5.1. Self-similarity and length scales

The analysis of the DNS data starts with mean profiles in order to determine the self-similar region where the investigation of the TNTI is to be conducted. In order to determine the time when the jet becomes self-similar, mean profiles of the streamwise velocity, turbulent kinetic energy and the $\langle u'v' \rangle$ component of the Reynolds stress are considered. Self-similarity means that statistics evolve with a time-local amplitude scaling and a time-local length scale, i.e. $\phi _0(t)$ and $\ell (t)$, so that the time-dependent $y$-profile of an $x$- and $z$-averaged quantity $\phi$ can be written in the form (Townsend Reference Townsend1976)

(5.1)\begin{equation} \phi = \phi_0(t)\,f(y / \ell (t)). \end{equation}

For the investigation of the self-similarity of the mean flow profiles, we start by normalising the profiles by using the jet half-width $\delta (t)$ (defined as the absolute value of $y$, where $U(y)$ is $U(0)/2$) as time-local length scale; see figure 2. In order to distinguish between self-similarity and scaling, the profiles are normalised in figure 2 by their maxima (Dairay et al. Reference Dairay, Obligado and Vassilicos2015).

Figure 2. Profiles of mean streamwise velocity $U$, streamwise velocity r.m.s. $u_{rms}$, Reynolds shear stress $\langle u'v' \rangle$, and turbulent kinetic energy $K$, normalized by the maximum values of the respective profiles and compared with experimental data from Cafiero & Vassilicos (Reference Cafiero and Vassilicos2019) (blue circles), Ramaprian & Chandrasekhara (Reference Ramaprian and Chandrasekhara1985) (green triangles) and Gutmark & Wygnanski (Reference Gutmark and Wygnanski1976) (red squares).

With similar DNS, da Silva & Pereira (Reference da Silva and Pereira2008) report that the self-similar regime starts at $t/T_{ref} \approx 20$, which is after the transition to turbulence has happened. In another study of the same flow, Van Reeuwijk & Holzner (Reference Van Reeuwijk and Holzner2013) report that the jet becomes fully turbulent at $t/T_{ref} \approx 30$. Looking at figure 2, it is observed that the mean flow, Reynolds stress, root-mean-square (r.m.s.) streamwise velocity and turbulent kinetic energy profiles collapse rather well as functions of $y/\delta (t)$ for $t/T_{ref} \geq 30$ in the present simulations; $t/T_{ref} = 30$ marks the beginning of the self-similar regime, and as shown in figure 1(a), it is also when the Taylor length Reynolds number starts remaining approximately constant in time. In figure 2, the self-similar profiles are also compared with the experimental data of Gutmark & Wygnanski (Reference Gutmark and Wygnanski1976), Ramaprian & Chandrasekhara (Reference Ramaprian and Chandrasekhara1985) and Cafiero & Vassilicos (Reference Cafiero and Vassilicos2019), showing good collapse between the present data and the profiles obtained in the experiments.

Figure 3(a) shows the time evolution of the normalized square of the jet half-width, i.e. $\delta ^2/H_J^2$.

Figure 3. (a) Time variation of the square of the jet half-width, $\delta ^2$. The red dashed line is the linear fit to the data for times when the jet is fully turbulent and mean profiles are self-similar. (b) Ratios $\lambda / \eta$ (solid line) and $\delta / \lambda$ (dashed line), demonstrating the similar time evolution of all length scales of the flow.

The data plotted in figures 2 and 3(a) are ensemble averages over the five simulations (as well as averages over the $x\unicode{x2013}z$ plane in every simulation, of course). A linear fit to the data for $t/T_{ref} \geq 30$ shows that $\delta ^2$ grows linearly with time, in agreement with the prediction in § 2. Figure 3(b) shows ratios of length scales, namely $\eta (t)/\lambda (t)$ and $\delta (t)/\lambda (t)$, where $\lambda$ and $\eta$ are calculated in terms of turbulent kinetic energy and dissipation rate at the centreplane $y=0$. It is observed that the turbulence length scales $\lambda$ and $\eta$ evolve similarly in time. In addition, the mean flow length scale $\delta (t)$ also evolves in the same way, leading to the confirmation of the conclusion in § 2 that all length scales grow identically with time.

To extract from the DNS data the scaling quantity $R_0$ of § 2, we identify it with $\langle u'v'\rangle _{max}$, the maximum value of the Reynolds shear stress profile in figure 2.

We find that the Townsend assumption $K_0 \sim R_0$ holds for times $t/T_{ref}=30$ to $t/T_{ref}=80$ (figure 3a). According to the scalings derived in § 2, $K_0$ should vary in time like $u_{0}^{2}$, where $u_{0} (t) \equiv U(y=0,t)$, and this is confirmed by our DNS data, as figure 4(a) makes clear over a range of times even greater than $K_0 \sim R_0$ (up to $t/T_{ref} =100$). This range of times is greater because the effects of the boundary conditions on the time-developing jet appear to be felt first by the Reynolds shear stress and later by other quantities such as $K_0$ and $u_0$. We chose to process our data from $t/T_{ref}=30$ to $t/T_{ref}=100$, where self-similarity holds and where the constancy of $u_0 \delta$, related to the volume flux (2.6), is definitely respected in our DNS (figure 4b). With the exception of figure 4(a) where $K_{0}/R_{0}$ starts deviating from its constancy in time after $t/T_{ref}=80$, all the figures where we plot quantities versus time do not show a drastic change after $t/T_{ref}=80$, which is why we chose to process our data until $t/T_{ref}=100$ rather than $t/T_{ref}=80$. There is no effect on our paper's conclusions.

Figure 4. (a) The ratios $K_0/R_0$ and $K_0/u_0^2$, and (b) constancy of the normalised volume flux, between $t/T_{ref}=26$ and $t/T_{ref}=98$.

5.2. Time dependence of scaling parameters and virtual origin

The time dependencies of the centreline streamwise velocity scale $u_0(t)$ and of the jet half-width $\delta (t)$, (2.19) and (2.20), are found to be power laws

(5.2)\begin{equation} \phi(t) = A(t - t_0)^b \end{equation}

in the theoretical analysis of § 2. It is important to note that these two power laws must combine properly to satisfy the governing equations, and that this can happen only if the virtual origin $t_0$ is the exact same one in (2.19) and (2.20) (Nedić Reference Nedić2013; Nedić, Vassilicos & Ganapathisubramani Reference Nedić, Vassilicos and Ganapathisubramani2013; Dairay et al. Reference Dairay, Obligado and Vassilicos2015; Cafiero & Vassilicos Reference Cafiero and Vassilicos2019).

There exist various methods for the determination of the exponent $b$ while taking proper account of the virtual origin $t_0$ (Nedić et al. Reference Nedić, Vassilicos and Ganapathisubramani2013; Dairay et al. Reference Dairay, Obligado and Vassilicos2015; Cafiero & Vassilicos Reference Cafiero and Vassilicos2019). In the present study, the method used in Cafiero & Vassilicos (Reference Cafiero and Vassilicos2019) is implemented on $u_{0}(t) \sim (t-t_{0})^b$ and $\delta (t) \sim (t-t_{0})^{-b}$.

The procedure starts with initial fits to the $u_0$ data in the form $u_0 \sim t^b$, and to the $\delta$ data in form $\delta \sim t^{-b}$, in agreement with volume flux conservation, (2.6). By this step, two approximate values for the exponent $b$ are obtained as initial guesses. Then the value of the exponent is varied in a certain range around the initial guess in order to find the corresponding $t_0$ values for every value of $b$. This procedure is carried out for both $u_0$ and $\delta$ separately. Plotting the resulting $(b, t_0)$ pairs yields the plot in figure 5, where blue and red colours are differentiating the values obtained from the $u_0$ and $\delta$ data. At the point where these two lines intersect, the best fit values $(b, t_0)$ are the ones that take into account that the virtual origin must be identical for both $u_0$ and $\delta$. These values are $b=-0.51$ and $t_0 = 11.7$. The time evolutions of $u_0$ and $\delta$ in the time range $t/T_{ref} = 30$ to $/T_{ref} = 100$, and their power-law fits with the pair $b = -0.51$, $t_0 = 11.7$, are shown in figure 6.

Figure 5. The optimal virtual origin $t_0$ as a function of exponent $b$ for the time evolutions of $u_0$ (blue circles) and $\delta$ (red squares). The dashed vertical lines show the best fit exponent $b$ for $t_0 = 0$ (blue for $u_0$, red for $\delta$), and the green diamond marks the one value of $b$ for which $t_0$ is the same for both (2.19) and (2.20).

Figure 6. Time variation of (a) $u_0$ and (b) $\delta$, with the best power-law fits obtained by the procedure based on figure 5.

At this point, we recall our result of § 2 that, unlike spatially developing turbulent jets (Cafiero & Vassilicos Reference Cafiero and Vassilicos2019), the evolutions (in time) of $u_0$ and $\delta _0$ in temporally developing turbulent jets are independent of the exponent $m$ in the turbulence dissipation law (2.18). The values found for $b$ and $t_0$ from the DNS data are compatible with the theoretical value $b=-0.5$ obtained in § 2 for any exponent $m$.

5.3. Identification of the turbulent jet and locating the TNTI

The TNTI is associated with the very high gradients of enstrophy observed between the rotational turbulent region and the irrotational outer flow. Thus it is the layer where isosurfaces of very different enstrophy values are spatially stacked very close to each other. In figure 7, we plot the turbulent jet volume $V_J$, defined as the volume where ${\omega }^{2} \geq \omega _{th}^{2}$, where ${\omega }^{2}$ is the enstrophy of the fluctuating velocity field, and $\omega _{th}^{2}$ is a threshold enstrophy. In this figure, $V_J$ is normalised by the domain volume $V_{tot}$, and plotted versus the normalised enstrophy threshold values $\omega ^2_{th}/\omega ^2_{ref}$, where the reference enstrophy $\omega _{ref}^{2}$ is the mean enstrophy value averaged over the centreplane. (Note that $\omega _{ref}^{2}$ evolves in time.)

Figure 7. Detected turbulent volume $V_J/V_{tot}$ obtained by varying the threshold values $\omega ^2_{th}/\omega ^2_{ref}$ for one of the simulations (PJ1).

Figure 7 reveals the presence of a plateau over a very wide range of threshold values at any time between $t/T_{ref}=30$ and $t/T_{ref}=90$. This is the range of enstrophies packed tightly together within the TNTI, leading to $V_J/V_{tot}$ being approximately constant for a wide range of $\omega ^2_{th}/\omega ^2_{ref}$ values, and thereby reflecting the sharp demarcation between the turbulent region and the outer non-turbulent region. Starting from the turbulent side of the TNTI and going through the interface, the enstrophy drops rapidly from its nearly homogeneous non-zero value in the inner region of the jet towards zero within a very short distance, which is typically of the order of $10\eta$ for the Reynolds numbers reachable by current DNS (Nagata, Watanabe & Nagata Reference Nagata, Watanabe and Nagata2018; Silva et al. Reference Silva, Zecchetto and da Silva2018).

The left-hand side of the plateau, corresponding to low enstrophy threshold values, is limited by the numerical noise. These numerical oscillations get significant as the threshold value goes to zero. The additional filtering that we introduced to reduce the numerical oscillations increases the $\omega ^2_{th}/\omega ^2_{ref}$ range of the plateau by extending its left-hand side to values closer to $\omega ^2_{th}/\omega ^2_{ref} = 0$, as the outer edge of the TNTI is cleaner in terms of noise.

Figure 8 shows a part of the computational domain that includes the turbulent jet for PJ1 at $t/T_{ref}=50$. The inset is the magnification of a small region around the TNTI and shows the isocontours $\omega ^2_{th}/\omega ^2_{ref} = 10^{-6}, 10^{-5}, 10^{-4}, 10^{-3}$. These threshold values are within the enstrophy range of the plateau in figure 7 and are therefore within the TNTI. Surfaces that are clean in terms of noise can be obtained for a very wide range of enstrophy thresholds from the simulation data.

Figure 8. Contour field of $\omega ^2/\omega ^2_{ref}$ and isocontours of certain $\omega ^2_{th}/\omega ^2_{ref}$ values to mark the TNTI layer. Simulation PJ1 at $t/T_{ref}=50$.

Following the determination of the $\omega ^2_{th}/\omega ^2_{ref}$ range defining the TNTI, we now determine the TNTI as shown in figure 9. The procedure starts by labelling the turbulent volume by the condition $\omega ^2(x,y,z) \ge \omega ^2_{th}/\omega ^2_{ref}$, and obtaining the binary field. The turbulent region corresponds to the blue region in figure 9(a), and the non-turbulent regions correspond to the white and red marked regions, where the engulfed regions (shown with red) are still present. Following this, the non-turbulent volumes are labelled in three dimensions by using the labelling function from the open-source SciPy library (Virtanen et al. Reference Virtanen2020), so that all independent non-turbulent volumes have their individual label number. At this stage, the connectivities of the non-turbulent regions are checked, leading to detection of engulfed non-turbulent volumes (with no connection in three dimensions with the external irrotational region). Some examples of these detected engulfed volumes can be seen in figure 9(a), marked in red. The white detached regions inside the turbulent area of figure 9(a) (in blue) are connected to the outer non-turbulent region in the three-dimensional field (out of the figure's plane). In order to consider only the outer surface, the engulfed volumes are suppressed in this study. To get the surface corresponding to a chosen $\omega ^2_{th}/\omega ^2_{ref}$ in three dimensions, a dilation procedure is used in three dimensions to expand the non-turbulent region into the turbulent region. Then by subtracting the original field from the dilated field, we end up with a field where the three-dimensional jet envelope is marked by the number 1, and all other data points are marked 0 in the entire simulation domain. A cut-section of the resulting field is shown in figure 9(b), as the dark line. This detection procedure is applied for various enstrophy threshold values to obtain the interface characteristics at different locations throughout the TNTI layer as in Van Reeuwijk & Holzner (Reference Van Reeuwijk and Holzner2013) and Krug et al. (Reference Krug, Chung, Philip and Marusic2017).

Figure 9. (a) The labelling of the turbulent, non-turbulent and engulfed regions. (b) Detected TNTI. For the instant $t/T_{ref}=50$ of simulation PJ1, $\omega ^2_{th}/\omega ^2_{ref} = 10^{-3}$.

5.4. Fractal dimensions of the TNTI

The theoretical analysis in § 2 relates the fractal dimension of the TNTI to the global Reynolds number scaling of the TNTI propagation velocity; see (3.6). It is therefore important to investigate the fractal/fractal-like properties of the TNTI.

The fractal/fractal-like nature of scalar isosurfaces relating to the TNTI has been reported in various studies (Sreenivasan et al. Reference Sreenivasan, Ramshankar and Meneveau1989; Miller & Dimotakis Reference Miller and Dimotakis1991; Sreenivasan Reference Sreenivasan1991; Lane-Serff Reference Lane-Serff1993; Dimotakis & Catrakis Reference Dimotakis and Catrakis1999; Mistry et al. Reference Mistry, Philip, Dawson and Marusic2016; Mistry, Dawson & Kerstein Reference Mistry, Dawson and Kerstein2018). However, these fractal/fractal-like characteristics are described somewhat differently in different studies. In some studies, a well-defined power law for the scale dependence of the surface area (thus constant fractal dimension) has been reported (Sreenivasan et al. Reference Sreenivasan, Ramshankar and Meneveau1989; Sreenivasan Reference Sreenivasan1991; Mistry et al. Reference Mistry, Philip, Dawson and Marusic2016, Reference Mistry, Dawson and Kerstein2018). This is the case where, when one covers the surface with boxes of size $r$, the number $N$ of boxes needed to fully cover the surface scales as $N(r) \sim r^{-D_f}$ (Mandelbrot Reference Mandelbrot1982), and the fractal dimension $D_f$ of the surface is independent of $r$ over a significant range of scales $r$. In other studies of isosurfaces, in flows such as turbulent jets and mixing layers, a scale-dependent fractal dimension is reported, i.e. $D_{f} = D_{f}(r)$, which means that there is no constant value for the fractal dimension $D_f$, but the fractal dimension varies with box size $r$ (Miller & Dimotakis Reference Miller and Dimotakis1991; Dimotakis & Catrakis Reference Dimotakis and Catrakis1999; Catrakis & Dimotakis Reference Catrakis and Dimotakis1999).

There is also the question of the enstrophy threshold used to define the TNTI because a strong threshold dependence of the fractal dimension of scalar isosurfaces has been reported in some studies (Miller & Dimotakis Reference Miller and Dimotakis1991; Lane-Serff Reference Lane-Serff1993; Flohr & Olivari Reference Flohr and Olivari1994). Varying the threshold within the range of thresholds where $V_J$ remains approximately constant is akin to sampling different inner iso-enstrophy surfaces within the TNTI layers’ inner structure (Van Reeuwijk & Holzner Reference Van Reeuwijk and Holzner2013). There may be not one single fractal dimension for the TNTI, but different fractal dimensions for different inner isosurfaces of enstrophy within the TNTI layer, an aspect of the problem that needs to be investigated.

We apply the box-counting procedure to obtain fractal dimensions of iso-enstrophy surfaces within the TNTI. Figure 10 shows typical ensemble-averaged box-counting results, these particular ones being for the isosurface $\omega ^2_{th}/\omega ^2_{ref} = 10^{-3}$ at time $t/T_{ref}=50$. Figure 10(a) is a log–log plot of the number $N$ of boxes needed to cover the iso-enstrophy surface versus the inverse box size $1/r$. The linear fit in orange is obtained by using all the points on the plot, and the slope of this fit is found to be $D_{f1}=2.161$ for this particular case. On the other hand, local slopes are also calculated by fits over nine consecutive data points on this plot. It is observed (see the example in figure 10b) that the local slope does not remain constant throughout all scales $r$. An approximately constant fractal dimension, seen as a plateau-like region in figure 10(b), appears to exist between $r = \delta$ and $r = \lambda$ for the entire range of isosurfaces of various enstrophy threshold values within the TNTI ($\omega _{th}^{2}/\omega ^2_{ref}$ between $10^{-6}$ and $10^{-3}$) and for all times where the jet is fully turbulent (local slope values marked by red square markers). Note that the constancy of this local fractal dimension is affected by the fact that it is calculated by using nine points around the value of $r$ where the local dimension is evaluated. This means that the highly non-constant values of the fractal dimension at scales $r$ larger than $\delta$ are responsible for deviations from constancy at scales close to but below $\delta$, and that the progressive decrease of the local slope towards $D_f =2$ as $r$ decreases at scales $r$ below $\lambda$ is responsible for the systematic deviation from constancy at scales close to yet larger than $\lambda$.

Figure 10. Ensemble-averaged results of the box-counting method applied to isosurface $\omega ^2_{th}/\omega ^2_{ref} = 10^{-3}$ at time $t/T_{ref} = 50$. (a) A plot of the number of boxes $N$ of size $r$ versus $1/r$ is shown in log–log scale, the orange line being the linear best fit for all data points on this plot. (b) The local slope calculated by the fits using nine consecutive data points, the value of the local slope being attributed to the centre point. The local slopes marked as red squares (as opposed to blue circles) are the points used to calculate $D_{f2}$. The dashed, dot-dashed and dotted vertical lines locate on the horizontal axis the length scales $\delta$, $\lambda$ and $\eta$, respectively (where $\lambda$ and $\eta$ are calculated on the centreplane.)

Throughout this study, the fractal dimension is calculated as the average value of the local slopes between box sizes $r = \delta$ and $r = \lambda$, and this fractal dimension is denoted $D_{f2}$. The first point with $r$ smaller than or equal to $\delta$ (i.e. the largest value of $r$ in the range $\lambda \leq r \leq \delta$) is excluded from this average so as to reduce the oscillation caused by less-converged values of $N$ at larger box sizes.

The fractal dimension $D_{f2}$ for different enstrophy threshold values in the TNTI range $\omega _{th}^2/\omega _{ref}^2\in [ 10^{-6}, 10^{-3}]$ is shown in figure 11 as a function of time. The fractal dimensions $D_{f2}$ of the TNTI may be considered to remain approximately constant in time for all these enstrophy thresholds, and the mean value around which $D_{f2}$ appears to fluctuate is shown by the dashed lines in the figure. For the threshold values $\omega _{th}^2/\omega _{ref}^2\in [ 10^{-6}, 10^{-3}]$, this fractal dimension value varies from $D_{f2}=2.09$ to $D_{f2}=2.18$. It can be observed that the values of $D_{f2}$ for different $\omega ^2_{th}/\omega ^2_{ref}$ get closer to each other towards the lower values of $\omega ^2_{th}/\omega ^2_{ref}$. It can also be argued that an objective definition of the viscous superlayer must include, within the superlayer, enstrophy iso-values for which the fractal dimension can be detected with a value larger than 2.

Figure 11. TNTI fractal dimensions $D_{f2}$ versus time $t/T_{ref}$ for different normalised enstrophy thresholds within the TNTI.

A significantly higher value, $D_{f2} = 2.36$, has been observed for the iso-enstrophy surface defined by the threshold $\omega _{th}^2/\omega _{ref}^2=10^{-2}$. This value is close to the fractal dimension $7/3 \approx 2.33$ reported in various studies (Sreenivasan et al. Reference Sreenivasan, Ramshankar and Meneveau1989; Sreenivasan Reference Sreenivasan1991; Mistry et al. Reference Mistry, Philip, Dawson and Marusic2016, Reference Mistry, Dawson and Kerstein2018). It must be noted that the enstrophy threshold $\omega _{th}^2/\omega _{ref}^2=10^{-2}$ rests on the turbulent side of the TNTI judging from the enstrophy range of the plateau shown in figure 7. However, it is also observed that the $\log _2 N$–$\log _2(1/r)$ plot obtained from the box-counting algorithm for this enstrophy threshold shows no evidence of a fractal dimension that is independent of $r$, i.e. there is no significant plateau region in figure 12(b), and the local slope varies significantly with $r$. The value $D_{f2} = 2.36$ is obtained by averaging over the local fractal dimensions (local slopes in figure 12b) from $r=\lambda$ to $r=\delta$, but these local fractal dimensions vary continuously with $r$ from 2.2 to over 2.45.

Figure 12. Same as figure 10, but for iso-enstrophy surface $\omega ^2_{th}/\omega ^2_{ref} = 10^{-2}$ at the same time $t/T_{ref} = 50$.

5.5. Propagation velocity of the interface

In § 2, we obtained (3.6) for the TNTI's mean propagation velocity on the basis of the fractal/fractal-like character of the TNTI. We now know, following the previous subsection, that the TNTI of our time-developing turbulent jet has a range of fractal dimensions $D_{f2}$ depending on the normalised enstrophy threshold $\omega _{th}^{2}/\omega _{ref}^{2}$, and that $D_{f2}$ is a fairly well-defined single number independent of box size $r$ in the range $\lambda \le r\le \delta$ if $\omega _{th}^{2}/\omega _{ref}^{2}$ is in the range $[ 10^{-6}, 10^{-3}]$. The question that arises naturally now is: Does (3.6) capture the time and enstrophy-threshold dependencies of the mean propagation velocity $v_n$? More specifically, can we use $D_{f2} = D_{f2} (\omega _{th}^{2}/\omega _{ref}^{2})$ defined in the range $\lambda \le r\le \delta$ as the fractal dimension in (3.6) to capture accurately the time and enstrophy-threshold dependencies of $v_n$? We stress that in this formula, $v_n$ depends on the enstrophy threshold only through $D_{f2}$, given that $A$ is defined in terms of quantities that are independent of enstrophy threshold, and $a$ in $V_{J}=2a\delta L_{x}L_{z}$ can be expected to have a negligibly weak dependence on enstrophy threshold.

To estimate $v_n$ independently from (3.6), we use (3.2), having first checked the validity of ${{\rm d}V_{J}}/{{\rm d}t} = 2a L_{x}L_{z}\,({{\rm d} \delta }/{{\rm d}t})$ (see figure 13), which is needed to go from (3.1) to (3.2). Figure 13 confirms that the dimensionless coefficient $a$ is approximately independent of time as it oscillates around the constant value $a=1.66$, and that it is also very weakly dependent on enstrophy threshold over at least four decades.

Figure 13. Validity of ${{\rm d} V_{J}}/{{\rm d}t} \sim 2 L_{x}L_{z}\,({{\rm d} \delta }/{{\rm d}t})$ over the time evolution of the fully turbulent jet.

To use (3.2), we need a reliable estimate of the TNTI surface area $S$ that is different from the fractal estimate (3.3). To obtain such an estimate of $S$, we plot $r^{2}\,N(r)$: as the box-counting algorithm's box size $r$ decreases and becomes small enough to resolve all the contortions of the iso-enstrophy surface, $r^{2}\,N(r)$ reaches a maximum and does not grow further with further decreasing $r$. We take this maximum as our estimate of $S$, i.e. $S=S_{R}\equiv \max _{r}[r^{2}\,N(r)]$. Of course, $S$ depends on the enstrophy threshold defining the chosen isosurface within the TNTI, and figure 14(a) shows an example of an $r^{2}\,N(r)$ versus $1/r$ log–log plot for $\omega _{th}^{2}/\omega _{ref}^{2} = 10^{-3}$ at $t/T_{ref}=50$, where the maximum $r^{2}\,N(r)$ is reached at $r$ close to $\eta$. In fact, figure 14(a) is quite typical of normalised enstrophy thresholds in the range $[ 10^{-6}, 10^{-3}]$ and times $t/T_{ref}$ in the range $[ 30, 100 ]$.

Figure 14. (a) Plot (using $\log _{2}$ values) of $r^{2}\,N (r)$ versus $1/r$, at time $t/T_{ref}=50$, for the threshold value $\omega _{th}^2/\omega _{ref}^2=10^{-3}$. The horizontal dashed line indicates the maximum value of $r^{2}\,N (r)$. (b) Plot of $S_{R}/(L_{x}L_{z}) \equiv \max _{r}[r^{2}\,N(r)]/(L_{x}L_{z})$ versus time $t/T_{ref}$ for different enstrophy threshold values.

In figure 14(b), we plot $S_{R}\equiv \max _{r}[r^{2}\,N(r)]$ as a function of $t/T_{ref}$ for various normalised enstrophy thresholds. Interestingly, the TNTI surface areas $S_R$ remain approximately constant in time for all thresholds $\omega _{th}^2/\omega _{ref}^2=10^{-6}$ to $10^{-4}$ from $t/T_{ref} = 40$ to $100$, and for threshold $\omega _{th}^2/\omega _{ref}^2=10^{-3}$ from $t/T_{ref} = 50$ to $100$. This is compatible with the fact that all length scales, large and small, grow together in this flow.

We now calculate the average TNTI propagation velocity $v_n$ by using (3.2) with $S$ obtained from $S_{R}\equiv \max _{r}[r^{2}\,N(r)]$, and we compare it with (3.6). First, in figure 15 we check the time-dependence of $v_n$, which, according to (3.6) and $\delta \sim \sqrt {U_{J}H_{J}(t-t_{0})}$, is the same as the time dependence of $u_{\eta }$ and of $u_{\lambda }$. In support of this prediction, figure 15 shows that $v_{n}/u_{\eta }$ and $v_{n}/u_{\lambda }$ oscillate around a constant as time proceeds for all $\omega _{th}^{2}/\omega _{ref}^{2}$ in the range $[ 10^{-6}, 10^{-3}]$.

Figure 15. Time dependence of (a) $v_{n}/u_{\eta }$ and (b) $v_{n}/u_{\lambda }$.

Second, we check the enstrophy threshold dependence of $v_n$, which, according to (3.6), should be $v_{n}/u_{\eta } \sim (Aa)^{1/(D_{f}(\omega _{th}^{2}/\omega _{ref}^{2})-1)} Re_{G}^{[2-D_{f}(\omega _{th}^{2}/\omega _{ref}^{2})]/[D_{f}(\omega _{th}^{2}/\omega _{ref}^{2})-1] +1/4}$ and equivalently $v_{n}/u_{\lambda } \sim (Aa)^{1/(D_{f}(\omega _{th}^{2}/\omega _{ref}^{2})-1)} Re_{G}^{[2-D_{f}(\omega _{th}^{2}/\omega _{ref}^{2})]/[D_{f}(\omega _{th}^{2}/\omega _{ref}^{2})-1] +1/2}$. We plot $v_{n}/u_{\eta }$ versus $\omega _{th}^{2}/\omega _{ref}^{2}$ for various time instants $t/T_{ref}$ in figure 16(a); and we take our measured $D_{f2} (\omega _{th}^{2}/\omega _{ref}^{2})$ (averaged over time for simplicity, this average being denoted $\overline {D_{f2}}$) to represent the fractal dimension $D_f$, and plot $(v_{n}/u_{\eta })(Aa)^{-1/(\overline {D_{f2}}-1)} Re_{G}^{-(2-\overline {D_{f2}})/(\overline {D_{f2}}-1)}$ versus $\omega _{th}^{2}/\omega _{ref}^{2}$ for various time instants $t/T_{ref}$ in figure 16(b). If (3.6) is able to capture the enstrophy threshold dependence of $v_n$, then $(v_{n}/u_{\eta })(Aa)^{-1/(\overline {D_{f2}}-1)} Re_{G}^{-(2-D_{f2})/(D_{f2}-1)}$ should be constant with varying $\omega _{th}^{2}/\omega _{ref}^{2}$ for all times $t/T_{ref}$ between $30$ and $100$, with $a \approx 1.66$ (as found already from figure 13) and $A\approx 0.058$ from figure 3(a).

Figure 16. (a) Average interface propagation velocity $v_n$ normalised by $u_\eta$ versus normalised enstrophy threshold for different times $t/T_{ref}$. (b) Plot of $v_n$ divided by $(Aa)^{1/(\overline {D_{f2}}-1)}Re_{G}^{(2-\overline {D_{f2}})/(\overline {D_{f2}}-1)}$ according to (3.6) (with $A=0.05777$ and $a=1.6574$) normalised by $u_\eta$ versus normalised enstrophy threshold for different times $t/T_{ref}$.

We can see clearly in figure 16(a) that, irrespective of time, $v_n$ decreases with increasing $\omega _{th}^{2}/\omega _{ref}^{2}$ in the TNTI normalised enstrophy range $[ 10^{-6}, 10^{-3}]$, which makes sense because $S$ increases with increasing $\omega _{th}^{2}/\omega _{ref}^{2}$. Indeed, we expect $Sv_n$ to be approximately independent of $\omega _{th}^{2}/\omega _{ref}^{2}$ in the TNTI range of enstrophy thresholds, judging from (3.1) and the approximate constancy of $V_J$ in that range (shown in figure 7).

Figure 16(b) shows that our formula (3.6) for the TNTI's mean propagation velocity $v_n$ with $D_f$ given by $\overline {D_{f2}}(\omega _{th}^{2}/\omega _{ref}^{2})$ – i.e. the time-averaged (from $t/T_{ref}=30$ to $98$) value of $D_{f2}(\omega _{th}^{2}/\omega _{ref}^{2})$ – captures the enstrophy threshold dependence of $v_n$ very well over the wide range of thresholds $10^{-6}\le \omega _{th}^{2}/\omega _{ref}^{2} \le 10^{-3}$, which is within the TNTI throughout the time range considered.

In the next section we explore the inconsistencies of the simple fractal model for $v_n$ presented in § 2, and investigate how they might be overcome.

5.6. A generalised Corrsin length for the TNTI

Our simple fractal model's formula (3.6) predicts quite well both the time dependence of the TNTI's mean propagation velocity $v_n$ and its enstrophy threshold dependence. However, our fractal model did not foresee the complex inner structure of the TNTI where different iso-enstrophy surfaces within the TNTI have different fractal dimensions.

Our model is based on: (i) ${{\rm d} V_{J}}/{{\rm d}t}= 2a L_{x}L_{z}\,({{\rm d}\delta }/{{\rm d}t})$ (needed to go from (3.1) to (3.2)), which our simulations rather support (see figure 13); (ii) $S = L_{x}L_{z} (\eta _{I}/\delta )^{2-D_{f}}$, where $\eta _{I} = \nu /v_{n}$ is the Corrsin length scale for the viscous superlayer's thickness; and (iii) a well-defined fractal dimension $D_f$ independent of $r$ over a significant range of $r$ values bounded from below by the smallest length scale on the TNTI. In the event, our DNS data have returned well-defined fractal dimensions $D_{f2}$ independent of $r$ in a range bounded from below by $\lambda$ but not by the smallest length scale on the TNTI, which appears to be $\eta$ as the maximum of $r^{2}\,N(r)$ is typically reached at $r$ close to $\eta$. The number $N$ of boxes needed to cover iso-enstrophy surfaces continues to increase faster than $r^{-2}$ as $r$ decreases from $\lambda$ to $\eta$, implying that these scales between $\lambda$ and $\eta$ contribute to the surface area, but not with a well-defined $r$-independent fractal dimension. Furthermore, in the range where an $r$-independent fractal dimension may be claimed, i.e. $\lambda \le r\le \delta$, this fractal dimension $D_{f2}$ is a decreasing function of enstrophy threshold $\omega _{th}^{2}/\omega _{ref}^{2}$, appearing to tend towards close to 2 as $\omega _{th}^{2}/\omega _{ref}^{2}$ tends to $0$.

In figure 17, we plot $S(\eta ) = L_{x}L_{z}(\eta /\delta )^{2-D_{f2}}$, $S(\lambda ) = L_{x}L_{z}(\lambda /\delta )^{2-D_{f2}}$ and $S(\eta _{I}) = L_{x}L_{z}(\eta _{I}/\delta )^{2-D_{f2}}$, all normalised by $S_{R}\equiv \max _{r}[r^{2}\,N(r)]$. These three quantities are plotted versus time for different enstrophy thresholds within the TNTI range of thresholds, i.e. $\omega _{th}^{2}/\omega _{ref}^{2}$ within $[ 10^{-6}, 10^{-3}]$. The fractal dimension $D_{f2}$ is our only possible choice of fractal dimension for the calculations of $S(\eta )$, $S(\lambda )$ and $S(\eta _{I})$ if we want to be consistent with our model's requirement that the fractal dimension should be well defined, i.e. $r$-independent over a significant $r$ range.

Figure 17. Plots of (a) $S(\eta ) = L_{x}L_{z}(\eta /\delta )^{2-D_{f2}}$, (b) $S(\lambda ) = L_{x}L_{z}(\lambda /\delta )^{2-D_{f2}}$ and (c) $S(\eta _{I}) = L_{x}L_{z}(\eta _{I}/\delta )^{2-D_{f2}}$ (where $\eta _{I} = \nu /v_{n}$, with $v_n$ values calculated in § 5.5), all normalised by $S_{R}\equiv \max _{r}[r^{2}N(r)]$, versus time $t/T_{ref}$ for various enstrophy thresholds within the TNTI.

First, figure 17 shows that $S(\eta )/S_R$, $S(\lambda )/S_R$ and $S(\eta _{I})/S_R$ are approximately constant in time for all TNTI enstrophy thresholds, which is not surprising given the approximate time constancies of $D_{f2}$ and $S_R$ and given that $\eta$, $\lambda$ and $\eta _{I}$ all have the same time dependence as $\delta$. Second, figure 17 shows that only $S(\eta )/S_R$ collapses for all enstrophy thresholds. This is not a trivial result because $S(\eta )$ is calculated in terms of a fractal dimension $D_{f2}$ that is not well-defined at scale $\eta$. The worse collapse is returned by $S(\lambda )/S_R$; and $S(\eta _{I})/S_R$ tends towards $S(\eta )/S_R$ with decreasing $\omega _{th}^{2}/\omega _{ref}^{2}$, which makes some sense because, in this limit, $D_{f2}$ decreases towards values close to $2$, and $\eta _{I}/\eta$ therefore approaches a value of order 1, extremely weakly dependent on $\omega _{th}^{2}/\omega _{ref}^{2}$ (see § 2). However, $S(\eta _{I})/S_R$ takes values between $1/5$ and $1/4$, which is different from 1 and therefore contradicts (3.5), which is a premise of our model. In fact, there is a dimensionless coefficient $b$ in (3.3), i.e. $S(r) = b L_{x}L_{z} (r/\delta )^{2-D_{f}}$. This coefficient $b$ is independent of enstrophy threshold because it is set by $S(r=\delta ) = bL_{x}L_{z}$. The only way to retrieve (3.5) is by writing $S=bL_{x}L_{z} (c\eta _{I}/\delta )^{2-D_{f}}$ with $bc^{2-D_{f}}=1$, which requires that the dimensionless coefficient $c$ is a function of $\omega _{th}^{2}/\omega _{ref}^{2}$. Without the arbitrary condition $bc^{2-D_{f}}=1$, the formula (3.6) predicted by our simple fractal model should be replaced by

(5.3)\begin{equation} \frac{v_n}{U_{J}}= \left(\frac{c^{D_{f}-2}}{b}\right)^{1/(D_{f}-1)}\left(Aa\right)^{1/(D_{f}-1)}\,\frac{H_J} {\delta}\, Re_G^{-(D_f -2)/(D_f-1)}. \end{equation}

The quantity ${c^{D_{f}-2}}/{b}$ is in fact the ratio $S(\eta _{I})/S_{R}$ (with $S(\eta _{I})$ given by $L_{x}L_{z}(\eta _{I}/\delta )^{2-D_{f2}}$) that we plot in figure 17, and from our data it transpires that $(S(\eta _{I})/S_{R})^{1/(\overline {D_{f2}}-1)}$ is a significantly decreasing function of $\omega _{th}^{2}/\omega _{ref}^{2}$ (see figure 18). Without setting ${c^{D_{f}-2}}/{b} =1$, our model does not return the right enstrophy threshold dependence of $v_n$, and ${c^{D_{f}-2}}/{b} =1$ does not agree with our DNS data, which show that $S(\eta _{I})/S_{R}$ (with $S(\eta _{I})$ given by $L_{x}L_{z}(\eta _{I}/\delta )^{2-D_{f2}}$) takes values between $1/5$ and $1/4$. We therefore need to explore how our model could be modified to be more realistic, and we do this by generalising the Corrsin length.

Figure 18. Plot of $(S(\eta _{I})/S_{R})^{1/(\overline {D_{f2}}-1)}$ as a function of $\omega _{th}^{2}/\omega _{ref}^{2}$ at different times $t/T_{ref}$.

The Corrsin length may be considered appropriate only for the viscous superlayer at the very lowest enstrophy thresholds where the generation of vorticity is viscosity-dominated and, consistently, $S(\eta _{I})/S_R$ and $S(\eta )/S_R$ appear to take similar values. To generalise this property to higher enstrophy thresholds, we introduce a generalised Corrsin length

(5.4)\begin{equation} \eta_T = \nu_T / v_n \end{equation}

in terms of a local turbulent viscosity $\nu _T$ (local to every iso-enstrophy surface within the TNTI) such that

(5.5)\begin{equation} S = b L_{x}L_{z} (c \eta_{T}/\delta)^{2-D_{f2}}, \end{equation}

where $b$ and $c=c (Re_{G}, \omega _{th}^{2}/\omega _{ref}^{2})$ are dimensionless coefficients independent of time.

The simple physical idea behind (5.4) is that the process of enstrophy production is increasingly dominated by vortex stretching rather than viscosity as the enstrophy threshold increases from the outer, viscous superlayer side of the TNTI to its inner, turbulent side. Studies over the past two decades have indeed shown that the TNTI has an inner structure that includes a viscous superlayer and a sort of buffer layer or turbulent sublayer where vorticity production dominates (da Silva et al. Reference da Silva, Hunt, Eames and Westerweel2014; Taveira & da Silva Reference Taveira and da Silva2014; Nagata et al. Reference Nagata, Watanabe and Nagata2018). Hence the turbulence viscosity $\nu _{T}=\nu _{T}(\omega ^2_{th}/\omega ^2_{ref})$ is expected to increase and become independent of the fluid's kinematic viscosity $\nu$ with increasing $\omega _{th}^{2}/\omega _{ref}^{2}$ within the TNTI.

We now ask whether (5.4), (5.5) and (3.2), which represent an attempt to improve the model for $v_n$ in § 2, are consistent with the requirement that $\nu _T$ must increase with $\omega ^2_{th}/\omega ^2_{ref}$. The three equations just mentioned imply

(5.6)\begin{equation} \nu_{T} = \frac{2a\delta}{c}\,\frac{{\rm d} \delta}{{\rm d}t} \left( \frac{S}{bL_x L_z} \right)^{-(D_{f2}-1)/(D_{f2}-2)}, \end{equation}

where the dimensionless constant $a$ is the one in $Sv_{n}=2aL_{x}L_{z}\,{\rm d}\delta /{\rm d}t$. It can be seen that $\nu _T$ depends on $\omega _{th}^{2}/\omega _{ref}^{2}$ through $S$ and $D_{f2}$ (and also $c$), but does not depend on time, in agreement with our observations in figures 3(a), 14(b) and 11. As $S/L_{x}L_{z}$ increases whereas $(D_{f2}-1)/(D_{f2}-2)$ decreases with increasing $\omega _{th}^{2}/\omega _{ref}^{2}$, it is not trivial to predict how $({S}/{L_x L_z})^{-(D_{f2}-1)/(D_{f2}-2)}$ behaves with varying $\omega _{th}^{2}/\omega _{ref}^{2}$. We therefore use time-averaged values of $S$ and $D_{f2}$ obtained in the previous section for different enstrophy thresholds, and plot in figure 19 the turbulent viscosity $\nu _T$ given by (5.6) with $c$ set to a constant independent of $\omega _{th}^{2}/\omega _{ref}^{2}$, and $\delta ({{\rm d} \delta }/{{\rm d}t}) = \tfrac {1}{2} ({{\rm d} \delta ^{2}}/{{\rm d}t})$ given by the DNS. The result shows that $\nu _{T}$ with $c={\rm const.}$ is a monotonically increasing function of $\omega _{th}^{2}/\omega _{ref}^{2}$, as required for our improved model to be physically viable. This means that $\eta _{T}=\nu _{T}/v_{n}$ is also a monotonically increasing function of $\omega _{th}^{2}/\omega _{ref}^{2}$ because (3.2) implies that $v_n$ is a decreasing function of $\omega _{th}^{2}/\omega _{ref}^{2}$. However, the result in figure 19 also suggests that $\nu _{T}$ and $\eta _T$ tend to $0$ as $\omega _{th}^{2}/\omega _{ref}^{2}$ decreases towards $0$, whereas $\nu _T$ should be tending towards the kinematic viscosity $\nu$ in that limit. In the next paragraph, we demonstrate how the model's dimensionless coefficient $c (Re_{G},\omega _{th}^{2}/\omega _{ref}^{2})$ can ensure that $\nu _T$ tends to $\nu$ as $\omega _{th}^{2}/\omega _{ref}^{2} \to 0$, i.e. as we move towards the outer edge of the TNTI.

Figure 19. The turbulent viscosity $\nu _{T}$ given by (5.6) with $a/c = 1$ and $b=1$ as a function of normalised enstrophy threshold.

We model $c$ as being a constant independent of both $Re_G$ and $\omega _{th}^{2}/\omega _{ref}^{2}$ for most enstrophy thresholds within the TNTI except the smallest ones, where we approximate it as $c (Re_{G},\omega _{th}^{2}/\omega _{ref}^{2}) \approx Re_{G}\,\tilde {c} (\omega _{th}^{2}/\omega _{ref}^{2})$, with $\tilde {c}$ being a function of $\omega _{th}^{2}/\omega _{ref}^{2}$ but not of $Re_G$. Given that $\delta ({{\rm d} \delta }/{{\rm d}t}) = ({A}/{2}) U_{J}H_{J}$ (from (2.20)), we can write $2a ({\delta }/{c}) ({{\rm d} \delta }/{{\rm d}t}) \approx Aa ({\nu }/{\tilde {c}})$ as $\omega _{th}^{2}/\omega _{ref}^{2}\to 0$, i.e.

(5.7)\begin{equation} \nu_{T} \sim Aa\,\frac{\nu}{\tilde{c}} \left( \frac{S}{bL_x L_z} \right)^{-(D_{f2}-1)/(D_{f2}-2)} \end{equation}

in that limit. For $\nu _{T}$ to tend to $\nu$ as $\omega _{th}^{2}/\omega _{ref}^{2}\to 0$, $\tilde {c}$ must tend to $0$ at the same rate as $({S}/{bL_x L_z} )^{-(D_{f2}-1)/(D_{f2}-2)}$, i.e.

(5.8)\begin{equation} \ln \tilde{c} \approx{-}\frac{D_{f2}-1}{D_{f2}-2} \ln \left(\frac{S}{bL_{x}L_{z}}\right) + {\rm const.} \end{equation}

as $\omega _{th}^{2}/\omega _{ref}^{2}\to 0$. It is not the goal of this paper's final part to determine the functions $\nu _{T}(Re_{G}, \omega _{th}^{2}/\omega _{ref}^{2})$ and $c(Re_{G}, \omega _{th}^{2}/\omega _{ref}^{2})$ in the improved model for $v_n$ based on (5.4), (5.5) and (3.2); the goal here is simply to demonstrate on the basis of our DNS and simple asymptotic arguments that such a model can be physically viable. The example of a choice of $c(Re_{G}, \omega _{th}^{2}/\omega _{ref}^{2})$ that we made at the start of this paragraph ensures that $\nu _T$ remains a monotonically increasing function of $\omega _{th}^{2}/\omega _{ref}^{2}$ while at the same time tending to $\nu$ as $\omega _{th}^{2}/\omega _{ref}^{2}$ tends to $0$. We now work out the consequences of this choice for $\eta _T$ and $v_n$.

The formulae for $v_n$ and $\eta _{T}$ that can be derived readily from our improved model are

(5.9)\begin{equation} v_{n}/u_{\eta}\sim \left(\frac{c^{(D_{f2}-2)}}{b}\right)^{{1}/({D_{f2}-1})} \left(Aa\right)^{{1}/({D_{f2}-1})} Re_{G}^{-({D_{f2}-2})/({D_{f2}-1})+{1}/{4}}\left(\nu_{T}/\nu \right)^{({D_{f2}-2})/({D_{f2}-1})} \end{equation}

and

(5.10) \begin{align} \eta_{T}/\eta &\sim \left(\frac{c^{(D_{f2}-2)}}{b}\right)^{-1/({D_{f2}-1})} (Aa)^{-{1}/({D_{f2}-1})} Re_{G}^{({D_{f2}-2})/({D_{f2}-1}) -{1}/{4}}\nonumber\\&\quad \times (\nu_{T}/\nu)^{-({D_{f2}-2})/({D_{f2}-1})}(\nu_{T}/\nu). \end{align}

Note that the original model of § 2 leads to $v_{n}/u_{\eta }\sim (Aa)^{{1}/({D_{f2}-1})} Re_{G}^{-({D_{f}-2})/({D_{f}-1})+{1}/{4}}$ and $\eta _{I}/\eta \sim (Aa)^{-{1}/({D_{f2}-1})} Re_{G}^{({D_{f}-2})/({D_{f}-1}) -{1}/{4}}$ without the extra powers of $c^{D_{f2}-2}/b$ and $\nu _{T}/\nu$ in (5.9) and (5.10).

Without these extra powers, the original model predicts the dependence of $v_n$ on $\omega _{th}^{2}/\omega _{ref}^{2}$ very well. In our improved model, $(\nu _{T}/\nu )^{({D_{f2}-2})/({D_{f2}-1})}$ is an increasing function of enstrophy threshold because $\nu _{T}/\nu$ is increasing and because the exponent $({D_{f2}-2})/({D_{f2}-1})$ is also increasing given that $D_{f2}$ is an increasing function of $\omega _{th}^{2}/\omega _{ref}^{2}$ as observed in our DNS. Our improved model is therefore capable of maintaining the original model's good prediction for $v_n$ if the increasing dependence of $(\nu _{T}/\nu )^{({D_{f2}-2})/({D_{f2}-1})}$ on $\omega _{th}^{2}/\omega _{ref}^{2}$ compensates the decreasing dependence of $(c^{D_{f2}-2}/b)^{1/(D_{f2}-1)}$ on $\omega _{th}^{2}/\omega _{ref}^{2}$. Indeed, $c^{D_{f2}-2}/b$ is not equal to 1, and $(c^{D_{f2}-2}/b)^{1/(D_{f2}-1)}$ is a decreasing function of enstrophy threshold, in agreement with our DNS observation in figure 17(c). The entire point of our improved model has been to show that by introducing the generalised Corrsin length and the turbulent viscosity $\nu _T$, it is possible to correct our original model's wrong assumption $c^{D_{f2}-2}/b =1$ without compromising its correct predictions.

We now show that the choice of $c$ that we made for $\nu _T$ to tend to $\nu$ as $\omega _{th}^{2}/\omega _{ref}^{2} \to 0$ also ensures that the generalised Corrsin length $\eta _T$ tends to a finite value in that limit. As we move within the TNTI from high to low iso-enstrophy levels – i.e. as we take the limit of $\omega _{th}^{2}/\omega _{ref}^{2}$ decreasing towards very small values close to 0 and we approach the outer edge of the viscous superlayer – $D_{f2}$ tends towards values close to $2$, and $\nu _{T}$ tends to $\nu$, assuming $c (Re_{G},\omega _{th}^{2}/\omega _{ref}^{2}) \approx Re_{G}\,\tilde {c} (\omega _{th}^{2}/\omega _{ref}^{2})$ in that limit. We are therefore left with

(5.11)\begin{equation} v_{n}/u_{\eta}\sim \tilde{c}^{({D_{f2}-2})/({D_{f2}-1})} Re_{G}^{{1}/{4}} \end{equation}

and

(5.12)\begin{equation} \eta_{T}/\eta \sim \tilde{c}^{-({D_{f2}-2})/({D_{f2}-1})} Re_{G}^{-{1}/{4}} \end{equation}

as we approach the outer edge of the viscous superlayer (we have omitted the unimportant factor $Aa/b$). Finally, (5.8) implies $\tilde {c}^{({D_{f2}-2})/({D_{f2}-1})} \sim L_{x}L_{y}/S$, and therefore our generalised model's predictions for the viscous superlayer where $D_{f2}$ is very close to $2$ and $\omega _{th}^{2}/\omega _{ref}^{2}$ is extremely small are

(5.13)\begin{equation} v_{n}/u_{\eta}\sim \frac{L_{x}L_{z}}{S_{\nu}}\,Re_{G}^{{1}/{4}} \end{equation}

and

(5.14)\begin{equation} \eta_{T}/\eta \sim \frac{S_{\nu}}{L_{x}L_{z}}\,Re_{G}^{-{1}/{4}}, \end{equation}

where $S_{\nu }$ is the finite surface area of the effectively smooth viscous superlayer of the TNTI. Our generalised model with $c (Re_{G},\omega _{th}^{2}/\omega _{ref}^{2}) \approx Re_{G}\, \tilde {c} (\omega _{th}^{2}/\omega _{ref}^{2})$, and (5.8) at the very smallest enstrophy levels and $c=1$ above those enstrophy levels, implies that $\eta _{T}$ is a monotonically increasing function of $\omega _{th}^{2}/\omega _{ref}^{2}$ with a finite value different from $\eta$ by a factor $Re_{G}^{-1/4}$ at the very smallest enstrophy thresholds. The exponent $1/4$ being small, this prediction is not easy to check as it requires numerical oscillation-free calculations at low enstrophy thresholds for many highly resolved DNS of temporally developing turbulent jets over a wide range of Reynolds numbers $Re_G$. (See Appendix B for some details about higher Reynolds number simulations and the importance of spatial resolution.) This is at, and perhaps even beyond, the very limit of the most powerful current computational capabilities and therefore beyond the present paper's scope. Such a computational check would also require a computable definition or surrogate for $\eta _T$, which we make a first attempt to give in the next couple of paragraphs. Before doing so, however, we point out that Silva et al. (Reference Silva, Zecchetto and da Silva2018) argued that the viscous superlayer thickness scales with the Kolmogorov length if $Re_{\lambda }$ is larger than approximately $200$, and that the TNTI layer's characteristic sizes may have different scalings at smaller values of $Re_{\lambda }$ depending on presence or absence of mean shear (see da Silva & Taveira (Reference da Silva and Taveira2010) and references therein). It must be stressed that the definition of the viscous superlayer used by Silva et al. (Reference Silva, Zecchetto and da Silva2018) does not necessarily include some low iso-enstrophy surfaces with fractal dimensions clearly larger than $2$ (see the discussion around figure 11 in § 5.4) and, more importantly, is not local in enstrophy threshold (i.e. it does not depend on the local position within the TNTI) and is therefore different from $\eta _{T}$, which is local in enstrophy threshold. The scaling (5.14) does not necessarily contradict the scalings in Silva et al. (Reference Silva, Zecchetto and da Silva2018) as they concern different quantities.

We close this section with an interpretation of the generalised Corrsin length $\eta _T$. As $\eta _T$ is local in terms of iso-enstrophy levels within the TNTI, and as it expresses some kind of thickness of iso-enstrophy surfaces, it appears natural to compare it with some average enstrophy length scale on the TNTI. To this end, we use enstrophy profiles conditioned on the interface location similar to Bisset, Hunt & Rogers (Reference Bisset, Hunt and Rogers2002). We define a local coordinate system with local coordinate $y_I$ chosen along the local unit normal ${\boldsymbol n} = -({\boldsymbol {\nabla } \omega ^2}/{|\boldsymbol {\nabla } \omega ^2|})$, which is pointing towards the non-turbulent region. The origin $y_I = 0$ of this local coordinate system is placed at a given location within the TNTI, for example on the isosurface defined by $\omega ^2_{th}/ \omega _{ref}^2 = 10^{-6}$, located at the very edge of the TNTI neighbouring the non-turbulent region. In this way, positive values of $y_I$ correspond to the very edge of the viscous superlayer and the non-turbulent region, whereas negative values of $y_I$ are within the TNTI and the turbulent region. Given such local coordinate systems on the TNTI, we calculate averages of any quantity $\phi$ at a given $y_I$ over all locations on the TNTI where the local $y_I$ axis does not cross the TNTI more than once in the range $y_I\in [-27\eta, +27\eta ]$. We use the notation $\phi _{I}$ to denote these average quantities, averaged conditionally on the specified isosurface location.

Figure 20 shows the vorticity magnitude, $\vert\omega\vert_I$, and the enstrophy profile, $\omega_I^2$, averaged conditionally on the distance from the enstrophy isosurface $\omega _{th}^{2}/\omega _{ref}^{2}=10^{-6}$; the profiles are normalised by the average values of the respective quantities at the centreplane. The drastic change of both vorticity and enstrophy values in a very short distance is visible, as shown previously in studies using similar methods, e.g. Nagata et al. (Reference Nagata, Watanabe and Nagata2018), Silva et al. (Reference Silva, Zecchetto and da Silva2018) and Watanabe, da Silva & Nagata (Reference Watanabe, da Silva and Nagata2019).

Figure 20. Vorticity magnitude and enstrophy values averaged conditionally on the distance from the iso-enstrophy surface defined by $\omega _{th}^{2}/\omega _{ref}^{2}=10^{-6}$ for the simulation PJ1.

We define the local length $\eta _{\omega } \equiv (({{\rm d}\omega ^2_{I}}/{{\rm d}y_{I}}) ({1}/{\omega ^2_{I}}))^{-1}$. In figure 21(a), we plot $\eta _{\omega }/\eta$ versus $\omega ^{2}_{I}/\omega _{ref}^{2}$. In agreement with $\eta _{T}$, $\eta _{\omega }$ is an increasing function of enstrophy, $\omega ^{2}_{I}/\omega _{ref}^{2}$ in this case: iso-enstrophy surfaces get further away from each other on average as $\omega ^{2}_{I}/\omega _{ref}^{2}$ increases within the TNTI. At the very smallest enstrophy thresholds, $\eta _{\omega }$ appears to tend to a finite value that is significantly smaller than $\eta$, which is also in agreement with $\eta _{T}$ at high enough $Re_G$ (see (5.14)).

Figure 21. (a) Plot of $\eta _{\omega }/\eta$ versus $\omega ^{2}_{I}/\omega _{ref}^{2}$ for $t/T_{ref}=50$, PJ1 simulation. This plot is typical of all times $t/T_{ref}$ between $30$ and $100$. (b) Profile of $\eta _{\omega }$ along $y_{I}/\eta$, with $y_{I}=0$ at $\omega ^2_{th}/ \omega _{ref}^2 = 10^{-6}$.

We also plot $\eta _{\omega }/\eta$ versus $y_{I}/\eta$ in figure 21(b), where $y_{I}=0$ corresponds to the iso-enstrophy surface $\omega ^2_{th}/ \omega _{ref}^2 = 10^{-6}$. We see that the profile of $\eta _{\omega }$ along $y_I$ is decreasing exponentially with increasing $y_I$. The linear region ends near $y_I/\eta \approx -2.5$. This is due to some points where the normal enstrophy profiles do not decrease monotonically to zero when going towards the non-turbulent region, even though the local enstrophy values always remain lower than the threshold value.