Nose cone design

General parameters used for constructing nose cone profiles.

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.

Nose cone shapes and equations

General dimensions

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]}

Conic

Conic nose cone render and profile with parameters shown.

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

Spherically blunted conic

Spherically blunted conic nose cone render and profile with parameters shown.

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

Bi-conic

Bi-conic nose cone render and profile with parameters shown.

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

Tangent ogive

Tangent ogive nose cone render and profile with parameters and ogive circle shown.

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

Spherically blunted tangent ogive

Spherically blunted tangent ogive nose cone render and profile with parameters shown.

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

Secant ogive

Secant ogive nose cone render and profile with parameters and ogive circle shown, ogive radius larger than for equivalent tangent ogive.

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

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

Alternate secant ogive render and profile which show a bulge due to a smaller radius.

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.

Elliptical

Elliptical nose cone render and profile with parameters shown.

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

Parabolic

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]

Half (K′ = 1/2)

Three-quarter (K′ = 3/4)

Full (K′ = 1)

Renders of common parabolic nose cone shapes.

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:

Parabola type	K′ value
Cone	0
Half	1/2
Three quarter	3/4
Full	1

Power series

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]

Graphs illustrating power series nose cone shapes

Half (n = 1/2) Three-quarter (n = 3/4)

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

Common values of n include:

Power type	n value
Cylinder	0
Half (parabola)	1/2
Three quarter	3/4
Cone	1

Haack series

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 ${\displaystyle x}$.

Graphs illustrating Haack series nose cone shapes

LD-Haack (Von Kármán) (C = 0) LV-Haack (C = 1/3)

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

Special values of C (as described above) include:

Haack series type	C value
LD-Haack (Von Kármán)	0
LV-Haack	1/3
Tangent	2/3

Von Kármán ogive

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]

Aerospike

Main article: Drag-reducing aerospike

An aerospike on the UGM-96 Trident I

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.

Nose cone drag characteristics

Influence of the general shape

Closeup view of a nose cone on a Boeing 737

Comparison of drag characteristics of various nose cone shapes in the transonic to low-mach regions. Rankings are: superior (1), good (2), fair (3), inferior (4).

General Dynamics F-16 with a nose cone very close to the Von Kármán shape

See also

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

Further reading

• 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).

References

1. Crowell Sr., Gary A. (1996). The Descriptive Geometry of Nose Cones (PDF) (Report). Archived from the original (PDF) on 11 April 2011. Retrieved 11 April 2011.
2. satyajit panigrahy (August 2020). "Improvement of Fire Power of Weapon System by Optimizing Nose Cone Shape and War Head Grouping". ResearchGate. doi:10.13140/RG.2.2.28694.36161.
3. Stoney, William E. (February 5, 1954). "TRANSONIC DRAG MEASUREMENTS OF EIGHT BODY-NOSE SHAPES" (PDF). Naca Research Memorandum. NACA-RM-L53K17 – via NASA Technical Reports Server.