Some characteristics of pressure fluctuations on low-ogee crest spillways relevant to flow induced structural vibrations

SOME CHARACTERISTICS OF PRESSURE

FLUCTUATIONS ON LOW-OGEE CREST

SPILLWAYS RELEVANT TO FLOW-INDUCED

STRUCTURAL VIBRATIONS

bY

irederick A. Locher

Sponsored by U.S. Corps of Engineers Contract No. DACW39-68-C-004

IIHR Report No. 130

Iowa Institute of Hydraulic Research

The University of Iowa

Iowa City, Iowa

February 1971

TABLE OF CONTLIITS

I. INTRODUCTION ₁

II. EXPERIMENTAL APPARATUS AND METHODS OF DATA ANALYSIS 4

Flume 4

Spillway Shape 4

Pressure Measurements 4

Data Analysis ₆

III. PRESSURE FLUCTUATIONS I. SPILLWAY CREST ₉

General Considerations ₉

RMS Value, Pressure Fluctuations, Point 1 13

Boundary Layer Measurements ₁₅

Spectral Density Function,

Spillway

Crest 16

Comparison with Prototype Data ₁₇

IV. PRESSURE FLUCTUATIONS II. SPILLWAY TOE 18

RMS Value, Pressure Fluctuations, Point 2 18

Spectral Density Function, Spillway Toe 19

Flow Visualization-Dye Tests 20

Discussion ₂₀

Spillway Toe Pressure Fluctuations, Hydraulic Jump 21

V. DISCHARGE COEFFICIENT 23

Dimensional Considerations . 23

Discharge Coefficient at the Design Head, Hd . . 25 Discharge Coefficient for He/Hd > 1.0 26

VI. SUMMARY AND DISCUSSION ₂₇

VII. CONCLUSIONS ₂₉

Figure 1 Definition sketch

Photographs of the experimental apparatus with discharges cor-responding to He/Ha

1.0, 1.3; 1.5,

and 1.8 . ,

Variation

of

the relative- rms value of the pressure fluctuations

at point 1 With He/Hd. No boundary layer trip . . . 55

Variation of the relative rms value of the pressure fluctuations

at point 1 With He/Hci. Boundary layer tripped . . .

.36

Spectrum of the pressure fluctuations

No boundary layer trip Spectrum of the pressure

Boundary layer tripped . .

Figure 7 Spectrum of the pressure fluctuations

No boundary layer trip.

Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Flgure Figure Figure Figure Figure Figure 13(c) .LIST, OF FIGURES

Figure 8 Spectrum of the pressure

Boundary layer tripped .

fluctuations

..

..

. ... .

fluctuations at point 1, H /H, = 1.0.

.... .

. . ? . . . at point 1, He/Hd = 1.0. at point 1, He/Hd = 1.3. at point 1, He/Hd = 1.3. 38 39

Figure

9

Spectrum Of the preSSure fluctuations at point 1, He/Ha =

1.63.

No boundary layer trip 41

Figure 10 Spectrum of the pressure fluctuations at point 1, Hi /HA =

Boundary layer tripped.. .

,

.

4

....

42

Figure 11 Spectrum of the pressure fluctuations at point 1, H /HA

= 1.8.

No boundary layer trip . . .

. .... .

.

?"

.

..

. . 43 Figure

12

Spectrum of the pressure fluctuations at point 1, H /Hd =

1.8.

e

Boundary layer tripped . 44

13(a) Oscillograms of the pressure fluctuations at point 1, no boundary layer trip

13(b) Oscillograms of the layer tripped

Oscillograms of the pressure fluctuations at point 2, no

boundary layer trip 47

Oscillograms of

the

pressure fluctuations at pOint 2, boundary

layer tripped . 48

OsOillograms of the pressure fluctuations at point

2,

hydraUlio

jump,

boundary layer tripped .

14

Variation Of the relative rMs Value of the pressure fluctuations

at point. 2 with He/Hd.

No boundary layer trip ...

. . 50

iii

45

pressure fluctuations at point 1, boundary 146

Figure

16

Spectrum'of the pressure fluctuations at point

2,

He/Hd No boundary layer trip

Figure 17 Figure

18

Figure Figure

20

Figure

21

Figure 22

Figure 23 Spectrum of the pressure

Boundary layer tripped .

Figure 24

Figure

Figure

Figure 30

Spectrum of the pressure No boundary layer trip

LIST OF FIGURES (continued)

Figure 15 Variation of the relative rms value of the pressure

fluctua-tionS.at point 2 With H /H

d.

. Boundary layer tripped. ,51

e

Spectrum of the pressure fluctuations

Boundary layer tripped 53

Spectrum of the pressure fluctuations No boundary layer trip

19 Spectrum of the pressure fluctuations Boundary layer tripped

\

Spectrum of the pressure fluctuations at pOint

No boundary layer trip .

56-Spectrum of the pressure fluctuations Boundary layer tripped

iv

at

28 Spectrum of the

Hydraulic jump, boundary layer tripped

at point 2, H /Hd = 1.0. e at point 2, He/Hd = 1.3. point 2, He/Hd = 1.3. . . .

2,

He/Hd =

1.5.

at point

2,

He/Ha

= 1.5.

57 fluctuations at point

2,

1%-/Hd =

1.8.

fluctuations at point 2, He/Hd

=1.8.

OOOOOO

Photographs of the spillway face-with dye injected at the piezo-meter located at point 1, for discharges corresponding to

He/Hd = 1.0, 1.3,

1.5,

and

1.8

Spectrum of the pressure fluctuations at point 2, Hydraulic jump, boundary layer tripped lie/

Spectrum of the pressure fluctuations at point

2,

He/H = 1.3

Hydraulic jump, boundary layer tripped

pressure fluctuations at point 2, He/Hd =

1.5.

27 Spectrum of the

Hydraulic jump, boundary layer tripped ... . . .

pressure fluctuations at point

2,

He/Hd

1.8.

54

55

58

6o

Figure 29 Photographs taken during the experimental rums with '8. hydraulic jump on the dOWnstream apron. Data plotted on Figs.

25-28

. 65 Flow With He/4d = 1.8 and separation, on the spillway face as

visualized by a' dye streak . . . OO O

..

. - 66

=1.0.

52 .Figure 25 Figure 26

63

64

LIST OF kGURES

(continued.)

Figure 31 Discharge coefficient as a function of He/lid . . . 67 Figure 32 Comparison of the discharge coefficient of the present

investi-gation with data published by Corps of Engineers

(1965).

Reproduced with permission of the Corps of Engineers . . . 68

Figure 33 Mean pressure at point 1 as a function of He/lid

..

69 Figure

34

Mean pressure at point 2 as a function of He/Hd 70

LIST OF SYMBOLS

Discharge coefficient, C = Q/LHe$12

d Discharge coefficient, Cd =

3/.2

- e

C(T)

Auto-covariance function.,

G(T) =

X(t)X(t+4)

Froude number

d Design bead for, the

spillway

.shape

H Head on the

epiiXway,

He= h

+ ha

Hm 'Maximum head, required to pass the

efdlivey

design flood. Length of the

spillway

crest in feet

.

SpiiIvall -height

P(f) Spectral Density function, P(f) = C( r) cos(2P0dT.

Discharge in cubic feet per second Reynolds number

11(T) Auto4-correlation function. normalized auto-covariance

function

Uo Velocity of the approach flow

111 Weber number

X(t) Ergodic random process (function of time)

Depth of water on the downstream apron, transducer face

diameter

Gravitational acceleration

Frequency in Hz.

Height of water over the spillway

ha Approach flow velocity head ha = UO2/2g

hd Distance between the line of total head and the free surface at the section where d is measured

Mean pressure

Instantaneous deviation of the pressure from the mean

pressure

1! = p'2 Root-Mean-Square value of the presSure fluctuations

X,Y Coordinates, origin

at

the spillway crest

0(f) Normalized Spectral Density function

0(f)

= 4

f

_R(T)

cos(2fT)dT

0

Geometrical parameters describing the spillway shape Geometric ratios describing the spillway shape

Boundary layer thickness

Boundary-layer diSpladetent thiCkness

Fluid Viscosity

. Fluid density Surface tension

Time delay

SOMEZHARACTERISTICS OF PRESSURE FLUCTUATIONS ON LOW-OGEE CREST SPILLWAYS RELEVANT TO FLOW-INDUCED

STRUCTURAL 'VIBRATIONS

I. inigODUCTION

The

primary

objective of the experimental iprogram reported

herein was to determine whether pressure fluctuations induced on the face of a low-ogee spillWay'under various flow conditions are a possible mechanism

for

the excitation of

spillway

vibration. Some of the more

important terms and concepts used to describe

spillways

will be defined

at the outset An the interest Of clarity.

With reference to figure 1, the spillway height p

is

the difference in elevation between the Spillway crest and-the approach

-channel. floor. The. head on the spillway He is the vertical distance

from

the

spillway crest to the line of total head; thatis, He is the sum of

the

head of Water over spiflway, h , and the approach

velocity head, ha = U162/2g , where Uc; Is- the approach flow velocity.

The design- head for- the spillway shareHd is the parameter used to

proportion the shape of -the Spillway such that for flaws with He -= Hd

the shape of the spillway will correspond to the shape of the lower time surface as determined from sharp-crested weir experiments (USBR,

1948).

The design head for the spillway shape should not be confused

with the maximUm head

Hin required to pass the spillway design flood.

It is convenient to group spillways into two classes,

high-overflow spillways and low-high-overflow spillways. High-overflow

spill-ways- are those spillways for which the approach flaw velocities are negligable. Low-overflow or loV-ogee spillways are characterized by an appreciable velocity of approach which affects the desired shape of the spillway cross.section and the discharge coefficient (USBR,

1948;

Although prototype measurements of pressure fluctuations on the.high-overflaw type spillway at Joseph Dmn. (Corps of 'Engineers,

1958)

did not reveal any fluCtUatiOng Which Could reasonably be.con-sidered a possible source of structural Vibrations, there have been

Several undocumented oral, reports of vibration of low-crest spillways

for flows with heads exceeding the Spillway height. High-overflow

spillways are usually constructed on good rock foundations. The structure

'and foundation has a-high modulus of elasticity" and the System as a whole is quite, rigid and missive. -PUrthermore, the mass Of

fluid in

motion over theepillway is small

in

comparison with the effective mass of the spillway and itefoundation.. Therefore, vibration of high-overflow

spillways has not been of 'particular concern. On .the basis of prototype

experience, satisfactory performnace has been obtained if the crest

height P is equal to or greater than .the head on the crest, provided

the foundationrock is good. (Corpsof Engineers,

1965).

On the other hand, low.-ogee crests are often'used in navigation

and conservation dams, and chute spillways of large structures; in many instances these structures are founded on alluvial soils. The spillway Is then a rigid structure on a relatively elastic foundation. Not only

is the elastic modulus comparatively, law, but ihe mass of water over the

spillway at the maximum head is of the same order of magnitude as the mass of the spillway itself. Thus a significant percentage of the system

comprised of the fluid flawing over the spillway, the spillway, and the spillway foundation is in motion, and should some flow-instability or. hydroelastic control be present, structural vibration of the low-ogee

crest becomes a distinct possibility as the head becomes progressively

greater.

Although there are several possible sources of spillway. vibra-tion (Campbell, 1969) including, for example, earthquake- forces and the .

effects of gates and gate piers-, this investigation was focused specific-ally the, pressure fluctuations generatedon the face of a low-ogee

spillway by the fluid flowing over the spillway at heads equal to or greater than the design head for the

Spillway

Shape,, Hd . The shape

3

Chosen Air study was a low-ogee shape. with a 45° upstream face pro

.-portioned according to data on plates 26, 29, and

34

(Corps of Engineers,

1965),

refer to figure 1. Two points were selected for measurement of

the fluctuating pressure.; the pressure fluctuations at point 1 (see

figure 1) are representative of pressure fluctuations at locations on

the spillway crest, and those at point 2 are indicative, of pressure

fluctuations at the Spillway

toe.,

Pressure fluctuations at the spillway toe

(point

2) were measured with and without a 'hydraulic jump on the

downstream apron floor.

The quantities Measured Were the

root-Meansquare

(RMS) valUe and spectral density function of the pressure fluctuations. The spectral density functions were Obtained Only:fOr ratiOt Of head on the spillway . to.design head for the spillway shape

of

1.0, 1.33, 1.5, and 1.8..

The principal objectives of the Operthental program were:

(1).

To determine whether the pressure fluctuations induced on the face of al.OW-dgee spillway at heads equal to

or

greater than the design head

fOr the spillWay shape,

without

the action of

a

hydraulic jump on the downstream apron, 'are a

possible Source of excitation Of spillway vibration.

To ascertain the effects of a

hydraulic

jut') on

the characteristics of

the

pressure

fluctuations 'at the spillWay toe, and their possible significance

with respect to Spillway

vibratiOn-To establish the discharge characteristics for the

spillway

shape.

.To review the estabitahed hydraulic deign Criteria

concerned with the

Vibration

of low-ogee spillways and recamMend possible modifications based upon

The experimental techniques, the analysis of the data, and the pertinent results are presented in the following sections.

II. ENPERMENTAL APPARATUS AND NETHODS OF -DATA. ANALYSIS.

Flume. All of the experiments reported-herein were con-ducted in the one-foot wide flume located on the first floor of the Iowa Institute of Hydraulic Research laboratory. The flume has been described in detail by Rouse

(1961).

Satisfactory flow conditions pre-vailed at discharges corresponding to heads less than or equal to the

design head, Hd For higher discharges, the transition from the head box to the flume produced a standing wave on the free surface. Submerged

screens in the head box and a wave suppressor minimized the undesirable free-surface fluctuations upstream from the model. False floors were installed in the flume to obtain the proper relationship between the

upstream ,approach channel elevation and the downstream apron elevation.

The maximum discharge available in

5.35

cfs, which corresponds to

H /Hd = '1.8. e

Abillitay'Shave. The shape of the spillway cross-section

used in this investigation is one of three low-ogee crest shapes suggested

-for generaX.study by the U.S. Arpy Corps of 'Engineers (1965,p. 52). The

ratio of the approach

flow

velocity head to the design head for the

spillwayfithape

is

ha/Hd = 0.12, and the ratio of the spillway height to

the design head for the spillway shape is P/Hd

= OA4.

It has been suggested that the results from a,study of this shape are applicable to

shapes in the range 0.3 < P/Hd <-0.57.- The 'upstreamquadrant was 'proportioned according to data on plate 26 (Corps of Engineers,

1965),

the doWnstream quadrant according to plate29, and the circular toe curve

according to plate 314 (see figure 1). Photographs of flow over the model

spillway

for, several flow conditions are Shown on figure 2. The Spillway

model height P is

0.1 'ft.,

and. the design head. Rd is

0.683

ft.

study was a Statham Model 131TC, a 2.5 psi differential pressure transducer with a one-half inch diameter diaphragm. Two points were selected for the measurement of the fluctuating pressure. Point 1,

representative of spillway crest pressure fluctuations, was located

at x/Hd = 0.276 and y/Hd = -0.056, with the origin of coordinates

at the spillway crest as shown in figure 1. The location of point 1 was based on observations on a preliminary model. Dye tests in preliminary model showed some disturbances in this region although these disturbances did noireappear in the model used for the pressure measurements. If separation were to occur, it was felt that the

pressure fluctuations at point 1 would be considerably more significant than those measured by a transducer at the spillway crest. Point 2,

representative of pressure fluctUations at the spillway toe, was

located at x/Hd = 1.78 and y/Hd = Measurements of the fluctuating pressure at point 2 were obtained with and without a hydraulic jump on the downstream apron to obtain comparative effects of a hydraulic jump on the pressure fluctuations on the spillway toe. A piezameter opening 1/16th inch in diameter was located at the same

elevation as the center of the pressure transducer so that mean pressure data could be obtained independent of drift problems associated with

the pressure transducer.

'Since the magnitude of the measured pressure fluctuations was

smallihigh

quality sighal amplification was necessary. The output of

the transducer was first amplified

2500

times by a Dana Model 2850 D.C.,

amplifier, and further amplified at the IBM

1800

Data Acquisition and Control System located

on

the third floor of the laboratory., Temperature Changes caused same drift in the mean value of the voltage from the

transducer. This drift was well-within the transducer specifications, but became. noticablebecause of the high amplification required to

analyze the fluctuating component of the pressure. The mean value

(D.C. voltage) was filtered out to yield a nearIY zero mean Which

facil-itated spectrum analysis. It was not necessary to eliminate the mean for the two minute averaging timetsed to obtain the EMS value of the

by sealing the flume tailgate, filling the flume with water, and then layering the water level in a series of steps by means of a drain

valve. The calibration coefficient remained within one percent of the average value throughout the course of the investigation.

D. Data Analysis. Analysis of the pressure fluctuations was accomplished through use of an IBM 1800 Data Acquisition and Control System which has been described in detail by Glover (1968). No attempt will be made here to be cOMplete in discussing the methods used for

analyzing the data. Suffice it to say that the pressure fluctuations can be described by the theory of ergodic random processes. The objective here is to define and make clear what interpretation may be ascribed to the quantities measured in this study.

-'Pressure fluctuations on the.spillivay-face- are a random phenomenon and cannot be' described adequatelyby. assigning A specific

amplitude orparticulat frequency to the fluctuation, as anyone who has etaminedstrip-chart recordings or,oscillograms

is

well aware, (see. figure 134

for

example). Instead, statistical Parameters which provide - a picture of the.average:behavior of the fluctuations are required*

A random process contains a distribution of amplitudes, with some occuring more frequently than others. Specifying the mean-square or root-mean-square (E(S) value is somewhat analogous to specifying the amplitude of a deterministic process (a sine wave for example). For an ergodic random process X(t) p where t is time, the mean-square value can be expressed as

/2

3771t= lim

=

IT

Itox(t)dt

T+=

The ENS value is the square root of the mean-square and is sometimes.

'referred to AS a measure of the Intensity of the fluatUations:

just as it contains a distribution of amplitudes. The spectral density function is a statistical function which displays some of the frequency characteristics of a random process, and indicates, on the average, the frequencies that are most likely to be encountered. The spectral density function can be obtained indirectly by first measuring the auto covariance

function

C( T) = X(t)X(t+T)

where the bar over the quantity denotes temporal average. Notice that

C(0) is simply the mean-square of the processCTET2-) . The random process is correlated with itself displaced in time by the lag time, T .

The auto-correlation function is'the auto-covariance function normalized

by C(0).

R(T)

c(I).

x(t)x(t+T)

'

c(o)

It Can also

be

dt0Onstrated"that the spectral density function and the al4O-coVatiance function are related by Fourier transforms (Wiener, 1930).

C(T) = P(f) -cos 27rfTdf

Rri72-P(f) = 41 C(T) cos 271...frdT

0

The normalized form of these relationships is used to present the experi-mental data obtained in this study

flt( _{= cos} fce 1)(f)

-77-

= 0(f) cos 2Tuftdf ' C 0 k )

car,.

44. 41

R(r) cos (2TrfT)dT

One interpretatioh of the spectral density fUnction

can

be Obtained by

8

Hence P(f)df represents the contribution to the mean-square from fluctuations with frequencies between f and f + df . The quantity

(f)df = P(f)/C(0)df therefore represents the intensity of fluctuations of frequencies between f and f + df relative to the total intensity

of the fluctuations i7777= C(0) .

Another property of the spectral density function can be derived from the fact that a sine wave of frequency f = f appears in the spectrum as a Dirac delta function at f = fo; that is, as a

rectangle or "spike" infinitely high, infinitesmally wide and having a

finite area. In practice, periodic functions appear as sharp, high

peaks in the spectrum. Thus, not only does the spectral density function

t(f) present the distribution of the relative intensity of the

fluctua-tions 'with respect to frequency, but it may also indicate the presence

of periodic or almost periodic components as well.

The auto-correlation function was determined by use of the IBM 1800 Data Acquisition and Control System. The computer program has been described by Locher

(1969),

and some references on the theoretical and practical aspects of methods for obtaining the spectral density

function are Blackman and.,Tukey (1958), Bendat and Piersol

(1966),

and Jenkins and Watts (1968).

In sunmary, the descriptors of the random pressure fluctuation's utilized are the BNB value and the spectral density function, which

are measures of the amplitude and frequency characteristics, respectively.

These two statistical 'parameters form the basis for interpretation of the characteristics of the pressure fluctuations measured in this study.

where

9

III. PRESSURE FLUCTUATIONS I. SPILLWAY CREST

A. General Considerationa. In order better to understand

the prOblam of eiperimental measurements and interpretation of the data associated With the fluctuating presture, consider first the simpler situation posed

by

the mean pressure. The general functional

relationship for the. mean pressure at agy point on the spillway face it

= f, (p g, Op Uo, P, h ai) where = fluid density = gravitational acceleration p = fluid viscosity a -= surface tension

U = velocity of the approach flow P = spillway height

h height of water over the spillway Crest a. = geometric parameters describing the spillway

1

shape, including k , the equivalent sand grain

.roughness of the spillway surface, and lid

the design head for the $141lway Shape

An application of the IT theorem yields the following relationship:

P gd

, h-, 131) ( 5 )

1F

=/417

a Froude number

= Uohp/p ,

a

Reynolds number

NW

Duo/Ph ,

a Weber ftudber

If the approach flow

is

restricted to being subcritids1 (Froude numbers) less than unity), then the Froude number is no:longer.an independent variable, but depends upon the remaining parsdeters On the tight side of

(5).: .(See Rouse, 1938, for further discussion of this point.) Further,

tined He := 2(Vo, h), the functional relationship

can_ba

written as

1 1;p110 Nis

2---1

2 2 0 -P He et: H H e e d

\

and,

after aode-manipulation, in the more familiar form

PH

)

(6)

(7)

PH k

e 6 -4/ H H H ' d d d

(8)

If the model

scale

is sUffiCientlY-.large, the viscous and surface tension

effects have negligdble effects On the gross flow pattern For a smooth

surface and a particular -spillway geometry, the final form of the

relationship

for

the Mean pressure is

.P _{A '}

(1-1

_pU

2 (1)

`,17d% H.

.2

0.

The validity- Of the., simplifying assumptionshas.been demonstrated

prac-tically

by

model-Prototype Correlations and theoretically by the

appli-cation

of potential flow calculations (Cassidy,

1964)i

.

Turning to consideration of the fluctuating pressure,

p'

similar

reasoning leads to the follotring relationship- fOr':the EMS value . of the pressure fluctuatione

However, the effects represented by-F,P, andIW .cannot be dismissed

-immediately as in the case of the mean pressure, just discussed..-The key .

effects on the gross flow pattern; pressure fluctuations depend on the fine scale characteristics of the flaw as well.

AMOng the possible sources of pressure fluctuations on the spillway face are 1) waves on the free surface, 2) tutbnience

in

the approach flow, and 3) turbulence originating in the boundary layer. Free surface waves may be either

a

gravitational or surface tension

effect, and hence they are Influenced

by both theTroude,and-Wehet

numbers. - .In general, the effects of free dUrface waves _may be discounted

if there are no waves observed in the model or prototype. It should be noted that unusual free-surface effects can be induced

by

upstream

conditions', as was observed In model tests for the Kaysinger Bluff Dam (Pickering,

1968).

No such effects were observed in the present study. However, the influence of

all

perturbations in. the free surface

cannot be completely ignored

in

the model', as Will be discussed later.

In the absence

of

toticablefreesutface effects, and provided that the model scale is sufficiently

large,

the Froude and Weber numbers need not be considered, as was diteussed previously, fOt the case of the mean

pressure,. p. The relationship for the EMS:Value of the ptesdure-fluctuations for a smoOth spillway and a. particular spillway geometry,

(8)

then reduces to

A

2

- T7

(9).

The Crux Of the problem may be seen from 4 comparison Of'

₍₇₎

and

(9)

above.. Whereas the mean pressure depends primarily on effects of gravity

and inertia, And henCe scales according to the familiar FroUde ctitetion,

the fluctuating 'pressure also depends strongly On effects of viscosity,

bringing in an additional parameter, the Reynolds number. Therefore,

the Model and prototype are not dynatioally similar, if Water is used

in

both cases, as fat as the fluctuating pressure is concerned.

The lack of dynamic similarity in the viscous forces is quite evident when the boundary layers in the model and prototype are compared.

12

Since the model Reynolds number is generally much less than the prototype Reynolds number, the proportion of the spillway face over which the boundary layer remains laminar will be much larger in the model than in the prototype. If the boundary layer is assumed to develop from the line determined by the intersection of the

45°

upstream slope and the approach channel floor, then in the present study the distance Reynolds number for point I (see figure 1) is

of the order 105. An acceptable value of the transition Reynolds number for about two percent turbulence intensity of the approach

flow is

3.2

x 105 (Schlichting,

1960).

Thus, transition from a

laminar to turbulent boundary layer probably takes place near point 1

in the model. If a 401 prototype spillway is considered, for example, then the Reynolds number at point 1 in the prototype would be about of the order 107 to 108, and the boundary layer in the prototype would be turbulent over all but the first few feet of the spillway

surface. It is therefore very doubtful that measurements of the pressure fluctuations at point 1 for a smooth spillway model would have been representative of the prototype characteristics, had special

steps not been taken.

A solution to this dilemma is to trip the boundary layer with artificial roughness elements, thereby making the boundary layer

turbulent over most of the surface of the spillway model. A review of data obtained by several investigators (Bull,

1963;

Serafini,

1963;

Willmarth,

1963;

Blake,

1969)

of pressure fluctuations under turbulent boundary layers on a flat plate in zero-pressure-gradient flows reveals

two significant features; 1) The ratio of the EMS value of the pressure

fluctuations to pU.2/2

where Uw

is the free stream velocity, is almost constant with varying distance along the plate; and 2) The

spectra of the pressure fluctuations superpose when normalized with the free stream velocity 'Um and the boundary layer displacement

thickness, (5*. Therefore, the RIC value of the pressure fluctuations

is practically independent of the distance Reynolds number, and if the boundary layer is turbulent on the face of the spillway model, representative measurements of the characteristics of the pressure fluctuations can be obtained that are adequate for most practical

engineering purposes. Utilizing the experimental evidence cited above, (9) can be reduced to the simple relationship

/17

P H

1 2 H

Hd

pUo

(10)

B. EMS Value, Presdure Fluctuations, Point 1. The boundary layer was tripped by two tots of artifical roughness elements 1/8 inch square and 1/32 inch high with a center-to-center spacing of 1/2 inch and located near the beginning of the Upstream quadrant (see figure 1). Experimental measurements obtained at point 1 included:

The EMS value\of the pressure fluctuations relative

to 1/2 pUO2. As a function of the 'ratio of head on

the spillway H to design head for the spillway

shape, _Rd

-Spectra'of the preasUre flUctuatiOns at ratios of He/Rd

equal

to 1.0,

l.33,

1.5, and 1.8.

. Boundary layer characteristics With the turbulence :stimulator* in place.

For

purposes of comparison, measurements of the REB values and spectra of the. pressure fluctuations were obtainedvith and without the turbulence

stinUlatOts.

Boundary

layer veldcity ptofiles were measured with a stagnation probe, following the procedure described by Cassidy (1964). No.sttempt was made to measure bOundaty layer Characteriatics Without

the tuibulence.stimuiatora.

Figures 3 and 4 depict the variation of the relative HMS value of the presaute.fluctuationeWith %/Hd with and without the turbulence

stiMulatorS; Notice that the HMS

Values

of' the pressure fluctuations.

for the smooth spillway are higher than corresponding results With the

roughness elements in place. This unsettling disagreement between the

the distance Reynolds number for point 1 is of the order of 105

indicating that transition from a laminar to turbulent boundary layer

should take place close to point 1. Transition does not occur at a.

well defined location, but oscillates back and forth about some mean

position on the spillway face. This random oscillation and the fact that the turbulence characteristics in the transition region are

different from those in a fully turbulent boundary layer are the most likely causes of the difference in the observed values. Furthermore, eddies are shed from the small separation region just upstream from the spillway, and may be convected downstream along the smooth boundary and influence the pressure fluctuations at point 1. The presence of the artificial roughnesses would tend to break up any large scale

eddies near the boundary and superpose a much finer scale of turbulence on the mean flow pattern, which in turn decreases the fluctuations that the 1/2-inch diameter transducer is able to resolve. Furthermore,

as H

e/Hd increases, the Reynolds number also increases and the trend

of the data from the smooth spillway is toward the data obtained with the turbulence stimulators in place, with almost the same results at the highest ratio of He/Hd = 1.8. That _is, the data for the two cases

tend to agree more closely at high values of the Reynolds number. It may therefore be concluded that the _differences in the results with and

without the turbulence stimulators is influenced by the Reynolds

number.

The trend toward higher relative BNB values at the upper and lower limits of the He/Hd values investigated may be explained as

follows. At He/Hd = 1.0 , the absolute magnitude of the pressure

fluctuations is small. The fact that 2/(1/2 p1102) ,is almost a constant

means that g is proportional to the square of the approach velocity U . Therefore, noise from the amplifier system, vibrations, and small

free-surface fluctuations begin to become significant with respect to the magnitude of the fluctuations, resulting in the observed upward trend at the lower values of He/Hd . With He/Hd = 1.8 , the wave suppressor was not as effective in reducing free-surface fluctuations as for lower discharges, and the approach flow appeared to be more turublent.

15

Both of thesefactors would contribute to the observed increase in the

RMS:valUes. It ,seems reasonable to state that for practical purposes,

the value Of p/(I/2

pu02)

at point i with the turbulence stimulators in place is a constant equal to 0..055 in the interval'i.0 g H

_ea

/H. 5 LT,.

C. Boundary Layer Measurements. The velocity profiles normal to the boundary at point I were obtained for He/Rd equal to 1.0, 1.33,

1.5 and 1.8. The displacement thickness 0 and velocity just outside

the boundary layer were required for comparison with published results.

It should be made clear, that this phase of the investigation was not a

study of wall pressure fluctuations under turbulent boundary layers on

curved surfaces. The data were obtained to determine whether the measured characteristics of the Pressure fluctuations were reasonable

in the light of the available information.

Let us consider the effect of transducer size on measurements

of the pressure fluctuations. It is obvious that those pressure

fluctuations which are derived from the smallest eddies in a turbulent flow cannot be resolved ifthe transducer size is larger than some

characteristic scale of the turbulence. Therefore, the larger the transducer diameter relative to some characteristic length, say ,

the less high-frequency fluctuations will be sensed and the lower the

HMS value will be. Spectra showing the drop in high-frequency fluctu-ations measured with successviely larger transducers have been published by Willmarth and Wooldridge (1963) for example. Because the

high-frequency fluctuations are not of significance in this study, the size of the transducer (within limits of course) is not important except

to permit comparison with other data.

One shOuldstrive to make the ratio of transducer face diameter to displacement thickness as small as possible, so that the high,frequenCy content of the speCtrum can be analyzed.. Some of the more recent data were obtained with d/0

of

0.1 (Blake,

1969);

in

-the present investigation d/0 was of -the order of 20. With this parameter determined, some Comparison with published data is possible.

For a smooth boundary, Serafini (1963) reports a value of

2/(1/2 pUO2) of about 5 x 10-3, whereas others report values from

5 to 8 x 10-3. A rough boundary also influences the magnitude of the pressure fluctuations, as has been reported by Blake

(1969).

Using several roughness patterns, Blake obtained values of 2/(1/2 pUO2) of

the order of 15 x 10-3 with d/d* = 0.1. The values of 2/(1/2 PU2) of the order of

7.5

x 10-3 obtained in this study are shown in figure 4. The scale at the left refers to data normalized with U0 whereas the

scale at the right refers to data normalized with U , the velocity

just outside the boundary layer. These values appear reasonable in

comparison with those quoted above. The trend of the data in figure 4 is the same as the data with

Uo as the reference velocity. The explanation for the higher valueS af 2/(1/2 pD02) at He/Hd = 1.0

and

1.8

has already been discussed. These data indicate that the pressure fluctuations measured in this study are primarily a boundary

layer phenomenon. Approach flaw turbulence and free-surface perturba-tions are present but are of minor importance, and as a first approxima-tion the characteristics of the pressure fluctuaapproxima-tions at point 1 should not differ appreciably from wall-pressure fluctuations under a turbulent

boundary layer.

D. Spectral Density Function, Spillway Crest. Spectral Density functions of the pressure fluctuations obtained at point 1

for values of He/Hd equal to 1.0,

1.33, 1.5,

and

1.8

with and without the turbulence stimulators are presented in figures 5 through 12.

These data are in qualitative agreement with published spectra (Bull,

1963;

Serafini,

1963;

Willmarth,

1963;

Blake,

1969)

of pressure fluctuations under a turbulent boundary layer in a zero pressure

gradient. That is, with increasing frequency, the spectra decrease rapidly at low frequencies, level off with perhaps a very slight rise, and then decrease continuously at higher frequencies. There is no tendency toward periodicity discernable

in

any of the spectra obtained

at point 1. A quantitative comparison of these data with spectra of wall-pressure fluctuations under turbulent boundary layers was not

17

An elimination of corresponding spectra with and without the turbulencestiMulators indicates that there are more fluctuations

at frequencies above 16 Et. (fP/UO > 1 approximately)

for

the Cues

wwithout the turbulence. stimulators than for cases with the stimulators.

This increase in the Spectral density is ofa "broad band" character, and IS probably the Consequence of fluatuations caused by the unsteady movement of the transition point from a laminar to turbulent boundary layer, discuised pretioubly. Both the low BNB values and the absence

of any

tendency toward peiiodicity.displaied

by

the spectral density functiOns of the pressure fluctuations at point 1 indiCate that the pressure fluctuations at the spillway crest are not

a

likely cause Of

spillway vibrations.

E.. _Comparison With Prototype Data. _{No pUblished prototype} spectra of pressure fluctuations were, found in the course of the

-extensive literature survey, conducted for this study.. However, the

prototype- tests conducted.at Chief Joseph Dam (Corps of Engineers, 1958)

cOntain:some.estimates of a "typical" amplitude of the pressure fluctuations at a point located approximately in the same position

.relative to the spillway crest as point 1 in the present model. Because of the fortunate circumstance that. 2/(1/2 pUO2) _{iSpractically,a}

-constant, and recalling that the BNO-value is -a measure of the amplitude

Of the fluctuations, the amplitude of the fluctuations

in

_{the model May} be scaled accordingto.the Ifroude criterion. Although the Chief Joseph dpillvay4s.a high.-overflow spillway, a reasonable estimate of the

amplitude 'can be 'obtained using the ratio of'designheada for the

spill-way shapes as the scale ratio. _{A "typical" amplitude fOrflow at the} design head Was selected from the oscillograms shown In 'figure 13 of

this report and compared with data in Table 13, test No. 21 (Corps of Engineers,

1958):

_{The amplitudeof the fluctuations recorded at Chief} Joseph .Dam was 0.3 ft.'of.Water; When reduced to the present model

(approximately 1161 scale ratio) the predicted amplitude is 0.0021 psi. The measured amplitude at the design head for the

Spillway

_{shape was}

0.0038 psi. In view of the uncertainties

in

_{picking a "typical"} amplitude from such records, the agreement is pleasantly _surprising.

Data were also available from a model study conducted by Copp (1962), in which strip chart records similar to those of figure 13

were available. Scaling the amplitudes measured by Copp (1962) to

the present model yielded an amplitude of 0.0047 psi. Thus, the present model data, a previous set of model data, and a prototype test all

produce results of the same order of magnitude. The prototype estimate is lowest, as expected, because small extraneous fluctuations due to the experimental setup are much more significant in the model than in

the prototype. The above result also reinforces the conclusion that the pressure fluctuations at point 1 with the artificial roughnesses in place are the most representative of prototype performance.

IV. PRESSURE FLUCTUATIONS II., SPILLWAY TOE

A. RMS_ Value, Pressure Fluctuations, Point 2. The data Measured at point2 (refer to figure 1) included:

The EMS.:Value of the pressure fluctuations relative to 1/2 pUb7

as

a function of He/Hd for flow canditions With and without turbulence stimulators:

Spectra of the pressure fluctuations with and Without turbulence stimulators at ratios of _He/Hd equal to

1.33., 1.5, and 1..8.

3

The relative BM

as a function of with and without

from both series

H /Hd > 1.2, the

18

. Spectra and EMS values of the pressure fluctuations

with a hydraulic jump on the downstream apron for ratios of He/Hd equal to 1.0, 1.33, 1.5, and 1.8. value of the pressure fluctuations at the spillway toe

H /H are depicted in figures 14 and 15 for conditions

e d

the turbulence stimulators, respectively. The data of tests superpose very well for H

e/d H < 1.2; for

initial decrease for which both sets superpose, each case has a range

of fairly constant values. This range extends over a larger interval for the case without the stimulators; the data with turbulence

stimulators is slightly higher in the interval 1.2 He/Hd -5- 1.5 in

comparison with data obtained without the stimulators, as is reasonable

to expect. The most significant differences occur for He/Hd > 1.5.

Clearly some effect of the turbulence stimulators is responsible for the increasing intensity of the pressure fluctuations recorded for

He/Hd > 1.5 (figure 15).

In comparison with the experimental results observed at point 1

with turbulence stimulators, the RMS. value of the pressure fluctuations

for both sets of data obtained at point 2 are consistently higher. Part

of this increase is no doubt a result of the adverse pressure gradient

in

the spillway toe curie. Schloemer

(1966)

has shown that the RMB value of the pressure fluctuations on a flat plate with an adverse pressure gradient is larger than at corresponding flaws with a zero

pressure gradient. Although the pressure fluctuations at point 1 and point 2 are still of the same order of magnitude, spectral analysis shows that the characteristics of the pressure fluctuations at point 2

are quite different from those at point 1.

B. Spectral Density Fuhction, Point 2. As tight have been anticipated from the agreement of the RMS value for He/Hd 1.2, the spectra obtained with and without turbulence stimulators for He/Hd = 1

(figures 16 and 17) are quite similar. The shape of both these spectra differs Slightly from data obtained at point 1 with turbulence stimulators

(figure 5) under the same flow conditions, with a minor rise near

fP/Uo =

0.8.

For flows with H /Hd = 1.3, (figures 18 and 19), the

e

minor rise has developed further, and the shape of the spectra, part-icularly those obtained with the stimulators present is now significantly different from corresponding data at point I (figure

8).

Spedtra obtained with He/Hd = 1.5 (figures 20 and 21) depict a small but definite peak near fP/U0

= 0.45,

which definitely does not

20

occur.

in

the corresponding data Obtained at point 1. Finally with

H /Hd =

1.8,

the peaking occurs

at a

Still laver value Of

fP/Uo 0.35

e

-or 0.4. Because of the already large values of the spectral density at low frequencies, this peak is not as apparent as at He/lid .= 1.5.

The spectra of the pressure fluctuations at the spillway toe thus indicate that there is some change in the flow pattern with

increasing values of He/Hd , and the peaking of the spectra indicate a tendency toward periodicity in the fluctuations. In order to obtain a better physical picture of the flaw pattern, flow visualization with

dye was attempted.

C. .Flow Visualization .-- Dye. Tests, 'Dye was injected through

the static piezometer.tap located:at point 1, and its -motion

along

the

Spillway face was observed closely. The dye plume was photographed for

flows with

He/Hd

= 1,0, 1.3,. 145, and .1.8 with turbulence stimulators

. glued to the spillway crest (figure l).

Figures 24a and 24b depict flaw conditions corresponding to

He/Hd = 1.0 and 1.3, respectively. The dye streak remained close to the spillway face and could be observed well downstream from the model.

For flow with He/Hd = 1.5 (figure 24c) the plume became more diffused in the lateral direction, and with some disturbances visible in the circular toe curve which augmented vertical diffusion of the plume. Finally with He/lid = 1.8 the dye spread out laterally well upstream from point 2 (figure 24d) and backflow along the spillway face was

clearly discernible. This observation indicates that local separation

was taking place. These tests show that the turbulence at point 2

should have different characteristics than at point 1, as was indicated by a comparison of the spectra of the pressure fluctuation at the toe

points discussed previously.

D. Discussion! The presence of an adverse pressure, gradient over approximately the upstream one-third of. the circular toe curve

(refer to Rouse, 1950, P. 539 for an illustration) explains the. above observation. At adverse pressure gradient promotes separation.

Indeed, as the dye test's have shown, separation does take place at H /Hd =

1.8.

The tendency toward periodicity in the pressure

fluctu-e

ations

is

a direct consequence of this separation..

Some investigations (ratinclaux,

1966;

Locher, 1967) of separated flaws behind idealized gate shapes have shown that the reattachment point of the separation zone is unstable and oscillates

with a dominant frequency. In this case, both the point of separation

and reattachment are not fixed, but oscillate. The peaking observed In the spectra at point 2 is more probably a result of this unstable oscillation. 'Furthermore, the mean position of the separation point

is dependent on the boundary layer characteristics. Thus, changing the boundary layer development along the spillway should change the mean position of the local separation zone and thereby change the

characteristics of the pressure fluctuations at point 2. One might

also speculate that the turbulence stimulators bring the velocity profile into closer agreement with the irrotational velocity profile, and that the mean position of the separation point moves downstream In a manner similar to the shift in separation point on a sphere as

its boundary layer becomes turbulent. The reattachment point would thus move closer to point 2, and continue to do so with increasing

Reynolds number. The turbulence, and hence the pressure fluctuations at point 2, would therefore be more intense with the turbulence

stimulators present than without, as was observed (figures 14 and 15). A series of careful measurements would be required to verify this

hypothesis.

E. Spillway,TOe_PressUre Fluctuations, Hydraulic Jump.

Spectra and RIC Values of. the pressure fluctuations Were also measured

with

a hydraulic

jump on the downstream apron floor

for

.He/Hd = 1.0,

1.3, 1.5, and 1.8. 'The. jump Was formed by raising a tailgate at the downstream end of the flume. Some difficulty was encountered in

22

reproducing the data because the pressure fluctuations were quite sensitive to the position of the jump with respect to the spillway

toe. Therefore, the results of these tests should be regarded as more qualitative than quantitative, and indicative of trends rather

than absolute values. These data are, however, of the same order of magnitude as those reported by Vasiliev (1967), but a direct

comparison cannot be made because the Froude number for the two cases

is not the same. The EMS values have been plotted on figure 15 for comparison with the data obtained without a jump. Notice the highly significant difference for He/Hd =

1.8.

Spectra of the pressure fluctuations at point 2 are shown in figures 25 through

28.

Photographs of the spillway model were

taken during the measurement of the spectral density 'function and are shown in figure 29. Furthermore, dye tests were run during, between, and after the measurement for the determination of the spectrum. The dye showed that separation would occur if the lead portion of the jump was above the downstream third of the circular toe curve for He/Hd = 1.3

and 1.5. By moving the jump so that the position is as shown in figure 29, no clearly visible separation occurred. Spectra at H

e/Hd = 1.3 and 1.5 were obtained with no apparent separation.

Sep-aration did not occur with H

_ea

/H, = 1 until the jump was positioned well up on the spillway. However, with He/Rd =

1.8,

separation

occurred even if the jump were not on the spillwv. The separation for

He/Rd =

1.8

was made visible by dye as shown in figure 30. Notice how far upstream the separation point is located on the spillway face.

The spectra measured with a hydraulic jump, on the spillway

toe are generally similar to corresponding spectra without the jump. Despite the rather large increase in RMS value at He/Rd = 1, the

spectra have a similar shape. Apparently the increase in RMS is a

consequence of the wave action of the jump which is randomly distributed

in both amplitude and frequency. Again, a peaking occurs in the

spectra for H /Hd = 1.3 and 1.5. Because the relative height of the

e

23

the jump augments the dominant fluctuations slightly. The tendency toward periodicity seems to disappear almost entirely from the spectra

at H /Hd = 1.8. The change in position of the separation point to a

e

location well upstream from point 2 explains this observation. A

significant increase in free surface fluctuations, as well as a rise in the water surface upstream from the spillway model' was also noted with He/Hd = 1.8. The significant increase in BNB value recorded is

thus a consequence of the change .in flow pattern, increased free-surface

fluctuations; the action of the hydraulic jump, and increased turbulence

in the approach flow. The change in head on the spillway with the jump present indicates a partially submerged condition as well.

V. DISCHARGE COEFFICIENT

A. :Dimensional Considerations. Measurements to obtain the discharge coefficient as a function of the ratio He/Hd were a routine procedure; the data have been plotted in figure 31. In contrast to results fram high-overflow spillways, the discharge coefficient for this model does not increase monotonically, with increasing ratio of H /Hd' but increases to a maximum near H /Hd = 1.35 and then decreases

e e

Slightly. Notice that there is only a 1.3 percent increase in C from H

e/Hd = 1.0 to "H /Ed = 1.35.e This behavior of the discharge coefficients

for low-ogee crest shapes has

been

observed

by others as well (Bradley,

1952).

Before proceeding to an explanation of the observed variation

of C _{with He/Hd}

'

let us quickly review some elementary concepts associated with the determination of spillway discharge coefficients. Since the discharge per unit width q depends upon the same variables

as the pressure at a point p , a general form of the relationship for the discharge coefficient may be written as

224

PH

(

H ' a

4/T

d d -d

The parameter .P/Hd represents the effects of the approach velocity, He/Hd. the effects of,headt_other than the design head, k/Hd the

relative roughness, and IR and IW the viscous and surface tension effects, respectively.- Included in the terms $i. are such effects as the slope of the upstream face of the spillway, (45° in this case), the position

of

the downstream apron, and the radius of the spillway toe

curve, for example.'

Experiments by Eisner (1933) (see also Cassidy, 1964, p. 11) showed that the discharge coeffiCient decreased as the relative

rough-ness increased. This effect was observable in the present study. Dis-charge coefficients obtained with the turbulence stimulators were less than corresponding values obtained without the turbulence stimulators. All of the data shown in figure 31 were therefore computed from

measure-ments taken without the artificial roughness on the spillway crest. Effects of viscosity and_surface tension on the discharge coefficient

can be ignored if the model scale is large enough. A more complete discussion is given by Cassidy (1964, 1970).

The effects

of

the approach depth, heads differing from the

design lead, .upstream face slope, and downstream apron floor position

have beeneXtentively investigated by the U.S..Army Corps

of

Engineers

(1965) and the U.S.E.R. (1948, 1960). The results

Of

these tests

together with some physical reasoning explains the behavior of the

discharge -coefficient Observed during the present investigation.

For-a given spillway geometry, the functional relationship

reduced to

C =

_{0 (H(g)}

d 1/. -He3/2 e d

C Q/LHea/Z

there results the relationship

C = (He/Hd)

which has been plotted in figure 31.

B. Discharge Coefficient at the Design Head,

Hd. A

com-parison of the value of the discharge coefficient obtained at the design head for the spillway shape Hd with the suggested design

curve shown on plate 32 (Corps of Engineers,

1965)

reproduced herein as figure 32 shows that the value of Cd

3.85

is in better agreement

with the

45°

weir curve than with the suggested curve. The suggested design curve for the

45°

crest face shown on figure 32 was obtained by

increasing the data for the

45°

weir experiments by three percent. This increase was obtained by a comparison of the 90° weir data with

900 spillway model studies (Corps of Engineers,

1965, p. 54).

However,

the increase in C for the inclined upstream face is not a constant,

but varies with P/Hd (OBB,

1960).

At a value of P/Hd =

0.44,

the

ratio of C for a vertical upstream slope to C for a

45°

slope is

only about 1.017 according to USBR

(1960, p. 277).

Multiplying the value of C for a vertical upstream slope given by USBR

(1960)

by

the ratio 1.017 yields an expected discharge coefficient of 3.89 for flow at the design head. The value C = 3.85 obtained in this study with flaw at the design head is in good agreement with the calculated

value of

3.89

as well as with data for similar shapes presented by Bradley

(1952).

Flow over a spillway may be classified as free or submerged. An overflow weir is said to be submerged when the water level on the

downstream side of the weir begins to affect the discharge characteristics

Of the weir. Similarly, flow over a spillway is said to be free if at the design head, Hd , the discharge coefficient does not differ

26

lower nappe surface corresponds to the shape of the spillway crest. Submergence effects reduce the discharge coefficient and may be

caused by water levels in the downstream channel exceeding the normal depth for the channel, or by raising the elevation of the downstream

apron floor. Free flow is a term which refers to flow conditions for which submergence effects cannot be detected; there is no sharp

demarcation between free flow and submerged flow.

A check for effects of the downstream apron elevation shows that the present shape is a "borderline case for flow with He/Hd = 1.0. It has been suggested by the USER (1948) that (hd + d)/He be greater than 1.7 if there is to be no submergence effects due to the apron

floor elevation. The quantities' /id

and d

are defined on figure 1.

At the design head Hd , the value of (hd + d)/He for the present

shape is 1.68; the influence of this factor on C at the design head according to the Corps of Engineers (1965, plate 33) is about 0.1

per-cent, or practically negligable.

C. Dischar:ke Coefficient for He/Hd > 1.0. However, as the

head on the spillway He increases, the elevation of the apron floor becomes an important factor. 'For free flow over the spillway at the

design head, the pressure on the spillway crest is equal to or slightly greater than atmospheric pressure as experimental results reported by the USBR (1960) have shown and as indicated by the results at point 1 for the present study (figure 33). If free flow occurs for all heads,

then as _{H /Hd} increases, the pressure at the spillway crest decreases,

e

the minimum value being limited by cavitation.

Figure 33, shows that the mean pressure at point 1 does not

decrease monotonically as He/Hd increases. Therefore, free flow does not exist at all heads for this particular spillway shape, and the value of the discharge coefficient will.be affected accordingly. As _He

increases, the pressure in the spillway toe curve increases in proportion to the square of the velocity, since the pressure in the toe curve

27

depends upon the centrepital acceleration v2/r _{(point 2, figure 34).} The effects of this pressure increase reach farther and farther upstream with increasing H, and finally begin to affect the pressure

distri-e

bution on the spillway crest, as may be seen from the data of figure 33. Notice that as He/Hd increases, the mean pressure at point I first

decreases in the interval from He/Hd = 1.0 to about He/Hd = 1.35 as

would be expected, but then increases for He/Hd > 1.3. This change in the pressure distribution is reflected in the behavior of the

dis-charge coefficient, which attains its maximum value between He/Hd _{= 1.3}

and 1.4. _{Based upon these observations, it is clear that changes in}

the pressure distribution on the spillway crest attributable to the

effects of the spillway toe curve and the apron floor elevation are

responsible for the observed variation of the discharge coefficient at

heads{ greater than the design head. _{Furthermore, the relatively small}

decrease in the pressure on the spillway Crest from He/Hd = 1.0 to 1.35 is reflected in the very small increase in discharge _{coefficient for}

these heads.

VI. SUMMARY AND DISCUSSION

The following statements summarize the results, obtained from

measurements of the pressure fluctuations on the spillway Model _at

point's 1 and 2.

1, _{The pressure fluctuations. on the spillway face}

,at-pint 1 are primarily a consequence of turbulence generated

in

_{the boundary}

layer. _{Turbulence in the approach flow and free surface effects are}

Data obtained with the turbulence stimulators in place are probably most indicative of prototype characteristics.

Spillway-crest pressure fluctuations at point I did not

exhibit a "dominant" frequency for any of the flow conditions observed during the investigation, although there was a significant increase in the intensity of the fluctuations for _H

28

I. The pressure fluctuations at the spillway toe (point 2)

do have a "dominant" frequency at heads greater than the design head

for the spillway shape, Hd . There is no appreciable tendency toward periodicity for flow at the design head Hd .

The pressure fluctuations at point 2 are complicated by the presence

of

an adverse pressure gradient occurring over the upstream one-third of the spillway toe curve. The MB value of the pressure

fluctuation is of the same order as those at point 1.

A hydraulic jump on the downstream apron significantly increases the intensity of the pressure fluctuations at point 2. Although there is no significant trend toward periodicity for flows

with He = Hd , some peaking in the spectra does appear for He/Hd = 1.3

and

1.5.

Not one of the measurements

of

the pressure fluctuations on the spillway face showed an unusually high BNB value

or

tendency toward periodicity With flow at the design head

for

the spillway shape.

The results of a study of the discharge coefficient data may

be summarized as follows:

The discharge coefficient, C =

3.85,

obtained with flow at the design head for the spillway shape is good agreement with predictions based on USBR data and with discharge coefficients of

similar shapes reported in the literature.

The discharge coefficient for the spillway shape investi-gated during this study does not increase continuously with increasing head on the spillway, but reaches a maximum at H

e/Hd =

1.35

and then

decreases. This behavior is attributable to the influence of the

spillway toe-curve and a6ron floor elevation of the pressure distri-bution at the spillway crest as the head on the spillway increases.

These results are significant with respect to present hydraulic design criteria, because the discharge coefficient for high-overflow spillways continues to increase for He/Hd > 1.0. The crest shape is often proportioned for a design head H of

0.75

times the maximum head

29

Such Ei _{crest is called an underdesigned shape. Therefore, the discharge}

coefficient for the underdesigned shape for flows at the maximum head will be greater than a crest shape designed with _{Hd = Hm . The} under-designed shape thus passes the design flood with _{a shorter crest length}

than

a

shape designed with Hm = Hd .

Tests with low-ogee crest shapes and free flow conditions also

show that the discharge coefficient continues to increase _as

He/lid

increases (Corps of Engineers, 1965, plate 6). _{Thus, there is a trend}

toward underdesigning low-ogee crest shapes (Campbell, 1969) _although present practice of the U.S. Army Corps of Engineers does not permit low-ogee crests to be underdesigned (Corps of Engineers, 1965, p. 13). As

far as the present model is concerned, the experimental evidence

presented in figure 31 indicates that there is practically _{no advantage}

gained by using an underdesigned shape for this _case.

VIII. CONCLUSIONS

On the basis of the results summarized in the preceding section, the principal conclusions of this study are:

Pressure fluctuations on the spillway crest with flow at

the design head for the spillway shape did not display an unusually high RMS value or alarming trend in the spectra and _{are probably not a source}

of excitation of spillway vibration.

The discharge coefficient for this spillway _{configuration} does not increase significantly for ratios of the head on the spillway

to design head for the spillway shape, He/Hd , greater than 1.0.

In View of 2 above, and the trends in the _{spectra of the} pressure fluctuations at point 2 with flows greater than the design head

for the spillway shape, it is recommended that _{the underdesign of}

low-ogee crests proportioned according to data on plates 26, 29, _{and 34, Corps} of Engineers (1965) with 0.3 P/Hd 0.57 and a geometrically similar

30

location of the downstream apron floor not be permitted.

In closing;, it Should

be recalled that the above conclusions . .

are applicable only to uncontr011ed ogee .crests, since the effects of gate. piers, gate slots, etc. were not considered.

Glover, J.R. & Giaquinta, A.R

Unsteady Flow Variables. 31

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₁₉₆₄

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w.w, 4

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Measurement's

of

the Correlation

Between the Fluctuating Velocities and Fluctuating Wall-Pressure

APPROACH CHANNEL FLOOR ROUGHNERS, ELEMENTS N.

SPILL WAY MODEL

Figure 1.

Definition sketch

X.. 1.893 He

1.324 Ha

hd

Hd Hd Hd H /H = 1.0 He Hd = 1'3 HeHd = 1-5. H /Hd = 1-8 Figure 2.

Photographs of the experimental apparatus with discharges corresponding to H /H

= 1,0, 1.3,

e

d

0.06'

0.04

on2

0.8

0

_ees

RMS VALUE OF THE

PRESSURE FLUCTUATIONS

POINT

1

NO BOUNDARYLAYER TR IP

o

Different 'Experimental

a Ruins

He

Hd

Figure 3.

Variation of the relative rms value of the

pressure

fluctuations at point 1 with.He/Nci.

No boundary layer trip,.

0:10

_0.08

1.0

1.2

1.4

1.6

1.8

2.0