Robust and efficient identification of optimal mixing perturbations using proxy multiscale measures

Understanding and optimizing passive scalar mixing in a diffusive fluid flow at finite Péclet number [Formula: see text] (where [Formula: see text] and [Formula: see text] are characteristic velocity and length scales, and [Formula: see text] is the molecular diffusivisity of the scalar) is a fundamental problem of interest in many environmental and industrial flows. Particularly when [Formula: see text] , identifying initial perturbations of given energy that optimally and thoroughly mix fluids of initially different properties can be computationally challenging. To address this challenge, we consider the identification of initial perturbations in an idealized two-dimensional flow on a torus that extremize various measures over finite time horizons. We identify such ‘optimal’ initial perturbations using the ‘direct-adjoint looping’ method, thus requiring the evolving flow to satisfy the governing equations and boundary conditions at all points in space and time. We demonstrate that minimizing multiscale measures commonly known as ‘mix-norms’ over short time horizons is a computationally efficient and robust way to identify initial perturbations that thoroughly mix layered scalar distributions over relatively long time horizons, provided the magnitude of the mix-norm’s index is not too large. Minimization of such mix-norms triggers the development of coherent vortical flow structures which effectively mix, with the particular properties of these flow structures depending on [Formula: see text] and also the time horizon of interest. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.