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Chapter 4
Steady and transient analyses at
Mach 0.775, buffet investigation
4.1 Simulation series at Mach equal to 0.775
The series of conducted analyses are specified in Table 2.6. Figure 4.1 shows contour of static pressure in each condition starting from right to left with increasing incidence angle:
Figure 4.1 Contours of static pressure at Mach = 0.775 (from left to right at 2.4, 2.5, 2.525, 2.550 degrees).
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Figure 4.2 Normalised residuals in steady analysis at Mach = 0.775 and α = 2.4°.
Figure 4.3 Normalised residuals in unsteady analysis at Mach = 0.775 and α = 2.4°.
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Figure 4.5 Normalised residuals in unsteady analysis at Mach = 0.775 and α = 2.5°.
Figure 4.6 Normalised residuals in steady analysis at Mach = 0.775 and α = 2.525°.
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Figure 4.8 Normalised residuals in steady analysis at Mach = 0.775 and α = 2.550°.
Figure 4.9 Normalised residuals in unsteady analysis at Mach = 0.775 and α = 2.550°.
As can be noted from previous figures normalized residual are higher for an incidence angle of 𝛼 = 2.525° giving a first indication of buffet onset point. Next analyses confirm this last observation furthermore showing a little less unsteadiness at 2.550 degrees.
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.
4.1.1 Simulation at 𝛂 = 2.4° and 𝛂 = 2.5°
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Figure 4.10 Lift, drag and moment coefficients charts at Mach = 0.775 and α = 2.4°.
In this and next condition, i.e. 2.4 and 2.5 degrees, PSDs were not shown because they were all null since it was achieved a stationary solution.
Figure 4.11 Lift, drag and moment coefficients charts at Mach = 0.775 and α = 2.5°.
0,42415 0,4242 0,42425 0 0,5 1 1,5 2 2,5 3 cl , l if t co ef fi ci e n t time [s]
Cl time history
0,031980 0,031990 0,032000 0 0,5 1 1,5 2 2,5 3 cd , dr ag co ef fi ci en t time [s]Cd time history
-0,013380 -0,013360 -0,013340 0 0,5 1 1,5 2 2,5 cm , momen t co ef fi ci en t time [s]Cm time history
0,453700 0,453750 0,453800 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 cl , l if t co ef fi ci e n t time [s]Cl time history
0,03486 0,03488 0,0349 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 cd , d ra g co ef fi ci en t time [s]Cd time history
-0,014280 -0,014260 -0,014240 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 cm, m ome n t co ef fi ci e n t time [s]Cm time history
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4.1.2 Simulation at 𝛂 = 2.525°
Next figures show results of analysis conducted at 2.525 degrees. In this case was not reached a steady solution, but lift, drag and moment coefficient started to oscillate indicating the onset of unsteadiness. PSDs inspection revealed frequencies involved with their super and sub harmonics.
Figure 4.12 Lift, drag and moment coefficients charts at Mach = 0.775 and α = 2.525° and a
magnification of lift coefficient time history to visualize limit cycle oscillations.
0,45620000 0,45622000 0,45624000 0,45626000 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
0,035220 0,035230 0,035240 0,035250 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
-0,014310 -0,014300 -0,014290 -0,014280 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
0,45624550 0,45624600 0,45624650 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 coe ff ici en t time [s]95
Figure 4.13 Lift, drag and moment coefficients PSD charts at Mach = 0.775 and α = 2.525°.
Last figure gives the indication that phenomenon is governed by a frequency around 98.2 Hz accompanied by all sub and super harmonics derived from a frequency of about 19.54 Hz.
A further investigation on the locus of origin of unsteadiness was conducted using the image of the RMSE (root mean square error) of static pressure shown in Figure 4.14; this last in association with the separated analysis of upper and lower surfaces lift coefficient of
19,54 39,57 59,11 78,65 98,19 1,00E-17 1,00E-15 1,00E-13 1,00E-11 1,00E-09 1,00E-07 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-18 1,00E-16 1,00E-14 1,00E-12 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-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 P SD o f cm [ 1/Hz] frequency [Hz]PSD of Cm
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the airfoil demonstrate that most of the phenomenon, in this condition, was concentrated on the upper surface. Figure 4.15 indeed shows that lift coefficient oscillation amplitude on lower surface is one order of magnitude lower than that on the upper one.
Figure 4.14 RMSE of static pressure at Mach = 0.775 and α = 2.525°.
An inspection of next Figure 4.15 show indeed that PSD of lift upper surface coefficient is three order of magnitude higher than the lower surface; furthermore Figure 4.16 demonstrates a big difference in LCO.
Unlike the Mach = 0.76 series in this case was not found a sudden drop of the unsteadiness as the analysis at α = 2.550° demonstrates; oscillation amplitude indeed tends to decrease, but in a mild way. Next analysis show these results.
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Figure 4.15 Lift coefficient PSD charts of upper and lower airfoil surfaces at Mach = 0.775 and α = 2.525°.
Figure 4.16 Lift coefficient LCO charts of upper and lower surfaces at Mach = 0.76 and α = 2.525°.
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 upper surface
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 cl _l ow [ 1/Hz ] frequency [Hz]PSD of Cl lower surface
0,6100905 0,6100915 0,6100925 0,6100935 0,6100945 -0,0015 -0,0005 0,0005 0,0015 cl _u p Δ(cl_up)/ΔtΔ(cl_up)/Δt
-0,153849 -0,153848 -0,153847 -0,153846 -0,153845 -0,0015 -0,0005 0,0005 0,0015 cl _l ow Δ(cl_low)/ΔtΔ(cl_low)/Δt
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4.1.3 Simulation at 𝛂 = 2.550°
Figure 4.17 Lift, drag and moment coefficients charts at Mach = 0.775 and α = 2.550° and a
magnification of lift coefficient time history to visualize limit cycle oscillations.
0,4562 0,45621 0,45622 0,45623 0,45624 0,45625 0,45626 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 coe ff ici en t time [s]
Cl time history
0,035225 0,03523 0,035235 0,03524 0,035245 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 co ie n t time [s]Cd time history
-0,014310 -0,014305 -0,014300 -0,014295 -0,014290 -0,014285 0 1 2 3 4 5 6 7 8 9 10 cm, momen t co ef fi ci e n t time [s]Cm time history
0,4562441 0,4562442 0,4562443 0,4562444 0,4562445 0,4562446 9,5 9,515 9,53 9,545 9,56 9,575 9,59 9,605 9,62 9,635 9,65 9,665 9,68 9,695 9,71 cl , l if t co ef fi ci en t time [s]99
Figure 4.18 Lift, drag and moment coefficients PSD charts at Mach = 0.775 and α = 2.550°.
These results are very similar to the previous analysis at 𝛼 = 2.525°, in case of separated surfaces too indicating an unsteadiness concentrated on the upper surface;
1,00E-16 1,00E-14 1,00E-12 1,00E-10 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-18 1,00E-16 1,00E-14 1,00E-12 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-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
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practically found values were the same in terms of frequencies and LCO, then charts were not reported. The only parameter of importance was the RMS value, slightly lower than in the previous case.
4.2 Results of simulation series at Mach = 0.775
An overview of the overall results shows a sudden increase in the RMS value of lift, drag and moment coefficient followed by an unstable region indicating a band of unstable incidence values and not a narrow zone like in the Mach = 0.76 simulations. Root mean square of power spectral density of lift, drag and moment coefficient, change in time rate of lift coefficient and maximum modulus of lift coefficient variance indicate a critical incidence of 2.525 degrees in accordance with results given by NASA in Ref. [5]. In Table 4.1 are reported root mean square of power spectral density values and in Table 4.2 are shown results of maximum oscillation and variance of time rate of lift coefficient found in each simulation. RMS values of PSD of α Cl Cd Cm 2,4 0 0 0 2,5 0 0 0 2,525 0,000136928 1,81274E-05 7,87308E-05 2,55 0,000136743 1,8109E-05 7,87234E-05
Table 4.1 Results of RMS of PSD values obtained
in each simulation at Mach = 0.775. α Δ(Δcl/Δt) Δcl 2,4 0 0 2,5 0 0 2,525 0,00065565 4,17E-07 2,55 0,00065565 4,17E-07
Table 4.2 Results of Limit Cycle Oscillation and maximum
modulus of oscillation of lift coefficient at Mach = 0.775.
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Figure 4.19 LCO amplitude and maximum oscillation of lift coefficient as a function of incidence at Mach = 0.775.
Figure 4.20 Comparison between amplitude of Limit Cycle Oscillations of lift coefficient at Mach = 0.775.
0,E+00 1,E-04 2,E-04 3,E-04 4,E-04 5,E-04 6,E-04 7,E-04 2,375 2,4 2,425 2,45 2,475 2,5 2,525 2,55 2,575 Δ (Δ cl /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 2,375 2,4 2,425 2,45 2,475 2,5 2,525 2,55 2,575 Δ cl α, incidence [deg]Δcl
green and blue superimposed 0,422000 0,427000 0,432000 0,437000 0,442000 0,447000 0,452000 0,457000 -0,0015 -0,001 -0,0005 0 0,0005 0,001 0,0015 cl Δcl/Δt
Δcl/Δt
2.550 deg 2.525 deg 2.500 deg 2.400 deg102
Figure 4.21 Amplitude and shape of Limit Cycle Oscillation charts of lift coefficient at Mach = 0.775.
0,424214 0,424215 0,424215 0,424216 0,424216 0,424217 0,424217 0,424218 0,424218 -0,0015 -0,0005 0,0005 0,0015 cl Δcl/Δt
Δcl/Δt at 2.4º
0,453776 0,453777 0,453777 0,453778 0,453778 0,453779 0,453779 0,453780 0,453780 -0,0015 -0,0005 0,0005 0,0015 cl Δcl/ΔtΔcl/Δt at 2,5º
0,456244 0,456245 0,456245 0,456246 0,456246 0,456247 0,456247 0,456248 0,456248 -0,0015 -0,0005 0,0005 0,0015 cl Δcl/ΔtΔcl/Δt at 2,525º
0,456243 0,456244 0,456244 0,456245 0,456245 0,456246 0,456246 0,456247 0,456247 -0,0015 -0,0005 0,0005 0,0015 cl Δcl/ΔtΔcl/Δt at 2,550º
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Figure 4.22 Root Mean Square of Power Spectral Density of lift, drag and moment coefficients charts at
Mach=0.775 as a function of incidence.
-1,0E-05 1,0E-05 3,0E-05 5,0E-05 7,0E-05 9,0E-05 1,1E-04 1,3E-04 1,5E-04 2,375 2,4 2,425 2,45 2,475 2,5 2,525 2,55 2,575 R M S(P SD( cl )) α, incidence [deg]
RMS of PSD of Cl at Mach=0.775
0,E+00 5,E-06 1,E-05 2,E-05 2,E-05 2,375 2,4 2,425 2,45 2,475 2,5 2,525 2,55 2,575 R M S(P SD( cd )) α, incidence [deg]RMS of PSD of Cd at Mach=0.775
0,E+00 2,E-05 4,E-05 6,E-05 8,E-05 1,E-04 2,375 2,4 2,425 2,45 2,475 2,5 2,525 2,55 2,575 R M S(P SD( cm )) α, incidence [deg]RMS of PSD of Cm at Mach=0.775
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As in previous series of simulations, last step was the study in MATLAB® of static pressure fluctuations on the airfoil surface by use of colours-map. Results of this last analysis, conducted in buffet onset condition, are reported in following figures.
Figure 4.23 Colours-map of upper (on top) and lower airfoil surfaces static pressure oscillations
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Figure 4.24 Colours-map of upper and lower airfoil surfaces static pressure fluctuations in plain view
at Mach = 0.775 and 𝛼 = 2.525°.
As can be noted from last two images, there is a repetitiveness of signals oscillations and although in a worse manner with respect to previous analysis it is possible to estimate the motion direction of the disturbance. On both surfaces is recognisable a “v” whose cusp can be located at the shock wave position, indicating the origin of disturbance on shock itself and propagation inside the shear layer both to leading edge and to trailing edge of the airfoil.
