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VE:litOCU.ff DI,SflUB'U!IO.J I:I SfiEl? iOU"~I OB:Alil\JIL ...
1'1
Chiang isung-fl#I ...
thesis su.bmi;tte4 to ,th• 'lrat.itta:te 'fae·ulty o.t th•
f1rgi:n.te Pol7t.1cbnicr: lnstlt~te
iXJ. cantU.dat>Y f:or thta d•glte• 0f
MASfil 61 SOl'.Dtll
:in
01v1l liigin~e:ring
lie~$mbel" 1963 ltla:cksburg, V1rg1n!a
2
Table of Oontent s
pr.:-i_ge
I. Int roduct 1011 • • • • g • • • • • • 4
II. Re-view of L:l.'tersture • • • • • • • • • 6
III. !iaboratory Equipment • • • • • • • • • 11
IV. Theoretical Oonsiderat ion. • • • • • • • • • 1-')
'V. :!?re sent EJ.-t 1 o~n of Data • • • • • • • • • • • • 19
VI. Velocity Distribution in T:ranqu.11 e.nd Rep id Regime • • • • • • • • • • • • • 23
VII. Velocity Distribution and Velocity Coef.fi-cients in Tumbling Jl'low Regime • • • • -;.7
j
VIII. Joncluslon • • • • • • • • • • • • 51 IX. Glossary • • • • • • • • • • • 53
x. Acknowledgment • • • • • • • • • • • • • 55
XI. Bibliograpl"'iY • • • • • • • • • • • • 56
XII. Vita • • • • • • • • • • • • 58
XIII. Appendix • • • • • • • • • • • • • • • 59
3
Figure Page
3-1
5-1
5-2 6-1
The ~'3h<-;pe of roug:tme:Js elements •••• ~ . . . Ii;low regime diagre.m for L/K ::: 5, K :::: 2t1 x 2 1'. •
Defini t :ton sketches • • • • • • • • • • • • • •
Velocity distribution f.1.fi'ected by :roughness ;:11'1spe i11 t1•anqu1.l flmr ~:.tt control depth • • •
6-2 ])im~Jfl!3ioule ru:; velocity di st r:lbut ion for squore
6-3
6-5
6-6
6-7
6-9
b::::.rs a.t cont :rol depth in tranquil :si11d rapid . . . . . . . ~ . . . • • • • • •
Dimonsionlei:rn velocity dlst:cibut1ol1 :tn t:ccinqull <;nd rapid regti:e for d.:tfferen:t s1Jape of rouslmess olements at cont:rol depth •••••
Velocity clist1"'ibution affected by roughnecs shape 1.n rapid flow a.t control depth ••• • •
The velocity distribu·tion around square bars ••
~rhe velocity distribution around triangular :r~ou.gh.ness oleme1rts e ••• •· •••••• ~ .
~l11:rn -1Telocity distrlbution around semi-circular roughness elements in tranquil a:rHl rapid fl.Olr
Velocity distribution i:tround 'the pt::i..:rallelogrsm b<.,,rs in tranquil and rapid flm·r regime • • •
Velocity distribution in trs.nquil a.nd rapid flow in the middle of a cycle • • • • • • • •
13
'.20
21
25
26
27
09 ·~.
50
31
32
34
6-10 Inflection po:i.nts between two roughness elements 35
'7-1 Velocity along the top of the roughness elements in r:!rpid and stable tumblin.g • • • .. • • • • 38
'7-2 Velocity distribution. around the par;oi1lelogreJn roughness elements in tumbling regime • • • • 39
7-3 The velocity distribution around triangular b8.rs in tm:nblin.t::; flo1:;r regime • • • • • • • • 40
3a
Figure p __ age
7-4 The velocity distribution around semi-circular bars in tumbling flo·w regime • • • • • • • • • 41
7-5 Velocity along the bed 1H~'tween two co:nsecuti·ve roughness elements • • • • • • • • • • • • • • 42
Velocity distribution ai'tected by roughness sh0-pe s in tumbling regime at control depth • •
7-· 7 Dimen.sionle ss velocity di st ribut ion a.t control
44.
depth for parallelogram bars in t.umbling flow. 47
'7-8 1nm.e11sionle ss velocity di st ribution at control denth for semi ... ci.rcule.r bsrs in tumbllnti flow. 48 ,. ~
I. IJ:TT3.0IJUCTI01'r
The gross :resist:':lnoe coeffic:l.ents, in rough cl12,nnels,
are ca.used by the combined effects of fr:i.ctional and form
resistances. Frictional re~:lstm1ce 2.nd form resistnnces are
intimately :related to veloclty distribution and to &.ll~lyse a.
flow, the friction factor must be determined.
Up to the ;:n·ese11t there has bee11 hardly any imrestiga-
tion 011 the distri'bu.tion of velocity in Steep Hough Ch.f1nnels.
'.I'he aim of this thesis is to r:rovide info?.'1'!ation concerning
the V(zlocity tiistTibutio:n. in steen channel with artificial
roughness.
In 19:"59, Peterson. and Moha..'11.ty classified the flow in
steep, rou{~h chBnnels into three regimes, wtd.ch e.:re tranquil,
tumbling nn.d. rapid regime. In 1961, Al-Khafai,ji and Peterson
extended th'L s classificEtion into seven reglmes: tranquil,
rapid, stable and. m1stable tumbling, trf'i.:nsitional rapid, and
tra:asitio:nal stable and t~cansitional unstable tumbling. In
this thesis the study of velocity dlstributlon. will be made
only in the three regimes, tranquil regime, stable tumbling
regime, ond rc:,pid regime. Throughout this thesis, un,le ss
otherwise stated, the term Htumblingn will denote stable
t umbliug.
'..i?he data this thesis is ha.sed on is talrn:n from I)roject
l}05 of the Civil F.in.gineer:tng J)epartment which is spnn.sored
by the Virginie. State Highwt:ty e.nd the u.s. :Bureau of Pu.blic
Roads.
5
The object of this thesis is to provide an information
concerning the velocity distribution in a. steep channel with
artificial roughness elements of various sizes a..nd several
dif:fere:nt shapes. The objectives of this study may be sum-·
marized in the following:
1. To study the velocity distributic:>n in the stream
under va.rious conditions of flow and roughn.ens geometry.
2. To examine the applicability of the logarithmic la:w
form of velocity distribution under conditions of extreme
roughness in the three major regimes.
3. To study the inflection-point of velocity distribu-
tion curve in tranquil and rapid flow regime.
4. To determine the velocity ooefficients in·tumbling
flow regime.
5. To study the relation of velocity distributions to
the flow cla.ssificetion proposed by :Peterson. Mohanty and
. Al-Khaf a.j 1.
6
II. REVIEW OF LI~BBJTURE
The nwn t.,tho first recognized the effect of boundary
roughno ss on fluld flow· was a J?rench engineer, Antclne Ohezy.
In 1775, sccording to Genguillet and Kutter (1) the first
recognized formula, giving the mei::rn velocity acrot~s P 1rerti-
ce.1 pection, for open eh~:nnels is in form of:
2-1
where, V J. s the mee.n ~;~~loci ty of flow, R ·is the hy-dr.::culic
rc-l.dtus of chs.nric1, 8 :1. s thr:: e:nergy slo-re, e.nd C ir:~ Ohezy' s
coefficie:u.t. ~rhis :formula is usually called Chezy' s :formula.
'Ehls formula does not tnke into eecount the velocity distri-
butiori. in. the section. H<mever, it W£u3 the generP.1 belief
tlrnt C is dependent on R e.nd B, as well fJ_s upon the degree
of roughness of the ChPri .. nel. It w:::s also 'believed that
veloclty distribution was p2ri;tbol.ic. For determining Ohe%:y's
O, many e(1uat:lo:ns have; been derived,. The mont widely used
formulf.l, is ~fa,nning' l::~ formul~t, published i.!1 1890 in the form
of:
where a
v ~ l,li.86 n 2-2
::- J. .. 4a§. nli6, h ~ d h 1 ;..,. .I."\ w ere n aepen s on c .. ecnne rougm1ess. n
In 1904 a:nd 1925 Prccndtl (2) presented his boundB.ry
layer theory ri.nd a mJ.xin.g length concept;, respectl'irely.
7
Prandtl h:;;;;.c1 given the e:xpre;::;;::;ion for the turbulent shea.r
stress fat any point in s, fluid moYing ;3ESt e. solid as follo1vs:
This c21 n be written in. the form of
rr7- ·- 0 d.'V l'fT j o,.p ··· y -d · J -of?: y
2-3
which is an approximate law of yelooity distribution in the
neighborhood of the wall.
Here
'{ --p -v ·-y --
the shee.ri:ng stress at the point
ths density of the fluid
the velocity at 1;he r·oint
the distance of the point f :rom the wall
·t;he so-ci:;.lled rn5.xing length of' momentum exchsn.ge
°2': = the shear in the fluid. at the wall 0
The fi1·,st rational logarithmic Vt'~locity distribution
formula. of t.he concepts of turbulent flow which were nnalyt i-
cally derlved by Prandtl a:nd. Von KS.rm.an (3} was in the form:
v 1 Vi ::: K . 1 :n (y /Yo)
This is t.he so-called Ke.rman' s law of veloclty d.ist:ri'bntion
in the n.eighbOrh.ood Of a solid 1 . .vall, Where V~} is .shear Velo-
city, Yo is a constant of intergrf:l.tion, K is univerB!f:l con-
stant.
8
In 1933, 1'Tiku.rdse {l}), B.fter performin.g a series of ex-
peri:roents using sand-coa.ted circuls.r "?ipes, oonfi:rmed the law
of velocity distribution. in the vicinity of a surface covered
with closely packed sand gn!tins. The equa:ticm. is
Ks is the me3n height of the sand grains forming the rQUghness
elements.
In 1936 Schlic.hting's i:nvestigati.on (5) of. roughness for
regular geometrical f<::>rms showed that ·the velocity distribu-
tion la.w in the region where the que.d.ratio resist~rnce la.w
holds is given by
v . ~- ,.. ... r~ 7i:~ loa (_...,.,.111r) v; ... UCa '" ,.. .. e ...I c:l " ~i. ·::i 7 4.-.-
which is of the same fo:rw. i-...s Eq_. 2-5. Here tlz vr;:ries both
with the shape find distribution of the rough:n.ess elements.
For a. r;urfe.ce covered with s2.nd a?:,,= 8.5, "chen Eq. 2-7
equals Eq .. 2-6.
In 1938, Keulega.n (6) analysed :Bazin's experiments and
applied the Prandtl-Karma.n concepts of h,ydra.ulio :r.esistcmce
to open oha~nels and was led to the formula
2-8
-where V is mean velocity of flo·w·, V->~ is mean shea.r velocity,
R is hy'draulic radius and K8 is the mean height o:f' the sand
gruins.
(1) The form of Hak·-
( f{) '~""'' ..,·t "l 0:0 WA'~ r•1,., 1° ''"' ''.~,._,_' .'f. fJ' lJ __ Q.,.,ff"'_., •. \. - "-"" \1_ \...J(. .:;:.... ,. • • • , P• J,J..-- •J-< .\. • l,.) Cl. - - 1: -
;r1 from wall rf::speci;i1rely, may be a1::ipl:i.ed. to the velocity d.is-
trlbutton ne::;,r the bed of an o:oen cl1ann:2l, y being me1:.1sured
2Jlong s. llne o~t'tl'.:wgonal to the J.lnes of equ~.l velocity. (2)
appec:.rs to be d.ependen.t upon the geo'n.et:ry of chc:rrnels.
arithmio velocity d:lstril)lrtion law .for pipe;3, (V ~· V:m 0z)/JT.lp = ~ log (y/r0 ) 9 in rectangula:r open chr!.1'.Ulel for uniform ·two
K dimensionaJ. flow, estnblish8d 'tha.t:
V - Vmex g_~ I -- ···- ~ • lOD' (yd) ~ rr 0 •· Jg_S "'"
2-10
"'!There d i->.:1 the depth of the flow, S is the slope of the chan-
nel, nnd. g is the aoce1er::-:.t lon of z:ravi ty.
fo'?cently, :1.:n 1961, T.racy 2,110. Lester (10} stud:ted smooth
rectnngu.l:s:r. ch;:m:nels r:tnd suggel':~ted the fol1.('nri.ng equ.r:ction!
v - Ve -. Y·l~
::: 2.5 4: 5.75 log y/yc 2-11
where v is time-average11 velocity component in x directlons.
V0 is average velocity in central -region of flow·, if'?-1- is
10
11
III. L.ABORATORY EQUIPM1~NT
l'he equipment and a.pparB.tus used. in this study i11cluded
water supply, test flume, two dimensiona.l roughness elements,
pitot tube, point gauge and equipmen.t to measure the dis-
charge, temperature and slope of the flume.
yfater .supply
Water flow to the experi:rnental site is through a closed
:pumping system. Water is pumped to a head tank, which is
approximately 50 feet above the experimental flume, then
baok down to the flume through six-inch ma.in pipe line.
Experimental Flum~
The wooden channel used for this study is 30 feet long
and 2 .feet deep by 2 feet wide. The channel is fastened to
a structural steel fre.me of' bolted constTuction, which rests
on a hinge in such a i--Tay that the flume lends itself to
slopes ranging from zero to 30 percent.
Head Tank
A structura.l steel head tank (rJhich receives flow from
the di.ffusor contains sufficient guide vances, stream-lining
fillets, and baffles to assure a uniform flow a.p-r1roach:ing
the head gate.
Head and Tail Gate
The head tenk is provided with a worm-end-roc1i:- driven
12
hand-operated headgate. .A tail gate is the same type ss the
head gi?.te, also hand-operated with worm-a:nd-roclt drive. Both
o:f these ga.tes pe·rmit .flexibility of the typ~~ of flow desired.
]._qughne ss ltllement s
Different sizes of artificial wooden roughness bars
square in cross-section were used for this study. '.l.~he length
of each bar is ex~~.ctly two feet so that it fits ~.:in.ugly in the
flume perpendicular t;o the walls of the f'l ume. JU so the two
inches high rough.~ess elements ur.;ed :for this study have semi-
circular, triangular and pB.rallelogre.m in oross-sect1o:n, as
shown in Figure 3-1.
Pitot Tube
Arra.ngement was marle to hold two plastic tubes, connected
with a sta.ndard pitot tube mounted on a. hand ... operated movable
carriage, hdd on a frr?dn13 which :i.nclined at an nngle of 300
to fpoilit1:;i,te the reading of the difference between ·the static
and dynamic head.
Po:tnt frgue;~
The depth of flow at any pe.rt icular no int, norm.al to the
bed of the flum.e or th<~ roughness surface, we,s me8.sured by
point gauge. The point gauges used for this study were ac-
curate enough to messure depth up to one-thousandth o.f a foot.
,, t7
N ' \
----s1i2 - ,,
WOJOOjaJIDJDd
-----91; ,,
14
~..§:surement of fno·giq
A cathotometer wa.s used to meium.re a.ocure.tely the slope
of the flume. This device consists of a level, a ~.,ne-meter
high steel stand i:i...nd a. base having three small legs. The
level rides on the meter steel stand ~tnd can be moved up ~11d
dovm conv·eniently. The stHnd. cn.n rotate to a:ay horizontal
angle. The flume bed slope can he con:routed frmn the differ-
ence of levels on the sr.rnle read d:irectly by the level.
Mee.sureme.n.:t_of Teumerature
.& sts,ndard thermometer \';i<:<.s used to [email protected]•e the tempera-
ture of the flow.
,Me.¥,surem.ent of Dischargs
The discharge w·a.s obtained from re~.dings of a manometer
tha.t had previously be~:n CfJlib:rated by standard ·weighing
methods. The total ranp;e of flow rates 1.11 the expe:d.mental
program ranged .from a minimum. of o.132 c.f .. s. to a ma.ximum
of 1.90 c.f.s.
15
IV. THEOHETICAL CONSIDER.A'.rION
The momentum transfer theory o:f turbulent flow developed
by l'randtl, by '3.ssuming that the momentum of ea.ah fluid :par-
ticle remains constant during movement from one region to
o.nother, lea.tis ·to an equation for two-d.imens:lonal flow:
4-1
where j is the mixlng length, ?"is tractive force, and f is
fluid density. Ey assuming that the mixing length is pro-
portional to the d:lstance from the we.11, y, that is f = ky,
then equation 4-1 reduces to:
integration of equation 4-2 results in the following:
v f!f = 1 1n y -r o k
4-2
4-3
Prom Von Karman 1 s investiga.tion in 1931, that 'tis a constant
·and equal to 'r0 , the v-ra.11 shear stress, "then equation 4-3
can be written
V == l 1n y + O :r"ToJp le 4-4
su·bstituting V* for Jaz:o/f , Eq. 4-4 reduces to:
4-5
16
For open channel flow, regardless of the effect of free
surface, it m.&~Y be ai.:::sumed t h:3.t the maximum velocity occurred
at the flow surface. Under this assumption, the integration
constant C iTJ. equation 4-5 bec::.;mes Vmax l VT -E 1n d, '/h.ere d is
the; depth of the flm,r. Therefore, equation 4-5 ca.n be writ-
ten as:
V - Vmax :::: l 1n (y/d) v* k
For rough chen:n.el, the depth, d, may be expressed by
Y1 + (K - Y1) where K is the height of roughness elem.ent, Yi
if:: the height of inflection point measured from flume bed,
and Y1 is the corJ.trol depth. This effective depth is a funQ-
tion of bed-slope, flow regime, discharge, the length of a.
cycle, and the shape of rougbne ss element. Since y 1 is also
a function of bed-slope, flow :cegi:me, discharge the length
of a cycle and the sh&.pe of roughness element, then y 1 +
{K - Yj) msy be replaced by 01Y1, here 01 is a proportional
const.llllt. i.:;q. 4-6 may be rewritten, by substituting Y-1 in-
stead o.f d, as
V Vmax 1 V-.1- = v~· + if1" 1n y/y1 4-7
K1 is a constant •
.Also, for a const<:'.nt slo:Je, a O.efini te roughness, in
sc..me regime Y1 is always proportions.l ·to the length of a
cycle. Therefore, Eq. 4-7 m~w be written a.s
17
4-8
where K2 is e.!l arbitrary constant.
From dimensional consideration, the general relationship
that exists may be stated c.s
4·-9
dimensions.l analysis yields.
~ = ¢2!n/K. L/K, V/[gY1. V:rt ('~ • Ks) 4-10
where
,t(_ -~ fluicl viscos1t7
L ~ the length of a
y = unit weight of
g -· acceleration of
-1.... ·- a. Froude num:oer Jg;r1
oyole
fluid
gravity
'!Jl..Jf. = a Reynolds number ,,.u. Ks = the shape tacrtor of roughness element.
Equation 4-10 may be rearranged as
Equating 4-11 and 4-8 yield
4-11
18
or
4-J .. 2
A is constant equal to VmHx/V~<t. So A B,nd K2 are also a
function of' y 1/k, L/Kt Np, Na, Ks. S:iri.ce the parameters are
so complex, theoretical Enalysis seems difficult.
(~uc.-.litative study will be m0de of the v2.rious pa:rn-
meters affect;:'!.ng the velocity distrihutlon in tranquil,
tumbling and rt::<.pid flow regines.
19
In tnis ch::~:oter tho seope cf this thesis ·will be outlined
and briefly d.ismlssed. 8inc0 the velocity dist:ctbutio:u is
stu.died iu t.h.1~ee flow regimeB, it seems sdvise.ble to orient
the rG:ader about these regimes.
According to the flow classification proposed by I'eter-
son, Piohanty and .A1-Khaff~ j 1. ( 11) the flo-vr rep.;i1rtes are !I. func-
tion of channel slope, discharge, a.nd roughness elements.
Figure 5-1 is a typical class.ifica.tion curve. From these
fig11re s, it obviously indicated that for a given v~1lue of
roughness p.a.r2meter, tranquil regime occurred only st a
slope ()f very small value, up to certnin vnlue of slope,
:rapid flow occurred at high discharge, tumbling flow occurred
2.t low d:1.scha:ege. The water surface patterns also differ as
shown. in Figure 15-~2.
Measu1:ernent; of Da;i:,a
Byste:u.8.t:lc measu:r0ments o.f velocity traverses were made
at the oontrol depth, middle of the cycle, along the channel
bed, along ths top of roughness elements and tht~ downstream
surf1:,~cc of the roughne us. The veloci·cy distribution of the
upstreEf.m stu·fsce of roughness elements, because of the length
of the pi tot tube, could not be measured very close to the
roughness elements. at learft two and one-half inches a.part.
05 Rapid CT
:c I- 0-4 0
~
t--z 0.3 ::J Unstable Tumbling a:: :::>
CT ~ c Q.. 0.2 IC
f\) ~
t- 0 w (!) 0:: <( 0.1 Sta be Tumbling :I: u (/) -0
0.0 0 2 5 10 ·15
2 FLUME SL OPE s, VALLI ES OF SI NS 10
.. .. FI G. 5 - I F L 0 W REG I ME D l AGRAM F 0 R LI K = 5, K = 2 X 2
One . ·1 fl ow Sluface A~ Tranq~1 flow surface B: Tumbling C: Control depth
Fig. 5-2 Definition
21
,•
Skete'hes
Po:'l.nt ve1oci ty :ment::inrement B were :n.;:·.o.e over 3_ sect ion ln the
middle e.nd neat' the downst rears. e11d o:r the flume~ 1' normal to
the fJ.ot-r.. The velocity tre.vcn:1es are shown in the tables
in the A1n;iendix. Slnca tr1e flow nea:r the d.o·wnst ream end is
fully developed and t..lso the we.11 effect cannot affect the
flow in th1; mlddle of the flume, at leB.st 'the affection is
lo·west in the w·hole cro:3s-sect:lo:n.
Ue.fl:n:tt:lon of Snecie.1. Terms
C-:nrtrol :De!r'(,h! The depth at the upstre~.m crest of the
roughness elements a~: r;'.l-1own :l.n 'Pig. S-2.
Length of e. Cycle: ~'he dj. :::'ta.nee between t·ii'rn neighbor
roughness elements from. center ·to cen-
ter, ~s shown in Fig. 5-2.
I11fl0ction :r'oint: The :point of the volocity distribu-
tion curve below which the velocj.ty is
const€.nt or :ne.s,rly so, above which the
velocity chD.nged :?'."ap:idly with the chG.nge
of depth.
Flume: Rectaue;ular open chs"nnel 30 feet long and two
feet i·:ride.
23
VI. VELOCITY DISTRIDUTIO:ti· IM TR.AN"QUIL AND RAPID FLOW REGIME
In this chapter an e,ttempt will be made to describe
velocity di,strihution a.round and between roughness elements
in tranquil and rapid flow regime in steep rough channel.
Velooi ty Distribution at Control De·oth. - Velocity distribu-
tion was made directly over the upstream edge of the rough-
ness elements in the centra.1 region for a.11 regimes.
The velocity distribution in tranquil regime was found
to be logarithmic regs.rdle ss of the configuration of rough-
ness geometry- except the semi-circular roughness elements
as shown in Figures 6-1, 6-2, 6-3. Plotting the velocity
on semi-logarithmic paper, testing for logarithmic distribu-
tion, it was found that the velocity varies linearly against
the depth. For semi-circular roughness, the velocity is
constant distributed within a certain distance from rough-
n$ss elements, then changes slightly a.t the region near
water surface. But it may be ta1cen as a constant through-
out the section, since the change is very sma.11.
For ra.pid flow regime, a.s shown in Figures 6-2, 6-3,
6-4, the velocity was found to be logarithmic for sme.11 size
of roughness and. small spacing, L/K = 2. 5, for larger spac-
ing; however, this distribution tends to deviate slightly
from logarithmic.
2
0
_,2 (fl Cl>
-5 I c
0
.c. +-a. Cl)
0
2
0
Fig. 6-1
24
L/K= 5 1 q = o.327 cfs
I Flow_ K = 2 in. ' ' s = 0.433°/o
3 4
~ i t L/K=5 i
Flow I K = 2 in. q = O .32 7 c ts s = o.433°/o
0 2 3 4
L/K=S K = 2 in. L/K = 5 K = 2 in. q=0.327 cfs q = 0.327 cfs s: 0.433 °/o S=0.433°/o
; ' '
) I Flow --
2 3 Velocity (ft/sec)
Velocity Distribution Effected Shape In Tranquil Flow At
By Ro!,Jghness Control Depth
... _·:T~-~ __ ~/L ..
3
~ r: ~~:: · d 1- -7 f---- _; f) ~-. -·· .. .
!Si-·- ..... ·- _.;_
2
1 f· 9·- -·
':':~ t
.. ' -1
1 i
t;,.· ..
7:
--:l ~ .
3 ~ .
' i-· . -·· -
+·A: L/K = 1.ls K = '•l" · 7 -i·-1 . s = 1.848 %
------L t : q = -o.2z ·cfs · B : L/k = 5 : K = 4 II
s • 1.187 % .10 q .. 0.327 cfs
C:. L/K c 5 i . -1- ---- K = §". r:·· ··::.
_,J __ s-~ .. s-%----·-·-t ~ - q • 0.3~5 cfs : . -- ; --~ . ·- --~-
l _J
.01
' j
L ..... .
. ·-· s
- Rapid FlQW
.10
. . . .. ~ !
A: K • 2" 'L/K • 5 s = 8 ~ 66'6 % ci =
B: K .. 4" L/k = 5 S= 3.65 i.. q = 1.47 cfs
C : K = 4" L/K = 5 .... $. - _5.1.l % -_q • 1.47 cfs
! ;·· ..... -·-··-· - -l .
!--· -: .: .. i
',JH - - - i ..... r.-~ _-_ -3 i
' 5 r .• - ·r --!- : ·--·-·-· - ..... --
.567
c ' I
;
cfs '.
A / / _,-B /
-- i ~ .
7
'9
i . . ·--+~ -: ,_ j- : ... :. ___ , -:
-.. ·- . - ... -- ..... ·-· -· ·- ---· i - ··-·. -·- .. --- - ... :_ •·· -· ... --~
-- t--- .. -·. -- - -·----D: L/K • 2. 5 - --- - · ·
K•4"_;_ __ . s = 0.767 % -q = o.Jr cfs
E: L/K • 5 K = . 2" . . :s .. 0.433% 1q • 0.327 cis
' ..... i
10· ·i I
I
11 . -- V/~~ --· -i
· 12 - ·- ·'t . ~-: ..... ;
(' ,,
K=.l_" Lf'K a 7. 5 ; ::· , ____ 7 __ j
S er 12.004 % ···· q• '0.465 cfs
B• ·K· • ·111 -' ··• .. ··· :•' . , ... - '
. L/K 1• i~ 5 : , , ~ ... .ii. 01 cz·\.:
ct.:··· Q. 615 -cfs K • 1" L/K = 7. 5 s = 12:004 % .... q • 0.708 cfs
i __ : . _ V/V* _
6 1 ~- .. ·-· -·- --
--i i
. t· _!_... . •. ..i. .••. --- ---- !--~ -- .. ·----- - ··-- - . --- ·r ·- -- ----b. ·Fig. 6-2 -- Di,mens·ionless Velocity Distribution For· Square '·Baes- ·· ·····-·
At : Contiol De~th In Tratjquil hid Rtp·id Flow ;.~~~~-~J-~ :-----s [- -- ·--- -- .. - . l 4 L. . . ·; ---i- -
i 3 .; ·t -
-~---- -- .. - ···-·- ·-··--··
1' ......
, ---r :·
' .. --·· -· -·-· -- -- r·----· -I l..
·-- __ ! ·-'- - - - r i i -·---+---
., --'--·'---:-~--'--,-----~- +---,------ ' I : . . ! .
; I . , ·-. _;_ __ ---- ·-- --!---------~- -· ··------------··
!
--- -______ , ________________ , _____ . --·---·-- -- ~ ·- _L ___ ~---- -~--
./::
.-.if---
' 3f-
2;
1 r --~- j-· -- -· 8;
7
6 ~-.
5; -
4f --
d'.
1' 91
7;
3 .-
3'
' 2 ~ ·-· ---·- - ·-
1 I ---
5
3:
;~ { --
1 ' - -
--- Y/T,.
- !
.10
.61
.10
.01 ' ' i
_ .... i-i - -i. (. ··- --
-- t -
26 i
Tranquil Flow ' --
K: = 2" ,L/~= 5 s = 0.43~ %
-- Parallelogram Baes
11
f ·- --t··-
= .27 = q • . 327
1-2 9
- Rapid Flow
10
Triangulae Bars 'B
A; q=.326 cfs, _B: -q "' .403
1l 12
K "" 2"- L/K e: 5 S = 8. 666 %
Parallelogram Bars
Semi-circular Bars
A A - ,i
/
10 l
'B
A: _q.,, ,327 ~: 4 •. 411 i
V/V*; 1-1 12
Triangular Bars
.J
A: q = ... j.
B: q_= .66 . 70
cf s A:q=0.70·cfs B: q = 0.745 cfs
I -f-i
-· --- L.
.8 10 5 6 1- . --
L L I
_i V/V*
-- : Fig. 6-3 Dimensionless Velocity Distribution In Tranquil abd Rapid Regime For Different Shape of ~ughne8s Ele~nts I ... --- -T
At Control Depth 1
j···
- - -· --- -- -I
. - _ _J - -- -- ____ J_ -
-~
2-
0
2 -If>
~ I -(.) c - 0
.c -c. Cl)
Cl
2
0
3
3
3
L/K = 5 K = 2 in. q = 0.7 cfs s: 8 ·666 °/o
L/K= 5 K =2 in. q = 0.7 cfs s: 8.666 °/o
Flow ---
27
Flow ---4
4
4 Velocity (ft/sec)
------- _J
L/K = 5 K =2 in. q = 0.567 cfs S=S.666 %
5
5
5
·'
Fig. 6-4 Velocity Distribution Effected By Roughness Shape - In Rapid Flow At Control Depth
28
Velocit;y_ J?i§tribution Around Roughne,.a,!li - Velocity distribu-
tion around roughness surface ·ws,s found to be a function of
flow rBgh1.1e a:nd the size B.nd shape of :r.ou.ghne ss elements~
Within one rE~gime and given roughness elements, it is a fu.nc-
tion of discharge and slope. For squa.re roughness elements,
the maximum velocity around roughness surface occurred at
the downstream crest of the roughness and was constant at
downstream. surface both for tranquil pnd ra.pid flow regime
as shown in Figure 6-5. For tria.ngula.r cross-section rough-
ne~rn as shown in Figure 6-6 the mc1ximum Yelocity occurred a.t
control de1)th; this is true also for semi-circular roughness,
and the velocity is nearly const~n1t at dov-mst ream s1.tri'aoe
both for tranquil and ra.pid regime. For sem.i-oircu.lar rough-
ness eleme1'lts, the velocity at downstream is increased from
zero or nearly ~10 t::<t bott.om to the maximum at the control
depth as sh.own in Figure 6-7. As .for p.srallelogre.m rough-
ness the maximum velocity ocour:red at downstream crest for
tranquil flow, upstream crest for r.e.pid flow. The velocit;y
at downstream sux'fs.ce is formed t-:.s an u:pwe.rd curve as shown
in Figure 6-8. :inuctus.tion in velotlity is due to the oocur-
re:uce of sepa.ration. an.cl eddying.
Veloci .. ~y .£:].ong Flume Bed - Velocity along t;he bed between
roughnesi:1 elements and about one-eighth of 0n inch above
:f'lume bed wag meE<..sured for different reg:i.me s. The velocity
along flume bed in tranquil and rapid flow regime w.s.s found.
4
3
-(.) Q)
"' :::: 2 -->--(,)
..2 ~
_, \
\ 0
\ \
\
\
0
29
I I
I f
I I
/ /
/ ,,,,_,.. /
K = 2" L/K = 5 A: Tranquil Flow
s =.433 % q= .327 cfs
8: Rapid Flow s: 8.666 °/o q = 0. 7 cfs
I • /~/ I /. I I I I I ,,,. I· / / . /.
/ Flow /
I
L.---~ " /
" ---- ----T---1 ="' ' ,. ;,
~ I ,! I I Roughness ,ef I
Element ,,; I .'-
"" I / •I
'-...., • / /I/ :- / --- "' 2 3 4 5 6
Distance (inches)
Fig. 6 -5 The Velocity Distribution Around Square Bars
4
3
-0 Q)
~2 ----0 0
~I
0
-I
Flow
' i : I r i I I
/11 I j ;
30
K=2" L/K=5 A: S=0.433°10
q=0-326 cfs. Tranquil Flow
B: S=8.666 °1~ q= 0. 7 cfs. Ra 'id Flow
I - I / / ~------...-·- B / / L-----'
- --~r"' A !
I --+---- ---1---\ I \ I
I '-:Y
""' 0
Fig. 6 - 6
2 v
/ I
I I
/
3 4 5 6 7 8 9 Distance (inches)
The Velocity Distribution Around Triangular Roughness Elements
41--
31--
-~ 2 (/)
........ ----u 0 Q)
> 0
0
0
Fig, 6 - 7
31
Flow
2.. 3 4 Distance (in ch es)
The Velocity Distribution
Roughness Elements In
K = 2'' L/K = 5 A: Tranquil Flow
s : 1 4 3 3 °/o q : , 3 2 7 B: R '1Pid Flow
-(\J
S = 8 • 6 6 6 °/o q = O. 78 6 cfs
.
Around Semi- circular Tranq. And Rap. Flow
-(..~
4
3
I I I
32
!~~ I ,.
Flow
K-= 2 in. L/K= 5 A: Tranquil Flow
S = · 4 3 3 °/o q = . 3 2 7 c fs 8: Rapid Flow
s: 8. 6 66 °/o Q=0.7 cfs
~ 2 f------ I I
I
' --->. +-0 0 CP
I
> I,___-I J
I t f
0
,,' j. / /
·\ / 7 """ / ......__~
1--- - -\ \ U/S \
\
0 2
Top
3 4
9---; J
I _J //
I I/ I I ,,./" i ·-5 7 Distance (inches)
8
Fig. 6 - 8 Velocity Distribution /\round The Parallelogram Bars In Tranquil And Rapid Flow Regimes
to vc.1.ry V'2ry slightly. The v2rir,.tion was due t,o the co.nfu£Jed
pattern of eddies.
!.!?!.lQ£!:tv_J!!.§i.tlb1t'l!on 1rL.il!~ .. ~~lftd!LQ.f....Qlpl£.. Velocity dis-tribution at the middle of 0. cycle was inacle to shoi;ir the order
oi' magnitude c)f velocities below and above the top of rough-
ness r:;1lemer.rtis.
'J?he VE~locity distribution at 'the middle of oyole in
rapid and tranquil r~.:iglme mey 1>e said to have the sa.me char-
acteri st; ic s. :Pigure 6-9 shows a typical curve. From this
diagr&m lt ls seen tha·t v~locity distribution in the trenqu.11
flow ls logarithmically distributed above ·the inflection
-ooint. Belov the :i.nfleotio11 point the velocity is constant.
'.!.'his figure Plso shows t~tiat :i.n. cas(:i of rar.iid flow the velo-
city dist:r.ibutio11 e.bove inflection point devi~·.tes slightly
from the 1ogarit:b.mio distribution. Belo·w the i11flection
poh1t the velocity is neo.rl;v constant.
111fl§C]J;.on ,I)oin;p. As to the inflection point, it wai::i found
to be a functian of flo~ regime, shape of rouglmess, length
of cycle, slope of bed,, and also a function of loc.rntion of
the yelocity t ?:'':'\Ye :.rse in the c;yrcle. The effect of d.i scharge
on the inflection point is very small. It was found to be
ahr;;;;ys higher :near the down st :r:-eam. surface of roughness ele-
ments fc;J1.d. to decrease rapidly within one e.n.d a. hs.lf inches
<ind th0n to a constant. Figure 6-10 shows the inflection
Inflection point ~
34
Flow
/
Tranquil It
_J
-Flume Bed Roughness Element
K =.3 Ft .
. 3 X ·3 Bar
F-i g. 6 - 9 Velocity Distribution In Tranquil And
Rapid Flow In The Middle Of A Cycle
0 ·- 0 -0 ex: IP If) ~ c:
..&; C'I :J 0
ex: "C c: c -c: 0 'l.. c: 0 -0 (1) -c: - 0 ~ ........ ~
35
A: Tranquil Row S = 0. 433 % B: Rapid Row S ~ 8.666 °/o C: Tranquil Flow S =0.433 % D : Rapid Flow S = 8.666 °/o E : Tranquil Flow S = 0.433 % F: Rapid Flow S = 8.666°/o G : Tranquil Flow S = O· 433-0/0
q =0.327 cfs q = o. 70 q = 0.327 q =O. 70 q = 0.326 q = 0.10 q = 0-327
I~-- ~~o_:_ .. -~- A ---_....____ B
Flow
Flow
~ <::;::::::: • ---- E ----------- F
Flow
-----+-----.--+-- G
Length of a Cycle for L/ K = 5_, K = 2 11
cf s cfs cfs cfs cfs cf s
I I
I /
·'
Fig. 6- 10 Inflection Points Between Two Roughriess Elements
36
points curve between roughness elements for all shapes of
roughness in rapid and tranquil regime. From this diagram,
it is shown that the height of inflection points, Yi• of
rapid flow always less than the height of inflection points
of tranquil flow.
Since the range of slope in t·ranquil flow regime is
only a few percent, the influence may be neglected. By sta-
tistical analysis, the height of inflection po~nts in tran-
quil flow regime for L/K = 5, for square, triangular, paral-
lelogram roughness elements, is o.65 K. It is very near
2/3 K. As to semi-circular bar the ratio of the height of
inflection point to the height of roughness, Yi/K, is about
o.42. In rapid flow this value for L/K = 5 ranged from 0.35 to 0.55.
Applicability of the Logarithmic Law - From the velocity
distribution curves it can be concluded that in tranquil
flow the velocity is logarithmically distributed. In rapid
£low, although the velocity deviates slightly from loga-
rithmic distribution, the logarithmic law may still be
applicable.
It is interesting to note that the constant A and k2
in the logarithmic law, Equation 4-8, are not equal to 8.5'
and o.40, respectively. They did not assume any regularity
values but seem to be functions of L/K, Yi/K, NF, NR, Ks.
In this chapter the ire1ocity distribu.tion around and be-
tween roughn<:Hrn elc:!ment rJ in tu.m.'blin'.s flow· ree;ime will be de-
scribed. .An l=Xttempt will be rnE.'.de ~'lso to deGcribe velocity
.~oefftcicnt s ~~n tumbling flow rGgi:w.e.
Owing to the s~µarat:ton., eddy and vortex, there is no regu-
larity of velocity dist:r:Lbutlon at'ound roughness elements in
·the tumbling fl(1w regime. The only thing worth mentioning
is that the 1naximum ·'.rnloclty around TOUf:hne0s element cc-
curred at the control depth fol'.' triangular Bnd sem1-circule.r
-nar·s ~md at the downstream crest for squgre s:nd -p2ralle1o-
gram roughness as s'.1ow:u in Figures 7-1, 7-2, 7-3, and 7-4.
V'Blocity Alon.\r. Flume Bed: ·~ The V•JlO<?:tty along the flu.me bed
varied with distance in the cycle as shown in 1!1igure 7-5.
Thls is because of the hydraulic jump which occurred.
o.t the mi(ldle of ::..:.. cycle in. tumbling regime depends m:lon the
position of the hydraulic jump. The position of the jump is
related to the discharge, the height of the roughness ele-
ments, the length of cycle, ri,nd the slope of the channel. It
is diff:lcu1t to give any regulrrr form of the veloc:i.t;r d:tstr1-
bution at the midrlle of a cycle in the tumbling :regime.
~ c 0 (,) cP
Cf)
'-Q)
a.. -Q) cP
LL
c ->-~ 0 0 cu >
5---
4---
3---
2
I I (
2 I
o---
c 2
38
• B j !
3
UK=5 K =. 3 F-t. Square bar A- Rapid B- Stable Tumbling
4 Distance Along The Top Of The Roughneo:;s Element - Inches
Fig. 7- I Velocity Along The Top Of The Roughness Element In Rapid And Stable Tumbling
·I
-0 CD fl)
4
3
~ 2 ~
-u 0 CD >
~ I I '-..1 I I
I I
I I
I
39
Flow
I~ ' \
0 t r-,----
\ 1 Top
\/ ~
'-..) \
' -1--
0
U/S D/S
2 3 4
Tumbling Flow
K = 2 in. L/K = 5
$: 8.666 °/o
q = o. 2 7 4 4 c fs
-----~
I I
I /
/
5 6 7 8 Distance (Inches)
Fig. 7-2 Velocity Distribution Around The Parallelogram
Roughness Elements In Tumbling Regime
(.) 0 cu >
4
3
0
-1
Flow
____ ) \ /
\ \ \
"' "'
40
K = 2 inches L/K = 5 q = 0.327 cfs S=6.863%
- - - -r-
I I
1 I
0 2 3 4 5 6 7 8 9 Distance {inches)
Fig. 7 - 3 The Velocity Oistri bution Around Triangular Bars In Tumbling Flow Regime
41
4
K = 2·' L/K = 5 s: 8• 66 6 °/o q =0.326 cfs
3t---
-0
~2 ...... ---0 0 a> I >
0
-I
Flow
0 2 3 4 Distance (inches}
Fig. 7 - 4 The Velocity Distribution Around Semi -circular Bars In Tumbling Flow Regime
"C c 0 0 Cl)
en ~
Q) a.. -Q) Q)
LL
c
0 0 •;ol
>
3----
2----
... ------- -. -..•. •
"-Roughness Element
•
Akng
42
~-~ ~ --· ,,,, . -.
• I
!
'Bed Of Flume
L/K = 5
K= .3 Ft.
Tumblin() flow
The Bed o~ flume
Fig. 7 - '.) Velocity / .. long Ti.c Ee:J f etween Two Conse cu 1 ive Ro u thnes::> Elements
43
Velocity Distribution at Control Depth: - The only regularity
of velocity in tumbling flow is the velocity at control depth.
Figure 7-6 shows typical curve for different shapes of rough-
ness.
1. Square roughness. - The velocity distribution for
square bars in tumbling flow varies from logarithmic to para-
bolic. For partially developed tumbling flow the velocity
approaches logarithmic distribution. For fully developed
tumbling flow the velocity was found to be parabolic regard-
less of the configuration of roughness elements. It was ob-
served that the velocity just above the top of the roughness
was low, but within a fraction of about 0.17 y 1 to 0.26 y 1
above the roughness it increases rapidly to a maximum value
(refer to Table 7-1). Then above this depth the velocity
begins to decrease. An equation which shows the statistical
average of the velocity with depth is:
' v = (k 1 )5/4 + y 7-1
where V is the velocity at a point a distance y above rough-
ness element. K is the height of roughness.
It should be mentioned that the dimension of this equa•
tion is not dimensionally homogeneous, and the equation is
valid only for points which are above the point of maximum
velocity.
2
0
2
0
-en CD
..c c.> c: -.c -· a. Q)
0
2.
0
0
44
L/K = 5 K = 2 in.
~ Flow
-----------------. q = 0.326 cfs. $:;:a. 666 °/o
L/K = 5 K = 2 in. '
2
'· ~
q = 0.331 S = 8. 66 6°/of='low _.
2
L/K = 5 K = 2 in. q = 0-259 cfs s : 14. 9 3 2 °/o
Flow
3 4
.3 4
L/K = 5 K = 2 in. q = 0.327 cfs. s: 6.863 °/o
-------~ ~
2
Fig. 7 - 6
__,_ ... --
3 2 3 Velocity (Ft I Sec)
Ve1ocity Distribution Effected by Roughness Shapes in Turnblir.g Regime at Control Depth
45
Table 7-1. The Point of lfa.ximum Velocity in Tumbling Flow at Control Depth
I- 3 ··~-·· Y1~----Y-!7Y1 - Roughness in. L/K, rJf . cfs ft. ft. Shane f)
.... .... • Zl ,
1 '7 .5 12.00.l~ .119 .095 .021 .220 Square
1 7.5 12.004 .242 .154 .033 • i?.14 tt
2 5.0 14.932 .208 .165 .033 .200 II
2 5.0 14.932 .233 .176 .033 .188 If
2 s.o llt. 932 .259 18·~ .038 .206 tl • . c_
4 5.0 10.600 .265 .170 .030 .177 tt
4 5.0 13.500 .. 265 .150 .029 .193 ti
4 7.5 7.890 .600 .250 .055 .220 II
4 10.0 a.350 .785 .310 .065 .210 II
6 5.0 10.734 .302 .212 .037 .170 ii
.6 5.o 13.160 .301 .208 .o::;6 .173 fl
2 5.0 6.863 .175 .138 .021 .152 Triangular
2 5.0 6.863 .218 .159 .024 .151 II
2 5.0 6.863 .271 .184 .034· .185 ti
2 5.0 6.863 • 327 .209 .035 .168 It
2 5.0 8.666 .154 .113 .010 Parallelo-p;ram
2 5.0 8.666 .228 • J.'+5 .010 ti ..
2 5.0 8.666 "7h. .166 .010 H • L.. •
2 5.0 8.666 .331 .183 .010 ti
46
2. ParalJ.elog}:-am Roughness Elements. ·- The velocity at
control depth for parallelogram roughness, just above the
roughness, is low, but it increases rapidly to maximum velo-
city within a fr2ction of one-Jcenth of an in.ch and then be-
gins to deorease. Figure 7-7 shows the curve.
3. ~Criangular Roughness Elements. - For t rlangular
bars, the velocity distrlb11tion te:nds to d.evhtte slightly
from parabolic. The maximum velocity occur:red lower tha.n
for square bars. Table 7-1 shows the ratios of YIY't •
4. Semi-circul::'r Roughness Elements. - :&"or semi-
circular ba.rs, the me,ximum velocity occurred just above the
roughness. For a constant slope and same roughness size
and constEnt length of cycle the velocity is proportional
to discharge. The decreasing velocity curves are in the
seme f'orm, as shown in Figure 7-8 • . . Velocitl_Qoefficients in Tumbling Flow: - For non-uniform
distribution of velocity of water flowing in open channels,
it is customary to use the velocity head bB.sed on the mean
velocity and a coefficient. This coefficient is the ra.tio
of el ther the mean of the squares of the loca.1 ·velocities
to the squere of the mean which is 111omentum coefficient or
Boussinesq coefficient (12) (3 , or the mean of the cubes of
the locB.l velocities to the cube of the meA.n velocity which
is called energy coefficient or Coriolis Ooefficien.t (13) ~ •
6 5 4
3 4
5 6
78
91
2
3 :r;r~4 7--:~~~:
'•
T I
•-;~
:++1
3 ·=-m~1~~
.;:;!·
i·t~!"
li4 lj
till _
_:_L
$44,
. ...
. . ... _
. . .
·-'j.
. '
.: ::
....
:!l
4 56789, 2 ] 4 567891
3~:'.' 1
L/1..
2 3 4 S671J91
VUiiJJ.JJli. . • ·-. ····'·+---···-~ ... !L J .. .J.--~·; .· , : r-+!'1 tH .. .s+·S":tic.,8~ 6,66tt
-.l j ~- . ~ -.. 'i ~ 'f. 1 ·; '~ ·•
2 3 4 567991
2' I : ; j ·! .. ,. .... 1 1 • , ; · 1· . · ... ,. ,.. . ' ' , ' " I·•~ ' ~ - -I .! • ,, ---~ j -· , • t . ltutl. . , . ···.•±·. llUll 2 .. . . : : '. • . 279 c(a ·:
T"l" TTl.un l i I ·: :
; · l J:'.:T:-l !"lhii-: 5 . ,:~; .409
0.1 9 8 7 6 5:
4
3
2i
. 01 9 8· 7[ 6; ~;
4
3
z.
.001
.q., ~1r;f~~ q• ~\·rcit,· '.;i . \ . l t" ' ~ ;•· - ) . • ! • 1 . ~ ~ . ! .
.•.•. ; ..... T' ··r1jj. ... ., ..... T .. l~1-· . ··.--+1,. -:·,t~:fJ~'fl~~ -.. · ; i· - 'H -1·--i--\.: . i
··'·i·: i··i+i : + rx:1
.. _,·.·······1". t,· ::j.· ~ 1 ' • 1
HT\-1 '.! - -~ -~-· .
I .. L
. ! ·1
7"·'"::·· · 1 ·rvb . ~;ij·m~ • ,, 1 l ' ·} t l ' ··-~--~-- 't. 1' .,... ,.,,,. ' . ..... ~~l~i :j:\! t;', .... -1·- f'' 4" t'::a-. . ' !J4Ji;_ U~t~i d .! ; ~~:
7-8 i · M.ttn~f · 1 i~~,-] Vel~t~· ~ta~rt ··· :- .. ~.a._, .. ciTC'u lar: Bai. .. ,.,..,.-... ' ....
,. (
I .i
. ~--; ; . -
1 .• ·; ~-1-·-~1
-! + r i-1 i j ,. ;_4 . ' ) ; : ""; 'l·: ;·-~ i __ LiTJ . l.'L l;;;:J '"' 1;: 1- /'i.:'l hl! ~,j: :;,::i:1
: ~ ' ,
'
"l
'' ,q ~ .. ···~.+;At Contro. t. ~th.· .
11 F~t';'
'.In'TQmbling Flow . : · ' .!:. ; ,.,~-·~j , I .. ,,, · r :.~ : '
~ co
49
The velocity ooeffic:lG11t s, ~ end ;'!; , for a length of
cycle equal to 5 1n tumbling flow ·was given in Table 7-2.
It shows that for square and parallelogram roughness ol varied
from 1.5 to 2.4, (3 varied :from 1.3 to 1.8; for tria:rigul.a,r
bars, c\ ve.:r1ed from 2.7 to 3.3, f> verled from 1.9 to 2.2;
for semi-clraular roughness, ~ varied from 2.1 to 2.5, f V13.ried. from 1.65 to 1.85. The coefficient is higher for
t:!'."iangular roughness and low·er for par1.:i.llelogram bars.
50
Table 7-2~ Velocity Coefficient in Tumbling Flow at Control ])e;pth.
~~~ . . -~...-... .. Rol"{i;hnes'S Y1 yr s ~r -·bv"'" ~· .m
cfs ft. :tn. L/K <(, "'tL~ec f't/s"'O _.§l+ape - ~-· . ., l; n u .., ..... ~ ... ~,. .. ____ -· .
• 30;! .216 6 i::· J 9.420 1.398 1.801 2.LW 1.6~)9 sql,,(are
.302 .212 6 5 10.734 1. 14.f'.5 1.745 1.858 1.500
• 302 .208 6 ,... 13.160 1.4152 1 .-..~5 1.706 1.42B :'.) . • I .? );
• :265 .170 l} 5 10.600 1.559 1.801 1.541 1. 3(?5 H
• 265 .1~)0 l~ 5 13.500 1.766 ?.05~~ 1. :)25 1. ;y::54
,.., f' • r..:O; .158 ') e .. 5 14 .. 932 1.316 1.730 ~-:. 268 1 706 .. t-. ,_ \ '1\
.23} .172 2 r.-'.) 14.932 1.355 l.7T3 2.316 1.711 ii
.259 .182 2 5 14.932 1.423 l.H91 c.:. 3'-'t-7 1.766 "
.154 .113 2 5 8.666 1.)63 l.;569 1. 5:.·?5 1. ~525 Pa.ra.llelo-gram
.228 ., !: ~~ r . h 8 .. 666 1.575 l.72rJ 1.561 l. 3li.6 • .r.r~J ~ _,, "
.274 .166 2 5 B.666 1.653 1. 9~"8 l. :i85 1.300 !i
.331 .183 2 5 8.666 1.807 2.161 1.706 1.4-28 H
.175 1 7.fl e:-;.• .,I 2 5 60863 l. ~~68 1.eo6 2.889 2.029 Triangular
• 2121 l~Ci 2 C:-, 6.863 l.:?72 2.023 3.202 .. , 172 • .,i.J . ./ :c:. . . ~ :
• ~>.71 .18J{. '"' (~ G .., 6.863 1.1r7~, ~?. 091 ~:. f~62 a.016 \•'
7?7 • __.!,~ .2C9 2 5 6.H63 1.566 2.185 2.719 1.948 "
.167 .102 ? ·~ 5 8.666 1.639 2.181 2 ""1[;'"7 '. _.;_) 1.771 Semi-cir·-
cular .207 .115 '"' 5 8.666 1.800 2.:348 2.219 1.101 r.:
.279 .140 ~~ ... 8.666 l~ 99i+ ~ ·-77 <~.159 1.671 :-; C.-· :> :t
7.·:06 .158 ,..., ~ 3.666 .-, "'6..,. 2.679 ;2 .190 , 6<17 • ./i: .... .;_ -" r:.. • 1.) ;; ... . () .409 .186 2 i;:, 8.666 ;;?.198 2.966 2.1~57 1. 8:?6 _,, j')
.51
VIII. CONCI1USION
After a CRreful study and i:malysis of the experiment8,l
data of velocity distribution, using .v2.rious sizes and dif-
ferent shapes of art ificia.l roughness elements in rectHngu-
lar open channel, the following conclusions may be drawn:
1. l!"lor t r~n1quil and rapid flow regime, the logarithmic
law m<:i.Y be a.pplied with modification in the constant for all
shB.pes of roughness e:x.oept a.t the control depth of serni-
ci:rcule.r one.
2. The velocity a.t control deµth of sem1-ci:rcular
roughness may be taken a.s e. constant velocity throughout
the section.
3. For tumbling .flow· regime, the velocity varied every
inch between roughness elements. At control depth, it is
nearly pa.rabolic distributed for tri;::.ngul~1r and rectangular
roughness. .As to PB-r&.llelogram and semi-circular roughness
elements the velocity is a dee re a.sing velocity.
4. The ef'fecti ve depth between roughness elements
which is equal to the total de:pth, measured from flume bed
to water surface, minus the height of inflection point, may
be used.
5. The height of inflection point is a function of the
height of roughness, the shape of roughness, the length of
a cycle, the slope of flume bed-and the flow regime.
52
6. ]1or tranquil flow, when L/K rB.nged within 5 to 10
for rectangular, t:rl2ngular and parallelogram roughness ele-
ments, t.he height of inflection point may be used as 2/3
times the height of roughness elements above the flume bed.
For semi·-circular bars it may be used as 2/5 times the rough-
ness element height.
7. For rapid flow, the height of inflection point is
less than that for tranquil flow, ranging from 0.35 to 0.55
times the height of roughness for rectangula.r, triangular
and parallelogram roughness elements.
8. The energy- and momentum coefficients for tumbling
flow are higher for triangulo.r bars and lower for parallelo-
gram roughness elements. It ranged from lc5 to 3.3 for ener-
gy ooefficient snd from 1.3 to 2.2 for momentum coefficient.
53
IX. GLOSSAHY
The letter symbols in this thesis are defined where they
first ra.p:pc.-:1ar and are assembled for conveniEmce of reference
in the following:
A
0
i"I "'1
d
g
k ~ ..
k1 t k 2 K
K~
'"£
L
1n
log
Np
NR q
s v Vm
Vma.-:
V*
constant, Vmr1Y:/Vi~
integrati11g cox1stB.nt:
arbitrary constant
depth of flow, measured fro~ flume bed. ft.
a.ccelers.tion of gravity, v~, ft./sec .. 2
unive:r.sal const>-1nt for K0,r:m.an' s equation, .·hO
arbitrary constant
height of roughness elements, in.
sha.pe fa.ct or of roughnesE~ elements
mixing length
length of a cycle, from roughness center to center
logarithm symbol base of e
1ogad.thm. symbol ba.<:1e of 10
Froude num.bor, V/JfsY1
Reynolds numbi.-Jr,, Vy/7 /µ discharge per unit 11.ridth, c.f.s.
slope of flume bed
velocity of flow, ft./sec.
mean velocity of flow, Q/A, ft./sec.
maximum velocity of flow, ft./sec. -shear velocity, J62;/e , :ft./sec.
Yi
D/S
do'Pth :mea:.:mrBd :f.:r.om f1u:rne b•3d or from. the top of- roughness elements, f·t.
control depth, measured from the upst:ret'rm crest of ror;ighnf3Sf:~ elesn(rnt to ws~ter sur:ra.ce, ft.
the hl':dgh.t of i11:f.'leet:ton point 1;, m.ei~.sured from flume bed to the inflection :potnt, ft.
fluid mass dens1.ty-, lb.sec.2/ft.4
fluid dy:numic vi soo sity, lb ;-sec ./ft. 2
unit -~1eight of fluid, 1b./ft.3
·tr'"''"'"' 1. • ..,,.e fo:r·ce · 1 b- <"'"'C /"""t 2 c·~~.,, V fl · . • - _, f. ...... •. ,,..)1;;; •, .. L •
,,.,,11 '""h""""r ·s·tre <:! lb-.:·e,.. /ft 2 -~i....~, ... ,._ •:J ... _.~4,_.~·- ,), S..:;;i, .. ei-3· \..re .. •
:momentum coef.f1.cient' v2/vm2 dow11st ream surf' ace
U/S upstream surface
55
thesis advisor, Dr. Al-Kha:f~ji, not on.ly for his Jl::ind. en-
oouragement snd generous guidsnce in the ~reparation of this
thes1is, but also for his cereful lectures .s1nd :oe:rF~ona.1 con-
ts.ct which have g:lve:n the author the~ necessary knowledge.
"''" jor pro fe r-rnor, J)r. ., .. M. Mor:r.is, He~1d, Den.":rtment cf Civil ,u,~c .. t . 11.
. hngine e ring, ~.nd Dr. s. rirr. \·ligggrt, who, through their olass
complete his graduate stud;~r. Many than1cs are extended to
Dr. a.nd ~1rs. Edmond O. Ku for their financial su·~1po:rt during
the first year of his study.
He io grateful to ~r. s. ~Tones, :Bugen.e Y .. Koo, tHld Y. M.
Ob.en for their unforgettable efforts an.a. coo:p~~s.t~.on in dolng
the resee.roh work.
Many t;hanks to Mrs. R. D. Walk('}r who typed. this thesis
in a. llmi·ted tJ.me.
'litT J • .1 ••
mu.la for u.n:lf'orm flow of wate:·~f3 in riv8rs 2:r.td other
'.) ·-. J dct e tu:rbulcnz. 14
Mechan:Lce. Stockho1m. pp. 85-93.
4. ,j.' 1933. der turbu.-
ten.ten Stromun:s :i.11 gln.tten. Hohrcn" Ver. Dant. I11g.,
Rauhir;l:rnit s problem. 11 Ing. -Arcbol.v. 7, 1936.
6. Keulegpn, G. H., 1938. i 11ew of turbuJ.ent flow in open
channels. 11 tr. s. National :aureau of Stand~:,,rds Journal c)f
Her:;earch, Vol. 21, pp. 707-741.
7. Ta;srlr.11", B., R., 1939. etVeloc:tty Distribution in Open
Oh8n.nels. 11 limerlcsn Geophysioal Union '.i:rans. pp. 641-643.
Princeton Univ. :Press, 1936.
9 •. Vanoni, V • . A., Assoc. H., A.'.3.C.E., '1VHlocity Di~rtribu-
tion 1n Open t:J!:iam1elr~1." Clvil E'ngin:eering, Vol. 11, 1941,
pp •. 356-357.
5T
ficient s tJ.!.l.d Velocity Dist :ributionf Smooth R.ectc:J"lp;ular
OhnXl:t1.el .. 1' iJ. s. G·eological Survey Hat er-Su:pply ~?ape:r
1592-A.
11. ,eJ.~Khn.fa.;ji, A .. M., "The J~irnamics of Two-Dim.ensiona.l
12.
Plow in Steep, .Rough, Open Chs.nnels." Ph.D. Thesis,
Ute.h St<-:;,te U:niversi.ty, Logr:n, Utah, pp. 132, 1961 ..
;:::ur la theorie des eau:r.: couJ'."antes
{On the Theory of FloH·i.ng Haters). M~moires :pr~sent~s
:par di Ye rs sevants a 1:~.c2d~mie des Science, Paris,
1877. 13. C·n·iolis; G., '1Sur · !e~::tiotb1issement de la formule qui
t.lonne lc::; :figure des remou.s, et sur la. correction o.u'ori
doit y i:o.t.roduire pour te:nir compte des dtff~rences de
vitesl!ie dan~1 les d:tvers points dune :ngme 3ection d 'un
oours.nt. 11 (On the bGJ.?1\:wgter-curve equation and the cor-
r.ections to be introduced to account for ·the difference
of the velocities at. dif/erent points on the same cross-
ssction.) M~m.oire, .Mo. 268, Ann.ales de;21 nont s et
chaussaes, Vol. 11, Ser. 1, pp. 314-335, 1836. I
Table
59 APPENDIX
Page
I Velocity Distribution in Open Channel with Artificial Roughness Element at Control "epth • • • • • • • • • • 60
II Velocity Distribution in Open Channel with Artificial Roughness Element at Middle of Cycle • • • • • • • • • 78
TABU: I
Velocity Distribution in Open Channel with Artificial Roughness Bar of Square Cross-Section at G<>n.trol Depth
A TRANOllIL FLOW v y q Yl s K L v. p•
Run Pt. Vel. Depth Above Disch. Control Bed Slope Roughness Length Shear Y/L V/V* Temp Ft/sec. Roughness per ft. Depth in Height of a Vel.
In. width Ft. percent In. Cycle Ft/sec. c.£.s.
1 2 3 4 5 6 1 8 9 10 11 12 1 .76 0 .066 .074 1.848 l 7.5K • 198 0 •
.80 .14 .019
.82 .24 .032
.83 .40 .053
.84 .56 .075
.845 .60 .080
2 • 71 0 .144 .115 1.848 1 7.5K .262 0 • .95 .10 .013
l.18 .31 .041 1.36 .61 .081 1.55 1.0 .133
3 1.36 0 .220 .160 1.848 l 7 .5K .308 0 • 1.56 .20 .027 l. 70 .42 .056 1.79 .64 .085 1.90 • 91 .121 1.98 1.09 .145 2.06 1.38 .184
l 1.12 0 .1543 .137 .433 2 SK .138 0 .6·
0\ 0
TABLE I (Continued)
1 2 3 4 5 6 7 8
.1543 • 137 .433 2 5K
2 .2364 .186 .433 2 SK
3 .3272 .242 .433 2 5K
4 .435 .275 .433 2 5K
1 .327 .245 .36 4 5K
1.80 1.80
9 10 11
.138 .025 9.710 .050 10.507 .080 10.941 .12 11.304
.161 0 7.640 .025 9.192 .060 10.248 .16 11.490
.184 0 8.316 .03 9.674 .07 10.653 .12 11.305 .18 11.685 .20 '11.631
.196 0 8.622 .030 9.388 .070 10.255 .115 10.918 .16 11.480 .21 11.939 .254 12.347
.168 0 8.333 .03 9.404 .06 9.999 .09 ' 10. 714
12
6 •
8•
8•
5•
0\ ....,
TABLE I (Continued)
1 2 3 4 5 6
1.86 2.40 .327 .245 .36
2 1.59 0 .327 .208 1.187 1.79 .so 1.94 1.10 2.04 1.9
3 1.61 0 .327 .189 2.082 1.83 .4 2.0 .8 2.17 1.4
4 1.64 0 .256 .159 2.082 1.81 .4 1.95 .8 2.09 1.3
5 1.56 0 .181 .136 2.082 1.69 .4 1.79 .8 1.85 1.1
1 1.27 0 .163 .112 .767 1.59 .2 1.81 .5 1.98 1.0
2 1.23 0 .227 .140 .767 1.5 .20 1.81 .50 2.03 .so 2.23 1.20
3 1.4 0 .310 .173 .767 1. 78 .30 2.02 .60
7 8 9 10
4 SK .168 .12
4 5K .282 0 .025 .055 .095
4 SK .356 0 .02 .04 .07
4 5K .326 0 .02 .04 .65
4 SK .302 0 .02 .04 .055
4 2.5K .166 0 .02 .05 .10
4 2.5K .186 0 .02 .OS .08 • 12
4 2.5K .206 0 .03 .06
11
11.071
5.638 6.347 6.879 7.234
4.522 5.140 5.618 6.096
5.031 5.552 5.981 6.411
5.165 5.596 5.927 6.125
7.650 9.578
10.903 11.928
6.612 8.064 9.731
10.913 11.988
6.796 8.640 9.805
12
78.s•
1a.5•
78.s•
79•
79•
73.5•
74•
14.5•
O'\ F\)
TABLE I (Continued)
1 2 3 4 5 6
2.21 1.0 .310 .173 .767 2.34 1.6
4 1.36 0 .461 .225 .767 1.78 .30 2.11 .60 2.43 1.10 2.61 1.60 2.67 2.10
1 l.16 0 .160 .153 .50 1.26 .4 1.33 .8 1.37 1.2 1.39 1.5
2 1.30 0 .227 .19S .5 1.41 .5 1.49 1.0 l.54 l.S 1.59 2.0
3 1.47 0 .305 .243 .s 1.54 .5 l.58 1.0 1.60 1.5 1.61 2.0 L62 2.4
7 8 9 10
4 2 .51C. .206 .10 .16
4 2.51C. .235 0 .03 .06 .11 .16 .21
6 5K .1S7 0 .013 .027 .040 .050
6 SK 1.77 0 .017 .033 .050 .067
6 SK .198 0 .017 .033 .OS .067 .080
11
10.727 11.3S8
5.787 7.574 8.978
10.340 11.106 11.361
7.389 8.025 8.471 8.726 8.853
7.345 7.966 8.418 8.701 8.983
7.424 7.778 7.980 8.081 8.131 8.182
12
75•
77•
77•
77'
0\ VI
TABLE I (Continued)
1 2 3 4 s 6
4 1.23 0 .305 .250 .11 1.36 .6 1.43 1.2 1.46 1.8 1.47 2.4 1.48 3.0
5 1.69 0 .30S .20 1.443 1. 7S .5 1.80 1.0 1.85 l.S 1.89 2.0
B RAPID FLC W
1 3.54 0 .61S .21S 8.128 3.77 .3 3.95 .6 4.10 .90 4.23 1.20 4.35 l.S.5 4.36 2.0
2 4.00 0 .615 2.02 11.07 4.41 .30 4.54 .60 4.69 •• 90 4.81 1.35 4.9 1.75
7 8 9 10
6 SK .094 0 .02 .03 .06 .08 .10
6 SK .304 0 .017 .033 .OS .067
1 7.SK .76S 0 .04 .08 .12 .16 .207 .267
1 7.5K .847 0 .04 .08 .12 .180 .233
11
13.085 14.468 lS.212 1S.S31 lS.638 lS.744
s.SS9 5.7S7
.5.921 6.086 6.217
4.627 4.928 5.163 5.360 5.595 5.686 S.699
4.722 5.206 5.360 S.537 5.679 5.785
12
77•
77•
1a.s•
78.6.
0\ •
TABIB I {Continued)
l 2 3 4 5 6
1 2.8 0 .465 .170 12.004 3.5 .10 3.81 .30 3.94 .so 4.01 .78
2 3.44 0 .527 .183 12.004 3.95 .20 4.18 .40 4.33 .60 4.44 .80 4.52 1.0
3 3.84 0 .615 .199 12.004 4.47 .30 4.79 .60 5.00 .9 5.11 1.1
4 4.18 0 .708 .220 12.004 4.49 .20 4.73 .40 4.91 .60 5.05 .80 5.22 1.10 5.36 1.40 5.49 1.70
1 3.42 0 .567 .185 8.666 3.65 .30 3.82 .60 4.04 1.00 4.22 1.40
7 8 9 10
1 7.SK .81 0 .013 .04 .067 .104
1 7.5K .839 0 .027 .053 .08 .107 .133
1 7.5K .875 0 .04 .08 .12 .147
1 7.SK .924 0 .027 .053 .080 .107 .147 .187 .227
2 5K • 717 0 .03 .06 .10 .14
11
3.306 4.132 4.498 4.652 4.734
4.100 4.708 4.982 5.161 5.292 5.387
4.388 5.108 5.474 5.714 5.840
4.524 4.860 5.119 5.314 5.466 5.650 5.801 5.942
4.770 5.091 5.328 5.635 5.886
12
79•
79•
79•
79.1
11•
•
0\ \J1
TABLE I (Continued)
1 2 3 4 5 6 7
1 2.15 0 1.47 .35 3.65 4 2.63 .6 3.08 1.20 3.47 1.80 3.80 2.40 4.05 3~00 4.25 3.60
2 3.08 0 1.48 .315 5.11 4 3.56 .50 3.96 1.00 4.30 1.50 4.57 2.00 4.80 2.50 5.00 3.00
C TUMBLING FLOW -
1 1. 72 0 .119 .095 12.004 l 1.93 .15 1.95 .25 1.89 .40 1.80 .50 1.65 .60 1.25 .70
2 1.16 0 .242 .154 12.004 1 2.08 .10 2.38 .20 2.50 .35 2.50 .40
8 9 10
5K .641 0 .03 .06 .09 .12 .15 .18
SK • 718 0 .03 .05 .08 .10 .13 .15
7.5K .606 0 .020 .033 .053 .067 .080 .093
7.5K • 77 0 .013 .027 .047 .053
11
3.354 4.103 4.805 5.414 5.928 6.318 6.630
4.290 4.958 5.515 5.989 6.365 6.685 6.964
2.838 3.185 3.218 3.119 2.970 2. 723 2.063
1.506 2.701 3.091 3.247 3.247
12
76°
76.3°
1r
77•
0\
°'
i 2 3 4 5
2.45 .50 .242 .154 2.24 • 65 l.92 • 75 1.85 .80
1 2.04 0 .208 .158 2.70 .20 2.86 .40 2.60 .60 2.31 .so 1.96 1.00 l.48 1.30
2 2.12 0 .233 .172 2.81 .20 2.95 .40 2.70 .60 2.42 .80 2.10 1.00 1.61 1.30 1.22 1.50
3 2.22 0 • 259 .182 2.60 .20 3.06 .40 2.87 .60 2.63 .80 2.38 1.00 1.97 I 1.30 1.36 l.60
TABLE I (Continued)
6 7 8 9
12.004 1 7.5K .11
14. 932 2 5K .886
14.932 2 5K .92
14. 932 2 SK .935
10
.067
.087
.100
.107
0 .020 .040 .060 .080 .10 .130
0 .020 .040 .060 .080 .100 .130 .150
0 .02 .04 .06 .08 .10 .13 .16
11
3.182 2.909 2.494 2.403
2.302 3.047 3.228 2.934 2.607 2.212 1.670
2.302 3.051 3.203 2.932 2.628 2.280 1. 748 1.325
2.374 2.781 3.273 3.069 2.813 2.545 2.107 1.455
12
rn. 1•
12•
n•
0\ -:a
1 2 3 4 5
1 1.59 0 .216 .17 2.07 .15 2.25 .30 2.28 .46 2.11 .70 1.90 1.00 1.61 1.40 1.30 l. 70
2 1.74 0 .216 .15 2.05 .15 2.21 .35 2.17 .10 1.97 1.10 1.68 l.60
l 1.92 0 .302 .216 2.24 .15 2.42 .35 2.35 .so 2.18 .80 2.00 1.10 1.81 1.40 1.59 1.70 1.42 2.00 1.24 2.30
2 l.83 0 .302 .212 2.12 .15 2.22 .30
TABLE I (Continued)
6 7 8 9
10.6 4 SK .78
13.5 4 SK .807
9.42 6 SK .81
10.734 6 5K .856
10
0 .008 .015 .023 .035 .05 .07 .085
0 .008 .018 .035 .055 .080
0 .005 .012 .017 .027 .037 .047 .057 .067 .077
0 .005 .010
11
2.108 2.744 2.983 3.023 2.797 2.519 2.134 1. 723
2.156 2.540 2.738 2.689 2.441 2.082
2.370 2.765 2.987 2.901 2.691 2.469 2.234 1.963 1. 753 1.531
2.138 2.477 2.593
12
10•
10•
75•
75:!
0\ co
1
3
A
Run
1 1
TABLE I (Continued}
2 3 4 5 6 7 8 9 10 11 12
2.26 .so .302 .212 10. 73' 6 51( .856 .017 2.640 2.19 .10 .023 2.558 .. 2.09 .90 .030 2.442 1.93 1.20 .040 2.255 1.71 1.50 .050 1.998 1.37 1.90 .063 1.600 1.01 2.30 .077 1.180
1.36 0 .302 .208 13.16 6 SI< .'/39 0 1.4'.8 75• 1.97 .15 .005 2.098 2.25 .40 .013 2.396 2.19 .70 .023 2.332 2.01 1.00 .033 2.205 1.89 1.30 .043 2.013 1.71 1.60 .053 1.821 1.33 2.10 .010 1.416
Velocity Distribution in Open Channel with Artificial Roughness Bar of Parallelogram Croes-Section at Control Depth
'.J!AH9!1L l'L<M v y q
Point Depth Disch. Vehtty above per.ft. Ft/sec. Rough- width
- .... - - -2 3 4
1.28 0 .175 1.41 .30 1.50 .65 1.57 1.00 1.54 1.40
Y1 S It Contro 1 Bed Rough-Depth Slope DUB rt. in Haight
1!f -5 6 7
.137 .433 2
L Length of a
cycle
8 SK
v. Shear y/L
Velocity Ft/sec.
9 10 .138 0
.03
.065
.10
.14
v1v.
11 9.275
10.217 10.869 11.376 ll.159
r• Temp.
12 68.s•
°' '°
TABLE I (Continued)
l 2 3 4 5 6 7
2 1.45 0 .270 .178 • 433 2 1.61 .40 1.70 .95 1.74 1.45 • 1.70 1.80
3 1.69 0 .327 .20 .433 2 1.85 .40 1.96 .80 2.02 1.20 2.05 1.60 2.02 2.00
4 1.54 0 .405 .234 .433 2 1.80 .60 2.06 1.20 2.12 1.60 2.15 2.10
L~D FIA>W I
1 4.04 0 .6 .184 8.666 2 4.23 .40 4.39 .80 4.55 1.20 4.6 1.40 4.58 1.50
2 4.04 0 .70 .204 8.666 2 4.37 .40 4.62 •• so 4.79 1.20 4.85 1.50 4.80 1.60
8 9 10
5K .158 0 .04 .09S .145 .180
SK .167 0 .04 .08 .12 .16 .20
SK .180 0 .06 .12 .16 .21
5K .482 0 .04 .08 .12 .14 .15
SK .569 0 .04 .08 .12 .15 .16
11
9.177 10.190 10. 759 11.012 10.759
10.120 11.078 11. 736 12.096 12.275 12.096
8.556 lO.ifZ 11.444 u. 778 11.944
8.382 8. 776 9.108 9.440 9.544 9.502
7.100 7.680 8.120 8.418 8.524 8.436
12
69 •
69.
69°
70.7
70.5
•
•
-..;a 0
C TUMBLING FLOW 1 2 3 4 5
1 2.11 0 .154 .113 2.65 .10 2.11 .20 1.72 .40 1.45 .60 1.12 .90
2 2.21 0 .228 .145 2.46 .10 2.30 .20 2.12 .40 1.94 .60 1.79 .ao 1.65 1.00 1.48 1.30
3 2.75 0 .331 .183 2.89 .10 2.79 ' .20 2.s; .40 2.46 .60 2.33 .80 2.20 1.00 2.02 1.30 1.73 1.80
TABLE I (Continued)
6 7 8 9 10
8.666 2 5K .315 0 .01 .02 .04 .06 .09
8.666 2 51. .404 0 .01 .02 .04 .06 • oa .10 .13
8.666 2 Sit .510 0 .01 .02 .04 .06 .08 .10 .13 .18
11
6.698 8.413 6.698 5.460 4.603 3.556
5.618 6.089 5.693 5.247 4.802 4.430 4.084 3.663
5.392 5.667
_,5.471 5.039 4.824 4.569 4.314 3.961 3.392
12
11•
10.5•
10.s•
~ .....
TABLE I (Continued)
Velocity Distribution in Open Channel with Artificial Roughness Bar of Triangular Cross-Section At Control Depth
A TRANQUIL FLOW v y
Run Point Depth Velocity above
Ft/sec Rough-ness In.
l 2 3
l 1.42 0 1.58 .30 1.67 .60 1.69 .90 1. 70 1.30
2 1.56 0 1.71 .40 1.83 .80 1.92 1.30 1.98 1.80
3 1.85 0 2.05 .50 2.19 1.00 2.30 1.50 2.40 2.00
4 1.64 0 1.87 .50 1.99 1.00 2.06 1. so 2.12 2.10
q Yl s Disch Control Bed per.ft. Depth Slope Width Ft. in c.f.s. Percent
4 5 6
.170 .141 .433
.257 .184 .433
.326 .21 .433
.403 .263 .433
I l j
K L v. Fo Rough- Length Shear y/L V/V* Temp. ness of a Velocity Height cycle Ft/sec. In.
7 8 9 10 11 12
2 SK .139 0 10.215 70° .03 11.367 .06 12.014
I .09 12.158 .13 12.230
2 SK .160 0 9.750 70.2° .04 10.688 .08 11.438 .13 12.000 .18 12.375
2 5K .171 0 10.819 70.3° .05 11. 988 .10 12.807 .15 }3.45(; .20 14.035
2 5K .191 0 8.586 70.3° .05 9.791 .10 10.419 .15 10.785
I .21 11.099
~ I\)
B RAPID FLOW l 2 3 4 5
l 2.99 0 .10 .227 3.43 .30 3.70 .60 4.02 .99 4.28 1.20 4.57 1.60 4.81 2.00
2 3.16 0 .745 .236 3.58 .40 4.02 .80 4.38 1.20 4.60 1.60 4.78 2.10
c Tllt.BLillG J IDW
l 1.94 0 .175 .138 2.11 .10 2.17 .20 2.17 .30 2.03 .60 1.84 .90 1.69 1.10
2 2.18 0 .218 .159 2.28 .20
TABLE l (Continued)
6 7 8 9
8.666 2 51<. .661
8.666 2 5K .658
6.863 2 SK .305
6.863 2 5K .351
10
0 .03 .06 .09 .12 .16 .20
0 .04 .08 .12 .16 .21
0 .01 .02 .03 .06 .09 .11
0 .02
11
4.523 5.189 5.597 6.081 6.475 6.913 7 .277
4.803 5.441 6.110 6.657 6.991 7.265
6.459 6.918 7 .115 7 .115 6.656 6.033 5.541
6.211 6.496
12
73•
73•
72•
72•
~ ~
TABLE I (Continued)
l 2 3 4 5 6 7 8 9 10 11 12
2.27 .40 .218 .159 6.863 2 SK .351 .04 6.467 2.18 .60 .06 6.211 2.09 .so .08 5.954 2.02 1.00 .10 S.755 1.93 1.20 .12 5.499 1.85 1.40 .14 5.271
3 2.16 0 .271 .184 6.863 2 5K .406 0 5.320 72• 2.32 .20 .02 5. 714 2.34 .40 .04 5.764 2.30 .60 .06 5.665 2.22 .80 .08 5.468 2.17 1.00 .10 5.345 -J 2.06 1.30 .13 5.074 -I> 1.98 1.50 .15 4.877
4 2.18 0 .327 .209 6.863 2 SK .462 0 4. 719 73° 2.40 .10 .01 5.195 2.50 .20 .02 5.411 2.52 .40 .04 5.455 2.49 .60 .06 5.390 2.46 .80 .08 5.325 2.38 1.10 .11 5.152 2.16 1.60 .16 4.675 2.01 1.80 .18 4.J'il
TABLE I (Continued) Velocity Distribution in Open Channel with Artificial Roughness Bar of Semi-Circular Cross-Section
At Control Depth
A TRANQUIL FLOW
Run
1
1
2
3
v Point Velocity Ft/sec.
2
1.45 1.45 1.45 1.45 1.47
1.45 1.45 1.45 1.45 1.49 1.58 1.62 1.59
1.83 1.83 1.83 1.87 1.90 1.91 1.90 1.83
Y q Yl S Depth Disch. Control Bed Above per;' ft Depth Slope Rough- Width Ft. in neaa c.f.s. percent Inches -·-·-
3 4 5 6
0 .160 .130 .433 .30 .60
1.00 1.30
0 .214 • 159 .433 .30 .60 .95
1.10 1.30 1.50 1.65
0 .327 .21 .433 .40 .70
1.00 1.40 1.70 2.05 2.15
K L Rough- Length neas of a Height Cycle Inches
7 8
2 Sit
2 Sit
~
2 5K
'Y* Shear y/L Velocity Ft/sec.
9 10
.135 0 .03 .06 .10 .13
.149 0 .Oll .060 .095 .110 .130 .150 .165
.171 0 .04 .01 .10 .14 .17 .205 .215
V/V*
11
10.74 10.74 10.74 10.7'1+ 10.888
9.732 9. 73.2 9.732 9.732
10.00 10.604 10.872 10.671
10. 702 10.702 10.702 10.936 11.111 11.169 11.111 10.702
p• Temp.
12
69.
69 •
10•
~ \11
TABLE I (Continued)
1 2 3 4 5 6 7
4 1.97 0 .411 .245 .433 2 1.97 .40 1.97 .• 80 1.97 1.20 2.07 1.50 2.13 1.80 2.17 2.20 2.19 2.50 2.17 2.80
B TlllBLDG FIDW
l 2.94 0 .167 .102 8.666 2 2.91 .10 2.66 .30 2.37 .so 2.10 .70 1.81 .90
2 3.18 • .207 .115 8.666 2 3.09 .20 2.81 .40 2.47 .60 1.96 1.00 1.80 1.15
3 3.62 0 .279 .14 8.666 2 3.54 .20 J.20 .40 2.70 .70 2.44 .90 2.07 1.20
8 9 10
5K .185 0 .040 .080 .120 .150 .180 .220 .250
I .280
SK .533 0 .010 .030 .050 .•1• .090
5K .566 0 .020 .040 .060 .100 .115
SK .641 0 .02 .04 .01 .09 •• 120
11
10.649 10.649 10.649 10.649 11.189 11.514 ll.730 11.838 11. 730
S.515 5.546 4.990 4.446 3.940 3.396
5.618 5.459 4.964 4.364 3.463 3.180
5.647 5.522 4.992 4.212 3.806 J. 229
12
10•
10.s•
10.5•
10.5•
~
°'
TABLE I (Continued}
1 2 3 4 5 6 7 8 9 10 11 12
l.89 1.40 .279 .14 8.666 2 5K .641 .140 2.948
4 3.93 0 .326 .158 8.666 2 SK .664 0 5.919 71• 3.82 .20 .020 5.753 3.49 .40 .040 5.256 2.96 .70 .070 4.458 2.54 LOO .100 3.825 2.05 1.40 .140 3.087 1.90 1.60 .160 2.861
5 4.52 0 .409 .186 8.666 2 SK . 719 0 6.287 11• 4.46 .20 .020 6.204 4.20 .40 .040 5.842 3.62 .70 .070 5.035 -.1
-.1 3.12 1.00 .100 4.340 2.64 1.30 .130 3.672 2.21 1.60 .160 3.074 1.98 1.90 .190 2.754
TABLE II
Velocity Distribution in Open Channel with Artificial Roughness Bar of Square Cross-Section At Middle of Cycle
A TRANQUIL FLOW v y q yi s K L v*
Run Point Depth Disch. Height Bed Rough- Length Shear Y/L V/V* Y/K Velocity Above per.ft. of In- Slope ness of a Velocity Ft/sec Flume width flecdm in Height cycle Ft/sec.
Bed c.f.s. Point percent In. In. In.
1 2 3 4 5 6 7 8 9 10 11 12
l .55 0 .066 1.848 1 7.5K .198 0 2. 778 .55 .28 .037 2. 778 .55 .58 .on 2. 778 .71 .79 .105 3.586 .81 .98 .131 4.091 .82 1.10 .147 4.141 .84 1.40 .187 4.242 .85 1.58 .587 .211 4.293 .587
2 .90 0 .144 1.848 1 7.5.K .262 0 3.435 .91 .30 .040 3.473 .92 .58 .077 3.512
1.12 .80 .107 4.275 1.23 l.00 .133 4.695 1.38 l.34 .179 5.267 1.49 1.65 .220 5.687 1.61 2.00 .586 .267 6.145 .586
3 .62 0 .220 1.848 l 7.5K .308 0 2.013 .62 .34 .045 2.013 .62 .58 .077 2.013 ;86 .83 .111 2.792
~ O>
l 2 3 4 5
1.12 1.08 .220 1.35 1.42 1.54 1. 74 1.69 2.08 l. 77 2.39 .585
1 0 0 .154 0 .70 0 1.30
.40 1.50
.76 1.80 1.03 2.20 l.2S 2.70 1.43 3.20 1.35
2 .50 0 .226 .50 .60
.• 50 1.30 .64 1.50 .96 1.90
1.27 2.30 1.48 2.80 1.65 3.65 1.34
3 .501 0 .327 .501 .so
TABLE II (Continued)
6 7 8 9
1.848 l 7 .SK .308
.433 2 SK .138
.433 2 SK .161
.433 2 SK .184
10
.144
.189
.232
.277
.319
0 .010 .130 .150 .180 .220 .270 .320
0 .060 .130 .150 .190 .230 .280 .365
0 .050
11
3.636 4.382 4.999 5.486 s. 745
0 0 0 2.898 5.507 7.463 9.058
10.362
3.106 3.106 3.106 3.975 5.963 7.888 9.192
10.248
2.723 2. 723
12
.58
.67
.67
5
5 ~ \D
l 2 3 4 5
.501 l.00 .327
.501 1.35
.82 1.55 1.18 1.90 1.44 2.35 1.65 2.90 1.60 3.55 1.87 4.00 1.82 4.07 1.35
4 -.29 0 .435 -.29 .70 -.29 1.34 0.01 1.50 0.40 1.75 0.75 2.10 1.08 2.50 1.43 3.00 1.67 3.50 1.87 4.20 1.97 4.80 1.34
l .29 0 .327 .29 .so .29 1.60 .29 2.60 .63 3.00 .91 3.60
1.15 4.20 1.36 4.80 1.58 5.60 1.74 6.40 2.6
TABLE II (Continued)
6 7 8 9
.433 2 5K .184
.433 2 SK .196
.36 4 SK .168
I I
10
0 .135 .155 .190 .235 .290 .355 .400 .407
0 .010 .134 .150 .175 .210 .250 .JOO .350 .420 .480
0 .040 .080 .130 .150 .180 .210 .240 .280 .320
11
2. 723 2. 723 4.457 6.413 7.826 8.968 9.783
10.163 9.892
-1.480 -1.480 -1.400
.357 2.041 3.827 5.510 7.296 8.520 9.511
10.051
1.726 1. 726 1.726 1.726 3.750 5.416 6.845 8.095 9.404
10.356
12
5
0
co 0
TABLE II (Continued)
l 2 3 4 5 6 1
2 .51 0 .327 1.187 4 .51 .90 .S1 1.80 .51 2.50 .88 3.20
1.18 3.80 1.52 4.50 1.78 5.10 2.08 5.90 2.5.
3 .501 0 .327 2.082 4 .501 .90 .501 1.80 .501 2.20 .57 2.60 .98 3.10
1.38 3.60 1.72 4.10 2.02 4.70 2.32 5.40 2.a
4 .71 0 .256 2.082 4 .71 .9 .71 1.8 • 74 2.6 .89 3.0
1.17 3.6 1.50 4.0 1.95 4.5 2.43 5.2 2.54
8 9 10
5K .282 0 .045 .090 .128 .160 .190 .225 .255 .295
5K .356 0 .045 .090 .110 .130 .155 .180 .205 .235 .270
SK. .326 0 .045 .090 .130 .150 .180 .200 .225 .260
11
2.021 2.021 2.021 2.021 3.120 4.184 5.390 6.312 7.376
1.407 1.047 1.04_7 1.047 1.601 2.753 3.876 4.831 5.674 6.517
2.178 2.178 2.178 2.270 2.730 3.589 4.601 5.981 7.454
12
.6'
.6J
.635
O> ....
1 2 3 4 5
5 .65 0 .181 .65 .9 .65 1.8 .70 2.6 • 91 3.1
1.09 3.6 1.53 4.1 1. 77 4.6 1.93 5.1 2.55
l • 71 0 .163 • 71 1.0 .71 2.0 • 71 3.0 .71 3.5
1.00 3.9 1.47 4.1 1.75 4.4 2.01 5.0 3.5
2 .65 0 .227 .65 1.0 .65 2.0 .65 3.0 .65 3.4 .78 3.6
TABLE II (Continued)
6 7 8 9
2.082 4 5K .302
.767 4 2.5K .166
.767 4 2.5K .186
10
0 .045 .090 .130 .155 .180 .205 .230 .255
0 .10 .20 .30 .35 .39 .41 .44 .so 0 .10 .20 .30 .34 -'36
11
2.152 2.152 2.152 2.318 3.013 3.609 5.066 5.860 6.390
4.277 4.277 4.277 4.277 4.277 6.024 8.855
10.542 12.108
3.494 3.494 3.494 3.494 3.494 4.193
12
5
CX> I\)
TABLE II (Continued)
1 2 3 4 5 6 1
1.18 4.0 .227 .767 4 1.60 4.3 1.88 4.7 2.05 5.2 3.45
3 .64 0 .310 .767 4 .64 1.0
.• 64 2.0 .64 3.0 .64 3.35 .82 3.6
1.26 3.9 1.66 4.2 2.05 4.8 2.30 5.6 3.37
4 .51 0 .461 .767 4 .51 1.0 .51 2.0 .51 3.0
';11 3.1 .54 3.3 .74 3.6
1.01 3.8 1.50 4.0 1.89 4.2 2.28 4.6 2.56 5.2 2.73 6.1 3.24
8 9 10
2.5K .186 .40 .43 .47 .52
2.5K .206 0 .10 .20 .30 .335 .36 .39 .42 .48 .56
2.5K .235 0 .10 .20 .30 .31 .33 .36 .38 .40 .42 .46 .52 .61
11
6.344 8.602
10.107 11.021
3.107 3.107 3.107 3.107 3.107 3.980 6.116 8.059 9.951
11.164
2.170 2.170 2.170 2.170 2.170 2.298 3.149 4.298 6.383 8.042 9.701
10.978 11.616
12
3
(l) ~
T"1JLE II (Continued)
1 2 3 4 5 6 7
1 .57 0 .160 .s 6 .57 1.5 .57 I 3.0 .57 I 4.0 j .65 4.6 .74 5.2 .87 5.9
1.04 6.5 1.22 7.1 1.38 1.5 4.03
2 .30 0 .227 .5
I 6
.30 1.5
.30 3.0
.30 4.0
.56 4.5
.79 5.2
.96 5.9 1.12 6.6 1.24 7.3 1.36 8.05 4.0
3 .41 0 .305 .5 6 .41 1.5 .41 3.0 .41 3.95 .51 4.6 • 71 5.2 .88 5.8
1.02 6.4 1.17 7.2
8 9 10 ,- I j SK I I I I I ' l
i I
-· 7 0
.05
.10
.133
.153
.173
.197
.217
.15
.237
.250
SK .17 7 0 .05 .10 .133 .150 .173 .197 .2.20 .243 .268
5K .19 8 0 .05 .10 .132 .153 .173 .193 • 213 .240
11 12 ·- -· .___
3.631 3.631 3.631 3.631 4.140 4. 113 I 5. 541 I 6.624 I 7. 771 s.190 1 .67
1.695 1.695 1.695 1.695 3.164 4.463 5.424 6.328 7.006 7.684 • 66
I 2.071 2.071 2.071 2.071 2.576 3.586 4.444 5.152 5.909
I
2
7
cc •
TABLE II (Continued)
~-·-...-- , .. 1 2 3 4 5 6 7 8 9 10 11
I . 267 l 1.29 8.0 .305 .5 i
6 SK .198 I 6. 515 1.33 8.6 3.98 . 287 6. 717
I ! 4 0 0 .305 .11 6 SK .094 0 0
0 1.5 .05 0 0 3.0 • 10 0 0 4.1 .137 0
.20 4.5 .150 l 2.128
.50 5.0 .167 5.319
.83 5.7 .190 8.830 1.05 6.5 • 217 11.170 1.19 7.4 .247 12.659 1.27 8.1 .270 13. 510 1.33 9.0 4.14 .300 14.149
5 .62 0 .305 1.443 6 SK .304 0 2.039 .62 1.5 .050 2.039 • 62 3.0 .100 2.039 .62 3.9 .130 2.039 .84 4.6 .153 2.763
1.12 5.4 .180 3.684 1.40 6.3 • 210 4.605 1.63 7.1 .237 5.362 Ji. 78 8.0 3.<J() .267 5.855
B RAPID FWW
1 ,- 1.94 0 .761 I I 5.015 I 1 I 7.5K .624 -, 0 3.110 1. 96 . 3 • 040 3. 142
12
.663 I
I I
.69
. 65
CX> VI
1 2 3 4 . 5
2.02 .5 .761 2.22 .7 2.52 1.0 3.08 1.4 3.61 1.8 3.92 2.1 4.35 2.6 4.74 3.3 .475
2 1. 77 0 .835 1.79 .30 1.81 .48 1.92 .60 2.19 .90 2.51 1.20 2.98 1.50 3.40 2.00 3.80 2.50 4.16 3.00 4.32 3.60 .48
3 1.68 0 .863 1.74 .30 1.83 .so 2.17 .70 2.60 1.00 2.96 1.30 3.40 1. 70 3.75 2.10 4.10 2.60 4.33 3.10 .475
TABLE II (Continued)
6 7 8 9
5.015 1 7.5K .624
5.015 1 7.SK .642
5.015 1 7.5K • 65
10
.067
.093
.133
.187
.240
.280
.347
.440
0 .040 .064 .080 .120 .160 .213 .267 .333 .400 .480
0 .040 .067 .093 .133 .173 .227 .280 .347 .413
11
3.238 3.559 3.956 4.937 5.787 6.284 6.973 7.598
2.757 2.788 2.819 2.991 3.411 3.910 4.642 5.296 5.919 6.480 6.729
2.585 2.677 2.815 3.339 4.000 4.554 5.231 5.769 6.308 6.662
12
.475
.48
.475
O> 0\
TABLE II (Continued)
1 2 3 4 5 6 7 8 9 10 11 12
1 1.36 0 . 615 8.128 1 7.5K . 765 0 1. 778 1.40 .30 .040 1.830 1.42 .41 .055 1.856 1.72 .60 .080 2.248 2.18 .90 .120 2.850 2.62 1.20 .160 3.425 3.25 1.50 .200 4.248 3.86 2.00 .267 4.579 4.36 2.50 .333 5.699 4.70 3.00 .412 .400 6.144 2
I
2 1.55 0 .615 11.07 1 7.51'. .847 0 1.830 1.58 .20 r, .oa7 1.865 ~ l.~62 .40 .• 953 1.913 2.10 .60 ·* 2.479 2.6, .If ~ .1ie 3.140 . 3.17 1.20 ~ .
.160 3.743 3.'A i.so I{ .. 200 4 .• 345 4 .• z~ 2.QI. dF
.267. 5.029 4.51 2.50 .400 I• .• 333 5.325 00
3 1. 74 0 .615 15.01 1 7.5K .955 0 1. 749 1.80 .30 .040 1.809 1.85 .40 .053 1.859 2.03 .60 .080 2.040 2.67 .90 .120 2.683 3.65 1.20 .160 3.668 4.33 1.50 .200 4.352 4.93 2.0Q .35 • 267 4.955
TABLE 11 (Continued)
1 2 3 4 5 6 7
1 ·1.33 0 .465 12.004 1 1.38 .20 1.44 .40 2.e5 .10 2.80 1.00 3.40 1.40 3.75 1.80 .40
2 1.51 0 .528 12.004 l 1.58 .20 1.fl .39 lJ. .50 2.48 .80 3.t5 1.18 3.77 1.40 4.19 1.80 4.28 2.00 .395
.,
3 1.2.6 0 .615 12.004 1 1.32 .ae 1.41 .40 1.83 .60 2.34 .80 2.92 1.00 3.58 1.30 4.06 1.60 4.60 2.00 4.0
4 1.02 0 .709 12.004 1 1.13 .20 1.26 .39
8 9 10
7.SK .81 0 .027 .053 .093 .133 .187 .240
7.SK .839 0 .027 .052 .067 .101 .147 .187 .240 .267
7.Slt .875 0 .927 .053 .080 .107 .133 .173 .213 .267
7 .51{ .924 0 .027 .052
11
1.642 1.704 l. 778 2.531 3.457 4.197 4.629
1.800 1.883 1.943 2.169 3.056
- 3. 754 4.493 4.970 5.101
1.440 1.508 1.611 2.091 2.674 3.337 4.091 4.640 5.257
1.104 1.223 1.364
12
.40
.395
4.0
O> O>
1 2 3 4 5
2.00 .60 .709 2.56 .80 3.08 1.00 3.84 1.30 4.35 1.60 4.69 2.00 s.oo 2.60 .39S
l .SOl 0 .S67 .67 .so .86 .90
l.2S 1.10 1.82 1.40 2.4S 1.70 2.99 2.00 3.70 2.50 4.25 3.00 4.59 3.40 .567 .92
2 .501 0 .635 .61 .so .82 .90
1.10 l.OS 1.45 1.20 2.10 1.50 2.67 1.80 3.12 2.10 3.67 2.so 4.12 2.90 4.41 3.20 .925
3 .os 0 .70
TABLE II (Continued)
6 7 8 9
12.004 l 7.5K .924
8.666 2 SK • 718
8.666 2 5K .74S
8.666 2 5K • 77
10
.080
.107
.133
.173
.213
.267
.347 0 .OS .09 .11 .14 .17 .20 .2S .30 .34 0 .05 .09 .lOS .120 .150 .180 .210 .250 .290 .320 0
11
2.165 2.771 3.333 4.156 4.708 5.076 5.412
.698 •• 933 1.198 1.741 2.S3S 3.412 4.164 5.153 5.919 6.392
.672
.819 1.101 1.477 l.~946 2.819 3.S84 4.188 4.926 5.530 S.920
.065
l2
.395
.46
.46J
CJ) \0
TABLE II (Continued)
1 2 3 4 5 6 7 8 9 10 11 12
.22 .5 .10 8.666 2 SK. .77 .05 .286
.501 .92 .092 .651 1.01 1.2 .12 1.312 1.69 1.5 .15 2.195 2.5 1.8 .18 3.247 3.1 2.1 .21 4.026 3.7 2.4 .24 4.805 4.2 2.1 .93 .27 5.455 .46 5
4 .03 0 • 775 8.666 2 SK .797 0 .038 .20 .5 .05 •• 251 .41 .92 .092 .514 .54 1.1 .u .678
1.36 1.3 .13 1.706 1.94 1.5 .15 2.434 8 2.93 1.8 .18 3.676 3.37 2.1 .21 4.218 4.07 2.4 .24 5.107 4.6 2.7 .93 • .27 5.772 .4E 5
l .40 0 1.48 3.65 4 SK .641 0 .624 .52 .6 .03 .811 .63 1.2 .06 .983 .80 1.8 .09 1.248 .92 2.0 .10 1.435
1.17 2.4 .12 1.825 2.06 3.0 .15 3.214 2.67 3.6 .18 4.165 3.22 4.2 .21 5.024 3.7 4.8 .24 5.772 4.16 5.4 .21 6.490
1 2 3 4 5
4.65 6.0 1.48 5.20 6.6 5.50 6.9 1.99
2 -.15 0 1.48 0 .6
.17 1.2
.40 1.9 1.15 2.6 2.30 3.2 3.15 3.8 3.74 4.4 4.2 5.0 4.62 5.6 5.00 6.0 1.9
TABLE II (Continued)
6 7 8 9 10
3.65 4 SK .641 .30 .33 .345
5.11 4 SK • 718 0 .03 .06 .095 .130 .160 .190 .22 .25 .28 .30
11
7.254 8.113 8.581
-.209 0
.237
.557 1.602 3.203 4.387 5.209 5.850 6.435 6.964
12
0.4
.47
98
5 \0 ....
TABLE II (Continued) Velocity Distribution in Open Channel with Artificial Roughness Bar of Parallelogram Cross-
Section At Middle Of Cycle A TRANQUIL FLOW
v y Run Point Depth
Velocity above Ft/sec. Flume
Bed In.
1 2 3
1 .40 0 .40 .80 .40 1.40 .so 1.65 .77 1.80
1.00 2.05 1.19 2.40 1.34 3.00 1.39 3.50
2 .43 0 .43 .60 .43 1.20 .so 1.40 .10 1.60
1.08 2.00 l.40 2.40 1.62 3.00 1.69 3.80
3 .58 0 .58 .600
q Disch. per ft. width c.f.s.
4 .175
.270
.327
Yi S K Height Bed Rough· of In- Slope ness flection in Height Point percent In. In 5 6 7
.433 2
1.42 .433 2
1.34 .433 2
L Length of a Cycle
8 5K
5K
5K
v. Shear y/L Velocity Ft/sec.
9 10 .138 0
.080
.140
.165
.180
.205
.240
.300
.350 .158 0
.060
.120
.140
.160
.200
.240
.300
.380 .167 0
.600
v /v.,,
11
2.898 2.898 2.898 3.623 5.579 7.246 8.623 9.710
10.072 2.721 2.721 2.721 3.165 4.430 6.835 8.861
10.530 10.696 3.473 3.473
Yt/K
12
• 71
• 67
\0 f\)
TABLE '11 (Continued)
1 2 3 4 5 6
3 .58 1.30 .327 .433 .61 1.45
l.06 1. 75 1.53 2.00 1.62 2.40 1.84 2.80 1.96 3.15 2.05 3.80 1.3
4 .65 0 .405 .433 .65 .60 .65 1.25
1.06 1.50 1.46 1.90 1.64 2.40 2.25 3.00 2.48 3.50 2.70 4.10 l.25
B RAPID F r l'lL1
l .72 .so .6 8.666 .75 .70 .80 .90 .85 1.05
1.14 1.30 1.80 1.60 2.88 2.00 3.64 2.40 4.33 3.00 4.93 3.80 1.08
7 8 9 10 2 5K 1.67 .130
.145
.175
.200
.240
.280
.315
.380
2 5K .180 0 .06 .125 .150 .190 .240 .300 .350 .410
2 SK .482 .os .07 .09 .105 .130 .160 .200 .240 .300 .380
11 3.473 3.653 6.347 9.162 9.701
11.018 11. 736 12.275 3.611 3.611 3.611 5.889 8.111 9.111
12.500 13. 778 15.000
1.494 l.556 1.660 l. 763 2.365 3.734 5.975 7.552 8.983
10.228
12
.65
.625
.54
'° VI
TABLE II (Continued)
1 2 3 4 5 6 7 8 9 10 11 12
2 .70 .so .677 8.666 2 SK .547 .OS 1.280 .73 .70 .07 1.335 .77 .90 .09 1.408 .82 1.05 .105 1.499
1.04 1.20 .120 1.901 1.51 1.50 .150 2.761 2.17 1.80 .180 3.967 2.86 2.0 .20 5.229 3.49 2.2 .22 6.380 4.02 2.5 .25 7.349 4.43 3.0 .30 8.100 4.63 3.5 .35 8.465 4.74 4.0 1.06 .40 8.666 .5 "° 3 .....
3 .43 .s .7 8.666 2 SK .569 .05 .756 .50 .7 .07 .879 .66 1.0 .10 1.160 .88 1.2 .12 1.547
1.51 1.5 .15 2.654 2.15 1.7 .u 3.779 3.01 2.0 .20 5.290 4.01 2.4 .24 7.048 4.49 2.9 .29 7.891 4.62 3.6 1.06 .36 8.120 .5 2S
4 .71 .5 .785 8.666 2 5K .608 .05 1.168 .73 .7 .07 1.201 • 77 .9 .09 1.266 .80 1.0 .10 1.316 .98 1.2 .12 1.612
1.45 l.S .15 2.38S
TABLE II (Continued)
1 2 3 4 5 6 7 8 9 10 11 12
2.28 1.8 .785 8.666 2 5K .608 .18 3.750 3.24 2.1 .21 5.329 4.23 2 • .5 .25 6.957 4.?4 3.0 .30 7.796 S.04 3 • .5 .35 8.289 S.13 3.8 1.04 .38 8.437 .52
Velocity Dietr1'ution ia Open Cluaanel with Artificial llougbnea• Bar of Triangular Croes-Section at Middle of Cycle
A. TrU.111111cptl ________ ~--- ________________ ---~ y'•n: --------.. - y -
Run Pqi!'t Def 9' Velci~ltJ ~· .rt{,ec. ,1.._
-~ 1a.:: 1 I• 2 ·3
V•·
1 .501 0 .soi .so .501 1.10 .501 1.30 .720 1.70 .910 2.00
1.10() 2.30 1.280 2.60 1.440 3.00 1.560 3.40
2 .470 0 .470 .50
;
;
* I•
y
ct Dia ch. per ft. ... c.f.e.
4 ;,,
.~10
.257
Yi S K Height Bed SlOfe lough-of In• ia •••• f lectien percent Height Pol at 111. Ill. -
.5 6 7 ! -
.433 2
1.33
.433 2
L Length of a Cycle
8
SK
SK
v. Shear y/f.. Velocity · 'ft/aee.
9 ·1• .;
.139 0 • .050
• tli» .130 .170 .200 .230 .260 .300 .340
.160 ~050
-vJY. Yi/K
11 12
3.604 3.604 3.604 3.604 5.180 6.547 7.913 9.208
10.359 11.223 .665
l·.\~\
\0 \JI
TABLE II (Continued)
1 2 3 4 5 6 7 8 9 10 11 12 .470 1.10 .257 .433 2 SK .160 .110 2.938 .500 1.30 .130 3.125 .600 1.50 .150 3.750 .730 1.70 .170 4.563 .880 1.90 .190 5.500 1.150 2.30 .230 7.188 1.360 2.70 .270 8.500 1.610 3.10 .310 10.063 1.970 3.80 1.3 .380 12.313 .65
3 .501 0 .326 .433 2 5K .171 0 2.930 .501 .60 .060 2.930 .501 1.20 .120 2.930 .570 1.40 .140 3.333 .840 1.70 .170 4.912 1.160 2.10 .210 6.784 1.450 2.50 .250 8.479 1.710 3.00 .300 10.000 1.910 3.50 .350 11.169 2.080 4.10 1.28 .410 12.164 .64
~
4 .650 0 .403 .433 2 5K .191 0 3.403 .650 .40 .040 3.403 .650 .90 .090 3.403 .650 1.25 .125 3.403 .780 1.50 .150 4.084 1.060 1.80 .180 5.550 1.340 2.10 .210 7.016 1.700 2.50 .250 8. 901 2.100 3.00 .300 10.995
TABLE II (Continued)
1 2 3 4 5 6 7 8 9 10 11 12
2.440 3.50 .403 .433 2 SK .191 .350 12. 775 2.750 4.00 1.27 .400 14.398 .635
B RAPID FLOW ~-
1 -.64 0 .70 8.666 2 SK .661 0 -.968 -.49 .60 .060 -. 741 -.38 1.00 .100 -.875
.10 1.20 .120 .151
.80 1.50 .150 1.210 1.48 1.80 .180 2.239 2.05 2.00 .200 3.101 ~ 2.88 2.30 .230 4.357 3.41 2.60 .260 5.159 3.90 3.00 .300 5.900 4.28 3.40 .340 6.475 4.67 4.00 1.04 .400 7.065 .52
2 -.64 0 .745 8.666 2 SK .658 0 -.973 -.52 •. so .050 -.790 -.48 1.00 .100 -.730 -.os 1.10 .no -.122
.40 1.30 .130 .608 1.00 1.60 .160 1.520 1.70 1.90 .190 2.584 2.60 2.30 .23d 3.951 3.13 2.60 .260 4.757 3.70 2.90 .290 5.623 4.19 J.20 .320 6.363 4.58 3.50 .350 6.961 4.92 4.00 1.04 .400 7.477 .52
TABLE II (Continued)
1 2 3 4 - 5 6 7 8 9 10 11 12
3 -.64 0 .80 8.666 2 SK .689 0 -.929 -.56 .50 .050 -.813 -.40 p 1.05 .105 -.581 -.02 1.10 .12.0 -.029
.69 1.50 .150 1.001 1.40 1.80 .180 2.032 2.31 2.20 .220 3.353 3~07 2.60 .260 4.456 3.75 3.00 .300 5.443 4.28 3.50 1.06 .350 6.212 .53
4 -.sa 0 .86 8.666 2 5K .733 0 -.791 -.49 .so .500 -.668 -.31 1.05 .105 -.423 -.03 1.20 .120 -.041 .so 1.50 .150 .682
1.31 1.80 .180 1.787 2.01 2.10 .210 2.-742 2.82 2.50 .250 3.847 3.59 3.00 .300 4.898 4,.15 3.50 .350 5.662 4.46 4.00 1.07 .4oct- 6.085 .53 5
Velocity Distribution in Open Channel with Artificial Roughness Bar of Semi-Circular Cross-Section at Middle of Cycle
A TRANQVIL PLOW -- - -
V _ y q y i S K L V* Run Point Depth Disch. Height Bed Slope Rough-Length Shear y/L V/V Yi/K
Velocity Above per ft. of In- in neas of a Velocity lJ:./sec. Flume width flection percent HeightCycle Ft/sec.
Bed c.f.s. Point In. 1 In. In.
~ I -~681 03 I .~6 I 5 I .:33 I I 1 s: ,-~.r~rio-r-~~r rz--
= (.,
TABLE II (Continued)
1 2 3 4 5 6 7
-.68 .4 .16 .433 2 -.68 '~8 -.56 1.0 -.28 1.2
.18 1.5
.50 1.8
.77 2.2
.97 2.6 l.13 3.0 1.23 3.3 l.18 3.4 .85
2 .29 0 .214 .433 2 .29 .4 .29 .8 .36 l.O .48 1.2 .63 1.5 .79 l.8 .98 2.2
1.17 2.6 1.30 3.0 1.38 3.4 1.42 3.8 .85
3 .08 0 .327 .433 2 .12 .4 .16 .8
8 9 10
5K .135 .:040 .080 .100 .120 .150 .180 .220 .260 .300 .330 .340
SK .149 0 .04 .08 .10 .12 .15 .18 .22 .26 .JO .34 .38
SK .171 0 .04 .08
11
-5.037 -5.037 -4.148 -2.074
1.333 3.704 5.704 7.185 8.370 9. lll 8.741 1.946 1.946 1.946 2.416 3.221 4.228 5.302 6.577 7.852 8. 725 9.262 9.530
.468
.702 • 936
12
.425
.415
\0 U)
TABLE 11 (Cont:inued)
1 2 3 4 5 6 7 8 9 10 11 12
.43 1.0 .327 .433 2 SK .171 .10 2.515
.66 1.3 .13 3.860
.91 l. 7 .17 5.322 1.08 2.1 .21 6.316 l.22 2.5 .25 7.135 1.33 3.0 .30 7. 778 1.42 3.5 .35 8.304 1.48 4.0 .40 8.655 1.51 4.3 .43 8.830 1.49 4.5 .82 .45 8.714 .4 1
4 -.24 0 .411 .433 2 SK .185 0 -1.297 - .19 .4 .04 -1.029
...... 8
- .14 .8 .08 -.757 .20 1.0 .10 1.081 .56 1.3 .13 3.027 .79 1.6 .16 4.270
1.02 2.0 .20 5.514 1.25 2.5 .25 6.757 1.42 3.0 .30 7.676 1.56 3.5 .35 7.892 1.70 4.0 .40 9.189 1. 79 4.4 .44 9.676 1.86 4.7 .81 .47 10.546 .'4 OS
.Abstract
VELOCITY DISTRIBUTION IB STEEP ROUGH OHA!lNEL
by
Ohiang Tsung-fing
This thesis consists of-an experimental study of the
velocit1 distribution in tranquil, stable tumbling and rapid
flow regimes in a steep rectangular channel with artificial
roughness elements.
Four shapes of roughness elements, rectangular, parallel-
ogram, triangular, and semi-circular, were use4. Effects on
velocity distribution due to variations ill discharge, flWl.e
slope and roughness geometry were studied. for each shape of
roughness element. The applicability of logar1tha1c law was
examined and the inflection points 1n tranquil and rapid flew
regime were studied. Also the velocity coefficients 1n tu.mb-
11ng regime were studied.
The findings were oont1rme4 through the analysis of
data taken from project 405 of the 01v11 Engineering Depart-
ment.
A review of literature on this sub3eat and a biblio-
graphy are included.
