Trans RINA, Vol 157, Part A3, Intl J Maritime Eng, Jul-Sep 2015
and dynamic lift coefficient at a water flow velocity of 1.82 m/s, frequency of 1.5 Hz and demand T-Foil incidence (α) range of ±15º. Although this graph shows a good agreement between experimental lift coefficient and quasi-static lift coefficient, there is a deviation at the peaks when the T-Foil reaches the maximum and minimum angle of attack.
As the results presented in Figure 17 were obtained through the experiments, it was decided to compare these results
Theodorsen’s Unsteady Thin Airfoil Theory [20] was applied to conduct the theoretical calculations dynamic lift effects.
V= 1.82 m/s ; f= 1.5 Hz ; Demand α Range= ±15º
0.2 0.4 0.6 0.8
-0.8 -0.6 -0.4 -0.2 0
012 Time (s)
Figure 17: Comparison between experimental lift
coefficient and Quasi-static lift coefficient at water flow velocity of 1.82 m/s, frequency of 1.5 Hz and demand T- Foil incidence (α) range of ±15º.
According to the Theodorsen’s theory, the lift which contains both circulatory and non-circulatory terms is given by [20] in the form:
2
(7)
where the function C(k) is a complex-valued function of the reduced frequency k, given [20] by
(8)
where are Hankel functions of the second kind,
(9)
The function C(k) is called Theodorsen’s function and is real and equal to unity for the steady case (i.e., for k=0) [20].
Three different values for the unsteady lift coefficient are now considered: steady lift curve
of 2.45 per radian, ©2015: The Royals Institution of Naval Architects
the Theodorsen prediction based on a slope
direct
which can be represented in terms of Bessel functions of the first and second kind, given [20] by
Experiment Quasi-Static
with theoretical calculations. Therefore, for
experimental data from the unsteady water tunnel tests and quasi-static values obtained from the static water tunnel tests. The latter would only be expected to apply at low frequencies. These results are to be compared at two different water flow velocities of 1.82 m/s and 2.70 m/s and three different frequencies of 0.5 Hz, 1.5 Hz and 2.5 Hz. In addition, results are presented for three different demand T-Foil
incidence (α) ranges of ±15º,
±10º and ±5º for the flow velocity of 2.7 m/s. Figures 18 to 21 show these results as well as the amplitude response of the foil angle of attack for each case. The left panel of each figure shows the comparison of the demand angle of attack with measured angle of attack. The selected frequencies (0.5, 1.5 and 2.5 Hz) are in the range for which model testing will take place and relatively small magnitude and phase errors are expected between actual movement and demand movement within this as previously discussed.
The results shown for the response amplitude of the angle of attack show a small deviation in the measured range of the angle of attack compared with the demand range of the angle of attack. These results are expected based on the frequency response presented in Figure 12. Figures 18 and 19 show that Theodorsen’s theory somewhat under predicts the magnitude of unsteady lift at the highest frequency, but the quasi-static calculation over predicts the magnitude
of unsteady lift at all
frequencies. These results show that the Theodorsen’s theory prediction is relatively close to the experimental magnitude of unsteady lift at a frequency of 1.5 Hz which is the frequency that the INCAT Tasmania model encounters peak motions during tank tests [18]. As can be seen, there is no significant difference in the results between the two different flow velocities (Figures 18 and 19). Thus, it was decided to focus on high water flow velocity of 2.7 m/s as it is close to the design speed of the catamaran model.
Figure 20 shows the results at frequencies of 0.5 Hz, 1.5 Hz and 2.5 Hz, water flow velocity of 2.70 m/s and demand T-Foil incidence (α) range of ±10º. The results shown for the response amplitude of the angle of attack (left panel) show 5% to 10% deviation in the measured range of the angle of attack compared with the demand range of the angle of attack which was expected based on the frequency response presented in Figure 12. The results shown for the unsteady lift coefficient (right panel) show that the Theodorsen theory somewhat over predicts the magnitude of unsteady lift at the lowest frequency, however the quasi-static calculation again over predicts the magnitude
of unsteady lift at all
frequencies. These results show that the Theodorsen theory prediction is relatively close to the experimental magnitude of unsteady lift at frequency of 1.5 Hz with a discrepancy of 3% at the peak and 11% at the trough.
Figure 21 shows the results at frequencies of 0.5 Hz, 1.5 Hz and 2.5 Hz, water flow velocity of 2.70 m/s and demand T-Foil incidence (α) range of ±5º. The results
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