Abstract
It is well known that plane Couette flow for an Oldroyd-B fluid is linearly stable, yet, most numerical methods predict spurious instabilities at sufficiently high Weissenberg number. In this paper we examine the reasons which cause this qualitative discrepancy. We identify a family of distribution-valued eigenfunctions, which have been overlooked by previous analyses. These singular eigenfunctions span a family of non-modal stress perturbations which are divergence-free, and therefore do not couple back into the velocity field. Although these perturbations decay eventually, they exhibit transient amplification during which their "passive" transport by shearing streamlines generates large cross-stream gradients. This filamentation process produces numerical under-resolution, accompanied with a growth of truncation errors. We believe that the unphysical behavior has to be addressed by fine-scale modelling, such as artificial stress diffusivity, or other non-local couplings.
Original language | English |
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Pages (from-to) | 169-190 |
Number of pages | 22 |
Journal | Journal of Non-Newtonian Fluid Mechanics |
Volume | 127 |
Issue number | 2-3 |
DOIs | |
State | Published - 1 May 2005 |
Bibliographical note
Funding Information:I am grateful to Frank Baaijens and Martien Hulsen for introducing me to this problem. Martien Hulsen and Raanan Fattal have contributed continual advice and a critical reading of the manuscript. Michael Renardy’s comments on the original manuscript have led to significant improvements in the paper. I have benefited from many stimulating discussions with G.I. Barenblatt, Alexandre Chorin, Ole Hald and John Neu. This research was carried out while I was visiting the Department of Mathematics at the Lawrence Berkeley National Laboratory. This research was funded in part by the Director, Office of Science, Computational and Technology Research, U.S. Department of Energy under Contract No. DE-AC03-76SF00098.
Keywords
- Couette flow
- Generalized functions
- Linear stability
- Non-normal operators
- Oldroyd-B model
- Stress diffusion