Nose cone design

Given 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.

General parameters used for constructing nose cone profiles.

Nose cone shapes and equations

General dimensions

Source:^[1]

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.^[2]

Conic

Conic nose cone render and profile with parameters shown.

${\displaystyle y={xR \over L}}$

${\displaystyle \phi =\arctan \left({R \over L}\right)}$ and ${\displaystyle y=x\tan(\phi )\;}$

Spherically blunted conic

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

${\displaystyle x_{t}={\frac {L^{2}}{R}}{\sqrt {\frac {r_{n}^{2}}{R^{2}+L^{2}}}}}$

${\displaystyle y_{t}={\frac {x_{t}R}{L}}}$

${\displaystyle x_{o}=x_{t}+{\sqrt {r_{n}^{2}-y_{t}^{2}}}}$

${\displaystyle x_{a}=x_{o}-r_{n}}$

Bi-conic

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

${\displaystyle L=L_{1}+L_{2}}$

For ${\displaystyle 0\leq x\leq L_{1}}$ : ${\displaystyle y={xR_{1} \over L_{1}}}$

For ${\displaystyle L_{1}\leq x\leq L}$ : ${\displaystyle y=R_{1}+{(x-L_{1})(R_{2}-R_{1}) \over L_{2}}}$

Half angles:

${\displaystyle \phi _{1}=\arctan \left({R_{1} \over L_{1}}\right)}$ and ${\displaystyle y=x\tan(\phi _{1})\;}$

${\displaystyle \phi _{2}=\arctan \left({R_{2}-R_{1} \over L_{2}}\right)}$ and ${\displaystyle y=R_{1}+(x-L_{1})\tan(\phi _{2})\;}$

Tangent ogive

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

${\displaystyle \rho ={R^{2}+L^{2} \over 2R}}$

The radius y at any point x, as x varies from 0 to L is:

${\displaystyle y={\sqrt {\rho ^{2}-(L-x)^{2}}}-R+\rho }$

Spherically blunted tangent ogive

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

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

Secant ogive

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

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

${\displaystyle \rho >{R^{2}+L^{2} \over 2R}}$ and ${\displaystyle \alpha =\arccos \left({{\sqrt {L^{2}+R^{2}}} \over 2\rho }\right)-\arctan \left({R \over L}\right)}$

Then the radius y at any point x as x varies from 0 to L is:

${\displaystyle y={\sqrt {\rho ^{2}-(\rho \cos(\alpha )-x)^{2}}}-\rho \sin(\alpha )}$

${\displaystyle {\frac {L}{2}}<\rho <{R^{2}+L^{2} \over 2R}}$

Elliptical

Elliptical nose cone render and profile with parameters shown.

${\displaystyle y=R{\sqrt {1-{x^{2} \over L^{2}}}}}$

Parabolic

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

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

Graphs illustrating Haack series nose cone shapes

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

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

For ${\displaystyle 0\leq \theta \leq \pi }$.

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

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]

Parabolic Series

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 series variable. ^[3]

Haack Series

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

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 a 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

Influence of the fineness ratio

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).