Chapter 1

10

Chapter 1

Model creation and validation

1.1

Introduction

It was selected, in this study, to use the NACA0012 aerofoil because of its high number of available data, thus increasing the confidence in the obtained results from CFD analysis after they are compared with data from wind tunnel tests.

In subsequent paragraphs will be explained, in detail, how the sample was built, specifying computer programs used and references from which were extracted setting data and are shown validation results.

1.2 Shape definition

To obtain the desired shape it was used Ref. [1] to generate a *.dat file containing the coordinates of points defining the aerofoil, selecting the option of a closed contour with a sharp trailing edge. The *.dat file was than imported in Windows Excel® to generate a macro, used to plot points and lines (splines) connecting them, in Catia v5 r20® _{; points}

were divided in three columns representing the coordinates on ‘x’, ‘y’, and ‘z’ axes and multiplied by 1000 to obtain a convenient chord of 1 meter, very easily scalable. All the ‘z’ coordinates were set to zero to ensure a 2-d sample. In Catia® environment was added a surface closing upper and lower boundary of the aerofoil as can be seen in Figure 1.1; then the file was saved in *.stp format.

Figure 1.1 NACA0012 aerofoil generated with Catia v5 r20® _{on the left hand of the figure.}

11

1.3 Working in ANSYS Workbench

®

The *.stp file was imported in Ansys Workbench® _{environment with the Geometry}

tool and opened in with Space Claim tool, to verify measures correctness and then exported in ICEM CFD tool to generate a computational grid. Were considered two kinds of far field: first of all was considered a rectangular shape to build an H-grid, then an arc shape ahead of the aerofoil to build a ‘C-grid’ as shown in Figure 1.2.

Figure 1.2 Images of far field building options.

The second model gave better results in terms of computation especially at higher Mach number, so the last was selected.

1.3.1 ICEM CFD environment

1.3.1.1 H-grid

The first step was the importation of the geometrical model of the aerofoil, after that it was built the far field by defining six points shown in Table 1.1 in the next page (points were defined with respect to leading and trailing edge). The dimension of far field was selected to ensure a sufficient length in all directions to practically do not compromise the results due to disturbances introduced by the aerofoil. It was built the vertical line connecting points 1, 2, 3 and named ‘INLET’, the vertical line connecting points 4, 5, 6 named ‘OUTLET’ and the two horizontals connecting points 2, 4 and points 3, 5 named ‘FARFIELD_UP’ and ‘FARFIELD_DOWN’ respectively. After creating the body named ‘FLUID’, the latter was splitted to build the blocking shown in Figure 1.3; association was

12

made to outer lines of the far field, and to the aerofoil by using some points conveniently created on its surface to ensure orthogonality of flux and cells, but then were associated to curve because of penetrating elements troubles. It was created an ‘O-grid’, around the section of the airfoil, closing at the trailing edge, and an ‘H-grid’ containing the latter and the whole far field.

Point x y z 1 -10 0 0 2 -10 15 0 3 -10 -15 0 4 26 15 0 5 26 -15 0 6 26 0 0

Table 1.1 Coordinates of point defining far field .

13

The resultant structured mesh has 231182 nodes and 229786 cells. The number of nodes was matched to ensure a good quality of mesh and stability of solution. Due to the aim of this study, it was selected to have a non-dimensional distance, 𝑦+, equal to at least one making sure that the laminar sublayer would be taken into account during calculation. Used formula to estimate the wall distance, 𝑊𝑑, defining the spacing of the first cell near the aerofoil is:

𝑊𝑑 = 𝑦

+_{∗ 𝜇}

𝑈𝑓𝑟𝑖𝑐𝑡∗ 𝜌

where:

μ is the dynamic viscosity of the fluid, ρ is its density and:

𝑈_{𝑓𝑟𝑖𝑐𝑡} = √𝜏𝑤𝑎𝑙𝑙 𝜌 𝜏𝑤𝑎𝑙𝑙 = 1 2∗ 𝜌 ∗ 𝑈∞ 2 _{∗ 𝐶} 𝑓 𝐶_𝑓 = 0.026 𝑅𝑒_𝑐 1 7 𝑅𝑒𝑐 = 𝜌 ∗ 𝑈_∞∗ 𝑐 𝜇

𝜏_{𝑤𝑎𝑙𝑙} is the friction force per unit surface, c is the aerofoil chord length, 𝑅𝑒_𝑐 is the Reynolds number referred to the chord and 𝑈_∞ is the fluid speed at infinity distance from the air foil (practically at sufficient distance from the aerofoil to be not affected by the presence of the latter). Values of 𝑊𝑑 used, were adapted to specific case studied: to validate the sample were calculated the polar curves at three different Mach numbers and at a fixed 𝑅𝑒𝑐 resulting in three values of 𝑊𝑑 summarized in the following Table 1.2:

Mach number 𝑾𝒅 [m] 0.6 9.88 ∗ 10−7

0.7 9.93 ∗ 10−7

0.8 9.95 ∗ 10−7

14

Due to these infinitesimal distance the long wake mesh zone behind the air foil would be too smashed, so it was decided to open the grid spacing gradually to improve mesh quality; otherwise, going forward in the wake, since the grid spacing in x direction (from left to right) increases remaining constant in the y direction, would have turned out in an excessive crush in the latter direction.

Everyone edge around the solid surface of the air foil was linked to the curvature of the latter and the two edge near the leading edge were splitted in order to improve mesh quality and make the cells as orthogonal as possible to the aerofoil.

In Figure 1.4 is shown a detail of cells near the aerofoil.

Figure 1.4 Mesh distribution near the air foil for H-grid.

The structured mesh was then converted in unstructured to be managed in Ansys Fluent®.

1.3.1.2 C-grid

The same procedure was used to build the C-grid; was imported the geometry in Icem CFD tool and then and created six points to define the shape of the far field. The points used are signposted in Table 1.3; points 1, 2, 3 were connected by an arc to create ‘INLET’ zone, points 2, 4 and 3, 5 were connected by straight lines to form ‘FARFIELD’ zone and finally points 4, 6, 5 were connected with another straight line to draw ‘OUTLET’ zone.

15

After creating the body named ‘FLUID’, the last was splitted into an ‘O-grid’ ahead of the aerofoil and terminating at the trailing edge of the latter and an ‘H-grid’ from the trailing edge to the ‘OUTLET’; the resultant two blocks were further splitted to form three bands around the air foil to have a better control on the nodes number and on the direction of resultant cells (to ensure orthogonality cell-flow) and finally almost all edges were curved to improve cells quality and mitigate changes of edges direction.

Point x y z 1 -15 0 0 2 0 15 0 3 0 -15 0 4 26 15 0 5 26 -15 0 6 26 0 0

Table 1.3 Points defining far field shape.

Resultant ‘Blocking’ is shown in Figure 1.5; Figure 1.6 shows in detail the points created on the aerofoil and in fluid to ensure symmetry of mesh whilst Figure 1.7 shows bands created near the solid surface of the air foil.

16

Figure 1.6 Points used to split and curve edges of C-grid, keeping their symmetry.

17

The resulting structured mesh has 343924 nodes and 345084 cells. These numbers were obtained after lots of run that have demonstrated the insensibility to further increase of nodes number, the capability of capture with very good resolution the formed shocks and good results in predicting drag. The last is the reason why was done an increase in node number in the region behind the trailing edge that makes possible to capture the wake also when the flow runs over the aerofoil at angles of attack quite big. Next Figure 1.8 shows the mesh near the air foil.

Figure 1.8 Mesh in proximity of the air foil of the C-grid.

Concerning the wall distance, 𝑊𝑑,were performed two tests choosing 𝑦+ equal to 30 and 1 respectively; the former causes 𝑊𝑑 exceeds the boundary layer and the latter ensures the presence of at least ten nodes in the boundary layer making possible it’s computation. The latter was selected. 𝑊𝑑 used are tabulated in next Table 1.4.

Mach number 𝑾𝒅 [m] 0.6 6.49 ∗ 10−6 0.7 6.49 ∗ 10−6 0.8 6.49 ∗ 10−6 Table1.4 Wall distances used in computations for C-grid.

18

Values are different from the other grid cause was chosen to use other values of pressure and temperature, so different values of density and velocity at constant Mach number; furthermore, was selected to use the same value of wall distance computed for Mach number equal to 0,8 ensuring an 𝑦+ value lower than one in all three cases.

Particular emphasis was devoted to increase quality factors of mesh cells; in following Figure 1.9 are shown some example of quality of this mesh elements.

Figure1.9 First graph represents quality; second graph represents skewness; third graph represents aspect ratio (Fluent); fourth graph represents minimum angle. All referring to C-grid.

As can be seen, aspect ratio is limited to about 1 ∗ 103 _{that is a reasonable value}

considering the small cells height near the air foil surface and in the wake region. As was done for the other grid, wake was opened at a certain distance from the trailing edge to avoid an excessive crush of cells that are otherwise growing only in the x direction (coincident with wake direction).

1.3.2 Fluent environment

The sample was imported in Fluent tool (2-D, double precision) and scaled to the right dimension. Wall distances used were magnified in Icem tool, by dividing the numbers obtained from formula of 𝑊𝑑 by the chord dimension of test experiment for which 𝑊𝑑 was computed, to achieve the right values after the scaling operation. It was given the command ‘solve/initialize/repair wall-distance’ to avoid some issues due to possible

mesh-19

wall interface imperfections, but was just a precaution, results indeed didn’t change in a run devoted to find out influence of that command. It was selected to use the solver type ‘Density-based’ cause of its better results in high subsonic and transonic flows than ‘Pressure-based’ type. Validation was performed with stationary analysis at constant Mach and Reynolds numbers; first grid (the H-grid) was validated using Ref. [2], while the second one (C-grid) was validated with Ref. [3] and [4]. Input settings were obtained from following formulas: 𝑀 =𝑈∞ 𝑎 𝑅𝑒_𝑐 = 𝜌 ∗ 𝑈∞∗ 𝑐 𝜇 𝑎 = √𝛾 ∗ 𝑅 ∗ 𝑇 𝑝𝑡 𝑝 = ( 𝛾 − 1 2 ∗ 𝑀 2₎𝛾−1𝛾 𝑇𝑡 𝑇 = ( 𝛾 − 1 2 ∗ 𝑀 2₎ 𝑝 𝜌= 𝑅 ∗ 𝑇 𝜇 = 𝜇𝑟𝑒𝑓∗ ( 𝑇 𝑇𝑟𝑒𝑓 ) 3 2 ∗𝑇𝑟𝑒𝑓+ 𝑆 𝑇 + 𝑆

where the last is the Sutherland law for viscosity and the constants values inside this formula are: 𝜇_𝑟𝑒𝑓= 1.716 ∗ 10−5 [ 𝑘𝑔 𝑚 ∗ 𝑠] 𝑇_𝑟𝑒𝑓 = 273.11° [𝐾] 𝑆 = 110.56° [𝐾] 1.3.2.1 H-grid validation

In materials was selected nitrogen, a perfect gas law for density to account for compressibility effect on the gas due to very high velocity involved, and was defined a

20

constant dynamic viscosity calculated from Reynolds number because it was the only free parameter to match with tests data. Were examined three values of Mach number to obtain polar curves and all used values are tabulated in Figure 1.10 a)-b)-c) in next pages.

Using compressible gas made necessary the use of the energy law in ‘Models’ task, where was selected to use a ‘𝐾 − 𝜀 ‘ model and in particular the ‘realizable ‘𝐾 − 𝜀’ model with ‘enhanced wall treatment’ for Near-wall treatment. Were used also the options: ‘pressure gradient effects’, ‘thermal effects’, ‘viscous heating’, ‘curvature correction’, ‘compressibility effects’ and ‘production limiter’.

Fluid operating pressure was set to ‘0’ Pa as suggested by Fluent Help®_{, making the}

gauge pressure, defined in boundary conditions, equal to the static pressure of the far field. In boundary conditions were specified the aerofoil and fluid characteristics of the specific prescribed condition. The air foil boundary condition was set to ‘wall’ with options of ‘stationary wall’ and ‘no slip condition’ and was defined a roughness height, 𝑅𝐻.

Done fluid conditions were ‘pressure farfield’ type on inlet zones and ‘pressure outlet’ in outlet region; were defined the static pressure, the Mach number, the velocity direction (cos and sin directors), turbulence was specified by ‘turbulent intensity’ and ‘turbulent viscosity ratio’ (default values), eventually was set the static temperature.

All values were extracted from Ref. [2], used to validate the sample, and scored in an excel file to calculate necessary data to set the analysis and compare results with reference values obtained by Fluent when computed from ‘inlet’. ‘Reference values’ of area and length were set to match the test sample dimensions of Ref. [2].

Were selected an ‘implicit formulation’ for ‘solution method’, a ‘Roe-FDS’ for ‘flux type’, a ‘least squares cell 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 to 10−6_.

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 maximum value 200) and calculation was made in five steps (left with default numbers) with increasingly finer grids to accelerate convergence.

21

Figure 1.10 a) Input settings for H-grid validation.

22

Figure 1.10 c) Input settings for H-grid validation.

Convergence was achieved fast enough between six and nine hundredth iteration, but it was decided to leave calculation going on until one thousandth and five hundredth iteration and was taken the mean value of parameters on at least one hundred iterations.

Results are shown in Tables 1.5-1.6-1.7 and Figures 1.11 a)-b)-c) for Mach number equal to 0.6-0.7-0.8 respectively. Test ( Ref [2] ) : α [deg] Cn Cd Cl -4,0120 -0,4768 0,00733 -0,47746 -1,9551 -0,2448 0,00679 -0,24471 0,0087 -0,0144 0,00661 -0,01440 2,0460 0,2113 0,00683 0,21119 4,0392 0,4393 0,00712 0,43989 6,0620 0,6644 0,00984 0,66709 7,0771 0,7631 0,01607 0,76696 8,0648 0,8412 0,02739 0,84572

23 Fluent: α [deg] Cn Cd Cl -4,0120 -0,57125 0,00885 -0,57203 -1,9551 -0,27721 0,00702 -0,27713 0,0087 0,00120 0,00647 0,00112 2,0460 0,29039 0,00717 0,29031 4,0392 0,57381 0,00888 0,57461 5,0000 0,70454 0,01227 0,70616 6,0620 0,82477 0,02152 0,82712 7,0771 0,90432 0,03381 0,90707

Table 1.5 Dataand results of H-grid at Mach=0.6.

Test ( Ref [2] ): α [deg] Cn Cd Cl -3,0039 -0,4053 0,00985 -0,40534 -1,9755 -0,2702 0,00712 -0,27011 -0,9776 -0,1392 0,00700 -0,13910 0,0161 -0,0171 0,00696 -0,01710 2,0466 0,2326 0,00701 0,23250 3,0441 0,3599 0,00813 0,35998 4,0430 0,4974 0,01367 0,49767 5,0507 0,6147 0,02470 0,61491 6,0579 0,6811 0,04119 0,68055 Fluent: α [deg] Cn Cd Cl -3,0039 -0,49666 0,01415 -0,49660 -1,9755 -0,32767 0,00779 -0,32759 -0,9776 -0,16091 0,00690 -0,16081 0,0161 0,00261 0,00667 0,00261 0,9776 0,16093 0,00693 0,16084 2,0466 0,34055 0,00824 0,34048 3,0441 0,49731 0,01373 0,49728 4,0430 0,64483 0,02621 0,64459 5,0507 0,73231 0,04203 0,73145

24 Test ( Ref [2] ) : α [deg] Cn Cd Cl -3,0039 -0,4053 0,00985 -0,40534 -1,9755 -0,2702 0,00712 -0,27012 -0,9776 -0,1392 0,00700 -0,13910 0,0161 -0,0171 0,00696 -0,01710 2,0466 0,2326 0,00701 0,23250 3,0441 0,3599 0,00813 0,35998 4,0430 0,4974 0,01367 0,49767 5,0507 0,6147 0,02470 0,61491 6,0579 0,6811 0,04119 0,68055 Fluent: α [deg] Cn Cd Cl -3,0039 -0,49666 0,01415 -0,49660 -1,9755 -0,32767 0,00780 -0,32759 -0,9776 -0,16091 0,00690 -0,16081 0,0161 0,00261 0,00667 0,00261 0,9776 0,16093 0,00693 0,16084 2,0466 0,34055 0,00824 0,34048 3,0441 0,49731 0,01373 0,49728 4,0430 0,64483 0,02622 0,64459 5,0507 0,73231 0,04203 0,73145

25

Figure 1.11 a) Polar curves obtained from H-grid at Mach=0.6.

-0,80 -0,60 -0,40 -0,20 0,00 0,20 0,40 0,60 0,80 1,00 -5,0 -4,0 -3,0 -2,0 -1,0 0,0 1,0 2,0 3,0 4,0 5,0 6,0 7,0 8,0 9,0 Cl , l if t co ef fi ci e n t α, [deg]

Cl-α curve

TEST CFD -0,80 -0,60 -0,40 -0,20 0,00 0,20 0,40 0,60 0,80 1,00 0,0050 0,0100 0,0150 0,0200 0,0250 0,0300 0,0350 Cl , l if t co ef fi ci e n t Cd, drag coefficient

Cl-Cd curve

TEST CFD 0,0000 0,0050 0,0100 0,0150 0,0200 0,0250 0,0300 0,0350 0,0400 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 Cd , d ra g co ef fi ci e n t α, [deg]

Cd-α curve

TEST CFD

26

Figure 1.11 b) Polar curves obtained by H-grid at Mach=0.7.

-0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 -4 -3 -2 -1 0 1 2 3 4 5 6 7 Cl , l if t co ef fi ci e n t α, [deg]

Cl-α curve

TEST CFD -0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8 0,0050 0,0100 0,0150 0,0200 0,0250 0,0300 0,0350 0,0400 0,0450 Cl , l if t co ef fi ci e n t Cd, drag coefficient

Cl-Cd curve

TEST CFD 0,0000 0,0050 0,0100 0,0150 0,0200 0,0250 0,0300 0,0350 0,0400 0,0450 -4 -3 -2 -1 0 1 2 3 4 5 6 7 Cd, dr ag coe ff ici en t α, [deg]

Cd-α curve

TEST CFD

27

Figure 1.11 c) Polar curves obtained by H-grid at Mach=0.8. Pressure component represent the pressure Cd component.

-0,5 -0,4 -0,3 -0,2 -0,1 0,0 0,1 0,2 0,3 0,4 0,5 -3 -2 -1 0 1 2 3 4 5 6 Cl , l if t co ef fi ci e n t α, incidence [deg]

Cl-α curve

TEST CFD -0,5 -0,4 -0,3 -0,2 -0,1 0,0 0,1 0,2 0,3 0,4 0,5 0,010 0,020 0,030 0,040 0,050 0,060 0,070 0,080 Cl , l if t coe ff ici en t Cd, drag coefficient

Cl-Cd curve

TEST CFD CFD (pressure component) 0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 -3 -2 -1 0 1 2 3 4 5 6 Cd , d ra g co ef fi ci e n t α, incidence [deg]

Cd-α curve

TEST CFD CFD (pressure component)

28

As can be seen from previous tables and images, results from CFD analysis are in good accordance with the TEST one at Mach numbers lower than 0.80, but not at the last one as Figure 1.11 c) shows (there are considerable gaps in the curves); considering that one of the aims of this thesis is studying transonic buffet, that develops at Mach numbers higher than 0.80, for small angles of attack at least, it was considered the other kind of mesh: the C-grid.

1.3.2.2 C-grid validation

According to Ref. [3] was selected air as fluid material and chosen a perfect compressible gas law for density; were modified values of specific heat at constant pressure, 𝑐_𝑝, thermal conductivity, 𝜆, and dynamic viscosity, 𝜇.

Again, energy equation was enabled cause of the choice of a perfect compressible gas law; was then selected the ‘Realizable k-ε’ model with ‘enhanced wall treatment’ with the options ‘pressure gradient effects’, ‘thermal effects’, ‘viscous heating’, ‘curvature correction’, ‘compressibility effects’ and ‘production limiter’.

Operating pressure was set to 0 [Pa] 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 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. [3]. All values used were extracted from Ref. [3], stored in an excel file and 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 10−6.

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 maximum value 100) and

29

calculation was made in three steps with increasingly finer grids to accelerate convergence; the first and the second one of 200 iterations, the third one of 100 iterations and the remaining iterations with the original grid.

Were examined two different Mach numbers to plot the polar curves: 0.6 and 0.8 and for the second one were considered two different Reynolds numbers (4000000 and 10000000); in next Tables 1.8, 1.9 and 1.10 are shown the settings used in calculations.

DATA: VELOCITY DIRECTION:

Rec 10000000 α dir X dir Y M 0,6 -2 0,999390827 -0,034899497 Tt [K] 273 -1 0,999847695 -0,017452406 RH [m] Laminato a freddo 0,0000016 0 1 0 R [j/kg*K] 287 1 0,999847695 0,017452406 γ 1,4 2 0,999390827 0,034899497 c [m] 0,21 3 0,998629535 0,052335956 4 0,99756405 0,069756474 DERIVED: 5 0,996194698 0,087155743 6 0,994521895 0,104528463 P [Pa] 294341,3717 7 0,992546152 0,121869343 T [K] 254,6641791 Pt [Pa] 375433,5311 ρ [kg/m^3] 4,027184725 a [m/s] 319,8813329 U [m/s] 191,9287998

μ [Pa*s] Sutherland law 1,62316E-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 1.8 Settings at Mach=0.6 for C-grid validation.

DATA: (see next page ↓) VELOCITY DIRECTION:

Rec 4000000 α dir X dir Y

M 0,8 -2 0,999390827 -0,034899497

30 RH [m] Laminato a freddo 0,0000016 0 1 0 α [deg] 0 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 ρ [kg/m^3] 1,189347509 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 1.9 Settings at Mach=0.8 and Reynolds=4000000 for C-grid validation.

DATA: VELOCITY DIRECTION:

Rec 10000000 α dir X dirY 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] 0 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

DERIVED: (see next page ↓)

P [Pa] 206530,5112

T [K] 242,0212766

31

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

Table 1.10 Settings at Mach=0.80 and Reynolds=10000000 for C-grid validation.

Convergence was achieved near one thousand iterations, later than the other grid probably because of the increased node number; in this case was not necessary to take a mean value on a large number of iterations because convergence was better and values was the same from one iteration to the next one.

Results are shown in Tables 1.11, 1.12, 1.13 and in Figures 1.12 a), b), c):

TEST( Ref [3] ): CFD: α cl cd α cl cd -1,75 -0,203 0,0071 -2 -0,28043 0,00821 -0,08 0,002 0,0067 -1 -0,1401 0,00782 1,61 0,196 0,0068 0 -7,5E-05 0,00772 3,38 0,416 0,0077 1 0,140096 0,00782 5,08 0,621 0,0101 2 0,280425 0,00821 5,93 0,722 0,0152 3 0,4213 0,00893 6,8 0,807 0,0237 4 0,56263 0,0101 7,7 0,863 0,037 5 0,69336 0,01312 6 0,805269 0,02114

TEST ( Ref [3]uncorrected): 7 0,8855 0,03366

α cl cd Cd (components):

-1,75 -0,206 0,0071 α pressure viscous total

-0,08 0,003 0,0067 -2 0,00229 0,00593 0,00821 1,61 0,199 0,0068 -1 0,00186 0,00596 0,00781 3,38 0,421 0,0077 0 0,00174 0,00597 0,00772 5,08 0,629 0,0101 1 0,00186 0,00596 0,00781 5,93 0,729 0,0152 2 0,00229 0,00593 0,00821

32 6,8 0,81 0,0237 3 0,00307 0,00586 0,00893 7,7 0,859 0,0369 4 0,00422 0,00576 0,00997 5 0,00760 0,00553 0,01312 6 0,01595 0,00519 0,02114 7 0,02899 0,00466 0,03366

Table 1.11 Data and results given by C-grid at Mach=0.6.

TEST ( Ref [3] ): CFD: α Cd Cl Cz α cl cd -1,56 0,0207 -0,30932 -0,31 -2 -0,33258 0,03547 -0,72 0,012 -0,13684 -0,137 -1 -0,20721 0,02345 0,03 0,0106 0,00499 0,005 0 -0,00002 0,01713 0,72 0,0137 0,14882 0,149 1 0,20721 0,02345 1,25 0,02 0,29949 0,3 2 0,33258 0,03547 2 0,0308 0,43366 0,435 3 0,36198 0,04539 2,93 0,0402 0,52825 0,531 4 0,35472 0,05306 3,78 0,051 0,58635 0,591 5 0,35266 0,06100 4,76 0,0709 0,64985 0,658

TEST ( Ref [3] uncorrected): Cd (components):

α cl α pressure viscous total

-2,13 -0,137 -2 0,02945 0,00602 0,03547 -0,95 -0,141 -1 0,01621 0,00605 0,02227 0,05 0,005 0 0,01078 0,00634 0,01713 1,04 0,153 1 0,01621 0,00605 0,02227 2,02 0,307 2 0,02945 0,00602 0,03547 3,03 0,441 3 0,03957 0,00582 0,04539 4,02 0,534 4 0,04749 0,00557 0,05306 4,99 0,592 5 0,05566 0,00534 0,06101 6,02 0,656

33 TEST ( Ref [3] ) : CFD: α Cd Cl Cz α Cd Cl -1,56 0,0207 -0,30932 -0,31 -2 0,03592 -0,36454 -0,72 0,012 -0,13684 -0,137 -1 0,02239 -0,20770 0,03 0,0106 0,00499 0,005 0 0,01575 0,0000017 0,72 0,0137 0,14882 0,149 1 0,02240 0,20770 1,25 0,02 0,29950 0,3 2 0,03592 0,36454 2 0,0308 0,43366 0,435 3 0,04596 0,38890 2,93 0,0402 0,52825 0,531 3,78 0,051 0,58635 0,591 4,76 0,0709 0,64985 0,658

TEST (uncorrected): Cd (components):

α Cl α pressure viscous total

-2,13 -0,137 -2 0,03073 0,00519 0,03592 -0,95 -0,141 -1 0,01696 0,00543 0,02239 0,05 0,005 0 0,01025 0,00551 0,01576 1,04 0,153 1 0,01696 0,00543 0,02239 2,02 0,307 2 0,03073 0,00519 0,03592 3,03 0,441 3 0,04103 0,00492 0,04596 4,02 0,534 4,99 0,592 6,02 0,656

34

Figure 1.12 a) Polar curves given by C-grid at M=0.6 and Re=10000000.

-0,4 -0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 -2 -1,5 -1 -0,5 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 Cl , l if t co ef fi ci en t α [degrees]

Cl-α curve

CFD TEST -0,4 -0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0,005 0,0075 0,01 0,0125 0,015 0,0175 0,02 0,0225 0,025 0,0275 0,03 0,0325 0,035 0,0375 0,04 Cl , l if t co ef fi ci e n t Cd, drag coefficient

Cl-Cd curve

TEST CFD 0 0,005 0,01 0,015 0,02 0,025 0,03 0,035 0,04 -2 -1,5 -1 -0,5 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 Cd , d ra g co ef fi ci e n t α [degrees]

Cd-α curve

TEST CFD

35

Figure 1.12 b) Polar curves given by C-grid at M=0.80 and Re=4000000.

-0,4 -0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 -2,5 -2 -1,5 -1 -0,5 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 Cl , l if t co ef fi ci en t α, incidence [deg]

Cl-α curve

CFD TEST TEST (uncorrected) -0,4 -0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,005 0,01 0,015 0,02 0,025 0,03 0,035 0,04 0,045 0,05 0,055 0,06 0,065 0,07 0,075 Cl , l if t co ef fi ci e n t Cd, drag coefficient

Cl-Cd curve

TEST CFD CFD (pressure component) 0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 -2 -1,5 -1 -0,5 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 Cd , d ra g co ef fi ci e n t α, incidence [deg]

Cd-α curve

TEST CFD CFD (pressure component)

36

Figure 1.12 c) Polar curves given by C-grid at M=0.80 and Re=10000000.

-0,5 -0,4 -0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 -2,5 -2 -1,5 -1 -0,5 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 Cl , l if t co ef fi ci e n t α, incidence [deg]

Cl-α Curve at Mach=0.8

TEST Ref [3] (uncorrected) CFD

TEST Ref [3] (corrected) TEST Ref [4]

TEST Ref [4] (corrected)

-0,5 -0,4 -0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,005 0,01 0,015 0,02 0,025 0,03 0,035 0,04 0,045 0,05 0,055 0,06 0,065 0,07 0,075 C l, lif t co ef fi ci en t Cd, drag coefficient

Polar curve Cl-Cd at Mach=0.8

TEST Ref [3] CFD CFD (pressure component) TEST Ref [4] 0 0,005 0,01 0,015 0,02 0,025 0,03 0,035 0,04 0,045 0,05 0,055 0,06 0,065 0,07 0,075 -2,5 -2 -1,5 -1 -0,5 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 Cd ,d ra g co ef fi ci e n t α, incidence [deg]

Cd-α curve at Mch=0.8

TEST CFD CFD (pressure component)

37

As can be seen from previous Figure 1.12 a) there is a good accordance with test data though the angular coefficient of Cl-α is greater in CFD analysis; the last discrepancy is due to the fact that test analysis was conducted in an aerodynamic tunnel whereas CFD data are extrapolated in freestream conditions. In Figures 1.12 b) and c) can be seen a very good accordance of Cl-α values only for low incidence angles (<1°), reflecting in the second graph of this figure, and at the same time a strange trend in both test curves of ref [3]; the angular coefficient of Cl-α test curves changes indeed from negative incidence angles to the positive ones in contradiction with the physics of the sample that is symmetric, consequently curves should have to be symmetric too with respect to abscissa axis like CFD analyses highlight. This fact led to nourish doubts about accuracy of test data given in ref [3]. In the second and third graph were extrapolated the Cd components, in ref [3] indeed is reported the wake drag coefficient calculated only by pressure component ( neglecting the viscous contribution to drag force); it can be appreciated a good estimation of polar curves especially in the third graph where there are not lift effects, while in the second one as in the first one results seems to be accurate only for low incidences, finally these two last test curves are not symmetrical too.

Further research in the literature brought to know a most recent work conducted by Charles D. Harris, mentioned as Ref [4], and used a posteriori to compare with CFD data; was found a complete accordance with Cl-α curve as can be seen in Figure 1.12 c) resulting in a good accordance with polar curve at higher angles of attack meanwhile showing a discrepancy at angles lower than 1.5 degrees, typical of CFD analyses, but maintaining the trend. Analyses were not conducted over 3 degrees incidence cause buffet onset starts for lower angles of attack, so was decided to stop at 3°.