A note on acoustic turbulence

Erik Lindborg

doi:10.1017/jfm.2019.523

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 $(\unicode[STIX]{x1D716}d)^{1/3}$ and the $p$th-order structure function scales as $(\unicode[STIX]{x1D716}d)^{p/3}r/d$, $\unicode[STIX]{x1D716}$ 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, $\unicode[STIX]{x1D716}$ should be replaced by $\unicode[STIX]{x1D716}+\unicode[STIX]{x1D712}$, where $\unicode[STIX]{x1D712}$ is associated with entropy production due to heat conduction. In particular, the third-order longitudinal structure function scales as $\langle \unicode[STIX]{x1D6FF}u_{r}^{3}\rangle =-C(\unicode[STIX]{x1D716}+\unicode[STIX]{x1D712})r$, where $C$ takes the value $12/5(\unicode[STIX]{x1D6FE}+1)$ in the weak shock limit, $\unicode[STIX]{x1D6FE}=c_{p}/c_{v}$ being the ratio between the specific heats at constant pressure and constant volume.

Highlights

A random acoustic field in which many modes are excited may be characterised as acoustic turbulence
We showed that similar scaling relations hold for an acoustic field that is dissipated by weak shocks as for forced Burgers turbulence, with the difference that should be replaced by + χ, where χ is associated with entropy production due to heat conduction
Apart from γ, the third-order structure function of acoustic turbulence depends on two parameters – unlike Burgers turbulence, where there this is a single parameter dependence

Summary

Introduction

A random acoustic field in which many modes are excited may be characterised as acoustic turbulence. Zakharov & Sagdeev (1970) used weak turbulence theory to derive an energy wavenumber spectrum of the form ∼k−3/2 in three dimensions This approach was criticised by Kadomtsev & Petviashvili (1973), who argued that shocks will always develop in an acoustic field if the Reynolds number is sufficiently large, and give rise to an energy spectrum of the form ∼k−2. As discussed in Frisch (1995), Landau (Landau & Lifshitz 1987) made the objection against Kolmogorov (1941b) that structure functions cannot be universally dependent on and r, since such a law would not be invariant under superposition of two fields This objection cannot be raised against (1.8) and (1.10), since the prefactors Kp will adjust in such a way that this invariance is fulfilled. We will consider the case of three-dimensional shock-dominated acoustic turbulence

Scaling relations for shock-dominated acoustic turbulence

Findings

Conclusions