Problem of transonic buffet and current approach ... 2

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I

Index

Abstract ... 1

Introduction ... 2

Problem of transonic buffet and current approach ... 2

Aim of thesis ... 4

Extended Abstract ... 5

1 Model creation and validation ... 10

1.1 Introduction ... 10

1.2 Shape definition ... 10

1.3 Working in ANSYS Workbench

^®

... 11

1.3.1 ICEM CFD environment ... 11

1.3.2 Fluent environment ... 18

2 CFD settings, further validation and method of analysis of results ... 38

2.1 Introduction ... 38

2.2 Fluent analyses ... 39

2.2.1 Further validation ... 39

2.2.2 CFD settings: steady and unsteady analyses ... 45

2.3 Method of analysis of results ... 54

3 Steady and transient analyses at Mach 0.76, buffet investigation ... 57

3.1 Results at Mach equal to 0.76 ... 57

3.1.1 Simulation at 𝛂 = 2.975° ... 65

3.1.2 Simulation at 𝛂 = 3° ... 67

3.1.3 Simulation at 𝛂 = 3.01° ... 71

3.1.4 Simulation at 𝛂 = 3.025° ... 74

3.1.5 Simulation at 𝛂 = 3.03° ... 79

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II

3.1.6 Simulation at 𝛂 = 3.05° and 𝛂 = 3.2° ... 80

3.2 Results of simulation series at Mach = 0.76 ... 81

4 Steady and transient analyses at Mach 0.775, buffet investigation ... 89

4.1 Simulation series at Mach equal to 0.775 ... 89

4.1.1 Simulation at 𝛂 = 2.4° and 𝛂 = 2.5° ... 92

4.1.2 Simulation at 𝛂 = 2.525° ... 94

4.1.3 Simulation at 𝛂 = 2.550° ... 98

4.2 Results of simulation series at Mach = 0.775 ... 100

5 Steady and transient analyses at Mach 0.80, buffet investigation ... 106

5.1 Simulation series at Mach equal to 0.80 ... 106

5.1.1 Simulation at 𝛂 = 0.5° ... 111

5.1.2 Simulation at 𝛂 = 0.80° ... 113

5.1.3 Simulation at 𝛂 = 0.85° ... 115

5.1.4 Simulation at 𝛂 = 0.925° ... 117

5.1.5 Simulation at 𝛂 = 0.975° ... 119

5.1.6 Simulation at 𝛂 = 0.99° ... 122

5.1.7 Simulation at 𝛂 = 1° ... 125

5.1.8 Simulation at 𝛂 = 1.025° ... 127

5.1.9 Simulation at 𝛂 = 1.1° ... 127

5.1.10 Simulation at 𝛂 = 1.2° ... 130

5.1.11 Simulation at 𝛂 = 1.3° ... 132

5.1.12 Simulation at 𝛂 = 1.4° ... 134

5.1.13 Simulation at 𝛂 = 1.5° ... 138

5.1.14 Simulation at 𝛂 = 4° ... 143

5.2 Results of simulation series at Mach = 0.80 ... 152

6 Steady and transient analysis at Mach 0.816 and 0.1 degrees of incidence, buffet investigation ... 163

6.1 Introduction ... 163

6.2 Simulation at Mach = 0.816 and 𝛂 = 0.1° ... 164

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7 Steady and transient analyses at null incidence angle and Mach above 0.80, buffet

investigation ... 173

7.1 Simulation series at null incidence angle and Mach above 0.80 ... 173

7.1.1 Simulation at Mach = 0.816 ... 175

7.1.2 Simulation at Mach = 0.82 ... 180

7.1.3 Simulation at Mach = 0.825 ... 184

7.1.4 Simulation at Mach = 0.86 ... 192

7.2 Results of simulation series above Mach 0.80 ... 199

8 Conclusions and comments ... 205

8.1 Overview of obtained results ... 205

9 Appendix ... 212

9.1 Used symbols ... 212

9.2 Used formulas ... 214

References... 217

Acknowledgements ... 219

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IV

Index of Figures and Tables

Figure 1.1

NACA0012 aerofoil generated with Catia v5 r20^® on the left hand of the figure

on the right-hand side there’s a detail of the sharp trailing edge. ……… …… …… …… ……….….10

Figure 1.2

Images of far field building options……… ……… …… ……… … .….11

Figure1.3

‘Blocking' around NACA0012 aerofoil and a detail near the air foil for H-grid…… ……….…..….……..12

Figure 1.4

Mesh distribution near the air foil for H-grid……… ……… …… …...……..14

Figure 1.5

‘Blocking’ of C-grid……… ……….……..……..15

Figure 1.6

Points used to split and curve edges of C-grid, keeping their symmetry………… ……….….…….16

Figure 1.7

‘Blocking’ bands near the air foil of the C-grid……… ……….……16

Figure 1.8

Mesh in proximity of the air foil of the C-grid……… ………17

Figure1.9

First graph represents quality; second graph represents skewness; third graph represents aspect ratio (Fluent); fourth graph represents minimum angle. All referring to C-grid. ……… ……….……….….18

Figure 1.10 a)

Input settings for H-grid validation……… ………21

Figure 1.10 b)

Input settings for H-grid validation……… ……… ……… ………… ………… ………..….21

Figure 1.10 c)

Input settings for H-grid validation……… ………22

Figure 1.11 a)

Polar curves obtained from H-grid at Mach=0.6……… …… … ……… ……… ………..….25

Figure 1.11 b)

Polar curves obtained by H-grid at Mach=0.7……… ……… … …… …… … … … ………...26

Figure 1.11 c)

Polar curves obtained by H-grid at Mach=0.8. Pressure component represents the pressure Cd component. ……… ………… … ……… …… ….. ...…..27

Figure 1.12 a)

Polar curves given by C-grid at M=0.6 and Re=10000000…… ……… … …… ……… ………….………...….34

Figure 1.12 b)

Polar curves given by C-grid at M=0.80 and Re=4000000……… ………...………..35

Figure 1.12 c)

Polar curves given by C-grid at M=0.80 and Re=10000000… …… ………… ……… ……… ……….…..36

Figure 2.1

Comparison of cp distribution between Ref [4] and numerical simulation data near 2° of incidence, Mach = 0.80……… …… ………… ……… …… …………43

Figure 2.2

Comparison of cp distribution between Ref [4] and numerical simulation data near 3° of incidence, Mach = 0.80……… ……….…..44

Figure 2.3

Comparison of cp distribution between Ref [5] and numerical simulation data at -0.1° of incidence, Mach = 0.80……… ………..….44

Figure 2.4

Comparison of cp contours between Ref [5] and numerical simulation data at 0.05° of incidence, Mach = 0.80……… ……….……..45

Figure 2.5

The two graphs represent lift coefficient as functions of time analysed with time step of 0.001 [s] on the left-hand while 0.0005 [s] on the right respectively……… ………..….47

Figure 2.6

Residual plots of steady analyses at Mach = 0.76 with an incidence of 2.975 and 3.025 degrees respectively… … …. 53

Figure 2.7

Residual plots of steady analyses at Mach = 0.80 with an incidence of 0.5 and 1.5 degrees respectively……….53

Figure 2.8

Axis direction and chord length used to adimensionalize node position along the air foil section………..……56

Figure 3.1

Contours of static pressure at Mach = 0.76 (from left to right at 2.975, 3, 3.01, 3.025, 3.03, 3.05, 3.2 degrees)……….57

Figure 3.2

Normalised residual in steady analysis at Mach = 0.76 and α = 2.975°……… ………..58

Figure 3.3

Normalised residual in unsteady analysis at Mach = 0.76 and α = 2.975°……… ……….……….58

Figure 3.4

Normalised residual in steady analysis at Mach = 0.76 and α = 3.0°……… … …… … ………..59

Figure 3.5

Normalised residual in unsteady analysis at Mach = 0.76 and α = 3.0°……… ………..59

Figure 3.6

Normalised residual in steady analysis at Mach = 0.76 and α = 3.01°……… ……… …… … …….60

Figure 3.7

Normalised residual in unsteady analysis at Mach = 0.76 and α = 3.01°……… ………60

Figure 3.8

Normalised residual in steady analysis at Mach = 0.76 and α = 3.025°……… ………..61

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Figure 3.9

Normalised residual in unsteady analysis at Mach = 0.76 and α = 3.025°………… ………..…61

Figure 3.10

Normalised residual in steady analysis at Mach = 0.76 and α = 3.03°……… ……….62

Figure 3.11

Normalised residual in unsteady analysis at Mach = 0.76 and α = 3.03°……… ……….62

Figure 3.12

Normalised residual in steady analysis at Mach = 0.76 and α = 3.05°… ……… ……… …… ………..63

Figure 3.13

Normalised residual in unsteady analysis at Mach = 0.76 and α = 3.05°……… ……….63

Figure 3.14

Normalised residual in steady analysis at Mach = 0.76 and α = 3.2°…… ……… …… …… ………64

Figure 3.15

Normalised residual in unsteady analysis at Mach = 0.76 and α = 3.2°…… ………...64

Figure 3.16

Lift, drag and moment coefficients charts at Mach = 0.76 and α = 2.975°………65

Figure 3.17

Lift, drag and moment coefficients PSD charts at Mach = 0.76 and α = 2.975°………..66

Figure 3.18

Lift, drag and moment coefficients charts at Mach = 0.76 and α = 3.0°…… ………...67

Figure 3.19

Lift and drag coefficients PSD charts at Mach = 0.76 and α = 3.0°……… ………..68

Figure 3.20

Moment coefficient PSD chart at Mach = 0.76 and α = 3.0°……… …… …… …………..69

Figure 3.21

Lift coefficient of upper and lower airfoil surfaces charts at Mach = 0.76 and α = 3.0°… ……… ………..69

Figure 3.22

Lift upper and lower coefficient PSD charts at Mach = 0.76 and α = 3.0°…… …… ……… ………… ……… ..…70

Figure 3.23

Lift coefficient Limit Cycle Oscillation of upper and lower surfaces at Mach = 0.76 and α = 3.0°… … … …… ……71

Figure 3.24

Lift, drag and moment coefficients charts at Mach = 0.76 and α = 3.01°………… …… …… …… … ………..…..72

Figure 3.25

Lift, drag and moment coefficients PSD charts at Mach = 0.76 and α = 3.01°. … …… …… ……….…..73

Figure 3.26

Lift, drag and moment coefficients charts at Mach = 0.76 and α = 3.025°…… …… ……… …… …… …………..……74

Figure 3.27

Lift and drag coefficients PSD charts at Mach = 0.76 and α = 3.025°…… ………..75

Figure 3.28

Moment coefficient PSD charts at Mach = 0.76 and α = 3.025°… ………..…..76

Figure 3.29

Lift and moment coefficients Limit Cycle Oscillation at Mach = 0.76 and α = 3.025° ………… ……… ……… … .….76

Figure 3.30

RMSE of static pressure at Mach = 0.76 and α = 3.025°……… ………...77

Figure 3.31

Upper and lower surface lift coefficient at Mach = 0.76 and α = 3.025°……… ………..77

Figure 3.32

Lift coefficient of upper and lower surfaces PSD charts at Mach = 0.76 and α = 3.025°… … …… …… ……… …….78

Figure 3.33

Lift coefficient Limit Cycle Oscillation of upper and lower surfaces at Mach = 0.76 and α = 3.025°… ………79

Figure 3.34

Lift, drag and moment coefficients charts at Mach = 0.76 and α = 3.03°… ……….79

Figure 3.35

Lift, drag and moment coefficients charts at Mach = 0.76 and α = 3.05°……… ……….80

Figure 3.36

Lift, drag and moment coefficients charts at Mach = 0.76 and α = 3.2°… … ……… ……… …………81

Figure 3.37

LCO amplitudes and maximum oscillation of lift coefficient as a function of incidence at Mach = 0.76…… ……..…83

Figure 3.38

Comparison between amplitudes of LCO of lift coefficient at Mach = 0.76……… ……….84

Figure 3.39

Comparison between amplitude and shape of Limit Cycle Oscillation of lift coefficient at Mach = 0.76…… ………..85

Figure 3.40

Root Mean Squares of Power Spectral Density of lift, drag and moment coefficients at Mach = 0.76 as a function of incidence…… ……… ……… …… ……… ………..86

Figure 3.41

Colours-map of upper (on top) and lower airfoil surfaces static pressure fluctuations at Mach = 0.76 and 𝛼 = 3.025°……… ………87

Figure 3.42

Colours-map of upper airfoil surface static pressure fluctuations in plain view at Mach = 0.76 and 𝛼 = 3.025° ……… ………… ……… … ………88

Figure 4.1

Contours of static pressure at Mach = 0.775 (from left to right at 2.4, 2.5, 2.525, 2.550 degrees). … .… ………89

Figure 4.2

Normalised residuals in steady analysis at Mach = 0.775 and α = 2.4°…… …… …… … ………..……… ……….90

Figure 4.3

Normalised residuals in unsteady analysis at Mach = 0.775 and α = 2.4°……… ………...………90

Figure 4.4

Normalised residuals in steady analysis at Mach = 0.775 and α = 2.5°……… … ………… ………90

Figure 4.5

Normalised residuals in unsteady analysis at Mach = 0.775 and α = 2.5°………91

Figure 4.6

Normalised residuals in steady analysis at Mach = 0.775 and α = 2.525°………91

Figure 4.7

Normalised residuals in unsteady analysis at Mach = 0.775 and α = 2.525°………...……91

Figure 4.8

Normalised residuals in steady analysis at Mach = 0.775 and α = 2.550°………..….92

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Figure 4.9

Normalised residuals in unsteady analysis at Mach = 0.775 and α = 2.550°……… ………..92

Figure 4.10

Lift, drag and moment coefficients charts at Mach = 0.775 and α = 2.4°……… ……….…93

Figure 4.11

Lift, drag and moment coefficients charts at Mach = 0.775 and α = 2.5°… …… ……… …… …… …… ……… …. …93

Figure 4.12

Lift, drag and moment coefficients charts at Mach = 0.775 and α = 2.525° and a magnification of lift coefficient time history to visualize limit cycle oscillations……… ……….…94

Figure 4.13

Lift, drag and moment coefficients PSD charts at Mach = 0.775 and α = 2.525°……… ………...95

Figure 4.14

RMSE of static pressure at Mach = 0.775 and α = 2.525°……… …… ……… ……… ………96

Figure 4.15

Lift coefficient PSD charts of upper and lower airfoil surfaces at Mach = 0.775 and α = 2.525°………97

Figure 4.16

Lift coefficient LCO charts of upper and lower surfaces at Mach = 0.76 and α = 2.525°……….….97

Figure 4.17

Lift, drag and moment coefficients charts at Mach = 0.775 and α = 2.550° and a magnification of lift coefficient time history to visualize limit cycle oscillations……….…..98

Figure 4.18

Lift, drag and moment coefficients PSD charts at Mach = 0.775 and α = 2.550°… ……… …… ………...….99

Figure 4.19

LCO amplitude and maximum oscillation of lift coefficient as a function of incidence at Mach = 0.775… … … …..101

Figure 4.20

Comparison between amplitude of Limit Cycle Oscillations of lift coefficient at Mach = 0.775……….101

Figure 4.21

Amplitude and shape of Limit Cycle Oscillation charts of lift coefficient at Mach = 0.775……….102

Figure 4.22

Root Mean Square of Power Spectral Density of lift, drag and moment coefficients charts at Mach=0.775 as a function of incidence……….103

Figure 4.23

Colours-map of upper (on top) and lower airfoil surfaces static pressure oscillations at Mach =0.775 and 𝛼 = 2.525°……… …… ……… …… ………104

Figure 4.24

Colours-map of upper and lower airfoil surfaces static pressure fluctuations in plain view at Mach = 0.775 and 𝛼 = 2.525°……….105

Figure 5.1

Contours of static pressure at Mach = 0.80 (from left to right at 0.5, 0.8, 0.85, 0.925, 0.975, 0.99, 1, 1.025, 1.1, 1.2, 1.3, 1.4, 1.5, 4 degrees)……….106

Figure 5.2

Normalised residuals at Mach = 0.80 and α = 0.5°; steady analysis on left and unsteady on right………….………….107

Figure 5.3

Normalised residuals at Mach = 0.80 and α = 0.8°; steady analysis on left and unsteady on right……….….107

Figure 5.4

Normalised residuals at Mach = 0.80 and α = 0.85°; steady analysis on left and unsteady on right … …… ….… …..107

Figure 5.5

Normalised residuals at Mach = 0.80 and α = 0.925°; steady analysis on left and unsteady on right……….108

Figure 5.6

Figure 5.7

Normalised residuals at Mach = 0.80 and α = 0.99°; steady analysis on left and unsteady on right………...108

Figure 5.8

Normalised residuals at Mach = 0.80 and α = 1°; steady analysis on left and unsteady on right……….…109

Figure 5.9

Figure 5.10

Figure 5.11

Figure 5.12

Figure 5.13

Figure 5.14

Normalised residuals at Mach = 0.80 and α = 1.5°; steady analysis on left and unsteady on right… … … ……. ….111

Figure 5.15

Normalised residuals at Mach = 0.80 and α = 4°; steady analysis on left and unsteady on right… … … … … .… 111

Figure 5.16

Lift, drag and moment coefficients charts at Mach = 0.80 and α = 0.5°……….…….…112

Figure 5.17

Lift, drag and moment coefficients charts at Mach = 0.80 and α = 0.8°……….……….…113

Figure 5.18

Lift, drag and moment coefficients PSD charts at Mach = 0.80 and α = 0.8°……….………...114

Figure 5.19

Lift, drag and moment coefficients charts at Mach = 0.80 and α = 0.85°… ……… ……… ……… ………….………..115

Figure 5.20

Lift, drag and moment coefficients PSD charts at Mach = 0.80 and α = 0.85°…… ……… ……… ……… …… ….…116

Figure 5.21

Lift, drag and moment coefficients charts at Mach = 0.80 and α = 0.925°… …… ……… …. …117

Figure 5.22

Lift, drag and moment coefficients PSD charts at Mach = 0.80 and α = 0.925°……… ………...118

Figure 5.23

Lift, drag and moment coefficients charts at Mach = 0.80 and α = 0.975°…… ……… …… ……… ………….…119

Figure 5.24

Lift and drag coefficients PSD charts at Mach = 0.80 and α = 0.975°……… ………120

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Figure 5.25

Moment coefficient PSD chart at Mach = 0.80 and α = 0.975°……….121

Figure 5.26

LCO charts of lift and moment coefficients at Mach = 0.80 and α = 0.975º………...121

Figure 5.27

RMSE of static pressure at Mach = 0.80 and α = 0.975º……… ……… ………… ………… …………122

Figure 5.28

Lift, drag and moment coefficients charts at Mach = 0.80 and α = 0.99°………123

Figure 5.29

Lift, drag and moment coefficients PSD charts at Mach = 0.80 and α = 0.99°………..124

Figure 5.30

Lift, drag and moment coefficients charts at Mach = 0.80 and α = 1°……….…125

Figure 5.31

Lift, drag and moment coefficients PSD charts at Mach = 0.80 and α = 1°………126

Figure 5.32

Lift, drag and moment coefficients charts at Mach = 0.80 and α = 1.025°… ……… ……… …… ……….127

Figure 5.33

Lift, drag and moment coefficients charts at Mach = 0.80 and α = 1.1°…… ……….…128

Figure 5.34

Lift, drag and moment coefficients PSD charts at Mach = 0.80 and α = 1.1°……… ……… ………… … …129

Figure 5.35

Lift, drag and moment coefficients charts at Mach = 0.80 and α = 1.2°……… …… ……… …… …… ……… … … .130

Figure 5.36

Lift, drag and moment coefficients PSD charts at Mach = 0.80 and α = 1.2°……….131

Figure 5.37

LCO charts of lift and moment coefficients at Mach = 0.80 and α = 1.2º………132

Figure 5.38

Lift, drag and moment coefficients charts at Mach = 0.80 and α = 1.3°………..133

Figure 5.39

Drag coefficient PSD chart at Mach = 0.80 and α = 1.3°……… …… …… …… ……… ……… …… …… …..…133

Figure 5.40

Lift, drag and moment coefficients charts at Mach = 0.80 and α = 1.4°………..134

Figure 5.41

Lift, drag and moment coefficients PSD charts at Mach = 0.80 and α = 1.4°……….135

Figure 5.42

Upper and lower surface lift coefficients charts at Mach = 0.80 and α = 1.4°… …… …… …… ………… ……..… ..136

Figure 5.43

Upper and lower surface lift coefficients PSD charts at Mach = 0.80 and α = 1.4°……… … …… ……. 136

Figure 5.44

Lift coefficient Limit Cycle Oscillation charts of upper and lower surfaces at Mach = 0.80 and α = 1.4°… ………..137

Figure 5.45

RMSE of static pressure at Mach = 0.80 and α = 1.4º……….………137

Figure 5.46

Lift, drag and moment coefficients charts at Mach = 0.80 and α = 1.5°……….….……138

Figure 5.47

Lift, drag and moment coefficients PSD charts at Mach = 0.80 and α = 1.5°……….………139

Figure 5.48

LCO charts of lift and moment coefficients at Mach = 0.80 and α = 1.5º……… …… …… ……….…..140

Figure 5.49

Upper and lower surface lift coefficients charts at Mach = 0.80 and α = 1.5°… …… …… ……….…….….140

Figure 5.50

Upper and lower surface lift coefficients PSD charts at Mach = 0.80 and α = 1.5°…… … …… … … …… ….141

Figure 5.51

Lift coefficient Limit Cycle Oscillation charts of upper and lower surfaces at Mach = 0.80 and α = 1.5°. ….… ..…142

Figure 5.52

RMSE of static pressure values at Mach = 0.80 and 𝛼 = 1.5°……… ……… …… …….… ....142

Figure 5.53

RMSE of velocity magnitude values at Mach = 0.80 and 𝛼 = 1.5°………...143

Figure 5.54

Lift, drag and moment coefficients charts at Mach = 0.80 and α = 4°……….144

Figure 5.55

Lift, drag and moment coefficients PSD charts at Mach = 0.80 and α = 4°… … …… …… …… … … …… .… …...145

Figure 5.56

LCO charts of lift and moment coefficients at Mach = 0.80 and α = 4º… …… …… …… … …… … …… .……… ..146

Figure 5.57

Upper and lower surface lift coefficients charts at Mach = 0.80 and α = 4°…… …… …… ………..146

Figure 5.58

Upper and lower surface lift coefficients PSD charts at Mach = 0.80 and α = 4°… … ……… …… … ………… … .147

Figure 5.59

Lift coefficient Limit Cycle Oscillation charts of upper and lower surfaces at Mach = 0.80 and α = 4°………….….147

Figure 5.60

RMSE of static pressure at Mach = 0.80 and α = 4º showing whole flow field……….….148

Figure 5.61

RMSE of static pressure at Mach = 0.80 and α = 4º showing flow field around the airfoil…… ………… …… … …148

Figure 5.62

Contours of Mach number at different time steps increasing in time from left to right; refence times are defined in Fig. 2.137 which shows an interval of lift coefficient time history at Mach = 0.80, 𝛼 = 4°… ………… …… ……….149

Figure 5.63

Contours of static pressure at different time steps increasing in time from left to right; refence times are defined in Fig. 2.137 which shows an interval of lift coefficient time history at Mach = 0.80, 𝛼 = 4°… …… ………… ……….150

Figure 5.64

Time sampling at Mach = 0.80 and 𝛼 = 4°, referred to Fig. 2.135 and Fig. 2.136… ……… …… ……… … …… … 151

Figure 5.65

Asymmetric grid for capture eventual shock wave motion……… …… …… ……… ………..…152

Figure 5.66

LCO amplitude and maximum oscillation of lift coefficient as a function of incidence at Mach = 0.80…...… …..….154

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Figure 5.67

Comparison between amplitudes of Limit Cycle Oscillation of lift coefficient at Mach = 0.80………...155

Figure 5.68

Amplitude and shape of LCOs of lift coefficient at Mach = 0.80………..156-158

Figure 5.69

Root Mean Squares of Power Spectral Density of lift, drag and moment coefficients at Mach=0.80 as a function of incidence………158

Figure 5.70

Root Mean Squares of Power Spectral Density of lift, drag and moment coefficients at Mach=0.80 as a function of incidence……… ……… ……… ……… ………159

Figure 5.71

Variation of value of first harmonic at buffet onset points with incidence angle at Mach = 0.80… … … … … … ..160

Figure 5.72

Colours-map of upper airfoil surface representing static pressure variance at Mach = 0.80 and 𝛼 = 0.975°…… …160

Figure 5.73

Colours-map of lower airfoil surface representing static pressure variance at Mach = 0.80 and 𝛼 = 0.975°……….161

Figure 5.74

Colours-map of upper airfoil surface in plain view representing static pressure variance at Mach = 0.80 and 𝛼 =0.975°.………..……….161

Figure 5.75

Colours-map of lower airfoil surface in plain view representing static pressure variance at Mach = 0.80 and 𝛼 = 0.975°……… ………..162

Figure 6.1

Static pressure contours at Mach = 0.816 and 𝛼 = 0.1°……….……163

Figure 6.2

Normalised residuals of steady (on the left side) and transient (on the right side) analysis at Mach = 0.816 and 𝛼 = 0.1°……….……164

Figure 6.3

Lift, drag and moment coefficient at Mach = 0.816 and 𝛼 = 0.1°……….……165

Figure 6.4

Lift coefficient PSD chart at Mach = 0.816 and 𝛼 = 0.1°………..165

Figure 6.5

Drag and moment coefficients PSD charts at Mach = 0.816 and 𝛼 = 0.1°……….……166

Figure 6.6

LCO charts of lift and moment coefficients at Mach = 0.816 and 𝛼 = 0.1°………166

Figure 6.7

Lift coefficient time history chart sampling at Mach = 0.816 and 𝛼 = 0.1°………167

Figure 6.8

Lift coefficient time history and polynomial charts at Mach = 0.816, 𝛼 = 0.1°, t = [5 ; 5.511] s………..168

Figure 6.9

Lift coefficient after deducted its local mean value time history chart at Mach = 0.816, 𝛼 = 0.1°, t = [5 ; 5.511] s……….…..168

Figure 6.10

PSD chart of lift coefficient after deducted its local mean value, at Mach = 0.816, 𝛼 = 0.1°, t = [5 ; 5.511] s……….…..169

Figure 6.11

LCO chart of lift coefficient after deducted its local mean value, at Mach = 0.816, 𝛼 = 0.1°, t = [5 ; 5.511] s……….…..169

Figure 6.12

PSD chart of lift coefficient after deducted its local mean value, at Mach = 0.816, 𝛼 = 0.1°, t = [5.256 ; 5.511] s……….…..170

Figure 6.13

LCO chart of lift coefficient after deducted its local mean value, at Mach = 0.816, 𝛼 = 0.1°, t = [5.256 ; 5.511] s………..….170

Figure 6.14

LCO charts of lift coefficient of upper and lower surfaces at Mach = 0.816 and 𝛼 = 0.1°… … ……… …… …….….171

Figure 6.15

Static pressure RMSE chart of whole flow field at Mach = 0.816 and 𝛼 = 0.1°……….171

Figure 6.16

Static pressure RMSE chart around the airfoil at Mach = 0.816 and 𝛼 = 0.1°………..…172

Figure 7.1

Contours of static pressure at 𝛼 = 0° and increasing Mach number from left to right (Mach = 0.816; 0.82; 0.825; 0.86)……….….173

Figure 7.2

Normalised residuals at Mach = 0.816 and α = 0°; steady analysis on left and unsteady on right……… ………….….174

Figure 7.3

Normalised residuals at Mach = 0.820 and α = 0°; steady analysis on left and unsteady on right…… ……….….174

Figure 7.4

Normalised residuals at Mach = 0.825 and α = 0°; steady analysis on left and unsteady on right… ……….….174

Figure 7.5

Normalised residuals at Mach = 0.86 and α = 0°; steady analysis on left and unsteady on right… … … … ……….…175

Figure 7.6

Lift, drag and moment coefficients charts at Mach = 0.816 and α = 0° and ∆t = 0.001 s………… ………..176

Figure 7.7

Lift, drag and moment coefficients PSD charts at Mach = 0.816 and α = 0° and ∆t = 0.001 s………… ……….177

Figure 7.8

Lift, drag and moment coefficients charts at Mach = 0.816 and α = 0° and ∆t = 0.0001 s……… …………178

Figure 7.9

Lift, drag and moment coefficients PSD charts at Mach = 0.816 and α = 0° and ∆t = 0.0001 s……… ……..179

Figure 7.10

RMSE of static pressure at Mach = 0.816, 𝛼 = 0° and ∆t = 0.0001 s… … ………… ……… ………..…180

Figure 7.11

Lift, drag and moment coefficients time histories charts and a particular of the lift coefficient one at 𝛼 = 0°, Mach = 0.82 and ∆t = 0.001 s……… …… …… ……… ………181

Figure 7.12

Lift, drag and moment coefficients PSD charts at Mach = 0.82, α = 0° and ∆t = 0.001 s…… ………..182

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Figure 7.13

Lift coefficient of upper and lower airfoil surfaces time histories charts at Mach = 0.82, α = 0°

and ∆t = 0.001 s……….183

Figure 7.14

Lift coefficient PSDs of upper and lower airfoil surfaces charts at Mach = 0.82, α = 0° and ∆t = 0.001 s………183

Figure 7.15

RMSE of static pressure at Mach = 0.82, α = 0° and ∆t = 0.001 s..……….………184

Figure 7.16

Lift coefficient total time history simulated at Mach = 0.825, 𝛼 = 0°, ∆t = 0.001 s………..185

Figure 7.17

Lift, drag and moment coefficients time history at Mach = 0.825, 𝛼 = 0°, ∆t = 0.001 s. ranging from 7.059 to 7.57s………185

Figure 7.18

Lift, drag and moment coefficients time history at Mach = 0.825, 𝛼 = 0°, ∆t = 0.001 s, ranging from 8.652 to 10.7s. ,………186

Figure 7.19

Lift coefficient time history at Mach = 0.825, 𝛼 = 0°, ∆t = 0.001 s, ranging from 11.993 to 12.505 s……….186

Figure 7.20

Lift, drag and moment coefficients PSD charts sampled from 7.059 to 7.57 s at Mach = 0.825, α = 0° and ∆t = 0.001 s………. …………..187

Figure 7.21

Lift, drag and moment coefficients PSD charts sampled from 8.052 to 10.7 s at Mach = 0.825, α = 0° and ∆t = 0.001 s……….….……..188

Figure 7.22

Lift coefficient PSD chart sampled from 11.993 to 12.505 at Mach = 0.825, α = 0° and ∆t = 0.001 s………...……..189

Figure 7.23

LCO charts of lift and moment coefficients at Mach = 0.825, α = 0º and ∆t = 0.001 s………...….189

Figure 7.24

Upper and lower surface lift coefficient charts at Mach = 0.825, 𝛼 = 0°, ∆t = 0.001 s………190

Figure 7.25

Upper and lower surfaces lift coefficient PSD charts at Mach = 0.825, 𝛼 = 0°, ∆t = 0.001 s……….191

Figure 7.26

Upper and lower surfaces LCO charts at Mach = 0.825, 𝛼 = 0°, ∆t = 0.001 s………..191

Figure 7.27

Sampling of lift coefficient time history at Mach = 0.86, 𝛼 = 0°………192

Figure 7.28

Flux behaviour around a NACA0012 airfoil at Mach = 0.86, 𝛼 = 0°. ……….…193

Figure 7.29

Lift, drag and moment coefficients time history at Mach = 0.86, 𝛼 = 0°, ∆t = 0.001 s……….…194

Figure 7.30

Lift, drag and moment coefficients PSD charts at Mach = 0.86, α = 0° and ∆t = 0.001 s………...195

Figure 7.31

LCO charts of lift and moment coefficients at Mach = 0.86, α = 0º and ∆t = 0.001 s………..196

Figure 7.32

Lift coefficient of upper and lower airfoil surfaces time histories charts at Mach = 0.86 and 𝛼 = 0°……….………..196

Figure 7.33

Lift coefficient PSDs of upper and lower airfoil surfaces charts at Mach = 0.86 and 𝛼 = 0°……… ………….197

Figure 7.34

LCO charts of upper and lower lift coefficients at Mach = 0.86 and 𝛼 = 0°……… …………..197

Figure 7.35

RMSE of static pressure at Mach = 0.86, 𝛼 = 0° and ∆t = 0.001 s……… ……..…198

Figure 7.36

Frequency and reduced frequency involved at buffet onset beyond Mach = 0.80……… ……….200

Figure 7.37

Root mean square of lift coefficient found in each simulation beyond Mach 0.80 as a function of Mach number……….200

Figure 7.38

Amplitude of lift coefficient LCO and oscillation amplitude of lift coefficient charts per each simulation beyond Mach 0.80 at 𝛼 = 0° as a function of Mach number……… …… ………..…… …… ……….201

Figure 7.39

Comparison between amplitude of limit cycle oscillation of each simulation beyond Mach 0.80…..…… ………201

Figure 7.40

Amplitude and shape of limit cycle oscillation of each simulation beyond Mach 0.80………..202

Figure 7.41

Upper (on top) and lower surface static pressure variance at Mach = 0.86 and 𝛼 = 0°………..203

Figure 7.42

Upper and lower surface static pressure variance in plant view at Mach = 0.86 and 𝛼 = 0°……… ………..…..204

Figure 7.43

Upper surface pressure variance in plant view at Mach = 0.86 and 𝛼 = 0°……… ………..…204

Figure 8.1

Buffet boundary with thesis and literature results……… …………...206

Figure 8.2

Reduced frequencies as a function of Mach number………207

Figure 8.3

Qualitative representation of subcritical Hopf bifurcation conditions……….210

Table 1.1

Coordinates of point defining far field………...…12

Table 1.2

𝑊𝑑,wall distances, used for computations………..13

Table 1.3

Points defining far field shape………..……….15

Table1.4

Wall distances used in computations for C-grid………..………17

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Table 1.5

Dataand results of H-grid at Mach=0.6………..22-23

Table 1.6

Dataand results of H-grid at Mach=0.7………23

Table 1.7

Dataand results of H-grid at Mach=0.8………24

Table 1.8

Settings at Mach=0.6 for C-grid validation………29

Table 1.9

Settings at Mach=0.8 and Reynolds=4000000 for C-grid validation……….29-30

Table 1.10

Settings at Mach=0.80 and Reynolds=10000000 for C-grid validation………..30-31

Table 1.11

Data and results given by C-grid at Mach=0.6………31-32

Table 1.12

Data and results given by C-grid at Mach=0.80 and Reynolds=4000000………32

Table 1.13

Data and results given by C-grid at Mach=0.80 and Reynolds=10000000……….33

Table 2.1

Used settings for cp contours (Ref [4]) with a Re number of 4*10⁶, Mach = 0.80………39-40

Table 2.2

Used settings for cp contours (Ref [4]) with a Re number of 10⁷, Mach = 0.80………..40

Table 2.3

Used settings for cp contours (Ref [5]) with a Re number of 10⁷ and a total pressure of 1.4*10⁵ Pa, Mach = 0.80 ………..………..41

Table 2.4

Used settings for cp contours (Ref [5]) with a Re number of 10⁷ and a total pressure of 2.8*10⁵ Pa, Mach = 0.80………..………42

Table 2.5

Settings for steady and transient analyses at fixed Mach number equal to 0.76……….48

Table 2.6

Settings for steady and unsteady analyses at fixed Mach number equal to 0.775……… ………48-49

Table 2.7

Settings for steady and unsteady analyses at fixed Mach number equal to 0.80……… ………..49-50

Table 2.8

Settings for steady and unsteady analyses at fixed Mach number equal to 0.816……… …………..50

Table 2.9

Settings for steady and unsteady analyses at fixed incidence angle of zero degrees and variable Mach number……… ………51

Table 2.10

Settings for steady and unsteady analyses at fixed incidence angle of zero degrees and variable Mach number……….……..52

Table 3.1

Results of RMS of PSD values obtained in each simulation at Mach = 0.76……….……..82

Table 3.2

Results of LCO amplitude and maximum modulus of oscillation of lift coefficient at Mach = 0.76……….……..82

Table 4.1

Results of RMS of PSD values obtained in each simulation at Mach = 0.775………..100

Table 4.2

Results of Limit Cycle Oscillation and maximum modulus of oscillation of lift coefficient at Mach = 0.775……….……100

Table 5.1

Results of LCO amplitudes and maximum modulus of oscillation of lift coefficient at Mach = 0.80……….…..153

Table 5.2

Results of RMS of PSD values obtained in each simulation at Mach = 0.80………..…..153

Table 7.1

Root mean square values, frequency and reduced frequency of conducted simulations beyond Mach 0.80 and 𝛼 = 0°………199

Table 7.2

Amplitude of lift coefficient Limit Cycle Oscillation and oscillation amplitude of lift coefficient per each simulation over Mach 0.80……….199

Table 8.1

Frequency and reduced frequency involved in buffet onset………..…207