A spectral chart method for estimating the mean turbulent kinetic energy dissipation rate

Abstract

We present an empirical but simple and practical spectral chart method for determining the mean turbulent kinetic energy dissipation rate \( \left\langle \varepsilon \right\rangle \) in a variety of turbulent flows. The method relies on the validity of the first similarity hypothesis of Kolmogorov (C R (Doklady) Acad Sci R R SS, NS 30:301–305, 1941) (or K41) which implies that spectra of velocity fluctuations scale on the kinematic viscosity ν and \( \left\langle \varepsilon \right\rangle \) at large Reynolds numbers. However, the evidence, based on the DNS spectra, points to this scaling being also valid at small Reynolds numbers, provided effects due to inhomogeneities in the flow are negligible. The methods avoid the difficulty associated with estimating time or spatial derivatives of the velocity fluctuations. It also avoids using the second hypothesis of K41, which implies the existence of a −5/3 inertial subrange only when the Taylor microscale Reynods number Rλ is sufficiently large. The method is in fact applied to the lower wavenumber end of the dissipative range thus avoiding most of the problems due to inadequate spatial resolution of the velocity sensors and noise associated with the higher wavenumber end of this range.The use of spectral data (30 ≤ Rλ ≤ 400) in both passive and active grid turbulence, a turbulent mixing layer and the turbulent wake of a circular cylinder indicates that the method is robust and should lead to reliable estimates of \( \left\langle \varepsilon \right\rangle \) in flows or flow regions where the first similarity hypothesis should hold; this would exclude, for example, the region near a wall.