 | Rooij van A. Aeroelastic limit-cycle oscillations resulting from aerodynamic non-linearities: proefschr. ter verkrijging van de graad van Doctor / Deutsches Zentrum für Luft- und Raumfahrt, Institut für Aeroelastik, Göttingen. - Köln: DLR, 2017. - xvi, 199 p.: ill. - (Forschungsbericht; 2017-26). - Bibliogr. at the end of the chapters. - ISSN 1434-8454
Шифр: (Pr 1120/2017-26) 02
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Summary ....................................................... vii
Samenvatting ................................................... xi
Preface ........................................................ xv
1 Introduction ............................................... 1
1.1 Motivation of this thesis .................................. 2
1.2 Limit-cycle oscillations ................................... 3
1.3 Previous investigations on limit-cycle oscillations ........ 5
1.3.1 Computational methods ............................... 5
1.3.2 Limit-cycle oscillation bifurcation behaviour ....... 6
1.3.3 Conclusions and open questions ...................... 8
1.4 Objectives ................................................ 10
1.5 Outline of the thesis ..................................... 11
References ................................................ 11
2 Aeroelastic problem description and solution strategies ... 17
2.1 Introduction .............................................. 17
2.2 Aeroelastic problem description ........................... 17
2.2.1 NLR7301 airfoil .................................... 19
2.2.2 Structural model ................................... 20
2.2.3 Fluid model ........................................ 21
2.3 Fluid-structure coupling in the time domain ............... 28
2.4 Linear flutter in the frequency domain .................... 29
2.4.1 Conventional p-k method ............................ 29
2.4.2 Frequency response functions ....................... 31
2.4.3 Linear flutter ..................................... 33
2.5 Limit-cycle oscillations in the frequency domain .......... 33
2.5.1 Amplitude-dependent p-k method (ADePK) ............. 34
2.5.2 Harmonic forced motion oscillations ................ 36
2.5.3 Limit-cycle oscillations ........................... 39
2.5.4 Sampling and interpolation techniques .............. 41
2.5.5 The ADePK method in perspective .................... 41
References ................................................ 43
3 Energy budget analysis of limit-cycle oscillations ........ 49
3.1 Introduction .............................................. 49
3.2 Computational methods ..................................... 50
3.2.1 CFD code and set-up ................................ 50
3.2.2 Structural model ................................... 51
3.2.3 Fluid-structure interaction ........................ 51
3.2.4 Forced motion oscillations ......................... 52
3.3 Results and discussion .................................... 53
3.3.1 Van der Pol-oscillator ............................. 54
3.3.2 Fluid-structure interaction ........................ 58
3.3.3 Single harmonic forced motion oscillations ......... 62
3.3.4 Non-linear aerodynamic power defect ................ 64
3.4 Conclusions ............................................... 69
References ................................................ 70
4 Verification and validation of the amplitude-dependent
p-k method ................................................ 73
4.1 Introduction .............................................. 73
4.2 Computational methods and set-up .......................... 75
4.2.1 CFD code and set-up ................................ 75
4.2.2 Two DoF airfoil system ............................. 75
4.2.3 Fluid-structure coupling ........................... 75
4.2.4 Conventional p-k method ............................ 76
4.2.5 Amplitude-dependent p-k method ADePK ............... 76
4.3 Results and discussion .................................... 77
4.3.1 Van der Pol-oscillator ............................. 77
4.3.2 Two DoF airfoil system ............................. 84
4.3.3 Structural parameter variation ..................... 91
4.4 Conclusions ............................................... 94
References ................................................ 94
5 Bifurcation behaviour of limit-cycle oscillation
solutions ................................................. 97
5.1 Introduction .............................................. 97
5.2 Sources of aerodynamic non-linearity ...................... 98
5.2.1 Transonic inviscid flow ............................ 99
5.2.2 Transonic flow with trailing-edge separation ...... 100
5.2.3 Subsonic flow with trailing-edge separation ....... 102
5.2.4 Subsonic flow with free boundary-layer
transition ........................................ 105
5.2.5 Conclusions ....................................... 107
5.3 Mach number variation in inviscid flow ................... 110
5.3.1 Flutter behaviour ................................. 110
5.3.2 LCO bifurcation behaviour ......................... 112
5.3.3 Conclusions ....................................... 116
5.4 Structural parameter variation ........................... 116
5.4.1 Structural frequency ratio ........................ 117
5.4.2 Mass ratio ........................................ 126
5.4.3 Structural damping ................................ 129
5.4.4 Elastic axis location ............................. 131
5.4.5 Conclusions ....................................... 133
5.5 Response surface analysis ................................ 135
5.5.1 One DoF van der Pol-oscillator .................... 135
5.5.2 Two DoF van der Pol-oscillator .................... 137
5.5.3 Two DoF airfoil system ............................ 142
5.5.4 Conclusions ....................................... 151
5.6 Conclusions .............................................. 152
References ............................................... 154
6 Conclusions and outlook .................................. 157
6.1 Energy budget analysis ................................... 157
6.2 Verification and validation of the amplitude-dependent
p-k method ............................................... 158
6.3 Bifurcation behaviour of limit-cycle oscillation
solutions ................................................ 159
6.4 Outlook .................................................. 160
A Experimental validation .................................. 161
References ............................................... 162
В Mesh- and time step convergence .......................... 163
B.1 Mesh ..................................................... 163
B.l.l Euler simulations .................................. 163
B.1.2 RANS simulations ................................... 165
B.2 Time step ................................................ 169
B.2.1 Euler simulations .................................. 169
B.2.2 RANS simulations ................................... 170
С Interpolation methods .................................... 175
C.l Polynomial interpolation ................................. 175
C.2 Cubic spline interpolation ............................... 176
C.3 Linear interpolation ..................................... 178
References ............................................... 178
D Additional Results of the amplitude-dependent p-k
method ................................................... 179
D.l Structural parameter variation ........................... 179
D.l.l Mass ratio ......................................... 179
D.1.2 Structural damping ................................. 186
D.1.3 Elastic axis location .............................. 190
D.2 Response surface analysis ................................ 194
Curriculum Vitae .............................................. 197
List of Publications .......................................... 199
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