Page 232 - Process Modelling and Simulation With Finite Element Methods

Geometric Continuation 219

singularity as the particle approaches scraping the duct wall. (6.4) would suggest
a second order singularity, 0(i2), on dimensional analysis alone for the thin disc
in broadside motion by analogy with pressure loss and drag for 2-D or
axisymmetric gaps. The sphere problem is not amenable to dimensional
analysis, as the gap width changes with polar angle relative to the sphere’s
center. Bungay and Brenner [9] computed that the singularity for the drag on the
sphere is O(U-~~>. Using finite element methods, Harlen [lo] found convergence
difficulties with close-fitting spheres in a cylindrical duct, indicating the extreme
difficulty in resolving large scale differences in numerical computations, even
with linear models, when small length scales dominate the dynamics of the flow.
It is my guess that much of the dynamics of close fitted particles with small gap
width can be found by extrapolation of solutions for larger gap width.
In this section, we have proposed first solving for the additional pressure
drop Ap due to the presence of the orifice plate with blockage factor E
obstructing the flow over the pressure drop for laminar flow in a channel without
the orifice plate. The gap radius is related to the blockage factor, a=l-i. The
difference between this problem and the drag on a sedimenting particle is
conceptually very small. For instance Shail and Norton [I I] calculated both for
the thin disc in broadside motion in a cylindrical duct, as well as the couple - the
induced force that opposes rotation of a stationary disc. As these quantities are
linearly related due to the linearity of (6.2), it is expected that the singular
behavior of one mirrors that of the other as the gap width is squeezed.
Model of an Ori$ce Plate Inserted in a 2-0 Channel
Launch FEMLAB and in the Model Navigator.

0 Select 2-D dimension
Select Physics modes*Incompressible Navier-Stokes >>

Pull down the options menu and select Add/Edit constants. The AddEdit
constants dialog box appears.

Name of constant: rho0 Expression: 0
Name of constant: mu0 Expression: 1
Name of constant: Umean Expression: 1
APPlY

The inlet boundary condition is fully developed Hagen-Poiseuille flow in a 2-D
channel, with Urn,,, as the single parameter characterizing the inlet condition.