Scale-by-scale kinetic energy budget near the turbulent/nonturbulent interface

A scale-by-scale kinetic energy budget is analyzed near the turbulent/nonturbulent interfacial (TNTI) layer with direct numerical simulations (DNSs) of a local turbulent front evolving without mean shear (shear-free turbulence). A local volume average is used to decompose the flow variables into their large-scale and small-scale components near the TNTI layer. The kinetic energy and interscale energy flux from large to small scales of motion are shown to be severely depleted for small scales within the viscous superlayer. The forward interscale energy transfer from large to small scales near the TNTI layer is mostly caused by the velocity gradient in the interface normal direction while the velocity gradient in the tangential direction transfers, on average, the energy from small to large scales. The velocity gradients that cause the forward energy transfer near the TNTI layer are associated with a compressive motion in the interface normal direction and a shearing motion due to the velocity in the tangential direction. The pressure diffusion increases the kinetic energy near the interface except at small scales within the TNTI layer. The averaged pressure diffusion term at the small scales within the TNTI layer has negative values, which are consistent with the presence of small-scale vortices within the TNTI layer. The transports by turbulent diffusion and interaction between large and small scales are negatively correlated even near the TNTI layer, and their effects are locally canceled by each other as also observed in other turbulent flows.

Figure 1

Local coordinate ${\zeta }_I$ used for calculating conditional statistics. A color represents ${\log }_{10}(\mathit{\omega }·\mathit{\omega })$ while an isosurface of $\left|\mathit{\omega }\right|={\omega }_{th}$ is shown with white lines (SFT1). The coordinates shown with broken lines are excluded from samples of conditional statistics.

Figure 2

Sketch outlining the computation of the conditional spherical averaging for spheres whose center is located in (a) the turbulent region and (b) the nonturbulent region. The position on the interface coordinate ${\zeta }_I$ is denoted by ${\mathit{x}}_I$ in the global coordinate system.

Figure 3

(a) Conditional rms velocity fluctuations in SFT1. (b) Variation of ${\xi }_b$ and ${\eta }_b$ across the TNTI layer in SFT1. (${\xi }_b,{\eta }_b$) obtained at the center of a fully developed turbulent mixing layer [38] is also plotted for comparison. Three thin lines in panel (b) correspond to an axisymmetric state with one large eigenvalue, ${\eta }_b={\xi }_b$, or one small eigenvalue, ${\eta }_b=-{\xi }_b$, and a two-dimensional state ${\eta }_b={(1/27+2{\xi }_b^3)}^{1/2}$.

Figure 4

Conditional rms vorticity in normal and tangential directions of the irrotational boundary: $\sqrt{2}{\omega }_{NI}=\sqrt{2}{\langle {\omega }_N^2\rangle }_I^{1/2}$ and ${\omega }_{TI}={\langle {\omega }_T^2\rangle }_I$ with ${\omega }_N=\mathit{n}·\mathit{\omega }$ and ${\omega }_T^2={\omega }^2-{\omega }_N^2$.

Figure 5

Properties of the filter used near the TNTI layer: (a) the effective filter size ${\langle R_f\rangle }_I$ divided by $r$; (b) the location ${\langle {\zeta }_f\rangle }_I$ at which filtered quantities are defined. These results are taken from SFT1.

Figure 6

Cumulative kinetic energy budgets as functions of ${\zeta }_I/\eta$ for SFT1, showing all the terms in Eq. (2), i.e., pressure diffusion ${\langle D_P\rangle }_I$, viscous diffusion ${\langle D_{\nu }\rangle }_I$, turbulent diffusion ${\langle D_T\rangle }_I$, diffusion by large- and small-scale interactions ${\langle D_S\rangle }_I$, viscous dissipation ${\langle {\varepsilon }_{\nu }\rangle }_I$, and energy flux ${\langle \mathrm{\Pi }\rangle }_I$ for (a) $r=6\eta$ and (b) $r=24\eta$. The results are presented for $|{\zeta }_I|\ge r$.

Figure 7

(a) Cumulative kinetic energy near the TNTI layer and conditional averages of (b) viscous dissipation $-{\langle {\varepsilon }_{\nu }\rangle }_I$ and (c) energy flux ${\langle \mathrm{\Pi }\rangle }_I$, which are plotted as functions of ${\zeta }_I$ and $r$ in SFT1. Black regions represents $|{\zeta }_I|<r$.

Figure 8

Correlation coefficients between $D_T$ and $D_S$, $C_I(D_T,D_S)$, and between $-\mathrm{\Pi }$ and $D_T$, $C_I(D_T,-\mathrm{\Pi })$, plotted against $r/\eta$ in SFT1. The results are presented for $|{\zeta }_I|\ge r$.

Figure 9

Conditional averages of diffusion terms of the cumulative kinetic energy as functions of ${\zeta }_I$ and $r$ in SFT1: (a) pressure diffusion ${\langle D_P\rangle }_I$; (b) viscous diffusion ${\langle D_{\nu }\rangle }_I$; (c) turbulent diffusion ${\langle D_T\rangle }_I$. The results are presented for $|{\zeta }_I|\ge r$ while black represents $|{\zeta }_I|<r$.

Figure 10

(a) Coordinate within the TNTI layer (${\zeta }_I/\eta$) and scale ($r/\eta$) at which the mean pressure diffusion term ${\langle D_P\rangle }_I$ attains its minimum, plotted against the Reynolds number on the centerline of the shear-free turbulence. (b) Comparison between the mean pressure diffusion ${\langle D_P\rangle }_I$ and the energy dissipation ${\langle {\varepsilon }_{\nu }\rangle }_I$ (both normalized by the energy flux ${\langle \mathrm{\Pi }\rangle }_I$) plotted against the Reynolds number, where the values are taken from the locations where ${\langle D_p\rangle }_I$ attains its minimum shown in panel (a).

Figure 11

Conditional averages of decomposed terms of the interscale energy flux, Eqs. (13, 14, 15, 16, 17), in SFT1: (a) $r/\eta =6$; (b) $r/\eta =24$. The results are presented for $|{\zeta }_I|\ge r$.

Figure 12

PDF of the kinetic energy flux $\mathrm{\Pi }$ for $r/\eta =4$ within the VSL (${\zeta }_I/\eta =-4$), TSL (${\zeta }_I/\eta =-10$), and turbulent core region (${\zeta }_I/\eta =-30$).

Figure 13

Joint PDF between $\mathrm{\Pi }$ and decomposed terms in Eqs. (13, 14, 15, 16, 17): (a) ${\mathrm{\Pi }}_{n,n}$, (b) ${\mathrm{\Pi }}_{n,t}$, (c) ${\mathrm{\Pi }}_{t,n}$, (d) ${\mathrm{\Pi }}_{t,tl}$, (e) ${\mathrm{\Pi }}_{t,tt}$. (f) Joint PDF between ${\mathrm{\Pi }}_{n,n}$ and ${\mathrm{\Pi }}_{t,n}$.

Figure 14

Dependence of conditional statistics on the different number of samples in SFT1: (a) cumulative kinetic energy; (b) averaged interscale energy flux. These results are obtained at ${\zeta }_I=-15\eta$ in SFT1.

Figure 15

Conditional statistics calculated with the filter that does not distinguish turbulent and nonturbulent regions: (a) cumulative kinetic energy; (b) averaged interscale energy flux. These results are obtained in SFT1.