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 nonturbulent flows, the socalled turbulent/nonturbulent (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 strainvorticity 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 Richardsonlike 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 smallscale 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/nonturbulent interface
 DNS
 Lagrangian statistics
1. Introduction
Interfaces dividing turbulent and nonturbulent 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/nonturbulent (T/NT) interfaces. Turbulence is generated by shear motions in various circumstances, where the turbulent fluids are surrounded by nonturbulent fluids. This locally generated turbulence often plays an important role in the relevant phenomena. For example, oceanmixing 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 nonturbulent region with the mass, momentum, and energy exchanges across the T/NT interface.
Corrsin and Kistler [3], in laboratory experiments with hotwire probes, found that the essential feature of the turbulent regions is the high vorticity, and the turbulent and nonturbulent regions can be distinguished by the vorticity. They also predicted that a very thin layer where the nonturbulent 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 highresolution 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 nonturbulent 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 nonturbulent 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 multiscale 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
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 smallscale 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 (
The initial velocity field is obtained by superimposing the statistically homogeneous and isotropic velocity fluctuations onto the mean velocity, which is given by
Here,
Run  ML04  ML08  PJ50  PJ90 

Flow type  Mixing layer  Mixing layer  Planar jet  Planar jet 
Re  400  800  5000  9000 

16 
16 
2.4 
2.6 

16 
16 
4.8 
3.8 

8 
8 
2.4 
1.3 

512  1 024  512  1 024 

500  700  850  1 150 

256  512  512  512 
Time step 
0.08 
0.04 
0.012 
0.006 
Δ_{x} = Δ_{z}  1.5 
1.2 
1.5 
1.4 
Δ_{y} ( 
1.0 
1.1 
1.2 
1.2 
Re_{λ}  105  151  94  158 

0.064 
0.041 
0.0096 
0.0059 
20.8 
23.3 
14.8 
20.3 
The Reynolds numbers Re are defined by
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 selfsimilar profiles of mean velocity and rms velocity fluctuations. The present DNS reproduces well the selfsimilar profiles of these statistics in previous studies. Figure 2 shows the onedimensional longitudinal spectrum on the centerline with the experimental plots. We can see the overlap of the spectrum in small scales, and the smallscale turbulent fluctuations are well resolved in the DNS.
3. Analysis on turbulent and nonturbulent 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 
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,
Figure 5 shows the conditional mean enstrophy profiles. The mean enstrophy is matched in the layer with the thickness of ≈ 15
The vorticity evolution near the interface is studied by the enstrophy transport equation:
where the first term on the righthand side is the enstrophy production
Figure 7 gives the conditional plots of passive scalar
The strainrate tensor
where
3.3. Lagrangian statistics of entrained fluid particles
The Lagrangian particle tracking is used for investigating the vorticity growth during the entrainment of nonturbulent fluids. Once the flows have reached the selfsimilar regime, 140,000 particles are seeded in the nonturbulent regions near the irrotational boundary. The particles are tracked with a thirdorder RungeKutta 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
The Lagrangian statistics are calculated for the fluid particles, conditioned on the time
Figure 9(b) shows the Lagrangian conditional average of
The separation vector
where
The relative velocity can be decomposed into the two components: the irrotational boundary propagation velocity (
For small
Integration of Eq. (11) yields
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
where
Figure 10(a) shows
Figure 10(c) shows the pdf of the cosine of the angle between the separation vector
Figure 11 shows the Lagrangian conditional mean enstrophy
Figure 12 shows the Lagrangian conditional statistics of the enstrophy budget, where again the Lagrangian statistics are plotted against
Figure 13(a) compares the Eulerian and Lagrangian conditional averages of the second invariant of velocity gradient tensor
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 nonturbulent region of the selfsimilar 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
The Lagrangian statistics of the entrained particle and the marker of the irrotational boundary showed that it takes about 7
Acknowledgments
This work was supported by JSPS KAKENHI Grant numbers 25289030 and 16K18013. C.B. da Silva acknowledges IDMEC, under LAETA projects PTDC/EMEMFE/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).
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