A note on acoustic turbulence

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 $ (εd)^{1/3} $ and the $ p $:th order structure function scales as $ (εd)^{p/3} r/d$, $ ε$ 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 $ ε$ should be replaced by $ ε+ χ$, where $ χ$ is associated with entropy production due to heat conduction. In particular, the third order longitudinal structure function scales as $ \langle δu_r^3 \rangle = -C(ε+ χ) r $, where $ C $ takes the value $ 12/5(γ+1) $ in the weak shock limit, $ γ= c_p/c_v$ being the ratio between the specific heats at constant pressure and constant volume.