The numerical method developed by 
Kerwin and Lee (8) is based on a 
linearized lifting-surface theory in the 
time domain. The propeller blades are 
represented by a spanwise and chord- 
wise distribution of discrete line 
vortex and source elements located on 
the exact camber surface of the blade. 
Thus the geometric complications of 
skew, rake and radial variation in pitch 
are readily accommodated. The trailing 
vortex wake is permitted to contract and 
roll up, and the effect of vortex sheet 
separtation from the blade tip is taken 
into consideration. The inflow velocity 
to the propeller may have radially and 
circumferentially varying axial, 
tangential, and radial components, and 
may therefore give rise to both steady 
and unsteady blade loading. This 
method, like the method of Tsakonas et 
al, assumes that the axis of the 
propeller slipstream coincides with the 
propeller axis. 
Kerwin (9) developed a refinement 
to the method of Kerwin and Lee (8) for 
operation in inclined flow where the 
slipstream is not axisymmetric about the 
propeller shaft. This refinement 
entails a more realistic representation 
of the path of the propeller 
slipstream. In this method the axis of 
the slipstream coincides with the 
propeller axis immediately behind the 
propeller, and coincides with the 
direction of the mean inflow far 
downstream in the ultimate wake. A 
simple function is assumed for the 
slipstream axis in the wake region 
between the propeller and the ultimate 
wake. Due to the asymmetry, the 
position of the trailing vortex wake 
relative to a blade oscillates with a 
once-per-revolution fundamental fre- 
quency, thus giving rise to unsteady 
induced velocities normal to the blade 
surface and thereby unsteady blade 
loadings of the same frequency. The 
strength of the vorticity in the wake, 
and thus the induced once-per-revolution 
variation of loading on the blades, is 
dependent upon the time-average load- 
ing of the propeller. All other 
characteristics of the method of Kerwin 
and Lee except for roll-up are retained, 
including the flexibility for the 
trailing vortex wake to contract, and 
allowance for the effect of vortex sheet 
separation from the blade tip. 
Correlations in Tangential Wakes 
Figures 9 and 10 present the 
amplitudes and phases, respectively, of 
the first harmonic loads on Propellers 
4661, 4710, and 4402 operating in the 
various tangential wakes over a range of 
JEL 
advance coefficient J. These figures 
present both experimental results and 
predictions based on the various 
theoretical methods described in the 
preceding section. For Propeller 4661 
with 10 degrees shaft inclination, the 
first harmonic components of Fy, My, 
Fy, and My are presented. For other 
conditions only the Fy component is 
shown. The results in Figure 9 and 10 
are summarized in Table II for design J. 
In the tangential wakes, a 
consistent variation from the 
experimental data occurred in the 
theoretical predictions for the three 
propellers evaluated. With few isolated 
exceptions, all of the theoretical 
methods underpredicted all of the 
loading components throughout the range 
of conditions evaluated. In general, 
the correlations are better for the Fy 
and My components than for the Fy 
and My components. The Fy and M 
components are the more important 
components since, in general, these are 
larger than the Fy and My components. 
The quasi-steady method of McCarthy 
(5) underpredicted the amplitude of 
(Fy)1 by approximately 10 to 30 
percent of the experimen tal values with 
closer agreement at higher values of J. 
The quasi-steady predictions main- 
tained a similar trend to the 
experimental data for Fy and My 
components for all conditions in 
tangential wakes; i.e., both the 
experimental and predicted values of 
K(Px) 1 and (Mx) decrease with 
increasing J. or a given propeller the 
predicted slope of the K(Px)1 and 
K(Mx) 1 Curves (not shown) with J 
increases with increasing amplitude of 
the tangential wake. No predictions of 
(Fy)1 and (My)1] were made with 
the quasi-steady procedure since this 
method does not predict the radial 
center of the load. These components 
could be predicted by the quasi-steady 
method by assuming a radial position of 
application of the load. 
The phases presented in Figure 10 
show that the quasi-steady method of 
McCarthy predicts that the maximum 
values of the loading components will 
occur at approximately 10 to 20 degrees 
of blade angular position before the 
experimentally determined angles. 
Predictions by the unsteady theory 
of Tsakonas et al (6,7) did not agree 
well with experimental results in 
tangential flow. This theory predicts 
that the amplitudes of all four loading 
coefficients increase with increasing J, 
