Shape optimization of stirring rods for mixing binary fluids

Abstract

Mixing is an omnipresent process in a wide range of industrial applications, which supports scientific efforts to devise techniques for optimizing mixing processes under time and energy constraints. In this endeavour, we present a computational framework based on nonlinear direct-adjoint looping for the enhancement of mixing efficiency in a binary fluid system. The governing equations consist of the nonlinear Navier–Stokes equations, complemented by an evolution equation for a passive scalar. Immersed and moving stirrers are treated by a Brinkman penalization technique, and the full system of equations is solved using a Fourier-based pseudospectral approach. The adjoint equations provide gradient and sensitivity information which is in turn used to improve an initial mixing strategy, based on shape, rotational and path modifications. We utilize a Fourier-based approach for parameterizing and optimizing the embedded stirrers and consider a variety of geometries to achieve enhanced mixing efficiency. We consider a restricted optimization space by limiting the time for mixing and the rotational velocities of all stirrers. In all cases, non-intuitive shapes are found which produce significantly enhanced mixing efficiency.

Mixing by stirring: Optimizing shapes and strategies

“…For an argument for this set of constraints, the reader is referred to Ref. [33]. The full optimization problem can then be stated as min θ mix | t=T , (7) subject to Eqs.…”

“…and their embedding into an optimization scheme, was presented in Ref. [33], and we refer the reader to this reference for further details. Here, we will briefly touch upon the main concepts.…”

Mixing enhancement in binary fluids using optimised stirring strategies

“…Lastly, with a physical problem this rich in possibilities and with a computational approach to match, there is an abundance of extensions and opportunities. Besides obvious explorations of other parameter combinations, the optimisation of the stirrers' shape is certainly within the capabilities of the computational framework; a preliminary study in this direction can be found in Eggl & Schmid (2020) using cycloids and trochoids as cross-sectional stirrer geometries. The path of the stirrers (in our study, concentric circles) can also be optimised; a collision-avoiding constraint may pose an additional challenge in this case.…”

A parallel-in-time approach for accelerating direct-adjoint studies

“…This information can then be used as part of a gradient-based optimization routine to find the extremum of the objective functional via repeated applications of this direct-adjoint loop. Recent examples include, among others, the optimization of mixing in binary fluids [13,42,7,9,8], finding the minimal seed that triggers turbulence in pipe flow [46] and determining the optimal place to ignite a diffusion flame [47].…”