On the normalized dissipation parameter in decaying turbulence

Abstract

The Reynolds number dependence of the non-dimensional mean turbulent kinetic energy dissipation rate$C_{\unicode[STIX]{x1D716}}=\overline{\unicode[STIX]{x1D716}}L/u^{\prime 3}$(where$\unicode[STIX]{x1D716}$is the mean turbulent kinetic energy dissipation rate,$L$is an integral length scale and$u^{\prime }$is the velocity root-mean-square) is investigated in decaying turbulence. Expressions for$C_{\unicode[STIX]{x1D716}}$in homogeneous isotropic turbulent (HIT), as approximated by grid turbulence, and in local HIT, as on the axis of the far field of a turbulent round jet, are developed from the Navier–Stokes equations within the framework of a scale-by-scale energy budget. The analysis shows that when turbulence decays/evolves in compliance with self-preservation (SP),$C_{\unicode[STIX]{x1D716}}$remains constant for a given flow condition, e.g. a given initial Reynolds number. Measurements in grid turbulence, which does not satisfy SP, and on the axis in the far field of a round jet, which does comply with SP, show that$C_{\unicode[STIX]{x1D716}}$decreases in the former case and remains constant in the latter, thus supporting the theoretical results. Further, while$C_{\unicode[STIX]{x1D716}}$can remain constant during the decay for a given initial Reynolds number, both the theory and measurements show that it decreases towards a constant,$C_{\unicode[STIX]{x1D716},\infty }$, as$Re_{\unicode[STIX]{x1D706}}$increases. This trend, in agreement with existing data, is not inconsistent with the possibility that$C_{\unicode[STIX]{x1D716}}$tends to a universal constant.