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equation computationally, but the numerical model would require significant
computer resources and not be useful for real-time engineering design.
Thus we ask if a simple approach embodying three-dimensionality can be
developed with the convergence speed of faster two-dimensional methods. The
key is an integral approach not unlike the integral methods and momentum
models used years ago to solve the viscous equations for aerodynamic drag,
which is fast yet rigorous mathematically. The procedure is straightforward.
We multiply Equation 7.3.1 by 2Sr, integrate ³dr radially over the radial
limits R
i
< r < R
o
and introduce the area-averaged velocity potential
)(x,T) = A
-1
³I(x,r,T) 2Sr dr (7.3.2)
Here, R
i
is the inner hub radius at the bottom of a siren lobe, R
o
is the inner drill
collar radius at the top of the lobe, and A is a reference area to be defined. Then
R
o
R
o
)
xx
+ 2SA
-1
(rI
r
)µ + A
-1
³1/r
2
I
TT
2Sr dr = 0
(7.3.3)
R
i
R
i
Now approximate the r in 1/r
2
by the mean value R
m
= ½ (R
i
+ R
o
) to obtain
R
o
)
xx
+ 1/R
m
2
)
TT
#
- 2SA
-1
(rI
r
)µ
#
2SA
-1
{R
i
I
r
(x,R
i
,T) - R
o
I
r
(x,R
o
,T)}
R
i
(7.3.4)
Next observe that I
r
(x,R
i
,T)/I
x
(x,R
i
,T) represents the ratio of radial to
streamwise velocities at the inner surface; kinematically, it must equal the
geometric slope V
i
(x,T). Thus, I
r
(x,R
i
,T) = V
i
I
x
(x,R
i
,T) at the inner radial
surface. Similarly, I
r
(x,R
o
,T) = V
o
I
x
(x,R
o
,T) for the outer surface, so that
)
xx
+ 1/R
m
2
)
TT
#
2SA
-1
{R
i
V
i
I
x
(x,R
i
,T) - R
o
V
o
I
x
(x,R
o
,T)} (7.3.5a)
From thin airfoil theory, the complete horizontal speed I
x
is approximated
by the oncoming flow speed U
f
, which is permissible away from the siren lobes
themselves where flow blockage is significant, so that, in the absence of
azimuthal variations, Equation 7.3.5a becomes
)
xx
+ 1/R
m
2
)
TT
#
2SA
-1
U
f
{R
i
V
i
- R
o
V
o
} (7.3.5b)
Note that the right side represents a non-vanishing distributed source term when
general annular convergence or divergence is allowed. In spaces occupied by
solid lobes, the flow speed above is increased by the ratio of total annular area to
total “see through” port area, as will be explained in greater detail later.
7.3.2 Pressure integral.
Bernoulli's equation applies in the absence of viscous losses and arises as
an exact integral for inviscid irrotational flow. In cylindrical radial coordinates,
Equation 7.2.3 takes the form
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