Invariants of the velocity-gradient, rate-of-strain, and …...Invariants of the velocity-gradient,...

Invariants of the velocity-gradient, rate-of-strain, and rate-of-rotationtensors across the turbulent/nonturbulent interface in jets

Carlos B. da Silvaa and José C. F. PereiraIDMEC/IST Technical University of Lisbon, Pav. Mecânica I, 1° andar/esq./LASEF,Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Received 14 August 2007; accepted 20 March 2008; published online 2 May 2008

The invariants of the velocity gradient R and Q, rate-of-strain RS and QS, and rate-of-rotationQW tensors are analyzed across the turbulent/nonturbulent T/NT interface by using a directnumerical simulation DNS of a turbulent plane jet at Re120. The invariants allow a detailedcharacterization of the dynamics, geometry and topology of the flow during the entrainment. Theinvariants Q and QS are almost equal and negative outside the turbulent region close to the T/NTinterface, which shows the existence of high values of strain product hence viscous dissipation ofkinetic energy at that location. Right at the T/NT interface, the invariants QW and QS show thatvirtually all flow points there are characterized by irrotational dissipation, with no discernible signof the coherent structures which are known to exist deep inside the turbulent region. Moreover, theinvariants of the velocity gradient tensor Q and R show that the classical “teardrop” shape of theirassociated phase map is not yet formed at the T/NT interface. All the invariants rapidly change afterthe T/NT interface is crossed into the turbulent region. For instance, the enstrophy density,proportional to QW, is zero in the irrotational flow region and high and more or less constant insidethe turbulent region, after it undergoes a sharp jump near the T/NT interface. Inside the turbulentregion, at a distance of only 1.7 from the T/NT interface, where is the Kolmogorov microscale,the invariants QW and QS suggest that large scale coherent vortices already exist in the flow.Furthermore, the joint probability density function of Q and R already displays its well knownteardrop shape at that location. Moreover, the geometry of the straining or deformation of the fluidelements during the turbulent entrainment process is preferentially characterized by biaxialexpansion with S :S :S=2:1 :−3, where S, S, and S are the eigenvalues of the rate-of-straintensor arranged in descending order. Based on an analysis of the invariants, many aspects of the flowtopology inside the turbulent region at a distance of only 1.7 from the T/NT interface are alreadysimilar to those observed deep inside the turbulent region. © 2008 American Institute of Physics.DOI: 10.1063/1.2912513

I. INTRODUCTION

The study of the invariants of the velocity gradient, rate-of-strain, and rate-of-rotation tensors in turbulent flows hasattracted much attention since the seminal papers by Chonget al.,1 Cantwell,2,3 and Perry and Chong.4 The invariants arescalar quantities whose values are independent of the orien-tation of the coordinate system and contain information con-cerning the rates of vortex stretching and rotation, and on thetopology and geometry of deformation of the infinitesimalfluid elements. Furthermore, the analysis of the invariantspermits the understanding of these issues using a relativelysmall number of variables, e.g., the second and third invari-ants of the velocity gradient tensor combined Q and R al-low to assess the topology of the flow enstrophy dominatedversus strain dominated or the enstrophy production vortexstretching versus vortex compression.

The invariants have been extensively used in severalflow configurations such as isotropic turbulence,5–7 turbulentmixing layers,8 and turbulent channel flows.9 Important

information is also obtained by analyzing the volume inte-gral of the invariants, as shown by Soria et al.10 In thesestudies, several “universal” features of turbulent flows wereobserved. An example of such a result is the well known“teardrop” shape of the joint probability density functionPDF of R and Q.

An interesting line of research using the invariants con-sists in writing transport equations for each one of theinvariants.7 Cantwell2 analyzed these equations for the in-variants of the velocity gradient tensor by using the so-called“restricted Euler model” proposed by Vieillefosse11 wherethe pressure Hessian and viscous terms are neglected. Thesolutions to these equations showed some of the flow fea-tures observed in isotropic turbulence such as the preferredalignment of the vorticity vector with the eigenvector corre-sponding to the intermediate eigenvalue of the rate-of-straintensor, as well as the tendency for a state corresponding totwo positive and one negative eigenvalues of this tensor.12

Another line of active research involving the invariantshas been the identification of the coherent structures in tur-bulent flows, either to visualize the flow regions associatedwith the presence of vortex tubes13–16 or to identify the re-gions responsible for the most important dissipation rates ofaElectronic mail: [email protected].

PHYSICS OF FLUIDS 20, 055101 2008

1070-6631/2008/205/055101/18/$23.00 © 2008 American Institute of Physics20, 055101-1

Downloaded 02 May 2008 to 193.136.128.14. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

http://dx.doi.org/10.1063/1.2912513

http://dx.doi.org/10.1063/1.2912513

http://dx.doi.org/10.1063/1.2912513

kinetic energy.17,18 Recently, the invariants have been usedalso in the context of subgrid-scale modeling.19,20

The present work uses the invariants in the context of theturbulent entrainment that exists in free shear flows such asmixing layers, wakes, and jets. In these flows, the flow fieldcan be divided into two regions. In one region, the flow isturbulent T and its vorticity content is high, while in theother region, the flow consists of largely irrotational nontur-bulent NT flow.21 The two flow regions are divided by theturbulent/nonturbulent T/NT interface where the turbulententrainment mechanism takes place, by which a given fluidelement from the irrotational zone becomes turbulent. ThisT/NT interface is very sharp and is continually deformedover a wide range of scales. Its thickness is of the order ofthe Taylor microscale.22

The mechanism of turbulent entrainment is still involvedin a great deal of mystery, despite the great number of worksdevoted to it see, e.g., Townsend23 for a review on the clas-sical ideas of turbulent entrainment. Understanding of thephysical mechanisms taking place at the T/NT interface isimportant in many natural and engineering flows since im-portant exchanges of mass, momentum, and passive or activescalar quantities take place across the T/NT interface.24 Itwas assumed in the past that the turbulent entrainmentmechanism is mainly driven by “engulfing” motions causedby the large scale flow vortices,23 but recent experimentaland numerical works give more support to the original modelof Corrsin and Kistler21 where the entrainment is primarilyassociated with small scale “nibbling” eddy motions.25,26

Nevertheless, it is still argued that the entrainment andmixing rates are largely determined by the large scales ofmotion.

Recently, the study of the turbulent entrainment saw verysignificant advances with the results from direct numerical,simulations DNSs of plane wakes by Bisset et al.,27,28 theDNSs of round jets by Mathew and Basu,25 the experimentalmeasurements of round jets by Westerweel et al.,26,29 andwith results from the flow generated by an oscillating grid byHolzner et al.30,31 Many new and sometimes quite unex-pected results were obtained in these works. One of the mostsurprising results see Mathew and Basu25 and Westerweelet al.26 consisted in the realization that the total amount offluid entrained due to the small scale motions near the T/NTinterface i.e., nibbling is more important than the amountof fluid entrained by large scale motions engulfing, as sug-gested more than 50 years ago by Corrsin and Kistler.21

Other interesting results are the existence of a finite jump inthe tangential velocity hence also in the z vorticity com-ponent at the T/NT interface26,28 as anticipated byReynolds,32 the existence of a positive contribution to theenstrophy by the viscous enstrophy diffusion term,28,31 theexistence of a region of high strain product in the irrotationalflow region close to the T/NT interface,31 and the estimationof the characteristic scales of motion at the T/NT interface asbeing of the order of the Taylor scale.26

The goal of the present work is to analyze the evolutionof the invariants of the velocity gradient, rate-of-strain, andrate-of-rotation tensors across the T/NT interface in turbulentjets in order to clarify the kinematics, dynamics, and topol-

ogy of the flow during the entrainment process. For this pur-pose, a DNS of a turbulent plane jet will be used. The tur-bulent plane jet is a well known canonical free shear flowwith many common features with other free shear flows suchas mixing layers, wakes, round, and coaxial jets. Namely, theflow statistics are dominated by the presence of an inhomo-geneous mean streamwise velocity profile and by the pres-ence of quite similar long lived large scale flow vortices,originated from well known instabilities e.g., Kelvin–Helmholtz instability, and whose “footprints” are still dis-cernible at the far field fully developed turbulent state.Therefore, it is expected that the results obtained in thepresent study display some universal qualitative features re-lated to the entrainment in free shear flows.

This article is organized as follows. In Sec. II, we de-scribe the plane jet DNS used in the present work. Section IIIdescribes the procedure used to detect the T/NT interface andthe data bank used in the subsequent analysis. The mainresults are described in Sec. IV. In Sec. V, the work ends witha review of the main results and conclusions.

II. DNS OF TURBULENT PLANE JETS

A. Statement of the problem

In this work, we are interested in analyzing the invari-ants of the velocity gradient, rate-of-strain, and rate-of-rotation tensors in connection with the mechanism of turbu-lent entrainment in jets. A typical spatially developing planejet issuing from a nozzle into a tank filled with the samefluid, as obtained in a laboratory experiment, is sketched inFig. 1a. Recent DNS of spatially evolving turbulent planejets include the works of Stanley et al.33 and da Silva andMétais.34

It is well known that numerical simulations of spatiallydeveloping flows, i.e., spatial simulations, can be very de-

(a)

(b)

FIG. 1. Color online Sketch of a a spatially developing jet as produced ina laboratory setup and b a temporal jet simulation as the one adopted in thepresent work. Notice the coordinate system used in the present work indi-cating the streamwise x, normal y, and spanwise z directions.

055101-2 C. B. da Silva and J. C. F. Pereira Phys. Fluids 20, 055101 2008


manding both in terms of computer time and memory due tothe need to simulate the flow inside a very large computa-tional domain containing all the length scales of the flow.

In the present work, a temporal simulation of a plane jet,as sketched in Fig. 1b, was used in order to limit the com-putational cost. The sketch also shows the coordinate systemused in the present work indicating the streamwise x, nor-mal y, and spanwise z directions. DNSs of temporallyevolving turbulent plane jets were carried out by Akhavanet al.35 and da Silva and Pereira.36 In these simulations, thecomputational domain is periodic in the three spatial direc-tions, which allows the use of very fast and accurate pseu-dospectral methods. Thus, one studies the temporal evolutionof the flow generated by an initial plane jet velocity profile,instead of the flow of a spatially developing jet. This factsubstantially limits the size of the computational domain re-quired by the simulations, which reduces the computationalcost.

Because in temporal simulations periodic boundary con-ditions are used in the streamwise direction, the feedbackeffects caused by the pressure field that are known to influ-ence the details of the transition to turbulence e.g., seeThomas and Chu37 are absent from temporal simulations.Another drawback of temporal simulations is that no rigor-ous comparison with experimental results can be made.However, since the flow field in the fully developed turbulentstate is to a great extent independent of the details of thetransition to turbulence e.g., see Refs. 33–36, temporalsimulations are a useful tool to analyze the flow at the farfield of a turbulent jet. Also, the large scale flow structuresthat are typical of jets and other free shear flows, such as theKelvin–Helmholtz and streamwise vortices, are well cap-tured in temporal simulations. The use of temporal simula-tions is also justified by the fact that despite their limitations,they represent valid solutions to the Navier–Stokes equationsand are relevant since the goal of the present work is toanalyze those features of the entrainment mechanism that aregenerally true in Navier–Stokes turbulence. Another justifi-cation comes from the fact that turbulent entrainment ismainly dominated by small scales, as shown by Mathew andBasu25 and Westerweel et al.26,29 Temporal DNSs of turbu-lent flows were previously used to analyze the mechanics ofthe T/NT interface in plane wakes27,28 and in round jets.25,38

B. Numerical method

The numerical code used here is a standard pseudospec-tral code collocation method39 in which the temporal ad-vancement is made with an explicit third order Runge–Kuttatime stepping scheme.40 The simulation was fully dealiasedby using the 3

2 rule.39 This code was used recently by theauthors in DNSs of turbulent plane jets described in da Silvaand Pereira.36

C. Physical and computational parameters

The DNS was started by using, as initial condition, anhyperbolic-tangent velocity profile,

Ux,y,z =U1 + U2

2−

U1 − U2

2tanh H

401 −

2yH

,

1

where 0 is the initial momentum thickness, H is the inletnozzle of the jet, and we used U1=1 and U2=0. The meanprofile defined by Eq. 1 was also used as the inlet conditionin the spatial plane jet simulations of Stanley et al.33 andSilva and Métais34 and as the initial condition in the temporalsimulations of Akhavan et al.35 and da Silva and Pereira.36

Many spatial and temporal simulations of turbulent roundjets use similar inlet or initial mean velocity profiles see,e.g., da Silva and Métais41 and Mathew and Basu25 since itis recognized that it represents a very good approximation tothe inlet velocity profile found in measured experimentaljets.42

A three-component velocity fluctuating “spectral noise”was superimposed to the mean velocity profile defined byEq. 1 through a proper convolution function that imposesthe velocity fluctuations in the initial shear layer region ofthe jet. The spectral noise used here is very similar to the oneused in da Silva and Métais34 and virtually equal to the oneused in da Silva and Pereira.36 This numerical noise is verysimilar to the standard noise used to initialize simulationsof decaying isotropic turbulence e.g., Lesieur et al.43. Inshort, each velocity component of the noise is prescribedin such a way that its energy spectrum is given byEkks exp−s /2k /k02, where the exponent s is s4 asin decaying isotropic turbulence. However, here, the peakwave number k0 is chosen to give an energy input which isdominant at small scales high k0 to allow the simulation toevolve “naturally” by “selecting” its natural instabilitymodes.

As in Stanley et al.,33 da Silva and Métais,34 and da Silvaand Pereira,36 a relatively high amplitude spectral noise wasadded 8% to the mean profile defined by Eq. 1 in order tospeed up the transition mechanism and allow the flow toquickly reach a fully developed turbulent state. It is impor-tant to stress that, as shown in previous works, the additionof a high H /0 ratio and initial amplitude noise, althoughfavoring a faster transition to turbulence, does not affect thedynamics of the self-similar fully developed turbulentstate.33,34,36,44

In the present work, we are interested in reaching aslightly higher Reynolds number than in Ref. 36 and at thesame time in having a resolution closer to the one used inisotropic turbulence, for which the optimal value is x /2.1 Pope45 and which corresponds to having kmax=1.5,where kmax is the maximum resolved wave number and isthe Kolmogorov microscale. For this purpose in the presentplane jet simulation, we use an initial Reynolds numberequal to ReH= U1−U2H /=3200,46 where H is the planejet inlet slot width, and we reduce the extent of the compu-tational box to Lx ,Ly ,Lz= 4H ,6H ,4H. The grid size con-sists now in NxNy Nz= 256384256 grid points,i.e., the number of grid points along the streamwise x andnormal y directions was retained from Ref. 36, while thespanwise z resolution was doubled. This was done in order

055101-3 Invariants of the velocity-gradient, rate-of-strain, and rate-of-rotation tensors Phys. Fluids 20, 055101 2008


to improve the degree of convergence of the statistics overone single instantaneous field. Notice, however, that the gridremains isotropic and we have x=y=z=0.0156H in-stead of 0.02H as in Ref. 36. Finally, in order to furtherreduce the time spent during the transition phase, the initialH /0 ratio was raised to H /0=35.

It is important to ensure that the new computational boxsize does not constrain the jet in its development and particu-larly its spreading rate. Recall that the use of classical pseu-dospectral schemes for spatial discretization implies that theboundary conditions used here are periodic in the three spa-tial directions. Concerning the box size in the streamwisedirection, it is important to ensure that the size of the com-putational domain in this direction is big enough to allow forthe development of the primary shear layer Kelvin–Helmholtz instability. Therefore, the box size has to begreater than the Kelvin–Helmholtz instability length scale.The Strouhal number of the primary Kelvin–Helmholtz in-stability is equal to Ssl= fsl0 /UC=0.033, where fsl is the fre-quency of the shear layer mode and UC=0.5U1+U2 is theconvection velocity.47 Since the initial H /0 ratio used hereis H /0=35, the associated instability length scale is equal tosl=0 /Ssl=0.87H. Thus, the box size in the streamwise di-rection Lx=4H is more than four times bigger than theKelvin–Helmholtz instability length scale.

Concerning the normal direction, it is important to em-phasize that numerous previous works have showed thatthere is no particular problem with the use of periodicboundary conditions provided that the box size is big enoughsee da Silva and Métais34 and Mathew and Basu25 and ref-erences therein. The box size along the normal directionused here is Ly =6H. This is the same lateral size used in thesimulations of Mathew and Basu.25 As shown in numerousvisualizations, including the one shown below in Fig. 5, thecontours that delimit the T/NT interface are on average at adistance of more than 1.5H from the normal box boundaries.This is more than the distance used in previous numericalsimulations used to analyze the turbulent entrainment, e.g.,Mathew and Basu25 had less than 1D D is the jet diameter,and this still posed no problems. Moreover, all the one pointstatistics obtained in the present simulation are in excellentagreement with the results from previous experimental andnumerical works see below. Thus, the lateral extent orboundary conditions in the normal direction do not influencethe plane jet development. Finally, concerning the box sizein the spanwise direction, we now have Lz=4H. Thisis larger than any of the spanwise box sizes used in thespatial or temporal simulations cited above, and since thelength scales, of the secondary spanwise instabilities aresmaller than the primary instabilities estimated above assl=0.87H, with Lz=4H, the box size used in the presentsimulation is indeed sufficient.

D. DNS assessment

The scenario of transition to turbulence is very similar tothe one described in previous numerical simulations, e.g.,Stanley et al.33 da Silva and Métais,34 and da Silva andPereira,36 and is characterized by the emergence of Kelvin–

Helmholtz vortices, both at the upper and lower shear layers,and is followed by the appearance of pairs of streamwisevortices connecting each two consecutive pairs of Kelvin–Helmholtz rollers. Shortly after, the streamwise vorticesbreak up into smaller structures with no preferential direc-tion, which is a sign of a fully developed turbulent stage.

Figure 2 shows the temporal evolution of the enstrophyand its components averaged over the whole computationaldomain. We denote these averages by B, e.g.,

x2t2

B=

1

NxNyNzi=1

Nx

j=1

Ny

k=1

Nz x2

2x,y,z,t . 2

As in Ref. 36, these curves show an increase until T /Tref

20, where Tref=H / 2U1−U2, followed by a decrease ata more or less constant rate.

The coherent structures of the flow were visualized byusing the “Q criteria”13,14 and the pressure field. FromT /Tref15 onward these structures are qualitatively similarto the ones shown in Figs. 2a and 2b from Ref. 36, i.e.,the small scale structures, which are more easily visualizedthrough positive values of Q, do not show the existence ofany particular spatial orientation, whereas the low pressureisosurfaces, which highlight the bigger structures, still showremnants of the Kelvin–Helmholtz rollers as in Ref. 36,there is, of course, some overlap between the structures fromthe two visualization criteria—see Dubief and Delcayre15.The self-similar regime is obtained at T /Tref20 againstT /Tref30 in Ref. 36, which corresponds to an equivalentstreamwise location of x /H= U1−U2T /H=T / 2Tref=10.Recall that in the spatial simulations of Stanley et al.33 andda Silva and Métais,34 x /H=10 marked also the beginning ofthe self-similar regime. Finally, at this station, the Reynoldsnumber based on the Taylor microscale and on the rootmean square of the streamwise velocity u is equal toRe=u /120 across the shear layer.

Figures 3a–3d show one point statistics for the planejet DNS. The statistics were obtained with a single instanta-neous field from the self-similar region where spatial aver-aging in the two homogeneous directions x and z was fol-lowed by “folding” of the mean profiles in relation to the jet

T/Tref

0 5 10 15 20 25 30 350

1

2

3

4

5 <ΩiΩi/2>B

<Ω2x/2>B

<Ω2y/2>B

<Ω2z/2>B

FIG. 2. Color online Temporal evolution of the mean enstrophy and itscomponents for the present turbulent plane jet DNS.



centerline y=0 to take advantage of the symmetry of thejet. We denote these averages by ,

uy = uy,t =1

NxNzi=1

Nx

k=1

Nz 1

2ux,y,z,t

+ ux,− y,z,t . 3

Due to the relatively small number of samples obtained foreach y coordinate 2NxNz=131 072, the mean profilesshow some wiggles. This limitation is a well known featureof temporal simulations and similar wiggles can be foundalso in the mean profiles from temporal simulations ofmixing layers,48 wakes,49 and round jets.50 Although the de-gree of convergence of the statistics is not perfect, theynevertheless allow a comparison to results from experimentaland numerical works from the literature. In Figs. 3a–3d,the present results are compared to experimental resultsfrom Gutmark and Wygnansky,50 Ramparian andChandrasekhara,51 and Thomas and Prakash,52 and theDNSs, from Stanley et al.,33 da Silva and Métais,34 and

da Silva and Pereira.36 Although the scatter between the ex-periments and computations is high, the mean streamwisevelocity profile and Reynolds stress profiles from the presentDNS agree with the data available well.

Figures 4a and 4b show the spatial three-dimensionalkinetic energy and kinetic energy dissipation spectra, respec-tively, at several instants. The kinetic energy spectrum has a−5 /3 region followed by a smooth decay at high wave num-bers. Notice that the product of the maximum resolved wavenumber to the Kolmogorov microscale is kmax1.5. Thatthe dissipative scales are indeed being well resolved is at-tested by the small upturns at the end of the wave numberrange. The dissipation spectrum shown in Fig. 4b peaks atk0.3, which marks the start of the dissipation region. Theshape and magnitudes of these spectra are very similar to theones obtained by Akhavan et al.,35 also in temporal simula-tions of turbulent plane jets, with similar resolution and simi-lar Reynolds number based on the Taylor microscale. Addi-tional resolution tests are given in Appendix A.

These results show that the present DNS is both accu-

y/δ0.5

<u(y

)>/U

C

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

Gutmark 1976Ramparian 1985Stanley 2002da Silva 2002da Silva 2004Present work

y/δ0.5

<u’2 (y

)>1/

2 /UC

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5 Gutmark 1976Ramparian 1985Thomas 1991Stanley 2001da Silva 2002da Silva 2004Present work

(a) (b)

y/δ0.5

<v’2 (y

)>1/

2 /UC

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5Gutmark 1976Ramparian 1985Thomas 1991Stanley 2001da Silva 2002da Silva 2004Present work

y/δ0.5

<w’2 (y

)>1/

2 /UC

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

Gutmark 1976Stanley 2001da Silva 2002da Silva 2004Present work

(c) (d)

FIG. 3. Color online Profiles of several one point statistics at the self-similar region from the present temporal plane jet direct numerical simulation presentwork compared to the experimental results from Gutmark and Wygnansky Ref. 50, Ramparian and Chandrasekhara Ref. 51, and Thomas and PrakashRef. 52 and with the DNSs of Stanley et al. Ref. 33, da Silva and Métais Ref. 34, and da Silva and Pereira Ref. 36: a Mean streamwise velocity, bstreamwise Reynolds stresses, c normal Reynolds stresses, and d spanwise Reynolds stresses. Uc is the streamwise mean centerline velocity and U is thehalf-width of the jet.



rate, at the large and small scales, and representative of afully developed turbulent plane jet. This completes the vali-dation of the turbulent plane jet DNS.

III. DATA BANK DESCRIPTION

A. Detection algorithm for the T/NT interface

The T/NT interface can be defined by using either thevorticity norm = ii1/2, where i is the vorticity field asin Bisset et al.,27,28 or using a passive scalar or concentrationfield as in Westerweel et al.26,29 The vorticity norm was usedin the present work, where it was observed that the detectionthreshold of =0.7U1 /H best delineated the vorticalregions. This is exactly the same value used by Bisset et al.28

and a similar level was used by Mathew and Basu.25

Figure 5 shows contours of vorticity modulus correspondingto this detection threshold in an x ,y plane of the jet atT /Tref=27. As in previous works, it can be seen that theT/NT interface is strongly contorted and some irrotationalfluid is engulfed.

In the present work, we analyze conditional statistics inrelation to the location of the interface envelope by using aprocedure similar to the one described in previous works,e.g., Bisset et al.27,28 and Westerweel et al.26,29

A detailed description of this procedure is given herewith the aid of a sketch shown in Fig. 6. The sketch repre-sents the T/NT interface separating the T from the irrota-tional or NT flow regions, at the upper shear layer of theplane jet. The vorticity surface defined by the selectedthreshold is indicated by a solid line, while the T/NT inter-face envelope is represented by gray dashed lines. The sketchdepicts events of large scale engulfment and small scale nib-

bling and also the original coordinate system x ,y used inthe numerical simulation of the turbulent plane jet.

Since the plane jet is homogeneous in the streamwise xand spanwise z directions, each x ,y plane is indepen-dently treated. Consider the upper shear layer depicted inFig. 6. The procedure starts with the determination of theT/NT interface envelope location YIx, for each one of theNx grid points in the original coordinate system along the xdirection. YIx is obtained through a linear interpolationalong the y direction, using the vorticity norm threshold in-dicated above to detect the T/NT interface.

In order to make conditional statistics in relation to thelocation of the interface envelope, we start by defining a newlocal coordinate system xI ,yI with the lines tangent xI andnormal yI, respectively, to the interface envelope see Fig.6. In this new coordinate system, the T/NT interface is ex-actly at xI ,yI= 0,0.

After the determination of the interface envelope YIx,one determines the coordinates of the axis line yI, from thenew local coordinate system xI ,yI, in the old coordinatesystem x ,y. Notice that this line is normal to the envelopefor each one of the Nx grid points along the x direction.Along both sides of this yI axis line for yI 0 and yI0,we define NI=80 points, starting at yI=0 and equally spaced

kη

E(k

)/(ε

ν5 )1/4

10-1 10010-4

10-2

100

102

104

106

108T/Tref=11.2T/Tref=14.4T/Tref=17.3T/Tref=20.2T/Tref=23.6T/Tref=27.1T/Tref=30.8T/Tref=34.9(-5/3)

(a)

kη

2νk2 E

(k)/

(εν)

3/4

10-1 10010-1

100

101

T/Tref=11.2T/Tref=14.4T/Tref=17.3T/Tref=20.2T/Tref=23.6T/Tref=27.1T/Tref=30.8T/Tref=34.9

(b)

FIG. 4. Color online Three-dimensional spatial kinetic energy and ki-netic energy dissipation spectra from the plane jet DNS at several instants:a Kinetic energy spectrum and b kinetic energy dissipation spectrum.

x/H

y/H

-2 -1 0 1 2-3

-2

-1

0

1

2

3 Ω=0.7(U1/H)

FIG. 5. Color online Contours of vorticity modulus corresponding to=0.7U1 /H in the x ,y plane of the jet at T /Tref=27.

FIG. 6. Color online Sketch of the T/NT interface for the plane jet indi-cating the vorticity surface solid line and the interface envelope graydashed lines. The sketch also shows the coordinate system used in thecomputation of the plane jet x ,y and the one used to analyze the T/NTinterface xI ,yI. In particular, the coordinate of the interface envelope isdenoted by YI. The three holes represent regions of irrotational fluid insidethe turbulent region.



with a distance equal to the mesh spacing x. For instance,the coordinates of these points in the turbulent region aregiven by xI ,yI= 0,yIl= 0, l−1x with l=1, NI. Thisdefines a total of 2NI−1=159 grid coordinates for the axisyI, which will be used to describe the mean conditional sta-tistical profiles. Then, for each coordinate yI, the mean con-ditional profile of a given flow variable Px ,y ,z , t, in theupper shear layer, for the plane x ,y corresponding to z=Zand at time t=T is obtained through

PyI IUp =

1

NLl=1

NL

PyIx,y,z = Z,t = T , 4

where PyIx ,y ,z=Z , t=T is the value of Px ,y ,z , t interpo-lated into the point xI ,yI= 0,yI in the plane x ,y ,z=Zand at time t=T. A simple bilinear interpolation is used forthis purpose. NL is the total number of samples used to makethe conditional mean for a given coordinate yI. The maxi-mum possible value for NL is NL=Nx; however, during thecomputation of the mean conditional profile defined in Eq.4, all points inside “holes” of “ambient fluid” that appear inthe jet, such as the three holes represented in Fig. 6, areremoved from the statistical sample.

The procedure just described is also used for the lowershear layer and the resulting profiles are averaged in the end.Thus, the maximum number of samples used to make a con-ditional mean for a given coordinate point yI in a given x ,yplane is then equal to 2Nx. The same procedure is usedfor each one of the x ,y planes available and the final resultis once again averaged over all the existing Nz planes. Themaximum possible number of samples corresponding to asingle instantaneous field is then equal to 2NxNz

=131 072.Finally, to further improve the degree of convergence of

the conditional statistics, several instantaneous fields takenfrom the fully developed turbulent regime were also used.Although the plane jet evolves in time and thus each instan-taneous field is, in its details, different from the others, se-lecting fields separated by a very small time interval reducesthese differences. In particular, in the far field, self-similar,fully developed turbulent regime, each instantaneous field isstatistically equivalent. Here, we used NT=11 instantaneousfields taken from T /Tref=20.2 to T /Tref=27.0. As can be seenin the kinetic energy and kinetic energy dissipation spectrashown in Figs. 4a and 4b, there are no appreciable differ-ences concerning these statistics for all the instantaneousfields from T /Tref=20.2 to T /Tref=27.0, a fact confirmed bycomparing the visualization of the fields from T /Tref=20.2and T /Tref=27.0. Thus, the use of these NT instantaneousfields from the self-similar regime allows us to increase thetotal number of samples used in the conditional averageswithout “mixing” different regimes of the plane jet develop-ment.

Therefore, except for the data points which are removedfrom the averaging procedure for being part of holes of irro-tational fluid in the turbulent zone and likewise for beingislands of turbulent fluid in the irrotational zone, the totalnumber of samples used to make a conditional mean inrelation to the distance from the T/NT interface is equal to

2NTNxNz1.4106. This is more than enough toobtain well converged statistics of the quantities in study inthis work.

With this procedure, conditional statistics of any flowquantity can be made in relation to the distance from theT/NT interface. We denote these conditional interface statis-tics by I to differentiate them from the spatial statisticsdone along the homogeneous directions of the plane jet,which are denoted by , and from the spatial statistics donein the whole computational box, which we denote by B.

Figure 7 shows the PDF of the interface distance fromthe center of the jet YI for the lower shear layer nondimen-sionalized by H. The mean, variance, skewness, and flatnessof the interface distance are YI /H =−1.14, YI /H2 =0.12, YI /H3 / YI /H2 3/2=0.09, and YI /H4 / YI /H2 2=2.95, respectively. As can be seen by the shapeof the PDF and by these values, the PDF of YI /H is nearGaussian. The upper shear layer displays similar shapes andvalues the mean is, of course, positive YI /H = +1.16. Bis-set et al.28 and Westerweel et al.26 also observed near Gauss-ian PDFs of the vertical distance of the T/NT interface.

B. Conditional mean vorticity in relationto the distance from the interface position

Figure 8 shows conditional mean profiles of I,x I, y I, z I, and z I nondimensionalized byU1 /H in relation to the distance from the T/NT interface YI,which is nondimensionalized by the value of the Kolmog-orov microscale at the jet shear layer. The vorticity modulusshows a sharp transition from the irrotational to the turbulentzone, where it reaches a plateau with I4.5U1 /H, andboth z I and z I show a peak very close to the interface.The shape and magnitudes of these profiles are in very goodagreement with the results of Bisset et al.27,28 and Wester-weel et al.26,29

In the remainder of this work, the conditional mean pro-file of z I will be used as a reference to indicate severallocations near the T/NT interface. Notice that in a plane jet,this is the only nonzero mean vorticity component, i.e.,zx ,y ,z = zy 0, while xx ,y ,z = yx ,y ,z

YI/H

PD

F

-2 -1 00

1

FIG. 7. Color online PDF of the vertical distance of the T/NT interface YI

for the lower shear layer.



=0. Moreover, as shown in Fig. 8, the gradient of this vor-ticity component is slightly steeper than either x I ory I, which is in agreement with the predictions byReynolds32 experimentally confirmed by Werterweelet al.26 since this component is connected with the existenceof a finite jump in the tangential velocity at the T/NT inter-face. However, it must be stressed that any other vorticitycomponent might be used instead, particularly since, as re-marked by a referee, there is almost no difference betweenthe conditional mean profiles of the three vorticity compo-nents shown in Fig. 8. Finally, in the present work, threeparticular locations in relation to the distance from the T/NTinterface will be frequently considered: yI /=0.0, which isexactly at the T/NT interface, yI /=1.7, which is close to thepoint of maximum z I, and yI /=8.6, which is alreadywell inside the turbulent zone.

IV. INVARIANTS OF THE VELOCITY GRADIENT,RATE-OF-STRAIN, AND RATE-OF-ROTATIONTENSORS ACROSS THE T/NT INTERFACE

In this section, the invariants of the velocity gradient,rate-of-strain, and rate-of-rotation tensors are used to studythe dynamics, topology, and geometry of the flow during theturbulent entrainment process. We start by recalling the defi-nitions of the invariants, their relationships, and physicalmeanings Sec. IV A. The next sections successively ana-lyze the geometry of the dissipation through the invariantsQW and QS Sec. IV B, the geometry of the straining of thefluid elements through the invariants QS and RS Sec. IV C,and, finally, the relation between the flow topology and theproduction/dissipation of enstrophy by vortex stretching/compression through the invariants Q and R Sec. IV D. Theanalysis uses conditional mean profiles of the invariants inrelation to the distance from the T/NT interface, “trajecto-ries” of these conditional mean values in their associatedphase maps, and joint PDFs of the invariants at several dis-tances from the T/NT interface.

A. Definitions of the invariants and associatedphysical meanings

The goal of this section is to review some of the back-ground material related to the invariants of the velocity gra-dient, rate-of-strain, and rate-of-rotation tensors. Extensivereviews of this material can be found in Chong et al.,1

Cantwell,3 Soria and Cantwell,16 Perry and Chong,4 Soriaet al.,8 Blackburn et al.,9 Ooi et al.,7 and Wang et al.20

The velocity gradient tensor Aij =ui /xj can be split intoa symmetric and a skew-symmetric component,

Aij = Sij + ij , 5

where Sij =12 ui /xj +uj /xi and ij =

12 ui /xj −uj /xi

are the rate-of-strain and rate-of-rotation tensors, respec-tively.

The velocity gradient tensor Aij has the following char-acteristic equation:

i3 + Pi

2 + Qi + R = 0, 6

where i are the eigenvalues of Aij. The first, second, andthird invariants of the velocity gradient tensor are

P = − Aii = − Sii, 7

Q = − 12AijAji = 1

4 ii − 2SijSij , 8

and

R = − 13AijAjkAki = − 1

3SijSjkSki + 34i jSij , 9

respectively, where i=ijkuj /xk is the vorticity fieldP=0 in incompressible flow.

Similarly, the invariants of the rate-of-strain tensor aredefined by its characteristic equation. The independent in-variants of Sij are

QS = − 12SijSij 10

and

RS = − 13SijSjkSki. 11

Finally, the only invariant of the rate-of-rotation tensor is

QW = 12ijij = 1

4ii. 12

Notice that the invariants of Sij are obtained by settingij to zero in Eqs. 8 and 9, while the only invariant of ij

is obtained by setting Sij to zero in Eq. 8. It is important torecall the physical meaning of these invariants see, e.g.,Soria et al.,8 Blackburn et al.,9 Wang et al.,20 and alsoDavidson53.

Starting with QW=ii /4, note that this invariant is re-lated to the second invariants of Aij and Sij through QW=Q−QS. Therefore, QW is proportional to the enstrophy density2 /2=ii /2. Regions of intense enstrophy tend to beconcentrated in tubelike structures in many turbulent flowssuch as isotropic turbulence,54 mixing layers,55 and jets.41

The second invariant of Sij, QS=−SijSij /2, is propor-tional to the local rate of viscous dissipation of kinetic en-ergy since =2S2=−4QS, where S2 /2=SijSij /2 is thestrain product. In isotropic turbulence, intense values of vis-cous dissipation tend to be concentrated in structures with

yI/η0 10 200

2

4

6<Ω>I

<Ωz>I

<|Ωx|>I

<|Ωy|>I

<|Ωz|>I

FIG. 8. Color online Mean conditional profiles of I, x I, y I,z I, and z I all normalized by U1 /H.



the form of sheets or ribbons.56 On the other hand, the thirdinvariant of Sij, RS=−SijSjkSki /3, is proportional to the strainskewness SijSjkSki. This invariant has two important physicalmeanings. It appears as part of the production term in thestrain product transport equation,53

D

Dt1

2SijSij = − SijSjkSki −

1

4i jSij − Sij

2p

xixj

+ Sij2Sij . 13

As can be seen, a positive value of RS is associated with theproduction of strain product and thus of viscous dissipa-tion, whereas RS0 implies a destruction of strain product.Moreover, in can be shown that RS=−S

3+S3+S

3 /3=−SSS, where SSS are the three eigenvalues ofSij arranged in descending order. Due to incompressibilityS+S+S=0, therefore, RS 0 implies that S ,S 0,S0 and the associated flow structure is sheetlike. IfRS0, then S 0, S ,S0, which implies a tubelikestructure.

Finally, the physical meaning of the invariants ofAij depends on the sign of Q. If Q 0, then the enstrophy2 /2=ii /2 dominates over strain product S2 /2=SijSij /2, whereas if Q0, the opposite occurs. In a Bur-gers vortex flow, for instance, the center of the vortex ischaracterized by Q 0, while in the region around it, Q0,implying that the strain product and hence viscous dissipa-tion of kinetic energy dominates. The meaning of R dependson the sign of Q. If Q0, then R−i jSij /4 and R0implies a predominance of vortex stretching over vortexcompression, and if R 0, vortex compression dominates.On the other hand, if Q0, then R−SijSjkSki /3=−SSS

and, therefore, R 0 is associated with a sheetlike structure,whereas R0 is associated with a tubelike structure.

The invariants defined above are usually analyzed injoint PDFs combining two invariants. The most commoncombinations consist on the maps of R ,Q, RS ,QS, andQW ,−QS. Figures 9a–9c show sketches of each one ofthese maps with the physical meaning associated with eachparticular location.

The R ,Q map Fig. 9a allows us to infer about therelation between the local flow topology enstrophy or straindominated and the enstrophy production term vortexstretching or vortex compression and associated geometryof the deformation of the fluid elements contraction or ex-pansion. The line defined by the discriminant DA=27 /4RA

2

+QA3 divides the map into two regions. If DA 0, Eq. 6 has

one real and two complex-conjugate roots, while if DA0,the equation has two real distinct roots. In many turbulentflows, the R ,Q map displays a strong anticorrelation be-tween R and Q in the region R 0,Q0 associated withsheetlike structures and also although not as strong in theregion R0,Q 0 associated with a predominance of vor-tex stretching. This gives the R ,Q map its characteristicteardrop shape that has been observed in a great variety ofdifferent turbulent flows such as isotropic turbulence,7 mix-ing layers,8 and channel flows.9

The RS ,QS map is particularly useful to analyze thegeometry of the local straining or deformation of the fluid

elements since RS=−SSS and S+S+S=0 due to in-compressibility. Defining the ratio of the second to the firsteigenvalue of the rate-of-strain tensor a=S /S, each valueof a represents a line in the RS ,QS map defined by

RS = − QS3/2a1 + a1 + a + a2−3/2, 14

where each line is associated with a given flow geometry:S :S :S=2:−1:−1 axisymmetric contraction, 1 :0 :−1two-dimensional flow, 3 :1 :−4 biaxial stretching, and1:1 :−2 axisymmetric stretching. The discriminant for theRS ,QS map is defined by the line DS=27 /4RS

2+QS3. Figure

9b shows a sketch of the RS ,QS map with several lines ofconstant a and their associated local flow geometry. More-over, since QS=− /4, large negative values of QS are asso-ciated with regions of intense kinetic energy dissipation. Inmany turbulent flows, the RS ,QS map shows a strong pref-

R

Q

Vortexstretching

Vortexcompression

Sheetstructure

Tubestructure

DA>0

DA<0

DA=0DA=0

0(a)

RS

QS

1:0:-13:1:-4

1:1:-22:-1:-1

2D flow

(Expansion)RS>0

(Contraction)RS<0

Axisym.expan.DS=0

Axisym.contrac.DS=0

2:1:-3

0(b)

QW

-QS

Irrotationaldissipation

Vortexsheets

Vortextubes

00

QW=-QS

(c)

FIG. 9. Color online Sketch of the invariant maps of R ,Q, RS ,QS, andQW ,−QS with the physical/topological features associated with each zone:a R ,Q map, b RS ,QS map, and c QW ,−QS map.



erence for the zone RS 0,QS0, indicating a predomi-nance of sheet structures. The most probable geometry ob-served in several turbulent flows corresponds to a geometryof 3 :1 :−4 or 2:1 :−3.7–9,12

Finally, the QW ,−QS map Fig. 9c is useful to ana-lyze the topology associated with the dissipation of kineticenergy. The horizontal line QW=ijij /2 represents pointswith high enstrophy but very small dissipation as in the solidbody rotation at the center of a vortex tube “vortex tubes”.On the other hand, the vertical line −QS=SijSij /2 representspoints with high dissipation but little enstrophy density.Thus, it represents points of strong dissipation outside andaway from the vortex tubes “irrotational dissipation”. Theline making 45° with the vertical and horizontal lines, QW

=−QS, represents points of both high dissipation and highenstrophy density, as occurs in vortex sheet structures. Incompressible mixing layers, it has been shown that thesmallest scale motions associated with the highest local val-ues of dissipation but with relatively small amounts of thetotal dissipation tend to be aligned with the QW=−QS line.8

However, generally, the regions of high dissipation are notcorrelated with regions of concentrated enstrophy.7–9

B. Analysis of the invariants QW and QSacross the T/NT interface

In this section, we investigate the second invariants ofthe rate-of-strain and rate-of-rotation tensors QW and QS nearthe T/NT interface in order to analyze the geometry of thedissipation. Conditional mean profiles of the invariants inrelation to the distance from the T/NT interface are shown inFig. 10 the invariant RS I is also shown. The conditionalmean profile of the vorticity component z I is also shownand the symbols mark three particular locations: yI /=0.0T/NT interface, yI /=1.7 point of maximum of z I,and yI /=8.6 deep inside the turbulent region.

As can be seen, all the invariants display quick changesnear the interface, as does z I. We start with the invariant

of the rate-of-rotation tensor QW=ijij /2. As expected, themean profile of this invariant QW I is similar to the evolu-tion of the conditional mean profile of I shown before inFig. 8, i.e., this invariant, like the vorticity modulus, is neg-ligible in the irrotational flow region yI /0 and steeplyrises once the T/NT interface has been crossed at yI /=0. Inthe turbulent region yI / 0, QW I like I is high andmore or less constant, in agreement with Corrsin andKistler,21 and with the recent numerical simulations of Bissetet al.28 and the experimental results of Westerweel et al.26,29

The first interesting observation comes from the condi-tional mean profile of QS I. This invariant is negligible inthe irrotational flow region and far away from the T/NTinterface, yI /−8. However, QS I begins to increasein modulus in the irrotational region from yI / −8 on-ward until very close to the T/NT interface, at yI /−1.7,the invariant QS I displays values of the order ofQS I−0.2U1 /H2. Since QS I=−SijSij I /2 is propor-tional to the dissipation rate, this means that non-negligibleviscous dissipation of kinetic energy occurs in the irrota-tional flow region near the T/NT interface. At first sight, thisresult seems surprising. How can it be that viscous dissipa-tion of kinetic energy takes place in regions with almost novorticity? More work on this subject has to be done in orderto understand this mechanism. What can be said, however, isthat this result is consistent with the existence of irrotationalvelocity fluctuations and Reynolds stresses outside the turbu-lent region near the T/NT interface, as observed by Wester-weel et al.26 The origin of this viscous dissipation in theirrotational region is analyzed in Appendix B following aremark made by a referee. Non-negligible values of the strainproduct and thus of kinetic energy dissipation have beenrecently observed by Holzner et al.31 outside the turbulentregion and close to the T/NT interface in experimental resultsfrom a turbulent front generated by an oscillating grid.

Returning to Fig. 10, we notice that once the T/NT in-terface is crossed, QS I increases in modulus at a fasterrate than it did before in −8yI /0 and reaches a more orless constant turbulent value of QS I−7U1 /H2 fromyI / 5 onward. It is interesting to see that the viscous dis-sipation of kinetic energy, i.e., −4QS I, needs more timethan the enstrophy density, which is proportional to QW I, toreach its turbulent value. Indeed, we see that after yI /=0,the transition into the fully developed turbulent state is fasterfor QW I than for QS I: QS I needs the space between0yI /5 to reach its turbulent value, while QW I reachesits peak at about yI /2.5. This suggests that the mecha-nism of viscous dissipation needs more time to start workingto its fullest than other physical mechanisms driving thegrowth of vorticity during the turbulent entrainment process.

Figures 11a and 11b show the “trajectory” of themean values of QW I and −QS I in their associated phasemap. The mean values of the invariants taken from the irro-tational region are represented by solid triangles, while solidinverted triangles represent points from the turbulent region.The trajectories connecting the mean values are representedby a solid line irrotational region and a dashed line turbu-lent region. Again, the symbols mark three particular loca-

yI/η-10 0 10-8

-4

0

4

8 <|Ωz|>I<QS>I<RS>I<QW>I

FIG. 10. Color online Conditional mean profiles in relation to the distancefrom the T/NT interface for the invariants of the rate-of-strain tensor QS I

and RS I and rate-of-rotation tensor QW I. The conditional mean profilez I is also shown and the symbols mark three particular locations: yI /=0 T/NT interface, yI /=1.7 point of maximum z I, and yI /=8.6 well inside the turbulent region.



tions: yI /=0.0 T/NT interface, yI /=1.7 z I maxi-mum, and yI /=8.6 deep inside the turbulent region. Inthe irrotational region, the mean values of the invariants arenear the vertical line defined by QW I=0. This line marksflow points with high dissipation but little enstrophy; thus,the mean geometry of the dissipation at the T/NT interface ischaracterized by irrotational dissipation. The interface regionis crossed at QW I0.2 and −QS I1.4. After this, the flowtopology detaches from the vertical line QW I=0 and ap-proaches the line QW I= −QS I, indicating a predominanceof vortex sheets in the flow inside the turbulent region. Itsinteresting to see how the mean flow topology moves in theQW ,−QS phase map as the flows goes from the T/NT inter-face into the point of maximum z I. It starts to be roughlyat the middle of the lines QW I=0 and QW I= −QS I, beforeturning into the region between the lines QW I= −QS I and−QS I=0. This may imply a mixed tendency for irrotationaldissipation and vortex sheets, followed by a mixed tendencyfor vortex sheets and vortex tubes. After the point ofmaximum z I has been crossed, at QW ,−QS7.5,5.0,the mean flow topology falls suddenly into the lineQW I= −QS I.

Figures 12a–12c show joint PDFs of QW ,−QS at thethree particular locations used before: yI /=0.0, 1.7, and8.6. Following common practice, the invariants were nondi-mensionalized with SijSij . At the T/NT interface yI /=0.0,the joint PDF of QW ,−QS shows a marked tendency to bealigned with the vertical line defined by QW=0, which attestsa strong predominance of dissipation strain product overenstrophy, i.e., irrotational dissipation dominates at the T/NT

interface, for all the scales of motion. Notice that the prob-ability of having points with QW / SijSij 0.14 is virtuallyzero, i.e., virtually all the points from the T/NT interface areassociated with irrotational dissipation, a fact that was al-ready observed for the “mean” values of this invariant map.As described in the previous section, the large scale flowvortices are characterized by near solid body rotation withlittle dissipation. Thus, the shape of this joint PDF suggeststhe total absence of large scale vortex tubes at the T/NTinterface. This result is not surprising considering that thesetube structures need some time to be generated and also

<QW>I

<-Q

S>I

0 0.5 1 1.5 20

0.5

1

1.5

2

IrrotationalTurbulent

QW=-QS

(a)

<QW>I

<-Q

S>I

0 2 4 6 80

2

4

6

8


QW=-QS

(b)

FIG. 11. Color online Trajectories of the conditional mean values of thesecond invariants of the rate-of-rotation tensor QW I and rate-of-strain ten-sor QS I in their associated phase map for a 0 QW I2 and0 −QS I2 and b 0 QW I9 and 0 −QS I9. The solid line andsolid triangles indicate points in the irrotational region yI /0, while thedashed line and solid inverted triangles indicate points in the turbulent re-gion yI / 0. The symbols mark yI /=0, yI /=1.7, and yI /=8.6.

QW/<SijSij>

-QS/<

SijS

ij>

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(a)

QW/<SijSij>-Q

S/<

SijS

ij>0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(b)

QW/<SijSij>

-QS/<

SijS

ij>

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(c)

FIG. 12. Color online Joint PDFs of QW and −QS at a yI /=0, byI /=1.7, and c yI /=8.6. The contour levels are 0.01, 0.03, 0.1, 0.3, 1, 3,10, and 30.



some room to occupy. At the point of maximum z I,yI /=1.7 as in deep inside the turbulent zone yI /=8.6, theshape of the joint PDFs is quite similar and resembles theshape of the joint PDFs obtained in isotropic turbulence byOoi et al.7 and in a turbulent channel flow by Blackburnet al.9 Notice that here, for the contour lines associated withthe most frequent values and with the scales responsible forthe bulk of the dissipation, there is still some tendency ofthe lines to be aligned with the vertical line QW=0; however,the most intense values, associated with rare events at thesmallest scales of motion but with a relatively small amountof dissipation seem to be aligned with the horizontal line−QS=0 associated with vortex tubes. Indeed, as in Ooiet al.,7 the contour lines of intense values of QW are slightlyskewed toward the axis −QS=0, which suggests that intensevalues of QW correspond to much smaller values of −QS, i.e.,the high QW regions are associated with solid body rotationwith little energy dissipation. This suggests that at bothyI /=1.7 and yI /=8.6, the flow already has some of itscharacteristic large scale coherent vortices. However, sinceQW and −QS are related to local not global features of theflow, the examination of the coherent vortices in relation tothe distance from the T/NT interface should be addressedcarefully in a future study. Finally, notice that in contrast tothe mean values of these invariants shown before in Fig. 11,there is here no discernible tendency for an alignment alongthe line, QW=−QS, associated with vortex sheet structures.This implies that in this case, the mean result obtained beforein the QW ,−QS phase map is just a consequence of theaveraging procedure, i.e., there is no clear tendency for theflow to be dominated by vortex sheet structures. The resultspoint instead to a topology inside the turbulent region whereboth vortex tubes, vortex sheets, and zones of irrotationaldissipation exist.

C. Analysis of the invariants QS and RSacross the T/NT interface

In this section, we investigate the second and third in-variants of the rate-of-strain tensor QS and RS near the T/NTinterface in order to analyze the geometry of straining of thefluid elements. Conditional mean profiles of these invariantsin relation to the distance from the T/NT interface wereshown before in Fig. 10 with the conditional mean profile ofthe vorticity component z I and the symbols markingthree particular locations: yI /=0.0, 1.7, and 8.6.

The evolution of the mean invariant QS I was alreadyanalyzed in Sec. IV B. As for the invariant RS I, one can seethat it is negligible for yI /0, which implies that the ob-served growth of viscous dissipation observed in that regionis not caused by the strain product production term −SijSjkSki

see Eq. 13. Indeed, RS I starts to grow only after theT/NT interface has been crossed. Notice that like QS I, itseems that RS I needs more time to reach its turbulent valueof RS I6.5U1 /H3 than the enstrophy. Finally, note thatthroughout the flow including the irrotational region whereRS I is very small, we always have RS I 0, implying thatthe mean flow structure is always sheetlike.

Figures 13a and 13b show the trajectory of the meanvalues of QS I and RS I in their associated phase map.Again, the mean values of the invariants taken from the ir-rotational region are represented by solid triangles, whilesolid inverted triangles represent points from the turbulentregion. The trajectories connecting the mean values are rep-resented by a solid line irrotational region and a dashed lineturbulent region, and the symbols mark the T/NT interface,the point of maximum z I, and a location placed deepinside the turbulent region. In the entire flow region, we seethat the invariants are in the region RS I 0 and QS I0,i.e., the mean flow geometry is associated with the expansionof the fluid elements. In the irrotational region, the meanflow topology is 3 :1 :−4 and changes to 2:1 :−3 between theT/NT interface and the point of maximum z I. Shortlybefore the point of maximum z I, the flow geometry turnsagain into 3:1 :−4, where it stays some time. Finally, themean flow topology becomes somewhere in the middle ofthese two lines, i.e., near 5

2 :1 :− 72 . Recall that the most

probable eigenvalue ratios observed in several works are3:1 :−4 and 2:1 :−3.

Figures 14a–14c show joint PDFs of RS ,QS at thethree particular locations used before: yI /=0.0, 1.7, and8.6. The three joint PDFs are not fundamentally different,

<RS>I

<QS>

I

0 0.2 0.4 0.6 0.8 1-2

-1.5

-1

-0.5

0


3:1:-4 1:1:-2

2:1:-3

(a)

<RS>I

<QS>

I

0 5 10-8

-6

-4

-2

0IrrotationalTurbulent

1:0:-13:1:-4

1:1:-2

2:-1:-1

2:1:-3

(b)

FIG. 13. Color online Trajectories of the conditional mean values of theinvariants of the rate-of-strain tensors RS I and QS I in their associatedphase map for a 0 RS I1 and −2 QS I0 and b 0 RS I10 and−8 QS I0. The solid line and solid triangles indicate points in the irro-tational region yI /0, while the dashed line and solid inverted trianglesindicate points in the turbulent region yI / 0. The symbols mark threeparticular locations: yI /=0 T/NT interface, yI /=1.7 point ofmaximum z I, and yI /=8.6 well inside the turbulent region.



i.e., all show a clear preference for the region RS 0 andQS0 associated with extensive straining of the fluid ele-ments, although contractive straining also exists at somemuch fewer points. The alignments of the contour lines ofthe joint PDF at the T/NT interface shows that the mostfrequent values seem to show a tendency toward 2:1 :−3see Fig. 14a, while intermediate values seem to be closerto 1:1 :−2, i.e., the smaller scales of motion at the T/NTinterface are associated with near axisymmetric extension. Atthe point of maximum z I at yI /=1.7 and deep insidethe turbulent region at yI /=8.6, the joint PDFs are quitesimilar compare Figs. 14b and 14c. Here, the contourlines are aligned with 2:1 :−3 for the most frequent, inter-mediate, and less frequent values.

D. Analysis of the invariants Q and Racross the T/NT interface

This section analyzes the second and third invariants ofthe velocity gradient tensor Q and R near the T/NT interfacein order to analyze the relation between the flow topologyand dynamics. Conditional mean profiles of the invariants inrelation to the distance from the T/NT interface are shownin Fig. 15. The conditional mean profile of the vorticitycomponent z I is also shown and the symbols markyI /=0.0, 1.7, and 8.6.

The invariant Q= ii−2SijSij /4 shows that in the ir-rotational flow region, Q IQS I0, as expected since inthis region there is virtually no vorticity and, therefore, theevolution of Q I is dominated by the increase of strain prod-uct and viscous dissipation described before. Q I reaches aminimum of Q I−1.3U1 /H2 at yI /=0.5 shortly afterthe T/NT interface Q I begins to grow and becomes positiveat yI /1.2, implying that from that point onward the en-strophy density dominates over strain product. Recall that thestrain product and thus viscous dissipation increases at asmaller rate than the vorticity and enstrophy density in theregion 0yI /5. This invariant attains a maximum atabout yI /2.5 shortly after the maximum of z I, be-fore falling to Q I0 for yI / 8. Notice that Q =0 inisotropic turbulence and, as shown by da Silva and Pereira,36

and also as confirmed in the present work, the plane jet is, inmany ways, close to isotropic at the center of the shear layer.The evolution of Q I just described underlines the interplaybetween enstrophy and strain during the first stages of theturbulent entrainment. Strain dominates over enstrophy inthe irrotational region near the T/NT interface enstrophy isvirtually zero there, while enstrophy dominates over strainat the start of the turbulent region, where, as discussedabove, strain product and also strain skewness do not in-crease as fast as the enstrophy. Deep inside the turbulentregion, the enstrophy and strain product are comparable, asin isotropic turbulence.

The conditional mean of the third invariant of the veloc-ity gradient tensor R=−SijSjkSki+ 3 /4i jSij /3 is zero or

RS/<SijSij>3/2

QS/

<SijS

ij>

-0.5 0 0.5

-1

-0.5

0

2:-1:-1

1:0:-1 3:1:-4

1:1:-2

2:1:-3

(a)

RS/<SijSij>3/2

QS/

<SijS

ij>

-0.5 0 0.5

-1

-0.5

0

2:-1:-1

1:0:-1 3:1:-4

1:1:-2

2:1:-3

(b)

RS/<SijSij>3/2

QS/

<SijS

ij>

-0.5 0 0.5

-1

-0.5

0

2:-1:-1

1:0:-1 3:1:-4

1:1:-2

2:1:-3

(c)

FIG. 14. Color online Joint PDFs of RS and QS at a yI /=0, byI /=1.7, and c yI /=8.6. The contour levels are the same as in Fig. 12.

yI/η-10 0 10-2

0

2

4<|Ωz|>I<Q>I<R>I

FIG. 15. Color online Conditional mean profiles in relation to the distancefrom the T/NT interface for the invariants of the velocity gradient tensorQ I and R I. The conditional mean profile z I is also shown and thesymbols mark yI /=0, yI /=1.7, and yI /=8.6.



negligible in the whole irrotational region, R I0, in yI /0. Once the T/NT interface has been crossed, R growsand reaches a positive maximum of R I0.5U1 /H3 atyI /0.5. Since in this region Q I is high and negative, thisimplies that R I−SijSjkSki I /3=−SSS I. Therefore,the associated flow structure is sheetlike, consistent withthe discussion about RS I described above. R I decreasesafter this, reaching a minimum of R I−1.0U1 /H3 atyI /1.6. At this point, Q I is large and positive and thusR I−i jSij I /4 0, i.e., positive enstrophy productionvortex stretching dominates the flow. Finally, for yI / 8,R I0, as expected since the flow is close to isotropic andR I=0 in isotropic turbulence.

Figure 16 shows the trajectory of the mean values ofQ I and R I in their associated phase map. The mean valuesof the invariants taken from the irrotational region are repre-sented by solid triangles, while solid inverted triangles rep-resent points from the turbulent region, and the trajectoriesconnecting the mean values are represented by a solid lineirrotational region and a dashed line turbulent region.Again, the symbols mark yI /=0.0, 1.7, and 8.6. In the irro-tational flow region, far away from the T/NT interface, themean invariants are at the origin, i.e., R I0 and Q I0.

As the flow approaches the T/NT interface, the mean invari-ants move away from the origin and become more and moredistant from the origin along the line, DA=0, for R I 0 andQ I0, which is associated with straining of fluid elements.Notice that the T/NT interface is very close to the point ofmaximum R I and to the point of minimum Q I. After theT/NT interface, the mean flow topology moves very quicklyinto the region, R I0 and Q I 0, associated with a pre-dominance of vortex stretching. It is interesting to notice thatthe point of maximum z I is located near the point that ismost distant from the origin in this quadrant, indicating thatthe point of maximum z I is connected with the maxi-mum mean values of vortex stretching that occur during theturbulent entrainment process. Finally, the mean invariantsfall into the region near the origin of the R ,Q phase map,as expected since the mean values of both R I and Q I inthe center of the jet shear layer are near zero, as in isotropicturbulence.

Figures 17a–17c show joint PDFs of R ,Q at thethree particular locations used before: yI /=0.0 T/NT inter-face, yI /=1.7 point of maximum z I, and yI /=8.6deep inside the turbulent zone. The first important observa-tion concerns the general shape of the PDFs. At the T/NTinterface, the teardrop characteristic shape of the R ,Qphase map cannot be seen yet see Fig. 17a. The values ofR and Q exist only below the lines defined by the discrimi-nant DA=0. This is consistent with the results described be-fore, i.e., at the T/NT interface, strain product dominatesover enstrophy and thus Q0 for virtually all the points ofthe T/NT interface, and not only in the mean, as we sawbefore. Notice, however, that even at the T/NT interface, thecontour lines of the joint PDFs for R 0 are already alignedwith the line, DA=0. At the points of maximum z I anddeep inside the turbulent region, the joint PDFs of the R ,Qmap already show the well known teardrop shape see Figs.17b and 17c, where Q and R are correlated in two par-ticular regions: R 0 and Q0 representing a predominanceof biaxial stretching of the fluid elements and R0 andQ 0 associated with a predominance of enstrophy produc-tion by vortex stretching. It is impressive to observe howquickly and how so close to the T/NT interface this teardropshape appears: the flow needs a length of less than 1.7 toform the classical teardrop shape. Also, note that the shape ofthe joint PDFs in yI /=1.7 and yI /=8.6 shown in Figs.17b and 17c is very similar and is similar also to the jointPDFs of these quantities in numerous works.3,7–9,19,20 Theonly small difference between Figs. 17b and 17c is thatthe alignment of the contours with the line DA=0 for R 0 isstronger at yI /=8.6 than in yI /=1.7 and also that theintermediate contour lines are a bit more squeezed in thehorizontal direction near the origin for yI /=1.7 than foryI /=8.6. This fact again suggests that there are still someadjustments going on within the flow between yI /=1.7 andyI /=8.6, although the overall shape of the joint PDFs aresimilar in both locations.

<R>I

<Q> I

-0.5 0 0.5-1.5

-1

-0.5

0

0.5

1IrrotationalTurbulent

DA=0DA=0

(a)

<R>I

<Q> I

-1 -0.5 0 0.5 1-2

0

2


DA=0DA=0

(b)

FIG. 16. Color online Trajectories of the conditional mean values of theinvariants of the velocity gradient tensors R I and Q I in their associatedphase map for a −0.6 R I0.6 and −1.5 Q I1 and b −1 R I

1 and −2 Q I3. The solid line and solid triangles indicate points inthe irrotational region yI /0, while the dashed line and solid invertedtriangles indicate points in the turbulent region yI / 0. The symbols mark yI /=0 T/NT interface, yI /=1.7 point of maximum z I,and yI /=8.6 well inside the turbulent region.



V. CONCLUSIONS

The invariants of the velocity gradient R and Q, rate-of-strain RS and QS, and rate-of-rotation QW tensors wereanalyzed near the T/NT interface, which is present in manyflows such as mixing layers, wakes, and jets, using a DNS ofa turbulent plane jet at Re120. The invariants were ana-lyzed by using conditional mean values in relation to thedistance from the T/NT interface, their associated trajectoriesin the classical phase maps, and joint PDFs at several dis-tances from the T/NT interface.

The mean and instantaneous value of all the invariants is

zero in the irrotational flow region far away from the T/NTinterface. As the T/NT interface is approached from the irro-tational flow region, the mean and instantaneous value ofmost of the invariants remains zero. However, we see thatQ I= QS I0 and these invariants are seen to increase inmodulus rapidly. This implies the existence of viscous dis-sipation of kinetic energy outside the turbulent region. Asimilar result was recently obtained by Holzner et al.31 nearthe turbulent front generated by an oscillating grid. More-over, we observed also that not only the mean value of Q isnegative Q I0 but also that the same is true for all pointslocated at −5yI /0, i.e., Q0 everywhere, thereby im-plying that strain product dominates over enstrophy in allflow points from this region. The physical mechanism re-sponsible for this irrotational dissipation still has to be ex-plained; however, the analysis of the invariant RS shows thatthe strain product production term—see Eq. 13—is negli-gible outside the turbulent region and thus cannot explain thehigh level of viscous dissipation found there. Either viscouseffects or nonlocal effects related to the pressure field areresponsible for this irrotational viscous dissipation. Prelimi-nary results discussed in Appendix B seem to imply that thisirrotational dissipation is caused by instantaneous localpure shear induced by the large scale entraining motions.Finally, although RS I0, its value is always positive,which implies that the mean flow geometry is already char-acterized by the straining as opposed to the contraction ofthe fluid elements. In particular, the mean values of RS I andQS I show a preference for a geometry characterized byS :S :S=3:1 :−4 in this region, where S, S, and S arethe eigenvalues of the rate-of-strain tensor arranged in de-scending order.

Right at the T/NT interface, the enstrophy density is stillnegligible as attested by the local and mean values of theinvariant of the rate-of-rotation tensor QW0, but on theother hand, the local and mean values of the strain product,proportional to −QS I, are very high. The joint PDF of QW

and −QS shows that at the T/NT interface, all the flow pointsare characterized by irrotational dissipation, i.e., there is stillno sign of the coherent vortices that are known to exist in theturbulent region. Moreover, the analysis of the invariants Rand Q show that the classical teardrop shape of the R ,Qphase map is not yet formed at the T/NT interface.

All the invariants display rapid changes shortly after theT/NT interface. In particular, the invariants show that thegeometry and topology of the flow rapidly evolves from theT/NT interface until the point of maximum z I located atyI /=1.7. The invariant QW I rapidly grows and reachesvalues which stay more or less constant afterward throughoutthe whole turbulent region. The invariants QS I and RS I,proportional to the viscous dissipation rate and strain skew-ness, respectively, also increase during this time, although ata smaller rate. Indeed, these invariants only reach their tur-bulent values long after the point of maximum z I. No-tice, however, that the geometry associated with the viscousdissipation changes quite dramatically from the T/NT inter-face to the point of maximum z I, as can be appreciatedin the joint PDF of QW and −QS. The contour lines of these

R/<SijSij>3/2

Q/<

S ijSij>

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

(a)

R/<SijSij>3/2

Q/<

S ijSij>

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

(b)

R/<SijSij>3/2

Q/<

S ijSij>

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

(c)

FIG. 17. Color online Joint PDFs of R and Q at three particular locations:a yI /=0 T/NT interface, b yI /=1.7 point of maximum z I, andc yI /=8.6 well inside the turbulent region. The contour levels are thesame as in Fig. 12.



curves at yI /=1.7 are already very similar to the ones ob-served in isotropic turbulence by Ooi et al.7 and also in aturbulent channel flow by Blackburn et al.9 There is stillsome tendency for the contour lines associated with the mostfrequent values to be aligned with the vertical line−QS=SijSij /2 associated with irrotational dissipation, but inmost of the contour lines, no correlation can be observedbetween QW and −QS. The smaller values of the contour linesseem to be tilted to the horizontal line −QS=0, which impliesthat the highest values of QW, representing points with veryhigh values of enstrophy, are associated with little viscousdissipation, as in the case of a solid body rotation. This sug-gests that at this point yI /=1.7, large scale coherent vorti-ces already exist in the flow. Furthermore, the contour linesof the joint PDF between RS and QS show that during theinitial entrainment phase i.e., for 0yI /1.7, the localflow topology is characterized by S :S :S=2:1 :−3. Themost interesting result observed at yI /=1.7 is related to theanalysis of the R and Q invariants and their phase map. Thejoint PDF of these invariants already shows the classicalteardrop shape observed in experimental and numerical stud-ies of many turbulent flows. It is remarkable that the flowneeds less than 1.7, since crossing of the T/NT interface, toform the teardrop shape completely. Moreover, the mean val-ues of these invariants show that the point of maximumz I is very close to the point of maximum Q I and thepoint of minimum R I. This implies that the maximum ofz I is near the point of maximum mean vortex stretch-ing.

Finally, from yI /=1.7 to yI /=8.6, few things seem tochange during the turbulent entrainment process. Indeed,several invariants and joint PDFs at yI /=1.7 and yI /=8.6 are very similar, e.g., the joint PDFs of QW ,−QS,RS ,QS, and R ,Q. However, a closer look into the invari-ants shows that this is not really the case. For instance, themean invariants QS I and RS I still increase to their turbu-lent values after the point of maximum z I has beencrossed at yI /=1.7. This suggests that between yI /=1.7and yI /=8.6, there are still some physical adjustment pro-cesses going on within the flow. The nature of these pro-cesses should be analyzed in future works.

ACKNOWLEDGMENTS

The authors would like to acknowledge an anonymousreferee for many interesting and important remarks madeduring the revision of this work. In particular, the authorsfeel indebted to this referee for an idea about the origin of theirrotational dissipation that we explored in Appendix B.C.B.d.S. is supported by the Portuguese Minister of Scienceand Technology MCTES under “Ciência 2007.”

APPENDIX A: RESOLUTION TESTS

The present work analyzes quantities associated with ex-tremely small and thus very intermittent scales of motionsuch as the invariants of the velocity gradient tensor Q andR. Therefore, it is useful to provide some additional reso-lution checks to the original DNS data bank. This is thepurpose of this appendix. The kinetic energy and kinetic en-

ergy dissipation spectra from the present DNS data were al-ready discussed in Sec. II D with respect to Figs. 4a and4b, respectively. A referee suggested an additional test thatwe carry out here.

We start by defining three filter sizes: k=0.8, 0.9, and1.0. Close inspection of Fig. 4b shows that these filters areplaced well after the peak in dissipation, which is located atk0.3. Therefore, if the present simulation is well re-solved, high pass filtering of the DNS fields at k=0.8, 0.9,and 1.0 will not cause any significant changes to the results.

Figure 18 shows the mean conditional profiles of theinvariants Q I and R I in the region −6yI /6, wherethe DNS data was high pass filtered before the invariantswere computed by using a cutoff filter at k=0.8, 0.9, and1.0, respectively. Results without the application of any filter“no filter” are also shown for comparison. Here, in contrastto Fig. 15, only one single instantaneous field was used tocompute the statistics. As can be seen, no significant differ-ence can be observed between the four conditional meanprofiles.

Figures 19a and 19b show the PDFs of Q and R,respectively, obtained without filtering the DNS data and byhigh pass filtering the data at k=0.8, 0.9, and 1.0 prior tothe computation of the invariants. The PDFs are nondimen-sionalized by the root mean square of the respective variable,e.g., Q= Q2 1/2 in order to highlight the tails of the PDFs.The shape of the PDFs for the three cases are virtually equalfor almost all their values. The zoom of the tails of the PDFsshows that only for PDF values below about 110−6 can westart discerning some small differences between the fourcases, which shows that the differences between the resultsobtained with and without filtering are indeed very small.Thus, we conclude that the invariants are indeed well cap-tured in the present simulation.

APPENDIX B: THE ORIGIN OF THE VISCOUSDISSIPATION IN THE IRROTATIONAL REGION

In this appendix, we analyze a pertinent question raisedby a referee: Is the viscous dissipation found inside the irro-tational NT flow region induced by instantaneous values of

yI/η

<Q> I,

<R> I

-5 0 5-2

-1

0

1

2

3 (no filter)kη=1.0kη=0.9kη=0.8

FIG. 18. Color online Conditional mean profiles of the invariants Q I andR I in the region −6yI /6, obtained after high pass filtering the DNSdata, using a cutoff filter at k=0.8, 0.9, and 1.0. Results obtained withoutfiltering no filter are also shown and the statistics were obtained using onesingle instantaneous field.



pure shear caused by nearby large scale engulfing/entrainingmotions, or is it caused by incoherent irrotational velocityfluctuations near the T/NT interface?

If the dissipation is caused by small velocity fluctua-tions, then we expect them to be mainly associated withsmall scale motions, maybe with velocity and length scalescharacteristic of the nibbling motions associated with the en-trainment mechanism. In this case, we expect them to benear isotropic due to the well known tendency to isotropy ofsmall scale motions in turbulent flows. These small scalemotions could be, for instance, originated by nonlocal effectscaused by the fluctuating pressure field in the nearby turbu-lent region.

However, if the dissipation is caused by nearby largescale engulfing motions, we expect it to be associated withanisotropic velocity fluctuations since it seems plausible forthe most frequent and the most intense of these entrainingmotions to be caused by the larger scale flow vortices in thejet, which are anisotropic and originate in the initial jet in-stabilities arising from the inlet or initial highest meanshear in the jet.

In order to investigate this problem, we decomposed thestrain product SijSij into its six components,

SijSij = S112 + S22

2 + S332 + 2S12

2 + 2S132 + 2S23

2 . B1

In isotropic turbulence, we have the followingrelations:45

u

x2 = v

y2 = w

z2 , B2

ui

xj2 = 2 ui

xi2 no summation , B3

ui

xj uj

xi = −

1

2 ui

xi2 no summation ,

B4

implying that S112 = S22

2 = S332 , S12

2 = S132 = S23

2 , and2S12

2 = 32 S11

2 .Figure 20a shows the conditional mean profiles of the

six terms defined in Eq. B1 in the region −10yI /5,where i , j=1, 2, and 3 represent the x, y, and z direc-tions, respectively. We start by analyzing the resultsfrom the turbulent region at yI /=4, where we have S11

2 I

S222 IS33

2 I1.8U1 /H2 and 2S122 I2S13

2 I2S232 I

2.8U1 /H2. By using these conditional mean values, weobtain 2S12

2 I / S112 I=2.8 /1.81.55, i.e., very close to the

isotropic value of 32 , which again confirms that the plane jet

is statistically very nearly isotropic inside the turbulentregion.

On the other hand, in the irrotational region atyI /=−4, we have S11

2 IS222 IS33

2 I0.05U1 /H2,2S12

2 I0.1U1 /H2, 2S132 I0.06U1 /H2, and 2S23

2 I

0.075U1 /H2, which give 2S122 I / S11 I2.0,

2S132 I / S11 I1.2, and 2S23

2 I / S11 I1.5.Although not very far from isotropic, these values are

inconsistent with isotropic velocity fluctuations inside theNT region near the T/NT interface. Moreover, notice that theterms with the highest conditional mean, i.e., 2S12

2 I and2S23

2 I, are precisely the ones associated with /y, i.e., thedirection of the highest mean shear in a plane jet. Thus, thepresent results seem to give support to the suggestion madeby an anonymous referee in that the existence of a non-

Q/Q’

Pdf

20 30 40 50 6010-7

10-6

10-5

10-4

(no filter)kη=1.0kη=0.9kη=0.8

0 20 40 6010-6

10-4

10-2

100

(a)

R/R’

Pdf

20 40 60 80 10010-7

10-6

10-5

10-4

(no filter)kη=1.0kη=0.9kη=0.8

0 50 10010-6

10-4

10-2

100

(b)

FIG. 19. Color online Joint PDFs of R and Q obtained after high passfiltering the DNS data with a cutoff filter at k=0.8, 0.9, and 1.0. Resultsobtained without filtering no filter are also shown.

yI/η-10 -5 0 5

10-2

10-1

100

101

102 <S112>I

<S222>I

<S332>I

<2S122>I

<2S132>I

<2S232>I

<SijSij>I

FIG. 20. Color online Conditional mean profiles of the six strain productterms defined in Eq. B1 nondimensionalized by U1 /H2 near the T/NTinterface. For simplicity, these conditional mean profiles were made usingonly 20 coordinate points in −10yI /5, while the other conditionalmean profiles used in this work e.g., in Fig. 10 use about 60 points in thesame interval.



negligible viscous dissipation outside the turbulent region iscaused by instantaneous local pure shear motions inducedby the large scale entraining motions.

However, in rigor, the present results do not really ex-clude the other possibility, i.e., nothing in the present resultscontradicts the possibility that both processes may exist, i.e.,the irrotational dissipation may be caused both by instanta-neous pure shear motions and by another small scale process.Clearly, this issue needs further study and should be ad-dressed in future works.

1M. Chong, E. Perry, and B. Cantwell, “A general classification of three-dimensional flow fields simulations of turbulence,” Phys. Fluids A 2, 7651990.

2B. Cantwell, “Exact evolution of a restricted Euler equation for the veloc-ity gradient tensor,” Phys. Fluids A 4, 782 1992.

3B. Cantwell, “On the behavior of velocity gradient tensor invariants indirect numerical simulations of turbulence,” Phys. Fluids A 5, 20081993.

4A. Perry and M. Chong, “Topology of flow patterns in vortex motions andturbulence,” Appl. Sci. Res. 53, 357 1994.

5J. Martín, A. Ooi, S. Chong, and J. Soria, “Dynamics of the velocitygradient tensor invariants in isotropic turbulence,” Phys. Fluids 10, 23361998.

6K. Nomura and G. Post, “The structure and dynamics of vorticity and rateof strain in incompressible homogeneous turbulence,” J. Fluid Mech. 377,65 1998.

7A. Ooi, J. Martin, J. Soria, and M. Chong, “A study of the evolution andcharacteristics of the invariants of the velocity-gradient tensor in isotropicturbulence,” J. Fluid Mech. 381, 141 1999.

8J. Soria, R. Sondergaard, B. Cantwell, M. Chong, and A. Perry, “A studyof the fine-scale motions of incompressible time-developing mixing lay-ers,” Phys. Fluids 6, 871 1994.

9H. Blackburn, N. Mansour, and B. Cantwell, “Topology of fine-scale mo-tions in turbulent channel flow,” J. Fluid Mech. 310, 269 1996.

10J. Soria, A. Ooi, and M. Chong, “Volume integrals of the QA−RA invari-ants of the velocity gradient tensor in incompressible flows,” Fluid Dyn.Res. 19, 219 1997.

11P. Vieillefosse, “Internal motion of a small element of fluid in an inviscidflow,” Physica A 125, 150 1984.

12W. Ashust, A. Kerstein, R. Kerr, and C. Gibson, “Alignment of vorticityand scalar gradient with strain rate in simulated Navier-Stokes turbu-lence,” Phys. Fluids 30, 2343 1987.

13J. Weiss, The Dynamics of Enstrophy Transfer in Two-Dimensional Hy-drodynamics La Jolla Institute, San Diego, CA, 1981.

14J. C. R. Hunt, A. A. Wray, and P. Moin, Annual Research Briefs Centerfor Turbulence Research, Stanford, 1988.

15I. Dubief and F. Delcayre, “On coherent-vortex identification in turbu-lence,” J. Turbul. 1, 11 2000.

16J. Soria and B. Cantwell, “Topological visualisation of focal structures infree shear flows,” Appl. Sci. Res. 53, 375 1994.

17K. Horiuti, “A classification method for vortex sheet and tube structures inturbulent flows,” Phys. Fluids 13, 3756 2001.

18K. Horiuti and Y. Takagi, “Identification method for vortex sheet struc-tures in turbulent flows,” Phys. Fluids 17, 121703 2005.

19F. van der Bos, B. Tao, C. Meneveau, and J. Katz, “Effects of small-scaleturbulent motions on the filtered velocity gradient tensor as deduced fromholographic particle image velocimetry measurements,” Phys. Fluids 14,2456 2002.

20B. Wang, D. Bergstrom, J. Yin, and E. Yee, “Turbulence topologies pre-dicted using large eddy simulations,” J. Turbul. 7, 34 2006.

21S. Corrsin and A. L. Kistler, “Free-stream boundaries of turbulent flows,”NACA Technical Report No. TN-1244, 1955.

22J. C. R. Hunt, I. Eames, and J. Westerweel, “Mechanics of inhomogeneousturbulence and interfacial layers,” J. Fluid Mech. 554, 499 2006.

23A. A. Townsend, The Structure of Turbulent Shear Flow Cambridge Uni-versity Press, Cambridge, 1976.

24J. S. Turner, “Turbulent entrainment: The development of the entrainmentassumption, and its application to geophysical flows,” J. Fluid Mech. 173,431 1986.

25J. Mathew and A. Basu, “Some characteristics of entrainment at a cylin-drical turbulent boundary,” Phys. Fluids 14, 2065 2002.

26J. Westerweel, C. Fukushima, J. M. Pedersen, and J. C. R. Hunt, “Mechan-ics of the turbulent-nonturbulent interface of a jet,” Phys. Rev. Lett. 95,174501 2005.

27D. K. Bisset, J. C. R. Hunt, X. Cai, and M. M. Rogers, Annual ResearchBriefs CTR, Stanford, 1998.

28D. K. Bisset, J. C. R. Hunt, and M. M. Rogers, “The turbulent/non-turbulent interface bounding a far wake,” J. Fluid Mech. 451, 383 2002.

29J. Westerweel, T. Hofmann, C. Fukushima, and J. C. R. Hunt, “Theturbulent/non-turbulent interface at the outer boundary of a self-similarturbulent jet,” Exp. Fluids 33, 873 2002.

30M. Holzner, A. Liberzon, M. Guala, A. Tsinober, and W. Kinzelbach,“Generalized detection of a turbulent front generated by an oscillatinggrid,” Exp. Fluids 41, 711 2006.

31M. Holzner, A. Liberzon, N. Nikitin, W. Kinzelbach, and A. Tsinober,“Small-scale aspects of flows in proximity of the turbulent/nonturbulentinterface,” Phys. Fluids 19, 071702 2007.

32W. C. Reynolds, “Large-scale instabilities of turbulent wakes,” J. FluidMech. 54, 481 1972.

33S. Stanley, S. Sarkar, and J. P. Mellado, “A study of the flowfield evolutionand mixing in a planar turbulent jet using direct numerical simulation,” J.Fluid Mech. 450, 377 2002.

34C. B. da Silva and O. Métais, “On the influence of coherent structuresupon interscale interactions in turbulent plane jets,” J. Fluid Mech. 473,103 2002.

35R. Akhavan, A. Ansari, S. Kang, and N. Mangiavacchi, “Subgrid-scaleinteractions in a numerically simulated planar turbulent jet and implica-tions for modelling,” J. Fluid Mech. 408, 83 2000.

36C. B. da Silva and J. C. F. Pereira, “The effect of subgrid-scale models onthe vortices obtained from large-eddy simulations,” Phys. Fluids 16, 45062004.

37F. O. Thomas and H. C. Chu, “An experimental investigation of the tran-sition of the planar jet: Subharmonic suppression and upstream feedback,”Phys. Fluids A 1, 1566 1989.

38A. Basu and R. Narashima, “Direct numerical simulation of turbulentflows with cloud-like off-source heating,” J. Fluid Mech. 385, 199 1999.

39C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Meth-ods in Fluid Dynamics Springer-Verlag, Berlin, 1987.

40J. H. Williamson, “Low-storage Runge-Kutta schemes,” J. Comput. Phys.35, 48 1980.

41C. B. da Silva and O. Métais, “Vortex control of bifurcating jets: A nu-merical study,” Phys. Fluids 14, 3798 2002.

42P. Freymuth, “On transition in a separated laminar boundary layer,” J.Fluid Mech. 25, 683 1966.

43M. Lesieur, S. Ossia, and O. Métais, “Infrared pressure spectra in 3D and2D isotropic incompressible turbulence,” Phys. Fluids 11, 1535 1999.

44S. Stanley and S. Sarkar, “Influence of nozzle conditions and discreteforcing on turbulent planar jets,” AIAA J. 38, 1615 2000.

45S. B. Pope, Turbulent Flows Cambridge University Press, Cambridge,2000.

46Note that the initial Reynolds number used in Ref. 36 was in fact 3000 andnot 1500 as was by mistake written in that article.

47P. A. Monkewitz and P. Huerre, “Influence of the velocity ratio on thespatial instability of mixing,” Phys. Fluids 25, 1137 1982.

48B. Vreman, B. Geurts, and H. Kuerten, “Large eddy simulation of theturbulent mixing layer,” J. Fluid Mech. 339, 357 1997.

49S. Ghosal and M. Rogers, “A numerical study of self-similarity in a tur-bulent plane wake using large-eddy simulation,” Phys. Fluids 9, 17291997.

50E. Gutmark and I. Wygnansky, “The planar turbulent jet,” J. Fluid Mech.73, 465 1976.

51R. Ramparian and M. S. Chandrasekhara, “LDA measurements in planeturbulent jets,” ASME J. Fluids Eng. 107, 264 1985.

52F. O. Thomas and K. M. K. Prakash, “An investigation of the naturaltransition of an untuned planar jet,” Phys. Fluids A 3, 90 1991.

53P. A. Davidson, Turbulence, An Introduction for Scientists and EngineersOxford University Press, New York, 2004.

54J. Jimenez, A. Wray, P. Saffman, and R. Rogallo, “The structure of intensevorticity in isotropic turbulence,” J. Fluid Mech. 255, 65 1993.

55E. Balaras, U. Piomelli, and J. Wallace, “Self-similar states in turbulentmixing layers,” J. Fluid Mech. 446, 1 2001.

56F. Moisy and J. Jiménez, “Geometry and clustering of intense structures inisotropic turbulence,” J. Fluid Mech. 513, 111 2004.



http://dx.doi.org/10.1063/1.857730

http://dx.doi.org/10.1063/1.858295

http://dx.doi.org/10.1063/1.858828

http://dx.doi.org/10.1007/BF00849110

http://dx.doi.org/10.1063/1.869752

http://dx.doi.org/10.1017/S0022112098003024

http://dx.doi.org/10.1017/S0022112098003681

http://dx.doi.org/10.1063/1.868323

http://dx.doi.org/10.1017/S0022112096001802

http://dx.doi.org/10.1016/S0169-5983(97)00034-8

http://dx.doi.org/10.1016/S0169-5983(97)00034-8

http://dx.doi.org/10.1016/0378-4371(84)90008-6

http://dx.doi.org/10.1063/1.866513

http://dx.doi.org/10.1007/BF00849111

http://dx.doi.org/10.1063/1.1410981

http://dx.doi.org/10.1063/1.2147610

http://dx.doi.org/10.1063/1.1472506

http://dx.doi.org/10.1017/S002211200600944X

http://dx.doi.org/10.1017/S0022112086001222

http://dx.doi.org/10.1063/1.1480831

http://dx.doi.org/10.1103/PhysRevLett.95.174501

http://dx.doi.org/10.1007/s00348-002-0489-5

http://dx.doi.org/10.1007/s00348-006-0193-y

http://dx.doi.org/10.1063/1.2746037

http://dx.doi.org/10.1017/S0022112072000813

http://dx.doi.org/10.1017/S0022112072000813

http://dx.doi.org/10.1017/S0022112002002458

http://dx.doi.org/10.1017/S0022112099007582

http://dx.doi.org/10.1063/1.1810524

http://dx.doi.org/10.1063/1.857333

http://dx.doi.org/10.1017/S0022112099004280

http://dx.doi.org/10.1016/0021-9991(80)90033-9

http://dx.doi.org/10.1063/1.1506922

http://dx.doi.org/10.1017/S002211206600034X

http://dx.doi.org/10.1017/S002211206600034X

http://dx.doi.org/10.1063/1.870016

http://dx.doi.org/10.1063/1.863880

http://dx.doi.org/10.1017/S0022112097005429

http://dx.doi.org/10.1063/1.869289

http://dx.doi.org/10.1017/S0022112076001468

http://dx.doi.org/10.1063/1.857867

http://dx.doi.org/10.1017/S0022112093002393

http://dx.doi.org/10.1017/S0022112004009802

Invariants of the velocity-gradient, rate-of-strain, and …...Invariants of the velocity-gradient,...

Documents

Transcript of Invariants of the velocity-gradient, rate-of-strain, and …...Invariants of the velocity-gradient,...