Finite Element Method - Application using Fastflo
Viscoelastic flow around a sphere
Contact personnel: X.-L.
Luo
A sphere of radius a falls in a cylindrical tube
of radius 2a filled with a viscoelastic fluid described by the
Upper-Convected Maxwell (UCM) model. In 1988, this was proposed as a
benchmark problem at the 5th International Workshop on Numerical
Methods in Non-Newtonian Flows. Calculations with the UCM model proved
to be very difficult then, even for a moderate Weissenberg number which is
a measure of elasticity in the flow. For this particular problem, the
Weissenberg number is defined as
where l is the relaxation
time in the UCM model and U is the speed of the falling sphere.
Although significant progress has been made since 1988, most calculations
in the literature still diverge beyond Wi = 2.
Apart from theoretical interest, an important practical aspect of the
problem is to understand the drag reduction mechanism of polymer
solutions. Calculations have predicted a significant drag reduction for
the UCM fluid in the sphere-in-tube problem, as compared to Newtonian
fluid. Accurate and efficient calculations for the sphere-in-tube problem
will improve our understanding of drag reduction, and aid the development
of better constitutive equations describing viscoelastic fluids. The
sphere-in-tube problem remains as the premier benchmark problem for
numerical simulation of non-Newtonian flow.
Pressure
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Ur velocity
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Uz velocity
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Srr extra stress
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Srz extra stress
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Szz extra stress
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Computations
This problem was solved using a decoupled transient
algorithm with a Streamline Upwind Petrov-Galerkin (SUPG) technique. The
algorithm incorporated a pre-conditioned conjugate gradient (PCG) method
for solving viscoelastic flows. This algorithm is provided in the
non-Newtonian module in Fastflo
The high efficiency of the decoupled algorithm enabled problems with
more than 105 DOF, on a mesh of 17,000 nodes, to be solved on a
133 MHz DEC workstation in about 7 hours. Good convergence was obtained
for Wi up to 2.8; this is the highest value yet reported. In fact, little
change was observed in going beyond a coarser mesh of 4,000 nodes, which
took less than 1 hour of cpu time. The calculation used an unstructured
triangular mesh, relying on azimuthal invariance, which was generated by Fastflo's
own mesh generator.
Results
The results presented above, computed for Wi = 2.5, show that the
velocity and pressure fields have a significant down-stream shift, as
compared to the upstream-downstream symmetry in the pure viscous creeping
flow case. The extra stress field is characterised by large gradients and
thin boundary layers adjacent to the sphere surface.
Reference
X.-L. Luo, Operator splitting algorithm for
viscoelastic flow and numerical analysis for the flow around a sphere in a
tube, Journal of Non-Newtonian Fluid Mechanics, 63
121-140 (1996).
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