Master's Thesis

MASTER'S THESIS

Implementation of a Computational Fluid Dynamics Code for Propellant Sloshing Analysis

Tiago Rebelo 2013

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

Luleå University of Technology Department 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

Implementation of a Computational Fluid Dynamics 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 - Lule˚aUniversity 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 “Sêtodo em cada coisa. Põequanto és No m´ınimoque 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 implementation 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 numerical and experimental results. The final test case is defined to reduce the gap between the simple test cases performed 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 Lule˚aUniversity 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 R˚aback 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 Lule˚aUniversity 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, Joãoand 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 7.2.3 Results & Evaluation ...... 62

x Contents

7.3 Test C ...... 69 7.3.1 Test Deﬁnition ...... 69 7.3.2 Implementation ...... 70 7.3.3 Results & Evaluation ...... 73

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 9.1.3 Results & Evaluation ...... 88 9.2 Test B ...... 94 9.2.1 Test Definition ...... 94 9.2.2 Implementation ...... 95 9.2.3 Results & Evaluation ...... 95 9.3 Test C ...... 98 9.3.1 Test Definition ...... 98 9.3.2 Implementation ...... 99 9.3.3 Results & Evaluation ...... 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 tank8 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

9.6 ESA tank test A - MON-3 (50% ﬁll ratio & fext = 0.70 Hz): PSD plot 2 ...... 92

9.7 ESA tank test A - MON-3 (50% ﬁll ratio & fext = 1.50 Hz): PSD plot 1 ...... 93

9.8 ESA tank test A - MON-3 (50% ﬁll ratio & fext = 1.50 Hz): PSD 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 B.4 Rect. tank - test C - h = 0.150m longer dir.: free surface shape at t = 0s ...... 124 B.5 Rect. tank - test C - h = 0.150m longer dir.: CoG x-coord. Vs time . 124 B.6 Rect. tank - test C - h = 0.150m longer dir.: max. wave amplitude . 124 B.7 Rect. tank - test C - h = 0.200m longer dir.: free surface shape at t = 0s ...... 125 B.8 Rect. tank - test C - h = 0.200m longer dir.: CoG x-coord. Vs time . 125 B.9 Rect. tank - test C - h = 0.200m longer dir.: max. wave amplitude . 125 B.10 Rect. tank - test C - h = 0.250m longer dir.: free surface shape at t = 0s ...... 126 B.11 Rect. tank - test C - h = 0.250m longer dir.: CoG x-coord. Vs time . 126 B.12 Rect. tank - test C - h = 0.250m longer dir.: max. wave amplitude . 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 B.16 Rect. tank - test C - h = 0.150m shorter dir.: free surface shape at t = 0s ...... 128 B.17 Rect. tank - test C - h = 0.150m shorter dir.: CoG x-coord. Vs time 128 B.18 Rect. tank - test C - h = 0.150m shorter dir.: max. wave amplitude . 128 B.19 Rect. tank - test C - h = 0.200m shorter dir.: free surface shape at t = 0s ...... 129 B.20 Rect. tank - test C - h = 0.200m shorter dir.: CoG x-coord. Vs time 129 B.21 Rect. tank - test C - h = 0.200m shorter dir.: max. wave amplitude . 129 B.22 Rect. tank - test C - h = 0.250m shorter dir.: free surface shape at t = 0s ...... 130 B.23 Rect. tank - test C - h = 0.250m shorter dir.: CoG x-coord. Vs time 130

xiv List of Figures

B.24 Rect. tank - test C - h = 0.250m shorter dir.: max. wave amplitude . 130

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 C.4 Cylindrical tank test - h = 0.150m: free surface shape at t = 0s ... 133 C.5 Cylindrical tank test - h = 0.150m: CoG x-coordinate Vs time . . . . 133 C.6 Cylindrical tank test - h = 0.150m: max. wave amplitude ...... 133 C.7 Cylindrical tank test - h = 0.200m: free surface shape at t = 0s ... 134 C.8 Cylindrical tank test - h = 0.200m: CoG x-coordinate Vs time . . . . 134 C.9 Cylindrical tank test - h = 0.200m: max. wave amplitude ...... 134

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

D.2 ESA tank test A - MON-3 (25% ﬁll ratio & fext = 1.50 Hz): CoG plots136

D.3 ESA tank test A - MON-3 (25% ﬁll ratio & fext = 0.70 Hz): sloshing amplitude plot ...... 137

D.4 ESA tank test A - MON-3 (25% ﬁll ratio & fext = 1.50 Hz): sloshing amplitude plot ...... 137

D.5 ESA tank test A - MON-3 (25% ﬁll ratio & fext = 0.70 Hz): PSD plot 1 ...... 138

D.6 ESA tank test A - MON-3 (25% ﬁll ratio & fext = 0.70 Hz): PSD plot 2 ...... 138

D.7 ESA tank test A - MON-3 (25% ﬁll ratio & fext = 1.50 Hz): PSD plot 1 ...... 139

D.8 ESA tank test A - MON-3 (25% ﬁll ratio & fext = 1.50 Hz): PSD plot 2 ...... 139

D.9 ESA tank test A - MON-3 (50% ﬁll ratio & fext = 1.50 Hz): CoG plots140

D.10 ESA tank test A - MON-3 (50% ﬁll ratio & fext = 1.50 Hz): sloshing amplitude plot ...... 140

D.11 ESA tank test A - MON-3 (75% ﬁll ratio & fext = 0.70 Hz): CoG plots141

D.12 ESA tank test A - MON-3 (75% ﬁll ratio & fext = 1.50 Hz): CoG plots141

D.13 ESA tank test A - MON-3 (75% ﬁll ratio & fext = 0.70 Hz): sloshing amplitude plot ...... 142

D.14 ESA tank test A - MON-3 (75% ﬁll ratio & fext = 1.50 Hz): sloshing amplitude plot ...... 142

D.15 ESA tank test A - MON-3 (75% ﬁll ratio & fext = 0.70 Hz): PSD plot 1 ...... 143

D.16 ESA tank test A - MON-3 (75% ﬁll ratio & fext = 0.70 Hz): PSD plot 2 ...... 143

xv List of Figures

D.17 ESA tank test A - MON-3 (75% ﬁll ratio & fext = 1.50 Hz): PSD plot 1 ...... 144

D.18 ESA tank test A - MON-3 (75% ﬁll ratio & fext = 1.50 Hz): PSD plot 2 ...... 144

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

D.20 ESA tank test A - MMH (25% ﬁll ratio & fext = 1.50 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

D.24 ESA tank test A - MMH (25% ﬁll ratio & fext = 0.70 Hz): PSD plot 2147

D.25 ESA tank test A - MMH (25% ﬁll ratio & fext = 1.50 Hz): PSD plot 1148

D.26 ESA tank test A - MMH (25% ﬁll ratio & fext = 1.50 Hz): PSD plot 2148

D.27 ESA tank test A - MMH (50% ﬁll ratio & fext = 0.70 Hz): CoG plots 149

D.28 ESA tank test A - MMH (50% ﬁll ratio & fext = 1.50 Hz): CoG plots 149

D.29 ESA tank test A- MMH (50% ﬁll ratio & fext = 0.70 Hz): sloshing amplitude plot ...... 150

D.30 ESA tank test A - MMH (50% ﬁll ratio & fext = 1.50 Hz): sloshing amplitude plot ...... 150

D.31 ESA tank test A - MMH (50% ﬁll ratio & fext = 0.70 Hz): PSD plot 1151

D.32 ESA tank test A - MMH (50% ﬁll ratio & fext = 0.70 Hz): PSD plot 2151

D.33 ESA tank test A - MMH (50% ﬁll ratio & fext = 1.50 Hz): PSD plot 1152

D.34 ESA tank test A - MMH (50% ﬁll ratio & fext = 1.50 Hz): PSD plot 2152

D.35 ESA tank test A - MMH (75% ﬁll ratio & fext = 0.70 Hz): CoG plots 153

D.36 ESA tank test A - MMH (75% ﬁll ratio & fext = 1.50 Hz): CoG plots 153

D.37 ESA tank test A- MMH (75% ﬁll ratio & fext = 0.70 Hz): sloshing amplitude plot ...... 154

D.38 ESA tank test A - MMH (75% ﬁll ratio & fext = 1.50 Hz): sloshing amplitude plot ...... 154

D.39 ESA tank test A - MMH (75% ﬁll ratio & fext = 0.70 Hz): PSD plot 1155

D.40 ESA tank test A - MMH (75% ﬁll ratio & fext = 0.70 Hz): PSD plot 2155

D.41 ESA tank test A - MMH (75% ﬁll ratio & fext = 1.50 Hz): PSD plot 1156

D.42 ESA tank test A - MMH (75% ﬁll ratio & fext = 1.50 Hz): PSD plot 2156

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.8 ESA tank test C - MMH (20s simulation): wave amplitude at t = 0.52s164 F.9 ESA tank test C - MMH (20s simulation): wave amplitude at t = 19.58s165 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

GP L General Public License

GUI Graphical User Interface

HSO Human Spaceﬂight and Operations

ILU Incomplete LU

xix List of Abbreviations

LAP ACK Linear Algebra Package

LGP L 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

T F QMR Transpose-Free Quasi-Minimal Residual

UMF P ACK Unsymmetric Multifrontal Sparse LU Factorization Package

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 satellites 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 questions 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 generate unstable attitude and orbit control.

This means that it is of major importance to accurately predict the behaviour 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 influence 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 examples;

• Validate the simulation environment for the selected examples.

1.3 Outline

• Chapter1 introduces the topic of this M.Sc. thesis, presents its aim and the general objectives expected to be achieved during its development.

• Chapter2 presents a detailed but not exhaustive literature review about the sloshing and CFD topics.

• Chapter3 brieﬂy introduces the requirements for the project.

• Chapter4 presents the deep state-of-the-art investigation developed to choose the most suitable CFD software for the purposes of this project.

• Chapter5 gives a brief introduction to the chosen CFD software - Elmer.

3 1. Introduction

• Chapter6 presents the setup of the complete simulation environment.

• Chapter7 introduces, deﬁnes, explains the implementation, presents the results and evaluates the tests performed for the rectangular tank test case.

• Chapter8 follows the same path of Chapter7 and introduces the cylindrical tank test case.

• Chapter9 similarly to chapters7 and8 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 concept - section 2.1, followed by a general overview of the theory behind CFD - section 2.2.

2.1 Sloshing

As mentioned in chapter1, 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 phenomenon 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 simple tank geometries subjected to small external excitations under accelerated 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 2. Literature Review

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 respective 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 2. Literature Review

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

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 important 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 propellant 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 2. Literature Review

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]

g h ω2 = π(2n − 1) tanh π(2n − 1) (2.1) n a a

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]

g h ω2 = 2mπ tanh 2mπ (2.2) m a a

Where: m: is the mode number

9 2. Literature Review

Equations (2.1) and (2.2) present the natural frequencies for the antisymmetric 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]

2 ωmn = gKmn tanh (Kmnh) (2.3)

Where: q (2m)2 (2n)2 Kmn = π a2 + 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 antisymmetric 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):

! r5 g r5 h ω2 = 2 tanh 2 (2.4) 1 2 L 2 L

Where: L: is the length of the rectangular tank along the direction of excitation

10 2. Literature Review

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):

! r27 g r27 h ω2 = 2 tanh 2 (2.5) 1 8 D 8 D

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 2. Literature Review

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.

ρvL Re = (2.6) µ

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 2. Literature Review

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 examples 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 introduce 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 predominance 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 Eötvös) number - equation (2.7), is used. This dimensionless number is normally used as the most common comparison ratio for gravity and surface tension forces.

ρaL2 Bo = (2.7) γ

Where: ρ: is the density of the ﬂuid

13 2. Literature Review

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 effects. 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 dynamics. 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 become 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 computer science. The dynamics of a ﬂuid can be characterized by mathematical 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 2. Literature Review

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, incompressible and isothermal are presented here for reference - further in the development 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 2. Literature Review

• Continuity equation:

∇ · (ρV~ ) = 0 (2.8)

• Momentum equations (or Navier-Stokes equations):

∂p ∂τ ∂τ ∂τ ∇ · (ρuV~ ) = − + xx + yx + zx + ρf (2.9) ∂x ∂x ∂y ∂z x

∂p ∂τ ∂τ ∂τ ∇ · (ρvV~ ) = − + xy + yy + zy + ρf (2.10) ∂y ∂x ∂y ∂z y

∂p ∂τ ∂τ ∂τ ∇ · (ρwV~ ) = − + xz + yz + zz + ρf (2.11) ∂z ∂x ∂y ∂z z

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

~ ∂u τxx = λ∇ · V + 2µ ∂x : is the the shear stress xx component

~ ∂v τyy = λ∇ · V + 2µ ∂y : is the the shear stress yy component

~ ∂w τzz = λ∇ · V + 2µ ∂z : is the the shear stress zz component

∂v ∂u τxy = τyx = µ ∂x + ∂y : is the the shear stress xy or yx component

∂u ∂w τxz = τzx = µ ∂z + ∂x : is the the shear stress xz or zx component

∂w ∂v τyz = τzy = µ ∂y + ∂z : is the the shear stress yz or zy component

µ: is the molecular viscosity

λ: is the bulk viscosity

16 2. Literature Review

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 ρf · 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 governing 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 related with velocity and pressure. The velocity components and/or the pressure are deﬁned at the boundary.

• Neuman boundary conditions - give the boundary conditions related 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 characterized 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 2. Literature Review same number of neighbours. The unstructured ones instead, have an irreg- ular connectivity where each point can have a diﬀerent number of neighbours. [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. Literature Review

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 methods 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 2. Literature Review

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 necessarily 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 2. Literature Review

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 element 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 instead 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 2. Literature Review

Some important advantages of FEM over FVM are the ease to obtain high order accuracy results and also the ease to implement boundary conditions. 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 compressible 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 viscous 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 2. Literature Review

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 obtained 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, preconditioning is related with the reduction of a condition number of the problem. Iterative methods are usually used to solve preconditioned problems. 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 2. Literature Review

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) represents 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 equations. 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 properties 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 2. Literature Review

For temporal problems the existence of stability guarantees that the method yields a bounded solution whenever the exact solution is bounded. For iterative 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 calculations of the algebraic equations need to be considered. These require attentive analysis and if they are to be minimized, systematic steps to perform numerical analysis, such as grid independence veriﬁcation and validation 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. Literature Review

2.3 Summary

The literature review presented in this chapter (chapter2) 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 parameters 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 decrease 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 important 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 governing equations are based on the mass, the momentum and the energy conservation principles - subsection 2.2.1

26 2. Literature Review

• 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 equations. 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 conclusions taken from the literature review presented in chapter2.

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 accelerations;

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 models (accurate for laminar Reynolds regimes); 6. The results of the CFD software shall be accurate in 1-g conditions; 7. The results of the CFD software shall be validated for the following 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 accelerations; 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 dominant in the liquid’s behaviour).

3.2 System Requirements

• Basic success

1. The CFD software shall be based on available open source software; 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 operating 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 4. CFD Software Selection

• Mesh elements type (support for at least hexahedral and/or tetrahedral);

• 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 disadvantages 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] • DUNS [24]

• arb [16] • Dolfyn [25]

• CFD2D [17] • Edge [26]

• CFD2k [18] • Elmer [27]

• cfdpack [19] • Featﬂow [28]

• Channelﬂow [20] • FEniCS Project [29]

• CLAWPACK [21] • freeFEM++ [30]

• Code Saturne [22] • Fluidity [31]

• COOLFluiD [23] • HiFlow3 [32]

32 4. CFD Software Selection

• Gerris Flow Solver [33] • OpenLB [44]

• hit3d [34] • OpenFVM [45]

• iNavier [35] • PartenovCFD [46] • ISAAC [36] • PETSc-FEM [47] • Kicksey-Winsey [37] • SLFCFD [48] • MFIX [38] • SSIIM [49] • NaSt2D-2.0 [39] • SU2 [50] • NEK5000 [40] • Tochnog [51] • NUWTUN [41]

• OpenFlower [42] • TYCHO [52]

• OpenFOAM [43] • 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 commercial 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 another software).

33 4. CFD Software Selection Support Documentation Installation Guide Theory Guide Practical User’s Guide Tutorials Course materials Models, Solvers, Parameters, Grid, GUI and MATC Manuals Tutorials Course materials Book Tutorials Course materials Manual User’s Guide Developer’s Guide Course materials User Guide Course materials Tutorials User’s Guide Developer’s Guide Tutorials Course materials User Manual and Description of Algorithms Book User Manual User’s technical support Post- Processing ParaView Paraview Gnuplot Octave Matlab ParaView ParaView ParaView Fluent Tecplot3D ParaView Tecplot None (Output as text ﬁle) EnSight Tecplot Mesh input format I-deas universal Gmsh ANSYS Gmsh Abaqus I-deas universal Gmsh Abaqus Exodus Gmsh Exodus ANSYS Fluent Gmsh I-deas universal CGNS (data standard) None No info Programming Languages Fortran C Python scripts Fortran (Solver) C/C++ (GUI) C++ Python UFL C/C++ Python Fortran C++ C++ Python scripts Fortran N/A License GPL GPL LGPL LGPL GPL GPL Astrium’s in-house use only Commercial Copyright Table 4.1: Second phase: satisfactory codes - codes and characteristics. Maturity Development started: 1997 Main developer: EDF Open source released: 2007 Development started: 1995 Main developer: CSCOpen - source IT released: Center 2005 for Science Development started: 2003 Main developer: SimulaOpen Research source Laboratory released: 2003 Development started: 1990s Main developer: ImperialOpen College source London released: No info Development started: 1980s Main developer: OpenFOAMOpen foundation source released: 2004 Development started: 2012 Main developer: StanfordOpen University source released: 2012 Development started: 1970s Main developer: Astrium Open source released: N/A Development started: 1980 Main developer: FlowOpen Science, source Inc. released: N/A Saturne 2 Code Elmer FEniCS Fluidity OpenFOAM SU Astrium SLOSHC FLOW-3D

34 4. CFD Software Selection

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.

Numerical Mesh Elements Supported Surface Tension Used for Sloshing Methods Type Parallelization Modelled Purposes Hexahedral None found Code Saturne FVM Yes No info Tetrahedral Hexahedral Similar purposes Elmer FEM Yes Yes Tetrahedral None found FEniCS FEM Tetrahedral Yes No info Hexahedral None found Fluidity FVM Yes No info Tetrahedral Hexahedral Yes OpenFOAM FVM Yes Yes Tetrahedral Hexahedral None found SU2 FVM Yes No info Tetrahedral Astrium Rectangular Yes FEM N/A No SLOSHC Triangular FEM Hexahedral Yes FLOW-3D Yes Yes FVM Tetrahedral

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 4. CFD Software Selection

Elmer:

• Advantages: • Disadvantages: - FEM based; - Programming language: Fortran; - Supports hexahedral mesh elements; - Not so good for high Reynolds. - Surface tension modelled; - Very accurate for low Reynolds; - On going development; - Used for similar purposes; - Support from developers.

FEniCS:

• Advantages: • Disadvantages: - FEM based; - Does not support hexahedral mesh - Programming language: elements; C/C++, Python, UFL; - Surface tension not modelled; - Support from developers. - Never used for similar purposes.

OpenFOAM:

• Advantages: • Disadvantages: - Supports hexahedral mesh elements; - FVM based; - Surface tension modelled; - Not so good for low Reynolds. - Programming language: C++; - Used for liquid sloshing analysis; - Support from developers.

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 4. CFD Software Selection

Based on the extensive literature review presented in chapter2, 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 5. Elmer - Open Source Finite Element Software

• 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 residual free bubbles and Streamline-Upwind Petrov-Galerkin (SUPG);

• Adaptivity, particularly in 2D;

• Boundary Element Method (BEM) solvers (without multipole acceleration).

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. Elmer - Open Source Finite Element Software

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 systems. These are brieﬂy introduced here together with the time discretization 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 5. Elmer - Open Source Finite Element Software

• 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 categories, 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 5. Elmer - Open Source Finite Element Software

Multilevel methods

Even though multilevel methods (introduced in chapter 2.2.5) can be applied 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 characteristics 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 approach 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. Elmer - Open Source Finite Element Software

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 5. Elmer - Open Source Finite Element Software

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 parameters 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 5. Elmer - Open Source Finite Element Software

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. Elmer - Open Source Finite Element Software

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 generator and mesh manipulation utility. It has the capability to read meshes generated by other programs and manipulate and convert them to a format 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 5. Elmer - Open Source Finite Element Software

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 capability 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 simulation 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 machine player - VMware R PlayerTM , compatible with several operating systems, 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 chapter5, 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 6. Simulation Environment Setup

As seen in figure 6.2, the pre-processing part of the simulation flow is responsible 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 responsible for presenting the results and developing further analysis. In the following sections, the used pre- and post- processing methods are described.

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 provided 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 6. Simulation Environment Setup

• 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 numerical 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 primarily 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 P a · s

– ρ− = 1000.0 kg/m3

• Fluid 2:

– µ+ = 0.01 P a · 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. Test case 1: Rectangular Tank

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 discussed 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 appendixA.

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 chapter6 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 7. Test case 1: Rectangular Tank

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 P a 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 7. Test case 1: Rectangular Tank

(a) t = 0.6s (b) t = 1.2s

(d) t = 2.4s (e) t = 3.0s Figure 7.2: Rectangular sloshing tank - test A a): free surface shape evolution.

57 7. Test case 1: Rectangular Tank

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 7. Test case 1: Rectangular Tank

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 % • c) ≈ 2 %

• b) ≈ 12 % • 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 frequencies obtained for tests A a), b), c) and d).

Table 7.1: Results - rectangular tank: test A. First natural antisymmetric Test sloshing frequency [Hz] a) 0.279 b) 0.283 c) 0.279 d) 0.283

59 7. Test case 1: Rectangular Tank

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 antisymmetric sloshing frequency [Hz] Abramson [5] 0.282 Fries [69] 0.279 Rasthofer et al. [68] 0.274 Elmer 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

7.2.1 Test Deﬁnition

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 7. Test case 1: Rectangular Tank

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.

7.2.2 Implementation

Test B was implemented using two diﬀerent techniques: one more experimentally oriented - a) and another more convenient for numerical simulations - 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 amplitude 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 antisymmetric 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 P a·s) and the fluid at the free surface was defined to be air (µair = 1.0·10−5 P a · 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 7. Test case 1: Rectangular Tank

It is important to note that with this approach only the first natural antisymmetric 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 2 A0 = 0.5 m/s 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 convenience, 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 sloshing tools. Using the PSD, it was possible to recognize the diﬀerent sloshing 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.

7.2.3 Results & Evaluation

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 pressure 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 presented 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 P a, was obtained for both cases.

62 7. Test case 1: Rectangular Tank

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 maximum 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 7. Test case 1: Rectangular Tank

Figure 7.7 presents the CoG x-displacement for the three excitation frequencies 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 diﬀerent excitation 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 position of the CoG of the liquid is presented in ﬁgure 7.9.

64 7. Test case 1: Rectangular Tank

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-direction 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 = 1.20 Hz.

65 7. Test case 1: Rectangular Tank

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 7. Test case 1: Rectangular Tank

Figure 7.12: Rectangular tank - test B b): PSD plot (measured at the top left point of the tank).

Figure 7.13: Rectangular tank - test B b): PSD plot (measured at the top center point 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 7. Test case 1: Rectangular Tank

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 1 steps (N) or the time step (∆t) itself - frequency resolution ∆f = N·∆t .

As a summary, the obtained results for the sloshing frequencies are presented 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 0.857 m=1 - 1.238 n=2 - 1.524 m=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 7. Test case 1: Rectangular Tank

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 / 0.86 m=1 1.249 1.26 1.253 1.238 n=2 1.530 1.54 1.545 1.524 m=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

7.3.1 Test Deﬁnition

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 P a · s – ρwater = 1000.0 kg/m3

• Air:

– µair = 1.0 · 10−5 P a · s – ρair = 1.2 kg/m3

69 7. Test case 1: Rectangular Tank

Slip boundary conditions were deﬁned along the tank walls and a no-slip 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

7.3.2 Implementation

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

2 • A0 = 0.2 m/s ;

• fext = 1.20 Hz to fext = 1.28 Hz;

• Mesh elements: 20 × 10 × 7.

70 7. Test case 1: Rectangular Tank

ii) h = 0.100 m

2 • A0 = 0.2 m/s ;

• fext = 1.50 Hz to fext = 1.58 Hz;

• Mesh elements: 20 × 10 × 14.

iii) h = 0.150 m

2 • A0 = 0.3 m/s ;

• fext = 1.60 Hz to fext = 1.68 Hz;

• Mesh elements: 20 × 10 × 21.

iv) h = 0.200 m

2 • A0 = 0.3 m/s ;

• fext = 1.64 Hz to fext = 1.72 Hz;

• Mesh elements: 20 × 10 × 28.

v) h = 0.250 m

2 • A0 = 0.3 m/s ;

• fext = 1.64 Hz to fext = 1.72 Hz;

• 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

2 • A0 = 0.6 m/s ;

• fext = 2.14 Hz to fext = 2.22 Hz;

• Mesh elements: 10 × 20 × 7.

71 7. Test case 1: Rectangular Tank

ii) h = 0.100 m

2 • A0 = 0.6 m/s ;

• fext = 2.32 Hz to fext = 2.40 Hz;

• Mesh elements: 10 × 20 × 14.

iii) h = 0.150 m

2 • A0 = 0.7 m/s ;

• fext = 2.34 Hz to fext = 2.42 Hz;

• Mesh elements: 10 × 20 × 21.

iv) h = 0.200 m

2 • A0 = 0.7 m/s ;

• fext = 2.34 Hz to fext = 2.42 Hz;

• Mesh elements: 10 × 20 × 28.

v) h = 0.250 m

2 • A0 = 0.7 m/s ;

• fext = 2.34 Hz to fext = 2.42 Hz;

• Mesh elements: 10 × 20 × 35.

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 7. Test case 1: Rectangular Tank

7.3.3 Results & Evaluation

The results obtained for test C are presented and discussed in this subsection. 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 appendixB.

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

(d) h = 0.200m (e) h = 0.250m Figure 7.14: Rectangular tank - test C: pressure at t = 0s for diﬀerent water depths.

73 7. Test case 1: Rectangular Tank

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 surface shape 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 surface shape 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 direction 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 7. Test case 1: Rectangular Tank

(a) t = 0s to t = 5s (250 time steps) (b) Zoom: t = 2.82s 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: CoG x-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: maximum wave amplitude (t = 2.82s).

75 7. Test case 1: Rectangular Tank

Figure 7.20: Rectangular tank - test C - h = 0.050m shorter direction: maximum wave 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 2.15 0.100 1.53 2.33 0.150 1.62 2.35 0.200 1.66 2.35 0.250 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 of First natural antisymmetric sloshing frequencies [Hz] water [m] Jaiswal et al. [6] Housner [4] Jaiswal et al. [6] Elmer - Experimental - ANSYS R 0.050 1.238 1.26 1.23 1.23 0.100 1.549 1.50 1.53 1.53 0.150 1.656 1.60 1.62 1.62 0.200 1.690 1.64 1.66 1.66 0.250 1.701 1.70 1.68 1.68

76 7. Test case 1: Rectangular Tank

Table 7.7: Comparison of results - rectangular tank: test C - along shorter direction. Depth of First natural antisymmetric sloshing frequencies [Hz] water [m] Jaiswal et al. [6] Housner [4] Jaiswal et al. [6] Elmer - Experimental - ANSYS R 0.050 2.191 2.17 2.15 2.15 0.100 2.390 2.33 2.33 2.33 0.150 2.410 2.38 2.35 2.35 0.200 2.412 2.40 2.35 2.35 0.250 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 solutions 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 chapter7, 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 P a · s – ρwater = 1000.0 kg/m3

• Air:

– µair = 1.0 · 10−5 P a · s – ρair = 1.2 kg/m3

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 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 • h3 = 0.150 m

• h2 = 0.100 m • h4 = 0.200 m

8.2 Implementation

Similarly to test C performed for the rectangular tank, an experimental approach 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 solutions 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

2 • A0 = 0.6 m/s ;

• fext = 1.96 Hz to fext = 2.02 Hz;

• Mesh elements: 6144.

79 8. Test case 2: Cylindrical Tank

ii) h = 0.100 m

2 • A0 = 0.6 m/s ;

• fext = 2.16 Hz to fext = 2.30 Hz;

• Mesh elements: 12288.

iii) h = 0.150 m

2 • A0 = 0.6 m/s ;

• fext = 2.20 Hz to fext = 2.32 Hz;

• Mesh elements: 18432.

iv) h = 0.200 m

2 • A0 = 0.6 m/s ;

• fext = 2.20 Hz to fext = 2.32 Hz;

• 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 subsection. 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 appendixC.

The procedure adopted in tests B a) and C of the rectangular tank test case (chapter7) 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 8. Test case 2: Cylindrical Tank

(a) h = 0.050m (b) h = 0.100m

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 8. Test case 2: Cylindrical Tank

The same resulting plots presented for tests B a) and C of the rectangular tank test case (chapter7) 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 maximum wave sloshing amplitude is marked and zoomed to identify the correspondent external frequency, which is then equal to the ﬁrst natural antisymmetric sloshing frequency.

(a) t = 0s to t = 2s (100 time steps) (b) Zoom: t = 1.48s 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 8. Test case 2: Cylindrical Tank

Table 8.1: Results - cylindrical tank test. First natural antisymmetric Depth of water [m] sloshing frequency [Hz] 0.050 1.99 0.100 2.18 0.150 2.22 0.200 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 of First natural antisymmetric sloshing frequencies [Hz] water [m] Jaiswal et al. [6] Housner [4] Jaiswal et al. [6] Elmer - Experimental - ANSYS R 0.050 2.064 2.07 1.99 1.99 0.100 2.287 2.30 2.18 2.18 0.150 2.314 2.33 2.22 2.22 0.200 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 complex 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 solutions for the liquid sloshing parameters. Nevertheless, rough results for some of the parameters can be obtained using approximated tank geometries.

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 commonly used liquid propellants sloshing inside the tank;

• Test B: Accurately determine the ﬁrst antisymmetric sloshing frequency 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

9.1.1 Test Deﬁnition

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 P a · s

• ρ = 3.2312 kg/m3

85 9. Test case 3: ESA Tank

Two diﬀerent propellant liquids were studied, their properties were:

• MON-3:

– µ = 3.967441 · 10−4 P a · s – ρ = 1433.401 kg/m3

• MMH:

– µ = 7.78024 · 10−4 P a · 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 condition 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.

9.1.2 Implementation

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 antisymmetric 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 9. Test case 3: ESA Tank

• 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 cylindrical tank was matched with that of a rectangular tank with the same height of liquid - Abramson [5] (chapter 2.1.1). The corresponding dimension of the rectangular tank along the excitation direction was found to be a = 0.972 m. With this dimension as a reference, the different sloshing 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 9. Test case 3: ESA Tank

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 2 2 found to be A0 = 0.35 m/s for a 25 % fill ratio and A0 = 0.7 m/s 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 unstructured 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 elements. 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 elements 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

• 50 % ﬁll ratio - 14528 elements

• 75 % ﬁll ratio - 12600 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.

9.1.3 Results & Evaluation

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 9. Test case 3: ESA Tank

(a) 25% (b) 50%

89 9. Test case 3: ESA Tank

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 obtained for all the other developed tests can be found in appendixD.

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): CoG plots.

90 9. Test case 3: ESA Tank

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): sloshing amplitude 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 9. Test case 3: ESA Tank

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 9. Test case 3: ESA Tank

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 9. Test case 3: ESA Tank

A deep evaluation of the presented PSD plots allowed to draw some conclusions 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

9.2.1 Test Deﬁnition

This test was defined to accurately determine the first antisymmetric natural sloshing frequency for the different liquid propellants sloshing inside the

94 9. Test case 3: ESA Tank

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.

9.2.2 Implementation

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 2 acceleration was deﬁned to be A0 = 0.981 m/s .

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.

9.2.3 Results & Evaluation

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 appendixE.

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 frequency, which corresponds to the maximum sloshing wave without rotation - see chapter 7.2.3 for details.

95 9. Test case 3: ESA Tank

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.

(a) t = 0s to t = 4s (200 time steps) (b) Zoom: t = 3.46s 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 propellant.

96 9. Test case 3: ESA Tank

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. First antisymmetric Liquid propellant natural sloshing frequency [Hz] MON-3 0.86 MMH 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 9. Test case 3: ESA Tank

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 results 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

9.3.1 Test Deﬁnition

Test C had two main objectives, being the ﬁrst one obtaining the liquid propellants’ damping ratio and the second one identifying the mass of propellant 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 9. Test case 3: ESA Tank

9.3.2 Implementation

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 amplitude 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 % .

9.3.3 Results & Evaluation

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 appendixF.

Similarly to what was done in chapter7 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 9. Test case 3: ESA Tank

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 at t = 30s.

100 9. Test case 3: ESA Tank

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 at t = 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 9. Test case 3: ESA Tank

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 example, 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 validated 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 9. Test case 3: ESA Tank

were numerically obtained are most probably overestimated.

This overestimation of the results can be explained by the mass loss phenomenon. 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 appendixF.

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 acceleration was abruptly removed.

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

103 9. Test case 3: ESA Tank

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.

Figure 9.19: ESA tank test C - MON-3 (20s simulation): wave amplitude at t = 19.60s.

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 9. Test case 3: ESA Tank

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 accurate. This is so because, as mentioned before, some waves might be artiﬁcially introduced in the sloshing dynamics at the begin of the simulation. 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 provided 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 after 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 behaviour. 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 extended 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 sloshing parameters, such as the liquid sloshing modes natural frequencies, the sloshing wave amplitudes, the liquid’s damping and the mass of liquid participating 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 validated 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 liquid 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 mission 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 natural sloshing frequencies were obtained for the different propellants and fill ratios. The damping ratio and the mass of fluid participating in the sloshing 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 chapter3, several tests need to be performed to evaluate the limits of the solver for problems related with liquid sloshing inside spacecraft tanks. More complex 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 surface 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.

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114 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

(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

(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

(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 surface shape at t = 0s.

(a) t = 0s to t = 5s (250 time steps) (b) Zoom: t = 3.56s 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: maximum wave amplitude (t = 3.56s).

123 h = 0.150 m

Figure B.4: Rectangular tank - test C - h = 0.150m longer direction: free surface shape at t = 0s.

(a) t = 0s to t = 5s (250 time steps) (b) Zoom: t = 3.06s Figure B.5: Rectangular tank - test C - h = 0.150m longer direction: CoG x-coord. Vs time for diﬀerent excitation frequencies.

Figure B.6: Rectangular tank - test C - h = 0.150m longer direction: maximum wave amplitude (t = 3.06s).

124 h = 0.200 m

Figure B.7: Rectangular tank - test C - h = 0.200m longer direction: free surface shape at t = 0s.

(a) t = 0s to t = 5s (250 time steps) (b) Zoom: t = 3.30s Figure B.8: Rectangular tank - test C - h = 0.200m longer direction: CoG x-coord. Vs time for diﬀerent excitation frequencies.

Figure B.9: Rectangular tank - test C - h = 0.200m longer direction: maximum wave amplitude (t = 3.30s).

125 h = 0.250 m

Figure B.10: Rectangular tank - test C - h = 0.250m longer direction: free surface shape at t = 0s.

(a) t = 0s to t = 5s (250 time steps) (b) Zoom: t = 2.66s Figure B.11: Rectangular tank - test C - h = 0.250m longer direction: CoG x-coord. Vs time for diﬀerent excitation frequencies.

Figure B.12: Rectangular tank - test C - h = 0.250m longer direction: maximum wave amplitude (t = 2.66s).

126 h = 0.100 m

Figure B.13: Rectangular tank - test C - h = 0.100m shorter direction: free surface shape at t = 0s.

(a) t = 0s to t = 2.5s (125 time steps) (b) Zoom: t = 1.48s Figure B.14: Rectangular tank - test C - h = 0.100m shorter direction: CoG x-coord. Vs time for diﬀerent excitation frequencies.

Figure B.15: Rectangular tank - test C - h = 0.100m shorter direction: maximum wave amplitude (t = 1.48s).

127 h = 0.150 m

Figure B.16: Rectangular tank - test C - h = 0.150m shorter direction: free surface shape at t = 0s.

(a) t = 0s to t = 2.5s 125 time steps) (b) Zoom: t = 1.26s Figure B.17: Rectangular tank - test C - h = 0.150m shorter direction: CoG x-coord. Vs time for diﬀerent excitation frequencies.

Figure B.18: Rectangular tank - test C - h = 0.150m shorter direction: maximum wave amplitude (t = 1.26s).

128 h = 0.200 m

Figure B.19: Rectangular tank - test C - h = 0.200m shorter direction: free surface shape at t = 0s.

(a) t = 0s to t = 2.5s (125 time steps) (b) Zoom: t = 1.26s Figure B.20: Rectangular tank - test C - h = 0.200m shorter direction: CoG x-coord. Vs time for diﬀerent excitation frequencies.

Figure B.21: Rectangular tank - test C - h = 0.200m shorter direction: maximum wave amplitude (t = 1.26s).

129 h = 0.250 m

Figure B.22: Rectangular tank - test C - h = 0.250m shorter direction: free surface shape at t = 0s.

(a) t = 0s to t = 2.5s (125 time steps) (b) Zoom: t = 1.26s Figure B.23: Rectangular tank - test C - h = 0.250m shorter direction: CoG x-coord. Vs time for diﬀerent excitation frequencies.

Figure B.24: Rectangular tank - test C - h = 0.250m shorter direction: maximum wave amplitude (t = 1.26s).

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.

(a) t = 0s to t = 2s (100 time steps) (b) Zoom: t = 1.58s 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

Figure C.4: Cylindrical tank test - h = 0.150m: free surface shape at t = 0s.

(a) t = 0s to t = 2s (100 time steps) (b) Zoom: t = 1.78s Figure C.5: Cylindrical tank test - h = 0.150m: CoG x-coordinate Vs time.

Figure C.6: Cylindrical tank test - h = 0.150m: maximum wave amplitude (t = 1.78s).

133 h = 0.200 m

Figure C.7: Cylindrical tank test - h = 0.200m: free surface shape at t = 0s.

(a) t = 0s to t = 2s (100 time steps) (b) Zoom: t = 1.78s Figure C.8: Cylindrical tank test - h = 0.200m: CoG x-coordinate Vs time.

Figure C.9: Cylindrical tank test - h = 0.200m: maximum wave amplitude (t = 1.78s).

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): CoG plots.

Figure D.2: ESA tank test A - MON-3 (25% ﬁll ratio & fext = 1.50 Hz): CoG plots.

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

Figure D.4: ESA tank test A - MON-3 (25% ﬁll ratio & fext = 1.50 Hz): sloshing amplitude 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 MON-3 - 50 % ﬁll ratio

Figure D.9: ESA tank test A - MON-3 (50% ﬁll ratio & fext = 1.50 Hz): CoG plots.

Figure D.10: ESA tank test A - MON-3 (50% ﬁll ratio & fext = 1.50 Hz): sloshing amplitude plot.

140 MON-3 - 75 % ﬁll ratio

Figure D.11: ESA tank test A - MON-3 (75% ﬁll ratio & fext = 0.70 Hz): CoG plots.

Figure D.12: ESA tank test A - MON-3 (75% ﬁll ratio & fext = 1.50 Hz): CoG plots.

141 Figure D.13: ESA tank test A - MON-3 (75% ﬁll ratio & fext = 0.70 Hz): sloshing amplitude plot.

Figure D.14: ESA tank test A - MON-3 (75% ﬁll ratio & fext = 1.50 Hz): sloshing amplitude plot.

142 Figure D.15: ESA tank test A - MON-3 (75% ﬁll ratio & fext = 0.70 Hz): PSD plot (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): PSD plot (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): PSD plot (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): PSD plot (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): CoG plots.

Figure D.20: ESA tank test A - MMH (25% ﬁll ratio & fext = 1.50 Hz): CoG plots.

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

Figure D.22: ESA tank test A - MMH (25% ﬁll ratio & fext = 1.50 Hz): sloshing amplitude 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 Figure D.25: ESA tank test A - MMH (25% ﬁll ratio & fext = 1.50 Hz): PSD plot (measured at the top left point of the tank).

Figure D.26: ESA tank test A - MMH (25% ﬁll ratio & fext = 1.50 Hz): PSD plot (measured at the top center point of the tank).

148 MMH - 50 % ﬁll ratio

Figure D.27: ESA tank test A - MMH (50% ﬁll ratio & fext = 0.70 Hz): CoG plots.

Figure D.28: ESA tank test A - MMH (50% ﬁll ratio & fext = 1.50 Hz): CoG plots.

149 Figure D.29: ESA tank test A - MMH (50% ﬁll ratio & fext = 0.70 Hz): sloshing amplitude plot.

Figure D.30: ESA tank test A - MMH (50% ﬁll ratio & fext = 1.50 Hz): sloshing amplitude plot.

150 Figure D.31: ESA tank test A - MMH (50% ﬁll ratio & fext = 0.70 Hz): PSD plot (measured at the top left point of the tank).

Figure D.32: ESA tank test A - MMH (50% ﬁll ratio & fext = 0.70 Hz): PSD plot (measured at the top center point of the tank).

151 Figure D.33: ESA tank test A - MMH (50% ﬁll ratio & fext = 1.50 Hz): PSD plot (measured at the top left point of the tank).

Figure D.34: ESA tank test A - MMH (50% ﬁll ratio & fext = 1.50 Hz): PSD plot (measured at the top center point of the tank).

152 MMH - 75 % ﬁll ratio

Figure D.35: ESA tank test A - MMH (75% ﬁll ratio & fext = 0.70 Hz): CoG plots.

Figure D.36: ESA tank test A - MMH (75% ﬁll ratio & fext = 1.50 Hz): CoG plots.

153 Figure D.37: ESA tank test A - MMH (75% ﬁll ratio & fext = 0.70 Hz): sloshing amplitude plot.

Figure D.38: ESA tank test A - MMH (75% ﬁll ratio & fext = 1.50 Hz): sloshing amplitude plot.

154 Figure D.39: ESA tank test A - MMH (75% ﬁll ratio & fext = 0.70 Hz): PSD plot (measured at the top left point of the tank).

Figure D.40: ESA tank test A - MMH (75% ﬁll ratio & fext = 0.70 Hz): PSD plot (measured at the top center point of the tank).

155 Figure D.41: ESA tank test A - MMH (75% ﬁll ratio & fext = 1.50 Hz): PSD plot (measured at the top left point of the tank).

Figure D.42: ESA tank test A - MMH (75% ﬁll ratio & fext = 1.50 Hz): PSD plot (measured at the top center point of the tank).

156 Appendix E

Test case 3 - Results: Test B

157 MMH - 50 % ﬁll ratio

(a) t = 0s to t = 4s (200 time steps) (b) Zoom: t = 3.46s 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 MON-3 - 50 % ﬁll ratio

Figure F.1: ESA tank test C - MON-3: PSD plot (measured at the top right point of the tank) for the 15 s period after the abrupt removal of a stabilized 0.1−g lateral perturbation.

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

161 MMH - 50 % ﬁll ratio

Figure F.3: ESA tank test C - MMH: PSD plot (measured at the top right point of the tank) for the 15 s period after the abrupt removal of a stabilized 0.1 − g lateral perturbation.

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