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Chapter 7
Steady and transient analyses at null
incidence angle and Mach above
0.80, buffet investigation
7.1 Simulation series at null incidence angle and Mach above
0.80
Last series of conducted analyses, at fixed incidence angle and variable Mach number, are specified in Tables 2.9 and 2.10. Figure 7.1 shows contour of static pressure in each condition starting from right to left with increasing incidence angle:
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).
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Figure 7.2 Normalised residuals at Mach = 0.816 and α = 0°; steady analysis on left and unsteady on right.
Figure 7.3 Normalised residuals at Mach = 0.820 and α = 0°; steady analysis on left and unsteady on right.
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Figure 7.5 Normalised residuals at Mach = 0.86 and α = 0°; steady analysis on left and unsteady on right.
As can be noted from previous figures, in this case too normalised residuals tend to increase were an unsteady phenomenon is present. Analysis at Mach equal to 0.86 aside, which is well beyond buffet onset condition, other simulations were performed trying to catch critical Mach number for buffet inception in a condition of 𝛼 = 0. Many authors like Crouch, Garbaruk, Magidov and Travin as well as Xiong, Liu, Luo, Ren, Zhao, Gao in their works (Ref. [7], [8] and [10], [12]) made transient analyses starting from a steady one at an incidence angle different from zero to simulate a perturbation in the flow field. Unlike of that works in this study was decided to start from a steady simulation at zero incidence angle considering more than enough the airfoil itself as a perturbation.
As next results will show, in this series of analyses, was found a critical Mach number in good accordance with experimental data once again showing a well-defined buffet onset accompanied by repetitive oscillations a quite clear frequency content.
7.1.1 Simulation at Mach = 0.816
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.
First value studied was 𝑀 = 0.816°. As wrote in Table 2.9, were used two different time steps; first of all was attempted to use one thousandth of a second that worked very well for transient analyses at Mach 0.80 but mean value of lift, drag and moment coefficient has not stabilised, thus was decided to use a value of a tenth of one thousandth of a second. In this last case results shown a stabilization of mean value of previous parameter making possible to have clearer charts and leading to consider flow quasi-steady in this condition. It was written ‘quasi’ because a sort of oscillation exists but it’s too little
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to bring macroscopic consequences. Next Figure 7.6 shows time history of lift, drag and moment coefficient for time step ∆𝑡 = 0.001 𝑠, while in Figure 7.7 are presented PSDs found in this case that are not sufficiently clear. Figure 7.8 and 7.9 represent time histories and PSDs of same parameters for a ∆𝑡 = 0.0001 𝑠, showing their frequency content.
Figure 7.6 Lift, drag and moment coefficients charts at Mach = 0.816 and α = 0° and ∆t = 0.001 s.
2,00E-06 2,20E-06 2,40E-06 3 3,1 3,2 3,3 3,4 3,5 3,6 3,7 3,8 3,9 4 4,1 4,2 4,3 4,4 cl , l if t co ef fi ci e n t time [s]
Cl time history
1,70E-02 1,70E-02 1,70E-02 3 3,1 3,2 3,3 3,4 3,5 3,6 3,7 3,8 3,9 4 4,1 4,2 4,3 4,4 cd , d ra g co ef fi ci e n t time [s]Cd time history
-7,30E-07 -7,20E-07 -7,10E-07 -7,00E-07 3 3,1 3,2 3,3 3,4 3,5 3,6 3,7 3,8 3,9 4 4,1 4,2 4,3 4,4 cm, m ome n t co ef fi ci e n t time [s]Cm time history
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Figure 7.7 Lift, drag and moment coefficients PSD charts at Mach = 0.816 and α = 0° and ∆t = 0.001 s.
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 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-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 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 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
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Figure 7.8 Lift, drag and moment coefficients charts at Mach = 0.816 and α = 0° and ∆t = 0.0001 s.
2,10E-06 2,10E-06 2,10E-06 2,10E-06 2,10E-06 1,57 1,77 1,97 2,17 2,37 2,57 cl , l if t co ef fi ci e n t time [s]
Cl time history
1,70E-02 1,70E-02 1,70E-02 1,70E-02 1,70E-02 1,70E-02 1,70E-02 1,57 1,77 1,97 2,17 2,37 2,57 cd , d ra g co ef fi ci e n t time [s]Cd time history
-6,95E-07 -6,95E-07 -6,95E-07 -6,95E-07 -6,95E-07 -6,95E-07 1,57 1,77 1,97 2,17 2,37 2,57 cm , m om en t co ef fi ci e n t time [s]Cm time history
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Figure 7.9 Lift, drag and moment coefficients PSD charts at Mach = 0.816 and α = 0° and ∆t = 0.0001 s.
1E-20 1E-19 1E-18 1E-17 1E-16 1E-15 1E-14 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 PS D of cl [ 1/Hz ] frequencies [Hz]
PSD of Cl (last 4096 time steps)
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 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 PS D of cd [ 1/Hz ] frequency [Hz]
PSD of Cd (last 4096 time steps)
1,E-20 1,E-19 1,E-18 1,E-17 1,E-16 1,E-15 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 PS D of cm [1/Hz ] frequency [Hz]
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Although a certain frequency content does exist energies quantity are so little that practically can be neglected. Moreover, inspection of LCO charts shows a point only with respect to next analyses leading to previous consideration. Was made a separated analysis too, but no other evidences were found. RMSE of static pressure in Figure 7.10 shows a flow field everywhere disturbed but with neglectable values of disturbances.
Figure 7.10 RMSE of static pressure at Mach = 0.816, 𝛼 = 0° and ∆t = 0.0001 s.
7.1.2 Simulation at Mach = 0.82
Having found an unsteadiness although of very little importance it would be expected that increasing a little bit incidence angle it would be appeared an oscillation but was not so. Simulation conducted at 𝛼 = 0.82° resulted in non-stabilisation of medium values of lift, drag and moment coefficients. Analysis was carried on far long to try to reach a stable mean value, moreover, obtained power spectral densities, although affected by last problem, were quite clear and did not revealed oscillations of appreciable dimensions. Next Figures 7.11 and 7.12 show lift, drag and moment coefficients time histories and PSDs of same parameters for this simulation.
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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. -6,00E-05 -4,00E-05 -2,00E-05 0,00E+00 2,00E-05 4,00E-05 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cl , l if t co ef fi ci e n t time [s]
Cl time history
1,70E-02 1,70E-02 1,70E-02 1,70E-02 1,70E-02 1,70E-02 1,70E-02 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cd , d ra g co ef fi ci e n t time [s]Cd time history
-1,00E-05 -5,00E-06 0,00E+00 5,00E-06 1,00E-05 1,50E-05 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cm, m ome n t co ef fi ci e n t time [s]Cm time history
2,44E-05 2,44E-05 2,44E-05 2,44E-05 2,44E-05 2,45E-05 12,5 12,6 12,7 12,8 12,9 13 13,1 13,2 13,3 13,4 13,5 13,6 13,7 13,8 13,9 14 cl , l if t co ef fi ci e n t time [s]Cl time history
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Figure 7.12 Lift, drag and moment coefficients PSD charts at Mach = 0.82, α = 0° and ∆t = 0.001 s.
1,E-18 1,E-17 1,E-16 1,E-15 1,E-14 1,E-13 1,E-12 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-21 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 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
1,E-20 1,E-19 1,E-18 1,E-17 1,E-16 1,E-15 1,E-14 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
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As can be noted from previous charts, oscillations found were of too much little level leading to think that in this case the lasts could be associated to software errors or anyway, be neglectable phenomena. In meanwhile was conducted also an analysis of separated upper and lower airfoil surfaces and results are presented in next Figures 7.13 and 7.14. as can be seen no oscillations were found in lift, drag and moment coefficients time histories, moreover power spectral densities of lift coefficient do not show frequencies associable to some physical behaviour.
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.
Figure 7.14 Lift coefficient PSDs of upper and lower airfoil surfaces charts at Mach = 0.82, α = 0° and ∆t = 0.001 s.
0,3632 0,3634 0,3636 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 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
-3,64E-01 -3,64E-01 -3,63E-01 -3,63E-01 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cl _l ow ,l o w e r su rf ace lif t co ef fi ci e n t time [s]
Cl lower surface time history
1,E-20 1,E-18 1,E-16 1,E-14 1,E-12 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-20 1,E-16 1,E-12 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
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Although root mean square error of static pressure seems to indicate a higher level of concentration of unsteadiness as happen in condition and about condition of buffet onset as next Figure 7.15.
Figure 7.15 RMSE of static pressure at Mach = 0.82, α = 0° and ∆t = 0.001 s.
7.1.3 Simulation at Mach = 0.825
Simulation performed in this condition had similar problems related to previous analyses, i.e. stabilisation of mean value of parameters but differently from the other solutions, started to oscillate. During analysis, oscillations appeared and disappeared many times due to smallness of values involved but, in the end, mean values stabilised and oscillations too reached a condition in which a repetitive cycle was recognisable although repetitiveness was not perfect. This last problem, probably due to numerical errors, led to make three analyses of extracted data in three different time intervals and of different length.
Next Figure 7.16 shows whole time history of lift coefficient, while in Figures 7.17-7.18-7.19 are reported charts of time intervals used in each of the three made analyses.
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Figure 7.16 Lift coefficient total time history simulated at Mach = 0.825, 𝛼 = 0°, ∆t = 0.001 s.
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. -3,E-05 -2,E-05 -1,E-05 0,E+00 1,E-05 2,E-05 3,E-05 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 10,5 11 11,5 12 12,5 cl , l if t co ef fi ci e n t time [s]
Cl time history
3,75E-06 4,00E-06 4,25E-06 4,50E-06 7,059 7,109 7,159 7,209 7,259 7,309 7,359 7,409 7,459 7,509 7,559 cl , l if t coe ff ici en t time [s]Cl time history ranging from 7.059 to7.57 s
3,11E-02 3,11E-02 3,11E-02 3,11E-02 3,11E-02 3,11E-02 3,11E-02 7,059 7,109 7,159 7,209 7,259 7,309 7,359 7,409 7,459 7,509 7,559 cd , d ra g co ef fi ci e n t time [s]
Cd time history ranging from 7.059 to 7.57 s
-1,00E-06 -7,50E-07 -5,00E-07 -2,50E-07 7,059 7,109 7,159 7,209 7,259 7,309 7,359 7,409 7,459 7,509 7,559 cm, m ome n t co ef fi ci e n t time [s]
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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.
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. 3,90E-06 4,10E-06 4,30E-06 4,50E-06 4,70E-06 8,652 8,852 9,052 9,252 9,452 9,652 9,852 10,052 10,252 10,452 10,652 cl , l if t co ef fi ci e n t time [s]
Cl time history ranging from 8,652 to 10,7 s
3,11E-02 3,11E-02 3,11E-02 3,11E-02 8,652 8,852 9,052 9,252 9,452 9,652 9,852 10,052 10,252 10,452 10,652 cd , d ra g co ef fi ci e n t time [s]
Cd time history ranging from 8,652 to 10,7 s
-8,00E-07 -7,00E-07 -6,00E-07 -5,00E-07 -4,00E-07 8,652 8,852 9,052 9,252 9,452 9,652 9,852 10,052 10,252 10,452 10,652 cm, momen t co ef fi ci en t time [s]
Cm time history ranging from 8,652 to 10,7 s
4,00E-06 4,20E-06 4,40E-06 4,60E-06 11,993 12,043 12,093 12,143 12,193 12,243 12,293 12,343 12,393 12,443 12,493 cl , l if t co ef fi ci e n t time [s]
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In Figure 7.19 is reported lift coefficient only, because this last analysis involved cl alone; the aim of the last was to validate results found in second analysis made on an interval of 2048 time steps. Results shown criticality of null incidence condition because lift, drag and moment fluctuations are of the order of 6 ∗ 10−7 thus, numerical errors, due to software or grid dimensions, would affect too much extracted data. Keeping in mind these problems is hard to define a specific frequency characterising phenomenon. First harmonic involved, from conducted analyses, ranges from about 50 to 75 Hz as can be seen in following figures. In Figure 7.20 are reported PSDs referred to Figure 7.17, in Figure 7.21 is shown PSDs pertaining to Figure 7.18 and last Figure 7.22 is relevant to Figure 7.19.
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. 54,79 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 0 20 40 60 80 100 120 140 160 180 200 220 240 PS D of cl [ 1/Hz ] frequency [Hz]
PSD of Cl (sampled from 7,059 s to 7,57 s)
1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 0 20 40 60 80 100 120 140 160 180 200 220 240 PS D of cd [ 1/Hz ] frequency [Hz]PSD of Cd (sampled from 7,059 s to 7,57 s)
1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 0 20 40 60 80 100 120 140 160 180 200 220 240 PS D of cl [ 1/Hz ] frequency [Hz]PSD of Cm (sampled from 7,059 s to 7,57 s)
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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. 74,26 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
74,26 1,E-16 1,E-15 1,E-14 1,E-13 1,E-12 1,E-11 1,E-10 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
74,26 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 di Cm
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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.
Last two analyses seem to indicate a frequency about 74 Hz, however, as previously wrote it was not possible to define with enough accuracy this value, moreover last two figures show involvement of other lower frequencies not well-defined. It was selected 54.79 Hz frequency because is first responsible of LCO charts in next figures. Only certainty is that, in this condition, a repetitive phenomenon starts with a well-defined limit cycle oscillation shown in Figure 7.23, normalised residual are higher than in previous analyses and oscillate, moreover root mean square values are higher of some order of magnitude than analyses at lower incidence; these consideration led to think that this value the buffet onset for NACA 0012 airfoil at null incidence.
Figure 7.23 LCO charts of lift and moment coefficients at Mach = 0.825, α = 0º and ∆t = 0.001 s.
74,36 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 (last 512 time steps)
3,80E-06 3,90E-06 4,00E-06 4,10E-06 4,20E-06 4,30E-06 4,40E-06 4,50E-06 4,60E-06 4,70E-06 -0,0008 -0,0004 0 0,0004 0,0008 cl Δcl/Δt
Δcl/Δt
-9,00E-07 -8,00E-07 -7,00E-07 -6,00E-07 -5,00E-07 -4,00E-07 -3,00E-07 -2,00E-07 -1,00E-07 -0,0008 -0,0004 0 0,0004 0,0008 cm Δcm/ΔtΔcm/Δt
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Further analysis of obtained data on upper and lower surfaces separately gave an unexpected result. It was predicted a symmetrical behaviour since airfoil is symmetrical and incidence is zero degrees; it was obtained an asymmetrical configuration instead. This fact could be due to microscopic error in model geometry or to complications similar to that connected with frequency locating. Normally a dynamic system responds to excitations with a first symmetrical mode, but in this case seems to respond with an asymmetrical one, thus it was argued that software could not be capable of find this first mode or actually could be an excitation of a second mode. Last considerations were derived from inspection of following Figures 7.24-7.25-7.26.
Figure 7.24 Upper and lower surface lift coefficient charts at Mach = 0.825, 𝛼 = 0°, ∆t = 0.001 s.
0,42094642 0,42094642 0,42094643 0,42094643 0,42094644 0,42094644 0,42094645 0,42094645 0,42094646 12 12,05 12,1 12,15 12,2 12,25 12,3 12,35 12,4 12,45 12,5 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
-0,4209425 -0,4209424 -0,4209423 -0,4209422 -0,4209421 -0,4209420 -0,4209419 -0,4209418 -0,4209417 12 12,05 12,1 12,15 12,2 12,25 12,3 12,35 12,4 12,45 12,5 cl _l ow ,l o w e r su rf ace lif t co ef fi ci e n t time [s]
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Figure 7.25 Upper and lower surfaces lift coefficient PSD charts at Mach = 0.825, 𝛼 = 0°, ∆t = 0.001 s.
Figure 7.26 Upper and lower surfaces LCO charts at Mach = 0.825, 𝛼 = 0°, ∆t = 0.001 s.
74,74 184,66 1,E-18 1,E-16 1,E-14 1,E-12 1,E-10 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 _u p [ 1/Hz ] frequency [Hz]
PSD of Cl upper surface
187,59 199,80 372,25 400,10 1,E-13 1,E-12 1,E-11 1,E-10 1,E-09 1,E-08 1,E-07 1,E-06 1,E-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 _l ow [ 1/Hz ] frequency [Hz]PSD of Cl lower surface
0,420937 0,420938 0,420939 0,42094 0,420941 0,420942 0,420943 0,420944 0,420945 0,420946 -0,0008 -0,0003 0,0002 0,0007 cl _u p Δ(cl_up)/ΔtΔ(cl_up)/Δt
-0,4209380 -0,4209379 -0,4209378 -0,4209377 -0,4209376 -0,4209375 -0,4209374 -0,4209373 -0,4209372 -0,4209371 -0,0008 -0,0003 0,0002 0,0007 cl _l ow Δ(cl_low)/ΔtΔ(cl_low)/Δt
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7.1.4 Simulation at Mach = 0.86
Last simulation of this series at 𝛼 = 0° was performed at 𝑀𝑎𝑐ℎ = 0.86, thus, in a region well beyond buffet boundary i.e. a condition of fully developed unsteadiness. Found results accordingly with literature predict very large of oscillations accompanied by low frequencies. This fact is probably due to shock wave large oscillations that govern phenomenon, despite of other series of analyses in which no macroscopically move of shock were noted and frequencies involved were much higher. In that cases indeed, unsteadiness was concentrated in very little zones and its behaviour depends on little moves of high frequency. As many authors wrote it was found a phase opposition phenomenon on the two side of the airfoil: when the shock on the upper surface reach the minimum distance from trailing edge, shock on lower surface reach its maximum distance. Currently shear layer downstream first shock wave separates from the airfoil forming a separated bubble of flux recirculation, consequently the unfavourable positive pressure gradient downstream the shock tends to increase encouraging further separation of shear layer and increasing separated bubble dimension. This fact is felt by flux as a thickening of the airfoil thus decreasing Mach number on the upper side; therefore, an oblique shock forms near wall and shock tends to go towards trailing edge diminishing its intensity.
Figure 7.27 Sampling of lift coefficient time history at Mach = 0.86, 𝛼 = 0°.
3,666 3,683 3,7 3,717 3,734 3,751 3,768 3,785 3,802 3,819 3,836 -0,06 -0,04 -0,02 0 0,02 0,04 0,06 3,65 3,665 3,68 3,695 3,71 3,725 3,74 3,755 3,77 3,785 3,8 3,815 3,83 3,845 3,86 cl , l if t co ef fi ci e n t time [s]
Cl time history
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Figure 7.28 Flux behaviour around a NACA0012 airfoil at Mach = 0.86, 𝛼 = 0°.
At a certain point shear layer returns attached to airfoil wall and shock comes back increasing newly its strength. On the lower surface it happens the same but in a mirrored way. Previous Figure 7.28 shows this process whose pictures were taken sampling time steps by 0.017 s as reported in Figure 7.27. As clarified later in this section, every about 0.170 s a complete cycle closes, thus ten pictures represent the behaviour in previous figure.
Results of simulation are presented in next Figures 7.29-7.30-7.31 in which are shown lift, drag and moment coefficients time histories, power spectral densities of previous parameters and limit cycle oscillations of lift and moment coefficient respectively. As can be noted drag coefficient mean value was not stabilised, but this was irrelevant for purpose of detection of frequencies involved. Furthermore, frequency content is very well defined.
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Figure 7.29 Lift, drag and moment coefficients time history at Mach = 0.86,
𝛼 = 0°, ∆t = 0.001 s. -0,06 -0,04 -0,02 0 0,02 0,04 0,06 0 0,5 1 1,5 2 2,5 3 3,5 4 cl , l if t co ef fi ci e n t time [s]
Cl time history
0,0635 0,064 0,0645 0,065 0,0655 0,066 0 0,5 1 1,5 2 2,5 3 3,5 4 cd , d ra g co ef fi ci e n t time [s]Cd time history
-0,06 -0,04 -0,02 0 0,02 0,04 0,06 0 0,5 1 1,5 2 2,5 3 3,5 4 cm , m om en t coe ff ici en t time [s]Cm time history
195
Figure 7.30 Lift, drag and moment coefficients PSD charts at Mach = 0.86, α = 0° and ∆t = 0.001 s.
5,86 17,59 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 1,E+04 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
5,86 11,72 23,45 1,E-08 1,E-07 1,E-06 1,E-05 1,E-04 1,E-03 1,E-02 1,E-01 1,E+00 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
5,86 17,59 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 1,E+04 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 cm [ 1/Hz] frequency [Hz]PSD of Cm
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Figure 7.31 LCO charts of lift and moment coefficients at Mach = 0.86, α = 0º and ∆t = 0.001 s.
Figure 7.29 shows that oscillation was arisen although transient analysis started from a condition of zero incidence angle, demonstrating that presence of solid airfoil itself is an enough perturbation to cause buffet onset, and that in first transient time oscillations start and increase very rapidly reaching a stable interval of variation in a short time. Moreover, buffet is a non-linear phenomenon thus starting from a different condition would not have brought surely to same results.
Simulation was performed dividing upper and lower surfaces to make sure of flux symmetry. Actually, results a perfect symmetry as next Figures 7.32-7.33-7.34 show.
Figure 7.32 Lift coefficient of upper and lower airfoil surfaces time histories charts at Mach = 0.86 and 𝛼 = 0°.
-0,05 -0,04 -0,03 -0,02 -0,01 0 0,01 0,02 0,03 0,04 0,05 -2 -1 0 1 2 cl Δcl/Δt
Δcl/Δt
-0,05 -0,04 -0,03 -0,02 -0,01 0 0,01 0,02 0,03 0,04 0,05 -2 -1 0 1 2 cm Δcm/ΔtΔcm/Δt
0,48 0,5 0,52 0,54 0,56 0 0,5 1 1,5 2 2,5 3 3,5 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 lift coefficient
-0,56 -0,54 -0,52 -0,50 -0,48 0 0,5 1 1,5 2 2,5 3 3,5 4 cl _l ow ,l o w e r su rf ace lif t co ef fi ci e n t time [s]
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Figure 7.33 Lift coefficient PSDs of upper and lower airfoil surfaces charts at Mach = 0.86 and 𝛼 = 0°.
Figure 7.34 LCO charts of upper and lower lift coefficients at Mach = 0.86 and 𝛼 = 0°.
5,86 11,72 17,59 1,E-07 1,E-05 1,E-03 1,E-01 1,E+01 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
5,86 11,72 17,59 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 _l ow [ 1/Hz ] frequency [Hz]PSD of Cl lower surface
0,49 0,5 0,51 0,52 0,53 0,54 0,55 -1,5 -0,5 0,5 1,5 cl _up Δ(cl_up)/ΔtΔ(cl_up)/Δt
-0,55 -0,54 -0,53 -0,52 -0,51 -0,5 -0,49 -1,5 -0,5 0,5 1,5 cl _l ow Δ(cl_low)/ΔtΔ(cl_low)/Δt
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Figures 7.30 and 7.33 show that frequencies involved are very clear whit an energy content of some order of magnitude larger than other values in the spectrum, probably since shock wave oscillation is the most energetic phenomenon involved in this condition and capable of hide all other frequencies. Comparison of the two previous figures show that lift and moment coefficients of whole airfoil do not have the second frequency about 11.72 Hz despite of power spectral density of drag coefficient of whole airfoil. Probably the waves linked to that frequency are in exact phase opposition on upper and lower surfaces resulting in a null wave by sum of the two.
Root mean square error of static pressure of this simulation, in Figure 7.35, confirm a symmetrical behaviour.
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7.2 Results of simulation series above Mach 0.80
An overview of obtained results in simulations at Mach above 0.80 and fixed incidence angle of zero degrees shows an increase of root mean square values while approaching critical condition. From normalised residuals behaviour, root mean square values and limit cycle oscillations, it turned out that critical Mach number for zero incidence angle is 0.825. In this condition as wrote in paragraph 7.1.3 it was not possible to define a specific frequency because mean value of parameters analysed were not enough stable to permit of obtain reliable data, although it was possible to estimate an interval of value ranging from about 50 to 75 Hz. Two of three analyses of time histories of lift coefficient showed a frequency of about 74 Hz, thus probably this last can be taken as a good value but keeping in mind previous consideration. Although problems occurred it was been possible to locate this critical condition from behaviour of solution similar in every peculiarity to other onset conditions found in other series of simulations.
Apart condition at Mach equal to 0.86, also in this case found oscillations were very small e.g., at the onset, lift coefficient oscillations are of the order of 10−7, moreover this
symmetric condition on a symmetric airfoil, accompanied with extremely small perturbations highlight software criticality in predict a solution even if it was possible to recognise presence on an unsteady buffet phenomenon.
Tables 7.1 and 7.2 shows numerical results obtained from simulations:
Mach RMS (cl) frequency reduced frequency time step
0,816 1,49E-05 none none 0,001 s
0,816 2,11E-07 none none 0,0001 s
0,82 9,61E-07 none none 0,001 s
0,825 1,32E-04 74,26 3,90E-01 0,001 s
0,86 3,15E+01 5,86 2,95E-02 0,001 s
Table 7.1 Root mean square values, frequency and reduced frequency of conducted simulations beyond Mach 0.80 and 𝛼 = 0°.
Mach Δ(Δcl/Δt) Δcl 0,816 1,75E-07 2,39E-10 0,82 1,46E-08 1,68E-09 0,825 8,90E-04 5,54E-07 0,86 3,47E+00 8,75E-02
Table 7.2 Amplitude of lift coefficient Limit Cycle Oscillation and oscillation amplitude of lift coefficient per each simulation over Mach 0.80.
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Next Figures 7.36-7.37-7.38 show charts of that value while Figures 7.39-7.40 show amplitude of LCOs and amplitude and shape of each LCO found.
Figure 7.36 Frequency and reduced frequency involved at buffet onset beyond Mach = 0.80.
Figure 7.37 Root mean square of lift coefficient found in each simulation beyond Mach 0.80 as a function of Mach number.
M = 0,825 f~ = 3,90E-01 M = 0,86 f~ = 2,95E-02 M = 0,825 f = 74,26 Hz M = 0,86 f = 5,86 Hz 0,E+00 1,E+01 2,E+01 3,E+01 4,E+01 5,E+01 6,E+01 7,E+01 8,E+01 0,82 0,83 0,84 0,85 0,86 0,87 fr e q u e n cy [Hz], re du ced fr equ enc y Mach number
Frequency and reduced frequency
beyond Mach = 0,80
Reduced frequency first harmonic 0,00E+00 2,00E-05 4,00E-05 6,00E-05 8,00E-05 1,00E-04 1,20E-04 1,40E-04 0,815 0,817 0,819 0,821 0,823 0,825 R M S of cl [ 1/Hz ] MachRoot mean square of cl
beyond Mach = 0,80
201
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.
Figure 7.39 Comparison between amplitude of limit cycle oscillation of each simulation beyond Mach 0.80.
0,E+00 2,E-04 4,E-04 6,E-04 8,E-04 1,E-03 0,81 0,815 0,82 0,825 0,83 Δ (Δc l/ Δ t) α, incidence [deg]
Δ(Δcl/Δt)
0,E+00 1,E-07 2,E-07 3,E-07 4,E-07 5,E-07 6,E-07 0,81 0,815 0,82 0,825 0,83 Δ cl α, incidence [deg]Δcl
0,00E+00 2,50E-06 5,00E-06 7,50E-06 1,00E-05 1,25E-05 1,50E-05 1,75E-05 2,00E-05 2,25E-05 2,50E-05 2,75E-05 -0,0004 -0,0002 0 0,0002 0,0004 0,0006 0,0008 cl Δcl/ΔtΔcl/Δt
Mach 0.816 Mach 0.82 Mach 0.825202
Figure 7.40 Amplitude and shape of limit cycle oscillation of each simulation beyond Mach 0.80.
Apart Figure 7.40, other charts do not report values at 𝑀𝑎𝑐ℎ = 0.86 because their values were too high to be compared to the rest. Obviously in Figure 7.40 all charts, but that showing 𝑀𝑎𝑐ℎ = 0.86 case, have the same length scale. As expected it can be noted a decrease in buffet frequency while increasing Mach number and an increase in amplitude of oscillations.
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, performed in condition of 𝑀𝑎𝑐ℎ = 0.86, are reported in following figures.
0,0000017 0,0000019 0,0000021 0,0000023 0,0000025 -0,0008 -0,0003 0,0002 0,0007 cl Δcl/Δt
Δcl/Δt at Mach=0,816
2,40E-05 2,42E-05 2,44E-05 2,46E-05 2,48E-05 -0,0008 -0,0003 0,0002 0,0007 cl Δcl/ΔtΔcl/Δt at Mach=0,82
0,0000038 0,000004 0,0000042 0,0000044 0,0000046 -0,0008 -0,0003 0,0002 0,0007 cl Δcl/ΔtΔcl/Δt at Mach=0,825
-0,05 -0,03 -0,01 0,01 0,03 0,05 -2 -1 0 1 2 cl Δcl/ΔtΔcl/Δt at Mach=0,86
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Figure 7.41 Upper (on top) and lower surface static pressure variance at Mach = 0.86 and 𝛼 = 0°.
As can be noted both charts are equal in magnitude and wave shape, but they are in phase opposition as expected. Next Figure 7.42 shows this consideration. Furthermore, looking at direction of propagation seems that pressure perturbations move from trailing edge to the shock wave as Figure 7.43 clearly shows following colours (for purpose of brevity it was reported only the figure referred to the upper surface, the other one indeed is equal and as told in phase opposition with respect to first one).
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Figure 7.42 Upper and lower surface static pressure variance in plant view at Mach = 0.86 and 𝛼 = 0°.
