Cavitation in Chutes and Spillways. Engineering monograph nº 42, USA Bureau of Reclamation

8/7/2019 Cavitation in Chutes and Spillways. Engineering monograph n 42, USA Bureau of Reclamation

1/164

A WATER RESOURCES TECHNICAL PUBLICATIONENGINEERING MONOGRAPH NO. 42

CAVITATIONIN CHUTESAND SPILLWAYS


2/164

i

4. TITLE AND SUBTITLECavitation in Chutes and Spill ways

Bureau o f Reclamation TECHNICAL RE

7. AUTHOR(S)Henry T. Falvey

9. PERFORMING ORGANIZATION NAME AND ADDRESSBureau of ReclamationDenver OfficePO Box 25007Denver, Colorado 80225

2. SPONSORING AGENCY NAME AND ADDRESS

Same

5. SUPPLEMENTAR Y NOTESMicrofich e and/or hardcopy available at the Denver Office

EPORT STANDARD TITLE PAG3. RECIPIENTS CATALOG NO.

5 . REPORT DATEApril 1990

6. PERFORMIN G ORGANIZATION CODE

6. PERFORMIN G ORGANIZATIONREPORT NO.

Engineering Monograph No. 4210. WORK UNIT NO.

D-375011. CONTRACT OR GRANT NO.

13 . TYPE OF REPORT AND PERIODCOVERED

14. SPONSORING AGENCY CODEDIBR

Editor: RNW

6. ABSTRACT

The purpose of this monograph is to give the designer of hydrauli c structures both an understandingof cavitation and the design tools neces sary to eliminate or reduce the damaging effects of cavitationin chutes and spillw ays. The monograph discu sses basic concepts, cavitation damage, pra ctical methodsof coping with cavitation damage, design recommendations, the influence of geometry and aeration, andBureau of Reclamation field experience. Two executable and one source and data diskettes are (included

specific structures. They will run on IBM AT compatible computers. Includes5 apps., 159 equations, and 145 pp.

herein) used to analyze3 disk s, 93 figs., 232 refs.,

7. KEY WORDS AND DDCUMENT ANALYSISI. DESCRIPTORS-- / *aerators/ *air entrainment/ *cavitation/ *computer programs/ *damage/ *designcriteria/ *invert geometry/ *open channels/ *surface irregulari ties/ *theory/ spillw ays/ high veloc ity flow/tunnel spillw ays/ erosion damage

1. IDENTIFIERS-- / Blue Mesa Dam/ Flaming Gorge Dam/ Glen Canyon Dam/ Hoover Dam/ KortesDam/ McPhee Dam/ Yellow tail Dam:. COSATI Field/Group 1300 COWR R: 1407 SRIM:6. DISTRIBUTION STATEMENT 19. SECURITY CLASS 21 . NO. OF PAGE

Available from the National Technical InformatIon Service. Operations ITHIS REPORT) 145Division, 5285 Port Royal Road, Springfield, Virgrnia 22161. UNCLASSlFlED20. SECURITY CLASS 22. PRICE(THIS PAGE)


3/164

A WATER RESOURCES TECHNICAL PUBLICATIONENGINEERING MONOGRAPH NO. 42

CAVITATION IN CHUTESAND SPILLWAYS

bYHenry T. FalveyResearch EngineerApril 1990Denver OfficeDenver, Colorado 80225

UNITED STATES DEPARTMENT OF THE INTERIOR*

BUREAUOFRECLAMATION


4/164

Mission: As the Nations principal conservation agency, theDepartment of the Interior has respons ibility for most of ournationally-owned public lands and natural and culturalresources. This includes fostering wise use of our land andwater resources, protecting our fish and wildlife, preservingthe environmental and cultural values of our national parksand historical places, and providing for the enjoyment of lifethrough outdoor recreation. The Department assesses ourenergy and mineral resources and works to assure that theirdevelopment is in the best interests of all our people. TheDepartment also promotes the goals of the Take Pride inAmerica campaign by encouraging stewardship and citizenrespons ibility for the public lands and promoting citizenparticipation in their ca re. The Department also has a majorrespons ibility for American Indian reservation communitiesand for people who live in Island Territories under U.S.administration.

ENGINEERING MONOGRAPHS are published in limitededitions for the technical staff of the Bureau of Reclamationand interested technical circ les in Government and privateagencies. Their purpose is to record developments, innovations,and progress in the engineering and scient ific techniques andpractices which are used in the planning, design, construction,and operation of water and power structures and equipment.

First Printing A pril 1990-1,500 copies


5/164

FRONTISPIECE.-Glen Canyon Dam. Left Spillway Tunnel Sept. 1983. The "big hole" in the spillway invert was11 meters deep. Photo C557-400-690NA


6/164

iv

Mission of the Bureau of ReclamationThe Bureau of Reclamation of the U.S. Department of the

Interior is responsible for the development and conservationof the Nations water resources in the Western United States.

The Bureaus original purpose to provide for the reclama-tion of arid and semiarid lands in the West today covers awide range of interrelated functions. These include providingmunicipal and industrial water supplies; hydroelectric powergeneration; irrigation water for agriculture; water qualityimprovement; flood control; river navigation; river regulationand control ; fish and wild life enhancement; outdoor recreation;and research on water-related design, construction, materials,atmospheric management, and wind and solar power.

Bureau programs most frequently are the result of closecooperations with the U.S. Congress, other Federal agencies,States, local Governments, academic institutions, water-userorganizations, and other concerned groups.

The research covered by this report was funded under theBureau of Reclamation Program Related Engineering andScientifi c Studies (PRESS) allocation No. DR-203 Preventionof Cavitation to Flow Surfaces.

When Congress authorized the reclamation of arid and semiaridlands in the West with the Reclamation Act of 1902, theSecretary of the Interior established the Reclamation Servicewithin the United States Geological Survey. In 1907, theReclamation Service became an independent agency within theDepartment of the Interior. By action of the Secretary of theInterior in 1923, the Reclamation Service became the Bureauof Reclamation. In November 1979, the name was changed tothe Water and Power Resources Service by Secretarial Order,and in May 1981, the Secretary of the Interior changed theagencys name back to the Bureau of Reclamation.

Reprinting or republication of any portion of this publicationshould give appropriate credit to the Bureau of Reclamation-U.S. Department of the Interior.

The information contained in this report regarding commercialproducts or firms may not be used for advertising orpromotional purposes and is not to be construed as anendorsement of any product or firm by the Bureau ofReclamation.


7/164

PrefaceThis monograph is not intended to he a comprehensive treatise on cavitation

in hydraulic structures. Instead, only highlights of cavitation theory arepresented to give the reader a sense for the basic mechanisms that must heunderstood before engineering problems and solutions can be discussed. Manyof the basic cavitation concepts are treated in much more detail by Knapp,Daily, and Hammitt in their class ic book Cavitation which is is available throughthe Institute of Hydraulic Research, University of Iowa, Iowa City, Iowa.

The purpose of this monograph is to give the designer of hydraulic structuresboth an understanding of cavitat ion and the design tools necessary to eliminateor reduce the damaging effects of cavitation in hydraulic chute and spillwaystructures. In the monograph:l Chapter 1 discusses highlights of cavitation theory.l Chapter 2 presents cavitation characterist ics of typical surface irregularities .

(The material in the first two chapters is applicable to both open andclosed conduit systems.)

l Chapter 3 is a discussion of cavitation damage.l Chapter 4 treats the influence of geometry.l Chapter 5 addresses aeration and aerators.l Chapter 6 contains recommendations for designers.

(This chapter should help in specifying surface tolerances and decidingwhen special designs such as aerators are necessary.)

l Chapter 7 concludes with selected Bureau of Reclamation field experiences.Several appendixes are included in the monograph which describe computer

programs useful in the analysis of specific geometries:Appendix A contains the algorithms to calculate the cavitation and damageindexes for open channel flow situations.Appendix B describes a program to plot the cavitation and flowcharacteristics.Appendix C contains descriptions of two programs to investigate the effectof changes in the invert geometry on the cavitation index.Appendix D contains a description of a computer code used to designaerators.Appendix E contains the description of a program to determine the damagepotential of a structure using historical data.

All programs in this monograph are written in Microsoft-Fortran. Theplotting program is written in PLOT882 which is compatible with Microsoft-Fortran. They have been compiled to run on an IBM PC compatible computer.The programs are included on three diskettes.

The firs t, Source and Data contains the source codes and example data forthe executable programs. The source codes and their respective program namesare as follows:l Flow and cavitation properties WS77.FORl Plot of cavitation and flow properties PLOT77.FORl Damage index from historical data DINDX.FORl Trajectories from aerators TRAJ.FORl Constant cavitation number profile ECAVNO.FORl Controlled pressure profile CONSTP.FOR

1 Registered trademark of Microsoft, Inc., 16011 NE 36th Way, Bo x 97017, Redmond WA 98073-9717.2 Registered trademark of Plotwork s, Inc., PO Box 12385, La Jolla CA 92037-0635.


8/164

vi

The data files and their names are:l Input file for WS77.EXE GLENINl Output files from WS77.EXE GLENOUT

GLENGEOMGLENPLOT

l Input file for PLOT.EXE GLENPLOTl Input file for DINDX.EXE GLENHYl Input file for TRAJ.EXE GLENINl Input file for ECAVNO.EXE ECNDAT

The contents of this diskette can be examined by booting the diskette andtyping the command, RUN. A list of the files on the diskette are displayed.The contents of any file can be seen by typing the command, TYPE .The contents of any file can be printed by typing the command, PRINT. For more details, see the DOS manual. This diskette is includedso that the user has the opportunity to examine the code and to make changes,if desired. If changes are made, the user will be required to recompile theprogram.

The other two disket tes, Executable Code I and Executable Code II, containexecutable codes of the programs.n Executable Code I contains the executable codes for determining:l flow and cavitation characteris tics WS77.EXE. trajectories from aerators TRAJ.EXEl damage indexes DINDX.EXEn Executable Code II contains executable codes for:l plotting PLOT77.EXEl determining constant cavitation number profile ECAVNO.EXEl controlled pressure spillway profile CONSTP.EXE

Both diskettes can be executed by booting the diskettes and typing, RUN.This will cause a menu to be displayed from which the appropriate programcan be selected.

The diskettes are not copy protected as they are in the public domain.However, proper credit should be given to the Bureau of Reclamation if anyof the code is used in other programs or is published in any context.

It is recommended that the executable programs be run from a copy ofthe diskettes. Several programs write to the diskette being used. Therefore,write-protect tabs should not be placed on the copies. The plot program requiresconsiderable scratch space on the diskette. Thus, only the executable versionof the plot program should be instal led on the copy.

The Bureau of Reclamation does not guarantee the performance of thecomputer programs nor help external users with problems other than minordifficulties of adaptation to their purposes. As with all programs, the usershould veri fy the proper functioning of programs through independent checks.

Suggestions for revising and improving the programs should be submittedto:

Bureau of ReclamationAttn: Code D-3750PO Box 25007Denver CO 80225-0007


9/164

vii

AcknowledgmentsCavitation in Chutes and Spillways has been in preparation for over eight

years. To acknowledge all who influenced the direction of this effort duringits writing is impossible. Nevertheless, some contributors deserve specialrecogni tion. Many thanks to T. J. Rhone, who undertook the task of colla tingthe editorial comments into a final manuscript following my retirement inJuly 1987. Appreciation is expressed to the following individuals who reviewedthe manuscript:

Task Committee on Cavitation in HydraulicStructures, Hydraulics Division,American Society of Civil Engineers . . . . ..I......... A. F. Babb

R.E.A. ArndtF. DeFazio

Water Resources Planning Commission,Taipei , Republic of China . . . . . . . . . . . . . . . . . . . . . . . . C. M. Wu

Sinotech, Taipei, Republic of China . ..I............. T. L. ChungY. ChengUniversity of Wisconsin ............................ J. A. HoopesSnowy Mountain Engineering Corporation ......... E. J..Lesleighte rPacifi c Gas and Electr ic Company . . . . . . . . . . . . . . . . . . . W. EnglishHarza Engineering Company . . . . . . . . . . . . . . . . . . . . . . . . C. Y. WeiBureau of Reclamation . . . . . ..I..................... P. H. Burgi

K. W. FrizellE. W. Holroyd

J. H. LaBoonB. F. MeffordT. J. RhoneR. V. Todd

J. C. WadgeR. N. Walters

The careful scrutiny of the manuscript by these individuals eliminated manyerrors and helped to explain areas which were not too clear. The review byProfessor Hoopes was especially appreciated for its detailed thoroughness.

The plotting program was revised by K.W. Frizel l to make it more flexible.The method of determining the radius of curvature from three points in thewater surface program was contributed by E. W. Holroyd. C. M. Wu notedthe simila rity of the cavitation inception data for away-from-the-flow offsetswith into-the-flow offsets. Technical editing and continuity for publication wasperformed by R. N. Walters.

Finally, I would like to thank my mentor, D. Mike Colgate, who firstintroduced me to the subject of cavitation. We spent many happy hours discussingvarious details on this fascinating subject.

Should the effort here result in value, it is because, as Sir Isaac Newtonsaid:

If I have seen further it is by standing upon the shoulders of giants.


10/164

. . .VU,1

Letter Symbols and QuantitiesSYMBOL QUANTITY

One-half amplitude of maximum negativepressure fluctuation

Cross-sectional area of jetConstant for a specific gas, mass, andtemperature

Dimensionless pressure ratioCross-sectional area of upstream pipeJet trajectory coefficientArea of cross sectionCross-sectional area of air ductExperimental coefficientBoussinesq numberBulk modulus of gasWidth at lower end of spillwayBulk modulus of mixtureWidth at upper end of spillwayBulk modulus of waterExperimental coefficientExperimental coefficientMean air concentration of developed

aerationBorda loss coefficientStabilized damage depth factorMean resistance coefficientCentrifugal acceleration factorSummation of loss coefficients through

the air ductPressure factorMean air concentrat ion at beginning of

aerationPressure coefficientMost negative value of Euler number on

bodyMinimum pressure coefficient of

roughnessAir concentration at wallMean air concentration at distance(Lx - Li)ConstantConstantExperimental coefficientWidth or depth of sectionDiameter of pipeDamage indexDiameter of jetFlow depth upstream of offsetDamage potentialStabi lized depth of damageFlow depth normal to invertFlow depth at beginning of constant

cavitation number profileFlow depth from constant cavitation

number profileEffective flow depth at first station

SYMBOL QUANTITYdidkdndoEEdEEO;FOFI;,ff(S,f(N)foflf2G

ii

H,Hb

Hd

HtI&

&IHAHhhihhchohphwI,klK,KSKtkkr

Initial dimensionless flow depth forconstant cavitation number profile

Critical depthDimensionless flow depthRamp height or offset heightEiitviis numberDepth of cavita tion damagePotential energy of flowPotential energy at reference point2.71828. . .Froude numberFroude number of flow at rampCosine of invert slopeResistance factorDarcy-Weishach friction factorA specified functionA function in terms of NValue of function at point 0Value of function at point 1Value of function at point 2Iteration functionGravitational constant (acceleration)Characteristic dimension, offset height,

length of cylinder, etc.Atmospheric headReservoir elevation minus elevation at

point hPressure head downstream of valve or

orificeInitial head at beginning of constant

cavitation number profileTotal head upstream of valveHead loss due to frictionPressure head upstream of valve or orificeVapor pressure (absolute) in terms of

headNegative pressure head pulsation at offsetMaximum time-average negative pressure

head on the irregularityFlow depth downstream of impact pointHead loss from crest to first station Critical depthFlow depth at end of rampPool depthWall pressure headCavitation damage intensityDamage coefficientCurvature normal to streamlineCavitation index in terms of relative headCurvature of streamlineCavitation index in terms of total headIdeal gas constantConstant to control the radius of

curvature


11/164


12/164

x LETTER SYMBOLS and QUANTITIES-Continued

SYMBOL QUANTITYElevationElevation at point aEnergy gradeline elevation

Henrys law, coefficientKinetic energy coefficientGas concentrationAir flow ratioBoundary layer thicknessDisplacement thicknessMomentum thicknessPressure intensity coefficient, one-half

amplitude of maximum negativepressure fluctuation divided byaverage velocity head

Interfacial surface tensionVorticityDimensionless headAngle between tangent to inver t andhorizontalDeflection angle of a vertical curveAngle between the tangent to nappe at

ramp and horizontalAngle between downstream chute invert

and horizontalAngle between centerline of jet at impact

point and invert of chuteAngle between ramp and horizontalAngle between upstream chute inver t and

horizontalEffective ramp angleTangent to jet trajectoryDimensionless convergence ratioKinematic viscosityDimensionless distance horizontalInitial dimensionless distance horizontal3.14159.. .Density of waterDensity of air

SYMBOL QUANTITYInvert elevation of first pointElevation of reservoirReference elevation

Density of gasDensity of mixtureCavitation indexCavitation index based on free stream

condition in plane of offsetCavitation index in circular conduitCavitation index in constant cavitation

number spillwayCavitation index of flowCavitation index of flow based on

conditions at height of offsetCavitation index based on conditions

at height of offsetIncipient cavitation indexCavitation index for a submerged jetCavitation index based on average veloc ity

over height of offsetCavitation index of a body having an

isolated roughnessCavitat ion index when damage beginsCavitation index based on free stream

condition upstream of start ofboundary layer

Collapse time of a bubbleVolumeVolume of gasOriginal volumeVolume of waterChange in volumeAngle between ramp and invertVolume of air per volume of mixturePotential of external forceAngular velocityVaries as


13/164

xi

ContentsPreface ........................ ......................... ........................ ..Acknowledgments .............. ......................... ........................ ..Letter Symbols and Quantit ies ...................... ......................... .......Chapter 1: Bas ic Concepts

Descr iption of Cavitat ion ..................... ......................... ..........Derivation of Cavitat ion Parameter ....................... ........................Formation of Cavitat ion ....................... ........................ ..........

Theory ......................................................................Empirical verification .........................................................

Collapse Dynamics ..............................................................Sonic Veloc ity of Gas Mixture ....................................................Bibliography ...................................................................

Chapter 2: Cavitation Characteristics of Flow SurfacesCategories of Surface Roughness ..................................................Singular Asperities ..............................................................

Into-the-flow offsets ........................ ......... . ........................Away-from-the-flow offsets ....................................................Holes in boundary ............................................................Transverse grooves ............................................................Grooves parallel with flow .....................................................Uniform roughness ....................... ......................... ...........Combination of uniform and isolated (singular) roughnesses ......................

Superposition Principle .........................................................Bibliography ...................................................................

Chapter 3: Cavitat ion DamageMode of Damage ................................................................Factors Affecting Cavitation Damage on a Surface ..................................

Cause of cavitation . . . .Damage location . . . . . .Cavitation intensity . . .Velocity effect . . . . . . . .Air content effect . . . . .Surface resistance to cavExposure time effect . .

.......................................................

............................................................................................................................................................................................................................tation damage ................................................................................................

Parameters for Predicting Damage ...............................................Damage potential .............................................................Damage index ................................................................Design values .................................................................

Recognition of Cavitation Damage ................................................Texture ......................................................................Symmetry ....................................................................Origin .......................................................................

Bibliography ...................................................................Chapter 4: Control of Cavitation Index by Geometry

Control Methods ................................................................Equations of Motion ............................................................Velocit y and Pressure at Flow Boundary ..........................................Geometric Relationships .........................................................Cavitation Parameter ............................................................Constant Cavitation Number Spillway ............................................Controlled Pressure Spillway Profile ..............................................

PageV

vii. . .VII1

125568

1013

1516162122232323232628

29292930313334343535373738393941414849495052525253


14/164

xii

CONTENTS-ContinuedChanging Invert Curvature-Example Page............................................ 54Constant cavitation number profile ............................................ 54

Controlled pressure profile .................................................... 55Changing Surface Roughness-Example ........................................... 55Bibliography ............................................ ....................... 57Chapter 5: Aerator DesignJustification for Aerators ........................................................ 59

Typesof Aerators .......................................... ..................... 60Location of Aerator ............................................................. 60RampDesign ................................................................... 60AirVent Design ....................... ........................... ............... 63

Air vent configurations ....................... ................................. 63Air flow relationships in ducts ................................................. 64Estimation of air entrainment capacity of the jet ................................ 64

OffsetDesign ................................................................... 66AeratorSpacing.. ............................................................... 67DesignProcedure ........................... ........................... ......... 69Bibliography ......................... ........................... ............... 71

Chapter 6: Design RecommendationsBackground .................................................................... 73Specifications of Surface Tolerances .............................................. 74

Offset ....................................................................... 74Slope ........................................................................ 74Uniformly distributed roughness ........................ ........................ 74Flow surface tolerance: definition and specifications ........................... 74

Geometric Considerations ........ .......................... ..................... 76DesignProcedure ............................................................... 77Bibliography ................................................................... 78

Chapter 7: Spillways - Field ExperienceCase Studies ....................... ........................... ..................

Blue Mesa Dam ........... ........................... .........................Flaming Gorge Dam .......................... ........................... .....Glen Canyon Dam ........................ ....................................Hoover Dam .................................................................Kortes Dam ..................................................................Yellowtail Dam ...............................................................

Bibliography ...................................................................

7979808183848590

Appendixes:A. Cavitation Characteristics for Open Channel Flow Program

Program Description ............ :. ........................................Algorithms Used in Program ...............................................

Friction factor .........................................................Curvilinear flow ........................................................Pressure distribution on steep slopes .....................................Kinetic energy correction factor ..........................................Critical depth ..........................................................Cavitation computations ................................................Airentrainment ........................................................

Profile Types .............................................................Computational Problems ..................................................

Starting station .........................................................Starting depth ..........................................................Impossible flow conditions ..............................................

9191919293939394949494949496


15/164

. . .X l .11

CONTENTS-Continued PageCross Section Shapes ...................................................... 96

Trapezoidal ............................................................ 96Circular ................ ..................... ...................... .... 96Egg shape ...................... ...................... .................. 96Horseshoe .......... ................................................... 97Composite .................... ....................... .................. 97Modified horseshoe ................... ....................... ........... 97User defined circula r-arc cross section .................................... 97

Input File ..................... ...................... ..................... 98Glen Canyon Left Tunnel Spillway-Input ..................... ............. 103Typical Output .................... ...................... ................. 104

Hydraulic properties of Glen Canyon left tunnel spillway - output ......... 104Cavitation properties of Glen Canyon left tunnel spillway - output ......... 105Formatted geometry file of Glen Canyon left tunnel spillway - output ...... 106Plot file of Glen Canyon left tunnel spillway - output ..................... 107

Bibliography ............................................................. 108B. Cavitation and Flow Characteristics Plotting I?rogram

Program Description ...................................................... 109Communications ports .................................................. 109Model of output device ................................................. 109

Program Output ......... ....................... ...................... .... 110C. Geometry Programs

Program Description ...................................................... 113Constant Cavitation Number Spillway ...................................... 113Controlled Pressure Spillway ................................................ 114Constant Cavitat ion Number Input ......................................... 115Constant Cavitation Number Output - Solid Body Rotation Assumption ...... 116Constant Cavitation Number Output - Irrotational Flow Assumption ......... 118Controlled Pressure Profile - Input ....................................... 121Controlled Pressure Profile - Output ...................................... 122

D. Aerator Trajectory ProgramProgram Description ...................................................... 125Program Input ........................................................... 126Program Output .......................................................... 127

E. Damage Index - From Histori cal Data ProgramProgram Description ...................................................... 129ProgramInput ........................................................... 130Program Output .......................................................... 132

Cavitation References .............................................................. 135

TABLESTable

l-l Properties of pure water ..................................................... 31-2 Influence of surface tension and air content on incipient cavitation index ........ 82-l Triangular and circular arc irregulari ty coefficients ............................ 192-2 Cavitation characteri stics of holes in a boundary ............................... 223-l Length of cavitation cavities in Glen Canyon Dam left tunnel spillway

- station 760.70 (m) ...................................................... 303-2 Length of cavitation cavities in Glen Canyon Dam left tunnel spillway- station 739.38 (m) ...................................................... 303-3 Damage indexes for Glen Canyon Dam left tunnel spillway .....................3-4 Design values of damage potential and damage index ...........................

z35-l Location of aerator and criti cal discharge ..................................... 615-2 Maximum pressure drop and air duct velocity ................................. 646-1 Grinding tolerances for high velocity flow ..................................... 736-2 Flowsurfacetolerances ......................................................6-3 Specification of flow surface tolerance ........................................

??


16/164

xiv

CONTENTS-Continued

NumberFIGURES

l-l1-21-3l-41-51-6l-71-81-9l-101-111-121-132-12-22-32-42-52-62-72-82-92-102-113-13-23-33-43-53-63-73-83-93-103-113-123-133-143-153-163-174-l4-24-34-44-54-64-7

Phasediagramforwater ....................................................Pressure distribution on hemispher ical rod ..................................Definition of pressure scales ................................................Reference veloc ity definition ...............................................Reference conditions for shear flow .........................................Developmentofcavitation ..................................................Equilib rium conditions for vapor bubbles containing air ......................Henrys law coefficients ....................................................Effect of veloc ity on vaporous and gaseous cavitation ..........................Bubble history for flow around a 1.5 caliber ogive ............................Collapse mechanisms of bubbles ............................................Cavitation of baffle p iers in Bonnev ille Dam model ...........................Sonic veloc ity of air-water mixtures ..........................................Typical isolated roughness elements found in hydrau lic structures .............Incipient cavitation charac teristics of offsets .................................Incipient cavitation charac teristics of chamfered and ellip tical ly rounded

offsets ..................................................................Cavitation charac teristics of chamfers ........................................Cavitation charac teristics of isolated irregu larities ............................Pressure distribution at a gate slot ..........................................Cavitation charac teristics of a backward facing step, sudden expans ions,

and submerged jet .......................................................Cavi tation damage at a bug hole .................................. .........Cavitation charac teristics of holes ...........................................Cavitation characteristics of gate slots ........................................Cavitation damage in longitudinal grooves ...................................Glen Canyon Dam, left tunnel spillw ay - station 760.70 .......................Glen Canyon Dam, left tunnel spillw ay - station 739.38 .......................Hoover Dam, Nevada tunnel spillw ay - Christmas-tree pattern of damage ......Cavitation damage rate .....................................................Cavitation damage with respect to cavitation index ............................Comparative cavitation resistance of various materials ........................Cavitation damage rate .....................................................Damage experience in spillways .............................................Variation in damage index with discharge and time ...........................Glen Canyon Dam, left tunnel spillway ......................................Glen Canyon Dam, left tunnel spillway - damage at station 760.20 .............Texture of cavitation damage in steel ........................................Cavitation damage in concrete ..............................................Bradbury Dam, California, damage produced in 254-mm-dia. butterfly valve

by sediment-laden water .................................................Kortes Dam, Wyoming, freeze-thaw damage ..................................Palisades Dam, symmetrical damage in outlet structure .......................Palisades Dam, outlet works-vortex caused damage downstream of slide gate ...Definition sketch of intrinsic coordinate system ..............................Definition sketch for integration along a streamline ..........................Definition sketch for geometry ..............................................Glen Canyon Dam, left spillway tunnel-cavitation index for flow of 475 m3/s ...Glen Canyon Dam, equal cavitation number spillway profile - cavitation

index for flow of 475 ms/s ................................................Glen Canyon Dam, controlled pressure sp illway profiles - cavitation index

forflowof475m3/s ......................................................Effect of rugosity on cavitation characteristics ................................

Page12234557789

12121516171820212224242526313233333435353638404142434445464750515254555656


17/164

XV

CONTENTS-Continued

NumberFIGURES

5-15-25-35-45-55-65-75-85-95-10

Page60616263656666676869

6-16-2 75767-17-27-37-47-57-67-77-8

7-117-127-137-147-15

Typesofaerators ..........................................................Lengthofjettrajectory .....................................................Jet trajectory coefficients ...................................................Airsupplytoaerators ......................................................Air vent criti cal pressure ratio ..............................................Ai r entrainment under nappe ...............................................Pool depth under nappe ....................................................Converging offset downstream of aerator ...... ......................... ......Development of self aerated flow ...................................... ......Air concentration at wall ...................................................Hoover Dam, Nevada, spillway - concrete surface near station 994.00 ..........Lombardi cres t ........ ......................... ........................ ...Blue Mesa Dam, tunnel spillway - aeration slot ..............................Flaming Gorge Dam, tunnel spillway - aeration slot ..........................Glen Canyon Dam, tunnel spillways - aeration slots ..........................Glen Canyon Dam, tunnel spillways - major damage .........................Glen Canyon Dam, tunnel spillways - damage profiles ........................Hoover Dam, tunnel spillways - aeration slots ...............................Hoover Dam, Arizona tunnel spillway - major damage (1942) ..................Hoover Dam, Arizona tunnel sp illway - misalignment that caused

major damage ...................... ........................ ............Kortes Dam, tunnel spillway - profile .......................................Kortes Dam, tunnel spillway - invert cavitation damage ( left side) caused

by freeze-thaw popout (right) ........................ .....................Kortes Dam, tunnel spi llway - springline damage ............................Yellowtail Dam, tunnel spillway - profile ...................................Yellowtail Dam, tunnel spi llway - major damage downstream of elbow ........Yellowtail Dam, tunnel spillway - damage in elbow ..........................Yellowtail Dam, tunnel spillway - aeration slot ..............................

79818182838384

7-97-108586878889898989

A-l Piezometric pressure with curvilinear flow ................................... 92A-2 Pressure distribution on steep slope ......................................... 93A-3 Profiles of gradually varied flow - classifica tion ............................. 95A-4 Trapezoidal and rectangular cross sections ................................... 96A-5 Circular cross section ...................................................... 96A-6 Egg-shape cross section ..................................................... 97A-7 Horseshoecrosssection .................................................... 97A-8 Composite cross section .................................................... 97A-9 Modfied horseshoe cross section ............................................ 97A-10 User defined circular -arc cross section ....................................... 97B-l Glen Canvon Dam. left spillwav tunnel - comparisons of outnuts ............._ -_-- --.-. -, 111


18/164


19/164

Basic Concepts Chapter 1DESCRIPTION OF CAVITATION

Cavitation is a process that usually is associatedwith damage to a surface or marked by intense noise.Roth phenomena may occur during cavita tion, butactually cavitation is neither of these. Instead,cavitation is defined as the formation of a bubbleor void within a liquid. If the void is filled primari lywith water vapor, the process is further classifiedas vaporous cavitation. If the void is filled primari lywith gasses other than water vapor, the process iscalled gaseous cavitation. The distinction betweenthese two types of cavitation are explained furtherin Formation of Cavitation - Theory.

An understanding of the cavitation process canbe obtained by examining the process of boiling.When water is heated the temperature increaseswhich results in increases in its vapor pressure.When the vapor pressure equals the local pressure,boiling will occur. At the boiling point, water ischanged into water vapor. This change fi rst occursat localized points within the water and it is observedas small bubbles.

The temperature at which boiling occurs is afunction of pressure. As pressure decreases, boilingwill occur at lower and lower temperatures. Sincethe pressure is a function of elevation, boiling occursat lower temperatures at higher elevations as notedon figure l-l. If it were possible to go to a highenough elevation, boiling would occur at roomtemperature.

Although cavities are formed in water by boilingand the process occurs when the local pressureequals the water vapor pressure, a technicaldifference between boiling and cavitation exists.Boiling is the process of passing from the liquidto the vapor state by changing temperature whileholding the local pressure constant. Cavitation isthe process of passing from the liquid to the vaporstate by changing the local pressure while holdingthe temperature constant (fig. l-l) . The localpressure reductions associated with cavitation canbe caused by turbulence or vortices in flowing water.

An example of bubble formation within a liquid,which occurs by reductions in pressure, can be seenwhen a bottle containing a carbonated liquid isopened. Upon opening the bottle, bubbles formwithin the liquid and rise to the surface. While inthe capped bottle, the liqu id is under enoughpressure to keep the carbon dioxide in solution.However, as the bottle is opened, the pressure isreduced and the liquid becomes supersaturatedrelative to the carbon dioxide. Therefore, the carbondioxide begins to diffuse out of the liquid. It shouldbe noted that this example is one of so calledgaseous cavitation in which vapor pressure of theliquid was never reached.

SOLID(Ice)

IT,=-5C atP = 60 300 kPaT-T~=()oc atP =bol.3kPa

LIQUID(Water)

0 0 50 100TEMPERATURE, C

FIGURE 1-l.-Phase diagram for water,


20/164

2 Cavitation in Chutes and Spillways

In hydraulic structures, water contains air bubblesand various types of impurities of many differentsizes. As will he seen later, microscop ic air bubblesor impurities in the water are necessary to initiatecavitation. However, once started, vaporization isthe most important factor in the cavitation bubblegrowth. The presence of air bubbles in the flow alsohas an effect on damage and noise produced by thecavitation.

In addition to describing cavitation by thecontents of the void; that is, by vaporous or gaseous,cavitation also can be described by its occurrence.For instance, if the pressure of flowing water i sdecreased through increases in the flow velocity, acriti cal condition is reached when cavitation willjust begin. This criti cal condition is called incipientcavitation. Similarly, if cavitation exists and the flowvelocity is decreased or the pressure is increased,a criti cal condition is reached when the cavitationwill disappear. This condition is called desinentcavitation. Incipient cavitation and desinentcavita tion often do not occur at the same flowconditions. The distinction is especially importantin laboratory investigations, but can usually beignored for all practical purposes in hydraulicstructures.

Finally, a set of criti cal flow conditions existsfor which the individual cavitation bubblessuddenly transition into one large void. Condi-tions under which the large void occurs is calledvariously as cavity flow, developed cavitation, orsuperca vita tion.

DERIVATION OF CAVITATIONPARAMETERThe preceding section alluded to the existenceof a criti cal combination of the flow velocity, flow

pressure, and vapor pressure of the water at whichcavitation will appear, disappear, or spontaneouslytransition into supercavitation. A parameter existswhich can be used to define these various occur-rences of cavitation. The parameter is known as thecavitation index and is derived below.

The equation for steady flow between two pointsin a flow stream is known as the Bernoulli equation.It can be written:

,&2 + Pi+ zpg = 2fl+p+zpg 1.1where:

P = pressure intensityPO = reference pressureV = flow velocity

V, = reference velocity2 = elevation2, = reference elevationg = gravitational constant (acceleration)p = density of water(The subscript o refers to upstream flow locationas noted on fig. l-2. )

In dimensionless terms, the comparable equationresults in a pressure coefficient, C,:

1.2

c, = ( P + Zpp) - (PO + ZOpg-)pv32 ++-(g)'0where:

Ef = potential energy of flow (defined by valuesin parentheses)E, = potential energy at reference pointIn many cases, the gravitational terms are small

relative to the pressure term or they are about equal;thus, the pressure coefficient can be written:

c = p-p0P pv32 1.3

The pressure coefficient is also known as thepressure parameter or Euler number.

The value of the Euler number, at any point ona body, is a constant as long as the minimum pressureon the body is greater than the vapor pressure ofwater. For example, on figure 1-2, the pressure atany point on a rod-having a hemispherical end-is predictable in terms of the upstream conditions.However, if the pressure at the location thatcorresponds to the minimum Euler number dropsto vapor pressure, then the pressure at that pointwill not decrease any further. The upstreamconditions that correspond to the onset of cavitationcan be calculated by replacing the boundarypressure in equation 1.2 with vapor pressure andsetting the value of the ratio equal to the minimumEuler number. The resulting parameter is knownas the cavitation index, u.

u= E,-Zpg -pv _p v,2/2 - - (Cp)min 1.4

Stream I In0

O&Scale for CpFIGURE 1-2.-Pressure distribution on hemispherical

rod.


21/164

Basic Concepts

If elevations 2 and Z, are equal, the cavitationindex is expressed as:

E-P,u= - =-(c&inP e/2where P, is the vapor pressure of water.

1.5

For smooth streamlined bodies, the most negativepressure occurs on the boundary. For these cases,the cavitation index can be estimated from pressuremeasurements made on the surface. However, if thebody is not streamlined, the flow will separate fromthe body and the most negative pressures will occurwithin the flow. In these cases, the cavitation indexwill be less than the absolute value of the minimumEuler number on the body.

To avoid ambiguities, both vapor pressure andreference pressure are referenced to absolute zeropressure as shown on figure 1-3. For example, inabsolute units, the reference pressure is given by:

PA = Pi + Pg 1.6In the literature, sometimes one notes a reference

to a reduced or vacuum pressure. This i s done toavoid expressing the gauge pressure as a negativenumber when the reference pressure is less thanthe atmospheric pressure. When using values of thevacuum pressure, the appropriate relationship forthe reference pressure, P, , is given by:

where:P, = Pa - P, 1.7

Pa = atmospheric pressurePg = gauge pressureP, = reduced pressure (vacuum)Reference7 pressure, PO

1 ~~Eiure. P,1 Absolute zero1 pressure

Note: A ll arrows are defined to point inpositive direction.FIGURE 1-3.-Definition of pressure scales.

For example, the cavitation index can be calcu-lated for the following conditions by using equations1.5 and 1.6: V = 30.0 m/sT, = 10C Pg = 9.8 kPa (1.0 m waterP, = 1.23 kPa column)

Pa = 101.0 kPa p = 999.7 kg/m3

u= ( 101.0 + 9.8 - 1.23) X lo3 = o 244999.7 [(30)2/2] *Table 1-l provides values of density, vapor pressure,and viscosity of water as a function of temperature.

Table I-l.-Proper ties of pure waterTemper- Density Vapor Kinematic

ature kg/m3pressure viscosity

C kPa m2/s X lo60 999.868 0.61 1.7875 999.992 0.87 1.51910 999.726 1.23 1.30715 999.125 1.70 1.140

20 998.228 2.33 1.00425 997.069 3.16 0.89330 995.671 4.23 .80135 994.055 5.62 .72440 992.238 7.38 .65845 990.233 9.58 .60250 988.052 12.3 .55360 983.20 19.9 .47570 977.77 31.1 .41380 971.80 47.3 .36590 965.31 70.1 .326100 958.36 101.3 .294

In practical situations, flow conditions are not asideal as those shown on figure 1-2. For instance,for a bluff body within a boundary layer, thedefinition of the cavita tion index depends upon thereference location as shown on figure l-4. In thiscase, three different reference locations are noted;the three locations and corresponding cavitationindexes are:

1. Far upstream and outside of the effects ofthe boundary layer;

1.8

2. Immediately upstream of an offset and at themaximum height of the offset;

Uh = ph - pvp G/2 1.9


22/164

4 Cavitation in Chutes and Spil lways

-

Shear zone/

4

_----_---_--- _------._PedcY L -,3=- ,r=I,- ---_ ----_b--

---

FIGURE l-4.-Reference veloc ity definition.

3. Outside of the boundary layer and in theplane of the offset;

ub= ph- pvp v,/z 1.10where:Ph = pressure in free stream in plane of offsetP, = pressure in free stream upstream of

beginning of boundary layerP, = vapor pressure of watervb = free stream velocity in plane of offsetvh = veloc ity at height of offsetV, = free stream veloc ity upstream of beginning

of boundary layerFor shear flows, severa l forms of the cavitation

parameter have been proposed. Each is based uponeasil y measurable reference conditions. For exam-ple, the cavitation index for a submerged jet, oj,shown on figure l-5a is given by:

Pe - P"Uj - p vv2 1.11where:P, = pressure in core of jetV, = velocity in core of jetOther types of shear flow include discharges

through gates, valves , and orifice s as shown on figurel-5b. For these, to define an appropriate referenceflow and veloc ity is often difficult. In these cases,another form of the cavitation index, Kt , had beenrecommended by Bal l [l]. It is:

K t = &-f&vf&-Hiwhere:Hd = pressure head downstream of valve

(12 pipe diameters [l])

1 Number within brackets refers to the bibliography.

1.12

a. Submerged jet

r------L ------ ---- _____-------L.~~~~--~-~ v*/2g

2D 12D------- -b. Inline orfice

FIGURE 1-5.-Reference conditions for shear flow.Ht = total head (pressure plus velocity)

upstream of valve (2 pipe diameters [l])H, = vapor pressure expressed in head relativeto atmospheric pressure

An alternate form of this equation is:a-i%Kr = H, - Hd 1.13

where H, equals pressure head upstream of valve.These expressions can be shown to

equation 1.11 by:

and

~~ = Uj + 1 - C$ - (Aj/A,)2(Aj/Ao)2 + G

K r = Uj f 1 - cb - (Aj/A,)2cbwhere:

1.14

1.15

be related to

A, = cross-sectional area of upstream pipecb = Borda loss coefficient = (H, - &) / ( K2/2g)In these equations, energy loss between the

upstream section and the vena contracta has beenneglected.

Although one parameter such as the cavitationindex-in whatever form it is expressed-cannot


23/164

Basic Concepts 5

describe many of the complexities of cavitation, itis an extremely useful parameter to indicate the stateof cavitation in a hydraulic structure. For example,for flow past a sudden into-the-flow offset, cavitationwill not occur if the cavitation index (defined byequation 1.8) is greater than about 1.8 as noted onfigure 1-6. As the cavitation index decreases below1.8, more and more cavitation bubbles form withinthe flow. To the naked eye, the cavita tion appearsto be a fuzzy white cloud. However, a flashphotograph reveals that the cloud consists ofindividual bubbles. For even lower values of thecavitation index, the cloud suddenly forms one longsupercavitating pocket.

FORMATION OF CAVITATIONTheory

Water does not spontaneously change from theliquid to the vapor state as either the temperature

Flow-

a. No cavitation, u = 3.0

OcoasIonal cavltatlonbubbles observed

b. Incipient cavitation, u = 1.8

Many small bubblesFlow /

c. Developed cavitation, 0.3 < u < 1.8

One large cavlty3 CD&ached cavtty

d. Supercavitation, u < 0.3FIGURE I-&-Development of cavitation.

is raised or the pressure is decreased. Water whichhas been distilled and filtered many times cansustain extremely large negative pressures withoutcavitating. Cavitation and boiling are both observedto begin at the location of impurities in the flowor at minute cracks on a smooth boundary. It isnot known if particles of dirt serve as nuclei forthe vaporization. However, Katz [2] observed thatthe appearance of visible cavitation in flowing waterwas always preceded by the occurrence of a swarmof microscopic bubbles in a small region of the flowfield. The importance of bubbles-as cavitationnuclei-has been known for a long time and all ofthe theory, for the formation of cavitat ion, has beenbuilt up around the existence of microscop icbubbles in the flow.

Spherical bubble stability was studied by Dailyand Johnson [3]. The force balance on a hemi-spherical section of a bubble shown on figure 1-7containing water vapor and a gas is given by:

7rrz(Pv+Pp) = 2rrre+7rr2Poor

P+ Pp = $ + PO

1.16

1.17

6

8. 0 ri8. 0 ri

1.0 -1.0 -

6.0 -6.0 -

5.0 -5.0 - LSurfaoe tenslon.

4.0 -4.0 - I x IO-kgMass of air In bubbleI)I)44,:,: 3.0-3.0-

9-9- To =To = 20 C (293. 16K)20 C (293. 16K)aa

Large free gas contentLarge free gas content

Locus of neutral .stabllItyLocus of neutral .stabllIty

EOUIL IBRlW BUBBLE DIAMETER. m m

FIGURE I-7.-Equilibrium conditions for vapor bubblescontaining air.


24/164


where: -pt LPp =P =Em=?T=

radius of bubblereference pressure (of fluid surrounding

the bubble)partial pressure of gasvapor pressure of waterinterfacial surface tension3.14159. . .

Assuming gas in the bubbles obeys the perfectgas law:

Pp = nm & Tk = nmRu Tk = m R, Tkvg rrr3 rrr3 1.18where:m=n, =M=Rg =R, =Tk =vg =

mass of gas, kgnumber of moles = m/Mmass of one mole of gas, kgengineering gas constantr, J/(kg*K) for airuniversal gas constant, J/(mole*K)temperature (absolute), Kelvin3volume of gas

Equation 1.17 becomes:p,-p, = 2 - :

where Ag is constant for a specific of gas, mass, andtemperature; i.e., Ag = rnRiTk/r.

Equation 1.19 gives the solution of a family ofcurves for various quantities of air in the bubblesas shown on figure l-7.

These curves represent the equilibrium conditionfor the vapor bubble containing air. The stabilityof the bubble can be investigated by examining thefull differential equation of the bubble [4]. Theequation is:

1.20,[r$ + i($)2] =- (fi-Pv+$-$)

If the right side of the equation is positive, thebubble will grow. Conversely, if the right side ofthe equation is negative, the bubble will decreasein diameter. Thus, the right side of the equationrepresents a force that causes the bubble size tochange. The bubble stability is determined by thealgebraic sign of the right side of the equation. Thecondition of neutral stability-known as the locusof neutral stability-occurs when the right side ofequation 1.20 is equal to zero. This curve, shownon figure 1-7, is defined by:

pd=po-p= -4 L3 ( 1rc

where:Pd = difference between minimum fluidpressure and vapor pressure

rc = critical radius = mi (m:ET*)

This theoretical development provides insightinto the mechanics of the formation of cavitation.For example, if a bubble with a small free gas contentis convected into a low-pressure region, the bubblediameter will remain about constant. However, ifthe pressure decreases enough for the bubbledimensions to reach the criti cal radius, the bubblesize will increase explosively (fig. l-7). The rapidexpansion is only possible through vaporization ofthe water with little contribution from theexpansion of the gas. Therefore, the process is oneof vaporous cavitation.

Conversely, if a bubble containing a large freegas content is convected into a low-pressure region,the bubble diameter will continually expandwithout ever reaching a criti cal radius. In this case,the bubble expansion is primari ly due to expansionof the gas. This nonexplosive growth of the bubblein which vapor pressure is never reached is char-acteristic of gaseous cavitation.Empirical Verification

A theory should not only provide insights intothe mechanics of a phenomenon, but it also shouldbe experimentally verifiable. Ho11 [S] initiated anempirical study of the effects of air content oncavitation by rewriting equation 1.17 in thefollowing form:

P-P, _ P-P,PC/2 - pv32 +p*2 - 2E/r 1.22o p vi/2

where:P=-

;z

pressure (from which the bubble begins todecrease in size)

Henrys law coefficientgas concentration (moles of gas per mole of

solute)In comparing equation 1.22 with equations 1.3 and

1.5, observe that:

. the first term corresponds to u,l the second term corresponds to -C, ,. the third term represents the effects of the gas

in the bubble, and. the fourth term represents the effects of surfacetension.

2 The properties of dry air (molecular weight = 28.97 kg) are:engineering gas constant 287 J/(kg*K), anduniversal gas constant 8314 J/(mole-K).3 Kelvin = 273.16 -I C

In equation 1.22, the partial pressure of the gasin the bubble has been replaced by the saturationpressure of the gas. For small gas concentrations,


25/164

Basic Concepts 7

the saturation or partial pressure of the gas is givenby Henrys law: ps=ap 1.23

The value of Henrys law coefficient is a functionof both the type of gas and the water temperatureas noted on figure 1-8. The coefficient is expressedin terms of pressure per mole fraction. The molefraction is defined as the number of moles of gasper mole of solute.

Ho11 [5] found good agreement between the onsetof cavitation predicted by equation 1.22 and theinception of cavitation on a hydrofoil for airconcentrations between 7 and 17 moles of air permillion moles of water. One example of the testsis that of investigations on a hydrofoil shown onfigure 1-9. The test velocities ranged between 12 and23 meters per second. The observed bubblediameters were on the order of 0.25 millimeter. Thepoints denoting area1 cavitation correspond to

vaporous cavitation. The points denoting spots ofcavitation were individual bubbles on the hydrofoilsurface that correspond to gaseous cavitation. Forthis hydrofoil, the theoretical minimum Eulernumber is 0.8. Test data indicate a cavitation indexof 0.45. The reason for the difference between thesetwo values is unknown.

NACA 16012 HYDROFOILSt MA)( =0.12 c0.5 c

7-d 1L

5040

30

20Carbon Dloxlde

I I I I Values from Internatlonal I

0 20 40 60 SO 100

I I I I I I I I I

0 15 30FREE-STREAM VELOCITY, V, m/s

LEGENDr = 0.0625 mmT, = 23.6 C= 17.3 x I;= 7.1 x IO +8 kPa/moleE = 0.0740 n /mp = 997.39 kg /m

TEMPERATURE, =C

FIGURE 1-8.-Henrys law coefficients.FIGURE 1-9.-Effect of velocity on vaporous and

gaseous cavitation.


26/164


The order of magnitude of the gaseous and surfacetension terms in equation 1.22 and their relativeimportance in the experiments of Ho11 is estimatedin table 1-2. The cavitation indices are calculatedfrom equation 1.22 and the minimum value of theEuler number observed from the experiments. Thevalue of the air concentration is not significant forvaporous cavitation nor is surface tension significantfor gaseous cavitation.Table I-2.-Influence of surface tension and air content

on incipien t cavitation index

V=21.3 m's

Temperature, CHenrys coefficient, kPa/mole fraction 23.67.1 x lo6Air concentration, mole fraction 17.3 x 1o-6Water density, kg/m3 997.39Surface tension, N/m 0.0740Bubble radius, mm 0.0625Mass of gas in bubble, kg 1.1 x lo-l5Minimum Euler number 0.45

Velocity, m/sdissolved gas effects apcavitation index for P vo2/2

gaseous cavitationsurface tension effects 26/r-p v:/2cavitation index for

vaporous cavitation

15.0 30.00.85 0.212

1.30 0.66- 0.048 - 0.005

0.402 0.445

The experiment of Ho11 indicates that themagnitude of the flow velocity and the dissolvedair concentrations have a major effect on theoccurrence of cavitation. The significance is that ifmodel studies are to predict accurately the cavita-tion potential of flows in the prototype, the modelstudies must be conducted with either highvelocities or low dissolved air concentrations.

The requirement for high velocities means thatEuler or Reynolds scaling may not be possible forclosed conduit models. Similarl y, Froude scalingmay not be possible for open channel models ifaccurate simulation of the cavitat ion phenomenonis to be obtained.

The influence of air content and surface tensionon the magnitude of the cavita tion index are notthe only factors which can be empirical ly verified.Plesset [6] investigated the change in size of acavitation bubble as it passed through a pressuregradient as shown on figure l-10. The theory, asexpressed by equation 1.20, predicts reasonably wellthe growth and decay portions of the bubble history.

COLLAPSE DYNAMICSIn the previous section, growth and decay of

bubbles was considered using a theory that assumedwater to be incomp ressible. In addition, the vapor

/* Exm-ImentaIDoInts'

60 o4

NUMBER OF FRAMES ( I franc = 5 x 10m5second)Note: One frame is approximately equivalent to 1 mm

of displacement along the body of revolution.Theoretical curve: solution for a cavi ty ofconstant vapor density and temperature inan incompressible liquid with a variablepressure field.

FIGURE I-lO.-Bubb le history for flow around a1.5-caliber ogive (Plesset [6]).

pressure, surface tension, and temperature were allconsidered to be constant. In reali ty, as a bubblecollapses these assumptions are not valid. Tosimulate the collapse dynamics it is necessary toconsider the compressibil ity of water, the compres-sib ili ty of the gas in the bubble, and the enthalpychanges. These considerations result in six differ-ential equations and four algebra ic equations thatmust be solved simultaneously [4].

Numerical solution of the equations reveals thatbubble collapse consists of phases in which thebubble diameter decreases, reaches a minimumvalue, and then grows or rebounds as shown onfigure l-11. The process is repeated for several cycles


27/164

Basic Concepts 9

-TimeOriginal bubble sizea. Collapse of an individua l bubble in a quiscent fluid

,High velocity jet of watercd Torus Pressurek

b. Collapse of an individua l bubble in a hydraulic gradient

Primary shock waveSecondary shock wave

////////////////////////c. Collapse of an individu al bubble near a boundary

Shock wave fromrebounding bubble0

Bubbles collapsedMassive shock wave fromrebound of bubble swarm

d. Collapse of a swarm of bubblesFIGURE 1-ll.-Collapse mechanisms of bubbles.


28/164


with the bubble diameter decreasing during eachcycle until it finally becomes microscop ic in size.

During the reversal or rebound phase, a shockwave forms. The shock wave velocity-as it radiatesoutward from the center of collapse- is equal tothe speed of sound in water. Assuming that wateris incompressible, Hickling and Plesset [7] foundthat the shock wave intensity varies inversely withthe distance from the collapse center. At a distanceof two times the initial bubble radius from thecollapse center, the pressure intensity is about 200times the ambient pressure at the collapse site. Thefollowing example illustrates a method of estimatingthe collapse pressure.

Assume a bubble diameter of 0.1 millimeter anda flow depth of 2.0 meters, thenWater pressure ==Ambient pressure =

==Pressure intensity 0.1==

2.0 m (999 kg/m3) 9.8 m/s219 580 Pawater pressure + barometricpressure19.58 kPa -I 101.3 kPa120.9 kPamm from collapse center200 X 120.9 kPa24 200 kPa

These computations ignore the effect of the initialbubble diameter and the water temperature oncollapse pressure. Fujikawa and Akamatsu [8] showthat both parameters have a significant influence.They found that the collapse pressure increased aseither the initial bubble diameter or the watertemperature decreases.

Theory also allows estimating the time for abubble to collapse. The collapse time, 7, is givenapproximately by [4]:

i- = Ro ( )e wPOwhere:PO = reference pressure (at collapse site)R, = initial radius of bubblep = density of water

For a bubble having an initial diameter of 0.1millimeter and an ambient pressure correspondingto 2.0 meters of water, the collapse time is:

7 = 0.0001(~> = 9.1 microseconds

Several factors modify the collapse mechanism ofa purely spherical bubble. For instance, if thebubble collapses in the presence of a pressuregradient, the shape of the bubble does not remainsymmetrical. Pressure gradients exist in flow aroundsubmerged bodies. If the bubble collapses near aboundary, the boundary restric ts the flow towardthe bubble which also causes an asymmetric collapse.

Both cases cause one side of the bubble to deforminto a jet which penetrates the opposite side of thebubble as depicted on figure l-11. The jet formedby the unsymmetrical collapse of a single bubbleis called a microjet. The velocities of microjets arelarge. Hammitt [9] concluded that in most casescavitation damage was due primari ly to the liquidmicrojet impact on the surface and not to thespherical shock waves which eminate from therebounding bubble. However, more recent photo-elastic studies [8] have shown that the shock wavegenerates much higher pressure impulses than thejet.

If more than one bubble is present, the collapseof the first will produce shock waves that radiateto other bubbles. These shock waves will cause thesudden unsymmetrical collapse of neighboringbubbles. The jet formed by the unsymmetricalcollapse of a bubble caused by shock waves is calledan ultrajet. The velocities produced by ultrajets areon the order of one-half the sonic veloci ty of theliquid, Tomita and Shima [lo]. Ultrajets generatehigher pressure intensities than either sphericalshock waves or microjets.

Unfortunately, theory does not exist to predictthe magnitude of the pressures generated by thecollapse of a swarm of bubbles. One can hypothesize:if one bubble in a swarm collapses, the shock wavethe bubble produced during rebound will causeother bubbles in the vicin ity to collapse. The processwill continue in the form of a chain reaction untilthe remainder of the swarm effectively collapsessimultaneously. It is logical that the synchronouscollapse of a bubble swarm would produce a higherpressure intensity than the random collapse ofindividual bubbles in the swarm.

Another important factor influencing collapse ofthe cavitation bubble is the presence of vorticeswithin the flow. Shear flows generate vortices thatcollect bubbles on their axes. Depending upon theproximity of bubbles, they may remain near eachother in a swarm or they may coalesce into onefilament shaped bubble as shown on figure 1-12.

Damage caused by the group of bubbles-trappedon the axis of a vortex-can be many times greaterthan that caused by the collapse of an individualbubble or even a swarm of bubbles. For example,an individual bubble having a 2.7-millimeterdiameter can create a depression in aluminum whichis about 0.2 millimeter in diameter [4]. However,cavitation occurring in the vortices of the shear layerformed by a partially opened slide gate (operatingsubmerged) has caused depressions up to I6millimeters long and measured pressure intensitiesof 1500 megapascals, Lesleighter [ 111.

SONIC VELOCITY OF GAS MIXTUREThe forgoing noted the sonic velocity of the fluid

surrounding the collapsing bubble is a significant


29/164

Basic Concepts 11

parameter with respect to the pressure intensitiesgenerated by the collapse . The most importantfactor influencing the sonic veloc ity of the surround-ing fluid is the presence of an undissolved gas inthe liquid. In this case, the gas is air in the formof air bubbles. For this development, the bubblemixture is considered to be a uniform distributionof air bubbles in water.In solids , disturbances produce longitudinalwaves, transverse shear waves, and gravity waves.However, fluids cannot sustain a shear wave. Thelongitudinal waves in a liquid propagate at the speedof sound. The veloc ity of sound in an elastic mediumis inversely proportional to the square root of thedensity of the material and direc tly proportionalto the square root of the bulk modulus. The sonicvelocity, V, , is mathematically expressed as:

v, = (Jywhere:

B, = bulk modulus of mixture = - AP/(Au/uJAP = change in pressureAu = change in volume

u0 = original volumePm = density of mixtureThe density, pm , of a homogeneous bubble mix-

ture is given by:

where:P=

Pb =*=

Pm = PQ-llr)+ Pgrlr

density of waterdensity of gas

1.26

volume of air per volume of mixtureThe gas density, pg , is determined from:

P-- - Rg Tk 1.27

where:Rg = engineering gas constantTk = temperature (absolute), Kelv inThe bulk modulus, B,, of a homogeneous mix-

ture can be obtained from:

Bm = (Au,/,) ::A,,/,,or

1.28

1.29whqe:Bw = bulk modulus of waterBg = bulk modulus of gas

The bulk modulus of water varies with temper-ature. However, the variation is less than 4 percentfrom a value of 1.07 gigapasca ls over the temperaturerange 0 to 15 C.

The bulk modulus of air for an adiabatic processis given by: B,=kP 1.30where:k = ideal gas constantP = pressure intensity

Substitution of equations 1.26, 1.29 and 1.30 intoequation 1.25 gives;

K=( kB,P ) l/2[(l - $)kP+ 1crBw1[(I 3,)~ + hl 1.31This relationship has an unusual characteristic.

At atmospheric pressure and an air concentrationof only 0.1 percent, the sonic velocity of the mixtureis equal to the sonic velocity in air as shown onfigure 1-13. For higher air concentrations-atatmospheric pressure-the sonic velocity is less thanthe sonic velocity in air! At higher pressures, a largerair concentration is needed for the sonic veloc ityof the mixture to equal the sonic veloc ity of air.


30/164

12 Cavitation in Chutes and Spil lways

FIGURE 1-12.-Cavitation of baffle piers in Bonnev ille Dam model[US.Army Corps of Engineers].

14oc

1200

400

200

0

- Asymptote 1440m/s

Sonic velocitySonic velocityof air= 285 m/sof air= 285 m/s

0.001 0.01 0.1AIR CONCENTRATION, volume air/volume mixtureFIGURE I-13.-Sonic veloc ity of air-water mixtures.


31/164

Basic Concepts 13

BIBLIOGRAPHY[l] Ball, J. W., Cavitation Characteristics of Gate

Valves and Globe Valves Used as FlowRegulators Under Heads up to About 125 ft.,Transactions of the American Society ofMechanical Engineers, vol. 79, No 9, August1957.

[2] Katz, J., Cavitation Phenomena WithinRegions of Flow Separation, Journal ofFluid Mechanics, vol. 140, pp. 397-436, 1984.

[3] Daily, J. W., Johnson, V. E. Jr., Turbulenceand Boundary Layer Effects on CavitationInception From Gas Nuclei, Transactionsof the American Society of MechanicalEngineers, vol. 78, pp. 1695-1706, 1956.

[4] Knapp, R. T., Daily, J. W., Hammitt, F. G.,Cavitation, McGraw-H ill, Inc., New York,1970.

[5] Holl, J. W., An Effect of Air Content on theOccurrence of Cavitation, Journal of BasicEngineering, Transactions of the AmericanSociety of Mechanical Engineers, series D,vol. 82, pp. 941-946, December 1960.

[6] Plesset, M. S., The Dynamics of CavitationBubbles, Journal of Applied Mechanics,American Society of Mechanical Engineers,vol. 16, pp. 277-282, 1949.

[7] Hickling, R., Plessett, M. S., Collapse andRebound of a Spherical Bubble in Water,Physi cs of Fluids, vol. 7, pp. 7-14, 1964.

[8] Fujikawa, S., Akamatsu, T., On the Mecha-nisms of Cavitation Bubble Collapse,International Association for HydraulicResearch, 10th Symposium of Section forHydraulic Machinery, Equipment andCavitation, Tokyo, pp. 91-102, 1980.

[9] Hammitt, F. G., Cavitation Erosion: TheState of the Art and Predicting Capability,Applied Mechanics Reviews, vol. 32, No. 6,pp. 665-675, June, 1979.

[lo] Tomita, Y., Shima, A., Mechanisms ofImpulsive Pressure Generation and DamagePit Formation by Bubble Collapse, Journalof Fluid Mechanics, vol. 169, pp. 535-564,1986.

[ll] Lesleighter, E., Cavitation in High-HeadGated Outlets - Prototype Measurementsand Model Simulation, InternationalAssociation for Hydraulic Research, 20thCongress, Moscow, vol. 3, sec. b., pp. 495-503, September 1983.


32/164


33/164

15

Cavitation Chapter 2Characteristics of Flow SurfacesCATEGORIES OF SURFACE ROUGHNESS

Upon examining the flow surface of a hydraulicstructure, the flow surface irregular it& or thesurface roughness can be characterized usually asbelonging to one of two main categories (1) singular(isolated) roughnesses or (2) uniformly distributedroughnesses. Singular roughnesses are irregulari tiesin a surface that are large relative to the surfaceirregulari ties from where they protrude. A uni-formly distributed roughness is a surface texturethat does not contain singular roughnesses. Betweenthese two extremes is a category of moderate heightsingular roughnesses in combination with a uni-formly distributed roughness which has not beensystematically studied.Sometimes, singular roughnesses are referred toas local asperities. Typical examples of these inhydraulic structures include:l offset into-the-flow, fig 2-la;l offset away-from-the-flow, fig 2-lb, c, d;l voids or grooves, fig 2-le; andl protruding joints fig 2-lg.

In all these cases, cavitation is formed byturbulence in the shear zone; the action is producedby the sudden change in flow direction at theirregularity . The location of the shear zone can bepredicted from the shape of the roughness.Depending upon the shape of the roughness,cavitation bubbles will collapse either within theflow or near the flow boundary.

Figure 2-lf depicts cavitation above a distributedroughness. Cavitation occurs within the flow becauseof turbulence generated by the roughness of theboundary. The cavitation location is not predictable;however, cavitation always occurs within the bodyof the flow for distributed roughnesses.

This chapter contains compilations of cavitationcharacteri stics for various types of roughnesselements. The correlations were derived, for themost part, from tests conducted with either asingular roughness or a distributed roughness. Anexample is given for tests conducted with a typicalconcrete surface eroded by sand abrasion. Thisexample illustrates the need for further studies ofthe combined effect of singular roughnesses on largescale distributed roughnesses.

FLOW- [Vapor cavities

a. Offset into flow

[Vapor cavities

b. Offset away fromflow/Vapor cavities

;_..,c. Abrupt curvature

away from flow__ /Vapor cavities

wiE?&.-:e. Void or transversegroove

d. Abrupt slope awayfrom flow- (Vapor cavities\

f. Roughened surface

A/soar cavities

g. Protruding jointFIGURE 2-l.- Typical isolated roughness elements foundin hydraulic structures.


34/164

16 Cavitation in Chutes and Spillway s

Flow over curved boundaries that contain singularroughnesses exhibit different cavitation charac-teristi cs than flow over a flat plate having the samelocal asperities. Therefore, a method is given toestimate cavitation characteri stics for flow overcurved boundaries. This method is known as thesuperposition principle.

SINGULAR ASPERITIESTwo types of singular asperities exist: misalign-

ments and isolated irregularities. Misalignmentsinclude both sudden and gradual changes in thealignment (figs. 2-la, b, c, and d). The height,dimension of misalignments is many times smallerthan the length dimension. Isolated irregulari tieshave approximately equal magnitudes of length andheight (see fig. 2-5). For each of these specific classesof surface roughness types, cavitation indexes wil lbe discussed in the following sections.Into-the-Flow Offsets

The earliest studies of singular asperities concen-trated on cavitation characteris tics at misalign-ments. Investigations of into-the-flow offsets werepresented as design curves by Colgate and Elder [l]

and are shown on figure 2-2. (Many of these curveswere used by Ball [2]). Data were obtained in a testfacili ty which was 102 millimeters high by 152millimeters wide. The test section was locatedimmediately downstream from a speciall y designedtransition that produced an extremely thin bound-ary layer. For the larger offsets, the data are in errorbecause blockage effects were not taken intoaccount. The families of curves were corrected forblockage and expressed in a more compact form in1977 by Colgate [3]. The condensed form was madepossible by expressing the curves in terms of acavita tion index referenced to the plane of theoffset, see equation 1.10. The technique permittedall the characterist ics of the sudden offsets to bedrawn on a single plot as shown on figure 2-3a. Forchamfers having length to height ratios less than5:1, the cavitation characteris tics are a function ofthe offset height. A similar plot, shown on figure2-3b, was made for rounding the leading edge ofthe offset to an ellipse.

Jin, et al. [4] also conducted studies of offsets andchamfers in a water tunnel. The test section was200 by 200 millimeters and located immediatelydownstream of a convergent section. The boundarylayer was extremely thin in this facility -as withthe tests by Colgate. Jin found that the incipientcavitation index, oi, for chamfers could be expressed

100

90

8070605040

I Vapor pressureO-! I0 IO 20 30 40 50 60 70 80

I I I - , ^

AVERAGE VELOCITY, V, AT OFFSET, ft / sFIGURE 2-2.-Incipient cavitation characteristics of offsets (Colgate and Elder [lfl.


35/164

Cavitation Characteristics of Flow Surfaces 17

Point ofv inclplent cavltatlon

$.Y &,se//'

I. 8I. 6

0. 80.6

01234567 0 I 2 3 4 5 6SLOPE, run / r lse a/ba. Chamfers b. Ellip tical rounding

I. 8

1.6

I. 4

0. 8

0. 6

FIGURE 2-3.-Incipient cavitation charac teristic s of chamfered and ellip tical ly rounded offsets.

according to the following empirical relationshipfor L,/H > 5.0:

I I \ - 0.7u; = 1.8 (%) 2.1

where:H = height of offsetL, = run of chamferoj = incipient cavitation indexZharov [5] conducted experiments wherein the

boundary layer th ickness was 9 to 11 millimeters.Offset heights of 3, 6, 10, and 15 mill imete rs weretested. By using a cavitation index referenced tothe height of the offset, as in equation 1.9,invest igations showed the index was independentof the relat ive height and Reynolds number. TheReynolds numbers tested ranged from lo3 to 106.

A comparison of results, by the three investigators,shown on figure 2-4 reveals that the experimentsof Colgate and Jin agree closely and actually com-pliment each other. For small chamfers (L,/H< l),Zharovs results also agree with Colgate and Jin.However, for large chamfers, the cavitation char-acteristi cs differ. One reason for the difference in

the results might be that Zharov tested with asignificant boundary layer thickness.

Another method of accounting for the velocitydistribution within the boundary layer was basedon the frozen streaml ine theory of Ho11 [6]. Heproposed that the relat ionship between the cavita-tion index without a boundary layer was related tothe cavitation index with a boundary layer by:

Uk = Uu 2.2where:

V, = characteristic velocityVu = reference free stream velocityUk = cavitation index based on average ve locityover height of offsetUU = cavitation index without a boundary layerThe characteristi c velocity is the average velocity

over the height of the offset and is defined as:


36/164


\\

-


37/164

Cavitation Characteristics of Flow Surfaces 19

where:H = height of offsetV, = velocity at height yV, = characteristic velocityy = vertical coordinate (distance from

boundary)

For the one-fifth power law distribution ofvelocity, equation 2.2 results in:0.4

2.4where 6 equals boundary layer thickness.

Clear ly, equation 2.4 demonstrates that the dataof Zharov should lie below that of Colgate and Jin,when all three are compared using a cavitation indexthat is based on a thin boundary layer.

This approach, by Holl, should permit tests bothwith and without a boundary layer to be referencedto a common base. For example, the cavitat ion indexwith a thin boundary layer should be- able to bepredicted from equation 2.4 using data taken fromtests with a boundary layer. However, this approachonly yielded fair results. Therefore, attempting tocharacterize the cavitation index only by somerepresentative velocity within ,the boundary layerappears too simplis tic.

Other studies of isolated asperities used a bettermethod to account for the velocity distribution inthe boundary layer, Arndt et al. [7]. They concludedthat cavitat ion index should be a function of therelative thickness of the boundary layer, theReynolds number of the flow, and a boundary layershape factor. This is expressed as:

uj o= f ( 5, Rh , s)where:

Rh = Reynolds number based on height of offsetand free stream velocity

S = boundary layer shape parameterThe boundary layer shape parameter, S, is defined

as the ratio of the displacement thickness to themomentum thickness:

s= + 2.6mThe displacement thickness, ad, is defined as:

6d =6(l- E) dy 2.7

and the momentum thickness, 6,) is defined as:

where I$, equals free stream velocity in the planeof the offset.

For a fully developed boundary layer over a flatsmooth surface, the shape parameter, 8, is approx-imately equal to 1.3.

To correlate the data for several types of isolatedroughnesses, equation 2.5 is assumed to have thefunctional form:

2.9

where u equals kinematic viscos ity and B , 6, C, andCare experimentally determined coefficients. It wasnot possib le to calculate the value of the shapeparameter for all the tests reported by Arndt etal. [7]. Therefore, for the majority of the tests,correlations were formed by ignoring the shapeparameter term and using an equation of the form:

2.10

The values of the coefficients are shown in thetable on figure 2-5. The range of the relat ive offsetheights for the two-dimensional offsets was about0.015 < H/6 < 3.4.

The relative importance of the boundary layershape parameter can be discerned in table 2-1 fora triangular and a circular arc irregularity. In thesetwo cases, sufficient measurements were made sothat the boundary shape parameter could bedetermined.

Table 2-l.-Triangular and circu lar arcirregularity coefficients

Irregularity B b c cTriangular 0.361 0.196 - 0.152

0.40 0.20 0.45 0.182Circular arc 0.344 0.267 - 0.0410.30 0.15 0.70 0.171

The following correlations are recommended fordesign purposes.

Into-the-flow misalignments:Elliptically rounding . . . . . . . . . fig. 2-3bChamfered leading edge . . . . . . . fig. 2-4Isolated irregularities:Triangles, circular arcs, etc. . . . . fig. 2-5

Since the into-the-flow misalignments do notinclude the effect of boundary layer, cavitationindexes will tend to be too large. If desired, part ofthe boundary layer effect can be approximated byusing the values of the cavitation index taken fromfigures 2-3b and 2-4 together with equation 2.4.


38/164


Flow DataSymbol Irregularity dimensions source- ..--__ 0 bn Triangles 2 Holi, 1960 0,361 0. I96 0.152

0 Circular arcs 2 Hall, 19

Documents

Cavitation in Chutes and Spillways. Engineering monograph nº 42, USA Bureau of Reclamation