Hamiltonian Interface Dynamics for Reduced-Order Optimization of Incompressible Mixing

TL;DR

This paper introduces a Hamiltonian-based reduced-order framework for optimizing mixing in 2D incompressible flows by maximizing interface length, leading to faster mixing and computational efficiency.

Contribution

It develops a novel Hamiltonian control approach with adjoint equations for efficient mixing optimization, validated through numerical experiments.

Findings

Optimized Hamiltonians achieve near-exponential interface stretching.

Faster decay of the $ abla^{-1}$ mix-norm compared to stationary flows.

Interface-based controls outperform Eulerian Sobolev-norm optimizers in mixing speed.

Abstract

We develop a reduced-order framework for optimizing mixing in two-dimensional incompressible flows. Instead of optimizing the full transport PDE, the method maximizes the length of advected material interfaces, leading to a finite-dimensional Hamiltonian control problem based on parametrized stream functions. We derive the continuous adjoint equations and reduced gradients, and discretize the forward and adjoint dynamics with the implicit midpoint rule. The resulting discrete adjoint is algebraically consistent with the derivative of the fully discrete objective, up to the tolerance of the nonlinear midpoint solves. The approach applies to bounded two-dimensional domains with smooth finite-dimensional stream-function parametrizations. Numerical experiments on cellular-flow and Doswell frontogenesis benchmarks show that the optimized time-dependent Hamiltonians generate near-exponential interface stretching and substantially faster decay of the $\dot{H}^{-1}$ mix-norm, in contrast with the polynomial behavior observed for stationary flows. When evaluated on a common reference transport solver, the interface-based controls produce faster $\dot{H}^{-1}$ decay than a Eulerian Sobolev-norm optimizer under a matched setup, while substantially reducing computational cost. We also identify a limitation of the reduced model: increasing the control basis may further improve the interface-length objective without yielding proportional gains in $\dot{H}^{-1}$ mixing, confirming that interface length is an effective but not fully faithful proxy for mixing in geometrically complex regimes.