Implement the non-circulatory lift calculation of Section 4.6 and use it to draw Figure 4 .40.

Data from Section 4.6

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Data from Figure 4.40

Transcribed Image Text:

Theodorsen's function was developed using the assumption that the wake is flat. Numer- ical modelling with a free wake is a tool that can be used in order to investigate the effect of this assumption on the aerodynamic loads. In Example 3.13, we calculated the amplitude of Theodorsen's function from the ratio of the circulatory unsteady lift to the circulatory quasi-steady lift at different reduced frequencies. We will do the same here using lift responses obtained from Basu and Hancock's method. It is not easy to obtain the circulatory lift directly from this numerical method, but we can estimate it from the difference between the total lift and the non-circulatory lift. Theodorsen theory and Basu and Hancock's approach share the same basic ingredients: the non-circulatory lift is calculated by imposing impermeability using a source distribution while the circulatory lift is calculated by imposing the Kutta condition using a vortex dis- tribution. This means that the non-circulatory lift can be readily calculated by ignoring all the vortices, both on the surface and in the wake. The impermeability boundary condition of Eq. (4.154) simplifies to An(t) = b(tk) where b(t) is given by Eq. (4.173), um(tk) and vm(tk) being calculated from Eqs. (4.77) and (4.78), respectively. Solving for (tk) we can then calculate the tangential velocities on the surface from u(t) = A(t)o(tk) + um (tk) oTx (tk) + Vm (tk) 0 Ty (tk)