Chapter 2

38

Chapter 2

CFD settings, further validation and

method of analysis of results

2.1 Introduction

Transonic buffet is an unsteady physical phenomenon characterized by large pressure oscillations more or less near (depending on velocity and incidence angle) and directly on the surface of a body immersed in the flux, in completely developed conditions at least; under the last assumption, the unsteadiness seems to be associated to whole flow field, showing in addition to extremely large pressure fluctuations, a movement of the shock wave back and forth. In this situation, the whole flow field is perturbed. These large pressure fluctuations, affecting the shear layer thickness too, result in large lift and drag forces variations limiting de facto the flight envelope of airplanes at high Mach numbers, near transonic region, therefore obliging most of commercial aircrafts to stay below a certain speed.

The phenomenon affecting wings of aircrafts is due to the coupling of the dynamic response of the elastic fluid to a perturbation (the presence of a solid body) that results in signals, for example of lift coefficient versus time, oscillating with natural frequencies of the fluid itself, and the dynamic response of the elastic solid body to the perturbing forces resulting from the fluid response; natural frequencies of a wing, being low, are near to buffeting fluid frequencies and this is the reason why the flux, while fluid entering energy in the body at that frequencies, results in high probability of exciting some natural modes of the wings; furthermore being energy extracted from fluid theoretically infinite, the buffet phenomenon could be very dangerous for aircraft wings.

In this work will be analysed a particular condition: the buffet onset. Fluid will be considered as an elastic body that moving in the flow direction encounters a perturbation, the solid body i.e. the aerofoil. To capture the essence of the phenomenon was chosen the use of a rigid body, analysing then the main physical cause of buffet. The problem was faced by use of RANS and URANS equations for steady and unsteady analyses respectively. Using of Reynolds averaged Navier-Stokes equations is legitimated by the fact that buffet frequencies are quite low, so time intervals to be considered are sufficiently long to capture the physical phenomenon even if equations are averaged on each time step.

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2.2 Fluent analyses

All analyses were conducted in Fluent® environment and data exported in Excel® and MATLAB® _{files to be visualized and studied.}

2.2.1 Further validation

The model validated in the previous chapter was rescaled to the right dimension, that related to Ref. [5]. First approach was devoted to reply the same conditions of Ref. [5] experiments, in which Mc Devitt and Okuno found a buffet boundary by wind tunnel tests measuring total pressure fluctuations, but in this case by using a numerical simulation approach, then was analysed the neighbourhood of the buffet onset points to better understand the unsteadiness behaviour. As a further proof of validity of the model were constructed the 𝑐_𝑝 contours, at Mach equal to 0.80 and for different incidence angles, and compared with that of Ref [4] and Ref [5]; used settings for comparison with Ref [4] are reported in Tables 2.1 and 2.2 whereas that for comparison with Ref [5] in Tables 2.3 and 2.4 (used formulas are the same wrote in chapter 1 page n.10). In the first case were considered two Reynolds numbers 4000000 and 10000000, in the second was used 𝑅𝑒_𝑐 = 10000000 only, but two different values of total pressure, one equal to twice the other. These data were extracted from steady analyses following the same methods used for validation and described in first chapter.

DATA: VELOCITY DIRECTION:

Rec 4000000 α dir X dir Y M 0,8 -2 0,999390827 -0,034899497 Tt [K] 273 -1 0,999847695 -0,017452406 RH [m] Laminato a freddo 0,0000016 0 1 0 α [deg] 2-3 1 0,999847695 0,017452406 R [j/kg*K] 287 2 0,999390827 0,034899497 γ 1,4 3 0,998629535 0,052335956 c [m] 0,21 4 0,99756405 0,069756474 5 0,996194698 0,087155743 DERIVED: P [Pa] 82612,20447 T [K] 242,0212766 Pt [Pa] 125929,0885

40

a [m/s] 311,8399412

U [m/s] 249,471953

μ [Pa*s] Sutherland law 1,55772E-05

Lift (-dirY,dirX,0) Drag (dirX,dirY,0) quarter chord 0,0525 cp [J/kg*K] 1003,5 KAl [W/m*K] 204 Kair [W/m*K] 0,0226

Table 2.1 Used settings for cp contours (Ref [4]) with a Re number of 4*106, Mach = 0.80.

DATA: VELOCITY DIRECTION:

Rec 10000000 α dir X dirY M 0,8 -2 0,999391 -0,0349 Tt [K] 273 -1 0,999848 -0,01745 RH [m] Laminato a freddo 0,0000016 0 1 0 α [deg] 2-3 1 0,999848 0,017452 R [j/kg*K] 287 2 0,999391 0,034899 γ 1,4 3 0,99863 0,052336 c [m] 0,21 DERIVED: P [Pa] 206530,5112 T [K] 242,0212766 Pt [Pa] 314822,7214 ρ [kg/m^3] 2,973368771 a [m/s] 311,8399412 U [m/s] 249,471953

μ [Pa*s] Sutherland law 1,55772E-05

Lift (-dirY,dirX,0) Drag (dirX,dirY,0) quarter chord 0,0525 cp [J/kg*K] 1003,5 KAl [W/m*K] 204 Kair [W/m*K] 0,0226

41

DATA: VELOCITY DIRECTION:

Rec 10000000 α dir X dir Y M 0,8 -0,1 0,9999985 -0,00175 Pt [Pa] 140000 0,05 0,9999996 0,000873 RH [m] 0,0000040640 α [deg] -0,1 ; 0,05 R [j/kg*K] 287 γ 1,4 c [m] 0,2032 DERIVED: P [Pa] 91843,02657 T [K] 122,6690755 Tt [K] 138,3707171 ρ [kg/m^3] 2,608730394 a [m/s] 222,0099874 U [m/s] 177,60799

μ [Pa*s] (from Rec) 9,41489E-06

dir X 0,999998477 dir Y -0,001745328 Lift (-dirY,dirX,0) Drag (dirX,dirY,0) quarter chord 0,0508 KAl [W/(m*K)] 204 Kair [W/(m*K)] 0,01128123 cp [J/(kg*K)] 1019,028529

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DATA: VELOCITY DIRECTION:

Rec 10000000 α dir X dir Y M 0,80 -0,1 0,999998 -0,00175 Pt [Pa] 280000 0,05 0,9999996 0,000873 RH [m] 0,0000040640 α [deg] -0,1 ; 0,05 R [j/kg*K] 287 γ 1,4 c [m] 0,2032 DERIVED: P [Pa] 183686,0531 T [K] 213,5792655 Tt [K] 240,9174115 ρ [kg/m^3] 2,996644312 a [m/s] 292,9439347 U [m/s] 234,3551478

μ [Pa*s] (from Rec) 1,42703E-05 dir X 0,999999619 dir Y 0,000872665 Lift (-dirY,dirX,0) Drag (dirX,dirY,0) quarter chord 0,0508 KAl [W/(m*K)] 204 Kair [W/(m*K)] 0,019588911 cp [J/(kg*K)] 1006,388861

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Results of comparison with Ref [4] are shown in Figures 2.1, 2.2 while that of comparison with Ref [5] are in Figures 2.3, 2.4. As can be seen there is a good accordance between test data and numerical simulation data.

Figure 2.1 Comparison of cp distribution between Ref [4] and numerical simulation data near 2° of incidence, Mach = 0.80.

-1,2 -1,05 -0,9 -0,75 -0,6 -0,45 -0,3 -0,15 0 0,15 0,3 0,45 0,6 0,75 0,9 1,05 1,2 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5 0,55 0,6 0,65 0,7 0,75 0,8 0,85 0,9 0,95 1 p re ssu re co ef fi ci e n t x\c

Comparison of c

distributions near α=2°

Ref [4] Harris upper surface (1.86 deg) Ref [4] Harris lower surface (1.86 deg) cp upper surface 2 deg

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Figure 2.2 Comparison of cp distribution between Ref [4] and numerical simulation data near 3° of incidence, Mach = 0.80.

Figure 2.3 Comparison of cp distribution between Ref [5] and numerical simulation data at -0.1° of incidence, Mach = 0.80.

-1,35 -1,2 -1,05 -0,9 -0,75 -0,6 -0,45 -0,3 -0,15 0 0,15 0,3 0,45 0,6 0,75 0,9 1,05 1,2 1,35 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5 0,55 0,6 0,65 0,7 0,75 0,8 0,85 0,9 0,95 1 p re ssu re co ef fi ci e n t x\c

Comparison of c

distributions near α=3°

CFD upper surface (3 deg) CFD lower surface (3 deg)

Ref [4] Harris upper surface (2.86 deg) Ref [4] Harris lower surface (2.86 deg)

-1 -0,85 -0,7 -0,55 -0,4 -0,25 -0,1 0,05 0,2 0,35 0,5 0,65 0,8 0,95 1,1 1,25 1,4 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5 0,55 0,6 0,65 0,7 0,75 0,8 0,85 0,9 0,95 1 p re ssu re co ef fi ci e n t x/c

C

p

at Mach=0.8 and α=-0,1° (Ref.[5] settings)

CFD upper surface (Pt) CFD lower surface (Pt) CFD upper surface (2 Pt) CFD lower surface (2 Pt) Ref [5] upper surface Ref [5] lower surface

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Figure 2.4 Comparison of cp contours between Ref [5] and numerical simulation data at 0.05° of incidence, Mach = 0.80.

This good accordance between test and numerical simulation data, even though the first refer to wind tunnel and the last to free air conditions, suggests that CFD data could be considered trustworthy.

2.2.2 CFD settings: steady and unsteady analyses

Completed this further validation, was decided to use, in next calculations, a fixed Reynolds number, 𝑅𝑒_𝑐 = 107, because, in Ref [5], the authors wrote that their results are trustworthy only for this value of the previous parameter, and a total pressure of 280000 Pa because it was found no appreciable differences between results of the two pressure values as can be easily seen from Figure 2.3; furthermore a higher pressure could help flux to remain more attached to the air foil surface encouraging the finding of more suitable numerical solutions because software can have criticality in prediction of high level of flux separation. It was decided as well to do three series of simulations at fixed Mach number, 𝑀 = 0.76 ; 0.775 ; 0.80; 0.816, varying of a very little quantity the incidence angle

-1,05 -0,9 -0,75 -0,6 -0,45 -0,3 -0,15 0 0,15 0,3 0,45 0,6 0,75 0,9 1,05 1,2 1,35 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5 0,55 0,6 0,65 0,7 0,75 0,8 0,85 0,9 0,95 1 p re ssu re co ef fi ci e n t x/c

Cp at Mach=0.8 and α=0,05° (Ref.[5] settings)

Ref [5] TEST upper surface Ref [5] TEST lower surface CFD upper surface

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simulation by simulation, inspecting so the neighbourhoods around the buffet onset points reported by Ref [5], Ref [7] and Ref [8]. For the peculiar condition of zero incidence angle, was decided to keep fix the incidence angle and varying of a little quantity the Mach number simulation by simulation; in this case were used 𝑀 = 0.816 ; 0.82 ; 0.825 ; 0.86.

Two simulations accompanied every condition analysed: first of all was performed a steady state calculation to obtain a value nearer as possible to the final solution then was used that value as initial input to the unsteady calculation.

Used settings for the two kind of analysis are the following (for more details about following terms see Ref. [9]):

• For steady state calculations were used a ‘density based’ solver type and a ‘k-e’ viscous model; in particular was used the ‘k-e realizable’ method with ‘enhanced wall treatment’ and the following options activated: pressure gradient effects, thermal effects, viscous heating, curvature correction, compressibility effects and production limiter. Material was defined with values done in next tables valid for steady and unsteady too calculations. In cell zone conditions was defined an operating pressure equal to zero [Pa] corresponding with an initially empty field, so that gauge pressure would be equivalent to the freestream static pressure. Boundary conditions of ‘pressure farfield’ and ‘pressure outlet’ were set up with the static freestream pressure, Mach number, velocity direction vector, static freestream temperature and static freestream pressure and temperature respectively, eventually turbulence was specified by ‘turbulent intensity’ and ‘turbulent viscosity ratio’ (default values) for both; in walls conditions, i.e. upper and lower contours of the aerofoil, were defined an heat exchange behind fluid and solid by heat convection, setting thermal conductivity of aluminum and an initial wall temperature, in equilibrium with freestream static temperature. A roughness height, 𝑅𝐻, was given accordingly to Ref. [6] considering an aerofoil made of cold laminated aluminum sheet. Reference values were adjusted to match area and chord length of the tests of Ref. [5]. All values used were extracted from Ref. [5], stored in an excel file and then were calculated the interesting parameters. In solution methods were selected an ‘implicit formulation’ for ‘solution method’, a ‘Roe-FDS’ for ‘flux type’, a ‘Green-Gauss node based’ for ‘gradient’, and ‘second order upwind’ for ‘flow’, ‘turbulent kinetic energy’ and ‘turbulent dissipation rate’ and all other options were left as default. Solution controls were left as default values cause were not convergence issues; residuals convergence criteria were set equal to 10−7. Monitors were defined to control convergence history of lift, drag and moment coefficient with appropriate direction vectors and centre of moments placed at 25% of chord. The solution was initialized with ‘standard initialization’ from inlet; then in ‘run calculation’ was selected the option ‘solution steering’ with ‘FMG initialition’, so was selected to leave control of CFL number to Fluent (with minimum value 0.75 and maximum value 100) and calculation was made in three steps with increasingly finer grids to accelerate convergence; the first and the second one of 200 iterations, the third

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one of 100 iterations and the remaining iterations with the original grid for a total number of five thousand iterations. This number was decided based on experience to ensure a stable solution and a shorter transient time of settlement in unsteady analysis. All values used in settings are reported in next Tables 2.5, 2.6, 2.7, 2.8, 2.9, 2.10.

• For unsteady calculations was imported the ‘case’ file and the final solution of the corresponding steady analysis; the only changes made to the case were obviously switching to ‘transient density based’ solver and in solution methods was selected a second order implicit transient formulation. Were then defined the report definitions giving the appropriate direction vector and moment centre and axis to have .dat files for lift, drag and moment coefficient with a larger number of significant figures with respect to the history ones; furthermore these last was set to be reported every time step. Was chosen a ‘fixed time stepping method’ so that calculations were made in all grid nodes with the same time step making significant all the statistic analyses. Time step size was set initially to a common value of 10-3 [s], then was adjusted based on consideration on the shape of lift coefficient oscillation; when the contour of this last parameter was not continuous with its derivatives indeed, was decided to lower the time step size obviously in all the series of analyses at a fixed Mach number to have coherence of results (see Figure 2.5). Was given a number of ten thousand time steps with twenty iterations per time step. Were activated data sampling for time statistics and added data file quantities of interest to be visualized in CFD Post®. All values used in settings are reported in next Tables 2.5, 2.6, 2.7, 2.8, 2.9, 2.10.

Figure 2.5 The two graphs represent lift coefficient as functions of time analysed with time step of 0.001 [s] on the left hand while 0.0005 [s] on the right respectively.

0,51543900 0,51543950 0,51544000 5,29 5,3 5,31 5,32 5,33 5,34 5,35

Cl history

Cl history

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DATA: VELOCITY DIRECTION: Rec 10000000 α [deg] dir X dir Y

M 0,76 2,975 0,998652 0,0519 Pt [Pa] 280000 3 0,99863 0,052336 RH [m] 0,000004064 3,01 0,99862 0,05251 R [j/kg*K] 287 3,025 0,998607 0,052772 γ 1,4 3,03 0,998602 0.052859 c [m] 0,2032 3,05 0,998583 0,053207 3,2 0,998441 0,055822 DERIVED: P [Pa] 190979,7267 T [K] 216,2341384 Tt [K] 241,2135061 ρ [kg/m^3] 3,077379884 a [m/s] 294,7590148 U [m/s] 224,0168513 μ [Pa*s] 1,40083E-05 Lift (-dirY,dirX,0) Drag (dirX,dirY,0) quarter chord 0,0508 KAl [W/(m*K)] 204 Kair [W/(m*K)] 0,019588911 cp [J/(kg*K)] 1006,388861 time step [s] 0,0005

Table 2.5 Settings for steady and transient analyses at fixed Mach number equal to 0.76.

DATA: VELOCITY DIRECTION: Rec 10000000 α [deg] dir X dir Y

M 0,775 2,4 0,999123 0,041876

Pt [Pa] 280000 2,5 0,999048 0,043619

RH [m] 0,000004064 2,525 0,999029 0,044055

R [j/kg*K] 287 2,55 0,99901 0,044491

γ 1,4

49 DERIVED: P [Pa] 188245,8086 T [K] 216,2341384 Tt [K] 242,2092643 ρ [kg/m^3] 3,033326493 a [m/s] 294,7590148 U [m/s] 228,4382365 μ [Pa*s] 1,40803E-05 Lift (-dirY,dirX,0) Drag (dirX,dirY,0) quarter chord 0,0508 KAl [W/(m*K)] 204 Kair [W/(m*K)] 0,019588911 cp [J/(kg*K)] 1006,388861 time step [s] 0,001

Table 2.6 Settings for steady and unsteady analyses at fixed Mach number equal to 0.775.

DATA: VELOCITY DIRECTION: Rec 10000000 α [deg] dir X dir Y

M 0,803 0,5 0,999962 0,008726535 Pt [Pa] 280000 0,8 0,999903 0,01396218 RH [m] 0,0000040640 0,85 0,99989 0,014834754 R [j/kg*K] 287 0,925 0,99987 0,016143594 γ 1,4 0,975 0,999855 0,017016139 c [m] 0,2032 0,99 0,999851 0,0172779 1 0,999848 0,017452406 DERIVED: 1,025 0,99984 0,017888671 1,1 0,999816 0,019197442 P [Pa] 183138,9273 1,2 0,999781 0,02094242 T [K] 131,6280671 1,3 0,999743 0,022687334 Tt [K] 148,6030595 1,4 0,999701 0,024432178 ρ [kg/m^3] 4,847862189 1,5 0,999657 0,026176948 a [m/s] 229,9742537 4 0,997564 0,069756474 U [m/s] 184,6693257

50 Lift (-dirY,dirX,0) Drag (dirX,dirY,0) quarter chord 0,0508 KAl [W/(m*K)] 204 Kair [W/(m*K)] 0,019588911 cp [J/(kg*K)] 1006,388861 time step [s] 0,001

Table 2.7 Settings for steady and unsteady analyses at fixed Mach number equal to 0.80.

DATA: VELOCITY DIRECTION: Rec 10000000 α [deg] dir X dir Y

M 0,816 0,1 0,999998477 0,001745328 Pt [Pa] 280000 RH [m] 0,000004064 R [j/kg*K] 287 γ 1,4 c [m] 0,2032 DERIVED: P [Pa] 180768,8864 T [K] 216,2341384 Tt [K] 245,0302981 ρ [kg/m^3] 2,912846008 a [m/s] 294,7590148 U [m/s] 240,5233561 μ [Pa*s] 1,42363E-05

Lift (-dirY, dirX, 0)

Drag (dirX, dirY, 0)

quarter chord 0,0508

KAl [W/(m*K)] 204

Kair [W/(m*K)] 0,019588911

cp [J/(kg*K)] 1006,388861

time step [s] 0,001

51 DATA: DATA: Rec 10000000 Rec 10000000 M 0,816 M 0,82 Pt [Pa] 280000 Pt [Pa] 280000 RH [m] 0,000004064 RH [m] 0,000004064 α [deg] 0 α [deg] 0 R [j/kg*K] 287 R [j/kg*K] 287 γ 1,4 γ 1,4 c [m] 0,2032 c [m] 0,2032 DERIVED: DERIVED: P [Pa] 180768,8864 P [Pa] 180040,0303 T [K] 216,2341384 T [K] 216,2341384 Tt [K] 245,0302981 Tt [K] 245,3133054 ρ [kg/m^3] 2,912846008 ρ [kg/m^3] 2,901101478 a [m/s] 294,7590148 a [m/s] 294,7590148 U [m/s] 240,5233561 U [m/s] 241,7023922

μ [Pa*s] 1,42363E-05 μ [Pa*s] 1,42484E-05

dir X 1 dir X 1

dir Y 0 dir Y 0

Lift (-dirY,dirX,0) Lift (-dirY,dirX,0)

Drag (dirX,dirY,0) Drag (dirX,dirY,0)

quarter chord 0,0508 quarter chord 0,0508

KAl [W/(m*K)] 204 KAl [W/(m*K)] 204

Kair [W/(m*K)] 0,019588911 Kair [W/(m*K)] 0,019588911

cp [J/(kg*K)] 1006,388861 cp [J/(kg*K)] 1006,388861

time step [s] 0,001 ; 0,0001 time step [s] 0,001

Table 2.9 Settings for steady and unsteady analyses at fixed incidence angle of zero degrees and variable Mach number.

52 DATA: DATA: Rec 10000000 Rec 10000000 M 0,825 M 0,86 Pt [Pa] 280000 Pt [Pa] 280000 RH [m] 0,000004064 RH [m] 0,000004064 α [deg] 0 α [deg] 0 R [j/kg*K] 287 R [j/kg*K] 287 γ 1,4 γ 1,4 c [m] 0,2032 c [m] 0,2032 DERIVED: DERIVED: P [Pa] 179129,298 P [Pa] 172769,6098 T [K] 216,2341384 T [K] 216,2341384 Tt [K] 245,6690105 Tt [K] 248,2194922 ρ [kg/m^3] 2,886426259 ρ [kg/m^3] 2,783948489 a [m/s] 294,7590148 a [m/s] 294,7590148 U [m/s] 243,1761872 U [m/s] 253,4927527

μ [Pa*s] 1,42628E-05 μ [Pa*s] 1,434E-05

dir X 1 dir X 1

dir Y 0 dir Y 0

Lift (-dirY,dirX,0) Lift (-dirY,dirX,0)

Drag (dirX,dirY,0) Drag (dirX,dirY,0)

quarter chord 0,0508 quarter chord 0,0508

KAl [W/(m*K)] 204 KAl [W/(m*K)] 204

Kair [W/(m*K)] 0,019588911 Kair [W/(m*K)] 0,019588911

cp [J/(kg*K)] 1006,388861 cp [J/(kg*K)] 1006,388861

time step [s] 0,001 time step [s] 0,001

Table 2.10 Settings for steady and unsteady analyses at fixed incidence angle of zero degrees and variable Mach number.

Residuals obtained from steady analyses show a good convergence level as can be seen in Figures 2.6 and 2.7. In these figures are shown residuals of steady analyses for two different Mach number and for two different incidence angles too. First graphs in each figure represent a steady situation while the second ones an unsteady behaviour of the flux; can be noticed that residuals of a steady analysis begin to oscillate and remain higher in a situation in which buffet is present even if a steady solution was reached in the steady analysis at least.

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Figure 2.6 Residual plots of steady analyses at Mach = 0.76 with an incidence of 2.975 and 3.025 degrees respectively

Figure 2.7 Residual plots of steady analyses at Mach = 0.80 with an incidence of 0.5 and 1.5 degrees respectively

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2.3 Method of analysis of results

The procedure implemented to analyse data is common for all simulations. First of all it was done a steady calculation to reach a good first solution for the transient one, then were acquired lift, drag and moment coefficient files as Fluent® report file. It was decided to use the last file type because, in this kind of file only, all parameters are reported with sixteen significant figures (comprehensive of the zero before point, the point itself and eventually the sign) giving then significantly more accurate results than that given by Fluent® _{history files.}

These data were then imported in an Excel® file to draw charts of them. As told in the previous paragraph the report interval chosen was the time step so it was created a matrix whose columns represent the number of time steps, the elapsed time, the parameter values and other columns defined later in this section. Were created charts to visualize the behaviour of each parameter versus time; it was noted, as expected, that after a transition time (for this model typically about 3.5 s) solution reached ‘a value’: a steady value in case of steady solution otherwise began to oscillate if steady solution did not exist, suggesting then an unsteady flux behaviour. Time sampled in most analyses was 10 seconds except for some that reached a significant solution in less time; from about seven seconds until the end of the simulation the fluctuating solutions reached a condition in which they are bounded by a maximum and a minimum because the average value has settled (like a LCO).

In some cases was noted a repetitive cycle of the order of 10-2 to 10-1 seconds, decreasing frequencies while growing Mach number, these cycles being perfectly repetitive at peculiar incidence angles: the angles then identified as belonging to the buffet boundary.

Having activated ‘data sampling’ in Fluent® _{during transient analyses, it was possible} to visualize contours of unsteady statistics, in particular the RMSE (root mean square error) of static pressure, velocity magnitude and velocity components. This parameter represents a discrete value of how static pressure, velocity magnitude and so on are spread out respect to their local time average value. It was then possible to visualize were the flow field was most perturbed; it was noted that, in conditions of buffet onset, perturbations were not spread throughout the domain, but concentrated in very little portions of fluid. This fact led to the hypothesis that buffet phenomenon does not starts up as a global unsteadiness of flux as reported in Ref. [7] and Ref.[8], but only as a local phenomenon. Furthermore was not noted a movement of the shock wave at buffet onset at least; only when the unsteadiness was completely developed, e.g. at 𝑀 = 0.80 and 𝛼 = 4° degrees was observed the behaviour described in literature, i.e. shock movement correlated to shear layer thickening and thinning, moreover unsteadiness was in this case affecting the whole considered domain.

As in previous cited references, shock wave tends to move to the trailing edge with growing Mach number; peculiar behaviour will be discussed in next sections accompanied by figures referring to each condition.

Returning to the Excel® file, taking into account the chart representing a parameter versus time, was selected a time interval of 2n values (in most cases 2048) in which the

55

mean value of the variable itself was stabilised and was made a FFT (fast Fourier transform) analysis to identify frequencies involved. Using results of last calculation was defined the PSD (power spectral density) of the parameter considered and in a chain calculation was then found the RMS (root mean square) of the power spectral density. Used relations are:

𝑃𝑆𝐷_𝑖 =2 ∗ |𝐹𝐹𝑇𝑖| 2 𝑇 𝑅𝑀𝑆 = √[2 ∗ (∑ 𝑃𝑆𝐷_𝑖 𝑁𝑡𝑖𝑚𝑒𝑠𝑡𝑒𝑝 2 −1 𝑖=3 ) + 𝑃𝑆𝐷₂ + 𝑃𝑆𝐷𝑁𝑡𝑖𝑚𝑒𝑠𝑡𝑒𝑝 2 ] ∗𝛥𝑓 2

were 𝑁_{𝑡𝑖𝑚𝑒𝑠𝑡𝑒𝑝} is the number of time step considered (in most cases 2048 as told before), 𝑇 is the time period sampled, 𝛥𝑓 the frequency sampling defined as:

𝑇 = 𝑡𝑡𝑖𝑚𝑒𝑠𝑡𝑒𝑝∗ (𝑁𝑡𝑖𝑚𝑒𝑠𝑡𝑒𝑝− 1) [s]

𝛥𝑓 =1 𝑇 [Hz]

were 𝑡_{𝑡𝑖𝑚𝑒𝑠𝑡𝑒𝑝} is the time step length. In PSD formula, 𝑖 ranges from 2 to 𝑁_{𝑡𝑖𝑚𝑒𝑠𝑡𝑒𝑝}/2 and values obtained were used to get an idea of how much energy each frequency carries with it, while the RMS value was used to get an idea of the total amount of energy involved.

Furthermore were calculated ∆𝑐𝑙/∆𝑡, ∆𝑐𝑚/∆𝑡, max (

∆𝑐𝑙

∆𝑡), min (

∆𝑐𝑙

∆𝑡), max (𝑐𝑙), min (𝑐𝑙) and then ∆(∆𝑐_𝑙) and ∆𝑐_𝑙. These last parameters were used to plot charts of LCO, that visualise amplitude of oscillations, rate of change of the considered physical quantities, repetitiveness of signals nonetheless, giving indeed a sort of pile the rounder the more the signal is repetitive and are reported in next section.

The last step was the analysis in MATLAB®: were exported from Fluent® static pressure data for a sufficient number of time steps of upper and lower surfaces of the aerofoil in separate files; data acquired were then imported in excel to form three matrices: first the position matrix, [x/c], representing the dimensionless position along the x axis starting from leading edge to trailing edge as shown in Figure 2.8 in next page; then the time matrix, [time], representing elapsed time; finally the delta static pressure matrix, [deltap], representing local fluctuations of static pressure with respect to a local mean value of the static pressure itself. Practically was defined a mean value of static pressure, one for each of the nodes dividing upper and lower air foil section, averaging values of static pressure taken at different times in the same position along the solid surface. The local delta static pressure was then calculated by subtracting from the value of static pressure at current time step the mean value previously computed.

56

Figure2.8 Axis direction and chord length used to adimensionalize node position along the air foil section.

For example:

∆𝑝𝑖,𝑗 = 𝑝𝑖,𝑗 − [∑ 𝑝𝑖,𝑗(𝑡𝑘) 𝑛

𝑘=1

] /𝑛

Obtaining a matrix whose rows are formed by j static pressure fluctuations around its mean value in dimensionless x-position i.

Matrices have all the same dimension [i×j], were i represents the rows number corresponding to the nodes number whereas j represents the columns number corresponding to the number of time steps sampled. [x/c] matrix is then formed by a series of j equal columns each one defining the nodes positions made dimensionless diving its x-value by the aerofoil chord length. [time] matrix is formed by a series of i equal rows representing times, time step by time step.

Resultant charts are surfaces having in x-axis the dimensionless position, in y-axis the time, in z-axis the static pressure fluctuation. This kind of diagrams, associating different colours to different values of static pressure fluctuations, can show the repetitiveness of signals and, inspecting the charts in the position-time plane, is possible to identify the direction of propagation of fluctuations looking at the angular coefficient of areas having the same colour in the colours-map. In addition, these charts are reported in next sections referred to its peculiar simulation.