Mud Siren Torque and Erosion Analysis - Measurement While Drilling Signal Analysis, Optimization and Design - page 197

This derivation extends the classic derivation (Ashley and Landahl, 1965) for
the streamfunction < to problems with non-zero /. If we now differentiate
Equation 7. 3.15c with respect to y and Equation 7. 3.15d with respect to x, and
eliminate the velocity potential, we obtain the governing equation
< xx + < yy = 0 (7.3.15e)
This takes the same form as Equation 7 .3.15a so that the same solution
algorithm applies. The boundary conditions used are obtained from Equations
7

.3.15c and 7 .3.15d since ) is already known. The normal Neumann derivative
< y is applied along the horizontal upper and lower box boundaries, while the
normal derivative < x is applied along the vertical left and right boundaries.
How is < used to trace streamlines? Note that streamline slope dy/dx is
kinematically equal to the velocity ratio ) y /) x , that is,
dy/dx = ) y /) x = - < x /< y + /y/< y (7.3.15f)

Then the total differential satisfies
d< = < x dx + < y dy = /y dx (7.3.15g)
When annular convergences are small, / can be neglected so that d< = 0. Thus,
< is constant along a streamline. In this limit, streamlines are easily constructed
by using a contour plotter for the converged numerical field <(x,y).

Figure 7.12. Periodic cascade of upstream and downstream lobes.

Force

Figure 7.13. Periodic boundary value problem flow domain.