This seminar will be held at the Rutherford Appleton Laboratory near Didcot.
Abstract:
Numerical calculations of laminar flow in a two-dimensional channel with a sudden
expansion exhibit a symmetry-breaking bifurcation at Reynolds number 40.45 when the
expansion ratio is 3:1. In the experiments reported by Fearn, Mullin and Cliffe [1]
there is a large perturbation to this bifurcation and the agreement with the numerical
calculations is surprisingly poor. Possible reasons for this discrepancy are explored
using modern techniques for uncertainty quantification.
When experimental equipment is constructed there are, inevitably, small manufacturing
imperfections that can break the symmetry in the apparatus. In this work we considered a
simple model for these imperfections. It was assumed that the inlet section of the
channel was displaced by a small amount and that the centre line of the inlet section
was not parallel to the centre line of the outlet section. Both imperfections were
modelled as normal random variables with variance equal to the manufacturing tolerance.
Thus the problem to be solved is the Navier-Stokes equations in a geometry with small
random perturbations. A co-ordinate transformation technique was used to transform the
problem to a fixed deterministic domain but with random coefficient appearing in the
transformed Navier-Stokes equations. The resulting equations were solved using a
stochastic collocation technique that took into account the fact that the problem has a
discontinuity in parameter space arising from the bifurcation structure in the problem.
The numerical results are in the form of an approximation to a probability measure on
the set of bifurcation diagrams. The experimental data of Fearn, Mullin and Cliffe are
consistent with the computed solutions, so it appears that a satisfactory explanation
for the large perturbation can be provided by manufacturing imperfections in the
experimental apparatus.
The work demonstrates that modern methods for uncertainty quantification can be applied
successfully to a bifurcation problem arising in fluid mechanics. It should be possible
to apply similar techniques to a wide range of bifurcation problems in fluid mechanics
in the future.
References:
[1] R M Fearn, T Mullin and K A Cliffe Nonlinear flow phenomena in a symmetric sudden
expansion, J. Fluid Mech. 211, 595-608, 1990.