The anomalous scaling exponents of turbulence in general dimension from random geometry

Abstract: We propose an analytical formula for the anomalous scaling exponents of inertial range structure functions in incompressible fluid turbulence. The formula is a Knizhnik-Polyakov-Zamolodchikov (KPZ)-type relation and is valid in any number of space dimensions. It incorporates intermittency in a novel way by dressing the Kolmogorov linear scaling via a coupling to a lognormal random geometry. The formula has one real parameter γ that depends on the number of space dimensions. The scaling exponents satisfy the convexity inequality, and the supersonic bound constraint. They agree with the experimental and numerical data in two and three space dimensions, and with numerical data in four space dimensions. Intermittency increases with γ, and in the infinite γ limit the scaling exponents approach the value one, as in Burgers turbulence. At large n the nth order exponent scales as n$$ \sqrt{n} $$. We discuss the relation between fluid flows and black hole geometry that inspired our proposal.