Aerodynamics of Axisymmetric Counterflowing Jets

Aerodyna

m

ics of ax

i

sy

m

metric co

un

te

r

flowing jets

Ja

i

m

e

Carp

i

o

a

. Amab

l

e

Li

fia

n

b

, Anto

ni

o

L.

Sanchez

c

·.

Fo

r

ma

n

A.

Will

i

ams

c

aE.T.S.I. Industriales, Universidad Politécnica de Madrid, Madrid 28006, Spain bE.T.S.I. Aeronáuticos, Pl. Cardenal Cisneros 3, Madrid 28040. Spain

c Department of Mechanical and Aerospace Engineering. University of California San Diego, La Jolla, CA 92093-0411, USA

A B S T R A C T

The laminar flow resulting from impingement of two steadily fed low-Mach-number gaseous jets issuing into a stagnant atmosphere from coaxial cylindrical ducts at moderately large Reynolds numbers, often used in combustion experiments, is studied through numerical integrations of the Navier-Stokes equations. In the Reynolds-number range addressed, 50-1000, the flow of the approaching jets is nearly inviscid, with viscous effects and mixing being restricted to the thin mixing layers surrounding the jets and to a thin layer located at the separating stream surface. The analysis of the

main inviscid flow shows that only two parameters, based on the scales associated with the radius and the velocity profiles of the two feed streams, are needed to characterize the flow, namely, the ratio of the inter-jet separation dis- tance to the duct radius and the ratio of momentum fluxes of the jets. The numerical results for uniform and Poiseuille velocity profiles provide, in particular, the value of the strain rate at the stagnation point for use in the analysis of experimental studies of counterflow premixed and diffusion flames.

Keywords

Counterflow combustor Strain rate

Impinging jets

1.

Introduction

Counterflow burners have been widely used by the combus-

tion community for over 50 years in investigations of premixed, partially premixed, and non-premixed flames [1,2]. Slot-jet coun

-terflow combustors producing planar flows are employed by some researchers [3], but most experimental arrangements involve ax-

isymmetric flow, addressed in the present paper. Although the

computations pertain to an apparatus employing screen-free noz-

zles and ducts, as sketched in Fig. 1, much of the following discus-

sion also applies to other experimental setups, such as opposing flows through porous plates of honeycombs or screens that pro- vide significant flow resistance. In the figure, two opposed steady round jets issue into a stagnant atmosphere from aligned nozzles of the same radius

R

separated a distance 2H. Although perfectly symmetric configurations involving identical jets are of interest, for example in twin-flame studies, in many situations the properties of the two streams, identified by subscripts 1 and 2, are different, with different densities ρ1 and ρ2, viscosities µ1 and µ2, and volume fluxes Q1 and Q2. The mean jet velocity Um = [Q1/ (πR

2

)]~[Q2/ (πR 2

)] used in experiments is typically much lower than the sound speed, resulting in two constant-density jets having spatial variations of pressure small compared with the ambient pressure,

•Corresponding author

E-mail addressalsp@eng ucsdedu (AL sanchez )

and such that Um2>>gR to minimize buoyancy effects, not considered here. In addition, in laminar combustion experiments the typical values of the relevant Reynolds number

Re

=

ρ1

Q

1

/

(πRµ1) ~ ρ2

Q

2

/

(πRµ2) range from about a hundred to about a thousand, large enough that the flow in the approaching streams is nearly inviscid . Mixing and viscous effects are confined to layers, of small thickness

R/Re

1/2

<< 1

[

4], one localized between the two opposing streams and the others at the two fluid surfaces originating at the rim of the ducts. This justifies the use of the Euler equations outside the mixing layer.

This description is simplified in the central region near the stagnation point, where the flow is self-similar both outside and in- side the mixing layer. There the radial velocity grows linearly with the radial distance r' in the form

A

1

r'/2

in the inviscid

stream 1 and

A

2

r'/2

in the inviscid stream

2.

The strain rates

A

1

and

A

2,both of order

U

m

/ R,

are related by the condition of equal pressure across the mixing layer, so that ρ2

A

22 = ρ1

A

12

.

The flow in the reactive mixing layer in this near-stagnation-point region is also self-similar, with the temperature and composition varying with the distance to the stagnation plane, leading to a one-dimensional problem [5], amenable to numerical integration by standard commercial codes (eg. [6]) and the resulting one-dimensional counterflow model has been the basis for studies of flame-flow interactions in nonpremixed and premixed combustion [7]. At leading order in the limit

R

»

1 the flame structure in the vicinity of the

(i.e.U₁ =U₂ =1andU₁ =U₂ =2

(

1−r2

)

foruniformflowandfor Poiseuille flow, respectively). These velocity distributions are

im-posedin thenumericalcalculationsata distanceoftheorder of,

although moderatelylargerthan,R upstreamfromthe ductexits,

outsidetheregion wheretheflowintheductsisaffected by

up-streamperturbationscomingfromthecollisionregion.

BesidesthePrandtlnumberandSchmidtnumbers,theproblem

dependsonfourparameters,namely,thegeometricparameterH/R,

the Reynolds numberRe,and thedensity andvolume-flux ratios

ρ

2 /

ρ

1 andQ2 /Q1 .The dependencecanbe reducedbynotingthat

theReynoldsnumberReismoderatelylargeintypicalapplications,

so that theflow ofthe counterflowing streamsis nearly inviscid,

withsignificant effectsofmoleculardiffusionoccurringonlynear

thejetboundariesandintheseparatingmixinglayer.

2.1. Inviscidsteadyflow

For steady flow in the limit Re 1, Eqs. (3) and (4) reduce

tov·

∇

T=0andv·

∇

Yi=0,respectively, revealingthat the

tem-perature andcomposition,andtherefore thedensityaccordingto

(5),remainconstantalongthestreamlines,therebyreducing (1)to

∇

·v=0.The descriptionoftheresultingsolenoidalvelocity field

can be simplified by introducing the stream function

ψ

, derived

fromthesteadymassconservationequationwithconstantdensity

ineachofthecounterflowingstreams,sothat

v

x=

1

r

∂ψ

∂

r and

v

r=−

1

r

∂ψ

∂

x. (7)

Forthisaxisymmetricflowthevorticityisazimuthal,with

magni-tude

ω

θ=

∂

_∂

v

_xr−

∂

_∂

v

_rx (8)

givenintermsofthestreamfunctionby

−r

ω

_θ =

∂

_∂

2

ψ

x2 +r

∂

∂

r

1 r

∂ψ

∂

r

. (9)

Since the flow is inviscid and steady, both the stagnation

pres-surep+

ρ

v2 /2andtheratio

ω

_θ/rremainconstantalonganygiven streamline [19],sothat

p+1

2

ρ

|∇

ψ

|

2 /r2 =H

(

ψ

)

(10)

and

ω

θ/r=

(

ψ

)

. (11)

Eq.(10)can be evaluatedalong theaxis toprovidethe

overpres-sure p_o at the stagnation point, which must be equal for both

streams,sothat

p_o=p₁ +1

2 U1 2

(

0

)

=p2 +1 2

2 U2 2

(

0

)

, (12)

where p₁ and p₂ are theoverpressures inthefeedstreams, with

theparameter

2 ₌

ρ

2

ρ

1

_Q 2 Q₁

2 (13)

measuring theratioofthemomentum fluxesofthetwo jets.The

function (

ψ

) can be evaluated by using the boundary

distribu-tionsinthefeedstreams

=−1

r

dU1

dr ,

ψ

=

r

0 U1 rdr, (14)

=

Q_Q2 1

₁

r

dU2

dr ,

ψ

=−

_Q

2

Q₁

r

0 U2 rdr, (15)

associatedwith (6).Inparticular, =0foruniformflow,whilefor

Poiseuilleflow isconstantinbothstreams; =4for0≤

ψ

≤

1/2,and =−4

(

Q₂ /Q₁

)

for−1

2

(

Q2 /Q1

)

≤

ψ

≤0.

Theproblemreducestotheintegrationof

∂

2

_ψ

∂

x2 +r

_∂

∂

_r

1 r

∂ψ

∂

r

=−r2

(

ψ

)

, (16)

obtainedbyusing (11)in (9).Inthefeedstreamsthestream func-tionisgivenfor0≤r≤1by

ψ

=r

0U1 rdrasx→−∞andby

ψ

= −

(

Q₂ /Q₁

)

₀ rU₂ rdrasx→∞.Thecondition

ψ

=0appliesatr=0 for_−∞<x_<+∞,while

ψ

₌1_/2atr₌1for_−∞<x_{≤ −}H_/Rand

ψ

=−1

2

(

Q2 /Q1

)

atr=1forH/R≤x<+∞.Onthejetfree

bound-ariesr₁ (x)andr₂ (x)separatingthejetflowfromtheouterstagnant gas,corresponding to

ψ

₌₋1_/2 and

ψ

₌₋1 ₂

(

Q_{2 /}Q₁

)

_, the condi-tionofconstantpressurecanbewrittenfrom (10)and (12)inthe form

1 2

|∇

ψ

|

2

r2 =po−

1 2[U

2

1

(

0

)

−U1 2

(

1

)

] at r=r1

(

x

)

(17)

1 2

_ρ

₂

ρ

1

_|∇

_ψ

_|

2

r2 =po−

1 2

2 _[U2

2

(

0

)

−U2 2

(

1

)

] at r=r2

(

x

)

(18)

involving the unknown overpressure p_o at the stagnation point,

whoselocation x=xo along theaxis isto be determined aspart

ofthesolution.Thestreamsurface

ψ

=0originatingatthe

stag-nationpoint,whichseparatesthetwoopposingfluids, isan

addi-tionalunknown surfacex=xs

(

r

)

,tobedeterminedfromthe

con-ditionofequalpressureonbothopposingflows

|∇

ψ

|

2

−=

_ρ

2

ρ

1

|∇

ψ

|

2

+ (19)

wherethesubscripts−and+denotethederivativesat

ψ

=0−and

ψ

=0+,respectively.

2.2.Reductiontothecaseofequaldensities

Theproblemcanbesimplifiedconsiderably,removingtheneed

to consider specifically the separating surface and reducing the

parametric dependence, by introducing alternative functions

ψ

ˆ

and ˆ defined by

ψ

ˆ₌

ψ

and ˆ ₌ for

ψ

_> 0 and by

ψ

ˆ₌

(

ρ

2 /

ρ

1

)

1 /2

ψ

and =

(

ρ

2 /

ρ

1

)

1 /2 for

ψ

< 0.Withthisreduced

formulationoftheinviscid problem,thecaseofunequaldensities

takestheformof theequal-density case. Unlike|

∇

ψ

|,the gradi-entof

ψ

ˆ remains continuousacrosstheseparatingsurface

ψ

ˆ=0, ascanbe inferredfrom (19),andthevorticity Eq.(16)reducesto

∂

2

_ψ

ˆ

∂

x2 +r

∂

∂

r

1 r

∂

_ψ

ˆ

∂

r

=−r2

(

ˆ

ψ

ˆ

)

, (20)

whichsimplifiesforuniformfeedvelocities,leadingto ˆ =0,and for Poiseuille flow, leading to =4 for 0_≤

ψ

ˆ_≤1_/2_, and =

−4

for−

/2≤

ψ

≤0. The boundary conditionsfor integration

include

⎧

⎨

⎩

ˆ

ψ

=0 atr=0 for − ∞ < x< +∞ ˆ

ψ

=1/ 2 atr=1 for − ∞ < x≤ −H/R

ˆ

ψ

=−

/ 2 atr=1 forH/R≤x < +∞,

(21)

alongwith

ˆ

ψ

=1/ 2; 1 2

|∇

_ψ

ˆ

_|

2

r2 =po−

1 2[U

2

1

(

0

)

−U12

(

1

)

] at r=r1

(

x

)

(22)

ˆ

ψ

=−

/ 2; 1 2

|∇

_ψ

ˆ

_|

2

r2 =po−

1 2

2 _[U2

2

(

0

)

−U2 2

(

1

)

] at r=r2

(

x

)

(23)

atthefree surfacesseparatingthe jetsfromthestagnant airand

theboundaryvelocitydistributions

ˆ ₌₋1

r

dU₁

dr ,

ψ

ˆ=

r

ˆ ₌

r

dU2

dr ,

ψ

ˆ=−

r

0 U2 rdr asx→∞, (25)

for0≤r≤1.Thislastequationprovidesanimplicitrepresentation forthefunction

(

ˆ

ψ

ˆ

)

,neededtointegrate (20).

The integration woulddetermine, inparticular, theaxial

loca-tion xo of the stagnation point, along with the associated local

stream-functiondescription

ˆ

ψ

=−Ar2 [x−xs

(

r

)

]/ 2, (26)

whereA=A₁ /[Q₁ /

(

π

R3

)

]isthenondimensionalvalueofthestrain rateatthestagnationpoint,andxs(r)isthesurfaceseparatingboth streams,givenby

xs=xo+

r2

2rc

(27)

nearthe stagnation point, where rc is the local radius of

curva-ture. While this curvature contributes a term of order r4 to the

expansionofthestreamfunctionaboutr=0in (26),therealsois acontributionoforderr4 proportionalto

(

x−xo

)

,sothat,through termsoforderr3 ,thecorrespondingnondimensional radial veloc-ityis

v

r=Ar/ 2+Br3 / 8, (28)

thepressuredecreasing locallywithincreasing rinproportionto

(Ar/2)2 attheaxis.Thesensitivities ofthevalues ofxo,rc,A,and

Btotheshapesofthevelocityprofilesintheapproachflowareof

interest.

Theformulationgivenin (20)–(25)isattractivefortworeasons. Firstofall,itremovestheneedtoconsidertheseparatingsurface

ˆ

ψ

=0 asa free surfacein numerical integrations of the inviscid

flow.Secondly,itdemonstratesthatthelow-Mach-numberflow

in-ducedbyopposedgasjetsatmoderatelylargeReynoldsnumbers,

foundincounterflowburners,dependsonlyonthe shapesofthe

velocity profiles U₁ (r) and U₂ (r) in the feed streams and on the twoparameters H/R and

=

(

ρ

2/

ρ

1

)

1 /2

(

Q2/Q1

)

, thelatter

effec-tivelyembodyingthedependencesondensityandvelocityratio.

3. ResultsofnumericalintegrationsoftheNavier–Stokes

equations

The reducedparametric dependenceidentified abovewas

con-sideredin defining the conditions for integration of the Navier–

Stokes Eqs.(1)–(4)forvaluesoftheReynoldsnumberintherange

50≤ Re≤ 1000.In particular, accordingto the previous

reason-ing,underthe large-Reynolds-numberconditionsconsidered here

thetemperatureandcompositionofthefeedstreamsenterinthe

outer nearly inviscid solution only through the factor (

ρ

₂ /

ρ

₁ )1/2 in

₌

(

ρ

_{2 /}

ρ

₁

)

1/2

₍

_Q

2 /Q1

)

,sothatsolutionswithdifferent

compo-sitionand differentfeedtemperatures but equalvalues of

ex-hibit small relative differences of order Re−1 /2 outside the

mix-inglayers bounding thetwo jet streams. Forsimplicity, the

inte-grations below pertain to isothermal jets of two different gases

ofmass fractions Y₁ and Y₂ with a Schmidt number ofSc=0.7,

appropriateforair,andcorresponding boundaryconditionsinthe

feedstreamsaway fromthe pipeexitgivenbyY₁ −1=Y₂ =0as

x→−∞ andY₁ =Y₂ −1=0 asx → ∞. The densityof the

am-bient gas, also with the same Schmidt number, is taken to be

equaltothatofjet1,therebyreducingtheequationofstate (5)to

ρ

[1−Y₂ +Y₂ /

(

ρ

2 /

ρ

1

)

]=1,whichcanbeusedin (4)togive

∂ρ

∂

t +v·

∇

ρ

=

ρ

ScRe

∇

·

ρ

−1

_∇

_ρ

₍₂₉₎

asaconservationequationforthedensity.Thenumerical

integra-tionsof (1),(2),and (29)employeda cylindricaldomainofradius

r=5extending axiallybetweenx=−4 andx=4. Constant

pres-surewasimposed at theambient-gas boundaries, although

com-putationswithexternalco-flows,atvelocitieslessthanmaximum

jet velocities, to delayinstabilities, exerted no observable center-lineeffects,supportingassumptionsofnegligibleinfluencesof

ex-perimental protective blanket streams. A nonslip conditionv=0

wasemployedatthepipewallinintegrationswithPoiseuilleflow

inthefeedstreams.Foruniformflow,however,aslip-flow

condi-tion

v

r=0atthewallwasusedinstead,tomaketheresultsofthe integrationsindependentoftheaxialextentoftheintegration do-main,therebyprovidingresultsforanoppositelimitingcase,with

flows inother configurations,such asnozzle-shaped jets, tending

tofallbetweenthesetwolimits.

The geometrical parameter H/R was varied in the range 0.1

≤ H/R ≤ 2.0 representative of counterflow-flame experiments.

Opposed-jetconfigurationsinvolvingidenticaljetswithlarger val-uesofH/R,ofinterest inindustrialapplications, areknownto be pronetooscillatoryinstabilities [20],resultinginthemixinglayer

becomingattachedtooneortheotherjet,an aspectofthe

prob-lem not investigated here. Different values of the density ratio

ρ

2 /

ρ

1 and of the volume-flux ratio Q2 /Q1 were computed, with

resultingvalues of

=

(

ρ

2 /

ρ

1

)

1 /2

(

Q2 /Q1

)

varying from

=1to

=2.Resultsofintegrations,obtainedwithusemadeofa

previ-ouslydevelopednumericalcode [21],aresummarizedin Figs.1–4.

The general flow structure is shown on the right-hand side

of Fig. 1, which includes two snapshots for isopycnic flow with

Re=1000 and H/R=1. Isocontours ofmass fraction are used to

revealtheboundariesoftheshearlayersboundingthejetsandof

theseparatingmixinglayer.Itisseenfromthisfigurethat,atthis

Reynoldsnumber,themixinglayeroccupiesa verysmallfraction

oftheflowfield, althoughthisfractionwillincrease asaresultof

displacementeffectsthroughthedensitydecreaseassociatedwith

theheat releasewhenthereisa flameintheboundarylayer [4].

BesidesthesymmetriccaseQ₂ /Q₁ =1,thecaseQ₂ /Q₁ =2isused

in thisfigure to illustrate themixing-layer displacement andthe

jet deflection occurring for configurations with unbalanced

mo-mentumflux.ForthisrelativelylargeReynoldsnumberthe

exter-nalboundingsurfacesareseentobeaffectedbyKelvin–Helmholtz

instabilitiesthatprecludetheestablishmentoffullysteady condi-tionsintheradialjet,butitisnoteworthythat,despitethese insta-bilities,theseparatingmixinglayerremains virtuallyunperturbed

anddisplaysin thevicinity of theaxis asteady one-dimensional

structurewithparallelisosurfaces.

Theshapeoftheseparatingstreamlinex=xs

(

r

)

thatoriginates atthestagnation point

(

x,r

)

=

(

xo,0

)

isinvestigated in Fig.2 for

configurationswithunbalancedmomentumflux,variationsofthe

momentum-flux ratio

being imposed computationallyby

vary-ingthevolume-fluxratioQ₂ /Q₁ .For

=1,theseparating stream-lineistheplane xs

(

r

)

=0,by symmetry.Sincethestreamlinex=

xs

(

r

)

obtainedwitha givenvalue of

andthat obtainedwithits

reciprocal 1/

are mirror images of each other about the plane

x₌0_, only results for

_> 1 are considered in the figure.

Be-sides separating streamlines for both uniform and Poiseuille

ve-locity profiles withH/R=1, the figure includes the dependence

on H/R of the stagnation-point location xo and of the curvature

1/rc defining in (27) the local shape of x=xs

(

r

)

near the axis.

Clearly, for the symmetric case

=1 these two quantities are

simply xo=1/rc=0 irrespective of the value of H/R. As can be

seen,theunbalancedmomentumfluxresultsinadisplacementof

thestagnation point towardsthe jetwith lowermomentum,

giv-ing relativedisplacements that increase withincreasing valuesof

andH/R. The separating streamline remains relativelyflat near

thestagnationpoint,withvaluesofthelocaldimensionless curva-ture1/rc thatremainbelow0.5inthecasesconsidered.The

com-parison betweenthe results withuniform andPoiseuille velocity

Fig.4. ThevariationofA/1/2_{withH/Rforuniformandparabolicboundary} veloc-itiesandthreedifferentvaluesof.

displacements andflatter separatingstreamlines. In addition,the

valueofBin (28)(notshown)hasoppositesignsfortheuniform

andparabolic inletprofiles, beingpositiveinthe formercasebut

negativein the latter, as might be expected from the shapes of

theseprofiles.

The flowstructurenearthestagnationpointisfurther

investi-gatedin Fig.3byplottingthedistributionsof

v

x

(

x,0

)

and

v

r

(

0,r

)

in the symmetric case H/R=1,

ρ

2 /

ρ

1 =1, Q2 /Q1 =1, andU1 =

U_{2 =}1forRe₌

(

50_,250_,1000

)

.Theaxialandradialvelocity

com-ponentsdisplaynearthestagnationpointtheexpectedlinear

de-pendence

v

x=−Ax and

v

r=Ar/2corresponding to the

potential-flow stagnation-point solution. In the right-hand figure, for Re₌

1000 the Kelvin–Helmholtz instabilities of the outer shear layers causetheradial velocity to be unsteady atlarge radial distances,

theprofileshownbeingarepresentativeone,althoughthesolution

remains steady, nearly unperturbed, in the near-stagnation-point

region,aspreviouslyindicated.

The resultsrevealthat thelinearvariationoftheaxialand

ra-dial velocity components with the distance from the stagnation

point applies in a fairly large region. The linear variation of the

radialvelocity isassociated witha constantvalue ofthe reduced

pressuregradient₋

(

1_/r

)

∂

p_/

∂

r₌A2 _/4attheseparatinginterface,

as is needed to ensure the validity of the one-dimensional

so-lution for the mixing layer [5]. Although these one-dimensional

solutions,applicable atall Reynolds numbers,exhibit axial varia-tionsof

∂

v

r/

∂

rthat aredifferent forcodeswithdifferent bound-aryconditions,all suchvariationsoccurwithintheverythin mix-ing layers seen in Fig.1, the presentvalues of

∂

v

r/

∂

r=A/2 and

∂

v

r/

∂

r=

(

ρ

1 /

ρ

2

)

1 /2 A/2inthetwostreamsessentiallybeingthose

justoutsidethemixinglayer.Thedeparturesfromthislinear vari-ation,measured bythe lasttermin (28),are seenin Fig.3tobe relativelysmall,sothatthemixinglayercanbeexpectedto

main-tain a one-dimensional structure up to distances of abouthalf a

pipe radius, as was found for all of the cases investigated. The

computationsrevealedthat,whileforthecaseU_{1 =}U_{2 =}1shown in Fig.3 theresultingradial velocity nearthe stagnationpoint is largerthan

v

r=Ar/2,correspondingtopositivevaluesofBin (28),

theopposite isfoundforPoiseuille flowinthe feedstreams.The

negativevalue ofB fortheparabolicinlet profileexhibitsa

mag-nitudewhich, unlike that for the uniforminlet (which decreases

with increasing H/R and depends little on

) is practically

in-dependent of H/R but increases substantially with increasing

,

reachingbeyond−20at

=2.Thisbehaviorisconsistentwiththe

reduction in the effectiveflow restrictionwith increasing radius,

associatedwiththePoiseuilleprofile,andtheincreaseinthe

mag-nitudewithincreasing

isaconsequenceofthedecrease ofthe

relative momentumofstream 1.Thesedifferencesemphasizethe

very noticeableeffectsof thetwo extremeinlet-profiles selected;

in intermediate cases,such asthose ofnozzle-type counterflows,

theinitialdeparturefromlinearityintheright-handpanelmaybe eitherpositiveornegative,dependingonthespecificdesignofthe

experiment. Thisisthelargestinfluenceoftheexperimental

con-figurationfoundinthepresentstudy.

The velocity profiles shown in Fig. 3 indicate that the

stagnation-point strain rate, nondimensionalized only on the

ba-sis of stream 1, A₌A_{1 /}[Q_{1 /}

(

π

R3

)

] is very similar for the three

Reynolds numbers, withnoticeable differences appearing only at

thelowestReynoldsnumber.Thisis quantitativelyinvestigatedin

theupperinsetof Fig.3,whichgivesthecorrespondingvalueofA

obtainedfor differentH/Rand Re=

(

50,250,1000

)

with

ρ

₂ /

ρ

1 =

1,Q₂ /Q₁ =1, andU₁ =U₂ =1.The resultingdifferencesin strain

rateareseentobecomesmallerastheReynoldsnumberincreases,

consistent with the predicted order of magnitude Re−1 /2 of the

departures from the inviscid limit. Viscous effects are more

sig-nificant forlarger internozzle distances, when thegrowth of the

boundingshearlayersdownstreamfromthejetexitinfluencesthe

resultingimpingementregion.Itisofinterestthatthedifferences

in A between the cases Re=250 and Re=1000 remain smaller

than5%overthewholerangeofvaluesofH/Rconsidered.

According to thereduced inviscid formulationintroduced

ear-lierthevalueofthestrainrateA₁ =A₂ √

ρ

₂ /

ρ

1 expressedinthe

di-mensionlessformA=A1[Q1/

(

π

R3

)

]dependson

ρ

2/

ρ

1 andQ2/Q1

through thesingle parameter

=

(

ρ

2 /

ρ

1

)

1 /2

(

Q2 /Q1

)

,so that

re-sults with different density and different volume-flux ratios but

identical values of

should give the same dimensionless strain

rate. This is testedin the lower inset in Fig. 3, which compares theaxialvelocitydistribution

v

x

(

x,0

)

obtainedforRe=1000with

ρ

2 /

ρ

1 =1 and Q2 /Q1 =1 to that obtained with

ρ

2 /

ρ

1 =4 and

Q_{2 /}Q_{1 =}0_.5_,both caseshaving

₌1.For

ρ

_{2 /}

ρ

_{1 =}4thejumpin densityacrossthemixinglayerresultsinajumpinvelocity gradi-ent,inagreementwiththeconditionA₁ =A₂ √

ρ

₂ /

ρ

1 .Theplotalso

showsthatthevelocitygradientstotheleftofthestagnationpoint forthetwocomputations

ρ

₂ /

ρ

1 =1and

ρ

2 /

ρ

1 =4,corresponding

tothevaluesofA=A₁ [Q₁ /

(

π

R3

)

],areindistinguishable,in

agree-ment withtheprevious predictions. Furthermore,whenthe axial

velocity

v

x

(

x,0

)

obtainedfor

ρ

2 /

ρ

1 =4isscaledwith

ρ

1/2 ,giving

the distribution represented withthe empty circles in the inset,

theresulting curvefallson topofthe velocitydistribution ofthe uniform-densitycase,alsoinagreementwiththeinviscidresults.

Figure4isacorrelationofallofthenumericalresultsobtained

at high Reynolds numbers for the strain rates right outside the

mixinglayer,whichisthequantityofinterestincombustion.These plots,whichextendoverawiderangeofH/Rthatencompasses es-sentiallyallcombustionapplications,revealexcellent scalingwith

A

1 /2 =

(

A1 A2

)

1 /2

(

Q1 Q2

)

1 /2 /

(

π

R3

)

, (30)

which shows that the geometric mean of the strain rate across

the mixing layer scales withthe geometric mean of the volume

fluxes. Whenthere isinterestinA,thestrain rateonone sideof

thelayer, nondimensionalizedby thetime scaleconstructedfrom

thevolumeflowrateonthatside,then itmaybe assumedto

in-creasewiththe momentum-flux ratioin proportionto itssquare

root. The resultsobtained with Poiseuillevelocity profiles in the

feed streams are particularlysimple, in that A/

1/2 ࣃ 4.5for all

three valuesof

inthe whole range0.1_≤ H/R_≤ 2explored in

thecomputations.Forthelimitofauniformvelocityprofile,onthe other hand,not onlyisthevalueofA/

1/2 lessthanhalfthat for Poiseuille,butalsotheslowdecreasewithincreasingH/Rismore

pronounced. The shape ofthe velocity profile exerts the greatest

thecurvesin Fig.4togetherwiththeconditionA₁ =A₂ √

ρ

₂ /

ρ

1

en-ableevaluationofthestrainratesincounterflowburnersinvolving

opposedroundjetsissuingfromscreen-freenozzlesprovidedthat

theassociatedjetReynoldsnumbersaresufficientlylarge.

4. Conclusions

Computations have now produced scaling relations for flow

fields and strain rates in axisymmetric counterflows that extend

to conditionsinwhich themomentum fluxesofthetwo streams

are not balancedandthat includethe full rangeofratiosof

sep-aration distances to duct radii. Althoughthe relevance of the

ra-tioofmomentumfluxesofthetwostreamsisconsistentwiththe

inviscid formulation for large Reynolds numbers presented here,

the virtual independence ofthe strain ratejust outsidethe

mix-ing layer from the nondimensional inter-jet distance, as well as

itsgeometric-meanscaling (30),arecomputationalresultsthatare

notimmediateconsequencesofthatformulation.Ofthetwo

phys-ical dimensions involved,R and Hof Fig. 1,the stagnation-point

strain rateincreases asthe formerdecreases butremains

practi-callyindependentofthelatter,asmightbeexpectedfromthe

ob-servations that, astheratio H/R approacheszerothe external

in-viscidflowapproachesthatfromringsource(theringsink

empty-ingtheflows fromthejets)independentofthewidthofthering,

while asthisratioapproachesinfinity alsoa limitingvalue

inde-pendentofHisapproachedifanyinstabilityoftheshearlayersat

the jet exitsis suppressed.The very weak dependenceonH isa

consequenceofthesetwolimitingvaluesnotdifferingmuch,with

the strainratesforintermediate valuesofHlyingbetweenthem.

Although flow-field aspects such as the development of Kelvin–

Helmholtz instabilities inthe outer shear layers also maybe

ex-pected to occur in the presence of significant flow resistance at

theductexits,thevirtuallackofdependenceofthestrainrateon

Hwill failto apply,withit becominginverselyproportional to H

andindependentofRwhenH/Rapproacheszero [22].Thesenew

computationalresultshelptoclarifycharacteristicsof

axisymmet-ric open-ductand nozzle-flowtypesof laminarcounterflows and

mayaidinconsiderationsofexperimentaldesignsofsuchdevices.

Acknowledgments

This work was supported by the Spanish MCINN through

Projects #CSD2010-00010 and MTM2015-67030-P. FAW is

sup-ported by the US National Science Foundation through Award

#CBET-1404026.

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