Second order theory for steady or unsteady subsonic flow past slender lifting bodies of finite thickness

Abstract

The well-known Munk-Jones-Miles slender theory predicts that the lift distribution and aerodynamic stability derivatives of slender bodies depend only on shape, but not upon thickness or Mach number (to a first approximation) for steady or unsteady flow. The present paper calculates corrections for thickness and compressibility from a second approximation to the nonlinear, time-dependent velocity potential (subject to corrected boundary conditions) and a refined pressure equation. The theory employs the Munk-Jones-Miles crossflow and Adams- Sears axial flow as the first approximation in a successive approximation scheme. The boundary conditions at infinity are satisfied by asymptotic matching of inner and outer flows. Calculated pressures for closed bodies show agreement with a known independent method for incompressible flow, and with experimental data for compressible flow, and suggest lower fineness ratio limits of 7 and 4, respectively, for first- and second-order theories. ECOND-ORDER corrections to the lift distribution on slender bodies in unsteady, subsonic flow are calculated from the general, nonlinear velocity potential equation that governs nonviscous, isentropic, time-dependent subsonic flow, as given by Miles 1 and formulated in a dimensionless axes coordinate system by the author.2 The present paper is restricted to small amplitude rigid motion of thick bodies, at low reduced frequencies, including steady flow (zero frequency) for which some experimental pressure data by Swihart and Whitcomb3 are available. Because of finite thickness of the body, coupling between the axial flow and crossflow is present as a consequence of nonlinearity of the velocity potential equation. The present theory complements the wind axis formulation of the author4 in his Ph.D. dissertation at UCLA which is now regarded as applicable to bodies (or for elastic incremental deformations from the axis reference frame); the present axis solution supersedes Ref. 4 for the rigid portion of any general time-dependent motion (up to and including cubic terms in reduced frequency for oscillatory harmonic motion). The high-frequency aspects and the wind axis body theory of Ref. 4 will be dealt with in a future paper since they are lengthy topics, and since there appears to be no experimental data on cambered slender bodies in subsonic flow. It can be said, however, that second-order cambered results have been obtained, via the theory of Ref. 4 (where of necessity, the lateral displacement is zero at the ends of a closed body). These second-order corrections are of the same order of magnitude as those for rigid bodies which are discussed herein, though naturally, they depend on the entire displacement mode shape.