Internal layers in turbulent free-shear flows

The characteristics of the internal layers of intense shear are examined in a mixing layer and in a jet, in the range of Reynolds numbers $134<{\text{Re}}_{\lambda }<275$. Conditionally averaged profiles of streamwise velocity conditioned on the identified internal layers present strong velocity jumps, which account for approximately $10\%$ of the characteristic large-scale velocity of the flow. The thickness $\langle {\delta }_{\text{w}}\rangle$ of the internal layers from the combined analysis of both the mixing layer and the jet scales with $\langle {\delta }_{\text{w}}\rangle /\lambda \sim {\text{Re}}_{\lambda }^{-1/2}$, which suggests a scaling with the Kolmogorov length scale ($\eta$), analogous to recent observations on the turbulent/nonturbulent interface (TNTI). The thickness of the internal shear layers within the mixing layer is found to be between $9\eta$ and $11\eta$. The concentration of a passive scalar across the internal layers is also examined, at the Schmidt number $\text{Sc}=1.4$. The scalar concentration does not show any jumps across the internal layers, which is an important difference between the internal layers and the TNTI. This can be explained from the analysis of the internal layers of intense scalar gradient, where the flow topology node/saddle/saddle dominates, associated with strain, whereas the internal layers of intense shear are characterized by a prevalence of focus/stretching. A topological content analogous to that obtained in layers of intense scalar gradient is found in proximity to the TNTI, at the boundary between the viscous superlayer and the turbulent sublayer. These observations evidence that the TNTI and the internal layers of intense scalar gradient are similar in several respects.

Figure 1

Two-dimensional cut of the scalar field at Schmidt number 1.4 in the turbulent mixing layer obtained from direct numerical simulations. The three regions delimited by squares of thick dashed lines identify the three subdomains analyzed in the present paper.

Figure 2

Snapshots of the streamwise velocity at constant $z$ within the mixing layer, obtained at the three different downstream locations shown in Fig. 1, corresponding to the three different Reynolds numbers ${\text{Re}}_{\lambda }\approx 200$, 250, and 275. The regions characterized by intense shear and obtained with $K=1.5$ in Eq. (2) are shown in red. The streamwise and crosswise coordinates, respectively $x$ and $y$, are nondimensionalized by the local momentum thickness ${\delta }_{\theta }$.

Figure 3

Snapshot of ${\omega }_{\text{SH}}{(x,y,z)|}_{il}$ at constant $z$, within the mixing layer at ${\text{Re}}_{\lambda }\approx 275$, obtained with $K=1.5$ in Eq. (2). The streamwise and crosswise coordinates, respectively $x$ and $y$, are nondimensionalized by the local momentum thickness ${\delta }_{\theta }$. The red dots are local maxima of ${\omega }_{\text{SH}}{(x,y,z)|}_{il}$ along the crosswise direction $y$, and indicate points constituting the internal layers of intense shear.

Figure 4

Snapshot of ${\omega }_{\text{SH}}{(x,y,z)|}_{\text{IL}}$ at constant $x$, within the jet, obtained as explained in Eq. (4). The Cartesian coordinates $y$ and $z$ are nondimensionalized by the equivalent orifice diameter $D_{\text{e}}$. The red circles are examples of ellipses that were fitted to the zones at intense ${\omega }_{\text{SH}}{(x,y,z)|}_{\text{IL}}$, while the black segments orthogonal to the long axes of the fitted ellipses are examples of profiles of the streamwise velocity that are retained in the statistics, thus contributing to the conditionally averaged velocity profiles. Note that profiles of streamwise velocity were determined in each of the points constituting the zones at intense ${\omega }_{\text{SH}}{(x,y,z)|}_{\text{IL}}$, and not only where the black segments are drawn. Here, only one segment per zone is presented for clarity.

Figure 5

(a) Conditionally averaged profiles of the streamwise velocity in the near vicinity of all detected internal layers within the range $0.25<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0.50$. In (b) and (c), only the profiles characterized by negative and positive values of $\frac{du}{dy}{|}_{\text{IL}}$, respectively, were retained and contribute to the conditionally averaged profiles of streamwise velocity. Red full $•$ symbols refer to ${\text{Re}}_{\lambda }\approx 200$, black $+$ symbols refer to ${\text{Re}}_{\lambda }\approx 250$, and blue empty $▹$ symbols refer to ${\text{Re}}_{\lambda }\approx 275$.

Figure 6

Averaged conditional statistics for the streamwise velocity across the internal layers of intense shear after distinguishing into (a) negative and (b) positive crosswise gradients of the streamwise velocity, $\frac{du}{dy}{|}_{\text{IL}}$. The quantities $\langle \delta u\rangle , \partial \langle u-u_{\text{i}}\rangle {/\partial y|}_{\text{max}}$, and $\langle {\lambda }_{\text{W}}\rangle$ correspond to the jump in the averaged streamwise velocity, the maximum local gradient of the averaged profile, and the characteristic thickness of the internal layer, respectively. In this figure, the profiles at ${\text{Re}}_{\lambda }\approx 250$ from Fig. 5 are used.

Figure 7

Black continuous lines show the average values of the instantaneous streamwise velocities $\overline{u_{\text{i}}}$ of all the points representing the internal layers as detected from Eq. (2), with $K=1.5$, at different nondimensional crosswise locations $y/\lambda$. The gray region bounds the instantaneous streamwise velocities within the range $\overline{u_{\text{i}}}\pm \sigma \left(u_{\text{i}}\right)/\mathrm{\Delta }U$, where $\sigma$ is the standard deviation. Blue dashed lines represent the same as black continuous lines, but the coefficient in Eq. (2) is $K=2.5$. Red full $•$ symbols are the average streamwise velocity of the flow. Vertical continuous lines identify the geometric centerlines, while dashed lines identify the crossing points within the flow, therefore the physical centerlines. These features are investigated at (a) ${\text{Re}}_{\lambda }\approx 200$, (b) ${\text{Re}}_{\lambda }\approx 250$, and (c) ${\text{Re}}_{\lambda }\approx 275$.

Figure 8

(a), (b) Conditionally averaged profiles of crosswise velocity in the near vicinity of (a) all the points representing the internal layers, and (b) only those points characterized by positive values of $\frac{du}{dy}{|}_{\text{IL}}$ detected within the ranges (blue line) $-0.25<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0$, (red line) $0<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0.25$, (green line) $0.25<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0.50$, and (black line) $0.50<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0.75$. (c) Conditionally averaged profiles of crosswise velocity with respect to the local crosswise velocity at the internal layers, within the range $0.25<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0.50$, at ${\text{Re}}_{\lambda }\approx 200$, 250, and 275 (symbols defined as in Fig. 5).

Figure 9

Velocity jump at the internal layers identified by setting $K=1.5$ in Eq. (3) for (a), (c) positive and (b), (d) negative values of $\frac{du}{dy}{|}_{\text{IL}}$ detected within the ranges (green symbols) $-0.25<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0$, (magenta) $0<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0.25$, (blue) $0.25<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0.50$, and (black) $0.50<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0.75$. $•$ symbols refer to ${\text{Re}}_{\lambda }\approx 200, \blacksquare$ symbols refer to ${\text{Re}}_{\lambda }\approx 250$, and $▸$ symbols refer to ${\text{Re}}_{\lambda }\approx 275$.

Figure 10

Thickness of the internal layers identified by setting $K=1.5$ in Eq. (3) for (a), (c) positive and (b), (d) negative values of $\frac{du}{dy}{|}_{\text{IL}}$ detected within the ranges (green symbols) $-0.25<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0$, (magenta) $0<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0.25$, (blue) $0.25<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0.50$, and (black) $0.50<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0.75$. $•$ symbols refer to ${\text{Re}}_{\lambda }\approx 200, \blacksquare$ symbols refer to ${\text{Re}}_{\lambda }\approx 250$, and $▸$ symbols refer to ${\text{Re}}_{\lambda }\approx 275$.

Figure 11

Conditionally averaged profiles of the streamwise velocity in the near vicinity of the internal layers of intense shear, in which only the profiles characterized by (a) negative and (b) positive values of $\frac{du}{ds}{|}_{\text{IL}}$ are retained and contribute to the calculation. Red $•$ symbols refer to the circular jet, while black $⧫$ symbols refer to the fractal jet.

Figure 12

Thickness of the internal layers identified in the jet (blue $\blacksquare$) and in the mixing layer (red $•$) as explained in Sec. 2, and normalized by the Taylor microscale $\lambda$, as a function of the Reynolds number ${\text{Re}}_{\lambda }$.

Figure 13

(a) Conditionally averaged profiles of scalar concentration at Schmidt number $\text{Sc}=1.4$ in the near vicinity of all detected internal layers within the range $0.25<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0.50$, in the DNS mixing layer. In (b) and (c), only the negative and positive values of $\frac{du}{dy}{|}_{\text{IL}}$, respectively, are retained and contribute to the conditionally averaged profiles of streamwise velocity. Symbols are defined in Fig. 5.

Figure 14

Snapshots showing the magnitude of $\nabla c(x,y,z)$ on planes at constant $z$, in a streamwise region where ${\text{Re}}_{\lambda }\approx 250$. The streamwise and crosswise coordinates, respectively $x$ and $y$, are nondimensionalized by an averaged momentum thickness ${\delta }_{\theta }$ in this region.

Figure 15

Conditionally averaged profiles of the streamwise velocity in the near vicinity of crosswise gradients of the passive scalar concentration at Schmidt number $\text{Sc}=1.4$ within the range (a) $-0.25<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0$, (b) $0<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0.25$, (c) $0.25<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0.50$, and (d) $0.50<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0.75$. $•$ symbols indicate ${\text{Re}}_{\lambda }\approx 200, +$ symbols indicate ${\text{Re}}_{\lambda }\approx 250$, and $▹$ symbols indicate ${\text{Re}}_{\lambda }\approx 275$.

Figure 16

Local flow topologies associated with the invariants of the velocity gradient tensors ($Q$ and $R$) for incompressible flow. Region I corresponds to stable focus/stretching $[R>0, Q>-{(27R^2/4)}^{1/3}]$, region II corresponds to unstable focus/stretching $[R<0, Q>-{(27R^2/4)}^{1/3}]$, region III corresponds to stable node/saddle/saddle $[R>0, Q<-{(27R^2/4)}^{1/3}]$, and region IV corresponds to unstable node/saddle/saddle $[R<0, Q<-{(27R^2/4)}^{1/3}]$. Figure adapted from Ooi et al. [56].

Figure 17

Joint probability density functions of the second ($Q$) and third ($R$) invariants of the velocity gradient tensor (a) in random points of the turbulent portion of the domain, (b) in points of intense shear vorticity ${\omega }_{\text{SH}}(x,y,z)$, and (c) in points of intense gradient of the scalar concentration $\left|\nabla c\right(x,y,z\left)\right|$ [with $H=0.6$ in Eq. (5)], at ${\text{Re}}_{\lambda }\approx 275$. (d) $QR$ scatter plot at $y_{\text{I}}=4\eta$ from the TNTI of the low velocity side of the flow inside the turbulent region, at ${\text{Re}}_{\lambda }\approx 250$. The blue continuous lines are the null-discriminant lines $Q=-{(27R^2/4)}^{1/3}$ (Chong et al. [55]). In each panel, four regions can be identified, and their physical meaning is illustrated in Fig. 16. $Q_{\text{W}}={(\nabla \times \mathbf{u})}^2/4$.

Figure 18

Covariance between the shear vorticity magnitude ${\omega }_{\text{SH}}$ and the scalar gradient magnitude $\left|\nabla c\right|$ in the range of crosswise positions $y=[5.1,6.5]$, at ${\text{Re}}_{\lambda }\approx 275$.

Figure 19

Covariance between the shear vorticity magnitude ${\omega }_{\text{SH}}$ and the scalar gradient magnitude $\left|\nabla c\right|$ after removing points characterized by unstable focus/stretching, in the range of crosswise positions $y=[5.1,6.5]$, at ${\text{Re}}_{\lambda }\approx 275$.

Figure 20

(a) Thickness of the internal layers of intense shear identified in the mixing layer at different values of the constant $K$ in Eq. (2), within the range $0.25<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0.50$. Only the profiles characterized by positive values of $\frac{du}{dy}{|}_{\text{IL}}$ are retained and contribute to the averages. Red full $•$ symbols refer to ${\text{Re}}_{\lambda }\approx 200$, while blue empty $▹$ symbols refer to ${\text{Re}}_{\lambda }\approx 275$. (b) Thickness of the internal layers of intense shear identified in the jet at different values of the constant $C$ in Eq. (4). Only the profiles characterized by positive values of $\frac{du}{dy}{|}_{\text{IL}}$ are retained and contribute to the averages. Blue full $\square$ symbols refer to the jet with circular orifice, while red asterisks $*$ refer to the jet with fractal orifice.

Figure 21

Conditionally averaged profiles of streamwise velocity identified in the mixing layer in the vicinity of internal layers of intense shear, at different values of the constant $K$ in Eq. (2), within the range $0.25<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0.50$, at (a) ${\text{Re}}_{\lambda }\approx 200$ and (b) ${\text{Re}}_{\lambda }\approx 275$. Only the profiles characterized by positive values of $\frac{du}{dy}{|}_{\text{IL}}$ are retained and contribute to the averages.

Figure 22

Conditionally averaged profiles of the streamwise velocity in the near vicinity of the internal layers of intense shear, at different values of the constant $C$ in Eq. (4), for (a) the jet with fractal orifice and (b) the jet with round orifice. Only the profiles characterized by positive values of $\frac{du}{dy}{|}_{\text{IL}}$ are retained and contribute to the calculation.

Figure 23

Conditionally averaged profiles of streamwise velocity identified in the mixing layer in the vicinity of internal layers of intense shear, at different values of the constant $K$ in Eq. (2), within the range $0.25<(\mathrm{\Delta }U-u_{\text{i}})/u_{\text{rms}}<0.50$, at (a) ${\text{Re}}_{\lambda }\approx 200$ and (b) ${\text{Re}}_{\lambda }\approx 275$. Only the profiles characterized by negative values of $\frac{du}{dy}{|}_{\text{IL}}$ are retained and contribute to the averages.

Figure 24

Conditionally averaged profiles of the streamwise velocity in the near vicinity of the internal layers of intense shear, at different values of the constant $C$ in Eq. (4), for (a) the jet with fractal orifice and (b) the jet with round orifice. Only the profiles characterized by negative values of $\frac{du}{dy}{|}_{\text{IL}}$ are retained and contribute to the calculation.

Figure 25

Joint probability density functions of the second ($Q$) and third ($R$) invariants of the velocity gradient tensor (a) in points of intense scalar gradient with $H=0.4$ in Eq. (5), (b) in points of intense scalar gradient with $H=0.5$ in Eq. (5), (c) in points of intense shear vorticity with $K=2$ at ${\text{Re}}_{\lambda }\approx 200$, (d) in points of intense shear vorticity with $K=3$ at ${\text{Re}}_{\lambda }\approx 275$. The regions within the different JPDFs are described in the caption of Fig. 17. $Q_{\text{W}}={(\nabla \times \mathbf{u})}^2/4$.