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.
Self-preservation (SP) solutions on the axis of a turbulent round jet are derived for the transport equation of the second-order structure function of the turbulent kinetic energy (k), which may be interpreted as a scale-by-scale (s.b.s.) energy budget. The analysis shows that the mean turbulent energy dissipation rate, , evolves like x −4 (x is the streamwise direction). It is important to stress that this derivation does not use the constancy of the non-dimensional dissipation rate parameter C = u 3 /L u (L u and u are the integral length scale and root mean square of the longitudinal velocity fluctuation respectively). We show, in fact, that the constancy of C is simply a consequence of complete SP (i.e. SP at all scales of motion). The significance of the analysis relates to the fact that the SP requirements for the mean velocity and mean turbulent kinetic energy (i.e. U ∼ x −1 and k ∼ x −2 respectively) are derived without invoking the transport equations for U and k. Experimental hot-wire data along the axis of a turbulent round jet show that, after a transient downstream distance which increases with Reynolds number, the turbulence statistics comply with complete SP. For example, the measured agrees well with the SP prediction, i.e. ∼ x −4 , while the Taylor microscale Reynolds number Re λ remains constant. The analytical expression for the prefactor A for ∼ (x − x o ) −4 (where x o is a virtual origin), first developed by Thiesset et al. (J. Fluid Mech., vol. 748, 2014, R2) and rederived here solely from the SP analysis of the s.b.s. energy budget, is validated and provides a relatively simple and accurate method for estimating along the axis of a turbulent round jet.
A numerical simulation based on the lattice Boltzmann method is carried out in the wake of a square cylinder with the view to investigating possible surrogates for the instantaneous turbulent kinetic energy dissipation rate, ε, as well as its mean value, \documentclass[12pt]{minimal}\begin{document}$\overline{\epsilon }$\end{document}ε¯. Various surrogate approximations of ε, based on local isotropy (εiso), local axisymmetry along the streamwise direction x (εa, x) and the transverse direction y (εa, y), local homogeneity (εhom), and homogeneity in the transverse plane, (ε4x), are assessed. All the approximations are in agreement with \documentclass[12pt]{minimal}\begin{document}$\overline{\epsilon }$\end{document}ε¯ when the distance downstream of the obstacle is larger than about 40 diameters. Closer to the obstacle, the agreement remains reasonable only for \documentclass[12pt]{minimal}\begin{document}$\overline{\epsilon }_{a,x}$\end{document}ε¯a,x, \documentclass[12pt]{minimal}\begin{document}$\overline{\epsilon }_{hom}$\end{document}ε¯hom and \documentclass[12pt]{minimal}\begin{document}$\overline{\epsilon }_{4x}$\end{document}ε¯4x. The probability density functions (PDF) and joint PDFs of ε and its surrogates show that ε4x correlates best with ε while εiso and εhom present the smallest correlation. The results indicate that ε4x is a very good surrogate for ε and can be used for correctly determining the behaviour of ε.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.