Direct numerical simulation of nonisothermal turbulent wall jetsgeert/papers/noniso_jet.pdfDirect numerical simulations of plane turbulent nonisothermal wall jets are performed and

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

PHYSICS OF FLUIDS 21, 035101 �2009�

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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.

035101-2 Ahlman et al. Phys. Fluids 21, 035101 �2009�


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

035101-3 Direct numerical simulation of nonisothermal Phys. Fluids 21, 035101 �2009�


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.



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.



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.



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.



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.



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�.



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.



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�.



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.



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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