NACA 0012 Hydrofoil

The turbulent flow around a NACA 0012 hydrofoil with 2.5 m chord length is simulated at Re

= 2.8 x 11⁶ with grid sizes of 70x14x10, 99x20x14, 140x28x20, 198x40x28 and 0, 6, 10, 15 degrees of incidence (a grid size of 70x14x10 implies that, 70 control volumes existed in the 𝜉 direction, 14 in the η direction and 10 in the 𝜁 direction). The profile of NACA 0012 hydrofoil is shown in Fig. 11.3.

(a) (b)

Fig. 11.3: Profile of NACA 0012: (a) 2D and (b) 3D

91 The obtained pressure coefficients, experimental and numerical results for each angle and a grid size of 198x40x28 are shown in Fig. 11.4 to 11.7 respectively.

Fig. 11.4: Pressure coefficients for NACA 0012 hydrofoil at 0-degree angle of attack

Fig. 11.5: Pressure coefficients for NACA 0012 hydrofoil at 6-degree angle of attack

92 Fig. 11.6: Pressure coefficients for NACA 0012 hydrofoil at 10-degree angle of attack

Fig. 11.7: Pressure coefficients for NACA 0012 hydrofoil at 15-degree angle of attack

93 From Fig. 11.4 to 11.7, it can be seen that obtained results show good agreement with experimental and numerical result available in the literature. At 0 and 6-degree angle of attack, the result agrees very well with the experimental result by Ladson, Hill and Johnson (1987) and numerical result obtained by Mursaline (2017). On the other hand, discrepancy is visible at the leading edge of the hydrofoil at 10- and 15-degree angle of attack which may be occurred partly due to inadequate turbulence modeling because k-epsilon turbulence model gives poor performance at higher angle of attack when flow separation takes place. Moreover, the structured mesh may not be able to capture the very strong gradients. At 10-degree angle of attack, although Mursaline (2017) worked with the CAFFA code of 2D version, the numerical results obtained by him show good agreement with the experimental result. However, the obtained results show discrepancy at the leading edge at 10-degree angle of attack compared to the numerical result of Mursaline (2017). For the 198x40x28 grid, effect of angle of attack is studied by plotting the pressure coefficients at 0, 6, 10 and 15 degree on the same axes as shown in Fig. 11.8.

Fig. 11.8: Effect of angle of attack on pressure coefficients of NACA 0012 hydrofoil.

For NACA 0012 (symmetrical hydrofoil), at zero angle of attack the pressure distribution over the upper surface is similar to the pressure distribution on the lower surface, this implies that no lift is generated at zero-degree angle of attack. The flow accelerates on the upper side of the hydrofoil and the velocity of flow decreases along the lower side and according to Bernoulli’s theorem the upper surface will experience lower pressure than the lower surface. The

94 distribution of pressure coefficient under different angles of attack is shown in Fig. 11.8. The pressure coefficient of the hydrofoil’s upper surface is negative and the lower surface is positive, thus the lift force of the airfoil is in the upward direction. It is found that larger the attack angle, greater is the difference of pressure coefficient between the lower and upper surface. The coefficient of pressure difference is much larger on the front edge than the rear edge, thus indicates that the lift force of the hydrofoil is mainly generated from the front edge.

Using the pressure distribution, lift coefficients are calculated at different angle of attack to obtain the stall angle. The lift coefficient curve for NACA 0012 hydrofoil is shown in Fig. 11.9.

Fig. 11.9: Lift coefficient curve of NACA 0012

From Fig. 11.9, it can be seen that the obtained lift coefficients show good agreement with the experimental result. Discrepancy is notable at higher angle of attack. Highest lift coefficient is obtained at 16-degree angle of attack whose value is 1.48 and the experimental value of highest lift coefficient is 1.55 which is also occurred at 16-degree angle of attack.

C type structured mesh with four different sizes are used to investigate the influence on surface pressure coefficients. The first mesh consists of 70 control volumes in the 𝜉 direction, 14 in the η direction and 10 in the 𝜁 direction (70x14x10) with 9800 cells in total. The subsequent finer meshes are of sizes 99x20x14, 140x28x20 and 198x40x28 with 27720, 78400 and 221760 control volumes respectively. At a given angle of attack, pressure coefficients are computed at

95 each of the four grids and plotted on the same axes. Results for 0, 10 and 15 degree of incidence are shown in Fig. 11.10, Fig. 11.11 and Fig. 11.12 respectively. By observing Fig. 11.10, Fig.

11.11, Fig. 11.12 and comparing with experimental data in Fig. 11.4, Fig. 11.6 and Fig. 11.7, it is evident that, refinement of grid gives increased numerical accuracy in the present case.

Fig. 11.10: Grid dependency of pressure coefficients for NACA 0012 hydrofoil at 0-degree incidence.

Fig. 11.11: Grid dependency of pressure coefficients for NACA 0012 hydrofoil at 10-degree incidence.

96 Fig. 11.12: Grid dependency of pressure coefficients for NACA 0012 hydrofoil at 15-degree

incidence.

The improvement in result is particularly marked at the point of lowest and highest-pressure coefficient near the leading edge. There is a reduction in discrepancy because the steep gradients can be better resolved by the finer mesh used.

To compare the grid dependency tests of 2D and 3D methods, the obtained results are plotted against the numerical results of Mursaline (2017) at same angle of attack and same grid sizes.

The comparison is shown in Fig. 11.13, 11.14 and 11.15.

(a)

97 (b)

(c)

Fig. 11.13: Grid dependency of pressure coefficients for NACA 0012 hydrofoil at 0-degree incidence of 2D and 3D methods

98 (a)

(b)

(c)

Fig. 11.14: Grid dependency of pressure coefficients for NACA 0012 hydrofoil at 6-degree incidence of 2D and 3D methods

99 (a)

(b)

(c)

Fig. 11.15: Grid dependency of pressure coefficients for NACA 0012 hydrofoil at 10-degree incidence of 2D and 3D methods

100 From Fig. 11.13-11.15, it is seen that the convergence of the result is faster for 2D method at coarse (50x14) and medium (88x20) grid sizes in comparison with 3D method where grid sizes are 50x14x10 and 88x20x14 for coarse and medium grid respectively at 0, 6 and 10-degree angle of attack. Although, at finer grid size the 2D method (176x40) and 3D method (176x40x20) can capture the strong gradients of pressure at small angle of attack, 3D method struggles to attain the pressure gradients at the leading edge at high angle of attack. So, it can be said easily that 2D method works better than 3D method.

Uncertainty analysis

Uncertainty analysis in the present work is done according to the verification methodology described in Stern et al. (2001). Numerical uncertainty is obtained from grid uncertainty (𝑈𝐺) and iterative uncertainty (𝑈_𝐼), according to the following equation.

𝑈_𝑆𝑁 ² = 𝑈_𝐺²+ 𝑈_𝐼² (11.1.1)

The convergence condition (𝑅_𝑘) of the different solutions (at least three) is defined as:

𝑅_𝑘 = 𝑆₂ − 𝑆₁

𝑆₃ − 𝑆₂ (11.1.2)

where 𝑆1, 𝑆2 𝑎𝑛𝑑 𝑆₃ correspond to solutions with fine, medium, and coarse grid respectively.

According to this formulation, three convergence conditions are possible:

(i) 0 < 𝑅_𝑘 < 1 : Monotonic Convergence (ii) 𝑅_𝑘< 0 : Oscillatory Convergence (iii) 𝑅𝑘 >1 : Divergence

For condition (i), generalized Richardson Extrapolation (RE) is used to estimate grid uncertainty 𝑈_𝐺 . For condition (ii), uncertainties are estimated simply by attempting to bound the error based on oscillation maximums 𝑆_𝑈 and minimums 𝑆_𝐿, i.e. 𝑈_𝐺 = ¹

2(𝑆_𝑈− 𝑆_𝐿). For condition (iii), errors and uncertainties cannot be estimated.

For convergence condition (i), generalized RE is used to estimate the error 𝛿_𝑅𝐸^∗ _𝑘1due to selection of the kth input parameter and order-of-accuracy𝑝_𝑘.

𝛿_𝑅𝐸^∗ _𝑘1 =𝑆₂ − 𝑆₁

𝑟_𝑘^𝑝𝑘− 1 (11.1.3)

where 𝑟_𝑘 is the grid refinement ratio.

101 𝑝_𝑘=

𝑙𝑛 (𝑆₃ − 𝑆₂ 𝑆₂ − 𝑆₁) 𝑙𝑛 (𝑟_𝑘)

(11.1.4) Correction of Eq. 11.1.3 through a multiplication factor 𝐶_𝑘 accounts for effects of higher-order terms and provides a quantitative metric to determine proximity of the solutions to the asymptotic range.

𝛿_𝑘1^∗ = 𝐶_𝑘𝛿_𝑅𝐸^∗ _𝑘1 =𝐶_𝑘𝑆₂ − 𝑆₁

𝑟_𝑘^𝑝𝑘− 1 (11.1.5)

where the correction factor is given by

𝐶_𝑘 = 𝑟_𝑘^𝑝𝑘− 1

𝑟_𝑘^{𝑝𝑘𝑒𝑠𝑡}− 1 (11.1.6)

and 𝑝_{𝑘𝑒𝑠𝑡}is an estimate for the limiting order of accuracy as spacing size goes to zero and the asymptotic range is reached so that 𝐶𝑘 ⟶ 1.Solution uncertainty estimates are given by,

𝑈_𝐺 = {[9.6(1 − 𝐶_𝑘)²+ 1.1]⎸𝛿_𝑅𝐸^∗ _𝑘1⎸, ⎸1 − 𝐶_𝑘⎸ < 0.125

[2 ⎸1 − 𝐶_𝑘⎸ + 1]⎸𝛿_𝑅𝐸^∗ _𝑘1⎸, ⎸1 − 𝐶_𝑘⎸ ≥ 0.125 (11.1.7) In the present study, four grids with a refinement ratio of √2 in each direction were carried out for convergence analysis. The results of grid uncertainty are studied in Table 11.1.

Table 11.1. Grid uncertainty results for lowest pressure coefficient of NACA 0012 at x/c=0.05 for 15 degree angle of attack

Grid ID Grid Size C_P Error Error ( with respect

to experimental value) Experimental

Value

-6.50 …… ……

Fine 𝑆₁ 198x40x28 -5.50 15.38(% Exp

Value)

15.38%

Medium 𝑆₂ 140x28x20 -5.40 1.81(% Fine

Grid)

16.92%

Coarse 𝑆₃ 99x20x14 -4.50 20.00 (% S2 ) 30.76%

Very Coarse 70x14x10 -3.00 33.33( % S3 ) 53.84%

𝑅_𝑘 0.11

102

𝑈_𝐺 0.45(% S1 )

Convergence Type

Monotonic

According to Table 11.1, the error percentages compared to experimental findings for very coarse, coarse, medium and fine grid size are found to be 53.84%, 30.76%, 16.92% and 15.38%

respectively which means that the error percentage is decreasing with increasing grid size. So, the results have good convergence and convergence type is monotonic. At medium and fine grid, error percentages are 16.92% and 15.38% respectively, which means that increasing grid density after medium grid has very little impact on pressure coefficient. So, solution becomes grid independent after medium grid. The grid uncertainty of pressure coefficient (Cp) is only 0.45(%S1). Also, according to Stern et al. (2001) iterative uncertainty can be neglected while calculating numerical uncertainty, since 𝑈𝐼 is very small.