365 



nominally one percent of this channel's signal 

 energy consisted of harmonics or noise. 



The methods used in data reduction are now 

 described. The mean value, y , and the standard 

 deviation, a , were calculated in the usual manner. 

 The sine wave amplitudes and frequencies, and the 

 transfer functions were obtained using operations 

 on measured autospectra and cross spectra. These 

 spectra were obtained using overlapped fast Fourier 

 transform (FFT) processing of windowed data segments, 

 Nuttall (1971), where the following reduction 

 parameters were used: FFT size of 1024, 50 percent 

 overlap ratio, and full cosine data window. The 

 true autospectrum of a sine wave is an impulse, 

 0.5A 6(f - f ) ; the measured autospectrum is this 

 true spectrum convolved with the spectral window. 

 The spectral window associated with the cosine has 

 the form: 



/ sin Tif \ 

 V fTT(l-f2) I 



The wave frequency is, in general, not sampled at 

 a rate which is an integral multiple of the sampled 

 frequency. Thus, the measured spectrum consists 

 of this spectral window sampled at evenly spaced 

 frequencies where the location of the samples 

 relative to the sine wave frequency or spectral 

 window maximum is unknown. The sine wave frequency 

 and amplitude are found by fitting the spectral 

 window shape to the three largest samples that are 

 closest to where the sine wave is expected. The 

 transfer functions are given by the cross spectra 

 between the input and output data channels divided 

 by the autospectra of the input channel . The 

 transfer functions were evaluated at the frequency 

 of oscillation of the foil. Quadratic interpolation 

 between spectral samples was used to obtain the 

 cross and autospectrum values. Once evaluated, the 

 complex transfer functions were converted to magni- 

 tudes and phases. The transfer function magnitude 

 is then the output sine wave amplitude, and the 

 transfer function phase is the phase angle of this 

 output sine wave. Except for data runs when cavi- 

 tation was present, the cross spectra coherency 

 was always greater than 0.98; this high coherency 

 implies low noise and high linearity at the foil 

 oscillation frequency. 



of providing adequate information to analyze unsteady 

 cavitation inception. 



The foil was pitched about an axis at h chord 

 length from the leading edge. The instantaneous 

 foil angle a is given by 



o + ai 

 ^ 



(1) 



where a , Uj, and co are the mean foil angle, pitch 

 amplitude, and circular frequency of pitch oscil- 

 lation. Let Cp(t), Cpg, and Cpu(t) denote the 

 total pressure coefficient, the magnitude of the 

 steady pressure coefficient at the foil mean 

 angle, and the magnitude of the dynamic pressure 

 coefficient, respectively. At a given location on 

 the foil, it is assumed that: 



C (t) = 



P(t) - P„ 



P + P (t) - P 

 s u « 



where 



and 



h p V 



= c + c (t) 



ps pu 



p - p 



s « 



h p v„ 



(2) 



-ps Jj p V 2 



P (t) 



C (t) 

 pu 



(3) 



(4) 



!s p V_ 



where P(t), Pg , P^(t), and p are the local total 

 pressure on the foil, static pressure on the foil, 

 dynamic pressure on the foil, and the fluid density, 

 respectively; P^ and V^ denote the freestream 

 pressure and freestream velocity. We have: 



-C (t) 



pu 



- AC 



pu 



sin (ut + (fi) 



(5) 



where AC 



pul 



and if are the amplitude of dynamic 

 pressure response and phase angle, respectively. 

 A positive value of (j) means that the pressure 

 response leads the foil angle. 



Let the Reynolds number, Rn , and the reduced 

 frequency, K, be defined by 



V C 



(6) 



3. UNSTEADY HYDRODYNAMICS IN FULLY WETTED FLOW 



Basic knowledge in the general field of unsteady 

 aerodynamics has been compiled, condensed, and 

 presented by several authors [for example see 

 Abramson (1967)]. Available experimental hydro- 

 dynamics information for oscillating wings and 

 foils is very limited, especially at high values 

 of Reynolds number. Most of the available experi- 

 mental data concern lift, drag, and moment 

 coefficients from flutter and craft control 

 investigations. For cavitation inception studies, 

 accurate determination of the pressure distribution, 

 especially around the leading edge, is of major 

 importance. In the present investigation, three 

 pressure gage transducers were installed on the 

 foil to measure the unsteady surface pressures. 

 Experimental data were then correlated with an 

 available unsteady flow theory with the intent 



and 



K = 



2 Voc 



(7) 



where C, v, and id are the chord length, kinematic 

 viscosity of the fluid, and the circular frequency 

 of the oscillating foil, respectively. Fully wetted 

 experiments covered the range of Reynolds number 

 Rn = 1.2 to 3.7 x 10^ and reduced frequency K = 0.23 

 to 2.30. The test results are given in Tables la 

 to Ic. The phase angles and the amplitude of 

 dynamic pressure response per radian of pitch 

 oscillation are given in Figures 4 and 5 at values 

 of Oi = 0.5, 1.0, and 2.0°. 



An xansteady potential flow theory for small- 

 amplitude motion recently developed, Giesing (1968) , 

 is used here to correlate the experimental results. 

 The unsteady part of the pressure coefficient is 



