Mud Siren Torque and Erosion Analysis - Measurement While Drilling Signal Analysis, Optimization and Design - page 185

pressures below 100 psi at the source, which is not necessarily bad from a
telemetry viewpoint. This is so because continuous wave signaling schemes
permit lower probabilities of bit error and more efficient data transmission.

7.2.1 Inviscid aerodynamic model.
If viscous stresses are ignored, fluid motions are governed by Euler's
equation D q /Dt = - 1/Up, where D/Dt is the convective derivative, q is the
Eulerian velocity vector, U is the constant mass density, and p is the static
pressure. This applies to all coordinate systems. In practical applications,
physical quantities related to directions perpendicular to the oncoming flow,
e.g., lift on airfoils, radial forces on engine nacelles, and torque on turbine and
compressor blades - and torque on siren stator and rotor stages - can be
modeled and successfully predicted using inviscid models.
Parallel forces associated with viscous shear, however, require separate
“boundary layer flow” analyses where inviscid pressures are impressed across
thin viscous zones. Such are the approaches used in calculating drag when
flows are streamlined. Even when flows are strongly separated, fluid
characteristics upstream of the separation point can be qualitatively studied by
inviscid flow models. Of course, details related to the separated region must be
examined by alternative, often empirically-based methods, not considered here.
When viscosity is neglected and the far upstream flow is uniform, the fluid
motion is said to be irrotational and satisfies the kinematic constraint u q = 0.
This allows q to be represented as the gradient of a total velocity potential I,
q = I (7.2.1)

so that
2 I = 0 (7.2.2)
by virtue of mass conservation, that is, x q = 0. A consequence of Euler's
equation is “Bernoulli's pressure integral,” which takes the general vector form
p + ½ U | I | 2 = p 0 (7.2.3)
where p 0 is the stagnation pressure determined completely from upstream
conditions.
Again, these equations apply three-dimensionally to all coordinate systems.
In inviscid single-airfoil formulations, the boundary value problem associated
with Equation 7.2.2 is first solved subject to geometric constraints, and
corresponding surface pressures are later calculated using Equation 7.2.3. This
equation, we emphasize, does not apply to the viscosity-dominated downstream
wake. For further details about inviscid flow modeling, its applications and
limitations, consult the classic topic by Batchelor (1970).