PHYSICAL MODELING FOR SIDE-CHANNEL WEIRS by Ka-Leung Lee and E. R. Holley Center for Research in Water Resources The University of Texas at Austin Austin, TX 78712 USA CRWR Online Report 02-2 http://www.crwr.utexas.edu/online.shtml prepared for Harris Flood Control District Suite 200 9900 Northwest Freeway Houston, TX 77092 USA April 15, 2002 i TABLE OF CONTENTS 1 - Introduction.................................................................................................................................1 1.1 - Regional Basins .................................................................................................................. 1 1.2 - Objectives ........................................................................................................................... 2 2 - Background.................................................................................................................................5 2.1 - Side Weir Flow Conditions ................................................................................................ 5 2.2 - No Weir Flow ..................................................................................................................... 6 2.3 - Reverse Flow ...................................................................................................................... 6 2.4 - Forward Flow ..................................................................................................................... 7 2.4.1 - Water Surface Profiles................................................................................................. 7 2.4.2 - Previous Work of Others ............................................................................................. 8 2.5 - Previous Work at CRWR on Hydraulic Computations.................................................... 10 2.5.1 - Purpose ...................................................................................................................... 10 2.5.2 - Forward Weir Flow.................................................................................................... 11 2.5.2.1 - Method of Analysis in the Previous Project ....................................................... 11 2.5.2.2 - Flow Asymmetry ................................................................................................ 12 2.5.2.3 - Downstream Depth ............................................................................................. 13 2.5.2.4 - Kinetic Energy and Momentum Correction Factors........................................... 14 2.5.2.5 - Weir Discharge ................................................................................................... 15 2.5.2.6 - Upstream Depth.................................................................................................. 15 2.5.3 - Reverse Weir Flow .................................................................................................... 16 2.5.3.1 - General Approach............................................................................................... 16 2.5.3.2 - Discharge Coefficients........................................................................................ 16 2.5.3.3 - Submergence Correction Factor ......................................................................... 17 2.5.3.4 - Channel Depths and Additional Head Change ................................................... 17 2.6 - Previous Experimental Work at CRWR........................................................................... 18 2.6.1 - Introduction................................................................................................................ 18 2.6.2 - Results for Unsubmerged Forward Flow................................................................... 18 2.6.3 - Results for Submerged Forward Flow....................................................................... 19 2.6.4 - Unsubmerged Flow in Tapered Channels.................................................................. 20 2.6.5 - Results for Weirs Downstream of Bends................................................................... 21 2.6.6 - Reverse Flow ............................................................................................................. 21 2.7 - Valves on Drainage Culverts............................................................................................ 22 2.7.1 - Tideflex Valves.......................................................................................................... 22 2.7.1.1 - General................................................................................................................ 22 2.7.1.2 - Manufacturer's Information ................................................................................ 22 2.7.1.3 - Calculation Method for Submerged Valves ....................................................... 26 2.7.1.4 - Calculation Method for Unsubmerged Valves ................................................... 28 2.7.2 - Flap Gates .................................................................................................................. 31 ii 3 - Experimental Facilities.............................................................................................................33 3.1 - Physical Model for Previous Project ................................................................................ 33 3.2 - Modification of the Physical Model to 4H:1V Side Slopes ............................................. 36 3.3 - Measurement of Discharges ............................................................................................. 38 3.3.1 - Calibration of Venturi Meter ..................................................................................... 38 3.3.2 - Calibration of V-Notch Weir..................................................................................... 39 3.3.3 - Calibration of Flow Sensor........................................................................................ 40 3.4 - Measurement of Velocities............................................................................................... 41 3.5 - Measurement of Water Surface Elevations ...................................................................... 43 4 - Re-Evaluation of Side Weir Discharge Coefficients ................................................................45 4.1 - Derivation of Equation for Changes in Depth .................................................................. 45 4.1.1 - Unsubmerged Flow in Prismatic Channels................................................................ 49 4.1.2 - Submerged Flow in Prismatic Channels.................................................................... 51 4.1.3 - Unsubmerged Flow in Tapered Channels.................................................................. 51 4.2 - Optimization and Regression Analysis for Channel with 2.5H:1V Side Slopes.............. 51 4.2.1 - Unsubmerged Flow in Prismatic Channels with 2.5H:1V Side Slopes..................... 51 4.2.1.1 - Constant Discharge Coefficient.......................................................................... 51 4.2.1.2 - Variable Discharge Coefficient .......................................................................... 52 4.2.2 - Submerged Flow in Prismatic Channels with 2.5H:1V Side Slopes......................... 53 4.2.3 - Unsubmerged Flow in Tapered Channels.................................................................. 54 4.3 - Comparison between Measured and Calculated Values for 2.5H:1V Side Slopes .......... 55 4.4 - Effects of Channel Slope and Roughness for 2.5H:1V Side Slopes ................................ 69 5 - Discharge and Head Loss Experiments for 4:1 Side Slopes.....................................................73 5.1 - Introduction ...................................................................................................................... 73 5.2 - Model results .................................................................................................................... 73 5.2.1 - Analysis of Data using Method A.............................................................................. 73 5.2.2 - Analysis of Data using Method B.............................................................................. 76 6 - Flow Asymmetry ......................................................................................................................85 6.1 - Introduction ...................................................................................................................... 85 6.1.1 - Background................................................................................................................85 6.1.2 - Related Literature ...................................................................................................... 85 6.1.3 - Objective.................................................................................................................... 87 6.2 - Equations for the Channel Flow ....................................................................................... 87 6.3 - Flow Conditions ............................................................................................................... 89 iii 6.4 - Velocity Measurements .................................................................................................... 90 6.4.1 - Measurement Procedures........................................................................................... 91 6.4.2 - Integrations ................................................................................................................ 92 6.4.3 - Case A........................................................................................................................ 92 6.4.4 - Case B........................................................................................................................ 93 6.4.5 - Case C........................................................................................................................ 94 6.4.6 - Case D........................................................................................................................ 96 6.4.7 - Case E ........................................................................................................................ 97 6.4.8 - Case F ........................................................................................................................ 97 6.4.9 - Case G........................................................................................................................ 98 6.4.10 - Case H...................................................................................................................... 99 6.5 - ? and ? Values at Downstream End of Weir.................................................................. 100 6.6 - Components of ? and ?.................................................................................................. 102 6.6.1 - Variation with Flow Distance.................................................................................. 102 6.6.2 - Variation with Diversion ......................................................................................... 106 6.7 - Length for Flow Re-Establishment................................................................................. 106 6.8 - Momentum and Energy Balances................................................................................... 109 6.9 - Application ..................................................................................................................... 112 7 - Diversion Culverts..................................................................................................................115 7.1 - Introduction .................................................................................................................... 115 7.2 - The physical model......................................................................................................... 115 7.3 - Model results .................................................................................................................. 117 7.3.1 - Unsubmerged flow................................................................................................... 117 7.3.2 - Submerged flow....................................................................................................... 118 7.4 - Calculation Procedure..................................................................................................... 120 8 - Conclusions ............................................................................................................................121 9 - References...............................................................................................................................125 iv 10 - Appendices ...........................................................................................................................127 Appendix 1 - Data from previous project (Tynes, 1989).........................................................127 Appendix 1.1 - Unsubmerged Flow Conditions ..................................................................127 Appendix 1.2 - Submerged Flow Conditions.......................................................................133 Appendix 1.3 - Tapered Channels........................................................................................134 Appendix 2 - Weir and channel geometries investigated in previous project for unsubmerged flow .............................................................................................137 Appendix 3 - Results of simulation of side weir flow for different slopes and roughness......139 Appendix 3.1 - Results of simulation using Method A .......................................................140 Appendix 3.2 - Results of simulation using Method B........................................................151 Appendix 4 - Summary of model data for 4H:1V side slopes.................................................163 Appendix 5 - Components of ? and ? .....................................................................................165 Appendix 5.1 - Variation of components of ? and ? with distance for 2.5H:1V side slopes...........................................................................................................165 Appendix 5.2 - Variation of components of ? and ? with distance for 4H:1V side slopes...........................................................................................................166 Appendix 5.3 - Components of ? and ? just downstream of weir for 2.5H:1V side slopes...........................................................................................................166 Appendix 5.4 - Components of ? and ? just downstream of weir for 4H:1V side slopes...........................................................................................................167 Appendix 6 - Momentum and energy balances .......................................................................169 Appendix 6.1 - Momentum balance for Case A ..................................................................169 Appendix 6.2 - Energy balance for Case A..........................................................................169 Appendix 6.3 - Momentum balance for Case B...................................................................170 Appendix 6.4 - Energy balance for Case B..........................................................................170 Appendix 6.5 - Momentum balance for Case C...................................................................171 Appendix 6.6 - Energy balance for Case C..........................................................................171 Appendix 6.7 - Momentum balance for Case D ..................................................................172 Appendix 6.8 - Energy balance for Case D..........................................................................172 Appendix 6.9 - Momentum balance for Case F...................................................................173 Appendix 6.10 - Energy balance for Case F ........................................................................173 Appendix 7 - Data for diversion culverts.................................................................................175 Appendix 7.1 - Results for diversion culverts with three barrels, unsubmerged flow.........175 Appendix 7.2 - Results for diversion culverts with three barrels, submerged flow.............176 Appendix 7.3 - Results for diversion culverts with two barrels, unsubmerged flow...........176 Appendix 7.4 - Results for diversion culverts with two barrels, submerged flow...............177 v LIST OF FIGURES Fig. 2.1 - Schematic diagram of side-channel weir..........................................................................5 Fig. 2.2 - Water surface profiles in a channel beside a side-weir ....................................................8 Fig. 2.3 - Definition sketch for side-channel weirs........................................................................12 Fig. 2.4 - Assumed velocity distributions and effective flow area.................................................14 Fig. 2.5 - Tideflex valves (from Red Valve Co., Inc. catalog).......................................................23 Fig. 2.6 - Sample of curves obtained from Red Valve Co., Inc. for Tideflex valves.....................24 Fig. 2.7 - Q half for Tideflex valves .................................................................................................25 Fig. 2.8 - Scaled downstream areas for Tideflex valves ................................................................25 Fig. 2.9 - Head loss coefficients for Tideflex valves .....................................................................26 Fig. 2.10 - Height of Tideflex valves.............................................................................................29 Fig. 2.11 - Assumed open shape of Tideflex valves ......................................................................29 Fig. 3.1 - Plan and elevation views of the model weir...................................................................34 Fig. 3.2 - A photograph of the model and weir..............................................................................35 Fig. 3.3 - Longitudinal profiles of channel invert and weir crest (arbitrary datum).......................37 Fig. 3.4 - Low flow calibration of Venturi meter...........................................................................39 Fig. 3.5 - Calibration of V-notch weir ...........................................................................................40 Fig. 3.6 - Low-flow calibration of V-notch weir ...........................................................................40 Fig. 3.7 - Calibration of flow sensor ..............................................................................................41 Fig. 3.8 - Effects of averaging time on longitudinal velocities......................................................42 Fig. 4.1 - Definition sketch for channel with lateral flow..............................................................46 Fig. 4.2 - Comparison of measured and calculated model values of side-weir discharge for unsubmerged flow using Method A.......................................................................57 Fig. 4.3 - Comparison of measured and calculated model values of upstream head on the weir for unsubmerged flow using Method A .........................................................57 Fig. 4.4 - Comparison of measured and calculated model values of side-weir discharge for submerged flow using Method A...........................................................................58 Fig. 4.5 - Comparison of measured and calculated model values of upstream head on the weir for submerged flow using Method A .............................................................58 Fig. 4.6 - Comparison of measured and calculated model values of side weir discharge for tapered channels using Method A..........................................................................59 Fig. 4.7 - Comparison of measured and calculated model values of upstream head on the weir for tapered channels using Method A ............................................................59 Fig. 4.8 - Comparison of measured and calculated model values of side weir discharge for unsubmerged flow using Method B.......................................................................60 Fig. 4.9 - Comparison of measured and calculated model values of upstream head on the weir for unsubmerged flow using Method B .........................................................60 Fig. 4.10 - Comparison of measured and calculated model values of side weir discharge for submerged flow using Method B .....................................................................61 Fig. 4.11 - Comparison of measured and calculated model values of upstream head on the weir for submerged flow using Method B .............................................................61 Fig. 4.12 - Comparison of measured and calculated model values of side weir discharge for tapered channels using Method B ....................................................................62 vi Fig. 4.13 - Comparison of measured and calculated model values of upstream head on the weir for tapered channels using Method B ............................................................62 Fig. 4.14 - Comparison of measured and calculated model values of side weir discharge for unsubmerged flow using Method C .................................................................63 Fig. 4.15 - Comparison of measured and calculated model values of upstream head on the weir for unsubmerged flow using Method C .........................................................63 Fig. 4.16 - Comparison of measured and calculated model values of side weir discharge for unsubmerged flow using Method D .................................................................64 Fig. 4.17 - Comparison of measured and calculated model values of upstream head on the weir for unsubmerged flow using Method D .........................................................64 Fig. 5.1 - Observed and calculated C e values.................................................................................74 Fig. 5.2 - Measured and calculated h u using Method A.................................................................75 Fig. 5.3 - C 1 from regression equation and from optimization with C 2 = 0.85 .............................77 Fig. 5.4 - Measured and numerically optimized h u with C 2 = 0.85 ...............................................78 Fig. 5.5 - C 1 from regression equation and from optimization ......................................................79 Fig. 5.6 - Values of h u from regression equation and from optimization ......................................80 Fig. 5.7 - Measured and calculated Q w using Method B................................................................81 Fig. 5.8 - Measured and calculated h u using Method B.................................................................82 Fig. 6.1 - Typical locations of velocity measurements looking downstream.................................92 Fig. 6.2 - Longitudinal distributions of velocity at downstream end of weir crest (Case A).........93 Fig. 6.3 - Longitudinal distributions of velocity 4.3 ft downstream from end of weir crest (Case A) .................................................................................................................93 Fig. 6.4 - Longitudinal distributions of velocity 22.3 ft downstream from end of weir crest (Case A) .................................................................................................................94 Fig. 6.5 - ? and ? values for Cases A - C ......................................................................................94 Fig. 6.6 - Longitudinal distributions of velocity 4.4 ft downstream from end of weir crest (Case C) .................................................................................................................95 Fig. 6.7 - Left boundaries of regions from which weir flow comes...............................................96 Fig. 6.8 - ? and ? values for Case D..............................................................................................97 Fig. 6.9 - Longitudinal distributions of velocity 2.5 ft from downstream end of weir crest (Case F)..................................................................................................................98 Fig. 6.10 - Variation of ? and ? (Case F) ......................................................................................99 Fig. 6.11 - Variation of ? and ? (Case G)....................................................................................100 Fig. 6.12 - ? and ? values at end of weir for 2.5H:1V side slopes ..............................................101 Fig. 6.13 - ? and ? values at end of weir for 4H:1V side slopes .................................................102 Fig. 6.14 - Assumed velocity distribution for calculating B s .......................................................103 Fig. 6.15 - Components of ? and ? for 54% diversion (Cases A and B).....................................103 Fig. 6.16 - Components of ? and ? for 25% diversion (Case C).................................................104 Fig. 6.17 - Components of ? and ? for forced separation zone (Case D)....................................104 Fig. 6.18 - Components of ? and ? for forced separation zone (Case F).....................................105 Fig. 6.19 - Components of ? and ? for forced separation zone (Case G)....................................105 Fig. 6.20 - Variation of components of ? and ? near end of weir with diversion for 2.5H:1V side slopes .............................................................................................107 vii Fig. 6.21 - Variation of components of ? and ? near end of weir with diversion for 4H:1V side slopes ............................................................................................................107 Fig. 6.22 - Exponential decay of excess ? for 2.5H:1V side slopes ............................................108 Fig. 6.23 - Exponential decay of excess ? for 4H:1V side slopes ...............................................109 Fig. 6.24 - Length of flow re-establishment region......................................................................110 Fig. 7.1 - Schematic diagram of diversion culverts in model (not to scale) ................................115 Fig. 7.2 - Diversion culverts.........................................................................................................116 Fig. 7.3 - Adjusted loss coefficients for flow from point 0 to point 2 .........................................119 ix LIST OF TABLES Table 3.1 - Effects of turbulent averaging time on velocities........................................................42 Table 4.1 - t-statistics for coefficients in Eq. (4.23) ......................................................................52 Table 4.2 - t-statistics for coefficients in Eq. (4.24) ......................................................................53 Table 4.3 - t-statistics for coefficients in Eq. (4.25) ......................................................................54 Table 4.4 - t-statistics for coefficients in Eq. (4.26) ......................................................................54 Table 4.5 - t-statistics for coefficients in Eq. (4.27) ......................................................................54 Table 4.6 - Methods of calculating Q w and h u ...............................................................................55 Table 4.7 - Statistics of differences between measured and calculated model values of Q w and h u (Method A) .................................................................................................65 Table 4.8 - Statistics of differences between measured and calculated model values of Q w and h u (Method B)..................................................................................................66 Table 4.9 - Statistics of differences between measured and calculated model values of Q w and h u (Method C) .................................................................................................67 Table 4.10 - Statistics of differences between measured and calculated model values of Q w and h u (Method D) ...........................................................................................67 Table 4.11 - Comparison between Method A and Method B ........................................................68 Table 4.12 - R 2 (Q w ) and R 2 (h u ) for comparison between measured and calculated values of Q w and h u ...........................................................................................................70 Table 4.13 - Geometric conditions used in simulation ..................................................................70 Table 4.14 - Largest differences between values of h u calculated from Methods A and B ...........71 Table 4.15 - Largest ratios between values of Q w calculated from Methods A and B ..................71 Table 5.1 - rms of ?Q w , ?Q w /Q w(mea) , ?h u and ?h u /h u(mea) ...........................................................83 Table 6.1 - Flow conditions for Type 1..........................................................................................90 Table 6.2 - Flow conditions for Type 2..........................................................................................91 Table 6.3 - Summary of errors in balancing momentum and energy equations ..........................111 xi ACKNOWLEDGMENTS This project was supported by the Harris County Flood Control District. Mr. Steve Fitzgerald was very helpful with both the technical and administrative aspects of this project. The project was conducted at the Center for Research in Water Resources of the Univer- sity of Texas at Austin. Kevin Wei assisted with some of the experimental work. Part of the project work is reported by Burgin and Holley (2002). That report is a user's manual for the computational scheme that has been developed for watershed hydrology, channel and side-weir hydraulics, and filling and emptying of detention basins. Red Valve Co., Inc., Pittsburgh, Pennsylvania, provided information on their Tideflex check valves as presented in Section 2.7.1. 1 PHYSICAL MODELING FOR SIDE-CHANNEL WEIRS By Ka-Leung Lee and E. R. Holley Center for Research in Water Resources The University of Texas at Austin Austin, TX 78712 1 - INTRODUCTION 1.1 - REGIONAL BASINS As watersheds become urbanized, the additional impervious cover and land improve- ments produce an increase in the volume and speed of storm water runoff. In consequence, downstream flooding becomes more recurrent and more severe, motivating the affected property owners to demand that restrictions be placed on the further land development in the watershed. The conflict between upstream and downstream interests has led many jurisdictions to adopt regulations allowing new development only when it causes no increase in the maximum discharge downstream. Developers can satisfy the regulations by using detention basins to reduce peak flow rates. Onsite detention provides temporary storage for excess discharges near their source, serving to redistribute the excess runoff from a single development. A more comprehensive solution employs one or more regional detention basins to consolidate the capacity of a number of separate, small detention facilities into fewer and larger facilities. Storm runoff is allowed to enter a receiving channel. If flow in the channel approaches that which will cause flooding, a portion of the flow is diverted into a regional deten- tion basin for temporary storage. When the flow in the channel has decreased sufficiently on the falling limb of the hydrograph, the water stored in the regional detention basin is released back into the stream. A side-channel weir can be used as the structure that diverts excess discharges from the main channel into the regional detention basin. This report presents a method to assist in designing side-channel weir and detention systems. To model the performance of a trial design, the method connects a hydrologic model, a channel hydraulics model, and a side-discharge hydraulics model into a recursive system that adjusts assumed diversions until they are matched by calculated diversions. 2 1.2 - OBJECTIVES In a previous project conducted at the Center for Research in Water Resources (CRWR) and sponsored by the Harris County Flood Control District (HCFCD), experiments were conducted (Tynes, 1989) to determine the hydraulic characteristics of embankment-shaped side weirs, and a design and modeling method (Davis and Holley, 1988) was developed for side weirs. The previous method used manual iteration between HEC-1, HEC-2, and a new program, SIDEHYDR, which was developed specifically for the task of modeling flow beside and over side-discharge weirs. The present project has built directly on the work done in the previous project. The objectives of the present project were as follows: 1. Develop a computer program to automatically perform the iterations between the programs HEC-1, HEC-2, and SIDEHYDR for the design of side-channel diversion weirs; 2. Add "pop-up" screens for input and for graphical display of the results of the iterations on the computer monitor; 3. Identify the source of computational oscillations in the computer program SIDEHYDR and change the program to remove the oscillations; 4. Prepare a user?s manual for the entire computational package of programs, including an improved treatment of the potential pitfalls and error messages in the SIDEHYDR program; 5. Expand the SIDEHYDR program to calculate culvert drainage of water stored in the deten- tion basin below the weir crest; 6. Modify the SIDEHYDR program to allow the choice of either side weirs or culverts for flow diversion; 7. Conduct hydraulic model experiments to evaluate the effects of channel side slopes on side weir hydraulics; 8. Modify the existing side weir physical model and conduct experiments to determine the size and hydraulic effects of the separation zone created in the main channel by the side weir diversion flow; 9. Reanalyze data from the previous project and use computations of water surface profiles along side weirs to evaluate the potential effects of channel slope and roughness on weir hydraulics; 10. Conduct experiments to evaluate the effects of channel flow on the hydraulics of culverts used for diversion and basin drainage at detention facilities, 3 11. Change the method used in the computational program for flow from the channel into the detention basin based on the results from Tasks 7 and 8, 12. Extend the work of Task 8 to include channels with 4H:1V side slopes. Only subcritical channel flows are considered in the computational methods and experiments in this report. Tasks 7 ? 10 and 12 are all related to experimental work and are addressed in this report. The other tasks are related to the computational scheme and are addressed in a companion report (Burgin and Holley, 2002). The computational scheme presented in that report uses the empirical results given in this report. 5 2 - BACKGROUND As the name implies, side-channel weirs (Fig. 2.1) are placed along the side of a channel parallel (or at a small angle relative to) the flow in the channel. The crest elevation, the crest length, and the length of the weir can be designed to control the operating characteristics of the weir. weir channel flow Fig. 2.1 - Schematic diagram of side-channel weir 2.1 - SIDE WEIR FLOW CONDITIONS There are three general types of flow conditions that can exist with side weirs: (a) During the rising and falling parts of the hydrograph when the water level in the channel is lower than the weir crest, gradually varied flow exists in the channel section where the weir is located. (b) When the water level in the channel is above the weir crest and above the water level in the basin, forward flow takes place from the channel into the basin. (c) If the basin fills to the point that the water level in the basin is above the weir crest, reverse flow from the basin back into the channel will occur when the water level in the channel fall below the water level in the basin during hydrograph recession. Depending on the relative values of the heads on the weir from both the channel and the basin sides, the weir flow in both directions may have either free or submerged flow conditions. A brief summary is given first for no weir flow and for reverse flow from the basin into the river channel, and then a more detailed treatment is given for forward flow from the channel into the basin. 6 2.2 - NO WEIR FLOW When there is no flow in either direction over the weir, HEC-2 could be used for compu- tations as if the weir were not present. However, the overall computational scheme is based on using HEC-2 in the sections of the channel with no weir for diversion and using the program SIDEHYD, which is a revised version of SIDEHYDR from the previous project (Davis and Holley, 1988), for computations in the channel where the weir is located. Thus, when there is no flow over the weir, the depth at the downstream end of the weir is taken from the HEC-2 com- putations. Then SIDEHYD computes the water surface profile in the part of the channel where the weir is located. This computation is based on the differential momentum equation, which can be written for gradually varied flow (Yen and Wenzel, 1970) as ? ? ? y x = ? ? SS 1Fr of 2 (2.1) where y = flow depth in the channel, x = longitudinal distance which is positive in the flow direc- tion, S o = bed slope, S f = friction slope, ? = momentum correction factor, and Fr = channel Froude number which is defined as T A g U Fr = (2.2) where U = average channel velocity (Q/A), A = channel flow area and T = top width of flow. These computations in SIDEHYD give the water surface elevation at the upstream end of the weir. This elevation is put into the HEC-2 input file for restarting the HEC-2 calculations for the channel upstream of the weir. 2.3 - REVERSE FLOW For reverse flow from the basin back into the channel, the weir behaves as a normal weir rather than as a side weir. The discharge equation for normal broad-crested weirs can be written as 2/3 snw h)x(g 3 2 3 2 CCQ ?= (2.3) where Q w = weir discharge, C n = discharge coefficient for a broad-crested weir, C s = submer- gence correction factor, g = acceleration due to gravity, h = head on the weir, and ?x = increment of length along the weir crest. Eq. (2.3)assumes that the approach velocity is small, as it should be since the flow back over the weir is coming from the detention basin. The sign convention is 7 that reverse flow from the basin to the channel is positive (Eq. (2.3)) while forward flow from the channel to the basin is negative (Eq. (2.4), Eq. (2.7)). If the weir crest is inclined (e.g., parallel to the invert of an improved channel), then the head on the weir will decrease in the upstream direc- tion since the water level in the detention basin will normally be horizontal. For reverse flow with subcritical channel flow, both the head loss in the channel due to the disturbance caused by the flow coming over the weir and the increasing discharge in the downstream direction mean that the depth in the channel decreases in the downstream direction. As a result of this change of depth in the channel, C s can vary along the length of a weir when submerged flow conditions exist. 2.4 - FORWARD FLOW Flow over side weirs depends on the head on the weir, among other factors. The head depends on the water surface profile along the channel where the diversion is taking place. While the primary factor affecting the water surface profile is the diversion itself, the channel slope and roughness also have an effect on the water surface profile just as they do in a channel without a side weir. Depending on the flow conditions and the channel geometry, the flow over the side weir will cause the flow remaining in the channel to develop a lateral distribution of velocity that is asymmetrical and may cause the flow to separate from the side of the channel opposite the weir. 2.4.1 - Water Surface Profiles Some of the possible longitudinal water surface profiles in a channel along a side weir for forward flow from the channel into the basin are illustrated in Fig. 2.2, which has been adapted from Henderson (1966). There are several things that are illustrated or implied in this figure that have a direct bearing on the flow diversion problem. One is that, for subcritical flow (Fig. 2.2a), the water surface elevation usually increases in the downstream direction. The second thing is that it is possible to have a hydraulic jump (Fig. 2.2c) in the channel because of the outflow. The possible occurrence of the jump depends on the hydraulics of the outflow and does not require supercritical flow in the channel upstream of the weir. Thus, in a channel with subcritical flow, it is possible for the outflow itself (even on a horizontal or very mild slope channel) to cause the flow in the channel to pass through critical depth at the upstream end of the weir giving super- critical flow, then a hydraulic jump, and finally subcritical flow again. Since the calculation of water surface profiles for subcritical flows depends on knowing a downstream boundary condi- tion (depth), a third thing implied by the first two is that it is impossible to correctly calculate the depths and water surface profile in the channel upstream of a weir without first considering the 8 details of the flow over the weir and the type of profile which exists at the weirs. Only subcriti- cal flow along the full length of the weir is considered in this design procedure. (a) Subcritical flow throughout (b) Jump due to side discharge with subcritical flow upstream and downstream (c) Supercritical flow throughout (d) Jump due to supercritical flow upstream and subcritical flow downstream Fig. 2.2 - Water surface profiles in a channel beside a side-weir 2.4.2 - Previous Work of Others The water that remains in the channel experiences the normal frictional head losses as the channel flow occurs along the weir, and these losses tend to reduce the total head in the flow direction. Hager (1987) discussed the fact that, when one-dimensional analysis is used, the hydraulic characteristics of side weir flow cause an additional head change that may be either positive or negative, depending on the flow conditions. This condition is also discussed by Idel- chik (1986) in conjunction with flow bifurcations in ducts. However, it was found in this project 9 that using the kinetic energy correction factor (?) eliminated the need for including an additional head change in the energy equation (Section 6.8). Most of the previous work has considered only forward flow from the channel into the basin. The earliest studies of the hydraulic characteristics of side-channel weirs were concerned primarily with the analytical prediction of the effects of the weirs on the longitudinal water surface profile in the channel for the idealized case of a rectangular channel with a vertical weir plate and a constant discharge coefficient (Forchheimer, 1930; de Marchi, 1934; Ackers, 1957; Collinge, 1957; Frazer, 1957; Chow, 1959; Henderson, 1966; Bos, 1976). Some other studies on evaluation of the discharge over the side-channel weir are those of Mostafa and Chu (1974), Subramanya and Awasthy (1972), and Hager (1987). Part of Hager's (1987) analysis was based on the side weir discharge per unit length of the weir (q w ) written as 2/3 1w hg 3 2 3 2 C dx dQ q =?= (2.4) where Q is the flow rate in the channel, C 1 is an empirical coefficient, and h is the head at any point along the weir. C 1 may be constant or variable along the weir. Hager wrote C 1 as C n ? where C n is a discharge coefficient for a normal weir of the same geometry as the side weir and ? is a lateral flow coefficient given by 2/1 3 2 3 2 )C1(2Fw )C1()2Fw( ? ? ? ? ? ? ? ? ?+ ?+ =? (2.5) where Fw is defined as gh U Fw = (2.6) and is called a weir Froude number since it is based on the head on the weir rather than the flow depth. C 3 is a residual pressure coefficient that is related to the pressure distribution at the con- trol section for the weir flow and is less than unity. Hager used a value of 2/3 for C 3 in Eq. (2.5). Apparently C 3 should depend on the particular type of weir under consideration. The effective discharge coefficient C n ? is variable along the weir. 10 2.5 - PREVIOUS WORK AT CRWR ON HYDRAULIC COMPUTATIONS 2.5.1 - Purpose Engineers designing side weir and detention basin facilities have to determine the side weir and basin dimensions necessary to reduce the channel flow depth and discharge to accept- able levels for given channel characteristics and a given storm. Although HEC-1 and HEC-2 contain some capabilities for modeling diversions, neither program is flexible enough to repre- sent some of the essential hydraulic features of side weir flows. For example, experimental work indicates that side weir discharge coefficients vary with channel velocity and head on the weir as they change during the passage of the hydrograph, but HEC-1 and HEC-2 cannot represent these changes. Also, the programs cannot predict when submergence of side weirs occurs as the basin fills, nor can they model the flow of water from the basin back to the channel as the channel water level drops. Thus, a program originally called SIDEHYDR was developed in the previous project to model side weir hydraulic characteristics. SIDEHYD used in the modeling presented later in this report is a revision of SIDEHYDR. SIDEHYDR represents side weir flow including the effects of channel flow character- istics, possible submergence as the basin fills, the discharge characteristics for an embankment- shaped weir, and possible reverse flow over the weir as the channel water level drops during the recession limb of the hydrograph. The channel flow and flow over a side weir interact in such a way that trial and error computations are normally required to determine the side discharge and all of the depths in the channel (at the weir and both upstream and downstream of the weir and in the basin). The side discharge depends on the depths in the channel, but the depths are controlled from downstream for subcritical flow and these depths depend on the discharge, which cannot be known until the side discharge is known. In addition, the depth at the downstream end of one weir can depend on other weirs downstream of it, and the discharge at the upstream end of a weir depends on other upstream weirs. Because of all of these interdependencies, it is necessary to iterate between HEC-1, HEC-2, and SIDEHYDR. In the previous project, these iterations were done manually. The procedure was to (1) run HEC-1 with an assumed diversion hydrograph at the weir to obtain hydrographs in the channel, (2) run HEC-2 for times throughout the hydrograph to obtain stage hydrographs at the weirs, (3) run SIDEHYDR using the discharge hydrographs from HEC-1 and the stage hydrographs from HEC-2 and weir discharge characteristics from the experimental part of the project to calculate the weir diversion hydrograph and the stage hydrograph in the basin, (4) run HEC-1 again using the calculated diversion hydrograph, and 11 (5) continue looping through these programs until the diversion hydrographs at the beginning and end of an iteration loop agreed within a specified tolerance. The manual iterations were extremely time consuming. Thus, part of the present project has been to automate the itera- tion process. This section gives a summary of the general computational approach that is used for the hydraulic parts of the problem for various flow conditions. Only subcritical flow along the entire weir length (Fig. 2.2a) is considered. Thus, the computations to determine the weir discharge and the water surface profile along the weir (or the depth change between the downstream (sub- d) and upstream (sub-u) ends of the weir) begin with the downstream water level and the down- stream head on the weir. 2.5.2 - Forward Weir Flow 2.5.2.1 - Method of Analysis in the Previous Project In this section, the method of analysis used in the previous project is reviewed. All of Tynes' (1989) test data are listed in Appendix 1 of this report. The tests were conducted in a channel with a trapezoidal cross section with 2.5H:1V side slopes. For side-channel weirs, the head and the discharge coefficient vary along the length of the weir crest. However, side weirs can be calibrated so that the total side discharge (Q w ) can be written in terms of a bulk discharge coefficient (C e ). For broad-crested weirs, this expression is QCC gAh wes w =? 2 3 2 3 12/ (2.7) where A w = a representative flow area (e.g., Lh in Eq. (2.3) for normal weir flow) and C e = bulk discharge coefficient. The sign convention is that flow into the channel is positive while flow out of the channel is negative. Thus, Eq. (2.3) is positive while Eq. (2.4) and Eq. (2.7) have a negative sign. In Eq. (2.7), some convention must also be established for defining h since the head varies along the length of the weir. Likewise, a convention is needed for defining A w .In the previous project, h was taken at the downstream (sub-d) end of the part of the weir crest parallel to the channel invert and A w was taken as h d times the average length of the flow area over the weir (Fig. 2.3). Thus, Eq. (2.7) can then be written as [] 3/2 d dsew hEShLg 3 2 3 2 CCQ +?= (2.8) where h = height of the water surface above the side weir crest, sub-d = downstream end of side weir crest, L = length of the weir crest parallel to the bed slope (Fig. 2.3), and ES = slope of the 12 ends of the side weir (e.g., ES = 6 for a 6H:1V slope). The subscript u, which will appear later, denotes the upstream end of the weir crest. 2.5.2.2 - Flow Asymmetry For forward flow from the channel into the basin, the flow in the channel develops an asymmetrical velocity profile compared to the one that would exist with no diversion (Chapter 6). Frequently, as part of the flow goes toward and over the weir into the detention basin, a sepa- ration zone forms in the channel on the side opposite to the weir. The flow going over the weir effectively pulls the flow that remains in the channel away from the opposite side of the channel. When separation occurs, the flow in the channel at the downstream end of the weir may be con- centrated on the side of the channel next to the weir, as shown by the velocity profiles in Section 6.4. Thus, the true velocity head and true momentum flux cannot be obtained from the average velocity given by Q d divided by the channel area. This condition needs to be taken into account in determining the actual depth in the channel at the downstream end of the weir from the depth indicated by HEC-2 calculations. Fig. 2.3 - Definition sketch for side-channel weirs As pointed out in the HEC-2 User?s Manual (US Army Corps Of Engineers, 1984) in conjunction with the flow conditions downstream of bridges, some channel length is required for flow expansion to take place downstream of a separation zone. Nevertheless, it was assumed in the previous project that the channel length for this flow expansion is negligible. Thus, there were two cross sections essentially adjacent to each other at the downstream end of the weir. Cross section db corresponds to the conditions calculated by HEC-2 for the downstream end of the weir but is actually at the downstream end of the zone of flow expansion since HEC-2 inher- ently assumes that the flow fills the entire cross section. Cross section da corresponds to the 13 actual conditions at the downstream end of the weir including the flow separation. The designa- tions da and db are used as subscripts. 2.5.2.3 - Downstream Depth The energy equation can be used to relate the depths at cross section da and db since () 2g U y 2g UU K 2g U y 2 db db 2 dbe E 2 da dada += ? ??+ (2.9) where U da =Q d /A da = the apparent velocity at cross section da, ? = the kinetic energy correction factor, U e = the effective velocity at cross section da, i.e., the average velocity in the part of the cross section in which flow is actually taking place, and K E = expansion loss coefficient. This form for head loss term (Henderson, 1966) is more appropriate for this type of flow expansion than the form used in HEC-2. From Eq. (2.9), () 2g UU K 2g U 2g U yy 2 dbe E 2 da da 2 db dbda ? +??+= (2.10) The velocity head at cross section da (the third term on the right-hand side of Eq. (2.10)) is always greater than at cross section db (the second term) for subcritical flow, and K E is less than unity. The result is that y da is less than y db and that the flow asymmetry and resulting flow re- establishment at the downstream end of the weir suppress the head on the weir relative to y db . Although Eq. (2.10) may be helpful toward understanding why y da ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? += (2.39) Since this equation is undefined at Q/Q half = 0, it was used only for Q/Q half > 0.036, which corre- sponds to the second smallest value read from the manufacturer's curves. For lower values, a linear interpolations was used, giving () () 036.0 Q Q for Q Q 44.6 036.0 Q/Q 0.036 09059.0 0.036log9277.09127.0 A A halfhalf half 5967.0 half valve ?= ? ? ? ? ? ? ? ? ? ? ? ? ? ? ++= (2.40) The curves for Eq. (2.39) and Eq. (2.40) are shown in Fig. 2.8. Fig. 2.5 - Tideflex valves (from Red Valve Co., Inc. catalog) 24 Fig. 2.6 - Sample of curves obtained from Red Valve Co., Inc. for Tideflex valves 25 A valve =A half (ft 2 ) 02468101214161820 Q half (c f s ) 0 50 100 150 200 250 D(in.) 847260483624 Fig. 2.7 - Q half for Tideflex valves As noted above, the manufacturer supplied both the head loss in the valve itself (H L )and the "total head loss", which was defined as the head loss in the valve plus the velocity head in the pipe. However, the actual loss associated with a submerged valve is the loss in the valve plus exit loss or the velocity head of the jet leaving the valve (not the velocity head in the pipe), Thus, only the head loss in the valve was used in calculating a loss coefficient. The exit loss was included in the calculations separately. The head loss coefficient (K L ) was defined by Q/Q half 0123456 A va l v e /A half 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 + + + + + + + + + X X X X X X X X X X D(in.) 24 36 48 60 72 84 Eq. Fig. 2.8 - Scaled downstream areas for Tideflex valves () g2 VV KH 2 pipevalve L valve L ? = (2.41) Eq. (2.40) Eq. (2.39) Eq. (2.38) 26 where V valve is the exit velocity from the valve and V pipe is the velocity in the culvert. The head loss coefficients defined in this manner are shown in Fig. 2.9. The mean value of K L is 1.0 with a standard deviation of 3%. There is a large amount of scatter for small values of Q/Q half , but the velocities and therefore the head losses are small for these small flow rates. Q/Q half 0123456 K L 0.90 0.95 1.00 1.05 1.10 1.15 + + + + + + + + + X X X X X X X X X X D(in.) 24 60 36 72 48 84 Fig. 2.9 - Head loss coefficients for Tideflex valves 2.7.1.3 - Calculation Method for Submerged Valves This section summarizes the calculations used when the Tideflex valve is submerged. The valve is considered to be submerged when the tailwater is above the downstream soffit of the culvert pipe. In the calculations, the tailwater depth (TW) is measured from the downstream invert of the pipe (not the valve). As described by Burgin and Holley (2002), the culvert flow is calculated by trial and error based on assumed flows. The assumed flows are adjusted until the head required for the assumed flow matches the head available in the detention pond. For the assumed flow rate (Q), the calculation procedure is as follows, with all values being in feet and cfs: 1) Calculate the culvert cross sectional area in ft 2 from A pipe = ?D 2 /4. 2) Calculate half of the culvert cross sectional area in ft 2 from A half =A pipe /2. 3) Calculate the flow rate in cfs corresponding to having an open area in the Tideflex valve equal to A half from Eq. (2.38). 4) Calculate the open area (A valve ) of the Tideflex valve corresponding to the specified Q from Eq. (2.39) or Eq. (2.40). 5) Since the valve will flow full for submerged conditions, calculate the valve exit velocity head from 27 g2 1 A Q g2 V 2 valve 2 valve ? ? ? ? ? ? ? ? = (2.42) where Q = assumed flow rate in the culvert. 6) Since the pipe will also flow full for submerged conditions, calculate the pipe velocity head from g2 1 A Q g2 V 2 pipe 2 pipe ? ? ? ? ? ? ? ? = (2.43) 7) Since the valve loss coefficient based on the difference in the entrance and exit velocity heads (Eq. (2.41)) is 1.0, the sum (H L ) of the valve and exit head losses is g2 V g2 V 2 g2 V g2 V g2 V H 2 pipe 2 valve 2 valve 2 pipe 2 valve L ?=+ ? ? ? ? ? ? ? ? ? ? ?= (2.44) 8) It is now possible to calculate the head at the end of the culvert (i.e., at the upstream end of the valve). Because of the way that the program for culvert flow does its calculations, the equivalent tailwater (TW equivalent ) at the downstream end of the culvert without the valve is needed. That is, a culvert with TW equivalent and with no valve would have the same flow as the culvert with the valve. TW equivalent to account for the valve and exit losses is ? ? ? ? ? ? ? ? ? ? ?+= ?+= g2 V g2 V 2TW g2 V HTWTW 2 pipe 2 valve actual 2 pipe Lactualequivalent (2.45) 9) Calculate the headwater (HW) based on TW equivalent and the assumed flow rate (Q). HW is relative to the upstream invert of the culvert. 10) Store TW and the final Q and HW for each time step in the calculations. Only the values for the most recent time step are kept in storage. These values are needed for unsubmerged conditions discussed in the next section. 11) At the end of the calculations for a given time step, check to determine whether TW is below the downstream soffit. If so, the calculations then shift to those described in the next section. 28 2.7.1.4 - Calculation Method for Unsubmerged Valves This section summarizes the calculations for unsubmerged conditions that are assumed to exist when the tailwater is below the downstream soffit. An approximate method of calculation is needed since information on the flow area of the valve and depth of flow in the valve outlet could not be obtained for unsubmerged conditions. Even when the valve outlet is not submerged, the head loss in the valve may cause the downstream end of the culvert to still be flowing full. At the end for the calculations for the assumed unsubmerged conditions, a check is made to determine if the downstream end of the culvert is full. 1) At the end of the first time step with the tailwater below the downstream soffit, a linear inter- polationisusedtodetermineQ soffit and HW soffit corresponding to having the TW at the downstream soffit. 2) For subsequent time steps, it is assumed that the water depth (y exit ) in the downstream valve opening decreases linearly in proportion to the decreasing headwater in the detention pond unless the tailwater is higher than the value indicated by this linear interpolation. Thus, ? ? ? ? ? ? ? ? = TWD, HW HW maxy soffit exit (2.46) where y exit is measured from the pipe invert, not from the bottom of the valve. The depth in the valve opening (y v exit ) is measured from the bottom of the valve, so 2 DH yy exitv exit ? += (2.47) where H is the height of the downstream end of the valve (Fig. 2.10) and is given by H = 1.64D. 3) From Q soffit , the valve open area (A soffit ) corresponding to having the tailwater at the soffit is obtained from Eq. (2.39) or Eq. (2.40). 4) For each subsequent tailwater which is below the downstream soffit, it is assumed that the valve open area decreases linearly, i.e., H y AA exit v soffitvalve = (2.48) 29 D = culvert diameter (ft) 01234567 H = va lv e hei ght ( f t) 0 2 4 6 8 10 12 H = 1.64D Fig. 2.10 - Height of Tideflex valves 5) It is further assumed that the open area of the valve is composed of two triangles with coinci- dent bases and with heights equal to half of the valve height, as shown in Fig. 2.11. The base width of the two triangles is given by Fig. 2.11 - Assumed open shape of Tideflex valves H 2A b valve = (2.49) 6) The surface width (T exit ) and the flow area in the valve opening (A exit ) are calculated from b T y ve xi t water surface 30 2 H yfor 2 y TA H/2 y bT 2 H yfor 2 y-H T-AA H/2 y-H bT exit exit exit exit exit exit v v exitexit v exit v v exitvalveexit v exit ? ? ? ? ? ? ? ? = = > ? ? ? ? ? ? ? = = (2.50) 7) The velocity head at the valve outlet is calculated from () g2 A/Q g2 V 2 exit 2 exit = (2.51) 8) Since K L = 1, the head loss in the valve is () 2g Q/A g2 V H 2 flow 2 exit L valve ?= (2.52) where A flow is the flow area corresponding to the flow depth (y pipe )attheendoftheculvert pipe. 9) TW equivalent is calculated from as follows: The energy equation between the end of the pipe and the valve exit is () () () 2g V 2y 2g Q/A 2y 2g V y 2g Q/A - 2g V 2g Q/A y 2 exit exit 2 flow pipe 2 exit exit 2 flow 2 exit 2 flow pipe +=+ += ? ? ? ? ? ? ? ? ?+ (2.53) This equation is solved by trial and error to obtain y pipe ,whichisTW equivalent if y pipe 0.5is ()[] 97.3 d 50.4 d t 1 s1 1Fwln62.55.0 h h 5.1900.1 C C +? ? ? ? ? ? ? ? ? ??= (4.25) For h t /h d ? 0.5, C 1s =1. The t-statistics for the coefficients in Eq. (4.25) are shown in Table 4.3. R 2 for Eq. (4.25) is 0.965. All the coefficients are significant at the 5% level except -5.62. The term involving Fw d is retained because of the improved correlation. However, Eq. (4.25) may give negative values of C 1s /C 1 . When this situation happens, Eq. (4.26) below is used instead of Eq. (4.25) to determine C 1s /C 1 . The regression equation with h t /h d as the only parameter is 76.4 d t 1 s1 5.0 h h 6.22887.0 C C ? ? ? ? ? ? ? ? ??= (4.26) 54 The t-statistics for the coefficients in Eq. (4.26) are shown in Table 4.4. R 2 for Eq. (4.26) is 0.799, which is substantially lower than that of Eq. (4.25). Although the coefficient -22.6 is not significant at the 5% level, the term involving h t /h d is needed because the plot of C 1s /C 1 vs. h t /h d shows a definite relationship between the two variables. Table 4.3 - t-statistics for coefficients in Eq. (4.25) Coefficient t-statistic 1.00 -19.5 4.50 -5.62 3.97 24.24 -2.82 8.95 -1.35 3.79 4.2.3 - Unsubmerged Flow in Tapered Channels The values of C 1 were obtained for the 65 tests for tapered channels using the same procedure described in Section 4.2.1. C 2 wasadjustedto0.70tominimizetheerrorsinh u .The best regression equation for C 1 is Table 4.4 - t-statistics for coefficients in Eq. (4.26) Coefficient t-statistic 0.887 -22.6 4.76 13.7 -1.15 3.83 ? ? ? ? ? ? += P h ln105.0101.1C d 1 (4.27) The t-statistics for the coefficients in Eq. (4.27) are shown in Table 4.5. Both coefficients are significant at the 5% level but R 2 for Eq. (4.27) is only 0.395. Again R 2 for Q w (calculated) vs. Q w (measured) is much higher. Values are given in Section 4.3. Table 4.5 - t-statistics for coefficients in Eq. (4.27) Coefficient t-statistic 1.101 0.105 34.0 6.41 Recall that w Ucos? =C 2 (?/P)U (Eq. (4.19)). A smaller value of C 2 for tapered channels seems to suggest that the longitudinal component of the velocity of side weir discharge is smaller 55 in tapered channels than in prismatic channels. This speculation can be evaluated only by detailed velocity measurements in the channel as well as in the region of the outflow. 4.3 - COMPARISON BETWEEN MEASURED AND CALCULATED VALUES FOR 2.5H:1V SIDE SLOPES Based on empirical coefficients from regression equations in Section 2.6 and on the equa- tions in Section 4.2, the various methods of calculating Q w and h u described in the previous sec- tions are summarized in Table 4.6. Method A is the method used in the previous project (Tynes, 1989). Method B uses a C 1 that is constant along the weir but varies with the hydraulic condi- tions at the downstream end of the weir and also uses a variable head along the length of the weir; while Method C uses a C 1 that varies along the length of the weir based on ? and a variable head. In Method D, the average of the values of ' 1 C obtained from the 238 tests was used with a variable head along the weir. For unsubmerged flow, the calculated flow depth at the upstream end of the weir using Method A was supercritical for Test A3B19N (identification code used by Tynes, 1989) and no solution could be obtained for h u for Tests A1C20W and A5C18N. This condition presumably resulted from a critical or supercritical solution not being found. When Method B was used, supercritical flow depths were obtained in the computed water surface profile for Test A5C18N. The flow conditions calculated at the upstream end of the weir using Method C and Method D were subcritical for all of the 238 tests. Table 4.6 - Methods of calculating Q w and h u Method Physical equations Empirical equations ? Unsub- merged Sub- merged Tapered Unsub- merged Sub- merged Tapered A Eq. (2.8), Eq. (2.23) Eq. (2.8), Eq. (2.23) Eq. (2.8), Eq. (2.36) Eq. (2.28), Eq. (2.29) Eq. (2.28), Eq. (2.29), Eq. (2.32), Eq. (2.33) Eq. (2.34), Eq. (2.35) not used B Eq. (2.4), Eq. (4.18) Eq. (2.4), Eq. (4.18) Eq. (2.4), Eq. (4.22) Eq. (4.19), Eq. (4.23) Eq. (4.19), Eq. (4.23), Eq. (4.25) Eq. (4.19), Eq. (4.27) not used C Eq. (4.18), Eq. (4.21) Eq. (4.19), Eq. (4.24) Eq. (4.20) D Eq. (4.18), Eq. (4.21) Eq. (4.19), ' 1 C = 1.063 Eq. (4.20) Note: Also see Table 4.11. 56 The calculated values of Q w and h u are plotted against the measured values in Fig. 4.2 to Fig. 4.17. The averages (avg), standard deviations (stdev) and root-mean-square values (rms) of the differences between measured and calculatedvaluesareshowninTable4.7toTable4.10,in terms of both absolute values and relative values. (If the sum in the calculation of the standard deviation is divided by N instead of N - 1, then rms = (avg) 2 + (stdev) 2 ). From Table 4.7 to Table 4.10, the differences between stdev and rms of either ?Q w , where ?Q w =Q w(mea) -Q w(cal) as shown in Table 4.7 through Table 4.10, or ?Q w /Q w(mea) were at most 2%, except for tapered channels using Method A. This result is illustrated in the figures by the fact that the points are scattered close to the 1:1 line. Similar observations were obtained for the upstream head on the weir. The differences between stdev and rms were generally larger for ?h u ,where?h u =h u(mea) -h u(cal) as shown in Table 4.7 through Table 4.10, and for ?h u /h u(mea) than for ?Q w and ?Q w /Q w(mea) . The largest difference was still only 8%, except for submerged flow using Method A and for tapered channels. The root-mean-square values were used to compare the different methods of analysis. For meaningful comparison, the statistics had to be based on the same number of tests. There- fore, when comparing Methods A and B, Tests A1C20W, A3B19N and A5C18N were excluded and when comparing Methods B, C and D, Test A5C18N was excluded. Method B gave smaller rms for the side weir discharge than Method A in terms of both ?Q w and ?Q w /Q w(mea) . For unsubmerged flow, the rms of ?Q w /Q w(mea) was reduced only by 2% when Method B was used instead of Method A. However the reduction was 57% for sub- merged flow and 44% for tapered channels. For unsubmerged flow, Method B gave smaller rms for ?h u but larger rms for ?h u /h u(mea) than Method A. For submerged flow, Method B gave smaller rms for both ?h u and ?h u /h u(mea) than Method A whereas for tapered channels, Method B gave larger rms than Method A for both ?h u and ?h u /h u(mea) . Nevertheless, the differences between the rms for ?h u using either method were only about 0.001 ft. Therefore, compared with Method A, Method B gave improved results or results of comparable accuracy. Method B takes into consideration more details of the side weir hydraulics, specifically the water surface profile along the weir. Hence the head correction required in Method A (Eq. (2.29), Eq. (2.33), and Eq. (2.35)) is eliminated in Method B. Moreover, Method B explicitly accounts for the channel slope and roughness so that it is applicable for different slopes and roughnesses. However, the improvement for submerged flow is partly due to the use of two parameters in the regression equation instead of one (Eq. (4.25) vs. Eq. (2.33)). Table 4.11 summarizes the differences between Method A and Method B. 57 Q w (measured) (cfs) 0123456 Q w ( c alcu l a t ed) (c f s ) 0 1 2 3 4 5 6 Fig. 4.2 - Comparison of measured and calculated model values of side- weir discharge for unsubmerged flow using Method A h u (measured) (ft) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 h u (c a l c u l a t e d) (f t) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Fig. 4.3 - Comparison of measured and calculated model values of upstream head on the weir for unsubmerged flow using Method A 58 Q w (measured) (cfs) 0123456 Q w ( c alc u lat ed) ( c f s ) 0 1 2 3 4 5 6 Fig. 4.4 - Comparison of measured and calculated model values of side- weir discharge for submerged flow using Method A h u (measured) (ft) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 h u ( c a l cul a te d) (f t) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Fig. 4.5 - Comparison of measured and calculated model values of upstream head on the weir for submerged flow using Method A 59 Q w (measured) (cfs) 0123456 Q w ( c al cul a te d) (c fs) 0 1 2 3 4 5 6 Fig. 4.6 - Comparison of measured and calculated model values of side weir discharge for tapered channels using Method A h u (measured) (ft) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 h u (calcula ted) ( ft) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Fig. 4.7 - Comparison of measured and calculated model values of upstream head on the weir for tapered channels using Method A 60 Q w (measured) (cfs) 0123456 Q w ( c alculat ed) (cf s ) 0 1 2 3 4 5 6 Fig. 4.8 - Comparison of measured and calculated model values of side weir discharge for unsubmerged flow using Method B h u (measured) (ft) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 h u (c alculat ed) (f t ) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Fig. 4.9 - Comparison of measured and calculated model values of upstream head on the weir for unsubmerged flow using Method B 61 Q w (measured) (cfs) 0123456 Q w (c alcu l a ted ) ( c f s ) 0 1 2 3 4 5 6 Fig. 4.10 - Comparison of measured and calculated model values of side weir discharge for submerged flow using Method B h u (measured) (ft) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 h u (c alculat ed) (f t ) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Fig. 4.11 - Comparison of measured and calculated model values of upstream head on the weir for submerged flow using Method B 62 Q w (measured) (cfs) 0123456 Q w ( c alc u l a te d) ( c f s ) 0 1 2 3 4 5 6 Fig. 4.12 - Comparison of measured and calculated model values of side weir discharge for tapered channels using Method B h u (measured) (ft) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 h u (c alculat ed) (f t) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Fig. 4.13 - Comparison of measured and calculated model values of upstream head on the weir for tapered channels using Method B 63 Q w (measured) (cfs) 0123456 Q w ( c alcu l a ted) (c fs ) 0 1 2 3 4 5 6 Fig. 4.14 - Comparison of measured and calculated model values of side weir discharge for unsubmerged flow using Method C h u (measured) (ft) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 h u (c alculat ed) (f t) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Fig. 4.15 - Comparison of measured and calculated model values of upstream head on the weir for unsubmerged flow using Method C 64 Q w (measured) (cfs) 0123456 Q w ( c alc u late d) (c fs ) 0 1 2 3 4 5 6 Fig. 4.16 - Comparison of measured and calculated model values of side weir discharge for unsubmerged flow using Method D h u (measured) (ft) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 h u ( c al cu la te d ) ( ft) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Fig. 4.17 - Comparison of measured and calculated model values of upstream head on the weir for unsubmerged flow using Method D 65 Table 4.7 - Statistics of differences between measured and calculated model values of Q w and h u (Method A) ?Q w (cfs) ?Q w /Q w(mea) ?h u (ft) ?h u /h u(mea) Unsubmerged N = 238 avg stdev rms -0.00474 0.124 0.123 -0.0112 0.0680 0.0688 Unsubmerged N = 235 (1) avg stdev rms 0.000983 0.112 0.112 -0.00819 0.0628 0.0632 0.00327 0.00877 0.00934 0.0123 0.0662 0.0672 Submerged N=35 avg stdev rms -0.0737 0.328 0.332 -0.0751 0.298 0.303 0.00280 0.00382 0.00469 0.0118 0.0177 0.0210 Tapered N=65 avg stdev rms -0.114 0.161 0.197 -0.0898 0.127 0.155 0.00280 0.00452 0.00529 0.0185 0.0372 0.0413 Notes: ?Q w =Q w(mea) -Q w(cal) , ?h u =h u(mea) -h u(cal) , mea = measured, cal = calculated, N = number of tests. (1) Tests A1C20W, A3B19N and A5C18N excluded In the tests for tapered channels, large values of Q u and small diversions were associated with small values of both h d and h u that were two to three times those of h d . Under these condi- tions Fw d was high and C e was very large. As a result, C e was found to be proportional to 3 d Fw . The relationship between C e and Fw d in Eq. (2.34) is an artifact of calculating Q w in terms of h d in Method A. In Method B, the variation of the head along the weir is considered directly in the computation. Fw d does not enter into the regression equation of C 1 for tapered channels. When Method B was used, the errors in the estimation of the side weir discharge aver- aged about 6%, 13% and 9% for unsubmerged flow, submerged flow and flow in tapered chan- nels respectively. Even though the error is larger for submerged flows, the flows themselves are normally small because of the small head difference across the weir. The errors in the estimation of the upstream head on the weir at prototype scale were on average about 0.20 ft, 0.10 ft and 66 0.14 ft for unsubmerged flow, submerged flow and flow in tapered channels respectively for a 1:25 scale model. Method B and Method C had comparable accuracy in predicting Q w while Method D was less accurate. The rms values for ?Q w and ?Q w /Q w(mea) from Method D were about 50% higher than those from Method B. Although Eq. (4.24) has a low R 2 , using this equation to estimate ' 1 C is better than simply using the average value of ' 1 C . The root-mean-square values of ?h u and ?h u /h u(mea) obtained from Methods B, C, and D were all about 0.008 ft and 0.1 respectively at model scale. Therefore these three methods were comparable in accuracy in predicting h u . Table 4.8 - Statistics of differences between measured and calculated model values of Q w and h u (Method B) ?Q w (cfs) ?Q w /Q w(mea) ?h u (ft) ?h u /h u(mea) Unsubmerged N = 237 (1) avg stdev rms 0.00173 0.106 0.106 -0.00361 0.0620 0.0619 -0.00165 0.00807 0.00822 -0.0338 0.101 0.107 Unsubmerged N = 235 (2) avg stdev rms 0.00306 0.105 0.105 -0.00296 0.0618 0.0617 -0.00167 0.00797 0.00813 -0.0339 0.100 0.106 Submerged N=35 avg stdev rms -0.00334 0.136 0.135 0.00181 0.132 0.130 -0.00100 0.00403 0.00410 -0.00554 0.0200 0.0205 Tapered N=65 avg stdev rms -0.00017 0.120 0.119 -0.00840 0.0865 0.0862 -0.00154 0.00531 0.00549 -0.0265 0.0541 0.0598 Notes: ?Q w =Q w(mea) -Q w(cal) , ?h u =h u(mea) -h u(cal) , mea = measured, cal = calculated, N = number of tests. (1) Test A5C18N excluded. (2) Tests A1C20W, A3B19N and A5C18N excluded. 67 Table 4.9 - Statistics of differences between measured and calculated model values of Q w and h u (Method C) ?Q w (cfs) ?Q w /Q w(mea) ?h u (ft) ?h u /h u(mea) Unsubmerged N = 238 avg stdev rms 0.00250 0.120 0.120 -0.00936 0.0639 0.0644 -0.00233 0.00799 0.00831 -0.0390 0.100 0.107 Unsubmerged N = 237 (1) avg stdev rms 0.00362 0.119 0.119 -0.00875 0.0633 0.0638 -0.00236 0.00799 0.00831 -0.0393 0.100 0.107 Notes: ?Q w =Q w(mea) -Q w(cal) , ?h u =h u(mea) -h u(cal) , mea = measured, cal = calculated, N = number of tests. (1) Test A5C18N excluded. Table 4.10 - Statistics of differences between measured and calculated model values of Q w and h u (Method D) ?Q w (cfs) ?Q w /Q w(mea) ?h u (ft) ?h u /h u(mea) Unsubmerged N = 238 avg stdev rms 0.0305 0.153 0.156 -0.0134 0.0913 0.0921 -0.00251 0.00871 0.00904 -0.0362 0.0959 0.102 Unsubmerged N = 237 (1) avg stdev rms 0.0311 0.154 0.156 -0.0132 0.0915 0.0922 -0.00278 0.00764 0.00812 -0.0373 0.0946 0.102 Notes: ?Q w =Q w(mea) -Q w(cal) , ?h u =h u(mea) -h u(cal) , mea = measured, cal = calculated, N = number of tests. (1) Test A5C18N excluded. 68 Table 4.11 - Comparison between Method A and Method B Method A Method B Bulk discharge equation Discharge equation in terms of discharge per unit length Discharge coefficient related to Fw d and L/B Discharge coefficient related to Fw d and h d /P Q w and h u obtained from separate equa- tions, h u from energy equation but esti- mated Q w required to calculate h u Q w and h u obtained from the same analysis based on momentum equation ? obtained from simplified assumption of velocity distribution in separation zone ? obtained from interpolation (in some cases extrapolation) of empirical results and assumed to vary linearly between up- stream and downstream ends of weir Details of hydraulics along weir not con- sidered Water surface profile along weir computed h c given by empirical equation h c not used Assumed ()UP/CcosU 2w ?=? Method C uses a simpler predictive equation for the discharge coefficient. However the derivation of the lateral flow coefficient does not seem to have a sound theoretical basis. The use of C 3 = 2/3 in the present calculation also lacks strong justification. Nevertheless, the lateral flow coefficient does seem to account for at least part of the variation of the discharge coefficient along the weir. The results should be regarded as preliminary and further study is needed. The four methods of calculation were also compared using values of R 2 (Q w )andR 2 (h u ) defined as () () () ? ? ? ? ?= N 1 2 )mea(w)mea(w N 1 2 )cal(w)mea(w w 2 QQ QQ 1QR (4.28) and () () () ? ? ? ? ?= N 1 2 )mea(u )mea(u N 1 2 )cal(u)mea(u u 2 hh hh 1hR (4.29) 69 where Q w(mea) and h u(mea) are the averages of Q w(mea) and h u(mea) respectively. R 2 (Q w )and R 2 (h u ) are not coefficients of determination for regression equations; rather they are defined using the concept of the coefficient of determination. A value of R 2 (Q w ) close to unity indicates that there is good agreement between the values of Q w(mea) and Q w(cal) and similarly for h u . Although the values of R 2 for Eq. (4.23), Eq. (4.24), and Eq. (4.27) are low at 0.682, 0.337 and 0.395 respectively, the values of R 2 (Q w )andR 2 (h u ) are all close to unity (Table 4.12). Method B gave higher values of R 2 (Q w ) than Method A for all flow conditions and higher values of R 2 (h u ) than Method A for both unsubmerged and submerged flows in prismatic channels. Both Methods A and B gave the same R 2 (h u ) for flow in tapered channels. Method B gave higher R 2 (Q w ) than both Method C and D and Methods B, C and D gave the same values of R 2 (h u ). In view of the above comparison using rms, R 2 (Q w )andR 2 (h u ), Method B is the best and is recommended for the calculation of side weir discharge and upstream head on the weir. 4.4 - EFFECTS OF CHANNEL SLOPE AND ROUGHNESS FOR 2.5H:1V SIDE SLOPES Simulations for prototype conditions with unsubmerged flow were performed to evaluate the effects of slope and roughness on Q w and h u . The geometric conditions were selected from those used in the model study and are shown in Table 4.13 for a 1:25 model. From the model test data, the maximum and minimum Q u for each geometric condition and the maximum and minimum h d for each Q u were selected and scaled to the prototype values. Four values of Manning's n (0.0125, 0.02, 0.03 and 0.04), and four values of channel slope (0.000385, 0.0008, 0.0012 and 0.0016) were used in the simulation so that there were 16 combinations of different slopes and roughnesses. The model had a Manning's n of 0.0125 and a slope of 0.000385. Manning's n of 0.04 and slope of 0.0016 are probably maximum limits of prototype conditions in Harris County. Results of simulation of Q w and h u for different geomet- ric and flow conditions using Methods A and B are presented in Appendix 3. In Method A, the calculation of Q w does not involve the slope and roughness. Therefore, only one value of Q w was obtained for each particular geometry and flow. However, for calcula- tion of h u , Eq. (2.23) takes into account the roughness and slope. The largest differences between values of h u calculated from Methods A and B are shown in Table 4.14 and the largest ratios between values of Q w calculated from Methods A and B are shown in Table 4.15. The worst cases give a 75% difference in Q w anda1.5ftdifferenceinh u . In general, larger slope and smaller roughness give larger values of h u (B)-h u (A) but smaller values of Q w (B)/Q w (A). The following observations were obtained from the results of the simulation. For a particular slope, h u increased as roughness was increased since higher head was required to over- come the increased frictional resistance. For a particular roughness, h u decreased as the slope 70 was increased. In most cases, a major contribution to the decrease in h u was the increase in the elevation of the upstream end of the weir crest relative to the downstream end. For flows with low velocities and thus negligible frictional loss, the decrease was entirely due to the change in elevation. Table 4.12 - R 2 (Q w )andR 2 (h u ) for comparison between measured and cal- culated values of Q w and h u Method Flow condition Number of tests R 2 (Q w )R 2 (h u ) A Unsubmerged Unsubmerged Submerged Tapered 238 235 (2) 35 65 0.988 0.990 0.868 0.969 0.988 0.985 0.992 B Unsubmerged Unsubmerged Submerged Tapered 237 (1) 235 (2) 35 65 0.991 0.991 0.978 0.989 0.991 0.991 0.989 0.992 C Unsubmerged Unsubmerged 238 237 (1) 0.989 0.989 0.991 0.991 D Unsubmerged Unsubmerged 238 237 (1) 0.980 0.980 0.989 0.991 Notes: (1) Test A5C18N excluded. (2) Tests A1C20W, A3B19N and A5C18N excluded. Table 4.13 - Geometric conditions used in simulation Weir length (ft) Invert width (ft) Weir height (ft) model prototype model prototype model prototype 23.91 598 3.4 85 0.52 13.0 10.00 250 3.4 85 0.52 13.0 15.00 375 1.8 45 0.52 13.0 10.00 250 1.8 45 0.52 13.0 20.00 500 3.4 85 0.70 17.5 10.00 250 3.4 85 0.70 17.5 20.00 500 1.8 45 0.70 17.5 10.00 250 1.8 45 0.70 17.5 71 Table 4.14 - Largest differences between values of h u calculated from Methods A and B Parameter Case 1 Case 2 L (ft) 500 250 B (ft) 45 85 P (ft) 17.5 13.0 Q u (cfs) 30012 30022 h d (ft) 3.00 3.55 n 0.04 0.02 S o 0.000385 0.0012 Q w (A) (cfs) 3977 3280 Q w (B) (cfs) 6962 2856 Q w (B)/Q w (A) 1.75 0.87 h u (A) (ft) 4.48 0.73 h u (B) (ft) 3.92 2.20 h u (B)-h u (A) (ft) -0.56 1.47 Table 4.15 - Largest ratios between values of Q w calculated from Methods AandB Parameter Case 1 Case 3 L (ft) 500 598 B (ft) 45 85 P (ft) 17.5 13.0 Q u (cfs) 30012 10156 h d (ft) 3.00 0.97 n 0.04 0.0125 S o 0.000385 0.0016 Q w (A) (cfs) 3977 1082 Q w (B) (cfs) 6962 430 Q w (B)/Q w (A) 1.75 0.40 h u (A) (ft) 4.48 -0.36 h u (B) (ft) 3.92 -0.10 h u (B)-h u (A) (ft) -0.56 0.26 Since the side weir discharge is primarily a function of the head on the weir (Eq. (2.4)), the same trends of variation with roughness and slope were observed for the side weir discharge. That is, for a particular slope, Q w increased as roughness was increased and for a particular roughness, Q w decreased as the slope was increased. In addition to the effects of the method of calculation (Table 4.14 and Table 4.15), the amount of variation of Q w and h u for different slopes and roughnesses depends on the geometric and flow conditions. For example, for L = 598 ft, B = 85 ft, P = 13 ft, Q u = 30144 cfs and h d = 72 4.85 ft, the difference between the maximum and minimum Q w was about 7000 cfs and the difference between the maximum and minimum h u was about 3 ft for the different S o andnfor which calculations were done. All of these results indicate that it is definitely beneficial to use Method B to account for different channel roughnesses and slopes. 73 5 - DISCHARGE AND HEAD LOSS EXPERIMENTS FOR 4:1 SIDE SLOPES 5.1 - INTRODUCTION The objective of the work reported in this section was to conduct hydraulic model experi- ments to evaluate the effects of channel side slope on weir hydraulics. The channel and weir were modified to have 4H:1V side slopes (Section 3.2). Slopes of 4H:1V are the expected extreme of flatter slopes, as contrasted to the previous experiments at the opposite extreme of steeper slopes at 2.5H:1V. 5.2 - MODEL RESULTS Twenty-four tests were conducted for unsubmerged flow for the same general hydraulic conditions as some of the previous experiments with 2.5H:1V side slopes. The model height of the weir was 0.5 ft. Two weir lengths of 5 ft and 10 ft were investigated. There were 15 tests for the 10 ft weir and 9 tests for the 5 ft weir. Six of the 15 tests for the 10 ft weir were duplicate measurements that confirmed the reproducibility of the results. The test data are tabulated in Appendix 4. The results of these tests were compared with the previous results to determine the effect of side slope. The methods of analysis have been described in Sections 4.1 - 4.3. 5.2.1 - Analysis of Data using Method A In Fig. 5.1, values of C e obtained from experimental results (C e (observed)) are plotted against values calculated from Eq. (2.28) (C e (regression)); the conditions for each test (A1, A2, etc.) are given in Appendix 4. The data for the previous tests with 2.5H:1V side slopes, B = 1.8 ft and P = 0.52 ft are also shown in the figure. For this particular geometry, the figure shows a positive bias in the coefficients calculated from the regression equation, i.e., the values of C e (regression) are all larger than the values of C e (observed). The bias is observed in the data for the tests with 4H:1V side slopes as well as for the tests with 2.5H:1V side slopes. It is noted that the data for unsubmerged tests in the previous project (Tynes, 1989) as a whole do not show a bias. The measured values of h u are plotted against the values calculated using Method A in Fig. 5.2. The measured side weir discharges were used in this calculation. Due to the small hump in the channel invert, Eq. (2.23) was modified to be cdu 2 d 2 u 2 u fuddu h)ELEL( A A 1 g2 U hPPhh +?? ? ? ? ? ? ? ? ? ??+?+= (5.1) 74 C e (r egre ssio n ) 0 . 4 0 .5 0. 6 0 .7 0.8 0 .9 1.0 C e ( o bs e r v e d) 0.40.50.60.70.80.91.0 X5 1 V : 2 .5 H 5 ft 1V: 2 .5 H 1 0 f t 1 V :4 H 5 ft 1V: 4 H 1 0 f t A1 A2 A3 B1 B2 B3 C1 C2 C3 AA3 BB 1 BB3 CC1 X1 X2 X3 X4 Y1 Y2 Y3 Z1 CC2 CC3 Fi g . 5.1 - Observed and cal cul a t e d C e values 2.5H:1V 5 f t 2.5H:1V 1 0 f t 4H:1V 5 f t 4H:1V 5 f t 75 h u ( c alculated ) ( ft) 0.0 0 0.05 0.1 0 0.15 0 . 20 0 . 25 0.30 h u (m easure d )( ft) 0.0 0 0.0 5 0.1 0 0.1 5 0.2 0 0.2 5 0.3 0 X5 1 V :2 .5 H 5 ft 1V :2 .5H 1 0 f t 1 V :4 H 5 ft 1V : 4 H 1 0 f t A1 A2 A3 B1 B2 B3 C1 C2 AA3 BB1 BB3 CC1 X1 X2 X3 X4 Y1 Y2 Y3 Z1 CC2 C3 CC3 Fi g . 5.2 - Measured and cal cul a t e d h u using M ethod A 2.5H:1V 5 f t 2.5H:1V 1 0 f t 4H:1V 5 f t 4H:1V 5 f t 76 where P u and P d are the weir heights at the upstream and downstream ends of the weir and EL u and EL d are the channel invert elevations at the upstream and downstream ends of the weir. The figure shows that there are larger discrepancies between measured and calculated values of h u for the tests with 4H:1V side slopes than for the tests with 2.5H:1V side slopes. Since the calculation of the side weir discharge by Method A does not take into account the longitudinal slope and roughness, it is more appropriate to analyze the data using Method B. 5.2.2 - Analysis of Data using Method B In the analysis using Method B, the local invert slope and a local weir height were used for each of the computational step. Fig. 5.3 shows the comparison between values of C 1 obtained from numerical optimization with C 2 = 0.85 and values of C 1 calculated from Eq. (4.23). Super- critical flow depths were obtained in the computed water surface profiles for Tests B3, BB3, C2, CC2, C3 and CC3. Therefore, no results are presented for these tests. There is also a positive bias in the coefficients calculated from the regression equation for the tests with 2.5H:1V side slopes. However, the data points for the tests with 4H:1V side slopes exhibit a different pattern with about half of them above the 1:1 line. The values of h u are shown in Fig. 5.4. While most of the points for the tests with 2.5H:1V side slopes lie close to the 1:1 line, the points for the tests with 4H:1V side slopes are all above the 1:1 line. In the analysis of the data of the previous pro- ject, C 2 was adjusted to minimize the discrepancies between the measured values of h u and the values from the numerical optimization. Hence the larger discrepancies between the measured values of h u and the values from the numerical optimization for the tests with 4H:1V side slopes suggested that C 2 = 0.85 was inappropriate for this set of data. C 2 was changed to 1.10 and the results are shown in Fig. 5.5 and Fig. 5.6. Supercritical flow depths were obtained in the computed water surface profiles only for Tests C3 and CC3. For C 2 = 1.10, the values of h u from the numerical optimization are in good agreement with the measured values for the tests with 4H:1V side slopes. Moreover, most of the points in the plot of C 1 (optimization) vs. C 1 (regression) are below the 1:1 line and follow a pattern similar to that for the tests with 2.5H:1V side slopes. Nevertheless, data points for a few tests (A3, AA3, B3, BB3 and X5) are still above the 1:1 line. These five tests had only about 10% diversion. In Fig. 5.7 and Fig. 5.8, the measured values of Q w and h u are compared with the values calculated using discharge coefficients calculated from Eq. (4.23). Fig. 5.7 illustrates that even though the results for A3, AA3, B3, BB3 and X5 do not follow the general trend in Fig. 5.5, this behavior should not be of concern in terms of the estimated side weir discharge because the anomaly occurs only for low discharges. All the data points in Fig. 5.8 lie close to the 1:1 line indicating good agreement between measured and calculated values of h u . 77 C 1 (regr es s i on) 0. 6 0 . 7 0.8 0 . 9 1. 0 1 . 1 1. 2 1 .3 C 1 (opti m iz ation) 0.60.70.80.91.01.11.21.3 X5 A1 A2 A3 B1 B2 C1 AA 3 BB1 CC 1 X1 X2 X3 X4 Y1 Y2 Y3 2.5H: 1 V 5 ft 2.5H :1V 1 0 f t 4H :1V 5 f t 4H: 1 V 1 0 f t Fi g. 5 . 3 - C 1 f r o m r e g r e ssion e qua tion a nd f r o m optimiza tion w ith C 2 =0 . 8 5 78 h u ( o pt im i z at io n) 0. 0 0 0. 05 0 . 1 0 0 . 1 5 0. 20 0. 2 5 0 . 3 0 h u (m eas ur ed) 0. 00 0. 05 0. 10 0. 15 0. 20 0. 25 0. 30 X5 2. 5 H : 1 V 5 f t 2.5 H :1V 1 0 f t 4H:1 V 5 ft 4H :1V 1 0 f t A1 A2 A3 B1 B2 C1 AA3 BB 1 CC 1 X1 X2 X3 X4 Y1 Y2 Y3 Fi g . 5.4 - Measured and num eri cal l y opt i m i z ed h u with C 2 =0 . 8 5 79 C 1 (regres s i on) 0.6 0 . 7 0. 8 0 . 9 1. 0 1 . 1 1. 2 1 .3 C 1 ( o p t im iz a t i o n) 0. 6 0. 7 0. 8 0. 9 1. 0 1. 1 1. 2 1. 3 X5 A1 A2 A3 B1 B2 C1 C2 AA3 BB1 CC1 X1 X2 X3 X4 Y1 Y2 Y3 Z1 CC2 2.5H:1V, 5 f t , C 2 = 0 .85 2.5H:1V, 10 ft , C 2 = 0 .85 4H:1V, 5 ft, C 2 = 1 .10 4H:1V, 10 ft, C 2 = 1 .10 Fi g. 5 . 5 - C 1 f r o m r e g r e ssion e qua tion a nd f r o m optimiza tion 80 h u (optimization) 0. 00 0. 05 0. 10 0 . 15 0 . 2 0 0. 25 0 . 3 0 h u ( m ea s u r ed) 0. 00 0. 05 0. 10 0. 15 0. 20 0. 25 0. 30 X5 A1 A2 A3 B1 B2 C1 C2 AA 3 BB 1 CC 1 X1 X2 X3 X4 Y1 Y2 Y3 Z1 CC2 2 . 5H:1 V , 5 ft, C 2 =0 . 8 5 2 . 5H:1 V , 10 ft, C 2 =0 . 8 5 4 H :1V , 5 ft, C 2 =1 . 1 0 4 H :1V , 1 0 ft, C 2 =1 . 1 0 F i g . 5.6 - Values of h u f r o m r e g r e ssion e qua tion a nd f r o m optimiza tion 81 Q W ( c a l c u lat ed) 0 . 0 0 . 5 1. 0 1 . 5 2. 0 2 .5 3. 0 3 .5 4 . 0 4 .5 Q W ( m easur ed) 0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0 3. 5 4. 0 4. 5 X5 1 V :2 .5H , 5 f t, C 2 =0 . 8 5 1 V :2 .5H , 10 f t , C 2 =0 . 8 5 1V :4H , 5 f t, C 2 =1 . 1 0 1 V :4 H , 10 ft , C 2 =1 . 1 0 A1 A2 A3 B1 B2 B3 C1 C2 AA 3 BB1 BB3 CC 1 X1 X2 X3 X4 Y1 Y2 Y3 Z1 CC 2 Fi g . 5.7 - Measured and cal cul a t e d Q w using M ethod B 2.5H:1V 5 f t , C 2 =0 . 8 5 2.5H:1V 1 0 f t, C 2 =0 . 8 5 4H:1V 5 f t , C 2 =1 . 1 5 4H:1V 5 f t , C 2 =1 . 1 5 82 h u (cal cula ted) 0.0 0 0.05 0.10 0 . 15 0.20 0.25 0.30 h u ( m eas ur ed) 0.000.050.100.150.200.250.30 X5 A1 A2 A3 B1 B2 B3 C1 C2 AA3 BB1 BB3 CC 1 X1 X2 X3 X4 Y1 Y2 Y3 Z1 CC2 1 V : 2 .5 H, 5 f t, C 2 =0 . 8 5 1 V :2 .5 H , 1 0 f t , C 2 =0 . 8 5 1 V : 4 H , 5 ft, C 2 =1 . 1 0 1 V :4 H, 1 0 ft, C 2 =1 . 1 0 Fi g . 5.8 - Measured and cal cul a t e d h u using M ethod B 2.5H:1V 5 f t , C 2 =0 . 8 5 2.5H:1V 1 0 f t, C 2 =0 . 8 5 4H:1V 5 f t , C 2 =1 . 1 5 4H:1V 5 f t , C 2 =1 . 1 5 83 Table 5.1 shows the root-mean-square values of ?Q w , ?Q w /Q w(mea) , ?h u and ?h u /h u(mea) for the data presented in Fig. 5.7 and Fig. 5.8. The accuracy of the prediction of Q w for the tests with 2.5H:1V side slopes was similar to that for the tests with 4H:1V side slopes. The rms of ?h u for the tests with 4H:1V side slopes was about half that of the tests with 2.5H:1V side slopes. Table 5.1 - rms of ?Q w , ?Q w /Q w(mea) , ?h u and ?h u /h u(mea) ?Q w (cfs) ?Q w /Q w(mea) ?h u (ft) ?h u /h u(mea) 2.5H:1V 0.155 0.119 0.00942 0.0590 4H:1V 0.172 0.130 0.00961 0.0839 The above discussion shows that the regression equation for discharge coefficient for channels with 2.5H:1V side slopes is applicable for channels with 4H:1V side slopes. However, C 2 should be increased to 1.1 for channels with 4H:1V side slopes. For slopes between 2.5H:1V and 4H:1V, linear interpolation may be used to estimate values for C 2 . 85 6 - FLOW ASYMMETRY 6.1 - INTRODUCTION 6.1.1 - Background Flow diversion at side-channel weirs causes an asymmetry in the velocity distribution for the flow that remains in the channel. For higher diversions and/or flatter channel side slopes, a separation zone is formed on the side of the channel opposite to the weir. Downstream of the weir, the velocity distribution gradually re-establishes itself to the conditions that would exist for the discharge downstream of the weir if there were no diversion. The asymmetry and possible separation zones are important because they can cause the momentum and kinetic energy correc- tion factors (? and ?, respectively) for the channel flow at the downstream end of the weir to be significantly greater than unity. For subcritical flows with downstream control, the result is that the head at the downstream end of the weir can be significantly lower than would be calculated if the flow were assumed to be symmetrical with ? and ? values near unity. When a separation zone is formed, it is also important for the traditional reasons such as sediment deposition. Although there have been several papers on the hydraulics of side-channel weirs, most of them have addressed discharge coefficients for side weirs and/or the water surface profile in the channel along a side weir. Very few publications have addressed flow asymmetry and related considerations. Related literature is summarized in Section 6.1.2 below. Because of the sparsity of literature on flow asymmetry, the importance of this phenomenon was not recognized at the beginning of the previous project and experiments for the previous project were planned without making provisions for measurement of the effects of the separation zone. As a result, the effects of the separation zone could be included in the previous project only by estimating the size of the separation zone, not by directly measuring either its size or its effects on the channel and weir hydraulics. For some situations, that approximate analysis indicated that the flow asymmetry can cause a head decrease of one foot or more on the weir compared to the water level at the down- stream end of the region of flow asymmetry. An effect with such a magnitude should be based on direct measurements, not on inferred or estimated characteristics of the separation zones, as was done previously. 6.1.2 - Related Literature Subramanya and Awasthy (1972) conducted experiments in a rectangular channel with a side weir. They stated simply that a separation zone was observed on the side of the channel opposite the weir, but they did not give any quantitative information or further details. The velocity profiles in their Fig. 4 show no evidence of separation zones, so they apparently were not 86 present for all of the test conditions. They said that ? =1.02and? = 1.04 at the upstream end of the weir. They did not give any values for the downstream end and said that the values could be taken as unity without appreciable error. El-Khashab and Smith (1976) presented a figure with velocity contours showing no sepa- ration zone in a rectangular channel for Q w /Q u = 56%, where Q w = weir discharge and Q u = channel discharge at the upstream end of the weir. They stated that a separation zone existed for Q w /Q u ? 70% for subcritical flow. They also stated that they used ? = 1. More of their results are considered in Section 6.8 and Section 6.9. Balmforth and Sarginson (1983) did experiments in a rectangular channel for the five flow types given by Frazer (1957). The flow types were identified by Balmforth and Sarginson as flows in a mild slope channel with (I) a low weir and no downstream throttle, (II) higher weirs and a downstream throttle, and (III) low weirs and a downstream throttle. Type IV and V were said to be similar to types I and III but for steep channels. The authors did not give their ranges of flow conditions and diversions. They said that ? had only small deviations from 1.05, but they evaluated ? for a tapered channel with a width that decreased in the flow direction beside the weir. This type of tapering can keep separation zones from forming, even in trapezoidal channels (Tynes, 1989). For nearly prismatic, rectangular channels with small slopes, Hager (1981) gave ? as ? =1+ Q'B Q 2 ? ? ? ? ? ? (6.1) where Q' = -dQ/dx, x = longitudinal coordinate, and Q = discharge in the channel. This equation was for channels with side weirs and also with side and bottom orifices. It effectively gives not only ? but also its variation along a side weir since both Q' and Q vary in a channel along a weir. Hager stated that the effects of ? on the hydraulic calculations are limited to subcritical flows, and later Hager and Volkart (1986) concluded that the effects of ? for rectangular channels are negligible in comparison to other effects and assumed ? = 1. Hager and Volkart show velocity profiles which indicate asymmetry but no separation zone for Q w /Q u = 50%. Cheong (1992) used ? = 1 for calculations for comparison with his experimental results for trapezoidal channels and for both subcritical and supercritical flows upstream of the side weir. (Note that Cheong?s paper uses the symbol ? for something other than the momentum correction factor.) Even though he had Q w /Q u values as high as almost 90%, he did not mention anything about separation zones. 87 6.1.3 - Objective The objective of the experimental work and associated analysis presented in this chapter was to determine the effects of the flow asymmetry and separation zones on the channel hydrau- lics and the weir flow. The primary focus was on the asymmetry, ?, and ?, but limited informa- tion was also obtained on the length of the flow re-establishment region downstream of side weir. 6.2 - EQUATIONS FOR THE CHANNEL FLOW Either the momentum equation or the energy equation can be used to calculate the change in stage in the channel due to flow asymmetry. The momentum equation was written for the prismatic channel used in the experiments described in this report as ()()()+??= ?????? ? A h h F QU QU 21 2 2 1 (6.2) where ? = fluid density, Q = flow rate, U = x component of cross sectional average velocity, ? = fluid specific weight, A = flow area, h = water surface elevation = piezometric head, F ? =x component of shear force on the channel bed, x is horizontal (not parallel to the flow, so there is no weight component in the equation) and positive in the downstream direction, 1 and 2 are respectively the upstream and downstream cross sections (e.g., at the downstream end of the weir and the downstream end of the flow re-establishment region), and h 1 -h 2 is small. The term ?A 2 (h 1 -h 2 ) in Eq. (6.2) accounts for the pressure forces at cross sections 1 and 2 plus the x component of the pressure force on the channel bed and sides. In applying Eq. (6.2), it was assumed that the channelslopeissmallsothatF ? is essentially horizontal and so that it is not necessary to distinguish between the direction normal to the bed and vertical. Dividing Eq. (6.2) by ?A 2 gives () () A A U g h F A U g h 1 2 2 1 1 2 2 2 2 ? ? ? ? +? = + (6.3) where g = gravitational acceleration. The momentum correction factor (?) is defined as 2 A 2 U dAu A 1 ? =? (6.4) where u is the x component of the total point velocity and U is the cross sectional average veloc- ity. The instantaneous value of u at a point can be written as 'uuu += (6.5) 88 where u = time-averaged velocity and u' = turbulent fluctuation. An overbar on any quantity is used to indicate time averaging. Substitution of Eq. (6.5) into Eq. (6.4) gives 321321 2 2 2 1 2 2 2 2 U 'u U u U u ? + ? ==? (6.6) and the brackets indicate an average over the cross sectional area. Note that ? and therefore the momentum flux include the turbulent flux. The components ? 1 and ? 2 in Eq. (6.6) are consid- ered in Section 6.4.8. F ? was calculated from Fx ? ?? = (P)+(P) 12 2 ? (6.7) where ?x is the channel length for which F ? is being calculated, P = wetted perimeter, and ? in English units is (Henderson, 1966) 3/1 h 2 R21.2 UUn? =? (6.8) where R h = hydraulic radius. Since F ? was frequently the smallest term in Eq. (6.2), it was not necessary to include the effects of the flow asymmetry in calculating ?. The energy equation can be written as H 1 -h f =H 2 (6.9) where h f is the head loss due to boundary friction and H is the total head given by H=h+? U g 2 2 (6.10) where ? is the kinetic energy correction factor defined as 3 2 3 A 2 3 A 2 U uV U dAuV A 1 U AdVV A 1 == ? =? ?? rr (6.11) where V is the magnitude of the point velocity vector ( r V )and r A is the area vector pointing downstream. Using Eq. (6.5), similar expressions for v and w, which are the y (transverse) and z (vertical) components of velocity, and V 2 =u 2 +v 2 +w 2 , ? in terms of the components of the velocity is 89 3 2 33 2 3 3 3 3 2 3 2 3 2 3 2 3 2 3 3 U 'w'u U 'w'uw2 U 'v'u U 'v'uv2 U 'u U 'wu U wu U vu U 'vu U 'uu3 U u ++++ ++++++=? (6.12) As with ?, ? includes the turbulent flux of kinetic energy. The experimental results showed that several of the terms in Eq. (6.12) contributed less than 1% to ? so that ? could be evaluated from { 32132143421 4 3 2 3 3 2 2 3 2 1 3 3 U vu U 'vu U 'uu3 U u ? + ? + ? + ? ?? (6.13) The various components of ? in Eq. (6.13) are considered in Section 6.4.8. In English units, the head loss due to boundary friction was approximated as ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? = 2 3/4 h 2 2 1 3/4 h 2 22 f RA Q RA Q 42.4 xn h (6.14) Note that the only head loss in Eq. (6.9) is that due to boundary friction, i.e., there is no expan- sion loss. As will be discussed in Section 6.9, Eq. (6.9) does not need to include any other head loss terms for the region of flow re-establishment downstream of a side weir if appropriate values of ? are used in defining the total head. 6.3 - FLOW CONDITIONS The experiments for studying the flow asymmetry were organized into two types. In Type 1, velocities and water surface elevations were measured for at several cross sections in the region of flow re-establishment for various flow conditions (Cases A - D for 2.5H:1V side slopes and Case F and G for 4H:1V side slopes, Table 6.1). For Cases A, B, C, and F, there was diver- sion over the side weir. Since there was only about 27.2 ft of channel length downstream of the weir, the channel was not long enough for laterally symmetrical velocity distributions to be re- established for Cases A, B, and F. For Cases D and G, the side weir was blocked and the flow conditions at the downstream end of the weir for Cases A and F were recreated at the upstream end of the channel so that about 60 ft of flow length could be used for measurements. As shown in Fig. 6.22, this attempt to reproduce the separation zone at the upstream end of the channel was not successful for Case D because of the way in which the separation zone was created; the 90 problem was corrected for Case G. For Cases E and H, there was no separation zone and the weir was blocked in order to determine ? and ? for symmetrical velocity distributions. Measurements were made near the downstream end of the channel. For Cases A, B, and C, the first cross section (x = 0) was at the bottom of the downstream sloped access ramp. The analysis showed significant transverse velocities at this cross section. Therefore, for Case F the first cross section was 2 ft farther downstream; the transverse velocities were much smaller at this cross section. Also, for the Type 2 measurements (Table 6.2), measurements were made at the down- stream end of the access ramp (x = 2 ft). Measurements showed that for a total discharge of 8.9 cfs with 54% and 26% diversion, there was only a 0.001 ft to 0.002 ft difference between the water surface elevations at the cross sections at x = 0 and x = 2 ft. The purpose of the tests in Type 2 was to get additional information on the flow asymmetry, ?,and? at the downstream end of the weir. Table 6.1 - Flow conditions for Type 1 Case Channel side slope Total dis- charge (cfs) Diver- sion (%) Remarks A 2.5H:1V 8.9 54 Side weir diversion B 2.5H:1V 3.0 54 Side weir diversion C 2.5H:1V 8.9 25 Side weir diversion D 2.5H:1V 4.1 N/A Separation zone at the upstream end of the channel E 2.5H:1V 4.1 N/A No flow asymmetry F 4H:1V 6.1 54 Side weir diversion G 4H:1V 4.5 N/A Separation zone at the upstream end of the channel H 4H:1V 4.6 N/A No flow asymmetry 6.4 - VELOCITY MEASUREMENTS Velocities were measured using an acoustic Doppler velocimeter (Section 3.4). This section discusses the measured velocities and the ? and ? values (Appendix 5) that were obtained from those measurements. The downstream end of the weir is x = 0 for Cases A, B, C, and F while zero distance is at the downstream end of the flow straighteners for Cases D, E, and G. The components of the ? and ? values are discussed in Section 6.6. 91 Table 6.2 - Flow conditions for Type 2 Channel side slope Total dis- charge (cfs) Diver- sion (%) 2.5H:1V 8.9 54 2.5H:1V 8.9 40 2.5H:1V 8.9 25 2.5H:1V 6.0 55 2.5H:1V 6.1 39 2.5H:1V 3.0 54 2.5H:1V 3.0 40 2.5H:1V 3.0 25 4H:1V 9.0 54 4H:1V 9.0 40 4H:1V 9.0 25 4H:1V 6.0 54 4H:1V 6.0 40 4H:1V 6.0 25 4H:1V 3.0 54 4H:1V 3.0 40 4H:1V 3.0 25 6.4.1 - Measurement Procedures The measurements were taken on seven verticals spaced at approximately 1.1 ft intervals. The middle vertical was at the centerline of the channel. The two outer verticals on each side were above the side slope. For smaller flow depths, the outermost vertical on each side was less than 3.3 ft from the centerline in order to maintain a sufficient distance between the sampling volume and the boundary. The other verticals were not moved for the smaller depths. Measurements were taken at three points on each of the middle three verticals. The top set of measurements was about 2.5 in. below the water surface. The bottom set of measurements was about 0.4 in. above the invert of the channel. The middle set of measurements was midway between the top and bottom measurements. Only one measurement was made on the outermost vertical on each side at about the same elevation as the top set of measurements on the middle three verticals. Two measurements were taken on the next-to-outermost vertical on each side at about the same elevations as the top and middle sets of measurements on the middle three verticals. Hence 15 points were measured in a cross section. The flow depth was measured at 0.4 ft to the right of the centerline. Water surface elevations were measured at the seven verticals for each cross section. 92 Transverse position (ft) -5 -4 -3 -2 -1 0 1 2 3 4 5 +++ +++++ +++++++ Left Right weir Fig. 6.1 - Typical locations of velocity measurements looking downstream Due to the time-consuming nature of the measurements, several days were required to complete the measurements for a particular combination of total discharge and percentage diver- sion. The water surface elevations were measured in one day. The flow rate and percentage diversion quoted for each case in Table 6.1 referred to those during the day of the measurement of the water surface elevations. The flow rates on different days were within 2% of each other, thereby indicating that there was also good reproduction of the flow depths from day to day. 6.4.2 - Integrations Numerical integrations of the measured velocities over the flow area had to be done to calculate ? and ? values (Eq. (6.4) and Eq. (6.11)). The velocities were also integrated to obtain the flow rate as a check against the flow rate from the flow meters. For all of the integration, each measured velocity was assumed to represent an area defined laterally and vertically by the midpoints between the measurements, the channel boundary, or the water surface. The area inte- grations were done first vertically then laterally using a trapezoidal rule in both directions. The integrands were assumed to be the same at the water surface as for the top measurement, while they were assumed to be zero at both the lateral boundary and the bottom boundary. 6.4.3 - Case A Case A had Q u = 8.9 cfs with 54% diversion for the channel with 2.5H:1V side slopes. Velocities were measured at x = 0, 4.3 ft, 10.2 ft, 17.3 ft and 22.3 ft.. Fig. 6.2 to Fig. 6.4 show the longitudinal velocity distributions for the cross sections at x = 0, 4.3 ft, and 22.3 ft. In Fig. 6.2, the presence of a separation zone is indicated by the upstream flow at the top on the left side of the channel and by the higher velocities on the right for the middle and bottom measurements; the weir is on the right side. At x = 4.3 ft, there was still a region of average upstream velocity but it had diminished in size (Fig. 6.3). Along the length of the weir, the flow on the right side of the channel next to the weir had a strong transverse velocity component due to the flow over the weir. Immediately downstream of the weir, the residual transverse velocity continued to pull the water to the right side of the channel. As a result, the velocity distribution was more skewed at x 93 = 4.3 ft than for x = 0. Fig. 6.4 shows that even at the end of the channel, the velocities were higher on the right than the left. Transverseposition(ft) -6-5-4-3-2-101234 u ( ft/s) -0.5 0.0 0.5 1.0 1.5 top middle bottom _ Fig. 6.2 - Longitudinal distributions of velocity at downstream end of weir crest (Case A) Transverse position (ft) -6-5-4-3-2-101234 u ( ft/s) -0.5 0.0 0.5 1.0 1.5 top middle bottom _ Fig. 6.3 - Longitudinal distributions of velocity 4.3 ft downstream from end of weir crest (Case A) The ? and ? values for Case A are shown in Fig. 6.5. From both Fig. 6.4 and Fig. 6.5 it is evident that the channel was not long enough downstream of the weir for the velocities distribu- tions to return to symmetry and for ? and ? to reach their asymptotic values for Case A. 6.4.4 - Case B Case B had Q u = 3.0 cfs with 54% diversion for the channel with 2.5H:1V side slopes. Velocities were measured at x = 0, 4.4 ft, 10.3 ft, 17.4 ft and 22.3 ft. The velocity distributions resembled those at the corresponding cross sections in Case A. The velocity measurements indi- cate that the relative velocity distributions depend primarily on the percent diversion without a strong dependence on the flow rate. This conclusion is also supported by the ? and ? values that are shown in Fig. 6.5 in comparison with the values for Case A. There are small differences in 94 the ? and ? values immediately downstream of the weir, but for x ? 10 ft, the values are much closer together. Transverse position (ft) -6-5-4-3-2-101234 u ( ft /s ) -0.5 0.0 0.5 1.0 1.5 top middle bottom _ Fig. 6.4 - Longitudinal distributions of velocity 22.3 ft downstream from end of weir crest (Case A) ? 1.0 1.2 1.4 1.6 1.8 2.0 X X X X X X x = Longitudinal distance (ft) 0 5 10 15 20 25 ? 1.0 1.5 2.0 2.5 3.0 3.5 X X X X X symmetrical velocity distribution Case A Case B Case C Fig. 6.5 - ? and ? values for Cases A - C 6.4.5 - Case C Case C had Q u = 8.9 cfs with 25% diversion for the channel with 2.5H:1V side slopes. Velocities were measured at x = 0, 4.4 ft, 10.2 ft, 17.4 ft and 22.2 ft. Fig. 6.6 shows the longitu- dinal velocity distributions at x = 4.4 ft. The velocities were higher on the left than on the right, in contrast to Cases A and B. This characteristic continued further downstream. The measured velocities were all positive, even at x = 0. The lateral flow over the weir was apparently not 95 strong enough to pull the bulk of the flow to the right side of the channel to create a separation zone. Dye tests showed that there were no separation zones for any diversions on the order of 30% or less for 2.5H:1V side slopes. For trapezoidal channels, specification of the conditions for initiation of a separation zone is somewhat subjective. Even very small diversions may cause a region of nearly zero or upstream velocity for a width of one to two inches in the model at the edge of the channel opposite to the weir. Also, upstream flow in a separation zone may be inter- mittent for low diversions. A separation zone was said to exist in the model when there was con- sistent upstream flow at the left most velocity measurement position (y = -3.3 ft). Transverse position (ft) -6-5-4-3-2-101234 u ( ft/ s ) 0.0 0.5 1.0 1.5 2.0 _ top middle bottom Fig. 6.6 - Longitudinal distributions of velocity 4.4 ft downstream from end of weir crest (Case C) While there is no absolutely conclusive explanation for the change in the velocity distri- butions for Case C relative to Cases A and B, the following comments are offered as speculation. For several flow conditions, dye tests were done to determine the parts of the channel from which the flow over the weir was coming. Dye was injected into the flow at the upstream end of the weir. The injection tube was moved laterally until the dye streak at the downstream end of the weir was split about evenly with half going over the weir and half going downstream in the chan- nel. This procedure was done for injections at the surface and at the bed. These dye tests indi- cated that the weir flow comes from farther away from the weir at the bottom of the channel than at the top for most diversions (Fig. 6.7). This behavior is reasonable because the flow has higher velocities near the surface than near the bed so the flow near the water surface has more down- stream momentum and a resulting stronger tendency to continue down the channel rather than go over the weir. Applying this rationale to Case C, it is possible that the lower velocities on the side of the channel near the weir are a result of water being drawn from the lower regions of the approach flow to fill in the region vacated by water going over the weir. (Fig. 6.7 is for the chan- nel after it had been modified to have 4H:1V side slopes. Earlier qualitative tests with the 2.5H:1V side slopes showed the same trends.) 96 Left Right Transverse Position (ft) -4-3-2-101234 Diversion 54% 40 % 30 % 9% weir Fig. 6.7 - Left boundaries of regions from which weir flow comes For Case C with Q u = 8.9 cfs and 25% diversion, the ? and ? values are shown in Fig. 6.5. The values immediately downstream of the weir are only slightly greater than their asymp- totic values, and they rapidly reach their asymptotic values. 6.4.6 - Case D The channel downstream of the side weir was not long enough for the flow to completely re-establish itself for diversions on the order of 50%. To allow more channel length to study the re-establishment for a 54% diversion with a separation zone, the side weir was blocked and a separation zone was created at the upstream end of the channel after measurements had been made for separation zones created by outflow over the side weir. The discharge into the channel was adjusted to be the same as the flow rate downstream of the side weir for Q u = 8.9 cfs with 54% diversion. For Case D, the left side of the channel cross section upstream of the flow straighteners at the head box was blocked to create a separation zone. The flow straighteners are thin vertical sheets 2 ft long and 2.4 in apart. Also, vertical wood strips were used to adjust the velocity distribution at the upstream end of the channel to be similar to that downstream of the side weir in the case with flow diversion. Even though the time-averaged velocities for the ?forced? separation had reasonable agreement with the actual separation zone, analysis of the measurements (Section 6.7, Fig. 6.22) showed that there were significant differences in the turbu- lence for the two cases so that the forced separation zone did not accomplish the desired objective for Case D. The straighteners apparently had a significant effect on the turbulence created by the flow separation. Since this difference was not discovered until after the channel had been modified to have 4H:1V side slopes, it was not possible to repeat the measurements. For Case G (Section 6.4.9), blocking of part of the channel was done downstream of the flow straighteners. The agreement between the results for Cases F and G is much better than between Cases A and D (Section 6.7, Fig. 6.23) . Case D had Q = 4.1 cfs with a forced separation zone at the upstream end of the channel with 2.5H:1V side slopes. This flow condition was similar to that downstream of the weir for Q u = 8.9 cfs with a 54% diversion; the flow of 4.1 cfs is 46% of 8.9 cfs. Velocity measurements 97 were made at the cross sections 7.4 ft, 12.5 ft, 25.5 ft, 37.2 ft, 48.9 ft and 60.6 ft from the down- stream end of the flow straighteners. The values of ? and ? are shown in Fig. 6.8. Even with 60 ft of channel length, the ? and ? still do not reach their asymptotic values but they get much closer than for Cases A and B. Longitudinal distance from headbox (ft) 0 10203040506070 ? 1.0 1.5 2.0 2.5 3.0 3.5 ? 1.0 1.2 1.4 1.6 1.8 2.0 symmetrical velocity distribution Fig. 6.8 - ? and ? values for Case D 6.4.7 - Case E Case E had Q = 4.1 cfs with no separation zone in the channel with 2.5H:1V side slopes. With the side weir blocked with a thin metal sheet and with no flow modification at the headbox (other than the packed bed and flow straighteners to remove the large scale eddies generated in the headbox), velocity measurements were made at the cross section 60.6 ft from the downstream end of the flow straighteners to determine ? and ? values for established flow. These velocity distributions were essentially symmetrical about the channel centerline. 6.4.8 - Case F Case F had Q u = 6.1 cfs with 54% diversion for the channel with 4H:1V side slopes. Velocities were measured at x = 2.5 ft, 8.1 ft, 13.6 ft, and 19.2 ft.. Fig. 6.9 shows the longitudi- nal velocity distributions at x = 2.5 ft for two sets of measurements made on different days. For the top and middle measurements, the agreement is good. The bottom measurements are indica- tive of a problem in many of the measurements for the 4H:1V side slopes, namely that it was difficult to obtain good reproducibility of the bottom measurements. Because of the steep gradi- 98 ents of velocity near the bottom of the channel, small differences in the vertical position of the velocity probe could make a significant difference in the velocities. Nevertheless, it is difficult to believe that this potential problem is the source of the different velocities since the position of the probe relative to the boundary was measured with the acoustic probe itself. Except for this problem with the bottom set of measurements, the velocity distributions for the 4H:1V side slopes were very similar to those for the 2.5H:1V side slopes. Fortunately, the problem with the bottom set of measurements did not greatly affect the results for ? and ?. For the two sets of measurements in Fig. 6.9 the two ? values were 1.99 and 1.91 (4% difference) and the two ? values were 3.83 and 3.80 (less than 1% difference). Transverse position (ft) -6 -5 -4 -3 -2 -1 1 2 3 40 u ( ft/ s ) -0.8 -0.4 0.4 0.8 1.2 1.6 0.0 2.0 top-1 middle-1 bottom-1 top-2 middle-2 bottom-2 _ Fig. 6.9 - Longitudinal distributions of velocity 2.5 ft from downstream end of weir crest (Case F) The ? and ? values are shown in Fig. 6.10. The values immediately downstream of the weir are a little larger than for Cases A and B (Fig. 6.5), and the values decrease more rapidly than for Cases A and B. The average values for Cases A and B are shown by the dashed lines in Fig. 6.10. The faster decrease is presumably a result of a greater influence of boundary shear with the flatter side slopes. 6.4.9 - Case G Case G for the channel with 4H:1V side slopes is similar in purpose to Case D for 2.5H:1V side slopes in that a separation zone was created artificially at the upstream end of the channel to allow additional channel length for re-establishment of the flow. However, this time the flow was blocked on the downstream side of the flow straighteners. The flow rate was 4.5 cfs, which corresponds to the flow downstream of a weir with Q u = 9.0 cfs and 50% diversion. The agreement between the results for Cases F and G is much better than between Cases A and D, as discussed in Section 6.7. Velocity measurements were made at 15 ft, 17.2 ft, 19.3 ft, 21.5 99 ft, 23.6 ft, 25.8 ft, 28 ft, 32.3 ft, 34.5 ft, 46.5 ft and 58.5 ft from the downstream end of the flow straighteners. The large number of measurements was due to the variation of ? and ? (Fig. 6.11) being somewhat irregular and the desire to try to determine the variation correctly. 0 5 10 15 20 25 ? 1.0 1.5 2.0 2.5 3.0 3.5 x = Longitudinal distance (ft) ? 1.0 1.2 1.4 1.6 1.8 2.0 symmetrical velocity distribution Case F Cases A&B Fig. 6.10 - Variation of ? and ? (Case F) Visual observation of the flow indicated that the flow immediately downstream of the obstruction blocking part of the channel to create the separation zone was not similar to the flow conditions at the downstream end of the weir. Thus, it was decided to start the measurements 15 ft downstream of the headbox. As the initial increases in ? and ? (Fig. 6.11) indicate, the asym- metry did not start to decrease until almost 20 ft downstream of the headbox. 6.4.10 - Case H Case H had Q = 4.6 cfs with no separation zone in the channel with 4H:1V side slopes. With the side weir blocked with a thin metal sheet and with no flow modification at the headbox (other than the packed bed and flow straighteners to remove the large scale eddies generated in the headbox), velocity measurements were made at the cross section 62.5 ft from the downstream end of the flow straighteners to determine ? and ? values for established flow. These velocity distributions were essentially symmetrical about the channel centerline. 100 Longitudinal distance from headbox (ft) 0 10203040506070 ? 1.0 1.5 2.0 2.5 3.0 3.5 ? 1.0 1.2 1.4 1.6 1.8 2.0 symmetrical velocity distribution Fig. 6.11 - Variation of ? and ? (Case G) 6.5 - ? AND ? VALUES AT DOWNSTREAM END OF WEIR The parameters ? and ? at the downstream end of the weir are important since at least one of them is needed to relate the head at the downstream end of the weir to the water level downstream of the flow re-establishment region. The importance of having reasonable values for ? or ? increases as the flow velocity increases. Eq. (6.3) shows that there is a linear relationship between ?U 2 /g and water surface elevation (or flow depth). If U is 6 ft/s at the downstream end of the weir for prototype conditions, then assuming that ? = 1 when the actual value is 1.75 will produce an error of 0.84 ft in the head on the weir while the error is only 0.05 ft when U = 1.5 ft/s. Eq. (6.10) shows a linear relationship between ?U 2 /2gandhforagivenHsoassuming? = 1 when the actual ? = 3 for U = 6 ft/s gives an error of 1.12 ft in h while the error is only 0.07 ft when U = 1.5 ft/s. In Fig. 6.12, measured ? and ? values at the end of the weir (Appendix 5.3) are plotted as functions of Q w /Q u for 2.5H:1V side slopes As mentioned earlier, the primary dependence of both ? or ? is on Q w /Q u , or equivalently Q u /Q d since Q u /Q d =1/(1-Q w /Q u ). The secondary variation (scatter about the curve) comes from the fact that both ? and ? decrease slightly as Q u increases. Several attempts were made to find a suitable dimensionless parameter to represent this variation, but none could be found. The best relationships that could be found for 2.5H:1V side slopes using dimensionless parameters are 101 ?= ?+ ? ? ? ? ? ? ? ? ? ? ? 108 0 991 0 301 0 298 2 . Q Q Q Q u d u d .. . for 1 Q Q u d ??125. for 1.25 Q Q u d <<2. (6.15) ?= ?+ ? ? ? ? ? ? ? ? ? ? ? 123 166 170 108 2 . .. Q Q Q Q u d u d . for 1 Q Q u d ??125. for 1.25 Q Q u d <<2. (6.16) where Q u /Q d =1/(1-Q w /Q u ). To obtain these relationships, polynomials were first fitted to the data points for flows with diversions. Then the intersections of the polynomials with the ? and ? values for undisturbed flow (? =1.08and? = 1.23) were found. The values for undisturbed flow were assumed to apply below the intersections, both of which occurred at Q u /Q d =1.25orQ w /Q u =0.2. Q w /Q u =Diversion 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ? and ? 1.5 2.0 2.5 3.0 3.5 1.0 ? ? Fig. 6.12 - ? and ? values at end of weir for 2.5H:1V side slopes The results for 4H:1V side slopes are given in Appendix 5.4. Curve fitting to the data points gave ? ? ? ? ? +? =? d u Q Q 987.0073.0 12.1 20.1 Q Q 1for d u ?? 2.2< Q Q <1.20for d u (6.17) 102 ? ? ? ? ? +? =? d u Q Q 93.228.2 17.1 18.1 Q Q 1for d u ?? 2.2< Q Q <1.18for d u (6.18) as the best fit equations. Q w /Q u = Diversion 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ? and ? 2.0 3.0 4.0 5.0 1.0 ? ? Fig. 6.13 - ? and ? values at end of weir for 4H:1V side slopes In the absence of a detailed investigation, Tynes (1989) assumed that the velocity was zero in the separation zone and uniform in the effective flow area. The ratio of the effective flow area to the cross-sectional area was taken to be Q d /Q u . Accordingly, immediately downstream of the side weir, ? = ? ? ? ? ? ? Q u Q d 2 and ? = Q u Q d (6.19) The assumption by Tynes (1989) about the velocity distribution in the separation zone leads to overestimation of ? and ?. For example, depending on the flow rates, for 54% diversion, ? is overestimated by 40% to 70% and ? is overestimated by 20% to 30%. 6.6 - COMPONENTS OF ? AND ? 6.6.1 - Variation with Flow Distance Appendix 5.1, Appendix 5.2, Fig. 6.15, Fig. 6.16, Fig. 6.17, Fig. 6.18 and Fig. 6.19 give the components of ? and ? (Eq. (6.4) and Eq. (6.13)) for the tests for which velocities were measured at different longitudinal distances in the channel. The components were evaluated to determine the relative significance of the various terms in the momentum and kinetic energy transport. The downstream end of the weir is x = 0 for Cases A, B, C, and F while zero distance is at the downstream end of the flow straighteners for Cases D, E, and G. The results for some of 103 the components of ? and ? were inaccurate for Case F, so there are some missing values in Appendix 5.2 and Fig. 6.18. The longitudinal distances were normalized with respect to a transverse length scale (B s ) associated with the asymmetrical velocities at the end of the weir. To calculate B s ,itwas assumed that all of the flow at the downstream end of the weir is in an effective area (A e ,Fig. 6.14) and that the velocity is zero in the remainder of the channel that has a width of B s .The velocity (u e ) in the effective area was calculated so that A e u e = Q. Since Q also is equal to AU, u e /U = A/A e . Then, from Eq. (6.4) neglecting the turbulent transport of momentum, e 2 2 e 2 e A 2 e 2 A 2 A A A Q A A Q A 1 U dAu A 1 U dAu A 1 = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ===? ? ? (6.20) A e B s u y u=0 u e Fig. 6.14 - Assumed velocity distribution for calculating B s x/B s = Dimensionless distance 012345678 Co mpon ents o f ? and ? 3 6 3 6 3 6 0.001 0.01 0.1 1 3 ? 1 ? 2 ? 1 ? 2 ? 3 ? 4 Fig. 6.15 - Components of ? and ? for 54% diversion (Cases A and B) 104 B s was then calculated as the value needed to give A e so that ? in Eq. (6.20) is equal to the empirical ? value at the downstream end of the weir. The results are shown in Appendix 5.1 and Appendix 5.2. Since Case D with the forced separation zone at the upstream end of the channel was supposed to represent the same flow conditions as Case A, B s was also taken to be the same for Case D as for Case A. For the same reason, B s forCaseGisthesameasforCaseF. There were no B s values for Cases E and H. The locations of the measurement cross sections for Cases A - C were essentially the same; the larger dimensionless distances for Case C are the result of the smaller B s value, not larger x values. x/B s = Dimensionless distance 0 5 10 15 20 25 30 Com pone nts o f ? and ? 3 6 3 6 3 6 0.001 0.01 0.1 1 3 ? 1 ? 2 ? 1 ? 2 ? 3 ? 4 Fig. 6.16 - Components of ? and ? for 25% diversion (Case C) x/B s = Dimensionless distance 0 5 10 15 20 25 Compon ents o f ? an d ? 6 3 6 3 6 3 6 0.001 0.01 0.1 1 3 ? 1 ? 2 ? 1 ? 2 ? 3 ? 4 Fig. 6.17 - Components of ? and ? for forced separation zone (Case D) 105 x/B s = Dimensionless distance 01234567 Comp onen ts of ? an d ? 3 6 3 6 3 6 3 6 0.001 0.01 0.1 1 ? 1 ? 2 ? 1 ? 2 ? 3 ? 4 Fig. 6.18 - Components of ? and ? for forced separation zone (Case F) x/B s = Dimensionless distance 0 5 10 15 Co mponen ts of ? and ? 6 3 6 3 6 3 6 3 6 0.001 0.01 0.1 1 ? 1 ? 2 ? 1 ? 2 ? 3 ? 4 Fig. 6.19 - Components of ? and ? for forced separation zone (Case G) Several trends are apparent from the measurements: (1) Most of the measurements at the end of the weir for the 2.5H:1V side slopes (Fig. 6.15 and Fig. 6.16) do not fit the trends downstream from the weir. Thus, the trend lines are not extended to x = 0. The apparent reason is the relatively strong transverse time-averaged velocity component toward the weir at this cross section. One of the clearest indications of this behavior is that ? 4 , which includes the time-averaged transverse velocity ( 2 v ), is almost two orders of magnitude larger at the end of the weir for Cases A and B than would be indi- cated by extrapolating the trend of the other points back to the weir. At the end of the weir, most of the other components are a little smaller than would be indicated by extrapolation from the downstream points. The same type of problem does not exist for Case F since the first measurement cross section was at x = 2.5 ft, not x = 0. 106 (2) Downstream from the weir, ? and ? and each of their components decrease with increasing longitudinal distance as the flow asymmetry decreases. (3) For Case C with the smaller diversion, the relative magnitudes of the turbulence terms are smaller than for Cases A, B, and F. There are no trend lines for Case C (Fig. 6.16) since the components for the first cross section were affected by the transverse velocities and since asymptotic conditions were reached upstream of the last two cross sections. (4) The relative magnitudes of the turbulence terms are smaller for Case D with the forced sepa- ration zone for the channel with 2.5H:1V side slopes than for Cases A and B even though the time-averaged normalized longitudinal velocity distributions were nearly the same for all three cases. This comparison indicates that the manner in which the separation zone was created for Case D was not a good reproduction of the effects of the weir, as mentioned earlier. The same problem does not exist for Case G where the area was blocked downstream of the flow straighteners to create the separation zone. (5) For Case E, the values of ? and ? were found to be 1.08 and 1.23. These values agree with the downstream values for Case C, so they were adopted as the values for undisturbed flow in the channel with 2.5H:1V side slopes. The corresponding values for the 4H:1V side slopes were ? =1.12and? = 1.17. 6.6.2 - Variation with Diversion Appendix 5.3, Appendix 5.4, Fig. 6.20, and Fig. 6.21 give the components of ? and ? at the downstream end of the weir. All components for both ? and ? increase as the diversion increases, but ? 2 and ? 2 through ? 4 increase at approximately the same rate and more rapidly than ? 1 and ? 1 . Similar rates of increase might be expected for ? 2 , ? 2 ,and? 3 since all of these terms have the mean of a squared turbulent velocity, but ? 4 has about the same rate of increase with diversion even though it includes only time-averaged velocities and is the smallest of the terms. The relative importance of the turbulent transport increases as the diversion increases; ? 2 varies from 1% to 10% of ? while ? 2 supplies 2% to 15% of ?, ? 3 supplies 0.2% to 2.4%, and ? 4 supplies only 0.1% to 1.5%. All of the other terms that come from a complete expansion of (Eq. (6.12)) are less that 1% of ?, even at the highest diversion rates. 6.7 - LENGTH FOR FLOW RE-ESTABLISHMENT Downstream of a side weir, the velocity distribution gradually returns toward symmetry. At the beginning of the measurements and analysis, it was not clear how to best quantify the asymmetry of the velocity distributions. Thus, three parameters (in addition to ? and ?)were used and were calculated separately for the top, middle and bottom sets of measurements at each cross section. These parameters represented (a) the root-mean squared variation of differences 107 between the velocities on the right side of the channel and the velocities at the corresponding points on the left side, (b) the skewness of the velocity distributions, and (c) the area under the velocity distribution curve in the right half of the channel and that in the left half. None of these parameters proved to be significantly more informative than ? and ?,so? and ? were used as the primary parameters to represent the amount of asymmetry in the velocity distributions. Q w /Q u = Diversion (%) 15 25 35 45 5520 30 40 50 Co mpone nts o f ? an d ? 3 6 3 6 3 6 0.001 0.01 0.1 1 3 ? 1 ? 2 ? 1 ? 2 ? 3 ? 4 Fig. 6.20 - Variation of components of ? and ? near end of weir with diversion for 2.5H:1V side slopes Q w /Q u = Diversion (%) 15 25 35 45 5520 30 40 50 Compon ents of ? and ? 3 6 3 6 3 6 3 6 0.001 0.01 0.1 1 ? 1 ? 2 ? 1 ? 2 ? 3 ? 4 Fig. 6.21 - Variation of components of ? and ? near end of weir with diversion for 4H:1V side slopes The variations of ? - ? o ,where? o is the value for symmetrical velocity distributions, with distance were used to determine the flow length (L s ) required for the flow to return to symmetry. 108 For 2.5H:1V side slopes, ? o =1.08 while the value is 1.12 for 4H:1V side slopes. The variations of ? - ? o were also studied, but they gave slightly different results. It was decided to use ? since it is used in the computer program rather than ?. The variations of ? - ? o with distance for 2.5H:1V side slopes are given in Fig. 6.22 on a semi-logarithmic plot since decay processes fre- quently have an exponential decay as they approach their asymptotic values. For this figure, the results for Case D were treated as if the section at 12.5 ft from the flow straighteners were 4.3 ft downstream from the end of the weir crest. This matching is based on the fact that the measured velocities for these two cross sections were essentially the same. x/B s = Dimensionless distance 024681012141618 ? -1 . 0 8 0.01 0.1 1 Case A B C D Fig. 6.22 - Exponential decay of excess ? for 2.5H:1V side slopes One of the prime objectives for Case D was to get a direct indication of the length required for the flow asymmetry to disappear. Since the turbulence for Case D was less intense than for Case A (Section 6.4.8), the asymmetry disappeared more slowly for Case D than for Case A. Nevertheless, Case D has data over a larger distance and indicates that there is indeed an exponential decay of ? - ? o . Based on this type of behavior, the best fit line through the points for Cases A and B (except for x = 0, for reasons discussed earlier) is extrapolated to a value of 0.05 to represent the point at which ? decays to within 5% of its asymptotic value. This process gives L s /B s = 12.5 for 54% diversion. For Case C, the last two points are not plotted since ? had reached its asymptotic value upstream of these points. The best-fit line through the remaining points gives L s /B s = 0.6 for 25 % diversion. However, it must be recognized that these values of L s come from extrapolation of the measurements, that different values of L s would be obtained if different parameters other than ? were used, and that the results for L s are very limited. Thus, these values must be viewed as only an indication of the length of the flow re-establishment dis- 109 tance. Fortunately, as is illustrated in Section 6.9, it is normally not necessary to know L s with high accuracy. Similar results for 4H:1V side slopes are given in Fig. 6.23. For these measurements, the blocked area to create a separation zone was downstream of the flow straighteners. It can be seen that the trends for Cases F (with actual diversion) and G (with a forced separation zone) are essentially the same. For these measurements L s /B s =7.3. x/B s = Dimensionless distance 024681012 ? -1 . 1 2 0.01 0.1 1 Case F G Fig. 6.23 - Exponential decay of excess ? for 4H:1V side slopes Assuming that L s /B s would be zero for no diversion and would remain small for diver- sions less than 30% since there is essentially no separation zone for those conditions, an esti- mated variation of L s /B s is given in Fig. 6.24. This figure, as well as a comparison of Fig. 6.22 and Fig. 6.23, shows that the flow conditions for 4H:1V side slopes return to symmetry more rapidly than for 2.5H:1V side slopes. These figures give the results in terms of dimensionless distances (x/B s and L s /B s ), but the same conclusion applies for actual distances (x and L s ). Because of the very limited data, Fig. 6.24 needs to be used with caution. 6.8 - MOMENTUM AND ENERGY BALANCES The measurements allowed all terms in Eq. (6.3) except F ? to be calculated, and F ? could be found from Eq. (6.7) and Eq. (6.8). Thus, to investigate the accuracy of the measurements, the momentum equation (Eq. (6.3)) was written as ( ) ( ) 2 2 2 M 2 1 1 2 2 1 h g U h A F h g U A A + ? =?? ? ?+ ? ? (6.21) 110 where ?h M is a residual term to account for any inaccuracies in the measurements in balancing the momentum equation. The term ?h M was evaluated from the measurements with cross sections 1 and 2 in Eq. (6.21) being successive measurement cross sections. The measurements also allowed determination of a residual term (?h E ) for the energy equation written as H 1 -h f - ?h E =H 2 (6.22) Q w /Q u =Diversion(%) 0 102030405060 L s /B s = D ime n si onl es s l e ngth of f l ow r e - e stal i s hme n t r egi on 0 2 4 6 8 10 12 14 Empirical 2.5H:1V 4H:1V Estimated Estimated Fig. 6.24 - Length of flow re-establishment region From the measurements, ?h M (Eq. (6.21)) and ?h E (Eq. (6.22)) were calculated for each pair of consecutive cross sections (Appendix 6.1 to Appendix 6.10). The rate at which momen- tum was transported across a section was calculated as ??Q d 2 /A, where Q d is the measured flow rate, for all cross sections except x = 0. ? A dAu wasusedinlieuofQ d for the cross section at the downstream end of the weir crest because of the outflow over the downstream ramp. The velocity head was calculated as ?Q d 2 /(2gA 2 ). In each case, Q d during the water surface elevation measurements was used for the cross sections downstream from the side weir. In the calculation of ? A dAu , a parabolic distribution was assumed for u below the bottom measurement point. Between the water surface and the top measurement point, u was assumed to be the same as that of the top measurement; u was assumed to vary linearly between the top and middle measurement points and between the middle and bottom measurement points. The vertical integration was done first using the assumed distribution of u and then laterally with u = 0 at the sides of the channel. For the cross section at the downstream end of the weir crest for Case A, the difference between ? A dAu and Q d was 8%. For all the other measurements, the differences were less than 6%. The discrepancies were considered acceptable given that veloci- ties were measured only at 15 points in a cross section. There was lateral flow over the ramp at the downstream end of the side weir. The higher discrepancy for the cross section at the down- stream end of the weir crest for Case A was probably due to the larger flow over the ramp. 111 As mentioned in Section 3.4, the distances from boundary reported by the data acquisition program were inaccurate for measurements above the side slopes. It was estimated that the errors in the calculation of ? A dAu , ? and ? due to the inaccuracy in the distance measurements were less than 1%. The water surface elevations were measured at the same cross sections and the same seven transverse locations where velocities were measured. In the momentum and energy balances, the water surface elevation for each cross section was taken as the average of the meas- urements for that cross section. For the cross sections other than that at the downstream end of the weir crest, the differences between water surface elevations for the same cross section were at most 0.003 ft. For the section at the downstream end of the weir crest, there were larger differ- ences because of the drawdown due to the flow over the weir crest. Therefore, for that section, the measurements at y = 1.1 ft, 2.2 ft, and 3.3 ft (i.e., the measurements on the weir side of the channel) were not included in the average for Case A, and the measurements at y = 3.3 ft were not included in the averages for Cases B and C; the drawdown at the weir was much smaller for Cases B and C than for Case A. The largest values of ?h M and ?h E were between the cross section at the end of the weir and the next cross section since there was still a significant transverse velocity at the end of the weir. Excluding the values at the end of the weir, the residuals in balancing the equations are smaller. The residuals in Table 6.3 are rather small given that the measurement accuracy for water surface elevations was on the order of 0.001 ft to 0.002 ft and only 15 velocities were measured in each cross section. It was essential to include the turbulent fluxes of momentum and energy to obtain this good degree of closure for the momentum and energy equations. An indi- cation of the magnitude of the turbulence flux terms is given in Section 6.4.8. The momentum and energy balances were not done for Case G; the excellent results for the other cases indicated that accurate measurement techniques were being used. Table 6.3 - Summary of errors in balancing momentum and energy equations ?h M ?h E (ft) (ft) All cross sections Exclude end of weir All cross sections Exclude end of weir maximum 0.0009 0.0009 0.0023 0.0023 average -0.0010 -0.0004 -0.0003 0.0001 minimum -0.0077 -0.0016 -0.0045 -0.0015 standard deviation 0.0019 0.0006 0.0014 0.0009 112 The discussion above on the momentum and energy balances relates to the flow re-estab- lishment region downstream of the weir. El-Khashab and Smith (1976) made detailed velocity and depth measurements in a rectangular channel in the region beside a side weir. The velocity and flow depth upstream of the weir were 3.9 ft/s and 0.85 ft. The diversion was 70%. (These values are only approximate since they had to be read from a graph or calculated from values read from a graph. Also, the 70% diversion is based on an estimated ? of 3 at the downstream end of the weir.) They found a large imbalance in trying to close the energy equation for the section of the channel along the weir. The imbalance was approximately 0.8 in. between the upstream and downstream ends of the weir; this 0.8 in. was 16% of ?U 2 /2g at the upstream end of the weir and 1.7 times ?U 2 /2g at the downstream end of the weir. El-Khashab and Smith (1976) included the lateral and vertical velocities in their kinetic energy terms but not the turbu- lent transport of kinetic energy. If the estimated diversion is approximately correct, then there was little or no separation zone for this flow condition in their rectangular channel. For their channel, it is difficult to imagine that this 0.8 in. is head loss due only to flow asymmetry. Omit- ting the turbulence terms apparently accounted for at least part of the excess head loss. 6.9 - APPLICATION To illustrate the importance of flow asymmetry and ?, consider an improved trapezoidal channel with a bed slope of 0.0008, a base width of 85 ft, side slopes of 2.5H:1V, a Manning?s n of 0.035, Q u = 25,000 cfs, and a 50% diversion so that Q d = 12,500 cfs. The flow depth (d 2 )at the downstream end of the flow re-establishment region is the normal depth of 16.1 ft. From Eq. (6.15), ? 1 = 1.58 at the downstream end of the weir. Eq. (6.2) or Eq. (6.3) with ? 2 =1.08gives the depth (d 1 ) at the downstream end of the weir as 15.4 ft. The calculation of d 1 must be done by iteration or by using a solver since B s (64.6 ft) depends on d 1 and ? as described previously. From Fig. 6.24, L s /B s ? 9 giving L s ? 580 ft. For this situation, the flow asymmetry at the end of the weir causes the flow depth and therefore the head on the downstream end of the weir to be 0.7 ft smaller than would be indicated by the downstream flow depth. Increasing or decreasing L s by 50% gives essentially no change in d 1 . Even though the change in L s gives a significant change in F ? , the change in L s also gives a compensating change in the water surface elevations at the two cross sections (h in Eq. (6.3)). Thus, it is not necessary to know L s with a high degree of accuracy. If the downstream controls give d 2 = 12.1 ft (half way between the critical and normal depths), then d 1 = 11.4 ft so that the depth at the downstream end of the weir is still 0.7 ft smaller than farther downstream. The previous examples assume that Q w is known. However, another level of iteration is required in most calculations since Q w cannot be determined until the head on the weir is known. For river channels, the flow (Q u ) approaching a side weir is determined by the hydrology of the 113 watershed upstream of the weir but the flow depths are controlled from downstream for subcriti- cal flows. Thus, iterative calculations are required to determine the flow depth (d 1 ) at down- stream end of a weir and therefore to determine the flow (Q w ) over a weir since Q w depends on the head which depends on the downstream depth which depends on the downstream flow (Q d ) which is equal to Q u -Q w . A typical computational approach would be to assume Q w then use one-dimensional gradually varied flow calculations to obtain the water surface elevation at the downstream end of the flow re-establishment zone for Q d =Q u -Q w and thereby to obtain the right-hand side of Eq. (6.3) using ? for established flow. From this depth, the head on the weir and then Q w can be calculated. This process can be continued until the assumed and calculated values of Q w agree. This is the type of calculation that is done in SIDEHYD (Burgin and Holley, 2002). If the energy equation is used rather than the momentum equation, the approach is basi- cally the same except that ?, Eq. (6.9) and Eq. (6.10) are used rather than ? and Eq. (6.3). It is important to recognize that the only head loss which is needed in Eq. (6.9) applied to the flow re- establishment region is the head loss due to the boundary shear stress, provided that appropriate ? values are used. There is no additional head loss needed downstream of the weir to account for the flow asymmetry or even the separation zones, for those flows with separation zones. This condition is indicated by the excellent closure of the energy equation for the laboratory measure- ments when using only Eq. (6.9) with no additional head loss terms. If ? = 1 were assumed throughout the flow re-establishment region, then the energy equation might be written as h U g hK U g h U g fL + ? ? ? ? ? ? ? ? ?? =+ ? ? ? ? ? ? ? ? 2 1 2 22 2 22 (6.23) where K L would appear to be a head loss coefficient. However, the laboratory measurements indicate that K L would need to be negative to balance the energy equation so K L could not actu- ally be a head loss coefficient. Rather, it would be due to the fact that U 1 2 /2g is too small to account for the true velocity head, which is ? U 1 2 /2g. These observations are consistent with those of Idelchik (1986) for other types of branching flows. Some of the results presented here are dependent on the channel geometry. El-Khashab and Smith (1976) did experiments in a rectangular channel 1.51 ft wide with heights of 0.33 ft to 0.82 ft for thin plate weirs. They found that separation zones formed only for diversions of 70% or greater. The experimental work for this project was done in a trapezoidal channel with 2.5H:1V side slopes, and separation zones formed for diversions of about 30% and greater. Flow visualization was done in a trapezoidal channel with 4H:1V side slopes. For this channel, sepa- ration zones formed for diversions of 20% and greater. This type of trend seems reasonable since 114 flatter side slopes give larger regions of low velocity on the side opposite to the weir, thereby making it easier for a separation zone to form. 115 7 - DIVERSION CULVERTS 7.1 - INTRODUCTION For some small diversions, it may be beneficial to use culverts for diversion rather than side weirs. Just as the discharge coefficients for side weirs depend on the channel flow charac- teristics as well as the normal weir parameters, it is to be expected that the flow through diver- sion culverts will also depend on the channel flow. Therefore the analysis of diversion for culverts needs to be modified to account for the effects of the channel flow. The objective of these experiments was to evaluate the effects of channel flow on the hydraulics of culverts at diversion facilities. 7.2 - THE PHYSICAL MODEL The diversion culvert model (Fig. 7.1) was built in the channel with 4H:1V side slopes. Part of the side weir was blocked leaving an opening 1.271 ft long for the culverts. The culvert model was made of 3/4 in. plywood with a base sitting on the weir crest of the side weir and with vertical walls at the upstream and downstream ends of the culverts. The vertical walls had a trapezoidal shape matching the 4H:1V side slopes of the embankment of the channel. Two verti- cal walls, 0.563 in. long, divided the culvert into three barrels 0.38 ft wide. The culvert model did not have a top so that the flow in the culvert could be observed more clearly. The invert of the culvert was about 0.55 ft from the invert of the channel. Manning?s n for the plywood culvert model was assumed to be 0.0012 (Henderson, 1966). Some tests were done with flow in all three barrels. Tests were also done with only two barrels. For these tests, a false wall was installed parallel to the end wall and in line with one of the walls separating the barrels. An inclined cover in line with the sloped side of the channel was then placed over the opening between the two ver- tical walls so that the geometry for the two operating barrels was similar to that for the three bar- rels. Fig. 7.1 - Schematic diagram of diversion culverts in model (not to scale) support model weir 4 1 0.563 ft 0.55 ft 1.625 ft 0.406 ft channel vertical end wall wall between culvert barrels point 0 1 2 3 4 5 116 Photographs of the model are shown in Fig. 7.2. For all flow conditions, large eddies developed upstream of the first and last barrel. Both eddies were caused by flow over the end walls; there was flow in the upstream direction, back into the culverts, over the downstream wall. The flow plunged over the walls, causing the eddies. The eddy on the most upstream barrel was frequently more pronounced than the one at the most downstream barrel since gravity added to the channel velocity as the flow plunged over the upstream wall while the gravity-induced flow and the channel flow were in opposite directions at the downstream wall. a) Higher channel velocity b) Lower channel velocity Fig. 7.2 - Diversion culverts channel flow channel flow 117 7.3 - MODEL RESULTS Experiments were conducted for four different upstream discharges. For each of four upstream discharge, different diversions were investigated. The total number of tests was 38. The results are given in Appendix 7. For most of the tests, there was free or unsubmerged flow at the downstream end of the culverts (cross section 4 in Fig. 7.1). A few tests were done with submerged conditions. 7.3.1 - Unsubmerged flow Discharges and water levels were measured as described in Sections 3.3 and 3.5. In most cases, the water surface elevations upstream and downstream of the culvert were the same; the largest difference was only 0.002 ft. Flow through the culvert was very complex. The simplified analysis of the flow was done as follows. Although there were multiple barrels, an average flow depth both upstream and downstream of the culvert barrels was used. Critical flow was assumed to exist at the down- stream end of the barrels for unsubmerged flow conditions. Gradually varied flow calculations were performed from the downstream end of the barrels to the upstream end of the barrels. The entrance loss at the upstream end of the barrels, h entr , was calculated as ( ) g2 VV 3.0h 2 2 2 3 entr ? = (7.1) where V 3 is the average velocity downstream of the entrance to the barrels and V 2 is the average velocity upstream of the entrance; Fig. 7.1 shows the numbering of the various cross sections in the flow. Hence g2 V h g2 V 2 3 3entr 2 2 2 +?=?+? (7.2) where ? 2 and ? 3 are the average flow depths immediately downstream and upstream of the entrance. The total head immediately upstream of the barrels was related to the total head in the channel by g2 V z g2 V K g2 V WS 2 2 22 2 2 E 2 o o ++?=?+ (7.3) where z 2 = invert elevation immediately upstream of the entrance of the barrels, WS o = water surface elevation in the channel measured at the centerline, V o = average velocity in the channel, and K E = head loss coefficient for flow from the channel to cross section 2 just before the entrance into the barrels. 118 The velocity head in the channel was calculated using ? the average velocity in the channel upstream of the culvert ? the average velocity in the channel downstream of the culvert ? the mean of the average velocities in the channel upstream and downstream of the culvert, and ? no velocity head for the channel (V o = 0 in Eq. (7.2)). Different values of K E were obtained using different methods. Including the velocity head in the channel gave better correlations than not including it. However the results for the three velocity heads were very similar. The downstream velocity head was preferred because the calculations proceed in the upstream direction for subcritical flows. K E obtained using the downstream velocity for V o was correlated with the upstream channel Froude number (F u ), downstream chan- nel Froude number (F d ), upstream weir Froude number (Fw u ), downstream weir Froude number (Fw d ), and Q w /NQ u, where Q w = discharge through the culverts, N = number of barrels, and Q u = channel discharge upstream of the culverts. The correlations with the downstream Froude num- ber and with Q w /NQ u were slightly better than the other correlations, but the channel Froude number does not include any parameters related to the flow through the culverts and the correla- tion with Q w /NQ u produced a more complicated predictive equation than the correlation with Fw d . Thus, the correlation with the downstream weir Froude number is recommended. The results are shown by the symbols that are capital letters in Fig. 7.3. The regression equation for thedatainFig.7.3ais d Fw50.2 c 4 E 10248.0K ? ? = (7.4) For unsubmerged flow, ? 4 /? c =1since? 4 , which is the depth at cross section 4, is equal to the critical depth (? c ) for flow in the culverts. The coefficient of determination (R 2 ) for Eq. (7.4) is 0.965 (versus 0.992 using the channel Froude number and 0.972 with Q w /NQ u in Fig. 7.3b). As stated above, the correlations with the weir Froude number is preferred since the correlation with Q w /NQ u , requires a more complicated equation, namely ()[] ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = 2 uw NQ/Qlog504.0 675.0 u w c 4 E 10 NQ Q 020.1K (7.5) 7.3.2 - Submerged flow For submerged outflow from the culverts, the loss coefficients for the flow from the chan- nel to cross section 2 were larger than for unsubmerged flow. It was found that multiplying the 119 loss coefficients by ? c /? 4 caused them to follow the same general trend as for unsubmerged flow, as shown in Fig. 7.3. Thus, Eq. (7.4) can be used for both unsubmerged and submerged flows. For submerged conditions, the water level in the basin can be used to obtain ? 4 ,whichis assumedtobeequalto? 5 . Fw d 0.0 0.1 0.2 0.3 0.4 0.5 0.6 1 10 E E E F F F G G H H e f g h A A A A B B B B C C C C D D D D a ab b c c d d 0.5 K E ( ? c / ? 4 ) = Adju st e d he ad l o ss co eff i cie n t Q u (cfs) 1.6 3.2 6.6 9.1 3 barrels Free flow A B C D 3 barrels Submerged a b c d 2 barrels Free flow E F G H 2 barrels Submerged e f g h Q w /NQ u 0.006 0.03 0.06 0.20.01 0.1 0.6 3 6 1 10 E E E F F F G G H H e f g h A A A A B B B B C C C C D D D D a a b b c c d d K E ( ? c / ? 4 ) = Ad justed h e a d l o ss coe f fici e n t Q u (cfs) 1.6 3.2 6.6 9.1 3barrels Free flow A B C D 3 barrels Submerged a b c d 2 barrels Free flow E F G H 2 barrels Submerged e f g h Fig. 7.3 - Adjusted loss coefficients for flow from point 0 to point 2 Eq. (7.4) Eq. (7.5) a) d 4 c E Fwvs.K ? ? b) u w 4 c E NQ Q vs.K ? ? 120 7.4 - CALCULATION PROCEDURE The experimental results and the resulting correlations are valid for 0.007 < Q w /NQ u < 0.12 for 2 or 3 barrels. For calculation of the flow through diversion culverts, it is assumed that the known information includes the culvert geometry, the flow conditions in the channel at the downstream end of the culverts, and the water level in the detention basin. For these conditions, the major steps in the calculation procedure for diversion culverts are as follows: 1) Assume a value of Q w . 2) From Q w and N, calculate the critical depth (? c ) in the culvert barrels. 3) Using Q w /N, use gradually varied flow computations to get the water surface profile through the culvert barrel to obtain ? 3 , which is the depth in the barrels at the upstream end of the barrels. These calculations start with ? 4 = ? c if the water level in the basin is below ? c . Otherwise, ? 4 comes from the water level in the basin. 4) Use Eq. (7.1) and Eq. (7.2) to calculate V 2 and ? 2 just upstream of the culvert entrance. 5) Use Eq. (7.3) and Eq. (7.4) to calculate the water surface elevation (WS o ) in the channel. 6) If WS o agrees with the know water level in the channel, then the assumed Q w is correct. If not, assume another Q w and repeat the previous steps until agreement is obtained. 7) Because Q w /Q u is small for diversion culverts, the water surface elevation in the channel at the upstream of the culverts can be assumed to be the same as at the downstream end. When the stage in the channel first rises above the culvert invert, the headwater will be very small giving a very large Fw d . The flow through the culverts will be very small giving a small Q w /NQ w . Using either Eq. (7.4) or Eq. (7.5), the calculated value of K E can be extremely large. In the computer program for diversion culverts (Burgin and Holley, 2002), the value of K E was taken as 10 if the calculated value was larger than 10. (See Fig. 7.3.) For these very small flows, any inaccuracy in the value of K E does not have any practical significance. 121 8 - CONCLUSIONS Side-channel weirs provide a viable means of flood control by diverting flow from chan- nels into detention basins. The hydraulics of the weirs have been studied and improved methods have been developed for calculating the flow over side-channel weirs. Hydraulic information on flap gates and Tideflex valves for drainage culverts has been obtained and analyzed to provide relationships for calculating the culvert drainage from detention basins. The valves prevent flow from the channels into the detention basins but, to varying degrees, they also slow the drainage flow by restricting the flow area at the downstream end of the culverts. This report addresses the experimental part of the project work. Another part of the pro- ject work is reported by Burgin and Holley (2002). That report is a user's manual for the com- putational scheme that has been developed for watershed hydrology, channel and side-weir hydraulics, and filling and emptying of detention basins. The computational scheme uses the results of the experimental work presented in this report. An improved computational scheme was developed for the estimation of side weir discharge and upstream head on the weir. Of the four calculation methods tested in Chapter 4, Method B is recommended. In Method B, the side discharge per unit length of weir is calculated from a differential equation for changes in water surface elevation in the channel along the weir. This equation was derived from the momentum principle for spatially varied flow in prismatic and tapered channels with trapezoidal cross sections. The computational scheme involves the calculation of the water surface profile along the weir. The calculations explicitly account for channel roughness and slope since the results of simulations showed that the roughness and slope can have a significant effect on the side weir discharge and upstream head on the weir. The physical equations to be used for calculating the side discharge and the changes in flow depth in the channel for different situations are listed in Table 4.6 for the recommended Method B as well as for the other methods that were tested. Regression equations were obtained for the empirical coefficients required in the computation; both previously published and new experimental results were used to develop the equations. Accuracy of the estimated values of side weir discharge and upstream head on the weir was generally good and was comparable to or better than that using the method of analysis in the previous project. The regression equations for the various empirical coefficients to be used in the calculations are also listed in Table 4.6. The effects of channel side slope on weir hydraulics were investigated. Experiments were conducted in a channel with 4H:1V side slopes for unsubmerged conditions. The results were compared with those obtained in Chapter 4 for tests in a channel with 2.5H:1V side slopes. Using discharge coefficients (C 1 ) predicted by the regression equation obtained for tests in the 122 channel with 2.5H:1V side slopes, the agreement between measured and calculated values of the side weir discharge and upstream head on the weir for tests with 4H:1V side slopes was comparable to that for tests with 2.5H:1V side slopes. Therefore the same regression equation is applicable for side slopes of both 2.5H:1V and 4H:1V. However, the coefficient (C 2 )intherela- tionship between the cross-sectional average velocity in the channel and the velocity of the lateral flow changes from 0.85 for 2.5H:1V side slopes to 1.10 for 4H:1V side slopes. For slopes between 2.5H:1V and 4H:1V, linear interpolation may be used to estimate values for C 2 . For channels with side weirs, flow over the weir creates an asymmetrical velocity distribution in the channel. For diversions of 30% or more of the approach flow in a trapezoidal channel with 2.5H:1V side slopes or 20% or more for 4H:1V side slopes, a separation zone is created on the side of the channel opposite to the weir. Downstream of the weir, there is a region of flow re-establishment as the asymmetry is eliminated. As a result of the asymmetry, the flow depth at the end of the weir is usually less than at the downstream end of the re-establishment region. The flow depth at the end of the weir can be determined from the depth at the end of the flow re-establishment region using the momentum or energy equation with a momentum correc- tion factor (?) or a kinetic energy correction factor (?) to account for the flow asymmetry. Sec- tion 6.5 gives the empirical ? and ? values for trapezoidal channels with 2.5H:1V and 4H:1V side slopes. Linear interpolation may be used to estimate ? and ? values for other side slopes between these two values. The ? and ? values depend primarily on the ratio of the weir discharge (Q w ) to discharge (Q u ) upstream of the weir and increase as Q w /Q u increases, where Q u /Q d =1/(1-Q w /Q u ). The ? and ? values at the end of the weir are 1.6 and 2.6, respectively, for 50% diversion for a channel with 2.5H:1V side slopes and 1.9 and 3.6 for 50% diversion for a channel with 4H:1V side slopes. The ? and ? values in this report include the turbulent fluxes of momentum and kinetic energy. Including the turbulent fluxes was important in obtaining good closure of the momentum and energy balances for the measurements. The turbulent flux of momentum was as much as 10% of the total momentum flux, while the turbulent flux of kinetic energy was as much as 17% of the total. The type of results presented in this report depends on channel geometry. The limited data obtained for the length of the flow re-establishment region were used to give an approximate relationship for this length. Fortunately, it is not necessary to know this length with high accuracy since it does not have a strong influence of the depth calculated at the downstream end of the weir. Some experiments were done using culverts rather than weirs for diversion. The experi- ments were used as a basis for developing a method for calculating flow rates for diversion cul- 123 verts. This method is included in the computer programs (Burgin and Holley, 2002). Since the results are rather limited, this method for diversion culverts should be used with caution. 125 9 - REFERENCES Ackers, P. (1957) ?A Theoretical Consideration of Side Weirs as Stormwater Overflows,? Pro- ceedings, Institute of Civil Engineers, London, vol. 6, pp. 250, February. Balmforth, D. J. and Sarginson, E. J. 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Army Corps of Engineers, Davis, CA, 40 p. + appendices. 127 10 - APPENDICES APPENDIX 1 - DATA FROM PREVIOUS PROJECT (TYNES, 1989) Appendix 1.1 - Unsubmerged Flow Conditions Test L B P Q u Q w h u h d C e Fw d F d h c (ft) (ft) (ft) (cfs) (cfs) (ft) (ft) (ft) A1A10W 23.91 3.40 0.52 3.250 0.293 0.019 0.039 0.978 0.293 0.512 -0.0156 A1A20W 23.91 3.40 0.52 3.243 0.813 0.040 0.061 0.611 0.225 0.724 -0.0149 A1A40W 23.91 3.40 0.52 3.232 1.199 0.054 0.076 0.443 0.180 0.766 -0.0150 A1A50W 23.91 3.40 0.52 3.237 1.713 0.072 0.092 0.292 0.129 0.820 -0.0125 A1B10W 23.91 3.40 0.52 6.516 0.595 0.036 0.062 1.474 0.547 0.517 -0.0313 A1B20W 23.91 3.40 0.52 6.493 1.199 0.046 0.089 1.037 0.453 0.603 -0.0389 A1B30W 23.91 3.40 0.52 6.471 1.810 0.062 0.111 0.780 0.375 0.651 -0.0409 A1B40W 23.91 3.40 0.52 6.527 2.554 0.082 0.130 0.591 0.304 0.723 -0.0386 A1B50W 23.91 3.40 0.52 6.535 3.142 0.099 0.140 0.468 0.247 0.795 -0.0302 A1C20W 23.91 3.40 0.52 9.566 2.374 0.080 0.161 0.904 0.507 0.485 -0.0570 A1C40W 23.91 3.40 0.52 9.614 3.612 0.102 0.180 0.688 0.403 0.623 -0.0539 A1C50W 23.91 3.40 0.52 9.646 4.823 0.126 0.194 0.519 0.313 0.742 -0.0474 A4A10W 10.00 3.40 0.52 3.287 0.281 0.041 0.054 0.818 0.285 0.711 -0.0102 A4A20W 10.00 3.40 0.52 3.205 0.666 0.075 0.087 0.502 0.216 0.814 -0.0084 A4A30W 10.00 3.40 0.52 3.199 0.961 0.096 0.108 0.383 0.182 0.842 -0.0087 A4A40W 10.00 3.40 0.52 3.188 1.307 0.117 0.129 0.282 0.144 0.871 -0.0083 A4A50W 10.00 3.40 0.52 3.205 1.580 0.133 0.143 0.225 0.120 0.897 -0.0064 A4B10W 10.00 3.40 0.52 6.497 0.669 0.073 0.100 1.054 0.485 0.660 -0.0216 A4B20W 10.00 3.40 0.52 6.472 1.328 0.117 0.145 0.705 0.379 0.738 -0.0220 A4B30W 10.00 3.40 0.52 6.428 1.984 0.152 0.176 0.520 0.302 0.815 -0.0187 A4B40W 10.00 3.40 0.52 6.417 2.484 0.177 0.199 0.414 0.253 0.842 -0.0169 A4B50W 10.00 3.40 0.52 6.414 3.170 0.203 0.230 0.300 0.193 0.855 -0.0210 A4C10W 10.00 3.40 0.52 9.607 0.926 0.074 0.142 1.238 0.665 0.532 -0.0427 A4C20W 10.00 3.40 0.52 9.586 1.984 0.145 0.200 0.797 0.487 0.667 -0.0355 A4C30W 10.00 3.40 0.52 9.613 3.005 0.184 0.241 0.586 0.384 0.753 -0.0386 A4C40W 10.00 3.40 0.52 9.573 4.005 0.228 0.274 0.437 0.300 0.818 -0.0328 A4C50W 10.00 3.40 0.52 9.590 4.852 0.258 0.300 0.340 0.241 0.858 -0.0305 A4D10W 10.00 3.40 0.52 1.560 0.154 0.028 0.036 0.488 0.141 0.720 -0.0045 A4D20W 10.00 3.40 0.52 1.548 0.351 0.050 0.058 0.312 0.112 0.796 -0.0044 A4D30W 10.00 3.40 0.52 1.555 0.500 0.062 0.072 0.239 0.095 0.816 -0.0063 A4D40W 10.00 3.40 0.52 1.575 0.622 0.074 0.084 0.195 0.083 0.802 -0.0063 A4D50W 10.00 3.40 0.52 1.583 0.779 0.086 0.093 0.153 0.068 0.859 -0.0033 A5A10W 5.00 3.40 0.52 3.191 0.324 0.075 0.084 0.601 0.258 0.810 -0.0080 A5A20W 5.00 3.40 0.52 3.194 0.636 0.115 0.127 0.398 0.204 0.829 -0.0105 A5A30W 5.00 3.40 0.52 3.180 0.964 0.151 0.161 0.286 0.161 0.860 -0.0085 A5A40W 5.00 3.40 0.52 3.195 1.256 0.177 0.189 0.219 0.132 0.865 -0.0103 A5A50W 5.00 3.40 0.52 3.197 1.552 0.200 0.213 0.167 0.105 0.879 -0.0112 A5B09W 5.00 3.40 0.52 6.446 0.557 0.104 0.122 0.929 0.466 0.774 -0.0145 A5B10W 5.00 3.40 0.52 6.546 0.724 0.126 0.144 0.808 0.434 0.773 -0.0177 A5B20W 5.00 3.40 0.52 6.534 1.240 0.174 0.193 0.586 0.356 0.825 -0.0170 A5B30W 5.00 3.40 0.52 6.542 1.929 0.227 0.241 0.416 0.274 0.891 -0.0122 128 Appendix 1.1 - Unsubmerged Flow Conditions (continued) Test L B P Q u Q w h u h d C e Fw d F d h c (ft) (ft) (ft) (cfs) (cfs) (ft) (ft) (ft) A5B31W 5.00 3.40 0.52 6.347 1.957 0.231 0.247 0.389 0.259 0.868 -0.0145 A5B40W 5.00 3.40 0.52 6.527 2.652 0.276 0.288 0.296 0.208 0.910 -0.0105 A5B50W 5.00 3.40 0.52 6.586 3.170 0.304 0.319 0.235 0.171 0.915 -0.0131 A5B51W 5.00 3.40 0.52 6.431 3.199 0.305 0.322 0.220 0.161 0.909 -0.0147 A5C10W 5.00 3.40 0.52 9.547 0.899 0.154 0.175 1.061 0.624 0.701 -0.0232 A5C11W 5.00 3.40 0.52 9.639 1.011 0.148 0.185 0.968 0.574 0.721 -0.0206 A5C20W 5.00 3.40 0.52 9.695 1.857 0.226 0.255 0.673 0.453 0.781 -0.0235 A5C30W 5.00 3.40 0.52 9.679 3.055 0.298 0.327 0.444 0.326 0.846 -0.0241 A5C40W 5.00 3.40 0.52 9.603 3.980 0.347 0.369 0.332 0.254 0.896 -0.0188 A5C50W 5.00 3.40 0.52 9.599 4.987 0.395 0.406 0.245 0.193 0.952 -0.0097 A6A10W 2.00 3.40 0.52 3.317 0.300 0.126 0.131 0.452 0.234 0.819 -0.0042 A6A20W 2.00 3.40 0.52 3.322 0.614 0.188 0.190 0.300 0.180 0.879 -0.0013 A6A30W 2.00 3.40 0.52 3.305 0.975 0.234 0.242 0.208 0.137 0.905 -0.0071 A6A40W 2.00 3.40 0.52 3.287 1.244 0.274 0.279 0.159 0.110 0.890 -0.0042 A6A50W 2.00 3.40 0.52 3.192 1.638 0.315 0.320 0.105 0.077 0.907 -0.0042 A6B10W 2.00 3.40 0.52 6.575 0.622 0.189 0.200 0.630 0.386 0.813 -0.0090 A6B20W 2.00 3.40 0.52 6.499 1.260 0.279 0.289 0.394 0.276 0.844 -0.0085 A6B30W 2.00 3.40 0.52 6.355 1.919 0.342 0.351 0.273 0.205 0.892 -0.0077 A6B40W 2.00 3.40 0.52 6.455 2.573 0.396 0.400 0.207 0.162 0.931 -0.0031 A6C10W 2.00 3.40 0.52 9.552 0.940 0.247 0.254 0.733 0.491 0.799 -0.0054 A6C20W 2.00 3.40 0.52 9.578 1.973 0.354 0.364 0.450 0.342 0.856 -0.0080 A6C30W 2.00 3.40 0.52 9.611 2.826 0.416 0.430 0.334 0.267 0.889 -0.0116 A6D20W 2.00 3.40 0.52 1.636 0.356 0.137 0.140 0.182 0.097 0.867 -0.0022 A6D30W 2.00 3.40 0.52 1.625 0.495 0.161 0.164 0.142 0.080 0.918 -0.0022 A6D40W 2.00 3.40 0.52 1.620 0.580 0.180 0.184 0.118 0.070 0.879 -0.0032 A6D50W 2.00 3.40 0.52 1.741 0.810 0.218 0.222 0.090 0.057 0.879 -0.0032 A3A22N 15.00 1.80 0.52 3.271 0.692 0.067 0.091 0.759 0.356 0.532 -0.0261 A3A31N 15.00 1.80 0.52 3.281 0.996 0.081 0.108 0.593 0.300 0.590 -0.0272 A3A39N 15.00 1.80 0.52 3.268 1.236 0.093 0.122 0.480 0.256 0.607 -0.0280 A3A49N 15.00 1.80 0.52 3.247 1.571 0.107 0.137 0.361 0.202 0.646 -0.0279 A3B19N 15.00 1.80 0.52 6.307 1.219 0.099 0.142 1.064 0.603 0.475 -0.0491 A3B31N 15.00 1.80 0.52 6.314 1.990 0.115 0.177 0.750 0.464 0.552 -0.0549 A3B40N 15.00 1.80 0.52 6.336 2.554 0.130 0.207 0.569 0.374 0.556 -0.0623 A3B46N 15.00 1.80 0.52 6.328 3.005 0.141 0.210 0.493 0.326 0.640 -0.0573 A3B51N 15.00 1.80 0.52 6.331 3.281 0.154 0.221 0.431 0.291 0.645 -0.0560 A4A10N 10.00 1.80 0.52 3.199 0.312 0.060 0.068 1.008 0.412 0.555 -0.0107 A4A13N 10.00 1.80 0.52 3.232 0.413 0.069 0.082 0.866 0.385 0.552 -0.0142 A4A23N 10.00 1.80 0.52 3.206 0.739 0.094 0.111 0.608 0.308 0.621 -0.0162 A4A31N 10.00 1.80 0.52 3.217 0.986 0.110 0.129 0.490 0.265 0.657 -0.0174 A4A41N 10.00 1.80 0.52 3.227 1.298 0.130 0.148 0.379 0.217 0.699 -0.0160 A4B12N 10.00 1.80 0.52 6.411 0.754 0.098 0.130 1.234 0.669 0.496 -0.0322 A4B21N 10.00 1.80 0.52 6.391 1.350 0.124 0.174 0.863 0.527 0.565 -0.0393 A4B31N 10.00 1.80 0.52 6.390 1.984 0.152 0.211 0.634 0.417 0.613 -0.0448 A4B40N 10.00 1.80 0.52 6.404 2.535 0.194 0.228 0.517 0.350 0.693 -0.0293 A4B50N 10.00 1.80 0.52 6.377 3.259 0.211 0.257 0.371 0.262 0.737 -0.0371 A5A08N 5.00 1.80 0.52 3.185 0.256 0.077 0.083 0.910 0.409 0.652 -0.0092 A5A20N 5.00 1.80 0.52 3.187 0.627 0.127 0.135 0.552 0.305 0.742 -0.0093 129 Appendix 1.1 - Unsubmerged Flow Conditions (continued) Test L B P Q u Q w h u h d C e Fw d F d h c (ft) (ft) (ft) (cfs) (cfs) (ft) (ft) (ft) A5A31N 5.00 1.80 0.52 3.187 0.989 0.166 0.174 0.383 0.235 0.779 -0.0085 A5A41N 5.00 1.80 0.52 3.205 1.320 0.196 0.204 0.284 0.186 0.802 -0.0080 A5A51N 5.00 1.80 0.52 3.191 1.628 0.222 0.227 0.213 0.145 0.830 -0.0050 A5B11N 5.00 1.80 0.52 6.457 0.709 0.133 0.172 1.011 0.617 0.569 -0.0336 A5B23N 5.00 1.80 0.52 6.457 1.454 0.209 0.236 0.657 0.453 0.695 -0.0262 A5B30N 5.00 1.80 0.52 6.439 1.935 0.250 0.268 0.521 0.376 0.749 -0.0193 A5B35N 5.00 1.80 0.52 6.392 2.235 0.271 0.290 0.443 0.329 0.758 -0.0196 A5C18N 5.00 1.80 0.52 9.568 1.723 0.269 0.295 0.822 0.613 0.568 -0.0267 A6A09N 2.00 1.80 0.52 3.204 0.289 0.130 0.131 0.632 0.344 0.789 -0.0023 A6A10N 2.00 1.80 0.52 3.263 0.364 0.141 0.148 0.578 0.332 0.806 -0.0059 A6A18N 2.00 1.80 0.52 3.262 0.595 0.186 0.193 0.423 0.270 0.829 -0.0059 A6A20N 2.00 1.80 0.52 3.197 0.644 0.201 0.205 0.377 0.245 0.806 -0.0043 A6A30N 2.00 1.80 0.52 3.284 0.940 0.239 0.243 0.299 0.208 0.866 -0.0032 A6A32N 2.00 1.80 0.52 3.189 1.007 0.254 0.258 0.258 0.183 0.832 -0.0040 A6A39N 2.00 1.80 0.52 3.177 1.236 0.282 0.287 0.206 0.152 0.839 -0.0048 A6A40N 2.00 1.80 0.52 3.287 1.294 0.280 0.289 0.213 0.158 0.867 -0.0078 A6A49N 2.00 1.80 0.52 3.202 1.552 0.314 0.317 0.158 0.120 0.875 -0.0028 A6A50N 2.00 1.80 0.52 3.283 1.633 0.316 0.321 0.158 0.121 0.899 -0.0041 A6B10N 2.00 1.80 0.52 6.407 0.633 0.204 0.216 0.813 0.539 0.722 -0.0125 A6B11N 2.00 1.80 0.52 6.576 0.706 0.210 0.230 0.789 0.538 0.719 -0.0133 A6B19N 2.00 1.80 0.52 6.418 1.215 0.290 0.302 0.524 0.393 0.750 -0.0119 A6B20N 2.00 1.80 0.52 6.498 1.320 0.291 0.301 0.530 0.398 0.820 -0.0077 A6B30N 2.00 1.80 0.52 6.490 1.913 0.348 0.355 0.391 0.310 0.871 -0.0055 A6B40N 2.00 1.80 0.52 6.482 2.560 0.395 0.412 0.282 0.235 0.875 -0.0140 A6B50N 2.00 1.80 0.52 6.433 2.977 0.426 0.442 0.228 0.194 0.887 -0.0134 A6C09N 2.00 1.80 0.52 9.615 0.902 0.243 0.271 0.982 0.708 0.681 -0.0193 A6C10N 2.00 1.80 0.52 9.626 0.961 0.274 0.274 0.979 0.711 0.711 -0.0020 A6C14N 2.00 1.80 0.52 9.639 1.294 0.291 0.332 0.758 0.587 0.669 -0.0275 A6C20N 2.00 1.80 0.52 9.615 1.697 0.345 0.364 0.657 0.526 0.736 -0.0118 C2A09W 20.00 3.40 0.70 3.201 0.296 0.028 0.038 0.683 0.180 0.639 -0.0051 C2A21W 20.00 3.40 0.70 3.213 0.677 0.047 0.061 0.451 0.149 0.713 -0.0084 C2A30W 20.00 3.40 0.70 3.189 0.961 0.059 0.073 0.355 0.127 0.770 -0.0081 C2A39W 20.00 3.40 0.70 3.217 1.227 0.070 0.084 0.290 0.111 0.794 -0.0079 C2A53W 20.00 3.40 0.70 3.192 1.693 0.087 0.101 0.193 0.080 0.826 -0.0075 C2B07W 20.00 3.40 0.70 6.347 0.472 0.039 0.052 1.150 0.353 0.634 -0.0154 C2B13W 20.00 3.40 0.70 6.343 0.794 0.052 0.069 0.915 0.320 0.694 -0.0176 C2B17W 20.00 3.40 0.70 6.338 1.074 0.064 0.083 0.772 0.294 0.708 -0.0184 C2B30W 20.00 3.40 0.70 6.325 1.935 0.094 0.116 0.515 0.228 0.764 -0.0191 C2B41W 20.00 3.40 0.70 6.351 2.658 0.115 0.139 0.381 0.182 0.794 -0.0199 C2B50W 20.00 3.40 0.70 6.327 3.199 0.132 0.153 0.300 0.150 0.824 -0.0165 C2C11W 20.00 3.40 0.70 9.525 1.055 0.063 0.089 1.187 0.467 0.625 -0.0334 C2C19W 20.00 3.40 0.70 9.537 1.836 0.088 0.123 0.866 0.394 0.662 -0.0368 C2C30W 20.00 3.40 0.70 9.552 2.935 0.115 0.158 0.620 0.314 0.719 -0.0403 C2C39W 20.00 3.40 0.70 9.569 3.798 0.141 0.180 0.489 0.261 0.760 -0.0356 C2C50W 20.00 3.40 0.70 9.545 4.890 0.169 0.207 0.353 0.200 0.787 -0.0334 C3B08W 15.00 3.40 0.70 6.498 0.522 0.051 0.068 0.994 0.346 0.617 -0.0177 C3B18W 15.00 3.40 0.70 6.497 1.112 0.079 0.102 0.689 0.288 0.705 -0.0214 130 Appendix 1.1 - Unsubmerged Flow Conditions (continued) Test L B P Q u Q w h u h d C e Fw d F d h c (ft) (ft) (ft) (cfs) (cfs) (ft) (ft) (ft) C3B25W 15.00 3.40 0.70 6.486 1.566 0.102 0.124 0.550 0.251 0.734 -0.0197 C3B28W 15.00 3.40 0.70 6.463 1.779 0.109 0.131 0.504 0.235 0.766 -0.0194 C3B42W 15.00 3.40 0.70 6.418 2.718 0.146 0.166 0.334 0.173 0.808 -0.0165 C3B50W 15.00 3.40 0.70 6.338 3.266 0.164 0.183 0.257 0.138 0.833 -0.0151 C3C10W 15.00 3.40 0.70 9.610 0.947 0.056 0.099 1.131 0.467 0.629 -0.0423 C3C17W 15.00 3.40 0.70 9.629 1.643 0.096 0.140 0.819 0.393 0.638 -0.0412 C3C21W 15.00 3.40 0.70 9.611 1.995 0.121 0.155 0.724 0.363 0.661 -0.0326 C3C30W 15.00 3.40 0.70 9.637 2.921 0.141 0.184 0.559 0.302 0.739 -0.0386 C3C40W 15.00 3.40 0.70 9.611 3.921 0.181 0.215 0.418 0.240 0.775 -0.0301 C3C50W 15.00 3.40 0.70 9.611 4.948 0.209 0.232 0.321 0.190 0.867 -0.0201 C4A09W 10.00 3.40 0.70 3.270 0.269 0.044 0.050 0.600 0.181 0.754 -0.0350 C4A18W 10.00 3.40 0.70 3.254 0.580 0.072 0.081 0.398 0.150 0.773 -0.0061 C4A31W 10.00 3.40 0.70 3.269 0.979 0.107 0.115 0.274 0.122 0.755 -0.0052 C4A42W 10.00 3.40 0.70 3.262 1.328 0.128 0.136 0.202 0.096 0.786 -0.0047 C4A51W 10.00 3.40 0.70 3.272 1.638 0.144 0.149 0.160 0.079 0.839 -0.0017 C4B11W 10.00 3.40 0.70 6.368 0.663 0.080 0.092 0.781 0.312 0.725 -0.0115 C4B20W 10.00 3.40 0.70 6.358 1.269 0.120 0.132 0.543 0.254 0.788 -0.0106 C4B30W 10.00 3.40 0.70 6.369 1.924 0.154 0.167 0.399 0.206 0.822 -0.0109 C4B39W 10.00 3.40 0.70 6.391 2.535 0.184 0.196 0.305 0.169 0.837 -0.0096 C4B44W 10.00 3.40 0.70 6.377 2.826 0.197 0.208 0.267 0.152 0.848 -0.0085 C4B49W 10.00 3.40 0.70 6.365 3.142 0.210 0.222 0.230 0.134 0.848 -0.0092 C4C11W 10.00 3.40 0.70 9.591 1.011 0.102 0.124 0.958 0.437 0.693 -0.0224 C4C12W 10.00 3.40 0.70 9.526 1.175 0.107 0.133 0.887 0.417 0.721 -0.0251 C4C18W 10.00 3.40 0.70 9.593 1.723 0.150 0.166 0.709 0.366 0.743 -0.0163 C4C32W 10.00 3.40 0.70 9.589 3.084 0.208 0.225 0.459 0.269 0.814 -0.0155 C4C41W 10.00 3.40 0.70 9.587 4.048 0.241 0.258 0.347 0.214 0.854 -0.0148 C4C46W 10.00 3.40 0.70 9.580 4.497 0.256 0.278 0.298 0.190 0.838 -0.0189 C5A09W 5.00 3.40 0.70 3.204 0.281 0.072 0.076 0.459 0.169 0.789 -0.0029 C5A19W 5.00 3.40 0.70 3.210 0.611 0.116 0.120 0.301 0.136 0.821 -0.0027 C5A31W 5.00 3.40 0.70 3.209 0.979 0.153 0.158 0.211 0.107 0.834 -0.0035 C5A39W 5.00 3.40 0.70 3.203 1.252 0.176 0.182 0.166 0.089 0.841 -0.0045 C5A52W 5.00 3.40 0.70 3.213 1.653 0.209 0.213 0.117 0.067 0.848 -0.0024 C5B10W 5.00 3.40 0.70 6.408 0.649 0.119 0.127 0.641 0.296 0.795 -0.0083 C5B20W 5.00 3.40 0.70 6.377 1.265 0.176 0.184 0.430 0.233 0.834 -0.0076 C5B31W 5.00 3.40 0.70 6.392 1.962 0.227 0.235 0.305 0.182 0.850 -0.0072 C5B40W 5.00 3.40 0.70 6.382 2.580 0.264 0.273 0.229 0.145 0.859 -0.0079 C5B50W 5.00 3.40 0.70 6.420 3.229 0.300 0.308 0.172 0.114 0.868 -0.0068 C5C10W 5.00 3.40 0.70 9.599 0.940 0.149 0.162 0.805 0.413 0.768 -0.0143 C5C17W 5.00 3.40 0.70 9.591 1.624 0.205 0.219 0.582 0.338 0.795 -0.0142 C5C21W 5.00 3.40 0.70 9.601 1.995 0.229 0.245 0.505 0.307 0.803 -0.0156 C5C29W 5.00 3.40 0.70 9.575 2.851 0.279 0.291 0.383 0.248 0.848 -0.0117 C2A15N 20.00 1.80 0.70 3.258 0.472 0.038 0.054 0.765 0.252 0.598 -0.0105 C2A19N 20.00 1.80 0.70 3.246 0.595 0.042 0.060 0.683 0.236 0.643 -0.0122 C2A22N 20.00 1.80 0.70 3.262 0.709 0.052 0.070 0.597 0.222 0.606 -0.0120 C2A28N 20.00 1.80 0.70 3.239 0.882 0.058 0.075 0.527 0.202 0.679 -0.0108 C2A41N 20.00 1.80 0.70 3.248 1.294 0.073 0.092 0.382 0.161 0.729 -0.0123 C2A46N 20.00 1.80 0.70 3.230 1.463 0.080 0.099 0.328 0.143 0.736 -0.0122 131 Appendix 1.1 - Unsubmerged Flow Conditions (continued) Test L B P Q u Q w h u h d C e Fw d F d h c (ft) (ft) (ft) (cfs) (cfs) (ft) (ft) (ft) C2A50N 20.00 1.80 0.70 3.239 1.604 0.084 0.103 0.295 0.131 0.760 -0.0121 C2B22N 20.00 1.80 0.70 6.327 1.376 0.070 0.104 0.889 0.396 0.642 -0.0275 C2B30N 20.00 1.80 0.70 6.340 1.935 0.091 0.129 0.677 0.332 0.649 -0.0297 C2B39N 20.00 1.80 0.70 6.386 2.516 0.106 0.144 0.548 0.281 0.712 -0.0292 C2B52N 20.00 1.80 0.70 6.333 3.372 0.128 0.172 0.365 0.202 0.724 -0.0335 C2C13N 20.00 1.80 0.70 9.604 1.231 0.082 0.120 1.357 0.644 0.461 -0.0321 C2C20N 20.00 1.80 0.70 9.593 1.899 0.087 0.146 1.078 0.556 0.526 -0.0404 C2C31N 20.00 1.80 0.70 9.470 3.027 0.123 0.187 0.741 0.424 0.571 -0.0435 C2C34N 20.00 1.80 0.70 9.613 3.334 0.133 0.190 0.713 0.411 0.613 -0.0392 C2C41N 20.00 1.80 0.70 9.594 4.048 0.132 0.211 0.576 0.347 0.632 -0.0519 C2C51N 20.00 1.80 0.70 9.570 5.046 0.161 0.228 0.439 0.272 0.698 -0.0455 C3A10N 15.00 1.80 0.70 3.256 0.300 0.037 0.049 0.861 0.271 0.585 -0.0081 C3A16N 15.00 1.80 0.70 3.243 0.517 0.051 0.065 0.668 0.240 0.655 -0.0096 C3A18N 15.00 1.80 0.70 3.257 0.580 0.055 0.070 0.626 0.233 0.656 -0.0105 C3A28N 15.00 1.80 0.70 3.245 0.875 0.071 0.086 0.484 0.198 0.722 -0.0102 C3A42N 15.00 1.80 0.70 3.219 1.345 0.092 0.110 0.323 0.148 0.759 -0.0127 C3A51N 15.00 1.80 0.70 3.245 1.643 0.107 0.122 0.257 0.123 0.790 -0.0097 C3B09N 15.00 1.80 0.70 6.402 0.546 0.053 0.074 1.321 0.504 0.567 -0.0190 C3B18N 15.00 1.80 0.70 6.373 1.112 0.076 0.110 0.908 0.414 0.628 -0.0272 C3B32N 15.00 1.80 0.70 6.386 2.039 0.118 0.149 0.599 0.312 0.718 -0.0243 C3B41N 15.00 1.80 0.70 6.365 2.645 0.138 0.177 0.448 0.251 0.711 -0.0304 C3B48N 15.00 1.80 0.70 6.383 3.099 0.153 0.187 0.378 0.216 0.764 -0.0263 C3C11N 15.00 1.80 0.70 9.641 1.093 0.087 0.125 1.345 0.650 0.506 -0.0286 C3C20N 15.00 1.80 0.70 9.668 1.968 0.101 0.152 1.046 0.549 0.672 -0.0339 C3C30N 15.00 1.80 0.70 9.659 2.949 0.149 0.209 0.703 0.421 0.610 -0.0406 C3C38N 15.00 1.80 0.70 9.659 3.684 0.166 0.231 0.573 0.358 0.650 -0.0443 C3C45N 15.00 1.80 0.70 9.680 4.407 0.183 0.239 0.491 0.310 0.736 -0.0387 C4A09N 10.00 1.80 0.70 3.168 0.271 0.045 0.054 0.794 0.262 0.675 -0.0062 C4A18N 10.00 1.80 0.70 3.174 0.580 0.075 0.086 0.529 0.216 0.705 -0.0078 C4A32N 10.00 1.80 0.70 3.174 1.018 0.103 0.113 0.364 0.168 0.807 -0.0066 C4A42N 10.00 1.80 0.70 3.181 1.328 0.126 0.137 0.272 0.137 0.777 -0.0074 C4A51N 10.00 1.80 0.70 3.182 1.633 0.143 0.155 0.207 0.109 0.786 -0.0083 C4B11N 10.00 1.80 0.70 6.406 0.706 0.080 0.100 1.049 0.459 0.678 -0.0165 C4B18N 10.00 1.80 0.70 6.388 1.139 0.106 0.131 0.796 0.392 0.716 -0.0199 C4B30N 10.00 1.80 0.70 6.418 1.919 0.151 0.176 0.543 0.303 0.754 -0.0197 C4B35N 10.00 1.80 0.70 6.310 2.265 0.168 0.193 0.452 0.262 0.767 -0.0197 C4B41N 10.00 1.80 0.70 6.367 2.665 0.185 0.217 0.375 0.228 0.746 -0.0256 C4B46N 10.00 1.80 0.70 6.387 2.956 0.202 0.225 0.336 0.208 0.780 -0.0180 C4B50N 10.00 1.80 0.70 6.393 3.221 0.212 0.232 0.303 0.189 0.809 -0.0154 C4C10N 10.00 1.80 0.70 9.597 1.015 0.094 0.137 1.259 0.632 0.594 -0.0272 C4C20N 10.00 1.80 0.70 9.592 1.889 0.135 0.193 0.861 0.500 0.639 -0.0373 C4C30N 10.00 1.80 0.70 9.594 2.963 0.201 0.230 0.638 0.397 0.754 -0.0211 C4C37N 10.00 1.80 0.70 9.590 3.636 0.230 0.258 0.516 0.336 0.767 -0.0206 C5A09N 5.00 1.80 0.70 3.250 0.291 0.076 0.080 0.643 0.255 0.753 -0.0033 C5A21N 5.00 1.80 0.70 3.269 0.663 0.125 0.128 0.408 0.200 0.801 -0.0020 C5A31N 5.00 1.80 0.70 3.243 0.982 0.156 0.161 0.297 0.161 0.811 -0.0037 C5A40N 5.00 1.80 0.70 3.256 1.265 0.180 0.184 0.235 0.134 0.834 -0.0027 132 Appendix 1.1 - Unsubmerged Flow Conditions (continued) Test L B P Q u Q w h u h d C e Fw d F d h c (ft) (ft) (ft) (cfs) (cfs) (ft) (ft) (ft) C5A51N 5.00 1.80 0.70 3.245 1.624 0.207 0.211 0.170 0.103 0.847 -0.0026 C5B10N 5.00 1.80 0.70 6.421 0.644 0.116 0.130 0.895 0.441 0.759 -0.0134 C5B21N 5.00 1.80 0.70 6.421 1.328 0.181 0.196 0.570 0.334 0.786 -0.0137 C5B30N 5.00 1.80 0.70 6.411 1.935 0.227 0.238 0.423 0.268 0.820 -0.0100 C5B39N 5.00 1.80 0.70 6.406 2.516 0.265 0.274 0.323 0.216 0.832 -0.0081 C5B47N 5.00 1.80 0.70 6.438 3.055 0.294 0.304 0.254 0.177 0.840 -0.0088 C5C10N 5.00 1.80 0.70 9.634 0.961 0.153 0.177 1.057 0.594 0.676 -0.0200 C5C20N 5.00 1.80 0.70 9.614 1.924 0.224 0.247 0.703 0.451 0.764 -0.0196 C5C25N 5.00 1.80 0.70 9.624 2.386 0.253 0.280 0.588 0.397 0.760 -0.0225 133 Appendix 1.2 - Submerged Flow Conditions Compared L B P Q u Q d h u h d h b Fw d C es h cs Test to Test (ft) (ft) (ft) (cfs) (cfs) (ft) (ft) (ft) (ft) C3C21WS C3C40W 15.0 3.40 0.70 7.152 1.486 0.192 0.212 0.202 0.421 0.300 -0.016 C3C26WS C3C40W 15.0 3.40 0.70 7.686 2.006 0.197 0.217 0.204 0.414 0.391 -0.016 C3C32WS C3C40W 15.0 3.40 0.70 8.310 2.665 0.191 0.216 0.192 0.413 0.523 -0.020 C3C35WS C3C40W 15.0 3.40 0.70 8.696 3.106 0.183 0.211 0.176 0.417 0.633 -0.023 C3C37WS C3C40W 15.0 3.40 0.70 9.016 3.387 0.184 0.214 0.173 0.415 0.675 -0.025 C3C38WS C3C40W 15.0 3.40 0.70 9.193 3.548 0.180 0.211 0.155 0.421 0.723 -0.025 C3C40WS C3C40W 15.0 3.40 0.70 9.430 3.848 0.187 0.215 0.152 0.410 0.761 -0.023 C3B32WS C3B50W 15.0 3.40 0.70 4.566 1.473 0.163 0.178 0.170 0.264 0.393 -0.010 C3B45WS C3B50W 15.0 3.40 0.70 5.437 2.434 0.168 0.185 0.165 0.249 0.610 -0.012 C3B30NS C3B48N 15.0 1.80 0.70 4.724 1.414 0.166 0.185 0.174 0.384 0.355 -0.015 C3B38NS C3B48N 15.0 1.80 0.70 5.290 2.017 0.163 0.185 0.165 0.380 0.506 -0.018 C3B46NS C3B48N 15.0 1.80 0.70 6.073 2.812 0.161 0.188 0.145 0.373 0.688 -0.022 C3C10NS C3C38N 15.0 1.80 0.70 6.641 0.652 0.212 0.230 0.223 0.577 0.116 -0.016 C3C21NS C3C38N 15.0 1.80 0.70 7.683 1.643 0.203 0.231 0.207 0.580 0.290 -0.025 C3C31NS C3C38N 15.0 1.80 0.70 8.703 2.745 0.188 0.229 0.182 0.576 0.491 -0.034 C4B28WS C4B49W 10.0 3.40 0.70 4.499 1.248 0.216 0.224 0.216 0.230 0.332 -0.005 C4B36WS C4B49W 10.0 3.40 0.70 5.169 1.873 0.211 0.222 0.207 0.235 0.505 -0.008 C4B44WS C4B49W 10.0 3.40 0.70 5.828 2.586 0.210 0.222 0.191 0.231 0.698 -0.009 C4C19WS C4C41W 10.0 3.40 0.70 6.833 1.320 0.252 0.261 0.252 0.342 0.273 -0.006 C4C36WS C4C41W 10.0 3.40 0.70 9.164 3.221 0.240 0.256 0.218 0.375 0.688 -0.013 C4B33NS C4B50N 10.0 1.80 0.70 4.746 1.542 0.219 0.233 0.217 0.305 0.384 -0.011 C4B42NS C4B50N 10.0 1.80 0.70 5.420 2.295 0.217 0.230 0.202 0.300 0.584 -0.010 C4B49NS C4B50N 10.0 1.80 0.70 6.148 3.005 0.214 0.233 0.162 0.299 0.749 -0.015 C4C07NS C4C30N 10.0 1.80 0.70 7.128 0.524 0.222 0.233 0.221 0.628 0.131 -0.010 C4C19NS C4C30N 10.0 1.80 0.70 8.220 1.552 0.193 0.232 0.190 0.636 0.390 -0.032 C5B15WS C5B40W 5.0 3.40 0.70 4.430 0.636 0.264 0.268 0.263 0.233 0.219 -0.003 C5B25WS C5B40W 5.0 3.40 0.70 5.050 1.244 0.265 0.271 0.258 0.231 0.420 -0.005 C5B34WS C5B40W 5.0 3.40 0.70 5.806 2.006 0.268 0.275 0.245 0.228 0.659 -0.006 C5C12WS C5C29W 5.0 3.40 0.70 7.704 0.923 0.286 0.292 0.280 0.385 0.273 -0.005 C5C23WS C5C29W 5.0 3.40 0.70 8.685 1.973 0.281 0.290 0.259 0.383 0.590 -0.008 C5B12NS C5B39N 5.0 1.80 0.70 4.471 0.537 0.269 0.273 0.266 0.328 0.179 -0.003 C5B23NS C5B39N 5.0 1.80 0.70 5.084 1.175 0.271 0.278 0.261 0.320 0.379 -0.006 C5B32NS C5B39N 5.0 1.80 0.70 5.712 1.830 0.262 0.272 0.235 0.324 0.613 -0.009 C5C05NS C5C25N 5.0 1.80 0.70 7.676 0.415 0.275 0.278 0.270 0.594 0.134 -0.005 C5C16NS C5C25N 5.0 1.80 0.70 8.610 1.350 0.262 0.275 0.236 0.600 0.444 -0.013 134 Appendix 1.3 - Tapered Channels Test L ?B' PQ u Q w h u h d F d Fw d C e h c (ft) (ft) (ft) (cfs) (cfs) (ft) (ft) (ft) C2A08T 20.0 0.080 0.70 3.205 0.263 0.024 0.024 0.288 1.289 1.137 -0.005 C2A20T 20.0 0.080 0.70 3.189 0.647 0.044 0.051 0.232 0.723 0.895 -0.005 C2A24T 20.0 0.080 0.70 3.194 0.779 0.050 0.058 0.216 0.635 0.886 -0.004 C2A32T 20.0 0.080 0.70 3.189 1.044 0.060 0.071 0.186 0.497 0.873 -0.004 C2A41T 20.0 0.080 0.70 3.172 1.324 0.070 0.084 0.155 0.384 0.857 -0.005 C2A51T 20.0 0.080 0.70 3.179 1.619 0.081 0.095 0.127 0.298 0.868 -0.003 C2B11T 20.0 0.080 0.70 6.339 0.683 0.058 0.032 0.542 2.111 1.912 -0.012 C2B13T 20.0 0.080 0.70 6.321 0.836 0.062 0.042 0.512 1.751 1.551 -0.011 C2B21T 20.0 0.080 0.70 6.445 1.324 0.077 0.073 0.442 1.165 1.061 -0.010 C2B28T 20.0 0.080 0.70 6.347 1.758 0.090 0.096 0.374 0.871 0.928 -0.008 C2B40T 20.0 0.080 0.70 6.327 2.612 0.112 0.132 0.278 0.562 0.845 -0.007 C2B51T 20.0 0.080 0.70 6.351 3.341 0.131 0.153 0.214 0.407 0.860 -0.001 C2C15T 20.0 0.080 0.70 9.694 1.473 0.103 0.035 0.782 2.916 3.602 -0.022 C2C16T 20.0 0.080 0.70 9.663 1.491 0.104 0.041 0.765 2.645 2.870 -0.021 C2C22T 20.0 0.080 0.70 9.601 2.096 0.116 0.090 0.621 1.488 1.221 -0.012 C2C29T 20.0 0.080 0.70 9.595 2.785 0.126 0.130 0.511 1.041 0.922 -0.011 C2C43T 20.0 0.080 0.70 9.573 4.195 0.154 0.182 0.358 0.633 0.825 -0.003 C2C53T 20.0 0.080 0.70 9.552 5.085 0.170 0.203 0.284 0.480 0.843 0.004 C3A10T 15.0 0.107 0.70 3.274 0.298 0.045 0.046 0.275 0.900 0.639 -0.006 C3A21T 15.0 0.107 0.70 3.285 0.661 0.067 0.074 0.226 0.592 0.687 -0.006 C3A30T 15.0 0.107 0.70 3.279 0.947 0.081 0.091 0.192 0.459 0.716 -0.006 C3A40T 15.0 0.107 0.70 3.271 1.281 0.096 0.110 0.157 0.343 0.723 -0.007 C3A52T 15.0 0.107 0.70 3.273 1.643 0.112 0.126 0.124 0.255 0.751 -0.005 C3B11T 15.0 0.107 0.70 6.411 0.663 0.079 0.053 0.522 1.597 1.146 -0.008 C3B19T 15.0 0.107 0.70 6.358 1.195 0.100 0.094 0.423 0.994 0.860 -0.007 C3B29T 15.0 0.107 0.70 6.316 1.830 0.123 0.131 0.336 0.682 0.788 -0.007 C3B41T 15.0 0.107 0.70 6.298 2.612 0.149 0.165 0.255 0.469 0.784 -0.004 C3B51T 15.0 0.107 0.70 6.298 3.311 0.170 0.191 0.195 0.338 0.790 -0.003 C3C12T 15.0 0.107 0.70 9.503 1.215 0.118 0.036 0.786 2.892 3.781 -0.011 C3C13T 15.0 0.107 0.70 9.496 1.223 0.117 0.044 0.769 2.569 2.807 -0.014 C3C14T 15.0 0.107 0.70 9.601 1.328 0.119 0.067 0.725 1.988 1.606 -0.020 C3C15T 15.0 0.107 0.70 9.485 1.298 0.119 0.053 0.743 2.275 2.245 -0.013 C3C21T 15.0 0.107 0.70 9.426 2.051 0.138 0.123 0.563 1.174 0.974 -0.011 C3C30T 15.0 0.107 0.70 9.567 2.928 0.164 0.175 0.449 0.806 0.801 -0.009 C3C39T 15.0 0.107 0.70 9.561 3.848 0.188 0.213 0.355 0.589 0.772 -0.006 C3C51T 15.0 0.107 0.70 9.559 5.006 0.215 0.248 0.263 0.410 0.788 0.000 C4A11T 10.0 0.160 0.70 3.180 0.337 0.057 0.056 0.255 0.762 0.794 -0.003 C4A20T 10.0 0.160 0.70 3.212 0.619 0.082 0.084 0.217 0.537 0.780 -0.002 C4A32T 10.0 0.160 0.70 3.197 1.022 0.108 0.114 0.169 0.365 0.799 -0.002 C4A39T 10.0 0.160 0.70 3.191 1.240 0.120 0.127 0.147 0.303 0.818 -0.002 C4A49T 10.0 0.160 0.70 3.191 1.556 0.139 0.146 0.118 0.229 0.823 0.000 C4B11T 10.0 0.160 0.70 6.430 0.698 0.096 0.074 0.492 1.290 1.070 -0.006 C4B20T 10.0 0.160 0.70 6.427 1.236 0.127 0.121 0.397 0.835 0.880 -0.005 C4B30T 10.0 0.160 0.70 6.427 1.919 0.159 0.164 0.312 0.576 0.844 -0.003 C4B41T 10.0 0.160 0.70 6.423 2.652 0.190 0.201 0.240 0.408 0.841 0.000 C4B49T 10.0 0.160 0.70 6.390 3.177 0.211 0.223 0.195 0.317 0.851 0.003 C4C10T 10.0 0.160 0.70 9.508 0.961 0.133 0.054 0.772 2.343 2.395 -0.010 135 Appendix 1.3 - Tapered Channels (continued) Test L ?B' PQ u Q w h u h d F d Fw d C e h c (ft) (ft) (ft) (cfs) (cfs) (ft) (ft) (ft) C4C12T 10.0 0.160 0.70 9.512 1.183 0.140 0.080 0.704 1.782 1.608 -0.008 C4C20T 10.0 0.160 0.70 9.539 1.946 0.172 0.151 0.542 1.034 0.976 -0.003 C4C29T 10.0 0.160 0.70 9.541 2.851 0.207 0.210 0.418 0.696 0.842 -0.001 C4C39T 10.0 0.160 0.70 9.547 3.831 0.238 0.254 0.325 0.503 0.829 0.001 C5A10T 5.0 0.320 0.70 3.149 0.324 0.079 0.075 0.247 0.640 0.929 -0.001 C5A13T 5.0 0.320 0.70 3.149 0.409 0.092 0.090 0.231 0.551 0.876 -0.002 C5A22T 5.0 0.320 0.70 3.142 0.689 0.127 0.128 0.188 0.384 0.833 -0.001 C5A31T 5.0 0.320 0.70 3.147 0.996 0.154 0.156 0.154 0.290 0.867 0.000 C5A39T 5.0 0.320 0.70 3.192 1.236 0.174 0.178 0.134 0.237 0.862 -0.001 C5A51T 5.0 0.320 0.70 3.196 1.628 0.205 0.209 0.100 0.166 0.863 0.001 C5B10T 5.0 0.320 0.70 6.442 0.658 0.129 0.109 0.464 1.018 1.034 -0.006 C5B20T 5.0 0.320 0.70 6.462 1.244 0.179 0.174 0.360 0.645 0.901 -0.004 C5B31T 5.0 0.320 0.70 6.433 1.968 0.229 0.232 0.271 0.432 0.871 -0.001 C5B41T 5.0 0.320 0.70 6.453 2.619 0.268 0.274 0.213 0.318 0.866 0.002 C5B50T 5.0 0.320 0.70 6.439 3.214 0.298 0.306 0.168 0.241 0.874 0.003 C5C12T 5.0 0.320 0.70 9.531 1.135 0.188 0.146 0.617 1.191 1.103 -0.007 C5C20T 5.0 0.320 0.70 9.496 1.878 0.234 0.219 0.475 0.775 0.919 -0.003 C5C27T 5.0 0.320 0.70 9.516 2.619 0.276 0.275 0.382 0.570 0.861 0.000 137 APPENDIX 2 - WEIR AND CHANNEL GEOMETRIES INVESTIGATED IN PREVIOUS PROJECT FOR UNSUBMERGED FLOW Weir height Weir length Channel invert width Number of tests PLB (ft) (ft) (ft) 0.52 23.91 3.4 12 0.52 15.00 1.8 9 0.52 10.00 3.4 20 0.52 10.00 1.8 10 0.52 5.00 3.4 19 0.52 5.00 1.8 10 0.52 2.00 3.4 16 0.52 2.00 1.8 21 0.70 20.00 3.4 16 0.70 20.00 1.8 17 0.70 15.00 3.4 12 0.70 15.00 1.8 16 0.70 10.00 3.4 17 0.70 10.00 1.8 16 0.70 5.00 3.4 14 0.70 5.00 1.8 13 139 APPENDIX 3 - RESULTS OF SIMULATION OF SIDE WEIR FLOW FOR DIFFERENT SLOPES AND ROUGHNESS The results for Method A begin on page 140 while the results for Method B begin on p. 151. For Method A, there is no way to specify different channel slopes when calculating Q w . Therefore, all calculated values of Q w are the same for each case. Notes for the tables: All results are for prismatic channels. (1) in the table means supercritical condition at upstream end. (2) in the table means uw QQ > and iteration stopped. (3) in the table means negative flow depth and iteration stopped. (4) in the table means uw Q6.0Q > in final solution. ?Max. diff.? is the largest difference between the values in the row or column. 140 A ppe ndix 3.1 - R e s ult s of sim u lat i on using M e t hod A Input geom etr y data: L = 598 f t B = 85 f t P = 13.0 f t I nput f l ow c onditions : Q u = 10156 c f s h d =0 . 9 7f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 1082 1082 1082 1082 0 0 .0125 0.47 0.19 - 0 .09 - 0.36 0.83 0.02 1082 1082 1082 1082 0 0 .02 0 .58 0 .31 0 .04 - 0.23 0.81 0.03 1082 1082 1082 1082 0 0 .03 0 .80 0 .53 0 .28 0 .03 0 .77 0.04 1082 1082 1082 1082 0 0 .04 1 .08 0 .83 0 .59 0 .35 0 .73 M a x . d i f f . 0000 M a x . d i f f . 0 . 6 1 0 . 6 4 0 . 6 8 0 . 7 1 Input geom etr y data: L = 598 f t B = 85 f t P = 13.0 f t I nput f l ow c onditions : Q u = 10116 c f s h d =2 . 3 0f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 4905 4905 4905 4905 0 0 .0125 1.67 1.40 1.14 0.87 0.80 0.02 4905 4905 4905 4905 0 0 .02 1 .73 1 .46 1 .20 0 .94 0 .79 0.03 4905 4905 4905 4905 0 0 .03 1 .84 1 .58 1 .33 1 .08 0 .76 0.04 4905 4905 4905 4905 0 0 .04 1 .99 1 .74 1 .50 1 .26 0 .73 M a x . d i f f . 0000 M a x . d i f f . 0 . 3 2 0 . 3 4 0 . 3 6 0 . 3 9 Input geom etr y data: L = 598 f t B = 85 f t P = 13.0 f t I nput f l ow c onditions : Q u = 29894 c f s h d =4 . 0 3f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 9829 9829 9829 9829 0 0 .0125 ( 1 ) ( 1) ( 1 ) ( 1) 0.02 9829 9829 9829 9829 0 0 .02 1 .06 ( 1) ( 1 ) ( 1) 0.03 9829 9829 9829 9829 0 0 .03 2 .92 2 .54 2 .15 1 .76 1 .16 0.04 9829 9829 9829 9829 0 0 .04 4 .20 3 .91 3 .63 3 .35 0 .85 M a x . d i f f . 0000 M a x . d i f f . 3 . 1 4 1 . 3 7 1 . 4 8 1 . 5 9 ( S ee notes on p. 139.) 141 A ppe ndix 3.1 - R e s ult s of sim u lat i on using M e t hod A ( c ont i nue d) Input geom etr y data: L = 598 f t B = 85 f t P = 13.0 f t I nput f l ow c onditions : Q u = 30144 c f s h d =4 . 8 5f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 14305 14305 14305 14305 0 0 .0125 0.97 ( 1 ) ( 1) ( 1 ) 0.02 14305 14305 14305 14305 0 0 .02 2 .36 1 .78 1 .08 ( 1) 1.28 0.03 14305 14305 14305 14305 0 0 .03 3 .54 3 .18 2 .82 2 .44 1 .10 0.04 14305 14305 14305 14305 0 0 .04 4 .55 4 .26 3 .98 3 .70 0 .85 M a x . d i f f . 0000 M a x . d i f f . 3 . 5 8 2 . 4 8 2 . 9 0 1 . 2 6 Input geom etr y data: L = 250 f t B = 85 f t P = 13.0 f t I nput f l ow c onditions : Q u = 30022 c f s h d =3 . 5 5f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 3280 3280 3280 3280 0 0 .0125 ( 1 ) ( 1) ( 1 ) ( 1) 0.02 3280 3280 3280 3280 0 0 .02 1 .71 1 .31 0 .73 ( 1) 0.98 0.03 3280 3280 3280 3280 0 0 .03 2 .82 2 .61 2 .40 2 .18 0 .64 0.04 3280 3280 3280 3280 0 0 .04 3 .76 3 .60 3 .45 3 .29 0 .47 M a x . d i f f . 0000 M a x . d i f f . 2 . 0 5 2 . 2 9 2 . 7 2 1 . 1 1 Input geom etr y data: L = 250 f t B = 85 f t P = 13.0 f t I nput f l ow c onditions : Q u = 29969 c f s h d =7 . 5 0f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 14482 14482 14482 14482 0 0 .0125 6.09 5.94 5.79 5.65 0.44 0.02 14482 14482 14482 14482 0 0 .02 6 .19 6 .04 5 .90 5 .76 0 .43 0.03 14482 14482 14482 14482 0 0 .03 6 .39 6 .25 6 .11 5 .97 0 .42 0.04 14482 14482 14482 14482 0 0 .04 6 .64 6 .50 6 .38 6 .25 0 .39 M a x . d i f f . 0000 M a x . d i f f . 0 . 5 5 0 . 5 6 0 . 5 9 0 . 6 0 ( S ee notes on p. 139.) 142 A ppe ndix 3.1 - R e s ult s of sim u lat i on using M e t hod A ( c ont i nue d) Input geom etr y data: L = 250 f t B = 85 f t P = 13.0 f t I nput f l ow c onditions : Q u = 4875 c f s h d =0 . 9 0f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 523 523 523 523 0 0 .0125 0.73 0.63 0.52 0.42 0.31 0.02 523 523 523 523 0 0 .02 0 .74 0 .64 0 .53 0 .43 0 .31 0.03 523 523 523 523 0 0 .03 0 .76 0 .66 0 .55 0 .45 0 .31 0.04 523 523 523 523 0 0 .04 0 .79 0 .68 0 .58 0 .48 0 .31 M a x . d i f f . 0000 M a x . d i f f . 0 . 0 6 0 . 0 5 0 . 0 6 0 . 0 6 Input geom etr y data: L = 250 f t B = 85 f t P = 13.0 f t I nput f l ow c onditions : Q u = 4947 c f s h d =2 . 3 3f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 2431 2431 2431 2431 0 0 .0125 2.12 2.02 1.92 1.81 0.31 0.02 2431 2431 2431 2431 0 0 .02 2 .13 2 .02 1 .92 1 .82 0 .31 0.03 2431 2431 2431 2431 0 0 .03 2 .14 2 .03 1 .93 1 .83 0 .31 0.04 2431 2431 2431 2431 0 0 .04 2 .15 2 .05 1 .95 1 .84 0 .31 M a x . d i f f . 0000 M a x . d i f f . 0 . 0 3 0 . 0 3 0 . 0 3 0 . 0 3 Input geom etr y data: L = 375 f t B = 45 f t P = 13.0 f t I nput f l ow c onditions : Q u = 10222 c f s h d =2 . 2 7f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 2657 2657 2657 2657 0 0 .0125 1.54 1.34 1.15 0.95 0.59 0.02 2657 2657 2657 2657 0 0 .02 1 .68 1 .49 1 .30 1 .11 0 .57 0.03 2657 2657 2657 2657 0 0 .03 1 .94 1 .75 1 .58 1 .40 0 .54 0.04 2657 2657 2657 2657 0 0 .04 2 .26 2 .09 1 .92 1 .76 0 .50 M a x . d i f f . 0000 M a x . d i f f . 0 . 7 2 0 . 7 5 0 . 7 7 0 . 8 1 ( S ee notes on p. 139.) 143 A ppe ndix 3.1 - R e s ult s of sim u lat i on using M e t hod A ( c ont i nue d) Input geom etr y data: L = 375 f t B = 45 f t P = 13.0 f t I nput f l ow c onditions : Q u = 10147 c f s h d =3 . 4 3f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 5595 5595 5595 5595 0 0 .0125 2.77 2.58 2.40 2.22 0.55 0.02 5595 5595 5595 5595 0 0 .02 2 .84 2 .66 2 .48 2 .31 0 .53 0.03 5595 5595 5595 5595 0 0 .03 2 .99 2 .81 2 .64 2 .47 0 .52 0.04 5595 5595 5595 5595 0 0 .04 3 .17 3 .01 2 .84 2 .68 0 .49 M a x . d i f f . 0000 M a x . d i f f . 0 . 4 0 0 . 4 3 0 . 4 4 0 . 4 6 Input geom etr y data: L = 375 f t B = 45 f t P = 13.0 f t I nput f l ow c onditions : Q u = 19709 c f s h d =3 . 5 5f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 4720 4720 4720 4720 0 0 .0125 ( 1 ) ( 1) ( 1 ) ( 1) 0.02 4720 4720 4720 4720 0 0 .02 1 .19 ( 1) ( 1 ) ( 1) 0.03 4720 4720 4720 4720 0 0 .03 2 .92 2 .64 2 .36 2 .06 0 .86 0.04 4720 4720 4720 4720 0 0 .04 4 .05 3 .85 3 .65 3 .44 0 .61 M a x . d i f f . 0000 M a x . d i f f . 2 . 8 6 1 . 2 1 1 . 2 9 1 . 3 8 Input geom etr y data: L = 375 f t B = 45 f t P = 13.0 f t I nput f l ow c onditions : Q u = 19784 c f s h d =5 . 5 3f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 11534 11534 11534 11534 0 0 .0125 3.80 3.49 3.17 2.82 0.98 0.02 11534 11534 11534 11534 0 0 .02 4 .11 3 .84 3 .57 3 .28 0 .83 0.03 11534 11534 11534 11534 0 0 .03 4 .61 4 .38 4 .16 3 .93 0 .68 0.04 11534 11534 11534 11534 0 0 .04 5 .15 4 .96 4 .77 4 .57 0 .58 M a x . d i f f . 0000 M a x . d i f f . 1 . 3 5 1 . 4 7 1 . 6 0 1 . 7 5 ( S ee notes on p. 139.) 144 A ppe ndix 3.1 - R e s ult s of sim u lat i on using M e t hod A ( c ont i nue d) Input geom etr y data: L = 250 f t B = 45 f t P = 13.0 f t I nput f l ow c onditions : Q u = 9997 c f s h d =1 . 7 0f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 1080 1080 1080 1080 0 0 .0125 1.23 1.09 0.96 0.83 0.40 0.02 1080 1080 1080 1080 0 0 .02 1 .34 1 .21 1 .08 0 .95 0 .39 0.03 1080 1080 1080 1080 0 0 .03 1 .56 1 .43 1 .31 1 .19 0 .37 0.04 1080 1080 1080 1080 0 0 .04 1 .84 1 .72 1 .60 1 .49 0 .35 M a x . d i f f . 0000 M a x . d i f f . 0 . 6 1 0 . 6 3 0 . 6 4 0 . 6 6 Input geom etr y data: L = 250 f t B = 45 f t P = 13.0 f t I nput f l ow c onditions : Q u = 10084 c f s h d =3 . 7 0f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 4465 4465 4465 4465 0 0 .0125 3.15 3.03 2.91 2.79 0.36 0.02 4465 4465 4465 4465 0 0 .02 3 .20 3 .07 2 .96 2 .84 0 .36 0.03 4465 4465 4465 4465 0 0 .03 3 .29 3 .17 3 .06 2 .94 0 .35 0.04 4465 4465 4465 4465 0 0 .04 3 .42 3 .30 3 .19 3 .08 0 .34 M a x . d i f f . 0000 M a x . d i f f . 0 . 2 7 0 . 2 7 0 . 2 8 0 . 2 9 Input geom etr y data: L = 250 f t B = 45 f t P = 13.0 f t I nput f l ow c onditions : Q u = 20034 c f s h d =3 . 2 5f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 2666 2666 2666 2666 0 0 .0125 ( 1 ) ( 1) ( 1 ) ( 1) 0.02 2666 2666 2666 2666 0 0 .02 ( 1) ( 1 ) ( 1) ( 1 ) 0.03 2666 2666 2666 2666 0 0 .03 2 .82 2 .60 2 .37 2 .13 0 .69 0.04 2666 2666 2666 2666 0 0 .04 3 .88 3 .72 3 .57 3 .41 0 .47 M a x . d i f f . 0000 M a x . d i f f . 1 . 0 6 1 . 1 2 1 . 2 0 1 . 2 8 ( S ee notes on p. 139.) 145 A ppe ndix 3.1 - R e s ult s of sim u lat i on using M e t hod A ( c ont i nue d) Input geom etr y data: L = 250 f t B = 45 f t P = 13.0 f t I nput f l ow c onditions : Q u = 19928 c f s h d =6 . 4 2f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 10657 10657 10657 10657 0 0 .0125 5.12 4.95 4.79 4.62 0.50 0.02 10657 10657 10657 10657 0 0 .02 5 .26 5 .10 4 .94 4 .79 0 .47 0.03 10657 10657 10657 10657 0 0 .03 5 .52 5 .37 5 .23 5 .08 0 .44 0.04 10657 10657 10657 10657 0 0 .04 5 .84 5 .70 5 .57 5 .44 0 .40 M a x . d i f f . 0000 M a x . d i f f . 0 . 7 2 0 . 7 5 0 . 7 8 0 . 8 2 Input geom etr y data: L = 500 f t B = 85 f t P = 17.5 f t I nput f l ow c onditions : Q u = 10003 c f s h d =0 . 9 5f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 998 998 998 998 0 0 .0125 0.64 0.42 0.21 0.00 0.64 0.02 998 998 998 998 0 0 .02 0 .66 0 .45 0 .24 0 .03 0 .63 0.03 998 998 998 998 0 0 .03 0 .72 0 .51 0 .30 0 .10 0 .62 0.04 998 998 998 998 0 0 .04 0 .81 0 .60 0 .39 0 .19 0 .62 M a x . d i f f . 0000 M a x . d i f f . 0 . 1 7 0 . 1 8 0 . 1 8 0 . 1 9 Input geom etr y data: L = 500 f t B = 85 f t P = 17.5 f t I nput f l ow c onditions : Q u = 9975 c f s h d =2 . 5 3f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 5068 5068 5068 5068 0 0 .0125 2.15 1.93 1.73 1.52 0.63 0.02 5068 5068 5068 5068 0 0 .02 2 .16 1 .95 1 .74 1 .54 0 .62 0.03 5068 5068 5068 5068 0 0 .03 2 .19 1 .98 1 .77 1 .57 0 .62 0.04 5068 5068 5068 5068 0 0 .04 2 .23 2 .02 1 .82 1 .62 0 .61 M a x . d i f f . 0000 M a x . d i f f . 0 . 0 8 0 . 0 9 0 . 0 9 0 . 1 0 ( S ee notes on p. 139.) 146 A ppe ndix 3.1 - R e s ult s of sim u lat i on using M e t hod A ( c ont i nue d) Input geom etr y data: L = 500 f t B = 85 f t P = 17.5 f t I nput f l ow c onditions : Q u = 29766 c f s h d =2 . 2 3f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 2964 2964 2964 2964 0 0 .0125 1.21 0.90 0.60 0.29 0.92 0.02 2964 2964 2964 2964 0 0 .02 1 .50 1 .21 0 .93 0 .64 0 .86 0.03 2964 2964 2964 2964 0 0 .03 2 .03 1 .77 1 .52 1 .26 0 .77 0.04 2964 2964 2964 2964 0 0 .04 2 .67 2 .43 2 .20 1 .97 0 .70 M a x . d i f f . 0000 M a x . d i f f . 1 . 4 6 1 . 5 3 1 . 6 0 1 . 6 8 Input geom etr y data: L = 500 f t B = 85 f t P = 17.5 f t I nput f l ow c onditions : Q u = 29828 c f s h d =5 . 1 7f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 14738 14738 14738 14738 0 0 .0125 4.05 3.80 3.54 3.29 0.76 0.02 14738 14738 14738 14738 0 0 .02 4 .16 3 .91 3 .66 3 .42 0 .74 0.03 14738 14738 14738 14738 0 0 .03 4 .37 4 .13 3 .90 3 .66 0 .71 0.04 14738 14738 14738 14738 0 0 .04 4 .65 4 .42 4 .19 3 .97 0 .68 M a x . d i f f . 0000 M a x . d i f f . 0 . 6 0 0 . 6 2 0 . 6 5 0 . 6 8 Input geom etr y data: L = 250 f t B = 85 f t P = 17.5 f t I nput f l ow c onditions : Q u = 10219 c f s h d =1 . 2 5f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 842 842 842 842 0 0 .0125 1.06 0.95 0.85 0.74 0.32 0.02 842 842 842 842 0 0 .02 1 .07 0 .97 0 .86 0 .76 0 .31 0.03 842 842 842 842 0 0 .03 1 .10 1 .00 0 .89 0 .79 0 .31 0.04 842 842 842 842 0 0 .04 1 .14 1 .04 0 .93 0 .83 0 .31 M a x . d i f f . 0000 M a x . d i f f . 0 . 0 8 0 . 0 9 0 . 0 8 0 . 0 9 ( S ee notes on p. 139.) 147 A ppe ndix 3.1 - R e s ult s of sim u lat i on using M e t hod A ( c ont i nue d) Input geom etr y data: L = 250 f t B = 85 f t P = 17.5 f t I nput f l ow c onditions : Q u = 10225 c f s h d =3 . 7 2f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 5215 5215 5215 5215 0 0 .0125 3.48 3.37 3.27 3.17 0.31 0.02 5215 5215 5215 5215 0 0 .02 3 .49 3 .38 3 .28 3 .18 0 .31 0.03 5215 5215 5215 5215 0 0 .03 3 .50 3 .39 3 .29 3 .19 0 .31 0.04 5215 5215 5215 5215 0 0 .04 3 .52 3 .41 3 .31 3 .21 0 .31 M a x . d i f f . 0000 M a x . d i f f . 0 . 0 4 0 . 0 4 0 . 0 4 0 . 0 4 Input geom etr y data: L = 250 f t B = 85 f t P = 17.5 f t I nput f l ow c onditions : Q u = 29972 c f s h d =3 . 1 0f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 3023 3023 3023 3023 0 0 .0125 2.39 2.25 2.11 1.97 0.42 0.02 3023 3023 3023 3023 0 0 .02 2 .50 2 .37 2 .23 2 .10 0 .40 0.03 3023 3023 3023 3023 0 0 .03 2 .73 2 .60 2 .47 2 .34 0 .39 0.04 3023 3023 3023 3023 0 0 .04 3 .03 2 .90 2 .78 2 .65 0 .38 M a x . d i f f . 0000 M a x . d i f f . 0 . 6 4 0 . 6 5 0 . 6 7 0 . 6 8 Input geom etr y data: L = 250 f t B = 85 f t P = 17.5 f t I nput f l ow c onditions : Q u = 29938 c f s h d =6 . 9 5f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 13816 13816 13816 13816 0 0 .0125 6.21 6.09 5.97 5.85 0.36 0.02 13816 13816 13816 13816 0 0 .02 6 .25 6 .13 6 .01 5 .89 0 .36 0.03 13816 13816 13816 13816 0 0 .03 6 .32 6 .20 6 .09 5 .97 0 .35 0.04 13816 13816 13816 13816 0 0 .04 6 .43 6 .31 6 .20 6 .08 0 .35 M a x . d i f f . 0000 M a x . d i f f . 0 . 2 2 0 . 2 2 0 . 2 3 0 . 2 3 ( S ee notes on p. 139.) 148 A ppe ndix 3.1 - R e s ult s of sim u lat i on using M e t hod A ( c ont i nue d) Input geom etr y data: L = 500 f t B = 45 f t P = 17.5 f t I nput f l ow c onditions : Q u = 10181 c f s h d =1 . 3 5f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 1499 1499 1499 1499 0 0 .0125 0.92 0.69 0.47 0.25 0.67 0.02 1499 1499 1499 1499 0 0 .02 0 .99 0 .76 0 .55 0 .33 0 .66 0.03 1499 1499 1499 1499 0 0 .03 1 .13 0 .91 0 .69 0 .48 0 .65 0.04 1499 1499 1499 1499 0 0 .04 1 .31 1 .10 0 .89 0 .68 0 .63 M a x . d i f f . 0000 M a x . d i f f . 0 . 3 9 0 . 4 1 0 . 4 2 0 . 4 3 Input geom etr y data: L = 500 f t B = 45 f t P = 17.5 f t I nput f l ow c onditions : Q u = 10122 c f s h d =2 . 5 7f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 4604 4604 4604 4604 0 0 .0125 2.09 1.86 1.65 1.43 0.66 0.02 4604 4604 4604 4604 0 0 .02 2 .13 1 .90 1 .69 1 .48 0 .65 0.03 4604 4604 4604 4604 0 0 .03 2 .21 1 .99 1 .78 1 .57 0 .64 0.04 4604 4604 4604 4604 0 0 .04 2 .31 2 .10 1 .89 1 .69 0 .62 M a x . d i f f . 0000 M a x . d i f f . 0 . 2 2 0 . 2 4 0 . 2 4 0 . 2 6 Input geom etr y data: L = 500 f t B = 45 f t P = 17.5 f t I nput f l ow c onditions : Q u = 30012 c f s h d =3 . 0 0f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 3977 3977 3977 3977 0 0 .0125 ( 1 ) ( 1) ( 1 ) ( 1) 0.02 3977 3977 3977 3977 0 0 .02 1 .46 0 .85 0 .00 ( 1) 1.46 0.03 3977 3977 3977 3977 0 0 .03 3 .10 2 .76 2 .43 2 .08 1 .02 0.04 3977 3977 3977 3977 0 0 .04 4 .48 4 .22 3 .96 3 .70 0 .78 M a x . d i f f . 0000 M a x . d i f f . 3 . 0 2 3 . 3 7 3 . 9 6 1 . 6 2 ( S ee notes on p. 139.) 149 A ppe ndix 3.1 - R e s ult s of sim u lat i on using M e t hod A ( c ont i nue d) Input geom etr y data: L = 500 f t B = 45 f t P = 17.5 f t I nput f l ow c onditions : Q u = 29906 c f s h d =5 . 7 0f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 15172 15172 15172 15172 0 0 .0125 3.78 3.40 3.01 2.59 1.19 0.02 15172 15172 15172 15172 0 0 .02 4 .13 3 .79 3 .45 3 .09 1 .04 0.03 15172 15172 15172 15172 0 0 .03 4 .72 4 .42 4 .14 3 .84 0 .88 0.04 15172 15172 15172 15172 0 0 .04 5 .37 5 .11 4 .86 4 .61 0 .76 M a x . d i f f . 0000 M a x . d i f f . 1 . 5 9 1 . 7 1 1 . 8 5 2 . 0 2 Input geom etr y data: L = 250 f t B = 45 f t P = 17.5 f t I nput f l ow c onditions : Q u = 9900 c f s h d =1 . 3 5f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 835 835 835 835 0 0 .0125 1.11 1.00 0.89 0.78 0.33 0.02 835 835 835 835 0 0 .02 1 .15 1 .03 0 .93 0 .82 0 .33 0.03 835 835 835 835 0 0 .03 1 .22 1 .10 1 .00 0 .89 0 .33 0.04 835 835 835 835 0 0 .04 1 .31 1 .20 1 .09 0 .99 0 .32 M a x . d i f f . 0000 M a x . d i f f . 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 1 Input geom etr y data: L = 250 f t B = 45 f t P = 17.5 f t I nput f l ow c onditions : Q u = 9944 c f s h d =3 . 8 8f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 5175 5175 5175 5175 0 0 .0125 3.57 3.46 3.36 3.25 0.32 0.02 5175 5175 5175 5175 0 0 .02 3 .59 3 .48 3 .37 3 .27 0 .32 0.03 5175 5175 5175 5175 0 0 .03 3 .61 3 .51 3 .40 3 .30 0 .31 0.04 5175 5175 5175 5175 0 0 .04 3 .65 3 .54 3 .44 3 .34 0 .31 M a x . d i f f . 0000 M a x . d i f f . 0 . 0 8 0 . 0 8 0 . 0 8 0 . 0 9 ( S ee notes on p. 139.) 150 A ppe ndix 3.1 - R e s ult s of sim u lat i on using M e t hod A ( c ont i nue d) Input geom etr y data: L = 250 f t B = 45 f t P = 17.5 f t I nput f l ow c onditions : Q u = 29991 c f s h d =3 . 4 3f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 2925 2925 2925 2925 0 0 .0125 1.76 1.43 1.05 0.52 1.24 0.02 2925 2925 2925 2925 0 0 .02 2 .31 2 .06 1 .80 1 .52 0 .79 0.03 2925 2925 2925 2925 0 0 .03 3 .11 2 .92 2 .74 2 .54 0 .57 0.04 2925 2925 2925 2925 0 0 .04 3 .94 3 .79 3 .63 3 .48 0 .46 M a x . d i f f . 0000 M a x . d i f f . 2 . 1 8 2 . 3 6 2 . 5 8 2 . 9 6 Input geom etr y data: L = 250 f t B = 45 f t P = 17.5 f t I nput f l ow c onditions : Q u = 29969 c f s h d =6 . 4 5f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 10778 10778 10778 10778 0 0 .0125 5.14 4.98 4.82 4.66 0.48 0.02 10778 10778 10778 10778 0 0 .02 5 .28 5 .13 4 .97 4 .82 0 .46 0.03 10778 10778 10778 10778 0 0 .03 5 .55 5 .40 5 .26 5 .11 0 .44 0.04 10778 10778 10778 10778 0 0 .04 5 .89 5 .75 5 .62 5 .48 0 .41 M a x . d i f f . 0000 M a x . d i f f . 0 . 7 5 0 . 7 7 0 . 8 0 0 . 8 2 ( S ee notes on p. 139.) 151 A ppe ndix 3.2 - R e s ult s of sim u lat i on using M e t hod B Input geom etr y data: L = 598 f t B = 85 f t P = 13.0 f t I nput f l ow c onditions : Q u = 10156 c f s h d =0 . 9 7f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 951 739 564 430 521 0.0125 0.66 0.41 0.16 - 0 .10 0 .76 0.02 1039 821 637 485 554 0.02 0.76 0.51 0.28 0.03 0.73 0.03 1216 991 795 622 594 0.03 0.95 0.72 0.49 0.27 0.68 0.04 1461 1226 1020 832 629 0.04 1.18 0.97 0.76 0.55 0.63 Max . dif f . 510 487 456 402 Max . dif f . 0.52 0.56 0.60 0.65 Input geom etr y data: L = 598 f t B = 85 f t P = 13.0 f t I nput f l ow c onditions : Q u = 10116 c f s h d =2 . 3 0f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 4801 4348 3924 3523 1278 0.0125 1.81 1.55 1.29 1.03 0.78 0.02 4875 4429 4010 3612 1263 0.02 1.87 1.61 1.35 1.10 0.77 0.03 5019 4587 4179 3787 1232 0.03 1.97 1.72 1.48 1.24 0.73 0.04 5203 4789 4399 4018 1185 0.04 2.10 1.87 1.64 1.41 0.69 Max . dif f . 402 441 475 495 Max . dif f . 0.29 0.32 0.35 0.38 Input geom etr y data: L = 598 f t B = 85 f t P = 13.0 f t I nput f l ow c onditions : Q u = 29894 c f s h d =4 . 0 3f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 6407 ( 2 ) ( 3) ( 3 ) 0 .0125 1.39 ( 2 ) ( 3) ( 3 ) 0.02 8104 7287 6478 5642 2462 0.02 2.29 1.97 1.65 1.32 0.97 0.03 10257 9598 8987 8382 1875 0.03 3.28 3.05 2.84 2.63 0.65 0.04 12380 11819 11266 10715 1665 0.04 4.20 4.02 3.83 3.64 0.56 Max . dif f . 5973 4532 4788 5073 Max . dif f . 2.81 2.05 2.18 2.32 ( S ee notes on p. 139.) 152 A ppe ndix 3.2 - R e s ult s of sim u lat i on using M e t hod B ( c ont i nue d) Input geom etr y data: L = 598 f t B = 85 f t P = 13.0 f t I nput f l ow c onditions : Q u = 30144 c f s h d =4 . 8 5f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 10338 9104 ( 3 ) ( 2) 1234 0.0125 1.95 1.45 ( 3 ) ( 2) 0.50 0.02 11947 10916 9875 8950 2997 0.02 2.79 2.39 1.96 1.66 1.13 0.03 14069 13247 12401 11675 2394 0.03 3.85 3.52 3.16 2.93 0.92 0.04 15895 15268 14656 14034 1861 0.04 4.69 4.48 4.27 4.04 0.65 Max . dif f . 5557 6164 4781 5084 Max . dif f . 2.74 3.03 2.31 2.38 Input geom etr y data: L = 250 f t B = 85 f t P = 13.0 f t I nput f l ow c onditions : Q u = 30022 c f s h d =3 . 5 5f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 2845 2661 2469 2243 602 0.0125 2.13 1.83 1.46 0.85 1.28 0.02 3174 3014 2856 2692 482 0.02 2.63 2.42 2.20 1.96 0.67 0.03 3721 3583 3449 3314 407 0.03 3.33 3.18 3.03 2.87 0.46 0.04 4368 4238 4112 3986 382 0.04 4.05 3.92 3.79 3.66 0.39 Max . dif f . 1523 1577 1643 1743 Max . dif f . 1.92 2.09 2.33 2.81 Input geom etr y data: L = 250 f t B = 85 f t P = 13.0 f t I nput f l ow c onditions : Q u = 29969 c f s h d =7 . 5 0f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 15596 15380 15170 14960 636 0.0125 6.51 6.37 6.23 6.09 0.42 0.02 15699 15487 15280 15073 626 0.02 6.59 6.46 6.32 6.19 0.40 0.03 15903 15696 15496 15295 608 0.03 6.75 6.63 6.50 6.37 0.38 0.04 16170 15971 15778 15586 584 0.04 6.96 6.84 6.73 6.61 0.35 Max . dif f . 574 591 608 626 Max . dif f . 0.45 0.47 0.50 0.52 ( S ee notes on p. 139.) 153 A ppe ndix 3.2 - R e s ult s of sim u lat i on using M e t hod B ( c ont i nue d) Input geom etr y data: L = 250 f t B = 85 f t P = 13.0 f t I nput f l ow c onditions : Q u = 4875 c f s h d =0 . 9 0f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 481 437 397 358 123 0.0125 0.78 0.67 0.57 0.47 0.31 0.02 485 441 400 362 123 0.02 0.79 0.68 0.58 0.48 0.31 0.03 493 449 408 370 123 0.03 0.81 0.70 0.60 0.51 0.30 0.04 504 460 419 380 124 0.04 0.83 0.73 0.63 0.53 0.30 Max . dif f . 23 23 22 22 Max . dif f . 0.05 0.06 0.06 0.06 Input geom etr y data: L = 250 f t B = 85 f t P = 13.0 f t I nput f l ow c onditions : Q u = 4947 c f s h d =2 . 3 3f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 2451 2367 2287 2208 243 0.0125 2.17 2.07 1.96 1.86 0.31 0.02 2454 2370 2290 2211 243 0.02 2.18 2.07 1.97 1.87 0.31 0.03 2460 2376 2296 2218 242 0.03 2.19 2.08 1.98 1.88 0.31 0.04 2469 2385 2305 2227 242 0.04 2.20 2.09 1.99 1.89 0.31 Max . dif f . 18 18 18 19 Max . dif f . 0.03 0.02 0.03 0.03 Input geom etr y data: L = 375 f t B = 45 f t P = 13.0 f t I nput f l ow c onditions : Q u = 10222 c f s h d =2 . 2 7f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 2604 2422 2251 2085 519 0.0125 1.64 1.47 1.30 1.13 0.51 0.02 2715 2535 2367 2203 512 0.02 1.75 1.59 1.43 1.27 0.48 0.03 2927 2753 2590 2430 497 0.03 1.97 1.82 1.67 1.52 0.45 0.04 3201 3029 2871 2717 484 0.04 2.23 2.09 1.96 1.83 0.40 Max . dif f . 597 607 620 632 Max . dif f . 0.59 0.62 0.66 0.70 ( S ee notes on p. 139.) 154 A ppe ndix 3.2 - R e s ult s of sim u lat i on using M e t hod B ( c ont i nue d) Input geom etr y data: L = 375 f t B = 45 f t P = 13.0 f t I nput f l ow c onditions : Q u = 10147 c f s h d =3 . 4 3f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 5708 5479 5261 5042 666 0.0125 2.77 2.62 2.45 2.28 0.49 0.02 5776 5550 5336 5122 654 0.02 2.82 2.69 2.53 2.36 0.46 0.03 5910 5688 5481 5276 634 0.03 2.93 2.81 2.67 2.52 0.41 0.04 6085 5869 5667 5471 614 0.04 3.07 2.95 2.84 2.71 0.36 Max . dif f . 377 390 406 429 Max . dif f . 0.30 0.33 0.39 0.43 Input geom etr y data: L = 375 f t B = 45 f t P = 13.0 f t I nput f l ow c onditions : Q u = 19709 c f s h d =3 . 5 5f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 ( 2 ) ( 3) ( 3 ) ( 3) 0.0125 ( 2 ) ( 3) ( 3 ) ( 3) 0.02 3969 3578 3167 ( 3 ) 802 0.02 1.99 1.71 1.43 ( 3 ) 0 .56 0.03 5199 4925 4661 4394 805 0.03 2.97 2.80 2.64 2.47 0.50 0.04 6342 6083 5863 5646 696 0.04 3.79 3.63 3.50 3.38 0.41 Max . dif f . 2373 2505 2696 1252 Max . dif f . 1.80 1.92 2.07 0.91 Input geom etr y data: L = 375 f t B = 45 f t P = 13.0 f t I nput f l ow c onditions : Q u = 19784 c f s h d =5 . 5 3f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 10427 10025 9610 9156 1271 0.0125 3.91 3.66 3.39 3.08 0.83 0.02 10785 10424 10061 9680 1105 0.02 4.17 3.97 3.75 3.51 0.66 0.03 11387 11069 10766 10453 934 0.03 4.49 4.39 4.27 4.10 0.39 0.04 ( 4 ) 11796 11513 11237 559 0.04 ( 4 ) 4 .74 4 .65 4 .57 0 .17 Max . dif f . 960 1771 1903 2081 Max . dif f . 0.58 1.08 1.26 1.49 ( S ee notes on p. 139.) 155 A ppe ndix 3.2 - R e s ult s of sim u lat i on using M e t hod B ( c ont i nue d) Input geom etr y data: L = 250 f t B = 45 f t P = 13.0 f t I nput f l ow c onditions : Q u = 9997 c f s h d =1 . 7 0f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 1064 999 939 880 184 0.0125 1.34 1.22 1.11 0.99 0.35 0.02 1116 1051 990 931 185 0.02 1.44 1.33 1.21 1.10 0.34 0.03 1219 1155 1094 1035 184 0.03 1.63 1.52 1.42 1.31 0.32 0.04 1360 1294 1233 1174 186 0.04 1.87 1.77 1.67 1.57 0.30 Max . dif f . 296 295 294 294 Max . dif f . 0.53 0.55 0.56 0.58 Input geom etr y data: L = 250 f t B = 45 f t P = 13.0 f t I nput f l ow c onditions : Q u = 10084 c f s h d =3 . 7 0f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 4728 4608 4492 4376 352 0.0125 3.21 3.09 2.97 2.84 0.37 0.02 4764 4645 4530 4416 348 0.02 3.25 3.13 3.02 2.90 0.35 0.03 4835 4718 4606 4494 341 0.03 3.34 3.22 3.11 3.00 0.34 0.04 4929 4816 4707 4598 331 0.04 3.46 3.34 3.24 3.13 0.33 Max . dif f . 201 208 215 222 Max . dif f . 0.25 0.25 0.27 0.29 Input geom etr y data: L = 250 f t B = 45 f t P = 13.0 f t I nput f l ow c onditions : Q u = 20034 c f s h d =3 . 2 5f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 ( 3 ) ( 3) ( 3 ) ( 3) 0.0125 ( 3 ) ( 3) ( 3 ) ( 3) 0.02 2343 2155 1940 ( 2 ) 403 0.02 1.93 1.66 1.19 ( 2 ) 0 .74 0.03 3087 2942 2800 2654 433 0.03 2.96 2.80 2.64 2.46 0.50 0.04 3797 3678 3563 3448 349 0.04 3.77 3.65 3.54 3.42 0.35 Max . dif f . 1454 1523 1623 794 Max . dif f . 1.84 1.99 2.35 0.96 ( S ee notes on p. 139.) 156 A ppe ndix 3.2 - R e s ult s of sim u lat i on using M e t hod B ( c ont i nue d) Input geom etr y data: L = 250 f t B = 45 f t P = 13.0 f t I nput f l ow c onditions : Q u = 19928 c f s h d =6 . 4 2f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 10959 10768 10582 10394 565 0.0125 5.31 5.17 5.04 4.89 0.42 0.02 11082 10898 10718 10536 546 0.02 5.41 5.30 5.17 5.03 0.38 0.03 11319 11143 10974 10803 516 0.03 5.58 5.49 5.40 5.29 0.29 0.04 11624 11455 11293 11133 491 0.04 5.81 5.72 5.64 5.56 0.25 Max . dif f . 665 687 711 739 Max . dif f . 0.50 0.55 0.60 0.67 Input geom etr y data: L = 500 f t B = 85 f t P = 17.5 f t I nput f l ow c onditions : Q u = 10003 c f s h d =0 . 9 5f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 887 729 593 476 411 0.0125 0.72 0.51 0.31 0.10 0.62 0.02 909 750 612 492 417 0.02 0.74 0.54 0.34 0.14 0.60 0.03 954 793 652 527 427 0.03 0.80 0.60 0.40 0.20 0.60 0.04 1017 853 708 578 439 0.04 0.88 0.68 0.49 0.29 0.59 Max . dif f . 130 124 115 102 Max . dif f . 0.16 0.17 0.18 0.19 Input geom etr y data: L = 500 f t B = 85 f t P = 17.5 f t I nput f l ow c onditions : Q u = 9975 c f s h d =2 . 5 3f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 5099 4768 4456 4152 947 0.0125 2.21 2.00 1.79 1.59 0.62 0.02 5117 4786 4475 4172 945 0.02 2.22 2.01 1.81 1.60 0.62 0.03 5152 4823 4515 4213 939 0.03 2.25 2.04 1.84 1.64 0.61 0.04 5199 4875 4569 4270 929 0.04 2.29 2.09 1.89 1.69 0.60 Max . dif f . 100 107 113 118 Max . dif f . 0.08 0.09 0.10 0.10 ( S ee notes on p. 139.) 157 A ppe ndix 3.2 - R e s ult s of sim u lat i on using M e t hod B ( c ont i nue d) Input geom etr y data: L = 500 f t B = 85 f t P = 17.5 f t I nput f l ow c onditions : Q u = 29766 c f s h d =2 . 2 3f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 2785 2503 2243 1995 790 0.0125 1.59 1.34 1.09 0.84 0.75 0.02 3050 2768 2505 2253 797 0.02 1.83 1.59 1.36 1.12 0.71 0.03 3567 3286 3023 2768 799 0.03 2.25 2.04 1.83 1.62 0.63 0.04 4254 3966 3699 3445 809 0.04 2.76 2.57 2.38 2.19 0.57 Max . dif f . 1469 1463 1456 1450 Max . dif f . 1.17 1.23 1.29 1.35 Input geom etr y data: L = 500 f t B = 85 f t P = 17.5 f t I nput f l ow c onditions : Q u = 29828 c f s h d =5 . 1 7f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 14698 14130 13581 13032 1666 0.0125 4.08 3.83 3.59 3.33 0.75 0.02 14885 14327 13789 13251 1634 0.02 4.18 3.94 3.70 3.46 0.72 0.03 15246 14708 14191 13674 1572 0.03 4.37 4.15 3.93 3.70 0.67 0.04 15712 15198 14705 14213 1499 0.04 4.62 4.41 4.21 4.00 0.62 Max . dif f . 1014 1068 1124 1181 Max . dif f . 0.54 0.58 0.62 0.67 Input geom etr y data: L = 250 f t B = 85 f t P = 17.5 f t I nput f l ow c onditions : Q u = 10219 c f s h d =1 . 2 5f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 787 736 687 641 146 0.0125 1.11 1.01 0.91 0.80 0.31 0.02 794 742 694 647 147 0.02 1.13 1.02 0.92 0.82 0.31 0.03 808 756 708 661 147 0.03 1.16 1.05 0.95 0.85 0.31 0.04 828 776 727 679 149 0.04 1.20 1.09 0.99 0.89 0.31 Max . dif f . 41 40 40 38 Max . dif f . 0.09 0.08 0.08 0.09 ( S ee notes on p. 139.) 158 A ppe ndix 3.2 - R e s ult s of sim u lat i on using M e t hod B ( c ont i nue d) Input geom etr y data: L = 250 f t B = 85 f t P = 17.5 f t I nput f l ow c onditions : Q u = 10225 c f s h d =3 . 7 2f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 5243 5133 5028 4923 320 0.0125 3.53 3.43 3.33 3.22 0.31 0.02 5248 5138 5033 4928 320 0.02 3.54 3.43 3.33 3.23 0.31 0.03 5257 5148 5043 4939 318 0.03 3.55 3.45 3.34 3.24 0.31 0.04 5271 5161 5057 4953 318 0.04 3.57 3.46 3.36 3.26 0.31 Max . dif f . 28 28 29 30 Max . dif f . 0.04 0.03 0.03 0.04 Input geom etr y data: L = 250 f t B = 85 f t P = 17.5 f t I nput f l ow c onditions : Q u = 29972 c f s h d =3 . 1 0f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 2921 2823 2729 2637 284 0.0125 2.60 2.47 2.35 2.22 0.38 0.02 2998 2900 2807 2715 283 0.02 2.70 2.58 2.46 2.33 0.37 0.03 3154 3056 2963 2872 282 0.03 2.91 2.79 2.67 2.55 0.36 0.04 3364 3268 3175 3084 280 0.04 3.17 3.06 2.95 2.84 0.33 Max . dif f . 443 445 446 447 Max . dif f . 0.57 0.59 0.60 0.62 Input geom etr y data: L = 250 f t B = 85 f t P = 17.5 f t I nput f l ow c onditions : Q u = 29938 c f s h d =6 . 9 5f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 14199 14019 13846 13673 526 0.0125 6.29 6.17 6.06 5.94 0.35 0.02 14243 14064 13892 13720 523 0.02 6.33 6.21 6.10 5.98 0.35 0.03 14331 14154 13984 13813 518 0.03 6.40 6.29 6.17 6.06 0.34 0.04 14453 14278 14109 13941 512 0.04 6.50 6.39 6.28 6.16 0.34 Max . dif f . 254 259 263 268 Max . dif f . 0.21 0.22 0.22 0.22 ( S ee notes on p. 139.) 159 A ppe ndix 3.2 - R e s ult s of sim u lat i on using M e t hod B ( c ont i nue d) Input geom etr y data: L = 500 f t B = 45 f t P = 17.5 f t I nput f l ow c onditions : Q u = 10181 c f s h d =1 . 3 5f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 1493 1291 1116 955 538 0.0125 1.01 0.80 0.60 0.39 0.62 0.02 1552 1349 1171 1007 545 0.02 1.07 0.86 0.67 0.47 0.60 0.03 1672 1468 1283 1115 557 0.03 1.20 1.00 0.80 0.61 0.59 0.04 1835 1629 1442 1266 569 0.04 1.36 1.17 0.98 0.80 0.56 Max . dif f . 342 338 326 311 Max . dif f . 0.35 0.37 0.38 0.41 Input geom etr y data: L = 500 f t B = 45 f t P = 17.5 f t I nput f l ow c onditions : Q u = 10122 c f s h d =2 . 5 7f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 4907 4573 4255 3948 959 0.0125 2.13 1.91 1.69 1.48 0.65 0.02 4952 4622 4307 4002 950 0.02 2.16 1.95 1.73 1.52 0.64 0.03 5042 4717 4409 4108 934 0.03 2.24 2.03 1.82 1.61 0.63 0.04 5160 4844 4545 4249 911 0.04 2.33 2.13 1.94 1.73 0.60 Max . dif f . 253 271 290 301 Max . dif f . 0.20 0.22 0.25 0.25 Input geom etr y data: L = 500 f t B = 45 f t P = 17.5 f t I nput f l ow c onditions : Q u = 30012 c f s h d =3 . 0 0f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 2902 2386 ( 2 ) ( 3) 516 0.0125 1.14 0.66 ( 2 ) ( 3) 0.48 0.02 3866 3446 3043 2635 1231 0.02 1.96 1.67 1.38 1.07 0.89 0.03 5412 5033 4670 4309 1103 0.03 3.02 2.80 2.58 2.37 0.65 0.04 6962 6631 6315 6003 959 0.04 3.92 3.75 3.59 3.43 0.49 Max . dif f . 4060 4245 3272 3368 Max . dif f . 2.78 3.09 2.21 2.36 160 A ppe ndix 3.2 - R e s ult s of sim u lat i on using M e t hod B ( c ont i nue d) Input geom etr y data: L = 500 f t B = 45 f t P = 17.5 f t I nput f l ow c onditions : Q u = 29906 c f s h d =5 . 7 0f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 13613 12799 12011 11239 2374 0.0125 3.38 2.94 2.53 2.14 1.24 0.02 14273 13566 12841 12106 2167 0.02 3.78 3.44 3.06 2.68 1.10 0.03 15337 14746 14163 13559 1778 0.03 4.38 4.12 3.86 3.58 0.80 0.04 16472 15965 15473 14974 1498 0.04 4.95 4.77 4.58 4.38 0.57 Max . dif f . 2859 3166 3462 3735 Max . dif f . 1.57 1.83 2.05 2.24 Input geom etr y data: L = 250 f t B = 45 f t P = 17.5 f t I nput f l ow c onditions : Q u = 9900 c f s h d =1 . 3 5f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 825 772 723 675 150 0.0125 1.17 1.06 0.96 0.85 0.32 0.02 840 787 738 690 150 0.02 1.20 1.09 0.99 0.89 0.31 0.03 872 818 769 720 152 0.03 1.27 1.16 1.06 0.96 0.31 0.04 916 862 812 763 153 0.04 1.35 1.25 1.15 1.05 0.30 Max . dif f . 91 90 89 88 Max . dif f . 0.18 0.19 0.19 0.20 Input geom etr y data: L = 250 f t B = 45 f t P = 17.5 f t I nput f l ow c onditions : Q u = 9944 c f s h d =3 . 8 8f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 5427 5315 5208 5102 325 0.0125 3.59 3.49 3.38 3.28 0.31 0.02 5437 5326 5219 5113 324 0.02 3.61 3.50 3.40 3.29 0.32 0.03 5457 5346 5240 5134 323 0.03 3.63 3.53 3.42 3.32 0.31 0.04 5485 5375 5270 5165 320 0.04 3.66 3.56 3.46 3.36 0.30 Max . dif f . 58 60 62 63 Max . dif f . 0.07 0.07 0.08 0.08 ( S ee notes on p. 139.) 161 A ppe ndix 3.2 - R e s ult s of sim u lat i on using M e t hod B ( c ont i nue d) Input geom etr y data: L = 250 f t B = 45 f t P = 17.5 f t I nput f l ow c onditions : Q u = 29991 c f s h d =3 . 4 3f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 2596 2448 2302 2152 444 0.0125 2.13 1.90 1.65 1.37 0.76 0.02 2859 2722 2588 2453 406 0.02 2.54 2.35 2.16 1.95 0.59 0.03 3325 3201 3081 2961 364 0.03 3.18 3.03 2.88 2.73 0.45 0.04 3881 3763 3653 3544 337 0.04 3.84 3.71 3.59 3.47 0.37 Max . dif f . 1285 1315 1351 1392 Max . dif f . 1.71 1.81 1.94 2.10 Input geom etr y data: L = 250 f t B = 45 f t P = 17.5 f t I nput f l ow c onditions : Q u = 29969 c f s h d =6 . 4 5f t Calc ulated Q w (cf s ) C a l cu l a te d h u (f t) Slope Slope Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . Manning' s n 0.000385 0.0008 0.0012 0.0016 Max . dif f . 0.0125 10689 10481 10280 10077 612 0.0125 4.92 4.76 4.60 4.43 0.49 0.02 10846 10643 10447 10250 596 0.02 5.06 4.91 4.75 4.60 0.46 0.03 11150 10956 10768 10580 570 0.03 5.33 5.19 5.05 4.91 0.42 0.04 11541 11356 11178 11000 541 0.04 5.67 5.54 5.41 5.28 0.39 Max . dif f . 852 875 898 923 Max . dif f . 0.75 0.78 0.81 0.85 ( S ee notes on p. 139.) 163 AP P E NDIX 4 - S UMMARY OF MODEL DATA F OR 4H:1V S IDE S LOP E S Te s t L B P u P d Q u Q w h u h d Fw d F d C e (f t) (f t) (f t) (f t) (cf s ) ( cf s) (f t) (f t) A1 10 1.8 0 .513 0.506 3.208 1.685 0.138 0.168 0.216 0.137 0.717 A2 10 1.8 0 .513 0.506 3.208 0.992 0.092 0.126 0.403 0.226 0.666 A3 10 1.8 0 .513 0.506 3.208 0.246 0.027 0.060 0.928 0.377 0.522 B1 10 1.8 0 .513 0.506 6.608 3.172 0.201 0.237 0.351 0.253 0.775 B2 10 1.8 0 .513 0.506 6.608 1.915 0.140 0.201 0.566 0.383 0.614 B3 10 1.8 0 .513 0.506 6.608 0.783 0.077 0.133 1.016 0.583 0.485 C1 10 1.8 0 .513 0.506 9.165 3.714 0.222 0.268 0.490 0.369 0.742 C2 10 1.8 0 .513 0.506 9.165 2.379 0.164 0.237 0.694 0.499 0.581 C3 10 1.8 0 .513 0.506 9.165 0.983 0.125 0.162 1.201 0.748 0.443 AA3 1 0 1 .8 0.513 0.506 3.200 0.256 0.026 0.060 0.922 0.375 0.543 BB1 1 0 1 .8 0.513 0.506 6.570 3.196 0.202 0.237 0.345 0.248 0.781 BB3 1 0 1 .8 0.513 0.506 6.570 0.794 0.078 0.132 1.010 0.579 0.495 CC1 10 1.8 0 .513 0.506 9.094 3.742 0.218 0.278 0.462 0.352 0.697 CC2 10 1.8 0 .513 0.506 9.094 2.463 0.167 0.237 0.678 0.488 0.602 CC3 10 1.8 0 .513 0.506 9.094 0.980 0.125 0.160 1.204 0.746 0.451 X 1 5 1 .8 0.513 0.490 3.184 1.627 0.215 0.236 0.165 0.120 0.717 X 2 5 1 .8 0.513 0.490 3.184 1.339 0.190 0.213 0.217 0.152 0.703 X 3 5 1 .8 0.513 0.490 3.184 0.943 0.151 0.174 0.320 0.207 0.696 X 4 5 1 .8 0.513 0.490 3.184 0.591 0.108 0.132 0.472 0.273 0.689 X 5 5 1 .8 0.513 0.490 3.184 0.286 0.064 0.085 0.743 0.357 0.678 Y 1 5 1 .8 0.513 0.490 6.428 1.491 0.207 0.239 0.518 0.377 0.643 Y 2 5 1 .8 0.513 0.490 6.428 1.021 0.161 0.196 0.690 0.467 0.617 Y 3 5 1 .8 0.513 0.490 6.428 0.546 0.110 0.138 1.030 0.608 0.592 Z1 5 1 .8 0.513 0.490 8.989 0.886 0.157 0.190 1.066 0.713 0.565 165 APPENDIX 5 - COMPONENTS OF ? AND ? Appendix 5.1 - Variation of components of ? and ? with distance for 2.5H:1V side slopes Case Q u (cfs) Diver- sion (%) B s (ft) s B x ? ? 1 = 2 2 U u ? 2 = 2 2 U 'u ? ? 1 = 3 3 U u ? 2 = 3 2 3 U 'uu ? 3 = 3 2 U 'vu ? 4 = 3 2 U vu A 0.00 1.689 1.554 0.135 3.180 2.466 0.365 0.0483 0.2782 8.9 1.16 1.636 1.494 0.142 2.859 2.376 0.395 0.0612 0.0135 54 2.75 1.520 1.420 0.100 2.541 2.165 0.311 0.0477 0.0040 3.72 4.66 1.326 1.250 0.076 1.986 1.711 0.230 0.0346 0.0034 5.98 1.287 1.225 0.062 1.876 1.640 0.198 0.0305 0.0041 B 0.00 1.794 1.630 0.165 3.483 2.669 0.461 0.0689 0.2593 3.0 1.18 1.813 1.681 0.132 3.309 2.779 0.426 0.0750 0.0048 54 2.77 1.545 1.437 0.108 2.588 2.171 0.338 0.0605 0.0026 3.72 4.67 1.388 1.304 0.084 2.166 1.855 0.262 0.0438 0.0012 6.00 1.324 1.246 0.078 2.006 1.694 0.260 0.0385 0.0027 C 0.00 1.142 1.133 0.010 1.396 1.341 0.0273 0.0026 0.0241 8.9 2.60 1.107 1.098 0.009 1.295 1.258 0.0281 0.0029 0.0050 25 6.13 1.091 1.083 0.008 1.245 1.213 0.0249 0.0027 0.0035 1.67 10.41 1.082 1.075 0.006 1.219 1.191 0.0202 0.0024 0.0040 13.30 1.084 1.077 0.007 1.227 1.199 0.0210 0.0024 0.0041 D 1.98 1.516 1.500 0.016 2.428 2.361 0.0481 0.0067 0.0081 4.1 3.35 1.437 1.420 0.017 2.225 2.166 0.0448 0.0070 0.0045 N/A 6.84 1.273 1.261 0.012 1.802 1.754 0.0394 0.0056 0.0009 3.72 9.99 1.221 1.206 0.015 1.639 1.579 0.0482 0.0070 0.0018 13.13 1.186 1.174 0.013 1.531 1.481 0.0417 0.0057 0.0012 16.28 1.149 1.137 0.013 1.421 1.372 0.0409 0.0059 0.0011 E N/A 1.083 1.077 0.006 1.229 1.203 0.0200 0.0034 0.0014 166 Appendix 5.2 - Variation of components of ? and ? with distance for 4H:1V side slopes Case Q u (cfs) Diver- sion (%) B s (ft) s B x ? ? 1 = 2 2 U u ? 2 = 2 2 U 'u ? ? 1 = 3 3 U u ? 2 = 3 2 3 U 'uu ? 3 = 3 2 U 'vu ? 4 = 3 2 U vu F 0.51 1.989 1.833 0.156 3.828 3.164 0.532 0.168 0.0080 6.1 0.51 1.908 1.763 0.145 3.800 3.211 0.481 0.083 0.0108 54 2.02 1.695 1.584 0.112 3.283 2.755 0.431 0.118 0.0105 3.99 2.02 1.728 1.617 0.112 3.226 2.784 0.354 0.0669 0.103 3.41 1.398 1.377 2.166 2.063 4.80 1.215 1.198 1.630 1.554 G 0.00 1.885 1.832 0.053 3.276 3.094 0.177 0.0253 0.0024 4.5 0.54 1.960 1.910 0.050 3.497 3.346 0.130 0.0169 0.0019 N/A 1.08 1.934 1.876 0.059 3.450 3.236 0.197 0.0232 0.0044 3.99 1.63 1.818 1.765 0.053 3.240 3.065 0.147 0.0206 0.0037 2.16 1.733 1.672 0.062 3.000 2.785 0.198 0.0236 0.0042 2.71 1.612 1.555 0.057 2.724 2.521 0.185 0.0243 0.0037 3.26 1.427 1.390 0.037 2.128 1.995 0.115 0.0186 0.0046 4.34 1.305 1.280 0.026 1.874 1.783 0.0785 0.0079 0.0017 4.89 1.236 1.209 0.028 1.671 1.565 0.0953 0.0107 0.0031 6.52 1.185 1.167 0.018 1.516 1.444 0.0571 0.0107 0.0044 7.89 1.201 1.184 0.017 1.536 1.495 0.0505 0.0052 0.0031 10.90 1.132 1.118 0.014 1.371 1.317 0.0443 0.0112 0.0018 H N/A 1.123 1.115 0.008 1.172 1.153 0.0133 0.0022 0.0035 Appendix 5.3 - Components of ? and ? just downstream of weir for 2.5H:1V side slopes Case Q u (cfs) Diver- sion (%) B s (ft) s B x ? ? 1 = 2 2 U u ? 2 = 2 2 U 'u ? ? 1 = 3 3 U u ? 2 = 3 2 3 U 'uu ? 3 = 3 2 U 'vu ? 4 = 3 2 U vu 25 8.91 1.122 1.111 0.011 1.330 1.292 0.030 0.0029 0.0041 25 3.01 1.119 1.106 0.012 1.310 1.265 0.037 0.0055 0.0016 39 6.13 1.336 1.281 0.056 1.871 1.710 0.129 0.0216 0.0067 40 8.92 1.268 1.213 0.055 1.732 1.554 0.145 0.0204 0.0144 40 3.01 1.341 1.291 0.050 1.874 1.717 0.124 0.0215 0.0055 54 8.91 1.617 1.455 0.162 2.737 2.195 0.421 0.0633 0.0403 54 3.01 1.839 1.692 0.147 3.333 2.740 0.475 0.0803 0.0222 55 5.97 1.785 1.620 0.165 3.184 2.657 0.413 0.0615 0.0376 167 Appendix 5.4 - Components of ? and ? just downstream of weir for 4H:1V side slopes Diver- sion (%) Q u (cfs) ? ? 1 = 2 2 U u ? 2 = 2 2 U 'u ? ? 1 = 3 3 U u ? 2 = 3 2 3 U 'uu ? 3 = 3 2 U 'vu ? 4 = 3 2 U vu 25 3.0 1.262 1.238 0.024 1.692 1.618 0.0652 0.0077 0.0017 25 6.0 1.140 1.104 0.036 1.394 1.267 0.122 0.0355 0.0011 25 9.0 1.297 1.208 0.089 1.823 1.538 0.275 0.0562 0.0057 40 3.0 1.704 1.631 0.074 2.837 2.581 0.211 0.0304 0.0040 40 6.0 1.504 1.301 0.203 2.393 1.794 0.576 0.1278 0.0080 40 9.1 1.561 1.448 0.113 2.501 2.131 0.327 0.0739 0.0083 53 3.0 2.513 2.322 0.191 6.016 4.995 0.817 0.1347 0.0234 53 6.1 1.989 1.833 0.156 3.828 3.164 0.531 0.1683 0.0080 53 9.1 2.137 1.882 0.255 4.366 3.356 0.944 0.2023 0.0137 53 2.9 2.524 2.309 0.215 5.844 4.749 0.880 0.1793 0.0051 55 6.0 1.908 1.763 0.145 3.800 3.211 0.481 0.0831 0.0108 169 AP P E NDIX 6 - M OMENTUM AND ENERG Y BALANCES Appendix 6.1 - M om entum b alance f o r C ase A Dis t anc e D epth dA u A ? dA u A ? 2 ? ? QU h () 2 1 2 h h gA ? ? ? F M h ? (f t) (f t) (cf s ) ( f t 4 /s 2 ) (f t 4 /s 2 ) ( ft ) ( ft 4 /s 2 )( f t 4 /s 2 )( f t ) 0.0 0 .986 4.284 5.361 1.69 5.361 1.2449 - 1 .158 0.051 - 0 .003 4.3 0 .991 4.040 4.584 1.64 4.730 1.2511 - 0 .585 0.066 - 0 .002 10.2 0 .997 4.040 4.222 1.52 4.356 1.2542 - 0 .364 0.078 0.001 17.3 1 .007 4.068 3.681 1.33 3.747 1.2561 - 0 .255 0.052 - 0 .001 22.3 1 .022 4.017 3.414 1.29 3.564 1.2574 Appendix 6.2 - E nergy b alance f o r C ase A Dis t anc e D epth dA u A ? dA u V A ? 2 ? g U 2 2 ? hH f h E h ? (f t) (f t) (cf s ) ( f t 5 /s 3 ) (f t) (f t) (f t) (f t) (f t) 0.0 0 .986 4.284 7.476 3.18 0.0271 1.2449 1.2720 0.0003 - 0 .001 4.3 0 .991 4.040 5.555 2.86 0.0220 1.2511 1.2731 0.0004 - 0 .001 10.2 0 .997 4.040 4.854 2.54 0.0192 1.2542 1.2734 0.0004 0.002 17.3 1 .007 4.068 3.764 1.99 0.0146 1.2561 1.2707 0.0003 - 0 .000 22.3 1 .022 4.017 3.282 1.88 0.0132 1.2574 1.2706 170 Appendix 6.3 - M om entum b alance f o r C ase B Dis t anc e D epth dA u A ? dA u A ? 2 ? ? QU h () 2 1 2 h h gA ? ? ? F M h ? (f t) (f t) (cf s ) ( f t 4 /s 2 ) (f t 4 /s 2 ) ( ft ) ( ft 4 /s 2 )( f t 4 /s 2 )( f t ) 0.0 0 .853 1.418 0.764 1.79 0.764 1.1205 - 0 .092 0.008 - 0 .000 4.4 0 .860 1.345 0.686 1.81 0.721 1.1211 - 0 .062 0.011 0.000 10.3 0 .865 1.367 0.600 1.54 0.610 1.1215 - 0 .109 0.012 - 0 .000 17.4 0 .871 1.353 0.523 1.39 0.543 1.1222 - 0 .129 0.008 - 0 .001 22.3 0 .893 1.342 0.474 1.32 0.500 1.1230 Appendix 6.4 - E nergy b alance f o r C ase B Dis t anc e D epth dA u A ? dA u V A ? 2 ? g U 2 2 ? hH f h E h ? (f t) (f t) (cf s ) ( f t 5 /s 3 ) (f t) (f t) (f t) (f t) (f t) 0.0 0 .853 1.418 0.445 3.48 0.0049 1.1205 1.1254 0.0001 - 0 .000 4.4 0 .860 1.345 0.353 3.31 0.0043 1.1211 1.1254 0.0001 0.001 10.3 0 .865 1.367 0.285 2.59 0.0033 1.1215 1.1248 0.0001 - 0 .000 17.4 0 .871 1.353 0.228 2.17 0.0027 1.1222 1.1249 0.0001 - 0 .000 22.3 0 .893 1.342 0.192 2.01 0.0023 1.1230 1.1253 171 Appendix 6.5 - M om entum b alance f o r C ase C Dis t anc e D epth dA u A ? dA u A ? 2 ? ? QU h () 2 1 2 h h gA ? ? ? F M h ? (f t) (f t) (cf s ) ( f t 4 /s 2 ) (f t 4 /s 2 ) ( ft ) ( ft 4 /s 2 )( f t 4 /s 2 )( f t ) 0.0 0 .899 6.403 9.224 1.14 9.224 1.1603 - 0 .593 0.158 - 0 .008 4.4 0 .906 6.278 8.503 1.11 9.736 1.1639 - 0 .050 0.220 - 0 .000 10.2 0 .912 6.368 8.537 1.09 9.500 1.1642 0.135 0.263 0.000 17.4 0 .917 6.458 8.642 1.08 9.352 1.1634 - 0 .310 0.173 - 0 .002 22.2 0 .934 6.369 8.210 1.08 9.136 1.1652 Appendix 6.6 - E nergy b alance f o r C ase C Dis t anc e D epth dA u A ? dA u V A ? 2 ? g U 2 2 ? hH f h E h ? (f t) (f t) (cf s ) ( f t 5 /s 3 ) (f t) (f t) (f t) (f t) (f t) 0.0 0 .899 6.403 14.218 1.40 0.0345 1.1603 1.1948 0.0010 - 0 .005 4.4 0 .906 6.278 12.170 1.30 0.0345 1.1639 1.1984 0.0013 0.000 10.2 0 .912 6.368 11.975 1.24 0.0325 1.1642 1.1967 0.0016 0.000 17.4 0 .917 6.458 12.047 1.22 0.0313 1.1634 1.1947 0.0010 - 0 .001 22.2 0 .934 6.369 11.050 1.23 0.0300 1.1652 1.1952 172 Appendix 6.7 - M om entum b alance f o r C ase D Dis t anc e D epth dA u A ? dA u A ? 2 ? ? QU h () 2 1 2 h h gA ? ? ? F M h ? (f t) (f t) (cf s ) ( f t 4 /s 2 ) (f t 4 /s 2 ) ( ft ) ( ft 4 /s 2 )( f t 4 /s 2 )( f t ) 7.4 0 .993 3.962 4.075 1.52 4.448 1.2537 - 0 .244 0.058 - 0 .000 12.5 0 .992 4.033 4.008 1.44 4.223 1.2550 - 0 .446 0.144 0.000 25.5 1 .015 4.195 3.719 1.27 3.622 1.2573 0.000 0.127 - 0 .000 37.2 1 .007 4.138 3.509 1.22 3.512 1.2573 - 0 .039 0.128 - 0 .000 48.9 1 .011 4.094 3.317 1.19 3.392 1.2575 - 0 .261 0.123 - 0 .001 60.6 1 .039 4.121 3.133 1.15 3.161 1.2588 Appendix 6.8 - E nergy b alance f o r C ase D Dis t anc e D epth dA u A ? dA u V A ? 2 ? g U 2 2 ? hH f h E h ? (f t) (f t) (cf s ) ( f t 5 /s 3 ) (f t) (f t) (f t) (f t) (f t) 7.4 0 .993 3.962 4.428 2.43 0.0189 1.2537 1.2726 0.0003 - 0 .000 12.5 0 .992 4.033 4.291 2.22 0.0174 1.2550 1.2724 0.0008 0.001 25.5 1 .015 4.195 3.664 1.80 0.0132 1.2573 1.2705 0.0007 0.000 37.2 1 .007 4.138 3.270 1.64 0.0123 1.2573 1.2696 0.0007 0.000 48.9 1 .011 4.094 2.924 1.53 0.0113 1.2575 1.2688 0.0006 - 0 .000 60.6 1 .039 4.121 2.562 1.42 0.0097 1.2588 1.2685 173 Appendix 6.9 - M om entum b alance f o r C ase F Dis t anc e D epth dA u A ? dA u A ? 2 ? ? QU h () 2 1 2 h h gA ? ? ? F M h ? (f t) (f t) (cf s ) ( f t 4 /s 2 ) (f t 4 /s 2 ) ( ft ) ( ft 4 /s 2 )( f t 4 /s 2 )( f t ) 2.5 0 .776 2.765 3.995 1.99 4.304 1.1920 - 0 .6095 0.0642 0.000 8.1 0 .776 2.546 2.887 1.70 3.670 1.1970 - 0 .7260 0.0646 - 0 .001 13.6 0 .773 2.618 2.534 1.40 3.045 1.2030 - 0 .6237 0.0632 - 0 .002 19.2 0 .787 2.791 2.430 1.22 2.570 1.2080 Appendix 6.10 - E nergy b alance f o r C ase F Dis t anc e D epth dA u A ? dA u V A ? 2 ? g U 2 2 ? hH f h E h ? (f t) (f t) (cf s ) ( f t 5 /s 3 ) (f t) (f t) (f t) (f t) (f t) 2.5 0 .776 2.765 5.588 3.83 0.0338 1.1920 1.2258 0.0005 - 0 .001 8.1 0 .776 2.546 3.740 3.28 0.0290 1.1970 1.2260 0.0005 0.003 13.6 0 .773 2.618 2.718 2.17 0.0194 1.2030 1.2224 0.0005 0.000 19.2 0 .787 2.791 2.336 1.63 0.0137 1.2080 1.2217 175 APPENDIX 7 - DATA FOR DIVERSION CULVERTS Appendix 7.1 - Results for diversion culverts with three barrels, unsubmerged flow Upstream discharge Diversion Upstream head on weir Down- stream head on weir Down- stream Froude number Down- stream weir Froude number Ratio of critical depth to depth at culvert outlet Loss coeffi cient from 0to2 Q u Q w h u h d F d Fw d ? c /? 4 K E (cfs) (cfs) (ft) (ft) 1.6 0.560 0.324 0.330 0.113 0.143 1 0.642 1.6 0.438 0.277 0.283 0.128 0.171 1 0.751 1.6 0.303 0.215 0.221 0.149 0.218 1 0.804 1.6 0.173 0.148 0.154 0.174 0.294 1 1.050 3.2 0.574 0.336 0.342 0.178 0.223 1 0.876 3.2 0.459 0.291 0.297 0.194 0.255 1 1.006 3.2 0.320 0.229 0.235 0.218 0.312 1 1.213 3.2 0.185 0.162 0.168 0.249 0.406 1 1.823 6.6 0.482 0.317 0.323 0.278 0.356 1 1.943 6.6 0.376 0.274 0.280 0.299 0.401 1 2.449 6.6 0.273 0.230 0.236 0.323 0.461 1 3.360 6.6 0.185 0.185 0.191 0.350 0.542 1 4.758 9.1 0.412 0.306 0.312 0.338 0.437 1 3.476 9.1 0.355 0.282 0.288 0.351 0.467 1 4.023 9.1 0.257 0.241 0.247 0.376 0.528 1 5.597 9.1 0.180 0.207 0.213 0.398 0.592 1 7.950 176 Appendix 7.2 - Results for diversion culverts with three barrels, submerged flow Upstream discharge Diversion Upstream head on weir Down- stream head on weir Down- stream Froude number Down- stream weir Froude number Ratio of critical depth to depth at culvert outlet Loss coeffi cient from 0to2 Q u Q w h u h d F d Fw d ? c /? 4 K E (cfs) (cfs) (ft) (ft) 1.6 0.584 0.341 0.347 0.109 0.135 0.887 0.816 1.6 0.187 0.187 0.193 0.162 0.251 0.583 1.838 3.2 0.546 0.333 0.339 0.179 0.225 0.983 1.080 3.2 0.188 0.192 0.198 0.236 0.360 0.645 3.145 6.5 0.457 0.317 0.323 0.277 0.355 0.946 2.257 6.6 0.190 0.212 0.218 0.334 0.492 0.703 6.273 9.1 0.408 0.315 0.321 0.334 0.427 0.923 3.734 9.2 0.188 0.224 0.230 0.388 0.560 0.760 8.731 Appendix 7.3 - Results for diversion culverts with two barrels, unsubmerged flow Upstream discharge Diversion Upstream head on weir Down- stream head on weir Down- stream Froude number Down- stream weir Froude number Ratio of critical depth to depth at culvert outlet Loss coeffi cient from 0to2 Q u Q w h u h d F d Fw d ? c /? 4 K E (cfs) (cfs) (ft) (ft) 1.6 0.381 0.334 0.337 0.121 0.153 1 0.672 1.6 0.273 0.267 0.270 0.139 0.190 1 0.722 1.6 0.194 0.214 0.217 0.156 0.230 1 0.850 3.2 0.372 0.329 0.332 0.186 0.236 1 0.776 3.2 0.261 0.263 0.266 0.209 0.287 1 0.997 3.2 0.190 0.217 0.220 0.228 0.335 1 1.300 6.6 0.314 0.317 0.320 0.283 0.363 1 1.894 6.6 0.169 0.229 0.232 0.326 0.470 1 3.503 9.1 0.284 0.312 0.315 0.337 0.435 1 3.065 9.1 0.178 0.252 0.255 0.371 0.516 1 5.044 177 Appendix 7.4 - Results for diversion culverts with two barrels, submerged flow Upstream discharge Diversion Upstream head on weir Down- stream head on weir Down- stream Froude number Down- stream weir Froude number Ratio of critical depth to depth at culvert outlet Loss coeffi cient from 0to2 Q u Q w h u h d F d Fw d ? c /? 4 K E (cfs) (cfs) (ft) (ft) 1.6 0.202 0.296 0.299 0.137 0.180 0.473 0.662 3.2 0.178 0.284 0.287 0.205 0.274 0.454 1.109 6.6 0.182 0.298 0.301 0.294 0.386 0.461 2.692 9.1 0.198 0.312 0.315 0.338 0.437 0.482 3.764