Sears–Haack body

The Sears–Haack body is the shape with the lowest theoretical wave drag in supersonic flow, for a slender solid body of revolution with a given body length and 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][2][3]

Sears–Haack body

The Kármán–Moore theory indicates that the wave drag scales as the square of the second derivative of the area distribution, ${\displaystyle D_{\text{wave}}\sim [S''(x)]^{2}}$ (see full expression below), so for low wave drag it is necessary that ${\displaystyle S(x)}$ 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

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

its volume is

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

its radius is

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

the derivative (slope) is

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

the second derivative is

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

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,
• ${\displaystyle R_{\text{max}}}$ is the radius at its maximum (occurs at x = 0.5, center of the shape),
• V is the volume,
• L is the length.

From Kármán–Moore theory, it follows that:

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

alternatively:

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

These formulae may be combined to get the following:

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

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

where:

• ${\displaystyle D_{\text{wave}}}$ is the wave drag force,
• ${\displaystyle C_{D_{\text{wave}}}}$ is the drag coefficient (normaled by the dynamic pressure and frontal area),
• ${\displaystyle \rho }$ is the density of the fluid,
• U is the velocity.

Derivation

According to Kármán–Moore theory, the wave drag force is given by

${\displaystyle F=-{\frac {\rho U^{2}}{2\pi }}\int _{0}^{l}\int _{0}^{l}S''(\xi _{1})S''(\xi _{2})\ln |\xi _{2}-\xi _{1}|d\xi _{1}d\xi _{2}}$

where ${\displaystyle S(x)}$ is the cross-sectional area of the body perpendicular to the body axis; here ${\displaystyle x=0}$ represents the leading edge and ${\displaystyle x=l}$ is the trailing edge, although the Kármán–Moore theory does not distinguish these ends because the drag coefficieint is independent of the direction of motion in the linear theory. Instead of ${\displaystyle S(x)}$, we can define the function ${\displaystyle f(x)=S'(x)}$ and expand it in series

${\displaystyle f=-l\sum _{n=2}^{\infty }A_{n}\sin n\theta ,\quad x={\frac {l}{2}}(1-\cos \theta )}$

where ${\displaystyle 0\leq \theta \leq \pi }$. The series starts from ${\displaystyle n=2}$ because of the condition ${\displaystyle S(0)=S(l)=0}$. We have

${\displaystyle S(x)=\int _{0}^{x}f(x)dx,\quad V=\int _{0}^{l}S(x)dx={\frac {\pi }{16}}l^{3}A_{2}.}$

Note that the volume of the body depends only on the coefficient ${\displaystyle A_{2}}$.

To calculate the drag force, first we shall rewrite the drag force formula, by integrating by parts once,

${\displaystyle F=\mathrm {p.v.} {\frac {\rho U^{2}}{2\pi }}\int _{0}^{l}\int _{0}^{l}f(\xi _{1})f'(\xi _{2}){\frac {d\xi _{1}d\xi _{2}}{\xi _{1}-\xi _{2}}}}$

in which ${\displaystyle \mathrm {p.v.} }$ stands for Cauchy principal value. Now we can substitute the expansion for ${\displaystyle f}$ and integrate the expression using the following two identities

${\displaystyle \mathrm {p.v.} \int _{0}^{\pi }{\frac {\cos n\theta _{2}}{\cos \theta _{2}-\cos \theta _{1}}}d\theta _{2}={\frac {\pi \sin n\theta _{1}}{\sin \theta _{1}}},\quad \int _{0}^{\pi }\sin n\theta _{1}\sin m\theta _{1}d\theta _{1}={\frac {\pi }{2}}{\begin{cases}1\,\,(m=n),\\0,\,\,(m\neq n).\end{cases}}}$

The final result, expressed in terms of the drag coefficient ${\displaystyle C_{d}=F/\rho U^{2}l^{2}/2}$, is simply given by^[4]

${\displaystyle C_{d}={\frac {\pi }{4}}\sum _{n=2}^{\infty }nA_{n}^{2}.}$

Since ${\displaystyle V}$ depends only on ${\displaystyle A_{2}}$, the minimum value of ${\displaystyle F}$ is reached when ${\displaystyle A_{n}=0}$ for ${\displaystyle n\geq 3}$.

Thus, setting ${\displaystyle A_{n}=0}$ for ${\displaystyle n\geq 3}$, we obtain ${\displaystyle S=(1/3)l^{3}A_{2}\sin ^{3}\theta }$,

${\displaystyle C_{d}={\frac {128}{\pi }}\left({\frac {V}{l^{3}}}\right)^{2}={\frac {9\pi }{2}}\left({\frac {S_{\mathrm {max} }}{l^{2}}}\right)^{2},\quad R(x)={\frac {8{\sqrt {2}}}{\pi }}\left({\frac {V}{3l^{4}}}\right)^{1/2}[x(l-x)]^{3/4},}$

where ${\displaystyle R(x)}$ is the radius as a function of ${\displaystyle x}$.

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 in NACA Report 1284.^[5] In this extension, the area ${\displaystyle S(x)}$ is defined on the Mach cone whose apex is at location ${\displaystyle x}$, rather than on the ${\displaystyle x={\text{constant}}}$ 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
• Lemon (geometry)

References

1. Haack, W. (1941). Geschossformen kleinsten wellenwiderstandes. Bericht der Lilienthal-Gesellschaft, 136(1), 14-28.
2. Sears, W. R. (1947). On projectiles of minimum wave drag. Quarterly of Applied Mathematics, 4(4), 361-366.
3. 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.
4. Landau, L. D., & Lifshitz, E. M. (2013). Fluid mechanics: Landau And Lifshitz: course of theoretical physics, Volume 6 (Vol. 6). Elsevier. page 473-474.
5. NACA Report 1284, Theory of Wing-Body Drag at Supersonic Speeds, by Robert T. Jones, 8 July 1953

External links

• Haack Minimum Drag Rifle Bullet Site down – https://web.archive.org/web/20160306044740/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