WAVERIDER DESIGN AND ANALYSIS USING COMPUTATIONAL FLUID DYNAMICS

Marcus Lobbia* and Kojiro Suzuki†

University of Tokyo 7-3-1 Hongo, Bunkyo, Tokyo 113-8656, Japan

ABSTRACT A waverider is a supersonic or hypersonic aircraft designed such that an attached shock wave is present along its entire leading edge; this prevents leakage of the lower surface flow into the upper surface region, providing the potential for high lift-to-drag ratios relative to conventional designs. Waveriders are designed inversely from a generating flowfield; therefore the use of both analytically-derived and CFD-derived flowfields was investigated. Comparison for a cone-derived waverider showed near identical results between the two methods; in addition several examples highlighting the expanded design space available through the use of CFD were generated. The application of CFD to the analysis of waverider flowfields was also investigated for various designs by solving the three-dimensional Euler and Navier-Stokes equations. The predicted lift-to-drag ratio of several designs was verified, and the relationship between leading edge bluntness and aerodynamic heating was also investigated. Only moderate degradation in lift-to-drag ratio for designs with relatively blunt leading edges occurred, while substantial reductions in the maximum heat flux were observed. These results help demonstrate the potential of waveriders in the design of realistic aerospace vehicles.

INTRODUCTION High-Speed Aerospace Design

In today’s society, fast and efficient transport of products and passengers from one location to another is an important ingredient in globalization. Current subsonic transport jets fly close to 90% of the speed of sound, and can deliver goods from one side of the earth to the other in less than one day. However, attempts to increase the speed of such transports over Mach 1 (e.g., supersonic transports such as the Concorde) have resulted in economic failure. Although concerns such as pollution and sonic boom effects are major reasons for this failure, the reduction of aerodynamic efficiency in the supersonic/hypersonic regime can also be considered one of the causes of the lack of success of high-speed civil transports.

One measure of the aerodynamic efficiency of an aircraft can be obtained by looking at its lift-to-drag ratio (L/D). The benefits of increasing the L/D of the design can be seen from the Breguet equation1 for cruise range:

(1) where mf is the total fuel mass, ms is the mass of the vehicle structure, mpay is the payload mass, d is the cruise range, Isp is the specific impulse, and U∞ is the cruise velocity. From Eq. (1), it can be observed that a low L/D can exponentially increase the fuel required to cover the same distance. Thus, for a high-speed cruise vehicle to have any chance of economic success, attaining a high L/D should be an important objective.

Waveriders One class of aerospace vehicles that have demonstrated the ability to attain a higher L/D compared to conventional designs is waveriders.2 A waverider is defined as a supersonic or hypersonic vehicle with an attached shock wave along its entire leading edge (see Fig. 1 for an example of a waverider designed from a two-dimensional wedge flowfield). The attached shock wave keeps the high-pressure flow contained below the waverider lower surface, thus allowing the potential for a high L/D. Because these vehicles appear to be riding

*Graduate Student, Dept. of Aeronautics and Astronautics, Graduate School of Engineering, email: mlobbia@sonic.t.u-tokyo.ac.jp †Associate Professor, Dept. of Advanced Energy, Graduate School of Frontier Sciences, University of Tokyo

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their own shock wave, they are referred to as “waveriders.” Research Objectives

The main objective of this research is to investigate the use of computational fluid dynamics (CFD) in both the waverider design process and the aerodynamic analysis of waveriders. The use of CFD in the design phase allows an expanded range of generating flowfields to be used in the inverse design of waveriders; through the use of this method, flowfields in which no analytical solution is possible (e.g., blunt-body flows) can be used to create waveriders. In order to validate this CFD-based design method, the aerodynamic performance and waverider characteristics (e.g., attached shock wave) of these designs are investigated by performing three-dimensional flow simulations around completed designs. Finally, waveriders are characterized by sharp leading edges (necessary for an attached shock wave), which can lead to large aerodynamic heating at hypersonic speeds. Therefore the tradeoffs between aerodynamic performance and heating resulting from blunt leading edges are also investigated.

METHODOLOGY

Waverider Design Method The inverse design process is commonly used to create waveriders. There are three main steps in this

method: 1) solution of the generating flowfield, 2) specification of the lower surface base curve of the waverider (used to describe a unique design for a given generating flowfield), and 3) the tracing of streamlines to form the upper/lower surfaces of the waverider.

CFD in the Waverider Design Process The first step in designing a waverider is to obtain the solution to a generating flowfield used in the inverse design process. In much of the research regarding waveriders, conical flows are commonly used for this purpose. The analytical solution for supersonic conical flow is expressed by the Taylor-Maccoll equation, which is an ordinary differential equation that can be integrated numerically using a fourth-order Runge-Kutta method. Similarly, other supersonic/hypersonic analytical flowfields (e.g., wedge/oblique shock wave) can be used to provide a relatively quick and simple method for obtaining the generating flowfield for waverider design.

Alternatively, CFD can be applied to obtain the solution of the generating flowfield. The Euler equations, in particular, are well suited to the waverider design philosophy – the tangential flow surface boundary condition in such a solution corresponds well with the idea of tracing streamlines to form the surfaces of waveriders. The solution of the two-dimensional axisymmetric Euler equations provides the potential of an expanded design space relative to analytical flow solutions; power-law or blunt-body flowfields, for example, can be used to create waveriders.3

Once the generating flowfield has been obtained, the use of the lower surface base curve of the waverider specifies a unique design for a given flowfield. The lower surface of the waverider is created by tracing the streamlines of the generating flowfield upstream; intersection with the shock wave specifies the leading edge of the waverider. The upper surface is then constructed by tracing back in the freestream direction. The lower surface of the waverider is parallel to the streamlines in the generating flowfield; thus theoretically it should create the same shock wave from its own leading edge.

Design Optimization

Although waveriders can be designed using the previous technique, it is desirable to find a design that is optimal based on several specified parameters. For example, the objective function:

(2)

can be minimized in order to generate an optimal design based on the aerodynamic performance, the volumetric

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efficiency (where V is the volume and A is the wetted area of the waverider), and the the length/width ratio of the waverider. The weights a, b, and c are arbitrary constants describing the strength of each factor in the optimization process.

The optimization algorithm used is the Nelder-Mead Downhill Simplex method.4 In this technique, 1) N+1 configurations are initially generated, 2) a new configuration is generated based on the best N configurations, and 3) the new configuration replaces the worst. Steps 2 and 3 are repeated until the difference between the best and worst designs reaches a specified tolerance. For this research, the N parameters varied during the optimization process describe the lower surface base curve (and thus the shape) of the waverider.

Computational Fluid Dynamics

When a CFD-based waverider design methodology is employed, the two-dimensional axisymmetric unsteady Euler equations are integrated using a diagonalized ADI algorithm.5 Yee’s Symmetric TVD scheme6 (second-order spatial accuracy) is used to provide accurate shock capturing; a local time step method7 based on the local grid size is also used to accelerate convergence to the steady-state solution. For these flow solutions, the grids were created by solving the two-dimensional Poisson equations.

In addition, CFD is also used to investigate the three-dimensional flowfield characteristics of completed waverider designs. For this purpose, the three-dimensional Euler equations were solved using the same numerical techniques as described above. Several simulations were also performed to investigate the heating environment of waveriders with and without blunt leading edges; in these cases the three-dimensional Navier-Stokes equations were integrated using the LU-SGS technique8 (other details of the numerical algorithm remain unchanged). The grids for these simulations were generated using algebraic interpolation; the grid points near the surface were modified to provide orthogonality at the surface (which results in reduced error for heat flux computations).

RESULTS AND DISCUSSION

Blunt-Body-Derived Waveriders By using CFD instead of the Taylor-Maccoll equation for waverider design, waveriders can be easily generated from non-conical flowfields. Since CFD is a shock-capturing technique (versus the shock-fitting approach of the Taylor-Maccoll equation), the method used to determine the shock wave location can influence the final waverider design. For this research, the streamlines were traced from the base region upstream in the generating flowfield; it was assumed that the shock wave was reached when the local pressure was within a specified tolerance of the freestream pressure of the flow. One of the difficulties in waverider design is creating configurations with a large usable volume (i.e., high volumetric efficiency). Therefore, the use of blunt-body-derived flowfields was investigated to see if any advantages resulting from the different generating flowfield were present. Eight optimized waveriders were generated from conical and blunt-body flowfields (all solved using CFD); four different flow conditions were used: M=6, δ=10 deg., M=6, δ =16 deg., M=10, δ =10 deg., and M=10, δ =16 deg. The lengths of each design were scaled to 60 m, and the freestream conditions corresponded to an altitude of 30 km (the relationship of the cone half-angle to the blunt-body shape is shown in Fig. 2). For each case, both a conical and blunt-body flowfield were solved. Each design was optimized with the same optimization constants and started from the same initial parameters; in Eq. (2) a=6, b=2, and c=0 were used; these constants were chosen arbitrarily to maintain a balance between aerodynamic performance and volumetric efficiency in the optimization of the various designs.

The waveriders are shown in Fig. 3. In general, cone-derived waveriders seem to provide the best overall characteristics (e.g., higher L/D and volumetric efficiency). For waveriders designed from large-angle generating flowfields, blunt-body flowfields can provide a higher L/D (relative to cone-derived waveriders). However, if volumetric efficiency is of primary importance, then using conical flowfields to generate waveriders can result in more efficient designs. These results are summarized by comparing L/D and volumetric efficiency of each

configuration (see Fig. 4). Part of the reason for the less-favorable characteristics of the blunt-body-derived waveriders might lay in the optimization process – for all eight cases, the same optimization constants were used to create a design. Additionally, in this research, the half-angle of the generating flowfield was fixed during the optimization process (due to the time constraints resulting from the use of CFD to obtain the generating flowfield). The half-angle is a major factor influencing the inviscid performance of the waverider, therefore it is desirable to include this as a variable in the optimization process. Finally, it should be noted that conical and blunt-body flowfields are substantially different; some variation of the generating flowfield (e.g., half-angle) is required to provide a more accurate comparison of the two types of designs.

Waverider Flowfield Investigation

Because waveriders exhibit an oblique shock wave attached to its leading edge, it is possible that the lower surface flowfield on a blunt-body-derived waverider may not mirror that of the generating flowfield, where a normal shock wave exists near the nose of the blunt-body, and the flow exhibits a rapid change from high pressure (just after the normal shock) to low pressure (near the base region of the blunt body). On the other hand, the waverider is designed using flow properties well away from the nose region of the blunt-body, thus the characteristics of this region might be similar to that of an oblique shock flow. Therefore, it is of interest to numerically simulate a waverider derived from a blunt-body flowfield to assess the correctness of this design method. A 61×31×31 grid was generated around a Mach 6, δ=10 deg. blunt-body-derived waverider; the flow properties were investigated by solving the three-dimensional Euler equations around the configuration at design conditions (Eckert’s reference temperature method9 was also applied to the design and Euler results to estimate the skin friction).

The density contours in the base plane are shown in Fig. 5; the design results are also shown for comparison on the right side of this figure. The overall shock location seems to be accurate (i.e., the shock is attached to the leading edge of the waverider as expected); however, some differences in the density contours are present. The design results predicted a L/D of 3.58, whereas the Euler results indicated the L/D to be 4.07, a relative difference of 13.7%. Some of the discrepancies might be attributed to coarseness of the grid used in the CFD simulation. The agreement of the aerodynamic properties and the attachment of the shock wave to the leading edge of the design (Fig. 6 shows the CFD results at several cross-sections) indicate that the concept of using a blunt-body flowfield to design waveriders is an acceptable methodology.

Leading-Edge Heating

The effects of non-sharp leading edges were investigated by numerically simulating the laminar viscous flow around four different versions of a Mach 5 waverider (length=0.6 m, altitude=20 km). The first configuration uses sharp leading; the other configurations use varying degrees of leading edge bluntness (with leading edge radii of 0.15, 0.3, and 0.6 cm). The grids used in these simulations ranged from approximately 100,000 to 300,000 cells; an example of the grid around the configuration with the largest bluntness is shown in Fig. 7. Although a high level of clustering was used for the computational grids in each case in order to allow accurate prediction of viscous effects, prediction of heating requires even finer resolution. Therefore, a new grid was generated around the nose region of each waverider, and the maximum heat flux was obtained on this grid separately from the full-configuration L/D results.

The maximum heat flux and L/D for each configuration is summarized in Fig. 8. It can be observed that, although the use of blunt leading edges can dramatically reduce the aerodynamic heating on the vehicle (e.g., even for the smallest leading edge radius simulated, qw is less than 10% of that in the sharp configuration), the aerodynamic performance is also penalized. One cause of the L/D reduction is due to the blunt nose; waveriders tend to be slender vehicles and thus this bluntness can cause a significant increase in the drag of the vehicle. An additional cause of the decrease in aerodynamic performance can be seen in Fig. 9, which shows the density contours in the base plane for the configurations with leading edge radii of 0.0 cm and 0.6 cm. The blunt leading edge allows leakage of the high-pressure lower surface flow into the upper surface region, causing a reduction in

the lift of the waverider. A smaller leading edge radius lessens the effect, but a reduction in L/D is still observed. Although only four levels of leading edge bluntness were investigated, it can be observed from Fig. 8 that

the configuration with the minimum leading edge radius of 0.15 cm provided an optimum balance between heat flux reduction and L/D penalty. Additionally, the waverider simulated was a relatively short 60 cm in length – a realistic aerospace configuration might be 100 times larger, therefore the same leading edge radius applied to such a vehicle should have a substantially less-significant impact on the aerodynamic performance.

CONCLUSIONS

In this research, the use of CFD in the design of waveriders was demonstrated. Blunt-body flowfields (for which no analytical solution is available) were used to generate waveriders; comparison with cone-derived waveriders showed that the generating flowfield has significant effects on the final optimized designs. Additionally, CFD investigation of a blunt-body-derived waverider successfully verified the attached shock wave characteristics of the design; comparison with the design predictions for the configuration indicate that blunt-body flowfields can be used successfully in the design of waveriders.

Finally, viscous flow simulations were performed to investigate the effects of leading edge bluntness on waveriders. It was found that even minimal leading edge bluntness resulted in substantially reduced maximum heat flux; however, the L/D was also decreased. For application to realistic aerospace missions, it can be inferred that a tradeoff between aerodynamic heating and performance is required for an optimal design.

REFERENCES

[1] Raymer, D. P., Aircraft Design: A Conceptual Approach, AIAA Education Series, 1999 [2] Bowcutt, K. G. and Anderson, J. D., Jr., “Viscous Optimized Waveriders,” AIAA 87-0272, 1987 [3] Lobbia, M. and Suzuki, K., “Design of Waveriders from Conical and Blunt-Body Flowfields,” 22nd ISTS

Proceedings, ISTS 2000-e-07, 2000 [4] Nelder, J. A. and Mead, R., “A Simplex Method for Function Minimization,” Computer Journal, Vol. 7, pp.

308-313, 1965 [5] Obayashi, S., Matsushima, K., and Fujii, K., “Improvements in Efficiency and Reliability for Navier-Stokes

Computations Using the LU-ADI Factorization Algorithm,” AIAA Paper 86-0338, 1986 [6] Yee, H. C., “A Class of High-Resolution Explicit and Implicit Shock-Capturing Methods,” NASA

TM-101088, 1989 [7] Pulliam, T. H., and Steger, J. L., “Recent Improvements In Efficiency, Accuracy, and Convergence for

Implicit Approximate Factorization Algorithms,” AIAA Paper 85-0360, 1985 [8] Jameson, A. and Yoon, S., “Lower-Upper Implicit Schemes with Multiple Grids for the Euler Equations,”

AIAA Journal, Vol. 25, No. 7, pp. 929-935, 1987 [9] Eckert, E. R. G., “Engineering Relations for Heat Transfer and Friction in High-Velocity Laminar and

Turbulent Boundary-Layer Flow Over Surfaces with Constant Pressure and Temperature,” Transactions of the American Society of Mechanical Engineers, Vol. 78, No. 6, pp. 1273-1283, 1956

Fig. 1: Waverider generation from wedge flowfield

Fig. 2: Relationship of cone half angle to axisymmetric blunt body

Fig. 3: Optimized waverider configurations generated from cone and blunt-body flowfields

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Fig. 4: L/D and volumetric efficiency comparison for optimized waveriders

Fig. 5: CFD and design density contours in baseplane of blunt-body-derived waverider

Fig. 6: CFD density contours at various cross-sections of blunt-body-derived waverider

Fig. 7: Grid used for Navier-Stokes heating analysis of waverider with blunt leading edges

Fig. 8: L/D and maximum heat flux for waveriders with sharp and blunt leading edges

Fig. 9: Baseplane density contours of waveriders with a) sharp and b) blunt leading edges

a) b)