APPLICATION OF PARTICLE IMAGE VELOCIMETRY TO THE HYDRAULIC JUMP

Transcription

1 The Pennsylvania State University The Graduate School College of Engineering APPLICATION OF PARTICLE IMAGE VELOCIMETRY TO THE HYDRAULIC JUMP A Thesis in Civil Engineering by Justin M. Lennon c 2004 Justin M. Lennon Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science May 2004

2 We approve the thesis of Justin M. Lennon. Date of Signature David F. Hill Assistant Professor of Civil Engineering Thesis Adviser Arthur C. Miller Professor of Civil Engineering Kendra V. Sharp Assistant Professor of Mechanical Engineering Andrew Scanlon Professor of Civil Engineering Head of the Department of Civil and Environmental Engineering

3 iii Abstract Hydraulic jumps are regions of rapidly varied flow connecting supercritical and subcritical free-surface or interfacial flows. The jumps arise in a variety of natural and engineered environments and are characterized by intense mixing, turbulence and aeration. Initial research into hydraulic jumps focused on bulk parameters such as roller and jump lengths and depth ratios. Subsequent research began to investigate the mean and turbulent flow fields through the use of pitot tubes, hot films and acoustic and laser velocimeters. The present work investigates the application of Particle Image Velocimetry to hydraulic jumps with two main goals. The first goal is to determine the degree to which relevant technical challenges, such as two-phase flow field, can be overcome. The second is to provide an extensive and spatially dense set of data on mean and turbulent flow characteristics.

4 iv Table of Contents List of Tables vi List of Figures viii Acknowledgments xiii Chapter 1. Introduction Chapter 2. Literature Review Introduction Water Surface Profile Research Hydraulic jump research Undular Jumps Internal Hydraulics Research Hydraulic Jumps Undular Jumps Other Analyses Pertaining to Undular and Hydraulic Jumps Chapter 3. Experimental Methods Introduction Experiment Setup Flume Setup Free Surface Measurement Setup

5 v PIV Setup Data Collection Free Surface Data Collection PIV Data Collection Data Analysis Chapter 4. Experimental Results Free Surface Analysis Mean Velocity Analysis Turbulent Flow Analysis Introduction Shear Stress from Mean Velocity Profiles Shear Stress by the Darcy Friction Factor Shear Stress from Reynolds Stresses Shear Stress from Root Mean Square Velocity Fluctuations. 85 Chapter 5. Conclusion Discussion of Technical Aspects of Data Collection Discussion of the Mean Velocity Analysis Discussion of the Boundary Shear Stress Analysis Future Considerations References

6 vi List of Tables 4.1 Crest and trough magnitudes for Hydraulic Condition Summary of the supercritical flow hydraulic properties for each hydraulic condition Summary of shear velocity, Coles wake parameter and boundary shear stress as determined from velocity profiles. Data presented is from the supercritical flow region for all three hydraulic conditions Summary of the boundary shear stress, Reynolds numbers and friction factors generated in the Colebrook-White and Darcy friction factor shear stress analyses. Case AF-1 reproduced from Chanson (2000) is provided to verify the validity of this analysis. The measured boundary shear stress for case AF-1 is 3.45 Pa Summary of boundary shear stress for the supercritical flow region for each hydraulic condition. The boundary shear stress for each condition was determined using a linear regression of the outer layer Reynolds stress data. The threshold depth values for the outer layer also are presented Summary of the shear velocity and boundary shear stress as determined from curve fitting the universal laws of turbulence intensity for both horizontal and vertical velocity fluctuations. The boundary shear stress data were calculated using the definition of shear velocity presented as Equation

7 vii 5.1 Summary of the boundary shear stress values as computed using the mean velocity profiles, Darcy s friction factor, the Reynolds stress profile and the universal laws of turbulence intensity

8 viii List of Figures 1.1 Classical hydraulic jump profile Classical undular jump profile Theoretical water surface profile for any hydraulic jump superimposed upon experimental data. Figure reproduced from Rajaratnam and Subramanya (1968) Non-dimensional plot of mean channel velocity versus horizontal distance of a free hydraulic jump and a wall jet. Figure reproduced from Long et al. (1990) Classification of undular jumps. Figure reproduced from Chanson and Montes (1995) Dimensionless bed shear distributions beneath an undular jump for (a) transverse profile for Froude number = 1.25 (b) longitudinal profile for Froude number =1.48. Figure reproduced from Chanson (2000) Line drawing of recirculating flume setup Sketch of the flow straightener relative to the direction of flow Simplified schematic of the interaction between a capacitance wave gage and the surrounding fluid Wave gage wiring schematic PIV wiring schematic

9 ix 3.6 Profile view of laser and laser light sheet orientation relative to flume Plan and profile view of the camera orientation relative to flume. Views also show orientation of the camera mount and the location of the exposed sections of the acrylic channel bottom that are utilized for laser light sheet access Typical layout of location changes within a given hydraulic condition Cross-sectional view of the camera viewable area relative to an undular crest. Setup depicts situations under which it is not possible for the camera to capture the entire depth of flow under a crest Manually measured free surface profiles for (a) Hydraulic Condition 1, (b) Hydraulic Condition Free surface profile generated from the PIV data superimposed upon measured free surface data for (a) Hydraulic Condition 1, (b) Hydraulic Condition 2, (c) Hydraulic Condition Generalized sketch of the interaction between the flow field and a smooth free surface as seen from the 2-dimensional perspective of a camera looking up at the free surface. Sketch illustrates the mirroring of seed particles onto the free-surface, creating a non-physical flow field above the free surface Hydraulic Condition 1 mean velocity field. Velocity fields are presented as (a) spatially complete contour map of streamwise velocity (b) select two-dimensional velocity vectors

10 x 4.5 Hydraulic Condition 2 mean velocity fields. Velocity fields are presented as (a) spatially complete contour map of streamwise velocity (b) select two-dimensional velocity vectors Hydraulic Condition 3 mean velocity fields. Velocity fields are presented as (a) spatially complete contour map of streamwise velocity (b) select two-dimensional velocity vectors Vorticity profile for (a) Hydraulic Condition 1, (b) Hydraulic Condition 2, (c) Hydraulic Condition Vertical velocity profiles at several streamwise distances; x. Hydraulic Condition 3; (a) superimposed profiles showing boundary layer development throughout the supercritical inflow, (b) superimposed velocity profiles throughout the smoothing reach Vertical profile of shear stress components and velocity profile Non-dimensional plot of streamwise velocity profiles superimposed upon the Prandtl-Von Karman logarithmic overlap layer inner law figure reproduced from White (1991) Comparison of theoretical velocity profiles versus experimental data for estimation of shear velocity. Hydraulic Condition 1 (a) Velocity profiles at x = 5.11 cm, (b) Velocity profiles at x = cm, (c) Non-dimensional velocity profiles at x = 5.11 cm, (d) Non-dimensional velocity profiles at x = cm

11 xi 4.12 Comparison of theoretical velocity profiles versus experimental data for estimation of shear velocity. Hydraulic Condition 2 (a) Velocity profiles at x = 0.13 cm, (b) Velocity profiles at x = cm, (c) Non-dimensional velocity profiles at x = 0.13 cm, (d) Non-dimensional velocity profiles at x = cm Comparison of theoretical velocity profiles versus experimental data for estimation of shear velocity. Hydraulic Condition 3 (a) Velocity profiles at x = 1.12 cm, (b) Velocity profiles at x = cm, (c) Non-dimensional velocity profiles at x = 1.12 cm, (d) Non-dimensional velocity profiles at x = cm Shear velocity and boundary shear stress predicted using Coles law of the Wake. Values are plotted at every tenth cross section across the entire profile for (a) Hydraulic Condition 1, (b) Hydraulic Condition 2, (c) Hydraulic Condition Vertical profiles of Reynolds stress superimposed upon a linear curve fit of the total stress. Plots represent data spatially averaged over 1 cm for: Hydraulic Condition 1 (a) supercritical flow, (b) crest of 1st undulation; Hydraulic Condition 2 (c) supercritical flow, (d) crest of 1st undulation; Hydraulic Condition 3 (e) supercritical flow, (f) peak of the roller

12 xii 4.16 Vertical profiles of horizontal and vertical velocity fluctuations superimposed upon theoretical curves for horizontal and vertical velocity fluctuations. Plots represent data spatially averaged over 1 cm for: Hydraulic Condition 1 (a) supercritical flow, (b) crest of 1st undulation; Hydraulic Condition 2 (c) supercritical flow, (d) crest of 1st undulation; Hydraulic Condition 3 (e) supercritical flow, (f) peak of the roller

13 xiii Acknowledgments I would like to thank Professor David Hill for all of his hard work and expert guidance in aiding in the completion of this task and Professor Kendra Sharp for her programming skills. Finally I would like to thank and my wife Juliana for all of her loving support.

14 1 Chapter 1 Introduction The purpose of the research detailed herein is to update current knowledge pertaining to the hydraulics within a stationary hydraulic jump by applying state of the art technology to this classical problem. The hydraulic jump is a naturally occurring phenomenon that is commonly associated with the hydraulics seen in white water rivers. Clearer examples of hydraulic jumps can be seen at the outlet structures of gravity dams and downstream of bridges and culverts during periods of high flow. Hydraulic jumps also can occur in non-liquid flows as demonstrated by cloud formations downwind of mountain ranges. A hydraulic jump occurs when a supercritical flow rapidly transitions to a subcritical flow, as seen in Figure 1.1. A hydraulic jump is characterized as a highly turbulent flow that involves large energy losses over a finite distance. Hydraulic jumps are commonly associated with air entrainment in the roller region, creating the so-called white water appearance. However, air entrainment is not a defining characteristic of a hydraulic jump. In Figure 1.1 it can be noted that as the upstream supercritical flow rapidly evolves into subcritical flow, a roller region is developed at the jump location. Little is known about the actual hydraulics of the roller region due to the many technical difficulties in making measurements in this region. Theoretically the roller has been modeled as a recirculating volume of fluid, suggesting a region of flow reversal. In a

15 2 Fig Classical hydraulic jump profile. classical hydraulic jump the roller dissipates large amounts of energy to transform the flow from supercritical to subcritical across a small distance. This roller region is a defining characteristic of a hydraulic jump. The transformation from supercritical flow to subcritical flow can manifest itself in a form other than a hydraulic jump. This other form is the undular jump. An undular jump, which is pictured in Figure 1.2, is similar to a hydraulic jump in that it serves as a transition from supercritical to subcritical flow. However, an undular jump does not transition as rapidly and does not have a roller. Additionally, an undular jump exhibits a wavy structure in the transitional region that serves to dissipate energy in place of the roller. The dimensionless quantity known as the Froude number generally is used to define the characteristics of any hydraulic jump. The Froude number for any rectangular channel is defined as: F r u gy (1.1)

16 3 Fig Classical undular jump profile. where u is the depth average velocity, y is the corresponding flow depth and g is gravity. The strength of the inflow Froude number can be used to predict whether an undular or hydraulic jump will occur. The Froude number boundary has been determined experimentally to be about 1.7 for jumps that have fully developed supercritical inflow (Ohtsu et al. 2001). The phenomenon of the hydraulic jump was first sketched and described by Leonardo DaVinci ( ) almost five centuries ago (Narayanan 1975). Since this early beginning, the understanding of the external workings of the hydraulic jump has been studied closely. The well-known sequent depth equation credited to Belanger (Bakhmeteff and Matzke 1935) for hydraulic jumps is one of the equations produced through those studies. This equation is derived from the one-dimensional momentum principle in conjunction with the continuity equation with the absence of frictional losses. The sequent depth equation is as follows: y 2 = y 1 2 [ F r 2 ] (1.2) 1

17 4 where F r is the Froude number and y is any flow depth with the subscripts 1 and 2 corresponding to the supercritical and subcritical flows, respectively. More recent research into the external shape of hydraulic jumps has been focused on numerical computer models that can predict the location, length and sequent depth ratio of a hydraulic jump in any type of channel. However, research into the internal workings of the hydraulic jump has not kept pace with the external modeling. The oldest studies date to the late 1950s when researchers investigated the internal structure of the hydraulic jump using an air-flow model (Rouse et al. 1958) instead of water-flow to create the jump. Later water-flow studies involved high speed photography of the internal fluid structure to qualitatively model the internal structure of the fluid. The turbulent nature of hydraulic jumps, including the reversal of the flow direction in the roller, the two-phase nature of strong jumps and the un-stable/oscillating nature of jumps have made quantitative measurement of flow structure using primitive fluid mechanics measurement tools such as pitot tubes very difficult. However, technological innovations in the past couple of decades have allowed researchers to attempt to quantitatively measure the internal structure. These technological innovations include Hot Film Anemometry, Acoustic Doppler Velocimetry (ADV), Laser Doppler Velocimetry (LDV) and Particle Image Velocimetry (PIV). The research described herein utilizes the PIV technique for velocity vector mapping to update previous work done to map the internal hydraulics in a hydraulic jump. This research appears to be the first time that PIV technology has been applied to the study of a free, stationary hydraulic jump. Application of PIV technology will allow

18 5 for the development of spatially extensive vector maps for various hydraulic conditions. This technology will allow for a detailed look at the flow structure that had previously not been possible. Additionally, measurements will be made of the free surface using point gages and capacitance wave gages. These measurements will be used to test the validity of the free surface locations produced by the PIV analysis. The flow field data collected in the PIV analysis will be used to analyze the mean flow structures present within the various hydraulic conditions, the vorticity structure with each condition, and the boundary shear stress structure. This data will be of value to other researchers as a tool for validating numerical models of hydraulic jumps.

19 6 Chapter 2 Literature Review 2.1 Introduction This review of the literature is a summary of the most relevant research articles pertaining to the measurement and analysis of the internal workings of hydraulic and undular jumps. Also summarized are select articles pertaining to other aspects of hydraulic jump research including free-surface measurement and analysis, numerical modeling techniques and other experimental works pertaining to the internal hydraulics of a hydraulic or undular jump. The summaries are separated into three sections: analysis of the free-surface of a hydraulic or undular jump, analysis of the flow structure in a hydraulic or undular jump, and other analyses pertaining to hydraulic or undular jumps. Within each section the summaries are organized in chronological order. 2.2 Water Surface Profile Research Hydraulic jump research The original work done on the hydraulic jump was believed to have been performed by Bidone in 1820 and included as part of his memoirs (Bakhmeteff and Matzke 1935). The work done by Bidone as well as the work done by Belanger in developing the sequent depth equation is not clearly outlined in any of the available literature on this subject. Since these early beginnings, the research related to hydraulic jumps has outlined most of

20 7 the qualities related to the water surface profile. In some of the earliest works and some of the newest works, researchers were concerned with measuring and developing equations that relate to the length of the hydraulic jump being studied. Recent research has also been concerned with classifying different types of jumps, investigating the oscillating characteristics of jumps and checking the applicability of the Belanger equation with various channel properties. Research pertaining to the water surface profile commonly is conducted using either point gages to get an instantaneous measurement of the water surface or a capacitance wave gage that captures a time series of the water surface elevation. The first published research on the investigation of the water surface profile was by Bakhmeteff and Matzke (1935). In this publication the authors credit Bidone as the first individual to describe the hydraulic jump and Belanger for developing the sequent depth equation, though neither Bidone nor Belanger have a referenced work from which this credit is given. The research done by Bakhmeteff and Matzke utilized a combination point gage and fixed scale to measure and map the water surface profile of 11 different hydraulic conditions. The authors developed a curve that correlates the dimensionless quantity of the jump length divided by the subcritical flow depth to the kinetic flow factor. The kinetic flow factor as defined by Bakhmeteff and Matzke is equivalent to the Froude number. The authors concluded that this curve should be capable of predicting the length of any jump being analyzed in a hydraulic design. Rajaratnam and Subramanya (1968) collapsed the water surface profile data collected by Bakhmeteff and Matzke (1935) and Rajaratnam (1965) with the intention of creating one general non-dimensional profile for any hydraulic jump. The profile was

21 8 to be generated by combining vertical and horizontal characteristic lengths collected by both of the credited authors. The authors presented the coordinates for this general profile in terms of the dimensionless values η and λ. In this profile η is the quantity y/y, where Y is the vertical scale which was computed as 0.75 (y 2 y 1 ) and y corresponds to the depth at that location in the profile. Additionally, λ is the quantity x/x, where X is the horizontal scale which is equivalent to x at the section where y = 0.75(y 2 y 1 ) and x is the horizontal location along the profile measured from the toe of the jump. This horizontal scale was chosen due to discrepancies in the definition of the length of the roller between Bakhmeteff and Matzke (1935) and Rajaratnam (1965); the author acknowledges that the ideal length scale would be the length of the jump. The general profile is presented as Figure 2.1. Fig Theoretical water surface profile for any hydraulic jump superimposed upon experimental data. Figure reproduced from Rajaratnam and Subramanya (1968).

22 9 Sarma and Newnham (1973) performed laboratory research on the water surface profile of hydraulic jumps with Froude numbers less than four. Their work improved upon previous works by providing a detailed analysis of low Froude number jumps that had previously been lacking. The authors present the data collected in the experiment and several plots of the data that utilize the same dimensionless quantities as Bakhmeteff and Matzke (1935). The authors also collapsed their data with the data of Bakhmeteff and Matzke (1935) and Rajaratnam (1965) to produce a new sequent depth equation for hydraulic jumps. The collapsed data shows a significant degree of scatter; to counter this, Sarma and Newnham developed a conservative prediction of the sequent depth values with an equation that lies above the data. This equation is based on the Belanger equation and has the form: y 2 = y [ ] F 2. (2.1) 2 r1 In Equation 2.1, the empirical constant 40 is referred to as the momentum coefficient. Sarma and Newnham compared this momentum coefficient to the constant 8 given by Bidone in Equation 1.2. The authors concluded that for hydraulic jumps with Froude numbers less than four, the momentum coefficient in the sequent depth equation should be significantly larger than previously thought. This conclusion accounts for the factor of five difference between the coefficient in Belanger s equation and Sarma and Newnham s coefficient. Mehrotra (1976) investigated the length of the hydraulic jump by compiling the data collected by Bakhmeteff and Matzke (1935) and Nagaratnam et al. The goal of the

23 10 author was to develop a set of equations to predict the length of any hydraulic jump based on the compiled data. Mehrotra postulated in this analysis that the length of any hydraulic jump is defined by the characteristic length of the eddies contained within the roller. The author produced an equation of the form: L r,j y 2 = ( ) y1 2 [ ( )] k r,j y 1 y 2 2 y F r 2 1 y y 2 2 F 4 r 1 (1 + y 1 y 2 ) 6 ( y1 y 2 ) 2 ( ) y (2.2) y 1 In Equation 2.2 k r,j is a constant that differentiates between the length of the roller and the length of the jump and L r,j is the length of the roller or the length of the jump depending on the k r,j value chosen. Mehrotra presents a graphical comparison of his equation versus the compiled data. In this graphical comparison, Mehrotra s equation seems to reasonably model the trends of the compiled data. However, the k constant utilized by the equation is an empirical coefficient that has no definitive trend defined by the author. Hughes and Flack (1984) investigated the sequent depth characteristics of a jump occurring over a rough bed. The researchers goal was to present data on hydraulic jumps occurring over rough beds and to compare that data to a theoretical sequent depth equation developed by Leutheusser and Kartha (1972). The research was conducted utilizing five different artificially roughened test beds inserted into a flume. With the test beds in place, measurements of the sequent depths were made on several test runs with each test bed using a point gage. The flow rate was varied during these test runs such that data were collected for discharges ranging from 0.34 to 0.5 cfs with Froude numbers ranging

24 11 from 3 to 10. The authors concluded that the data collected reasonably agreed with a simplified version of Leutheusser and Kartha s sequent depth equation on a quantitative level. Leutheusser and Kartha s (1972) sequent depth equation was simplified to account for variables that could not be measured with the available equipment. The authors attempted to develop a theoretical agreement between the two, but were unable to provide satisfactory results. Hager et al. (1990) conducted a thorough investigation into the length of the roller occurring in various hydraulic jumps. The research was conducted using photographs of the free-surface and point gage measurements on jumps created in three different channels. The authors present the roller length data collected during this experiment and concede that the definition of the roller length is arbitrary. They concluded that the data collected are insufficient for the definition of a generalized length equation. Zhuo (1991) constructed a new theoretical model for sequent depths using data collected by Rouse et al. (1958) and Resch et al. (1974). The theoretical model developed by Zhuo improves upon the Belanger equation by accounting for turbulence, boundary friction and non-uniform velocity distributions up and downstream of the jump. Zhuo s model consists of two equations that form an upper and lower boundary encompassing the compiled data, as shown: ( y ( ) 2 ) 3 + ɛ F 2 y F 2 = 0 y r 1 1 y r 1 1 Lower Boundary ( y 2 y 1 ) 3 + ( ɛ F 2 r 1 ) y2 y F 2 r 1 = 0 Upper Boundary. (2.3)

25 12 In Equation 2.3, ɛ is a dimensionless factor that accounts for boundary friction and the empirical constants 1.88, 2.28 and 2.16 account for the turbulent intensities and non-uniform velocity distributions. Ohtsu and Yasuda (1991) investigated the properties of type B and D jumps with the goal of developing sequent depth and roller length equations for each of these conditions. These types of jumps are formed at the intersection of a steep slope and a relatively shallow slope. This slope change is generally associated with spillways and stilling basins. A type-b jump is a jump that occurs at the intersection of the slopes. A type-d jump is a jump that occurs just up slope of the slope intersection. The experiment was conducted in a horizontal flume with plywood planks installed in the flume to create a dam spillway type simulation in the flume. The jumps measured in this experiment were created at the break in slope occurring at the junction of the planks and the horizontal flume bottom. The planks were installed at angles varying from 8 to 60 with Froude numbers ranging from 4 to 14. Data were collected using point gages and Prandtl type pitot tubes, though no data on the velocity characteristics were presented. The authors developed length and sequent depth equations for each of these jump types. Hager (1993) developed an equation for the free surface profile of any hydraulic jump. This equation is based upon six experiments with inflow Froude numbers ranging from 2 to 10. The equation was developed by curve fitting dimensionless graph of depth versus horizontal distance of the free surface measurements. The free surface equation developed by Hager is: y y 1 y 2 y 1 = tanh[1.5(x/l r )] (2.4)

26 13 where y corresponds to any depth throughout the profile, x the horizontal location corresponding to y, and L r the length of the roller. Husain et al. (1994) investigated the sequent depth ratio and length of various forced and free hydraulic jumps. A forced hydraulic jump is defined by the authors as any jump occurring due to the use of an energy dissipator, such as up-steps or blocks. A free jump would then be any jump occurring without the use of an energy dissipator. The characteristics of each jump were measured by first tracing the profile of the jump onto tracing paper attached to the flume wall; from this sketch lengths and depths were measured directly. The forced jumps analyzed in the research were created by installation of steps at a given distance downstream of the head gate. Hydraulic jumps with Froude numbers ranging from 4 to 12 were analyzed with slopes ranging from 2.5 percent to 7.5 percent and steps ranging from 1 to 10 cm installed in the flume. The authors developed a set of non-dimensional equations in terms of a profile coefficient for jumps occurring in a sloping channel with or without steps Undular Jumps Though several authors have investigated the free surface of hydraulic jumps, far fewer have investigated undular jumps. Reviewing the literature reveals that the difference between undular and hydraulic jumps probably was not considered by early researchers. As previously stated, from a qualitative standpoint undular jumps, which are characterized by numerous undular crests and troughs, and hydraulic jumps, which are characterized by a single roller, are quite different.

27 14 The first published work on undular jumps was by Anderson (1978). Anderson developed an equation for the entire free surface profile across an undular jump. His equation is based on the Boussinesq energy equation and has the form: dy dx = 3h s y 2 2y 3 3 ln(y) 3h s y y ln(y 1 ) (2.5) where h s is the total specific head that is held constant throughout the transition reach. The transition reach is defined in the literature as the region between the fully developed supercritical inflow and the fully developed subcritical outflow. In an undular jump the transition reach contains all of the undular crest and troughs. The y values correspond to flow depths at any location and the subscript 1 corresponds to the supercritical inflow. Anderson s equation was compared to four data points that were produced from an unspecified source. This comparison showed good agreement between experimental data and Equation 2.5. Reinauer and Hager (1995) investigated the characteristics of undular jumps using photographs to record the shape of the jump and point gage measurements. The goal of their research was to describe the main flow characteristics that are present in an undular jump, outline the strong scale effects by which the undular jump is governed and outline design relations, by which the undular jump can be described. The authors outlined four different types of undular jumps and their bounding Froude numbers. The jump types are separated by the characteristics of the shockwaves that propagate at 45 to the mean channel flows which are used to visualize the transition from 2-D to 3-D flow.

28 15 Steinrück et al. (2003) developed a numerical model for low Froude number undular jumps with fully developed turbulent inflow. The result of their analysis is a set of first order differential equations that predict the variations in water surface elevation, wave length and wave amplitude. The theoretical basis for this numerical model was a variation of the Prandtl-von Karman logarithmic overlap equation. The equations developed by Steinrück et al. were validated using water surface profile data collected by Chanson and Montes (1995) for inflow Froude numbers ranging from 1.15 to 1. Steinrück et al. concluded that the water surface profile computed using their numerical model reasonably agrees with the data collected by Chanson. 2.3 Internal Hydraulics Research The earliest studies into the internal hydraulics of any type of jump generally were concerned with the condition of the supercritical inflow and the impact of this inflow on the shape and nature of the jump occurring. Other works on internal hydraulics were concerned with mapping mean flow and turbulence characteristics, the size and shape of the vortices within a jump and the boundary shear stress beneath a jump Hydraulic Jumps The first noted research into the internal workings of hydraulic jumps was performed by Rouse et al. (1958). This research team made measurements using a double sided pitot tube and hot film anemometry in an air-flow model for Froude numbers of 2, 4 and 6. The pitot tube was used to measure an average velocity at various locations while the hot film was used to measure turbulence characteristics at the same locations.

29 16 Hot Film Anemometry is a velocity measurement method that works by the concept that the resistance through a wire is proportional to the temperature of the wire. The wire is generally a 1 to 2 mm long wire or thin film that is heated by a voltage bridge supplied by supports at either end of the wire. The hot wire is exposed to a fluid flow; as the flow passes the wire thermodynamic laws describe the interaction between the two that attempts to cool the wire. The voltage bridge across the wire is increased to counter this effect and maintain a steady resistance across the wire. The voltage changes across the wire can be monitored and calibrated to obtain a velocity measurement at the probe. Rouse et al. (1958) utilized an air-flow model for analysis instead of a water-flow model due to the complications air entrainment in the roller region causes with hot film anemometry. The air-flow model was constructed using the free-surface profile measured from a hydraulic jump occurring in a water flume. This free-surface profile was used to construct a rigid boundary in the shape of a hydraulic jump. This boundary was inserted as the bottom boundary in a wind tunnel and the action of the supplied air-flow was measured. The largest drawback of this model as noted by the authors is the lack of free-surface interaction at the hydraulic jump. This interaction is responsible for the entrainment of air in a water-flow jump and allows for oscillation of the jump. Nevertheless the authors concluded that the hydraulic jumps produced in their experiment follow the basic equations of motion and that their data are of both qualitative and quantitative value when describing the motion of a water-flow hydraulic jump.

30 17 The first notable work on the internal hydraulics of a water-flow hydraulic jump was done by Rajaratnam (1965), whose goal was to collect pressure, velocity and boundary shear stress data to support the theory that a hydraulic jump is analogous to a wall jet. The wall jet as defined by Rajaratnam is a jet that impinges tangentially on a boundary, surrounded by relatively stationary fluid. The pressure and velocity measurements were made with a Prandtl-type pitot static tube. The measurements were made only in the forward flow region downstream of the roller, due to the previously discussed complications with making measurements in the aerated roller region. A Preston tube was used to make measurements of the shear stress occurring on the boundary below the jump. A Preston tube is essentially a boundary mounted pitot tube that is calibrated to calculate shear stress based on measurement of stagnation pressure and the static pressure. In this experiment, data were collected for nine different hydraulic conditions, with Froude numbers ranging from 2.68 to The author presents select data related to dimensionless velocity and depth factors and graphically relates this data to a theoretical wall jet. Rajaratnam concluded that the velocity distribution in the free mixing portion of the hydraulic jump closely relates to the distribution in a wall jet. However, the jet-like nature of a hydraulic jump falls off faster than in a wall jet. Rajaratnam (1965) also developed a theoretical momentum equation that models the data acquired in the research more accurately that the Belanger equation. This equation, unlike Belanger s equation, accounts for energy loss due to boundary friction, and is given as Equation 2.6. ( y2 y 1 ) 3 y ) 2 (1 ɛ + 2F 2r1 + 2F 2 = 0 (2.6) y r 1 1

31 18 The ɛ value in this equation is a correction factor that accounts for the boundary friction occurring in an open channel flow. Rajartnam presents a graphical relationship between the ɛ value and the Froude number based on the data collected in this experiment. An ɛ value of 0 represents a frictionless condition. This condition satisfies the assumptions of the Belanger equation, and Equation 2.6 thereby reduces to the Belanger equation. Research using water-flow hydraulic jumps continued with the work of Leutheusser and Kartha (1972), who investigated the supercritical inflow into a hydraulic jump. The team used a pitot tube to make several velocity measurements throughout a given supercritical cross section; these measurements were used to define the development of the inflow into the jump. The researchers goal was to use experimental data to test the validity of the general hydraulic jump equation given undeveloped and fully-developed supercritical inflow. Leutheusser and Kartha concluded that the condition of the inflow into a hydraulic jump has significant effects on the hydraulics within the hydraulic jump. Their experiment determined that a hydraulic jump having fully developed inflow would be lower and longer than its undeveloped counterpart. Through this work, Leutheusser and Kartha derived a new equation to predict the sequent depths across a jump, accounting for the shear development in the supercritical inflow. This new equation, later tested by Hughes and Flack (1984), is: ( ) y 2 y2 y 1 y 1 1 2F 2 r 1 = 1 + ɛ. (2.7)

32 19 The ɛ coefficient in this equation is a summation of independent coefficients that account for a non-uniform velocity distribution, skin friction and turbulence. This equation reduces to the Belanger equation if the assumptions of that equation are met, i.e., uniform velocity distribution, negligible skin friction and the absence of turbulence. The ɛ coefficient is also affected by the development of the shear inflow. Narayanan (1975) produced a theoretical analysis of Rajaratnam s (1965) theory that a free hydraulic jump in the region beneath the roller is similar to a two dimensional wall jet. The author presents several graphical comparisons between the theoretical properties of a wall jet and hydraulic jump data collected by Rajaratnam (1965) and Rouse et al. (1958). Narayanan concluded that the wall jet analogy could be improved upon with a better understanding of the turbulent stresses acting on a jump. The research team of Hoyt and Sellin (1989) investigated the internal workings of a hydraulic jump by photographing the profile of the jump. The intent of the researchers was to use the entrained air bubbles as tracers for flow visualization. A polymer was introduced into the flow to control the fine structure within the jump itself and presumably control the amount of air entrained within a jump. From this experiment the researchers were able to sketch a theoretical model of vortex formation and braiding within a hydraulic jump. Long et al. (1990) performed the first measurements within a hydraulic jump using a non-intrusive measurement method. The researchers made measurements of both horizontal and vertical mean velocities, turbulent shear stress and turbulence intensities on a submerged hydraulic jump using Laser Doppler Velocimetry, an non-intrusive measurement method that uses the Doppler shift created by laser beams reflecting off

33 20 seed particles to calculate instantaneous fluid velocities for a discrete volume of flow. A LDV system is capable of sampling data from a very small sampling volume with large temporal resolution. The sampling volume for an LDV system is on the order of cubic millimeters (Long et al. 1990). Long et al. (1990) also made free surface measurements using an unspecified method. The authors view the submerged hydraulic jump as being a transitional phenomenon between a free jump and a wall jet. A submerged jump is essentially a hydraulic jump that is formed downstream of a sluice gate that is bounded on the downstream side by ponded water. Generally the toe of a submerged jump will be located at the exit from the sluice gate and the entire jump will occur under the free surface. The study of a submerged jump is advantageous because the roller does not interact with the free surface and there is no two-phase flow. The authors intend the study to serve as a comparison of these three classes of flows. The study was performed on jumps with Froude numbers ranging from 3 to 8 and submergence factors (S) ranging from 0.2 to 1.7. The submergence factor is defined by Long et al. as: S = (y 1 y 2 ) y 2. (2.8) The jumps were formed in a 7.5 meter long glass and aluminum flume with adjustable head and tail gates. The flow was seeded with latex paint to enhance the backscatter signal being measured by the LDV system. The authors present the seed particle size as being approximately 0.1 mm with a concentration of about 1 ppm; however, they do not provide a specific gravity for the seed particle. The authors present a graph (see

34 21 Figure 2.2) that shows the deviation of a free jump from a wall jet with respect to a non-dimensional velocity scale versus a non-dimensional length scale. In Figure 2.2, the velocity scale is defined as the mean velocity at any cross-section (u) divided by the inlet velocity (u 1 ). The length scale is defined as the horizontal distance from the toe (x) divided by the distance at which the velocity scale equals 0.5(L). The data used consists of data produced by Rajaratnam (1965) for the free jump and Rajaratnam and Subramanya (1968) for the wall jet, along with the data collected during this experiment for a submerged jump. From Figure 2.2, Long et al. observed that for length scale values less than 1.5 the decay of the velocity scale is similar for free jumps, submerged jumps and wall jets. However, as the length scale increases beyond 1.5, the submerged jump continues to follow the decay of a free jump while both deviate from the wall jet. The authors concluded that much like free hydraulic jumps, submerged jumps are threedimensional in nature in the roller region and two-dimensional as the flow progresses downstream. Another photographic study of the internal structure was conducted by Long et al. (1991). An 0.47 meter wide flume was used in the research; however, the photographs taken were focused on the side of the flume. Four different inflow Froude numbers were analyzed using cameras capturing photos at a rate of 2000 per second. Each inflow Froude number was monitored for a period of several seconds with the resultant photographs being spliced into a video. The researchers analyzed the video at various rates of speed and observed and mapped the vertical flow structure within each jump guided by the motion of entrained air within the jump. Using the mapping, measurements of the magnitude of the vortices relative to the depth of flow in the channel were made. The

35 22 Fig Non-dimensional plot of mean channel velocity versus horizontal distance of a free hydraulic jump and a wall jet. Figure reproduced from Long et al. (1990).

36 23 researchers reported that based on these measurements, the vertical magnitude of the vortices is roughly 1.5 times the size of the horizontal magnitude for vortices occurring in the larger parts of the jump. Hornung et al. (1995) performed the first published research into a traveling hydraulic jump, or bore, using Particle Image Velocimetry techniques. Particle Image Velocimetry (PIV) is the latest technological innovation in fluid mechanics that nonintrusively measures fluid motion. PIV relies on the interrogation of pairs of images of a region of the flow taken at two different times to generate velocity vectors. The major advantage of PIV over LDV is its ability to generate a much better spatial distribution of data, e.g. develop velocity maps over the entire fluid flow field. However, the disadvantage of PIV is that the temporal resolution is not as good as LDV. Whereas it is trivial to obtain tens or hundreds of thousands of realizations of velocity at a single point with LDV, current storage limitations limit PIV ensembles to about a thousand realizations. Hornung et al. (1995) were interested in obtaining mean vorticity information downstream of a moving hydraulic jump. Data was collected on 10 different jumps with Froude numbers ranging from 2 to 6. The PIV data were analyzed on a frame-by-frame basis to give information about vorticity generation as the bore approached and passed the test section. The measurements verified the control-volume analysis conducted by the authors and provided data about the development of a shear layer at the toe of the bore. Gunal and Narayanan (1996) investigated the mean velocity properties within a hydraulic jump using hot film anemometry techniques. Their research was conducted in a 6 meter flume with slopes adjusted between 0 to 10 percent and data were collected

37 24 for various Froude numbers. In order to collect data within the roller region of the hydraulic jump, the researchers rotated the hot film probe 180 to allow measurement of the velocities within the region of flow reversal. The researchers collected data including water surface profiles, roller lengths and mean channel velocities under seven different slope and Froude number conditions. Gunal and Narayanan concluded that the length of the roller as determined by visual observation is as much as 1.6 times the length of the roller as determined by analysis of the mean velocity. They also concluded that the maximum turbulent shear stress is a function of the maximum mean velocity relative to the mean reverse flow velocity at a defined cross-section. Veeramony and Svendsen (1997) performed LDV measurements with the intention of analyzing the roller, and equating the roller motion to the breaking of waves. Three different hydraulic conditions were analyzed in this research, with Froude numbers varying from 1.28 to 1.6. The authors were able to obtain values for the roller thickness, shear stress, vorticity, and eddy viscosity distributions along the lower limit of the roller. They concluded that these measured properties can be consistently scaled to uniform distributions for all of the Froude numbers investigated. A study by Svendsen et al. (2000) obtained internal hydraulics and free surface data on several hydraulic jumps using LDV and capacitance wave gages. The jumps analyzed in this research were relatively weak jumps having Froude numbers of 1.38, 1.46 and Weak hydraulic jumps were used so that the impact of air entrainment on the data collection would be minimal. Data were taken along the centerline of the flow and the authors present results on mean velocity, turbulence intensity, roller thickness, momentum flux, shear stress, vorticity and eddy viscosity. Svendsen et al. (2000) found that

38 25 the distribution of vorticity and shear stress near the free-surface confirm the hypothesis that the breaking at the roller resembles a shear-layer, but that it deviates from the flow in an ordinary shear layer. They also found that the dimensionless forms of shear stress, vorticity and eddy viscosity vary independent of the Froude number variations. The most recent available research on hydraulic jumps was completed by Liu et al. (2003). They used a micro Acoustic Doppler Velocimeter (micro ADV) to make measurements within weak hydraulic jumps having Froude numbers of 2, 2.5 and Acoustic Doppler Velocimetry (ADV) is an intrusive measurement method that can account for the flow reversal and two-phase properties of hydraulic jumps, particularly in the roller region. ADV measures the average velocity within a discreet test section by measuring the Doppler shift that occurs when emitted sound waves reflect off seed particles within the test section. The test section utilized in ADV is generally on the order of about 0.3 cubic centimeters (SonTek 1997), limiting the ability of an ADV system to measure very fine turbulent structures within a fluid flow. The micro ADV used in Liu et al.ś research is a version of ADV this has better accuracy and resolution than a standard ADV. However, micro ADV is still an intrusive method that will alter the hydraulics occurring within a jump. The results of the ADV analysis were processed using the phase-space thresholding method that filtered out data spikes created by air bubbles entering the sampling area. This processing produced better results for the researchers. The results of this experiment included mean velocities, turbulence intensities, Reynolds stresses and power spectra of velocities for many points within the flow. Liu et al. (2003) concluded that the maximum turbulent kinetic energy

39 26 decreases linearly with longitudinal distance within the jump and levels off in the transition region reaching a constant value of about 3 percent of the upstream supercritical kinetic energy Undular Jumps Chanson and Montes (1995) conducted a study of the free surface and internal hydraulics of several undular jumps. Seventy-four different hydraulic conditions were analyzed with Froude numbers ranging from 1.05 to The free surface measurements were made using a rail-mounted point gage and the internal hydraulics were measured with a pitot tube. The authors used these measurements as well as photography of the free surface to identify five different classes of undular jumps, as shown in Figure 2.3. The authors present graphical data on the center line velocity distribution, pressure distribution and specific energy at consecutive wave crests in a type C undular jump. They do not present limiting Froude numbers for each of the types described in the research. It is acknowledged that additional work will be necessary to understand the mechanisms of lateral shock waves present in high Froude number undular jumps and the related limiting Froude numbers. Montes and Chanson (1998) conducted a study of several undular jumps by measuring velocity, pressure and energy distributions, wavelengths and wave amplitudes. They show that the jumps each have strong three-dimensional features that differentiate an undular jump from an undular bore. Montes and Chanson used the data collected in this research and the data from Chanson and Montes (1995) to validate a theoretical Boussinesq-type solution for the free surface presented by the authors. The theoretical