We propose a novel analytical framework for incompressible Navier-Stokes (NS) turbulence, revealing a duality between classical fluid dynamics and one-dimensional nonlinear dynamics in loop space. This reformulation leads to a universal momentum loop equation, which excludes finite-time blow-ups, establishing a \textit{No Explosion Theorem} for turbulent flows with finite initial noise. Decaying turbulence emerges as a solution to this equation and is interpreted as a solvable string theory with a discrete target space composed of regular star polygons. The derived decay spectrum exhibits excellent agreement with experimental data and direct numerical simulations (DNS), replacing classical Kolmogorov scaling laws with universal functions derived from number theory. These results suggest a deeper mathematical structure underlying turbulence, uniting fluid dynamics, quantum mechanics, and number theory.

Decaying turbulence, characterized by energy dissipation from an initial high-energy state, remains a fundamental challenge in classical physics. This work presents an exact analytical solution to the Navier-Stokes (NS) equations for incompressible fluid flow in the context of decaying turbulence, introducing the novel framework of the \textit{Euler ensemble}. This framework maps turbulent dynamics onto discrete states represented by regular star polygons with rational vertex angles in units of 2π. A key feature of the Euler ensemble is a duality between classical turbulence and a hidden one-dimensional quantum system, analogous to the AdS/CFT correspondence in quantum field theory. This duality enables the derivation of exact turbulence statistics, replacing traditional heuristic scaling laws with universal results derived directly from the NS equations. For example, the decay law for turbulent kinetic energy is predicted as $ E(t)∼t^{−5/4}$, with quantitative agreement to within 1% standard deviation in experimental and numerical data. The framework is validated using Direct Numerical Simulations (DNS) and experimental results, including grid turbulence and large-tank experiments. Additionally, the Euler ensemble predicts novel macroscopic quantum-like effects, such as oscillations in the decay index as a function of the scaling variable $r/\sqrt t$. These predictions highlight new avenues for experimental and numerical exploration of turbulence. This work addresses long-standing challenges in turbulence theory, providing a rigorous, universal description of decaying turbulence with applications across fluid dynamics, geophysics, and engineering.

A new approach to strong turbulence based on ideas of dynamical geometry and topological conservation laws is developed. In terms of the quantum field theory, this is another example of the duality between a fluctuating vector field and fluctuating geometry. Some new exact solutions of Navier-Stokes and Euler equations are found. The loop equation suggested in the early 90-ties is investigated in detail. The loop equation plays the same role in our theory as the Boltzmann kinetic equation in statistical physics. It has the form of the Schrödinger equation with a complex Hamiltonian in loop space. The viscosity plays the role of Planck's constant. Strong turbulence corresponds to the WKB limit of the loop equation. In this limit, we find a fixed point of the loop equation we call Kelvinon. Kelvinon has a conserved circulation for a fixed loop in space, generalizing Kelvin's theorem. Clebsch field of this solution has nontrivial topology with two winding numbers. These topological conservation laws allow us to compute the PDF of circulation in a WKB limit (large circulation in the viscosity units). This PDF perfectly matches the results of numerical simulations of the conventional forced Navier-Stokes equation.

A new approach to strong turbulence based on ideas of dynamical geometry and topological conservation laws is developed. In terms of the quantum field theory, this is another example of the duality between a fluctuating vector field and fluctuating geometry. Some new exact solutions of Navier-Stokes and Euler equations are found. The loop equation suggested in the early 90-ties is investigated in detail. The loop equation plays the same role in our theory as the Boltzmann kinetic equation in statistical physics. It has the form of the Schrödinger equation with a complex Hamiltonian in loop space. The viscosity plays the role of Planck's constant. Strong turbulence corresponds to the WKB limit of the loop equation. In this limit, we find a fixed point of the loop equation we call Kelvinon. Kelvinon has a conserved circulation for a fixed loop in space, generalizing Kelvin's theorem. Clebsch field of this solution has nontrivial topology with two winding numbers. These topological conservation laws allow us to compute the PDF of circulation in a WKB limit (large circulation in the viscosity units). This PDF perfectly matches the results of numerical simulations of the conventional forced Navier-Stokes equation.