11,444 research outputs found
Optimal bounds with semidefinite programming: an application to stress driven shear flows
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 corresponds to a subset of 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.Comment: 17 pages; typos removed; extended discussion of linear matrix
inequalities in Section III; revised argument in Section IVC, results
unchanged; extended discussion of computational setup and limitations in
Sectios IVE-IVF. Submitted to Phys. Rev.
- …