Exact Two-Point Correlation Functions of Turbulence Without Pressure in Three-Dimensions

TL;DR

This paper derives exact two-point correlation functions for pressureless isotropic turbulence in three dimensions, revealing specific power-law behaviors and energy spectrum scaling, supported by numerical and experimental validation.

Contribution

It provides the first exact analytical expressions for density correlation functions and energy spectrum scaling in 3D turbulence without pressure effects.

Findings

Density-density correlator scales as |x_1 - x_2|^{-rac{12 + 2\sqrt{33}}{6}}

Energy spectrum E(k) scales as k^{-(2 - rac{\sqrt{33}}{12})}

Results are confirmed by numerical simulations and experiments.

Abstract

We investigate exact results of isotropic turbulence in three-dimensions when the pressure gradient is negligible. We derive exact two-point correlation functions of density in three-dimensions and show that the density-density correlator behaves as $ |{x_1 - x_2}|^{-\alpha_3}$, where $\alpha_3 = 2 + \frac{\sqrt{33}}{6}$. It is shown that, in three-dimensions, the energy spectrum $E(k)$ in the inertial range scales with exponent $ 2 - \frac {\sqrt{33}}{12} \simeq 1.5212$. We also discuss the time scale for which our exact results are valid for strong 3D--turbulence in the presence of the pressure. We confirm our predictions by using the recent results of numerical calculations and experiment.