4.1 Bubbles and drops

The first step is to define the interface separating the flow into turbulent and non-turbulent regions. This is not as straightforward as the previous section may suggest. Figure 2(b) shows that the vorticity isocontour is not usually a singly connected surface. Depending on the threshold, there may be a few or several thousands of disconnected components of the isosurface, but one of them is typically much larger than the rest and divides the computational box into two large disjoint regions. The remaining isosurface components can be classified as envelopes of low-vorticity bubbles within the turbulent region (figure 5 c), or of high-vorticity drops in the free stream. It will be shown in § 5 that, although there can be a large number of bubbles, they are usually too small to contribute significantly to most quantities related to the T/NT interface. There are generally very few drops. In consequence, the rest of this paper defines the interface as the largest singly connected component of the vorticity isosurface that separates ‘smoothed’ irrotational and vortical regions from which drops and bubbles have been eliminated.

Figure 5.

Sketch of the three basic geometrical features in the vorticity isosurface: (a) handles, (b) pockets and (c) bubbles.

The algorithm to obtain this largest component is sketched in figure 6. We first decompose the computational domain into computational cells (voxels). The flow properties are defined at their vertices. We next obtain the set ${\it\Omega}_{{\it\omega}>}$ of voxels for which at least one vertex has a vorticity higher than the threshold (figure 6 a). This set contains the drops and the bulk of the turbulent flow. Similarly, we obtain the set ${\it\Omega}_{{\it\omega}<}$ for which at least one vertex has a vorticity lower than the threshold, containing the bubbles and the bulk of non-turbulent flow (figure 6 b). Each of these sets has a connected component many times larger than the rest (approximately seven orders of magnitude in our case). In the case of ${\it\Omega}_{{\it\omega}>}$ , this set represents the bulk of the turbulent flow, ${\it\Omega}_{t}$ , while in the case of ${\it\Omega}_{{\it\omega}<}$ it represents the bulk of the free stream, ${\it\Omega}_{n}$ . The sets of voxels containing the drops, ${\it\Omega}_{d}$ , and the bubbles, ${\it\Omega}_{b}$ , are obtained by subtracting these largest components from their respective sets. Thus, ${\it\Omega}_{d}={\it\Omega}_{{\it\omega}>}-{\it\Omega}_{t}$ and ${\it\Omega}_{b}={\it\Omega}_{{\it\omega}<}-{\it\Omega}_{n}$ . The final step is to define the set of voxels of the cleaned T/NT interface as (figure 6 c)

It should be noted that the two terms of this equation are effective definitions of the smoothed turbulent region (the bulk of the turbulent flow plus the bubbles) and the smoothed non-turbulent region (the bulk of the non-turbulent flow plus the drops).

Figure 6.

(a) The set ${\it\Omega}_{{\it\omega}>}$ of voxels in which at least one vertex is ${\it\omega}>{\it\omega}_{0}$ . (b) The set ${\it\Omega}_{{\it\omega}<}$ of voxels where ${\it\omega}<{\it\omega}_{0}$ . (c) The set ${\it\Omega}_{i}$ of voxels that contain the interface, from (4.1).

Drops and bubbles should not be confused with other complications of the interface, such as the handles and overhangs or ‘pockets’ represented in figure 5(a,b). The former complicate the topology of the flow and cannot be eliminated. The latter are topologically neutral, but may be important from the experimental or dynamical point of view. They hide part of the surface to some observational procedures, and may be precursors of large-scale engulfing. At this point, the interface is still a set of voxels that has to be converted into a surface, but this representation is sufficient for the analysis in the next two sections.

4.2 Fractal dimension

Mandelbrot (Reference Mandelbrot1975) was the first to suggest that the hierarchy of turbulent eddies can be approximated by a fractal when the Reynolds number is large enough. This was first verified by Sreenivasan & Meneveau (Reference Sreenivasan and Meneveau1986) for the bulk of the flow, and by Sreenivasan et al. (Reference Sreenivasan, Ramshankar and Meneveau1989) for the T/NT interface. The latter also proposed a simple theory to relate the two results. The fractal dimension of the vorticity isosurface measures how contorted it is, and is a useful statistical measure of its complexity. The most widely used definition is the box-counting Kolmogorov capacity: if $N_{b}$ is the number of boxes of size $r$ required to cover a set ${\it\Omega}$ , such as the interface, the fractal dimension $D$ is defined by $N_{b}\propto r^{D}$ .

In practice, the computation of fractal dimensions is complicated because turbulence is only self-similar in a limited range of scales. Vorticity is smooth at scales of the order of the Kolmogorov microscale, and the largest eddies responsible for the energy input are not self-similar. In cases in which an extended power law is not immediately obvious, a reasonable redefinition of the box-counting dimension is

where ${\it\varsigma}$ stands for the smallest possible box size at which the data set remains self-similar, or, in the present case, for the size of the computational voxels. This requirement is difficult to define, and it is hard to speak of a fractal unless the self-similar range extends over a reasonably wide range.

Sreenivasan et al. (Reference Sreenivasan, Ramshankar and Meneveau1989) found a clear power law from two-dimensional sections of the interface, and measured a constant dimension $D$ away from the saturation caused by the shortest and longest scales. They concluded that the interface is a monofractal in that range. Moisy & Jiménez (Reference Moisy and Jiménez2004) computed the fractal dimension of three-dimensional enstrophy isosurfaces in homogeneous turbulence, using the full three-dimensional field instead of cross-sections, and three-dimensional boxes instead of two-dimensional ones. They found that the self-similar range observed by Sreenivasan et al. (Reference Sreenivasan, Ramshankar and Meneveau1989) is only an approximation, and defined a local dimensional exponent to account for the dependence on the box size,

This definition includes the previous two. If $D_{b}(r)$ is constant and the T/NT interface is a monofractal, $\dim _{b}=D=D_{b}(r)$ .

The local exponent (4.3) of our ‘cleaned’ interface is presented in figure 7 as a function of ${\it\omega}_{0}^{\ast }$ for several Reynolds numbers. Figure 7(a) plots $D_{b}$ for the smallest possible value of $r$ , and tries to approximate (4.2). Figure 7(b) plots the maximum value $D_{b}$ over the whole range of $r$ . The differences between the two figures quantify how far from a monofractal the T/NT interface is. The good collapse of the different Reynolds numbers when parametrised with ${\it\omega}_{0}^{\ast }$ should be noted. The black horizontal bar near the peaks of both figures is the variation of ${\it\omega}^{\ast }/{\it\omega}^{+}$ in our range of Reynolds number. A similar bar is included in all later figures that make a Reynolds number comparison, and measures how much the collapse of the different curves would deteriorate if the data had been normalised with ${\it\omega}_{0}^{+}$ instead of with ${\it\omega}_{0}^{\ast }$ .

Figure 7.

Estimation of the box-counting fractal dimension, obtained in (a) from the local fractal exponent in the limit of small box size, and in (b) from the maximum of the local exponent over $r$ . The symbols represent different values of ${\it\delta}_{99}^{+}$ : ○, 1100; $\times$ , 1300; ▿, 1500; ▵, 1700; $\star$ , 1900. The horizontal bar is the variation of ${\it\omega}^{\ast }/{\it\omega}^{+}$ in our range of ${\it\delta}_{99}^{+}$ .

The dependence of $D_{b}$ on ${\it\omega}_{0}^{\ast }$ confirms the visual impression from figure 2 that the threshold has a dramatic effect on the interface. At low thresholds, the dimension approaches the smooth limit $D=2$ , but, at higher ones, the T/NT interface is significantly more convoluted. Sreenivasan et al. (Reference Sreenivasan, Ramshankar and Meneveau1989) predicted $D=7/3$ for the T/NT interface, precisely the value observed by de Silva et al. (Reference de Silva, Philip, Chauhan, Meneveau and Marusic2013) in a more recent experiment, which is within the range of the present results. It would correspond to ${\it\omega}_{0}^{\ast }\simeq 1$ in figure 7(a), and to the lowest possible dimension in figure 7(b).

Regardless of the differences in their absolute values, the two estimations of the fractal dimension in figure 7 behave similarly with respect to ${\it\omega}_{0}^{\ast }$ . There is a transition between ${\it\omega}_{0}^{\ast }\simeq 0.2$ and ${\it\omega}_{0}^{\ast }\simeq 2$ , across which the geometrical complexity of the interface increases significantly. This suggests that across this range the isosurface moves inside the turbulent core, where it reflects the geometrical features of the turbulent vorticity itself. It also follows from figure 4 that ${\it\omega}_{0}^{\ast }\simeq 0.2$ corresponds to the threshold for which the average location of the interface $\langle y_{I}\rangle$ decreases fastest as the threshold increases.

The decrease of the dimension beyond ${\it\omega}_{0}^{\ast }\simeq 2$ was already observed by Moisy & Jiménez (Reference Moisy and Jiménez2004), who used thresholds equivalent to ${\it\omega}_{0}^{\ast }\simeq 2{-}12$ to study the geometry of the volume of the vorticity in isotropic turbulence. There is no simple relation between the fractal dimensions of a set and of its surface, but Moisy & Jiménez (Reference Moisy and Jiménez2004) noted that in the limit of very high thresholds the vorticity would be reduced to a discrete cloud of points for which $D\simeq 0$ . A similar argument can be applied to the interface.

4.3 Genus

The geometric complexity of an object can also be characterised by its topological properties. The genus $g$ is a topological invariant of any connected orientable surface, and measures the number of its ‘handles’ (figure 5 a). A sphere has genus zero, a torus has genus one, and two connected tori have genus two. To the best of our knowledge, the genus was first used to characterise turbulent structures in homogeneous turbulence by Leung, Swaminathan & Davidson (Reference Leung, Swaminathan and Davidson2012), who cite instances of its earlier use in disciplines such as astrophysics. In most of those cases, the genus is obtained by integrating the mean and Gaussian curvatures over the interface, which requires a careful triangulation of the surface. This step is time consuming and prone to errors, and we bypass it by computing the genus directly from the Euler characteristic ${\it\chi}$ of the numerically defined contour. The algorithm is described in Lozano-Durán & Borrell (Reference Lozano-Durán and Borrell2016), and is optimised to exploit discrete data in structured grids.

Briefly, any numerical isosurface in a Cartesian grid is a polyhedron of stacked parallelepipeds. If $V$ is the number of vertices, $E$ is the number of edges and $F$ is the number of faces, its Euler characteristic is given by the Euler–Poincaré formula,

and the genus is

The genus is a measure of complexity, like the fractal dimension, but the two are not equivalent. A wrinkled piece of paper has genus zero, independently of the amount of wrinkling. A regular Brownian surface is defined as a fractal single-valued map on the real plane. Its fractal dimension is $D=2.5$ , but it has no handles (Russ Reference Russ1994).

Figure 8.

(a) Genus normalised with its maximum over ${\it\omega}_{0}$ . The horizontal bar is the variation of ${\it\omega}^{\ast }/{\it\omega}^{+}$ in our range of ${\it\delta}_{99}^{+}$ . The horizontal bar is the ratio of ${\it\omega}^{\ast }/{\it\omega}^{+}$ over the present range of ${\it\delta}_{99}^{+}$ . (b) Maximum genus per cubic boundary-layer thickness, occurring in all cases at ${\it\omega}_{0}^{\ast }\simeq 2$ . Both axes are logarithmic. The solid line is a power law fit, $\max (g)/{{\it\delta}_{99}^{+}}^{3}\propto {{\it\delta}_{99}^{+}}^{1.6}$ . The dashed one is the Kolmogorov limit, $\max (g)/{{\it\delta}_{99}^{+}}^{3}\propto {{\it\delta}_{99}^{+}}^{9/4}$ . The symbols represent values of ${\it\delta}_{99}^{+}$ , as in figure 7.

As in the previous section, we compute the genus for the largest connected component of the vorticity isosurface, which is shown in figure 8(a) normalised by its maximum over ${\it\omega}_{0}$ . There is a topological transition in which handles begin to appear over roughly the same range, ${\it\omega}_{0}^{\ast }\simeq (0.2{-}2)$ , as the growth of the fractal dimension. Around ${\it\omega}_{0}^{\ast }\simeq 1$ , handles are the dominant feature of the surface, and there are hundreds or thousands of them in a volume $O({\it\delta}_{99}^{3})$ . We suggested in the discussion of the fractal dimension that the T/NT interface at these high thresholds is basically a reflection of the internal geometry of the turbulent vorticity, and the reasons for the decrease of the dimension beyond the end of the transition also apply here. Some turbulent features disappear for very large thresholds, causing the genus to decrease. The maximum genus occurs at the end of the topological transition ${\it\omega}_{0}^{\ast }\simeq 2$ , and figure 8(b) shows that it increases with the Reynolds number as $\max (g)/{{\it\delta}_{99}^{+}}^{3}\propto {{\it\delta}_{99}^{+}}^{1.6}$ . This exponent is somewhat lower than that for the number of Kolmogorov-size structures per cubic integral scale ${{\it\delta}_{99}^{+}}^{9/4}$ , which sets an upper bound for the scaling of the possible complexity. It is tantalisingly close to the corresponding number of ${\it\lambda}$ -sized structures, ${{\it\delta}_{99}^{+}}^{3/2}$ . One should note again the good collapse provided by ${\it\omega}_{0}^{\ast }$ for the Reynolds number dependence of the genus.

This proliferation of handles will become important for the conditional analysis of the flow in the next section. When the analysis of a surface with handles is carried out using a lower-dimensional section, such as a two-dimensional plane or a line, the results can be subject to interpretation artefacts. For example, the planar section of a torus across its principal axis is two circles, giving the impression of two disconnected geometrical objects. Up to a point, the same is true for pockets such as those in figure 5(b). For example, the interface shown below in figure 12(b) is a section of a singly connected isosurface, although it appears to contain many unconnected irrotational bubbles within the turbulent region. Another effect of the handles has to do with values conditioned to the direction normal to the interface. The usual assumption in this case is that a normal defined from high towards low values of the vorticity points into the free stream. In a handle, or in a narrow pocket, this is only true over distances of the order of the feature thickness, and becomes an issue if handles and pockets are dominant. The problem is less pressing when the threshold is chosen below ${\it\omega}_{0}^{\ast }\simeq 0.2$ , where the T/NT interface is smoother, but figure 4 shows that a great deal of the published work uses thresholds within the topological transition, characterised by non-trivial fractal dimensions and, presumably, large genera.

The main conclusion from this section is that the properties of the fully turbulent flow appear gradually in the geometry of the interface as the threshold traverses the topological transition, and that the handles, folds and high fractal dimensions are probably the reflection of the internal structure of the flow.

5.1 The signed distance field

The discrete fields obtained with the minimum and vertical distances are called the ball-distance field ${\it\Delta}_{b}(x,y,z)$ and the vertical-distance field ${\it\Delta}_{v}(x,y,z)$ respectively. The symbol ${\it\Delta}$ denotes distance regardless of the particular definition.

The concept of a distance field is also found in Mellado, Wang & Peters (Reference Mellado, Wang and Peters2009), who use the length of the trajectories along lines of maximum gradient of an advected scalar to measure the distance with respect to the interface. While their definition can also be used regardless of the complexity of the surface, the gradient lines of the vorticity magnitude become very contorted away from the interface on the turbulent side, and less suitable for conditional analysis than either of the two definitions mentioned above.

Sections of the two distance fields of the same snapshot of the flow are shown in figure 12, each computed for two different thresholds. The isosurface ${\it\Delta}=0$ is our effective definition of the interface, but note that the two distance definitions generate different surfaces. The first observation is that the two distances give fairly different results on the turbulent side, particularly for the higher vorticity thresholds. On the non-turbulent side, where the interface is more convex, the differences are not as important. When the threshold is within the topological transition, such as ${\it\omega}_{0}^{\ast }=0.5$ in figure 12(b,d), the contortions of the ‘ball’ interface are so intense that there are very few points on the turbulent side for which ${\it\Delta}_{b}>100{\it\eta}$ . We emphasise that ${\it\Delta}_{b}=0$ in figure 12(b) corresponds to a single connected surface from which bubbles have been removed, and that the apparently isolated contours within the turbulent side are artefacts of the two-dimensional section. Comparison of the results of the two thresholds for each distance definition shows that the vertical-distance field in figure 12(c,d) is less sensitive to the contortions than the ball distance in figure 12(a,b), and also less sensitive to the choice of the threshold. Because of this, it misses most of the interface complexity and the existence of the topological transition.

Away from the wall, the two distance definitions also behave differently. Because of its connection with the Eikonal, the ball distance can be visualised as a wavefront moving away from the interface at a uniform velocity. As is does, the interface irregularities are eliminated by successive mergings of caustics, and the ${\it\Delta}_{b}$ isosurface becomes smoother. Roughly speaking, a ${\it\Delta}_{b}$ isosurface only retains wavelengths larger than $O(|{\it\Delta}_{b}|)$ . The vertical distance does not share this smoothing property. Because the ${\it\Delta}_{v}$ isosurfaces are vertical translations of the interface, they retain its irregularities at all distances. It should be noted also that neither distance is additive. Because a ${\it\Delta}\neq 0$ isosurface is not an iso-vorticity surface, even if the ${\it\Delta}=0$ surface is defined as one, it is impossible to define a distance between interfaces with different thresholds such that ${\it\Delta}(p\rightarrow {\it\omega}_{0})={\it\Delta}(p\rightarrow {\it\omega}_{1})+{\it\Delta}({\it\omega}_{1}\rightarrow {\it\omega}_{0})$ . This will later lead to ambiguities in the definition of the thickness of the T/NT interface layer.

The angle ${\it\theta}$ between the normal to the ${\it\Delta}_{b}$ interface and the vertical can be estimated by $\text{d}{\it\Delta}_{v}/\text{d}{\it\Delta}_{b}|_{{\it\Delta}_{b}=0}=1/\cos {\it\theta}$ , but there is no simple way to evaluate the orientation of the ${\it\Delta}_{v}$ interface in this manner.

Figure 13.

(a) Mean position of the T/NT interface as a function of the vorticity threshold. No symbols, vorticity isosurface as in figure 4; ▵, interface defined as ${\it\Delta}_{b}=0$ ; ○, ${\it\Delta}_{v}=0$ . The threshold in (b) is the vertical dashed line. (b) The PDFs of the vertical positions of the three isosurfaces for ${\it\omega}_{0}^{\ast }=0.19$ . The lines are as in (a); ${\it\delta}^{+}=1500$ .

None of the interfaces defined by the above distance criteria exactly coincide with a vorticity isosurface. In the case of ${\it\Delta}_{b}$ the only difference is the absence of the bubbles and drops discarded in the smoothing step, and the deviations are relatively minor. The vertical distance misses substantial parts of the isosurface, and may deviate significantly from it. Figure 13(a) shows the mean positions of the two interfaces as a function of ${\it\omega}_{0}^{\ast }$ , compared with the mean position of the vorticity isosurface. The mean $\langle y_{I}({\it\Delta}_{b})\rangle$ deviates little from the position $\langle y_{I}\rangle$ of the vorticity isosurface (figure 4), but $\langle y_{I}({\it\Delta}_{v})\rangle$ remains close to the edge of the boundary layer even when the vorticity isosurface moves closer to the wall. This is confirmed by the PDFs of the heights of the three isosurfaces, given in figure 13(b). For low thresholds (not shown), the PDFs of the two interfaces and of the vorticity isosurface roughly coincide, and are approximately Gaussian (Corrsin & Kistler Reference Corrsin and Kistler1955). However, for the higher threshold in figure 13(b), $y_{I}({\it\Delta}_{b})$ follows the isosurface into the turbulent core of the boundary layer substantially better than $y_{I}({\it\Delta}_{v})$ . As a consequence, $y_{I}({\it\Delta}_{b})$ results in a much better representation of the intermittency parameters of the boundary layer, such as ${\it\gamma}$ . It should be noted that the vorticity threshold used in figure 13(b), ${\it\omega}_{0}^{\ast }=0.19$ , although relatively high, is below the topological transition, and in the range of most of the studies collected in table 2.

5.2 Conditional analysis of distance and vorticity

The properties of the vorticity conditioned to its position with respect to the interface can be analysed using the joint PDF of the vorticity magnitude and the distance, $F_{{\it\omega},{\it\Delta}}$ . Figure 14 shows four examples corresponding to the thresholds and distance definitions in figure 12. Similar PDFs were obtained by Taveira & da Silva (Reference Taveira and da Silva2014) using ${\it\Delta}_{v}$ in a planar jet, and by da Silva et al. (Reference da Silva, Taveira and Borrell2014b ) for jets, a shearless interface and a subset of the present boundary layer. The analysis in the present paper has been carried out for five Reynolds numbers in the range ${\it\delta}_{99}^{+}\in (1100{-}1900)$ , and 10 thresholds in ${\it\omega}_{0}^{\ast }\in (0.01{-}0.5)$ , each of them computed for the two distance definitions described above.

Figure 14.

Premultiplied joint PDF of vorticity and distance, ${\it\omega}F_{{\it\omega},{\it\Delta}}$ . Each panel corresponds to the thresholds, Reynolds number and distance definitions in figure 12 ((a,b) ${\it\Delta}_{b}$ , (c,d) ${\it\Delta}_{v}$ , (a,c) ${\it\omega}_{0}^{\ast }=0.01$ , (b,d) ${\it\omega}_{0}^{\ast }=0.5$ ). The contours contain 50 %, 90 % and 99 % of points.

The joint PDF can be divided into four quadrants (figure 14 b), separated by the axes ${\it\Delta}=0$ and ${\it\omega}={\it\omega}_{0}$ , marked with dashed lines in figure 14. Given that the flow field is the same in the four figures, the differences in the joint PDF are due to the different distance definitions and thresholds.

The first quadrant, which contains points classified as turbulent and with a relatively high vorticity, represents the core turbulent flow. As already seen in figure 12, the minimum and vertical distances behave similarly for low thresholds (figure 14 a,c), but very differently for thresholds within the topological transition. The field of vertical distances depends only slightly on the threshold (figure 14 c,d), but there are few points at distances beyond ${\it\Delta}_{b}=100{\it\eta}$ for the higher threshold in figure 14(b).

The second quadrant contains different geometrical objects depending on the distance definition. It contains bubbles for ${\it\Delta}_{b}$ , and bubbles, handles and pockets for ${\it\Delta}_{v}$ . For the ball distance, the weight of the second quadrant is always small compared with the first one, and contributes little to the averaged vorticity on the free-stream side of the interface (figure 14 a,b). This is not the case for the vertical distance, and it is clear from figure 14(c,d) that the weight of this quadrant increases as the interface becomes more complex at high thresholds. This quadrant, with special reference to the properties of the pockets, will be studied in more detail in § 5.4.

The third quadrant contains points of low vorticity classified as non-turbulent. It represents the bulk of the free stream which, in the case of ${\it\Delta}_{b}$ , also includes the irrotational pockets. It depends only weakly on the threshold and on the distance definition, except for ${\it\omega}\approx {\it\omega}_{0}$ .

The fourth quadrant, with ${\it\omega}>{\it\omega}_{0}$ and negative distances, corresponds to the objects defined in § 4.1 as drops. It is almost empty for all of the cases considered in this study, confirming that the smoothing of the free stream described in § 4.1 does not affect the results of the conditional analysis.

Figure 15.

(a) Sketch of the discontinuity of the vertical distance at the edge of a pocket. The hatched line marks points where ${\it\omega}={\it\omega}_{0}$ and ${\it\Delta}_{b}=0$ , but ${\it\Delta}_{v}>0$ . The non-turbulent region just outside $A$ has ${\it\Delta}_{b}\simeq 0$ but ${\it\Delta}_{v}<0$ . (b) Premultiplied PDF, ${\it\Delta}_{v}F_{{\it\omega}_{0}^{-},{\it\Delta}_{v}}$ , of the vertical distance of the non-turbulent points with ${\it\omega}^{\ast }=0.25{-}0.5$ , whose vorticity is close to the threshold ${\it\omega}_{0}^{\ast }=0.5$ : ○ (blue), ${\it\delta}_{99}^{+}=1100$ ; $\times$ (green), 1300; ▵ (red), 1500; ▿ (black), 1700; $\star$ (magenta), 1900.

The influence of the distance definition on the joint PDF is most visible far from the horizontal axis in the neighbourhood of ${\it\omega}={\it\omega}_{0}$ . These are points in which the vorticity is close to the identification isosurface, but which are incorrectly identified as being far from the interface. The range of possible ball distances for ${\it\omega}\simeq {\it\omega}_{0}$ (figure 14 a,b) is very narrow, $|{\it\Delta}_{b}|<100{\it\eta}$ , especially in the second quadrant, and can be interpreted as a typical distance to the interface of the irrotational bubbles that have been labelled as turbulent by the smoothing process. On the other hand, the vertical distances in the same region can be as large as $250{\it\eta}$ on both sides of the interface (figure 14 c,d). We denote by ${\it\omega}_{0}^{-}$ the vorticities just below the threshold. The wide ${\it\Delta}_{v}$ tails of $F_{{\it\omega}_{0}^{-},{\it\Delta}_{v}}$ have several causes, sketched in figure 15(a). On the positive side, ${\it\Delta}_{v}>0$ in $Q_{2}$ , all of the points represented with a hatched line in that figure are on the ${\it\omega}={\it\omega}_{0}$ and ${\it\Delta}_{b}=0$ isosurfaces, but not on the ${\it\Delta}_{v}$ interface, which is only the top of the overhang. Points near the hatched line have vorticities close to ${\it\omega}_{0}$ , but are counted as being deep within the turbulent region by ${\it\Delta}_{v}$ . The ${\it\Delta}_{v}<0$ tail of $F_{{\it\omega}_{0}^{-},{\it\Delta}_{v}}$ in $Q_{3}$ contains points whose vorticity is slightly below that of the threshold, but which are classified by ${\it\Delta}_{v}$ as being far within the irrotational region. They correspond to points such as $A$ in figure 15(a), where the orientation of the interface is vertical and induces a discontinuity in the height of the ${\it\Delta}_{v}$ interface. Such discontinuities are clearly visible in figure 12(c,d). These tangencies are less common than the overhung surfaces, and the mass in the negative tail of $F_{{\it\omega}_{0}^{-},{\it\Delta}_{v}}$ is typically smaller than in the positive one, especially in the convoluted interfaces at the higher thresholds (only 15 % as many in figure 12 d).

It is clear from figure 15(a) that the negative tail of $F_{{\it\omega}_{0}^{-},{\it\Delta}_{v}}$ contains information about the ‘depth’ of the pockets, rather than about the thickness of the interface. The premultiplied probability distribution ${\it\Delta}_{v}F_{{\it\omega}_{0}^{-},{\it\Delta}_{v}}$ , integrated over the band ${\it\omega}\in ({\it\omega}_{0}/2,{\it\omega}_{0})$ , is presented in figure 15(b) for a relatively high threshold. It is well approximated by a power law $F_{{\it\omega}_{0}^{-},{\it\Delta}_{v}}\propto {\it\Delta}_{v}^{-1}$ for ${\it\Delta}_{v}\lesssim 0.2{\it\delta}_{99}$ . Although the reason for this particular power is not completely clear, it suggests a regular structure for the ${\it\Delta}_{v}$ interface. That interface has no overhangs, and renders pockets as holes with steep sides. If we assume that the interface is covered with pockets of size ${\it\Delta}$ , the contribution of each hole to the PDF in figure 15(b) would be proportional to the $O({\it\Delta})$ length of its lip. Their number would be proportional to ${\it\Delta}^{-2}$ and the total lip length would be proportional to ${\it\Delta}^{-1}$ , as in the figure.

Even if this explanation turns out to be oversimplified, the fact that the distribution of pocket heights satisfies a power law is consistent with a fractal interface, and suggests that the discontinuities represent a self-similar hierarchy of overhangs. For the threshold in figure 15(b), the self-similar range ends around ${\it\Delta}_{v}\approx 0.2{\it\delta}_{99}$ , and the probability of finding pockets deeper than that limit is very low. This is approximately three times the standard deviation of the position of the vorticity isosurface for this threshold (figure 4). At lower thresholds such as those in figure 14(a,c), the self-similar range disappears, and the ‘pocket’ distribution is concentrated around ${\it\Delta}_{v}=10{\it\eta}$ .

Figure 16.

Conditional profiles: – –▫– –, $\overline{{\it\omega}}$ ; – –○– –, $\overline{{\it\omega}_{1}}$ ; – –▵– –, $\overline{{\it\omega}_{2}}$ . The figures follow the same arrangement as in figure 12 ((a,b) ${\it\Delta}_{b}$ , (c,d) ${\it\Delta}_{v}$ , (a,c) ${\it\omega}_{0}^{\ast }=0.01$ , (b,d) ${\it\omega}_{0}^{\ast }=0.5$ ). The black dashed lines correspond to the values of the threshold (vertical) and zero distance (horizontal). The inset in each figure corresponds to the same plot, using linear coordinates for the vorticity magnitude. The two solid lines with crosses in (c) and (d) correspond to $\overline{{\it\omega}}(3.1{\it\Delta}_{b})$ (see § 5.3.1).

5.3 Conditional averages

The averaged vorticity conditioned to the distance to the interface can be computed from $F_{{\it\omega},{\it\Delta}}$ as

It is given by the solid lines with squares in figure 16(a–d), and is equivalent to the conditional vorticity profiles in Westerweel et al. (Reference Westerweel, Hofmann, Fukushima and Hunt2002). One should note the use of the bar over the symbol to distinguish (5.1) from the more usual mean profile $\langle {\it\omega}\rangle$ at a given distance from the wall, defined as

We will use the notation $\overline{{\it\omega}}({\it\Delta}_{b})$ and $\overline{{\it\omega}}({\it\Delta}_{v})$ to distinguish between conditional profiles obtained with each definition of distance.

The conditional vorticity in all of the panels of figure 16 increases monotonically to its expected fully turbulent level, ${\it\omega}^{\ast }=O(1)$ , within a few Kolmogorov lengths from the interface, except for the plateau at ${\it\Delta}_{v}/{\it\eta}=15{-}40$ in figure 16(d).

The existence of a plateau or of a maximum in the conditional vorticity profile near the T/NT interface has been mentioned in wakes (Townsend Reference Townsend1976; Bisset et al. Reference Bisset, Hunt and Rogers2002) and reported in jets (Westerweel et al. Reference Westerweel, Fukushima, Pedersen and Hunt2009; da Silva et al. Reference da Silva, dos Reis and Pereira2011). Its presence has sometimes been used to define the thickness of the interface layer (da Silva & Taveira Reference da Silva and Taveira2010), and taken as the basis for theoretical models in which the interface is maintained by the presence of a strong localised shear (Hunt & Durbin Reference Hunt and Durbin1999). Similar models have been used to suggest similarities between the T/NT interface in jets (Westerweel et al. Reference Westerweel, Fukushima, Pedersen and Hunt2009) and strong internal vortex layers in homogeneous turbulence (Ishihara, Kaneda & Hunt Reference Ishihara, Kaneda and Hunt2013). Chauhan et al. (Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014) report a strong conditional vorticity peak in boundary layers, but their interface is defined in terms of the velocity fluctuations, and we will argue below that it is probably different from the one discussed here. Moreover, not all of these papers use the same definition of the interface or even the same thresholded scalar. In fact, when da Silva et al. (Reference da Silva, Hunt, Eames and Westerweel2014a ) compiled conditional vorticity statistics for a variety of flows, the only obvious peak was found in the early stages of the evolution of a shearless mixing layer (da Silva & Taveira Reference da Silva and Taveira2010). Bisset et al. (Reference Bisset, Hunt and Rogers2002) also find strong-vorticity peaks for some high-vorticity thresholds in their wake, but attribute them to the presence of isolated vorticity patches, and discard them in favour of a lower threshold $({\it\omega}^{\ast }\approx 0.1)$ for which the maximum is barely noticeable. Although we will find and discuss below comparable peaks in other variables, we note at this stage that, if the vorticity magnitude were particularly intense close to the interface, a plateau analogous to the one in figure 16(d) should also appear in the $\overline{{\it\omega}}({\it\Delta}_{b})$ profile in figure 16(b), but this is not the case. An alternative explanation is that the vorticity close to the interface is not particularly intense, but that, when the conditional profiles are obtained as a function of ${\it\Delta}_{v}$ at a sufficiently high threshold, some non-turbulent flow is counted as being turbulent within the inner part of the interface, lowering the local average vorticity (Bisset et al. Reference Bisset, Hunt and Rogers2002).

To differentiate between the two hypotheses we split the conditional profile $\overline{{\it\omega}}({\it\Delta}_{v})$ into contributions from the high-vorticity first quadrant, $Q_{1}$ , and the mislabelled non-turbulent points in $Q_{2}$ . Equation (5.1) is split into

where

are the conditional averages for $Q_{1}$ and $Q_{2}$ , and

are the corresponding weights. The profiles of $\overline{{\it\omega}}$ , $\overline{{\it\omega}}_{1}$ and $\overline{{\it\omega}}_{2}$ are given in figure 16. In the case of low thresholds (a,c), $\overline{{\it\omega}}_{1}\simeq \overline{{\it\omega}}$ , and the contribution of the second quadrant is small, regardless of the distance definition.

The only case in which $\overline{{\it\omega}}_{1}$ is clearly different from the overall average is figure 16(d), in which the contribution of the handles and pockets is significant. In this figure, the maximum relative weight of $Q_{2}$ is $W_{2}\simeq W_{1}/4$ at ${\it\Delta}_{v}=20{\it\eta}$ . At the even higher thresholds at which the interface reaches its maximum geometrical complexity near the end of the topological transition, the weights of the two quadrants are comparable. This has a noticeable effect on the conditional profiles, and it is clear from figure 16(d) that the plateau is a consequence of the negative contribution from $\overline{{\it\omega}}_{2}$ . If we consider this contribution as a spurious effect of ${\it\Delta}_{v}$ , the ‘true’ conditional vorticity $\overline{{\it\omega}}_{1}$ in figure 16(d) increases monotonically near the interface. In essence, the conditional vorticity remains constant or decreases away from the interface because ${\it\Delta}_{v}$ misclassifies some weakly vortical pockets as part of the turbulent flow.

Figure 17.

Conditional vorticity profiles for ${\it\delta}_{99}^{+}=1900$ , computed as in figure 16, as functions of the threshold: ○, ${\it\omega}_{0}^{\ast }=0.17$ ; ▫, 0.29; ▵, 0.52; ▿, 0.88; (a) ${\it\Delta}_{v}$ , (b) ${\it\Delta}_{b}$ .

This effect is clearer in figure 17, which presents conditional vorticities for several interface thresholds. Figure 17(a) is computed with ${\it\Delta}_{v}$ , and develops a plateau and eventually a peak as the threshold increases. As in figure 16(d), it can be shown that this is a due to the increasingly negative contribution from the pockets as the complexity of the interface increases. Figure 17(b) presents the same cases computed for ${\it\Delta}_{b}$ , and shows no trace of an interface peak.

It should be noted that the distances in figure 17(b) are much lower than in figure 17(a), while the conditional vorticities are higher. In fact, similar conditional vorticities are found when the horizontal axis of figure 17(a) is extended to ${\it\Delta}_{v}\simeq 400$ , carrying the plot to the neighbourhood of the wall. The plot of $\overline{{\it\omega}}({\it\Delta}_{v})$ for these large distances is very similar to a shifted version of ${\it\omega}^{\prime }(y)$ (figure 3 a). The vorticity isosurface at these high thresholds permeates the whole boundary layer, and occasionally comes very close to the wall. The ball distance recognises this fact and brings the strong near-wall vorticity closer to the interface, while the vertical distance misses that complexity.

This discussion suggests that the apparent strongly vortical interface layer observed in some of the studies mentioned above either is an artefact of how a one-dimensional definition of distance interacts with a fully three-dimensional geometry or is only manifested in variables different from the vorticity magnitude. In the present case, the above arguments show that the interface peak is due to the neglect of the effect of irrotational pockets on the conditional quantities. We next discuss the relevance of these pockets in the entrainment process.

5.4 The relevance of pockets

We saw in figure 15 that pockets form a self-similar hierarchy of many different sizes, and it has been conjectured that their formation signals the large-scale engulfment of irrotational fluid before it is finally entrained by small-scale ‘nibbling’. Their abundance has been used to quantify the relative importance of the two processes (Sandham et al. Reference Sandham, Mungal, Broadwell and Reynolds1988; Mathew & Basu Reference Mathew and Basu2002).

Figure 18.

Flow within pockets of the interface at ${\it\omega}_{0}^{\ast }=0.5$ : – – –, ${\it\delta}_{99}^{+}=1100$ ; ——, ${\it\delta}_{99}^{+}=1900$ . (a) Joint PDF of ${\it\Delta}_{v}$ and ${\it\Delta}_{b}$ within the region. The two contours for each Reynolds number contain 60 % and 98 % of the points. (b) Average vorticity within the pockets as a function ${\it\Delta}_{v}$ and ${\it\Delta}_{b}$ within the region that contains 98 % of the points. The contours are ${\it\omega}^{\ast }=0.15$ , 0.25, 0.35. The two red diagonals are ${\it\Delta}_{v}={\it\Delta}_{b}$ .

We can define pockets as regions identified by the ball distance as part of the free stream, ${\it\Delta}_{b}<0$ , and by the vertical distance as turbulent, ${\it\Delta}_{v}>0$ . For the purpose of this section, they include the undersides of handles as well as simple folds of the interface. Figure 18(a) shows the joint PDF of the two distances in the range corresponding to pockets. The figure is drawn for the relatively high vorticity threshold of figure 12(b,d), guaranteeing both the presence of abundant pockets and the possibility of observing how the vorticity diffuses into the irrotational flow. It includes the two extreme Reynolds numbers in our simulation, allowing some scaling comparisons. For instance, the agreement between the two profiles of $\langle y_{I}\rangle$ suggests that the thickness of the intermittent region is not expected to change substantially with higher Reynolds numbers. It turns out that the size of the pockets, as measured by the maximum ${\it\Delta}_{b}$ , scales best in terms of the Taylor microscale, while their depth within the layer, as measured by ${\it\Delta}_{v}$ , scales better with the boundary-layer thickness. The joint PDF is roughly triangular. It is bounded on the left by the trivial limit ${\it\Delta}_{b}\leqslant {\it\Delta}_{v}$ , plotted for each Reynolds number as a thick inclined straight line, and on the right by a roughly hyperbolic curve which can be interpreted to mean that deeper pockets (large ${\it\Delta}_{v}$ ) tend to be smaller (small ${\it\Delta}_{b}$ ), presumably because they have been broken down by the turbulence while being entrained.

The question of whether being entrained into a pocket also promotes the diffusion of vorticity is tested in figure 18(b), which shows the distribution of the conditionally averaged vorticity in the same parameter space as figure 18(a). It should be noted that all of the vorticity levels in this figure are below the interface threshold, so that the band of higher vorticities along the top of figure 18(b) portrays how vorticity diffuses into the irrotational fluid. It should be noticed that the size of the pockets in this figure is normalised with ${\it\eta}$ . Comparison with figure 18(a) shows that the difference between scaling it with ${\it\eta}$ or ${\it\lambda}$ is not great, but the collapse of the vorticity band at the top of figure 18(b) is considerably better with ${\it\eta}$ than with ${\it\lambda}$ . Its width, approximately $5{-}10{\it\eta}$ , strongly suggests a viscous origin (van Reeuwijk & Holzner Reference van Reeuwijk and Holzner2014), and it is clear from the figure that the vorticity is correlated with the ball distance, but not with the vertical position with respect to the interface. The only exceptions are points near the line ${\it\Delta}_{v}={\it\Delta}_{b}$ , where the two measures coincide.

The implication is that the fluid within pockets is sensitive to how close it is to the interface, but not to how deep it is within the turbulent layer. If engulfment were an important mechanism to promote the diffusion of vorticity into the irrotational fluid, for example by preferentially straining it, one would expect some correlation between ${\it\Delta}_{v}$ and the width of the diffusion band at the top of figure 18(b), but there is little evidence for that. Apparently, whether the fluid is within a pocket or not is immaterial to its behaviour, although the breakup of the deeper pockets into smaller sizes should enhance the overall effect of viscous diffusion. We will only use ${\it\Delta}_{b}$ from now on in our analysis.

Figure 19.

Premultiplied joint PDF, ${\it\omega}F_{{\it\omega},{\it\Delta}_{b}}$ , of the vorticity and ball distance on the turbulent side of the interface for (a) a low threshold, ${\it\omega}_{0}^{\ast }=0.01$ , and (b) a moderate one at the beginning of the topological transition, ${\it\omega}_{0}^{\ast }=0.09$ . Two Reynolds numbers are presented in each panel, ${\it\delta}_{99}^{+}=1100$ (—— black) and ${\it\delta}_{99}^{+}=1900$ (– – – blue). The vertical solid line is ${\it\omega}_{0}^{\ast }$ . (c,d) Same as (a,b), but with the vorticity in wall units and the distance normalised with the boundary-layer thickness. The horizontal bar is the variation of ${\it\omega}^{\ast }/{\it\omega}^{+}$ in our range of ${\it\delta}_{99}^{+}$ . The curves with markers correspond to the average vorticity magnitudes for each Reynolds number, ${\it\delta}_{99}^{+}=1900$ (green ▵) and ${\it\delta}_{99}^{+}=1100$ (magenta ○). The contours contain 50 %, 90 % and 99 % of points.

5.5 The interface layer

While the previous sections deal with the properties of the interface surface, it is also interesting to characterise the properties of the interface layer, understood as the part of the turbulent flow that is directly influenced by its proximity to the free stream. As a first step, figure 19(a,b) reproduces the first and second (turbulent) quadrants of the joint PDFs of the vorticity and distance in figure 14(a,b). The distance axis is now logarithmic, to emphasise the region close to the interface, and each panel includes the two extreme Reynolds numbers in our data set.

Three regions can be distinguished in order of increasing distance from the interface. The first and closest to the interface contains the strongest vorticity gradients. A precise definition of the thickness of this layer will be given in § 5.7, but, if we define the limit of this layer by the intersection of two straight lines tangent to the probability isocontours near and far from the interface, its thickness is of the order of a few Kolmogorov units for the different Reynolds numbers, suggesting a viscous origin. However, this thickness depends on the identification threshold. It is approximately $10{\it\eta}$ in figure 19(a) $({\it\omega}_{0}^{\ast }=0.01)$ , $5{\it\eta}$ in figure 19(b) $({\it\omega}_{0}^{\ast }=0.09)$ , and almost vanishes at the beginning of the topological transition, ${\it\omega}_{0}^{\ast }=0.2$ (not shown). In the cases in which this region can be identified in the joint PDF, its limit is $\overline{{\it\omega}}^{\ast }\approx 1$ , which we have seen above to be the level of fully developed turbulence.

The viscosity-dominated region just outside the interface has been recently studied by van Reeuwijk & Holzner (Reference van Reeuwijk and Holzner2014) and Taveira & da Silva (Reference Taveira and da Silva2014) in temporally evolving turbulence fronts. They identify it with the ‘superlayer’ conjectured by Corrsin & Kistler (Reference Corrsin and Kistler1955), and find that its characteristic thickness is the Kolmogorov microscale computed with the energy dissipation rate of the core flow. The enstrophy level in this viscous layer depends somewhat on the definition, but is typically very low. The viscous region in figure 19 is probably not the superlayer, whose observation requires a higher numerical resolution and a quieter free stream than those in our simulation (van Reeuwijk & Holzner Reference van Reeuwijk and Holzner2014). We will see below that both the rate of strain and the vortex stretching remain high in the viscous layer of figure 19, and that this region is probably best interpreted as part of the ‘buffer layer’ defined by van Reeuwijk & Holzner (Reference van Reeuwijk and Holzner2014) in the range ${\it\omega}^{\ast }\in (0.1{-}1)$ . In analogy to the similarly named layer in wall-bounded turbulence, both nonlinear and viscous effects are important in this region. It is interesting that such a hybrid mechanism was proposed by Townsend (Reference Townsend1976), who noted that viscous diffusion of vorticity and its tangential transport should be comparable near the interface. On the assumption of homogeneity, the magnitude of the rate of strain is proportional to the enstrophy, and the predicted result of this mechanism is also a thickness of $O({\it\eta})$ .

The region beyond the buffer interface layer is self-similar, in the sense that both the conditionally averaged vorticity and the probability isocontours follow power laws in ${\it\Delta}_{b}$ . This implies that there is no intrinsic length or vorticity scale, which is consistent with the intuitive idea of an inertial layer but difficult to interpret dynamically. In the first place, different parts of the PDF scale differently. The strong-vorticity isocontours to the right of figure 19(a) are almost vertical $({\it\omega}\propto {\it\Delta}_{b}^{0})$ , but those corresponding to weak vorticity on the left of the figure follow ${\it\omega}\propto {\it\Delta}_{b}^{1}$ , and the conditional mean enstrophy approximately satisfies ${\it\omega}\propto {\it\Delta}_{b}^{1/2}$ . In the second place, these slopes change with the detection threshold, and it is hard to distinguish any power law in figure 19(b) or at higher thresholds. We mentioned with regard to (3.2)–(3.3) that $\langle {\it\omega}\rangle \propto y^{-1/2}$ is a consequence of the local energy equilibrium and the logarithmic profile in the logarithmic layer, but the same argument cannot be used here. The interface is not an impermeable boundary that limits the size of eddies as the wall does, although it could be argued that the size of the eddies defines the position of the interface. The trend in figure 19(a), that larger eddies have more intense vorticity, is contrary to the inertial relation of homogeneous turbulence, ${\it\omega}\propto {\it\Delta}^{-2/3}$ (Kolmogorov Reference Kolmogorov1941), and the most plausible explanation is that larger eddies reach closer to the wall and are therefore stronger. The apparent self-similarity in figure 19 may be coincidental.

The width of this intermediate region depends on the identification threshold, but scales with the boundary-layer thickness. It extends to the hockey stick at the top of the PDFs, which contains the points with the highest vorticity and farthest from the interface. This last region is mostly formed by points near the wall. When ${\it\omega}$ is scaled in wall units and ${\it\Delta}_{b}$ is normalised with the boundary-layer thickness, as in figure 19(c,d), the two Reynolds numbers collapse well for long distances and high vorticities. At the two Reynolds numbers in figure 19, ${\it\delta}_{99}/{\it\eta}\approx 250$ and 450.

5.6 Other velocity gradients

In the previous sections, we have discussed the properties of the vorticity field near a vorticity isosurface, and it is perhaps not surprising that they may be special. For example, an interesting question is whether the vorticity within the interface layer has different properties from the core of the turbulent flow, such as perhaps being weaker because it is less strained, but such questions are hard to answer if the interface is defined by the magnitude of the vorticity itself. It is useful for that purpose to determine the conditional properties of other quantities besides the one being thresholded. In this section we study the properties of the strain-rate tensor $\unicode[STIX]{x1D64E}$ in the neighbourhood of the vorticity interface, as well as the behaviour of the vorticity in the neighbourhood of an interface defined in terms of the strain. We define $S$ as the Euclidean norm of the rate-of-strain tensor, $S=\Vert \unicode[STIX]{x1D64E}\Vert$ . In analogy to (3.4), and taking into account that

in homogeneous flows, the star units for $S$ are defined as

Equation (5.8) then becomes $\langle {\it\omega}^{2}\rangle ^{\ast }=\langle S^{2}\rangle ^{\ast }$ , and suggests that ${\it\omega}^{\ast }$ and $S^{\ast }$ should be of the same order. The joint PDFs of $S$ and $y$ , and of ${\it\omega}$ and $y$ , are presented in figure 20.

Figure 20.

Premultiplied joint PDFs: —— (black), $S{\it\Gamma}_{y,S}$ ; – – – (blue), ${\it\omega}{\it\Gamma}_{y,{\it\omega}}$ . The contours contain 50 %, 90 % and 99 % of points. The dashed diagonal is the exponential decay of the Fourier modes of irrotational strain with a wall-parallel wavelength $2{\it\delta}_{99}$ .

Both PDFs agree within the turbulent region in the right-lower corner of figure 20, supporting the normalisation (5.9), but the vorticity in the free stream on the left-hand side of the figure is almost two orders of magnitude lower than the rate of strain. This is not unexpected in a nominally irrotational part of the flow but, since (5.8) is a kinematic relation whose only condition is spacial homogeneity, the mismatch between the two magnitudes implies that the strain in the free stream is an inhomogeneous residual effect of the vortical flow within the boundary layer.

Any solenoidal velocity field can be written as

where the potentials satisfy ${\rm\nabla}^{2}{\it\phi}=0$ and ${\rm\nabla}^{2}\boldsymbol{B}=-{\bf\omega}$ (Batchelor Reference Batchelor1967). In the irrotational free stream, both potentials satisfy Laplace’s equation and, if they are expanded in terms of wall-parallel Fourier harmonics, decay away from the wall as $\exp (-ky)$ , where $k^{2}=k_{x}^{2}+k_{z}^{2}$ is the magnitude of the wall-parallel wavevector. All of the velocity components and the rate-of-strain tensor decay exponentially at the same rate, and the slowest decay corresponds to the largest horizontal wavelengths. It is known that this results in an algebraic decay of the velocity fluctuations for $y\gg {\it\delta}_{99}$ , because different distances are dominated by different wavenumbers in the long-wavelength end of the spectrum (Phillips Reference Phillips1955; Stewart Reference Stewart1956), but the near field is controlled by the peak of the $v$ spectrum in the turbulent region. The thick dashed diagonal in figure 20 is $S\propto \exp (-{\rm\pi}y/{\it\delta}_{99})$ , corresponding to the decay of irrotational velocity fluctuations due to structures within the boundary layer whose shortest dimension is $O(2{\it\delta}_{99})$ . This is the order of magnitude of the largest structures in boundary layers (Sillero et al. Reference Sillero, Jiménez and Moser2014).

The vorticity is unrelated to the velocity potentials, and decays much faster than the rate of strain as it enters the free stream. In fact, this was one of the reasons why we originally chose vorticity over other quantities to characterise the T/NT interface.

It should be noted that the vorticity also decays exponentially with $y$ in the free stream, although at a much lower absolute level than $S$ . This is not a kinematic result, but a consequence of the numerical inflow conditions, which determine the three velocities at the inflow but not their derivatives with respect to $x$ . The result is that there is a residual vorticity in the free stream due to terms like $\partial _{x}v$ , which inherits the exponential decay of the velocity potentials at the inflow plane.

Figure 21.

(a) Fractal dimension and (b) genus of an interface defined by thresholding the norm of the strain-rate tensor: ○, ${\it\delta}_{99}^{+}=1100$ ; $\times$ , 1300; ▿, 1500; ▵, 1700; $\star$ , 1900. The horizontal bar is the variation of ${\it\omega}^{\ast }/{\it\omega}^{+}$ in our range of ${\it\delta}_{99}^{+}$ .

A consequence of the relatively high strain levels in the free stream is that the separation between its characteristic values on the turbulent and non-turbulent sides is not as clear-cut as in the case of the vorticity. Even so, the complexity transition happens at comparable thresholds. The geometrical properties of the strain interface are presented in figure 21. The fractal dimension in figure 21(a) should be compared with figure 7(b) for the vorticity interface. The strain isosurface is smoother, with minimum values close to the non-fractal value, $D\approx 2$ . The maximum dimension is also somewhat lower than for the vorticity, in agreement with the observation by Moisy & Jiménez (Reference Moisy and Jiménez2004) that strong dissipation structures are less fractal (plate-like) than those of vorticity (string-like). The evolution of the genus in figure 21(b) is also similar to the case of the vorticity, although the maximum genus and fractal dimension are reached for slightly higher thresholds, $S_{0}^{\ast }\approx 2.5$ instead of ${\it\omega}_{0}^{\ast }\approx 1.5$ . The topological transition is also narrower for the strain interface, especially for the genus in figure 21(b), which starts to increase at $S_{0}^{\ast }\approx 1$ instead of at ${\it\omega}_{0}^{\ast }\approx 0.3$ , as it did in figure 8(a). The reason is probably that, while the maximum dimension and genus mark the threshold for which the interface has fully moved into the core turbulent flow, the slower decay of the strain fluctuations with $y$ means that low-strain isosurfaces are farther from the wall than for similar enstrophy thresholds, and the corresponding interface becomes regular much faster.

The evolutions of the conditional statistics of the flow across the vorticity and strain isosurfaces are compared in figure 22. The two thresholds chosen are $S_{0}^{\ast }=0.1$ and ${\it\omega}_{0}^{\ast }=0.01$ , both of which are within the plateau that separates the values of turbulent and non-turbulent flow in their joint PDFs, and well below the beginning of the respective topological transitions. The average height of the resulting interfaces is $\langle y_{I}\rangle \approx 1$ in both cases. Figure 22 shows the conditionally averaged enstrophy and strain for each of the two interfaces. They are plotted as functions of the respective ball distances, which are denoted by ${\it\Delta}_{b}^{{\it\omega}}$ and ${\it\Delta}_{b}^{S}$ respectively.

The interesting question to be answered is whether the sharp vorticity gradient across the vorticity interface is a statistical artefact of the thresholding procedure or a true physical effect. The former is a possibility, because enstrophy is fixed at the interface, and moving slightly away from the geometrically complex isosurface could sample flow regions that are unrelated to it and representative of the bulk of the turbulent and irrotational regions. In a related example, Chauhan et al. (Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014) find a sharp velocity jump across an interface defined in terms of the velocity magnitude, raising similar questions. In both cases, the sharp jump at the interface is what makes the criterion useful, and the reason why an interface can be defined at all (Corrsin & Kistler Reference Corrsin and Kistler1955). The lines without symbols in figure 22 refer to the vorticity interface. Within ${\it\Delta}_{b}=O(10{\it\eta})$ of ${\it\Delta}_{b}^{{\it\omega}}=0$ the conditional vorticity (dashed) drops by three orders of magnitude, and a similarly sharp gradient is seen for the conditional strain (solid).

Figure 22.

Conditional mean profiles of the vorticity and rate of strain, as functions of the distance ${\it\Delta}_{b}^{{\it\omega}}$ to the ${\it\omega}_{0}^{\ast }=0.01$ interface or of the distance ${\it\Delta}_{b}^{S}$ to $S_{0}^{\ast }=0.1$ for ${\it\delta}_{99}^{+}=1900$ : ——, $\overline{S}$ ; – – –, $\overline{{\it\omega}}$ . Lines without symbols are with respect to ${\it\Delta}_{b}^{{\it\omega}}$ . Those with symbols are with respect to ${\it\Delta}_{b}^{S}$ .

The behaviour is different for the strain interface, represented by the lines with circles in figure 22, whose vorticity and strain cross the level ${\it\Delta}_{b}^{S}=0$ relatively smoothly. The difference between the two behaviours strongly suggests that while a sharp vorticity jump is a dynamically significant feature separating distinct regions of the flow, that of the strain is not.

We are now ready to define a ‘natural’ interface as an enstrophy isosurface below the topological transition, such as ${\it\omega}_{0}^{\ast }\approx 0.01$ . This threshold is somewhat lower than most of those compiled in table 2, with the result that the turbulent region contains part of the buffer and viscous superlayers defined by van Reeuwijk & Holzner (Reference van Reeuwijk and Holzner2014).

Figure 23.

(a) Premultiplied joint PDFs at ${\it\delta}_{99}^{+}=1900$ , with respect to the interface ${\it\omega}_{0}^{\ast }=0.01$ : – – – (blue), ${\it\omega}F_{{\it\omega},{\it\Delta}_{b}}$ ; —— (black), $SF_{S,{\it\Delta}_{b}}$ . Symbols: ○, $\overline{{\it\omega}}^{\ast }$ ; ●,  $\overline{S}^{\ast }$ . The vertical line is ${\it\omega}^{\ast }=S^{\ast }=1$ . (b) Premultiplied joint PDF of ${\it\Delta}_{b}$ and – – – (blue), vortex stretching; —— (black), vortex compression. The two horizontal lines ——, ${\it\Delta}_{b}=100{\it\eta}$ ; – – –, ${\it\Delta}_{b}=7{\it\eta}$ are used in (c,d). The three contours in (a,b) contain 50 %, 90 % and 99 % of points. (c) Premultiplied PDF of ▵, normalised vortex stretching; ▫, compression. (d) Premultiplied PDFs of the absolute values of the normalised individual eigenvalues of the rate-of-strain tensor: ♢, most positive; ▿, most negative; ▵, positive intermediate; ▫, negative intermediate. The abscissae in (c,d) are normalised with $\overline{S}$ . In both cases, the PDFs are compiled at – – –, ${\it\Delta}_{b}=7{\it\eta}$ ; ——, ${\it\Delta}_{b}=100{\it\eta}$ .

The structure of the flow with respect to this interface is displayed in figure 23. The joint PDFs of the vorticity and rate of strain with ${\it\Delta}_{b}$ are shown in figure 23(a). The sharp decay of the vorticity below $\overline{{\it\omega}}^{\ast }\approx 1$ is clearly visible, spanning a thickness of approximately $20{\it\eta}$ . The strain decays slowly as it gets closer to the interface and far from the wall, but shows no special behaviour within the buffer region. The vorticity in the buffer layer lives in essentially the same straining environment as in the core turbulent flow.

This is seen more clearly in figure 23(b), which shows the PDF of the vortex-stretching component of the strain, ${\bf\omega}\unicode[STIX]{x1D64E}{\bf\omega}/{\it\omega}^{2}$ . The positive (stretching) and negative (compression) PDFs are plotted separately to allow a logarithmic representation. Both decay slowly and apparently self-similarly as they approach the interface, but do not change appreciably as they enter the buffer layer. The different rates of decay of the vorticity and the strain rate was also mentioned by Holzner et al. (Reference Holzner, Liberzon, Nikitin, Kinzelbach and Tsinober2007). Figure 23(c) shows the same result in the form of one-dimensional PDFs of the vortex-stretching term at two distances from the interface, one within the buffer region and another one in the core of the flow. The normalisation with $\overline{S}$ absorbs most of the differences between the two levels. An even more detailed comparison is figure 23(d), which shows the PDFs of the individual eigenvalues of the rate-of-strain tensor. The PDFs at the two distances also collapse well. In both cases, the implication is that the straining environment within the buffer layer is essentially the same as in the core of the flow. Enstrophy is viscously diffused into the free stream, but it keeps being stretched as it does, in agreement with the model proposed in Townsend (Reference Townsend1976). It should be noted that the magnitude of the vorticity within this inhomogeneous region is of the order of the isosurfaces that were shown in § 4 to be within the topological transition, and that the geometry of the flow in this layer is therefore intermediate between the irrotational free stream and the turbulent core, but much more complex than the former.

Other authors have explored higher-order quantities close to the interface, such as the invariants of the velocity gradient tensor (da Silva & Pereira Reference da Silva and Pereira2008), and the different terms of the vorticity equation (Holzner et al. Reference Holzner, Liberzon, Nikitin, Kinzelbach and Tsinober2007, Reference Holzner, Liberzon, Nikitin, Lüthi, Kinzelbach and Tsinober2008). The study of these quantities for the present boundary layer is unfortunately beyond the scope of this study, but our data are openly accessible from our web site, and interested researchers are encouraged to use them to test their ideas.

5.7 The thickness of the interface layer

We have normalised our lengths up to now in terms of ${\it\eta}$ , ${\it\lambda}$ or ${\it\delta}_{99}$ , according to which of these scales appears to collapse better the different Reynolds numbers in each particular figure, or arbitrarily in figures involving a single Reynolds number. We saw in the introduction that the thickness of the T/NT interface layer has been the subject of much discussion, and we mentioned that comparisons are difficult because of the variety of definitions used by investigators. In general, there is some consensus that the properties of the flow change across the interface over distances of the order of the Taylor microscale, even if the narrow range of Reynolds numbers makes the scaling ambiguous in some cases. However, Gampert et al. (Reference Gampert, Narayanaswamy, Schaefer and Peters2013) showed that the thickness of the mixing interface of a passive scalar in a jet scales with ${\it\lambda}$ over a range of $Re_{{\it\lambda}}$ somewhat wider than ours. This result is surprising to us, because there are relatively few examples in which the Taylor microscale appears in fully developed turbulence (see, however, the correlation length along the strong vortices of isotropic turbulence in Jiménez & Wray Reference Jiménez and Wray1998). However, the T/NT interface is not fully developed turbulence, and Hunt & Durbin (Reference Hunt and Durbin1999) and Hunt, Eames & Westerweel (Reference Hunt, Eames and Westerweel2006) have given theoretical arguments as to why the thickness of strong vortex layers within a turbulent flow should scale with ${\it\lambda}$ . They propose that the interface is one such layer (see also Eisma et al. Reference Eisma, Westerweel, Ooms and Elsinga2015). Here, we examine the scaling of the thickness of our interface, defined as a layer in which the enstrophy and rate of strain do not satisfy the homogeneity constraints. We will find that the thickness scales with the Taylor microscale, but it is unclear whether the reasons are those in Hunt & Durbin (Reference Hunt and Durbin1999). They argue that strong shear layers are only subject to the rate of strain of the largest turbulent scales because the smaller ones are excluded by the shear, and that their thickness is controlled by viscosity. The question of whether there is an effectively high shear at the interface will be discussed in the next section, but we have seen above that the rate of strain and the stretching eigenvalues within the interface layer are similar to those in the bulk of the flow, which would imply viscous lengths of the order of ${\it\eta}$ in an equilibrium flow. We also saw that the geometry of the vorticity within this layer is complex, and not immediately consistent with an equilibrium viscous process. Our interface layer is probably not the same one as analysed by Hunt & Durbin (Reference Hunt and Durbin1999). Other dynamical models reach different conclusions about the scaling of the thickness of the interface layer starting from different assumptions. For example, Teixeira & da Silva (Reference Teixeira and da Silva2012) show that the initial decay of a shear-free synthetic turbulent interface should have thickness of order  ${\it\eta}$ .

Figure 24.

(a) Ratio of the conditional vorticity and rate-of-strain magnitude as a function of the ball distance to the vorticity interface, normalised with the Taylor microscale at $y/{\it\delta}_{99}=0.6$ : ○, ${\it\omega}_{0}^{\ast }=0.01$ ; ▿, 0.02; ▵, 0.04. (b) Sketch of the definition of the interface vorticity thickness, ${\it\Delta}_{{\it\omega}}$ . For other symbols, see text. (c) Length scales as functions of ${\it\delta}_{99}^{+}$ , normalised with respect to ${\it\delta}_{99}^{+}=1100$ . The lines with symbols are ${\it\Delta}_{{\it\omega}}$ for interface thresholds as in (a). The lines without symbols are flow length scales: – - – - –, ${\it\eta}$ ; ——, ${\it\lambda}$ ; – – –, ${\it\delta}_{99}$ . (d) Conditional profiles of ——, $\overline{{\it\omega}}$ ; – – –, $\overline{S}$ ; for ${\it\omega}_{0}^{\ast }=0.01$ and the two extreme Reynolds numbers. Open symbols are for ${\it\delta}_{99}^{+}=1100$ and closed ones are for ${\it\delta}_{99}^{+}=1900$ : ○, ${\it\Delta}_{out}$ in (b); ▵, ${\it\Delta}_{in}$ ; ▿, position ${\it\Delta}_{u}$ of the maximum gradient of the velocity magnitude in figure 25(b).

Since we have defined the interface by the difference between the conditional vorticity and the rate of strain, the ratio ${\it\varpi}=\overline{{\it\omega}}^{\ast }/\overline{S^{\ast }}$ is a useful indicator of its location. It is shown in figure 24(a) for three different thresholds. Because the geometry of the interface changes with ${\it\omega}_{0}^{\ast }$ , the indicator also changes, but it always undergoes a smooth change between ${\it\varpi}\approx 1$ in the turbulent core and ${\it\varpi}\ll 1$ in the free stream. It is interesting that the limit at the turbulent end is ${\it\varpi}=0.9$ rather than the homogeneous result ${\it\varpi}=1$ , but conditional and volume averages are not equivalent, and the observed ratio is robust across thresholds and Reynolds number. The ratio eventually climbs to approximately unity at distances from the interface of the order of ${\it\delta}_{99}$ , probably because far from the interface the ${\it\Delta}_{b}$ isosurfaces become approximately flat, and $\overline{{\it\omega}}\approx \langle {\it\omega}\rangle$ . The mean enstrophy and dissipation agree very well at all of the wall distances within the boundary layer (Sillero et al. Reference Sillero, Jiménez and Moser2013).

Our definition of thickness is sketched in figure 24(b) as the distance ${\it\Delta}_{{\it\omega}}$ between the intersections with ${\it\varpi}=0$ and ${\it\varpi}=1$ of a tangent drawn through the steepest point of the indicator. Because ${\it\Delta}_{b}$ is only defined with respect to a particular isosurface and is not an additive property (see § 5.1), any definition of thickness depends on the detection threshold, but ${\it\Delta}_{{\it\omega}}$ scales well the whole indicator profile for a given ${\it\omega}_{0}^{\ast }$ , as a function of the Reynolds number. The ratio of ${\it\Delta}_{{\it\omega}}({\it\delta}_{99})$ to its value at ${\it\delta}_{99}^{+}=1100$ is shown in figure 24(c). It should be noted that the Reynolds number dependence is the same for the three thresholds in the figure, even if the thickness at the highest threshold is approximately 1.5 times narrower than at the lowest one (not shown). The figure also includes the Reynolds number dependence of the three candidate length scales, and it is clear that the Taylor microscale is the best match.

The scaling with ${\it\lambda}$ extends to the conditional profiles of $\overline{{\it\omega}}$ and $\overline{S}$ , shown in figure 24(d) for our two extreme Reynolds numbers. This figure also displays the inner and outer limits of the vorticity interface layer, defined as in figure 24(b). They span a thickness ${\it\Delta}_{{\it\omega}}\approx 0.66{\it\lambda}$ for this particular ${\it\omega}_{0}^{\ast }$ . The peak of the velocity gradient interface discussed in the next section is included in figure 24(d) for reference. It is always deeper into the turbulent region than the vorticity interface.