# A note on acoustic turbulence

## TL;DR

This paper explores the scaling relations of acoustic turbulence in an ideal gas with entropy production confined to weak shocks, showing similarities to Burgers turbulence and deriving specific structure function scalings.

## Contribution

It demonstrates that acoustic turbulence exhibits similar scaling laws to Burgers turbulence, with modifications accounting for heat conduction effects.

## Key findings

- Shock amplitude scales as (εd)^{1/3}
- Third order structure function scales linearly with r as -C(ε+χ)r
- Scaling relations hold for acoustic fields with entropy production confined to weak shocks

## Abstract

We consider a three-dimensional acoustic field of an ideal gas in which all entropy production is confined to weak shocks and show that similar scaling relations hold for such a field as for forced Burgers turbulence, where the shock amplitude scales as $ (\epsilon d)^{1/3} $ and the $ p $:th order structure function scales as $ (\epsilon d)^{p/3} r/d$, $ \epsilon $ being the mean energy dissipation per unit mass, $ d $ the mean distance between the shocks and $ r $ the separation distance. However, for the acoustic field $ \epsilon $ should be replaced by $ \epsilon + \chi $, where $ \chi $ is associated with entropy production due to heat conduction. In particular, the third order longitudinal structure function scales as $ \langle \delta u_r^3 \rangle = -C(\epsilon + \chi) r $, where $ C $ takes the value $ 12/5(\gamma +1) $ in the weak shock limit, $ \gamma = c_p/c_v$ being the ratio between the specific heats at constant pressure and constant volume.