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

In turbulent flows, energy flux, the cornerstone of turbulence theory, refers to the transfer of kinetic energy across different scales of motion. The direction of net energy flux is prescribed by the dimensionality of the fluid system: Energy cascades to smaller scales in three-dimensional flows but to larger scales in two-dimensional (2D) flows. Manipulating energy flux is a formidable task because the energy at any scale is not localized in the physical space. Here, we report a theoretical framework that enables control over energy flux direction. On the basis of this framework, we conducted experiments and direct numerical simulations, producing a 2D turbulence with forward energy flux, contrary to classical expectations. Beyond theory, 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 10⁶ meters, including enhanced mixing of microfluidic devices, biologically generated turbulence, breaking persistent coastal transport barriers, and ocean energy budget.