APPLICATION OF PARTICLE IMAGE VELOCIMETRY TO THE HYDRAULIC JUMP · 2004-02-09 · The Pennsylvania...

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 Fulfillmentof the Requirements

for the Degree of

Master of Science

May 2004

We approve the thesis of Justin M. Lennon.

Date of Signature

David F. HillAssistant Professor of Civil EngineeringThesis Adviser

Arthur C. MillerProfessor of Civil Engineering

Kendra V. SharpAssistant Professor of Mechanical Engineering

Andrew ScanlonProfessor of Civil EngineeringHead of the Department of Civil and Environmental Engineering

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 aer-

ation. 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.

iv

Table of Contents

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 2. Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Water Surface Profile Research . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Hydraulic jump research . . . . . . . . . . . . . . . . . . . . . 6

2.2.2 Undular Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Internal Hydraulics Research . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Hydraulic Jumps . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.2 Undular Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Other Analyses Pertaining to Undular and Hydraulic Jumps . . . . . 30

Chapter 3. Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Experiment Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2.1 Flume Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2.2 Free Surface Measurement Setup . . . . . . . . . . . . . . . . 36

v

3.2.3 PIV Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.1 Free Surface Data Collection . . . . . . . . . . . . . . . . . . 45

3.3.2 PIV Data Collection . . . . . . . . . . . . . . . . . . . . . . . 46

3.4 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Chapter 4. Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1 Free Surface Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Mean Velocity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3 Turbulent Flow Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3.2 Shear Stress from Mean Velocity Profiles . . . . . . . . . . . . 68

4.3.3 Shear Stress by the Darcy Friction Factor . . . . . . . . . . . 80

4.3.4 Shear Stress from Reynolds Stresses . . . . . . . . . . . . . . 82

4.3.5 Shear Stress from Root Mean Square Velocity Fluctuations . 85

Chapter 5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.1 Discussion of Technical Aspects of Data Collection . . . . . . . . . . 89

5.2 Discussion of the Mean Velocity Analysis . . . . . . . . . . . . . . . 92

5.3 Discussion of the Boundary Shear Stress Analysis . . . . . . . . . . . 93

5.4 Future Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 94

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

vi

List of Tables

4.1 Crest and trough magnitudes for Hydraulic Condition 2. . . . . . . . . . 55

4.2 Summary of the supercritical flow hydraulic properties for each hydraulic

condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3 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. . . . . . . . . 78

4.4 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. . . . . . . . . . . . . . . . . . . . . . . . 81

4.5 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. 83

4.6 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 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

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. . . . . . . . . . . . . . . . 95

viii

List of Figures

1.1 Classical hydraulic jump profile. . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Classical undular jump profile. . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Theoretical water surface profile for any hydraulic jump superimposed

upon experimental data. Figure reproduced from Rajaratnam and Sub-

ramanya (1968). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 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). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Classification of undular jumps. Figure reproduced from Chanson and

Montes (1995). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 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). . . . . 29

3.1 Line drawing of recirculating flume setup. . . . . . . . . . . . . . . . . . 33

3.2 Sketch of the flow straightener relative to the direction of flow. . . . . . 35

3.3 Simplified schematic of the interaction between a capacitance wave gage

and the surrounding fluid. . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4 Wave gage wiring schematic. . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5 PIV wiring schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

ix

3.6 Profile view of laser and laser light sheet orientation relative to flume. . 41

3.7 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.8 Typical layout of location changes within a given hydraulic condition. . 47

3.9 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. . . . . . . . . . 48

4.1 Manually measured free surface profiles for (a) Hydraulic Condition 1,

(b) Hydraulic Condition 3. . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2 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 3. . . . . . . . . . . . . . . . . . . 58

4.3 Generalized sketch of the interaction between the flow field and a smooth

free surface as seen from the 2-dimensional perspective of a camera look-

ing up at the free surface. Sketch illustrates the mirroring of seed par-

ticles onto the free-surface, creating a non-physical flow field above the

free surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4 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. . . . . . . . . . . . . . . . . . . . . . . 61

x

4.5 Hydraulic Condition 2 mean velocity fields. Velocity fields are presented



4.6 Hydraulic Condition 3 mean velocity fields. Velocity fields are presented



4.7 Vorticity profile for (a) Hydraulic Condition 1, (b) Hydraulic Condition

2, (c) Hydraulic Condition 3. . . . . . . . . . . . . . . . . . . . . . . . . 66

4.8 Vertical velocity profiles at several streamwise distances; x. Hydraulic

Condition 3; (a) superimposed profiles showing boundary layer devel-

opment throughout the supercritical inflow, (b) superimposed velocity

profiles throughout the ‘smoothing’ reach. . . . . . . . . . . . . . . . . . 69

4.9 Vertical profile of shear stress components and velocity profile. . . . . . 70

4.10 Non-dimensional plot of streamwise velocity profiles superimposed upon

the Prandtl-Von Karman logarithmic overlap layer inner law figure re-

produced from White (1991). . . . . . . . . . . . . . . . . . . . . . . . . 73

4.11 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 = 48.09 cm, (c) Non-dimensional

velocity profiles at x = 5.11 cm, (d) Non-dimensional velocity profiles at

x = 48.09 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

xi





x = 26.78 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75





x = 39.48 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.14 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 3. . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.15 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. . . . 84

xii

4.16 Vertical profiles of horizontal and vertical velocity fluctuations superim-

posed upon theoretical curves for horizontal and vertical velocity fluctua-

tions. 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. . . . . . . . . . 87

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.

1

Chapter 1

Introduction

The purpose of the research detailed herein is to update current knowledge per-

taining 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 as-

sociated 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

2

Fig. 1.1. 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:

Fr ≡u√

gy(1.1)

3

Fig. 1.2. 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 experi-

mentally 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 (1508-1513) 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:

y2 =y12

[−1 +√

1 + 8F 2r1

] (1.2)

4

where Fr 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 re-

searchers 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, in-

cluding 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 measure-

ment 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 Ve-

locimetry (ADV), Laser Doppler Velocimetry (LDV) and Particle Image Velocimetry

(PIV).

The research described herein utilizes the PIV technique for velocity vector map-

ping 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

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.

6

Chapter 2

Literature Review

2.1 Introduction

This review of the literature is a summary of the most relevant research arti-

cles 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: anal-

ysis 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

2.2.1 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

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 col-

lected 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

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 pro-

file 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 ∗ (y2 − y1) 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(y2− y1)

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. 2.1. Theoretical water surface profile for any hydraulic jump superimposed uponexperimental data. Figure reproduced from Rajaratnam and Subramanya (1968).

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:

y2 =y12

[−1 +

√1 + 40F 2

r1

]. (2.1)

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

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:

Lr,j

y2=

√√√√√√√√kr,j

(y1y2

)2[1− y2

y1+ 1

2F 2r1

(1−

y21

y22

)]2

F 4r1

(1 + y1

y2

)6 −(

y1y2

)2(y2y1− 1

)2. (2.2)

In Equation 2.2 kr,j is a constant that differentiates between the length of the roller and

the length of the jump and Lr,j is the length of the roller or the length of the jump

depending on the kr,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 equa-

tion 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

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 at-

tempted 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:

(y2y1

)3 +(

ε− 1− 1.88F 2r1

)y2y1

+ 2.16F 2r1

= 0 Lower Boundary

(y2y1

)3 +(

ε− 1− 2.88F 2r1

)y2y1

+ 2.16F 2r1

= 0 Upper Boundary. (2.3)

12

In Equation 2.3, ε 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 − y1y2 − y1

= tanh[1.5(x/Lr)] (2.4)

13

where y corresponds to any depth throughout the profile, x the horizontal location

corresponding to y, and Lr 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.”

2.2.2 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.

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=

√3hsy

2 − 2y3 − 3 ln(y)− 3hsy21 + 2y3

1 + 3 ln(y1) (2.5)

where hs 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.

15

Steinruck et al. (2003) developed a numerical model for low Froude number un-

dular 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 de-

veloped by Steinruck 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. Steinruck

et al. concluded that the water surface profile computed using their numerical model rea-

sonably 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.

2.3.1 Hydraulic Jumps

The first noted research into the internal workings of hydraulic jumps was per-

formed 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.

16

Hot Film Anemometry is a velocity measurement method that works by the con-

cept 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 en-

trainment 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.

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 bound-

ary 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 measure-

ments 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 pres-

sure. In this experiment, data were collected for nine different hydraulic conditions, with

Froude numbers ranging from 2.68 to 9.78. 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. (y2y1

)3− y2

y1

(1− ε + 2F 2

r1

)+ 2F 2

r1= 0 (2.6)

18

The ε 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 ε value and the Froude number based on the data collected in this experiment. An

ε 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 su-

percritical 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, ac-

counting for the shear development in the supercritical inflow. This new equation, later

tested by Hughes and Flack (1984), is:

y2y1

(y2y1− 1

)2F 2

r1

= 1 + ε. (2.7)

19

The ε coefficient in this equation is a summation of independent coefficients that

account for a non-uniform velocity distribution, skin friction and turbulence. This equa-

tion 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

ε 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 pre-

sumably 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 us-

ing a non-intrusive measurement method. The researchers made measurements of both

horizontal and vertical mean velocities, turbulent shear stress and turbulence intensi-

ties 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

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 0.001

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 phe-

nomenon 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 =(y1 − y2)

y2. (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

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 (u1). 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 three-

dimensional 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

22

Fig. 2.2. Non-dimensional plot of mean channel velocity versus horizontal distance ofa free hydraulic jump and a wall jet. Figure reproduced from Long et al. (1990).

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 hy-

draulic jump, or bore, using Particle Image Velocimetry techniques. Particle Image

Velocimetry (PIV) is the latest technological innovation in fluid mechanics that non-

intrusively 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 disadvan-

tage 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

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 inten-

tion 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 num-

bers 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 an-

alyzed in this research were relatively weak jumps having Froude numbers of 1.38, 1.46

and 1.56. 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, mo-

mentum flux, shear stress, vorticity and eddy viscosity. Svendsen et al. (2000) found that

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 3.32.

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.s 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 bet-

ter 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

26

decreases linearly with longitudinal distance within the jump and levels off in the tran-

sition region reaching a constant value of about 3 percent of the upstream supercritical

kinetic energy.

2.3.2 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 2.83. 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 mea-

suring 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

27

Fig. 2.3. Classification of undular jumps. Figure reproduced from Chanson and Montes(1995).

28

model presented is valid for undular jumps produced by fully developed shear inflow at

Froude numbers less than 1.4. The theoretical model agrees with the experimental data

with “fair accuracy”.

Chanson (2000) investigated the spatial variations of boundary shear stress oc-

curring beneath an undular jump. Chanson collected shear stress data on four different

undular jumps using Prandtl type pitot tube calibrated and used as a Preston tube.

The undular jumps analyzed in this research had Froude numbers ranging from 1.26

to 1.51. Resultant boundary shear stress values for two different hydraulic conditions

are presented as Figure 2.4, where the minimum boundary shear stress values occur be-

neath the undular crests and the maximum values occur beneath the undular troughs.

Chanson concluded that this distribution suggests the formation of three-dimensional

standing wave bed forms beneath undular jumps.

An upper Froude number limit for undular jumps was charted by Ohtsu et al.

(2001). The goal of the research was to chart this limit and determine the variability

of the limit depending on the development stage of the incoming supercritical inflow.

The development of the inflow was determined using LDV techniques. Through this

work the researchers determined that a Froude number of 1.7 was the upper limit for

undular jump formation with fully developed inflow. They also determined that the

limit deviated from as low as 1.3 to as high as 2.3, depending on the development of the

supercritical flow.

29

Fig. 2.4. Dimensionless bed shear distributions beneath an undular jump for (a) trans-verse profile for Froude number = 1.25 (b) longitudinal profile for Froude number =1.48.Figure reproduced from Chanson (2000).

30

2.4 Other Analyses Pertaining to Undular and Hydraulic Jumps

Farhoudi and Smith (1982) conducted experiments on the scour created by a

hydraulic jump. They uses a flume with an artificial mobile bed installed downstream

of a spillway. Farhoudi and Smith were able to validate Equation 2.9 for the time scale

of the scour downstream of the jump.

dmaxdo

=(

t

to

)α(2.9)

in which dmax is the maximum scour depth downstream of the hydraulic jump, do is a

characteristic depth of scour equal to the depth of flow above the undisturbed channel

bed, t is the time for the maximum scour depth to be reached and to is the amount of

time for the characteristic depth of scour to be reached. The alpha value is a constant

that varies depending on the type of jump and the material present in the movable bed.

Farhoudi and Narayanan (1991) conducted an experiment to measure the mean

and fluctuating forces exerted on a slab beneath a free hydraulic jump. They showed that

the intensity of the force fluctuations depends on the length and width of the measuring

device and its position relative to the toe of the jump.

Liu and Drewes (1994) developed a numerical model of the turbulence character-

istics in free and forced hydraulic jumps. The model was based upon the k-epsilon model

adapted for a moving free surface as is present in any hydraulic jump. The results of the

model were compared to the turbulence characteristics published by Long et al. (1990).

The results of this numerical model show that the energy dissipation rate is proportional

31

to the inflow Froude number, and that the turbulence energy is mainly dissipated in the

roller region.

Chanson and Qiao (1994) measured and developed a theoretical model for gas

transfer within hydraulic jumps. They investigated the air bubble entrainment in hy-

draulic jumps with partially developed shear inflow. The results of the analysis allowed

Chanson and Qiao to conclude that the theoretical model developed allows for adequate

prediction of the dissolved gas content and water quality downstream of any hydraulic

jump.

Yamashiki et al. (1997) developed a numerical model for the two-dimensional

structure of large eddies and erosion occurring in and under a hydraulic jump. Their

numerical model is based upon the two-dimensional mixed flow Navier-Stokes equations.

Mossa and Tolve (1998) investigated the bubbly two-phase properties of several

hydraulic jumps. Flow visualization techniques were used by the researchers to develop

relationships between the length of the jump and the average air fraction at a section.

Mossa and Tolve concluded that the bubbly structure captured in the photographs per-

mits one to observe the turbulence structure within the flow and that the greatest areas

of air concentration within a jump existed in two regions within the roller.

Mossa (1999) investigated the oscillating characteristics of hydraulic jumps using

free surface measurements. Her interest in this aspect of hydraulic jumps stems from

difficulties that had been encountered by other researchers while attempting to measure

the scour being caused by hydraulic jumps. The water surface profile was measured

using an electronic hydrometer type point gage with electronic integrators which allowed

for estimation of the time-averaged depth of the flow. Video cameras also were used

32

to analyze in detail each hydraulic jump and provide visual evidence of the oscillations

occurring in the flow. Measurements were made in a 7.72 meter long flume with an

irregular rigid plywood channel installed for several conditions; several runs were made

with a forced jump in a rectangular cross-section. During this research 70 hydraulic

conditions were analyzed and Mossa concluded that the oscillations of hydraulic jumps

are not dependent on whether the bed is made out of erodible or non-erodible material.

The oscillations occurring in the analyzed jumps occurred with regularity such that a

time scale for the oscillations may be defined, though this article did not define it.

33

Chapter 3

Experimental Methods

3.1 Introduction

The experiments conducted in the present research were conducted using a 16

foot long recirculating flume. The flume has an adjustable head gate, tail gate and

channel slope that allows for control of the location and strength of the hydraulic jumps

to be studied. Water is recirculated through the flume using two centrifugal pumps that

combined supply roughly 100 GPM (Hampden). A simplified line drawing of the flume

is presented in Figure 3.1.

Head-boxTail-box

Pumps

Screw DrivenActuator

Water Storage Tanks

Flume Channel

Fig. 3.1. Line drawing of recirculating flume setup.

34

Measurements of the internal hydraulics were made using Particle Image Ve-

locimetry techniques. The PIV system used in the research consists of a double-pulsed

90 mJ Nd:YAG laser and a digital camera with a 2048 pixel by 2048 pixel CCD array

that records 4096 different shades of grey into 12-bit tiff images (TSI 2001). The PIV

system is described in more detail in Section 3.2.3.

3.2 Experiment Setup

3.2.1 Flume Setup

The flume used in this research is a 16 foot long acrylic construction flume with a 1

foot wide by 1.5 foot tall rectangular cross-section constructed by Hampden Corporation.

Water is recirculated through the flume by two 1.5 hp centrifugal pumps, each operating

at about 50 GPM (Hampden). As water leaves each pump it travels through 10 feet of 2

inch diameter reinforced flexible hose and into 2 inch diameter PVC pipe. The flow rate

throughout the system can be monitored using a 1 inch diameter orifice plate installed

in each of the PVC pipes. The orifice plates are monitored by non-recording electronic

differential manometers that report a three second average pressure differential in psi.

The recirculation water is stored inside three 89-gallon plastic tanks located under the

flume, as seen in Figure 3.1. The flume is supported by a pivot on one side and a screw

driven actuator on the other that allows for fine adjustment of the flume slope. The

flume slope is output to the flume control panel in units of feet per hundred feet and can

be read to three decimal places.

35

During initial trials the flow down the flume had several cross-currents being

produced by irregularities in the transition from the head-box to the flume channel.

These irregularities most likely are due to the mechanism that allows for movement of

the head-gate and could not be permanently fixed. In order to develop a more uniform

two-dimensional flow throughout the flume channel, a flow straightener was installed

just downstream of the head gate. The flow straightener (Figure 4.1) is constructed with

polycarbonate, is 12 inches square at the base and 4 inches tall, and consists of 1/4”

diameter honeycomb cells produced by Plascore Inc.

12"

12"

Fig. 3.2. Sketch of the flow straightener relative to the direction of flow.

Installation of the flow straightener appeared to reduce the cross-currents that

had been present in the flume, yielding a more uniform 2-dimensional flow across the

channel.

The exact flow rate being analyzed within the flume could not be measured di-

rectly by the orifice plates since they had not been calibrated by the manufacturer. The

36

orifice plates were calibrated during the research using the average velocity profile mea-

sured using the PIV. The average velocity profile was combined with a known water

surface elevation using the continuity equation to determine the flow rate in the channel.

Using this flow rate and a manually time averaged measurement from each of the digital

manometers, a calibration coefficient was determined. The equation used to predict the

flow rate based on the manometer readings is:

Q = CA

√2g

(P1γ

+ z1 −P2γ

+ z2

). (3.1)

In order to calibrate the orifice plate, it was assumed that each orifice plate would

have similar headloss across the plate; thus, the C coefficient for each plate was assumed

to be equivalent. This assumption is made since both orifice plates were constructed

and installed at the same manufacturing facility. The assumption is necessary because a

flow rate down the channel supplied by each pump independently has not been measured

at this time. With this assumption, a C coefficient of 0.6302 was determined. This C

coefficient closely agrees with the C coefficient of a sharp-edged orifice plate with a C of

0.61 (Street et al. 1996).

3.2.2 Free Surface Measurement Setup

Two different types of free surface measurement gages were utilized during this

experiment: a digitally recording wave gage and a manual point gage. Both gages were

mounted on a 4 inch by 4 inch wooden plank that spanned the top of the flume. The

devices were positioned such that all of the measurements were taken along the centerline

37

of the flume channel. The horizontal location of each measurement was determined using

a cloth measuring tape attached to the top of the flume.

The point gage used in the experiments is a stainless steel point gage that records

distances in feet and is accurate up to 0.001 feet. Prior to any measurement being

made with the point gage, the instrument was calibrated by measuring the distance to

the bottom of the flume. The depth of any water flow then was determined by simply

subtracting the calibration distance from the measured distance to the water surface.

The wave gage used in this experiment was a capacitance type wave gage on loan

from the Mathematics Department at The Pennsylvania State University. This type of

wave gage works similar to a standard two plate capacitor. The gage supplies a negative

voltage to the surrounding water while a positive voltage is supplied to a wire sealed

inside an insulator. A simplified schematic of this interaction is presented as Figure 3.3.

NEGATIVELY CHARGEDWATER ( - )

POSITIVELY CHARGEDWIRE ( + )

INSULATOR

Fig. 3.3. Simplified schematic of the interaction between a capacitance wave gage andthe surrounding fluid.

38

Similar to a standard capacitor, a charge is built up across the insulator that is

proportional to the amount of voltage supplied and the area of the charged elements.

In the case of the wave gage, as the area of the charge elements (e.g., the depth of the

water changes) the amount of voltage supplied to the elements must change to maintain

a constant charge across the insulator (Holman 1989). Hence a relationship between

water depth and supplied voltage can be developed.

To outline a more detailed description of how the wave gage works, a wiring

schematic is presented in Figure 3.4. The wave gage is wired such that a voltage is

supplied to the shore side electronics which in turn supplies a voltage to the sea side

electronics and the wave gage. The voltage being supplied to the wave gage is recorded by

the shore-side electronics, and transferred to the National Instruments Data Acquisition

Board. After the data is digitized, it is transferred to the computer. The National

Instruments LabVIEW program then is used to automate the acquisition. The voltage

data being read into LabVIEW is recorded over a set interval of time, providing detailed

measurements of the voltage changes occurring across the wave gage. The wave gage was

calibrated using a bucket of water and several known depth values. Using the voltage

values recorded by LabVIEW and the known depths of waver corresponding to the

voltages, a linear curve fit with an R2 value of 0.9997 was generated for the instrument.

This curve fit then was used to convert any voltage reading registered by the wave gage

into a flow depth.

Both the point gage and the wave gage were positioned within the flume such

that all depth measurements were made along the centerline of the flow.

39

COMPUTER

12 VOLTPOWERSUPPLY

SHORE SIDEELECTRONICS

ANALOG OUTPUT BOARD

+

-

SEA SIDEELECTRONICS WAVE GAGE

INSULATOR ENCASEDWIRE

METAL BRACKET

+-

Fig. 3.4. Wave gage wiring schematic.

40

3.2.3 PIV Setup

The Particle Image Velocimetry system used in this research project consists of

a double-pulsed 90 mJ Nd:YAG lasers encased in a single housing, a 12 bit 2048 x 2048

pixel digital camera (TSI 2001) and a synchronizer, all manufactured by TSI corporation.

A simple wiring schematic of the PIV system is presented as Figure 3.5.

The system is wired such that the laser fires at the same time as the camera

takes a picture; this action is controlled by the synchronizer through the Flashlamp and

Q-switch ports for the laser and the synchronizer port for the camera. The pictures

are loaded onto the computer through the frame grabber which is wired to the camera.

The port A wire interfaces the synchronizer with the computer allowing the computer

to control the timing settings of the synchronizer. The laser umbilical cord powers and

controls both laser heads interfacing with the power supply and recirculates cooling water

to prevent the laser heads from overheating.

Cylindrical and spherical lenses were attached to the front of the laser to manip-

ulate the laser beam such that a flat plane of laser light would intersect the test section.

The cylindrical lens spreads the laser beam into a sheet while the spherical lens narrows

the waist of the sheet, such that in the test section there is a wide narrow sheet of light

rather than the small cylindrical beam emitted by the laser. The laser sheet then is redi-

rected by a mirror such that the laser can sit horizontally while the laser sheet vertically

enters the test area within the flume. The path of the laser sheet is shown in Figure 3.6.

Due to the large size of the light sheet being redirected, a high-energy mirror was

not a cost effective method for redirecting the sheet. In this research, three different

41

LASER

POWERSUPPLY

COMPUTER

DIGITALCAMERA

SYNCRONIZER

Frame Grabber

COM #1

CoolingWaterTubing

LaserUmbilical

Port A

Flashlamp #1 & #2

Q-Swicth #1 & #2

Fig. 3.5. PIV wiring schematic.

FLUME CHANNEL BED

CameraViewable Area

Laser Mirror

Spherical and CylindricalLenses

Laser Mounting Board

LaserLight Sheet

Fig. 3.6. Profile view of laser and laser light sheet orientation relative to flume.

42

types of reflective surfaces were utilized to address this issue. The first type of reflective

instrument used was a pair of 90 degree prisms, stacked on top of each other to create a

flat surface for perfect reflection. The prisms worked well for flows that were sufficiently

shallow such that the light sheet did not need to be very wide due to the small cam-

era viewable area. The second type of reflective instrument used was a standard front

reflecting mirror purchased from a hardware store. The front reflecting mirror worked

fairly well for a short period of time. The mirror’s reflective surface was burned through

by the laser beam after about 400 pulses from the laser. Data were taken using this

mirror by taking half of a data set, then slightly moving the mirror and finishing the

data set. This method work well until the mirror was burned to the point that it could

no longer be used. The third type of reflective instrument used was a piece of stainless

steel with a #7 mirror finish (McMaster-Carr). Much like the front-reflecting mirror,

the reflective surface of the stainless steel was burned by the laser after a period of time.

However, the reflective surface on the steel mirror lasted significantly longer such that

an entire data set could be taken without the mirror being moved.

The digital camera used in the data collection was mounted to the side of the

flume. The mount consisted of an aluminum mounting board, several steel spacers,

hollow rectangular steel conduit and steel cable added for support. The plan and profile

view of the camera mount are displayed in Figure 3.7.

43

Las

er L

ight

She

et

Flume

Flume

Camera

Profile View

Plan View

Cam

era

Mou

nt

Camera Mount Rails

Camera Mount Rails

Cam

era

LaserLight Sheet

Light SheetAccess Panels

Fig. 3.7. Plan and profile view of the camera orientation relative to flume. Views alsoshow orientation of the camera mount and the location of the exposed sections of theacrylic channel bottom that are utilized for laser light sheet access.

44

The mounting board contained rails that allowed for motion with the horizontal

plane of the flume. The steel spacers allowed for vertical motion of the camera, and the

steel conduit allowed for the camera to be moved closer to / farther from the flume. The

mounting board can be raised by placing 1 inch and 3 inch spacers under the mounting

frame; and 1/8 inch spacers can be placed under the camera to raise the camera off the

mounting board.

As seen in the plan view in Figure 3.7, the bottom of the flume has two areas that

provide access for the laser to enter the flume. The two areas are limited and separated

by structural members that support the flume. The first area is 24 inches by 8.5 inches

and the second area is 20 inches by 8.5 inches. The data maps created in each of the

three hydraulic conditions contain a hole in the data that is explained by this limitation.

3.3 Data Collection

In this research, data were collected for three different hydraulic conditions with

Froude numbers ranging from 1.37 to 2.99. The range of Froude numbers that could be

experimentally analyzed was restricted by the experimental equipment. Hydraulic jumps

with Froude numbers higher than 2.99 could not be analyzed because the flume pumps

lacked sufficient power to produce a stronger jump. Additionally, it is unlikely that a

significantly strong hydraulic jump would produce adequate PIV data in the roller region

due to the large amount of air entrainment that would be expected. The complications

involved in collecting PIV data within a two phase flow are discussed in detail in section

3.3.3.

45

3.3.1 Free Surface Data Collection

Free surface data were collected for each of the hydraulic conditions. The methods

used to measure the free surface of each hydraulic condition varied depending on the

nature of the jump.

The first experiment analyzed in this research, herein referred to as Hydraulic

Condition 2, involved an oscillatory undular jump. The jump oscillated such that the

location of the jump toe as well as the crests and troughs of the undulations shifted by

as much as 5 centimeters in the horizontal direction, over a period of about 1 minute.

The intrusion of the wave gage apparatus into the flow of Hydraulic Condition 2 caused

the magnitude of the oscillations to increase with the mean location of the jump toe

shifting. Due to this consequence, only the point gage could be used to make free

surface measurements for this condition. However, the oscillations prevented a detailed

mapping of the free surface using the point gage. Thus, only the magnitudes of the toe,

crests and troughs could be measured for this condition.

The second experiment, herein referred to as Hydraulic Condition 1, consisted

of a weak undular jump that was much steadier than Hydraulic Condition 2. However,

much like Hydraulic Condition 2, the intrusion of the wave gage into the flow disrupted

the hydraulics of the system causing oscillations in the jump location. The point gage

thus was used to make detailed free surface measurements for Hydraulic Condition 1.

Using the point gage, data were collected over varying distance intervals throughout the

length of the test area.

46

The final experiment, referred to as Hydraulic Condition 3 , consisted of a stronger

hydraulic jump that had no visible oscillations. The intrusion of the point gage into the

subcritical and roller regions of this flow did not seem to seriously disrupt the hydraulics

of this condition. Thus, free surface data were gathered in these regions using the wave

gage, with point gage data collected to verify the wave gage data. The wave gage

apparatus contained a base that was too thick to collect data in the thin supercritical

flow; thus, the point gage only was used to collect data in this region.

3.3.2 PIV Data Collection

During this research, three different hydraulic conditions were analyzed by the

PIV, with each condition containing between 9 and 12 different streamwise data collection

locations. At each of these locations the camera was realigned and the mirror was

relocated. These relocations were necessary so that the desired output for the entire

length of the hydraulic condition could be produced. The desired output from the data

collected involves an image area that is large enough to encompass the full depth of flow,

yet small enough to view the fine turbulence characteristics present in a hydraulic jump.

A typical alignment and sizing of the image areas is presented in Figure 3.8.

As can be seen in Figure 3.8, the viewable area from one location to another can

change in magnitude. This change occurs to accommodate the change in the depth of

the water flow. The magnitude of the viewable area is controlled by the location of the

camera relative to the flume, e.g. the closer the camera is to the flume, the smaller

the viewable area. The water surface of the hydraulic jumps occurring in some of the

cases herein researched have both streamwise variations and cross-channel variations.

47

Location #1

Location #2

Location #3

Location #4

Location #5

Location #6

Viewable Area perCamera Placement

Fig. 3.8. Typical layout of location changes within a given hydraulic condition.

When placing the cameras this cross-channel variation in the water surface profile must

be considered. The variation occurs such that the flow in the center of the flume rises

up higher than the water at the edge of the flow. This variation is handled by placing

the camera low enough so that the camera is viewing up into the flow while still slightly

looking down onto the channel bottom, as seen in Figure 3.7. The placement is such

that the focal axis is just above the bottom of the channel. However, under certain

conditions it was not possible to place the camera so that the entire depth of flow could

be analyzed. If the ratio between the flow depth at the edge of the flume and the depth

of the centerline was large enough, the camera could not capture data near the crest of

an undulation. This situation is presented in Figure 3.9.

The location of each test section was tracked using a 1.3 meter-long rigid plastic

ruler. The ruler was placed in the flume before and after data collection at each location

and images were recorded to accurately track the data location. The location of the ruler

48

Las

er L

ight

She

et

Cross-Sectional Viewof the Water Surface

Profile of an Undular Crest

FlumeChannel

Camera

Fig. 3.9. 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 theentire depth of flow under a crest.

49

was calibrated using a set location on the flume to ensure proper placement each time the

ruler was placed in the flume. The set location of the ruler also was calibrated to coincide

with the horizontal measurement locations from the free surface measurement. The

flume was filled with water before images of the ruler were taken to negate magnification

differences that exist when viewing the test section in water instead of air. With the

ruler in place and the flume filled with water, the camera would be repositioned so that

a small overlap in the test sections existed. This overlap can be seen in Figure 3.8.

The overlap was used because the best data generally exist in the central portion of

the captured images. This situation exists because the intensity of the light emitted by

the laser has a Gaussian distribution. This distribution causes the central region of the

images to be brighter than the edges.

Once the camera placement was set and the mirror and lens settings corrected,

data were taken. The Insight software provided by TSI Corporation that complements

the PIV hardware was used to collect the data. Data were collected in ensembles of

400 pairs of images, with each image pair captured across a set time step. The time

step between the first and second images in a pair is a parameter set by the user and

is dependent on several other parameters, including the velocity of the particle motion

within the test section, the amount of seed particle present in the fluid flow and the

size of the interrogation window used in the data analysis. The time step used in this

research ranged from 75 to 200 microseconds. Insight stores each of the images collected

as 12-bit tiffs with an a-b file name suffix to differentiate between the first and second

image within a pair.

50

The particles used for the first two hydraulic conditions were 10 micron white

spherical particles from Potters Industries, Inc. These particles have a specific gravity

very close to 1; this, combined with the small diameter, ensures a very short response

time, which is important as the central assumption to PIV is that the particles follow

the flow. The particles were added to the storage tanks prior to taking data and the

flume was run for at least 10 minutes to ensure that the recirculating flow had an even

distribution of the particles. A direct relationship exists between the number of particles

in the test section and the amount of light that will enter the camera. The amount of

light entering the camera also is controlled by the F-stop setting on the Nikkor lens that

is attached to the digital camera. A low F-stop allows more light to enter the CCD array

in the camera, but reduces the depth of focus and increases the risk of a reflected laser

beam entering the camera and damaging the CCD array. To minimize this risk, sufficient

particles were added to the flow so that the F-stop could be set to a higher value while

still producing adequate vector interrogations. Additional particles were introduced into

the flow after long periods of time to account for any particles that may have settled

out or adhered to the flume or tank walls. This was done to ensure that the optimal

amount of information could be obtained from all of the images. The water used in the

flume was replaced between each hydraulic condition to ensure that excessive particles

were not contained within the system.

This procedure worked well for the first two hydraulic conditions analyzed. How-

ever, the third hydraulic condition contained a significantly stronger jump and contained

two-phase flow, unlike the first two conditions. The reflective properties of air bubbles

in a two-phase flow are a danger when using standard PIV measurements. Under these

51

conditions, it is possible for the light sheet to reflect off a bubble and directly into the

CCD array in the camera. The laser light is intense enough that it can do damage to

the sensor array. This problem was resolved by using fluorescent particles to change the

wavelength at which the images are being recorded.

The practice of using fluorescent particles is referred to in the literature as fluores-

cent Particle Image Velocimetry (fPIV). The particles used in this research are 10 micron

particles coated in fluorescene and rhodamine and were obtained from the research group

of Dr. Joseph Katz at Johns Hopkins University. The rhodamine on the particles emits

light at wavelengths between 560 and 584 nm with a peak emission wavelength of 572

nm (Katz, personal correspondence) when excited by the laser. The laser light and the

reflections off of standard seed particles and bubbles have a wavelength that peaks at

532 nm (TSI 2001). The camera was protected from reflected laser light by a standard

#22 red-orange filter (B+W Filter Company) attached to the Nikkor lens. This filter is

a long-pass filter that blocks all wavelengths of light below about 560 nm (Kodak). This

method, therefore, protects the CCD array from large reflections but still allows for the

acquisition of seed particle images. If the bubble phase has too high a concentration,

the optical path from the laser sheet to the CCD array will be too obstructed to yield

satisfactory images. The downside to this type of analysis is that the cameras were not

intended to record images from light emitting at 572 nm; consequently the images pro-

duced were much dimmer than the ones produced in the first two hydraulic conditions.

However, the images did contain enough information for a good vector interrogation

when the source data was left as 12-bit images.

52

3.4 Data Analysis

After the data were collected, they were analyzed using various programs on a

location-by-location basis. After all locations had been interrogated with vectors pro-

duced and validated, the velocity fields were averaged and spliced together to form mosaic

vector maps representative of each hydraulic condition.

First the images from Insight were rescaled to 8-bit images. This rescaling was

necessary since the subsequent programs used in the data analysis currently are not

capable of handling 12-bit images. The scaling of the images reduces the file size associ-

ated with the images and the grey scale, but does not affect the spatial resolution of the

image.

Next the images were interrogated using PIV Sleuth, version 1.21 (Christensen et

al. 2000). Based upon a user specified window size, Sleuth subdivides each image into a

rectangular grid of sub-images or windows. Each window in image ‘a’ is cross-correlated

with its respective window in image ‘b’ to determine the average particle displacement in

that window. The result is returned as a two-dimensional displacement vector, centered

in the window. In this fashion, the spatial resolution of the images is reduced from

4.2 mega pixels to a grid of several thousand vectors. A universal set of interrogation

parameters for Sleuth does not exist, as system optimization depends upon the specific

flow field. However, a 64 by 64 pixel window, with a 50 percent overlap commonly was

used in the present study.

Next, the raw vector fields were validated, or “cleaned” to remove outlier vectors.

The “cleaning” of the vector map was accomplished by loading the raw vector file into

53

CleanVec (Christensen et al. 2000). CleanVec allows the user to filter out vectors based

upon a number of criteria. Successful validation, defined as the removal of non-physical

vectors without the removal of likely physical vectors, is as much art as science. As such,

the appropriate selection of filter parameters requires a trial-and-error process. CleanVec

then will look to replace removed vectors with suitable alternatives provided by Sleuth.

CleanVec also can ‘fill’ holes in the data using a linear interpolation between adjacent

points. Use of this interpolation feature should be limited as it essentially creates data.

Using CleanVec to validate the vector map of the current experiments typically produced

between 8 percent and less than 1 percent ‘bad’ vectors and, therefore, empty grid points

that contained no acceptable alternative within the vector map of any given location for

all of the hydraulic conditions.

Once all 400 vector files for each location had been validated using CleanVec, an

ensemble average of each grid point within the vector maps was developed. The ensemble

averaged data files then were scaled and spliced together to form a mosaic velocity profile

for each hydraulic condition. The averaging and scaling of the data transformed the large

amounts of data collected throughout the analysis into a one-dimensional profile for each

condition that is representative of the hydraulics existing in that condition. The mosaic

vector maps produced in this fashion all contain 1220 vertical columns of vector data;

the horizontal data ranges from 60 to 110 rows or between 73,000 and 134,000 vectors.

As discussed in the data collection section, the data collected in the third condition

needed to remain 16 bit images for an adequate vector interrogation due to the dimness of

the images. Due to this restriction, PIV Sleuth was not used to interrogate the vectors for

the third hydraulic condition. TSI’s Insight program was used for the vector interrogation

54

in this condition, with the results being imported into CleanVec and processed normally

from there. Insight uses the same FFT algorithm as PIV Sleuth for interrogation of the

images; however, the controls in Insight are not as user friendly. Thus the program was

not used for the first two hydraulic conditions.

55

Chapter 4

Experimental Results

4.1 Free Surface Analysis

The free surface data from Hydraulic Condition 1 and Hydraulic Condition 3 are

presented as Figure 4.1. The measured toe, crest and trough magnitudes for Hydraulic

Condition 2 are presented in table 4.1.

Table 4.1.Crest and trough magnitudes for Hydraulic Condition 2.

Location Depth

(centimeters)

Supercritical Flow 3.078

Crest of Wave 1 6.949

Trough of Wave 1 3.962







56

x (cm)

Flow

dept

h(c

m)

0 50 1000

1

2

3

4

5

6

7

8

9

10

Wave gage dataPoint gage data

b.)

x (cm)

Flow

dept

h(c

m)

50 1000

1

2

3

4

5

Point gage data

a.)

Fig. 4.1. Manually measured free surface profiles for (a) Hydraulic Condition 1, (b)Hydraulic Condition 3.

57

For each hydraulic condition, an attempt also was made to locate the free surface

using a ‘quality of data’ threshold. As discussed in Section 3.4, CleanVec removes vectors

that appear to be non-physical. Each image contained a region in space that was above

the free surface. Not surprisingly, most of the vectors at these grid points were removed.

In the averaging of the ensemble realizations, a criteria of 50 percent missing vectors was

applied. Thus, any grid point containing less that 50 percent valid vectors was assumed to

lie above the free surface. The free surface profiles for each hydraulic condition generated

in this fashion are shown in Figure 4.2.

A comparison between the directly measured free surface and the free surface

determined as discussed above shows that for Hydraulic Conditions 1 and 2, discrepancies

exist. The third hydraulic condition, however, shows little difference between the data

sets, suggesting that this sort of analysis holds promise for free surface mapping. A

possible source of the discrepancies in Hydraulic Conditions 1 and 2 could be reflection of

the laser illuminated seed particles onto the underside of the free surface. A generalized

schematic of the flow field reflected onto the free surface is presented in Figure 4.3.

Presumably this reflection is not present in Hydraulic Condition 3 because this condition

was much more turbulent with a ‘choppy’ free surface. Hydraulic Conditions 1 and 2

each had a much smoother free surface that was likely capable of producing a clear

reflection of the seed particles in the water, with Condition 1 being smoother. Visual

inspection of the images in these data sets did indeed reveal the presence of obvious

reflections. These reflections are coherent enough to be interrogated by Sleuth and will

result in over-estimation of the free surface.

58

x (cm)

Flow

dept

h(c

m)

50 1000

1

2

3

4

5

Point Gage DataPIV Free Surface Data

a.)

x (cm)

Flow

dept

h(c

m)

0 50 1000

2

4

6

8

Wave Gage DataPoint Gage DataPIV Free Surface Data

c.)

x (cm)

Flow

Dep

th(c

m)

0 50 1000

1

2

3

4

5

6

7

Point Gage Magnitude MeasurementsPIV Free Surface Data

b.)

Fig. 4.2. Free surface profile generated from the PIV data superimposed upon measuredfree surface data for (a) Hydraulic Condition 1, (b) Hydraulic Condition 2, (c) HydraulicCondition 3.

59

Laser Light Sheet

Camera Capture AreaNon-Physical Flow Field Reflected onthe Free Surface

Physical FlowField mirrored by the Free Surface

Fig. 4.3. Generalized sketch of the interaction between the flow field and a smooth freesurface as seen from the 2-dimensional perspective of a camera looking up at the freesurface. Sketch illustrates the mirroring of seed particles onto the free-surface, creatinga non-physical flow field above the free surface.

60

Additionally in Hydraulic Condition 3, the PIV generated free surface as shown

in Figure 4.2 is significantly lower than the directly measured free surface in the vicinity

of the roller. The deviation is due to the strong 3-dimensional shape of the roller in this

condition. The roller was shaped such that the water surface at the wall of the flume

was significantly lower than the water surface at the centerline of the flume. This shape

made it impossible for the camera to gather data up to the free surface. The physical

limitations resulting in this situation are discussed in Section 3.3.2 and presented in

Figure 3.9. Thus the camera was positioned to gather data from the channel bottom to

the highest point possible under the roller.

In all future analyses of this data involving the free surface, the directly measured

free surface data will be used.

4.2 Mean Velocity Analysis

The mean velocity fields for each of the three hydraulic conditions are presented

in Figures 4.4 through 4.6 as both contour plots of the streamwise velocity and as vertical

profiles of the two-dimensional velocity vectors. In the velocity vector mapping, every

fifth vector in the vertical direction and every thirtieth to fortieth vector in the horizontal

direction are shown. This is done solely for visual clarity as each vector mapping contains

1220 streamwise cross sections with between 30 and 110 vectors per cross section.

Several observations can be made from the presented velocity contour and vector

maps. Each hydraulic condition can be seen to have a clear boundary layer present in the

supercritical approach flow. The height of this layer appears steady as the supercritical

flow approaches the toe of the jump, leading to the conclusion that the flow is fully

61

x(c

m)

y(cm)

050

100

5

100

cmse

c-1(b

)

x(c

m)

y(cm)

050

100

5

Hor

izon

talV

eloc

ity;u

(cm

sec-1

):2

2038

5674

(a)

Fig

.4.

4.H

ydra

ulic

Con

diti

on1

mea

nve

loci

tyfie

ld.

Vel

ocity

field

sar

epr

esen

ted

as(a

)sp

atia

llyco

mpl

ete

cont

our

map

ofst

ream

wis

eve

loci

ty(b

)se

lect

two-

dim

ensi

onal

velo

city

vect

ors.

62

x(c

m)

y(cm)

050

100

5

100

cmse

c-1

x(c

m)

y(cm)

050

100

5

u:3

2344

6484

Fig

.4.

5.H

ydra

ulic

Con

diti

on2

mea

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ocity

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ors.

63

x(c

m)

y(cm)

050

100

0510

100

cmse

c-1(b

)

x(c

m)

y(cm)

025

5075

100

510

Hor

izon

talV

eloc

ity;u

(cm

sec-1

):-1

420

5488

122

(a)

Fig

.4.

6.H

ydra

ulic

Con

diti

on3

mea

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ocity

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sar

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64

developed for all three jump conditions. As the flow enters the undular formations of

Hydraulic Conditions 1 and 2, a ‘waviness’ can be observed in the velocity field. The

‘waviness’ of the velocity field mimics the undular free surface such that under the free

surface crests, the velocity gradient at the boundary significantly decreases. The decrease

in the velocity gradient in this region has implications when analyzing the scour caused

by an undular jump.

Classical hydraulic jump behavior can be seen in the velocity field for Hydraulic

Condition 3. The velocity near the free surface in the roller region can be seen to signif-

icantly decrease, though no flow reversal is evident in the measured velocity data. This

observation contradicts visual observations made during experimentation. During the

experimental period, flow reversal clearly could be seen at the free surface for this con-

dition. The discrepancy between these observations can be explained by the deviations

in the free surface of the jump at the roller seen in Figure 4.2, as explained the Section

4.1.

The Froude number for each of the hydraulic conditions was determined using the

mean velocity profiles discussed above. As previously discussed, the Froude number is

defined as Fr ≡um√gy , where um is the mean cross-sectional velocity, y is the flow depth

and g is gravity. The Froude numbers for each hydraulic condition were determined by

first spatially averaging a number of velocity profiles from the supercritical portion of

the mean velocity field. This was done to compile a smooth velocity profile that was

representative of the entire supercritical inflow. From this spatially averaged velocity

profile, a mean velocity was determined by depth averaging the profile. Combining

this mean velocity with the measured supercritical flow depth, Froude numbers were

65

generated. The Froude numbers determined for the analyzed hydraulic conditions are

presented in Table 4.2.

Table 4.2.Summary of the supercritical flow hydraulic properties for each hydraulic condition.

Hydraulic Flow Rate Inflow Froude Supercritical Flow Depth Averaged

Condition Number Depth Velocity

(GPM) (centimeters) (cm sec−1)

1 114.68 1.37 3.139 76.2

2 130.7 1.65 3.079 90.7

3 125.50 2.99 1.966 131.3

The rotation rate of discrete volumes of fluid in each of the hydraulic conditions

was quantified by calculating the vorticity of the fluid at those points. Vorticity is defined

as ω ≡ ∇ × u, and is the curl of the velocity vector u (Kundu 1990). The quantitative

value of vorticity determined in this manner theoretically is twice the angular velocity

of the fluid normal to the plane of rotation. In this research the vorticity was calculated

using the PIV Pack program created by Steven Soloff and packaged with PIV Sleuth and

CleanVec. The vorticity was calculated on a location-by-location basis, with the results

being scaled and compiled into aggregate vorticity profiles for each hydraulic condition.

Contours of vorticity for each hydraulic condition are presented in Figure 4.7.

In Figure 4.7, a strong clockwise vorticity can be seen near the channel boundary

for each hydraulic condition. This vorticity exists due to the significant velocity gradient

66

x(c

m)

y(cm)

050

100

51015vo

rtici

ty(r

adse

c-1):

-40

-30

-20

-10

010

2030

40

(c)

x(c

m)

y(cm)

050

100

51015vo

rtici

ty(r

adse

c-1):

-30

-22

-15

-70

715

2230

(b)

x(c

m)

y(cm)

5010

0

51015vo

rtici

ty(r

adse

c-1):

-20

-14

-9-3

28

1319

(a)

Fig

.4.

7.V

orti

city

profi

lefo

r(a

)H

ydra

ulic

Con

diti

on1,

(b)

Hyd

raul

icC

ondi

tion

2,(c

)H

ydra

ulic

Con

diti

on3.

67

present near the channel boundary. In Hydraulic Conditions 2 and 3, the vorticity at the

channel boundary can be seen to decrease under the crest/roller as the velocity gradient

near the boundary at these locations lessens. Additionally in Hydraulic Condition 3, a

strong counter-clockwise vorticity can be seen in the roller region. This high vorticity

suggests a significantly changing velocity gradient as the fast ‘jet-like’ supercritical flow

is adjusted to the flow reversal that was observed at the roller.

4.3 Turbulent Flow Analysis

4.3.1 Introduction

Turbulent flow is characterized as a fluid flow containing significant random fluctu-

ations in velocity and pressure, rotational fluid packets known as eddies, mixing between

shear layers and the ability to self-sustain turbulence production (White 1991) .

In this analysis, the boundary shear stress throughout each hydraulic condition

was determined using turbulent boundary layer theory. The boundary shear stress (τo)

can be defined as the frictional force between the channel bottom and the fluid flow

per unit contact area (Street et al. 1996). In this analysis the boundary shear stress

is determined as a function of the fluid shear velocity, defined by Equation 4.1 (Kundu

1990):

u∗ ≡√

τoρ

(4.1)

where ρ is the density of water. The boundary shear stress for each condition was

determined in four different ways. The first method involved curve fitting a theoretical

boundary layer profile to the measured mean velocity profile at select cross-sections. The

68

second method involved calculation of boundary shear stress using Darcy friction factor

theory. The third method involved linear regression of spatially averaged Reynolds stress

profiles at select locations. The final method involved curve fitting the exponential laws

of turbulence intensities to measured horizontal and vertical velocity fluctuations.

Additionally, in several of these analyses the compiled data were ‘smoothed’ by

spatially averaging over a horizontal distance of about 1 centimeter. The underlying

assumption that allows for this smoothing is that the supercritical inflow has a fully

developed boundary layer. A fully developed boundary layer is essentially a state where

the boundary layer thickness and velocity profile shape are steady with respect to down-

stream velocity profiles. The development of the boundary layers for each of the hydraulic

conditions were determined by simply superimposing vertical velocity profiles at several

streamwise distances (see Figure 4.8), and observing their agreement with each other.

In Figure 4.8, note that the superimposed profiles covering the entire supercritical

inflow reach show a small degree of deviation in the velocity profiles with respect to

downstream distance. This observation suggests that the flow is slightly slowing as it

approached the toe of the jump. However, analysis of the superimposed profiles within

the 1 centimeter long ‘smoothing’ reach, shows that locally the boundary layer is fully

developed and steady. Since the boundary layer can be shown to be locally steady, the

‘smoothing’ process thus seems valid.

4.3.2 Shear Stress from Mean Velocity Profiles

The curve fitting of the theoretical boundary layer velocity profile involved com-

parison of two theoretical equations to the measured data. The equations involved in

69

Velocity (cm sec-1)

Dep

th(c

m)

0 50 100 150

0.25

0.5

0.75

1

1.25

1.5

1.75

X = 1.12 cmX = 1.52 cmX = 1.93 cmX = 2.13 cmAverage

(b)

Velocity (cm sec-1)

Dep

th(c

m)

0 50 100 150

0.25

0.5

0.75

1

1.25

1.5

1.75

X = 0 cmX = 5 cmX = 10 cmX = 15 cmX = 20 cm

(a)

Fig. 4.8. Vertical velocity profiles at several streamwise distances; x. Hydraulic Con-dition 3; (a) superimposed profiles showing boundary layer development throughout thesupercritical inflow, (b) superimposed velocity profiles throughout the ‘smoothing’ reach.

70

the curve fitting were the Prandtl-von Karman logarithmic overlap law and Cole’s law

of the wake.

According to Prandtl and von Karman, the velocity profile of any wall-bounded

turbulent flow consists of three parts: an inner layer, an outer layer and an overlap layer

(White 1991). Figure 4.9 depicts the relationship between the different shear stresses

and the velocity profile in an open channel flow.

Shear Stress Profiles Typical Velocity Profile

Total Shear Stress

TurbulentShearStress

Outer Layer

Inner Layer

Overlap Layer

Laminar Shear Stress

Fig. 4.9. Vertical profile of shear stress components and velocity profile.

Figure 4.9 outlines the three regions defined by Prandtl and von Karman, where

each region is loosely defined by the type of shear stress present in that region. The

inner layer is the region of the profile near the boundary where the total shear stress

is dominated by laminar/viscous shear (White 1991). The outer layer consists of the

majority of the velocity profile where the total stress is dominated by Reynolds/turbulent

71

shear stress. The overlap layer involves both types of shear, laminar and turbulent, and

connects the inner and outer layer.

The properties of the inner layer were theorized by Prandtl. Prandtl deduced a

theoretical velocity profile that was dependent largely on molecular viscosity and thus

was valid only in the region dominated by laminar stress. This profile is known as the

law of the wall, and is presented as Equation 4.2,

u

u∗= f

(yu∗

υ

)(4.2)

where u∗ is the shear velocity, u is the horizontal velocity and υ is the kinematic viscosity.

The properties of the outer layer were similarly theorized by von Karman. Von

Karman deduced a theoretical velocity profile that neglected molecular viscosity. This

profile is known as the velocity defect law and is presented as Equation 4.3:

Ue − u

u∗= g

(y

δ, ξ

)(4.3)

where Ue is the free stream velocity, u is the velocity at any depth y, δ is the boundary

layer thickness and ξ is a constant. The ξ constant is defined as ξ ≡ δτo

dPedx with dPe

dx

being the streamwise external pressure gradient (White 1991).

Using the solutions for the inner and outer regions of the velocity profile, a the-

oretical profile for the overlap region was defined. This theoretical profile is known as

the Prandtl-von Karman logarithmic overlap equation and is presented as Equation 4.4

72

(White 1991):

u

u∗=

1κ

ln

(yu∗

υ

)+ B (4.4)

where κ and B values are constants for smooth walls. The von Karman constant κ is

universally accepted to be 0.40 − 0.41, while the range accepted for B is more broad,

∼ 5.0−5.5. Experimental data (Figure 4.10) are found to collapse to the overlap equation

extremely well. In this Figure the non-dimensional data are presented in terms of y+

and u+, where y+ are known as wall units and defined y+ ≡ yu∗υ and u+ is defined as

u+ ≡ uu∗ . It should be noted from the figure that the data agree with the logarithmic

overlap equations only in the range of 30 to 350 wall units. Beyond 350 wall units, wakes

can be seen in the data. A wake is a generic term used to describe the deviation of a

velocity profile from the logarithmic overlap layer equation in the outer layer. Wall unit

values of 30 and 350 are generally viewed as the threshold values separating the inner,

overlap and outer layers (White 1991).

Measured velocity profile data in the supercritical approach flow and under the

first ‘crest’ for each condition are plotted both dimensionally and non-dimensionally in

Figures 4.11 through 4.13. Superimposed on each plot is the overlap equation. Note

that the overlap equation is only plotted in the range of applicability on the dimensional

plot, i.e. up to 350 wall units. The smooth logarithmic overlap equation was curve fit

to the experimental data using a nonlinear least-squares optimization treating the shear

velocity as a free parameter.

73

Fig. 4.10. Non-dimensional plot of streamwise velocity profiles superimposed upon thePrandtl-Von Karman logarithmic overlap layer inner law figure reproduced from White(1991).

74

Velocity, u (cm sec-1)

Flow

Dep

th,y

(cm

)30 40 50 60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

b.)


Flow

Dep

th,y

(cm

)

60 70 80

0.5

1

1.5

2

2.5

3

Smooth Regime Logarithmic Overlap LayerColes' Law of the WakeRaw Data

a.)

y+ = y * u* / ν

u+=

u/u

*

200 400 600 800

14

16

18

20

22

24

26

28

30

32

d.)

y+ = y * u* / ν

u+=

u/u

*

500 1000

17

18

19

20

21

22

23

24

Data Normalized using Coles' u*

Coles' Law of the WakeOverlap Layer

c.)

Fig. 4.11. Comparison of theoretical velocity profiles versus experimental data forestimation of shear velocity. Hydraulic Condition 1 (a) Velocity profiles at x = 5.11 cm,(b) Velocity profiles at x = 48.09 cm, (c) Non-dimensional velocity profiles at x = 5.11cm, (d) Non-dimensional velocity profiles at x = 48.09 cm.

75


Flow

Dep

th,y

(cm

)10 20 30 40 50 60 70

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

b.)


Flow

Dep

th,y

(cm

)

60 70 80 90 100

0.5

1

1.5

2

2.5


a.)

y+ = y * u* / ν

u+=

u/u

*

100 300 5005

15

25

35

45

55

d.)

y+ = y * u* / ν

u+=

u/u

*

500 1000141516171819202122232425

Raw Data, Normalized by Coles' u*


c.)


76


Flow

Dep

th,y

(cm

)10 20 30 40 50 60 70

0.5

1

1.5

2

2.5

3

3.5

4

4.5

b.)


Flow

Dep

th,y

(cm

)

100 110 120 130 140 1500.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


a.)

y+ = y * u* / ν

u+=

u/u

*

100 200

20

30

4050

6070

80

90100

110

120

d.)

y+ = y * u* / ν

u+=

u/u

*

500 100016

17

18

19

20

21

22

23

Data Normalized using Coles' u*


c.)


77

Figures 4.11 through 4.13 all show a clear wake above the limit of about 350 wall

units. Looking at the non-dimensional plots of the supercritical approach flow, there

is clearly a wake above 350 wall units. Note that, for purposes of illustration only, the

overlap curve fit has been extended over the full range of the dimensionless plots. This

unsuitability of the overlap equation to the full depth of open channel flow has been

pointed out by Nezu and Nakagawa (1993). Therefore, Coles’ law of the wake was also

fit to the data. Coles theorized that there exists a wake or shape function defined in

terms of η ≡ y/δ (White 1991). The addition of the wake function into the logarithmic

overlap law produces Coles’ law of the wake, which is presented as Equation 4.5:

u

u∗=

1κ

ln

(yu∗

υ

)+ B +

2Πk

f(η) f(η) ≈ 3η2 − 2η3 (4.5)

where the Π parameter is known as Coles’ wake parameter and κ and B were chosen to be

0.40 and 5.5, respectively, as per the smooth Prandtl-von Karman overlap case. As can

be seen in Figures 4.11 through 4.13 Coles’ law of the wake seems to fit the experimental

data in the supercritical flow better than the Prandtl-von Karman smooth logarithmic

overlap law. Additionally, Coles’ law of the wake improves upon the logarithmic overlap

law in that it considers the entire depth of the flow. In this experiment this is advan-

tageous due to the limited number of data points available in the applicable range of

the logarithmic overlap law. The shear velocity at every tenth cross-section throughout

each hydraulic condition was determined by curve fitting Coles’ law of the wake to the

velocity profiles at each corresponding location. The curve fitting in this process utilized

the same nonlinear least-squares optimization previously discussed. The resultant shear

78

velocities and corresponding boundary shear stresses for each hydraulic condition are

presented in Figure 4.14. The boundary shear stress corresponding to the calculated

shear velocities was computed at each cross-section using the definition of shear velocity

presented in Equation 4.1. Table 4.3 summarizes shear velocity, Coles’ wake parameter

and boundary shear stress in the supercritical region as determined above.

Table 4.3.Summary of shear velocity, Coles’ wake parameter and boundary shear stress asdetermined from velocity profiles. Data presented is from the supercritical flow regionfor all three hydraulic conditions.

Hydraulic Shear Velocity Coles’ Wake Boundary

Condition Parameter Shear Stress

(cm sec−1) (Pa)

1 3.614 0.345 1.31

2 4.058 0.546 1.65

3 6.163 0.456 3.80

Analysis of the centerline boundary shear stress data presented in Figure 4.14

shows that the minimum values of boundary shear stress in each hydraulic condition

are occurring under the first undular crest or under the roller. Conversely, the highest

values of boundary shear stress occur in the supercritical flow upstream of the jumps.

The ‘wavy’ structure of Hydraulic Conditions 1 and 2 shows that all undular troughs

throughout the analyzed profile have relatively high values of boundary shear stress,

while the corresponding crests have relatively low shear stress values. These results

from a qualitative standpoint agree with the results published by Chanson (2000) and

79

x (cm)

Shea

rVel

ocity

(cm

sec-1

)

Shea

rStre

ss(P

a)

50 1000

0.5

1

1.5

2

2.5

3

3.5

0

0.2

0.4

0.6

0.8

1

1.2Shear VelocityShear Stress

(a)

x (cm)

Shea

rVel

ocity

(cm

sec-1

)

Shea

rStre

ss(P

a)

0 50 1000

0.5

1

1.5

2

2.5

3

3.5

4

0

0.4

0.8

1.2

1.6

(b)

x (cm)

Shea

rVel

ocity

(cm

sec-1

)

Shea

rStre

ss(P

a)

0 50 1000

1

2

3

4

5

6

0

1

2

3

(c)

Fig. 4.14. Shear velocity and boundary shear stress predicted using Coles’ law of theWake. Values are plotted at every tenth cross section across the entire profile for (a)Hydraulic Condition 1, (b) Hydraulic Condition 2, (c) Hydraulic Condition 3.

80

presented earlier in Figure 2.4. This ‘wavy’ structure has implications with respect

to erosion, deposition and bed-form formation beneath undular jumps. The ‘wavy’

structure of the shear stress suggests that ‘wavy’ bed-forms, with shapes similar to the

undular free surface, could be formed by an undular jump.

Analysis of Hydraulic Condition 3 as presented in Figure 4.14 shows that the

boundary shear stress is a minimum under the roller and increases to a relatively high

secondary peak just downstream. Downstream of this secondary peak the boundary

shear stress decays almost linearly with distance downstream of the jump. The impli-

cations of this structure with respect to sediment transport suggest that the maximum

amount of erosion will be occurring upstream of the hydraulic jump with deposition

possibly occurring under the roller.

4.3.3 Shear Stress by the Darcy Friction Factor

The boundary shear stress values for each hydraulic condition were determined

from mean hydraulic properties for each hydraulic condition through use of the Darcy

Friction factor. This method involved calculation of the hydraulic diameter and Reynolds

number, calculation of the friction factor, and calculation of boundary shear stress using

the Darcy friction factor theory. The friction factors in this analysis were calculated

using the Colebrook-White equation for a smooth boundary, presented as Equation 4.6

(White 1999):

1

f1/2= −2.0 log

(ε/d

3.7+

2.51

Redf1/2

)(4.6)

81

where f is the friction factor, Red is the Reynolds number, d is the hydraulic diameter

and ε is the pipe roughness coefficient. Since the flume is constructed of acrylic, a smooth

boundary with an ε value of 0 was chosen for this analysis.

The friction factor calculated as above was used in the theory for the Darcy friction

factor to calculated the boundary shear stress. The Darcy friction factor is presented as

Equation 4.7:

f =8τoρV 2 (4.7)

where ρ is the fluid density and V is the average cross-sectional velocity. The results of

Colebrook-White and Darcy friction factor this analysis are presented in Table 4.4.

Table 4.4.Summary of the boundary shear stress, Reynolds numbers and friction factors generatedin the Colebrook-White and Darcy friction factor shear stress analyses. Case AF-1reproduced from Chanson (2000) is provided to verify the validity of this analysis. Themeasured boundary shear stress for case AF-1 is 3.45 Pa.

Hydraulic Hydraulic Reynolds Colebrook-White Boundary

Condition Diameter number Friction Shear

Measured Factor Stress

(m) (Pa)

1 0.1039 78, 970 0.0189 1.37

2 0.1022 92, 703 0.0182 1.88

3 0.0697 91, 553 0.0183 3.96

AF − 1 0.191 247, 000 0.015 3.12

82

Analysis of the Chanson (2000) boundary shear stress data presented in Table

4.4 shows good agreement between the measured shear stress and the calculated shear

stress. This agreement in measured and calculated values of shear stress shows that this

form of analysis is a valid way of determining boundary shear stress.

4.3.4 Shear Stress from Reynolds Stresses

The Reynolds stress in any turbulent flow is defined as −u′v′ and is related to

turbulent shear stress (τ) by Equation 4.8:

τ = −u′v′ρ (4.8)

where u′ is the ensemble streamwise velocity fluctuations, v′ is the ensemble vertical

velocity fluctuations; the overbar denotes time averaging. The role of turbulent shear

stress in the shear stress profile for a wall-bounded shear flow is depicted in Figure 4.9.

As previously discussed, turbulent shear stress dominates a large portion of the shear

stress profile, particularly in the outer layer. As defined by von Karman, the total shear

stress in the outer layer is accounted for almost solely by the turbulent shear stress.

The Reynolds stress for each of the hydraulic conditions throughout the entire

data field was calculated in this experiment. Reynolds stress was calculated as the en-

semble average of the cross-product of the horizontal and vertical velocity fluctuations.

A vertical profile for the supercritical flow and the first crest/roller of each hydraulic con-

dition was developed by spatially averaged vertical profiles across ∼1 cm. The boundary

shear stress in the supercritical flow was deduced from these profiles by applying the

83

above discussed relationship between Reynolds stress, turbulent shear stress and total

stress. A linear regression was applied to the Reynolds stress data in the outer layer,

above 350 wall units. Based on the knowledge that the total shear stress profile is lin-

ear in a fully developed open-channel flow, and that the total shear stress is composed

almost solely of turbulent shear stress in the overlap layer, the linear regression of the

Reynolds stress data in the outer layer is used as a surrogate for the total stress pro-

file. Extrapolating this linear regression to the boundary, the boundary shear stress is

obtained. This analysis was applied only to the Reynolds stress profiles from the super-

critical flow region, as the flow under the crest was not fully developed. The vertical

profiles of Reynolds stress for the supercritical flow and crest locations are presented as

Figure 4.15, and the boundary shear stress data are presented in Table 4.5.

Table 4.5.Summary of boundary shear stress for the supercritical flow region for each hydrauliccondition. The boundary shear stress for each condition was determined using a linearregression of the outer layer Reynolds stress data. The threshold depth values for theouter layer also are presented.

Hydraulic Reynolds Stress Outer Layer

Condition Profile Depth

Shear Stress Threshold

(Pa) (cm)

1 1.21 0.97

2 1.50 0.86

3 3.08 0.57

84

Shear Stress (Pa)

Flow

Dep

th(c

m)

0 0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

2.5

3

Total Shear StressMeasured Reynold's Stress

(a)

Shear Stress (Pa)

Flow

Dep

th(c

m)

0 0.2 0.4 0.6 0.8

1

2

3

4

5(b)

Shear Stress (Pa)

Flow

Dep

th(c

m)

-1 0 1 2 3

1

2

3

4

5

6

(d)

Shear Stress (Pa)

Flow

Dep

th(c

m)

-15 -10 -5 0

1

2

3

4

5

6

(f)Shear Stress (Pa)

Flow

Dep

th(c

m)

0.5 1 1.5

0.5

1

1.5

2

2.5

3(c)

Shear Stress (Pa)

Flow

Dep

th(c

m)

0 1 2 3

0.25

1

1.75

(e)

Fig. 4.15. Vertical profiles of Reynolds stress superimposed upon a linear curve fitof the total stress. Plots represent data spatially averaged over ∼1 cm for: HydraulicCondition 1 (a) supercritical flow, (b) crest of 1st undulation; Hydraulic Condition 2 (c)supercritical flow, (d) crest of 1st undulation; Hydraulic Condition 3 (e) supercriticalflow, (f) peak of the roller.

85

Analysis of Figure 4.15 shows that the Reynolds stress profiles in the supercritical

flow regions have the expected shape and correspond with the total stress surrogate

nicely. The Reynolds stress profiles in the crest/roller regions make good qualitative

sense with respect to the velocity vector profiles; however, no models currently exist to

evaluate these profiles quantitatively. The Reynolds stress profiles in the crest/roller

regions appear to qualitatively agree with similar plots produced by Liu et al. (2003),

from their ADV data.

4.3.5 Shear Stress from Root Mean Square Velocity Fluctuations

Vertical profiles of the root mean square values of horizontal and vertical velocity

fluctuations were compiled in the same ensemble and spatial averaging fashion as the

Reynolds stress data discussed in Section 4.3.3. The boundary shear stresses for each

of the three hydraulic conditions were derived from the universal laws of turbulence

intensity and the definition of shear velocity. The universal laws of turbulence intensity

for horizontal and vertical velocity fluctuations are presented as Equations 4.9 and 4.10,

respectively (Nezu and Rodi 1986).

u′

u∗= Du exp

(−λu

y

h

)(4.9)

v′

u∗= Dv exp

(−λv

y

h

)(4.10)

In Equations 4.9 and 4.10, the quantities λu and λv are empirical constants that equal

1.0 and 0.67, respectively. The empirical coefficients Du and Dv are equal to 2.30 and

1.27, respectively. All of the empirical coefficients in these laws have been proven to be

86

independent of the Reynolds and Froude numbers and thus should apply to all three

data sets (Nezu and Nakagawa 1993).

The universal laws of turbulence intensity were curve fit to the root mean squared

values of measured horizontal and velocity fluctuations using the same non-linear least

squares optimization routine as previously discussed. The resultant curve fits from this

analysis are presented as Figure 4.16 and a summary of the shear velocity and boundary

shear stress is presented as Table 4.6.

Table 4.6.Summary of the shear velocity and boundary shear stress as determined from curvefitting the universal laws of turbulence intensity for both horizontal and vertical velocityfluctuations. The boundary shear stress data were calculated using the definition ofshear velocity presented as Equation 4.1

Hydraulic u′ Profile u′ Profile v′ Profile v′ Profile

Condition Shear Velocity Boundary Shear Velocity Boundary

Shear Stress Shear Stress

(cmsec−1) (Pa) (cmsec−1) (Pa)

1 4.892 2.40 5.739 3.29

2 4.056 1.65 4.302 1.85

3 6.206 3.85 5.438 2.96

Figure 4.16 shows very good agreement for the universal law of turbulence in-

tensity for the streamwise velocity fluctuations in the supercritical flow data set for

Hydraulic Conditions 2 and 3. However, Hydraulic Condition 1 does not show reason-

able agreement between the data and both the horizontal and vertical universal laws of

turbulence intensity. The reasons for this inadequacy are not clear at this point in time.

87

RMS Velocity Fluctuations (cm sec-1)

Flow

Dep

th(c

m)

0 5 10

1

2

3

4

5

6

d.)


Flow

Dep

th(c

m)

3 4 5 6 7 8

0.5

1

1.5

2

2.5

3c.)


Flow

Dep

th(c

m)

0 5 10 15 20 25 30

1

2

3

4

5

6

7

f.)


Flow

Dep

th(c

m)

0 5 10

1

2

3

4

5b.)


Flow

Dep

th(c

m)

3 4 5 6 7 8 9

0.5

1

1.5

2

2.5

3a.)


Flow

Dep

th(c

m)

0 5 10

0.25

0.5

0.75

1

1.25

1.5

1.75

Horizontal Velocity Fluctuations (u')Vertical Velocity Fluctuations (v')Theoretical Curve of u'Theoretical Curve of v'

e.)

Fig. 4.16. Vertical profiles of horizontal and vertical velocity fluctuations superimposedupon theoretical curves for horizontal and vertical velocity fluctuations. Plots representdata 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 1stundulation; Hydraulic Condition 3 (e) supercritical flow, (f) peak of the roller.

88

Since the turbulence laws allegedly are not influenced by Reynolds and Froude numbers,

the strength of the jumps should not be an issue in this analysis. The agreement be-

tween the turbulence law and the vertical velocity fluctuations in the supercritical flow

is satisfactory for Hydraulic Conditions 2 and 3. The shape of the root mean square

vertical velocity fluctuations suggests that the free surface is generating a comparable

amount of turbulence with respect to the turbulence generated by the boundary in the

vertical direction, causing some deviation from the expected curves. This result concurs

with the observations of Nezu and Nakagawa (1993) concerning an open channel flow.

Analysis of the crest/roller profiles in Figure 4.16 show several trends. The pro-

files for Hydraulic Condition 1 show that the distribution of both horizontal and vertical

velocity fluctuations has not really changed from the supercritical distribution. The

shapes of the horizontal and vertical velocity fluctuations under the crest/roller in Hy-

draulic Conditions 2 and 3 suggest that turbulence is being generated in these locations

by factors other than the boundary.

89

Chapter 5

Conclusion

5.1 Discussion of Technical Aspects of Data Collection

In this research the relatively new technique of Particle Image Velocimetry has

been applied for the first time to stationary undular and hydraulic jumps. Three different

hydraulic conditions were analyzed with moderate inflow Froude numbers ranging from

1.37 to 2.99. Additionally, free surface data were collected using a point gage and/or a

wave gage for each hydraulic condition.

Throughout this research several different technical challenges were encountered.

The first challenge involved collection of data throughout hydraulic jumps having signifi-

cant aspect ratios with respect to the depth of flow and the desired horizontal distance to

be analyzed. This situation was handled by collecting data at several locations along the

centerline. These locations were spliced together carefully, forming a mosaic of the flow

field throughout the jump. Though great care was taken to produce a seamless mosaic

of the flow field, gaps between the locations exist. These gaps are likely a perimeter

distortion common with single camera PIV. This distortion is due to the perspective of

the camera and the relatively longer distance from the edge of the camera lens to the

edge of the capture area as compared to the distance from the camera lens to the capture

area along the focal axis (Hill et al. 2000).

90

The two-phase nature of hydraulic jumps was another complication encountered

in this research. The complications with air bubble entrainment and PIV are discussed

in detail in Section 3.3.2. This complication was dealt with in Hydraulic Condition 3

by use of fluorescent particles and fluorescent Particle Image Velocimetry. In Condition

3, the use of fluorescent particles mitigated the influence of air bubbles on the data

collected; however, larger problems could be anticipated for stronger hydraulic jumps.

As significantly stronger jumps are analyzed, a significantly higher ratio of air to water

can be expected in the roller. As this ratio increases, the clarity of the flow structure

will significantly decrease due to the so-called “white water” effect. This effect will likely

obstruct the research into hydraulic jumps using any type of flow visualization or image

capture technique along the centerline of the flow. Analyses of submerged hydraulic

jumps are likely the only application of PIV techniques that will yield satisfactory results

for high Froude number hydraulic jumps.

The oscillations of Hydraulic Condition 2, as discussed in Section 3.3.1, created

several complex issues in this research. This problem could not be overcome while still

producing a jump in the Froude number range of 1.65. Fortunately these oscillations only

affected Hydraulic Condition 2, as Hydraulic Conditions 1 and 3 remained quite stable

without the intrusion of measurement instruments into the flow. The likely affect of the

oscillations in Hydraulic Condition 2 is the ‘washing-out’ of maximum and minimum flow

under crests and troughs. This ‘washing-out’ effect occurs as areas of higher and lower

velocity oscillate across the mean locations of the crest and trough. In a similar fashion,

this ‘washing-out’ effect will smear all of the other data points within the oscillating

portion of the profile.

91

Ideally, several stronger hydraulic jumps would have been analyzed in this work,

however, this was not possible. The strength of jumps analyzed in this research was

limited by the available facilities. The flume used in the research was incapable of

producing hydraulic jumps with inflow Froude numbers significantly larger than 3. The

size and power of the centrifugal pumps supplying the flow to the flume is likely the

limiting factor causing this situation. Additionally, the aforementioned air entrainment

issues would also likely have hampered the progress of the research had the physical

development of stronger hydraulic jumps been possible.

The final technical difficulty encountered in this research involved measurement

of the free surface, which proved challenging due to the previously discussed profile

oscillations. Ideally a high-frequency digital device, such as a capacitance wave gage,

would be used for collection of temporally dense free-surface data. However, the intrusion

of such a device into the flow caused significant complications in the first two hydraulic

conditions. As shown in Figure 4.2 the use of image analysis in charting the free surface

profile shows promise. However, as seen in Hydraulic Conditions 1 and 2, problems

with this system can arise with a smooth free surface capable of generating reflections of

the seed particles. A system that could utilize image analysis calibrated by free surface

data collected digitally would be the best solution to this complication. To this end a

better form of digital free surface data collection would be needed or stronger jumps not

affected by intrusion of a measuring instrument would need to be analyzed.

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5.2 Discussion of the Mean Velocity Analysis

The mean velocity fields presented in Figures 4.4 through 4.6 give a clearer picture

of the flow field inside hydraulic jumps. The data produced in this study is much more

spatially dense than data produced by all previous studies. The density of the data is

significant in that it allows for numeric integration of various fluid mechanics theories,

such as the universal laws of turbulence intensity and the total shear stress profile as

presented in Sections 4.3.5 and 4.3.4. This sort of analysis was not possible previously,

given the spatial deficiency of data points in previous studies.

The structure of the mean velocity fields has implications when considering the

mechanisms of erosion and deposition. The ‘wavy’ structure of undular jumps shows

areas of relatively low velocity near the boundary under the undular crests and areas of

relatively high velocity near the boundary under the undular troughs. This structure

suggests that a movable channel bed beneath an undular jump will erode under the

troughs and deposit under the crests. These sediment transport activities likely will

cause bed-form development that mirrors the free surface shape of an undular jump.

The structure of the flow field beneath the hydraulic jump suggests similar bed-

form development. Beneath the hydraulic jump roller, a relatively low velocity exists

near the boundary and relatively higher velocities elsewhere. This suggests that the

largest amounts of erosion beneath a hydraulic jump do not occur below the roller.

The vorticity structure developed using the mean velocity data shows a few as-

pects of the fluid rotation beneath the undular and hydraulic jumps. The structure of

the vorticity field beneath the undular jumps suggests that the only areas of significant

93

fluid rotation are occurring along the channel boundary. The structure of the vorticity

fields beneath the hydraulic jump show that there is significant fluid rotation in the roller

region as well as at the channel boundary.

5.3 Discussion of the Boundary Shear Stress Analysis

The boundary shear stress occurring in the supercritical flow region of each hy-

draulic condition was deduced in four different ways in this experiment. These four

methods included analysis of mean velocity profiles, Darcy’s friction factor, Reynolds

stress profiles and the universal laws of turbulence intensities. The respective boundary

shear stresses as determined by each method are presented in Table 5.1.

The shear stresses provided in Table 5.1 show that each of these analyses produced

comparable results with respect to one another. The only significant deviation in these

results comes from the analysis of Hydraulic Condition 1 using the universal laws of

turbulence intensity. The cause of this deviation is not known at this time and perhaps

further investigation of this type will prove or disprove the validity of the turbulence

intensity laws for low Froude number jumps. Additionally, the quantitative values of the

boundary shear stress data collected generally agree with published results on directly

measured boundary shear stress for undular jumps (Chanson 2000).

Overall, these forms of analysis seem to validate the use of PIV for the measure-

ment of turbulent statistics such as the Reynolds stress and turbulence intensities. Based

on the assumption that PIV produces data sufficient for evaluation of the boundary shear

stress, profiles of the centerline shear stress for all three hydraulic conditions were com-

piled using Coles’ law of the wake. Analysis of these shear stress profiles suggests similar

94

Table 5.1.Summary of the boundary shear stress values as computed using the mean velocityprofiles, Darcy’s friction factor, the Reynolds stress profile and the universal laws ofturbulence intensity.

Hydraulic Mean Velocity Darcy’s Reynolds stress

Condition Profile Friction Factor Profile

Shear Stress Shear Stress Shear Stress

(Pa) (Pa) (Pa)

1 1.31 1.37 1.21

2 1.65 1.88 1.50

3 3.80 3.96 3.08

Hydraulic Horizontal Turbulence Vertical Turbulence

Condition Intensity Profile Intensity Profile

Shear Stress Shear Stress

(Pa) (Pa)

1 2.40 3.29

2 1.65 1.85

3 3.85 2.96

95

properties of erosion and deposition as discussed in the mean velocity profile section.

Additionally, from a qualitative standpoint, the shear stress profiles produced in this

analysis agree with directly measured shear stress profiles published by Chanson (2000)

and presented as Figure 2.4.

5.4 Future Considerations

The analysis of hydraulic jumps using PIV technology could be improved upon

in several ways. The largest improvement in this area involves development of a method

that allows for data acquisition with significantly stronger hydraulic jumps than the ones

discussed herein.

Further improvements could be in the analysis of undular and hydraulic jumps

by utilizing different forms of PIV analysis. Among these other forms are stereoscopic

PIV techniques. Stereoscopic PIV would allow investigation of the three-dimensional

nature of undular and hydraulic jumps. This sort of analysis could provide a useful

contour map of the shear stress occurring beneath undular and hydraulic jumps. A PIV

analysis of undular and hydraulic jumps occurring over a movable bed would provide

valuable information about bed-form formation. This sort of analysis could be used to

validate the bed-form development characteristics suggested by the mean velocity profile

and the boundary shear stress analysis. Additionally, analysis of a hydraulic jump using

a multi-camera PIV system, with each camera operating independently, could provide a

‘real-time’ view of vortex development in hydraulic jumps. This form of analysis would

provide detail about the rotational aspects of fluid flow in a hydraulic jump beyond the

detail provided by the vorticity analysis.

96

In consideration of the inability to curve fit the Reynolds stress and turbulence

intensity profiles and the marginal quality of the velocity profile data fit under the crests

and roller, compilation of data from significantly more hydraulic conditions may allow

for models of these distributions to be developed. A calibrated model of the boundary

shear stress being produced by hydraulic jumps would be a useful tool in the design of

stilling basins and culvert/bridge outlet structures.

97

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