This paper presents a theory of the position dependence of the statistical average $\langle \varepsilon \rangle$ of the energy dissipation rate ε per unit mass in the inertial sublayer of turbulent channel flow. The theory gives $\langle \varepsilon \rangle y/u_{\tau }^3\sim 1/{\kappa }_{\varepsilon }+C_p\left(y/h\right)+C_v\left(l_{\tau }/y\right)$ for small but finite ratios $y/h$ and $l_{\tau }/y$, at large but finite friction Reynolds number ${\mathrm{Re}}_{\tau }=h/l_{\tau }$, where $y$ is the distance from the wall, $h$ is the channel half-width, $u_{\tau }$ and $l_{\tau }$ are the friction velocity and length respectively, and ${\kappa }_{\varepsilon }$, $C_p$, and $C_v$ are nondimensional constants. The theory agrees well with the data of a series of direct numerical simulations of turbulent channel flow with ${\mathrm{Re}}_{\tau }$ up to approximately 8000. The data suggest ${\kappa }_{\varepsilon }\approx 0.44$, which is distinctively different from the widely accepted value ($\approx 0.40$ or so) for the von Kármán constant for the mean velocity in the log-law region of wall-bounded flows.

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• Turbulence simulations
• Turbulent boundary layers