3.1. Universality of the normalised conditional mean enstrophy profiles

When analysing the dynamics of the flow in the vicinity and within the TNTI layer, the vorticity magnitude $|\boldsymbol {\omega }| = (\omega _i\omega _i)^{1/2}$ or, alternatively, the enstrophy $\omega = \omega _i \omega _i/2$ are key important quantities since in the IR (or NT) region $|\boldsymbol {\omega }|=\omega =0$, whereas in the T region $|\boldsymbol {\omega }| \ne 0$ (and $\omega \ne 0$), so that the evolution of these quantities is an important measure of the transition of the flow from the NT into the T region, and can be used to quantify the small-scale features of the entrainment process (‘nibbling’). In the present work, enstrophy is defined by $\omega = \omega _i \omega _i$, instead of the definition $\omega = \omega _i \omega _i/2$ used by some authors. Without loss of generality, the definition of enstrophy as $\omega = \omega _i \omega _i/2$ is only used in connection with the conditional budgets across the TNTI layer, which are analysed in § 3.2.

As described in the introduction, conditional mean profiles of both $|\boldsymbol {\omega }|$ and $\omega$, with minor differences in the averaging procedure, have been obtained in many experimental and numerical works, where the normalisation has been done in several different ways (Silva et al. Reference Silva, Zecchetto and da Silva2018). In earlier investigations of the TNTI layer the normalisation was typically done using the flow parameters (e.g. initial/outer characteristic scales), as in figure 2 of the present work, or using large-scale flow parameters (e.g. centreline mean velocity, half-width, integral scale) (Bisset et al. Reference Bisset, Hunt and Rogers2002). However, it is difficult to compare conditional mean profiles using these quantities and many authors have used either the Taylor scale or the Kolmogorov micro-scale, computed at some suitable location, to normalise the distances within the TNTI layer. In light of recent developments the normalisation of the distances with the Kolmogorov micro-scale makes more sense, since the thickness of the TNTI layer scales with this scale at high Reynolds numbers (Silva et al. Reference Silva, Zecchetto and da Silva2018), and this normalisation has been used in many recent works (e.g. Watanabe et al. Reference Watanabe, Zhang and Nagata2018; Breda & Buxton Reference Breda and Buxton2019). In these cases the reference Kolmogorov scales are taken from a region inside the turbulent core region of the flow, adjacent to the TNTI layer. Figures 3(a) and 3(b) show conditional mean profiles of the Kolmogorov micro-scale $\eta$ and of the Kolmogorov velocity $u_{\eta } = (\nu \varepsilon )^{1/4}$, respectively, for all the simulations used in the present work. It is clear that at a distance relatively close to the TNTI layer ($y_I/\langle \eta \rangle _T \gtrsim 60\text {--}70$), both the Kolmogorov micro-scale and the Kolmogorov velocity are virtually constant and equal to the value measured inside the turbulent core region, i.e. $\langle \eta \rangle _I / \langle \eta \rangle _T = 1.0$, for (recall that the thickness of the TNTI layer is $\langle \delta _{\omega } \rangle / \langle \eta \rangle _T \approx 15$). The Kolmogorov scale has no meaning outside the T region ($y_I/ \langle \eta \rangle _T \le 0$) and, even within the VSL region of the TNTI layer ($y_I/\langle \eta \rangle _T \lesssim 5$), changes only slightly over the entirety of the flow, displaying a maximum at the IB ($y_I/\langle \eta \rangle _T =0$) of $\langle \eta \rangle _I / \langle \eta \rangle _T \approx 1.5$, the observed variation ($50\,\%$) happening only in the smallest possible space, in a fraction of the VSL, which typically has a size of about $\approx 5 \eta$. Otherwise, the Kolmogorov micro-scale (like the Kolmogorov velocity scale) is virtually constant in the entirety of the turbulent region i.e. for $y_I>0$.

Figure 3. Conditional mean profiles of Kolmogorov micro-scale $\langle \eta \rangle _I$ (a) and Kolmogorov velocity $\langle u_{\eta } \rangle _I$ (b) for all the simulations considered in this work, normalised by their mean values at the turbulent core region of the flow, $\langle \eta \rangle _T$ and $\langle u_{\eta } \rangle _T$, respectively. The distance from the IB ($y_I$) is normalised with the mean Kolmogorov micro-scale computed at the turbulent core region, $\langle \eta \rangle _T$. (a) Conditional mean Kolmogorov micro-scale $\langle \eta \rangle _I$ and (b) conditional mean Kolmogorov velocity $\langle u_{\eta } \rangle _I$.

Thus, the normalisation of the conditional mean enstrophy profiles by the Kolmogorov (velocity and length) scales is justified by the uniform value these quantities take in the turbulent core region near the TNTI layer, and by the recently observed Kolmogorov scaling of this interface and its sublayers at high Reynolds numbers.

Figure 4 compares the mean conditional enstrophy profiles normalised by the reference (turbulent core) Kolmogorov velocity and length scales, $\langle \omega _i \omega _i \rangle _I / ( \langle u_{\eta } \rangle _T / \langle \eta \rangle _T)^2$, for all the simulations used in the present work. As in da Silva & Pereira (Reference da Silva and Pereira2008), Silva et al. (Reference Silva, Zecchetto and da Silva2018), Holzner et al. (Reference Holzner, Liberzon, Luthi, Nikitin, Kinzelbach and Tsinober2008, Reference Holzner, Luthi, Tsinober and Kinzelbach2009), Watanabe et al. (Reference Watanabe, Zhang and Nagata2018), the Kolmogorov scales used for the normalisation are taken from deep inside the turbulent region (here the reference values are taken at $y_I/\langle \eta \rangle _I = 250$).

Figure 4. Conditional mean enstrophy profile $\langle \omega _i \omega _i \rangle _I$ for all the simulations considered in this work, normalised with the mean Kolmogorov velocity and length scale, computed at the turbulent core region of the flow, $\langle \eta \rangle _T$ and $\langle u_{\eta } \rangle _T$, respectively. The inset shows the amplified region near the IB ($y_I=0$).

This comparison between the conditional mean profiles of enstrophy has seldom been done using data from completely different flow types. The compilation of conditional mean vorticity and enstrophy profiles shown in figure 6 in da Silva et al. (Reference da Silva, Hunt, Eames and Westerweel2014) is not a fair comparison because of differences in the data (enstrophy, spanwise vorticity, vorticity magnitude) and the details of the procedure used to obtain the conditional statistics (e.g. using a one-dimensional vertical line, two-dimensional normal, 3-D normal). More importantly, the Reynolds numbers of some of these simulations were simply too low, as the analysis of interface thickness scaling described in Silva et al. (Reference Silva, Zecchetto and da Silva2018) has shown.

The curves in Figure 4 are somehow similar, with a sharp enstrophy jump after the IB position, followed by a slow rise into the turbulent core region. However, it is clear that the conditional profiles do not collapse because of differences in the normalised enstrophy magnitude and the way the enstrophy moves into the turbulent core region, i.e. different magnitudes of the profiles at a given distance $y_I/\langle \eta \rangle _T$ from the IB position. The thickness of the vorticity jump is of course of the same order in all cases, with $\langle \delta _{\omega } \rangle / \langle \eta \rangle _T \approx 10\text {--}20$, in agreement with Silva et al. (Reference Silva, Zecchetto and da Silva2018). The different evolution of the enstrophy for $y_I / \langle \eta \rangle _T \gg 20$, i.e. large distances from the IB is connected with large-scale inhomogeneities within the flow and not with the dynamics of the flow within the TNTI layer (as will be shown below), as it is unlikely that the small-scale dynamics at the interface can be affected by small-scale events taking place at distances of more than 200 Kolmogorov micro-scales away. The approach towards $\langle \omega _i \omega _i \rangle _I / ( \langle u_{\eta } \rangle _T / \langle \eta \rangle _T)^2 \rightarrow 1$ for $y_I / \langle \eta \rangle _T \gg 1$ is explained by kinematic constraints, since in the turbulent core region $\langle \omega _i \omega _i \rangle / \langle (u_{\eta }/\eta )^2 \rangle = 1$, as in homogeneous turbulence.

The non-collapsing nature of the profiles of conditional mean enstrophy for different flows, or even for the same flow at different Reynolds numbers, when these profiles are normalised by a reference Kolmogorov velocity and length scale, has been reported before in several works; see, e.g. figures 5 and 16(b) of Watanabe, Nagata & da Silva (Reference Watanabe, Nagata and da Silva2017b), Watanabe et al. (Reference Watanabe, Zhang and Nagata2018), respectively. The particular shape of these mean enstrophy profiles can be justified by the simple arguments described in appendix B.

In light of the above discussion, since the Kolmogorov velocity, length (and, therefore, time) scales are clearly the characteristic scales of the TNTI layer, it is natural to normalise all small-scale quantities with the mean local Kolmogorov scales, i.e. to plot $\langle \omega _i \omega _i \rangle _I$ normalised by $(\langle u_{\eta } \rangle _I / \langle \eta \rangle _I )^2$, as shown in figure 5. In contrast to figure 4, here all the conditional enstrophy profiles collapse nicely into a single curve. That the collapse is indeed perfect can be attested by the very small separation distance between the individual curves, which is comparable to the resolution of the simulations, of the oder of one Kolmogorov micro-scale. As before, the normalised enstrophy tends to $\langle \omega _i \omega _i \rangle _I / ( \langle u_{\eta } \rangle _I / \langle \eta \rangle _I )^2 = 1$ away from the TNTI layer, but unlike in figure 4, here this value is attained very quickly, at $y_I / \langle \eta \rangle _I \approx 30$, which is shortly after the end of the TNTI layer, which extends at most until $y_I / \langle \eta \rangle _I \approx 20$. Inside the TNTI layer, again in marked contrast to figure 4, the normalised enstrophy exhibits a peak that reaches $\langle \omega _i \omega _i \rangle _I / ( \langle u_{\eta } \rangle _I / \langle \eta \rangle _I )^2 \approx 1.3$, and is located at $y_I/\langle \eta \rangle _I \approx 15$. Close to the IB the normalised enstrophy displays a very sharp rise, as in figure 4. Note that the conditional profile of $\langle \omega _i \omega _i / ( u_{\eta } / \eta )^2 \rangle _I$ is theoretically ‘similar’ but is affected by considerably more intermittency, and, therefore, would need more samples/instants to attain an equivalent degree of convergence. The use of the local Kolmogorov scales to normalise the enstrophy can be questioned at first sight because the definition of these scales arises from the classical description of the energy cascade mechanism, which does not take place very close to the IB (Watanabe et al. Reference Watanabe, da Silva and Nagata2019); however, it can also be argued that the Kolmogorov scales are a natural definition of the viscous scales at work within the flow, and in this sense, the variable Kolmogorov scales near the IB have a clear physical meaning. Appendix C shows that the spatial or temporal nature of the simulations/flow type does not affect the present results; Reynolds number effects are discussed in appendix D.

Figure 5. Conditional mean enstrophy profile $\langle \omega _i \omega _i \rangle _I$ for all the simulations considered in this work, normalised with the mean local Kolmogorov velocity and length scale, computed at each coordinate $y_I, \langle \eta \rangle _I$ and $\langle u_{\eta } \rangle _I$, respectively. The inset shows the amplified region near the IB ($y_I=0$).

The shape of the normalised enstrophy depicted in figure 5 and, in particular, the deviation from the asymptotic value of $1$ can be understood as a consequence of a deviation of the flow from local homogeneity near the TNTI layer. To see this, consider the expansion of the enstrophy and the normalising Kolmogorov velocity and length scale ratio, i.e.

(3.1)\begin{equation} \omega_i \omega_i=\epsilon_{ijk}\frac{\partial u_k }{\partial x_j}\epsilon_{ilm}\frac{\partial u_m }{\partial x_l}=\left(\frac{\partial u_k}{\partial x_j}\right)^2-\frac{\partial u_k}{\partial x_j}\frac{\partial u_j}{\partial x_k} \end{equation}

and

(3.2)\begin{equation} \left(\frac{u_\eta}{\eta}\right)^2=\frac{\varepsilon}{\nu}=2S_{ij}S_{ij}=\frac{1}{2} \left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)^2 = \left(\frac{\partial u_i}{\partial x_j}\right)^2+ \frac{\partial u_i}{\partial x_j}\frac{\partial u_j}{\partial x_i}. \end{equation}

Using the above relations, the conditional mean enstrophy normalised by the local Kolmogorov scales can be written as

(3.3)\begin{equation} \frac{\left\langle \omega_i\omega_i\right\rangle_I}{\left\langle \dfrac{u_\eta^2}{\eta^2}\right\rangle_I}= \frac{ \left\langle \left(\dfrac{\partial u_i}{\partial x_j}\right)^2\right\rangle_I-\left\langle \dfrac{\partial u_i}{\partial x_j}\dfrac{\partial u_j}{\partial x_i}\right\rangle_I}{ \left\langle \left(\dfrac{\partial u_i}{\partial x_j}\right)^2\right\rangle_I+\left\langle \dfrac{\partial u_i}{\partial x_j}\dfrac{\partial u_j}{\partial x_i}\right\rangle_I}. \end{equation}

For an homogeneous incompressible flow,

(3.4)\begin{equation} \left\langle \frac{\partial u_i}{\partial x_j}\frac{\partial u_j}{\partial x_i}\right\rangle_I = \frac{\partial }{\partial x_j}\left\langle u_i\frac{\partial u_j}{\partial x_i}\right\rangle_I = 0, \end{equation}

and, therefore, the ratio in (3.3) is only different from $1$ in flow regions where the flow is not (locally) homogeneous. Figure 5 therefore shows that, for $y_I/\langle \eta \rangle _I \lesssim 35$, which already covers a small region outside the TNTI layer, the flow is locally inhomogeneous. This result is interesting because it shows that there is a length of about $10 \eta$, outside the TNTI layer, where the small-scale features of the flow are already characteristic of the turbulent core region, i.e. the flow is still adjusting from the presence of the nearby TNTI layer.

The value of $\langle \omega _i \omega _i \rangle _I/ \langle (u_\eta /\eta )^2 \rangle _I > 1$ observed for $10 \lesssim y_I/\langle \eta \rangle _I \lesssim 35$ can be explained by the bigger density of intense vortices here compared with inside the turbulent core region of the flow, because the IB is defined around these structures (da Silva et al. Reference da Silva, dos Reis and Pereira2011; Watanabe et al. Reference Watanabe, Jaulino, Taveira, da Silva, Nagata and Sakai2017a). Indeed, (3.3) implies that a region with values of the normalised conditional enstrophy with $\langle \omega _i \omega _i \rangle _I/ \langle (u_\eta /\eta )^2 \rangle _I > 1$ is only possible if $\langle ({\partial u_i}/{\partial x_j})({\partial u_j}/{\partial x_i})\rangle _I < 0$ in those regions. This term can be related to the Laplacian of conditional mean pressure $\langle p \rangle _I$, through the Poisson equation,

(3.5)\begin{equation} \nabla^2\left\langle p/\rho \right\rangle_I={-}\left\langle \frac{\partial u_i}{\partial x_j}\frac{\partial u_j}{\partial x_i}\right\rangle_I, \end{equation}

where $\rho$ is the (constant) flow density. Therefore, the observed maximum of $\langle \omega _i \omega _i \rangle _I/ \langle (u_\eta /\eta )^2 \rangle _I$ indicates the existence of a mean (local) pressure minimum. The conditional mean profile of $\langle \nabla ^2 p \rangle _I$ can be also connected with the second invariant of the velocity gradient tensor $Q = \frac {1}{4} ( \omega _i \omega _i - 2 S_{ij}S_{ji} )$, since $\nabla ^2p = 2 Q$ (Davidson Reference Davidson2004). Conditional mean profiles of $\langle Q \rangle _I$, normalised with the Kolmogorov micro-scales at the turbulent core region, are shown in figure 6(a) for all the simulations used in the present work. These profiles are similar to that shown in da Silva & Pereira (Reference da Silva and Pereira2008, Reference da Silva and Pereira2009), and show the existence of double (positive/negative) peaks of $Q$ associated with a predominance of enstrophy/strain near the IB position in the T and NT regions, respectively. The mean value of $\langle Q \rangle _I = 0$ observed in the turbulent core and irrotational core regions is again explained by the homogeneity of the flow in these regions.

Figure 6. Normalised conditional mean profiles of the second invariant of the velocity gradient tensor $\langle Q \rangle _I$ for all the simulations used in the present work, normalised with the reference Kolmogorov scales (a) and with the local Kolmogorov scales (b).

Figure 6(b) shows the same profiles normalised with the local mean Kolmogorov scales, again showing a better collapse of all the curves and the emergence of the universal shape of the conditional $Q$ in the TNTI layer. The maximum is $\langle Q \rangle _I / \langle (u_\eta /\eta )^2 \rangle _I \approx 0.06$ and is located at the same location corresponding to the maximum of $\langle \omega _i \omega _i \rangle _I / \langle (u_\eta /\eta )^2 \rangle _I$, while the value of $ < Q > _I/\langle (u_\eta /\eta )^2 \rangle _I \approx -1/4$ observed in the IR region is simply a consequence of the definition of $Q$ in a region of zero enstrophy, and of the scaling of strain $S_{ij} S_{ij} \sim (u_{\eta }/\eta )^2$. The conditional mean profiles of the third invariant of the velocity gradient $R = -1/3 (S_{ij} S_{jk} S_{ki} - 3/4 \omega _i \omega _j S_{ij})$ (not shown) display similar trends, i.e. while the conditional mean profiles of $\langle R \rangle _I$ obtained from the several simulations normalised with the Kolmogorov scales from the turbulent core region display considerable differences, the same profiles normalised with the local Kolmogorov scales exhibit the same universal profile in all the simulations used in the present work.

It is clear that the particular shape of $\langle \omega _i \omega _i \rangle _I/\langle (u_\eta /\eta )^2 \rangle _I$ and $\langle Q \rangle _I / \langle (u_\eta /\eta )^2 \rangle _I$ is a consequence of the local low pressure associated with the presence of a large number of eddies, because the IB is defined by the flow eddies bounding the TNTI layer, and by their particular alignment (tangential to this layer), as shown in numerous works, e.g. da Silva et al. (Reference da Silva, dos Reis and Pereira2011), Mistry, Philip & Dawson (Reference Mistry, Philip and Dawson2019).

The results presented so far are related, totally or in part, to the vorticity (or the enstrophy), but it is important to show that the observed universality is also present in the rate-of-strain tensor, which is another important small-scale turbulence quantity. Figure 7 shows the conditional mean rate-of-strain magnitude $\langle S_{ij} S_{ij} \rangle _I$, normalised in the usual way, i.e. with the reference Kolmogorov velocity and length scales from the turbulent core region. Clearly with this normalisation no universality can be observed, as the several profiles are different within the TNTI layer and beyond, even for $y_I / \langle \eta \rangle _T \gtrsim 250$. That this fact is associated with the effects of the large-scale inhomogeneities of the flow near the TNTI layer is easily shown by writing

(3.6)\begin{equation} \langle S_{ij} S_{ij} \rangle_I = \frac{1}{2 \nu} \left\langle \frac{C_{\varepsilon} u'^3}{L} \right\rangle_I, \end{equation}

where the non-dimensional dissipation $C_{\varepsilon } = \varepsilon L / u'^3$ is defined with the integral scale of turbulence $L$ and the root mean square of the velocity fluctuations $u'$. Since, as shown by Watanabe et al. (Reference Watanabe, da Silva and Nagata2019), both $C_{\varepsilon }$ and $L$ are approximately constant for $y_I/\langle \eta \rangle _T \gtrsim 15$, the continuous rise of $\langle S_{ij} S_{ij} \rangle _I$ observed in figure 7 must be explained by variations of the Reynolds stresses when moving away from the IB position. However, the universality or the rate-of-strain magnitude, $S_{ij} S_{ij}$, (and of the viscous dissipation rate $\varepsilon$) when using the local normalisation is trivially demonstrated from (3.2),

(3.7)\begin{equation} \frac{ \langle S_{ij} S_{ij} \rangle_I}{ \langle ( u_{\eta} / \eta ) \rangle_I^2} = \frac{1}{2} \end{equation}

for all $y_I$ (including the NT region), i.e. the mathematical definition of the Kolmogorov velocity and length scales implies the similarity of the rate-of-strain magnitude, when normalised with the local mean Kolmogorov scales. This has been confirmed for all the simulations used in the present work (not shown), and explains also why $\langle S_{ij} S_{ij} \rangle _I$ in figure 7 (normalised with the turbulent core Kolmogorov scales) is seen approaching the constant $1/2$ as $y_I /\langle \eta \rangle _T$ increases.

Figure 7. Normalised conditional mean profiles of the rate-of-strain magnitude $\langle S_{ij} S_{ij} \rangle _I$ for all the simulations used in the present work, normalised with the mean Kolmogorov velocity and length scale, computed at the turbulent core region of the flow, $\langle \eta \rangle _T$ and $\langle u_{\eta } \rangle _T$, respectively.

It is noteworthy that the perfect collapse observed above for the conditional means of the vorticity and strain (and related small-scale quantities – see below) is not observed for typical large-scale quantities. As expected, the conditional Reynolds stresses (when normalised with the local Kolmogorov scales) do not collapse into the same curve. On the contrary, very different curves are obtained for these statistics, even for different instants of the same flow, except right at the IB (not shown).

The degree to which the small scales of turbulence can satisfy the classical isotropic turbulence statistics can be appreciated by analysing the skewness of the longitudinal velocity gradient, $S_0$, defined by Davidson (Reference Davidson2004),

(3.8)\begin{equation} S_0 = \left\langle \left( \frac{\partial u'}{\partial x} \right)^3 \right\rangle \left/ \left\langle \left( \frac{\partial u'}{\partial x} \right)^2 \right\rangle^{3/2}\right., \end{equation}

where the averaging is performed in the homogenous directions of the flow, and is strongly linked to the production of enstrophy,

(3.9)\begin{equation} \langle \omega_i \omega_j S_{ij} \rangle ={-} \frac{ 7 }{ 6 \sqrt{15} } S_0 \langle \omega_i \omega_i \rangle ^{3/2}. \end{equation}

Clearly, a negative value of $S_0$ is necessary for the mean of the enstrophy production to be positive, as is always the case in fully developed turbulence (Davidson Reference Davidson2004). Ishihara et al. (Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2007) showed that $S_0$ is approximately constant, with only a mild-Reynolds-number dependence ($0.51 \lesssim - S_0 \lesssim 0.67$ for $200 \lesssim Re_{\lambda } \lesssim 300$).

Figure 8 shows the mean conditional $S_0$ for all the simulations used in the present work, computed as

(3.10)\begin{equation} \langle S_0 \rangle_I ={-} \frac{ 6 \sqrt{15} }{ 7 } \frac{ \langle \omega_i \omega_j S_{ij} \rangle_I }{ \langle \omega_k \omega_k \rangle_I^{3/2} }. \end{equation}

Figure 8. Conditional mean profile of the skewness of the velocity gradient $\langle S_0 \rangle _I$ for all the simulations considered in this work, where the distance to the IB ($y_I=0$) is normalised by the mean local Kolmogorov scale $y_I/\langle \eta \rangle _I$, and the inset shows the amplified region near the IB.

Here $\langle S_0 \rangle _I$ is constant in the turbulent core and has the same value moving into the TNTI layer until about $y_I / \langle \eta \rangle _I \approx 20\text {--}30$ where it displays a small increase. The inset shows that values of $\langle S_0 \rangle _I$ are between $0.52 \lesssim - S_0 \lesssim 0.62$, in good agreement with the isotropic values given the degree of convergence of the profiles. These profiles contrast with some conclusions of Breda & Buxton (Reference Breda and Buxton2019) and show that the small scales of the flow can be represented by classical isotropic relations impressively close to the IB.

Finally, in order to document the universal shape of the normalised conditional mean enstrophy displayed in figure 5, we have averaged all the profiles in this figure. Figure 9 shows the resulting ‘averaged’ mean profile, and table 2 lists some of its coordinates for future reference. This profile of normalised conditional mean enstrophy is possibly a self-similar solution of a (locally) normalised enstrophy equation budget. Since this profile is virtually equal for the three very different flow types analysed here, a similar mean conditional enstrophy profile should be observed in virtually all Newtonian and incompressible TNTI layers. For completeness, appendix E discusses the small differences one gets when using the interface defined by the local IB introduced in § 2.2, instead of the envelope that is used throughout the present work.

Figure 9. The self-similar conditional mean enstrophy profile $\langle \omega _i \omega _i \rangle _I$ normalised with the local Kolmogorov velocity and length scale, $\langle \eta \rangle _I$ and $\langle u_{\eta } \rangle _I$, respectively, obtained by averaging all the curves displayed in figure 5. The inset shows the amplified region near the IB ($y_I=0$).

Table 2. Coordinates of the self-similar conditional enstrophy profile $\langle \omega _i \omega _i \rangle _I / \langle u_{\eta } / \eta \rangle _I^2$, obtained by averaging over all the conditional enstrophy profiles in figure 5.

We end this section with a small consideration on the importance of the results described above. Indeed, all the results convincingly show that the (mean) local Kolmogorov velocity and length scales are the natural scales characterising the conditional profiles of several small-scale quantities within the TNTI layer. At first sight one might be tempted to think that this is a trivial result since (i) it is well known that in fully developed turbulence the characteristics of the small scales of motion can be linked to the Kolmogorov micro-scale, e.g. the production of enstrophy can be estimated as $\omega _i \omega _j S_{ij} \sim (u_{\eta }/\eta )^3$, and (ii) it has recently been shown, in agreement with (i), that the thickness of the TNTI layer scales with the Kolmogorov micro-scale at sufficiently high Reynolds numbers.

However, we believe this is not a trivial result for two main reasons. First, the very definition of the Kolmogorov scales arises from the concept of an ongoing energy cascade, which is clearly atypical within the TNTI layer, as recently shown in Watanabe et al. (Reference Watanabe, da Silva and Nagata2019). In short, the small scales are strongly depleted in a big part of the TNTI layer, and are strongly unbalanced (in relation to the energy flux they receive from the large scales). Moreover, the non-dimensional dissipation $C_{\varepsilon }$ displays anomalous power laws $C_{\varepsilon } \sim Re_{\lambda }^{\alpha }$ within the TNTI layer (Watanabe et al. Reference Watanabe, da Silva and Nagata2019). Furthermore, the flow within the VSL is clearly not characteristic of fully developed turbulence, because viscosity strongly dominates this flow region no matter how high the Reynolds number may be. The second reason has to do with the supposed link between enstrophy $\rightarrow$ velocity gradients $\rightarrow$ viscous dissipation. Although this link certainly exists in a statistical sense in homogeneous turbulence, locally the relation between vorticity and strain is rather complex, as described at length in Tsinober, Ortenberg & Shtilman (Reference Tsinober, Ortenberg and Shtilman1999), Tsinober (Reference Tsinober2019), and, moreover, the TNTI layer has distinct regions where it cannot be considered to be homogeneous. In short, enstrophy and strain are locally quite different, and this is even more so within the TNTI layer, where the flow is not statistically homogeneous.

To summarise, one should remember that the thickness of the TNTI layer is associated with just one feature of the conditional enstrophy profile in the TNTI layer (the length of the vorticity jump or the location of its peak), while present analysis of the conditional enstrophy profile focuses on the Kolmogorov scaling along the entirety of this profile. In this sense, the recent results demonstrating the Kolmogorov scaling of the TNTI thickness can be seen as a particular result encompassed in the present results on the shape of the conditional enstrophy profiles.

3.2. Universality of conditional mean enstrophy budgets

We now move into the analysis of the conditional mean enstrophy budgets. Applying a conditional mean at the enstrophy transport equation, the conditional mean enstrophy budget is written as

(3.11) \begin{equation} \underbrace{\left\langle \frac{D \omega^2/2}{D t} \right\rangle_I}_{{\scriptsize \textrm{Total variation},\ T_{\omega}}} = \underbrace{\langle \omega_i \omega_j S_{ij} \rangle_I}_{{\scriptsize \textrm{Production},\ P_{\omega}}} +\underbrace{\langle \nu \nabla^2(\omega^2/2) \rangle_I}_{{\scriptsize \textrm{Diffusion},\ D_{\omega}}} - \underbrace{\langle \nu\boldsymbol{\nabla} \omega_i\boldsymbol{\cdot} \boldsymbol{\nabla} \omega_i \rangle_I}_{{\scriptsize \textrm{Dissipation},\ E_{\omega}}}, \end{equation}

where the left-hand side represents the total variation of enstrophy ($T_{\omega }$), whereas the terms on the right-hand side represent the production ($P_{\omega }$), viscous diffusion ($D_{\omega }$) and viscous dissipation ($E_{\omega }$), respectively. Note that the application of this averaging procedure assumes that the IB moves at constant speed, otherwise, new terms would have to be added to this conditionally averaged equation, as discussed in Westerweel et al. (Reference Westerweel, Fukushima, Pedersen and Hunt2009). This assumption is somehow consistent with the very small values observed for the local IB velocity (Holzner et al. Reference Holzner, Luthi, Tsinober and Kinzelbach2009; Watanabe et al. Reference Watanabe, Sakai, Nagata, Ito and Hayase2014). The shape of these terms within the TNTI layer has been analysed in detail in numerous works (e.g. Bisset et al. Reference Bisset, Hunt and Rogers2002; Holzner et al. Reference Holzner, Liberzon, Luthi, Nikitin, Kinzelbach and Tsinober2008; Taveira & da Silva Reference Taveira and da Silva2014; Watanabe et al. Reference Watanabe, da Silva, Sakai, Nagata and Hayase2016b). Figure 10 shows the conditional mean terms for WAKE$_{259}$ normalised in the usual way, i.e. using the reference Kolmogorov variables taken from the turbulence core region (very far from the IB) to normalise the whole conditional profiles.

Figure 10. Conditional mean enstrophy budgets (3.11) for WAKE$_{259}$, normalised with the mean Kolmogorov velocity and length scale, computed at the turbulent core region of the flow, $\langle \eta \rangle _T$ and $\langle u_{\eta } \rangle _T$, respectively: enstrophy production ($P_{\omega }$), enstrophy diffusion ($D_{\omega }$) and enstrophy dissipation ($E_{\omega }$). The figure also displays the location of the VSL and the TSL, which comprise the TNTI layer, separating the NT (or IR) and the turbulent core (TR) regions.

The VSL consists of a subregion where viscous diffusion $D_{\omega }$ dominates the enstrophy growth within the TNTI layer ($D_{\omega } > P_{\omega }$), while in the TSL the enstrophy growth is dominated by the enstrophy production ($D_{\omega } < P_{\omega }$). The total extent of the TNTI layer (sum of the VSL and TSL) goes from the IB ($y_I/\eta =0$) until the point at which enstrophy ceases to grow steeply ($y_I/\langle \eta \rangle \approx 15$), and separates the IR region from the turbulent core region. Clearly, the enstrophy production, diffusion and dissipation are all still slowly evolving (increasing) outside the TNTI layer because the enstrophy is still adjusting to large-scale inhomogeneities within the flow. Similar conditional enstrophy budgets have been observed in TNTIs from very different flow types such as mixing layers (Watanabe et al. Reference Watanabe, da Silva, Sakai, Nagata and Hayase2016b), plane wakes (Bisset et al. Reference Bisset, Hunt and Rogers2002), SFT (Holzner et al. Reference Holzner, Liberzon, Nikitin, Kinzelbach and Tsinober2007, Reference Holzner, Liberzon, Luthi, Nikitin, Kinzelbach and Tsinober2008), boundary layers (Chauhan, Philip & Marusic Reference Chauhan, Philip and Marusic2014a; Borrell & Jiménez Reference Borrell and Jiménez2016; Watanabe et al. Reference Watanabe, Zhang and Nagata2018) and plane jets (Silva & da Silva Reference Silva and da Silva2017; Silva et al. Reference Silva, Zecchetto and da Silva2018; Taveira & da Silva Reference Taveira and da Silva2014; Watanabe et al. Reference Watanabe, Sakai, Nagata, Ito and Hayase2014).

The next two figures show the conditional mean enstrophy budgets for all the simulations used in the present work, normalised by the turbulent core (figure 11$a$) and by the local (figure 11$b$) Kolmogorov velocity and length scales. It is clear that the curves in figure 11(a) show the same trend observed above, and do not collapse, while the curves in figure 11(b) not only collapse nicely into the same profiles, but are also constant for $y_I/\langle \eta \rangle _I \gtrsim 25$, i.e. shortly after the end of the TNTI layer.

Figure 11. Normalised conditional mean profiles of the governing enstrophy equation terms (3.11) for all the simulations used in the present work, normalised with the Kolmogorov velocity and length scales, computed at the turbulent core region of the flow, $\langle \eta \rangle _T$ and $\langle u_{\eta } \rangle _T$ (a), and with the local Kolmogorov velocity and length scale, computed at the turbulent core region of the flow, $\langle \eta \rangle _I$ and $\langle u_{\eta } \rangle _I$ (b): enstrophy production ($P_{\omega }$), enstrophy diffusion ($D_{\omega }$) and enstrophy dissipation ($E_{\omega }$).

In order to assess the detailed shape of each one of the conditional mean enstrophy terms obtained with the new normalisation, figure 12 shows these profiles for WAKE$_{259}$ near the TNTI layer. The evolution of enstrophy dynamics can be qualitatively interpreted in a similar manner as before with an initial part where enstrophy viscous diffusion is the leading term (VSL), followed by a region where enstrophy production takes over (TSL), until, moving further into the turbulent core, the viscous diffusion becomes negligible and production balances dissipation (turbulent core). Figure 12 also indicates the location of the VSL and TSL within the TNTI layer, and it is instructive to compare this figure with figure 10 that shows the same profiles with the standard normalisation. In short, incipient peaks of several quantities in the classical normalisation (figure 10) become much more clearly observed in the new normalisation (figure 12), which allows for a much more clear identification of the two sublayers within the TNTI layer.

Figure 12. Conditional mean enstrophy budgets (3.11) for WAKE$_{259}$, normalised with the local Kolmogorov velocity and length scale, computed at the turbulent core region of the flow, $\langle \eta \rangle _I$ and $\langle u_{\eta } \rangle _I$, respectively: enstrophy production ($P_{\omega }$), enstrophy diffusion ($D_{\omega }$) and enstrophy dissipation ($E_{\omega }$). The figure also displays the location of the VSL and the TSL, which comprise the TNTI layer, separating the NT (or IR) and the turbulent core (TR) regions.

In order to detect the end of the TSL (and of the TNTI layer) Silva et al. (Reference Silva, Zecchetto and da Silva2018) used a procedure based on the local maximum enstrophy, while Watanabe, Riley & Nagata (Reference Watanabe, Riley and Nagata2017c) developed a method based on the derivative of $\langle |\omega | \rangle _I$, in respect to the distance from the IB, where the end of the TNTI layer is defined by $d \langle |\omega | \rangle _I / d y_I = 0.25 \textrm {Max} ( d \langle |\omega | \rangle _I / d y_I$, with the two methods giving similar results. Figure 12 shows that using the new normalisation with the mean local Kolmogorov scales, this point simply coincides with the mean (normalised) enstrophy peak, which is located at $y_I/\langle \eta \rangle _I \approx 13$. The new normalisation thus allows for a much easier detection of this point and also a more direct estimation of the size of the TNTI layer. Table 3 lists the values and coordinates of the enstrophy governing terms at several key points in the new (local) normalisation, showing a small variation for these quantities for all the simulations used in the present work.

Table 3. Details of the values and coordinates at several key points of the locally normalised enstrophy equation terms (3.11) in all the simulations used in the present work: enstrophy production ($P_{\omega }$), enstrophy diffusion ($D_{\omega }$) and enstrophy dissipation ($E_{\omega }$). The terms are normalised with the local conditional mean Kolmogorov velocity and length scales, $\langle u_{\eta }\rangle _I$ and $\langle \eta \rangle _I$, respectively (e.g. $\langle P_{\omega } \rangle _I / (\langle u_{\eta } \rangle _I / \langle \eta \rangle _I )^3$), while the corresponding coordinates (indicated inside brackets) are normalised by the local mean Kolmogorov micro-scale, $y_I / \langle \eta \rangle _I$.

As in the ‘classical’ normalisation the peak of enstrophy production ($P_{\omega }$) also appears during the fast increase of $\langle |\omega | \rangle _I$ in the TSL ($y_I/\langle \eta \rangle _I \approx 8.0$), however, in contrast to figure 10, in figure 12 the enstrophy production ($P_{\omega }$) quickly attains a constant value once the turbulent core region is reached. The magnitude of the normalised production in the turbulent core region $\langle P_{\omega }^{\ast } \rangle _I = \langle P_{\omega } \rangle _I / \langle (u_{\eta }/\eta )^3 \rangle _I$, which by inspection of figure 12 is $\langle P_{\omega }^{\ast } \rangle _I \approx 0.17$, can again be easily explained by the isotropic relations mentioned above, by remarking that normalising with $(u_{\eta }/\eta )$ is equivalent to normalising with the Kolmogorov time scale $\tau _{\eta } = (\nu /\varepsilon )^{1/2}$, which is equal to $\tau _{\eta } = 1 / \langle \omega _i \omega _i \rangle _I^{1/2}$ using the homogeneous flow relation $\langle 2 S_{ij} S_{ij} \rangle = \langle \omega _i \omega _i \rangle$. Therefore, $\langle P_{\omega }^{\ast } \rangle _I = \langle P_{\omega } \rangle _I \langle \tau _{\eta }^3 \rangle _I = \langle P_{\omega } \rangle _I / \langle \omega _i \omega _i \rangle _I^{3/2}$, which by using (3.9) with a constant value of $S_0=-0.55$ yields $\langle P_{\omega }^{\ast } \rangle _I = 0.166$.

The enstrophy diffusion $D_{\omega }$ exhibits a profile which is similar in both normalisations (compare figures 10 and 12), however, in figure 12 the maximum of $D_{\omega }$ and $P_{\omega }$ is now comparable, and the maximum of $D_{\omega }$ is attained a bit sooner, by $y_I/\langle \eta \rangle _I \approx 2.0$. The crossing between the enstrophy production and diffusion ($D_{\omega }=P_{\omega }$), which marks the end of the VSL, is located at $y_I/\langle \eta \rangle _I \approx 4.0$ in both cases, consistently with the direct measurements of this layer undertaken by Taveira & da Silva (Reference Taveira and da Silva2014), with the smaller value obtained for the local normalisation ($y_I/\langle \eta \rangle _T = 4.2$ and $y_I/\langle \eta \rangle _I = 3.1$).

The enstrophy dissipation term ($E_{\omega }$) is the only term that displays a very different shape compared with the usual normalised profile (compare $E_{\omega }$ in figures 10 and 12). The profile also displays a constant value once the end of the TNTI layer is approached, roughly coinciding with a second minimum at $y_I/\langle \eta \rangle _I \approx 12.1$, however, the profile also exhibits another minimum very close to the IB, at $y_I/\langle \eta \rangle _I \approx 2.7$, coinciding with the end of the VSL, and a local maximum within the TSL at $y_I/\langle \eta \rangle _I \approx 5.1$, i.e. slightly before (but very close to) the point of maximum enstrophy production and minimum of enstrophy diffusion. This local maximum in $E_{\omega }$ roughly coincides with the minimum of enstrophy diffusion $D_{\omega }$ and indicates that these two viscous effects are interrelated. The JPDF between these two variables shows indeed a very strong (anti)correlation between the two variables at this location (not shown).

To summarise, the new normalisation of the enstrophy transport equation terms using the local mean Kolmogorov velocity and length scales shows that the conditional mean profiles of these terms exhibit a universal shape, with very small variations in the simulations used in the present work, and allows for a much more clear assessment of the position of the several sublayers within the TNTI layer, as well as the limits of this layer.

This fact has an important implication on the statistics of the small-scale ‘nibbling’ mechanism as measured in different flow types. The ‘nibbling’ mechanism, as defined by Corrsin & Kistler (Reference Corrsin and Kistler1955), consists of the small-scale diffusion of vorticity (or enstrophy) into the NT flow region, which occurs mainly at the VSL, where the enstrophy diffusion $D_{\omega }$ dominates the enstrophy dynamics. The total amount of this effect can be computed in several different ways, e.g. by computing a ‘flux’ of $D_{\omega }$ across the edge of the VSL, or by integrating $D_{\omega }$ over the entire VSL. Whatever the definition used, the fact that the mean profile of $D_{\omega }$, normalised with the local mean Kolmogorov velocity and length scales, is equal in very different flow types strongly suggests that the statistics of the ‘nibbling’ mechanism are universal, i.e. equal in jets, wakes and mixing layers.

The next section will show how the instantaneous values associated with the small-scale turbulence dynamics, as observed by the shape of the probability density functions, also collapse nicely into the same contours, provided these quantities are properly normalised.

3.3. Probability density functions at several locations within the TNTI layer

In order to assess the degree of collapse provided by the new normalisation we now analyse joint probability density functions (JPDFs) of several quantities at fixed coordinates within the TNTI layer. These JPDFs were computed by using the values of several flow variables taken at the same (fixed) distance $y_I$ from the IB position, using the same procedure described in § 2.2. Thus, the approach used to obtain these JPDFs is quite different to that developed in Pope (Reference Pope1985), which to the authors knowledge has never been used in the context of TNTIs. Figure 13 shows the JPDF of the total enstrophy variation $T_{\omega }$ and enstrophy viscous diffusion $D_{\omega }$ at four locations inside the TNTI layer. The two variables are again normalised with the local, i.e. conditional mean Kolmogorov velocity and micro-scale. In agreement with Taveira et al. (Reference Taveira, Diogo, Lopes and da Silva2013), the correlation between $T_{\omega }$ and $D_{\omega }$ is very strong inside the VSL at the point of maximum $D_{\omega }$ (figure 13$a$) and decreases inside the TSL and beyond (figure 13$b$–$d$). The universality of the enstrophy dynamics within the TNTI layer can be confirmed by the strong near collapse of the isolines of the JPDFs for three very different flow types, particularly for the contours associated with the most frequent values of the variables. The smaller degree of collapse observed for high values of $T_{\omega }$ and $D_{\omega }$ in the first location (figure 13$a$) can be explained by the very fast evolution of the small-scale characteristics at this point of Max[$D_{\omega }$], and the smaller collapse between the curves for the contours associated with less frequent events may also be affected by the use of the interface envelope, instead of the interface itself, in the procedure of the conditional statistics used in the present work.

Figure 13. Joint probability density function of total enstrophy variation $T_{\omega }$, and enstrophy diffusion, $D_{\omega }$, normalised with the (mean) local Kolmogorov velocity and length scale, for three different flow types (SFT$_{202}$ – dash/blue, JET$_{193}$ – dash–dot/magenta, WAKE$_{266}$ – solid line/red) at several locations within the TNTI layer: (a) peak of enstrophy diffusion Max[$D_{\omega }$]; (b) peak of enstrophy production Max[$P_{\omega }$]; $(\textit {c})$ peak of enstrophy Max[$\omega ^2/2$]; $(\textit {d})$ turbulent core region ($y_I/\langle \eta \rangle _I = 250$). In all cases the isolines correspond to the same levels ($1.0, 0.1, 0.01$ and $0.001$).

Figure 14 shows the JPDF of the total enstrophy variation $T_{\omega }$ and enstrophy production $P_{\omega }$ at the same four locations of the TNTI layer as in figure 13. Again, and in agreement with Taveira et al. (Reference Taveira, Diogo, Lopes and da Silva2013), we see that the correlation between $T_{\omega }$ and $P_{\omega }$ is strong in the entire TSL and in the turbulent core region (figure 13$b$–$d$) but is somehow much smaller inside the VSL (figure 14$a$). Again the collapse between the isolines of the JPDF of $T_{\omega }$ and $P_{\omega }$ is very impressive, particularly for the contours associated with the most frequent values, and the fact that these are obtained from very different flows underscores the universality of the enstrophy dynamics within the TNTI layer.

Figure 14. Joint probability density function of total enstrophy variation $T_{\omega }$, and enstrophy production, $P_{\omega }$, normalised with the (mean) local Kolmogorov velocity and length scale, for three different flow types (SFT$_{202}$ – dash/blue, JET$_{193}$ – dash–dot/magenta, WAKE$_{266}$ – solid line/red) at several locations within the TNTI layer: (a) peak of enstrophy diffusion Max[$D_{\omega }$]; (b) peak of enstrophy production Max[$P_{\omega }$]; $(\textit {c})$ peak of enstrophy Max[$\omega ^2/2$]; $(\textit {d})$ turbulent core region ($y_I/\langle \eta \rangle _I = 250$). In all cases the isolines correspond to the same levels ($1.0, 0.1, 0.01$ and $0.001$).

The evolution of small scales of turbulence inside the TNTI layer is further explored by analysing the JPDFs for the second $Q$, and third $R$, invariants of the velocity gradient tensor, $\partial u_i /\partial x_j$. The invariants are interesting variables to analyse in this context because they characterise many of the well-know universal features of the small-scale dynamics of turbulence, including the geometry of the small-scale straining motions as well as the small-scale details of the strain and vorticity generation (Cantwell Reference Cantwell1993; Davidson Reference Davidson2004). This universality is manifested in the well-know ‘tear-drop’ shape of the JPDF of $Q$ and $R$ that has been observed in virtually all turbulent flows, such as in experimental turbulent boundary layers (Elsinga & Marusic Reference Elsinga and Marusic2010), round jets (Breda & Buxton Reference Breda and Buxton2019) and turbulence generated by fractal grids (Gomes-Fernandes, Ganapathisubramani & Vassilicos Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2014), and in numerical simulations of turbulent mixing layers (Soria et al. Reference Soria, Sondergaard, Cantwell, Chong and Perry1994), turbulent channel flows (Blackburn, Mansour & Cantwell Reference Blackburn, Mansour and Cantwell1996), turbulent boundary layers (Chong et al. Reference Chong, Soria, Perry, Chacin, Cantwell and Na1998), forced and decaying isotropic turbulence (Ooi et al. Reference Ooi, Martin, Soria and Chong1999), shear-free turbulence (Watanabe et al. Reference Watanabe, Jaulino, Taveira, da Silva, Nagata and Sakai2017a) and turbulent plane jets (da Silva & Pereira Reference da Silva and Pereira2008).

However, the so-called ‘tear-drop’ shape is not completely formed in the IR (or NT) region and the build-up of this shape across the TNTI layer has been analysed in detail by da Silva & Pereira (Reference da Silva and Pereira2008) and, more recently, by Watanabe et al. (Reference Watanabe, Jaulino, Taveira, da Silva, Nagata and Sakai2017a). Figure 15 shows the evolution of the JPDF of $Q$ and $R$ from the IB to the turbulent core region for the three different flow configurations analysed in figures 13 and 14. The invariants are again normalised using the local (conditional mean) Kolmogorov velocity and length scales. As in da Silva & Pereira (Reference da Silva and Pereira2008), Watanabe et al. (Reference Watanabe, Jaulino, Taveira, da Silva, Nagata and Sakai2017a), the tear-drop starts forming in the fourth quadrant, inside the VSL (figure 15$a$), and expands later in the second quadrant when crossing the TSL (figure 15$b$), and is completely formed and virtually equal to deep inside the turbulent region by the time the end of the TNTI layer is reached (figure 15c,d). Recall that using the classical ‘normalisation’ differences in these maps can be observed while moving away from the IB (for the same flow), e.g. da Silva & Pereira (Reference da Silva and Pereira2008), and this is clearly not the case here. On the contrary, the almost perfect collapse observed for these JPDFs at three radically different flow configurations, and at different locations, even for the most frequent events, again demonstrates the universality of the small-scale geometry, and of the dynamics of strain and vorticity within the TNTI layer.

Figure 15. Joint probability density function of the second $Q$ and third $R$ invariants of the velocity gradient tensor, normalised with the (mean) local Kolmogorov velocity and length scale, for three different flow types (SFT$_{202}$ – dash/blue, JET$_{193}$ – dash–dot/magenta, WAKE$_{266}$ – solid line/red) at several locations within the TNTI layer: (a) IB ($y_I/\langle \eta \rangle _I$); (b) peak of enstrophy production Max[$P_{\omega }$]; $(\textit {c})$ peak of enstrophy Max[$\omega ^2/2$]; $(\textit {d})$ turbulent core region ($y_I/\langle \eta \rangle _I = 250$). In all cases the isolines correspond to the same levels ($1.0, 0.1, 0.01$ and $0.001$). The lines represent the zero discriminant $4Q^3 + 27R^2 = 0$ (Cantwell Reference Cantwell1993; Davidson Reference Davidson2004) (note that the line is actually plotted for each case but are indistinguishable due to the collapsing).