turbulent dispersion from elevated line sources in channel and couette flow

Turbulent Dispersion from Elevated Line Sourcesin Channel and Couette Flow

Phuong M. LeSchool of Chemical Engineering and Materials Science, The University of Oklahoma, Norman, OK 73019

Dimitrios V. PapavassiliouSarkeys Energy Center, The University of Oklahoma, Norman, OK 73019

DOI 10.1002/aic.10507Published online June 17, 2005 in Wiley InterScience (www.interscience.wiley.com).

Turbulent dispersion of a scalar emitted from a source in homogeneous turbulence hasbeen placed in a solid theoretical context by Taylor, Saffman, and Batchelor. However,the case of source diffusion in the near-wall region, where the turbulence is not Gaussian,and the effects of the molecular Prandtl number (Pr) on the effective dispersion have notbeen explored to similar depth. The present work studies the behavior of elevated sourcesin turbulent channel flow and in turbulent-plane Couette flow. The trajectories of heatmarkers are monitored in space and time as they move in a hydrodynamic field createdby a direct numerical simulation. The fluids span several orders of magnitude of Pr (orSchmidt number), Pr � 0.1, 0.7, 3, 6, 10, 100, 200, 500, 1000, 2400, 7500, 15,000, and50,000 (liquid metals, gases, liquids, lubricants, and electrochemical fluids). It is foundthat the molecular Pr has negligible effects in the evolution of the marker cloud for Pr �3, when the point of marker release is away from the viscous wall sublayer. However,when the markers are released close to the wall, the molecular effects on dispersion arestrong. It is also found that total effective dispersion is higher in the case of plane Couetteflow, where the total stress across the channel is constant. © 2005 American Institute ofChemical Engineers AIChE J, 51: 2402–2414, 2005Keywords: turbulent transport, Lagrangian methods, source diffusion

Introduction

The prediction of turbulent dispersion of a scalar contami-nant emitted from sources above a surface is a problem thatgained importance recently because of its application in atmo-spheric pollution and in the dispersion of bioagents in the caseof an act of terrorism. The statistical description of turbulentdispersion from a Lagrangian perspective was introduced byTaylor,1 who described the rate of dispersion of fluid particlesfrom a point source in homogeneous, isotropic turbulence as

dXf2

dt� 2u2 �

0

t

RL��d� (1)

where Xf is the displacement of a fluid particle relative toits source, u2 is the mean square of the x-component of thevelocity of the fluid particles, and RL is the Lagrangian corre-lation coefficient. Taylor’s equation can be seen as an exten-sion of Einstein’s relation2 for the dispersion of particles withBrownian motion, given as

dXp2/dt � 2D (2)

where Xp is the displacement of a particle relative to its sourceand D is the molecular diffusivity. Saffman3 studied the effects

D. V. Papavassiliou is also affiliated with the School of Chemical Engineering andMaterials Science, The University of Oklahoma, Norman, OK 73019.

Correspondence concerning this article should be addressed to [email protected].

© 2005 American Institute of Chemical Engineers

2402 AIChE JournalSeptember 2005 Vol. 51, No. 9

of molecular diffusion on turbulent dispersion and developed arelation for dispersion in this case by defining a materialautocorrelation function, which correlated fluid velocity com-ponents along the trajectories of scalar markers instead of fluidparticles. Saffman argued that scalar markers can move off afluid particle as a result of molecular diffusion, and thus theeffect of molecular diffusion is to diminish turbulent diffusionbecause the markers do not follow the chaotic turbulent fluidmotion. With respect to anisotropic turbulent flows, Batchelor4

developed a theory for the prediction of the statistical behaviorof a source in a turbulent boundary layer. Batchelor4 argued(based on similarity) that the Lagrangian velocity within theconstant stress region depends only on the friction velocity u*and time so that

Vy �dY�

dt� bu* (3)

where b has to be an absolute constant, independent of molec-ular diffusion effects.

Laboratory measurements for turbulent dispersion have beenreported for the case of continuous elevated sources of apassive scalar.5,6 Shlien and Corrsin5 examined turbulent dis-persion of heat in a wind tunnel, downstream from a heatedwire, and Fackrell and Robins6 studied turbulent mass disper-sion with emphasis on the concentration fluctuations usingpropane as a tracer gas. More recently, direct numerical sim-ulations (DNS) of turbulent flows in conjunction with trackingof scalar markers have been used for the investigation of scalardispersion in anisotropic turbulent flows.7-10 However, empha-sis has been given to sources located at the solid surface.Papavassiliou11 studied the effects of the molecular Prandtlnumber (Pr) on the evolution of a cloud of markers releasedinstantaneously from a line source at the wall of a channel (thatis, a puff) for 0.1 � Pr � 50,000. The puff was found todevelop in the following three stages:

● Zone I, in which molecular diffusion dominates disper-sion.

● Zone II, which is a transition zone.● Zone III, in which turbulent convection dominates disper-

sion.The extent of Zones I and II depends on the Pr; it becomeslonger as the Pr increases. Mitrovic and Papavassiliou9 calcu-lated the turbulent transport properties for the plume that re-sults from a continuous line source at the wall, and modeled itsbehavior for 0.1 � Pr � 50,000. They found that the behaviorof the plume exhibits three zones of development, which cor-respond directly to the three stages of development of a puff.The behavior of both puffs and plumes that are emitted fromwall sources was found to be Pr dependent.

The present study explores the effects of molecular diffusionon turbulent transport for the case of elevated sources from thewall, and the effects of turbulence structure on turbulent dis-persion. The effects of molecular diffusion are explored bychanging the Pr of the fluid, and the effects of the turbulencestructure are investigated by placing the sources in a planechannel flow and in a plane Couette flow. In plane Couetteflow, the driving force for the flow is the shear effect of the twochannel walls moving in directions opposite to each other. Thetotal stress is constant across the whole channel, creating a very

extensive constant stress region, similar to the logarithmicregion in Poiseuille channel flow. Thus, we can achieve a widelogarithmic layer that is computationally difficult to obtainotherwise (that is, with a DNS of plane channel flow). Atracking algorithm is used to monitor the trajectories of scalarmarkers in space and time as they move in the hydrodynamicfield created by a DNS. The fluids span several orders ofmagnitude of Pr [or Schmidt number (Sc)], Pr � 0.1, 0.7, 3, 6,10, 100, 200, 500, 1000, 2400, 7000, 15,000, 50,000 (liquidmetals, gases, liquids, lubricants, and electrochemical fluids).

Lagrangian Scalar Tracking

The basic concept of Lagrangian scalar tracking (LST) isthat passive scalar transport in turbulent flow is the result of thecombined behavior of an infinite number of continuous sourcesof heat or mass markers, such as the markers discussed bySaffman.3 Hanratty12 used this concept to describe the transferof heat from a hot to a cold wall in turbulent channel flow. Thestudy of heat or mass transfer using such an approach hasbecome feasible with the development of appropriate numeri-cal methodologies and the appearance of supercomputers.13

Papavassiliou and Hanratty8,14 used a DNS in conjunction withthe tracking of scalar markers to study heat transfer based ondirect calculations of the behavior of such wall sources. Moreabout the implementation and validation of the LST method-ology, which includes the stochastic tracking of heat or massmarkers in a turbulent flow field, and the statistical postpro-cessing of the results to obtain scalar profiles, can be foundelsewhere.8,10,15,16 Heat transfer across the interface between aturbulent gas and a turbulent liquid has also been simulatedusing LST.17,18

For the present study, the trajectories of heat or mass mark-ers in the flow field created by a DNS were calculated using thetracking algorithm of Kontomaris et al.13 The marker motionwas partitioned into a convective part and a molecular diffusionpart. The convective part was calculated from the fluid velocityat the particle position yielding for the equation of particlemotion V� (x�o, t) � [�X� (x�o, t)]/�t, where the Lagrangian velocityof a marker released at location x�o is given as V� (x�o, t) �U� [X� (x�o, t), t] (U� is the Eulerian velocity of the fluid at thelocation of the marker at time t). The effect of moleculardiffusion follows from Einstein’s theory for Brownian motion(see Eq. 2); the diffusion effect was simulated by adding athree-dimensional random walk on the particle motion at theend of every convection step. The size of the random jump ineach one of the three space directions took values from aGaussian distribution with zero mean and a standard deviationof � � �2�t/Pr in wall units. Effects of Pr on the process canthus be studied by modifying �. This allows the LST method-ology to simulate fluids within an extensive range of Pr. Incontrast, Eulerian simulations for heat transfer in the case ofanisotropic turbulence (that is, channel or Couette flow) havebeen limited to a narrower range of fluids with molecular Pr orSc between 0.025 and 10,† because, to resolve all the scales ofmotion and temperature, the number of grid points has to beanalogous to Pr3/2.19-26 In the case of homogeneous and isotro-

† To our knowledge, there is no published Eulerian DNS work for anisotropic flowand Pr � 10, even though Tiselj’s group (2002) at the Institute Jozef Stefan hasaccomplished DNS with at least Pr � 54.

AIChE Journal 2403September 2005 Vol. 51, No. 9

pic turbulence, however, recent Eulerian DNS have examinedcases of relatively high Sc.27 Yeung et al.28 examined the caseof 0.125 � Sc � 64, and Nieuwstadt and Brethouwer29 andBrethouwer et al.30 studied the range of 0.04 � Sc � 144.Finally, the markers used here are assumed to be passive andthus have no effect on the flow.

Direct Numerical Simulations

The behavior of a scalar source was determined by followingthe paths of a large number of scalar markers in a flow fieldcreated by a DNS (see Lyons et al.31 and Gunther et al.32 for thevalidation of the channel flow DNS used in this study; andPapavassiliou and Hanratty33 for the methodology imple-mented for the Couette flow DNS used in this study). The flowwas for an incompressible Newtonian fluid with constant phys-ical properties. In the case of channel flow, it was driven by aconstant mean pressure gradient, and for the case of planeCouette flow it was driven by the shear motion of the twomoving walls of the channel. The Reynolds number, definedwith the centerline mean velocity and the half-height of thechannel for the Poiseuille flow channel, and defined with halfthe velocity difference between the two walls and the halfchannel height for the Couette flow channel, was 2660 for both.For the Poiseuille channel, the simulation was conducted on a128 � 65 � 128 grid in x, y, z, and the dimensions of thecomputational box were 4�h � 2h � 2�h, where h � 150 inwall units.‡ For the Couette flow channel, the simulation wasconducted on a 256 � 65 � 128 grid, and the dimensions of thecomputational box were 8�h � 2h � 2�h, where h � 153. Theflow was regarded as periodic in the x and z directions, with theperiodicity lengths equal to the dimensions of the computa-tional box in these directions. The Couette flow channel waschosen to be longer than the Poiseuille channel to minimize theeffects of the large-scale structures known to be present inCouette flow simulations.33,34 First- and second-order turbu-lence statistics for the flow fields are presented in Figure 1. Themean velocity profile is shown in Figure 1a, and the turbulenceintensities are shown in Figure 1b for the x, y, and z directions.Experimental and numerical data obtained in other previousstudies for the case of Couette flow are included in these figuresto demonstrate the behavior of that simulation. The time stepfor the calculations of the hydrodynamic field and the Lagrang-ian tracking was �t � 0.25 and �t � 0.2 for the Poiseuille andCouette channels, respectively. Both simulations were firstallowed to reach a stationary state before the heat markers werereleased.

Results and Discussion

Table 1 presents a summary of the runs conducted for thecurrent study. Runs P1 to P14 tracked markers in a Poiseuilleflow channel with Pr � 0.1, 0.7, 3, 6, 10, 100, 200, 500, 1000,2400, 7500, 15,000, and 50,000 up to time t � 300. In each oneof these runs, a total of 16,129 markers were released instan-taneously from a uniform rectangular grid that covered the x–zplane of the computational box at different distances from thewall. The choices of these distances were the edge of the

thermal sublayer (that is, yo � 5 for Pr � 1000, yo � 1 for Pr �1000), the region of transition between the viscous sublayerand the logarithmic velocity layer (that is, yo � 15), a point atyo/h � 0.2 to compare with the experimental measurements ofFackrell and Robins6 for elevated sources at Pr � 0.7, a pointin the logarithmic region (that is, yo � 75), and a point at thecenter of the channel. Runs C1 to C12 tracked markers forsimilar conditions in a plane Couette flow, where the maindifference was that 145,161 markers were tracked.

The initial marker positions were on a uniform 127 � 127grid and a 381 � 381 grid for the Poiseuille flow and Couetteflow, respectively. For plane channel flow, Mitrovic and Pa-pavassiliou9,35 found that using more markers than the 127 �127 case (one order of magnitude more) in the flow improvesthe calculation of statistics only slightly, so the use of 16,129markers is sufficient for the plane channel case. The computa-tional box is twice as large for the plane Couette flow, and oneorder of magnitude more markers are used in that case.

Instantaneous line source behavior

The cloud that results from an instantaneous source of ascalar is usually called a puff and the cloud that results from acontinuous source of a scalar is called a plume. Figure 2apresents the mean puff trajectory in the normal direction for themarker cloud of runs P2 and P3, and Figure 2b for the markercloud of run C2 (Pr � 0.7 in these cases, corresponding todispersion of heat in air). Physically, this is the trajectory of thecentroid of a puff of markers released from an instantaneous

‡ All quantities in this work are expressed in viscous wall units unless otherwisespecified.

Figure 1. Turbulent flow statistics.(a) Mean velocity profile, (b) turbulence intensity in thestreamwise, normal, and spanwise flow directions. [P&H �Papavassiliou and Hanratty33; L&K � Lee and Kim44;A&L � Aydin and Leutheusser45; R&J � Robertson andJohnson.46]


line source located at different distances yo from the channelwall. It is observed that the marker clouds tend to move awayfrom the wall in both cases. The reason for this effective

diffusion of markers away from the wall, even though theprobability of moving toward the wall or away from it is thesame for each marker, is that there are initially many moremarkers close to the wall. Thus, the number of markers that cango away from the wall is larger than those going toward thewall from the outer region because there are not that manymarkers in the outer region of the flow. As a result, the netnumber of markers moving away from the wall is positive, andthe cloud centroid moves away from the wall. It is also ob-served that the centroid of the puff in the Couette flow envi-ronment is moving away from the wall much faster than in thePoiseuille flow environment. Figure 1b shows that the rootmean square of the vertical velocity fluctuations is higher inCouette flow than that in channel flow, meaning that strongerfluctuations can take markers away from the wall. Na et al.36

studied large-scale structures that produce Reynolds stressesand look like sheets extending well into the logarithmic region(which they called “super-bursts”). Because Couette flow is avery large constant stress region, such structures are alsopresent in Couette flow, resulting in higher rates of dispersionaway from the wall. This appears to be a difference in thefundamental mechanism of heat convection in the log layer andin channel flow.

The density of the cloud of the markers is represented by theprobability, P1(x� , t � x�o, to), of a marker to be at a location x� �(x, y, z) in the flow field at time t, given that it was released atlocation x�o � (xo, yo, zo) at time to. This probability can beinterpreted physically as concentration3 and thus as a snapshotof a cloud of contaminants released instantaneously from x�o.Figures 3 and 4 present contours of the puff concentration forPoiseuille and Couette flows, respectively, and for Pr equal to0.7 and to 200 (runs P2, P8, C2, and C6). The point of release

Table 1. Summary of the Conditions Applied to the Simulation Runs Used in This Work*

Run Number Pr

Source Elevation from the Wall in Viscous Wall Units

Numbers of MarkersCase a Case b Case c Case d Case e

P1 0.1 5 15 28.5 75 150 16,129P2 0.7 5 15 28.5 75 150 16,129P3 0.7 2 38.5 50 96 125 16,129P4 3 5 15 28.5 75 150 16,129P5 6 5 15 28.5 75 150 16,129P6 10 5 15 28.5 75 150 16,129P7 100 5 15 28.5 75 150 16,129P8 200 5 15 28.5 75 150 16,129P9 500 2 15 28.5 75 150 16,129P10 1000 5 15 28.5 75 150 16,129P11 2400 1 15 28.5 75 150 16,129P12 7500 1 15 28.5 75 150 16,129P13 15,000 1 15 28.5 75 150 16,129P14 50,000 1 15 28.5 75 150 16,129C1 0.1 1 15 28.5 75 150 145,161C2 0.7 1 15 28.5 75 150 145,161C3 6 1 15 28.5 75 150 145,161C4 10 1 15 28.5 75 150 145,161C5 100 1 15 28.5 75 150 145,161C6 200 1 15 28.5 75 150 145,161C7 500 1 15 28.5 75 150 145,161C8 1000 1 15 28.5 75 150 145,161C9 2400 1 15 28.5 75 150 145,161C10 7500 1 15 28.5 75 150 145,161C11 15,000 1 15 28.5 75 150 145,161C12 50,000 1 15 28.5 75 150 145,161

Each run was for a different Pr and a different flow field (the letter P in the run number indicates Poiseuille flow and the letter C indicates Couette flow). Passivemarkers were released at five different elevations in each of the flow fields, indicated by the letters a–e. Each simulation was run for 300 viscous time units.

Figure 2. Mean puff normal position for different sourceelevations as a function of time for Pr � 0.7.(a) Poiseuille flow; (b) Couette flow.


is yo � 28.5 in both cases. In both types of flow the effect ofincreasing the Pr is similar; the cloud of markers is moreconcentrated around the point of release. For small Pr themarkers disperse faster to other areas of the flow field arisingfrom bigger molecular jumps, whereas for higher Pr, a largerpercentage of markers stay close to the point of release forlonger times. However, for sources at higher elevations (notshown here) there are no major differences in the puff concen-tration for different Pr number fluids. The behavior of thePoiseuille flow markers and the Couette flow markers is similar

in this respect. Figure 4 shows that the dispersion of the puff inthe normal direction is higher for Couette flow than for planechannel flow. The reason is the same as for the case of thefaster Y� movement away from the wall observed in Figure 2,that is, larger-velocity fluctuations in the vertical direction forCouette flow and larger-scale flow structures extending in theouter region result into higher dispersion.

The mean cloud trajectories in the normal direction areshown in Figure 5 for different Pr. Figures 5a and 5b presentthe trajectories of clouds released at the edge of the thermalconductive sublayer for channel and Couette flows, respec-tively. The effects of the fluid Pr are evident in these figures:the plane channel puffs become Pr independent for Pr � 6 andthe plane Couette flow puffs for Pr � 1000. Figures 5c and 5d

Figure 3. Contour plot for the concentration profile re-sulting from a puff in channel flow for at t� �300 and source elevation of yo � 28.5.(a) Pr � 0.7; (b) Pr � 200.

Figure 4. Contour plot for the concentration profile re-sulting from a puff in Couette flow for at t� �300 and source elevation of yo � 28.5.(a) Pr � 0.7; (b) Pr � 200.

Figure 5. Mean puff normal position for different Pr as afunction of time.(a) Poiseuille flow (* yo � 1, no asterisk yo � 5); (b) Couetteflow, yo � 1, (c) Poiseuille flow, yo � 28.5; (d) Couette flow,yo � 28.5.


show the trajectories of clouds released within the transitionregion. For both channel flow and Couette flow, the effects ofPr are negligible for release locations in the outer region of theflow. Only in the case of Pr � 0.1 in channel flow does itbehave in a manner different from that of the other cases. Thesame behavior is found for puffs released at locations fartherfrom the wall (not shown here). The reason for this behaviorcan be explored, if one considers the relative magnitude of theeddy diffusivity and the molecular diffusivity. Fluids withmolecular Pr on the order of magnitude of �1 have moleculardiffusivities that are comparable to or higher than the eddydiffusivity, and thus Pr effects are expected to be observed. Toillustrate this point, we can review published values of theturbulent Pr and estimate the order of magnitude of the eddydiffusivity using them. Results from direct numerical simula-tions for turbulent heat transfer in a channel,20-22,37,38 from largeeddy simulations,39 and from modeling correlations40,41 suggestthat the value of the turbulent Pr is on the order of one fordifferent molecular Pr. The turbulent Pr for wall turbulence isa function of the distance from the wall (as are the eddyviscosity and the eddy diffusivity), but in terms of order ofmagnitude it varies only slightly for y � 10. The eddy viscosityfor the flow field under consideration here is on the order of10,8 in the outer region of the flow (y � 30), which means thatthe eddy diffusivity is on the order of 10 as well. Both DNS20

and experiments42 agree with this estimation (in terms of orderof magnitude). Because the fluid viscosity is one, a fluid withmolecular Pr � 1 has diffusivity of one and a fluid withmolecular Pr � 0.1 has diffusivity of 10, which is comparableto the eddy diffusivity.

The mean streamwise velocity of the puff V� x is shown inFigure 6 for different elevations and Pr � 0.7. This velocity isexpected to reach the value of the bulk mean velocity of theflow field (which is 15.1 in wall units for channel flow and 0 forCouette flow) at long times because the markers are expectedto disperse uniformly across the channel at long times. Figure6a shows that the mean puff velocity drops for elevationshigher than yo � 10 because the markers start to disperse toregions where the mean velocity is smaller than the meanvelocity at the point of release. Similarly, for Couette flow, themean puff velocity shows a maximum for elevations betweenthe viscous sublayer and the center of the channel because themarkers disperse to areas of higher mean velocity before uni-formly covering the channel.

Figures 7a and 7b present the mean streamwise velocity as afunction of Pr for the case of Couette flow and for sourcelocation in the viscous sublayer and in the outer region, respec-tively. The conclusions reached above based on the puff tra-jectories also apply here; the effects of Pr are important forsources within the viscous sublayer, as seen in Figure 7a,whereas they are not for sources farther away unless the Pr issmall (Pr 10). Figure 7a shows the first two zones of plumedevelopment for high Pr, similar to those observed by Mitrovicand Papavassiliou,9 for sources on the channel wall. In the firstzone, the markers move as a result of molecular dispersion andin the second zone the contributions of convection begin toappear. The transition point between these zones is Pr depen-dent.

Figures 8a and 8b present the mean normal velocity as afunction of Pr for the Couette flow case and for source loca-tions in the viscous sublayer (yo � 1) and in the outer region (yo

Figure 6. Mean puff streamwise velocity for differentsource elevations as a function of time for Pr �0.7.(a) Poiseuille flow; (b) Couette flow.

Figure 7. Mean puff streamwise velocity for different Pras a function of time for Couette flow.(a) yo � 1; (b) yo � 75.


� 75). Figure 8 shows that the velocity V� y is a function of timeand of Pr. It is observed that V� y reaches a maximum when themovement of the markers is restricted by the channel wall, andbeyond that point tends to the value of 0. It is also observed thatthe assumption that the Lagrangian velocity of scalar markersis constant within the constant stress region and independent ofPr is not accurate, and therefore a universal constant b (see Eq.3) cannot describe all dispersion cases.

Continuous line source behavior and comparison toexperiments

The behavior of a plume originating at xo� and emittingmarkers from time to to time tf can be simulated by integratingthe probability density function that describes the behavior ofthe puffs

P2�X � xo, Y, tf� � �t�to

tf

P1�X � xo, Y, t�xo�, to� (4)

The probability P1 was calculated for each Pr using a grid thatcovers the flow domain and counting the number of markersthat are present in each grid cell.16 The grid in the normaldirection was constructed by uniformly dividing the width ofthe channel into 300 bins. In the streamwise direction, the gridwas stretched to take measurements at long distances down-stream from the source. The stretching in the streamwise di-rection followed the relation �xn � 1.06n�xo with �xo � 5.

The DNS/LST methodology has been validated with exper-imental data for the case of continuous wall sources.9,11 Here,we present comparisons with experiments for the case of ele-

vated scalar sources. Fackrell and Robins6 measured meanconcentration profiles for a passive plume from an elevatedsource within a turbulent boundary layer. The source locationwas at yo/h � 0.2 (note that h is the boundary layer thicknessat the source location for the Fackrell–Robins experiments) andthe source gas consisted of a mixture of propane and helium,the former being used as a trace gas for concentration mea-surements. They calculated the plume half-width (y), which isdefined to be the distance from the location of maximumconcentration at which the concentration falls to half of itsmaximum. Figure 9 presents a comparison of the experimentalmeasurements with the DNS/LST results showing very goodagreement between the two cases. Figure 10a presents themean concentration profile normalized with the maximum con-centration at different distances downstream from the source.The agreement between the experiments and the DNS/LSTresults is also quite good. Figure 10b shows a comparison withthe experiments of Shlien and Corrsin,5 in which dispersionwithin a turbulent boundary layer was measured downstream ofa heated wire located at different elevations from the wall. Heatwas supplied to the wire at a constant rate, so that their case isequivalent to the calculation of P2 profiles using LST. Theyscaled the distance from the wall and the distance downstreamfrom the source with the displacement thickness of the mo-mentum turbulent boundary layer (d) at the location of thetagging wire. To calculate the appropriate length scale for thechannel flow DNS, the mean centerline velocity of the channelis used in place of the free stream velocity for the calculationof the boundary layer thickness. The calculated value for theDNS is d � 23.2. Figures 9 and 10 indicate that the DNS/LSTdata are reliable for the calculation of the properties of elevatedscalar sources.

Figures 11a–11c show the plume half-width for plane chan-nel flow and different source elevations for all Pr considered. Itappears that the Pr affects the development of the plume forsources within the viscous wall region, but the plume becomesPr independent for sources farther from the wall. The reasonfor this is that turbulent dispersion is dominant in the outerregion relative to the molecular diffusion. The development ofy follows a relation of the form

y /h � a� x/h�c (5)

According to the statistical theory of turbulent dispersion43 thedispersion of the plume in the normal direction is predicted to

Figure 8. Mean puff normal velocity for different Pr as afunction of time for Couette flow.(a) yo � 1, (b) yo � 75.

Figure 9. Comparison of the plume half width computedwith the DNS/LST method and experiments.The value of R2 for the line shown is 0.945.


be Y2� � [v2� /(U� )2]x2 for small times and Y2� � 2�L[v2� /(U� )2]x for

long times, where �L is the Lagrangian timescale and v2� is themean square of the velocity fluctuations in the normal direc-tion. Thus, one would expect the exponent in Eq. 5 to be c �1/2 (because in the present case the plume is allowed todevelop for long times). However, the turbulence in channel

flow is not homogeneous, and v2� /(U� )2 can be assumed tochange with the distance from the wall according to a function(y/h)b with b � 0 up to y � 50, implying that c � 1/(2 b) �1/2.

The results show that for sources within the conductivesublayer, the development of y is initially proportional to(x/h)1/2 and proportional to (x/h)3/2 at larger distances from thesource. It is also noted that the (x/h)1/2 region is longer forsmaller Pr (Figure 11a). For sources within the transitionregion and for Pr � 10, the behavior of y is better describedwith two equations (Figure 11b): one for the region near thesource (x/h 10) and one for the region far from the source(x/h � 10). For sources outside the viscous wall region, thebehavior of y can be described with one equation for Pr � 3(see Figure 11c). Figures 12a–12c present the plume half-widthfor the Couette flow channel. Similar to Poiseuille flow, Eq. 5is found to describe the data. For sources within the conductivesublayer, y � (x/h)2/3 initially, and y � (x/h)3/2 at largerdistances from the source (Figure 12a). The half-plume growthrate, for Couette flow sources outside the viscous layer, exhib-its an exponential dependency on x/h similar to that for the

plane channel case. However, as seen from Figures 12b and12c, the values of the parameter a are higher for Couette flow,indicating a higher dispersion rate for the Couette flow plumes.

Correlation coefficients and “material timescales”

The material autocorrelation coefficient can be calculatedsimilarly to the material correlation defined by Saffman3 as

RViVj�t, to� �V�i�t � to�V�j�to�

�V�i2�t � to��

1/ 2�V�j2�to��

1/ 2i, j � x, y, z

(6)

The marker velocity at the time of marker release is used forthe calculations of the autocorrelation coefficient presented inFigure 13. The overbar denotes ensemble average over the total

Figure 10. Comparison of the concentration profile re-sulting from an elevated source with theDNS/LST method and experimental measure-ments.(a) Comparison to the mass transfer experiments of Fackrelland Robins6; (b) comparison to heat transfer experiments byShlien and Corrsin.5

Figure 11. Plume half-width for Poiseuille channel flow.(a) yo � 5; (b) yo � 15; (c) yo � 28.5.


number of markers in the flow field and the prime denotes

Lagrangian velocity fluctuations, Vi � Vi(t) Vi� (t). Note here

that Vi(t) is the velocity of a fluid particle that is located at thesame point as the scalar marker, as mentioned above in the LSTsection.

The streamwise-streamwise, RVxVx(t, to); the normal–normal,

RVyVy(t, to); and the spanwise–spanwise, RVzVz

(t, to), correla-tions are shown in Figures 13a, 13b, and 13c, respectively, forthe case of channel flow and a source within the logarithmicregion (yo � 75). Figure 13 shows that the material autocor-relation coefficient does not depend on Pr for Pr � 3 forsources in the logarithmic region of the channel flow. A mea-sure of the difference between the material autocorrelationcoefficient and the usual Lagrangian coefficient that can becalculated along the trajectories of fluid particles can be ob-tained by comparing the values in Figure 13 with the values ofRViVi

for the highest Pr shown (Pr � 50,000) because fluid

particles behave like markers with Pr 3 . For markers withsmall Pr, the coefficients are smaller, meaning that moleculardiffusion quickly moves these markers off large fluid flowstructures. In this respect, the effectiveness of turbulence mix-ing is diminished because the markers do not follow the tur-bulent flow eddies. It is important to note here that, althoughthe effectiveness of turbulence to mix is diminished, the overalleffective dispersion is enhanced when the Pr is small. Theconcept proposed by Saffman,3 that the overall effective dis-persion will be diminished as a result of molecular effects, isapplicable to very small times, that is, t 3 0. Also, Figure 13shows that total dispersion in the normal direction is morerapid, given that RVxVx

� RVzVz� RVyVy

. Similar results arefound for the Couette flow correlation coefficients, shown inFigure 14. However, the correlation coefficients are smaller forthe Couette flow case, indicating more efficient total disper-sion.

Figure 12. Plume half-width for Couette flow.(a) yo � 5; (b) yo � 15; (c) yo � 28.5.

Figure 13. Material correlation coefficients as a functionof Pr for markers released in Poiseuille flow.(a) RVxVx

, yo � 75; (b) RVyVy, yo � 75; (c) RVzVz

, yo � 75.


Figures 15a and 15b present the Lagrangian timescale as afunction of source elevation for dispersion in the y-directionthat can be calculated as

�Ly � �to

RVyVy�t, to�dt (7)

for Poiseuille and Couette flow, respectively. This timescalemay be called the material timescale to differentiate it from theterm Lagrangian timescale, which is usually reserved for thetimescale of fluid particles moving without molecular diffu-sion. The material timescale in this case (that is, inhomoge-neous, anisotropic turbulence) is expected to depend on thesource elevation and, to some degree, on the Pr. The materialcorrelation coefficient can be written as the material autocor-

relation coefficient for the case of homogeneous turbulence,(RVyVy

)H, modified by a factor representing the effects of inho-mogeneity. This factor may be assumed to have two compo-nents, one that depends on the location of the source, (RVyVy

)yo,

and a second one that includes the effects of the Pr, (RVyVy)Pr.

The material correlation coefficient may then be written as

RVyVy � �RVyVy�H�RVyVy�yo�RVyVy�Pr (8)

The factor (RVyVy)H is usually assumed to follow an expo-

nential function12,43

�RVyVy�H � et/� (9)

which gives � � �Ly when combined with Eq. 7 for the case of

homogeneous turbulence. Considering that the term [Vj2�to��

]1/2

in the denominator of Eq. 6 is equal to the root mean square ofthe velocity fluctuations in the direction normal to the walls ofthe channel, v, and keeping a first-order dependency on thesource location of the modification factor that accounts for theelevation of the source, one may write

�RVyVy�yo � A1 A2�yo

h � �1

��yo

(10)

where A1 and A2 are constants. The physical meaning of A1 isthat (RVyVy

)yo�0 � A1, so that Eq. 8 does not yield a zero valueat yo � 0. Note also that v depends on the distance from the

Figure 14. Material correlation coefficients as a functionof Pr for markers released in Couette flow.(a) RVxVx

, yo � 75; (b) RVyVy, yo � 75; (c) RVzVz

, yo � 75.

Figure 15. Material timescale as a function of the eleva-tion of the point of release.(a) Poiseuille flow (MH � data from Mito and Hanratty10);(b) Couette flow.


wall, in the form v � (y/h)s for small y (that is, y 30, whereasfor y 3 0 incompressibility implies s � 2), and v is almostconstant for large y (that is, y � 50; see Figure 1b). Equation10 can then be written as

�RVyVy�yo � A1 A2�yo

h �1s

(11)

where s is zero at large yo. The Pr-dependent factor can beassumed to follow an exponential

�RVyVy�Pr � b1Prb2 (12)

The exponential form can be justified because a Lagrangiancorrelation coefficient is related to the eddy diffusivity, and theeddy diffusivity in wall turbulence has an exponential depen-dency on the Pr (see Churchill41), especially for small Pr.Substituting Eqs. 9, 11, and 12 into Eq. 8 yields

RVyVy � b1Prb2et/��A1 A2�yo/h�1s� (13)

Mito and Hanratty10 studied the behavior of markers re-leased at different elevations in a Poiseuille flow channel tocalculate the associated material timescales (Pr � 0.1, 0.3, 1,and ). Their goal was to use these timescales to solve amodified Langevin equation for the prediction of the velocityfield along the trajectories of the markers, and subsequently usethat velocity field to predict the marker trajectories (in otherwords, they substituted the DNS velocity in an LST procedurewith the velocity field resulting from the solution of the Lan-gevin equation). The results in Figure 15a are in agreementwith the findings of Mito and Hanratty10 that there is a Pr effectfor small Pr and that this effect is more pronounced close to thewall. The Couette flow results exhibit similar qualitative be-havior. However, the values of the material timescale aresmaller than those for Poiseuille flow. Substitution of Eq. 13into Eq. 7 shows that the values of �Ly can be modeled with anequation of the form

�Ly � B1Prb2�A1 A2�yo/h�1s� (14)

where B1, b2, A1, A2, and s are parameters that can be deter-mined using regression. For medium and high Pr (Pr � 3) thereis no Pr dependency (b2 � 0), and the following relations areobtained:

Poiseuille Flow

�Ly � 4.59 33.04� yo/h�0.87 R2 � 0.988 (15a)

Couette Flow

�Ly � 1.83 24.85� yo/h�0.58 R2 � 0.993 (15b)

For small Pr, the material timescale exhibits a dependencyon Pr, and the following relations are obtained:

Poiseuille Flow

�Ly � �0.98 Pr0.13��4.59 33.04�yo/h�0.87� R2 � 0.997

(16a)

Couette Flow

�Ly � Pr0.083�1.83 24.85�yo/h�0.58� R2 � 0.992

(16b)

The regression results are shown in Figure 15. Note that the Prdependency is stronger for Poiseuille flow (b2 � 0.13 in Eq.16a) than for Couette flow (b2 � 0.083 in Eq. 16b). ComparingEq. 14 with the above, the value of the exponent s is 0.13 forPoiseuille flow and 0.42 for Couette flow (a power approxima-tion to the v profile yields values of s equal to 0.11 and 0.36 forchannel flow and Couette flow, respectively, based on the datashown in Figure 1b).

The differences observed in Figure 15 between the two typesof flow can be investigated further by examining the spectrumof the Lagrangian velocity covariance C(t),

Cij�t� � RViVj�t, to��V�i2�t � to��

1/ 2�V�j2�to��

1/ 2 i, j � x, y, z

(17)

which can be defined as

E�w� �1

� �

�

C�t�eiwtdt (18)

The spectra E(w) of the Cyy material covariance for markersreleased inside the viscous wall region are shown in Figures16a and 16b for the cases of channel and Couette flows,respectively. Figures 17a and 17b present the spectra for thecase of markers released in the outer region of the channel. Asmentioned earlier regarding Eq. 9, the Lagrangian autocorre-lation coefficient is usually assumed to follow an exponentialfunction12 of the form Ri � exp(t/�i) that is appropriate forhomogeneous, isotropic turbulence and gives Ri � 0.368 whent � �i. The spectrum that results using this function in Eq. 17is also shown in Figures 16 and 17 (designated as “analytical”)to show the effects of the turbulence structure on heat transfer.For release close to the wall (Figure 16), the differences in thespectra for different Pr are significant for small and medium Pr.The spectra become Pr independent for high Pr (�100). Thereare also differences between the actual spectrum and the ex-ponential function arising from the presence of the wall and theanisotropies introduced by the wall.

For release in the logarithmic region (Figure 17), the spec-trum shows differences at small wavenumbers for different Pr,indicating differences in the contribution to heat transfer aris-ing from large turbulence scales. It is seen that the reason forwhich the mechanism of heat transfer depends on the Pr, whenit does so, is that the turbulence velocity structures that con-tribute to heat transfer are different in each case. Integrating thespectrum E(w) up to the first 10 frequencies (w � 0.078125),and dividing this integral to the value of the same integral for


Pr � 50,000, shows a ratio of 0.84 and 0.78 for channel flowand Pr � 100 and 0.7, respectively, when the point of releaseis at yo � 75. For release in the center of the channel (yo �150), the value of the ratio is 0.88 and 0.84 for Pr values of 100and 0.7, respectively. However, such differences are muchsmaller for Couette flow. In Couette flow, for release at yo �75, the ratio is 1.0 and 0.96 for Pr values of 100 and 0.7,respectively. For release at yo � 150, the value of the ratio is1.0 and 0.97 for Pr values of 100 and 0.7, respectively. Itappears that within the constant-stress region (which coversalmost the whole channel in the Couette flow case) all thevelocity scales contribute almost equally to the dispersion ofheat, apart from the case of low Pr, that is, Pr � 0.1. Further-more, the values of the spectral function are higher for theCouette flow case, showing that similar frequency structurescan be more dispersive in Couette flow than in channel flow.The fundamental reason for the observed differences in disper-sion between different types of flow appears to be the intensityof the large-scale velocity events (that is, the magnitude of thefluctuations), but not their duration.

Conclusions

The behavior of instantaneous and continuous line sources ofheat or mass at different locations of turbulent plane Poiseuilleand plane Couette flow has been investigated in this work usingLagrangian scalar tracking. The effect of the turbulence struc-ture and of different Pr in turbulent transport has also beenstudied. The range of the fluid molecular Prandtl numberextended from 0.1 to 50,000. The numerical results agreed well

with previous experimental measurements for the case of aplume, as well as with previous Lagrangian simulations for thecase of the Lagrangian timescale for channel flow.

With respect to the turbulent dispersion dependency on Pr, itwas found that it is important when the molecular Pr is com-parable to the turbulent Pr, in agreement with previous workand theoretical expectations. The material autocorrelation func-tion and the associated material timescale were Pr independentfor Pr � 3. However, when the source location was close to thewall and, more specifically, within the viscous wall sublayer,the effects of Pr were significant. The assumption that theLagrangian velocity of scalar markers is independent of Prwithin the constant stress region was found to be invalid.Descriptive correlations for the material timescale have beencalculated (Eqs. 15 and 16) for low and high Pr. It was alsofound that dispersion is stronger in the plane Couette flow caserelative to the plane channel flow, indicating enhancement ofturbulent dispersion in the constant stress region and thus in thelogarithmic region. The difference is attributed to the large-scale velocity events that contribute to heat transfer. In general,turbulent dispersion is different in different turbulent velocityfields because the large scale structures of these turbulentvelocity fields are different.

AcknowledgmentsThe support of NSF under CTS-0209758 is gratefully acknowledged.

This work was also supported by the National Computational ScienceAlliance under CTS990021N, used the NCSA IBMp690 and the NCSASGI/CRAY Origin2000. Computational support was also offered by the

Figure 16. Spectrum of the material autocorrelation co-efficient RVyVy

for high Pr and markers re-leased inside the viscous wall region (yo � 1).(a) Poiseuille flow; (b) Couette flow. The lines marked“Analytical” show the spectrum of RVyVy

� exp(t/�Ly).

Figure 17. Spectrum of the material autocorrelation co-efficient RVyVy

for high Pr and markers re-leased inside the logarithmic region (yo � 75).(a) Poiseuille flow; (b) Couette flow. The lines marked“Analytical” show the spectrum of RVyVy

� exp(t/�Ly).


University of Oklahoma Center for Supercomputing Education and Re-search (OSCER).

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Manuscript received Feb. 11, 2004, and revision received Jan. 12, 2005.


turbulent dispersion from elevated line sources in channel and couette flow

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