Empirical observations show that turbulence exhibits a broad range of scaling exponents, characterizing how the velocity gradients diverge in the inviscid limit. These exponents are thought to be linked to singular solutions of the Euler equations. In this work, we propose a dynamic approach to construct concept of these solutions directly from the fluid equations, using a reversible framework and introducing the efficiency $\mathcal{E}$, a nondimensional number that quantifies the amount of energy stored within the flow due to an applied force. To circumvent the computational burden of tracking singularities at finer and finer scale, we test this approach on fluids on log-lattices, which allow for high effective resolutions at a moderate cost, while preserving the same symmetries and global conservation laws as ordinary fluids. We observe a phase transition at a given efficiency, separating regular, viscous solutions (hydrodynamic phase), from singular, inviscid solutions (singular phase). The singular solutions experience self-similar blowups with exponents corresponding to nondissipative solutions. By applying a stochastic regularization, we are able to go past the blowup and show that the resulting solutions converge to power-law solutions with exponents characterizing dissipative solutions. Overall, the range of scaling exponents observed for log-lattice solutions is comparable to those of ordinary fluids.

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• Turbulence
• Euler equation
• Navier-Stokes equation