Mechanisms for Optimal Heat and Momentum Transport in Fluid Flows

Understanding the core mechanisms behind how turbulent flows achieve extreme properties, including its wide range of interacting scales, enhanced mixing, rapid momentum transport, and high rates of energy dissipation, is crucial both for engineering applications and for explaining fundamental physical processes. This comes into play when identifying the key mechanisms and principal flow structures responsible for efficient turbulent transport, which can guide the development of active and passive strategies for improved heat transfer. One central goal of this project is to determine optimal heat transport strategies for a given amount of pumping power used to stir a fluid. A primary motivation comes from the growing energy demands of AI data centers which require intensive cooling for densely packed GPUs and other computing hardware. The second part of the project is inspired by anomalous dissipation, one of the defining characteristics of turbulent flows. This phenomenon, in which fluid continues to dissipate energy even as viscosity becomes very small, is intrinsically linked to rapid transfer of energy and momentum across scales. Motivated by this, the project aims to gain deeper insight into how energy cascades to small scales in turbulent flows and to make the underlying mechanisms more quantitative. Such insights will advance our understanding of turbulence and the principles governing energy dissipation in fluid flows. This project also supports training graduate students in this research domain. This project is motivated by the overarching goal of uncovering how turbulent flows achieve high scalar/momentum transport and energy dissipation rates at a detailed mechanistic level. The project brings together tools from mathematical analysis, variational methods, and computational optimization to investigate the principles of optimal heat and momentum transport in incompressible flows. The first part of the project focuses on maximizing heat transport in two key scenarios of practical significance: internally heated convection and convection between rough boundaries. The project seeks to provide optimal flow strategies and rigorously establish scaling laws for heat transport. The second part explores anomalous dissipation of energy in incompressible flows, which is intimately linked with fast transport of momentum. Constructing solutions to the Navier-Stokes Equation with various turbulent statistics is a long-sought-out but extremely challenging goal in mathematical fluid dynamics. To make progress, the project focuses on the advection-diffusion of a divergence-free passive vector, a simpler setting than the full Navier-Stokes Equation, yet more challenging than scalar diffusion due to the presence of pressure. The project aims to use computational optimization and rigorous constructions to produce examples of anomalous dissipation in this setting. These insights will then be extended to a hierarchy of partial differential equation based systems that progressively move closer to the Navier-Stokes Equation. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.