4.1 Local spherical volume averages

In order to compute the turbulent characteristics near the TNTI layer, we use local volume averages in a sphere, denoted by an overbar, where for each sphere of radius $r$ centred at point $\boldsymbol{x}$ , a spherical volume average of a variable $\unicode[STIX]{x1D719}(\boldsymbol{x})$ is defined by (Watanabe & Nagata Reference Watanabe and Nagata2016),

(4.1) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D719}}(\boldsymbol{x},r)=\frac{\displaystyle \iiint G(\boldsymbol{x},\boldsymbol{x}^{\prime },r)\unicode[STIX]{x1D719}(\boldsymbol{x}^{\prime })\,\text{d}\boldsymbol{x}^{\prime }}{V_{R}}=\frac{1}{N_{p}}\mathop{\sum }_{n=1}^{N_{p}}\unicode[STIX]{x1D719}[\boldsymbol{x}^{(n)}], & & \displaystyle\end{eqnarray}$$

where $V_{R}(r)$ is the volume of the sphere and $G(\boldsymbol{x})$ is a kernel function which is equal to 1 and 0 inside and outside of the sphere, respectively. This average corresponds to a low-pass filter operation with a cutoff length proportional to $r$ (Pumir, Shraiman & Chertkov Reference Pumir, Shraiman and Chertkov2001), whereby $\unicode[STIX]{x1D719}(\boldsymbol{x})$ can be decomposed into large-scale ( $\overline{\unicode[STIX]{x1D719}}$ ) and small-scale components ( $\unicode[STIX]{x1D719}^{\prime \prime }$ ), i.e. $\unicode[STIX]{x1D719}(\boldsymbol{x})=\overline{\unicode[STIX]{x1D719}}(\boldsymbol{x},r)+\unicode[STIX]{x1D719}^{\prime \prime }(\boldsymbol{x},r)$ . Here, a single length scale $r$ is used without distinguishing orientations.

In practice, the computation is carried out by taking $N_{p}$ fluid points randomly placed within the sphere, and by interpolating the value of the fluid variable $\unicode[STIX]{x1D719}(\boldsymbol{x})$ on each point $\boldsymbol{x}^{(n)}$ using a tri-linear interpolation, as indicated in the last term of the equation above. The number of points $N_{p}$ , used for each sphere of radius $r$ , is set as $N_{p}=\text{min}[(r/20\unicode[STIX]{x1D702})^{3},N_{lim}]$ , where the parameter $N_{lim}$ is chosen to limit the computational cost while $(r/20\unicode[STIX]{x1D702})^{3}$ ensures that the computational method used here is equivalent to the definition of the local volume average. When $(r/20\unicode[STIX]{x1D702})^{3}$ is smaller than 20, $N_{p}=20$ is used instead of $(r/20\unicode[STIX]{x1D702})^{3}$ . Additionally, an ensemble average of spheres, $\langle \overline{\unicode[STIX]{x1D719}}\rangle (r)$ , can be taken by using a large number of equally sized spheres located at different points $\boldsymbol{x}$ within the computational domain.

Figure 3.

(a) Cumulative kinetic energy $\langle k_{r}\rangle$ normalized by the turbulent kinetic energy $k_{T}$ in cases HIT1 and HIT2 for several values of $N_{lim}$ . The theoretical curves $\unicode[STIX]{x1D6FD}\langle \unicode[STIX]{x1D700}\rangle ^{2/3}r^{2/3}$ (with $\unicode[STIX]{x1D6FD}=2$ ) are also shown. (b) Contribution to the kinetic energy from scale $r$ , $rE_{r}$ , normalized by the turbulent kinetic energy $k_{T}$ .

We start by showing that using the new spherical volume average metric, one recovers the classical results observed in homogeneous isotropic turbulence. Specifically, we consider the kinetic energy averaged in a sphere, $k(\boldsymbol{x},r)=\overline{u_{i}(\boldsymbol{x})u_{i}(\boldsymbol{x})}/2$ , split into two contributions resulting from the spherical average: $k(\boldsymbol{x},r)=k_{m}(\boldsymbol{x},r)+k_{r}(\boldsymbol{x},r)$ , where $k_{m}(\boldsymbol{x},r)=\overline{u}_{i}\overline{u}_{i}/2$ involves the contribution from the sphere’s centre-of-mass (mean) motion, whereas $k_{r}(\boldsymbol{x},r)=\overline{u_{i}^{\prime \prime }u_{i}^{\prime \prime }}/2$ represents the contribution from (relative) motion in relation to the sphere’s centre of mass. From these definitions, one can interpret the ensemble average of $k_{r}(\boldsymbol{x},r)$ as the cumulative kinetic energy held in scales smaller than $r$ , such that $\langle k_{r}\rangle$ for large $r$ is equal to the turbulent kinetic energy $k_{T}=\langle u_{i}^{\prime }u_{i}^{\prime }\rangle /2$ . The brackets $\langle \,\,\rangle$ represent a spatial average. For computing $\langle k_{r}\rangle$ in homogeneous isotropic turbulence, the spatial average is taken with a large number of spheres (centred at different locations in the computational domain). Furthermore, a kinetic energy density at scale $r$ can be defined by $E_{r}=\unicode[STIX]{x2202}\langle k_{r}\rangle /\unicode[STIX]{x2202}r$ , whose integral gives the total kinetic energy as $k_{T}=\int _{0}^{\infty }E_{r}\,\text{d}r$ (Townsend Reference Townsend1976).

Figure 3(a) shows $\langle k_{r}\rangle$ in homogeneous isotropic turbulence (cases HIT1 and HIT2) for three different values of $N_{lim}$ ; $\langle k_{r}\rangle$ increases with the radius of the sphere $r$ as expected from the definition, and for an intermediate range of $r$ , it closely follows the inertial range law $\langle k_{r}\rangle =\unicode[STIX]{x1D6FD}\langle \unicode[STIX]{x1D700}\rangle ^{2/3}r^{2/3}$ (with $\unicode[STIX]{x1D6FD}=2$ ) predicted by Kolmogorov for the second-order velocity structure function (Davidson Reference Davidson2004), which is also a measure of the cumulative kinetic energy (Townsend Reference Townsend1976). Here, the broken line that represents $\langle k_{r}\rangle =\unicode[STIX]{x1D6FD}\langle \unicode[STIX]{x1D700}\rangle ^{2/3}r^{2/3}$ is obtained with $\langle \unicode[STIX]{x1D700}\rangle$ computed from the DNS results. The scale range with $\langle k_{r}\rangle =\unicode[STIX]{x1D6FD}\langle \unicode[STIX]{x1D700}\rangle ^{2/3}r^{2/3}$ is wider for HIT2, which has a higher Reynolds number than HIT1. It is clear that $\langle k_{r}\rangle$ is quite insensitive to the value of $N_{lim}$ provided that this number is sufficiently high, and a value of $N_{lim}=6000$ is therefore used for the rest of the paper.

Figure 3(b) shows $rE_{r}$ in semi-log coordinates, which represents the contribution to the turbulent kinetic energy from scale $r$ . It is expected that $rE_{r}$ peaks at the scale $r=L$ (Davidson Reference Davidson2004), where $L$ is the integral scale defined as $L=\int _{0}^{\infty }f(r)\,\text{d}r$ , where $f(r)=\langle u^{\prime }(x)u^{\prime }(x+r)\rangle /\langle {u^{\prime }}^{2}(x)\rangle$ is the longitudinal auto-correlation function. In the present simulations, $rE_{r}$ attain their maxima at $r=140\unicode[STIX]{x1D702}$ and $r=270\unicode[STIX]{x1D702}$ , for cases HIT1 and HIT2, respectively, which compare well with $L=140\unicode[STIX]{x1D702}$ and $L=265\unicode[STIX]{x1D702}$ (represented by the vertical lines in figure 3 b) directly obtained by integrating $f(r)$ for cases HIT1 and HIT2, respectively. Thus, we confirm using the present DNS that the integral scale can be obtained as the scale $r=L$ where $rE_{r}$ attains its maximum.

4.2 Conditional spherical volume averages

The local spherical volume averages are computed near the TNTI layer in relation to the position of the irrotational boundary. A volume average of ‘turbulent fluids’ on the local coordinate system is defined by extending the spherical volume average as

(4.2) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D701}_{I},r)=\frac{\displaystyle \iiint I(\boldsymbol{x}^{\prime })G(\unicode[STIX]{x1D701}_{I},\boldsymbol{x}^{\prime },r)\unicode[STIX]{x1D719}(\boldsymbol{x}^{\prime })\,\text{d}\boldsymbol{x}^{\prime }}{\displaystyle \iiint I(\boldsymbol{x}^{\prime })G(\unicode[STIX]{x1D701}_{I},\boldsymbol{x}^{\prime },r)\,\text{d}\boldsymbol{x}^{\prime }}, & & \displaystyle\end{eqnarray}$$

where $I(\boldsymbol{x})$ is equal to 1 and 0 when the coordinate $\boldsymbol{x}$ is taken from the turbulent and non-turbulent regions, respectively. Introducing $\unicode[STIX]{x1D701}_{I}$ and $I(\boldsymbol{x})$ allows us to investigate the statistical properties of turbulence removing the contribution from non-turbulent fluid regions. An ensemble average is calculated with a large number of spheres (with radius $r$ ) whose centres are located at a fixed distance $\unicode[STIX]{x1D701}_{I}$ from the irrotational boundary positions, where this average is denoted by $\langle \,\,\rangle _{RI}$ . The cumulative kinetic energy held below scale $r$ at a distance of $\unicode[STIX]{x1D701}_{I}$ from the irrotational boundary is denoted by $\langle k_{r}\rangle _{RI}(\unicode[STIX]{x1D701}_{I},r)$ . The advantage of the present metric is that it allows us to simultaneously assess the turbulent statistics as a function of the scale $r$ and of the position $\unicode[STIX]{x1D701}_{I}$ in relation to the TNTI. A classic low-pass filtering operation could also be used to perform a scale separation, however the filtering procedure would include flow points from both the turbulent and non-turbulent regions. Here we are interested in analysing the turbulent fluid near or within the TNTI layer while a sphere of radius $r$ at some location $\boldsymbol{x}$ may intersect the irrotational boundary at several points. Therefore, here the non-turbulent fluid is removed from these spherical averages. Thus, we can focus on the turbulent characteristics of the turbulent fluid near the irrotational boundary.

Figure 4.

Conditional mean vorticity magnitude $\langle |\unicode[STIX]{x1D74E}|\rangle _{I}$ and its derivative with respective to $\unicode[STIX]{x1D701}_{I}$ , $\unicode[STIX]{x1D714}_{I}^{\prime }=\text{d}\langle |\unicode[STIX]{x1D74E}|\rangle _{I}/\text{d}\unicode[STIX]{x1D701}_{I}$ in PJET. The thickness of the TNTI layer is denoted by $\unicode[STIX]{x1D6FF}_{I}$ .

Figure 5.

Conditional averages of enstrophy production term $\langle P_{\unicode[STIX]{x1D714}}\rangle _{I}$ and diffusion term $\langle D_{\unicode[STIX]{x1D714}}\rangle _{I}$ in PJET. The thicknesses of the turbulent sublayer and viscous superlayer are denoted by $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ and $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ , respectively.

5.1 Conditional statistics near the TNTI layer

Figure 4 shows the conditional average of vorticity magnitude $\langle |\unicode[STIX]{x1D74E}|\rangle _{I}$ and its derivative $\unicode[STIX]{x1D714}_{I}^{\prime }=\text{d}\langle |\unicode[STIX]{x1D74E}|\rangle _{I}/\text{d}\unicode[STIX]{x1D701}_{I}$ with a negative sign in PJET; $\langle |\unicode[STIX]{x1D74E}|\rangle _{I}$ rapidly increases from the non-turbulent region ( $\unicode[STIX]{x1D701}_{I}>0$ ) toward the turbulent region ( $\unicode[STIX]{x1D701}_{I}<0$ ), and a peak of $-\unicode[STIX]{x1D714}_{I}^{\prime }$ appears at $\unicode[STIX]{x1D701}_{I}=-0.016H$ . The TNTI layer can be defined as the region across which the vorticity magnitude rapidly increases from the non-turbulent region toward the turbulent region. Following a previous study (Nagata, Watanabe & Nagata Reference Nagata, Watanabe and Nagata2018), the mean extent of the TNTI layer on the local coordinate can be quantified based on $\unicode[STIX]{x1D714}_{I}^{\prime }$ . Here, the outer edge of the TNTI layer is located at $\unicode[STIX]{x1D701}_{I}=0$ for the present choice of the threshold. The inner edge of the TNTI layer can be identified as the location at which $-\unicode[STIX]{x1D714}_{I}^{\prime }$ is equal to 10 % of the maximum value of $-\unicode[STIX]{x1D714}_{I}^{\prime }$ , $[-\unicode[STIX]{x1D714}_{I}^{\prime }]_{max}$ . The length between the outer and inner edges of the TNTI layer yields the thickness of the TNTI layer, $\unicode[STIX]{x1D6FF}_{I}$ , as shown in figure 4, where $\unicode[STIX]{x1D6FF}_{I}=0.055H$ is obtained. This method for estimating the TNTI layer thickness $\unicode[STIX]{x1D6FF}_{I}$ was compared with another method based on fitting an error function to conditional statistics in Watanabe et al. (Reference Watanabe, Riley, Nagata, Onishi and Matsuda2018a ), where both methods were shown to give a similar value of $\unicode[STIX]{x1D6FF}_{I}$ .

The inner structures of the TNTI layer, called the viscous superlayer and turbulent sublayer, are often discussed with respect to vorticity dynamics (van Reeuwijk & Holzner Reference van Reeuwijk and Holzner2014; da Silva et al. Reference da Silva, Hunt, Eames and Westerweel2014; Jahanbakhshi & Madnia Reference Jahanbakhshi and Madnia2018; Silva et al. Reference Silva, Zecchetto and da Silva2018). These inner layers can be detected by analysing the enstrophy transport equation

(5.1) $$\begin{eqnarray}\displaystyle \frac{D\unicode[STIX]{x1D714}^{2}/2}{Dt}=\unicode[STIX]{x1D714}_{i}S_{ij}\unicode[STIX]{x1D714}_{j}+\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FB}^{2}(\unicode[STIX]{x1D714}^{2}/2)-\unicode[STIX]{x1D708}\unicode[STIX]{x1D735}\unicode[STIX]{x1D714}_{i}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D714}_{i}=P_{\unicode[STIX]{x1D714}}+D_{\unicode[STIX]{x1D714}}+\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D714}}, & & \displaystyle\end{eqnarray}$$

where $P_{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D714}_{i}S_{ij}\unicode[STIX]{x1D714}_{j}$ is the production term, $D_{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FB}^{2}(\unicode[STIX]{x1D714}^{2}/2)$ is the viscous diffusion term and $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D714}}=-\unicode[STIX]{x1D708}\unicode[STIX]{x1D735}\unicode[STIX]{x1D714}_{i}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D714}_{i}$ is the viscous dissipation term. When non-turbulent fluid is entrained into the TNTI layer, $\unicode[STIX]{x1D714}^{2}/2$ increases on the entrained fluid path through $P_{\unicode[STIX]{x1D714}}$ and $D_{\unicode[STIX]{x1D714}}$ . The conditional averages of $P_{\unicode[STIX]{x1D714}}$ and $D_{\unicode[STIX]{x1D714}}$ are shown for PJET in figure 5. Note that the conditional averages of $|\unicode[STIX]{x1D74E}|$ , $P_{\unicode[STIX]{x1D714}}$ and $D_{\unicode[STIX]{x1D714}}$ in a mixing layer and shear free turbulence are also similar to those in a planar jet (Watanabe et al. Reference Watanabe, Sakai, Nagata, Ito and Hayase2015, Reference Watanabe, da Silva, Nagata and Sakai2017c ). Near the irrotational boundary ( $\unicode[STIX]{x1D701}_{I}\approx 0$ ), there is a region with $\langle P_{\unicode[STIX]{x1D714}}\rangle _{I}\leqslant \langle D_{\unicode[STIX]{x1D714}}\rangle _{I}$ , where the enstrophy growth is dominated by the viscous diffusion. This region is called the viscous superlayer (Taveira & da Silva Reference Taveira and da Silva2014). On the other hand, $\langle P_{\unicode[STIX]{x1D714}}\rangle _{I}$ exceeds $\langle D_{\unicode[STIX]{x1D714}}\rangle _{I}$ in the inner part of the TNTI layer, which is called turbulent sublayer (Silva et al. Reference Silva, Zecchetto and da Silva2018). The mean thickness of the viscous superlayer $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ can be obtained as the distance from $\unicode[STIX]{x1D701}_{I}=0$ to the location of $\langle P_{\unicode[STIX]{x1D714}}\rangle _{I}=\langle D_{\unicode[STIX]{x1D714}}\rangle _{I}$ . Then, the mean thickness of the turbulent sublayer, $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ , can be obtained as $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D6FF}_{I}-\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ . Values of $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ and $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ divided by Kolmogorov scale $\unicode[STIX]{x1D702}$ on the centreline ( $y=0$ ) are shown in table 1. Similar values of $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}/\unicode[STIX]{x1D702}$ and $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D702}$ were also reported in various flows (Silva et al. Reference Silva, Zecchetto and da Silva2018; Zhang et al. Reference Zhang, Watanabe and Nagata2018; Nagata et al. Reference Nagata, Watanabe and Nagata2018), and agree with the scaling of the TNTI layer at high Reynolds numbers and of its sublayers, recently analysed in Silva et al. (Reference Silva, Zecchetto and da Silva2018).

Based on $\unicode[STIX]{x1D6FF}_{I}$ , $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$ and $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ obtained in this way, the TNTI layer, viscous superlayer and turbulent sublayer, can be identified in the plots of conditional statistics as shown in figures 4 and 5. Furthermore, the turbulent region for $\unicode[STIX]{x1D701}_{I}\leqslant -\unicode[STIX]{x1D6FF}_{I}$ is called a turbulent core region in figure 5. Then, the local coordinate $\unicode[STIX]{x1D701}_{I}$ is divided into four regions: non-turbulent region, viscous superlayer, turbulent sublayer and turbulent core region. Identifying these regions is useful to interpret various statistics obtained near the TNTI layer. For example, the boundary between the viscous superlayer and turbulent sublayer was shown to have a large scalar dissipation rate and a high production rate of the scalar dissipation rate (Watanabe et al. Reference Watanabe, Sakai, Nagata, Ito and Hayase2015). The movement of entrained fluid particles with respective to the irrotational boundary location was also shown to have different scalings between the turbulent sublayer and viscous superlayer (Watanabe et al. Reference Watanabe, da Silva, Sakai, Nagata and Hayase2016b ). The turbulent sublayer also exhibits different statistical properties from the turbulent core region (e.g. alignment between the vorticity vector and eigenvectors of the strain tensor (Watanabe et al. Reference Watanabe, Sakai, Nagata, Ito and Hayase2014)). For this reason, the conditional statistics are presented with marks that represent the boundaries between these four regions. It should be noted that the viscous terms are not dominant in the kinetic energy budget within the viscous superlayer. Pressure diffusion of kinetic energy also makes an important contribution to the kinetic energy budget within the TNTI layer (Taveira & da Silva Reference Taveira and da Silva2013; Terashima et al. Reference Terashima, Sakai, Nagata, Ito, Onishi and Shouji2016; Watanabe et al. Reference Watanabe, da Silva and Nagata2016a ).

Figure 6.

Conditional average of relative kinetic energy $\langle k_{r}\rangle _{RI}$ as a function of the distance $\unicode[STIX]{x1D701}_{I}$ from the irrotational boundary and the size of the sphere $r$ for shear free turbulence: (a) dependence on $r$ for several distances from the irrotational boundary, $-20\leqslant \unicode[STIX]{x1D701}_{I}/\unicode[STIX]{x1D702}\leqslant -0.5$ (the conditional energy density $rE_{rI}$ is also shown); (b) contours of $\langle k_{r}\rangle _{RI}$ on $(\unicode[STIX]{x1D701}_{I},r)$ for $-20\leqslant \unicode[STIX]{x1D701}_{I}/\unicode[STIX]{x1D702}\leqslant -0.5$ , and $10^{0}\leqslant r/\unicode[STIX]{x1D702}\leqslant 10^{3}$ . $\langle k_{r}\rangle _{RI}$ is normalized by the turbulent kinetic energy on the centreline of turbulence, $k_{T0}$ .

Figure 7.

Conditional average of relative kinetic energy $\langle k_{r}\rangle _{RI}$ and conditional energy density $rE_{rI}$ in cases (a) PJET and (b) ML.

5.2 Conditional statistics with volume average near the TNTI layer

The conditional statistics are computed with the volume average (4.2) in the shear free turbulence, planar jet and mixing layer cases. Figure 6(a) plots the cumulative kinetic energy $\langle k_{r}\rangle _{RI}$ against $r/\unicode[STIX]{x1D702}$ obtained at different locations on $\unicode[STIX]{x1D701}_{I}$ in shear free turbulence while figure 6(b) shows a colour contour of $\langle k_{r}\rangle _{RI}$ as a function of $(r/\unicode[STIX]{x1D702},\unicode[STIX]{x1D701}_{I}/\unicode[STIX]{x1D702})$ . For $\unicode[STIX]{x1D701}_{I}/\unicode[STIX]{x1D702}>-10$ , $\langle k_{r}\rangle _{RI}$ for small $r$ notably decreases with $\unicode[STIX]{x1D701}_{I}$ , which indicates a depletion of kinetic energy at small scales within the TNTI layer. da Silva & Pereira (Reference da Silva and Pereira2008) showed that the irrotational motions outside the irrotational boundary are dominated by the large scales. However, the cumulative kinetic energy further shows that the energy distribution across scales begins to change within the TNTI layer. As in homogeneous isotropic turbulence, the integral scale of turbulence near the TNTI layer is estimated using the kinetic energy density $E_{rI}=\unicode[STIX]{x2202}\langle k_{r}\rangle _{RI}/\unicode[STIX]{x2202}r$ premultiplied by $r$ as shown in figure 6(a). The peak location of $rE_{r}$ that yields the integral length scale near the TNTI layer, $L_{I}$ , is insensitive to $\unicode[STIX]{x1D701}_{I}$ , in agreement with the rapid distortion theory applied for the TNTI layer (Teixeira & da Silva Reference Teixeira and da Silva2012).

Figure 7 shows the cumulative kinetic energy $\langle k_{r}\rangle _{RI}$ and kinetic energy density $rE_{r}$ near the TNTI layer for cases PJET and ML. Here, the range of $r$ is chosen to present the results for the length scales from the Kolmogorov scale $r/\unicode[STIX]{x1D702}=1$ to the length scale of the peak in $rE_{r}$ . The dependence of $\langle k_{r}\rangle _{RI}$ and $rE_{r}$ on $r$ and $\unicode[STIX]{x1D701}_{I}$ is similar for all flows. The cumulative kinetic energy $\langle k_{r}\rangle _{RI}$ in small scales decreases as $\unicode[STIX]{x1D701}_{I}$ becomes close to 0, and small-scale motions are less active as the irrotational boundary $\unicode[STIX]{x1D701}_{I}=0$ is approached from the turbulent region. The scale at which $rE_{r}$ peaks hardly depends on $\unicode[STIX]{x1D701}_{I}$ in the planar jet and mixing layer. The integral length scale $L_{I}$ at each $\unicode[STIX]{x1D701}_{I}$ can be obtained as the length $r$ at which $rE_{r}$ peaks, where $rE_{r}$ is computed for a wide range of $r$ and $\unicode[STIX]{x1D701}_{I}$ . However, $L_{I}$ does not change with $\unicode[STIX]{x1D701}_{I}$ as confirmed from the peak in $rE_{r}$ in figures 6(a) and 7. The values of $L_{I}$ in the shear free turbulence, planar jet and mixing layer are $L_{I}=1.9L_{0}$ , $L_{I}=1.0H=2.0L_{0}$ and $L_{I}=9.9h_{0}=2.3L_{0}$ , respectively, where $L_{0}$ is the integral length scale at $y=0$ , obtained by integrating the longitudinal auto-correlation function of the velocity fluctuation in the $x$ direction.

Figure 8.

Conditional statistics in relation to the irrotational boundary: (a) averaged viscous dissipation rate $\unicode[STIX]{x1D700}$ ; (b) turbulent kinetic energy $k_{TI}$ ; (c) turbulent Reynolds number $Re_{\unicode[STIX]{x1D706}}$ ; (d) non-dimensional dissipation rate $C_{\unicode[STIX]{x1D700}}$ . Different lines highlight the two sublayers that exist within the TNTI layer between the turbulent core (TC) and the non-turbulent region (NT): the viscous superlayer (VSL) and turbulent sublayer (TSL). Subscript 0 denotes the value taken at the centre of turbulence ( $y=0$ ).

5.3 Non-dimensional dissipation rate near the TNTI layer

The non-dimensional dissipation rate in free shear flows has been studied on the centreline of the flow, where it is calculated with the averaged energy dissipation rate and turbulent kinetic energy (or root-mean-squared streamwise velocity fluctuation) defined as one-point statistics and with the integral length scale computed with an auto-correlation function (Antonia et al. Reference Antonia, Satyaprakash and Hussain1980; Deo et al. Reference Deo, Mi and Nathan2008, Reference Deo, Nathan and Mi2013; Takamure et al. Reference Takamure, Sakai, Ito, Iwano and Hayase2019). Similarly, the non-dimensional dissipation rate near the TNTI layer is calculated with the averaged kinetic energy dissipation rate $\langle \unicode[STIX]{x1D700}\rangle _{I}$ , turbulent kinetic energy $k_{TI}=(\langle u_{j}u_{j}\rangle _{I}-\langle u_{j}\rangle _{I}\langle u_{j}\rangle _{I})/2$ and the integral length scale $L_{I}$ , where $\langle \unicode[STIX]{x1D700}\rangle _{I}$ and $k_{TI}$ are obtained as one-point statistics on the local coordinate. Figure 8(a–c) shows $\langle \unicode[STIX]{x1D700}\rangle _{I}$ , $k_{TI}$ and the turbulent Reynolds number $Re_{\unicode[STIX]{x1D706}}=(2k_{TI}/3)^{1/2}\unicode[STIX]{x1D706}_{I}/\unicode[STIX]{x1D708}$ near the TNTI layer in the shear free turbulence, mixing layer and planar jet, where $\unicode[STIX]{x1D706}_{I}=\left(10\unicode[STIX]{x1D708}k_{TI}/\langle \unicode[STIX]{x1D700}\rangle _{I}\right)^{1/2}$ is the Taylor microscale. Note that the volume average (4.2) is not used for these quantities, and the conditional averages are simply taken on the local coordinate. Both $\langle \unicode[STIX]{x1D700}\rangle _{I}$ and $k_{TI}$ decrease when approaching the irrotational boundary from the turbulent region. However, $Re_{\unicode[STIX]{x1D706}}$ sharply increases in the viscous superlayer region and in the non-turbulent region, in agreement with Teixeira & da Silva (Reference Teixeira and da Silva2012).

The results from figures 8(a–c), 6(a) and 7 allow us to observe for the first time the non-dimensional energy dissipation rate $C_{\unicode[STIX]{x1D700}}$ near the TNTI layer. Figure 8(d) shows the profile of $C_{\unicode[STIX]{x1D700}}$ , which is computed directly from its definition as $\langle \unicode[STIX]{x1D700}\rangle _{I}L_{I}/(2k_{TI}/3)^{3/2}$ . Here, $C_{\unicode[STIX]{x1D700}}$ is shown for the turbulent region since the integral length scale is obtained as a length scale of turbulence. In the turbulent core region, $C_{\unicode[STIX]{x1D700}}$ is roughly constant in the three flows with $1.0\lesssim \langle C_{\unicode[STIX]{x1D700}}\rangle _{I}\lesssim 1.5$ , which is close to the value of $C_{\unicode[STIX]{x1D700}}\approx 1.0$ observed for decaying isotropic turbulence (Sreenivasan Reference Sreenivasan1984). However, $C_{\unicode[STIX]{x1D700}}$ exhibits a peculiar evolution near the TNTI in all the flows studied here, rising into a peak roughly at the middle of the turbulent sublayer at $\unicode[STIX]{x1D701}_{I}/\unicode[STIX]{x1D702}\approx -10$ , before sharply decreasing inside the viscous superlayer.

Figure 9.

Normalized energy dissipation rate $C_{\unicode[STIX]{x1D700}}$ as a function of the turbulent Reynolds number $Re_{\unicode[STIX]{x1D706}}$ in the viscous superlayer and turbulent sublayer: (a) a logarithmic plot of $(C_{\unicode[STIX]{x1D700}},Re_{\unicode[STIX]{x1D706}})$ ; (b) a linear plot of $(C_{\unicode[STIX]{x1D700}}Re_{\unicode[STIX]{x1D706}}^{2},Re_{\unicode[STIX]{x1D706}})$ . Different lines are used for the turbulent sublayer (TSL) and viscous superlayer (VSL).

The spatial variation of $C_{\unicode[STIX]{x1D700}}$ is investigated in relation to the turbulent Reynolds number within the TNTI layer. Figure 9(a) plots $C_{\unicode[STIX]{x1D700}}$ against $Re_{\unicode[STIX]{x1D706}}$ in the turbulent sublayer and viscous superlayer. As $Re_{\unicode[STIX]{x1D706}}$ decreases from the viscous superlayer toward the turbulent sublayer, $C_{\unicode[STIX]{x1D700}}$ increases, where a power law $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-2}$ is clearly found for all flows. Figure 9(b) plots $C_{\unicode[STIX]{x1D700}}$ premultiplied by $Re_{\unicode[STIX]{x1D706}}^{2}$ against $Re_{\unicode[STIX]{x1D706}}$ to confirm the relation $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-2}$ , which appears as a horizontal line, and shows that the relation $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-2}$ holds from the viscous superlayer to the middle of the turbulent sublayer. Clearly the relation $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-2}$ arises mainly from the viscous superlayer, where viscous effects dominate enstrophy. The fact that the same relation is observed in a part of the turbulent sublayer is not easy to explain. The concepts of the viscous superlayer and turbulent sublayer are defined in relation to the enstrophy, which is a small-scale quantity, unlike the velocity, which impacts in the definition of $C_{\unicode[STIX]{x1D700}}$ . Clearly, from its definition, the viscous superlayer is dominated by enstrophy viscous diffusion. On the other hand, enstrophy production is dominant in the enstrophy growth in the turbulent sublayer. However, in practice there is of course a small overlap between the two effects – viscous diffusion and production – at the boundary between the viscous superlayer and turbulent sublayer, i.e. viscous effects do not end abruptly when we move from the viscous superlayer into the turbulent sublayer. To complicate matters, $C_{\unicode[STIX]{x1D700}}$ depends not only on the small scales but also on the velocity fluctuations associated with the large scales of motion with no direct link to the definition of the two sublayers, which is related to the small scales. The origin of $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-2}$ is discussed further below and in the Appendix.

Figure 10(a) plots $C_{\unicode[STIX]{x1D700}}$ against $Re_{\unicode[STIX]{x1D706}}$ in the turbulent sublayer and turbulent core region. At the local minimum of $Re_{\unicode[STIX]{x1D706}}$ found in the turbulent sublayer (in figure 8 c), the relation between $C_{\unicode[STIX]{x1D700}}$ and $Re_{\unicode[STIX]{x1D706}}$ is changed, and $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-1}$ follows in the rest of the turbulent sublayer. Figure 10(b) plots $C_{\unicode[STIX]{x1D700}}Re_{\unicode[STIX]{x1D706}}$ against $Re_{\unicode[STIX]{x1D706}}$ for confirming the relation $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-1}$ , and thus $C_{\unicode[STIX]{x1D700}}Re_{\unicode[STIX]{x1D706}}$ tends to be constant in the turbulent sublayer and turbulent core region. The relation $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-1}$ was also observed in non-equilibrium turbulence (Vassilicos Reference Vassilicos2015).

Figure 10.

Normalized energy dissipation rate $C_{\unicode[STIX]{x1D700}}$ as a function of the turbulent Reynolds number $Re_{\unicode[STIX]{x1D706}}$ in the turbulent sublayer and turbulent core region: (a) a logarithmic plot of $(C_{\unicode[STIX]{x1D700}},Re_{\unicode[STIX]{x1D706}})$ ; (b) a linear plot of $(C_{\unicode[STIX]{x1D700}}Re_{\unicode[STIX]{x1D706}},Re_{\unicode[STIX]{x1D706}})$ . Different lines are used for the turbulent core (TC) and turbulent sublayer (TSL).

In previous studies of non-equilibrium turbulence, the scaling $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-1}$ has been explained via the influence of the large-scale coherent structures (Goto & Vassilicos Reference Goto and Vassilicos2016) or as a non-equilibrium perturbation correction to the Kolmogorov equilibrium spectrum (Bos & Rubinstein Reference Bos and Rubinstein2017). In the analysis of the TNTI layer, the relation $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-1}$ is observed here for three very different flows, which have different large-scale structures and it is unclear how spectral spikes described in Goto & Vassilicos (Reference Goto and Vassilicos2016) could explain this scaling. Recall that in the shear free turbulence case, the typical large-scale eddies are somehow different from the ones observed in other flows, such as free shear flows. It is noteworthy that the energy spectra of streamwise velocity fluctuations in the temporally evolving mixing layer and planar jet, observed in previous studies, did not show any clear spikes at the large scales (Watanabe, Nagata & da Silva Reference Watanabe, Nagata and da Silva2017b ; Watanabe & Nagata Reference Watanabe and Nagata2017a ), but we cannot completely rule out the possible influence of these spikes that might exist in the early times of the simulations. On the other hand, the analysis considered in Bos & Rubinstein (Reference Bos and Rubinstein2017) is for relatively small perturbations, whereas the perturbations found within the turbulent sublayer are not likely small, since there are many possible reasons for perturbations being induced in the vicinity of the TNTI layer. For example, it was shown that the kinetic energy near the TNTI layer is transferred in physical space by pressure diffusion, whose contribution depends on the associated length scales (Taveira & da Silva Reference Taveira and da Silva2013; Terashima et al. Reference Terashima, Sakai, Nagata, Ito, Onishi and Shouji2016; Watanabe et al. Reference Watanabe, da Silva and Nagata2016a ). As suggested by an anonymous referee it is also possible that none of these explanations can be used to describe the observed relation $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-1}$ . What is clear at the moment is that this scaling law is robust and is consistently observed in the turbulent sublayer of very different flow configurations.

Figure 11.

Influences of the threshold $\unicode[STIX]{x1D714}_{th}$ used for detecting the irrotational boundary in PJET. $C_{\unicode[STIX]{x1D700}}$ plotted against (a)  $\unicode[STIX]{x1D701}_{I}/\unicode[STIX]{x1D702}$ and (b)  $Re_{\unicode[STIX]{x1D706}}$ for $\unicode[STIX]{x1D714}_{th}=0.9\unicode[STIX]{x1D714}_{th0}$ , $\unicode[STIX]{x1D714}_{th0}$ and $1.1\unicode[STIX]{x1D714}_{th0}$ , where $\unicode[STIX]{x1D714}_{th0}=0.18(U_{J}/H)$ is used in the present analysis.

Figure 6 shows that the integral length scale $L_{I}$ is almost constant near the TNTI layer. In figures 8(a) and 8(b), although both averaged energy dissipation rate $\langle \unicode[STIX]{x1D700}\rangle _{I}$ and turbulent kinetic energy $k_{TI}$ decrease in the turbulent region toward $\unicode[STIX]{x1D701}_{I}=0$ , $k_{TI}$ only weakly depends on $\unicode[STIX]{x1D701}_{I}$ within the viscous superlayer. The assumption of constant $L_{I}$ and $k_{TI}$ yields $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-2}$ from the definitions of $C_{\unicode[STIX]{x1D700}}$ and $Re_{\unicode[STIX]{x1D706}}$ , where $C_{\unicode[STIX]{x1D700}}=\langle \unicode[STIX]{x1D700}\rangle _{I}L_{I}/(2k_{TI}/3)^{3/2}$ and $Re_{\unicode[STIX]{x1D706}}=(20/3\unicode[STIX]{x1D708})^{1/2}k_{TI}\langle \unicode[STIX]{x1D700}\rangle _{I}^{-1/2}$ can be combined to give $C_{\unicode[STIX]{x1D700}}Re_{\unicode[STIX]{x1D706}}^{2}=CL_{I}\sqrt{k_{TI}}/\unicode[STIX]{x1D708}$ with constant $C$ . Therefore, the power law $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-2}$ observed in the viscous superlayer and turbulent sublayer is related to the faster decay of $\langle \unicode[STIX]{x1D700}\rangle _{I}$ with $\unicode[STIX]{x1D701}_{I}$ , than those of $L_{I}$ and $k_{TI}$ , as discussed in the Appendix.

Finally, figure 11 shows $C_{\unicode[STIX]{x1D700}}$ plotted with $\unicode[STIX]{x1D701}_{I}/\unicode[STIX]{x1D702}$ and $Re_{\unicode[STIX]{x1D706}}$ obtained for different thresholds $\unicode[STIX]{x1D714}_{th}$ for PJET. Here, the results are compared for $\unicode[STIX]{x1D714}_{th}=0.9\unicode[STIX]{x1D714}_{th0}$ , $\unicode[STIX]{x1D714}_{th0}$ and $1.1\unicode[STIX]{x1D714}_{th0}$ , where $\unicode[STIX]{x1D714}_{th0}=0.18(U_{J}/H)$ is used in the other figures. The threshold dependence is very small and we observe the same tendency in all these plots.