Missile Performance

The missile performance equations were taken from a http://www.dbf.gatech.edu/performance.ppt The equations estimate the missile's normal force and parasite drag (tangential force) coefficients. The coefficients have been non-dimensionalized with respect to the reference area based on the missile diameter.

Parasite Drag Coefficient

$CD_{0, body, wave} = \left( 1.59 + \frac{1.83}{M^2} \right) \left\{ tan^{-1} \left[ \frac{0.5}{\left( \frac{l_N}{d} \right)} \right] \right\}^{1.69}$ for M > 1, $tan^{-1}$ in rad

$CD_{0, base, coast} = \frac{0.25}{M}$ if M > 1, and

$CD_{0, base, coast} = 0.12 + 0.13 M^2$ if M < 1

$CD_{0, base, powered} = \left( 1 – \frac{A_e}{S_{ref}} \right) \left( \frac{0.25}{M} \right)$ if M > 1, and

$CD_{0, base, powered} = \left( 1 – \frac{A_e}{S_{ref}} \right) \left( 0.12 + 0.13 M^2 \right)$ if M < 1

$CD_{0, body, friction} = 0.053 \left( \frac{l}{d} \right) \left[ \frac{M}{ql} \right]^{0.2}$ q in psf, l in ft.

$CD_{0, body} = CD_{0, body, wave} + CD_{0, base} + CD_{0, body, friction}$

Note: $CD_{0, body, wave}$ = body zero-lift wave drag coefficient, $CD_{0, base}$ = body base drag coefficient, $CD_{0, body, friction}$ = body skin friction drag coefficient, $CD_{0, body}$ = body zero-lift drag coefficient, $ l_N $ = nose length, d = missile diameter, l = missile body length, $ A_e $ = nozzle exit area, $S_{ref}$ = reference area, q = dynamic pressure

$CD_{0, wing, friction} = n_W \left\{ 0.0133 \left[ \frac{M}{q c_{mac}} \right]^{0.2} \right\} \left( \frac{2 S_{wet}}{S_{ref}} \right)$ q in psf, cmac in ft

$CD_{0, wing, wave} = n_W \left[ \frac{2}{\gamma M_{LE}^2} \right] \left\{ \left\{ \frac{( \gamma + 1 ) M_{LE}^2 }{2} \right\} ^{\frac{\gamma}{\gamma - 1}} \left\{\frac{\gamma + 1}{2 \gamma M_{LE}^2 – (\gamma - 1)}\right\}^{\frac{1}{\gamma - 1}} – 1 \right\} sin^2 \delta_{LE} cos \Delta_{LE} t_{mac} \frac{b}{S_{ref}}$ based on Newtonian impact theory.

$CD_{0, wing} = CD_{0, wing, wave} + CD_{0, wing, friction}$

$ n_W $ = number of wings ( cruciform = 2 ), q = dynamic pressure in psf, $c_{mac}$ = length of mean aero chord in ft, $\lambda$ = Specific heat ratio = 1.4, $M_{LE} = M cos \Delta_{LE}$ = Mach number perpendicular to the leading edge, $\delta_{LE}$ = leading edge section total angle, $\Delta_{LE}$ = leading edge sweep angle, $t_{mac}$ = max thickness at mean aerodynamic chord, b = span

Normal Force Coefficient

$|C_{N, body}| = \left[|\frac{a}{b} cos \phi + \frac{b}{a} sin \phi |\right] \left[| sin ( 2\alpha ) cos ( \frac{\alpha}{2} ) | + 2 \frac{l}{d} sin^2 \alpha \right]$

$|C_{N, wing}| = \left[ \frac{4|sin \alpha’ cos \alpha’|}{(M^2 – 1)^{\frac{1}{2}}} + 2 sin^2 \alpha’ \right] \left( \frac{S_w}{S_{ref}} \right)$ , if $M > \left\{ 1 + \left[ \frac{8}{ \pi A } \right]^2 \right\}^{\frac{1}{2}}$

$|C_{N, wing}| = \left[ \frac{\pi A}{2}|sin \alpha’ cos \alpha’| + 2 sin^2 \alpha’ \right] \left( \frac{S_w}{S_{ref}} \right)$ , if $M < \left\{ 1 + \left[ \frac{8}{ \pi A } \right]^2 \right\}^{\frac{1}{2}}$

Note: Linear wing theory applicable if $M > \left\{ 1 + \left[ \frac{8}{ \pi A } \right]^2 \right\}^{\frac{1}{2}}$ , slender wing theory applicable if $M < \left\{ 1 + \left[ \frac{8}{ \pi A } \right]^2 \right\}^{\frac{1}{2}}$ , A = Aspect Ratio, $S_{w}$ = Wing Planform Area, $S_{ref}$ = Reference Area