Non-dimensional dissipation at strong unsteady transitions in isotropic turbulence

Exact mathematical expressions are derived to predict the exponent $p$ observed in non-equilibrium turbulence, where the classical dissipation law is replaced by a new dissipation scaling law $C_{\varepsilon } \sim \textit{Re}_{\lambda }^p$ . Here, $ \textit{Re}_{\lambda }$ is the Taylor-based Reynolds number and $C_{\varepsilon } = \varepsilon L_{11} / u^{\prime 3}$ is the non-dimensional dissipation rate, defined by the viscous dissipation rate, $\varepsilon$ , longitudinal integral scale, $L_{11}$ , and root-mean-square of the velocity fluctuations $u^{\prime} = \sqrt {\overline {u^{\prime 2}}}$ (Vassilicos, Annu. Rev. Fluid Mech. , vol. 47, 2015, pp. 95–114). Assuming homogeneous and isotropic turbulence, it is shown that the exact value of $p$ involves only first-order derivatives of these variables; however, at very high Reynolds numbers, and under particularly strong changes in the power input of the external forcing (without changing the shape of the forcing spectrum), the exact expression simplifies to $p = 3\pi / 4\alpha L_{110} - 5 / 2$ , where $L_{110}$ is the initial value of the longitudinal integral scale and $\alpha$ represents an effective forcing wavenumber. Thus, the main finding is that only large-scale effects are involved in the imposition of the non-equilibrium dissipation scaling law. The results are compared with direct numerical simulation (DNS) results of isotropic turbulence under abruptly changing forcing conditions and with experimental data of non-equilibrium decaying isotropic turbulence, showing consistent results.