Hypersonic Slender Body Theory for All Angles of Attack

Euler's equations, the shock wave relations, and the surface boundary condition are used to derive a hypersonic slender body theory applicable to angles of attack from zero to 90 degrees. The theory is not based on small disturbances, but it gives Van Dyke's theory for small angles of attack, Sychev's similarity for moderate angles of attack, and strip theory for large angles of attack. The new similarity parameters are Meo~, 0 cos a/~, and y (where ~=o cos a + sin a). They are shown to be a combination of Sychev's parameters and can be used to modify equations for pressure and force coefficients at small angles of attack for moderate and large angle of attack applications. B(X,y,Z) CA CA CD CD CN CN Cp Cp d D Db A n p,p S(x,y,Z) T Nomenclature scaled function for body geometry axial force coefficient, based on ref. area ::: 2Ld scaled axial force coefficient given by eq. (50) drag coefficient, based on ref. area ::: 2Ld scaled drag coefficient given by eq. (51) normal force coefficient based on ref. area ::: 2Ld scaled normal force coefficient given by eq. (19) pressure coefficient, eq. (47) ... scaled pressure coefficient given by eq. (47) base height of lower surface, see Fig. 1 parameter defined by eq. (2) parameter defmed by eq. (44) unit vectors in x, y, z directions, respectively similarity parameter, 0 cos a~ similarity parameter, Moo~ body length freestream Mach number unit vector normal (inner) to surface static pressure and scaled static pressure, eq. (25) scaled function for shock surface temperature * Professor of Mechanical and Aerospace Engineering, Associate Fellow AlAA. "Copyright© 1992 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved." 542 u,v,w u,v,w ~ Veo,VOQ x,y,z x,y,z a ~ y o ~ Cartesian velocity components in x,y,z directions, respectively scaled (nondimensional) velocity components freestream velocity vector and magnitude Cartesian coordinates, see Fig. 1 x/L, y/d, z/d, respectively angle of attack shockwave angle ratio of specific heats, 1.4 for air lower surface thickness ratio, d/L, see Fig. 1 --'> lower surface thickness ratio relative to V 00' see Fig. 1 and eq. (15) p,p density and scaled density, eq. (22) circumferential angle, =0 in windward plane SUbscripts c Cone b body s shock wave 00 freestream value Introduction Analysis of inviscid, hypersonic flow past slender bodies has been developed by Van Dyke l for small angles of attack and by Sychev2 for large angles of attack. The small angle of attack analysis is based on small disturbance theory and the similarity parameters are Moo 0, a/o, and y. At large angles of attack the Sychev similarity parameters are Moo sin a, 0 cot a, and y. Hemsch3 showed that the large angle of attack similarity parameters can be combined to yield the small angle of attack parameters when a is small, i.e. (MOQ sin a) (0 cot a) --'> Moo 0 and 0 cot a --'> o/a. Cox and Crabtree4 and Hayes and Probstein5 discuss hypersonic small disturbance theory and Sychev similarity. BarnweU6 showed that Sychev's analysis is applicable to all slender-body flows with crossflow Mach numbers greater than sonic and hence is not restricted to hypersonic values of the cross flow Mach number. In addition, he showed that Sychev similarity applied to a number of slender-body flows with subsonic crossflow Mach numbers, including incompressible flow. Hemsch3 demonstrated empirically that Sychev similarity holds for any value of the cross-flow Mach number if the axial flow component is 4