Three-dimensional closure of the passage-averaged vorticity-potential formulation

Abstract

with 40 panels. Also shown is the result from thin wing, small-amplitude theory. 14 Apart from a mild transient in the first cycle, the results rapidly converge to a steady-state solution, and with use of Eq. (3), this is within 2.5% of the thin wing solution. The linear variation of potential on the first wake panel made the calculation relatively insensitive to the time step size. 12 With a Kutta condition in the form of Eq. (1), the lift coefficient values were underpredicted by about 30%. Figure 2 shows the pressure distributions over the section for this motion at the nondimensional time t n = 20: Fig. 2a is with the Kutta condition in the form of Eq. (1); Fig. 2b is with use of Eq. (3). At this time step, the wing is descending with a nosedown pitch and lift is in an upward direction. Use of Eq.(1) leads to a pressure coefficient difference at the trailing edge of about 0.17 and generally lower absolute values of the pressure coefficient (and lower circulation) compared with the calculation done using Eq. (3), which insures equality of pressure on upper and lower surfaces at the trailing edge. Pressure coefficient differences at the trailing edge varied over the range ± 0.3 for this calculation.Large differences in the pressure coefficient at the trailing edge again became predominant when calculations were done for conditions where the angle of attack of the airfoil was large (above about 15 deg) over a portion of the cycle. However, in real flow, separation would occur in these conditions, so this does not cause a limitation on the method in practice.