Nose cone design

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

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

Spherically blunted conic

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

Bi-conic

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

Tangent ogive

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

Spherically blunted tangent ogive

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

Secant 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:

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

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

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

Parabolic

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

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

K′ can vary anywhere between 0 and 1, but the most common values used for nose cone shapes are:

Power series

A power series nosecone is defined by ${\displaystyle r=x^n}$ where ${\displaystyle (0\le x\le 1)}$. ${\displaystyle n<1}$ will generate a concave geometry, while ${\displaystyle n>1}$ will generate a convex (or "flared") shape.

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

Common values of n include:

Haack series

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

• r is the radius divided by the maximum radius at a given θ or x,
• x is the distance from the nose divided by the total nose length.

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

Special values of C (as described above) include:

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. An ogive with minimal drag for a given length and volume can be called an LV-Haack series, defined by ${\displaystyle C=\frac{1}{3}}$. 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 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.