Mixing by stirring: Optimizing shapes and strategies

The mixing of binary fluids by stirrers is a commonplace procedure in many industrial and natural settings, and mixing efficiency directly translates into more homogeneous final products, more enriched compounds, and often substantial economic savings in energy and input ingredients. Enhancements in mixing efficiency can be accomplished by unorthodox stirring protocols as well as modified stirrer shapes that utilize unsteady hydrodynamics and vortex-shedding features to instigate the formation of fluid filaments which ultimately succumb to diffusion and produce a homogeneous mixture. We propose a PDE-constrained optimization approach to address the problem of mixing enhancement for binary fluids. Within a gradient-based framework, we target the stirring strategy as well as the cross-sectional shape of the stirrers to achieve improved mixedness over a given time horizon and within a prescribed energy budget. The optimization produces a significant enhancement in homogeneity in the initially separated fluids, suggesting promising modifications to traditional stirring protocols.

Figure 1

Sketch of the winding number concept: for a point (green symbol) outside the closed curve (a), the ray connecting the point and any points on the curve will sweep over a zero net angle, i.e., ${\sum }_i{\varphi }_{i,i-1}=0;$ for points inside the closed curve (b), we sweep through a net angle of $2k\pi$ with nonzero integer $k.$ In our sketched case, $k=1$.

Figure 2

Comparison of final snapshots at the end of the optimization horizon ($t=T=12$), visualized by the passive scalar $\theta$: (a) unoptimized base case [37], (b) area-preserving shape optimization of stirrers' cross-sections [38], (c) path-velocity optimization of the shape-modified stirrers along circular paths [39], (d) composite optimization of cross-sectional shapes and path-velocities [40].

Figure 3

Optimization of the cross-sectional shape of both stirrers, leaving the velocity along the circular paths unchanged. The red and green curves signify the initial and final configuration, respectively, with gray shapes denoting the intermediate steps of the optimization procedure. Blue arrows have been added to indicate the travel direction of the two stirrers. Black arrows pinpoint features of interest, as they develop during the optimization process. (a) The outer stirrer shows edges on the leading edge, up to the location perpendicular to the travel direction, and a pronounced splitter-plate-like appendage at the trailing edge. (b) The inner stirrer shows a more complex shape, with a marked appendage perpendicular to its travel direction, causing a significant increase in the cross-sectionally exposed area.

Figure 4

Selected snapshots of the passive scalar $\theta$ after 11 iterations of the direct-adjoint looping, optimizing the cross-sectional stirrer shapes to enhance the mix-norm at $T=12.$ The corresponding snapshots for the base case with cylindrical stirrers [37] are juxtaposed to the lower right of each snapshot (in reduced size) for comparison.

Figure 5

Examples of increased passive scalar transportation with shape-optimized stirrer. (a) Modification of the inner (low-velocity) stirrer, showing an elongation perpendicular to its direction of motion. (b) Interaction of shed vortices generated by the perpendicular edge of the outer stirrer and the trailing edge protrusion, respectively, with arrows indicating the path of the vortices. (c) Generation of thin filaments and vortex interaction in the wake of the enhanced shape of the outer stirrer. See Ref. [38] for an animation of the entire mixing process.

Figure 6

Velocity trace of the two stirrers, visualized by position samples at equispaced intervals in time. Each time the stirrers change direction, the radius is reduced by a small percentage to avoid visual overlay. The true stirrers, of course, remain on their circular paths. The position is colored from blue (starting position) to red (stopping position).

Figure 7

Selected snapshots of the final velocity optimization for the modified stirrer shapes. (a) Initial plunging through the interface by the outer stirrer, while the inner stirrer generates start-stop vortices. (b) Oscillatory motion of the outer stirrer and acceleration of the inner stirrer. (c, d) Reversal of both stirrers, allowing interaction of the stirrers with the generated vortices. (e, f) Small-amplitude rapid oscillation of the outer stirrer, while the inner stirrer gradually comes to rest. See Ref. [39] for an animation of the entire mixing process.

Figure 8

Closeups of key features of the velocity-optimized stirring protocal. (a) Shedding of stop-vortex by the outer stirrer, aimed for impact on the inner stirrer. (b) Breakdown of large-scale vortical elements via the appendage of the outer stirrer. (c) Generation of shed vortices by oscillatory motion of the outer cylinder, near the end of the optimization horizon.

Figure 9

Optimal cross-sectional shapes for both stirrers, using a combined shape-velocity optimization. The red and green curves signify the initial and final configuration, respectively, with gray shapes denoting the intermediate steps of the optimization procedure. Black arrows pinpoint features of interest, as they develop during the optimization process. (a) The outer stirrer develops an irregular pentagon shape, with one edge more pronounced. (b) The inner stirrer shows similar shape with one conspicuous extended blade.

Figure 10

Velocity trace of the two stirrers for the combined shape-velocity optimization, visualized by position samples at equispaced intervals in time. Each time the stirrers change direction, the radius is reduced by a small percentage to avoid visual overlay. The true stirrers, of course, remain on their circular paths. The position is colored from blue (starting position) to red (stopping position).

Figure 11

Selected snapshots of the final combined shape-velocity optimization. (a) Initial approach of the interface by both stirrers, with deceleration-acceleration near the interface. (b) Acceleration of outer stirrer, deceleration of inner cylinder. (c) Oscillatory motion of the outer stirrer and near stagnation of inner stirrer. (d–f) Multiple reversals of outer stirrer and slow motion of inner stirrer, as the stirrers come to their respective resting position. See Ref. [40] for an animation of the entire mixing process.

Figure 12

Closeups of key features of the shape-velocity-optimized stirring protocol. (a) Narrow passage of two stirrers, creating a strong shear layer and a subsequent roll-up of the interface. (b) Guiding and deflection of vortices by the inner stirrer. (c) Breakdown of vortices as they interact with the sharp protrusions of the outer stirrer.