Importance of small-scale anisotropy in the turbulent/nonturbulent interface region of turbulent free shear flows

There has been much debate over the past decade or so over the scaling of the thickness of the turbulent/nonturbulent (TNT) interface for turbulent shear flows. It is generally considered to consist of the outer viscous superlayer, in which viscous processes are significant, and an inner turbulent sublayer which is dominated by inertial processes. Various authors have stated that the interface thickness scales with the Taylor length scale $\lambda$ while others state that it scales with the Kolmogorov length scale $\eta$ [Buxton et al., Phys. Fluids 23, 061704 (2011)]. Frequently, only self-similar turbulent flows are considered in which a single value of either $\lambda$ or $\eta$ is sufficient to scale various phenomena, including the thickness of the TNT interface. In this paper we show that for flows which are not self-similar the local Kolmogorov length scale increases quite significantly as one moves closer to the boundary between the turbulent and nonturbulent fluid. We find that the variation of this local Kolmogorov length scale with normal distance from the TNT boundary may be collapsed by the local Kolmogorov length scale computed from the TNT boundary itself. We subsequently show that this variation in local Kolmogorov length scale occurs concurrently with an increase in the small-scale anisotropy in the TNT interface. This anisotropy peaks at the boundary between the viscous superlayer and the turbulent sublayer, suggesting that these two sublayers are in fact quite distinct from one another. We show that these results hold for a number of different TNT interfaces from various flows and with the bulk turbulence being in a variety of states of development. The local viscous scaling that we obtain, along with an increase in anisotropy primarily driven by an increased magnitude of velocity gradients in the TNT-boundary-normal direction, leads us to draw an analogy between TNT interfaces and the wall in wall-bounded turbulence.

Figure 1

DNS data set. (a) Percentage of data points identified as being turbulent as a function of enstrophy threshold for all three time steps considered. The chosen threshold is denoted by the dashed line. (b) Example of the turbulent region defined according to the chosen threshold. The red line denotes the high-speed TNT interface and the blue line denotes the low-speed TNT interface.

Figure 2

Illustration of the coordinate transform from the Cartesian set $(x,y)$ to the TNT interface parallel or normal set $(x^{*},y^{*})$.

Figure 3

Conditional (a) mean enstrophy $\langle {\omega }^2\rangle$ and (b) dissipation rate $\varepsilon$ as a function of interface-normal distance $y^{*}$ for both interfaces of the mixing layer. The dashed black line in (a) indicates the Taylor length scale as computed from the centerline of the flow and the dash-dotted black line indicates a thickness of $17\eta$, where $\eta$ is the Kolmogorov length scale computed along the centerline.

Figure 4

Conditional mean value for the terms ${\mathcal{T}}_{\omega }$ of the enstrophy transport equation as a function of interface-normal distance $y^{*}$ for both (a) the low-speed interface and (b) the high-speed interface of the mixing layer.

Figure 5

DNS data set. (a) Variation of the local Kolmogorov length scale ${\eta }_l$ with $y^{*}$ normalized by the bulk mean Kolmogorov length scale $\langle \eta \rangle$ for both interfaces. (b) Variation of the local Kolmogorov length scale with $y^{*}$ normalized by the value at the interface ${\eta }_l^{*}={\eta }_l(y^{*}=0)$.

Figure 6

DNS data set. (a) Variation of local Taylor length scale ${\lambda }_l$ with $y^{*}$ normalized by the bulk mean Kolmogorov length scale $\langle \eta \rangle$ for both interfaces. (b) Variation of the local Taylor length scale with $y^{*}$ normalized by the local Kolmogorov length scale ${\eta }_l$. (c) Variation of the local Taylor Reynolds number ${\text{Re}}_{\lambda }$ with $y^{*}$ (solid lines) and variation of ${({\lambda }_l/{\eta }_l)}^2$, multiplied by some constant to match ${\text{Re}}_{\lambda }$ at $y^{*}=0$, with $y^{*}$.

Figure 7

DNS data set. Anisotropy ratio $\mathrm{\Sigma }$ and local Kolmogorov length scale ${\eta }_l$, normalized by the bulk Kolmogorov length scale $\langle \eta \rangle$ as a function of $y^{*}$ for (a) the low-speed TNT interface and (b) the high-speed TNT interface.

Figure 8

Tomographic PIV data set. (a) Local Kolmogorov length scale ${\eta }_l$, normalized by the bulk Kolmogorov length scale $\langle \eta \rangle$, and (b) anisotropy ratio $K_2$ as a function of radial distance $r_i$ from the TNT interface. The $\circ$ denote the round jet and $*$ the fractal jet. Here OTNTI refers to the outer TNT interface and ITNTI refers to the inner TNT interface at the measurement station $x/D=2$.