Manipulating the direction of turbulent energy flux via tensor geometry in a two-dimensional flow

In turbulent flows, energy flux refers to the transfer of kinetic energy across different scales of motion, a concept that is a cornerstone of turbulence theory. The direction of net energy flux is prescribed by the dimensionality of the fluid system. According to Kolmogorov's 1941 scaling theory, three-dimensional turbulence has a net energy flux toward smaller length scales, while in two-dimensional turbulence, energy transfers toward larger scales, as described in Kraichnan and Batchelor's seminal works. Manipulating energy flux across different scales with localized physical perturbations in flow systems is a formidable task because the energy at any scale is not localized in physical space. Here, we report a theoretical framework that enables the manipulation of energy flux direction in weakly turbulent flows. Based on this framework, we successfully manipulated a flow system to achieve the desired directions of net energy flux through both electromagnetically driven thin-layer flow experiments and direct numerical simulations. Significantly, we generated a type of turbulent flow that has never been produced before—two-dimensional Navier-Stokes turbulence with a net forward energy flux. Apart from theoretical interest, we discuss how our theoretical framework can have profound applications and implications in natural and engineered systems across length scale ranges from $10^{-3}$ to $10^{6}$ m, including enhanced mixing of microfluidic devices, biologically generated turbulence for a better understanding of the biogeochemical structure of water columns, breaking persistent coastal transport barriers for better coastal ecological health, and ocean energy budget in facing of climate change.