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Basics

1.1 Introduction

Slender body configuration is a geometrical shape with one dimension much larger than the other two dimensions. Rocket shape is a typical example for slender body configuration. A slender wing shape is shown in Fig. 1.1.

For a slender wing, the height is very large compared to the base , that is, , and the aspect ratio (equal to the square of the wing span () divided by the wing area ) is much smaller than unity. That is,

or

where is the projected surface area of the wing normal to the flow direction. Equation (1.1) can be applied to geometries having aspect ratio as high as unity (). Hence, may be taken as the upper limit. Such a configuration is used for supersonic civil transport aircraft.

Some of the typical slender body shapes, which are practically possible shapes and come under the study in this book are the Ogee and Gothic bodies as shown in Fig. 1.2.

The Ogee curve is a double S-shaped curvature along the cheeks that gives the face contour and dimension (a distinctive pattern with two continuous S-shaped curves narrowing at the nose and widening at the base, as in Fig. 1.2(a)). It is best appreciated when looking at a shape of the cheek1 from a angle. Gothic shape is a pointed arch, in which the two sides of the arch are joined by a sharp angle, as illustrated in Fig. 1.2(b).

Figure 1.1 A slender wing.

Figure 1.2 (a) Ogee shape and (b) Gothic shape.

Figure 1.3 Rocket configuration: (a) Older type and (b) New type.

Two typical rocket configurations, which are essentially slender bodies, are shown in Fig. 1.3. In the older type, the rockets had the control surfaces, namely, the fins only at the rear of it, as shown in Fig. 1.3(a). But in the new type of rockets, the control surface runs almost up to the nose, as shown in Fig. 1.3(b).

Example 1.1 If the aspect ratio of a triangular slender body of height 6 m is 0.3, determine the base length.

Solution

Given, m, and .

By Eq. (1.1),

where the projected surface area . Therefore, the base length, , becomes

1.2 Supersonic Transport Aircraft

The popular versions of supersonic transport aircraft are the Russian TU 144, the French-British Concorde, and Boeing 2707 of the United States of America. Among these, TU 144 was designed to cruise at Mach 2.2, the Concorde at Mach 2, and the Boeing 2707 at Mach 2.7. Two prototype Concorde were made, one each in France and England. It's cruising altitude is 10 km, whereas the cruising altitude of Boeing 2707 is 13 km, which is the earth's curvature. Though TU 144 and Concorde aircraft were successfully tested and brought to practical use, the noise produced by them were around 120 dB, which is not acceptable to humans, especially when they fly at low altitudes, while approaching for landing or climbs after takeoff. This noise pollution added to the fuel economy issues forced the companies to ground them.

Currently, the American company Boom Technology, Inc. is developing supersonic transport aircraft.2 Boom Technology is designing Mach 1.7, 65-88-passenger supersonic airliner. It is claimed that the maximum noise level of this aircraft is only around 60 dB. Therefore, this is expected to become a feasible supersonic passenger aircraft and come to commercial use in the year 2029.

Some agencies designed and tested hypersonic wings and wave riders, which are essentially slender body flying machines.

1.3 Wings of Supersonic Aircraft

The wings meant for supersonic flight should have the flow characteristics such that they are suitable for supersonic aircraft. To have an idea about this specific requirement, let us consider inviscid subsonic and supersonic flows over a flat plate of chord at an angle of attack, as illustrated in Fig. 1.4.

If the flow is assumed to be inviscid, the plate in the subsonic flow would not experience any wave over it. Also, there won't be a skin friction acting over the body. Therefore, there won't be any drag acting on the plate. But, for the plate in the supersonic flow there will be compression and expansion waves at the leading and trailing edge, as illustrated in Fig. 1.4(b). Due to the presence of these waves, a drag force will be acting on the plate. This drag caused by the waves is termed as wave drag.3 It is to be noted that the presence and absence of drag over the plate in supersonic (Fig. 1.4(b)) and subsonic (Fig. 1.4(a)) flows, respectively, are because the flow is assumed to be inviscid. But in actual situation, the flow will be viscous and, hence, the drag due to the skin friction4 will be acting on the plate in subsonic and supersonic flows.

Figure 1.4 Flat plate in (a) subsonic flow and (b) supersonic flow.

Figure 1.5 Flow past a slender wing.

The center of pressure, c.p. (the point at which the line of action of the resultant of the forces due to pressure acts) is at the quarter chord point for the plate in subsonic stream and at the midpoint in the supersonic stream. Therefore, a slender body vehicle of large aspect ratio flying at supersonic speed would encounter severe trim problems. But a slender wings' behavior is independent of freestream Mach number, therefore, there is no trim problem. It is the component of the freestream Mach number that is normal to the leading edge, as shown in Fig. 1.5, that determines the behavior of the wing.

Designing slender wings having subsonic leading edges5 is advantageous from an operation point of view. The advantage of this kind of leading edge is the following:

1. The center of pressure (the point of application of total pressure force exerted by the flow on the object), c.p., is independent of the freestream Mach number . That is, better trim.
2. No additional wave drag. In other words, because of the leading edge being subsonic, the wing has better drag characteristic.
3. Simple theory for design condition. This includes optimization of configuration.

Figure 1.6 A slender wing at an angle of attack: (a) vortex sheets over the wing and (b) lift variation with angle of attack.

The disadvantage associated with the slender wings with subsonic leading edges is the following:

Off-design: In off-design conditions, there will be the leading edge separation, that is, when the angle of attack is high, the flow has to go over the leading edge very quickly and it can no more go along the surface. The flow will get separated, curl and form small vortices behind the leading edge. These vortices will result in the formation of two big vortex sheets that would stay well above the upper surface, as illustrated in Fig. 1.6(a).

Due to the shedding of vortices and leading edge separation, the flow is attached and is different from a normal flow. The flow behavior is nonlinear, causing the lift variation with angle of attack as nonlinear, as illustrated in Fig. 1.6(b).

The vortex breakdown experienced is a stability problem. At some point, the vortex breaks down and this is a troublesome phenomenon because the breakdown point is far ahead of the trailing edge, as illustrated in Fig. 1.6(a). The area of the region over which the vortex breakdown takes place touches the upper surface of the wing. The flow behind this region is turbulent and the pressure distribution is changed fully. Therefore, it is necessary to make sure that the breakdown point is far behind the leading edge so that the area of this flow does not touch the wing's upper surface. Vortex breakdown point is very sensitive to yaw. There will be an additional unsymmetry introduced under yawed condition.

At the design condition, there will be no vortices when the flow hits the leading edge tangentially (this is achieved by cambering the wing).

Example 1.2 What should be the nose angle of a slender triangular wing flying at Mach 5 to have a subsonic leading edge.

Solution

Given, . For the triangular wing to have Mach number at its nose less than 1, the semi-nose angle the condition is , for hypersonic similarity. Therefore,

Thus, the nose angle has to be

Example 1.3 A slender delta wing of nose angle cruises at supersonic speed. Determine the limiting speed up to which the wing will be shock wave-free.

Solution

The semi-angle of the nose (Fig. 1.5) is the angle by which the freestream flow with Mach number, , will be turned to make it flow parallel to the leading edge of the wing. For the wing to be free of shock waves, the flow Mach number, , normal to its leading edge should be less than 1. That is, up to , the wing will be shock wave free. For the given wing, the semi-nose angle is , therefore,

That is, for a freestream Mach number up to 11.43, the wing will fly without any shock wave over it.6

1.3.1 Sharp Leading Edge

Providing sharp leading edge to a slender wing with the aim of making the wing not to experience separation at the leading edge. This has to be done to ensure...