Energy cascade at the turbulent/nonturbulent interface

Interscale energy transfers at the vicinity of the turbulent/nonturbulent interface are from small to large scales in directions close to the interface's tangent plane where motions are predominantly stretching, but from large to small scales in the other directions where motions are predominantly compressive and significantly correlated with square angular momentum. An important role in this predominance is played by the extreme compressive motions which can be significantly more likely than extreme stretching motions even where motions are on average stretching. The most intense interscale transfer rates and dissipation occur when the interface is as far as possible from the wake centerline.

Figure 1

Plots in the ($r_n/\delta$, $r_t/\delta$) plane (wake width $\delta$ at $x/L_b=60$) of (a) $\langle \delta \mathit{u}·\widehat{\mathit{r}}|{\mathit{X}}_I\rangle$, (b) $\langle \delta \mathit{u}·\widehat{\mathit{r}}\left|\delta \mathit{u}{|}^2\right|{\mathit{X}}_I\rangle$, (c) $\langle {(\delta \mathit{u}·\widehat{\mathit{r}})}^3|{\mathit{X}}_I\rangle$, (d) $\langle \delta \mathit{u}·\widehat{\mathit{r}}|\widehat{\mathit{r}}\times \delta \mathit{u}{|}^2|{\mathit{X}}_I\rangle$, (e) the Pearson correlation coefficient $C$ between $\delta \mathit{u}·\widehat{\mathit{r}}$ and $|\widehat{\mathit{r}}{\times \delta \mathit{u}|}^2$ conditional on ${\mathit{X}}_I$ and (f) the flatness of $\delta \mathit{u}·\widehat{\mathit{r}}$ conditional on ${\mathit{X}}_I$. The magenta line in plots (a) to (f) is where $\langle \delta \mathit{u}·\widehat{\mathit{r}}|{\mathit{X}}_I\rangle =0$. The black line in plots (b) to (e) is where the quantity plotted is 0.

Figure 2

Plots in the $(r_n/\delta ,r_t/\delta )$ plane (wake width $\delta$ at $x/L_b=60$) of (a) $\langle \mathrm{\Pi }|{\mathit{X}}_I\rangle /\langle {\varepsilon }_r|{\mathit{X}}_I\rangle$ and (b) $\langle T_u|{\mathit{X}}_I\rangle /\langle {\varepsilon }_r|{\mathit{X}}_I\rangle$.

Figure 3

Averages conditional on $R_I$ of (a) ${\mathrm{\Pi }}^a$, (b) $\varepsilon$, (c) $\omega$ as functions of $R_I$ and $R$ and (d) summary schematic of the three intercale transfer regions at the TNTI.