Mud Siren Torque and Erosion Analysis - Measurement While Drilling Signal Analysis, Optimization and Design - page 186

7.2.2 Simplified boundary conditions.
Geometric complexities arising from the three-dimensional blunt body
character of the siren lobes preclude exact analysis. Thus we are led to examine
approximate but accurate “mean surface” approaches for fluid-dynamical
modeling. One successful method used in aerodynamics is “thin airfoil theory,”
which is classically used to predict lift, the force perpendicular to the direction
of the oncoming flow. This model, summarized in Figure 7.5, is discussed and
solved in the topic of Ashley and Landahl (1965), with Marten Landahl at
M.I.T., this author's doctoral thesis advisor, being one of its principal
developers.
In modeling flows past two-dimensional airfoils, as noted in the upper
diagram, an exact tangent flow kinematic condition applies at the geometric
surface itself, while a “Kutta condition” related to smooth downstream flow
applies at the trailing edge (this mimics viscous “starting flow” effects). In the
approximate model, shown in the lower diagram of Figure 7.5, the tangent flow
condition is replaced by a simpler boundary condition (setting the ratio of
vertical to horizontal velocities to the local airfoil slope), and applied along a
mean surface y = 0 (y is the vertical coordinate perpendicular to the oncoming
flow). These assumptions form the basis of thin airfoil theory, used successfully
for most of the twentieth century in aerodynamic design. When planform (e.g.,
wing areal layout) effects are important, three-dimensional geometric boundary
conditions are traditionally evaluated on a mean flat surface; this method forms
the basis for classical lifting surface theory. In all these methods, the evaluation
of forces perpendicular to the direction of flow is very accurate, while parallel
forces (related to viscous effects) require separate boundary layer or empirical
separated flow corrections.
Correspondingly, an analogous model can be designed for nacelles, which
house the engine turbomachinery components installed beneath airplane wings.
It is known that the presence of the engine can improve or degrade the
aerodynamics of an optimally designed “wing alone” flow. Thus, one objective
of nacelle design is favorable aerodynamic performance, if possible, to offset
any unfavorable interference effects. In this approach, outlined in Figure 7.6 for
baseline axisymmetric nacelles, exact tangent flow kinematic conditions applied
on the nacelle surface are replaced by approximate conditions along a mean
cylinder with constant radius r = R, noting that r is the radial coordinate
perpendicular to the oncoming flow. Not shown, for clarity only, are the
internal actuator disks used to model energy addition and pressure increase due
to the presence of engine turbomachinery.