AN UNSTRUCTURED WAVE DRAG CODE FOR PRELIMINARY DESIGN OF FUTURE SUPERSONIC AIRCRAFT Sriram K. Rallabhandi¤ and Dimitri N. Mavrisy Aerospace Systems Design Lab, Georgia Tech, Atlanta Abstract programs, analysis of supersonic flight needs pres- 4 In a preliminary design environment, the designer sure wave propagation routines. The linearized sonic 5 needs to have freedom to quickly evaluate different boom theory is mostly based on the work of Whitham 6 configurations and come up with the most promising and Walkden. Various concepts of sonic boom theory 7¡8 configuration. In the supersonic regime, most lin- have been introduced and sonic boom minimiza- 9¡10 earized codes that are available today can only handle tion has also been investigated. specific shapes and configurations. These codes only Even though it is desired to use computational fluid aid in optimizing conventional configurations and do dynamics much earlier in the design, today’s computa- not span the entire space of possible shapes, which tional power does not allow a quick and easy prediction include revolutionary and unconventional configura- of the required aerodynamic data. This is the reason tions. This paper proposes using a set of GNU libraries people still rely on fast linearized models. However, and analyses codes to overcome the shortcomings of as the aircraft configuration assumes unconventional the legacy codes. It is known that any surface can be shapes in the design environment, most of the lin- discretized into triangles using efficient Delaunay tri- earized legacy codes fail to produce aerodynamic data angulation algorithms. The proposed method involves to the necessary accuracy or detail. Moreover, input to creating a triangulated aircraft from a generic CAD many of these legacy codes is generally in the form of environment, using the set of geometric libraries and awkward control cards and many times the actual ge- then performing necessary surface operations for the ometry needs to be tweaked for the code to handle the desired result, which in our case is the calculation of geometry. Since geometry holds the key to designing the wave drag. Linearized methods for wave drag esti- better aircraft, a tool or analysis code is needed that mation call for the calculation of the intercepted areas can perform the aerodynamic analysis without any as- of the aircraft with a Mach cone and the GNU libraries sumptions pertaining to the geometry of the aircraft. help us in obtaining these areas. Finally, in order to The linearized codes have to be constantly upgraded to validate the code, the new code is used to compute the be able to correctly model, at-least to the first order, wave-drag of a Sears-Haack body and F-16 and the re- the effects of revolutionary aircraft shapes. To increase sults are compared to the results from AWAVE, the the fidelity of the analysis, any proposed tool should Harris Wave Drag program. be capable of doing complex three dimensional geo- metric operations rather than being limited to planar approximations. In this study we attempt to improve Introduction the capability of the traditional wave drag method so The design of an efficient aircraft is dependent on that the new code can handle arbitrary geometries ac- the availability of rapid and sufficiently accurate the- curately. In doing so, we also obtain other important oretical and computational methods for aerodynamic parameters which will prove useful in the later stages analysis. Proper effect of the aerodynamic influence of the aerodynamic design. in the initial phases of design should be accurately Before proceeding to discuss a new program, let us predicted so that the best configuration can be se- look further into the shortcomings of the traditional lected. Traditionally, preliminary supersonic aerody- wave drag code that is used in most analyses today. namic analysis has relied on various linearized and Harris wave drag code was developed by Boeing and modified linearized1¡3 methods. In addition to these NASA Langley for the determination of fuselage cross- sections which yield minimum wave drag. This is ¤ Graduate Research Assistant,Student member AIAA achieved by enforcing the supersonic area rule, which yAssociate Professor, Boeing chair for Advanced systems de- sign, Associate Fellow AIAA employs Von-Karman’s slender body formula. Apart Presented at the 33rd Fluid Dynamics Conference and Exhibit, from approximating the geometry in terms of awk- Orlando, Florida, June 23-26, 2003. Copyright °c 2003 by Sriram Rallabhandi, D.N. Mavris, Published by American Institute of Aero- ward control cards, this program does not perform a nautics and Astronautics, Inc. with permission. smooth transition between the fuselage and wing root 1 of 8 American Institute of Aeronautics and Astronautics Paper chord causing holes or doubly accounted areas. The used in many of the preliminary aerodynamic codes. extreme left part of figure 1 shows the actual wing- The wave drag of a slender body12 is given by equation fuselage combination. The next two parts show the 1 approximation in AWAVE which causes gaps or dupli- Z Z cates area. The reason for this is that the program can ¡½U 2 l l only handle straight wing sections and has no strategy D = S00(x )S00(x )Lnjx ¡ x jdx dx w 4¼ 1 2 1 2 1 2 to deal with the component intersections. This leads 0 0 (1) to an incorrect area and volume distribution resulting in an over-prediction of the wave drag. Since AWAVE where ½ is the free stream density, U is the free stream 00 00 deals with crude approximations of conventional ge- velocity, S (x1);S (x2) are the second derivatives of ometries, coke bottled fuselages as well as interesting the area distribution with respect to the integration concepts like joined-wing, channel-wing and oblique- parameters, x1; x2, which are the locations along the wing cannot be handled with this setup. axis of the aircraft. To obtain an estimate of the wave drag of any configuration, we need an accurate repre- sentation of the second derivative of the intercepted areas. Although the area distribution is continuous along the fuselage axis, numerically we obtain the area only at certain finite number of points along the axis. It is important to maintain a sufficient resolution of the points along this axis. Failure to do so may lead to nu- merical round-off errors due to increased error in the second derivative calculation. As in any approxima- tion theory, a very fine resolution may cause numerical precision errors. Thus, the resolution should be fixed so as to reduce precision errors and avoid round-off errors. Fig. 1 Poor Aircraft Numerical Definition Once the second derivative of the area distribu- Using the same theoretical basis as used in AWAVE, tion has been obtained, we can perform a numerical we have developed a code which accepts any arbitrary quadrature to obtain the wave drag provided we know configuration as input and computes the wave drag of the flight conditions of the vehicle. In addition to the the configuration. The process involves triangulating wave drag, we can also compute the F-function which the surface to obtain a surface grid, calculating the is defined as: Mach-cone intersected areas of the configuration, ob- Z taining the second derivatives of the area distribution 1 x S00(t) F (x) = ; dt 1=2 (2) and performing numerical gaussian quadrature to ob- 2¼ 0 (t ¡ x) tain the wave drag and F-function. The surface grid Calculation of F-function is important because it has can be used as a first step to create volume grids and been shown that the disturbance pressure away from used for higher fidelity analysis. The F-function can be a supersonic aircraft can be obtained in terms of the used for shape optimization studies and Sonic Boom F-function as: analysis. Formulation γM 2F (x ¡ ¯r) ±p = p0 (3) Using linearized theory, it has been shown7 that the (2¯r)1=2 wave drag of an aircraft in supersonic flight is the same where ±p is the disturbance pressure, p0 is the undis- as the wave drag of an equivalent body of revolution turbed ambient pressure, x is the axial co-ordinate, M having the same cross sectional area distribution as the is the Mach number, ¯ is the Prandtl-Glauert factor, aircraft. The equivalent body has contributions from F is the F-function defined in equation 2 and r is the the volume as well as lift. Estimation of the equivalent radius vector of the point of interest from the aircraft. area due to volume involves the area of the aircraft in- Thus, F-function acts as an acoustic source term for tercepted by the Mach cone. Equivalent area due to the pressure signature propagation. Gaussian quadra- lift can be estimated by using a linearized lift analysis ture has been used to carry out most of the numerical program like Mach-box method or accurate non-linear integrations in this study because of its simplicity. CFD methods. However, in this study we are con- cerned only with the equivalent area due to volume. Geometry generation In the next few paragraphs we present the formula- With the procedure now well understood, we should tion of a wave drag prediction environment which uses be able to create and analyze a gamut of configura- a higher degree of geometric abstraction than has been tions in a short amount of time. This calls for efficient 2 of 8 American Institute of Aeronautics and Astronautics Paper surface parameterization techniques which can lead As each additional component is read, its holes are to efficient surface generation and automation. In filled and the surface is triangulated and combined this study, however, we use Rapid Aircraft Modeler with the already triangulated components using sur- (RAM), which is a conceptual level CAD environment face boolean operations to form a single unit.