2.1 Numerical methods and computational parameters

Direct numerical simulations are performed for temporally evolving mixing layers and planar jets (Watanabe et al. Reference Watanabe, da Silva, Sakai, Nagata and Hayase2016). The computation domains with a size of $L_{x}\times L_{y}\times L_{z}$ are represented by $N_{x}\times N_{y}\times N_{z}$ grid points. The flows are periodic in the streamwise ( $x$ ) and spanwise ( $z$ ) directions, and spread in the cross-streamwise ( $y$ ) direction, where slip boundary conditions are applied. The origin of the coordinate system is located at the centre of the computational domain. The DNS code is an incompressible Navier–Stokes solver based on the fractional step method, and was used in our previous studies (Watanabe & Nagata Reference Watanabe and Nagata2016; Watanabe et al. Reference Watanabe, da Silva, Sakai, Nagata and Hayase2016). The governing equations are solved by using a fully conservative finite-difference method for spatial discretization and a third-order Runge–Kutta method for temporal advancement. Fourth-order and second-order central difference schemes are used in the periodic and cross-streamwise directions respectively. The Poisson equation for pressure is solved by using the Bi-CGSTAB method. For the mixing layers, a passive scalar $\unicode[STIX]{x1D719}$ , whose evolution is described by the convection–diffusion equation with the Schmidt number $Sc=\unicode[STIX]{x1D708}/D=1$ , is also simulated as in Watanabe et al. (Reference Watanabe, Sakai, Nagata, Ito and Hayase2015), where $\unicode[STIX]{x1D708}$ is the kinematic viscosity and $D$ is the molecular diffusivity. The grid is equidistant in the $x$ and $z$ directions, while in the $y$ direction, a fine grid is used near the centre of the mixing layers and planar jets. In addition to solving the Navier–Stokes equations, the fluid particles are simultaneously tracked using a third-order Runge–Kutta scheme for temporal advancement and the trilinear interpolation scheme as in previous Lagrangian studies (Schumacher Reference Schumacher2009; Watanabe et al. Reference Watanabe, da Silva, Sakai, Nagata and Hayase2016).

The simulations are initialized by statistically homogeneous and isotropic velocity fluctuations, which are superimposed onto the mean streamwise velocity given by $\langle U\rangle =0.5U_{M}\tanh (2y/\unicode[STIX]{x1D703}_{M})$ in mixing layers and $\langle U\rangle =0.5U_{J}+0.5U_{J}\tanh [(H-2|y|)/4\unicode[STIX]{x1D703}_{J}]$ in planar jets. Here, $U_{M}$ is the velocity difference in the mixing layers, $U_{J}$ is the jet velocity, $H$ is the width of the jet inlet and $\unicode[STIX]{x1D703}_{M}$ ( $\unicode[STIX]{x1D703}_{J}$ ) is the initial shear layer thickness in the mixing layers (planar jets). The angular bracket denotes the averaged value in an $x{-}z$ plane. We set $\unicode[STIX]{x1D703}_{J}=0.015H$ . The initial scalar profile in the mixing layer is given by $\unicode[STIX]{x1D719}=0.5\tanh (2y/\unicode[STIX]{x1D703}_{M})$ . The Reynolds numbers $Re$ are defined by $U_{M}\unicode[STIX]{x1D703}_{M}/\unicode[STIX]{x1D708}$ and $U_{J}H/\unicode[STIX]{x1D708}$ . The DNS are performed for the planar jets at $Re=10\,000$ and 20 000 and for the mixing layers at $Re=2000$ and 6000. The physical and computational parameters are listed in table 1 ( $\unicode[STIX]{x1D706}$ is the Taylor microscale, $\unicode[STIX]{x1D702}$ is the Kolmogorov scale and $Re_{\unicode[STIX]{x1D706}}$ is the turbulent Reynolds number). As shown in table 1, the spatial resolutions, $\unicode[STIX]{x1D6E5}_{i}$ , are small and comparable to the Kolmogorov scale on the centreline, and are therefore small enough to capture the small-scale fluctuations.

2.2 Lagrangian particle tracking

Once the mixing layers and planar jets have reached a self-similar state, we seed 40 000 tetrahedra consisting of four particles each and start to track $4\times 40\,000$ particles. Two cases are considered in this study: (i) tetrahedra seeded inside the turbulent region and (ii) tetrahedra seeded in the non-turbulent region near the TNTI. The turbulent region is detected as the region with $|\unicode[STIX]{x1D74E}|>\unicode[STIX]{x1D714}_{th}$ (da Silva et al. Reference da Silva, Hunt, Eames and Westerweel2014), where $|\unicode[STIX]{x1D74E}|$ is the magnitude of the vorticity vector, which is defined as the curl of the velocity field $\unicode[STIX]{x1D714}_{i}=\unicode[STIX]{x1D700}_{ijk}\unicode[STIX]{x2202}u_{k}/\unicode[STIX]{x2202}x_{j}$ . The threshold $\unicode[STIX]{x1D714}_{th}$ is determined so that the isosurface $|\unicode[STIX]{x1D74E}|=\unicode[STIX]{x1D714}_{th}$ is located near the outer edge of the TNTI layer using a well-known feature of the dependence of the turbulent volume on $\unicode[STIX]{x1D714}_{th}$ (Taveira et al. Reference Taveira, Diogo, Lopes and da Silva2013). It has been shown that the interface boundary location computed using this procedure, as well as the resulting conditional statistics, is insensitive to the particular value of the $\unicode[STIX]{x1D714}_{th}$ thus obtained. In the present DNS, $\unicode[STIX]{x1D714}_{th}$ is 4 % of the mean vorticity magnitude on the centreline (Watanabe et al. Reference Watanabe, Sakai, Nagata, Ito and Hayase2015). This isosurface is referred to as the irrotational boundary hereafter (Watanabe et al. Reference Watanabe, Sakai, Nagata, Ito and Hayase2015). The tetrahedra are seeded with the method used by Schumacher (Reference Schumacher2009) where an initial tetrahedron consists of four particles located at $\boldsymbol{x}_{1}=\boldsymbol{x}_{0}$ , $\boldsymbol{x}_{2}=\boldsymbol{x}_{0}+r_{0}\boldsymbol{e}_{x}$ , $\boldsymbol{x}_{3}=\boldsymbol{x}_{0}+r_{0}\boldsymbol{e}_{y}$ and $\boldsymbol{x}_{4}=\boldsymbol{x}_{0}+r_{0}\boldsymbol{e}_{z}$ ( $\boldsymbol{x}_{3}=\boldsymbol{x}_{0}-r_{0}\boldsymbol{e}_{y}$ is used when $y_{0}<0$ ), where $\boldsymbol{e}_{i}$ is the unit vector in the $i$ direction. Here, $\boldsymbol{x}_{0}$ is randomly chosen from the turbulent region in the case (i) and from the non-turbulent region near the TNTI in the case (ii). When the tetrahedra are seeded inside the turbulent region, all particles of the tetrahedra are located inside the turbulent region, whereas in case (ii), all particle locations $\boldsymbol{x}_{1}$ , $\boldsymbol{x}_{2}$ , $\boldsymbol{x}_{3}$ and $\boldsymbol{x}_{4}$ are in the non-turbulent region. The simulations are performed for tetrahedra with initial side lengths of $r_{0}=2\unicode[STIX]{x1D702},4\unicode[STIX]{x1D702},8\unicode[STIX]{x1D702},16\unicode[STIX]{x1D702},32\unicode[STIX]{x1D702}$ and $48\unicode[STIX]{x1D702}$ . We calculate two-particle statistics from the pairs of ( $\boldsymbol{x}_{1}$ , $\boldsymbol{x}_{2}$ ), ( $\boldsymbol{x}_{1}$ , $\boldsymbol{x}_{3}$ ) and ( $\boldsymbol{x}_{1}$ , $\boldsymbol{x}_{4}$ ), while the tetrahedra are used for four-particle statistics.

The Lagrangian multi-particle statistics are computed as a function of time $\unicode[STIX]{x1D70F}$ from the above DNS results. In the case (i), $\unicode[STIX]{x1D70F}$ is defined as the time after the particles are released. The case (ii) uses the time elapsed after one of the particles making pairs or tetrahedra crosses the irrotational boundary. We denote the ensemble average of pairs or tetrahedra by $\langle \ast \rangle _{\unicode[STIX]{x1D70F}}$ . The Lagrangian studies of the entrainment showed that the entrained fluid particles stay in the VSL for $0\leqslant \unicode[STIX]{x1D70F}\lesssim 7\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ , and it takes more than $20\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ for the particles to cross the entire TNTI layer (Watanabe et al. Reference Watanabe, da Silva, Sakai, Nagata and Hayase2016), where $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}=(\unicode[STIX]{x1D708}/\unicode[STIX]{x1D700})^{1/2}$ is the Kolmogorov time scale ( $\unicode[STIX]{x1D700}$ is the mean kinetic energy dissipation rate). In the present paper, the Lagrangian statistics are presented for large enough $\unicode[STIX]{x1D70F}$ to observe the entrained particle movements in the entire TNTI layers. It has been confirmed that the small-scale characteristics of the entrainment are qualitatively similar in mixing layers and planar jets (Watanabe et al. Reference Watanabe, da Silva, Sakai, Nagata and Hayase2016). Therefore, we discuss differences between the two flow types when the results are related to large-scale quantities.

3.1 Two-particle dispersion

The relative location of two particles is described by the separation vector $\boldsymbol{R}(\unicode[STIX]{x1D70F})=\boldsymbol{x}_{2}(\unicode[STIX]{x1D70F})-\boldsymbol{x}_{1}(\unicode[STIX]{x1D70F})$ . The classical Richardson scaling $\langle R^{2}\rangle _{\unicode[STIX]{x1D70F}}\sim \unicode[STIX]{x1D70F}^{3}$ is only valid in statistically stationary flows. In non-stationary decaying turbulence, a modified Richardson law is obtained for the separation vector as (Larcheveque & Lesieur Reference Larcheveque and Lesieur1981)

where $L$ is the integral scale and $\unicode[STIX]{x1D700}(t)$ is the time-dependent mean kinetic energy dissipation rate. The classical scaling for the energy dissipation rate, $\unicode[STIX]{x1D700}\sim u^{\prime 3}/L$ ( $u^{\prime }$ is the root mean square velocity), supports the power-law decay $\unicode[STIX]{x1D700}(t)\sim t^{-n}$ in the self-similar regime of temporally evolving free-shear flows, where $n=1$ in mixing layers and $n=2$ in planar jets. The present DNS also confirmed these decay exponents. Integration of (3.1) with $\unicode[STIX]{x1D700}(t)\sim t^{-n}$ yields

where $g_{m}$ is defined as the modified Richardson constant.

Figure 1(a,b) shows $\langle (R^{2}-r_{0}^{2})/\unicode[STIX]{x1D700}\rangle _{\unicode[STIX]{x1D70F}}^{1/3}/(r_{0}^{2}/\unicode[STIX]{x1D700}_{0})^{1/3}$ divided by $\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ for particles released in the turbulent region and in the non-turbulent regions near the TNTI in PJ20, where $\unicode[STIX]{x1D700}_{0}$ is the kinetic energy dissipation rate at the centreline at the time when the particles are released. Richardson-like scaling is expected to be observed for times larger than $(r_{0}^{2}/\unicode[STIX]{x1D700}_{0})^{1/3}$ (Bourgoin et al. Reference Bourgoin, Ouellette, Xu, Berg and Bodenschatz2006). According to studies on the relative dispersion in stationary turbulence (Ott & Mann Reference Ott and Mann2000; Ishihara & Kaneda Reference Ishihara and Kaneda2002), equation (3.2) is also examined by using the plot of $\langle R^{2}/\unicode[STIX]{x1D700}\rangle _{\unicode[STIX]{x1D70F}}^{1/3}$ against $\unicode[STIX]{x1D70F}$ in the insets. Although the separation distance hardly changes before the particles cross the irrotational boundary ( $\unicode[STIX]{x1D70F}\leqslant 0$ ), the modified Richardson law (3.2) is satisfied in the turbulent core regions. The other jet and mixing layer simulations in the present study also yielded similar results. The modified Richardson constant is obtained by applying the least-squares method to $\langle R^{2}/\unicode[STIX]{x1D700}\rangle _{\unicode[STIX]{x1D70F}}^{1/3}$ , and is plotted as a function of the initial separation $r_{0}$ in figure 1(c,d). For the particles released in the turbulent region, $g_{m}$ increases slightly with increasing $r_{0}$ for small $r_{0}$ , as in forced homogeneous isotropic turbulence. For $r_{0}/\unicode[STIX]{x1D702}\geqslant 32$ , $g_{m}$ hardly depends on the initial separation, and $g_{m}\approx 0.65{-}0.75$ , with small variations with the Reynolds number and flow configuration. This value of $g_{m}$ is very close to the Richardson constant in statistically stationary turbulence. For the entrained particle pairs, $g_{m}$ shows a variation depending on the flow, and $g_{m}$ decreases as $Re_{\unicode[STIX]{x1D706}}$ increases. It is found that the cases PJ20 and ML02, where $Re_{\unicode[STIX]{x1D706}}\approx 200$ , give similar values of $g_{m}$ .

Figure 1. (a,b) Plots of $\langle (R^{2}-r_{0}^{2})/\unicode[STIX]{x1D700}\rangle _{\unicode[STIX]{x1D70F}}^{1/3}$ normalized by $(r_{0}^{2}/\unicode[STIX]{x1D700}_{0})^{1/3}$ ( $\unicode[STIX]{x1D700}_{0}$ is the kinetic energy dissipation rate at the centreline at the time when the particles are released) and compensated by $\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ in PJ20 for two-particle pairs released (a) in the turbulent region and (b) in the non-turbulent region near the TNTI, with initial separations equal to $r_{0}/\unicode[STIX]{x1D702}=2,4,\ldots ,48$ , where the Richardson-like scaling appears as a plateau. Here, $r_{0}^{2}/\unicode[STIX]{x1D700}_{0}$ and $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ used in the normalization are taken at the jet centreline at the time when the particles are released. The insets show $\langle R^{2}/\unicode[STIX]{x1D700}\rangle _{\unicode[STIX]{x1D70F}}^{1/3}$ against $\unicode[STIX]{x1D70F}$ , where the Richardson-like scaling appears as a straight line. The symbols denote the time $(r_{0}^{2}/\unicode[STIX]{x1D700}_{0})^{1/3}$ . (c,d) The modified Richardson constant $g_{m}$ plotted against $r_{0}/\unicode[STIX]{x1D702}$ for two-particle pairs released (c) in the turbulent region and (d) in the non-turbulent region near the TNTI. The Richardson constant $g$ in forced homogeneous isotropic turbulence (HIT) (Ishihara & Kaneda Reference Ishihara and Kaneda2002) is also shown for comparison.

3.2 Motions of tetrahedra

We consider the motions of tetrahedra, which consist of four particles with the location $\boldsymbol{x}^{(n)}$ and the velocity $\boldsymbol{u}^{(n)}$ ( $n=1,\ldots ,4$ ). The kinetic energy of the motion of a tetrahedron (Pumir et al. Reference Pumir, Shraiman and Chertkov2001) is defined by $E=\overline{(\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{u})/2}$ , where the overbar denotes the averaged value of the four particles of a tetrahedron. Since the velocity of the particle relative to the centre-of-mass motion is given by ${\boldsymbol{u}^{\prime }}^{(n)}=\boldsymbol{u}^{(n)}-\overline{\boldsymbol{u}}$ , the kinetic energy of the particle motion can be rewritten as $E=E_{m}+E_{r}$ , where $E_{m}=(\overline{\boldsymbol{u}}\boldsymbol{\cdot }\overline{\boldsymbol{u}})/2$ and $E_{r}=(\overline{\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{u}^{\prime }})/2$ are the kinetic energies of the mean and relative motions respectively. Figure 2(a,b) shows the averaged energies of the mean and relative motions of tetrahedra during the entrainment in the planar jets. For both values of $Re$ , the kinetic energy of the tetrahedra evolves similarly with time. Once a part of the tetrahedron crosses the irrotational boundary, both $\langle E_{m}\rangle _{\unicode[STIX]{x1D70F}}$ and $\langle E_{r}\rangle _{\unicode[STIX]{x1D70F}}$ increase rapidly, while they show a slower increase in the non-turbulent region. For a tetrahedron with a larger initial size, the relative motions have larger kinetic energy. However, when the total kinetic energy $\langle E\rangle _{\unicode[STIX]{x1D70F}}$ is plotted as in figure 2(c), the dependence on the initial size is hardly seen during the entrainment. Thus, in the planar jets, although the tetrahedra gain kinetic energy almost independent of their initial size, their size affects the ratio between the mean and relative kinetic energy. The relative particle motions are mainly associated with the small scales of motion, and are therefore relatively flow-independent. Thus, the growth rate of the energy in the relative motions during entrainment is similar in jets and mixing layers. However, the energy of the mean motions is flow-dependent because of the different mean velocity profiles; in the temporal mixing layers, $\langle E_{m}\rangle _{\unicode[STIX]{x1D70F}}$ rapidly decreases during the entrainment.

Figure 2. Averaged kinetic energy of mean ( $E_{m}$ ) and relative ( $E_{r}$ ) motions of tetrahedra during entrainment in planar jets: (a) PJ10 and (b) PJ20. (c) Averaged kinetic energy ( $E=E_{m}+E_{r}$ ) in PJ20.

The equations governing the kinetic energy of the mean and relative motions of a tetrahedron are written as

where $p$ is the pressure divided by the constant density and $\unicode[STIX]{x1D61A}_{ij}$ is the strain tensor. The first, second and third terms are the pressure diffusion, viscous diffusion and dissipation terms respectively. Figure 3(a) compares $\langle \text{d}E_{m}/\text{d}t\rangle _{\unicode[STIX]{x1D70F}}$ and $\langle \text{d}E_{r}/\text{d}t\rangle _{\unicode[STIX]{x1D70F}}$ among different initial sizes of the tetrahedra released in the non-turbulent region for PJ20. The tetrahedra gain more energy in relative motions for larger initial sizes. For $r_{0}/\unicode[STIX]{x1D702}\geqslant 16$ , $\langle \text{d}E_{r}/\text{d}t\rangle _{\unicode[STIX]{x1D70F}}$ has a peak value around $\unicode[STIX]{x1D70F}\approx 10\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ , at which more than one particle is passing the TNTI layer (Watanabe et al. Reference Watanabe, da Silva, Sakai, Nagata and Hayase2016), and the relative motions grow rapidly during the entrainment process. For small tetrahedra, the growth rate of the relative motions is kept low in the TNTI layer. This is highlighted in figure 3(b), where $\langle \text{d}E_{r}/\text{d}t\rangle _{\unicode[STIX]{x1D70F}}$ at $\unicode[STIX]{x1D70F}=0$ is plotted against $r_{0}$ for all DNS. Moreover, it is found that $\langle \text{d}E_{r}/\text{d}t\rangle _{\unicode[STIX]{x1D70F}}$ is significantly decreased as $r_{0}$ becomes small for $r_{0}\lesssim 10\unicode[STIX]{x1D702}$ . Indeed, figure 3(c) shows the probability density functions (PDFs) of $\text{d}E_{r}/\text{d}t$ during the entrainment of tetrahedra in PJ20. These PDFs display a large peak at $\text{d}E_{r}/\text{d}t=0$ for $r_{0}\lesssim 10\unicode[STIX]{x1D702}$ , and this peak significantly decreases from $r_{0}=8\unicode[STIX]{x1D702}$ to $32\unicode[STIX]{x1D702}$ . Since these PDFs are positively skewed and the relative motion grows during the entrainment process, the statistics of $\text{d}E_{r}/\text{d}t$ imply that the four-particle cluster being entrained crosses the TNTI layer nearly along with the motion of the mass centre when the length scale of the volume defined by the particles is less than ${\sim}10\unicode[STIX]{x1D702}$ .

Figure 3. (a) Averaged growth rates of the kinetic energies of the mean and relative motions of tetrahedra during entrainment in a planar jet (PJ20). (b) Dependence of $\langle \text{d}E_{r}/\text{d}t\rangle _{\unicode[STIX]{x1D70F}}$ at $\unicode[STIX]{x1D70F}=0$ on the initial size of the tetrahedra. (c) The PDF of $\text{d}E_{r}/\text{d}t$ for PJ20 at $\unicode[STIX]{x1D70F}=14\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ .

Figure 4(a) shows the average of each term in (3.3) for two initial sizes of the tetrahedra. The tetrahedra being entrained gain the kinetic energy of the mean motions by the pressure diffusion. Even in the non-turbulent region, the kinetic energy of the mean motions is increased by the pressure diffusion. The growth rate of the energy of the mean motions becomes large when the tetrahedra are passing the TNTI $(\unicode[STIX]{x1D70F}\approx 10\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}})$ , which is consistent with a mean velocity jump observed in the TNTI layer (Westerweel et al. Reference Westerweel, Fukushima, Pedersen and Hunt2009). Unlike what is observed in planar jets, the energy of the mean motions in mixing layers is reduced during the entrainment because of differences in the mean velocity profiles (not shown). However, we observed that the pressure diffusion dominates the mean motion in both mixing layers and jets. Figure 4(b,c) shows the average of each term in (3.4). For small tetrahedra with $r_{0}=2\unicode[STIX]{x1D702}$ , the initial growth of the relative motions is caused by the viscous diffusion, $D_{vr}$ . For larger tetrahedra, the pressure diffusion has a non-negligible contribution to the energy of relative motion even in the non-turbulent region. The pressure effects on the large tetrahedra cause rapid growth of the energy of relative motions during the entrainment.

Figure 4. (a) Averaged budget of kinetic energy for the mean motion (3.3) for the initial sizes $r_{0}/\unicode[STIX]{x1D702}=2$ and 32 in PJ20. (b,c) Averaged budget of kinetic energy for the relative motion (3.4) for (b) $r_{0}/\unicode[STIX]{x1D702}=2$ and (c) $r_{0}/\unicode[STIX]{x1D702}=32$ . The inset in (b) shows the profiles around $\unicode[STIX]{x1D70F}=0$ .

The decomposition $\boldsymbol{u}^{(n)}=\overline{\boldsymbol{u}}+{\boldsymbol{u}^{\prime }}^{(n)}$ can be considered as a scale decomposition: the large-scale velocity $\overline{\boldsymbol{u}}$ by a low-pass filtering and the small-scale velocity $\boldsymbol{u}^{\prime }$ by a high-pass filtering (Pumir et al. Reference Pumir, Shraiman and Chertkov2001). Then, the $r_{0}$ dependence of $\text{d}E_{r}/\text{d}t$ implies the scale dependence of the kinetic energy transfer. Since the volume of the tetrahedra hardly changes at $\unicode[STIX]{x1D70F}\lesssim 0$ , the initial length $r_{0}$ can be used as the length of the scale decomposition, and $E_{r}$ represents the kinetic energy contained in the scales smaller than $r_{0}$ . Thus, the kinetic energy at small scales ( $E_{r}$ with small $r_{0}$ ) grows by the viscous diffusion, while as $r_{0}$ increases the pressure diffusion becomes dominant. In the Eulerian kinetic energy budget, the pressure diffusion was also found to be important in the energy transfer near the TNTI (Taveira & da Silva Reference Taveira and da Silva2013). Furthermore, the mean and relative motions of tetrahedra provide the scale dependence of this energy transfer in the fluid motions. The $r_{0}$ dependence of $D_{pr}$ indicates that most energy transferred by the pressure diffusion is contained in scales larger than ${\sim}10\unicode[STIX]{x1D702}$ . This is likely to be caused by the presence of coherent eddy motions near the interface, with length scales of the order of ${\sim}10\unicode[STIX]{x1D702}$ , since these structures display a pressure minimum that is long lived and thus able to imprint long-lasting effects on the nearby particle dynamics. The tetrahedra in the non-turbulent regions have non-trivial mean kinetic energy, whereas relative motions are absent especially for small tetrahedra. This can be related to the large-scale irrotational motions, which were also observed in the Eulerian statistics in planar jets (da Silva & Pereira Reference da Silva and Pereira2008).

3.3 Shape evolution of tetrahedra

The shape of tetrahedra is well described by the position of the particles relative to the centre position $\boldsymbol{r}^{(n)}=\boldsymbol{x}^{(n)}-\overline{\boldsymbol{x}}$ . The volumetric tensor (Robert et al. Reference Robert, Roux, Harvey, Dunlop, Daly and Glassmeier1998) is defined by $\unicode[STIX]{x1D619}_{ij}=\overline{r_{i}r_{j}}$ , and has three eigenvalues denoted by $R_{1}$ , $R_{2}$ and $R_{3}$ , ranging from the largest to the smallest ( $R_{1}\geqslant R_{2}\geqslant R_{3}$ ); the corresponding eigenvectors are $\boldsymbol{e}_{1}$ , $\boldsymbol{e}_{2}$ and $\boldsymbol{e}_{3}$ . Three parameters can be defined by the eigenvalues of the volumetric tensor (Robert et al. Reference Robert, Roux, Harvey, Dunlop, Daly and Glassmeier1998). One is the characteristic size of the tetrahedron: $L=2\sqrt{R_{1}}$ . The shape is described by the other two parameters: elongation $E$ and planarity $P$ , defined by $E=1-\sqrt{R_{2}/R_{1}}$ and $P=1-\sqrt{R_{3}/R_{2}}$ . When four particles lie nearly on a straight line, $R_{2}\approx R_{3}\approx 0$ , resulting in $E\approx 1$ , and the direction of elongation is given by $\boldsymbol{e}_{1}$ . When a tetrahedron is squashed in one direction and four particles lie nearly on a plane, $R_{2}\gg R_{3}$ and $P\approx 1$ , and the normal of the planarity is given by $\boldsymbol{e}_{3}$ . The volumetric tensor is also related to the moment-of-inertia tensor, and the normalized eigenvalues of the moment-of-inertia tensor are obtained by $I_{i}=R_{i}/(R_{1}+R_{2}+R_{3})$ .

Figure 5(a) shows $\langle I_{i}\rangle _{\unicode[STIX]{x1D70F}}$ during the entrainment in PJ10 and ML06, which have the smallest and the largest $Re_{\unicode[STIX]{x1D706}}$ in the present DNS datasets respectively. Once the tetrahedra cross the irrotational boundary, they rapidly deform while passing the TNTI layers. The time scale of the deformation is very similar for $r_{0}/\unicode[STIX]{x1D702}=2$ and 8 in both PJ10 and ML06, and scales with the Kolmogorov time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}=(\unicode[STIX]{x1D708}/\unicode[STIX]{x1D700})^{1/2}$ , indicating that the local strain rate, of order $(\unicode[STIX]{x1D700}/\unicode[STIX]{x1D708})^{1/2}$ , dominates the deformation. The mean values of $I_{i}$ for large $\unicode[STIX]{x1D70F}$ become close to those observed in other flows (Schumacher Reference Schumacher2009). Figure 5(b) shows the mean trajectory on the $E{-}P$ map after a part of a tetrahedron crosses the irrotational boundary. Once the tetrahedron moves within the TNTI layer, the shape changes into a knife-blade shape, which is both elongated and flattened, and a tetrahedron with $r_{0}/\unicode[STIX]{x1D702}=2$ reaches a quasi-stationary shape $(E,P)=(0.62,0.80)$ at $\unicode[STIX]{x1D70F}=20\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ , at which the particles are located in the TSL (Watanabe et al. Reference Watanabe, da Silva, Sakai, Nagata and Hayase2016). This shape evolution is very similar in planar jets and mixing layers.

Figure 5. (a) Mean normalized eigenvalues of the moment-of-inertia tensor during entrainment in a mixing layer (ML06) and a planar jet (PJ10). (b) Averaged shape evolutions of tetrahedra with $r_{0}=2\unicode[STIX]{x1D702}$ across the TNTI on the $E{-}P$ map. Examples of potatoes ( $P=0.52$ , $E=0.6$ ) and knife blades ( $P=0.84$ , $E=0.82$ ) are also shown in the figure. Tetrahedron shapes are characterized by elongation $E$ and planarity $P$ (Robert et al. Reference Robert, Roux, Harvey, Dunlop, Daly and Glassmeier1998). Symbols mark the specific times of $\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}=0$ , 5, 10, 20 and 40. (c) The PDFs of the cosines of the angles between the vorticity vector ( $\unicode[STIX]{x1D74E}$ ) and the elongation and planarity-normal directions ( $\boldsymbol{e}_{1}$ , $\boldsymbol{e}_{3}$ ). The PDF is shown for the tetrahedra being entrained with $r_{0}/\unicode[STIX]{x1D702}=2$ at $\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}=2$ , 4 and 14 in ML02.

Figure 5(c) shows the PDFs of $|\text{cos}\,\unicode[STIX]{x1D703}_{1\unicode[STIX]{x1D714}}|$ and $|\text{cos}\,\unicode[STIX]{x1D703}_{3\unicode[STIX]{x1D714}}|$ , where $\unicode[STIX]{x1D703}_{i\unicode[STIX]{x1D714}}$ is the angle between $\boldsymbol{e}_{i}$ and $\unicode[STIX]{x1D74E}$ at one of the four particle locations ( $\boldsymbol{x}_{1}$ ). The PDF shows that the direction of elongation ( $\boldsymbol{e}_{1}$ ) tends to align with $\unicode[STIX]{x1D74E}$ , and the tetrahedron is flattened in the direction perpendicular to $\unicode[STIX]{x1D74E}$ . This indicates that the vortical structures are circulated around by the entrained tetrahedra which are elongated in the direction of $\unicode[STIX]{x1D74E}$ . The elongation in the vorticity direction can be related to the strain field stretching the vorticity vector. As shown from the deformation time scale, the entrained fluid volume is deformed by the local strain $S^{\prime }\sim (\unicode[STIX]{x1D700}/\unicode[STIX]{x1D708})^{1/2}$ , which acts on the small-scale eddy structures, whose radius sizes are $4{-}5\unicode[STIX]{x1D702}$ (da Silva, Dos Reis & Pereira Reference da Silva, Dos Reis and Pereira2011). Thus, the elongation of the fluid volume during the entrainment is related to the small-scale eddies, which are sustained by the strain $S^{\prime }$ within the TSL.

The growth rate of the energy of the relative motions clearly changes around $r_{0}\approx 10\unicode[STIX]{x1D702}$ , as shown in figure 3. Considering the small energy in the relative motions for $r_{0}\leqslant 10\unicode[STIX]{x1D702}$ during the entrainment (figure 2), we can anticipate that a lump of non-turbulent fluid with a size of approximately $10\unicode[STIX]{x1D702}$ is entrained undergoing the deformation by the local strain acting on the small-scale eddy in the TSL, resulting in the thin-slab structures of the entrained fluid. Figure 6 confirms this shape evolution of the small entrained fluid volume. Along a fluid particle, $\unicode[STIX]{x1D719}$ changes only by molecular diffusion. Therefore, the entrained fluid near the TNTI has $\unicode[STIX]{x1D719}$ close to non-turbulent values (Watanabe et al. Reference Watanabe, Sakai, Nagata, Ito and Hayase2015). In figure 6(a), we can see that the spanwise vortical structures with large $\unicode[STIX]{x1D714}_{z}$ have a core radius of approximately $5\unicode[STIX]{x1D702}$ , like the intense vorticity structures studied by da Silva et al. (Reference da Silva, Dos Reis and Pereira2011). In figure 6(b), thin-slab structures of the entrained fluid with $\unicode[STIX]{x1D719}\approx -0.5$ (white) are found around the vortical structures with large $\unicode[STIX]{x1D714}_{z}$ , and are flattened so that the planarity normal is perpendicular to the vorticity vector ( $z$ ) direction. This result agrees with the previous Lagrangian statistics, showing that the entrained fluid circumvents the strong-vorticity regions near the TNTI (Watanabe et al. Reference Watanabe, da Silva, Sakai, Nagata and Hayase2016). In figure 6(b), the size of the entrained thin-slab structures is comparable to the vortical structures ( ${\sim}5{-}10\unicode[STIX]{x1D702}$ ), as expected from the small kinetic energy of the relative motion of entrained tetrahedra with $r_{0}\leqslant 10\unicode[STIX]{x1D702}$ . Thus, some of the thin-slab structures observed in free-shear flows are formed during the entrainment process. It should be noted that this generation process of thin-slab structures inside fully developed turbulence (Brethouwer et al. Reference Brethouwer, Hunt and Nieuwstadt2003) is also possible in the thin shear layer between intense and weak turbulent regions, which was observed in high- $Re$ turbulent flows (Ishihara, Kaneda & Hunt Reference Ishihara, Kaneda and Hunt2013).

Figure 6. Visualization of (a) the spanwise vorticity squared, $\unicode[STIX]{x1D714}_{z}^{2}$ , and (b) the scalar $\unicode[STIX]{x1D719}$ near the TNTI in ML06. The irrotational boundary, $|\unicode[STIX]{x1D74E}|=\unicode[STIX]{x1D714}_{th}$ , is shown by a thin line. The vortical structures with large $\unicode[STIX]{x1D714}_{z}$ are marked in (a), while the entrained fluid around the spanwise vortical structures is marked in (b).