Navier-Stokes predictions of pitch damping for finned projectiles using steady coning motion

Abstract

Previous theoretical investigations have proposed that the side force and moment acting on a body of revolution in steady coning motion could be related to the pitch-damping force and moment. In the current research effort, this approach has been applied to produce the first known Navier-Stokes predictions of the pitch damping for finned projectiles. The flow field about finned kinetic energy projectiles in steady coning motion has been successfully computed using a parabolized Navier-Stokes computational approach. The computations make use of a rotating coordinate frame in order to solve the steady flow equations. From the computed flow field, the side moment due to coning motion is used to determine the pitchdamping coefficient. The computational predictions of the slope of the side moment coefficient with coning rate normalized by the sine of the angle of attack have been compared with pitch damping coefficients determined from range firings for two finned projectile configurations. The predictions show good agreement with the range data. This computational approach provides a significant predictive capability for the design of kinetic energy projectiles whose terminal ballistic performance can be degraded by moderate levels of yaw at the target. Nomenclature a00 freestream speed of sound c m pitching moment coefficient c m , slope of the pitching moment coefficient with angle of attack Cm, + Cm, pitch damping moment coefficient c n side moment coefficient * Senior Member, AIAA t Associate Fellow, AIAA Cn, slope of the side moment coefficient mith angle of attack c n slope of the side moment coefficient with coning rate c n , Magnus moment coefficient Chra slope of the normal force coefficient with angle of attack CN, + CN, pitch damping force coefficient slope of the side force coefficient with coning rate Magnus force coefficient projectile diameter total energy per unit volume flux vectors in transformed coordinates source term in Navier-Stokes eqs. jacobian characteristic length, typically D freestream Mach number pressure, as used in N-S eqs. spin rate, as used roll equations freestream static pressure Reynolds number, amp, Dip, distance downrange center of gravity shift, calibers viscous flux vector reference area of projectile, 7 r ~ ~ / 4 time velocity components in x,y,z directions freest ream velocity Cartesian coordinates w.r.t. body axial location of body center of gravity Note: Force coefficients are scaled, F / ; ~ , a& M& Srej ; Moment coefficients are scaled, M / f p, a& M& DS,,~