by Dr. S.Kinnas, MIT

The authors presented an interesting application of inviscid flow methods for practical design. It would be very interesting also if they show convergence tests of the involved methods, in particular, with number of panels and time step size for different values of reduced frequency of the incoming gust.

Author's Reply

The numerical convergence for steady flow calculations has been reported in reference (2) of the main text. Numerical errors, defined as the root-mean-square values of the differences between the calculated Cp and the exact Cp distributions, were presented based on different panel numbers for both circular cylinder and NACA 0010 airfoil. It was concluded that the present approach generates results with an accuracy better than the second order methods and the panel number in the order of 100 for airfoil geometries is adequate to drive the error down to 10 −5.

For unsteady flows, the panel size is also governed by two other factors. First, the size of the panels near the foil trailing edge should not be larger than the distance between two successive vortices shed from the trailing edge.

That is

Δs=Δt•Uo (A1)

where Δt=2π/ωnt and nt is the number of time steps in one period. And Eq. (A1), in terms of reduced frequency f=ωc/2uo, becomes


Second, when downstream foils cutting through a vortical wake shed from upstream foils, the size of the panels adjacent to the wake vortices is limited to the distance traveled by the vortices in one time step during the close encounter between the vortex and the foil. This condition requires a similar constrain as shown in Eq. (A2) for panels in the front one-third of the foil.

For a typical calculation with a reduced frequency of 10, a normalized unity chord length and freestream velocity, and 25 time steps in one calculation period, c/Δs is about 80. On the average, we use 25 to 50 time steps and 160 to 300 panels to represent periodic foil motion in unsteady flows.

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