Nonlinear geometric decomposition of airfoils into the thickness and camber contributions

Abstract

In the thin airfoil theory, the camber line and the thickness distribution of general airfoils are mainly extracted by a linear combination of the upper and lower surfaces, giving rise to geometric distortions at the leading edge. Furthermore, despite the recent effort to obtain analytic expressions for the zero-lift angle of attack and quarter-chord moment coefficient, analytic generalizations are needed for the camber line component in the trigonometric series coefficients. In this sense, the present paper proposes a straightforward algorithm to extract the camber line and thickness distribution of general-shaped airfoils based on a finite difference method and the Bézier curve fitting. Integrals in the thin airfoil theory involving a Bernstein basis are performed, leading to series coefficients related to Gegenbauer polynomials. The algorithm is validated against analytical expressions of the NACA airfoils without introducing or adapting geometric parameters, and the results demonstrate good accuracy. In addition, the proposed algorithm indicated a significantly different geometric behavior for the SD7003 and E387 airfoils’ camber slope at the leading edge in contrast with the classical linear approximation. Moreover, the method can be coupled conveniently in recent unsteady aerodynamic models established on the thin airfoil theory to obtain closed-form expressions for general airfoils.

Appendices

Appendix

A. Thickness problem integrals

From Eqs. (18), (17) and (42) the induced velocity by the source sheet at $y=0$ has the form

$$\begin{aligned} u_i(x)= & {} \dfrac{\beta _0 U}{2\pi }\int _0^c \dfrac{dx_0}{\sqrt{x_0}(x-x_0)} + \dfrac{U}{\pi }\sum _{k=0}^{n-1} (k+1) \beta _{k+1} \nonumber \\{} & {} \quad \int _0^c \dfrac{x_0^k }{x-x_0}dx_0. \end{aligned}$$

(A1)

The first integral in the right-side hand in (A1) can be evaluated by the change of variable $x_0=u_0^2$:

$$\begin{aligned} \int _0^c \dfrac{dx_0}{\sqrt{x_0}(x-x_0)}&= 2\int _0^c \dfrac{du_0}{u^2-u_0^2} \nonumber \\&= \dfrac{1}{u}\int _0^c \dfrac{du_0}{u-u_0} + \dfrac{1}{u}\int _0^c \dfrac{du_0}{u+u_0}. \end{aligned}$$

(A2)

The formula given by Brandão [47] is helpful in performing principal value integrals as the one in Eq. A2:

$$\begin{aligned} \int _a^b \dfrac{f(x_0)}{x-x_0}dx_0 = f(x) \ln {\left( \dfrac{x-a}{b-x}\right) } - \int _a^b \dfrac{f(x)-f(x_0)}{x-x_0}dx_0. \end{aligned}$$

(A3)

Thus, Eq. A2 becomes:

$$\begin{aligned} \int _0^c \dfrac{dx_0}{\sqrt{x_0}(x-x_0)} = \dfrac{1}{\sqrt{x}}\ln {\left( \dfrac{\sqrt{c}+\sqrt{x}}{\sqrt{c}-\sqrt{x}}\right) }. \end{aligned}$$

(A4)

The second integral in Eq. A1 is computed by using the identity $x^k-x_0^k=(x-x_0)\sum _{m=1}^k x^{k-m}x_0^{m-1}$:

$$\begin{aligned} \int _0^c \dfrac{x_0^k }{x-x_0}dx_0&= x^k \ln {\left( \dfrac{x}{c-x}\right) } - \int _0^c \dfrac{x^k-x_0^k}{x-x_0}dx_0 \nonumber \\&= x^k\left[ \ln {\left( \dfrac{x}{c-x}\right) } - \sum _{m=1}^k \dfrac{(c/x)^m}{m}\right] . \end{aligned}$$

(A5)

B. Camber-slope integral evaluation

By making use of the identity $\cos \varphi = (1/2)\left( e^{i\varphi } + e^{-i\varphi }\right)$, $\varphi \in {\mathbb {C}}$, the integral (46) is evaluated as

$$\begin{aligned} I = -(-2)^{-(m+1)} \int _0^\pi \left( e^{i\varphi } - 2\varepsilon + e^{-i\varphi } \right) ^m \left( e^{in\varphi } + e^{-in\varphi }\right) d\varphi . \end{aligned}$$

(B6)

The change of variable $z=e^{i\varphi }$ yields

$$\begin{aligned} I = i(-2)^{-(m+1)} \int _{C} \left( z^2 - 2\varepsilon z + 1 \right) ^m \left( z^{n-m-1} + z^{-n-m-1}\right) dz, \end{aligned}$$

(B7)

where C is the positively oriented arc of the circle $|z|=1$, where $\mathfrak {arg}(z)$ goes from $\theta _i = 0$ to $\theta _f = \pi$.

One can expand the left term in parenthesis on the integrand in a Gegenbauer series:

$$\begin{aligned} \left( z^2 - 2 \varepsilon z + 1 \right) ^m = \sum _{j=0}^\infty z^j C_{(j)}^{(-m)}(\varepsilon ), \end{aligned}$$

(B8)

where $C^\lambda _\upsilon (x)$ are the Gegenbauer polynomials [48].

Replacing (B8) into Eq. B7 gives

$$\begin{aligned} I = i(-2)^{-(m+1)} \int _{C}f(z) dz, \end{aligned}$$

(B9)

where

$$\begin{aligned} f(z)=\sum _{j=0}^\infty \left( z^{n-m-1+j} + z^{-n-m-1+j}\right) C_{(j)}^{(-m)}(\varepsilon ). \end{aligned}$$

(B10)

By making use of the residue theory [49] and the fact that f(z) is a Laurent series with a singular point at $z=0$, we have

$$\begin{aligned} \underset{z=0}{\text {Res}}\, f(z) = C_{(m+n)}^{(-m)}(\varepsilon ) + C_{(m-n)}^{(-m)}(\varepsilon ). \end{aligned}$$

(B11)

The $N^\lambda _{\upsilon }(x)$ polynomials are introduced:

$$\begin{aligned} N^\lambda _{\upsilon }(x) = C_{(\lambda \pm \upsilon )}^{(-\lambda )}(x). \end{aligned}$$

(B12)

The integral in the complex plane may be evaluated as

$$\begin{aligned} \int _C f(z) dz = i(\theta _f - \theta _i)\sum \text {Res}\, f(z) = 2\pi i N^m_n(\varepsilon ). \end{aligned}$$

(B13)

In this way, (B13) is replaced into Eq. B9 to give

$$\begin{aligned} I = \pi (-2)^{-m}N^m_n(\varepsilon ). \end{aligned}$$

(B14)

C. Gegenbauer polynomials

The Gegenbauer series coefficients can be expressed as [50]

$$\begin{aligned} C_\upsilon ^\lambda (x)= & {} \left( {\begin{array}{c}\upsilon +2\lambda -1\\ \upsilon \end{array}}\right) \sum _{p=0}^\upsilon \left( {\begin{array}{c}\upsilon \\ p\end{array}}\right) \dfrac{(2\lambda + \upsilon )_p}{(\lambda +1/2)_p} \left( \dfrac{x-1}{2}\right) ^p, \nonumber \\{} & {} \quad \upsilon =0,1,2,\ldots , \quad x\in [-1,1], \end{aligned}$$

(C15)

where the Pochhammer symbol $(a)_n$ is defined by

$$\begin{aligned} (a)_n = \left\{ \begin{array}{ll} a(a+1)\cdots (a+n-1), &{} n=1,2,3,\ldots \\ 1, &{} n=0. \end{array} \right. \end{aligned}$$

(C16)

In special, we have $(1)_n=n!$.

Hence, the Gegenbauer coefficients in Eq. B12 may be written as

$$\begin{aligned} N^m_n(x)= & \, C_{(m\pm n)}^{(-m)}(x) \nonumber \\= & {} \left( {\begin{array}{c}-m\pm n-1\\ m\pm n\end{array}}\right) \sum _{p=0}^{m\pm n} \left( {\begin{array}{c}m\pm n\\ p\end{array}}\right) \dfrac{(-m\pm n)_p}{(-m+1/2)_p} \left( \dfrac{x-1}{2}\right) ^p. \end{aligned}$$

(C17)

The binomial coefficient for a negative integer k and integer n is given by [51]

$$\begin{aligned} \left( {\begin{array}{c}k\\ n\end{array}}\right) = \left\{ \begin{array}{ll} (-1)^n \left( {\begin{array}{c}-k+n-1\\ n\end{array}}\right) &{} (n\ge 0) \\ (-1)^{k-n} \left( {\begin{array}{c}-n-1\\ k-n\end{array}}\right) &{} (n\le k) \\ 0 &{} otherwise. \end{array} \right. \end{aligned}$$

(C18)

For complex $\lambda$ and integer $n\ge 0$, the binomial coefficient can be written as [50, 52]

$$\begin{aligned} \left( {\begin{array}{c}\lambda \\ n\end{array}}\right) = \dfrac{(\lambda - n + 1)_n}{(1)_n} = \dfrac{(-1)^n(-\lambda )_n}{n!}. \end{aligned}$$

(C19)

Furthermore, the following properties of the Pochhammer symbol are useful:

$$\begin{aligned} (\lambda )_{-n}&= \dfrac{(-1)^n}{(1-\lambda )}_n, \end{aligned}$$

(C20)

$$\begin{aligned} (\lambda )_{n+m}&= (\lambda )_{n}(\lambda +n)_{m}, \end{aligned}$$

(C21)

$$\begin{aligned} (\lambda )_{2n}&= 2^{2n}\left( \dfrac{\lambda }{2}\right) _n\left( \dfrac{1+\lambda }{2}\right) _n, \end{aligned}$$

(C22)

$$\begin{aligned} (-\lambda )_n&= \left\{ \begin{array}{ll} (-1)^n \dfrac{\lambda !}{(\lambda -n)!} &{} (0 \le n \le \lambda ) \\ 0 &{} (n > \lambda ). \end{array} \right. \end{aligned}$$

(C23)

The Pochhammer symbols in Eq. C17 need especial attention:

$$\begin{aligned} (-m\pm n)_p&\overset{(C23)}{=} \left\{ \begin{array}{ll} (-1)^p \dfrac{(m \mp n)!}{(m \mp n -p)!} \overset{(C19)}{=} (-1)^p p! \left( {\begin{array}{c}m\mp n\\ p\end{array}}\right) &{} (0 \le p \le m \mp n) \\ 0 &{} (p > m \mp n), \end{array} \right. \end{aligned}$$

(C24)

$$\begin{aligned} (-m + 1/2)_p&\overset{(23)}{=} \left\{ \begin{array}{ll} (-1)^p \dfrac{(m -1/2)!}{(m -1/2-p)!} \overset{(19)}{=} (-1)^p p! \left( {\begin{array}{c}m-1/2\\ p\end{array}}\right) &{} (0 \le p \le m -1/2) \\ 0 &{} (p > m -1/2). \end{array} \right. \end{aligned}$$

(C25)

In this way, for a non-zero value in Eq. C17 we must have $m \ge n$ and $p\le m-n$. Thus, the binomial coefficient exterior to the summation in Eq. C17 is

$$\begin{aligned} \left( {\begin{array}{c}-m\pm n-1\\ m\pm n\end{array}}\right)&\overset{(C19)}{=} (-1)^{m\pm n} \left( {\begin{array}{c}2m\\ m\pm n\end{array}}\right) \nonumber \\&= \dfrac{(-1)^{m + n} (2m)!}{(m + n)!(m - n)!}. \end{aligned}$$

(C26)

Then, the Gegenbauer coefficients in Eq. C17 can be written conveniently as

$$\begin{aligned} N^m_n(x) = \left\{ \begin{array}{ll} (-1)^{m + n} \left( {\begin{array}{c}2m\\ m + n\end{array}}\right) \sum _{p=0}^{m - n} \left( {\begin{array}{c}m + n\\ p\end{array}}\right) \left( {\begin{array}{c}m - n\\ p\end{array}}\right) \dfrac{p! (1-x)^p}{2^p(1/2-m)_p}, &{} 0\le n \le m; \\ 0, & \text{otherwise}. \end{array} \right. \end{aligned}$$

(C27)