Structure of turbulence in temporal planar jets

A detailed analysis of the structure of turbulence in a temporal planar turbulent jet is reported. Instantaneous snapshots of the ﬂow and three-dimensional spatial correlation functions are considered. It is found that the ﬂow is characterized by large-scale spanwise vortices whose motion is felt in the entire ﬂow ﬁeld. Superimposed to this large-scale motion, a hierarchy of turbulent structures is present. The most coherent ones take the form of quasi-streamwise vortices and high and low streamwise velocity streaks. The topology of these interacting structures is analyzed by quantitatively addressing their shape and size in the different ﬂow regions. Such information is recognized to be relevant for a structural description of the otherwise disorganized motion in turbulent free-shear ﬂows and can be used for the assessment of models based on coherent structure assumptions. Finally, the resulting scenario provides a phenomenological description of the elementary processes at the basis of turbulence in free-shear ﬂows


I. INTRODUCTION
Research on turbulence generally aims at providing a description of the phenomenon that is simpler than that given by the Navier-Stokes equations.To this purpose, almost every approach to turbulence is essentially based on a decomposition of the turbulent field.Historically, famous examples are the Reynolds decomposition in a mean and fluctuating flow and the spectral decomposition in a hierarchy of scales of motion.More recently, research efforts often rely on the decomposition of turbulence in random processes and coherent structures. 1 The interest is in describing turbulence in terms of a small number of relevant elementary processes.
In the present work, we aim at identifying and characterizing such relevant structures of turbulence in free-shear flows by using direct numerical simulation (DNS) data of a temporal planar turbulent jet.In the context of free-shear flows, the assessment of the statistically relevant structure of turbulence [2][3][4][5][6][7] is of importance for a number of scientific and applicative issues.An example is the study of turbulent entrainment. 8,9It is well-known that entrainment is not only diffusion of vorticity at the turbulent interface.1][12] In particular, the anisotropy induced by large-scale structures was found in Cimarelli and Boga 13 to represent a fundamental phenomenon for the effective sustainment of the entrainment and mixing processes.5][16] As shown in da Silva and Me `tais, 17 the most intense energy exchanges between large and small scales are related to the presence of coherent structures in the flow, such as spanwise and longitudinal vortices.From an application point of view, the identification of the coherent structure of turbulence is highly relevant for flow control.A structural representation of turbulence is indeed known to be effective in identifying the action mechanisms to manipulate specific turbulence properties. 18,19he study of the structure of turbulence in free-shear flows dates back some time ago.Jim enez 20 showed that superimposed on the primary spanwise vortex structure of plane mixing layers, a second instability develops in the form of an undulation of the primary vortex core.This picture was later refined by Bernal and Roshko, 21 where this deformation was shown to evolve in longitudinal vortex structures, see also Liepmann and Gharib, 4 Nogueira et al. 22 The structure of turbulence in free-shear layers has been also analyzed in flows with separation.In this context, it is shown that the flow first develops spanwise vortices as a result of the two-dimensional shear instability.Perturbations of the flow field lead to a lift up and stretching of the primary spanwise vortices, thus forming hairpin-like structures 23 that by flowing downstream undergo stretching giving rise to streamwise vortices. 24As shown by Lasheras and Choi, 25 streamwise vortices are then found to induce entrainment and streamwise velocity streaks.More recently, this picture has been quantitatively assessed in Cimarelli et al. 26 by means of a statistical analysis of three-dimensional spatial velocity correlation.
The present work is devoted to the statistical characterization of the structure of turbulence in the plane shear layers developed in a temporal jet.We start by reporting the numerical settings adopted for the direct numerical simulation in Sec.II A and by briefly describing the main flow features in Sec.II B. Important considerations on the domain size are reported in Sec.II C. The structure of turbulence is analyzed first in Sec.III by addressing the instantaneous flow topology.Then, in Sec.V, a characterization of the statistically relevant flow structures is reported by means of a detailed analysis of the threedimensional spatial velocity correlation functions that in turn are defined in Sec.IV.The work is closed by final remarks in Sec.VI.

II. TURBULENT TEMPORAL JET A. Direct numerical simulation
In the present work, we consider direct numerical simulation (DNS) data of a temporal plane jet performed by solving the Navier-Stokes equations coupled with the evolution equation of a passive scalar, Equation ( 1) is written in a dimensionless form by using the initial jet half-width H, the initial jet centerline velocity U 0 and the initial centerline scalar concentration H 0 .Here, Re ¼ U 0 H= is the Reynolds number, where is the kinematic viscosity while Sc ¼ =a is the Schmidt number where a is the diffusivity of the scalar.The indices i ¼ 1, 2, 3 correspond to the streamwise, spanwise, and cross-flow velocities (u, v, w) and directions (x, y, z), respectively.The numerical method used to solve the problem is based on a finite volume discretization of the governing equations, see Craske and van Reeuwijk 27 for details.The symmetry-preserving method of Verstappen and Veldman 28 is employed, thus allowing for mass, momentum, and energy conservation and for a fourth-order accuracy.To advance the solution in time a third-order variable-time step, Adams-Bashforth scheme is finally used.Periodic boundary conditions are imposed in the streamwise and spanwise directions, free-slip on the cross-flow boundaries.The initial condition is a fluid layer that is quiescent and characterized by a null concentration of the scalar except for a thin region ÀH < z < H where the streamwise velocity and the scalar are non-zero and homogeneously distributed in the streamwise and spanwise directions, uðx; y; z; where r 0 ¼ 2H=35 is the initial momentum thickness.In order to facilitate a rapid transition to turbulence, a perturbation is superimposed at the initial conditions (2) consisting in a uniform random noise with an intensity that is 4% the maximum initial velocity.
The flow problem considered here is for Re ¼ 1000 and Sc ¼ 1.The computational domain is a cuboid of size 96H Â 96H Â 48H and has been discretized by using 2304 Â 2304 Â 960 uniformly spaced grid points.By using the smallest value of the Kolmogorov scale g min obtained at the centerline of the jet for t ¼ 40, the corresponding resolution is Dx ¼ Dy % 1:66g min and Dz % 2g min .By considering that the Kolmogorov scale increases from this minimum in time as ffiffi t p and with the cross-flow position z since dissipation is maximum at the centerline, the resolution adopted is considered appropriate for a DNS.Finally, a variable time step was used for the temporal integration in order to obtain a condition CFL < 0.3.The total integration time for each simulation is t ¼ 160.
Starting from the same initial conditions (2) except for the perturbation superimposed, three simulations have been performed in order to provide three statistically independent samples for the ensemble average.In fact, average quantities denoted as hÁi are computed by performing an ensemble average between the different flow realizations, a spatial average in the statistically homogeneous streamwise and spanwise directions and by taking advantage of the statistical symmetry of the flow in the cross-flow direction.Accordingly, the average of a generic quantity b is defined as where the sum and difference of the two integrals are imposed by the symmetric/antisymmetric nature of the considered variable, L x and L y are the domain dimensions in the homogenous directions, and N ¼ 3 is the number of independent flow samples used.
In what follows, the customary Reynolds decomposition of the flow in a mean and fluctuating field will be adopted, i.e., u i ¼ U i þ u 0 i and h ¼ T þ h 0 .In order to take into account the self-similar behavior of the flow, variables will be presented dimensionless by using the mean centerline velocity U cl ðtÞ ¼ Uð0; tÞ and the jet half-width h X ðtÞ, defined as the jet centerline distance where the mean enstrophy profile reaches 2% of its centerline value.The related non-dimensional flow variables will be denoted with a superscript Ã.

B. Main flow features
As shown in van Reeuwijk and Holzner 29 and da Silva and Pereira, 30 after an initial transient when the system develops turbulence, the flow approaches a self-similar regime while decaying.This dynamic equilibrium is characterized by an increase in the jet length scales as $ ffiffi t p and by a decrease in the jet velocity scales as $1= ffiffi t p .As shown in Cimarelli et al. 31 where the same data-set is used, all the flow observables are found to follow this self-similar scaling for t > 60.Accordingly, flow statistics are here reported for t ¼ 120, thus allowing us to characterize the flow structure in the self-similar regime.A constant Taylor Reynolds number, Re k ¼ ffiffiffiffiffiffiffiffiffiffiffiffi 2k cl =3 p k cl = ¼ 50, is also achieved in the self-similar regime, where k cl is the centerline turbulent kinetic energy and k ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 10k cl =e cl p is the Taylor microscale with e cl the centerline turbulent dissipation.3][34] It is, however, sufficiently high to observe and study the large-scale coherent structures of the flow and their interactions as demonstrated in the rest of the work.
In Fig. 1(a), the mean velocity and turbulent kinetic energy profiles, U Ã ðzÞ and k Ã ðzÞ, respectively, are shown.The inner part of the jet is characterized by high levels of velocity and turbulent kinetic energy.Moving away from the jet centerline, the mean velocity decreases while the turbulent kinetic energy increases reaching a maximum for z Ã % 0:3.This region of the flow is also characterized by the highest values of mean shear and, as shown in Cimarelli et al., 31 of turbulence production.By further increasing the jet centerline distance, both the mean shear and the turbulent kinetic energy decrease asymptotically reaching zero values in the external region of the jet.
The one-dimensional streamwise turbulent spectrum evaluated in the production region at z Ã ¼ 0:35 is shown in Fig. 1(b).As shown in the main plot, the spectrum of fluctuations exhibits a small inertial range, less than one decade, where the power law k À5=3 x is followed.This range takes origin from very large length scales of the order of k Ã x ¼ 2p=k Ã x % 1:5 and is followed by a viscosity-dominated scaling for k Ã x < 0:2.As shown in the inset panel of Fig. 1(b), the premultiplied streamwise turbulent spectrum highlights that the length scale k Ã x % 1:5 origin of the inertial range scaling is also the most energetic one.It is worth noting that this relevant length scale is larger than the jet half-width h X , thus highlighting the relevance of using large computational domains in order to correctly capture the turbulence dynamics of temporal jets.This aspect is further addressed in Sec.II C.

C. Effect of the domain size
The study of the topology of turbulence requires a detailed verification of the influence of the numerical domain extent on the flow structures.To address this issue, in the present section flow variables will be presented dimensionless by using the initial jet half-width H for lengths and the initial jet centerline velocity U 0 for velocities, i.e., the numerically solved flow variables presented in Sec.II A. Indeed, in numerical simulations, H and U 0 are prescribed with the initial conditions and, hence, can be used to define a suitable dimensionless form for the equations of motion and for the domain dimensions.
The computational domain used in the present work has dimensions 96 Â 96 Â 48, see again Sec.II A. This domain is significantly larger than the 24 Â 24 Â 36 domain used in van Reeuwijk and Holzner 29 and even more than the 8 Â 8 Â 12 domain used in da Silva and Pereira. 30o verify the need of such a very large domain, we have performed two additional simulations with the same numerical settings but different domain lengths, specifically 24 Â 24 Â 36 and 72 Â 48 Â 36.
In Fig. 2, the instantaneous vorticity pattern of the simulations with the larger and smaller domain is shown for t ¼ 120.The simulation with the larger domain (top plots) clearly highlights that the fine scale of turbulence is superimposed to very-large motions.As better shown in Sec.III, these motions induce large-scale meandering of the turbulent jet region.On the contrary, the simulation with the smaller domain (bottom plots) does not allow to reproduce such a large-scale spatial modulation of the turbulent jet core.The spatial interaction mechanisms between consecutive large-scale vortex motions cannot be reproduced in the smaller domain.In this case, large-scale motions emerge only through large-scale unsteadiness in time, thus, suppressing the spatial instability mechanisms observed in the simulation with the larger domain.On the contrary, the small scale motion does not appear to be significantly affected by the domain size.
The suppression of large-scale mechanisms due to a limited numerical domain extent strongly impacts the structure of the flow.To analyze this issue, we consider the spatial correlation function of streamwise, spanwise, and cross-flow velocity, R uu , R vv , and R ww , respectively, evaluated at the jet centerline for t ¼ 120.See Sec.IV for the definition.As shown in Fig. 3, significant differences in terms of correlation lengths and shapes by varying the domain size are present.In particular, only the largest domain size appears to show a clear decorrelation of the velocity field.Also the intermediate domain extent is found to allow for a decorrelation of the velocity field with small differences with respect to the behavior observed in the larger domain extent.On the contrary, the smaller domain extent is found to produce a substantially different large-scale flow.In particular, the streamwise velocity fluctuations do not decorrelate in the streamwise direction and shows an artificial positive peak for spanwise scales r y % 8.
Also the spanwise velocity fluctuations exhibit artificial negative and positive peaks for streamwise scales r x % 3 and r x % 8, respectively, and do not exhibit decorrelation both in the streamwise and spanwise directions.Finally, the cross-flow fluctuations show an unnatural negative peak of correlation for spanwise scales r y % 7.
These results clearly highlight the need of using large domain extents in order to capture the statistically relevant large-scale structures of the flow.In order to highlight the relevance of this information for the common practice, let us recall that the smaller domain extent analyzed here is the same used in van Reeuwijk and Holzner 29 that is FIG. 2. Instantaneous vorticity pattern x for the simulations with the larger (top) and smaller (bottom) domain extent evaluated at t ¼ 120.The left plots show a y-plane view of the spanwise vorticity, x y 2 ½À0:2; 0:2.The right plots show a x-plane view of the streamwise vorticity, x x 2 ½À0:2; 0:2.FIG. 3. Spatial correlation functions evaluated at the jet centerline for t ¼ 120 for the simulations with large (solid line), intermediate (dashed line), and small (dashed-dotted line) domain.The spatial correlation is computed for the streamwise (left plots), spanwise (central plots), and cross-flow (right plots) velocity fluctuations, i.e., R uu , R vv , and R ww , respectively.The spatial correlation is shown as a function of the streamwise separation r ¼ ðr x ; 0; 0Þ (top plots) and of the spanwise separation r ¼ ð0; r y ; 0Þ (bottom plots).three times larger in each direction than that adopted in da Silva and Pereira. 30Finally, let us emphasize that the use of very large domains also allows to substantially improve the statistical convergence.

III. INSTANTANEOUS FLOW PATTERN
In this section, we further analyze the instantaneous structures of the flow.Figure 4 shows iso-surfaces of the three-dimensional swirl x Á u=juj of the temporal jet.A very complex pattern appears.As highlighted by the inset, these swirling motions are predominantly aligned with the axial direction.In particular, they appear to be superimposed on larger scale motions.As shown in Figs.5(a) and 5(b), the lateral and front views of the iso-contours of the scalar field highlight that these large scale motions take the form of spanwise rolls (lateral view) and of plume-like structures (front view).The common picture of these large scale motions is the following.The flow first develops spanwise vortex tubes as a result of a two-dimensional shear instability.The strain field associated with this spanwise vortex pattern amplifies flow field perturbations leading to the formation of vortex tubes aligned with it (see the swirling motions reported in Fig. 4).In turn, these predominantly streamwise vortices interact with the primary spanwise vortices that created them inducing an undulation of their axis of rotation. 20The lift-up of spanwise vortices within pairs of counter rotating streamwise vortices leads to the formation of spanwise arranged array of mushroom-like structures, see panel 5(b), and, consequently to high-and low-speed streamwise velocity streaks. 21,25nterestingly, the scalar field highlights that the mixing processes are not able to fill the entire space even within the turbulent core.As shown by the top views in Figs.5(c) and 5(d), regions of zero concentration of the scalar are observed also within the turbulent core.Obviously, these regions of unmixed flow are more prevalent in the outer regions of the jet.

IV. THREE-DIMENSIONAL SPATIAL CORRELATION FUNCTIONS
The statistical signature of the previously described flow pattern can be studied by means of two-point statistics, such as the velocity correlation function in physical space.This observable allows us to identify the statistically dominant three-dimensional structures of the flow and to quantitatively assess their topology.For the symmetries of the flow, the spatial correlation function for a generic quantity b can be defined as R bb ðr x ; r y ; r z ; z; tÞ ¼ hb 0 ðx; y; z; tÞb 0 ðx þ r x ; y þ r y ; z þ r z ; tÞi hb 0 b 0 iðz; tÞ : Note that in inhomogeneous turbulence, the two-point correlation function lacks an unique definition.The related technical details are reported in the Appendix.Equation ( 4) highlights that the correlation function, for a fixed time, is defined in a four-dimensional space of separations ðr x ; r y ; r z Þ and positions (z).For each position (z) within the flow, the spatial correlation function allows us to define the ðr x ; r y ; r z Þ of the statistically dominant coherent motions.Due to statistical inhomogeneity in the cross-flow direction and to the presence of a mean flow in the streamwise direction, the threedimensional spatial correlation function (4) among the space of separations is symmetric only in the r y -direction, i.e., R bb ðr x ; r y ; r z ; z; tÞ ¼ R bb ðr x ; Àr y ; r z ; z; tÞ: (5) The following analysis of the behavior of the three-dimensional spatial correlation function allows us to describe the shape of statistically relevant flow structures.In order to give a more quantitative description of their size, we will use the quantity ' j defined as the maximum distance between points of a given three-dimensional correlation iso-surface along the j direction and d j to denote the distance between peaks of positive and negative correlation along the j direction.Accordingly, ' j describes the size of fluctuating velocity structures while d j both the distance between alternating fluctuating velocity structures or the diameter of vortical structures. 35

V. STATISTICALLY RELEVANT FLOW STRUCTURES
For reasons of compactness, the analysis of the threedimensional spatial correlation function of the velocity field is reported in this section for four fixed reference z-locations.As shown in Fig. 1(a) with vertical lines, the selected locations have been chosen in order to characterize the structures of the flow in the inner, production, outer, and external regions of the jet, see Cimarelli et al. 31 for a FIG. 4. Iso-surface of swirl, x Á u=juj ¼ 63:6 colored with streamwise velocity u for t ¼ 120.
description.In the following, these regions will be addressed separately.
It is finally useful to recall that, as shown in Cimarelli et al. 31 and discussed in Secs.II A and II B, the flow observables exhibit a clear self-similar behavior for t > 60.For this reason, the following analysis of the statistically relevant flow structures is reported in self-similar units denoted with a superscript Ã, see their definition in Sec.II A, and for t ¼ 120, so well within the self-similar regime.Accordingly, the reported self-similar size and shape of the three-dimensional correlation functions are expected to be time invariant.

A. Inner region
We start by considering the most relevant turbulent structures populating the jet inner core by studying the behavior of the spatial correlation function at z Ã ¼ 0. As shown in Fig. 6, the most relevant feature of this region is given by the fact that the intensity of the negative peaks of the correlation function R uu and R vv is very weak, of the order of 5%-7% the maximum value of correlation.Furthermore, these negative peaks are at very large separations far exceeding the jet half-width.Hence, we argue that no vortical structures nor streaky pattern are statistically relevant in this region of the flow.On the contrary, the spatial correlation function R ww shows a significant negative peak of correlation in the streamwise direction of the order of the 17% of the maximum correlation.This behavior is the statistical signature of the large-scale spanwise vortices and will be further analyzed in Secs.V B and V C.
Overall, the scenario described by the spatial correlation function for the jet inner core is similar to isotropic turbulence.No turbulent structures are detected, thus suggesting that turbulent vortices and structures are randomly oriented in this region of the flow such as in isotropic turbulence.The only exception is given by a large-scale signature given by the spanwise vortex structures related to the shear instability.Hence, we argue that randomly oriented turbulent eddies are superimposed on this anisotropic large-scale motion without being significantly modulated.

B. Production region
We now consider the structure of turbulence in the production region by considering the behavior of the spatial correlation function at z Ã ¼ 0:35.As reported in Fig. 7, the spatial correlation function R uu shows the presence of high and low streamwise velocity fluctuations alternated in the spanwise direction with a spacing d Ã y % 0:64.The shape of these structures is elongated in the streamwise direction, ð' Ã x ; ' Ã y ; ' Ã z Þ % ð1:5; 0:7; 0:9Þ.As anticipated in Sec.III, streamwise velocity streaks are associated with the presence of quasistreamwise vortices, which lift up high velocity fluid.The presence of these structures is unveiled by the behavior of the spatial correlation function R vv .Indeed, as shown in Fig. 7, R vv highlights the presence of vortical motions inclined with respect to the symmetry plane of the jet with an angle u % À31 .These vortical structures have an elongated shape whose principal axis are ðr s ; r y ; r g Þ.The size of these structures is ð' Ã s ; ' Ã y ; ' Ã g Þ % ð1:34; 0:92; 0:32Þ, and the diameter of the corresponding vortical motion is d Ã g % 0:4.Contrary to R uu and R vv , a completely different flow pattern induced by the large-scale spanwise vortex structures is unveiled by the spatial correlation function R ww .As shown in Fig. 7, the shape of these cross-flow structures is elongated in the cross-flow direction, ð' Ã x ; ' Ã y ; ' Ã z Þ % ð0:7; 1:16; 1:52Þ, while the diameter of the corresponding vortical motion is d Ã x % 0:64.Hence, this vortical motion involves the entire jet region.The strength of this motion cancels out the appearance of negative peaks of correlation in the spanwise direction that we are expecting to see as related to the previously observed inclined vortical motions.This is probably due to the fact that the latter occurs at smaller scales and, hence, is probably less intense with respect to the former.Overall, we observe that in the production region quasistreamwise vortical structures inclined À31 with respect to the symmetry plane of the jet exist and are associated with the appearance of a streamwise velocity streak pattern.Considering that this region is also characterized by high levels of mean shear, this structure of turbulence is classically associated with turbulence production mechanisms.The interactions between these structures are superimposed on a largerscale spanwise vortex motion.The clear matching of the spatial correlation functions reported in Fig. 7 with those reported in Balamurugan et al. 36 for the case of a spatially evolving mixing layer actually suggests a possible universality of the type of structures involved in the production of turbulence in free shear flows.

C. Outer region
The structure of turbulence in the outer region at z Ã ¼ 0:82 is similar to that observed in the production region, i.e., streamwise velocity streaks and inclined vortical motions superimposed on a verylarge spanwise vortex motion.The main differences are the size of the structures and their regions of influence.As reported in Fig. 8, the spatial correlation function R uu shows that the streaky pattern is not localized at the same distance from the jet centerline.Streamwise fluctuations in this region are correlated with streamwise fluctuations with opposite sign at inner locations.As it will be shown in Sec.V D, x ; r Ã y ; r Ã z Þ-space by means of two iso-surfaces of positive and negative correlation, i.e., for R bb ¼ 0:3R max bb (red) and for R bb ¼ 0:7R min bb (blue), respectively.Two-dimensional section of the iso-levels of velocity correlations for r Ã x ¼ 0, central panels ( , and for r Ã y ¼ 0, bottom panels (c), (f), and (i).Solid lines represent positive values, while dashed lines denote negative values.
this behavior is associated with the cross-flow mushroom-like shape taken by the very-large spanwise vortex structures that, in turn, is due to the lift-up phenomenon induced by pairs of counter-rotating quasistreamwise vortices.On the other hand, the spatial correlation function R vv shows that the inclined quasi-streamwise vortices in the outer region have a larger inclination, u % À45 , and a larger size, d Ã g % 0:58.The spatial correlation function R ww is again dominated by the signature of the spanwise vortex motions.The main difference with respect to the production and inner region is given by the fact that the width of the statistically relevant cross-flow fluctuations is larger and that are correlated with cross-flow fluctuations of opposite sign at inner locations.
The overall scenario of the outer region is similar to that reported in the production region, i.e., inclined vortical structures and streamwise velocity streaks superimposed on a larger-scale spanwise vortical motion.The main difference is given solely to the larger size and inclination of these turbulent structures.We argue that the larger inclination is given by the fact that in this region the direction of strain induced by pairs of spanwise vortices is more inclined than in the production region.The intensity of the production mechanisms in the outer region are smaller than in the production region despite the structures of turbulence are almost the same.This is probably due to the lower mean shear levels reached in the outer region.

D. External region
We consider here the structure of turbulence in the external region of the jet at z Ã ¼ 1:17.As shown in Fig. 9, the spatial correlation function R uu does not show any more a streaky pattern for the streamwise velocity fluctuations.Streamwise fluctuations are very wide structures ' Ã y % 3:5 and are correlated with streamwise fluctuations of opposite sign at inner locations, thus highlighting that in the external region of the flow, the streamwise velocity field is actually dominated by the spanwise vortex motion resulting from the shear instability of the flow.The inclination of these streamwise velocity structures with respect to the symmetry plane of the jet is u % 38 .Interestingly, R uu exhibits a mushroom-like shape that appears to be the result of a lift-up phenomenon of the very-large spanwise vortex motion of the jet.We argue that the lift-up of very-large spanwise vortices is induced by the action of pairs of counter-rotating quasi streamwise vortices populating the inner regions of the flow as shown in Secs.V B and V C. Accordingly, the mushroom-like shape can be clearly associated with the interaction of the flow motion induced by quasi-streamwise vortices with that of the very-large spanwise vortices.
All the spatial correlation functions show a peak of correlation at inner locations with respect to the detection point, i.e., for negative values of r z .This shift is particularly large for the spanwise and cross-flow turbulent structures, R vv and R ww , respectively.In particular, we measure that the peak of correlation occurs at ðz þ r z Þ Ã % 0:64 and ðz þ r z Þ Ã % 0:52 for R vv and R ww , respectively.Hence, the spanwise and cross-flow velocity fluctuations in the external region are not a result of local turbulent structures but are given by the motions induced by turbulent structures centered at inner locations.In particular, the spanwise velocity motion in the external region appears as the spanwise velocity spreading induced by the inclined vortical motions populating the outer and production regions.On the other hand, the x ; r Ã y ; r Ã z Þ-space by means of two iso-surfaces of positive and negative correlation, i.e., for R bb ¼ 0:2R max bb (red) and for R bb ¼ 0:8R min bb (blue), respectively.Two-dimensional section of the iso-levels of velocity correlations for r Ã x ¼ 0, central panels (b), (e), and (h), and for r Ã y ¼ 0, bottom panels (c), (f), and (i).Solid lines represent positive values, while dashed lines denote negative values.x ; r Ã y ; r Ã z Þ-space by means of two iso-surfaces of positive and negative correlation, i.e., for R bb ¼ 0:5R max bb (red) and for R bb ¼ 0:5R min bb (blue), respectively.Two-dimensional section of the iso-levels of velocity correlations for r Ã x ¼ 0, central panels (b), (e), and (h), and for r Ã y ¼ 0, bottom panels (c), (f), and (i).Solid lines represent positive values while dashed lines denote negative values.
cross-flow velocity motion in the external region appears as a result of the flow circulation induced by the spanwise vortex rolls.
Overall, it appears that the motions in the external regions of the jet are induced by the flow circulation generated by turbulent structures populating inner locations.Indeed, the peaks of correlation are reached for significantly negative cross-flow separations r z < 0. In particular, the streamwise and cross-flow fluctuations in the outer region are the result of the interaction of very-large spanwise vortices with quasi-streamwise vortices populating the inner jet regions.Accordingly, the streamwise fluctuations have a mushroom-like shape ' Ã y % 3:5 wide and u % 38 inclined.This is in agreement with the presence of lift up phenomena of the spanwise vortex rolls that is induced by the flow circulation of pairs of quasi-streamwise vortices whose influence is felt also in the external region of the jet as shown by the spatial correlation function of spanwise velocity.

VI. CONCLUDING REMARKS
Based on direct numerical simulation data of a temporal planar turbulent jet, the structure of turbulence in free-shear flows was analyzed in detail.The instantaneous flow pattern revealed the presence of large-scale motions induced by the flow instability and of smallscale turbulent structures superimposed on it.To characterize the topology of the statistically relevant structures populating the different flow regions, a study of the three-dimensional spatial correlation function was performed.The presence of large spanwise vortices was highlighted.The induced motion is found to be felt in the entire flow field, especially for the cross-flow fluctuations.Superimposed to this large-scale motion, the presence of quasi-streamwise vortices is unveiled in the production and outer regions of the jet.The length of these vortices is ' Ã s % 1:4 while their inclination and diameter is found to increase by moving from the core to the outer regions of the jet.In particular, we measure, u % À31 and d Ã g % 0:4 at z Ã ¼ 0:35 and u % À45 and d Ã g % 0:58 at z Ã ¼ 0:82.The flow motion induced by pairs of quasi-streamwise vortices is responsible for the lift-up of the large-scale spanwise rolls and for their deformation forming mushroom-like flow structures flanked from below by streamwise velocity streaks.To summarize, the flow topology can be schematized as in Fig. 10.It consists of a spanwise arranged array of intense spanwise vortices whose diameter is d Ã x % 0:64 and whose width changes with the jet centerline distance from ' Ã y % 1:1 to ' Ã y % 3:5 due to their cross-sectional mushroom-like shape.These spanwise vortices are flanked from below by counter-rotating pairs of quasi-streamwise vortices that created them.This hairpin-like flow circulation is in turn responsible for the generation of high and low streamwise velocity streaks whose length and width are ' Ã x % 1:5 and ' Ã y % 0:7.In conclusion, the present statistical analysis is shown to provide a rigorous description of the coherent structures populating free-shear flows.The emerging qualitative picture is in agreement with several previous observations, such as those of Perry and Tan, 3 Liepmann and Gharib, 4 Bernal and Roshko, 21 Brancher et al., 37 Kantharaju et al. 38 On the other hand, the quantitative description given by the correlation functions can be eventually used to assess and calibrate large-scale eddy models, such as that of Philip and Marusic 7 to give a structural prediction of the otherwise disorganized motion in turbulent free-shear flows.As a side result, it is also found that the very large separation of scales occurring in free-shear flows calls for the adoption of very large computational domains in order to capture the interactions between large-scale instability and small-scale turbulence.By increasing the area spanned by the two homogeneous directions, larger domains are also found to improve the statistical convergence of the data.

FIG. 1 .
FIG. 1.(a) Mean velocity and turbulent kinetic energy profiles, U Ã ðzÞ and k Ã ðzÞ, respectively.The vertical lines denote the locations used for the analysis of the three-dimensional spatial correlation function reported in Sec.V. (b) Streamwise turbulent spectrum E xÃ kk evaluated at z Ã ¼ 0:35.The dashed line reports the k À5=3 x law.The inset panel shows the premultiplied streamwise turbulent spectrum k Ã x E xÃ kk as a function of the wavelength k Ã x evaluated at the same position and time.The vertical lines denote the value k Ã x ¼ 2p=k Ã x ¼ 1:5.

FIG. 5 .
FIG.5.Iso-contours of the scalar field at t ¼ 120 evaluated for a fixed y-plane (a), x-plane (b) and for two fixed z-planes in the core of the jet (c) for z ¼ 0 and in the outer region of the jet (d) for z Ã ¼ 0:5.