riprap at the entrance to horizontal culverts...
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Riprap at the entrance to horizontal culverts
Item Type text; Thesis-Reproduction (electronic)
Authors Patton, John Lawrence, 1946-
Publisher The University of Arizona.
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Link to Item http://hdl.handle.net/10150/554545
RIPRAP AT THE ENTRANCE TO HORIZONTAL CULVERTS
by
John Lawrence P atto n
A T hesis Subm itted to th e F acu lty o f th e
DEPARTMENT OF CIVIL ENGINEERING AND ENGINEERING MECHANICS
In P a r t i a l F u lfillm e n t o f th e Requirements For th e Degree o f
MASTER OF SCIENCE WITH A MAJOR IN CIVIL ENGINEERING
>In th e Graduate College
THE UNIVERSITY OF ARIZONA
1 9 7 3
STATEMENT BY AUTHOR
This th e s is has been subm itted in p a r t i a l fu lf i l lm e n t o f re» quirem ents fo r an advanced degree a t The U n iv e rs ity o f Arizona and i s deposited in th e U n iv e rs ity L ib ra ry to be made a v a ila b le to borrowers under ru le s o f th e Library® .
B rie f q u o ta tio n s from t h i s th e s is a re allow able w ithout s p e c ia lperm ission , provided th a t accu ra te acknowledgment o f source i s made® Requests f o r perm ission f o r extended q u o ta tio n from o r rep roduction o f t h i s m anuscript in whole o r in p a r t may be g ran ted by th e head o f th e major departm ent o r th e Dean o f th e Graduate College when in h is judgment, th e proposed use o f th e m a te r ia l i s in th e in te r e s t s o f sch o la rship®' In a l l o th e r in s ta n c e s , however, perm ission must be ob tained from th e author®
SIGNED; I w - . (^ Q llX erv v ^
APPROVAL BY THESIS DIRECTOR
This th e s is has been approved on th e d a te shown below;
Be M. LAURSEN P ro fe sso r o f C iv il Engineering
% Date "
ACKNOWLEDGMENT
The w r i te r w ishes to express h is s in c e re a p p re c ia tio n to Dr,
Emmett M, Laursen who suggested th e genera l to p ic o f t h i s s tudy and
ac ted as th e th e s is d ire c to rs The a s s is ta n c e given by Mr* Louis
Gemson, Technician o f th e C iv il E ngineering L aboratory , in c o n s tru c t
in g th e la b o ra to ry apparatus and by Mr* Alan M* Hawkins in help ing
c o lle c t much o f th e d a ta was invaluab le* To my w ife Caryn I give a
sp e c ia l thanks f o r h e r help and encouragement s
i i i
TABLE OF CONTENTS
Page
LIST OP ILLUSTRATIONS © © © © © © © © © © c o s e o © © © © v
LIST OP TABLES © o © © e © 6 © e o © © © c e ® o © o e © e V I X
ABSTRACT © o o e o c e e o o c o o o o o o c e i o c o G © © V X 3 .5 ,
I q INTRODUCTION o © o B © o o o © © © e c o © o s © o e e © o I
S cope e s o e e e o o G e e o s e o o e e o e e e e o e 1.
2© BACKGROUND INFORMATION © © © # © © 6 .© © © o © o © © © © © 3Energy R ela tio n sh ip in Open Channel Flow © * @ * 3F lu id Forces on th e Bed M ate ria l © e © ® ® ® 0 © © © © 4In c ip ie n t Motion o e © © © © © © © © © © © © © © © © ©
F lu id Forces on P a r t ic le on S loping Bed e © e e 0 © © 8R atio o f C r i t ic a l Shear on Slope to C r i t ic a lShear ox F la t Bed ©©©©. © © © © © © © c © © © © © © © 10
3© APPROXIMATE FLOW ANALYSIS * © o , , * * , . , o , * . . / . 13
Flow A nalysis © © © e © © © © © © © © ® © © © © © © © 13
4© EXPERIMENTAL APPARATUS * * . „ o * * * © .. o . © , * , * 20
Hydraul i c Model © © © © © © © © © © © © © © © © v©© <d0
5^ INTERPRETATION OF RESULTS 27S iz in g ox Riprap © © © © © © © © © © - © o © © © © © © 46S i l t in g o f th e B arre l © © © o © © © © ® © © © ^ © © © © ^2
6o CONCLUSIONS © © © © © 6 © o © e © © © © © © e © s o o © © 33
NOMENCLATURE © © © © © © © © © © © © © o © © © « © © © © 37
LIST OF REFERENCES © , * * © © 60
iv
LIST OF ILLUSTRATIONS
F igure Page
1. D e p re ss -in le t c u lv e r t e e e e « 6 e e 0 « o e e e » « » e 2
2 o Energy r e la t io n s h ip between two se c tio n sXn Open channel flow e o o e o e e e e o a o . . e o c e o o e 3
3. R e la tio n sh ip o f head lo s s to energy s lope (ex ag g e ra te d )e „ 5
4 0 Forces a c tin g on a f lu id mass e 6 o c o 8 o 6 o o « o e e 6
5© Forces a c tin g on a submerged p a r t i c le © e 0 © © e © © « © 9
6 © P lo t o f " ^ g / ^ versus d ep re sse d « in le t s lo p e , 0 8f o r d i f f e r e n t ang les o f repose © © © © © « © © © © © © © © 12
7® Experim ental appara tu s © © © © © © © © © , © © © © © © © © 21
8 © Experim ental apparatus-photograph © © © © © © © © « © © © 22
9= Plywood stream tube-photograph © © © © © © © © © © © « © © 22
10© V eloc ity m easuring apparatus-photographs © « © © © © © © © 24
H e 90 V«-notch wexr © © © o © © © © © © . © © © © © © © © © © © 23
12. Photographs o f r ip ra p . © , © © . , 26
13© Depth p r o f i le f o r Run 1 © © © © © © © . © © © © © © © © e 28
14® Depth p r o f i le f o r Run 2 © © » © © © © © © © © © © © © , , 29
15® Depth p r o f i le f o r Run 3 © © © ©.© © © © © © © © I © © © © 30
16© Depth p r o f i le f o r Run 4 © « © © © © © © © © © © © © © © © 31
17© Depth p r o f i le f o r Run 5 © © © © . © © © © © © © © © , © » 32
18© Depth p r o f i le f o r Run 6 « © . © © © © © © © . © © © © © © 33
v
v i
LIST OF ILLUSTRATIONS$ continued
F igure Page
19® V elocity p r o f i le s f o r Run 5 e o e e e e e e e e e . e . e ' 34
20e V elocity p r o f i le s f o r Run 6 e e 0 8 e 0 e e e » = „ e 0 37
21o V elocity p r o f i le o f submerged j e t = 0 = 0 0 0 0 = 0 0 = 41
22® V elo c ity p r o f i le 3" in s id e c u lv e r t 0 » . o » e e . o e . 43
23® Pool e le v a tio n versus f lo w ra te . e e o - e e e ® ® . ® . ® ® 48
24® H ead-discharge r e la t io n fo r a d e p re sse d - in le t c u lv e r t0 . 49
t
LIST OF TABLES
Table Page
5. e VHXXIOS Of 0^ © e c c o o c e . e o o e o o o e o o J, X.
2e Comparison o f sh ea r s tr e s s e s f o r Runs 9~12 e 0 e e 9 © 9 44
3 o Typxcal values o f C © © & © © © © © © © © * © © ©■ © - © © ©
4© Typical values o f © © © © © © © © © © 8 © © © © © © © 51
y
v i i
ABSTRACT
The h o r iz o n ta l c u lv e rt concept has been found to be hydrau-
l i c a l l y advantageous, In o rd er th a t th e concept be accepted# th e e ro
sio n o f th e depressed cone slope must be p reven ted . An approximate
a n a ly s is o f th e flow c h a r a c te r is t ic s in th e depressed cone was made to
p re d ic t th e s iz e r ip ra p req u ired to l in e th e depressed cone. To v e r i fy
th e a n a ly s is a h y d rau lic model was co n stru c ted s im u la tin g f i e ld condi
t io n s .
The r e s u l t s of th e h y d rau lic model t e s t in g v e r i f ie d th e approx
im ate a n a ly s is in p re d ic tin g th e flow c h a r a c te r is t ic s . The an a ly s is
was th en used to o b ta in a design equation f o r computing th e s iz e r ip ra p
req u ired on th e depressed cone.
v i i i
CHAPTER 1
INTRODUCTION
The h o r iz o n ta l c u lv e r t concept has "been found to be advanta
geous (2 ) when th e a llow able head-pool e le v a t io n .to a c u lv e r t i s lim
i te d , The slope and len g th o f th e conventional c u lv e r t must be such
th a t th e i n l e t to th e h o r iz o n ta l c u lv e r t can be depressed a t l e a s t one
h a lf th e c u lv e r t d iam eter. At low flow s as w ell as h igh flows th e
h o r iz o n ta l c u lv e rt w il l flow f u l l th ereb y using th e e n t i r e c u lv e rt
a rea and f a l l a v a i la b le . A h o r iz o n ta l c u lv e r t of th e same s iz e as a
conventional c u lv e r t then has th e c a p a b il i ty o f handling a r a r e r ,
la r g e r f lo o d o r f o r a given design frequency and magnitude th e h o r i
zo n ta l c u lv e r t allow s a red u c tio n o f th e c u lv e r t s iz e .
Several problems do a r i s e when th e h o riz o n ta l c u lv e r t i s used.
Due to the. high v e lo c i t ie s o f th e flow in th e cone shaped en trance
hole (F igure l ) , th e re i s a tendency f o r th e bed o f th e cone to scour
o r e ro d e , Another p o ss ib le problem i s th e s i l t i n g of th e b a r re l of th e
c u lv e rt w ith sediment which has been c a r r ie d by th e flow ,t
Scope
The a n a ly s is and experim ents p resen ted have been performed to
s u b s ta n tia te th e h o r iz o n ta l c u lv e r t concept and e i th e r e lim in a te th e
problems noted above o r prove th a t th e e f f e c ts a re n e g lig ib le .
2
Flow
Plan
Lip o f ConeEmbankment
-D epressed Cone SlopeFlow
///-''/// oVz/rV/ /^D *= .Z
^ ___________________A--' ///o~ / / /v' ' - //vV//<-'/// /// v"E leva tion
F igure 1 . D ep ressed -in le t c u lv e r t.
CHAPTER 2
BACKGROUND INFORMATION
When th e n a tu ra l path of flow in a channel i s in te r ru p te d , high
v e lo c i t ie s u su a lly occur causing scour o f th e bed m a te r ia l . The eng i
neering design of any s tru c tu re th a t changes th e flow may, th e re fo re ,
need to include some method o f p reven ting scou r. Methods of p reven ting
scour vary from g ra s s - lin e d to c o n c re te -lin e d channels. This study i s
an in v e s tig a tio n of a p a r t ic u la r case of r ip ra p l in in g which provides
much more p ro te c tio n than th e g ra s s - lin e d channel w hile being much le s s
expensive than th e co n c re te -lin ed channel, A r ip ra p - l in e d channel i s a
channel lin e d w ith a la y e r o r two of rocks on th e channel bottom an d /o r
s id es which a re la rg e enough to r e s i s t th e dynamic fo rc e s of the flow
v e lo c i t ie s , thereby p reven ting sco u r. In o rd er to make a p re d ic tio n o f
th e degree of p ro te c tio n req u ired to prevent scour, an a n a ly s is of th e
magnitude o f th e dynamic fo rces i s needed.
Energy R ela tio n sh ip 3.n Open Channel Flow
The usual energy equation fo r open channel flow , assuming u n i
form v e lo c ity d is t r ib u t io n , h y d ro s ta tic p ressu re d i s t r ib u t io n , and
flow alm ost h o r iz o n ta l i s
as shown in F igure 2 where and V0 a re th e mean v e lo c i t ie s a t sec
t io n s 1 and 2 re sp e c tiv e ly , and y0 a re th e depths o f w ater, and
z^ a re th e h e ig h ts o f th e bed above some datum, and h^ i s the head
lo ss o r energy lo s s from se c tio n 1 to se c tio n 2 .
The slope o f th e energy l in e a t s e c tio n 1, g ives an ap
proxim ation of th e head lo s s between s e c tio n s 1 and 2 when th e slope
i s m u ltip lie d by th e d is ta n c e between th e two s e c tio n s , ^ r , assuming
th a t th e d is ta n ce ^ r i s sm all (F igure 3)« The head lo s s could a lso be
approximated by using th e slope o f th e energy l in e a t s e c tio n 2 ,
tim es ^ r and in f a c t th e a c tu a l head lo s s hj, f a l l s between th e l im its
Se l O y ) < hf < Se2 (Ar ) . (2 )
Since i t i s e a s ie r to use an ev a lu a tio n o f th e head lo s s a t e i th e r se c
t io n 1 o r se c tio n 2 than some average v a lu e , and s in ce th e d iffe re n c e
between th e so lu tio n s i s sm all i f th e d is ta n c e between th e sec tio n s i s
sm all, th e computer program was w ritte n to analyze th e flow c h a ra c te r
i s t i c s f i r s t u sing th e downstream energy s lo p e , S ^ , and then in a
second a n a ly s is u s in g th e upstream energy s lo p e , . A b e t te r so lu tio n
should f a l l between th ese two l im i t s .
F lu id Forces on th e Bed M aterial /
When uniform flow e x is ts th e s lo p es of th e energy l in e , th e
w ater su rfa c e , and th e channel bottom a re equal (S = = S^) and
th e depth and v e lo c ity o f th e f lu id a re co n s ta n ts . There i s a r e s i s
tance to th e flow c a lle d boundary drag th a t i s described by th e shear
5
S e c t . - l Sect.O S e c t . l S e c t .2_THL
Datum
F igure 2 , Energy r e la t io n s h ip between two se c tio n s in open channel flow .
TIE,
W.S
e2
///x-'-'/A -V /A -V //
Sect.O S e c t . l S e c t .2F igure 3« R ela tio n sh ip o f head lo s s to
energy slopes (exaggera ted ).
s t r e s s T^, F igure 4 shows th e fo rces a c tin g on a f lu id mass where th e
component of th e weight of th e f lu id p a r a l le l to th e flow i s equal to
th e sh ea r fo rce which a c ts over th e e n t i r e w etted p e rim e te r. The sh ear
s t r e s s can "be shown to equal ( 3 » p. 91)
= YRS (3)
where y i s th e s p e c if ic weight o f th e f lu id and R = A/P which i s the
h y d rau lic rad iu s when A i s th e cross s e c tio n a l a rea and P i s th e w etted
p erim e te r.
W s in 6W cos 0F1
F igure 4 , Forces a c tin g on a f lu id mass.
Uniform flow fo r open channel flow i s g en e ra lly described by
th e f a m il ia r Manning equation
V = p2/3 s l /2 (4 )n , x z
where n i s the Manning c o e f f ic ie n t of roughness. In n a tu ra l r iv e rs th e
roughness c o e f f ic ie n t n depends on channel geometry, bed m a te r ia l,
v e g e ta tio n , and channel meandering. In a s t r a ig h t a r t i f i c i a l channel
th e roughness c o e f f ic ie n t n i s sim ply a fu n c tio n o f th e bed m ate ria l
s iz e . The Manning n can then he r e la te d to th e mean d iam eter, d , o f
th e bed m a te ria l by th e expression
n = C1 d1/ 6 ( 5)
which was f i r s t suggested by S t r ic k le r in 1923 ( in 3» Po98). Anderson,
P a in ta l , and Davenport ( l , p .6 ) have suggested th e value o f to be
0,0395 and Equation 5 th e re fo re becomes
n = 0.0395 d1/ 6 .
In c ip ie n t Motion
The c r i t i c a l t r a c t iv e fo rc e , i s th e average hydrodynamic
fo rce over space and tim e f o r which th e most exposed p a r t i c le ju s t
moves. The amount o f movement i s somewhat ambiguous, but one must
agree th a t th e re i s a th e o re t ic a l p o in t where th e re i s no movement and
i f th e flow i s in creased ju s t s l ig h t ly th e re i s movement. The sh ear
fo rce a t t h i s p o in t i s c a lle d th e c r i t i c a l t r a c t iv e fo rc e .
Many experim enters have t r i e d to r e l a t e th e c r i t i c a l shearing
s t r e s s to th e mean d iam eter o f th e p a r t i c le , y ie ld in g
% - c2 V ( 7 )
Various in v e s tig a to rs have suggested values o f C^, Anderson e t a l .
( l , p .9 ) c o rre la te d sev e ra l experim enters* work and suggested a value o f
= 5 f o r ro adside d i tc h e s . This type o f an equation assumes th a t f o r
la rg e p a r t ic le s th e drag i s independent o f th e lo c a l p a r t i c le Reynolds•
number, Anderson, a lso suggested a more conservative value o f = 4 ,
whereby Equation ? becomes
% - 4 a 50 ( 8 )
and i t i s Equation 8 th a t has been used in t h i s in v e s tig a tio n to r e la te
th e c r i t i c a l sh ear to th e p a r t ic le d iam eter.
When th e p a r t ic le d iam eter i s sm all, th e c r i t i c a l shear i s r e
la te d to th e p a r t ic le Reynolds* number as S h ie ld s showed in 1935 when
he p lo tte d fy ^ y d versus th e p a r t ic le Reynolds * number, R = u*d/v ,
where u = «/7j/P and v i s th e k inem atic v is c o s i ty . In 1968 Ward ( 5 )
showed q u ite a s c a t te r of h is and W hite’s d a ta ( 6 ) above th e S h ie lds
curve. Ward rev ised th e S h ie ld s curve to include a f a c to r f o r the
dynamic beginning o f movement which included a mass r a t io term . The
Ward p lo t o f 7^/^yd [ l + 1 ,25(ys /a y )] versus u*d/v shows a much b e t te r
convergence of t h i s d a ta .
The minimum value o f f o r th e S h ie ld s curve ( fo r medium
values of th e p a r t ic le Reynolds number) i s 3, 1 and fo r th e Ward curve
6,2c, The value of f o r high p a r t ic le Reynolds numbers i s 6 ,2 fo r
S h ie ld s and 9*3 f o r Ward's curve ($ )0
F lu id Forces on P a r t ic le on S loping Bed/
The fo rces a c tin g on a p a r t ic le on th e s lo p in g bed o f a stream
a re shown in F igure 5* The angle 0 equals th e slope o f th e bed, <|> i s
th e angle o f repose o f th e bed m a te r ia l, and ^y i s th e submerged
s p e c if ic weight o f th e p a r t ic le which equals yo - y. The volume o f
3 3th e p a r t ic le i s OKr and i t s weight under w ater i s t t ^ y i o The a c tu a l
apparen t sh ear s t r e s s i s d is t r ib u te d unevenly among th e in d iv id u a l
p a r t ic le s and th e fo rce on th e most exposed p a r t ic le due to th e flowP p
can be taken as 3d where 3d" i s th e e f fe c t iv e su rface a rea o f th e
p a r t ic le exposed to th e sh ea r s t r e s s , When th e sh ea r s t r e s s ,
equals th e c r i t i c a l sh ear s t r e s s , 7^, th e p a r t ic le i s a t th e p o in t o f
in c ip ie n t motion.
I t should be noted th a t th e l i f t due to th e hydrodynamic fo rces
on th e p a r t i c le has been neg lec ted in F igure 5* When th e l i f t fo rce
i s in c lu d ed , th e m anipulation o f th e equation f o r e v a lu a tin g th e
sh earin g s t r e s s becomes q u ite awkward* As a f i r s t approxim ation, th e
l i f t i s g e n e ra lly om itted ( ? , p . 292). This i s a llow ab le s in ce th e
co n stan ts o f th e equation must be determ ined experim en ta lly and s in ce
th e l i f t depends on th e same v a r ia b le s as th e drag*
Plane o f bed
/O
P o in t o f support
F igure 5* Forces a c tin g on a submerged p a r t i c l e .
10
Ratio of Critical Shear on Slope to Critical Shear on Flat Bed
The sum o f the moments of th e two fo rces shown in F igure 5
about th e p o in t of support must equal ze ro , y ie ld in g (? )
- CCAYd sin(<j> - 0 ) P T gd2 cos(<t>)J?2 = 0 (9 )
where 7^^ i s th e c r i t i c a l sh ear fo rce on th e s lo p e . Solv ing fo r
g ives
V/hen 6 = 0, Equation 10 becomes th e c r i t i c a l shear on th e f l a t bed, o r
T c b = f (AY)d ta n * ( 11)
which i s very s im ila r to th e White equation (6 , p , 324) when equals
The r a t io o f th e c r i t i c a l shear on th e slope to th a t on th e bed
can then be shown to be
The so lu tio n o f Equation 12 i s given in Table 1 and F igu re 6 g iv ing
ty p ic a l values o f C_ f o r d i f f e r e n t ang les o f <p and 9 , When T = 4dJ cb
Equation 12 can be re w rit te n to be
r cs - c3 ” c3 ( ^ ) = =4 d . (13)
11
Table 1„ Typical va lues o f C y
25° 30° 35° 40°
0 1,000 1*000 1*000 1,000
5° 0*809 0*845 .0,872 0,892
.7 .1 3 ° 0,726 0*777 0,815 0,844
9«46° 0*634 0,702 0,752 0,790
I 4 e04° 0,450 0,550 0,624 0,681
1 8 ,# ° 0,271 0,401 0,497 0,572
20° 0,206 0*347 0,451 0,532.
25° 0,000 0*174 0,303 0,403
30° - 0,000 0,152 0,270
35° 00 0,000 0,136
U0° 0,000
12
o
o
6 , degrees
F igure 6 . P lo t o f ^ s /^ v 0 versus d e p re sse d -in le ts lo p e , 9 , f o r d i f f e r e n t ang les o f rep o se0
CHAPTER 3
APPROXIMATE FLOW ANALYSIS
The general approach used in t h i s s tudy was to f i r s t w rite an
approxim ate a n a ly s is u s in g th e usual open channel flow assum ptions
which was programmed to be solved by th e computer. The flow charac
t e r i s t i c s ob tained were th e dep th , v e lo c i ty , slope o f energy l i n e , and
th e boundary sh ea r along th e e n t i r e p r o f i l e . This a n a ly s is was then
v e r i f ie d by se le c te d measurements on th e h y d rau lic model.
Flow A nalysis
At th e l i p o f th e cone (F igure l ) th e re i s a t r a n s i t io n from a
mild slope to a s tee p s lo p e . At t h i s p o in t c r i t i c a l flow w ill occur
(3 , p . 10?) and
where yQ and y^ a re th e depths a t th e l i p , Vq and a re th e v e lo c i t ie s
a t th e l i p , and q ~ Q/Knro where r Q i s th e rad iu s o f th e cone, and K i s
a constan t le s s than 2 which v a r ie s w ith the geometry o f th e depressed-
i n l e t slope and embankment s lo p e . From co n tin u ity
9 = V o = Vl yl • ( 15)
13
14
S olving f o r in Equation 15 gives
r o Vo\ „ r0 / qv - a < * «
and
2 f To\/ qvi = r ) i - i • ( 17)
When Equation 1? i s s u b s ti tu te d in to Equation 1 th e genera l energy
equation becomes
vi 2 (k)( 4 )li" + + Ss Ar = ---- z E--- — + y2 + Se2 A r (l8)
where 2 - ~ ^ r , and th e slope o f th e energy l in e a t s e c tio n 2
i s sued to p re d ic t th e head lo s s a t s e c tio n 2 . S ince r^ = r Q - ^ r
and r 2 = r o - 2 ( ^ r ) , then r^ = r^ - a ( /\r ) o r a = (r^ - r a ) /^ r where
a equals th e number o f s te p s o f le n g th / \ r taken from th e l i p o f th e
cone. I f r a = 0, then a = r o//^r.
V/hen v a ry in g from a = 0 to a = r^/% r, Equation 18 becomes
( 9 W W , (9 r ) ( —+ ya + =
+ ya+l + Se(a+1 ) A’* • (19)
15
The term can be ob ta ined from th e Manning eq u atio n , assuming
uniform flow over th e d is ta n c e ^ r , which i s
ITV l (Ra+l) 2/3 (Se(a+1 )^ /2 * (20>
Where A = K t r r ^ ya+^ , Q = KnTo (q ) , and (v a lid s in ce th e
depth i s sm all compared to th e len g th o f th e l i p 2rfr^) Equation 20
then becomes
q2 n2 / r Q \ 2 / 1 ^
e(a+l) (l.^ )2 V 0-(a+1) Ar y Va+J * (21)
N on-dim ensionalizing Equation 19 by d iv id in g by y^ and s u b s t i tu t in g
Equation 21 fo r gives
2 ' ^ \ 2 q2 , r , 2
ya 'yc V o -(a ) A r/ + f a , ^ M fo ya+ l"yc \ r o -(a+1)2g yc " yc ” 2g
ya+l I 2 1,2 / r 0 \ 2 Ar
^ < K >
A fu r th e r red u c tio n o f Equation 22 i s p o ss ib le by s u b s t i tu t in g
q2/g y c ” yc2 and n *= (0 e04) d1^ y ie ld in g
16
V / r o \ 2y j \ V (a) . ya . _' ' T 1 T D
Ar \ r . a+t ro-(a+l)s ' r o / yc
1 /3
a+1g(O.CA-)^ —
(1 .4 9 )W 3 r„-(a+l) Ar/ 7,
(23)
and f u r th e r red u c tio n g ives
2 (0.0232)
/
17
By a s im ila r an a ly s is^ ag a in using th e slope o f th e energy l in e
a t th e downstream s e c tio n , th e equation o f th e flow c h a ra c te r is t ic s
upstream of th e l i p o f th e cone becomes
, l /3
w , , , , , , M - f IVI y„ / / \ 0 £
a+i
\1 + (a+ l)
£ r+ -S±l + s. fAr \ ?
ro / yc(25)
ro /
The a n a ly s is was then rep ea ted using th e slope o f th e energy
l in e a t th e upstream se c tio n g iv in g th e equation
/y,
0.0232a . „ K \ rQ
^ V /3
ll - (a) ^ x 0 /
yc J /
+ ^ + S s t e i i - r ; : w 3 "
Ar
1 -
a+l
2 I -Istl) V - (a+l) —( 26)
fo r th e se c tio n s downstream of th e l i p .
18
S im ila r ly , f o r th e se c tio n s upstream o f th e l i p th e equation
becomes
l+ (a + i)
l+ (a+ l)
Equations 24, 25* 26, and 2? were then programmed f o r th e
computer and f o r a given f lo w ra te , Q, ra d iu s o f cone, r ^ , slope o f
bed, Sb , s lope o f cone, S^, and d iam eter o f r ip ra p , d , th e program
computed th e dep th , v e lo c i ty , s lope o f th e energy l i n e , and th e bound
a ry sh ea r a t s e le c te d p o in ts along th e channel. The maximum boundary
sh ear on th e cone was then compared to th e c r i t i c a l t r a c t iv e fo rc e ,
^ = c3 (to).The v a l id i ty o f Equations 24, 25» 26, and 27 must be proven.
Before a model can be co n stru c ted , th e dynamic s im i la r i ty between th e
model and p ro to type must be made. S ince th e flow over th e l i p o f th e
cone i s governed by g ra v ity , th e Froude number o f th e model and th e
p ro to type must be eq u a l. A lso, th e r e la t iv e roughness and th e r a t io
o f th e p a r t ic le sh ear to th e c r i t i c a l t r a c t iv e fo rce o f th e model and
19
th e p ro to type must he equal in o rd e r th a t th e model p re d ic t th e
a c tu a l flow c h a r a c te r is t ic s o f th e p ro to ty p e .
From geometry (y / r ) must equal (y / r ) , where m and pc# o in c o ps ig n ify model and p ro to ty p e . From th e c r i t i c a l flow equation3 2 /yc = q /g and i t can he shown th a t
q2 3 “ j p ' (28)
6 r o
which i s a type o f a Froude number. By Froude number modeling
3
(29)
A s im ila r a n a ly s is between y^ and th e mean d iam eter o f the
r ip ra p m a te r ia l , d , can be made r e la t in g th e r a t io o f yc to d o f th e
model and th e p ro to type l in e a r ly . Then s in ce y^ and r Q, and y^ and d
a re each l in e a r r e la t io n s h ip s , d and r Q must a lso have a l in e a r
r e la t io n s h ip between th e model and th e p ro to ty p e . The r a t io o f
d / r Q i s th e r e la t iv e roughness and th e re fo re ( d / r Q)m ** ( d / r Q)^ ,
In o rd e r th a t th e model p re d ic t th e p o in t o f in c ip ie n t motion
on th e p ro to ty p e , th e r a t io o f T ' / 7 o f th e model and th e p ro to type
must be eq u a l. Since th e c r i t i c a l t r a c t iv e fo rce term has a submerged
s p e c if ic weight o f th e p a r t ic le term a s so c ia te d w ith i t , modeling w ith
various types o f bed m a te ria l i s p o s s ib le . I f a s t r i c k ly geom etric
model i s co n s tru c ted , th e Froude number, th e r e la t iv e roughness, and
th e p a r t ic le sh ear to c r i t i c a l sh ea r r a t io o f th e model and th e
p ro to type w ill each be eq u a l0
CHAPTER 4
EXPERIMENTAL APPARATUS
In th e H ydraulic Laboratory o f th e C iv il Engineering Department
a t The U n iv e rs ity o f Arizona th e h y d rau lic model was co n stru c ted to
v e r ify th e approxim ate flow a n a ly s is th a t was derived in Chapter 3®
The r e s u l t s from each run made in th e la b o ra to ry were th en compared to
th e approximate a n a ly s is e
H ydraulic Model
The d e ta i l s o f th e h y d rau lic model a re shown in F igu res 78 8 ,
and 9. The 16* x 8 * x 4* b as in was used as th e head-w ater pool a t th e
en trance to th e c u lv e r t . The i n i t i a l e ig h t runs were made on a con
c re te su rfa c e . The purpose o f th e se runs was to v e r i fy th e depths and
v e lo c i t ie s of th e a n a ly s is . The l a s t fo u r runs were made w ith a r ip r a p -
l in e d channel and were used to compare th e sh ear s t r e s s e s o f th e analy
s i s and h y d rau lic model.
A ty p ic a l stream tube 1 .7 f e e t wide a t th e l i p was construc ted
o f plywood on th e concrete su rface so t h a t th e depths and v e lo c i t ie s
would be o f s u f f ic ie n t magnitude th a t measurements could be made accu
r a te ly without in te r fe re n c e from backw ater from th e c u lv e r t , Since th e
w id th /dep th r a t io a t th e l i p was about 16 to 1 , th e f r i c t i o n a l e f fe c ts
of th e plywood should have had minimal e f f e c ts on th e r e s u l t s , except
20
ToUnderground
R eservoir
8" diam eter G .I, Supply Line
Culvert— Plywood StreamtubeHead Water
R eservo ir .Constant
HeadR eservoir
Sand D etentior BasinControl
Valve
IF Return Channel
90 V-notch Weir Plan
8" d iam eter G .I. Supply Line
Constant Head R eservoir —-
Depressed Cone —, Culvert
PointGuage Sand D etention
Basin Pump90 V-notch
Weir — 7//A* ///<*///oV// //A -V /A V /A -'///
Return Channel
E levation Underground----------------------------------- R eservoir
Figure 7, Experimental apparatus,
22
Figure 8 . Experimental apparatus-photograph
Figure 9# Plywood streamtube-photograph.
23
p o ssib ly near th e v e rtex o f th e cone. Depth measurements in th e
stream tube were made w ith a p o in t gauge and th e v e lo c ity measurements
w ith a P i to t tu b e .
The P i to t tube shown in F igure 10 was used to measure th e ve
lo c i t i e s and was a ttach ed to th e g la ss s tandp ipes as shown in F igure
10, Poin t gauges were mounted above each standpipe and th e e le v a tio n
o f th e w ater su rfaces recorded a t each p o in t . In one standpipe was
th e s t a t i c w ater le v e l and in th e o th e r , th e t o t a l energy le v e l . The
d iffe re n c e between th ese two w ater le v e ls i s th e v e lo c ity head, h, and
th e v e lo c ity a t th a t p o in t can be c a lc u la te d from th e equation
h = — o r V = V^gh. (30)2g
The sand d e ten tio n basin was constru c ted f o r th e so le purpose
o f keeping sand and rocks from th e r e c i r c u la t in g w ater supply used in
th e H ydraulic L aboratory ,
A 90° V-notch w eir (F igure l l ) was p laced in th e re tu rn channel
and i t was used to c a lc u la te th e flow r a te through th e c u lv e r t0 A
p o in t gauge was used to measure th e head-w ater pool, H^, o f th e re tu rn
channel. The d ischarge was c a lc u la te d by th e fo llow ing re la t io n s h ip
Q = 2.48 H12,48 ( 31)
where Q i s th e d ischarge in cubic f e e t p e r second.
In o rd e r to make th e runs w ith th e r ip ra p , th e concrete was
removed from th e en trance cone. Sand was placed in th e en trance simu
la t in g th e normal bed m a te ria l and th e r ip ra p was placed over th e sand.
I
(a ) P i to t tu b e . (b) Glass s tandp ipes.
Figure 10, V elocity measuring apparatus-photographs,
25
Two d if f e r e n t s iz e s o f r ip ra p were used, d = O.O345 f e e t and d = 0.055
fee t* F igure 12 shows photographs o f th e a c tu a l r ip ra p used . The
experim ents were performed on th re e d i f f e r e n t cone s lo p es t 113, 1 *4 ,
and 1 :6 .
HE
12 " - 5/ 16"
Plan
-L.i JL11
S ection A-A
Tie P la te -
E leva tion
F igure l l . 90° V-notch w eir.
26
K % #
(a ) d = 0.0345 f e e t .
w *,K
¥ t W ;
(b) d = 0.055 f e e t .
Figure 12. Photographs o f riprap.
CHAPTER 5
INTERPRETATION OF RESULTS
The f i r s t v e r i f ic a t io n o f th e approximate a n a ly s is was th e
measurements made o f th e flow c h a r a c te r is t ic s on th e concrete surface®
The concrete su rface was somewhat im precise and had th e te x tu re approx
im ating a sand surface® F or values o f th e Manning n rang ing from
0.® 012? to 0,0159 (which i s th e same as in c re a s in g th e mean d iam eter of
th e "bed m a te ria l fo u r tim es from 0e001 to 0,004) th e approxim ate ana ly
s i s p r o f i le i s changed very l i t t l e , A mean value of n = 0,014 was
th e re fo re used in th i s a n a ly s is .
F igures 13 through 18 show p lo ts o f th e measured depth d iv ided
by th e c a lc u la te d depth a t th e l i p , y^, versus th e ra d iu s to th e p o in t
in q u es tio n , r , d iv ided by th e ra d iu s o f th e cone, r Q, f o r runs 1
through 6 , These s ix p lo ts show th a t th e a n a ly s is p re d ic tin g th e depth
in th e en trance cone i s a very good approxim ation based on th e measured
d a ta .
Runs 5 and 6 were a lso used to measure th e v e lo c i t ie s a t sev -/
e ra l se c tio n s in th e en trance cone. F igu res 19 and 20 show p lo ts of
th e v e lo c ity p r o f i l e s 6 The dashed v e r t i c a l l in e s a re th e values o f th e
average v e lo c ity as computed by th e a n a ly s is a t each s e c t io n . G enerally ,
th e measured values of v e lo c ity as th e s e c tio n g e ts c lo se r to th e v e rtex
of th e cone were h ig h er than th e average values ob ta ined from th e
2?
y/y computed w ith headless evaluated c from downstream energy s lo p e ,
y /y computed w ith headloss evaluated from upstream energy s lope .
Measured y / computed y
2 .0
y
c
0 .5
0 ,01.25 0.25
r
Figure 13. Depth p r o file fo r Run 1,roco
y /y computed w ith headloss evaluated 0 from downstream energy s lo p e ,
y /y computed w ith headloss evaluated c from upstream energy s lo p e .
Measured y / computed y
Q = 0, 287cfs
2 .0
y
y,c
0 .5
0 ,00.751.00
r1.25 0.25
roFigure 14, Depth p r o file fo r Run 2. ro
xO
--------------- y /y computed w ith headless evaluatedc from downstream energy slo p e ,
--------------- y/y computed w ith headloss evaluated0 from upstream energy slo p e .
Q = 0,35^cfs
C) Measured r / computed y^
//f/
C)- .0
O
_____ ______ (
yy
y
/ /y
O
° y
-
1 .5 0 1 .25 l .o o 0 .75 0.50 0 .25r
Figure 15. Depth p ro file fo r Run 3.
2 .0
1 .5
1.0
y /y computed w ith headless evaluated from downstream energy s lo p e ,
------------- y /y computed w ith headloss evaluatedfrom upstream energy s lo p e .
Q = 0,104cfs
O Measured >
o -......................................... -
r / computed y^
n
()
O
() //
j
//
/ /
°
/X
O
o ■
0 .5
0 .01.50 1 .25 1.00 0 .75 0.50 0.25
Figure 16, Depth p ro file for Run 4 , V)
2 .0
1 .5
i .o
--------- y/y computed w ith headloss evaluatedfrom downstream energy s lo p e .
-------- y /y computed w ith headloss evaluatedfrom upstream energy s lo p e .
Q = 0 , 26Oofs
O Measured ]{ / computed y^
if
C)
- O ——
o/
/ //
/b
c)9 ^
-
0.5
0.01.50 1.25 1.00 0.75 0.50 0.25
Figure 17. Depth p r o file fo r Run 5. %
y /y computed w ith headless evaluated C from downstream energy s lo p e .
-------------- y /y computed w ith headloss evaluatedc from upstream energy s lo p e .
Q = 0.329cfs
C) Measured jr / computed y^
faf
/
() ~~~~o -
c)
(
/y
Z
\ o o
° z
-
1 .50 1.25 l .o o 0 .7 5 0 .50 0 .25r
VFigure 18, Depth p r o file fo r Run 6,
Water Surface
Average v e lo c ity between th e two computed approximate a n a ly s is v e lo c itie :o
V elocity measured w ith P i to t tube
-pON
o oPeak = 0,80fpsXi
%A
O0.5 2 .0
V elo c ity , fp s .
(a ) S ection a t r = 4 ,8 f t .
oo
•Peak = l,0 5 fp s
o o
0 .5 2 .0
V elo c ity , fp s .
(b) Section a t r = 4 ,3 f t .
Figure 19. V elocity p r o file s fo r Run 5 .
Dep
th,
ft.
Dep
th,
ft.
Dep
th0,
05
M „
o 0.
05
o.l
0
1,15 fpsO Peak = 1,19 fps
0.5 l . o 1 .5V elo c ity , fp s ,
(c ) S ec tion a t r = 3 .8 f t .
2 . 0 2 .5
Jll ,? 0 fp sPeak = 1,86 fps
2 .51.0 1.5 V elo c ity , fp s ,
(d) S ec tion a t r = 3 .3 f t . ( l i p of cone).
4 .2 5 fp s— h O ^ P eak = 4 ,66fps O
T 4V elo c ity , fp s ,
(e ) S ection a t r = 2 .3 f t .Figure 19. Run 5, continued.
36
Peak =4,76 fps 5*49 fps
2 3V elo city , fp s .
( f ) S ec tion a t r = 1 ,8 f t .
oo
Peak = 6 ,0 8 fp s
<Dn
o
V elo c ity , fp s .
(g) S ection a t r = 1 ,3 f t .
F igure 19 , Run 5, continued.
37
Water Surface
Average v e lo c ity between th e two computed approxim ate a n a ly s is v e lo c i t ie s
V elocity measured with P i to t tubeo
Peako
OCV5 1
V elo city , fps (a ) S ection a t r = 4 ,8 f t .
O'
o ..o
Peak = 1 ,1 4 fps
o - t —-------- ------------ 1---------------- 1—o 0 , 5 l , o i , 5 2 . 0
V elocity , fp s ,(b) S ection a t r = 4 ,3 f t .
Figure 20, V elocity p r o file s fo r Run 6 ,
Depth, ft.
Depth, ft.
Depth, ft
0.05
0.1
0
0.05
0.1
0
0.05
38
-Peak = 1 . ps1 e24fps—
§
0.5 l . o 1.5Velocity, fps,
(c) Section at r = 3.8 ft.
2 .0
i v = 1.84 fps
O Peak = 2,11 fps
2 .51.0 1.5 Velocity, fp s ,
(d) S ection a t r = 3 .3 f t . ( l i p o f cone).
-pPeak = 3.58 fpso
o
Oo0 1 43 62 5
Velocity, fps.(e) Section at r = 2.8 ft.Figure 20. Run 6 , continued.
+>o
■Peak = 4 ,67 fpso
o
o
V elo c ity , fp s ,f t .( f ) S ec tion a t r = 2 ,3 f tS ec tion a t 32r #
o
Peak = 5.59 fp s
o
V elo c ity , fp s , (g) S ec tion a t r = 1 ,8 f t .
5.40 fps
ch Peak = 6 ,20 fp s o
3 4V elo c ity , fp s ,
(h) S ec tion a t r = 1 ,3 f t .
Figure 20, Run 6, continued•
40
a n a ly s is . S ince th e v e lo c i t ie s were measured in th e c e n te r o f th e
stream tuhe, and s in ce th e stream tuhe narrows toward th e v e r te x , th e
v e lo c ity a t th e c e n te r should "be h ig h er due to th e f r i c t i o n a t th e s id e s
o f the.plyw ood stream tuhe8 The average v e lo c ity , though, would he very
c lo se to th e a n a ly s is e
Run 7 was performed to determ ine whether th e re was a submerged
j e t o r h y d rau lic jump on th e cone when th e flow pooled behind th e cu l
v e r t e n tran ce , F igure 21 shows a p lo t o f th e v e lo c i ty p r o f i le a t two
se c tio n s below th e submergence. I t can be re a d ily seen th a t th e re i s .
a d e f in i te submerged j e t and th a t th e magnitude o f th e v e lo c ity o f th e
j e t i s s u b s ta n t ia l ly le s s th an what th e v e lo c ity would have been had
th e re been no submergence w ith th e same flow r a te , (This would be th e
case when th e c u lv e r t d iam eter i s in c rea sed , a l l o th e r th in g s rem aining
th e same, th e reb y conveying th e same flow w ithout a head w ater p o o l,)
The sm a lle r v e lo c i t ie s w ith submergence in d ic a te th a t th e boundary
sh ear would be considerab ly le s s a lso when th e re i s submergence,
A q u estion a r i s e s here as to whether sediment w il l d ep o sit in
th e head-pool and block th e c u lv e r t en tran ce . Run 8 was performed to
see i f t h i s were in fact, th e case . When th e c u lv e r t was flow ing , sand
was thrown in a t se v e ra l lo c a tio n s to see where i t d e p o s ite d . There
was a submergence near th e c r e s t , Upstream o f th e l i p very l i t t l e sand
moved. Some sand th a t was p laced in th e submergence (on th e cone)
s e t t l e d , but a l l sand placed a t th e en trance o f th e c u lv e r t was c a rr ie d
through th e c u lv e r t , The v e lo c ity a t a p o in t approxim ately th re e
Dep
th,
ftNotes
Plywood stream tube was used Q = 0,645 c fs .Jump lo ca ted a t r = 2 .8 f t .CM
o
V elocity p ro f i le a t r = 2 .3 f t . -----
T - i
o V elocity p ro f i le a t r = 1,8 f t . -----
o0 1 2
V elocity , fp s .
F igure 21, V elocity p ro f i le o f submerged j e t .
42
inches in s id e th e c u lv e r t was measured and th e v e lo c ity p r o f i le o f t h i s
se c tio n i s shown in F igure 22,
The v e lo c ity in th e p ipe i s V = Q/C^A where 0^ i s a c o e f f ic ie n t
o f c o n tra c tio n and Q = 0,509$i Gc " 0,7$ and A - Tf(6e25/48)^ g ives V =
3 ,4 f p s e The v e lo c ity o f 1 ,05 fp s measured seems to be very r e a l i s t i c
s in ce th e flow in th e p ipe i s not f u l ly developed a t t h i s s e c tio n .
A lso, t h i s v e lo c ity ju s t th re e inches in s id e th e c u lv e r t i s g re a te r th an
th e v e lo c i t ie s 1 ,5 f e e t upstream o f th e l i p .
The o v e ra ll r e s u l t s o f th e e ig h t runs d iscussed in d ic a te th a t
th e a n a ly s is p re d ic ts very w ell th e a c tu a l flow c h a r a c te r is t ic s measured
in th e h y d ra u lic model.
The next s te p was th e comparison o f th e p o in t o f in c ip ie n t mo
t io n o f th e most exposed p a r t ic le to th e sh ear s t r e s s a t th e po in t of
movement as c a lcu la ted from th e a n a ly s is . This was done on th e r ip r a p -
l in e d en tran ce . For each run th e flow was increased u n t i l th e re was a
general movement o f th e r ip ra p m a te r ia l . When th i s happened th e flow -
r a te was measured and th e sh ear ob ta ined from th e a n a ly s is was compared
to th e th e o r e t ic a l and em pirica l values o f as shown in 'T a b le 2 ,
For comparison, Table 2 shows values o f 7^^(Anderson)$ /^ (S h ie ld s -
High R ) and 7% (Ward - High R ) , 7% " 5 i was used 'here in8 CS 8 CS
Anderson6s equation s in ce 5d a c tu a l ly d e sc rib e s th e p o in t o f in c ip ie n t
motion b e t t e r than th e conservative value o f 4d which i s used in th e
design eq u a tio n s.
The la r g e r values o f 7^g f o r th e S h ie ld s curve and th e Ward
curve a re a t t r ib u te d to th e f a c t th a t th e p a r t ic le Reynolds6 number in
O Measured V e lo c itie s0.6 -
0 .2 -Depressed Gone
/// ///^Y/%/7// / / / s / // <■'///<?///<?■///0.5V elocity , fp s .
F igure 22, V elocity p ro f i le 3" in s id e c u lv e r t .
V)
Table 2» Comparison o f sh e a r s tr e s s e s f o r Runs 9=12%
Run number 9 10 11 12
Radius o f coneg r Q» f t* 4 .9 3 e0 4=0 2=5
Depressed cone s lo p e , 1:6 1:4 1 :6 1:3
Riprap s iz e , d , f t . 0.055 0=055 0=0345 0*055
Angle o f repose , <f> 35° 35° 33° 35°
D ischarge, Q, e f s e 0.460 0el?9 0*086 0*169
L ocation o f movement, r , f t . - 1=8 1 @6 2=0
Tq a t p o in t o f movement 0 S146 0*240 0=137 0=215
Tcs (Anderson) = C^($d) 0 . 20? 0=1?1 0=127 0=137
t CB (S h ie ld s ) » 0^(6 .2d) 0.255 0.211 0.157 0=169
r cs (Ward) = C3 (9 .3d ) 0.384 0=318 • 0=236 0=255
7 ^ 0
C3( l l 4 ) y ^ " d 2/Z= 1=373 1=049. 1 = 523
(Anderson)/cs
610 .1=404 1 = 079 1 = 569
(S h ie ld s)/cs
1=137 0=873 1.272
TI f T (Ward)
. /cs0=755 0=581 0=843
each o f runs 9 through 12 was very high and a c tu a l ly g r e a te r than th e
maximum p a r t ic le Reynolds® number shown in e i th e r p lo t ( 5 ) , . A lso, th e
S h ie lds and Ward ana lyses have la rg e depth o f flow to p a r t i c le s iz e
r a t i o s , In th e la b o ra to ry runs d iscu ssed in t h i s in v e s t ig a t io n , the
flow o fte n was a c tu a l ly flow ing around th e p a r t ic le s and th e r a t io o f
th e depth o f flow to p a r t ic le s iz e was sm all compared to S h ie ld s8 and
Ward8s a n a ly se s . This then in d ic a te s th a t th e p a r t ic le s w il l a c tu a l ly
move sooner than S h ie ld s and Ward p re d ic t .
In run 9 th e re was no. genera l movement in d ic a tin g th a t th e r i p
rap was s ta b le . The boundary sh ear as c a lc u la te d from th e a n a ly s is f o r
th e design d ischarge was le s s th an th e c r i t i c a l shears o f Anderson e t a l e
S h ie ld s , and Ward,
Runs 10 and 11 both had gen era l movements and th e boundary
shears as c a lc u la te d in d ic a te d th a t th e re should have been movement.
In run 11 th e re was a d e f in i te channeling o f th e flow over a low spo t
on th e l i p o f th e cone. I t was estim ated th a t e ig h ty p ercen t o f th e
t o t a l flow o f Q - 0,086 cfs o r Q = 0,069 was going over a fo u r fo o t
len g th on th e c r e s t . The f o r t h i s case was increased , from 0,13? p s f
to 0,180 p s f . The l a t t e r value was in f a c t a more accu ra te estim ate o f
th e boundary sh ear on th e p a r t ic le s which f i r s t moved, /
In run 12 a t a very low flo w ra te (0 ,065 c fs ) th e re was a major
s lip p ag e o f th e e n t i r e cone. The sand bed m a te ria l was probably a t o r
near i t s submerged angle o f repose and i t s e t t l e d to a more n a tu ra l
s lo p e . The slope o f th e cone when th e re was a general movement o f th e
r ip ra p m a te r ia l was a c tu a l ly something le s s than one to th re e . This
im plies th a t even though th e r ip ra p i s designed p ro p erly and i s s ta b le ,
th e re s t i l l can he a f a i lu r e i f th e bed m ate ria l i s u n s ta b le .
The r a t io o f th e p a r t ic le sh ea r to th e c r i t i c a l t r a c t iv e fo rce
was f i r s t in troduced by Laursen (4 , p . l 68)0 The r a t io tak es th e form
where ^ = ^d. For th e c r i t i c a l t r a c t iv e fo rce on th e s lo p e , Equation
32 becomes
Equation 33 was evaluated usin g th e v e lo c i ty , dep th , and p a r t ic le s iz e
f o r th e p o in t o f in c ip ie n t motion f o r each ru n . This value was com-
th e th re e and approxim ates th e value o f Equation 33 th e n e a re s t.
The h y d rau lic model v e r i f ic a t io n o f th e a n a ly s is new allow s an
adequate d e sc r ip tio n o f any flow case , A s im p lif ic a tio n i s needed fo r
design p r a c t ic e , however.
( 32 )
#
(33)
pared in Table 2 to th e value o f 7 ^ / when 7^ comes from th e approxi
mate a n a ly s is and 77 i s evaluated from Anderson, S h ie ld s , and Ward
re sp e c tiv e ly . The Anderson value o f 7 ^ / i s th e more co n se rv a tiv e .o f
S iz in g o f R irran
4?
The f i r s t design c r i t e r i a needed i s th e design flovrrate fo r
th e r ip ra p s iz in g . This flovrrate i s le s s than th e design flow rate fo r
th e c u lv e r t s iz in g . I f th e head-w ater pool on th e c u lv e r t drowns out
th e l i p o f th e cone, th e v e lo c ity a t a s e c tio n on th e depressed slope
and th e re fo re th e "boundary sh ea r a t th a t s e c tio n w il l a c tu a l ly be le s s
than th e v e lo c ity and- boundary sh ear a t th a t same se c tio n fo r l e s s e r
flow s. I t i s suggested th a t th e design flow fo r th e s iz in g o f th e
r ip ra p should be th e d ischarge where th e submergence i s ju s t a t th e
l i p o f th e cone. F igure 23 shows a p lo t o f th e h e ig h t o f w ater above
th e f l a t bed versus d ischarge and th e pool heigh t on th e c u lv e rt versus
d isch a rg e . The p lo t shows ( fo r th e c u lv e r t used in th e la b o ra to ry ) th e
design flo w ra te as 0,450 c f s . The f lo w ra te i s dependent upon th e c u l
v e r t d iam eter, le n g th , and c o e f f ic ie n t o f f r i c t io n as w ell as th e en
tra n c e and e x i t c o e f f ic ie n ts . When any o f th ese a re changed a new
design flo w ra te must be determ ined, A ty p ic a l graph r e la t in g th e
a v a ila b le head, H^, to a c u lv e r t d iam ete r, D, versus d isch a rg e , Q, i s
shown in F igure 24, The design flo w ra te i s where H^/D = .
The s iz e r ip ra p req u ired must be r e la te d to some param eter to
enable i t to be fu n c tio n a lly u su ab le . To do th i s th e r a t io
J ^ . = Or was determ ined where , i s th e maximum shear on th e' max c re s t b maxslope o f th e cone and th e sh ea r a t the l i p o r c re s t o f the
cone. Since
^ a . / 3
00
o
Measured pool e lev a tio n fo r d if f e r e n t flow ratesMeasured depths of f l a t bed f o r d if f e re n t flow rates
Riprap design flow Q = 0,45 c fs .
0 .40.10 0.60.2 0.3 0 .5F low rate , c f s .
Figure 23. Pool e levation versus flow rate.
&
49
Length o f c u lv e r t *= 50 f t
4 -Length of c u lv e r t = 30 f t
Length o f c u lv e r t = 10 f t
2 -
Riprap design flo w ra te f o r th e 30 f t , long c u lv e r t ,
0.60.2 0 .8
Q
eV a d5/2
F igure 24, H ead-discharge r e la t io n f o r ad e p re s se d - in le t c u lv e r t . ( 2 , p 6 18)
50
then i t may be shown th a t
Tmax G T ' u5 c re s t
Y V ^(0.04)22 „„ 173
,1/3 (35)
The constan t can be obtained from th e approximate a n a ly s is fo r
many d i f f e r e n t cases . When ^ , Equation 13 becomesmax cs'
(36)
For a given slope o f th e depressed cone, , and th e angle o f
repose o f th e m a te r ia l, 4>, Equation 35 becomes
V - c5YVf,2 ( 0 . 0 4 ) 2
(1 .4 9 )2,1/3 (37)
where and can be ob ta ined from Tables 3 and 4 re sp e c tiv e ly . By2
s u b s t i tu t in g gy^ fo r and so lv in g f o r th e d iam eter, d , th e equation
becomes
o r
Y (0 .0 4 )2 g 3/2[ V
3/2
(1 .4 9 )2 . .
d = 1.7433/2
yc •
(38)
(39)
51
Table 3« Typical values o f C^,
25° 30° 35° 40°
1:8 2 e 91 3,11 3,26 3.38
1 $6 2,54 2,81 3.01 3.16
1 :4 1,80 2,20 2,50 2,72
Table Typical va lues o f G^e
Ss ' C5
1:8 7
1:6 8
1 :4 9
/
52
The critical depth at the lip of the cone, yc, can be deter
mined from the discharge, Q, and the geometry of the entrance by the
equation
3/(|a / l )2 ' ,yc ^ V -------
where L is the length of the lip of the cone.
The values of are actually very conservative. This con
servatism is warranted, though, since the design is for uniform flow
over the lip of the cone and in actual field installations, the flow
will probably not be uniform, but actually one or more low points will
channel a large percent of the total flow down the depressed cone.
Silting of the Barrel
Silting of the barrel proved to be of no significant problem in
any of the test runs. Run 8 specifically looked at this possible
problem. The velocities in the culvert were large enough to remove
all fine sediment and assuming the riprap is placed upstream far
enough to prevent scour of the natural bed material, nothing but the
very fine sediment should enter the entrance.
This suggests the question of how far upstream is the riprap
needed. At a radius of r, where the riprap ends, the depth over the
natural bed and the depth over the riprap are assumed to be approxi
mately equal. Also, the velocities on each side of this section are
53
assumed to "be approxim ately eq u a l. The shears a t t h i s s e c tio n a re
then
'h'c ( r r )
and
^c(mat)
YVr 2(0 .0 4 )2
( 1 M ) 2
YVr 2(0 .0 4 ) '
( r r )1 /3 (41)
(1 .4 9 )2 yr 1/3, ' 1 /3
(mat) (42)
The r a t io o f i s then
^ c ( r r )^c(m at)
(mat)( r r )
1/3(43)
and when ^ 4 d^rnri+>kf Equation 43 becomes
^ c ( r r ) = 4 d (m a t) 2 /3 d ( r r ) l / 3 ' (44)
For very sm all d iam eters o f bed m a te ria l and vary ing s iz e s o f
r ip ra p m a te r ia l, th e lo c a tio n o f th e s e c tio n where Equation 44 i s t ru e
was in v e s tig a te d using th e approximate a n a ly s is . This d is ta n c e up
stream o f th e v e rte x o f th e cone was always le s s than (3 /2 ) r^ . The
r ip ra p th e re fo re does not need to be p laced upstream o f th e cone any
g re a te r d is ta n c e than 2 ( r^ ) .
The g e n e ra lly accepted design p ra c tic e s fo r th e number o f la y e rs
o f r ip ra p req u ired to p revent th e f in e m a te ria l beneath th e r ip ra p from
being picked up and c a r r ie d o f f o f f by th e tu rb u len ce o f th e flow
should be follow ed. This phenomenon i s c a lle d leach in g . Anderson e t
a l . ( l , pp. 16-1?) suggests th e p lac in g o f a f i l t e r la y e r o f m a te ria l
beneath th e r ip ra p . I f th e d iffe re n c e between th e mean d iam eters o f
th e bed m a te ria l and th e r ip ra p i s g re a t (g re a te r than 4 0 i1 ) , then a
f i l t e r la y e r i s re q u ire d . The f i l t e r la y e r should be such th a t
d r0 ( F i l t e r ) d <n (R iprap)-------------- < 40 and < 4 0 . (45 )
d^Q (Base) d ^ ( F i l t e r )
CHAPTER 6
CONCLUSIONS
The approximate analysis was shown to predict very adequately
the flow characteristics in the depressed-inlet to the horizontal
culvert. From this analysis design procedures were developed to
choose the size riprap required for the given geometry of the inlet.
When the culvert size is determined, the inlet should be
designed such that it is on as mild a slope as possible. During one
run in this investigation, when the slope of the depressed cone was
1:3, there was a major failure of the bed material at a very low flow-
ratee This was probably caused by the sand bed material being at or
near its submerged angle of repose. The weight of the riprap on the
bed material could also have been a factor in causing this failure.
In any case, the characteristics of the bed material are just as im
portant as those of the riprap material if the slope of the depressed-
inlet is largeo
The riprap design flowrate can be determined from the discharge
when the head-pool on the culvert equals the depression of the culvert0
From this the critical depth at the lip of the cone can be determinedp
and thence from
th e s iz e r ip ra p req u ired can be determ ined6 The s iz e r ip ra p i s r e
la te d d i r e c t ly to th e depth o f flow over th e l i p , and i f th e flo w ra te
i s in c reased , (by u sing a la r g e r c u lv e r t ) th e depth w il l in c rease over
th e l i p , and th e re fo re , th e s iz e r ip ra p req u ired w il l a lso in c re a se e
The values o f given in Table 4 a re c o n se rv a tiv e = This con
serva tism i s w arranted , though, because th e a n a ly s is assumes uniform
flow over th e e n t i r e l i p o f th e cone and in a c tu a l f i e ld in s ta l l a t io n s
th e flow w il l no t be e n t i r e ly uniform .
The r ip ra p should be p laced approxim ately a t a d is ta n c e o f
tw ice th e cone ra d iu s upstream o f th e v e r te x o f th e cone to p reven t
scou r o f th e n a tu ra l r iv e r bed m a te r ia l . I f t h i s i s done th e problem
o f s i l t i n g in th e b a r re l o f th e c u lv e r t has been m inim ized. The
v e lo c i t ie s in th e c u lv e r t b a r re l were s u f f ic ie n t ly h ig h , enab ling th e
b a r re l to remain c le a r o f f in e sedim ents even a t low flow r a te s .
I f th e p o s s ib i l i ty o f s tan d in g w ater in th e h o r iz o n ta l c u lv e r t
might cause a nu isance, th e c u lv e r t , can be la id on a sm all s lope
a llow ing dra inage w ithout a f f e c t in g th e advantages o f th e h o riz o n ta l
c u lv e r t ,
NOMENCLATURE
Number o f s te p s taken o f le n g th ^ r f o r th e l i p o f th e cone
Cross s e c tio n a l a re a ( f t . ^ )
C o effic ien t o f co n trac tio n
Constant r e la t in g Manning8s n to r ip ra p d iam eter
Constant r e la t in g c r i t i c a l t r a c t iv e fo rce to r ip ra p d iam eter
Constant r e la t in g c r i t i c a l sh ea r on s lope to c r i t i c a l sh ea r on f l a t bed
\Constant equal to 4(C^)
Constant r e la t in g maximum sh e a r on depressed cone to sh ear a t c r e s t o r l i p
Mean d iam eter o f p a r t i c le ( f t . )
Diameter o f c u lv e r t ( f t . )
H y d ro sta tic p re ssu re fo rce
Froude number
G ra v ita tio n a l a c c e le ra tio n ( f t . / s e c .
V eloc ity head measured w ith P i to t tube ( f t . )
Head lo s s ( f t . )
A vailab le head ( f t , ) /
E lev a tio n o f pool above V-notch w eir ( f t . )
Constant r e la t in g len g th o f l i p o f cone to wcq
Moment-arm le n g th ( f t . )
Length o f l i p o f cone ( f t , )
Manning6s re s is ta n c e c o e f f i c i e n t .
Wetted p erim e te r ( f t 6)
D ischarge p e r fo o t o f l i p o f cone ( f t ^ / s e c .
T o ta l d ischarge ( c f s )
D istance from in v e r t o f depressed cone ( f t , )
Radius o f cone (ft® )
Increm ental d is ta n c e along channel ( f t , )
H ydraulic rad iu s (ft® )
Slope f o r normal flow (S = S, *= S - S )* b e w
Slope o f bed
Slope o f energy l in e
Slope o f depressed cone
Slope o f w ater su rface
Shear v e lo c ity equal to ( f t® /s e c ,)
Average v e lo c ity o f flow ( f t , / s e c , )
C r i t ic a l v e lo c i ty a t l i p o f cone ( f t , / s e e , )
Weight o f f lu id mass ( l b s , )
Depth o f flow ( f t ®)
C r i t ic a l depth a t l i p o f cone ( f t , )
E lev a tio n above datum ( f t , )
T o ta l dep ression o f cone ( f t , ) >
e © e e © e © e o @ o o © © e e o e e
Volum etric c o e f f ic ie n t o f p a r t i c le
Area c o e f f ic ie n t o f p a r t ic le
S p e c if ic weight o f f lu id ( l b s , / f t , ^ )
S p e c if ic weight o f p a r t ic le ( l b s . / f t . )
AY
e
v
Tf
To
% '
%T i b
T eresi
Tc(mat)
T c (rr)/ yZcs
/max
Submerged s p e c if ic w eight o f p a r t ic le (yg - y) ( l b s , / f t
R atio o f dep ress io n to c u lv e r t d iam eter
Angle of depressed cone to h o r iz o n ta l (d e g ,)
Kinematic v is c o s i ty
R atio o f c i r c u la r circum ference to d iam eter
D ensity o f f lu id ( lb s ,= s e c 0 / f t e^)
Average boundary sh ea r ( lb s 0/ f t 0^)*
Boundary sh ear a s so c ia te d w ith p a r t ic le ( i b s . / f t , ^ )
C r i t ic a l t r a c t iv e fo rce ( l b s , / f t ,
C r i t ic a l t r a c t iv e fo rce on f l a t bed ( l b s , / f t ,
Shear s t r e s s a t l i p o f cone ( l b s , / f t ,
C r i t ic a l t r a c t iv e fo rce o f n a tu ra l bed m a te r ia l ( l b s , / f t / )
C r i t i c a l t r a c t iv e fo rce o f r ip ra p ( l b s , / f t ,
C r i t i c a l t r a c t iv e fo rce on depressed cone ( l b s , / f t / )
Maximum boundary sh ea r on depressed cone ( l b s , / f t / )
Angle of repose, of r ip ra p
LIST OP REFERENCES
. Anderson» Alvin G ., Arareek S a P a in ta lg and John T, Davenportt "T en ta tiv e Design Procedures f o r R iprap Lined C h an n e ls /’N ational Coorperative Highway Research Program Report 108« 1970o
2 9 B h a tta ra ij Lava R aj, "Flow Through H orizon ta l C u lv e r ts / ’ M,Se T h esis , The U n iv e rs ity o f A rizona, 1972,
3» Henderson, P.M. Open Channel Flow, New York; M acmillan, 1966,
4-0 Laursen, Emmett M», "Scour a t Bridge C ro ss in g s /’ T ran sac tio n s , American' S o c ie ty o f C iv il E ng ineers, Vol. 127, P a r t 1, 1962,p p 8 l66-»80a
5® Ward, Bruce Douglas, "Surface Shear a t In c ip ie n t Motion o f Uniform S a n d s /’ Ph.D. D is s e r ta t io n , The U n iv e rs ity o f A rizona, 1968.
6 , White, C.M., "The Equilibrium o f G rains on th e Bed o f a S tream ," P roceedings, Royal S o c ie ty o f London, Vol. 17 -A, 1940, pp. 332-38.
7. Task Committee on P rep a ra tio n o f Sedim entation Manual, Committee on Sedim entation , "Sediment T ran sp o rta tio n Mechanicss I n i t i a t i o n o f M otion," Jo u rn a l o f th e H ydraulics D iv isio n , Proceedings o f th e American S o c ie ty o f C iv il E ng ineers, Vol. 92, P a r t 1 , March 1966.
60
