Results
5.2 Aerodynamic Analysis Results
With the Aerodynamic model now geometrically defined and appropriately discretized, it is now possible to take some results from a VII Aerodynamic analysis with the Aerodynamic Module here implemented for the two cases conditions previously defined . Figure 5.8 shows the convergence of the aerodynamic coefficients with the iteration number.
It is evident that, regarding the case whereα= 3.5o, from the 5th iteration the result can be consid- ered converged, only demonstrating small oscillations related to the Direct VII procedure implemented.
From then on, the results always oscillate bellow1%, as can be seen in Figure 5.9. For the other test case, whereα= 6.95o, as already observed in the previous subsections, the convergence of results is not as good with the increase of the iteration number, with oscillations around10%. This may also be due to the incapacity of the implemented model to deal with boundary layer separation that may occur at such angles of attack.
The obtained solutions can now be compared to the inviscid solution obtained with APAME for each flow condition and to the solution obtained from CFD analysis of the geometry for the same conditions as well. The used geometry is presented in Figure
0.256 0.258 0.26 0.262 0.264 0.266 0.268 0.27 0.272 0.274
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Lift Coefficient
Iteration Number
(a) Convergence of the lift coefficient withα= 3.5o.
0.52 0.54 0.56 0.58 0.6 0.62 0.64
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Lift Coefficient
Iteration Number
(b) Convergence of the lift coefficient withα= 6.95o.
0.01 0.0105 0.011 0.0115 0.012 0.0125 0.013 0.0135 0.014 0.0145
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Drag Coefficient
Iteration Number
(c) Convergence of the drag coefficient withα= 3.5o.
0.01 0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.018 0.019 0.02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Drag Coefficient
Iteration Number
(d) Convergence of the drag coefficient withα= 6.95o.
-0.34 -0.335 -0.33 -0.325 -0.32 -0.315 -0.31
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Moment Coefficient
Iteration Number
(e) Convergence of the moment coefficient withα= 3.5o.
-0.86 -0.84 -0.82 -0.8 -0.78 -0.76 -0.74 -0.72
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Moment Coefficient
Iteration Number
(f) Convergence of the moment coefficient withα= 6.95o.
Figure 5.8: Convergence of the aerodynamic results with the iteration number for both test cases.
These results were obtained through the implementation of the geometry and flow conditions in the CFD commercial package StarCCM+. The model employed was a Reynolds Averaged Navier-Stokes (RANS) model with an SST (Menter’s Shear Stress Transport)k−ωturbulence model. The comparison of results from the VII code, CFD and APAME are presented in Table 5.9.
First of all, it is possible to observe that the best results obtained with the VII code were for Case
#2, with the lower angle of attack. For this case every aerodynamic coefficient was calculated to an error smaller than3%when compared to the CFD solution. This is a very good approximation, taking into account that the computational effort was almost 50 times lower for the VII code computation when compared with the CFD analysis (see Table 5.10). Drag was particularly well estimated, with an error of
−0.8%, meaning that the VII code slightly underestimated the Drag for this case.
0.001%
0.010%
0.100%
1.000%
10.000%
100.000%
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Relative Error (log scale)
Iteration Number
Cl error Cd error Cm error
(a) Relative error of the aerodynamic coefficients forα= 3.5o.
0.001%
0.010%
0.100%
1.000%
10.000%
100.000%
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Relative Error (log scale)
Iteration Number
Cl error Cd error Cm error
(b) Relative error of the aerodynamic coefficients forα= 6.95o.
Figure 5.9: Evolution of the Relative Error of the aerodynamic coefficients with the iteration number for both test cases.
Figure 5.10: Wing geometry modeled on StarCCM+ for CFD aerodynamic analysis.
The method that presents consistently bigger errors accross all aerodynamic coefficients for this case is the inviscid Panel Code, APAME. This was also expected partly because the shear loads on the wing skin are not taken into account in inviscid computations, especially for the estimation of Drag, which is underestimated presenting an error of −46.4%. Inviscid methods are only able to account for
Table 5.9: Comparison of aerodynamic coefficient results between StarCCM+ CFD solution, the VII code implemented and APAME inviscid solution.
Case #1: APAME VII code StarCCM+ Error APAME Error VII code CL 0.6206 0.5667/0.5844 0.5614 10.5% 0.9%/4.1%
CD 0.0123 0.0171/0.0185 0.0233 −46.4% −20.6%/−26.8%
Cm0 −0.8457 −0.7886/−0.7625 −0.7554 12% 0.9%/4.4%
Case #2: APAME VII code StarCCM+ Error APAME Error VII code
CL 0.272 0.258 0.252 8.2% 2.7%
CD 0.00524 0.01385 0.01397 −62.5% −0.8%
Cm0 −0.3387 −0.31679 −0.31019 9.2% 2.1%
Table 5.10: Comparison of Nomalized CPU computational times for the solution obtained with Star- CCM+, the VII code implemented and APAME inviscid code.
CPU Time(hours) APAME VII code StarCCM+
CASE #1 0.022 2.15 114.67
CASE #2 0.022 2.326 112.4
induced drag.
The Lift Coefficient is most overestimated by the inviscid solver APAME as was expected due to the displacement effect of the Boundary Layer not being considered, with an error of8.2%for Case #2. The VII code here implemented presents an error of2.7%when compared with StarCCM+ solution, standing in between the CFD and the Panel Method solutions.
Looking now at the results obtained for CASE #1, the errors are visibly bigger, both for the APAME and VII codes. Nevertheless, all aerodynamic coefficients calculated with the VII code fell within the results from APAME and the CFD analysis. Lift was calculated to a maximum error of4.1%and Moment to a maximum error of4.4%. On the other hand, the Drag computation presented significantly bigger errors, topped at−26.8%. This difference is justified by the fact that the VII code is still lower in fidelity when compared to a CFD approach, which is capable of modeling with better fidelity boundary layer separation effects that may occur at such an high angles of attack, as well as accounting for other phenomena only possible to model when solving the higher order RANS equations in a volume domain.
However, the error committed by the VII method is still less than half than the Drag estimated with the inviscid code, which presented a result underestimated by62.5%.
As intended, the VII code is able to successfully account for the Boundary Layer displacement and Shear Stress effects with a good agreement to the results obtained by a CFD code, especially for low angles of attack.