Waverider Design

Hypersonic Flow
Vehicle Design
We have already explored various theories used to predict and solve for hypersonic flowfields. We have also seen the general characteristics of hypersonic vehicles and various issues the designer must be aware of. We can now learn how waveriders are optimized based on the characteristics of the assumed flowfield.

Caret Wing:

The waverider owes its origins to work done on winged atmospheric reentry vehicles in the 1950s. In 1959, Terence Nonweiler proposed that a three-dimensional hypersonic vehicle could be constructed from a known flowfield, like those described earlier. In particular, Nonweiler chose the flowfield behind a planar oblique shock (the shock formed on a 2D wedge) and used the stream surfaces behind the shock to generate a body shape. Viewed from overhead, the shape looks like a delta wing, but it is actually a concave pyramid shape, known as a caret, as viewed from the aft. The advantage of the caret wing is that the body appears to be riding on top of the attached shock wave at design Mach number, as illustrated below.

Caret wing
Caret wing showing the attached shock plus a three-view construction [from Corda and Anderson, 1988 and Starkey and Lewis, 1998]

Because the planar shock surface is attached at the leading edge, there is no flow spillage from the lower to upper surface. The flow contained beneath the vehicle is at high pressure because of the shock and therefore results in compression lift being generated on the wing’s lower surface.

Conical-Flow Waverider:

In addition to the simple 2D tangent-wedge flowfield used to design the caret wing, more complicated flowfields can also be used to generate hypersonic bodies. In fact, the flowfield over any axisymmetric body can be used to design a waverider with a shock surface attached along the leading edge. Probably the most commonly used flowfield is that based on conical shapes. Conical shapes typically used include the right circular cone, inclined circular and elliptical cones, yawed circular and elliptical cones, and bodies with longitudinal curvature. Some investigators have gone further using calculus of variations and hypersonic small disturbance theory to search for optimum waverider shapes with high lift-to-drag ratios (see Bowcutt, Anderson and Capriotti, 1987 and Corda and Anderson, 1988). The following figure illustrates in greater detail how conical-flow waveriders are designed. Typically, numerical techniques are used to solve for the inviscid conical shock wave about a cone-shaped surface. The lower surface of the body is defined by the intersection of the conical flow stream surface with the shock wave; this intersection produces the body’s leading edge. The upper surface of the body is usually designed assuming it is a freestream surface or an expansion surface. In the latter case, some small increase in L/D can be gained at the price of a more complicated and time-consuming construction method. The result of this process is a three-dimensional inviscid flow waveriding body, as illustrated below.

Conical-flow waverider
Construction of a body producing a conical shock; (inset) sample conical-flow waverider [from Cockrell, Huebner and Finley, 1995 and Corda and Anderson, 1988]

Osculating Cone Waverider:

A modification of conical-flow techniques, mentioned previously, is the osculating cones method. Although conical-flow derived waveriders are relatively easy to design, they tend to be prohibitively long and difficult to develop into practical flight vehicles. The advantage of the osculating cones method is that multiple cones are used to design non-axisymmetric shock patterns allowing greater flexibility in vehicle design. For example, a designer can use varying conical shock shapes for different portions of the vehicle to improve performance after the incorporation of engines, crew canopies, control surfaces, and other high-drag items. An example of an osculating cones derived waverider is illustrated below.

Mach 6 waverider designed using the osculating cones method
Mach 6 waverider designed using the osculating cones method [from Miller, Argrow, Center, and Brauckmann, 1998]

Viscous Waveriders:

The waveriders discussed thus far were all designed assuming inviscid flow (no friction). Since viscous forces play such a large role in hypersonic flight, we might expect that inclusion of these forces should have a significant impact on vehicle shape. Starting with conical flow waveriders based on the inviscid analysis discussed above, researchers have included viscous effects into the design process using integral boundary layer relationships. The designer need only vary the cone shock angle (theta s) to produce an entire family of optimized conical waveriders for a given Mach number. An example of such a series is illustrated below. Note how the shape of each vehicle changes to optimize its L/D performance as the assumed shock angle changes.

Viscous optimized waveriders at Mach 6
Series of viscous optimized waveriders at Mach 6 [from Bowcutt, Anderson and Capriotti, 1987]

In addition, the performance of each vehicle derived for a given design Mach number can be compared to locate a so-called "optimum of the optimums," or the optimum shape for that design Mach number. This is done in the graph below (a), and this plot indicates that the highest L/D value is obtained by the waverider designed for a shock angle of 12°. The resulting vehicle is also shown below (b).

Optimum Mach 6 waverider
(a) Comparison of waverider performance; (b) optimum Mach 6 waverider (theta s=12) [from Bowcutt, Anderson and Capriotti, 1987]

Note that the optimum shape is composed of very complex curves indicating that the optimization routine is carefully adjusting the vehicle shape to reduce both wave and skin friction drag. In fact, the authors noted that the optimum shape designed for any Mach number exhibited roughly equal values of both types of drag. Shock angles less than the optimum resulted in greater skin friction while the reverse was true of higher shock angles.

In addition, this optimization technique also allows the designer to optimize for different performance criterion and to apply different types of viscous boundary layers. The following figure shows the different types of configurations that can result from optimizing for maximum lift-to-drag ratio as opposed to minimum drag coefficient (C D). Also, note how the vehicle edges become more rounded as the assumed boundary layer profile becomes more turbulent.

Optimized Mach 6 and Mach 14 waveriders
Mach 6 and Mach 14 waveriders optimized for different conditions [from Bowcutt, Anderson and Capriotti, 1987]

The authors also applied this optimization technique to a Mach 25 waverider to produce a waverider body at the upper extreme of hypersonic flight. The resulting shape illustrated below, makes an interesting comparison with the Mach 6 waverider.

Optimum Mach 25 waverider
Optimum Mach 25 waverider [from Bowcutt, Anderson and Capriotti, 1987]

Note how much greater the wing sweep is at the higher Mach number due to the much smaller Mach cone produced at such a high speed. The raised spline centerbody apparent in the Mach 25 vehicle is also noteworthy. The feature results from a volume requirement placed on the design and indicates a compromise between minimizing skin friction and maintaining sufficient vehicle volume.

Returning briefly to the aerodynamic characteristics of hypersonic vehicles, recall the "L/D barrier" so important to hypersonic vehicle research:

Hypersonic lift-to-drag ratios
Maximum lift-to-drag ratios for hypersonic vehicles and the "L/D barrier" [from Anderson, 2000]

These viscous optimized designs (solid data points) appear to break that barrier and actually more closely obey the following relationship (represented as the dashed line in the above figure):

Thus, these results indicate that waveriders show great promise in breaking the L/D barrier and deserve great consideration in future hypersonic vehicle design. The viscous optimization techniques outlined here have also been applied to power law bodies since such bodies represent minimum drag shapes. See Sabean and Lewis, 1998, for more information about viscous-optimized power law bodies.

Star Bodies:

A final noteworthy waverider concept is another variation on the conical-flow design known as the star body. Instead of exhibiting the wedge-like shape of the concepts discussed thus far, the star body is actually composed of multiple conical-flow or power law body waveriders superimposed and joined at the leading edge. An example star body showing two joined power law shapes is shown below.

Power law body star body
¾ power law star body designed for Mach 6.4 [from Sabean and Lewis, 1998]

The theoretical advantage of this concept is that an attached shock will be formed between each of the star "fins" reducing wave drag. The cost, however, is greater surface area resulting in greater skin friction drag. Nonetheless, research indicates that a properly designed star body configuration can see up to a 20% reduction in drag over an equivalent conical waverider, as indicated below.

Star body versus conical waverider
Comparison of drag coefficients for a star body and conical waverider [from Sabean and Lewis, 1998]

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