Optimal stirring strategies for passive scalar mixing

Abstract

We address the challenge of optimal incompressible stirring to mix an initially inhomogeneous distribution of passive tracers. As a quantitative measure of mixing we adopt the $H^{-1}$ norm of the scalar fluctuation field, equivalent to the (square-root of the) variance of a low-pass filtered image of the tracer concentration field. First we establish that this is a useful gauge even in the absence of molecular diffusion: its vanishing as $t-->\infty$ is evidence of the stirring flow's mixing properties in the sense of ergodic theory. Then we derive absolute limits on the total amount of mixing, as a function of time, on a periodic spatial domain with a prescribed instantaneous stirring energy or stirring power budget. We subsequently determine the flow field that instantaneously maximizes the decay of this mixing measure---when such a flow exists. When no such `steepest descent' flow exists (a possible but non-generic situation) we determine the flow that maximizes the growth rate of the $H^{-1}$ norm's decay rate. This local-in-time optimal stirring strategy is implemented numerically on a benchmark problem and compared to an optimal control approach using a restricted set of flows. Some significant challenges for analysis are outlined.

10 pages, 3 figures. PDFLaTeX with JFM style (included)

References in corpus (3)

• Topology, Braids, and Mixing in Fluids
• Multiscale Mixing Efficiencies for Steady Sources
• Optimizing the Source Distribution in Fluid Mixing

Cited by in corpus (39)

• Frontiers of chaotic advection
• Using multiscale norms to quantify mixing and transport
• Suppression of chemotactic explosion by mixing
• Relative periodic orbits form the backbone of turbulent pipe flow
• Lower bounds on the mix norm of passive scalars advected by incompressible enstrophy-constrained flows
• Wall to Wall Optimal Transport
• Maximal mixing by incompressible fluid flows
• Dissipation Enhancement by Mixing
• Almost-sure enhanced dissipation and uniform-in-diffusivity exponential mixing for advection-diffusion by stochastic Navier-Stokes
• Diffusion-limited mixing by incompressible flows
• Invariant measures for passive scalars in the small noise inviscid limit
• Phase separation in the advective Cahn-Hilliard equation
• A quantitative theory for the continuity equation
• Mixing enhancement in binary fluids using optimised stirring strategies
• Mixing by stirring: optimizing shapes and strategies
• New Lagrangian diagnostics for characterizing fluid flow mixing
• Optimal mixing in two-dimensional stratified plane Poiseuille flow at finite Péclet and Richardson numbers
• Quantification of mixing in vesicle suspensions using numerical simulations in two dimensions
• Enhanced diffusivity and skewness of a diffusing tracer in the presence of an oscillating wall
• Optimal heat transfer and optimal exit times
• Complete Chaotic Mixing in an Electro-osmotic Flow by Destabilization of Key Periodic Pathlines
• A gradient-based framework for maximizing mixing in binary fluids
• Nonuniform mixing
• On mix-norms and the rate of decay of correlations
• Robust and efficient identification of optimal mixing perturbations using proxy multiscale measures
• Regularity estimates for the flow of BV autonomous divergence free vector fields in ${\mathbb{R}}^2$
• Enhanced dissipation for stochastic Navier-Stokes equations with transport noise
• Mutual information as a measure of mixing efficiency in viscous fluids
• Optimal initial condition of passive tracers for their maximal mixing in finite time
• Shape optimisation of stirring rods in mixing binary fluids
• Bounding the scalar dissipation scale for mixing flows in the presence of sources
• Open-flow mixing and transfer operators
• Front speed enhancement by incompressible flows in three or higher dimensions
• Péclet-number dependence of optimal mixing strategies identified using multiscale norms
• Optimal perturbations for nonlinear systems using graph-based optimal transport
• A bound on the mixing rate of 2d perfect fluid flows
• Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps
• On the Littlewood--Paley spectrum for passive scalar transport equations
• On the energy-constrained optimal mixing problem for one-dimensional initial configurations