Polymeric turbulence, the flow of fluid with polymer additives at large Reynolds numbers, exhibits striking deviations from the Kolmogorov-like behavior of Newtonian turbulence. Recent experiments and simulations have uncovered a robust self-similar energy spectrum scaling as $k^{-2.3}$, in sharp contrast to the $k^{-5/3}$ scaling of Newtonian flows. The origin of this novel scaling, however, has remained unresolved. In this work we identify the physical mechanism underlying this emergent behavior using fundamental governing equations aided by scaling arguments. We show that the fluid energy cascade is depleted by polymers across a wide range of scales at a constant rate, which emerges as a second invariant alongside the total energy flux. The new invariant sets the novel scaling behavior, and the two invariants together allow for a minimal model description of polymeric turbulence. Concisely, our results reveal that polymeric turbulence is governed by two simultaneous invariants, unlike the single-invariant structure of Newtonian turbulence, and hint at turbulence control through suitably engineered and targeted polymer design.

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• Complex fluids
• Energy cascade
• Non-Newtonian fluids
• Three-dimensional turbulence
• Turbulence
• Turbulence simulations
• Turbulent multiphase flows
• Direct numerical simulations
• Eulerian flow field
• Navier-Stokes equation
• Scaling methods
• Statistical methods