The Anomalous Scaling Exponents of Turbulence in General Dimension from Random Geometry

TL;DR

This paper introduces an exact formula for turbulence scaling exponents across dimensions using a gravitational KPZ-type relation, linking intermittency to random geometry and matching empirical data.

Contribution

It presents a novel analytical formula for turbulence exponents valid in any dimension, incorporating intermittency through a gravitational dressing approach.

Findings

The formula matches experimental and numerical data in 2D, 3D, and 4D.

Intermittency increases with the parameter γ, approaching Burgers turbulence in the limit.

At large order n, the exponents scale as the square root of n.

Abstract

We propose an exact analytical formula for the anomalous scaling exponents of inertial range structure functions in incompressible fluid turbulence. The formula is a gravitational Knizhnik-Polyakov-Zamolodchikov (KPZ)-type relation, and is valid in any number of space dimensions. It incorporates intermittency by gravitationally dressing the Kolmogorov linear scaling via a coupling to a random geometry. The formula has one real parameter $\gamma$ 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 $\gamma$, and in the infinite $\gamma$ limit the scaling exponents approach the value one, as in Burgers turbulence. At large $n$ the $n$th order exponent scales as $\sqrt{n}$. We discuss the relation between fluid flows and black hole geometry that inspired our proposal.