Optimal bounds with semidefinite programming: An application to stress driven shear flows
File(s)
Accepted version
OA Location
Author(s)
Fantuzzi, G
Wynn, A
Type
Journal Article
Abstract
We introduce an innovative numerical technique based on convex optimization to solve a range of
infinite dimensional variational problems arising from the application of the background method to
fluid flows. In contrast to most existing schemes, we do not consider the Euler-Lagrange equations
for the minimizer. Instead, we use series expansions to formulate a finite dimensional semidefinite
program (SDP) whose solution converges to that of the original variational problem. Our formulation
accounts for the influence of all modes in the expansion and the feasible set of the SDP is strictly
contained within the feasible set of the original problem. Moreover, SDPs can be easily formulated
when the fluid is subject to imposed boundary fluxes, which pose a challenge for the traditional
methods. We apply this technique to compute rigorous and near-optimal upper bounds on the
dissipation coefficient for flows driven by a surface stress. We improve previous analytical bounds
by more than 10 times, and show that the bounds become independent of the domain aspect ratio
in the limit of vanishing viscosity. We also confirm that the dissipation properties of stress driven
flows are similar to those of flows subject to a body force localized in a narrow layer near the surface.
Finally, we show that SDP relaxations are an efficient method to investigate the energy stability of
laminar flows driven by a surface stress.
infinite dimensional variational problems arising from the application of the background method to
fluid flows. In contrast to most existing schemes, we do not consider the Euler-Lagrange equations
for the minimizer. Instead, we use series expansions to formulate a finite dimensional semidefinite
program (SDP) whose solution converges to that of the original variational problem. Our formulation
accounts for the influence of all modes in the expansion and the feasible set of the SDP is strictly
contained within the feasible set of the original problem. Moreover, SDPs can be easily formulated
when the fluid is subject to imposed boundary fluxes, which pose a challenge for the traditional
methods. We apply this technique to compute rigorous and near-optimal upper bounds on the
dissipation coefficient for flows driven by a surface stress. We improve previous analytical bounds
by more than 10 times, and show that the bounds become independent of the domain aspect ratio
in the limit of vanishing viscosity. We also confirm that the dissipation properties of stress driven
flows are similar to those of flows subject to a body force localized in a narrow layer near the surface.
Finally, we show that SDP relaxations are an efficient method to investigate the energy stability of
laminar flows driven by a surface stress.
Date Issued
2016-04-08
Date Acceptance
2016-03-07
Citation
Physical Review E, 2016, 93
ISSN
1550-2376
Publisher
American Physical Society
Journal / Book Title
Physical Review E
Volume
93
Subjects
Fluids & Plasmas
01 Mathematical Sciences
02 Physical Sciences
09 Engineering
Publication Status
Published
Article Number
043308