106
Chapter 5
Steady and transient analyses at
Mach 0.80, buffet investigation
5.1 Simulation series at Mach equal to 0.80
The series of conducted analyses are specified in Table 2.7. Figure 5.1 shows contour of static pressure in each condition starting from right to left with increasing incidence angle:
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).
107
Figure 5.2 Normalised residuals at Mach = 0.80 and α = 0.5°; steady analysis on left and unsteady on right.
Figure 5.3 Normalised residuals at Mach = 0.80 and α = 0.8°; steady analysis on left and unsteady on right.
108
Figure 5.5 Normalised residuals at Mach = 0.80 and α = 0.925°; steady analysis on left and unsteady on right.
Figure 5.6 Normalised residuals at Mach = 0.80 and α = 0.975°; steady analysis on left and unsteady on right.
109
Figure 5.8 Normalised residuals at Mach = 0.80 and α = 1°; steady analysis on left and unsteady on right.
Figure 5.9 Normalised residuals at Mach = 0.80 and α = 1.025°; steady analysis on left and unsteady on right.
110
Figure 5.11 Normalised residuals at Mach = 0.80 and α = 1.2°; steady analysis on left and unsteady on right.
Figure 5.12 Normalised residuals at Mach = 0.80 and α = 1.3°; steady analysis on left and unsteady on right.
111
Figure 5.14 Normalised residuals at Mach = 0.80 and α = 1.5°; steady analysis on left and unsteady on right.
Figure 5.15 Normalised residuals at Mach = 0.80 and α = 4°; steady analysis on left and unsteady on right.
From previous figures, it can be noted that residuals tend to increase near some incidence angles.
Unlike other series of simulations, there are more than a single analysis with higher residuals indicating a plurality of critical incidence values or a range of the last were unsteadiness exists. Next analyses will highlight a probable existence of multiple solutions.
5.1.1 Simulation at
𝛂 = 0.5°
In following figures are shown charts obtained from transient analyses; for each incidence condition are shown lift coefficient, drag coefficient, moment coefficient versus time and power spectral density charts of previous parameters.
112
Figure 5.16 Lift, drag and moment coefficients charts at Mach = 0.80 and α = 0.5°.
As can be seen no oscillations were found at this incidence angle, so PSD charts are null and are not reported.
0,115200 0,115300 0,115400 0,115500 0,115600 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 8,5 9 9,5 10 cl , l if t co ef fi ci e n t time [s]
Cl time history
0,018725 0,018730 0,018735 0,018740 0,018745 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 8,5 9 9,5 10 cd , d ra g co ef fi ci e n t time [s]Cd time history
-0,009700 -0,009650 -0,009600 -0,009550 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 8,5 9 9,5 10 cm, m ome n t co ef fi ci e n t time [s]Cm time history
113
5.1.2 Simulation at
𝛂 = 0.80°
Next charts refer to simulation made at 𝛼 = 0.80°.
Figure 5.17 Lift, drag and moment coefficients charts at Mach = 0.80 and α = 0.8°.
Although oscillations are very small (about 𝑂(10−7) for lift coefficient), results
cannot be due to numerical errors of the software because they are too repetitive, thus hardly associated to a kind of background noise. PSD of the coefficients indeed, show a well-defined frequency content indicating born of an unsteady phenomenon.
0,1807084 0,1807085 0,1807086 0,1807087 0,1807088 0,1807089 9,95 9,96 9,97 9,98 9,99 10 10,01 10,02 10,03 10,04 10,05 10,06 10,07 10,08 cl , l if t co ef fi ci e n t time [s]
Cl time history
0,02137974 0,02137975 0,02137976 0,02137977 0,02137978 9,95 9,96 9,97 9,98 9,99 10 10,01 10,02 10,03 10,04 10,05 10,06 10,07 10,08 cd , d ra g co ef fi ci e n t time [s]Cd time history
-0,01520785 -0,01520775 -0,01520765 -0,01520755 9,95 9,96 9,97 9,98 9,99 10 10,01 10,02 10,03 10,04 10,05 10,06 10,07 10,08 cm, m ome n t co ef fi ci e n t time [s]Cm time history
114
Figure 5.18 Lift, drag and moment coefficients PSD charts at Mach = 0.80 and α = 0.8°.
1,E-14 1,E-13 1,E-12 1,E-11 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 P SD o f cl [ 1/Hz] frequency [Hz]
PSD of Cl
1,E-17 1,E-16 1,E-15 1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cd [ 1/Hz ] frequency [Hz]PSD of Cd
1,E-17 1,E-16 1,E-15 1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 1,E-07 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cm [1/Hz ] frequency [Hz]PSD of Cm
115
5.1.3 Simulation at
𝛂 = 0.85°
Next simulation refers to an incidence angle of 0.85 degrees. Found results are summarised in following figures.
Figure 5.19 Lift, drag and moment coefficients charts at Mach = 0.80 and α = 0.85°.
Found oscillation are of the same order of magnitude with respect to previous analysis at 𝛼 = 0.8°. In next Figure 5.20 are reported PSD charts that still indicate a well-defined frequency content but can be noted an involvement of super-harmonics widespread in all frequency band analysed.
0,1913632 0,1913633 0,1913634 0,1913635 0,1913636 0,1913637 9,7 9,725 9,75 9,775 9,8 9,825 9,85 9,875 9,9 9,925 9,95 9,975 10 cl , l if t co ef fi ci e n t time [s]
Cl time history
0,0219229 0,021923 0,021923 0,021923 0,021923 0,021923 0,021923 9,7 9,725 9,75 9,775 9,8 9,825 9,85 9,875 9,9 9,925 9,95 9,975 10 cd , d ra g co ef fi ci e n t time [s]Cd time history
-0,016157 -0,016157 -0,016157 -0,016157 -0,016157 -0,016157 9,7 9,725 9,75 9,775 9,8 9,825 9,85 9,875 9,9 9,925 9,95 9,975 10 cm, m ome n t co ef fi ci e n t time [s]Cm time history
116
Figure 5.20 Lift, drag and moment coefficients PSD charts at Mach = 0.80 and α = 0.85°.
1,E-16 1,E-15 1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cl [ 1/Hz ] frequency [Hz]
PSD of Cl
1,E-19 1,E-18 1,E-17 1,E-16 1,E-15 1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cd [ 1/Hz ] frequency [Hz]PSD of cd
1,E-17 1,E-16 1,E-15 1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cm [1/Hz ] frequency [Hz]PSD of Cm
117
5.1.4 Simulation at
𝛂 = 0.925°
Simulation done at 𝛼 = 0.925° shows a random and very little oscillation of lift, drag and moment coefficients, thus indicating as previous analyses an unsteady phenomenon. This time instead, frequency content is not clear because there is not a well-defined repetitiveness of signals giving completely full spectra.
Figure 5.21 Lift, drag and moment coefficients charts at Mach = 0.80 and α = 0.925°.
0,2038889 0,2038891 0,2038893 0,2038895 0,2038897 0,2038899 9,8 9,81 9,82 9,83 9,84 9,85 9,86 9,87 9,88 9,89 9,9 9,91 9,92 9,93 9,94 9,95 9,96 9,97 9,98 9,99 10 cl , l if t co ef fi ci e n t time [s]
Cl time history
0,02261500 0,02261502 0,02261504 0,02261506 0,02261508 0,02261510 0,02261512 9,8 9,81 9,82 9,83 9,84 9,85 9,86 9,87 9,88 9,89 9,9 9,91 9,92 9,93 9,94 9,95 9,96 9,97 9,98 9,99 10 cd , dr ag coe ff ici en t time [s]Cd time history
-0,0172047 -0,0172046 -0,0172045 -0,0172044 -0,0172043 -0,0172042 9,8 9,81 9,82 9,83 9,84 9,85 9,86 9,87 9,88 9,89 9,9 9,91 9,92 9,93 9,94 9,95 9,96 9,97 9,98 9,99 10 cm , m om en t coe ff ici en t time [s]Cm time history
118
Figure 5.22 Lift, drag and moment coefficients PSD charts at Mach = 0.80 and α = 0.925°.
1,00E-14 1,00E-13 1,00E-12 1,00E-11 1,00E-10 1,00E-09 1,00E-08 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 P SD o f cl [ 1/Hz] frequency [Hz]
PSD of cl
1,00E-16 1,00E-15 1,00E-14 1,00E-13 1,00E-12 1,00E-11 1,00E-10 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 P SD o f cd [ 1/Hz] frequency [Hz]PSD of cd
1,00E-15 1,00E-14 1,00E-13 1,00E-12 1,00E-11 1,00E-10 1,00E-09 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cm [1/Hz ] frequency [Hz]PSD of cm
119
5.1.5 Simulation at
𝛂 = 0.975°
Next simulation was that at 𝛼 = 0.975°. As can be noted from following charts there is an increase in the order of magnitude of perturbations testified by coefficients time histories and magnitude of PSDs. A further inspection of obtained data, led to indicate a buffet onset in this condition as will be made clear at the end of the paragraph, when will be made an overall consideration of results.
Figure 5.23 Lift, drag and moment coefficients charts at Mach = 0.80 and α = 0.975°.
0,216536 0,216537 0,216537 0,216538 0,216538 0,216539 0,216539 9 9,05 9,1 9,15 9,2 9,25 9,3 9,35 9,4 9,45 9,5 9,55 9,6 9,65 9,7 9,75 9,8 9,85 9,9 9,95 10 cl , l if t co ef fi ci en t time [s]
Cl time history
0,0233567 0,0233568 0,0233569 0,0233570 0,0233571 0,0233572 9 9,05 9,1 9,15 9,2 9,25 9,3 9,35 9,4 9,45 9,5 9,55 9,6 9,65 9,7 9,75 9,8 9,85 9,9 9,95 10 cd , d ra g co ef fi ci e n t time [s]Cd time history
-0,018310 -0,018310 -0,018309 -0,018309 -0,018308 -0,018308 -0,018307 9 9,05 9,1 9,15 9,2 9,25 9,3 9,35 9,4 9,45 9,5 9,55 9,6 9,65 9,7 9,75 9,8 9,85 9,9 9,95 10 cm, m ome n t co ef fi ci e n t time [s]Cm time history
120
It was found a quasi-perfect repetitiveness of oscillations, moreover testified by root mean square analyses.
Figure 5.24 Lift and drag coefficients PSD charts at Mach = 0.80 and α = 0.975°.
89,89 182,22 272,11 364,44 453,83 1,00E-15 1,00E-14 1,00E-13 1,00E-12 1,00E-11 1,00E-10 1,00E-09 1,00E-08 1,00E-07 1,00E-06 1,00E-05 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cl [ 1/Hz ] frequency [Hz]
PSD of Cl
1,00E-17 1,00E-16 1,00E-15 1,00E-14 1,00E-13 1,00E-12 1,00E-11 1,00E-10 1,00E-09 1,00E-08 1,00E-07 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cd [ 1/Hz ] frequency [Hz]PSD di Cd
121
Figure 5.25 Moment coefficient PSD chart at Mach = 0.80 and α = 0.975°.
First natural frequency involved is about 88.89 Hz, as can be seen from previous figures 5.24 and 5.25, accompanied by its super-harmonics. Moreover, looking at Figure 5.24 it can be noted that PSD of lift coefficient chart presents some other frequencies with less energy content despite PSD of moment coefficient chart. This last consideration is demonstrated by LCO charts in Figure 5.26; as can be noted 𝑐𝑙 limited cycle oscillation is not a perfect circle like the 𝑐𝑚 one.
Figure 5.26 LCO charts of lift and moment coefficients at Mach = 0.80 and α = 0.975º.
89,89 182,22 272,11 364,44 453,83 1,00E-13 1,00E-12 1,00E-11 1,00E-10 1,00E-09 1,00E-08 1,00E-07 1,00E-06 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cm [1/Hz ] frequency [Hz]
PSD di Cm
0,216536 0,2165365 0,216537 0,2165375 0,216538 0,2165385 0,216539 -0,002 -0,001 0 0,001 0,002 cl Δcl/ΔtΔcl/Δt
-0,01831 -0,01831 -0,018309 -0,018309 -0,018308 -0,018308 -0,018307 -0,002 -0,001 0 0,001 0,002 cm ΔCm/ΔtΔCm/Δt
122
A further investigation of the RMSE of static pressure highlighted a singular situation: unsteadiness is concentrated as in previous series of analyses in a little portion of flow field, but downstream of shock wave in the lower surface of the airfoil. This last is an absolute newness. Next figure testifies this consideration:
Figure 5.27 RMSE of static pressure at Mach = 0.80 and α = 0.975º.
5.1.6 Simulation at
𝛂 = 0.99°
It was decided to analyse conditions with incidence angles near 0.975 degrees to inspect if disturbances, obtained in last simulation, would be amplified or not. Thus, next presented results refer to an incidence angle of 𝛼 = 0.99°. Data obtained reveal a clear decrease in magnitude of unsteadiness; next figures show analysis of results.
123
Figure 5.28 Lift, drag and moment coefficients charts at Mach = 0.80 and α = 0.99°.
As can be noted from previous charts, signals repetitiveness was lost although oscillations are limited in a narrow band but accompanied by a lesser rate of oscillation and a lesser amplitude of oscillation as testified by LCO and maximum oscillation charts (reported with that of other analyses of this series at the end of this paragraph).
PSD charts show more distributed frequencies and a lesser energy content than case at 0.975 degrees; reported charts are in next Figure 5.29.
0,2193256 0,2193256 0,2193257 0,2193257 0,2193258 11,1 11,125 11,15 11,175 11,2 11,225 11,25 11,275 11,3 11,325 11,35 cl , l if t coe ff ici en t time [s]
Cl time history
0,02353190 0,02353191 0,02353192 0,02353193 11,1 11,125 11,15 11,175 11,2 11,225 11,25 11,275 11,3 11,325 11,35 cd , dr ag coe ff ici en t time [s]Cd time history
-0,0185257 -0,0185256 -0,0185256 -0,0185255 11,1 11,125 11,15 11,175 11,2 11,225 11,25 11,275 11,3 11,325 11,35 cm, mome n t co ef fi ci en t time [s]Cm time history
124
Figure 5.29 Lift, drag and moment coefficients PSD charts at Mach = 0.80 and α = 0.99°.
1,00E-17 1,00E-16 1,00E-15 1,00E-14 1,00E-13 1,00E-12 1,00E-11 1,00E-10 1,00E-09 1,00E-08 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cl [ 1/Hz ] frequency [Hz]
PSD of Cl
1,00E-19 1,00E-18 1,00E-17 1,00E-16 1,00E-15 1,00E-14 1,00E-13 1,00E-12 1,00E-11 1,00E-10 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 P SD o f cd [ 1/Hz] frequency [Hz]PSD of Cd
1,00E-18 1,00E-17 1,00E-16 1,00E-15 1,00E-14 1,00E-13 1,00E-12 1,00E-11 1,00E-10 1,00E-09 1,00E-08 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cm [1/Hz ] frequency [Hz]PSD of Cm
125
5.1.7 Simulation at
𝛂 = 1°
Next figures refer to simulation at incidence angle of 1 degree. It was found a clear repetitive cycle although one order of magnitude smaller than simulation at 0.975º degrees. Resultant PSD charts, reported in Figure 5.31, are very clear and frequencies involved are practically the same of last cited analysis.
Figure 5.30 Lift, drag and moment coefficients charts at Mach = 0.80 and α = 1°.
0,2211552 0,2211553 0,2211554 0,2211555 0,2211556 0,2211557 8,1 8,11 8,12 8,13 8,14 8,15 8,16 8,17 8,18 8,19 8,2 cl , l if t coe ff ici en t time [s]
Cl time history
0,02364654 0,02364656 0,02364658 0,02364660 0,02364662 8,1 8,11 8,12 8,13 8,14 8,15 8,16 8,17 8,18 8,19 8,2 cd , d ra g co ef fi ci e n t time [s]Cd time history
-0,01865925 -0,01865920 -0,01865915 -0,01865910 -0,01865905 -0,01865900 -0,01865895 8,1 8,11 8,12 8,13 8,14 8,15 8,16 8,17 8,18 8,19 8,2 cm, m ome n t co ef fi ci e n t time [s]Cm time history
126
Figure 5.31 Lift, drag and moment coefficients PSD charts at Mach = 0.80 and α = 1°.
90,86 181,73 273,08 363,95 454,81 1,E-16 1,E-15 1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 1,E-07 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cl [ 1/Hz ] frequency [Hz]
PSD of Cl, at M=0.80, α=1°, time step=0.001s
1E-19 1E-18 1E-17 1E-16 1E-15 1E-14 1E-13 1E-12 1E-11 1E-10 1E-09 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cd [ 1/Hz ] frequency [Hz]PSD of Cd, at M=0.80, α=1°, time step=0.001s
1,E-16 1,E-15 1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cm [1/Hz ] frequency [Hz]PSD of Cm, at M=0.80, α=1°, time step=0.001s
127
Anyway, found values of power spectral density are two orders of magnitude smaller than that in condition at 0.975 degrees.
5.1.8 Simulation at
𝛂 = 1.025°
Another simulation was conducted at an incidence angle of 1.025 degrees for two different time steps, i.e. 0.001s and 0.0001s, but no oscillations were found leading to a stable condition. For brevity were reported only time history of lift, drag and moment coefficient neglecting the null PSDs.
Figure 5.32 Lift, drag and moment coefficients charts at Mach = 0.80 and α = 1.025°.
5.1.9 Simulation at
𝛂 = 1.1°
In next analysis, conducted for 𝛼 = 1.1°, would be expected to find no oscillation as occurred in other series of simulations at different Mach numbers. Unusually were found a new growth of unsteady phenomena leading to think that other instability onset could exist at 𝑀𝑎𝑐ℎ = 0.80 for NACA 0012 airfoil. This hypothesis was confirmed by other analyses conducted for greater incidence angles. Results found for 𝛼 = 1.1° are shown in next figures reporting time histories of lift, drag and moment coefficients and PSDs of the previous. Power spectral density shown presence of some very high and very energetic frequencies, which cannot be associated to buffet phenomenon that normally involve lower frequencies. 0,226750 0,226800 0,226850 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 8,5 9 9,5 10 cl ,l if t co ef fi ci e n t time [s]
Cl time history
0,023970 0,023980 0,023990 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 8,5 9 9,5 10 cd , d ra g co ef fi ci e n t time [s]Cd time history
-0,019300 -0,019280 -0,019260 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 8,5 9 9,5 10 cm,mom e n t co ef fi ci e n t time [s]Cm time history
128
Figure 5.33 Lift, drag and moment coefficients charts at Mach = 0.80 and α = 1.1°.
0,2403503 0,2403504 0,2403504 0,2403505 0,2403505 0,2403506 0,2403506 9,8 9,825 9,85 9,875 9,9 9,925 9,95 9,975 10 10,025 10,05 10,075 10,1 cl , l if t co ef fi ci e n t time [s]
Cl time history
0,0249054 0,0249054 0,0249054 0,0249054 0,0249054 0,0249054 9,8 9,825 9,85 9,875 9,9 9,925 9,95 9,975 10 10,025 10,05 10,075 10,1 cd , d ra g co ef fi ci e n t time [s]Cd time history
-0,0202971 -0,0202970 -0,0202970 -0,0202970 -0,0202970 -0,0202970 -0,0202969 -0,0202969 -0,0202969 9,8 9,825 9,85 9,875 9,9 9,925 9,95 9,975 10 10,025 10,05 10,075 10,1 cm, m ome n t co ef fi ci e n t time [s]Cm time history
129
Figure 5.34 Lift, drag and moment coefficients PSD charts at Mach = 0.80 and α = 1.1°.
1,E-17 1,E-16 1,E-15 1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 1,E-07 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 PS D of cl [ 1/Hz ] frequency [Hz]
PSD of Cl
1E-19 1E-18 1E-17 1E-16 1E-15 1E-14 1E-13 1E-12 1E-11 1E-10 1E-09 1E-08 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 PS D of cd [ 1/Hz ] frequency [Hz]PSD of Cd
1E-18 1E-17 1E-16 1E-15 1E-14 1E-13 1E-12 1E-11 1E-10 1E-09 1E-08 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 PS D of cm [1/HZ ] frequency [Hz]PSD of Cm
130
5.1.10 Simulation at
𝛂 = 1.2°
Analysis at 𝛼 = 1.2° revealed a considerable development of instability as suggested by previous simulation. Results shown a non-perfect repetitiveness of signals even if a repetitive cycle is recognisable.
Figure 5.35 Lift, drag and moment coefficients charts at Mach = 0.80 and α = 1.2°.
Further investigation of PSDs of lift, drag and moment coefficients in a suitable time interval revealed a quite clear frequency content and a remarkable energy level involved. LCO charts show presence of an unsteady phenomenon not completely well defined due to imperfect repetitiveness of signals, but quite large compared with other results.
0,2585472 0,2585477 0,2585482 0,2585487 0,2585492 0,2585497 3,5 3,55 3,6 3,65 3,7 3,75 3,8 3,85 3,9 3,95 4 cl , l if t coe ff ici en t time [s]
Cl time history
0,0262204 0,0262205 0,0262206 0,0262207 0,0262208 0,0262209 3,5 3,55 3,6 3,65 3,7 3,75 3,8 3,85 3,9 3,95 4 cd , d ra g co ef fi ci e n t time [s]Cd time history
-0,0217904 -0,0217902 -0,0217900 -0,0217898 -0,0217896 -0,0217894 3,5 3,55 3,6 3,65 3,7 3,75 3,8 3,85 3,9 3,95 4 cm, m ome n t co ef fi ci e n t time [s]Cm time history
131
Figure 5.36 Lift, drag and moment coefficients PSD charts at Mach = 0.80 and α = 1.2°.
41,10 84,15 125,24 166,34 209,39 250,49 291,59 332,68 375,73 416,83 457,93 499,02 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 1,E-07 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cl [ 1/Hz ] frequency [Hz]
PSD of Cl(sample range:3,5-4 s;512 time steps)
1E-15 1E-14 1E-13 1E-12 1E-11 1E-10 1E-09 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cd [ 1/Hz ] frequency [Hz]
PSD of Cd(sample range:3,5-4 s;512 time steps)
1,00E-15 1,00E-14 1,00E-13 1,00E-12 1,00E-11 1,00E-10 1,00E-09 1,00E-08 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cm [1/Hz ] frequency [Hz]
132
Figure 5.37 LCO charts of lift and moment coefficients at Mach = 0.80 and α = 1.2º.
5.1.11 Simulation at
𝛂 = 1.3°
Analysis at 𝛼 = 1.3° did not reveal oscillation giving thus a stable solution and the only power spectral density different from zero is that of drag coefficient; values of the last are very little and have not a physical meaning.
0,258547 0,258548 0,258548 0,258549 0,258549 0,258550 0,258550 -0,0015 -0,0005 0,0005 0,0015 cl Δcl/Δt
Δcl/Δt
-0,021792 -0,021791 -0,021791 -0,021790 -0,021790 -0,021789 -0,021789 -0,0015 -0,0005 0,0005 0,0015 cm Δcm/ΔtΔcm/Δt
133
Figure 5.38 Lift, drag and moment coefficients charts at Mach = 0.80 and α = 1.3°.
For total clarity purpose of drag coefficient in this case is reported in next Figure 5.39, even if, as told before, their values have not a physical meaning.
Figure 5.39 Drag coefficient PSD chart at Mach = 0.80 and α = 1.3°.
0,275920 0,275940 0,275960 0,275980 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 8,5 9 9,5 10 cl , l if t co ef fi ci e n t time [s]
Cl time history
0,027585 0,027590 0,027595 0,027600 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 8,5 9 9,5 10 cd , d ra g co ef fi ci e n t time [s]Cd time history
-0,023210 -0,023205 -0,023200 -0,023195 -0,023190 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 8,5 9 9,5 10 cm, m ome n t co ef fi ci e n t time [s]Cm time history
1,E-20 1,E-19 1,E-18 1,E-17 1,E-16 1,E-15 1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cd [ 1/Hz ] frequency [Hz]134
5.1.12 Simulation at
𝛂 = 1.4°
Last analysis shown a sink into a steady situation, but following simulation performed at incidence angle of 1.4 degrees reveal a new activation of unsteady phenomena. Charts of extracted data (Figure 5.40) exhibit signals of lift, drag and moment coefficient again variable in time and a quasi-perfect repetitiveness of shape and amplitude oscillations. Thus, power spectral density charts, shown in Figure 5.41, are quite clear. Was also conducted a separated analysis to investigate loci of unsteadiness onset; results of this last, reported in Figures 5.42-5.43-5.44, locate instability origin on lower surface again. This result was justified by inspection of root mean square error of static pressure on airfoil surface whose picture is shown if Figure 5.45.
Figure 5.40 Lift, drag and moment coefficients charts at Mach = 0.80 and α = 1.4°.
0,2914092 0,2914094 0,2914096 0,2914098 0,2914100 0,2914102 9,5 9,55 9,6 9,65 9,7 9,75 9,8 9,85 9,9 9,95 10 cl , l if t co ef fi ci e n t time [s]
Cl time history
0,02894990 0,02895000 0,02895010 0,02895020 9,5 9,55 9,6 9,65 9,7 9,75 9,8 9,85 9,9 9,95 10 cd , d ra g co ef fi ci e n t time [s]Cd time history
-0,02423545 -0,02423535 -0,02423525 -0,02423515 9,5 9,55 9,6 9,65 9,7 9,75 9,8 9,85 9,9 9,95 10 cm,mom e n t co ef fi ci e n t time [s]Cm time history
135
Figure 5.41 Lift, drag and moment coefficients PSD charts at Mach = 0.80 and α = 1.4°.
1,E-16 1,E-15 1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 1,E-07 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cl [ 1/Hz ] frequency [Hz]
PSD of Cl
1,E-17 1,E-16 1,E-15 1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cd [ 1/Hz ] frequency [Hz]PSD of Cd
1,E-18 1,E-17 1,E-16 1,E-15 1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cm [1/Hz ] frequency [Hz]PSD of Cm
136
Figure 5.42 Upper and lower surface lift coefficients charts at Mach = 0.80 and α = 1.4°.
Figure 5.43 Upper and lower surface lift coefficients PSD charts at Mach = 0.80 and α = 1.4°.
0,52851108 0,52851118 0,52851128 9,8 9,825 9,85 9,875 9,9 9,925 9,95 9,975 10 cl _u p ,u p p e r su rf ace lif t co ef fi ci e n t time [s]
Cl upper surface time history
-0,2371023 -0,2371018 -0,2371013 9,8 9,825 9,85 9,875 9,9 9,925 9,95 9,975 10 cl _ lo w ,l o we r su rf ace lif t co ef fi ci en t time [s]
Cl lower surface time history
1,E-16 1,E-15 1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 1,E-07 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cl _u p [ 1/Hz ] frequency [Hz]
PSD of Cl_up
1,E-16 1,E-15 1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 1,E-07 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D o f cl _ lo w [ 1 /Hz ] frequency [Hz]PSD of Cl_low
137
Figure 5.44 Lift coefficient Limit Cycle Oscillation charts of upper and lower surfaces at Mach = 0.80 and α = 1.4°.
Figure 5.45 RMSE of static pressure at Mach = 0.80 and α = 1.4º.
From previous figures clearly appears that unsteadiness onset is on the lower surface; its magnitude, indeed, is very higher on lower surface than on upper one.
0,52851008 0,52851058 0,52851108 0,52851158 0,52851208 0,52851258 0,52851308 -0,0015 -0,0005 0,0005 0,0015 cl _up Δ(cl_up)/Δt
Δ(cl_up)/Δt
-0,23710340 -0,23710290 -0,23710240 -0,23710190 -0,23710140 -0,23710090 -0,23710040 -0,0015 -0,0005 0,0005 0,0015 cl _l ow Δ(cl_low)/ΔtΔ(cl_low)/Δt
138
5.1.13 Simulation at
𝛂 = 1.5°
Last simulation in buffet onset region was made at an incidence angle of 1.5 degrees. In this condition was found a further increase of unsteady phenomena with respect to previous analysis. Obtained charts of lift, drag and moment coefficient show a larger oscillation amplitude and a more complex shape of repetitive element; this last, moreover, reflects on power spectral density charts that has a larger number of harmonics involved. Furthermore, this multiplicity of frequency which have a comparable energy quantity could be deduced from LCO charts whose shape looks like a series of overlapping circles.
In Figure 5.46 are shown lift, drag and moment coefficient time history, highlighting their repetitiveness, while in Figure 5.47 are reported related power spectral density charts.
Figure 5.46 Lift, drag and moment coefficients charts at Mach = 0.80 and α = 1.5°.
0,305955 0,305956 0,305957 0,305958 9,5 9,55 9,6 9,65 9,7 9,75 9,8 9,85 9,9 9,95 10 cl , l if t co ef fi ci e n t time [s]
Cl time history
0,030326 0,030326 0,0303261 0,0303261 0,0303262 0,0303262 9,5 9,55 9,6 9,65 9,7 9,75 9,8 9,85 9,9 9,95 10 cd , d ra g co ef fi ci en t time [s]Cd time history
-0,0252 -0,025199 -0,025198 9,5 9,55 9,6 9,65 9,7 9,75 9,8 9,85 9,9 9,95 10 cm,mom e n t co ef fi ci e n t time [s]Cm time history
139
Figure 5.47 Lift, drag and moment coefficients PSD charts at Mach = 0.80 and α = 1.5°.
22,47 27,85 44,94 67,42 1,E-15 1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 1,E-07 1,E-06 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cl [ 1/Hz ] frequency [Hz]
PSD of Cl (last 2048 time steps)
1,E-18 1,E-17 1,E-16 1,E-15 1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D o f cd [ 1 /Hz ] frequency [Hz]
PSD of Cd (last 2048 punti)
27,85 1,E-15 1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 1,E-07 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cm [1/Hz ] frequency [Hz]PSD of Cm (last 2048 punti)
140
First frequency involved is about 27.47 Hz, which has its super-harmonics, but there is another frequency about 27.85 Hz that seems to be isolated.
LCO charts are presented in Figure 5.48 and refers to lift and moment coefficients.
Figure 5.48 LCO charts of lift and moment coefficients at Mach = 0.80 and α = 1.5º.
A further inspection, separating contributes of upper and lower surfaces, shows a difference in lift coefficient of one order of magnitude between the two in favour of lower surface as can be seen from next Figure 5.49.
Figure 5.49 Upper and lower surface lift coefficients charts at Mach = 0.80 and α = 1.5°.
0,3059552 0,3059557 0,3059562 0,3059567 0,3059572 0,3059577 0,3059582 -0,0015 -0,0005 0,0005 0,0015 cl ΔCl/Δt
ΔCl/Δt
-0,0252009 -0,0252004 -0,0251999 -0,0251994 -0,0251989 -0,0251984 -0,0251979 -0,0015 -0,0005 0,0005 0,0015 cm ΔCm/ΔtΔCm/Δt
0,5377856 0,5377858 0,5377860 0,5377862 1 1,05 1,1 1,15 1,2 1,25 1,3 1,35 1,4 cl _u p ,u p p e r su rf ace lif t co ef fi ci e n t time[s]Cl upper surface time history
-0,231832 -0,231831 -0,231830 -0,231829 -0,231828 1 1,05 1,1 1,15 1,2 1,25 1,3 1,35 1,4 cl _l ow ,l o w er su rf ace lif t co ef fi ci e n t time [s]
141
Power spectral densities too reflect this showing same difference between values; PSD charts are reported in Figure 5.50. From inspection of these charts can be noted a clearer frequencies content on the lower surface and many harmonics carrying a comparable energy content. Furthermore, the origin of the spurious frequency of about 27.85 Hz highlighted in Figure 5.47 seems to be located on the upper airfoil surface; even if modulated in the overall analysis the difference of 0.5 Hz with respect to frequency found in separated simulation, of about 28.35 Hz (highlighted in Figure 5.50), permits a safe identification of that value.
Figure 5.50 Upper and lower surface lift coefficients PSD charts at Mach = 0.80 and α = 1.5°.
22,48 28,35 1,E-16 1,E-15 1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 0 50 100 150 200 250 300 350 400 450 500 PS D of cl _u p [ 1/Hz ] frequency [Hz]
PSD of cl upper surface
22,48 1,E-15 1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 1,E-07 0 50 100 150 200 250 300 350 400 450 500 PS D of cl _l ow [ 1/Hz ] frequency [Hz]PSD of cl lower surface
142
LCO charts indicate the same fact that unsteadiness is concentrated on the lower airfoil section (see Figure 5.51).
Figure 5.51 Lift coefficient Limit Cycle Oscillation charts of upper and lower surfaces at Mach = 0.80 and α = 1.5°.
Inspection of Figures 5.50 and 5.51 led to think that, in this case too, fluctuations are more concentrated on the lower airfoil side; this last was confirmed by RMSE analyses: in next Figures 5.52 and 5.53 are shown RMSE of static pressure and RMSE of velocity magnitude. First image despite what could seems, does not indicate a major unsteadiness on upper surface because is spread off more than on the lower one. Furthermore, on lower surface, oscillations are highly concentrated near the airfoil surface, while on the other side are at a certain distance over separated shear layer zone thus, less affecting pressure distribution around the airfoil and consequently giving lesser contributes to lift, drag and moment fluctuations. A better clarification of this fact is given in Figure 5.53 which shows RMSE velocity magnitude indicating on the lower side of airfoil the most of fluctuations.
Figure 5.52 RMSE of static pressure values at Mach = 0.80 and 𝛼 = 1.5°.
0,5377842 0,5377847 0,5377852 0,5377857 0,5377862 0,5377867 0,5377872 0,5377877 0,5377882 -0,0015 -0,001 -0,0005 0 0,0005 0,001 0,0015 cl _up Δcl_up/Δt
Δcl_up/Δt
-0,231832 -0,231831 -0,231831 -0,23183 -0,23183 -0,231829 -0,231829 -0,231828 -0,0015 -0,001 -0,0005 0 0,0005 0,001 0,0015 cl _l ow Δcl_low/ΔtΔcl_low/Δt
143
Figure 5.53 RMSE of velocity magnitude values at Mach = 0.80 and 𝛼 = 1.5°.
5.1.14 Simulation at
𝛂 = 4°
Last analysis, performed in the 𝑀𝑎𝑐ℎ = 0.80 series is at 𝛼 = 4°,was not made to identify buffet onset, being indeed far beyond onset condition and in a situation of completely developed buffet phenomenon; its purpose was to observe if simulation method would been capable of capture classical behaviour described in literature, i.e. shock wave move, vortex shedding and separation accompanied by thinning and thickening of turbulent shear layer.
Found results indicate a global instability and an animation of the dynamic problem show that used method, i.e. κ-ε model, can give good results even if in a condition of very large flux separation. Next Figure 5.54 shows lift, drag and moment coefficient time history. Evidently, fluctuations magnitude is enormous compared to that found in previous analyses. From inspection of Figure 5.54 can be noted that unsteadiness, after a transient time, starts to increase very rapidly reaching large amplitudes oscillating in a very short period as shown in Figure 5.55 through frequency content. Thus, in a real situation, an aircraft wing does not reach that situation because would brake much earlier. Figure 5.56 shows LCO charts; even if a clear repetitiveness was not found, in this condition length scale hides the non-perfect repetition of oscillation cycles, thus charts appear to be quite regular circles formed by bands of curves.
144
An exam of separated airfoil surfaces shows that instabilities is not located in a peculiar region but spreads out all around the airfoil surfaces, even if a little bit smaller on the lower surface as literature reports. Next Figures 5.57-5.58-5.59 report extracted data.
Figure 5.54 Lift, drag and moment coefficients charts at Mach = 0.80 and α = 4°.
0,35 0,36 0,37 0,38 0,39 0,4 0,41 0,42 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 cl , l if t co ef fi ci e n t time [s]
Cl time history
0,0445 0,0495 0,0545 0,0595 0,0645 0,0695 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 cd , d ra g co ef fi ci e n t time [s]Cd time history
-0,02 -0,01 0,00 0,01 0,02 0,03 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 7,5 8 cm, m ome n t co ef fi ci e n t time [s]Cm time history
145
Figure 5.55 Lift, drag and moment coefficients PSD charts at Mach = 0.80 and α = 4°.
1,E-06 1,E-05 1,E-04 1,E-03 1,E-02 1,E-01 1,E+00 1,E+01 1,E+02 1,E+03 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cl [ 1/Hz ] frequency [Hz]
PSD of Cl
1,E-06 1,E-05 1,E-04 1,E-03 1,E-02 1,E-01 1,E+00 1,E+01 1,E+02 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cd [ 1/Hz ] frequency [Hz]PSD of Cd
1,E-06 1,E-05 1,E-04 1,E-03 1,E-02 1,E-01 1,E+00 1,E+01 1,E+02 1,E+03 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cm [1/Hz ] frequency [Hz]PSD of Cm
146
Figure 5.56 LCO charts of lift and moment coefficients at Mach = 0.80 and α = 4º.
Figure 5.57 Upper and lower surface lift coefficients charts at Mach = 0.80 and α = 4°.
0,35 0,36 0,37 0,38 0,39 0,4 0,41 0,42 -20 -10 0 10 20 cl Δcl/Δt
Δcl/Δt
-0,03 -0,02 -0,01 0 0,01 0,02 0,03 0,04 -20 -10 0 10 20 cm Δcm/ΔtΔcm/Δt
0,57 0,58 0,59 0,6 0,61 0,62 0,63 2,6 3,1 3,6 4,1 4,6 5,1 5,6 6,1 6,6 7,1 7,6 8,1 cl _u p ,u p p e r su rf ace lif t co ef fi ci e n t time [s]Cl upper surface time history
-0,235 -0,230 -0,225 -0,220 -0,215 -0,210 -0,205 -0,200 2,6 3,1 3,6 4,1 4,6 5,1 5,6 6,1 6,6 7,1 7,6 8,1 cl _l ow , l ow e r su rf ace lif t co ef fi ci e n t time [s]
147
Figure 5.58 Upper and lower surface lift coefficients PSD charts at Mach = 0.80 and α = 4°.
Figure 5.59 Lift coefficient Limit Cycle Oscillation charts of upper and lower surfaces at Mach = 0.80 and α = 4°.
1,E-06 1,E-05 1,E-04 1,E-03 1,E-02 1,E-01 1,E+00 1,E+01 1,E+02 1,E+03 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cl _u p [ 1/Hz ] frequency [Hz]
PSD of Cl upper surface
1,E-06 1,E-05 1,E-04 1,E-03 1,E-02 1,E-01 1,E+00 1,E+01 1,E+02 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 PS D of cl _l ow [ 1/Hz ] frequency [Hz]PSD of Cl lower surface
0,57 0,58 0,59 0,6 0,61 0,62 0,63 -20 -15 -10 -5 0 5 10 15 20 cl _up Δ(cl_up)/ΔtΔ(cl_up)/Δt
-0,25 -0,24 -0,23 -0,22 -0,21 -0,2 -0,19 -20 -15 -10 -5 0 5 10 15 20 cl _l ow Δ(cl_low)/ΔtΔ(cl_low)/Δt
148
In this case too were visualized RMSE of static pressure highlight loci of unsteadiness but Figures 5.60 and 5.61 indicate a whole perturbed field.
Figure 5.60 RMSE of static pressure at Mach = 0.80 and α = 4º showing whole flow field.
149
Figure 5.62 Contours of Mach number at different time steps increasing in time from left to right; refence times
150
Figure 5.63 Contours of static pressure at different time steps increasing in time from left to right; refence times
151
Figure 5.64 Time sampling at Mach = 0.80 and 𝛼 = 4°, referred to Fig. 2.135 and Fig. 2.136.
Previous figures show contours of Mach and static pressure while oscillatory value of lift coefficient goes from a maximum to a minimum and newly to next maximum. Can be appreciated vortex shedding and a sort of wind probably due to pressure waves going back to the flux current and sweeping from behind to forward the whole field; this last seems to be the reason of shock wave oscillation as hypothesised by Lee in Ref. [14].
Results shown that there is not a great shock oscillation near airfoil wall but quite far away; this last could be a limit of the chosen turbulence method or a software limit however obtained data reveal optimum accordance with the experimental ones.
To dispel any doubt on the symmetric grid capability, which was used in previous calculations, of capture shock oscillation phenomenon was build an asymmetrical one too. Asymmetric grid has about 500000 nodes (the upper limit for ANSYS® Workbench academic teaching license) and two simulations were performed with same settings and using a k-ω model. Obtained data were practically the same given by the other mesh, thus was deduced that a finer grid dimension would not have led to different results. Furthermore, a change in turbulence model seemed to not have advantages in this case at least. Due to larger time waste required for calculations with a so increased node number and due to the fact that no differences were found between results of the two meshes was decided that the symmetrical grid was sufficient to accurate describe the unsteady phenomenon in this condition at least.
An image of the asymmetric grid is shown in Figure 5.65.
7,977 7,978 7,979 7,98 7,981 7,982 7,983 7,984 7,985 7,986 0,34 0,35 0,36 0,37 0,38 0,39 0,4 0,41 0,42 0,43 7,965 7,9675 7,97 7,9725 7,975 7,9775 7,98 7,9825 7,985 7,9875 7,99 7,9925 7,995 7,9975 8 cl , l if t co ef fi ci e n t time [s]
Cl time history
152
Figure 5.65 Asymmetric grid for capture eventual shock wave motion.
5.2 Results of simulation series at Mach = 0.80
An overview of obtained results at 𝑀𝑎𝑐ℎ = 0.80 shows an increase in RMS values as in previous cases while increase the incidence angle. Despite other series of analyses, after the drop down of RMS values it was noted a further increase, a new drop down and an increase again. Charts of root mean square in this case look like saw teeth indicating, in combination with oscillation amplitude, limited cycle oscillation and normalised residuals increasing, multiple solution existence. Analyses performed indicate three solutions of instability onset at least: 𝛼𝑐𝑟1= 0.975°, 𝛼𝑐𝑟2 = 1.2°, 𝛼𝑐𝑟3 = 1.5°, furthermore the most of instability at buffet onset conditions was found on the lower surface. Thus, the fact that, for Mach equal to 0.80 for a NACA 0012, buffet instability could be triggered by lower surface instability seems to be a constant. Found values of maximum oscillation and variance of time rate of lift coefficient found in each simulation are reported in next page in Table 5.1, while Table 5.2 shows root mean square values of power spectral density for each incidence angle. Next Figure 5.66 displays charts of variance of time rate and maximum oscillation of lift coefficient, while Figure 5.67 contains a chart of amplitude of variance of time rate of lift coefficient oscillations. Figure 5.68 defines both amplitude and shape of variance of time rate oscillation of lift coefficient, eventually Figures 5.69-5.70 exhibits root mean square charts of lift, drag and moment coefficients of each simulation.
153 α Δ(Δcl/Δt) Δcl 0,5 0,00E+00 0,00E+00 0,8 7,15E-04 4,62E-07 0,85 7,30E-04 4,62E-07 0,925 1,55E-03 1,01E-06 0,975 2,29E-03 2,44E-06 0,99 2,83E-04 1,94E-07 1 8,34E-04 4,47E-07 1,025 1,49E-05 1,49E-08 1,1 4,47E-04 2,53E-07 1,2 3,07E-03 2,65E-06 1,3 0,00E+00 0,00E+00 1,4 5,96E-04 8,34E-07 1,5 1,82E-03 2,47E-06
Table 5.1 Results of LCO amplitudes and maximum
modulus of oscillation of lift coefficient at Mach = 0.80.
RMS values of PSD of
α Cl Cd Cm
0,5 0,00E+00 0,00E+00 0,00E+00 0,8 2,67E-04 1,63E-05 1,30E-04 0,85 1,42E-04 1,64E-05 6,11E-05 0,925 2,19E-04 2,30E-05 1,14E-04 0,975 7,90E-04 1,47E-04 7,76E-04 0,99 5,22E-05 7,02E-06 3,77E-05 1 1,55E-04 1,85E-05 9,77E-05 1,025 5,66E-06 0,00E+00 0,00E+00
1,1 1,16E-04 6,82E-06 7,61E-05 1,2 3,49E-04 3,54E-05 1,86E-04 1,3 0,00E+00 6,85E-07 0,00E+00 1,4 2,07E-04 7,12E-05 5,48E-05 1,5 5,01E-04 5,41E-05 2,60E-04
Table 5.2 Results of RMS of PSD values obtained
154
Figure 5.66 LCO amplitude and maximum oscillation of lift coefficient as a function of incidence at Mach = 0.80.
0,E+00 5,E-04 1,E-03 2,E-03 2,E-03 3,E-03 3,E-03 4,E-03 0,5 0,55 0,6 0,65 0,7 0,75 0,8 0,85 0,9 0,95 1 1,05 1,1 1,15 1,2 1,25 1,3 1,35 1,4 1,45 1,5 1,55 Δ (Δc l/ Δ t) α, incidence [deg]
Δ(Δcl/Δt)
0,E+00 5,E-07 1,E-06 2,E-06 2,E-06 3,E-06 3,E-06 0,5 0,55 0,6 0,65 0,7 0,75 0,8 0,85 0,9 0,95 1 1,05 1,1 1,15 1,2 1,25 1,3 1,35 1,4 1,45 1,5 1,55 Δcl α, incidence [deg]Δcl
155
Figure 5.67 Comparison between amplitudes of Limit Cycle Oscillation of lift coefficient at Mach = 0.80.
0,11 0,115 0,12 0,125 0,13 0,135 0,14 0,145 0,15 0,155 0,16 0,165 0,17 0,175 0,18 0,185 0,19 0,195 0,2 0,205 0,21 0,215 0,22 0,225 0,23 0,235 0,24 0,245 0,25 0,255 0,26 0,265 0,27 0,275 0,28 0,285 0,29 0,295 0,3 0,305 0,31 -0,0015 -0,001 -0,0005 0 0,0005 0,001 0,0015
Δcl/Δt
1.5 deg 1.4 deg 1.3 deg 1.2 deg 1.1 deg 1.025 deg 1 deg 0.99 deg 0.975 deg 0.925 deg 0.85 deg 0.80 deg 0.5 deg156 Continue… 0,115249 0,1152495 0,11525 0,1152505 0,115251 0,1152515 0,115252 -0,0015 -0,0005 0,0005 0,0015
Δcl/Δt at 0,5 deg
0,1807079 0,1807084 0,1807089 0,1807094 0,1807099 0,1807104 0,1807109 -0,0015 -0,0005 0,0005 0,0015Δcl/Δt at 0,80 deg
0,1913627 0,1913632 0,1913637 0,1913642 0,1913647 0,1913652 0,1913657 -0,0015 -0,0005 0,0005 0,0015Δcl/Δt at 0,85 deg
0,2038883 0,2038888 0,2038893 0,2038898 0,2038903 0,2038908 0,2038913 -0,0015 -0,0005 0,0005 0,0015Δcl/Δt at 0,925 deg
0,216536 0,2165365 0,216537 0,2165375 0,216538 0,2165385 0,216539 -0,0015 -0,0005 0,0005 0,0015Δcl/Δt at 0,975 deg
0,2193251 0,2193256 0,2193261 0,2193266 0,2193271 0,2193276 0,2193281 -0,0015 -0,0005 0,0005 0,0015Δcl/Δt at 0,99 deg
157 Continue… 0,22115465 0,22115515 0,22115565 0,22115615 0,22115665 0,22115715 0,22115765 -0,0015 -0,0005 0,0005 0,0015
Δcl/Δt at 1,0 deg
0,2268006 0,2268011 0,2268016 0,2268021 0,2268026 0,2268031 0,2268036 -0,0015 -0,0005 0,0005 0,0015Δcl/Δt at 1,025 deg
0,2403498 0,2403503 0,2403508 0,2403513 0,2403518 0,2403523 0,2403528 -0,0015 -0,0005 0,0005 0,0015Δcl/Δt at 1,1 deg
0,2585475 0,258548 0,2585485 0,258549 0,2585495 0,25855 0,2585505 -0,0015 -0,0005 0,0005 0,0015Δcl/Δt at 1,2 deg
0,2759690 0,2759695 0,2759700 0,2759705 0,2759710 0,2759715 0,2759720 -0,0015 -0,0005 0,0005 0,0015Δcl/Δt at 1,3 deg
0,2914082 0,2914087 0,2914092 0,2914097 0,2914102 0,2914107 0,2914112 -0,0015 -0,0005 0,0005 0,0015Δcl/Δt at 1,4 deg
158
Figure 5.68 Amplitude and shape of LCOs of lift coefficient at Mach = 0.80.
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. 0,305955 0,3059555 0,305956 0,3059565 0,305957 0,3059575 0,305958 -0,0015 -0,0005 0,0005 0,0015
ΔCl/Δt at 1,5 deg
0,E+00 1,E-04 2,E-04 3,E-04 4,E-04 5,E-04 6,E-04 7,E-04 8,E-04 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1,5 1,6 R M S(P SD( cl )) α, incidence [deg]RMS of PSD of Cl at Mach=0.80
159
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.
By inspection of first frequency involved in buffet onset solutions it can be noted, as expected, a decrease in first harmonic value at growing incidence angle shown in Figure 5.71 0,E+00 2,E-05 4,E-05 6,E-05 8,E-05 1,E-04 1,E-04 1,E-04 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1,5 1,6 R M S(P SD( cd )) α, incidence [deg]
RMS of PSD of Cd at Mach=0.80
-1,E-05 9,E-05 2,E-04 3,E-04 4,E-04 5,E-04 6,E-04 7,E-04 8,E-04 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1,5 1,6 R M S(P SD( cm )) α, incidence [deg]RMS of PSD of Cm at Mach=0.80
160
Figure 5.71 Variation of value of first harmonic at buffet onset points with incidence angle at Mach = 0.80.
As in previous series of simulations, last step was the study in MATLAB® of static pressure fluctuations on airfoil wall by using colours-map. Results of this last analysis, conducted at first buffet onset condition, i.e. at 𝛼 = 0.975° are reported in following figures.
Figure 5.72 Colours-map of upper airfoil surface representing static pressure variance at Mach = 0.80 and 𝛼 = 0.975°.
0,975º; 89,88 Hz 1,2º; 41,1 Hz 1,5º; 22,47 Hz 0 10 20 30 40 50 60 70 80 90 100 0,9 0,95 1 1,05 1,1 1,15 1,2 1,25 1,3 1,35 1,4 1,45 1,5 1,55 fr qu enc y [Hz] α, incidence [deg]
First harmonic variation with
incidence angle
161
Figure 5.73 Colours-map of lower airfoil surface representing static pressure variance at Mach = 0.80 and 𝛼 = 0.975°.
From previous Figures 5.72-5.73 it can be noted that static pressure fluctuations about a medium local value, although very little, are larger on lower surface from an amplitude point of view. Moreover, can be deduced, from Figures 5.73 and 5.75, that pressure disturbances travel from shock wave, were it seems they could have their origin, to both leading edge and trailing edge direction.
Figure 5.74 Colours-map of upper airfoil surface in plain view representing static pressure variance at Mach = 0.80
162
Figure 5.75 Colours-map of lower airfoil surface in plain view representing static pressure variance at Mach = 0.80 and 𝛼 =
0.975°.
This observation seems to confirm, at least in part, Lee’s theory of shock oscillation self-sustaining (see Ref. [14]); furthermore, in accordance with results at α = 4° indicating pressure fluctuations going up to stream, previous theory seems to have found a confirmation.
Figures 5.72 and 5.74 do not well clarify unsteadiness behaviour even if is recognisable, following peaks in first figure and colours in second one, a tendency to travelling from shock wave to trailing edge of pressure disturbances.
