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## Material Information- Title:
- Boundary shear stress through meandering river channels
- Creator:
- Randle, Timothy James (
*author*) - Place of Publication:
- Denver, CO
- Publisher:
- University of Colorado Denver
- Publication Date:
- 2014
- Language:
- English
- Physical Description:
- 1 electronic fiole (261 pages). : ;
## Thesis/Dissertation Information- Degree:
- Doctorate ( Doctor of Philosophy)
- Degree Grantor:
- University of Colorado Denver
- Degree Divisions:
- Department of Civil Engineering, CU Denver
- Degree Disciplines:
- Civil engineering
## Subjects- Subjects / Keywords:
- Turbulent boundary layer -- Mathematical models ( lcsh )
Shear flow ( lcsh ) Meandering rivers -- Mathematical models ( lcsh ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Abstract:
- As water moves through a meandering river channel, flow velocity and shear stress both vary along the river banks as a result of channel curvature. An improved predictive relationship of how bank shear stress increases with channel curvature is needed for the prediction of river bank erosion and channel migration. The erosion of meandering river banks is of interest to people living near a river and to agencies planning or maintaining infrastructure within or along a river. A numerical, three-dimensional hydraulic model was used to simulate boundary shear stress along a wide range of meandering river channels of the type (width and depth) found in nature. A total of 72 virtual meandering channels were developed from sine-generated curves. The hydraulic conditions of these channels spanned three orders of magnitude of bank full discharge and four orders of magnitude in longitudinal slope. Channel sinuosity (ratio of channel length to valley length) ranged from 1.1 to 3.0. Trapezoidal cross sections of constant width were used define each channel. The hydraulic data sets, generated from the numerical model simulations, were used to develop empirical relationships between magnitude and location of maximum dimensionless shear stress and dimensionless channel properties: ratio of channel width to minimum radius of curvature, width-depth ratio, and sinuosity. Maximum dimensionless near-bank shear stress in meandering channels was found to increase linearly with increases in the ratio of channel width to radius of curvature and decrease with increases in the width-depth ratio. The empirical equations can be used to estimate shear stress for the design of streambank protection in meandering channels and an example is provided.
- Thesis:
- Thesis (Ph.D.)--University of Colorado Denver. Civil engineering
- Bibliography:
- Includes bibliographic references,
- System Details:
- System requirements: Adobe Reader.
- General Note:
- Department of Civil Engineering
- Statement of Responsibility:
- by Timothy james Randle.
## Record Information- Source Institution:
- University of Colorado Denver
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- |Auraria Library
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- 892857912 ( OCLC )
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BOUNDARY SHEAR STRESS THROUGH MEANDERING RIVER CHANNELS
by TIMOTHY JAMES RANDLE, P.E., D.WRE. B.S. in Civil Engineering, University of Utah, 1981 M.S. in Civil Engineering, University of Colorado, 2004 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Civil Engineering Program 2014 This thesis for the Doctor of Philosophy degree by Timothy James Randle has been approved for the Civil Engineering Program by James C.Y. Guo, Advisor Rajagopalan Balaji, Chair Edmund D. Andrews John P. Crimaldi David Mays Zhiyong Ren March 11, 2014 Randle, Timothy James (Ph.D., Civil Engineering) Boundary Shear Stress Through Meandering River Channels Thesis directed by Professor James C.Y. Guo ABSTRACT As water moves through a meandering river channel, flow velocity and shear stress both vary along the river banks as a result of channel curvature. An improved predictive relationship of how bank shear stress increases with channel curvature is needed for the prediction of river bank erosion and channel migration. The erosion of meandering river banks is of interest to people living near a river and to agencies planning or maintaining infrastructure within or along a river. A numerical, three-dimensional hydraulic model was used to simulate boundary shear stress along a wide range of meandering river channels of the type (width and depth) found in nature. A total of 72 virtual meandering channels were developed from sine-generated curves. The hydraulic conditions of these channels spanned three orders of magnitude of bankfull discharge and four orders of magnitude in longitudinal slope. Channel sinuosity (ratio of channel length to valley length) ranged from 1.1 to 3.0. Trapezoidal cross sections of constant width were used define each channel. The hydraulic data sets, generated from the numerical model simulations, were used to develop empirical relationships between magnitude and location of maximum dimensionless shear stress and dimensionless channel properties: ratio of channel width to minimum radius of curvature, width-depth ratio, and sinuosity. Maximum dimensionless near-bank shear stress in meandering channels was found to increase linearly with increases in the ratio of channel width to radius of curvature and decrease with increases in the width-depth ratio. The empirical equations can be used to estimate shear stress for the design of streambank protection in meandering channels and an example is provided. The form and content of this abstract are approved. I recommend its Publication. Approved: James C.Y. Guo IV DEDICATION I dedicate this work to my wife, Kathy Randle, who has been extremely supportive of my efforts to pursue an advanced degree, and to my father, Kenneth W. Randle, for encouraging me to pursue a career in engineering. v ACKNOWLEDGMENTS I would like to thank Professor James C.Y. Guo and the dissertation committee for their advice and encouragement. I also wish to thank Dr. Yong G. Lai for his gracious help to teach me the proper application of his 2D and 3D hydraulic models. VI TABLE OF CONTENTS CHAPTER I. INTRODUCTION..................................................................1 Thesis Statement..............................................................1 Research Objective............................................................2 General Research Methodology..................................................2 Motivation and Need for Research..............................................4 Background Literature.........................................................8 Causes of Channel Meandering................................................9 Planform Evolution of Erodible Channels....................................11 Nature of Channel Bend Instability.........................................14 Channel Meander Bend Migration.............................................17 Existing Numerical Models of River Channel Migration.......................19 II. RESEARCH STRATEGY...........................................................24 Open Channel Hydraulics......................................................25 Three-Dimensional Hydraulic Model............................................39 Governing Equations........................................................41 Discretization.............................................................44 Boundary Conditions........................................................47 Model Verification...........................................................50 University of Iowa Physical Model..........................................51 vii Massachusetts Institute of Technology Physical Model 63 Colorado State University Physical Model.................................64 III. NUMERICAL MODEL SIMULATIONS.............................................99 Regime Channel Geometry Regions...........................................100 Range of Numerical Model Simulations......................................102 Model Boundary Conditions.................................................118 Meandering Channel Alignments.............................................120 Channel Entrance and Exit Conditions....................................126 Number of Meander Curves................................................130 Model Mesh Sizes........................................................132 Discussion of Meandering Channel Alignments.............................140 Conclusions of Numerical Model Simulation...............................143 Dimensionless Parameters..................................................144 Dimensionless Hydraulic Parameters......................................144 Dimensionless Channel Parameters........................................145 Pre- and Post-Processing Programs.........................................147 IV. MODEL SIMULATION RESULTS................................................150 Meandering Channel Alignments.............................................150 Example 2D Model Results..................................................152 Example 3D Model Results..................................................156 viii Example 3D Model Mesh 157 Example 3D Velocity Results 159 Example 3D Model Shear Stress Results....................................164 Discussion of Model Simulation Results.....................................174 V. HYDRAULIC RELATIONSHIPS...................................................178 Maximum Shear Stress Magnitude.............................................178 Maximum Shear Stress Location..............................................184 Maximum Near-Bank Velocities...............................................187 VI. VALIDATION AND UNCERTAINTY................................................189 Maximum Shear Stress through Meandering Channels...........................191 Shear Stress Phase Lag.....................................................198 Maximum Near-Bank Velocities...............................................202 Limitations.................................................................203 VII. EXAMPLE APPLICATION......................................................205 Basic Channel Design Example 1.............................................207 Solution 1, Part A.........................................................208 Solution 1, Part B.........................................................211 Basic Channel Design Example 2.............................................215 Solution 2.................................................................215 VIII. CONCLUSIONS.............................................................219 Application of Numerical Models for Research...............................219 Empirical Equations........................................................220 Model Boundary Conditions..................................................221 IX Suggestions for Future Research.........................................222 REFERENCES................................................................223 x LIST OF TABLES Table 1. Channel dimensions and hydraulic properties of the physical model reported by Yen (1965)....................................................................52 2. Summary of Physical Model dimensions (Heintz, 2002)........................66 3. Physical model hydraulic data...............................................66 4. Summary comparison of measured and model results at cross section 6 for a model mesh size of 492 x 42 x 32...............................................92 5. Summary comparison of measured and model results at cross section 10 for a model mesh size of 492 x 42 x 32...............................................92 6. Summary comparison of measured and model results at cross section 16 for a model mesh size of 492 x 42 x 32...............................................93 7. Simulation set matrix summary..............................................105 8. Model simulation matrix, part 1............................................110 9. Model simulation matrix, part 2............................................114 10. CSU Physical Model Dimensions.............................................133 11. Numerical model mesh size used to simulate the flow through the CSU physical model....................................................................133 12. Example 3D model mesh size for three consecutive meander bends............134 13. Range of numerical model mesh densities...................................135 14. Validation data from simulation sets 1, 2, 3, 10,11, and 12...............190 15. Typical permissible shear stresses for bare soil and stone linings........206 XI LIST OF FIGURES Figure 1. Dungeness River near Sequim, Washington where channel migration has destroyed homes along the floodplain..................................................6 2. Dungeness River near Sequim, Washington where channel migration has eroded property and a well that was formerly on an adjacent terrace is in the channel center.................................................................7 3. Hoh River near Forks, Washington where channel migration threatened the Olympic National Park road..........................................................7 4. Elwha River near Port Angeles, Washington where channel migration could threaten homes on top of a high glacial terrace......................................8 5. Super elevation of flow around a meander bend causes a pressure imbalance, which induces a secondary flow current..............................................12 6. Planform phase lag definition sketch............................................14 7. Example velocity and shear stress profiles for steady, turbulent, uniform open channel flow with water depth H, streamwise velocity u at depth z, depth-averaged velocity U, and bed roughness height ks (modified from Garcia, 2007)...26 8. Relationship between channel width, depth, and hydraulic radius for trapezoidal channels...................................................................32 9. Example distributions of boundary shear stress in a trapezoidal channel with a 2:1 side slope using membrane analogy (Olsen and Florey, 1952).................33 XII 10. Variation in relative bed (zb) and bank (rs) shear stress with the channel bottom width-to-depth ratio (b/D)....................................................34 11. Definition sketch for flow in a curved channel (Chang, 1988)..................35 12. Boundary shear stress distributions in curved trapezoidal channels measured by Ippen and Drinker (1962)......................................................37 13. Application of the three-dimensional, U2RANS model, (modified from Lai et al., 2003a) to the physical model experiment by Yen (1965)........................52 14. Comparison of U2RANS model depth results; using coarse, medium, and fine meshes; with physical model data (modified from Lai et al., 2003a)...........53 15. Velocity profile at-0.461 B...................................................55 16. Velocity profile at SO, -0.307 B..............................................55 17. Velocity profile at 0.000 B...................................................55 18. Velocity profile at +0.307 B..................................................56 19. Velocity profile at +0.461 B..................................................56 20. Simulated boundary shear stress is presented for the case of a fine mesh.....57 21. Plan-view alignment is presented for the physical model by Yen (1965). Velocity profile measurements are reported at the cross section labeled SO. Boundary shear stress measurements are reported at the cross sections labeled Cl 10, ti/16, n/8, and ti/4.................................................................58 22. Comparison of simulated and measured shear stress is presented for cross section CIIO. The channel cross section is also plotted along with the location of high velocity (within 90 percent of the maximum velocity).....................59 xiii 23. Comparison of simulated and measured shear stress is presented for cross section ti/16. The channel cross section is also plotted along with the location of high velocity (within 90 percent of the maximum velocity)..................60 24. Comparison of simulated and measured shear stress is presented for cross section ti/8. The channel cross section is also plotted along with the location of high velocity (within 90 percent of the maximum velocity)..................61 25. Comparison of simulated and measured shear stress is presented for cross section ti/4. The channel cross section is also plotted along with the location of high velocity (within 90 percent of the maximum velocity)..................62 26. Comparison of measured and simulated boundary shear stress from the physical model reported by Ippen and Drinker (1962)................................64 27. Photograph of the large physical model of flow through two channel bends at Colorado State University.................................................65 28. Colorado State University model plan-view configuration for flow through two channel bends of constant radius..........................................66 29. Plan-view location of cross section velocity measurements is shown from the upstream most cross section 1 to the downstream most cross section 18. The naming convention of the vertical velocity measurements at each cross section is a, b, c, d, e,f, and g, left to right looking downstream.....................68 30. The two-dimensional numerical model has a structured mesh with 326 cells in the downstream direction and 28 cells in the cross-stream direction 70 31. A close-up view of the two-dimensional numerical model mesh (326 x 28) is shown through the upstream channel bend........................................71 32. The centerline water surface profiles from the numerical and physical models are compared over the reach of the physical model............................72 33. The numerical model mesh boundary (meters) extends ten channel widths both upstream and downstream from the physical model boundaries...............73 34. A plan-view close up is presented of the three-dimensional structured mesh (492 x 42 x 32) along the straight transition reach.............................75 35. The right, cross-sectional half of the three-dimensional structured mesh (492 x 42 x 32) is presented at cross section 6 through the upstream bend............76 36. Simulated (blue) and measured (red) velocity vectors are compared in plan view for the entire region of the physical model..................................77 37. Simulated (blue) and measured (red) velocity vectors are compared in a close-up plan view of the first (upstream) channel bend...........................78 38. Simulated (blue) and measured (red) velocity vectors are compared in a close-up plan view of the downstream portion of the first channel bend and straight transition reach.....................................................79 39. Simulated (blue) and measured (red) velocity vectors are compared in a close-up plan view of the upstream portion of the second channel bend.............80 40. Simulated (blue) and measured (red) velocity vectors are compared in a close-up plan view of the downstream portion of the second channel bend...........81 xv 41. Comparisons of seven measured and simulated stream-wise velocity profiles using four different mesh sizes at cross section 6 in the upstream channel bend (measurement locations "a" through "d" are positioned left to right looking downstream)........................................................82 42. Comparisons of seven measured and simulated stream-wise velocity profiles using four different mesh sizes at cross section 10 at the downstream end of the straight transition reach (measurement locations "a" through "d" are positioned left to right looking downstream)..........................................84 43. Comparisons of seven measured and simulated stream-wise velocity profiles using four different mesh sizes at cross section 16 in the downstream channel bend (measurement locations "a" through "d" are positioned left to right looking downstream)........................................................86 44. Comparison of measured and simulated cross-stream velocity profiles at cross section 6 for a range of three-dimensional model mesh sizes............89 45. Comparison of measured and simulated cross-stream velocity profiles at cross section 10 for a range of three-dimensional model mesh sizes...........90 46. Comparison of measured and simulated cross stream velocity profiles at cross section 16 for a range of three-dimensional model mesh sizes...........91 47. Comparison of depth-averaged measured and simulated stream-wise velocity along the channel for a model mesh size of 492 x 42 x 32. Measurement locations c" and "e" are at the toe of the left and right channel banks. Measurement location "d" is at the center...............................................94 XVI 48. Simulated dimensionless boundary shear stress for the physical model at Colorado State University.........................................................98 49. The graph of the relationship between bankfull discharge, channel slope, and median-sediment grain size includes three distinct regions (Chang, 1988). Solid contour lines indicate the channel surface width while dashed contour lines denote the bankful channel depth............................................100 50. Numerical model simulation sets are presented on the graph of bankfull discharge, channel slope, and median-sediment grain size. Each simulation set includes nine meandering channels.....................................................104 51. Discharge, valley slope, and grain size..................................107 52. A plot of relative grain roughness versus Reynolds number show that model simulations are in the turbulent range....................................109 53. Range of hydraulic radius, width-depth ratio, and Froude number of the model simulation matrix.........................................................118 54. Example meandering river channel alignment, generated using a sine-generated curve, for simulation set 5 with a sinuosity of 2.75 and five consecutive curves.. 123 55. Example channel alignment points and the variable radii of curvature are plotted for simulation set 5 with a sinuosity of 2.75 and five consecutive curves.....124 56. A close-up view of channel alignment points and variable radii of curvature show that their alignments are similar, but they do not exactly coincide.......125 57. Meandering channel alignments of simulation Set 6 with three consecutive meandering bends 126 58. Simulation Set 6, Sinuosity of 3.00. 127 59. Simulated vertical velocity profiles for the staight channel reach after channel length equal to 20 channel widths........................................128 60. Simulated bottom shear stress long the centerline of the straight trapezoidal channel..................................................................129 61. Channel plan view and location of simulated maximum velocity through five consecutive meander bends................................................131 62. Relative discharge on the left and right sides of the channel through five consecutive meander bends................................................132 63. Simulated vertical velocity profiles at a meander bend cross section for a coarse model mesh size of 542 x 25 x 14........................................ 136 64. Simulated vertical velocity profiles at a meander bend cross section for a medium model mesh size of 684 x 31 x 17........................................ 137 65. Simulated vertical velocity profiles at a meander bend cross section for a fine model mesh size of 860 x 39 x 21.............................................. 137 66. Simulated cross section shear stress and location of high velocity at a meander bend for a coarse model mesh size of 542 x 25 x 14........................... 138 67. Simulated cross section shear stress and location of high velocity at a meander bend for a medium model mesh size of 684 x 31 x 17........................... 138 68. Simulated cross section shear stress and location of high velocity at a meander bend for a fine model mesh size of 860 x 39 x 21. XVIII 139 69. Simulated velocity and shear stress are compared with the total model mesh size. ...........................................................................140 70. Relationship between channel curvature and radius of curvature.............147 71. Model channel alignment for a sinuosity of 1.10............................151 72. Model channel alignment for a sinuosity of 1.25............................151 73. Model channel alignment for a sinuosity of 1.50............................151 74. Model channel alignment for a sinuosity of 1.75............................151 75. Model channel alignment for a sinuosity of 2.00............................151 76. Model channel alignment for a sinuosity of 2.25............................151 77. Model channel alignment for a sinuosity of 2.50............................152 78. Model channel alignment for a sinuosity of 2.75............................152 79. Model channel alignment for a sinuosity of 3.00............................152 80. Plan view of the structured 2D model mesh domain for Simulation Set 5 with a sinuosity of 2.75..........................................................153 81. Close-up plan view of the structured 2D model mesh for Simulation Set 5 with a sinuosity of 2.75..........................................................154 82. Channel- bottom elevation contours of the 2D model mesh for Simulation Set 5 with a sinuosity of 2.75........................................................155 83. Water surface elevation contours from the 2D model results for Simulation Set 5 with a sinuosity of 2.75...................................................156 84. Elevation contours of the 3D model mesh for Simulation Set 5 with a sinuosity of 2.75.......................................................................158 XIX 85. Concept of the 3D structured mesh is presented for the right half of the cross section......................................................................159 86. Simulated 3D velocity contours at the water surface for Simulation Set 5 with a sinuosity of 2.75..............................................................160 87. Close-up view of simulated 3D velocity contours at the water surface...........161 88. Close-up 3D view of cross sectional slices of velocity contours................162 89. Simulated thread of maximum velocity for Simulation Set 5 with a sinuosity of 2.75. ...............................................................................163 90. Simulated velocities near the left and right-channel banks for simulation set 5 with a sinuosity of 2.75..............................................................164 91. Channel bottom shear stress contours for simulation set 5 with a sinuosity of 2.75. ...............................................................................165 92. Longitudinal channel bank and centerline shear stress for simulation set 5 with a sinuosity of 2.75..............................................................166 93. Plan-view alignment of the meandering channel from simulation set 5 with a sinuosity of 2.75 and positions of selected cross sections 112, 702, 810, and 950. ...............................................................................167 94. Lateral shear stress distribution and zone where velocity is at least 90 percent of the maximum velocity at cross section 112 (upstream straight reach)..............168 95. Lateral shear stress distribution and zone where velocity is at least 90 percent of the maximum velocity at cross section 702 (3rd meander bend)...................168 xx 96. Lateral shear stress distribution and zone where velocity is at least 90 percent of the maximum velocity at cross section 810 (3rd meander bend)....................169 97. Lateral shear stress distribution and zone where velocity is at least 90 percent of the maximum velocity at cross section 950 (4th meander bend)....................169 98. Vertical streamwise velocity profiles at cross section 112 (upstream straight reach) for simulation set 5 with a sinuosity of 2.75...............................170 99. Vertical streamwise velocity profiles at cross section 702 (3rd meander bend) for simulation set 5 with a sinuosity of 2.75...................................170 100. Vertical streamwise velocity profiles at cross section 810 (3rd meander bend) for simulation set 5 with a sinuosity of 2.75...................................171 101. Vertical streamwise velocity profiles at cross section 950 (4th meander bend)for simulation set 5 with a sinuosity of 2.75...................................171 102. Longitudinal channel bank and centerline shear stress for simulation set 7 with a sinuosity of 2.00 and a width-depth ratio of 7.3............................172 103. Longitudinal channel bank and centerline shear stress for simulation set 8 with a sinuosity of 2.00 and a width-depth ratio of 9.1............................173 104. Longitudinal channel bank and centerline shear stress for simulation set 9 with a sinuosity of 2.00 and a width-depth ratio of 35.6...........................174 105. Summary of 3D model simulation results for minimum and maximum shear stress along the toe of the left and right banks through meandering channels.......179 106. Correlation between the maximum dimensionless shear stress in a channel bend with the ratio of channel width to minimum radius of curvature..............180 XXI 107. Residual maximum shear stress computed by empirical Equation 63 and the maximum shear stress simulated by the 3D model is plotted against the channel width-depth ratio..................................................181 108. 3D plot of maximum shear stress against the ratio of channel width to minimum radius of curvature and the width-depth ratio.........................183 109. Comparison of maximum dimensionless bank shear stress computed by empirical Equation 64 and simulated by the 3D model....................................184 110. Correlation between the shear stress phase lag and channel sinuosity........185 111. Relationship between the maximum near bank velocity and the ratio of channel width to minium radius of curvature for Simulation Sets 4 through 9.........188 112. Validation of the relationships between maximum dimensionless shear stress in a channel bend with the ratio of channel width to minimum radius of curvature. New regression lines, using all data, are compared with the initial regression lines using data from one-half of the simulation sets..........................191 113. Validation of the residual relationships between maximum shear stress computed by empirical Equation 68 and the maximum shear stress simulated by the 3D model as a function of the channel width-depth ratio...........................193 114. 3D plot of maximum dimensionless shear stress, computed by Equation 69, compared with shear stress simulated by the 3D model........................195 115. Comparison of maximum dimensionless bank shear stress computed by empirical Equation 69 and simulated by the 3D model. 196 116. Comparison of maximum dimensionless shear stress from empirical estimates with physical model measurements. Empirical estimates are based on Equation 17 and Equation 69..........................................................197 117. Testing with validation data of the correlation between the shear stress phase lag and channel sinuosity......................................................199 118. Comparison of channel phase lag computed by empirical Equation 70, Equation 71, and Equation 72 with the simulated phase lags from the 3D model............201 119. Relationship between the maximum near bank velocity and the ratio of channel width to minium radius of curvature for Simulation Sets 1 through 12.......202 120. Relationship between the total channel curvature and the ratio of channel width to radius of curvature. 204 CHAPTER I INTRODUCTION The migration of meandering river channels across their floodplains and the occasional erosion of terrace banks is a natural process (Leopold et al. 1964, Yang 1971, Dunne and Leopold 1978, Leopold 1994, and Thorne 1992 and 2002). This process can become especially important to people living near a river and to agencies planning or maintaining infrastructure within or along a river. An improved understanding of river hydraulics that cause channel bank erosion and migration is needed. Thesis Statement Application of a numerical, three-dimensional, hydraulic model to a wide range of meandering river channels of the type (width and depth) found in nature provides detailed data sets of flow velocity and boundary shear stress from which important empirical relationships can be developed and tested. The boundary shear stress exerted by moving water (fluid) increases through a meander bend, relative to a straight channel reach, as the result of the channel curvature. The maximum shear stress along the banks of a meandering river channel can be linearly related to the ratio of channel width to radius of curvature and to the width-depth ratio. 1 Research Objective The objective of this research is to empirically determine dimensionless relationships between hydraulic parameters of meandering river channels and the maximum shear stress caused by the channel curvature. General Research Methodology A numerical, three-dimensional hydraulic model was used like a computational flume to generate data sets representing a wide range of meandering river channels of the type (width and depth) found in nature. These data sets were used to develop empirical relationships between dimensionless shear stress and geometric properties of the channel cross section and alignment. Natural meandering river channels include a wide range of sinuosities (ratio of channel length to valley length) and width-depth ratios. Field and laboratory settings provide opportunities for time-varying measurements of the velocity flow field. However, data sets of detailed hydraulic conditions are not readily available from a wide range of natural channels. Natural river channels often contain irregularities in alignment and stream bank properties. In addition, natural stream flow rates are often highly variable. Therefore, obtaining a wide range of hydraulic data sets from natural river channels is 2 difficult for the purposes of developing a general understanding of shear stress distributions along meandering river channels. Channel dimensions and stream flow rates can be controlled in laboratory physical models. Research utilizing physical models with curved channels typically contain one or two curves of constant radius (Professor Pierre Julian, Colorado State University, Oral Communication, June 1, 2007). However, natural river channels typically have multiple channel bends and variable rates of curvature. Physical modeling of a wide range of meandering channels is time consuming and expensive and there can be model scaling issues. The use of different horizontal and vertical physical model scales can affect the quantitative results, but the qualitative results are not normally affected. Numerical model results may be affected by mesh size, but model scales are not a relevant issue for numerical models. Numerical model simulations of channel hydraulics were found to replicate physical measurements quite well. Therefore, an existing three-dimensional numerical model was used to simulate velocity and shear stress through a wide range of meandering river channels of the type found in nature. This three-dimension model is more fully described in Chapter II. There are relatively few reports of studies where a three-dimensional (3D) numerical model was used to simulate primary and secondary currents through meander bends that are similar to natural river channels (Wilson, et al., 2003). The difficulties in 3 acquiring good spatial resolution data of high quality in rivers have meant that the majority of numerical model development and subsequent testing has focused on the study of simplified river geometries in controlled environments. Most models have been applied to rectangular curved channels or trapezoidal channels with steep side slopes and with simple alignment geometries. For this research, meandering trapezoidal channels with 2:1 side slopes were simulated with variable rates of curvature. Sine-generated curves were used to define the meandering channel alignments with variable radius of curvature and three or five consecutive meander curves. For each meandering channel, fixed-bed hydraulics was simulated at the bankfull discharge. A non-erodible channel boundary was assumed (fixed-bed hydraulics) in the model simulations to preserve the channel dimensions and alignment during simulation. Estimate of the maximum shear stress can only be assured if the channel banks and bed are not allowed to erode. Model simulation results were then used to develop general relationships between dimensionless bank shear stress and channel geometric parameters. Motivation and Need for Research This research is motivated by the need to simulate how river channels migrate across their floodplain. Hydraulic and Sediment transport models have existed for many years 4 to simulate the sediment erosion and deposition along the bed of a river channel. The U.S. Army Corps of Engineers' HEC-6 (1993) and HEC-RAS (2010) models and the U.S. Bureau of Reclamation's STARS (Orvis and Randle, 1987), GSTARS (Yang, et al., 2005), and SRH-1D (Greimann and and Huang, 2006) models are examples of one-dimensional hydraulic and sediment transport models. These models have important application to degrading and aggrading stream channels. However, these models do not simulate the bank erosion that results during the lateral migration of a meandering river channel. Many river channels are in a dynamic equilibrium where there is a balance between the upstream sediment supply and the downstream transport. These channels are neither significantly aggrading nor degrading over time, but they may migrate laterally across their floodplain. Two and three-dimensional hydraulic and sediment transport models can be applied to meandering river channels, but such models are not widely available and are difficult to apply over long time and space scales (kilometers of river length over years of time) because of computational time requirements and model stability issues. The research presented in this dissertation could be applied to a one-dimensional model to simulate river channel migration along tens of kilometers of length and over decades of time. For streambank protection designs, the research presented in this disseration will help in estimating the maximum shear stress along curved channel banks. 5 The migration of meandering river channels can sometime pose hazards to people, property, and infrastructure along a river. Some examples of channel-migration hazards are shown in Figure 1 through Figure 4. Natural rates of channel migration can be accelerated by human land-use activities. For example, the clearing of riparian vegetation can accelerate the rate of channel migration because cohesion provided by vegetation root structure is no longer available (Beeson and Doyle 1995). Figure 1. Dungeness River near Sequim, Washington where channel migration has destroyed homes along the floodplain. 6 Figure 2. Dungeness River near Sequim, Washington where channel migration has eroded property and a well that was formerly on an adjacent terrace is in the channel center. Figure 3. Hoh River near Forks, Washington where channel migration threatened the Olympic National Park road. 7 Figure 4. Elwha River near Port Angeles, Washington where channel migration could threaten homes on top of a high glacial terrace. Background Literature Natural geomorphic features of a river channel result from the continuous dynamic interaction between the motion of a sediment-carrying fluid and an erodible boundary of the channel (Seminara, 2006). The major factors affecting alluvial stream channel forms are listed by Lagasse, et al. (2004): 1. stream discharge (magnitude, duration, and frequency), temperature, viscosity; 2. sediment load and grain size; 3. longitudinal valley slope; 4. bank and bed resistance to erosion, including the effects of riparian vegetation; 8 5. local geology, including bedrock outcrops, clay plugs, changes of valley slope; and 6. human development and activities. Theoretical analyses and laboratory experiments have shown that the observed channel patterns are mostly related to fundamental instability mechanisms. The mobile interface between the fluid and the erodible channel boundary is unstable, rather than the flow itself (Seminara, 2006). A meandering river channel is characterized by a sequence of smooth bends. There tends to be persistent erosion along the outer bank, forming pools, and deposition of point bars along the inner bank. This erosion and deposition leads to the migration of meander bends both across the floodplain and along the downstream valley axis. The meander migration process has been observed in the field by Leopold et al. (1964), Hickin (1974), Hickin and Nanson (1975), Hickin and Nanson (1984), Parker (1975), as well as in the laboratory by Friedkin (1945) and Chang et al. (1971). Causes of Channel Meandering Meandering river channel alignments are observed in natural-river channels at all scales, all continents and latitudes, and in ice (Leopold et al. 1964, Thorne 1992, and Leopold 9 1994). Meandering channels have even been reported on the planet Mars (De Hon, 2007) and Saturn's moon Titan (Stofan, et al, 2008). The width, depth, and slope of a meandering river channel continually adjust over time to achieve an equilibrium condition with the changing upstream water discharge and sediment load and within the constraints of the valley slope, local geology, and human structures (Lane, 1955; Simons and Senturk, 1992; and Yang, 1986). Yang (1971) attributes river meandering to the minimum rate of energy dissipation, subject to the constraints applied to the system. The precise cause of meandering remains undefined. Ashworth, et al. (1996) suggests that the process begins with coherent structures in the flow field that are on the order of a channel width. Large eddies produced in the flow field of a straight channel induce a sinuous path in the streamlines of maximum velocity. Seminara (2006) used a linear model of flow and bed topography to show that meanders behave as linear oscillators which may resonate at some distinct values of the channel width-to-depth ratio and of the ratio of channel width to meander wavelength. Resonance excites a natural mode of oscillation in the form of stationary alternate bars. When the resonance barrier is crossed, the direction of meander migration is reversed. Lai (2006) has demonstrated with numerical modeling that the sinuous path of the maximum velocity streamline begins with the transport of the alluvial bed material. The 10 surrounding flow field along these sinuous streamlines is subsequently strengthened by positive feedback between curvature of the flow and skew-induced secondary currents of Prandtl's first kind (Bathurst, et al., 1979). A meander-bend cutoff occurs when one migrating channel bend intercepts another (neck cutoff) or when flood flows erode a shorter channel path across the meander bend (chute cutoff). The prediction of chute cutoffs has not yet been resolved. A chute cutoff is a process whereby a meander can sometimes be abandoned well before a neck cutoff (Seminara, 2006). Planform Evolution of Erodible Channels Meandering rivers shift laterally and migrate longitudinally through a process of erosion at concave banks and deposition at convex banks (Seminara, 2006). As the river flow is conveyed through a meander bend, the centrifugal force causes the water surface elevation to be slightly higher on the outside of the bend (Figure 5). This super elevation of the water surface causes a pressure imbalance perpendicular to the main flow direction, which then causes a secondary flow current (Edwards and Smith, 2002 and Seminara, 2006). The secondary current is described by flow traveling in a downstream helical motion, which is downward near the outer bank, across the deep portion of the channel bottom toward the shallow inner bank, and then back across the 11 channel near the water surface. At the upper portion of the outer bank, a small cell of reverse circulation often exists that forces flow upward along the bank. According to Thorne and Hey (1979) and Thorne (1992), this reverse circulation is highly significant to the flow processes acting to erode the outer bank. Figure 5. Super elevation of flow around a meander bend causes a pressure imbalance, which induces a secondary flow current. In channels with non-erodible banks, flow along the inner bend accelerates relative to the outer bend. Farther downstream, the secondary flow transfers momentum towards the outer bend, moving the thread of high velocity from the inner bank to the outer bank (Seminara, 2006). The establishment of a bar-pool pattern creates an additional component of the secondary flow, which further modifies the bed topography. 12 For rivers with a lateral channel-bed slope, gravitational forces tend to cause sediment particles to move downhill (toward the outside bend), and bedload deviates from the direction of the local average bottom shear stress (Seminara, 2006). Dietrich and Smith (1983) analyzed several sets of laboratory and field data and documented that shoaling over the point bar in the upstream part of the bend concentrates the high-velocity core of flow toward the pool. The channel curvature-induced component of boundary shear stress is confined to 20 to 30 percent of the channel width at the pool. Dietrich and Smith (1984) found that cross-stream bed-load transport varied with particle grain size and was affected by point bar topography, trough-wise flow along obliquely oriented bed forms, rolling of particles on bed form lee faces, and mass sliding on steep point bar slopes. If the river bed-material is mobile, asymmetry in the velocity and boundary shear stress distributions rapidly leads to the generation of pools, riffles, and alternate bars (Smith, 1987). There is a lag effect of the hydraulics through a meander bend where the effects of channel curvature develop some distance downstream from where the channel curvature begins. This distance is referred to as the planform phase lag and can cause downstream migration of the meander bed (Figure 6). If the phase lag were zero, the channel would migrate only laterally across the floodplain and increase its meander bend amplitude. If the phase lag were equal to one-quarter of the meander wave 13 length, then the meander bend migration would be entirely downstream along the valley alignment without an increase in meander bend amplitude. Figure 6. Planform phase lag definition sketch. Nature of Channel Bend Instability Erodible channel banks are a necessary condition for the initiation of a meander bend, although they are not the direct cause of meandering (Rhoads and Welford, 1991). In streams with erodible banks, the sinuous path of the maximum velocity streamlines initiates a pattern of bank erosion that marks the onset of channel meandering (Friedkin, 1945; Keller, 1972; and Hakanson, 1973). Where the maximum velocity streamline converges with the channel bank, local scour of the channel bed undercuts the bank and promotes erosion, instability, and rapid bank 14 retreat (Thorne and Lewin, 1979 and Thorne, 1982). The rapid retreat of the outer bank that enables active meanders to migrate within the floodplain. The rate of bank deposition depends on the upstream sediment supply. Channel width can remain relatively constant over time only if the rates of bank erosion and deposition are in balance. Otherwise, the channel will widen or narrow over time. Brice (1982) discovered that channels that do not vary significantly in width were relatively stable, whereas channels that were wider at bends were more active. High sinuosity and equal-width streams were the most stable, whereas other equal-width streams of lower sinuosity were less stable, and wide bend streams had the highest erosion rates. Channel bend instability theory predicts that any small random perturbation of a straight channel alignment eventually grows, leading to a meandering pattern (Seminara, 2006). However, this is difficult to confirm from field and laboratory observations. Cohesionless sediments are typically employed in laboratory experiments with an initially straight channel. The channel typically undergoes a sequence of processes (Federici and Paola 2003) associated with a continuous widening of the channel, driven by the erosion of cohesionless banks. Though, at an intermediate stage, an apparent meandering channel forms, it continues to evolve through further widening, the occurrence of chute cutoffs, and the emergence of bars until a braided pattern is formed. 15 The persistence of a coherent meandering pattern requires a cohesive floodplain (Seminara, 2006). Vegetation plays an important role in stabilizing the banks, constraining channel migration, and allowing deeper and narrower channels to develop. Alternate bars are not the precursor to river meandering for the following reasons (Seminara, 2006): First, alternate bars would have to migrate fairly fast to be responsible for the localized erosion of cohesive banks driving meander formation. Secondly, and more importantly, the typical dimensionless wave number of developed alternate bars falls within the stable bend range and would not grow. Thirdly, alternate bars are observed to coexist with and migrate through weakly meandering channels, an observation which contradicts the idea that they would evolve into the fixed-point bars of river meanders. Finally, in the experiments of Smith (1998), meandering developed in the absence of alternate bars and the chain of events typically observed did resemble the initial stage of meander bend instability. "In conclusion, the 'bend' mechanism appears at this stage the only rational scheme able to explain various features observed in the field." Bend instability is the process whereby a perturbation of an initially straight channel alignment grows, driven by bank erosion, and leads eventually to the development of a meandering pattern. 16 Seminara (2006) argues that, in order to understand meander formation in purely cohesive and erosional channels, "the classical bar-pool scheme must be abandoned in favor of a purely erosional mechanism driven by the three-dimensional structure of the flow field." Channel Meander Bend Migration The morphology and behavior of a given river reach is strongly determined by the amount and rate of sediment and water discharge supplied from upstream. Therefore, any significant modification of sediment load and water discharge will impact local rates of channel migration and evolution (Lagasse, et al., 2004). Locally, the distribution of velocity and shear stress and the characteristics of bed and bank sediment will control channel behavior. The direction and rate of channel migration are a function of the following channel factors (Lagasse, et al., 2004): local curvature and sinuosity, channel shape (width-depth ratio), longitudinal slope, discharge, and bank strength. Channel migration can be slowed or limited by floodplain vegetation, alluvial deposits, geologic controls, and human structures (e.g. bank protection). The bank strength from 17 vegetation depends on the root depth and density (Pollen, et al, 2007). Large woody vegetation can provide recruitment for large woody debris (log jams). Alluvial deposits frequently include oxbows, meander scrolls and scars, and clay plugs, each of which has different erodibility characteristics. Geologic controls include bedrock outcrops and erosion resistant features along the margins of the floodplain such as terrace banks. The amplitude of a single meander bends tends to increase with time up to a maximum and then subsequently decreases while rate of migration decreases (Nanson & Hickin 1983). In the absence of geological constraints, meanders often evolve continuously until a neck cutoff occurs (Seminara, 2006). The alignment of the neck cutoff creates a geometric discontinuity, which is smoothed out through additional planform evolution and channel migration. The abandoned meander bend (oxbow lake) is slowly filled up through the deposition of fine suspended sediments (often clay size) during floods. As the deposited sediments consolidate overtime, the future resistance to local erosion will increase, which may further affect the planform evolution on time scales of the order of centuries. Stream-bank erodibility depends on the previous history of the floodplain formation, as well as on the presence of vegetation, geological constraints, and anthropogenic effects (Seminara, 2006). In some cases, a chute cutoff may occur before long before the meander bend achieves a neck cutoff. This process occurs in wide bends with fairly large curvatures, high 18 discharges, noncohesive and unvegetated banks, and steep longitudinal channel slopes (Howard & Knutson 1984). Existing Numerical Models of River Channel Migration Previous models have assumed the river discharge as well as channel width, river slope, and grain size to be constant (Seminara, 2006). These assumptions are acceptable when the spatial scale of the reach as well as the temporal scale of the process is not too large. Models do not yet simulate chute cutoffs and neck cutoff are modeled in a schematic way. The process of river degradation or aggradation, driven by the cutoffs, is ignored and unsteady flow is not accounted for. The bank erosion equation by Ikeda et al. (1981) initiated empirical and theroretical study to predict meander bend migration. Parker (1976) and Odgaard (1989a and 1989b) developed meander migration models by utilizing similar approaches as Ikeda et al (1981). However, these approaches do not consider the sediment load of the stream. Hasegawa (1989) pointed out that bank erosion is caused by an imbalance between sediment supply and bank material entrainment. Hooke (1975) described the distribution of sediment transport and shear stresses in a meander bend. Smith and Mclean (1984) and Mclean and Smith (1986) were the first to develop dynamically correct two-dimensional models and explain the mechanics of flow 19 through a meander bend. Nelson and Smith (1989) combined a bed-load transport algorithm with the numerical model for flow and boundary shear stress fields to investigate point bars in meandering channels and alternate bars in straight channels. National Cooperative Highway Research Program commissioned a study on the Methodology for Predicting Channel Migration (Lagasse, et al., 2004). This study concluded that it is very difficult to predict the magnitude, direction, and rate at which channel migration will occur and that no good practical models of channel migration presently existed. The study, therefore, concludes that the "best model of the river is the river itself" and recommends the river channel migration history be analyzed in order to predict the potential for future migration. This method is very useful, but a predictive model is still needed to account for proposed hydraulic structures or land use change. A study by Johns Hopkins University (Cherry, 1996) for the U.S. Army Corps of Engineers Waterways Experiment Station concluded that both the accuracy and applicability of the bend-flow meander migration model by Garcia, et al. (1994) are limited by a number of simplifying assumptions. Among the most important of these are the use of a single discharge and the assumption of constant channel width, both of which prevent the model from successfully forecasting the spatial and temporal variability that appears to be inherent in the process of bend migration. 20 The study also concluded that much of the discrepancy between the predicted and observed distributions of erosion can be accounted for by the fact that meander migration is modeled as a smooth, continuous process. In reality, erosion occurs predominantly in discrete events (i.e., during flood flows), and varies greatly both temporally and spatially along the channel from bend to bend. The Johns Hopkins study noted that the identification of local factors that influence the amount of bank erosion that occurs is a subject that will require further investigation and that further refinements in bend-flow modeling will not improve our predictive capability until we find a more rational way to wed the flow model to a bank erosion model. Three-dimensional models are appropriate for a length scale of few meander curves and a time scale of days to weeks (Huang et al. 2001, Lai et al. 2003a, and Lai et al. 2003b). Two-dimensional models are appropriate for a length scale of a few kilometers and a time scale of months to years (Duan, 2004 and Lai, 2006). Analytical models can be applied to spatial scales of tens of river kilometers and time scales of decades to centuries. Analytical models are typically based on certain simplifying assumptions: The meandering channel only has long radius meander bends. Channel width is constant with time. There is no channel aggradation or degradation. 21 Only the model by Huang, Greimann, and Randle (2007) considers channel aggradation or degradation. Bank erosion rate (BE) is a function of a dimensionless bank erosion coefficient (E0) and the increased bank velocity (Vc) due to channel curvature (BE = E0 Vc). E0 is on the order of 10"7 to 10"8. Johannesson and Parker (1989) proposed an analytical method to predict the increase in channel velocity through a meander based on a linearization of the two-dimensional equations for depth-averaged flow and a sediment transport equation to predict the cross-channel slope. The deviation is assumed to be linearly related to the maximum curvature of the channel. For their linearization analysis, the flow variables are the sum of two parts: (1) flow in straight channel and (2) flow deviation due to a slightly curved channel. These perturbed flow variables are substituted into the 3D flow equations. The equations are then simplified and grouped into the terms responsible for the straight channel solution and those due to the channel curvature. The equations become ordinary differential equations and can be solved analytically or through relatively simple numerical methods. The sediment transport is assumed to be a function of the local velocity and shear stress and is used to compute the cross-channel slope around the bed. The method proposed 22 by Johannesson and Parker (1989) has been widely accepted in the literature, but is only applicable to long radius curves. Sun et al. (2001a and 2001b), re-derived the equations to account for multiple sediment grain sizes. Seminara (2006) provides some very interesting theoretical results about meander bend processes. Even though the results from analytical models have been qualitatively substantiated through field or laboratory observations, the simplifying assumptions are not always applicable for predictive models in engineering applications. 23 CHAPTER II RESEARCH STRATEGY An existing set of two- and three-dimensional numerical models were used to simulate depth, velocity, and shear stress through a wide range of meandering river channels of the type found in nature. Even though each numerical model simulation takes hours of computer time, a large number of model simulations can be performed at much less time and cost than the construction and application of physical models or measurements from natural channels. In addition, numerical model results include three-dimensional velocities at each mesh cell of the model flow domain and shear stress at each boundary cell. The numerical model results are repeatable and there are no scaling issues. The numerical model was first validated with measurements from physical models. Sine-generated curves were used to define the meandering channel alignments with three or five consecutive curves. Fixed-bed channels were simulated at the bankfull discharge. Model simulation results were then used to develop general relationships between dimensionless river hydraulic parameters and dimensionless geometric channel properties. 24 Open Channel Hydraulics The American Society of Civil Engineers, Sedimentation Manual (Garcia, 2007) provides a very good basic description of the velocity and shear stress profiles for the case of steady, turbulent, and uniform open channel flow (Figure 7). Open channel flow exerts a tangential force per unit area on the stream bed known as the bed shear stress. For the case of a wide channel, the bottom shear stress can be expressed as: Equation 1 Tb = pgHS where Tb = bed shear stress, P = water density, g = acceleration of gravity, H = channel depth, s = longitudinal energy slope Equation 1 is also the one-dimensional, conservation of momentum equation of steady uniform flow. 25 Velocity and Shear Stress Profiles Figure 7. Example velocity and shear stress profiles for steady, turbulent, uniform open channel flow with water depth H, streamwise velocity u at depth z, depth-averaged velocity U, and bed roughness height ks (modified from Garcia, 2007). The shear velocity is commonly defined from the boundary shear stress as: Equation 2 The shear velocity and boundary shear stress provide a direct measure of the flow intensity and its ability to transport sediment particles (Garcia, 2007). For steady, 26 uniform flow of a wide channel, the shear stress (r) varies linearly with flow depth (Figure 7): Equation 3 The vertical flow velocity distribution is well represented by the logarithmic distribution or "law of the wall" (Garcia, 2007; Schlichting, 1979; and Nezu and Rodi, 1986): Equation 4 u u* Where u = time-averaged flow velocity at a distance z above the stream bed, z0 = bed roughness height where the flow velocity goes to zero, and k = von Karman's constant, equal to 0.41. Garcia (2007) reports that von Karman's constant is not affected by the presence of suspended sediment as previously believed and can be considered to be a universal constant (Smith and McLean, 1977; Coleman, 1981 and 1986; Lyn, 1991; Soulsby and Wainright, 1987; Wright and Parker 2004). Equation 4 applies only in a relative thin vertical layer near the bed (z/H < 0.2) (Nezu and Nakagawa, 1993). However, it is commonly used as a reasonable approximation throughout the flow depth in many streams and rivers (Garcia, 2007). Equation 4 is not 27 exact because the vertical velocity distribution can be affected by wake effects near the free surface, sediment stratification, and bed forms. The surface roughness on the channel bed and banks is a function of the sediment particle size. This surface roughness affects the velocity and shear stress distributions in the channel. If the channel bed is smooth, turbulence will be drastically suppressed in a very thin layer near the bed, known as the viscous sublayer: Equation 5 v 8V = 11.6 it* Where 6V = height of the viscous sublayer and v = kinematic viscosity of water. The boundaries of natural stream channels are mostly rough (Garcia, 2007). Typically, muddy bottoms as well as channel beds composed of silt and fine sand can be hydraulically smooth. However, stream channel beds composed of coarse sands and gravel are typically rough. Let ks denote the effective roughness height. No viscous sublayer will exist when ks is greater than 5V because the sediment particles will protrude through the viscous sublayer. The roughness Reynolds number, Re*, can be used to define whether a channel is smooth, rough, or in transition (Garcia, 2007): 28 Equation 6 If Re. < 5 If 5 < /?e* < 70 If 70 < Re. tt*/Cc D _ A r^e* V hydraulically smooth bottom bottom in transition hydraulically rough bottom For rough channels, the logarithmic velocity profile is predicted by: Equation 7 It 1 lz\ 1/ z\ = -in +8.5 =-ln 30 u* k \ks/ k \ ks/ The equation for depth-averaged velocity is derived by integration of Equation 7: Equation 8 U it* l rH\i (z\ H Jk Kln \ks) + 8.5 dz After integration, Equation 8 becomes: Equation 9 U it* A resistance equation can be developed for the bed shear stress using the shear velocity and the above equation for depth-averaged velocity: 29 Equation 10 tb= pCf U2 Where C/ is the friction coefficient, which is a function of the water depth and the effective grain roughness height: Equation 11 Cf = \1 In K .,-2 The friction coefficient can be related to the Chezy and Manning's roughness coefficients: Equation 12 Cz = g_ 9f V2 Equation 13 n = h Where Cz = Chezy's roughness coefficient and n = Manning's roughness coefficient. 30 Most authors (Garcia, 2007; Chang, 1988, Simons and Senturk, 1992) present equations with the assumption of a wide channel where the hydraulic radius is nearly equal to the maximum channel depth: Equation 14 A R=-~D Where R = hydraulic radius, A = cross-sectional area of the wetted channel, P = wetted perimeter of the cross section, and D = flow depth. The relationship between channel width, depth, and hydraulic radius for trapezoidal channels is presented in Figure 8. This relationship indicates that a trapezoidal channel (with a 2:1 side slope) needs to have a width-depth ratio greater than 14 before the hydraulic radius is within 20 percent of the maximum depth. The hydraulic radius is within 10 percent of the maximum depth when the width-depth ratio is at greater than 26. 31 Relationship between Channel Width, Depth, and Figure 8. Relationship between channel width, depth, and hydraulic radius for trapezoidal channels. Equation 1 was developed under the assumption of one-dimensional, steady flow through a wide channel. The lateral distribution of shear stress across the channel is not uniform (Chang, 1988), especially for channels with a width-depth ratio less than 14. The distribution of boundary shear stress in straight trapezoidal channels was studied by Olsen and Florey (1952) using membrane analogy (Figure 9). These results were reported again by Chang (1988) and Simons and Senturk (1992). The results represented in Figure 9 are unusual because they indicate that boundary shear stress is zero at the bank toe of a trapezoidal channel. This work was expanded by the Highway 32 Research Board (1970), which estimates the maximum boundary shear stress on the channel bed and banks, relative to the average shear stress (t = y/?5). These results are presented in Figure 10 and show that bank shear stress increases as the channel becomes narrower. The results indicate that the bed shear stress reaches a maximum when the bottom width-to-depth ratio is between 2 and 3. According to the figure, both bank and bed shear stress increase as the side slopes become less steep. (a) Distribution of boundary shear stress Figure 9. Example distributions of boundary shear stress in a trapezoidal channel with a 2:1 side slope using membrane analogy (Olsen and Florey, 1952). 33 18 Figure 10. Variation in relative bed (?<,) and bank (rs) shear stress with the channel bottom width-to-depth ratio (b/D). The flow in curved channels is under the influence of centrifugal acceleration, which produces secondary currents and super elevation in water surface (Chang, 1988). Secondary currents (spiral motion) are due to the difference in centrifugal acceleration {u2/r; where u is the local longitudinal velocity and r is the radius of curvature) along the vertical velocity profile (Figure 11). Flow through a curved channel tends to have vertical velocity profile with greatest velocity near the water surface. Streamwise velocity tends to be greater near the outside of the bend. Cross-stream velocity tends 34 to be toward the outside bank in the upper portion of the cross section and toward the inside bank in the lower portion of the cross section. Figure 11. Definition sketch for flow in a curved channel (Chang, 1988). Chang (1988) provides an equation for the radial shear stress (ry) in fully developed transverse flow through a curve: Equation 15 Tr 1 + m D _ ----p-U2 (2 + m)m r Where / = friction factor 35 Secondary currents grow in strength upon entering a channel curve (Chang, 1988). In a prismatic curved channel of sufficient length, the secondary currents will reach an equilibrium conditions and are considered to be fully developed. Because of the variable radius of curvature in natural meandering channels, secondary currents grow and decay with longitudinal channel length (Chang, 1988). These secondary currents modify the boundary shear stress in curved channels. Measured shear stress distributions in curved trapezoidal channels (both smooth and rough) are provided by Ippen and Drinker (1962) in Figure 12. In these experimental examples, shear stress is initially highest along the inside of the curve (near the bank toe) and then the zone of highest shear stress transitions to the outside of the curve (near the bank toe) just downstream from where the curve ends and transitions again to a straight channel. 36 Figure 12. Boundary shear stress distributions in curved trapezoidal channels measured by Ippen and Drinker (1962). Ippen and Drinker (1962) concluded that the boundary shear stress patterns they obtained experimentally "cannot be predicted quantitatively from the gross characteristics of the flow." Measurements indicate that local shear stress was more than twice the mean shear stress for a straight channel. The distribution and relative 37 magnitudes of shear stress appear to be functions of the stream geometry. The locations of local maximums of shear stress were found near the inside bank in the curve and near the outside bank downstream from the end of the curve. Relatively high shear stress was found to persist for a considerable distance downstream from the curve. Guidelines for the protection of stream banks from maximum hydraulic conditions are based on distributions of velocity and boundary shear stresses. The Federal Highway Administration (2005) used the research by Young et al. (1996), who proposed a ratio of the maximum channel bend shear stress to the shear stress at the beginning of the curve: Equation 16 T-max Kb To Where Kb = Ratio of channel bend shear stress to approach shear stress Tmax = Maximum channel bend shear stress t0 = Approach shear stress (yRS) The Federal Highway Administration (2005) describes Kb as a piecewise, mathematical function: 38 Equation 17 Rc Kb = 2.00, for ^ < 2 (R \ / R \^ R + 0.0073 (-7) for 2 < 7 < 10 WJ \W/ w Rc Kb = 1.05, for 10 < Where Rc = Channel radius of curvature W = Wetted channel top width Three-Dimensional Hydraulic Model The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid. These equations were derived by applying Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term, which is proportional to the velocity gradient, and a pressure term. The Navier-Stokes equations dictate not position but rather velocity. The Navier-Stokes equations describe the velocity of the fluid at a given point in space and time. The equations are presented below in Cartesian coordinates for momentum: 39 Equation 18 /du du du du\ p{m + ual+v^+w^> Equation 19 (dv dv dv dv\ p{-dt + uTx + vdj + wfc) Equation 20 (dw dw dw dw\ P ~X~ + uh v h w I H\dt dx dy dzJ dp fd2u d2u d2u dx + ^ 1 dx2 + dy2 + dz2 + P9 dp (d2 v d2 v d2 v dy + fi 1 dx2 + dy2 + dz2 + P9 dp fd2w d2w d2w dz + ^ ydx2 + dy2 + dz2 + pg The equation for conservation of mass is given by: Equation 21 dp d(pu) d(pv) d(pw) dt dx dy dz When the flow is incompressible, the fluid density, p, does not change and = 0. dt Therefore, the continuity equation is reduced to: Equation 22 du dv dw dx + dy+ dz 40 Where u, v, and w are the fluid velocities in the x, y, and z directions. These four nonlinear partial differential equations have to be solved numerically for most practical problems, such as flow through a river channel. The three-dimensional numerical model, U2RANS, was developed to predict the turbulent flows through a wide variety of open channels (Lai et al. 2003a). The U2RANS model was used for the research described in this dissertation to simulate flow through straight and meandering river channels. U2RANS is an implicit, finite volume model that has been proven to provide accurate results and has been successfully applied to a wide variety of practical problems (Lai et al. 2003b). Applying a cell-centered storage scheme to discretized finite volumes, the model solves the three-dimensional Reynolds-averaged Navier-Stokes equations for the distributions of flow velocity, static pressure, and shear stress each mesh cell. The Reynolds-averaged Navier-Stokes equations are solved for steady, incompressible, turbulent flows using the standard k-s turbulence model. This model was developed without making the assumption of a hydrostatic pressure distribution in the governing equations. Governing Equations The three-dimensional equations for the conservation of mass and momentum, in the U2RANS model (Lai, et al., 2003a), are written in tensor form: 41 Equation 23 V (pU
Equation 24 |

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PAGE 1 BOUNDARY SHEAR STRESS THROUGH MEANDERING RIVER CHANNELS by TIMOTHY JAMES RANDLE P.E., D.WRE. B.S. in Civil Engineering University of Utah, 1981 M.S. in Civil Engineering, University of Colorado, 2004 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Civil Engineering Program 201 4 PAGE 2 ii This thesis for the Doctor of Philosophy degree by Timothy James Randle has been approved for the Civil Engineering Program by James C.Y. Guo, Adv is o r Rajagopalan Balaji, Chair Edmund D. Andrews John P. Crimaldi David Mays Zhiyong Ren March 11 2014 PAGE 3 iii Randle, Timothy James ( Ph.D., Civil Engineering) Boundary Shear Stress Through Meandering River Channels Thesis directed by Professor James C.Y. Guo ABSTRACT As water moves through a meandering river channel, flow velocity and shear stress both vary along the river banks as a result of channel curvature. An improved predictive relationship of how bank shear stress increases with channel curvature is needed for the prediction of river bank erosion and channel migration The erosion of meandering river banks is of interest to people living near a river and to agencies planning or maintaining infrastructure within or along a river. A numerical, three dimensional hydraulic model was used to simulate boundary shear stress along a wide range of meandering river channels of the type (width and depth) found in nature. A total of 72 virtual meandering channels were developed from sine generated curves. The hydraulic conditions of these channels spanned three orders of magnitude of bankfull discharge and four orders of magnitude in longitudinal slope. Channel sinuosit y (ratio of channel length to valley length) ranged from 1.1 to 3.0. T rapezoidal cross section s of constant width w ere used define each channel. The hydraulic data se ts generated from the numerical model simulations, were used to develop empirical relationships between magnitude and location of maximum PAGE 4 iv dimensionless shear stress and d imensionless channel properties: ratio of channel width to minimum radius of curvatu re, width depth ratio, and sinuosity. Maximum d imensionless near bank shear stress in meandering channels was found t o increase linear ly with increases in the ratio of channel width to radius of curvature and decrease with increases in the width depth rat io Th e empirical equation s can be used to estimate shear stress for the design of streambank protection in meandering channels and an example is provided The form and content of this abstract are approved. I recommend its P ublication. Approved: James C.Y. Guo PAGE 5 v DEDICATION I dedicate this work to my wife, Kathy Randle, who has been extremely supportive of my efforts to pursue an advanced degree, and to my father, Kenneth W. Randle, for encouraging me to pursue a career in engineering. PAGE 6 vi ACKNOWLEDGMENTS I would like to thank Professor James C.Y. Guo and the dissertation committee for their advice and encouragement. I also wish to thank Dr. Yong G. Lai for his gracious help to teach me the proper application of his 2D and 3D hydraulic models. PAGE 7 vii TABLE OF CONTENTS CHAPTER I. INTRODUCTION ................................ ................................ ................................ ............... 1 Thesis Statement ................................ ................................ ................................ ............ 1 Research Objective ................................ ................................ ................................ ......... 2 General Research Methodology ................................ ................................ ..................... 2 Motivation and Need for Research ................................ ................................ ................. 4 Background Literature ................................ ................................ ................................ .... 8 Causes of Channel Meandering ................................ ................................ ................... 9 Planform Evolution of Erodible Channels ................................ ................................ .. 11 Nature of Channel Bend Instability ................................ ................................ ........... 14 Channel Meander Bend Migration ................................ ................................ ............ 17 Existing Numerical Models of River Channel Migration ................................ ............ 19 II. RESEARCH STRATEGY ................................ ................................ ................................ .... 24 Open C hannel Hydraulics ................................ ................................ .............................. 25 Three Dimensional Hydraulic Model ................................ ................................ ............ 39 Governing Equations ................................ ................................ ................................ 41 Discretization ................................ ................................ ................................ ............. 44 Boundary Conditions ................................ ................................ ................................ 47 Model Ve rification ................................ ................................ ................................ ........ 50 University of Iowa Physical Model ................................ ................................ ............ 51 PAGE 8 viii Massachusetts Institute of Technology Physical Model ................................ ........... 63 Colorado State University Physical Model ................................ ................................ 64 III. NUMERICAL MODEL SIMULATIONS ................................ ................................ ............. 99 Regime Channel Geometry Regions ................................ ................................ ........... 100 Range of Numerical Model Simulations ................................ ................................ ..... 102 Model Boundary Conditions ................................ ................................ ....................... 118 Meandering Channel Alignments ................................ ................................ ............... 120 Channel Entrance and Exit Conditions ................................ ................................ .... 126 Number of Meander Curves ................................ ................................ .................... 130 Model Mesh Sizes ................................ ................................ ................................ .... 132 Discussion of Meandering Channel Alignments ................................ ...................... 140 Conclusions of Numerical Model Simulation ................................ .......................... 143 Dimensionless Parameters ................................ ................................ ......................... 144 Dimensionless Hydraulic Parameters ................................ ................................ ...... 144 Dimensionless Channel Parameters ................................ ................................ ........ 145 Pre and Post Processing Programs ................................ ................................ ............ 147 IV. MODEL SIMULATION RESULTS ................................ ................................ .................. 150 Meandering Channel Alignments ................................ ................................ ............... 150 Example 2D Model Results ................................ ................................ ......................... 152 Ex ample 3D Model Results ................................ ................................ ......................... 156 Example 3D Model Mesh ................................ ................................ ......................... 157 PAGE 9 ix Example 3D Velocity Results ................................ ................................ ................... 159 Example 3D Model Shear Stress Results ................................ ................................ 164 Discussion of Model Simulation Results ................................ ................................ ..... 174 V. HYDRAULIC RELATIONSHIPS ................................ ................................ ....................... 178 Maximum Shear Stress Magnitude ................................ ................................ ............. 178 Maximum Shear Stress Location ................................ ................................ ................. 184 Maximum Near Bank Velocities ................................ ................................ ................. 187 VI. VALIDATION AND UNCERTAINTY ................................ ................................ ............... 189 Maximum Shear Stress through Meandering Channels ................................ ............. 191 Shear Stress Phase Lag ................................ ................................ ................................ 198 Maximum Near Bank Velocities ................................ ................................ ................. 202 Limitations ................................ ................................ ................................ ................... 203 VII. EXAMPLE APPLICATION ................................ ................................ ............................ 205 Basic Channel Design Example 1 ................................ ................................ ................. 207 Solution 1, Part A ................................ ................................ ................................ ........ 208 Solution 1, Part B ................................ ................................ ................................ ........ 211 Basic Channel Design Example 2 ................................ ................................ ................. 215 Solution 2 ................................ ................................ ................................ .................... 215 VIII. CONCLUSIONS ................................ ................................ ................................ .......... 219 Application of Numerical Models for Research ................................ .......................... 219 Empirical Equations ................................ ................................ ................................ .... 220 Model Boundary Conditions ................................ ................................ ....................... 221 PAGE 10 x Suggestions for Future Research ................................ ................................ ................ 222 REFERENCES ................................ ................................ ................................ .................... 223 PAGE 11 xi LIST OF TABLES Table 1. Channel dimensions and hydraulic properties of the physical model reported by Yen (1965). ................................ ................................ ................................ ....................... 52 2. Summary of Physical Model dimensions (Heintz, 2002). ................................ ............ 66 3. Physical model hydraulic data. ................................ ................................ .................... 66 4. Summary comparison of measured and model results at cross section 6 for a model mesh size of 492 x 42 x 32. ................................ ................................ ....................... 92 5. Summary compar ison of measured and model results at cross section 10 for a model mesh size of 492 x 42 x 32. ................................ ................................ ....................... 92 6. Summary comparison of measur ed and model results at cross section 16 for a model mesh size of 492 x 42 x 32. ................................ ................................ ....................... 93 7. Simulation set matrix summary. ................................ ................................ ................ 105 8. Model simulation matrix, part 1. ................................ ................................ ............... 110 9. Model simulation matrix, part 2. ................................ ................................ ............... 114 10. CSU Physical Model Dimensions. ................................ ................................ ............. 133 11. Numerical model mesh size used to simulate the flow t hrough the CSU physical model. ................................ ................................ ................................ ..................... 133 12. Example 3D model mesh size for three consecutive meander bends. .................... 134 13. Range of numerical model mesh densities. ................................ ............................. 135 14. Validation data from simulation sets 1, 2, 3, 10, 11, and 12. ................................ .. 190 15. Typical permissible shear stresses for bare soil and stone linings. ......................... 206 PAGE 12 xii LIST OF FIGURES Figure 1. Dungeness River near Sequim, Washington where channel migration has destroyed homes along the floodplain. ................................ ................................ ....................... 6 2. Dungeness River near Sequim, Washington where channel migration has eroded property and a well that was formerly on an adjacent terrace is in the channel center. ................................ ................................ ................................ ......................... 7 3. Hoh River near Forks, Washington where channel migration threatened the Olympic National Park road. ................................ ................................ ................................ ..... 7 4. Elwha River near Port Angeles, Washington where channel migration could threaten homes on top of a high glacial terrace. ................................ ................................ ...... 8 5. Super elevation of flow around a meander bend causes a pressure imbalance, which induces a secondary flow current. ................................ ................................ ............ 12 6. Planform phase lag definition sketch. ................................ ................................ .......... 14 7. Example velocity and shear stress profiles for steady, turbulent, uniform open channel flow with water depth H streamwise velocity u at depth z depth averaged velocity U and bed roughness height k s (modified from Garcia, 2007). .................. 26 8. Relationship between channel width, depth, and hydraulic radius for trapezoidal channels. ................................ ................................ ................................ ................... 32 9. Example distributions of boundary shear stress in a trapezoidal channel with a 2:1 side slope using membrane analogy (Olsen and Florey, 1952). ............................... 33 PAGE 13 xiii 10. Variation in relative bed ( b ) and bank ( s ) shear stress with the channel bottom width to depth ratio (b/D). ................................ ................................ ....................... 34 11. Definition sketch for flow in a curved channel (Chang, 1988). ................................ .. 35 12. Boundary shear stress distributions in curved trapezoidal channels measured by Ippen and Drinker (1962). ................................ ................................ ......................... 37 13. Application of the three dimensional, U 2 RANS model, (modified from Lai et al., 2003a) to the physical model experiment by Yen (1965). ................................ ........ 52 14. Comparison of U2RANS model depth results; using coarse, medium, and fine meshes; with physical model data (modified from Lai et al., 2003a). ...................... 53 15. Velocity profile at 0.461 B. ................................ ................................ ........................ 55 16. Velocity profile at S0, 0.307 B. ................................ ................................ .................. 55 17. Velocity profile at 0.000 B. ................................ ................................ ......................... 55 18. Velocity profile at +0.307 B. ................................ ................................ ....................... 56 19. Velocity profile at +0.461 B. ................................ ................................ ....................... 56 20. Simulated boundary shear stress is presented for the case of a fine mesh. ............. 57 21. Plan view a lignment is presented for the physical model by Yen (1965). Velocity profile measurements are reported at the cross section labeled S0. Boundary shear ................................ ................................ ................................ ..................... 58 22. Comparison of simulated and measured shear stress is presented for cross section CIIO. The channel cross section is also plotted along with the location of high velocity (within 90 percent of the maximum velocity). ................................ ............ 59 PAGE 14 xiv 23. Comparison of simulated and measured shear stress is presented for cross section velocity (within 90 percent of the maximum velocity). ................................ ............ 60 24. Comparison of simulated and measured shear stress is presented for cross section velocity (within 90 percent of the max imum velocity). ................................ ............ 61 25. Comparison of simulated and measured shear stress is presented for cross section section is also plotted along with the location of high velocity (within 90 percent of the maximum velocity). ................................ ............ 62 26. Comparison of measured and simulated boundary shear stress from the physical model reported by Ippen and Drinker (1962). ................................ ......................... 64 27. Photograph of the large physical model of flow through two channel bends at Colorado State University. ................................ ................................ ........................ 65 28. Colorado State University mod el plan view configuration for flow through two channel bends of constant radius. ................................ ................................ ............ 66 29. Plan view location of cross section velocity measurements is shown from the upstream most cross section 1 to the downstream most cross section 18. The naming convention of the vertical velocity measurements at each cross section is a b c d e f and g left to right looking downstream. ................................ ............... 68 30. The two dimensional numerical model has a structured mesh with 326 cells in the d ownstream direction and 28 cells in the cross stream direction. .......................... 70 PAGE 15 xv 31. A close up view of the two dimensional numerical model mesh (326 x 28) is shown through the upstream channel bend. ................................ ................................ ....... 71 32. The centerline water surface profiles from the numerical and physical models are compared over the reach of the physical model. ................................ ..................... 72 33. The numerical model mesh boundary (meters) extends ten channel widths both upstream and downstream from the physical model boundaries. .......................... 73 34. A plan view close up is presented of the three dimensional structured mesh (492 x 42 x 32) along the straight transition reach. ................................ ............................. 75 35. The right, cross sectional half of the three dimensional structured mesh (492 x 42 x 32) is presented at cross section 6 through the upstream bend. ............................ 76 36. Simulated (blue) and measured (red) velocity vectors are compared in plan view for the entire region of the physical model. ................................ ................................ .. 77 37. Simulated (blue) and measured (red) velocity vectors are compared in a close up plan view of the first (upstream) channel bend. ................................ ...................... 78 38. Simulated (blue) and measured (red) velocity vectors are compared in a close up plan view of the downstream portion of the first channel bend and straight transition reach. ................................ ................................ ................................ ........ 79 39. Simulated (blue) and measured (red) velocity vectors are compared in a close up plan view of the upstream portion of the second channel bend. ............................ 80 40. Simulated (blue) and measured (red) velocity vectors are compared in a close up plan view of the downstream portion of the second channel bend. ....................... 81 PAGE 16 xvi 41. Comparisons of seven measured and simulated stream wise velocity profiles using four different mesh sizes at cross section 6 in the upstream channel bend downstream). ................................ ................................ ................................ ............ 82 42. Comparisons of seven measured a nd simulated stream wise velocity profiles using four different mesh sizes at cross section 10 at the downstream end of the straight right looking downstream). ................................ ................................ ...................... 84 43. Comparisons of seven measured and simulated stream wise velocity profiles using four different mesh sizes at cross section 16 in the downstream channel bend downstream). ................................ ................................ ................................ ............ 86 44. Compariso n of measured and simulated cross stream velocity profiles at cross section 6 for a range of three dimensional model mesh sizes. ................................ 89 45. Comparison of measured and simulated cross stream velocity profiles at cross section 10 for a range of three dimensional model mesh sizes. .............................. 90 46. Comparison of measured and simulated cross stream velocity profiles at cross section 16 for a range of three dimensional model mesh sizes. .............................. 91 47. Comparison of depth averaged measured and simulated stream wise velocity along channel banks. Measurement location ................................ ................................ ................................ ... 94 PAGE 17 xvii 48. Simulated dimensionless boundary shear stress for the physical model at Colorado State University. ................................ ................................ ................................ ........ 98 49. The graph of the relationship between bankfull discharge, channel slope, and median sediment grain size includes three distinct regions (Chang, 1988). Solid contour lines indicate the channel surface width while dashed contour lines denote the bankful channel depth. ................................ ................................ ..................... 100 50. Numerical model simulation sets are presented on the graph of bankfull discharge, channel slope, and median sediment grain size. Each simulation set includes nine meandering channels. ................................ ................................ ............................. 104 51. Discharge, valley slope, and grain size. ................................ ................................ .... 107 52. A plot of relative grain roughness versus Reynolds number show that model simulations are in the turbulent range. ................................ ................................ .. 109 53. Range of hydraulic radius, width depth ratio, and Froude number of the model simulation matrix. ................................ ................................ ................................ ... 118 54. Example meandering river channel alignment, generated using a sine generated curve, for simulation set 5 with a sinuosity of 2.75 and five consecutive curves. 123 55. Example channel alignment points and the variable radii of curvature are plotted for simulation set 5 with a sinuosity of 2.75 and five consecutive curves. .................. 124 56. A close up view of channel alignment points and variable radii of curvature show that their alignment s are similar, but they do not exactly coincide. ..................... 125 57. Meandering channel alignments of simulation Set 6 with three consecutiv e meandering bends. ................................ ................................ ................................ 126 PAGE 18 xviii 58. Simulation Set 6, Sinuosity of 3.00. ................................ ................................ ......... 127 59. Simulated vertical velocity profiles for the staight channel reach after channel length equal to 20 channel widths. ................................ ................................ ........ 128 60. Simulated bottom shear stress long the centerline of the straight trapezoidal channel. ................................ ................................ ................................ ................... 129 61. Channel plan view and location of simulated maximum velocity through five consecutive meander bends. ................................ ................................ .................. 131 62. Relative discharge on the left and right sides of the channel through five consecutive meander bends. ................................ ................................ .................. 132 63. Simulated vertical velocity profiles at a meander bend cross section for a coarse model mesh size of 542 x 25 x 14. ................................ ................................ .......... 136 64. Simulated vertical velocity profiles at a meander bend cross section for a medium model mesh size of 684 x 31 x 17. ................................ ................................ .......... 137 65. Simulated vertical velocity profiles at a meander bend cross section for a fine model mesh size of 860 x 39 x 21. ................................ ................................ ..................... 137 66. Simulated cross section shear stress and location of high velocity at a meander bend for a coarse model mesh size of 542 x 25 x 14. ................................ ...................... 138 67. Simulated cross section shear stress and location of high velocity at a meander bend for a medium model mesh size of 684 x 31 x 17. ................................ ................... 138 68. Simulated cross section shear stress and location of high velocity at a meander bend for a fine model mesh size of 860 x 39 x 21. ................................ .......................... 139 PAGE 19 xix 69. Simulated velocity and shear stress are compared with the total model mesh size. ................................ ................................ ................................ ................................ 140 70. Relationship between channel curvature and radius of curvature. ........................ 147 71. Model channel alignment for a sinuosity of 1.10. ................................ ................... 151 72. Model channel alignment for a sinuosity of 1.25. ................................ ................... 151 73. Model channel alignment for a sinuosity of 1.50. ................................ ................... 151 74. Model channel alignment for a sinuo sity of 1.75. ................................ ................... 151 75. Model channel alignment for a sinuosity of 2.00. ................................ ................... 151 76. Model channel alignment for a sinuosity of 2.25. ................................ ................... 151 77. Model channel alignment for a sinuosity of 2.50. ................................ ................... 152 78. Model channel alignment for a sinuosity of 2.75. ................................ ................... 152 79. Model channel alignment for a sinuosity of 3.00. ................................ ................... 152 80. Plan view of the structured 2D model mesh domain for Simulation Set 5 with a sinuosity of 2.75. ................................ ................................ ................................ ..... 153 81. Close up plan view of the structured 2D model mesh for Simulation Set 5 with a sinuosity of 2.75. ................................ ................................ ................................ ..... 154 82. Channel bottom elevation contours of the 2D model mesh for Simulation Set 5 with a sinuosity of 2.75. ................................ ................................ ................................ .. 155 83. Water surface elevation contours from the 2D model results for Simulation Set 5 with a sinuosity of 2.75. ................................ ................................ .......................... 156 84. Elevation contours of the 3D model mesh for Simulation Set 5 with a sinuosity of 2.75. ................................ ................................ ................................ ........................ 158 PAGE 20 xx 85. Concept of the 3D structured mesh is presented for the right half of the cross section. ................................ ................................ ................................ .................... 15 9 86. Simulated 3D velocity contours at the water surface for Simulation Set 5 with a sinuosity of 2.75. ................................ ................................ ................................ ..... 160 87. Close up view of simulated 3D velocity contours at the water surface. ................. 161 88. Close up 3D view of cross sectional slices of velocity contours. ............................. 162 89. Simulated thread of maximum velocity for Simulation Set 5 with a sinuosity of 2.75. ................................ ................................ ................................ ................................ 163 90. Simulated velocities near the left and right channel ba nks for simulation set 5 with a sinuosity of 2.75. ................................ ................................ ................................ ..... 164 91. Channel bottom shear stress contours for simulation set 5 with a sinuosity of 2.75. ................................ ................................ ................................ ................................ 165 92. Longitudinal channel bank and centerline shear stress for simulation set 5 with a sinuosity of 2.75. ................................ ................................ ................................ ..... 166 93. Plan view alignment of the meandering channel from simulation set 5 with a sinuosity of 2.75 and positions of selected cr oss sections 112, 702, 810, and 950. ................................ ................................ ................................ ................................ 167 94. Lateral shear stress distribution and zone where velocity is at least 90 percent of the maximum velocity at cross section 112 (upstream straight reach). ...................... 168 95. Lateral shear stress distribution and zone where ve locity is at least 90 percent of the maximum velocity at cross section 702 (3 rd meander bend). ................................ 168 PAGE 21 xxi 96. Lateral shear stress distribution and zone where velocity is at least 90 percent of the maximum velocity at cross section 810 (3 rd meander bend). ................................ 169 97. Lateral shear stress distribution and zone where velocity is at least 90 percent of the maximum velocity at cross section 950 (4 th meander bend). ................................ 169 98. Vertical streamwise velocity profiles at cross section 112 (upstream straight reach) for simulation set 5 with a sinuosity of 2.75. ................................ .......................... 170 99. Vertical streamwise velocity profiles at cross section 702 (3 rd meander bend) for simulation set 5 with a sinuosity of 2.75. ................................ ............................... 170 100. Vertical streamwise velocity profiles at cross section 810 (3 rd meander bend) for simulation set 5 with a sinuosity of 2.75. ................................ ............................... 171 101. Vertical streamwise velocity profiles at cross section 950 (4 th meander bend)for simulation set 5 with a sinuosity of 2.75. ................................ ............................... 171 102. Longitudinal channel bank and centerline shear stress for simulation set 7 with a sinuosity of 2.00 and a width depth ra tio of 7.3. ................................ ................... 172 103. Longitudinal channel bank and centerline shear stress for simulation set 8 with a sinuosity of 2.00 and a width depth ratio of 9.1. ................................ ................... 173 104. Longitudinal channel bank and centerline shear stress for simulation set 9 with a sinuosity of 2.00 and a width depth ratio of 35.6. ................................ ................. 174 105. Summary of 3D model simulation results for minimum and maximum shear stress alon g the toe of the left and right banks through meandering channels. ............. 179 106. Correlation between the maximum dimensionless shear stress in a channel bend with the ratio of channel width to minimum radius of curvature. ......................... 180 PAGE 22 xxii 107. Residual maximum shear stress computed by empirical Equation 63 and the maximum shear stress simulated by the 3D model is plotted against the channel width depth ratio. ................................ ................................ ................................ ... 181 108. 3D plot of maximum shear stress against the ratio of channel width to minimum radius of curvature and the width depth ratio. ................................ ...................... 183 109. Comparison of maximum dimensionless bank shear stress computed by empirical Equation 64 and simulated by the 3D model. ................................ ........................ 184 110. Correlation between the shear stress phase lag and channel sinuosity. .............. 185 111. Relationship between the maximum near bank velocity and the ratio of channel width to minium radius of curvature for Simulation Sets 4 through 9. ................. 188 112. Validation of the relationships between maximum dimensionless shear stress in a channel bend with the ratio of channel width to minimum radius of curvature. New regression lin es, using all data, are compared with the initial regression lines using data from one half of the simulation sets. ................................ .................... 191 113. Validation of the residual relationships between maximum shear stress computed by empirical Equation 68 and the maximum shear stress simulated by the 3D model as a function of the channel width depth ratio. ................................ ..................... 193 114. 3D plot of maximum dimensionless shear stress, computed by Equation 69, compared with shear stress simulated by the 3D model. ................................ ...... 195 115. Comparison of maximum dimensionless bank shear stress computed by empirical Equation 69 and simulated by the 3D model. ................................ ........................ 196 PAGE 23 xxiii 116. Comparison of maximum dimensionless shear stress from empirical estimates with physical model measurements. Empirical estimates are based on Equation 17 and Equation 69. ................................ ................................ ................................ ............ 197 117. Testing with validation data of the correlation between the shear stress phase lag and channel sinuosity. ................................ ................................ ............................ 199 118. Comparison of channel phase lag computed by empirical Equation 70, Equation 71, and Equation 72 with the simulated phase lags from the 3D model. .................... 201 119. Relationship between the maximum near bank velocity and the ratio of channel width to minium radius of curvature for Simulation Sets 1 through 12. ............... 202 120. Relationship between the total channel curvature and the ratio of channel width to radius of curvature. ................................ ................................ ............................ 204 PAGE 24 1 CHAPTER I INTRODUCTION The migration of meandering river channels across their floodplains and the occasional erosion of terrace banks is a natural process (Leopold et al. 1964, Yang 1971, Dunne and Leopold 1978, Leopold 1994, and Thorne 1992 and 2002). This process can become especially important to people living near a river and to agencies planning or maintaining infrastructure within or along a river. An improved understanding of river hydraulics that cause channel bank erosion and migration is needed. Thesis State ment Application of a numerical, three dimensional, hydraulic model to a wide rang e of meandering river channels of the type (width and depth) found in nature provides detailed data sets of flow velocity and boundary shear stress from which important empi rical relationships can be dev elop ed and tested. The boundary shear stress exerted by moving water (fluid) increases through a meander bend relative to a straight channel reach, as the result of the channel curvature. Th e maximum shear stress along the banks of a meandering river channel can be linearly related to the ratio of channel width to radius of curvature and to the width depth ratio PAGE 25 2 Research Objective The objective of this research is to empirically determine dimensionless relationships between hydraulic parameters of meandering river channels and the maximum shear stress caused by the channel curvature. General Research Methodology A numerical, three dimensional hydraulic model was used like a computational flume to generate data sets representing a wide range of meandering river channels of the type (width and depth) found in nature. These data sets were used to develop empirical relationships between dimensionless shear stress and geometric properties of the channel cross section an d alignment. Natural m eandering river channels include a wide range of sinuosities (ratio of channel length to valley length) and width depth ratios. Field and laboratory settings provide opportunities for time varying measurements of the velocity flow f ield. However, data sets of detailed hydraulic conditions are not readily available from a wide range of natural channels Natural river channels often contain irregularities in alignment and stream bank properties In addition, natural stream flow rates are often highly variable. Therefore, obtaining a wide range of hydraulic data sets from natural river channels is PAGE 26 3 difficult for the purposes of developing a general understanding of shear stress distributions along meandering river chan nels. Channel dimensions and stream flow rates can be controlled in laboratory physical models Research utilizing physical models with curved channels typically contain one or two curves of constant radius (Professor Pierre Julian, Colorado State Unive rsity, Oral Communication, June 1, 2007). However, natural river channels typically have multiple channel bends and variable rates of curvature. Physical m odeling of a wide range of meandering channels is time consuming and expensive and there can be m odel scaling issues. The use of different horizontal and vertical physical model scales can affect the quantitative results, but the qualitative results are not normally affected. Numerical model results may be affected by mesh size, but model scales are not a relevant issue for numerical models. Numerical model simulations of channel hydraulics were found to replicate physical measurements quite well. Therefore, an existing three dimensional numerical model was used to simulate velocity and shear stress through a wide range of meandering river channels of the type found in nature. This three dimension model is more fully described in Chapter II. There are relatively few reports of studies where a three dimensional (3D) numerical model was used to simulate primary and secondary currents through meander bends that are similar to natural river channels (Wilson, et al., 2003). The difficulties in PAGE 27 4 acq uiring good spatial resolution data of high quality in rivers have meant that the majority of numerical model development and subsequent testing has focused on the study of simplified river geometries in controlled environments. Most models have been appl ied to rectangular curved channels or trapezoidal channels with steep side slopes and with simple alignment geometries. For this research, meandering trapezoidal channels with 2:1 side slopes were simulated with variable rates of curvature. Sine generate d curves were used to define the meandering channel alignments with variable radius of curvature and three or five consecutive meander curves. For each meandering channel, f ixed bed hydraulics was simulated at the bankfull discharge A non erodible chann el boundary was assumed (fixed bed hydraulics) in the model simulations to preserve the channel dimensions and alignment during simulation. Estimate of the maximum shear stress can only be assured if the channel banks and bed are not allowed to erode. Model simulation results were then used to develop general relationships between dimensionless bank shear stress and channel geometric parameters. Motivation and Need for Research This research is motivated by the need to simulate how river channels mi grate across their floodplain. Hydraulic and Sediment transport models have existed for many years PAGE 28 5 to simulate the sediment erosion and deposition along the bed of a river channel. The 6 ( 19 93 ) and HEC RAS ( 20 10) models and the U.S. and SRH 1D (Greimann and and Huang, 2006) models are examples of one dimensional hydraulic and sediment transport models. These model s have important applica tion to degrading and aggrading stream channels. However, these models do not simulate the bank erosion that results dur in g the lateral migration of a meandering river channel. Many river channels are in a dynamic equilibrium where there is a balance between the upstream sediment supply and the downstream transport. These channels are neither significantly aggrading nor degrading over time, but they may migrate laterally across their floodplain. Two and three dimensional hydraulic and sediment transport models can be applied to meandering river channels, but such models are not widely availab le and are difficult to apply over long time and space scales (kilometers of r iver lengt h over years of time) because of computational time requirements and model stability issues The research presented in this dissertation could be applied to a one dimensional model to simulate river channel migration along tens of kilometers of length a nd over decades of time For streambank protection designs, t he research presented in this disseration will help in estimating the maximum shear stress along curved channel banks PAGE 29 6 The migration of meandering river channels can sometime pose hazards to pe ople, property, and infrastructure along a river. Some examples of channel migration hazards are shown in Figure 1 through Figure 4 Natural rates of channel migration can be accelerated by human land use activities For example, the clearing of riparian vegetation can a ccelerate the rate of channel migration because cohesion provided by vegetation root structure is no longer available (Beeson and Doyle 1995). Figure 1 Dungeness River near Sequim, Washington where channel migration has destroyed homes along the floodplain. PAGE 30 7 Figure 2 Dungeness River near Sequim, Washington where channel migration has eroded property and a well that was formerly on an adjacent terrace is in the channel center Figure 3 Hoh River near Forks, Washington where channel migration threatened the Olympic National Park road. PAGE 31 8 Figure 4 Elwha River near Port Angeles, Washington where channel migration could threaten homes on top of a high glacial terrace. Background Literature Natural geomorphic features of a river channel result from the continuous dynamic interaction between the motion of a sediment carrying fluid and an erodible boundary of the channel (Seminara 2006). The major factors affecting alluvial stream channel forms are listed by Lagasse, et al. ( 2004): 1. stream discharge (magnitude, duration, and frequency), temperature, viscosity; 2. sediment load and grain size; 3. longitudinal valley slope; 4. bank and bed resistance to erosion, including the effects of riparian vegetation; PAGE 32 9 5. local geology, including bedrock outcrops, clay plugs, changes of valley slope; and 6. human development and activities. Theoretical analyses and laboratory experiments have shown that the observed channel patterns are mostly related to fundamental instability mechanisms. The mobile interface between the fluid and the erodible channel boundary is unstable, rather than the flow itself (Seminara, 2006). A meandering river channel is cha racterized by a sequence of smooth bends. There tends to be persistent erosion along the outer bank, forming pools, and deposition of point bars along the inner bank. This erosion and deposition leads to the migration of meander bends both across the flo odplain and along the downstream valley axis. The meander migration process has been observed in the field by Leopold et al. (1964), Hickin (1974), Hickin and Nanson (1975), Hickin and Nanson (1984), Parker (1975), as well as in the laboratory by Friedkin (1945) and Chang et al. (1971). Causes of Channel Meandering Meandering river channel alignments are observed in natural river channels at all scales, all continents and latitudes, and in ice (Leopold et al. 1964, Thorne 1992, and Leopold PAGE 33 10 1994). Meand ering channels have even been reported on the planet Mars (De Hon, The width, depth, and slope of a meandering river channel continually adjust over time to achieve an equilibrium condition with the changing upstream water discharge and sediment load and within the constraints of the valley slope, local geology, and human structures (Lane, 1955; Simons and Sentrk, 1992; and Yang, 1986). Yang (1971) attributes river meandering to the minimum rate of energy dissipation, subject to the constraints applied to the system. The precise cause of meandering remains undefined. Ashworth, et al. (1996) suggests that the process begins with coherent structures in the flow field that are on the order of a channe l width. Large eddies produced in the flow field of a straight channel induce a sinuous path in the streamlines of maximum velocity. Seminara (2006) used a linear model of flow and bed topography to show that meanders behave as linear oscillators which m ay resonate at some distinct values of the channel width to depth ratio and of the ratio of channel width to meander wavelength. Reson ance excites a natural mode of oscillation in the form of stationary alternate bars. When the resonance barrier is cross ed, the direction of meander migration is reversed. Lai (2006) has demonstrated with numerical modeling that the sinuous path of the maximum velocity streamline begins with the transport of the alluvial bed material. The PAGE 34 11 surrounding flow field along thes e sinuous streamlines is subsequently strengthened by positive feedback between curvature of the flow and skew induced secondary currents of Prandtl's first kind (Bathurst, et al., 1979). A meander bend cutoff occurs when one migrating channel bend interc epts another (neck cutoff) or when flood flows erode a shorter channel path across the meander bend (chute cutoff). The prediction of chute cutoffs has not yet been resolved. A chute cutoff is a process whereby a meander can sometimes be abandoned well b efore a neck cutoff (Seminara, 2006). Planform Evolution of Erodible Channels Meandering rivers shift laterally and migrate longitudinally through a process of erosion at concave banks and deposition at convex banks (Seminara, 2006). As the river flow i s conveyed through a meander bend, the centrifugal force causes the water surface elevation to be slightly higher on the outside of the bend ( Figure 5 ). This super elevation of the water surface causes a pressure imbalance perpendicular to the main flow direction, which then causes a secondary flow current (Edwards and Smith, 2002 and Seminara, 2006). The secondary current is described by flow traveli ng in a downstream helical motion, which is downward near the outer bank, across the deep portion of the channel bottom toward the shallow inner bank, and then back across the PAGE 35 12 channel near the water surface. At the upper portion of the outer bank, a small cell of reverse circulation often exists that forces flow upward along the bank. According to Thorne and Hey (1979) and Thorne (1992), this reverse circulation is highly significant to the flow processes acting to erode the outer bank. Figure 5 Super elevation of flow around a meander bend causes a pressure imbalance, which induces a secondary flow current. In channels with non erodible banks, flow along the inner bend accelerates relative to the outer bend. Farther down stream, the secondary flow transfers momentum towards the outer bend, moving the thread of high velocity from the inner bank to the outer bank (Seminara, 2006). The establishment of a bar pool pattern creates an additional component of the secondary flow, which further modifies the bed topography. PAGE 36 13 For rivers with a lateral channel bed slope, gravitational forces tend to cause sediment particles to move downhill (toward the outside bend), and bedload deviates from the direction of the local average bottom shear stress (Seminara, 2006). Dietrich and Smith (1983) analyzed several sets of laboratory and field data and documented that shoaling over the point bar in the upstream part of the bend concentrates the high velocity core of flow toward the pool. The channel curvature induced component of boundary shear stress is confined to 20 to 30 percent of the channel width at the pool. Dietrich and Smith (1984) found that cross stream bed load transport varied with particle grain size and was affected by point b ar topography, trough wise flow along obliquely oriented bed forms, rolling of particles on bed form lee faces, and mass sliding on steep point bar slopes. If the river bed material is mobile, asymmetry in the velocity and boundary shear stress distributi ons rapidly leads to the generation of pools, riffles, and alternate bars (Smith, 1987). There is a lag effect of the hydraulics through a meander bend where the effects of channel curvature develop some distance downstream from where the channel curvatur e begins. This distance is referred to as the planform phase lag and can cause downstream migration of the meander bed ( Figure 6 ). If the phase lag were zero, the channel would migrate only laterally across the floodplain and increase its meander bend amplitude. If the phase lag were equal to one quarter of the meand er wave PAGE 37 14 length, then the meander bend migration would be entirely downstream along the valley alignment without an increase in meander bend amplitude. Figure 6 Planform phase lag definition sketch. Nature of Channel Bend Instability Erodible channel banks are a necessary condition for the initiation of a meander bend, although they are not the direct cause of meandering (Rhoads and Welford, 1991). In streams with erodible banks, the sinuous path of the maximum velocity s treamlines initiates a pattern of bank erosion that marks the onset of channel meandering (Friedkin, 1945; Keller, 1972; and Hakanson, 1973). Where the maximum velocity streamline converges with the channel bank, local scour of the channel bed undercuts the bank and promotes erosion, instability, and rapid bank PAGE 38 15 retreat (Thorne and Lewin, 1979 and Thorne, 1982). The rapid retreat of the outer bank that enables active meanders to migrate within the floodplain. The rate of bank deposition depends on the u pstream sediment supply. Channel width can remain relatively constant over time only if the rates of bank erosion and deposition are in balance. Otherwise, the channel will widen or narrow over time. Brice (1982) discovered that channels that do not var y significantly in width were relatively stable, whereas channels that were wider at bends were more active. High sinuosity and equal width streams were the most stable, whereas other equal width streams of lower sinuosity were less stable, and wide bend streams had the highest erosion rates. C hannel bend instability theory predicts that any small random perturbation of a straight channel alignment eventually grows, leading to a meandering pattern (Seminara, 2006) However, this is difficult to confirm from field and laboratory observations Cohesionless sediments are typically employed in laboratory experiments with an initially straight channel. The channel typically undergoes a sequence of processes (Federici an d Paola 2003) associated with a continuous widening of the channel, driven by the erosion of cohesionless banks. Though, at an intermediate stage, an apparent meandering channel forms, it continues to evolve through further widening, the occurrence of chu te cutoffs, and the emergence of bars until a braided pattern is formed. PAGE 39 16 The persistence of a coherent meandering pattern requires a cohesive floodplain (Seminara, 2006). Vegetation plays an important role in stabilizing the banks, constraining channel migration, and allowing deeper and narrower channels to develop. Alternate bars are not the precursor to river meandering for the following reasons (Seminara, 2006): First, alternate bars would have to migrate fairly fast to be responsible for the local ized erosion of cohesive banks driving meander formation. Secondly, and more importantly, the typical dimensionless wave number of developed alternate bars falls within the stable bend range and would not grow. Thirdly, alternate bars are observed to coe xist with and migrate through weakly meandering channels, an observation which contradicts the idea that they would evolve into the fixed point bars of river meanders. Finally, in the experiments of Smith (1998), meandering developed in the absence of alt ernate bars and the chain of events typically observed did resemble the initial stage of meander bend the only rational scheme able to explain various features observed in the Bend instability is the process whereby a perturbation of an initially straight channel alignment grows, driven by bank erosion, and leads eventually to the development of a meandering pattern. PAGE 40 17 Seminara (2006) argues that, i n order to understand meander formation in purely cohesive and erosional channels the classical bar pool scheme must be abandoned in favor of a purely erosional mechanism driven by the three dimensional structure of the flow field. Channel Meander Bend Migration The morphology an d behavior of a given river reach is strongly determined by the amount and rate of sediment and water discharge supplied from upstream. Therefore, any significant modification of sediment load and water discharge will impact local rates of channel migratio n and evolution (Lagasse, et al., 2004). Locally, the distribution of velocity and shear stress and the characteristics of bed and bank sediment will control channel behavior. The direction and rate of channel migration are a function of the following ch annel factors (Lagasse, et al., 2004): local curvature and sinuosity, channel shape (width depth ratio), longitudinal slope, discharge, and bank strength. Channel migration can be slowed or limited by floodplain vegetation, alluvial deposits, geologic controls, and human structures (e.g. bank protection). The bank strength from PAGE 41 18 vegetation depends on the root depth and density (Pollen, et al, 2007). Large w oody vegetation can provide recruitment for large woody debris (log jams). Alluvial deposits frequently include oxbows, meander scrolls and scars, and clay plugs, each of which has different erodibility characteristics. Geologic controls include bedrock outcrops and erosion resistant features along the margins of the floodplain such as terrace banks. The amplitude of a single meander bends tends to increase with time up to a maximum and then subsequently decreases while rate of migration decreases (Nanso n & Hickin 1983). In the absence of geological constraints, meanders often evolve continuously until a neck cutoff occurs (Seminara, 2006). The alignment of the neck cutoff creates a geometric discontinuity, which is smoothed out through additional planf orm evolution and channel migration. The abandoned meander bend (oxbow lake) is slowly filled up through the deposition of fine suspended sediments (often clay size) during floods. As the deposited sediments consolidate over time, the future resistance to local erosion will in crease, which may further affect the planform evolution on time scales of the order of centuries. Stream bank erodibility depends on the previous history of the floodplain formation, as well as on the presence of vegetation, geolog ical constraints, and anthropogenic effects (Seminara, 2006). In some cases, a chute cutoff may occur before long before the meander bend achieves a neck cutoff. This process occurs in wide bends with fairly large curvatures, high PAGE 42 19 discharges, noncohesi ve and unvegetated banks, and steep longitudinal channel slopes (Howard & Knutson 1984). Existing Numerical Models of River Channel Migration Previous models have assumed the river discharge as well as channel width, river slope, and grain size to be c onstant (Seminara, 2006). These assumptions are acceptable when the spatial scale of the reach as well as the temporal scale of the process is not too large. Models do not yet simulate chute cutoffs and neck cutoff are modeled in a schematic way. The pr ocess of river degradation or aggradation, driven by the cutoffs, is ignored and unsteady flow is not accounted for. The bank erosion equation by Ikeda et al. (1981) initiated empirical and theroretical study to predict meander bend migration. Parker (19 76) and Odgaard (1989a and 1989b) developed meander migration models by utilizing similar approaches as Ikeda et al (1981). However, these approaches do not consider the sediment load of the stream. Hasegawa (1989) pointed out that bank erosion is caused by an imbalance between sediment supply and bank material entrainment. Hooke (1975) described the distribution of sediment transport and shear stresses in a meander bend. Smith and Mclean (1984 ) and Mclean and Smith ( 1986) were the first to develop dynamically correct two dimensional models and explain the mechanics of flow PAGE 43 20 through a meander bend. Nelson and Smith (1989) combined a bed load transport algorithm with the numerical model for flow and boundary shear stres s fields to investigate point bars in meandering channels and alternate bars in straight channels. National Cooperative Highway Research Program commissioned a study on the Methodology for Predicting Channel Migration (Lagasse, et al., 2004). This study concluded that it is very difficult to predict the magnitude, direction, and rate at which channel migration will occur and that no good practical models of channel migration iver is order to predict the potential for future migration. This method is very useful, but a predictive model is still needed to account for proposed hydraulic structure s or land use change. A study by Johns Hopkins University (Cherry, 1996) for the U.S. Army Corps of Engineers Waterways Experiment Station concluded that both the accuracy and applicability of the bend flow meander migration model by Garcia, et al. (1994) are limited by a number of simplifying assumptions. Among the most important of these are the use of a single discharge and the assumption of constant channel width, both of which prevent the model from successfully forecasting the spatial and temporal v ariability that appears to be inherent in the process of bend migration. PAGE 44 21 The study also concluded that much of the discrepancy between the predicted and observed distributions of erosion can be accounted for by the fact that meander migration is modeled a s a smooth, continuous process. In reality, erosion occurs predominantly in discrete events (i.e., during flood flows), and varies greatly both temporally and spatially along the channel from bend to bend. The Johns Hopkins study noted that the identific ation of local factors that influence the amount of bank erosion that occurs is a subject that will require further investigation and that further refinements in bend flow modeling will not improve our predictive capability until we find a more rational wa y to wed the flow model to a bank erosion model. Three dimensional models are appropriate for a length scale of few meander curves and a time scale of days to weeks (Huang et al. 2001, Lai et al. 2003a, and Lai et al. 2003b). Two dimensional models are appropriate for a length scale of a few kilometers and a time scale of months to years (Duan, 2004 and Lai, 2006). Analytical models can be applied to spatial scales of tens of river kilometers and time scales of decades to centuries. Analytical models are typically based on certain simplifying assumptions: The meandering channel only has long radius meander bends. Channel width is constant with time. There is no channel aggradation or degradation. PAGE 45 22 Only the model by Huang, Greimann, and Randle (2007) c onsiders channel aggradation or degradation. Bank erosion rate (B E ) is a function of a dimensionless bank erosion coefficient (E o ) and the increased bank velocity (V c ) due to channel curvature (B E = E o V c ). E o is on the order of 10 7 to 10 8 Johannesson and Parker (1989) proposed an analytical method to predict the increase in channel velocity through a meander based on a linearization of the two dimensional equations for depth averaged flow and a sediment transport equation to predict the cross channel slope. The deviation is assumed to be linearly related to the maximum curvature of the channel. For their linearization analysis, the flow variables are the sum of two parts: (1) flow in straight channel and (2) flow deviation due to a slightly curved channel. These perturbed flow variables are substituted into the 3D flow equations. The equations are then simplified and grouped into the terms responsible for the straight channel solution and those due to the channel curvature. The equations become ordinary differential equations and can be solved analytically or through relatively simple numerical methods. The sediment transport is assumed to be a function of the local velocity and shear stress and is used to compute the cross channel slope aroun d the bed. The method proposed PAGE 46 23 by Johannesson and Parker (1989) has been widely accepted in the literature, but is only applicable to long radius curves. Sun et al. (2001a and 2001b), re derived the equations to account for multiple sediment grain sizes. Seminara (2006) provides some very interesting theoretical results about meander bend processes. Even though the results from analytical models have been qualitatively substantiated through field or laboratory observations, the simplifying assumptions a re not always applicable for predictive models in engineering applications. PAGE 47 24 CHAPTER II RESEARCH STRATEGY A n existing set of two and three dimensional numerical model s w ere used to simulate depth, velocity and shear stress through a wide range of meandering river channels of the type found in nature. Even though each numerical model simulation take s hours of computer time, a large number of model simulations can be performed at much less time and cost than the construction and application of physical models or measurements from natural channels In addition, numerical model results include three dimensional velocit ies at each mesh cell of the model flow domain and shear stress at each boundary cell. The n umerical model results are repeatable and there are no scaling issues. The numerical model was first validated with measurements from physical model s Sine generated curves were used to define the meandering channel alignments with three or five consecut ive curves. Fixed bed channel s were simulated at the bankfull discharge Model simulation results were then used to develop general relationships between dimensionless river hydraulic parameters and dimensionless geometric channel properties PAGE 48 25 Open Channel Hydraulics The American Society of Civil Engineers, Sedimentation Manual (Garcia, 20 0 7) provides a very good basic description of the velocity and shear stress profiles for the case of steady, turbulent, and uniform open channel flow ( Figure 7 ). Open channel flow exerts a tangential force per unit area on the stream bed known as the bed shear stress. For the case of a wide channel, the bottom shear stress can be expressed as: Equation 1 where b = bed shear stress, = water density, g = acceleration of gravity, H = channel depth, S = longitudinal energy slope Equation 1 is also the one dimensional, conservation of momentum equation of steady uniform flow. PAGE 49 26 Figure 7 Example velocity and shear stress profiles for steady, turbulent, uniform open channel flow with water depth H streamwise velocity u at depth z depth averaged velocity U and bed roughness height k s (modified from Garcia, 20 07 ). The shear velocity is commonly defined from the boundary shear stress as: Equation 2 The shear velocity and boundary shear stress provide a direct measure of the flow intensity and its ability to transport sediment particles (Garcia, 2007). For steady, PAGE 50 27 uniform flow of a wide channel, the shear stress ( ) varies linearly with flow depth ( Figure 7 ): Equation 3 The vertical flow velocity distribution is well represented by the logarithmic distribution (Garcia, 2007; Schlichting, 1979; and Nezu and Rodi, 1986): Equation 4 Where u = time averaged flow velocity at a distance z above the stream bed, z 0 = bed roughness height where the flow velocity goes to zero, and = Garcia (2007) reports that von suspended sediment as previously believed and can be considered to be a universal constant (Smith and McLean, 1977; Coleman 1981 and 1986; Lyn, 1991; Soulsby and Wainright, 1987; Wright and Parker 2004). Equation 4 applies only in a relative thin vertical layer near the bed ( z / H < 0.2) (Nezu and Nakagawa, 1993). However, it is commonly used as a reasonable approximation throughout the flow depth in m any streams and rivers (Garcia, 2007). Equation 4 is not PAGE 51 28 exact because the vertical velocity distribution can be affected by wake effects near the free surface, sediment stratification, and bed forms. The surface roughness on the channel be d and banks is a function of the sediment particle size. This surface roughness affects the velocity and shear stress distributions in the channel. If the channel bed is smooth, turbulence will be drastically suppressed in a very thin layer near the bed, known as the viscous sub layer: Equation 5 Where v = height of the viscous sublayer and = kinematic viscosity of water. The boundaries of natural stream channels are mostly rough (Garcia, 2007). Typically, muddy bottoms as well as channel beds composed of silt and fine sand can be hydraulically smooth. However, stream channel beds composed of coarse sands and gravel are typically rough. Let k s denote the effective roughness height. No viscous sublayer will exist when k s is greater than v because the sediment particles will protrude through the viscous sublayer. The roughness Reynolds number, R e* can be used to define whether a channel is sm ooth, rough, or in transition (Garcia 2007 ) : PAGE 52 29 Equation 6 If hydraulically smooth bottom If bottom in transition If hydraulically rough bottom For rough channels, the logarithmic velocity profile is predicted by: Equation 7 The equation for depth averaged velocity is derived by integration of Equation 7 : Equation 8 After integration, Equation 8 becomes: Equation 9 A resistance equation can be developed for the bed shear stress using the shear velocity and the above equation for depth averaged velocity : PAGE 53 30 Equation 10 Where C f is the friction coefficient which is a function of the water depth and the effective grain roughness height : Equation 11 coefficients: Equation 12 Equation 13 Where C z = roughness coefficient and n = PAGE 54 31 Most authors (Garcia, 2007; Chang, 1988 Simons and Sent rk 19 92 ) present equations with the assumption of a wide channel where the hydraulic radius is nearly equal to the maximum channel d epth: Equation 14 Where R = hydraulic radius, A = cross sectional area of the wetted channel, P = wetted perimeter of the cross section, and D = flow depth. The relationship between channel width, depth, and hydraulic radius for trapezoidal channels is presented in Figure 8 This relationship indicates tha t a trapezoidal channel (with a 2:1 side slope) needs to have a width depth ratio greater than 14 before the hydraulic radius is within 20 percent of the maximum depth. The hydraulic radius is within 10 percent of the maximum depth when the width depth ra tio is at greater than 26 PAGE 55 32 Figure 8 Relationship between channel width, depth, and hydraulic radius for trapezoidal channels. Equation 1 was developed under the assumption of one dimensional, steady flow through a wide channel. The lateral distribution of shear stress across the channel is not uniform (Chang, 1988) especiall y for channel s with a width depth ratio less than 14 The distribution of boundary shear stress in straight trapezoidal channels was studied by Olsen and Florey (1952) using membrane analogy ( Figure 9 ). These results were reported again by Chang (1988) and Simons and Sentrk (1992). The results represented in Figure 9 are unusual because they indicate that boundary shear stress is zero at the bank toe of a trapezoidal channel. This work was expanded by the Highway PAGE 56 33 Research Board (1970), which estimate s the maximum boundary shear stress on the channel bed and banks relative to the average shear stress ( ) These results are presented in Figure 10 and show that bank shear stress increases as the channel becomes narrower The results indicate that the bed shear stress reaches a maximum when the bottom width to depth ratio is between 2 and 3. According to the figure, b oth bank and bed shear s tress increase as the side slopes become less steep. Figure 9 Example d istributions of boundary shear stress in a trapezoidal channel with a 2:1 side slope using membrane analogy (Olsen and Florey, 1952). PAGE 57 34 Figure 10 Variation in relative bed ( b ) and bank ( s ) shear stress with the channel bottom width to depth ratio (b/D). The flow in curved channels is under the influence of centrifugal acceleration, which produces secondary currents and super elevation in water surface (Chang, 1988). Secondary currents (spiral motion) are due to the difference in centrifugal acceleration ( u 2 / r ; where u is the local longitudinal velocity and r is the radius of curvature ) along the vertical velocity profile ( Figure 11 ). Flow through a curved channel tends to have vertical velocity profile with greatest velocity near the water surface. Streamwise velocity tends to be greater near the outside of the bend. Cross stream velocity tends PAGE 58 35 to be toward the outside bank in the upper portion of the cross section and toward the inside bank in the lower portion of the cross section. Figure 11 Definition sketch for flow in a curved channel (Chang, 1988). Chang (1988) provides an equation for the radial shear stress ( r ) in fully developed transverse flow through a curve: Equation 15 Where f = friction factor PAGE 59 36 Secondary currents grow in strength upon entering a channel curve (Chang, 1988). In a prismatic curved channel of sufficient length, the secondary currents will reach an equilibrium conditions and are considered to be fully developed. Because of the vari able radius of curvature in natural meandering channels, secondary currents grow and decay with longitudinal channel length (Chang, 1988). These secondary currents modify the boundary shear stress in curved channels. Measured shear stress distributions i n curved trapezoidal channels (both smooth and rough) are provided by Ippen and Drinker (1962) in Figure 12 In these experimental examples, shear stress is initially highest along the inside of the curve ( near the bank toe) and then the zone of highest shear stress transitions to the outside of t he curve ( near the bank toe) just downstream from where the curve ends and transitions again to a straight channel. PAGE 60 37 Figure 12 Boundary shear stress distributions in curved trapezoidal channels measured by Ippen and Drinker (1962). Ippen and Drinker (1962) concluded that the boundary shear stress patterns they obtain ed s more than twice the mean shear stress for a straight channel. The distribution and relative PAGE 61 38 magnitudes of shear stress appear to be functions of the stream geometry. The locations of local maximum s of shear stress were found near the inside bank in the curve and near the outside bank downstream from the end of the curve. Relatively high shear stress was found to persist for a considerable distance downstream from the curve. Guidelines for the protection of stream banks fr om maximum hydraulic conditions are based on distributions of velocity and boundary shear stresses. The Federal Highway Administration (200 5 ) use d the research by Young et al. (1996), who proposed a ratio of the maximum channel bend shear stress to the sh ear stress at the beginning of the curve: Equation 16 Where K b = Ratio of channel bend shear stress to approach shear stress max = Maximum channel bend shear stress o = Approach shear stress The Federal Highway Administration (200 5 ) describes K b as a piecewise mathematical function: PAGE 62 39 Equation 17 Where R c = Channel radius of curvature W = Wetted channel top width Three Dimensional Hydraulic Model The Navier Stokes equations, named after Claude Louis Navier and George Gabriel Stokes, describe the motion of fluid. These equations were derived by applying Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term, which is proportional to the velocity gradient, and a pressure term The Navier Stokes equations dictate not position but rather velocity. The Navier Stokes equations describe the velocity of the fluid at a given point in space and time. The equations are presented below in Cartesian coordinates for momentum : PAGE 63 40 Equation 18 Equation 19 Equation 20 The equation for conservation of mass is given by: Equation 21 When the flow is incompressible, the fluid density, does not change and Therefore, the continuity equation is reduced to: Equation 22 PAGE 64 41 Where u v and w are the fluid velocities in the x y and z directions. These four nonlinear partial differential equations have to be solved numerically for most practical problems, such as flow through a river channel The three dim ensional numerical model, U 2 RANS was developed to predict the turbulent flows through a wide variety of open channels ( Lai et al. 2003a) The U 2 RANS model was used for the research described in this dissertation to simulate flow through straight and meandering river channels. U 2 RANS is an implicit finite volume model that has been proven to provide accurate results and has been successfully applied to a wide variety of practical problems (Lai et al. 2003b) Applying a cell centered storage scheme t o discretized finite volumes, the model solves the three dimensional Reynolds averaged Navier Stokes equations for the distributions of flow velocity, static pressure, and shear stress each mesh cell. The Reynolds averaged Navier Stokes equations are solv model. This model was developed without making the assumption of a hydrostatic pressure distribut ion in the governing equations. Governing Equations The three dimensional equations for the conservation of mass and momentum in the U 2 RANS model (Lai, et al., 2003a) are written in tensor form: PAGE 65 42 Equation 23 Equation 24 where and are the mean and fluctuating velocity vectors, P = mean static pressure, g = acceleration of gravity, = fluid density, and = dynamic fluid viscosity. Lai, et al. (2003 a ) used the standard k turbulence model (Launder and Sp alding 1974) for the Reynolds stress tensor Th e k turbulence model relates the Reynolds stresses to the mean strain rate through an eddy viscosity as described in Equation 25 Equation 25 T I is the unit tensor. In this model, the turbulent eddy viscosity is obtained from the following equation Equation 26 PAGE 66 43 where k is the turbulence kinetic energy per unit mass and is the turbulence dissipation r ate. The transport equations for k and for non buoyant flows are expressed as Equation 27 and Equation 28 where the generation of k The standard turbulence model coefficients are given below: Equation 29 Wormleaton et al. (2006) state that the k model described by Launder and Spalding known, commonly used and fastest 3 D turbulence This k model assumes turbulent isotropy and, therefore, cannot model secondary currents in straight channels that are drive n by turbulent anisotropy. Large eddy simulation models are needed to simulate turbulent anisotropy. However, the secondary currents resulting from turbulent stress anisotropy are relatively small. PAGE 67 44 In meandering channels, dominate secondary currents are generated by centrifugal pressure variation through the curve. Therefore, the standard k model is expected to perform better for meandering channels than for a straight channel. Demuren (1993) and Wilson et al. (2003) successfully applied standard k models to flow through a meandering river with and found good agreement with observed laboratory data for both primary and secondary velocities. Therefore, the U 2 RANS model is capable of simulating secondary flow through a curved channel, but not sec ondary flow through a straight channel caused by turbulent anisotropy. Discretization The governing equations were discretized by Lai, et al. (2003 a ) using the finite volume approach, following the work of Lai (1997 and 2000). The solution domain is def ined by a number of mesh cells with all dependent variables stored at the cell geometric centers. The governing equations are integrated over cells using the Gauss theorem. The shapes of cells and cell faces must be uniquely defined before discretization can be carried out. The polyhedron of each mesh cell consists of a number of enclosing faces. The shape of a polyhedron is uniquely defined if each face is defined. Geometric PAGE 68 45 quantities (e.g., cell volume, face area, and face unit normal vector) are ca lculated for each cell. Lai, et al. (2003 a ) derived and utilized the following discretized governing equation for element P (center of mesh cell), which combines the diffusion and convection terms : Equation 30 P nb to all neighbor elements associated with the mesh cell element P any dependent variable. The terms of the abo ve equation are defined below: Equation 31 Equation 32 Equation 33 PAGE 69 46 Equation 34 Equation 35 where diffusivity across the cell face, fluid density across the cell face, fluid velocity across the cell face, cell face area, d d the values at the edge center, indicates the summation over all cell faces, and indicates the summation o ver all edges of the cell face. The terms D n D c edge C edge are defined below: Equation 36 PAGE 70 47 Equation 37 Equation 38 where and the vectors from the center of each adjacent cell to the center of the adjoining cell face, cell face unit normal vector, the distance vector along the edge of the cell face. Boundary Conditions The 3D model of Lai, et al. (200 3a ) is able to accommodate the following boundary conditions: inlet, outlet, no slip wall, plane of symmetry, and free surface. At the inlet and at no slip wall boundaries, 3D Cartesian velocity components are specified at the boundary cell face centers. Pressure is extrapolated from the interior to the boundary using either first or second order extrapolations in order to solve the momentum equations. The mass fluxes on these boundar ies are specified and remain unchanged during the model simulation so no pressure boundary condition is required for the solution of the pressure correction equation. The turbulence quantities, k and are specified at the inlet s. PAGE 71 48 Th e standard wall function approach by Launder and Spalding (1974) is used at a solid wall. For this approach, l et z p denote the normal distance between the wall boundary and the center point, p of the mesh cell adjacent to the wall. k p and p are the tur bulence kinetic energy and dissipation rate at the cell center. The turbulence kinetic energy is assumed to be zero at the wall ( k w = 0). The shear stress is computed iteratively using the following logarithmic law : Equation 39 Where: the kinematic viscosity B is the roughness coefficient, which is zero in the case of a smooth wall. For a rough wall, the roughness coefficient is governed by the equation: E quation 40 PAGE 72 49 Where k s is the effective roughness height of the rough wall. Turbulence values k and can be computed from the following equations: 8 At an outlet, pressure is specified at the cell face center, while Cartesian velocity components and turbulence quantities are extrapolated from the interior to the outlet boundary using a second order extrapolation. For the pressure correction equation, the pressure increment is set to zero at the outlet, as the pressure should not change during the model simulation At a plane of symmetry, the velocity component normal to the pla ne is set to zero, while the tangential components, pressure, and turbulence quantities are extrapolated from the interior using the first or second order extrapolations. For the pressure correction equation, no pressure condition is needed at the symmet ry as zero mass flux is enforced. At a free surface, a rigid lid free slip condition is assumed (Lai, et al., 2003a) For many steady state applications in natural rivers, the flow is subcritical (Froude number is less PAGE 73 50 than one) and changes in free sur face elevation are relatively small. For such problems, the rigid lid approximation is a good first order approach. With the rigid lid free slip method, the free surface elevation is either known before the computation, imbedded within the grid, or a fla t surface is assumed. For the research in this dissertation, the SRH 2D, two dimensional hydraulic model (Lai, 2006) was used to compute the water surface elevations as the rigid lid free surface in the U 2 RANS three dimensional model. The depth averaged SRH 2D model was developed by Lai (2006) from the U 2 RANS model so the two models have a consistent formulation. A bankfull discharge was assumed for the upstream boundary and normal depth was assumed for the downstream boundary water surface in the two dimensional model. Steady state conditions and fixed channel geometry were assumed for each two and three dimensional model simulation. Model V er i fic ation The U 2 RANS model was model was verified by comparing simulations with measurements from three separate physical models: University of Iowa (Yen, 1965) Massachusetts Institute of Technology (Ippen and Drinker, 1962) PAGE 74 51 Colorado State University (Heintz, 2002 and Schmidt, 2005) University of Iowa Physical Model Lai, et al. (2003a) v erifi ed the U 2 RANS model by comparing simulat ed water depth and velocity profiles with measurements from a physical model at the University of Iowa reported by Yen (1965). These simulations were repeated for this dissertation to also The physical model channel consists of two identical 90 bends of opposite direction connected by a short straight channel ( Figure 13 ). The upstream and downstream most portions of channel also have short, straight alignments. The channel of the physical model has smooth wall s and a constant trapezoidal cross section with 1H:1V side slopes. The channel dimensions and hydraulic properties are presented in Table 1 Model s imulations were performed using coarse, medium, and fine meshes: 45 x 24 x 12 90 x 48 x 24 180 x 96 x 48 PAGE 75 52 Figure 13 Application of the three dimensional, U 2 RANS model, (modified from Lai et al., 2003a) to the physical model experiment by Yen (1965). Table 1 Channel dimensions and hydraulic properties of the physical model reported by Yen (1965). Channel bottom width, B 1.829 m Average channel depth 0.156 m Average wetted top width 2.141 m Width depth ratio 13.7 Upstream straight reach length 2.134 m Upstream curve radius at centerline 8.535 m Middle straight reach length 4.267 m Downstream curve radius at centerline 8.535 m Downstream straight reach length 2.134 m Longitudinal channel slope 0.00072 Mean velocity 0.692 m/s Froude number 0.58 Reynolds number 9 x 10 4 PAGE 76 53 The longitudinal slope of the physical model slope was mild enough that the slope was neglected and a flat rigid lid was assumed for the water surface. The 3D, U 2 RANS model simulates variations in water surface elevation from the pressure head at the rigid lid. The super elevation of s imulated water depth s (using coarse, medium, and fine meshes ) were compared with depths measured at four channel cross sections located d ownstream from the upstream channel curve ( Figure 14 ). Simulated water depths from all three mesh sizes compared well with the measured depths. Figure 14 Comparison of U2RANS model depth results; using coarse, medium, and fine meshes; with physical model data (modified from Lai et al., 2003a). PAGE 77 54 For this dissertation, t he model simulations of Lai, et al. (2003a) were repe ated to simulate vertical velocity profiles and to boundary shear stress. Simulated velocity profiles for coarse, medium, and fine meshes were compared with velocity profiles measured at the downstream end of th e first (upstream) channel curve. Comparisons of measured and simulated velocity profiles at cross section S0 ( Figure 13 ), are presented in Figure 15 through Figure 19 For the fine mesh, the overall mean error between measured and simulated velocities was +1.0% (+0.009). The mean error at individual velocity profiles ranged from 10.2% to +6.9% to ( 0. 091 to +0.071). For the fine mesh, the overall RMS error between measured and simulated velocities was 6.8% (0.067). The RMS error at individual velocity profiles ranged from 7.3% to 10.6% (0.076 to 0.094). PAGE 78 55 Figure 15 Velocity profile at 0.461 B. Figure 16 Velocity profile at S0, 0.307 B. Figure 17 Velocity profile at 0.000 B. PAGE 79 56 Figure 18 Velocity profile at +0.307 B. Figure 19 Velocity profile at +0.461 B. Simulated velocity profiles from all three mesh sizes compared well with the measured profiles, but the simulated velocities from the finest model mesh matched best near the channel bottom. In addition to the simulation of velocity, the 3D model also simulates the boundary shear stress ( Figure 20 ). The plan view of the physical model alignment and measurement locations are presented in Figure 21 PAGE 80 57 Figure 20 Simulated boundary shear stress is presented for the case of a fine mesh. PAGE 81 58 Figure 21 Plan view alignment is presented for the physical model by Yen (1965). Velocity profile measurements are reported at the cross section lab eled S0. Boundary The model simulation s indicate that boundary shear stress is initially greatest along the inside bank of the first ( upstream ) channel curve a nd then transfers to the outside bank at the downstream end of the first curve Shear stress is at a maxium along the inside bank near the upstream end of the second channel curve. Measured and simulated shear stress were compared at four channel cross sections identified in Figure 21 Figure 22 through Figure 25 ). For the PAGE 82 59 fine mesh, the overall mean error between measured and simulat ed shear stress was 2.6% ( 0.00010). The mean error at individual cross sections ranged from 1.3% to 3.9% ( 0.00005 to 0.00016). For the fine mesh, the overall RMS error between measured and simulated shear stress was 7.0% (0.00026). The RMS error a t individual velocity profiles ranged from 5.1% to 11.2% (0.00021 to 0.00045). Figure 22 Comparison of simulated and measured shear stress is presented for cross section CIIO. The channel cross section is also plotted along with the location of high velocity (within 90 percent of the maximum velocity). PAGE 83 60 Figure 23 Comparison of simulated and measured shear stress is presented for cross with the location of high velocity (within 90 percent of the maximum velocity). PAGE 84 61 Figure 24 Comparison of simulated and measured shear stress is presented for cross th the location of high velocity (within 90 percent of the maximum velocity). PAGE 85 62 Figure 25 Comparison of simulated and measured shear stress is presented for cross h the location of high velocity (within 90 percent of the maximum velocity). There was no apparent local decrease in the magnitude of measured shear stress at the bottom corners of the trapezoidal cross section. The 3D model is not able to provide accurate shear stress results at the corners of the trapezoid so simulated shear stress results were not used at or nea r the corner mesh cells. Otherwise, the simulated shear stress matched measured shear stress quite well. Simulated shear stress was not sensitive to mesh density along the channel bottom. However, simulated shear stress on the channel banks did sometime s vary with mesh density. PAGE 86 63 Massachusetts Institute of Technology Physical Model Shear stress results from the physical model at the Massachusetts Institute of Technology were reported by Ippen and Drinker (1962) in the form of contour plots. This physical model consisted of a trapezoidal cross section (2H:1V side slopes) with one 60 degree bend. The width depth ratio was 12 and the Froude number was 0.53. The 3D model was used to simulate th is case for a qualitative comparison ( Figure 26 ). Again, the longitudinal slope of this physical model slope was mild enough that the slope was neglec ted and a flat rigid lid was assumed for the water surface. No information was available from the journal article on the density of shear stress measurements that were used to develop the contour plots. The simulated shear stress contours match the pat terns of contours from the measurements reasonably well. PAGE 87 64 Figure 26 Comparison of measured and simulated boundary shear stress from the physical model reported by Ippen and Drinker (1962). Colorado State University Physical Model For this dissertation, the U 2 RANS model (Lai, et al., 2003 a ) was v erifi ed again by comparing simulated velocity profiles with measurements from the more complex physical model at Colorado State University (Heintz, 2002 an d Schmidt, 2005). This physical model consist s of a large laboratory flume with two channel bends ( Figure 27 Figure 28 and Table 2 ) The channel cross section is trapezoidal with 3H:1V side slopes. The re are two separate channel bends of constant curvature separated by a short, Relative Shear Stress PAGE 88 65 straight reach that constricts the downstream channel width (at a 10 :1 ratio over a distance of 6.355 m ) to 74 percent of the upstream width. The channel width and radius of curvature are constant through each channel bend, but they are different be tween the two channel bends. The channel width to depth ratios r ange from 12 to 20. Normal depth was computed for the trapezoidal geometry of both channel bends for a discharge of 0.566 m 3 /s (Table 2). Figure 27 Photograph of the large physical model of flow through two channel bends at Colorado State University. PAGE 89 66 Figure 28 Colorado State University model plan view configuration for flow through two channel bends of constant radius. Table 2 Summary of Physical Model dimensions (Heintz, 2002). Bend Type Top Width (m) Bottom Width (m) Radius of Curvature (m) Bend Angle (degrees) Relative Curvature (R/W) Channel Length (M) 1 5.843 3.100 11.811 125 2.02 25.8 3 4.572 1.829 20.065 73 4.39 25.5 Table 3 Physical model hydraulic data. Channel Curve Dimensions (m) Upstream Curve Downstream Curve Flow depth (m) 0.255 0.325 Wetted cross sectional area (m 2 ) 0.985 0.912 Wetted perimeter (m) 4.712 3.886 Manning's n roughness 0.018 0.018 Normal Discharge (m 3 /s) 0.566 0.566 Flow velocities in the physical model were measured with an acoustic doppler velocity meter. The plan view of the laboratory physical model, along with the velocity profile PAGE 90 67 measurement locations, is presented in Figure 29 Measurement cross sections are numbered from 1, at the upstream end, to 18, at the downstream end. Velocity profiles were measured at seven locations along each cross section and those mea surement bank side slope. bank e channel. bank bank side slope. PAGE 91 68 Figure 29 Plan view location of cross section velocity measurements is shown from the upstream most cross section 1 to the downstream most cross section 18. The naming convention of the vertical velocity measurements at each cross section is a b c d e f and g left to right looking downstream. At cross section 1, measured velocities were highest on the right side (inside of first channel bend). By cross section 4 measured velocities were more uniform. By cross section 8 (downstream end of channel bend), measured velocity was greatest on the left side of the channel (outside bend) At cross section 10 (downstream end of straight transition), flow velocity was still greatest on the left side of the channel. By cross section 1 8 (downstream end of the secon d channel bend), flow velocity was greatest on the right side (outside of the 2 nd channel bend). PAGE 92 69 The longitudinal slope of this physical model was steep enough and the channel length long enough, that the water surface needed to be simulated. Therefore, c hannel hydraulics w ere first simulated with the SRH 2D model (Lai, 2006). The simulated water surface elevation s from the two dimensional model (including superelevation along curves) w ere used to define the rigid lid elevations of 3D model grid. The 2 D model mesh was structured with 326 cells in the longitudinal direction and 28 cells in the cross stream direction for a total of 9,128 mesh cells ( Figure 30 ). For the cross stream direction, 20 cells represented the channel bottom, 4 cells represented the left side slope, and 4 cells represented the right side slope ( Figure 31 ). The numerical model channel was mathematically extended (at the same slope) for the equivalent of ten channel widths upstream and downstream of the physical model boundaries so that the nu merical model boundaries would not influence the solution domain representing the physical model. The upstream numerical model boundary was specified by a constant discharge rate and the assumption of uniform velocity distribution parallel to the channel. The downstream boundary was specified as a level water surface elevation corresponding to normal depth PAGE 93 70 Figure 30 The two dimensional numerical model has a structured mesh with 326 cells in the downstream direction and 28 cells in the cross stream direction. PAGE 94 71 Figure 31 A close up view of the two dimensional numerical model mesh (326 x 28 ) is shown through the upstream channel bend. The SRH 2D numerical model was calibrated by adjusting the water surface elevation at the downstream boundary, and the channel roughness, so that the simulated water surface profile match ed the measurements fr om the physical model ( Figure 32 ). The calibrated, downstream boundary, water surface elevation was 29.868 m. This elevation corresponds to a flow depth of 0.32 m, which was maintained through the PAGE 95 72 entire downstream channel bend. The channel roughness was calibrated to 0.0186, which is consistent with a smooth concrete channel bottom. Figure 32 The centerline water surface profiles from the numerical and physical model s are compared over the reach of the physical model. The 3D model mesh like the 2D model mesh, extends at the same slope for a distance of ten channel widths both upstream and downstream from t he physical model boundaries ( Figure 33 ). The top surface of the 3D mesh corresponds to the simulated water surface from the 2D model. A uniform velocity distribution (laterally and vertically) was specified at the upstream model boundary. PAGE 96 73 Figure 33 The numerical model mesh boundary (meters) extends ten channel widths both upstream and downstream from the physical model boundaries. Both the 2D and 3D models are very robust and compute reasonable results for correct input data. However, simulated velocity profiles of the U 2 RANS model can be sensitive to the mesh size. The refore, the sensitivity of the 3D numerical model mesh density was tested using four different mesh sizes. The number of mesh points along the downstream ( I ) direction, cross stream ( J ) direction, and vertical ( K ) direction are listed below for each of the four mesh sizes: Physical Model (m) PAGE 97 74 1. 327 x 21 x 16 for a total of 109,872 mesh cells 2. 492 x 21 x 16 for a total of 1 65 31 2 mesh cells 3. 492 x 32 x 24 for a total of 377 ,8 56 mesh cells 4. 492 x 42 x 32 for a total of 661 248 mesh cells At least one mesh dimension was kept the same between two consecutive meshes. A close up plan view of the 492 x 42 x 32 mesh size is presented in Figure 34 and a cross section view is presented in Figure 35 The cross sectional mesh lines in the K direction are vertical at the channel center and then gradually transition to match the channel bank boundar ies o f the left and right side slopes. PAGE 98 75 Figure 34 A plan view close up is presented of the three dimensional structured mesh (492 x 42 x 32) along the straight transition reach. PAGE 99 76 Figure 35 The right, cross sectional half of the three dimensional structured mesh (492 x 42 x 32) is presented at cross section 6 through the upstream bend. The numerical model velocity results (U, V, W) at each mesh cell were imported into the Tecplot 10 graphi cs program. The measured velocity data (U, V, W) at each cross section were also imported into the Tecplot 10 graphics program. This includes between five and nine separate velocity measurements at each vertical profile. The Tecplot 10 graphics program was then used to plot velocity vectors at all vertical points at each measurement location and interpolate simulated velocity vectors at these same measurement locations from the numerical model results. The measured (red) and simulat ed (blue) velocity ve ctors were then directly compared in plan view plots. An overview comparison of these velocity vectors (based on the densest mesh) is presented in Figure 36 Close up views of the velocity vector comparisons are presented in Figure 37 through Figure 40 PAGE 100 77 Near the upstream boundary of the physical model at cross section 1, measured velocity vectors are greater along both the left and right channel banks than at the channel center ( Figure 37 ). By cross section 6 the greatest velocity magnitude was at the center measurement location. Figure 36 Simulated (blue) and measured (red) velocity vectors are compared in plan view for the entire region of the physical model. PAGE 101 78 Figure 37 Simulated (blue) and measured (red) velocity vectors are compared in a close up plan view of the first (upstream) channel bend. 1 5 PAGE 102 79 Figure 38 Simulated (blue) and measured (red) velocity vectors are compared in a close up plan view of the downstream portion of the first channel bend and straight transition reach. 8 5 10 PAGE 103 80 Figure 39 Simulated (blue) and measured (red) velocity vectors are compared in a close up plan view of the upstream portion of the second channel bend. 10 14 PAGE 104 81 Figure 40 Simulated (blue) and measured (red) velocity vectors are compared in a close up plan view of the downstream portion of the second channel bend. The simulated velocity results, from the four different model mesh sizes, are similar. T he stream wise velocity profile s simulated by the numerical model, under the four different mesh sizes are plotted and compared with the measured velocity profiles from the physical model at cross sections 6, 10, and 16 ( Figure 41 Figure 42 and Figure 43 ). 14 16 18 PAGE 105 82 Figure 41 Comparisons of seven measured and simulated stream wise velocity profiles using four different mesh sizes at cross section 6 in the upstream channel downstream). PAGE 106 83 Figure 41 (cont inued ). Comparisons of seven measured and simulated stream wise velocity profiles using four different mesh sizes at cross section 6 in the upstream looking do wnstream). PAGE 107 84 Figure 42 Comparisons of seven measured and simulated stream wise velocity profiles using four different mesh sizes at cross section 10 at the downstream end of the straight transition reach d left to right looking downstream) PAGE 108 85 Figure 42 (continued). Comparisons of seven measured and simulated stream wise velocity profiles using four different mesh sizes at cross section 10 at the downstream position ed left to right looking downstream). PAGE 109 86 Figure 43 Comparisons of seven measured and simulated stream wise velocity profiles using four different mesh sizes at cross section 16 in the downstream channel downstream). PAGE 110 87 Figure 43 (continued). Comparisons of seven measured and simulated stream wise velocity profiles using four different mesh sizes at cross section 16 in the downstream lookin g downstream). PAGE 111 88 In addition, the cross stream velocity profiles are plotted and compared with the measured cross stream velocity profiles from the physical model at cross sections 6, 10, and 16 ( Figure 44 Figure 45 and Figure 46 ). A summary comparison of measured and simulated stream wise and cross stream velocities at cross section 6 is presented in Table 4 for the densest model mesh size. Similar summary comparisons are presented for cross section 10 in Table 5 and for cross section 16 in Table 6 PAGE 112 89 Figure 44 Comparison of measured and simulated cross stream velocity profiles at cross section 6 for a range of three dimensional model mesh sizes. Mesh size: 327 x 21 x 16 Mesh size: 492 x 21 x 16 Mesh size: 492 x 3 2 x 24 Mesh size: 492 x 42 x 32 3 2 34 PAGE 113 90 Figure 45 Comparison of measured and simulated cross stream velocity profiles at cross section 10 for a range of three dimensional model mesh sizes. Mesh size: 327 x 21 x 16 Mesh size: 492 x 21 x 16 Mesh size: 492 x 3 2 x 24 Mesh size: 492 x 4 2 x 32 PAGE 114 91 Figure 46 Comparison of measured and simulated cross stream velocity profiles at cross section 16 for a range of three dimensional model mesh sizes. Mesh size: 327 x 21 x 16 Mesh size: 492 x 21 x 16 Mesh size: 492 x 3 2 x 24 Mesh size: 492 x 4 2 x 32 PAGE 115 92 Table 4 Summary comparison of measured and model results at cross section 6 for a model mesh size of 492 x 42 x 32. Measure ment Location Average Measured Lateral Velocity, VL/Um Average Simulated Lateral Velocity, VL/Um Ratio of simulated to measured Average Measured Down stream Velocity, VD/Um Average Simulated Down stream Velocity, VD/Um Ratio of simulated to measured a 0.052 0.052 0.99 0.870 0.722 0.83 b 0.061 0.043 0.71 0.909 0.847 0.93 c 0.079 0.034 0.43 0.931 0.908 0.97 d 0.105 0.014 0.14 0.983 1.059 1.08 e 0.114 0.053 0.47 0.879 0.794 0.90 f 0.084 0.056 0.66 0.772 0.696 0.90 g 0.064 0.031 0.48 0.616 0.507 0.82 Table 5 Summary comparison of measured and model results at cross section 10 for a model mesh size of 492 x 42 x 32. Measure ment Location Average Measured Cross stream Velocity, V L /U m Average Simulated Cross stream Velocity, V L /U m Ratio of simulated to measu red Average Measured Stream wise Velocity, V D /U m Simulated Stream wise Velocity, V D /U m Ratio of simulated to measured a 0.048 0.049 1.02 0.919 0.747 0.81 b 0.056 0.052 0.93 1.016 0.873 0.86 c 0.068 0.051 0.75 1.022 0.962 0.94 d 0.110 0.011 0.10 0.911 0.965 1.06 e 0.128 0.039 0.31 0.804 0.827 1.03 f 0.135 0.040 0.30 0.775 0.791 1.02 g 0.123 0.032 0.26 0.665 0.687 1.03 PAGE 116 93 Table 6 Summary comparison of measured and model results at cross section 16 for a model mesh size of 492 x 42 x 32. Measure ment Location Average Measured Cross stream Velocity, V L /U m Average Simulated Cross stream Velocity, V L /U m Ratio of simulated to measu red Average Measured Stream wise Velocity, V D /U m Average Simulated Stream wise Velocity, V D /U m Ratio of simulated to measured a 0.033 0.019 0.57 0.568 0.607 1.07 b 0.045 0.024 0.53 0.757 0.854 1.13 c 0.051 0.028 0.55 0.897 0.909 1.01 d 0.055 0.005 0.10 1.029 1.060 1.03 e 0.063 0.022 0.34 0.983 0.992 1.01 f 0.073 0.035 0.47 0.923 0.984 1.07 g 0.079 0.044 0.55 0.807 0.935 1.16 Simulated and measured relative velocity magnitudes are compared longitudinally in Figure 47 PAGE 117 94 Figure 47 Comparison of depth averaged measured and simulated stream wise velocity along the channel for a model mesh size of 492 x 42 x 32. Measurement Measurement The upstream boundary of the physical model is at the start of channel curvature for the upstream bend. The measured stream wise velocities at cross section 1 are greater along both channel banks than the middle of the channel ( Figure 37 ). However, the numerical model has a straight channel for ten channel widths upstream from the first channel bend. This straight channel reach allows the simulated velocities to be greatest near the channel center. PAGE 118 95 A nother model simulation was performed where the upstream boundary of 3D model was moved to coincide with the upstream boundary of the physical model. The upstream boundary velocity distribution was specified to match the patt ern of measured velocities at cross section 1 ( the upstream most measurement section ) However, this model simulation did not significantly improve the comparison with measured velocities and was not pursued father At cross section 6, the measured and simulated stream wise velocity profiles agree reasonably well at measurement locations b c d e and f ( Figure 41 ). However, the agreement is not as good at measurement locations a and g on the shallow portions of the left and right bank side slopes ( Table 4 ). The measured and simulated cross stream velocity profiles are generally in the same direction, but the average of the simulated velocity magnitudes at the measure ment locations tend to be lowe r than the measurements ( Figure 44 ). At cross section 10 (downstream end of the straight transition reach), the measured and simulated stream wise velocity profiles agree well at measurement locations b through g ( Figure 42 and Table 5 ). The patterns of the measured and simulated cross stream velocity profiles match best on the right side of the channel while there is poor agreement on the left side and channel center ( Figure 45 ). The numerical model predicts that the cross stream velocities at the left and right banks are toward the channel center, which is consistent with a constricting channel. However, the measured PAGE 119 96 cross stream velocities are generally toward the left side of the channel, indicating the continuing influence of the secondary currents from the upstream channel bend. At cross section 16 (near the dow nstream end of the second channel bend), the measured and simulated stream w ise velocity profiles agree well at measurement locations b through g ( Figure 43 and Table 6 ). The patterns of the measured and simulated cross stream velocity profi les agree well, but their magnitudes still differ ( Figure 46 ). The patterns of measured and simulated streamwise velocities compare reasonably well along the toe of both channel banks and along the channel center ( Figure 47 ). Along the left (outside) bank of the upstream bend (measurement location c ), the measured stream wise velocities are fairly constant from cross sections 1 through 8. However, t he simulated stream wise velocities increase from cross sections 1 through 5 and then remain nearly con stant through cross section 8. T he measured depth averaged stream wise velocities along the right (inside curve ) bank of the channel bend (measurement location e ) are initially greater in magnitude than the left (outside curve) bank and then decrease in m agnitude through cross sections 8 Simulated velocities along the right bank tend are lower than along the left bank and channel center. The simulated right bank velocities match the magnitude of the measurements reasonably well from cross sections 7 thr ough 18. PAGE 120 97 The stream wise velocities remain greater along the left bank than the right bank through the straight transition reach and through most of the second channel bend even though the curvature is reversed. Both the measured and simulated stream wise velocities along the right bank (outside curve) increase through the second channel bend, indicating that the strength of the secondary currents is still increasing. Boundary shear stress was also simulated by the 3D model ( Figure 48 ). The zone of highest shear stress was initially along the inside bend (right bank) of the upstream curve. This zone of highest shear then transitions to the left bank through the straight transition reach. The maximum shear stress was located near the left bank at the beginning of downstream curve. The zone of hightest shear stress continues along the inside bend (left bank) through the downstream curve and then transitions to the right bank at the end of the curve. Based on verifications by the model developer and the Ph.D. student, the two and three dimensional numerical models (SHR 2D and U 2 RANS) are considered suitable for predicting the location and ma gnitude of increase bank velocity through channel bends relative to straight reaches. The 3D numerical model simulations of velocity and shear stress compare well with physical model measurements, especially in the stream wise direction. For the cross st ream velocity profiles, the numerical model results match the patterns of the measured profiles. The 3D model is capable of simulating the secondary currents in a pattern similar to the physical model measurements. The locations of PAGE 121 98 maximum bank velocity from the 3D model simulations match the measurement locations through the two channel bends of the physical model. Figure 48 Simulated dimensionless boundary shear stress for the physical model at Colorado State University. PAGE 122 99 CHAPTER III NUMERICAL MODEL SIMULATIONS A model simulation matrix was created to include a wide range of virtual meandering river channels of the type found in nature : Discharge, longitudinal slope, width, and depth The regime channel geometry graph presented by Chang (1988) was used to define the range of meandering river channels ( Figure 49 ) to be simulated by the 3D model. The graph defines three region s in the slope discharge plane that covers a wide range of discharge and slope conditions over a few orders of magnitude. The slope axis of the graph also includes the median sediment particle grain size (d 50 ) Regions 1, 2, and 3 of the graph are bounded between threshold lines. The lowest threshold line corresponds to the critical slope for incipient motion of the median sediment particle PAGE 123 100 Figure 49 The graph of the relationship between bankfull discharge, channel slope, and median sediment grain size includes three distinct regions (Chang, 1988) Solid contour lines indicate the channel surface width while dashed contou r lines denote the bankful channel depth. Regime Channel Geometry Regions Model simulations sets were defined in region s 1 and 3 but not in region 2 Region 1 includes channels with gentle longitudinal slopes, low velocity, small bed material loads, and flow resistance in the lower regime of ripples and dunes (Chang, 1988). The channels are typically meandering and have width depth ratios generally in the range of 4 to 20. The channel surface width in region 1 varies with discharge, slope, and grain size (Chang, 1988) : Region 1 Region 2 Region 3 PAGE 124 101 Equation 41 where B is the surface channel width (m) d = median particle grain size (mm) S = longitudinal channel slope S c = critical slope for incipient motion of the median sediment particle size Q = bankfull river discharge (m 3 /s) The channel center depth in region 1 also varies with discharge, slope, and grain size (Chang, 1988) : Equation 42 where D is the center channel depth (m) at the bankfull discharge. Region 2 is the narrowest on the graph ( Figure 49 ) River channels in this region typically have a large width depth ratio and are often braided and straight. Therefore, no simulations of meandering channels were defined in region 2. PAGE 125 102 Region 3 includes rivers that may be braided, but may also include sinuous point bar rivers with riffle s and pools. The channel surface width and centerline depth in region 3 are a function of discharge, slope, and grain size (Chang, 1988) : Equation 43 Equation 44 Range of Numerical Model Simulations A model simulation strategy was developed to determine the mesh density, length of the channel entrance and exit conditions, and the number of meander bends. The two dimension model, SRH 2D (La i, 2006), was used to compute the water surface elevations throughout each virtual channel. These water surface elevation results were then used as the rigid lid surface of the 3D, U 2 RANS model (Lai, 2003a and Lai, 2003b). The 3D model was then used to s imulate the channel hydraulics through the meandering channels 2D and 3D model simulations of each meandering river channel required about 16 hours of computer simulation time. A FORTRAN post p rocessing program was developed for this dissertation to cre ate a series of asci output files, which were imported to a spreadsheet template. Post processing of model results from each simulation required another 1 hour of effort. PAGE 126 103 The graph by Chang (1988) was used to define twelve numerical model simulations sets ( Figure 50 and Table 2 ) that include the following discharge and channel parameters: Bankfull discharges of 2.83 2 m 3 /s, 28.3 2 m 3 /s, 283 .2 m 3 /s, and 2,83 2 m 3 /s. River valley slopes ranging from 7 11 x 10 6 to 2 69 x 10 2 Trapezoidal channel cross section shapes, with 2:1 side slopes, were assumed for each model simulation. Channel width depth ratios ranged from 4.5 to 92 Nine separate meandering c hannel alignments were defined for each simulation set using a sine generated curve with the following channel sinuosities : 1.10, 1.25, 1.50, 1.75, 2.00, 2.25, 2.50, 2.75, and 3.00. PAGE 127 104 Figure 50 Numerical model simulation set s are presented on the graph of bankfull discharge, channel slope, and median sediment grain size. Each simulation set includes nine meandering channels. The first three simulation sets (1, 2, and 3) correspond to the lowest discharge and steep est channel slopes while the last three sets (10, 11, and 12) correspond to the highest discharge and most mild channel slopes. Empirically based equations of maximum shear bank stress are based only on results from simulation sets 4 through 9. Simulation sets 1 to 3 and 10 to 12 are used to validate the equations. PAGE 128 105 Table 7 Simulation set matrix summary. Simulation Set Discharge (m 3 /s) Valley Slope Median Sediment Grain Size (mm) Region 1 2.832 6.820E 04 1 1 2 2.832 6.178E 03 2 1 3 2.832 2.693E 02 4 3 4 28.32 1.490E 04 0.5 1 5 28.32 1.231E 03 1 1 6 28.32 5.885E 03 2 3 7 283.2 3.257E 05 0.25 1 8 283.2 2.454E 04 0.5 1 9 283.2 1.286E 03 1 3 10 2,832 7.115E 06 0.125 1 11 2,832 4.890E 05 0.25 1 12 2,832 2.811E 04 0.5 3 Two thirds of the simulation sets (eights sets) were designed to be within region 1 and one third (four sets) within region 3. For the eight simulation sets within region 1, one half of the sets (four sets) include one meandering river c hannel at the critical slope for incipient motion of sediment particles and a sinuosity of 3.00. The other meandering rivers in the simulation set increase in channel slope as the sinuosity decreases. The other half of the simulation sets in region 1 includes one meandering r iver channel at the upper boundary of the region with a sinuosity of 1.1 T he other meandering rivers in the simulation set decrease in slope as the sinuosity increases. Each of the four simulation sets in region 3 includes one meandering river channel a t the lower boundary of the region with a sinuosity of 3.00 The other meandering rivers in the simulation set increase in slope as the sinuosity decreases. PAGE 129 106 The vertical axis of the graph ( Figure 49 and Figure 50 ) is the ratio of channel slope to the square root of the median grain s ize. Therefore, the median grain size of the bed material must be assumed for the simulation matrix. According to van den Berg (1995), median bed material grain size has a general tendency to increase with increases in channel slope and decrease with inc reasing discharge. In addition, the median bed material grain size is most frequently between 0.1 and 1.0 mm (sand range) or between 10 and 100 mm (gravel and cobble range). Median grain sizes between 1 and 10 mm are infrequent. A range of m edian bed ma terial grain sizes were assumed for the river cases of the simulation matrix. The grain sizes of the simulation matrix were distributed by valley slope and discharge ( Figure 51 ). PAGE 130 107 Figure 51 Discharge, valley slope, and grain size. The channel properties of the model simulation matrix ( bankfull discharge, slope, width, flow depth, roughness, and velocity) are presented in Table 8 and Table 9 Channel side slopes of 2:1 were assumed for all simulations compute the normal flow depth and velocity. The Reynolds number is ratio of inertial forces t o viscous forces and can be compute from : PAGE 131 108 Equation 45 Where R e = Reynolds number U = mean velocity R = hydraulic radius of the cross section = kinematic viscosity (assumed to be 1.30 x 10 6 m 2 /s at 10 o C) The Reynolds numbers for these simulations were ranged from 1.19 x 10 5 to 1.27 x 10 7 which means they are in the turbulent range and dominated by inertial forces ( Figure 52 ). PAGE 132 109 Figure 52 A plot of relative grain roughness versus Reynolds number show that model simulations are in the turbulent range. PAGE 133 110 Table 8 Model simulation matrix, part 1. Simulation Set Discharge Channel Sinuosity Channel Slope Slope/grain Top Width Depth Width Depth Ratio Bottom Width Q, m 3 /s S (S / d0.5) x 10 3 T, m D, m W/D B, m 1 2.83 3.00 2.27E 04 0.2273 7.00 1.349 5.2 1.633 2.83 2.75 2.48E 04 0.2480 7.47 1.170 6.4 2.785 2.83 2.50 2.73E 04 0.2728 7.59 1.109 6.8 3.149 2.83 2.25 3.03E 04 0.3031 7.66 1.060 7.2 3.424 2.83 2.00 3.41E 04 0.3410 7.73 1.015 7.6 3.665 2.83 1.75 3.90E 04 0.3897 7.78 0.971 8.0 3.896 2.83 1.50 4.55E 04 0.4546 7.83 0.926 8.5 4.130 2.83 1.25 5.46E 04 0.5455 7.89 0.877 9.0 4.380 2.83 1.10 6.20E 04 0.6199 7.92 0.844 9.4 4.544 2 2.83 3.00 2.06E 03 1.4563 8.10 0.642 12.6 5.535 2.83 2.75 2.25E 03 1.5886 8.12 0.622 13.0 5.630 2.83 2.50 2.47E 03 1.7475 8.14 0.601 13.5 5.734 2.83 2.25 2.75E 03 1.9417 8.16 0.577 14.1 5.848 2.83 2.00 3.09E 03 2.1844 8.18 0.551 14.8 5.974 2.83 1.75 3.53E 03 2.4964 8.20 0.522 15.7 6.116 2.83 1.50 4.12E 03 2.9125 8.23 0.488 16.9 6.278 2.83 1.25 4.94E 03 3.4950 8.26 0.449 18.4 6.466 2.83 1.10 5.62E 03 3.9716 8.28 0.422 19.6 6.596 3 2.83 3.00 8.98E 03 4.4885 7.80 0.400 19.5 6.203 2.83 2.75 9.79E 03 4.8965 8.41 0.389 21.6 6.850 2.83 2.50 1.08E 02 5.3862 9.11 0.378 24.1 7.596 2.83 2.25 1.20E 02 5.9847 9.95 0.366 27.2 8.489 2.83 2.00 1.35E 02 6.7328 11.0 0.352 31.2 9.579 2.83 1.75 1.54E 02 7.6946 12.3 0.336 36.6 10.95 2.83 1.50 1.80E 02 8.9770 14.0 0.317 44.1 12.72 2.83 1.25 2.15E 02 10.7724 16.3 0.296 55.1 15.12 2.83 1.10 2.45E 02 12.2414 18.2 0.281 64.7 17.03 PAGE 134 111 Table 8 (continued) Simulation Set Discharge Channel Sinuosity Channel Slope Slope/grain Top Width Depth Width Depth Ratio Bottom Width Q, m 3 /s S (S / d0.5) x 10 3 T, m D, m W/D B, m 4 28.3 3.00 4.97E 05 0.0703 19.5 3.918 5.0 3.802 28.3 2.75 5.42E 05 0.0766 21.5 3.454 6.2 7.706 28.3 2.50 5.96E 05 0.0843 21.9 3.273 6.7 8.775 28.3 2.25 6.62E 05 0.0937 22.1 3.128 7.1 9.578 28.3 2.00 7.45E 05 0.1054 22.3 2.996 7.4 10.29 28.3 1.75 8.51E 05 0.1204 22.4 2.866 7.8 10.96 28.3 1.50 9.93E 05 0.1405 22.6 2.733 8.3 11.65 28.3 1.25 1.19E 04 0.1686 22.7 2.587 8.8 12.39 28.3 1.10 1.35E 04 0.1916 22.8 2.490 9.2 12.87 5 28.3 3.00 4.10E 04 0.4104 23.3 1.956 11.9 15.48 28.3 2.75 4.48E 04 0.4478 23.4 1.898 12.3 15.76 28.3 2.50 4.93E 04 0.4925 23.4 1.834 12.8 16.07 28.3 2.25 5.47E 04 0.5473 23.5 1.764 13.3 16.41 28.3 2.00 6.16E 04 0.6157 23.5 1.686 14.0 16.78 28.3 1.75 7.04E 04 0.7036 23.6 1.599 14.8 17.20 28.3 1.50 8.21E 04 0.8209 23.7 1.499 15.8 17.68 28.3 1.25 9.85E 04 0.9851 23.8 1.383 17.2 18.24 28.3 1.10 1.12E 03 1.1194 23.8 1.303 18.3 18.62 6 28.3 3.00 1.96E 03 1.3871 24.8 1.127 22.0 20.30 28.3 2.75 2.14E 03 1.5132 26.7 1.097 24.3 22.30 28.3 2.50 2.35E 03 1.6645 28.9 1.066 27.1 24.65 28.3 2.25 2.62E 03 1.8495 31.6 1.030 30.7 27.46 28.3 2.00 2.94E 03 2.0807 34.9 0.991 35.2 30.91 28.3 1.75 3.36E 03 2.3779 39.0 0.946 41.2 35.22 28.3 1.50 3.92E 03 2.7742 44.4 0.895 49.6 40.82 28.3 1.25 4.71E 03 3.3290 51.7 0.834 62.1 48.41 28.3 1.10 5.35E 03 3.7830 57.6 0.791 72.8 54.45 PAGE 135 112 Table 8 (continued) Simulation Set Discharge Channel Sinuosity Channel Slope Slope/grain Top Width Depth Width Depth Ratio Bottom Width Q, m 3 /s S (S / d0.5) x 10 3 T, m D, m W/D B, m 7 283 3.00 1.09E 05 0.0217 54.7 11.7 4.7 8.09 283 2.75 1.18E 05 0.0237 62.0 10.2 6.1 21.28 283 2.50 1.30E 05 0.0261 63.0 9.66 6.5 24.40 283 2.25 1.45E 05 0.0289 63.7 9.23 6.9 26.76 283 2.00 1.63E 05 0.0326 64.2 8.84 7.3 28.84 283 1.75 1.86E 05 0.0372 64.7 8.46 7.6 30.82 283 1.50 2.17E 05 0.0434 65.1 8.06 8.1 32.84 283 1.25 2.61E 05 0.0521 65.5 7.63 8.6 35.00 283 1.10 2.96E 05 0.0592 65.8 7.35 9.0 36.42 8 283 3.00 8.18E 05 0.1157 67.0 5.96 11.3 43.20 283 2.75 8.92E 05 0.1262 67.2 5.78 11.6 44.04 283 2.50 9.82E 05 0.1388 67.3 5.59 12.0 44.95 283 2.25 1.09E 04 0.1542 67.5 5.39 12.5 45.95 283 2.00 1.23E 04 0.1735 67.7 5.16 13.1 47.05 283 1.75 1.40E 04 0.1983 67.9 4.90 13.9 48.30 283 1.50 1.64E 04 0.2314 68.1 4.60 14.8 49.71 283 1.25 1.96E 04 0.2776 68.4 4.25 16.1 51.37 283 1.10 2.23E 04 0.3155 68.6 4.02 17.1 52.52 9 283 3.00 4.29E 04 0.4286 69.0 3.46 20.0 55.20 283 2.75 4.68E 04 0.4676 84.7 3.09 27.4 72.32 283 2.50 5.14E 04 0.5143 91.7 3.00 30.6 79.73 283 2.25 5.71E 04 0.5715 100.2 2.90 34.5 88.62 283 2.00 6.43E 04 0.6429 110.7 2.79 39.6 99.49 283 1.75 7.35E 04 0.7347 123.8 2.67 46.4 113.1 283 1.50 8.57E 04 0.8572 140.9 2.52 55.9 130.8 283 1.25 1.03E 03 1.0286 164.2 2.35 69.9 154.8 283 1.10 1.17E 03 1.1689 182.8 2.23 82.0 173.9 PAGE 136 113 Table 8 (continued) Simulation Set Discharge Channel Sinuosity Channel Slope Slope/grain Top Width Depth Width Depth Ratio Bottom Width Q, m 3 /s S (S / d0.5) x 10 3 T, m D, m W/D B, m 10 2,832 3.00 2.37E 06 0.0067 154 34.5 4.5 15.86 2,832 2.75 2.59E 06 0.0073 179 30.1 5.9 58.52 2,832 2.50 2.85E 06 0.0080 182 28.5 6.4 67.69 2,832 2.25 3.16E 06 0.0089 184 27.2 6.7 74.60 2,832 2.00 3.56E 06 0.0101 185 26.1 7.1 80.69 2,832 1.75 4.07E 06 0.0115 186 25.0 7.5 86.52 2,832 1.50 4.74E 06 0.0134 188 23.8 7.9 92.45 2,832 1.25 5.69E 06 0.0161 189 22.5 8.4 98.79 2,832 1.10 6.47E 06 0.0183 190 21.7 8.7 103.0 11 2,832 3.00 1.63E 05 0.0326 193 18.1 10.6 120.3 2,832 2.75 1.78E 05 0.0356 193 17.6 11.0 122.8 2,832 2.50 1.96E 05 0.0391 194 17.0 11.4 125.5 2,832 2.25 2.17E 05 0.0435 194 16.4 11.8 128.4 2,832 2.00 2.45E 05 0.0489 195 15.7 12.4 131.7 2,832 1.75 2.79E 05 0.0559 195 15.0 13.0 135.4 2,832 1.50 3.26E 05 0.0652 196 14.1 13.9 139.6 2,832 1.25 3.91E 05 0.0782 197 13.1 15.1 144.5 2,832 1.10 4.45E 05 0.0889 197 12.4 16.0 147.9 12 2,832 3.00 9.37E 05 0.1325 250 8.95 27.9 214.1 2,832 2.75 1.02E 04 0.1445 269 8.72 30.9 234.0 2,832 2.50 1.12E 04 0.1590 291 8.46 34.4 257.4 2,832 2.25 1.25E 04 0.1767 318 8.18 38.9 285.5 2,832 2.00 1.41E 04 0.1988 351 7.87 44.6 319.9 2,832 1.75 1.61E 04 0.2271 393 7.51 52.3 363.0 2,832 1.50 1.87E 04 0.2650 447 7.10 63.0 419.0 2,832 1.25 2.25E 04 0.3180 521 6.62 78.8 494.9 2,832 1.10 2.56E 04 0.3614 581 6.28 92.4 555.4 PAGE 137 114 Table 9 Model simulation matrix, part 2. Simulation Set Area Wetted Perimeter Hydraulic Radius Manning's Roughnes s Cofficient Flow Velocity Reynolds number Froude Number A, m 2 P, m R, m n V, m/s R e F 1 5.84 7.7 0.76 0.0259 0.48 2.84E+05 0.170 6.00 8.0 0.75 0.0275 0.47 2.72E+05 0.168 5.95 8.1 0.73 0.0282 0.48 2.69E+05 0.171 5.88 8.2 0.72 0.0290 0.48 2.67E+05 0.176 5.78 8.2 0.70 0.0299 0.49 2.65E+05 0.181 5.67 8.2 0.69 0.0308 0.50 2.64E+05 0.187 5.54 8.3 0.67 0.0319 0.51 2.63E+05 0.194 5.38 8.3 0.65 0.0332 0.53 2.62E+05 0.204 5.26 8.3 0.63 0.0341 0.54 2.62E+05 0.211 2 4.38 8.4 0.52 0.0454 0.65 2.59E+05 0.281 4.28 8.4 0.51 0.0456 0.66 2.59E+05 0.291 4.17 8.4 0.49 0.0458 0.68 2.59E+05 0.303 4.04 8.4 0.48 0.0458 0.70 2.58E+05 0.318 3.90 8.4 0.46 0.0457 0.73 2.58E+05 0.336 3.73 8.4 0.44 0.0455 0.76 2.58E+05 0.359 3.54 8.5 0.42 0.0449 0.80 2.57E+05 0.389 3.31 8.5 0.39 0.0438 0.86 2.57E+05 0.432 3.14 8.5 0.37 0.0429 0.90 2.57E+05 0.467 3 2.80 8.0 0.35 0.0465 1.01 2.73E+05 0.539 2.97 8.6 0.35 0.0511 0.95 2.54E+05 0.512 3.16 9.3 0.34 0.0564 0.90 2.35E+05 0.486 3.37 10.1 0.33 0.0626 0.84 2.15E+05 0.461 3.62 11.2 0.32 0.0699 0.78 1.95E+05 0.436 3.90 12.4 0.31 0.0788 0.73 1.75E+05 0.411 4.24 14.1 0.30 0.0899 0.67 1.54E+05 0.387 4.65 16.4 0.28 0.1038 0.61 1.32E+05 0.364 4.94 18.3 0.27 0.1140 0.57 1.19E+05 0.351 PAGE 138 115 Table 9 (continued) Simulation Set Area Wetted Perimeter Hydraulic Radius Manning's Roughness Cofficient Flow Velocity Reynolds number Froude Number A, m 2 P, m R, m n V, m/s R e F 4 45.6 21.3 2.14 0.0188 0.62 1.02E+06 0.130 50.5 23.2 2.18 0.0221 0.56 9.41E+05 0.117 50.1 23.4 2.14 0.0227 0.56 9.30E+05 0.119 49.5 23.6 2.10 0.0234 0.57 9.24E+05 0.122 48.8 23.7 2.06 0.0241 0.58 9.20E+05 0.125 47.9 23.8 2.01 0.0249 0.59 9.16E+05 0.129 46.8 23.9 1.96 0.0258 0.61 9.12E+05 0.134 45.4 24.0 1.90 0.0268 0.62 9.09E+05 0.141 44.4 24.0 1.85 0.0275 0.64 9.07E+05 0.146 5 37.9 24.2 1.57 0.0366 0.75 8.99E+05 0.187 37.1 24.2 1.53 0.0368 0.76 8.98E+05 0.193 36.2 24.3 1.49 0.0370 0.78 8.97E+05 0.201 35.2 24.3 1.45 0.0372 0.81 8.97E+05 0.210 34.0 24.3 1.40 0.0372 0.83 8.96E+05 0.221 32.6 24.4 1.34 0.0371 0.87 8.95E+05 0.236 31.0 24.4 1.27 0.0368 0.91 8.93E+05 0.255 29.1 24.4 1.19 0.0361 0.97 8.92E+05 0.282 27.7 24.5 1.13 0.0355 1.02 8.91E+05 0.303 6 25.4 25.3 1.00 0.03981 1.11 8.60E+05 0.352 26.9 27.2 0.99 0.0436 1.05 8.01E+05 0.335 28.5 29.4 0.97 0.0479 0.99 7.41E+05 0.319 30.4 32.1 0.95 0.0530 0.93 6.79E+05 0.303 32.6 35.3 0.92 0.0591 0.87 6.16E+05 0.287 35.1 39.5 0.89 0.0666 0.81 5.52E+05 0.271 38.1 44.8 0.85 0.0757 0.74 4.86E+05 0.256 41.8 52.1 0.80 0.0872 0.68 4.18E+05 0.241 44.3 58.0 0.76 0.0957 0.64 3.76E+05 0.233 PAGE 139 116 Table 9 (continued) Simulation Set Area Wetted Perimeter Hydraulic Radius Manning's Roughness Cofficient Flow Velocity Reynolds number Froude Number A, m 2 P, m R, m n V, m/s R e F 7 366.0 60.2 6.08 0.0142 0.77 3.62E+06 0.096 424.7 66.9 6.35 0.0177 0.67 3.26E+06 0.081 422.2 67.6 6.25 0.0183 0.67 3.22E+06 0.083 417.4 68.0 6.13 0.0188 0.68 3.20E+06 0.085 411.2 68.4 6.01 0.0194 0.69 3.19E+06 0.087 403.8 68.7 5.88 0.0200 0.70 3.17E+06 0.090 394.9 68.9 5.73 0.0208 0.72 3.16E+06 0.093 383.7 69.1 5.55 0.0217 0.74 3.15E+06 0.097 375.6 69.3 5.42 0.0223 0.75 3.14E+06 0.101 8 328.3 69.8 4.70 0.0294 0.86 3.12E+06 0.124 321.6 69.9 4.60 0.0297 0.88 3.12E+06 0.128 314.0 70.0 4.49 0.0299 0.90 3.11E+06 0.133 305.5 70.0 4.36 0.0301 0.93 3.11E+06 0.139 295.8 70.1 4.22 0.0302 0.96 3.11E+06 0.146 284.4 70.2 4.05 0.0302 1.00 3.10E+06 0.155 271.0 70.3 3.86 0.0301 1.04 3.10E+06 0.167 254.7 70.4 3.62 0.0297 1.11 3.09E+06 0.184 243.1 70.5 3.45 0.0293 1.16 3.09E+06 0.198 9 214.7 70.7 3.04 0.0329 1.32 3.08E+06 0.239 242.8 86.1 2.82 0.0370 1.17 2.53E+06 0.220 257.5 93.2 2.76 0.0406 1.10 2.34E+06 0.210 274.2 101.6 2.70 0.0449 1.03 2.14E+06 0.199 293.4 112.0 2.62 0.0499 0.97 1.95E+06 0.189 315.9 125.1 2.53 0.0561 0.90 1.74E+06 0.179 342.6 142.1 2.41 0.0637 0.83 1.53E+06 0.169 374.8 165.3 2.27 0.0733 0.76 1.32E+06 0.160 397.6 183.9 2.16 0.0803 0.71 1.18E+06 0.154 PAGE 140 117 Table 9 (continued) Simulation Set Area Wetted Perimeter Hydraulic Radius Manning's Roughness Cofficient Flow Velocity Reynolds number Froude Number A, m 2 P, m R, m n V, m/s R e F 10 2,93 5 170.4 17.23 0.0106 0.96 1.28E+07 0.071 3,57 1 193.1 18.50 0.0142 0.79 1.13E+07 0.057 3,554 195.2 18.21 0.0147 0.80 1.12E+07 0.058 3,51 7 196.4 17.90 0.0151 0.81 1.11E+07 0.059 3,467 197.4 17.57 0.0156 0.82 1.10E+07 0.060 3,407 198.2 17.19 0.0162 0.83 1.10E+07 0.062 3,333 198.9 16.76 0.0168 0.85 1.10E+07 0.064 3,241 199.6 16.24 0.0175 0.87 1.09E+07 0.067 3,17 4 199.9 15.87 0.0180 0.89 1.09E+07 0.070 11 2,837 201.3 14.09 0.0236 1.00 1.08E+07 0.083 2,782 201.5 13.81 0.0238 1.02 1.08E+07 0.086 2,720 201.7 13.49 0.0241 1.04 1.08E+07 0.089 2,650 201.9 13.13 0.0243 1.07 1.08E+07 0.092 2,5 70 202.1 12.71 0.0244 1.10 1.08E+07 0.097 2,476 202.3 12.24 0.0245 1.14 1.08E+07 0.103 2,365 202.6 11.67 0.0245 1.20 1.08E+07 0.110 2,2 30 202.9 10.99 0.0243 1.27 1.07E+07 0.120 2,13 3 203.1 10.50 0.0241 1.33 1.07E+07 0.129 12 2,07 6 254.2 8.17 0.0288 1.36 8.57E+06 0.151 2,191 273.0 8.03 0.0314 1.29 7.98E+06 0.145 2,32 2 295.3 7.86 0.0344 1.22 7.38E+06 0.138 2,470 322.1 7.67 0.0379 1.15 6.76E+06 0.131 2,64 1 355.1 7.44 0.0421 1.07 6.13E+06 0.125 2,84 1 396.6 7.16 0.0472 1.00 5.49E+06 0.118 3,07 8 450.7 6.83 0.0535 0.92 4.83E+06 0.112 3,364 524.5 6.41 0.0615 0.84 4.15E+06 0.106 3,567 583.5 6.11 0.0673 0.79 3.73E+06 0.102 PAGE 141 118 The simulation matrix includes a wide range of flow depths (hydraulic radius), width depth ratios, and Froude numbers ( Figure 53 ). All model simulations are in the subcritical range, which is typic al for natural river channels. Figure 53 Range of hydraulic radius, width depth ratio, and Froude number of the model simulation matrix. Model Boundary Conditions For each model simulation, the upstream boundary is specified by the bankfull discharge and the assumption of a uniform velocity distribution across the channel entrance The PAGE 142 119 downstream boundary was specified by the water surface elevatio n corresponding to normal depth: Equation 46 Where Q = bankfull discharge (m 3 /s), n = channel roughness coefficient, A = cross section area of the flow (m 2 ), B = channel bottom width (m), z = ratio of horizontal side slope distance per unit distance of vertical rise, D = normal depth or maximum channel depth R = hydraulic radius (m), P = wetter perimeter of the channel cross section (m), and S = longitudinal slope of the channel bottom. St raight channel alignments with lengths equal to 10 or 15 channel widths were included at the upstream and downstream ends of the meandering channel alignments so that the specified boundary conditions are far enough away from the meandering portion of the simulated channel. These straight channels segments were assigned the PAGE 143 120 same slope as the meandering portion of the channel, so the longitudinal channel slope is constant over the entire channel length of a given simulation. Meandering Channel Alignments A sine generated curve was used to de fine meandering channel alignments with variable radius curves ( Figure 54 ). Langbein and Leopold (1966) found that the sine generated curve will best approximate the alignment of a natural meandering river channel : Equation 47 where = angle of alignment along the channel relative to the downstream valley axis, = maximum angle of channel alignment L = distance along the river channel centerline, = meande r wavelength, and channel sinuosity. The horizontal coordinates of the channel alignment can be determined by integrating Equation 47 to prod uce the following equations: PAGE 144 121 Equation 48 Equation 49 where x = horizontal coordinate parallel to the downstream valley axis, y = horizontal coordinate perpendicular to the downstream valley axis, and L = distance along the channel centerline at positions n and n +1. A regression equation ( Equation 50 ), developed by Randle (2004), w as be used to relate the maximum angle of the meander bend ( : Equation 50 The average channel slope ( S o ) is computed from the valley slope ( S v : Equation 51 as applied to determine the bankfull depth ( D ) of each trapezoidal channel ( E quation 52 ) for a given of a bankfull discharge ( Q b ). In this form of the equation channel bank roughness c an be different from the bed roughness : PAGE 145 122 E quation 52 where Q b = bankfull discharge [L 3 /T], D = bankfull water depth [L], B = bottom width of the trapezoidal channel [L], z = horizontal component of the channel bank slope for a unit vertical rise, n b n roughness coefficient for the channel bottom, and n s n roughness coefficient for the channel bank. The bankfull channel width ( W ) w as computed using the equation for a trapezoidal channel : Equation 53 An example of t he meandering river channel alignment, generated using a sine generated curve is presented in Figure 54 for simulation S et 5 with five consecutive meander bends Four lines were generated to represent the top left bank, left bottom edge, right bottom edge, and top right bank of the channel. E ach of these lines were defined by a series of three dimensional coordinates. 2D and 3D model mesh es were generated using these four lines ( Figure 55 ). Channel cross sections can be delineated PAGE 146 123 from the model mesh cells The cross section alignments from these mesh cell s are similar to, but not the same as, the alignments for the variable radii of curvature ( Figure 56 ). The radii of curvature lines are perpendicular to the flow, so the model velocity data were projected along and normal to the radii alignments. Figure 54 Example meandering river channel alignment, generated using a sine generated curve for simulation set 5 with a sinuosity of 2 75 and five consecutive curves PAGE 147 124 Figure 55 Example channel alignment points and the variable radii of curvature are plotted for simulation set 5 with a sinuosity of 2.75 and five consecutive curves PAGE 148 125 Figure 56 A close up view of channel alignment points and variable radii of curvature show that their alignments are similar, but they do not exactly coincide. For simulation S et 6 a series of meandering channel alignments were computed using a sine generated curve over 1.5 meander wavelengths ( Figure 57 ). The sine generated curves result in variable radius of curvature so the radius is a minimum a t the meander bend apex and infinite where the direction of channel curvature is reversed PAGE 149 126 Figure 57 Meandering channel alignments of simulation S et 6 with three consecutive meandering bends Channel Entrance and Exit Conditions The simulated channel alignments include straight channel segments at the upstream entrance to the meandering reach and at the downstream exit of the meandering reach ( Figure 57 ). Each of the straight channel segments has a length e quivalent of 10 or 15 channel widths. The straight channel segments each connect to the meandering reach at points where there is no curvature. Otherwise, additional bank shear stress would be PAGE 150 127 generated where the channel transitioned from curvature to a tangential alignment. An example three dimensional view of a meandering channel is presented in Figure 58 Figure 58 Simulation Set 6, Sinuosity of 3.00. A length of 10 or 15 channel widths for the straight channel segments was assum ed to be long enough so that the specified boundary conditions would not influence the hydraulic conditions in the middle curves of the m eandering channel reach. The hydraulic simulation through the straight upstream channel segment was used to represent the reference bottom shear stress for the meandering channel. PAGE 151 128 S imulated vertical velocity profiles for the center and bottom edges of the straight trapezoidal channel are presented in Figure 59 for the S imulation S et 6 wi th a sinuosity of 3.00 The vertical velocity profiles are identical at the channel center, middle left, and middle right portions of the channel bottom. The vertical velocity profiles at the left and right bottom corners of the trapezoidal cross section are also identical, but less in magnitude than the middle of the channel. Figure 59 Simulated vertical velocity profiles for the staight channel reach after channel length equal to 20 channel widths. The proper length of the straight channel segments was verified by simulating the channel hydraulics through a straight channel with a length equal to 80 channel widths PAGE 152 129 ( Figure 60 ) Th is straight channel simulation enable s the computation of bottom shear stress, without the influence of channel curvature, as a reference shear stress for the meandering channels. Th e straight channel w as simulated with the same bottom width and longitudinal slope as the meandering channel of Simulation Set 6 with a sinuosity of 3.00 The simulated bottom shear stress is highest at the upstream boundary (because of the uniform velocity distribution) and then dimi nishes to a near constant value after a distance of 10 to 15 channel widths. Figure 60 Simulated bottom shear stress long the centerline of the straight trapezoidal channel. PAGE 153 130 Number of Meander Curves For a series of meandering channel bends, the simulated hydraulics through the upstream most and downstream most bends can be influenced by the hydraulics through upstream and downstream straight reaches. In the case of three consecutive channel meander bends (1.5 meander wave len gths), the assumption is made that hydraulics through the middle meander bend are not influenced by the upstream boundary conditions or the straight reach segment. In other words, the hydraulics of the middle bend would be repeated if there were multiple meander bends between the upstream and downstream meander bends. This was found to be the case for Simulation Set 6 ( Figure 61 ), but five consecutive meander bends were simulated for all other sets. PAGE 154 131 Figure 61 Channel p lan view and location of simulat ed maximum velocity through five consecutive meander bends. For simulation set 6 with a sinuosity of 3.00, t he location of maximum velocity along the channel (blue line) shifts from right to left and back again as the fluid travels through the five meander bends. The relative discharge, left and right of the channel centerline, fluctuates as the fluid passes through the me ander bends ( Figure 62 ). PAGE 155 132 Figure 62 Relative discharge on the left and right sides of the channel through five consecutive meander bends. Model Mesh Sizes The U 2 RANS model can simulate up to about 1 million mesh cells. The model mesh size used to simulate hydraulic flow conditions throu gh the physical model at Colorado State University can be used to guide the model mesh sizes for the meandering river channel cases of the simulation matrix. The dimensions of the Colorado State University physical model are presented in Table 10 The dimensions of the densest model mesh used to simulate the physical model are presented in Table 11 A unit mesh size can be computed using the dimensionless ratios of channel length to width and channel width PAGE 156 133 to depth. The unit mesh size dimensions are then used to determine the mesh sizes for the river channel cases of the simulati on matrix. Table 10 CSU Physical Model Dimensions. Channel Segment Channel Length (m) Channel Top Width (m) Channel Depth (m) Length / Width Width / Depth Upstream bend 25.8 4.57 0.24 5.6 19.0 Straight transition 6.5 4.57 0.31 1.4 14.7 Downstream bend 25.6 3.7 0.31 6.9 11.9 Table 11 Numerical model mesh size used to simulate the flow through the CSU physical model. Channel Segment i j K Straight entrance 176 32 24 13.4 1. 68 457.0 Upstream bend 75 32 24 13.3 1. 68 457.0 Straight transition 20 32 24 14.1 2.2 353.8 Downstream bend 75 32 24 10.8 2.7 286.5 Straight end segment 146 32 24 10.8 2.7 286.5 Based on the limiting values of Table 11 the following ratios were used to determine the model mesh size: PAGE 157 134 Equation 54 Equation 55 Equation 56 Where i j and k are the model mesh sizes in the longitudinal ( i ), transverse ( j ), and vertical ( k ) dimensions, L W and D are the channel dimensions of length ( L ), width ( W ), and depth ( D ). The use of Equation 54 Equation 55 and Equation 56 produces a total mesh size of 704,340 cells for the simulation of three meander bends ( Table 12 ) Approximately, 1 millon mesh cells were used to simulate five consecutive meander bends. Table 12 Example 3D model mesh size for three consecutive meander bends. Model Reach Segment i cells j cells k cells Straight Entrance Reach 68 39 21 1 st Meander Bend 241 39 21 2 nd Meander Bend 241 39 21 3 rd Meander Bend 242 39 21 Straight Exit Reach 68 39 21 Entire Channel Mesh Dimensions 860 39 21 Total Mesh Size 704,340 PAGE 158 135 In order to evaluate the effects of the model mesh density, three different mesh densities were simulated ( Table 13 ). The fine mesh density size was computed using Equation 54 Equation 55 and Equation 56 The medium mesh density was determined to be one half the total number of mesh cells of the fine mesh. The coarse mesh density was determined to be one half the total number of mesh cells of the medium mesh. Table 13 Range of numerical model mesh densities Mesh Density i cells j cells k cells Total Mesh Size Coarse 542 25 14 189,700 Medium 684 31 17 360,468 Fine 860 39 21 704,340 Simulated vertical velocity profiles, at the downstream portion of the middle meander bend, are presented in Figure 63 Figure 64 and Figure 65 for the three different mesh sizes. The simulated shear stress at the same cross section location is presented in Figure 66 Figure 67 and Figure 68 PAGE 159 136 Figure 63 Simulated v ertical velocity profiles at a meander bend cross section for a coarse model mesh size of 542 x 25 x 14. PAGE 160 137 Figure 64 Simulated v ertical velocity profiles at a meander bend cross section for a medium model mesh size of 684 x 31 x 17. Figure 65 Simulated v ertical velocity profiles at a meander bend cross section for a fine model mesh size of 860 x 39 x 21. PAGE 161 138 Figure 66 Simulated c ross section shear stress and location of high velocity at a m eander bend for a coarse model mesh size of 542 x 25 x 14. Figure 67 Simulated c ross section shear stress and location of high velocity at a meander bend for a medium model mesh size of 684 x 31 x 17. PAGE 162 139 Figure 68 Simulated c ross section shear stress and location of high velocity at a meander bend for a fine model mesh size of 860 x 39 x 21. The comparison of simulated model results from the three mesh sizes is presented in Figure 69 This comparison shows that the simulated velocity and shear stress tend to increase with model mesh density or the total number of model mesh cells. PAGE 163 140 Figure 69 Simulated velocity and shear stress are compared with the total model mesh size. Discussion of Meandering Channel Alignments The length of the straight entrance and exit reaches was tested by simulating a single straight long reach to determine the longitudinal length where the cross sectional velocities reach a near constant value. By inspection, this appears to occur at a longitudinal length equal to between 10 and 15 10 channel widths for Simulation Set 6 ( Figure 60 ). However, simulation of longer straight channels was sometimes need ed to determine the reference bottom shear stress. PAGE 164 141 A total of five meander bends were simulated ( Figure 61 ) to insure that results of the middle three bends are repeatable and not influenced by the boundary conditions. The hydraulic co nditions through the upstream and downstream most meander bends are influenced to some degree by the straight entrance and exit reaches. However, the hydraulic conditions through the second, third, and fourth meander bends are identical for Simulation Set 6 with a sinuosity of 3.00. Five consecutive meander bends were simulated for all other simulation sets. The sensitivity of the model mesh size to simulated hydraulic properties (velocity and shear stress) was tested by simulating the same channel geome try with three different mesh sizes ( Table 13 ). Of the three model mesh sizes, the densest mesh size (860 x 39 x 21) is believed to produce the most accurate results. The comparison of simulated velocity and shear stress with the total model mesh size ( Figure 69 ) indicates that a total mesh size of about 1,000,000 cells will produce sufficiently accurate results. Therefore, Equation 57 Equation 58 and Equation 59 were used to develop the model mesh size s : Equation 57 Equation 58 PAGE 165 142 Equation 59 The model mesh cell at the bottom corner has a fixed boundary on the bottom and side of the mesh cell. Therefore, shear stress results from this corner mesh cell, and the cells immediately adjacent to the corner cells, may not be accurate are are n ot used in this research The maximum boundary shear stress was found to occur on the channel bottom near the toe of the bank slope and, occasionally, on the channel bank above the bank toe. A theoretical analysis of the boundary shear stress distribution in straight trapezoidal channels was performed by Olsen and Florey (1952) who utilized a membrane analogy. These results indicated that the shear stress would be zero at the bottom corners o f a trapezoidal channel. However, data presented by the ASCE Task Committee on Hydraulics, Bank Mechanics, and Modeling of River Width Adjustment (1998) indicate that the shear stress is somewhat higher near the bottom corners of a trapezoidal channel and not zero. Ippen and Drinker (1962) measured the boundary shear stress in curved trapezoidal channels and found that the maximum shear stress was near the bottom corners of the trapezoidal channel, either along the inside curved bank or along the outside bank at the end of the curve ( Figure 12 ) PAGE 166 143 Conclusions of Numerical Model Simulation The straight entrance and exit reach segments should have a lengt h of at least 10 channel widths. However, a length equal to 15 channel widths would further reduce, if not eliminate, the hydraulic effects of the upstream and downstream boundaries on the meandering portions of the river channel, especially the boundary condition influence through the middle meander curves. At least three or five consecutive meander bends need to be simulated and model results should only be used from the middle curves Model results from the first meander bend could be used to represent cases where the lateral distribution of velocit ies is symmetrical about the channel centerline at the upsteam end of the meander bend Of the three model mesh sizes that have been tes ted so far, the finest or densest mesh should be used for about 1 million total mesh cells to produce sufficiently accurate results. The maximum boundary shear stress simulated by the model is near the bottom corners of the trapezoidal channel which is supported by measurements found in the literature. PAGE 167 144 Dimensionless Parameters Dimensionless parameters were formulated to describe hy draulic and channel properties for use in empirical analysis of 3D model simulation results. Dimensionless Hydraulic Parameters Maximum boundary s hear stress along curved or meandering channel s is the main focus of this dissertation In post processing of the 3D model results, d imensionless shear stress wa s comput ed at each boundary mesh cell using the following equation: Equation 60 Where R = dimensionless shear stress at the boundary mesh cell w = shear stress at the boundary mesh cell and 0 = reference shear stress along the channel centerline of a long and prismatic straight channel. Shear stress simulated by the 3D model was used as the reference shear stress for consistency. The shear stress at the channel centerline ( 0 ) was chosen to coincide with the maximum depth of the trapezoidal channel rather than the hydraulic depth or hydraulic radius. PAGE 168 145 The cross sectional distribution of f low velocity is asymmetrical through a meandering or curved channel. Therefore, the loction of maximum velocity and the near bank velo city is of interest The post processing program comp ut ed the r elative maximum and near bank velocities along both channel banks for each model simulation: Equation 61 Where V R = the relative velocity, V max = the maximum near bank velocity in any model mesh cell between the bank and channel centerline, and V ave = mean flow velocity of the channel ( V ave = Q/A) T he vertical and lateral position of maximum velocity may be different for each channel that is modeled. A standard relative channel position will not capture the location of maximum velocity. Dimensionless Channel Parameters Ippen and Drinker (1962) concluded that the spatial distribution and relative magnitudes of shear stress appear to be functions of the stream geometry. Therefore, PAGE 169 146 correlation of dimensionless shear stress and near bank velocity was tested against several dimensionless channel parameters: The radius to w idth ratio ( r c / W ) is computed from the channel radius of curvature ( r c ) and the channel wetted top width ( W ) Correlations were attempted with the radius to width ratio and relative bank shear stress (Ippen and Drinker, 1962 and Yen, 1965). Nason and Hic kin (1983) have attempted to cor relate the radius to wi d th ratio with the rate of channel migration. The radius of curvature i s infinite for straight channels and in meandering channels at the point in the alignment where there is change in the direction of curvature. Curvature ( C radians or degrees of curvature per unit length of stream) can be related to the radius of curvatur e ( r c ) ( Figure 70 ) : Equation 62 The product of curvature ( C ) and channel depth ( D ) is another dimensionless ratio 1 to describe the channel geometry and is a ratio used in Equation 15 to compute the radial shear stress. 1 Curvature ( C ) is expressed as radians or degrees per unit length of stream and channel depth (D) has units of length. The product ( C D ) is di mensionless because an angle is measured as the ratio of arc length to radius, which cancels out any units. Angles measured in degrees are also dimensionless because they are defined as the ratio of the arc length to 1/360 of a circle, which results in th e need for a multiplier. PAGE 170 147 Width depth ratio is also another important ratio to describe the cross sectional geometry of the channel. Figure 70 Relationship between channel curvature and radius of curvature. Pre and Post Processing Program s Although the 2D and 3D hydraulic model s had already been developed, new pre and post processing program s had to be developed for this research. Both of t hese PAGE 171 148 programs were written in FORTRAN The pre processing program used Equation 48 and Equation 49 to develop four meandering channel alignments for each simulation: Alignment of top left channel bank Alignment of left channel bank toe Alignment of right channel bank toe Alignment of top right channel bank The post processing program computed the following parameters along longitudinal alignment of channel mesh cells: Longitudinal alignment of the thread of maximum velocity Relative maximum near bank velocity along both channel banks. Di mensionless shear stress along both channel banks and the channel bottom centerline. In addition, the post processing program computes lateral distributions of dimensionless shear stress and identifies zones of high velocity at selected cross sections. Vertical, streamwise velocity profiles are also computed at these selected cross sections at the following locations: Left bank toe Left middle of channel bottom (half way between the left bank toe and the channel centerline) Channel centerline PAGE 172 149 Right middl e of channel bottom (half way between the channel centerline and right bank toe) Right bank toe PAGE 173 150 CHAPTER IV MODEL SIMULATION RESULTS The 2D and 3D hydraulic models (SRH 2D and U2 R ANS) were first applied to Simulation Sets 4 through 9 Model results from these simulation s ets were used to develop empirical ly based relationships of dimensionless shear stress as a function dimensionless channel parameters. Meandering Channel Alignments Each simulation set includes nine virtual stream channels with a range of sinuosity from 1.10 to 3.00. E xample channel alignments from simulation set 4 are presented in Figure 71 through Figure 79 These plots also show the simulated alignment for the maximum thread of velocity along each channel (dark blue line) The location of maximum velocity is difficult to define along the upstream straight reach because the 3D velocity is specified as uniform at the upstream model boundary. PAGE 174 151 Figure 71 Model channel alignment for a sinuosity of 1.10. Figure 72 Model channel alignment for a sinuosity of 1.25. Figure 73 Model channel alignment for a sinuosity of 1.50. Figure 74 Model channel alignment for a s inuosity of 1.75. Figure 75 Model channel alignment for a sinuosity of 2.00. Figure 76 Model channel alignment for a sinuosity of 2.25. PAGE 175 152 Figure 77 Model channel alignment for a sinuosity of 2.50. Figure 78 Model channel alignment for a sinuosity of 2.75. Figure 79 Model channel alignment for a sinuosity of 3.00. Example 2D Model Results Complete model output from 54 separate meandering river channels is too large to display in this dissertation report. However, graphs of model results from simulation set 5, with a channel sinuosity of 2 75 are presented as an example of the model result s available for all 54 simulations. PAGE 176 153 A plan view of the 2D model mesh is presented in Figure 80 where flow enters at the southwest end. A close up view of this mesh is presented in Figure 81 Figure 80 Plan view of the structured 2D model mesh domain for Simulation Set 5 with a sinuosity of 2 75 PAGE 177 154 Figure 81 C lose up plan view of the structured 2 D model mesh for Simulation Set 5 with a sinuosity of 2 75 The uniform longitudinal slope of the channel bottom is depicted in Figure 82 where the bed elevation contours are presented. Water surface elevation contours are plotted from the 2D model results ( Figure 83 ). PAGE 178 155 Figure 82 Channel bottom elevation contours of the 2D model mesh for Simulation Set 5 with a sinuosity of 2 75 PAGE 179 156 Figure 83 Water surface elevation contours from the 2D model results for Simulation Set 5 with a sinuosity of 2 75 Example 3D Model Results Example 3D model results for simulation set 5 (with a sinuosity of 2 75 ) include the 3D mesh and spatial distributions of velocity and shear stress. PAGE 180 157 Example 3D Model Mesh The structured 3D model mesh ( Figure 84 ) is similar to the 2D mesh, but the vertical mesh lines radiate from along the channel bottom and tend to follow the angle of channel side slopes away from the channel center ( Figure 85 ). For the example simulation, t he structured 3D model mesh has 1,587 cells in the longitudinal streamwise direction ( i di mension) 31 cells in the cross stream direction ( j di mension) and 2 0 cells in the vertical direction ( k di mension) for a total of 983 940 mesh cells. PAGE 181 158 Figure 84 Elevation contours of the 3D model mesh for Simulation Set 5 with a sinuosity of 2 75 PAGE 182 159 Figure 85 Concept of the 3D structured mesh is presented for the right half of the cross section. Example 3D Velocity Results A contour shading of surface velocities from the 3D model results is presented in Figure 86 A close up view of velocity contours is presented in Figure 87 A close up view of cross sectional slices of velocity magnitude through a meander bend is presented in Figure 88 PAGE 183 160 Figure 86 Simulated 3D velocity contours at the water surface for Simulation Set 5 with a sinuosity of 2 75 PAGE 184 161 Figure 87 Close up view of simulated 3D velocity contour s at the water surface. PAGE 185 162 Figure 88 Close up 3D view of cross sectional slices of velocity contours. The thread of maximum velocity alignment is presented in Figure 89 A u niform veloc ity distribution is specified a t the upstream model boundary so identifying the thread of maximum velocity is difficult through the upstream straight reach. At the beginning of each meander bend the thread of maximum velocity is near the inside curve and gradually migrates to the channel centerline near the apex (minimum radius of PAGE 186 163 curvature) of the meander bend. The thread of maximum velocity is along the outside curve thro ugh the downstream portion of the meander bend and remains along that bank until the curvature is reversed and the magnitude of curvature is significant. Figure 89 Simulated thread of maximum velocity for Simulation Set 5 with a sinuosity of 2.75 The maximum dimensionless velocity for each model cross section was determined for the left and right sides of the channel. In most cases, the maximum velocity was at the water surface, but not always. The maximum velocity for the left and right sides of each model cross section was then plotted against the dimensionless channel curve length ( Figure 90 ). The maximum velocity near the channel bottom ( 5 th mesh cell above the channel bottom) is also plotted versus the longitudinal channel length Longitudinal PAGE 187 164 channel length is made dimensionless by dividing it by the longitudinal length of one meander bend. Figure 90 Simulat ed velocities near the left and right channel banks for simulation set 5 with a sinuosity of 2.75 Example 3D Model Shear Stress Results Boundary shear stress for the entire channel is plotted in Figure 91 The shear stress distribution patterns follow the velocity distribution patterns ( Figure 86 ). PAGE 188 165 Figure 91 Channel bottom shear stress contours for simulation set 5 with a sinuosity of 2 75 Shear stress as a function of dimensionless channel length is plotted in Figure 92 The lateral distribution of s hear stress and maximum velocity is plotted at three selected cross sections The plan view location of these three cross sections is presented in Figure 93 Cross section plots of shear stress and zones of maximum velocity are presented in Figure 94 Figure 95 Figure 96 and Figure 97 Vertical velocity profiles PAGE 189 166 plots are p resented for these four cross sections are presented in Figure 98 Figure 99 Figure 100 and Figure 101 Figure 92 Longitudinal channel bank and centerline shear stress for simulation set 5 with a sinuosity of 2.75 PAGE 190 1 67 Figure 93 Plan view alignment of the meandering channel from simulation set 5 with a sinuosity of 2.75 and positions of selected cross sections 112 7 02 810, and 9 5 0 PAGE 191 168 Figure 94 Lateral shear stress distribution and zone where velocity is at least 90 percent of the maximum velocity at cross section 112 (upstream straight reach) Figure 95 Lateral s hear stress distribution and zone where velocity is at least 90 percent of the maximum velocity at cross section 702 (3 rd meander bend) PAGE 192 169 Figure 96 Lateral shear stress distribution and zone where velocity is at least 90 percent of the maximum velocity at cross section 810 (3 rd meander bend) Figure 97 Lateral shear stress distribution and zone where velocity is at least 90 percent of the maximum velocity at cross section 9 5 0 (4 th meander bend) PAGE 193 170 Figure 98 Vertical streamwise velocity profiles at cross section 112 (upstream straight reach) for simulation set 5 with a sinuosity of 2.75. Figure 99 Vertical streamwise velocity profiles at cross section 702 (3 rd meander bend) for simulation set 5 with a sinuosity of 2 75 PAGE 194 171 Figure 100 Vertical streamwise velocity profiles at cross section 810 (3 rd meander bend) for simulation set 5 with a sinuosity of 2 75 Figure 101 Vertical streamwise velocity profiles at cross section 9 5 0 (4 th meander bend) for simulation set 5 with a sinuosity of 2 75 PAGE 195 172 The longitudinal patterns of bank shear stress vary depending on the channel width depth ratio. Example patterns of longitudinal shear stress are presented in Figure 102 Figure 103 and Figure 104 Figure 102 Longitudinal channel bank and centerline shear stress for simulation set 7 with a sinuosity of 2.00 and a width depth ratio of 7 .3 PAGE 196 173 Figure 103 Longitudinal channel bank and centerline shear stress for simulation set 8 with a sinuosity of 2.00 and a width depth ratio of 9.1. PAGE 197 174 Figure 104 Longitudinal channel bank and centerline shear stress for simulation set 9 with a sinuosity of 2.00 and a width depth ratio of 35.6. Discussion of Model Simulation Results The model simulation matrix represents a wide range of channel sizes as expresse d by the range in bankfull discharge, longitudinal slope, channel width, channel depth, and median sediment grain size. Application of the 2D and 3D models to S imulation S ets 4 through 9 produced hydraulic data from 54 different meandering river channels (6 simulation sets with 9 meandering channels per set). These m odel results were used to PAGE 198 175 develop empirically based equations while model results from a subset of simulation sets 1, 2, and 3 and sets 10, 11, and 12 are used to validate the equations. Model results of velocity and shear stress are different through the first meander bend than subsequent meander bends, which tend to be similar to each other. Therefore, model results are not used from the first meander b end, but from the second to last m eander bend in the series The lateral distributions of velocity and boundary shear stress are symmetrical about the channel centerline through the upstream straight reach. The maximum bank shear stress through the first meander bend tends to be lower th an through subsequent meander bends because of the symmetrical velocity distribution entering the first meander bend. By the downstream end of the first meander bend, velocity and shear stress have increased along the left (outside) bank of the first be nd, which is the inside bank at the beginning of the second meander bend. The zone of high velocity and shear stress continue along the inside bank of the next meander bend and reach a maximum before the bend apex where the radius of curvature is at a mim inum. By the downstream end of the second menander bend (curving to the left), s econdary currents have begun to increase velocity and shear stress along the opposite, right bank The transfer of high velocity and shear stress from the inside bank to the outside bank is repeated through subsequent meander bends ( Figure 86 through Figure 92 and Figure 102 through Figure 104 ) PAGE 199 176 The longitudinal distance required for the thread of high velocity and shear stress to be transferred from the inside bank to the outside bank is known as the planform phase lag which is found in na tural channels. This longitudinal distance is long enough that the zone of highest velocity and shear stress is along the outside bank near the downstream end of the meander bend, which is also along the inside bank at the beginning of the next ( downstrea m ) meander bend. The laboratory experiments by Yen (1965) and Ippen and Drinker (1962) also demonstrate that the zone of highest shear stress is along the inside channel bank through the curve and then transfers to the outside bank near the downstream end of the curve. The plan and cross section plots ( Figure 93 through Figure 97 ) show an increase in shear stress along the channel bottom near the bank s At the channel crossing location where the zone of high velocity and shear stress is transferring from the inside bank to the outside bank, the shear stress is somewhat uniform across the channel bottom, but the velocity distribution is not uniform because of the secondary currents ( Figure 95 through Figure 97 ). The vertical veloc ity profiles show the model is capable of computing the maximum velocity at a given profile, that is often at the water surface but sometimes below the water surface ( Figure 99 Figure 100 and Figure 101 ). The zones of high velocity and shear stress alternating along the channel bank s, would tend to erode the channel along those locations and from pools with point bars depositing along the opposite, inside bank. This would result in a transverse channel PAGE 200 177 bed slope. However, the model simulations have fixed bed and bank b oundaries and erosion is not simulated. Seminara (2006) points out that for channels with non erodible banks, flow velocity tends to b higher along the inner b ank Farther downstream, the secondary currents transfer momentum towards the outer bend, moving the thread of high velocity from the inner bank to the outer bank The model results from all simulation sets indicate a similar longitudinal distribution of shear stress. The longitudinal increase in sh ear stress along a channel bank is largely parabolic. The longitudinal locations where the shear stress along both banks is the same corresponds closely with the beginning and ending of each meander curve. This is illustrated by the examples from Simulat ion Sets 7, 8, and 9 for a sinuosity of 2.00 ( Figure 102 Figure 103 and Figure 104 ). Model results indicate that the location of maximum shear stress is one quarter to one half way through the next meander bend, but upstream of the bend apex where the radius of curvature reaches a minimum. The maximum shear stress tends to reduce with an incease in the width depth ratio. PAGE 201 178 CHAPTER V HYDRAULIC RELATIONSHIPS For meandering river channels, th ere is intere st in knowing how much the bank shear stress and near bank velocity increase through a curve relative to a straight channel reach of the same cross sectional dimensions and longitudinal slope. There is also interest in knowi ng the location of maximum bank shear stress and near bank velocity and the pattern s of these hydraulic parameters through the curve. Maximum Shear Stress Magnitude For each meandering river channel simulation, the centerline equilibrium shear stress from a long straight reach was used to compute the dimensionless shear stress along every boundary mesh cell of the channel. The dimensionless shear stress along both channel banks was then plotted against the dimens ionless curve length ( Figure 102 Figure 103 and Figure 104 ). The maximum and minimum dimensionless bank shear stresses through the meander curves are summarized in Figure 105 for model simulation sets 4 through 9. For these simulations, maximum dimensionless shear stresses ran ged from 1.21 to 1 54 while minimum dimensionless shear st resses ranged from 0. 46 to 0.86 The dimensionless shear stress did not correlate well with channel sinuosity but maximum dimensionless shear stress tended to be greatest for sinuosit ies between sinuosities of 1. 50 and 2.25 ( Figure 105 ) PAGE 202 179 Figure 105 Summary of 3D model simulation results for minimum and maximum shear stress along the toe of the left and right bank s through meandering channels. T he maximum dimensionless near bank shear stress was plotted against the ratio of minimum radius of curvature to the wetted channel top width ( Figure 106 ) The data were segragated by the channel width depth ratio (W/D) into three categories: 5 < W/D < 10 (Simulations Sets 4 and 7) 10 < W/D < 20 (Simulations Sets 5 and 8) 20 < W/D < 84 (Sim ulations Sets 6 and 9) PAGE 203 180 Initial, l inear relaitonships were found for each category with correlation coefficients (R 2 ) raging from 0.60 to 0.94. All three linear regression lines had similar positive slopes Maximum shear stress was found to decrease with an increase in the width depth ratio. Figure 106 Correlation between the maximum dimensionless shear stress in a channel bend with the ratio of channel width to minimum radius of curvature The initial, linear regression equation for width depth ratios between 4 and 10 had the best correlation: PAGE 204 181 Equation 63 Therefore, this initial equation was used to compute the maximum shear stress for all 54 simulations. The residual between the maximum shear stress computed by empirical Equation 63 and the maximum shear stress simulated by the 3D model was plotted against the channel width depth ratio ( Figure 107 ). Figure 107 Residual maximum shear stress computed by empirical Equation 63 and the maximum shear stress simulated by the 3D model is plotted against the channel width depth ratio PAGE 205 182 The residuals were low ( 0.015 to +0.030) and seemed to be random for channels with width depth ratios between 4 and 10. A series of linear relationships were found for channels with width depth ratios greater than 10 The regression equation slope s were found to decrease with increases in the widt h depth ratio and approached a a slope of zero for width depth ratios beyond 40. When the residual regression equations are combined with Equation 63 the following initial, general equation can be used to estimate the maximum shear stress in a trapezoidal channel: Equation 64 Where The dimensionless maximum shear can be ploted against the ratio of channel width to minimum radius of curvature (W/Rc) and the width depth ratio (W/D) in a 3D plot ( Figure 108 ). PAGE 206 183 Figure 108 3D plot of maximum shear stress against the ratio of channel width to minimum radius of curvature and the width depth ratio. Equation 64 was applied to the data from simulation sets 4 through 9 to estimate the maximum dimensionless bank shear stress These maximum shear stress values were then comp ar ed with the shear stress v alues simulated by the 3D model ( Figure 109 ). The regression slope of the comparison ( 1.000 ) matches the line of perfect agreement and correlation is good (R 2 = 0. 971 ). PAGE 207 184 Figure 109 Comparison of maximum dimensionless bank shear stress computed by empirical Equation 64 and simulated by the 3D model. Maximum Shear Stress Location The location of maximum shear stress is a function of the meandering channel alignment and the channel width depth ratio. In all 3D model simulations, the location of maximum shear stress occurred downstream from the end of the meander bend and through the beginning of the next meander bend where the there is a reversal of channel curvature. The bank shear stress phase lag is defined as the channel length PAGE 208 185 between the point of maximum curvature ( minimum radius at the bend apex) and the downstream location of maximum bank shear stress. This phase lag is made d imensionless by dividing the channel length by the length of a complete meander bend ( one half of the channel length through one meander wavelength). The dimensionless shear phase lag was correlated with channel sinuosity ( P ) ( Figure 110 ) Phase lags ranged from 0.845 to 1.075. The phase lag data were also segragated by the channel width depth ratio (W/D) into three categories: 5 < W/D < 10 (Simulations Sets 4 and 7) 10 < W/D < 20 (Simulations Sets 5 and 8) 20 < W/D < 84 (Simulations Sets 6 and 9) Figure 110 Correlation between the shear stress phase lag and channel sinuosity. PAGE 209 186 In general, the 3D model simulation results show a trend of decreasing phase lag with increases in channel sinuosity. The initial regression results also indicate that, for a given sinuosity, the phase lag is longest for channels with width ratios less th an 10. Phase lags tend to decrease with increases in width depth ratio. Initial regression equations were developed using 3D model results from Simulation Sets 5 through 9. For width depth ratios between 4 and 10, a second order polynomial regression e quation was found to have a correlation coefficient (R 2 ) of 0.87: Equation 65 This polynomial empirical equation predicts that the phase lag is longest for a sinuosity of 1.1 and at a minimum for a sinuosity of 2.25. For width depth ratios between 10 and 20, a linear regression equation was found to have a correlation coefficient (R 2 ) of 0.89: Equation 66 This linear empirical equation predicts that the phase lag is longest for a sinuosity of 1.10 and decreases with increases in sinuosity. PAGE 210 187 For width depth ratios between 20 and 84, the phase lag is nearly constant with sinuosity and the average phase lag is 0.908. Because the phase lag only varied between 0.8 78 and 0.9 42 (between 3 3 % and + 3 7 % of the mean), the slope of the linear regression equation is relatively flat, so the correlation coefficient (R 2 ) was only 0.32: Equation 67 Maximum Near Bank Velocit ies For each meandering river channel simulation, dimensionless velocities were computed for each 3D model cell by dividing the simulated velocity magnitude by the mean channel velocity. The maximum dimensionless velocity, for both the left and right sides of each model cross section, was plotted against the dimensionless channel curve length ( Figure 90 ). The maximum dimensionless velocity that occurs downstream from the meander bend apex was ploted against the ratio of channel width to minimum radius of curvature ( Figure 111 ). The maximum relative velocities from all channels ranged 1. 205 to 1. 272 which is within 3 percent of the average value of 1. 232 Ther efore, e mpirical regression equations are not better than just using the average average dimensionless velocity PAGE 211 188 Figure 111 Relationship between the maximum near bank velocity and the ratio of channel width to minium radius of curvature for Simulation Sets 4 through 9 PAGE 212 189 CHAPTER VI VALIDATION AND UNCERTAINTY T he empirical regression s equations presented in Chapter V w ere tested by comparison of 3D model results from the vert ual river channels of simulati on sets 1, 2, 3, 10, 11, and 12. For these sets, hydraulics of river channels with sinuosities of 1.75, 2.25, and 2.75 were modeled for validation purposes ( Table 14 ) The rivers channels from these simulation sets have discharges longitudinal slopes, and median sediment grain sizes that are both substantially smaller and larger than the channels of sim ulation sets 4 through 9. Model results of maximum shear stress, phase lag, and maximum velocity were compared for all 12 simulation sets for validation purposes New regression equations were developed using data from all model simulation sets, but th e equations coefficients are all similar. PAGE 213 190 Table 14 Validation data from simulation sets 1, 2, 3, 10, 11, and 12. Simu lation Set Dis charge Channel Sinuosity Channel Slope Width Depth Ratio Manning's Roughness Coefficient Ratio of Channel Width to Minimum Radius of Curvature Maximum Dimen sionless Shear Stress Dimen sionless Phase Lag Q m 3 /s S W / D n W/R c max 1 2.832 2.75 2.48E 04 6.5 0.0275 0.336 1.414 0.872 1 2.832 2.25 3.03E 04 7.4 0.0290 0.381 1.479 0.898 1 2.832 1.75 3.90E 04 8.3 0.0308 0.426 1.544 0.944 2 2.832 2.75 2.25E 03 13.4 0.0456 0.343 1.384 0.841 2 2.832 2.25 2.75E 03 14.6 0.0458 0.388 1.410 0.889 2 2.832 1.75 3.53E 03 16.3 0.0455 0.434 1.466 0.937 3 2.832 2.75 9.79E 03 21.5 0.0511 0.343 1.311 0.896 3 2.832 2.25 1.20E 02 26.8 0.0626 0.387 1.329 0.910 3 2.832 1.75 1.54E 02 36.1 0.0788 0.431 1.353 0.907 10 2,832 2.75 2.59E 06 6.0 0.0142 0.335 1.413 0.986 10 2,832 2.25 3.16E 06 6.9 0.0151 0.379 1.451 0.978 10 2,832 1.75 4.07E 06 7.7 0.0162 0.426 1.527 0.975 11 2,832 2.75 1.78E 05 11.3 0.0238 0.342 1.479 0.883 11 2,832 2.25 2.17E 05 12.2 0.0243 0.387 1.500 0.916 11 2,832 1.75 2.79E 05 13.6 0.0245 0.433 1.516 0.955 12 2,832 2.75 1.02E 04 29.6 0.0314 0.342 1.335 0.916 12 2,832 2.25 1.25E 04 37.8 0.0379 0.386 1.346 0.900 12 2,832 1.75 1.61E 04 51.4 0.0472 0.432 1.376 0.900 PAGE 214 191 Maximum Shear Stress through Meandering Channels The modeled maximum bank shear stress through the meandering channels using data from all simulation sets is presented in Figure 112 The new linear regression lines (solid lines), developed using all data, match very closely with the initial regression lines (dashed lines), developed using data from one half of the simulation sets. Figure 112 Validation of the relationship s between maximum dimensionless shear stress in a channel bend with the r atio of channel width to m in imu m radius of curvature. New regression lines, using all data, are compared with the initial regression lines using data from one half of the simulation sets. PAGE 215 192 The linear regression equation for width depth ratios between 4 and 10 was again used as the basis to predict the maximum shear stress. Model results from Simulation Sets 1, 4, 7, 9, and 12 were used in the regression analysis: Equation 68 This equation was used to compute the maximum shear stress for all 72 simulations. The residual between the maximum shear stress, computed by empirical Equation 68 and the maximum shear stress simulated by the 3D model was plotted against the channel width depth ratio ( Figure 107 ). PAGE 216 193 Figure 113 Validation of the r esidual relationships between maximum shear stress computed by empirical Equation 68 and the maximum shear stress simulated by the 3D model as a function of the channel width depth ratio. The new regression lines, developed using all data, are very similar to the regression lines, developed using data from one half of the data sets. When the new residual regression equations are combined with Equation 68 the following new general equation can be used to estimate the maximum shear stress in a trapezoidal channel: PAGE 217 194 Equation 69 Where A 3D plot of t he dimensionless maximum shear stress, computed from Equation 69 is compared with the maximum shear stress simulated by the 3D model ( Figure 114 ). PAGE 218 195 Figure 114 3D plot of maximum dimensionless shear stress, computed by Equation 69 compared with shear stress simulated by the 3D model. Equation 69 was applied to th e data from simulation sets 1 through 12 to estimate the maximum dimensionless shear stress. These empirical maximum shear stress values were then compared with the maximum shear stress values simulated by the 3D model ( Figure 115 ). The regression slope of the comparison (1.000) still matches the line of PAGE 219 196 perfect agreement and correlation is good (R 2 = 0.9 6 1 ). The confidence interval abou t the estimated value s is 0.031, which represents the 5 and 95 percent confidence limit s. Figure 115 Comparison of maximum dimensionless bank shear stress computed by empirical Equation 69 and simulated by the 3D model. Equation 69 was used to compute the maximum dimensionless shear stress for the laboratory experiments by Ippen and Drinker (1962) ( Figure 116 ). Equation 17 from the Federal Highway Administration (2005) was also used to compute the maximum dimensi onless shear stress and the results are compared to reported measurements. PAGE 220 197 For Equation 69 the average difference between the empirical estimate and the laboratory measurement is 0.10 (underestimate) and the RMS error is 0.17. For Equation 17 the average difference between the empirical estimate and the laboratory measurement is +0.18 (overestimate) and the RMS error is 0.20. Figure 116 Comparison of maximum dimensionless shear stress from empirical estimates with physical model measurements. Empirical estimates are based on Equation 17 and Equation 69 The laboratory flume had a 60 degree channel bend of constant curvature and then a straight channel reach The transition from the curve to the straight channel represents PAGE 221 198 an abrupt change in the derrivitive of the channel alignment. The virtual channels alignments for this dissertation had variable radi i of curvature and very gradual transitions in alignment and the rate of curvature. Therefore, the near bank shear stress predicted by the empirical equation are expected to be somewhat less than the laboratory measurements. Shear Stress Phase Lag The modeled shear stress phase lag through the meandering channels, using data from all simulation sets, is presented in Figure 117 The new linear regression lines (solid lines), developed using all data, match very closely with the initial regression lines (dashed lin es), developed using data from one half of the simulation sets. PAGE 222 199 Figure 117 Testing with validation data of the correlation between the shear stress phase lag and channel sinuosity. Phase lags ranged from 0.841 to 1.075. The phase lag data were again segragated by the channel width depth ratio (W/D) into three categories: 5 < W/D < 10 (Simulations Sets 1, 4, 7, and 10) 10 < W/D < 20 (Simulations Sets 2, 5, 8, and 11) 20 < W/D < 84 (Simulations Sets 3, 6, 9, and 12) PAGE 223 200 New regression equations were developed using 3D model results from all simulation sets. For width depth ratios between 5 and 10, a second order polynomial regression equation was found to have a correlation coefficient (R 2 ) of 0.68: Equation 70 For width depth ratios between 10 and 20, a linear regression equation was found to have a correlation coefficient (R 2 ) of 0.88: E quation 71 For width depth ratios between 20 and 84, the phase lag is nearly constant with sinuosity and the average phase lag is 0.907. Because the phase lag only varied between 0.878 and 0.942 (between 3.2% and +3.9% of the mean), the linear regression equation has a relatively mild slope and the correlation coefficient (R 2 ) was only 0.26: Equation 72 Equation 70 E quation 71 and Equation 72 were applied to the data from simulation sets 1 through 12 to estimat e the phase lag for the maximum dimensionless shear stress. These empirical phase lag channel lengths were then compared with the phase lag values simulated by the 3D model. The regression slope of the comparison ( 0 82 0) PAGE 224 201 somewhat lower than the line of p erfect agreement with a reasonable correlation coefficient (R 2 = 0. 822 ). The confidence interval about the estimated value s is 0.03 7 which represents the 5 and 95 percent confidence limits Figure 118 Comparison of channel phase lag computed by empirical Equation 70 E quation 71 and Equation 72 with the simulated phase lags from the 3D model. PAGE 225 202 Maximum Near Bank Velocities Using data from all simulation sets, the maximum dimensionless velocity that occurs downstream from the meander bend apex was ploted against the ratio of channel width to minimum radius of curvature ( Figure 119 ). The maximum relative velocities from all channels ranged 1. 199 to 1.27 9 which is within 3 .8 percent of the average value of 1.232 0.047 Therefore, e mpirical regression equations are not better than just using the average dimensionless velocity Figure 119 Relationship between the maximum near bank velocity and the ratio of channel width to minium radius of curvature for Simulation Sets 1 through 12 PAGE 226 203 Limitations The limitations of empirical the equations to predict dimensionless near bank shear stress magnitude ( Equation 69 ) and phase lag ( Equation 70 E quation 71 and Equation 72 ) of meandering channels is primarily related to the idealized geometry of the simulated river channels. All the channels simulated by the 3D model have a trapezoidal cross section with 2V:1H side slopes and a flat, transverse bottom slope The width and depth of each simulated channel are constant along the channel length. The simulation of erodible channels or channels with a variable transverse bottom slope was beyond the scope of this research because it would have added another dimension of complexity. Friedkin (1945) conducted laboratory experiments with erodible and meandering channel s. For constructed channels with a constant trapezoidal cross section, zones of high shear stress would be expected to erode the channel bed and banks Over time, this erosion would be expected to create pools, point bars, and possibly riffles. Depending on sediment supply, c hannel width could vary with longitudinal distance. This evolved channel geometry is expected to have some effect on the distribution and magnitude of velocity and shear stress. For example, the formation of point bars and pools would be expected concentrate flow and increase velocity and shear stress along the outside curve of meander bends. Therefore, additional validation of the empirical equations with field data is recommended before their use in design PAGE 227 204 An assumed range of channel sinuosity (1.1 to 3.0) was used to develop the virtual meandering channels for this research. The total curvature of these meandering channel s anged from 69 to 207 degrees ( Figure 120 ) Equation 69 through Equation 72 should work best when applied to meandering channels that are consistent with the relationship presented in Figure 120 Figure 120 Relationship between the total channel curvature and the ratio of channel width to radius of curvature. PAGE 228 205 CHAPTER VII EXAMPLE APPLICATION Stable channel design can be based on the concept of permissible shear stress where the maximum boundary shear stress (times a factor of safety) is less the shear stress necessary to cause erosion. Chow (19 5 9) defined permissible shear stress as the maximum shea r stress that will not cause serious erosion of channel bed material. T he Federal Highway Administration (2005) provides guidelines for the design of r oadside c hannels with f lexible l inings Design examples are provided for a straight channel and a curve d channel. The guideline includes a table of typical permissible shear stresses for bare soil and stone linings ( Table 15 ). PAGE 229 206 Table 15 Typical permissible shear stresses for bare soil and stone linings. Lining Category Lining Type Permissible Shear Stress (N/m 2 ) Cohesive Bare Soil (PI = 10) Clayey sands 1.8 to 4.5 Inorganic silts 1.1 to 4.0 Silty sands 1.1 to 3.4 Cohesive Bare Soil (PI > 20) Clayey sands 4.5 Inorganic silts 4.0 Silty sands 3.5 Inorganic clays 6.6 Non cohesive Bare Soil (PI < 10) Finer than coarse sand ( d 75 < 1.3 mm) 1.0 Fine gravel ( d 75 = 7.5 mm) 5.6 Gravel ( d 75 = 15 mm) 11 Gravel Mulch Coarse gravel ( d 50 = 25 mm) 19 Very coarse gravel ( d 50 = 50 mm) 38 Rock Riprap d 50 = 150 mm 113 d 50 = 300 mm 227 The Federal Highway Administration (2005) guidelines outline the following procedure: 1. Determine the design discharge ( Q ) channel slope ( S ) and cross section shape 2. Initially select the channel bed material or lining. 3. Estimate the channel flow depth ( D ), at the design discharge ( Q ), and compute the hydraulic radius ( R ). 4. n ) and verify the depth assumption by computing the discharge. 5. If the computed discharge differs from the design discharge by more than 5 percent, then assume another flow depth and repeat steps 3 and 4. PAGE 230 207 6. Calculate the maximum shear stress ( d ) for a strai ght channel select and apply a safety factor (S.F.) and the determine permissible shear stress ( p ) based on the bed material or lining type from Table 15 Also, c ompute the maximum shear stress for the curved reach. 7. If the maximum shear stress ( S.F. d ) through the straight or curved reach is less then the permissible shear stress ( p ), then the lining is acceptable for that reach Otherwise, a new lining has to b e selected and steps 2 through 7 are repeated. Equation 10 and Equation 11 for depth averaged velocity, can be used to compute the boundary shear stress in rough, straight channel with normal depth. A safety factor ( between 1.0 and 1.5 ) is applied to the computed shear stress for channel design purposes. For curved channels, either Equation 16 and Equation 17 ( Federal Highway Administration 2005 ) or Equation 69 (from the research of this dissertation) are applied to determine the increase in shear stress relative to straight channels. Basic Channel Design Example 1 The design example from the Federal Highway Administration (2005 ) is used but with a curve channel, to illustrate the procedures of steps 1 to 7 For this example, e valuate PAGE 231 208 the stability of a proposed gravel mulch lining for a curved trapezoidal channel, given the following conditions: Design d ischarge, Q = 0.42 0 m 3 /s Channel bottom width B = 0.4 00 m Channel side slope, z = 3 .00 Longitudinal channel slope, S o = 0.008 00 Median bed material grain size, d 50 = 25 .0 mm Radius of curvature R c = 6.84 m The above channel conditions satisfy steps 1 and 2. Solution 1, Part A Step 3, e stimate that the channel flow depth ( D ) at the design discharge, is 0. 344 m T hen c ompute the hydraulic radius ( R ) from the cross sectional area ( A ) and wetted perimeter ( P ): PAGE 232 209 Step 4, e from Table 2 2 of the Federal Highway Administration guideline ( 2005) to be 0.035 T hen compute the mean flow velocity and discharge Step 5, check to see if the computed discharge is within 5% of the design discharge. Indeed, the computed discharge is nearly the sames as the design discharge, so proceed to Step 6. Step 6, c ompute the maximum shear stress in a straight channel using Equation 1 : Assume a safety factor of 1. 3 PAGE 233 210 The maximum shear stress for the straight reach exceeds the permissible shear stress for the gravel mulch lining For the Federal Highway Administration (2005) method, use Equation 16 and Equation 17 to compute the increased shear stress due to the channel curve. where, For the method developed in this dissertation, u s e Equation 69 to compute the increased shear stress due to the channel curve. PAGE 234 211 For design purposes, add 0.031: The maximum shear stress through the curved reach is estimated at Step 7, the permissible shear stress (from Table 15 ) for the gravel multch lining, with a median grain size of 25 mm, is 19 N/m 2 T he maximum shear stress with a safety factor, through the curved reach (5 3 N / m 2 or 6 6 N / m 2 ) is greater than permissible shear stress of 19 N/m 2 Therefore, a more erosion resistant lining must be chosen (Step 2) for both the straight and curved sections : A r ock riprap lining with a median rock size ( d 50 ) of 150 mm is chosen and Steps 3 through 7 are then repeated. Solution 1, Part B Step 3, e stimate that the channel flow depth ( D ) at the design discharge, is 0. 461 m, then compute the hydraulic radius ( R ) from the cross sectional area ( A ) and wetted perimeter ( P ): PAGE 235 212 Step 4, e from Table 2 2 of the Federal Highway Administration guideline ( 2005) to be 0.0 69 T hen compute the mean flow velocity and discharge Step 5, the computed discharge matches the design discharge so proceed to Step 6. Step 6, c ompute the maximum shear stress in a straight channel using Equation 1 : Assume a safety factor of 1.3. PAGE 236 213 The maximum shear stress for the straight reach is less than the permissible shear stress for the r ock riprap lining For the Federal Highway Administration (2005) method, use Equation 16 and Equation 17 to compute the increased shear stress due to the channel curve. where, For the method developed in this dissertation, u s e Equation 69 to compute the increased shear stress due to the channel curve. PAGE 237 214 For design purposes, add 0.031: The maximum shear stress through the curved reach is estimated at This computed shear stress is 83 percent of the shear str ess computed using the Federal Highway Administration equation. Step 7, the permissible shear stress (from Table 15 ) for the rock riprap lining, with a median grain size of 150 mm, is 1 13 N/m 2 T he maximum shear stress, with a safety factor, through the curved reach ( 78 N / m 2 or 94 N / m 2 ) is less than the permissible shear stress Therefore, the chosen riprap lining is acceptable for both the straight and curved section s PAGE 238 215 Basic Channel Design Example 2 A different example is presented with more discharge, milder longitudinal slope, and a wider channel Evaluate the stability of a proposed gravel mulch lining for a curved trapezoidal channel, given the following conditions: Design d ischarge, Q = 4 2 0 m 3 /s Channel bottom width B = 8 00 m Channel side slope, z = 3 .00 Longitudinal channel slope, S o = 0.0 0 08 00 Median bed material grain size, d 50 = 25 .0 mm Radius of curvature, R c = 33 6 m The above channel conditions satisfy steps 1 and 2. Solution 2 Step 3, e stimate that the channel flow depth ( D ) at the design discharge, is 0. 686 m T hen compute the hydraulic radius ( R ) from the cross sectional area ( A ) and wetted perimeter ( P ): PAGE 239 216 Step 4, e from Table 2 2 of the Federal Highway Administration guideline ( 2005) to be 0.03 1 5 T hen compute the mean flow velocity and discharge Step 5, the computed discharge matches the design discharge, so proceed to Step 6. Step 6, c ompute the maximum shear stress in a straight channel using Equation 1 : Assume a safety factor of 1.3. The maximum shear stress for the straight reach is less than the permissible shear stress for the r ock riprap lining and also less than the permissible shear stress (11 N/m 2 ) for the finer gravel lining (d 75 = 15 mm) PAGE 240 217 For the Federal Highway Administration (2005) method, use Equation 16 and Equation 17 to compute the increased shear stress due to the channel curve. where, For the method developed in this dissertation, u s e Equation 69 to compute the increased shear stress due to the channel curve. PAGE 241 218 For design purposes, add 0.031: The maximum shear stress through the curved reach is estimated at T his computed shear stress is 76 percent of the shear stress computed using the Federal Highway Administration equation, which does not consider the channel width depth ratio. Step 7, the permissible shear stress (from Table 15 ) for the gravel multch lining, with a median grain size of 25 mm, is 19 N/m 2 T he maximum shear stress, with a safety factor, through the curved reach is (9.86 N / m 2 or 13 N / m 2 ) Therefore the chosen riprap lining is acceptable. Based on Equation 69 of this dissertation, a finer g ravel lining (d75 = 15 mm) could be used for both the straight and curved sections. PAGE 242 219 CHAPTER VII I CONCLUSIONS Conclusions are provided on the application of numerical models, empirical equations, model boundary conditions, and suggestions for future research. Application of N umerical M odels for R esearc h The application of a verified 3D numerical hydraulics model to develop detailed data sets representing a wide range of river channels is a useful tool for studying hydraulic relationships. Numerical modeling does not replace the need for detailed hyd raulic measurement from natural channels However, the logistic s of t iming such field measurements during a discharge of interest and finding reaches without anomalies is challenging. Detailed hydraulic measurements laboratory flumes is also valuable. However, t he construction of a wide range of different channel geometries in the laboratory can be cost prohibitive The use of a 3D numerical model provides another useful tool for the study of hydraulics. The numerical simulation of river h ydraulics for this dissertation produced data sets representing a wide range of meandering river channels of the type (width and depth) found in nature The range i n bankful discharge for the model simulation sets spanned three orders of magnitude (2.82, 28.3, 283, and 2,830 m 3 /s) The range in channel slope PAGE 243 220 of the data sets spanned four orders of magn i tude (2.37 x 10 6 to 2.45 x 10 2 ). Channel w idth depth ratios ranged from 5 to 84 and median sediment grain sized ranged from 0.125 to 4 mm. Empirical Eq uations Empirical analysis of the 3D model data sets for meandering channels indicates that the maximum dimensionless near bank shear stress is linearly correlated with the ratio of channel width to minimum radius of curvature and the channel width depth ratio. Shear stress linearly increases with an increase in the ratio of width to radius of curvature. Shear stress for a given ratio of width to radius linearly decreases w ith and increase in the width depth ratio. For non erodible meandering channels of constant trapezoidal cross section, the location of maximum bank shear stress is downstream from the end of the meander bend and within the next meander bend where there is a reversal of channel curvature. The phas e lag between the location of minimum radius of curvature and maximum shear stress is correlated with channel sinuosity and the width depth ratio. For wide channels (width depth ratios greater than 10), the phase lag linearly decreases with increases in sinuosity. For wider channels (width depth ratios greater than 2 0), the phase lag is nearly constant (0.93). For narrow channels (width depth ratios less than PAGE 244 221 10), the phase lag parabolically varies with sinuosit y, with minimum phase lag near a sinuosity of 2.25. The relative maximum velocity in meandering channels does not vary substantially from an average value of 1.23. The maximum velocity is typically at the water surface and away from the channel boundari es. Because the maximum velocity does not vary much with channel curvature boundary shear stress is likely a better predictor of stream bank and bed erosion. Model Boundary Conditions Evaluation of 3D model results indicates that the user specified up stream and downstream boundary conditions need to be far away from the areas of interest. The downstream boundary water surface has the potential to influence depths through the downstream meander bend. The assumed velocity distribution at the upstream b oundary will influence the velocity and shear stress distribution through the first meander bend. Therefore, laboratory flume studies of meandering channels need successive downstream channel bends to demonstrate the repeatability of the measurements. La boratory experiments of a single channel curve may not produce results that are repeatable through a second downstream curve At least three curves may be necessary to achieve repeatable results PAGE 245 222 Suggestions for Future Research Maximum, dimensionless near bank shear stress though meandering river channels has been found to linearly correlate with the ratio of channel width to radius of curvature and the width depth ratio However, more research is needed to determine the influence of a variable transv erse bottom slope and variable bank slope on the maximum shear stress through a meander bend. The application of a 3D model to investigate the influence of variable transvers bed and bank slopes on shear stress would be useful. Laboratory flume experimen t s would also be very useful and could be guided by numerical model results. More research on the effect of the total curvature through a channel bend on the maximum shear stress also may be needed. 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