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Because of the problem of theaerodynamicdesign of thenose cone section of any vehicle or body meant to travel through acompressible fluid medium (such as arocket oraircraft,missile,shell orbullet), an important problem is the determination of thenose cone geometrical shape for optimum performance. For many applications, such a task requires the definition of asolid of revolution shape that experiences minimal resistance to rapid motion through such a fluid medium.

Source:^[1]

In all of the following nose cone shape equations,L is the overall length of the nose cone andR is the radius of the base of the nose cone.y is the radius at any pointx, asx varies from0, at the tip of the nose cone, toL. The equations define the two-dimensional profile of the nose shape. The fullbody of revolution of the nose cone is formed by rotating the profile around the centerlineC⁄L. While the equations describe the "perfect" shape, practical nose cones are often blunted or truncated for manufacturing, aerodynamic, or thermodynamic reasons.^[2]

Conic nose cone render and profile with parameters shown.

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

${\displaystyle \varphi =\arctan \left(\frac{R}{L}\right)}$ and${\displaystyle y=x\tan (\varphi )}$

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 nose cone render and profile with parameters shown.

${\displaystyle L=L_1+L_2}$

For${\displaystyle 0\le x\le L_1}$ :${\displaystyle y=\frac{x{R}_1}{L_1}}$

For${\displaystyle L_1\le x\le L}$ :${\displaystyle y=R_1+\frac{(x-L_1)(R_2-R_1)}{L_2}}$

Half angles:

${\displaystyle {\varphi }_1=\arctan \left(\frac{R_1}{L_1}\right)}$ and${\displaystyle y=x\tan ({\varphi }_1)}$

${\displaystyle {\varphi }_2=\arctan \left(\frac{R_2-R_1}{L_2}\right)}$ and${\displaystyle y=R_1+(x-L_1)\tan ({\varphi }_2)}$

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

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

The radiusy at any pointx, asx varies from0 toL is:

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

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

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.

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

Then the radiusy at any pointx asx varies from0 toL is:

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

Elliptical nose cone render and profile with parameters shown.

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

Half (K′ = 1/2)

Three-quarter (K′ = 3/4)

Full (K′ = 1)

Renders of common parabolic nose cone shapes.

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 between0 and1, but the most common values used for nose cone shapes are:

Half (n = 1/2) Three-quarter (n = 3/4)

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

Common values ofn include:

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

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

Special values ofC (as described above) include:

A power series nosecone is defined by${\displaystyle r=x^n}$ where${\displaystyle (0\le x\le 1)}$. will generate a concave geometry, while${\displaystyle n>1}$ will generate a convex (or "flared") shape^[3]

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 series variable.^[3]

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${\displaystyle x}$.

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 withC = 0, commonly called theVon Kármán orVon Kármánogive. A cone with minimal drag for a given length and volume can be called a LV-Haack series, defined with${\displaystyle C=\frac{1}{3}}$.^[3]

An aerospike can be used to reduce the forebody pressure acting on supersonic aircraft. The aerospike creates adetached shock ahead of the body, thus reducing the drag acting on the aircraft.

• Index of aviation articles
• Bullet-nose curve
• Nose bullet

• Haack, Wolfgang (1941)."Geschoßformen kleinsten Wellenwiderstandes"(PDF).Bericht 139 der Lilienthal-Gesellschaft für Luftfahrtforschung:14–28. Archived fromthe 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).

• Aerodynamics
• Rocketry

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