We computed the case ±0% (Sβ = 0kg m) for inlet velocity 7.5 m/s in free area for investigation of the influence of walls of the channel on the aero-elastic stability of the system. The computation was done on the mesh shown in Figure 9. We obtained the results given in Figure 62. The computed
pressure field, the magnitude of velocity field are shown in Figures 68 - 69 at several time instants.
The comparison with the flow in the channel is shown in Figure 63 and gives an evidence of the influence of a wall of a tunnel on the aero-elastic stability. The test shows that closer walls to an airfoil drive the system more unstable.
Figure 9: Anisotropically adapted mesh for NACA 0012 airfoil inserted into free area
-10 -8 -6 -4 -2 0 2 4 6 8 10
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-h [mm]
time [s]
-400 -300 -200 -100 0 100 200 300
-10 -8 -6 -4 -2 0 2 4 6 8 10
- h
• [mm/s]
-h [mm]
• S
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
α [°]
time [s]
-500 -400 -300 -200 -100 0 100 200 300 400 500
-5 -4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-20 -15 -10 -5 0 5 10 15 20
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
β [°]
time [s]
-2000 -1500 -1000 -500 0 500 1000 1500 2000
-20 -15 -10 -5 0 5 10 15 20
β • [°/s]
β [°]
• S
Figure 10: Functionsh, α, β and their phase diagrams for case ±0% and for inlet velocity 2.5 m/s.
72
0 0.2 0.4 0.6 0.8 1 1.2
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 0.5 1 1.5 2 2.5 3 3.5 4
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 11: Spectral analysis for case ±0% and for inlet velocity 2.5m/s.
-10 -8 -6 -4 -2 0 2 4 6 8 10
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-h [mm]
time [s]
-400 -300 -200 -100 0 100 200 300
-10 -8 -6 -4 -2 0 2 4 6 8 10
- h
• [mm/s]
-h [mm]
• S
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
α [°]
time [s]
-500 -400 -300 -200 -100 0 100 200 300 400
-5 -4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-20 -15 -10 -5 0 5 10 15 20
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
β [°]
time [s]
-2000 -1500 -1000 -500 0 500 1000 1500 2000
-20 -15 -10 -5 0 5 10 15 20
β • [°/s]
β [°]
• S
Figure 12: Functionsh, α, β and their phase diagrams for case ±0% and for inlet velocity 5 m/s.
74
0 0.1 0.2 0.3 0.4 0.5 0.6
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 13: Spectral analysis for case ±0% and for inlet velocity 5m/s.
-8 -6 -4 -2 0 2 4 6 8 10 12
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-h [mm]
time [s]
-400 -300 -200 -100 0 100 200 300
-8 -6 -4 -2 0 2 4 6 8 10 12
- h
• [mm/s]
-h [mm]
• S
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
α [°]
time [s]
-500 -400 -300 -200 -100 0 100 200 300 400 500
-5 -4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-20 -15 -10 -5 0 5 10 15 20
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
β [°]
time [s]
-2000 -1500 -1000 -500 0 500 1000 1500 2000
-20 -15 -10 -5 0 5 10 15 20
β • [°/s]
β [°]
• S
Figure 14: Functionsh, α, β and their phase diagrams for case ±0% and for inlet velocity 7.5 m/s.
76
0 0.1 0.2 0.3 0.4 0.5 0.6
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.5 1 1.5 2 2.5
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 1 2 3 4 5 6 7 8
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 15: Spectral analysis for case ±0% and for inlet velocity 7.5m/s.
-8 -6 -4 -2 0 2 4 6 8 10 12
0 0.2 0.4 0.6 0.8 1 1.2
-h [mm]
time [s]
-400 -300 -200 -100 0 100 200 300 400
-8 -6 -4 -2 0 2 4 6 8 10 12
- h
• [mm/s]
-h [mm]
• S
-8 -6 -4 -2 0 2 4 6 8
0 0.2 0.4 0.6 0.8 1 1.2
α [°]
time [s]
-600 -400 -200 0 200 400 600
-8 -6 -4 -2 0 2 4 6 8
α • [°/s]
α [°]
• S
-30 -20 -10 0 10 20 30
0 0.2 0.4 0.6 0.8 1 1.2
β [°]
time [s]
-2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500
-30 -20 -10 0 10 20 30
β • [°/s]
β [°]
• S
Figure 16: Functionsh, α, β and their phase diagrams for case ±0% and for inlet velocity 10 m/s.
78
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.5 1 1.5 2 2.5
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 1 2 3 4 5 6 7 8 9 10
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 17: Spectral analysis for case ±0% and for inlet velocity 10 m/s.
-10 -8 -6 -4 -2 0 2 4 6 8 10 12
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
-h [mm]
time [s]
-400 -300 -200 -100 0 100 200 300 400
-10 -8 -6 -4 -2 0 2 4 6 8 10 12
- h
• [mm/s]
-h [mm]
• S
-6 -4 -2 0 2 4 6
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
α [°]
time [s]
-600 -400 -200 0 200 400 600
-6 -4 -2 0 2 4 6
α • [°/s]
α [°]
• S
-30 -20 -10 0 10 20 30
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
β [°]
time [s]
-2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500 3000
-30 -20 -10 0 10 20 30
β • [°/s]
β [°]
• S
Figure 18: Functionsh, α, β and their phase diagrams for case ±0% and for inlet velocity 12.5 m/s.
80
0 0.5 1 1.5 2 2.5
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 1 2 3 4 5 6 7 8 9 10
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 19: Spectral analysis for case ±0% and for inlet velocity 12.5 m/s.
-10 -5 0 5 10 15
0 0.05 0.1 0.15 0.2 0.25
-h [mm]
time [s]
-500 -400 -300 -200 -100 0 100 200 300 400 500
-10 -5 0 5 10 15
- h
• [mm/s]
-h [mm]
• S
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 0.05 0.1 0.15 0.2 0.25
α [°]
time [s]
-400 -300 -200 -100 0 100 200 300 400
-5 -4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-30 -20 -10 0 10 20 30
0 0.05 0.1 0.15 0.2 0.25
β [°]
time [s]
-2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500
-25 -20 -15 -10 -5 0 5 10 15 20 25
β • [°/s]
β [°]
• S
Figure 20: Functionsh, α, β and their phase diagrams for case ±0% and for inlet velocity 15 m/s.
82
0 0.5 1 1.5 2 2.5 3 3.5
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 1 2 3 4 5 6 7 8 9
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 21: Spectral analysis for case ±0% and for inlet velocity 15 m/s.
-15 -10 -5 0 5 10 15
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
-h [mm]
time [s]
-600 -400 -200 0 200 400 600
-15 -10 -5 0 5 10 15
- h
• [mm/s]
-h [mm]
• S
-4 -3 -2 -1 0 1 2 3 4 5
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
α [°]
time [s]
-400 -300 -200 -100 0 100 200 300 400
-4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-25 -20 -15 -10 -5 0 5 10 15 20 25 30
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
β [°]
time [s]
-2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500
-25 -20 -15 -10 -5 0 5 10 15 20 25 30
β • [°/s]
β [°]
• S
Figure 22: Functionsh, α, β and their phase diagrams for case ±0% and for inlet velocity 17.5 m/s.
84
0 0.5 1 1.5 2 2.5
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 1 2 3 4 5 6 7 8 9
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 23: Spectral analysis for case ±0% and for inlet velocity 17.5 m/s.
-15 -10 -5 0 5 10 15
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
-h [mm]
time [s]
-800 -600 -400 -200 0 200 400 600 800 1000
-15 -10 -5 0 5 10 15
- h
• [mm/s]
-h [mm]
• S
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
α [°]
time [s]
-400 -300 -200 -100 0 100 200 300 400
-5 -4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-30 -20 -10 0 10 20 30
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
β [°]
time [s]
-2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500
-30 -20 -10 0 10 20 30
β • [°/s]
β [°]
• S
Figure 24: Functionsh, α, β and their phase diagrams for case ±0% and for inlet velocity 20 m/s.
86
0 0.5 1 1.5 2 2.5 3 3.5 4
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 1 2 3 4 5 6 7
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 25: Spectral analysis for case ±0% and for inlet velocity 20 m/s.
-10 -8 -6 -4 -2 0 2 4 6 8 10
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-h [mm]
time [s]
-400 -300 -200 -100 0 100 200 300 400
-10 -8 -6 -4 -2 0 2 4 6 8 10
- h
• [mm/s]
-h [mm]
• S
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
α [°]
time [s]
-500 -400 -300 -200 -100 0 100 200 300 400 500
-5 -4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-20 -15 -10 -5 0 5 10 15 20
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
β [°]
time [s]
-1500 -1000 -500 0 500 1000 1500
-20 -15 -10 -5 0 5 10 15 20
β • [°/s]
β [°]
• S
Figure 26: Functionsh, α,β and their phase diagrams for case +5% and for inlet velocity 2.5 m/s.
88
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 0.5 1 1.5 2 2.5 3
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 27: Spectral analysis for case +5% and for inlet velocity 2.5 m/s.
-10 -8 -6 -4 -2 0 2 4 6 8 10
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-h [mm]
time [s]
-400 -300 -200 -100 0 100 200 300 400
-10 -8 -6 -4 -2 0 2 4 6 8 10
- h
• [mm/s]
-h [mm]
• S
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
α [°]
time [s]
-500 -400 -300 -200 -100 0 100 200 300 400 500
-5 -4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-20 -15 -10 -5 0 5 10 15 20
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
β [°]
time [s]
-1500 -1000 -500 0 500 1000 1500
-20 -15 -10 -5 0 5 10 15 20
β • [°/s]
β [°]
• S
Figure 28: Functionsh, α,β and their phase diagrams for case +5% and for inlet velocity 5 m/s.
90
0 0.1 0.2 0.3 0.4 0.5 0.6
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 0.5 1 1.5 2 2.5 3 3.5 4
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 29: Spectral analysis for case +5% and for inlet velocity 5 m/s.
-8 -6 -4 -2 0 2 4 6 8 10 12
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-h [mm]
time [s]
-400 -300 -200 -100 0 100 200 300
-8 -6 -4 -2 0 2 4 6 8 10 12
- h
• [mm/s]
-h [mm]
• S
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
α [°]
time [s]
-500 -400 -300 -200 -100 0 100 200 300 400
-5 -4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-20 -15 -10 -5 0 5 10 15 20
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
β [°]
time [s]
-2000 -1500 -1000 -500 0 500 1000 1500 2000
-20 -15 -10 -5 0 5 10 15 20
β • [°/s]
β [°]
• S
Figure 30: Functionsh, α,β and their phase diagrams for case +5% and for inlet velocity 7.5 m/s.
92
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 1 2 3 4 5 6
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 31: Spectral analysis for case +5% and for inlet velocity 7.5 m/s.
-8 -6 -4 -2 0 2 4 6 8 10 12
0 0.2 0.4 0.6 0.8 1 1.2
-h [mm]
time [s]
-400 -300 -200 -100 0 100 200 300
-8 -6 -4 -2 0 2 4 6 8 10 12
- h
• [mm/s]
-h [mm]
• S
-8 -6 -4 -2 0 2 4 6 8
0 0.2 0.4 0.6 0.8 1 1.2
α [°]
time [s]
-800 -600 -400 -200 0 200 400 600 800
-8 -6 -4 -2 0 2 4 6 8
α • [°/s]
α [°]
• S
-30 -20 -10 0 10 20 30
0 0.2 0.4 0.6 0.8 1 1.2
β [°]
time [s]
-2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500
-30 -20 -10 0 10 20 30
β • [°/s]
β [°]
• S
Figure 32: Functionsh, α,β and their phase diagrams for case +5% and for inlet velocity 10 m/s.
94
0 0.2 0.4 0.6 0.8 1 1.2
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.5 1 1.5 2 2.5 3
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 1 2 3 4 5 6 7 8 9 10
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 33: Spectral analysis for case +5% and for inlet velocity 10 m/s.
-8 -6 -4 -2 0 2 4 6 8 10 12
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
-h [mm]
time [s]
-400 -300 -200 -100 0 100 200 300
-8 -6 -4 -2 0 2 4 6 8 10 12
- h
• [mm/s]
-h [mm]
• S
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
α [°]
time [s]
-400 -300 -200 -100 0 100 200 300 400
-5 -4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-30 -20 -10 0 10 20 30
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
β [°]
time [s]
-2000 -1500 -1000 -500 0 500 1000 1500 2000 2500
-25 -20 -15 -10 -5 0 5 10 15 20 25
β • [°/s]
β [°]
• S
Figure 34: Functionsh, α,β and their phase diagrams for case +5% and for inlet velocity 12.5 m/s.
96
0 0.5 1 1.5 2 2.5 3
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 1 2 3 4 5 6 7 8 9
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 35: Spectral analysis for case +5% and for inlet velocity 12.5 m/s.
-10 -8 -6 -4 -2 0 2 4 6 8 10 12
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
-h [mm]
time [s]
-500 -400 -300 -200 -100 0 100 200 300 400
-10 -8 -6 -4 -2 0 2 4 6 8 10 12
- h
• [mm/s]
-h [mm]
• S
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
α [°]
time [s]
-400 -300 -200 -100 0 100 200 300 400
-5 -4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-25 -20 -15 -10 -5 0 5 10 15 20 25 30
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
β [°]
time [s]
-2000 -1500 -1000 -500 0 500 1000 1500 2000 2500
-25 -20 -15 -10 -5 0 5 10 15 20 25 30
β • [°/s]
β [°]
• S
Figure 36: Functionsh, α,β and their phase diagrams for case +5% and for inlet velocity 15 m/s.
98
0 0.5 1 1.5 2 2.5
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 1 2 3 4 5 6 7 8
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 37: Spectral analysis for case +5% and for inlet velocity 15 m/s.
-15 -10 -5 0 5 10 15
0 0.1 0.2 0.3 0.4 0.5 0.6
-h [mm]
time [s]
-600 -400 -200 0 200 400 600
-15 -10 -5 0 5 10 15
- h
• [mm/s]
-h [mm]
• S
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 0.1 0.2 0.3 0.4 0.5 0.6
α [°]
time [s]
-500 -400 -300 -200 -100 0 100 200 300 400
-5 -4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-25 -20 -15 -10 -5 0 5 10 15 20 25 30
0 0.1 0.2 0.3 0.4 0.5 0.6
β [°]
time [s]
-2000 -1500 -1000 -500 0 500 1000 1500 2000 2500
-25 -20 -15 -10 -5 0 5 10 15 20 25 30
β • [°/s]
β [°]
• S
Figure 38: Functionsh, α,β and their phase diagrams for case +5% and for inlet velocity 17.5 m/s.
100
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 1 2 3 4 5 6 7
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 39: Spectral analysis for case +5% and for inlet velocity 17.5 m/s.
-15 -10 -5 0 5 10 15
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
-h [mm]
time [s]
-800 -600 -400 -200 0 200 400 600 800
-15 -10 -5 0 5 10 15
- h
• [mm/s]
-h [mm]
• S
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
α [°]
time [s]
-400 -300 -200 -100 0 100 200 300 400
-5 -4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-25 -20 -15 -10 -5 0 5 10 15 20
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
β [°]
time [s]
-2000 -1500 -1000 -500 0 500 1000 1500 2000
-25 -20 -15 -10 -5 0 5 10 15 20
β • [°/s]
β [°]
• S
Figure 40: Functionsh, α,β and their phase diagrams for case +5% and for inlet velocity 20 m/s.
102
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 1 2 3 4 5 6
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 41: Spectral analysis for case +5% and for inlet velocity 20 m/s.
-10 -8 -6 -4 -2 0 2 4 6 8 10
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-h [mm]
time [s]
-400 -300 -200 -100 0 100 200 300
-10 -8 -6 -4 -2 0 2 4 6 8 10
- h
• [mm/s]
-h [mm]
• S
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
α [°]
time [s]
-500 -400 -300 -200 -100 0 100 200 300 400 500
-5 -4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-20 -15 -10 -5 0 5 10 15 20 25
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
β [°]
time [s]
-2000 -1500 -1000 -500 0 500 1000 1500 2000
-20 -15 -10 -5 0 5 10 15 20 25
β • [°/s]
β [°]
• S
Figure 42: Functionsh, α, β and their phase diagrams for case −5% and for inlet velocity 2.5 m/s.
104
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 0.5 1 1.5 2 2.5 3 3.5 4
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 43: Spectral analysis for case −5% and for inlet velocity 2.5m/s.
-10 -8 -6 -4 -2 0 2 4 6 8 10
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-h [mm]
time [s]
-400 -300 -200 -100 0 100 200 300
-10 -8 -6 -4 -2 0 2 4 6 8 10
- h
• [mm/s]
-h [mm]
• S
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
α [°]
time [s]
-500 -400 -300 -200 -100 0 100 200 300 400
-5 -4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-20 -15 -10 -5 0 5 10 15 20 25
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
β [°]
time [s]
-2000 -1500 -1000 -500 0 500 1000 1500 2000
-20 -15 -10 -5 0 5 10 15 20 25
β • [°/s]
β [°]
• S
Figure 44: Functionsh, α, β and their phase diagrams for case −5% and for inlet velocity 5 m/s.
106
0 0.1 0.2 0.3 0.4 0.5 0.6
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.2 0.4 0.6 0.8 1 1.2
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 0.5 1 1.5 2 2.5 3 3.5 4
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 45: Spectral analysis for case −5% and for inlet velocity 5m/s.
-8 -6 -4 -2 0 2 4 6 8 10 12
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-h [mm]
time [s]
-400 -300 -200 -100 0 100 200 300
-8 -6 -4 -2 0 2 4 6 8 10 12
- h
• [mm/s]
-h [mm]
• S
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
α [°]
time [s]
-500 -400 -300 -200 -100 0 100 200 300 400
-5 -4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-20 -15 -10 -5 0 5 10 15 20 25
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
β [°]
time [s]
-2000 -1500 -1000 -500 0 500 1000 1500 2000
-20 -15 -10 -5 0 5 10 15 20 25
β • [°/s]
β [°]
• S
Figure 46: Functionsh, α, β and their phase diagrams for case −5% and for inlet velocity 7.5 m/s.
108
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 47: Spectral analysis for case −5% and for inlet velocity 7.5m/s.
-10 -8 -6 -4 -2 0 2 4 6 8 10 12
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
-h [mm]
time [s]
-400 -300 -200 -100 0 100 200 300 400
-10 -8 -6 -4 -2 0 2 4 6 8 10 12
- h
• [mm/s]
-h [mm]
• S
-8 -6 -4 -2 0 2 4 6 8
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
α [°]
time [s]
-800 -600 -400 -200 0 200 400 600 800
-8 -6 -4 -2 0 2 4 6 8
α • [°/s]
α [°]
• S
-30 -20 -10 0 10 20 30
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
β [°]
time [s]
-3000 -2000 -1000 0 1000 2000 3000
-30 -20 -10 0 10 20 30
β • [°/s]
β [°]
• S
Figure 48: Functionsh, α, β and their phase diagrams for case −5% and for inlet velocity 10 m/s.
110
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.5 1 1.5 2 2.5 3
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 1 2 3 4 5 6 7 8 9 10
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 49: Spectral analysis for case −5% and for inlet velocity 10 m/s.
-10 -8 -6 -4 -2 0 2 4 6 8 10 12
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
-h [mm]
time [s]
-400 -300 -200 -100 0 100 200 300 400
-10 -8 -6 -4 -2 0 2 4 6 8 10 12
- h
• [mm/s]
-h [mm]
• S
-6 -4 -2 0 2 4 6
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
α [°]
time [s]
-600 -400 -200 0 200 400 600
-6 -4 -2 0 2 4 6
α • [°/s]
α [°]
• S
-30 -20 -10 0 10 20 30
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
β [°]
time [s]
-2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500
-30 -20 -10 0 10 20 30
β • [°/s]
β [°]
• S
Figure 50: Functionsh, α, β and their phase diagrams for case −5% and for inlet velocity 12.5 m/s.
112
0 0.5 1 1.5 2 2.5
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.5 1 1.5 2 2.5
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 1 2 3 4 5 6 7 8 9 10
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 51: Spectral analysis for case −5% and for inlet velocity 12.5 m/s.
-15 -10 -5 0 5 10 15
0 0.05 0.1 0.15 0.2 0.25
-h [mm]
time [s]
-500 -400 -300 -200 -100 0 100 200 300 400 500
-15 -10 -5 0 5 10 15
- h
• [mm/s]
-h [mm]
• S
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 0.05 0.1 0.15 0.2 0.25
α [°]
time [s]
-500 -400 -300 -200 -100 0 100 200 300 400
-5 -4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-30 -20 -10 0 10 20 30
0 0.05 0.1 0.15 0.2 0.25
β [°]
time [s]
-2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500
-30 -20 -10 0 10 20 30
β • [°/s]
β [°]
• S
Figure 52: Functionsh, α, β and their phase diagrams for case −5% and for inlet velocity 15 m/s.
114
0 0.5 1 1.5 2 2.5 3 3.5
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 1 2 3 4 5 6 7 8 9 10
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 53: Spectral analysis for case −5% and for inlet velocity 15 m/s.
-15 -10 -5 0 5 10 15
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
-h [mm]
time [s]
-600 -400 -200 0 200 400 600 800
-15 -10 -5 0 5 10 15
- h
• [mm/s]
-h [mm]
• S
-4 -3 -2 -1 0 1 2 3 4 5
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
α [°]
time [s]
-400 -300 -200 -100 0 100 200 300 400
-4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-25 -20 -15 -10 -5 0 5 10 15 20 25 30
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
β [°]
time [s]
-2000 -1500 -1000 -500 0 500 1000 1500 2000 2500
-25 -20 -15 -10 -5 0 5 10 15 20 25 30
β • [°/s]
β [°]
• S
Figure 54: Functionsh, α, β and their phase diagrams for case −5% and for inlet velocity 17.5 m/s.
116
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 1 2 3 4 5 6 7 8
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 55: Spectral analysis for case −5% and for inlet velocity 17.5 m/s.
-20 -15 -10 -5 0 5 10 15
0 0.05 0.1 0.15 0.2 0.25
-h [mm]
time [s]
-800 -600 -400 -200 0 200 400 600 800 1000
-20 -15 -10 -5 0 5 10 15
- h
• [mm/s]
-h [mm]
• S
-4 -3 -2 -1 0 1 2 3 4 5
0 0.05 0.1 0.15 0.2 0.25
α [°]
time [s]
-500 -400 -300 -200 -100 0 100 200 300 400
-4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-25 -20 -15 -10 -5 0 5 10 15 20 25 30
0 0.05 0.1 0.15 0.2 0.25
β [°]
time [s]
-2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500
-25 -20 -15 -10 -5 0 5 10 15 20 25
β • [°/s]
β [°]
• S
Figure 56: Functionsh, α, β and their phase diagrams for case −5% and for inlet velocity 20 m/s.
118
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α
0 1 2 3 4 5 6 7 8 9
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β
Figure 57: Spectral analysis for case −5% and for inlet velocity 20 m/s.
-8 -6 -4 -2 0 2 4 6 8 10 12
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-h [mm]
time [s]
-400 -300 -200 -100 0 100 200 300
-8 -6 -4 -2 0 2 4 6 8 10 12
- h
• [mm/s]
-h [mm]
• S
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
α [°]
time [s]
-500 -400 -300 -200 -100 0 100 200 300 400 500
-5 -4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-20 -15 -10 -5 0 5 10 15 20
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
β [°]
time [s]
-2000 -1500 -1000 -500 0 500 1000 1500 2000
-20 -15 -10 -5 0 5 10 15 20
β • [°/s]
β [°]
• S
Figure 58: Functionsh, α, β and their phase diagrams for case ±0% and for inlet velocity 7.5 m/s and half time step.
120
-8 -6 -4 -2 0 2 4 6 8 10 12
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-h [mm]
time [s]
τ* = 0.01 τ* = 0.005
0 0.1 0.2 0.3 0.4 0.5 0.6
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum h, τ* = 0.01 spectrum h, τ* = 0.005
-6 -4 -2 0 2 4 6
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
α [°]
time [s]
τ* = 0.01 τ* = 0.005
0 0.5 1 1.5 2 2.5
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum α, τ* = 0.01 spectrum α, τ* = 0.005
-20 -10 0 10 20
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
β [°]
time [s]
τ* = 0.01 τ* = 0.005
0 1 2 3 4 5 6 7 8
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
spectrum β, τ* = 0.01 spectrum α, τ* = 0.005
Figure 59: Comparison of half time step test and original case: Functions h, α,β and their spectra diagrams for case ±0% and for inlet velocity 7.5 m/s.
-8 -6 -4 -2 0 2 4 6 8 10 12
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-h [mm]
time [s]
-300 -200 -100 0 100 200 300
-8 -6 -4 -2 0 2 4 6 8 10 12
- h
• [mm/s]
-h [mm]
• S
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
α [°]
time [s]
-500 -400 -300 -200 -100 0 100 200 300 400
-5 -4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-20 -15 -10 -5 0 5 10 15 20
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
β [°]
time [s]
-2000 -1500 -1000 -500 0 500 1000 1500 2000
-20 -15 -10 -5 0 5 10 15 20
β • [°/s]
β [°]
• S
Figure 60: Functionsh, α, β and their phase diagrams for case ±0% and for inlet velocity 7.5 m/s and for turbulence model.
122
-8 -6 -4 -2 0 2 4 6 8 10 12
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-h [mm]
time [s]
laminar model turbulent model
0 0.1 0.2 0.3 0.4 0.5 0.6
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
laminar model turbulent model
-6 -4 -2 0 2 4 6
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
α [°]
time [s]
laminar model turbulent model
0 0.5 1 1.5 2 2.5
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
laminar model turbulent model
-20 -10 0 10 20
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
β [°]
time [s]
laminar model turbulent model
0 1 2 3 4 5 6 7 8
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
laminar model turbulent model
Figure 61: Comparison of Spalart-Allmaras turbulence model and laminar model (original case): Functions h, α, β and their spectra diagrams for case
-8 -6 -4 -2 0 2 4 6 8 10 12
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-h [mm]
time [s]
-400 -300 -200 -100 0 100 200 300
-8 -6 -4 -2 0 2 4 6 8 10 12
- h
• [mm/s]
-h [mm]
• S
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
α [°]
time [s]
-500 -400 -300 -200 -100 0 100 200 300 400
-5 -4 -3 -2 -1 0 1 2 3 4 5
α • [°/s]
α [°]
• S
-20 -15 -10 -5 0 5 10 15 20
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
β [°]
time [s]
-2000 -1500 -1000 -500 0 500 1000 1500 2000
-20 -15 -10 -5 0 5 10 15 20
β • [°/s]
β [°]
• S
Figure 62: Functionsh, α, β and their phase diagrams for case ±0% and for inlet velocity 7.5 m/s and for free flow.
124
-8 -6 -4 -2 0 2 4 6 8 10 12
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-h [mm]
time [s]
tunnel free area
0 0.1 0.2 0.3 0.4 0.5 0.6
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
tunnel free area
-6 -4 -2 0 2 4 6
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
α [°]
time [s]
tunnel free area
0 0.5 1 1.5 2 2.5
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
tunnel free area
-20 -10 0 10 20
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
β [°]
time [s]
tunnel free area
0 1 2 3 4 5 6 7 8
0 5 10 15 20 25 30 35 40 45 50
amplitude
frequency [Hz]
tunnel free area
Figure 63: Comparison of free flow and original case (flow in a tunnel):
Functions h, α, β and their spectra diagrams for case ±0% and for inlet
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
t=−0.002s t= 0.014s
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
t = 0.03s t= 0.046s
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
t= 0.062s t= 0.078s
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
t= 0.094s t = 0.11s
Figure 64: Pressure field by laminar model for case±0% and for inlet velocity 7.5 m/s .
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
t=−0.002s t= 0.014s
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
t = 0.03s t= 0.046s
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
t= 0.062s t= 0.078s
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
t= 0.094s t = 0.11s
Figure 65: Magnitude of dimensionless velocity by laminar model for case
±0% and for inlet velocity 7.5 m/s .
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
t=−0.002s t= 0.014s
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
t = 0.03s t= 0.046s
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
t= 0.062s t= 0.078s
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
t= 0.094s t = 0.11s
Figure 66: Pressure field by turbulent model for case ±0% and for inlet velocity 7.5 m/s .
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
t=−0.002s t= 0.014s
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
t = 0.03s t= 0.046s
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
t= 0.062s t= 0.078s
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
t= 0.094s t = 0.11s
Figure 67: Magnitude of dimensionless velocity by turbulent model for case
±0% and for inlet velocity 7.5 m/s .
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
t=−0.002s t= 0.014s
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
t = 0.03s t= 0.046s
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
t= 0.062s t= 0.078s
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
pressure[Pa]: -18 -12 -9 -6 -3 0 3 6 9
t= 0.094s t = 0.11s
Figure 68: Pressure field by laminar model for free flow for case ±0% and for inlet velocity 7.5 m/s .
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
t=−0.002s t= 0.014s
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
t = 0.03s t= 0.046s
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
t= 0.062s t= 0.078s
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
7 6 5 4 3 2 1 0 1 2 3 4 5 6 7
velocity: 0 0.20.40.60.8 1 1.2
t= 0.094s t = 0.11s
Figure 69: Magnitude of dimensionless velocity by laminar model for free flow for case ±0% and for inlet velocity 7.5 m/s .
5 Conclusion
This work was devoted to the numerical simulation of the interaction of viscous, incompressible flow with a vibrating airfoil with three degrees of freedom. The governing system consists of the Navier-Stokes equations and continuity equation formulated with the aid of the ALE method, considered in a time-dependent computational domain and equipped with initial and boundary conditions, and the structure second-order ordinary differential equations for the displacement h in the vertical direction, the rotation angle α of the airfoil and the rotation angle β of the flap.
The main attention was paid to the development of a sufficiently ac-curate, efficient and robust numerical method, based on the application of the stabilized finite element method for the flow problem combined with the Runge-Kutta scheme for the structure problem. The method allows the solu-tion of laminar as well as turbulent flow. Due to the complexity and difficulty of the problem, the work is not concerned with the analysis of qualitative properties of the continuous problem or with theoretical investigation of the computational process.
The developed numerical technique was programmed in the language C and tested on a technically relevant problem of flow past an airfoil inserted into a wind tunnel, for which NASTRAN results are available.
We obtained a good agreement with NASTRAN results in main and flut-ter frequencies for lower inlet velocities and also in showing the tendency of lowering of the main frequency of the motion of the airfoil in dependence on increasing the inlet flow velocity. Both our computations and the NASTRAN analysis show that the system becomes unstable at the inlet velocity 10 m/s.
The improvement of the accuracy of the computational process by including the Spalart-Allmaras turbulence model in our simulations is significant. The half time step test shows the accuracy of the method. The computation of flow past an airfoil in free area shows that the existence of tunnel walls drives the airfoil more unstable.
Finally, the algorithm proves to be very useful for the prediction of behav-ior of systems in instabilities for large vibration amplitudes, when aero-elastic stability is lost. The most significant contribution consists in the fact that it is very difficult to detect experimentally a loss of aero-elastic stability, because of the high price of aerodynamic models used in experiments and because of the possibility of the destruction of such models. On the other hand, the developed numerical technique allows the analysis of a number of cases during relatively short time and at low expenses.
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