Multhopp's subsonic lifting-surface theory of wings in slow pitching oscillations

Abstract

A draft of this theory was completed by H. Multhopp during 1950, before he left the Ministry of Supply. It has been edited by the writer, who is responsible for the calculated examples. This report is an extension of Multhopp's subsonic lifting-surface theory (Ref. 1) from steady flow to harmonic pitching oscillations of low frequency. The method is applicable to wings of arbitrary plan-form. The basis of the method is to calculate the local lift and pitching momenf at a number of chordwise sections from a set of linear equations satisfying the downwash conditions at two points of each section. By neglecting terms of second order in frequency, the oscillatory problem is related to the corresponding steady one with changed boundary conditions. The evaluation of these conditions involves chordwise integrations, which require two new influence functions. Complete tables of these functions as well as the original functions i and j, occurring in steady motion (Ref. 1), are obtainable from the Aerodynamics Division, National Physical Laboratory (Ref. 11). With the aid of these tables the derivatives of lift and pitching moment become calculable by a straightforward routine. The limitations imposed by assuming only two terms in the chordwise loading cannot be evaluated at this stage. The theory is easily generalized to include any number of ehordwise terms, but each additional term introduces two further influence functions. The theory is outlined in sections 2 to 5. Section 6 describes calculations of pitching derivatives for circular, arrowhead and a family of delta wings; promising comparisons are obtained, when the number of spanwise terms is varied. In sections 7 and 8 these results are compared with other theories; a development of vortex-lattice theory (Ref. 5) is shown to give satisfactory agreement, and the deficiencies of a purely steady theory are evaluated. The available wind-tunnel data for oscillating wings of the selected plan-forms are discussed in section 9. The theory is remarkably consistent with the pitching derivatives measured at low speeds and predicts fairly well the effect of compressibility up to a Mach number of about 0.9. Appendix II gives instructions for the computer.