2017/12/19 by Christopher J. Miles, Christopher Miles, Charles R. Doering
Engineering · Mathematics · Physics and Astronomy · #Advection #Compressibility #Diffusion #Enstrophy #Flow (mathematics) #Incompressible flow #Lattice Boltzmann Simulation Studies #Markov Chains and Monte Carlo Methods #Mathematics #Mechanics #Mixing (physics) #Numerical diffusion #Physics #Scale (ratio) #Statistical physics #Stochastic processes and statistical mechanics #Thermodynamics #Turbulence #Vortex #Vorticity #Wake #physics.flu-dyn
paper · pdf · doi:10.1088/1361-6544/aab1c8
14 pages
arxiv created 2017/12/19 · openalex publication_date 2018/04/16 · arxiv updated 2018/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Abstract Incompressible flows can be effective mixers by appropriately advecting a passive tracer to produce small filamentation length scales. In addition, diffusion is generally perceived as beneficial to mixing due to its ability to homogenize a passive tracer. However we provide numerical evidence that, in cases where advection and diffusion are both actively present, diffusion may produce negative effects by limiting the mixing effectiveness of incompressible optimal flows. This limitation appears to be due to the presence of a limiting length scale given by a generalised Batchelor length (Batchelor 1959 J. Fluid Mech . 5 113–33). This length scale limitation may in turn affect long-term mixing rates. More specifically, we consider local-in-time flow optimisation under energy and enstrophy flow constraints with the objective of maximising the mixing rate. We observe that, for enstrophy-bounded optimal flows, the strength of diffusion may not impact the long-term mixing rate. For energy-constrained optimal flows, however, an increase in the strength of diffusion can decrease the mixing rate. We provide analytical lower bounds on mixing rates and length scales achievable under related constraints (point-wise bounded speed and rate-of-strain) by extending the work of Lin et al (2011 J. Fluid Mech . 675 465–76) and Poon (1996 Commun. PDE 21 521–39).