Mud Siren Torque and Erosion Analysis - Measurement While Drilling Signal Analysis, Optimization and Design - page 196

and close the valve, should that be chosen as the rotor. Next consider the right
downstream lobe. If the indicated force is positive, it will tend to open the valve
if it were chosen as the rotor; if it is negative, on the other hand, the lobe will
tend to close the valve. If F single-lobe is the force acting on one lobe, as calculated
by numerical integration, the torque associated with that lobe is given by

T single lobe = R m F single-lobe (7.3.14)
The square and trapezoidal shapes shown are for illustrative purposes only
- in the computer program, different lobe aspect ratios and taper angles may be
assigned. In addition, lobe taper slopes at top and bottom need not be equal and
opposite; they can take arbitrary values and hold identical signs. In more
general engineering designs, when identical signs are taken, the resulting rotors
can be stable-closed, stable-open, or self-rotating, drawing upon the kinetic
energy of the oncoming mudstream; see, e.g., refer to the patent descriptions in
Chin and Ritter (1996, 1998) or U.S. Patents 5,586,083 and 5,740,126. One
analogy to airfoil analysis should be mentioned. In Figure 7.4, both positive and
negative lifts can be obtained depending on the angle-of-attack. Similarly,
positive and negative forces on siren lobes are in principle possible. However,
experimentally and numerically, it has never be possible to achieve a stable open
design for upstream rotors. Downstream rotors, depending on the taper chosen,
may be stable open, stable closed, neutrally stable in both positions, and also
stable in the partially-open position. Also, unlike the flow past turbine blade
rows (with upward lift), for which the flow deflection is downward,
computations show that the downstream deflection can be downward or upward,
accordingly as the total force on both rotor and stator is upwards or down.
7.3.9 Streamline tracing.
The streamline pattern assumed by any particular flow sheds insight on
locations prone to flow separation and those likely to induce surface erosion. It
can be constructed by tracing velocity vectors, by integrating “dy/dx = (vertical
velocity)/(horizontal velocity),” but this procedure is very inaccurate. For
example, particle locations easily “fly off” the computational box whenever high
surface speeds are encountered in the numerical integration. An alternative
method applicable to weak annular convergence can be implemented. The
theory is developed by first writing Equation 7. 3.13a in the more concise form
) xx + ) yy # / (7.3.15a)
where / denotes the right side of Equation 7 .3.13a, and then re-expressing it as
w{) x }/wx + w{) y - /y}/wy = 0 (7.3.15b)

This “conservation form” implies the existence of a function < satisfying

) x = < y (7.3.15c)

) y - /y = - < x (7.3.15d)