Theoretical subsonic derivatives for an oscillating M-wing

Abstract

The report describes a theoretical investigation in support of measurements being made on an oscillating half-wing model of the plan-form shown in Fig. 1 in the National Physical Laboratory 25 in. by 20 in. Wind Tunnel fitted with slotted liners. Little is known about the steady or unsteady characteristics of M-wings. Results are obtained by low-frequency theory at Mach numbers 0 and 0.8 and by general theory for frequency parameters 0.3 and 0.6 (based on mean chord) at the Mach number 0- 8. The calculations cater for rigid pitching about an arbitrary axis and rigid bending about the wing root, the latter mode being used experimentally to estimate forces on a complete rolling M-wing. The sharp kinks located at the root and mid-semi-span of the leading and trailing edges subject the theories to a severe test. The calculated steady characteristics reveal a very slow rearward trend in aerodynamic centre as Math number increases and large discrepancies in the local aerodynamic centre over the outer panel of the M-wing. The oscillatory characteristics are summarized in tables of the calculated pitching and bending derivatives, the former being given numerically for the three pitching axes for which provision is made in the experiments. The figures show how the derivatives vary with axis position, frequency parameter and Math number. As compared with conventional delta or arrowhead plan-forms, the M-wing has a high minimum pitching damping at low speeds, which occurs for a pitching axis close to the aerodynamic centre. Although a change in frequency parameter from 0 to 0.6 reduces the damping derivative about rearward axes by roughly 30 per cent, the pitching oscillation shows no likelihood of becoming undamped. The error in the calculated values of the damping about all practical pitching axes may be as much as 15 per cent of its minimum value; only half this inaccuracy is incurred in the other derivatives. Nevertheless, exceedingly laborious calculations would probably be needed to establish their values to two places of decimals. The comparison between calculated and measured values of the pitching derivatives is good for the in-phase lift and moment and somewhat less satisfactory for the damping derivatives. The significance of the differences is doubtful, as slotted-wall interference effects in unsteady flow are unknown and there is reason to suppose that they may be large. The symmetrical rigid-bending mode is highly damped for the range of frequency parameter. Some calculations with an anfisymmetrical rolling mode have been made in order to estimate the corrections which must be applied to the experiments. It is shown that a factor of about 0.89 is necessary to convert the rigid-bending damping of the half-wing model to the rolling damping of the complete M-wing.