Non-dimensional dissipation at strong unsteady transitions in isotropic turbulence

Abstract

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
_ε ∼Re
^p
_λ. Here, Re
_λ is the Taylor-based Reynolds number and C
_ε =εL
₁₁/u
³ is the non-dimensional dissipation rate, defined by the viscous dissipation rate, ε, longitudinal integral scale, L
₁₁, and root-mean-square of the velocity f luctuations (Formula presented) (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/4αL110 −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.