1. Introduction and Background

2. Methodology

2.2. Class Shape Transformation (CST) Parametrization Technique

CST (Class-Shape Transformation) or Kulfan airfoils constitute a group of airfoil profiles characterized by a parameterized representation of their upper and lower surfaces. The CST approach is a versatile parameterization strategy for crafting both two-dimensional and three-dimensional forms. Utilizing a class function alongside a shape function, the CST method encapsulates an airfoil’s form, ensuring a closed trailing edge. Introduced by Kulfan [8], this parameterization technique delineates a two-dimensional figure through the integration of a class function C($\overline{x}$) and a shape function S($\overline{x}$), augmented by a component that delineates the trailing edge’s thickness. The construction of an airfoil via the CST method involves aggregating the contributions from its foundational function, crafted using Bernstein polynomials. For initial airfoil design and optimization, CST employing lower-order polynomials is preferred due to its minimal parameter requirement for defining specific airfoil shapes. Although intricate, CST can be adapted to a wide array of aerodynamic body shapes without necessitating a predefined geometry. This methodology offers precise control over critical features such as the leading-edge radius, boat-tail angle, and trailing edge thickness, making it applicable in both aerodynamic studies and optimization endeavors.

2.2.1. CST Formulation

The general form of the mathematical expression that represents a typical airfoil geometry is:

$$\zeta \left(\psi \right)=\sqrt{\psi }(1-\psi )\sum _{i=0}^N{A}_i{\psi }^i+\psi {\zeta }_T$$

(1)

where $\psi =x/c$, $\zeta =z/c$ and ${\zeta }_T=\Delta Z_{TE}/c$. The term $\sqrt{\psi }$ corresponds to the round nose of the airfoil. Similarly, the term $(1-\psi )$ corresponds to the sharp trailing edge of the airfoil. The term $\psi {\zeta }_T$ controls the thickness of the trailing edge, and the summation term is a general function that describes the shape of the airfoil between the round nose and the sharp trailing edge.

The CST parameters are given as follows (Adapted from [12]),

Figure 1. CST Parameters

Figure 1. CST Parameters

2.2.2. Airfoil Shape and Class Function

The shape function $S\left(\psi \right)$ is given by the relation,

$$S\left(\psi \right)=\sum _{i=0}^N\left[A_i{\psi }^i\right]$$

(2)

The term $\psi [1-\psi ]$ defines the class function $C\left(\psi \right)$ given by,

$$C_{N2}^{N1}\left(\psi \right)\triangleq {\left(\psi \right)}^{N1}{[1-\psi ]}^{N2}$$

(3)

Typically, the parameters N1 and N2 are set to 0.5 and 1 for an airfoil with a rounded nose. The class function delineates broad categories of geometric forms, while the shape function specifies the exact contours within a particular geometric category.

Defining an airfoil shape function and specifying its geometry class is equivalent to defining the actual airfoil coordinates, which can then be obtained from the combination of the shape and class functions from the relation,

$$\zeta \left(\psi \right)=C_{N2}^{N1}\left(\psi \right)S\left(\psi \right)+\psi {\zeta }_T$$

(4)

The airfoil is then defined using the Bernstein polynomials, which are defined for the order n as follows,

$$S_{r,n}\left(x\right)=K_{r,n}{x}^r{(1-x)}^{n-r}$$

(5)

The binomial coefficients are defined as,

$$K_{r,n}=\left(\begin{array}{c}n\\ r\end{array}\right)=\frac{n!}{r!(n-r)!}$$

(6)

The first term of the polynomial defines the leading-edge radius, and the last term is the boat-tail angle. The terms that fall in between serve as the "shaping terms". Here, a Bernstein polynomial of the eighth order is employed, resulting in eight weighted coefficients upon completion of the CST process. The upper surface of the airfoil is defined by the equations,

$${\left(\zeta \right)}_{upper}=C_{N2}^{N1}\left(\psi \right)Sl\left(\psi \right)+\psi \Delta {\xi }_{upper}$$

(7)

$$Su\left(\psi \right)=\sum _{i=1}^{N}A{u}_i{S}_i\left(\psi \right)$$

(8)

The lower surface of the airfoil is defined by the equations,

$${\left(\zeta \right)}_{lower}=C_{N2}^{N1}\left(\psi \right)Sl\left(\psi \right)+\psi \Delta {\xi }_{lower}$$

(9)

$$Sl\left(\psi \right)=\sum _{i=1}^{N}A{l}_i{S}_i\left(\psi \right)$$

(10)

where,

$$\Delta {\xi }_U=\frac{z{u}_{TE}}{C}and\Delta {\xi }_L=\frac{z{l}_{TE}}{C}$$

(11)

The selection of an eighth order Bernstein polynomial was justified by its drag predictions and pressure distribution which perfectly matched experimental observations.

2.2.3. Key Observations

Throughout the process of airfoil parameterization, several key findings emerged that deserve attention. Notably, when the leading edge point deviated from the origin (0,0) with a y-offset, the airfoils generated through the CST method exhibited a marked increase in error percentage. This deviation significantly altered the airfoil’s curvature, rendering it beyond the capture capability of the ’blockMesh’ utility. Additionally, a significant portion of the airfoils had their trailing edges positioned away from (1,0), challenging the meshing process, which necessitates a trailing edge at (1,0). Consequently, airfoils not meeting this criterion were excluded from the preliminary airfoil database. The comparison between the original and the derived airfoils showed only a minimal error margin. This approach was developed utilizing MATLAB^® building on the work of Pramudita Satria Palar [13].

2.2.4. Shape Generation

Figure 2 illustrates several different airfoil sections created through the CST framework.

3. Data Description

4. Data validation

5. Conclusions

References

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