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 Cε ~ Reλᵖ . Here, Reλ is the Taylor-based Reynolds number and Cε = ε L₁₁ / u³ is the non-dimensional dissipation rate, defined by the viscous dissipation rate, ε , longitudinal integral scale, L₁₁ , and root-mean-square of the velocity fluctuations u^ = √ u² (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 = 3π / 4α L₁₁₀ - 5 / 2 , where L₁₁₀ is the initial value of the longitudinal integral scale and α 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.