Fluid-network relations: Decay laws meet with spatial self-similarity, scale invariance, and control scaling.

Diverse implicit structures of fluids have been discovered recently, providing opportunities to study the physics of fluids applying network analysis. Although considerable work has been devoted to identifying the informative network structures of fluids, we are limited to a primary stage of understanding what kinds of information these identified networks can convey about fluids. An essential question is how the mechanical properties of fluids are embodied in the topological properties of networks or vice versa. Here, we tackle this question by revealing a set of fluid-network relations that quantify the interactions between fundamental fluid-flow properties (e.g., kinetic energy and enstrophy decay laws) and defining network characteristics (e.g., spatial self-similarity, scale-invariance, and control scaling). We first analyze spatial self-similarity in its classic and generalized definitions, which reflect, respectively, whether vortical interactions or their spatial imbalance extents are self-similar in fluid flows. The deviation extents of networks from self-similar states exhibit power-law scaling behaviors with respect to fluid-flow properties, suggesting that the diversity among vortices is an indispensable basis of self-similar fluid flows. Then, the same paradigm is adopted to investigate scale-invariance using renormalization groups, which reveals that the breaking extents of scale-invariance in networks, similar to those of spatial self-similarity, follow power-law scaling with respect to fluid-flow properties. Finally, we define a control problem in networks to study the propagation of perturbations through vortical interactions over different ranges. The minimum cost of controlling vortical networks scales exponentially with range diameters (i.e., control distances), whose growth rates experience temporal decays. We show that this temporal decay speed is fully determined by fluid-flow properties in power-law scaling behaviors. In sum, all these discovered fluid-network relations sketch a picture in which we can study the implicit structures of fluids and quantify their interactions with fluid dynamics.