Test Case 3
As discussed in Section 4.6, the unsteady panel method uses steady-state conditions to
initiate the time-dependent inviscid flow solutions. This procedure is quite acceptable at low
frequencies but not at higher frequencies. To illustrate this with the inviscid panel method, we
show the variation of the lift coefficient with angle of attack for a NACA 0012 airfoil undergoing
a rotational harmonic motion,
  5  11sin t
Figure 10.3 shows the results for three cycles for   0.001 . As can be seen from Fig.
10.3a, the solutions that originate at   5o (t  0) show a small spike which continues up to
11o. After that the solutions are free from the initial conditions. Figure 10.3b shows that at the
beginning of the first cycle, the value of the lift coefficient is different than its value at the end of
the third cycle. The difference is small but increases with increasing frequency. This is
illustrated in Fig. 10.4 for   0.01 where the spike in the lift coefficient is much more different
than its value at the end of the third cycle. Figures 10.5, 10.6 and 10.7 show the results for
  0.1, 0.25, 0.5 respectively. As can be seen, in order to obtain truly unsteady flow solutions,
it is necessary to run at least two cycles. This is an important point to remember in performing
inviscid-viscous interactions since the spike due to steady-state effects can cause the breakdown
of the boundary-layer solutions.
Fig 10.3 Variation of lift coefficient cl with (a) angle of attack a and (b) cycle,   0.001
Fig 10.4 Variation of lift coefficient cl with (a) angle of attack a and (b) cycle,   0.01
6/28/2016 3:12 AM
1
Fig 10.5 Variation of lift coefficient cl with (a) angle of attack a and (b) cycle,   0.1
Fig 10.6 Variation of lift coefficient cl with (a) angle of attack a and (b) cycle,   0.25
Fig 10.7 Variation of lift coefficient cl with (a) angle of attack a and (b) cycle,   0.5
6/28/2016 3:12 AM
2