Relationship between the energy dissipation function and the skin friction law in a turbulent channel flow

Abstract

Integrals of the mean and turbulent energy dissipation rates are examined using direct numerical simulation (DNS) databases in a turbulent channel flow. Four values of the Kármán number ($h^{+}=180$, 395, 640 and 1020;$h$is the channel half-width) are used. Particular attention is given to the functional$h^{+}$dependence by comparing existing DNS and experimental data up to$h^{+}=10^{4}$. The logarithmic$h^{+}$dependence of the integrated turbulent energy dissipation rate is established for$300\leqslant h^{+}\leqslant 10^{4}$, and is intimately linked to the logarithmic skin friction law,viz.$U_{b}^{+}=2.54\ln (h^{+})+2.41$($U_{b}$ is the bulk mean velocity). This latter relationship is established on the basis of energy balances for both the mean and turbulent kinetic energy. When$h^{+}$is smaller than 300, viscosity affects the integrals of both the mean and turbulent energy dissipation rates significantly due to the lack of distinct separation between inner and outer regions. The logarithmic$h^{+}$dependence of$U_{b}^{+}$is clarified through the scaling behaviour of the turbulent energy dissipation rate$\overline{{\it\varepsilon}}$in different parts of the flow. The overlap between inner and outer regions is readily established in the region$30/h^{+}\leqslant y/h\leqslant 0.2$for$h^{+}\geqslant 300$. At large$h^{+}$(${\geqslant}$5000) when the finite Reynolds number effect disappears, the magnitude of$\overline{{\it\varepsilon}}y/U_{{\it\tau}}^{3}$approaches 2.54 near the lower bound of the overlap region. This value is identical between the channel, pipe and boundary layer as a result of similarity in the constant stress region. As$h^{+}$becomes large, the overlap region tends to contribute exclusively to the$2.54\ln (h^{+})$dependence of the integrated turbulent energy dissipation rate. The present logarithmic$h^{+}$dependence of$U_{b}^{+}$is essentially linked to the overlap region, even at small$h^{+}$.