Panels Definition

0 0.2 0.4 0.6 0.8 1 -0.15

-0.1 -0.05 0 0.05 0.1

Fitted vs Exact

1^st and 3^rd bezier curve 2^nd and 4^th bezier curve Airfoil data points

Figure 4.6: Airfoil NACA 0010 fitted by bezier curves

Figure 4.7 illustrates the imposed constraints during the airfoil NACA 0010 parametrization. The control points in red,AandGwere not allowed to move, corresponding to the trailing and leading edge, respectively. Similarly, the points D and J were also fixed, corresponding to the points of maximum thickness. The points in yellowBandL, were allowed to move in any direction. The points in green,C, E,I andK are constrained to move relatively toDandJ in the horizontal direction, as suggested by the horizontal red lines. Similarly, the points in blue,F andH were constrained to move in the vertical direction relatively to the pointG, as suggested by the vertical red lines.

The applied constraints are also necessary to parametrize acceptable airfoil shapes during the opti- mization process. However, to obtain shapes with camber and different thicknesses, the pointsD and J must be free to move in any direction. Moreover, control about these features may be achieved by bounding the control points appropriately.

Figure 4.7: Airfoil NACA 0010 defined by bezier polygons and respective control points constraints

there is no guarantee that four nearest points belonging to the wing’s geometry are coplanar. Therefore, it is necessary to define an average plane where the element lies on, bounded by the edges formed by its vertexes. To achieve this task, a MATLAB^R function calledpanels.mwas developed. A detailed list of the inputs and outputs are presented in Table 4.2.

Outputs Inputs

Panels Corner Points (X1,X2,X3,X4) Panels Areas (DS) Input Points (WP1,WP2,WP3,WP4) Collocation Points (CP) Panels Basis Vectors (l,m,n)

Table 4.2: List of inputs and outputs of functionpanels.m

Typically, when the wing has no twist, the taper ratio is small and the airfoil shape is the same for the all wing, it is possible to assume that four wing points closest to each other are almost coplanar.

Therefore, there is no need to define an average plane to the panel. In the tool developed by Cardeira [17], the latter approach was assumed and thus, modified in this work. The followed methodology is based on the work of Hess [61].

Before proceeding further some nomenclature needs to be introduced. The generated set of wing points are represented by the vectorWP, as already suggested in Table 4.1. It is defined as

WP=h

WP₁^T WP₂^T WP₃^T WP₄^T i^T

(4.35)

with e.g. WP1given by WP₁=h

∪

W P

∪

W P

∪

W P

T

,∀i∈ {1, ..., M},∀j∈ {1, ..., N} (4.36) where∪ij means concatenation, a compact way to indicate that the indexes are unrolled, andM and N are the chordwise and semi spanwise number of panels. The same variableWP₁, with indexes, is defined as

WP_1ij =h

W P1_ij1 W P1_ij2 W P1_ij3

iT

(4.37) whereW P1_ij1,W P1_ij2andW P1_ij3 are the three dimensional coordinates of the pointWP1ij, belonging to the mesh location (i, j), as stated by Equation (4.37). Points WP1ij to WP4ij are the inputs to form the panel(i, j)as depicted in Figure 4.8. The expressions forWP_2ijuntilWP_4ij are obtained by changing the variable name in Equation (4.37). When no indexes are present in a panel related variable, one is assuming that the quantity in question is represented for all the computational mesh, with all the indexes(i, j)unrolled, as stated by Equation (4.36). This nomenclature holds for both scalar quantities such as the panel area, or vectors (represented in bolt) such as position vectors, and it will be used from here forward.

Consider the input points from the wing’s discretization WP_1ij, WP_2ij, WP_3ij, WP_4ij, and the panel(i, j), depicted in Figure 4.8. Two parallel edges are formed by connecting the pointsWP_1ij with

Figure 4.8: Panel construction through a set of four non-coplanar nearest points.

WP2ij andWP3ij withWP4ij. Thus, four auxiliary vectors may be defined as

Pfij=WP2ij−WP1ij (4.38a)

P_sij =WP_3ij−WP_4ij (4.38b)

X_fij= 1

2 WP_1ij+WP_2ij

(4.39a) X_sij = 1

2 WP_3ij+WP_4ij

(4.39b) Using the definitions from Equation (4.38a) and Equation (4.38b), the first basis vector, which lies on the panel’s plane, can be defined as

l_ij= P_fij+P_sij

kPfij+Psijk (4.40)

In this manner, the panel’s corner points X1ij, X2ij, X3ij and X4ij are automatically defined as a function the vectorsXfij,Xsij,Pfij,Psij andlijas

X1,2ij=Xfij∓1

2 kPfijk.lij

(4.41a)

X3,4ij=Xsij±1

2 kPsijk.lij

(4.41b) where the minus sign corresponds to the assignment ofX_1ij andX_4ij, and the plus sign toX_2ij and X_3ij, in Equations (4.41a) and (4.41b).

Next, the unit normal of the panel is defined as nij= Nij

kNijk (4.42)

whereNijis a normal vector which is a function of the corner points N_ij= X_3ij−X_1ij

× X_4ij−X_2ij

(4.43)

As already said, a local frame of reference is required to measure the velocity field in each panel.

Two perpendicular unit vectors are already defined, thus, the last one may be determined based on the last two as

m_ij=n_ij×l_ij (4.44)

Finally, it is necessary to calculate the panel’s area and place the collocation point in the center of the panel. Both can easily be calculated as a function of the corner points as

CPij =1

4 X1ij+X2ij+X3ij+X4ij

(4.45) and

DS_ij= 1 2

hkX_Bij×X_Aijk+kX_Cij×X_Bijki

(4.46) where XAij = X2ij −X1ij, XBij = X3ij −X1ij, XCij = X4ij −X1ij. Additionally, CPij is the collocation point location andDSij is the panel’s area.