Longitudinal Dispersion Coefficient in Estuaries

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1 Journal of Coastal Research SI ICS (Proceedings) Brazil ISSN 79-8 Longitudinal ispersion Coefficient in Estuaries E. Jaari ; R. Bozorgi anda. Etemad-Shahidi College of Ciil Engineering Iran Uniersity of Science and Technology, Tehran, Narmak, Iran jaari@iust.ac.ir ABSTRACT JABBARI, E.; BOZORGI, R.; ETEMA-SAII A., 6. Longitudinal dispersion coefficient in estuaries. SI 9 (Proceedings of the 8th International Coastal Symposium), Itajaí, SC, Brazil, ISSN The determination of the longitudinal distriution of a sustance in natural channels is ased on the mass alance equation. In this equation the dispersion term accounts for the longitudinal mixing which results from the comined effects of turulent diffusion and the shear-induced elocity distriution in oth the transerse and ertical directions. In estuaries, due to the tidal action and large scale graitational circulation associated with salinity intrusion the process is more complex.two data collection campaigns were carried out during a neap tide and during a spring tide in the Scheldt estuary. Fie neasurement points were situated on a cross-section of the estuary and two others at the upstream and downstream of the cross-section. The measured quantities were elocity and its direction, conductiity, salinity, temperature and concentrations of sand and mud. The results of these measurements were used to inestigate whether the increase in the Taylor-Elder dispersion coefficient found y Fischer in riers due to the transerse elocity distriution is applicale to estuaries with usual large width to depth ratio. This research was also extended to find out the effect of flow oscillation in estuaries on dispersion and the possiility of prediction of the magnitude of dispersion coefficients in tidal waterways. AITIONALINEX WORS: Salt intrusion, mass alance equation, dispersion models. INTROUCTION In his expression for the dispersion coefficient in a twodimensional flow, Elder considers only the elocity ariation in the ertical direction. Natural channel flow is not twodimensional and it has een shown that the existence of transerse ariation of elocity can result in dispersion coefficients that are times igger than those gien y Elder's expression ( OLLEY et al. 97). The difference etween the predicted alues through the Taylor-Elder's approach and the osered alues of dispersion coefficient hae een studied extensiely. In fact the complexities of estuaries withstand ery rigorous analysis and the only promising approach is the empirical and the oserational one. The modified Elder model for dispersion has een used y ARLEMAN (97) for seeral estuaries. The predicted dispersion coefficients hae een consistently lower than coefficients osered in nature. e stated that the larger coefficients osered in estuaries are caused y a net stea ertical circulation induced y the longitudinal density gradient known as graitational circulation. FISCER (967, 97), stuing the dispersion of salt in the Mersey estuary introduced a model for prediction of dispersion coefficient. e showed that the difference etween the osered alues of dispersion coefficient and that from Elder analysis was more due to the transerse circulation which is induced partly y the oundary geometry and partly y the longitudinal density gradient. e stated that this transerse circulation was more important in causing dispersion than the ertical circulation. SMIT (976) has inestigated the case where the interaction of shear and longitudinal density gradient induced force produces secondary flows that significantly alter the magnitude of the dispersion coefficient. e proposed an equation for the dispersion coefficient ased on the solution of the salt transport equation and the momentum equation which includes the longitudinal density gradient terms. As analytical approaches, the Taylor-Elder's relation, the Prych's equation and the Smith's formulation are explained here and the alues of the dispersion coefficients oer a tidal cycle are presented and compared graphically. The approach of Fischer is applied to a cross-section of the Scheldt estuary, for calculation of the dispersion coefficient directly from salinity and elocity profiles in the ertical direction and with consideration of the effect of transerse shear on the dispersion coefficients y taking the ariations of salinity and elocity into account in the transerse direction. STUY AREA The Scheldt estuary is situated partly in Belgium and partly in the Netherlands. The water leel in this estuary is determined y the tidal ariations in the North sea. The tides penetrate up to Gent at aout 5Km from the mouth of the estuary. From the iewpoint of salinity structure, this estuary may e considered as a well mixed to partially stratified estuary. A one dimensional analysis of slat transport was performed oer the region Oosterweel-Baalhoek The model was calirated for the dispersion coefficient y checking the salinities produced y the simulation with the aailale data at the prosperpolder cross-section. This comparison showed that the dispersion coefficient required y the model was much higher than those otained from Taylor-Elder's relation. This relation is a asis for the inestigation of dispersion coefficient. The effect of transerse circulations in estuaries and longitudinal density ariatin on the dispersion coefficient was highlighted y using the aailale data of salinity and elocity distriution oer two cross-sections of the Scheldt estuary. It should e noted that since the aim of the measurement campaigns was somewhat different from the ojecties of this stu the aailale data resulting from the measurements were not as extensie as it should hae een for such an inestigation. Neertheless, it was tried to make the est use of the aailale information and it can e said that the results of this stu were conclusie with regard to the aailaility of data. ANALYTICAL STUIES OF ISPERSION COEFFICIENT The analytical approaches used for determining the dispersion coefficients are the Taylor-Elder's and Prych's relations for the dispersion due to the ertical circulation and the Smith's relation for the dispersion due to the ertical and horizontal circulations.

2 58 Jaari et al. Taylor-Elder's Relation The asic principles of dispersion in shear flow were first explained y Taylor in his pulication in 95 (Fischer 967, olley et al. 97, and yer 97). Taylor found that, after an initial transitional period, the dispersion process can e descried y the so-called Fickian diffusion equation. For a conseratie material dispersing in unidirectional flow n a straight circular tue, he showed that dispersion is caused y the shear elocity oer the wall of the pipe and the ariation of the longitudinal adection with depth. e relate the dispersion coefficient in a pipe to the radius of the pipe and the shear elocity as In which a is the radius of the pipe and u * / is the shear elocity and and are the wall shear stress and flow density respectiely. Elder ( BELTAOS 98) later applied Taylor analysis to the stea open channel flow. For a channel with trapezoidal cross-section, assuming a logarithmic elocity profile and constant flow conditions across the channel he found that In terms of elocity and Manning coefficient this equation is written as where n is the Maning's roughness coefficient. ARLEMAN (966) expressed this equation in a dimensionless form as in which R is the Reynolds numer. Elder's analysis depends on the effect of ertical elocity profile and assumes that flow conditions are constant across the channel. By assuming that the flow profile is logarithmic ut oscillatory in time, Bowden found analytically that, under certain assumptions, in oscillatory flow the dispersion coefficient should e one half of that in the equialent unidirectional flow (YER 97). For a tidal current of amplitude the longitudinal dispersion coefficient is The dispersion model of Elder was applied to a one dimensional analysis of the salt transport in a reach of the Scheldt estuary. Figure shows the ariations of the dispersion coefficient and of water-depth with time for a point situated in the middle of the reach. Prych's Equation From a similarity solution for the ertical elocity and salinity profiles ased on two dimensionless parameters (estuarine Richardson numer and estuarine Froude numer), which are empirically related to the parameters of the estuary, Prych otained a elation for the longitudinal dispersion coefficient as in which.au () * 5.9 u () * (5/ 6) 8.57 nu h () m n.57 / 6 R 8 ().5U (5) z x.( )(.K.KU f.9u f ) (6) K N t z and N g d dx and is the ed-iscosity coefficient for momentum in ertical direction and Uf is the fresh water current elocity. The equation of Prych was applied to the cross-section of the Scheldt estuary, where the measurement campaign was carried out. The results are shown in Figure. Smith's Formulation Through the analytical solution of the momentum and salt transport equation, S MIT (976) deried a formulation for the longitudinal dispersion coefficient in which he considered the secondary flow produced y the interaction of shear and the longitudinal density gradient (GUYMER and W EST, 99). The resulting equation for dispersion coefficient is Q C QQ K x K C Q Q K (7) A x K K C Q Q K... x K in which is width of the channel and K ispersion coefficient m /s ispersion coefficient m /s z t y 9 g yz tz 5 K. Q F y U y y () Q g z ispersion coefficients epthes Figure. ispersion coefficients from the Taylor-Elder's relation. Neap tide Taylor-Elder's relation 6 8 ispersion coefficients epthes Spring tide Talor-Elder's relation epth m epth m

3 Longitudinal ispersion Coefficient 59 ispersion coefficient m /s ispersion coefficient m /s ispersion coefficients epthes Figure. ispersion coefficients from the Prych's relation. In the aoe equations is the coefficient in the equation of state which relates the salt concentration to the difference etween the fresh water density and the density of salty water. () The parameters F and z are functions of channel shape, discharge, and ed iscosity and are defined as F AU The aoe formulation was applied to the cross-section where measurement were carried out and the results are shown in Figure for neap and spring tide. ETERMINATION OF TE LONGITUINAL SIPERSION COEFFICIENT etermination of the longitudinal dispersion coefficient was carried out y calculating it directly from the measured alues of salinity and elocity oer a cross-section of the Scheldt rier. The approach of F ISCER (97) was applied with the difference that Fischer'sformulation gies a gloal dispersion coefficient oer the whole tidal cycle y considering the deiation of the elocity and salinity from a mean cyclic alue. This part of the dispersion was not considered here. Only the two components of the dispersion coefficient due to the ertical and transerse shears were calculated at any time oer the tidal cycles where a complete data set oer the cross-section were aailale (GUYMER and W EST, 99). This was applied to the measurements of neap tide and spring tide. Fischer'sApproach (), F ISCER (97) studied the effect of transerse shear due to the lateral ariation of elocity and salinity y decomposing these ariales into a mean cross-sectional alue and the deiation from the mean alues in the ertical and in the transerse directions. First he decomposed the total mass transport in an estuary into the parts caused y the tidal aeraged motions and into the parts caused y the fluctuations from them. Since here the alues of the dispersion coefficient at one z Neap tide Prych's relation 6 8 ispersion coefficients epthes Spring tide Prych's relation epth m epth m ispersion coefficient m /s ispersion coefficient m/s Figure. ispersion coefficients from the Smith's relation. point in time in a cross-section of the channel is considered, this decomposition has not een performed. Considering the mass conseration equation for transport of a conseratie sustance, the instantaneous alues of elocity and salinity at a point in a cross-section may e diided into turulent mean terms and the turulent fluctuation terms for the diffusion part of the dispersion process. These longitudinal turulent fluctuations may e neglected due to the fact that compared to other mechanisms there is small. Taylor has shown that the turulent diffusion term has a alue less than % of the total dispersion ( OLLEY et al.97). The turulent mean term is written in terms of a depth mean alue and a depth deiation term as u u s s u s (8) The depth mean terms are written as a width mean alue and a width deiation term as u s u s t t u s t t (9) Integration of the mass alance equation oer a cross-section of the channel after sustitution of (7) and (8) gies ( As A) ( AU AsA) A ( utst us) () t x x The dispersion coefficient due to the ertical shear may e written as ispersion coefficients epthes us s x Neap tide Smith's relation 6 8 ispersion coefficients epthes Spring tide Smith's relation 6 8 () in which is the ertical shear dispersion and is the mean cross-sectional concentration. The dispersion coefficient associated with the transerse shear ( t) is expressed as utst t s () x s epth m epth m

4 5 Jaari et al. Calculation Procedure To inestigate the alue of the dispersion coefficient, field measurements of elocity and salinity on a cross-section of the Scheldt were used to ealuate the temporal ariation of the ertical and transerse shear induced components of the longitudinal dispersion coefficient. As alrea stated, detailed measurements of the two dimensional, ertical and transerse flow structure in a cross-section of the Scheldt estuary during a neap and a spring tide were aailale. Velocity and salinity distriutions were measured at fie stations across the stu cross-section. The data otained were processed in seeral aspects, efore any calculation was performed. A temporal correction was necessary, ecause the measurement at the stations were not carried out with the same time-interal, mainly ecause at the deeper locations more points oer the ertical were measured. The time of the measurement at all points were compared and only those were selected where the time of measurements at all fie stations almost corresponded to each other. It should e noted that the duration of a measurement oer a ertical was not the same in all the stations due to the different numer of points oer the ertical. The maximum duration did not exceed more than 5 minutes. At seeral time-interals the data at all points had to e discarded ecause the measurements in one station was suspended due to mechanical prolems. It was necessary to descrie the ertical ariations of elocity and salinity in detail in order to determine the turulent diffusion coefficient and the components of longitudinal dispersion coefficient due to the ertical shear effects. At some locations the reading of elocity and salinity were not performed at the same leels of the flow. Assuming that salinities at the surface and at the ottom of the estuary were the same as the salinities at the highest and lowest data points oer the ertical respectiely, the salinity oer the ertical were distriuted linearly y m space interal. It was tried to determine the elocities with the same space interal and to estalish a logarithmic profile etween two successie points y inserting the information at the points as oundary conditions. This did not succeed, ecause mostly the shape of the measured elocity profile was not logarithmic. Therefore, the elocity was also distriuted y linear interpolation or extrapolation for the surface with the assumption of zero elocity at the ottom A computer program was deeloped to make the calculation of the aoe procedure. This procedure was applied to the data of neap and spring tide measurements. The alues of the dispersion coefficients for oth tides are presented in the Figure. COMPARISON BETWEEN ISPERSION COEFFICIENTS FROM IFFERENT APPROACES ifferent alues of the dispersion coefficient from different approaches were shown graphically at different times during a tidal cycle. The dispersion coefficients from the Taylor-Elder analysis were ery low compared to the reality. In fact, in the original form of the Taylor's equation, the dispersion coefficient is proportional to the inerse of the momentum diffusion coefficient. Bowden suggested that the difference etween the dispersion coefficient from the Taylor's relation and the osered alue can e explained y the fact that the momentum iffusion coefficient is reduced y stratification to aout / of the alue one would expect in a homogeneous flow ( YER, 97). ence, if the dispersion coefficient shown in the graph is multiplied y the alue, the real alues of dispersion coefficient should e produced. By doing so, the alues of the coefficient get ery close to oth the osered alues and the alues from Prych's equation. Prych's equation produced much closer alues although his ispersion coefficient m /s ispersion coefficient m /s ispersion coefficients epthes Figure : ispersion coefficients using the Fischer's approach. Equation was deried considering only the shear induced y the ertical elocity gradients. This higher alues of dispersion coefficients, and indeed more correct alues, should e attriuted to the real alues of momentum diffusion coefficient inoled in Prych's equation. Consequently, the Prych's equation was highly sensitie to the momentum diffusion coefficient as can e seen from the equation (6). A good ealuation of oth momentum and solute diffusion coefficient is essential in order to gie a good estimation of the dispersion coefficient. Generally, in the asence of sufficient information to ealuate these coefficients, they are related to the product of the surface elocity and the flow depth ( YER, 97, GUYMER and W EST, 99). The proportionality factor differs from estuary to estuary. The alue of this factor has een suggested y S MIT (976) as.5 in the ertical direction. The alues of the dispersion coefficients from the Fischer's approach were, as expected, generally higher than those produced y Prych's one due to the consideration of the transerse shear induced y the the transerse elocity gradients. Unlike the alue determined y the equations of Taylor-Elder and Prych, the alues calculated y Fischer's approach, for spring tide were not higher than those of neap tide. From Figure it is seen that the alues of dispersion coefficient are rather scattered and do not follow a general inclination. This is due to the fact that the elocity profiles oer the width of the channel were estalished y linear interpolation of the depth-aeraged elocities from the measurement at fie points. Considering the aerage width of the estuary at this cross-section (around m), this is a rough approximation. There should hae existed data in at least ten points so that one could rely on the elocity profiles produced through interpolation. CONCLUSIONS Neap tide Fischer's approach 6 8 ispersion coefficients epthes Spring tide Fischer's approach 6 8 ue to the fact that the alue of the dispersion coefficient seemsto e much larger than the one predicted y the wellknown relation of Elder (order of times) in this stu seeral equations deeloped for prediction of this coefficient were tested in the cross-section of the case stu. Finally, it was tried to ealuate the dispersion coefficient in the longitudinal direction of the estuary y descriing and determining the magnitudes of the mechanisms responsile for causing the rates epth m epth m

5 Longitudinal ispersion Coefficient 5 Of longitudinal salt transport in partially stratified estuaries following Fischer's approach. Although it would e more conenient to take a gloal alue for the dispersion coefficient oer the whole tidal cycle and ery reasonale results were otained y modelling the salt transport in this way, all the approaches confirmed temporal ariation of the dispersion coefficient for different flow structure during a tide. Salt motion simulation with a constant dispersion coefficient throughout the tidal cycle should e carried out with caution. Generally, the predicted dispersion coefficient increases with tidal amplitude. They do not follow the same trend as there is a time lag etween the elocity ariations and the tidal ariations. Taylor and Elder hae assumed in their analysis that the ertical graitational circulation is the most important mechanism for dispersion. The graitational circulations in the transerse direction, howeer, can e as or een more important. Their relatie importance, howeer, was not as high as that predicted y Fischer as. Comparison of the dispersion coefficients predicted for two cross-sections of the estuary proed that the longitudinal dispersion coefficient increased towards the mouth of the estuary. All the approaches for determining the dispersion coefficient produced higher alues of the coefficient for spring tide than for neap tide. The main reason is that the deiation of the elocities and salinities from the mean depth alue during spring tide are larger than during neap tide. LITERATURE CITE BOZOGI, R.,. Stu of salt mixture condition in the tidal parts of the Bahmanshir rier. Tehran: Iran Uniersity of Science and Technology, Master's thesis, 96p. YER, K.R., 97. Estuaries: A Physical Introduction, John Wiley, p. FISCER,.B., 967. The mechanics of dispersion in Natural streams. Journal of the ydraulic iision, ASCE, 9(6), FISCER,.B., 97. Mass transport mechanism in partially stratified estuaries. Journal of Fluid Mechanics, 5(), GUYNER, I. and WEST, R., 99, Longitudinal dispersion coefficients in estuaries. Journal of ydraulic Engineering, ASCE, 8(5), ARLEMAN,.R.F. and TATCER, M.L., 97. Longitudinal dispersion and unstea salinity intrusion in estuaries. La ouille Blanche, (), 5-. OLLY, E.R.; ARLEMAN,.R.F. and FISCER,.B., 97. ispersion in homogeneous estuary flow. Journal of the ydraulic iision, ASCE, 96(8), SING, U.P.; GARE, R. J., and RANGA RAJU, K.G., 99. Longitudinal dispersion in open-channel flow, International Journal of Sediment Transport, 7(), SMIT, R., 976. Longitudinal dispersion of a uoyant contaminant in a shallow channel, Journal of Fluid Mechanics, 78(),

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