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.

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.

y={xR \over L}

\phi =\arctan \left({R \over L}\right)

and

y=x\tan(\phi )\;

Spherically blunted conic

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

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

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

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

x_{a}=x_{o}-r_{n}

Bi-conic

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

L=L_{1}+L_{2}

For

0\leq x\leq L_{1}

:

y={xR_{1} \over L_{1}}

For

L_{1}\leq x\leq L

:

y=R_{1}+{(x-L_{1})(R_{2}-R_{1}) \over L_{2}}

Half angles:

\phi _{1}=\arctan \left({R_{1} \over L_{1}}\right)

and

y=x\tan(\phi _{1})\;

\phi _{2}=\arctan \left({R_{2}-R_{1} \over L_{2}}\right)

and

y=R_{1}+(x-L_{1})\tan(\phi _{2})\;

Tangent ogive

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

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

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

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.

{\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.

\rho >{R^{2}+L^{2} \over 2R}

and

\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:

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

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

Elliptical

UNlike the other equations on this page, here x goes from 0 to L,

AND the nose of the nosecone is at max x, not at min x.

Therefore the 2 images for this section are pointing in the wrong direction, they are left-to-right flipped, for this equation.

Elliptical nose cone render and profile with parameters shown.

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

Parabolic

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

Half (

K' = 1/2

)

Three-quarter (

K' = 3/4

)

Full (

K' = 1

)

Renders of common parabolic nose cone shapes.

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

$K'$ 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 $r=x^{n}$ where $(0\leq x\leq 1)$ . $n<1$ will generate a concave geometry, while $n>1$ will generate a convex (or "flared") shape.^[3]

Half (

n = 1/2

)

Three-quarter (

n = 3/4

)

For

0\leq n\leq 1

:

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

LD-Haack (Von Kármán) (

C = 0

)

LV-Haack (

C = 1/3

)

{\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 $0\leq \theta \leq \pi$ .

Special values of $C$ (as described above) include:

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

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 $C={\tfrac {1}{3}}$ .^[3]

Aerospike

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

Influence of the fineness ratio

References

^ 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.
^ 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.
^ ^a ^b ^c ^d 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.

[satyajit-1] 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.

[crowell-2] 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.

[:0-3] 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.

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