Analytical study of the mechanics of scour for two-dimensional jet

•

•

April

8

,

19t0

AHALY'l'ICAL S'i'UDY OF THE MEGF.JJITCS OF SCOUR FOR T,lO-DIMEUSIONAL JET

by

Lucien Duc~tcin, Yuichi I·,,ro.cDY.i, Georg~ L. Snith,

and

Maurice L. Albertson

}Tepared for

U. S. Buree.u of fublic Roods

t

Under Contract /,~I'Rll-5504

Enginccri_n0 Rcse~ch Colorado State University

Fort Collins, ColorQdo

CER()CGLSJ2

llllll~llllllll~lllllllllllllllllllllllllllllllll/1111111111111111111111 U18401 0592279

-ACKlIOWLEDG11ENT'3

The analytic::.l ::;tuc]y of the :..,echo.nics of scour for both t:1e three -dimensional anu. tvo dinenzional jet wo.s r.,o.de ut Color~dc Stc.tc U:u.vcrsity W1der the s1,onsorshi; ot' the U. S. Bu=cau of }·ublic Roe.1ls. The rcn:ri.ssion of Nr. Carl Izz:i.:rcl, Cl1ief of tl1e Di visio:1 of Hydraulic Resec..rch, U. ~. Bureau of Fublic Ro.;.dz, to ::i:res!:nt these stuc1..ics is gratefully o..ciz.'1. cW'-ledgcd. The -...Titer::. o.lso wish t o thunl: I{r. Eric I late for r.is review, critic ism, and :J.Ssist.:'..Ilce in cle.rificc.tion of t:1e busic theory and of the conccr,t of sccii,::c!1-: tr::ms:r,ort for cliffercnt conu.i tions of scour w::<l dc1,o -sition; md Dr. !,.

r:

.

ChG...-:1'!:>erlc.in, Actinc Deo.n anu Chief, En.:;inecr::.r-c: Research, Colorado Stc•.tc Uni VE;rsi ty f'.Jr :1.is encouragement in r,:-cr:lrc.tion•

of this report.

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CONTZN'l'S

ACKNOwLEDGHEI:fTS

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LIST OF SY1 li30LS

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LI

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OF .FIGURES

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ABSTRACT

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INTRODUC'l'IOH

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EQUATIOK

OF

co

r

i'l'TI TUI':'Y OF

NA

S

S

SEDIJ.~fl' TR!'.JTSPORT

. . . .

IMi DrG.El-~fi' O}' A 7\W-Dil,2i:fSI0IfAL JE."'l' OF IDEAL ~ TJID

orr

A

i i i i i ... ~A 1 2 2~

UORNAL, 1-' LUTE BED • • • • • • • • • • • • • • • • • • • • 11 IMHI!GEr-IBI;'l' OF .'\. IT0N-SlJnI-1ERG-ED, ':;:.TO-DD-NfSIONAL JM OF REAL

FLUID OH A l-L:J,IB Brill • • • • • • • • • • • • • • • • • • • 14

Dll'IITGE?-iEIIT OF A SUBMERGED, 'l":!0-DH'.ElTSIOiiAL JE.'T OF REAL

FLUID on A IrOPJ.lJ\.L, HAl;E BED • • • • • • • • • • • • • • • 23

INF-DiGfil•iEii'I' OF A ITO:f-SU3HZRGSD, '.i:'i-/0-DH::::IiSIOIUJ.., VERTICAL

JEI' OF RZ!J.. FLUID

on

AN ERODIDLS :DSD • • • • • • ₂₇

Il,:IINGZr-Zif.i.' O? l StJ3?:ERGw, '?.!O-DI?-::;rsror;JJ., Vl::HTICAL JEI'

OF REAL FLUID OH AN ERODIBLE I3ZD • • • • • • • • • • • 31~ DlITtfG~-ZJ:;.:

oi~

A '::'\i0-DTI-£ITSI01IAL, rrcLII;m ,TE7 OE' REAL

FLUID Olf IJ-T :C:10DIELE D~ • • • • • 39

SiJ!.l-lARY lufD COiICLUSIOITS

REF&Eife35 • • • •

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ii

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5

0

Symbol

ao

.

e. a' A Ao Al b Ft b' Ft B Bo .Ft Bl C c1, c2, c3, c4 c5 C C' d Ft ds Ft

F

Fo

Unit

SYMBOLS

Definition

Coefficient of pro::portionn.li ty bet ... 1een

U and X or V and

z.

Function defined by Eq 80. Fu..'1.ction defined by Eq 81.

Ratio of rn.onentum thicaiess of bou..'1.d~

layer to thic!mess of bou.."ldary layer

o •

Constant in sediment transport equation.

Function of t~~e having positive value.

Tailwo.tcr depth or height of outlet from

normal bounde.ry or simulated strcru:i bed.

b/sin

e.

Ratio of displacenent thic~JJ.ess of boundary layer to thickness of bou."ldary layer 5. Width of two-di~£nsionlll jet at outlet or

tailwater surface •

.Function of time having positive value. Function of tine having positive value. Intei;ral constants.

Constc.nt.

Function defined by Eq 112. Function defined by Eq 113,

DiS!:leter of three-dimensional jet at outlet or tailwo.ter surface.

Mean dia.r.i.cter of sediment particle.

Function defined by Eq 82.

Function defined by Eq 83.

G g h k L m

m'

m" M N n p F

F'

Q'

r R

R'

Symbol Unit Ft/sec2 Ft Ft Ft Ft3/scc/Ft Ft SYMBOLS (cont'd) Definition

Function.defined by Eq 114. Function defined by Eq

ll5.

Acceleration due to gravity.

Thickness of deflected jet in co.se of ideal

fluid.

Constant, 0.0225 for smooth boundary and m' =

1/7.

Fu."'lction defined b:, Eq

58

.

ExJ?onent in scdi~ent transport cquc.tion.

E..xponent in power-luv velocity clistribution.

Constant.

Coordinate point relative to

x

,

z

ruds.

Coordinate point relative tc

X,

z

ruds.

Ex:ponent defined by 2;n I / ( 1 +!:l I ) and equ.a.l to

1/4 for m' =

1/7.

Fressurc intensity at u FOint.

Pressure o.t tl1e r,oint of stagnation. Function defined by Eq

147.

Derivative of

F

•,nth respect to

(X/b).

Mass rate of sedi:::ent trunsport 11e:r- uait width.

Function defined by Eq

17

6

.

Derivative of Q with resrect to

(x/o).

Radial coordinate :pc.rallel to noIT.JD.1 boundc.ry

Function defined by Eq

77

.

Derivative of R with respect to (x/b).

Re s

s

S'

t u

u

u

0 U-l<c V V m

V'

V'

₀ V' _b Symbol Unit

Sec

Sec Ft/sec Ft/sec Ft/sec Ft/sec Ft/sec Ft/sec Ft/sec Ft/sec Ft/cec Ft/sec Ft/sec Ft/sec Ft/sec Ft/sec SYMBOLS (cont'd) Definition

Reynolds nurr,.ber of the flow.

Ratio

z/o.

Function defined by Eq 109.

Derivative of S with res:r,cct to (X/b).

Ti.11e.

Velocity in the boundary lnyer.

Hori~ontnl co~~onent of velocity of deflected jet at a :roint in ideal fluid..

Horizontnl component of velocity of deflected jet at n point N in Fig.

7

.

Horizontal co~ponent of velocity at a point for cc..sc of viscous flow along X-axis. Shear velocity.

Critical shear velocity.

Vertical com~onent of velocity at a point in ideal fluid.

Value of V at Z =

o

.

Vertical comr-0nent of jet velocity nt outlet or tail\rn..ter surface.

Ma.xillUl~ velocity at ce~terline of jet.

Value of Vm at Z = O

Velocity component of inclined jct ut a

point.

Value of V'

at

outlet or tailwntcr surface.

Value of

V'

at Z = b.

Symbol

V'

_m

vbm

vbmo

Wo

X

X'

z

Z'

Z' _Soo

a,

~

a•, ~·

Unit . Ft/sec Ft/sec Ft/sec Ft/sec Ft/sec Ft Ft Ft Ft Ft Ft Ft Ft Ft SYMBOLS (cont'd) vi Definition

Ma;d.r.ium velocity of jet o.t centerline of

inclined jet.

Value of V~ corresponding to a point

along the X-axis.

Vo.lue of

Vbm

at Z = b and Y. = 0

Vertical con:ponent of velocity for three-d.inensional jet at outlet or tail~ater surface.

MrucinUia velocity at center of three -dimensioncl jet.

Horizontal coordinate parallel to norr:ial

bmm.da.ry or sinuluted o.lluvio.l stream bed.

X

corresponding to a certain vo.luc of the integral constant.

Horizontal coordinate fron centerline of

inclined jet.

Vertical coordinate perpend.iculc..r to

deflecting boundary or bed level.

Distance froo bed level, :pc.rallel to center line of inclined jet.

Vertical distances from X-cxis to points

M and .N in Fig.

7.

Depth of scour (Zs=

-Z).

Final depth of scour. Inclined value of Z_{5 ,}

Inclined vc.J.ue of Zso-o·

Constants in the ex:pression which chara c-terizes the velocity function at every

section within th<! diffusion reeion of a

jet issuing fron a non-suboerecd outlet.

Value o-f

c,

~ for the case of a submerged

Symbol 1 1'

·.;r

V

e

p (1 1'

£'

Unit Ft ") Ft'-/sec Dcerecs 2

4

Lb-sec /ft 2 1~ Lb-sec /ft

Lb

I

lrt

2

Lb/ft2

SYMBOLS

(cont'd) Definition

Coefficient in cqtution for botU1da.17 layer

thickness for ideal flow.

Coefficient in equation for boundar,J lo.ycr

thicmess for viscous flm,.

Thic1::iess of botmc1.3.ry layer. 'o/X.

Function C)..'}1rcsscd by Eq

45.

vat

z

=

o

.

Deri vo.ti Yc of -. with res r.ect to

Function Jcfincd by Eq

143.

Porosity of scdir:1ent.

KineliUl.tic viscosity of va.tcr.

Angle of jct with rcsr,cct to t~1c ta.ilvc,tcr

surfo.ce.

Hn.ss density of vo.ter.

Mass density of sedincnt~ specific ga.Yity

of s cdincnt

=

2.

65.

Intensity of shea.r.

Intensity of shear along bou..'1C..:U-.f or

sinulutcd o.lluvi~l strcom becl.

Critico.l intenzity of shear.

Function Jefined by Eq

79,

:F'u.'1.ction defined by Eq 106. Function defined by Eq 111.

Function defined by Eq 180.

Unit (l) I (l) SYMBOLS (cont'd) Definition Function defined by Eq 123. Function defined by Eq 159. Function defined by Eq 171. viii

,.. ~

..

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•

I

I

I Fi~ure ; - I 1 2 3

4

5

G

7

Rectilinear coordinates for theoretical devclop:1ent

of the continuity eqi..l..'.ltion • • • • • • • • • • • •

5

Dfotributi ons of _~ in the CC.SC of no scour

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Distributioru, of _'ls in the C['.SC of u:ufor::;. scour

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₉

Distributior.s of _~ in the c~e of unifor:.1

depooition

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₉

Distributions of 'ls and

O

z/

0

t in t!1c cc.sc of

linea.r sccur ancl line:1_r de:)o::;ition • • • . • • • • • 10

Distributions of O..z o.nd

o

z/

o

t

in the CC'.Se of

unifon:1 scour, linear scour, :ind lineer de:rosition • • • lO

Sche;"il.'.:!.tic Jrc.,r2.nz of the impincm e:.t of a two

-clincnsionc.l jet of ideal fluid on a nornal,

plane bed . . . 11 8 Distribution of the velocity ccrJ~oncnt U ulo~g

9

10

the bed • . . . 13

Schc::-.o.tic d.l·a.i;.ring of the i:::riir.r;c;::.ent of n. non -su':)ncri:;ctl, t;:o-di:..ension2.l jct of :::-e1l flui(~

on c. nor:-:-~l pla.nc bed • • • • . . . . _• , 1· _{- r}

1

6

11 Deceleration of m.axi~um velocity of diff~ed jet • • • . • . 20

12 Schen::-.tic d.rc.:;r:.r.g of i.r.;pinp,encnt of a su;:rccrc;ec.,

two-cJ..i.:J.Cl"~iom:!.. jct of :::·c.:::.l :'luic. on a no1-:-,_al,

plcnc bed • • . . • . • . • •

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23

13

14

15

16

Horizontal vcloci ty distribut::.on alone the bed • • • • •

Va.rfo.tion of L ·with X/b • • •

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Shcc.r vclocit:,r distri1mtion o.lof16 the plane bed

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Coordinntc sys'tc:-:: for un i12clincc. jct

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ix

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ABS7RACT

Using the rectilinear coordinc.te systcn the equation of continuity

of I:1.USs sedinent tr::-.."lfjport is derivcu, and t~e continuity eqUD.tion is used

to help in describing mather:llltically t:1c pl,enonenon of scour by the

con-tinuity eq_U'.).tion. The relo.tionsh.i? between the sho.p! of the scour hole

and its vn.rio.tion ·.nth tine is invcstigo.ted for different conditions of

scour n.nd d.::!r,osition. Expressions for d.istrioution 'Jf se<l.ir:icnt tran.sr;ort

along the bed are derived for ec.ch condition.

T'nc irapin[;cncnt of a two-dbcnsiono.l jct on o. nor!ll[;.l bou."ld.ar:,r is anal;:rzcd by r:i.al:i:ic the asstL':lption that the Dcrnou.lli Equation is v-alid in

the neighborhocd of the st~nn.tion :point. Plo.ne, potential flow is co

n-sidered first, followed by flow of o. fluid with viscosity. For flaH Vith

viscosity fro~ suoner~cd and non-subccr5ed outlets, eA7ressioru:; for the horizontul vclocit:r and shco.r distribution o.long t:1e botmdary are developed

using Bernoulli's theoreo o.nd the bounun.ry layer theory.

The vo.rio.tio!'l of the de:pt:1 of scour is detcmined for tvo cond.i tions

of outlets by n..'3Gu.-:iing o. 1'1W of open channel flow for sediment transpor

-tation and by using the previously deternined shcc.r distributioru:; c.nd the

continuity equation. In particular, tnc vo.riation of scour depth vi.th

respect to time o.nd the final depth 01 scour urc described theoretically

in te!"r'..s of dincnsionless parar:icters. It is then shown that the develop -ment of the scotu· hole vith respect to ti!.:ic follows the pm.er lav end t:ie

logari thoic lc.v for the subr:tc_{r 0cd un<l.} r..on-:,ub!':'lcrgcd outlet:; rcs:pecti vely

before the fin~l state is reached.

-1-The influence o:f t:1c o.nglc of the jet is c.n.a.lyzcd in the ::rn . .:;.c :;w.reJ.cr

as in the cnse of c. vcr".:.ical jet. In t~i:::; c~c, e~qiressions for the v2.ri

a-tion.:::; of the de:r:;th o:f :::;cour wl th respect to tir.le and the final dei:;th of

scour o.re c.lso dcvclor,cd

Ilfl'RODUC?ION

Various asr,ccts of the pheno::iena of scour ho.s concerned engineers

since the £?.dvent of the construction of hydraulic structures in or nc2.r

· alluvial watervG.ys. Clo..ssical studies on scour anu. scour control incli.i.de

those of Rouse

(1)*,

Doddia.."1

(

2),

Thonn.s

(3),

Ib.lb1arl~

(4),

Albertson

(5),

and Snith ( 6) to nn.-::e n fe'...'. However, these inves tie:;o.tions ·.;ere concern::!d

vi th only in.di vidU2.l fc.cets of the rher..or.:.cmc..

In

1

957

,

Dr. Yuic:1:i. I;.ragaki, Kyoto Unive:-:::;ity, Kyoto, Jo.:pnn, c~""lC to Colorado State Uni vcr:::;i ty c.s c. visiting ?rofcssor in hyd!·aclics. Bec:1usc

of his interest in the 1-:nowlcd.ge of scour in c.lluvial mo.terial, E. L.

Albertson s ucr:ested t hc.t Dr. Iv'2gal:i sLtd-:r t:1c t:1eoretical asr;ects of scour caused by a jc'.:. of u.:.tcr from the view_)oi:-1.t of the continuity, .:10:::icntun,

and cner.;y eq_UJ.tions.

In dovelorinc; the theorJ, Dr. h ~ci:i :1e.de use of the Berno'.lll.i

theore.1, exreri:7tcnte.l a..'1.cl ti::.coreticnl studies by Schlichting a..'1.d Truc::cn

-brodt

(7),

and. 'I':-ac}:cn"orodt ( o) of n jct deflecting on o. nornn.l bounuc:J, experimental anc. theorcticf'.l studies by Albertson end others ( 9) on t:1c

diffus::.o:i. of c ~uor.:e:::·.::;ccl jet, ancl of the scclinent trarwport eq_t.:ntion

dcvelo1ed by I3:-o,m o.ncl L:turs en ( 10) • Iwc:-._;a}d 's develo;_,;:1ento.l wor~: Wa.5

based lare;ely on the funcl.::tr:1entc..l ideo.s ;ut fo:-th by Shields ( 11), Ko.lin-s ke ( 12), oncl E:mcr (

13)

.

Concurrently vi th the devclopnent of the

three-d.incnsion.:i.l t}1cor:r, L. Duc1:stcin, unclcr the supervision of

*Hum'::lers refer to c.:r,1~:1c.cd references

-2-Dr. IwB.3e.ki, wrote the two-d.inensiorw.l forn o:'..'the theory. Editing of

both forms of the theory W!lS done by Dr. Hn.uricc L. Albertso:i n.nd

George

L.

Snith

The theory in its tva- and threc-c.i.."'1Cnsion:!l forn is nci ther cooplcte nor applicn.blc to an V..."1l'.lys is cf o.11 :;i~1~cs of the scour :phenoneru:?.. This

theory does n.tten1-t to consolidate all of the v:.;.rious studies made on this type of scour. noteworthy is the fact that I,r~aki 's wor:<:. has ltclv-cd to focus attention on the c.reas where more funda.':lcntal studies arc needed

before ~n adeq_U:."'.te "'.:.heor-J co.n be develo:r,cd. In ~ticular, tr.ere is the

need for dcterminir.e tl,c cri ticnl s:ieo..r stress developed alone c.

deflect-ing hydraulicn.11:.r sr::ooth boundary by o. no1T.UJ.l jct over a w.:cle ra..12;c of flo,r

concli tions .

In uddition to t!1e shear stress, it is essential to knov t~,(! velocity profile v:i. thin the 0ou.'1<lnrJ layer for cliffcrent bed roughnessc:::;. O::icc

t he vcloci ty rrofile is 1~oi-r.1, the rate or c:10.:1.ee of mome:1ttt:-, c.lor..g t:le

boun<lnrf can be dctcr-::1incd. At rrescnt, it sccns desirable to dcscr:.bc

the scour of :.;.llu·,rial material in terr:,.;:; of the momentu.r:i change occurrinc

nlong the crodine surface of the alluvie.l bed.

In light of t:1c :1ced of d.eterr.unir:c the inter-relationship bctucen

3hcnr stress, vcJ.ocit:,· :profile, o.nd ra.tc of cllo.ngc of mo::,ent\;::: o.lons n

deflecting bound..."'.r:.r, u funili1Jr.ental study en the flow characte::::-::.stics of c. circular, s ul>ncrccd JCt in~inging nor::1.c.lly on c. s::iooth bour..clc..r-J

,.-:::z

initia-c;cd o.t Colo:::o.c.o Sto.te University "uy ti~c Association :)f J\."':'.crice.n

Railroncls. Rcnu.lts of the study are given in a. rer,ort by H. forc:1 (11~).

The results indicc.te:cl th.:J.t ex::::,onents ir. the thcol"'J of scour by Ii:.:::_;:;o..1:i

would. ne2d so::-.c :::odi.fice.tio::.- Furtl:cn·.orc, since this stucly su:-~:o~cc"!.

-3-lwagaki' s theory, c. I!:Ore cor.11)rehensi ve fu:1d.::ur.cntul study on the dyaa.-:-.ics of e. jet of air i.!~1:i~ins ur,on a nor1'iU:.l, s:::ooth botL~.cbrJ vo.s ini tinted under sr,o~orshi~ of" the !Iution:.:..l Science Fm.L"ldn.tion.

At this tine, the c4eriraental eq_ui}:~~nt for the National Science Foundation su:prortccl study hM been c::m:;tructcd ancl tested. The 1iri!".'~ oojectives of t:!·~ first J.,ho..se of this rcsc2.rch 1.roe;;rar:1 2.re, for given

fluid charE:..cteristics a:1d boundary c;eor..ct!"J, c..s follows:

1. To neCt.Sm·e, for a :r-rc-detcrr..incd range of flow c:1ar<.!.cteristics, t he si1car stress, vcloci ty rrofilc, and rate of ch.:L.,gc cf nonc:i.tu::: c..long a hyJre.ulic~lly smooth bouI1d.ary.

2. To ~:tc~m·c, for a :i:,re-cl0~tc:!::-::incd ra1~c of flow che..ructcristics, t:1c sh-::!:2: stress, v~loci '.-y ~ rofilc, c.nd rate of chs.n:::;c of

::i.c:::c::tu: .. e.lor1£.: c.n e.rt ifj_,:;ie1.lly-roug)'i.ened bou."1•i~J, v:,crc the

__rou,s.lmcss is of a rru.J.or.. vr statistical nature.

EQUA7IO!i OF C0i;.:'I2WI' r;_ FCR S2DD-Z:T '2.Tu\.~l~}C~'.i.':

Thecretic2.l_ Dcvc o~·;.,c-:t - To ckter:-J.::c

-:::1c

c~u:,.tion of c.-:,ntinui ty

0

-::

::. :u'i ni tes ir.K!.l e:l!:!:~. :rt ,\i,' 3' J, Fis. 1, <:cfL!cd uy li ncs J\.,'-..' c..nd. 3 • i3 o.nc.

li:1cs _·

!.'

_·

~

_-

'

·_-''ncl /."'_~ , .·_-·;-_,_-r ~_.., 'c co~,._. s1·(1_·" ~_rc?._- . r.,_.:.· _{:1e c;_u.:i.n..,1} ... t

_:r

_o f _Se di _{':-J.cnt. 1...::-2.ns}....

-r-ortcd -'.::1rou0h

t:,

c

~ccti on AA' ~er w1i t -..;iJ.tI1 c,e:r unit of tir.!c is

(C1,s)(l) and throll,2;h section i 'Il (See Fi₆. 1) i~

0

(1,.

c~

+

I

I

!

Jet

z

1

_of--

-Fig. l R·~ctil ·, ::::i~ cocr 1_{i:1nt~s 1'o~ :~hcorcticf'.l dcvelo:r}

-nc~~ of ~nc continuity CQtt.:.~io:1.

-I

·1

in which ~ is the nass rate of' scdiocnt transport per unit width from

Z

=

0 to Z

=

·Y: •

I I

The difference between these qUlliltities is equal to

0~

oX

dX

(1)

The qU!l.ntity of scdincnt scoured by the jet per unit of tine can be expressed

by

-;.t

(Z dX) (1 - )..) (1) ₍₂₎

in which ).. i:::; the porosity of the :::;edil:lcnt.

T'ne equation of continuity of mass seclincnt truru;port in rectilineo.r

coordi-~

nates is obtained by equuting the quo.nti ties given by Eqs 1 e..'1d 2, that is,

(1 - )..) Otc) (Z dX)

+'ox~

dX = 0

t

..

\' or

(1 - )..) ~'Z + ~ -= 0

0t

c;X

This equation has been given by Exnc= (13) and IwngoY..i (15)

(3)

Conditions of Ap~lication of the Continuity Equution - The relation

betveen the ::;~IX! of' the scour hole und the distribution of the scoured

sediment will nO',T be considered for the follm-r.i.ng conditions of scour e.nd

deposition.

1. The condition of

oz

/

o

t = O (No scour). For this condition Eq

3

beco.ies

(4)

Intcc;:ration of Eq 4 gives

(5) in which _{c1 is the} integrnl constant.

-2. The condition of

'?)z/ot

= - c (t), C

>

0 (lfaiforn scour). Fran Eq

3

or

O%

= (1 -,)..) C

ox

~ = (1 - ~)ex+ c₂

in .t.iich c₂ is t:ie intcgl'al conste.::it.

(6)

3.

T'hc conditio:1 of

OZ/CJ

t = c (t), c

>

0 (Unifom .:.:.c:;:io:::;ition). 1'he solution for thiz co:iG.i tion is oot~ir.eC::. by c:~:::.i:.;in[:; the sign of c in Eq

6

,

o~

in -..r:~ich

a _-s = -(1 ~)ex+ c

3

is the intcsrc'..l constant (Se0 Fi;;.

4).

(7)

4

.

The conJ.itio:i of

o

z/

Ot = -(A₁ - B₁X), (A_{1 ,} J

1 ~ O, X

<

A1/B1) .

(Line.'.:!i· scour). Fron Eq 3

Intesru.ting Eq 8 ylcld.G

in vhich is the intce;r2.l constc.nt.

+ C1 ~

(8)

(9)

cquatio:1 for this corn.ti tiun is the si:.:ne 3.S for ti1e co:1di tion

of linc.'.1!' scour defincc. oy Eq

9

.

Fi_;.

5

.s.lso r~:;:,rc:::;c:.ts this

CC.SC.

6

.

The conclitio:is of U..'1if'or::: zcour, linear sco~, a::.1d linco..r de:;:,o:::;i tio!1 are cor:;::iir.cC.. ;:i.s follows:

oz

/

ot

= - c (t) fo;: X ~ X

0 (U:iifor::. scour).

?Jz

/

c>t

= - _(A1 - D1X) for X

0

<

X

<

A1/31 (Li:1c1::.Y GCour).

-7-Tr1c c;-:in-ccsions civinc lls in this c~c o.rc c..s follo,r., : CLs = (1

<ls

-· (

1

;\.)ex~

for X

>

X₀

The distribution of seci.i.ncnt tro.ns:port for condi tio!i:; stip u-lated i::i 1 through

6

ere ilhwtrated :::;cher:'..:>.ticclly in

Fir;::;. 2,

3, 4, 5,

n.nd

6.

'.i'hc direction of sccliJ:lc:1t ::::over.:cnt

is in t::-ie direction of the arrmr on each figure.

Limi tnt ions oi' An>lico.tion_ of t::c Continuity Eql.!:ltion - T:.:c forc13oing

ma.ther:""taticc.l o.nc.lysis lw.s considercu. botn the pi1ysice.lly i:osniblc and the

physically inpos:::;iblc co..zes of scour, dc:position, and scour c.r.d der~cition.

'.i'herefore, i t is CG5cntial thnt the r,hysico.lly possible cases, ·..rhich hc.ve o.

bee.ring on the o.nalysis of scour o.s ~;resented in this report :)c idcntificc..

First, sir..c2 is a funct ion cf c _n -- i_-ntcr:ro.l c_.... onstant -- the

origin muzt be c:-:cludcd uz o. zon~ 0f conzidcro.tio:1; for if the co:--..sto.nt of integration l1..e1, c. finite value, en

>

O, then

<ls

i1as a finite vn.luc -.1t

the orig.i:;,. 7o circu.71"cnt this discre1~nc:.r it is n.ssU:P.cd t:.i.::.t in the vicinity of t:i.c :::;tc..::,-nn.tion :point (X ::: 0) t::c co:istant en chD.I1[;eS rapidly fron 0 at X: :: 0 to X = X •

0 This assu::rption is vc..lid since

the const2-.'1t of intcc~ct ion ci.cl'!otes the ini tio.l scclimcnt dische.rsc cc_rricd

into the zone U.'1d:~r consid2rc.tio!1 by the inco::un.s flow. Due t o tl:e finite vidth of tl1e jct, the clisclu:i.rt1e co.nnot be introduced ut c. si1ielc :point.

Furthernore, ::;incc "t:!.c ori.sin is a. point of stue;nation, the scrlir::c:1t di s-cho.rge r!.t the oric;::.n :::ust be zero 'occ~usc the velocity is zero.

-

8