CHAPTER 5
ANALYSIS OF THE BUBBLE SCREEN FLOW FIELD IN FLOWING WATER
5.1 The Problem
In order to describe the bubble screen shape in flowing water,
is necessary to know the vertical and horizontal velocities o, the
bubbles at every instant.
This task is complicated because the size oi the bubbIes is not constant, not in horizontal plane and not in vertical plane the buobles expand when they rise!. Another problem is the mutual action between the bubb'! es when they rise close to each other.
In this chapter, an approximation for the bubble screen shape in
flowing water is made.5.2 Anal sis of the Bubble Screen Sha e in Flowin Water
The air bubbles move faster than the local fluid velocity by an amount referred to as slip velocity, U . There are strong indications
s
for the constancy of U in both stagnant and moving liquids. The vertical
5
water velocity at the centerline of the bubble screen is also constant for a given air discharge. These results lead to a conclusion tnat the bubbles rising velocity should be closed to constant. The average rising velocity of the bubbles was given by Helmut '.obus as a result of
experiments:
B-43
8-44
T' = ~g H ~ 0.48 ~ L ]
b o
v g 0
.I!
Basing on this equation it is possible to write:
.2!
In which yb is measured from the bed.
The horizonta1 velocity profile in the ~ough turbulent range is:
u = u ~ 2.5 ln
k
where
.4!
using equation .2!, the following can be obtained:
K ~ t
xb t! uf ~ 25 In 297 k dt
0
t 29.7 Vb t
uf ~ 2.5 ln , + i uf 2.5 ln + dt
f 0
29.7 Vb
= 2.5 uf ln
kt + 2.5 uf t ln t - t!
or
29.7 Vb
x t = 2.5 uf t ln 1 + t ln t!
.5!centerline is:
2.5 uf y 29.7 V y y
ln k - 1+ ln-
Vb Vb Vb
.6!
xb-
The distribution of the vertical velocities in stagnant water is given by Charlton eouation .8!
2
= V -40 e -! x
Vmax
In flowing water, this distribution may approximately be written:
-40 b 2
V = e j
max
.7!
where xb is given by equation .6!.
Figure 5.1 describes the bubble screen shape in flowing water.
This figure has been done for the following parameters:
I| = 0.01 m sec3 -1
0
uf = 0.01 m sec
-1k = 0.005 m H =10m
0
Using equations .2! and .5!, the equation for the bubble screen
Y F 0
Og 0.01 ~' s
Figure 5.1 - Pporoxinateo Shape of' Air Suable Screen in '.0,'in' '..'ater.
5.3 Streamlines and Particles movement
ln order to get the streamline of a bubble screen in flowing water, i t is necessary to superpose the horizontal velocity profile in the channel and the velocity field generated by the bubble screen. To get the pathlines o particles in this flow. field, the settling velocity of the particles
has to be superposed to this flow field. Figures 5.2 and 5.3 describe the streamline pattern for different values of flow velocity in the channel. The streamline. pattern is characterized by the dimensionless
number
As can be seen from. the streamline pattern, the bubble screen is
inefficient for stopping propagation of pollutants in flowing water, but it is possible to lift the particles in suspension and bed load to the
water surface.
~ I E oO O 'll
0 O
U
U1 'I
rg 5
C!
C 0
l t
B--" 9
O C
ZiQ 0
U Q c
tvQJ
V1 5
C CO
O O O
n0
Figure 5.4 - Air Bubble Screer. Idiht I ow Yelocity in the Channel.
~ ~ a
V
i,
V
.' ~
Figure 5.5 - Air Bubble Screen !.'i;h High Yelocity in the Channel
CHAPTER 6
APPLICATIONS FOR CONTROL OF MARINA'S EiiVIROiiMEfiTAL IMPACT
6.1 Anal sis of Air Bubble Screen to Prevent Surface Trans ort of Pol utants
Pievious studies have been done about tnis subject. The principal was to generate surface velocity which is higher than the spreading
velocity of the pol'Iution. A different technique that requires less
energy is presented.Nhen pollution propagates in a channel on the water surface, it is possible not only to stop it, but also to collect it, using a diagonal pipe as descri bed in Figure 6.1. The necessary condition for the pol- lutants to travel along the oipe without crossing it is given in
the equation: u u cos 8
tg g
u sin g
max
u = u cos
max p
where u is the spreading velocity of the oollution on the suri ace.
The pollution is carried out from the channel with a vei y thin layer of water, through wei rs at the channel banks.
6-61
B-5Z
6.1 - .ir =.='ole creen S.'s;e,", .o Con.rol S.~ .ce Poi''-=~on.
8-53
The air discharge per unit length, to create surface velocity which
is equal to the pol'Iution spreading velocity is:u + 3 H !
o H
Q- 0
1. 46 ~ g
.3!
and the total air discharge is:
L .4!
o p
u cos 8! 3 + !H p H
Q0
1.46 ~ g
and the total air discharge is:
u cos 3 3 6 + !H
Q D
P HT2 os .-
1.46 ~ gComparison bet veen equations .6! and .4! yields:
Q = Q cos 2 2
.7!
Similar technique is presented in Figure 6.2. A diagonal pipe
from one bank of the channel to the other create a bubble screen which is effective for flow in both directions.where L is the length of the pipe. The air discharge required to push
p
the pollutants along a diagonal oipe is:
'o
.g H
b o
vg 0
.1!
Basing on this equation it is possible to write:
t! = Vb
. Z!In which yb is measured from the bed
u = uf . 2.5 In .~!
where
.~!
~s~~g equation .2!, the following can be obtained:
V. ~
t! = uf 2.5 ln Z9.7 k dt
0
t 29.7 Vb t
uf ~ 25 ln, + ~ uf 251n+dt
f 0
29.7 Vb
= 2.5 uf ln k t + ~,5 u t In t - t!
,he hori zonta1 velocity profile in the rough turbulent range is:
or
29.7 Vb
xbz = 2.5 uf t ln k - 1 + t ln t!
.5!Using equations .2! and .5!, the equation for the bubble screen
centerline is:
2.5 uf yb 29.7 Vb yb yb ln k -.1 + ln-
Vb Vb Vb
.6!
xb
The distribution of the vertical velocities in stagnant water is
g i ven by C ha rl ton eoua ti on . 8!
V -40 x!
2'max
In flowing water, this distribution may approximately be written:
-40 x - xb
.7!This figure has been done for the following parameters:
Q = 0.01 m sec3
0
uf = 0.01 m sec
-1k = 0.005 m H =10m
0
where x is given by equation .6!.
Figure 5.1 descri bes the bubble screen shape in flowing water.
Y
W S Y= H
Y~P L
o 0.01 I
Figure 5.1 - approximated Shape of Air Suoa1e Screen in F1o,,ing '. ater.
5.3 Streamlines and Particles Movement
In order to get the streamline of a bubble screen in flowing water,
it is necessary to superpose the horizontal velocity profile in the channel and the velocity field generated by the bubble screen. To get the pathlines o particles in this flow. field, the settling velocity of the particles
has to be suoerposed to this flow field. Figures 5.2 and 5.3 describe the streamline pattern for different values of flow velocity in the channel. The streamline pattern is characterized by the dimensionless
number - u
m
As can be seen from. the streamline pattern, the bubble screen is
inefficient for stopping propagation of pollutants in flowing water, but i t is possible to l.ift the particles in suspension and bed load to the
water surface.
a-as
Il
'o
0 O
IJ
ulC!
CJ C.'1
Q
5
'I
0I>
Ni8
I IO
I t/IQ 'I
E O o 0 0 Q 0 1
Figure 5.4 - Air Bubble Screer, <Jiht Low lleloc1ty in the Channel.
Figure 5.5 - Air Bubble Screen !.'ith High jlelocity in .he Channel
CHAPTER 6
APPLICATIONS FOR CONTROL OF MARINA'S EiVVIRONNENTAL IMPACT
6.1 Anal sis of Air Bubble Screen to Prevent Surface Trans ort of Pollutants
Previous studies have been done about this subject. The principal
was to generate surface velocity which is higher than the spreading velocity of the pollution. A different technique that requires less
energy is presented.l/hen pollution propagates in a channel on the water surface, it is possible not only to stop it, but also to collect i t, usi ng a diagonal pipe as described in Figure 6.1. The necessary condition for the pol- lutants to travel along the oipe without crossing it is given in
the equation:
. 1!
U u cos
u sin g max
or
U = U cos
nax p
where u is the spreading velocity of the pollution on the surface.
p
The pollution is carried out from the channel with a very thin
layer of water, through weirs at the channel banks.
8-52
=icicle 6.1 - ".ir =a':ole 5creen 5's<e.".o Con.ro1 S.r-,.ce I-'o i', .ioo.
The air discharge per unit length, to create surface velocity which
is eaual to the pollution spreading velocity is:u + 3 H !
o H 0
1.46 ~ g3
.3!
and the total air discharge is:
Q =Q 4!
o p
u cos 8! + H !
3 HP 0
1.46 ~ g
and the total air discharge is:
3 3 H
p cos B + !
H
1.46 ~ g
Comparison between equations .6! and .4! yields:
QT QT
22 1
. 7!
Similar technique is presented in Figure 6.2. A diagonal pipe
from one bank of the channel to the other create a bubble screen which is effective for flow in both directions.
where L is
the 1 ength of the pipe . The air di scharge required to push
P
the pollutants along a diagonal oipe is:
Figure 6.2 - Air Bubble Screen SysteI",, to Control Surface Pollution
for Flo:I in ~oth directions.
B-55
6.2 Anal sis of Air Bubble Screen Filter to Reduce Trans art of Pollutants in Suspension and Bed Load
Air bubble screens properly installed and operated at a marina
This chapter represents a design method for a bubble screen filter at the
marina entrance.
Figure 6.4 descri bes the solution suggested. The particles are lifted up to the water surface by the vertical currents. They are pushed from one bubble screen to another whi'Ie being carried forward by the
channel flow. Finally, they are collected by two wei rs at the channel
banks.
The water that flows out from the channel through the weirs is directed to a settling basin where the particles will settle dovrn.
The distance between one bubble screen to another is suggested to be 0.5H. After this distance the streamlines start curving daven causing the particles to settle faster.
Figure 6 . 4 ind~cates the longest path of a particle in the bubble screen velocity field. The time for this particIe to travel from point A
to point D is approximately:0.5H
0.5 umax
0.5H max
..B!
where 0.5 u /6 is approximately the average velocity between points A
max
and B, 0 .5 V , - V } is the particle average velocity between B and C,
max s
and 0.5 u is the average velocity between C and 0.
max
The same equatior. may be written;
7
'max
entrance can reduce transport of poIIutants in suspension and in bed load.
3 > IJ 1 ip , 3 w1 l 3fJi!Q ' e Scr en F~ 'i 'sr '.o PeCuce Tr ans oar . o, a'i 1 utants
~n a~s~ensior. an" e. i oa..
8- ~7
Q
C S P C
ED
7
Using equations .12! and .7! for u and
maxV
maA' '1/3
~.S + !
H .10!
g Qo g Qo
The length of each pipe should be:
=u, T .11!
p m
wher e u i s the mean fl oar ve loci ty in the channel .
u
TIle total discharge is:
Q L ~ Q =u T ~ Q .l2!
p 0 0
or
Q = u .2 I + ! Q + !
2 Q.13!
Tm~o12 Q!/ y
Q should be designed to create surface velocity which is higher than the
0
lateral velocity fluctuations in the channel and to give reasonable value
for L . P
CHAPTER 7
EXTENSION OF RESEARCH
Flow in and out the marina occurs when the water level in the sea
is changing due to tide. In thi s chapter, a basi c model is descri bed to approximate the flow velocity grenerated at the marina entrance.
7.1 0 en Channel Transient Flow
equations for unsteady open channel flow are;
0 V V
~y + -gs~na+2VV + A + A +V =0
x -R x A x .1!
in which y is the water depth, V is the flow velocity, A is the cross- section area and ~ is the bed slope.
Continuit E uation:
-2!
"A +A +A V =0
x t x
no lateral inflow or outflow exists!
A marina can be modeled by an hori zontal open channel in which one end is a dead-end, and at the other end the water level is changing according to the tide. One may modify this basic model by making the channel wider at certain locations as indicated in Figure 7.1.
The method of' characteristics wi th specified time interval is pre- sented i n order to solve this unsteady state problem. The differential
B-60
Figure 7.1 - Basic Model for Floiv Velocity Approximation in the Marina.
8-,61
The solution by method of character~sties yields:
for C character~sties:t
V -V,g -dy+
1 PLg S-S!j «=0
p R C 0
R
.S!
t P
x - x = v + c! dt p R
R
.C!
where c is the wave celerity in the channel
for C characteristics:
IP1 P
V - V - g I dy + g S - S ! dt = 0
p s 3 C Jy t
0a
.6!
P
x - x = v - c! dt
p s
s
.6!
See Figure 7.2 for definition of the points p, S, R.
eauations:
for C oharacteristics:
V - V " g/C y - y ! = g S - S ! at = 0
.7!p R R p R r o
- xR = VR + CR! at
P
.8!
The method of specified-time interval leads to simple algebraic
B-62
C
Figure 7.2 - Unsteady Open Channel Flora - Solution by,'lethod of
Characteristics viith Specified Tirade Interval.
B-63
for C characteri sti cs:
V -V -g/C y -y!+g S -S !~t=0
p s s p S s 0 9!
x - x = V - C ! at
p s s s .10!
y from previous calculations.
Solving equations .7!
and .9!the solution of y and V for
p p
interior points can be found:
VR - V
~C+ ~y R ~R R ~ R s
.11!
v =v -g~-g~~Is
- yR-sI
p R CR
R o .12!for downstream boundary condition it is necessary to solve equation .7!
C characteristic! with the boundary condition:
Y p 0
for upstream boundary condition equation .9! C characteristic! should
be solved with the condition:
V = 0 P
Hy linear interpolation it is possible to find the values of V, C and
y for the points S and R, basing on known values of V, VB, C, CB, y and
8-,64
7.2 litotes on the Oon uter Pro ram and the Results
The comouter program which is listed in Appendix A-4, calculates the
water depth and the flow velocity in the channel as a function of time,for different sections along the channel.
The solution given in Appendix A-4 is for a channel, 2000 m long, 20 m wide, and 3 in deep. The rate of tide is 1 m in 6 hours. The flow velocity oscillates between 0.0 m/s and 0.046 m/s. One should remember that higher velocity may be generated due to winds effect. Hodification of' thi s program can be made to describe di fferent shapes of marinas.
Two recommendations for design can be made:
1! It is possible to fit the air discharge in the bubble screen to the velocity oscillations in t'h e channel, and in this way, to reduce the amount of energy required.
2! It is recommended to locate the bubble screen as far as possible from the channel mouth, where the flow velocity
is the highest.
CHAPTER 8
CONCLUSIONS AND RECONiENDATIONS
Experimental measurements as well as analytical investigations discussed in the previous chapters have led to the following conclusions:
1. CEI water current meter was found to be a good tool for investigations in bubble screens. Flow velocity measure-
ments can be done with good accuracy.
2. It, was found possible to describe the flow field generated by a bubble screen in stagnant and flowing water by
streamlines. Basing on velocity measurements and linear superposition, a computer program was built to make these
streamlines.
3. The behavior of particles in the bubble screen flow field may be described by superposing the settling velocity of
the particles to the velocity field.
4. A technique to control marinas environmental impact was
described. A bubble screen installed at a marina entrance
can stop further propagation of surface pollution and can be an efficient filter for pollutants in suspension and bed load. The advantages of the bubble screen is the low energy consumption and the capability of boats to use the channel without interference.
B-65
5. An approximation for the flow velocity at the marina
entrance can be made by using the method of characteristic.
APPENGIX A-1
COMPUTER PROGRAN FOR CURVE FITTING USING GAUSE ELIt;INATION METHOO
70
40 50
90 100
DOUBLE PRECISION A,X,C,B,ZABS
::,'iE«sro< A c,o},x o},c c!
READ,A READ,C READ,X N=O
SUBROUTIHE GAUSSR WILL SOLVE N Srt1ULTANEOUS EQUATIQ«S
OF THE FORM A*X=C, A ROW INTERCHANGE IS PERMITTED SUCH THAT A K,K!
WILL BE LARGEST POSSIBLE IN ABSOLUTE VALUE FROi't ALl ROWS BELOW ROM
K NNl =N=1
DO 50 K=1,f1511 KP1=K+1
oo 50 I=KPi,N 00 69 L=KPl,N
IF DABS A L,K!!.LE.DABS A K,K!!! GO TO 60
DO 70 J=K,N
OJ=A L,J!'
A L,J!=A K,J}
A K,J!=DJ CONTINUE T=C L!
C L!=C K!
C K!=T CONTINUE
Bi-A I,K!/A K,K!
00 40 J=KPl,N
A I,J!=B"A K,J!+A I,J!
c r!=B*c K!+c I!
X «!=C N!/A N,N!
DO 100 K=2,Jl I=N-K+1
x I!=C I!
Kl'51=K-1 DO 90 J=l,Kfll
x r!=x r!-'A r, r+J! x r J!
X I!-X I!/A I, I!
00 1 10 I=1, 4
pRrNT, x r!
STOP
$| 0
APPENDIX A-2
COMPUTER PROGRAM FOR STREAMLINES OF A 8USQLE SCREEN FLOH FIELD IN STAGNANT HATER
2 3 4 5 6 7 8 9 10
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24.
25 26 27 28 29 30 31 32 33 34 35 36 37 50 38 39 40 41 99 42
T=O
3=0. 0014394 E=2. 72 G=9. 81 H=O. 578 L-1. 2
UYiM=1 . 46* . +H/ 10. !** -O. 333 !* G*g !* Q. 333
READ., X, Y DQ 50 I=1, 10
IF Y. GT. H! GQT TO 99 IF X. LT. 0.2! THEN
IF Y. LT. 4. 21*Q**0.67! THEN VftAX=O. 5* G*Y!**0. 5
ELSE
VftAX=l. 2" G*Q!**0. 333
EffD IF
V=VftAX*E"" -40. * S/Y!**2. !
ELSE
IF Y.GT. 0. 75+H! THEN
V= H/5. -. * Y-0. 75*H!**2. !/H!*Uftl4l 0. 5/L
ELSE
V= UftH*Y!,. *L!
END IF V=-V END IF
IF Y. GT. 0. 707*H! THEN
UI1=Uf+1*4.~Y/H-3. *UftH ELSE
Uf'1=-Uf'tf'<*0. 172 El'fD IF
IF X. GT. 0. 2! THEN
U=-UM*X+l. 2*Uft
U=3. 6429* X/H
ELSE!**3. -24. 005* X/H882. +8 . 2360* X/H !+0. 0246!*Uf't
EffD IF X=X+U/10 Y=Y=V/10 PRINT, X, Y T=T+1
GO TQ 10 STOP END
B-70
APPENDIX A-3
COMPUTER PROGRAj'1 FOR STREAj'1LINES OF BUBBLE SCREEN FLOlrl FIELD IN FLOWING LlATER
REAL K,L,ft,N READ,Q,H G=9. 81.
E-2. 718 HO=10 UF=O. 01 K=O. 001
UV= Q/ G*HO**3. 0!"*O. 5!~~ l. 0/6, 0!*0. 48 G*HO""O. 5
UM=Z. 5*UF*ALOG0. 9*H/K!
TG=UV/UPI LOOP READ,X,Y AT END S OP
PRINT 5
5 FORrtAT '1, ttEXT PARTICLE' !
3 PRINT 10,X, Y
10 FORMAT '','X=',F7. 3,8X, 'Y=',F7. 3!
00 20 I=1, 10
C=9.0*Q*". 0/3. 0!/G**. 0/3. 0!
IF Y .LT. "! THEN Vtt=0. 5* G"Y!'"0. 5
ELSE
V"'I 1. Z* G*Q!"* l. 0/3. 0!
END IF
V=vra* E -50* X-V/TG!**2/Y* 3!!
H=.46*. 0+H/HO!** -1. 0/3. 0!* G*Q!"*. 0/3. 0!!
D=~*0. 5/2! *H IF Y .LT. 0! THEN
8=-M*. 0-2. 0*2. 0**0. 5!
ELSE
6=4. 0*M* Y/H-3. 0/4. 0!
END IF
IF X. GT. Y/TG!! THEN
U=B* X-Y/TG!/ H/*TG!+0.45*H-Y/TG!+UF*2. 5*ALOG9. 7*Y/K!
U=-B"
ELSEY/TG-X !/ H* . 45-1. 0/ . 0*TG ! !+Y/TG }+UF*2. 5*ALOG *
Nr0 IF X=X+U/20 Y=Y+V/20 ZO CONTINUE
IF ABS X! .LT. 0. 45*H! THEN IF Y ,LT. 1. 5~H! GO TO 3
END IF END LOCi' STOP END
29. 7*Y/K!
6-72
APPENDIX A-4
COMPUTER PROGRAM FOR SOLUTION OF UNSTEADY OPEN CHANNEL FLOLl
APPEf<DIÃ A-4
CQf1PUTER PROGRAM FOR SOLUTION OF UNSTEADY OPEN CHANNEL FLOH
A computer program. for solution of unsteady open channel flow is presented. Sy chaning the boundary conditions, large variety of problems may be solved. The program uses the method of characteristics with
specified time interval for solution.
length of channel
B - width of channel
bank slope RN - Manning's n
N - number of reaches along the channel
zpR - number of time increment iterations between printouts
9- 74 Input Data:
XL
S
N
[1AX
bed slope
normal depth at steady state condition
maximum time For computation
DIMENSION Y1 ! V< 31 ! > YP 1 ! > VP 1 ! C 1 ! > ID!
AREA YY! «YY» < Il»YV»Z 1 !
PER YY!ee+2»YY»SORT<Zl»Zl+l. !
CEL<YY!~SGRT <G»YY ~ <rt»YY»Z! ! / e»2. «71*YY! !
SLOPE YY> VV>e RN/CM >! ~ »2« PER YY> /<YV»<D»YY«71 ! ! !eel. 3333 2«VV»ABS<VV
CMAe l.
READ, XL, B. Zl, RN, SO, Yr<. TMAX. N, IPR Ce9 806
00~ 4R EA < Yk ! eel. 6666eSQR T SO ! / RNe P ER YN > ee0. 6666 ! ka~tle 1
NP~N/5
VOeQO/4REA<YN!
DO 24 Iel NS V< I !RVO Y I !eYN
24 C I ! eCEl Y I ! ! YOe Y h<S!
DXeX1 /N
DT«>DX/ V NS!+C NS!!
THO Ke0
YL»YN+VO»VG ~ . 5/C L<R I TE . 25 > QO
25 FQRM4T < ' 00e ' ~ F10. 2. / 'TIME Iht MINUTES '// ' T!HE U. S. G'
2> ' X/Le .0 .2 .4 , 6 ,8 1. D. S. Q ' !
30 TN eT/60.
Olkev< 1 >»4RE4 Y 1 ! ! 04~AREA<Y<NS!>»V<NS!
L'R! TE <6 j5> TM. GIN. V I !. I»1 NS> NP ! > Y I ! > Iil ~ NS. NP ! ~ QL<
35 FQRM4T 1H ~ F9 3 F9. 2> 3H Ve> dF8. 3/19X 3H Ye> bFB. 3. F8, 2!
40 IF < T. GT. Tt14X! CO TO 99 KeKef
DX X%0.
DO 45 I el ~ NS
DXIe<ABS<v< 1 ! !+C< I! !»DT 45 IF DX I. GT. DXX! DXX~DX!
ZETA~DXX/DX DTeDT/ZETA T~T»DT THeDT/DX C INTERIOR POINTS
DO 50 Ie2, ht CAec I >-C< I-1!
VR~ V< I !»TH» <C < I >»V I 1! V I ! eC l 1! ! !/ l. »TH» V I !-V I I >+ A! ! CRe C l!wR»TH»C4! / l. »THeCA>
YRe Y I !-TH»< VR+CR! ~ Y I ! Y I 1! ! CR&CEL<YR!
SReSLOPE<YR. VR!
CB>C I !-C I » 1 !
VS V I >-TH» <V I > »C I»l !-C <! >«V< I+I ! ! !/ l. -THe V l !-V le! !MB! ! CSe C I !+VS» TH»CB ! / I. » TH«CB >
YSeY I >+TH» VS-CS! e Y I!-Y I+1! ! CSeCE <YS!
YP< I!e YS»CR+YR»CS«CR»CS« VR-VS!/0-DT« SR SLOPE YS. VS!!!!/ CR+CS!
50 VP I!evR<»< YP<l>-YA!/CR»DT»<SR-SO!!
C UPSTREAM BOUND4RY
CB~C l!-C!
VS~ V 1!-TH» V 1 ! «C !-C ! «V! ! !/, -THe V!-V!<B! ! CSe C I ! »VS«TH»CB > / I . +TH«CB !
YSe Y < 1 >»1H» < VS-CS! ~ < Y 1 !-Y ! ! C2eC/CEL < YS !
CM>VS-C2«YS-C«DTe <SLOPE YS. VS!-SO!
ALFAel. /36GO.
YLeYN+VO»VO» 5/C-4LF4»T R AD~2. »C2« C M»C2» YL > /C 65 VP 1 >w0.
YP<!!eYS»CS»DT« SLOPE YS> VS!-SO!-VS«CS/C C DOLINSTREeM BOUNDARY
66 CAeC t'<S!~ N!
VRa V kSTH» C NS! »V N!-V h'S! «C k! ! !/. +TH@ V NS! V ht!+CA! !
CRe C<NS!-VR»>H»C4!/. »TH»C4>
YR~ Y<NS!- TH ~ <VR»CR ! e Y <NS!-Y ht! ! ALFAe! /<36GG »O. >
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