Optimal Wall-to-Wall Transport by Incompressible Flows

Area of Science:

• Fluid Dynamics
• Transport Phenomena
• Materials Science

Background:

• Understanding passive tracer transport in fluid flows is crucial for various scientific and engineering applications.
• Previous research established upper bounds for transport rates constrained by enstrophy (a measure of vorticity).
• The "ultimate" heat transport scaling (Nu∼Ra^{1/2}) is a key benchmark in buoyancy-driven convection.

Purpose of the Study:

• To construct steady, two-dimensional, divergence-free velocity fields that maximize wall-to-wall passive tracer transport.
• To determine the achievable transport rates under a given enstrophy budget (Pe).
• To compare the transport efficiency of constructed flows with theoretical limits and buoyancy-driven flows.

Main Methods:

• Construction of steady, two-dimensional, divergence-free velocity fields (u) with a specified enstrophy budget (⟨|∇u|^{2}⟩≤Pe^{2}).
• Analysis of the transport rate (Nu) of a passive tracer in these flows, particularly in the large enstrophy limit.
• Exploitation of a mathematical connection between optimal transport problems and singularly perturbed variational problems.

Main Results:

• Constructed flows achieve transport rates Nu(u)≳Pe^{2/3}/(logPe)^{4/3} in the large enstrophy limit.
• Maximally transporting flows satisfy Nu∼Pe^{2/3}, aligning with theoretical upper bounds up to logarithmic corrections.
• While flows approaching the ultimate scaling Nu∼Ra^{1/2} are theoretically possible, they are not always realizable via buoyancy-driven convection.

Conclusions:

• The study establishes the scaling of maximally efficient passive tracer transport under enstrophy constraints.
• It highlights a discrepancy between theoretically optimal transport and the practical limitations of buoyancy-driven flows.
• The findings link fluid transport optimization to pattern formation problems in materials science.