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

Abstract

Turbulent-kinetic-energy (TKE) production $\mathscr{P}_{k}=R_{12}(\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}y)$ and TKE dissipation $\mathscr{E}_{k}=\unicode[STIX]{x1D708}\langle (\unicode[STIX]{x2202}u_{i}/x_{k})(\unicode[STIX]{x2202}u_{i}/x_{k})\rangle$ are important quantities in the understanding and modelling of turbulent wall-bounded flows. Here $U$ is the mean velocity in the streamwise direction, $u_{i}$ or $u,v,w$ are the velocity fluctuation in the streamwise $x$- direction, wall-normal $y$- direction, and spanwise $z$-direction, respectively; $\unicode[STIX]{x1D708}$ is the kinematic viscosity; $R_{12}=-\langle uv\rangle$ 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 $\int _{0}^{\unicode[STIX]{x1D6FF}}\mathscr{P}_{k}\,\text{d}y=U_{b}u_{\unicode[STIX]{x1D70F}}^{2}-\int _{0}^{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D708}(\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}y)^{2}\,\text{d}y$ for channel flows, where $U_{b}$ is the bulk mean velocity, $u_{\unicode[STIX]{x1D70F}}$ is the friction velocity and $\unicode[STIX]{x1D6FF}$ is the channel half-height. Using inner scaling, the identity for the global integral of the TKE production in channel flows is $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\text{d}y^{+}=U_{b}^{+}-\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{+})^{2}\,\text{d}y^{+}$. 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 $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}\approx U_{b}^{+}-9.13$. (ii) At sufficiently high Reynolds number, the integrals of TKE production and dissipation are equally partitioned around the peak Reynolds shear stress location $y_{m}:\,\int _{0}^{y_{m}}\mathscr{P}_{k}\,\text{d}y\approx \int _{y_{m}}^{\unicode[STIX]{x1D6FF}}\mathscr{P}_{k}\,\text{d}y$ and $\int _{0}^{y_{m}}\mathscr{E}_{k}\,\text{d}y\approx \int _{y_{m}}^{\unicode[STIX]{x1D6FF}}\mathscr{E}_{k}\,\text{d}y$. (iii) The integral of the TKE production ${\mathcal{I}}_{\mathscr{P}_{k}}(y)=\int _{0}^{y}\mathscr{P}_{k}\,\text{d}y$ and the integral of the TKE dissipation ${\mathcal{I}}_{\mathscr{E}_{k}}(y)=\int _{0}^{y}\mathscr{E}_{k}\,\text{d}y$ exhibit a logarithmic-like layer similar to that of the mean streamwise velocity as, for example, ${\mathcal{I}}_{\mathscr{P}_{k}}^{+}(y^{+})\approx (1/\unicode[STIX]{x1D705})\ln (y^{+})+C_{\mathscr{P}}$ and ${\mathcal{I}}_{\mathscr{E}_{k}}^{+}(y^{+})\approx (1/\unicode[STIX]{x1D705})\ln (y^{+})+C_{\mathscr{E}}$, where $\unicode[STIX]{x1D705}$ is the von Kármán constant, $C_{\mathscr{P}}$ and $C_{\mathscr{E}}$ are addititve constants. The logarithmic-like scaling of the global integral of TKE production and dissipation, the equal partition of the integrals of TKE production and dissipation around the peak Reynolds shear stress location $y_{m}$ and the logarithmic-like layer in the integral of TKE production and dissipation are intimately related. It is known that the peak Reynolds shear stress location $y_{m}$ scales with a meso-length scale $l_{m}=\sqrt{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}}$. The equal partition of the integral of the TKE production and dissipation around $y_{m}$ underlines the important role of the meso-length scale $l_{m}$ in the dynamics of turbulent wall-bounded flows.