Tracer density discontinuities in turbulent flows: simple model and scaling laws

TL;DR

This paper introduces a simple 1D model based on Baker's map to analyze scalar discontinuities in turbulent flows, deriving scaling laws and distribution functions that align with previous numerical and experimental findings.

Contribution

It presents a novel 1D model capturing the core mechanism of scalar discontinuities and derives analytical expressions for their distribution and scaling exponents.

Findings

Derived distribution function for discontinuity fronts

Established scaling exponents $z_p$ for turbulence

Model results agree with prior simulations and experiments

Abstract

Mixing in fully developed incompressible turbulent flows is known to lead to a cascade of discontinuity fronts of passive scalar fields. A one-dimensional (1D) variant of Baker's map is developed, capturing the main mechanism responsible for the emergence of these discontinuities. For this 1D model, expressions for the height-distribution function of the discontinuity fronts and structure function scaling exponents $\zeta_p$ are derived [for Kolmogorov turbulence, $\zeta_p=\frac 23\log_3(p+1)$]. These analytic findings are in a good agreement with both our 1D simulations, and the results of earlier numerical and experimental studies.