Integral properties of turbulent-kinetic-energy production and dissipation in turbulent wall-bounded flows

Turbulent-kinetic-energy (TKE) production P₊=R₁₂ (x2202U/x2202y) and TKE dissipation E₊=x1D708 (x2202u₈/x₊) (x2202u₈/x₊) are important quantities in the understanding and modelling of turbulent wall-bounded flows. Here U is the mean velocity in the streamwise direction, u₈ or u, v, w are the velocity fluctuation in the streamwise x - direction, wall-normal y - direction, and spanwise z -direction, respectively; x1D708 is the kinematic viscosity; R₁₂=- uv is the kinematic Reynolds shear stress. Angle brackets denote Reynolds averaging. This paper investigates the integral properties of TKE production and dissipation in turbulent wall-bounded flows, including turbulent channel flows, turbulent pipe flows and zero-pressure-gradient turbulent boundary layer flows (ZPG TBL). The main findings of this work are as follows. (i) The global integral of TKE production is predicted by the RD identity derived by Renard & Deck (J. Fluid Mech. , vol. 790, 2016, pp. 339–367) as ₀^x1D6FFP₊\, dy=U₁uₗ₁₃₇₀₅^2- ₀^x1D6FFx1D708 (x2202U/x2202y) ^2\, dy for channel flows, where U₁ is the bulk mean velocity, uₗ₁₃₇₀₅ is the friction velocity and x1D6FF is the channel half-height. Using inner scaling, the identity for the global integral of the TKE production in channel flows is ₀^x1D6FF^{+}P₊^+dy^+=U₁^+- ₀^x1D6FF^{+} (x2202U^+/x2202y^+) ^2\, dy^+. In the present work, superscript + denotes inner scaling. At sufficiently high Reynolds number, the global integral of the TKE production in turbulent channel flows can be approximated as ₀^x1D6FF^{+}P₊^+\, dy^+ U₁^+-9. 13. (ii) At sufficiently high Reynolds number, the integrals of TKE production and dissipation are equally partitioned around the peak Reynolds shear stress location y₌: \, ₀^y₌P₊\, dy ₘ_₌^x1D6FFP₊\, dy and ₀^y₌E₊\, dy ₘ_₌^x1D6FFE₊\, dy. (iii) The integral of the TKE production I₏_₊ (y) = ₀^yP₊\, dy and the integral of the TKE dissipation I₄_₊ (y) = ₀^yE₊\, dy exhibit a logarithmic-like layer similar to that of the mean streamwise velocity as, for example, I₏_₊^+ (y^+) (1/x1D705) (y^+) +C₏ and I₄_₊^+ (y^+) (1/x1D705) (y^+) +C₄, where x1D705 is the von Kármán constant, C₏ and C₄ are addititve constants. The logarithmic-like scaling of the global integral of TKE production