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- https://ufdc.ufl.edu/UF00098181/00001
## Material Information- Title:
- Dispersive mass transport in oscillatory and unidirectional flows
- Creator:
- Taylor, Robert Bruce, 1942-
- Publication Date:
- 1974
- Copyright Date:
- 1974
- Language:
- English
- Physical Description:
- xvi, 143 leaves. : illus. ; 28 cm.
## Subjects- Subjects / Keywords:
- Diffusion coefficient ( jstor )
Flow coefficients ( jstor ) Fluid shear ( jstor ) Fourier coefficients ( jstor ) Pressure gradients ( jstor ) Shear flow ( jstor ) Sine function ( jstor ) Transport phenomena ( jstor ) Velocity ( jstor ) Velocity distribution ( jstor ) Civil and Coastal Engineering thesis Ph. D ( lcsh ) Dissertations, Academic -- Civil and Coastal Engineering -- UF ( lcsh ) Fluid dynamics ( lcsh ) Hydraulics ( lcsh ) One-dimensional flow ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis -- University of FLorida.
- Bibliography:
- Bibliography: leaves 140-142.
- General Note:
- Typescript.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 000580815 ( AlephBibNum )
14087798 ( OCLC ) ADA8920 ( NOTIS )
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DISPERSIVE MASS TRANSPORT IN OSCILLATORY AND UNIDIRECTIONAL FLOWS by ROBERT BRUCE TAYLOR III A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY ACKNOWLEDGMENTS The author would like to thank Dr. Robert G. Dean, Professor of the Department of Civil and Coastal Engineering, who supervised the research. Dr. Dean's many hours of encouragement and patient teaching will not be forgotten. Appreciation is also extended to Dr. Wayne C. Huber, Associate Professor of the Department of Environmental Engineering Sciences, whose interest and suggestions were of a great help to the author. Ms. Pat Hulit and Mrs. Wanda Smith worked many arduous hours on short notice to type a highly professional final draft of the manuscript. Special thanks are given to Mrs. Evelyn Hill, who assisted in many ways including the typing of the rough draft of the manuscript, and to Ms. Denise Frank, who did the drafting. This research was sponsored by the Florida Power and Light Company through a grant to the Coastal and Oceanographic Engineering Laboratory, University of Florida. Both the funds made available by the Sponsor and the use of the COE Laboratory facilities are greatly appreciated. All computer work associated with this project was done on the Northeast Regional Data Center's IBM System/370-165. TABLE OF CONTENTS Acknowledgments . . . . . . List of Tables . . . . . . . List of Figures . . . . . . Key to Symbols and Abbreviations . . Abstract . . . . . . . . Page . iii . vi , vii Six Sxv CHAPTER I INTRODUCTION . . . . . . . . . . . . I-A General Background . . I-B Discussion of Previous Work Mass Transport . . . on D spearss .ve on Dispersive . . . . . . . I-B.1 Unidirectional Flow . . . . . . I-B.2 Oscillatory Flow . . . . . . . . I-C Description and Scope of the Present Work . . . II CONSIDERATIONS FOR APPLICATION OF TAYLOR'S DISPERSION ANALYSIS TO FREE SURFACE FLOWS. . . . . II-A General . . . . II-B Unidirectional Flow . . II-C Oscillatory Flow . . . II-D Applications of Dispersion by Taylor's Method . . 22 . . . . . 22 . . . . . . 27 . . . . . . 30 Coefficients Predicted . . . . . . 33 III DISPERSIVE MASS TRANSPORT IN AN INFINITELY WIDE RECTANGULAR CHANNEL . . . . . . . . . III-A Steady Unidirectional Flow . . . . . . III-A.1 Velocity Distribution. . III-A.2 Concentration Distribution III-A.3 Dispersive Mass Transport. III-B Oscillatory Flow. . . . . . . . . . III-B.1 Velocity Distribution. . III-B.2 Concentration Distribution III-B.3 Dispersive Mass Transport. III-C Discussion of Two Dimensional Shear Flow Results. III-C.I Dispersion in Unidirectional Flow as a Limit of the Oscillatory Flow Case . . III-C.2 Characteristics of Dispersive Mass Transport in Oscillatory Flow . . . ' I I f I I f I I I ) I I I I I CHAPTER Page IV DISPERSIVE MASS TRANSPORT IN RECTANGULAR CHANNELS OF FINITE WIDTH. . . . . . . . .. 66 IV-A Oscillatory Flow. . . . . . . . ... 66 IV-A.1 Velocity Distribution . . . . .. 67 IV-A.2 Concentration Distribution. . . . .. 71 IV-A.3 Dispersive Mass Transport . . . ... 74 IV-B Steady Unidirectional Flow . . . . ... 90 IV-B.1 Velocity Distribution . . . . .. 90 IV-B.2 Concentration Distribution. . . . ... 93 IV-B.3 Dispersive Mass Transport . . . ... 95 V COMPARISON OF RESULTS WITH FIELD AND 100 LABORATORY DATA . . . . . . . . . . . V-A Unidirectional Flow. . . . . . . . ... 100 V-B Oscillatory Flow . . . . . . . .... 106 VI SUMMARY AND CONCLUSIONS . . . . . . .... 116 APPENDIX A DERIVATION OF NORMALIZING EXPRESSIONS FOR TWO DIMENSIONAL OSCILLATORY SHEAR FLOW SOLUTION . . . . .... 120 A-1 Surface Amplitude of Velocity, umax . . . .. 120 A-2 Surface Excursion Length, L . . . . . . 124 B MATHEMATICAL DETAILS FOR THREE DIMENSIONAL OSCILLATORY SHEAR FLOW DISPERSION . . . . .... 125 B-1 Velocity Solution . . . . . . . .. 125 B-2 Concentration Solution . . . . . . .. 128 B-3 Dispersive Mass Transport. . . . . . ... 134 C DERIVATION OF umax NORMALIZING EXPRESSION FOR THREE DIMENSIONAL SHEAR FLOW SOLUTION . . . ... .137 REFERENCES. . . . . . . . . ... ....... 140 BIOGRAPHICAL SKETCH . . . . . . . .... .... 143 LIST OF TABLES Table Page 1 PREDICTED RESONANT PEAK DATA FOR LONGITUDINAL DISPERSION COEFFICIENT IN THREE DIMENSIONAL OSCILLATORY SHEAR FLOW. . . . . . . .. 83 2 LIMITING WIDTH-TO-DEPTH RATIOS FOR APPLICATION OF THE TWO DIMENSIONAL OSCILLATORY SHEAR FLOW DISPERSION COEFFICIENT. . . . . . . . 87 3 OBSERVED DISPERSION DATA AND PREDICTED VALUES USING (111-59) . . . . . . .... 106 4 OBSERVED DISPERSION DATA AND PREDICTED VALUES FOR THE MERSEY NARROWS . . . . . .. 110 5 OBSERVED DISPERSION DATA AS REPORTED BY SEGALL AND CORRESPONDING PREDICTED LIST OF FIGURES Figure Page 1 DEFINITION SKETCH. .. . . . . . . .22 2 DEFINITION SKETCH FOR TWO DIMENSIONAL SHEAR FLOW . . . . . . . .. .. . 34 3 NON-DIMENSIONAL DISPERSION COEFFICIENT 4 DISPERSION IN UNIDIRECTIONAL FLOW AS A LIMIT OF THE OSCILLATORY FLOW CASE . . . . . .. 55 5 COMPARISON OF DISPERSION FOR DIURNAL AND SEMI-DIURNAL PERIODS OF OSCILLATION WITH SAME umax . . . ... 58 6 COMPARISON OF DISPERSION FOR DIURNAL AND SEMI-DIURNAL PERIODS OF OSCILLATION WITH SAME SURFACE EXCURSION LENGTH . . . . . . . . . .. . 60 7 NON-DIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT E', AS A FUNCTION OF T' FOR TWO DIMENSIONAL x z OSCILLATORY SHEAR FLOW . . . . . . . .. 61 8 NON-DIMENSIONAL VELOCITY AND CONCENTRATION VARIATION COMPONENT PROFILES FOR T' = 0.5. . . . . . 63 9 NON-DIMENSIONAL VELOCITY AND CONCENTRATION VARIATION COMPONENT PROFILES FOR T' = 1.58 . . . . . 64 10 NON-DIMENSIONAL VELOCITY AND CONCENTRATION VARIATION COMPONENT PROFILES FOR T' = 5.0. . . . . . 65 11 DEFINITION SKETCH FOR THREE DIMENSIONAL SHEAR FLOW 67 12 NON-DIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT E',AS A FUNCTION OF T' FOR FULL RANGE OF T' VALUES ... 79 x z C 13 NON-DIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT E', AS A FUNCTION OF T' FOR T' = 1.0, 0.67, 0.33 xz c AND 0.1. . . . . . . . . . . . 81 14 NON-DIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT E' AS A FUNCTION OF T' FOR T' = 0.1, 0.01, 0.001, x z c 0.0001, and <1 x 10-6. . . . .............. 82 LIST OF FIGURES (Continued) Figure Page 15 NON-DIMENSIONAL VERTICAL MIXING TIME, T', AT WHICH MAXIMUM E' OCCURS AS A FUNCTION OF T' . 84 16 PEAK Ex AS A FUNCTION OF T'. . . . . .... 84 17 SYMMETRIC DISPLAY OF ISOLINES OF E' x 105 x IN THREE DIMENSIONAL OSCILLATORY SHEAR FLOW. ... 89 18 NON-DIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT, E', AS A FUNCTION OF T' FOR THREE DIMENSIONAL UNIDIRECTIONAL SHEAR FLOW. . . . . . ... 98 19 COMPARISON OF PREDICTED AND OBSERVED LONGITUDINAL DISPERSION IN THREE DIMENSIONAL UNIDIRECTIONAL SHEAR FLOW . . . . . . . . ... . 101 20 COMPARISON OF PREDICTED AND OBSERVED LONGITUDINAL DISPERSION FOR THE ENGLISH CHANNEL, LIVERPOOL BAY, AND CUMBERLAND COAST . . . . . . . .. 109 21 COMPARISON OF PREDICTED AND OBSERVED LONGITUDINAL DISPERSION FOR THE MERSEY ESTUARY. . . . .. 112 22 COMPARISON OF PREDICTED AND OBSERVED LONGITUDINAL DISPERSION USING SEGALL'S DATA (19). . . . .. 114 viii LIST OF SYMBOLS AND ABBREVIATIONS a Pipe radius an,a ,amn, Fourier coefficients am' a~k A Flow cross-sectional area Ai General function used by Bowden B General function used in examination of conditions for which concept of dispersion is valid in oscillatory free surface flow B1 General function used by Bowden c Concentration of substance c Mean concentration of substance over flow cross section c" Variation of substance concentration from cross-sectional mean value c" Separation variable representing spatially dependent s portion of c" cND Non-dimensional form of c" cm Centimeters D Molecular diffusion coefficient in radial direction Dt Molecular diffusion coefficient in direction transverse to principal flow axis e Vertical molecular or eddy diffusivity used by Elder EL Longitudinal dispersion coefficient for unidirectional flow Ex Longitudinal dispersion coefficient for oscillatory flow, temporally dependent E Time averaged longitudinal dispersion coefficient for oscillatory flow with an infinitely large period of oscil- lation, used by Holley et al. LIST OF SYMBOLS AND ABBREVIATIONS (Continued) E' Non-dimensional time averaged longitudinal dispersion x coefficient for oscillatory flow period of oscillation E' Non-dimensional longitudinal dispersion coefficient for unidirectional flow f Summation index fDW Darcy-Weisbach friction factor fl General function used in examination of conditions for which concept of dispersion is valid in oscillatory free surface flow f, General function used in examination of conditions for which concept of dispersion is valid in unidirectional free surface flow ft Feet F Function used to simplify umax normalization of two dimensional oscillatory shear flow solution FI Functional distribution of turbulent pipe flow used by Taylor g Function used to simplify velocity solution for three dimensional oscillatory shear flow g, General function used in examination of conditions for which concept of dispersion is valid in oscillatory free surface flow g2,G General functions used in examination of conditions for which concept of dispersion is valid in unidirectional free surface flow h Water depth i /5- 11,12 Functions used to simplify derivation of umax normalization for three dimensional oscillatory shear flow solution j Summation index k Summation index LIST OF SYMBOLS AND ABBREVIATIONS (Continued) K Pressure gradient modulus K Amplitude of eddy diffusion coefficient used by Fukuoka Kz Vertical coefficient of eddy diffusion K Lateral coefficient of eddy diffusion K Mean cross-sectional value of K z z K Mean cross-sectional value of K k Summation index Distance from point of maximum surface velocity to most distant bank, used by Fischer L Surface excursion length m Summation index m Mass transport M Meters n Summation index osc.- Oscillatory p Summation index P Pressure P1 General function used by Bowden q Summation index q" Depth integrated u" used by Fischer Qi General function used by Bowden r Cartesian coordinate in radial direction rl,r2 Functions used to simplify presentation of E' for three dimensional oscillatory shear flow Re Real part of complex function LIST OF SYMBOLS AND ABBREVIATIONS (Continued) RH Hydraulic radius RI,R2 Functions used to simplify development of three dimensional oscillatory shear flow solution for u", c", and Rmn Rpq Functions used to simplify development of three dimensional Rp unidirectional shear flow solution for u", c", and EL. iRk sec Seconds t Time t' Variable of integration T Period of oscillation Tcy Lateral mixing time Tcz Vertical mixing time T Semi-diurnal tidal period, 44,712 sec. T' Non-dimensional vertical mixing time used by Holley et al. T' Relative mixing time T' Relative mixing time for which two dimensional oscillatory Cm E' solution is valid x T' Non-dimensional lateral mixing time y T' Non-dimensional vertical mixing time u Velocity component along principal flow axis, x uf Unidirectional flow component used by Fukuoka umax Temporal and spatial maximum of u u Separation variable representing spatially dependent portion of u ut Oscillatory flow component used by Fukuoka u Mean value of u over flow cross section u" Variation of u from cross-sectional mean value LIST OF SYMBOLS AND ABBREVIATIONS (Continued) UND Non-dimensional form of u" u" Separation variable representing spatially dependent s portion of u" u, Shear velocity Uo Surface amplitude of unidirectional flow component used by Okubo unidir Unidirectional Vo Surface amplitude of oscillatory flow component used by Okubo w Width of channel x Cartesian coordinate directed along principal flow axis of the channel y Cartesian coordinate directed in lateral direction across width of channel z Cartesian coordinate directed in vertical direction over depth of channel z' Variable of integration a Constant al, 02, a3 Phase angles B Shear wave parameter in two dimensional oscillatory shear flow solution 61,62,63,64,6s Functions used to simplify three dimensional shear flow solutions ez Coefficient of vertical eddy viscosity ey Coefficient of lateral eddy viscosity Ez Mean value of e over flow cross section y Mean value of cy over flow cross section e,602 Functions used to simplify development of three dimensional oscillatory shear flow LIST OF SYMBOLS AND ABBREVIATIONS (Continued) v Locus of channel perimeter X Function used to simplify development of three dimensional oscillatory shear flow Pkk Factor used to properly account for ao0, abk, and aj' terms in three dimensional shear flow solutions. Cartesian coordinate directed along principal flow axis of the channel as seen by point traveling with velocity u p Fluid density a Angular frequency of oscillation, T S Summation symbol T Non-dimensional time used by Aris 4,,'- Functions used to simplify umax normalization of three dimensional shear flow solutions X General function i Function used to simplify presentation of E' for three dimensional oscillatory shear flow solution dimensional oscillatory shear flow solution Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DISPERSIVE MASS TRANSPORT IN OSCILLATORY AND UNIDIRECTIONAL FLOWS By Robert Bruce Taylor August, 1974 Chairman: Robert G. Dean Major Department: Civil and Coastal Engineering Four sets of boundary value problems for the equations of fluid motion and transport diffusion are solved to obtain expressions for the spatial variations of velocity and concentration in free surface flows. The four problems investigated treat unidirectional and oscillatory uniform shear flows in rectangular channels of infinite and finite width. The solutions are then used to determine longitudinal dispersive mass transport. Analytical expressions for the longitudinal dispersion coefficient associated with infinitely wide channels are presented as functions of the temporal and spatial maximum of velocity, channel depth, period of oscillation oscillatoryy flow only), and depth mean vertical coefficient of eddy diffusivity. For channels of finite width, analytical expressions for the longitudinal dispersion coefficient are presented as functions of the temporal and spatial maximum of velocity, channel depth, channel width, period of oscillation oscillatoryy flow only), and cross- sectional mean values of vertical and lateral coefficients of eddy dif- fusivity. It is found that for infinitely wide channels the longitudinal dis- persion coefficient in oscillatory flow is described by a type of resonant interaction between the period of oscillation and the time scale of vertical mixing. The maximum longitudinal dispersive mass transport occurs when the ratio of the vertical mixing time to the period of oscillation equals 1.58. This functional behavior is shown to be unique and distinctly different from the dispersive process in unidirectional flow. For rectangular channels of finite width it is shown that the dis- persive mass transport process is symmetric, such that a channel geometry skewed in width produces the same dispersive mass transport as a channel geometry skewed in depth, provided that specific requirements regarding the maximum velocity, period of oscillation, and vertical and lateral mixing times are satisfied. The longitudinal dispersion coefficient for oscillatory flow in rectangular channels of finite width is presented as a family of resonance curves described by the resonant interaction of the period of oscillation with the vertical and lateral mixing times. For unidirectional flow in channels of finite width it is found that the predicted non-dimensional longitudinal dispersion coefficient increases without bound as the skewness of the channel geometry is corres- pondingly increased. Thus, for unidirectional flow the three dimensional solution does not collapse in the limit to the two dimensional case as is found for oscillatory flow. Predicted values for the longitudinal dispersion coefficient are compared with values based upon field and laboratory data. The compari- son shows that when reliable measurements are available for all of the functional parameters included in the predictive solutions the agreement between observed and predicted values is good. CHAPTER I INTRODUCTION I-A General Background A growing awareness in recent years of the need for a rational approach to the planned use and development of coastal regions has significantly contributed to a concern for maintaining an acceptable level of water quality within our river, estuarine, and bay system waters. To achieve this a fundamentally sound understanding of the physical processes which govern the mass transport characteristics of these waters is essential. The mass transport of a substance as it applies here may be defined as "the time averaged net transfer of substance along the principal flow axis of the watercourse or channel." With the aid of this definition one can intuitively reason that the mass transport mechanism for a conservative substance is governed by the kinematic description of the flow field and the distribution of the substance within the flow field. Unfortunately, the complexities associated with turbulent shear flows in natural water- courses have thus far precluded a complete analytical description of either the flow kinematics or the distribution of the substance. However, the analytical treatment of longitudinal mass transport in steady uniform axisymmetric shear flows by Taylor (1,2) provided for the first time a reasonable method for predicting these phenomena. In his analysis Taylor made the distinction between three mechanisms of longitudinal mass transport: (1) Convective transport of the cross-sectional mean concentration of substance by the mean cross-sectional velocity, (2) Convective transport due to the correlation over the cross section of the spatial variations from the mean of velocity and concentration, (3) Diffusion of the substance along the principal flow axis (mole- cular diffusion in laminar flow and eddy diffusion in turbulent flow). The first of these mechanisms has long been recognized as a mass transport phenomenon and is the most amenable of the three to an analytical treat- ment. In the absence of diffusive transport due either to molecular or turbulent motion this mechanism may be visualized as the transport along the principal flow axis of a slug of substance uniformly mixed over the flow cross section and undistorted by boundary shear. The third mechanism, that due to longitudinal diffusive transport, was shown by Taylor to be less than 1 per cent of the second which he defined as dispersive mass transport. It is this second mechanism, longitudinal dispersive mass transport, which is the subject of the present work. Following Taylor's initial formulation of the dispersion mechanism investigators began to extend his analysis to unidirectional two dimen- sional open channel flows for infinitely wide channels and subsequently to the more realistic three dimensional case of unidirectional flow in a channel of finite width. More recently, the emphasis of the research effort has shifted from steady unidirectional flow to oscillatory flow in an effort to apply the knowledge obtained from the former to develop a predictive capability for dispersive mass transport for the latter. A brief summary of previous efforts follows. I-B Discussion of Previous Work on Dispersive Mass Transport I-B.1 Unidirectional Flow As mentioned previously, Taylor (1) was the first investigator to define the mechanism of dispersive mass transport. In his paper Taylor considered the case of a passive, conservative substance transported by a laminar steady flow in a circular tube. A passive substance is one which is present in small enough concentrations so as not to alter either the velocity flow field or the magnitude of the diffusion coefficient; a conservative substance is one which has a fixed total mass present for all time in the system being considered. All substances referred to in the present work should be considered to be both passive and conservative. To begin his analysis, Taylor transformed the Eulerian form of the trans- port diffusion equation to one with the origin moving with the cross- sectional mean velocity by defining = x ut (I-1) u = u + u" (1-2) where u is the cross-sectional mean velocity, u" is the variation from the mean of the velocity in the radial direction due to the shear profile, and x is the coordinate along the axis of the tube positive in the direction of flow. Neglecting molecular diffusion along the axis of the tube he was then able to write D ( U ) + C- (r ) (1-3) at'S, u 5 r Sr Pr where, (-) = time rate of change of concentration as seen from moving coordinate system ac ac -t x Dr = molecular diffusion coefficient in radial direction. If (1-3) is integrated over the cross-sectional area of the tube and the concentration, c, decomposed into its cross-sectional mean and variation component, by the expression c = E + c" (1-4) then (1-3) reduces to - T (I-5) For standardization throughout the text overbars shall be considered to represent cross-sectional averaging whereas the symbol < >T shall be considered to represent temporal averaging over the periodic interval T. The form of (1-5) led Taylor to define the convective transport shown as a Fickian flux which he called dispersion, or E 1 u"c"dA (1-6) EL where EL is the longitudinal dispersion coefficient for steady unidirec- tional flow and A is the flow cross-sectional area. As indicated by (1-6) the heart of Taylor's analysis lies in the determination of the spatial variations of velocity and concentration, u" and c" respectively. The velocity variation was readily determined from the parabolic velocity profile. To determine c" Taylor returned to (1-3), decomposed c by (1-4) and made the following assumptions: (1) Steady state conditions exist with respect to the moving coor- dinate system, i.e. ( ) = 0. (2) c" is not a function of 5. (3) is constant. DE, These assumptions reduced (1-3) to u r -3c ) (1-7) which Taylor then solved for c", performed the integration of u"c" in (1-6) and produced the following expression for the longitudinal disper- sion coefficient a2u2 E max (1-8) EL -192 D where a is the radius of the tube, and umax is the centerline velocity. It is important to note that the dispersion coefficient varies directly with the ratio a2/Dr which represents the time scale of mixing of the substance over the flow cross section due to transverse diffusivity. Using the definition given in (1-6) for EL, (1-5) may then be written as D ELC 2 c (1-9) t ) = L S2 Equation (1-9), with EL given by (I-8), shows that the cross-sectional mean concentration of a substance is effectively diffused longitudinally about a point traveling with the mean velocity by the combined effect of the velocity shear profile and the transverse diffusivity. Taylor correctly cautioned, however, that in applying (1-9) the time scale of convective transport associated with u" must be significantly greater than the time scale of cross-sectional mixing due to transverse diffusivity. A further discussion of the assumptions made by Taylor and the application of dis- persion coefficients obtained in this manner will be presented in Chapter II. Having looked at laminar flows Taylor (2) extended his analysis to axisymmetric turbulent shear flow in a pipe. For u" Taylor used an exper- imentally verified velocity distribution of the form umax -u(r) ax = Fi(r) (1-10) where ux is the centerline velocity, u, the shear or friction velocity, and Fi(r) an experimentally determined function. Then by assuming Reynolds analogy which states that the transfer of mass, heat, and momentum by turbulence are exactly analogous, and the same conditions required to obtain (1-7) he obtained a solution for c" and numerically performed the integration of u"c" shown in (1-6). The result was the following expression for the longitudinal dispersion coefficient in turbulent pipe flow EL = 10.06 a u, (1-11) where a is the radius of the pipe. By assuming isotropy of the turbulent fluctuations he was then able to estimate the additional longitudinal transport due to eddy diffusion along the flow axis. Adding this to (1-11), the corrected dispersion coefficient becomes EL = 10.1 a u, (1-12) Shortly after Taylor's discovery and formulation of dispersive mass transport, Aris (3) presented a very elegant analysis which supported Taylor's theory for unidirectional steady flows and rigorously defined the conditions under which Taylor's assumptions were valid. Rather than obtaining solutions for u" and c" to compute the dispersive mass trans- port, Aris chose to apply a moment analysis of the longitudinal distribution of the cross-sectional mean concentration of substance about a point moving with the mean velocity. The significance of Aris' analysis lies in its generality, and the insight that it yields for looking at the precise conditions under which an ordered correlation of u"c" over the cross-sectional flow area may be represented by the implied stationary random process of a Fickian flux. Aris' development begins with the basic Eulerian form of the transport diffusion equation for an arbitrary flow geometry and an arbitrary initial distribution of substance. He then proceeds to develop expressions for the moments up through the third of the cross-sectional mean concentration about with no assumptions regarding steady state, or the functional form of the concentration c. This generality allows him to obtain time dependent solutions which show that after an initial transient period, steady state conditions will be asymptotically approached in which the cross-sectional mean concentration, c, disperses about a point traveling at the mean speed of the flow in a Gaussian manner and that the rate of growth of the variance is constant with time. In other words the critical assumption in Taylor's analysis is the one concerning steady state conditions with respect to the moving coordinate system, i.e. (-) = 0, which by Aris' analysis must be satis- fied if the definition of the convective mass transport of a substance as a Fickian flux about the mean speed of flow is to hold. The dimensionless time parameter governing the dispersive process is defined by Aris as Dtt i a (1-13) where t represents time; Dt is the molecular diffusion coefficient in the transverse direction; and a is a characteristic length of the flow cross section. The ratio a2/Dt, as noted previously, represents the time scale required for the substance to mix over the flow cross section due to trans- verse diffusion. Thus, steady state conditions are approached for large t, or when the convective flow time is large compared to the time required for cross-sectional mixing. This supports Taylor's comments regarding the application of dispersion coefficients to predict concentration distributions. In Chapter II it will be shown how the assumption of steady state condi- tions supports Taylor's other assumption regarding the functional dependence of c for both steady unidirectional and oscillatory flows. Aris' method of analysis has been used by other investigators to obtain expressions for the dispersion coefficient and for brevity later in the text will be referred to as the "method of moments." Although it remains a very useful method for determining expressions for longitudinal dispersion coefficients it does not provide the physical insight into the kinematic flow field and cross-sectional distribution of substance that Taylor's method does. Further support of the general applicability of Taylor's dispersion analysis to different flow geometries was given by Elder (4) who applied Taylor's analytical technique to obtain a predictive expression for the dispersion coefficient in two dimensional unidirectional shear flows in infinitely wide open channels. In his analysis Elder obtains the following expression for the longitudinal dispersion coefficient: h z z EL I uh v e1 u" dz'dz'dz' (1-13) 0 0 0 where z' is a variable of integration; z is the vertical coordinate; h is the depth of flow; and e is either the vertical molecular diffusivity or the vertical eddy diffusivity depending upon whether the flow is laminar or turbulent. This expression indicates, in a manner similar to (1-8), that in either laminar or turbulent unidirectional shear flow the longi- tudinal dispersion varies directly with the time scale of cross-sectional mixing. To obtain an expression for u" Elder assumed the velocity distri- bution to be logarithmic which upon substitution into (1-13) and correcting for longitudinal eddy diffusion as done by Taylor, yields EL = 5.93 uh (1-14) where u, is the shear velocity. The importance of the flow cross section geometry on the velocity shear profile, transverse diffusivity, and ultimately the longitudinal dispersion is demonstrated by (1-12) and (1-14) which if expressed in terms of the hydraulic radius, RH, for the respective flows yield values of EL/RHu, of 20.2 for the pipe and 5.9 for the infinitely wide channel. Bowden (5) developed expressions for EL using several different velocity and eddy diffusivity distributions with depth and showed that EL/u,h ranged from 5.9 to 25 for the cases analyzed. Thus, the particular velocity shear profile used along with the cross-sectional eddy diffusivity have a significant effect on the magnitude of the predicted longitudinal dispersion coefficient. While the works of Taylor, Aris, and Elder proved the existence of an effective dispersive mass transport mechanism and there was reasonable agreement between laboratory experimental data for pipe and infinitely wide open channel flow geometries, other investigators were finding that observed values of the longitudinal dispersion coefficient for natural rivers and streams were considerably higher than those predicted by theory. As reported by Fischer (6) observed values for EL in natural watercourses ranged from 50 to 700 hu, as compared with the 5.9 to 25 hu, range predicted by Elder and Bowden. To account for this increased effect Fischer concluded that the dominant mechanism in longitudinal dispersion was the interaction of the velocity shear profile and turbulent mixing time scales across the finite width of the channel as opposed to the vertical variations treated by the previous investigators for infinitely wide channels. The effect of asymmetrical flow cross sections on longitudinal dispersion was first examined by Aris (3) who showed that the dispersion in a circular tube is less than that in an elliptical one of the same area. This in addition to the fact that all real watercourses have a finite width and that width- to-depth ratios for natural channels are usually significantly greater than unity makes Fischer's hypothesis very reasonable. For wide channels the transverse mixing time, k /Ky, over some characteristic zs would tend to be greater than the corresponding vertical mixing time scale, h2/Kz. Fischer thus argued that the velocity shear profile across the channel would have a greater longitudinal dispersive effect since an increased mixing time would produce larger values of the spatial correlation u'c". Using this as the basis for his analysis Fischer applied Taylor's technique to a rectangular coordinate system and assumed that vertical variations in c" were negligible compared to the transverse variations to obtain u"- a (K (1-15) where y is the coordinate axis across the channel. Fischer then integrated (1-15) over the cross-sectional area of the channel to obtain an expression for c" which he then correlated with u" for the dispersive mass transport and using Taylor's definition of a Fickian flux obtained a longitudinal dispersion coefficient of w y EL = q"(y)dy K hy T dy q"(y)dy (I-16) 0 0 0 where the depth, h, is a function of y; K is the transverse eddy diffusi- vity; w is the width of the channel; and q"(y) is the depth integrated flow defined by h(y) q"(y) = S u"(y,z) dz (1-17) 0 To obtain a more useful expression for EL Fischer defines a Lagrangian time scale for cross-sectional mixing which he obtains through an extension of Taylor's work on diffusion by continuous movements (7), and relates it to the Eulerian time scale for cross-sectional mixing previously discussed. Then, using Elder's (4) experimental determination that the transverse eddy diffusivity in an infinitely wide channel with a steady unidirectional flow can be expressed as K = 0.23 hu, (1-18) Fischer arrives at an alternate expression for EL given by 2 EL = 0.3 u" (1-19) L RHU. where s in a natural channel is the distance from the point of maximum surface velocity to the most distant bank. To verify his theory Fischer conducted a detailed set of laboratory experiments. Using width-to-depth ratios ranging from 9 to 15.7 Fischer obtained good agreement between predicted and observed values of EL using (1-16). However, (1-19) over- predicted in each case, with a maximum error of 75 per cent. Fischer (8) also applied his analysis to data obtained by other investigators in 6 natural watercourses including Copper Creek, Va. and Clinch River, Tenn. (2 locations each), Powell River, Tenn., and Coachella Canal, Cali- fornia. Width-to-depth ratios for these sites range from 15 to 62 with most being less than 40. To carry out his analysis Fischer numerically integrated (1-16) using measured data and found that in uniform channels the predicted values for EL were within 30 per cent of the observed values whereas in non-uniform channels the predicted value varied from the observed by as much as a factor of 4. Thus, while Fischer's hypothesis regarding the dominance of lateral effects on longitudinal dispersion in unidirectional flows yields reason- able results his analysis provides very little insight into the physics of the three dimensional problem. Both (1-16) and (1-19) make no attempt to describe the velocity distribution over the flow cross section thereby requiring detailed measurements for their application. Moreover, by neg- lecting the vertical shear effects it remains impossible to obtain from his analysis the conditions for which either the vertical or lateral effects would be the dominant mechanism for longitudinal dispersion. I-B.2 Oscillatory Flow The analysis of dispersive mass transport in oscillatory flow is more complex than the corresponding problem in steady unidirectional flow because of the unsteady nature of the governing equations of motion and transport diffusion. Since, however, many important water quality problems involve the predicted distribution and transport of substances within tidal waters the analysis of dispersive processes in oscillatory flows has attracted considerable interest. There is in fact a reasonable doubt that one is justified in representing the convective mass transport u"c" as a Fickian flux in an oscillatory flow. However, it will be shown in Chapter II that under the assumptions of periodicity and uniform flow the analyses of Taylor and Aris apply. Bowden (5) was the first investigator to look at longitudinal dis- persive mass transport in oscillatory flow. Bowden generally followed Taylor's original analytical technique and began by assuming a monochro- matic flow in an infinitely wide channel in which u" can be expressed as u"'z,t) = Ai(z) cos at + Bi(z) sin at (1-20) He likewise assumed that c" is simple harmonic in time and of the form c"(z,t) = P1(z) cos at + Q1(z) sin at (1-21) conditional upon satisfying the transport diffusion equation expressed as ( ) + u" (Kz c) (1-22) (d t=a z (z z S9z ) This is a very proper formulation of the problem, allowing for the periodic variation of u" and c" with time and the existence of temporal phase shifts for u" and c" as a function of position within the water column. Unfortunately, Bowden proceeded by means of his assumptions to reduce the problem to one which applies only to the case for which the period of oscillation is infinitely long, (- = 0, and for which the temporal phase shift as a function of position within the water column has been eliminated. For these conditions Bowden found that the longitu- dinal dispersion coefficient in an oscillatory flow is one-half the value of the same coefficient for a corresponding unidirectional flow having the same surface velocity, the same shear profile, and the same vertical eddy diffusivity. Holley and Harleman (9) treated the case of oscillatory turbulent pipe flow and obtained an expression for the longitudinal oscillatory flow dis- persion coefficient, Ex, using Aris' method of moments. In their analysis they assumed that at each instant in time the lateral distribution of velocity and eddy diffusivity are the same as they would be if the flow were steady with the same cross-sectional mean velocity. This assumption effectively removes the oscillatory nature of the flow and reduces the problem to essentially the same one solved by Aris (3) and Taylor (2). Thus, it is not surprising that Holley and Harleman predicted a longitu- dinal dispersion coefficient of Ex (t) = 10.la u, (t) (1-23) where, as before, a is the radius of the pipe, and u, (t) is the friction velocity now periodic in time. Predicted values of Ex were compared with observations made from a series of laboratory experiments in which an oscillatory flow was produced in a 1-1/2 inch pipe by a piston genera- tor. Superimposed upon the oscillating flow was a small unidirectional flow to simulate the net fresh water discharge of an estuarine type flow. The experimental data, however, showed that due to the small magnitude of the unidirectional flow compared to the velocity amplitude of the oscilla- tory flow, its effect on longitudinal dispersion could be neglected. The data also showed that (1-23) generally underpredicted the dispersion coefficient with better agreement between predicted and observed values coming with higher Reynolds number flows. The authors attributed this behavior to laminar sublayer effects. Okubo (10) obtained a less restrictive solution which for the first time demonstrated that in a two dimensional oscillatory shear flow the period of oscillation and the time scale of vertical mixing played an important role in the dispersive process. Okubo confined his analysis to vertically bounded and unbounded flows with no transverse shear. Only the formerwill be discussed. He assumed a linear, monochromatic flow superimposed upon a linear unidirectional flow, or u(z,t) = Uo (1-) + Vo (1- ) sin at (1-24) where Uo and Vo are the surface amplitudes of the unidirectional and oscillatory flow, and a is the angular frequency of oscillation. Okubo then assumed a constant eddy diffusivity both in the direction of flow and over the depth, and zero diffusive flux conditions at the surface and bottom. Using the method of moments as applied by Aris and the assumptions discussed he obtained an expression for the longitudinal dispersion coeffi- cient, Ex, functionally expressed as Ex = X (Uo, Vo, h, T, -) (1-25) x z h2 where T is the period of oscillation, and the time scale of vertical z mixing. To interpret his results Okubo considered two extreme cases. (1) ->> T Neglecting longitudinal eddy diffusion his solution becomes U h2 V2 T2K -x z) + (- ) (I-26) unidirectional oscillatory From this Okubo correctly concluded that for situations in which the verti- cal mixing time is large compared with the period of oscillation the con- tribution to the longitudinal dispersive mass transport due to the oscilla- tory flow is small compared with the contribution due to the unidirectional flow. However, (1-26) also suggests that the functional behavior of dispersion in an oscillatory flow is significantly different from the same process in unidirectional flow. One of the principal findings of the present work concerns this point and will be treated in depth in Chapters III and IV. h2 (2) T >> Again neglecting longitudinal eddy diffusion Okubo's result may be written as U0 h2 V h2 Ex ) +2 ( (1-27) z z This case is comparable to Bowden's analysis for a very long period of oscillation and supports his finding that Ex 1/2 EL for large T. Holley et al. (11) successfully applied Taylor's original method of analysis to obtain an analytical expression for periodic, uniform shear flow. For their analysis they assumed the spatial variation component of velocity to be u" (z,t) = a z sin at (1-28) where a is a constant. This is identical to the linear oscillatory flow profile assumed by Okubo. This expression for u" is then used to force the following form of the transport diffusion equation to obtain a solution for c" (z,t) (=) Kz, =-" (1-29) The solutions obtained for u" and c" are then used to determine the depth integrated time mean convective mass transport with respect to the depth mean velocity. The resulting expression for the mean longitudinal dispersion coefficient over the period of oscillation is a2 T2 h4 1 TK where T' = -h. Holley et al. then proceed to relate flow to the dispersive process in an oscillating flow of infinite period with the same velocity profile as given by (1-28). Solving for c" using the steady state form of (1-29) and proceeding in the usual manner they obtain an expression for is one-half the corresponding value for EL in agreement with Bowden (5), and is subsequently used to compare unidirectional flow coefficient. Apparently the investigators' primary goal was not to describe longitudinal dispersion in an oscillatory flow as a separate distinct physical process but rather to describe it in terms of a "corresponding" process in a unidirectional flow. To do this the solution for TK the non-dimensional time, T' = --, suggested by Okubo's (10) results. Numerical simulations were also carried out for other velocity profiles in which the velocity was again assumed to be temporally in phase over depth during the period of oscillation. Based upon their analytical and numerical results Holley et al. concluded that: (1) T'>1 The longitudinal dispersion coefficient in an oscillatory flow is independent of T' and equal to one-half the value of the dispersion coefficient in a "corresponding" unidirectional flow. (2) T'<0.1 The longitudinal dispersion coefficient in an oscilla- tory flow rapidly becomes insignificant compared to the dispersion coefficient in a "corresponding" unidirectional flow and is functionally described by These results have been subsequently applied by Fischer (12, 13), Awaya (14), and Fischer and Holley (15) in analyses of dispersion in oscillatory flows. Fischer (12) attributes the functional behavior of oscillatory flow dispersion to a temporal phase shift between u" and c" as a function of T'. However, it will be shown in the present work that while these conclusions are applicable for the non-dimensional parameter they are incorrect and misleading in their description of the longitudinal dispersive mass transport process in oscillatory flow. Fukuoka (16) obtained analytical expressions for E from several cases of two dimensional and axisymmetric oscillatory shear flows using the method of moments. For his analysis he assumed a velocity of the general form u = us (z) + ut (z) sin at (1-32) similar to Okubo (10), implying once again no temporal phase shift of the velocity over the flow cross section. In specifying the forms of us (z) and ut (z),Fukuoka assumed that the spatial variation at each instant of time is the same as that for an equivalent unidirectional flow. Specific cases treated in depth are: (1) linear velocity profile over depth with constant K (2) parabolic velocity profile for axisymmetric flow with constant Kz (3) linear velocity profile over depth with Kz = Ko sin ot . In interpreting his results Fukuoka follows Holley et al. by plotting Ex vs. T' and arrives at the same conclusion regarding the functional be- havior of dispersion in oscillatory flows. For Case 3 he shows analytically that the dispersion coefficient corresponding to a time varying eddy diffu- sivity differs by a factor of 8/72 from one that is considered to be 2 constant, provided the constant Kz is Ko. This small effect of a periodic eddy diffusivity supports results obtained numerically by Holley et al. It is interesting to note that none of the analyses dealing with oscillatory flow discussed thus far have attempted to solve analytically the velocity distribution over the flow cross section. Moreover, it has been further assumed that the velocity profile at each instant in time is the same as an equivalent unidirectional flow profile and that no temporal phase shift as a function of spatial position exists. These assumptions are particularly unrealistic when one considers the definition of dispersive mass transport as defined by Taylor, i.e.A u"c" dA. By restricting the form of u" in this manner one is also restricting the form of c" since by (1-29) it is seen that the distribution of substance is in part forced by the convective flux. Thus, any artificiality regarding the assumed form of u" appears in the dispersion coefficient through both u" and c". The significantly different nature of a truly oscillatory flow as compared with a unidirectional one is illustrated bySchlichting (17) in his treatment of the fluid motion above an oscillating flat plate, and Lamb (18) in his treatment on the effect of bottom friction on long waves in two dimensional shear flow. Although the forcing functions in these two cases are different (oscillating boundary shear inSchlichting as opposed to an oscillating pressure gradient in Lamb), the result of a shear wave propagating through the fluid in a direction normal to the axis of flow is similar, causing a continuously changing phase shift of the flow over the cross-sectional area. Awaya (14), Segall (19), and Segall and Gidlund (20) have obtained analytical expressions for the longitudinal dispersion coefficient in oscillatory flow situations using realistic velocities obtained directly from solutions to the appropriate forms of the equations of motion. Awaya addresses the case of an oscillatory laminar flow in a circular cross section for which he obtains an expression for 20 normalizes it by E, after Holley et al. His results are similar to those of Holley et al. and his interpretation the same. Segall and Segall and Gidlund use a simplified version of Lamb's previously mentioned solution for u corresponding to small values of the vertical eddy viscosity, ez which they apply to the transport diffusion equation and solve using the method of moments. In both of these analyses the interpretation of the functional behavior of the dispersive process in oscillatory flows is the same as that of previous investigators although the matter is not pursued in any detail. I-C Description and Scope of the Present Work In the chapters that follow four sets of boundary value problems will be solved yielding analytical solutions for u", c", EL, and following cases: (1) Steady unidirectional flow in an infinitely wide rectangular channel, Ez and Kz constant (2) Monochromatic periodic flow in an infinitely wide rectangular channel, ez and Kz constant (3) Steady unidirectional flow in a rectangular channel of width w; EZ, E KZ, and K constant (4) Monochromatic periodic flow in a rectangular channel of width w; EZ, Ey, K, and K constant. These solutions deal exclusively with dispersive mass transport in homo- geneous fluids and uniform or nearly uniform flows. In Chapter II a discussion is presented of the conditions under which Taylor's definition of dispersive mass transport is valid for unidirectional and oscillatory flows. Chapter III treats Cases 1 and 2 above yielding solutions for EL and extends the analysis to three dimensional shear flows through Cases 3 and 4. In Chapter V predicted values of EL and in Chapters III and IV are compared with available data. Chapter VI presents a summary of the present work and some conclusions drawn from it. The mathe- matical details associated with portions of the boundary value problem solutions are reserved for Appendices A, B, and C. CHAPTER II CONSIDERATIONS FOR APPLICATION OF TAYLOR'S DISPERSION ANALYSIS TO FREE SURFACE FLOWS II-A General In order to provide a solid base upon which the analyses presented in Chapters III and IV are built, it is necessary to set forth at this juncture a detailed development of Taylor's dispersion analysis as it applies to both unidirectional and oscillatory free surface flows. By doing this it is hoped that all assumptions made in obtaining solutions for u", c", Ex and EL, and the conditions under which a dispersion coefficient may be defined and applied consistent with Aris (3) and Taylor (1,2), will be clearly understood. Consider a three dimensional shear flow, either unidirectional or oscillatory, in a rectangular channel of width w and depth h. Let the coordinate system be selected as shown in Figure 1 with the x coordinate taken along the longitudinal axis of the channel denoting the direction of the u (y,z,t) z z=0 yrw/2 \Y = f/2 S= -h" FIGURE 1 DEFINITION SKETCH velocity component, u(y,z,t); the y coordinate denoting lateral position across the channel; and the z coordinate acting positive upward from the free surface. To begin the analysis it shall be assumed that: (1) Longitudinal mass transport due to eddy diffusion along the flow axis is small compared to convective transport mechanisms. (2) Eddy diffusivities K and Kz can be represented by their mean values over the flow cross section. (3) Uniform flow conditions along the principal axis of the channel exist as indicated by the functional notation for u. Applying these assumptions to the channel shown in Figure 1 the general form of the transport diffusion equation for a conservative sub- stance may be written as ac aC 2 + 2(I-l) + u K K(11-1) at 3x y y2 Z az2 After Taylor (1,2), transform (II-1) to a coordinate system traveling with the cross-sectional mean velocity using u = u + u" (11-2) S= x ut (11-3) so that the functional transformation of the concentration c is given by c(x,y,z,t) + c(S(x,t),y,z,t). The transformation relationships are then 3c c +3c C =t (1) + ac ac 95 ax 3E ax -c ac and 2 unchanged 3y az Using (11-3), (11-4) (11-5) Equations (11-2), (11-4), yield and (11-5) are then substituted into (II-1) to + C 1 ac 32C 92C at)c + ull K y K Z@Z K-+Ky2 a2 (11-6) Next, decompose c into its cross-sectional mean and variation components by the expression c = c + c" and substitute into (11-6). For uniform flow this produces (11-7) ac ac c1 a2 2c" (u"c")_ u" 3W -8) at t y ay2 Z 2 a which is the form of the transport diffusion equation as seen from a coordinate system traveling with the mean cross-sectional velocity. Proceeding with the analysis, (11-8) is integrated over the cross- sectional area, Sw/2 h w/2 h w/2 ( ) dydz + ( a) dydz K dydz -w/2 0 -w/2 0 -w/2 h w/2 h w/2 h w/2 -z dydz (u"c") dydz u" dydz (11-9) 0 -w/2 0 -w/2 0 -w/2 It is next assumed that: (1) The time rate of change of the depth is small. ac (9) af at aca ac Dc 5 T TE (2) There is no turbulent diffusion of substance across the enclosing boundaries of the flow cross section. To apply the condition of zero diffusive flux across the boundaries of the channel it is helpful to use Green's Theorem in the plane which for the third and fourth terms on the left hand side of (11-9) states a__ ac" a c" ac" c '(K -) + z(K dA = dzdy} (II-10) ay y 9y at y ay z A u where u is the perimeter of the area A. Since by assumption (2) it is required that there be no diffusive flux across u, the left hand side of (II-10) is equal to zero. Using this result, assumption (1), and noting that u" dA = c" dA = 0 A A by definition, (11-9) is reduced to ( = (u ) (II-11) As noted in I.B.1 the form of (II-11) induced Taylor to define the spa- tially averaged convective flux u"c" as a Fickian flux thus transforming (II-11) to the one dimensional heat equation form (-C) = EL (11-12) where, Su"c" (11-13) EL _c ac Referring again to I.B.I, Aris showed that (11-12) and (11-13) are valid formulations if the following necessary and sufficient conditions are satisfied: (1) steady uniform flow; and (2) steady state concentration with respect to a point traveling at the mean velocity, i.e. (-c-) = 0. at t In the sections that follow it will be shown that these conditions are necessary and sufficient not only for unidirectional flows but for oscilla- tory flows as well provided that steady state is defined in the periodic sense. It will also be shown that Taylor's assumptions regarding the functional dependence of c and c" on 5, y, and z as stated in I.B.1 follow directly from the more general requirements of steady state u and c, and uniform flow. II-B Unidirectional Flow To obtain an expression for EL as defined by (11-13) it is necessary first to have solutions for u" and c". Ideally a solution for u"(y,z) is obtained from the governing equation of motion in the x direction. This solution is then used to force a simplified form of (11-8) to obtain a solution for c" which is then correlated with u" according to (11-13) for the determination of the longitudinal dispersion coefficient. As stated in I.B.1 Taylor reduced an expression similar to (II-8) to a solvable form by assuming: (1) c" is not a function of 5. (2) is constant. (3) Steady state conditions with respect to the moving coordinate system exist, i.e. (3c-) 0. 3t E Aris, however, showed that assumption (3) along with the requirements of steady uniform flow were necessary and sufficient for the definition of a dispersion coefficient as stated by (11-13), and it will now be demonstrated that assumptions (1) and (2) follow from these conditions. For steady state conditions to exist (II-11) reduces to 3C A = I u"c" dA A Assuming the channel to be prismatic, the order of differentiation and integration may be reversed yielding 0 = A T (u"c") dA A which for uniform flow becomes 0 = 5 u" dA (11-14) A To examine the functional dependence of c", let U" = g2 (y,z) 3c" = f2 (y,z,S) Equation (11-14) may then be written as 0 = : g2 (y,z)-f2 (y,z,S) dA (11-15) A which upon integration yields G() = 0 (11-16) From (II-16) it may be inferred that Tc= 0 or c" is not a function of thereby justifying Taylor's first assumption restated as c(x,y,z) = c(x) + c"(y,z) (11-17) The above result is then used to substantiate Taylor's second assump- tion that is constant. To do this (11-8) is differentiated with respect to E. For steady state conditions and uniform flow the result is 32-= 0 (11-18) which upon integration yields j-= constant (11-19) ot. The application of (11-17) and (11-19) reduces (11-8) to K + z = u"- (11-20) where c is constant. Thus it is shown that the definition of the longitudinal dispersion coefficient as stated by (11-13), which implies the use of solutions for u" and c" obtained from the governing equation of motion and (11-20) respectively, is valid for free surface flows provided the necessary and sufficient conditions of uniform steady flow and steady state concentration distributions with respect to a coordinate system traveling with the mean velocity are satisfied. Taylor's assumptions as stated by (11-17) and (11-19) follow from these conditions. II-C Oscillatory Flow The same analysis carried out in II-B for steady unidirectional flow can be extended to the oscillatory flow case if it is assumed that steady state conditions in the oscillatory sense can be satisfied by requiring both the velocity, u, and the concentration, c, to be temporally periodic. This, of course, implies that u, c, u", and c" are also temporally periodic. To show that Taylor's same assumptions apply to dispersion in oscilla- tory flows it is again necessary only to require that the flow be uniform and that the velocity and concentration be steady state functions in the oscillatory sense, i.e.,periodic. If this can be shown to be true then it is reasonable to assume that Aris' argument could be extended to periodic functions thereby justifying the definition of a longitudinal dispersion coefficient in oscillatory flow using Taylor's method. The analysis is begun by averaging (II-11) over one full period of oscillation which for a periodic c yields <- (u"c > = 0 (11-21) where C is now defined by C = x- \ (t') dt' (11-22) 0 Next, it is assumed once again that the time rate of change in depth is small so that the implied order of integration and differentiation in (11-21) may be reversed h w/2 T h w/2 5 u"c" dt dy dz = 0 (11-23) S0 -w/2 0 Since it is required that the flow be uniform and that u and c be temporally periodic, assume u" = g1(y,z) cos at c" = fl(y,z,') cos(at -a,) where ai is a constant. It then follows that the temporal integration of u"c" is of the form T u"c"dt = B(y,z,S) cos ct (11-24) 0 Substituting (11-24) into (11-23) h w/2 h Sw B(y,z,S) cos ai dy dz = 0 0 -w/2 which upon integration reduces to the form cos ai x) = 0 (11-25) From this it is seen that X() is a constant and thus c" is not a function of . This is the same result obtained in II-B and is consistent with Taylor's first assumption. However, for the oscillatory case there is an additional distinction to be made. Recognizing that uniform flow is required and c" is not a function of E, (II-11) reduces to ( = 0 (11-26) and (11-8) becomes (at KKz 2 =- u" 3 (11-27) Differentiating (11-27) with respect to E yields u" 0 32 or, D= constant, which is again consistent with Taylor's assumption for the unidirectional flow case. Equation (11-27) with -constant is the comparable form of (11-20) for oscillatory flow. It is from the solution of this equation that c"(y,z,t) is obtained for determination of the longitudinal dispersion coefficient, Ex. Based upon the above discussion it is concluded that Taylor's analysis, under the conditions of uniform flow and temporal periodicity of u and c, is capable of being extended for the determination of longitudinal dispersion coefficients in oscillatory flows and remains consistent with the condi- tions for which a dispersion coefficient may be defined as shown by Aris. II-D Application of Dispersion Coefficients Predicted by Taylor's Method It is not the purpose of the present work to provide a detailed anal- ysis of the conditions under which values of EL and Ex predicted by Taylor's method may be applied for the determination of concentration distributions. However, a comment regarding this matter is considered useful here in order to retain a proper perspective on the work presented. Equation (11-12) indicates that in applying values of the dispersion coefficient, steady state conditions for the cross-sectional mean concen- tration are not satisfied whereas in Taylor's analysis for the determina- tion of the dispersion coefficient it is assumed that these steady state conditions exist. Aris (3) has shown that the steady state conditions required for the convective mass transport to behave as a Fickian flux are approached asymptotically at a rate determined by the ratio of the time of convective flow to the time of turbulent cross-sectional mixing. This interpretation was also given by Taylor (1) on a qualitative basis. Fischer (21) then reasoned that a necessary condition for the application of coeffi- cients obtained in this manner for the prediction of concentration is that c" be everywhere much less than c. This in effect strikes a compromise between the strict steady state requirement for the definition of a dis- persion coefficient and the conditions for which it may be used. Fischer (21) among others has developed detailed criteria for the application of dispersion coefficients in (11-12). CHAPTER III DISPERSIVE MASS TRANSPORT IN AN INFINITELY WIDE RECTANGULAR CHANNEL III-A Steady Unidirectional Flow The coordinate system has been selected (Figure 2) with origin at the bed of the channel, z coordinate positive upwards, and x coordinate positive in the direction of flow, u. z=h _^_---____ z- u(z) c(x, z) FIGURE 2. DEFINITION SKETCH FOR TWO DIMENSIONAL SHEAR FLOW The general procedures used throughout Chapters III and IV will be to first obtain analytical solutions for u" and c" and then use these to calculate the dispersive mass transport u"c" from which the dispersion coefficient is determined. III-A.1 Velocity Distribution It is assumed that the turbulent shear stress may be expressed in terms of the Boussinesq approximation and that the eddy viscosity, ez, is constant and equal to its mean value over depth. The equation of motion for two dimensional steady uniform shear flow is then 0 P U (III-1) p ax z z2 where p is the density of the fluid and P is pressure. It will be assumed in all analyses that turbulent fluctuations have been averaged and incorporated into the remaining terms. The pressure gradient term for this case may then be expressed as 1 2P K (III-2) p Tx where K is constant. Substituting (III-2) into (III-1) and requiring the boundary conditions of no slip at the bed and zero shear stress at the surface the flow boundary value problem is then DE: d2u K Sdz2 E (111-3) z BC's: u(0) = 0 du Tu7 z=h = 0 The velocity shear profile is obtained by integration of (111-3) and the application of the boundary conditions. The resulting expression is u(z) = -- (z2 hz) (III-4) The spatial variation of the velocity as defined by (11-2) is then obtained by averaging (III-4) over depth and subtracting out the depth mean component to yield u"(z) = K( hz +2) (III-5) Cz The portion of the unidirectional flow analysis remaining in this section and portions of the analyses in later sections treat the expres- sion of the variations of velocity and concentration and the dispersion coefficient as functions of normalizing parameters other than the pressure gradient modulus, K. In some cases these alternate formulations are considered to be more useful for application purposes, whereas in other cases they are presented for the comparison of longitudinal dispersive mass transport in oscillatory and unidirectional flow or in oscillatory flows of differing periods of oscillation. Therefore, before proceeding further a brief discussion follows describing these alternate formulations and the rationale used in their selection. As noted in Chapter I the approach taken by most investigators in describing dispersive mass transport in oscillatory flows has been to first assume a velocity profile that is at each instant in time similar in form to a corresponding unidirectional flow, and then use this to obtain an expression for the dispersion coefficient, Ex, either directly through the method of moments or indirectly through Taylor's method of describing the convective mass transport uc" as a Fickian flux. The dispersion coefficients thus obtained have then been analyzed only as they relate to a "corresponding" unidirectional flow coefficient. The approach taken here is that it is not realistic to assume a priori that the dispersive mechanism in oscillatory flow is functionally similar to the same mechanism in unidirectional flvo because of the distinct dif- ferences in the flow characteristics of the two cases. This argument may also be applied to the more general case of comparing dispersive mass transport characteristics of oscillatory flows of different periods. It will later be shown that such comparisons are easily misleading and if done at all they must be done with a full understanding of the mechanism involved. To illustrate this point, there are many ways in which unidirectional and oscillatory, or two oscillatory mass transport cases of different periods, can be considered to be "corresponding." Normal- izing parameters considered in the present analysis for comparison of "corresponding" dispersion coefficients are: (1) pressure gradient modulus, K, (2) maximum cross-sectional velocity, umax' which for the most general case of three dimensional oscillatory shear flow would represent the amplitude of the periodic velocity function located on the centerline of the channel at the surface, (3) surface excursion length, L, experienced by a particle on the free surface over a period of time equal to one-half the period of oscillation. For the case of a unidirectional flow, such as that being considered in this section, the concept of an excursion length as described above has little physical meaning. Thus, excursion length normalization of u", c", and Ex solutions will be made for oscillatory flows only. The expression of (111-5) in terms of umax is easily accomplished by solving (111-4) for K at z = h, or 2e u K = 2 max (111-6) h which upon substitution into (111-5) yields 2u 2 h2 u,,(z) max z2 h2 u z2 hz + ( ) (111-7) III-A.2 Concentration Distribution A solution for c"(z) as a function of K is obtained by using the solution for the spatial variation of velocity, u"(z), given by (III-5), to force the one dimensional form of (11-20). Thus, the concentration variation boundary value problem may be stated as DE: d2c" E dzK u (111-8) dc" BC's: z = O,h dc"= 0 dz where, ac = constant u"(z) is given by (III-5). Substituting for u", integrating once with respect to z, and applying the boundary condition at z = 0 produces dc" K BE z3 hz2 h2z (11-9) d *- (- 6- -[ + ] (III-9) dz -- (JL6 2 3 SzKz It should be noted that (III-9) implicitly satisfies the boundary condi- h tion at z = h since by definition S u"(z) dz = 0. Equation (III-9) is 0 then integrated once more over z and the constant of integration is evaluated using the requirement that c"(z) must have a zero mean over depth. The resulting solution for the concentration variation is then c"(z) = K c z4 hz3 h2z2 h4 c"(z) ( 2 + l- (III-10) c K 24 6 6 t5 zKz Using (III-6) to substitute for K in (III-10) the expression for c"(z) as a function of umax is c"(z = a 4 hz3 h2z2 h' c"(z) = max (T ) r-4- -6 + T- (III-ii) K h2 z III-A.3 Dispersive Mass Transport The dispersive mass transport over the depth of flow is given by S= S u"c"dz (III-12) Substituting for u" and c" using (III-5) and (III-10) respectively, (I1I-12) becomes K2 aBE z h2 z4 hz3 h2z2 h 1 = -- () (2- hz + 3 24 6 6 45 dz ezKZ 0 Carrying out the integration the mass transport is then =(- ) 2K2h (III-13) 945EzKz It shall be assumed for this analysis and for the analyses that follow that Reynolds analogy relating the transfer of mass and momentum by turbulent processes is applicable so that E = K (III-14) z z This assumption is not necessary to obtain solutions for EL and Ex using the method presented here; however as will be seen later its use simpli- fies the forms of solutions considerably and facilitates interpretation of the physical processes involved. Thus, with the aid of (III-14) and the introduction of the vertical mixing time defined by T h2 (III-15) K z Equation (111-13) may be written as 2K2T3 h S( z4 (III-16) Assuming next that the required conditions for defining a longitudinal dispersion coefficient as discussed in II-B are satisfied, then m L a)h (- - 2K2T' cz EL =~9 (III-17) Using (III-6) and (111-15) in (111-17) the longitudinal dispersion coefficient as a function of uax is 8u2 T max cz EL ----4 (111-18) The functional form of EL given by (111-18) is the same as that obtained by Taylor for laminar unidirectional flow in a tube,(I-8). In both cases the longitudinal dispersion varies directly as the product of the square of the velocity and the cross-sectional mixing time. Thus, for a given unidirectional velocity profile the longitudinal mass transport of a substance increases linearly with the time required to mix that substance over the flow cross section by diffusive processes. III-B Oscillatory Flow III-B.1 Velocity Distribution The same assumptions used in III-A regarding selection of coordinate system, and the use of the Boussinesq approximation for expressing the viscous term in the equation of motion will be applied here. The govern- ing equation of fluid motion for uniform unsteady two dimensional shear flow is then au 1 a2u (III-19) at px+ E z az2 For the oscillatory flow case it will be assumed that long wave phenomena are of primary interest and therefore the pressure distribution over depth is hydrostatic. It is then reasonable to assume that the pressure gradient in (III-19) is periodic and of the form 1 P KeiOt (III-20) p ax where K is constant, as before, and a is the angular frequency of oscilla- tion. Note that only the real part of (III-20) has physical meaning. To obtain a solution assume that u(z,t) is periodic in time and of the form u(z,t) = us(z)eiot (III-21) where u (z) is a complex function. The boundary value problem is then formulated by incorporating (III-20) and (III-21) into (III-19) which along with the boundary conditions of no slip at the channel bed and zero shear stress at the surface becomes dau DE: z dS ius = K (III-22) z dz2 s BC's: u (0) = 0 du Iz z=h = This problem has been previously solved by Lamb (18) and Segall (19) who obtained u(z,t) iK El cosh B(l+i)(h-z) iat (111-23) u(,t ac 1 cosh B(l+i)h where, B= =/ (111-24) with only the real part of (111-23) having any physical meaning. Its form is clearly that of a damped progressive shear wave propagating upward through the water column and causing a temporal phase shift in the velocity as a function of z. It was also pointed out by Segall and Gidlund (20) that the solution as expressed by (111-23) correctly predicts flow reversals in the lower momentum layers of the fluid near the bed prior to a shift in flow direction of the higher momentum layers near the free surface. These effects are extremely important to the disper- sive transport mechanism. The spatial variation of the velocity, u"(z,t), is obtained by averaging (111-23) over depth and subtracting out the mean from the total velocity according to u"(z,t) = u(z,t) u(t) The result is then u"(z,t) = iKeiot { sinh (l+i)h cosh B(1+i)(h-z)} (111-25) a cosh B(1+i)h (l+i)h For reasons that will become obvious in the next section the velocity variation solution as given by (111-25) will now be expanded in a Fourier cosine series of the form u"(z,t) = an cos nT eiat (III-26) n=l where coefficients an are complex. The a0 term in the series has been omitted to satisfy the zero mean requirement for u". It is noted by Hildebrand (22) that any piecewise differentiable function may be com- pletely represented by a cosine series of the form a cos z- over the closed interval 0 < z < h. Proceeding then, h S2 iK sinh B(1+i)h nrz d a h (1+i)h cosh B(1+i)h cos dz 0 h 2 ( iK niz h a cosh (1+i)h cosh B(1+i)(h-z) cos n dz o The first term on the right hand side integrates to zero while the second term must be integrated by parts twice which yields for an, 2K Bh(1-i) sinh B(l+i)h n o cosh B(1+i)h [(ni)2 + i2(Bh)2] (III-27) Since only the real part of u"(z,t) is of any interest, (III-27) will now be put in polar form for incorporation into (III-26). Considering each complex term separately, (1-i) = v2 e'T' (11-28) sinh B(1+i)h = (cosh 26h cos 2Bh) eial (111-29) cosh B(l+i)h = (cosh 2Bh + cos 2gh) e1i2 (111-30) [(nT)2 + i2(Bh)2] = [(nr)4 + 4(Bh)4]2 eia3 (111-31) where, et = tan (cosh 0h sin h) (111-32) sinh Ah cos Bh 2 = tan-1 (sinh h sin Bh) (111-33) cosh h cos 6h a3 = tan-1 C2 )] (111-34) (n) 2 44 To express u"(z,t) in its final form it is again assumed that Reynolds analogy is applicable allowing the use of (III-15). A non- dimensional time relating the period of oscillation to vertical mixing time is then introduced by T cz (III-35) z T With the aid of Reynolds analogy and (III-15), (III-24), and (III-35) the Bh arguments appearing throughout an become h =V/irT (III-36) The final form for u"(z,t) as a function of the pressure gradient is arrived at by combining (III-26) through (III-36) to yield, 2T' (cosh 2 V71 cos 2/VrT) ) u"(z,t) = KT { Z z z 7 (cosh 2V/7rT + cos 2viT'/ ) Scos 2 e(t + L t a3(n)) n=1 [(nT)4 + (2Tz)2] (III-37) where, cosh v rT sin VT ai = tan-I { (III-38) sinh I/TT cos /i z z Ssinh vV/T sin vv/T~ ap = tan {i z (III-39) cosh v/TT cos ViT Z Z i 2Trr a3(n) = tan-' { z2} (111-40) Expressions for u"(z,t) as a function of the surface amplitude, umax, of the periodic velocity function and the surface excursion length, L, experienced by a particle during one-half of the period of oscilla- tion are obtained by solving for K as a function of umax and L using max (III-23). Details of this development are presented in Appendix A, the results of which are VT Tr(cosh 2 v i + cos 2 vN/ T )2 K zZ u (III-41) T (cosh VTJ cos A/_F?) max and 2 I L (cosh 2 2;' + cos 2 VJiTf)4 K =- z z (III-42) T2 (cosh vTYT cos /'i) z z Substitution for K in (111-37) using these two expressions then yields 2u (~T' (cosh 2 v/T' cos 2 V;- )} u"(z,t) max z (cosh -vTT cos /TTz') nc z ei(ot T + al a2 a3) )h] (III-43) [(nt)" + (2rTz)2 n=1 and, 2'iL { TzT' (cosh 2 v/' cos 2 /7T)} u"(z,t) = - T (cosh cos n ei(ot -+ 0i-C 2 03) (III-44) n=1 [(nT)4 + (2T )2]'2 III-B.2 Concentration Distribution For temporally periodic u and c, uniform flow, and constant eddy diffusivity it was shown in II-C that the transport diffusion equation as described from a coordinate system moving with the cross-sectional mean velocity could be written as ac" K c" = -u"c (111-45) 3t z azz 'l where the rate of change with respect to C will hereafter be implied. The concentration variation, c"(z,t), shall be assumed to be of the form iat cos mnz iat c"(z,t) = c1(z)eit a cos eit (III-46) m=l where a' is complex. This expression for c" is seen to satisfy the require- ments of periodicity and zero diffusive flux of substance across the flow boundaries. Moreover, if (III-46) is substituted into (III-45) and operated on, the left hand side is composed of cos nz- terms only. Thus, to satisfy the conditions of equality, the expansion of (III-25) for u" in a cos -z series is justified. The formal statement of the boundary value problem for c"(z) is then Sd2c" DE: K 2 ic = u" (III-47) z dz2 s s - dc" BC's: dc ,h = 0 dz z=0,h where is constant and u (z) is defined by (III-26). Substitution for c" and u" transforms (III-47) to s s { + iuala cos z(z ()C) cos niz Sm h 3 mn h m=1 n=1 which due to the orthogonal properties of cos mT and cos n requires h d cos requires that m=n for a non-trivial solution. The Fourier coefficient a' can m then be written as S h2a a' m ( m (III-48) SKz (mr)2 + i(h2)] Kz The required solution for c"(z,t) is arrived at by: (1) using (III-26) and (III-37) to determine the polar form of am; (2) transforming the denominator of (III-48) to polar form; (3) applying the definitions for T and T' where necessary; and (4) substituting the resulting expression into (III-46). The variation component of concentration as a function of the pressure gradient is then, 3 2(cosh 2 r/TP cos 2 v/iT) c"(z,t)= (- C) KT T'T {- -- z rz . T z ir(cosh 2 Vn 27 + cos 2 vi/T ) Z cos i(t 1 + ai 2a (m)) -Cos --- e 4 (III-49) m=1 [(mr)" + (2TTz)2 where ai, a2, and a3(m) are given by (III-38) through (III-40) with m replacing n in (III-40). Solutions for c"(z,t) as a function of umax, and L are obtained by applying (111-41) and (111-42) to (III-49), to yield c"(z,t) = 2uax T T() (cosh 2 cos 2 max z - (cosh v T cos VIT) z z Scos z i(at + c -1 z2 2ca3(m)) [(mr)4 + (23rT')2 (III-50) m=1 and 3 (cosh 2 v/TT cos 2 VTi) c"(z,t) = 2L(TT) ) z z (cosh VnTV cos v'7 ) mo z ei(at : + c~ 2 2a3(m)) C(m+)* + (2nTz.)2 (III-51) m=1 III-B.3 Dispersive Mass Transport For the oscillatory flow case the mass transport considered will be the time mean transport averaged over one period of oscillation. Thus, the dispersive mass transport is given by <> r = Re(u") Re(c") dz T (III-52) Using (III-37) and (III-49) for u"(z,t) and c"(z,t), the integra- tion over depth of the product R (u") R (c") reduces the implied double summation over m and n to a single sum through the orthogonal properties of the functions cos nz- and cos TZ- since, h h h n h m cos hr cos -- dz = 00 n m Performing this integration and averaging the results over T yields, -h K2T3T12 (cosh 2 V7i cos 2 2vj) T 27r (cosh 2 V/ + cos 2 VTV) cos a3(n) > Cos a-(n) (111-53) n=l [(nT)4 + (2+ T)2 2 The longitudinal dispersion coefficient is next introduced using Taylor's analogy defining the convective mass transport Fickian flux so that S <- E h()> (111-54) Noting from (111-40) that cos a3 (n) = (nr)2 [(ni)4 + (2rT )23] and combining (111-53) and (111-54) the time mean longitudinal dispersion coefficient as a function of the pressure gradient is expressed as SKT3T'2 (cosh 2 V/vz cos 2 /'T) S (cosh 2 VTr/ + cos 2 VJ/~i) >J (nir)2 [(nir)4+(2rT)22] (III-55) n=l Solutions for substituting directly for K2 in (III-55) using (III-41) and (III-42). The resulting expressions are as follows: (cosh 2 VT7 cos 2 V"_7) x T max z (cosh vi cos I I)2 z z > (nn)'2 (III-56) n [(n) 4+(2rTz)212 n=1 and, iT L2T'2 (cosh 2 /rT cos 2 /vT') x T T (cosh /T7 cos /-'TT)2 (nm)2 (III-57) [(nr)4 + (2Tz)22 n=1 Equation (111-56) is considered to be the most useful and descrip- tive of the solutions presented for shear flow. Therefore, to facilitate a discussion of these results in the next section, (111-56) is non-dimensionalized by defining E x (III-58) x 2T max 50 and restating it as ST'2(cosh 2 V T'z. cos 2 Vi )T' E; = ----- -- -- . X (cosh v/rTTz cos v rT')2 z z (ns)2 n=1 [(nw)' + (2wTz)2]2 (III-59) III-C Discussion of Two Dimensional Shear Flow Results III-C.1 Dispersion in Unidirectional Flow as a Limit of the Oscillatory Flow Case This discussion shall begin by relating the oscillatory flow disper- sion coefficient, coefficient, EL, as has been previously done by Holley et al., Awaya, and Fukuoka; and shall then proceed to show how the unidirectional process is in fact a limiting case of an oscillatory process which has its own distinct characteristics. Thus, solutions obtained in III-B.3 for of the velocity are normalized by the "corresponding" solutions for EL obtained in III-A.3. Dividing (111-55) by (111-17) and (111-56) by (III-18) yields Z (n-)2 (111-60) n=1 [(ni)" + (2T)2]2 for the same K, and A I z z EL 8 (cosh viT- T- cos / iT)2 z z Z (nT)2 n=1 [(nT)4 + (27TT)2]2 (111-61) for the same umax. All calculations involving infinite summations for two dimensional shear flow dispersion use 200 terms in each sum, well within fourth place accuracy. The expressions given by (111-60) and (111-61) are plotted in Figure 3 vs. the non-dimensional time 1/Tl corresponding to T' defined by z Holley et al. Also included in Figure 3 is the solution obtained by Holley et al. for discussed in I-B.2. It is noted that all three solutions approach the limiting value of one-half for situations when the period of oscillation is greater than the vertical mixing time in accordance with the findings of Okubo and Bowden. However, the difference in behavior of the two solutions given by (III-60) and (III-61) is dramatic and immediately raises the question of whether or not a plot such as Figure 3 is the most meaningful method of illustrating the characteristics of dispersive mass transport in oscillatory flow. For the case of the pressure gradient normalized solutions the ratio with increasing Tz because: (1) For unidirectional flow the pressure gradient and viscous forces are in equilibrium so as to produce a constant unidirec- tional shear flow for the dispersive transport of substance. (2) In oscillatory flow the pressure gradient is assumed to be simple harmonic in time and therefore is in constant balance with the varying inertia and friction forces. As the period of oscillation decreases the inertial effects become very large such that in the limiting case little or no flow would be induced. This results in little or no dispersive transport. This effect of the period of oscillation on velocity can be seen from (III-23). The solutions normalized by umax exhibit a much slower decrease of A 1 L t I0" I0" 10 T' TK- h2 FIGURE 3 NON-DIMENSIONAL DISPERSION COEFFICIENT velocities of the unidirectional and oscillatory flow are required to be the same. In terms of pressure gradients this would correspond to the situation where Kosc >> Kunidir Based upon these results it becomes apparent that due to the very different hydrodynamic characteristics of unidirectional and oscillatory flow systems, the interpretation of the nature of dispersive mass trans- port in an oscillatory system as compared to the same process in a uni- directional system is misleading and can vary widely depending upon how one chooses to relate the two systems. This is demonstrated by Figure 4. In this figure the longitudinal dispersion coefficient, by (111-56) is plotted against the vertical mixing time, Tcz, for four periods of oscillation ranging from 22,536 to 223,560 seconds and a umax of 1 ft/sec. Also plotted is the solution for the unidirectional flow coefficient, EL, as given by (III-18). As shown by Figure 4 the behavior of EL and coefficient varies directly with the vertical mixing time which for little or no turbulent mixing over the water column allows the shear flow to transport higher concentrations of substance far downstream. The behavior of the oscillatory flow coefficient, however, is governed by a type of resonant interaction between the period of oscillation and the vertical mixing time. As the period of oscillation is increased the resonant peak shifts to the right and likewise increases until the limiting case is reached where the peak is infinitely large and the values predicted for flow coefficient EL. - 600 ..D -A - 5I S500 -, 400 I 400- C 81 0 I l \ 10 102 104 10106 10 c 300- 447 (sec)sec ^ (Semi- Diurnal Tide) I FIGURE 4 DISPERSION IN UNIDIRECTIONAL FLOW AS A LIMIT OF THE OSCILLATORY FLOW CASE 200- 100- 10 102 10o 104 10e 10e 107 FIGURE 4 DISPERSION IN UNIDIRECTIONAL FLOW AS A LIMIT OF THE OSCILLATORY FLOW CASE The resonant characteristic of the abrupt decrease in the ratio value of T' at which the knee of the curve occurs in Figure 3 corresponds to the value of Tcz in Figure 4 at which diverge from the unidirectional solution. At this point EL continues to increase whereas III-C.2 Characteristics of Dispersive Mass Transport in Oscillatory Flows The physical reasoning behind the resonant behavior of straightforward and is begun by considering the extreme cases of T << Tc and T >> Tcz: (1) T < small during one period of oscillation that an elemental volume initially residing in the water column at elevation z, and containing an initial concentration c, would remain at this elevation thus being transported over the closed pathline of flow, and returned to its initial position with no net longitudinal dispersion having occurred. Conversely,in a unidirectional flow with little or no vertical mixing the longitudinal dispersion would be very large. (2) T >> Tcz Here the rate of vertical mixing is so rapid that there is no time for the velocity shear profile to transport the sub- stance longitudinally before it loses its identity through vertical mixing. In this case the oscillatory and unidirectional flow dispersive processes behave in a similar manner and are both small. Thus, it is seen that for an oscillatory flow the longitudinal dispersive mass transport becomes small for both T << T and T >> T, cz cz whereas for a unidirectional flow the longitudinal dispersive mass trans- port varies directly with Tcz. This resonant nature of further explained on physical grounds by beginning with Case 1 above and gradually allowing the vertical mixing time to decrease. In this example the velocity shear profile in conjunction with the increased vertical mixing causes a net longitudinal mass transport to occur over a period of oscillation. This effect continues to increase as Tcz is decreased until the optimum ratio between the vertical mixing time and the period of oscillation is reached. At this point the longitudinal dispersive mass transport has reached its maximum value. A further decrease in Tcz begins to introduce the effect noted in Case 2 where the vertical mixing is now too rapid thus causing Additional physical insight into the behavior of tained by considering the relative longitudinal dispersion associated with two oscillatory flows, one having twice the period of oscillation of the other. The two periods considered are 44,712 and 89,424 seconds corresponding to the periods of a semi-diurnal and diurnal tide respectively. The first of the two comparisons is illustrated by Figures 5a and 5b. Once again the expression used for umax = 1 ft/sec for both the semi-diurnal and diurnal cases. As shown in Figure 5a after an elapsed time of t = 89,424 sec the semi-diurnal tide will have completed two full cycles of excursion length L,, while the diurnal tide will have completed one full cycle of excursion length L2 = 2L,. In Figure 5b the favorable effect of the longer excursion length on longitudinal dispersion is seen for flows of different periods of oscillation with the same umax. The second compari- son between the semi-diurnal and diurnal period flows uses excursion length as a normalizing parameter and is illustrated by Figures 6a and 58 x+ ---______-------------- T=To; Umx I ft/sec ST=2To Uax = I ft/sec L2= 2L. (a.) Tcz = h (sec) (b.) FIGURE 5 COMPARISON OF DISPERSION FOR DIURNAL AND SEMI-DIURNAL PERIODS OF OSCILLATION WITH SAME umax 6b. In this comparison the velocity for the diurnal tide is scaled downward so as to produce the same excursion length as the semi-diurnal tide. Figure 6a shows that after an elapsed time of t = 89,424 sec the semi-diurnal tide will have completed two full cycles of excursion length Lias before. However, the diurnal tide, because of the reduced velocity, has completed one full cycle of the same excursion length. Values of seen from this plot the semi-diurnal dispersion has remained unchanged from Figure 5b whereas the diurnal curve is now everywhere less than the semi-diurnal curve. This is not surprising since from (III-57) the fixing of excursion length and vertical mixing time leaves only the decrease in the period of oscillation as a means of increasing longitudinal dispersion. The results of this analysis on the predicted behavior of in two dimensional oscillatory shear flow are summarized in non- dimensional form by Figure 7. In this figure the non-dimensional longitudinal dispersion coefficient as defined by (III-59) is plotted vs. Tz. Thus, by knowing the period of oscillation, the surface amplitude of velocity, the water depth, and the vertical eddy diffusi- vity, one could use Figure 7 to obtain a predicted value for the longitudinal dispersion coefficient. The resonant value of Tz for which T' = 1.58 (11-62) This relationship is applicable to all oscillatory flows in in- finitely wide rectangular channels provided the assumptions made in this analysis are reasonably well satisfied. A look at the kinematic structure of the flow field and the T = To; L To/ Umax + T= 2To; L=TO/r Umax itllf/if -/_/l//ff/F//////lrr ^ 10 I 106 I0. FIGURE 6 COMPARISON OF DISPERSION FOR DIURNAL AND SEMI-DIURNAL PERIODS OF OSCILLATION WITH SAME SURFACE EXCURSION LENGTH 61 350 I ii I I 300- 250 x 150 - 100 50- II I T = 1.58 10 I0 10 I 10 102 l Tc z T- TZ T FIGURE 7 NON-DIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT E;, AS A FUNCTION OF T' FOR TWO DIMENSIONAL OSCILLATORY SHEAR FLOW concentration distribution over depth is presented for three values of Tz in Figures 8, 9, and 10. In these figures, non-dimensional forms of the solutions for u"(z,t) and c"(z,t) given by (III-43) and (III-50) respectively are plotted over the non-dimensional depth, z/h. The non- dimensional forms used are UND Umax C1 =- (III-64) (- )u T DS) max Each figure shows the u" and c" profiles for maximum flow and slack water. Several interesting points are illustrated here. First, the superior correlation of u" and c" over depth in Figure 9 when compared to the same correlation in Figures 8 and 10 demonstrates qualitatively why considered. Second, the phase dependency of the velocity with depth is clearly shown in all figures near slack water with flow reversals occurring in the lower momentum layers near the channel bed prior to a shift in the upper higher momentum layers. Finally, it is interesting to note that the shear profile is confined to a decreasingly thinner layer near the bed with the upper portion of the profile becoming flatter as T' is increased (increasing eddy viscosity by Reynolds analogy). analogy). 1.0 lI I 1.0 0.8- '-0.8 NOD NO 0.6- CN-- / C -- -.6 z z. 0.4 / 0.4 0.2- -.2 WlO -0.75 -0.5 -0.25 0 0.25 0.5 0.25 1.0 .-1.0 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1.0 I I I-I I I I D I I I I I I L I I -8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8 CN1 X 102 CND X 102 a. Maximum Flow (ot = (o-t)max) b. Slack Water (crt = (-t)max + 7'/2) FIGURE 8 NON-DIMENSIONAL VELOCITY AND CONCENTRATION VARIATION COMPONENT PROFILES FOR T' = 0.5 z U NO "UNO-- - CN' -- II I / / /I iI UND CNo / I /2/ // / K -10 -0.75 -05 -0.25 0 0.25 0.5 0.75 1.0 -1.0 -0.75 -05 -0.25 0 0.25 0.5 0.75 1.0 -8 -6 -4 -2 CO 10 2 4 6 8 -8 -6 -4 -2 0 22 4 6 Maximum Flow b. Slk Wter a. Maximum Flow ( 0-t = (0t)max) b. Slack Water (ot = (0t)max + ) FIGURE 9 NON-DIMENSIONAL VELOCITY AND CONCENTRATION VARIATION COMPONENT PROFILES FOR T' = 1.58 Z 1.0 0.8 0.6 z h 0.4 0.2 0, I I I ' I I I 1.0- \ 1.0 \ II 0.8- 0.8 uND \ UNO / 0.6- CND.-- N C' / 0.6 z -/ z h 0.4- / 0.4h 0.2- l -0.2 / / O0 I I 0 -1.0 -075 -0.5 -0.25 0 0.25 05 075 1.0 -1.0 -075 -0.5 -0.25 0 025 0.5 0.75 1.0 U" U " I I I I IND I I I I I I IND I I I -8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8 CN, X 10 CN X 102 a. Maximum Flow (ot = (ot)max) b. Slack Water (oat = (ot)max + "/2) FIGURE 10 NON-DIMENSIONAL VELOCITY AND CONCENTRATION VARIATION COMPONENT PROFILES FOR T' = 5.0 z CHAPTER IV DISPERSIVE MASS TRANSPORT IN RECTANGULAR CHANNELS OF FINITE WIDTH IV-A Oscillatory Flow The physical problem and the general analytical approach used to solve it are the same in this chapter as those discussed in Chapter III with the added feature of a finite channel width. The effect of this feature is to introduce a three dimensional shear flow with vertical and transverse velocity variations which are used to force the transport diffusion equation to obtain a solution for c"(y,z,t). Solutions for u"(y,z,t) and c"(y,z,t) are then correlated over the flow cross section and, in the case of oscillatory flow, averaged over the period of oscillation to obtain the longitudinal dispersive mass transport. The coordinate system selected has its origin at the center of a rectangular section whose sides correspond to z = h, and y = w/2. As shown in Figure 11, the open rectangular channel is mathematically represented by the lower half of the section. Selection of the coordinate system in this manner preserves the symmetry of the problem about the origin and, as will be shown later, correctly predicts the same dispersive mass transport for equal degrees of skewness in channel geometry in either the vertical or lateral directions. z=h u(y, z,t) zz y=-w/2 y=w/2 ^ /^ y Iy=-w 20 z=-h ,,ry=w/2 z=-h FIGURE 11. DEFINITION SKETCH FOR THREE DIMENSIONAL SHEAR FLOW IV-A.1 Velocity Distribution Using the same assumptions that were made in Chapter III regarding eddy viscosity, time averaging of turbulent fluctuations, and the form of the pressure gradient term, the equation of motion for uniform flow in the x direction may be written as au iot 2u + 2u =- Ke +E +e (IV-1) at z z2 3y2 If a solution of the form u(y,z,t) = us(y,z) eiat (IV-2) is assumed to be valid for (IV-1) then the three dimensional shear flow boundary value problem for oscillatory flow may be formally stated as: a2u a2u PDE: E + E iau = K (IV-3) Sz2 y aSy s BC's: u (y,h) = 0 us( ,z) = 0 It is noted that physical considerations require that u (y,z) be an even function in y and z. Thus, a solution to (IV-3) of the form us(y,z) a' cos (1 (n 2m+1 1) y (IV-4) m=0 n=O is assumed. An examination of (IV-4) with the statement of the boundary value problem above shows that the assumed solution satisfies the no slip conditions on the perimeter of the section. Moreover, (IV-4) also satisfies the condition of zero vertical shear at the free surface, and zero lateral shear at the centerline of the channel. The assumed solution is then substituted into (IV-3) and the Fourier coefficients, amn, are determined in the usual manner through the use of the orthogonal proper- ties of the cosine function. Details of this analysis are presented in Appendix B. Solving for amn and thus determining the form of us(y,z), the complete solution for the velocity function u(y,z,t) using (IV-4) and (IV-2) is then Z V m+nc (16K2m+1)6rz s( (2n+l)gTy igt u(y,z,t) = 2-16K(- hmc 2 cos w w (IV-5) j (2m+1) ( rCo [2m+1h2 + C (2n+1 )T, m=O n=O (2m+1)(2n+l)Tr{ E 2i 1 2 + + io} To obtain the spatial variation, u"(y,z,t), the cross-sectional mean velocity, u(t), must be subtracted out of (IV-5). The mean component of u is defined by 2 iot u(t) = e2h us(y,z) dy dz (IV-6) -h -w/2 Performing the integration in (IV-6) and subtracting the result from (IV-5) yields u"(y,z,t) = n {cos (2m+)z cos(2n+1)y g(m,n)}eit (IV-7) m=O n=O where, m+n g(m,n) = 4(-1)m '(2m+1)(2n+1) and, 16K (-1)m+n amn n 2(2m+1)(2n+l) { z [(2E 12 + Ey 2n+ 1 i (IV-8) (IV-9) Since only the real portions of these expressions have any physical meaning, (IV-7) is transformed to polar form as was done in Chapter III for the two dimensional case. In addition, it will be assumed once again for simplification of results that Reynolds analogy applies thereby allowing the introduction of the vertical mixing time Tcz and the defini- tion of a comparable lateral mixing time as T = (/2)2 cy K y The solution for u"(y,z,t) as a function of the pressure then be written as u(yzt) 8K ()m+n ei(ct-ai(m,n)) u"(y,z,t) = 7-.) m = 0 n = 0 (2 m+1)(2 n+l) R0 (m,n) (2 m+)z cos (2 n+l)y g(mn) (IV-10) gradient may K (IV-11) where, T m+l)2 T +I +)712) R1(m,n) = T [ (2 m+1)12 2 T E (2 n+l1)72 + 1 1 c cy al(m,n) = tan { [(2 mlh12 T [ ( nlL 1 T(mn) = tan(T 2 m+l)721 T (2 n+1)Tn 2 2T ^ 2 2] T T L 2 J (IV-12) (IV-13) In order to put (IV-11) in a more usable form, (IV-5) is used to obtain a relationship between the pressure gradient, K, and umax, where umax is now defined as the amplitude of the periodic velocity function at the free surface centerline. Thus, from (IV-5) u(0,0,t) = . m = 0 n= 0 -16 K(-)m+ ei (-14) T2(2 m+l)(2 n+1) {Ez[2 m+) 2+ [(2 n+l)l 2 i w which upon maximization with repsect to time yields the relationship umaxir3 K -max (IV-15) 8T where, S = ,-)m+n(R(m,n)-l) m 0 n 0 (2 m+l)(2 n+l)Rl(m,n + (-)m+n (IV-16) m = 0 n = (2 m+l)(2 n+l) R(m,n) The details of developing (IV-15) from (IV-14) are presented in Appendix C. Substitution of (IV-15) into (IV-11) for K yields the following expression for u" as a function of umax metUm)max) u"(y,z,t) umax (-Im+ni(t-(mn)) m = 0 n = 0 (2 m+1)(2 n+l)R (m,n) cos (2 m+) cos (2 n+l)y g(m,n) (IV-17) 2h w IV-A.2 Concentration Distribution The form of the transport diffusion equation used to solve for c"(y,z,t) is identical to (11-27), or c" a2cl' a2c a3 2 K z K = u" (IV-18) at z z2 Y y2 aT where u"(y,z,t) is given by (IV-7). It is assumed that c"(y,z,t) is of the form c"(y,z,t) = c" (y,z) eiot (IV-19) s where c"(y,z) is complex following the same approach used with u"(y,z,t). Substituting (IV-19) into (IV-18) and requiring zero diffusive flux across the boundaries of the flow, the boundary value problem is then stated as a2c" 92c" PDE: K s + K ic" = u" (IV-20) z @2 y 9y2 s s 9E ac" BC's: I = 0 z =h y = w/2 Now, c" must also be an even function in y and z based on physical arguments and is therefore assumed to be of the form c (y,z) = a'k COST z cos 2kY (IV-21) Z=0 k=0 Equation (IV-21) is seen to satisfy the conditions of zero diffusive flux across the flow boundaries. Substituting for c" using (IV-21) and for u" using (IV-4) and (IV-7), (IV-20) becomes Kz ()2 + ( )2+ ic} a cos z cos 2 ry Z (Lh. y w Zk h w a =0 k=O -7 (2 m+l),z (2 n+l) g(m,n)) ( ) amn {cos (2 m+1),z cos ( n 1)ny m = 0 n = 0 (IV-22) where amn is given by (IV-9). The unknown Fourier coefficients ask are solved for in the same manner as was used to determine amn. The detailed mathematics of the solution are presented in Appendix B, the result of which is given by at 256K 2i (IV-23) aik = 4{_- 2 -) 2 O 2 Kz() y+ K( + i p = 0 q = 0 (-1)-(P+q) [(2 p+1) -(2Z)2][(2 q+l)2-(2k)2]{[E (2 ]2 r(2 q+1z12 + io} Thus, the solution for c"(y,z,t) includes four infinite summations arising from the non-orthogonal nature of the cosine functions representing u"(y,z) and c'(y,z). Again introducing Reynolds analogy and the expression for the vertical and transverse mixing times, the solution for c"(y,z,t) in polar form as a function of the pressure gradient is c"(y,z,t) = (c) 64KT = 0 k = 0p= 0 q =0 pk(-)-(+k)cos _z cos 2kRy e ie2 h w (IV-24) [(2 p+1)2-(2P)2][(2 q+1)2-(2k)2]R (p,q)R (l,k) where, 68 = at a((p,q) az(k,k) (IV-25) R,(p,q) is given by (IV-12) with p replacing m and q replacing n. R,(,k) = {T (T)2 + 2T- (k7)2}2 + 1 (IV-26) cz cy ac(p,q) is given in (IV-13) with p replacing m and q replacing n. c2(Z,k) = tan' { T 1 T (IV-27) 2 F 2rT (kiT)2 cz cy (0, k = = =k = , k = 0 or R = 0 (IV-28) 1, k f 0,, 0 The factorPZik is necessarily included in (IV-24) to properly account for the a6k and a'o terms, and to insure that c"(y,z,t) has a zero mean over the flow cross-sectional area. Detailed mathematics for arriving at (IV-24) are given in Appendix B. The expression of c"(y,z,t) as a function of umax is determined by substitution for K in (IV-24) using (IV-15) to yield, c"(y,z,t) = (- aI 8 k=0 k=0 p=O q=0 S(-l)(+k)cos 7z cos 2ktry ei,2 h w -(IV-29) [(2 p+1)2-(2)2][(2 q+1)2-(2k)2]R (p,q)R,(R,k) IV-A.3 Dispersive Mass Transport The time mean dispersive mass transport averaged over one period of oscillation for the full rectangular section as shown in Figure 11 is given by w/2 T = R R[U"(y,z,t)].Re["(y,z,t)] dz dy > T (IV-30) -w/2 -h The integration in (IV-30) are carried out in Appendix B and it is noted that the non-orthogonality of the cosine functions representing u" and c", arising from the form of their respective arguments, once again produces two additional infinite summations. Thus, using (IV-11) and (IV-24) for u" and c" the following expression for the longitudinal dispersive mass transport over the region -h < z < h and -w/2 < y < w/2 is determined = 2048 wh K2T> S0 k= 0 m= 0 n= 0 p 0 q = 0 Upk{(R2(o,k)-l) cos X + sin X} 61( ,k,m,n,p,q) (IV- where, X = al(m,n) al(p,q) (IV-32) 61(a,k,m,n,p,q) = [(2p+l)2-(22)2][(2q+1)2-(2k)2][(2n+l)2- (2k)2] (IV-33) [(2m+l ) _(2P)2] R I 1(m,n)R'lj'(p,q)R2(P,k) Now the dispersive mass transport associated with the open rectangu- lar channel comprising the lower half of the full box is the Fickian flux representation of the dispersive mass transport in the open rectangular channel is simply - Replacing for =0 k= 0 m= 0 n= 0 p = 0 q = 0 uZk{(R2(,k)-l)2 cos X + sin X} (IV-35) 61(Z, k,m,n,p,q) The longitudinal dispersion coefficient is expressed as a function of umax by substituting for K in (IV-35) using (IV-15) to give C=O k=O m=O n=O p=O q=O T l = 0 k = 0 m = 0 n = 0 p = 0 q = 0 PZk(R2(z,k)-l)2 cos A + sin X} (IV-36) 61(Z,k,m,n,p,q) To facilitate the presentation of results and the application of (IV-36) for predictive purposes a non-dimensional relative mixing time, Tc, is introduced as T' T Tc =T' (IV-37) y cy where, T T' = cy (IV-38) y T Thus, as shown by (IV-37), T' is a measure of the relative effects of vertical and lateral shear for a given width-to-depth ratio, and vertical and lateral eddy diffusivities. With the aid of (IV-37) the following definitions can then be made: ri(m,n) = (2nTI )i(m,n) = {[ 2m + T [ ]2}+(2T)2 (IV-39) r2(z,k) = (2TTT)2R2(Z,k) = {( +)2 +Tc(ki)2}2+(21T')2 (IV-40) c,(m,n) = tan [2ml 2]2+Ti[ ]2 (IV-41) 2 2 (-)m+n [rl(m,n)-(2,Tz)2 1 (2 2= 2iT (2m+l)(2n+l)r1(m,n) I m = O n = 0 + 21z T 2m+l (2n r(mn (IV-42) z2aT (2m+l)(n Zn)r)1 ,) m =0 n= 0 These relationships are then applied to (IV-36) to obtain an expression for the non-dimensional longitudinal dispersion coefficient, E defined by (111-58). This expression represents the final form of the solution for the case of three dimensional oscillatory shear flow and is there- fore presented along with a summary of the associated terms as follows: max = 0 k = m = 0 n = p = q = sk{[r2(Z,k)-(2Tz )2] cos x + (2rT') sin x} z(IV-43) 62(Z,k,m,n,p,q) where, 62(k,k,m,n,p,q) = (2rTz)461(t,k,m,n,p,q) = [(2p+l)2-(2Z)2][(2q+l)2-(2k)2][(2m+l)2-(2)2] [(2n+l)2-(2k)2]r(m,n)rpq)r( 2(,k) (IV-44) r1(m,n) {[(2m+1)1]2 T 2+ 2}2 + (2TT') r1(p,q) = r1(m,n) with p replacing m and q replacing n r2(.,k) ={(pr)2+Tc(k7)2}2 + (2iT )2 27rT' 1(m,n) = tan -'gi+l){ al(p,q) = al(m,n) with p replacing m and q replacing n (-l)m+n[rc(mn)-(21T)2] 1 2 2 = (2m+l)(2n+l)r1(m,n) + 2 0 nz = (2m+I)(2n+l)r1(m,n) m = 0 n = S= a1(m,n) ca(p,q) 1 k f 0, 0 k! = k = 0 or Z = 0 0 k = 0 The symmetry of this solution can now be seen. If, for example, T, had been inversely defined as Tcy /Tcz, then (IV-36) would have reduced to (IV-43) with T' replacing T' everywhere, and the newly defined y z T' multiplying all complementary terms in rl, r2, and a,. This form of (IV-43) would predict the same value for E' as the original formulation x provided that u and T were the same and that the new T' equaled the max y old T' and the new T' equaled the old T'. Stated another way, for the z- c c same umax and T, a channel whose half-width was twice its depth would produce the same dispersive mass transport as a channel whose depth was twice its half-width provided that the ratio K /K for the first case equaled K /K in the second case. The functional behavior of E' as given by (IV-43) for three dimen- x sional oscillatory shear flow is illustrated by Figure 12. In this figure E' is plotted against the non-dimensional vertical mixing time, x Tz, for fixed values of the relative mixing time, Tc. Because the solution for E' contains six infinite summations nested in series some x limitations were necessary in carrying out the required computations. By varying the upper bound on each of the sums it was demonstrated that the solution converges upward to its limiting value. The convergence occurs reasonably rapidly; however, above an upper limit of 10 terms for each sum the rate of convergence is slowed considerably. The results 400 350 - 300 =0.01 S= 1.0 250- 5 / \\ c ----Tc = 0.67 S= 0.33 200- c// \\ =0 0.1 x c - X w 0 0.0c 50 - T' X 0 10 and X1 Tc': 0.001 T0 0 o - 50 15)3 102 1(' 10 102 T :Tcz TT = FIGURE 12 NON-DIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT E', AS A FUNCTION OF Tz FOR FULL RANGE OF T' VALUES x C presented in Figure 12 and in subsequent figures were computed using an upper bound of 10 on each sum; therefore, each value of E' includes approximately 1.2 x 106 terms. An estimate of the error between the com- puted values of E' and the values predicted by the complete series x represented by (IV-43) is obtained by comparing the limiting curve for a very wide channel in Figure 12 with the corresponding curve for an infinitely wide channel shown in Figure 7. The limiting curve in Figure 12 is identified by T' < 1 x 10.6 on the left hand side of the peak and c - by T' < 0.001 on the right hand side of the peak. A comparison of the two C -6 curves suggests that for T' < 1 x 10- the solution for E' given by ix (IV-43) very nearly approximates for all T' the solution given by (III-59) for E' in an infintely wide channel. The general shapes of the two curves are nearly identical with both peaks occurring at T' = 1.58. For the infinitely wide channel the peak value of Ex is 3.27 x 10-3 whereas for the limiting curve in Figure 12 it is 3.07 x 10-3 or 6 per cent below the infinitely wide case. Thus, it appears that the three dimensional solution for E' approaches the two dimensional solution for E' in the x x limit as T' 0, and it is estimated that values predicted for E' in c x Figure 12 are low but generally within 10 per cent of their actual values as determined by (IV-43). This does not apply to the range T' > 100 where the Figure 12 values are as much as 70 per cent low. However, this is not a region of practical significance since values of Tz in nature are generally <100. To aid in the interpretation of results, the eight curves shown in Figure 12 have been separated into two groups and are replotted in Figures 13 and 14. Figure 13 shows the E' resonance curves for T' = 1.0, S 0.67, 0.33, and 0.1 while Figure 14 shows the corresponding curves for 81 400 I I 350 - 300- 250- - 200 - -x A D- x 150- Tc-' 0. I 100 Tc= 0.33 Tc'.= 0.67 50- Tc= 0 10" 102 I I I0 I02 103 Tcz Tz- T FIGURE 13 NON-DIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT E' AS A FUNCTION OF T; FOR T' = 1.0, 0.67, 0.33 AND 0.1 To' 0 .- -x 100 Tc' =0.01---j (C = 0.001\ 50- TC' =0.0001 -6 Tc' IXIO 103 102 10-1 I 01 102 03 S Tcz Tz T FIGURE 14 NON-DIMENSIONAL LONGITUDINAL DISPERSION COEFFICIENT E' AS A FUNCTION OF T' FOR T' = 0.1, 0.01, 0.001, x z c 0.0001, and <1 x 10-6 -6 T' = 0.1, 0.01, 0.001, 0.0001, and < 1 x 10-6. Note that the curve for c T' = 0.1 is repeated to provide a common reference. If for the purposes c of interpretation it is assumed that K = K then T' may be written as z y c T' = [ h]2 (IV-45) c (w/2r which is simply the inverse square of the channel half-width to depth ratio. Referring to Figures 12, 13, and 14 with the aid of (IV-45) it is seen that as the half-width of the channel is increased from a value equal to the depth the resonant peak shifts from T' = 4.5 leftward toward Tz = 1.58 z at which it occurs when T' < 0.001. Concurrent with the shift is an ini- C - tial increase in the peak value of E' as T' is decreased from 1.0 to 0.25 x c followed by a decrease in the peak value as Tc is further decreased to 0.001. Decreasing Tc below this value has no effect on EL for T' > 1.2. c x z A summary of the maximum value of E' achieved and the value of T' at which x z it occurs for the corresponding T' is given in Table 1. These data are also presented graphically in Figures 15 and 16. TABLE 1 PREDICTED RESONANT PEAK DATA FOR LONGITUDINAL DISPERSION COEFFICIENT IN THREE DIMENSIONAL OSCILLATORY SHEAR FLOW Non-Dimensional Peak Value S. Vertical Mixing of Non-Dimensional Relative Mixing Time, T' at which Longitudinal Dispersion Time, T' z Coefficient, E' x 105 c Maximum Occurs x 1.00 4.50 339 0.67 3.75 340 0.40 2.40 350 0.33 2.20 353 0.25 1.95 354 0.20 1.80 352.5 0.10 1.65 346 0.01 1.60 325 0.001 1.58 310 0.0001 1.58 307 0.00001 1.58 307 0.000001 1.58 307 84 4.0 3.0 Tz 2.0 .2 0T z = 1.58 1.0- 0 10-I 10-2 10-3 i0o 10 5 Td FIGURE 15 NON-DIMENSIONAL VERTICAL MIXING TIME, T;, AT WHICH MAXIMUM E' OCCURS AS A FUNCTION OF T' x C 360- o 340 0 -x Li FIGURE 16 PEAK E' AS A FUNCTION OF T' x C |