Anomalous dissipation and kinetic-energy distribution in pipes at very high Reynolds numbers

A symmetry-based theory is developed for the description of (streamwise) kinetic energy $K$ in turbulent pipes at extremely high Reynolds numbers (Re's). The theory assumes a mesolayer with continual deformation of wall-attached eddies which introduce an anomalous dissipation, breaking the exact balance between production and dissipation. An outer peak of $K$ is predicted above a critical Re of ${10}^4$, in good agreement with experimental data. The theory offers an alternative explanation for the recently discovered logarithmic distribution of $K$. The concept of anomalous dissipation is further supported by a significant modification of the $k-\omega$ equation, yielding an accurate prediction of the entire $K$ profile.

Figure 1

(a) $\theta$ by experimental $K^{+}$ (divided by $W^{+}\approx r$) compared with (5) in solid lines. Dashed lines indicate anomalous scaling ${\gamma }_b\approx 0.05$ and ${\gamma }_m\approx -0.09$ in (5). (b) Predictions of $K^{+}$ by (6) (solid line) compared with data (symbols) show good agreement in the entire outer flow. The dashed-dotted line shows the log distribution $K^{+}=-1.25ln(y^{+}/{\text{Re}}_{\tau })+1.61$ in [5]. (c) Diagnostic function $-y^{+}d{K}^{+}/d{y}^{+}$ shown by experimental data (symbols) compared with (9) (solid line) and the log profile slope $-1.25$ (dashed line). Symbols for experimental data are ${\text{Re}}_{\tau }=20251$ (square); $37450$ (circle), $68371$ (up triangle), $98187$ (down triangle) from [5].

Figure 2

(a) A comparison between data and the predictions by the Wilcox's $k-\omega$ equation [6] (dashed-dotted line) and the modified $k-\omega$ equation (solid line), respectively. (b) Predictions by the modified $k-\omega$ equation (solid line) for a series of Re's. Dashed lines show the log distribution as in Fig. 1.