PROJECTE O TESINA D’ESPECIALITAT
Tí ol
Es udio del lujo espacialmen e a iado en canales de
ecogida de plu iales
Au o /a
Elena Pons Minguillón
Tu o /a
Manuel Gó
mez Valen ín
Depa amen
Enginye ia h
id áulica
, ma í ima i ambien al.
In ensi icació
Hid áulica i Hid ologia.
Da a
28/06/2013
INDEX
FIGURES INDEX ........................................................................................................ 5
TABLE INDEX ............................................................................................................ 7
ABSTRACT ................................................................................................................. 9
RESUMÉ .................................................................................................................... 10
I. INTRODUCTION ............................................................................................... 11
II. OBJECTIVES .................................................................................................. 12
III. HYDRAULIC ENGINEERING APPLICATIONS ........................................... 13
Inc easing Spa ially Va ied Flow ............................................................................. 13
Side O e low collec ing canal in dams spillways ................................................. 13
Road and S ee D ainage Canals ........................................................................ 14
Roo Rain Gu e s ................................................................................................ 15
Dec easing Spa ially Va ied Flow ............................................................................ 15
Side wei s ............................................................................................................. 15
Bo om ou le canal each .................................................................................... 16
IV. MATHEMATICAL DESCRIPTION OF THE SPATIALLY VARIED FLOW 17
One-Dimensional Me hod o Flow analysis .............................................................. 17
Mean eloci y ....................................................................................................... 17
Boussinesq Coe icien β....................................................................................... 18
Co iolis Coe icien α............................................................................................ 18
Con inui y Equa ion ................................................................................................ 18
Ene gy Equa ion ...................................................................................................... 19
Momen um Equa ion ............................................................................................... 19
F ic ion Losses......................................................................................................... 19
Go e ning Equa ions o In low and Ou low .......................................................... 20
Flow wi h Inc easing Discha ge ............................................................................ 20
Flow wi h Dec easing Discha ge ........................................................................... 21
Gene al Exp ession .............................................................................................. 22
V. NUMERICAL APPROACH OF THE SPATIALLY VARIED FLOW ............. 23
Con ol Poin .......................................................................................................... 23
C i ical Dep h Line .............................................................................................. 25
T ansi ional P o ile .............................................................................................. 26
Con ol Poin o a ec angula canal ................................................................... 27
In eg a ion by 4
h
O de Runge-Ku a ..................................................................... 28
VI. VALIDATION TEST ....................................................................................... 30
Tes Case 1 ............................................................................................................. 31
Tes Case 1.A.: .................................................................................................... 31
Tes Case 1.B.:..................................................................................................... 33
Tes Case 2. ............................................................................................................ 34
Tes Case 3. ............................................................................................................ 36
Discussion ................................................................................................................ 39
VII. WATER PROFILE: ONE SLOPE CASES ...................................................... 40
Small bed slope canal .............................................................................................. 40
Fla slope canal ....................................................................................................... 41
“U” Shaped canal .................................................................................................... 43
VIII. WATER PROFILE: TWO SLOPES CASES ................................................ 46
Summa y Table ....................................................................................................... 47
Ini ial Case .......................................................................................................... 48
Case 1 .................................................................................................................. 49
Case 2 .................................................................................................................. 51
Case 3.1. .............................................................................................................. 52
Case 3.2. .............................................................................................................. 53
Case 4 .................................................................................................................. 54
Case 5.1. .............................................................................................................. 55
Case 5.2. .............................................................................................................. 58
Case 6 .................................................................................................................. 60
Case 7 .................................................................................................................. 62
Case 8 .................................................................................................................. 64
Impossible Cases .................................................................................................. 65
IX. CONCLUSIONS AND FUTURE DEVELOPMENT ........................................ 66
REFERENCES ........................................................................................................... 67
FIGURES INDEX
Figu e 1: Hoo e Dam, be ween Ne ada and A izona, USA. ....................................... 13
Figu e 2. Fuensan a Rese oi , Albace e, Spain. ......................................................... 14
Figu e 3. S ee D ainage Canals ISVF. ...................................................................... 14
Figu e 4. O e low in a s ee d ainage canal. ............................................................. 14
Figu e 5. Roo Rain Gu e ISVF. ............................................................................... 15
Figu e 6. Ele a ion iew o a Side Wei DSVF. .......................................................... 15
Figu e 7. Bo om ou le DSVF. ................................................................................... 16
Figu e 8. Veloci y Dis ibu ion. ................................................................................... 17
Figu e 9. Inc easing Spa ially Va ied Flow. ................................................................. 18
Figu e 10. Momen um Con ol Volume. ...................................................................... 19
Figu e 11. Inc easing SVF Con ol Volume. ................................................................ 20
Figu e 12. In eg a ion di ec ion i Con ol Poin loca ed downs eam. ........................ 24
Figu e 13. In eg a ion di ec ion i Con ol Poin Loca ed in he canal each. .............. 24
Figu e 14. In eg a ion di ec ion i Con ol Poin Loca ed Ups eam ........................... 25
Figu e 15. Ill-condi ioned T ansi ional P o ile. ............................................................ 27
Figu e 16. Tes Case 1.B. C i ical Dep h and T ansi ional p o ile. .............................. 32
Figu e 17. Tes Case 1.A. Wa e P o ile and Bo om Slope. ....................................... 32
Figu e 18. Tes Case 1.A. F oude Numbe . ................................................................. 33
Figu e 19. Tes Case 1.B., C i ical Dep h and T ansi ional p o ile. ............................. 34
Figu e 20. Tes Case 2. C i ical Dep h and T ansi ional p o ile. ................................. 35
Figu e 21. Tes Case 2. Wa e P o ile and Bo om Slope. ........................................... 35
Figu e 22. Tes Case 2. F oude Numbe ...................................................................... 36
Figu e 23. Tes Case 3. C i ical dep h and T ansi ional P o ile. ................................. 37
Figu e 24. Tes Case 3, Wa e P o ile and Bo om Slope. ........................................... 38
Figu e 25.Tes Case 3, F oude Numbe . ...................................................................... 38
Figu e 26. Small Slope Case C i ical dep h and T ansi ional p o ile............................ 40
Figu e 27. Small Bed Slope. Wa e P o ile and Bo om Slope. .................................... 41
Figu e 28. Small Bed Slope Case 4. F oude Numbe . .................................................. 41
Figu e 29. Fla Slope Case. Wa e P o ile. .................................................................. 42
Figu e 30. Fla Slope Case. F oude Numbe . .............................................................. 43
Figu e 31. C oss Sec ions used by ACO Company. ..................................................... 43
Figu e 32. “U” Shaped C oss Sec ion. ......................................................................... 44
Figu e 33. U” Shaped. Wa e P o iles and bo om slope. ............................................ 45
Figu e 34. Two Slope Cases. ....................................................................................... 47
Figu e 35. Two slope Ini ial Case Wa e P o ile .......................................................... 49
Figu e 36. Two slope Case1 Wa e P o ile. ................................................................. 50

Figu e 37. Two slope Case 2 Wa e P o ile. ................................................................ 52
Figu e 38. Two slope Case 3.1. Wa e P o ile. ............................................................ 53
Figu e 39. Two slope Case4 Wa e P o ile. ................................................................. 55
Figu e 40. Two slope Case5.1. Independen Wa e P o iles. ........................................ 56
Figu e 41. Two slope Case5.1. Hyd aulic Jump in R2. ................................................ 57
Figu e 42. Two slope Case 5.1. Hyd aulic Jump in R1. ............................................... 58
Figu e 43. Two slope Case 5.1. Wa e P o ile. ............................................................ 58
Figu e 44. Two slope Case 5.2 Hyd aulic Jump in S1 o S2. ....................................... 59
Figu e 45. Two slope Case 5.2 Final Wa e P o ile wi h bo om slope. ....................... 60
Figu e 46. Two slope Case 6 Independen Wa e P o iles............................................ 61
Figu e 47. Two slope Case5.2 Hyd aulic Jump in S1 ................................................... 61
Figu e 48. Two slope Case5.2 Final Wa e P o ile....................................................... 62
Figu e 49. Two slope Case 6 Final Wa e P o ile wi h bo om slope. .......................... 62
Figu e 50. Two slope Case 7 Wa e P o ile wi h bo om slope. ................................... 63
Figu e 51. Two slope Case 8 Wa e P o ile wi h bo om slope. ................................... 64
TABLE INDEX
Table 1. Tes Case 1 Da a .......................................................................................... 31
Table 2. Tes Case 2 ISP loca ion Li e a u e. ............................................................. 31
Table 3. Tes Case 2 ISP loca ion Li e a u e. ............................................................. 31
Table 4. Tes Case 1.B. ISP loca ion Li e a u e. ......................................................... 33
Table 5. Tes Case 1.B. ISP loca ion. .......................................................................... 33
Table 6. Tes Case 2 Da a. ......................................................................................... 34
Table 7. Tes Case 2 ISP loca ion Li e a u e. ............................................................. 34
Table 8. Tes Case 2 ISP loca ion. .............................................................................. 35
Table 9. Tes Case 2. Wa e Le el P o ile Compa ison, Ups eam. ............................ 36
Table 10. Tes Case 2. Wa e Le el P o ile Compa ison, Downs eam. ....................... 36
Table 11. Tes Case 3 Da a. ........................................................................................ 37
Table 12. Tes Case 3 ISP loca ion li e a u e. ............................................................. 37
Table 13. Tes Case 3 ISP loca ion. ............................................................................ 37
Table 14. Tes Case 3. Wa e Le el P o ile Compa ison, Ups eam. ........................... 39
Table 15. Tes Case 3. Wa e Le el P o ile Compa ison, Downs eam. ....................... 39
Table 16. Small Slope Case Da a. ............................................................................... 40
Table 17. Small Slope Case ISP loca ion. .................................................................... 40
Table 18. Fla Slope Case Da a. .................................................................................. 42
Table 19. Fla Slope Case ISP loca ion Li e a u e....................................................... 42
Table 20. “U” Shaped Da a......................................................................................... 44
Table 21. U” Shaped Cases ISP loca ion. .................................................................... 45
Table 22. Two slopes. Main legend. ............................................................................ 48
Tabla 23. Two slopes. Ex a wa e p o iles o Hyd aulic Jump legend. ...................... 48
Table 24. Two slope Ini ial Case da a ......................................................................... 49
Table 25. Two slope Ini ial Case ISP loca ion. ............................................................ 49
Table 26. Two slope Case1 da a. ................................................................................. 50
Table 27. Two slope Case1 ISP loca ions. ................................................................... 50
Table 28. Two slope Case 2 Da a. ............................................................................... 51
Table 29. Two slope Case2 ISP loca ions. ................................................................... 51
Table 30. Two slope Case 3.1. Da a. ........................................................................... 52
Table 31. Two slope Case 3.1. ISP loca ions. .............................................................. 52
Table 32. Two slope Case3.2. Da a. ............................................................................ 53
Table 33. Two slope Case 3.2. ISP loca ions. .............................................................. 53
Table 34. Two slope Case 4 Da a. ............................................................................... 54
Table 35. Two slope Case4 ISP loca ions. ................................................................... 54
Table 36. Two slope Case 5.1. Da a. ........................................................................... 55
Table 37. Two slope Case5.1. ISP loca ions. ............................................................... 56
Table 38. Two slope Case 5.1. Hyd aulic Jump Loca ion. ........................................... 58
Table 39. Two slope Case 5.2 Da a. ............................................................................ 59
Table 40. Two slope Case 5.2. ISP loca ions. .............................................................. 59
Table 41. Two slope Case 6 Da a. ............................................................................... 60
Table 42. Two slope Case6 ISP loca ions. ................................................................... 61
Table 43. Two slope Case 6 Hyd aulic Jump Loca ion. ............................................... 61
Table 44. Two slope Case 7 Da a. ............................................................................... 63
Table 45. Two slope Case 7 ISP loca ions. .................................................................. 63
Table 46. Two slope Case 8 Da a. ............................................................................... 64
Table 47. Two slope Case 8 ISP loca ions. .................................................................. 64
ABSTRACT
This documen aims o ind a good app oxima e solu ion o he wa e p o ile ha
occu s in open canals wi h a spa ially a ied low conce ning an inc easing discha ge.
An in oduc ion o he Spa ially Va ied Flow and i s applica ions in na u al and
a i icial canals is p esen ed and a de ailed deduc ion o he go e ning equa ion is done.
The equa ion ha desc ibes his low condi ion is an O dina y Di e en ial Equa ion.
A nume ical code is hen w i en in o de o sol e he ODE ha de ines he wa e
p o ile. The loca ion o he in e nal bounda y condi ion s ands as a key s ep o a
success ul compu a ion and a sys ema ically p ocedu e o de e mine i is desc ibed. All
physically possible loca ions and hei associa ed implica ions a e also discussed.
All possible si ua ions ega ding he Inc easing Spa ially Va ied Flow occu ing in a
wo slope ec angula canal a e s udied and he impossible si ua ions a e b ie ly
explained. In he cases whe e a Hyd aulic Jump occu s u he compu a ions a e
ca ied ou in o de o ob ain a inal wa e p o ile ha ul ills bo h in e nal bounda y
condi ions.
Addi ionally, he wa e p o ile ob ained in a s ee d ainage canal is analyzed o
unde s and he way i commonly ails.
Key wo ds: spa ially a ied low, inc easing discha ge, con ol poin , side spillway,
s ee d ainage canals.
Bo om ou le canal each
A di e en dec easing SVF is he low occu ing o e a bo om ou le .
Figu e 7. Bo om ou le DSVF.
In his case he amoun o ou low a e also needs o be de e mined.

III. MATHEMATICAL DESCRIPTION OF THE
SPATIALLY VARIED FLOW
Gene al laws ha desc ibe he low along an open-canal a e b ie ly de ined in his sec ion.
Once he cons i u i e equa ions a e in oduced, he go e ning equa ion o he Spa ially
Va ied Flow can be deduced.
One-Dimensional Me hod o Flow analysis
The p esence o he bounda ies in open canals causes he eloci y ec o s o he low o ha e
componen s in he h ee coo dina e di ec ions, no only on he longi udinal di ec ion bu also
in he wo no mal di ec ions o he low. The same happens o he p essu e g adien .
In Figu e 8 eloci y dis ibu ions in di e en ype o c oss sec ions can be obse ed. In any kind
o c oss-sec ion i is known ha he eloci y is ze o a he solid bounda ies and g adually
inc eases as he dis ance o he bounda y ge s g ea e .
Figu e 8. Veloci y Dis ibu ion.
A Th ee-Dimensional Analysis conside s hese eloci y p o iles, esul ing in a e y complex
analysis. Howe e , i can be simpli ied and ye gi e meaning ul esul s by conside ing a One-
Dimensional app oach, he mean o he c oss sec ion eloci ies and p essu e g adien s o
each c oss sec ion a e aken as ep esen a i e alues.
Mean eloci y
The mean eloci y  in e ms o he  di ec ion can be de ined as:
=1
··
.

Conside ing he Discha ge as: =· This leads o a new de ini ion o i :
=·
.

The e o e, wo coe icien s a e in oduced o adequa ely ep esen he non-uni o mi y o
he eloci y dis ibu ion h ough he ans e sal sec ion.
Boussinesq Coe icien 
This coe icien is de ined so ha he momen um p inciple can be exp essed in e ms o
V:
The momen um lux is w i en as:=

 besides, i ’s known ha : =



.
,
Thus i ’s deduc ed ha : =



.



Co iolis Coe icien 
This coe icien is de ined so ha he Kine ic Ene gy can be exp essed in e ms o V:
The Kine ic Ene gy is w i en as: ..=




besides, i ’s known ha : ..=






.
, Thus i ’s deduc ed ha : =


!
."
#

!
These wo coe icien s a e equal o 1 i he eloci y dis ibu ion is indeed uni o m and
a e g ea e ha one in any o he case. As he eloci y dis ibu ion ge s less uni o m he
coe icien s ge g ea e han 1.
Con inui y Equa ion
I is he exp ession ha ep esen s he law o conse a ion o mass applied in an open
canal low.
Conside ing a s eady s a e low o he SVF i ’s no iced ha he olume ic a e o low
is no cons an along he canal since he e is a la e al addi ion o wi hd awal. Figu e 9
illus a es he case o an inc easing SVF.
Figu e 9. Inc easing Spa ially Va ied Flow.
Being 
$
he ini ial discha ge and %
∗
 he a e o addi ion o subs ac ion o
discha ge: ()=
$
+%
∗
·
*
$
[1]
Ene gy Equa ion
The Be nouilli equa ion is used o ep esen he ene gy equa ion in he 1D Analysis o
s eady open canal low:
+=,+-·./01+·

23
In his pape 1≤0.15 and hen he assump ion./01~1 is always alid.
Hence,
+=,+-+·


8
[2]
Which can be w i en as well as: +=,+-+·
9

8

[3]
Momen um Equa ion
The momen um p inciple in luid mechanics is based on New on’s second law o
mo ion, s a ing ha he ex e nal o ces applied in a con ol olume a e equal o he
a e o change o he momen um.
Figu e 10 is used o deduce he exp ession o he momen um:
Figu e 10. Momen um Con ol Volume.
Being :; he ex e nal o ces applied o each conside ed sec ion:
<:=:;1−:;2=(

−

)
:;1=

·ℎ
?
·@ and :;2=

·ℎ
?
·@
No e ha @=·3 and ℎ
?
is he cen oid and can be de ined as: ℎ
?
=
A·

Then he momen um can be w i en as: =·ℎ
?
+
9

8
[4]
F ic ion Losses
I ’s conside ed ha he ic ion losses a e adequa ely ep esen ed by Manning’s
o mula:
=

B
C
A/
·E
$/
[5]
This exp ession links mean eloci y, hyd aulic adius and bo om slope making use o
he Manning numbe , which is conside ed cons an along he canal.
Go e ning Equa ions o In low and Ou low
In o de o deduce he cons i u i e equa ions o he SVF, some assump ions a e
conside ed:
 Incomp essible luid
 Hyd os a ic p essu e, s eady s a e solu ion.
 1D Analysis
 Manning ic ion looses
 P isma ic canals, hen
F
F*G
=0.
In low and Ou low cases will be s udied sepa a ely; di e en equa ions a e used o
desc ibe he low in he canal.
Flow wi h Inc easing Discha ge
In his ype o Spa ially Va ied Flow, an app eciable po ion o he ene gy loss is due
o he u bulen mixing o he added wa e and he wa e lowing in he canal. In mos
cases, hese losses a e ela i ely high and unce ain. Fo his eason he momen um
equa ion is mo e con enien han he ene gy equa ion when modeling he low.
Figu e 11 shows a con ol olume ep esen ing a ISVF:
Figu e 11. Inc easing SVF Con ol Volume.
F om he Figu e 11, on he di ec ion pa allel o he bo om slope i can be w i en
ha : 

−

=H

−H

+I·0JK1−:L [6]
Whe e: 

,

is he momen um a each c oss sec ion,
H

, H

is he p essu e o ce,
I·0JK1 is he weigh o con ol olume ∆ in he  di ec ion , I·0JK1=
@E
$
·∆.
:L is he ic ion o ce, :L=@E
O
·∆.
Thus, ∆=−∆H+I·0JK1−:L [7]
Di iding equa ion [7] by ∆ and aking limi s as ∆→0:
Q
*
=−
R
*
+@E
$
−@E
O
[8]
wi h
R
*
=
STGUV
*
=@
G
*
Besides, om he 1D analysis i ’s known ha : =

=

/,
De i ing his exp ession:
Q
*
=·W
9
*
·
9

−

·

*
·



X [9]
By de ini ion:
9
*
=%
∗

and

*
=Y·
FG
F*
+
F
F*G
, being a p isma ic canal:

*
=Y·
FG
F*
Thus equa ion [9 ] esul s in:
Q
*
=·W%
∗
·
9

−
9



·Y·
FG
F*
X [10]
Using equa ion [10], equa ion [8] can be ew i en as:
G
*
=
Z
[
Z
]
^
_[`∗
a
 ^
_[
a
b
[11]
Equa ion [11] is he go e ning equa ion o he ISFV, i can be seen ha is an ODE
whe e
G
*
is de ined in e ms o - and .
Flow wi h Dec easing Discha ge
This ype o low can be unde s ood as a low di e sion whe e he di e ed wa e does
no a ec he ene gy head. The speci ic ene gy is no a ec ed by he wa e lea ing he
main low. The e o e he ene gy equa ion is sui able o sol e his kind o low.
This can be sum up as an ex a assump ion:
 The wi hd aw o wa e does no a ec he ene gy con en pe uni o
mass o he wa e in he canal.
Di e en ia ing he ene gy equa ion [3] wi h espec o :

c
*
=
d
*
+
G
*
+
e
8
W
9


·
9
*
−
·9



·

*
X [12]
Conside ing again ha :
9
*
=%
∗
 he la e al discha ge, and

*
=Y·
FG
F*
o p isma ic
canals.
By de ini ion:
c
*
=−E
O
and
d
*
=−E
$
.
Finally equa ion [12] can be simpli ied and ew i en as:
G
*
=
Z
[
Z
]
e
_[`∗
a
 e
_[
a
b
[13]
Equa ion [13] is he go e ning equa ion o he DSFV, i can be seen ha is again an
ODE whe e
G
*
is de ined in e ms o - and .
Gene al Exp ession
The same s uc u e can be seen by aking a close look o bo h go e ning equa ions. A
gene al exp ession is hen w i en ep esen ing bo h cases:
G
*
=
Z
[
Z
]
 g
_[`∗
a
 g
_
a
b
[14]
Whe e:
Fo a Dec easing SVF : µ =  and h=1
Fo an Inc easing SVF: µ =  and h=2
Al hough bo h cases a e ep esen ed in equa ion [14 ], his equa ion is no he ODE
aimed o sol e by nume ical me hods. Assump ions aken in each case di e a lo om
he o he . A clea example o hese di e ences is ound in %
∗
: he amoun o in low is
known while he amoun o ou low depends on he wa e le el -, he sough a iable.
The e o e i is belie ed ha analyzing each ype o Spa ially a ied Flow sepa a ely
would lead o simple compu a ions, deal wi h bo h lows as a gene al case would no
be a p ac ical app oach.
In his pape he subjec o s udy is he Spa ially Va ied Flow wi h Inc easing
discha ge, so he equa ion deal wi h is equa ion [11].
IV. NUMERICAL APPROACH OF THE SPATIALLY
VARIED FLOW
Once he low has been cha ac e ized by a go e ning equa ion, equa ion [11], a
nume ical code is w i en, which compu es he wa e p o ile o any gi en c oss
sec ion’s canal geome y as well as o any gi en bo om slope, as long as he
assump ion ./01~1 is accep able.
The wa e p o ile can be simply compu ed by in eg a ing he go e ning equa ion along
he canal, conside ing he app op ia ed in e nal bounda y condi ions. This in eg a ion
is app oxima ed using a nume ical me hod, which is a disc e e me hod ha compu es
an app oxima e solu ion using a ce ain ∆ in a g ea amoun o disc e e poin s.
The key s ep o ca y ou he in eg a ion success ully is o de e mine he In eg a ion
S a ing Poin , he nume ical in eg a ion should be s a ed he e. I he canal has wo
eaches an In eg a ion S a ing Poin is independen ly ound o each o hem.
The In eg a ion S a ing Poin is usually loca ed in he so called Con ol Poin .
The Con ol Poin is he only posi ion whe e he c i ical dep h -
?
can occu , which is a
wa e le el easily o be compu ed and i does no depend on he each slope. I s
loca ion depends on he combina ion o all he pa ame e s a ec ing he low such as
he slope, he c oss-sec ion geome y, he oughness coe icien .
I he Con ol Poin is ound along he canal each he In eg a ion S a ing Poin is
loca ed he e and he wa e le el in his posi ion is he c i ical dep h. I he Con ol
Poin is loca ed elsewhe e, he In eg a ion S a ing Poin is ound in he ups eam o
downs eam ex eme o he each and he alue o he wa e le el has o be
de e mined.
Con ol Poin
The possible loca ions o he Con ol Poin a e he e discussed, hei associa ed
In eg a ion S a ing Poin loca ion and alue a e also p esen ed.
The Con ol Poin is loca ed whe e he T ansi ional P o ile line and he C i ical Dep h
line c oss. Each condi ion is ep esen ed by one equa ion, bo h equa ions a e
in oduced in he nex sec ion.
The e a e basically h ee possible si ua ions:
1) Con ol Poin Loca ed Downs eam he Canal.
In his case he T ansi ional p o ile does no each he C i ical dep h line in he
canal leng h and he posi ion o he Con ol Poin is ound downs eam he
canal.
In his case he In eg a ion S a ing Poin is loca ed in he downs eam
ex eme, he alue o he wa e le el migh be he C i ical dep h in he case o a
single slope canal. I ano he canal each is ound downs eam, he alue o he
wa e le el is gi en by he ini ial wa e le el o he second each which is a
subc i ical low. In any case he low is subc i ical along he each s udied.
Figu e 12. In eg a ion di ec ion i Con ol Poin loca ed downs eam.
2) Con ol Poin Loca ed In he Canal
In his case i exis s a poin om he canal each  such ha bo h lines c oss.
The Con ol Poin is loca ed along he canal and so he In eg a ion S a ing
Poin is loca ed a his posi ion. The in eg a ion is ca ied ou ups eam and
downs eam he Con ol Poin .
The low is subc i ical a he beginning o he canal o each, goes h ough a
c i ical dep h, and p oceeds being supe c i ical.
No e ha since he amoun o wa e inc eases along he canal, i ’s no in ui i e
o de e mine i he supe c i ical dep h is highe o lowe han he subc i ical
low occu ed some me e s ups eam.
Figu e 13. In eg a ion di ec ion i Con ol Poin Loca ed in he canal each.
3) Con ol Poin Loca ed Ups eam he Canal
In his case he C i ical dep h line eaches he T ansi ional opo ile ups eam he
each domain, his can only occu i he Ini ial Discha ge 
i
is no ze o. The
Con ol Poin is loca ed ups eam he canal, in o he wo ds. The S a ing
In eg a ion Poin is he ini ial posi ion o he canal o each and i s condi ion is
de e mined by he p e ious each in he case o wo slope canal. The low is
supe c i ical along he canal.
Figu e 14. In eg a ion di ec ion i Con ol Poin Loca ed Ups eam
C i ical Dep h Line
Gi en a canal c oss sec ion’s geome y and a ce ain discha ge, he c i ical dep h is
de ined as he wa e dep h such ha he wa e is lowing wi h he minimum speci ic
ene gy. O he canal p ope ies such as he bed slope and he oughness do no in luence
he c i ical low condi ion.
This condi ion can be exp essed as:
j
G
=0, being =-+
9



 hen
j
G
=1−
e9

8

·Y
The F oude Numbe associa ed o he low is de ined as: :k

=
e9

8

·Y
Thus,
:k= 
l3
Y
Hence, he c i ical low condi ion can be desc ibed as he wa e le el - such ha he
F oude Numbe is one. :k
?
=
9
m8

n
no
= 1
Rea anging he e ms a new exp ession ha cha ac e izes he c i ical low is ob ained:


−
8

b
=0 [15]
No e ha equa ion [15] depends on , which is a iable along he canal in Spa ially
Va ied low condi ions. In opposi ion o he G adually Va ied Flow i can be seen ha
o he SVF he C i ical dep h line is no a s aigh line pa allel o he bed slope.
No e m ega ding he bo om slope o he oughness coe icien does appea in
equa ion [15], he c i ical dep h line is independen on he bo om slope canal.
The in e ac ion be ween he ansi ional p o ile and he c i ical dep h line a e shown in
he Figu e 16:
Figu e 16. Tes Case 1.B. C i ical Dep h and T ansi ional p o ile.
This leads o a di e ence o 2.7| on he, which can be also exp essed as:
.œ
•
·100=
6.9%.
By analogy on he - axis i leads o a di e ence o 0.24|, which can be also exp essed
as:
$.—
.
·100=9.23%.
These di e ences a e no ole a ed. I ’s conside ed ha he abula solu ion o he
Con ol Poin , which is in his case he In eg a ion S a ing Poin , is a oughe
app oxima ion han he nume ical one since he disc e iza ion is done conside ing
bigge space s eps. Howe e , a u he analysis compa ing he me hod p esen ed in his
pape wi h o he echnical li e a u e is needed.
The wa e p o ile and he alue o he F oude numbe associa ed can be seen in he
g aphs ob ained om he nume ical code:
Figu e 17. Tes Case 1.A. Wa e P o ile and Bo om Slope.

Figu e 18. Tes Case 1.A. F oude Numbe .
No e ha o gua an ee s abili y when in eg a ing, i ’s needed o s a om - a li le
bi highe o lowe han -
?
=2.36|:
Ups eam sec ion: -
¡R
=-
?
·(1+0.06)=2.5|
Dows eam sec ion: -
¡R
=-
?
·(1−0.03)=2.28|
This ope a ion hough causes small dis u bances ha can be ound Figu e 17 and
Figu e 18.
Figu e 18 gi es ele an in o ma ion abou he ype o low ha occu s in each posi ion
 o he canal. Fi s he low is subc i ical, F oude <1, un il i ge s o he Con ol
Poin whe e he low ge s c i ical, F oude =1. Finally i con inues as supe c i ical low,
F oude>1, un il he end o he canal.
Tes Case 1.B.:
The case is epea ed o %
∗
=3
£

¤£
⁄
, in his case only he esul s conce ning he
In eg a ion S a ing Poin a e p esen ed since i ’s conside ed ha a u he analysis
would ge o he same conclusions as he p e ious case.
By analogy Sub amanya (1982) ob ains he In eg a ion S a ing Poin a :
In eg a ion S a ing
Poin
x
56

|
y
4

|
Table 4. Tes Case 1.B. ISP loca ion Li e a u e.
Simila esul is ob ained by using he nume ical code:
In eg a ion S a ing
Po
in
x
51
.
7

|
y
3
.
61

|
Table 5. Tes Case 1.B. ISP loca ion.
Which co espond again o he Con ol Poin loca ion.
Figu e 19. Tes Case 1.B., C i ical Dep h and T ansi ional p o ile.
Again, a signi ican di e ence is p esen ed when compa ing he  alue.
This leads o a di e ence o 4.3| on he, which can be also exp essed as:
—.
¦
·100=
7.7%.
By analogy on he - axis i leads o a di e ence o 0.39|, which can be also exp essed
as:
$.•
—
·100=12.25%.
Again hese di e ences a e conside ed oo g ea o be accep ed due o he me hod used
by Sub amanya (1982).
Tes Case 2.
Jain (2001) inds he Con ol Poin loca ion by gi ing andom alues o  and checking
i bo h sides o he equa ion ma ch by a ial an e o p ocedu e. The go e ning
equa ion is also sol ed wi h he same p ocedu e.
Jain (2001) uses he ollowing da a:
Cana
l Da a
Geome y
T apezoidal
Canal
Bo om
Wid h
ƒ
=
3

|
Tangen In
. XS
|
=
0
.
5
Bed Slope
E
i
=
0.15
Up.
Ini ial Discha ge

i
=

0

|

/
0
La e al Discha ge
%
=

4


|

/
0
/
|
C
ana
l Leng h
˜
=
122

|
Boussinesq coe icien

=
1
Manning Coe icien
K
=
0
.
015
Table 6. Tes Case 2 Da a.
Jain (2001) inds he In eg a ion S a ing Poin a :
In eg a ion S a ing
Poin
x
55

|
y
5
.
88

|
Table 7. Tes Case 2 ISP loca ion Li e a u e.
The In eg a ion S a ing Poin using he nume ical code is loca ed a :
In eg a ion S a ing
Poin
x
53
.
06

|
y
5
.
81

|
Table 8. Tes Case 2 ISP loca ion.
Figu e 20. Tes Case 2. C i ical Dep h and T ansi ional p o ile.
This leads o a di e ence o 0.04| on he coo dina e, which can also be exp essed as:
.—
¦¦
·100=2.54%.
By analogy on he - axis, i leads o a di e ence o 0.02|, which can also be exp essed
as:
$.$œ
¦.§§
·100=1.19%.
The low is Subc i ical un il i eaches he Con ol Poin a =49.94| hen i u ns
Supe c i ical.
Small di e ences can be obse ed compa ing he wo me hods, hese di e ences can be
accep ed, i can be conside ed ha bo h me hods lead o he same esul .
The wa e p o ile and he alue o he F oude numbe associa ed o each wa e le el
ca e shown:
Figu e 21. Tes Case 2. Wa e P o ile and Bo om Slope.
Figu e 22. Tes Case 2. F oude Numbe .
No e ha o gua an ee s abili y when in eg a ing, i ’s needed o s a om - a li le
bi highe o lowe han -
?
=5.81|, in o he wo ds, i ’s needed o ge away om he
c i ical dep h.
Ups eam sec ion: -
¡R
=-
?
·(1+0.01)=5.87|
Downs eam sec ion: -
¡R
=-
?
·(1−0.01)=5.75|
The low is subc i ical un il i eaches he Con ol Poin a =53.06| and hen i
u ns supe c i ical.
A Wa e P o ile Compa ison
Once he Con ol Poin in ound, Jain (2001) p oceeds wi h he wa e p o ile
compu a ion o ce ain posi ions . The esul s a e compa ed:
Values loca ed ups eam he In eg a ion S a ing Poin :
x [m]
Jain (2001
) [m]
Nume ical c
ode
[m]
Di e ence [m]
x = 0
y = 2.6
y = 2.92
0.32
x = 25
y = 4.7
y = 4.76
0.06
Table 9. Tes Case 2. Wa e Le el P o ile Compa ison, Ups eam.
Values loca ed downs eam he In eg a ion S a ing Poin :
x [m]
Jain (2001
) [m]
Nume ical code
[m]
Di e ence [m]
x = 90
y = 6.74
y = 6.75
0.01
x = L =122
y = 7.38
y
= 7.38
0.00
Table 10. Tes Case 2. Wa e Le el P o ile Compa ison, Downs eam.
Compa ing he alues ob ained by he wo me hods, i ’s obse ed ha he g ea es
di e ence is: 0.32|which can also be exp essed in a pe cen age as:
$.
.
·100=12.3%.
Tes Case 3.
The same case is p esen ed in Chow (1959) and F ench (1985). In his case a simila
abula p ocedu e used by Sub amanya (1982) is used o compu e he Con ol Poin as
well as he wa e p o ile.
The da a used in his case is:
C
ana
l Da a
Geome y
T apezoidal
Canal
Bo om
Wid h
ƒ
=
3

|
Tangen In . XS
|
=
0
.
5
Bed Slope
E
i
=
0.1505
Up.
Ini ial Discha ge

i
=

0

|

/
0
La e al Discha ge

i
=

3
.
72


|

/
0
/
|
Cana
l Leng h
˜
=
122

|
Boussinesq coe icien

=
1
Manning Coe icien
K
=
0
.
015
Table 11. Tes Case 3 Da a.
The au ho s ind he In eg a ion S a ing Poin a :
In eg a ion S a ing
Poin
x
49
.
98

|
y
5
.
39

|
Table 12. Tes Case 3 ISP loca ion li e a u e.
Simila esul is ob ained by using he Nume ical Code:
In eg a ion S a ing
Poin
x
49
.
94

|
y
5
.
41

|
Table 13. Tes Case 3 ISP loca ion.
The compu a ions a e shown in Figu e 23:
Figu e 23. Tes Case 3. C i ical dep h and T ansi ional P o ile.
This leads o a di e ence o 0.04| on he, which can be also exp essed as:
$.$—
—•.—§
·100=0.08%.
By analogy on he - axis i leads o a di e ence o 0.02|, which can be also exp essed
as:
$.$
¦.•
·100=0.37%.

Really small di e ences can be obse ed compa ing he wo me hods; bo h me hods
lead o he same esul .
The wa e p o ile and he alue o he F oude numbe associa ed can be seen in he
ollowing igu es:
Figu e 24. Tes Case 3, Wa e P o ile and Bo om Slope.
Figu e 25.Tes Case 3, F oude Numbe .
No e ha , again, o gua an ee s abili y when in eg a ing i ’s needed o s a om - a
li le bi highe o lowe han-
?
=5.41|:
Ups eam sec ion: -
¡R
=-
?
·(1+0.02)=5.52|
Downs eam sec ion: -
¡R
=-
?
·(1−0.01)=5.36|
This ope a ion is no no iced in Figu e 24 nei he in Figu e 25 because he de ia ion in
e y small, 2%.
Again he low is subc i ical un i i eaches he Con ol Poin a =49.94| o inally
low in a supe c i ical egime.
Wa e P o ile Compa ison
Once he In eg a ion S a ing Poin in ound, Chown (1959) p oceeds wi h he wa e
le el compu a ion. The esul s a e compa ed.
Values loca ed ups eam he In eg a ion S a ing Poin :
x [m]
Chow (1959
) [m]
Nume ical code
[m]
Di e ence [m]
x = 0
y = 2.
62
y = 2.70
0.08
x = 3
y = 3.01
y = 3.05
0.04
x = 7.6
y = 3.43
y = 3.47
0.04
x = 15.3
y = 3.95
y = 4.00
0.05
Table 14. Tes Case 3. Wa e Le el P o ile Compa ison, Ups eam.
Values loca ed downs eam he In eg a ion S a ing Poin :
x [m]
Chow (1959
) [m]
Nume ical c
ode
[m]
Di e ence [m]
x = 60.9
y = 5.68
y = 5.72
0.04
x = 76.2
y = 6.07
y =
6.12
0.05
x = 91.5
y = 6.41
y = 6.44
0.03
x = 106.8
y = 6.72
y = 6.75
0.03
x = L = 122
y = 6.96
y = 7,02
0.06
Table 15. Tes Case 3. Wa e Le el P o ile Compa ison, Downs eam.
Compa ing he alues ob ained by he wo me hods, i ’s obse ed ha he g ea es
di e ence is: 0.06|which can also be exp essed in a pe cen age as:
$.$
œ.$
·100=0.85%.
I is a eally small di e ence, he nume ical code compu es he same app oxima ion as
Chow (1959).
Discussion
Once he esul s implemen ing he Tes Cases on he nume ical code a e compa ed
wi h he esul s ound by he echnical li e a u e, i can be concluded ha he
nume ical code wo ks co ec ly and i inds a good app oxima ion o he ODE ha
go e ns he Inc easing SVF.
Howe e , some sensi i i y is de ec ed when unning he code in Ma lab ega ding he
ini ial app oxima ions used o ze o unc ions.
All he cases shown un il now ep esen he same ype o open canal: a apezoidal
canal wi h a g ea slope and a leng h o 100 m o mo e.
Hence o h he nume ical code is used o compu e new cases no ound in li e a u e
ha ep esen di e en kind o si ua ions wi h ci il enginee ing applica ions.
VI. WATER PROFILE: ONE SLOPE CASES
Small bed slope canal
He e i ’s conside ed he case whe e he bo om slope ,So, is such ha he Con ol Poin
is loca ed downs eam he gi en canal, i.e.
¡R
>˜ . Hence, he S a ing In eg a ion
Poin is loca ed a 
¡R
=˜ and he in eg a ion can be ca ied ou ups eam co e ing
he whole canal leng h.
Fo his case Tes Case 3 is modi ied by diminishing E
$
so ha he condi ion
men ioned abo e is ul illed. All he o he Inpu s a e conside ed he same as Tes Case
3.
The da a in his case is:
Canal
Da a
Geome y
T apezoidal
Canal
Bo om Wid h
ƒ
=
3

|
Tangen In . XS
|
=
0
.
5
Bed Slope
E
i
=
0.09
Up.
Ini ial Discha ge

i
=

0

|

/
0
La e al Discha ge
%
i
=

3
.
72


|

/
0
/
|
C
anal
Leng h
˜
=
122

|
Boussinesq coe icien

=
1
Manning Coe icien
K
=
0
.
015
Table 16. Small Slope Case Da a.
In eg a ion S a ing Poin loca ion:
In eg a ion S a ing
Poin
x
122

|
y
8
.
56

|
Table 17. Small Slope Case ISP loca ion.
Figu e 26. Small Slope Case C i ical dep h and T ansi ional p o ile.
I can be seen ha he Con ol Poin would be loca ed downs eam ˜=122|, bu
since he canal is sho e , he in eg a ion is s a ed a =122|. . I is conside ed ha
he ou le is ee, he wa e le el eaches he C i ical dep h a he end o he canal.
The wa e p o ile is compu ed in eg a ing ups eam he In eg a ion S a ing Poin . The
esul ing wa e p o ile as well as he alue o he F oude numbe associa ed o each
wa e le el can be seen in he g aphs ob ained om he nume ical code, Figu e27 and
Figu e 28:
Figu e 27. Small Bed Slope. Wa e P o ile and Bo om Slope.
Figu e 28. Small Bed Slope Case 4. F oude Numbe .
The low is Subc i ical along he en i e canal and he F oude numbe inc eases
con inuously in he low di ec ion eaching he alue o 1 a ˜=122|.
Fla slope canal
Finally o conclude wi h he T apezoidal Canals an ex eme case whe e he bed slope is
ze o is p esen ed he e.
Fo his case Tes Case 3 is modi ied using E
$
=0. All he o he inpu s a e conside ed
he same as Tes Case 3.
The g ey cells co espond o impossible si ua ions which a e b ie ly explained while he
possible cases a e co e ed in de ail.
Fo he ollowing cases some no a ion needs o be men ioned o a good unde s anding
when looking a he esul s p esen ed in his documen :
Reach 1. Subc i ical Flow.
Reach 1. Supe c i ical Flow.
Reach 2. Subc i ical Flow.
Reach 2. Supe c i ical Flow.
Bo om Slope Canal.
Table 22. Two slopes. Main legend.
Con ol Poin igu es as well as F oude numbe igu es a e no p esen ed in his
sec ion, i is conside ed ha a his s age o he p oceedings he low cha ac e is ics can
be unde s ood wi hou his suppo g aph. Howe e , bo h igu es can be ound in
Appendix 4 in needed.
A complemen a y legend is he e shown which is used in some o he cases p esen ed in
his sec ion, when a Hyd aulic Jump needs o be compu ed:
Reach 2. Supe c i ical Flow
om R1.
Reach 1
and 2
. Subc i ical
Flow om R2.
Reach
1 and
2. Conjuga ed
Supe c i ical Flow om R2.
Tabla 23. Two slopes. Ex a wa e p o iles o Hyd aulic Jump legend.
Ini ial Case
Table 24 shows he da a used o compu e he Ini ial Case.
C
anal
Da a
Geome y
Rec angula
Canal
Wid h
ƒ
=
4
|
Bed Slope
E
i
=
0.05
Up.
Ini ial Discha ge

i
=

0

|

/
0
La e al Discha ge
%
∗
=

0
.
5


|

/
0
/
|
C
ana
l Leng h
˜
=
100
|
Boussinesq coe icien

=
1
Manning Coe icien
K
=
0
.
015

Table 24. Two slope Ini ial Case da a
The Ini ial Case consis s simply on a subc i ical low along he en i e canal, he
Con ol Poin is loca ed downs eam he end o he canal.
Hence he In eg a ion S a ing Poin is loca ed a :
In eg a ion S a ing Poin
x
100
m
y
2.52 m
Table 25. Two slope Ini ial Case ISP loca ion.
In eg a ing he canal go e ning equa ion ups eam he In eg a ion S a ing Poin he ollowing
wa e p o ile is ob ained:
Figu e 35. Two slope Ini ial Case Wa e P o ile
F om his case he ollowing cases a e de i ed by modi ying he bo om slope.
The canal is di ided in wo equal eaches 50 m long.
Case 1
Fi s ly he slope o he Reach 2 is dec eased. The ollowing esul s a e hus expec ed:
- The Con ol Poin 1 posi ion is expec ed o be ound downs eam Reach 1 since
Reach 1 can be seen as a sho e e sion o he Ini ial Case.
- Meanwhile, he posi ion o Con ol Poin 2 is expec ed o be ound downs eam
since Reach 2 can be seen as a sho e and la e e sion o he Ini ial Case.
The da a used o his case is:
Canal
Da a
Geome y
Rec angula
Canal
Wid h
ƒ
=
4

|
Bed Slopes
E
i
1
=
0.05
E
i
2
=
0.01
Up.
Ini ial Discha ge

i
=

0

|

/
0
La e al Discha ge
%
∗
=

0
.
5


|

/
0
/
|
C
ana
l Leng h
˜
=
100
|
Boussinesq coe icien

=
1
Manning Coe icien
K
=
0
.
015
Table 26. Two slope Case1 da a.
The In eg a ion S a ing Poin s a e ound a :
In eg a ion S a ing
Poin s
x1
50 m
y1
-
x2
100 m
Y2
2.52 m
Table 27. Two slope Case1 ISP loca ions.
Rema k ha no in o ma ion is gi en ega ding he wa e le el - a =50|, he
posi ion o In eg a ion S a ing Poin 1. This is so since he wa e le el a his posi ion
is no -
?
as i would be i i Reach 1 was an independen canal. The wa e le el is gi en
by he in eg a ion o he wa e p o ile in Reach 2. In o he wo ds, Reach 1 wa e le el
depends on Reach 2, he low along he canal depends on a bounda y condi ion ound
a he end o he canal.
The in eg a ion o he en i e canal is done in he ups eam di ec ion.
Hence he wa e p o ile is:
Figu e 36. Two slope Case1 Wa e P o ile.
The low is subc i ical along he whole canal.
Case 2
This case is he opposi e case o Case 1, he slope o Reach 1 is dec eased while he
slope o Reach 2 is kep equal o he Ini ial Case’s slope.
By analogy o Case 1, he same assump ions can be made. The canal is expec ed o be
subc i ical in all i s leng h since he only modi ica ion done om he Ini ial Case is
dec easing he slope o he each o he canal.
The da a used o his case is:
Cana
l Da a
Geome y
Rec angula
Canal
Wid h
ƒ
=
4
|
Bed
Slopes
E
i
1
=
0.01
E
i
2
=
0.05
Up.
Ini ial Discha ge

i
=

0

|

/
0
La e al Discha ge
%
∗
=

0
.
5


|

/
0
/
|
C
ana
l Leng h
˜
=
100
|
Boussinesq coe icien

=
1
Manning Coe icien
K
=
0
.
015
Table 28. Two slope Case 2 Da a.
The in eg a ion S a ing Poin s a e again ound a :
In eg a ion S a ing
Poin s
x1
50 m
y1
-
x2
100 m
Y2
2.52 m
Table 29. Two slope Case2 ISP loca ions.
Again he low is subc i ical along he canal and he wa e le el a he T ansi ion Slope
Poin depends on he downs eam condi ion. Al hough he In eg a ion S a ing Poin is
he same, he Wa e P o ile akes a di e en shape:
Figu e 37. Two slope Case 2 Wa e P o ile.
Case 3.1.
The p e ious case, Case 2, is sligh ly modi ied by inc easing he bo om slope o Reach
2 such as he Con ol Poin 2 is now loca ed a Reach 2.
In his case he da a used is:
C
ana
l Da a
Geome y
Rec angula
Cana
l
Wid h
ƒ
=
4
|
Bed Slopes
E
i
1
=
0.01
E
i
2
=
0.06
Up.
Ini ial Discha ge

i
=

0

|

/
0
La e al Discha ge
%
∗
=

0
.
5


|

/
0
/
|
C
ana
l Leng h
˜
=
100
|
Boussinesq coe icien

=
1
Manning Coe icien
K
=
0
.
015
Table 30. Two slope Case 3.1. Da a.
The in eg a ion S a ing Poin s a e loca ed a :
In eg a ion S a ing
Poin s
x1
50 m
y1
-
x2
75.9
m
Y2
2.
09
m
Table 31. Two slope Case 3.1. ISP loca ions.
The in eg a ion is also s a ed a Reach 2, ups eam and downs eam he In eg a ion
S a ing Poin 2, which is in his case Con ol Poin 2. A e ha he in eg a ion can
be con inued in Reach 1 using as he In eg a ion S a ing Poin (bounda y condi ion)
he wa e le el ound om Reach 2.
The wa e p o ile is hen:
Figu e 38. Two slope Case 3.1. Wa e P o ile.
The low is subc i ical un il i eaches he Con ol Poin 2 a =76.3| o go on as a
supe c i ical low.
Case 3.2.
The same si ua ion could be gi en being Reach 1 slope highe han Reach 2 Slope.
Howe e , he ange o slope alues ha gi e such si ua ion is e y na ow. One o hese
cases is:
C
ana
l Da a
Geome y
Rec angula
Canal
Wid h
ƒ
=
4
|
Bed Slopes
E
i
1
=
0.065
E
i
2
=
0.057
Up.
Ini ial Discha ge

i
=

0

|

/
0
La e al Discha ge
%
∗
=

0
.
5


|

/
0
/
|
C
ana
l Leng h
˜
=
100
|
Boussinesq coe icien

=
1
Manning Coe icien
K
=
0
.
015
Table 32. Two slope Case3.2. Da a.
The Posi ions o he In eg a ion S a ing Poin s a e:
In eg a ion S a ig
Poin s
x1
50 m
y1
-
x2
91 m
Y2
2.36 m
Table 33. Two slope Case 3.2. ISP loca ions.
In his case, i he en i e canal would ha e Reach’s 1 Slope he Con ol Poin would be
loca ed be ween he T ansi ion Slope Poin and he end o he canal. In o he wo ds
Con ol Poin 1 is loca ed along Reach 2 and so he In eg a ion S a ing Poin 1 is he
T ansi ion Slope Poin .

Reach 2 slope is e y simila o Reach 1 slope and so Con ol Poin 2 is loca ed a li le
bi downs eam he supposed Con ol Poin 1, bu s ill in Reach 2.
The compu a ions o his case would be e y simila o he ones o Case 3.1.
Case 4
He e he p e ious case, Case 3.1., is sligh ly modi ied by inc easing he bo om slope o
Reach 2 such as he Con ol Poin 2 in ound ups eam he T ansi ion Slope Poin .
In Case 4 he da a used is:
Canal Da a
Geome y
Rec angula
Cana
l
Wid h
ƒ
=
4
|
Bed Slopes
E
i
1
=
0.01
E
i
2
=
0.07
Up. Ini ial Discha ge

i
=

0

|

/
0
La e al Discha ge
%
∗
=

0
.
5


|

/
0
/
|
Canal
Leng h
˜
=
100
|
Boussinesq coe icien

=
1
Manning Coe icien
K
=
0
.
015
Table 34. Two slope Case 4 Da a.
In eg a ion S a ing Poin s 1 and 2 a e loca ed a :
In eg a ion S a ing
Poin s
x1
50 m
y1
1.59 m
x2
50 m
Y2
1.59 m
Table 35. Two slope Case4 ISP loca ions.
The low in he whole Reach 2 in hen supe c i ical while he low in Reach1 is
subc i ical. Tha is o say, he In eg a ion S a ing Poin 1 and 2 a e he same poin ,
he T ansi ion Slope Poin . This si ua ion is possible since Reach 1 is la enough so
ha Con ol Poin 1 would be loca ed downs eam he T ansi ion Slope Poin while
Reach 2 is s eep enough so Con ol Poin 2 would be loca ed ups eam he T ansi ion
Slope Poin .
In he T ansi ion Slope Poin he wa e le el is known, i is he C i ical Dep h -
?
, in
his pa icula case -
?
=1.59|.
The e o e he wa e p o ile is compu ed ups eam (Reach 1) and downs eam (Reach 2)
his In eg a ion S a ing Poin :
Figu e 39. Two slope Case4 Wa e P o ile.
Case 5.1.
In his case he Ini ial Case is modi ied so ha he loca ion o Con ol Poin 1 is
loca ed along Reach 1, so ha i co esponds o he In eg a ion S a ing Poin 1. This
can be done by inc easing he bo om slope. The In eg a ion S a ing Poin 2 is kep a
he end o he canal; in his case is chosen a la e slope han he Ini ial Case’s slope
so ha he esul s a e clea e .
The ollowing esul s a e expec ed:
- Reach 1 is in eg a ed ups eam and downs eam In eg a ion S a ing Poin 1.
- Reach 2 is in eg a ed ups eam he In eg a ion S a ing Poin 2.
The e o e bo h wa e p o iles do no ma ch a he T ansi ion Slope Poin and a u he
analysis needs o be ca ied ou . Fo his eason an ex a igu e is shown o make he
easoning clea e . The wa e p o ile is shown, and once he conclusion is ob ained, he
bo om slope is shown o a gene al idea o he whole p oblem.
The da a used o his case:
Canal Da a
Geome y
Rec angula
Cana
l
Wid h
ƒ
=
4
|
Bed Slopes
E
i
1
=
0.09
E
i
2
=
0.04
Up. Ini ial Discha ge
%
∗
=

0

|

/
0
La e al Discha ge

i
=

0
.
5


|

/
0
/
|
Canal Leng h
˜
=
100
|
Boussinesq coe icien

=
1
Manning Coe icien
K
=
0
.
015
Table 36. Two slope Case 5.1. Da a.
The In eg a ion S a ing Poin s a e ound a :
In eg a ion S a ing
Poin s
x1
20
.
11
m
y1
0.86
m
x2
100 m
Y2
2.52 m
Table 37. Two slope Case5.1. ISP loca ions.
The wa e p o ile conside ing each each is hen:
Figu e 40. Two slope Case5.1. Independen Wa e P o iles.
I can be seen ha wa e p o iles om bo h eaches do no ma ch a all. Mo eo e , he
low ge s supe c i ical in Reach 1 while he low in Reach 2 is subc i ical. I can be
deduced hen ha a Hyd aulic Jump is occu ing in o de o ma ch he condi ions om
bo h eaches, he Hyd aulic Jump is he hyd aulic solu ion o o e come his
discon inui y.
The posi ion o he Hyd aulic Jump can be ound in Reach 1 o in Reach 2.
Hyd aulic Jump in Reach 2
Fi s ly i is assumed ha he loca ion o he Hyd aulic Jump is in Reach 2:
- The conjuga e dep h line o he wa e p o ile in Reach 2 is compu ed, which
co esponds o a supe c i ical low.
- A new wa e p o ile is compu ed in Reach 2 using as he In eg a ion S a ing
Poin he wa e le el a he T ansi ion P o ile gi en by Reach 1, since he
In eg a ion S a ing poin co esponds o a supe c i ical low, he wa e P o ile
compu ed is also in supe c i ical low condi ions.
Hence, i he Hyd aulic Jump does exis in Reach 2, i is loca ed whe e bo h lines c oss.
The ollowing igu e shows he jus men ioned compu a ions:
Figu e 41. Two slope Case5.1. Hyd aulic Jump in R2.
In Figu e 40 i can be seen ha he wa e p o ile compu ed om Reach 1, he g een
line, su e s some ins abili ies, so his beha io lacks o physical sense. This senseless
endency can be explained as ollows: he wa e le el inc eases un il eaching he
C i ical Dep h (a =53.1|) om ha poin he in eg a ion loses s abili y.
The de i a e is
FG
F*
→∞ , his can be easily seen in equa ion [16], when =
?
he
denomina o is ze o and so he de i a e ends o ∞. Taking a ca e ul look o he
Runge-Ku a Nume ical Me hod i can be seen ha once
FG
F*
→∞, hen 
•
→∞, and
ha ’s he eason why he in eg a ion loses i s physical sense.
S a ing om he supe c i ical condi ion gi en by Reach 1 he low has he endency o
inc ease un il i eaches he c i ical condi ion, i wouldn’ ha e any physical sense ha
he low would u n mo e supe c i ical in o de o c oss he conjuga e line so ha he
Hyd aulic Jump could occu .
Thus he Hyd aulic jump is no loca ed in Reach 2.
Hyd aulic Jump in Reach 1
The Hyd aulic Jump is hen sough is Reach 1 using he same p ocedu e:
- The wa e p o ile om Reach 2 is leng hened in Reach 1. This is done
in eg a ing ups eam using he condi ion gi en by he wa e p o ile in Reach 2,
which is a subc i ical low.
- The conjuga e o his new wa e p o ile is compu ed, ha co esponds o a
supe c i ical low.
Hence, i he Hyd aulic Jump does exis in Reach 1, i is loca ed whe e he conjuga e
wa e p o ile c osses he supe c i ical wa e p o ile om Reach 1. The ollowing able
shows he jus men ioned compu a ions:
Case 8
A simila case o Case 7 in shown he e, he same cha ac e is ics a e sough while
pe mu ing he bo om slopes.
In his Case:
Canal Da a
Geome y
Rec angula
Cana
l
Wid h
ƒ
=
4
|
Bed Slopes
E
i
1
=
0.07
E
i
2
=
0.09
Up. Ini ial Discha ge
%
∗
=

0

|

/
0
La e al Discha ge

i
=

0
.
5


|

/
0
/
|
Canal Leng h
˜
=
100
|
Boussinesq coe icien

=
1
Manning Coe icien
K
=
0
.
015
Table 46. Two slope Case 8 Da a.
The In eg a ion S a ing Poin s a e loca ed a :
In eg a ion
S a ing
Poin s
x1
44.94 m
y1
1.47 m
x2
50 m
Y2
-
m
Table 47. Two slope Case 8 ISP loca ions.
No e ha he same easoning is used o jus i y he lack o in o ma ion ega ding he
wa e le el in In eg a ion S a ing Poin 2, i s alue depends on he ups eam low, in
Reach 1.
The wa e p o ile is shown in Figu e 51:
Figu e 51. Two slope Case 8 Wa e P o ile wi h bo om slope.

As well as Case 7, in his case he low is subc i ical he i s me e s o Reach 1 un il i
eaches he Con ol Poin 1 and i u ns supe c i ical.
Impossible Cases
Finally he g ey cells ha ep esen impossible cases ha canno occu in eali y a e
b ie ly p esen ed in he nex sec ion.
I-1
Being So1>So1, his case would ep esen a Subc i ical low in Reach 1 and a
Supe c i ical low in Reach 2. This would be an ex eme case o Case 3.2. , wi h
Con ol Poin 1 loca ed a he T ansi ion Slope Poin and Con ol Poin 2 also loca ed
a he T ansi ion Slope Poin , so So1=So2, which is no conside ed he e since i ’s
supposed So1>So2.
I-2
Being So1<So1, his case would ep esen a Subc i ical-Supe c i ical low in Reach 1
and a Subc i ical low in Reach 2.
Assuming ha he low is Subc i ical-Supe c i ical is Reach 1, a s eepe slope, So2,
would lead o a mo e Supe c i ical low, he e is no chance ha gi en hese condi ions
a Subc i ical low could occu in Reach 2.
I-3
Being So1<So1, his case would ep esen a Subc i ical-Supe c i ical low in Reach 1
ollowed by a Subc i ical-Supe c i ical low in Reach 2.
Again, assuming he low is Subc i ical-Supe c i ical is Reach 1, a s eepe slope, So2,
would lead o a mo e Supe c i ical low whe e i s Con ol Poin would be loca ed
ups eam he T ansi ion Slope Poin .
VIII. CONCLUSIONS AND FUTURE DEVELOPMENT
This documen inds an app oxima e solu ion o he wa e p o ile o one slope and
wo slope canals o a spa ially a ied low wi h an inc easing discha ge. The p ocedu e
has been kep as gene al as possible.
The nume ical code is eady o be implemen ed on any kind o geome y c oss-sec ion,
he only s ep ha should be ca ied ou would be w i ing he unc ion ha exp esses
he a ea, he canal wid h and he we pe ime e in e ms o wa e le el.
The nume ical code has been success ully alida ed using some cases ound in echnical
li e a u e.
The same s uc u e could be used o analyze he spa ially a ied low wi h dec easing
discha ge. In his case some assump ions ega ding he amoun o ou low a e should
be s udied and inally he nume ical code could be modi ied o ind he wa e p o ile.
The nume ical code used o sol e he O dina y Di e en ial Equa ion wo ks co ec ly
despi e o he g ea sensi i i y shown ega ding i s app oxima ions. The nume ical
code could be imp o ed o a oid such incon eniences. Howe e , i canno be o go en
ha he nume ical code is conside ed a ma hema ical ool o sol e a physical p oblem,
hus he code i sel is no conside ed as he main objec i e o his documen .
This documen can be seen as a kick-o s udy o he spa ially a ied low wi h
inc easing discha ge. The gene al app oach o wo slope cases can be conside ed he
i s s ep o analyze mul iple slope cases wi h his ype o low. The in e ac ion
be ween bo h eaches is desc ibed and i s possible implica ions a e ca e ully explained.
A u he s udy migh ind he app oxima e solu ion o he wa e p o ile o a K−slope
canal. The challenge would be ound in w i ing he code ha conside s independen ly
K−1 in e ac ions o la e compu e a inal wa e p o ile. All physical concep s could be
ex ac ed om his documen as he p oblem was deal as gene al as possible.
REFERENCES
- Sub amanya, K.(1982). Flow in Open Channels. Ta a McG aw-Hill Publishing
Company Limi ed, New Delhi
- F ench,R. (1985). Open-Channel Hyd aulics. McG aw-Hill Publishing, New Yo k.
- Jain, S. 2001. Open-Channel Flow. Uni ed S a es o Ame ica: John Wiley & Sons, Inc,
New Yo k.
- Chow, V. (1959). Open-Channel Hyd aulics. McG aw-Hill Book Company Limi ed,
New Yo k.
- Jeppson, R. (2011). Open Channel Flow. CRC P ess, USA.
- Mon es, S. (1998). Hyd aulics o Open Channel Flow. ASCE P ess, USA.
- Smi h,K (1867). Con ol Poin in la e al spillway cannel. ASCE P ess, USA.
- May R. e al.(2003). Hyd aulic designs o side we is. Thomas Tel o d, London.
APPENDICES
APPENDIX 1. Con ol Poin compu a ion.
In his appendix he in e sec ion o T ansi ional P o ile and C i ical Dep h Line is
desc ibed, his compu a ions using Ma lab equi e a ca e ul look.
Ma lab doesn’ deal wi h explici exp essions, he e a e wo ways o ind he
in e sec ion be ween his wo unc ions: by plo ing hem disc e ely and inding an
app oxima ion o he in e sec ion poin (,) o by inding he wo adjacen poin s
be ween which he in e sec ion occu s and compu ing he in e sec ion poin as a poin
be ween he wo adjacen poin s
The second me hod is used in he nume ical code w i en. No e ha one o he
equa ions deal wi h, equa ion [xTP] exp esses  = (). Fo his eason he
disc e iza ion is conside ed using ∆ s eps.
De ining a ce ain ∆, wo ec o s 

and 

a e compu ed, which co espond o he
alue o  equa ions [xTP] and [CD] o di e en alues o .
A  ec o is c ea ed wi h all he alues o  whe e he compu a ion is ca ied ou .
The Di e ence ec o () = 

()−

() is compu ed as well as he p oduc ec o :
()= ()·( +1), whe e  deno es a gene ic componen o  ec o .
No e ha he ela i e posi ion o he wo unc ions is no know in ad ance, in o de o
keep gene ali y, he sign o he p oduc o he di e ences is he de e minan alue in
opposi ion o he alue o he di e ence i sel .
The i s  whe e his di e ence changes i s sign, i.e. he i s nega i e coo dina e o 
ec o , co esponds o 

. The in e sec ion has al eady occu ed. Figu e below
illus a es he si ua ion desc ibed.
Then wo new a iables a e de ined: 

=

and 

=

The 

is de ined as 

=
!
"#$
!
%&'
(
)
, 

(

) and 

(

) a e hen compu ed.
Then 

= 

(

)−

(

) is compu ed.
- I he absolu e alue o 

< + ,being + a ce ain ole ance p e iously de ined,
he compu a ions a e s op and i is conside ed he in e sec ion o be ound a


.
In his pape i is used + = 0.05/
- I 

·

< 0, hen 

=

, since he in e sec ion is ound be ween


and 

.
- I 

·

> 0, hen 

=

, since he in e sec ion is ound be ween


and 

.
In he case whe e 

≥ + a i e a ion p ocess is s a ed using he new de ini ion o


o 

un il he condi ion is ul illed.

APPENDIX 2. “U” Shaped A ea Compu a ions.
The A ea and he Wid h a e compu ed o an “U” shaped canal.
The c oss-sec ion o his canal can be analyzed as a ec angle plus a semi
ci cum e ence.
A ea Compu a ion
Fo  < 2 :
The main challenge o compu e he a ea o semici cula bo om elies on
exp essing he a ea in e ms o , he wa e dep h.
Fi s he a ea is exp essed in e ms o 3. F om igu e below and conside ing
ha he a ea o he low sec ion is he a ea o he sec ion minus he a ea o he
iangula po ion: 4 =
)2)·23−
)22·673 ·28963
The p e ious equa ion can be ew i en in e ms o  because:
8963 = :;!
;< which lead o 3 = =>8896:;!
;<
Finally: 4 = 
)2):2·=>8896:;!
;<−sin(2·=>8896:;!
;<)<
Fo  ≥ 2:
The a ea can be simply exp essed as: 4 = B;C
)+22 ·(−2)
Wid h Compu a ion
Fo  < 2 :
The wid h (D) can be compu ed in e ms o 3 : D = 22 ·673
Then: D = 22 ·sin(=>8896:;!
;<)
Fo  ≥ 2:
The wid h is: D = 22
APPENDIX 3. Main ins uc ions o he nume ical code.
APPENDIX 4. Con ol Poin Loca ions and F oude Numbe o
Two Slope Cases.
Case 1
Case 2