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A peer-reviewed article of this preprint also exists.
This version is not peer-reviewed
Submitted:
21 August 2023
Posted:
21 August 2023
You are already at the latest version
A peer-reviewed article of this preprint also exists.
This version is not peer-reviewed
Submitted:
21 August 2023
Posted:
21 August 2023
You are already at the latest version
1 | After excluding some inappropriate or almost identical designs, a base set with 1263 foil designs was identified. For each of these base designs, 5 random shape perturbations around them were generated, leading to $6\times 1263=7578$ foil designs. |
2 | excluding of course the case of a coincident pair of points at the trailing edge. |
3 | If we split the foil curve in two parts using the points corresponding to the minimum and maximum coordinate, each part should behave as a graph of function. |
4 | |
5 | recall that we need to use 17 parameters of the foil parametric model to represent the dataset within Kulfan tolerance; see also Section 2.1.1. |
6 | To a lesser extend when only second moments are included. |
${\mathit{\mu}}_{0,0}$ | ${\mathit{\mu}}_{1,0}$ | ${\mathit{\mu}}_{0,1}$ | ${\mathit{\mu}}_{0,2}$ | ${\mathit{\mu}}_{1,1}$ |
1 | 0 | 0 | 0.0036 | 7.1537E-04 |
${\mu}_{2,0}$ | ${\mu}_{0,3}$ | ${\mu}_{1,2}$ | ${\mu}_{2,1}$ | ${\mu}_{3,0}$ |
0.3260 | 3.2836E-05 | -4.6849E-04 | -0.0036 | 0.0499 |
${\mu}_{0,4}$ | ${\mu}_{1,3}$ | ${\mu}_{3,1}$ | ${\mu}_{2,2}$ | ${\mu}_{4,0}$ |
2.6985E-05 | 1.2810E-05 | 7.8598E-04 | -4.0455E-04 | 0.2297 |
Geometry only | $\mathit{p}\left({\mathit{\vartheta}}_{-1}\right)$ | ${\mathcal{U}}_{(-1)}$ |
Geometry and second moments | $\mathit{p}\left({\mathit{\vartheta}}_{2}\right)$ | ${\mathcal{U}}_{\left(2\right)}$ |
Geometry and third moments | $\mathit{p}\left({\mathit{\vartheta}}_{3}\right)$ | ${\mathcal{U}}_{\left(3\right)}$ |
Geometry and fourth moments | $\mathit{p}\left({\mathit{\vartheta}}_{4}\right)$ | ${\mathcal{U}}_{\left(4\right)}$ |
Geometry and second to third moments | $\mathit{p}\left({\mathit{\vartheta}}_{2-3}\right)$ | ${\mathcal{U}}_{(2-3)}$ |
Geometry and second to fourth moments | $\mathit{p}\left({\mathit{\vartheta}}_{2-4}\right)$ | ${\mathcal{U}}_{(2-4)}$ |
Discretization | Initial | ${\mathcal{U}}_{(-1)}$ | ${\mathcal{U}}_{\left(2\right)}$ | ${\mathcal{U}}_{\left(3\right)}$ | ${\mathcal{U}}_{\left(4\right)}$ | ${\mathcal{U}}_{(2-3)}$ | ${\mathcal{U}}_{(2-4)}$ |
---|---|---|---|---|---|---|---|
Uniform Spacing (Parametric Domain) | 2.0518 | 1.4176 | 1.6590 | 2.1210 | 2.1494 | 2.5657 | 2.7720 |
Cosine Spacing | 1.1562 | 1.2220 | 1.5515 | 2.0975 | 1.2719 | 1.4411 | |
Curvature-Based Spacing | 1.2967 | 1.8346 | 2.8000 | 2.7417 | 2.6835 | 2.2227 | |
Uniform Spacing (Physical Domain) | 1.1214 | 1.1368 | 1.2826 | 1.2210 | 1.1519 | 1.1816 |
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Zahid Masood
et al.
,
2023
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