Constrained and Unconstrained Flow about a Pitching Foil

Abstract

HIS paper presents a numerical method of evaluating the unsteady flows around lifting profiles in motion in a twodimensional perfect fluid. A particular study is made of quasisteady flow about a rigid foil of RAE 104 profile in pitching oscillation about the quarter-chord axis with a zero incidence mean. The effects of flow constraint from equidistant parallel walls is presented, together with corresponding results. Contents In an early analysis of two-dimension al unsteady motion,! a potential flow theory for small-amplitude pitching oscillations of thin aerofoils made the assumption that the wake vortices moved downstream in a single plane at the velocity of the undisturbed fluid. A later analytical method included the effects of aerofoil thickness and a possible oscillation of the rear stagnation point.2 A numerical method for unsteady two-dimensional flow of a perfect fluid around moving bodies of arbitrary cross section3'4 was developed from a steady flow method5'6 in which the lifting profile was represented by a distribution of source and vortex elements. In the following numerical model, a moving profile with a defined trailing edge is represented by n uniform straight source segments of strength ajf j=l...n, arranged segmentally around the profile. In general, a circulation TC occurs about the profile which is represented by a distribution of m vortices of strength yjf j=\,...,m, located within the profile. The flow is considered for successive instants at times / = 0, t = eAt, e = l,2,3..., during the motion, and the following conditions are imposed at each instant: 1) The normal flow velocity component at the center / of each line source segment is equal to the corresponding velocity component of the segment. 2) The flow velocity component at the trailing edge in the direction normal to the bisector of the trailing-edge angle equals the velocity component of the trailing edge in that direction. Hence no discontinuity is permitted between the upper and lower surface tangential velocities at the trailing edge, and the Kutta condition is satisfied at each instant. In unsteady flow, a change in the profile circulation TC is associated with net transport of vorticity from the profile boundary layers into the wake. This is represented in the model by the detachment of a discrete vortex element of constant strength TWJ from the trailing edge during each time interval between successive instants. Hence the Kutta condition essentially is relaxed during each time interval. After detachment, the vortex elements convect downstream to represent the wake and form an integral part of the flow calculation. The total circulation r on a closed path taken around the foil and wake is constant and equal to the circulation at the commencement of the motion. Hence,