Defra / Environment Agency FCERM Joint Science Programme. SC030218/SR Afflux Estimation System. August 2007 HYDRAULIC REFERENCE

Defra / Environment Agency FCERM Joint Science Programme SC030218/SR Afflux Estimation System August 2007 HYDRAULIC REFERENCE South Barn Broughton Hall SKIPTON North Yorkshire BD23 3AE UK t: +44 (0)1756 799 919 f: +44 (0)1756 799 449 www.jbaconsulting.co.uk

REVISION HISTORY Revision Ref./ Date Issued Technical notes: 5 December 2005 AES Hydraulic Reference Notes August 2007 Amendments Issued to Andrew Pepper, Mervyn Bramley, Technical Review Group CES/AES web site CONTRACT These research notes describe work commissioned by under Science Group contract Ref. SC030218. s representative for the contract was Andrew Pepper of ATPEC. Prof. Peter Mantz, Dr. Serter Atabay, Dr. Rob Lamb, Steve Rose, Claire Packett and Jeremy Benn of carried out the work. Prepared by:... Peter Mantz, BSc, MSc, PhD, CEng, CPhys, MICE, MASCE = = = = = = mêáååáé~ä=^å~äóëí= Reviewed by:... Rob Lamb, MA PhD = = = = = = qéåüåáå~ä=aáêéåíçê= Approved by:...... Jeremy Benn, MA, MSc, CEng, FICE, FCIWEM, MASCE = = = = = = j~å~öáåö=aáêéåíçê= DISCLAIMER This document has been prepared solely as a series of research notes for. JBA Consulting accepts no responsibility or liability for any use that is made of this document or associated software other than by the Client for the purposes for which it was originally commissioned and prepared. www.jbaconsulting.co.uk i

ACKNOWLEDGMENTS We are grateful to the Defra/Environment Agency joint flood management research programme (Engineering Theme) for funding this study. In particular, we would like to acknowledge the support given to the project by Dr Mervyn Bramley OBE (formerly Theme Leader) and Andrew Pepper (External Theme Advisor). We are grateful for expert comment and review by Prof. Donald Knight, Dr Les Hamill, Prof. Nigel Wright, Dr Paul Samuels and Dr John Riddell. We are also grateful to the above for helping with the provision of data on afflux, and for having the opportunity to support a number of MSc laboratory projects at Birmingham University through this project. We would also like to thank Dr Galip Seckin for assistance with data. www.jbaconsulting.co.uk ii

EXECUTIVE SUMMARY This document This technical reference note describes the background and development of the Afflux Estimator. It has been compiled for circulation to the project Technical Review Group. The Afflux Estimation System (AES) Accurate estimation of flood water level underpins almost all sectors of flood risk management. A new support tool - the Afflux Estimation System (AES) for bridges and culverts - has been commissioned by the Environment Agency as part of the joint Defra/EA flood research programme to ensure that best available methods relevant to conditions in the UK are used for flood risk management. The Afflux Estimation System will comprise two main outputs: a simple Afflux Advisor providing quick, approximate calculations and guidance in an accessible spreadsheet form a more rigorous Afflux Estimator, produced as open source code, and available to be implemented as a software application or an additional component of existing river modelling packages. Afflux Estimator Afflux Estimator (AE) is being developed to model one dimensional water surface profiles at varied depths above and below a bridge or culvert located in a stream. The profiles are generated using the standard step method, which is based upon the principle of conservation of energy. Additionally, the code can model single bridge piers using pier drag coefficients together with the principle of conservation of momentum. To date, the bridge code is in place and sufficient testing has been undertaken to enable the development of AE with the Conveyance Estimation System software (CES, 2004). The AE model is designed for simplicity, therefore only three bridge reaches are used. These are the two transition reaches from the undisturbed stream sections upstream and downstream of the structure to the bridge (the flow contraction and expansion reaches respectively) and the bridge reach itself. The alongstream lengths of these transition reaches are estimated using a Transition Calculator. The latter computes algorithms derived from the downstream flow Froude number, the bridge Blockage ratio (the ratio of the area of blockage cause by the bridge and the total flow area at the bridge), and the relative floodplain width. The algorithms were derived from recent laboratory transition length measurements (Atabay and Knight, 2002), and field transition length data attained by computer simulation (HEC, 1995A). The transition calculator also estimates the associated, transition energy loss coefficients from the same data. There are four arch bridge types modelled in the code and two beam bridge types. The arch types are semicircular, parabolic, elliptical and user-defined. The beam types are continuous pier (for which the pier extends beneath the full bridge width) and single pier (for which the pier does not extend beneath the full bridge width). These model types have been tested by AE using the recent laboratory data from the University of Birmingham (Atabay and Knight, 2002; Seckin et al, 2004), and field data from large rivers surrounded by densely wooded floodplains (USGS, 1978). www.jbaconsulting.co.uk iii

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CONTENTS Page REVISION HISTORY CONTRACT EXECUTIVE SUMMARY CONTENTS LIST OF FIGURES LIST OF TABLES NOTATION i i iii v vi vii viii 1 PRELIMINARY ANALYSIS FOR TRANSITION LENGTHS -------------------------------------- 1 1.1 Summary...1 1.2 The HRC method...2 1.3 Transition length determinations for the UB (2002) data...3 1.4 Transition length analyses for the UB (2002) data...8 1.5 Transition length analyses for the HEC (1995) data...13 1.6 Transition length analyses for combined UB (2002) and HEC (1995) data...16 1.7 Discussion of the HEC (1995) transition length analyses...19 1.8 Design tables for transition lengths...21 1.9 Appendices to Chapter 1...24 2 BACKWATER FLOW THROUGH A CONTRACTION-------------------------------------------29 2.1 Summary...29 2.2 The standard step method...30 2.3 Uniform flow for the UB(2002) Case 2 laboratory channel...32 2.4 Uniform flow for the CES (2004) River Main...33 2.5 Uniform flow for the USCS (1978) Yellow River...34 2.6 Contracted flow for the UB(2002) Case 1 and Case 5 laboratory channel...36 2.7 Contracted flow for the CES (2004) River Main...39 2.8 Contracted flow for the USGS (1978) Yellow River...40 3 FLOW PROFILES FOR A SINGLE ARCH AND BEAM BRIDGE -------------------------------41 3.1 Summary...41 3.2 The Bridge Data Entry worksheet...44 3.3 Conveyance estimation...47 3.4 The Backwater code...50 3.5 Rating Comparisons for a single parabolic arch and single beam bridge...51 4 PHYSICAL CONSTRAINTS FOR BRIDGE DIMENSIONS--------------------------------------54 4.1 Summary...54 4.2 Combined Weir and Orifice modes...55 4.3 Single Arch and Beam bridge of 20 m span and high road level...57 4.4 Single Arch bridge of 10 m span and low road level...58 4.5 Single Arch bridge of 10 m span and very low soffit level...59 4.6 Single Arch bridge of eccentric 20 m span...60 4.7 Summary of physical constraint data...61 5 MULTI-SPAN BRIDGES--------------------------------------------------------------------------62 5.1 Summary...62 5.2 Numerically testing the Multi-arch code...64 5.3 Numerically testing the Multi-beam code...66 5.4 Comparison of AE and HEC-RAS discharge ratings...67 www.jbaconsulting.co.uk v

5.5 Summary of Multi-span data...70 6 BEAM BRIDGES WITH SINGLE PIERS ---------------------------------------------------------73 6.1 Summary...73 6.2 CMM used in HEC-RAS...75 6.3 CMMR used in AE...76 6.4 Pier drag coefficients (Cd)...77 6.5 CMMR applied to a 20 m span beam bridge...78 6.6 CMMR applied to a piered beam bridge compared with HEC-RAS modelling...80 7 AE CALIBRATION FOR LABORATORY SCALE BRIDGES ------------------------------------83 7.1 Summary...83 7.2 The AE model...84 7.3 AE calibration with UB data...86 7.4 Comparison with HEC-RAS...90 7.5 AE examples with UB data...91 7.6 Appendix to Chapter 7 (summary of afflux data)...94 8 AE CALIBRATION FOR FIELD SCALE BRIDGES ----------------------------------------------98 8.1 Summary...98 8.2 The AE model...99 8.3 Calibration principles for the transition parameters...103 8.4 Transition reach lengths...104 8.5 Transition reach energy coefficients...105 8.6 AE and HEC-RAS comparison for the Yellow River (USGS, 1978, Survey 610)...106 9 DEVELOPMENT AND VERIFICATION OF AE FOR USE WITH FIELD SURVEYS---------- 111 9.1 Summary...111 9.2 Arch bridge verification...115 9.3 Arch bridge results...119 9.4 Beam bridge verification...121 9.5 Appendix 1: Summary of USGS (1978) data for modelling water surface profiles in AE...123 10 COMPARISON OF AE CULVERT CODE WITH HEC-RAS MODELS------------------------ 127 10.1 Summary...127 10.2 Design of the AE code for culverts...129 10.3 Shallow culvert modelling...130 10.4 Comparison with HEC-RAS models for the shallow culverts...134 10.5 Deep culvert modelling...135 10.6 Comparison with HEC-RAS models for deep culverts...139 www.jbaconsulting.co.uk vi

LIST OF FIGURES Figure 1-1: Side elevation at a bridge contraction...2 Figure 1-2: Summary of Afflux data plotted on HRC dimensionless coordinates...2 Figure 1-3: Estimation of contraction lengths for the UB experiments...5 Figure 1-4: Estimation of contraction lengths for the UB Beam experiments...6 Figure 1-5: Estimation of expansion lengths for the PBEAM3 experiments...7 Figure 1-6: Estimation of expansion lengths for the AMOSC and BEAM1 experiments...8 Figure 1-7: Plot of Contraction length (Lc) against Afflux (dh) with linear trend...10 Figure 1-8: Plot of Expansion length (Le) against Afflux (dh) with possible linear trend...10 Figure 1-9: Variation of dimensionless contraction length with F3 and J3, and predicted trended length against measured length...11 Figure 1-10: Variation of dimensionless expansion length with F3 and J3, and predicted trended length against measured length...12 Figure 1-11: River cross section and data used in the HEC (1995) computational model length...13 Figure 1-12: Variation of dimensionless contraction length with F3 and J3 and predicted trended length against measured length...14 Figure 1-13: Variation of dimensionless expansion length with F3 and J3, and predicted trended length against measured length...15 Figure 1-14: Predicted against measured dimensionless contraction lengths (upper curve) and expansion lengths (lower curve) for the UB (open symbols) and HEC data...17 Figure 1-15: Predicted against measured dimensionless contraction lengths (upper curve) and expansion lengths (lower curve) for combined UB and HEC data...18 Figure 2-1: Plan and profile for flow through a river contraction...29 Figure 2-2: River reach for the standard step water surface computations...31 Figure 2-3: Uniform flow for the UB Case 2 channel...33 Figure 2-4: Uniform flow for the River Main...34 Figure 2-5: Uniform flow for the Yellow River...35 Figure 2-6: AE profile and Section worksheets for the UB Case 1 channel...37 Figure 2-7: AE profile and Section worksheets for the UB Case 5 channel...38 Figure 2-8: AE profile and Section worksheets for the CES (2004) River Main...39 Figure 2-9: AE profile and Section worksheets for the Yellow River...40 Figure 3-1: Subsoffit flow profile for a single arch bridge...42 Figure 3-2: Supersoffit flow profile for a single arch bridge...43 Figure 3-3: Bridge Data Entry Worksheet...43 Figure 3-4: Cross section data for Section 2D...44 Figure 3-5: Bridge and Culvert surface roughness coefficients...46 Figure 3-6: Bridge and Culver surface roughness coeffcients...46 www.jbaconsulting.co.uk vii

Figure 3-7: Section 2D for a multi-arch bridge...47 Figure 3-8: Section 1 worksheet...48 Figure 3-9: Section 2D conveyance rating...49 Figure 3-10: Extrapolation methods for AE and HEC-RAS...50 Figure 3-11: Comparison ratings for a single parabolic arch bridge (AE is the Structure curve)...52 Figure 3-12: Comparison ratings for a single beam bridge (AE is the Structure curve)...53 Figure 4-1:The Affluc Estimator application...55 Figure 4-2: The four modes of flow modelled in AE...56 Figure 4-3: Combined Weir and Orifice mode...57 Figure 4-4: Rating curves at Section 4 for the unstructured (Rating 4) and structured flow...58 Figure 4-5: Rating curves at Section 4 for an arch bridge of 10m span (low road level)...59 Figure 4-6: Rating curves at Section 4 for an arch bridge of 10m span (low soffit level)...60 Figure 4-7: Rating curves at Section 4 for an asymmetrical arch bridge of 20m span...60 Figure 5-1: Multi-span bridge examples...63 Figure 5-2: Multi-arch bridge models...65 Figure 5-3: Multi-beam bridge models...66 Figure 5-4: Twin Arch Model and Ratings...67 Figure 5-5: Twin Beam Model and Ratings...68 Figure 5-6: 20 Arch model and ratings...69 Figure 5-7: 20 Beam model and ratings...70 Figure 5-8: HEC-RAS profiles for the 20 Beam bridge...70 Figure 6-1: Beam bridge with continuous and single piers...74 Figure 6-2: Cross section near and inside a bridge (from HEC-RAS Hydraulic Reference, Fig 5.3)...75 Figure 6-3: Discharge ratings for a 20m beam bridge with varied piers and openings...79 Figure 6-4: AE Bridge Data Entry worksheet for an 11 opening, 20m Beam bridge...80 Figure 6-5: AE and RAS 10 pier models for a 20m span Beam bridge...81 Figure 6-6: AE and RAS 19 pier models for an 80m span Beam bridge...81 Figure 7-1: Correlation of computed and measured afflux...84 Figure 7-3: Water surface profile for a multiple arch bridge (from UB, 2002)...85 Figure 7-4: Cross section data used in AE...85 Figure 7-5: Transitional lengths for the UB data...87 Figure 7-6: Afflux comparisons for the subcritical flow computations...88 Figure 7-7: Affflux comparisons for the critical flow computations...89 Figure 7-8: Afflux comparisons for all flow computations...90 Figure 7-9: HEC-Ras and AE afflux comparisons for ASOE data...90 Figure 7-10: Discharge ratings for the ASOE Case 2 experiment...91 Figure 8-2: Water surface profile for a multiple arch bridge (from UB, 2002)...101 Figure 8-3: Submerged orifice flow modelled in Afflux Estimator...102 Figure 8-4: River reach for the standard step water surface computations...103 Figure 8-5: AE Transition calculator...106 www.jbaconsulting.co.uk viii

Figure 8-6: Section 1 worksheet in AE showing friction slope calibrated for Q=6.6m 3 /s...108 Figure 8-7: Bridge Data Entry for the USGS (1978) Survey 610...108 Figure 8-8: Discharge ratings for the Q=56.6 m 3 /s flood event...109 Figure 8-8: Discharge ratings for the Q=187.0 m 3 /s flood event...110 LIST OF TABLES Table 1-1: Summary of the number of experimental discharges for the UB normal bridge flows with measured transition lengths....4 Table 1-2: HEC Contraction Ratios (from HEC, 1995, Table 15)...19 Table 1-3: Trended Contraction Ratios...19 Table 1-4: Difference between HEC and trended Contraction Ratios...20 Table 1-5: HEC Expansion Ratios (From HEC 1995, Table 15)...20 Table 1-6: Trended Expansion Ratios...20 Table 1-7: Difference between HEC and trended Expansion Ratios...20 Table 1-8: Simplified design for transition lengths...21 Table 1-9: Simplified design for transition lengths...22 Table 1-10: Detailed design for dimensionless expansion lengths (Le/D3)...23 Table 2-1: Comparison of cross section statistics for the UB Case 2 channel...33 Table 2-2: Comparison of cross section statistics for the River Main...34 Table 2-3: Comparison of cross section statistics for the Yellow River...35 Table 4-1: Summary of examples...61 Table 5-4: Multi-span examples...71 Table 6-1: Coefficients used for pier geometry (from HEC-RAS V3, 2001)...77 Table 6-2: Coefficients used for pier geometry and 6:1 aspect ratio (from Montes, 1998)...78 Table 6-4: Multi-pier examples...82 Table 7-4: Bridge data entry for the Case 3 examples...93 Table 8-1: AE Transition data examples for laboratory and field scales...99 Table 8-2: Mean and observed Water surface elevations modelled in HEC-RAs (metric units)...100 Table 8-3: Water surface elevations modelled in HEC-RAS (from HEC, 1995B, feet units)...100 Table 8-4: Bridge flow modes for different codes (Zd is the downstream water level at the bridge and Zu is the upstream water level)...103 www.jbaconsulting.co.uk ix

NOTATION Mathematical symbols and notation have been used in the same way as in the original publications and are introduced in the text at each relevant equation. Since there is considerable variation in the notation used by different authors and organisations, this document uses the original notation in all places. Some symbols therefore have different meanings in different parts of the text. www.jbaconsulting.co.uk x

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1 PRELIMINARY ANALYSIS FOR TRANSITION LENGTHS 1.1 Summary A recent series of laboratory experiments which measured afflux for bridges that are normal to a compound channel flow has been conducted at the University of Birmingham, UB (Atabay and Knight, 2002). A total of 145 afflux measurements were made for bridge types of single semicircular openings (20 experiments, abbreviated ASOSC), multiple semi-circular openings (15 experiments, abbreviated AMOSC), single elliptical openings (15 experiments, abbreviated ASOE) and straight deck beam bridges with or without piers (95 experiments, abbreviated PBEAM and BEAM). A preliminary analysis of the total data was made using a dynamic similarity method first invoked at the University of Roorkee (Raju et al, 1983), and later by Hydraulics Research at Wallingford (Brown, 1988). A simplified version of this method involved expressing bridge afflux (dh) in dimensionless terms as follows: dh/d3 = f(f3, J3) where D3 is the normal flow depth without the presence of the structure, F3 is the Froude number of the flow without the presence of the structure, and J3 is a blockage ratio to the flow caused by the structure, and is given by the quotient: J3 = area of flow blocked by the downstream bridge section total flow area in the undisturbed channel These dimensionless variables were successfully applied to the UB (2002) laboratory data and to USGS field data (USGS, 1978) for use with the Afflux Advisor. Since the data were from compound channels, the variables are named herein as HRC dimensionless variables. They are now used for determining transition lengths for a structure located in a compound channel. The HEC (1995) model for a structural contraction in a subsoffit stream flow is illustrated for a bridge in Figure 1.1. Channel sections differ in nomenclature to that of the HRC model, and the downstream normal flow depth, y1 (D3 in HRC), is located at Section 1. There are sequentially 4 sections to the location of maximum backwater or afflux at Section 4. The alongstream distances between Sections 1 and 2 and Sections 3 and 4 are called the transition lengths. The former is a flow expansion length (Le) and the latter is a flow contraction length (Lc). The following notes analyse the transition lengths in terms of the HRC dimensionless variables. It is shown that Lc was almost a constant length for the UB (2002) data, and Le increased with afflux. Quadratic trending of the HRC variables were used to give predicted values as: Lc/D3 = f(j3, F3) ; Le/D3 = f(j3,f3) The HEC (1995) computational data were similarly analysed, and it was found that the effect of river scale was also a dominant variable. This scale variable was subjectively taken as the floodplain width, and a final predictive algorithm was derived for both narrow floodplain laboratory data (UB, 2002) and wide floodplain computational data (HEC, 1995). www.jbaconsulting.co.uk 1

Lc Bridge Le Backwater Flow Afflux y 4 y 3 y 2 y 1 Section 4 Section 3 Section 2 Section 1 Figure 1-1: Side elevation at a bridge contraction 1.2 The HRC method The HRC method is preferred as a similarity model for afflux hydraulics because the independent variables used are explicit and the dimensionless variables are physically bounded. The blockage ratio, J3 is dependent on the structure blockage area as a fraction of the total flow area. It is therefore specific to the structure. The Froude number (V3/ (gd3) 0.5 ) applies to the flow without the structure. It is therefore explicit to the undisturbed stream. The HRC method is simply expressing the fractional afflux as a function of the fractional structure blockage area and the Froude number of the undisturbed stream. 1.2 dh/d3 1.0 0.8 0.6 0.4 UB Low friction UB Medium friction USGS High friction HR Arch field data J3=0.9 J3=0.7 J3=0.5 0.2 J3=0.3 J3=0.1 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 F3 Figure 1-2: Summary of Afflux data plotted on HRC dimensionless coordinates www.jbaconsulting.co.uk 2

In contrast, the BPR similarity method uses a scale velocity that is the average velocity through the structure at normal depth (D3). This scale velocity is now an implicit variable which is also dependent on the other dependent variable J3 (which is a function of the BPR opening area). The BPR similarity method thus involves implicit variables which may lead to spurious correlations. The afflux data previously considered for the Afflux Advisor is summarised in terms of HRC variables in Figure 1.2. Both J3 and F3 have values from zero to unity. The plot is bounded at the abcissa by J3 = 0 and at the ordinate by J3 = 1. Hypothesised curves from about F3 = 0.8 bound the right side of the plot from the maximum J3 afflux conditions to F3 = 1. (For the latter condition of critical flow the afflux is theoretically zero, as indicated by the approximate HR Arch bridge data.) If the symbol hs represents the height of a structure soffit above bed level, then supersoffit flow (called the soffit threshold) begins approximately when: hs = dh +D3 or dh/d3 = (hs/d3) - 1 An approximate estimate for hs can be made using bridge design criteria. It is sometimes recommended that the soffit level should exceed the 200 year flood level (TAC, 2004). This flood level may be approximated as about twice the normal depth. The soffit threshold is thus about dh/d3 = 1, and this line provides the upper bound for Figure 1.2. The HRC plot is therefore applicable for subsoffit flows only. Note that with the present Afflux Advisor, the J3 curves are discontinuous at F3 = 0.1(being optimised for USGS field data) and F3 = 0.25 (being optimised for the medium friction UB data). These discontinuities may be smoothed in the future with an associated small loss in the accuracy of afflux prediction. Since the HRC method has successfully predicted afflux for the entire subsoffit flow through a structure, it is now tested on UB transition length data. 1.3 Transition length determinations for the UB (2002) data The available flow discharges for the determination of transition lengths are summarised in Table 1.1. For many experiments, supercritical flow conditions occurred through the structure and thus a flow control was located at the structure as well as downstream. For some cases, a hydraulic jump occurred and the flow was also unsteady. An expansion length could not be measured for these conditions (AMOSC-Case 1 abbreviated to AMOSC1, ASOE1, ASOSC1). For the remaining flows, the transition levels were measured at 0.1 m increments downstream for large water surface gradients (such as at standing waves) and at 1 m increments downstream for small gradients. All water level measurements were made along the channel centreline. Section 3 was located at the 58.965 m mark from upstream and Section 2 at the 59.085 m mark. Both sections were therefore taken as the 59.0 m mark for simplicity in transition length estimation. The complete Lc measurements are illustrated in Figures 1.3 and 1.4. For objectivity, Section 4 was taken as the first maximum elevation upstream of the 59.0 m bridge section position. This method contrasted with the method used in HEC (1995), which used maximum water surface contours over the entire cross section. Representative examples for the expansion length measurements are given in Figures 1.5 and 1.6. The use of a 2 point moving average trend for the water levels was found adequate to dampen the standing waves in this reach, and to establish an objective intersect with the normal water level profile. Figure 1.5 (piered beam experiments) illustrates the smallest Le of about 2 m encountered for a high friction (and low afflux) condition. Figure 1.6 (AMOSC2 experiments) illustrate a medium Le of about 4 m, and the largest Le of about 7 m is illustrated for the lowest friction (and highest afflux) BEAM1 experiments. Note that for the latter measurements only, a moving average trend of more than 2 points was sometimes necessary for an objective assessment. This method for Le determination differed considerably from the HEC (1995) method. For the latter, velocity vector plots were used to establish uniformity of the flow for Section 1. Furthermore, the Le estimated herein have expansion ratios (defined in HEC, 1995 as Le/Lobs, where Lobs is the average obstruction length equal to 0.5*(floodplain width bridge width)) that are much larger than the HEC (1995) values. It is however considered below that the differences can be explained in terms of the different scale and friction of the floodplain widths. Note that the objective estimates for the UB (2002) transition length data used herein are considered unambiguous. www.jbaconsulting.co.uk 3

Table 1-1: Summary of the number of experimental discharges for the UB normal bridge flows with measured transition lengths. = Case 1 Case 2 Case 3 Case 4 Case 5 oçìöüåéëë= j~áå=åü~ååéä=j~ååáåöûë=å= MKMN= MKMNR= MKMOU= MKMOV= MKMPO= cäçççéä~áå=j~ååáåöûë=å= MKMMV= MKMP= MKMQT= MKMSV= MKMUM= Bridge Type Bridge code Multiple semi-circular AMOSC 5 (Lc) 5 (Lc, Le) 5 (Lc, Le) arch Single elliptical arch ASOE 5 (Lc) 5 (Lc, Le) 5 (Lc, Le) Single semi-circular ASOSC 5 (Lc) 5 (Lc, Le) 5 (Lc, Le) 5 (Lc, Le) arch Beam (Span = 0.398) BEAM-4 5 (Lc, Le) 5 (Lc, Le) 5 (Lc, Le) 5 (Lc, Le) Beam (Span = 0.498) BEAM-5 5 (Lc, Le) 5 (Lc, Le) 5 (Lc, Le) Beam (Span = 0.598) BEAM-6 5 (Lc, Le) 5 (Lc, Le) 5 (Lc, Le) Piered Beam (Span = PBEAM-4 5 (Lc, Le) 5 (Lc, Le) 5 (Lc, Le) 0.398) Piered Beam (Span = PBEAM-5 5 (Lc, Le) 5 (Lc, Le) 5 (Lc, Le) 0.498) Piered Beam (Span = PBEAM-6 5 (Lc, Le) 5 (Lc, Le) 5 (Lc, Le) 0.598) kçíéw== The total number of experiments with measured contraction length Lc is 145, and with measured expansion length Le is 130. www.jbaconsulting.co.uk 4

Figure 1-3: Estimation of contraction lengths for the UB experiments (The abcissa is the centreline channel position of measurement from upstream in metres, the ordinate is the water level elevation in millimetres. The small black square denotes the contraction length upstream limit.) www.jbaconsulting.co.uk 5

Figure 1-4: Estimation of contraction lengths for the UB Beam experiments (The abcissa is the centreline channel position of measurement from upstream in metres, the ordinate is the water level elevation in millimetres for both normal and bridge flows.) www.jbaconsulting.co.uk 6

Figure 1-5: Estimation of expansion lengths for the PBEAM3 experiments (The abcissa is the centreline channel position of measurement from upstream in metres, the ordinate is the water level elevation in millimetres for both normal and bridge flows.) www.jbaconsulting.co.uk 7

Figure 1-6: Estimation of expansion lengths for the AMOSC and BEAM1 experiments (The abcissa is the centreline channel position of measurement from upstream in metres, the ordinate is the water level elevation in millimetres for both normal and bridge flows.) 1.4 Transition length analyses for the UB (2002) data The total data for transition lengths against HRC variables are summarised in Appendix 1.1. As a first appraisal, both Lc and Le were compared against afflux (dh) in Figures 1.7 and 1.8. It was seen that Lc was approximately constant with afflux thus: Lc = 0.50 ± 0.02 m at the 95% confidence level Since the average obstruction lengths, Lobs were mainly about 0.4 m for these data, the contraction ratio (Lc/Lobs) was about 1.4 (This value was similar to those predicted in HEC, 1995). In contrast, Le increased with afflux and a probable power law trend would give a higher correlation than the possible linear trend shown. www.jbaconsulting.co.uk 8

Physically, these results imply that the main flow control is at the structure. The afflux at Section 4 may be considered as an increased reservoir level at an approximately constant distance from the structure. The downstream flow expansion occurs further downstream with increased afflux, as the structure control increasingly dominates the downstream control. In accord with the afflux method, the dimensionless contraction lengths, Lc/D3 and Le/D3, were tested against the HRC variables in Figures 1.9 and 1.10. Figure 1.9 first illustrates an approximate linear correlation of Lc/D3 increasing with F3 alone and decreasing with J3 alone. The F3 trend was meaningful since it was expected that Lc approached zero as F3 approached zero (i.e. still water). And the J3 trend indicated that Lc approached zero as J3 approached 1 (i.e. the flow was totally blocked). A quadratic surface trendline was applied to the 3 dimensionless variables, and the coefficient of determination was reasonably high at 0.60. The parity between the measured and predicted results was of the same magnitude. The results were similar for the Le/D3 analysis. Figure 1.10 illustrates an approximate linear correlation of Le/D3 increasing with F3 alone and decreasing with J3 alone. The F3 trend was meaningful since it was expected that Lc approached zero as F3 approached zero (i.e. still water). And the J3 trend indicated that Lc approached zero as J3 approached 1 (i.e. the flow was totally blocked). A quadratic surface trendline was applied to the 3 dimensionless variables, and the coefficient of determination was high at 0.88. The parity between the measured and predicted results was of the same magnitude. www.jbaconsulting.co.uk 9

0.8 0.7 0.6 0.5 0.4 0.3 0.2 Lc: m y = -11x + 0.5116 R 2 = 0.0258 0.1 0.0 dh: mm 0 10 20 30 40 50 60 70 80 Figure 1-7: Plot of Contraction length (Lc) against Afflux (dh) with linear trend 12 10 Le: m y = 0.1227x + 1.8321 R 2 = 0.5796 8 6 4 2 0 dh: mm 0 10 20 30 40 50 60 70 80 Figure 1-8: Plot of Expansion length (Le) against Afflux (dh) with possible linear trend www.jbaconsulting.co.uk 10

Lc/D3 Lc/D3 10 Case 3, 4, 5 Case 2 Case 1 9 8 7 y = 5.3292x + 2.9091 R 2 = 0.3851 6 5 4 3 2 1 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 F3 10 y = -9.9802x + 9.3834 R 2 = 0.2388 9 8 7 6 5 4 3 2 1 0 0.0 0.1 0.2 0.3 0.4 J3 0.5 0.6 0.7 0.8 0.9 1.0 10 9 8 y = 0.608x + 1.7063 R 2 = 0.608 Predicted 7 6 5 4 3 2 1 0 Measured 0 1 2 3 4 5 6 7 8 9 10 Figure 1-9: Variation of dimensionless contraction length with F3 and J3, and predicted trended length against measured length www.jbaconsulting.co.uk 11

140 120 y = 174.2x - 12.066 R 2 = 0.8762 100 Le/D3 80 60 40 20 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 F3 140 120 y = -81.056x + 69.1 R 2 = 0.064 100 Le/D3 80 60 40 20 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 J3 140 120 y = 0.8815x + 3.2342 R 2 = 0.8815 Predicted 100 80 60 40 20 0 0 20 40 60 80 100 120 140 Measured Figure 1-10: Variation of dimensionless expansion length with F3 and J3, and predicted trended length against measured length www.jbaconsulting.co.uk 12

1.5 Transition length analyses for the HEC (1995) data A 2DH computational procedure was used in HEC (1995) for estimating transition lengths. Firstly, 5 flood events from the USGS (1978) river afflux data were calibrated for use with the 2DH model. The associated variables and dimensions were then used to simulate a single, symmetrically shaped river model for various conditions of bridge opening width (b), discharge (Q), overbank and main channel (n = 0.04 throughout) Mannings coefficients and bedslopes (Figure 1.11). A total of 76 computations were conducted using spill-through abutments and 11 computations for vertical abutments. The variables used for the spill through abutments were converted to SI units and are enumerated in Appendices 1.2, 1.3 and 1.4. Afflux Advisor was then used to rapidly compute the associated HRC dimensionless variables, and these are also listed in the Appendices. An HRC analysis was then conducted as for the UB (2002) data above. The Lc/D3 linear correlation against F3 and D3 is illustrated in Figure 1.12. Note that the F3 range is smaller than that used for the UB laboratory data. However, Lc/D3 increased with F3 and approached zero as F3 approached zero (i.e. to the still water condition). The overall Lc/D3 correlation with J3 is more scattered than that for the UB data. This is because the data consisted almost equally of 3 different bridge opening widths (b). A further analysis showed that trendlines through each of the separate bridge width data approached zero as J3 approached zero (i.e. to complete blockage). A quadratic surface trendline was applied to the 3 dimensionless variables, and the coefficient of determination was moderate at 0.33. The parity between the measured and predicted results was of the same magnitude. Similar comments apply to the Le/D3 analysis (Figure 1.13), however the quadratic trending coefficient of determination was significantly improved at 0.53. Figure 1-11: River cross section and data used in the HEC (1995) computational model length www.jbaconsulting.co.uk 13

45 40 y = 72.691x + 18.261 R 2 = 0.2745 35 30 Lc/D3 25 20 15 10 5 0 F3 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 45 40 y = -3.483x + 28.754 R 2 = 46 35 30 Lc/D3 25 20 15 10 5 0 J3 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 45 40 y = 0.3251x + 17.957 R 2 = 0.3251 35 Predicted 30 25 20 15 10 5 0 0 10 20 30 40 50 Measured Figure 1-12: Variation of dimensionless contraction length with F3 and J3 and predicted trended length against measured length www.jbaconsulting.co.uk 14

140 120 y = 57.906x + 30.761 R 2 = 0.0557 100 Le/D3 80 60 40 20 0 F3 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 140 120 y = 53.41x + 4.4852 R 2 = 0.3467 100 Le/D3 80 60 40 20 0 J3 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 140 120 y = 0.5247x + 17.78 R 2 = 0.5247 100 Predicted 80 60 40 20 0 0 20 40 60 80 100 120 140 Measured Figure 1-13: Variation of dimensionless expansion length with F3 and J3, and predicted trended length against measured length www.jbaconsulting.co.uk 15

1.6 Transition length analyses for combined UB (2002) and HEC (1995) data Although the separate UB and HEC data used different methods for estimating transition lengths, an attempt was made to combine all data using the HRC variables. This gave coefficients of determination of 0.28 for Lc/D3 and 0.66 for Le/D3. It was clear that the data required a fourth dimensionless variable. The obvious additional variable was that of flow scale, since small laboratory flows were being combined with large field conditions. Since both data sets used a single floodplain width (B = 1.2 m and 305 m respectively), the scale variable B/Bmean (B/Bm) was considered appropriate. The combined transition data are first illustrated with their respective quadratic trends in Figure 1.14. The floodplain width scale is immediately apparent for the Lc/D3 plot. Although a correspondence of combined Le/D3 predictions with the HRC variables occurs, the curve is significantly extended with the low friction, laboratory results at Le/D3 = 100. The algorithm chosen to represent the combined data sets used a simple linear interpolation between the quadratic trend coefficients of each data set since there was no other available data. For Lc/D3, the algorithm was: b0 = 2.8831 * B/Bm - 8.8777 b1 = -32.569 * B/Bm + 50.768 b2 = 28.995 * B/Bm + 26.554 b3 = 86.445 * B/Bm - 36.168 b4 = -25.9 * B/Bm - 20.402 b5 = 63.93 * B/Bm - 39.097 Lc_D3 = b0 + b1 * F3 + b2 * J3 + b3 * F3 ^ 2 + b4 * J3 ^ 2 + b5 * F3 * J3 And for Le/D3, the algorithm was: b0 = 10.882 * B/Bm - 26.205 b1 = -44.927 * B/Bm + 197.8 b2 = 8.6719 * B/Bm + 15.198 b3 = -133.36 * B/Bm - 22.215 b4 = 2.6434 * B/Bm + 9.8698 b5 = 60.188 * B/Bm - 2.8248 Le_D3 = b0 + b1 * F3 + b2 * J3 + b3 * F3 ^ 2 + b4 * J3 ^ 2 + b5 * F3 * J3 The combined data are illustrated in Figure 1.15, and the respective coefficients of determination were 0.89 and 0.80 for contraction and expansion lengths respectively. The high correlations imply that either dimensionless transition length may be computed to be within about 4% of its mean value (at the 95% confidence level). These algorithms may therefore be applied to all floodplain widths between about 1 and 1000 m. Inevitably, there is a need for intermediate field data for further verification and analyses. www.jbaconsulting.co.uk 16

45 40 y = 0.3251x + 17.957 R 2 = 0.3251 35 30 Predicted 25 20 15 10 5 0 y = 0.608x + 1.7063 R 2 = 0.608 0 10 20 30 40 Measured 120 100 y = 0.8815x + 3.2342 R 2 = 0.8815 Predicted 80 60 40 20 y = 0.5247x + 17.78 R 2 = 0.5247 0 0 20 40 60 80 100 120 Measured Figure 1-14: Predicted against measured dimensionless contraction lengths (upper curve) and expansion lengths (lower curve) for the UB (open symbols) and HEC data www.jbaconsulting.co.uk 17

45 40 y = 0.9528x R 2 = 0.8954 35 30 Predicted 25 20 15 10 5 0 0 10 20 30 40 50 Measured 140 120 y = 0.9533x R 2 = 0.8015 100 Predicted 80 60 40 20 0 0 20 40 60 80 100 120 140 Measured Figure 1-15: Predicted against measured dimensionless contraction lengths (upper curve) and expansion lengths (lower curve) for combined UB and HEC data www.jbaconsulting.co.uk 18

1.7 Discussion of the HEC (1995) transition length analyses 1.7.1 Contraction lengths The procedure used in HEC (1995) to determine transition lengths involved a multiple regression analysis of dimensionless flow variables (Tables 14 and 15, HEC, 1995). Since the quadratic trending of 3 variables is a better method than multiple regression for interpolation and extrapolation, the HEC (1995) results are reproduced herein and discussed using quadratic trending. Initially, the independent variables were considered to be the flow discharge (Q), the bridge opening width (b), the average stream bedslope (S) and the overbank Mannings coefficient (nob). They are enumerated in Appendices 1.2, 1.3 and 1.4. The following relation (which included constants of the computations) was proposed for the contraction ratio (CR): CR = Lc/Lobs = f(q/qm, nob/nmc, S) = f(d, N, S) where Lobs is the average obstruction length equal to 0.5*(B-b) for a beam bridge, Qm is the average discharge, and nmc is the constant main channel friction coefficient (nmc = 0.04). Table 15 from HEC (1995) is reproduced in Table 1.3, and mean CR values are added to amplify the data. A quadratic trend analysis was first made on the variables for constant S, then the trend coefficients were varied with S to give the following algorithm for CR: b0 = 182310 * S ^ 2-590.51 * S + 1.6519 b1 = 122549 * S ^ 2-94.026 * S + 0.5384 b2 = -172997 * S ^ 2 + 370.97 * S - 0.5456 b3 = -43907 * S ^ 2 + 102.18 * S - 0.0367 b4 = 28340 * S ^ 2-48.574 * S + 0.088 b5 = 8133.3 * S ^ 2-73.402 * S - 0.0844 CR = b0 + b1 * D + b2 * N + b3 * D ^ 2 + b4 * N ^ 2 + b5 * D * N This CR algorithm was used to regenerate Table 1.2 as Table 1.3, and the small differences are enumerated in Table 1.4. Note that the differences average about 0.1 for the total data. Although this dimensionless representation is recommended by HEC (1995) for non-iterative use, it has the deficiency that Q is a function of S, and these variables are therefore not independent. Table 1-2: HEC Contraction Ratios (from HEC, 1995, Table 15) N=1 N=2 N=4 S D=0.3 D=1.0 D=1.7 D=0.3 D=1.0 D=1.7 D=0.3 D=1.0 D=1.7 02 1.0 1.7 2.3 0.8 1.3 1.7 0.7 1.0 1.3 1 1.0 1.5 1.9 0.8 1.2 1.5 0.7 1.0 1.2 2 1.0 1.5 1.9 0.8 1.1 1.4 0.7 1.0 1.2 Table 1-3: Trended Contraction Ratios N=1 N=2 N=4 S D=0.3 D=1.0 D=1.7 D=0.3 D=1.0 D=1.7 D=0.3 D=1.0 D=1.7 02 1.3 1.5 1.8 1.0 1.2 1.4 0.9 1.0 1.1 1 1.1 1.4 1.7 0.9 1.1 1.3 0.9 0.9 0.9 2 1.0 1.4 1.9 0.8 1.1 1.4 0.9 0.9 0.9 www.jbaconsulting.co.uk 19

Table 1-4: Difference between HEC and trended Contraction Ratios N=1 N=2 N=4 S D=0.3 D=1.0 D=1.7 D=0.3 D=1.0 D=1.7 D=0.3 D=1.0 D=1.7 02-0.3 0.1 0.5-0.2 0.0 0.3-0.2 0.0 0.2 1-0.1 0.0 0.2-0.1 0.0 0.2-0.2 0.0 0.3 2 0.0 0.0 0.0 0.0 0.0 0.0-0.2 0.0 0.3 1.7.2 Expansion lengths A similar analysis by HEC (1995) was conducted for expansion ratios, except that the added variable b/b was used to regress the data (Table 1.5), thus: ER = Le/Lobs = f(q/qbar, nob/nmc, S, b/b) = f(d, N, S, b/b) Since Lobs = 0.5*(B-b) = 0.5*B(1 b/b) and B was constant for the data, the addition of the b/b variable was unnecessary. The data were therefore trended as for the CR data, and gave the following algorithm: b0 = -258.06 * S + 1.4919 b1 = 946620 * S ^ 2-1950 * S + 2.0133 b2 = -199333 * S ^ 2 + 395.4 * S - 0.5157 b3 = -367172 * S ^ 2 + 533.22 * S - 0.2526 b4 = 37065 * S ^ 2-80.601 * S + 0.1063 b5 = 158.85 * S - 0.3047 ER = b0 + b1 * D + b2 * N + b3 * D ^ 2 + b4 * N ^ 2 + b5 * D * N This ER algorithm was used to regenerate Table 1.5 as Table 1.6, and the small differences are enumerated in table 1.7. Note that the differences average about 0.2 for the total data. Furthermore, Table 1.6 is simpler to use. Table 1-5: HEC Expansion Ratios (From HEC 1995, Table 15) N=1 N=2 N=4 b/b S D=0.3 D=1.0 D=1.7 D=0.3 D=1.0 D=1.7 D=0.3 D=1.0 D=1.7 b/b = 02-0.3 0.1 0.5-0.2 0.0 0.3-0.2 0.0 0.2 0.1 1-0.1 0.0 0.2-0.1 0.0 0.2-0.2 0.0 0.3 2 0.0 0.0 0.0 0.0 0.0 0.0-0.2 0.0 0.3 b/b = 02 1.6 2.3 3.0 1.4 2.0 2.5 1.2 1.6 2.0 0.25 1 1.5 2.0 2.5 1.3 1.7 2.0 1.3 1.7 2.0 b/b = 0. 5 2 1.5 1.8 2.0 1.3 1.7 2.0 1.3 1.7 2.0 02 1.4 2.0 2.6 1.3 1.6 1.9 1.2 1.3 1.4 1 1.3 1.7 2.1 1.2 1.4 1.6 1.0 1.2 1.4 2 1.3 1.7 2.0 1.2 1.4 1.5 1.0 1.2 1.4 Table 1-6: Trended Expansion Ratios N=1 N=2 N=4 S D=0.3 D=1.0 D=1.7 D=0.3 D=1.0 D=1.7 D=0.3 D=1.0 D=1.7 02 1.5 2.3 3.0 1.2 1.9 2.3 1.3 1.5 1.6 1 1.2 1.8 2.2 1.1 1.5 1.8 1.1 1.3 1.4 2 1.1 1.8 1.9 0.8 1.6 1.7 0.9 1.7 1.8 Table 1-7: Difference between HEC and trended Expansion Ratios www.jbaconsulting.co.uk 20

N=1 N=2 N=4 b/b S D=0.3 D=1.0 D=1.7 D=0.3 D=1.0 D=1.7 D=0.3 D=1.0 D=1.7 b/b = 0.1 02-0.1 0.2 0.6 0.1 0.3 0.7-0.1 0.1 0.5 1-0.2 0.0 0.3-0.3-0.1 0.2-0.3 0.1 0.6 2-0.1-0.2 0.3 0.0-0.2 0.3-0.1-0.3 0.2 b/b = 02 0.1 0.0 0.0 0.2 0.1 0.2-0.1 0.2 0.5 0.25 1 0.3 0.2 0.3 0.2 0.2 0.2 0.2 0.2 0.4 b/b = 0.5 2 0.4-0.1 0.1 0.5 0.1 0.3 0.4 0.1 0.3 02-0.1-0.3-0.4 0.1-0.3-0.4-0.1-0.1-0.1 1 0.1-0.1-0.1 0.1-0.1-0.2-0.1-0.3-0.2 2 0.2-0.2 0.1 0.4-0.2-0.2 0.1-0.4-0.3 1.8 Design tables for transition lengths 1.8.1 The HRC transition analyses In general, the HRC analyses accurately predict the transition lengths as: Lc/D3 or Le/D3 = f(j3, F3, B/Bm) However, the use of the flow scale (B/Bm) is considered herein as a mathematical convenience, since the general friction conditions for both data sets varied. For the UB (2002) data, the average main channel Mannings friction coefficient (nmc) was 0.025 and the floodplain coefficient (nfp) was 0.040. For the HEC (1995) data, the average nmc was 0.052 and the average nfp was 0.096. It was thus inevitable that the expansion ratio (Le/Lobs) was larger for the lower friction conditions of the UB data, since the decreased friction gave lower lateral turbulent viscosity coefficients. Further data coupled with computer analyses are required to improve the HRC physical model towards using a friction condition rather than the B/Bm variable. Meanwhile, the existing analyses are proposed for present design. 1.8.2 Simplified transition length design It was noted above that the UB (2002) contraction lengths (Lc) were near constant, with a 95% confidence limit (CL) of about 5% of the average magnitude. Similar average statistics were applied to both the UB and HEC data (Table 1.8). The CL for Lc were similar at 5%, and for Le they were about 10% of the absolute magnitudes. A simplified transition length, depending upon floodplain width (B) only, may thus be computed from these data using the prediction equations in Table 1.8. Note that these values are approximate, and a more detailed design is recommended below. Table 1-8: Simplified design for transition lengths B Lc Le Simple prediction 1.2 0.50 3.1 Lc = 0.38B + 0.04 305.0 117.0 171.0 Le = 0.55B + 2.4 95% CL 5% 10% 1.8.3 Detailed transition length design The HRC algorithms for dimensionless transition length design (Lc/D3 and Le/D3) are reproduced for floodplain widths (B) of 1 m, 10 m, 100 m and 1000 m in Tables 1.10 and 1.11. The scale of B = 1 m applies to laboratory flows, and that of B = 1000 m applies to flooded estuaries for the UK. Intermediate values may be obtained by linear interpolation, since a linear scale was used for the algorithm. For design cases where normal flow data are not readily available (such as that for the HEC, 1995 data), Afflux Advisor may be used to rapidly compute such flows from the Mannings friction conditions. The extrapolation of quadratic surface trends may sometimes lead to negative values at the extremities. Any tabulated negative data has simply been replaced herein with zeroes, since that is www.jbaconsulting.co.uk 21

the data trend. For the B = 1000 m expansion length table, most data for FR > 0.3 was negative, and this occurred because the HEC computations did not extend beyond this Froude number. Inevitably, high Froude number flows for such wide and deep floodplains lead to unrealistically extreme flows. It is finally noted that the HRC transition algorithms, or Tables 1.9 and 1.10, may be used without further detailed iteration such as required by the HEC (1995) method. Furthermore, the different laboratory and field transition lengths necessitate the use of a HEC-RAS model to determine expansion and contraction energy loss coefficients for all of the UB experiments. As a first approach, it is expected that the UB contraction energy loss coefficient (Cc) will be similar to that of the HEC (1995) value of Cc = 0.1, since Lc/Lobs is similar. Initial calibration may therefore begin using this value, and the expansion energy loss coefficient adjusted according to the HRC expansion length and associated afflux. Table 1-9: Simplified design for transition lengths B = 1.0 m B = 10 m J3 F3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 J3 F3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0 2 4 7 8 9 9 8 7 0.1 0 2 5 7 9 10 11 11 10 0.2 0 3 5 7 8 9 8 7 5 0.2 0 3 6 8 9 10 10 10 9 0.3 1 4 6 7 8 8 7 6 4 0.3 1 4 7 8 9 10 10 9 7 0.4 2 4 6 7 7 7 6 4 1 0.4 2 5 7 8 9 9 8 7 6 0.5 2 4 6 6 6 5 4 2 0 0.5 3 5 7 8 8 8 7 5 3 0.6 2 4 5 5 5 3 1 0 0 0.6 3 5 6 7 7 6 5 3 1 0.7 2 3 4 3 3 1 0 0 0 0.7 3 4 5 5 5 4 2 0 0 0.8 1 2 2 2 0 0 0 0 0 0.8 2 3 3 3 3 1 0 0 0 0.9 0 0 0 0 0 0 0 0 0 0.9 0 1 1 1 0 0 0 0 0 B = 100 m B = 1000 m J3 F3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 J3 F3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0 4 8 12 17 22 28 34 40 0.1 23 26 40 65 100 146 202 269 346 0.2 4 7 11 16 21 26 32 38 44 0.2 42 49 67 96 135 184 244 315 396 0.3 7 10 14 19 24 29 34 41 47 0.3 58 69 91 123 166 219 283 357 442 0.4 8 12 16 21 26 31 37 43 49 0.4 70 85 110 146 193 250 318 396 485 0.5 10 13 18 22 27 32 38 44 51 0.5 79 97 126 166 216 277 349 431 523 0.6 10 14 18 23 28 33 39 45 51 0.6 83 106 138 182 236 301 376 462 558 0.7 10 14 18 22 27 33 39 45 51 0.7 84 110 147 194 252 320 400 489 589 0.8 9 13 17 21 26 32 38 44 50 0.8 81 111 151 202 264 336 419 513 617 0.9 7 11 15 20 25 30 36 42 49 0.9 74 108 152 207 272 348 435 532 640 www.jbaconsulting.co.uk 22

Table 1-10: Detailed design for dimensionless expansion lengths (Le/D3) B = 1.0 m B = 10 m J3 F3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 J3 F3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0 14 33 51 68 86 102 119 134 0.1 0 14 32 49 66 82 98 112 127 0.2 0 16 34 52 70 87 104 120 136 0.2 0 16 34 51 68 84 99 114 129 0.3 0 18 36 54 72 89 106 122 138 0.3 0 18 36 53 70 86 102 116 131 0.4 1 20 38 56 74 91 108 124 140 0.4 2 20 38 56 72 88 104 119 133 0.5 3 22 41 59 76 93 110 126 142 0.5 4 23 41 58 75 91 107 121 136 0.6 6 25 43 61 79 96 112 129 144 0.6 7 25 43 61 78 94 109 124 138 0.7 9 28 46 64 82 99 115 131 147 0.7 10 28 46 64 81 97 112 127 141 0.8 12 31 49 67 84 101 118 134 150 0.8 13 31 49 67 84 100 115 130 145 0.9 15 34 52 70 87 104 121 137 153 0.9 16 35 53 70 87 103 119 134 148 B = 100 m B = 1000 m J3 F3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 J3 F3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0 13 25 34 42 47 50 51 49 0.1 38 5 0 0 0 0 0 0 0 0.2 2 16 28 38 46 52 55 56 55 0.2 50 21 0 0 0 0 0 0 0 0.3 5 20 32 42 51 56 60 62 61 0.3 62 37 0 0 0 0 0 0 0 0.4 8 23 36 47 55 62 66 68 67 0.4 75 54 16 0 0 0 0 0 0 0.5 12 27 40 51 60 67 71 74 74 0.5 89 72 37 0 0 0 0 0 0 0.6 16 31 45 56 65 72 77 80 80 0.6 103 90 59 10 0 0 0 0 0 0.7 20 36 50 61 71 78 83 86 87 0.7 117 108 81 37 0 0 0 0 0 0.8 24 40 54 67 76 84 90 93 94 0.8 133 127 104 63 5 0 0 0 0 0.9 28 45 60 72 82 90 96 100 102 0.9 148 147 128 91 36 0 0 0 0 www.jbaconsulting.co.uk 23

1.9 Appendices to Chapter 1 Appendix 1.1: Contraction and expansion lengths (Lc, Le) for the UB (2002) normal bridge flow data with HRC dimensionless variables C a s e Experiment Discharge F3 J3 dh/d3 Lc Le Lc/D3 Le/D3 Name (l/s) m m 1 AMOSC121 20.97 0.64 0.37 0.52 0.3 4.48 AMOSC124 24.02 0.65 0.40 0.61 0.4 5.69 AMOSC127 27.04 0.66 0.42 0.69 0.6 8.17 AMOSC130 29.98 0.67 0.44 0.76 0.5 6.56 AMOSC135 34.43 0.69 0.46 0.87 0.5 6.24 2 AMOSC218 18.03 0.46 0.42 0.24 0.6 4.7 8.30 64.98 AMOSC221 20.99 0.41 0.47 0.25 0.5 3.2 6.21 39.74 AMOSC224 24.08 0.38 0.50 0.25 0.5 4.0 5.60 44.82 AMOSC230 29.97 0.33 0.56 0.24 0.6 6.0 5.65 56.46 AMOSC235 34.28 0.31 0.59 0.25 0.5 7.0 4.20 58.81 3 AMOSC315 14.97 0.24 0.50 0.08 0.5 1.9 5.66 21.51 AMOSC318 18.00 0.23 0.54 0.09 0.4 2.5 4.02 25.13 AMOSC321 20.89 0.22 0.57 0.11 0.5 1.9 4.53 17.22 AMOSC324 24.04 0.20 0.60 0.11 0.6 2.3 4.91 18.81 AMOSC327 26.82 0.19 0.62 0.12 0.5 2.2 3.76 16.54 1 ASOE121 20.97 0.64 0.34 0.46 0.5 7.47 ASOE124 24.02 0.65 0.37 0.54 0.4 5.69 ASOE127 27.04 0.66 0.40 0.63 0.4 5.45 ASOE130 29.98 0.67 0.42 0.68 0.4 5.25 ASOE135 34.43 0.69 0.44 0.79 0.5 6.24 2 ASOE218 18.03 0.46 0.39 0.19 0.5 4.1 6.91 56.69 ASOE221 20.99 0.41 0.44 0.20 0.5 6.0 6.21 74.50 ASOE224 24.08 0.38 0.48 0.19 0.5 7.0 5.60 78.44 ASOE230 29.97 0.33 0.54 0.18 0.6 6.0 5.65 56.46 ASOE235 34.28 0.31 0.57 0.19 0.5 6.4 4.20 53.77 3 ASOE315 14.97 0.24 0.48 0.07 0.4 1.9 4.53 21.51 ASOE318 18.00 0.23 0.52 0.07 0.5 2.4 5.03 24.13 ASOE321 20.89 0.22 0.55 0.09 0.7 2.1 6.35 19.04 ASOE324 24.04 0.20 0.58 0.09 0.6 2.3 4.91 18.81 ASOE327 26.82 0.19 0.60 0.11 0.7 2.2 5.26 16.54 1 ASOSC121 20.97 0.64 0.34 0.46 0.6 8.96 ASOSC124 24.02 0.65 0.37 0.53 0.4 5.69 ASOSC127 27.04 0.66 0.40 0.62 0.6 8.17 ASOSC130 29.98 0.67 0.41 0.66 0.6 7.87 ASOSC135 34.43 0.69 0.44 0.75 0.5 6.24 2 ASOSC218 18.03 0.46 0.39 0.20 0.5 4.0 6.91 55.31 ASOSC221 20.99 0.41 0.44 0.20 0.7 6.7 8.69 83.20 ASOSC224 24.08 0.38 0.48 0.18 0.5 6.0 5.60 67.23 ASOSC230 29.97 0.33 0.52 0.17 0.6 6.0 5.65 56.47 ASOSC235 34.28 0.31 0.55 0.17 0.6 7.0 5.04 58.81 3 ASOSC315 14.97 0.24 0.47 0.07 0.3 1.3 3.40 14.72 ASOSC318 18.00 0.23 0.51 0.07 0.5 2.3 5.03 23.12 ASOSC321 20.88 0.22 0.53 0.08 0.7 2.3 6.35 20.86 ASOSC324 24.05 0.20 0.55 0.09 0.4 2.4 3.27 19.62 ASOSC327 26.79 0.19 0.57 0.09 0.4 2.2 3.01 16.55 4 ASOSC415 15.15 0.23 0.48 0.05 0.5 2.1 5.53 23.24 ASOSC421 20.92 0.20 0.54 0.05 0.5 3.8 4.36 33.16 ASOSC424 23.86 0.19 0.56 0.06 0.5 2.6 3.94 20.47 ASOSC427 26.81 0.18 0.58 0.07 0.5 2.8 3.58 20.07 ASOSC435 34.30 0.16 0.61 0.08 0.5 2.3 2.91 13.40 1 D398121P 20.98 0.64 0.42 0.48 0.3 7.7 4.48 114.96 b=398 D398124P 23.98 0.65 0.44 0.55 0.2 6.7 2.84 95.31 with piers D398127P 26.95 0.66 0.46 0.63 0.3 6.9 4.09 94.09 D398130P 30.02 0.67 0.48 0.70 0.3 8.0 3.93 104.91 D398135P 34.29 0.69 0.50 0.78 0.3 8.0 3.75 99.99 3 D398321 20.88 0.22 0.53 0.08 0.5 2.1 4.53 19.05 b=398mm D398324 24.05 0.20 0.55 0.09 0.5 2.3 4.09 18.81 no piers D398327 26.79 0.19 0.56 0.09 0.5 2.0 3.76 15.05 D398330 29.84 0.19 0.57 0.09 0.7 2.6 4.83 17.95 D398335 34.32 0.17 0.59 0.09 0.6 2.9 3.69 17.83 3 D398321P 20.88 0.22 0.58 0.10 0.4 2.2 3.63 19.95 b=398mm D398324P 24.05 0.20 0.60 0.10 0.5 2.6 4.09 21.26 with piers D398327P 26.79 0.19 0.61 0.09 0.5 2.7 3.76 20.32 D398330P 29.84 0.19 0.62 0.10 0.5 3.0 3.45 20.72 D398335P 34.32 0.17 0.63 0.09 0.6 3.0 3.69 18.44 3 D498321 20.88 0.22 0.46 0.08 0.5 2.7 4.53 24.49 b=498mm D498324 24.05 0.20 0.48 0.07 0.5 2.5 4.09 20.44 no piers D498327 26.79 0.19 0.49 0.07 0.5 2.3 3.76 17.31 D498330 29.84 0.19 0.50 0.07 0.6 2.4 4.14 16.57 D498335 34.32 0.17 0.51 0.06 0.5 2.9 3.07 17.83 3 D498321P 20.88 0.22 0.52 0.09 0.7 2.2 6.35 19.95 b=498mm D498324P 24.05 0.20 0.53 0.08 0.5 2.1 4.09 17.17 with piers D498327P 26.79 0.19 0.54 0.08 0.5 2.6 3.76 19.56 www.jbaconsulting.co.uk 24