orthogonality or skew-

ness that can cause non-physical solutions due to the transformation of

the governing equations. The transformed equations that accommodate

the non-orthogonality act as the link between the structured coordinate

system and the body-fitted coordinate system containing additional terms

and thereby augmenting the cost of the numerical calculations and the dif-

ficulties in programming. Consequently, the use of such a mesh may also

affect the accuracy and efficiency of the numerical algorithm that is being

applied.

Unstructured Grid

Unstructured grids are currently the prevalent and widespread grid type in

many CFD applications. In these grids the cells are allowed to be assembled

freely within the computational domain. The connectivity information for

each face thus requires appropriate storage in some form of table. The most

typical shape of an unstructured element is a triangle in two dimensions

or a tetrahedron in three dimensions. However, any other elemental shape

including quadrilateral or hexahedral elements can also be used. [8]

18


2.2.3.2 Mesh Elements

Within the various mesh elements supported by the different grids, the

hexahedral and the tetrahedral elements are by far the most used ones.

Even though, for most of the geometries the output results shall not differ

even if using different element types, there are in fact advantages and

disadvantages of using one or another type of elements.

Grids made of tetrahedral elements are known for having the capability to

easily discretize any complex geometry in a fast and simple way, almost

with no user intervention. The use of hexahedral elements instead may

require some more effort to mesh complex geometries. However, grids

made of hexahedral elements do have a significant advantage as they have

the capability to preserve the accuracy in the wall normal direction even

for highly stretched viscous grids. Also, grids composed by these elements

have a reduced number of elements, edges and faces when compared to a

grid made of tetrahedral elements. [9]

2.2.4 Numerical Methods

Analytical solutions obtained from the flow governing equations can only

be found for simple and geometrically well defined problems. Obviously,

this is not the case for most of the real problems. For these problems, the

solution of the governing equations, in a general sense PDEs, can only be

approximated using numerical methods. The most used numerical meth-

ods in CFD are the Finite Element Method (FEM) and the Finite Volume

Method (FVM). A brief overview of these methods is presented here. How-

ever, no extensive nor detailed explanations are given as these can be easily

found in many standard books, see [12, 13].

The non inclusion of the Finite Difference Method (FDM) in this study is

mainly due to its many constrains which make them rarely used for CFD

applications. The most important of its constraints deals with the difficulty

to handle complex geometries − a high degree of mesh regularity is needed.

Due to this and other constraints, only a very small number of engineering

codes rely on this method. [8]

An overview of the process of the computational solution procedure using

FEM or FVM is shown in figure 2.3.

19


Figure 2.3: Computational solution procedure process.

2.2.4.1 Finite Element Method

The FEM is one of the available techniques to solve PDEs. The following

fundamental characteristics can be defined for this method (from [7]):

• The continuum field or domain is subdivided into cells, called elements,

which form a grid. The elements can have tetrahedral or hexahedral

forms and can be rectilinear or curved. The grid itself does not nec-

essarily need to be structured. Using unstructured grids and curved

cells, complex geometries can be handled with ease.

• The solution of the discrete problem is assumed a priori to have a

prescribed form. The solution has to belong to a function space, which

is built by varying function values in a given way, for instance linearly

or quadratically, between values in nodal points. The nodal points or

nodes, are typically points of the elements such as vertices, mid-side

points, mid-element points, etc. Due to this, the representation of the

20


solution is strongly linked with the geometric representation of the

domain.

• The solution of the PDEs itself is not what FEM looks for, instead a

solution of an integral form of the PDE is what is looked into.

• The discrete equations are constructed from contributions on the ele-

ment level which afterwards are assembled.

2.2.4.2 Finite Volume Method

The FVM is the most used technique in CFD. The following fundamental

characteristics can be defined for this method (from [7]):

• The integral form of the equations are discretized in this method in-

stead of the differential form.

• The flow field or domain is subdivided into a set of non-overlapping

cells that cover the whole domain.

• The conservation laws are applied to determine the flow variables in

some discrete points of the cells, the nodes. These nodes are typical

locations of the cells, such as cell-centers, cell-vertices or mid-sides.

There is considerable freedom in the choice of the cells and nodes.

Cells can be triangular, quadrilateral, etc. They can form structured

or unstructured grids.

• There is geometric flexibility in the choice of the grid and also in

defining the discrete flow variables.

2.2.4.3 Comparison

Both the FEM and FVM have the same geometrical flexibility. Neverthe-

less, the link between the representation of the solution and the geometric

representation of the domain is not as strong in FVM as in FEM. The

FVM tries to combine the best of the FEM, i.e. the geometric flexibility,

with the best of the FDM, i.e. the flexibility in defining the discrete flow

field. [7]

21


Some important advantages of FEM over FVM are the ease to obtain high

order accuracy results and also the ease to implement boundary condi-

tions. Nevertheless, the FVM is best suited for flow problems in primitive

variables, where the viscous terms are absent (Euler equations) or are not

dominant (high Reynolds number in Navier-Stokes equations). Further,

curved cell boundaries as used in FEM is difficult to implement in FVM.

FVM is mostly only second-order accurate.

Although, FVM seems to have disadvantages when compared to FEM,

specially in terms of accuracy, historically FVM have been mostly used

in CFD instead of FEM. This is so because FEM is originated from the

field of structural mechanics in which the partial differential formulation

of a problem can be replaced by an equivalent variational formulation -

the minimization of an energy integral over the domain. However, in fluid

dynamics, in general, a variational formulation is less obvious to formulate.

This makes it less obvious to formulate FEM for fluid dynamic purposes.

Hence, most breakthroughs in CFD have first been made in the context of

the FDM or FVM techniques and it has always taken some considerable

time (often more than a decade) to incorporate the same idea into the

FEM. However, in the other hand, once a suitable FEM formulation has

been found, the FEM is almost exclusively used. Obviously, this is due

to the mentioned advantages in the treatment of complex geometries and

obtaining high order accuracy.

Currently, for the simplest problems such as potential flows, both compress-

ible and incompressible Navier-Stokes flows at low Reynolds numbers, the

FEM is already fully grown. Although, new evolutions, specially for Navier-

Stokes problems, are still to come. Complex problems, like compressible

flows governed by Euler or Navier-Stokes equations or incompressible vis-

cous flows at high Reynolds numbers, still form an area of active research.

2.2.5 Numerical Analysis

In this section the methods to obtain the solution of a system of equations

are briefly introduced and explained. Direct, Iterative, Preconditioned and

Multigrid methods are concisely defined inside this chapter. If more details

are needed, please refer to [14].

22


Direct Methods

The so-called direct methods compute the solution of a problem in a finite

number of steps. The precise results of a system of equations would be ob-

tained using these methods, if they were performed with infinite arithmetic

precision. As this is not possible, in practice, finite precision is used and

the result is an approximation of the true solution, assuming that stability

exists.

Iterative Methods

Iterative methods, oppositely to direct methods, do not have an expected

number of steps to terminate. The resolution of the system is made by

starting with an initial guess, which then using iterative methods form

successive approximations that will eventually converge to the (acceptable)

exact solution (in the limit). A criteria to define whether a solution is

accurate enough needs to be specified. In these methods, even the use of

infinite arithmetic precision would not allow to reach an exact solution in

a finite number of steps.

Preconditioned Methods

A preconditioning is a procedure in which a transformation called the pre-

conditioner is applied for a given problem. This changes the problem to a

form that is more suitable to obtain numerical solutions. Normally, pre-

conditioning is related with the reduction of a condition number of the

problem. Iterative methods are usually used to solve preconditioned pro-

blems. This is so because the rate of convergence for most of the iterative

solvers increases as the condition number of a matrix decreases as a result

of the preconditioning.

Multigrid Methods

In numerical analysis multigrid methods are considered a specialized group

of algorithms that are used to solve differential equations using a hierarchy

of discretizations. The use of this methods accelerate the convergence of a

basic iterative method by means of a global correction from time to time.

23


This is accomplished by solving a coarse problem. This principle is similar

to the interpolation between coarser and finer grids.

Multigrid methods are among the fastest solution techniques known. In

contrast to other methods, these methods are general in a way that they

can treat arbitrary regions and arbitrary conditions. They do not depend

on the separability of the equations or other special properties. A direct

application of these can be used for more complicated, non-symmetric and

non-linear systems of equations, such as the Navier-Stokes equations.

2.2.6 Solution Analysis

The analysis of a computational solution (or approximate solution) repre-

sents one of the most important steps to be performed when using CFD.

As it is known, the system of algebraic equations is solved using numerical

methods which provide an approximate solution of the governing equa-

tions. Because the solution is approximate and not exact, as it would be

if analytical means were used, there are certain properties that one should

care when evaluating the results of a CFD simulation. Some of these prop-

erties are briefly introduced here − more details can be found in literature,

see [8].

Consistency

The consistency of a method concerns the discretization of the PDEs where

the approximation performed should diminish or become exact if the finite

quantities, such as the time step and the mesh spacing, tend to zero. Thus,

for a numerical method to be consistent, the truncation error − the differ-

ence between the result of the discretized equation and the exact one −must become zero when the time step and the mesh spacing tend to zero.

Stability

The existence of stability for a method is related with the growth or decay

of the errors introduced at any stage of the computation. A numerical

solution method is considered to be stable if it does not magnify the errors

that appear in the course of the numerical solution process.

24


For temporal problems the existence of stability guarantees that the method

yields a bounded solution whenever the exact solution is bounded. For it-

erative methods existing stability ensures the existence of a solution which

does not diverge.

Convergence

If a numerical method satisfies the consistency and stability properties,

generally the numerical method procedure is also convergent. Convergence

of a numerical process happens when the solution of the system of algebraic

equations approaches the true solution of the PDEs having the same initial

and boundary conditions as the refined grid system.

Accuracy

Because a converged solution does not necessarily mean an accurate one,

some possible sources of solution errors resulting from the numerical cal-

culations of the algebraic equations need to be considered. These require

attentive analysis and if they are to be minimized, systematic steps to per-

form numerical analysis, such as grid independence verification and vali-

dation of numerical methods, are necessary. Possible sources of solution

errors are: discretization errors, round-off errors, iteration or convergence

errors, physical-modelling errors and human errors.

Efficiency

The efficiency of a method is not exactly a property used to evaluate the

results of a CFD study. Nevertheless, if increased efficiency is required

to perform a certain study, this can for example be achieved by means of

parallel computing (e.g. using Message Passing Interface - MPI). Normally,

an increase in efficiency represents a less computationally exigent method

which will thus achieve a solution in less time.

25


2.3 Summary

The literature review presented in this chapter (chapter 2) allows a deeper

understanding of both the liquid sloshing concept and CFD. The following

conclusions can be summarized:

• Liquid propellant sloshing in spacecraft tanks can be of critical in-

fluence to the dynamics of the system, as well as to the AOCS. The

generated sloshing forces and torques imposed by the liquid motion

shall then be carefully analysed - section 2.1

• Lateral sloshing is the most common way of liquid sloshing. Its para-

meters can be analytical evaluated using a simplified analytical a-

pproach - subsection 2.1.1

• For non-ideal fluids damping exists and has an important role in the

liquid behaviour. The amplitude of the excited surface wave will de-

crease once the external excitation decreases - subsection 2.1.2

• For small external perturbations non-linear effects can be neglected.

However, for large perturbations or other sloshing types, such as rotary

sloshing, these effects dominate the sloshing response and shall be

accounted for - subsection 2.1.3

• In micro-gravity environments, where body forces are small, surface

tension forces are dominant at the free surface of the liquid. These

forces shall be carefully modelled for problems occurring in micro-

-gravity environments - subsection 2.1.4

• CFD is a very interesting branch of fluid dynamics, having an impor-

tant role in engineering predictions. It integrates disciplines of fluid

mechanics, mathematics and computer science - section 2.2

• CFD results are only as valid as the physical models incorporated

in the governing equations and boundary conditions. Hence, they

are subject to various error sources, which can severely influence the

accuracy of the results - section 2.2

• A set of mathematical equations - governing equations, can be used to

describe a fluid flow, representing its physical behaviour. The govern-

ing equations are based on the mass, the momentum and the energy

conservation principles - subsection 2.2.1

26


• Boundary conditions are required to dictate the particular solutions

to be obtained from the governing equations. There are two main

types of boundary conditions, the Dirichlet and the Neuman ones -

subsection 2.2.2

• A mesh represents a discretized geometry, designating the elements on

which the flow is to be solved. Hexahedral elements are preferred as

they allow increased accuracy for more complex problems and have a

reduced number of elements when compared to tetrahedral elements

- subsection 2.2.3

• The fluid flow governing equations can only be solved (for the vast

majority of the problems) using numerical methods, which find an

approximated solution to the problem. From the available numerical

methods, the FEM guarantees an ease to obtain high order accuracy

results and to handle complex geometries. However, it is more difficult

to formulate for fluid dynamics purposes and hence FVM have been

mostly used in CFD - subsection 2.2.4

• There are several solution methods used to solve a system of equa-

tions. These are normally included in the following categories: direct,

iterative, preconditioned or multigrid methods - subsection 2.2.5

• Analysing the solution of a computational method is a very important

step to be performed when using CFD. There are certain properties

which help identifying the performance of the method and the validity

of its solution - subsection 2.2.6

From the conclusions drawn in this literature review, the requirements for

this project were determined, the CFD software selection process was ruled

and the test cases to be validated with the software were defined.

In the following chapters the requirements are presented, followed by the

selection process that led to the choice of the CFD software, the software

characterization and later the implementation, evaluation and validation

of the defined test cases.

27

Chapter 3

Requirements

“Requirements are the What. Design is the How.”

- a System Engineer’s saying

“A factor present in every successful project and absent in every

unsuccessful project is sufficient attention to requirements.”

- Suzanne & James Robertson

In this chapter the requirements for the implementation of this project

are presented. These requirements were defined based on EADS Astrium’s

objectives for this M.Sc. thesis project, but also on some important con-

clusions taken from the literature review presented in chapter 2.

The requirements are divided in functional and system requirements. The

functional requirements include two types of requirements, the ones which

are intended to be accomplished during this project and the ones which

shall be considered for the CFD software choice, but do not necessarily

need to be fulfilled during the project.

3.1 Functional Requirements

• Basic success

1. The CFD software shall be able to perform sloshing analysis for

problems in which the tanks are subjected to translational accele-

rations;

28

3. Requirements

2. The CFD software shall support arbitrary tank geometries;

3. The CFD software shall be able to model real physical problems;

4. The CFD software shall consider viscosity effects in its models;

5. The CFD software shall include linear sloshing effects in its mod-

els (accurate for laminar Reynolds regimes);

6. The results of the CFD software shall be accurate in 1-g condi-

tions;

7. The results of the CFD software shall be validated for the follow-

ing test cases:

(a) Rectangular laterally excited tank;

(b) Cylindrical laterally excited tank;

(c) Cylindrical w/ hemispherical domes laterally excited tank.

• Considered for future developments

1. The CFD software shall be able to perform sloshing analysis for

problems in which the tanks are subjected to rotational accele-

rations;

2. The CFD software shall include non-linear sloshing effects (caused

by high external excitations) in its models;

3. The results of the CFD software shall be accurate for all gravity

conditions, including micro-gravity (surface tension effects dom-

inant in the liquid’s behaviour).

3.2 System Requirements

• Basic success

1. The CFD software shall be based on available open source soft-

ware;

2. The CFD software shall be based on FEM or FVM;

3. The CFD software shall be integrated with EADS Astrium’s pre-

and post- processing environments;

4. The CFD software shall interface with MATLAB R©;

5. The resulting simulation environment shall be used in any oper-

ating system (OS).

29

Chapter 4

CFD Software Selection

“The world’s an exciting place when you know CFD.”

- John Shadid

“Computational fluid dynamics has certainly come of age in

industrial applications and academia research. In the beginning

this popular field of study was primarily limited to high

technology engineering areas of aeronautics and astronautics, but

now it is a widely adopted methodology for solving complex

problems in many modern engineering fields.”

- Tu, J., Yeoh, G.H. & Liu, C.

This chapter presents the extended state-of-the-art study that was made

to find the most suitable CFD software for the purposes of this project. In

section 4.1 the selection process is introduced and later in section 4.2 the

results of this selection are presented.

4.1 Selection Process

The extended selection process to find the most suitable CFD software

was divided in several phases. In a first step, starting from the system

requirements (defined in 3.2), a list of available software which satisfied

them was comprised. In a second phase, important characteristics of the

software were defined and a selection of the solvers, based on these, was

made. The third phase of this selection comprised the choice of the top 3

solvers based on a narrower set of characteristics. Finally, in a last step,

30

4. CFD Software Selection

the final selection was made based on a compromise solution that gave

guarantees of success.

The following subsections deeply define each step that was taken in the

selection process. The characteristics of the software that were looked

into, as well as the chosen approach are presented.

4.1.1 Available Codes

To successfully pass this phase the software had to satisfy the following

criteria, extrapolated from the project’s requirements:

• Be an open source software;

• Be a CFD solver.

4.1.2 Satisfactory Codes

The characteristics which were defined as crucial to be satisfied for the

solvers to pass this phase were:

• High level of maturity;

• Use of known programming languages;

• Licensed as GNU GPL or LGPL;

• Functionalities in-line with the defined requirements;

• Existing interfaces for pre- and post- processing;

• Sufficient support documentation.

4.1.3 Top 3 Codes

The narrower code characteristics that were studied in this third phase

were:

• FEM or FVM based;

31


• Mesh elements type (support for at least hexahedral and/or tetrahe-

dral);

• Parallelization supported;

• Surface tension modelled;

• Already used for liquid sloshing purposes.

4.1.4 Final Selection

From the top 3 codes a final selection was performed. At this point, the

codes were analysed individually and the main advantages and disadvan-

tages of each one were defined. A compromise solution was then made.

4.2 Results

The results of the selection process previously introduced (section 4.1) are

presented in the following subsections.

4.2.1 Available Codes

The following list of software passed the first phase:

• ADFC [15]

• arb [16]

• CFD2D [17]

• CFD2k [18]

• cfdpack [19]

• Channelflow [20]

• CLAWPACK [21]

• Code Saturne [22]

• COOLFluiD [23]

• DUNS [24]

• Dolfyn [25]

• Edge [26]

• Elmer [27]

• Featflow [28]

• FEniCS Project [29]

• freeFEM++ [30]

• Fluidity [31]

• HiFlow3 [32]

32


• Gerris Flow Solver [33]

• hit3d [34]

• iNavier [35]

• ISAAC [36]

• Kicksey-Winsey [37]

• MFIX [38]

• NaSt2D-2.0 [39]

• NEK5000 [40]

• NUWTUN [41]

• OpenFlower [42]

• OpenFOAM [43]

• OpenLB [44]

• OpenFVM [45]

• PartenovCFD [46]

• PETSc-FEM [47]

• SLFCFD [48]

• SSIIM [49]

• SU2 [50]

• Tochnog [51]

• TYCHO [52]

• Typhon solver [53]

4.2.2 Satisfactory Codes

Table 4.1 presents the CFD solvers that satisfied the conditions established

for the second phase of the selection process. The former EADS Astrium

tool used to analyse sloshing - SLOSHC [54], as well as the most used com-

mercial software for this purpose - FLOW-3D [55], are also characterised.

It is important to note that a set of functionalities was defined and the

solver could only pass this phase if these were satisfied. Besides any other

functionalities that the code might have, at least these had to exist:

• Compressible flows model;

• Incompressible flows model;

• Multiphase flows model (at least two-phase flows);

• Turbulent flows model;

• Fluid structure interaction capabilities (even if by coupling with an-

other software).

33


Tab

le4.

1:Sec

ond

phas

e:sa

tisf

acto

ryco

des

-co

des

and

char

acte

rist

ics.

Maturity

Lic

ense

Program

min

gLanguages

Mesh

input

form

at

Post-

Processin

gSupport

Docum

entatio

n

Code

Satu

rne

Develo

pm

ent

start

ed:

1997

Main

develo

per:

ED

FO

pen

sourc

ere

lease

d:

2007

GP

LFort

ran

C Pyth

on

scri

pts

I-deas

univ

ers

al

Gm

shP

ara

Vie

w

Inst

allati

on

Guid

eT

heory

Guid

eP

racti

cal

Use

r’s

Guid

eT

uto

rials

Cours

em

ate

rials

Elm

er

Develo

pm

ent

start

ed:

1995

Main

develo

per:

CSC

-IT

Cente

rfo

rScie

nce

Op

en

sourc

ere

lease

d:

2005

GP

LFort

ran

(Solv

er)

C/C

++

(GU

I)

AN

SY

SG

msh

Abaqus

I-deas

univ

ers

al

Para

vie

wG

nuplo

tO

cta

ve

Matl

ab

Models

,Solv

ers

,P

ara

mete

rs,

Gri

d,

GU

Iand

MA

TC

Manuals

Tuto

rials

Cours

em

ate

rials

FE

niC

SD

evelo

pm

ent

start

ed:

2003

Main

develo

per:

Sim

ula

Rese

arc

hL

ab

ora

tory

Op

en

sourc

ere

lease

d:

2003

LG

PL

C+

+P

yth

on

UF

L

Gm

shA

baqus

Exodus

Para

Vie

wB

ook

Tuto

rials

Cours

em

ate

rials

Flu

idit

yD

evelo

pm

ent

start

ed:

1990s

Main

develo

per:

Imp

eri

al

College

London

Op

en

sourc

ere

lease

d:

No

info

LG

PL

C/C

++

Pyth

on

Fort

ran

Gm

shE

xodus

Para

Vie

w

Manual

Use

r’s

Guid

eD

evelo

per’

sG

uid

eC

ours

em

ate

rials

Op

enF

OA

MD

evelo

pm

ent

start

ed:

1980s

Main

develo

per:

Op

enF

OA

Mfo

undati

on

Op

en

sourc

ere

lease

d:

2004

GP

LC

++

AN

SY

SF

luent

Gm

shI-

deas

univ

ers

al

Para

Vie

wF

luent

Tecplo

t3D

Use

rG

uid

eC

ours

em

ate

rials

Tuto

rials

SU

2D

evelo

pm

ent

start

ed:

2012

Main

develo

per:

Sta

nfo

rdU

niv

ers

ity

Op

en

sourc

ere

lease

d:

2012

GP

LC

++

Pyth

on

scri

pts

CG

NS

(data

standard

)P

ara

Vie

wT

ecplo

t

Use

r’s

Guid

eD

evelo

per’

sG

uid

eT

uto

rials

Cours

em

ate

rials

Ast

rium

SL

OSH

C

Develo

pm

ent

start

ed:

1970s

Main

develo

per:

Ast

rium

Op

en

sourc

ere

lease

d:

N/A

Ast

rium

’sin

-house

use

only

Fort

ran

None

None

(Outp

ut

as

text

file

)

Use

rM

anual

and

Desc

ripti

on

of

Alg

ori

thm

s

FL

OW

-3D

Develo

pm

ent

start

ed:

1980

Main

develo

per:

Flo

wScie

nce,

Inc.

Op

en

sourc

ere

lease

d:

N/A

Com

merc

ial

Copyri

ght

N/A

No

info

EnSig

ht

Tecplo

t

Book

Use

rM

anual

Use

r’s

technic

al

supp

ort

34


In terms of functionalities FLOW-3D states to feature all the above. As

for the former SLOSHC tool, it can only compute the parameters of an

equivalent mechanical model for lateral sloshing effects in axially symmetric

containers accelerated in the direction of the symmetry axis (being this only

valid for non-rotating satellites and ideal fluids).

4.2.3 Top 3 Codes

The various CFD software selected in the second phase were once again

evaluated. Table 4.2 presents the evaluation performed for all the selected

codes.

Table 4.2: Third phase - codes and evaluated characteristics.NumericalMethods

Mesh ElementsType

SupportedParallelization

Surface TensionModelled

Used for SloshingPurposes

Code Saturne FVMHexahedralTetrahedral

Yes No infoNone found

Elmer FEMHexahedralTetrahedral

Yes YesSimilar purposes

FEniCS FEM Tetrahedral Yes No infoNone found

Fluidity FVMHexahedralTetrahedral


OpenFOAM FVMHexahedralTetrahedral

Yes YesYes

SU2 FVMHexahedralTetrahedral


AstriumSLOSHC

FEMRectangularTriangular

N/A NoYes

FLOW-3D FEMFVM

HexahedralTetrahedral

Yes YesYes

As it is possible to observe in table 4.2, the choice of the top 3 codes is

not clear as they all have pros and cons. Nevertheless, it was decided

that the FEM based solvers were preferred due to the reasons already

stated in chapter 2.2.4. Thus, from the available FEM based methods,

Elmer and FEniCS were considered the most complete ones. Even though

OpenFOAM is a FVM based solver, it has already been used for liquid

sloshing analysis purposes and thus it was also considered for the top 3

codes.

4.2.4 Final Selection

In the last step of the selection process, the top 3 codes were individually

evaluated and a set of advantages and disadvantages was defined for each

one.

35


Elmer:

• Advantages:

- FEM based;

- Supports hexahedral mesh elements;

- Surface tension modelled;

- Very accurate for low Reynolds;

- On going development;

- Used for similar purposes;

- Support from developers.

• Disadvantages:

- Programming language: Fortran;

- Not so good for high Reynolds.

FEniCS:

• Advantages:

- FEM based;

- Programming language:

C/C++, Python, UFL;


• Disadvantages:

- Does not support hexahedral mesh

elements;

- Surface tension not modelled;

- Never used for similar purposes.

OpenFOAM:

• Advantages:

- Supports hexahedral mesh elements;

- Surface tension modelled;

- Programming language: C++;

- Used for liquid sloshing analysis;


• Disadvantages:

- FVM based;

- Not so good for low Reynolds.

The above mentioned advantages and disadvantages for each one of the

codes do not necessarily had the same weight in the final choice. Therefore,

the selection had to be based on a compromise solution that would allow

the choice of the most complete and suitable software for the purposes of

the project.

36


Based on the extensive literature review presented in chapter 2, it was

defined that a software based on FEM had to be used. The advantages are

clear, it allows accurater results for the tests planned to be implemented

(low Reynolds), as well as it has a bigger margin for future developments.

It was also defined that the software should support both types of mesh

elements − tetrahedral and hexahedral elements.

Concluding, Elmer was considered the most suitable software to be used for

the purposes of this project, the tests to be implemented and the expected

future developments. Elmer is based on FEM, supports tetrahedral and

hexahedral mesh element types, was already used for similar purposes, shall

give accurate results for the regime in which the fluid was intended to be

treated and it is also supported by a motivated and helpful development

team.

37

Chapter 5

Elmer - Open Source Finite

Element Software

“When it comes to software, I much prefer free software, because

I have very seldom seen a program that has worked well enough

for my needs, and having sources available can be a life-saver.”

- Linus Torvalds

Elmer - Open Source Finite Element Software for Multiphysical Problems

was the chosen software to be used in the implementation of this project.

This chapter presents a brief introduction of Elmer, its functionalities,

models, solvers, interfaces, pre- and post- processing tools.

5.1 Overview

Elmer is an open source finite element software package for multiphysical

problems. Basically, it is a software package used to solve partial differential

equations. Elmer can deal with a great number of different equations, which

can be coupled in a generic manner, making Elmer a versatile tool. Being

an open source software, Elmer gives the user the possibility to modify the

existing solution procedures and thus develop new solvers for equations of

its own interest. [56]

Elmer’s development started in 1995 as part of a national CFD technology

program funded by the Finnish Funding Agency for Technology and Inno-

vation. The initial 5 years project included several partners being CSC -

38

5. Elmer - Open Source Finite Element Software

IT Center for Science [57] the main developer and the one that after this

initial period kept the project under development. In 2005, the software

package was finally released as an open source. Since then, the user com-

munity widened and the number of international users grew together with

the software.

Being a multiphysical problems software package, Elmer contains solvers

for a variety of mathematical models. The following list summarizes Elmer’s

capabilities and integrated physical models for some of the main specialized

fields (from [56]):

• Heat transfer: models for conduction, radiation and phase change;

• Fluid flow: the Navier-Stokes, Stokes and Reynolds equations, k-η

model;

• Species transport: generic convection-diffusion equation;

• Elasticity: general elasticity equations, dimensionally reduced models

for plates and shells;

• Acoustics: the Helmholtz equation, linearized Navier-Stokes equations

in the frequency domain and large amplitude wave motion of an ideal

gas;

• Electromagnetism: electrostatics, magnetostatics, induction;

• Microfluidics: slip conditions, the Poisson-Boltzmann equation;

• Levelset method: Eulerian free boundary problems;

• Quantum Mechanics: density functional theory (Kohn-Sham).

In terms of numerical methods used for approximation and linear systems

solutions, Elmer includes a great number of possibilities. The list below

summarizes the most important ones (from [56]):

• All basic element shapes in 1D, 2D and 3D with the Lagrange shape

functions of degree k ≤ 2;

• Higher degree approximation using p-elements;

• Time integration schemes for the first and second order equations;

39


• Solution methods for eigenvalue problems;

• Direct linear system solvers;

• Iterative Krylov subspace solvers for linear systems;

• Multigrid solvers for some basic equations;

• ILU preconditioning of linear systems;

• Parallelization of iterative methods;

• The discontinuous Galerkin method;

• Stabilized finite element formulations, including the methods of resi-

dual free bubbles and Streamline-Upwind Petrov-Galerkin (SUPG);

• Adaptivity, particularly in 2D;

• Boundary Element Method (BEM) solvers (without multipole accel-

eration).

As most of the CFD software packages, Elmer is composed of three main

parts: the pre-processor, the solver and the post-processor. These are

separate executables that can also be used independently. Thus, the main

executables included in Elmer’s package are (more details [56]):

• ElmerGUI - graphical user interface (GUI) for Elmer;

• ElmerGrid - provides functionalities for the generation of simple

meshes and conversion of accepted file formats to the native format;

• ElmerSolver - main part of Elmer, the solver;

• ElmerPost - simple GUI post-processor.

Each one of the main parts of Elmer has its own characteristics, which could

be thoroughly explored. Nevertheless, the idea is only to provide the reader

with a general understanding of the software. Thus, the following sections

introduce Elmer’s Models, Solvers, Interfaces, Pre- and Post- processing

main characteristics. Hopefully, by the end of this chapter the reader will

have a general idea about Elmer.

40


5.2 Models / Solvers

Elmer is a multiphysical problems CFD software package solver composed

of several different modules or solvers. Being a very complete software,

a description of all its solvers is totally out of the scope for this M.Sc.

thesis. Thus, only the important solvers for the work to be performed are

briefly introduced below. More information about other solvers included

in Elmer, as well as more details on those introduced here, can be found

in its manuals [58].

• FlowSolve - solves the Navier-Stokes equations (already introduced

in chapter 2.2.1);

• MeshSolve - moves the current mesh nodes so that the mesh remains

intact when a boundary is moved. It updates the mesh after each time

step;

• FreeSurfaceSolver - allows the specification of a boundary as a free

surface, which can then be solved in combination with the Navier-

-Stokes equations (FlowSolve) and the mesh update solver (Mesh-

Solve).

Elmer includes several solution methods to solve linear and non-linear sys-

tems. These are briefly introduced here together with the time discretiza-

tion strategies included in Elmer. For completeness and extra information,

please refer to [59].

Methods for linear systems

For linear systems, the solution methods included in Elmer fall into two

large categories: direct methods and iterative methods (already briefly

introduced in chapter 2.2.5).

Direct methods

Elmer offers two possibilities to use direct methods:

• The Linear Algebra Package (LAPACK) collection of subroutines [60];

41


• The Unsymmetric Multifrontal Sparse LU Factorization Package (UMF-

PACK) set of routines [61].

Iterative methods

The iterative methods available in Elmer can be divided in two main cat-

egories, the preconditioned Krylov subspace methods and the multilevel

methods.

Preconditioned Krylov methods

Elmer’s solver includes the following set of Krylov subspace methods:

• Conjugate Gradient (CG);

• Conjugate Gradient Squared (CGS);

• Biconjugate Gradient Stabilized (BiCGStab);

• BiCGStab(l);

• Transpose-Free Quasi-Minimal Residual (TFQMR);

• Generalized Minimal Residual (GMRES);

• Generalized Conjugate Residual (GCR).

A deeper explanation on the integration of these methods in Elmer can

be found in [59]. For a detailed explanation on some of these methods

see [62, 63].

In terms of preconditioning strategies (already introduced in chapter 2.2.5),

Elmer includes some basic strategies (see [59] for details):

• ILU(N) preconditioners;

• ILUT preconditioners;

• Preconditioning by multilevel methods;

• Block pPreconditioning.

42


Multilevel methods

Even though multilevel methods (introduced in chapter 2.2.5) can be ap-

plied to define preconditioners for the Krylov subspace methods, they are

iteration methods on their own. In Elmer’s solver, two different multilevel

method approaches are available (for details see [59]):

• Geometric Multigrid (GMG);

• Algebraic Multigrid (AMG).

Methods for non-linear systems

The non-linearity of a system might be intrinsically related with the charac-

teristics of the equations to be solved, but it can also result from non-linear

material parameters that depend on the solution. Thus, in Elmer, the ap-

proach used in the linearization of non-linear systems changes from one

solver to another. More details on these strategies can be found in [58, 59].

As an example, for the Navier-Stokes solver there are two different methods

included in Elmer:

• Picard linearization;

• Newton linearization.

Time discretization strategies

The integration of time dependent systems can be performed in Elmer

using one of the following methods (see [59] for more details):

• Crank-Nicolson method;

• Backward Differences Formulae (BDF) of several orders.

43


5.3 Interfaces

As most open source CFD codes, Elmer can be used directly from the

command line by calling the solver executable - ElmerSolver. The pre- and

post- processing executables, respectively, ElmerGrid and ElmerPost, can

also be called from the command line. Nevertheless, oppositely to most

open source CFD software, Elmer also includes a modern programmable

graphical user interface - ElmerGUI.

In this section, a brief description of Elmer’s GUI capabilities is presented.

The basic concepts about how to run Elmer from the command line are

also introduced.

5.3.1 Graphical User Interface

ElmerGUI is a very complete program, capable of performing almost all

the tasks that can be performed when running Elmer from the command

line. The GUI is capable of importing finite element mesh files in several

formats, generate finite element partitionings for various geometry input

files, setup systems of PDEs to be solved and export model data and results

for ElmerSolver and ElmerPost to solve and post-process, respectively. [64]

Figure 5.1: ElmerGUI main window.

44


Figure 5.1 shows Elmer’s GUI main window. Several main menus exist in

this window, each one having its purposes. Below an introduction to these

menus (see [64] for more details):

• File - allows the user to load a saved project or to start a new one by

loading a mesh file. The GUI’s definitions and the save buttons are

also located in this menu;

• Mesh - allocates the mesh configuration buttons;

• Model - menu that allows the user to stipulate the model defini-

tions. The Setup, Equation, Material, Body force, Initial condition

and Boundary condition sub-menus are located here. Defining the pa-

rameters located inside each one of these sub-menus defines the model

to be simulated;

• View - allows the user to set view preferences;

• Sif - allows the generation of the Solver Input file (Sif) based on the

Model defined properties. The user can also manually edit the Sif;

• Run - used to start the solver or the post-processor (ElmerPost);

• Help - help menu.

5.3.2 Command Line

Elmer can also be run directly from the command line. To do this, a test

folder needs to be created and at least the following files need to exist:

• Mesh file;

• Solver input file.

The mesh can be generated using ElmerGrid or it can also be converted

from one of Elmer’s mesh accepted input files to the native Elmer mesh file

using ElmerGrid. More details about ElmerGrid are provided in section 5.4

together with Elmer’s post-processing - ElmerPost.

The .Sif file provides the solver with a precise description of the problem.

It contains user-prepared input data that specifies the location of the mesh

45


files, controls the selection of the physical models and defines the material

parameters, the boundary conditions, the initial conditions, the solver’s

stopping tolerances, etc. The file is organized into different sections which

can form the following general structure:

• Header

• Simulation

• Constants

• Body

• Material

• Body Force

• Equation

• Solver

• Boundary Condition

• Initial Condition

All the required model parameters are defined in the correspondent section

of the Sif. For details on each section, please refer to [59]. To note that,

as stated before, a mesh file representing the geometry of the problem is

also required to completely define the problem, this shall be called inside

the Sif header section.

To start ElmerSolver from the command line, the executable as to be called

together with the .Sif file:

$ ElmerSolver Test.sif

Issuing this command runs the solver and saves the results in the chosen

directory. The results can then be visualized and evaluated using Elmer’s

post-processing.

46


5.4 Pre- and Post- Processing

As stated before, Elmer includes its own pre- and post- processing tools −ElmerGrid and ElmerPost. In this section a brief introduction to these is

presented.

5.4.1 Pre-Processing

ElmerGrid is one of the executables included in Elmer’s software package.

It is responsible for the pre-processing, consisting of a simple mesh gener-

ator and mesh manipulation utility. It has the capability to read meshes

generated by other programs and manipulate and convert them to a for-

mat accepted by ElmerSolver. The following mesh formats are accepted

by ElmerGrid:

• .grd : ElmerGrid file format;

• .mesh.* : Elmer input format;

• .ep : Elmer output format;

• .ansys : Ansys input format

• .inp : Abaqus input format by Ideas;

• .fil : Abaqus output format;

• .FDNEUT : Gambit (Fidap) neutral file;

• .unv : Universal mesh file format;

• .mphtxt : Comsol Multiphysics mesh format;

• .dat : Fieldview format;

• .node,.ele: Triangle 2D mesh format;

• .mesh : Medit mesh format;

• .msh : GID mesh format;

• .msh : Gmsh mesh format;

• .ep.i : Partitioned ElmerPost format.

47


These formats can be converted using ElmerGrid to Elmer’s native mesh

format: .mesh.* .

For more details about ElmerGrid, its capabilities and how to use it, please

refer to its manual [65].

5.4.2 Post-Processing

ElmerPost is Elmer’s included post-processing executable. It has the ca-

pability to read the results output by Elmer in the .ep format. Using

this tool, the results can be visualized and evaluated. Figure 5.2 presents

ElmerPost’s main window and graphics window.

Figure 5.2: ElmerPost main window & graphics window.

Even though this simple post-processing tool is enough to evaluate the

simple test cases, other more advanced post-processing tools can also be

used. Therefore, it is important to note that the results of Elmer’s solver

can be saved in several formats. Currently, the supported output formats

include GiD, Gmsh, VTK legacy, XML coded VTK file bearing the suffix

VTU and Open DX. [58]

48

Chapter 6

Simulation Environment Setup

“A simulation is a concrete abstraction of the relevant features of

some real world problem.”

- Unknown

This chapter presents the complete simulation environment including a

description of the simulation flow and the used pre- and post- processing

tools.

As defined in the system requirements for this project (chapter 3.2), the

CFD software had to be integrated with EADS Astrium’s pre- and post-

processing tools, had to interface with MATLAB R© and the resulting sim-

ulation environment had to be used in any operating system.

In the subsequent sections, the final simulation environment which satisfies

the defined requirements is presented. The final software package includes

the CFD solver - Elmer, as the main part, but it also integrates specific

pre- and post- processing tools. First the simulation flow is introduced and

later the pre- and post- processing methods are described.

6.1 Simulation Flow

To satisfy this project’s system requirements (chapter 3.2), a virtual ma-

chine player - VMware R© PlayerTM

, compatible with several operating sys-

tems, was used. FedoraTM

18 OS was built in the virtual machine and the

CFD software - Elmer - was installed in this Linux based OS, figure 6.1.

49

6. Simulation Environment Setup

Figure 6.1: Software installation diagram.

As mentioned in chapter 5, Elmer includes its own pre- and post- pro-

cessing tools: a mesh converter and generator - ElmerGrid, and a simple

post-processing tool - ElmerPost. Nevertheless, some other pre- and post-

processing tools were also to be integrated together with Elmer. Thus, the

diagram of figure 6.2 presents the final, complete, software simulation flow.

Figure 6.2: Simulation flow.

50


As seen in figure 6.2, the pre-processing part of the simulation flow is re-

sponsible for the generation or conversion of a mesh file to Elmer’s native

mesh file format. In the other extrema, the post-processing part is re-

sponsible for presenting the results and developing further analysis. In

the following sections, the used pre- and post- processing methods are de-

scribed.

6.2 Pre-Processing Methods

As seen in chapter 5.4, ElmerGrid has the capability to generate meshes for

simple geometries. Thus, if a problem is simple enough, its geometry can

be meshed using ElmerGrid. However, if the geometry is rather complex,

more complete mesh generators can be used.

During this project’s implementation, Gmsh [66] and CubitTM

(exporting

the mesh as .unv) were used to generate some of the meshes. Nevertheless,

any other mesh generator may be used, as long as it can export in one

of the formats accepted by ElmerGrid (chapter 5.4) - which can act as a

converter to Elmer’s mesh native format.

6.3 Post-Processing Methods

In terms of post-processing, as mentioned in chapter 5.4, Elmer includes

its own post-processing tool - ElmerPost. However, this is a very simple

tool that did not completely serve the purposes of this project. Therefore,

ParaView [67] was found to be the most suitable tool to visualize and

analyse the results of the simulations.

Using ParaView, the results obtained with ElmerSolver can be directly

visualized and analysed. If needed, a specific sloshing post-processing pro-

vided by EADS Astrium and based on MATLAB R© can also be used. This

specific post-processing has the capability to receive inputs from ParaView,

process them and have as output the following sloshing parameters:

• Liquid sloshing natural frequencies;

• Sloshing modes recognition;

51


• Sloshing wave amplitudes;

• Mass of liquid participating in the sloshing movement;

• Damping ratio.

As this post-processing is based on MATLAB R©, it is located outside the

virtual machine for convenience. Therefore, figure 6.3 presents the final

simulation environment installation diagram.

Figure 6.3: Complete software installation diagram.

52

Chapter 7

Test case 1: Rectangular Tank

“Testing is a process of gathering information by making

observations and comparing them to expectations.”

- Dale Emery

In this chapter, the tests developed for the rectangular laterally excited

tank test case are introduced. As mentioned before, liquid sloshing inside

a laterally excited rectangular tank is one of the simplest cases of sloshing

that may occur. The simple geometry of the tank allows several relatively

accurate analytical solutions - chapter 2.1.1. Having as a basis the nume-

rical and experimental results available in literature [6, 68, 69, 70, 71] for

sloshing tests performed with rectangular tanks, three different tests were

defined to be implemented, their main objectives were:

• Test A: Evaluate the first antisymmetric sloshing frequency of a liquid

sloshing inside a 2D tank;

• Test B: Recognize the natural sloshing frequencies and modes of a

liquid sloshing inside a 2D tank;

• Test C: Obtain the first antisymmetric sloshing frequency of liquid

water sloshing inside a 3D tank.

53

7. Test case 1: Rectangular Tank

These tests are presented in the following sections. Each test is prima-

rily defined, then its implementation is explained and later the results are

presented, evaluated and validated.

7.1 Test A

7.1.1 Test Definition

The evaluation of the first natural antisymmetric sloshing frequency was

addressed in this test - similarly performed by [68, 69]. For that, a two-

-phase flow in a two-dimensional sloshing tank was considered. The tank

was subjected to a vertical acceleration g = −1 m/s2 and geometrically it

was defined with width a = 1 m and height H = 1.5 m.

The properties of the defined fluids [68, 69] were:

• Fluid 1:

– µ− = 1.0 Pa · s

– ρ− = 1000.0 kg/m3

• Fluid 2:

– µ+ = 0.01 Pa · s

– ρ+ = 1.0 kg/m3

A no-slip boundary condition was prescribed to the bottom of the tank

and slip boundary conditions were defined along the walls. Initially, the

velocity field was assumed to be zero.

The interface separating the two fluids was considered a free surface and it

was initially given as y = 0.26 + 0.1 sin(πx), where x and y have its origin

at the center of the tank. The simulation was performed for t = 20 s.

54


7.1.2 Implementation

To implement test A, two different approaches were used. In one approach,

the input mesh was defined as a rectangular tank in which the free surface

shape of the liquid was given by an initial condition - tests a) and c);

oppositely, in the other approach, the input mesh was already generated

with the intended skewed shape - tests b) and d). These two approaches

were implemented using both 2-dimensional meshes (tests a) and b)) and

3-dimensional meshes with single elements in the third direction (tests c)

and d)).

For tests b), c) and d), the origin of the coordinate system was defined to

be located in the center of the tank at a distance equal to 0.75 m from the

bottom. As for test a), its origin was defined to be located at the bottom

center of the tank.

The meshes for these tanks were generated with 40 × 40 elements for test

a), 32 × 32 elements for test b), 26 × 1 × 32 elements for test c) and 31

× 1 × 47 elements for test d).

For all the four tests, the time step sizes were defined to be 0.02 s and the

number of steps 1000. The total simulation time was, as intended, 20 s.

7.1.3 Results & Evaluation

The results obtained for tests A a), b), c) and d) are presented and dis-

cussed in this subsection. Due to their similarity, the results are presented

only for test A a). Nevertheless, the obtained results for tests A b), c) and

d) can be found in appendix A.

To give an impression of the computational effort required by the CFD

software to run these tests, it is important to state the architecture in

which the tests were run, as well as the time that was required by each

simulation. Therefore, the simulations were run in a Samsung Series 9

NP900X4C-A03PT notebook in which a virtual machine was mounted -

see chapter 6 for details. This system is composed by an Intel R© CoreTM

i7-3517U CPU processor (4M Cache, up to 3.00 GHz) and 8.0 GB DDR3

RAM memory. However, the virtual machine was running only on 6.0 GB

of RAM. Thus, for reference, the fastest test performed during this project

- test A a) required approximately 7 minutes to be simulated.

55


The first step on the evaluation of the obtained results was to verify the

correct implementation of the tests even before they had started, this is, at

t = 0s - when the fluids were still in rest. Figure 7.1 shows both the fluid

pressure distribution and the free surface position obtained for test A a). A

maximum pressure of about P = 1010 Pa was expected: P = ρgh, where

ρ = 1000kg/m3, g = 1m/s2 and h = 1.01m (non-disturbed fluid). For

the four tests, the obtained values approximately match the expected ones

(even if they are slightly different between tests). As intended, the initial

free surface position is the same for all the tests, even though different

implementations were used.

Figure 7.1: Rectangular tank - test A a): pressure at t = 0s.

The evolution of the free surface position is shown in figure 7.2. When

comparing the results of the different tests, it is possible to conclude that

these slightly differ from test to test. Nevertheless, the results are very

similar for the test pairs that had the same free surface initial position

implementation - tests a) & c) and b) & d). This validates the fact that

a 2D mesh implementation guarantees the same results as the use of a 3D

mesh with single elements in the third direction.

56


(a) t = 0.6s (b) t = 1.2s

(c) t = 1.8s

(d) t = 2.4s (e) t = 3.0s

Figure 7.2: Rectangular sloshing tank - test A a): free surface shape evolution.

57


The variation of the liquid’s CoG position is presented for test a) in fi-

gure 7.3.

Figure 7.3: Rectangular tank - test A a): CoG plots.

In the presented plot it is possible to visualize that, as the liquid sloshes, the

CoG position in the x-direction describes an harmonic behaviour, moving

together with the liquid slosh wave amplitude. It is possible to conclude

that in those tests where a 2D mesh was used (tests a) and c)) there is

no variation in the third direction, as this is non-existent. In tests b) and

d), where 3D meshes were used, the variation in the third dimension is,

as expected, zero - the visualized disturbances in test d) are only due to

computational errors and thus negligible.

In the vertical direction, a decrease in the CoG position is seen. This

decrease represents a loss of liquid’s mass that occurs due to the fact that

in FEM mass conservation is not guaranteed. For long simulations, the

accumulation of this error can lead to a significant loss of mass that might

not be acceptable. [72]

Therefore, this mass loss shall be kept within reasonable limits. In these

particular sloshing problems, the mass of liquid participating in the sloshing

behaviour was not immediately affected by the general loss of total mass in

the liquid. Hence, a reasonable maximum limit was defined to be located

58


below the 5 to 10 % mass loss, depending on the external perturbation and

the liquid height inside the tank.

In the above presented tests, the percentage of mass change was:

• a) ≈ 1, 2 %

• b) ≈ 12 %

• c) ≈ 2 %

• d) ≈ 12 %

The sloshing wave amplitude plot presented in figure 7.4 for test a) shows

the evolution of the sloshing wave amplitude measured in the most top left

point of the tank. From the results obtained in the different tests, it is

possible to conclude that approximately the same behaviour is reproduced

for all. Nevertheless, once again, the tests that had the same free surface

initial position implementation show a much identical behaviour.

Figure 7.4: Rectangular tank - test A a): sloshing amplitude plot.

Table 7.1 presents the resulting first natural antisymmetric sloshing fre-

quencies obtained for tests A a), b), c) and d).

Table 7.1: Results - rectangular tank: test A.

TestFirst natural antisymmetricsloshing frequency [Hz]

a) 0.279b) 0.283c) 0.279d) 0.283

59


As seen in table 7.1, the first natural antisymmetric sloshing frequencies

for the different tests are very similar. Nevertheless, due to the constrains

mentioned before, tests a) and c) are validated and assumed more accurate

than tests b) and d). Hence, for test A, the resulting first antisymmetric

sloshing natural frequency is f = 0.279 Hz.

A comparison of the obtained result with the analytical solution of Abram-

son [5] and the results of similar numerical tests available in literature [68,

69] is presented in table 7.2.

Table 7.2: Comparison of results - rectangular tank: test A.

First natural antisymmetricsloshing frequency [Hz]

Abramson [5] 0.282Fries [69] 0.279

Rasthofer et al. [68] 0.274Elmer 0.279

From the results presented in table 7.2 it is possible to verify that the

obtained result is in good agreement with the results available in literature

for similar tests.

7.2 Test B


The evaluation and recognition of the liquid’s natural sloshing frequencies

and modes was the objective of this test - similarly performed by [70, 71].

A two-phase flow in a two-dimensional tank was considered. The width of

the container was a = 1.0 m and the depth of liquid inside the tank was

also h = 1.0 m.

The tank was subjected to a vertical acceleration g = −9.81 m/s2 and

laterally excited. The liquid was to be considered non-viscous with density

ρ = 1000.0 kg/m3.

60


Slip boundary conditions were defined along the tank walls and a no-slip

boundary condition was assigned to the bottom. Initially, the velocity field

was assumed to be zero. The surface of the liquid was considered a free

surface initially in rest.


Test B was implemented using two different techniques: one more experi-

mentally oriented - a) and another more convenient for numerical simula-

tions - b). These different approaches are introduced below:

a) Experimental Approach

In sloshing experiments, the general technique used to measure the liquid’s

first antisymmetric natural frequency is to oscillate the tank at low ampli-

tude and record the frequency at which the undistorted wave shape reaches

the maximum amplitude without rotation (curl = 0). When the external

frequency matches the first antisymmetric sloshing natural frequency, the

amplitude of the wave reaches its maximum.

Therefore, this same approach was used to find the liquid’s first antisym-

metric natural frequency. A 2-dimensional mesh was used and after some

initial tests, the optimal amplitude of the lateral harmonic acceleration was

found to be 0.6 m/s2 for a time step equal to 0.02 s and a simulation time

of 5 s (250 time steps). The 2D mesh was defined to have 40 × 40 elements

and the origin of the coordinate system was defined to be the center of the

liquid.

Knowing that the expected analytical sloshing frequency was 0.88 Hz

(Abramson [5]), the external frequency was varied between fext = 0.84

Hz and fext = 0.90 Hz.

Because Elmer is not capable of solving the Euler’s equations for inviscid

fluids, the liquid fluid was considered to be liquid water (µwater = 1.0 ·10−3

Pa·s) and the fluid at the free surface was defined to be air (µair = 1.0·10−5

Pa · s and ρair = 1.2 kg/m3), both at Normal Temperature and Pressure

(NTP) conditions. Thus, the first antisymmetric sloshing frequency was

expected to be smaller than that of an ideal fluid.

61


It is important to note that with this approach only the first natural an-

tisymmetric frequency of the liquid sloshing could be found. To recognize

the different sloshing modes and respective natural frequencies, a different

approach had to be used.

b) Numerical Approach

In this approach, a random harmonic acceleration was used as the external

excitation, A0 · sin(2πfext · t). The frequency and amplitude of the signal

were chosen carefully to ensure convergence and diminished errors during

the simulation. After some initial tests the amplitude was defined to be

A0 = 0.5 m/s2 and the external frequency fext = 1.20 Hz for a simulation

that ran for 10.5 s using a time step equal to 0.01 s (1050 steps). For con-

venience, a 3-dimensional mesh with single elements in the third dimension

was used. The 3D mesh was defined to have 15 × 1 × 15 elements and the

origin of the coordinate system was defined at the center of the liquid.

The increased number of steps allowed the relatively accurate use of the

Power Spectral Density (PSD) post-processing included in Astrium’s slo-

shing tools. Using the PSD, it was possible to recognize the different slosh-

ing modes and respective natural frequencies.

As in approach a), the liquid fluid was also treated as liquid water and the

fluid at the free surface considered air. Therefore, slightly different natural

sloshing frequencies were, once again, expected.


The results obtained for test B using the two different approaches are

presented and discussed in this subsection.

Similarly to what was done for test A, the validation of the liquid’s pres-

sure distribution is the first point to be checked to evaluate a correct test

implementation. In figures 7.5 and 7.6, the pressure distributions are pre-

sented together with the free surface positions of the liquid still in rest. As

expected, a maximum pressure of about P = 1000 ·9.81 ·1 = 9810 Pa, was

obtained for both cases.

62


Figure 7.5: Rectangular tank - test B a): pressure at t = 0s.

Figure 7.6: Rectangular tank - test B b): pressure at t = 0s.

Seeing that for test a) an experimental approach was used, the results

obtained for each external frequency tested would have to be individually

post-processed to find the sloshing wave amplitude (using Astrium’s tool).

This procedure would have been very slow and inefficient. Hence, knowing

that the variation of the position of the liquid’s CoG in the x-direction

is proportional to the sloshing wave amplitude, the post-processing and

visualization of the results could be performed using ParaView. This tool

allows the simultaneous visualization of the results for different tests and

thus different external excitation frequencies can be evaluated at the same

time in a much more efficient process.

Thus, the maximum displacement of the CoG position of the liquid in the

x-direction is what is looked into. In relative terms, the external frequency

that corresponds to the maximum wave amplitude without rotation also

corresponds to the highest CoG displacement. Thus, to find the maxi-

mum wave amplitude without rotation happening, the following criteria

was used:

• Visual evaluation of the liquid behaviour to find the maximum sloshing

wave amplitude without rotation;

• Verification of the occurrence of a sudden increase in mass loss after

rotation occurs.

63


Figure 7.7 presents the CoG x-displacement for the three excitation fre-

quencies that gave the highest sloshing wave amplitudes.

(a) t = 0s to t = 5s (250 time steps) (b) Zoom: t = 3.48s

Figure 7.7: Rectangular tank - test B a): CoG x-coord. Vs time for differentexcitation frequencies.

In plot a), the entire simulation time is presented and the point considered

to be correspondent to the maximum sloshing wave amplitude without

rotation is marked with a vertical green bar crossing the time of occurrence.

In plot b), a zoom of this point is presented. As it is possible to see,

fext = 0.86 Hz gives the highest wave amplitude. Thus, the first natural

antisymmetric sloshing frequency for this test is found to be f = 0.86 Hz.

The position of the free surface at the time for which the maximum wave

amplitude is reached is shown in figure 7.8.

Figure 7.8: Rectangular tank - test B a): maximum wave amplitude (t = 3.48s).

For test b), where a more suitable numerical procedure was used, the po-

sition of the CoG of the liquid is presented in figure 7.9.

64


Figure 7.9: Rectangular tank - test B b): CoG plots.

It is possible to visualize in the plots that the CoG changes in the y-di-

rection are only due to computational errors and thus negligible. In the

vertical direction (z-direction), there is some mass loss, but very small and

also negligible for the simulation time. In the x-direction it is possible to

see that there is a repeatable pattern of the CoG oscillation. Figure 7.10

shows this pattern in higher detail.

Figure 7.10: Rectangular tank - test B b): CoG x-coordinate Vs time for fext = 1.20Hz.

65


It is possible to see that after each cycle, the sloshing movement stalls for

a small time and again the liquid starts sloshing. It is believed that this

happens due to a superposition of waves that lead to the sloshing stall and

restart.

The previously discussed plot of figure 7.7 a) presents only part of the

cycle that is repeated (as seen in figure 7.10). Nevertheless, because in the

experimental approach the objective was to reach the maximum sloshing

wave amplitude in the smallest simulation time, ensuring convergence, the

entire cycle and the repeated pattern were not of significant importance.

The plot presented in figure 7.11 intends to show the above mentioned

proportionality between the CoG displacement in the x-direction and the

sloshing wave amplitude.

Figure 7.11: Rectangular tank - test B b): sloshing amplitude plot.

As it is possible to see, the same behaviour occurs in both plots (figures 7.10

and 7.11) - the waves are inverted by 180 degrees.

In figure 7.12, the PSD plot that presents the antisymmetric modes sloshing

frequencies is shown. Figure 7.13 presents the same PSD plot but for the

symmetric modes. To find the antisymmetric modes sloshing frequencies

the PSD was measured in the top most left point of the tank, as to find

the symmetric modes it was measured in the top center point.

66


Figure 7.12: Rectangular tank - test B b): PSD plot (measured at the top leftpoint of the tank).

Figure 7.13: Rectangular tank - test B b): PSD plot (measured at the top centerpoint of the tank).

The following frequencies are highlighted in the PSD plot of figure 7.12:

• f = 0.857 Hz - 1st antisymmetric mode sloshing frequency;

• f = 1.238 Hz - 1st symmetric mode sloshing frequency - possible to

visualize because the sloshing wave is not completely symmetric and

thus the incidence of this frequency is also noticeable in the top most

left point of the tank;

• f = 1.524 Hz - 2nd antisymmetric mode sloshing frequency.

67


For the PSD plot of figure 7.13 the following frequencies are highlighted:

• f = 0.857 Hz - 1st antisymmetric mode sloshing frequency - less

stronger than when measured in the top most left point, but still

visible;

• f = 1.238 Hz - 1st symmetric mode sloshing frequency;

• f = 1.809 Hz - 2nd symmetric mode sloshing frequency;

• Other visible frequencies - correspond to the cycle repetition patterns.

The uncertainty of the obtained frequencies is approximately ±0.095 Hz

- increased resolution could be achieved by increasing the number of time

steps (N) or the time step (∆t) itself - frequency resolution ∆f = 1N ·∆t

.

As a summary, the obtained results for the sloshing frequencies are pre-

sented in table 7.3.

Table 7.3: Results - rectangular tank: test B.

Mode Sloshing natural frequencies [Hz]

Test a) Test b)

n=1 0.86 0.857m=1 - 1.238n=2 - 1.524m=2 - 1.809

As it was expected, the first natural antisymmetric sloshing frequencies are

approximately the same for both tests, even though the one obtained with

approach a) is more accurate. Nevertheless, because finding the different

sloshing frequencies and recognizing the different sloshing modes were the

objective of this test, test b) is considered for evaluation purposes.

Table 7.4 presents a comparison of the obtained results, with the calculated

analytical solutions from Abramson [5] and the numerical results of similar

tests available in literature [70, 71].

68


Table 7.4: Comparison of results - rectangular tank: test B.

Mode Slosh frequencies [Hz]

Abramson [5] N. C. Pal et al. [70] P. Pal et al. [71] Elmer

n=1 0.881 0.88 0.883 0.857 / 0.86m=1 1.249 1.26 1.253 1.238n=2 1.530 1.54 1.545 1.524m=2 1.767 1.80 1.786 1.809

The results presented above (table 7.4) attest a good agreement of the

obtained results with those available in literature. This is so, despite the

fact that the presented results are for tests in which idealized fluids were

used instead of liquid water. Thus, the small noticeable differences in the

results were already expected.

7.3 Test C


The objective of this test was to obtain the first antisymmetric sloshing

frequency along the longer and shorter directions of a three-dimensional

rectangular tank filled with water - similarly performed by [6].

The width of the container was a = 0.270 m, the breadth b = 0.135 m and

the height H = 0.300 m. The tank was subjected to a vertical acceleration

g = −9.81 m/s2 and laterally excited.

A two-phase flow with air and water fluids was considered. The material

properties of these fluids were:

• Water:

– µwater = 1.0 · 10−3 Pa · s

– ρwater = 1000.0 kg/m3

• Air:

– µair = 1.0 · 10−5 Pa · s

– ρair = 1.2 kg/m3

69



boundary condition was assigned for the bottom. Initially, the velocity

field was assumed to be zero. The surface of the liquid was also considered

a free surface initially in rest.

Different water depths were considered:

• h1 = 0.050 m

• h2 = 0.100 m

• h3 = 0.150 m

• h4 = 0.200 m

• h5 = 0.250 m


In this test the experimental approach introduced in test B was used -

approach a). Only the first natural antisymmetric sloshing frequency was

to be found for the different water depths. Therefore, this approach was

more experimentally oriented and less time consuming.

Once again, some initial tests were performed and the optimal external

amplitudes were found. Knowing the expected sloshing natural frequencies

from the analytical solutions of Housner [4], the frequency ranges were

chosen. For each test, the following amplitude, external frequency range

and number of mesh elements were used:

a) Longer Direction

i) h = 0.050 m

• A0 = 0.2 m/s2;

• fext = 1.20 Hz to fext = 1.28 Hz;

• Mesh elements: 20 × 10 × 7.

70


ii) h = 0.100 m

• A0 = 0.2 m/s2;



iii) h = 0.150 m

• A0 = 0.3 m/s2;



iv) h = 0.200 m

• A0 = 0.3 m/s2;



v) h = 0.250 m

• A0 = 0.3 m/s2;


• Mesh elements: 20 × 10 × 35 .

The simulations ran for 5 s using a time step equal to 0.02 s (250 steps).

b) Shorter Direction

i) h = 0.050 m

• A0 = 0.6 m/s2;



71


ii) h = 0.100 m

• A0 = 0.6 m/s2;



iii) h = 0.150 m

• A0 = 0.7 m/s2;



iv) h = 0.200 m

• A0 = 0.7 m/s2;



v) h = 0.250 m

• A0 = 0.7 m/s2;



The simulations ran for 2.5 s using a time step equal to 0.02 s (125 steps).

It is important to note that, for all the different liquid heights inside the

tank, the origin of the coordinate system was defined to be the center of

the liquid.

72



The results obtained for test C are presented and discussed in this subsec-

tion. Due to their similarity, the results are presented here only for the

cases in which the water depth inside the tank was h = 0.050 m. Never-

theless, the obtained results for the other considered water depths can be

found in appendix B.

Following the same procedure adopted in tests A and B, the first validation

step of test C was to check the liquid’s pressure distribution inside the

containers.

(a) h = 0.050m (b) h = 0.100m

(c) h = 0.150m

(d) h = 0.200m (e) h = 0.250m

Figure 7.14: Rectangular tank - test C: pressure at t = 0s for different waterdepths.

73


As it is possible to visualize in figure 7.14, the maximum pressure values

match the analytical expected values for which one of the different water

depths inside the tank.

As a reference, the free surface position for each one of the tests and water

depths inside the tank at t = 0s was evaluated. For the test in which

h = 0.050m and the external excitation occurs along the longer direction

of the tank, this is shown in figure 7.15.

Figure 7.15: Rectangular tank - test C - h = 0.050m longer direction: free surfaceshape at t = 0s.

For the same water depth, but with the excitation occurring along the

shorter direction, the results are presented in figure 7.16.

Figure 7.16: Rectangular tank - test C - h = 0.050m shorter direction: free surfaceshape at t = 0s.

Similarly to test B a), an experimental approach was also used to find the

first natural antisymmetric sloshing frequency. Therefore, the same results

as in test B a) were obtained for each one of the tests and water depths.

Hence, the explanations given in test B a) to describe these results are also

valid here.

In the plots of figures 7.17 (longer direction test) and 7.18 (shorter di-

rection test), the position of the CoG in the x-direction is presented and

the maximum wave sloshing amplitude marked and zoomed to identify the

external frequency that matches the first natural antisymmetric sloshing

frequency.

74



Figure 7.17: Rectangular tank - test C - h = 0.050m longer direction: CoG x-coord.Vs time for different excitation frequencies.

(a) t = 0s to t = 2.5s (125 time steps) (b) Zoom: t = 1.60s

Figure 7.18: Rectangular tank - test C - h = 0.050m shorter direction: CoGx-coord. Vs time for different excitation frequencies.

Figures 7.19 (longer direction test) and 7.20 (shorter direction test) present

the correspondent free surface positions at the time in which the maximum

sloshing wave amplitude occurred.

Figure 7.19: Rectangular tank - test C - h = 0.050m longer direction: maximumwave amplitude (t = 2.82s).

75


Figure 7.20: Rectangular tank - test C - h = 0.050m shorter direction: maximumwave amplitude (t = 1.60s).

A summary of the obtained first natural antisymmetric sloshing frequencies

is presented for all tests in table 7.5.

Table 7.5: Results - rectangular tank: test C.

Depth of water [m] First natural antisymmetric sloshing frequencies [Hz]

along longer direction along shorter direction

0.050 1.23 2.150.100 1.53 2.330.150 1.62 2.350.200 1.66 2.350.250 1.68 2.35

A comparison of these results with the analytical solutions of Housner [4]

and the numerical and experimental data available in literature [6] for

similar tests is presented in tables 7.6 for the longer direction tests and 7.7

for the shorter direction ones.

Table 7.6: Comparison of results - rectangular tank: test C - along longer direction.

Depth ofwater [m]

First natural antisymmetric sloshing frequencies [Hz]

Housner [4]Jaiswal et al. [6]- Experimental

Jaiswal et al. [6]- ANSYS R© Elmer

0.050 1.238 1.26 1.23 1.230.100 1.549 1.50 1.53 1.530.150 1.656 1.60 1.62 1.620.200 1.690 1.64 1.66 1.660.250 1.701 1.70 1.68 1.68

76


Table 7.7: Comparison of results - rectangular tank: test C - along shorter direction.

Depth ofwater [m]




0.050 2.191 2.17 2.15 2.150.100 2.390 2.33 2.33 2.330.150 2.410 2.38 2.35 2.350.200 2.412 2.40 2.35 2.350.250 2.412 2.40 2.35 2.35

As it is possible to see in the above presented tables (7.6 and 7.7), the

results obtained with Elmer for test C of the rectangular tank test case are

in good agreement with similar test results available in literature.

77

Chapter 8

Test case 2: Cylindrical Tank

“A test is an experiment designed to reveal information, or

answer a specific question, about the software or system.”

- Elisabeth Hendrickson

In this chapter, the tests developed for the cylindrical laterally excited

tank test case are introduced. Having a simple geometry, the cylindrical

tank similarly to the rectangular one also allows accurate analytical solu-

tions for simple sloshing problems - chapter 2.1.1. Based on numerical and

experimental results available in literature [6], a test was defined.

This test is presented in the following sections. Similarly to chapter 7, the

test is primarily defined, then its implementation is explained and later the

results are presented, evaluated and validated.

8.1 Test Definition

Similarly to what was done in test C of the rectangular tank test case,

the objective of this test was to obtain the first antisymmetric sloshing

frequency of a cylindrical tank filled with water - similarly performed by [6].

The tank had diameter D = 0.170 m and height H = 0.230 m. It was

subjected to a vertical acceleration g = −9.81 m/s2 and laterally excited.

A two-phase flow with air and water fluids was considered. The material

properties of these fluids were:

78

8. Test case 2: Cylindrical Tank

• Water:

– µwater = 1.0 · 10−3 Pa · s

– ρwater = 1000.0 kg/m3

• Air:

– µair = 1.0 · 10−5 Pa · s

– ρair = 1.2 kg/m3


boundary condition was assigned to the bottom. Initially, the velocity field

was assumed to be zero and the surface of the liquid was considered a free

surface initially in rest.

The first antisymmetric sloshing frequency was obtained for different water

depths:

• h1 = 0.050 m

• h2 = 0.100 m

• h3 = 0.150 m

• h4 = 0.200 m

8.2 Implementation

Similarly to test C performed for the rectangular tank, an experimental ap-

proach was used to find the first natural antisymmetric sloshing frequency

for different water depths inside the cylindrical tank. To find the optimal

external amplitude some initial tests were performed.

Knowing the expected sloshing frequency results from the analytical solu-

tions of Housner [4], the frequency ranges were chosen. For each test, the

following amplitude, external frequency range and number of elements per

mesh were used:

i) h = 0.050 m

• A0 = 0.6 m/s2;


• Mesh elements: 6144.

79


ii) h = 0.100 m

• A0 = 0.6 m/s2;



iii) h = 0.150 m

• A0 = 0.6 m/s2;



iv) h = 0.200 m

• A0 = 0.6 m/s2;


• Mesh elements: 24576

The simulations ran for 2 s using a time step equal to 0.02 s (100 steps).

To provide reference, it is important to note that the coordinate system

was defined to have origin at the bottom center of the tank.

8.3 Results & Evaluation

The results obtained for this test are presented and discussed in this sub-

section. Due to their similarity, the results are presented only for the

considered smaller water depth inside the tank (h = 0.050 m). Neverthe-

less, the obtained results for the other tested water depths can be found in

appendix C.

The procedure adopted in tests B a) and C of the rectangular tank test

case (chapter 7) was also adopted to evaluate the cylindrical tank tests.

Thus, the first step was to check the liquid’s pressure distribution inside

the containers at time t = 0s. In figure 8.1, the pressure distribution for

each one of the different water depths inside the tank is presented.

80


(a) h = 0.050m (b) h = 0.100m

(c) h = 0.150m (d) h = 0.200m

Figure 8.1: Cylindrical tank - test: pressure at t = 0s for different water depths.

As it is possible to visualize, the maximum pressure value clearly matches

the analytical expected ones.

The free surface position for each one of the different water depths inside

the tank still in rest (t = 0s) was studied - in figure 8.2 the results are

shown for the case in which the water depth was h = 0.050m.

Figure 8.2: Cylindrical tank test - h = 0.050m: free surface shape at t = 0s.

81


The same resulting plots presented for tests B a) and C of the rectangular

tank test case (chapter 7) are presented for this test case. Similarly, these

were used to find the first natural antisymmetric sloshing frequency for

which one of the tests.

Figure 8.3 presents the position of the CoG in the x-direction. The max-

imum wave sloshing amplitude is marked and zoomed to identify the cor-

respondent external frequency, which is then equal to the first natural

antisymmetric sloshing frequency.


Figure 8.3: Cylindrical tank test - h = 0.050m: CoG x-coordinate Vs time.

The correspondent free surface position at the time in which the maximum

sloshing wave amplitude occurred is shown in figure 8.4.

Figure 8.4: Cylindrical tank test - h = 0.050m: maximum wave amplitude (t =1.48s).

A summary of the obtained first natural antisymmetric sloshing frequencies

for the different water depths that were tested is shown in table 8.1.

82


Table 8.1: Results - cylindrical tank test.

Depth of water [m]First natural antisymmetricsloshing frequency [Hz]

0.050 1.990.100 2.180.150 2.220.200 2.22

The above presented results (table 8.1) are compared with the analytical

solutions of Housner [4] and the numerical and experimental data available

in literature [6] for similar tests −table 8.2.

Table 8.2: Comparison of results - cylindrical tank test.

Depth ofwater [m]




0.050 2.064 2.07 1.99 1.990.100 2.287 2.30 2.18 2.180.150 2.314 2.33 2.22 2.220.200 2.317 2.33 2.22 2.22

From table 8.2 it is possible to attest a good agreement of the results

obtained in this test with the results available in the literature.

83

Chapter 9

Test case 3: ESA Tank -

Cylindrical Tank w/

Hemispherical Domes

“Knowing a great deal is not the same as being smart;

intelligence is not information alone but also judgement, the

manner in which information is collected and used.”

- Carl Sagan

In this chapter, the tests developed for a cylindrical tank with hemispherical

domes (from now on called ESA tank) are introduced. This test case was

proposed by ESA’s HSO directorate and its main goal was to reduce the gap

between the previously presented test cases and the real sloshing problems

faced in space vehicles.

In test cases 1 and 2, the software was already validated for simple sloshing

problems where comparators were available. In this test case, a more com-

plex problem in which the liquid propellant tank has a geometry commonly

used in space vehicles is presented.

This type of tank do not have directly accessible accurate analytical so-

lutions for the liquid sloshing parameters. Nevertheless, rough results for

some of the parameters can be obtained using approximated tank geome-

tries.

Because no numerical or experimental data was available for comparison

and validation of the test results, the obtained results had to be critically

84

9. Test case 3: ESA Tank

judged and evaluated before being validated.

To accomplish the goals defined for this test case three different tests were

defined, their objectives were:

• Test A: Recognize the natural sloshing modes frequencies of com-

monly used liquid propellants sloshing inside the tank;

• Test B: Accurately determine the first antisymmetric sloshing fre-

quency of commonly used liquid propellants sloshing inside the tank;

• Test C: Determine the liquid’s damping ratio, as well as the mass

participating in the sloshing movement after the abrupt removal of a

0.1− g lateral acceleration.

In the following sections, the tests that were developed are presented. Each

test is primarily defined, then its implementation is explained and later the

results are presented and evaluated.

9.1 Test A


The objective of this test was to perform a sloshing analysis that would

allow the recognition of the natural sloshing modes frequencies. The test

was performed for different liquid propellants sloshing inside the tank.

The tank was subjected to a vertical acceleration g = −9.81 m/s2 and

laterally excited. A two-phase flow in a three-dimensional sloshing tank

was considered.

Geometrically, the tank was defined to have a main cylindrical part and an

hemispherical dome at each end. The radius of the tank was r = 0.569 m

and the height of the cylindrical part H = 1.206 m.

The fluid at the free surface of the liquid propellant was defined to be

pressurized Helium (P = 20 bar , T = 298 K) with properties:

• µ = 1.9786 · 10−5 Pa · s

• ρ = 3.2312 kg/m3

85


Two different propellant liquids were studied, their properties were:

• MON-3:

– µ = 3.967441 · 10−4 Pa · s

– ρ = 1433.401 kg/m3

• MMH:

– µ = 7.78024 · 10−4 Pa · s

– ρ = 870.372 kg/m3

Different propellant fill ratios were evaluated:

• 25 %

• 50 %

• 75 %

The boundary conditions along the tank walls were defined to be slip

boundary conditions. For the bottom hemisphere, a no-slip boundary con-

dition was prescribed. Initially, the velocity field was assumed to be zero

and the top surface of the liquid propellant was assumed a free surface,

initially in rest.


To implement this test, the numerical approach introduced in test B b) of

the rectangular tank test case was used - see chapter 7.2.2 for more details.

An approximated analytical approach was used to have a rough idea of

the expected natural sloshing frequencies. To start with, the first anti-

symmetric natural sloshing frequency of a cylindrical tank (Housner [4] -

chapter 2.1.1) with the same radius r = 0.569 m and variable liquid height

h - dependent on the fill ratio, was used:

• 25 % fill ratio - h = 0.681 m: f = 0.885 Hz

• 50 % fill ratio - h = 1.172 m: f = 0.895 Hz

86


• 75 % fill ratio - h = 1.663 m: f = 0.896 Hz

To have an estimate of the different antisymmetric and symmetric sloshing

frequencies, the first antisymmetric natural frequency found for the cylin-

drical tank was matched with that of a rectangular tank with the same

height of liquid - Abramson [5] (chapter 2.1.1). The corresponding dimen-

sion of the rectangular tank along the excitation direction was found to

be a = 0.972 m. With this dimension as a reference, the different slosh-

ing modes frequencies were then calculated and are believed to be good

estimations for the cylindrical tank with hemispherical domes:

• 25 % fill ratio - h = 0.681 m:

– n=1 - f = 0.885 Hz

– n=2 - f = 1.552 Hz

– m=1 - f = 1.267 Hz

– m=2 - f = 1.792 Hz

• 50 % fill ratio - h = 1.172 m:

– n=1 - f = 0.896 Hz

– n=2 - f = 1.552 Hz

– m=1 - f = 1.267 Hz

– m=2 - f = 1.792 Hz

• 75 % fill ratio - h = 1.663 m:

– n=1 - f = 0.896 Hz

– n=2 - f = 1.552 Hz

– m=1 - f = 1.267 Hz

– m=2 - f = 1.792 Hz

From these rough estimations it was possible to conclude that for the de-

fined fill ratios the sloshing natural frequencies would be approximately the

same.

87


Having as a basis test B b) of the rectangular tank test case, a simulation

with 20 s and 1000 time steps was defined. The external excitation was

defined to start only after 1 s.

After some initial tests, the optimal external acceleration amplitude was

found to be A0 = 0.35 m/s2 for a 25 % fill ratio and A0 = 0.7 m/s2 for

the other fill ratios. Two different external frequencies were tested for each

propellant and fill ratio: fext = 0.70 Hz and fext = 1.50 Hz.

The geometries of the liquids inside the tanks were meshed using unstruc-

tured tetrahedral elements. Even though hexahedral mesh elements are

normally preferred (chapter 2.2.3), for this test tetrahedral elements were

chosen - the meshing of the tank was much easily achieved using these ele-

ments. Moreover, given the relative simplicity of the geometries, the final

results are believed to have a similar accuracy as if hexahedral elements

were used.

After validating the mesh for the 75 % fill ratio tank (the number of ele-

ments was doubled and the same test was performed, having been obtained

similar results), the number of tetrahedral elements per mesh, for each fill

ratio, was settled as:

• 25 % fill ratio - 11816 elements



For reference, the coordinate system was defined to have its origin at the

center of the tank in the plane intersecting the bottom hemispherical dome

and the cylindrical part of the tank.


The results obtained for test A are presented, discussed and evaluated in

this subsection.

As it was done for the other test cases, the first step to validate the correct

implementation of the defined tests is to check the pressure distribution

and the free surface shape at t = 0s.

Figure 9.1 shows the pressure distribution for the MON-3 liquid propellant

tests. Figure 9.2 presents the same but for the MMH propellant case.

88


(a) 25% (b) 50%

(c) 75%

Figure 9.1: ESA tank - test A - MON-3: pressure at t = 0s for the different fillratios.

(a) 25% (b) 50%

(c) 75%

Figure 9.2: ESA tank - test A - MMH: pressure at t = 0s for the different fill ratios.

89


As it is possible to see in figures 9.1 and 9.2, the maximum pressure value

clearly matches the analytical expected values (P = ρ · g · h) for each

propellant and liquid height.

Validated the correct implementation of the test at time t = 0s, the next

step was to evaluate, for each liquid propellant and fill ratio, the results that

would allow the identification of the liquids’ natural sloshing frequencies.

To do this, the CoG variation, the sloshing amplitude measured at the top

most left point of the tank and the PSD plots for both the antisymmetric

and the symmetric modes were evaluated and are presented below.

The obtained results are very similar for the different propellants and fill

ratios. Therefore, in this section, only the most important results of the

MON-3 propellant with a 50 % fill ratio test are shown. The results ob-

tained for all the other developed tests can be found in appendix D.

The variation of the CoG of the liquid is shown in figure 9.3 for the case

in which the external frequency was fext = 0.70 Hz.

Figure 9.3: ESA tank test A - MON-3 (50% fill ratio & fext = 0.70 Hz): CoGplots.

90


A general evidence common to all the developed tests is the fact that in

the vertical direction (z-direction) there is significant decrease of the CoG

position. This is due to the already described mass loss problem, which

in this case is evident. However, because the mass of liquid participating

in the sloshing movement is not immediately affected, it was assumed that

this effect had no major impact in the obtained results. In the y-direction,

the noticeable variations are very small, as expected. In the x-direction,

the CoG position has the expected behaviour, moving together with the

liquid sloshing.

In figure 9.4 the sloshing wave amplitude measured at the top left most

point of the tank (along the excitation direction) is shown.

Figure 9.4: ESA tank test A - MON-3 (50% fill ratio & fext = 0.70 Hz): sloshingamplitude plot.

Figures 9.5 and 9.6 present the PSD plots for the antisymmetric and the

symmetric modes sloshing frequencies for the case in which fext = 0.70 Hz.

Figures 9.7 and 9.8 present the same plots but for fext = 1.50 Hz.

It is important to note that to recognize the antisymmetric modes sloshing

frequencies the PSD was measured in the top most left point of the tank,

and to find the symmetric modes it was measured in the top center point.

91


Figure 9.5: ESA tank test A - MON-3 (50% fill ratio & fext = 0.70 Hz): PSD plot(measured at the top left point of the tank).

Figure 9.6: ESA tank test A - MON-3 (50% fill ratio & fext = 0.70 Hz): PSD plot(measured at the top center point of the tank).

92


Figure 9.7: ESA tank test A - MON-3 (50% fill ratio & fext = 1.50 Hz): PSD plot(measured at the top left point of the tank).

Figure 9.8: ESA tank test A - MON-3 (50% fill ratio & fext = 1.50 Hz): PSD plot(measured at the top center point of the tank).

93


A deep evaluation of the presented PSD plots allowed to draw some conclu-

sions about the first two antisymmetric and symmetric sloshing frequencies.

The following frequencies are highlighted from the plots:

• f = 0.70 Hz - Excitation frequency;

• f = 1.50 Hz - Excitation frequency;

• f = 0.90 Hz - 1st antisymmetric mode sloshing frequency;

• f = 1.40 Hz - 1st symmetric mode sloshing frequency;

• f = 1.60 Hz - 2nd antisymmetric mode sloshing frequency;

• f = 1.95 Hz - 2nd symmetric mode sloshing frequency.

The obtained frequencies have an uncertainty of ±0.05 Hz.

It is important to note that both the 1st antisymmetric and symmetric

sloshing frequencies were obtained with relative confidence, as their power

in the PSD is clearly visible. As for the 2nd antisymmetric and symmetric

sloshing frequencies these were obtained by looking for the points in the

PSD that were in the vicinities of our initial rough analytical estimations.

Therefore, the confidence in these results is slightly diminished.

Regarding the other frequencies visible in the plots, these are not relevant

for this study. They are believed to result from a superposition of waves

originated in the initial time steps of the simulation and then propagated

over time on the top of the general sloshing motion. These waves are

thought to be introduced artificially due to imperfections in the numerical

simulations. Their almost negligible PSD attests their insignificance.

For the different fill ratios, as well as for the other liquid propellant (MMH),

the sloshing frequencies were found to be approximately the same as those

presented and discussed here.

9.2 Test B


This test was defined to accurately determine the first antisymmetric natu-

ral sloshing frequency for the different liquid propellants sloshing inside the

94


tank. Another objective of this test was to identify the maximum sloshing

wave amplitude for a lateral excitation of amplitude 0.1−g.

The tank was subjected to the same set of conditions defined for test A -

subsection 9.1.1.


In this test, the same experimental approach used in the rectangular tank

test case - tests 2 a) and 3, as well as in the cylindrical tank test case, was

used.

Using as a basis the initial rough estimations of the first antisymmetric

sloshing frequency, the external excitation frequency was varied between

fext = 0.82 Hz and fext = 0.92 Hz and the amplitude of the external

acceleration was defined to be A0 = 0.981 m/s2.

The same meshes previously defined for test A were also used here. The

tests were defined to run for 4 s with a time step equal to 0.02 s (200 time

steps).

After experimentally finding the first antisymmetric sloshing frequency, the

maximum sloshing amplitude measured at the left most point of the tank

was also evaluated. It is important to note that the used approach is only

valid as long as there is no wave rotation.

Because the results were expected to be approximately the same for the

three different fill ratios, this test was only performed for the case in which

the propellant liquid occupies 50 % of the tank.


The results achieved in test B are presented in this subsection for the MON-

-3 liquid propellant. The results obtained for the MMH liquid propellant

case can be found in appendix E.

The free surface position of the liquid inside the tank still in rest (t = 0s)

is presented together with a study of the CoG position in the x-direction.

From these, it is possible to identify the first antisymmetric sloshing fre-

quency, which corresponds to the maximum sloshing wave without rotation

- see chapter 7.2.3 for details.

95


Figure 9.9 presents the free surface position of the liquid propellants, MON-

-3 or MMH at time t = 0s (as expected, in rest, the free surface of the

liquids was the same).

Figure 9.9: ESA tank test B - MON-3 or MMH: free surface shape at t = 0s.

In figure 9.10, the position of the CoG in the x-direction is shown for

the MON-3 liquid propellant. The maximum wave sloshing amplitude is

also marked and zoomed to allow the identification of the corresponding

external frequency, which is then equal to the first antisymmetric sloshing

natural frequency.


Figure 9.10: ESA tank test B - MON-3: CoG x-coordinate Vs time.

The free surface position at the time in which the maximum sloshing wave

amplitude occurred is presented in figure 9.11 for the MON-3 liquid pro-

pellant.

96


Figure 9.11: ESA tank test B - MON-3: maximum wave amplitude (t = 3.46s).

In table 9.1, a summary of the obtained first antisymmetric natural sloshing

frequencies is presented.

Table 9.1: Results - ESA tank: test B.

Liquid propellantFirst antisymmetricnatural sloshing frequency [Hz]

MON-3 0.86MMH 0.86

Figure 9.12 presents the sloshing wave amplitude measured at the left most

point of the tank.

Figure 9.12: ESA tank test B - MON-3: sloshing amplitude plot.

97


The maximum wave amplitude for the case presented in figure 9.12 was

approximately 1.521 m, which means that from the initial rest position

of the free surface of the liquid at 0.603 m, the amplitude achieved by

the sloshing wave was 0.918 m. This measurement was performed at t =

3.5s - corresponding to the last wave peak before the rotation of the wave

happened.

For the case in which the MMH liquid propellant was used, the value

obtained was very similar, about 0.915 m. The slightest smaller result

was already expected seeing that the MMH liquid propellant has higher

viscosity than the MON-3.

It is important to note that for this specific test the amplitude of the wave

was only important to measure the sloshing frequency. Hence, the sloshing

wave amplitude values previously presented shall only serve to give an idea

of the expected sloshing wave amplitudes for the established conditions.

In the specific case of the tanks being used, at the referred time t = 3.5s,

the sloshing wave would have already hit the top hemisphere of the tank

and therefore wave rotation and turbulence would have already happened.

As explained in the implementation section, it was assumed that the re-

sults obtained for the tank containing 50% of propellant could be extended

to the tanks containing 25% and 75%. Nevertheless, if certainty in the

results needs to be ensured, the same procedure adopted here could also

be implemented for these two cases.

9.3 Test C


Test C had two main objectives, being the first one obtaining the liquid

propellants’ damping ratio and the second one identifying the mass of pro-

pellant participating in the sloshing phenomenon.

The same set of conditions to which tests A and B had been subjected

were also defined for this test - subsection 9.1.1.

98



To implement test C an external perturbation was laterally applied to the

fluid and abruptly removed after some time. The damping ratio of the

liquid was measured for the initial period at which the lateral acceleration

was still acting, as well as for the period from which the lateral perturbation

was removed.

Due to the assumptions considered in the models, the damping ratio is only

dependent on the viscosity of the liquid, the fill ratio and the tank shape.

Two different tests were implemented: one in which a constant 0.1 − g

lateral excitation was applied to the liquid and abruptly removed when

a quasi steady-state was reached, and another in which the same lateral

excitation was removed immediately after the maximum sloshing wave am-

plitude was reached.

For the first test, the simulation was run for 60 s, using a time step equal

to 0.02 s (3000 time steps). In the second test, the total time was reduced

to 20 s and the same time step equal to 0.02 s was used (1000 time steps).

The mass of liquid participating in the sloshing phenomenon was only

measured for the period after the removal of the lateral perturbation.

Following the same logic adopted in test B, this test was only performed

for a propellant fill ratio equal to 50 % .


The results obtained in test C are presented in this subsection for the MON-

-3 liquid propellant. For the MMH liquid propellant case, the resulting

plots can be found in appendix F.

Similarly to what was done in chapter 7 for the fastest test performed

in this project, it is also important to note how long did it take to run

the longest test performed. Therefore, for the 60 s test, approximately 47

hours were required to complete the simulation using the same computer

described in chapter 7.1.3.

The free surface of the liquid propellants still in rest was already presented

in test B - figure 9.9, and shall serve as a reference also in this case.

99


To start with, figure 9.13 presents the CoG x-coordinate development for

the case in which the simulation was run for 60s.

Figure 9.13: ESA tank test C - MON-3 (60s simulation): CoG x-coord. Vs time.

As it is possible to visualize in the figure, it takes about 30 seconds for the

liquid propellant to stabilize after the initial impact of the 0.1 − g lateral

perturbation. Figure 9.14 shows the free surface shape at this point in

time.

Figure 9.14: ESA tank test C - MON-3 (60s simulation): wave amplitude att = 30s.

100


After the abrupt removal of the lateral acceleration, it is visible in fi-

gure 9.13 that it takes about 15s (750 time steps) for the liquid surface

to stabilize around zero. As expected, the damping has an exponential

behaviour going, in the limit, to zero - please refer to [1] for more details.

Figure 9.15 presents the quasi-stabilized free surface at t = 45.5s.

Figure 9.15: ESA tank test C - MON-3 (60s simulation): wave amplitude att = 45.5s.

The variation of the x-coordinate of the CoG at this point (t = 45.5s) has

an amplitude of approximately 1.5 mm and the sloshing wave amplitude

is almost unnoticeable. Figure 9.16 presents the sloshing wave amplitude

measured at the top most right point of the tank.

Figure 9.16: ESA tank test C - MON-3 (60s simulation): sloshing amplitude plot.

101


As it is possible to see in the figure the behaviour is not that which was

expected. Nevertheless, the significant decrease in the amplitude of the

sloshing wave is easily explained by the mass loss phenomenon (already

addressed in chapter 7.1.3), which in this case is clearly visible.

Figure 9.17 presents the variation of the z-coordinate of the CoG. In this

figure it is possible to visualize that the liquid’s mass loss in this test is

significant, as mentioned before.

Figure 9.17: ESA tank test C - MON-3 (60s simulation): CoG z-coord. Vs time.

Besides the already mentioned mass loss phenomenon it was also noticed

that for some tests some other numerical issues would also arise. For ex-

ample, in some tests it was noticed that noise could appear in the vicinities

of zero. This noise is believed to be due to the time integration scheme

that was being used to run the tests using Elmer.

Having addressed all these different numerical constrains and carefully val-

idated the obtained results, the damping ratio was calculated. For the

initial 30s period in which the lateral acceleration was still acting, this was

found to be ζ = 0.023 for the MON-3 propellant and ζ = 0.025 for the

MMH one.

For the period after the abrupt removal of the lateral acceleration these

values were found to be higher, being about ζ = 0.0225 for the two liquid

propellants.

From [1] analytical results were obtained for the damping ratio of these

two liquid propellants. It was found that for the MON-3 propellant -

ζ = 0.00036 and for the MMH - ζ = 0.00064. Meaning that the values that

102


were numerically obtained are most probably overestimated.

This overestimation of the results can be explained by the mass loss phe-

nomenon. The damping ratio is calculated from the maximum wave peaks

of the sloshing wave amplitude, from which an exponential fit is performed

- see [1] for details. As seen in figure 9.16 the sloshing wave amplitude

is being decreased over time because of the liquid’s mass loss. Thus, the

fit being performed to these maximum sloshing wave peaks gives in fact a

damping ratio slightly higher than that which would have been obtained

if mass loss would not be present. In the second part of test the mass loss

phenomenon is even more dominant and therefore for evaluation purposes

the results obtained in the first 30s of the test are considered.

The PSD plots obtained for the 15s period after the abrupt removal of the

lateral acceleration attest the conclusions reached in test B, regarding the

first antisymmetric sloshing frequencies of the two liquids, and therefore

they are also included in appendix F.

For the second tested setup - simulation time equals to t = 20s and lateral

acceleration released after the first sloshing wave peak - the results were

found to be very similar to those obtained in the previously presented test.

Figure 9.18 shows the shape of the surface of the liquid at its maximum

wave amplitude - this corresponds to the point at which the lateral acce-

leration was abruptly removed.


103


The same results are presented in figure 9.19 for the time t = 19.6s. At

this point in time, the CoG x-coordinate was already in the vicinities of

zero.


Figures 9.20 and 9.21 present the variation of the CoG x-coordinate and

z-coordinate, respectively.

Figure 9.20: ESA tank test C - MON-3 (20s simulation): CoG x-coord. Vs time.

The damping ratio found in this test was approximately ζ = 0.027 for both

the propellants.

104


Figure 9.21: ESA tank test C - MON-3 (20s simulation): CoG z-coord. Vs time.

However, the results from the previous test are believed to be more ac-

curate. This is so because, as mentioned before, some waves might be

artificially introduced in the sloshing dynamics at the begin of the simu-

lation. Therefore, in a shorter simulation, these are more dominant and

might influence the damping results, specially when no time is given for

stabilization of the lateral acceleration.

The mass participating in the liquid sloshing was roughly estimated using

ParaView and looking at the mesh elements that were moving. The pro-

vided result shall be taken as a very rough estimate and is only intended

to give an approximated idea of the percentage of mass participating in

the sloshing movement. Thus, it was found that about 30 % to 40 % of

the liquid mass inside the tank participated in the sloshing movement af-

ter the abrupt removal of the lateral acceleration. From [1] an analytical

estimation was calculated and found to be about 26 % - this value is in

good agreement with the obtained results.

105

Chapter 10

Conclusions

“The logic of validation allows us to move between the two limits

of dogmatism and scepticism.”

- Paul Ricoeur

The effects of propellant liquids sloshing inside spacecraft tanks have been

defined long ago as being of critical influence to the dynamics of space

vehicles. The interaction of the disturbance forces and torques generated

by the moving fuel with the solid body and the control system can lead

to an increase in the AOCS actuators commands, which can degrade the

vehicle’s pointing performances and, in critical cases, generate unstable

attitude and orbit control.

During years, simplifying analytical models were the only possible way

to predict the dynamics of liquids sloshing inside rigid containers. How-

ever, the necessity of ensuring the correct functioning of any vehicle sent

to space, together with the development of science and technology led to

the development of new tools that can be used to predict the vehicles be-

haviour. Thus, for the cases in which analytical approximate solutions are

not reasonable, two possibilities exist to accurately predict the behaviour

of liquids sloshing inside rigid tanks: an experimental approach - which

brings increased expense and complexity to any project; and a numerical

approach using CFD techniques - which allows early testing and facilitates

the achievement of accurate solutions for complex problems.

This project was focused on the implementation of a CFD code in an

existing EADS Astrium simulation environment. This code was used to

assess the influence of liquid propellant sloshing for specific missions.

106

10. Conclusions

Starting with a defined set of functional and system requirements, an ex-

tended selection process was performed to choose the most suitable CFD

software that would suit the objectives of this project. From a list of

more than 35 available open source CFD software based on FEM or FVM,

Elmer - Open Source Finite Element Software for Multiphysical Problems

was selected.

The CFD solver was integrated with the available pre- and post- processing

environments and the resulting simulation environment allowed a deep and

complete testing of liquid propellants sloshing inside rigid containers.

The geometry of the tanks can be meshed using available meshing software,

such as Gmsh and CubitTM

. For simple geometries, simpler meshing tools,

such as Elmer’s included tool - ElmerGrid, can also be used.

The sloshing problems can be fully defined using Elmer’s simple solver

input file (.Sif ), which is then input to ElmerSolver - the most important

part of Elmer’s package.

The results obtained from the solver can be exported and visualized with

ParaView, which allows a direct visual analysis of the sloshing behaviour.

If a deeper analysis is required, a specific sloshing post-processing based

on MATLAB R© can be used to obtain some of the most important slosh-

ing parameters, such as the liquid sloshing modes natural frequencies, the

sloshing wave amplitudes, the liquid’s damping and the mass of liquid par-

ticipating in the sloshing movement.

To validate the CFD software for sloshing purposes, specific test cases

were defined. The first and the second test cases dealt, respectively, with

rectangular and cylindrical laterally excited tanks, and the final test case -

defined by ESA’s HSO directorate - dealt with a laterally excited cylindrical

tank with hemispherical domes.

For test case 1, three different tests were defined, implemented and va-

lidated against available numerical and experimental data. For all these

tests the obtained results were proven to be in good agreement with the

available comparators.

In test case 2, the first natural antisymmetric sloshing frequency of a li-

quid sloshing inside a laterally excited cylindrical tank was obtained. The

achieved results were also compared and validated against data available

in the literature for similar tests.

107

10. Conclusions

In test case 3, a typical liquid propellant tank was defined and real mis-

sion conditions were simulated. The cylindrical tank with hemispherical

domes was subjected to a 1− g vertical acceleration and laterally excited.

Two different liquid propellants were evaluated - MON-3 and MMH - for

three different liquid fill ratios (25 %, 50 % and 75 %). The liquid’s natu-

ral sloshing frequencies were obtained for the different propellants and fill

ratios. The damping ratio and the mass of fluid participating in the slosh-

ing movement were also estimated. The obtained results were evaluated

and validated against simplified analytical results - a good agreement was

found.

As a final conclusion, the validation of the above mentioned sloshing test

cases together with the fulfilment of the defined requirements allowed the

validation of the final simulation environment for the sloshing problems

that were addressed - chapter 3.1. Hence, it was possible to conclude that

the objectives stipulated for this project were successfully accomplished.

108

Chapter 11

Future Work

“Learn from yesterday, live for today, hope for tomorrow. The

important thing is to not stop questioning.”

- Albert Einstein

The research presented in this thesis addressed and gave answers to several

open questions. Nevertheless, it also raised and left behind many other

important issues that still lack answers. Thus, several lines of investigation

arise from this work and shall be pursued in the future.

Firstly, the several issues related with the numerical problems experienced

during the development of this work shall be addressed. Issues such as mass

loss, numerical noise and unrealistic behaviours appearing at the first time

steps of the simulation (believed to be due to the used time integration

schemes) need to be carefully addressed to increase the robustness of the

software and the validity of the results of sloshing problems.

Rising directly from the future functional requirements defined in chapter 3,

several tests need to be performed to evaluate the limits of the solver for

problems related with liquid sloshing inside spacecraft tanks. More com-

plex tank geometries with internal elements and different configurations

shall be tested; different types of sloshing, such as rotary sloshing, shall

be addressed; sloshing due to high external excitations needs to be tested,

evaluated and validated; tests in micro-gravity environments, where sur-

face tension effects are dominant, need to be performed; and finally, fluid

structure interaction approaches shall be considered to simulate different

types of sloshing problems (e.g. non-isothermal fluids), as well as allow the

simulation of liquids sloshing inside non-rigid tanks.

109

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114

http://www.paraview.org/

Appendix A

Test case 1 - Results: Test A

115

b) 2D Skewed Tank

Figure A.1: Rectangular tank - test A b): pressure at t = 0s.

(a) t = 0.6s (b) t = 1.2s

(c) t = 1.8s

(d) t = 2.4s (e) t = 3.0s

Figure A.2: Rectangular sloshing tank - test A b): free surface shape evolution.

116

Figure A.3: Rectangular tank - test A b): CoG plots.

Figure A.4: Rectangular tank - test A b): sloshing amplitude plot.

117

c) Simple 3D Tank

Figure A.5: Rectangular tank - test A c): pressure at t = 0s.

(a) t = 0.6s (b) t = 1.2s

(c) t = 1.8s

(d) t = 2.4s (e) t = 3.0s

Figure A.6: Rectangular sloshing tank - test A c): free surface shape evolution.

118

Figure A.7: Rectangular tank - test A c): CoG plots.

Figure A.8: Rectangular tank - test A c): sloshing amplitude plot.

119

d) Simple 3D Skewed Tank

Figure A.9: Rectangular tank - test A d): pressure at t = 0s.

(a) t = 0.6s (b) t = 1.2s

(c) t = 1.8s

(d) t = 2.4s (e) t = 3.0s

Figure A.10: Rectangular sloshing tank - test A d): free surface shape evolution.

120

Figure A.11: Rectangular tank - test A d): CoG plots.

Figure A.12: Rectangular tank - test A d): sloshing amplitude plot.

121

Appendix B

Test case 1 - Results: Test C

122

h = 0.100 m

Figure B.1: Rectangular tank - test C - h = 0.100m longer direction: free surfaceshape at t = 0s.


Figure B.2: Rectangular tank - test C - h = 0.100m longer direction: CoG x-coord.Vs time for different excitation frequencies.

Figure B.3: Rectangular tank - test C - h = 0.100m longer direction: maximumwave amplitude (t = 3.56s).

123

h = 0.150 m





124

h = 0.200 m





125

h = 0.250 m





126

h = 0.100 m

Figure B.13: Rectangular tank - test C - h = 0.100m shorter direction: free surfaceshape at t = 0s.


Figure B.14: Rectangular tank - test C - h = 0.100m shorter direction: CoGx-coord. Vs time for different excitation frequencies.

Figure B.15: Rectangular tank - test C - h = 0.100m shorter direction: maximumwave amplitude (t = 1.48s).

127

h = 0.150 m


(a) t = 0s to t = 2.5s 125 time steps) (b) Zoom: t = 1.26s



128

h = 0.200 m





129

h = 0.250 m





130

Appendix C

Test case 2 - Results

131

h = 0.100 m

Figure C.1: Cylindrical tank test - h = 0.100m: free surface shape at t = 0s.


Figure C.2: Cylindrical tank test - h = 0.100m: CoG x-coordinate Vs time.

Figure C.3: Cylindrical tank test - h = 0.100m: maximum wave amplitude (t =1.58s).

132

h = 0.150 m





133

h = 0.200 m





134

Appendix D

Test case 3 - Results: Test A

135

MON-3 - 25 % fill ratio

Figure D.1: ESA tank test A - MON-3 (25% fill ratio & fext = 0.70 Hz): CoGplots.


136

Figure D.3: ESA tank test A - MON-3 (25% fill ratio & fext = 0.70 Hz): sloshingamplitude plot.


137

Figure D.5: ESA tank test A - MON-3 (25% fill ratio & fext = 0.70 Hz): PSD plot(measured at the top left point of the tank).

Figure D.6: ESA tank test A - MON-3 (25% fill ratio & fext = 0.70 Hz): PSD plot(measured at the top center point of the tank).

138

Figure D.7: ESA tank test A - MON-3 (25% fill ratio & fext = 1.50 Hz): PSD plot(measured at the top left point of the tank).

Figure D.8: ESA tank test A - MON-3 (25% fill ratio & fext = 1.50 Hz): PSD plot(measured at the top center point of the tank).

139




140




141



142

Figure D.15: ESA tank test A - MON-3 (75% fill ratio & fext = 0.70 Hz): PSDplot (measured at the top left point of the tank).

Figure D.16: ESA tank test A - MON-3 (75% fill ratio & fext = 0.70 Hz): PSDplot (measured at the top center point of the tank).

143

Figure D.17: ESA tank test A - MON-3 (75% fill ratio & fext = 1.50 Hz): PSDplot (measured at the top left point of the tank).

Figure D.18: ESA tank test A - MON-3 (75% fill ratio & fext = 1.50 Hz): PSDplot (measured at the top center point of the tank).

144

MMH - 25 % fill ratio

Figure D.19: ESA tank test A - MMH (25% fill ratio & fext = 0.70 Hz): CoGplots.


145

Figure D.21: ESA tank test A - MMH (25% fill ratio & fext = 0.70 Hz): sloshingamplitude plot.


146

Figure D.23: ESA tank test A - MMH (25% fill ratio & fext = 0.70 Hz): PSD plot(measured at the top left point of the tank).

Figure D.24: ESA tank test A - MMH (25% fill ratio & fext = 0.70 Hz): PSD plot(measured at the top center point of the tank).

147



148




149



150



151



152




153



154



155



156

Appendix E

Test case 3 - Results: Test B

157



Figure E.1: ESA tank test B - MMH (50% fill ratio): CoG x-coordinate Vs time.

Figure E.2: ESA tank test B - MMH (50% fill ratio): maximum wave amplitude(t = 3.46s).

158

Figure E.3: ESA tank test B - MMH (50% fill ratio): sloshing amplitude plot.

159

Appendix F

Test case 3 - Results: Test C

160


Figure F.1: ESA tank test C - MON-3: PSD plot (measured at the top right pointof the tank) for the 15 s period after the abrupt removal of a stabilized 0.1−g lateralperturbation.

Figure F.2: ESA tank test C - MON-3 (20s simulation): sloshing amplitude plot.

161


Figure F.3: ESA tank test C - MMH: PSD plot (measured at the top right point ofthe tank) for the 15 s period after the abrupt removal of a stabilized 0.1− g lateralperturbation.

Figure F.4: ESA tank test C - MON-3 (60s simulation): CoG x-coord. Vs time.

162

Figure F.5: ESA tank test C - MMH (60s simulation): wave amplitude at t = 30s.

Figure F.6: ESA tank test C - MMH (60s simulation): wave amplitude at t =45.46s.

163

Figure F.7: ESA tank test C - MMH (60s simulation): CoG z-coord. Vs time.

Figure F.8: ESA tank test C - MMH (20s simulation): wave amplitude at t = 0.52s.

164

Figure F.9: ESA tank test C - MMH (20s simulation): wave amplitude at t =19.58s.

Figure F.10: ESA tank test C - MMH (20s simulation): CoG x-coord. Vs time.

165

Figure F.11: ESA tank test C - MMH (20s simulation): CoG z-coord. Vs time.

Figure F.12: ESA tank test C - MMH (60s simulation): sloshing amplitude plot.

Figure F.13: ESA tank test C - MMH (20s simulation): sloshing amplitude plot.

166

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