Determine the lift and drag coefficients of an infinitely-thin but mildly-cambered airfoil at zero angle of attack in a horizontal supersonic flow at speed M∞.

Question

Determine the lift and drag coefficients of an infinitely-thin but mildly-cambered airfoil at zero angle of attack in a horizontal supersonic flow at speed [latex]M_\infty[/latex].

Accepted Answer

Figure 14.13 shows an infinitely thin cambered foil at zero angle of attack. Use the x-y coordinates shown there, let the camber line of the foil be [latex]y_c(x)[/latex], and assume the foil extends from x = −2b to x = +2b. Here, zero angle of attack implies: [latex]y_c(\pm2b)=0[/latex]. To compute the lift coefficient, [latex]C_L[/latex], first determine the pressure coefficient starting from is definition, then eliminate the pressure difference using (15.68):

[latex]\frac{p-p_{\infty}}{p_{\infty}}=\frac{\gamma M_{\infty}^2 \delta}{\sqrt{M_{\infty}^2-1}}[/latex], (15.68)

[latex]C_p=\frac{p-p_{\infty}}{(1 / 2) \rho_{\infty} U_{\infty}^2}=\frac{\gamma M_{\infty}^2 \delta}{\sqrt{M_{\infty}^2-1}}\frac{p_{\infty}}{(1 / 2) \rho_{\infty} U_{\infty}^2}=\frac{\gamma M_{\infty}^2 \delta}{\sqrt{M_{\infty}^2-1}} \frac{2}{\gamma M_{\infty}^2}=\frac{2 \delta}{\sqrt{M_{\infty}^2-1}}[/latex].

For small flow deflection angles: δ ≈ sinδ ≈ tanδ = [latex]dy_c/dx[/latex] and cosδ ≈ 1. Thus:

[latex]C_L=\frac{1}{4 b}\overset{+2 b}{\underset{-2 b}{\int}}\left(C_{p,l}-C_{p, u}\right) \cos \delta d x \cong \frac{1}{4 b}\overset{+2 b}{\underset{-2 b}{\int}}\left(\frac{-2}{\sqrt{M_{\infty}^2-1}}-\frac{2}{\sqrt{M_{\infty}^2-1}}\right)\left(\frac{d y_c}{d x}\right) d x[/latex]

[latex]=\frac{-1}{b \sqrt{M_{\infty}^2-1}}\left[y_c\right]_{-2 b}^{+2 b}=0[/latex],

where the extra ’l’ and ’u’ subscripts indicate the lower and upper foil surfaces, and δ is measured from the horizontal and is positive when it leads to flow compression. So, a cambered foil at zero angle of attack does not produce lift in a supersonic flow.
To determine the coefficient of drag, the local component of the pressure force in the direction of flow must be considered:

[latex]C_D=\frac{1}{4 b}\overset{+2 b}{\underset{-2 b}{\int}}\left(C_{p, u}-C_{p, l}\right) \sin \delta d x \cong \frac{1}{4 b}\overset{+2 b}{\underset{-2 b}{\int}}\left(\frac{2}{\sqrt{M_{\infty}^2-1}}-\frac{-2}{\sqrt{M_{\infty}^2-1}}\right)\left(\frac{d y_c}{d x}\right)^2 d x=\frac{4}{\sqrt{M_{\infty}^2-1}} \overline{\left(\frac{d y_c}{d x}\right)^2}[/latex],

where the overbar on the last term implies a spatial average over the chord length. This result suggests that airfoil camber only leads to drag in supersonic flow. Thus, the wings and control surfaces of supersonic aircraft and missiles are nearly flat.

Question 15.9: Determine the lift and drag coefficients of an infinitely-th...

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