Finite Element Method - Application using Fastflo
Double roller coating
The flow between two adjacent rollers counter-rotating at
different speeds has been modelled. The main interest is the free surface
formed by the coating fluid as the rollers rotate. The shape of the free
surface is determined by the roller speeds, their radii, the gap thickness
and the physical properties of the liquid. In this case, the rollers have
radii 100 units and the gap width is 2 units. This problem mirrors example
24 in [1].
The upper and lower roller surfaces have speeds of 1, 0.5
units respectively, and a matching shear flow is imposed at the inlet gap.
At the two outlets, the flow moves freely under a zero stress boundary
condition. Slope conditions are imposed on the free surface -
at both ends, the free surface must be parallel to the roller surfaces,
with slopes of ± 28 degrees to the horizontal.
For this flow, the Reynolds number ~
1 and the capillary number ~ 0.1; the latter
measures the relative strength of surface tension to viscous forces. The
kinematic iteration can be used for this problem, but it is near the lower
practical limit of capillary number. The initial guess to the steady free
surface is comprised of three sections, with two straight sections and one
central circular section, as shown below.
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| Initial mesh |
Final mesh |
Computations
The kinematic iteration is a natural segregated solution method for
transient free surface problems. From some initial flow field and domain,
the mesh is advanced according to the flow across the free surface, a new
flow field is calculated and the mesh advanced again, and so on. At each
step the mesh is adjusted so as to comply with a kinematic condition that
matches the motion of the free surface to the velocity field. The
kinematic iteration can be applied to solve a full transient problem or
just to solve for a steady state. A mesh of 6-noded triangular elements
with 400 corner nodes was used, distributed uniformly over the initial
domain as shown in the above figure on the left.
Results
A steady state is reached after 155 steps using the
kinematic iteration. Convergence is asymptotically quite slow, reflecting
the marginal capillary number. The final domain and mesh at the steady
state are shown in the above figure on the right. The steady pressure
contours and streamlines are shown below. The steady flow field has a pair
of weak circulation cells near the centre of the free surface, with
anticlockwise circulation in the lower cell and clockwise circulation in
the upper cell. Three stagnation points on the free surface divide the
flow into the upper and lower cells and the upper and lower main flows.
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| Steady state pressure contours |
Steady state streamlines |
The same problem can be solved using the alternative normal stress
update. More details of computations and results can be found in [2],[3].
References
[1] Fluid Dynamics International, Inc., FIDAP Examples
Manual, Revision 7.0, 1st Edition (1993).
[2] J.R. Mooney and F.R. De Hoog, Fastflo steady state
free surface module, Technical Report DMS-C 95/30 (CSIRO Division of
Mathematics and Statistics, 1995).
[3] J.R. Mooney and F.R. de Hoog, Modelling steady
state free surface flows using Fastflo, Proceeding of the 12th
Australasian Fluid Mechanics Conference (University of Sydney, 1995), 407-
410.
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