MEMORANDUM

To

Through

From

Subject

Chief Engineer

Chief, Hydraulic Laboratory

Chian-Min Wu

Final Report

HYDRAULICS BRANCH OFFICIAL FILE COPY

Denver, Colorado October 14, 63

At the conclusion of my training program, I am requested to submit a final report in five copies.

Enclosed please find a copy of my final report, several technical articles are involved. Among which II Flow Characteristics of Spill­way Buckets," is an interesting one. Academically, this will lead to a better understanding of the hydraulic jump in Fluid Mechanics and practically summarize the studies of spillway bucket since construction of Grand Coulee Dam in 1933.

Prior to my departure from the Hydraulic Laboratory, I have solved another interesting problem," Hydraulics of Vertical Wells" • The result of the analysis is attached as Appendix 4 in my final report 0

I believe you will find this report interesting and informative, and will lead to comprehensive discussion of the extensive research program in the near future o

I ·am sure and you would agree that the training program will lead to a rewarding experience, and that my fellowship will effectively help me to contribute to the development of my country.

r

cJIAPYL, ~ l)JN\J

cc. Mr. Howard ,M.A., Head, Training and Visitors Section

Enclosure

,;, ' t, ,·

:,

t

;

...

Final Report

on training on assignmet with the United States Bureau of Reclamation, Hydraulic Laboratory for the United Nations Technical Assistance Operations.

by

Chian - Min Wu

September 1963

PAP 180,

0 co .--4

0.. ~ 0...

"

OPTION AL FORM NO , 10 5010-104

UNITED STATES GOVERNMENT

Memorandum Memorandum TO Chief, Hydraulics Branch

FROM C. M. Wu

'?p..? - PAP 18 0

Denver, Colorado DATE: October ll, 1963

-SUBJECT: Hydraulics of vertical stilling wells

Hydraulics of vertical stilling wells has been studied in the attached paper.

The investigation has proceeded since 19o6 and 'With aid of model tests, extensive studies have been conducted in this laboratory since 1949 to 1962 as one of the research problems.

I am encouraged by my preliminary theoretical analysis, which I submitted to you on September 27 and under the direction of Mr. P. F. Enger, I canpleted the draft in last 2 weeks. Hoping this will lead to a short cut in solving the problem.

Enclosure

" CONTENTS

SUMMARY

INTRODUCTION

HYDRAULIC STRUCTURE MODEL

MOVABLE BED HYDRAULIC MODELS AND SEDIMENT STUDY

SEDDAENT TRANSPORTATION

EI.EC 'IRONIC COMPUTOR

1

2

2

3

5

7

APPEND IX -- 1 1- : ,-v, e :, r· ,~ C ,,,.,:..., ~ ,J $ f ·-1.-1-U=,1 "E," ' ..;.1 8

APPENDIX - 2 ~cle...Q.. ~L~ "'G v~1..,...,: ti.R.. $~t'i w . .tl cd 53

APPENDIX - 3 T~ket.(} Or~-l, i,( ~~.1 ..... e...-ac,,{ 62 ~~-

A r1 P( N D 1 )< - '4 ~ toe,..~ "'( V ~.J2 ~l ... ea .... :, uJ.di

..

Final Report

on training on assignment with the United States Bureau of Reclamation, Hydraulic Laboratory for the United Nations Technical Assistance Operations.

by

Chian Min Wu

September 1963

This report presents technical articles on subjects studied during the 6 month's training program in the United states Bureau of Reclamation Hydraulic Laboratory as :f'ul.fillment of a fellowship :f':ran the United Nations Technical Assistance Operations.

The assigmnent covered the period fran March 28 to September 16, 1963.

1. Engineering science is an art rather than an exact science. Particularly in the field of engineering hydraulics, many empirical coefficients and constants are introduced, emphasizing the world­wide :fundamental need for hydraulic laboratories.

2. There are no borders for scientific studies and hydraulic studies are no exception. Accamplisbments of the United states in modern instruments, such as oscillographs, oscilloscopes, pressure cells, high speed photography, electronic computers, radio active isotopes as tracers, radar and sonar are worthy of mention and of further study.

3. Hydraulic laboratory practice should not be limited to experi­ment. A canbination of mathematical a.na.J.ysis and experimental techniques is the only way to obtain basically sound result.

4. Academic education combined with technical training is the f'undamental approach for the Asian countries to catch up with modern developnents. ·

JETRODUCTION

Engineering science is an art rather than an exact science. Espe­cially in the field of engineering hydraulics many empirical coeffi­cients and constants are introduced. Thus, there is a worldwide f'undamental need for bydraulic laboratories.

Considerable progress has been made in the past decade in the field of laboratory study, particularly the basic theory of model simi­larity. Many bydraulic model tests such as those for spillway struc­tures, scour, deposition and waves in harbors, flow in closed conduits, and movable and fixed-bed river models have been conducted. Perhaps the most outstanding contribution to the hydraulic laboratory study is the rapid development of instruments, such as oscillographs, oscil­loscopes, pressure cells, high speed photography and electronic com­puters~ These have made it possible to solve and observe the complex matter of hydraulic problems.

Because of the wide scope of the subject matter available, it has been necessary to limit this report to same of the problems in which the writer has experienced the need of more research and which must be introduced to aid hydraulic development in his mother country.

HYDRAULIC srRUaruRE MODEL

&)?illwa,.y; Model

Although spillways have been the subject of more research than perhaps any other hydraulic structure, there are still deficiencies in our knowledge of their functioning. There is more to be learned about the process of energy loss on the spillway face, pressure distribution, cavitation, and the effects of submergence of the spillway jet.

Basic information concerning the characteristics of spillway flow is not sufficient. The solution to the problem has been approached by means of hydraulic models.

Model investigations are sometimes extensive and time consuming and, in addition, a nontheoretically guided-model study will sanetimes introduce questionable results. Mod.el investigations can be supple­mented with mathematical analysis to obtain more understand.able results. Therefore, it is apparent that a combination of analytical and experimental study is desirable.

An example study of the theoretical and experimental approach has been conducted by the writer during his assignment and is shown in Appendix l ("Flow Characteristics of Spillway Buckets~).

2

Dimensional Analysis

The proper interpretation of a set of observations is much more dif­ficult than the mere accumulation of observations in both fixed and movable hydraulic models, since the interpretation involves consid­eration of the relative importance of all factors that cause varia­tions in the observations.

Dimensional analysis is a tool to obtain the proper interpretation and indicate the general trend or law. An example study of the dimensional analysis has been conducted and is shown in Appendix 2 ("Model study of Vertical stilling Well Basin").

MOVABLE-BED HYDRAULIC MODELS AND SEDIMENT gruny

The rivers in Taiwan, Republic of China, and similar Asian countries, are always subjected to serious sediment problems because of the action of "Earthquakes" and "Ty:phoon Rains'r on the weak and ~asily eroded rock formations. It is found that there are many areas in which the tools available to the engineer for solution of such problems are inadequate. The fields in which research is needed are herein discussed.

Empirical Approach of Model Similarity

Although the utility of movable-bed river models is generally accepted there remains many problems. The most fundamental but unsolved problem is the establishment of model similarity. Many laws have be~~ established among which the bottcm velocity, settl­ing veloci t~ , tractive force21, regime theorr/, and sediment transport theories~ are fa.milar to hydraulic engineers, yet the prototype record is the only criterion of movable-bed analogy law. Past prototype records must be repeatedly checked by model, other­wise, many quantitative similarities are not free from model effects.

The most direct and reliable method of selecting the various scales of a movable-bed. model is the so-called "Historical Verification

!./"Model studies of Sediment Control structures on Diversion Dams," by H. M-.. Martin and E. J. Carlson, August 195, 21 "Hydraulic Laboratory Practice," by J. R. Freeman, 1929

~"Scale Relations Among Sand-Bed Rivers Including Models, ,r by T. Blench, Proc. ASCE Vol. 81, Separate No. 667, April 1955 ~ "Similarity of Distorted River Models with Movable Bed," by H. A. Einstein and N. Chien, Proc. ASCE Vol. Bo, Separate No. 566, December 1954

,•

Method." It is to adjust the mode1 scales to simulate some knO'W?l prototype occurrences similar to that of the mode1. Upon application of the "Historical Verification Method" several tests were conducted and results showed that temperature change influenced both prototype and model.~ Hydraulic model tests using the "Historical Verifica­tion Method" as a criterion, should take into account this critical phenomenon. Model verifications which have been made with high temperatures are not necessarily valid at low water temperatures.

Further tests should be conducted to determine temperature effects on bed conditions and roughness.

As described above the first and foremost in a movable-bed study is the matter of reliable basic data. Unfortunately hydraulic and bydrologic data are limited, so that theoretical investigations of similarity play a great roll in movable-bed hydraulic studies as a supplementary tool to the empirical approach of movable-bed mode1 studies.

Theoretical Approach of Movable-bed s1w:JJarity There are many problems for which the basic equations are known but these are mathematically so complicated that their direct applica­tion becomes practically impossible. Many such problems can be solved by the use of models and the laws of similarity. The even more important method of applying similarity is by means of general relationships between dimensionless variables and dimensional homogeneity. Based upon these facts Einstein, Blench, and others have derived methods of determining similarity of distorted movable­bed models. Einstein's model law is more theoretical than those of others.

Einstein solved the f'low and sediment similarity equations by choosing one of the following ratios: (1) horizontal ratio, (2) vertical ratio, and (3) density ratio. Further investigations were made by the Hydraulic Laboratory, Taipei, Taiwan, and the result is shown in Appendix 3.

The similarity conditions for distorted river models with movable­bed materials are thus derived from the theoretical and empirical equations to describe the hydraulic and sediment phenomena in rivers. Rivers, in general, are not susceptible to mathematical or theoretical analysis except for certain assumptions. In the pre­ceding approach several assumptions were made. Further advanced study is necessary for more exact and sound approximation.

§)"Effective of Temperature on Sediment Transpcrtation 2," by E. w. Lane, E. J. Carlson, and o. s. Hanson

4

SEDIMENT TRANSPORMTION

Adequate basic data on sediment transportation are not available in many countries. Much more field and laboratory studies must be made for correlation and then for extension to expedite solutions.

Suspended Load Sampling

In past years, in Taiwan and several Asian countries, suspended sediment was sampled by using vertical-type samplers. ~ling is currently being done by using U S DH-48, D 49, and P-1'6 , horizontal­type samplers.

Because standard samplers were not available in the past, calibra­tion of the sampling apparatus previously used for collecting field data should be carried out in the hydraulic laboratory. Recently the efficiency of vertical-type suspended load samplers was deter­mined in a laboratory and a value of about Bo percent was obtained • .!/ Therefore, the efficiency of uncalibrated suspended sediment samplers should be determined in hydraulic laboratories. This would make much of the data collected in the past usable and would add greatly to the reliability of any data developed with nonstandard equipment.

Distribution study

In the determination of the total suspended load transported, it is usually necessary to combine field data with empirical computations or theoretical extrapolation, except when it is possible to sample through the total depth. Unfortunately, the sampling depth is usually limited and a theoretical correction of the nonsampled sedi­ment load is based on a vertical distribution formula of suspended sediment, which is fJJ

- = k 9..:-Z • ....Jl_ C7 (d )z ca y d- a

§)"The Design of Improved Types of Suspended Sediment Samplers," Report No. 6 of a study of Methods used in Measurement and Analysis of Sediment Loads in Streams, by Subcommittee on Sedimentation, Federal Inter-Agency River Basin Cammi ttee, June 1946

!/"On the Efficiency of Vertical Type Suspended Load Sampler," Hyd-Lab Report No. 5, National Taiwan . University, 1955 §/"The Present status of Research on Sediment Transport," by Ning Chien, Proc. ASCE, Separate No. 565, Vol. Bo, December 1954

5

where CY and Ca are the concentrations of a grain size with settling velocity w at distances y and a fran the bed, respectively; R is the van Ka1rnan universal constant; d is the depth of the flow; g is the gravitational acceleration; R is the hydraulic ra.dius;and S is the slope.

The controlling parameter z can be corrected frcm either field or laboratory data. As there are many disadvantages in field operations, laboratory studies of the above parameter, though several tests have been conductedfV, is still necessary.

Standard Sampler

A physical limitation of modern samplers is the distance between the sampling nozzle intake and the stream bottcm. To obtain sedi­ment samples near the bed, several samplers have been developed. A horizontal tube sampler for sampling close to hard stream bottcms and for large sediment has been widely applied. Drag flume studies of plastic tube samplers have contributed greatly to the development of standard samplers suitable for special river conditions such as in Taiwan.

The photoelectric cell is used in several American laboratories. Perhaps the use of a photocell will improve the sampling equipment more rapidly than sampling techniques will improve.

Bedloa.d Movement

On estimating the sediment loads of streams for reservoir projects, empirical formulae have been widely used. Most bedloa.d formulae,· are based on drag flume experiments and when applied to river prob­lems mavr not give as accurate results as expected.

Owing to a wide variation of physical aspects of sediments from one geographic area to another, study in the hydraulic laboratory, where flow conditions can be controlled, is regarded as a necessity.

The bedload sampler is a fundamental tool to sediment movement study. Unfortunately, existing bedloa.d samplers were developed in countries where the rivers were much more gentle than Taiwan. The nature of the rivers on Taiwan and other rugged countries will probably require new equipment for field measurement of bedloa.ds.

Development and calibration of such equipment can best be conducted under controlled conditions in the laboratory to simplify the problems.

6

ELECTRONIC COMPlJI'ER

Programing

Use of computers is a revolution in mathematic science. Laboratory experiments often produce so much data that a ccmputer becomes necessary, in order to apply correction factors, to eliminate invalid readings, to establish general trends, and to reduce the data to meaningful.

Fortran programing is one of the most simple languages in computer works. The method of programing had been studied and several examples are shown in Appendix 1.

7

APPENDIX - 1

~· .SubJect: Flow characteristics of spillway buckets

PURPOSE

To investigate the bucket flow characteristics considering the

radial effect of' the bucket and to develop theoretical and e~eri­

mental equations for the main-flow parameters.

CONCLUSIONS

1. A theoretical analysis and study of experimental data show a

pronounced centrifugal effect on the action of the radial f'low

through a roller bucket and suggest that the flow characteristics

in spillway buckets can be analyzed by use of the equations of

pressure plus momentum 'With the ad.d:f.tion of the centrifugal force.

2. A theoretical equation is developed to relate the bucket radius

to the flow characteristics of' two different flow conditions

described as a sweepout jump-f'low condition and a submerged jump-

flow condition. Also, a. theoretical e<1.uation is derived for the

energy loss in both. These equations a.re experinenta..lly verified -<

using data obtained by the USBR, Hydraulic Laborator.r, 'Within the

·----WSll!l'CL~:. -- a· =r.,_!!1911!5--!9!1!!!!!!!!!!!!!!!!!!-!!!!!!!!!!!!!!!!!!!!l!!!!!!!!!!!'!!!!!'.!!!!!!!!!!!!!!!'!!'!!!!!!!!!!!!!!!!!!!!!!!l!!!!!!!!l!!!!!!!!-!!!!!I!!!!---'"'!""'""!'""-------

limits of' Froude numbers up i;o 9 and backup :factors (ratio of' flow

depth above the invert, y , to the supercritical depth at the 3 .

invert, y1 ) to a:r>praximately 25. A theoretical equation is developed

for the backup depth factor at the bucket invert for the submerged

jump-flow condition. This equation is also experimentally verified

within these limits by applying a correction coefficient for air­

entrainment.

3. Curvilinear effect on Froude number and critical depth of bucket

flow is analyzed and it is shown that a correction factor must be V

applied to the originally defined Froude number, i.e., F = J;;;. and it is also shown that the hydraulic jum.p can be formed in the

bucket even though the Froude number is less than 1.

4. Considerable information concerning jum.p, surge length, surge

height, and other flow para.meters is analyzed., and it is shmm that

all of the flow characteristics may be ex:pressed as f"unctions of

the flow :parameters (Froude number, tail'water factor, submerged

factor) and the shape parameters ( abrupt factor, baffle factor, and

radius factor) •

5. Using the equations obtained, it is found that the energy loss

in a submerged jump 'Within a bucket is more than in the corre~:pond.­

ing free jum.p on a horizontal floor, bnt constantly .Less than tbat

2

of a corresponding sweepout jump-flow· condition ·within the bucket.

However, due to unstable characteristics of the sweepout jump-flow

condition, a slight submergence is recOilllllended.

6. Comparison of energ,J dissipation capacity of various hydxaulic

jump basins and the lengths of these basins with that of the bucket

shows that the bucket is more efficient in d.issj.pa.ting the ~ne1-gy

and does so in a shorter length of structure.

JN:I~RClDUtJ.rION

If an initial hydraulic jump depth, Yi., is to be formed at the toe

of a bucket invert for a supercritical stream discharging over a

spillway, the tailwater depth should be equal to the subcritical

sequent depth, y4

, given py the mcmentum equation. If the tailwater

depth, y5 , is less than y4 , the jump is swept out of the bucket.

This is known as a flip-bucket flow condition (Figure 1-a*). 1m,t:r y

5 = y 4 , the hydxaulic jump is on the verge of sweeping out of the

basin. · This is defined here as a sweepout jump-flow condition

(Figure 1-b). If, however, y5 is greater than y4 , the jump is

submerged or drowned as shown in Figure 1-c. This· is defined as a

submerged jump-fl0v1 condition.

Ba5ic inf'orma.tion concerning the chare.cter istics o-: the bucket flow

is published i'l many hydxaulic and fluid mechanics books, yet the

*All figures refer to figures in this appendix, only.

3

solution of' the problem has b1:!en awroached by means of' hydraulic

mod.els . Theref'ore, it is ap:pa.rent that an analytical study of

bucket- fl0tr phenomenon would be helpful to at least pro-vide a

theoretical guide f'or the mod.el study.

In 1933 and 1945 bucket-fl01.;r characteristics were studied by the

Bureau of Reclamation with the aid of hydraulic mod.els for the

Grand Coulee and Angostura Dam buckets.!/ g/ In 1.953-1954 extem,ive

hydraulic model tests were conducted by the Bureau of' Reclamation's

Hydraulic Laboratory to establish general design proced:ures • .:11

Similar studies were conducted by Lehigh University to d.e:f'ine the

general performance characteristics.!.!

The radial effect on the flOiv through the bucket for bot~ sweepout

jUillJ;)-flow and. submerged jump-fl01v conditions has not been conclusively

established. No doubt the CClllI.Plexity of the behavior of the rolling

body, including its turbulence and. surge, has discouraged the

earnest efforts of many investigators, and perhaps the perspective

on the main issues of the problem has been obstructed by minor

1/Jacob E. Warnock, "Spillway and. Outlets for Grand Coulee Dam," Hydraulic Laboratory Report No. 103, USBR, 1935

gJ "Hydraulic Model Studies on the Spillway for the Angostura ]~i1J':pt," Hydraulic Laboratory Report No. 192, USBR, 1946 , ·

5}.J G. L. Beichley and A. J. Peterka, "The ~Iydraulic Design of Slotted Spill,;.;ra.y Buckets, 11 ?roe. ASCE, Vol. 85, Mo. Hy-10, October 1959

i./M. B. l-!cPherson and. M. H. Karr, "A Study of Bucket-Type Energy Dissipato~ C'ha.ra.cteristics," Pree. ASCE Vol. 83, No. liy-3, Jl.ll1e l957

4

inconsistencies of flow behavior . It is suspected that difficulties

in the :pa.st have been experienced largely due to efforts to merge

the phenomenon into camplex theory, whereas , guidance by the simple

fundamental laws of mechanics and application of empirical formulae

based upon analyses of exper:i.menta.l. data would have :probably yielded

substantial progrE:ss.

In order to gain a more ratio:nnJ. understanding of this phenam,3non, •

a combination of theory and experiments are used in this study.

Notation: The letter symbols adopted for use in this study are

defined where they first appear and are arranged alphabetically in

-)(· Ap:pendi..x: A.

SWEEPOur JUMP-FLOW CONDITION

Review of Theory

The equations of pressure plus momentum are expressions of basic

theories and must be satisfied in the simultaneous algebraic calcu­

lations required for the solution of jump problems. Recently J)ij il

concept of the hydraulic horizontal jump was used for analysis of

the bucket flow where an apron extended horizontally from the

bucket invert.§./ Centrifugal force was not talcen into considera­

tion. This t'YPe of bucket is called a half ·oucket in this analysis .

Ftu~ther developments of solid anrl. slotted buckets have shown that

j}Serge Leliavsky, 11 Irrigation and Hydraulic Def'ign," Chapna.n and Hall., Ltd., London, Volume J., 1955

5

,, I

,. ,, ,,

the centrif'ugal effect on the pressures in the bucket (Figure 2)

may become so pronounced that it must be included in the analysis.

The sweepout jump-flow condition, Figure 1-b, is not a reconnnend.ed

operating condition since it creates much water surface roughness

and is unstable but is analyzed here only as preparation for

analysis of the submerged jump-flow condition.

Definition Sketch

Based on recent model studies!./; ·g; 5a/ :!:.I ~ a. definition sketch for

the bucket flow is presented.

Referring to Figure 1-b, the factors involved in the a.na.lysis of

the sweepout jump-flow conditions are defined as follows:

Sequent depth factor ('IV) j_s defined as the ratio of the seq_uent

depth y4 to the supercritical initial depth, y1

•

Tailwater factor (S1

) is defined as the ratio

(y - y) 4 2

ID Ats~"'D. 01::c.da c.nd Takeshi I~;hibashi, 11Ilyc'lrau:i.ic Model. Tests on Flood Spillway of Sakuma DamJ. 11 Central Research Institute, Electric Power Industry, Japan, June 1, 1956

6

11111

(l.)

(2)

where y2

is the subcritical seg_uent depth of normal free jump

on horizontal floor.

Abrupt factor (Z) is defined as the ratio of abl"U)?t rise, z, to

initial depth, y1

,

Baf'fle factor (k) is defined as the ratio of the total width of

baffle, bv to that of bucket width, b,

Radius factor (R) is defined as the ratio of the initial depth,

For slotted bucket similar to the Angostura S:pill,;ray bucket,

O < k < 1 and Z > o.

For solid bucket similar to the Grande Coulee bucket without

baffle, k = 1 and. Z > o.

For half bucket where a horizontal apr,,n exi;ends horizontally

from the bucket invert, either k or Z = o.

7

(3)

(4)

(~)

·------- ---

Dr ag ]'orces on Bucket Baffles

If' the shear stress along t he horizontal solid boundaries of a

solid bucket between Sections land 4, Figure 1-b, can be '.'leglected

and the mcmentum coefficients , r:,J. and (34

, are assumed as unity,

then by use of force and momentum theory, ·

in which Y is the specific weight of water, q is the discharge

per unit width of bucket, g is the acceleration due to gravity,

Yi~ is the static head correction factor of the bucket due to

grl.

the radial f'low. 'J../ §)

p is the force acting on the baffle unit of the solid bucket and

can be written in the form of the basic drag equation

where Cd is the coefficient of drag, p is the mass density of

1-ra.ter.

7...J Veu Te Cl1ow, 11 Open-Ciia.nnel Hydraulic:., , 11 ..:..959 8 / - D. B. GU!'.1::msky, "Design of Side Walls in Chute and Spill·way, 11

Trn.ns . Af~ Vol. 119, 1951~

8

(6)

(7)

Introducing Equation 7 into Bqua.tion 6

Equation 8 could be reduced to form

Cd=+ { 2 F 2 (1 - }) - y:2 + (l + F~ R)} F Z 1 'l'

l

·where FJ. is the Froude number for the corresponding initiaJ. fl,.:,w v1

condition for the horizontal hydraulic jump, i.e., P1

= ~ J.

Equation 9 is the drag coeff:lcient relationship for solid bucket .

In the case of slotted bucket (Figure 1-c), the drag equation is

written as

P v2 J.

p=Cd·C•k·~liZ

where Cd is the drag coefficient for the corresponding solid

bucket ,;,Ti th abrupt factor Z and C is the spacing parameter for

drag coefficient of slotted bucket. Then, the momentum equation

:for slotted bucket could be written as

9

(8)

(9)

(10)

qy ( q =- - -e \ Y1

g ) .i,..

and simplifying

C = k Cdl F2 Z { 2 F: ( l - i) ... 1jr 2 + l + F 2 R }

Equation 11 is the general expression of pressure plus mOll'~ntum

equation for spill1·ra.y bucket, and could be used in evaluating flow

parameters, such as :sequent depth factor, radius factor, etc .,

2%:Perimenta.l. Studies of Drag Coefficient

Experimental data, taken by the USBR Hydraulic Laboratory ~am

Grand Coulee,.!./ .Angostura,EI a..'11.d "general :perform.a.nee studies..J/

were used in evaluating the d:rag coefficient of the bucket baffles.

Plot of test data shows that the drag coefficient of the bucket

baffle is mainly controlled by dimensionless :para.meters; Z and.

k. The best fitted curves fo:,: different ty:pes of bucket are

shown i n Figure 3-a. From Figure 3-a, it is seen that Cd

decreases as 2. ·1ncreases, and for values of Z larger than 2;5,

the ei'i\"!ct of baffle is not s:Lcr...if'icant. The11 e::'ore, f'or design

purpos..:), a · .z · value of less than ·2 i s rec am:1endea .

10

(11)

(12)

- !: .. • / J

To evalua:ce spacing parameter, C, test data from Angostura mciielsg/

is plotted against baffle f actor, k, in Figure 3-b. It is seen

that C value increases from C = l at k = 100 percent to maximum

of 1.60 at k = 67 percent and then decreases to O at k = o. Tlm.s,

from the vie1-1:point of drag force, it is found that the slotted

bucket of k = 67 percent will give the best effect.

Evaluation of Sequent Denth F2.ctor

Substituting Equation l into Equation 11, using the equation of con­

tinuity, a cubic equation for sequent depth factor in a sweepout

jUlllJ? flow condition can be evaluated,

'ljr3 + { kC Cd F~ Z - (1 + F

2 R) - 2 F~} \jr + 2F~ = 0 (13)

.Y4 A mathematic solution of the sequent depth factor, --- , is possible

Yi. but CO!l11)licated, and. usually introduces some inlagina.ry root. This

maJ.~es the analysis of flow characteristics much more COill]?licated,

yet the solution of the equation shows

'Ir = f ( F1., R, k, and Z)

or simply the sequent depth fe.ctor in the sweepout jump can be

shown as a function of flow Jia.rameter, F 1 , and shape factors R,

k, z. This relationship can be used in solvirg for tail-:-ra.ter derth

in the sweepout for knmm valw.:F of R, K, a.:-.cl Z coi.:ditio1 .. s .

11

Again a factor of

F G=-~===~1======~== Yi{

le C Cd :B'i2

y1 6.Z_ - --­gr1.

-i + ----------

can be ado]?ted into Equation ll and simplifying

This is the general equation for hydraulic jump and from the above

analysis it is apparent that

G = f ( F1 , R, k and Z)

However, a Fortran program is given in Appendix B to simply the

Com.J?utation of Equation l3.

EvaJ.uation of Radius Factor

For knmm tail water condition, Equation 11 can be used in evaluating

the radius factor, however, a tail·water factor, 8i is adopted.

as a tool of comparison, then

l2

(l5)

(17)

• R = Yi = ~ - { kC Cd F 2 Z + '11 2 - 2 F 2 (1 - l) - 1 \

"/ F 1 1 t J 1 1

= ...!_ { k C Cd F 2 Z + (1 + S )2 0

2 - 2 F2 (1 - ( l ) \ - 1} (18)

~ 1. - 1 · 1 l+S1

cp) 1

where

y2 l J 2 cp = - = - (1 + 8 F - 1 ) y1 2 1

and

Equation 18 is the general radius factor equation for the bucket

flow in the slotted-type bucke,t.

For the solid bucket in the sweepout jump-flow condition, Equation 18

can be reduced to

R = : 2 { kCd F~ Z + t 2 - 2 F~c 1 - t) - 1}

l

and. can be expressed as function of F 1

., S1

, 1...nd Z ,

13

(19)

(20)

(2.1)

(22) .

(23)

If' either Z or K = o, as :for the 1'21.f bucket Eq_uation 18 again

can be reduced to

which Tl'!ea.ns that the radius factor can be expressed as a function

Energy Loss

The initial energy at the toe of the jUlllJ? or invert of the bucket

can be written as

= 2 2

y + y F l 1 1

Intro:l.uci:ng R,

(24)

(25)

(2:> )

(27)

The E::!e.r·gy at the end of i l~ 1:9 is

2 v, . 4 +- =

2g ( 1 J_. S 'V ' 1 / J2,.

2g ( l+s) 2 y~

in which v4

is the mean velocity a-;; the ta.ilvra.ter.

The energy los s EL is given by

(28)

E = E - E (29) L 1 4

su:b t · tuti E ... · 26 and "c.8 • .... E ... i rv-. d · l"' .,.._ · ..., S J. Ilg '::J.'UavJ.OnS J.IlvO · y_U,S.v on c:;1 ar.. S11Ilp uy-.LDg,

E r. E = 1-

1

that is

or either

a nd.

E ~ = f' (J\, R, z, k),

E ./' - f' ( S1 , R, z, k) ~

,

15

(30)

(31)

(32)

(33)

Thus f'or the relative loss of energy in various ty:pes of buckets,

Equation 30 has been shown to be a f'un:!tion of Froude number (F1

)

tailwa.ter factor (S1

) and the bucket shape factors (R, z, and. Jr).

II IJ I It is apparent that the enerey dissipating capacity of the bucket

is much more than that of the corresponding horizon~ jump. In

order to figure out the eff-lciency of energy dissipation, an

index n is used.,

C::) n=

b.j

where b.j represents the bucket flow in sweepout condition and f.j

represents the corresponding horizontal jump condition. All the

above-mentioned :flow para.meters can be obtained froru. the Fortran

program in Appendix C.

Curvilinear Effect on Froude Number and Critical Depth

The general specific energy eq_ua.tion at the toe of the jump or

invert of the bucket can be w.t"i tten a.s

~ v2 ~ a2 E=y+y-+-=y.,_·..,-+-gr 2g · g1-y · · 2gy2

The critical state of flc . h- ~ b~en de!'ir..ed ~·.s the state o-r f lew o.t

which ": '"'2 s :oecii'ic

·----------

'3 I enerzy is m:::.nimmn f or a giv<=n disch~~,:-e .·-

:·.J "'""'- · · ·· .,., ~ :,- •·· <·.--,,,...Cl" d "'r T.ra ~'"'e'·~·ii '""7ella ,.,.~ be.;m nc,.,·1"'e"' d .. -s _ -.;..1.:.J. .::>OS.,, ~·--"--- ~ --~...:, C 1, ,;,.:; .. ..:.~--"'.., ·.;;,C .t. I.-... ,_, ..._ ._;;

Fliesszustp.ndes," ( Computation of U:r~er Svx:face with Change of the Flow Type 11

), Springer-Verlag, :Czrlin, 1919, J?P 20 P.nd 52.

theoret i cal criterion for critical flmr nay be developed ·uy rlifrer­

entiatir!8 Equation 26 with respect to y and noting q as a con­

stant thus

dEdy .... = 1 - --9.:.. (1 + ii:\. = 1 gy3 T ) v2 --gy

At the critical state of flow the specific. er!ereni· is

the depth where min.1.!"'llUin energy occurs : :::: 0 L=t

a mir i.mJ.rn, or

Y ::i:-epresents cb

the critical depth for the bucket flow, then y00

must satisfy

the followi:ri.g condition, that is

or

3 q2 q2 0 't i' .. ....,. ,r "'- !:! "cb g T "cb g

The solution of y0

b from Equation 36 can be obtained, but is not

a matter of inter~ . However, the critical depth of the lix:u:i~J,!r

cbannel, y - ::: ·: 3 / q2

, is introduced in Equation 36 to get a better 0 , 1\J g

c omparison, that is

3 y .:.£.

3 y Cb

17

(35)

, (36)

(36')

(37)

Furthermore, the combined effect of gr2.vity and curva·tw..·e of bucket

upon the state of flow, or Froucle number for the bucket car. be

defined as

F = f v 2 ( 1 + "l. )

b \ gy i .

or

when Fb = l, the flow state is corresponding as defined by

Equation 36, and the relationship between the Froude number for

V linear channel, F = Jiy,, and linear channel is

J y ~b F =F t:+-=l b . . . y

y . where ::' varies o~~< l, thus fo1· -"Ghe critical state of t he bucket

11 r flow., the corresponding Froude number of the nonbucket, F:.: ~,

1 will vary from 1 to -,=-. Th i s means that hydraulic jump cm1 be "12

V formed even though the corresponding Froude number, F =--.:::-,

,Jgy

is l e2s t han l.

SUBtlERGED j UUP FLJ . ." CONDITION

~~ferrin~ to ~i~c l-c, t~e ~r ~::.ctors i ....... -" . .·:. in t he

an:tlysis of the subme~:c;,::!J. J1..1:1y-:::lo1.r condition are d .of':i ned

18

(39)

c:.s follows :

Bac~edup depth factcr (lib ) is defined as the ratio of

the depth of flow il:. the buc1:et above the in""rer'i;, y , 3

to the supercriticaJ. Qepth, Yi,

JI,, =

Submerged factor (S~) is defined as the ratio

s = 2

.:::.

Tailwater factor (s) is defined as the ratio

s = 3

,:, ~· {y - y .)

2 5

Surge factor (H) is defined as the ratio s

H = s

where y6

is the average surge peak depth .

T:"!e ::i.:bm.erged hydraulic Ju"::9 in a. bud:.c-c is a :3y::-:.te:;1 of two dis-

-cinct stre9JnS; one, the ~:::!. .:.~·.'"' : -- ~1 st;~ · -,1 C-:!Cl'.\:.rir,:; t:be lower portion

19

(4o)

of the jum:9 body, and the ot::K:r a rotating mixtui·e of' wakr and air

supported both in motion and position by forces imparted to it by

the princi:pa.l stream. By the presentation of J. H. Dorrm.a.,Y2./ the

depth of the princi:pa.l st1·e2.1n in a bucket invert, y 1 , can be shown q

as a f\mction of a dimensionless :para.meter, r::-7' (Appendix n.) r

1 ,.J2gh

Thus the radius factor, R, can be easily determined for i.:he

submerged jump-flow condition.

Then refer to Figure l~c, cut free body tlu·ough bucket invert, the

forces J?lUS momentum equation can be written as:

(

y v2 1 1. 1 z :2Y~ + er,

~

Substituting the equation of continuity and simplifying,

u = y 3 = J ,1)2 + k C Cd Z F2 R - 2 F 2 (1 - J:_) .u.. y 'l' l l ,•rt. I:) l. 'f

where y

(1 + S ) (1 + S ) ..3. 2 1 y

l

y = (1 + S ) 3, = ~

2., (1 -:· 8 ) ( .,_. :. ·.- 8 F ~ - 1) (1 + S . .)

3 Yi ' l "'

- - ·-------:v::i/ Di.::-."· ·s s ed by J . I-I. Do11i"!'.a , ":Jes i Gt. "':° Side Hall:;; in G'nut€s c.nd C:r,iJ..:.. ~. :.y.:; , 11 Trans . ASCE, Vol. 119; 19:;:.-.-

20

(44)

(45)

(46)

Hb can be shown to be functi on of Fi, S2

and shape fact ors,

R, k, z, that is

for a solid bucket

and for a half bucket,

To simplfy the computation, all flow r,arameters can be compared

with the horizontal jump, and the third tailwater factor S3

is

used in evaluation of the backedup depth factor, then Equations 47

to 49 can be expressed as

lilJ,i

(47)

(48)

(49)

(50)

Further more, the depth of the principal stream in a bucket invert,~/

y1 , can be approximately determined, and. backedup depth selected.

In5Uch a case, determination of radius factor, R, becO!ues necessary.

Then, by use of Equation 44, the radius factor, R, can b1:: expressed

as a function of Fl, S3

, Hb and shape factors k, z. That is

R = 2:_ ir· v; •2 + kC Cd Z !~ - rf:- - 2 F 2(\l - .1.)I'· }'2"" 0 J. \~ f •·

l

ill ' f II P . M. Stepanov, 0 :pod:to:plen:2c:;-:i pr:,·::;~;l~e vody, }idrotekl:.nika i 1-ielioratsiya, Vol. X, No . 1, 1958, J?? '.-;-'..,i3

21

(51)

or

When H. = l(y. = y,), Equation 51 can be reduced to Equation 18 . .:, 3 -

This means that the sweepout cc::iition is a special case of the

submerged condition. Eg_u.a:tion 2:.4 can be used in evaluating drae

coefficient for spillway buckets for subr.1e.,..3ed ju:mp-:f:i...m;- c0r.di tio"l .

The resulting equation is

1 {-2 •:, .... 2( 1"} c Cd = k. ~z .1!'1 R - 'l 1 ·- -;. lib + 2 F 1. , 1 - F )

Numerical values can be obtained from e:qx:!rimental data. Sowever,

due to air entraining facili t:r of the bucket roller and unstable

characteristics of water surface, it is reasonable to assume that

the drag coefficient for submerged jump-flow condition ,;.Till r emain

the sa.n:.e as for corresponding sweepout jump-flow condition.

In era.el' to V'e't'iry the a.ssu:op,;ion, e)...-pe:d.me:nta.l backup de)?th factors

taken by the USBR hydraulic laboratory;3/ are plotted against c0t1puted

values using Equation 45 and clrag coefficient values obtain from

swee:pout jump-flow condition. Because of air entraining facility

of the bucket roller, the experimental backup factor seems to be

higher than the computed value, Figure 4.

The best fit relationship, Figure 4, may b:: E:xp:::essed as

~ = R.. + 1.5 -~'exp) 0 (the)

22

(52)

(53)

(54)

This correlation is seen to be goo1 . A correctio~ factor of 1.5

takes care of the simplifications made in theoretic2.l analysis.

Hence, the theoretical Equations 45 ar.d 51 could be used to compute

the relationship between R, H0

, Fl, and 83

, ar.d a Fol"train progrrun

is given in Appendix E to simplify the cc..,1putation using thE:: prozram,

the required bucket radius :for any particular :'lcw cc>."' be cbt'-dned

and the backedup depth of mw special flmr conclition cr.n be easily

predicted. This infonr.a.tion is not important in hyd:raulic design

of buckets but also important phase in structure design of the walls

in chutes and spillways.~

Evt:i~luation o:f Surge Fe.cto:i_,:

Approxim2:ce surge-height characteristics obtained. by the USBR hydrau­

lic laborato~ are shown in Figure 5. The best fit cu.rye shows

that the equation of

l)[J/

can be f itted. This information is helpful in setting top eleva­

tions for training walls.

~!)(1-~;h of the Submerged J1;'111J;1 in Bucket

As tested by the USBR hydraulic laboratori.fl the hydra-ulic behavior

of -C!'le submerged bucket dis!; i:xi:cor is rr.9.nifested prirr::i.:.~ily by the

:for..-:,-2:cion of two rollers; one is on the snr:.:'ace m,Jvil 3 c0unter-

cloc::-::1.se and is contained. 1-:':i. tr..i:n the r03ior: above tl1e curved ·ouc~i:.et,

23

(55)

and t:he other is a ground. ::.·ollc:.~ :m: .. }\'.i-:.£_; a. sur.::;e moving in a. clock-

wise direction and is s:I:c.,.::::::~c:.. J.m-r..:;:;·c-1"c2.lli from the bucket. To i'ind

the length of the su-bm.erge.l j ....-:.,:p , the bucket invert was taken as

the ori.::;in, and the end. of the jum:p 1:-as taken to be the end of' the

said roller and surge, where t:1e dep·ch bec"lllle equal to the tailwater

depth, y5

•

is the length of ·;;ne sub::aergcd jump, the variation . of

L b. s.j.

with backedu:p depth fa.ctor, Hb, ta.ken 'by the US·3R hydraulic

laborator?f is shown in Figure 6a. It ·was found tilat the th~oretical

backedup factors could be satisfied. by the linear equation

= 2.47 - 0.035 a 1' (the .)

with a correlation coefficient of 0.87.

For experimental backedup factors, the same relation..,hip as given

by Equation 54 can be adopted.

If' Equation 45 is substituted. into Eq_u.ation 56, jtm1p length factor,

~ff_s.J. cai."1 be shown as a f'Lmction of ~, S

2, or S

3, and shape

factors R, z, k. That is

I

1 /y b.s.jyw5

= 2.47 - o. o-:5 / ., \, •;

24-

+ kC OJ. Z F2 1

Fe. R l

1 , - 'I ..._ - 1.'•I) ,, (57)

.-

or

L b.s.1.

For E. = l, that. is, when flo,7 begins to sueepout, ;,;.,,..;... w.:.ll o Ys L.

be as sr."1 ·r as 2.44 and after -~ > 1, 1' •

5 • J • will be decreo.se::1.,

Ys gradually. A compz.rison of t :-.e juml;l leri-3ths on the d~:f:22.cent types

of ener5-y dissipatorsW shews that the bucket energy d.:issipators

are the shortest.

The ler..gth f'rcm bucket invert to the average surge peak is taken L

as the length of the surge, L5

, and the relationship of -2. to Ys

Hb is shmm in Figure 6-b. It was found that all the best :?it

cu..."l"Ve t:..rough the points couJ.d be satisfied by the eg,u.ation

"Ir "5

= l.47 - 0.019 Rb

"ititr: , cc,:trelation coei'ficie:nt of o.8~-.

(59)

Co:nparison of equations 56 and 59 shovis that the length betvrnen average sur(;e peak and end of j1.1r.1p has a va1~iation of'

· ... , - L ) ... l {,Sc)

Ys - ,., ,-,...-~~~~~~~~~

;::.;.,::J r:liydrG.clic Design of s::;iJ.J.i:---:.3 E?.f,:'.. · ,~ and Other Enere;:w :Jh,si.pators, " E:r...giJ.c.eering £.!onogr,1.phs No. ~:j, US13R, _,;53

Due to the bucket action, t:1e l e:::;th o::., t.he roller ,::::.s ~:o-~ so

pronunced as in the horizor..".;:,.l. hydrz.ulic jump. T'.r..e relatior!ship

between -c11e ratio o-Z the roller length., "ii,.., to tailwa+.8r dr-:pth, ...

y5 , and backedup de;th factor, I{._, , is shmm in Figure 6-c. The

correlation is no"G e-ocd and a. 1rariat:i.on of o. 'tc, 0 .Li is obtained .

Energy Loss

Referring to Figure 1-c, the energy at the invert. section of the

bucket can be written as

or

The

or

E J.

E :,_

::y 3

;:; -= 0 Yi

2 v· + ...2..

2g

F2 J.

+ 2

y1 v~ +----

-i· 2

Fl R

energy at end of jump is

v2 ]'2 3

E 5 • .:t Yi

- y + 2g = Ys ;- ~ :"....,.. ......

5 5 ?y,,2 .. ~ . 5

E:;-: Ys p2,.,2 r;,2 , ~ .. , -· .... -- = + = \\r' ..1.. . : .._

Yi Yi 2 2 f ,- 12..

Y5 ,_.,

26

(61)

(62)

(63)

(64)

Then the ener[;'J loss

= E -l. R

5

E is .:;:L .. .r2n by L

Substitutin3 Equations 61 ar..:l 63 into Ec1_ur'.tion 65, simplifyins

Again it is shrnm that

1i1 '1"'

;: = f' ( F.l, S3

, R, Hb, Z, k)

and when R;. = 1, Eq\19.tion 66 can be reduced to Equation 30, as 0

stated above, similar conclusion can be made for the relationship

between sweepout and submerged jump condition in bucket flow.

Verification of Equation 66 is obtained by using experimental

losses computed from data ta1':en by the USBR hydraulic l a.borator~

ce:mpo.::..~ca. with theoretical lossed cCY...aputed. usir,g Equations 4'.} and 66

1rlth Cd value obtained from Figure 3. '.Ine results of the c0:npari­

son are shmm in Figure 7. The correlation is seen to be good.

(65).

(66)

(67)

The be..cl:edup depth f actor is 2. elei";18;.::c i"" th:! enerrsy J..oss Eq1,.ation 66,

yet, t'.:1.:: experble:nt results sbo~r -~b.at -~.l1e ~raria.tion due tu error in

27

energy loss and the rel2..tive loss curv2 _, ?:1::.,-u:-·e 7 shows a bccter

co:crela:tion t:ha.n that of the -,,.J.::!::sdup d'°'}.Ych factor cui-,.re, Fi3ure 4 .

Through uce of Equation 66, ·c:,2 e~::.~r ~r c:.is0ip::cing capacity of the

subr.ie:;.·ged bucket is seen to be ..::0:-::e or lec:s than that of the cor-

responding free jlunp on horizont"--..1 floor, dcp2~ding upcn. the par-

ticula.r vclue of Ti' -1. '

R: z, 1~ . /l.,'.;G.ill th8

index of' 1·elative dissipation, Equation 54 is used for E:V2J.uat:Lon

of dissi~~tion efficiency. For any given Fl, k, Z, and S, n could

be more or less than unity, de:pen::ling rn.::. R and S • ~

By differentiating Equation 31,. with respect to either S3

, F1

, or

R, and trea.ting the resulting equation eg_ual too, the~ a naximum

value o-S: n, for any particular flow condition can be theoretic3.lly

obtai r.ed . However, the resultiri..g eg_u.a:cion is a complicated .,:"'\.me-

tion of F1 , S, Rand shape factors k, z. A s a-inple ccm~utor com­

putation shows that the eg_v.at:i.on is an increasing function a."t1d

becomes i~linity as ~ incre:ases .

In the :preceeding consideratic:i of the energy loss, it should. be

always be ranembered that the co:n:puted ener5y loss in the submerged

Cc.':lr,::.risc:.1 o:: the energy c1issi:- ···· _ :x1 c:: - - . .:::Lty is co~c-t:::: .re~:· 1 01-,er

28

than th:1t of the sweepout cm::l:i.·::.ion. :i:":::..~'. ·c:·1ece co::::,::.c'.ero.tiorr...,

it is co:-icludcd that the suli:7!e::'ged jtT.:.:? cc:.:.:.'..C:. not .. oe }.,:.·cfc : .. ·--r2d. to

unl2s:c:, ·,·he ,IJ'

backeu.:u:_) .:actor is less th2n 2,::,11roxin:2.·~2ly 20 p::::.·c'.:!nt of the tail-

by USJ3R a..-.d the test results civc:.1 by i:cPherson ar.a.

Eence, the equations men~cio:t12d above c:::...'1 ".Je 3.do:pterl in buc!~e:ts

design, graphical or tabula:r.· re:pr-cscr.t2.·i;io..1 of' 0q_un.tions is :possi:OJ.e

but due to too :many variables included in t.he al.1E' lydis, ar1c1 in viei:r

oi' t l:.e E:xtensive and ra.pidly grmling use of digi,A:.. cumpute1·s.,

such -~[:.-Oles or graphs would not be of' ,;.ridespread interest. In::;tead,

a For·c:·c.n :9rogram was ·written for use on an IBM 7090 and is sho~m.

-;-y---. "rt7~---·siur1 0:1. E!.12r.:;3r Diss:i ... -:,J·l:c(-.. ~ :::::d ?o~ . ··; ::?', .. L.).::.ic.::.:. c:i.0~1 i."o . '{v,

29

" .~~ .. - I \-"':,:: "a. o::::_ I~.---,-­.{ .... ·--~:c.., .trL,eu~· ..: J_>__,··-

·----··-------

f

I

-...-···- .. ---------- ------~

'·· "

q.1 I J

I . ' ' \--,......:.."";!\ _:_ I ~

? r::.: rO" ' --: ', - ' ,,..i ' ' ••

A. Ffip Bu.cke.t oY-- Jet Flow Coy,dd-i'o r:.

--t-

c. nc, (k ~- r~,uc. l<eJ o, Su_t)}1')12. 11J Cd

f I ov,J Co·nd 1·+1· c1)1 .

J1.,l W\:f) 'I I

}4

t

® I l 1· .

cl·5" I . ± ' I

' i I ! I

' l-

"" 0.. >-I-0 I-0 a: ... I I-11.J . 11.J . ... z

.. ,. z 0 .. · ' .-:. . ;: c( >

-~' 11.J ..J

, , . . i la.I

f-\q-u RE 2.. J~ S.P o RT HyD.

1010 r----,-=-----,r""'----.-----~-----..------.

PIIOTIITTP[ Dl5CHA.IIK l,000.000 cla.

-..... __

1000 i-----+-----1------l-

oao

soo,ooocf

0 to 40

- ......... _._..111&· 11001 L ----i.•o- •

', ., ,,

10 10

DISTANCE IN FE: T • PROTOTYPE

1 N -,H c

' ·11

fi(C s~uRES ·err ANO (OLJLEE., SPILL\yA ' ( ·

B UGl~C:T

IOO llO

•

I I

I

, I :, I I i:

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r I

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d· I, • '•,

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l l• ••

··r:, .: 'f. '

r ·'· f .. ',, .L : . ,. !;· l /, • • ' 1

~ : '~ 1 ;' .,· ' i, I

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:1.-' '

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..

0,'.2.. ... ---- ...

o, I a.S 0.4-

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3 HYD

(" l (

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\

·---"---~-- -----------···----==-<-i

L5

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;- · \ ,•-, ''

2C

, I

fl GrU)( E REPORT

4 rlYD .

30

10

£ ~ 15

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co 6

s

+ l-l-r,! ..JT7J.. ' '.

-1...,-1- ' -+-LI 1--i

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... _2

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(c).

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1,-~

(B).

-1-1 +t I HTtt-1-tti-t ffl+'.! ++ - :tt - =!=++t " ' -l-h-,-44-1-+4-++++++++++'+++++++++H

+ IT -- H+ '" -- -H- ,

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15 '.W ? '-- J 2<l 35

Depi·h f o..e,to ,~ ', 11 11 l

(A) .

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~) j 1 11:'s10-:-t.J, Exreyi,11C111r ... /

Ji:;.,·ht... t i 2'' ·s \ dl. ri_

1 The.ov,~ i icr._/

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ctd·O~ I

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---Tl,•" I' •jC·\ 1' Uv \ \-!b E i-1...Lct .\,· esn ,

- - - - E'x fe.y,\1K\"\l-t:1'-\ Hi E-tt-°' ·h rS'r, ,

LE~NG TH J LJ [ '/ p

PA e f\ rv(E:TC r~s foR ~ u B Ht _R. c:..,· r~ 1) F L()W (C1NDl T! Cl\{ .

-!: (. -

.,') ,fl 0 _,

d -::i

-i= d

J: ,l

-1-;.-. r.l f" r--/j ~ -'Y,

,· )...!._ i

'r~ f SI 6tt ed. b~1_c. k ,,;;­

; ri'Slotid buc.k.c.+

J .. . ·,

A Cr,G\..n~ Couf{) c.::. :: ohd b1 ..... d<-eX, I .

£~>( PER l M Ef\\ TA L vc· i\ / F ! (j\/ I ct~(

SQ LJ A 1\ 0 \\JS. FOK~ '.1C\'···12 .. RGr~-D

b

b t

C

Cd

E

. ,.; .~ .• :1..

bucket width

wia.th of bc.ffle i:n 01.:cket

s:pacin~ :93,rc.:neter

drag coefficie~1·c

specific ene1·sy

El, E , E speci:fic energy e:c various sections 2 3

E specific energy lE:ss L

F l

F b

i.'

g

G

II

H s

L b.s.j.

L r

L s

n

p r

"O

d

p u

Js:oude number for line::i.r ch::nn,~l

Rroud.e number for b": .. lcket

i'u..11.ction

2.cceleration due to gravity

difference between reservoir elevation a.110. ta..ilwa.ter elevation

backedup de:pth fae:tor

surc;e factor

l ength of submergr:,d jump in bucket flow

~d;.gth- of roller in bucket

len.:,r.rth of sm·ge in bucket

relative efficiency of ener0y dissipation

con..i:ponent of roller weight :.,c·ci~,s dmmstre3.fil on the princip3.l strcCL.'11 of bucb::;t

floor reaction

floor reaction

30

q

R

r

s

s 1,2, 3

V

X

y

r

cp

p

--- - --------- - - - -· --------------------- ----

discharge intensity

radius factor

radius

radius at various sections

submerged factor or tailwa.ter factor

submerged factor o~ ta.ilwa.ter factor for various flows

mean velocity

mean velocity at different sections

longitudinal distance

de:ptn of flow

critical depth of linear channel

critical depth of bucket

depth of flow at different sections

specific weight of water

sequent depth ratio for free jump

sequent depth factor for bucket jump

momentum coefficient

density of water

31

16 04 FOR TRAN CODI G FORM ij NAME C. 1Y} . W..uJ . . " . PROGRAMAPPE1IDIX -B SEQUENT DEPI'H F~R BUCKETS PAGE V, ROUTINE DATE

T STATEMENT C SERIAL y 0 .

p NUMBER N FORTRAN STATEMENT NUMBER E T. 1 2 5 6 7 ' 72 73

C I I I I PR MR fl u Fa.R S'F.OTW.t l'T' . n~P'T'P I I I I I I

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I CK="BAFFLE FACT(l)R I I I I ' . C

I ' I H -ABRUPI' FACT¢R I I I I I I

(' C :::SPECI NG PARAMETER FR¢M FIG 8 I I I I I I

C I CD=DRAG CQEFFICIENT FR~M FIG 3 I I I I

C EO UAT i hlN - ~n-;i-:n Ja r r•v-i!{;*r.n>,ff~~ *H _ ( 1 () ...1. v * y* y ) '"' ., ('\ ·"'· v ."-V' ,,e n _, ') f'\ .>.1-V.>N I I I ' I ' ' I , --- - -

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(( - Oo5) * A2 + SO.RTF (( A2*A2/4.0) + (Al*Al*Al/2700)))-~~ 00333 A3 :: I I I I I I I '

I I I A4 = ( ( - 0 .5)-i~A2 - SQRTF ( (A2*A2/4oO ) + (Al*A~~1'.Ji27 oO)) ).;P',.-()0333 I I I I I I

I I I SD =A3 t A4

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I ,6 O(}l T~ 2

I I I I I

Fi'.Nrl . I I I I I I I I I

1604 FORTRAN. ·coDING FORM @ . . NAME e. m.w.v-'

PROGRAM APPENDIX C RADIUS FACT(/)R F<llR SWEEP<DUT BUCKRT. PAGE (/2

ROUTINE ' DATE \

T STATEMENT

C . SERIAL y 0 p NUMBER N FORTRAN STATEMENT ~=yMBER E T. 1 2 5 6 7 ) 72 73 80

C I I I FLQ)Vi PARAMETERS FqJR SVIEEP<l)UT BUCKET I I I I I I I

,1 F¢)RMAT ( 6 FB.h ) ' I I I I I I I I I

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C I

Y = FR())UDE N()> I I I I I

C I I I SD =TATIWATER / INITIAL DEfTH I . I I I I 1"- I

C CK =-BAFFLE FACT(/)R I I I I I I I I I I

C I I I H = ABRUPI' FACTdJR. I I I I I I I

C I

C ::- ·SPACING FACT¢R FR¢M FIG 3 I I I I

C I I CD ::: DR.AG C©EFFI CIENT FR.<DM FTn. 1 I I I I I

C X ~ RADIUS FACT¢R I I I I I I I I I I

C I I I ER =RELATIVE ENERGY L(/)S& I I I I I I I

C I

A = EFFICIENCY I I I I I I I

I I I 0 X = ( I. o / Y tY) ~ (SD t SD - 2. 0 ~ Y .iJ' Y ~ ( 1.0 - 1.·() /s n l - I. 0 t C. tC..D 1t Y ~ I I I I I I I

I I I 1 Y~H '*' e, K "') I I I I I I I

I I I EL~ 1.0 t- Y~Y t X + 0.5 ~Y*°Y-SD-(cJ.5*Yi:'Y)/(~DtSD') I I I I I I I

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I I r R=- ELIE I I I I I I I I

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I I I I t(2.otYtY;J I I I ' I

I I I A::. ER /E 2 I I I I I I I

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RA FFLF I

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ROUTINE I

SwEEf <{)UT f3U4ET DATE T C .

SERIAL y STATEMENT 0 p NUMBER N FORTRAN STATEMENT NUMfsER E T. 1 2 5 6 7 72 73 80

I I I W RTTF tbt IT PUT TA'PE ~ 4- X Y sn C. k'_ ,-1 F= S2 A I I I I I I

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I s x E 14.5 ~.J.0,5) ,I

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I I I I I I I I I I

I I I I I I I I I

APPENDIX D

D.e.t.e.rmination of Depth on the Princi:pa.l Stream in a Bucket Invert

According to previous model studies of flow in the spillwey buckets

of Grand Coulee, Pine Flat, and Popolopen Dam, J. H. Douma assumed

that the velocity distribution on the bucket would be th.at of an

irrotational. vortex; V = ! , in which A is a constant and r is r

the radius of any streamline. For this flow pattern, all the

streamlines are circular and concentric with the buckets. This

assumption has also been adopted for determining the depth of the

principal stream in a bucket invert.

FrOlll' ··nouma ' s streamline analysis, the following relationship was , .I /

obtained.:J, ·

r log ( ll. 1 = ---9 -2 r J°:?,..·H 2 I - --e

,.ffier.e r 1

is the radius of the bucket;

r is the radius of' the streamline in the principal stream 2 of bucket;

H is the difference between reservoir elevation and tail­water elevation.

Dividing both sides of Equation 1 by r 1 , a dimensionless equation

can be obtained, that is

~/Discussed by J. H. Douma, :,.oesign of Side Walls in Chutes and Spillways," Trans. ASCE, Vol. ll9, 1954-

32

(1}

(2)

.. .,.':.. ----...n:7

r or _g can be expressed as function of a dimensionless :parameter

r1

q

where r i, q, H are all known values.

The depth of the princi:pa.l stream in a bucket invert, Yi, can be

expressed in terms of r 1. and r2

, that is

Yi Again using a dimensionless parameter of and in terms of

. ·1 r i

:f.' ( q ) , a simple solution of y1 can be obtained by

ri·.J 2g'ti

using the relationship of

( Yi) ( ) - = f q '· :17 i r J 2gli

i

The result of 'Rqua.tion 5 is shown in Figure~

33

(4)

(5) .

~ I

/,C - - :.::i::: 1- >- -- -- ... ,__

'

O·l

0 ( ~ I I

I I

)

I

II 0,4-I

I

03

:;:;

0 . 2 V f

• I/

1,-~ ....

I ~

o. ,I-'

0 ~

0 O, I

I I

~

..... "' 1-.

" ' I -

/ J

,

-'

I/ I/

V I , -

• I\

I

'-1--

f\C,UR E / . APP ~ N'DI X D.

0

0 . 1

0 . 2.

0 ·3

o.~ :-:'i µ

. " C. ..:1

0-·..:i

o.(,

o.,

(.()

0.4

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1 2 5 6 7

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...

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ROUTINE DATE lb A 'b~ ' , T

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..

:c.

lt-----t--"-~-t---t---------------t---'___.___.___.___.__.___._ l...____.._.._______._ _______________________ --1-_.___.._---'---',.___,__~

OPTIONAL FORM NO. 10

UNITED STATES GOVER'NMENT

Memorandum Merorandum Denver, Colorado TO Chief, Hydraulics Branch DATE: September 27, 1963

FROM C. M. Wu

sunJECT: Model study of vertical stilling wells

Flow characteristics of vertical stilling wells have been restudied and the original draft, which will be Appendix 2 in my final report to the U.N. and was submitted to you on August 12, 1963, is rewritten.

A theoretical analysis and experirr:ental data show the problem can be classified into two categories: (1) well georr:etry and (2) fillet effect on wave action. Some preliminary conclusions and recommenda­tions are xrentioned in the draft.

..

Appendix 2

MClDEL aruDY OF VERrICAL &!'ILLmG WELL

Introduction

A vertical stilling well was considered. to be the most econcmical structure to dissipate kinetic energy of the high velocity flow, yet, there was little information regarding the dimensions required. for satisfactory operation. Recently, an equation for determining the , basin dimensions was developed. by using model tests data of' Wanshi~ and Soap Lake Siphon stilling well.~ However, the resulting equa­tion was unbalanced. in dimensions. Therefore, the original data were reanalyzed in this study.

Parameter Analysis

A definition sketch . is shown in Figure 1, Appendix 2. The variables involved. in the problem can be classified. into two categories: well geometry and shape factors.

As well geometry is concerned., previous investigationJr.l, 2/ had shown that well performance was sensitive to a number of variables. Further studies show t~t the effectiveness of fillets and their exact location, size, etc., is a matter of' shape factor and is investigated separately in this analysis.

For the overall well geometry and flow phenomenon in question, with effects of boundary friction and surface tension considered. negligible,

where

d = depth in well;

AP = area of pipe;

Aw = area of well;

Q = discharge;

H = total head at pipe outlet= ~/2g + p;

!./"Hydraulic Model studies of the stilling Well for the Blow-off Structure, Soap Lake Siphon, Columbia Basin Project," by D. Colgate, USBR Hydraulic Laboratory Report No. Hyd-277 2111Hydraulic Model studies of the Wanship Dam Vertical Stilling Wells," USBR Hydraulic Laboratory Report No. Hyd-481, 1962

VP = velocity in the pipe;

vw = velocity in the well.;

z = height of pipe outlet;

p = density of water;

p = static head of pipe;

1 = unit weight of water

in which several elements are not important and are neglected in the analysis.

One of the several combinations of dimensionless parameters which satisfy the above equation in determination of well dimensions is

d (J\ v: z ii= f ~, 2gH' ii, shape factor) . (2)

The parameter v2 /2g H can be equated to the function of the ratio of static head to velocity head, but in design computation it is moved directly usable as it stands.

All. the parameters mentioned above can be checked by applying force plus momentum equation in free body diagram as shown on Figure 2, Appendix 2.

Under certain conditions, if the slope of the -water surface is near horizontal and the effects of vertical well. boundary friction and surface tension can be neglected, then:

+ r i (v +v ) = o g p w (3)

where k1 and k are correction factors of static head due to the shape factor !or the pipe outlet and well. floor, respectively. This can be simplified into

2

~c AP~ +- J.+- =O g ~

(4)

or

Dividing both sides by total head, H, to make the equation dimension­less, . then

or

(6)

where k represents the shape~· factor.

Investigation

Previously constructed models and test data are used in this study. The model., a box 4 by 4 by 6 foot deep with 4-1/2-inch pipe was test~ with discharges between 1.5 and 7.13 cfs, velocities of 9.0 :.. 64.5 ft/sec, and total heads of 7.5 - 71.0 ft.

Using the parameters previously analyzed, the test results are presented in Figure 3, Appendix 2, which is a plot of V2-/2g H versus d/H and Z/H.

3

The equations

E;. 1.25(1 - L) H 2gH

16

are found to be good fits for the experimental data. This curve can be used in sizing other installations of

16

( ..2..)2 .! . 24 4

Further Investigation

The present study has investigated the relationship between flow conditions and one pipe-well area ratio, i.e.,

~ 16 ~ ~ _( ___ i....,)""""2 -l

only. Perhaps a family of design charts for different pipe-well area ratios is necessary for econCIIlical design of the vertical stilling well.

From what ba.s been investigated already, it is believed that as the well shape, i.e., fillet configuration, is concerned, no significant effects on energy dissipation and flow conditions are mentioned. By

4

placing fillets in favorable locations, rollers which contain the turbulence in the lower part of the well and reduce the wave action, are formed. Further studies may simplify design criteria by dividing the study into two categories, well geometry and flow condition, fillet parameters and wave height. Perhaps a correlation of the fillet parameters against significant wave heigh~ will result in the best vertical stilling well energy dissipator.

~ nwave Action Below Spillwra.ys," by Mohamed Mokhles Abou-Seida, Proc. ASCE Vol. 89, No. Hy 3, May 1963, Part 1, PP 133-152

5

. . .,.

I I I

I

I

I

( \

"

---

J

/

/

...... /

/ /

-r-. I

I .

r .I·

- ~ -- J_

f\Cq-URE I

/

/

I

/ ,, Fi !let

( )

___ L _____ _

DEFINITION SKETCH OF Vl=.R1t CAL­ST/ LL/NG,- 'w ELL BASIN

. .,,, '

-~,; , J

' 1 _ - ---

I j

I

F\~URF-2

l 1v; I . .

( .

\ \

\

fREE BODY DIAqR~M VERTICAL ST!LLfN4 WELL BA~tN

'.

\

t . I

J/H

o o.t o,2. 0.3 o.4 o.5 o. r:. o.1 09 o.C) 1.o /.0 . err

T ·r

o.1

Z/H

- --: -I

... -1 -~T .

0 . :1..

Vf: RT/ CAL 5TI LL/~/q WF LLS DESIGrN C/-/ART

FI C(.JR r:. 3 /,. r-: .... ro1 ·' - . r'r . , I·• A .,.

-------------

APPENDIX -- 3

..

Appendix 3

THEOREI'ICAL APPROACH OF MOVABLE-BED SIMILARITY

Introduction

Recently, the similarity conditions for distorted river models with movable bed were derived by H. A. Einstein, M. ASCE, and Ning Chien, A.M. ASCE.U In this paper same approach is ad.opted and further study is ma.de.

Similarity Conditions

.w Einstein expressed the relationship of velocity and friction as

where V is the horizontal flow velocity;

RT is the hydraulic radius of the total; section with the bottcm width as wetted perimenter;

is the grain size of the bed representative for its grain roughness;

g is the gravitational acceleration;

S8 is the slope,

C is the constant in the generalized Manning's equation; and

m the exponent in the generalized Manning's equation,

with the additional of Froude Law

v2 2g + h = constant

where h is the water depth.

(1)

(2)

J1 *"Similarity of Distorted River Models with Movable Bed," by Einstein and N. Chien, Proc. ASCE Vol. 80, Separate No. 566, December 1954

LI

His bedload :function is

and

ps . - Pr D y ! ( log 10.6 ) V* = p R1 S ~ e log 10.6 X/6

~ b e (4)

where ~* is the intensity of transport for individual grain size;

~* is the intensity of shear for individual grain size;

ib is the fraction of bed material in a given grain size range;

iB is the fraction of bedload in a given grain size range;

D is the grain size;

Pr is the density of the fluid;

p5

is the density of the solids;

~ is the hydraulic radius with respect to the grain;

qB is the bedload rate in weight under water per unit of time and width;

Y is the pressure correction in transition smooth-rough;

f is the hiding factor of grains in a mixture;

X is the characteristics grain size of a mixture; and

6 is the apparent roughness diameter.

He used the vertical distribution of suspended load equation

z

h~a)

where c7

and Ca are the concentrations of a grain size with setting velocity w at Points y and a frcm the bed, respectively, and

K and Z are the constant and exponent for the distri­bution equation.

2

'

The control factor in the distribution equation is

where w is the settling velocity of a sediment particle;

k is the von Karman universal constant of turbulent exchange,

and the continuity equation

where A denotes to the area of a cross section;

t denotes to time;

x denotes to distanc~; and

Q the discharge,

and continuity equation for sediment discharge

where ot denotes to the total-Load rate;

15

denotes to the unit weight of sediment; and

Z denotes to the depth change of river bed.

(5)

(6)

(7)

Einstein arrived at the solutions of the above similarity equations by choosing one of the following ratios: (1) horizontal ratio, (2) vertical ratio, and (3) density ratio. A simultaneous solution of a combination of two of the above ratios were obtained and the result is shown in Table l, Appendix 3.

3

.. . Table l--Appendix 3

Mocl.el Ratios for Open Channel Flows with Sediment Motion

Choose hr and Lr Choose Lr and (p5

-p.f')r Scale Ratio Symbol

Lr hr Cr B 6.V ~ 6S Lr (ps-pt)r hr B 6.V &fl'

Horizontal Lengths Lr Chosen Chosen

1 2m 2 1 -2 -1 Vertical Lengths hr Chosen 2m-l 2m-l 2in='r ~

l/2 1/2 _m._ ~ 1 2(m-1) -1/2 Velocity Vr 1 2m-l 2m-l 2{2m-1) 2m-l

m 2 l ___:g__ Slope Sr -1 l l 2m-l 2m-l 2m-l 2m-l

=1 2m+l 1 1 -1 1 2m+l 4 2 -4 -1 Sediment Size Dr 2in - 2m - 2m 1 ~ 2m-l 2m-l 2iii::i 2m m m

Sediment density under ( ) l-2m 2m-l =1 =1 1 2m-l Chosen ps-p:t' r 2in 2in - ~ 'Water m 2m m

Bedload ratio per unit .:2 3/2 3/2 ~ 6 --2- -6

-3/2 width and time by weight qBr 2 2m-l 2m-l 2m-l 2m-l under water

Total load rate per unit .:2. 3/2 3/2 ~ 6 l .....L -6 -3/2 width and time by weight 3 l 2m-l -q'tr - 2m-l 2m-l 2m-l 2

under water

-l -l 1/2 .....:!!L -=L 1 -g(m-1} 1/2 Hydraulic Time tr1

1 - 2m+l 2m-l 2{2m-1) 2m-1 2

2!1!!:1. -(2m+l) -l -1 ! -(:1 l 1/2 -(2m+1) ....±. -1. -2 4 1/2 Sedimentation Time tr - -l 2m 2m-l. 2m-1 2ni-I 2m-1 2 2m 2m m

)

HYDRAULICS OF VERrICAL srILLING WELL

Purpose

To investigate the flow characteristics and energy dissipation capacities of the vertical stilling we1l to provide the design performance of the basin as a high velocity energy dissipater.

Conclusions

1. Theoretical analysis and study of experimental data show that the manentum theorum can be applied in the vertical stilling we1l analysis.

2. A theoretical equation is developed to relate the well depth to known flow parameters. Also a theoretical equation is derived for the energy loss for the basin and experimental coefficients are given. For design purposes a simple equation is arrived at in this report.

Introduction

A vertical stilling well (Figure 1) was considered to be the most effective structure to dissipate kinetic ep.e:rgy of the high velocity flow. Many structures, such as Banninger,*' Kreuter,&! Poebing,.ail and Forchheimert' dissipaters were developed, yet there was little information regarding the dimensions required for satisfactory operation of the basin. In 1949 to 1962, with the aid of byd.ra.ulic models, an equation for detennining the basin dimensions was

il:sanninger, K., studien und Versuche uoer Unsetzung von Geschwindig Keit in Druck, usw, Zeits fur das gesamte Turbinenwesen, Vol. 3, 19o6, p 12 ii Kreuter, F., Hydra.ulische Bremse, Wasser kraft und Wasserwirtschaft, Vol. 15, 1920, p 65 .3/Poebing, o., Ein neuer Energievernichter, Wasserk.raft und Wasserwirtschaft, Vol. 19, 1924, 0.373 ~ Forchheimer, P. , Hydra.ulik, Teubner, Leipzig und Berlin, F.d.. 3, 1930, p 319

..

developed by the Bureau of Recl.JJ:mation.:Jt:Jz/a./tJ../ However, the resulting equation was unbalanced in dime:iasions and the energy dis­sipating capacity is not quite understood.

In order to gain a more rational understanding of this phenomenon, a combination of theory and experiments are used in this study.

Hydraulics of Stilling Well

Theoretical Well Depth

According to Newton's secc!>D.Ct. · l aw of motion, the change of manentum per unit of time in the body of water in a stilling well w1 th fil­lets is equal to the resultant of all the external forces acting on the body. Applying this principle to the free body shown in Figure 2, the following expression for the force plus manentum enclosed between Section l and 2, Figure 2,may be written:

-rk dA +f3 S,-v +f3 S,-v -F =O 2 w i g p 2 g w f

where p is the pressure head in the downspout; A and v a.re area and velocity with subscripts referring to pipe, well and fillet

§./Hydraulic Model Study of the Masonville Siphon Turnout Structure-­Horsetooth Feeder Canal, Hydraulic Laboratory Report No. Hyd-237, USBR, February 1949

!i/Hydraulic Model Study of a Proposed Outlet Works Design for Anchor Dam, Hydraulic La.boratory Report No. Hyd-244, USBR, January 1949

ZIHydraulic Model Studies of the Stilling Well for the Blow-off structure, Soap Lake Siphon, Columbia Basin Project, Hydraulic Laboratory Report No. Hyd-277, USBR, 1950

~/Hydraulic Model Studies of the Regulating Gate and Stilling Well-­Trenton Dam Canal Outlet Works--Missouri River Basin Project, Hydraulic Laboratory Report No. Hyd-300, USBR, 1951

!!/Hydraulic Model studies of the Wan.ship Dam Vertical stilling Wells, Weber Basin Project, utah, Hydraulic Laboratory Report No. Hyd- 481, USBR, 1962

2

(1)

sections; W is the weight of water enclosed between the pipe out­let and bottan of well floor; 1 is the unit weight of water, c4 is the drag coefficient for the fillets; d is the depth of the well; z is the height of the downspout outlet; Fr is the total external force of friction and resistance acting along the well; and 1s._ and ~ are pressure correction factors due to turbulence is Sections 1 and 2, respectively.

In short reach, the external force of friction along the well sur­face can be ignored. Thus, with Fr• 0 and assuming 131 = 132 = 1, Equation 1 becanes

2 A

+3?.g(1+ p) A -A w p

or simply can be shown as

where k represents shape factor.

In order to make the resulting equation dimensionless, dividing both sides of Equation 2 by total head, H, then becanes

3

(2)

(3)

VP p 2 A ) +gH(1+Ay-~

(4)

(5)

Thus the required depth of the vertical stilling well can be related to the total head. of the dO'WtlSpout and can be expressed as a func­tion of dimensionless parameters of area ratio (i.e., ~/Aw, Ar/Ay), velocity head. ratio of the downspout (i.e., v:/2&1), downspout height ratio (Z/H) and shape factors (Cd ana · k).

E:f'f1ciency of Energy Dissipation

According to the principle of conservation of energy, the total energy at the downspout outlet should be equal to the total energy head. at the channel section (Figure 2) plus the loss of energy, h f, between these two sections, then with respect to the well floor

4

(6)

.•

where v d is the velocity of flow in the channel section, a is the energy coefficient.

The loss of energy between well and channel sections can be evaluated by canbining Equations 2 and 6, then

where Ad is the area of channel section.

The efficiency of the vertical stilling well can be expressed as the ratio of the loss of energy between the well and channel intake sections to the total energy at the downspout outlet. Then

where n is the efficiency of the well.

5

(7)

(8)

In general, for well without fillets, Ad >>AP, A.,, >>AP, Is_ m:;: ka -, 1, then

2 V

limn= P 2

A ...R. + 0 A· d

( p +~)

and when v2 /2g >> p , limit n = lOO percent. Thus 1 t is shown that a. vertical stilling well is one of the most effective struc­tures to dissipate kinetic energy of the high velocity flow.

Jacperiments

Model

A well 6 feet in depth by 4- feet s·quare vra.s bull t to provide a testing fa.cili ty. The downstream side of the well emptied into a trapozoidal channel with movable sloping sides and an adjustable tailgate. The tailgate control provided a total basin depth up to 7-}/4 feet and channel flow depth of up to 1-3/4- feet.

Adjustable corner fillet frames and supports were attached to modify the square well configuration, so as to determine the optimum fillet position and dimensions.

An adjustable downspout of 4-1/2-inch-inside-diameter pipe was installed in the center of the well and the pipe was inserted into a larger 6-inch stand.a.rd pipe, thus providing adjustment from less than a diameter to approx:illla.tely 20 diameters above the well floor.

The flow into the system was supplied by 8-inch pipe and controlled through the Venturi meter system with maximum capacity of up to 7-1/2 cubic feet per second.

6

(9)

.•

Thirty piezaneter holes were located in a gridlike pattern in the downstream quadrant of the basin floor. With taps connected to a boa.rd of differential mercury mancmeters, the pressure distribution on the basin well can be obtained.

A ring of four piezcmeters was placed near the end of the downspout to obtain the pressure head in the inccming flow. Three piezometers were located on the channel floor to provide the pressure in the outgoing flow. A capacitance wave probe with half bridge recorder was employed in measuring and recording of surface roughness.

A 1:9.14 scale model (Figure 3) of the larger basin was constructed of clear plastic to provide a means of viewing flow conditions in a geanetrically similar well.

Test Procedures

Two separate procedures were used in the previous study. The first procedure consisted of a series of tests without corner fillets for various combination of well depths and heights of downspout. The second procedure consisted of a series of similar tests, but with the corner fillets. Fillets of 45, 50, 55, 60, and 65° (Figure 4) were constructed and tested. To evaluate the optimum location of the fillets, several fillet locatioil{:)were tested with the other conditions remaining the same as the first procedure so as to get a better ccmpa.rison.

In both procedures, the flow was adjusted to give a smooth water surface in the well. Pressure readings were taken to obtain the discharge, static head in the downspout, and pressure profile on the well floor.

The tests were repeated in the ranges of discharges frcm 1 to 7.38 cfs, total heads to 7 to 76 feet of water, and pipe velocities of 9 to 67 feet per second. Well depths of up to 7.90 feet and channel depths of up to 1.90 feet were util~zed.

Prior to these two procedures, the sma.11 clear plastic model was used to investigate visually the general flow patterns. Various similar fillets of scaled sizes and shapes were tested in the plastic model so as to obtain additional information about their 1\mctioning and validity.

Investigation

Previous investigations were based on dimensional analysis. Data were taken according to the result of analysis and were not suf­ficient to ca:nplete a general study. However, the main flow

7

parameters were evaluated by limited information available. Several curves for ma.in flow :parameters were presented. Despite appreci­able experimental scatter, the analytical curves are in reasonable agreement with the experimental data over the entire range of the independent variables. In order to save time and effort, the factors, which must be observed or measured experimentally were reccmmended in this study.

;[va.1.ua.tion of we11 constants As analyzed in Equations 4 and 6, the parameters involved in well hydraulics were mcmentum coefficients: f31,, {3

2, velocity-distribution

coefficients; exp, ad, drag coefficient for fillets; Cd, and pressure­

distribution coefficients ~ and k2

• Unfortunately the data for

analyzing pressure, velocity and drag coefficients were not avail­able. Pressure readings on the basin floor were available and were used in evaluating the pressure-distribution coefficients in the analysis. It was found that the pressure-distribution coefficient is critical to the well performance.

In order to evaluate the pressure-distribution coefficient on the well floor, k, the pressure readings on the floor were integrated by a ccmputer2 program which calculated a total force on the well fl.oor by Ep~a and required only mananeter readings as input data. Then the coefficients k

2 for each different conditions were

computed by

Plot of test data shows that

~ = r( ~, li' and shape factors) . w

The best fitted curves for different conditions are tabulated in Table 1 and a design chart for 60° fillets with size of 1/3 • b and height of ~r/z = o.26 is given in Figure 5. Where b is width of the well.

Three trends are noted: the ~ values will increase as z/H decreases; the k

2 values increase as the fillet angles increase

8

(10)

(11)

..

f'ran 45 to 60°; and k2 values increase UJ;> to O]?timmn fillet height and then decrease gradua.ll.y.

In order to evaluate pressure-distribution coefficient on the down­spout section, piezometer readings around the section are necessary. Due to the lack of readings the coefficient is not analyzed here, but it may be reasonably assumed to be near 1.

Energy and momentum coefficients have been analyzed in many open­channel f'lows, but limited information was available for vertical flow condition.!.21 Velocity-distribution measurements are necessary so that the equations

Ev3 6A a=·---

- V 3 A

and

2 t3 = Ev . M

. v2 A

can be applied.

Where V is the mean velocity. A is the total area. v and .6.A are the elementary velocity and area, respectively.

Total Force on Well Floor

Pressure distribution is essential for structure design of vertical stilling well. The dimensionless pressure distribution for dif­ferent conditions was plotted in Figure 6. It was found that the pressure profile could be correlated to total head in the downspout and well depth. The maximum pressure can be related to total head in the downspout and can be shown as a i'unction of' z/H. (Figure 7.) It occurs directly beneath the downspout. The profile gradually decreases to o.8 d for approximately 7 pipe diameters and recovers to 1 d f'or 4 diameters then becomes larger than 1 d at the edge of the well. For structural design purposes pressure distribution of 1 diameter total head on the center, 1 diameter half' the total head around the center of' the well, and k

2d the rest of the well section

is recommended.

Jet Expansion

The pressure distribution curve is a good index to the investigation of jet expansion. It -was found that the jet either side of the

iJiJ Chow, "Open-Channel Hydraulics," 1959, p 28

9

(12)

(13)

centerline varied from apprarlma.tely 52-l/2° with the downspout l do'WD.spout diameter frcm the floor down to approximately 10° with the do'WD.spout 7-l/4 diameters from the floor. The curve appeared to become asymptotic to 10° s.o that a further increase in downspout distance from the well floor will not affect the expansion angle of the jet. This information agrees with angles 12 to 14 ° on horizon­tal channels as studied by Hunter Rouse, etc.!!.!

Wave Amplitude

Measurements of wave amplitude were made at a point 3 feet down­stream from the well and on the channel centerline. It was found that wave amplitude can be correlated to pressure-distribution coefficient, fillet angle, and fillet height. It is shown that wave a.mplitude decreases as the pressure-distribution coefficient value increases, and the optimum conditions were determined as approximately a 60° fillet with a size of 14-l/2 inches (~b/3) and height of o.4 foot (0.052 d). -

Practical Application

Evaluation of Design Equation

Determination of coefficients frcm the data available was not pos­sible for a completely general study. However, for a particular well, the effect of fillets can be correlated to the pressure­distribution coefficient and the design equation can be simplified from Equation 2 to

The pressure-distribution coefficients obtained by the experiments were used in verification of Equation 2 1 • Despite appreciable experimental scatter, the analytical curve is in reasonable agree­ment with the measurement over the entire range of the independent variables (Figure 8).

ll./ Albertson, Dai, Jensen, and H. Rouse, ''Diffusion of Submerged Jets," Trans. ASCE, Vol. 115, 1950, pp 260-273

10

(2•)

:F\lrthermore Equation 8 is simplified into

n= [ ) ( A ) v

2 ( 2.A · A" 1 l ck -1 _! d _ ....P. 1 + p + 1

v2 2 AP g ~ - AP , Al. l? + J.

2g

and the equation is experimentally verified (Figure 9).

11

Table 1

1/4 dynamic* . Fillet : Tail-: Wave . Test • Fillet . . height : water : force on . aJil)?litude . k:i No. . angle . . . f't . f't . Floor No. ft . . . . . . .

1 650 : o.oo . 1.67: 139 0.023 . 1.072 . . 2 . 0.50 . 1.77 : 195 0.002 : 1.100 . . . 3 . . 1.00 1.68 : 176 . 0.011 : 1.091 . . . 4 . 1.50 . 1.67: 123 0.022 : i.o64 . . 5 . 60° . o.oo 1.67 : 128 o.oo8 : 1.o66 . . 6 . 0.25 . 1.67 : 138 : 0.003 : l.:.072 • . 7 . 0.50 . 1.67 : 126 . 0.0025 : 1.o65 . . . 8 . 0.75 . 1.67 : lll . 0.005 : 1.057 . . . 9 . . 1.00 1.67 : 93 o.oo65 : l.o48 . .

10 . 55° o.oo . 1.66: 118 . 0.011 : 1.o61 • . . 11 . . 0.25 . 1.66 : 116 0.0022 : 1.o6 . . . 12 . . 0.50 . 1.66 . 102 . 0.0042 : 1.05 . . . . . 13 . 0.75 . 1.67 : 97 . 0.0055 : 1.05 . . . 14 . . 1.00 1.67 : 97 . o.oo8o : 1.05 . . . 15 . 50° . 0.60 1.66 . 130 . 0.0023 : 1.07 . . . . 16 . . 0.25 . 1.66 . 111 . 0.0028 : 1.o6 . . . . . 17 . 0.50 . 1.66 . 102 . 0.0070 : 1.o6 . . . . 18 . 45° . o.oo . 1.66 . 114 . : 1.o6 . . . . . 19 . . 0.25 . 1.66 . 99 . : 1.05 . . . . . 20 : No . . 1.67 . 70 . : 1.03 . . . . . fillem: . . . . . . . . .

*l/4 dynamic force= (total force on 1/4 of well floor) minus (dead weight on 1/4.of well floor due to water level in well)= l/4(Ep~A - dA)

Q, = 5 -cfs d = 4-1/2 inches Fillet size= 16 inches Well area = J.6 f't2 Well depth .= 6 feet Height of downspout= 4-1/2 inches k:a = pressure-distribution coefficient of well floor

12

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