Mud Siren Torque and Erosion Analysis - Measurement While Drilling Signal Analysis, Optimization and Design - page 194

arising purely from inviscid considerations. This loss is small compared to the
loss that would be realized in the actual viscous flow, one that should be
estimated from wake models or orifice formulas. We emphasize that Equation
7.3.12b is not to be used in computing pressure drops through the downstream
wake. Finally, periodic velocities are assumed at the top and bottom of the
problem domain in Figure 7.11. It is computationally important that single-
valued velocities are modeled, noting that velocity potentials themselves are
multivalued if torques are nonzero and lead to programming complexities.
7.3.6 Lobe tangency conditions.
Equation 7.3.5a for the averaged potential is solved with flow tangency
conditions on lobe surfaces. These are easily derived. For example, the ratio of
the vertical to the horizontal velocity is kinematically R m -1 I T /I x = f x (T,x,r)
where f(T,x,r) represents the surface locus of points and the subscript x is the
streamwise derivative. We can rewrite this as R m -1 I T = f x (T,x,r) I x , multiply
through by 2Sr, carry out the former integration, and introduce our definitions
for ). If f x (T,x,r) is approximated by f x (T,x,R m ), we obtain R m -1 ) T /) x = f
x (T,x,R m ) to leading order, where ) x is roughly U f .
One significant modification to I y (x,0)/U f #”slope” in Figure 7.5 must be
made. In airfoil cascade analysis, as in thin airfoil theory, flow blockage is very
minimal, and the left side of the approximate tangency condition expresses the
ratio of the vertical to the horizontal velocity, taken to leading order as the
freestream speed itself. This treatment does not apply to mud sirens because
flow blockage in the neighborhood of port spaces is significant, nominally
amounting to half of the flow area. Thus, U f in the boundary condition must be
replaced by U hole , which can take on different values for rotor and stator. At x
locations not occupied by solid siren lobes, the velocity I x can be approximated
by U f . But at locations occupied by the siren, it is convenient to introduce a
“see through” area A hole . Mass flow continuity requires that A hole U hole =
A total U f , where A total = S (R o 2 - R i 2 ). Thus, the value of U hole is completely
determined everywhere along the streamwise direction.
7.3.7 Numerical solution.
The classical aerodynamic cascade problem, in summary, solves Poisson's
equation subject to (1) uniform flow far upstream, (2) approximate tangency
conditions evaluated on a mean line, (3) periodic velocities at the top and bottom
of the computational box in Figure 7.11, and finally, (4) selection of a deflection
angle D far downstream that is consistent with momentum conservation.
Although isolated closed form analytical solutions are available for simple
classical airfoil problems, the boundary value problem developed here is highly
nonlinear, owing to Equation 7.3.12a, and must be solved numerically by an
iterative method. The integral in our Equation 7.3.12a is evaluated over both
solid surfaces.