Nose cone design

Because of the problem of the aerodynamic design of the nose cone section of any vehicle or body meant to travel through a compressible fluid medium (such as a rocket or aircraft, missile, shell or bullet), an important problem is the determination of the nose cone geometrical shape for optimum performance. For many applications, such a task requires the definition of a solid of revolution shape that experiences minimal resistance to rapid motion through such a fluid medium.

In all of the following nose cone shape equations, L is the overall length of the nose cone and R is the radius of the base of the nose cone. y is the radius at any point x, as x varies from 0, at the tip of the nose cone, to L. The equations define the two-dimensional profile of the nose shape. The full body of revolution of the nose cone is formed by rotating the profile around the centerline C⁄L. While the equations describe the "perfect" shape, practical nose cones are often blunted or truncated for manufacturing, aerodynamic, or thermodynamic reasons.^[1][2]

$${\displaystyle {\begin{aligned}y&={\frac {xR}{L}}=x\tan(\phi )\\\phi &=\arctan \left({\frac {R}{L}}\right)\\\end{aligned}}}$$

$${\displaystyle {\begin{aligned}x_{t}&&&={\frac {r_{n}L^{2}}{R}}{\sqrt {\frac {1}{R^{2}+L^{2}}}}\\y_{t}&={\frac {x_{t}R}{L}}&&=r_{n}L{\sqrt {\frac {1}{R^{2}+L^{2}}}}\\x_{o}&=x_{t}+{\sqrt {r_{n}^{2}-y_{t}^{2}}}&&=r_{n}\left({\frac {L^{2}}{R}}{\sqrt {\frac {1}{R^{2}+L^{2}}}}+{\sqrt {1-{\frac {L^{2}}{R^{2}+L^{2}}}}}\right)\\x_{a}&=x_{o}-r_{n}&&=r_{n}\left({\frac {L^{2}}{R}}{\sqrt {\frac {1}{R^{2}+L^{2}}}}+{\sqrt {1-{\frac {L^{2}}{R^{2}+L^{2}}}}}-1\right)\\\end{aligned}}}$$

$${\displaystyle {\begin{aligned}L&=L_{1}+L_{2}\\\phi _{1}&=\arctan \left({\frac {R_{1}}{L_{1}}}\right)\\\phi _{2}&=\arctan \left({\frac {R_{2}-R_{1}}{L_{2}}}\right)\\y&={\begin{cases}{\frac {xR_{1}}{L_{1}}}=x\tan(\phi _{1}),&0\leq x\leq L_{1}\\R_{1}+{\frac {(x-L_{1})(R_{2}-R_{1})}{L_{2}}}=R_{1}+(x-L_{1})\tan(\phi _{2}),&L_{1}\leq x\leq L\end{cases}}\end{aligned}}}$$

$${\displaystyle {\begin{aligned}\rho &={\frac {R-{\frac {L^{2}}{R}}}{2}}\\y&={\sqrt {\rho ^{2}+x(2L-x)}}+\rho \\\end{aligned}}}$$

$${\displaystyle {\begin{aligned}x_{o}&=L-{\sqrt {\left(R-r_{n}\right)\left({\frac {L^{2}}{R}}-r_{n}\right)}}\\y_{t}&={\frac {r_{n}\left(L^{2}-R^{2}\right)}{L^{2}+R^{2}-2Rr_{n}}}\\x_{t}&=L-{\sqrt {\left(R-r_{n}\right)\left({\frac {L^{2}}{R}}-r_{n}\right)}}-r_{n}{\sqrt {1-\left({\frac {L^{2}-R^{2}}{L^{2}+R^{2}-2Rr_{n}}}\right)^{2}}}\\\end{aligned}}}$$

For a chosen ogive radius ρ greater than or equal to the ogive radius of a tangent ogive with the same R and L:

$${\displaystyle {\begin{aligned}\rho &\geq {\frac {R+{\frac {L^{2}}{R}}}{2}}\\\alpha &=\arctan \left({\frac {R}{L}}\right)-\arccos \left({\frac {\sqrt {R^{2}+L^{2}}}{2\rho }}\right)\\y&={\sqrt {\rho ^{2}-(x-\rho \cos \alpha )^{2}}}+\rho \sin(\alpha ),&&0\leq x\leq L\\\end{aligned}}}$$

A smaller ogive radius can be chosen; for ${\textstyle {\frac {1}{2}}\left(R+{\frac {L^{2}}{R}}\right)>\rho >{\frac {L}{2}}}$, you will get the shape shown on the right, where the ogive has a "bulge" on top, i.e. it has more than one x that results in some values of y.

$${\displaystyle y={\frac {R{\sqrt {x(2L-x)}}}{L}}}$$

A parabolic series nosecone is defined by ${\displaystyle r={\tfrac {2x-Kx^{2}}{2-K}}}$ where ${\displaystyle 0\leq x\leq 1}$ and ${\displaystyle K}$ is a series-specific constant.^[3]

For ${\displaystyle 0\leq K'\leq 1}$, ${\displaystyle y=R\left({2\left({x \over L}\right)-K'\left({x \over L}\right)^{2} \over 2-K'}\right)}$

K′ can vary anywhere between 0 and 1, but the most common values used for nose cone shapes are:

A power series nosecone is defined by ${\displaystyle r=x^{n}}$ where ${\displaystyle (0\leq x\leq 1)}$. ${\displaystyle n<1}$ will generate a concave geometry, while ${\displaystyle n>1}$ will generate a convex (or "flared") shape.^[3]

For ${\displaystyle 0\leq n\leq 1}$: ${\displaystyle y=R\left({x \over L}\right)^{n}}$

Common values of n include:

A Haack series nosecone is defined by ${\displaystyle r(x)={\sqrt {\frac {\theta -{\frac {1}{2}}\sin(2\theta )+C\sin ^{3}\theta }{\pi }}}}$ where ${\displaystyle \theta =\arccos \!\left(1-{\frac {2x}{L}}\right)}$.^[3] Parametric formulation can be obtained by solving the ${\displaystyle \theta }$ formula for ${\displaystyle x}$.

$${\displaystyle {\begin{aligned}x(\theta )&={L \over 2}\left(1-\cos(\theta )\right)\\y(\theta ,C)&={R \over {\sqrt {\pi }}}{\sqrt {\theta -{\sin(2\theta ) \over 2}+C\sin ^{3}(\theta )}}\\0&\leq \theta \leq \pi \end{aligned}}}$$

Special values of C (as described above) include:

The LD-Haack ogive is a special case of the Haack series with minimal drag for a given length and diameter, and is defined as a Haack series with C = 0, commonly called the Von Kármán or Von Kármán ogive. A cone with minimal drag for a given length and volume can be called an LV-Haack series, defined with ${\displaystyle C={\tfrac {1}{3}}}$.^[3]

An aerospike can be used to reduce the forebody pressure acting on supersonic aircraft. The aerospike creates a detached shock ahead of the body, thus reducing the drag acting on the aircraft.

• Index of aviation articles
• Bullet-nose curve
• Nose bullet

• Haack, Wolfgang (1941). "Geschoßformen kleinsten Wellenwiderstandes" (PDF). Bericht 139 der Lilienthal-Gesellschaft für Luftfahrtforschung: 14–28. Archived from the original (PDF) on 2007-09-27.
• U.S. Army Missile Command (17 July 1990). Design of Aerodynamically Stabilized Free Rockets. U.S. Government Printing Office. MIL-HDBK-762(MI).