 | Banuti D. Thermodynamic analysis and numerical modeling of supercritical injection / Deutsches Zentrum für Luft- und Raumfahrt, Institut für Aerodynamik und Strömungstechnik, Göttingen. - Köln: DLR, 2015. - xix, 195 p.: ill. - (Forschungsbericht; 2015-16). - Res. also Germ. - Bibliogr.: p.179-195. - ISSN 1434-8454
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Acknowledgments ............................................... iii
Nomenclature ................................................... xv
Abstract ...................................................... xxi
Zusammenfassung ............................................. xxiii
1 Introduction ................................................. 3
1.1 A New Era in Spaceflight ................................ 3
1.2 Experiments and Models .................................. 4
1.2.1 Injection ........................................ 4
1.2.2 Understanding and CFD ............................ 6
1.2.3 Gas Combustor .................................... 6
1.2.4 Lagrangian Multi Phase ........................... 6
1.2.5 Eulerian Multi Phase ............................. 7
1.2.6 Eulerian Real Gas ............................... 11
1.3 Is something still missing? ............................ 14
1.4 Outline ................................................ 16
2 Injection Phenomena ......................................... 17
2.1 Rocket Engine Performance .............................. 17
2.2 Jet Break-Up ........................................... 19
2.2.1 Classification of Jet Break-Up .................. 19
2.2.2 Jet Core Length ................................. 23
2.3 Thermodynamic Classification of Injection .............. 28
2.3.1 Injection in Rocket Engines ..................... 32
2.3.2 Subcritical Pressures ........................... 32
2.3.3 Supercritical Pressures ......................... 33
2.3.4 Mixture Critical Pressure ....................... 35
2.3.5 Multiphase Mixtures in Rocket Engines? .......... 37
2.4 Summary ................................................ 40
3 Modeling and Behavior of Real Fluids ........................ 41
3.1 Thermodynamic Description of Fluids .................... 41
3.1.1 Simple Models ................................... 41
3.1.2 Real Gas Equations of State ..................... 42
3.1.3 Corresponding States Principle .................. 45
3.1.4 Mixture Rules ................................... 47
3.1.5 Real Gas Departure Functions .................... 49
3.2 Properties of Supercritical Fluids ..................... 51
3.2.1 Pseudo Surface-Boiling in Heat Transfer ......... 51
3.2.2 Structure of the Supercritical State Space ...... 51
3.2.3 The Widom Line .................................. 53
3.2.4 Pseudo-boiling - Crossing the Widom Line ........ 55
3.2.5 Summary ......................................... 56
3.3 A Quantitative Theory of Pseudo-Boiling ................ 58
3.3.1 Choosing a Widom Line Definition ................ 58
3.3.2 Clausius-Clapeyron Equation in the Critical
Limit ........................................... 59
3.3.3 Extension of the Coexistence Line ............... 61
3.3.4 A New Equation for the Widom Line ............... 62
3.3.5 Supercritical Heat of Vaporization .............. 64
3.4 Implications for Transcritical Injection ............... 72
3.4.1 Transcritical Fluid Boundary .................... 72
3.4.2 Thermodynamic Meaning ........................... 73
3.4.3 A Thermal Break-Up Mechanism .................... 75
3.5 Conclusion ............................................. 79
4 The DLR CFD Code TAU for Ideal Gases ........................ 81
4.1 Overview ............................................... 81
4.2 Equations .............................................. 81
4.2.1 Navier-Stokes ................................... 81
4.2.2 Transport Coefficients .......................... 82
4.2.3 Chemical Reactions .............................. 83
4.2.4 Finite Volume Method ............................ 84
4.3 Thermodynamics Structure ............................... 86
4.4 The TAU Code for Combustion Applications ............... 87
4.4.1 Validation of TAU for Combustion of Ideal
Gases ........................................... 87
4.4.2 Turbulent Combustion ............................ 87
5 Real Gas Extension of TAU ................................... 89
5.1 Introduction ........................................... 89
5.2 Arbitrary Phase Euler-Euler Model ...................... 94
5.3 Thermodynamics Architecture ............................ 96
5.4 Multi-Fluid Mixing ..................................... 98
5.4.1 Mixture Rules for Coaxial Injection ............. 98
5.4.2 Main Functions ................................. 101
5.4.3 Supplemental Functions ......................... 102
5.5 Real Gas Library ...................................... 106
5.5.1 Computing the MBWR Equation of State ........... 106
5.5.2 Transport Coefficients ......................... 108
5.5.3 Library Function ............................... 109
5.5.4 Conclusion ..................................... 110
5.6 Verification and Validation ........................... 111
5.6.1 Verification ................................... 111
5.6.2 Validation ..................................... 114
5.7 Conclusion ............................................ 129
6 Applications ............................................... 131
6.1 Cryogenic Injection Experiment ........................ 131
6.1.1 The Experiment ................................. 131
6.1.2 The Experiment in CFD .......................... 136
6.1.3 Analysis ....................................... 139
6.1.4 Conclusion ..................................... 146
6.2 Exemplary CFD of Thermal Atomization .................. 147
6.2.1 Rationale ...................................... 147
6.2.2 Literature ..................................... 147
6.2.3 Numerical Setup ................................ 147
6.2.4 Results ........................................ 149
6.2.5 Conclusion ..................................... 151
6.3 Reactive Single Injector .............................. 153
6.3.1 Mascotte A60 Testcase .......................... 153
6.3.2 Numerical Setup ................................ 154
6.3.3 Experimental Results ........................... 155
6.3.4 Numerical Results .............................. 155
6.3.5 Comparison with other CFD Codes ................ 158
6.3.6 A Posteriori Assessment of MFM Assumptions ..... 161
6.3.7 Conclusion ..................................... 162
7 Summary and Conclusion ..................................... 163
7.1 Real Gas Thermodynamics and Mixture Model ............. 163
7.2 Supercritical State Transition and Injection .......... 163
7.3 Axial Density in Transcritical Injection .............. 165
Appendix ...................................................... 167
1 Modified Benedict-Webb-Rubin Equation ................... 167
2 Transport Coefficients .................................. 173
3 Thermodynamics Algorithm Schematics ..................... 175
4 Reaction Mechanism of H2 - O2 Combustion ................ 177
5 Quantitative A60 CFD Results ............................ 178
Bibliography .................................................. 179
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