 | Gauer M. Simulation of the sloshing behaviour of two-phase flows in cryogenic rocket upper stage / Deutsches Zentrum für Luft- und Raumfahrt, Institut für Aerodynamik und Strömungstechnik, Göttingen. - Köln: DLR, Bibliotheks- und Informationswesen, 2013. - 148 p.: ill. - Ref.: p.142-148. - (Forschungsbericht; 2013-07). - ISSN 1434-8454
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Acknowledgements ................................................ 2
List of figures ................................................. 5
List of tables ................................................. 10
Nomenclature ................................................... 11
Abstract ....................................................... 14
Zusammenfassung ................................................ 16
1 Introduction ................................................ 18
1.1 Background and motivation .............................. 18
1.2 Cryogenic propellants and their behaviour in upper
stages ................................................. 19
1.3 The DLR Digital-X philosophy ........................... 27
1.4 Thesis outline ......................................... 27
2 State-of-the-art ............................................ 29
2.1 Previous work and current status ....................... 29
2.2 Modelling techniques of two-phase flows ................ 32
2.2.1 Fixed grid methods .............................. 33
2.2.2 Reconstruction of the interface and advection ... 36
2.3 Contribution of thesis ................................. 41
3 Implementation of a two-phase flow infrastructure into the
DLR TAU code ................................................ 43
3.1 Numerical framework - The DLR TAU code ................. 43
3.1.1 The dual-grid approach .......................... 44
3.1.2 Volume-of-fluid initialisation .................. 45
3.2 Governing equations .................................... 47
3.2.1 Pressure Poisson equation ....................... 49
3.2.2 Indicator equation .............................. 57
3.2.3 Momentum equation ............................... 58
3.3 The CICSAM Scheme ...................................... 60
3.3.1 Computation of the face value ................... 61
3.3.2 Boundedness and corrector step .................. 64
3.3.3 The CFL constraint .............................. 66
3.4 Implementation of surface tension ...................... 67
3.4.1 Smoothing of the VOF values ..................... 67
3.4.2 Evaluation of the curvature ..................... 70
3.4.3 Surface tension source term ..................... 72
3.5 Boundary conditions .................................... 75
3.5.1 Pressure Poisson equation ....................... 75
3.5.2 Wall adhesion ................................... 77
4 Model test results .......................................... 80
4.1 VOF transport with the CICSAM scheme ................... 80
4.1.1 Transport of a circle with different CFL
numbers ......................................... 80
4.1.2 Transport of a circle on different grids ........ 82
4.1.3 Transport of a rectangle in an oblique
velocity field .................................. 88
4.2 Surface tension ........................................ 92
4.2.1 Capillary pressure rise in a droplet ............ 93
4.2.2 Wall adhesion - static surface .................. 98
4.2.3 Wall adhesion - dynamic surface ................ 101
5 Case studies ............................................... 103
5.1 2D sloshing in a box .................................. 103
5.2 2D breaking dam ....................................... 107
5.3 3D breaking dam ....................................... 110
6 Sloshing Analysis of the Ariane 5 ESC-A upper stage ........ 117
6.1 Sloshing behaviour - without baffles .................. 118
6.2 Sloshing behaviour with horizontal ring baffles ....... 125
6.3 Sloshing behaviour with horizontal and vertical
baffles ............................................... 128
6.4 Comparison ............................................ 130
6.5 Summary ............................................... 136
7 Closure .................................................... 137
7.1 Summary and conclusions ............................... 137
7.2 Possibilities and outlook for future work ............. 140
References .................................................... 142
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