Mobilising Optimal Scalar Transport using semidefinite programmes
Fluid flows can transport scalars such as heat, pollutants, and other quantities in complex ways. This project develops new mathematical and computational tools to identify globally optimal flows for scalar transport, by reformulating problems involving the Navier-Stokes equations as Semidefinite Programmes (SDPs). Unlike traditional non-convex approaches using brute-force numerical simulations, SDPs are convex optimization problems so one can solve them systematically. This offers an exciting new approach for both theoretical and applied investigation of optimal transport problems in fluid mechanics.
Team¶
Recommended reading¶
A general introduction for how SDPs can be used to solve theoretical problems in
fluid mechanics is given in [1], with applications including
studying convective heat transfer problems [2] and shear-driven
flows [3]. The aim of MOST:def is to use SDP analysis to
solve optimal transport problems, an introduction to which can be found in
[4, 5].
Giovanni Fantuzzi, Ali Arslan, and Andrew Wynn. The background method: theory and computations. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 380(2225):20210038, April 2022. doi:10.1098/rsta.2021.0038.
Ali Arslan, Giovanni Fantuzzi, John Craske, and Andrew Wynn. Bounds on heat transport for convection driven by internal heating. Journal of Fluid Mechanics, 919:A15, 2021. doi:10.1017/jfm.2021.360.
G. Fantuzzi and A. Wynn. Optimal bounds with semidefinite programming: An application to stress-driven shear flows. Physical Review E, 93(4):043308, April 2016. doi:10.1103/PhysRevE.93.043308.
Ian Tobasco and Charles R. Doering. Optimal Wall-to-Wall Transport by Incompressible Flows. Phys. Rev. Lett., 118(26):264502, June 2017. doi:10.1103/PhysRevLett.118.264502.
Ian Tobasco. Optimal cooling of an internally heated disc. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 380(2225):20210040, April 2022. doi:10.1098/rsta.2021.0040.
Ali Arslan, Giovanni Fantuzzi, John Craske, and Andrew Wynn. Internal heating profiles for which downward conduction is impossible. Journal of Fluid Mechanics, 993:A5, August 2024. doi:10.1017/jfm.2024.590.
G. Blekherman, P. Parrilo, and R. Thomas. Semidefinite Optimization and Convex Algebraic Geometry. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2012. doi:10.1137/1.9781611972290.
Sergei Chernyshenko. Relationship between the methods of bounding time averages. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 380(2225):20210044, April 2022. doi:10.1098/rsta.2021.0044.
Anuj Kumar, Ali Arslan, Giovanni Fantuzzi, John Craske, and Andrew Wynn. Analytical bounds on the heat transport in internally heated convection. Journal of Fluid Mechanics, 938:A26, May 2022. doi:10.1017/jfm.2022.170.
