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Direct numerical simulation of nonisothermal turbulent wall
jetsDaniel Ahlman, Guillaume Velter, Geert Brethouwer, and Arne V.
JohanssonLinné Flow Centre, KTH Mechanics, SE-100 44 Stockholm,
Sweden
�Received 20 November 2008; accepted 23 December 2008; published
online 4 March 2009�
Direct numerical simulations of plane turbulent nonisothermal
wall jets are performed andcompared to the isothermal case. This
study concerns a cold jet in a warm coflow with an ambientto jet
density ratio of �a /� j =0.4, and a warm jet in a cold coflow with
a density ratio of�a /� j =1.7. The coflow and wall temperature are
equal and a temperature dependent viscosityaccording to
Sutherland’s law is used. The inlet Reynolds and Mach numbers are
equal in all thesecases. The influence of the varying temperature
on the development and jet growth is studied as wellas turbulence
and scalar statistics. The varying density affects the turbulence
structures of the jets.Smaller turbulence scales are present in the
warm jet than in the isothermal and cold jet andconsequently the
scale separation between the inner and outer shear layer is larger.
In addition, acold jet in a warm coflow at a higher inlet Reynolds
number was also simulated. Although thedomain length is somewhat
limited, the growth rate and the turbulence statistics
indicateapproximate self-similarity in the fully turbulent region.
The use of van Driest scaling leads to acollapse of all mean
velocity profiles in the near-wall region. Taking into account the
varyingdensity by using semilocal scaling of turbulent stresses and
fluctuations does not completelyeliminate differences, indicating
the influence of mean density variations on normalized
turbulencestatistics. Temperature and passive scalar dissipation
rates and time scales have been computed sincethese are important
for combustion models. Except for very near the wall, the
dissipation time scalesare rather similar in all cases and fairly
constant in the outer region. © 2009 American Institute ofPhysics.
�DOI: 10.1063/1.3081554�
I. INTRODUCTION
A plane wall jet is obtained by injecting fluid along asolid
wall such that the velocity of the jet supersedes that ofthe
ambient flow. The resulting inner boundary layer andouter free
shear layer have different length and time scales,which has
implications for mixing and heat transfer. Ahlmanet al.1 studied an
isothermal wall jet using direct numericalsimulations �DNSs�. The
inner layer showed similarities to azero-pressure boundary layer,
while the outer layer in theDNS was found to resemble a plane jet
shear layer. Approxi-mate collapse of statistics in the near wall
and outer regionwas achieved by applying inner and outer scalings,
respec-tively, thereby revealing the self-similar development of
thejet, which was also observed in the experiments by Erikssonet
al.2 In the study a passive scalar was added in the jet inletto
study mixing properties of the wall jet. The scalar fluctua-tions
in outer scaling were found to correspond to valuesreported for
free plane jets. Streamwise and wall-normal sca-lar fluxes in the
outer layer were of comparable magnitude.Outer scaling also led to
a collapse of the scalar dissipationrate profiles.
In this paper we extend the work by Ahlman et al.1 andstudy a
warm jet in a cold surrounding and a cold jet in awarm environment
by fully compressible DNS. The study ofthe evolution and dynamics
of a cold and warm jet is ofrelevance for thin film cooling and
combustion applications.Of special interest is how the flow
development and mixingare influenced by the varying density.
Structural compressibility effects on turbulence are in
general not expected in fluid flow, as long as the
densityfluctuations are small. This is often referred to as
theMorkovin hypothesis.3 In this case turbulence statistics
ofcompressible flows become similar to incompressible flowsby
properly accounting for the mean density variations in thescaling.
This has been shown in a number of studies of com-pressible
wall-bounded flows.
In simulations of supersonic channel flow and super-sonic
boundary layers the van Driest transformation leads toa collapse of
the mean streamwise velocity profiles both forisothermal and
adiabatic boundary conditions.4–6 To accountfor the variation in
mean density near the wall in compress-ible flow, Huang et al.7
proposed a “semilocal” scaling. Thesemilocal velocity scale is also
consistent with the velocityscale proposed by Morkovin3 for
similarity of the Reynoldsstress in compressible flows. In the
supersonic boundarylayer simulation by Guarini et al.5 and the
channel flowsimulation by Coleman et al.4 and Morinishi et al.6 the
ve-locity fluctuation intensities and the shear stress in
semilocalscaling were reported to compare well with data for
incom-pressible flows. Semilocal scaling was also applied in a
com-pressible channel flow by Foysi et al.8 They reported thenormal
stress peak positions to collapse, but not the peakmagnitudes.
Scalar mixing is of interest in a range of areas includinge.g.,
combustion and atmospheric pollutant transport. A num-ber of
studies have been devoted to mixing in plane andround jet flows
�see, e.g., Refs. 9–11�. Mixing in a reactingenvironment has
recently been studied by means of DNS in a
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reacting shear layer by Pantano et al.12 and in a
reactingmethane-air jet by Pantano.13
Previous studies have mainly focused on compressibleeffects in
relatively high Mach number flows. Studies ofshear flows with
significant density gradient and low Machnumbers are scarce, and
the low-speed compressible effectsin these flows are not well
understood. To our knowledge nonumerical investigation concerning
turbulent nonisothermalwall jets has been published. In the present
study, we there-fore analyze the development and statistics of
plane noniso-thermal turbulent wall jets with low Mach numbers,
bymeans of three-dimensional DNSs. Cold and warm jets aresimulated
and the temperature differences between the ambi-ent and jet fluid
at the inlet are about 400 and 200 K, respec-tively. The aim of
this investigation is to study how the wall-jet development is
influenced by the varying density.Properties of the nonisothermal
jets are compared to resultsobtained in an isothermal jet.1 Proper
scaling approaches inthe respective inner and outer layers are
investigated. Theinfluence of the varying density on the mixing and
transportof scalars is also studied, and the self-similarity of the
veloc-ity and scalar fields is evaluated.
II. GOVERNING EQUATIONS
The governing equations in all jet simulations are thefully
compressible Navier–Stokes equations
��
�t+
��uj�xj
= 0, �1�
��ui�t
+��uiuj
�xj= −
�p
�xi+
��ij�xj
, �2�
��E
�t+
��Euj�xj
=�
�xj�� �T
�xj� + ��ui��ij − p�ij��
�xj, �3�
where � is the mass density, uj is the velocity vector, p is
thepressure, and E=e+ 12uiui is the total energy, being the sumof
the internal energy e and kinetic energy. Fourier’s law,where � is
the coefficient of thermal conductivity, is used toapproximate the
energy fluxes. The stress tensor is defined as
�ij = �� �ui�xj + �uj�xi � − 23��ij�uk�xk , �4�where � is the
dynamic viscosity.
The fluid is assumed to be calorically perfect and to obeythe
ideal gas law
e = cvT , �5�
p = �RT , �6�
and a ratio of specific heats of �=cp /cv=1.4 is used. Toaccount
for the substantial variations in density and tempera-ture a
temperature dependent viscosity is used in thenonisothermal cases.
The viscosity is determined throughSutherland’s law
�
� j= � T
Tj�3/2Tj + S0
T + S0, �7�
where T is the local temperature and Tj is the jet
centertemperature at the inlet. The reference coefficient
isS0=110.4K which is valid for air at moderate temperaturesand
pressures.
A transport equation for a passive scalar
���
�t+
�
�xj��uj�� =
�
�xj��D ��
�xj� , �8�
where D is the scalar diffusion coefficient is solved to
inves-tigate the mixing. For the energy equation a constant
Prandtlnumber Pr=�cp /�=0.72 is assumed and for the passivescalar a
constant Schmidt number Sc=� /�D=1 is used.This implies that the
heat conductivity � and the scalar dif-fusion �D have the same
temperature dependence as the dy-namic viscosity � �Sutherland’s
law� since we also assumeconstant cp.
III. NUMERICAL METHOD
The simulations are performed employing a sixth-ordercompact
finite difference scheme14 for the spatial discretiza-tion, and a
third-order low-storage Runge–Kutta scheme forthe temporal
integration.15 To minimize reflections at in- andoutlets, boundary
zones as described by Freund16 are applied.Within the boundary
zones the solution is smoothly forcedtoward target profiles and
spurious fluctuations are damped.The outlet target functions must
be constructed with care,especially in the nonisothermal cases.
Smaller runs were per-formed to compute the half widths and maximum
velocitypositions at the outlet to obtain target functions for the
finalsimulations.
The goal of the investigation is to study turbulent walljets,
hence high magnitude disturbances, urms� =0.065Uj,where Uj is the
inlet jet center velocity, are applied at theinlet to facilitate a
fast and efficient transition to turbulence.Three types of
disturbances are used; random but correlatedin time and space using
the method of Klein et al.;17 stream-wise vortices in the upper
shear layer and harmonic stream-wise disturbances. The disturbances
are superimposed at theinlet and added at every time step. For the
correlated distur-bances a correlation length of h /3 was used in
all three di-rections, except in the streamwise direction in the
cold andwarm jet simulations, where the correlation length was
about40% lower. The simulation method is presented in more de-tail
in Ahlman et al.1
IV. WALL-JET SIMULATION
An isothermal plane wall jet was investigated by Ahlmanet al.,1
and data from their DNS are used in this study forcomparison. We
have carried out DNS of three nonisother-mal wall jets. To generate
a cold jet in a warm surroundingand a warm jet in a cold
environment, the inlet energy anddensity profiles are varied
appropriately. The respective flowcases are characterized by the
density ratio of the ambient tojet fluid at the center of the inlet
��=�a /� j. In the cold jetcases a ratio of ��=0.4 is used and in
the warm ��=1.7.
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The simulation domain is a rectangular box with a no-slip wall
at the bottom. Periodic boundary conditions areused in the spanwise
direction. The streamwise, wall-normal,and spanwise directions are
denoted by x, y, and z, respec-tively. Above the jet a slight
coflow of Uc=0.10Uj is appliedfor computational reasons. In
particular, at startup large scalevortices may develop above the
jet. The coflow advects theselarge persistent vortices out of the
domain. The temperatureat the wall is constant and equal to the
ambient condition inall three cases. For the passive scalar a
no-flux boundarycondition is imposed, ��� /�y�y=0=0. At the top of
the domain
an inflow velocity Utop of 0.026Uj is applied for cases I2,
C2,and W2 while Utop=0.065Uj for C7 because the entrainmentis
larger in this case. The inlet profiles for the velocity andpassive
scalar, plotted in Fig. 4, are similar to the ones usedin the
previous investigation of the isothermal jet.1 The jetheight h is
defined as the velocity half width at the inlet, i.e.,h=y1/2�x=0�.
The velocity half width is the distance from thewall where the mean
velocity is equal to half the mean ex-
cess value, i.e., Ũ�x ,y1/2�x��=12 �Ũm�x�− Ũc�. The inlet
den-
sity profile is defined, using a thin near-wall layer Jy,w,
as
�in�y�� j
= ��� + �1 − ���tanh�Gwy� , 0 y
h
5
1 − �1 − ���1
2�1 + tanh�G�y − h�� ,
h
5 y Ly , �9�
where Gw=50 and G=15.6 define the density gradient in thenear
wall and outer layer, respectively. The outer layer gra-dient
coefficient is the same for the velocity, the passivescalar, and
the density. The temperature profile is then deter-mined by the
ideal gas law with constant pressure. We haveperformed simulations
of a warm and a cold jet with an inletReynolds number of Re=Ujh /�
j =2000, where � j is the ki-nematic viscosity at the inlet jet
center, and a correspondingMach number of M =Uj /c=0.5. These
values correspond tothose used in our previous isothermal jet
simulation.1 Sincethe resulting friction Reynolds number in the C2
case is low�see Sec. V A�, it is complemented with a cold jet
simulationwith Re=Ujh /� j =7000 and M =Uj /c=0.5. That Re is
cho-sen so that the resulting friction Reynolds number
becomessimilar to that for the isothermal case.
The inlet density ratios, initial temperatures, box dimen-sions,
and resolutions are presented in Table I. The table alsoincludes
the reference name for each case. As will be dis-cussed later, the
cold and warm jet flow differ significantly interms of the range of
scales present, and hence different reso-lutions and box sizes are
used in the four cases. The compu-tational grid is stretched in the
wall-normal direction using acombination of a hyperbolic tangent
and logarithmic func-tion to obtain a clustering of nodes near the
wall and keepinga sufficient number of nodes in the outer layer. In
the stream-
wise direction the grid is slightly stretched with the
highestresolution in the transition region and a gradually
reducedresolution downstream. The smallest scales in the jet
arefound close to the wall, which is natural since the
energydissipation attains its maximum at the wall. Wall units
aretherefore used in Table II to quantify the numerical reso-lution
in the four cases. Values in the region where the flowis fully
turbulent, x /h�15, are presented. The streamwisestretching
approximately follows the flow development. Theresolution in the
isothermal and warm jets is comparable toresolutions used in DNS of
channel flow simulations �see,e.g., del Álamo et al.18� The
resolution used in the cold jetsimulations is comparable or
significantly better.
Wall-jet statistics are computed by applying ensembleaveraging
over time and the periodic spanwise direction. The
TABLE I. Simulation cases. h is the inlet jet height.
Jet Case Re
Density ratio Temperatures Box dimensions �h� Resolution
Line style�a /� j Ta Tj LxLy Lz NxNy Nz
Isothermal I2 2000 1.0 293.0 293.0 47189.6 384192128 –—
Cold C2 2000 0.4 732.5 293.0 35177.2 384192160 −·−
Warm W2 2000 1.7 293.0 498.1 28147.2 448256160 – – –
Cold C7 7000 0.4 732.5 293.0 30168.0 256192128 ––
TABLE II. Simulation resolution in wall units at x /h=15 and x
/h=25.
Case �x+ �y1+ �z+
I2 10.7–11.8 0.865–1.30 5.49–8.26
C2 3.05–3.10 0.285–0373 1.28–1.68
W2 12.5–12.9 1.17–1.45 7.61–9.46
C7 11.6–12.7 1.05–1.12 7.48–7.99
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start time for the sampling after the startup of the
simula-tions, the time separation between samplings and the
totaltime over which averaging is carried out are presented inTable
III in terms of the inlet time scale tj =h /Uj and an outertime
scale to=y1/2 /Um, defined in the downstream regionwhere turbulent
statistics are acquired.
V. RESULTS
Statistics of the nonisothermal wall jets are presentedbelow and
compared to those of the isothermal jet. The linestyles defined in
Table I are used throughout this paper todiscriminate between
statistics in the warm, isothermal andcold jets. If not stated
otherwise, wall-normal profiles andcorrelations presented are
acquired at a downstream positionof x /h=22 in all cases.
Density-weighted or Favre decomposed statistics are of-ten used
in combustion models since the averaged equationstake the same form
as the incompressible Reynolds averagedones. The familiar
incompressible modeling approaches canthen be applied also for the
compressible case. In the presentstudy statistics using Reynolds
and Favre decomposition, ac-cording to
f = f̄ + f�, �10�
f = f̃ + f� =�f
�̄+ f�, �11�
will be presented. Reynolds averages and fluctuations aredenoted
by over bars and single accents while Favre aver-ages and the
corresponding fluctuations are denoted by tildesand double
accents.
A. Structures and mean flow development
To provide an overview of the turbulence structures inthe
different jets, snapshots of the passive scalar concentra-tion are
shown in Fig. 1. The turbulence structures are dis-tinctly
different in the three Re=2000 cases. The warm jetcontains the
largest range of scales and has significantlymore small scale
energy than the other two cases. Corre-spondingly the isothermal
case contains a larger range ofscales and more small scale energy
than the cold case. Thesame phenomenon is also seen in snapshots of
the fluctuatingvelocity components. The observed difference in
turbulencestructure will have a profound influence on, for
example,the heat transfer to the wall in cooling and
combustionapplications.
To understand the origins of the differences caused bythe
varying density, the friction Reynolds number
Re� =�
l+=
u��
�w=
�
�w1/2��dUdy �y=0 �12�
is examined in Fig. 2. When an appropriate outer length scale�
is used, the friction Reynolds number provides an estimateof the
outer to inner layer length scale ratio. In the fullyturbulent
region, Re� in the warm jet is about 4.5 timeshigher than in the
cold jet at Re=2000 which explains thepresence of smaller
structures in the warm jet. The frictionReynolds number can
therefore be considered to be the ef-fective Reynolds number of
nonisothermal wall jets, ratherthan the inlet Reynolds number. The
difference in Re� isrelated to the temperature dependence of the
viscosity at walltemperature. In the warm jet �w is low near the
relativelycold wall, whereas in the cold jet �w is high near the
rela-tively warm wall, which results in high and low Re�,
respec-
TABLE III. Time scales sampled in terms of the inlet time scale,
tj =h /Uj,and an outer time scale to=y1/2 /Um defined at a
downstream distance ofx /h=25 and x /h=22.
Case Start �tj� Separation �tj� Sampled �tj� Sampled
I2 185 0.500 309 70.4 �to,25�C2 152 0.568 366 93.7 �to,25�W2 153
0.600 367 85.5 �to,25�C7 242 0.647 363 101.0 �to,22�
(a) y/h
(b) y/h
(c) y/h
(d) y/h
x/h
FIG. 1. �Color online� Snapshots of the passive scalar
concentration � /� j inthe cold jet at Re=2000 �a� and Re=7000 �b�,
isothermal jet �c�, and warmjet �d�.
5 10 15 20 25 30 35 400
100
200
300
400
500
Reτ
x/h
FIG. 2. Downstream development of the friction Reynolds number
Re�=u�y1/2 /�w.
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tively. The higher inlet Reynolds number in case C7 leads toa
Re� about equal to that in case I2.
The resulting heat transfer to the wall in the warm andcold jets
is shown in Fig. 3 in terms of the Nusselt numberdefined for the
inner shear layer
Nui = qwym
�w�Tm − Tw�, �13�
where ym is the jet center position and Tm denotes the maxi-mum
mean temperature in the warm jet and the minimummean temperature in
the cold. Due to the different Reynoldsnumbers the Nusselt numbers
differ in cases C2 and C7, butin both the cold cases Nui is
significantly smaller than in thewarm jet.
Cross stream profiles of the velocity, temperature, andpassive
scalar concentration are shown in Fig. 4. The stream-wise velocity
and temperature profiles, normalized by theinlet conditions, show
that the warm jet has the fasteststreamwise decay, followed by the
isothermal and the coldjets, respectively. The passive scalar
concentration develop-ment has a different character. The decay of
the maximumconcentration at the wall in the warm and isothermal
jets arepractically the same, while the concentration at the wall
inthe cold jets decays significantly slower. The region near
thewall, where the concentration gradient develops, is also
vis-ibly larger in case C2. This is likely caused by the low
fric-tion Reynolds number in this case.
Figure 5 shows xz-plane snapshots of the streamwisevelocity
fluctuations u� /u� in the cold, isothermal and warmjet at y+=7.
Elongated streamwise streaks, typically presentin the viscous
sublayer of boundary layers, are seen in allfour jets. The width of
the streaks is however influenced bythe varying density and the
streaks with the smallest physicalwidth are seen in the warm jet.
Note, however, that the op-posite is true in viscous scaling. To
determine the size of theturbulence structures in the near-wall
layer, two-point veloc-ity correlations in the spanwise direction
at y+=7 are shownin Fig. 6�a�. The streamwise correlations contain
minima cor-responding to half the spanwise streak spacing. Using
thismeasure the streak spacings are approximately 46, 90, 94,and
140 wall units in the cold jet cases C2 and C7, isother-mal and
warm jet, respectively. In terms of wall units thewarm jet contains
the widest streaks. The streak spacing
does, therefore, not depend solely on the viscous lengthscale.
The streak spacing found in the isothermal wall jet andcase C7 is
similar to what is found in incompressible turbu-lent boundary
layers19 and channel flows,20 whereas in caseC2 it is significantly
smaller. Two-point correlations of thetemperature and passive
scalar acquired at the half-width po-sition are shown in Fig. 6�b�.
In the outer layer the tempera-ture and passive scalar correlations
are practically identical,except for case C7. This indicates that
temperature mixingand transport can be considered passive in this
region. In theinner region the correlations differ due to the
differentboundary conditions.
B. Mean density effects
According to the hypothesis by Morkovin3 �also de-scribed in
Refs. 21–23� the effects of density on the turbu-
5 10 15 20 25 30 35 400
2
4
6
8
10
Nui
x/h
FIG. 3. Downstream development of the Nusselt number of the
inner shearlayer.
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
y
h
Ũ/Uj
(a)
0.4 0.6 0.8 1 1.2 1.4 1.60
1
2
3
4
5
6
y
h
T̃/T̃w
(b)
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
y
h
Θ̃/Θj
(c)
FIG. 4. Cross stream profiles of the mean velocity �a�,
temperature �b�, andpassive scalar concentration �c� normalized by
the inlet conditions. Profilesat the inlet �thick solid� and at x
/h=22 shown.
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lence structure are small as long as the density
fluctuationintensity compared to the absolute density is small. A
com-mon measure is �� / �̄0.1. Following this assumption,
theturbulent statistics of compressible flows are similar to
sta-tistics of incompressible flows when a proper scaling,
ac-counting for the mean density variation, is used.
AlthoughMorkovin’s investigation concerned boundary layers, it is
of-ten applied to other types of shear flows as well.
Turbulencestatistics in compressible boundary layers with free
streamMach numbers of M 5 and compressible jets with Machnumbers of
M 1.5 generally compare well with the statis-tics of their
incompressible counterparts, provided that aproper scaling is
applied.22 However, ratios of turbulentquantities to mean flow
values may still be greatly affectedby density since Morkovin’s
hypothesis does not include theeffects of mean density
gradients.
Normalized density fluctuation intensities in the noniso-thermal
jets are shown in Fig. 7. When scaled by the meandensity, the
fluctuation intensities are approximately at thelevel where
compressible effects could become noticeable,according to the
Morkovin criteria above, but can still beconsidered small. The
higher intensity in the cold jets is pre-
sumably due to the higher relative difference in �a and �
j.Coleman et al.4 observed, in a compressible channel flowwith
isothermal walls, values up to �� / �̄=0.13 and con-cluded that
Morkovin’s hypothesis holds. In the warm jet theposition of the
outer fluctuation maximum is further awayfrom the wall than in the
cold jets. The pressure fluctuationintensity, scaled by the
density-weighted turbulent kinetic en-ergy at the half-width
position, is plotted in Fig. 8. Thescaled fluctuations are all of
the order of one, but the fluc-tuation intensity is affected by the
mean density differencesand Re�. Both in the warm and cold jet case
C2 the fluctua-tion levels are higher than in the isothermal case,
whereas incase C7 it is about equal. Comparing the full profiles
thefluctuation intensity is higher in the outer layer, reflecting
thehigher sensitivity of free shear layers to density effects
com-pared to boundary layers.
Mean density effects can also be studied through com-parison of
statistics using Favre and Reynolds decomposi-tion. The differences
between Reynolds and Favre averagedmean velocities and turbulent
stresses can be written as
Ūi − Ũi = ui� = −��ui�
�̄= −
��ui�
�̄, �14�
ui�uj� − ui�uj�˜ = ui� uj� −
��ui�uj�
�̄, �15�
i.e., differences are caused by correlations of the density
andvelocity fluctuations and by ensemble averages of the
Favrefluctuations. Mean velocities using Favre and Reynolds av-
(a) z/h
(b) z/h
(c) z/h
(d) z/h
x/h
FIG. 5. �Color online� Snapshots of streamwise velocity
fluctuations inxz-planes situated at y+=7 in the cold jet at
Re=2000 �a� and Re=7000 �b�,isothermal jet �c�, and warm jet �d�.
Light color represents positive fluctua-tions and dark
negative.
0 50 100 150
-1
0
1
2
3 (a)
R7y+
ui
Cold C2
Isothermal
Warm
Cold C7
z+0 1 2 3 4
-1
0
1
2
3 (b)
Ry1/2t
Ry1/2θ
Cold C2
Isothermal
Warm
Cold C7
z/h
FIG. 6. �Color online� Spanwise two-point correlations in a
near-wall posi-tion, y+=7, �a� and at the half-width position y
/y1/2=1 �b�. Velocity corre-lations in �a�; Ru �solid�, Rv
�dashed�, and Rw �dashed-dotted�. Temperatureand passive scalar
correlations in �b�; Rt �dashed� and R� �dashed-dotted�.
0 0.5 1 1.5 2 2.5 30
0.05
0.1
0.15
ρ′rmsρ̄
y/y1/2
FIG. 7. Density fluctuation intensity normalized by the local
mean density.
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
p′rms(ρ̄K̃)y1/2
y/y1/2
FIG. 8. Pressure fluctuation intensity normalized by �̄K̃ at the
half-widthposition.
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eraging are compared in Figs. 9�a� and 9�b�. Notable
differ-ences, from the averaging, are only visible in the
wall-normal component. This is due to the considerably
lowermagnitude of this component. Reynolds and Favre
averagedturbulent kinetic energy profiles are shown in Fig. 9�c�.
Thedifferences from averaging are small for this quantity aswell,
as is also the case for the individual normal stresses.
Inconclusion, the mean density effects on the mean and fluc-tuating
statistics, in terms of density correlations, are small.The
significant differences between the profiles of the differ-ent jets
and the notable density fluctuations in Fig. 7 thusresult from the
varying mean density.
C. Turbulence statistics
The visualizations and the Re� plot have shown that thejets are
fully turbulent for x /h�15. Here we present statis-tics obtained
in the turbulent region.
Mean streamwise velocity profiles at downstream posi-tions x
/h=20 and x /h=25 for cases C2, I2, and W2, and atx /h=17 and x
/h=22 for case C7 are shown in Fig. 10. Incase C7 different outflow
boundary conditions had to be usedthan in the other cases due to
the higher Reynolds number.These appear to affect the statistics
beyond x /h=22 in the C7case whereas in the other cases the
statistics at x /h=25 areunaffected. In Ahlman et al.1 the
near-wall layer of the iso-thermal wall jet was shown to closely
resemble a zero-pressure gradient boundary layer. Mean profiles are
thereforeshown in Fig. 10 using three types of inner scaling.
InFig. 10�a� conventional boundary layer scaling is used. Herey+=y
/ l+=yu� /�w and U+=U /u�, where the friction velocityis defined as
u�=��w /�w and the subscript w denotes condi-tions at the wall. The
marked differences between the veloc-ity profiles are consistent
with the different Re� in thesimulations.
Using wall variables all profiles collapse in the
viscoussublayer, which also has the same approximate width in
wallunits, y+5, in the four cases. The physical size on the
otherhand varies since the viscous length scale l+ varies with
tem-perature, implying that the near-wall region with
significantviscosity effects is largest in the cold jet case C2.
Furtheraway from the wall, an inertial sublayer in correspondence
toa boundary layer has been found in experiments and large-eddy
simulations at sufficiently high Reynolds numbers �see,
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3
y/y1/2
Ũ/Uj , Ū/Uj
(a)
-0.02 0 0.020
0.5
1
1.5
2
2.5
3
Ṽ /Vj , V̄ /Vj
(b)
0 0.02 0.040
0.5
1
1.5
2
2.5
3
K̃/12U2j , K̄/
12U2j
(c)
FIG. 9. Mean Favre �lines with sym-bols� and Reynolds averaged
�lines�streamwise velocity �a�, wall-normalvelocity �b�, and
turbulent kinetic en-ergy �c�.
100 101 102 10302468
101214161820
U+
(a)
y+
100 101 102 10302468
101214161820
U+vD
y+
(b)
100 101 102 10302468
101214161820
U∗
y∗
(c)
FIG. 10. �Color online� Mean streamwise velocity using
conventional wallscaling �a�, van Driest transformation �b�, and
semilocal scaling �c�. Profilesat x /h=20 and x /h=25 for cases C2,
I2, and W2. Profiles at x /h=17 andx /h=22 for case C7. Viscous,
U+=y+, and inertial sublayer, U+
= 10.38log�y+�+4.1 �Österlund et al. �Ref. 24��, profiles
added.
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e.g., Refs. 2, 25, and 26�. No such region is found in
thepresent study, presumably due to the moderate
Reynoldsnumbers.
As a result of the varying density near the wall, the
meanprofiles in conventional wall variable scaling do not
collapseoutside the viscous sublayer. This has also been observed
tooccur in compressible boundary layers or channel flows
withsignificant temperature variations near the wall.
Alternativescaling approaches have been developed which take into
ac-count the mean density variation. One of these is thevan Driest
transformation27 defined as
UvD+ = �
0
U+
��̄/�̄w�1/2dU+. �16�
Using the van Driest transformation, the mean velocity pro-files
of compressible flows and incompressible flows aresimilar �see,
e.g., Refs. 4–6�. However, as concluded byHuang and Coleman28 the
transformation does not lead tosimultaneous collapse in the
sublayer and logarithmic layer.Van Driest transformed wall-jet
profiles are shown in Fig.10�b�. As a result of the transformation
the profiles becomeslightly more similar, but they still do not
collapse outsidethe buffer layer. In the warm jet, which has the
highest Re�,a logarithmic region in accordance with the
incompressibleinertial sublayer starts to appear.
Another approach to account for the varying mean den-sity, the
semilocal scaling proposed by Huang et al.7 is usedin Fig. 10�c�.
In this scaling the wall variables are based onthe local mean
density and viscosity, and their relation to theconventional wall
units becomes, as a result,
u�� =��w
�̄=� �̄w
�̄u� �17�
l� =��̄/�̄�
u�� =
��̄/�̄w��̄/�̄w
l+, �18�
where u�=��w / �̄w and l+= �̄w /u� are the conventional
fric-tion velocity and length scales, respectively, and wall
condi-tions are denoted by a subscript w. In the results
presentedproperties scaled by semilocal quantities are denoted by
a
superscript star, hence U�= Ũ /u�� and y�=y / l�. The
semilocal
scaling provides the best scaling of both the position
andmagnitude of the jet center in the four jets. The warm,
iso-thermal and cold jet case C7 collapse out to the beginning
of
the logarithmic layer, while in the cold jet case C2 the
jetprofile deviates closer to the wall due to its very low Re�.
The Reynolds shear stress in the four jets is plotted inFig. 11
using inner scaling. When the conventional boundarylayer scaling is
applied, the differences between the profilesare significant, which
was also found for the mean velocityand kinetic energy profiles.
Both the inner minimum and theouter maximum have higher magnitudes
and extend tohigher y+ values for the warm jet. This development is
ex-pected on the basis of the variation of Re�. Figure 11�b�shows
shear stress profiles in semilocal scaling. The sum ofthe viscous
and shear stress must be equal to u�
�2 in the near-wall region. However, in case C7 it stays large
up to rela-tively large values of y� whereas in the other cases the
sumof the viscous and shear stress decays more rapidly awayfrom the
wall �results not shown here�. Correspondingly, inthe inner layer
the scaled shear stress is significantly morenegative in case C7
than in the other cases, as seen in Fig.11�b�. The correlation
coefficient of the shear stress alsoreaches larger negative values
in the cold jets than in theisothermal and warm jet.
The streamwise fluctuation intensity and the kinetic en-ergy in
all four cases are shown in Fig. 12. The semilocalscaling improves
the collapse but notable differences stillexist, also in the inner
region. This is the case also for thespanwise fluctuations. In
channel flow the scaled streamwisefluctuation intensity increases
with increasing Re�. We seethe same trend when the cases C2 and C7
are compared, butin our simulations the increase is much larger for
an equiva-lent span of Re�, also when using semilocal scales. In
caseC7 the scaled streamwise fluctuation intensity and the
kineticenergy are higher than in the isothermal jet, although Re�
isabout equal in both cases. The differences can thus not solelybe
attributed to Reynolds number effects. The turbulent in-tensity is
also affected by the mean density gradient.
The rate of viscous dissipation of turbulent kinetic en-ergy,
�=�ij��ui� /�xj� / �̄, is shown in Fig. 13. Similar to tur-bulent
channel flows, slight kinks are present in the innerregion
approximately at the position of maximum productionof kinetic
energy.29 In all four cases the scaled viscous dis-sipation at the
wall is higher than the values 0.16–0.17 at-tained on an isothermal
wall in the incompressible and com-pressible channel flows
simulated by Morinishi et al.6 In allfour jet cases the dissipation
rate magnitude in the outer
layer, in terms of the outer scaling �Ũm− Ũc�3 /y1/2 �not
100 101 102 103-1
-0.50
0.51
1.52
2.53
3.54
(a)
˜u′′v′′
u2τ
y+100 101 102 103
-1-0.5
00.5
11.5
22.5
33.54
(b)
˜u′′v′′
u∗τ2
y∗
FIG. 11. Reynolds shear stress, usingboundary layer scaling �a�
and semilo-cal scaling �b�. Profiles at x /h=20 andx /h=25 for
cases C2, I2, and W2. Pro-files at x /h=17 and x /h=22 for caseC7.
In �a� symbols mark the Ũm ���and y1/2 ��� positions.
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shown�, is lower than the value 0.015 found at the
half-widthposition in the plane jet simulation of Stanley et
al.10
D. Self-similarity
Self-similarity in the simulated wall jets is assessed
bystudying the downstream development and the application ofscaling
to collapse statistics in the inner and outer layers. InFig. 14 the
wall-normal growth and the streamwise decay inthe jets are
evaluated. The wall-normal growth is character-ized by the
development of the density-weighted half widthy1/2
� , which is the position in the outer shear layer where
thedensity-weighted velocity is equal to half its maximum ex-
cess value, i.e., 12 ���̄Ũ�m− �̄cŨc�. Incompressible wall jets
areknown to display a linear half-width growth, similar to
planejets, but the wall-jet growth rate is usually about
20%–30%lower than for plane jets.2,25,26,30 Linear growth was
previ-ously observed in the isothermal simulation and in Fig.14�a�,
this is also found to hold for the nonisothermal jets.The
streamwise decay is evaluated in Fig. 14�b� whichshows the
development of the maximum momentum excess
Me =��̄Ũ�m − �̄cŨc� jUj − �̄cŨc
. �19�
The streamwise momentum decay in the warm and cold jetsis also
seen to be of the same type as in the isothermal walljet. The
results show that the generality of the outer layerevolution of
isothermal wall jets also applies in the noniso-thermal cases. The
varying density wall jets can hence beconsidered self-similar, and
the characteristic scales of theouter layer are similar to those of
the isothermal wall jet.
Despite the similarity in development to isothermal con-ditions,
the imposed density variation slightly influences thewall-normal
growth and streamwise decay. The largest dif-ferences occur in the
transition phase after which the growthrate is similar. The same
variation is also seen in the standardhalf widths that are not
density weighted. This effect is prob-ably mainly due to the
variation in Re�. Differences in theentrainment process could also
play a role, but how this wasinfluenced by the varying density is
not clear. In wall-jetexperiments both the wall-normal growth and
the streamwisedecay appear to be Reynolds number dependent.25,31,32
Forincreasing Reynolds numbers the growth rates decreaseslightly,
but this trend cannot be observed when cases C2 andC7 are
compared.
Density differences influence the growth rate of turbu-lent free
shear layers.33,34 This can also play a role in oursimulations. In
the growth rate we cannot discern a cleareffect of the density
differences, but the momentum decayrates in the two cold cases are
somewhat higher than in theisothermal and warm jet for x /h�12. The
variation ingrowth and decay rates observed are, however, small
despitethe significant density differences.
In the isothermal jet, statistics in the near-wall regionwere
found to be self-similar using conventional wallvariables.1 The
extent of the collapsed region, however, var-ies between different
statistics. Furthermore, the inner layersin the four cases differ
due to mean density variation. Ac-counting for the mean density by
semilocal scaling leads to acollapse of the mean velocity profiles
in the inner layer, asseen in Fig. 10�c�. However, this scaling
does not lead to acollapse of the Reynolds stress profiles of the
four jets, as
100 101 102 1030
0.5
1
1.5
2
2.5
3
u′′rmsu∗τ
(a)
y∗100 101 102 103
02468
101214161820
K̃12 ρ̄u
∗τ2
(b)
y∗
FIG. 12. Streamwise fluctuation in-tensity �a� and turbulent
kinetic en-ergy �b� using semilocal normaliza-tion. Profiles at x
/h=20 and x /h=25 for cases C2, I2, and W2. Pro-files at x /h=17
and x /h=22 for caseC7.
103100 101 1020
0.05
0.1
0.15
0.2
0.25
�
u∗τ4/ν̄
y∗
FIG. 13. Viscous dissipation rate �=�ij��ui� /�xj� / �̄ in
semilocal scaling.
0 5 10 15 20 25 30 35 400
0.5
1
1.5
2
2.5
3
3.5
4
yρ1/2
h
x/h
(a)
10 20 30 40
10−0.3
10−0.2
10−0.1
100
Me
x/h
(b)
FIG. 14. Downstream development of the density-weighted half
width y1/2�
�a� where ��̄Ũ�y=y1/2� =12 ���̄Ũ�m− �̄cŨc� and the decay of
the maximum mo-
mentum excess �b�.
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seen in Figs. 11 and 12, which indicates that the mean den-sity
variation affects the near-wall turbulence.
To evaluate self-similarity in the outer region, profiles ofthe
four jets are shown in Fig. 15 in terms of the isothermalouter
scaling. The outer scaling leads to a collapse of thestreamwise
velocity profiles of cases I2, W2, and C7,whereas the profiles of
case C2 differ probably because ofthe low Re�. The streamwise
velocity fluctuation profiles ofthe warm and isothermal jet are
quite similar and the sameholds for the wall normal and spanwise
components. In con-trast, the maximum scaled streamwise fluctuation
intensity ofthe two cold jets in the outer layer is higher than in
theisothermal and warm jet which shows the relative high
tur-bulence intensity in the two former cases. On the other
hand,the scaled wall-normal scalar flux is lower in case C7 than
inthe isothermal and warm jet.
E. Temperature and passive scalar statistics
In Fig. 16, the fluctuation intensity and wall-normal fluxes of
the temperature and passive scalar are pre-sented. The fluctuations
are normalized by the maximum
temperature and scalar difference, ��m=�w�̄w, and�Tm=max��T̄w−
T̄��. The temperature flux is shown insemilocal scaling, where the
inner temperature scale is de-fined as
T�� =
qw�̄wcpu�
�=
�̄w
�̄w Pr u��� �T̄
�y�
y=0, �20�
where qw is the wall heat flux. The passive scalar flux isscaled
by the wall concentration since the passive scalar, as aresult of
the no-flux condition at the wall, lacks a naturalinner scale. The
different boundary conditions are evident inboth the fluctuation
intensities and the fluxes. Inner and outer
peaks are present in the temperature but not in the
passivescalar flux profiles.
Passive scalar fluctuations are in the cold jet atRe=7000 near
the wall larger than in the outer layer wherethe fluctuations
further decrease in intensity. In contrast, thepassive scalar
fluctuations are of comparable magnitude inthe inner and outer
layers in the other jets at Re=2000. In thewarm jet the intensity
even increases slightly for y�y1/2.Scaled with �Tm and �w, the
temperature and passive scalarfluctuations, respectively, have a
similar intensity in the innerregion in all simulations, with the
exception of case C2which shows small passive scalar fluctuations
there. In theouter layer, the intensity of the passive scalar
fluctuations islower than of the temperature using outer scaling,
in particu-lar, in case C7.
The maximum intensity of the temperature fluctuation inthe outer
layer scales approximately with the maximum tem-perature
difference. Near the wall, the fluctuations show dis-tinct peaks in
the two cold jet cases but not in the warm jet.The difference
between the inner and outer layer is thus lesspronounced in the
warm jet than in the cold jets. This issomewhat counterintuitive
because an increased scale sepa-ration in most cases acts to
pronounce differences betweenthe inner and outer regions. The
maximum value of the nor-malized passive scalar flux in the outer
layer appears to in-crease with increasing Re�. In the inner layer,
the magnitudeof the normalized wall-normal temperature flux is
larger incase C7 than in the warm jet but in the outer layer it
iscomparable.
Gradient-diffusion approaches are commonly used tomodel the
Reynolds shear stresses and scalar fluxes. Forplane flows the model
parameters describing the mixing andheat transfer are usually
referred to as the turbulent Schmidtand Prandtl numbers, which are
defined as
0 0.5 1 1.5 2 2.50
0.2
0.4
0.6
0.8
1
Ũ
Ũm
(a)
y/y1/2
0 0.5 1 1.5 2 2.50
0.020.040.060.080.1
0.120.140.16
u′′rmsŨm
(b)
y/y1/2
0 0.5 1 1.5 2 2.50
0.005
0.01
0.015
0.02
v′′θ′′
Ũm∆Θm
(c)
y/y1/2
FIG. 15. Streamwise velocity �a�,streamwise fluctuation
intensity �b�,and wall-normal scalar flux �c� in outerscaling.
Profiles at x /h=20 andx /h=25 for cases C2, I2, and W2. Pro-files
at x /h=17 and x /h=22 for caseC7.
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Sct =�tDt
=u�v�
v���
���̃/�y�
��Ũ/�y�, �21�
Prt =�t�t
=u�v�
v�t�
��T̃/�y�
��Ũ/�y�, �22�
respectively, where �t is the turbulent viscosity, and Dt and�t
are the turbulent passive scalar and heat diffusivities,
re-spectively. In most simple turbulent flows Prt and Sct are ofthe
order of one. Prt and Sct, evaluated a priori from thesimulations,
are shown in Fig. 17. In the near-wall region Prtand Sct differ due
to the different boundary conditions. TheSchmidt number goes to
zero due to the vanishing meangradient, whereas the turbulent
Prandtl number increases to-ward the wall. Outside the near-wall
region Prt and Sct de-crease slightly in all cases. Further out
both the shear stressand heat flux change sign. This takes place
before the corre-sponding mean temperature and velocity extremum
points,which causes an abrupt decline and negative Prt and Sct
val-
ues over a short distance. Throughout the outer region,
theturbulent Schmidt number is approximately constant andaround
0.7. Due to the similarity of the heat and passivescalar transport
in this layer Prt is very close to Sct andtherefore not shown. In
conclusion, significant variations inPrt and Sct exist only in the
near-wall region and in theregion separating the positions of
vanishing turbulent fluxesand vanishing mean gradients. Constant
values are good ap-proximations for the whole outer region.
The scalar dissipation rate is of interest since it corre-sponds
to the local dissipation of scalar fluctuations andhence describes
the small scale mixing. The dissipation rateis therefore an
important quantity in many mixing and com-bustion models. Figure 18
shows the dissipation rate of thepassive scalar and the
temperature, using an outer scaling forthe dissipation rate and the
wall distance.
The profiles of the passive scalar dissipation show veryclear
differences in the two cold jet cases. The effective Rey-nolds
number �Re�� is very low for the C2 case and thedissipation rate
curve exhibits a character different from that
0 1 2 30
0.1
0.2
0.3(a)
t′rms∆Tm
y/y1/2
0 1 2 30
0.1
0.2
0.3(b)
θ′rmsΘw
y/y1/2
100 101 102 103-1
0
1
2
3
4(c)
v′t′
u∗τT ∗τ
y∗100 101 102 103
0
0.1
0.2
0.3(d)
v′θ′
u∗τΘw
y∗
FIG. 16. Scalar fluctuation intensities��a� and �b�� and
wall-normal fluxes��c� and �d��. Profiles at x /h=20 andx /h=25 for
cases C2, I2, and W2. Pro-files at x /h=17 and x /h=22 for
caseC7.
0 20 40 60 80 100-1
-0.5
0
0.5
1
1.5
2(a)
Prt
Sct
y+0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
0
0.5
1
1.5
2(b)
Sct
y/y1/2
FIG. 17. Turbulent Schmidt andPrandtl numbers in the inner layer
�a�and in the outer layer �b�.
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of the other cases, with a maximum in the outer layer. In theC7
case the shape of the dissipation profile has a similarcharacter to
that in the isothermal and warm jet, but thedissipation rate in
terms of the outer scale is much lower.Also for the temperature,
the dissipation rate magnitude inthe outer layer is significantly
lower using outer scaling, ascompared to the other cases.
The ratio of the mechanical to passive scalar time scaleand the
corresponding temperature to passive scalar timescale ratio are
presented in Fig. 19. In the outer layer, noclear effect of the
density differences is perceivable in themechanical to passive
scalar time scale ratio. The time scaleratio is relatively constant
throughout most of the wall jet,with exception for the inner
region. For the passive scalar thetime scale ratio shows even less
variation in the outer region.The ratio of the temperature and
passive scalar time scale isless than one indicating a more intense
dissipation of tem-perature fluctuations than of passive scalar
fluctuations.
VI. CONCLUSIONS
DNSs of a warm wall jet in a cold environment and acold jet in a
warm environment have been carried out. Theinlet Reynolds and Mach
numbers are the same as in a pre-viously performed isothermal
wall-jet simulation.1 In addi-tion, a cold jet at a higher Reynolds
number has been simu-lated. The cold jet cases can be seen as an
idealized filmcooling configuration and the hot jet case mimics the
flow ofhot exhaust gases over a cold wall.
Statistics of the jet development, turbulence and mixingare
presented and the results are compared to statistics of
theisothermal jet. Due to the varying viscosity the friction
Rey-nolds numbers, here based on the half-width positions,
aredifferent. Correspondingly, in the warm jet smaller turbu-
lence structures are present and the scale separation betweenthe
inner and outer shear layer is larger than in the cold jet atthe
same inlet Reynolds number. As a result of the densityvariation,
conventional wall scaling fails to collapse thenonisothermal and
isothermal jets outside the viscous sub-layer. Applying the
semilocal scaling leads to a collapse ofthe mean profiles in the
inner layer. Semilocal scaling is notcapable of collapsing the
inner peak magnitudes of thestreamwise and spanwise fluctuations
intensities. Also in theouter layer, the profiles of the streamwise
velocity fluctua-tions in outer scaling do not collapse. The
normalized turbu-lence statistics thus appear to be influenced by
mean densityvariations. In the nonisothermal jets the development
of thedensity-weighted growth and streamwise momentum decayrate are
similar to the isothermal case.
Streamwise streaks are present in the viscous sublayer inall
cases, but their width, in terms of wall units, varies. Meandensity
effects are seen in the density and temperature fluc-tuations but
the levels are small, in a Morkovian sense, de-spite the fact that
for instance the cold jet inflow density is2.5 times the ambient
coflow density.
The profiles of the mean and fluctuating velocities
aresignificantly different in the four jets, but the differences
be-tween Favre and Reynolds averages are small. The com-pressible
effects are thus mainly a result of the mean densityvariations.
The turbulent Schmidt and Prandtl numbers vary signifi-cantly
only in the near-wall layer and below the jet center.When comparing
the scalar dissipation rates in the innerlayer, the temperature and
the passive scalar dissipation ratesare different due to the
different boundary conditions.
The difference in Reynolds number between the twocold jet
simulations results in large differences in the passive
0 0.5 1 1.5 20
0.004
0.008
0.012
0.016(a)
χθχθ,o
y/y1/2
0 0.5 1 1.5 20
0.05
0.1
0.15(b)
χtχt,o
y/y1/2
FIG. 18. Scalar ��=2�D����� /�xi������ /�xi� �a� and
tem-perature, �t=2���t�� /�xi���t�� /�xi� �b�dissipation rate using
outer scaling
where ��,o= �̄�w2 �Ũm− Ũc� /y1/2, and
�t,o= �̄cp�Tm2 �Ũm− Ũc� /y1/2.
0 0.5 1 1.5 20
0.4
0.8
1.2
1.6
2(a)
K̄/�
θ′′2/χθ
y/y1/2
0 0.5 1 1.5 20
0.4
0.8
1.2
1.6
2(b)
t′′2/χt
θ′′2/χθ
y/y1/2
FIG. 19. Mechanical to scalar �a� andtemperature to passive
scalar �b� timescale ratio.
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scalar dissipation profile. At higher Reynolds number,
thisprofile has a similar shape as in the isothermal and warm
jetbut normalized with an outer scale the dissipation rate ismuch
lower. In the outer layer, the mechanical to scalar timescale
behavior is approximately equal in the four jets. Theratio of the
temperature to passive scalar time scale is lessthan one indicating
a smaller temperature time scale andhence a more intense
dissipation of temperature fluctuations.
ACKNOWLEDGMENTS
Funding for the present work was provided by The Cen-tre for
Combustion Science and Technology �CECOST�. Thecomputations were
performed at the Center for ParallelComputers at KTH, using time
granted by the Swedish Na-tional Infrastructure for Computing
�SNIC�. ProfessorBendiks Jan Boersma is thanked for providing the
originalversion of the DNS code.
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