Physical and computational parameters of the DNS. The displayed turbulence characteristics are from the turbulent core regions.

## Abstract

Vorticity dynamics is studied near the interface between turbulent and non-turbulent flows, the so-called turbulent/non-turbulent (T/NT) interface, with the direct numerical simulations of planar jets and mixing layers. The statistics near the interface confirm that the T/NT interface consists of two layers: viscous superlayer and turbulent sublayer. The viscous superlayer with the thickness of four times of Kolmogorov length scale is found at the outer edge of the interface, where the vorticity grows with the viscous diffusion. In the turbulent sublayer between the viscous superlayer and the turbulent region, the strain-vorticity interaction becomes active. In the Lagrangian statistics for the fluid particles, the different scaling laws appear in the entrained particle movement depending on the layer: a ballistic evolution in the viscous superlayer and the Richardson-like scaling for relative dispersion in the turbulent sublayer. These scalings indicate that the change in the particle position in the viscous superlayer is governed by the outward viscous diffusion of vorticity, whereas it is governed by the inviscid small-scale eddy motions in the turbulent sublayer. The flow topology on the particle path line shows that the fluid being entrained tends to circumvent the core region of intense eddies near the T/NT interface.

### Keywords

- jet
- mixing layer
- turbulent/non-turbulent interface
- DNS
- Lagrangian statistics

## 1. Introduction

Interfaces dividing turbulent and non-turbulent regions appear in various canonical turbulences, such as boundary layers, jets, and mixing layers, where turbulence is generated from the shear due to the wall friction or mean velocity difference. These interfaces are called turbulent/non-turbulent (T/NT) interfaces. Turbulence is generated by shear motions in various circumstances, where the turbulent fluids are surrounded by non-turbulent fluids. This locally generated turbulence often plays an important role in the relevant phenomena. For example, ocean-mixing layers [1], generated in the stably stratified fluid, are sometimes responsible for the transport of heat, salinity, and plankton. The atmospheric boundary layer [2] is related to the cooling/heating of the ground surface and the transport of contaminant. In the flows with the T/NT interface, the turbulent region grows into the non-turbulent region with the mass, momentum, and energy exchanges across the T/NT interface.

Corrsin and Kistler [3], in laboratory experiments with hot-wire probes, found that the essential feature of the turbulent regions is the high vorticity, and the turbulent and non-turbulent regions can be distinguished by the vorticity. They also predicted that a very thin layer where the non-turbulent fluids acquire vorticity by the viscous diffusion is formed at the outer edge of the turbulent region. This thin layer, called the viscous superlayer, was confirmed with the recent high-resolution direct numerical simulations (DNSs) [4]. Furthermore, the statistical approach conditioned relative to the interface [5] clearly showed that the T/NT interface is the layer with a finite thickness. In addition to the viscous superlayer, an adjacent layer, turbulent sublayer, was found between the turbulent core region and the viscous superlayer [6]. One of the differences between the turbulent sublayer and the viscous superlayer is in the vorticity dynamics; the initial growth of vorticity of the non-turbulent fluid occurs by the viscous diffusion in the viscous superlayer with the absence of inviscid vortex stretching, whereas the vortex stretching plays an important role in the amplification of vorticity in the turbulent sublayer [7, 8].

The T/NT interface has been studied in particular attention to the entrainment process since this is where the non-turbulent fluid acquires vorticity and results in the transition to turbulence. Turbulent flows consist of the motions in a wide range of scales, and both small and large scales can cause the entrainment by nibbling [9] and engulfment [10], respectively. The experiments in the boundary layers indicated that the entrainment is the multi-scale process [11]. The entrainment across the interface was studied in [12] with the propagation velocity of the enstrophy isosurface. These analyses on the isosurface movement showed that the propagation velocity is of the order of the Kolmogorov velocity *v*_{η} = (*νε*)^{1/4} [13, 14], where *ν* is the kinematic viscosity and *ε* is the dissipation rate of turbulent kinetic energy, and the complex shape of the isosurface [15] relates the propagation velocity to the total entrainment rate, which can be written as a function of large-scale quantities [16]. The enstrophy isosurface is an infinitely thin surface located within the T/NT interface layer. Therefore, a more precise description of the entrainment process is the fluid movement across the entire T/NT interface layer than across the enstrophy isosurface. During the entrainment, the irrotational particles pass both the viscous superlayer and the turbulent sublayer. The Lagrangian analysis is useful for studying the entrainment, and both experiments and simulations have been used for tracking the fluid particles (tracers) being entrained from the non-turbulent regions [14, 17, 18]. These studies showed the evolution of turbulence characteristics during the entrainment. However, the relation between these Lagrangian statistics and the layer structures is not clear because the particle tracking does not show the location within the T/NT interface layer because the T/NT interface also moves with the convective fluid motion.

In this study, we explore the connection between the T/NT interface structure and the Lagrangian statistics during the entrainment process based on our recent DNS results [19]. The DNS is performed for mixing layers and planar jets, and used for tracking the fluid particles being entrained. In addition to the fluid particles, the outer edge of the T/NT interface layer, defined by the enstrophy isosurface, is also tracked with the Lagrangian markers, enabling us to examine the location of the fluid particle within the T/NT interface layer and to relate the Lagrangian statistics to the Eulerian counterparts. The roles of small-scale eddy structures in the entrainment are considered from the Lagrangian and Eulerian statistics. This chapter is organized as follows: Section 2 presents the numerical methods and parameters as well as the conventional statistics for the validation of the DNS data. Section 3 discusses the analysis on the T/NT interface, such as the interface detection, and the conditional analysis based on the Eulerian and Lagrangian statistics. Finally, Section 4 closes the chapter with the conclusion.

## 2. Direct numerical simulations

Direct numerical simulations are performed for temporally evolving mixing layers and planar jets [19]. These flows develop from the initial state in the computational domain, which is periodic in the mean flow (*x*) and spanwise (*z*) directions. The flows spread in the cross-streamwise (*y*) direction. We consider the computational box with the size of (*L*_{x} × *L*_{y} × *L*_{z}) represented by (*N*_{x} × *N*_{y} × *N*_{z}) grid points. The boundaries in the *y* direction are treated as the slip wall [7]. The origin of the coordinate system is at the center of the computational domain. The DNS code is an incompressible Navier-Stokes solver based on the fractional step method [8]. In addition to the flow field, a passive scalar *ϕ* is simulated with the convection-diffusion equation. The governing equations are spatially discretized with a fully conservative finite-difference method [20]. The second-order and fourth-order schemes are used in the cross-streamwise and the periodic directions, respectively. The governing equations are integrated in time with a third-order Runge-Kutta method. The Poisson equation is solved with the fast Fourier transform along the periodic directions and the diagonal matrix algorithm along the *x* direction.

The initial velocity field is obtained by superimposing the statistically homogeneous and isotropic velocity fluctuations onto the mean velocity, which is given by

Here, *U*_{M} is the velocity difference in the mixing layers, *U*_{J} the jet velocity, *H* the width of the jet inlet and *θ*_{M} (*θ*_{J}) the initial shear layer thickness in the mixing layers (planar jets). The angular bracket denotes the averaged value in a *x* − *y* plane. We set *θ*_{J} = 0.015*H*. The initial velocity fluctuations are generated by a diffusion process [21], where the characteristics length scales are 0.07*H* in the planar jets and 0.25*δ*_{M} in the mixing layers. The initial rms velocity is 0.04*U*_{J} for |*y*|/*H* ≤ 0.5 in the planar jets and is 0.025*U*_{M} for |*y*|/*δ*_{M} ≤ 3 in the mixing layer. Except these regions, the fluctuations are not imposed on the mean velocity. The initial scalar profiles are given by

Run | ML04 | ML08 | PJ50 | PJ90 |
---|---|---|---|---|

Flow type | Mixing layer | Mixing layer | Planar jet | Planar jet |

Re | 400 | 800 | 5000 | 9000 |

L _{x} | 16πθ_{M} | 16πθ_{M} | 2.4πH | 2.6πH |

L _{y} | 16πθ_{M} | 16πθ_{M} | 4.8πH | 3.8πH |

L _{z} | 8πθ_{M} | 8πθ_{M} | 2.4πH | 1.3πH |

N _{x} | 512 | 1 024 | 512 | 1 024 |

N _{y} | 500 | 700 | 850 | 1 150 |

N _{z} | 256 | 512 | 512 | 512 |

Time step dt | 0.08θ_{M}/U_{M} | 0.04θ_{M}/U_{M} | 0.012H/U_{J} | 0.006H/U_{J} |

Δ_{x} = Δ_{z} | 1.5η | 1.2η | 1.5η | 1.4η |

Δ_{y} (y = 0) | 1.0η | 1.1η | 1.2η | 1.2η |

Re_{λ} | 105 | 151 | 94 | 158 |

η | 0.064θ_{M} | 0.041θ_{M} | 0.0096H | 0.0059H |

λ | 20.8η | 23.3η | 14.8η | 20.3η |

The Reynolds numbers Re are defined by *U*_{M}*θ*_{M}/*ν* and *U*_{J}*H*/*ν*. We perform the DNS for the planar jets with Re = 5 000 and 9 000 and for the mixing layers with Re = 400 and 800. The Schmidt number of *ϕ* is *Sc* = *ν*/*D* = 1, where *D* is the molecular diffusivity. Table 1 shows the physical and computational parameters, such as the Kolmogorov scale *η*, Taylor microscale *λ*_{x}, and the turbulent Reynolds number Re_{λ}, where the Taylor microscale and turbulent Reynolds number are calculated from the streamwise velocity statistics. The computational grid size, Δ_{i}, is comparable to *η*, and is able to capture turbulent motions in very small scales.

The fundamental characteristics of the planar jets and mixing layers are compared with other DNS and experiments for validation of the DNS. Figure 1 compares the self-similar profiles of mean velocity and rms velocity fluctuations. The present DNS reproduces well the self-similar profiles of these statistics in previous studies. Figure 2 shows the one-dimensional longitudinal spectrum on the centerline with the experimental plots. We can see the overlap of the spectrum in small scales, and the small-scale turbulent fluctuations are well resolved in the DNS.

## 3. Analysis on turbulent and non-turbulent interface

### 3.1. Detection of the T/NT interface

The turbulent regions are characterized by high vorticity [3]. Therefore, following [5], we define the turbulent region as where the vorticity magnitude |**ω**| exceeds the threshold *ω*_{th}. Then, with an appropriate value of *ω*_{th}, the isosurface of |**ω**| = *ω*_{th} can be detected so that it is located near the outer edge of the T/NT interface layer. The specific value of *ω*_{th} is obtained from a well-known dependence of turbulent volume on *ω*_{th} [18]. Figure 3(a) shows the volume fraction of turbulent regions as a function of *ω*_{th}. We can see a plateau in the turbulent volume, and the isosurface location hardly changes with *ω*_{th} for the plateau. We choose *ω*_{th} = 0.04**ω**|_{C}, which is from the plateau shown in Figure 3(a). This value is chosen so that the isosurface is located at the outer edge of the T/NT interface layer. We call this isosurface as the *irrotational boundary* hereafter. Figure 3(b) and (c) show the enstrophy profile and the irrotational boundary. The irrotational boundary surrounds the high enstrophy region and is located at the outer edge of the turbulent fluids. Thus, the outer edge of the T/NT interface layer is well defined by thresholding the vorticity magnitude. The irrotational boundary is visualized in Figure 4. The T/NT interface has a very complex shape. The Re dependence is also clear; the higher Re mixing layer has smaller-scale structures because of the small length scales of turbulence.

### 3.2. Statistics conditioned on the location of the T/NT interface

The vorticity dynamics is studied with the statistics conditioned on the location from the irrotational boundary. This interface coordinate, *ζ*_{I}, is taken in the normal direction of the irrotational boundary **n** = − ∇*ω*^{2}/|∇*ω*^{2}|, where *ζ*_{I} = 0 is the location of the irrotational boundary. The non-turbulent region is indicated by *ζ*_{I} > 0. Because of the complicated shape of the T/NT interface, a turbulent (non-turbulent) fluid and an associated irrotational boundary can appear for *ζ*_{I} > 0 (*ζ*_{I} < 0). For separating the statistics into the turbulent and non-turbulent parts, the statistics are calculated solely from turbulent and non-turbulent regions in *ζ*_{I} < 0 and *ζ*_{I} > 0, respectively. When another irrotational boundary is found at *ζ*_{I} ≠ 0, the region within the distance of *λ* from this boundary is excluded from the statistics for preventing the T/NT interface layer from affecting the statistics for *ζ*_{I} ≫ 0 or ≪ 0. Note that previous studies have shown that the T/NT interface layer thickness is about 0.5*λ* [27]. Hereafter, _{I} denotes the conditional mean value.

Figure 5 shows the conditional mean enstrophy profiles. The mean enstrophy is matched in the layer with the thickness of ≈ 15*η*. The scaling of the thickness of the interface layer is examined in the plots of *ω*^{2}/2_{I} normalized by the value at *ζ*_{I} = 0, *ω*^{2}/2_{I0}, in Figure 5(b) and (c), where *ζ*_{I} is normalized by the Kolmogorov scale *η* and Taylor microscale *λ*, respectively. The plots tend to better collapse onto a single curve for *ζ*_{I}/*η* than *ζ*_{I}/*λ*, and thus the thickness of the T/NT interface layer, across which the enstrophy changes, is scaled with the Kolmogorov scales. It should be noted that the Taylor microscale can be the characteristics length scale of the T/NT interface when the large-scale coherent structures exist near the T/NT interface [28].

The vorticity evolution near the interface is studied by the enstrophy transport equation:

where the first term on the right-hand side is the enstrophy production *P*_{ω} (*S*_{ij}: strain-tensor), the second is the viscous diffusion *D*_{ω}, and the third is the viscous dissipation *ε*_{ω}. The conditional average of each term is plotted in Figure 6 for ML08 and PJ90. The plots are very similar for these flows in the T/NT interface layer; the enstrophy grows by the viscous diffusion near the outer edge of the T/NT interface, whereas the inviscid vortex stretching becomes important slightly inside the outer edge. The profile of *D*_{ω}_{I} exhibits negative and positive values in the T/NT interface layer, indicating the vorticity transport toward the non-turbulent region. Near the outer edge of the T/NT interface layer, *D*_{ω}_{I} is larger than *P*_{ω}_{I} in the region of − 4*η* ≤ *ζ* ≤ 0. This thickness, *δ*_{ν} = 4*η*, agrees with the direct observation of the viscous superlayer thickness [4]. Thus, from the conditional mean profiles of enstrophy and its budget, we can identify the viscous superlayer in − 4*η* ≤ *ζ*_{I} ≤ 0 and the adjutant layer, turbulent sublayer, with the thickness of *δ*_{ω} = 11*η* in − 15*η* ≤ *ζ*_{I} ≤ − 4*η*. This structure of the T/NT interfaces is observed in all DNS dataset. In the turbulent core region, the mean enstrophy production *P*_{ω}_{I} almost balances with the mean viscous dissipation *ε*_{ω}_{I}. This balance is absent in the T/NT interface layer; from *ζ*_{I} = − 15*η*, *ε*_{ω}_{I} becomes small toward the irrotational boundary, whereas *P*_{ω}_{I} hardly changes with the location for − 15*η* ≤ *ζ*_{I} ≤ − 9*η*.

Figure 7 gives the conditional plots of passive scalar *ϕ* and scalar dissipation rate *χ* = *D*∇*ϕ* ⋅ ∇*ϕ*. In the mixing layer, the conditional statistics are calculated from the upper interface, for which the non-turbulent fluid has *ϕ* = 0.5, where the upper interface is detected as the irrotational boundary with ∇*ω*^{2} ⋅ ∇*ϕ* < 0. The conditional mean scalar, *ϕ*_{I}, also changes in the T/NT interface layer, and is adjusted between the turbulent and non-turbulent regions. The jump in *ϕ*_{I} is very similar in all DNS, and the thickness of this jump scales with the Kolmogorov scale at *Sc* = 1 and in the Re range studied here. Because of the difference in *ϕ**ζ*_{I} = − 4.9*η* in the T/NT interface layer. This location is shown in Figure 7(a) by the vertical line, and is close to the inflection point of *ϕ*_{I} and to the boundary between the viscous superlayer and the turbulent sublayer.

The strain-rate tensor *S*_{ij} plays an important role in small-scale dynamics of turbulence. The interaction between strain and vorticity leads to the vortex stretching *ω*_{i}*S*_{ij}, and in turn to enstrophy production *ω*_{i}*S*_{ij}*ω*_{j}. The gradient of passive scalar, *G*_{i} = ∂*ϕ*/∂*x*_{i}, is also affected by the strain field via the straining term − *G*_{i}*S*_{ij}, which appears as the production term of *χ*. The effective strains acting on the vorticity and the scalar gradient are written as follows [29]:

where *α*_{ω} and *γ*_{χ} are the production rates of enstrophy and of scalar dissipation rate, respectively [30]. Note that the vortex stretching and the compression of the scalar gradient are denoted by positive *α*_{ω} and *γ*_{χ}, respectively, and positive values of the effective strains contribute to the amplification of enstrophy and of scalar dissipation rate. Figure 8 shows the conditional average of *α*_{ω} and *γ*_{χ} normalized by the strain product on the centerline *S*_{ij}*S*_{ij}_{C}. The profiles are almost independent of the flows. *α*_{ω}_{I} and *γ*_{χ}_{I} decrease toward the interface in the turbulent core region, but peaks can be found in the turbulent sublayer, where the amplification of enstrophy and scalar dissipation rate becomes more efficient. This results in a predominance of the enstrophy production over the viscous dissipation in the T/NT interface layer (*P*_{ω}_{I} > |*ε*_{ω}_{I}| in Figure 6).

### 3.3. Lagrangian statistics of entrained fluid particles

The Lagrangian particle tracking is used for investigating the vorticity growth during the entrainment of non-turbulent fluids. Once the flows have reached the self-similar regime, 140,000 particles are seeded in the non-turbulent regions near the irrotational boundary. The particles are tracked with a third-order Runge-Kutta method and a trilinear interpolation scheme [31]. The flow characteristics are changed depending on the location in the T/NT interface layer. Therefore, it is important to know the entrained fluid particle location within the T/NT interface layer for better understanding of the Lagrangian properties of the entrainment. Because of the T/NT interface movement, the entrained particle tracking does not show the location in the T/NT interface layer. Here, in addition to the fluid particles, the irrotational boundary is also tracked with a marker, which moves with the velocity of the enstrophy isosurface movement **u**_{I}. As in Figure 9(a), the marker is placed on the irrotational boundary where the fluid particle has crossed. **u**_{I} is the sum of the fluid velocity at the irrotational boundary **u**_{0} and the propagation velocity of the enstrophy isosurface **u**_{P} = *v*_{E}**n**, where *v*_{E} = (*Dω*^{2}/*Dt*)/|∇*ω*^{2}|. It was shown that only a negligible fraction of entrained fluid particles is trapped inside a non-turbulent region completely surrounded by turbulent fluids [18]. Because the irrotational boundary of this region disappears after it becomes turbulent, the markers of this irrotational boundary are no longer located on the enstrophy isosurface. Therefore, |**ω**| on the markers is monitored at every time step, and markers with |**ω**| > 2**ω**_{th} are excluded from the subsequent analysis.

The Lagrangian statistics are calculated for the fluid particles, conditioned on the time *τ* elapsed after a fluid particle has crossed the irrotational boundary, and the Lagrangian conditional average is denoted by _{τ}. A separation vector *δ***x** is introduced as in Figure 9(a), and is used for examining the particle location in the T/NT interface layer.

Figure 9(b) shows the Lagrangian conditional average of *δx* = |*δ***x**|, where *δx* and *τ* are normalized by the Kolmogorov length scale *η* and time scale *τ*_{η} at the time when the fluid particles are seeded. The plots are quite similar for small *τ* in all DNS. It takes about 7*τ*_{η} for the entrained particles to reach the turbulent sublayer by moving across the viscous superlayer. A difference in *δx* becomes clear in the turbulent sublayer; the time needed for the particles to move across the turbulent sublayer changes depending on the flow configuration and Reynolds number. The relation between *δx*_{τ} and *τ* is used for relating the Lagrangian statistics with the interface structure by plotting the Lagrangian statistics, which is a function of *τ*, against *δx*_{τ}.

The separation vector *δ***x**(*τ*), a fluid particle location relative to a marker of the irrotational boundary, changes as

where *δ***u** is the fluid particle velocity in relation to the velocity of the marker of the irrotational boundary, and is simply referred to as the relative velocity. The dot product of Eq. (8) with *δ***x** yields the following equation [32]:

The relative velocity can be decomposed into the two components: the irrotational boundary propagation velocity (**u**_{P}) and the fluid velocity difference (**u** − **u**_{0}) between the fluid particle and the location of the marker of the irrotational boundary:

For small *τ*, we can assume that the fluid particles are located in the proximity of the irrotational boundary [14], and the fluid velocity is almost the same between the locations of the fluid particle and the marker of the irrotational boundary. Then, |**u**_{P}| ≫ |**u** − **u**_{0}| ≈ 0, and the relative velocity can be approximated by *δ***u**(*τ*) ≈ − **u**_{P}(0) [33]. Thus, Eq. (9) is simply,

Integration of Eq. (11) yields *v*_{E} is taken at *τ* = 0. Thus, the Lagrangian conditional root-mean-squared distance changes with

It was shown that the propagation velocity scales with the Kolmogorov velocity [14]. By contrast, the fluid velocity difference between two points can be much larger in turbulent flows. Therefore, once the fluid particle has reached far away from the irrotational boundary, the fluid velocity difference can be large compared with the propagation velocity. Then, in the case of *δ***u** ≈ **u** − **u**_{0}, the fluid particle movement in relation to the irrotational boundary is described as the two-particles dispersion problem [34]. Similar to Richardson’s law for the relative diffusion, under the assumption that *δx* changes by eddies of size *δx*(*τ*) [35], we can obtain the following relationship in the self-similar regime:

where *C* is a constant, and the mean kinetic energy dissipation rate *ε* is time dependent. Here, we use *ε* obtained in the turbulent core region. This expression can be obtained with the modified Richardson’s law for decaying turbulence [36, 37] in the self-similar regime, where *ε*(*t*) decays as *ε*(*t*) ~ *t*^{− n}.

Figure 10(a) shows *τ*/*τ*_{η} ≲ 10, Eq. (12) well predicts *τ*/*τ*_{η} ≤ 7), *δ***x** is changed by the irrotational boundary propagation with only a negligible influence of the fluid velocity. Since the irrotational boundary is located at the outer edge of the T/NT interface, where the enstrophy grows by the viscous diffusion with only a negligible influence of vortex stretching, the outward enstrophy diffusion causes the fluid particles to reach the turbulent sublayer. Figure 10(b) shows *τ*. The plots of *τ* ≈ 0, Eq. (12) yields *τ*, Eq. (13), is satisfied from *τ*/*τ*_{η} ≈ 9, which is the time slightly after the particles reach the turbulent sublayer (see Figure 9(b)). Eq. (13) is valid for larger *τ*/*τ*_{η}, including the entire turbulent sublayer. The values of the constant *C*, obtained with the least-squares methods, are between 0.25 and 0.30 as displayed in Figure 10(b). The important assumption behind the relationship, Eq. (13), is that the fluid particle movement in relation to the irrotational boundary is caused solely by eddies of size *δx* without viscous effects nor eddies with different sizes. The entrained particles within the turbulent sublayer, *δ*_{ν} ≲ *δx* ≲ *δ*_{ν} + *δ*_{ω}, obey Eq. (13), indicating that the particle movement is caused by the small-scale eddy motions whose size is from *δ*_{ν} to *δ*_{ν} + *δ*_{ω}. These small-scale eddies with core radius of about 5*η* (≈*δ*_{ν}) were found within the turbulent sublayer as intense vorticity structures [38].

Figure 10(c) shows the pdf of the cosine of the angle between the separation vector *δ***x** and the irrotational boundary normal **n**. Because the particle location within the viscous superlayer changes with the irrotational boundary propagation, whose direction is given by **n**, the particle in the viscous superlayer stays in the normal direction of the irrotational boundary. This is confirmed by a large peak in the pdf associated with a parallel alignment of *δ***x** and **n**.

Figure 11 shows the Lagrangian conditional mean enstrophy *ω*^{2}/2_{τ}, where the inset plots *ω*^{2}/2_{τ} against *δx*_{τ} for comparison with the Eulerian statistics in Figure 5. Once the particle moves into the T/NT interface layer, the enstrophy begins to grow. The inset shows that even after the particle reaches deep inside the turbulent region, the mean enstrophy on the particle path is much smaller than the Eulerian conditional mean enstrophy in Figure 5(a). It should be noted that the Lagrangian statistics are obtained only from the fluid being entrained, whereas the Eulerian statistics contain the contributions from the entrained fluid and the fluid from the turbulent core region. This makes differences between the Lagrangian and Eulerian statistics, and the Eulerian statistics are not enough for studying the entrainment process across the T/NT interface layer.

Figure 12 shows the Lagrangian conditional statistics of the enstrophy budget, where again the Lagrangian statistics are plotted against *δx*_{τ} for comparison with the Eulerian statistics in Figure 6. Qualitative differences can be found between the Eulerian and Lagrangian conditional means of *D*_{ω}. The Eulerian *D*_{ω}_{I} displays both positive and negative values indicating an outward mean enstrophy transport, whereas the Lagrangian *D*_{ω}_{τ} is positive even for large *δx*_{τ}. Thus, although the fluid being entrained possesses an important level of enstrophy in the T/NT interface layer, the enstrophy transport toward the non-turbulent region is hardly associated with this entrained fluid. The Lagrangian enstrophy production and dissipation terms are smaller than their corresponding Eulerian counterparts. Note that these terms are proportional to the enstrophy, and this difference between Lagrangian and Eulerian statistics seems to be due to a smaller enstrophy level on the entrained particle path.

Figure 13(a) compares the Eulerian and Lagrangian conditional averages of the second invariant of velocity gradient tensor *Q* = (*ω*_{i}*ω*_{i} − 2*S*_{ij}*S*_{ij})/4. A large positive value of *Q* implies the predominance of vorticity over the strain while its negative value is related to where dissipation is dominant. The vortex core region of an eddy often has positive *Q* while negative *Q* appears around the core region [35]. The Eulerian *Q*_{I} has a negative peak near the irrotational boundary and a large positive peak inside the turbulent region. However, the Lagrangian *Q*_{τ} is negative even for large *δx*_{τ} in the turbulent core region. The third invariant of the velocity gradient tensor is defined by *R* = − (*S*_{ij}*S*_{jk}*S*_{ki}/3 − *ω*_{i}*S*_{ij}*ω*_{j}/4), and the joint pdf of *Q* and *R* has been used for investigating the local flow topology in various turbulent flows [39–41]. Figure 13(b) and (c) compares the joint pdf of *Q* and *R* obtained as the Eulerian and Lagrangian statistics in the turbulent sublayer. Both Eulerian and Lagrangian pdfs show a “teardrop” shape similar to various turbulent flows, but a difference is found for large positive *Q*; the probability of finding intense values of *Q* ≫ 0 is smaller in the Lagrangian pdf than in the Eulerian counterpart. These statistics of *Q* show that although there are regions with *Q* ≫ 0 within the T/NT interface layer, the fluid particles being entrained tend to circumvent these regions. The regions with *Q* ≫ 0 can be related to the core of the intense eddies. Thus, a circular motion induced by these eddies may explain this entrained particle path.

## 4. Conclusion

The DNS of planar jets and mixing layers was performed for investigating the vorticity dynamics near the T/NT interface. The outer edge of the T/NT interface layer, irrotational boundary, is detected as an isosurface of the vorticity magnitude. The Eulerian and Lagrangian statistics were investigated in this study. The former was calculated conditioned on the distance from the irrotational boundary. For investigating the Lagrangian properties of the entrainment, a large number of fluid particles are seeded in the non-turbulent region of the self-similar regime. The Lagrangian statistics were calculated as a function of time elapsed after the particle crosses the irrotational boundary. Furthermore, a marker of the irrotational boundary is also tracked with the velocity of the enstrophy isosurface movement, and is used for examining the fluid particle location within the T/NT interface layer.

The Eulerian conditional mean enstrophy and its budget showed that the T/NT interface is a layer with the thickness of about 15*η*, and consists of the viscous superlayer and the turbulent sublayer with the thickness of *δ*_{ν} ≈ 4*η* and *δ*_{ω} ≈ 11*η*, respectively. It was also found that the amplification of the vorticity and scalar gradient is efficient in the turbulent sublayer. The passive scalar exhibits a jump in the T/NT interface layer, where the large scalar dissipation rate appears near the boundary between the viscous superlayer and the turbulent sublayer in all simulations.

The Lagrangian statistics of the entrained particle and the marker of the irrotational boundary showed that it takes about 7*τ*_{η} for the entrained particle to pass across the viscous superlayer. The mean-squared distance exhibited two different scalings depending on the location: a ballistic evolution in the viscous superlayer and the Richardson-like scaling for the relative dispersion in the turbulent sublayer. These scalings indicate that the different mechanisms govern the entrained fluid movement between the two layers. A ballistic evolution was explained reasonably based on the irrotational boundary propagation, which arises from the viscous diffusion of vorticity. The Richardson-like scaling implies the importance of inviscid motions of small-scale eddies in the entrainment. The Lagrangian statistics also showed that although the fluid being entrained possesses an important level of enstrophy, it does not contribute, in a mean sense, to the viscous diffusion of vorticity to the irrotational region. Thus, the fluid existing in the turbulent core region plays an important role in the vorticity diffusion near the T/NT interface. The *Q* − *R* analysis indicates that the entrained fluid path appears around the core of the small-scale eddy structures related to very large *Q*.

## Acknowledgments

This work was supported by JSPS KAKENHI Grant numbers 25289030 and 16K18013. C.B. da Silva acknowledges IDMEC, under LAETA projects PTDC/EME-MFE/122849/2010 and UID/EMS/50022/2013. The authors acknowledge Prof. Yasuhiko Sakai (Nagoya University) for a number of insightful comments. A part of the results presented in this chapter was published in Physics of Fluids (http://scitation.aip.org/content/aip/journal/pof2/28/3/10.1063/1.4942959).