Supersonic and Hypersonic Minimum Drag for Bodies of Revolution

Forebody drag measurements have been acquired on a variety of nose geometries with fineness ratios ranging from 2 to 4, over an angle of attack range from 0 to 10 degrees, and at Mach numbers of 3, 4.5, and 6. Nose geometries tested included a tangent ogive, von Karman, power-series and cone. Test data indicates that the 0.7 power-series nose out-performs other geometries over the majority of the test conditions. Test data is compared to predictions made using the USAF Missile DATCOM and the Aeroprediction Inc./NSWC AP02 codes. In general both codes predict the drag within about 5-percent, however it is shown that the AP02 code more accurately estimates angle of attack effects on axial force. Nomenclature CAF Forebody axial force coefficient (body-fixed axis system) CAU Uncorrected forebody axial force coefficient (body-fixed axis system) CD Drag coefficient (wind axis system) Lref Reference length, diameter of model (2.214 in.=1 caliber) M Freestream Mach number q Freestream dynamic pressure, psi Sref Reference area based on reference length, (3.850 in) α Missile angle of attack, deg *Aerospace Engineer, Senior Member AIAA ** Aerospace Engineering Intern This paper is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Introduction The development of a tube-launched, hypervelocity missile that operates at sea level presents unique challenges in missile and projectile configuration design. From an aerodynamic standpoint, the minimization of total integrated drag during flight is critical to obtain maximum performance in range and lethality. As part of an extensive missile configuration trade study, public domain literature spanning over 50 years have been reviewed to determine if there is a nose planform that has minimum drag over the intended operational flight envelope (Mach 0 to 6). While the literature search provided a wealth of data, there was an apparent lack of directly comparable data. As a result, a study was undertaken to obtain forebody drag data on a wide range of nose planform shapes in an effort to identify the lowest drag nose planform. The objectives of this paper are to present recently acquired test data for a variety of nose planform geometries and to present comparisons of that data with predictions from engineering design codes. Historical Review In 1951, Perkins and Jorgensen published data indicating that the theoretical shapes for minimum diameter or given diameter and volume derived by von Karman and Haack do not have less drag than all other possible shapes having identical values of the same parameters. Their data covered a Mach range of 1.24 to 3.67 and also indicated that for a fixed fineness ratio of 3, the fore drag is reduced somewhat by small degrees of blunting, however for fixed cone angle, blunting always increased drag. In 1957, Eggers, Resnikoff and Dennis published data for various nose shapes over a Mach range of 2.73 to 6.28 that indicated that the 3⁄4 power-law body has the minimum forebody drag as re-presented in Figure 1. Figure 2 (also extracted from reference 3) indicates that modified Newtonian Impact theory predicts that the 0.7 power-law body may have the minimum drag for 3-caliber nose lengths. Perkins, Jorgensen, and Sommer, published data in 1958 that indicated that the paraboloid of revolution (1/2 power21st Applied Aerodynamics Conference 23-26 June 2003, Orlando, Florida AIAA 2003-3417 This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. 2 American Institute of Aeronautics and Astronautics law nose) had the least foredrag below M=1.5 and the hypersonic optimum shape (3/4 power-law nose) had the least foredrag above M=1.5. This report also presented data for tangent ogive, cone, power-series, ellipsoid, and LD, L-V, & D-V Haack noses planform geometries. Peckham conducted experiments on power-law bodies in 1965 and noted that minimum pressure drag is obtained by accepting higher pressures on a relatively small area of large slope near the nose, with a consequent reduction in pressure on a relatively large area towards the base of the nose. Building upon the work of Eggers, Resnikoff and Dennis, Miele uses Newtonian impact theory to show that the minimum drag for given constant diameter and length is achieved by a 0.75 power-law nose, and the minimum drag for given constant diameter and wetted area is achieved by a cone or 1.00 power-law nose. In 1966, Spencer and Fox published data at Mach 10 for a family of power-law bodies having values of the exponent of 0.25, 0.50, 0.66, 0.75, and 1.00, and a theoretical hypersonic minimum-wave-drag body. This study acquired data on 12 different body shapes with circular or elliptical cross-sections. The effects of increasing ellipticity for a given power-law body or the minimum-wave-drag body resulted in an almost constant incremental increase in maximum L/D, independent of body longitudinal contour. In 1989, Wesby and Regan presented hypersonic drag data for a series of 2-caliber power-law bodies at Mach numbers of 9.84 and 12.8, in the transitional rarefied flow regime (Reynolds numbers ranging from 1936 to 350000) that indicates that 0.75 is the exponent for minimum drag, and that ~0.64 is the exponent for minimum drag per unit volume given a fixed fineness ratio. In 1990, Mason and Lee published results from a numerical study that indicated that the minimum drag power-law body would have a 0.70 exponent. Experimental Test In the current study, a series of nose geometries were tested at Mach numbers of 3.0, 4.5, and 6.0 to characterize the drag. Tests were conducted at the Lockheed Martin Missiles and Fire Control (LMMFC) High Speed Wind Tunnel (HSWT) and at the NASA Langley Research Center (LaRC) Mach 6 facilities. Table 1 summarizes the test conditions. The model was 2.214 inches in diameter and was instrumented with a 6-component internal strain-gage balance housed inside a 4.719-caliber cylindrical afterbody. Figure 3 and Figure 4 depict the model hardware and nose geometry. Four pressure ports were used to measure base pressure. Table 2 presents the axial force balance accuracies in coefficient form for the given test conditions. Experimental Results Figure 5 presents a summary of the Forebody Drag Coefficient at zero Angle-of-Attack for various nose geometries on a standard cylindrical afterbody. This figure clearly illustrates the benefits of a high fineness ratio nose. It also clearly indicates that within a given fineness ratio, the power-series (specifically the 0.7 power-series) nose has the lowest planform wave drag for the majority of the test conditions. The Missile DATCOM predicted relative magnitudes of the drag components for a 3-caliber, 0.70 power-series nose are presented in Figure 6. This data is also for a clean, nose+body configuration. The significance of base drag in the subsonic through supersonic region is clearly shown for this optimized nose planform case. Base drag may be reduced by using a sustainer motor. If, however, significant base area is present, then the nozzle flow can act as an ejector to further lower the base pressure thereby increasing the base drag. However, since this base pressure acts only over the base minus nozzle area, the over all base pressure will be reduced in the majority of the cases. Comparisons with Design Code Predictions Figure 7 and 8 compare AP02 and Missile DATCOM Axial Force predictions, respectively, with wind tunnel test data at Mach 3 conditions. Here it is seen that both codes predict the zero lift drag within about 5%, however AP02 tends to more accurately predict the angle of attack effects. Figure 9 compares the measured Uncorrected Axial Force and the measured Forebody axial force to the values predicted by AP02 and Missile DATCOM for a 2-caliber, 0.70 power-law nose planform. This figure indicates that AP02’s modeling of angle of attack effects on base pressure is the likely reason that AP02 tends to better capture the AoA effects. Figure 10 presents the predicted axial force components (wave, skin friction, and base pressure) for a 2-caliber, 0.70 power-law nose planform. This figure indicates that the most significant difference between the predicted components below 10-degrees AoA, is the AoA dependency of base pressure as modeled in AP02. Figure 11, 12, and 13 present additional comparisons of AP02 and Missile DATCOM with test results. Table 3 through Table 6 present a summary of the codes predictive accuracies for both CAU and CAF. Here it is seen that AP02 tends to be slightly more accurate than Missile DATCOM, but both tend to predict body-alone drag within about 10 percent.