Geology Reference
In-Depth Information
With the aerodynamic formulation stated, emphasizing that our
“unwrapping” includes radial effects, we develop a boundary value problem
model appropriate to cascades of siren rotors and stators. We begin by
considering a portion of the unwrapped lobe structure, as shown in Figure 7.12.
Since the velocity field is periodic going from one set of rotor-stator lobes to
another, a smaller computational box can used in the flow domain at whose
upper and lower boundaries we invoke periodic disturbance velocities.
To be consistent with aerodynamic analysis, we focus on a primary blade
pair (arbitrarily) located along a (bottom) horizontal box boundary, as shown in
Figure 7.13. If the slopes associated with streamwise annular curvatures are
small, as they generally are in actual mud sirens, then Equation 7.3.12 applies to
leading order. If we note that setting R
m
T to y transforms Equation 7.3.10 to
classical form (e.g., see Figure 7.5), that is,
)
xx
+ )
yy
#
2{R
i
V
i
I
x
(x,R
i
,T) - R
o
V
o
I
x
(x,R
o
,T)}/(R
0
2
- R
i
2
) (3-13a)
#
2U
f
{R
i
V
i
- R
o
V
o
}/(R
0
2
- R
i
2
) (7.3.13b)
but modified by a non-zero right side Poisson term, it is clear that computational
methods developed for aerospace problems can be applied with minor
modification to mud siren torque analysis problems.
In our software, the source code and data structure of Chin (1978) was
modified to solve the foregoing problem. The flow in Figure 7.13 is solved by a
finite difference column relaxation procedure. The solution is initialized to an
appropriate guess for the flowfield, which can be taken as an approximate
analytical cascade solution or the flowfield to a slightly different siren problem
whose solution has been stored. The mesh is discretized into, say, 100
streamwise grids and 50 vertical grids. The solution along each column is
obtained, starting from the left and proceeding to the right, with one such sweep
constituting one iteration through the flowfield. Latest values of the potential
are always used to update all tridiagonal matrix coefficients. At the end of each
sweep, the deflection angle is updated based along the far right vertical
boundary. For the quoted 100
u
50 grid, as many as 10,000 sweeps may be
required for absolute convergence; this requires approximately three seconds on
typical computers for the efficiently coded velocity potential solution, and an
additional two seconds for the streamfunction streamline tracing solution. For
any particular siren design, torque characteristics are of interest for several
degrees of relative rotor-stator closure. If six or seven equally spaced azimuthal
positions are studied in order to define the torque versus closure curve, a
complete solution can be obtained in about one minute.
7.3.8 Interpreting torque computations.
How might computed torques be interpreted in terms of stable-opened and
stable-closed performance? Consider first the left upstream lobe in Figure 7.13.
If the indicated force vector is positive, the black lobe will tend to move upward
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