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… Click to show full 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.
               
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