2-D Particle Tracking to Determine Transport Characteristicsin Rivers with Dead Zones

Volker Weitbrecht1, Wim Uijttewaal2 & Gerhard H. Jirka1

1 Institute for Hydromechanics, University of Karlsruhe2 Hydromechanics Section, Technical University Delft

A Lagrangian-Particle-Tracking-Method (LPTM) has been developed to determine the influence of dead waterzones on the transport characteristics in natural rivers. Hereby, the local processes at single dead water zoneshave been investigated with the help of laboratory experiments. The resulting information, such as storagetime, velocity distribution, and distribution of the diffusivity has been implemented into the LPTM. The methodhas been tested by comparing the results of the simulation with analytical solutions to the one-dimensionaladvection-diffusion equation. It could be shown that in the presence of large dead water zones at the river banks,an equilibrium between longitudinal shear and transverse diffusion can be reached if the morphologic conditionsdo not change. The simulations are leading asymptotically to a Gaussian distribution of the cross sectionalaveraged concentration in longitudinal direction. The transport velocity resulting from LPTM corresponds to thetransport velocity predicted by the one-dimensional dead-zone-model only for certain geometrical conditions,which correspond to a fixed exchange coefficient. The transverse distribution of tracer material is influenced bythe presence of dead water zones.

1 IntroductionThe prediction of pollutant transport in rivers is im-portant for the appropriate management of water re-sources. In rivers with strong morphological hetero-geneities, the prediction is difficult and needs to beimproved. The River-Rhine-Alarm-Model (Spreaficoand van Mazijk 1993) has been developed by the”International Commission for the Hydrology of theRiver Rhine” (CHR) and the ”International Commis-sion for the Protection of the Rhine” (ICPR). For thiskind of predictive model, much effort and expensemust be spent on calibration by means of extensivein-situ tracer measurements (van Mazijk 2002). In thecase of the River Rhine Alarm Model, which usesa one-dimensional analytical approximation for thetravel time and concentration curve, a dispersion co-efficient and a lag coefficient have to be calibrated.The model works well for cases of similar hydro-logical situations. However, variations in discharge,and thus, changes in water surface levels, lead to in-creased errors if the same calibrated parameters areused for different hydrological situations. Insufficientknowledge about the relation of river morphology andtransport processes are the reason for these uncertain-ties. Hence, predictive methods that are appropriatefor variable flows and therefore morphological condi-

tions are needed.In the present work we focus on the influence of

dead water zones, such as groin fields, on the dis-persive mass transport in the far field of pollutant re-leases. Longitudinal dispersion in rivers that can betreated as shallow waters is controlled by two pro-cesses. First, the longitudinal stretching due to thehorizontal depth averaged velocity shear, and sec-ond, transverse homogenization by turbulent diffu-sion (Fischer et al. 1979). Therefore, detailed veloc-ity and concentration measurements have been per-formed in the laboratory in order to determine typ-ical flow patterns and local mass transport phenom-ena. Direct measurements of dispersion coefficientsare problematic because the dispersive character of atransport phenomena reaches its final behavior onlyafter a very long travel time (Fischer et al. 1979),which is determined by the width of the flow andthe intensity of the transverse turbulent diffusion. Inmost cases laboratory flumes are much too short, toexamine longitudinal dispersion in the far field. To ad-dress this problem, laboratory and numerical experi-ments have been combined in such a way that the ef-fect of local phenomena that have been measured, aretranslated into the behavior of tracer clouds in the farfield with the help of Lagrangian-Particle-Tracking-

1

Int. Symp. on Shallow Flows, Delft, 2003

Method (LPTM).The flow measurements have been performed in

a laboratory flume (20m long, 1.8m wide) with anadjustable bottom slope. Groins were places on oneside of the channel. With the help of Particle-Image-Velocimetry (PIV) measurements, mean flow veloc-ity profiles as well as the distribution of the turbulentflow properties across the channel cross section wereobtained. Due to the fact that the flow in this system isvery shallow (channel width / water depth ≈ 30), thevelocity information at the water surface is sufficient,to describe the mean behavior of the flow. Thereforethe PIV measurements have been performed at thewater surface, using floating particles that are homo-geneously distributed with the help of a particle dis-penser (Weitbrecht et al. 2002).

Concentration measurements were performed at asingle groin field in order to determine exchange ratesbetween groin field and main stream. These measure-ments are similar to the experiments presented by Ui-jttewaal et al. (2001). The results, obtained from thesemeasurements are now used in the LPTM, to simulatetracer clouds in the far field of the tracer source.

2 Method and ValidationA Lagrangian-Particle-Tracking-Method (LPTM) hasbeen developed, to analyze the influence of morpho-logic heterogeneities on the mass transport in natu-ral rivers. It represents a random walk problem (Sul-livan 1971) based on statistical mechanical transporttheories presented by Taylor (1921). The behavior ofdiscrete particles under the influence of advection inlongitudinal direction and of transverse diffusion isdetermined in a two-dimensional domain. The idea isto initiate a cloud of particles that is advected withina known mean flow profile. This advective movementis superimposed by a random movement in transversedirection representing turbulent diffusion.

The characteristic transport parameters, like disper-sion coefficient, transport velocity and skewness coef-ficient, can be determined by analyzing the statisticsof such a particle cloud. The influences of groin fieldsare included with the help of extra boundary condi-tions that represent the mean retention time of par-ticles in the area of dead water zones. Herewith, thismethod represents the key to transfer experimental re-sults, obtained in a short flume (short, with respectto the length of the advective zone) at a single groinfield to a system of many groin fields. In other words,the influence of different groin field geometries on themass transport, especially on the dispersion, in the farfield of a pollutant spill can be predicted.

The essential part of such a simulation is the track-ing of a sufficient large number of discrete particles,whose displacements are governed by the followingprinciple:

xnew = xold + (∆t · u(y))︸ ︷︷ ︸

deterministic

(1)

ynew = yold + Z√

2Dy(y)∆t︸ ︷︷ ︸

stochastic

(2)

Where xold, yold and xnew,ynew are the spatial loca-tions at times t and t + ∆t respectively, and Dy is adiffusion coefficient. The function u(y) denotes themean flow velocity in relation to the position in trans-verse direction. The random movement of each par-ticle lies in the properties of Z. In this case, Z isa normal distributed variate with a mean quantity ofzero and a variance equal to one. Thus the spread-ing in transverse direction y is scaled with

√

2Dyt de-scribing the standard deviation of mass displacement,which has been defined for a Fickian type of diffusion(Hathhorn 1997). Consequently, in every time step, aparticle moves convectively in x-direction dependingon the velocity profile and does a positive or negativediffusive step in transverse direction.

The reason why there is no diffusive step in the x-direction needed (Eq. 1), can be explained by thefact, that turbulent diffusion and longitudinal diffu-sion are additive processes (Aris 1959), which meansthat the final dispersion coefficient can be adjustedby adding the turbulent diffusion coefficient. Fischeret al. (1979) showed that in natural rivers the coef-ficient of longitudinal dispersion EL lies within therange of 30 < DL/(u?h) < 3000, while the longi-tudinal turbulent diffusion coefficient Dx is consid-erably smaller, approximately Dx ≈ (0.6u?h). Thus,turbulent diffusion in longitudinal direction can be ne-glected in this approach.

A problem in performing LPTM-simulations isgiven by the fact that in flows with inhomogeneousturbulent diffusion coefficients, particles segregateinto regions of low diffusivity. The reason for that canbe explained by looking at the governing equations. Inthe stochastic model there is no relation between par-ticles moving from regions with high diffusivity in re-gions with low diffusivity, which should be the case inorder to satisfy continuity. That means that the proba-bility of a particle to move from a region of high diffu-sivity to a region of low diffusivity is higher than viceversa. Thus, an extra advection term in y-direction hasto be included, to achieve consistency with the gov-erning advection-diffusion-equation. This extra termis called the noise-induced drift component (Dunsber-gen 1994), and can be determined as follows. First ithas to be decided if the deterministic step is done firstduring every time step, or the stochastic jump, whichis known as the Ito-Stratonovich dilemma (van Kam-pen 1981). In the current work the Ito interpretation

2

Int. Symp. on Shallow Flows, Delft, 2003

has been chosen, which means, that the deterministicstep is performed before the stochastic jump. Duns-bergen (1994) showed, that in this case, the noise-induced drift component ∆yn can be derived as fol-lows

∆yn =∂Dy

∂y∆t (3)

If Eq. 2 is extended with the given expression forthe noise-induced drift component (Eq. 3), it can bestated that the problem is described consistently withthe advection-diffusion-equation.

Of particular importance are the boundaries of ourcalculation domain and how they act on the particles.The inflow and outflow boundaries do not affect theparticles as in our case the domain has an infinitelength. In y-direction we can find two possibilitiesfor the boundaries. If we analyze the effect of verti-cal shear on dispersion, we have the water surface aswell as the channel bottom. In case of horizontal shearthe boundaries represent the channel banks. For bothsituations the boundaries act as reflective walls. Thismeans, that particles which would cross the upper orlower boundary at a certain time step are reflected intothe calculation domain (Fig. 1).

x,yold

Reflective-Boundary

a a

x,ynew

Figure 1: Schematic visualization of a particle tryingto cross a reflective boundary in one time step

One important issue of this work is to determine theinfluence of dead water zones given by groin fields(Fig. 2) on the mean transport characteristics of atracer cloud. A particle that enters a dead water zonedoes not move on average in x-direction, if we pre-sume that the longitudinal extension of the dead waterzone is small compared to the length of the modeledriver section. The particle remains in the dead waterzone for a certain period, which is called the retentiontime Ta, and after that period it gets back on averageinto the main stream.

In this approach we make the assumption, that theinfluence of dead water zones can be captured glob-ally by the retention time Ta, which can be imple-mented using a modified boundary condition (Fig. 3).Thus, the channels walls have to act as a transient-adhesion boundary, which means that particles thatreach such a boundary are fixed to that position un-til Ta has passed.

Main channelFlow direction

Channel center line

River bank

Groin fieldL

W

B

x

y

Figure 2: Definition of geometrical parameters in ariver section with groin fields

x,yold

x,yold

time = ti

time = t + Tai

x,ynew

Transient-Adhesion-Boundary

Transient-Adhesion-Boundary

Mean flow

Mean flow

x,ynew

Figure 3: Schematic visualization of a particle tryingto cross a transient adhesion boundary at time = ti andtime = ti + Ta, where Ta = mean residence time

The outcome of a LPTM-simulation are x and y-positions of every single particle at different timesteps, that can be used to retrieve different infor-mation about the transport characteristics. The one-dimensional longitudinal dispersion coefficient DL,which is a measure of the spatially averaged spread-ing rate of a tracer cloud, can be determined by cal-culating the change of the longitudinal variance of theparticle distribution as follows

DL =1

2

σ2

x(t2)− σ2

x(t1)

t2 − t1(4)

A second result will be the skewness Gt of the par-ticle cloud, which can be used as an indicator for thelength of the advective zone, in order to define whenit is acceptable to apply the Taylor solution to a pol-lutant transport problem.

Also, the transport velocity Ct of the tracer cloud,defined as the velocity of the center of mass, can bedetermined using this approach. In the case of regularchannel flow with ordinary reflective boundary condi-tions C is equal to the mean velocity.

The method has been validated with the help of dif-ferent analytical solutions of the advection-diffusion-equation, by comparing the dispersion coefficients,obtained from LPTM-simulations are compared withthe predicted dispersion coefficients from analyticalsolutions. As an example, the analytical result given

3

Int. Symp. on Shallow Flows, Delft, 2003

by Elder (1959) is compared to the results obtainedby LPTM-simulations.

In the case that has been described by Elder, ahomogeneous flow is assumed in an infinitely widechannel. The velocity profile over the vertical is de-scribed by the logarithmic law

u(y) = −u?

κlog(1− y/h) (5)

where u? is the shear velocity, given by√

τo/ρ, h isthe water depth and κ is the von Karman constant0.41. The distribution of the diffusivity is determinedby the velocity gradient leading to

Dy(y) = κu?y(1− y/h) (6)

By inserting Eq. 5 and 6 into the analytical expressionfound by Taylor(1954) for the determination of DL

DL =−1

h

∫ h

0

u′

∫ z

0

1

Dy

∫ z

0

u′dzdzdz (7)

the following expression for DL can be found by eval-uating the three integrals

DL = 5.86u?h (8)

To adjust the diffusive-noise drift component,which has been described above, for the varying dif-fusivity in space, it has to be derived by the differen-tiation of Eq. 6, leading to

∂Dy

∂y= κu?(1− 2y/h) (9)

0

0.5

1

y/h

i)

0 0.002 0.004 0.006 0.008 0.01 0.0120

0.05

0.1

x/h

Part

icle

den

sity

ii)

Figure 4: Result of LPTM-simulation after first timestep. i) Particle position, showing the log law; ii) par-ticle distribution in longitudinal direction.

In order to compare the results of the analytical so-lution with the output of the LPTM, a simulation hasbeen performed, where 10000 particles were released

0

0.5

1

y/h

i)

0 50 100 150 200 2500

0.05

0.1

x/h

Part

icle

den

sity

ii)

Figure 5: Result of LPTM-simulation after 1 ·105 timesteps, corresponding to a mean displacement of 120water depths, i) Particle position; ii) particle densityin longitudinal direction.

Table 1: Parameter values for the Elder simulationProperty Valuewater depth h [m] 1.0u(y) velocity profile (Eq. 5)No. of particles 10,000No. of time steps 100,000∆t length of time step [s] 0.0005Dy diffusivity [m2/s] (Eq. 6)

homogeneously distributed over the river cross sec-tion at x = 0. The boundary conditions for this simu-lation are summarized in Tab. 1.

In Fig. 4 and Fig. 5 the particle cloud and the dis-tribution of the particles in longitudinal direction arevisualized after the first and the last time step of thesimulation. In Fig. 4 the velocity profile is clearly vis-ible after the first time step because the diffusive stepin transverse direction is small compared to the ad-vective step in x-direction. It can be seen that the ini-tial distribution of the particles in longitudinal direc-tion is strongly negatively skewed, with a strong ris-ing limb and the typical tailing. In Fig. 5 the particlesare homogeneously distributed over the water depth.The velocity distribution has been smeared out. Theparticle distribution in longitudinal direction is closeto a Gaussian distribution.

In Fig. 6 the evolution of the longitudinal dis-persion coefficient and of the skewness Gt is plot-ted. This DL-curve shows, that the equilibrium be-tween longitudinal stretching and transverse diffu-sion, which corresponds to a linear growth of thetracer cloud is reached after approximately 12 wa-ter depths. At this distance, DL has reached its finalvalue, which is close to the analytical prediction byElder (Eq. 8) with a value of 5.86. The DL curve hasbeen smoothed, in order to show the mean behavior of

4

Int. Symp. on Shallow Flows, Delft, 2003

0 50 100 150−5

0

5

10

x/h

5.86

Gt

D

L/u

*h

DL

Gt

Figure 6: Evolution of the longitudinal dispersion co-efficient DL and the skewness Gt during the LPTM-simulation. DL is smoothed with a sliding average fil-ter of increasing window size.

the tracer cloud. Without filtering an increased scat-ter could be observed, due to the increasing error thatis produced by determining the standard deviation σ(Eq. 4) that gets larger with increasing width of thetracer cloud.

3 APPLICATION AND RESULTSIn this section the LPTM is applied to different flowfields, that have been investigated in the laboratoryin order to determine the influence of river hetero-geneities on the mass transport properties of a river.Three different cases will be analyzed in detail. Firstthe behavior of the dispersive character of pure chan-nel flow without the influence of groin fields. In a sec-ond step groin fields are implemented, and finally theinfluence of different residence times on the transportcharacteristics is determined.

3.1 Straight Open Channel FlowA LPTM-simulation has been performed, in orderto analyze transport phenomena in regular channelswithout groin fields. Therefore the measured velocitydistribution has been approximated with an analyticalfunction, that can be seen in Fig. 7 (i). The diffusioncoefficient in this case has been chosen to be constantover the whole river cross section, with a value ac-cording to Fischer et al. (1979) for regular channels

Dy = 0.15u?h (10)

In Tab. 2 the properties of the flow and the settingsof the LPTM-simulation for the case of pure channelflow are listed.

In that case the evolution of skewness and disper-sion coefficient is in principle very similar to Elder’scase, shown in Fig. 6. The equilibrium between lon-gitudinal stretching and transverse diffusion, wherethe dispersion coefficient does not change any moreis reached after 800 times the channel’s half width.The final value of DL/(u?h) is 175, and is therefore30 times larger than in Elder’s case.

Table 2: Parameter values for the pure channel flowsimulation, taken from the measurements

Property Valuewater depth h [m] 0.046Channel half width B [m] 1Mean Velocity [m/s] 0.19Velocity profile Fig. 7 (i)Channel slope I [%] 0.032No. of particles 5,000No. of time steps 40,000length of time step ∆t [s] 1diffusivity Dy [m2/s] (Eq. 10)

0

0.5

1

y/B

i)

LPTMMeasured Values

0 0.05 0.1 0.15 0.20

0.05

0.1

0.15

0.2

x/h

Part

icle

den

sity

ii)

Figure 7: Result of LPTM-simulation with pure chan-nel flow, after first time step. i) Particle position ac-cording to the fitted velocity profile and the measuredvelocities taken from the experiment; ii) particle den-sity in longitudinal direction.

3.2 Channel Flow with Groin FieldsThe influence of groin fields is simulated with thetransient-adhesion boundary condition (Sec. 2), thatrepresents the mean residence time of a particle inthe groin field. These mean residence times have beenmeasured in this study with two different approaches.Concentration measurements have been performed,where concentration decay in a single groin field hasbeen tracked with digital video analysis. Starting witha known homogeneous concentration in the groinfield and zero concentration in the main stream an ex-ponential decay could be observed, leading to a typi-cal time scale Ta describing the mean residence time.These experiments are in principle analogous to themeasurements that have been performed by Lehmann(1999). An improvement could be achieved by thedevelopment of a multi-port injection device, that isable to produce reproducible homogeneous concen-tration fields as initial condition for the concentrationmeasurement (Kurzke et al. 2002). Another possibil-ity to determine the residence times is given by usingthe velocity fields for the determination of the mass

5

Int. Symp. on Shallow Flows, Delft, 2003

exchange rate between groin field and main stream(Kurzke et al. 2002).

In order to determine the influence of groin fields, aLPTM-simulation has been performed with the sameflow properties as in the described case above (Tab.2). The difference is the transient-adhesion boundarycondition at the channel wall. This simulation repre-sents groin fields, where the ratio between the widthW of a groin field divided by the length L is 0.4 (Fig.2). These conditions correspond to laboratory mea-surements with a groin field length 1.25 m and a widthof 0.5m. As the water depth in the groin field and themain stream is the same, the ratio between the crosssectional area of the dead water zone and the mainstream is 0.5. The mean residence time Ta of a tracerparticle has been set to 90 seconds, which correspondsto the measured dimensionless exchange coefficient

k =W

TaU= 0.028 (11)

where W is the width of the groin field and U repre-sents the mean flow velocity in the main channel.

In the case of channel flow with groin fields the ve-locity profile is slightly changed compared to the purechannel flow, because in the presence of groin fieldsthe velocity profile has to represent the mixing layerbetween groin field and main stream and can be ap-proximated with a tanh function, where the velocityis greater than 0 at y = 0 (Fig. 2).

The distribution of the diffusivity in that case is notconstant over the channel cross section. Turbulencemeasurements showed, that the velocity fluctuationsin the region of the mixing layer between groin fieldand main channel are much stronger than in the undis-turbed main channel. Therefore the diffusivity givenby Eq. 10 is amplified in the region of the mixinglayer, proportional to the increasing transverse veloc-ity fluctuations. This has been done, by fitting a gaus-sian curve to the transverse rms-values of the chan-nel flow, such, that the diffusivity in the mixing layeris three times larger than in the main channel (Weit-brecht et al. 2002).

0 5 10 15 20 25 30 35 400

0.5

1

y/B

x/B

W/B = 0.5

Figure 8: Result of LPTM-simulation after 200 timesteps with a mean residence time of Ta = 90s, thatcorresponds to a width to length ratio of a groin fieldof 0.4.

In Fig. 8 the LPTM-simulation with groin fields isvisualized after 200 time steps. The main difference

with respect to pure channel flow are particle cloudsthat travel far behind the main tracer cloud, which isan phenomena, that can also be seen in the laboratoryflume. These small particle clouds arise by the effectof the dead water zones. Particles that have crossedthe lower boundary layer during the simulation, re-main at the same x-position for the mean residencetime Ta. After Ta has elapsed the particles get backto the flow. The mean distance between those cloudscorresponds to the mean residence time Ta.

The final stage of mixing in the case with groinsshows again, that after a long period the tracer cloudgets close to gaussian distribution in longitudinal di-rection Fig. 9.

0

0.5

1

y/B

i)

0 1000 2000 3000 4000 5000 6000 7000 80000

0.05

0.1

x/h

Part

icle

den

sity

ii)

Figure 9: Result of LPTM-simulation after 40000time steps with a mean residence time of Ta = 90s,that corresponds to a width to length ratio of a groinfield of 0.4 and W/B = 0.5; i) particle position; ii)particle density in longitudinal direction

Interesting properties of this simulation are alsothe evolution of the dispersion coefficient and of theskewness, that are visualized in Fig. 10. The fi-nal value of DL/(u?h) is in that case approximately24.800 which is of a factor 140 higher than in the caseof pure channel flow, and about ten times higher thanthe expected values for natural rivers without groinfields . This can be explained by the ratio between thecross sectional area of the dead water zone and thecross sectional area of the main channel, which is 0.5for the experiment and rather high compared to natu-ral rivers.

The equilibrium between longitudinal stretchingand transverse diffusion is achieved after approxi-mately 800 times the channel width (Fig. 10), whichmeans that the advective zone has the same lengthcompared to the case of pure channel flow.

In Elder’s case and in the case of pure channelflow, the transport velocity c, which is defined as thetranslation velocity of the center of mass of the tracercloud, is always the mean flow velocity in the chan-

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Int. Symp. on Shallow Flows, Delft, 2003

0 2000 4000 6000 8000

−2

0

2

x/h

Gt

EL/u

*h 10

−4

EL

Gt

Figure 10: Evolution of Dispersion coefficient DL ·

10−4 normalized with the u? and the water depth hand the evolution of the skewness. DL is smoothedwith a sliding average filter of increasing windowsize. (W/B = 0.5)

nel. In case of the groin field flow the transport ve-locity decreases during the travel of the tracer clouduntil an equilibrium between the particles in the deadwater zones and in the main channel is established asseen in Fig. 11. In this simulation the transport veloc-ity does not change further after the tracer cloud hastravelled approximately 1000 times the channel halfwidth. The final transport velocity is 64% of the meanflow velocity in the main channel.

0 1000 2000 3000 4000 5000 60000

0.5

1

x/B

W/B = 0.5

C/U

Figure 11: Evolution of transport velocity c of a tracercloud normalized with mean velocity of the mainchannel.

The transport velocity in the far field of transportprocess, according to the one-dimensional dead-zone-model (Valentine and Wood 1979; van Mazijk 2002)can be determined with the ratio between the crosssectional area of the dead water zone and the crosssectional area of the main stream as follows

c =U

1 + Ad/As

(12)

with As = cross sectional area of the main stream andAd the cross sectional area of the dead water zone. Inthe presented case this relation would lead to a valueof 67% of the mean flow velocity, which is very closeto the result of the LPTM-simulation.

Looking at the particle distribution in transverse di-rection of the river cross section, it can be stated thatthe initial homogeneous distribution (Fig. 12, i) doesnot shift to any direction in the case of pure chan-nel flow. This behavior changes under the influenceof groin fields (Fig. 12, ii, iii, iv). The particle dis-tribution during the LPTM-simulation shifts towards

0 0.50

0.5

1x/B = 0

i)

y/B

x/B = 17

ii)

Particle density

iii)

x/B = 154

0 0.5

iv)

x/B = 5020

Figure 12: Particle density distribution in river cross-section as a function of the distance x/B.

the dead water zones. In the final stage of mixing (Fig.12, iv) almost 40% of the tracer material is distributedin the region of the dead water zones. The remaining60% of the material is travelling in the main stream.

3.3 Groin Fields with Different Exchange BehaviorThe dimensionless exchange coefficient k (Eq. 11),can be found in the literature to be in the order0.02 ± 0.01 (Valentine and Wood 1979; Uijttewaalet al. 2001), with no clear dependency to the shapeof the dead water zone. With our laboratory experi-ments, it could be shown, that k varies with the widthto length ratio of the groin fields within the rangeof 0.015 − 0.035. Hereby, the longest groin fields(W/L = 0.3) are leading to the highest k-values andthe shortest groin fields (W/L = 3.5) correspond tothe lowest k-value (Weitbrecht and Jirka 2001). In allcases the groin field width W was constant. Accord-ing to Eq. 12 the transport velocity should alwaysbe the same in that case. In order to demonstrate theinfluence of the changing k-values two more LPTM-simulations have been performed, where the meanresidence time has been set to 73 respectively 129 sec-onds. That means, the k-value has been changed from0.028 to 0.020 and 0.035, respectively.

Table 3: Results of LPTM-simulations with differ-ent exchange coefficients and B/W = 0.5, comparedwith the case of pure channel flow (k = 0).

k = 0 k = 0.035 k = 0.028 k = 0.02DL/(u?h) 177 19800 24800 32400c/U 1 0.68 0.64 0.56

In Tab. 3 the results of the simulations with chang-ing k-values are summarized. It can be seen thatthe dimensionless dispersion coefficient DL/(u?h) isdirectly proportional to the residence time Ta. Thelonger the residence time, the higher the stretchingrate of the tracer cloud.

It can be seen that the influence of the differentk-values on the transport velocities is at maximum10% for the presented configuration. According tothese simulations, the results of the one-dimensionaldead-zone-model mentioned above predicts the trans-port velocity correctly, for a k-value of about 0.035,which corresponds to groin fields with an aspect ratio

7

Int. Symp. on Shallow Flows, Delft, 2003

of about W/L = 0.33.An equilibrium between longitudinal stretching

and transverse diffusion is reached in all cases afterapproximately 800 times the channel’s half width B.That indicates that the dominant time scale is not Ta,but, the diffusive time scale B2/(2Dy) that describesthe time needed for a particle to cross B. If we deter-mine the length x/B that corresponds to this time, weget

x

B=

cB

2D= 750 (13)

which fits very well to the observed behavior of theparticle clouds during the LPTM-simulations.

4 CONCLUSIONSThe combined approach, using laboratory experi-ments and the LPTM makes is possible to determinetransport characteristics, in the far field of a pollu-tant spill in shallow , strongly two-dimensional riverflows. Detailed velocity and concentration measure-ments to determine local flow and transport phenom-ena in the presence of groin fields, can be translatedvia the LPTM, into transport velocities, longitudinaldispersion and skewness coefficients of the cross sec-tional averaged pollutant cloud in the far field. Thisinformation can be used for the improved predictabil-ity of one-dimensional alarm-models, in order to re-duce the need of calibration with tracer experiments.

The fact, that the transverse tracer distribution is in-fluenced by the presence of dead water zones shouldbe taken into account for the planning of future fieldexperiments, and for the interpretation of existingdata.

The computationally simple two-dimensional ap-proach allows for studying near field effects as well.Point releases of contaminants near one of the banksand not fully mixed states are impossible to predictwith a one-dimensional approach. The same holds forstrongly varying flow geometries with confluences,weirs, bends etc.

5 ACKNOWLEDGEMENTSThe numerical part of the project was funded by the”German Research Council” (DFG) and the ”Nether-lands Organization for Scientific Research” (NWO)Grant No. Ji18/8-1. The laboratory measurementswere sponsored by the German ”Federal Ministry forEducation and Research” (bmb+f) Grant No. 02 WT9934/9.

REFERENCESAris, R. (1959). The longitudinal diffusion coeffiecient

in flow through a tube with stagnant pockets. Chem-ical Engineering Science 11, 194–198.

Dunsbergen, D. W. (1994). Particle Models for Trans-port in Three-Dimensional Shallow Water Flow. Ph.

D. thesis, TU-DELFT, Faculty of Civil Engineering.

Elder, J. W. (1959). The dispersion of marked fluid inturbulent shear flow. Journal of Fluid Mechanics 5,544–560.

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