Nose cone design

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Given the problem of the aerodynamic design of the nose cone section of any vehicle meant to travel through a compressible fluid medium (such as a rocket or aircraft), the main problem at hand is the determination of the nose cone geometrical shape. For many applications, such a task requires the definition of solid of revolution shape that experiences minimal resistance to rapid motion through a such a fluid medium, which consists of elastic particles.

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 2-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). Note that the equations describe the 'perfect' shape; practical nose cones are often blunted or truncated for manufacturing or aerodynamic reasons.

Conical

A very common nose cone shape is a simple cone. This shape is often chosen for its ease of manufacture, and is also often (mis)chosen for its drag characteristics. The sides of a conical profile are straight lines, so the diameter equation is simply

[y = ]

Cones are sometimes defined by their half angle, [\phi \;] :

[\phi = \arctan \left(\right)] and [y = x \tan\phi]

Bi-conic

A bi-conic nose cone shape is simply a cone with length L₁ stacked on top of a frustum of a cone (commonly known as a conical transition section shape) with lengh L₂, where the base of the upper cone is equal in radius R₁ to the top radius of the smaller frustum with base radius R₂.

L = L₁ + L₂

• for [0 \le x \le L_1 ] : [y = ]

half angle :

[\phi_1 = \arctan \left(\right)] and [y = x tan \phi_1 \;]

• for [L_1 \le x \le L] : [y = R_1 + ]

half angle :

[\phi_2 = \arctan \left(\right)] and [y = R_1 + (x - L_1) tan \phi_2 \;]

Tangent ogive

Next to a simple cone, the tangent ogive shape is the most familiar in hobby rocketry. The profile of this shape is formed by a segment of a circle such that the rocket body is tangent to the curve of the nose cone at its base; and the base is on the radius of the circle. The popularity of this shape is largely due to the ease of constructing its profile.

The radius of the circle that forms the ogive is called the Ogive Radius [\rho] and it is related to the length and base radius of the nose cone as expressed by the formula :

[\rho = ]

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

[y = \sqrt+R - \rho]

The nose cone length, L, must be equal to, or less than the Ogive Radius [\rho]. If they are equal, then the shape is a hemisphere.

Secant ogive

The profile of this shape is also formed by a segment of a circle, but the base of the shape is not on the radius of the circle defined by the ogive radius. The rocket body will not be tangent to the curve of the nose at its base. The Ogive Radius [\rho] is not determined by R and L (as it is for a tangent ogive), but rather is one of the factors to be chosen to define the nose shape. If the chosen Ogive Radius of a Secant Ogive is greater than the Ogive Radius of a Tangent Ogive with the same R and L, then the resulting Secant Ogive appears as a Tangent Ogive with a portion of the base truncated.

[\rho > ] and [\alpha = \arctan \left(\right) - \arccos \left( \over 2\rho}\right)]

Then the radius y at any point x as x varies from 0 to L is:

[y = \sqrt+\rho\sin\alpha]

If the chosen [\rho] is less than the tangent ogive [\rho], then the result will be a Secant Ogive that bulges out to a maximum diameter that is greater than the base diameter. The classic example of this shape is the nose cone of the Honest John. Also, the chosen ogive radius must be greater than twice the length of the nose cone.

[ < \rho < ]

Elliptical

The profile of this shape is one-half of an ellipse, with the major axis being the centerline and the minor axis being the base of the nose cone. A rotation of a full ellipse about its major axis is called a prolate spheroid, so an elliptical nose shape would properly be known as a prolate hemispheroid. This shape is popular in subsonic flight (such as model rocketry) due to the blunt nose and tangent base. This is not a shape normally found in professional rocketry. If R equals L, this is a hemisphere.

[y = R \sqrt}]

Parabolic

This nose shape is not the blunt shape that is envisioned when people commonly refer to a ‘parabolic’ nose cone. The Parabolic Series nose shape is generated by rotating a segment of a parabola around a line parallel to its Latus rectum. This construction is similar to that of the Tangent Ogive, except that a parabola is the defining shape rather than a circle. Just as it does on an Ogive, this construction produces a nose shape with a sharp tip. For the blunt shape typically associated with a parabolic nose, see the Power Series. (The parabolic shape is also often confused with the elliptical shape.)

For [0 \le K' \le 1] : [y = R\left() - K'()^2 \over 2 - K'}\right) ]

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

K’ = 0 for a cone

K’ = 0.5 for a 1/2 parabola

K’ = 0.75 for a 3/4 parabola

K’ = 1 for a full parabola

For the case of the full Parabola (K’=1) the shape is tangent to the body at its base, and the base is on the axis of the parabola. Values of K’ less than one result in a ‘slimmer’ shape, whose appearance is similar to that of the secant ogive. The shape is no longer tangent at the base, and the base is parallel to, but offset from, the axis of the parabola.

Power series

The Power Series includes the shape commonly referred to as a ‘parabolic’ nose cone, but the shape correctly known as a parabolic nose cone is a member of the Parabolic Series, and is something completely different. The Power Series shape is characterized by its (usually) blunt tip, and by the fact that its base is not tangent to the body tube. There is always a discontinuity at the nose cone / body joint that looks distinctly non-aerodynamic. The shape can be modified at the base to smooth out this discontinuity. Both a flat-faced cylinder and a cone are shapes that are members of the Power Series.

The Power series nose shape is generated by rotating a parabola about its axis. The base of the nose cone is parallel to the latus rectum of the parabola, and the factor n controls the ‘bluntness’ of the shape. As n decreases towards zero, the Power Series nose shape becomes increasingly blunt. At values of n above about 0.7, the tip becomes sharp.

For [0 \le n \le 1] : [y = R\left(\right)^n]

Where :

n = 1 for a cone

n = 0.75 for a 3/4 power

n = 0.5 for a 1/2 power (parabola)

n = 0 for a cylinder

Haack series

Unlike all of the nose cone shapes above, the Haack Series shapes are not constructed from geometric figures. The shapes are instead mathematically derived for the purpose of minimizing drag. While the series is a continuous set of shapes determined by the value of C in the equations below, two values of C have particular significance : when C = 0, the notation ‘LD’ signifies minimum drag for the given length and diameter, and when C = 1/3, ‘LV’ indicates minimum drag for a given length and volume. The Haack series nose cones are not perfectly tangent to the body at their base, however the discontinuity is usually so slight as to be imperceptible. Haack nose tips do not come to a sharp point, but are slightly rounded.

[\theta = \arccos \left(1 - \right)]

[y = + C \sin^3 \theta} \over \sqrt}]

Where :

C = 1/3 for LV-Haack

C = 0 for LD-Haack (also known as the Von Kármán or the Von Kármán Ogive)

References

• Gary A. Crowell Sr. (1996). The Descriptive Geometry Of Nose Cones. [link]
• [link]

 

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