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International Journal of Heat and Fluid Flow xxx (2009) xxx–xxx

ARTICLE IN PRESS

Contents lists available at ScienceDirect

International Journal of Heat and Fluid Flow

journal homepage: www.elsevier .com/ locate/ i jhf f

DNS of a spatially developing turbulent boundary layer with passive scalar transport

Qiang Li *, Philipp Schlatter, Luca Brandt, Dan S. HenningsonLinné Flow Centre, KTH Mechanics, Osquars Backe 18, SE-100 44 Stockholm, Sweden

a r t i c l e i n f o

Article history:Received 7 November 2008Received in revised form 12 May 2009Accepted 29 June 2009Available online xxxx

Keywords:Turbulent boundary layerPassive scalarDirect numerical simulation (DNS)Prandtl number

0142-727X/$ - see front matter � 2009 Elsevier Inc. Adoi:10.1016/j.ijheatfluidflow.2009.06.007

* Corresponding author. Tel.: +46 8 790 68 76.E-mail address: [email protected] (Q. Li).

Please cite this article in press as: Li, Q., et al. D(2009), doi:10.1016/j.ijheatfluidflow.2009.06.00

a b s t r a c t

A direct numerical simulation (DNS) of a spatially developing turbulent boundary layer over a flat plateunder zero pressure gradient (ZPG) has been carried out. The evolution of several passive scalars withboth isoscalar and isoflux wall boundary condition are computed during the simulation. The Navier–Stokes equations as well as the scalar transport equation are solved using a fully spectral method. Thehighest Reynolds number based on the free-stream velocity U1 and momentum thickness h isReh ¼ 830, and the molecular Prandtl numbers are 0.2, 0.71 and 2. To the authors’ knowledge, this Rey-nolds number is to date the highest with such a variety of scalars. A large number of turbulence statisticsfor both flow and scalar fields are obtained and compared when possible to existing experimental andnumerical simulations at comparable Reynolds number. The main focus of the present paper is on thestatistical behaviour of the scalars in the outer region of the boundary layer, distinctly different fromthe channel-flow simulations. Agreements as well as discrepancies are discussed while the influence ofthe molecular Prandtl number and wall boundary conditions is also highlighted. A Pr scaling for variousquantities is proposed in outer scalings. In addition, spanwise two-point correlation and instantaneousfields are employed to investigate the near-wall streak spacing and the coherence between the velocityand the scalar fields. Probability density functions (PDF) and joint probability density functions (JPDF) areshown to identify the intermittency both near the wall and in the outer region of the boundary layer. Thepresent simulation data will be available online for the research community.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

The understanding of the spreading of a passive scalar in turbu-lent flows was initially gained solely through wind tunnel experi-ments. The early studies of heat transfer were performed by e.g.Corrsin (1952) and Warhaft and Lumley (1978) in grid-generatedturbulence and homogeneous turbulence. Hishida and Nagano(1979) and Nagano and Tagawa (1988) measured various typesof moments of velocity and scalar fluctuations in fully developedpipe flow to investigate the transport mechanism in turbulenceand to correlate the transfer processes of momentum and scalarwith coherent motions. In particular, the importance of the coher-ent motions in the turbulent diffusion process of Reynolds-stresscomponents and scalar fluxes was demonstrated for the first time.Later, Mosyak et al. (2001) and Hetsroni et al. (2001) carried outexperiments to study the wall-temperature fluctuations underdifferent wall-boundary conditions and the thermal coherentstructure in a fully developed channel flow. For turbulent bound-ary-layer flows, Perry and Hoffmann (1976) examined the similar-ity between the Reynolds shear stress hu0v 0i and scalar flux hv 0h0iusing quadrant analysis. However, the Reynolds-stress hu0v 0i was

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NS of a spatially developing tur7

analysed in the ðu;vÞ plane and the turbulent scalar flux hv 0h0i inthe ðv ; hÞ plane. Therefore the correspondence between fluids mo-tions and scalar transport was not strictly specified. Subramanianand Antonia (1981) measured several quantities in a slightlyheated boundary layer to address the effect of Reynolds number.Krishnamoorthy and Antonia (1987) and Antonia et al. (1988)investigated the temperature dissipation and the correlation be-tween the longitudinal velocity fluctuation and temperature fluc-tuation in the near-wall region.

Due to the rapid progress in high-performance computers, di-rect numerical simulation (DNS) of turbulent flows involving pas-sive scalars, especially in channel geometry, has matured to animportant research tool during the past few decades. The first di-rect numerical simulations of passive scalar transport were per-formed by Rogers et al. (1986) in a homogeneous shear flow.Numerical simulations in channel geometry were pioneered byKim and Moin (1989) with Pr ¼ 0:1, 0.71 and 2.0 at Res ¼ 180where Res is the Reynolds number based on the friction velocityus and the channel half width h and Pr is the molecular Prandtlnumber. Heat is introduced by an internal source created and re-moved from both walls. A high correlation between the stream-wise velocity and temperature was found in the wall region.Later, Kasagi et al. (1992) and Kasagi and Ohtsubo (1993)performed DNS at Res ¼ 150 with Pr ¼ 0:71 and 0.025. The

bulent boundary layer with passive scalar transport. Int. J. Heat Fluid Flow

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Nomenclature

cp specific scalar capacityh channel half widthH12 shape factork scalar conductivityp pressurePe Péclet number = Re PrPr molecular Prandtl number or Schmidt number = m

aPrt turbulent Prandtl numberqw rate of the scalar transfer from the wall to the flow =

�k @h@y jy¼0

Re Reynolds numberRed�0

Reynolds number based on the inlet displacementthickness d�0

Reh Reynolds number based on the momentum thickness hRes Reynolds number based on the friction velocity usSt Stanton number = qw

qU1cpðhw�h1Þt timeui;u;v ;w instantaneous velocity components in the streamwise,

wall-normal and spanwise direction (in direction i)

us friction velocity =ffiffiffiffiffiswq

qU1 free-stream mean velocityxi; x; y; z Cartesian coordinates in the streamwise, wall-normal

and spanwise direction (in direction i)yL height of the domainO Landau symbol (order of)P production of turbulent energyUi Blasius laminar base flow

subscriptf properties of the fluidw properties at the wall1 properties in the free-streamrms root-mean-square value of the quantity

superscript^ dimensional term for the variable quantity0 fluctuating part+ scaling in viscous (wall) unitsout scaling in outer unitshi average over time and the homogeneous direction

Greek symbolsa scalar diffusivity = k

qcp

at eddy diffusivity = � hv0h0i@hhi@y

b pressure gradient coefficient = d�

sw

dpdx jfree�stream

d� displacement thicknessd�0 inlet displacement thicknessd99

h 99% local scalar boundary-layer thicknesse dissipation of turbulent energyN log-law diagnostic functionNh log-law diagnostic function for scalarh momentum thicknessh scalarhi scalar ihs friction scalar = qw

qcpus

hw scalar concentration at the wallh1 scalar concentration in the free streamj von Kármán constantjh von Kármán constant for the scalark fringe functionk streak spacingm kinematic viscosity = l

q

mt turbulent eddy viscosity = � hu0v 0 i@hui@yq density

sw shear stress at the wall

2 Q. Li et al. / International Journal of Heat and Fluid Flow xxx (2009) xxx–xxx

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scalar-fluxes budgets were shown and the low Pr number effectswere discussed. However, the Reynolds number of these simula-tions still remains at a low value. Wikström (1998) performed aDNS at a higher Reynolds number of Res ¼ 265 with Pr ¼ 0:71.Abe et al. (2004) reached up to Res ¼ 1020 and Pr ¼ 0:025 and0.71. All these simulations are done, however, with a Prandtl num-ber lower than two. This is due to the fact that the smallest scalesin the scalar fluctuation decrease with the increase of Pr. Thereforethe DNS becomes an even more difficult task when the Prandtlnumber is high. With the help of larger parallel computers,Kawamura et al. (1998) performed the DNS in periodic channelflow at Res ¼ 180 but for a wider range of Pr from 0.025 to 5.0. La-ter, Tiselj et al. (2001) performed a channel DNS at Res ¼ 150 withPr from 0.71 to 7. In his work, the ideal isoflux boundary conditionwas compared to the previous results and underestimated valuesfor the wall-temperature fluctuations of the previous simulationswere reported. Recently, Redjem-Saad et al. (2007) performed aDNS in a fully turbulent pipe flow to explore the impact of the wallcurvature on the turbulent heat transfer. The Reynolds numberbased on the pipe radius is 5500 ðRes ¼ 186Þ and the Pr varies from0.026 to 1. For pipe flows, slightly more intense temperature fluc-tuations than in channel flow were found.

However, for flat-plate boundary layers with zero pressure gra-dient (ZPG), which is a relevant canonical flow case for theoretical,numerical as well as experimental studies, relatively few numeri-cal results have been published for medium or high Reynolds num-bers. The direct numerical simulation by Spalart (1988) using aninnovative spatio-temporal approach provided valuable data at

Please cite this article in press as: Li, Q., et al. DNS of a spatially developing tur(2009), doi:10.1016/j.ijheatfluidflow.2009.06.007

Reh ¼ 300; 670; 1410. Later, Komminaho and Skote (2002) per-formed a true spatial DNS up to Reh ¼ 700. Concerning boundary-layer simulations with passive scalars, to our knowledge the firstDNS was performed by Bell and Ferziger (1993) up to a mediumReynolds number of Reh ¼ 700 with Pr being 0.1, 0.71 and 2.0. La-ter, a DNS was performed by Kong et al. (2000) up to a lower Rey-nolds number of Reh ¼ 420 and Pr ¼ 0:71 with isothermal andisoflux boundary conditions. Recently, Hattori et al. (2007) per-formed a DNS to study the buoyancy effects on the boundary layerstarting from Reynolds number of Reh ¼ 1000 to Reh ¼ 1200 andPr ¼ 0:71. A new DNS was performed by Tohdoh et al. (2008) upto a low Reynolds number of Reh ¼ 420, with Pr ¼ 0:71 and 2.0.

This paper is a study of passive scalar transport in a turbulentboundary layer spatially developing over a flat plate with zeropressure gradient (ZPG). The investigation is performed using di-rect numerical simulation (DNS). A spatial formulation (inflow/outflow setting) was adopted since it is the best model for a bound-ary layer which grows in the downstream direction rather than intime. Moreover, all the scalars were simulated simultaneously, i.e.one velocity field accommodates all the scalars. The Reynoldsnumber Red�0

based on the free-stream velocity U1 and the inletdisplacement thickness d�0 is 450 and Prandtl number Pr are chosento be 0.2, 0.71 and 2.0. Isoscalar and isoflux wall boundary condi-tions are employed for comparison. Since similarities exist be-tween the boundary layer and channel flow in the near-wallregion, this study mainly focuses on the outer region of the bound-ary layer, i.e. the wake region. Based on the present data, the sca-lings based on the Prandtl number are also proposed for various



Q. Li et al. / International Journal of Heat and Fluid Flow xxx (2009) xxx–xxx 3

ARTICLE IN PRESS

scalar quantities and also for the budgets of the scalar fluxes inboth inner and outer units. The goal of this paper is to extendour knowledge about the scalar transport in turbulent boundary-layer flows to a wider range of both Reynolds number and Prandtlnumbers. In addition, a data base for the research community isgenerated, which can be useful in particular for modellingpurposes.

2. Numerical methodology

2.1. Governing equations and numerical method

The three-dimensional, time-dependent Navier–Stokes equa-tions for incompressible flow as well as the transport equationfor the passive scalar in non-dimensional form using the summa-tion convention are given by

@ui

@xi¼ 0; ð1Þ

@ui

@tþ uj

@ui

@xj¼ � @p

@xiþ 1

Red�0

@2ui

@xj@xj; ð2Þ

@h@tþ ui

@h@xi¼ 1

Red�0Pr

@2h@xi@xi

¼ 1Pe

@2h@xi@xi

; ð3Þ

where ðx1; x2; x3Þ ¼ ðx; y; zÞ are the Cartesian coordinates in thestreamwise, wall-normal and spanwise direction, respectively.ðu1;u2;u3Þ ¼ ðu; v;wÞ are the corresponding instantaneous velocitycomponents, t represents the time, p is the pressure and h the scalarquantity. The Reynolds number Red�0

is based on the free-streamvelocity U1 and the inlet displacement thickness d�0 while Pr de-notes the molecular Prandtl (or Schmidt) number and Pe ¼ Red�0

Prfor the Péclet number.

In the present paper, the dimensional variables are non-dimen-sionalised as in Kong et al. (2000), i.e.

ui ¼ui

U1; xi ¼

xi

d�0; p ¼ p

qU21; t ¼ U1 t

d�0; ð4Þ

h ¼ hw � hhw � h1

; for the isoscalar boundary condition; ð5Þ

h ¼ 1� kðh� h1Þqwd�0

; for the isoflux boundary condition: ð6Þ

The dimensional reference quantities are defined as follows. U1is the free-stream velocity, d�0 the inlet displacement thickness, qthe density, k the scalar conductivity, qw the scalar flux at the wall,hw the scalar concentration on the wall and h1 the scalar concen-tration in the free-stream. The hat ^ denotes the dimensional termfor the variable quantities, i.e. ui; xi; p; t and h.

The simulation code (Chevalier et al., 2007) used in the presentstudy employs a pseudo-spectral method comparable to that usedby Kim et al. (1987). Fourier series expansion is used in streamwiseand spanwise directions (wall-parallel directions) assuming peri-odic boundary conditions. In the wall-normal direction, Chebyshevexpansions employing the Chebyshev-tau method are used. Thetime advancement uses a four-step third-order Runge-Kuttascheme for the nonlinear terms and a second-order Crank–Nicol-

Fig. 1. Instantaneous contour plot of the fluctuating scalar field in a x—y plane, starting frobox height is enlarged by a factor of two.


son scheme for the linear terms. The nonlinear terms are calculatedin physical space to avoid convolution sums. Aliasing errors are re-moved by using the 3/2-rule in the wall-parallel directions. Fig. 1shows the computational box together with a contour plot of aninstantaneous scalar fluctuation. To fulfil the periodic boundaryconditions in the streamwise direction, a ‘‘fringe region” (Bertolottiet al., 1992) is added at the downstream end of the domain. It isimplemented by adding a volume force Fi to the Navier–Stokesequations. The force is of the form

Fi ¼ kðxÞðUi � uiÞ ð7Þ

with Ui being the desired inflow condition. Note that the fringeforcing is also applied for the scalar field in a similar fashion. Thefringe function kðxÞ which has continuous derivatives of all ordershas a prescribed shape to minimise the upstream influence (Nord-ström et al., 1999) and is non-zero only in the fringe region. Inthe fringe region, the outflow is forced by the volume force to thelaminar inflow condition which is a Blasius boundary-layer profile.In addition, to trigger rapid (natural) laminar-turbulent transition, arandom volume forcing located at a short distance downstream ofthe inlet ðx ¼ 10; Rex ¼ 72;950Þ is used. The trip forcing can beused to generate noise at low amplitude or turbulence. The volumeforce is directed normal to the wall with a steady amplitude and atime-dependent amplitude. The generated noise has a uniform dis-tribution covering all the frequencies lower than the cutoff fre-quency corresponding to p

2 in the present DNS. The turbulent fieldgenerated by the trip force leads to very good quality at the expenseof a slightly enlarged transitional inflow region. In particular, thereare no streamwise correlations in the fluctuations as opposed to e.g.the rescaling and recycling method employed by Kong et al. (2000).

2.2. Boundary condition

The velocity and scalar fields are periodic in the horizontaldirections while boundary conditions at the wall and in the free-stream are needed to solve the governing equations.

At the solid wall, the no-slip boundary conditions for thevelocities

ujy¼0 ¼ 0; v jy¼0 ¼ 0; wjy¼0 ¼ 0;@v@y

��y¼0¼ 0; ð8Þ

are applied. For the boundary conditions in the free-stream, a Neu-mann condition, i.e.

@ui

@y

��y¼yL

¼ @Ui

@y

��y¼yL

; ð9Þ

is imposed with yL being the height of the domain and Uiðx; yÞ theBlasius laminar base flow.

In the present implementation for the scalar field, two types ofwall-boundary conditions are available. Similar to those used byKong et al. (2000), one is an isoscalar wall and the other an isofluxwall. These two kinds of wall-boundary conditions are given by

hjy¼0 ¼ 0 for the isoscalar boundary condition; ð10Þ@h@y

��y¼0¼ 1 for the isoflux boundary condition: ð11Þ

m Reh ¼ 175 to Reh ¼ 850. Note that the fringe region is not shown in the figure. The




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They corresponds to two limiting cases of the physical configu-ration (Tiselj et al., 2001), i.e. the scalar activity ratio being 0 for anisoscalar wall and 1 for an isoflux wall where the scalar activityratio is defined as

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqf cpf kf =qwcpwkw

qwith q; cp and k being den-

sity, scalar capacity and conductivity, respectively. The subscriptf corresponds to the properties of the fluid where subscript w cor-responds to the ones of the wall. For the boundary condition in thefree-stream, a Dirichlet condition is imposed. A Neumann condi-tion was also tested and the results turned out to be indistinguish-able to those obtained with the Dirichlet condition. The boundarycondition in the free-stream thus reads

hjy¼yL¼ 1: ð12Þ

2.3. Simulation parameters

The computational box has a dimension of 750d�0 � 40d�0 � 34d�0in the streamwise, wall-normal and spanwise directions, respec-tively. The corresponding resolution is 1024� 289� 128 whichgives a grid spacing of 17, 0.025–4.6 and 6.3 (in viscous unitsand based on the friction velocity at the x ¼ 150 or Reh ¼ 400) inthe three directions. The Reynolds number based on the free-stream velocity U1 and momentum thickness h at the inlet isReh ¼ 175 and Reh ¼ 830 at the outlet. A comparison of the wall-normal resolution with the corresponding Kolmogorov and Batch-elor scales for Pr ¼ 2:0 is shown in Fig. 2. It can be concluded that asufficient amount of grid points is used in the wall-normaldirection.

A summary of the molecular Prandtl number together with thewall-boundary condition and the boundary-layer thickness, basedon the 99% thickness, for all the scalars is listed in Table 1.

All results presented unless specified are averaged results. Theaverage, denoted by the angular brackets, was performed over bothtime and the homogeneous spanwise direction. The correspondingfluctuating part is denoted by a prime. The averaging time after theinitial transient is about 12,000 time units d�0

U1

� �corresponding to

tþ ¼ 13;500 in viscous units to make sure that all the statisticsare sufficiently converged.

3. Results

The present paper is focused on the results pertaining to thescalar transport. In the interest of space, only one representativeresult of the mean velocity profile compared with other DNS datais shown, other hydrodynamic results (stresses, budgets, etc.) arenot shown here. However, these hydrodynamic results were care-

y+

Δ y+

,η+

,ηθ+

10−2

100

102

10−2

10−1

100

101

102

Fig. 2. Resolution check. —— Wall-normal resolution Dyþ , - - - Kolmogorov scalegþ;� � � Batchelor scale gþh for Pr ¼ 2:0.


fully validated and compared with other available numerical andexperimental results and the agreements are very good in general,see Li et al. (2008).

3.1. Spatial evolution and flow quality

The shape factor H12, defined as the ratio between the displace-ment thickness d� and the momentum thickness h, provides a di-rect assessment of the flow field. As seen in Fig. 3a, a slowdecrease of H12 with the increase of the Reynolds number in theturbulent region is observed and the agreement with the experi-mental data is good in the fully turbulent region close to the outlet,Reh ¼ 830; H12 ¼ 1:49. The variation of the ratio of the integratedturbulent kinetic energy production P and dissipation e alongthe streamwise direction is shown in Fig. 3b. Based on this figure,the range of Reynolds number in which the boundary layer can beassumed to be in equilibrium is for the present DNS from aboutReh ¼ 350 to 830.

The non-dimensional parameter b related to the pressure gradi-ent is defined as

b ¼ d�

sw

dpdx

��free�stream

; ð13Þ

where d� is the displacement thickness and sw is the shear stress atthe wall. The b calculated from the simulation is of Oð10�9Þ indicat-ing that the flow far away from the wall is indeed subjected to zeropressure gradient.

Österlund et al. (1999) suggested a turbulent correlation for theskin-friction coefficient which reads,

cf ¼ 21

0:384ln Reh þ 3:75

� ��2

: ð14Þ

The present results of the skin-friction coefficient compare wellwith this correlation even at comparably low Reynolds numbers.

The Stanton number St for a scalar boundary layer is the coun-terpart of the skin-friction coefficient for a momentum boundarylayer. This non-dimensional scalar-transfer coefficient is definedby

St ¼ qw

qU1cpðhw � h1Þ; ð15Þ

where qw is the rate of the scalar transfer from the wall to the flow,q is the density of the fluid, U1 is the free-stream velocity, cp is thespecific scalar, and hw and h1 are the scalar concentrations at thewall and in the free-stream, respectively. The variations of the Stan-ton number with different downstream positions are shown inFig. 4. The turbulent correlation according to Kays and Crawford(1993) which reads

St ¼ 0:0125Pr�25Re

�14

h ; Pr > 0:5; ð16Þ

is included for comparison. The difference might be related to thefact that the correlations suggested by Kays and Crawford (1993)are for high Reynolds numbers. An overshoot of the peak is ob-served which also exists in the profile of the skin-friction coeffi-cient, see e.g. Brandt et al. (2004). The overshoot is consistentwith the general behaviour of spectral methods applied to transi-tional flows and it is slightly diminished with increased resolution.It is observed that the overshoot vanishes for the scalars with iso-flux wall-boundary condition.

3.2. Instantaneous fields

Fig. 5 shows the instantaneous streamwise disturbance velocityat yþ � 7 in the ðx; zÞ plane as well as the corresponding scalar fluc-tuations. All the plots are obtained at the same time instant, and



Table 1Parameters for the scalars. Note that the boundary-layer thicknesses dþ99 are in viscous units and measured at Reh ¼ 830. The corresponding velocity boundary-layer thickness isabout dþ99 ¼ 315 at Reh ¼ 830.

Scalar no. h1 h2 h3 h4 h5

Wall-boundary condition Isoscalar Isoscalar Isoflux Isoscalar IsofluxPrandtl number Pr 0.2 0.71 0.71 2.0 2.0Boundary-layer thickness dþ99 365 340 340 320 320

Reθ

H12

200 300 400 500 600 700 8001.4

1.6

1.8

2

2.2

2.4

Reθ

0 200 400 600 800 10000

1

2

3

4

5

(a)

(b)

Fig. 3. Streamwise evolution of the shape factor H12 and the ratio of the wall-normal integrated turbulent kinetic energy production P and dissipation e. ——Present DNS, � T3A (Roach and Brierley, 1992), � (Purtell et al., 1981). (a) shape

factor H12, (b)R1

0PdyR1

0edy

.

Reθ

St

200 300 400 500 600 700 800

1

2

3

4

5

x 10−3

Fig. 4. Streamwise evolution of the Stanton number St. + h1, — h2, -�- h3, - - -h4, .......h5, � (Kays and Crawford, 1993).


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the visualised box is centred around Reh ¼ 625 and has a length ofDxþ ¼ 2000 and width of Dzþ ¼ 700 in viscous units. The viscousunit is defined by the friction velocity us at the centre of the do-main, i.e. x ¼ 350; Rex ¼ 189;500 and Reh ¼ 625. Streaky struc-tures are clearly observed and a strong similarity exists betweenthe velocity field and the scalar field, mainly Pr ¼ 0:71. The regionsof low and high scalar concentrations are elongated in the stream-wise direction with a mean spanwise spacing similar to that of thestreamwise velocity fluctuation. Finer structures with strongerspanwise and wall-normal gradients are observed with increasingPr. In the case of the isoflux wall-boundary condition, a clear differ-ence can be observed: the scalar fluctuations are enhanced by theisoflux boundary condition in contrast to the ones with isoscalarboundary condition. It is also consistent with previous work bye.g. Kong et al. (2000) that the low-speed fluids are associated withlow scalar concentration region and high-speed fluids with highscalar concentration region.

3.3. Mean results and turbulence intensities

For the mean profile and fluctuations of the velocity field, goodagreements are observed with the DNS data from Spalart (1988)and the DNS data from Komminaho and Skote (2002) as seen inFig. 6. A small deviation can be observed for the mean streamwisevelocity profile when comparing to Spalart (1988). This is probablydue to the temporal approach used for the latter simulation and isalso seen in other boundary-layer simulations, e.g. Schlatter et al.(in press). The von Kármán constant j used in the log-law is 0.41and it gives good agreement for this comparably low Reynoldsnumber. The log-law diagnostic function N ¼ yþ d<u>þ

dyþ is plotted inFig. 7. This sensitive function N is supposed to approach a constantvalue of the inverse von Kármán constant 1

j in the overlap regionaccording to the log-law (Nagib et al., 2007). As seen from the plot,the present DNS does not yet have a well developed logarithmicoverlap region due to the comparably low Reynolds number.

The von Kármán constant for the mean scalar distributions jh, isassumed to be independent of the Prandtl number and the Rey-nolds number in the logarithmic region. Similar to the diagnosticfunction for the velocity field, a constant log-law diagnostic func-tion for scalar fields Nh, defined as yþ d<h>þ

dyþ , is plotted in Fig. 8a.The present jh is chosen to be 0.41 which is close to the chan-nel-flow results by Kawamura et al. (1999) of about 0.4 but smallerthan 0.47 suggested by Kader (1981).

The root-mean-square (RMS) of the fluctuations for differentscalars are shown in Fig. 8b which clearly demonstrates the effectsof the different boundary conditions. The data with isoscalarboundary condition go to zero as the wall is approached whereasthe cases with isoflux boundary condition remain finite. Far awayfrom the wall, the profiles of the RMS of the scalar fluctuationswith different wall-boundary conditions collapse with each otherindicating that the influence from the boundary conditions is onlyconfined to the near-wall region. The limiting value of the scalarvariance for h3 ðPr ¼ 0:71Þ with isoflux boundary condition was re-ported to be 2.0 and independent of the Reynolds number by Konget al. (2000). However, having a higher Reynolds number range forthe present DNS, a slight increase of the limiting wall value for



Fig. 5. Instantaneous flow and scalar fields at yþ � 7. (a) u0þ , (b) h0þ1 , (c) h0þ2 , (d) h0þ3 , (e) h0þ4 , (f) h0þ5 .

y+

<u>

+

100

101

102

103

0

5

10

15

20

25

Fig. 6. Comparison of the mean streamwise velocity huiþ with other numericalsimulations. —— Present DNS at Reh ¼ 670, � (Spalart, 1988) at Reh ¼ 670, �(Komminaho and Skote, 2002) at Reh ¼ 670, ...... huiþ ¼ 1

j ln yþ þ 5:2 with j ¼ 0:41.

y+

Ξ

100

101

102

103

0

1

2

3

4

5

6

Fig. 7. Profile of the mean velocity and the diagnostic function N at differentReynolds numbers. —— at Reh ¼ 390, - - - at Reh ¼ 620, � � � at Reh ¼ 830; j ¼ 0:41.


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increasing Reynolds number was observed for both h3 and h5,roughly following Re0:1

h . For example, a value of 2.12 at Reh ¼ 850was found for h3 ðPr ¼ 0:71Þ. However, to what extent this growthcontinues for high Reynolds number is still unclear due to the lim-ited maximum Reh in the present study. Comparing the wall RMSvalues for h3 ðPr ¼ 0:71Þ and h5 ðPr ¼ 2:0Þ, one can find that theyroughly scale as Pr0:6 over the present Reynolds number range. Itis also noticeable that the maximum RMS values of the scalar fluc-


tuation increase as Pr is increasing. The peak values of the RMS ofthe scalar fluctuations at Pr ¼ 2:0 are about twice as those atPr ¼ 0:71. It is also observed that the peak position moves awayfrom the wall as Pr is decreased. The present DNS results agree wellwith various results from Kader (1981), Kasagi et al. (1992), andKawamura et al. (1998) for both the mean profile and the RMS ofthe scalar fluctuation in the near-wall region.

The previous study by Kawamura et al. (1998) reported that inthe vicinity of the wall for isoscalar boundary condition, hhiþ and



y+

Ξ θ

100

101

102

103

0

2

4

6

8

10

y+

θ rms

+

10−1

100

101

102

103

0

1

2

3

4

5

(a)

(b)

Fig. 8. Profiles of the diagnostic function Nh with jh ¼ 0:41 and scalar variance hþrms

at Reh ¼ 830. + h1, — h2, -�- h3, - - - h4, ...... h5. Isoscalar wall: h1; h2 and h4. Isofluxwall: h3 and h5. (a) Nh , (b) hþrms .

y+

<u’

θ’>

+/P

r, <

u’θ’

>+

/Pr0.

6

10−1

100

101

102

10−4

10−2

100

102

y+

−<

v’θ’

>+

/Pr,

−<

v’θ’

>+

/Pr0.

6

10−1

100

101

102

10−5

100

(a)

(b)

Fig. 9. Profiles of the scalar fluxes hu0h0i and �hv 0h0i at Reh ¼ 830. + h1, — h2, -�- h3, - -- h4, ........ h5. Isoscalar wall: h1; Pr ¼ 0:2; h2; Pr ¼ 0:71 and h4; Pr ¼ 2:0. Isoflux wall:h3; Pr ¼ 0:71 and h5; Pr ¼ 2:0. � (Kasagi et al., 1992) at Res ¼ 150; Pr ¼ 0:71, �(Abe et al., 2004) at Res ¼ 1020; Pr ¼ 0:71, � (Kawamura et al., 1998) atRes ¼ 180; Pr ¼ 5:0. (a) hu0h0iþ , (b) �hv 0h0iþ .


ARTICLE IN PRESS

hþrms are proportional to Pryþ while hu0h0iþ and�hv 0h0iþ vary as Pryþ2

and Pryþ3, respectively, except for low Pr. This is also true for the

present DNS (see Fig. 9) and for the turbulent pipe-flow simulationby Redjem-Saad et al. (2007). For scalars with isoflux boundarycondition, using the above exponent for the wall fluctuations,hþrms is found proportional to Pr0:6yþ in the vicinity of the wall whilethe scaling for the mean scalar profile remains the same as for theisoscalar case. For the streamwise and wall-normal scalar fluxes,Kong et al. (2000) concluded that hu0h0iþ and �hv 0h0iþ are propor-tional to yþ and yþ2, respectively, based on near-wall asymptoticexpansion of u0; v 0 and h0. Since only one scalar with isoflux wallwas computed in their simulation, the relation with Pr is not clear.Based on the present data, hu0h0iþ and �hv 0h0iþ with isoflux bound-ary condition vary as Pr0:6yþ and Pr0:6yþ2, respectively.

An important parameter in scalar transfer is the turbulent Pra-ndtl number Prt which is defined as the ratio of the turbulent eddyviscosity mt to the eddy diffusivity at . mt and at are defined by

mt ¼ �hu0v 0i@hui@y

ð17Þ

and

at ¼ �hv 0h0i@hhi@y

; ð18Þ

respectively. The turbulent Prandtl number (see Fig. 10) is often as-sumed to be a constant value which is independent of the wall-nor-mal distance and the molecular Prandtl number. However, thedependence on the wall-normal position and Pr has long been asubject of many investigations, e.g. Antonia and Kim (1991). Forthe present DNS, Prt for the scalars h2 and h4 (isoscalar boundary


condition) approach approximately a constant value of about 1.1at the wall independent of the molecular Prandtl number. Bothreach their maxima at around yþ � 45. Similar values are also re-ported by previous studies, e.g. Kong et al. (2000) for turbulentboundary-layer flow and Antonia and Kim (1991) for fully devel-oped turbulent channel flow. However, the profile pertaining tothe scalar h1 with Pr ¼ 0:2 is different from the others throughoutthe boundary layer. The value of Prt at the wall for h1 is about 0:9and increases from the wall to the peak value of 1:2 at yþ � 45. Belland Ferziger (1993) obtained a similar peak value for Pr ¼ 0:1, buttheir peak position occurs slightly closer to the wall. In addition,similar to the observation by Bell and Ferziger (1993), the peak val-ues of Prt for all the Prandtl numbers are slightly lower than thosereported by Kim and Moin (1989) in channel flow. However,Kawamura et al. (1998) found even lower peak values ofPr ¼ 0:71 at Res ¼ 180. In the wake region all the Prt start increasingagain and reach a second peak near the boundary-layer edge. Thissecond peak is due to the intermittency in the wake region of theboundary layer. Comparing with the data from Bell and Ferziger(1993), they only observed the second peak for Pr ¼ 0:1, but notfor the other Prandtl numbers. Outside the boundary layer, in allcases, Prt decreases again.

To analyse the near-wall asymptotic behaviour of the turbulentPrandtl number, one can expand the velocity and scalar distribu-tions in Taylor series as discussed previously. Then the turbulentPrandtl number for the isoscalar wall is found to be constant nearthe wall while for the isoflux wall it has a linear behaviour withPr0:4yþ. These limiting behaviours are shown in Fig. 10b.



y/δ99θ

bud

get f

or <

u’ θ’

>

0 0.2 0.4 0.6 0.8 1 1.2−1

−0.5

0

0.5

1

Fig. 11. Budgets of normalised streamwise scalar flux in outer scaling at Reh ¼ 830pertaining to h4 with Pr ¼ 2:0. —— Production, - - - dissipation, � � � turbulent

y+

Pr t

0 100 200 300 4000

0.2

0.4

0.6

0.8

1

1.2

1.4

y+

Pr t,

Pr t/P

r0.4

100

101

102

10−2

10−1

100

101

(a)

(b)

Fig. 10. The wall-normal distribution of the turbulent Prandtl number Prt atReh ¼ 830. + h1, — h2, -�- h3, - - - h4, ........ h5. The boundary-layer thickness are rangingfrom yþ ¼ 320 to yþ ¼ 365 for different scalars.


ARTICLE IN PRESS

3.4. Budget equations

From the DNS, the full budgets of the Reynolds-stress and sca-lar-flux equations are obtained. The transport equations for Rey-nolds-stresses can be written as

@hu0iu0ji@t

þ Cij ¼ Pij þPij þ Gij þ Tij þ Dij � eij; ð19Þ

where the interpretations of the different terms can be found e.g. inPope (2000).

Similarly, the transport equations for the scalar fluxes are givenby

@hu0ih0i

@tþ Chi ¼ Phi þPhi þ Ghi þ Thi þDhi � ehi: ð20Þ

The different terms in the Eq. (20) are commonly expanded as

Chi � huli@hu0ih

0i@xl

;

Phi � �hu0lh0i @huii@xl� hu0iu0li

@hhi@xl

;

Phi � p0@h0

@xi

� ;

Ghi � �@hp0h0i@xi

;

Thi � �@hu0iu0lh

0i@xl

;


Dhi �@

@xl

1Pe

u0i@h0

@xl

� þ 1

Red�0

h0@u0i@xl

� !;

ehi �1

Red�0

þ 1Pe

!@u0i@xl

@h0

@xl

� ;

where Chi is denoted as the mean convection, Phi is the productionterm due to both the mean gradients of velocity and scalar. Phi isthe pressure scalar-gradient correlation term representing the in-ter-component redistribution of the turbulent energy between sca-lar-flux terms. Ghi is the divergence of the pressure-scalarcorrelation term which represents spatial redistribution of the en-ergy among different scalar-flux components due to inhomogenei-ties in the flow field. Phi þ Ghi is the scalar pressure-gradientcorrelation term also denoted as the pressure scrambling term. Thi

is the turbulent diffusion, which is the divergence of the triple cor-relation tensor, acting as a spatial redistribution term. Dhi is themolecular diffusion term and ehi the dissipation term.

All the terms in the budget equations are explicitly evaluatedincluding the pressure terms. Two scalings are used: First, a scalingin wall units (inner scaling), i.e. non-dimensionalised by u4

sm for the

Reynolds-stress budgets and u3shsm for the scalar-flux budgets where

hs is the friction scalar and defined by qwqcpus

. Secondly, an outer scal-

ing is used, i.e. quantities are non-dimensionalised by U31

d� for the

Reynolds-stress budgets and U21ðh1�hwÞ

d99h

for the scalar-flux budgets

where d� is the local displacement thickness and d99h the 99% local

scalar boundary-layer thickness. The residual for all the budgets isat most Oð10�3Þ in viscous scaling.

All the Reynolds-stress budgets compare very well with previ-ous studies in both inner and outer scalings, e.g. see Spalart(1988) and Komminaho and Skote (2002). Therefore, these budgetsare not discussed further in the interest of space.

In Fig. 11, the budget for the streamwise scalar flux ofh4 ðPr ¼ 2:0Þ is shown. All the terms are normalised such that thesum of the square of all terms is unity. One can clearly see thatthe mean convection term, which is negligible near the wall, be-comes a major balancing term near the boundary-layer edge, to-gether with the pressure-diffusion and the turbulent diffusionterms. The latter is noticeable in the near-wall region and nearthe boundary-layer edge, similar as for the Reynolds-stresses. Forthe case of h2ðPr ¼ 0:71Þ, the budget for the wall-normal scalar fluxhv 0h0i at Reh ¼ 830 compares well with the data from Hattori et al.(2007) at a higher Reynolds number Reh ¼ 1000.

To further investigate the Pr effects, the budgets of the stream-wise and wall-normal scalar fluxes are scaled such that all thedominant terms collapse for both inner and outer scalings. For

diffusion, ........ scalar pressure-diffusion, þmolecular diffusion, �mean convection.



Pr(1/4) y+

bud

get f

or <

u’ θ’

>

0 10 20 30 40 50−0.4

−0.2

0

0.2

0.4

Pr(1/4) y+

bud

get f

or <

v’ θ’

>

0 10 20 30 40 50

−0.1

−0.05

0

0.05

0.1

y/δ99θ

bud

get f

or <

u’ θ’

>

0 0.2 0.4 0.6 0.8 1−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

y/δ99θ

bud

get f

or <

v’ θ’

>

0 0.2 0.4 0.6 0.8 1−4

−3

−2

−1

0

1

2

3x 10

−3

(a) (b)

(c) (d)

Fig. 12. Prandtl-number dependent scaled budgets of the streamwise and wall-normal scalar fluxes at Reh ¼ 830. — h2, -�- h3, - - - h4, ........ h5, � production, � dissipation, �scalar pressure-diffusion, / molecular diffusion, . turbulent diffusion, �mean convection. Symbols are from Kawamura et al. (1999) at Res ¼ 395 with Pr ¼ 0:71. (a) Budget forhu0h0 i in inner scaling. (b) Budget for hv 0h0i in inner scaling. (c) Budget for hu0h0i in outer scaling. (d) Budget for hv 0h0i in outer scaling.


ARTICLE IN PRESS

the budgets in inner scaling as shown in Fig. 12a and b, the stream-wise budget hu0h0i normalised by Pr

12 and the wall-normal budget

hv 0h0i normalised by Pr14 are found to scale with Pr

14yþ. The channel

data from Kawamura et al. (1999) with Pr ¼ 0:71 at Res ¼ 395 isalso included, and very good agreement can be seen for both thestreamwise and wall-normal scalar fluxes. For the streamwise sca-lar fluxes of h3 and h5, due to the different boundary conditions theagreement with the channel data is only good except in the closevicinity of the wall. The production and dissipation terms are dom-inant and the molecular diffusion term is only noticeable in thenear-wall region. The scalar pressure-gradient correlation term al-ways lies on the loss side and becomes comparable with the dissi-pation term at yþ � 40. Further away from the wall, the scalarpressure-gradient correlation term becomes larger than the dissi-pation term (see Fig. 11). For the wall-normal scalar flux (seeFig. 12b), a larger but still insignificant difference is observed.The wall boundary condition seems to only influence the dissipa-tion and turbulent diffusion terms and it is less effective in thewall-normal scalar fluxes compared to the streamwise scalarfluxes. The production term is negative for the wall-normal scalarflux. Due to the isotropy in the dissipation scale, the dissipation isnegligible for fluids with Pr P 0:7 (Kawamura et al., 1998). This isalso true for the present DNS except in the close vicinity of thewall. Thus, the production is balanced mainly by the scalar pres-sure-gradient correlation term. According to Kawamura et al.(1998), in a low Prandtl-number fluid, the dissipation is dominantbecause it takes place in eddies of a larger scale. They reported thatthe scalar pressure-gradient term is dominant for Pr ¼ 0:4 and 5.0while the dissipation term is overwhelming for Pr ¼ 0:05. They


also reported that the scalar pressure-gradient and the dissipationterms become comparable at Pr ¼ 0:2 which is however not truefor the present DNS (not shown here).

For the outer scaling as shown in Fig. 12c and d, the budgets arenormalised empirically by Pr�

13 to collapse all the curves. The bud-

gets belonging to the scalar with Pr ¼ 0:2 are excluded due to lowPr effects. The present scalings are based on the range of Pr of thesimulation, therefore further investigations are needed to confirmthe validity of the results. In addition, except for the low Pr case, allthe budgets of the scalar fluxes hu0h0i and hv 0h0i look very much likethose of the Reynolds-stresses hu0u0i and hu0v 0i, respectively. Fur-ther discussions concerning low Pr effects can be found in Kasagiand Ohtsubo (1993).

3.5. Higher-order statistics

For a normally distributed random variable, the respectiveskewness and flatness factors are 0 and 3, respectively. In thenear-wall region, the behaviours of the skewness and flatness fac-tors for the velocity and pressure components are similar to thosein the channel-flow simulation by Kim et al. (1987). The wall val-ues of the skewness and flatness factors for the pressure fluctua-tion are �0.04 and 4.9 which are comparable to the results �0.1and 5.0 obtained by Kim (1989).

The skewness and flatness factors for the scalars are shown inFig. 13. The profiles of h2 and h4 compare well with the data by Toh-doh et al. (2008). The skewness factors for the isoflux wall are clo-ser to zero in the vicinity of the wall which implies moresymmetric fluctuations than those of the isoscalar wall (Kong



y+

S(θ

’)

0 50 100 150 200 250 300−2

−1.5

−1

−0.5

0

0.5

1

y+

F(θ

’)

0 50 100 150 200 250 3002

3

4

5

6

(a)

(b)

Fig. 13. Skewness and flatness-factor distributions of the scalars at Reh ¼ 830. þ h1,—— h2, � � � h3, - - - h4, ........ h5. (a) Skewness factor Sðh0Þ. (b) Flatness factor Fðh0Þ. Thescalar boundary-layer thicknesses at Reh ¼ 830 are ranging from yþ � 320 to 365.

θ’/θrms

P(θ

’/θ rm

s)

−6 −4 −2 0 2 40

0.2

0.4

0.6

0.8

1

1.2

Fig. 14. PDF of h04 with Pr ¼ 2:0 and isoscalar wall at Reh ¼ 830 with boundary-layeredge at yþ ¼ 320. yþ � 5, j yþ � 50, � yþ � 100, N yþ � 300, - - - Gaussiandistribution.


ARTICLE IN PRESS

et al., 2000). Within the conductive sub-layer, Sðh0Þ with isoscalarwall are positive which is consistent with the positive Sðu0Þ (Anto-nia and Danh, 1977). Redjem-Saad et al. (2007) reported that in aturbulent pipe simulation Fðh0Þ ¼ 7 at the wall for Pr ¼ 0:71 whichis higher than the present DNS results, which might be due to thesurface curvature in the pipe. In addition, Fðh0Þ reaches its first min-imum at about the wall-normal position where the correspondingRMS of the scalar fluctuations are maximum. A similar behaviourcan be observed for the streamwise velocity (Durst et al., 1987).An interesting observation is that the effects of the differentboundary conditions seem to influence not only the near-wall re-gion; a clear difference can be observed up to yþ � 150. On theother hand, the wall values of the higher-order terms for isoscalarboundary condition, i.e. about 1.0 for the skewness and 4.5 for theflatness, appear to be Prandtl-number independent. In the outer re-gion, the skewness and flatness factors increase rapidly which indi-cates the intermittent region similarly as for the velocity andpressure components. The maximum peak values of Sðh0Þ andFðh0Þ are however much higher than Sðu0Þ and Fðu0Þ which is alsoobserved in experiments by Antonia and Danh (1977).

3.6. Probability density functions

A different perspective on the characteristics of the fluctuationsof one or more variables is provided by analysing the probabilitydensity functions (PDF). The PDF distributions of the velocity andpressure fluctuations at various wall-normal positions, rangingfrom yþ � 5 in the viscous sub-layer to yþ � 500 in the free-stream(the boundary-layer thickness is about yþ ¼ 315 at Reh ¼ 830), areobtained from the simulation as well as the corresponding distri-butions of the different scalar fluctuations.


The general shape of Pðu0Þ agrees well with the experimentstudy by Durst et al. (1987). For the PDF of the wall-normal veloc-ity fluctuation, Nagano and Tagawa (1988) reported that the distri-butions are close to the Gaussian distribution in the near-wallregion and depart from the Gaussian distribution far away fromthe wall. However, the present DNS shows more pointy distribu-tions very close to the wall due to the large Fðv 0Þ values, a behav-iour which has been reported virtually for all simulations of wall-bounded turbulence. The PDF of the pressure fluctuation (notshown here) is negatively skewed throughout the boundary layeruntil the free-stream and the wall-pressure distribution compareswell with the data from Kim (1989).

The PDF of the scalar fluctuations of the isoscalar boundary con-dition are similar to the distributions of the streamwise velocitywhich reflects the overall similarity between the streamwise veloc-ity u and scalar h (Antonia et al., 1988). The PDF distribution ofh4ðPr ¼ 2:0Þ is shown in Fig. 14 at different wall-normal positions.A Gaussian distribution with zero mean and matching variance isalso included as a reference. All the fluctuations are normalisedby the corresponding RMS values and the probability density func-tions are normalised to unit area. The PDF for the other scalars withisoscalar boundary condition have a similar shape. In particular, along positive tail at yþ � 5 also exists as for streamwise velocity u.This tail is caused by the sweep-type motion of the low momen-tum fluids and low concentration of scalars (Nagano and Tagawa,1988). The positive tail shortens with increasing Pr and also withincreasing wall-normal distance as shown in Fig. 14. As the Pr orthe wall-normal distance is increased, the negative tail extendslonger. For the scalars with isoflux boundary condition (not shownhere), the PDF distributions are closer to the Gaussian distributionin the near-wall region. Far away from the wall ðyþ > 50Þ, there isno noticeable difference between the distributions with the differ-ent boundary conditions. At yþ � 300, all the PDF of the scalar fluc-tuations are extremely negatively skewed which indicates theexistence of the intermittent region. In the free-stream, as ex-pected, the profiles are again close to the Gaussian distributionwith extremely small variance.

The high correlation between streamwise velocity and the sca-lar is also illustrated by the joint probability density function(JPDF). The results of h02 at yþ � 5 are consistent with the previousresults from a channel DNS simulation by Kim and Moin (1989).JPDF of ðu0; h0Þ of Pr ¼ 2:0 at yþ � 5 and Reh ¼ 830 are shown inFig. 15. A less correlated relation between the streamwise velocityand the scalars with isoflux boundary condition is observed, indi-cating the influence of the different boundary conditions for uand h for these cases.



u’/urms

θ’/θ

rms

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

u’/urms

θ’/θ

rms

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

(a)

(b)

Fig. 15. JPDF of ðu0; h0Þ ðPr ¼ 2:0Þ at yþ � 5 with Reh ¼ 830. (a) Pðu0; h04Þ, (b) Pðu0; h05Þ.Contour levels are 0.025:0.025:0.5 for (a) and 0.016:0.016:0.32 for (b).

u’/urms

−v’

/vrm

s

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

u’/urms

θ’/θ

rms

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

(a)

(b)

Fig. 16. JPDF of ðu0 ;�v 0Þ and ðu0; h02Þ at yþ � 200 with Reh ¼ 830. (a) Pðu0;�v 0Þ, (b)Pðu0; h02Þ. Contour levels are 0.012:0.012:0.24 for (a) and from 0.01:0.01:0.2 for (b).


ARTICLE IN PRESS

The JPDF of ðu0; h02Þ and ðu0;�v 0Þ at yþ � 200 with Reh ¼ 830 areshown in Fig. 16. Far away from the wall, all the JPDF distribu-tions show mildly correlated shapes as expected. The scalarboundary condition has almost no effect at this wall-normal posi-tion. It is noticeable that not all the contours have their centreslocated at the origin and the peaks of ðu0; h0Þ moves from the thirdquadrant to the first quadrant as the wall-normal positionincreases.

Note that in JPDF Fig. 15a and b, one can observe a sharp cutofffor the scalar fluctuation h0. This is caused by the Dirichlet bound-ary condition, which bounds the scalar values to be between hw

and h1. In consequence, the scalar fluctuations have strict lowerand upper bounds, depending on the local mean values.

3.7. Spanwise two-point correlation

The spanwise two-point correlations of the velocity compo-nents and pressure at Reh ¼ 830 at several wall-normal positionsfrom yþ � 5 to yþ � 300 are obtained. In general the results areconsistent with the numerical results by Kim et al. (1987). As alsoobserved by Kim (1989), the spanwise two-point correlation coef-ficient of the pressure does not have the negative excursion.

The spanwise two-point correlations of the scalars near thewall are similar to those observed by Kim and Moin (1989). Onthe other hand, the scalars of the isoflux boundary conditionseem to be much more affected by the Prandtl number. In gen-eral, scalars with isoflux wall have larger mean spanwise spacingsthan those of the isoscalar wall (Kong et al., 2000). Far from thewall, the boundary conditions do not have an influence on theprofiles.


For the mean low-speed streak and scalar streak spacings ob-tained from twice the first minimum of the spanwise two-pointcorrelations, the results for h2 and h3 compare well with Kong etal. (2000). As also noted by Kong et al. (2000) that the low-speedstreak spacing is similar to that of h2 and the differences fromthe boundary conditions are only observable in the near-wall re-gion. One can also observe that the mean scalar streak spacingsare larger for smaller Pr near the wall. Due to the isoflux wall-boundary condition, the scalar spacings first decrease and then in-crease again as being away from the wall (not shown here). Thestreamwise evolution of the mean streak spacings at yþ � 7 fromReh ¼ 390 to Reh ¼ 830 is shown in Fig. 17. In general all the steakspacings grow slightly downstream with increasing Reynolds num-bers, however, the growth rate levels off towards larger Reynoldsnumbers. The large growth rate in the low Reynolds number rangemay be due to a low Reynolds-number effect (Antonia and Kim,1994). At Reh ¼ 830, the streaks have a spacing of about 110 in wallunit which is slightly larger than the usual value of about 100 (Kimet al., 1987). On the other hand, this spacing is in good agreementwith recent studies in turbulent boundary layer by Schlatter et al.(in press). These authors also found that the velocity streak spacingcontinues to grow up to Reh ¼ 1500 and then settles at a constantvalue around 115. The streaks pertaining to scalars with isofluxboundary condition are growing faster than those with isoscalarboundary condition at the same Pr. It is also suggested by the pres-ent results that the streak spacing with lower Pr grows faster. It hasalso been shown by Österlund (1999) and Abe et al. (2001)) thatthe spanwise two-point correlations of the wall-shear stress sw

and streamwise velocity fluctuation u0 have less prominent nega-tive peaks with increasing Reynolds number, which is confirmed



λz+

y+

101

102

103

100

101

102

103

λz+

y+

101

102

103

101

102

λz+

y+

101

102

103

100

101

102

103

(a)

(b)

(c)

Fig. 18. Premultiplied energy spectra of the streamwise velocity and scalars atReh ¼ 830. (a) — kzUuuðkzÞ=u2

s , - - - kzUh2h2 ðkzÞ=h2s , (b) kzUhhðkzÞ=h2

s at Pr ¼ 2:0, —isoscalar boundary condition, - - - isoflux boundary condition, (c) kzUhhðkzÞ=h2

s withisoscalar boundary condition, —— Pr ¼ 0:2, - - - Pr ¼ 0:71, ........ Pr ¼ 2:0. Contourlevels are 1:0.375:2.5 in (a); 2:1.5:8 in (b); and 0.3:0.3:0.9 for Pr ¼ 0:2, 1:0.75:2.5for Pr ¼ 0:71 and 2:3:8 for Pr ¼ 2:0 in (c).

λ+

Reθ

400 500 600 700 80090

100

110

120

130

140

150

160

Fig. 17. Streamwise variation of the mean spanwise streaks spacing kþ at yþ � 7. h

u, � h1, M h2, r h3, / h4, . h5.


ARTICLE IN PRESS

by the present DNS. This effect is due to the dominating large-scalestructures which scale with outer units, i.e. the boundary-layerthickness or the channel half width, see also Schlatter et al. (inpress). Schlatter et al. (in press) also showed that at leastReh ¼ 1500 is needed to clearly observe a second peak indicatingthe footprint of the large-scale structure in a contour plot of thetwo-point correlation of the wall-shear stress sw versus the Rey-nolds number. Concerning the scalar two-point correlations, Abeet al. (2004) showed that less prominent negative peaks withincreasing Reynolds number only exist for the surface heat fluxqw with Pr ¼ 0:71, but not for the case with Pr ¼ 0:025 due to alow Prandtl-number effect. However, for the Pr-range consideredhere, all scalars irrespective of the boundary condition feature adecreasing near-wall (inner) peak.

The spanwise two-point correlations not only contain the infor-mation about the spanwise organisation of the near-wall streaksbut also the large-scale motions in the outer region of the bound-ary layer. By showing the premultiplied spanwise spectra kzUuuðkzÞof the streamwise velocity u, both experiment (e.g. Hutchins andMarusic, 2007) and simulation (e.g. del Álamo and Jiménez,2003) showed evidence of the large-scale structures existing inthe outer layer. For the scalar field, (Abe and Kawamura, 2002) re-ported the existence of the large-scale structure in the outer layerof the channel for Res ¼ 640 with Pr ¼ 0:025 and 0.71, but no de-tailed discussion was provided. Fig. 18a shows the one-dimen-sional spanwise spectra kzUuuðkzÞ and kzUh2h2 ðkzÞ scaled with u2

sand h2

s , respectively, at Reh ¼ 830. In general, both spectra are sim-ilar in appearance with the inner peak located at yþ � 15 (see alsoFig. 8b) and kz � 110 indicating the near-wall streaks. However,differences can be observed in the outer region ðyþ > 100Þ wherethe outer peak of h2 is weaker than that of u. Fig. 18b shows thespanwise spectra for scalars h4 and h5 with different boundary con-ditions but same Pr ¼ 2:0. The influence of the different boundaryconditions is clearly seen. The spectrum of h5 with isoflux bound-ary condition has a constant value down to the wall whereas thespectrum of h4 is decreasing to zero to fulfil the boundary condi-tion. Above yþ ¼ 10, there is no clear difference. Compared toFig. 18a, it is important to note that the outer peak ðkþz � 400Þ doesnot exist for Pr ¼ 2:0. Fig. 18c shows Pr effects on the spanwisespectra. With increasing Pr, the spectral peaks move towards thewall and towards smaller spanwise wavelengths kz (see alsoFig. 17). Note that the inner peak for the case of Pr ¼ 0:2 residesat yþ � 50 which is also observed in the RMS value of the scalarvariance as shown in Fig. 8b. If the spanwise spectrum ofh1 ðPr ¼ 0:2Þ is scaled by h2

rms instead of h2s , the peak extends to-

wards yþ � 10, however the corresponding fluctuations hrms closeto the wall are extremely small.


4. Conclusions

A direct numerical simulation (DNS) of a spatially developingturbulent boundary layer with passive scalars over a flat plate un-der zero pressure-gradient (ZPG) has been carried out. The Rey-nolds number based on the inlet displacement thickness Red�0

is450, and Prandtl numbers are varying from 0.2 to 2 while twowall-boundary conditions, i.e. isoscalar and isoflux, are employed.The highest Reynolds number obtained is Reh ¼ 850 based on themomentum thickness h. The computed velocity and scalar fields




ARTICLE IN PRESS

are compared with existing data from the literature and the agree-ment is very good in general.

The main conclusions of the present study are summarised asfollows:

The mean scalar profiles are virtually independent of theemployed wall-boundary conditions whereas the effects on thescalar variances are obvious in the near-wall region. Furtheraway from the wall, the effects from different wall-boundaryconditions are negligible. The skewness and flatness profilesare different up to about 150 in viscous units, i.e. to the middleof the boundary layer for the present simulation parameters.

The results (mean scalar profiles, Prt , JPDF, two-point correla-tions etc.) for h2 with Pr ¼ 0:71 and isoscalar boundary conditionobtained from the present DNS and those from Kim and Moin(1989) using an internal heat source appear similar in thenear-wall region. This implies that these two boundary condi-tions might not have a significant influence on the low-orderstatistics in the near-wall region.

All the terms in the Reynolds-stress and scalar-flux budgets areexplicitly evaluated including the pressure terms. Far away fromthe wall, both the mean convection and the turbulent diffusionterm become balancing terms in the scalar-flux budgets,together with the pressure-diffusion term. This is in good agree-ment to the Reynolds-stress budgets.

Prandtl-number scalings for the scalar-flux budgets and severalother scalar quantities in both inner and outer units are pro-posed based on the present data, however, further investigationsat wider Pr range are needed to validate the results.

The intermittency in the outer region is identified and quantifiedvia higher-order statistics and PDF distributions.

The scalar with isoscalar boundary condition is highly correlatedwith the streamwise velocity component in the near-wallregion, however showing the influence of the wall-boundarycondition. Near the boundary-layer edge, a mild correlationbetween these two quantities was observed, especially indepen-dent of the boundary condition.

All the scalar streak spacings grow downstream with increasingReynolds numbers. The scalar streak spacings pertaining to theisoflux boundary condition and lower Pr grow faster.

The large-scale structures could be identified with the help ofthe premultiplied spanwise energy spectra. There certainly existlarge-scale structures for the velocity field and scalar withPr ¼ 0:71. For the higher Pr cases, no large-scale structures areidentified. The inner spectra peaks move away from the walland towards larger wavelengths kz as Pr decreases.

Even though the Reynolds number considered in the presentstudy is so far the highest with such a variety of scalars, it is stilllow compared to the experiments. Therefore, low Reynoldsnumber effects are important and play a role when interpretingthe results.

In the future, the large amount of data from the simulation willbe further processed and new and existing LES/RANS closures formodelling the scalar fluxes will be developed and critically evalu-ated against the present database.

The present database will be open to public access through thefollowing link: http://www.mech.kth.se/.

Acknowledgements

Computer time was provided by the Center for Parallel Comput-ers (PDC) at the Royal Institute of Technology (KTH) and the Na-tional Supercomputer Center (NSC) at Linköping University inSweden.


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