Chapter 6

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Chapter 6 Steady and transient analysis at

Mach 0.816 and 0.1 degrees of

incidence, buffet investigation

6.1 Introduction

Due to difficulties encountered in defining frequency correlated to buffet onset at null incidence angle for a Mach beyond 0.80 as will be clarified in next chapter showing results of that analyses, it was decided to simulate a condition very near to that of null incidence. It was hypothesised that problems may be due to smallness of forces involved; theoretically indeed, a symmetric airfoil at zero degrees of incidence should be subjected to lift and moment forces equal to zero in a steady situation, thus, since at buffet onset oscillations are very small, resultant forces and disturbances are of about the same order of magnitude. This fact made analyses uncertain even if is clearly recognisable if a steady solution exists or an unsteady one turns out. Then was selected to find a point on buffet onset curve shown in Figure 8.1 near at null incidence condition but accompanied by an average value of lift force different from zero, i.e. that for 𝛼 = 0.1°. In Figure 6.1 is shown static pressure contours obtained for this condition.

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6.2 Simulation at Mach = 0.816 and

𝛂 = 0.1°

From inspection of Figure 8.1 was selected present combination of values of Mach number and incidence angle since point defined by these coordinates belongs to buffet onset curve. Steady analysis normalised residuals shown immediately high values leading to think system was unsteady, hypothesis then confirmed by normalised residuals related to transient simulation both reported in Figure 6.2.

Figure 6.2 Normalised residuals of steady (on the left side) and transient (on the right side)

analysis at Mach = 0.816 and 𝛼 = 0.1°.

As can be noted from previous image, residuals not only remain quite high but also oscillate in transient case.

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Figure 6.3 Lift, drag and moment coefficient at Mach = 0.816 and 𝛼 = 0.1°.

Figure 6.4 Lift coefficient PSD chart at Mach = 0.816 and 𝛼 = 0.1°.

0,023500 0,023525 0,023550 0,023575 0,023600 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

0,023710 0,023720 0,023730 0,023740 0,023750 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 en t time [s]

Cd time history

-0,003290 -0,003280 -0,003270 -0,003260 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-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

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Figure 6.5 Drag and moment coefficients PSD charts at Mach = 0.816 and 𝛼 = 0.1°.

Figure 6.6 LCO charts of lift and moment coefficients at Mach = 0.816 and 𝛼 = 0.1°.

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 [1/Hz ] frequency [1/Hz]

PSD of Cd

1,E-17 1,E-15 1,E-13 1,E-11 1,E-09 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 [1/Hz]

PSD of Cm

0,023542 0,023544 0,023546 0,023548 0,02355 0,023552 0,023554 0,023556 0,023558 -0,001 -0,0005 0 0,0005 0,001 cl Δcl/Δt

Δcl/Δt

-0,003281 -0,003281 -0,00328 -0,00328 -0,003279 -0,003279 -0,003278 -0,003278 -0,003277 -0,003277 -0,003276 -0,001 -0,0005 0 0,0005 0,001 cm Δcm/Δt

Δcm/Δt

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As can be noted from previous charts, unsteadiness is clearly present. Although after an initial transient software was capable to capture repetitive cycles related to buffet onset condition, average value of lift, drag and moment coefficient did not stable. When the last reached a quite stable value permitting a frequency analysis more reliable than that unclear shown in Figures 6.4-6.5, numerical errors due, to smallness of forces involved comparable with themselves oscillations, had corrupted solution. For this reason, was decided to analyse only lift coefficient in a short time period located from 5 and 5.511 seconds whose chart is reported in next Figure 6.7.

Figure 6.7 Lift coefficient time history chart sampling at Mach = 0.816 and 𝛼 = 0.1°.

Results obtained by inspecting this interval shown some peaks although not enough clear. Was then decided to try to clean the signal by subtracting its local mean value point by point. To do this operation was created a polynomial tendency line of sixth order approximating average local values of lift coefficients whose formula is:

𝑐̅ = −0.0059369395 ∗ 𝑡_𝑙 6 + 0.0095443008 ∗ 𝑡5− 0.0058354612 ∗ 𝑡4

+ 0.0016774429 ∗ 𝑡3 − 0.0002275819 ∗ 𝑡2+ 0.0000230546 ∗ 𝑡 + 0.0235434056

were 𝑡 is time in seconds, 𝑐̅ is the local average value of lift coefficient. _𝑙

A picture of this function is reported in Figure 6.8 with lift coefficient curve for comparison. Following operation of subtracting this polynomial curve to lift coefficient curve led to a signal oscillating around zero as shown in Figure 6.9.

0,023543 0,023544 0,023545 0,023546 0,023547 0,023548 0,023549 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5 cl , l if t co ef fi ci e n t time [s]

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Figure 6.8 Lift coefficient time history and polynomial charts at Mach = 0.816, 𝛼 = 0.1°,t = [5 ; 5.511] s.

Figure 6.9 Lift coefficient after deducted its local mean value time history chart

at Mach = 0.816, 𝛼 = 0.1°, t = [5 ; 5.511] s.

Were conducted two separated analyses on this time interval: first was used whole interval, secondly was taken second half of the last. Results of that analyses are reported in next Figures 6.10-6.11-6.12-6.13.

From these figures it can be noted a substantial clearness of solution with respect to Figures 6.4 and 6.5 even if due to brevity of time interval sampled, values are only approximations of actual ones.

0,023543 0,023544 0,023545 0,023546 0,023547 0,023548 0,023549 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5 cl , l if t co ef fi ci e n t time [s]

Lift coefficient time history (sample 5-5,511 s)

-3,E-07 -2,E-07 -1,E-07 0,E+00 1,E-07 2,E-07 3,E-07 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5 cl , l if t co ef fi ci e n t time [s]

Lift coefficient time history after deducted its

local mean value(sample 5-5,511 s)

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Figure 6.10 PSD chart of lift coefficient after deducted its local mean value,

at Mach = 0.816, 𝛼 = 0.1°, t = [5 ; 5.511] s.

Figure 6.11 LCO chart of lift coefficient after deducted its local mean value,

at Mach = 0.816, 𝛼 = 0.1°, t = [5 ; 5.511] s. 5,87 11,74 84,15 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 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 (sample: 5-5,511 s)

0,0235430 0,0235440 0,0235450 0,0235460 0,0235470 0,0235480 0,0235490 -0,0005 -0,00025 0 0,00025 0,0005 cl Δcl/Δt

Δcl/Δt (sample: 5-5,511 s)

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Figure 6.12 PSD chart of lift coefficient after deducted its local mean value,

at Mach = 0.816, 𝛼 = 0.1°, t = [5.256 ; 5.511] s.

Figure 6.13 LCO chart of lift coefficient after deducted its local mean value,

at Mach = 0.816, 𝛼 = 0.1°, t = [5.256 ; 5.511] s. 7,84 86,27 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 (sampled from 5,256 to 5,511)

-0,0005000 -0,0004000 -0,0003000 -0,0002000 -0,0001000 0,0000000 0,0001000 0,0002000 0,0003000 0,0004000 0,0005000 -3E-07-2E-07-1E-07 0 0,00000010,00000020,0000003 cl Δcl/Δt

Δcl/Δt (5,256-5,511 s)

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From Figures 6.10-6.12 was noted presence of low frequencies similar to that obtained at 𝑀𝑎𝑐ℎ = 0.86 and 𝛼 = 0°. Was selected a frequency about 84.15 Hz due to larger amplitude in frequency shown by that value; this fact is associated to larger energy content. Separated analysis of upper and lower surfaces even if unclear on frequency point of view showed once again that most of unsteadiness content was located on lower surface of the airfoil. LCO charts related to last computation are reported in Figure 6.14.

Figure 6.14 LCO charts of lift coefficient of upper and lower surfaces at Mach = 0.816 and 𝛼 = 0.1°.

RMSE of static pressure of conducted analysis are shown in Figures 6.15-6.16.

Figure 6.15 Static pressure RMSE chart of whole flow field at Mach = 0.816 and 𝛼 = 0.1°.

0,4021472 0,4021473 0,4021474 0,4021475 0,4021476 0,4021477 0,4021478 0,4021479 0,4021480 -0,0006 -0,0001 0,0004 cl _u p Δ(cl_up)/Δt

Δ(cl_up)/Δt

-0,3785922 -0,3785921 -0,3785920 -0,3785919 -0,3785918 -0,3785917 -0,3785916 -0,3785915 -0,3785914 -0,0006 -0,0001 0,0004 cl _l ow Δ(cl_low)/Δt

Δ(cl_low)/Δt

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Figure 6.16 Static pressure RMSE chart around the airfoil at Mach = 0.816 and 𝛼 = 0.1°.

Finally, it was not possible to extract colours-map of static pressure due to large numerical errors that would conduct to not reliable results.