vortex ﬁlament simulation of the turbulent coﬂowing...

Vortex filament simulation of the turbulent coflowing jetPeter S. Bernarda�

Department of Mechanical Engineering, University of Maryland, College Park, Maryland 20742, USA

�Received 6 October 2008; accepted 8 January 2009; published online 27 February 2009�

A high resolution grid-free vortex filament scheme is applied to the prediction and simulation ofcoflowing round jets with a view to acquiring a new perspective on their physics and establishingthe validity of the numerical technique. Vortex loop removal at inertial range scales provides anondiffusive model of local dissipation that remains compatible with the presence of backscatter.Consistency with the Kolmogorov �5/3 inertial range spectrum is used to estimate the effectivedissipation rate and subsequently the local and global Reynolds numbers associated with theturbulent vortex field. New insights into the accuracy of the scheme are presented includingdependencies on numerical parameters. Comparisons of the computed statistics and structuralfeatures of the coflowing jet versus experiments show the method to provide an accurate renderingof the flow. Some consideration is also given to the dispersion of a scalar contaminant in the jet andits comparison to data. © 2009 American Institute of Physics. �DOI: 10.1063/1.3081559�

I. INTRODUCTION

Complex phenomena associated with turbulent jets, suchas sound generation1–3 and the dispersion of continuous ordiscrete contaminants,4–6 are often tied to the presence ofnonsteady turbulent vortical structures. Thus, the successfulsimulation of jet flows including complex physics requiresnumerical methodologies that offer a physically accurate rep-resentation of the continuously changing vortical aspects ofthe turbulent jet field. Under these circumstances, large eddysimulation �LES� represents a favored direction to take byvirtue of its practicality and reduced role for modeling.

A number of grid-based LES formulations have beenapplied to the prediction of turbulent jet flows. Among these,Olsson and Fuchs7 applied a dynamic model in studying thenear field of a circular jet noting the effects of grid reso-lution, modeling parameters, and Reynolds number. Webband Mansour8 considered a forced round jet at relatively lowReynolds number using a dynamic model and found that theturbulence scales agreed with experiments. A simulation ofcompressible subsonic jets by Zhao et al.9 with several vari-ants of the dynamic model also included prediction of farfield sound radiation. Borg et al.10 used LES without an ex-plicit subgrid model to simulate the entry region of a circularjet and included a concentration field so as to predict massfluxes. Debonis and Scott2 used a compressible form of theSmagorinsky model to simulate a high Reynolds number jet,with some success in matching the correct physics. The ef-fectiveness of different subgrid models in computing com-pressible jet flows was examined by Bogey and Bailly3,11

who noted the influence of the subgrid models in effectivelylowering the Reynolds number.

As in numerous other flows where LES has been ap-plied, many of the properties of jets observed in the afore-mentioned studies are sensitive to the particular form of theadopted subgrid models, as well as parameter choices andgrid resolution. These effects are in part symptomatic of the

persistent problem of crafting subgrid models that can accu-rately regulate the subtle physics governing local variationsin the two-way flow of energy between the resolved andunresolved scales. Any oversimplification of the energy fluxcreates an opening for the appearance of numerical diffusionthat can eliminate vortical flow features that are of centralimportance to the physics. For phenomena such as soundgeneration and particle dispersion that depend precisely onthe vortices, a degree of uncertainty is injected into the nu-merical predictions and the accuracy can be compromised.

Vortex methods,12 in which the computational elementsare freely convecting vortices, have minimal exposure to nu-merical diffusion and thus, in principle, an increased likeli-hood of predicting the natural motion of vortical structures.Grid-free schemes relying on vortex filaments have beenused in a number of prior jet studies that focused on thelaminar and early transitional regimes. Among these, Chungand Troutt4 considered particle dispersion in an axisymmetricjet in which the computational elements were vortex rings,while Agui and Hesselink13 used a single layer of vortexrings with streamwise periodicity to model the evolution ofthe vortex transition including the appearance of streamwisevorticity. Similarly, Martin and Meiburg14,15 used a vortexfilament scheme with periodic boundary conditions to simu-late the effect of perturbations on the vortical makeup of jets.In more recent times, Marzouk and Ghoniem16 used a variantof the vortex filament method to study the vortex structure ofa transitioning transverse jet including the initial breakdownof the flow into turbulence.

The adaptation of vortex methods to the direct numericalsimulation �DNS� of turbulence in which all flow scales in-cluding small viscous dissipative eddies are resolved requiresa formidable computational effort as well as a means for theaccurate representation of viscous forces over a Lagrangianfield of vortex elements. Some success in fulfilling the latterrequirement in the context of a simulation of decaying iso-tropic turbulence has been achieved by periodically redistrib-uting vortex particles.17 Alternatively, a traditional mesh canbe introduced to aid in vorticity redistribution and in thea�Electronic mail: [email protected].

PHYSICS OF FLUIDS 21, 025107 �2009�

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evaluation of the viscous and stretching terms. Such hybridschemes have been used to study isotropic turbulence18 andin reproducing qualitative features of viscous vortex flows infree space.19

For turbulent flows of some complexity, such as theround jets of interest in this study, the most practical imple-mentation of a vortex method is in the context of a LES andnot a DNS. In this regard it has been proposed20 to base aLES on the hybrid vortex particle method by appending atraditional diffusive subgrid model. However, in a LES thedirect treatment of viscous forces acting below the thresholdof resolved scales is not necessary so that alternatives tovortex particle methods, such as the vortex filament schemeused here, may be considered.

The success of a vortex method in simulating turbulentjets in the context of a LES depends primarily on how thefundamental representation of energy transfer to and from“subgrid” scales is handled. For a vortex filament scheme, inwhich the computational elements are short, straight, vortextubes linked end to end forming filaments, it has beensuggested21,22 and previous work verified23,24 that the exci-sion of vortex loops naturally forming from highly stretchedvortex filaments provides an elegant and effective means ofaccounting for the subgrid energy transfer process. In fact,removal of loops as they form provides local dissipation—since loops contribute primarily to the local velocity field—that would ordinarily be occurring at much smaller scales bythe action of viscosity. At the same time, unlike diffusivesubgrid models, loop removal does not inhibit local back-scatter where vortices congregate forming larger scale struc-tures. Computations have shown23,24 that the use of loop re-moval is sufficient to limit growth in the number of vortexelements to practical limits and that the simulated turbulentfield remains fully consistent with many fundamental prop-erties of turbulence.

The present work considers the capabilities of the fila-ment scheme in simulating coflowing turbulent round jetswith an interest in providing a new perspective on their phys-ics. For example, the vortex simulations prove useful in elu-cidating such aspects of the jet flow as the connection be-tween coflow, the potential core, and vortical structures. Inrecognition of the relative novelty of the filament scheme asapplied to turbulent flow, some attention is also directed atproviding further understanding of the numerical propertiesof the filament scheme, such as the importance of numericalparameters, and in more precisely characterizing the Rey-nolds numbers associated with the turbulent field of vortices.Finally, calculations including passive tracer particles areperformed that provide for additional comparisons with ex-periments and allow for a fresh look to be taken at differ-ences between the development of the velocity and scalarfields.

II. THE VORTEX FILAMENT SCHEME

In the vortex filament scheme, straight vortex tubesstrung end to end forming filaments are the principal com-putational element. The ith tube of N total is represented byits end points xi

1 and xi2 and a circulation, say, �i, that it

shares in common with other tubes on the same filament. Thecirculation is assigned to each filament at its inception—usually at a boundary—and remains fixed for all time con-sistent with Kelvin’s theorem and the relegation of viscouseffects to small, invisible scales.

The time evolution of the flow is accounted for by con-vecting the vortex tubes via their end points. Besides advec-tion, such a scheme identically models vortex stretching andreorientation. Calculations with a first-order Euler scheme aswell as fourth-order Runge–Kutta scheme have been imple-mented, with the necessity of using the latter most plainlyevident for applications containing solid boundaries wherethe heightened accuracy is beneficial in resolving the dy-namically important motions leading to vortex creation nearwalls. In the present case such considerations are of lesserimportance and the less costly Euler scheme is used for theresults shown here.

The vortex filaments follow material fluid elements andthus have a net tendency to stretch. To maintain the accuracyof the discretization, an upper bound, say, h, is placed on thetube lengths si��si� where si=xi

2−xi1 is the axial vector along

a tube. Tubes for which si�h are subdivided into segmentsof equal length. Where the flow is turbulent vortex stretchingleads to the creation of prodigious numbers of tubes. Follow-ing Chorin21,22 vortex loops are removed as they form. Thetighter the tolerance in defining a “loop,” the less often loopsare identified and the larger the number of tubes there are incalculations that have reached equilibrium. The criterion foridentifying loops that is used here is based on twice thelength of the tubes that are in close proximity to each other.Changes to the loop removal criterion affect the equilibriumnumber of tubes in a calculation but do not appear to havenoticeable effect on such statistics as the mean flow.

The velocity field, U�x , t�, at a point x in the flow do-main at time t is determined from the cumulative contribu-tions from the vortex tubes together with a potential flow,Up�x , t�, needed to enforce nonpenetration at solid bound-aries. Consequently,

U�x,t� = Up�x,t� −1

4��i=1

Nri � si

�ri�3�i��ri�/�� , �1�

where ri=x−xi and xi= �xi1+xi

2� /2. The ith term in the sumin Eq. �1� represents a simple midpoint rule approximation tothe velocity associated with an individual tube as given bythe Biot–Savart integral. The smoothing function12

��ri�/�� = 1 − �1 − 32 ��ri�/��3�e−��ri�/��3

�2�

is used to desingularize the approximation to the simpleBiot–Savart integral and thus prevent unphysical interactionsbetween vortices that are close to being coincident. The mag-nitude of the numerical parameter � affects how high upalong the singularity in the Biot–Savart function the velocityfield is allowed to go before being smoothed to zero. In fact,the effect of � is confined to the region �ri��2.34� since �is essentially unity beyond this point. The approximate ve-locity field produced by each individual tube is very accurateoutside of its own near field. The use of � compensates forthe fact that information about the vorticity within a tube is

025107-2 Peter S. Bernard Phys. Fluids 21, 025107 �2009�


limited to just the locations of the end points and the amountof circulation, so that the precise form of the detailed, localvelocity field cannot be reconstructed. In this aspect thescheme is similar to a LES in the sense that subgrid motionsare unknowable. Since h is associated with the magnitude ofthe velocity induced by individual vortices according to theapproximation in Eq. �1�, as it diminishes so too does theextent of the near field where local interactions are prone toerror, and the overall discretization becomes more accurate.This aspect of the approximation is likely to have had somerole in the observation24 that the use of relatively long tubescan lead to overprediction of the Reynolds stresses. An inter-esting aspect of this is that once h is set below a threshold thevortex filaments become smooth curves as against a se-quence of straight lines with discrete angular changes be-tween the elements. The Reynolds stresses in this case ap-pear to become independent of h and very close toexperimental trends. This suggests that errors produced bysudden changes in orientation between adjacent straighttubes may also have something to do with the potential tooverpredict the Reynolds stresses. Diminishing h lowers theangles and the errors that go with them.

A more controlled view of the errors produced by h maybe seen by applying Eq. �1� to the evaluation of the velocityfield corresponding to an infinitely long vortex filament. Inparticular, consider a vortex filament situated along the z axiswhose associated velocity field is independent of z and hasnonzero components U and V in the x and y directions, re-spectively. Subdividing the filament into tubes of equallength and applying Eq. �1� mimics the way in which thevelocity is evaluated in the current scheme. Figure 1 showsthe U component of velocity computed this way along a lineparallel to the z axis at the fixed position x=0, y=0.0125 forthree different values of h. It may be noticed that U is equalto the exact constant value �i.e., independent of z� when thefilament is subdivided into short tubes with h=0.005. Forlarge h=0.25 there is a noticeable oscillation of the com-puted velocity that coincides with the relative positions thatthe evaluation points have with respect to the locations of the

tubes. For tubes with h=0.0125, which equals the distancebetween the line where the velocity is computed and thefilament, the velocity is computed to be nearly constant withjust a small oscillation. This establishes the general rule thatEq. �1� is accurate at points further than h from the filaments.Consequently, in any computation for which nearby tubesremain separated by a distance greater than the maximumtube length, the discretization errors associated with Eq. �1�will not be significant. However, neighboring tubes within adistance shorter than the tube length are likely to be affectedby errors similar to those in Fig. 1. It may be imagined thatthis kind of error is exacerbated by sudden changes in orien-tation of adjacent tubes on the same filament.

Since the magnitude of h is a major factor determiningthe number of tubes in a calculation and hence its expense,there is a necessary trade-off between accuracy and effi-ciency in the present simulations very much the way there isin grid-based methods. Fortunately, it appears from this andprevious studies that even if it is not practical to work withthe optimal h in all cases, nonetheless the statistics and flowproperties are well predicted. This suggests that the inci-dence of very close encounters between vortex tubes forwhich the local errors may be large are not burdensomein reproducing the correct physics with the filamentmethodology.

III. NUMERICAL PROBLEM

The flow considered here consists of a round jet withcoflow entering an infinite volume at the origin as shown inFig. 2. It is assumed that length and velocities are scaled bythe jet diameter D and inlet centerline velocity UCL, respec-tively. Consistent with this, the incoming stream is taken tobe of the form

U�r� = �1 + Uc�/2 − �1 − Uc�/2 tanh��r − 1�/�22�� , �3�

where Uc�1 is the velocity of the coflow and parameters1�0 and 2�0 together determine the location and steep-ness of the velocity profile at the orifice edge—in this caser=1 /2. Equation �3� mimics the conditions in a high Rey-nolds number flow where the potential core occupies most ofthe inlet and U�r� drops monotonically from 1 to Uc in a thinboundary layer of thickness 0 at the edge of the jet. In thepresent study, the parameter choices 1=0.495 and 2=0.001are chosen which may be seen to imply that the scale 0 isroughly equal to 0.01.

−0.06 −0.04 −0.02 0 0.02 0.04 0.06−0.2

−0.19

−0.18

−0.17

−0.16

−0.15

−0.14

−0.13

−0.12

−0.11

−0.1

z

U

FIG. 1. Evaluation of U velocity component induced on the line x=0, y=0.0125 by a filament lying on the z axis: —, h=0.005; ¯, h=0.0125; - - -,h=0.025.

x

y

z

U = Uc

U = 1

FIG. 2. Geometry of jet simulation.

025107-3 Vortex filament simulation Phys. Fluids 21, 025107 �2009�


U�r� forms the basis for determining the circulation ofcircular filaments �i.e., rings� that enter the flow domain ateach time step �t. The boundary layer region at the edge ofthe orifice, here between 0.5−0�r�0.5, is subdivided intoN concentric rings through each of which a new filamententers the flow at each time step. The subdivisions of thisregion are demarcated by points ri, i=0,1 , . . . ,N, with r0

=0.5−0, U�r0�=1 and rN=0.5, U�rN�=Uc. The circulationof the ith filament is �i= �U�ri+1�2−U�ri�2��t /2. The pointsri are determined with a view toward achieving approxi-mately equal circulation for all filaments. The larger N is thelower the circulation of any one tube and the discretizationaccuracy is improved. On the other hand, the more ringsused in accommodating the incoming flow, the greater thenumerical expense. Experience suggests that five layersachieve a reasonable balance between the two counterpoisedtendencies.

Each of the circular filaments that enter the flow domainis originally defined as the sides of a polygon inscribed in theunit circle. Typically 20 subdivisions of the circle are madeand since for most calculations reported here h=0.04, thestraight segments immediately subdivide into several vortextubes as the computations begin. Each of the sides of theinscribed polygon are taken to be a “filament” insofar as thebook keeping in the code is concerned even though the sumtotal of these form one complete physical closed filament.The motivation for this procedure has to do with the need toremove filaments when they travel sufficiently far down-stream, as will be discussed below.

A potential flow, as included in Eq. �1� must be intro-duced to maintain the imposed velocity profile at the en-trance nozzle. In essence, the use of the Biot–Savart law incomputing velocities makes it necessary to account for theeffects of vorticity lying outside the computational domain.For simplicity, the potential flow introduced here is consis-tent with that produced by a semi-infinite unit diameter vor-tex tube with unit interior velocity lying upstream of x=0. Inpractical terms it is most efficient to compute this velocitycontribution by replacing the vortex cylinder with a unit di-ameter potential disk placed at the origin with sourcestrengths chosen in such a way as to be consistent with theimposition of the desired U�r� at the inlet. In other words, theflow from the potential disk together with that from the ringsnear the orifice exit combine to produce the desired inletvelocity distribution.

In this formulation there is no inclusion of a forcing tostimulate the jet to transition to turbulence, as is often nec-essary in grid-based schemes. In a vortex filament method,the tubes are sensitive to small perturbations intrinsic to thediscretization and transition without prompting. To some ex-tent the rate and amplitude of the vortex instability may becurtailed by the use of greater numbers of weaker, shortervortices that ameliorate the coarseness of the discretization,but in the absence of a viscous diffusion model to smoothperturbations, the occurrence of transition is to be expected.It should be noted that, as will be touched upon later, becauseof the energy dissipation associated with the use of loopremoval at inertial range scales, there is an implied presence

of viscous dissipation with an associated Reynolds number,even if the latter is not given explicitly.

For the computational problem to be practical it is nec-essary to impose a means for limiting the downstream extentof the vortex elements used in representing the jet. By re-moving vortices at a fixed distance, however, the flow that isinside the computational domain upstream of this point willbe affected. Compensation for lost vorticity in this case ismade difficult by the fact that the jet velocity decays withdistance, so that it not easy to create a simple model of themissing vorticity field. In fact, the least intrusive strategyappears to do nothing and accept the fact that some distortionin the flow properties exists in the region nearest the endplane of the calculation. By monitoring the flow statistics itis possible to trace the induced errors upstream to reveal thatpart of the solution domain that is affected by the boundary.A similar kind of error was observed in the mixing layerflow24 where it proved to not be a far reaching effect. In thepresent case, the effect of the downstream removal of vortic-ity is felt approximately two jet diameters upstream.

Vortex filaments are removed at a downstream boundaryby either of two different approaches. In the first they areremoved after all of the component tubes pass a given point,and, in the second, filaments are deleted if any one of theirtubes passes a given point. For similar sized computationsthe former method has an exit plane upstream of that of thelatter. On the other hand, the first method has the advantagethat the set of vortices in the computational domain is com-plete. The efficiency with which either of the boundary con-ditions works depends on how long the removed filamentsare in the streamwise direction. In the first case, the longerthe filaments are, the more vortex tubes they possess and thegreater the burden in un-needed computation beyond the exitplane. For the second boundary condition, there may be asignificant part of the computational domain with only a par-tial population of vortices.

For the jet calculation, the physical vortex rings that en-ter the flow tend to become very elongated and contain manythousands of tubes by the time they exit the flow domain.The rings are subdivided into numerical filaments as previ-ously discussed so as to reduce the effective lengths of thefilaments that are deleted. The penalty for removing parts ofthe physical vortex rings is that the filaments remaining inthe calculation contain free ends for which loop removal be-comes ineffective. In fact test computations have demon-strated that the presence of too many loose ends—such aswould happen if every tube passing a point was removedfrom the computation—can lead to unbounded growth in thenumber of tubes due to the failure of loop removal. Theapproach adopted here is a compromise between the twoextremes: it permits some tolerance of filaments having freeends so as to reap the advantage of reducing the streamwiseextent of the departing filaments, but it does not include somany as to hamper the effectiveness of loop removal.

It may be noted that a separate issue in having free endsis that at each of these points the zero divergence conditionof the vorticity field is violated. This presumably producesadditional local errors besides those associated with theBiot–Savart evaluation and the perturbations produced by



loop removal. While it is possible to modify the effectivevorticity at the free ends so that zero divergence ismaintained,12 the potential benefit of such an alteration doesnot appear to be sufficient to warrant the greater numericalexpense entailed in its implementation.

The presence of a significant positive jet coflow meansthat the number of vortices appearing in a calculation can beexpected to reach equilibrium since the support of the vor-ticity field is finite in this case. In addition, the higher thecoflow, the smaller the residence time of tubes in the flowdomain and the longer the computational domain can bemade for the same number of tubes. The calculations hereinclude coflows Uc=1 /10, 1/5, 1/4, 1/3, and 1/2 or, in termsof the parameter ��1−Uc� /Uc representing the ratio of theinitial velocity excess to the coflow speed, the cases consid-ered have =9, 4, 3, 2, and 1. Both of these means of char-acterizing the simulations will be used in the sequel. Thehigher the value of , the closer the coflowing jet is to atraditional round jet. The streamwise extent of the simula-tions varies from L=5.25 for Uc=1 /10 to L=15.8 forUc=1 /2. For most runs h=0.04, �t=0.04, and contain up to13�106 vortices at equilibrium. A limited simulation wasalso made for the 1/3 coflow case with h=0.015 and 20�106

vortices at equilibrium.The simulations were performed primarily on an HP Al-

pha EV7 parallel supercomputer at the Pittsburgh Supercom-puting Center. The most time consuming aspect of the algo-rithm is the evaluation of velocities at the locations of the Nvortices via Eq. �1�. The nominal O�N2� cost of this step isreduced to O�N� via a parallel implementation of the adap-tive fast-multipole method,25 thus creating the opportunity touse large numbers of vortices. Mean statistics for the jetsimulations are taken from velocity records accumulatedover a nondimensional time interval of 20–40 encompassing500–1000 time steps. For a case with approximately 12�106

vortices at equilibrium and using 16 processors, the compu-tations require between 2.5 and 5 days for execution. Suchtimings compare favorably to numerical costs reported forgrid-based LES simulations �e.g., Refs. 2 and 9�.

Depending on the coflow, from 1�106 to 3�106 addi-tional vortex tubes are needed to resolve each additional jetdiameter downstream in the fully turbulent region of the jet.For typical computer systems, this represents a significantobstacle in extending the calculations to very long distances.A practical consequence for the present study is that thecases with the lowest coflows and hence shortest computa-tional domains may show a comparatively greater influenceof the downstream boundary than the other simulations.

It is interesting to note that the computation of jet flowsscaled on orifice diameter is a substantially more ambitiouscalculation than a planar shear layer scaled to unit width. Infact, the jet inlet has a shear layer of length � that is morethan three times longer than that of the plane shear layer.Unless unusually large computer resources are available, thismeans that a choice has to be made between choosing large Lor small h. In this study, h is taken to be in a range that islarger than it would be if dictated solely by a desire to elimi-nate discretization errors. By this step it has been possible tocalculate the jets well into the turbulent region.

IV. VELOCITY STATISTICS

The coflowing jet has been well studied in physical ex-periments and many flow statistics are available with whichto assess the quality of the vortex filament simulations. Ofparticular interest is seeing if the method can capture trendsthat arise from changes in the coflow since these point to-ward the effectiveness of the approach in responding tosubtle physical differences that are not readily modeled.

To set the stage for making comparisons it is helpful tofirst consider fundamental quantities categorizing jets such asthe momentum excess,

J � 2��0

�

�U − Uc�Urdr

= 2��0

�

�U − Uc�Urdr + 2��0

�

u2rdr , �4�

where u2 is the streamwise Reynolds stress, and the massflux

M � 2��0

�

�U − Uc�rdr , �5�

that may be calculated at each fixed streamwise location.Integration of the momentum equation on planes normal tothe axial direction suggests that, barring any significant vis-cous effect or change in mean pressure, J is constant inde-pendent of x. In this case, the momentum radius, ��J /Uc,is useful in a number of contexts for characterizing the jetbehavior independent of conditions specific to a particularcase. According to the second equality of Eq. �4�, J may bedecomposed into the sum of a part depending on the meanvelocity and a contribution from the streamwise Reynoldsstress. Figure 3 shows an evaluation of Eq. �4� including thetwo terms in its decomposition for the cases with Uc=1 /5and 1/3. In both plots in the figure the approximate constancyof J is evident with the small variability most likely an arti-

0 2 4 6 8 100

0.2

0.4

0.6

x

0 2 4 6 8 100

0.2

0.4

0.6

x(b)

(a)

FIG. 3. Streamwise decomposition of J. —, J; - - -, contribution from U;- · -, contribution from u2. �a� Uc=1 /5; �b� Uc=1 /3.



fact of the limited averaging and possibly the influence of thedownstream boundary condition.

According to the figure, from initially laminar conditionsin which the momentum excess is entirely contained withinthe mean field, the contribution from the Reynolds stressgrows finally accounting for a significant share of the totalmomentum excess. The growth of the Reynolds stress con-tribution reflects the transition to turbulence in the jet—acentral part of which is the gradual diminishment of thelaminar potential core with streamwise distance, as will beobserved in detail below. Some tendency toward an equilib-rium in the division of J between the mean field and theReynolds stress terms appears to be occurring for the 1/3coflow. In both cases the streamwise extent is not longenough to see what the final trends will be. Between the twocases it appears that for the smaller coflow the Reynoldsstress grows to account for a majority of the velocity excess.The results shown in Fig. 3 fit in with similar analyses of theother coflows. Until such time as substantially longer jets arecomputed it is not possible to reach stronger conclusionsabout the asymptotic behavior of the momentum excess or toperhaps make comparisons with experiments.

The mass flux, shown in Fig. 4, rises initially and, inaccordance with the finite and relatively modest length of thecomputational domains, reaches a maximum and subse-quently falls. The simulations with the smallest coflows seethe greatest rise in mass flux indicating the presence of stron-ger transitional vortices that cause greater entrainment offluid into the jet. At the same time, since these are the short-est of the simulations, they are seen to undergo a relativelyrapid decrease in mass flux as the effects of the truncation ofthe vorticity field are felt. Liepmann and Gharib26 made acareful study of the normalized entrainment d�M /M0� /dx,where M0�M�0�, in the near field of a round jet withoutcoflow. Assuming that near the orifice the results for M inFig. 4 are untainted by downstream effects, then it is inter-esting to examine the consistency between the observed en-trainment in a round jet and the trend in the present data as itchanges with the coflow.

According to Fig. 4, M is linear near x=2 for each of thecoflows considered, so that it is possible to get an unambigu-ous estimate of the slope by least-squares fitting a straightline in each case. A cursory examination of the observeddependence of the predicted entrainment on Uc suggests thatit is exponential which motivates the semilog plot of entrain-ment versus Uc given in Fig. 5. Included in the figure is avalue for the round jet26 which is seen to be fully consistentwith the behavior seen in the simulations. Whether or not theresult shown in this figure is amenable to explanation interms of the dynamics of the near field vortices remains to beseen.

A representative example of the simulated mean velocityfield across the jet is shown in Fig. 6 corresponding to the1/3 coflow solution at x=8.75. The computed, normalized

mean velocity, Un�x ,r��U�x ,r�−Uc� /�U�x�, where

�U�x�� U�x ,0�−Uc, is compared to experimental values27

as well as the Gaussian form e−�r / �2. Here, the length scale

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

M

FIG. 4. Streamwise variation of M for different values of Uc: �, 1/10;+, 1/5; �, 1/4; �, 1/3; �, 1/2.

0 0.1 0.2 0.3 0.4 0.5 0.6

10−2

10−1

Uc

d(M

/M0)/

dx

FIG. 5. Dependence of entrainment on speed of the coflow: �, from Liep-mann and Gharib �Ref. 26�; �, computations.

−2 −1 0 1 20

0.2

0.4

0.6

0.8

1

r/δ

(U−

Uc)/

∆U

FIG. 6. �U−Uc� /�U: �, data of Chu et al. �Ref. 27� at several values of

�10; �, computed with =2 at x=8.75; —, e−�r / �2.



is determined as the value of r when Un=e−1. There appearsto be little doubt that the computed mean velocity is consis-tent with a Gaussian profile, a result which holds equallywell at any of the streamwise positions in this and the othercoflows where the flow is fully turbulent.

A more general definition of the jet width,28 that is notspecific to a Gaussian profile, may be defined via

� ��0

�

�U − Uc�r2dr�0

�

�U − Uc�dr . �6�

In the case that U is Gaussian, a calculation shows that

�/ = 1/2. �7�

Nickels and Perry28 fitted their measured values of Un for =2, 10, and 20 to the empirical function

e−0.677�m2 +0.364�m

3 −0.121�m4

, �8�

where �m�r /�, showing excellent agreement and collapseof the data. Figure 7 compares the same computed profileshown in Fig. 6 with Eq. �8�. The agreement in this case isnot quite as good as with the Gaussian in Fig. 6, suggestingthat at least for the present study, a Gaussian is the morelikely behavior of the mean velocity field. The reason for thediscrepancy between experiments is not apparently known.

Regardless of the details of the mean velocity distribu-tions, the predicted scales for the simulated jets show a sub-stantial agreement with both sets of experiments as is indi-cated in Fig. 8 for and Fig. 9 for �. In particular, it is seenthat the computed values merge smoothly into the physicalmeasurements after the transitional region. The data used inthe comparison for the scale include some measurementsof Chu et al.27 and a sampling of results from many differentexperiments assembled by Davidson and Wang.29 The data inthis case undergo a transition from slope 1 at small x /� toslope 1/3 at large values. The results from the simulationsappear to be largely compatible with this nonlinear trend.

Since the velocity profiles at the orifice are not Gaussian,it is interesting to see at what point Gaussianity is achieved.Some idea of where this occurs is furnished by Fig. 10 con-taining a plot of the streamwise variation of the ratio � / forthe different coflows. Despite some obvious noise in thedata, part of which is presumably due to the downstreamboundary condition, there seems to be a clear trend towardthe Gaussian value of 1 /2 at locations in the turbulent fieldimmediately after the end of the potential core region.

Accurate prediction of the length of the potential coreregion in a traditional round jet represents a significant chal-lenge for LES techniques.2 The additional influence that acoflow may have on this length makes the problem thatmuch more difficult. A quantitative indication of the coreregion for any particular jet may be deduced from plots ofthe velocity excess defined as �U /Uc since this is constantthrough the core region before subsequently dropping there-after. Results for the velocity excess for the present compu-

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

r/∆

(U−

Uc)/

∆U

FIG. 7. �U−Uc� /�U. �, computed with �J=2 at x=8.75; —, Eq. �8� fromNickels and Perry �Ref. 28� fitted to data with �J=2, 10, and 20 at x=30.

10−1

100

101

102

10−2

10−1

100

x/θ

δ/θ

FIG. 8. Streamwise dependence of . Computations at different Uc: �, 1/10;+, 1/5; �, 1/4; �, 1/3; �, 1/2. �, experiments by Chu et al. �Ref. 27�; ·, dataassembled by Davidson and Wang �Ref. 29�.

10−1

100

101

102

10−2

10−1

100

x/θ

∆/θ

FIG. 9. Streamwise dependence of �. Computations at different Uc:�, 1/10; +, 1/5; �, 1/4; �, 1/3; �, 1/2. ·, experiments from Nickels andPerry �Ref. 28�.



tations are plotted in Fig. 11 together with an agglomerationof experimental measurements from Chu et al.27 and Nickelsand Perry.28 With the exception of a small under predictionfor the smaller coflows the extent of the core region is wellaccounted for. While the effect of the downstream boundarymay have some influence on the discrepancies in this figure,they also may reflect differences in the upstream boundaryconditions between the simulations and experiments as wellas the strong perturbative effect of the vortex tube represen-tation that leads to a relative rapid transition. Both of thesekinds of influences have been seen previously in DNS jetsimulations by Boersma et al.30 In all cases the decay ofcenterline velocity immediately after the potential core isseen to be consistent with experiments both in slope and inthe fact that it collapses to a single curve.

V. REYNOLDS STRESSES

It has been observed,28 particularly insofar as amplitudesare concerned, that there is considerable variability from oneexperiment to another in the normalized Reynolds stressessuch as q�u2 / ��U�2. Evidently, it is necessary to mimic thedetails of a specific facility �e.g., upstream turbulence levels,nozzle design� in order make precise comparisons betweennumerical predictions of Reynolds stresses and data fromphysical experiments. For the present study there also tendsto be a significant discrepancy between the downstream lo-cations in the self-similar region where the Reynolds stressesare often measured in experiments and the points further up-stream where they are computed in this study. Other factorsthat inhibit direct comparison between experiment and com-putation are differences in Reynolds number and possibly theuse of vortex discretizations with h larger than optimal.Nonetheless, by focusing on some of the more general prop-erties of the Reynolds stresses it is possible to gain somesense of the intrinsic physicality with which they are ac-counted for by the vortex simulation.

The Reynolds stress amplitude measured in Ref. 28 hasq�0.08 which is in the low end of observations and approxi-mately half the amplitude of the present simulations. Usefulcomparisons can be made in this case by scaling by the localmaxima as shown in Fig. 12 where separate plots are givenof the streamwise, radial, and azimuthal normal Reynoldsstress components as well as the shear component. To obtainsmoother results, the numerical curves represent averagesover values taken between 3.7�x /��4.9. The radial scalinguses � averaged over the same interval and it is seen that thecomputations match the trends closely. The most significantdifference here lies in the greater isotropy of the computa-tions in which, unlike the experiments, v2 and w2 are almostidentical. For this reason the calculated w2 appears to beslightly higher than the trend in the data. The heightenedisotropy of the computations is likely to be a by-product ofthe use of a relatively large h.

Nickels and Perry28 showed that �uv� / �uv�max has a ten-

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

∆/δ

FIG. 10. Variation of � / with x. Computations at different Uc: �, 1/10;+, 1/5; �, 1/4; �, 1/3; �, 1/2. The horizontal line is at 1 /2.

10−1

100

101

102

10−1

100

101

102

x/θ

∆U

/U

c

FIG. 11. Streamwise dependence of nondimensional velocity excess. Com-putations at different Uc: �, 1/10; +, 1/5; �, 1/4; �, 1/3; �, 1/2; ·, experi-ments as given by Chu et al. �Ref. 27� and Nickels and Perry �Ref. 28�.

0 2 40

0.5

1

r/∆

u2/u

2m

ax

0 2 40

0.5

1

r/∆

v2/v

2m

ax

0 2 40

0.5

1

r/∆

w2/w

2m

ax

0 2 40

0.5

1

r/∆

uv/u

vm

ax

FIG. 12. Reynolds stress components. �, experiments �Ref. 28� with =2;—, computations with =2 and data averaged between 3.7�x /��4.9.



dency to spread wider near the origin where �U is relativelylarge than it does far downstream where the velocity deficitis small. A similar trend is found to occur here as shown inFig. 13 where the computed normalized shear stress for the1/5 coflow is seen to be spread wider than that for the 1/3coflow. It should be noted that in this figure the plottingdepends on the predicted value of � that may to some extent,particularly for the 1/5 coflow case, be influenced by prox-imity to the downstream boundary.

A different look at the normal streamwise Reynoldsstress is given in Fig. 14 for data from Antonia and Bilger31

that has considerably larger amplitude than that in Ref. 28. Infact, the computed stresses in this case are similar to theexperiments in regards to both magnitude and other qualita-tive features. The length scale h used in Fig. 14 is the truevelocity half-width defined by the requirement that

Un�x ,h�=1 /2. Shown in the figure is the computed Rey-

nolds stress distribution for Uc=1 /3 at position x /�=3.2 inthe vicinity of the end of the potential core and at x /�=3.9near the downstream boundary. In both cases and most mark-edly in the first position, the Reynolds stress has a localminimum on the jet axis. This is qualitatively similar to theexperimental measurements that are taken at the same valueof but very much more jet diameters downstream. Thenumerical results are seen to be spread slightly further thanthe data, an effect that could be tied to differences in h

among a number of other factors.

VI. VORTICITY FIELD

The population of vortex filaments that comprise thesimulated jets provides a direct view of the underlying vor-tical structure of the flow field that is helpful in understand-ing its development through transition to a turbulent state. Itis most useful to consider views of the vortex tubes selectedaccording to their having an end point within narrow slicesof the flow volume. For example, vortices with end pointxi

1= �xi1 ,yi

1 ,zi1� satisfying the condition �zi

1��0.01 provide aview of the jet structure on the x-y plane through the centralaxis. Figure 15 shows these collections of vortices for thethree coflows 1/4, 1/3, and 1/2. From this perspective thevortical shear layers leaving from the orifice are plainly vis-ible as they frame the potential jet core at the center.

The shear layers progress through a transition that isvery similar if not equivalent to that of a planar shear layer.Vortical structures, essentially ringlike roller vortices withconnecting streamwise rib vortices, form, grow, and mergeout of the incipient shear layers and represent the primarymechanism by which the potential core of the jet graduallyerodes giving way to a fully turbulent state.26,32 As in experi-mental visualizations in round jets33 a noticeable asymmetrydevelops to the vortex structure on either side of the jet.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

r/∆

uv/uv

max

FIG. 13. Shear component of Reynolds stress. �, like a small-excess jet�Ref. 28�; �, like a free jet �Ref. 28�. Computations: —, Uc=1 /3;- - -, Uc=1 /5.

−4 −3 −2 −1 0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

r/δh

√u

2/∆

U

FIG. 14. rms streamwise velocity scaled by velocity excess. Symbols: datafrom Antonia and Bilger �Ref. 31� at several locations in jet with =2.Computations with =2: - - -, x /�=3.2; —, x /�=3.9.

FIG. 15. Vortex elements intersecting within 0.01 of the x-y plane.�a� Uc=1 /4; �b� Uc=1 /3; �c� Uc=1 /2.



Exactly how far downstream the core region persists for agiven coflow is tied to the properties of the vortical struc-tures produced in the shearing. For the 1/4 coflow shownhere and the smaller coflows that were studied, the potentialcore extends to somewhat beyond six diameters downstreamand fits in with the core lengths in a range up to approxi-mately 6 seen in the classical round jet with no coflow.27 Forthe higher coflows, it is clear from the figure that there is asignificant increase in the length of the coflow region.

The initial development of the jet shear layers immedi-ately downstream of the orifice follows a sequence of eventsthat is controlled somewhat by the discrete representation ofthe flow in terms of filaments and tubes. In particular, asobserved in the planar mixing layer study,24 the initial per-turbations of the filaments are constrained to occur at thejunctions between straight tubes. In the present case, thistendency is heightened by the systematic arrangement of thetubes in the vortex rings that enter the computational do-main. In particular, the polygonal structure to the rings con-tains discontinuous changes in tube orientation that act asperturbations to the axial symmetry, in addition to whateverbiases are contained in Eq. �1�. The footprint of the polygo-nal structure is for the most part not evident beyond onediameter from the orifice where the shear layer thickens asseen in Fig. 15. Beyond this point the azimuthal distributionof rib vortices, for example, does not show any connection tothe artificial structure imposed at the inlet.

Figure 15 is useful for seeing how the length of the coreregion determined by the growth of the shear layers is con-nected to differences in the vortical transition correspondingto different coflows. Clearly, for the relatively low coflow of1/4, the growth rate of the turbulent vortices is much fasterthan the high coflow Uc=1 /2 case with the result that thepotential core is closed much earlier. It may also be seen thatthe relatively weaker vortices in the latter case undergo sub-stantially more vortex mergers than the lower coflow. Forexample, a visual inspection of Fig. 15 suggests no morethan five pairings for the 1/4 coflow and at least ten for the1/2 coflow.

A closer look at the three-dimensional �3D� structure ofthe transition is provided by Figs. 16 and 17 showing a viewfrom the top and bottom, respectively, of the vortex filamentscontained in the top and bottom halves of the streamwiseregion 2�x�4 for the Uc=1 /3 jet. For clarity, only everythird filament is displayed in these figures. These imagesmay be directly correlated with the structure seen in the cen-ter illustration of Fig. 15. In particular, the sequence of braid,roller, braid, and roller events in the lower shear layer cen-tered at the points x=2.4, 2.75, 3.1, and 3.5, respectively, isreproduced magnified in Fig. 17. Matching these up with theview in Fig. 16 establishes that the turbulent roller vorticesare more or less ringlike and contain streamwise rib vorticesof an equivalent scale. At the location visualized here thelarge roller and rib vortices are the direct result of the vortexmerger process in the growing shear layers. The roller vortexat 2.75 in Fig. 17 gives a good indication of how rollervortices are perturbed by the wrapping around them ofstreamwise rib vortices.

An end-on view of the vortices in the 1/3 jet at the four

specific streamwise locations referenced above are shown inFig. 18. It is interesting to note the difference in the thick-nesses of the vortex layers between the left and right imagescorresponding to the braids and rollers, respectively. Thisdichotomy is similar to that seen in dye marker viewed inbraid and roller positions in a transitioning jet.26 The lobesrepresent streamwise vortices and vortex pairs arrayedaround the jet perimeter. As in Figs. 16 and 17, the rib vor-tices often extend between several of the images in Fig. 18.As mentioned in regards to Fig. 17 the ribs are seen to have

2 2.5 3 3.5 4−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

z

x

FIG. 16. Roller and rib vortices in the top half-plane viewed from above for1/3 coflow jet in Fig. 15.

2 2.5 3 3.5 4−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

z

x

FIG. 17. Roller and rib vortices in the bottom half-plane viewed from belowfor 1/3 coflow jet in Fig. 15.



a significant effect in distorting the circularity of the rollervortex rings. The end result is to create the lobed appearanceof the vortices in the cross sections.

Figure 19 is a detail of a particularly well formed vortexlobe that is composed from counter-rotating rib vortices. Thevortex elements in the figure intersect the thin volume withinx=3.5�0.025 �at a different time than in Fig. 18�. Also plot-ted are velocity vectors showing the projected motion in theplane. It is evident that the vortices induce a significant out-flow as well as potential motion outside the support of thevorticity itself34 that would lead to entrainment of the fluidinto the jet. The kind of motion in this figure is not unlikethat described in experiments26 and is a central facet of themotion that produces the lateral spread of the jet. Figure 20shows contours of the streamwise velocity together with theplanar velocity vectors that leave little doubt that the actionof the streamwise vortices is to both cause the lateral ejection

outward of high speed fluid that is initially located near theaxis of the jet as well as to entrain the slower moving coflowtoward the jet center.

After transition and the end of the potential core, Fig. 15shows that the fully turbulent jets acquire structure encom-passing the full jet width. This has the appearance of a sinu-ous motion built up from agglomerations of the large vorti-ces produced at the end of the vortex merger and growth.Presumably this structure persists downstream some distancebefore forming the kinds of structures that define the self-similar region.

Another aspect of the jet structure is revealed by lookingat individual vortex filaments at different locations within thejet. These are the physical filaments representing the union ofthe numerical filaments forming complete vortex rings enter-ing the computational domain. A 3D view of filaments be-ginning at the orifice for the 1/5 coflow is given in Fig. 21with corresponding images from an end-on perspective inFig. 22. For the filament near x=1 the azimuthal perturba-tions going around the ring are seen to also have a significantstreamwise component that includes deformation both up-stream and downstream corresponding to inward and out-ward deviations from the nominal position. At this point theimprint of the initial polygonal structure is reflected in thenumber of distortions spread around the ring.

Beginning near x=2 and partially evident in the end-onview in Fig. 22�c� the influence of the inlet structure has

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

y

z

(a)

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

y

z

(b)

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

y

z

(c)

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

y

z

(d)

FIG. 18. End-on view of vortex elements for the Uc=1 /3 jet: �a� x=2.4�braid�; �b� x=2.75 �roller�; �c� x=3.1 �braid�; �d� x=3.5 �roller�.

0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

0.1

0.2

0.3

0.4

0.5

0.6

y

z

FIG. 19. Detail of velocity vectors and vortical structure on plane x=3.5 ata fixed time in the 1/3 coflow simulation.

y

z

0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

0.1

0.2

0.3

0.4

0.5

0.6

FIG. 20. Contours of axial velocity U together with velocity vectors for thesame event as in Fig. 19. U ranges from 1 �lightest shading� to 1/3 �darkestshading�.

0 2 4 6 8 −10

1−1

0

1

yx

z

FIG. 21. Individual vortex filaments in the 1/5 coflow jet.



been supplanted by larger scale disturbances associated withthe growth of the rib vortices. Here there begins long exten-sions in the streamwise direction, culminating in the fourthand fifth images that are in the late stage of transition. Thefilaments in this case wind through the transitional vortexstructure as seen in Figs. 16 and 17. The final vortex fila-ment, containing thousands of individual tubes, is entirely inthe fully developed part of the jet. This covers a substantialstreamwise region as well as the entire jet cross section. Itmay be assumed that loop removal has been felt in the elimi-nation of finer scale windings of the vortex filaments in thisand the other filaments in Figs. 21 and 22.

VII. REYNOLDS NUMBER

As noted previously, the coflowing jet begins to transi-tion to a turbulent state largely as the end result of perturba-tions intrinsic to the discrete representation of the vorticityfield. In this there is no explicit use of a viscosity tied to aReynolds number. Moreover, while a Reynolds number maybe implicit in the incoming flow—since boundary layerscontaining vorticity are put in place at the edge of theorifice—it makes no explicit appearance in the simulation.On the other hand, the simulation relies on loop removal tosupply what amounts to spatially intermittent energy dissipa-tion and where there is dissipation there must be a viscosityand hence local and global Reynolds numbers. The goal nowis to estimate the Reynolds numbers for the jet flow, bothbecause of their intrinsic interest but also to gain some in-sight into how the choice of numerical parameters associatedwith loop removal, primarily vortex length, affects these val-ues. This should be useful in problems containing solidboundaries where a Reynolds number is explicitly assignedand where one would like it to be consistent with the Rey-nolds number produced by the turbulent vortices.

A means of estimating Reynolds number follows fromseveral universal properties of homogeneous and isotropicturbulence that are herein assumed to prevail locally in an

approximate sense in the jet flow. In particular, the one-dimensional �1D� Kolmogorov inertial range spectrum,

E�k� = CK�2/3k−5/3, �9�

is assumed to apply, where k is the wave number and E�k� isthe energy spectrum density. A distillation of empirical re-sults from a diverse group of experiments35 suggests that0.53 is the best estimate for the Kolmogorov constant CK

with the qualification that the Reynolds number be suffi-ciently large. For example, DNS of isotropic turbulence36

suggests that CK=0.60 may be more appropriate if R��100.Here, the Reynolds number R�=u2� /�; u2 is the stream-wise normal Reynolds stress and � is the longitudinal Taylormicroscale. A preliminary calculation has shown that for ei-ther choice of CK it turns out that R� is below 100 so that thefollowing discussion is predicated on the use of CK=0.60. Inany event, use of a different value of this parameter leads torelatively small quantitative changes in the results withoutaffecting the overall conclusions.

To the extent that a �5/3 law of the form C1k−5/3 withconstant C1 is observed locally in the jet simulation, then thedissipation rate can be estimated from

� = �C1/CK�3/2. �10�

In fact, consistency of the vortex simulation with Eq. �9� wasreported previously23 and is also found to be present in thejet simulations considered here, so a determination of � canbe made as a first step toward acquiring a Reynolds number.

A further assumption may be made that the isotropicidentity

� =15�u2

�2 �11�

has at least some approximate local validity in relating thedissipation rate to � and u2. In this case, rearranging Eq. �11�and applying the scaling used in the jet problem yields

Re =UCLD

�=

15u2

�2�, �12�

and a further calculation gives

R� =u��

�= u��Re, �13�

where u��u2. In this way, an approximate determination ofReynolds numbers can be achieved.

For the purposes of the present study, the Reynolds num-ber associated with the Uc=1 /5 simulation at x=6.25 and the1/3 coflow simulation at x=8.25 are considered. These arerepresentative of the results for simulations at the othercoflows. To provide further contrast the simulation with 1/3coflow is taken to have h=0.015 instead of the nominalh=0.04 used in the 1/5 coflow calculation.

An estimate of � and the other quantities pertaining toEqs. �12� and �13� can be had from velocity data obtained oncompact regions of short streamwise extent. Appropriate tothe jet, whose mean field is axisymmetric, the velocity dataare computed on a cylindrical mesh whose appearance on across plane is shown in Fig. 23. The mesh in the figure con-

−1 0 1−1

0

1z

y

(a)

−1 0 1−1

0

1

y

(b)

−1 0 1−1

0

1

y

(c)

−1 0 1−1

0

1

z

y

(d)

−1 0 1−1

0

1

y

(e)

−1 0 1−1

0

1

y

(f)

FIG. 22. End-on views of the vortex filaments shown in Fig. 21. �a�–�f�correspond to the vortices in increasing x.



sists of 12 concentric circles of equally spaced points withthe outermost layer at r=0.5. Shown with it are the vortextubes in a thin slice 6.24�x�6.26 at one instant of time ofthe 1/5 coflow jet. At this streamwise location, more than sixdiameters from the origin, there is significant intermittencyas is evidenced by voids in the vortex field that penetrate towithin r=0.5 of the jet axis. The grid in the figure is limitedto a radius of 0.5 so as to avoid having to take intermittencyinto account in the analysis: it is assumed that the flow iscontinuously turbulent within the central core of the jet cov-ered by the mesh out to r=0.5.

The streamwise discretization of the grid in Fig. 23 con-sists of 201 points extending a distance 0.2 fore and aft of theplane x=6.25. A similar mesh centered on 8.25 is used forthe 1/3 coflow case. The spacing of points, �x=0.002, ismore than adequate to ensure a fine-grained representation ofthe velocity traces. To obtain statistics at any fixed r, aver-aging is performed over the data on fixed cylindrical shellsof grid points: 57 time steps separated by �t=0.4 represent-ing an elapsed time of 22.4 is used for the 1/5 coflow dataand 35 time steps covering an elapsed time of 13.6 for the1/3 coflow. Points further from the center in the mesh arenecessarily better averaged than those closer in. In view ofthe minimal amount of data at r=0, averaging is not done atthis point.

Figure 24 gives representative views of the 1D stream-wise energy spectrum at r=0.32 for the two coflows. Theseapply equally well to any of the other radial positions in themesh. In this, the 1D spectrum is determined by averagingbased on a finite Fourier transform of the individual stream-wise lines of data points in the grid. Included in the figure arefitted lines obeying a k−5/3 power law that are determined byaveraging over the region where E�k�k5/3 is approximatelyconstant. Small parallel shifts to the fitted lines have no ef-

fect on the subsequent discussion other than small, immate-rial, quantitative changes to the computed quantities.

The values of � determined by this technique as a func-tion of radial position for the two simulations are shown inFig. 25, where they are seen to be essentially constant acrossthe core region of the jet. Also shown in Fig. 25 are estimatesof the rms streamwise velocity variance u� and Taylor mi-croscale � determined from the same data sets. The latter iscomputed by fitting a parabola at s=0 to the calculated two-point longitudinal correlation function R11�x ,s��u�x�u�x+se1� /u2�x�. As in the case of �, the values of �and u� are also nearly constant along the central part of thejet.

Between the two cases considered here, the 1/3 coflowhas a smaller velocity variance, a larger �, and a significantlysmaller dissipation rate. Besides the somewhat obvious effectof coflow in producing different levels of shearing, the dif-ferences in Fig. 25 reflect the influence that numerical pa-rameters, such as tube length, have on establishing the small-est resolved scales in the simulation. Numerical parameters

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

y

z

FIG. 23. End-on view of the grid used in obtaining velocity statistics super-imposed on the vortex filaments within a thin slice of the 1/5 coflow jet atx=6.25.

101

102

10−7

10−6

10−5

10−4

10−3

k

E(k

)

FIG. 24. 1D energy spectrum at r=0.32: +, Uc=1 /5, x=6.25; ·, Uc=1 /3,x=8.25. Straight lines are �k−5/3.

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

ε

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

√u

2

0 0.1 0.2 0.3 0.4 0.50

0.05

r

λ

FIG. 25. Computed distribution of �, u2, and �: –, Uc=1 /5, x=6.25; - - -,Uc=1 /3, x=8.25.



must affect how the 1D energy spectrum and the two-pointcorrelation functions are realized, a connection that has yet tobe established more fully. As one example of the subtletiesinvolved, it may be noted that � is smaller than h for the 1/5coflow simulation yet larger than h in the case of the 1/3coflow.

Figure 26 displays the results of using the quantitieswhose values are shown in Fig. 25 to arrive at a determina-tion of Re and R�. Consistent with the absence of significantvariation in the constituent fields, the local and global Rey-nolds numbers are essentially unchanging over the jet core.Of course, one must expect that Re, which may be interpretedas the inverse of the viscosity, is constant everywhere in theflow domain. An average of the Reynolds number values inthe figure suggests that the 1/5 coflow jet at x=6.25 is char-acterized by R��60 and Re�6312 while the 1/3 coflow jetat x=8.25 has R��79 and Re�7653. These are relativelymodest values that imply that loop removal is quite effectivein replacing what would nominally be a model of inviscidEuler flow, with a model of turbulent flow at finite viscosity.That the higher Reynolds number is associated with thesmaller tubes is to be expected since the degree to which thesmall scales are resolved is the primary distinguishing fea-ture between any two simulations that have been scaled simi-larly in terms of UCL and D.

With the Reynolds number known it is straightforward toevaluate the Kolmogorov length and time scales given, re-spectively, by

� =1

Re3/4�1/4 �14�

and

� =1

��Re�1/2 . �15�

These have average values across the jet core given by �=0.0020 and �=0.026 for the 1/5 coflow and �=0.0022 and�=0.038 for the 1/3 coflow. The values of � are significantlysmaller than the respective � values as well as the spatial

resolution implied by the length of the vortex tubes. � falls inthe range of the time step �t=0.04 used in marching thesimulations forward in time. In a general sense the computedvalues of the scales are consistent with viewing the vortexsimulation with loop removal as a LES with dissipationscales lying in the subgrid.

A plot of the compensated spectrum in which E�k� ismultiplied by k5/3 and normalized by Kolmogorov variablesis given in Fig. 27 for the same cases shown in Fig. 24.Consistent with the choice of CK=0.60, the compensatedspectrum is approximately constant at this value. Moreover,the extent of the constant region is greater for the simulationwith the larger of the two Reynolds numbers. A subtle aspectof the figure, which also occurs at most of the r positionswhere the spectrum is computed, is the presence of a smallbump or plateau in the curve just to the right of the flat,inertial range region. Such a feature has been seen in a num-ber of DNS simulations of homogeneous isotropicturbulence.36–38 Finally, it can be construed from the figurethat the inertial range in the simulation is roughly in theinterval 0.1�k��0.2 for the 1/3 coflow, which is a slightlyhigher range than in DNS of isotropic turbulence36 for whichthe upper end is at 0.05.

For the inertial range scales, the structure functionS2�s�= �u�x+se1�−u�x��2 is expected to have s2/3 power lawbehavior given by

S2�s� = CS�s��2/3, �16�

with CS a constant. In fact, Eq. �16� provides an alternativemeans of estimating dissipation rate and consequently Rey-nolds number. Thus, equating an empirical fit, say, C2s2/3, toEq. �16� gives the estimate

� = �C2/CS�3/2. �17�

The Reynolds numbers that have been discerned for thesimulations may be regarded as not being high enough forthe flow to have a fully isotropic inertial range. In this case,according to DNS simulations,36 CS is likely to be substan-tially smaller than its isotropic value, CS=4.02 CK=2.13. Infact, the compatibility of the 1D energy spectra shown in Fig.

0 0.1 0.2 0.3 0.4 0.50

50

100

r

Rλ

0 0.1 0.2 0.3 0.4 0.50

5000

10000

r

Re

FIG. 26. R� and Re across the jet core region: —, Uc=1 /5, x=6.25; - - -,Uc=1 /3, x=8.25.

10−1

100

10−1

100

E(k

)(kη)5

/3R

5/4

e/ε1

/4

kη

FIG. 27. Compensated energy spectrum in Kolmogorov variables: —, Uc

=1 /5, x=6.25; - - -, Uc=1 /3, x=8.25. Dotted horizontal line representsCK=0.60.



27 with the structure functions computed from the same datasets can be tested by evaluating CS from Eq. �17� for given �and C2 and seeing how these compare to values at similar R�

studied in DNS.36 After determining C2 from the raw data forS2�s�, the plot of compensated structure functions S2�s��s��−2/3 shown in Fig. 28 may be made. The horizontallines in the plot correspond to the particular value of CS

=1.43 for the 1/5 coflow at R�=60 and CS=1.58 for the 1/3coflow at R�=79. The appearance of these results has con-siderable qualitative agreement with the shear-free isotropicflow,36 including the relative change between the two curves,though the numerical values are displaced lower by approxi-mately 0.24. Such discrepancies can be accounted for by themany factors affecting the estimates, such as the sensitivityto the � prediction. On the whole, it appears that the simu-lated turbulence in the round jet can be said to have physicalcharacteristics similar to those in other numerical simulationsin a similar Reynolds number range.

VIII. SCALAR FIELD

By adding passive tracer particles to the simulated jets,the discussion can be broadened to include the mixing of acontaminant laden jet with the environment. Either discreteparticles or concentration fields deduced from particle densi-ties in fixed volumes can be considered. In the latter case,turbulent diffusion of the scalar field is assumed to be suffi-ciently dominant so that it is not necessary to include anappropriate random walk to model molecular diffusion at afinite Schmidt number. The results computed here are com-pared to concentration fields determined in the experimentsof Chu et al.27

Particles are seeded into the developing jet at the orificefrom a unit disk of points similar in appearance to that shownin Fig. 23. Thus, each tracer may be viewed as being at thecenter of a small volume of fluid Vt=��t / �4Nt� that entersthe domain at each time step. Here, Nt is the number ofparticles covering the inlet. Since the incoming fluid has unitconcentration, each tracer is assigned mass m=Vt. The con-

centration at subsequent times at the ith collection volume Vi

is estimated to be ci=NiVt /Vi where Ni is the number ofparticles in Vi at a given time. The scalar field is computedfor the case of the 1/5 and 1/3 coflows. These have coarse�Nt=225� and extremely fine �Nt=5041� representations interms of tracers, respectively. The latter allows for smootherinstantaneous predictions of concentration field, though, withsuitable time averaging, the mean statistics appear to beequally well accounted for in both cases.

A view of the axisymmetric concentration field for thetwo coflows is shown in Fig. 29. At Uc=1 /3, the averageincludes 400 consecutive records covering an elapsed time of16, while for the 1/5 coflow 27 fields are averaged coveringa time interval of 5.4. While further averaging might be de-sirable, the figure makes clear the influence of the greatershear in the 1/5 coflow case in increasing the spreading rateof the scalar, besides the reduced length of the core region.The pure conical shape of the core region in both cases isapparent as well and has the same appearance as inexperiments.10

The maximum mean concentration field, located on thecenterline at every position forms the basis of the normalized

centerline dilution27 defined as 1 / � C�x ,0��. Here andhenceforth, to compensate for the reduced averaging at thejet center, the centerline mean scalar is taken to be an appro-priate weighted average of the mean scalar in the small re-gion r�0.167 surrounding the central axis. Comparison ofthe centerline dilution with data from a variety of experi-ments as described by Chu et al.27 is given in Fig. 30. Simi-lar to the results for the velocity excess in Fig. 11, the dilu-tion of the scalar field with downstream distance is seen tofall within a range that is consistent with the measuredvalues.

Similar to Figs. 6 and 7, Fig. 31 compares measured andcomputed distributions of the normalized mean scalar field

C�x ,r� / C�x ,0� across the jet width. The radial distance is

100

101

102

10−1

100

S2(s

)/(s

ε)2/3

s/η

FIG. 28. Compensated structure function: —, Uc=1 /5, x=6.25; - - -, Uc

=1 /3, x=8.25. Dotted lines correspond to CS=1.43 and 1.58, respectively,for the two cases. x

r

0 2 4 6 8 10−2

−1

0

1

2

x

r

0 2 4 6 8 10−2

−1

0

1

2(a)

(b)

FIG. 29. Axisymmetric mean contours of the scalar concentration. �a� Uc

=1 /5; �b� Uc=1 /3.



scaled in terms of the concentration width c, defined by the

condition that C�x ,c� / C�x ,0�=1 /e. The computed curve,which is typical of those in both of the scalar jet calculations,shows behavior that is somewhat off from a Gaussian. Inparticular, the lateral spread of the calculated concentration issomewhat less than that in the experiments while the distri-bution near the centerline compensates by being fuller thanthe Gaussian. From this figure it appears that the computedfield is in a middle stage of transitioning from a top hatprofile at the inlet to what may very well become Gaussianfurther downstream. While the experimental data here are atsubstantially higher values, in fact, close to 10, comparedto the simulation having =2, perhaps a more importantdifference is that the experiments are at many more orificediameters downstream, thus providing greater distance forGaussianity to develop. This behavior is distinctly differentthan the case of the mean velocity field that reaches Gaussi-anity well within the extent of the current simulations as seenpreviously.

As for the concentration width, Fig. 32 shows a plot ofthe streamwise growth in c compared to experiment27 whereit seen to have a similar growth rate but significantly greatermagnitude. To some extent this reflects the non-Gaussianityof the simulated field which has yet to achieve a form inde-pendent of the inlet distribution. The contrast between thisbehavior and that of the mean velocity field is highlighted inFig. 33 comparing the mean concentration and velocity pro-files at the identical position x=8.75 in the 1/3 coflow jet.These curves show that velocity and concentration have thesame extent of penetration into the ambient, coflowing fluid,a fact that is evident from instantaneous visualizations of thetracer particles and velocity contours. The significant differ-ence between the mean fields is in the more uniform spreadof the concentration field over the jet than the velocity field.

It has been often noted6 that the scalar concentrationtends to be relatively well mixed over individual large eddiesthat comprise the principal structure of the transitional andearly turbulent jet. This implies that only sufficiently fardownstream as the eddies break up into a finer scale turbu-lent structure can a Gaussian distribution of mean scalar

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x/θ

1/(Λ

C(x

,0))

FIG. 30. Normalized centerline dilution for the concentration field. �,physical experiments �Ref. 27�. Simulations: —, Uc=1 /3; ¯, Uc=1 /5.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

r/δc

C/C

max

FIG. 31. Normalized concentration across the jet. �, physical experiments�Ref. 27�; �, simulation with Uc=1 /3, x=10.86.

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x/θ

δ c/θ

FIG. 32. Scalar half width. �, physical experiments �Ref. 27�; —, simula-tion with Uc=1 /3.

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

r

FIG. 33. Normalized mean velocity and concentration across the 1/3 coflowjet at x=8.75: �, velocity; �, concentration.



emerge. In contrast, the velocity field is determined every-where as a summation over a distribution of turbulent vorti-ces of varying strengths and orientations that evidently pro-motes the early onset of Gaussianity.

These points are supported by Fig. 34 showing end-oncontour plots at x=8.75 in the 1/3 coflow jet of the velocityand concentration fields. Images �a� and �c� to the left are ofthe instantaneous normalized velocity, while �b� and �d� areof the concentration field. The top images, �a� and �b�, con-trast contours between 0.025 and 0.25 spaced at intervals of0.025, while the lower two figures, �c� and �d�, contain con-tours between 0.75 and 1, also at intervals of 0.025. Consid-ering �a� and �b� at the smaller contour values, there is adramatic difference between the concentration field—that isconfined to the edge of the domain of tracer particles—andthe far more dispersed velocity field. A different situationoccurs with images �c� and �d� in Fig. 34 for the upper rangeof contour levels. Here, the largest scalar contours are dis-tributed over most of the cross section of the jet, while thoseof the velocity are confined to the center. It was noted pre-viously in reference to Fig. 15 that the turbulence regimeprior to the downstream boundary contains many large ed-dies whose provenance is in the more organized upstreamstructures. Evidently, the further downstream dissolution ofthese into smaller scale eddies—within each of which thescalar contaminant is presumably well mixed—is a require-ment for the emergence of complete Gaussianity in the jetscalar field.

IX. CONCLUSIONS

This paper has considered the simulation of the turbulentcoflowing jet via a grid-free vortex filament scheme incorpo-rating loop removal as a de facto subgrid model. The coflow-ing jet flows produced by the method were seen to havelength scales and velocity statistics that fit in well with anumber of physical experiments. The vortical make up of thetransition was fully consistent with experiments including a

major role for the development and growth of streamwisevorticity. The special relationship of the latter to the lateralspread of the jet through entrainment and the ejection of fluidfrom the jet core was observed in detail. Through the use oftracer particles the range of the study was expanded to in-clude analysis of the spread of a scalar jet into an ambientcoflowing fluid. The rapid appearance of Gaussianity in themean velocity distribution was not matched by the meanconcentration field. The contrasting behaviors appear to re-flect different underlying mechanisms in their dispersion,with the concentration field depending on the complete dis-solution of large eddies within which the scalar field is wellmixed.

A significant interest of this work, motivated in part bythe relatively new status of the vortex filament approach as aLES scheme, was in exploring some of the dependencies ofthe method on numerical parameters. A major part of thiswas in establishing a connection between the Reynolds num-ber and the use of loop removal with its implied dissipationrate. For the Reynolds number range corresponding to thesimulations, the jet turbulence was observed to have statisti-cal properties agreeing well with established results for tur-bulence determined from DNS.

Finally, it may be concluded that the vortex filamentscheme achieves one of the primary goals of LES, namely, tocapture essential large scale features of the turbulence with-out the necessity of resolving small viscous scales or care-fully selecting a specific subgrid model and its parameters.For the coflowing jet this was particularly evident in thecapability of the scheme in reproducing subtle aspects of thephysics such as the effect of coflow on the length of thepotential core. This encourages future studies that will utilizethe approach in modeling particle dispersion, sound genera-tion, and other complex phenomena that depend on a robusttreatment of the underlying turbulent eddies. Future plansalso include advances in expanding the numerical efficiencyso that an order of magnitude increase in the number ofvortex elements is feasible. This will enable an expansion ofthe jet simulations to encompass the entire spatial develop-ment from the orifice through transition to the self-similar farfield.

ACKNOWLEDGMENTS

The author would like to thank Professor Joseph Hun-Wei Lee and Professor Mark Davidson for supplying experi-mental data used in the comparisons. The assistance of Dr. P.Collins and Dr. M. Potts in the development and use of theVorcat, Inc. implementation of the 3D vortex filamentmethod is appreciated. This research was supported in partby the National Science Foundation through TeraGrid re-sources provided by the Pittsburgh and San Diego Super-computing Centers.

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z

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(b)

−1 0 1

−1

0

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(d)

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−1

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(a)

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