Nose cone design

Two-dimensional drawing of an elliptical nose cone with dimensions added to show how L is the total length of the nose cone, R is the radius at the base, and y is the radius at a point x distance from the tip.

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.

Conic

File:Nose cone conical.svg

File:Conic Nose Cone Render.png

Conic nose cone render and profile with parameters shown.

$$\begin{array}{cc}y& =\frac{xR}{L}=x\tan (\varphi )\\ \varphi & =\arctan \left(\frac{R}{L}\right)\\ \end{array}$$

Spherically blunted conic

File:Spherically blunted cone geometry.svg

File:Spherically Blunted Conic Nose Cone Render.png

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

$$\begin{array}{cccc}{x}_t& & & =\frac{r_n{L}^2}{R}\sqrt{\frac{1}{R^2+L^2}}\\ y_t& =\frac{x_{t}R}{L}& & =r_{n}L\sqrt{\frac{1}{R^2+L^2}}\\ x_o& =x_t+\sqrt{r_n^2-y_t^2}& & =r_n(\frac{L^2}{R}\sqrt{\frac{1}{R^2+L^2}}+\sqrt{1-\frac{L^2}{R^2+L^2}})\\ x_a& =x_o-r_n& & =r_n(\frac{L^2}{R}\sqrt{\frac{1}{R^2+L^2}}+\sqrt{1-\frac{L^2}{R^2+L^2}}-1)\\ \end{array}$$

Bi-conic

File:Nose cone bi-conic.png

File:Bi-Conic Nose Cone Render.png

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

$$\begin{array}{cc}L& =L_1+L_2\\ {\varphi }_1& =\arctan \left(\frac{R_1}{L_1}\right)\\ {\varphi }_2& =\arctan \left(\frac{R_2-R_1}{L_2}\right)\\ y& =\{\begin{array}{cc}\frac{x{R}_1}{L_1}=x\tan ({\varphi }_1),& 0\le x\le L_1\\ R_1+\frac{(x-L_1)(R_2-R_1)}{L_2}=R_1+(x-L_1)\tan ({\varphi }_2),& L_1\le x\le L\end{array}\end{array}$$

Tangent ogive

File:Tangent ogive nose cone.svg

File:Tangent Ogive Nose Cone Render.png

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

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

Spherically blunted tangent ogive

File:Spherically blunted tangent ogive geometry.svg

File:Spherically Blunted Tangent Ogive Nose Cone Render.png

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

$$\begin{array}{cc}{x}_o& =L-\sqrt{(R-r_n)(\frac{L^2}{R}-r_n)}\\ y_t& =\frac{r_n(L^2-R^2)}{L^2+R^2-2R{r}_n}\\ x_t& =L-\sqrt{(R-r_n)(\frac{L^2}{R}-r_n)}-r_n\sqrt{1-{\left(\frac{L^2-R^2}{L^2+R^2-2R{r}_n}\right)}^2}\\ \end{array}$$

Secant ogive

File:Nose cone secant ogive 1.png

File:Secant Ogive Nose Cone Render.png

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:

$$\begin{array}{cc}\rho & \ge \frac{R+\frac{L^2}{R}}{2}\\ \alpha & =\arctan \left(\frac{R}{L}\right)-\arccos \left(\frac{\sqrt{R^2+L^2}}{2\rho }\right)\\ y& =\sqrt{{\rho }^2-(x-\rho \cos \alpha {)}^2}+\rho \sin (\alpha ),& & 0\le x\le L\\ \end{array}$$

File:Nose cone secant ogive 2.png

File:Secant Ogive Nose Cone (Bulge) Render.png

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

A smaller ogive radius can be chosen; for $\frac{1}{2}(R+\frac{L^2}{R})>\rho >\frac{L}{2}$, 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

File:Nose cone elliptical.svg

File:Elliptical Nose Cone Render.png

Elliptical nose cone render and profile with parameters shown.

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

Parabolic

A parabolic series nosecone is defined by $r=\frac{2x-K{x}^2}{2-K}$ where $0\le x\le 1$ and $K$ is a series-specific constant.

File:Parabolic (Half) Nose Cone Render.png

Half (K′ = 1/2)

File:Parabolic (Three-Quarter) Nose Cone Render.png

Three-quarter (K′ = 3/4)

File:Parabolic (Full) Nose Cone Render.png

Full (K′ = 1)

Renders of common parabolic nose cone shapes.

For $0\le K^{\prime }\le 1$, $y=R\left(\frac{2\left(\frac{x}{L}\right)-K^{\prime }{\left(\frac{x}{L}\right)}^2}{2-K^{\prime }}\right)$

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 $(0\le x\le 1)$. $n<1$ will generate a concave geometry, while $n>1$ will generate a convex (or "flared") shape.

File:Nose Cone Power Series.svg Graphs illustrating power series nose cone shapes

File:Power Series (Half) Nose Cone Render.png Half (n = 1/2) File:Power Series (Three-Quarter) Nose Cone Render.png Three-quarter (n = 3/4)

For $0\le n\le 1$: $y=R{\left(\frac{x}{L}\right)}^n$

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 $\theta =\arccos (1-\frac{2x}{L})$. Parametric formulation can be obtained by solving the $\theta$ formula for $x$.

File:Nose Cone Haack Series.svg Graphs illustrating Haack series nose cone shapes

File:LD-Haack Series (Von Kármán) Nose Cone Render.png LD-Haack (Von Kármán) (C = 0) File:LV-Haack Series Nose Cone Render.png LV-Haack (C = 1/3)

$$\begin{array}{cc}x(\theta )& =\frac{L}{2}(1-\cos (\theta ))\\ y(\theta ,C)& =\frac{R}{\sqrt{\pi }}\sqrt{\theta -\frac{\sin (2\theta )}{2}+C{\sin }^3(\theta )}\\ 0& \le \theta \le \pi \end{array}$$

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 $C=\frac{1}{3}$.

Aerospike

Main article: Drag-reducing aerospike

File:Aerospike detail.png

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

File:JET Airways at Mumbai Airport 2016.jpg

Closeup view of a nose cone on a Boeing 737

File:Nose cone drag comparison.svg

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 Fighting Falcon

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