On the normalised energy dissipation rate in homogeneous isotropic turbulence

The Reynolds number dependence of the normalised energy dissipation rate Cε=ε L/u³ is studied, where ε is the energy dissipation rate, L is the integral length scale and u is the root-mean-square velocity. We present the derivation of the exact relationship between the normalised energy dissipation rate and integrated form of the Kármán–Howarth equation in homogeneous isotropic turbulence. The present mathematical formulation is valid for both forced and decaying turbulence. The discussion of Cε is developed under the assumption that the term resulting from the nonlinear energy transfer appearing in Cε is constant at sufficiently high-Reynolds-number turbulence. The fact that the integrated term originating from nonlinear energy transfer is constant plays the role of a lower bound in Cε , implying that the energy dissipation rate is finite in high-Reynolds-number turbulence. Furthermore, the origin of the non-equilibrium dissipation law could be the imbalance between u and dL/ dt , the influence of external forces, or both. In decaying turbulence with forced turbulence as the initial condition, the imbalance between u and dL/ dt causes the non-equilibrium dissipation law. The validity of the theoretical analysis is investigated using direct numerical simulations of the forced and decaying turbulence.