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, Template:Mvar is the overall length of the nose cone and Template:Mvar is the radius of the base of the nose cone. Template:Mvar is the radius at any point Template:Mvar, as Template:Mvar varies from Template:Math, at the tip of the nose cone, to Template:Mvar. 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 Template:Frac. While the equations describe the "perfect" shape, practical nose cones are often blunted or truncated for manufacturing, aerodynamic, or thermodynamic reasons. Template:Clear

Conic

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

Spherically blunted conic

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

Bi-conic

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

Tangent ogive

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

Spherically blunted tangent ogive

Template:Multiple image $$\begin{array}{cc}{x}_o& =L-\sqrt{(R-r_n)(\frac{L^2}{R}-r_n)}\\ y_t& =\frac{r_n(L^2-R^2)}{L^2+R^2-2R{r}_n}\\ x_t& =L-\sqrt{(R-r_n)(\frac{L^2}{R}-r_n)}-r_n\sqrt{1-{\left(\frac{L^2-R^2}{L^2+R^2-2R{r}_n}\right)}^2}\end{array}$$ Alternatively, if blunted length is specified in the design process, noted by $N$, theoretical un-blunted length can be determined using: $$\begin{array}{cc}L& =\sqrt{R(\frac{{(N-r_n)}^2}{R-r_n}+r_n)}\end{array}$$ where $N=L-x_a$.

Secant ogive

For a chosen ogive radius Template:Mvar greater than or equal to the ogive radius of a tangent ogive with the same Template:Mvar and Template:Mvar:

$$\begin{array}{cc}\rho & \ge \frac{R+\frac{L^2}{R}}{2}\\ \alpha & =\arctan \left(\frac{R}{L}\right)-\arccos \left(\frac{\sqrt{R^2+L^2}}{2\rho }\right)\\ y& =\sqrt{{\rho }^2-(x-\rho \cos \alpha {)}^2}+\rho \sin (\alpha ),& & 0\le x\le L\end{array}$$ Template:Clear Template:Multiple image 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 Template:Mvar that results in some values of Template:Mvar.

Elliptical

Template:Multiple image $$y=\frac{R\sqrt{x(2L-x)}}{L}$$ Template:Clear

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. Template:Multiple image

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

Template:Math can vary anywhere between Template:Math and Template:Math, but the most common values used for nose cone shapes are:

Parabola type	Template:Math value
Cone	Template:Math
Half	Template:Math
Three quarter	Template:Math
Full	Template:Math

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.

Graphs illustrating power series nose cone shapes

Template:Multiple image

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

Common values of Template:Mvar include:

Power type	Template:Mvar value
Cylinder	Template:Math
Half (parabola)	Template:Math
Three quarter	Template:Math
Cone	Template:Math

Haack series

Graphs illustrating Haack series nose cone shapes

Template:Multiple image

A Haack series nosecone is defined by: $$\begin{array}{cc}r(\theta )& =\sqrt{\frac{\theta -\frac{1}{2}\sin (2\theta )+C{\sin }^3\theta }{\pi }}\\ \theta & =\arccos (1-2x)\\ 0& \le x\le 1\Rightarrow 0\le \theta \le \pi \end{array}$$ where

• Template:Mvar is the radius divided by the maximum radius at a given Template:Mvar or Template:Mvar,
• Template:Mvar is the distance from the nose divided by the total nose length.

Parametric formulation can be obtained by solving the Template:Mvar formula for Template:Mvar (here, Template:Mvar is now distance from the nose, separated from the total nose length Template:Mvar, and Template:Mvar is the radius). $$\begin{array}{cc}x(\theta )& =\frac{L}{2}(1-\cos (\theta ))\\ y(\theta ,C)& =\frac{R}{\sqrt{\pi }}\sqrt{\theta -\frac{\sin (2\theta )}{2}+C{\sin }^3(\theta )}\end{array}$$

Special values of Template:Mvar (as described above) include:

Haack series type	Template:Mvar value
LD-Haack (Von Kármán)	Template:Math
LV-Haack	Template:Math
Tangent	Template:Math

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 Template:Math, commonly called the Von Kármán or Von Kármán ogive. An ogive with minimal drag for a given length and volume can be called an LV-Haack series, defined by $C=\frac{1}{3}$. However, the LV-Haack series produces different values for radius as a function of x as opposed to the Sears-Haack body, which also attempts to provide a shape with minimal drag for a given length and volume. For example, the LV-Haack value for radius relative to maximum radius at x=0.5 is ≈ 0.7785, while a Sears-Haack body at the same point (halfway along the nose, which is 25% of the way along the body) has a radius relative to maximum radius of ≈ 0.8059.

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

• Template:Cite journal
• Template:Cite book