Thin-Airfoil Theory

The shock-expansion theory of the previous section provides a simple and general method for computing the lift and drag on a supersonic airfoil, and is applicable as long as the flow is not compressed to subsonic speeds, and the shock waves remain attached to the airfoil. However, the results of this theory cannot generally be expressed in concise analytic form. In fact, the theory is mostly used to obtain numerical solutions. However, if the airfoil is thin, and the angle of attach small, then the shocks and expansion fans attached to the airfoil become weak. In this situation, shock-expansion theory can be considerably simplified by using approximate expressions for weak shocks and expansion fans.

The basic approximate expression [cf. Equation (15.19)]

$$\frac{{\mit\Delta}p}{p}\simeq \left(\frac{\gamma\,{\rm Ma}^{\,2}}{\sqrt{{\rm Ma}^{\,2}-1}}\right){\mit\Delta}\theta,$$ (15.46)

specifies the relative change in pressure across either a weak oblique shock (see Section 15.4) or a weak expansion fan (see Exercise vi) that deflects flow of Mach number ${\rm Ma}$ through an angle ${\mit \Delta }\theta$ . Because, in the weak wave approximation, the pressure, $p$ , never greatly differs from the upstream pressure, $p_1$ , and the Mach number, ${\rm Ma}$ , never differs appreciably from the upstream Mach number, ${\rm Ma}_1$ , we can write

$$\frac{p-p_1}{p_1}\simeq \left(\frac{\gamma\,{\rm Ma}_1^{\,2}}{\sqrt{{\rm Ma}_1^{\,2}-1}}\right)\theta,$$ (15.47)

which is correct to first order in $\theta$ . Here, $\theta$ is the deflection angle relative to the upstream flow.

It is convenient to define a dimensionless quantity known as the pressure coefficient:

$$C_p =\frac{p-p_1}{q_1},$$ (15.48)

where $q_1=(1/2)\,\rho_1\,w_1^{\,2}$ , and $\rho_1$ and $w_1$ are the upstream density and flow speed, respectively. Given that the upstream sound speed is $c_1=\sqrt{\gamma\,p_1/\rho_1}$ , and ${\rm Ma}_1=w_1/c_1$ , we obtain

$$C_p =\frac{2}{\gamma\,{\rm Ma}_1^{\,2}}\,\frac{p-p_1}{p_1},$$ (15.49)

which yields

$$C_p =\frac{2\,\theta}{\sqrt{{\rm Ma}_1^{\,2}-1}}.$$ (15.50)

This is the fundamental formula of thin-airfoil theory. It states that the pressure coefficient is proportional to the local deflection of the flow from the upstream direction.

Consider the flat-plate airfoil shown in Figure 15.10. The deflection angle is $-\alpha_0$ on the upper surface of the airfoil, and $+\alpha_0$ on the lower surface. (A positive deflection angle corresponds to compression, and a negative deflection angle to expansion.) Thus, the pressure coefficients on the upper and lower surfaces are

$$C_{pU} = -\frac{2\,\alpha_0}{\sqrt{{\rm Ma}_1^{\,2}-1}}$$ (15.51)

and

$$C_{pL} = \frac{2\,\alpha_0}{\sqrt{{\rm Ma}_1^{\,2}-1}},$$ (15.52)

respectively. It is convenient to define the dimensionless coefficient of lift,

$$C_L = \frac{L}{q_1\,c},$$ (15.53)

and the dimensionless coefficient of drag,

$$C_D= \frac{D}{q_1\,c}.$$ (15.54)

It follows that

$$C_L = \frac{(p_L-p_U)\,c\,\cos \alpha_0}{q_1\,c}= (C_{pL}-C_{pU})\,\cos\alpha_0,$$ (15.55)

$$C_D = \frac{(p_L-p_U)\,c\,\sin \alpha_0}{q_1\,c}= (C_{pL}-C_{pU})\,\sin\alpha_0.$$ (15.56)

Making use of Equations (15.51) and (15.52), as well as the conventional small angle approximations $\cos\alpha_0\simeq 1$ and $\sin\alpha_0\simeq \alpha_0$ , we obtain

$$C_L = \frac{4\,\alpha_0}{\sqrt{{\rm Ma}_1^{\,2}-1}},$$ (15.57)

$$C_D = \frac{4\,\alpha_0^{\,2}}{\sqrt{{\rm Ma}_1^{\,2}-1}}.$$ (15.58)

The focus (see Section 9.3) of the airfoil is at the midchord. Moreover, the ratio $D/L^{\,2}=({\rm Ma}_1^{\,2}-1)^{1/2}/4$ is independent of $\alpha _0$ .

Figure: A diamond-section airfoil. ${\rm Ma}_1$ is the upstream Mach number. $p_2$ and $p_3$ denote pressures.

As a second example, consider the diamond-section airfoil pictured in Figure 15.11. This airfoil has a nose angle $2\,\epsilon$ , and zero angle of attack. The pressure coefficient on the front face of the airfoil is

$$C_{pF} = \frac{2\,\epsilon}{\sqrt{{\rm Ma}_1^{\,2}-1}},$$ (15.59)

whereas that on the rear face is

$$C_{pR} =- \frac{2\,\epsilon}{\sqrt{{\rm Ma}_1^{\,2}-1}}.$$ (15.60)

It follows that the pressure difference is

$$p_2-p_3 = \frac{4\,\epsilon}{\sqrt{{\rm Ma}_1^{\,2}-1}}\,q_1,$$ (15.61)

giving a drag

$$D = (p_2-p_3)\,t = (p_2-p_3)\,\epsilon\,c= \frac{4\,\epsilon^{\,2}}{\sqrt{{\rm Ma}_1^{\,2}-1}}\,q_1\,c,$$ (15.62)

where $t$ and $c$ are the thickness and chord-length of the airfoil, respectively. (See Figure 15.11.) Thus,

$$C_D = \frac{D}{q_1\,c}= \frac{4\,\epsilon^{\,2}}{\sqrt{{\rm Ma}_1^{\,2}-1}}=\frac{4}{\sqrt{{\rm Ma}_1^{\,2}-1}}\left(\frac{t}{c}\right)^2.$$ (15.63)

Figure 15.12: An arbitrary airfoil.

Figure 15.12 shows the cross-section of an arbitrary airfoil. The cross-section is assumed to be uniform in the $z$ -direction, with the upstream flow parallel to the $x$ -axis. The upper surface of the airfoil corresponds to the curve $y=y_U(x)$ , the lower surface to the curve $y=y_L(x)$ , and the camber line (i.e., the centerline) to the curve $y=y_C(x)$ . Furthermore, the leading and trailing edges of the airfoil lie at $x=x_L$ and $x=x_R$ , respectively. Hence, $y_U(x_L)=y_C(x_L)=y_L(x_L)$ and $y_U(x_R)=y_C(x_R)=y_L(x_R)$ . By definition,

$$y_C(x)=\frac{y_U(x)+y_L(x)}{2},$$ (15.64)

for $x_L\leq x\leq x_R$ . It is helpful to define the half-width of the airfoil,

$$h(x)= \frac{y_U(x)-y_L(x)}{2},$$ (15.65)

for $x_L\leq x\leq x_R$ . Note that $h(x_L)=h(x_R)=0$ . We can also define the mean angle of attack of the airfoil:

$$\alpha_0 = \frac{y_U(x_L)-y_U(x_R)}{x_R-x_L}= \frac{y_L(x_L)-y_L(x_R)}{x_R-x_L}=\frac{y_C(x_L)-y_C(x_R)}{x_R-x_L}.$$ (15.66)

Thus, we can write

$$y_C(x) = y_C(x_L) -\alpha_0\,(x-x_L) -\alpha_C(x),$$ (15.67)

where $\alpha_C(x_L)=\alpha_C(x_R)=0$ . Here,

$$\alpha(x)= \alpha_0+\alpha_C(x)$$ (15.68)

is the local angle of attack of the camber line. Thus, the camber function, $\alpha_C(x)$ , parameterizes the deviations of $\alpha(x)$ from $\alpha _0$ across the width of the airfoil. Equations (15.64), (15.65), and (15.67) yield

$$y_U(x) = y_C(x)+ h(x) = y_C(x_L) -\alpha_0\,(x-x_L) -\alpha_C(x)+h(x),$$ (15.69)

$$y_L(x) =y_C(x)- h(x) = y_C(x_L) -\alpha_0\,(x-x_L) -\alpha_C(x)-h(x).$$ (15.70)

Hence, the airfoil shape is completely specified by the thickness function, $h(x)$ , the camber function, $\alpha_C(x)$ , and the mean angle of attack, $\alpha _0$ .

The pressure coefficients on the upper and lower surfaces of the airfoil are [see Equations (15.50)]

$$C_{pU} = \frac{2}{\sqrt{{\rm Ma}_1^{\,2}-1}}\,\frac{dy_U}{dx},$$ (15.71)

$$C_{pL} =\frac{2}{\sqrt{{\rm Ma}_1^{\,2}-1}}\left(-\frac{dy_L}{dx}\right),$$ (15.72)

respectively. It follows from Equations (15.68)-(15.70) that

$$\frac{dy_U}{dx} = -\alpha(x)+\frac{dh}{dx},$$ (15.73)

$$\frac{dy_L}{dx} = -\alpha(x)-\frac{dh}{dx}.$$ (15.74)

Now, the lift and drag per unit transverse length acting on the airfoil are given by [cf., Equations (15.53)-(15.56)]

$$L = q_1\int_{x_L}^{x_R} \left(C_{pL}-C_{pU}\right)dx,$$ (15.75)

$$D = q_1\int_{x_L}^{x_R} \left[C_{pL}\left(-\frac{dy_L}{dx}\right)+C_{pU}\left(\frac{dy_U}{dy}\right)\right]dx.$$ (15.76)

Thus, it follows from Equations (15.71)-(15.74) that

$$D = \frac{4\,q_1}{\sqrt{{\rm Ma}_1^{\,2}-1}}\int_{x_L}^{x_R} \alpha(x)\,dx,$$ (15.77)

$$L = \frac{2\,q_1}{\sqrt{{\rm Ma}_1^{\,2}-1}}\int_{x_L}^{x_R}\left[\left(\frac{dy_L}{dx}\right)^2+\left(\frac{dy_U}{dx}\right)^2\right]dx$$

$$= \frac{4\,q_1}{\sqrt{{\rm Ma}_1^{\,2}-1}}\int_{x_L}^{x_R}\left[\left(\frac{dh}{dx}\right)^2+\alpha^{\,2}\right]dx.$$ (15.78)

It is helpful to define the chord-average operator:

$$\overline{y} \equiv \frac{1}{c}\int_{x_L}^{x_R} y(x)\,dx,$$ (15.79)

where $c=x_R-x_L$ is the chord-length. Taking the average of Equation (15.73), making use of Equation (15.66), as well as the fact that $h(x_L)=h(x_R)=0$ , we obtain

$$\overline{\alpha}= \alpha_0.$$ (15.80)

However, the average of Equation (15.68) yields

$$\overline{\alpha} = \alpha_0 + \overline{\alpha_C},$$ (15.81)

which implies that $\overline{\alpha_C}=0$ . We can also write

$$\overline{\alpha^{\,2}} = \overline{(\alpha_0+\alpha_C)^2}=\overl... ...lpha_C}+\overline{\alpha_C^{\,2}} = \alpha_0^{\,2} + \overline{\alpha_C^{\,2}}.$$ (15.82)

Hence, the coefficients of lift and drag, $L/(q_1\,c)$ and $D/(q_1\,c)$ , are written

$$C_L = \frac{4\,\overline{\alpha}}{\sqrt{{\rm Ma}_1^{\,2}-1}}=\frac{4\,\alpha_0}{\sqrt{{\rm Ma}_1^{\,2}-1}},$$ (15.83)

$$C_D = \frac{4}{\sqrt{{\rm Ma}_1^{\,2}-1}}\left[\,\overline{\left(\frac{dh}{dx}\right)^2}+\overline{\alpha^{\,2}}\right]$$

$$=\frac{4}{\sqrt{{\rm Ma}_1^{\,2}-1}}\left[\, \overline{\left(\frac{dh}{dx}\right)^2}+\alpha_0^{\,2}+\overline{\alpha_C^{\,2}}\right],$$ (15.84)

respectively. Thus, in thin-airfoil theory, the lift only depends on the mean angle of attack, whereas the drag splits into three components. Namely, a drag due to thickness, a drag due to lift, and a drag due to camber.