Physical and computational parameters of the DNS. The displayed turbulence characteristics are from the turbulent core regions.
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.
- mixing layer
- turbulent/non-turbulent interface
- Lagrangian statistics
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 , generated in the stably stratified fluid, are sometimes responsible for the transport of heat, salinity, and plankton. The atmospheric boundary layer  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 , 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) . Furthermore, the statistical approach conditioned relative to the interface  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 . 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  and engulfment , respectively. The experiments in the boundary layers indicated that the entrainment is the multi-scale process . The entrainment across the interface was studied in  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 . 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 . 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
|Flow type||Mixing layer||Mixing layer||Planar jet||Planar jet|
||512||1 024||512||1 024|
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 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 . Therefore, following , 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 right-hand side is the enstrophy production
Figure 7 gives the conditional plots of passive scalar
The strain-rate tensor
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 . 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
The relative velocity can be decomposed into the two components: the irrotational boundary propagation velocity (
Integration of Eq. (11) yields , where
It was shown that the propagation velocity scales with the Kolmogorov velocity . 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
Figure 10(a) shows for comparison between the DNS results and Eq. (12). For
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
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
The Lagrangian statistics of the entrained particle and the marker of the irrotational boundary showed that it takes about 7
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).
Thorpe SA. The near-surface ocean mixing layer in stable heating conditions. J. Geophys. Res. 1978; 83:2875–2885. DOI: 10.1029/JC083iC06p02875
Mahrt L. Stratified atmospheric boundary layers. Boundary-Layer Meteorol. 1999; 90:375–396. DOI: 10.1023/A:1001765727956
Corrsin S, Kistler AL. Free-stream boundaries of turbulent flows. NACA Technical Report No. TN-1244; 1955. DOI: 19930092246
Taveira RR, da Silva CB. Characteristics of the viscous superlayer in shear free turbulence and in planar turbulent jets. Phys. Fluids 2014; 26:021702. DOI: 10.1063/1.4866456
Bisset DK, Hunt JCR, Rogers MM. The turbulent/non-turbulent interface bounding a far wake. J. Fluid Mech. 2002; 451:383–410. DOI: 10.1017/S0022112001006759.
da Silva CB, Hunt JCR, Eames I, Westerweel J. Interfacial layers between regions of different turbulence intensity. Annu. Rev. Fluid Mech. 2014; 46:567–590. DOI: 10.1146/annurev-fluid-010313-141357
van Reeuwijk M, Holzner M. The turbulence boundary of a temporal jet. J. Fluid Mech. 2014; 739:254–275. DOI: 10.1017/jfm.2013.613
Watanabe T, Sakai Y, Nagata K, Ito Y, Hayase T. Turbulent mixing of passive scalar near turbulent and non-turbulent interface in mixing layers. Phys. Fluids 2015; 27:085109. DOI: 10.1063/1.4928199
Westerweel J, Fukushima C, Pedersen JM, Hunt JCR. Mechanics of the turbulent-nonturbulent interface of a jet. Phys. Rev. Lett. 2005; 95:174501. DOI: 10.1103/PhysRevLett.95.174501
Townsend AA. The Structure of Turbulent Shear Flow. Cambridge University Press; Cambridge 1976.
Philip J, Meneveau C, de Silva CM, Marusic I. Multiscale analysis of fluxes at the turbulent/non-turbulent interface in high Reynolds number boundary layers. Phys. Fluids 2014; 26:015105. DOI: 10.1063/1.4861066
Holzner M, Lüthi B. Laminar superlayer at the turbulence boundary. Phys. Rev. Lett. 2011; 106:134503. DOI: 10.1103/PhysRevLett.106.134503
Wolf M, Lüthi B, Holzner M, Krug D, Kinzelbach W, Tsinober A. Investigations on the local entrainment velocity in a turbulent jet. Phys. Fluids 2012; 24:105110. DOI: 10.1063/1.4761837
Wolf M, Holzner M, Lüthi B, Krug D, Kinzelbach W, Tsinober A. Effects of mean shear on the local turbulent entrainment process. J. Fluid Mech. 2013; 731:95–116. DOI: 10.1017/jfm.2013.365
de Silva CM, Philip J, Chauhan K, Meneveau C, Marusic I. Multiscale geometry and scaling of the turbulent-nonturbulent interface in high Reynolds number boundary layers. Phys. Rev. Lett. 2013; 111:044501. DOI: 10.1103/PhysRevLett.111.044501
Deo RC, Nathan GJ, Mi J. Similarity analysis of the momentum field of a subsonic, plane air jet with varying jet-exit and local Reynolds numbers. Phys. Fluids 2013; 25:015115. DOI: 10.1063/1.4776782
Holzner M, Liberzon A, Nikitin N, Lüthi B, Kinzelbach W, Tsinober A. A Lagrangian investigation of the small-scale features of turbulent entrainment through particle tracking and direct numerical simulation. J. Fluid Mech. 2008; 598:465–475. DOI: 10.1017/S0022112008000141
Taveira RR, Diogo JS, Lopes DC, da Silva CB. Lagrangian statistics across the turbulent-nonturbulent interface in a turbulent plane jet. Phys. Rev. E 2013; 88:043001. DOI: 10.1103/PhysRevE.88.043001
Watanabe T, da Silva CB, Sakai Y, Nagata K, Hayase T. Lagrangian properties of the entrainment across turbulent/non-turbulent interface layers. Phys. Fluids 2016; 28:031701. DOI: 10.1063/1.4942959
Morinishi Y, Lund TS, Vasilyev OV, Moin P. Fully conservative higher order finite difference schemes for incompressible flow. J. Comput. Phys. 1998; 143:90–124. DOI: 10.1006/jcph.1998.5962
Kempf A, Klein M, Janicka J. Efficient generation of initial-and inflow-conditions for transient turbulent flows in arbitrary geometries. Flow Turbul. Combust. 2005; 74:67–84. DOI: 10.1007/s10494-005-3140-8
Watanabe T, Sakai Y, Nagata K, Terashima O. Experimental study on the reaction rate of a second-order chemical reaction in a planar liquid jet. AIChE J. 2014; 60:3969–3988. DOI: 10.1002/aic.14610
Bell JH, Mehta RD. Development of a two-stream mixing layer from tripped and untripped boundary layers. AIAA J. 1990; 28:2034–2042. DOI: 10.2514/3.10519
Tanahashi M, Iwase S, Miyauchi T. Appearance and alignment with strain rate of coherent fine scale eddies in turbulent mixing layer. J. Turbulence 2001; 2:1–18. DOI: 10.1088/1468-5248/2/1/006
Kitamura T, Nagata K, Sakai Y, Sasoh A, Terashima O, Saito H, Harasaki T. On invariants in grid turbulence at moderate Reynolds numbers. J. Fluid Mech. 2014; 738:378–406. DOI: 10.1017/jfm.2013.595
Uberoi MS, Freymuth P. Turbulent energy balance and spectra of the axisymmetric wake. Phys. Fluids 1970; 13:2205–2210. DOI: 10.1063/1.1693225
Attili A, Cristancho JC, Bisetti F. Statistics of the turbulent/non-turbulent interface in a spatially developing mixing layer. J. Turbulence 2014; 15:555–568. DOI: 10.1080/14685 248.2014.919394
da Silva CB, Taveira RR. The thickness of the turbulent/nonturbulent interface is equal to the radius of the large vorticity structures near the edge of the shear layer. Phys. Fluids 2010; 22:121702. DOI: 10.1063/1.3527548
Smyth WD. Dissipation-range geometry and scalar spectra in sheared stratified turbulence. J. Fluid Mech. 1999; 401:209–242. DOI: 10.1017/S0022112099006734
Tsinober A. An informal conceptual introduction to turbulence. ***Springer; Berlin 2009. DOI: 10.1007/978-90-481-3174-7
Yang Y, Wang J, Shi Y, Xiao Z, He XT, and Chen S. Acceleration of passive tracers in compressible turbulent flow. Phys. Rev. Lett. 2013; 110:064503. DOI: 10.1103/PhysRevLett.110.064503
Batchelor GK. The application of the similarity theory of turbulence to atmospheric diffusion. Q. J. R. Meteorol. Soc. 1950; 76:133–146. DOI: 10.1002/qj.49707632804
Watanabe T, Sakai Y, Nagata K, Ito Y, Hayase T. Vortex stretching and compression near the turbulent/nonturbulent interface in a planar jet. J. Fluid Mech. 2014; 758:754–785. DOI: 10.1017/jfm.2014.559
Salazar JPLC, Collins LR. Two-particle dispersion in isotropic turbulent flows. Annu. Rev. Fluid Mech. 2009; 41:405–432. DOI: 10.1146/annurev.fluid.40.111406.102224
Davidson PA. Turbulence: An Introduction for Scientists and Engineers. Oxford ***University Press; Oxford 2004.
Larcheveque M, Lesieur M. The application of eddy-damped Markovian closures to the problem of dispersion of particle pairs. J. Méc. 1981; 20:113–134. DOI: 10.1007/978-3-540-32603-8_ 44
Nelkin M, Kerr RM. Decay of scalar variance in terms of a modified Richardson law for pair dispersion. Phys. Fluids 1981; 24:1754–1756. DOI: 10.1063/1.863597
da Silva CB, dos Reis RJN, Pereira JCF. The intense vorticity structures near the turbulent/non-turbulent interface in a jet. J. Fluid Mech. 2011; 685:165–190. DOI: 10.1017/jfm.2011.296
Soria J, Sondergaard R, Cantwell BJ, Chong MS, Perry AE. A study of the fine-scale motions of incompressible time-developing mixing layers. Phys. Fluids 1994; 6:871–884. DOI: 10.1063/1.868323
Blackburn HM, Mansour NN, Cantwell BJ. Topology of fine-scale motions in turbulent channel flow. J. Fluid Mech. 1996; 310:269–292. DOI: 10.1017/S0022112096001802
Zhou Y, Nagata K, Sakai Y, Suzuki H, Ito Y, Terashima O, Hayase T. Development of turbulence behind the single square grid. Phys. Fluids 2014; 26:045102. DOI: 10.1063/1.4870167