JounNeL on SpecEcnnrreNo Rocrers Vol. 33, No. 5, September-October 1996 Nomenclature a = vector of global parameters characterizing proj ectile-fl uid interaction a! , ai , a! , a! = parameters determining a specific localized interaction model C = integral aerodynamicforce coefficient Co = drag coefficient C1 = lift coefficient D = diameter ofprojectile base, m L = projectile length, m n" = local inner normal vector at a body's surface R = radius of a projectile base, m; seeFig. I Reo = Reynolds number basedon D Rp, Rr , RD, RL - parameters determining a specific iocalized interaction model &r = characteristic area, area of a projectile base, m' = exposed area,m2 = surface temperatureof a projectile, K = stagnationtemperature, K = freestreamvelocity, m/s = vector perpendicular to v-; seeFig. 1 = coordinates; seeFig. I = angle of attack = specific heats ratio = coordinates; seeFig. 1 = tangent vector aFa surface point in the plane determined by vectors n' and v'* = function determining a longitudinal contour of a projectile = functions determining a specific model of a proj ectile-fl ow interaction = angle between vectors no and u| Introduction n URING the last two decades various localized interaction mod- l-f els were developed andusedat different physical conditions for calculating the momentum and energy exchange between a moving projectile and the surrounding gaseous medium.l'2 In these models it is assumed that the total effect ofprojectile-fluid interaction can be represented as a sum of independentlocal interactions of small surfaceelementsof the projectile with a medium. These local inter- actions dependon the local geometric and kinematic characteristics of the surfaie and on global parameters, which are the same for all projectile surfaces. In the most widely usedversion ofthis approach, applicable for translational motion of a projectile, the only local pa- rameter is the angle between the main stream velocity vector and the local unit normal vector to the surface. Specific models ofthis type (e.g., seeRef. 1) were used in hypersonic aerodynamics in free molecular, intermediate, and continuum flow regimes. Becauseof the wide use of the local interaction models it is de- sirable to develop a calculation procedurethat is more efficient than direct integration of the local interactions over the exposed sur- face. This motivated the differential equations method,r-5 which reduces the problem to solving the recurrent system of ordinary differential equations. However, determining the particular solu- tions and constants in the general solutions of these differential equations is very cumbersome.A method suggested in this study appears to be more convenient in application since it allows the derivation ofexplicit formulas for aerodynamiccoefficients through one-dimensionalquadratures. Formulation of the Problem In general,localize9 interaction models can be represented math- ematically as follows' : cr : Qp(a, at)n' -f {2"(a, a)r' (l) where c6 is surfaceforce coefficient vector per unit surface area. The specific model is determinedby functions Qo and Q", and the parameters d have physical meaning (e.g., Mach number, Reynolds numbei, etc.). The integral force coefflcient is determined by inte- grating c p over the exposedsurface area of the body, where S* is determinedby the condition Method for Calculating Force Coefficients of Bodies of Revolution A. Dubinsky* and T. Elperinl Ben-Gurion University of theNegev, Beer-Sheva 84105, Israel Localized projectilefluid interaction models are widely used for calculating drag and lift coefficients in a high- velocity flow in free-molecular, intermediate, and continuum flow regimes. A fast procedure for evaluating drag and lift coelficients of bodies of revolution in the range of the angles of attack from 0 to 90 deg, which can be employed in these calculations, is suggested. The developed exact method reduces computation of integrals from local force coefficients over the exposed part of the projectile's surface at each value of angle of attack to the evaluation one-dimensional quadratures. Such simplification is achieved by analyzing all possible types of the exposed surface structure and derivation of the analytical formulas via integration over one of the coordinates. The developed procedure comprises a set of formulas and tables, which can be readily implemented in a computer code. Method is applied in an intermediate flow region with a localized interaction model often used in hypersonic aerodynamics and the obtained results are compared with experimental data. s- Tv1 Ts v" vo, '64 x,y,z a v p,0 Q@) oc) a Superscripts o = unit vector 0 = derivative with respectto p Received June 15, 1995; revision received Sept. 29, 1995; acceptedfor publication Dec. 15, 1995. C.opyright @ 1996 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. *Research Fellow, PearlstoneCenter for Aeronautical Engineering Stud- ies, Department of Mechanical Engineering, Faculty of Engineering Science, P.O. Box 653. tAssociate Professor of Mechanical Engineering, PearlstoneCenter for Aeronautical Engineering Studies, Department of Mechanical Engineering, Faculty of Engineering Science,P.O. Box 653. Member AIAA. c: + I 1,.,,0, (2) 665 cos a-r > 0 (3)