Control of mixing in a Stokes' fluid flow

Accepted versio

Exponential self-similar mixing by incompressible flows

We study the problem of the optimal mixing of a passive scalar under the action of an incompressible flow in two space dimensions. The scalar solves the continuity equation with a divergence-free velocity field, which satisfies a bound in the Sobolev space $W^{s,p}$, where $s \geq 0$ and $1\leq p\leq \infty$. The mixing properties are given in terms of a characteristic length scale, called the mixing scale. We consider two notions of mixing scale, one functional, expressed in terms of the homogeneous Sobolev norm $\dot H^{-1}$, the other geometric, related to rearrangements of sets. We study rates of decay in time of both scales under self-similar mixing. For the case $s=1$ and $1 \leq p \leq \infty$ (including the case of Lipschitz continuous velocities, and the case of physical interest of enstrophy-constrained flows), we present examples of velocity fields and initial configurations for the scalar that saturate the exponential lower bound, established in previous works, on the time decay of both scales. We also present several consequences for the geometry of regular Lagrangian flows associated to Sobolev velocity fields.Comment: To appear in Journal of the American Mathematical Society. Some results were announced in G. Alberti, G. Crippa, A. L. Mazzucato, "Exponential self-similar mixing and loss of regularity for continuity equations", C. R. Math. Acad. Sci. Paris, 352(11):901--906, 2014, arXiv:1407.2631v

Optimal mixing in two-dimensional plane Poiseuille flow at finite Peclet number

International audienceWe consider the nonlinear optimisation of the mixing of a passive scalar, initially arranged in two layers, in a two-dimensional plane Poiseuille flow at finite Reynolds and Péclet numbers, below the linear instability threshold. We use a nonlinear-adjoint-looping approach to identify optimal perturbations leading to maximum time-averaged energy as well as maximum mixing in a freely evolving flow, measured through the minimisation of either the passive scalar variance or the so-called mix-norm, as defined by Mathew, Mezić & Petzold (Physica D, vol. 211, 2005, pp. 23-46). We show that energy optimisation appears to lead to very weak mixing of the scalar field whereas the optimal mixing initial perturbations, despite being less energetic, are able to homogenise the scalar field very effectively. For sufficiently long time horizons, minimising the mix-norm identifies optimal initial perturbations which are very similar to those which minimise scalar variance, demonstrating that minimisation of the mix-norm is an excellent proxy for effective mixing in this finite-Péclet-number bounded flow. By analysing the time evolution from initial perturbations of several optimal mixing solutions, we demonstrate that our optimisation method can identify the dominant underlying mixing mechanism, which appears to be classical Taylor dispersion, i.e. shear-augmented diffusion. The optimal mixing proceeds in three stages. First, the optimal mixing perturbation, energised through transient amplitude growth, transports the scalar field across the channel width. In a second stage, the mean flow shear acts to disperse the scalar distribution leading to enhanced diffusion. In a final third stage, linear relaxation diffusion is observed. We also demonstrate the usefulness of the developed variational framework in a more realistic control case: mixing optimisation by prescribed streamwise velocity boundary conditions

Adjoint-based mixing enhancement for binary fluids

Mixing is a fundamental fluid process that dominates {a} great many natural phenomena and is present in a wide variety of industrial applications. Therefore, studying the characteristics and optimisation of this process may lead to a significant impact in many fields. This thesis presents an analytical and computational framework for optimising fluid mixing processes using embedded stirrers based on a non-linear direct-adjoint looping approach. The governing equations are the non-linear Navier-Stokes equations, augmented by an evolution equation for a passive scalar, which are solved by a Fourier-based spectral method. Stirrers are embedded in the computational domain by a Brinkman-penalisation technique, and shape and path gradients for the stirrers are computed from the adjoint solution. The relationship between this penalisation approach and the adjoint will be examined through the derivation of a dual system of equations, and three different optimisation scenarios of increasing complexity, each focusing on different optimisation parameters, are considered. Within the limits of the parameterisations of the geometry and the externally imposed bounds, significant improvements in mixing efficiency are achieved in all cases.Open Acces

Exponential self-similar mixing by incompressible flows

We study the problem of the optimal mixing of a passive scalar under the action of an incompressible flow in two space dimensions. The scalar solves the continuity equation with a divergence-free velocity field, which satisfies a bound in the Sobolev space Ws,p, where s ≥ 0 and 1 ≤ p ≤ ∞. The mixing properties are given in terms of a characteristic length scale, called the mixing scale. We consider two notions of mixing scale, one functional, expressed in terms of the homogeneous Sobolev norm H−1, the other geometric, related to rearrangements of sets. We study rates of decay in time of both scales under self-similar mixing. For the case s = 1 and 1 ≤ p ≤ ∞ (including the case of Lipschitz continuous velocities, and the case of physical interest of enstrophy-constrained flows), we present examples of velocity fields and initial configurations for the scalar that saturate the exponential lower bound, established in previous works, on the time decay of both scales. We also present several consequences for the geometry of regular Lagrangian flows associated to Sobolev velocity fields