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# SPTP GRNDWATER HYDRO ENG CVEN 489

Texas A&M

GPA 3.53

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This 176 page Class Notes was uploaded by Carolyn Kuhn on Wednesday October 21, 2015. The Class Notes belongs to CVEN 489 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 13 views. For similar materials see /class/226121/cven-489-texas-a-m-university in Civil Engineering at Texas A&M University.

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Date Created: 10/21/15

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S39WIX xz mks Vicki s Law 575DS 3969 3 use Twobr Salas E39xVamgima Xz gtltl gafx W WE IAW39 ov ar ng 0F ordev 0G oc DV lt45 CLC Ac 4 Ac Ax z J lgtogtlt 06 3Wb9w 3W mass emserva o nf 5 29 6 7 52 3 QC BE A 57 g1 Mp cx 42C me V 52 1 Alec Movie 1 M C JXJV 5 Combine All 3 D mm5 wsz Wiggga SL9 D 33 93E 92C at 3 Ms W J39 9 72 4 a t 9 93 93 9 9t a D 2 37 Swag 99 2 9C ale 3 C 0 5 Di RLSII M MHmt 33 52c 92C N t D 32c gt W 4 3y T 5 21 4 me mfe Laf a iam a chm35 13 In C i MFFW 2 bvmdaxy CHASfan Cnvzokmm Far 1 mat161m VamboLCc Pa w39al 4 Kerm Eqm m 5mm 09 We had eqm m ExamMe 1D 33414531224 ivm P gtez 4 L B 4 C 39 C2 Cxwl 0mm xed cmc SWy 351 i1 95 3amp7 32 smack ewe 9 2 O EDWAmFy Conan HMS C CD 3 Cl C L 7 C7 So uHm l quot ac fevgx C cfxcfo 967 BCS Co C ii0 C2 CLgtC2 139 Cl Cl 2 quot 7quot C C a CK C2 C L w C C CO 31 X CI C F1L Km 04 7 Water Quality Modeling Until now we have derived governing equations for and sought solutions to idealized cases where analytical solutions could be found Many problems in the natural world however are complex enough that simpli ed analytical solutions are inadequate to predict the transport and mixing behavior In these situations approximations of the governing transport equations such as nite difference must be made so that numerical solutions can be found These approximations can be simple or complex but often result in a large number of equations that must be solved to predict the concentration distribution Hence computer algorithms are used to make the numerical solutions tractable In this chapter we introduce the eld of water quality modeling based on computerized nu merical or digital tools This chapter begins by outlining how to select an appropriate numerical tool The next two section describe common computer approximations First simple numerical models based on plug ow and continuously stirred tank reactors are introduced Second an overview of numerical approximations to the governing equations is presented Because we are now dealing with approximate solutions new procedures are needed to assure that our results are acceptable The nal section outlines the crucial steps necessary to test the accuracy of a numerical result Although computer power is rapidly growing it remains important to use sim ple tools and thorough testing in order to understand and synthesize the meaning of numerical results 71 Systematic approach to modeling A model is any analysis tool that reduces a physical system to a set of equations or a reduced scale physical model Moreover all of the solutions in previous chapters are analytical models of natural systems Whether analytical or numerical the main question the modeler must answer is which model should I use 711 Modeling methodology The ASCE amp WPCF 1992 design manual Design and Construction of Urban Stormwater Management Systems outlines a four step selection processes for choosing a water quality analysis tool These steps are discussed in detail in the following and include 1 de ning project goals 2 describing an acceptable modeling tool 3 listing the available tools that could satisfy the goals and model description and 4 selecting the model to be used based on an optimal compromise between goals and available tools Copyright 2004 by Scott A Socolofsky and Gerhard H Jirka All rights reserved 126 7 Water Quality Modeling 1 De ne project goals It may sound like an obvious rst step but it is essential and regrettably often overlooked de ne the project goals before even choosing the model Fischer et al 1979 emphasize this step as well saying that the choice of a model depends crucially on what the model is to do Modeling goals are quite variable ranging from the practical provide the analysis necessary to get the client his discharge permit to the research oriented develop a new tool that overcomes some current modeling shortfall During this step as much as possible should be learned about the system to be modeled Fischer et al 1979 suggest that if possible the investigator should become personally familiar with the water body by going out on it in the smallest boat that is safe Then before venturing near a computer or a model basin he or she should make all possible computations being approximate where necessary but seeking a feel for what the model will predict Only after we understand our system can we formulate appropriate project goals In the words of the famous landscape photographer Ansel Adams Visualization is of utmost importance many failures occur because of our uncertainty about the nal image77 quoted in Fischer et al 1979 During this stage one begins to formulate the necessary attributes of the model This leads naturally the next step 2 Describe an acceptable modeling tool Before selecting the model for the analysis formulate a list of abilities and characteristics that the model must have These can include things like input output exibility common usage in the regulatory community and physical mixing processes the model must include Our simpli ed predictions from step 1 of how the system behaves are used in this step to formulate the model requirements For instance if we expect rapid near eld mixing we may suggest using a one dimensional model This step should keep in mind what models are available but not limit the analysis to known tools if they would be inadequate to meet the project goals In this stage the project goals may also need to be revised If the only acceptable modeling tool to meet a particular goal is too costly in terms of computation time and project resources perhaps that goal can be reformulated within a reasonable project scope The purpose therefore of this step is to optimize the modeling goals by describing practical requirements of the modeling tool 3 List applicable tools Once the analysis tool has been adequately described one must formulate a list of available tools that meet these requirements In engineering practice we must often choose an existing model with a broad user base Appendix D lists several public domain models Most of them are available free of charge from government sponsor agencies but some are also commercial The purpose of choosing an existing model is that it has been thoroughly tested by many previous users and that the regulatory agencies are accustomed to seeing and interpreting its output However available tools may not always be adequate to meet the project goals If existing tools are inadequate then a new tools must be developed and a list of existing methods is an important step Methods are the building blocks of models A onedimensional nite difference model employs two methods a one dimensional approximation and a nite difference numerical scheme Going a step deeper the nite difference method can have many attributes such as forward central or backward differencing implicit or explicit formulation 71 Systematic approach to modeling 127 and rst second or higher order solution algorithms Section 73 describes what some of these terms mean The point is that when designing a new tool there are many existing building blocks from which to choose and these can be quite helpful It may turn out that simply adding an unsteady algorithm to an existing steady state model will meet the project goals Hence knowing as much as possible about existing models and methods is essential to implementing andor designing an analysis tool 4 Make an optimal compromise between goals and available tools In the nal step of choosing a modeling tool we seek an optimal compromise between the project goals and the available tools This is where the decision to proceed with a given modeling tool is made The project goals are the guide to choosing the model It sounds simple but choose the best model to meet the project goals not just the best available model As computers become faster the tendency is to just pick the biggest boldest model and to force it to meet your needs However the enormous amount of output from such a model may be overwhelming and costly and unnecessary in the light of certain project goals Therefore choose the most appropriate simplest model that also satis es the scienti c rigor of the project goals and when necessary develop new tools 712 Issues of scale and complexity Throughout the process of choosing a modeling tool one is confronted with issues of system scale and complexity The world is inherently three dimensional and turbulent but with current computer resources we must often limit our analysis to one and two dimensional approximations with turbulence closure schemes that approximate the real world Hence we must make trade offs between prototype complexity and model ability We can evaluate these trade offs by doing a scale analysis to determine the important scales in our problem This is the essence of steps one and two above where we try to predict what the model will tell us and use this information to characterize the needed tool For transport problems we must consider the advective diffusing reaction equation For illustration consider a rst order reaction 8 0 3uC 8110 8wC D182C Dyt32C D1820 8t 8m 8y 82 8x2 8342 822 This equation has three unit scales mass time and length It also has three processes advection i k0 71 diffusion and reaction We would like to formulate typical scales of these three processes from the typical units in the problem For example the advection time scale is the time it takes for uid to move through our system If we are modeling a river reach of length L with mean velocity U then the advective time scale is Ta L U 7 2 Processes that occur on time scales much shorter than Ta can neglect advection We can use these scales to non dimensionalize the governing equation To do this we de ne the non dimensional variables using primes as follows 128 7 Water Quality Modeling m meg y Lyy z L12 u Uu U VUg w Wu C 000 t Tot k 1Trk 73 where the upper case variables are typical scales in the problem For example To is an external time scale such as a discharge protocol or the diurnal cycle and T7 is the reaction time scale such as the half life for a dyeoff reaction The L s are system dimensions and U V and W are average velocities Substituting into 71 gives i13 C U 31C V 310 W LuC To dt Lm 895 Ly 81 L1 82 DE 820 Dy 82Cquot Dz 82Cquot 1 L 8252 L 832 L 822 i TlZk 039 7394 Note that to make this equation fully non dimensional we must multiple each term by a time scale for example To We can now determine the relative importance of each term by considering their leading coef cients Comparing the convective terms Longitudinal advection 7 U L 7 y 75 Lateral advection V Lm Longitudinal advection 7 U Lz 7 6 Vertical advection T W Lm 39 If these ratios are much greater than one then longitudinal advection is the only advection term that we must keep in the equation Thus the importance of a given convection term depends on the velocity scales in our problem and the length of river we are considering In many river problems these ratios are much greater than one and we only keep the longitudinal advection term Likewise we can compare the diffusion terms to the advection term For longitudinal diffusion we have Longitudinal diffusion Dm 77 Longitudinal advection LgTa which is our familiar Peclet number For large Peclet numbers we only consider diffusion and for small Peclet numbers we only consider advection Hence the important terms in the equation again depend on the length of river we are considering As an example when might a one dimensional steady state model with dyeoff be an accept able model for a given river reach The governing model equation would be 7kC 78 u a a By comparing with the non dimensional equation above this equation implies several constraints on the river Consider rst the external time scale This model implies UTO L m gtgt 1 79 72 Simple water quality models 129 For the convective terms this model implies that ULy VLz ULZ WLz or alternatively 80 80 87 E Similarly for diffusion this model implies that UL LmDy UL Lsz or again alternatively 80 80 8 37 E 0 715 Finally for the reaction this model implies that E L12 Therefore by comparing the relevant scales in our problem to the approximations made by gtgt1 710 gtgt1 711 712 gtgt 1 713 gtgt 1 714 g 1 716 models we can determine just how complex the model must be to approximate our system adequately 713 Data availability As a nal comment on the selection and implementation of an analysis tool we discuss a few points regarding the data that are used to validate the model Section 74 below discusses how to test a model in more detail The only test available to determine whether the model adequately reproduces our natural system is to compare model output to data measurements taken from the prototype system In general as the model complexity increases the number of parameters we can use to adjust the model results to match the prototype also increases giving us more degrees of freedom The more degrees of freedom we have the more data we need to calibrate our model Hence the data requirements of a model are directly proportional to the model complexity If very limited data are available then complex models should be avoided because they cannot be adequately calibrated or validated 72 Simple water quality models Some simple water quality models can be developed for special cases where advection or diffusion is dominant As introduced in Chapter 2 the Peclet number is a measure of diffusion to advection dominance The Peclet number P5 is de ned as 130 7 Water Quality Modeling wave Fig 71 Schematic of a plug ow reactor D P5 E 717 D m 718 where the two de nitions are equivalent The Peclet number is small when advection is dominant and large when diffusion is dominant Two simple models can be developed for the limiting cases of P5 a 0 and P5 a 00 A third hybrid model is also introduced in this section for simpli ed application to arbitrary P5 721 Advection dominance Plug ow reactors For P5 a 0 we can neglect longitudinal diffusion and dispersion7 and we have the sO called plug ow reactor Shown in Figure 717 a slab of marked uid is advected with the mean ow7 perhaps undergoing reactions7 but not spreading in the lateral Taking D 07 the governing reactive transport equation becomes u iR 719 To solve this equation7 we make the familiar coordinate transformation to move our coordinate system with the mean ow That is7 Em7ut 720 T t 721 As demonstrated in Chapter 27 using the chain rule to substitute this coordinate transformation into 719 gives 80 E iR 722 which is easily solved after de ning an initial condition and the transformation reaction R For example7 consider a rst order die off reaction for a slab with initial concentration CO The solution to 722 is CT CO exp7k739 723 or in the original coordinate system7 we have the interchangeable solutions Ct Co exp7kt 724 Cm Co exp7kzu 725 The residence time for a plug ow reactor depends on the distance of interest L0 From the de nition of residence time 72 Simple water quality models 131 7 Fig 72 Schematic of a continuouslystirred tank reantor CSTR t 7 K 7 85 Q LoA Q where A is the cross sectional area of the channel and Q is the steady ow rate The uid 726 residence time7 the travel time for a slab to move the distance L07 can also be expressed using the same variables L tslab 0 LL 7 LOA Q 255 727 Hence7 the uid and species residence times are equal 722 Diffusion dominance Continuouslystirred tank reactors For P5 a 00 we can neglect advection and we have the so called continuously stirred tank reactor CSTR Shown in Figure 727 uid that enters the reactor is assumed to instantaneously mix throughout the full reactor volume To write the governing equation7 consider mass conservation in the tank mm 7 mm 728 The in ow provides the mass ux into the control volume7 Loss of mass7 mm is given by the out ow and possible dieoff reactions Writing the conservation of mass in concentrations and ow rates yields dCV dt where V is the volume of the tank and S VB is a source or sink reaction term Because the QC7 7 omiS 729 tank is well mixed7 we can assume that Com is equal to the concentration in the tank C Taking V as constant7 we can move it outside the derivative7 and the governing equation becomes 10 7 Q E 7 7 Substituting the de nition of the residence time7 we have nally Cm 7 C i R 730 132 7 Water Quality Modeling E L dt tm which is the governing equation for a CSTR Cm 7 C i R 731 Consider rst the conserving case where R 0 Taking the initial condition as a clean tank CO O the solution to 731 is Ct 0m 17 amp 7 t 732 tTBS Thus the concentration in the tank increases exponentially with a rate constant k 1tres The concentration in the tank reaches steady state asymptotically If we de ne steady state as the time 2595 until C 09900 then 25 46255 733 Therefore without reactions steady state is reached in about 46 residence times The solution for the reacting case is slightly more complicated because 731 is an inhomo geneous differential equation with forcing function iR Consider the case of a rst order die off reaction and an initial tank concentration of CO 0 Assuming a particular solution refer to Appendix C for an example of solving an inhomogeneous equation of the form Op AC the solution is found to be Ct 17 exp 7 25H 734 1 ktres tres Because of the reaction the steady state concentration in the tank is no longer the in ow concentration but rather Ci 1 95755 and the coef cient 11 ktres is one minus the removal rate The time to reach steady state 2595 is the time to reach 0990SS or 462955 1 km 099 735 736 tss 7 2 3 Tanksinseries models The simplest type of river ow model that incorporates some form of diffusion or dispersion is the tanks in series model which is a chain of linked CSTRs An example tanks in series model is shown in Figure 73 In the example each tank has the same dimensions and the ow rate is constant The method also works for variable volume tanks and under gradually varied ow conditions stage discharge relationships can be used to route variable ows through the tanks To see why the tanks in series model produces diffusion consider an instantaneous pulse injection in the rst tank The out ow from that tank would be the solution to the CSTR given by 732 The out ow from the rst tank is therefore exponential clearly not the expected Gaussian distribution But this out ow goes into the next tank At rst that tank is clean and the little bit of tracer entering the tank in the beginning is quickly diluted hence the out ow concentration starts at zero and increases slowly Eventually a large amount of the 72 Simple water quality models 133 Q A ampH A o o 0 CH CI CM 0 o 0 V V V k Ax H Fig 73 Schematic of a tanksinseries model The model is made up of several CSTR linked in series tracer in the rst tank has moved on to the second tank and the out ow from the second tank reaches a maximum concentration The in ow from the rst tank becomes increasingly cleaner7 and the out ow from the second tank also decreases in concentration Eventually7 all the tracer has owed through both tanks and the concentration is zero at the outlet of the second tank The concentration curve over time for the second tank started at zero7 increased smoothly to a maximum concentration7 then decreased slowly back down to zero These characteristics are very similar to the Gaussian distribution hence7 we expect that by dimensioning the tanks properly7 we should be able to reproduce the behavior of advective diffusion in the downstream tanks To nd the proper tank dimensions7 consider the mass conservation equation for a central tank In ow comes from the upstream tank7 and out ow goes to the downstream tank thus7 we have QCl1 7 0 i VB 737 For the remaining analysis we will neglect the reaction term Assuming each tank has the same length7 the tank volume can be written as V AAz Azi 7 771 Using this de nition to write M1 in concentration units gives 610739 7 CH 0739 dt 7 u l 7 141 738 which is the discrete equation describing the tanks in series model The difference term on the right hand side of 738 is very close to the backward difference approximation to dCdm 80 C39 7 C 1 11 739 8 mi 7 miil which has no error for Am a O For nite grid size7 the Taylor series expansion provides an estimate of the error The second order Taylor series expansion of C271 about 0 is 801 1 820139 0771 0739 l E i 771 ii1 02 740 as given in Thomann amp Mueller 1987 Rearranging this equation7 we can obtain 01quot 0171 807 1 820139 7 7 741 ml 7 mFl 8m 2 8x2 1 1 1 Multiplying this result by 71 gives 134 7 Water Quality Modeling C271 7 0 80 1820i 7 7 7 742 m 7mi1 8m 2 8x2 1 1 1 which can be substituted immediately for the right hand side of 738 leaving 7 i71 7 743 dt u 8m 2 8x2 Dropping the subscripts and recognizing Am 7 l 71 gives the governing equation 10 80 uAm 320 744 dt u 8m 2 8x2 Thus our derived governing equation for a tanks in series model has the same form as the advective diffusion equation with a diffusion coef cient of Dn quZ The effective diffusion coef cient D for a tanks in series model is actually a numerical error due to the discretization As the discretization becomes more course the numerical error increases and the numerical diffusion goes up For Am a O the numerical diffusion vanishes and we have the plug ow reactor Hence for a tanksin series model we choose the tank size such that Dn is equal to the physical longitudinal diffusion and dispersion in the river reach 73 Numerical models Although the tank in series model was shown to be a special discretization of the advective diffusion equation other numerical techniques speci cally set out to discretize the governing equation For our purposes a numerical model is any model that seeks to solve a differential equation by discretizing that equation on a numerical grid 731 Coupling hydraulics and transport To simulate chemical transport the velocity eld represented by u in the transport equation must also be computed The model that calculates u is called the hydrodynamic or hydraulic model Thus to simulate transport the hydrodynamic and transport models must be properly coupled Whether the hydrodynamic and transport models must be implicitly coupled or whether they can be run in series depends on the importance of buoyancy effects If the system is free from buoyancy effects the hydrodynamics are independent of the transport hence they can be run rst and their output stored Then many transport simulations can be run using the hydrodynamic data without re running the hydrodynamic code If buoyancy effects are present in the water body then the transport of buoyancy heat or salinity or both must be coupled with the hydrodynamics and both models must be run together Once the output from the coupled model is stored further transport simulations can be run for constituents that do not in uence the buoyancy these are called passive constituents Because the hydrodynamic portion of the model is computationally expensive the goal in transport modeling is to de couple the two models as much as possible 73 Numerical models 135 732 Numerical methods There are probably as many numerical methods available to solve the coupled hydrodynamic and advective transport equations as there are models however7 most models can be classi ed by a few key words There are three main groups of numerical methods nite difference7 nite volume7 and nite element For special selections of basis functions and geometries7 the three methods can all be made equivalent7 but in their standard applications7 the methods are all slightly different The nite difference method is built up from a series of nodes7 the nite volume method is built up from a group of cells7 and the nite element method is made up of a group of elements7 where each element is comprised of two or more grid points In the nite difference case7 the differential equation is discretized over the numerical grid7 and derivatives become difference equations that are functions of the surrounding cells In the nite volume case7 the uxes through the cell network are tracked and the differential equations are integrated over the cell volume For nite elements7 a basis function is chosen to describe the variation of an unknown over the element and the coef cients of the basis functions are found by substituting the basis functions as solutions into the governing equations Because nite difference methods are easier to implement and understand7 these methods are more widely used A numerical method may further be explicit or implicit An explicit scheme is the easiest to solve because the unknowns are written as functions of known quantities For instance7 the concentration at the new time is dependent on concentrations at the previous time step and at upstream known locations In an implicit scheme7 the equations for the unknowns are functions of other unknown quantities For instance7 the concentration at the new time may depend on other concentrations at the new time or on downstream locations not yet computed In the implicit case7 the equations represent a system of simultaneous equations that must be solved using matrix algebra The advantage of an implicit scheme is that it generally has greater accuracy Finally7 numerical methods can be broadly categorized as Eulerian or Lagrangian Eulerian schemes compute the unknown quantities on a xed grid based on functions of other grid quan tities Lagrangian methods use the method of characteristics to track unknown quantities along lines of known value For instance7 in a Lagrangian transport model7 the new concentration at a point could be found by tracking the hydrodynamic solution backward in time to nd the point where the water parcel originated and then simply advecting that concentration forward to the new time Because the Lagrangian method relies heavily on the velocity eld7 small er rors in the velocity eld particularly for elds with divergence can lead to large errors in the conservation of mass The advantage of the Lagrangian method is that it can backtrack over several hydrodynamic time steps hence7 there is no theoretical limitation on the size of the time step in a Lagrangian transport model An example of a one dimensional Lagrangian scheme is the Holly Preissman method By contrast7 for the Eulerian model7 the time step is limited by a so called Courant number restriction7 that says that the time step cannot be so large that uid in one cell advects beyond the next adjacent cell over one time step Mathematically7 this can be written as 136 7 Water Quality Modeling At g 745 U where At is the time step and Am is the grid size 733 Role of matrices In the case of explicit models matrices are not a necessity but for implicit models and models simulating many contaminant matrices provide a comfortable and often necessary means of solving the governing equations For an implicit scheme the equations for a given node at the new time are dependent on the solutions at other nodes at the same time This means that implicit schemes are inherently a system of equations sometimes non linear which are best solves with matrices The general matrix equation is 41 b 746 where A is an n x 71 matrix of equation coef cients z is an n x 1 vector of unknowns for example ow rates and b is an n x 1 vector of forcing functions Writing the equations in such a way makes derivation of the model equations manageable and implementation in the computer algorithm straightforward The solution of 746 is 1 A lb 747 where A71 is the matrix inverse Most computer languages have built in methods for solving matrices For the non linear case an iteration technique must be employed A common method is the Newton Raphson method 734 Stability problems One limitation already mentioned for an Eulerian transport scheme is the Courant number restriction In general all schemes have a range of similar restrictions that limit the allowable spatial grid size and time step such that the scheme remains stable If the time step is set longer than such a constraint the model is unstable and will give results with large errors that eventually blow up The full hydrodynamic equations are hyperbolic and generally have more stringent limitations than the parabolic transport equation Before implementing a model it is advised to seek out the published stability criteria for the model this can save a lot of time in getting the model to run smoothly 74 Model testing An unfortunate fact of numerical modeling is that implementation and calibration is very time consuming and little time is available for a thorough suite of model tests This does not excuse the fact that model testing is necessary but rather explains why it is often neglected Even when using well known tools the following suite of tests is imperative to ensure that the model is working properly for your application The following tests are speci c to transport models but apply in a generalized sense to all models 74 Model testing 137 741 Conservation of mass All transport water quality models must conserve mass This is a zeroth Order test that con rms whether the zeroth moment of the concentration distribution is accurately reproduced in the solution Clearly7 when reactions are present7 a given species may be loosing or gaining mass due to the reaction This test must con rm7 then7 that the total system mass remains constant and that a species only gains mass at the rate allowed by the reaction equation This test is often conducted in conjunction with the next test However7 it should always also be conducted for the complex real world case being simulated7 where analytical solutions are not available 742 Comparison with analytical solutions The model should be tested in idealized conditions to compare its results to known analytical solutions This test con rms whether the model actually solves the governing equation that it was designed to solve Deviations may be caused by many sources7 most notably programming errors and numerical inaccuracy Although most widely used models are free from programming errors7 this cannot be tacitly assumed In this author s experience7 programming errors have been found in well known7 government supported models by running this test The issue of numerical inaccuracy arises due to the discretization7 as in the case of numerical diffusion mentioned for the tanks in series model described above Hence7 the idealized case should have length and time scales as close to the prototype as possible in order to accurately assess the importance of numerical inaccuracy arising from the numerical method and the discretization This step can save a lot of time in applying the model to the prototype because the source of errors can often be identi ed faster in idealized systems First7 the analytical solution is a known result lfthe model gives another result7 the model must be wrong Second7 the complexity of the real world case makes it dif cult to assess the importance of deviations from measured results Once the model has been thoroughly tested against analytical results7 deviations can be explained by physical phenomena in the prototype not present or falsely implemented in the model Third7 this test helps determine the stability requirements for complex models 743 Comparison with eld data Only after it is certain that the model is solving the equations properly and within a known level of error can the model be compared to eld or laboratory measurements of the prototype The comparison of model results with these date serves two purposes First7 the model must be calibrated that is7 its parameters must be adjusted to match the behavior of the prototype Second the model must be validated This means that a calibrated model must be compared to data not used in the calibration to determine whether the model is applicable to cases outside the calibration data set These prototype measurements fall into two categories tracer studies and data collection of natural events 138 7 Water Quality Modeling Tracer studies In a tracer study dye is injected into the natural system and concentrations are measured in time and space to record how the dye is transported and diluted The advantage of a tracer study is that the source injection rate and location are known with certainty and that reactions can often be neglected Tracer studies help calibrate the model parameters such as diffusion and dispersion coef cients and turbulent closure schemes to the real world case These studies also help to con rm whether the model assumptions are met such as the one dimensional approximation and are good tests of both the hydrodynamic and water quality models Water quality data The nal set of data available for model testing is actual measurements of the modeled constituents in the prototype under natural conditions These measurements represent true values but are dif cult to interpret because of our incomplete description of the prototype itself We often do not know the total loading of constituent and all the model equa tions are approximations of the actual physical processes in the prototype These measurements further help to con rm whether the model assumptions are valid and to calibrate model parame ters Once the tests listed above are completed the modeler should have a good understanding of how the complex physical processes in the model combine to give the model results Deviations between the model and the eld measurements should then be explained through the physical insight available in the model It is important to point out that the water quality measurement campaign should compliment the output available from the model That is the data should be collected such that they can be used to calibrate and test the model If the model only outputs daily predictions then the measurements should be able to predict daily values instantaneous point measurements are only useful for a parameter that does not vary much over the diurnal cycle In summary the model is only as good as the data that support it and the data must be compatible with the model and ex the parts of the model that are the most uncertain Summary This chapter introduced the concept of water quality modeling A model is de ned as any analy sis tool that reduces a physical system to a set of equations or a reduced scale physical model A four step procedure was suggested to help select the appropriate model 1 de ne project goals 2 describe an acceptable modeling tool 3 list the available tools that could satisfy the goals and model description and 4 select the model to be used based on an optimal compromise between goals and available tools Because analytical solutions are not always adequate nu merical techniques were introduced These included tank reactor models and numerical solution methods for differential equations such as nite difference and nite element Because numerical solutions result in a large number of calculations a rigorous procedure for testing a numerical model was also suggested These steps include 1 con rming that model conserves mass 2 testing the model in idealized cases against analytical solutions and 3 comparing the model to eld data in the form of dye studies and the collection of water quality data Good modeling projects should follow all of these suggested procedures Ebrercises 139 Measured dye concentration breakthrough curve I I I I F n Concentration mgI 0 1 20 Time since injection start min Fi 74 Measured dye concentration for example dye study Dye uctuations are due to instrument uncertainty 39 s not due to turbulent uctuation Exercises 71 Equation scaling Non dimensionalize the one dimensional momentum equation 814 814 1 8p 8214 748 at uam p 8m V8352 using the non dimensional variable de nitions u Uou m Lm 749 t LUo p png 750 Divide the equation by the coef cient in front of 81 8t What familiar non dimensional number becomes the leading coef cient of the viscous term When is the viscous term negligible 72 Finite difference Write the explicit backward difference approximation to the reaction equa tion 10 k0 751 dt Program this solution in a computer and suggest a criteria for selecting the appropriate time step At by comparing to the analytical solution Ct Co expkt 752 73 Tanks in series model A river has a cross section of h 1 m deep and B 10 m Wide The mean stream velocity is 225 cms A dye study was conducted my injecting 225 gs of dye uniformly across the cross section 150 m upstream of a measurement point The measurements of dye concentration at L 150 m are given in Figure 74 F rom the figure7 determine the value of the dispersion coef cient Based on this value7 how many tanks in a tanks in series model would be needed to reproduce this level of dispersion in the numerical model 140 7 Water Quality Modeling m AAVcclivc dalmasicm equa n an Ram lake problem39 Ta 3 z 2 39 WermocHW oi 23 m 3 WW WW a 111 i G 8 I 55153 Givcu AQ2 3M 1194 m 9 u 39 as 93 6 Di L39smquot if s 141 ERIE 2 393m 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Accidenw 993m Flam View 1 oil spin 4 Acm h39m NVMAM Oz chmMae a I MIC 02 6mm vadik See Fiduch m web SAC chrcot Resmnesquot new meow C Pro le 539 Nut me am exam91 oom39vc wowMam Somewhat 3 Mixing in Rivers Turbulent Diffusion and Dispersion In previous chapters we considered the processes of advection and molecular diffusion and have seen some example problems with so called turbulent diffusion77 coef cients where we use the same governing equations but with larger diffusion mixing coef cients ln natural rivers a host of processes lead to a non uniform velocity eld which allows mixing to occur much faster than by molecular diffusion alone In this chapter we formally derive the equations for non uniform velocity elds to demonstrate their effects on mixing First we consider the effect of a random turbulent velocity eld Second we consider the combined effects of diffusion molecular or turbulent with a shear velocity pro le to develop equations for dispersion In each case the resulting equations retain their previous form but the mixing coef cients are orders of magnitude greater than the molecular diffusion coef cients We start by giving a description of turbulence and its effects on the transport of contaminants We then derive a new advective diffusion equation for turbulent ow and show why turbulence can be described by the regular advective diffusion equation derived previously but using larger turbulent diffusion coef cients We then look at the effect of a shear velocity pro le on the transport of contaminants and derive one dimensional equations for longitudinal dispersion This chapter concludes with a common dye study application to compute the effective mixing coef cients in rivers 31 Turbulence and mixing In the late 1800 s Reynolds performed a series of experiments on the transport of dye streaks in pipe ow These were the pioneering observations of turbulence and his analysis is what gives the Re number its name It is interesting to realize that the rst contribution to turbulence research was in the area of contaminant transport the behavior of dye streaks therefore we can assume that turbulence has an important in uence on transport In his paper Reynolds 1883 wrote taken from Acheson 1990 The experiments were made on three tubes They were all about 4 feet 6 inches 137 m long and tted with trumpet mouthpieces so that water might enter without disturbance The water was drawn through the tubes out of a large glass tank in which the tubes were immersed arrangements being made so that a streak or streaks of highly colored water entered the tubes with the clear water The general results were as follows Copyright 2004 by Scott A Socolofsky and Gerhard H Jirka All rights reserved 52 3 Mixing in Rivers Turbulent Diffusion and Dispersion 39 a L Q P C G L 1 a c rquot 2 m d Fig 31 Sketches from Reynolds 1883 showing laminar ow top turbulent ow middle and turbulent ow illuminated with an electric spark bottom Taken from Acheson 1990 1 When the velocities were suf ciently low the streak of colour extended in a beautiful straight line through the tube 2 If the water in the tank had not quite settled to rest at suf ciently low velocities the streak would shift about the tube but there was no appearance of sinuosity 3 As the velocity was increased by small stages at some point in the tube always at a considerable distance from the trumpet or intake the color band would all at once mix up with the surrounding water and ll the rest of the tube with a mass of colored water Any increase in the velocity caused the point of break down to approach the trumpet but with no velocities that were tried did it reach this On viewing the tube by the light of an electric spark the mass of color resolved itself into a mass of more or less distinct curls showing eddies Figure 31 shows the schematic drawings of what Reynolds saw taken from his paper The rst case he describes the one with low velocities is laminar ow the uid moves in parallel layers along nearly perfect lines and disturbances are damped by viscosity The only way that the dye streak can spread laterally in the laminar ow is through the action of molecular diffusion thus it would take a much longer pipe before molecular diffusion could disperse the dye uniformly across the pipe cross section what rule of thumb could we use to determine the required length of pipe The latter case at higher velocities is turbulent ow the uid becomes suddenly unstable and develops into a spectrum of eddies and these disturbances grow due to instability The dye which more or less follows the uid passively is quickly mixed across the cross section as the eddies grow and ll the tube with turbulent ow The observations with an electric spark indicate that the dye conforms to the shape of the eddies After some time however the eddies will have grown and broken enough times that the dye will no longer have strong concentration gradients that outline the eddies at that point the dye is well mixed and the mixing is more or less random even though it is still controlled by discrete eddies 31 Turbulence and mixing 53 Reynolds summarized his results by showing that these characteristics of the ow were de pendent on the non dimensional number Re ULl7 where U is the mean pipe ow velocity7 L the pipe diameter and 1 the kinematic viscosity7 and that turbulence occurred at higher values of Re The main consequence of turbulence is that it enhances momentum and mass transport 311 Mathematical descriptions of turbulence Much research has been conducted in the eld of turbulence The ideas summarized in the following can be found in much greater detail in the treatises by Lumley amp Panofsky 19647 Pope 20007 and Mathieu amp Scott 2000 In this section we will consider a special kind of turbulence homogeneous turbulence The term homogeneous means that the statistical properties of the ow are steady unchangingithe ow can still be highly irregular These homogeneous statistical properties are usually described by properties of the velocity experienced at a point in space in the turbulent ow this is an Eulerian description To understand the Eulerian properties of turbulence7 though7 it is useful to rst consider a Lagrangian frame of reference and follow a uid particle In a turbulent ow7 large eddies form continuously and break down into smaller eddies so that there is always a spectrum of eddy sizes present in the ow As a large eddy breaks down into multiple smaller eddies7 very little kinetic energy is lost7 and we say that energy is ef ciently transferred through a cascade of eddy sizes Eventually7 the eddies become small enough that viscosity takes over7 and the energy is damped out and converted into heat This conversion of kinetic energy to heat at small scales is called dissipation and is designated by E d1ss1pated kinetic energy 31 time which has the units LZTgl Since the kinetic energy is ef ciently transferred down to these small scales7 the dissipated kinetic energy must equal the total turbulent kinetic energy of the How this means that production and dissipation of kinetic energy in a homogeneous turbulent ow are balanced The length scale of the eddies in which turbulent kinetic energy is converted to heat is called the Kolmogorov scale LK How large is LK We use dimensional analysis to answer this question and recognize that LK depends on the rate of dissipation or7 equivalently7 production of energy7 e and on the viscosity7 17 since friction converts the kinetic energy to heat Forming a length scale from these parameters7 we have Vex4 LK X W 32 This is an important scale in turbulence Summarizing the Lagrangian perspective7 if we follow a uid particle7 it may begin by being swept into a large eddy7 and then will move from eddy to eddy as the eddies break down7 conserving kinetic energy in the cascade Eventually7 the particle nds itself in a small enough eddy one of order LK in size7 that viscosity dissipates its kinetic energy into heat This small eddy is also a part of a larger eddy hence7 all sizes of eddies are present at all times in the ow 54 3 Mixing in Rivers Turbulent Di usion and Dispersion Sample Laser DopplerVelocimeter LDV data l l 0 N N VeIOCIty ms 0 N l 2 lt l n vl W M 1 win W l lquot W W 39 U l l l l l 7 0 05 1 15 2 25 3 Time s ig 32 Schematic measurement of the turbulent uctuating Velocity at a point showing the average Velocity E and the uctuating component u t Because it is so dif cult to follow a uid particle with a velocity probe this is what we try to do with Particle Tracking Velocimetry PTV turbulent velocity measurements are usually made at a point and turbulence is described by an Eulerian reference frame The spectrum of eddies pass by the velocity probe transported with the mean ow velocity Large eddies produce long period velocity uctuations in the velocity measurement and small eddies produce short period velocity uctuations and all these scales are present simultaneously in the ow Figure 32 shows an example of a turbulent velocity measurement for one velocity component at a point If we consider a short portion of the velocity measurement the velocities are highly correlated and appear deterministic If we compare velocities further apart in the timeseries the velocities become completely uncorrelated and appear random The time scale at which velocities begin to appear uncorrelated and random is called the integral time scale 251 In the Lagrangian frame this is the time it takes a parcel of water to forget its initial velocity This time scale can also be written as a characteristic length and velocity giving the integral scales 741 and 11 Reynolds suggested that at some time longer than 251 the velocity at a point z could be decomposed into a mean velocity 77 and a uctuation such that WW7 75 WM WM t7 33 and this treatment of the velocity is called Reynolds decomposition 251 is then comparable to the time it takes for 77 to become steady constant One other important descriptor of turbulence is the root mean square velocity U39rms UU which since kinetic energy is proportional to a velocity squared is a measure of the turbulent kinetic energy of the ow ie the mean ow kinetic energy is subtracted out since u is just the uctuation from the mean 31 Turbulence and mixing 55 312 The turbulent advective diffusion equation To derive an advective diffusion equation for turbulence7 we substitute the Reynolds decompo sition into the normal equation for advective diffusion and analyze the results Before we can do that we need a Reynolds decomposition analogy for the concentration7 namely7 C957 t Cxi7t 35 Since we are only interested in the long term long compared to 251 average behavior of a tracer cloud7 after substituting the Reynolds decomposition7 we will also take a time average As an example7 consider the timeaverage mass ux in the z direction at our velocity probe7 w qz U 72 um 0 u iUu C u up 36 where the over bar indicates a time average 1 ttr uC quT 37 751 t For homogeneous turbulence7 the average of the uctuating velocities must be zero7 F O7 and we have wu CW 38 where we drop the double over bar notation since the average of an average is just the average Note that we cannot assume that the cross term 1420 is zero With these preliminary tools7 we are now ready to substitute the Reynolds decomposition into the governing advective diffusion equation with molecular diffusion coef cients as follows 80 81410 8 80 lt 890i 875 am 3m 86 0 8u7 u 0 7 8 85 0 875 835 87 D 835 39 3399 Next7 we integrate over the integral time scale t1 tt i I 8CC8ululCCi D8CC dT t t 8739 85 3 80775 7707 147 up i a 85 W 875 835 87 D 835 39 33910 Finally7 we recognize that the terms 1470 and F are zero7 and7 after moving the C term to the right hand side7 we are left with a 86 8140 a lt 86gt u 7 7 D 3m 82 try M a 87 33911 56 3 Mixing in Rivers Turbulent Di usion and Dispersion To utilize 311 we require a model for the term Since this term is of the form 140 we know that it is a mass ux Since both components of this term are uctuating it must be a mass ux associated with the turbulence Reynolds describes this turbulent component qualitatively as a form of rapid mixing thus we might make an analogy with molecular diffusion Taylor 1921 derived part of this analogy by analytically tracking a cloud of tracer particles in a turbulent ow and calculating the Lagrangian autocorrelation function His result shows that for times greater than 251 the cloud of tracer particles grows linearly with time Rutherford 1994 and Fischer et a1 1979 use this result to justify an analogy with molecular diffusion though it is worth pointing out that Taylor did not take the analogy that far For the diffusion analogy model the average turbulent diffusion time scale is At t1 and the average turbulent diffusion length scale is Am uItI 1 hence the model is only valid for times greater than 251 Using a Fick s law type relationship for turbulent diffusion gives 80 7120 Dt am 312 with 4902 D At 7411 313 Substituting this model for the average turbulent diffusive transport into 311 and dropping the over bar notation gives 80 80 8 80 8 80 As we will see in the next section D is usually much greater than the molecular diffusion 314 coef cient Dm thus the nal term is typically neglected 313 Turbulent diffusion coe icients in rivers How big are turbulent diffusion coef cients To answer this question we need to determine what the coef cients depend on and use dimensional analysis For this purpose consider a wide river with depth h and width W gt h An important property of threedimensional turbulence is that the largest eddies are usually limited by the smallest spatial dimension in this case the depth This means that turbulent properties in a wide river should be independent of the width but dependent on the depth Also turbulence is thought to be generated in zones of high shear which in a river would be at the bed A parameter that captures the strength of the shear and is also proportional to many turbulent properties is the shear velocity 741 de ned as u E 315 where To is the bed shear and p is the uid density For uniform open channel ow the shear friction is balanced by gravity and 31 Turbulence and mixing 57 Example Box 31 Turbulent diffusion in a room To demonstrate turbulent diffusion in a room a professor sprays a point source of perfume near the front of a lecture hall The room dimensions are 10 m by 10 m by 5 m and there are 50 people in the room How long does it take for the perfume to spread through the room by turbulent diffusion To answer this question we need to estimate the air velocity scales in the room Each person repre sents a heat source of 60 W hence the air ow in the room is dominated by convection The vertical buoyant velocity 11 is by dimensional analysis 70 BL1 3 where B is the buoyancy ux per unit area in L2T3 and L is the vertical dimension of the room here 5 m The buoyancy o the air increases with tem perature due to expansion The net buoyancy ux per unit area is given by H m where B is the coe 39icient of thermal expansion 000024 K71 for air H is the heat ux per unit area p is the density 125 kgm3 for air and CU is the speci c heat at constant volume 1004 JKgK for air 359 For this problem 7 50 pers 60 Wpers 102 m2 30 Wm2 H This gives a unit area buoyancy ux of 56 10 5 mZs3 and a vertical velocity of w 007 ms We now have the necessary scales to estimate the turbulent diffusion coe 39icient from 313 Taking m X 11 and I X h where h is the height of the room Dt olt nah z 035 m2s which is much greater than the molecular diffusion coe 39icient compare to Dm 1 5 s in air The mixing time can be taken from the standard deviation of the cloud width 2 tmix z For vertical mixing L 5 m and tmix is 1 minute for horizontal mixing L 10 m and tmix is 5 min utes Hence it takes a few minutes not just a couple seconds or a few hours for the students to start to smell the perfume 1L1 xghS 316 where S is the channel slope Arranging our two parameters 1 and 741 to form a diffusion coef cient gives D x u1h 317 Because the velocity pro le is much different in the vertical direction as compared with the transverse direction D is not expected to be isotropic ie it is not the same in all directions Vertical mixing Vertical turbulent diffusion coef cients can be derived from the velocity pro le see Fischer et a1 1979 For fully developed turbulent open channel ow it can be shown that the average turbulent log velocity pro le is given by 172 U1 1 1nzh 318 where a is the von Karman constant Taking a 04 we obtain Dm 00671141 319 This relationship has been veri ed by experiments for rivers and for atmospheric boundary layers and can be considered accurate to i25 58 3 Mixing in Rivers Turbulent Diffusion and Dispersion Example Box 32 Vertical mixing in a river A factory waste stream is introduced through a lateral diffuser at the bed of a river as shown in the following sketch 1 III At what distance downstream can the injection be considered as fully mixed in the vertical assumption of fully mixed77 can be de ned as the condition where concentration variations over the crosssection are below a threshold criteria Since the vertical domain has two boundaries we have to use an imagesource solution similar to 247 to compute the concentration distribution The results can be summarized by determining the appropriate value of a in the relationship 1040 where h is the depth and a is the standard devia tion of the concentration distribution Fischer et al 1979 suggest a 25 For vertical mixing we are interested in the ver tical turbulent diffusion coe 39icient so we can write h 25 2Dmt where t is the time required to achieve vertical mix ing Over the time t the plume travels downstream a distance L it We can also make the approxi mation u 01 Substituting these relationships together with 319 gives h 25 2 0067h01 L Solving for L gives L 1211 Thus a bottom or surface injection in a natural stream can be treated as fully vertically mixed af ter a distance of approximately 12 times the channel depth Transverse mixing On average there is no transverse velocity pro le and mixing coef cients must be obtained from experiments For a wealth of laboratory and eld experiments reported in Fischer et a1 1979 the average transverse turbulent diffusion coef cient in a uniform straight channel can be taken as Dm 015m 320 The experiments indicate that the width plays some role in transverse mixing however it is unclear how that effect should be incorporated Fischer et a1 1979 Transverse mixing deviates from the behavior in 320 primarily due to large coherent lateral motions which are really not properties of the turbulence in the rst place Based on the ranges reported in the experiments 320 should be considered accurate to at best i50 In natural streams the cross section is rarely of uniform depth and the fall line tends to meander These two effects enhance transverse mixing and for natural streams Fischer et a1 1979 suggest the relationship Dm 0611171 321 If the stream is slowly meandering and the sidewall irregularities are moderate the coef cient in 321 is usually found in the range 04708 Longitudinal mixing Since we assume there are no boundary effects in the lateral or longi tudinal directions longitudinal turbulent mixing should be equivalent to transverse mixing Dm DWI 322 32 Longitudinal dispersion 59 Side View of river UZ a b 0 Depthaverage concentration distributions 5 E E x x X Fig 33 Schematic showing the process of longitudinal dispersion Tracer is injected uniformly at a and stretched by the shear pro le at At Vertical di usion has homogenized the Vertical gradients and a depthaveraged Gaussian distribution is expected in the concentration pro les However7 because of non uniformity of the vertical velocity pro le and other non uniformities dead zones7 curves7 non uniform depth7 etc a process called longitudinal dispersion dominates longitudinal mixing7 and DH can often be neglected7 with a longitudinal dispersion coef cient derived in the next section taking its place Summary For a natural stream with width W 10 m7 depth h 03 m7 ow rate Q 1 mils7 and slope S 000057 the relationships 3197 3207 and 322 give Dm 64 104 mZs 323 Day 57 103 mZs 324 Dm 57 103 mZs 325 Since these calculations show that D in natural streams is several orders of magnitude greater than the molecular diffusion coef cient7 we can safely remove Dm from 314 32 Longitudinal dispersion In the previous section we saw that turbulent uctuating velocities caused a kind of random mixing that could be described by a Fickian diffusion process with larger7 turbulent diffusion coef cients In this section we want to consider what effect velocity deviations in space7 due to non uniform velocity7 or shear ow7 pro les7 might have on the transport of contaminants Figure 33 depicts schematically what happens to a dye patch in a shear ow such as open channel ow If we inject a contaminant so that it is uniformly distributed across the cross section at point a7 there will be no vertical concentration gradients and7 therefore7 no net 6O 3 Mixing in Rivers Turbulent Di usion and Dispersion diffusive ux in the vertical at that point The patch of tracer will advect downstream and get stretched due to the different advection velocities in the shear pro le After some short distance downstream the patch will look like that at point At that point there are strong vertical concentration gradients and therefore a large net diffusive ux in the vertical As the stretched out patch continues downstream turbulent diffusion will smooth out these vertical concentration gradients and far enough downstream the patch will look like that at point The amount that the patch has spread out in the downstream direction at point is much more than what could have been produced by just longitudinal turbulent diffusion This combined process of advection and vertical diffusion is called dispersion If we solve the transport equation in three dimensions using the appropriate molecular or turbulent diffusion coef cients we do not need to do anything special to capture the stretching ef fect of the velocity pro le described above Dispersion is implicitly included in threedimensional models However we would like to take advantage of the fact that the concentration distribution at the point is essentially one dimensional it is well mixed in the y and z directions In addition the concentration distribution at point is observed to be Gaussian suggesting a Fickian type diffusive process Taylor s analysis for dispersion as presented in the following is a method to include the stretching effects of dispersion in a onedimensional model The result is a one dimensional transport equation with an enhanced longitudinal mixing coef cient called the longitudinal dispersion coef cient As pointed out by Fischer et al 1979 the analysis presented by G 1 Taylor to compute the longitudinal dispersion coef cient from the shear velocity pro le is a particularly impressive example of the genius of G 1 Taylor At one point we will cancel out the terms of the equation for which we are trying to solve Through a scale analysis we will discard terms that would be dif cult to evaluate And by thoroughly understanding the physics of the problem we will use a steady state assumption that will make the problem tractable Hence just about all of our mathematical tools will be used 321 Derivation of the advective dispersion equation To derive an equation for longitudinal dispersion we will follow a modi ed version of the Reynolds decomposition introduced in the previous section to handle turbulence Referring to Figure 34 we see that for one component of the turbulent decomposition we have a mean velocity that is constant at a point z in three dimensional space and uctuating velocities that are variable in time so that umi t uxi t 326 For shear ow decomposition here we show the log velocity pro le in a river we have a mean velocity that is constant over the depth and deviating velocities that are variable over the depth such that U uz 327 32 Longitudinal dispersion 61 u z I I U39Z I I I I I I t I U u Reynolds Decomposition Reynolds Decomposition turbulence shearflow Fig 34 Comparison of the Reynolds decomposition for turbulent ow left and shear ow right where the over bar represents a depth average7 not an average 0 turbulent uctuations We explicitly assume that U and u are independent of z A main difference between these two equations is that 326 has a random uctuating component u mit whereas7 327 has a deterministic7 non random and fully knownl uctuating component 127 which we rather call a deviation than a uctuation As for turbulent diffusion above7 we also have a Reynold s decomposition for the concentrations Cz72 Cx7 z 328 which is dependent on m and for which C m7 z is unknown Armed with these concepts7 we are ready to follow Taylor s analysis and apply it to longi tudinal dispersion in an open channel For this derivation we will assume laminar ow and an in nitely wide channel with no ux boundaries at the top and bottom7 so that v w O The dye patch is introduced as a plane so that we can neglect lateral diffusion 8083 0 The governing advective diffusion equation is 80 80 820 820 E 14 me Dzw This equation is valid in three dimensions and contains the effect of dispersion The diffusion 329 coef cients would either be molecular or turbulent7 depending on whether the ow is laminar or turbulent Substituting the Reynolds decomposition for the shear velocity pro le7 we obtain 85 C 85 0 825 0 825 0 8t Hullquot 8m Dw 3m2 131 822 39 Since we already argued that longitudinal dispersion will be much greater than longitudinal 330 diffusion7 we will neglect the Dm term for brevity it can always be added back later as an additive diffusion term Also7 note that U is not a function of 2 thus7 it drops out of the nal Dz term As usual7 it is easier to deal with this equation in a frame of reference that moves with the mean advection velocity thus7 we introduce the coordinate transformation 62 3 Mixing in Rivers Turbulent Di usion and Dispersion 5 m 7 at 331 T t 332 z z 333 and using the chain rule the differential operators become 8 i 8 85 8 87 8 82 m mwm mm 8 8 5 334 2333333 6325 7 85 8t 87 8t 82 8t 8 8 7 E 7 738 5 335 333amp3 82 7 85 82 8739 32 82 82 8 7 336 Substituting this transformation and combining like terms and dropping the terms discussed above we obtain 85 C BuU C 820 8739 85 z 822 7 which is effectively our starting point for Taylor s analysis The discussion above indicates that it is the gradients of concentration and velocity in the 337 vertical that are responsible for the increased longitudinal dispersion Thus we would like at this point to remove the non uctuating terms terms without a prime from 337 This step takes great courage and profound foresight since that means getting rid of 8582 which is the quantity we would ultimately like to predict Fischer et a1 1979 As we will see however this is precisely what enables us to obtain an equation for the dispersion coef cient To remove the constant components from 337 we will take the depth average of 337 and then subtract that result from 337 The depth average operator is 1 h d 338 h A 2 lt gt Applying the depth average to 337 leaves 85 810 3 85 0 339 since the depth average of Cquot is zero but the cross term u C may not be zero This equation is the one dimensional governing equation we are looking for We will come back to this equation once we have found a relationship for MC Subtracting this result from 337 we obtain 80 8 80 7 810 D 82Cquot 5 a5 7quot a5 85 Z 822 340 32 Longitudinal dispersion 63 which gives us a governing equation for the concentration deviations C If we can solve this equation for C then we can substitute the solution into 339 to obtain the desired equation for 5 Before we solve 340 let us consider the scale of each term and decide whether it is necessary to keep all the terms This is called a scaleanalysis We are seeking solutions for the point in Figure 33 At that point a particle in the cloud has thoroughly sampled the velocity pro le and C ltlt 5 Thus 8C 86 u 85 ltltu 85 and 341 EmCquot 8 85 lt u 8 5 342 We can neglect the two terms on the left hand side of the inequalities above leaving us with 80 8 820 W t 7quot 8 5 Dz 822 39 This might be another surprise In the turbulent diffusion case it was the cross term u C that 343 became our turbulent diffusion term Here we have just discarded this term In turbulence as will also be the case here for dispersion that cross term represents mass transport due to the uctuating velocities But let us also take a closer look at the middle term of 343 This term is an advection term working on the mean concentration 6 but due to the non random deviating velocity Thus it is the transport term that represents the action of the shear velocity pro le Next we see another insightful simpli cation that Taylor made In the beginning stages of dis persion a and in Figure 33 the concentration uctuations are unsteady but downstream at point c after the velocity pro le has been thoroughly sampled the vertical concentration uctuations will reach a steady state there will be a balanced vertical transport of contami nant which represents the case of a constant timeinvariant dispersion coef cient At steady state 343 becomes 85 8 8C D 344 7quot 85 82 lt Z 82 gt l where we have written the form for a non constant Dz Solving for Cquot by integrating twice gives 85 Z 1 1 C 2 u dzdz 345 l 5 5 0 Dz 0 l l which looks promising but still contains the unknown E term Step back for a moment and consider what the mass ux in the longitudinal direction is In our moving coordinate system we only have one velocity thus the advective mass ux must be qa u Cl 346 To obtain the total mass ux we take the depth average 64 3 Mixing in Rivers Turbulent Di usion and Dispersion 1 h qa uC Cdz h 0 1 h uCdz h 0 MC 347 Recall that the depth average of 15 is zero Substituting the solution for Cquot from 345 the depth average mass ux becomes 1 h 8 394 1 394 u u dzdzdz 348 q 110 85 0 Dz 0 We can take 8585 outside of the integral since it is independent of 2 leaving us with 85 7D 349 gm L where D 1h 21Z ddd 350 7 u u z z z L h 0 0 Dz 0 and we have a Fick s law type mass ux relationship in 349 Since the equation for BL is just a function of the depth and the velocity pro le we can calculate DL for any velocity pro le by integrating thus we have an analytical solution for the longitudinal dispersion coef cient The nal step is to introduce this result into the depth average governing equation 339 to obtain 85 8 85 D 351 5 85 lt L 85 7 which in the original coordinate system gives the onedimensional advective dispersion equation 85 85 8 85 352 u lt L 8m gt E 825 a with DL as de ned by 350 322 Calculating longitudinal dispersion coe icients All the brilliant mathematics in the previous section really paid off since we ended up with an analytical solution for the dispersion coef cient D 1hZlZddd 7 U U222 L 710 oDzo In real streams it is usually the lateral shear in the y direction rather than the vertical shear that plays the more important role For lateral shear Fischer et a1 1979 derive by a similar analysis the relationship D 1 hy 1 y hd d d 3 53 7 u u L 1 0 0 Dyh 0 y y y where A is the crosssectional area of the stream and W is the width Irrespective of which relationship we choose the question that remains is how do we best calculate these integrals 32 Longitudinal dispersion 65 Example Box 33 River mixing processes As part of a dye study to estimate the mixing c0 e 39icients in a river a student injects a slug point source of dye at the surface of a stream in the mid le 0f the crosssection Discuss the mixing processes and the length scales affecting the injected tracer Although the initial vertical momentum of the dye injection general y results in good vertical mixing assume here that the student carefully injects the dye just at the stream surface Vertical turbulent diffusion will mix the dye over the depth and from Ebrample Box 32 above the injection can be treated as mixed in the vertical after the point Lz 1211 where h is the stream depth As the dye continues to move downstream lat eral turbulent diffusion mixes the dye in the trans verse direction Based on the discussion in Ebram le Box 32 the tracer can be considered well mixed lat erally after W2 311 where W is the stream width For the region between the injection and Lz the dye cloud is fully threedimensional and n0 simpli cations can be made to the transport equation Be yond Lz the cloud is vertically mixed and longitudi nal dispersion can be applied For distances less than Ly a twodimensional model with lateral turbulent diffusion and longitudinal dispersion is required For distances beyond Ly a onedimensional longitudinal dispersion model is acceptable Ly Analytical solutions For laminar ows analytical velocity pro les may sometimes exist and 350 can be calculated analytically Following examples in Fischer et al 1979 the simplest ow is the ow between two in nite plates where the top plate is moving at U relative to the bottom plate For that case UZdZ DL m 354 where d is the distance between the two plates Similarly for laminar pipe ow the solution is aZUZ D 0 355 L 19213 l where a is the pipe radius U0 is the pipe centerline velocity and D is the radial diffusion coef cient For turbulent How an analysis similar to the section on turbulent diffusion can be carried out and the result is that 350 keeps the same form and we substitute the turbulent diffusion coef cient and the mean turbulent shear velocity pro le for Dz and u The result for turbulent ow in a pipe becomes DL 10151 356 One result of particular importance is that for an in nitely wide open channel of depth h Using the log velocity pro le 318 with von Karman constant a 04 and the relationship 350 the dispersion coef cient is DL 593m 357 Comparing this equation to the prediction for longitudinal turbulent diffusion from the previous section Dim 01571143 we see that DL has the same form cx hm and that BL is indeed much greater than longitudinal turbulent diffusion For real open channels the lateral shear velocity pro le between the two banks becomes dominant and the leading coef cient for DL can range 66 3 Mixing in Rivers Turbulent Di usion and Dispersion from 5 to 7000 Fischer et a1 1979 For further discussion of analytical solutions see Fischer et a1 1979 Numerical integration In many practical engineering applications the variable channel ge ometry makes it impossible to assume an analytical shear velocity pro le In that case one alternative is to break the river cross section into a series of bins measure the mean velocity in each bin and then compute the second relationship 353 by numerical integration Fischer et a1 1979 give a thorough discussion of how to do this Engineering estimates When only very rough measurements are available it is necessary to come up with a reasonably accurate engineering estimate for DL To do this we rst write 353 in non dimensional form using the dimensionless variables denoted by de ned by y ng 1 Wu Dy D yD h Em where the over bar indicates a cross sectional average As we already said longitudinal dispersion in streams is dominated by the lateral shear velocity pro le which is why we are using y and Dy Substituting this non dimensionalization into 353 we obtain 22 DL W I 358 Dy where I 71 MW L u hdydydy 3 59 0 0 D17 0 39 39 As Fischer et a1 1979 point out in most practical cases it may suf ce to take I z 001t001 To go one step further we introduce some further scales measured by Fischer et a1 1979 From experiments and comparisons with the eld the ratio WUZ can be taken as 02i003 For irregular streams we can take D y 06du Substituting these values into 358 with I 0033 gives the estimate 2 2 DL 0011wa 360 14 which has been found to agree with observations within a factor of 4 or so Deviations are primarily due to factors not included in our analysis such as recirculation and dead zones Geomorphological estimates Deng et a1 2001 present a similar approach for an engineer ing estimate of the dispersion coef cient in straight rivers based on characteristic geomorpho logical parameters The expression they obtain is D 015 W 53 2 L lt gt a hm 860 h 73 where eto is a dimensionless number given by 1 a W 13938 0145 362 6 0 1 3520 h gt These equations are based on the hydraulic geometry relationship for stable rivers and on the assumption that the uniform ow formula is valid for local depth averaged variables Deng et a1 33 Application Dye studies 67 2001 compare predictions for this relationship and predictions from 360 with measurements from 73 sets of eld data More than 64 of the predictions by 361 fall within the range of 05 DLpredidionDlewmmmem 2 This accuracy is on average better than that for 360 however in some individual cases 360 provides the better estimate Dye studies One of the most reliable means of computing a dispersion coef cient is through a dye study as illustrated in the applications of the next sections It is important to keep in mind that since BL is dependent on the velocity pro le it is in general a function of the ow rate Hence a BL computed by a dye study for one ow rate does not necessarily apply to a situation at a much different ow rate In such cases it is probably best to perform a series of dye studies over a range of ow rates or to compare estimates such as 360 to the results of one dye study to aid predictions under different conditions 33 Application Dye studies The purpose of a dye tracer study is to determine a river s ow and transport properties in particular the mean advective velocity and the effective longitudinal dispersion coef cient To estimate these quantities we inject dye upstream measure the concentration distribution down stream and compare the results to analytical solutions The two major types of dye injections are instantaneous injections and continuous injections The following sections discuss typical results for these two injection scenarios 331 Preparations To prepare a dye injection study we use engineering estimates for the expected transport prop erties to determine the location of the measurement stations the duration of the experiment the needed amount of dye and the type of dye injection For illustration purposes assume in the following discussion that you measure a river cross section to have depth h 035 m and width W 10 m The last time you visited the site you measured the surface current by timing leaves oating at the surface and found Us 53 cms A rule of thumb for the mean stream velocity is V 085U9 045 cms You estimate the river slope from topographic maps as S 00005 The channel is uniform but has some meandering Measurement stations A critical part of a dye study is that you measure far enough down stream that the dye is well mixed across the cross section If you measure too close to the source you might obtain a curve for Ct that looks Gaussian but the concentrations will not be uni form across the cross section and dilution estimates will be biased We use our mixing length rules of thumb to compute the necessary downstream distance Assuming the injection is at a point conservative case it must mix both vertically and transversely The two relevant turbulent diffusion coef cients are Dm 0067dxgdS 97 104 mZs 363 68 3 Mixing in Rivers Turbulent Di usion and Dispersion Dm 06dgdS 87 103 mZs 364 The time it takes for diffusion to spread a tracer over a distance 1 is 2125D thus the distance the tracer would move downstream in this time is 2 U 125D There are several injection possibilities If you inject at the bottom or surface the dye must m 365 spread over the whole depth if you inject at middle depth the dye must only spread over half the depth Similarly if you inject at either bank the dye must spread across the whole river if you inject at the stream centerline the dye must only spread over half the width Often it is possible to inject the dye in the middle of the river and at the water surface For such an injection we compute in our example that Lmz for spreading over the full depth is 42 m whereas Lmy for spreading over half the width is 95 m Thus the measuring station must be at least Lm 100 m downstream of the injection The longitudinal spreading of the cloud is controlled by the dispersion coef cient Using the estimate from Fischer et al 1979 given in 360 we have UZWZ dxgdS 154 mZs 366 DL 0011 We would like the longitudinal width of the cloud at the measuring station to be less than the distance from the injection to the measuring station thus we would like a Peclet number P5 at the measuring station of 01 or less This criteria gives us 7 D 7 342 m 367 Lm Since for this stream the Peclet criteria is more stringent than that for lateral mixing we chose a measurement location of Lm 350 m Experiment duration We must measure downstream long enough in time to capture all of the cloud or dye front as it passes The center of the dye front reaches the measuring station with the mean river ow to LCU Dispersion causes some of the dye to arrive earlier and some of the dye to arrive later An estimate for the length of the dye cloud that passes after the center of mass is L 3 2DLLmU 525 m 368 or in time coordinates t0 1170 s Thus we should start measuring immediately after the dye is injected and continue taking measurements until 25 to 1 t0 30 min To be conservative we select a duration of 35 min 33 Application Dye studies 69 Amount of injected dye tracer The general public does not like to see red or orange water in their rivers so when we do a tracer study we like to keep the concentration of dye low enough that the water does not appear colored to the naked eye This is possible using uorescent dyes because they remain visible to measurement devices at concentrations not noticeable to casual observation The most common uorescent dye used in river studies is Rhodamine WT Many other dyes can also be used including other types of Rhodamine B 6G etc or Fluorescein Smart amp Laidlay 1977 discuss the properties of many common uorescent dyes In preparing a dye study it is necessary to determine the amount mass of dye to inject A common eld uorometer by Turner Designs has a measurement range for Rhodamine WT of 004 to 4010 2 mgl To have good sensitivity and also leave room for a wide range of river ow rates you should design for a maximum concentration at the measurement station near the upper range of the uorometer for instance CWm 4 mgl The amount of dye to inject depends on whether the injection is a point source or a continuous injection For a point source injection we use the instantaneous point source solution with the longitudinal dispersion coef cient estimated above M CmmA 47rDLLmU 54 g 369 For a continuous injection we estimate the dye mass ow rate from the expected dilution m UOATCmam 63 gs 370 These calculations show that a continuous release uses much more dye than a point release These estimates are for the pure usually a powder form of the dye Type of injection To get the best injection characteristics we dissolve the powder form of the dye in a solution of water and alcohol before injecting it in the river The alcohol is used to obtain a neutrally buoyant mixture of dye For a point release we usually spill a bottle of dye mixture containing the desired initial mass of dye in the center of the river and record the time when the injection occurs For a continuous release we require some tubing to direct the dye into the river a reservoir containing dye at a known concentration and a means of regulating the ow rate of dye The easiest way to get a constant dye ow rate is to use a peristaltic pump Another means is to construct a Marriot bottle as described in Fischer et al 1979 and shown in Figure 35 The idea of the Marriot bottle is to create a constant head tank where you can assume the pressure is equal to atmospheric pressure at the bottom of the vertical tube As long as the bottle has enough dye in it that the bottom of the vertical tube is submerged a constant ow rate Q0 will result by virtue of the constant pressure head between the tank and the injection We must calibrate the ow rate in the laboratory for a given head drop prior to conducting the eld experiment The concentration of the dye CO for the continuous release is calculated according to the equation 7O 3 Mixing in Rivers Turbulent Diffusion and Dispersion Vent tube 2 1 AWLjag Fig 35 Schematic of a Marriot bottle taken from Fischer et al 1979 Measured dye concentration breakthrough curve 9 01 4 N 401wa Concentration mgI 0 01 O 5 1O 15 20 25 30 35 Time since injection start min 0 F i F L P Fig 36 Measured dye concentration for example dye study Dye uctuations are due to instrument uncertainty not due to turbulent uctuations m 000 371 where Q0 is the ow rate from the pump or Marriot bottle With these design issues complete a dye study is ready to be conducted 332 River ow rates Figure 36 shows a breakthrough curve for a continuous injection based on the design in the previous section The river ow rate can be estimated from the measured steady state concen tration in the river C1 at t 35 min Reading from the graph we have C 315 mgl Thus the actual ow rate measured in the dye study was 2 20 m3s 372 34 Application Dye study in Cowaselon Creek 71 Notice that this estimate for the river ow rate is independent of the crosssectional area To estimate the error in this measurement we use the error propagation equation 6y i 87 Sm2 373 71 877 where 6y is the error in some quantity 39y estimated from 71 measurements mi Computing the error for our river ow rate estimate we have 00 2 Q0 2 000 2 6 6 6C 6C 374 Q Q0C o02 If the uncertainties in the measurements were 0 315 i 004 mgl CO 32 i 001 gl and Q0 02 i 001 ls then our estimate should be QT 20 i 01 m3s The error propagation formula is helpful for determining which sources of error contribute the most to the overall error in our estimate 333 River dispersion coe icients The breakthrough curve in Figure 36 also contains all the information we need to estimate an in situ longitudinal dispersion coef cient To do that we will use the relationship a2 2DLt 375 Since our measurements of a are in time we must convert them to space in order to use this equation One problem is that the dye cloud continues to grow as it passes the site so the width measured at the beginning of the front is less than the width measured after most of the front has passed thus we must take an average The center of the dye front can be taken at C 0500 which passed the station at t 1294 min and represents the mean stream velocity One standard deviation to the left of this point is at C 01600 as shown in the gure This concentration passed the measure ment station at t 835 min One standard deviation to the right is at C 08400 and this concentration passed the station at t 2012 min From this information the average velocity is U 045 ms and the average width of the front is 20 2012 7 835 1177 min The time associated with this average sigma is t 835 11772 1424 min To compute DL from 375 we must convert our time estimate of at to a spatial estimate using a Eat Solving for DL gives U20 DL 2t 148 mZs 376 This value compares favorably with our initial estimate from 350 of 154 mZs 34 Application Dye study in Cowaselon Creek In 1981 students at Cornell University performed a dye study in Cowaselon Creek using an instantaneous point source of Rhodamine WT dye The section of Cowaselon Creek tested has 72 3 Mixing in Rivers Turbulent Di usion and Dispersion a very uniform cross section and a straight fall line from the injection point through the mea surement stations At the injection site the students measured the cross section and ow rate obtaining Q06 m3s W107m U017 ms h03 m From topographic maps they measured the creek slope over the study area to be S 43 1041 The concentration pro les were measured at three stations downstream The rst station was 670 m downstream of the injection the second station was 2800 m downstream of the injection and the nal station was 5230 m downstream of the injection At each location samples were taken in the center of the river and near the right and left banks Figure 37 shows the measured concentration pro les For turbulent mixing in the vertical direction the downstream distance would be Lmz 12d 17 m This location is well upstream of our measurements thus we expect the plume to be well mixed in the vertical by the time it reaches the measurement stations For mixing in the lateral direction the method in Example Box 33 using Day 015dur for straight channels gives a downstream distance of Lmy 2500 m Since the rst measurement station is at L 670 m we clearly see that there are still lateral gradients in the concentra tion cloud At the second measurement station 2800 m downstream the lateral gradients have diffused and the lateral distribution is independent of the lateral coordinate Likewise at the third measurement station 5230 m downstream the plume is mixed laterally however due to dispersion the plume has also spread more in the longitudinal direction To estimate the dispersion coef cient we can take the travel time between the stations two and three and the growth of the cloud The travel time between stations is 6t 397 hr The width at station one is 01 236 m and the width at station two is 02 448 m The dispersion coef cient is 2 2 039 039 DE 2 1 2625 51 mZs 377 Comparing to 360 and 361 we compute Balm he 34 mZs 378 DLlpeng 54 11128 379 Although the geomorphological estimate of 54 mZs is closer to the true value than is 33 mZs for practical purposes both methods give good results Dye studies however are always helpful for determining the true mixing characteristics of rivers 34 Application Dye study in Cowaselon Creek 73 Measurement at station 1 A dv i i i Concentration pgI I i 1 Time hrs Measurement at station 2 i i i Concentration pgI 45 Time hrs Measurement at station 3 A Ju i Concentration pgI Time hrs Fig 37 Measured dye concentrations at tWo stations in COWaselon Creek for a point injection Measurements at each station are presented for the stream centerline and for locations near the right and left banks 74 3 Mixing in Rivers Turbulent Di usion and Dispersion Summary This chapter presented the effects of contaminant transport due to variability in the ambient velocity In the rst section turbulence was discussed and shown to be composed of a mean velocity and a random uctuating turbulent velocity By introducing the Reynolds decomposi tion of the turbulent velocity into the advective diffusion equation a new equation for turbulent diffusion was derived that has the same form as that for molecular diffusion but with larger turbulent diffusion coef cients The second type of variable velocity was a shear velocity pro le described by a mean stream velocity and deterministic deviations from that velocity Substitut ing a modi ed type of Reynolds decomposition for the shear pro le into the advective diffusion equation and depth averaging led to a new equation for longitudinal dispersion and an integral relationship for calculating the longitudinal dispersion coef cient To demonstrate how to use these equations and obtain eld measurements of these properties the chapter closed with an example of a simple dye study to obtain stream ow rate and longitudinal dispersion coef cient Exercises 31 Properties of turbulence The axial velocity u of a turbulent jet can be measured using a laser Doppler velocimetry LDV system Obtain a data le that contains only one column the u component velocity with a unit of ms the sampling rate is 100 Hz Do the following 0 Plot the velocity and examine whether the ow is turbulent by checking the randomness in your plot Comment on your observations 0 Use Matlab to calculate the mean velocity and create a variable that contains just the uctu ating velocity you do not have to turn in anything for this step 0 Plot the uctuating velocity 0 Calculate the mean value of the uctuating velocity 0 Write a program to compute the correlation function normalized so the maximum correlation is 10 and plot the function 0 Calculate the integral time scale integrate between 0 to 03 s only 0 Estimate the typical size of the eddies nd the integral length scale 32 Turbulent diffusion coef cients The Rhine river in the vicinity of Karlsruhe has width B 300 m and Manning s friction factor 002 The slope is 1 1041 Assume the river is always at normal depth and that the width is constant for all ow rates 0 For each ow rate in Table 31 compute the dispersion coef cient from the equation from Fischer et al 1977 2B2 DL 001112 380 Uak where U is the mean velocity and ur is the shear velocity Ebrercises 75 Table 31 Flow rates in the Rhine river near Karlsruhe Flow rate 1113 S 120 240 500 800 1200 o For each ow rate in Table 31 compute the dispersion coef cient from the equation from Deng et al 2001 D 015 B 53 2 L lt gt ltigt huT 860 h 1LT with 1 U B 138 0145 382 6 0 3520 h gt where h is the water depth and B is the channel width 0 Plot the dispersion coef cient for each method as a function of ow rate and comment on the trends Do you think it is adequate to do one dye study to evaluate DL Why or why not 33 Numerical integration Using the velocity pro le data in Table 32 from Nepf 1995 perform a numerical integration of 353 to estimate a longitudinal dispersion coef cient You should obtain a value of DL 15 mZs 34 Dye study This problem is adapted from Nepf 1995 A small stream has been found to be contaminated with Lindane a pesticide known to cause convulsions and liver damage Ground water wells in the same region have also been found to contain Lindane and so you suspect that the river contamination is due to groundwater in ow To test your theory you conduct a dye study using a continuous release of dye Based on the information given in Figure 38 what is the groundwater volume ux and the concentration of Lindane in the groundwater between Stations 2 and 3 The variables in the gure are Qd the volume ow rate of the dye at the injection Cd the concentration of the dye at the injection and at downstream stations 0 the concentration of Lindane in the river at each station W the width of the river and d the depth in the river Hint this is a steady state problem so you do not need to use diffusion coef cients to solve the problem other than to determine whether the dye is well mixed when it reaches Station 2 Due to problems with the pump the dye ow rate has an error of Q1 100 i 5 cm3s Assume this is the only error in your measurement and report your measurement uncertainty 76 3 Mixing in Rivers Turbulent Diffusion and Dispersion Table 32 Stream Velocity data for calculating a longitudinal dispersion coef cient Station Distance from Total depth Measurement Velocity number bank d depth zd u le le H CmS 1 00 0 0 00 2 300 14 06 30 3 584 42 02 60 0 8 6 4 4 813 41 02 168 0 8 176 5 1041 43 02 134 0 8 136 6 1372 41 02 136 0 8 142 7 1702 34 02 90 0 8 9 6 8 2032 30 02 50 08 5 4 9 2362 15 02 10 08 1 4 10 2692 15 02 08 08 1 2 11 3150 14 06 00 12 3607 0 0 00 35 Accidental kerosene spill A tanker truck has an accident and spills 100 kg of kerosene into a river The spill occurs over a span of 3 minutes and can be approximated as uniformly distributed across the lateral cross section of the river A sh farm has its water intake 25 km downstream of the spill location Refer to Figure 39 0 Use the following relationship to compute the longitudinal dispersion coef cient7 DL D 015 B 53 U 2 L 383 H ul 86t0 H ul B and H are the river width and depth7 U is the average ow velocity7 741 is the shear velocity7 and eto is a non dimensional number given by 1 U B 138 0145 384 6 0 1 3520 gt 229 o What is the length in the downstream direction that the spill occupies due to its 3 minute duration At what point downstream of the spill do you think it would be reasonable to approximate the spill using an instantaneous point source release Ebrercises 77 Station 2 X 70 m Cd 10 ugI Station 12 Cl 05 ugl Dye injected X 0 m Qd 100 cm3s Cd 50 mgI Station 3 River crosssection X 170 m W 1 m Cd 8 ugI d 05 m 0 09 ugI Fig 38 Dye study to determine the source of Lindane contamination in a small stream Fish farm Spill location inlet EEI U 40 cms 3 0 X25km H2m S00001 B25m X Fig 39 Schematic of the accidental spill with the important measurement Values B and H are the width and depth of the river7 Ui is the average river ow Velocity at the accident location7 and S is the channel slope 0 Plot the concentration in the river as a function of downstream distance at t 2 hr after the accident From the gure7 determine the location of the center of mass of the kerosene cloud7 the maximum concentration in the river7 and the characteristic width of the cloud in the z direction approximate the cloud using one standard deviation of the concentration distribution 0 Write the equation for the concentration as a function of time at the inlet to the sh farm Plot your equation and determine at what time the maximum concentration passes the sh farm 0 A dye study was conducted in the river at an earlier time and concluded that there is a ow of groundwater into the river along the stretch between the accident and the sh farm How would this information in uence the results reported in the previous steps of this problem 36 Ocean mixing This problem is adapted from Nepf 1995 Ten surface drogues are released into a coastal region at local coordinates my 07 0 The drogues move passively with the surface currents and are tracked using radio signals Their locations at the end of 251 1 and t2 20 days are given in the following table 78 3 Mixing in Rivers Turbulent Di usion and Dispersion Table 33 Drogue position data Drogue 031 1031 032 1032 Number km 1 2 5 02 5 3 8 1 2 4 6 14 23 1 0 3 2 3 1 2 66 3 9 4 3 1 0 4 67 2 8 5 1 5 0 8 05 4 2 6 1 4 2 1 101 3 6 7 4 7 2 1 66 1 4 8 2 7 0 2 60 2 9 9 1 5 26 32 2 1 10 4 9 23 4 0 1 7 1 Estimate the advection velocity and the lateral coef cients of diffusion Dm and Dy for this coastal region Using the radio links the positions of all ten drogues can be collected within ten min utes Suppose the radio link were to break down and the positions were instead determined through visual observation Even using a helicopter it requires nearly four hours to locate all ten drogues How does this change the accuracy of your data Can you still consider the to measurements to be synoptic Later a freight ship is caught in a winter storm off the coast where this drogue study was conducted High winds and rough seas cause several shipping containers to be washed over board One of the containers breaks open releasing its contents 29000 children s bathtub toys Estimate how long it will take for the toys to begin to wash up on shore assuming the same transport characteristics as during the drogue study and that the spill occurs 1 km off the coast This really happened in the Paci c Ocean and the trajectory of the bathtub toys plastic turtles and ducks were subsequently used to gain information about the current 03 system 37 Mixing ofa continuous point source in a river Many dye studies are conducted by injecting a continuous point source of dye at one station and measuring the dilution at downstream stations where the dye is assumed to be well mixed across the cross section The continuous point source solution in an in nite domain with constant uniform advection current in the z direction U is 7 2U 7 2U m exp 2 20 7 y yo 33985 47rzD1Dy 4Dzm 4Dym where m is the mass ux of dye at the injection z is the downstream distance Dz is the vertical diffusion coef cient Dy is the lateral diffusion coef cient 20 is the vertical position of the injection and yo is the lateral position of the injection To derive this solution we have assumed that longitudinal diffusion is negligible called the slender plume approximation Cm y 2 Ebrercises 79 In a river there are four boundaries that must be accounted for In the vertical direction there are boundaries at the channel bed and at the freewater surface In the lateral direction there are boundaries at each channel bank Write a Matlab function that solves this problem for an arbitrary injection point at 0 yo 20 It should take the vertical coordinate as positive upward with origin at the channel bed and the lateral coordinate as positive toward the right hand bank with origin at the channel center line Dimensional analysis can be used to estimate the distance Lm downstream of an injection at z 0 at which the injection may be considered well mixed in the vertical or lateral direction This relationship is LZU Lm X T 386 where L is the distance over which the dye must spread for vertical mixing this would be the water depth Lm is the downstream distance to the point where the cloud can be considered well mixed and D is the pertinent diffusion coef cient for vertical mixing this would be Dz To calibrate this relationship we de ne well mixed as the point at which the ratio of the minimum dye concentration 0mm to the maximum dye concentration Cm across the cross section reaches a threshold value Common practice is to de ne well mixed as OwnCm 095 Using this criteria we can calculate Lm and a proportionality constant 04 can be determined giving the relationship LZU Lm 1 387 Use this information to study a river with the following characteristics 0 Width B 107 m 0 Depth H 03 m o Slope S 43 10 4 0 Mean ow velocity U 017 ms 0 Dye injection rate m 1 gs Use enough image sources that your solution is independent of the number of images you are using and answer the following questions 1 What is the value of Manning s roughness coef cient 71 that corresponds to the given ow depth and channel slope Do you think that this value is reasonable If it seems too high or too low do you expect that the estimate for the shear velocity of ur gH S is an over prediction or under prediction for this stream Use your calibrated value of n for the remaining questions Plot the relative concentration Cz 0 HCz 0 05H for an injection at 0 0 05H versus downstream distance for z between z 1 m and z 45 m to 03 Calibrate the coef cient 04 in 387 for vertical mixing for an injection at yo 0 and 20 0 025H05H075H H for the criteria C392397m39nC2Wmm 095 Repeat your calculations in the previous step but move the injection to yo B2 Does this affect your results Why or why not F 80 3 Mixing in Rivers Turbulent Di usion and Dispersion U Calibrate the coef cient 04 in 387 for lateral mixing for an injection at 20 05H and yo 1327 1347 07B477B2 for the criteria CyWnCwmlm 095 DO you think your results would change if you moved the injection at 20 to a different elevation Why or why not 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