A new scaling is developed for the turbulent kinetic energy (TKE) and its dissipation in a turbulent wall-bounded flow. In the traditional dimensional analysis of wall-bounded turbulence, the control parameters for the near-wall region are the kinematic viscosity $\nu$ and the wall shear stress, resulting in the friction velocity $u_{\tau }$ as the characteristic velocity scale and the viscous length scale $\nu /u_{\tau }$ as the characteristic length scale. Although the mean streamwise velocity scales well with the friction velocity, the TKE, and, in particular, the TKE dissipation does not scale with the friction velocity. In the present paper, a new dimensional analysis is performed to identify a proper scaling for the TKE and its dissipation. The control parameters in the near-wall region are the kinematic viscosity and the TKE dissipation at the wall ${\epsilon }_{k,\mathrm{w}}$. The new inner velocity scale for the TKE budget equation is the Kolmogorov wall velocity $u_{\epsilon }\stackrel{def}{=}{\left(\nu {\epsilon }_{k,\mathrm{w}}\right)}^{1/4}$, and the proper length scale is the Kolmogorov wall length $\nu /u_{\epsilon }$. The profiles of the TKE and its dissipation in the near-wall region collapse well under the new scaling, and the TKE profiles in the outer layer also scale well with the new scaling. However, the TKE peak value $k_{\max }$ does not scale with the Kolmogorov wall velocity $u_{\epsilon }$ or the friction velocity $u_{\tau }$ but with a mixed scale. One mixed scale is $u_{\tau }{U}_{\infty }$ first proposed by DeGraaff and Eaton [J. Fluid Mech. 422, 319 (2000)], where $U_{\infty }$ is the mean velocity in the free stream or at the channel centerline. A new mixed scale developed in this paper for $k_{\max }$ is $\nu {\epsilon }_{k,\mathrm{w}}/u_{\tau }^2$, and justification is provided by adding a control parameter to the new dimensional analysis.

中文翻译：

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湍流壁面流动中湍动能和耗散的标度

针对湍动能（TKE）及其在湍流边界流中的耗散，开发了一种新的标度。在传统的壁面湍流尺寸分析中，近壁区域的控制参数是运动粘度$\nu$ 和壁面的剪切应力，导致摩擦速度 ${ü}_{\tau }$ 作为特征速度标度和粘性长度标度 $\nu /{ü}_{\tau }$作为特征长度标尺。尽管平均水流速度与摩擦速度成比例，但TKE，特别是TKE耗散与摩擦速度成比例。在本文中，进行了新的尺寸分析，以确定TKE及其耗散的合适比例。近壁区域的控制参数是运动粘度和壁上的TKE耗散${\epsilon }_{ķ，\mathrm{w}}$。TKE预算方程的新内部速度标度是Kolmogorov壁速${ü}_{\epsilon }\stackrel{dËF}{=}{（\nu {\epsilon }_{ķ，\mathrm{w}}）}^{\mathrm{1个}/4}$，正确的长度比例就是Kolmogorov壁长 $\nu /{ü}_{\epsilon }$。在新的缩放比例下，TKE的轮廓及其在近壁区域的耗散很好地塌陷，而在新缩放比例下，外层的TKE轮廓也很好的缩放。但是，TKE峰值${ķ}_{\mathrm{最高}}$ 与Kolmogorov壁速不成比例 ${ü}_{\epsilon }$ 或摩擦速度 ${ü}_{\tau }$但规模参差不齐。一种混合比例是${ü}_{\tau }{ü}_{\infty }$最早由DeGraaff和Eaton提出[ J. Fluid Mech。 422，319（2000）]，其中${ü}_{\infty }$是自由流中或通道中心线处的平均速度。本文针对${ķ}_{\mathrm{最高}}$ 是 $\nu {\epsilon }_{ķ，\mathrm{w}}/{ü}_{\tau }^2$，并通过向新的维度分析中添加控制参数来提供证明。