LARGE SMOOTH CYLINDRICAL ELEMENTS LOCATED IN A RECTANGULAR CHANNEL: UPSTREAM HYDRAULIC CONDITIONS AND DRAG FORCE EVALUATION by BENOIT TURCOTTE B.ING. École Polytechnique de Montréal, Québec, 2004 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) October 2008 © Benoit Turcotte, 2008 Abstract Classical approaches to evaluate the stability of large woody debris (LWD) introduced in streams for habitat restoration or flood management purposes are usually based on inappropriate assumptions and hydraulic equations. Results suggest that the physics of small cylindrical elements located in large channels cannot be transferred to the case of a large roughness elements placed in small channels. The introduction of LWD in a small channel can generate a significant modification of the upstream hydraulic conditions. This modification has direct implications on the stability of the LWD. Experiments were performed in a controlled environment: a small stream section was represented by a low roughness rectangular flume and LWD were modeled with smooth PVC cylinders. Direct force measurements were performed with a load cell and results were used to identify an equation that evaluates the drag force acting on a large cylindrical element place in a rectangular channel. This equation does not depend on a drag coefficient. Water depths were also measured during the experiments and results were used to develop an approach that evaluates the upstream hydraulic impacts of a large cylinder introduced in a rectangular channel. The effect of the variation of the unit discharge (discharge per unit of width), cylinder size, cylinder elevation from the channel bed, and downstream hydraulic conditions, could be related to the upstream hydraulic conditions with relative success. Dimensionless parameters were developed to increase the versatility of the approach. The application of this approach to field cases is expected to require adjustments, mainly because of the roughness of natural environments differs from the smoothness of the controlled environment described in this work. ii Table of Contents Abstract ...................................................................................................................................................... ii Table of Contents ...................................................................................................................................... iii List of Tables.............................................................................................................................................. vi List of Figures ............................................................................................................................................vii Nomenclature ............................................................................................................................................xi Acknowledgements..................................................................................................................................xiii 1.0 Introduction ................................................................................................................................... 1 1.1 Project’s Objectives.................................................................................................................... 2 1.2 Thesis Outline............................................................................................................................. 3 2.0 Literature Review ........................................................................................................................... 4 2.1 Evaluation of the Drag Force (Fd) ............................................................................................... 4 2.1.1 Evaluation of the Drag Coefficient (Cd) .............................................................................. 4 2.1.2 Back‐Calculated Drag Coefficients (Cd)............................................................................... 7 2.1.3 Definition of the Upstream Velocity (V)............................................................................. 7 2.1.4 Effect of the Blockage Ratio (BR) ....................................................................................... 8 2.1.5 Effect of the Cylinder Elevation from the Channel Bed (wsg) ........................................... 10 2.1.6 Conclusion on the Drag Force Evaluation ........................................................................ 11 2.2 Evaluation of the Lift Force (Fl) ................................................................................................ 13 2.3 Evaluation of Upstream Hydraulic Conditions ......................................................................... 13 2.3.1 Weir Flow ......................................................................................................................... 14 2.3.2 Orifice Flow ...................................................................................................................... 16 2.3.3 Weir and Orifice Flows ..................................................................................................... 18 2.3.4 Conclusion on the Evaluation of the Upstream Hydraulic Conditions............................. 19 iii 3.0 Experiments ................................................................................................................................. 20 3.1 Experimental Material.............................................................................................................. 20 3.1.1 Flume Description ............................................................................................................ 21 3.1.2 Test Cylinders ................................................................................................................... 21 3.1.3 Water Depth Measurements ........................................................................................... 22 3.1.4 Force Measurement Apparatus ....................................................................................... 22 3.1.5 Flow Meter ....................................................................................................................... 23 3.1.6 Experimental Setup .......................................................................................................... 24 3.2 4.0 Experimental Set 1 ................................................................................................................... 27 Results .......................................................................................................................................... 30 4.1 Streamwise Forces ................................................................................................................... 31 4.1.1 Signal Distribution ............................................................................................................ 32 4.1.2 Comparing Calculated Drag Forces to Measured Streamwise Forces ............................. 34 4.2 Hydraulic Data Set.................................................................................................................... 36 4.2.1 Curve Types ...................................................................................................................... 36 4.2.2 Influence of Known Parameters....................................................................................... 40 5.0 Result Analysis.............................................................................................................................. 45 5.1 Legend Convention .................................................................................................................. 45 5.2 Curve 1‐2‐3 Model ................................................................................................................... 45 5.3 Curve 1‐2‐5 Models .................................................................................................................. 51 5.4 Separation between Curve 2 and Curve 5 ............................................................................... 57 6.0 Comparison of Results ................................................................................................................. 60 6.1 Upstream Water Depth Evaluation.......................................................................................... 60 6.2 Drag Force Evaluation .............................................................................................................. 61 6.2.1 Drag Force Equation......................................................................................................... 62 iv 6.2.2 Drag Force Equation and Blockage Ratio ......................................................................... 64 6.2.3 Momentum Equation ....................................................................................................... 65 7.0 Applications and Limitations........................................................................................................ 67 7.1 Synthesis of the Approach ....................................................................................................... 67 7.2 Example of the Approach......................................................................................................... 69 7.3 Limitations of the Approach..................................................................................................... 72 8.0 Conclusions .................................................................................................................................. 74 References................................................................................................................................................ 76 Appendix A Load Cell Calibration ........................................................................................................ 78 A1 Streamwise Force Signal Calibration (Fx).................................................................................. 78 A2 Moment Signal Calibration (Mz)............................................................................................... 80 A3 Vertical Force Signal Calibration (Fy) ........................................................................................ 81 Appendix B Downstream Water Depth Correction............................................................................. 83 Appendix C Vertical Forces.................................................................................................................. 86 C1 Comparing Measured Lift Forces (Flm) to Theoretical Lift Forces (Fl)....................................... 86 C 2 Buoyancy forces (Fb)................................................................................................................. 89 C 3 Conclusion on the Lift Force Analysis....................................................................................... 89 Appendix D Calibration of the α1 Coefficient ...................................................................................... 93 Appendix E Conception of Figure 5.12 ................................................................................................ 95 v List of Tables Table 3.1. Table 3.2. Table 6.1. 15 experiments performed for each cylinder size (marked with an “X”) ........................ 28 Measured and recorded parameters for each experimental run .....................................28 List of experiments from Experimental Set 1 rejected from the drag force comparison analysis ..........................................................................................................62 Table 7.1. Input parameters of Data Set A and Data Set B ................................................................69 vi List of Figures Figure 2.1. Drag Coefficient (Cd) expressed as a function of the Reynolds number (Re) for infinite cylinders ..................................................................................................................5 Figure 2.2. Two‐dimensional representations of (a) towing tank or wind tunnel experiments and of (b) a large cylinder located in a small rectangular channel .....................................6 Figure 2.3. Two‐dimensional representation of a large cylinder located on the channel bed ...........14 Figure 2.4. Two‐dimensional representation of a large cylinder acting as a sluice gate .................. 16 Figure 3.1. Scheme of the flume and experimental setup..................................................................20 Figure 3.2. Experimental flume in the hydraulics laboratory of the Department of Civil Engineering at University of British Columbia ..................................................................21 Figure 3.3. PVC cylinders used for the experiments ...........................................................................22 Figure 3.4. Load cell used for the experiments...................................................................................23 Figure 3.5. Flow meter and sensors installed on the flume recirculation pipe ..................................24 Figure 3.6. Platform, load cell, and profiled vertical rods...................................................................25 Figure 3.7. Experimental setup fixed to the flume .............................................................................26 Figure 3.8. Vernier and Point Gauge at the downstream end of the flume .......................................27 Figure 4.1. Scheme of different hydraulic states as the downstream water level decreases: (a) low turbulence (b) contraction (c) submerged hydraulic jump (d) hydraulic jump separating from cylinder (e) hydraulic jump separated from cylinder (f) supercritical conditions. For experiments where the cylinder is located at higher elevations (wsg): (g) vortex formation upstream of cylinder and (h) cylinder acting as a sluice gate ..................................................................................................................31 vii Figure 4.2. Figure 4.3. Distribution of the streamwise force signals (Fx) for six distinct experimental runs ........33 Drag force (Fd) calculated using equation (2.4) and measured streamwise force (FFx) expressed as a function of the downstream water depth (Ydwn) for experiment D4wsg010Q10 ...................................................................................................................34 Figure 4.4. Drag force (Fd) calculated using equation (2.4) expressed as a function of the measured streamwise force (FFx) for 611 experimental runs ...........................................35 Figure 4.5. Drag force (Fd) expressed as a function of the downstream water depth (Ydwn) for experiment D4wsg010Q10, and two‐dimensional schemes and photos representing the hydraulic conditions for each of the 3 curves .......................................37 Figure 4.6. Drag force (Fd) expressed as a function of the downstream water depth (Ydwn) for experiment D4wsg076Q10, and two‐dimensional schemes and photos representing the hydraulic conditions for each of the 4 curves .......................................39 Figure 4.7. Upstream water depth expressed as a function of the downstream water depth for both experiments D4wsg010Q10 and D4wsg076Q10 (Note that Curve 4 is considered a being part of Curve 5.) .................................................................................40 Figure 4.8. Example of the influence of the cylinder size (D) on the relation between the upstream water depth (Yup) and the downstream water depth (Ydwn) .............................41 Figure 4.9. Example of the influence of the cylinder elevation (wsg) on the relation between the upstream water depth (Yup) and the downstream water depth (Ydwn).......................42 Figure 4.10. Example of the influence of the discharge (Q) on the relation between the upstream water depth (Yup) and the downstream water depth (Ydwn). The critical depths (Yc) are identified for each discharge ....................................................................43 Figure 5.1. Upstream water depth (Yup) expressed as a function of the downstream water depth (Ydwn) for 48 experimental results following a Curve 1‐2‐3 model. Critical water depths (Yc) and 1:1 line are presented as well........................................................46 viii Figure 5.2. Upstream water depth (Yup) expressed as a function of the downstream water depth (Ydwn) for 15 experiments during which the cylinder was located on the flume bed. Critical water depths (Yc) and 1:1 line are presented as well .........................47 Figure 5.3. Upstream water depth (Yup) expressed as a function of the downstream water depth (Ydwn) for 15 experiments during which the cylinder was located on the flume bed using compound parameter (5.1). 1:1 line is also presented ..........................48 Figure 5.4. Upstream water depth (Yup) expressed as a function of the downstream water depth (Ydwn) for 48 Experimental results following a Curve 1‐2‐3 model with compound parameter (5.1) on both axes. 1:1 line is also presented. The spread in the results is generated by the different cylinder elevation (wsg) values .........................49 Figure 5.5. Upstream water depth (Yup) expressed as a function of the downstream water depth (Ydwn) for 48 experimental results following a Curve 1‐2‐3 model with compound parameter (5.2) on both axes. 1:1 line is also presented ...............................51 Figure 5.6. Reference Graph for Curve 1‐2‐3 models with compound parameter (5.2) on both axes....................................................................................................................................52 Figure 5.7. Upstream water depth (Yup) expressed as a function of the downstream water depth (Ydwn) for 23 experimental results following a Curve 1‐2‐5 model with compound parameter (5.1) on both axes. 1:1 line is also presented ...............................53 Figure 5.8. Back‐calculated sluice gate coefficients (Csg) for 25 experimental results following a Curve 5 model expressed as a function of the downstream Froude Number (Frdwn).................................................................................................................................54 Figure 5.9. Results from Experimental Set 1 and Experimental Set 2 where the sluice gate coefficient (Csg) is expressed as a function of the downstream Froude number (Frdwn).................................................................................................................................55 Figure 5.10 Reference Graph for cylinders acting as sluice gates (Curve 5) ........................................56 Figure 5.11. Threshold between Curve 1‐2‐3 models and possible Curve 1‐2‐5 models .....................58 Figure 5.12. Theoretical thresholds for vortex formation upstream of a cylinder ...............................59 ix Figure 6.1. Calculated upstream water depths from Experimental Set 1 expressed as a function of the measured upstream water depths. The 1:1 line is also presented..........61 Figure 6.2 Calculated drag force (Fd) using equation (6.1) and a drag coefficient (Cd) of 1.0 expressed as a function of the measured streamwise force (FFx) using the load cell.......63 Figure 6.3. Calculated drag force (Fd) using equation (6.1) and a drag coefficient (Cd) based on equation (6.3) corrected for blockage ratios (BR) using equation (2.9) expressed as a function of the measured streamwise force (FFx) using the load cell.............................65 Figure 6.4. Calculated drag force (Fd) using the momentum equations (2.4) expressed as a function of the measured streamwise force (FFx) measured with the load cell................66 Figure 7.1. Diagram of the approach developed in this research project to calculate the upstream water depth (Yup) and the drag force (Fd) acting on a large cylindrical element .............................................................................................................................68 Figure 7.2. Downstream rating curves from Data Set A and Data Set B .............................................70 Figure 7.3. Measured and calculated upstream rating curves from Data Set A and Data Set B ........71 Figure 7.4. Calculated upstream water depth expressed as a function of the measured upstream water depth for Data Set A and Data Set B ......................................................71 x Nomenclature Aup Upstream cross‐section area (m2) α1 Coefficient defined in equation (5.3) α2 Coefficient defined in equation (5.4) B Channel or flume width (m) BR Blockage ratio Cd Drag coefficient CdBR Drag coefficient considering the blockage ratio C l Lift coefficient Csg Sluice gate coefficient Cw, Cw* Weir coefficient d1,d2,d3 Parameters used in equation (D1), Appendix D, to calibrate equation (5.3) D Cylinder diameter (inches, m) e1, e2 Parameters used in equation (E1), Appendix E E Energy (m) defined with equation (2.12) Ecyl Energy at the cylinder section (m) Eup Energy upstream of the cylinder (m) Fb Buoyancy force (N) Fbm Measured buoyancy force (N) Fd Drag Force (N) FFx Streamwise force based on Fx signal (N) FFy Vertical force based on Fy signal (N) Fl (Dynamic) Lift force (N) Flm Measured (dynamic) lift force (N) FMz Streamwise force based on Mz signal (N) Fr Froude number Frdwn Downstream Froude number Fx Streamwise signal (Volts) Fxini Initial streamwise signal (Volts), Appendix A Fy Vertical signal (Volts) Fyini Initial vertical signal (Volts), Appendix A xi g Gravity constant (m/s2) h Distance between load cell and streamwise force application point (mm), Appendix A hgate Weir gate elevation (cm) l Cylinder length (m) M Momentum (m3), defined with equation (2.5) Mdwn Downstream momentum (m3) Mup Upstream momentum (m3) MMz Moment (Nm) Mz Moment signal (Volts) Mzini Initial moment signal (Volts), Appendix A ν Fluid kinematic viscosity (m2/s) q Unit discharge (m2/s) qsg Orifice flow unit discharge (m2/s) qw Weir unit discharge (m2/s) Q Discharge (L/s, m3/s) Re Reynolds number ρ Density of fluid (kg/m3) ρcyl Density of PVC cylinders (kg/m3) ρlog Hypothetical log density (kg/m3), Appendix C ρw Density of water (kg/m3) V Flow velocity (m/s) Vcyl Flow velocity at the cylinder section (m/s) Vdwn Downstream flow velocity (m/s) Vup Upstream flow velocity (m/s) Y, Y1, Y2 Water depth (m) Yc Critical water depth Ycyl Water depth at the cylinder section (m) Ydwn Downstream water depth (cm, m) Ydwn’ Water depth at the vena contracta or contraction (m) Yup Upstream water depth (m) Yw Weir flow depth (m) wsg Cylinder elevation from the channel bed (mm, m) xii Acknowledgements The past 24 months in Vancouver have greatly contributed to my personal and professional development. This experience was enriched by the presence of professors, technicians, students, and close individuals who shared their knowledge, resources, and time with me. I would like to address special thanks to my supervisor, Dr. Robert G. Millar, for funding and for proposing a challenging and motivating research project. Dr. Millar also gave me the possibility to develop my own ideas and offered guidance and assistance when needed. The diversity of his interests added to the richness of my experience at UBC. I would like to express gratitude to Dr. Marwan Hassan from the department of Geography for constant support and for giving me opportunities to participate in research projects in Geomorphology. I believe that my engineering background was significantly improved by his presence. I owe Dr. Hassan my understanding of watershed processes and evolution. I would also like to thank Dr. Violeta Martin for her devotion to students, including myself. Her contribution made a great difference in all aspects of my laboratory experiments. I would like to thank Scott Jackson for electronic resources and computer assistance as well as Doug Hudniuk, Bill Leung, and Harald Schrempp who were able to convert an experimental setup design into a flawless final product. I want to highlight the important contribution of Stephen Rennick, who helped producing additional experimental data sets. The meticulousness of his work is reflected in my analysis results. Finally, I am grateful for presence and understanding of my fiancée Rima. Her constant support and language skills were essential to my success throughout my Master’s degree. xiii 1.0 Introduction Forest management practice in riparian zones and direct large woody debris (LWD) removal from streams have proven to generate negative effects on aquatic habitats (D’Aoust and Millar, 2000), and have also been linked to increased peak floods by decreasing the in‐stream travel time (Gippel 1995). The importance of LWD in streams is now recognized and channel restoration projects including LWD reintroduction have been widely investigated. However, the reported high failure rate of artificially introduced roughness elements (D’Aoust and Millar, 2000; Shields et al., 2004) suggests that our understanding of the hydraulics of LWD is not developed enough to create solid guidelines for stream restoration projects. Therefore, taking a step backwards seems necessary. Classic drag force experiments (Lindsey, 1938) were performed using cylinders (or other shaped elements) that were small compared to the channel cross‐section (Gippel et al., 1992). As a result, these elements exerted negligible effects on upstream flow conditions. Conversely, LWD are often of significant size relatively to streams cross‐sections and their effects on the local flow conditions are usually important. Consequently, direct application of the classical approach to evaluate the drag force on LWD in streams presents 2 fundamental limitations: (1) The flow velocity upstream and downstream of large cylinders in small channels can be substantially different and (2) the “text book” values of drag coefficients can underestimate the true drag coefficient of LDW by an order of magnitude. D’Aoust and Millar (2000) adopted a drag coefficient of 0.3 for LWD elements introduced in streams while Manners et al. (2007) obtained back‐calculated drag coefficient values as high as 9.0 for natural log jams. Several studies have analyzed the drag force acting on modeled cylindrical LWD (Gippel et al., 1992; Shields and Gippel, 1995; Gippel et al., 1996; Baudrick and Grant, 2000; D’Aoust and Millar, 2000; Wallerstein et al., 2001; Wallerstein et al., 2002; Hygelund and Manga, 2003; Alonso 2004). The dynamic lift force acting on cylindrical roughness elements has also been investigated (Alonso 2004). Inconsistencies were noticed regarding the choice of water velocities and drag coefficients considered for drag force evaluations. Moreover, several factors affecting the drag force, such as the cylindrical element elevation from the channel bed and the blockage ratio, have not been tested for the range of conditions that is representative of LWD in streams. Consequently, one objective of this research was to identify an adequate formulation using appropriate parameters to evaluate the drag force and the lift force acting on cylindrical roughness element of significant size. 1 The main challenge about LWD reintroduction practice however comes from the initially unknown hydraulic conditions created by the LWD. Ironically, these hydraulic conditions are required to evaluate the forces acting on the LWD in order to design proper anchoring systems. An approach that predicts the variation in hydraulic conditions induced by the introduction of a large cylindrical roughness element would be valuable for three reasons. (1) From a habitat restoration perspective, a given LWD size and position could be related to a desired water level and velocity. (2) From a flood management angle, a given LWD size and position could be related to a stored water volume and to an overbank flooding return period. (3) The predicted hydraulic conditions would facilitate a LWD stability investigation (Baudrick and Grant, 2000; D’Aoust and Millar, 2000) for a proper anchoring system design, which would limit the eventuality of premature entrainment during flooding events. Flume experiments were performed in the hydraulics laboratory at the University of British Columbia in 2008. LWD were modeled as cylinders (Gippel et al., 1992; Shields and Gippel, 1995; Gippel et al., 1996; Baudrick and Grant, 2000; Wallerstein et al., 2001; Wallerstein et al., 2002; Hygelund and Manga, 2003). A number of parameters (cylinder size, cylinder vertical position, discharge, initial water depth) that affect the stability of a roughness element were considered. Dimensionless relations were developed as a function of the channel width to express the variation in hydraulic conditions created by the introduction of a cylindrical roughness element of significant size in a channel. The motivation behind this study was to create guidelines for river restoration projects including the introduction of LWD, but it remains focused on the physics of smooth cylindrical elements introduced in rectangular channels. 1.1 Project’s Objectives The first objective of this work was to develop a methodology to evaluate the drag force acting on a large cylinder located in a channel. This approach only considers the channel hydraulic conditions and differs from the conventional method that refers to a drag coefficient. A complementary investigation was performed to identify an accurate equation to evaluate the dynamic lift force acting on cylindrical elements. These two dynamic forces represent the main uncertainties involved in the estimate of the cylindrical element stability. 2 The second objective of this work was to develop an approach that evaluates the hydraulic effect of a large cylindrical element. This approach assumed that apart from local disturbance of the velocity profile, large cylinders can only have an influence in the upstream direction (Gippel et al., 1996). Therefore, the downstream hydraulic conditions were assumed to be independent of the cylinder size and position, even when the Froude number was less than 1. The approach proposed in this work was developed to determine the upstream hydraulic conditions and the dynamic forces acting on a cylindrical roughness element for a range of channel discharges, channel widths, element diameters, element vertical positions from the channel bed, together with downstream hydraulic conditions. It would only be applicable for rectangular or wide channels. Cylindrical elements would have to be of constant diameter, to be the same length as the stream width, and to be placed perpendicularly to the flow direction (referred as a Yaw angle of 0° by Alonso, 2004) and parallel to or directly on the channel bed. 1.2 Thesis Outline The structure of this thesis is as follows. Chapter 2.0 presents a literature review of the methods used to evaluate the drag force and lift force affecting cylindrical elements exposed to flowing water. It also introduces hydraulic equations that could be used to evaluate the hydraulic conditions created upstream of a roughness element of significant size. Flume experiments are described in Chapter 3.0. Experimental results and drag force analysis are explained in Chapter 4.0. Chapter 5.0 presents the development of the upstream hydraulic conditions predictive approach. Chapter 6.0 compares the results from the adopted drag force evaluation equation to the classical evaluation approach, which depends on a drag coefficient. Chapter 7.0 synthesizes the approach developed in this research project, an example of data processing is given, and limits to adapt the approach to field cases including stream restoration projects are presented. Finally, a summary of the findings is listed in Chapter 8.0. 3 2.0 Literature Review The stability of a large cylindrical roughness element depends on two dynamic forces: the drag force Fd and the lift force Fl. Both forces depend on hydraulic conditions that are unknown before the roughness element is introduced in the channel. This chapter presents background to the drag force and lift force evaluation methods. It also introduces equations that could possibly be used for the prediction of the hydraulic conditions generated upstream of a large cylindrical element introduced in a channel. 2.1 Evaluation of the Drag Force (Fd) The generally accepted equation to describe the drag force Fd acting on a cylindrical is Fd = C d ρDlV 2 2 (2.1) Here, Cd represents an empirical drag coefficient, ρ is the density of the fluid, D is the diameter of the cylinder, l is the length of the cylinder, and V is a flow velocity. Two of these terms appear to be recurrently difficult to define in literature: the drag coefficient Cd and the flow velocity V. 2.1.1 Evaluation of the Drag Coefficient (Cd) The drag coefficient of a cylinder affected by an incident flow is often presented in a graph as a function of the Reynolds number Re. Re = DV ν (2.2) Here ν is the kinematic viscosity of the fluid. Figure 2.1 presents an example of this graph, which is usually presented in hydraulic handbooks. For Re values between 103 and 105, the corresponding drag coefficient values are typically 0.9 to 1.2 for smooth infinite cylinders. 4 1.E+02 1.E+01 Cd Relation for infinite cylinders 1.E+00 1.E‐01 1.E‐01 1.E+00 1.E+01 1.E+02 Re 1.E+03 1.E+04 1.E+05 1.E+06 Figure 2.1. Drag Coefficient (Cd) expressed as a function of the Reynolds number (Re) for infinite cylinders The results presented in Figure 2.1 are generally based on experiments performed in towing tanks or wind tunnels (Lindsey, 1938; Zahm et al., 1972). The cylinders tested are usually small and the flow can be considered of infinite extent (Gippel et al., 1992) or infinitely large (Hygelund and Manga, 2003). This means that the boundary effects are negligible and that the cylinder has no measurable influence on the upstream flow conditions. Therefore, the velocity upstream of the cylinder is comparable to the velocity at the cylinder cross‐section or downstream of the cylinder. Figure 2.2a presents a two‐dimensional scheme of a towing tank or a wind tunnel experiment. The drag coefficient evaluation based on Figure 2.1 could only be used to calculate the drag force Fd acting on LWD in a stream if the LWD would not have a measurable effect on the upstream hydraulic conditions. In reality, LWD in small channels usually generate significant back‐water effects (referred as afflux by Ranga Raju et al., 1983). In this case, the flow velocity upstream of the cylinder is different from the velocity at the cylinder cross‐section or downstream of the cylinder. Figure 2.2b presents a two‐dimensional scheme of a large cylinder placed in a small rectangular channel. 5 (a) Y1 Y2 D (b) Yw Yup D Ydwn wsg Figure 2.2. Two‐dimensional representations of (a) towing tank or wind tunnel experiments and of (b) a large cylinder located in a small rectangular channel The difference between LWD hydraulics in streams (Large D) and the classic drag force experiment (small D) presented in Figure 2.2 was recognized by Gippel et al. 1992 who proposed an alternative empirical approach to evaluate the drag coefficient of a cylindrical object based only on its geometry: ⎛ l ⎞ C d = 0.81⎜ ⎟ ⎝D⎠ 0.062 (2.3) Equation (2.3) is only valid when l/D < 21 and when the flow is perpendicular to the cylinder. In this case, drag coefficient values of 0.8 to 1.0 are obtained for l/D ratios ranging from 1 to 21. 6 2.1.2 BackCalculated Drag Coefficients (Cd) An alternative method to evaluate the drag coefficient is to back‐calculate it using equation (2.1) if the drag force is known. Hygelund and Manga (2003) used a torque wrench to measure the drag force acting on a cylinder. Wallerstein et al. (2001) measured the drag force on cylinders using a dynamometer. Gippel et al. (1996) and Manners et al. (2007) used the momentum approach to estimate the drag force acting on cylinders and log jams, respectively (e.g. Finnemore and Franzini, 2002). Fd = ρg (M up − M dwn ) (2.4) Here, g is the gravity constant and Mup and Mdwn are the momentum upstream and downstream of the roughness element respectively. The momentum is defined by the following equation: 2 q2B Y B M = + 2 gY (2.5) Here, B is the channel width, Y is the water depth, and q is the unit discharge defined as (for rectangular or wide channel sections only) q= Q B (2.6) where Q is the total discharge. The reliability of these techniques to measure the drag force is questionable since the back‐calculated drag coefficient is purely empirical and values significantly vary from a study to another. One reason that partly explains this variation is that the back‐calculation of the drag coefficient still depends on one undefined parameter in equation (2.1): the flow velocity V. 2.1.3 Definition of the Upstream Velocity (V) Because a cylindrical roughness element of significant size creates a back‐water effect, the flow velocity upstream of the element (Vup) differs from the flow velocity at the element (Vcyl) or downstream of the element (Vdwn). This can be straightforwardly appreciated in Figure 2.2b. For a given drag coefficient Cd, flow conditions, and object geometry (D and l), these three velocities used in equation (2.1) could provide drag force values that are considerably different. 7 D’Aoust and Millar (2000) and Alonso (2004) did not specify the streamwise location of the reach‐ averaged velocity V to consider in equation (2.1). Manners et al. (2007) suggested that V is the flow velocity “independent of the object” (log jam in their case). This would indicate that the flow velocity has to be measured upstream of the back‐water region. Wallerstein et al. (2001) and Wallerstein et al. (2002) proposed that V is the “approach velocity” but did not confirm if this velocity was measured in the back‐water region or further upstream where the influence of the roughness element fades. Gippel et al. (1996) proposed that V would be the “mean velocity at the section upstream of the object”. A similar description was given by Hygelund and Manga (2003) who proposed that V would be the depth‐averaged velocity of the water as it “approaches” the roughness element (as opposed to the local velocity, which is the velocity at the element cross‐section). The depth‐averaged velocity in the back‐water region Vup can be approximated using this equation for rectangular sections: Vup = Q Q q = = Aup BYup Yup (2.7) Here Aup is the upstream section area. For the purpose of this research, it was assumed that Aup, B, and Yup should be taken in the back‐water region, upstream of the flow contraction zone created by the object. This seems to be the best definition of an upstream section and thus the most suitable to use in equation (2.1) and equation (2.4). When the roughness element has a significant size (D and l) compared to the upstream cross‐section Aup, this element “feels” a velocity that is consequently higher than the approach velocity Vup. Therefore, this additional concern cannot be ignored when one calculates the drag force acting on large roughness elements (Gippel et al., 1992). 2.1.4 Effect of the Blockage Ratio (BR) The blockage ratio was proposed as an alternative to correct the drag coefficient obtained from towing tank or wind tunnel based‐experiments. The blockage ratio BR of a cylinder in a rectangular channel is evaluated as follow: BR = Dl Dl = Aup BYup (2.8) 8 If the upstream depth‐averaged velocity obtained from equation (2.7) was to be considered in equation (2.1), the drag coefficient Cd could be corrected for significant blockage ratios using this empirical equation proposed by Shields and Gippel (1995): C dBR = 0.997C d (1 − BR ) −2.06 (2.9) This equation was calibrated from 50 flume experiments for BR values ranging from 0.03 to 0.30 and an approach Froude number of 0.35. Gippel (1995) had suggested that the effect of the blockage on the drag coefficient should be considered in stream channels. Shields et al. (2004) referred to equation (2.9) to obtain corrected drag coefficient CdBR values of 0.7 to 0.9 for their LWDs structures but did not mention any blockage ratio value. Curran and Wohl (2002) also used equation (2.9) even if their blockage ratios ranged from 0.3 to 0.8 and that their Froude numbers could be “locally quite high”. They obtained CdBR values as high as 25. Hygelund and Manga (2003) collected drag force data from field experiments (Froude number Fr<0.2) using PVC cylinders with blockage ratios ranging from 0.1 to 0.7 and compared their results with a modified but similar version of equation (2.9). C dBR = C d (1 − BR ) −2 (2.10) Their CdBR values ranged from 1.0 to 3.3. They observed that as blockage increased, drag increases but that for blockage ratios greater than about 0.3, the drag (coefficient) CdBR becomes independent of the blockage (ratio) and seemed to stabilize at a value of CdBR ~2.1. This contradicts the assumption previously made by Curran and Wohl (2002). Wilcox et al. (2006) also considered equation (2.10) for their flume experiments on resistance partitioning. Equation (2.10) can easily be demonstrated (refer to Figure 2.2b) using continuity: Vup Yup = Vcyl Ycyl Vup Yup = Vcyl (Yup − D ) ⎛ D Vup = Vcyl ⎜1 − ⎜ Y up ⎝ ⎞ ⎟ ⎟ ⎠ 9 When l=B (cylinder length equals to the channel width), using equation (2.8) Vup = Vcyl (1 − BR ) Vup2 = Vup2 (1 − BR ) 2 ⎛ Vcyl ⎜ ⎜V ⎝ up ⎞ ⎟ ⎟ ⎠ −2 = (1 − BR ) −2 This proposes that equation (2.9) and equation (2.10) are simply adjusting Cd to account for the higher velocity at the cylinder section. However, this demonstration, also presented by Hygelund and Manga (2003), does not take the flow contraction into account (in reality, Yup is greater than Ycyl + D). The amplitude of the flow contraction at the cylinder cross‐section can be calculated using the energy (E) equation: Eup = Ecyl + D (2.11) V2 2g (2.12) (2.13) where E =Y + One could show that: Vcyl > Vup (1 − BR )2 This means that the larger the cylinder, the more significant the flow contraction. Therefore, the drag force estimation error using equation (2.1) and equation (2.10) should be greater for large cylinders. 2.1.5 Effect of the Cylinder Elevation from the Channel Bed (wsg) Gippel et al. (1992), Gippel et al. (1996), Wallerstein et al. (2002), Hygelund and Manga (2003), and Alonso (2004) considered the influence of the cylinder elevation (wsg) from the channel bed (refer to Figure 2.2b) on the drag coefficient. Gippel et al. (1996) observed that when the cylinder was located close to the bed (wsg ~ 0), a zone of near‐zero velocity developed upstream of the cylinder, causing the apparent drag coefficient CdBR to be high. They also reported that this zone dissipated and CdBR decreased as the model (i.e., the cylinder) was elevated from the channel bed. This disagrees with Figure 3 in Alonso (2004) that clearly shows a lower drag coefficient value when the cylinder is close to the channel bed (wsg/D ~ 0). 10 Gippel et al. (1996) also mentioned that the drag coefficient (CdBR) remained relatively constant with relative height (wsg/Yup) for larger flow depths (when D << Yup). This tends to agree with Figure 3 by Alonso (2004) but contradicts an observation made by Hygelund and Manga (2003) who stated that this constancy would be valid for large cylindrical elements (D > 30% Yup). Both observations by Gippel et al. (1996) and Hygelund and Manga (2003) were based on data sets of limited sizes. Gippel et al. (1992) and Gippel et al. (1996) additionally observed that as the cylindrical element was raised close to the water surface, CdBR dropped sharply. This is in contradiction with Wallerstein et al. (2002) who stated that the occurrence of stationary surface waves caused the drag coefficient (Cd) to increase considerably when a roughness element was positioned near the free surface. The vertical position of the cylindrical element in the water column is presented by different parameters in literature. Gippel et al. (1996) expressed the relative depth with the ratio wsg/Ydwn (Ydwn being the downstream water depth). Wallerstein et al. (2002) suggested a submergence parameter defined as Yw+D/2, which is the radius of the cylinder added to the weir flow above the cylinder. Hygelund and Manga (2003) proposed the depth ratio Yw/(Yw+wsg). Most of these expressions have no physical meaning in open channel hydraulics. As a result, it can be concluded that the interaction between the object submergence (or its elevation from the channel bed) and the drag force is still vague in literature. 2.1.6 Conclusion on the Drag Force Evaluation It seems that previous authors have performed a limited number of experiments on a restricted range of parameters and are thus not able to conclude on a global understanding of the effects of the blockage ratio (Gippel et al., 1992; Shields and Gippel, 1995; Hygelund and Manga, 2003) and log submergence or cylinder elevation (Gippel et al., 1992; Gippel et al., 1996; Hygelund and Manga, 2003; and Wallerstein et al., 2002; Alonso, 2004) on the drag coefficient. Even in controlled environments such as rectangular flumes with smooth boundaries, the conceptual drag coefficient has not been related to other parameters with great confidence. For example, equation (2.3) from Gippel et al. (1992) is only based on 5 data points. Also, equation (2.9) from Shields and Gippel (1995) only presents an R2 value of 0.81. This scatter could be explained by the interference with factors influencing the drag coefficient ratio that were not taken into account. 11 Curran and Wohl (2002) justified the adoption of equation (2.9) for their field experiments based on the fact that the flume study on which it is built is the closest available simulation of their study. Shields et al. (2004) adopted CdBR values of 0.7 and 0.9 to design the anchoring of their hydraulic structures. They confess that, during the second year following construction, 31% of the structures failed during high flows, probably due to inadequate anchoring. The drag coefficient (Cd) assumed by D’Aoust and Millar (2000) to evaluate the stability of artificially‐introduced single logs was 0.3 and no blockage ratio was considered in the drag force calculation. Baudrick and Grant (2000) adopted a drag coefficient of 1.0 for their flume experiments on modeled LWD entrainment. They mentioned that pieces in the larger diameter class generally moved at depths less than their predicted value. Manners et al. (2007) attempted to adapt the drag coefficient approach to natural log jams. They obtained back‐ calculated drag coefficient values reaching 9.0, one order of magnitude greater than the values adopted by Shields et al. (2004), D’Aoust and Millar (2000), and Baudrick and Grant (2000). Even the meaning of the drag coefficient is confusing in literature. Wallerstein et al. (2002) proposed that Cd is a function of the roughness, element geometry, submergence, element Froude number, and element Reynolds number. This is comparable to what was proposed by Alonso (2004). Manners et al. (2007) mentioned that Cd has no physical meaning on its own, while Curran and Wohl (2002) propose that objects in flow have an inherent drag coefficient. Equation (2.3) by Gippel et al. (1992) precedes this idea. The drag force equation (2.1) is found to be problematic for the following reasons: • No convincing equation exists to define the drag coefficient and values found in literature vary by more than one order of magnitude. • This approach does not seem to be compatible with LWD in small streams because of the importance of the back‐water effect and flow contraction. • The influence of factors such as the cylinder elevation from the channel bed on the drag force is unclear. An alternative approach to the drag force equation (2.1) is the momentum equation (2.4). This equation does not include any ambiguous coefficient. However, the momentum equation is referred with limited popularity in literature, possibly because it is less versatile and because it depends on a hydraulic parameter that is often neglected: the downstream water depth (ydwn), which represents the condition undisturbed by the cylinder. In the perspective of a river restoration project, Ydwn is initially known. The effectiveness of the momentum equation will be explored in Chapter 4.0. 12 2.2 Evaluation of the Lift Force (Fl) The dynamic lift force (Fl) applied on a cylindrical element exposed to flowing water has received little attention in literature compared to the drag force (Fd). Baudrick and Grant (2000), D’Aoust and Millar (2000) and Shields et al. (2004) did not consider the dynamic lift force acting on their LWD structures for stability calculation, assuming that this force would be negligible compared to the static buoyancy force (Fb). The dynamic lift force applied on a cylindrical roughness element can be evaluated using this generally accepted equation: Fl = Cl ρDlVup2 2 (2.14) The upstream water velocity (Vup) was defined in Subsection 2.1.3. The lift coefficient (Cl) can be estimated with the following empirical equation, calibrated for well‐submerged smooth cylinders over a flat bed, mentioned by Alonso (2004): C l = 0.5e ⎛ wsg ⎜ −4 ⎜ D ⎝ ⎞ ⎟ ⎟ ⎠ (2.15) Equation (2.14) presents the same inconvenience as the drag force equation (2.1): It depends on a coefficient that is difficult to define. Moreover, for significant cylindrical object sizes, the blockage ratio BR defined in equation (2.8) does not seem to have been adapted to equation (2.14). No alternative equation was found to evaluate the dynamic lift force acting on a cylindrical object exposed to flowing water. 2.3 Evaluation of Upstream Hydraulic Conditions In the perspective of a stream restoration project including LWD introduction, the anchoring system that supports the dynamic and static forces must be designed before the log is introduced in the stream. However, the hydraulic conditions created upstream of the log are initially unknown and equation (2.1), equation (2.4), and equation (2.14) cannot be used. If fact, the upstream hydraulic conditions represent the only unknown on the right hand side of the momentum equation (2.4). This section presents a review of open channel flow equations that could be used to evaluate the hydraulic conditions created upstream of a cylinder introduced in a rectangular channel. 13 A procedure to approach the hydraulics of a cylinder disposed horizontally at a given distance from the channel bed (wsg) would be to consider two interdependent unit discharges (refer to Figure 2.2b): the weir flow unit discharge (qw) above the element and the orifice flow unit discharge (qsg) under the element. q = q w + q sg (2.16) In this case, q is a known parameter while qw and qsg are not defined. 2.3.1 Weir Flow When the gap between the channel bottom and the cylinder (wsg) is negligible (Figure 2.3), one could assume that the total unit discharge q can be evaluated using a weir flow equation. If the approach velocity (Vup) is neglected, the weir flow equation is expressed as follow (based on Finnemore and Franzini, 2002): q w = C w* 3 2 2 g (Yup − D ) 2 3 (2.17) where Cw* is a weir coefficient. Other parameters have been defined previously. Yw Yup D Ydwn Figure 2.3. Two‐dimensional representation of a large cylinder located on the channel bed 14 In order to obtain a negligible upstream velocity in the back‐water region, the blockage ratio defined in equation (2.8) would have to be significant and the drag force acting on the element could be estimated using hydrostatic force equations. This assumption cannot be made for the entire range of hydraulic conditions and log sizes involved in a stream restoration project. Equation (2.17) takes the following form when the upstream energy defined in equation (2.12) is considered: q w = C w* 3 2 2 g (E up − D ) 2 3 (2.18) In order to evaluate the upstream water depth (Yup) for a given discharge (qw) and cylinder size (D), one has to solve a complex equation that includes the undefined weir coefficient (Cw*). A collection of empirical formulas characterizing Cw*for different weir crest shapes are available in literature and often depend on upstream hydraulic conditions as well. Indeed, a large proportion of the effort made in research to understand the hydraulics of weir gates have been made in the perspective to calculate a discharge for given upstream hydraulic conditions, and not the opposite. The closest weir crest commonly present in literature that approaches the shape of a cylindrical weir is the overflow spillway crest. Finnemore and Franzini (2002) presented a graph (Figure 11.33) where the weir coefficient (Cw) was defined by: C w = C w* 2 2g 3 (2.19) In equation (2.19), Cw would take a minimum value of 1.7 for low flows and a maximum value of 2.3 for high flows and significant spillway height. These limits could fluctuate if one considers the difference between a properly‐designed overflow spillway and a cylindrical element acting as a weir. To add to the complexity of the problem, equation (2.17) and equation (2.18) are only valid when the downstream water depth (Ydwn) does not affect the upstream water depth (Yup). This assumption cannot be made for a range of conditions considered in the present project. It is unclear in literature if the influence of the downstream hydraulic conditions can be corrected with the weir coefficient only or if a modification of equation (2.17) or equation (2.18) is required. The evaluation of Yup using common weir flow equations for given q, D, and Ydwn does not appear to find any simple solution. 15 2.3.2 Orifice Flow When the cylinder is not fully submerged, water is confined under the cylinder, which acts as a sluice gate (Figure 2.4). The closest sluice gate shape commonly presented in literature that compares to a cylinder is the radial gate. Based on Bernoulli (or the energy) equation, Shahrokhnia and Javan (2006) proposed this sluice gate (or orifice) flow equation for radial gates and rectangular channels: q sg = C sg wsg 2 g (Yup − Ydwn ') (2.20) In equation (2.20), Csg is the sluice gate coefficient and Ydwn’ is the downstream water depth taken at the vena contracta or minimum jet thickness (Clemmens et al., 2003), just downstream of the gate. This parameter depends on the sluice gate height and shape and is thus initially unknown. Shahrokhnia and Javan (2006) mentioned that, under free‐flow conditions, the downstream hydraulic conditions cannot affect the upstream water depth or passing discharge (qsg) and can therefore be ignored. In this case, equation (2.20) simplifies into: q sg = C sg wsg 2 g (Yup ) (2.21) Shahrokhnia and Javan (2006) calibrated equation (2.20) and (2.21) based on flume experiments with radial gates. They obtained Csg values of 0.89 and 0.57 for both submerged and free‐flow conditions respectively. D Yup Ydwn wsg Ydwn’ Figure 2.4. Two‐dimensional representation of a large cylinder acting as a sluice gate 16 Swamee (1992) presented a simple approach to calibrate vertical sluice gate coefficients. However, both free flow and submerged flow conditions where expressed with equation (2.21). The influence of the downstream water depth for submerged flow conditions was included in the equation that calculates the discharge coefficient. This corresponds to what was suggested by Finnermore and Franzini (2002): Csg can absorb the effects of the upstream velocity and downstream water depth. Clemmens et al. (2003) developed the more complex Energy‐Momentum (E‐M) method to calibrate radial gate discharges under both free‐flow and submerged flow conditions. They commented that the transition between free‐flow and submerged flow could be well captured but still needed refinement. Tony and Wahl (2005) used a large data set to improve the precision of the empirical relations developed by Clemmens et al. (2003). The approach proposed by Swamee (1992) and Clemmens et al. (2003) was based on an iterative method that is not convenient for the purpose of the present study. Here, qsg is known while Yup and Csg (which depends on Yup and Ydwn) have to be defined. Shahrokhnia and Javan (2006) unsurprisingly stated that the accurate estimation of the discharge coefficient is the main problem in evaluating the discharge for all kinds of sluice gates. They proposed that Csg is affected by viscosity, velocity, turbulence, velocity distribution, and the shape of the gate. A similar statement was proposed by Ferro (2000). In this study, the viscosity of water, and shape and roughness of the gate (cylinders) was assumed to be constant. Turbulence is probably the most difficult parameter to put into equation. Sheridan et al. (1997) performed flume experiments of flow past a cylinder close to a free surface focusing on the velocity distribution, vorticity distributions and turbulence of the flow. Based on visual records, they observed a link between Froude numbers and the cylinder elevation (wsg), velocity distribution, and vorticity distribution. Considering these results, it can be assumed that the flow velocity, velocity distribution and turbulence past a cylinder can be related to the downstream Froude number and cylinder elevation. This hypothesis is expressed by the following function: C sg = f (Frdwn , wsg ) (2.22) (2.23) Here, the downstream Froude number Frdwn is defined as: Frdwn = q g (Ydwn ) 1.5 17 One striking conclusion about orifice flow research is that the terms “free flow conditions” and “submerged flow conditions” are not defined precisely in literature, nor expressed as a function of the downstream Froude number. One could assume that when the sluice gate opening is smaller than the critical water depth (Yc, a property of the unit discharge) ⎛ q2 Yc = ⎜⎜ ⎝ g ⎞ ⎟⎟ ⎠ 0.33 (2.24) free‐flow conditions will occur if the downstream Froude number is lesser than 1 (supercritical conditions) and submerged flow conditions will be observed if the hydraulic jump reaches the downstream side of the gate (subcritical conditions). This hypothesis is comparable to a description made by Clemmens et al. (2003) who proposed that submergence occurs when the rising downstream water depth starts to affect the upstream water depth. However, for high gate elevations (wsg>Yc), this hypothesis is ambiguous. The importance of the critical water depth in the understanding of orifice flows is only known to have been mentioned by Ferro (2000) and Shahrokhnia and Javan (2006). Both studies were based on dimensional analysis and captured the importance and versatility the critical water depth. The dimensionless parameter Yc/wsg proposed by Ferro (2000) and Shahrokhnia and Javan (2006) and the hypothesis expressed in equation (2.22) inspired a series of analyses that will be described in Section 5.3. 2.3.3 Weir and Orifice Flows When a cylinder lies in mid‐flow (refer to Figure 2.2b), both weir and orifice flow unit discharges (equation (2.16)) have to be considered (qw and qsg respectively). Because both orifice and weir flows have a direct effect on the upstream water depth, the problem becomes complex. To our knowledge, Ferro (2000) is the only study based on hydraulic equations that attempted to solve this problem. A dimensional approach was adopted and empirical equations were calibrated to solve the problem. However, the downstream water depth was not considered and the influence of the downstream hydraulic conditions was not mentioned. Therefore, the equations proposed by Ferro (2000) are not compatible with subcritical flows observed in natural settings and were consequently not considered in the present work. 18 2.3.4 Conclusion on the Evaluation of the Upstream Hydraulic Conditions As for the drag force estimation using equation (2.1), it does not seem possible to evaluate the hydraulic conditions created upstream of a cylinder without having to deal with ambiguous coefficients. To our knowledge, a cylindrical element acting as a weir and a sluice gate has not yet been considered in literature. As previously mentioned, most of the research done on weirs and sluice gates try to evaluate the discharge with known upstream hydraulic conditions while the objective of this project is to evaluate the upstream hydraulic conditions knowing the total discharge and downstream hydraulic conditions. As a result, equations proposed in literature can only be considered in this work to a certain extent. As stated in Section 1.1, the second objective of this project was to develop a method that evaluates the upstream hydraulic conditions based on the following hypothesis: for a given combination of unit discharge (q), cylindrical element diameter (D), sluice gate opening (wsg), and downstream water depth (Ydwn), there is only one possible upstream water depth (Yup). This assumption would be valid for rectangular or wide channels where the channel width (B) is equal to the cylinder length (l), for constant element skin roughness, and for cylindrical elements disposed parallel to the channel bed and perpendicular to the flow direction. This method is developed in Chapter 5.0. 19 3.0 Experiments A series of flume experiments were performed in the hydraulics laboratory of the Department of Civil Engineering at University of British Columbia with the objective to determine the drag and lift forces acting on a range of “large” cylinders located at different elevations from the channel bed. The term “large” refers to significant blockage ratios. The hydraulic conditions upstream and downstream of the cylinders were also taken into account. This chapter presents a complete description of the flume experiments. 3.1 Experimental Material Figure 3.1 presents a scheme of the experimental setup. Each component of the setup is described in the following subsections. Figure 3.1. Scheme of the flume and experimental setup 20 3.1.1 Flume Description The Plexiglas flume (Figure 3.2) used for the experiments was 6.232m long (X or streamwise axis), 0.476m high (Y axis), and 0.152m wide (Z axis). The pump had a capacity of 0.020m3/s. The discharge was controlled by a butterfly valve. A small reservoir and series of flow conditioners located at the head of the flume (X=0.000m) ensured an initially uniform velocity distribution over the width of the channel. An adjustable thin plate weir gate located at the downstream end of the flume made the control of the water depth possible. The slope of the flume was set to 0% for all the experiments. 3.1.2 Test Cylinders Low roughness solid Polyvinyl Chloride (PVC) cylinders were used for the experiments (Shields and Gippel, 1995; Gippel et al., 1996; Hygelund and Manga, 2003). Their density (ρcyl) was 1 409 kg/m3. It was assumed that the hydraulics of PVC cylinders and LWD would be governed by the same dominant factors. Baudrick and Grant (2000) mentioned that cylinders represent a reasonable model for logs in streams. Five different cylinder sizes (Figure 3.3) where chosen: from 2 to 6 inches in diameter with a 1 inch increment. The length (l) of each cylinder was precisely adjusted to fit the width of the flume B (0.152 m). Figure 3.2. Experimental flume in the hydraulics laboratory of the Department of Civil Engineering at University of British Columbia 21 Figure 3.3. PVC cylinders used for the experiments 3.1.3 Water Depth Measurements Water depths were measured upstream (Yup) and downstream (Ydwn) of the cylinder location in the flume using rulers (0.300m long, 0.001m increments) fixed to the transparent flume walls. Depths were measured relative to the horizontal flume bed. Other rulers were used to measure the water level at the contraction in the cylinder region as well as the cylinder elevation (wsg) and the weir flow depth (Yw) on top of the cylinder. 3.1.4 Force Measurement Apparatus One objective of this research was to investigate the applicability of the momentum equation (2.4). Fd = ρg (M up − M dwn ) The right hand side of the equation depends on experimentally measured hydraulic parameters (Y and q) while the left hand side of the equation was measured independently. Gippel et al. (1996) used a dynamometer to measure the drag force applied on cylinders. Wallerstein et al. (2001) and Wallerstein et al. (2002) used an electronic balance to evaluate the drag force while Hygelund and Manga (2003) used a torque wrench for the same purpose. For this research project, a load cell (model MC3A from AMTI, Figure 3.4) was used. The load cell presented 6 degrees of freedom, but only three components 22 of interest were considered: the streamwise force FFx, the vertical force FFy, and the moment MMz (which is equivalent to FFx multiplied by a distance h). Output from the load cell was expressed as a voltage. A number of calibration tests (Appendix A) using weights were performed to convert the voltage signals into force and moment units. It was found that the voltage and the applied load were linearly proportional. Gippel et al. (1996) commented that vibration of the force‐measuring apparatus made accurate measurement difficult during their experiments. Similarly, Alonso (2004) suggested that vortex shedding due to flow separation from cylinders may lead to severe flow‐induced body vibration. To limit the errors related to this phenomenon, the load cell was connected to a computer and signals were recorded every second (1 Hz). The duration of each test varied from 15 sec to 329 sec with a typical test being performed for 90 sec. (Subsection 4.1.1). 3.1.5 Flow Meter The discharge was measured using a GE Panametrics AT 868 flow meter that was fixed to the recirculation pipe of the flume (Figure 3.5). The precision of this discharge‐measuring device was set to 0.001 L/s (10‐6 m3/s) but its true accuracy was not fully tested. Figure 3.4. Load cell used for the experiments 23 Figure 3.5. Flow meter and sensors installed on the flume recirculation pipe 3.1.6 Experimental Setup In order to measure the force acting on the cylinders with minimum interference with hydraulic conditions, the load cell was fixed to an aluminum platform (Figure 3.6) that was fixed to the rails along the top of the flume. The upper part of the load cell was connected to the test cylinder with two vertical rods, acting as two arms holding the cylinder in the water. The rods were adjustable in the vertical direction (Y axis). Therefore, it was possible to adjust the elevation of the cylinder for each experiment in order to vary the distance between the cylinder and the flume bed (wsg). The vertical rods were machined to present a more streamlined profile in order to minimize the residual drag. Their maximum thickness (Z axis) after machining was 0.002 m (originally 0.005m) and their width was 0.025m in the streamwise direction (X axis). The rods presented a lateral blockage of about 3% of the flume width for weir flows. 24 Figure 3.6. Platform, load cell, and profiled vertical rods The experimental setup is presented in Figure 3.7. The platform was located at 3.800m from the flume head (X = 3.800m), where the velocity profile appeared to be stable. The ruler measuring the upstream water elevation was located 1.100m upstream of the aluminum platform location (X=2.700m). This conforms to the upstream location definition mentioned in Subsection 2.1.3. The ruler measuring the downstream water depth was located 1.200m downstream of the aluminum platform location (X=5.000) in order to avoid the hydraulic instabilities produced at the cylinder location as well as to avoid the effect of the flow acceleration as it approaches the downstream weir gate (X=6.230m). Important water surface fluctuations were expected in the downstream part of the flume, especially when hydraulic jumps would form downstream of the cylinder. This could eventually produce unacceptable estimation errors of Ydwn using a conventional ruler described in Subsection 3.1.3. Therefore, it was decided to use a Point Gauge fixed to a Vernier (Figure 3.8) with a precision of 10‐4m to measure the downstream gate elevation (hgate). It was assumed that for given measured hgate and q, there is only one possible Ydwn. Point Gauge measurements would represent a reference that would be used to confirm or adjust suspicious Ydwn values. Results of this analysis are exposed in Appendix B. Point Gauge measurements could not be used to confirm or adjust supercritical downstream water depth values (Equation (2.17) is only valid for subcritical upstream water depths). 25 Figure 3.7. Experimental setup fixed to the flume Finally, in order to minimize cylinder‐wall interference, which could significantly affect the load cell force and moment measurements, grease was used to lubricate the ends of each cylinder as well as the profiled rod sides adjacent to the flume walls. Occasionally, when small cylinders (diameter of 2 and 3 inches) were tested, interference with the wall was evident in the load cell signal. In these circumstances, light tapping of the flume wall at the cylinder location would usually release the friction between the cylinder and the walls. 26 3.2 Figure 3.8. Vernier and Point Gauge at the downstream end of the flume Experimental Set 1 One objective of the experiments was to investigate the influence of the unit discharge (q), cylinder size (D), cylinder elevation (wsg), and downstream water depth (Ydwn) on the upstream water depth (Yup). This would be completed under a wide range of hydraulic conditions including blockage ratios (BR) and Froude numbers (Fr). A series of experiments were planned to investigate one of the four variables q, D, wsg, and Ydwn at a time, while holding the other three variables constant. Fifteen experiments for each cylinder (diameter of 2, 3, 4, 5, and 6 inches) were proposed (Table 3.1). Three flume discharges (Q) were chosen (5, 10, and 15L/s) and 6 different cylinder elevations from the channel bed (wsg) were selected (0, 10, 30, 48, 76, and 100 mm). The last 3 elevations correspond to the critical water depth (Yc from equation (2.24)) related to the 3 discharges (48mm, 76mm, and 100mm for 5L/s, 10L/s, and 15L/s respectively). Finally, each of the 15 experiments presented in Table 3.1 would include up to 11 distinct downstream hydraulic conditions, from a water depth Ydwn of about 0.3m to a maximum downstream weir gate opening (hgate=0.0m). As a result, a maximum of 825 hydraulic states (or runs) distributed over a total 27 of 75 experiments were initially planned for the Experimental Set 1. The measured or recorded parameters for each hydraulic state are presented in Table 3.2. For experiments when the cylinder was resting on the flume bed (wsg = 0), the load cell measurements were not collected because of the interference between the flume bed and the cylinder. In these cases (15 experiments), the drag and lift forces had to be estimated using theoretical equations and measured hydraulic parameters. Q = 5L/s Q= 10L/s Q= 15L/s wsg = 0 mm X X X wsg = 10 mm X X X wsg= 30 mm X X X wsg= 48 mm X X X wsg= 76 mm X X wsg= 100 mm X Table 3.1. 15 experiments performed for each cylinder size (marked with an “X”) Cylinder size D Distance from the channel bed wsg Discharge Q Downstream gate height hgate Downstream water level Ydwn Water depth at the contraction Ydwn’ Water depth on top of cylinder (weir flow) Yw Upstream water depth Yup Streamwise force signal Fx Streamwise moment signal Mz Vertical force signal Fy Table 3.2. Measured and recorded parameters for each experimental run 28 Since monitored discharge values were not constant over time, a tolerance range was defined for each discharge. If a monitored value fell outside of this ±2% range, the discharge was adjusted immediately with the valve. These corrections were mostly required at 5L/s and 15L/s while the discharge measurements appear to be stable at 10L/s. In order to simplify the experiment labeling, a notation that identifies each of the 75 experiments using the constant parameters D, wsg, and Q was adopted. For instance, the experiment for which the cylinder diameter (D) was 3 inches, the cylinder elevation from the channel bed (wsg) was 10 mm, and the flume discharge (Q) was 10 L/s, was labeled D3wsg010Q10. 29 4.0 Results Experimental Set 1 was completed between January and March 2008. This includes 75 experiments and 763 different hydraulic states. Subcritical downstream Froude numbers (Frdwn) ranged from 0.080 (D6wsg000Q05) to 0.859 (D2wsg076Q10), supercritical downstream Froude numbers ranged from 1.383 (D2wsg048Q15) to 3.171 (D6wsg000Q05), and the blockage ratio (BR) varied between 0.20 (D2wsg010Q05) and 0.83 (D6wsg030Q05). All experiments were initiated at relatively low Froude numbers (i.e., large water depth). Therefore, the water surface at the cylinders initially showed little disturbance (Figure 4.1a). As the downstream weir gate was lowered, a flow contraction appeared downstream of the cylinders and turbulence made downstream water depth measurements more difficult (Figure 4.1b). When the downstream water depth Ydwn reached a given level, this contraction (or trough) would collapse into a hydraulic jump that stands flat downstream of the cylinder location, and local turbulence would drastically decline (Figure 4.1c). As the downstream Froude number was set to higher values, turbulence intensified progressively (Figure 4.1d) and in most of the 75 experiments, the hydraulic jump separated from the cylinders (Figure 4.1e) and was finally flushed out of the flume (Figure 4.1f). These qualitative observations for each of the 763 hydraulic states are part of Experimental Set 1. At high cylinder elevation (wsg) and low downstream water depths (ydwn), instabilities were observed at the surface of the water just upstream of the cylinders and vortex pairs would form when the weir flow became shallow (Figure 4.1g). It was also observed that the appearance of these vortices would correspond to a sharp decline in the streamwise force signal (Fx) and a noticeable drop of the upstream water depth (Yup). The vortex pairs would fade as the downstream water depth was set to a lower level and they would disappear when the difference between the upstream and downstream water depth became small (Figure 4.1h). At this point, the cylinder would no longer be submerged and it would act as a sluice gate. 30 Vortex (a) (b) (c) (d) (e) (f) (g) (h) Figure 4.1. Scheme of different hydraulic states as the downstream water level decreases: (a) low turbulence (b) contraction (c) submerged hydraulic jump (d) hydraulic jump separating from cylinder (e) hydraulic jump separated from cylinder (f) supercritical conditions. For experiments where the cylinder is located at higher elevations (wsg): (g) vortex formation upstream of cylinder and (h) cylinder acting as a sluice gate. 4.1 Streamwise Forces One objective of this research was to compare the streamwise force measured with the load cell to the drag force calculated using hydraulic conditions and equation (2.4). Two independent signals were initially considered to measure the horizontal hydraulic forces: the streamwise force signal (Fx) and the moment signal (Mz). These two signals were converted into streamwise forces (refer to Appendix A) that turned out to be virtually identical for the majority of cases. However, the load cell was limited to measure moment values (MMz) less than about 9Nm (corresponding to forces ranging from 16.9N to 18.4N, depending on D and wsg). Since streamwise forces (FFx) as high as 29N were measured, it was decided to work primarily with the streamwise force signal. 31 4.1.1 Signal Distribution A total of 610 series of streamwise force data were recorded. This corresponds to the 60 experiments for which the load cell was in use, i.e, when wsg was greater than 0. Measurements were recorded at a frequency of 1 Hz, for durations ranging from 15 seconds to 329 seconds, with an average duration of 95 seconds. Series that presented a standard deviation smaller than 0.2N were 80 seconds long on average. Conversely, data sets with a standard deviation greater than 0.5N were about 170 seconds long on average. These high standard deviation values were observed for 12 different experiments, mostly with the 4 and 6 inch diameter cylinders. For 5 of these experiments, significant vibrations were observed during testing (D2wsg100Q15, D4wsg030Q10, D4wsg030Q15, D4wsg48Q15, D6wsg30Q10). This phenomenon was also reported by Gippel et al. (1996) and Alonso (2004). Instabilities as a consequence of vortex formation were noted in another 3 experiments (D6wsg030Q15, D6wsg048Q15, D6wsg100Q15), while during 4 of the 12 experiments (D4wsg076Q15, D6wsg048Q10, D6wsg076Q10, D6wsg076Q15), no irregularities were observed. The distribution of the signal records (126 in total) forming these 12 experiments was examined in detail. Since a normal distribution would usually be associated to representative data set sizes, the distribution shape of each record was investigated. Figure 4.2 presents 6 different distribution shapes, each divided into 9 intervals from the minimum to the maximum recorded values. Figure 4.2a and Figure 4.2d present relatively symmetric normal distributions and are thus associated to data sets of representative sizes. Figure 4.2e presents an apparently normal but asymmetric distribution. This data set was also considered to be of representative size. Figure 4.2c and Figure 4.2f present symmetric distributions that are the inverse of a normal distribution. This could be explained by the cylinders vibration tempo being in harmony with the signal recording rate (1Hz). Finally, Figure 4.2b presents a distribution that is neither normal nor symmetrical. However, this distribution shape could also be related to the signal recording rate. It was concluded that the signal distribution shape was a poor detector of anomalous load cell records. Consequently, the 610 average signal values were considered to be representative of the actual streamwise forces. 32 Fx Signal (Volt) 50 40 40 30 30 20 20 10 10 0 0 Fx Signal (Volt) [6.842‐7.410] 50 [6.274‐6.842[ 60 [5.706‐6.274[ 60 [5.138‐5.706[ 70 [4.570‐5.138[ (f) [4.002‐4.570[ [3.917‐4.120] [3.714‐3.917[ [3.511‐3.714[ [3.308‐3.511[ [3.105‐3.308[ [2.902‐3.105[ 160 140 120 100 80 60 40 20 0 [2.699‐2.902[ (d) [2.496‐2.699[ [3.889‐4.060] [3.718‐3.889[ [3.547‐3.718[ [3.376‐3.547[ [3.205‐3.376[ [3.034‐3.205[ [2.863‐3.034[ [2.692‐2.863[ 80 70 60 50 40 30 20 10 0 [3.434‐4.002[ D4wsg076Q15 ; Ydwn = 0.139m [2.350‐2.692[ [2.733‐2.750] [2.716‐2.733[ [2.699‐2.716[ [2.682‐2.699[ [2.665‐2.682[ [2.648‐2.665[ (b) [2.866‐3.434[ D4wsg030Q15 ; Ydwn = 0.066m [2.090‐2.496[ [4.70‐5.07] [4.33‐4.70[ [3.96‐4.33[ [3.59‐3.96[ [3.22‐3.59[ [2.85‐3.22[ [2.631‐2.648[ D2wsg100Q15 ; Ydwn = 0.127m [1.730‐2.866[ [2.878‐2.900] [2.856‐2.878[ [2.834‐2.856[ [2.812‐2.834[ [2.790‐2.812[ 70 [2.768‐2.790[ (e) [2.48‐2.85[ 45 40 35 30 25 20 15 10 5 0 [2.746‐2.768[ (c) [2.614‐2.631[ [2.580‐2.614[ Number of measurements 80 70 60 50 40 30 20 10 0 [2.11‐2.48[ [1.37‐2.11[ Number of measurements (a) [2.724‐2.746[ [2.680‐2.724[ Number of measurements D4wsg030Q10 ; Ydwn = 0.047m D4wsg048Q15 ; Ydwn = 0.140m D6wsg030Q10 ; Ydwn = 0.045m Figure 4.2. Distribution of the streamwise force signals (Fx) for six distinct experimental runs 33 4.1.2 Comparing Calculated Drag Forces to Measured Streamwise Forces The average value of each streamwise force data set (FFx) was compared to the drag force (Fd) calculated using the momentum equation (2.4). Figure 4.3 presents the measured streamwise force (FFx) and the calculated drag force (Fd) results for experiment D4wsg010Q10 expressed as a function of the downstream water depth (Ydwn). The 610 data points from 60 experiments are presented in Figure 4.4. Results show that measured streamwise forces acting on cylinders (FFx) are slightly smaller than drag forces (Fd) calculated using equation (2.4). It was expected that the indirectly calculated values of Fd based on the momentum equation would represent a conservative upper bound on the values of FFx measured by the load cell. The disparity between FFx and Fd in Figure 4.4 can be explained by (or related to): • The momentum correction factor β (Finnermore and Franzini, 2002) that was assumed to be equal to 1.0 in equation (2.4). This correction factor is usually greater than 1.0 because of the effect of the boundary (or side wall) drag and non‐uniform velocity distribution. • Hydraulic obstruction of the two profiled rods (interference with the force measurements) • The unknown precision of the flow meter • The load cell calibration, which is based on a data set of limited size (Appendix A). 16.0 Calculated drag force using equation (2.4) 14.0 Streamwise force measured by load cell 12.0 Fd and FFx (N) 10.0 8.0 6.0 4.0 2.0 0.0 0.00 0.05 0.10 0.15 Ydwn (m) 0.20 0.25 0.30 Figure 4.3. Drag force (F d) calculated using equation (2.4) and measured streamwise force (FFx) expressed as a function of the downstream water depth (Ydwn) for experiment D4wsg010Q10 34 35.0 Linear Regression 30.0 y = 1.0101x + 0.7269 R² = 0.9934 1:1 line Fd (N) 25.0 20.0 15.0 10.0 5.0 0.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 FFx (N) Figure 4.4. Drag force (Fd) calculated using equation (2.4) expressed as a function of the measured streamwise force (FFx) for 610 experimental runs The mean absolute error in Figure 4.4 is 0.80N. This error could not be related to the cylinder size or cylinder elevation. However, there is a clear influence of the discharge on the difference between Fd and FFx in Figure 4.4. For discharges of 5L/s, 10L/s and 15L/s, the average absolute difference between Fd and FFx is respectively 0.38N, 0.76N, and 1.16N. The first objective of this research (Section 1.1) was to indentify an equation that evaluates the drag force applied on a large cylinder. Results show small disparity between measured (FFx) and calculated drag forces (Fd) using the momentum equation (2.4). Because of this consistency, equation (2.4) is considered to be conservative as well as reliable. As a result, the drag force can be estimated without having to define the enigmatic drag coefficient (Subsection 2.1.6) required in equation (2.1). In the context of this project, the upstream water depth and the drag force are initially unknown. Consequently, equation (2.4) cannot be solved at this point. The second section of this chapter has for objective to identify patterns between known hydraulic parameters (D, wsg, Q, B, and Ydwn) and the unknown upstream water depth (Yup). 35 4.2 Hydraulic Data Set The hydraulic measurements from all 75 experiments from Experimental Set 1 (including the 15 experiments where the cylinder was resting on the channel bed and for which direct streamwise force measurements were not performed) were used to investigate the interaction between known hydraulic parameters and the upstream water depth. This section presents general trends and tendencies that were identified within the data. 4.2.1 Curve Types Drag force data from the 75 experiments were initially presented as a function of the downstream water depth (Ydwn). Figure 4.5 presents the drag force results for experiments D4wsg010Q10. These results were divided into 3 distinct curves. Starting from high downstream water depths and moving toward high Froude numbers (from right to left on the X axis), the 3 curves are described as follows: • Curve 1: For a Ydwn value higher than 0.195m, the relation between Fd and the Ydwn follows an upward curve (Figure 4.5). In this case, the water surface in the cylinder region presented a contraction of increasing amplitude as Ydwn was lowered. Hydraulic conditions in the cylinder regions were subcritical. This is referred to as the Subcritical Curve. • Curve 2: When Ydwn was lower than 0.195m, the contraction in the cylinder region collapsed into a submerged hydraulic jump and the relation between Fd and Ydwn followed a straight upward line. The hydraulic conditions in the cylinder region were critical but they remained subcritical downstream of the cylinder. This curve is called the Submerged Hydraulic Jump Curve. • Curve 3: When the Ydwn was lower than 0.115m, the hydraulic jump separated from the cylinder and was eventually flushed out of the flume. This was observed to be independent of Fd., which means that Ydwn could not longer affect Fd (and Yup). This has a straightforward explanation when one finds that the moment (equation (2.5)) at Ydwn=0.115m and Ydwn=0.046m is equivalent when q is equal to 0.066 m2/s. This horizontal line is referred to as the Supercritical Curve. During experiment D4wsg010Q10, Ydwn could not physically take a value between 0.115m and 0.046m. This explains the dashed line defining Curve 3. Note that the blue lines in Figure 4.5 represent the critical water depth (equal to 0.076m from 10L/s). Experiment D4wsg010Q10 is said to follow a Curve 1‐2‐3 model. Forty eight out of the 75 experiments from Experimental Set 1 followed an analogous model. 36 16.0 Critical depth Yc 14.0 D4wsg010Q10 Curve 3 12.0 Curve 2 Curve 1 Fd (N) 10.0 8.0 6.0 4.0 2.0 0.0 0.00 0.05 0.10 0.15 0.20 Ydwn (m) 0.25 0.30 0.35 Figure 4.5. Drag force (Fd) expressed as a function of the downstream water depth (Ydwn) for experiment D4wsg010Q10, and two‐dimensional schemes and photos representing the hydraulic conditions for each of the 3 curves 37 In Figure 4.6, the results from experiment D4wsg076Q10 present a different aspect. The trend of experiment D4wsg076Q10 was divided into 4 distinct curves that are described as follows (from right to left on the X axis in Figure 4.6): • Curve 1: The Subcritical Curve follows a similar trend that the one defined previously for experiment D4wsg010Q10 when Ydwn>0.197m. • Curve 2: When Ydwn ranged from 0.172m to 0.197m, a submerged hydraulic jump was also observed on the downstream side of the cylinder. • Curve 4: When Ydwn= 0.161m, a pair of vortex was observed upstream of the cylinder and Fd dropped sharply. The upstream water depth also dropped but the cylinder was still submerged (i.e. a weir flow was still observed on top of the cylinder). This is called the Vortex Curve. • Curve 5: When Ydwn<0.161m, the cylinder was no longer submerged and Fd continued to drop as Ydwn decreased. As a result, the cylinder acted as a sluice gate. This is referred to as the Sluice Gate Curve. Note that this curve was usually difficult to distinguish from the Vortex Curve. Therefore, the Vortex Curve will be considered as a transition between Curve 2 and Curve 5. Experiment D4wsg076Q10 is labeled to follow a Curve 1‐2‐5 model. Twenty three out of the 75 experiments from Experimental Set 1 followed a similar model. The five curves previously defined are less distinct in Figure 4.7 when Yup is expressed as a function of Ydwn for both experiments D4wsg010Q10 and D4wsg076Q10. Curve 1, which was described as an upward curve on Figure 4.5 and Figure 4.6, could be approximated as a straight line that is almost parallel to the 1:1 line for both experiments. Curve 2, which was a straight line in Figure 4.5, presents an arc that separates from the 1:1 line as Ydwn decreases (right to left in Figure 4.7). Curve 3 for experiment D4wsg010Q10 shows the same horizontal trend for low values of Ydwn as in Figure 4.5. The Sluice Gate Curve of experiment D4wsg076Q10 sharply separates from the Submerged Hydraulic Jump Curve and tends to shift toward the 1:1 line as Ydwn decreases. The blue lines in Figure 4.7 represent the critical water depth for both experiments (0.076m). The upstream water depth never reached the supercritical regime throughout all the tests (i.e. no data was observed under the horizontal blue line in Figure 4.7, independently of the parameters tested). When cylinders were located close to the channel bed, Curve 5 could not be obtained. Inversely, when the cylinder elevation (wsg) was high and the discharge (Q) was low, vortex pairs were often observed and cylinders usually acted as sluice gates once the downstream water depth was low. 38 16.0 Critical depth Yc 14.0 D4wsg076Q10 12.0 Curve 4 Curve 5 10.0 Fd (N) Curve 2 Curve 1 8.0 6.0 4.0 2.0 0.0 0.00 0.05 0.10 0.15 0.20 Ydwn (m) 0.25 0.30 0.35 Figure 4.6. Drag force (F d) expressed as a function of the downstream water depth (Ydwn) for experiment D4wsg076Q10, and two‐dimensional schemes and photos representing the hydraulic conditions for each of the 4 curves 39 0.35 Critical depth Yc D4wsg010Q10 D4wsg076Q10 1:1 line 0.30 Yup (m) 0.25 Curve 1 Curve 2 Curve 3 0.20 0.15 Curve 5 0.10 0.05 0.00 0.00 0.05 0.10 0.15 0.20 Ydwn (m) 0.25 0.30 0.35 Figure 4.7. Upstream water depth expressed as a function of the downstream water depth for both experiments D4wsg010Q10 and D4wsg076Q10 (Note that Curve 4 is considered a being part of Curve 5.) 4.2.2 Influence of Known Parameters This subsection presents partial results illustrating the effects of the main parameters D, wsg, Q, and Ydwn on Yup. Figure 4.8 presents an example of the influence of the cylinder size (D) using the results from experiments D2wsg048Q10, D3wsg048Q10, D4wsg048Q10, D5wsg048Q10, and D6wsg048Q10. All 5 experiments thus present a constant cylinder elevation (wsg) and discharge (Q). Results from experiments using small cylinders D2 and D3 (2 and 3 inch diameter respectively) follow a Curve 1‐2‐3 model (showing a transition from the 1:1 line to a horizontal line as Ydwn decreases). Conversely, the results from experiments with the largest cylinders D5 and D6 follow a Curve 1‐2‐5 model (showing a drop that tends to get closer to the 1:1 line as Ydwn decreases). The experiment with cylinder D4 is one of the rare examples (4 experiments out of 75 in total) during which a vortex pair was observed and where the cylinder was always submerged, independently of the downstream water depth. In this case, the results follow a Curve 1‐2‐4 model. 40 0.35 0.30 Yup (m) 0.25 0.20 0.15 Yc D2 D3 0.10 D4 D5 0.05 D6 1:1 0.00 0.00 0.05 0.10 0.15 0.20 Ydwn (m) 0.25 0.30 0.35 Figure 4.8. Example of the influence of the cylinder size (D) on the relation between the upstream water depth (Y up) and the downstream water depth (Ydwn) All 5 experiments presented in Figure 4.8 have the interesting aspect of reaching a downstream Froude number greater than 1 when Ydwn is low. This was not always the case for Curve 1‐2‐5 models. Results from experiment D4wsg076Q10 presented in Figure 4.7 supports this observation. Figure 4.9 presents an example of the effect of the cylinder elevation (wsg) variation on the upstream water depth using the results from experiments D4wsg000Q10, D4wsg010Q10, D4wsg030Q10, D4wsg048Q10, and D4wsg076Q10. All 5 experiments thus present a constant cylinder size (D) and discharge (Q). Figure 4.9 shows that experiments with cylinder elevation equal or smaller than 30mm follow the Curve 1‐2‐3 model (constantly separating from the 1:1 line as Ydwn decreases) while the experiment with a cylinder elevation of 76mm clearly follows a Curve 1‐2‐5 model (net separation from the Submerged Hydraulic Jump Curve and shift toward the 1:1 line). 41 0.35 0.30 Yup (m) 0.25 0.20 0.15 Yc wsg000 wsg010 0.10 wsg030 wsg048 0.05 wsg076 1:1 0.00 0.00 0.05 0.10 0.15 0.20 Ydwn (m) 0.25 0.30 0.35 Figure 4.9. Example of the influence of the cylinder elevation (wsg) on the relation between the upstream water depth (Y up) and the downstream water depth (Ydwn) In Figure 4.9, all Subcritical Curves (Ydwn > 0.20m) are superposed and the effect of the cylinder elevation becomes visible within the Submerged Hydraulic Jump Curve. A cautious investigation of the downstream water depth corresponding to the separation point of all the experimental results presented in Figure 4.9 reveals that Ydwn roughly coincides with the cylinder size (D) added to the critical water depth (Yc). This value is equal to 0.178m in Figure 4.9. The importance of the parameter (D+Yc) will be investigated in detail in Chapter 5.0. Figure 4.9 proposes that, the higher the cylinder position, the lower the upstream water depth for a given downstream water depth. This is in contradiction with observations by Wallerstein et al. (2002) about an increase of the drag force for low submergence (i.e. higher cylinder elevation) due to surface waves. Small waves were observed travelling upstream prior to vortex initiation in low submergence experiments but it was assumed that they were due to complex velocity distribution or oscillation of the hydraulic jump downstream of the cylinder. No measurable effect on the drag force was related to these waves. 42 The effect of the cylinder elevation on the upstream water depth can be explained with the following hypothesis: once critical hydraulic conditions are reached at the cylinder section, an orifice flow is more efficient than a weir flow of equivalent height for the same downstream water depth. This is supported by a comparison of equations (2.17) and (2.21) using coefficients suggested in literature. Finally, Figure 4.10 presents an example of the effect of the discharge (Q) variation on the upstream water depth using results from experiments D4wsg010Q05, D4wsg010Q10, D4wsg010Q15. All 3 series in Figure 4.10 thus present a constant cylinder size (D) and cylinder elevation (wsg). The experimental results presented in Figure 4.10 follow a Curve 1‐2‐3 model. Moreover, all 3 curves present similar but shifted trends. The vertical and horizontal lines in Figure 4.10 represent the critical water depth related to each discharge (Yc=0.048m, 0.076m, and 0.100m for Q=5L/s, 10L/s, and 15L/s respectively). It appears that the distance between the critical water depth lines is comparable to the vertical and horizontal shift of the three series. Therefore, Yc would represent a potential valuable parameter to develop a relation between Ydwn and Yup that would be applicable for any discharge. 0.35 0.30 Yup (m) 0.25 0.20 Yc (5 L/s) 0.15 Yc (10 L/s) Yc (15 L/s) 0.10 Q05 Q10 0.05 Q15 1:1 0.00 0.00 0.05 0.10 0.15 0.20 Ydwn (m) 0.25 0.30 0.35 Figure 4.10. Example of the influence of the discharge (Q) on the relation between the upstream water depth (Yup) and the downstream water depth (Ydwn). The critical depths (Yc) are identified for each discharge. 43 The next chapter of this work presents the development phases of a dimensionless approach that will be used to evaluate the upstream hydraulic conditions with any given value of cylinder size (D), cylinder elevation from the channel bed (wsg), discharge (Q), downstream water depth (Ydwn), and channel width (B). Analysis of the vertical forces (including lift force and buoyancy) is presented in Appendix C. 44 5.0 Result Analysis 5.1 Legend Convention In order to facilitate the comprehension of the figures included in the present chapter, a legend structure was defined to differentiate each of the 75 experiments from Experimental Set 1. The cylinder size (D) follows a color code: The cylinder elevation from the channel bed (wsg) can be identified with different markers. Finally, the discharge (Q) can be differentiated with the marker size and line thickness. 5.2 Curve 123 Model Of the 75 flume experiments from Experimental Set 1, 48 experiments follow a Curve 1‐2‐3 model, which was described in Chapter 4.0 as a relation between the upstream water depth and downstream water depth when cylinders are submerged independently of the downstream water depth. These 48 experimental results are presented in Figure 5.1. The 3 blue vertical and horizontal lines in Figure 5.1 represent the critical water depth (Yc) for each discharge. The 1:1 (Yup=Ydwn) line is also presented. 45 0.35 0.30 Yup (m) 0.25 0.20 0.15 0.10 0.05 0.00 0.00 0.05 0.10 0.15 0.20 Ydwn (m) 0.25 0.30 0.35 Figure 5.1. Upstream water depth (Yup) expressed as a function of the downstream water depth (Ydwn) for 48 experimental results following a Curve 1‐2‐3 model. Critical water depths (Yc) and 1:1 line are presented as well. 46 The objective of this section is to develop a unifying formulation that would collapse all 48 experimental results into one dimensionless model. This is achieved by developing a dimensionless parameter using known parameters, leaving the upstream water depth as the only unknown. As a first step, experiments during which the cylinder was located on the flume bed (wsg = 0) are initially considered. These 15 experiments are presented in Figure 5.2. The different series represent different values of D and Q. The list of possible parameters that can be used to develop a dimensionless compound parameter includes the unit discharge (q), critical water depth (Yc), cylinder size (D), gravity constant (g), and kinematic viscosity (ν). A formal dimensional analysis was not required since the solution was straightforwardly developed, based on the parameter (D+Yc) previously identified in Subsection 4.2.2. The dimensionless water depth takes the following form: Y (D + Yc ) (5.1) Here, Y is the water depth either upstream or downstream of the cylinder. 0.35 0.30 Yup (m) 0.25 0.20 0.15 0.10 0.05 D2wsg000Q05 D2wsg000Q10 D2wsg000Q15 D3wsg000Q05 D3wsg000Q10 D3wsg000Q15 D4wsg000Q05 D4wsg000Q10 D4wsg000Q15 D5wsg000Q05 D5wsg000Q10 D5wsg000Q15 D6wsg000Q05 D6wsg000Q10 D6wsg000Q15 0.00 0.00 0.05 0.10 0.15 0.20 Ydwn (m) 0.25 0.30 0.35 Figure 5.2. Upstream water depth (Yup) expressed as a function of the downstream water depth (Ydwn ) for 15 experiments during which the cylinder was located on the flume bed. Critical water depths (Yc) and 1:1 line are presented as well. 47 Figure 5.3 presents the collapsed results with dimensionless compound parameter (5.1) on each axis. Results from experiment D2wsg000Q15 are the only ones that separate from the general trend. Interestingly, water depth measurements were labeled as being highly unstable and difficult to perform during experiments D2wsg000Q15 and D3wsg000Q15. No difficulties or anomalies were detected during all 13 other experiments presented in Figure 5.2 and Figure 5.3. The average dimensionless upstream water depth for shallow downstream water depths in Figure 5.3 is 1.08. If one considers the energy equation (2.11) to evaluate Yup when the flow conditions are critical at the cylinder section and assuming no energy losses, the resulting dimensionless upstream water depths range from 1.11 to 1.24. One remarkable aspect of Figure 5.3 is that no coefficient was required to obtain such a compact collapse. It clearly presents the 3 different curves described in Subsection 4.2.1. Curve 1 is roughly linear for values above about 1.40 on the X axis. Curve 2 tends to separate from the 1:1 line for values ranging between 0.75 to 1.40 on the X axis and presents a smooth transition to Curve 3. For values smaller than 0.75, Curve 3 is essentially horizontal. 2.00 1.75 1.50 Yup / (D + Yc) 1.25 1.00 0.75 0.50 0.25 D2wsg000Q05 D2wsg000Q10 D2wsg000Q15 D3wsg000Q05 D3wsg000Q10 D3wsg000Q15 D4wsg000Q05 D4wsg000Q10 D4wsg000Q15 D5wsg000Q05 D5wsg000Q10 D5wsg000Q15 D6wsg000Q05 D6wsg000Q10 D6wsg000Q15 0.00 0.00 0.25 0.50 0.75 1.00 1.25 Ydwn / (D + Yc) 1.50 1.75 2.00 Figure 5.3. Upstream water depth (Yup) expressed as a function of the downstream water depth (Ydwn ) for 15 experiments during which the cylinder was located on the flume bed using compound parameter (5.1). 1:1 line is also presented. 48 The third investigated parameter, the cylinder elevation from the channel bed (wsg), was subsequently aggregated to the analysis. If one applies compound parameter (5.1) on all 48 experiments presented in Figure 5.1, the result is evidently less compact than what was presented in Figure 5.3. The residual influence of wsg is visible in Figure 5.4. The effect of the cylinder elevation could be considered with a second dimensionless compound parameter that takes the following form: Y (D + Yc − α1α 2 wsg ) (5.2) In Figure 4.9, it was observed that the effect of wsg on Yup was only noticeable when Ydwn<(D+Yc). A similar observation can be made in Figure 5.4 when Ydwn/(D+Yc)<1. Moreover, the greatest separation between the experimental results presented in Figure 5.4 is visible in the shallow downstream water depth region. Therefore, it was decided that the coefficient α1 in parameter (5.2) would be calibrated using the shallowest downstream water depth of each experiment. 2.00 1.75 1.50 Yup / (D + Yc ) 1.25 1.00 0.75 0.50 0.25 0.00 0.00 0.25 0.50 0.75 1.00 1.25 Ydwn / (D + Yc) 1.50 1.75 2.00 Figure 5.4. Upstream water depth (Yup) expressed as a function of the downstream water depth (Ydwn ) for 48 experimental results following a Curve 1‐2‐3 model with compound parameter (5.1) on both axes. 1:1 line is also presented. The spread in the results is generated by the different cylinder elevation (wsg) values. 49 In Figure 5.3, the minimum and maximum (excluding experiment D2wsg000Q15) values on the Y axis for the shallowest downstream water depth are 1.071 (D5wsg000Q15) and 1.105 (D2wsg000Q10) respectively. These two values were set as limits to calibrate coefficient α1. Details of this calibration based on 30 experiments out of 33 (48 experiments in total minus 15 experiments where wsg was 0) are presented in Appendix D. Experiments D2wsg010Q05, D5wsg010Q15, and D5wsg030Q15 were not considered in the calibration process for reasons listed in Table 6.1. The best‐fit resulting equation is: ⎛D⎞ α 1 = 0.6⎜⎜ ⎟⎟ ⎝ Yc ⎠ 0 .8 − 0.3 (5.3) Note that α1 cannot be negative and has to be set to 0 when the ratio D/Yc is smaller than 0.42. The second coefficient α2 in compound parameter (5.2) was then related to the applicability of α1. It takes a value of 0 when Ydwn/(Yc+D) < 1 and takes a value of 1 when curve 3 is reached. The transition between both curve sections is assumed to be linear. If Ydwn (D + Yc ) > 1.00 => α2 = 0.00 [0.75 , 1.00] => α2 = 4.00⎜⎜1 − < 0.75 => α2 = 1.00 ⎛ ⎝ Ydwn ⎞ ⎟ (D + Yc ) ⎟⎠ (5.4) The results using compound parameter (5.2), equation (5.3), and equation (5.4) are presented in Figure 5.5. The collapse in Figure 5.5 is visually more compact than what was presented in Figure 5.4. From these results, a reference graph was developed (Figure 5.6). This graph should be valid for Curve 1‐2‐3 models independently of the unit discharge (q), cylinder size (D), cylinder vertical position (wsg), and downstream water depth (Ydwn). To my knowledge, Figure 5.6 has no precedent in literature. Indications propose that considering small cylindrical elements in Figure 5.6 could slightly underestimate the upstream water depth. 50 2.00 1.75 Yup / (D + Yc ‐ α1α2wsg ) 1.50 1.25 1.00 0.75 0.50 0.25 0.00 0.00 5.3 0.25 0.50 0.75 1.00 1.25 Ydwn / (D + Yc ‐ α1α2wsg) 1.50 1.75 2.00 Figure 5.5. Upstream water depth (Yup) expressed as a function of the downstream water depth (Ydwn) for 48 experimental results following a Curve 1‐2‐3 model with compound parameter (5.2) on both axes. 1:1 line is also presented. Curve 125 Models The 23 experimental results following a Curve 1‐2‐5 model are presented in Figure 5.7 in terms of the dimensionless depth parameter (5.1). When Ydwn/(D+Yc) > 1.0, all experimental results collapsed in the same trend that was observed in Figure 5.4 for Curve 1‐2‐3 models. However, it was not possible to collapse these curves for values smaller than 1.0 on the x axis using compound parameter (5.2). Since Curve 5 refers to the Sluice Gate Curve in Subsection 4.2.1, the unifying method to describe Curve 5 was elaborated using the sluice gate equation (2.21). When qsg = q, equation (2.21) becomes q = C sg wsg 2 gYup (5.5) where Csg is a sluice gate coefficient. 51 2.00 1.75 Yup / (Yc+D‐α1α2wsg) 1.50 1.25 1.00 0.75 0.50 0.25 0.00 0.00 0.25 0.50 0.75 1.00 1.25 Ydwn / (Yc+D‐α1α2wsg) 1.50 1.75 2.00 Figure 5.6. Reference Graph for Curve 1‐2‐3 models with compound parameter (5.2) on both axes Taking into account the critical depth equation (2.24) and following a similar approach to what was proposed by Ferro (2000) and Shahrokhnia and Javan (2006), equation (5.5) takes this form: ⎛Y =⎜ c ⎜Y Yc ⎝ up wsg ⎞ ⎟ ⎟ ⎠ 0.5 1 C sg 2 (5.6) The left hand side of equation (5.6) presents a particularly interesting dimensionless parameter (wsg/Yc). On the right hand side, the upstream water depth Yup and the sluice gate coefficient Csg are initially unknown. It was proposed in Subsection 2.3.2 with equation (2.22) that Csg would depend on both downstream hydraulic conditions and cylinder elevation. Equation (5.6) was rearranged into C sg Y = c wsg ⎛ Yc ⎜ ⎜Y ⎝ up ⎞ ⎟ ⎟ ⎠ 0.5 1 2 (5.7) and the data set presented in Figure 5.7 was used to generate Figure 5.8. Two additional experimental results, presented in a distinct legend in Figure 5.8, were included to complete the overall picture of this analysis: D4wsg065Q10 and D5wsg065Q10. 52 2.00 1.75 Yup / (D + Yc ) 1.50 1.25 1.00 0.75 0.50 0.25 0.00 0.00 0.25 0.50 0.75 1.00 Ydwn / (D + Yc) 1.25 1.50 1.75 2.00 Figure 5.7. Upstream water depth (Yup) expressed as a function of the downstream water depth (Ydwn) for 23 experimental results following a Curve 1‐2‐5 model with compound parameter (5.1) on both axes. 1:1 line is also presented. The highest back‐calculated (cylindrical) sluice gate coefficient in Figure 5.8 is 0.86 (D6wsg030Q05), which is considerably higher than the maximum value of 0.6 proposed by Finnemore and Franzini (2002) for conventional sluice gates. Since Figure 5.8 only includes 115 data points, a limited number of conclusions were initially made. 53 1.00 0.90 0.80 wsg/Yc = 0.63 0.70 Csg wsg/Yc = 0.77 0.60 wsg/Yc = 0.86 0.50 wsg/Yc = 1.00 0.40 D4wsg065Q10 D5wsg065Q10 0.30 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 Frdwn Figure 5.8. Back‐calculated sluice gate coefficients (Csg) for 25 experimental results following a Curve 5 model expressed as a function of the downstream Froude Number (Fr dwn) The first evident observation in Figure 5.8 is that 4 groupings defined by different wsg/Yc ratios can be distinguished. The grouping identified as wsg/Yc = 0.63 is formed by 6 experiments and presents a certain scatter. The grouping identified as wsg/Yc = 0.77 is only formed by 3 experiments and is less distinct. Finally, the groupings identified as wsg/Yc = 0.86 and wsg/Yc = 1.00 are formed by 2 and 14 experiments respectively and are fairly consistent. The second observation in Figure 5.8 is the central effect of the downstream hydraulic conditions (Froude number) on the sluice gate coefficient. The hypothesis stated in equation (2.22) was modified for cylinders acting as sluice gates: w ⎛ ⎞ C sg = f ⎜ Frdwn , sg ⎟ Y c ⎠ ⎝ (5.8) An additional set of 25 experiments (Experimental Set 2) including a total 175 data point distributed over 9 wsg/Yc ratios was performed in July 2008 to improve the precision of Figure 5.8. Sluice gate coefficients were back‐calculated using equation (5.7) and values as high as 1.00 were obtained. 54 A family of curves categorized by different wsg/Yc ratios (from 0.4 to 1.2 with 0.1 increments) was manually interpolated within the results from both Experimental Set 1 and Experimental Set 2 (total of 290 data points). Results are presented in Figure 5.9. Consistent experimental results with low wsg/Yc ratios (lower than 0.6) were difficult to obtain, even in the controlled experimental environment described in Chapter 3.0. Supercritical conditions were seldom obtained downstream of the cylinders for data series with wsg/Yc ratios smaller than 0.80. It is well approved in the literature that supercritical downstream hydraulic conditions cannot affect upstream hydraulic conditions. Therefore, the supercritical extension of each manually interpolated curve, was assumed to be horizontal when the downstream Froude number (Frdwn) was greater than 1 (then, Yup would be independent of Frdwn). For wsg/Yc ratios greater than 0.8, the supercritical extension was not presented but our hypothesis is that this extension should be horizontal as well. Figure 5.9 was used to create the reference graph (Figure 5.10) for cylinders acting as sluice gates (Curve 5). To my knowledge, Figure 5.10 has no precedent in the literature. It was not obtained from a meticulous physical analysis, but it appears compatible with the orifice flow theory. 1.05 wsg/Yc = 0.40 wsg/Yc = 0.50 0.95 wsg/Yc = 0.60 0.85 Exp Set 2: wsg/Yc = 0.40 Exp Set 2: wsg/Yc = 0.50 Exp Set 2: wsg/Yc = 0.60 Exp Set 1: wsg/Yc = 0.63 Exp Set 2: wsg/Yc = 0.70 Exp Set 1: wsg/Yc = 0.77 Exp Set 2: wsg/Yc = 0.80 Exp Set 1: wsg/Yc = 0.86 Exp Set 2: wsg/Yc = 0.90 Exp Set 1: wsg/Yc = 1.00 Exp Set 2: wsg/Yc = 1.00 Exp Set 2: wsg/Yc = 1.10 Exp Set 2: wsg/Yc = 1.20 wsg/Yc = 0.70 Csg 0.75 wsg/Yc = 0.80 wsg/Yc = 0.90 0.65 wsg/Yc = 1.00 0.55 wsg/Yc = 1.10 wsg/Yc = 1.20 0.45 0.35 0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 Frdwn 1.75 2.00 2.25 2.50 2.75 Figure 5.9. Results from Experimental Set 1 and Experimental Set 2 where the sluice gate coefficient (Csg) is expressed as a function of the downstream Froude numbers (Frdwn). 55 Sluice Gate Curves (Curve 5) can be reconstructed if one considers Figure 5.10 (input data: Frdwn, wsg, and Yc) and the upstream hydraulic conditions are obtained using a transformation of equation (5.5): Yup = 1 Yc3 2C sg2 wsg2 (5.9) For a given cylinder size (D), cylinder elevation (wsg), discharge (Q), channel width (B), and downstream water depth (Ydwn), it is not initially possible to confirm whether the cylinder will be submerged or not (i.e., it is not initially possible to determine if this given case falls on a Curve 2, 3, or 5). The next section will answer this question: Which reference graph (Figure 5.6 or Figure 5.10) should be used in order to estimate the upstream water depth for given downstream hydraulic conditions, cylinder size and cylinder vertical position? 1.05 wsg/Yc = 0.4 wsg/Yc = 0.5 0.95 wsg/Yc = 0.6 0.85 wsg/Yc = 0.7 Csg 0.75 wsg/Yc = 0.8 wsg/Yc = 0.9 0.65 wsg/Yc = 1.0 wsg/Yc = 1.1 0.55 wsg/Yc = 1.2 0.45 0.35 0.25 0.00 0.25 0.50 0.75 1.00 1.25 Frdwn Figure 5.10. Reference Graph for cylinders acting as sluice gates (Curve 5) 1.50 1.75 2.00 56 5.4 Separation between Curve 2 and Curve 5 When the weir flow over the cylinder became shallow, vortices were observed upstream of the cylinders. This corresponds to the beginning of Curve 4, which is considered as a transition between Curve 2 and Curve 5. The first evidence that the cylinder would not be submerged for a given set of parameters is when the upstream water depth evaluated using Figure 5.6 would have a smaller value than the cylinder size (D) added to the cylinder elevation (wsg). Yup < (D + wsg ) (5.10) In this case, it is recommended to refer to Figure 5.10 to evaluate the upstream water depth. However, because of the complexity of the transition between Curve 2 and Curve 5, the condition proposed in (5.10) was not considered adequate to justify to use of Figure 5.6 or Figure 5.10. The cylinder submergence at the lowest downstream gate elevation (highest downstream Froude number) for all 75 experiments from Experimental Set 1 was used to develop an exclusive assessment. The most critical hydraulic state of each experiment was classified as: Cylinder surbmerged (Curve 3, 48 experiments), Vortex (Curve 4, 4 experiments), or Cylinder exposed (Curve 5, 23 experiments). An additional set of experiments (Experimental Set 3) providing 85 additional data points was completed in June 2008. These experiments were performed with a fully opened weir gate at the end of the flume. Figure 5.11 presents all 160 data points. The Envelope curve in Figure 5.11 represents a limit that is physically impossible to intersect. No data point was obtained in the proximity of this envelope since the smallest cylinder tested had a diameter of 2 inches. The Threshold curve represents a limit between submerged and possibly exposed cylinders when the hydraulic conditions downstream of the cylinder approach the critical state. This means that, independently of the downstream water depth Ydwn, a point located in the white markers region in Figure 5.11 automatically represents a Curve 1‐2‐3 model and Figure 5.6 should be used to evaluate the upstream water depth. If a point is located in the black markers region, a Curve 4 or a Curve 5 is possible but not certain. This will depend on the downstream hydraulic conditions. Note that the highest extremity of the Threshold in Figure 5.11 still requires additional investigation. 57 2.50 Envelope Experimental Set 1: Submerged Experimental Set 1: Vortex Experimental Set 1: Exposed Experimental Set 3: Submerged Experimental Set 3: Vortex Experimental Set 3: Exposed Threshold 2.00 Yc / (wsg+D) 1.50 Curve 1‐2‐3 model, Use Figure 5.6 1.00 0.50 Possible Curve 1‐2‐5 model 0.00 0.00 0.20 0.40 0.60 0.80 1.00 (wsg /Yc )2 1.20 1.40 1.60 1.80 Figure 5.11. Threshold between Curve 1‐2‐3 models and possible Curve 1‐2‐5 models The last analysis of this research project concerns the initiation of the vortex. The objective was to find a maximum upstream water depth under which the use of Figure 5.6 becomes inaccurate. As a complement to Figure 5.11, when a given set of parameters falls in the black markers region, this new condition would confirm if Figure 5.10 should be used to evaluate the upstream water depth. It was attempted to investigate the separation between Curves 2 and 5 (refer to Figure 4.7) with Experimental Set 1 including 23 experiments following a Curve 1‐2‐5 model. Since the water depths corresponding to vortex initiation were not specifically recorded in this data set, another set of experiments (Experimental Set 4) was prepared in July 2008. A total of 37 experiments were performed in the same flume as described in Section 3.1. The highest upstream water depth corresponding to the first vortex appearance was recorded for each experiment. This upstream water depth was then related to other parameters: the cylinder size (D), the cylinder elevation (wsg), and the critical water depth (Yc). A dimensionless correlation between these parameters was not obtained, the critical water depth being expressed in metric units in the legend of Figure 5.12. Details of this analysis are presented in Appendix E. 58 1.40 Yc = 0.050 m Yc = 0.075 m 1.20 Yc = 0.100 m No Vortex, Refer to Figure 5.6 to evaluate Yup ( Yup ‐ D ) / Yc 1.00 0.80 Possible Vortex, Refer to Figure 5.10 to evaluate Yup 0.60 0.40 0.20 0.00 0.20 0.30 0.40 0.50 0.60 0.70 wsg / Yc 0.80 0.90 1.00 1.10 Figure 5.12. Theoretical thresholds for vortex formation upstream of cylinders expressed as a function of the critical water depth (Y c) In Figure 5.12, a given set of parameters falling under its corresponding Yc curve would be related to vortices formation and Figure 5.10 would be recommended to evaluate Yup, even if condition (5.10) is not respected. Inversely, a given set of data located above its corresponding curve suggests that Figure 5.6 should be used to evaluate the upstream water depth. Because critical water depths between 0.050 m and 0.100 m were not extensively tested in Experimental Set 4, Figure 5.12 presents some limitations. An additional problem in Figure 5.12 comes from the upstream water depth on the Y axis, which initially needs to be estimated with Figure 5.6. Additional roughness could significantly affect the upstream water depth corresponding to vortex formation. Moreover, the happening of vortices on the upstream side of a water intake depends on complex parameters that were not tested and an exhaustive investigation on the physics of vortices was not performed. Therefore, Figure 5.12 will not be considered in the approach developed in this research project (Refer to Section 7.1). Additional efforts should be made to improve the precision and versatility of Figure 5.12. 59 6.0 Comparison of Results The approach developed in Chapter 5.0 can be used to evaluate the water depth upstream (Yup) of a cylindrical element of length (l) equal to the rectangular channel width (B) for given element size (D), element elevation from the channel bed (wsg), and discharge (Q). This approach is tested in Section 6.1 and then in Section 7.2 using independent verification data. Once the upstream water depth is known, various methods can be used to evaluate the drag force applied on the cylindrical element. One common method is to refer to the drag force equation (2.1), which requires an empirical drag coefficient (Cd). The second method, investigated in Section 4.1, is based on to the momentum equation (2.4), which considers the difference in water depths upstream and downstream of the element, and does not require a drag coefficient. Both approaches are compared in Section 6.2. 6.1 Upstream Water Depth Evaluation Figure 5.6 and Figure 5.10 were used to calculate the upstream water depths for the 763 hydraulic states included in Experimental Set 1. These values are presented as a function of the measured upstream water depth in Figure 6.1. Note that data points located on a Curve 1, 2, or 3 (white markers in Figure 6.1) were generally evaluated using Figure 5.6 and data points located on a Curve 4 or 5 (black markers in Figure 6.1) were mostly evaluated using Figure 5.10. The different curve identities (1 to 5) were initially unknown and their identification is part of the approach developed here. Unsurprisingly, the results in Figure 6.1 follow the 1:1 line quite accurately (Figure 5.6 was developed using the same data from Experimental Set 1, and Experimental Set 2 used to develop Figure 5.10 was developed using similar experimental conditions). Data from Curves 1, 2, or 3 present an absolute average error of 0.002 m and a maximum error of 0.019 m (experiment D6wsg048Q10). Data from Curves 4 and 5 present an absolute average error of 0.006 m and a maximum error of 0.038 m (experiment D6wsg048Q15). In general, the agreement is considered satisfactory. 60 0.35 Curve 1‐2‐3 (Figure 5.6) Curve 4‐5 (Figure 5.10) 0.30 1:1 line Calculated Yup (m) 0.25 0.20 0.15 0.10 0.05 0.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Measured Yup (m) Figure 6.1. Calculated upstream water depths from Experimental Set 1 expressed as a function of the measured upstream water depths Figure 5.12 was not used to differentiate Curve 2 from Curve 4 data in the results presented in Figure 6.1. Instead, the vortex initiation was simply ignored. The lowest upstream water depth value calculated from Figure 5.10 or Figure 5.6 would respectively dictate if a Curve 1‐2‐3 or a Curve 5 should be assumed. This identification technique was unsuccessful 15 times for the 763 hydraulic states presented in Figure 6.1. 6.2 Drag Force Evaluation The calculated upstream water depths obtained in Section 6.1 were then used to evaluate the drag force on the cylinders. Twenty‐two experiments (out of 75) were partially or entirely rejected for reasons listed in Table 6.1. 61 Reason for rejection: Experiments entirely or partially rejected (hydraulic states rejected): Cylinder vibration during experiments D2wsg100Q15 D4wsg048Q15 (8‐9) D4wsg030Q10 (9‐10) D6wsg030Q10 (8‐10) D4wsg030Q15 (8‐11) Unstable water depths during D2wsg000Q15 D6wsg030Q05 experiments D3wsg000Q15 Discharge variation over the 2% limit D5wsg010Q15 D5wsg076Q15 due to pump or valve instabilities D5wsg030Q15 D5wsg100Q15 D5wsg048Q15 Two apparent stable water depths for D3wsg030Q10 (10) D6wsg048Q15 given Q, D, and wsg D4wsg030Q05 D6wsg100Q15 D6wsg030Q15 (10) Wall interference (friction) with cylinder D2wsg010Q05 D2wsg010Q15 D2wsg048Q10 D3wsg010Q10 (6‐8) Table 6.1. List of experiments from Experimental Set 1 rejected from the drag force comparison Three different methods were compared to evaluate the drag force applied on cylinders. These methods all consider the upstream water depth obtained in Section 6.1 and the calculated drag force (Fd) is compared to the streamwise force (FFx) measured by the load cell (Section 4.1). The load cell was not operating when cylinders were located on channel bed. Therefore 15 experimental results (153 data points) are excluded from this analysis. The rejected experimental results listed in Table 6.1 (150 data points) are presented with a distinct marker color. 6.2.1 Drag Force Equation The first method considered to evaluate the drag force Fd is based on the drag force equation (2.1), which is commonly accepted in literature. Equation (2.1) can be rewritten for cylinders of length equal to the channel width (l=B): Fd = C d ρDBVup2 2 (6.1) 62 The upstream Reynolds numbers calculated for the 763 hydraulic states of Experimental Set 1 ranged from 6.54 X 103 to 1.12 X 105. Values of the drag coefficient (Cd) were set equal to 1.0, which corresponds to Reynolds numbers between 103 to 105 (Refer to Figure 2.1). The upstream velocity was considered to be the depth‐averaged velocity obtained using the following equation (from equation (2.7), Subsection 2.1.3): Vup = Q BYup (6.2) Figure 6.2 presents the results of 460 data points identified by black markers and 150 rejected data points identified by light blue markers. Equation (6.1) seriously underestimates the streamwise force applied on large cylinders when the drag coefficient is equal to 1.0. 35.0 Non‐rejected data from Experimental Set 1 Rejected data from Experimental Set 1 30.0 1:1 line 25.0 Fd (N) 20.0 15.0 10.0 5.0 0.0 0.0 5.0 10.0 15.0 20.0 FFx (N) 25.0 30.0 35.0 Figure 6.2. Calculated drag force (Fd) using equation (6.1) and a drag coefficient (Cd) of 1.0 expressed as a function of the measured streamwise force (F Fx) using the load cell 63 6.2.2 Drag Force Equation and Blockage Ratio The assumption of a drag coefficient equal to 1.0 for all experiments appears simplistic. In Subsection 2.1.1, an alternate empirical equation (2.3) (Gippel et al., 1992) was proposed to evaluate the drag coefficient Cd for cylinders characterized by a geometry respecting the condition l/D < 21. Here, l is equal to B and equation (2.3) can be rewritten as: ⎛B⎞ C d = 0.81⎜ ⎟ ⎝D⎠ 0.062 (6.3) The drag coefficient (Cd) values obtained from equation (6.3) using data from Experimental Set 1 ranges from 0.81 to 0.87, which is smaller than the value of 1.0 proposed in Subsection 6.2.1. However, it has been recommended that the blockage ratio (BR) should be considered in the drag force calculation for cylinders of significant size compared to the channel section (Gippel et al., 1992; Gippel, 1995; Shields and Gippel, 1995; Hygelund and Manga, 2003). The blockage was defined in equation (2.8) and represents an attempt to account for the higher water velocity at the cylinder section when there is significant blockage of the channel section. When the cylinder length (l) is equal to the stream width (B), equation (2.8) takes this form: BR = D Yup (6.4) The drag coefficient (Cd) can be corrected for cylinders of significant blockage ratios (CdBR) using equation (2.9). C dBR = 0.997C d (1 − BR ) −2.06 This equation was developed using data from 50 experiments (Shields and Gippel, 1995). Values of the corrected drag coefficient for blockage ratios (CdBR) obtained from equation (2.9) present an average of 4.02 and a maximum value of 25.94 (Experiment D6wsg030Q05). This maximum value is comparable to the maximum drag coefficient mentioned by Curran and Wohl (2002). Figure 6.3 displays the results of the same data presented in Figure 6.2, but it considers a cylinder geometry‐based drag coefficient (equation (6.3)), which is corrected for blockage ratios (CdBR) using equation (2.9). 64 35.0 Non‐rejected data from Experimental Set 1 Rejected data from Experimental Set 1 30.0 1:1 line 25.0 Fd (N) 20.0 y = 0.4137x R² = 0.0881 15.0 10.0 5.0 0.0 0.0 5.0 10.0 15.0 20.0 FFx (N) 25.0 30.0 35.0 Figure 6.3. Calculated drag force (Fd) using equation (6.1) and a drag coefficient (Cd) based on equation (6.3) corrected for blockage ratios (BR) using equation (2.9) expressed as a function of the measured streamwise force (FFx) using the load cell The data in Figure 6.3 presents a significant scatter and does not follow this 1:1 line. It seems that equation (2.9) used in equation (6.1) does not represent a suitable approach to evaluate the drag force acting on cylinders of significant sizes. The automatic linear interpolation (dashed line interpolated within the non‐rejected data only) presents a poor correlation factor. This could partially be explained by the fact that equation (2.9) was developed using results from experiments that are different from the experiments described in Chapter 3.0. 6.2.3 Momentum Equation The second method proposed is to evaluate the drag force using the momentum equation (2.4). Fd = ρg (M up − M dwn ) The calculated upstream water depths from Section 6.1 were used on 460 non‐rejected data points and 150 rejected data points from Experimental Set 1 to evaluate the drag force (Fd) with equation (2.4). Results are presented in Figure 6.4. 65 35.0 Non‐rejected data from Experimental Set 1 Rejected data from Experimental Set 1 30.0 y = 0.9847x + 0.8253 R² = 0.9652 1:1 line Fd (N) 25.0 20.0 15.0 10.0 5.0 0.0 0.0 5.0 10.0 15.0 20.0 FFx (N) 25.0 30.0 35.0 Figure 6.4. Calculated drag force (Fd) using the momentum equations (2.4) expressed as a function of the measured streamwise force (F Fx) measured with the load cell The agreement between the drag force obtained from equation (2.4) and the streamwise force measured by the load cell in Figure 6.4 is satisfactory. The mean absolute error for non‐rejected data is 0.998 N. The best fit linear trend (dashed line interpolated within the non‐rejected data only) presents a coefficient of determination R2 = 0.965. This trend is almost parallel to the 1:1 line. Clearly, the results show that for large cylinders located in a small channel, the momentum equation approach yields for superior drag force estimates. 66 7.0 Applications and Limitations This chapter presents a synthesis of the method developed in Chapter 5.0 to evaluate the hydraulic conditions upstream of a large cylindrical element in a rectangular channel. The second part of this chapter presents an example of input and output data. Finally, the limitations of the approach are listed in Section 7.3. 7.1 Synthesis of the Approach The synthesis of the approach developed in this work is applicable to large cylinder (large refers to elements that create a measurable upstream backwater region) of length (l) equal to the width of a rectangular channel (B), aligned perpendicular to the flow direction and parallel to the channel bed. One representative case would be a stream segment identified for a stream restoration project including reintroduction of LWD. Generally, the known parameters would include: • The channel width (B) and cylinder length (l) • The LWD diameter (D) • The LWD elevation from the channel bed (wsg) • The downstream water depth (Ydwn) corresponding to a design discharge (Q), or a rating curve relating the water depth (Ydwn) to the total channel discharge (Q). In the case of a log introduction into a stream, this rating curve would be taken at the stream cross‐section located immediately downstream of the log introduction location. The outcomes would be an estimate of the upstream water depth (Yup) or an upstream rating curve and a corresponding drag force (Fd) acting on the LWD. The maximum drag force corresponding to a given discharge would be appropriate for a stability analysis of the LWD. Figure 7.1 presents a diagram that synthesizes the approach developed in this work. If this approach is considered for stream restoration project including more than one log reintroduction location, the approach should be applied from the downstream‐most log introduction location moving upstream, especially if the back‐water region created by a log is expected to reach the downstream side of the next upstream log introduction location. 67 Input data: Q, B (l), D, wsg, and Ydwn Calculate: Ydwn (Yc + D ) Calculate: wsg > 0 wsg = 0 q Eq (2.6) Yc Eq (2.24) Ydwn >1 (Yc + D ) Calculate: (Yc + D) Ydwn <1 (Yc + D ) Calculate: Calculate: (wsg/Yc)2 Figure 5.11 Above Threshold α2 Eq (5.4) Under threshold Yc/(wsg+D) α1 Eq (5.3) Calculate: Calculate: Frdwn Eq (2.23) α1 Eq (5.3) (Yc+ D – αβwsg) Figure 5.6 Obtain: Yup α2 Eq (5.4) (Yc+ D – αβwsg) Figure 5.10 Figure 5.6 Obtain: Yup Calculate: Yup Eq (5.9) If Yup< (D+wsg) : Reject Yup Choose minimum Yup value Calculate: Fd Eq (2.4) Figure 7.1. Diagram of the approach developed in this research project to calculate the upstream water depth (Yup) and the drag force (Fd) acting on a large cylindrical element 68 Note that the X axis on Figure 5.6 presents a maximum value of 2.00. Any data point located above this value is not considered in this research and is not likely to represent a critical case for LWD stability. Also note that when a calculated upstream water depth is smaller than the downstream water depth (using Figure 5.10), which could be the case at low flows and high cylinder elevations, the upstream water depth should be assumed equal to the downstream water depth. Again, this is not likely to represent a critical case for LWD stability. 7.2 Example of the Approach Two additional data sets (A and B) were obtained from flume experiments using the same PVC cylinders. While the data was collected using the same experimental setup (Chapter 3.0), it had no implication in the development of the approach presented in this work. The input data is presented in Table 7.1 and the downstream rating curves are displayed in Figure 7.2. Note that the rating curve of Data Set A presents a drop for an increasing discharge. This represents a shift from subcritical to supercritical regime. In each case, the drag force acting on the cylinders was not measured with the load cell. Table 7.1 Input parameters of Data Set A and Data Set B 69 0.14 Data Set A 0.12 Data Set B Ydwn (m) 0.10 0.08 0.06 0.04 0.02 0.00 0.000 0.004 0.008 0.012 Q (m3/s) 0.016 0.020 0.024 Figure 7.2. Downstream rating curves data from Data Set A and Data Set B The data from Table 7.1 and rating curves from Figure 7.2 were converted into upstream rating curves using the approach presented in the diagram of Figure 7.1. The results were then compared to the measured upstream rating curves. This comparison is presented in Figure 7.3. The reconstructed upstream rating curve from Data Set A was based on both Figure 5.6 and Figure 5.9. The maximum error between the measured and reconstructed curves is 0.011m. This evaluation error occurred at the transition between subcritical to supercritical conditions. This point visually falls above the rating curve trend. The reconstruction of the upstream rating curve from Data Set B was only based on Figure 5.6 and shows a maximum error of 0.002m. Figure 7.4 present the reconstructed (calculated) upstream water depths expressed as a function of the measured upstream water depths. The average absolute error is 0.002m. The good agreement is not totally surprising given that the data was obtained from flume experiments performed under similar conditions than the ones from which the present approach was developed. The approach developed in this work should be tested with field data before further conclusions can be made about its applicability to stream restoration projects including log introduction. 70 0.30 Data Set A Data Set B 0.25 Calculated Data Set A Calculated Data Set B Yup (m) 0.20 0.15 0.10 0.05 0.00 0.000 0.004 0.008 0.012 Q (m3/s) 0.016 0.020 0.024 Figure 7.3. Measured and calculated upstream rating curves from Data Set A and Data Set B 0.30 Calculated Yup (m) 0.25 0.20 0.15 0.10 0.05 0.00 0.00 0.05 0.10 0.15 0.20 Measured Yup (m) 0.25 0.30 Figure 7.4. Calculated upstream water depth expressed as a function of the measured upstream water depth for Data Set A and Data Set B 71 7.3 Limitations of the Approach In a physical open channel flow context, the approach developed in this work presents reasonable limitations. The most important constraint is related to the one‐dimensional formulation scheme on which it is based: The channel considered should be rectangular or wide enough so that the rectangular approximation could be made. Then, the cylindrical roughness element should have a constant diameter, obstruct the full width of the channel, and be disposed parallel to the water surface or to the channel bed. Finally, the flow should be evenly distributed over the width of the channel and its direction should be perpendicular to the cylindrical element. The approach still presents a number of uncertainties that should be explored. Any upstream water depth evaluation based on coefficients α1 and α2 should be considered with skepticism. Empirical equation (5.3) was calibrated with a data set of limited range (See appendix D) and the linearity of coefficient α2 in equation (5.4) is questionable. However, when the cylindrical element elevation wsg is negligible, the trend in Figure 5.6 and parameter (5.1) appear reliable. The influence of the flume side walls on the velocity distribution was not explored and the momentum correction factor β in the momentum equation (Finnemore and Franzini, 2002) is assumed to be equal to 1.0. For this reason, the momentum equation (2.4) is expected to slightly over‐estimate the drag force acting on large cylinders. The application of this approach to field cases is expected to require a number of adjustments and the context on which it is calibrated should always be kept in mind. Natural channels generally present rough beds and banks. Moreover, LWD used for stream restoration projects are rougher than PVC cylinders and their section is rarely perfectly cylindrical. The natural rough features present in streams create a flow resistance that interacts with the additional resistance created by a LWD introduction. This interaction is complex (Wilcox et al., 2006) and is expected to have an effect on the trends of the reference graphs presented in Figure 5.6, Figure 5.10, and Figure 5.11. 72 In the long term, the local modification of an artificially‐made roughness element will generate a response of the local stream morphology. Deposition of sediment (bedload) should be expected in the back‐water region and local scour under the log and directly downstream of the log should also be anticipated. The drag force generated by a given discharge acting on the log may decrease with time, due to erosion of the channel boundaries (Wallerstein et al., 2001). However, the log could also act as a trap for smaller mobile debris (Gippel et al., 1992; D’Aoust and Millar 2000). This would tend to increase the load supported by the designed anchors. Therefore, a conservative anchor design should consider the long term possibility of partial to total orifice flow congestion. 73 8.0 Conclusions An approach that evaluates the water depth upstream of a large cylindrical roughness element in a rectangular channel was developed in this work. The drag force affecting this cylindrical element can be estimated using the momentum equation with relative confidence. However, a formulation scheme that calculates the dynamic lift force applied on the cylindrical element still needs to be found (Appendix C). Therefore, the analysis of the stability of a cylindrical roughness element introduced in a channel is not yet complete. Experimental results support the initial hypothesis based on Gippel et al. (1996) on which this project was developed: apart from local disturbance of the velocity profile, roughness elements only have an influence in the upstream direction. This assumption however is only valid in non‐erodible channels (Wallerstein et al., 2001). Additional efforts should confirm that Figure 5.6, Figure 5.10, and Figure 5.11 can be used in a natural setting context. The influence of parameters that were not considered in this work should be investigated. The cylindrical element skin roughness and the channel boundaries roughness appear to be the most critical parameters that could interfere with the applicability of the approach developed in this work to stream restoration projects. In other words, the applicability of the approach developed for smooth cylindrical elements should be tested and adapted for prototype logs and LWD. In a theoretical context, the following observations were made: • A large cylinder can be defined by: D> Y − Yc 2 When this condition is respected, the cylinder creates a significant increase of the upstream water level. In this case, the momentum equation (2.4) is suitable to calculate the drag force (Fd). Otherwise, the cylinder is too small to have a measurable influence in the upstream direction and the drag force equation (2.1) could present better approximations of the drag force. • The meaning of the blockage ratio BR is limited. The influence of the blockage ratio on the drag force depends on upstream and the downstream hydraulic conditions. The hydraulic impacts of a roughness element cannot be related to its geometric size or to its blockage ratio only. Therefore, the geometric definition of a LWD should consider the hydraulic context in which it is located. 74 • The drag force equation (2.1) appears unsuitable for large cylinders. Theoretical values of drag coefficients have proven to greatly underestimate the drag force applied on a cylindrical element of significant size (Figure 6.2). Considering the blockage ratio in the drag coefficient calculation could still lead to errors approaching one order of magnitude (Figure 6.3). Back‐calculated drag coefficient values from a given study should not be applied in a different context. In literature, the drag coefficient was only related to a limited range of hydraulic parameters and its physical meaning is still uncertain. • The momentum equation (2.4) was validated with load cell measurements in a one‐dimensional context. This equation proved to be reliable to evaluate the drag force applied on a cylindrical roughness element. • The influence of the cylindrical roughness element vertical position is independent of the drag force if the downstream water depth is greater than the element diameter added to the critical water depth (i.e., if Ydwn/(Yc+D)>1). This is only valid if the element is submerged. This has not been mentioned previously in literature and it partially contradicts observations from Gippel et al. (1996), Wallerstein et al. (2002), Hygelund and Manga (2003), and Alonso (2004). • The dimensionless ratio of the sluice gate opening divided by the critical water depth (i.e., wsg/Yc) is a key parameter in the orifice flow theory. It has been fitted in this research to a large data set and its meaning was relevant (Figure 5.9). A comparable observation can be made about the downstream Froude number Frdwn. • To our knowledge, Figure 5.6, Figure 5.10, and Figure 5.11 have no precedents in literature. Their applicability to other contexts has not been tested yet. 75 References Alonso, C. V. (2004), ‘‘Transport mechanics of stream‐borne logs’’, Riparian vegetation and fluvial geomorphology: Hydraulic, hydrologic, and geotechnical interactions, S. J. Bennett and A. Simon, eds., American Geophysical Union, Washington, D.C. Braudrick, C.A., Grant, G.E. (2000), “When do logs move in rivers?”, Water Resources Research, 36(2), 571–583. Clemmens, A. J., Strelkoff, T. S., and Replogle, J. A. (2003), “Calibration of submerged radial gates”, Journal of Hydraulic Engineering, 129(9), 680–687. Curran, J.H., Wohl, E.E. (2003), “Large woody debris and flow resistance in step‐pool channels, Cascade Range, Washington”, Geomorphology, 51, 141– 157. D’Aoust, S., Millar, R.G. (2000), “Stability of Ballasted Woody Debris Habitat Structures”, Journal of Hydraulic Engineering, 126, 810‐817. Ferro, V. (2000), “Simultaneous flow over and under a gate”, Journal of Irrigation and Drainage Engineering , 126(3), 190–193. Finnemore, E.J., Franzini, J.B. (2002), “Fluid Mechanics with engineering Applications”, McGraw Hill, New York. Gippel, C. J., O’Neill, I. C., and Finlayson, B. L. (1992), “The hydraulic basis of snag management”, Center for Environmental Applied Hydrology, Department of Civil and Agricultural Engineering, and Department of Geography, University of Melbourne: Australia. Gippel, C.J., O’Neill, I. C., Finlayson, B. L., and Schnatz, I. (1996), “Hydraulic guidelines for the re‐ introduction and management of large woody debris in lowland rivers”, Regulated Rivers: Research and Management, 12, 223‐236. Gippel, C. J. (1995), “Environmental hydraulics of large woody debris in streams and rivers”, Journal of Environmental Engineering, 121, 388–395. Hygelund, B., Manga, M. (2003), “Field measurements of drag coefficients for model large woody debris”, Geomorphology, 51, 175–185. Manners, R.B., Doyle, M.W., and Small, M.J. (2007), “The structure and hydraulics of natural woody debris jams”, Water Resources Research, 43(6), W06432, doi:10.1029/2006WR004910. Lindsey, W.F. (1938), “Drag of cylinders of simple shapes”, NACA Report 619, 169‐176. Ranga Raju, K. G., Rana, O. P. S., Asawa, G. L., and Pillai, A. S. N. (1983), “Rational assessment of blockage effect in channel flow past smooth circular cylinders”, Journal of Hydraulic Research, 21, 289– 302. 76 Shahrokhnia, M.A., Javan, M. (2006), “Dimensionless Stage–Discharge Relationship in Radial Gates”, Journal of Irrigation and Drainage Engineering, 132(2), 180‐184. Sheridan, J., Lin, J.‐C., Rockwell, D. (1997), “Flow past a cylinder close to a free surface”, Journal of Fluid Mechanics, 330, 1‐30. Shields, F. D., Jr., Gippel, C. J. (1995), “Prediction of effects of woody debris removal on flow resistance”, Journal of Hydraulic Engineering, 121(4), 341–354. Shields, F.D., Morin, N., and Cooper, C.M. (2004), “Large woody debris structures for sand‐bed channel”, Journal of Hydraulic Engineering, 130, 208‐217. Swamee, P.K. (1992), “Sluice‐gate discharge equations”, Journal of Irrigation and Drainage Engineering, 118(1), 56‐60. Tony, L., Wahl, P.E. (2005), “Refined Energy Correction for Calibration of Submerged Radial Gates”, Journal of Hydraulic Engineering, 131(6), 457‐466. Wallerstein, N., Alonso, C., Bennett, S., and Thorne C. (2001), “Distorted froude‐scaled flume analysis of large woody debris”, Earth Surface Processes and Landforms, 26, 1265‐1283. Wallerstein, N., Alonso, C., Bennett, S., and Thorne C. (2002), “Surface Wave Forces Acting on Submerged Logs”, Journal of Hydraulic Engineering, 128(3), 349‐353. Wilcox, A.C., Nelson, J.M., and Wohl, E.E. (2006), “Flow resistance dynamics in step‐pool channels: 2. Partitioning between grain, spill, and woody debris resistance”, Water Resource Research, 42, W05419, doi:10.1029/2005WR004278. Zahm, A.F., Smith, R.H., and Hill, G.C. (1972), “Point drag and total drag of navy struts No. 1 Modified”, NACA Report 137, 125‐139. 77 Appendix A Load Cell Calibration Calibration of the load cell was completed with a set of weights, a low friction pulley (Figure A1), and a rope. The three signals (Fx, Mz, and Fy) were calibrated independently. A1 Streamwise Force Signal Calibration (Fx) The streamwise force signal (Fx) was calibrated with different combinations of weights, from 0.00Kg to 1.20Kg. The forces were applied at different vertical locations from the load cell (h). It was found that h had a slight but non‐negligible effect on the streamwise force signal (Fx). Results of the tests are presented in Figure A2. A linear trend was interpolated within each of the 7 data series. The slopes of the trends were then expressed as a function of h. Results are presented in Figure A3. From Figure A3, the calibration equation that transforms the streamwise signal (Fx) expressed in volts into a streamwise force (FFx) expressed in N is: FFx = g FFx (Fx − Fxini ) (− 0.00014h + 0.54234) (F − Fxini ) = 7143g x (3874 − h ) (A1) Here, g is the gravity constant (m/s2), Fxini is the initial streamwise signal with no force applied (Volts), and h is the distance between the load cell center and the vertical point of streamwise force application (mm). Note that the data presented in Figure A3 presents a certain scatter. Figure A1. Weights and pulley used for the calibration 78 3.30 h = ‐93 mm y = 0.5198x + 2.5194 h = ‐43 mm y = 0.5441x + 2.5189 h = ‐28 mm y = 0.5423x + 2.5198 3.20 h = 137 mm y = 0.5677x + 2.5232 3.10 h = 265 mm y = 0.5732x + 2.5206 h = 407 mm y = 0.5952x + 2.5223 h = 541 mm y = 0.6160x + 2.5246 Fx (volt) 3.00 2.90 2.80 2.70 2.60 2.50 0.00 0.20 0.40 0.60 0.80 Weight (Kg) 1.00 1.20 1.40 Figure A2. Streamwise force signal (Fx) expressed as a function of the weight applied at different vertical locations (h) 0.64 Equation slope (Volt / Kg) 0.62 0.60 0.58 y = ‐0.00014x + 0.54234 R² = 0.96899 0.56 0.54 0.52 0.50 ‐600 ‐500 ‐400 ‐300 ‐200 h (mm) ‐100 0 100 200 Figure A3. Slopes of linear trends presented in Figure A2 expressed as a function of the vertical location of the streamwise force application point (h) 79 A2 Moment Signal Calibration (Mz) The moment signal (Mz) was calibrated with the same method described for the streamwise force signal (Fx). The effect of the distance between the load cell and the vertical streamwise force application point was expected to be more significant. Figure A4 shows the test results. The slopes of the trends from Figure A4 were expressed as a function of h in Figure A5. From Figure A5, the calibration equation that transforms the streamwise signal (Mz ) expressed in Volts into a streamwise force (FMz ) expressed in N is: FMz = g (M z − M zini ) (0.01296h − 0.09415) (M z − M zini ) FMz = 77.16 g h − 7.26 (A2) Here, Mzini is the initial momentum signal with no force applied (Volts). The precision of the linear trend in Figure A5 presents a fine correlation compared to the trend presented in Figure A3. However, the linear trend was expected to meet with the origin of the graph (0,0), which is not the case. 2.00 0.00 Mz (volt) ‐2.00 ‐4.00 h = ‐93 mm y = 1.1055x + 0.1629 h = ‐43 mm y = 0.2199x + 0.3005 h = ‐28 mm y = 0.1342x + 0.2670 ‐6.00 h = 137 mm y = ‐1.8895x + 0.1472 h = 265 mm y = ‐3.5291x + 0.1228 ‐8.00 h = 407 mm y = ‐5.3220x + 0.1083 h = 541 mm y = ‐7.0397x + 0.0801 ‐10.00 0.00 0.20 0.40 0.60 0.80 Weight (Kg) 1.00 1.20 1.40 Figure A4. Momentum signal (Mz) expressed as a function of the weight applied at different vertical locations (h) 80 2.00 1.00 Equation slope (Volt / Kg) 0.00 ‐1.00 y = 0.01296x ‐ 0.09415 R² = 0.99992 ‐2.00 ‐3.00 ‐4.00 ‐5.00 ‐6.00 ‐7.00 ‐8.00 ‐600 A3 ‐500 ‐400 ‐300 ‐200 ‐100 0 100 200 h (mm) Figure A5. Slopes of linear trends presented in Figure A4 expressed as a function of the vertical location of the streamwise force application point (h) Vertical Force Signal Calibration (Fy) The vertical signal (Fy) was calibrated by adding weights directly on the load cell. Three distinct tests were performed. The first test was achieved with no cylinder fixed to the experimental setup (refer to Figure 3.1 and Figure 3.7). The second and third tests were respectively performed with the 3 inch and 5 inch cylinders fixed to the experimental setup. Results of the calibration tests for the vertical signal (Fy) are presented in Figure A6. The 3 linear interpolations in Figure A6 were used to generate the calibration formula for the vertical force (FFy). The importance of each slope would depend on the size of its respective data set. The average slope obtained was 0.1408 Volt/Kg and the calibration equation is: FFy = g (F y − Fyini ) 0.1408 FFy = 7.102 g (Fy − Fyini ) (A3) Here, Fyini is the initial vertical signal with no force applied (Volts). 81 7.10 y = 0.145x + 6.111 No Cylinder 7.00 3 inches Cylinder y = 0.140x + 6.398 5 inches Cylinder y = 0.142x + 6.651 6.90 6.80 Fy (volt) 6.70 6.60 6.50 6.40 6.30 6.20 6.10 6.00 0.00 0.50 1.00 1.50 2.00 2.50 Weight (Kg) Figure A6. Vertical force signal (Fy) expressed as a function of the weight for 3 different initial weights 82 Appendix B Downstream Water Depth Correction The downstream water depth (Ydwn) was measured with a transparent ruler. In order to validate the precision of the measurements from Experimental Set 1, measured water depths for the three discharges (5L/s, 10L/s, and 15L/s) were plotted as a function of the downstream weir gate elevation (hgate). The gate elevation was measured with a Point Gauge fixed to a Vernier (See Figure 3.8). This measure is considered reliable. Figure B1, B2, and B3 present the data for 5L/s, 10L/s, and 15L/s respectively. The supercritical water depths are represented by distinct markers and are not expected to follow the subcritical trend. The data was compared with the theoretical weir flow equation (2.14), which was modified for the purpose of this analysis: q = C w* 3 2 2 g (Ydwn − hgate ) 2 3 (B1) The weir coefficient (Cw*) was estimated with this empirical equation presented by Finnemore and Franzini (2002) for thin plate weirs: C w* = 0.605 + 1 1000(Ydwn − hgate ) + 0.08 (Y dwn − hgate ) hgate (B2) From equation (B1) and equation (B2), the downstream water depth was back‐calculated and results are presented by a black line in Figures B1, B2, and B3. The weir flow equation was not expected to fit the data but served as a reference to detect any anomalies in the subcritical water depths measured with rulers. The conclusion from Figures B1, B2, and B3 was that the downstream water depth measurements of the subcritical hydraulic states of Experimental Set 1 were consistent. Therefore, none of them was corrected or rejected. The reliability of the supercritical water depth measurements could not be confirmed. 83 35.0 Subcritical water depths Supercritical water depths 30.0 Theoritical Weir gate equation Critical water depth Ydwn (cm) 25.0 20.0 15.0 10.0 5.0 0.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 hgate (cm) Figure B1. Downstream water depth (Ydwn) expressed as a function of the downstream gate elevation (hgate) for a discharge (Q) of 5L/s 35.0 Subcritical water depths Supercritical water depths 30.0 Theoritical weir flow equation Ydwn (cm) 25.0 Critical water depth 20.0 15.0 10.0 5.0 0.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 hgate (cm) Figure B2. Downstream water depth (Ydwn) expressed as a function of the downstream gate elevation (hgate) for a discharge (Q) of 10L/s 84 35.0 Subcritical water depths Supercritical water depths Theoritical weir flow equation 30.0 Critical water depth Ydwn (cm) 25.0 20.0 15.0 10.0 5.0 0.0 0.0 2.0 4.0 6.0 8.0 10.0 hgate (cm) 12.0 14.0 16.0 18.0 20.0 Figure B3. Downstream water depth (Ydwn) expressed as a function of the downstream gate elevation (hgate) for a discharge (Q) of 15L/s 85 Appendix C Vertical Forces The dynamic lift force represents an additional hydraulic component affecting the stability of roughness elements. It was mentioned in Subsection 3.1.4 and Section 3.3 that the vertical force (FFy) acting on cylinders was measured by the load cell. This section presents the results, which are part of the Experimental Set 1. C1 Comparing Measured Lift Forces (Flm) to Theoretical Lift Forces (Fl) The total vertical force measured by the load cell (FFy) is the sum of the buoyancy force (Fbm) and the lift force (Flm). FFy = Flm + Fbm (C1) The buoyancy force acting on the cylinder (D’Aoust and Millar 2000) or submerged weight of the cylinder (Alonso 2004) is defined by: 2 ⎛D⎞ Fb = π ⎜ ⎟ Bg (ρ cyl − ρ w ) ⎝2⎠ (C2) Here, D is the cylinder diameter, g is the gravity constant, ρcyl is the density of the cyinders, and ρw is the density of water. The density of the PVC cylinders is ρcyl = 1409 Kg/m3. The density of water was assumed to be ρw = 1000 Kg/m3. When the cylinder was completely submerged, equation (C2) was used in equation (C1). However, when the cylinder was not fully submerged (when it acted as a sluice gate), equation (C2) had to be converted to calculate the partially submerged cylinder weight. This parameter was obtained using the upstream water depth and the cylinder was separated in two parts of different relative densities. The downstream water depth was not taken into account for the submerged density calculation, even when supercritical downstream hydraulic conditions were observed. Before each of the 60 experiments for which the load cell was used, the vertical force signal (Fy) was recorded with no water in the flume as well as when the cylinder was fully submerged in steady water. These two signals were used to back‐calculate the buoyancy force measured by the load cell (Fbm). These 60 values were compared to the theoretical buoyancy force calculated with equation (C2). The results are presented in Figure C1. 86 14.0 D2 D3 D4 D5 D6 +/‐10% 12.0 10.0 Fb (N) 8.0 6.0 4.0 2.0 0.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 Fbm (N) Figure C1. Buoyancy force (Fb) calculated with equation (B2) expressed as a function of the buoyancy force measured by the load cell (Fbm) Two lines were plotted in Figure C1 to illustrate a 10% departure from the 1:1 line. The average absolute difference between Fb and Fbm was 0.41 N and the maximum difference was 1.24 N (Experiment D4wsg030Q15). The difference between the total averages was only 0.12 N. This value compares with the submerged weight of the aluminum rods that was not considered in the calculation. However, the wide range of values obtained for each cylinder size suggests that the initial load cell vertical force measurements were not as accurate as the streamwise force measurements described in Section 4.1. Equation (2.14) was adapted to this case study by replacing the cylinder length l with the channel width B. Fl = C l ρw 2 DBVup2 (C3) 87 Equation (C3) and equation (2.15) were used to calculate the theoretical lift force (Fl) that was compared to the measured lift force (Flm) obtained from equation (C1) and equation (C2) (Here, the theoretical buoyancy was considered in equation (C1) instead of the measured buoyancy presented in Figure (C1)). The results are plotted in Figure C2. One can straightforwardly conclude that empirical equation (C3) and equation (2.15) do not agree with the lift force measured by the load cell (Flm) for two reasons: the order of magnitude differs significantly and a majority of Flm values are negative. Note that the lift force values are of same order of magnitude as the measured drag forces (FFx) presented in Figure 4.4. 0.7 1:1 line 0.6 0.5 Fl (N) 0.4 0.3 0.2 0.1 0.0 ‐40.0 ‐35.0 ‐30.0 ‐25.0 ‐20.0 ‐15.0 Flm (N) ‐10.0 ‐5.0 0.0 5.0 10.0 Figure C2. Theoretical lift force (Fl) calculated with equation (C3) and equation (2.15) expressed as a function of the net lift force (Flm) (equation (C1) and equation (C2)) measured by the load cell 88 C 2 Buoyancy forces (Fb) In the perspective of a river restoration project, the 610 measured lift force (Flm) values from Experimental Set 1 were compared to the buoyancy forces (Fb) of hypothetically lighter cylinders. The original cylinder density (ρcyl) of 1409 kg/m3 was replaced by two commonly considered log densities (ρlog): one light density of 500 Kg/m3 and an average log density of 800 kg/m3. The buoyancy force (Fb) was then calculated using equation (C2). Results are presented in Figure C3. The data points located on the left hand side of the ‐1:1 line characterize cases where the log would be stable, i.e. the resulting force calculated using equation (C1) would be directed downward. Reversely, the markers located on the right hand side of the ‐1:1 line present a net upward force. As the density of the log increases, the number of unstable cases decreases. The hypothetical buoyancy force (Fb) and the measured lift force (Flm) roughly show a similar order of magnitude. If the measured lift force values from Experimental Set 1 could be confirmed, the results of this work would have an impact on the LWD structure stability approach presented by D’Aoust and Millar (2000) and Baudrick and Grant (2000). Indeed, their approach did not consider the dynamic lift force applied on LWD structures. However, the present study does not propose any reliable equation to evaluate the lift force. C 3 Conclusion on the Lift Force Analysis Some interesting trends were observed in the lift force data. Figure C4 shows the influence of the discharge on the measured lift force as a function of the downstream water depth for constant cylinder size and elevation. Results are difficult to justify using know hydraulic equations. Figure C5 presents the effect of the cylinder size on the measured lift force as a function of the downstream water depth for constant discharge and cylinder elevation. Even if the curves presented in Figure C5 are irregular, a general tendency for negative lift force can be observed as the cylinder size varies. However, the influence of the cylinder size does not find a simple solution in Figure C6, which presents a distinct set of data with constant discharge and cylinder position. 89 15.0 12.0 ‐1:1 line Stable region Fb (N) 9.0 Unstable region 6.0 3.0 Light density Hight density 0.0 ‐45.0 ‐30.0 ‐15.0 Flm (N) 0.0 15.0 Figure C3. Hypothetical buoyancy force (Fb) calculated with equation (C2) expressed as a function of the measured dynamic lift force (Flm) calculated from equation (C1) and equation (C2) 6.0 4.0 2.0 Flm (N) 0.0 ‐2.0 ‐4.0 ‐6.0 ‐8.0 D6wsg010Q05 D6wsg010Q10 ‐10.0 D6wsg010Q15 ‐12.0 0.00 0.05 0.10 0.15 Ydwn (m) 0.20 0.25 0.30 Figure C4. Example of the influence of the discharge (Q) on the measured lift force (Flm) expressed as a function of the downstream water depth (Ydwn) for the 6 inch cylinder 90 2.0 0.0 Flm (N) ‐2.0 ‐4.0 ‐6.0 ‐8.0 D2wsg010Q05 D3wsg010Q05 D4wsg010Q05 ‐10.0 D5wsg010Q05 D6wsg010Q05 ‐12.0 0.00 0.05 0.10 0.15 Ydwn (m) 0.20 0.25 0.30 Figure C5. Influence of the cylinder size (D) on the measured lift force (Fl) expressed as a function of the downstream water depth (Ydwn) when the discharge (Q) is 5L/s 10.0 5.0 Flm (N) 0.0 ‐5.0 ‐10.0 D2wsg030Q15 D3wsg030Q15 D4wsg030Q15 D5wsg030Q15 D6wsg030Q15 ‐15.0 ‐20.0 0.00 0.05 0.10 0.15 0.20 Ydwn (m) 0.25 0.30 0.35 Figure C6. Influence of the cylinder size (D) on the measured lift force (Fl) expressed as a function of the downstream water depth (Ydwn) when the discharge (Q) is 15L/s 91 The analysis of the lift force data could not be conducted any further. Results from Figure C2 were not satisfying and no alternative equation could be found to calculate theoretical lift force values. Since the measured lift forces were often negative, it was assumed that the vertical forces acting on the cylinders would have a limited effect on their stability in the range of parameters tested here. Additional experiments would be required to confirm or reject this hypothesis. 92 Appendix D Calibration of the α1 Coefficient The equation that estimates the correction coefficient α1 was calibrated by considering the last (supercritical) hydraulic state of the 30 (out of 48) experiments that followed a Curve 1‐2‐3 model. The 15 experiment during which the cylinder elevation was nil were not considered in the calibration process. Three experiments were rejected for uncertainties related to discharges (D5wsg010Q15 and D5wsg030Q15) or to other problems affecting the water depth (D2wsg10Q05). The last hydraulic state of two experiments (D3wsg030Q10 and D6wsg30Q15) was rejected because of water depth instabilities. However, data from these experiments was considered in this analysis. In Figure 5.3, values of Yup/(D+Yc‐α1wsg) (set equal to Yup/(D+Yc)) range from 1.071 to 1.105. Maximum and minimum values of α1 were back‐calculated for each experiment. The maximum and minimum resulting values for the 30 experiments are presented as a function of their respective D/Yc ratio in Figure D1. 2.00 Maximum value Minimum value Polynomial interpolation for maximum α1 values Equation (D2) 1.50 α1 1.00 0.50 Polynomial interpolation for minimum α1 values 0.00 ‐0.50 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 D/Yc Figure D1. Maximum and minimum values of the correction coefficient α1 expressed as a function of the cylinder size (D) divided by the critical water depth (Yc) 93 Second order polynomial functions (black and red curves for maximum and minimum α1 values respectively) were automatically interpolated in Figure D1. Then, a power function that takes the following form was considered: ⎛D⎞ α 1 = d 1 ⎜⎜ ⎟⎟ ⎝ Yc ⎠ d2 − d3 (D1) The parameters d1, d2, and d3 were manually adjusted until equation (D1) would be located between a majority of maximum and minimum α1 values. The resulting equation was: ⎛D⎞ α 1 = 0.6⎜⎜ ⎟⎟ ⎝ Yc ⎠ 0.8 − 0 .3 (D2) Equation (D2) is presented on Figure D1 as a blue line. Only 2 red markers are located above the equation trend and only three black markers are positioned slightly under the equation trend. It is expected that equation (D2) might not be compatible with data sets obtained from different experimental conditions. The influence of the cylinder elevation when Ydwn < (D+Yc) is complex and does not seem to be independent of the cylinder size (D) and the unit discharge (q). 94 Appendix E Conception of Figure 5.12 Experimental Set 4 was performed to complete the investigation related to the transition from Curve 2 to Curve 5. The maximum upstream water depth corresponding to vortex formation was measured for 37 experiments. Four different cylinder sizes were tested (3, 4, 5, and 6 inches), up to 4 different discharges were tested for each cylinder, and up to 3 different cylinder elevations were tested for each discharge. Each experiment included two independent evaluations of this water level. The 37 averaged results are presented in Figure E1. It was observed in Figure E1 that the data was organized as a function of the critical water depth (Yc), even if each axis included this parameter. Figure E2 presents the same results with a power equation interpolated within the data. The form of the equation is: (Y up − D) Yc ⎛ wsg = e1 ⎜⎜ ⎝ Yc e2 ⎞ ⎟⎟ ⎠ 0.40 0.50 (E1) 1.30 Yc = 0.086 Yc = 0.076 1.20 Yc = 0.066 Yc = 0.054 1.10 Yc = 0.041 (Yup‐D) / Yc Yc = 0.034 1.00 0.90 0.80 0.70 0.60 0.20 0.30 0.60 0.70 wsg /Yc 0.80 0.90 1.00 1.10 Figure E1. Results from Experimental Set 4: Dimensionless upstream water depth (Yup) corresponding to vortex formation for given values of cylinder size (D), cylinder elevation (wsg), and critical water depth (Yc) 95 In Figure E2, apart from the three data points defined by Yc = 0.034m, all interpolations are visually parallel to each other. Three automatically interpolated power equations are presented by extended dashed lines (Yc = 0.054m, Yc = 0.076m and Yc = 0.095m) and their respective equation is presented in Figure E2. These equations were slightly modified and associated to Yc values of 0.050m, 0.075m, and 0.100m respectively. These modifications were performed manually. The e1 and e2 values of equation (E1) presented in Figure 5.12 are defined in table E1. Note that vortex formation upstream of a sluice gate or a water intake has not been investigated in this research project. Trends in Figure 5.12 should be extrapolated cautiously and more experiments should be performed in order to confirm the tendencies presented in this figure. Roughness and water velocity distribution are expected to have non‐negligible effects on the trends presented in Figure 5.12. e1 in equation (E1) e2 in equation (E1) Yc = 0.050 m 1.260 0.630 Yc = 0.075 m 1.225 0.670 Yc = 0.100 m 1.190 0.735 Table E1. Values of parameters e1 and e2 for each value of the critical water depth (Yc ) in equation E1 96 1.30 y = 1.2591x0.6339 Yc = 0.086 Yc = 0.076 1.20 Yc = 0.066 y = 1.2153x0.6784 Yc = 0.054 1.10 y = 1.189x0.7362 Yc = 0.041 (Yup‐D) / Yc Yc = 0.034 1.00 0.90 0.80 0.70 0.60 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 wsg /Yc Figure E2. Results from Figure E1 with power equation interpolation within each data set corresponding to a distinct critical water depth (Yc) 97
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Large smooth cylindrical elements located in a rectangular channel : upstream hydraulic conditions and… Turcotte, Benoit 2008
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