I lxr. 1(, 1 l)ltitr,ttlo||.tl llo||rrurrlly2. IltcrlrvcIrrrrrr.rrl ol ltr.llr(,tivctl(.llx](l li)ren(,rlty (lissil)itti()l ill lltcoutlcli,l it lty(lrilltlir stlllcll ci.1. llrc t.rhlcliol ol r.t t(.rAy l( )ss tl tllc intakc structurc or rt thc trlnsition sectiotr:4. lhc rlcvclolrrcnl ol n cllicient, economic spillway or otfier type offlood_

rL.lritsiltE slrUCturc li)r a reser\oir;5. tltc detennination ofan average time oftravel in a temperatur control stuc_

ture, for example, in a cooling pond in a power plant;6. the determination for the best cross section, location, and dimensions of

the various componelts ofthe structure, such as the breakwater, the docks,and the locks, etc., in harbor and waterway design;

7. the determination ofthe dynamic behaviors ofthe floating, semi-immersible,and lloor installed structures in transportation and installation of ofTshorestructureS.

lliver models have also been extensively used in hydraulic engineering to determine1. the pattern a flood wave travels through a fiver channel;2. the et]ect of artificial structures, such as bends, levees, dikes, jetties, and

training walls, on the sedimentation movements in the channel reach- aswell as il the upstream and downstream channels:

3. the direction and force of currents in the channel o harbor and their effecton navigirtion and marine Iif'e.

10.1 Dimensional HomogeneityWhen a physical phenomenon is described by an equation or a set of equations, allterms in each of the equations must be kept dimensionally homogeneous.* In othewords, all terms in an equltion must be expressed in the same units.

In fact, to derive a relationship antong seveml plranetcrs involved in a physicalphenomenon, one should always check the equation for honogeneity of units. If allterms in an equation do not appear to have the same unit, then oDe can be sure thrt cer_tain important parameters may be missing or.misplacerl.

Based on the physical understanding of the phenomenon and the concept ofdimensional homogeneity, the solution of many hydraulic problems may be formu_lated. For example, we understand that the speed of surface wave propagution onwater, C, is related to the gravitational acceleration, g, a[d the water depth, rl. Gen_erally, we may wdte

c = f(g, d) (ro.l)The units of the physical quantities invoived are indicated in rhe brackets.

c = [LT_tls=lLT2ld = l,Ll

+ wilh a few exceptions. such as the empi cal equations (e.g.. Section 3.6).

Hydraul ic Simi I itude andModel Studies

Use of small models for studyirg the prototype hydraulic design can be dated at leslto Leonardo da Vinci.x But the method developed for using the results of experimL'nt\conducted on a scale(l model to predict quantitatively the behavior of a full sizL.hydraulic structure (or prototype) was realized only after the turn of this century. Tlxprinciple or which the model studies are based comprises the theory of trydrauli,similitude. The anrlysis of the basic relationship of the various physicai quantiricsinvolved in the static and dynamic behaviors of watcr flow in a hydraulic stl'ucturc i\known as dimensional analysis.

All important hydraulic structures are now designed and built after certain prcliminary model studies have been completed. Such studies may be conducted for anyone or more of the following purposes:

1. the determination of the discharge coefficient ofa large measurement structure, such as an overflow spillway or a weir;

* Leonardo da Vinci (t452 l5l9), a gcnius, Renaissance sciertist, engineer, archilect. painlcr.sculplor, ard musician.

296

2gA liyrlrlll: lilllrllllrl rl l\4rxlrrl !lrrln (:ltt, ll,

Sincc thc lcll-hllnrl si(lc ol lillritliou (lO. I ) ltts lltc ll its r]l l/,/' ll. tll(,sc l ts ll r Itltcn Ippcar oxplicitly on I hc riglll ltrrtrlsitlc. lhrrs,/irrtltrrust corrrlrirrr.its ir l)r(nlr, rand the f unction, /; must bc tl'rc squaro r'(xf . Wc lrvc

c=Jndas discussed in Chapter 6, (Equation 6.I l).

The dimensions of the physical quantities commonly uscd in hydllrrlir, crrl,rneering are listed in Tabte 10.1.

TABLE 10.1 Dimensions of Physical OuanttiesCommonly Used in Hydraulic Engineering

Quantity Dimensior Quantity

rrxr l(l 2 llhlilt'lndofllyrllrlllll'.tilllllllltlrl 200

An rrcir. 1. is llr('ro{lu(I ol lwo lrotlrtlofotts lettgllts; ltttttc. lltt l'tlio ol llteIrtttolrrltrtts itt'cr is itls() r aorlsllrrl itttrl r'itt lx r'xrtessctl as

(10.-1)

A volumc, Vol, s the product of three homologous lengths; the ratio of theIurrollAorrs volume can be expressed as

Volo _

Vol,, =L1 ( 10.4),jL3,,

Dinrcnsiol

LengthAreaVolumeAngle (radians)TimeDischargeLinear VelocityAngular VelocityAccelerationMassMoment of InertiaDensityViscosity

LL2L3NoneTL1T II,T-ITtMML2ML1MT:1T I

MLtMLIIMLtl'ML 1t'ML t',lM I L:tM',l )MLl'MLl',t )ML2't 'ML21 )ML27''

ForcePressureShear StressSpecific WeightModuhs of ElasticityCoefficient of CompressibilitySurface TensionMomentumAngular MomentumTorqueEnergyPowerKinematic Viscosity

l,rnnpllj l0.l\ ltr.(rrot ically similal open channel model is constructed with a 5:1 scale. lf the model mea-,urr.s l (lischarge of 0.2 ml/sec, rletermine the corresponding discharge in the prototype.\0hrli(lnI lrr' vclocity ratio between the prototype and the model is

lr lt!.ometric similarity, time in model and pototype remains unscaled. The area ratio betweenrh prototype and the model is

L,,vP

_i _ L'V,,, L,, L,,,

t2* * t",, - Ll =25

= (2s)(5) = t25

10.2 Principles of Hydraulic SimilitudeSimiladty between hydraulic models and prototypes may be achieved in three bsr,forms:

1 geometricsimilarity,2. kinematic similarity,3. dynamic similarity.

Geometrc simLarity implies similarity of form. The model is a geomcrr lreduction of the prototype and is accomplished by maintaining a fixed ratio fbr rllhomologous lengths between the model and the prototype.

The physical quantities involved in geometric similarity are length , area,,.l.and volume, Vol. To keep the homologous lengths in the prototype and the modcl :rconstant ratio, they can be expressed as

lhus, the coi'responding discharge in the prototype is

Q, = 125Q,, = 125 0.2 = 25 mr/sec

Knematic simllrry implies similarity in motion. Kinematic similarity betweenI model and the prototype is attained if the homologous moving particles have thesirme velocity ratio along geometrically similar paths. The kinematic similarityirvolves the scale of time as well as length. The ratio of times required for homologousl)ilrticles to trayel homologous distances in a model and its prototype is

/\(e(ndingly, the discharge ratio is

er _

o,,,

T.LtL^ (lo..rI ( l0.s)

JOo llyrlr.lrtl !;l||rlllt rl, r'Iillvl'rl',| :;t rlti'r,'l'hc vcloci(y V is r[.lirrr.rl i tr.lrrs 0l (lislill(.(. 1.t tlrt ltl(.i llIr\,

vclocities can bc cxr|csscil as

l'rlll, llrl,,',,,1 l ly,lhrlli',IL!IIIIIIILr

(10.H)

lrurtrrle ll).2\ lll I s( rlL. rx,(lcl is const-uctcd to study the flow motion in a coolig pond. lf the designedlr ,lr,ult( lionr thc rower plant is 200 ml/sec and the model can accommodate l milxilum

11 , , .,1,' ,,1 ll.lrr'/.(r.dcterrnrnelhelim(rclio.'i,h'l "rll! l(.rllth rntio between the prototype ard the modei is

,ilr,,,,1 rl,, ,li\ lurrr rlri,, i. ?..

':^'9 'noo , rr0tLi,

o -Q, =-=I',l Ir'I= r: r'' A", t:,' L"' 't,1n

',rlr\rilLrting the legth mtio into the discharge mtio gives the timc rato

7 = l' = l - (lo) - o'.,"- r.. - ?, - 2{fru -

7,,, = 27,\ rrit time period measured in the model is equivalent Lo two periods of time in thelr0l(fype pond.

Dynamic similorty implies similarity in lbrces involved in rnotior. Dynamic,rril rity between a model and its prototype is attained ifthe ratio of homologousl,,recs in the model and prototype is kept a[ a constant value, o

F.l -r:

F,,,

Many hydrodynamic phenomena may involve seyeral different kinds of lorcesrr rrction. Sice models are usually built to simulate the prototype on rcduced scales,rlr(.y usually are not capable of simulating all the forces simultaneously. In pt.lrctice, arrrr cl is designed to study the l:l'fects of

302 Hydrutlo Slmllltltda tnd Modrl Btudl..multiplied by accclcration, rr, lnd sillco llass is crrrrrl lo rklsily, 1. lilrtrsVol, the force ratio tray he exprcssctl as

Fo -

Mooo =

Pr\olu'',F^ M*a^ p^Yol,, a,,

Chp, 1(l

volulr.,

lr0, 10,2 lrlnolpl.r ol Hydrullo tlmlllludaI llr, l,\.vcr rrlio is, lir)Dr li(lurtli,r { ll, lS )

,, [',1., -

t ]5(rXli) -

304

10.2.t.

10.2.2.

I lyrlrdullc lil rlllllrIr nlll Mr,rll iit llfll

PROBLEMS

(:lrlr lrr | !rr l(l il I'ltllI rl{ (i vr, |l,rlly Vthr [h I 0ll r, n,y otrIi N(||rl,!r I w

10.2,5. If a I :5 scale mode, at I 200 rpm. is used to study the prototypc of a centrifugal prr,lthat produces I m3/scc al 30-m head when rot"ting ai400,pm, cleter.rnine the nr,,t,discharge and head.

10.2.6, A I:25 scale modcl is being dcsigncd to study a prototype hydralrlic str.ucture. It,vclocity ratio between the model and the prototype s l:5, and the neasurement l(.( ,racy is rcquired to be within I q. of the total force. Determine the accuracy of thc I0r,mcasLrement in the model ii the expected fbrce on the prototype is 45,000 N. l,/r/lluse Equation (8.9)1.

10.2.7. Sedimentation in a river secfion 5 km long is to be studied in a lab channei 20 n l0rA time ratio of4 will be used and the prototype discharye is 75 nf/sec. Select a suilirt,llength scale and determine the model discharge.

10.2.8. A l:20 scale model of a prototype energy dissipation structure is constructed ro str, tIorce distribution and water depths. A velocity ratio oi 7.75 is used. DeterniDc lt,fbrce ratio rnd prototype dschae if the model discharge s 3ou/sec.

10,2.9. A 1:50 scale model is used to study tlre power- requirements ofa prototype subn)ir rrThc model will betowedataspeed50timesgreaterthanthespe;dofthepr_orotyr,a tank tilled with sea watcr_ Detel-minc the conversion ratios lrom the prototype lo ttmodel for rhe following quanriries: (a) rime, (b) fbrce, (c) eneqy, (d) power.

t 0.3 Phenomena Coverned by Viscous Force-Reynolds Number Law

Water in motion always involves inertial fbrces. When the inertia forces and thc \ rcous foces can be considered to be the only fbrces that govem the motion, thc r:rlr,

n, it rclliirl lolcc"" - ;"..-:,.;.' f,,...

,r lrr.r'e r is the viscosity and ydenotes the velocitv.F.quating rulue: ol ,trorn Equation:r 10. l.t I and (10. g r. weget

p,Llr;2 = t,.tlr;1lrr l rr which

p,rlt,,=o]l _p,t,v, _ ,

P, Ll r ,' P,r, tt.lii'rrrlanging the above equation, we may write

( Prtrv,lL-i( p,,,1,,,v,, \t-l

PrLrv,, P,,,L^v,,lt P',

A l-ln lo0!!, l:ilo nlotlcl is usctl kr stutly llrr witvc l(,rcr otr:t In(,t(,tr-l)(,r)l lt sr.:r \\structure.llthe total wavc lbrcc lrtetslr.cd on thc I(xlul is 2.27 N irrl(l ii,.vct,r,.il, * ,i\ l:10. dete11nine thc tbr\.rper unI lerrth rrt rlrr. pr,,t,,tThe moment exerted on a gate s(ructure is studicd i r lirborrtor.y wulct.trnk \\rtrllI:l25scalemodel. Il the moment measured on the m, rdc I i\ l.5N ||r tlllth(j ll|rt,,rgate arn, determine the moment exerted on the prototypc.A l:100 scale model is constNcted to study a gate ptototype tht is dcsigned lo (lr, r,a reselvoir. Ifthe mociel rcservoir is dmined in 5.2 min, how long should iitake to rlr.,, ,the protu) pe?An overflow spillway is desjgned to have a 100-m hng crest to calTy the dischtr!,.I 1.50 m'/.ec under it pcrrniled rnr\imum head,,1'. m. fhe opcrlioir ol thc r,,t,,r spillway is studied by a I:50 scale rodel in a hytlraulic laborator.y.(a) Detenninc fhe model disclrarge based on Equation (g.9).(b) The model velocity mesurcd at the end (toe) of the spillway is 3 m/sec. Deter rr r r

the corresponding velocity in tlte prototype.(c) What are thc model and prototype F!.oude numbers at the toe,l(d) If the lorce on a bucker energy dissipator ar the toe of the spillway is nreasulr(t t,be 37.5 N, determine the corresponding tbrce on the protorpe.

rrl llr..rI lor'"s lrclilllt olr IorrroIr J,orr., |trtir,lcs ir ir Il(xl(,1 iIl(l ils l)t.ot0lypc is r[.firrcrllrr tlrr' l(t'yrlolrls ttrnlrr.l llrtv.

( 10.17 )

Fo

TABLE 10.2 Srrirlr) llirli()ri l(,r Il(,yI(,1(lri Nrrrlmr lrw (l,r WrrlrrrBoth Mo(lol an(l Protolylxr, I, l,ll, l)

,rx t{) 4 lilrrtrrrr lr (r,vxrlrllrllry lrllvllV ltllrlh| l,,riI r:rlr,, llr( rr. rr

l lyrl''tllx :;Irr,!Irrr'L' rrrl M,,rlnl iitrrlln {'lrrt, llr I[xlll Nrl|rl,i'r Idw

0.0-i2

Example 10.5Inordertostudyatransientpocess,amodelisconstructedatal0:lscale.Watcristl\rilrrrll,prototype, and it is known that viscous fbrces are the dominant ones. Compare lhc tirr,' , Ifbrce ratios, if the model uses

(a) water(b) oil 5 times more viscous than water, with poil = 0.8p*"t",.

Solution(a) From Table 10.2

r,.= t1 = (10), = 1)sv, = L,t - (10)r= 0.1F, = I

(b) From the Reynolds numbe law,P,L,V,

=

tl, l ,tl('ir-t)

'l his cxnrplc dor)x)ostmtes the impo(ance of selecting the model fluid- The prope(ies,rl llrc liquid used in th(] model, especially the viscosity, greatly al'l'ect the pe ormance in thel{cynolds numbcr models.

PROBLEMSl(1.3.1. A Reynolds number scale model is used to study the operation ofa prototype hydraulic

device. The rnodel is built on a t:5 scale and uses water at 2(fC The prototype dis-chages 1 1.5 mr/sec of water at 90'C temperature. Determine the model discharge'

l{1.3.2. The monent exefted o a ship's rudder is studied with a l:20 scale model in a watertunnel using the same temperature as the river water. If the torque measured on themodel is l0N m lbr a water tunel velocity o[20 m]/sec, determine the coespod-ing torque and speed for the prototype.

10,3.3. A l: l0 scale modelofa water supply piping system is to be tested at 2C to detenninethe iotal head loss in the prototype that carries water at 85oC. The prototype is designedto carry 5.0 m3/sec discharge with l-m diameter pipes. Determjne the model dischargeand model velocity. Discuss how losses determied from the model are converted to

. the prototype losses.A submerged vehicle moves at 5 m/sec in the ocean. At what theoretical speed must al:10 model be towed forthere to be dynamic similarify between the model and the pro-totype l Assume that the sea water and towing tank water are the same.A structure is built underwater o the ocean floor where a strong current of 5 n sec ismeasured. The structure is to be studied by a l:25 model in a water tunnel using seawater at the same temperature as that measured in the ocean. What speed must thewater tunnel provide in order to study the lbrce load on tho structure due to the current?lf the required tunnel velocity is judged to be impractical, can the study be pedormedin a wicle tunnel using air at 20'C? What would the comesponding air speed in the tun-nel need to be?

10.4 Phenomena Governed by Gravily Force-Froude Number Law

When inertial force and gravity force are considered to be the only dominant forces inthe fluid motion, the ratio of the inertial forces acting on the homologous elements ofthe fluid in the model and prototype can be defined by Equation (10.13)

/1 =qI-P, t.25GeomctlicSimilarily

KincnrillicSinrilality

l)yIillrirlii,rilirrity

LengthAreaVolume

Iincc IMass L|

L,LiLl.

I me l;,Velocity L,lAcceleration L;JDischarge L,.Angular velocity L;,Angularacceleration Ll

p,,L,,,v,"

we have

p, L,v,p,

Since the ratios of viscosity and density are, respectively,

= P*,,., _

- 1 .25

L!.,,

10.3.4.

10.3.5.

,,=xFrom the Reynolds number law

v = 4r= 'o2l =oorr' p,L, { 1.25 }{ I0)The time ratio is

-

L,'v,

(1.25)(lU):- = ,/:)(0.2)

p,Li F..;t = prLlT;" (10.13)

300 llyrlrnt|lh iiIIIrIIIIIII n|lrl M{)rll liltrld (:lltt,, 10 l,ir l(1.4 l'llntll,llrt rlr (lltvt,rlrttlllry (llvlly fl,ll)tl l rolllll Nlrrlll)l lnw 300

l,l\nllrl]l('lll.(r/\rr r,lxrr (lrirtllr(lltlt,(lrl l ( ) I I I t r i r r I ' r ', ltrrrll l() slrlsly tll( li1'lr(l( tlrrllll)( r litw Whrrl is tlli llt'wtl llr(.!|l(xk.l tin l| pr(fl'lyl)(. llo(nl ol /O0 lr/srr it'lhc scnlc usctl is21):tll)ctc|rrri t:rlsotltc

rhrlirnrlirrrr'l rblc 10.J, thc dsch rgerltio is

ll[rs, the model flow shoLlld beo,, =! =lr,

lhc Iblce ratio is

,, - n\, =

Ll = q2orr = ltooo

PROBLEMSll).4.1. An overflow spillway with a 300-m crest js designed to discharge 3600 rrl/sec A l :20

model of the cross section ot' the dam is built in the laboratory flume I m wide Calculate the requied laboratory flow rate. Neglect viscosity and surlace tension eflecls'

tl).4.2. If a I : 1000 scale tidal basin model is used to study the operation of a prototype \rtis-fying the Froude number law, what length of time in the modet rcPresents a pe od ofone day in the PrototYpeJ

l(1.4.3. A ship I00 rn long designcd to travel at a top speed of I m/sec is to be stLldied in a towing tank wlth a l:50 scle model. Detenine what speed the modcl mLlst he towed for(a) the Reynolds number law and (b) the Froude number law'

10.4.4. An overilow spillway is designed to be 100 m high and 120 m long, carying a dis-charge of 1200 m3/sec under an appioaching head of2 75 m The spillway operrtion isto be analyzed by a l:50 model i a hydraulic labortory'(a) Determine fhe model discharge.(b) If the discharge coefllcient at the model crest measures 2- 12, what is the prototype

c1est discharge coefTicient?(c) If the velocity at the outlet of the model spillway measures 25 m/sec' whf is the

prototype velocity?(d) If the U.S.B.R. Type II tiliing basin, 50 m wide is used to dissipate the energy at

the toe of the spillway, what is the energy dissipation in the model ard in the pro-totype as measured in units of kilowattsJ

(e) What is the efflciency of the dissipator in the prototype l10.4.5. A energy dissiptor is being designed to force a hydraulic jump at the end of a spill-

way channel di;charging 400 m3/sec. The initiai depth in the 20-m wide plototyPe isexiected to be 0.8 m. Detennine the discharge ofthe l: l0 scle model and the velocityand force ratios between prorotype and model.

anrl lltc lirlio ol glirvily liuc(.s, wllicll is ( I (' I ( . I I I i I I { , r I I,y tlt(. w(.iltllt ol tlr. lrolrlologrl r,,l luid clcntcnts itlvolvcrl.

F, -

Yt\ -

P,.\, lr' -

n,,,,F_ M.t,., p,,,t,,,1:,,Equating the values from Equations ( 10. I 3) and ( 10.2 I ), wc gcr

p,.Llr;2 = p,g,L:

(10.2II

Rearranging, we get

From which

t)tt

tLt----vt 1;

Q, = L5l1 = (20)15 = 1789

700 m3/sec = 0.391 m3/sec = 391 f/secl?89

v, _,

8:,, L:,,

(,,\\ s'/'t)!' )

(1O.2))

( v, )\;tr* )Hence,

vr v,,ti',:r'- |fi - '"" (Froude number) (10.23r

In other words, when thq inertial force and the gravity force are considercd to b(the only fbrces that clominate ihe{uid motion. the Froude nurber of the model an(lthe plototype should be kept at thelame value.

If the same fluid is used in both the model and the prototype, and they are borlrsubjected to the same gravitational force field, many physical quantities can bL,derived based on the Froude number law. These quantities are liste in .l.able 10.3.

TABLE f0,3 Ratios for the Froude Number Law (9r= .1, pr= 1)GeometdcSimilarity

KinematicSimilarity

DynamicSimilarity

Length L,AIC1 L;,Volume l].

Force L1.Mass LlWork Lr,Power Lltz

Time r ttlvelocitv t |1Acceleration 1Discharge Lt/,Argular velocity L )t2Angulaacceleration ;r

310 llyrllxrllr: I ; I I I I I I I I I I r I r r nlll Ml'r['l titurlto Cllnl| llt10.4.6. Al:f5nrrrlcl isl)ill l()stl|(lylstilli|)'l)irs rirl llr(.oUtit(t ir\tIr'l'r,Irll\\rryrlIrtt, r,

stilit)g b sin c(lr)sisls (,1 l lrorizo rirl lLr()r (ll)r(,Il witlr U S It l{.'ly|r. ll lrrr1,installed to stbilizc lhc lociltion ()l'tllc lty(lr lluli( iuntl). 'l ltr' l)t(t(fy(. llts il I r(lirl,lrl L,cross section 25 m wide dcsgcd lo cill.t-y it 75 ntr/scc (lis(.lrilr.p(.. 'l ll(. vel()(ily illllll,diately betbre thejump is I0 m/scc. Dete nire thc lirliwirr:(al th( moJel di5charge:(b) the depth downstream ofthejump in the prototype l thc dynrnric lir.ce nrcusllr, ,t

on the model baffles is 16.2 N:(c) the force on the baffles per unjt width ofthe prototype channel;(d) the energy dissipated in the basi.

10.5 Phenomena Coverned by Surface Tension-Weber Number Law

Surface tension is a measure of energy level on the suface of a liquid body. The 1ir r ,is ofprimary importance in hydraulic engineering practice in the study ofthe motiorrof small surface waves or control of evaporation from a large body of water, such :r,.a water storage tank or reservoir.

Sudace tension, denoted by q, is measured in terms of force per unit lengtlrHence, the force is F= o. The mtio ofanalogous surface tension forces itr prototylxand model is

r,,!, lll 11 l'lrrtrrrrntt (ovrllrtllrl lry llrrlll tllvlly nil(lVihr orlh lorrir',r

ll, r, r'.

lt:\:!L: =

u.';,;:' . - N,, (wcbcr nun,bcr) (10.27)

lo (fhc'w(nds, thc Wober numbel mut be kept at the same value in the model(l in thc prot(fypc fbr studying phenomena governed by surface tension force. Ifthelirnrc liqud is uscd in both model and prototype, then p,= 1.0, and o.= 1.0, and Equa-lpn ( 10.27) can be simplified to

vz,L, = 1

v, = Lt!'

Sil.ce v,= Lr/T. we may also write

:r 11

(r 0.28)

L, ,t2T,

ll)us,( 10.29)

PBOBLEMSll).5.1. A model is built to study the surfhce tension phenomena in a reservoir. Determine the

conversion mtios between the model and the prototype fbr the following quantities ifthe model is built with a 1:100 scale (a) rate offlow, (b) energy' (c) pressure, (d) power.The same fluid is used in the model and the prototype

ll).5.2. A measuing device includes certaill small glss tubes of a given geometry. To studythe sulface tesion effect, a 5: I scale model (larger than prototype) is built. Determinethe discharge and fbrce ratios.

10.5,3. Detennine the suiface tension of a liquid in the prototype ifa time ratio of2 is establishedwith a l: l0 scale model. The suface tension of the liquid in the model is 150 dyn/cm'What is the force ratio?

10.6 Phenomena Governed by Both Gravity and Viscous Forces

ln the case of suface vessels moving through water or the propagation of shallowwater waves in open channels, both gravity and viscous forces may be impoftant. Thestudy of these phenomena requires that both the Froude number and Reynolds num-ber laws be satisfied simultaneously. That is, N/r = Nr, or

prL,.V, =

V,p, (g,L,),t,

-

,3/2

F, o..L_.F..

- __

= _-!: J

= o"L.., F,, O,,L,,, (10.2J rEquating the sudace tension force ratio to the inertial force ratio [Equation ( 10.13 )lgives

L1e, ri

Rearranging gives

o,L, =

By substituting for f. the basic relationship of V,.= L,/T"be rearranged to give

,, L, I o, lrtz( P,\"',:,tz \ hLrt\q/

P,VI L,or

t ^ '

ll)r, =l\l L),,\ t)t/ (l0.2sl

the Equation (10.25) m)

( 10.2)

:112 llyrll rt( :it tlltrrli,, |llM|llllt :;t(Itt,r |trr, t0Assrrrrrirr.q llrrr )()rl rlr(.r)(l(.1 ro(l rlr( |r()r0r),1)(.:lI rllr.( rl'rllr| rlri.r,rrlr.s J,lirrirrrtionitl licld. t,.= I lrrril silrcc t, t/, tlrt rl,rri rt.l.rrr0lrsllllr rrrrrr lrr.rirrrrlilrr.rl l0

( 10. lorTlris requirenrent can only be met by cho.si,g a spccial n*rclcl I.lLrid wirrr ir k ir)rlrrrr,viscosity ratio to water equal to thc thrce-half power of thc scalc ral io. In gcncra l. rlrrrequirement is difficult to meet. For example, a I : l0 scrle nrrxlcl wou Id rccuir c llrrrrthe model fluid have a kinetic viscosity of 30 times lcss thiur thlt ol.wrtcr, whiclr robviously impossible.

However, two expedients may be available depending on thc reltivc inllx)rtance of the two forces il the particular phenomenon. In the case of ship rcsistancc. tlr,ship model may be built accoriling to the Reynolds model Iaw and may operatc irrtowing tank in accorlance with the Froude number law. In the case oflshallow wirl,.rwaves in open channels, empirical relationships snch as Manning,s formula [Equatil,rr(3.28)l may.be used as an auxiliary condition for the wave mea-surements, accor.tlirlto the Froude number lu

1 0.7 Models for Floating and Submerged BodiesModel studies fbr floating and subme ged botlies are pebnned to obtain intbmatiorr ol

L the fiiction drag along the bountlary ofthe moving vessel,2. the form drag rcsulting from flow separadon from the vessel boundrv,tr,

to the boundary shape,3. the fbrce expended in thc gene.ation of gravity waves,4. the stability of the body in withstanding the water waves and the wrrr,

forces on the body.

The lirst two forces are strictly viscous phenomena, anii, therefore, the mo(l(.1:l-"-rl1T *.,qr.,d ,rccording ro rhe R,.yotds nurnher tw Th.rhir,lr.c,.rgr,,,,,rorce go\,erned phenornenon and_ \ce, nlust be analyzed by applying the Frrrrl,number Iaw. All rhree measurcmenrsiay be perforrncj sirrioniu'rif '1,., rvot"l. ;,, .towing tank. In nalyzing thc data, however, the friction rn."", ora .,. fbrm rr,,.tbrces arc first computed fiom the measuremcnts by using known lbnnulas ancl tlr: ,coeicients. The remaining fbrce measured in towing th-e vessef tf,rorgi tlr"-*.,surface,i rhe brL.e expenrled in gcrrelating the grur itfu c. 1r,r ru. ,..,i.,ar.",. ,,,,,rl r\ \caled up Io the plotr)type ralues b rhe Froudc number lurt.

The analysis procedure is demostrated in Example ( 10.7). For substrfircc r lsels, such s submarines. the eflect of surfce waves on the ,"rr.t, rnofU. ,"gt".,, , rHence, the Froude number morlel is not needed. To study the stability ;n; wavc li )r, ,on stationary offhore structures, the eflect ol.inertial ttrce must b taken into c,,,,sideration. The inertial fbr.ce, defined a s F i = M, . e , can be clculated ircctly f .rt ,lthe prototypc dimersions. Here, M, is the mass of w,,t", airpl,,."-iy1t,e porr()r ,ithe structure immersed bclow the waterline (also known as te rirtoo ,riror", ,,,,rrl , ,the acceleration of the water mass

,,,r lll / Mrx|lln lrr lIrllllrll nrrrll'rlillr'1lUtl llrxllrtr

I ir:rrrrrk l{).7\slll|)lll(trlfl'\\,llllillll'l\llll(llll(lll\].\f(tllIlllltlll('ll()I()'/l{lll,illlltlcls((ll](.|(l$'lll(.wlll(llill(..lr:rsrrlrrrlrt'ttristit hrrlrllr('l o.rlltl lll(rrro(ltl isl()wc(l illllwllvclilllklllllcst)ccrlol'0'51ll/scc'I (', llrc l)ilieulrlr shirlx ()l llx v( ss(1. l is li)und thirl thc drag coell'icicnl cnD bo ilPpl-oxil1crlt,y (r,: (0.(x)/Nlri) l(n l0r l0" The Froudcrrrrrrrl^-.r' law is applied lin thc I :50 model. Dudrg the expe ment' a total lorce o1 0 40 N is mea-\ure(|. Dclotmine the total resistxce tbrce on the prototype vessel'sl,hrtionllirscd on the Froude number law, we may determine thc velocity ratio,

V, = Ll/2 = (50)tt1 = '7 07

llence, the conEspondig velocity ol the vessel is

v,'=v, v. = 05.'7'O'7 = 354nJsec

l'hc Reynolds number for the model is

N.. .V- 1,. _ 0.5 0.() 4.4(,. I0,v I r)01. l0 6irnd the drag coefficient for th- model is

,, -

u')6 o.ou2.1(4.4q IU )r/l'he dlag force on a vessel is delied as I) = Co(ipA y2) , where p is the water density' and,t is the-plojected ar.ea ofthe immersed portion ofthc vessel on a plane normal to lhe directio,rl the motion. Thus. the model dlag lbrce can be calculated as

D,,,= cD,,(+p,,,A* v;,) = 0.0023 i' 1000'078 05'] = 0'22'13 Nl,hc model wave resistl1ce is the difTerence betweel the nleasuled towing lbrce and the dfaglince

F,,. = 0.4 -0.2243 = 0 i757 N

I.n lhe prolul)pc. lhe Rc)ll!,1J" ntrrnber i'vt.Lt,

_

V,.L, L,,, = 1.59 Ios

lhedr g coeflicient ofthe prototype vessel is C/),= 0(X)18' and thc dlag fbrce tsD,, = c (1p,,A,,.v?) = cD,p,,A,,, L1' v;) = 21937N

'l'hc wave resistarlce on the prototype vessel is calculated by pplying the Froude number law( sce Table 10.3)

F-. = r*r F*, = L3, 4,r, + 21963 N

,.,ra. s t,,lJl rc\i\tanue lorcc on lhe prolotype i'P' = Dl, + F,,,. = 2193'l + 21963 = 43900 N

:Jt4

10_7 _1.

t0.7.2.

10.7.3.

A ship 100 rrl long nx)vcs ill 1.5 rr/se. ilr It'(.sltwrl(.t irl l\( A I lO(ls(;r[. rr,,l, t, !theprototypcshipistobctcslcLlinalowinl.ltiIrLri)ItitiIiDt'lli(tuirlll srrrrlr, ,r r,rr0.9. What viscosity must this li(luid hnvc lirl both ltr'yrrolils rrrrrl liru,l, rrrrrrlr r t,to be satisfied?A 1:250shipmodel is towed in a wave tank and ll w.lvc tcsistilltce ol l()./ N r. rr,.sured. Determine the corresponding ptototype wavc rcsistncc () tlle Irolr)t!lA barge model I m long is tested in a towing tank at a spcc(l ol' l rtr/scc. l)el.lrr rl,protolype velocity if the prototype is 150 m in length. The rnodol hts 2-c r (lrirlt . i,,I ,l0 cm wide. The drag coefTicient is Co= 0.25 forNr>5 l0r. and thc rowirr, r, ,,rcquired totow the model is 0.3 N. What fbrce is rcquircd to tow rhe bargc irr wirl, r\\A concrete caisson 60 m wide, 120 m long, and 12 m high is to bc towc( irr s(.ir s ,r ,in the longitudinal direction to an ollshore costrxction site where i will bc srrl. |,calculated floxting depth ofthe caisson is 8 m, with 4 m remaining above thc $,irl,.r ,,face. A 1:100 model is built to study the operation of the prototype. If thc r(rt, t rtowed in a wave tank using sea water, what is the model speed that correspol(l\ t,, rt,profype speed of 1.5 Jsec ? The model study considers both the skin drap rrxt t, i,,drxg (Reynolds number) and the resistace duc to the generation ofgravity \| r\, ,,motion (Froude number).

, l{li {)IrllIIIIIiIIII|tl| lrrlxl' :t1h

,Ir,,'llr( l\4iUrIll),:,t t11,Itttr',,ror'lltrtr'ttl rr " ^'/]/r

llrrttittiott 1(r. 1l w, rrr;rywttti'

( IO..B)llrt, nroLk l nriry lrc(lucrrlly r.slrll ill il lll()(lcl !clocity so small (or, conversely, tllerrr,rrI l rorrtltrtess will l)c so llrrS.c) lllat rc listic nrcasurement cannot be made; or alrrr lr'l wrrlcl t[.'rtlt nrly bc so shrllow that Lhc plrysical characteristics of the flow may1,, rll rcrl. Such situations may be rcsolved by using a distorted model in which the,, rtri rl scirlc lld tl]c horizontal scale do not have the same value; usually' a smaller' l trcirl sclc latio, X. > y. . This means that

s =!-L.r' s,,, x,I l, r!( ( , ,\,, > .ti,, and the result is a larger slope fbr the model. The use of the Manning, |lrirri(l rcquires that the t'low be fully turbulent in both model and prototype.

open channel models involving problems of sediment transpofi, erosion, or

'

h.rosit require movable bcd models. A movable channel bed consists of sand or otherl,xr\c rlrterial that can be moved in response to the forces ofthe culTent at the channellr,.ri. Nolmally, it is impractical to scale the bed material down to the modet scale. A\ , r lic l scale distortion is usually employed on a movable bed model in order to pro-\ r([. ir sufficient tractive force to induce bed material movement. Quantitative simi-lrily is difficult to attain in movable bed models. For any sedimentation studiesi.rlormed, it is impodant that the movable bed model be quantitatively verilied by aIrrl]er of field measurements.

l,lrurnple 10.8,\rr l)pen channel moclel is built to study the efTects oftidal waves on sedimentation movementirr rr lo-km river each (the reach meanders in an al-ea 7 km long). The mea depth and width, rl rbc rcach are 4 m and 50 m, respectively, and the discharge is 850 nl/sec. Malrning's roughrrr.ss coefficient n, = 0.035. If the model is to be constructed in a lorakny room l8 m long,,[.t:r-rine a convenient scale and the model discharge$,lttionlll surlace wave phenomenon the gravitational forces are dominant. The Froude number lawwill be used tbr the modeling. The laboratoiy length will limit the horizontal scale

r _!. =

Z9@ sq^' - L., - I,J -"w.. r,' ill u(e , - 4UU lor convenlence.

It isjudged reasonable to use a vertical scale of )'. = 80 (enough to measure surfce|radients). Recall that the hydraulic radius is the characteristic dmension in open channelflowiurd that for a large width-to depth ratio the hydraulic radius is roughly equal to the water depth'lhus, we can make the following approximation:

R = f,= 80

u -

tr =l'

g,,,, R i,.

Ilyr i,ri rllr i;lhrllllrrrl, l|rrlMrl.,l l;lr r ll rnPBOBLEMS

(10. 1Lr

10.7.4-

10,8 Open Channel ModelsOpen channel models may be used to study either the velocity discharge-slopc r, ,tionship or the effect of flow pattems on thc changes in bed configuration. lirr rt,,lbrmer applicatiol'ts, relatively long reaches of the river channel can be model( (l \special example is the U.S. Army Engincers Waterways Experiment Station in Vi( t.burg, Mississippi, where the Mississippi River is modeled on one site. In these irl)l,tLcations, where the influence of changes in bed configuration are only of seconrlir,concern, a fixed-bed model may be used.

.B-asically, this model is used in studyirrg rlr,velocity-slope rclationship in a particular $annel; therefore, the effect of bcd rorrr,lrncs ir mporlirnl.

An empirical relation, such as the Manning equation [Equation (3.28)], rrar t,used to assume the similarity between the prototype and the model

I olr" l:,,

v, nt,"h.."r I _2t,.1

"t-1, - --_-,\, o/vut I D2t1

J10

'lhsl

we hrve

I lyr[Iltr] strIltrrrh, llril Mrtrl,rt :;tl|ltrh

v, = n,i,' ' r'j'' ' is1'''Using Manninll's fi)nnula lor E(lu tion ( I0..1 I ) I,

' =*,- !oi':"':

' = ?

'fc' lll! llul'l llt',,r,rrrr il'l '

l!lll.,l, /\\lrrl,,llrrlrlr(lrrr,l, ll I'rIrItIII IIIII\ ,I (IIIIIt IIIiIIIItII (1!lrll('l rrr llr'In(lr'l)l !vlrl(llrslllrr \!r(I irrrl / \ lrr(Ill,irrul( rrrl( r';r rlrsr'lrrrrl't ,rl l{lO rrrL/st'(. l,(rr rr vcrlitjrl srrrl( r'lll,\irrr(lrrri'rrl,ltrr(s\L,r'llr,r,rrl ,rl /r/ O.0.1 (iclcr lillcllll thi('lhc rre'{l('(l rirlioslorllr( slrrlly rl ,,,, o.o.)

lll ll. t, A I 100 s(irl! rrr()(lrl is uxrslnrclc(l lo sludy thc Pttcrrt ol l-low ir) t rivel reach ll lhcr t :rr'lt hus a Mantrirrg s cocl'licicrrt / = {)-025, whal shotlld be thc corresponding valuer)l /r ir lltc ltx)dcl ? Discuss the errors that may result liom the study if the value ofn/.r( rr(f llx)dclcd.

lllf..l. l)clcrlnirc the value of n in the model in Ploblem 10 8 3 if the vefiical scaleisexag-g(.rirlcd to l:25 distoltion.

lll,N.s. A l:l(X) sccle rnodel is cotlstructed to study the clischalge depth rclationship in a iver|cach with Manning's cocfllcient ? = 0.031 . lf the nrodel discharges 52//scc and hasMaIning's coefficient n = 0 033, determine an adequate vertical scalc tio and thel'lowrate tbr the PlototyPe.

lll.() The Pi-Theoremt ,rrrrplcx hydraulic engineering problems often involve many variablcs Each ofrl sc variables usually contains one or more than one dimensions ln this section, thel'r thcorem is introduced asamcthodwhich iscapableof reducing the complexity ofir llroblcm by grouping a nulrber of inclependent variables to form dimensionless,,,,,,1rs through tlimensional analysis. Dimensional analysis is found particularlylr,.lpiul in hydraulic model studies because it provides indicatiotls ttl those Iactorsillrich have significrnt influence on the phcnomenon; thus it guides the direction inr lrich the experimental work should be conducted.

Based on a geleral understanding of a ceftain physical phenomenon, one lit'st

rredicts the physical parameters that will influence thc phenomenon Then theseirrrlameters aie combined into several dimensionless groups to simplify tlle problelnlr )r a better understanding of an otherwise cornplex flow phenomenorl'

Physical quantities in hydraulic engineering may be expressed either in theli ceength-time (F'LT) system orin the mass-length time (Ml'T) system Thesetwosystcms are connccted by Newton's second law. which statcs that force equals masstirrres acceleration, ot F - na. Through this reltion, conversion can be madc fiomonc system to the othe[.

The steps in dimensional analysis can bc demonstrated by examining a simplellow phcnomenon such as the drag force exerted on a sphere as it moYes through a vis-(oushuid. Our geueral understanding of the phenomenon is that the cirag filrce isrclated to the size of the sphere, D; the velocity of the sphere, y, the viscosity of theIluid, pi and the fluid density, p. Thus, wc may express the drag force as a function of/), Y, p and p, or we maY write:

F - f (D,V, p, U)A generalized approach to the dimensional analysis ofthe phenomenon may be

nrade thugh the Buckingham Pi-theorem. This theorem states that ifa physical Phe-nomenon involves n dimensional variables in a dimensionally homogeneous eqtlationrlescribed by ,? fundamental rlimensions, the variables may be grouped n't (n - m)

l:lrrtr tr

( -. \rj

--)\) I ).-

,c;",1" ' l], Y,

Y)" r 8(,)r/r=

,-J" = A,n = o'"t

, -',1,1, [rr*E oo,*

The discharge ratio is

Q,= A,.v, = X,y,.V, = fli,, = +OO.13O,12 = 286.21jThUs, the rodeldischarge requircd is

" 'Q' ^5n' Q-. 28r\J | 7_- - () 0{)'l In /'e = 3 //'e'To use the Manning lin.ntula- tDrhulenr l.low musf be ensured in the model. .I.o verity Ih,lLJrhulont t'lL,u c,,llitr,,r irr tl)c nuJel. i

nolds number. , ...__-,, , rs necessAt_y to calculate the vallle of the lnodel Ro\The horiz6l p,u velocrry i.

850 m3/sec= 4.25 m/sec4m.50m

Hece,

Hence,

v,,, -

The odel Reynolds number is

==0'475m/sec

= 21,598

vt,

V,

n, -

V.Y. 0.475. Ottsv l.t . t0 6

which is much greater than the criticat Rin the model :eynolds number (2,000). Hece, the flow is tLrbule,rt

PROBLEMS10.8.1. A new laboratory site is available for modelinB rhe hirn neJ ,rf E xintple l 0. tj so that fhclength is no longera restaiction. bul the ktughnesr coelfrcient ofthe material to be use(l

iliffiTilXll; t"'. "' = 0 0 r s Determiri " ",'",;;i;;; ;; ;;',r,t'"o,,..punar,g

318 llyrlrlrrllr : i I r I I I I I I I I I I r | Il Morllll Iilt I llrrh (lltlr llr ',,r I(l !l Ilrr l't Ilr(lrtrrditlcttsionlcss glotrrs firl irrrirlysis. l r l l r i s t. i r s t' ( ) l (llp lr)t( L olt lt rt()vil)l \l)ltrtr,.,rtotal ol'llve dimcllsionll valiahlcs llc involvctl. 'l'lrr' lt.viorrs ctrrirlir)ll nlty lllrs lr,expressed as

f '(.F,b D' V, P, l,) = oThese five variables (n

- 5) are described by the l'undamcntal rlirucnsious M, 1,, rrrl /

(nr = 3). Thus, n -

m = 2, and we can express the flnction using two I l-gr.orps:a( F) =0

The lext step is to determine the two dimensionless z-groups by lrrangilit irllfive dimensional parameters into them:

tll = p,, pb D,V,tn,

-- p" njvt'

The values of the exponents are determined by noting that the z groups are clinrt.Isionless, and they can be replaced by /101.

Because most hydraulic studies involve cenain common dimensionless gr.orrr,such as the Reynolds number, the Froude number, or the Weber number, as discusst.,Iin the previous sections in this chapter, one should always be on the look out for thL.rrin performing the dimensional analysis. To determine the lI1-group, we may thr,write:

forM:0=a+bfor L: 0 = -3a - b + c + dforT:0=-b-d

There are four unknowns in the three conditions given. We can always solve for thr.t lof them in terms of the fourth, say . We have:

a = b, d=-b,and.c = -bThus

n | = p b pb D-t r t = l-o *\' = ( ry ) b.

ie. the Reynotd s Numberr,,vo r.

Working in similar fashion with the n2-group, we get

Il,= r?,pD'.V'.

Finally, we return to the original condition that A{l1,Il) = 0, we may wrirr..\ = A') or(n2) = Z"(rI1). Thus,

ll rllrl lrr't rrlrlrtsru lrlllrl rlrrrr'tiotitl irturlysis rlrcs rurl rtovrlc solrrliorrs lorr rtolrlcttt. rirtlx t il rtr'irllr ir lrrrrI lo toittt orrl lltt tclitliorrshirs ru(nrg llrc l)iu-nnr-t'lr'rs llrirl iu('irrrlir.irlrlt lo llr( lolrl(||r. ll ()r]e o11)i1s tn inrportnt pllrnrotcl. thcrr',,rrlls rtc ileonrl,lc(c arrtl llrrrs cur lcod to incorleot conclusions. However, ifonerrr, lr[['s rararrrctc|s tllrt r'c uulolatsd to the problem, additional dimensionless1,rorrrs which alc illclcvant to the problem will result. Thus, successful application ofr lrr r l'r rsiorral a nalysis depends, to a certain degree, on the engineer's basic understand-rrr1l ol thc hydraulic phenomenon involved. These considerations can be demon-rtrirlr'(l hy thc lirllowing example problem.l,lrlltlple 10.9'\rr ovcll low spillway model is designed to study the discharge,4, per fbot of the pototype.,1)rllwiry. Assume that the overflow water sheet is relatively thick so that th(. surtace tension,Ixl llrus viscosity of the fluid can be neglected.Slllllir)nlh\cd on our general understandiDg ofthe phenomenon, we may assume that the discharge,4,\\,rld be affected by the head, H, the gravitational acceleration, g, and the spillway height, .tnrs, q = f(H,s, h\ or f'(q, H,s,h) = 0.

ln this case, = 4, and m = 2 because none of the terms involve mass. Accordig to theI'i thcolem, there are r? m=2dimensionlessgroups, and

u (nr, nr) = olirking 4 and 1as the basic repeating variables, we haveMo Lo ro - (#)' (#r)'

",,

(i)

'! . = a",r,,pD'.V',FD = /D2V2A" (NR)

lrrom the fll group, we have

lllus,

llDnce,

liom the n2 group, we have

l hus,

fl\ = q"'Hb'B'.'Il2 = q"'Hh'11"

/.,,r,, _ f , J, /.,1 i\TL1 \I',iL,O - 2a)+bt+clT:0 = at 2ct

,, - \o' b, - la,l. I ( ., \,

11, = ,1" H '''g t'' = ] i;i' ;;; I\8 H )

LaO = 2a2+ b2 + c2T:0 = -a.

/ rr \.L"r' = l;.) L"'1"and,

llyrirrrllr liIrilllrrl,, rrlMrx[,] :;lrrll ( lr,rt, Ir

Hrnec.

l:. ,i'H ,, =(,1,)Now that the two dimensionless groups are identificcl to bc

1..'r-- l

we mv rehnrl to

a$tt.,2) = r(--","o ur)=,

and

flr/r-

-rD-'8n

Despite their apparent similarity, the tetm lzyrlroft.rgy should not be confused with theterm hltdrauLics. As stated in Chapter l, hydraulics is a branch oI cngineering thatapplies fluid mechanics principles to problems dcaling with thc coileclion, storage,control, tlansport, regulation, measurcmelt, and use of watel. In contrast, hydrologyis a science dealing with the properties, distribution, and circulation of the earth'swater. Thus, hydrology generally refers to natural processes, whereas hydraulics gen-crally refers to man made or man-controlled processes.

Even though hydrology and hydraulics represent distinct disciplines, they areinexorably linked in engineering practice. Many hydruulic projects require a hydrologic study to establishthe designflow rae (O). Indeed, the design flow rate is criticalin establishing the appropriate size and design of many hydraulic structures. Forexainple, rainfall events result in water t'lowing over land to natural or man-madechannels. The design of storm sewers, drainage channels, levees, and dams is predi-cated on establishing an appropriate design flow rate fbr these structures.

Hydrology for Design

n = i''ot'tr'(*)Thc lesult indicates that thc discharge pel unit length of the spillway is proporrional to

^/,qHrl2. The flow rate is also influenced by the ratio of (r/H) as was shown in chapter 9.

PROBLEMS10,9,1. 11 viscosity, density, and suace tension olaliquid ale included as variables irr tlr

dimcnsional analysis of Example 10.9, other dimensionless groups would hr\resulted. Derive the expression of these new groups.

10.9.2. Derive the expressions of the dimensionless groups fbr the drag fbrce exelted on a s 1,maline. Tlre parameters involved are the length of the subrarine. llz, the velocity ol llr,submarine, y, the viscosity, !, and the density p of sea wter.

10.9.3. By dimensional analysis derive an expression lbr the powel developed by a molor rrrtens of the torque and rotatioal speed.

10.9.,1. Derive an expression for the velocity of an air bubble l.ising through a stationary liqui,l10.9.5. A li