Mud Siren Torque and Erosion Analysis - Measurement While Drilling Signal Analysis, Optimization and Design - page 183

7.1.5 Numerical modeling.
The above needs spurred the development of a three-dimensional
computational system that quantifies not only the relationships needed between
the taper, azimuthal width and rotor-stator gap constraints summarized above,
but also between new design parameters such as annular convergence and
divergence in the inner drill collar wall and central hub space, both leading
toward and away from the siren assembly. The matrix of experimental tests
needed to validate any mechanical design can be significantly reduced by
identifying important qualitative trends by computer simulation.
In very early work, the geometry shown, say in Figure 7.3, was
“unwrapped” azimuthally and solved by modeling the two-dimensional planar
flow past a row of periodic, block-like, rotor-stator cascades. While this
approach is standard and very successful in aircraft turbine and compressor
design, the method led to only limited success because the three-dimensional
character of the radial coordinate was ignored. In aircraft applications, the radial
extent of a typical airfoil blade is very small compared to the radius of the
central hub, e.g., much less than 10% and even smaller. In downhole tools,
geometrical constraints and mechanical packaging requirements increase this
ratio to approximately one-half, making the above “unwrapping” questionable at
best: centrifugal effects cannot be ignored. Thus, the method that is described in
this chapter was required to be accurate physically, as well as computationally
fast and efficient, while retaining full consistency with the experimental
observations identified in Chin and Trevino (1988). Here, the motivation,
mathematical model and numerical solution, together with detailed computed
results for streamline fields, torques, pressures, and velocities, are described.
7.2 Mathematical Approach
Flows past mud sirens, even stationary ones, are extremely complicated.
Present are separated downstream flows and viscous wakes even when the
bluntly shaped lobes are fully open and not rotating. Such effects cannot be
modeled without empirical information. Flow prediction in downstream base
regions is extremely challenging, e.g., the viscous flow behind a simple slender
cone defies rigorous prediction even after decades of sophisticated missile
research in the aerospace industry. Some physical insight into certain useful
properties is gained from classical inviscid airfoil theory (Ashley and Landahl,
1965). In calculating lift (the force perpendicular to the oncoming flow),
viscosity can be neglected provided the flow does not separate over the surface
of the airfoil. This assumption applies at small flow inclinations. However,
viscous drag (parallel to the flow) and separation effects cannot be modeled
without using the full equations; at small angles, of course, boundary layer
theory is used to estimate resistance arising from surface shear stresses.
Theoretical versus experimental lift results for the NACA 4412 airfoil in Figure
7.4, for example, show excellent agreement prior to aerodynamic stall.