Here we outline a method by Kinnas & Hsin (1990) for quite exact repre-sentation of propellers of arbitrary shape in a spatially non-uniform flow of an ideal, incompressible fluid. This procedure is a recent development by Kerwin and his colleagues at MIT of computer methods for design and analysis of propellers.
The basic equation for the velocity potential on the surface of the blade SB and that containing the shed vorticity in the blade wake Sw, d. Figure 18.6, is obtained from Green's second identity with the use of 1/R as the Green function, d. Section 3 of the Mathematical Compendium, p. 506.
This equation will have the form of (M3.7) where S is the boundary sur-face of the fluid volume, Le. S
=
SB+
S;t+
Sw+
a surface around the entire propeller but "far" away from it. This interpretation makes S a single closed surface. SB is now the actual surface of the propeller blade (and not just the fluid reference surface used previously). Normals are defined positive out of the fluid as indicated in the Figure 18.6.A Potential-Based Boundary-Element Procedure 369
since the distance between two points on S;t and Sw is infinitely small.
The velocity normal to the trailing-vortex sheet is continuous. To see this, note that the helical trailers will give a contribution analogous to the straight-line trailers of the wing,
d.
Equation (10.5). The transverse vor-tices in the wake, due to the change of blade loading with time, can be interpreted as those bound to a two-<iimensional wing of varying circula-tion and the velocity calculated according to Equation (2.38). We then' have that 81>+/On+= -
81>-/On- and the last integrand is zero. The first integrand contains the jump in potential across the wake cut l11>w=
1>+-1>_which is non-zero, d. Equation (2.60) (although this applies to the steady case). We now have the equation for the velocity potentialWe now can see that the surface potential 1>is the superposition of poten-tials of distributions of sources of strength
8<p/
On' and normal dipoles of strength 1>on the face and back surfaces of the blade plus a distribution of dipoles on the wake surface Sw of strength l11>w.As in two dimensions, the history of the flow must be included. In the present formulation it appears in the wake term l11>w. This can be put in evidence and an expression secured which is needed for the solution of the problem by the following argument:
First, note that the pressure jump across the wake must be zero because there is no physical barrier, such as the propeller blade, to maintain a pressure difference. The full pressure equation referred to fixed axes is given by Equation (1.27) (with f(t)
=
0). With respect to rotating axes fixed to the blade (which is restrained from translating in a superimposed stream) for which 'Y= 'Yo+
1/1= wt+
1/1,the relative velocity vector, qrel= (-U +
ua+
u1r, ur+
u'jY,ut+
u'f - wr), must be used in the pressure equation and the pressure is then given by370 LiftinrrSurface Theory
A Potential-Based Boundary-Element Procedure 371
372 LiftinfiSurface Theory
The solution is carried out iteratively. For t
<
0 the steady case is con-sidered and the unsteady inflow is then "turned on" at t =o.
At eachtime step the propeller is rotated and (18.73) is solved using the solution of the previous step until convergence. Further details of the numerical procedures can be found in Kinnas
&
Hsin (ibid.) as well as results showing consistency of their boundary-€lement method with analytical results for forces of a flat plate.Among the results reported were calculated unsteady forces on a highly skewed propeller shown in Figure 18.8. The wake was of a special type with large amplitudes of the 5th, 10th and 15th harmonics 5 to 15 times larger than those of typical ship wakes and each being 10 per cent of the mean axial velocity. This wake was used to determine the number of panels required at four frequencies to secure convergent results. It was found that 40 chordwise and 30 spanwise panels give converged results after about three rotations of the propeller.
Harmonic analysis of the total unsteady thrust on this 5-bladed propeller revealed amplitudes at each of the harmonics of 1, 5, 10 and 15 times, which was to be expected, according to linear theory, d. p. 367. But addi-tional amplitudes arising from the sum and difference of the spatial orders present in the wake were found. These amplitudes are explained as arising from the non-linear leading-€dge suction forces giving rise to forces at twice the inflow harmonic orders as well as contributions from the sum and difference of the wake orders. The results at frequencies other than the four in the imposed wake gave amplitudes varying from 5 to 15 per cent of those at the basic orders. We surmise that these fractions are unduly large because of the non-typical large values of the imposed am-plitudes at the 5th, 10th and 15th orders.
This highly sophisticated procedure gives very close approximation of the pressure distribution for use in cavitation inception and for close pre-diction of mean thrust, torque and efficiency. As hull-wake harmonics at blade order and multiples thereof are small, it is sufficient to use the lin-earized theory for unsteady forces. Correlations with some measurement, which are displayed in the next chapter, ,show that the predictions of un-steady forces are reasonable to good.
We may conclusively remark that although this procedure, as outlined here, may seem simple, in particular when compared with the detailed formulas for the velocity field using Fourier series, p. 350-361, the numeri-cal aspects such as generation of grids over the blades and placing of col-lation points are complicated. With the high accuracy and generality of this and similar methods it can be considered as a "numerical cavitation tunnel", and like the real cavitation tunnel the results obtained, e.g., field pressures or blade forces, contain the whole "signal" and must be further analyzed to give the harmonics. Moreover, in the opinion of the authors, computer programs, despite their completeness, do not provide as much
A Potential-Based Boundary-Element Procedure 373
physical insight to the various aspects of propeller flows as do the more lengthy, albeit approximate, mathematical analyses also presented in this chapter.
Several aspects of propeller hydrodynamics have not been addressed in the foregoing. Model measurements of unsteady shaft forces have revealed rapidly varying modulations in amplitudes by large factors frequently within 1jz-th revolution, always at blade frequency (and integer multi-ples). This requires a theory to account for flows having spatial harmonics with amplitudes varying stochastically with time.
Another problem area is presented by the coupling of propeller induc-tions with hull-wake vorticity. Calculations of the alteration of nominal-to-€ffective wake by this mechanism have been given, d. Huang & Groves (1980), Goodman (1979), FalcAo de Campos & van Gent (1981) and Breslin (1992). A single calculation by Van Houten (see Breslin, Van Houten, Kerwin
&
Johnsson (1982)) showed large alterations of the har-monics of importance to the prediction of unsteady forces as well as the extent of transient cavitation.Moreover, the effects of viscosity and turbulence, which are particularly significant in off--<lesign operation, have been virtually untouched. Un-doubtedly these aspects will be attacked in the future by computer-€ffect-ed representations analogous to those currently being applied to turbulent hull flows, d. Stern, Toda