Sears–Haack body

Sears–Haack body

The Sears–Haack body is the shape with the lowest theoretical wave drag in supersonic flow, for a given body length and given volume. The mathematical derivation assumes small-disturbance (linearized) supersonic flow, which is governed by the Prandtl-Glauert equation. The derivation and shape were published independently by two separate researchers: Wolfgang Haack in 1941 and later by William Sears in 1947.^[1]

The theory indicates that the wave drag scales as the square of the second derivative of the area distribution, $D_{{\text{wave}}}\sim [S''(x)]^{2}$ (see full expression below), so for low wave drag it's necessary that $S(x)$ be smooth. Thus, the Sears–Haack body is pointed at each end and grows smoothly to a maximum and then decreases smoothly toward the second point.

Useful Formulas

The cross sectional area of a Sears–Haack Body is:

$S(x)={\frac {16V}{3L\pi }}[4x(1-x)]^{{3/2}}=\pi R_{{max}}^{2}[4x(1-x)]^{{3/2}}$

The volume of a Sears–Haack Body is:

$V={\frac {3\pi ^{2}}{16}}R_{{max}}^{2}L$

The radius of a Sears–Haack Body is:

$r(x)=R_{{max}}[4x(1-x)]^{{3/4}}$

The derivative (slope) is:

$r'(x)=3R_{{max}}[4x(1-x)]^{{-1/4}}(1-2x)$

The second derivative is:

$r''(x)=-3R_{{max}}\{[4x(1-x)]^{{-5/4}}(1-2x)^{2}+2[4x(1-x)]^{{-1/4}}\}$

where:

• x is the ratio of the distance from the nose to the whole body length. This is always between 0 and 1.
• r is the local radius
• $R_{{max}}$ is the radius at its maximum (occurs at center of the shape)
• V is the volume
• L is the length
• $\rho$ is the density of the fluid
• U is the velocity

From Slender-body theory:

$D_{{\text{wave}}}=-{\frac {1}{4\pi }}\rho U^{2}\int _{0}^{\ell }\int _{0}^{\ell }S''(x_{1})S''(x_{2})\ln |x_{1}-x_{2}|{\mathrm {d}}x_{1}{\mathrm {d}}x_{2}$

alternatively:

$D_{{\text{wave}}}=-{\frac {1}{2\pi }}\rho U^{2}\int _{0}^{\ell }S''(x){\mathrm {d}}x\int _{0}^{x}S''(x_{1})\ln(x-x_{1}){\mathrm {d}}x_{1}$

These formulae may be combined to get the following:

$D_{{\text{wave}}}={\frac {64V^{2}}{\pi L^{4}}}\rho U^{2}={\frac {9\pi ^{3}R_{{max}}^{4}}{4L^{2}}}\rho U^{2}$

$C_{{D_{{\text{wave}}}}}={\frac {24V}{L^{3}}}={\frac {9\pi ^{2}R_{{max}}^{2}}{2L^{2}}}$

where:

• ${\displaystyle D_{\text{wave}}}$ is the wave drag

Generalization by R.T Jones

The Sears–Haack body shape derivation is correct only in the limit of a slender body. The theory has been generalized to slender but non-axisymmetric shapes by Robert T. Jones NACA Report 1284. In this extension, the area $S(x)$ is defined on the Mach cone whose apex is at location $x$, rather than on the $x={\text{constant}}$ plane as assumed by Sears and Haack. Hence, Jones's theory makes it applicable to more complex shapes like entire supersonic aircraft.

Area rule

A superficially related concept is the Whitcomb area rule, which states that wave drag due to volume in transonic flow depends primarily on the distribution of total cross-sectional area, and for low wave drag this distribution must be smooth. A common misconception is that the Sears–Haack body has the ideal area distribution according to the area rule, but this is not correct. The Prandtl-Glauert equation which is the starting point in the Sears–Haack body shape derivation is not valid in transonic flow, which is where the Area rule applies.

See also

• Prandtl–Glauert transformation
• Aerodynamics
• Anti-shock body
• Haack series nose cone
• Area rule

References

1. Palaniappan, Karthik (2004). Bodies having Minimum Pressure Drag in Supersonic Flow – Investigating Nonlinear Effects (PDF). 22nd Applied Aerodynamics Conference and Exhibit. Antony Jameson. Retrieved 2010-09-16.

External links

• Haack Minimum Drag Rifle Bullet Site down – http://web.archive.org/web/*/http://www.lima-wiederladetechnik.de/Englisch/Haack_minimum_drag_bullet.htm
• Geschoßformen kleinsten Wellenwiderstandes by W. Haack, Bericht 139 der Lilienthal-Gesellschaft (1941)
• Sears–Haack body calculator