This work aims to compute stability derivatives in the Newtonian limit in pitch when the Mach number tends to infinity. In such conditions, these stability derivatives depend on the Ogive’s shape and not the Mach number. Generally, the Mach number independence principle becomes effective from M = 10 and above. The Ogive nose is obtained through a circular arc on the cone surface. Accordingly, the following arc slopes are considered λ = 5, 10, 15, −5, −10, and −15. It is found that the stability derivatives decrease due to the growth in λ from 5 to 15 and vice versa. For λ = 5 and 10, the damping derivative declines with an increase in λ from 5 to 10. Yet, for the damping derivatives, the minimum location remains at a pivot position, h = 0.75 for large values of λ. Hence, when λ = −15, the damping derivatives are independent of the cone angles for most pivot positions except in the early twenty percent of the leading edge.

The analysis of supersonic/hypersonic of simple shapes like wedges, cones, and Ogives are of great interest when oscillating at high Mach numbers, and significant incidences are of great interest. Researchers have shown great interest owing to the advent of the space program and high-speed fighter aircraft because of the cost involved in conducting experiments at high Mach numbers. Therefore, simple but reasonably accurate techniques are needed to compute the aerodynamic load and stability derivatives during re-entry. And more so, such parametric computations are very valuable in the initial design process when various geometrical and inertia parameters are investigated.

The present work aims to evaluate pitch aerodynamic stiffness and damping derivatives for limiting cases. In this situation, flow parameters are not dependent on the inertia levels. However, flow parameters are a function of geometry. In the Newtonian limit, Mach numbers will tend to infinity. The specific heat ratio (γ) at constant pressure and volume, with a constant value of 1.4 (i.e., γ = 1.4) at a standard ballistic atmosphere, becomes unity. The knowledge of aerodynamic derivatives at the Newtonian limit can be handy for the space program. For launch vehicles, the flow Mach number becomes very high. In aerodynamic vehicles like hypersonic missiles and the space shuttle, the focus of study shifts from optimizing the aerodynamic shapes to reduce the drag force to focusing on aerodynamic heating. At supersonic flow, the primary concern is to minimize the drag of the projectiles and missiles. The easiest option is to have a blunt nose instead of a sharp nose, which will give immediate relief from the high-temperature build-up at the nose. Unique material is used for the nose portion of the hypersonic missiles and the aircraft to address the issue of high temperature at the design stage. For space shuttles, tiles are used to protect from aerodynamic heating.

Appleton [_{2} > 2.5). He ignored the impact of the Lee surface as the contribution from the Lee surface was negligible. He mainly focussed on the windward side accompanied by an oblique shock wave at the plate. A unified supersonic/hypersonic theory for delta wings and cones was developed by Ghosh [

An axisymmetric Ogive is obtained by the revolution of the plane Ogive of semi-nose angle δ =

As the Ogive move with a velocity of

where

The projection of point A (^{1} has a length of x_{o} from the apex.

Let α be the pitch angle at any instant, and q be the pitch rate. Therefore, the local piston Mach number at A is

If Ψ is the azimuthal angle (

After simplification, we obtain

In the limit α, q →0,

where

Hence

Putting

Differentiating

And from

Since

where

where

Expanding Binomially, and neglecting higher-order terms and simplifying, we obtain

From

On simplification, we get

where

Further simplifying, we obtain

and

Now expanding

This expression must be evaluated in the limit

Hence, from

And

Therefore

After simplification, we obtain

where

From

where

Hence

where

Substituting

Hence

And

The expression for the total pitching moment of the Ogive is given by

Here P_{b} is given by

The stiffness derivative,

where

S_{b} = Ogive surface area =

c = chord size of an Ogive

Since P_{b} is a function of M_{p},

Substituting from

where

Now, substituting for

Putting

where

Since

and

Putting

Substituting I_{1} and I_{2} in

Substituting

Hence on simplification, we obtain

where

The Damping derivative

where

On solving similarly like above, we obtain

where

and

For limiting case

By applying limit

where

Therefore, the Stiffness derivative in the limiting case is given by

Hence, the Damping derivative in the limiting case is given by

The results were computed and discussed based on the stiffness and damping derivatives formulations. Before analyzing the results, we must keep in mind that we evaluate stiffness & damping derivatives for a limiting case where M tends to infinity. The value of γ of air is typically 1.4, but it will increase to unity in the Newtonian limit. The following parameter is the very high Mach number. Because Mach becomes infinity, which implies that Mach number will no more be a variable, outcomes will indicate the impact of the geometric constraints alone in the present case. We are considering these circumstances while discussing the results.

Results for cone angles from 10 to 25 degrees & λ = −5 are shown in

When the Ogive arc λ = 10 and other parameters are the same, the outcomes of the stiffness derivatives are displayed in

Results for λ = −5 are seen in

When λ = 10, the outcomes of the present investigations are shown in

When λ = −10, the discrepancy of the damping derivatives with the pivot position is shown in

Results for λ = 15 are shown in

When λ = −15, damping derivatives for similar cone angles, as discussed earlier in

Because of the above deliberations, we may conclude our discussion as under:

The linear growth in the stiffness derivative is seen owing to the cone angle growth. With the rise in the Ogive slope from λ = 5 to 10, linear decrease in the stiffness derivative are seen, and it attains maximum values at h = 0, and the center of pressure is from h = 0.65 to 0.8 λ = 5 & from h = 0.52 to 0.75 for λ = 10.

For λ = 15, it is seen that the contour of the Ogive is such that the surface pressure allocation has augmented significantly. Hence, if λ = 15 is used, avoiding cone angles ten and twelve degrees is better. The reversal’s location in the results’ outline has shifted at h = 0.75. For this combination of the parameters, cone angles of 15 to 25 degrees are a better option.

A peculiar trend is seen in the results and the flow field for this case for λ = −5. The stability derivatives assume higher values than the higher cone angles, and the tendency reverses at h = 0.4. Based on the results obtained, it is recommended that cones with λ = −5 need not consider for design.

For λ = −15, it increases the Ogive surface for cone angles from 10 to 25 degrees. The magnitude of the increase in λ from −10 to −15 has increased significantly, hence the larger stiffness derivatives values. The variation of the center of pressure is limited to the range from h = 0.8 to 0. 86.

For λ = 5 and 10, the damping derivative declines with an increase in λ from 5 to 10. The location of minima remains at h = 0.75, which is also the center of pressure. For λ = 15, the form of Ogive is such the surface pressure distribution has augmented significantly, resulting in a changed pattern for lower cone angles, mainly ten degrees. Hence, avoiding cone angles ten and twelve degrees is better to have similar results. For this combination of the parameters, cone angles of 15 to 25 degrees are a better option.

When λ = −5, −10, and −15, there is a progressive increase in the stiffness cone angles that do not influence the damping derivatives as the shape of the surface is convex. Only a marginal impact is seen for a cone angle of twenty-five degrees. Hence, it can be stated that when λ = −15, damping derivatives are independent of the cone angles for most of the pivot places except in the early twenty percent of the leading edge.

The current theory for oscillating cones is limited to the quasi-steady case: the primary issue in considering the unsteady case would be the flow being non-uniform in the conical-annular space, even for the steady piston. The Mach waves’ velocity is variable, contrasting the waves in front of a plane piston equivalent to the oscillating wedge.

This research is supported by the Structures and Materials (S&M) Research Lab of Prince Sultan University. Furthermore, the authors acknowledge Prince Sultan University’s support for paying this publication’s article processing charges (APC).