(1.1) INTRODUCTION
Critical depth is an important parameter in open channel hydraulics. It can be
defined in four ways, each assuming hydrostatic pressure distribution (Heggen, 1991):
I. The depth at which final velocity equals velocity (celerity") of a gravity
wave of infinitesimal amplitude in a wide channel with a uniform
velocity profile (constant velocity at all depths) .... When water is flowing
at critical depth Yc, the flow velocity will be Vc ( = .Yg.Yc) and the flow
velocity and wave celerity will be almost equal in magnitude provided
1'1 YIY is very small (Daugherty et. al., 1985).
Where, llY = Wave height
·wave celerity is the speed at which an infinitesimally small wave travels in a fluid
(Roberson & Crowe, 1993). It can be computed by applying the basic equations (Fox &
Mcdonald, 1995)
• 14 •
For example, if a stone were dropped into still water, waves would
move with the velocity = ...Jg.Yc. This equation, however, should not be
applied to ocean waves or waves in very deep channels (Binder, 1962).
2. The depth at which specific energy and specific force, neither
corrected for velocity profile, are minimized.
3. The depth at which specific energy corrected for velocity profile
is minimized.
4. The depth at which specific force corrected for velocity profile IS
minimized.
For most designs, the profile-corrected specific-energy minimisation is the
definition of Y c.
Determination of critical depth for the g1ven value of discharge is often of
paramount importance to an engineer for prudent hydraulic design due to several reasons,
such as:-
• The relation of critical and normal depth determines the appropriate
family of gradually varied flow profiles.
• Critical depth represents the dividing point between the two entirely
different flow regimes: supercritical and subcritical.
• Wave problems occur in flow near critical depth (Heggen, 1991). The
U.S. Army Corps of Engineers (1970) suggests that flow is prone
toward instability when depth is (1.0 ± 0 l)Yc, Yc being without profile
correction.
• Control in many hydraulic structures is established at critical conditions.
• Critical depth can also serve as the characteristic length for studying
problems such as backwater profiles (Chen & Wang, 1969)
For non-exponential sections, there are no computational difficulties in solving
critical depth equations when the discharge is unknown However. when the cross-section
- 15 -
is unknown, the solution cannot be found explicitly. Simplified solutions, therefore, in the
form of rapidly converging iterative scheme have been devised so as to cut down the
rigorous supplementary mathematical operations which would otherwise be required for
available trial & error computations. This chapter, as such, will attempt at developing &
analysing alternative iterative procedures for quickly and accurately solving the implicit
problem of determining the critical-depth in circular sections (Section 1.2), compound
channel sections (Section 1.4), and standard-lined canals (Section 1.5). No iterative
solution has been presented in this report for trapezoidal channels because Pillai et. al.
(1989) have already given one (Annexure IV) and Author has not yet been able to make it
even better. Similarly, development of any algorithm is unwarranted for parabolic
channels because critical depth in such channels can be found explicitly.
Conditions have been developed that guarantee that each iterative procedure will
converge to a unique solution. The test results show that the iterative procedures
presented here meet the requirements of guaranteed convergence and computational
efficiency (speed & accuracy-wise). This would be helpful in contributing directly to the
development of computer simulation models by providing an efficient algorithm with
guaranteed convergence for critical-flow depth in circular, standard-lined, and compound
channels when necessary. Proposed algorithms can also be used over a simple
programmable calculator.
An alternative explicit mathematical solution for quickly solving the implicit
problem of critical depth calculations in circular sewers has also been developed in
Section ( 1 3). The test results will undisputedly show that the solution meets the
requirements (of computational efficiency with reasonable degree of accuracy) more
- 16-
effectively than that of available ones. This explicit solution can also be used as a starting
point for the iterative solution so as to reduce the number of iterations (Rathore, 1993 &
1998).
A Spatially Varied Flow (SVF) model for determining the location of critical-flow
section in a collector channel has been presented in Section (1.6). The analysis deals with
the case where the discharge increases downstream resulting in a change of flow regime
from subcritical to supercritical in an SVF. However, the critical flow section serves as a
control section.
• ••
- 17-
(1.2) CIRCULAR CHANNELS (ITERATIVE SOLUTION)*
Use of circular sewers is not uncommon for waste-water conveyance;
consequently it has drawn the attention of so many researchers. Two-parameter
relationship for the determination of critical-depths in circular channels was evolved by
Rajratnam & Thiruvengadam (1961). Swamee (1972) presented non-dimensionalised
tabular solutions for the two-parameter-relationship evolved by Rajratnam &
Thiruvengadam (I 961) for the determination of critical-depths in circular channels.
Subramanya (1972) reports an approximate solution for the same. Achyuthan (1974)
expressed critical-depth in terms of specific energy. Although the concepts behind all the
above methods are still valid, there is an imperative need for replacing these particular
approaches by simplified iterative solution.
• This solution is a composite and modified version of the papers published by Rathore
(1993), Rathore & Sen (1997b), and Rathore (1998a).
- 18-
Simple bisection method requires a larger code and hence not time-effective. Hence,
this method is not particularly recommended when greater computational accuracy is
needed. Rathore (1993) developed rapidly converging iterative solution which was
subsequently modified by Rathore (I 998a). Rathore & Sen (1997b) presented an
explicit solution for the same which reduces average absolute errors down to half when
compared to the solution presented by Hager (1991). Several trial & error solutions have
also been reported by Rathore (1992). Several other graphical solutions have also come
out of the efforts made in the directions of critical flow computations in circular channels.
Numerical Analysis
The critical depth in an open channel is determined for a given channel discharge Q
by solving the following general equation:
T g
or,
(iT Yc Yc (-f3
--- ........................................................ ............ (1.2.1) g A
Depending upon the relative values of radius (r) and discharge (Q), either of the
two conditions may possibly exist i e. Y c :;:: r or Y c :S r (Figures 1.2.1 & 1.2.2). When
circular channel flows half full under critical condition, corresponding critical discharge
can be computed by substituting:
A= and T = 2r 2
- I 9-
in equation (1.2.1) which gives,
Q, = 'If rsg
.y ( --------) 16
Or, in tenns of a constant "K.t which will be introduced later in this section,
condition Yc = 1 may be expressed as:
Therefore, if Q > Qh (or K. > V7d2), corresponding Y, is greater than r and vice-
versa. Top-width increases upto Y, :'0 r, then starts decreasing. In order to develop
conditions that guarantee that the iterative procedure will converge to the unique solution,
requisite iterative scheme for both these conditions may be developed separately as
hereunder:
(i) Y :S r (Q:S Qh or Y :S 1 or K4 :S .Yn/2):
From the geometry of figure (1.2.1), one gets:
....................................................... (1.2.2)
T = 2.r. Sin (<!>1) ............ ' ................................................. (1.2.3)
where,
-----------------------(r-Y c)
By substituting the values of 'A' and 'T' from equations (1.2.2) & (1.2.3)
respectively in equation ( 1.2. I), one gets·
- 20-
KLK2 y c ~ --------- ......................................................................... (1.2.4)
K3
where,
2 rQ2
(--------) 113
g
and,
r K3 = ---- .<j>J - (---- - l).r. Sin (<j>I)
Equation (1.2.4) is the requisite iterative scheme which can conveniently be used
upto any desired degree of convergence tolerance with an initial depth guess of Yc = r
when Q < Qh. It is to be noted that 'K1' is a lumped coefficient as it is required to be
calculated only once.
Non-dimensionalized version of equation (1.2.4) while preserving its converging
behavior can also be presented in the following form:
where,
2Q2 K. = ( -------) 116
g rs
' .. ' .. ' ................................................... ' ............ (12.5)
- 21 -
I K,; = ---- [ljl, - (1 -Yc) . Sin (ljl,)]t/2
Yc
and,
Tan·' ,I{Yc (2- Yc)} 4> I = -------------------------
(I - Yc)
ii) Yc > R (Q > Qh or Xc > 1 or K4 > ,ln;/2):
From the geometry of the Figure (1.2.2), one gets:
A= n.~- r.lj>z +(Y-R). r. Sin (<l>z) ........................................................... (1.2.6)
T "' 2.r. Sin 4>2 ...................................................... (1.2.7)
By substituting the values of 'A' and 'T' from equations (1.2.6) & (1.2.7)
respectively in equation (1.2.1), one gets:
Yc= --------... ( 1 2 8) ......................................................................................
Ks
where,
r
K, = (n- 4>z). -----+(I - -----).r.Sin 4>z Yo Y,
Equation (1.2.8) is the requisite iterative scheme which can be used with an initial
depth guess of Y, = r when Q > Ot,.
- 22-
Equation ( 1.2.8) can also be presented in its non-dimensionalized from as shown
hereunder:
Yc =
Where,
Tan·1 v{Yc (2- Yc)} Ql2 = --------------------------
(Yc- 1)
I
Kw = --- [1t- Ql2 + (Yc- !).Sin (cjJ2)] !6 Yc
Initial Flow-Depth Guess
................................... (1.2.9)
The number of iterations required for convergence up to desired degree of tolerance
is a measure of the numeric procedure speed. It was found during the test runs that the
speed (Table 1.2. I & 1.2.2) of the iterative procedure is quite sensitive to the starting
point necessitating thereby an appropriate well defined trial depth guess so as to optimise
the efficiency of the proposed computer simulation model.
Numeric procedure was found to converge for all the values of initial relative trial
depth (Yctd) in the range 0 < Yc :S I for condition I (i.e Yc :S 1, K.. :S ..J7!12, or Yc :S r).
and I <YeS 2 for condition II (i e L ~ I, K.. ~ ..J7!12, or Yc ~ r) respectively. Yc1d = I
(or, Y c~d = r) can conveniently be used as a starting point of the numeric procedure for
both the conditions (i.e Condition I & II), but the number of iterations required are
awkward as can be realized from Table (I 2.1) Moreover, while employing the values of
- 2J-
You~ as obtained from Eq. (1.3.8} as initial trial-depths, numeric procedure was found to
converge in the range 0 < Yctd::; 1.890647 except for the condition 1 <Ltd :S 1.00766234
as Yctd < I in this range. In order to develop the requisite condition of guaranteed
convergence so that 0 < Y.td::; 1 and I ::; Ycld < 2 for condition II, eq. (1.3.8) can be
modified as hereunder:
1 ""' < f + 0.0078556 ....................................................................... (1.2.1 0) 1
Where,
2Q2
K == < ------r g.r5
Alternatively, Author's eq. (1.3.9) can also be used as initial trial-depth guess.
However, while employing the values of Yc"' as obtained from Eq. (1.3.9) as initial trial-
depths, numeric procedure was found to converge in the range 0 < Yo1<1 :S 1.890647 except
for the condition 1 < Yctd :S 1.000813 as Yctd < I in this range. In order to develop the
requisite condition of guaranteed convergence so that 0 < Y ctd :S I and 1 ::; Yet<~ < 2 for
condition II, eq. (1.3.9) can be modified as hereunder:
I""' O -F •••••""•••"••••••••••••••• m•••••••••••••••••••••m• m•••m•••••••••• m••• m••••••••(1.2.11) I Equation (1.2.11) has the advantage that it involves lesser number of mathematical
operations that that of equation (1 .2.10).
- 24-
Alternatively, the following equation can also be used for the detemiination of
initial depth-guess:
Yctd= 0.012 + 0.7852K1 - 0.1552K12 + 0.0770SK/ - 0.01SK14
.................... (1.2.12)
Where,
[11.9856Q2] 0.281492
}(1 == -------------------
or,
(0.9986. KJ1.6889S2
Although equation (1.2.12) makes quite an accurate initial depth-guess except for
the range 0.9993895 < Yc ~ 1.0000001, it involves more number of mathematical
operations than that of equations (1.2.10) & (1.2.11). In the range 0.9993895 < Yc ~
1. 000000 I, the estimated value of Yctd comes out to be greater than one and hence the
algorithm does not converge. The equation (1.2.12) is to be modified if at all it is to be
used to calculate Y ctd so that it functions unfalteringly well throughout the range.
Yctd; 0.011387 + 0.7852K1 - 0.1552K12 + 0.07705K13
- 0.015K14
.............. (1.2.13)
Performance & Evaluation
Flow-chart for the optimized computational procedure has been presented in Figure
(1.2.3). Test runs were performed to evaluate the iterative numerical technique. It was
verified during the test runs that the algorithm converged for a convergence tolerance of
I o"" or more (Tables 1.2.1, 1.2.2. & 1.2.3.). Range of validity of the algorithm for various
convergence tolerances has been shown in Table (I 2.4) for the value of Y ctd = 1 .
- 25-
Similarly, range of validity of the algorithm for various convergence tolerances has been
shown in Tables (1.2.5), (1.2.6) and (1.2. 7) when the values of Y ctd are adopted as per
equations (1.2.10), (1.2.11) and (1.2.13) respectively. It can be verified from these tables
that the values of Y,td adopted as per equation (1.2.13) provide the highest ranges of
validity to the proposed algorithm for various convergence tolerances. However, for the
convergence tolerance of 10"1 alone, the range of validity of initial depth-guess Y,,d= 1 is
same as provided by eq. (1.2.13). Maximum, average, and minimum absolute errors for
various convergence tolerances have been shown in Table (1.2.8) for the condition Y,,d =
1 andY ctd adopted as per equation (1.2.1 0).
Limitations*
Author wishes to make it very clear that this algorithm (and also all the available
ones) have their practical limitations. For example, any of the numeric procedure would
not be valid for a I 0" corrugated metal pipe under certain conditions, because the
roughness height would cause the underlying assumptions to be invalid (average pipe
diameter constant, velocity is uniform). With this in mind, Author would say that the
tolerances listed herein do not necessarily reflect the expected degree of validity of the
equations (e.g. the procedure may give a critical depth with 7 significant figures when the
prediction is valid to only 2 significant figures) .
•••
* ASCE Review (June, 1994)
- 26-
Figure (1.2.1)
Condition 1: Y :S r (Q:S Qh or Y :S l or K4 ::; -./n/2)
I - ·-··- .. .,....-· / " . I .,
/ / ~ '·,_ ,.., 1 ', r
./ '· / '
T (r- Yc)
+ Yc
j~o~•----- 2rsin<j>1 •
_l
- 27-
Figure (1.2.2)
Condition II: Yc > R (Q > Qh or Yc > 1 or :K4 > Vn/2)
I~ 2rsinQ>2 ~I
' ' '· -+-- / ' I / ', r ~ ..... /
' ' ,tb_.·
T Yc-r
_j_ -~/~ ·- - ·- -- - ·-·
Yc
. 28-
Figure (1.2.3) Flow Chart of the Proposed Algorithm
YES
ctd) = K, K, I .K,;
IS \BS[f{Y:ctd)Y:ctd] >CT
?
NO
YES
IS K,~--/rr/2
?
Xctd = f (Yctd)
NO
WRITE f(Yctd)
YES
NO
COMPUTE 4l2, K,. K,
f(Yctd) = K. K1 I K,
IS ABS [ f(Y ctd) -Y:ctd] > CT
?
NO
Table (1.2.1). Maximum No. oflterations Required for Various Convergence Tolerances when Yctd = 1.
Maximum No. oflterations Required for Exact Value of the Convergence Tolerance of-Value K~ of Yc w·l w-2 w-3 w·• w-5 10-6
10-7 2.7589lx!0-5 2 2 2 2 3 03 w-5 6.014x1o-• 2 2 2 2 4 06 w-3 1.296!x1 o·2 2 2 3 5 7 09 0.10 0.2783035 2 4 6 8 10 12 0.20 0.4402862 3 4 7 9 11 13 0.30 0.5749725 3 5 7 9 11 13 0.40 0.6941425 3 5 7 9 11 03 0.50 0.8027145 2 5 7 9 11 13 0.60 0.9033525 2 4 7 9 11 13 0.70 0.9977206 2 4 7 9 11 03 0.80 1.086964 2 4 7 9 11 13 0.90 1.171934 l 4 6 8 ll 13 1.00 1.253314 1 1 l 1 1 01 1.10 1.332178 1 3 6 8 10 12 1.20 1.408172 2 4 6 9 11 13 1.30 1.482296 2 5 7 9 ll 13 1.40 1.555467 3 5 7 9 ll II 1.50 1.628763 3 5 7 9 11 13 1.60 1.704026 3 5 7 9 10 12 1.70 I. 784803 ~ 5 6 8 9 10 .)
1.80 1.879472 3 4 5 6 6 07 1.887583 I. 994739 3 5 10 14 19 25
- 30-
Table (1.2.2). Average No. of Iterations Required for Various Convergence Tolerances when Yctd is Adopted as per Eq. (1.2.10,
Average No. oflterations Required for Exact Value of the Convergence Tolerance of-Value ~lt of Yc w-1 w-2 w·' w-4 10-5 w-6
10-7 2. 7589lxl o-5 I I 2 2 2 02 w-5 6.014xl0·4 1 I 2 2 3 05 10"' 1.296lxl0-2 1 1 2 4 6 08 0.10 0.2783035 1 1 3 5 8 10 0.20 0.4402862 1 1 4 6 8 10 0.30 0.5749725 1 2 4 6 8 10 0.40 0.6941425 l 2 4 6 8 10 0.50 0.8027145 1 2 4 6 8 10 0.60 0.9033525 1 1 4 6 8 10 0.70 0.9977206 1 1 4 6 8 10 0.80 1.086964 I 1 4 6 8 10 0.90 1.171934 I 1 3 5 8 10 1.00 1.2533I4 I 1 1 1 1 01 1.10 1.332178 I 1 3 5 7 09 1.20 1.408I72 I I 3 6 8 10 1.30 1.482296 I 2 4 6 8 10 1.40 1.555467 1 2 4 6 8 10 1.50 1.628763 I 2 4 6 8 10 1.60 1.704026 . I 2 4 5 7 9 1.70 1.784803 1 1 2 4 5 7 1.80 1.879472 I 2 3 3 4 5 1.887583 1.994739 4 6 11 15 20 25
- 31 -
Table (1.2.3). Average No. oflterations Required for Various Convergence Tolerances when Yctd is Adopted as per Eq. (1.2.13)
Average No. oflterations Required for Exact Value of the Convergence Tolerance of-Value K" of Yc w·l w·2 w·3 w·• w·s 10"6
10-7 2. 75891xl0-5 I 2 2 2 2 3 10"' 6.014xlo·• I 2 2 2 3 5 w·' 1.2961xl0.2 I I 2 4 6 8 0.10 0.2783035 1 I 1 2 4 6 0.20 0.4402862 1 1 1 3 5 7 0.30 0.5749725 1 I 1 2 4 6 0.40 0.6941425 1 1 1 2 4 7 0.50 0.8027145 1 1 1 3 5 7 0.60 0.9033525 1 1 1 2 5 7 0.70 0.9977206 1 1 1 1 3 5 0.80 1.086964 1 1 I 2 4 6 0.90 1.171934 1 1 1 2 4 6 1.00 1.253314 1 I 1 1 1 1 1.10 1.332178 1 1 1 3 6 8 1.20 1.408172 1 1 1 4 6 8 130 1.482296 1 1 1 4 6 8 1.40 1.555467 1 1 1 3 6 8 1.50 1.628763 I I I 3 5 7 1.60 1.704026 1 1 1 2 4 6 1.70 1.784803 1 1 1 2 4 5 1.80 1.879472 1 1 2 2 3 3 1.8998 2.016459 1 1 6 15 24 30 1.9110 2.038614 1 3 61 118 176 --1.9127 2.042203 1 38 488 938 -- --1.9129 2.042629 I 144 2496 -- -- --1.9460 2.130950 I -- -- -- -- --
- 32-
Table (1.2.4). Range of Validity of the Iterative solution When Yctd = 1
S. Convergence Range of Validity No. Tolerance
1 10-l O<YcSL9300
2 10-2 O<YcSL9129
3 10-3 0 < Yc:::: 1.9129
4 w-• 0 < Yc :5 1.9127
5 10-5 O<YcS 1.9110
6 10-<; 0 < Yc :5 1.8998
- 33-
Table (1.2.5). Range of Validity of the Iterative solution When Yctd is taken as per Equation (1.2.10)
S. Convergence Range of Validity No. Tolerance
I w-1 0 < Yc:::; 1.8717
2 10-2 O<Yc:SI.8717
3 w·3 0 < Yc:::; 1.8717
4 10"" 0 < Yc s 1.8717
5 w-' 0 < Yc:::; 1.8717
6 10-6 0 < Yc:::; 1.8717
- 34-
Table (1.2.6). Range of Validity of the Iterative solution When Yctd is taken as per Equation (1.2.11)
S. Convergence Range of Validity No. Tolerance
1 10-1 0 < Yc::; 1.887583
2 10-2 0 < Yc::; 1.887583
3 w-3 0 < Yc::; 1.887583
4 w-4 0 < y c::; 1.887583
5 10_, 0 < Yc::; 1.887583
6 10-6 0 < y c::; 1.887583
- 35 -
Table (1.2. 7). Range of Validity ofthe Iterative solution When Yctd is taken as per Equation (1.2.13)
s. Convergence Range of Validity No. Tolerance
1 10'' 0 < Yc :S 1.9460
2 10-2 0 < Yc:; 1.9129
3 w-> 0 < Yc :S 1.9129
4 10-4 O<Yc:S !.9127
5 JOos O<Yc:S\.9110
6 10-6 0 < Yc :S 1.8998
0 36 °
TABLE -(1.2.8)
Maximum, Average, aud Minimum Absolute Errors for Various Convergence Tolerances When Ltd is adopted as per Equation (1.2.10)
Convergence Average No Absolute Error Tolerance of Iterations
Required Maximum Average Minimum
3.482187 X 10-2 4.461784 x w-3 0 10"1 1.03 (2.26)
(6.135428 x w-2) (2. 60252 I x 10-2) (0)
5.558491 X 10-3 2.463184 X 10-3 0 w-2 1.50 (4.19)
(5.722165 x w-3) (2.865238 x w-3
) (0)
5.699594 X 10-4 2.659594 X 10-4 0 w·3 3.40 (6.29)
(5.71847 X 10-4) (2.72I I 19 X 10-4) (0)
5.668402 X 10 -5 2. 755756 X 10-5 0 IO"" 5.41 (8.34)
( 5. 686283 X 10-5) (2.75172 X 10-5
) (0)
5.960465 x w- 6 2. 952958 X 10-6 0 w·5 7.42 (10.35)
(5.960465 X 10-6) (2.914059 X 10-6) (0)
1.072884 X 10-7 3.295092 X 10-7 0 10-6 9.42 (12.35)
(1.072884 X J0-7) (3.227845 x w-7
)_ 10)
Values in brackets correspond to the condition Y ctd = l
. 37 -
(1.3) Circular Channels: An Alternative Explicit Solution
Introduction
An explicit mathematical solution for quickly solving the implicit problem of
critical depth calculations in circular sewers has been developed by Rathore & Sen
(1997b ). The test results undisputedly show that the equation presented here meets the
requirements (of computational efficiency with reasonable degree of accuracy) more
effectively than that of available ones and, as such, can also be used as a starting point for
the proposed iterative solution so as to reduce the number of iterations (Rathore, 1993 &
1998) as " ... efficiency is measured by the number of iterations to obtain a good
approximation to the solution" (Gillet al, 1981). Ortega & Rheinboldt (1970) calls it "the
economy of the entire operation".
Subramanya (1972) has reported following direct solution for the rough
determination of critical depth in circular sewers:
- 38 -
Q 1 0206
[c ·· 2.02/--. --- j .J g ,p
or,
I Yc = 1.30811. K0·618
.......... (1.3.1) I
Where,
2Q2 K = ( _____ )1'6
g.rs
However; Eq (1.3.1) is claimed to be valid only for 0.04 :S Yc :S 1.70.
Uniform flow in circular, partially filled conduits were experimentally considered
by Sauerbrey (1969), among others. According to Sauerbrey, relative pipe filling under
normal-flow condition (Yn) cannot exceed 1.90, because unstable flow occurs otherwise.
The conduit then gets pressurized over limited reaches between which confined free
surface flow occurs. The experimental observations of Sauerbrey may be expressed as
(Hager, 1989):
nQ 3 7 ---------- - ---- Yn2(1- ---Yn2
), Yn < 1.90 ......................................... (1.3 2) S0
1h d8'3 16 48
An expansion of Eq. (1.3.2) for the friction slope Sr, for which Yn --~ Y states
that
- 39-
3 7 nQ -- I 2(1- ---I2
). In< 1.90 . . .. .. . . ....................... (1.3.3)
st t.fl3 16 48
The cross-sectional area A of partially filled conduit flow may be approximated as
(Hager, 1989) :
A y y_2 =0.47140P12 [I- __ ::- ---1 ............................................. (13.4)
uF 8 25
Q dA Now, the Froude number F = -------- can be obtained by substituting T = ----- :
(g.A/T)l/2 dP
SY 7Y2
Q2[ 1 - -----= - ------] 27 24 75
F2 = ----- . 2 y y2
. ..................................... (1.3.5)
d54 - -3
g p [ 1 - ---- - ----] 8 25
for 0.1 < Y < 1.90, the approximation
F
Q [ ---- ]112
g.d5
.................................................. (1.3.6)
deviates less then 6% from Eq. (1.3.5). Therefore, relative critical depth (Subscript << c
>>)may be expressed as (Hager, 1991):
.. ( 1.3. 7)
-40-
or,
Yc= ..................................................... (1.3.8)
where,
2Q2 K = ( -------) Ii6
g.~
Eq. (1.3.8) can explicitly be used for the approximate computation of critical depth
in circular channels.
Authors, however, have found that "Y c" can more appropriately be expressed as a
function of "K" in the following form with a slightly increased value of the exponent:-
............................................................ (1.3 •• ) I
Validation
For the validation of the proposed solution and to compare its efficiency (accuracy-
wise) as compared to the available ones, test runs were performed as follows:
• For a given value of relative critical depth, the angle"<!>" subtended at the centre
of the sewer by the free surface was directly calculated.
• As the non-dimensionalised constant "K" is a function of"<!>" & relative critical
depth, its value can subsequently be calculated
• Now, from the known value of"K", "Y c" was calculated from equations (I 3 7),
(13 8)&(13 9)
- 41 -
• The absolute value of the difference between the given relative critical depth and
the estimated critical depth was calculated which is a measure of numerical
procedure error. Extensive test computer simulation set-up was written in
ForTran77.
Equation (1.3.1) goes out of competition as it gives excessive maximum & average
absolute errors in the range 0.04 <; Y c <; 1.70. Table (1.3.1) shows these values. Also , the
standard deviation was found to be 0.48348 which is quite unsatisfactory. Now the only
comparable equation with the author's equation is Eq. (1.3.8). Table (1.3.2) shows the
comparative performance of these two equations in the range 0. I < Y c < 1. 90. By
performing 1798 test runs for each equation, reduction in standard deviation, & average
absolute errors was found to be 10.097% and 48.19% respectively by the use of author's
equation.
Performance & Evaluation
A large number of test runs were performed over the entire range to observe the
comparative performance of the proposed equation and its validation. Test runs, as such,
show that the proposed solution performs better than the available ones so far as the
average absolute errors, standard deviation, and variance are concerned. However;
maximum absolute error was found to increase by 37.92% as compared to Eq. (1.3.8).
It was also observed during the test runs that the maximum absolute errors go on
decreasing in the Author's equation as higher range of "Yc" is brought down. For
example, in the range 0 04 < Yc < 1.70, it was found to be only 2.46% which is less than
what is given by Eq ( 1 3 17) in the same range Also, in the range 0.0885 < Yc < I 7023,
- 42-
error It not greater than I% which seems to be quite satisfactory as compared to the
corresponding value of 3.876% given by equation (1.3.8). If Author's equation is
compared with equation (1.3.8) in the range 0.1 ~ Yc ~ 1.6947001, maximum absolute
error is found to be about 0.888% at the value ofYc = 1.6947001 for both the equations
(Table 1.3 .3). Beyond this value of Y c, absolute errors shoot up rather abruptly in the
Author's equation as compared to equation (1.3.8). Comparative performance of
Author's equation versus equation (1.3.8) has graphically been depicted in Figure (1.3.1).
No such graphically depiction has been considered necessary for the comparative
performance of Author's Equation versus equation (1.3.1) as it has already been declared
out of competition in the previous page.
Conclusively; Author's equation- if used in the range 0.1 :S Y, :S 1.81651- gives
maximum absolute error not exceeding 3.815% which is equal to what is given by
equation (1.3.8) in the same range (Table 1.3.3). However; Author's equation reduces the
average absolute error down to 0.59% as compared to equation (1.3.8) which generates
1.54% average absolute error in the range 0.1 :S Y, :S 1.81651. In fact, Author's equation
can be used over a comparatively wider range (i.e. 0.02128 :S X., :S 1.81651) without
exceeding the value of maximum absolute error which is generated by equation (1.3. 8).
Summary & Conclusions
A direct non-dimensionalised solution was presented for solving the implicit
problem of determining the critical depth in circular sewers rather quickly with
reasonable degree of accuracy over a wide range of "Y," or "Yc". Test runs show that
-43-
the proposed solution performs better than the available ones so far as the standard
deviation and absolute errors (average & maximum) are concerned .
• • •
- 44-
Table (1.3.1). Comparative Performance of Author's Equation Vs
Equation (1.3.1)
Efficiency Range 0.04::5 Yc ::5 1.70 Parameters
Equation ( 1) Author's Equation
Standard 0.4834846 3.99066 x w·3
Deviation
Variance 0.2337574 1.5825 x w·5
Maximum% Absolute Error 918.3254 2.461029
Average% Absolute Error I 17.3937 0.5201101
- 45 -
Efficiency Parameters
Standard Deviation
Average% Absolute Error
Maximum% Absolute Error
Table (1.3.2). Comparative Performance of Author's Vs
Equation (1.3.8)
Range OJ < Yc ~ 1.70 % Improvement in Author's
Equation (1.3.8) Author's Equation Equation
0.0129265 4.257333 X 10"3 + 67.065%
1.611292 0.4828513 + 70.033%
3.815703 0.9660659 + 74.681%
- 46-
Table (1.3.3). Comparative Chart of the Absolute Errors Given By Author's Equation Vs Equation (1.3.8)
Absolute Error S.No. Exact Value ofYc
Author's Eq. Eq. (L3.8)
1 0.02128 -3.815 +4.228 2 0.08850 -1.000 +3.876 3 0.100 -0.785 +3.815 4 0.16385 0.000 +3.480 5 0.20 +0.263 +3.289 6 0.30 +0.661 +2.762 7 0.40 + 0.787 +2.234
8 0.50 +0.765 +1.708 9 0.60 +0.651 +1.185 10 0.70 +0.477 +0.670 11 0.833431 +0.188 0.000 12 0.913401 0.000 --0.387 13 1.00 -0.204 -0.785 14 l.IO -0.373 -1.159 15 1.20 -0.569 -1.538 16 1.30 -0.698 -1.837 17 1.40 -0.717 -2.017 18 1.50 -0.554 -2.010 19 1.611014 0.000 -1.631 20 1.6947011 +0.888 --0.888 21 1.7023 +1.000 -0.790 22 1. 750724 +1.888 0.000 23 1.816151 +3.815 +1.765 24 1.90 +8.960 +6.595
-47-
Fig
ure (1.3.1)
%E
rrors
give
n b
y A
uth
or's
Eq
ua
tion
an
d
Eq
ua
tion
(1.3.8)
10 8 r···
..... , ........ , .......... , .......... , .......... , .................... . '
' '
--·-----··:-----------~----_____ ! _______ -----------------------------------------·---
6 ----·-
------·•· --------·------
___ ;_ -----------
----~ ----.......... -----~----
----·-:--------
-·--------
-:-------
--~------
--~-----~ -----
---···· -----·-·--·
: '
'
' '
' ' '
' ........ -------·····-------;··· --····-:·-------
-~----------~--------~----------:---------+ ------------------
-------------------~--------
--------~
4 2 ·--~-----
----~---------~------
.... : ..... -----~ .... ----~------· --~---
0
-2 ... -
-.... -.. -. --...... ·--------··-
----.... ·-···--· ·---!··---..... ~
-------
-4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 1.1
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
No
rma
lised
Critical D
epth >>
>
!•AUthOr's E
quation • E
quation ( 1.3.8) -I
(1.4) COMPOUND CANALS
As often as not non-exponential Compound Canal Sections (CCS) such as those in
cutting and filling are adopted for irrigation and power canals. This Section analyzes a
non-dimensionalised iterative procedure* for quickly and accurately solving the implicit
problem of determining the critical flow depth in CCS. The test results show that the
iterative procedure presented here meets the requirements of guaranteed convergence to a
unique solution and highest computational efficiency (accuracy and speed-wise).
* This procedure was presented by Rathore ( 1997a) in his paper entitled Computation
of Critical Flow Depth in Compound Channels in the (J) Indian Water Works
Association
-49-
Numerical Analysis
Consider a fixed location in CCS (Figure 1.4.1) for any given depth of flow
Y > De, the following geometric relations can be defined:
B, A(Y) = C;.Y2 + Y [B + B1 + 2C1Dc- 2Dc.C,] + Dc2[C,- C1- ----]. ................ (1.4.1)
De T(Y) = B + ZY(Z1 + Zz) + B1 - 2Dc.(Z3 + Z•)
= ... T(Y) = B + ZYC2 + B1- 2Dc.C ............................................................. (1.4.2)
\\>'here,
C1 and C2 are constants defined in terms of the side slopes in cutting (i.e. z, and
Ct = tl [Zt, Zz] = [Zt + Zz]w ................................................................ (1.4.3)
Cz = fz [Zt, Zz] = [Zt + Zz] ........................................................................ (1.4.4)
Similarly CJ and C 4 can be defined in terms of the side slopes in filling (i.e. ZJ, z.)
as:
c3 = r, [Z3,~] = [Z3+ z.t2.. .. ... . . ... ... ... ... ... ... . ........................... (145)
c.= fz [ZJ, z.) = [Z3 + 2 4] ..........................................................•....•......... (1.4.6)
By substituting the values of A(Y) and T(Y) from equations (l.4.1) and (1.4.2)
respectively in equation (1.2.1) and rearranging, one gets:
Ql
C3.Yc2 + 2Yc.Kt.C; + K~CJ = (---(
3 [B + 2Ye.C2 + B1 - 2Dc(Z3 + ~))1 G g
%ere,
K, ~ [B + B, + 2C,Dc- 2Dc.C,] f2C,
B, Kl ~ De2
[(',- C1 - ---] fC3
De -50-
Where,
1 Q2 2 113 K Y ] 113 (2C )113 K = ... Yc + 2K1Yc = ----(----) .[ 2+ c . 2 - s
c3 g
Adding K12 on both sides,
l ZC2Q2
= ... Y c2 + 2Kt Yc + Kt2 = ----(---------)w[K2 + Yc] 113
- Ks + Kt2
c3 g
I ZC2Q2
= ... (Yc +Kti = ----(---------)113 [K2+ Yc]l/3 - Ks + Kt2
c3 g
= ... I f(Yc)="[~ + {K3(K2+Y)}113]- K1 ............................................ (1.4.7)
where,
ZC2Q2
K3 = { ----------} , and gC33
K,= K/-Ks
Equation (1.4. 7) is the requisite iterative scheme which converges rapidly
Moreover, non-dimensionalised version of equation (I .4. 7) can be shown in the following
form while preserving its rapidly converging behaviour:
~· ~ , .. ""' ~ .,r. -t"' .- - .. <0\.. "'
., ~ r """' ~ • •· r ' ,. .·.·r )'1o
: • I ~i'l'(s;.~) \ 11! 1-• .... \ ) .. . . \ ) ~ j I I •
T ·,~-:'fiJ ,., ••••••• T 187~
f(Yc) = "!K.I + {Kl(fu+ X)} 113]- Kt .................................... (1.4.7A)
where,
Yc=Yc/B
Kz = [B + B1- 2Dc.C4]/2BCz
2CzQ2
K3 = { ------------} , and gB5C/
B1 K4 = K1 2
- Dc2 [CJ - C1- ----] /B2C3
De
For a given flow rate Qo > 0, ifY = f(Y), then,
Qo=Q(Y)
Given any initial Y1 > 0, Yn is iteratively defined by:
Yn+l = f]Yn] ... ... . . . . .. . . . . . . . . . . . . .. . .. .. . . . .. . .. . . . . . • . . . . ...................... (1.4.9)
If Yo denotes the true solution of Qo = Q(Yo); then Yo = fl:Yo). Also, the
following premises of fl:Y) can mathematically shown to be true:
PREMISE I. IfY <Yo, then Y < f(Y) <Yo.
PREMISE II. If Yo< Y, the Yo< f(Y) < Y.
It can be shown that the sequence Yn converges to Yo. If the initial value, Y1 is less
than or equal to Yo, then premise I says that Y n is an increasing sequence bounded above
by Yo. Such a sequence must converge to some value y, Eq. (1.4.9) and the continuity
of f give Y • = flY.), and hence Qo = Q(Y • ). Since the latter equation has Yo as its
unique solution, Y • = Yo. If the initial value Y, is greater or equal to Yo, then one gets
Yn converge downward to Yo using a similar argument with premise II
-52-
Equation (1.4.7) and (1.4.7A) offer monotonous convergence and have yet another
ingenuity that they can conveniently be used for normal depth calculations in trapezoidal
channel as well [i.e. Yn :S De] simply by substituting Bt =De= 0, C4 = Cz, and C3 =Ct.
By making such substitution, lumped-coefficients* used in equations (1.4.7) get reduced
to the following form:
B Kt = -----,and
2C,
B K2 = ----- , and
c2 2CzQ2
K3 = { ---------} , and gCt3
Performance & Evaluation
Test runs of the numerical procedure were performed as follows. Channel cross-
section as shown in Fig. (1.4.1) was selected for the performance study of the proposed
iterative procedure. Critical outflow rates were computed directly from the equations
using critical depths varying from O.oi to 3.00m in equal steps of O.Olm. The selected
cross-section and the computed critical outflow rate were then used in the numerical
• Lumped coefficients are calculated only once and hence the algorithm works faster
-53-
procedure to estimate the critical flow-depth at a specified convergence tolerance and an
initial critical flow-depth guess. The absolute value of the difference between the given
and the estimated critical flow-depths is a measure of the numerical procedure error. The
number of iterations required for requisite convergence is a measure of the numerical
procedure speed.
For each selected convergence tolerance and initial flow-depth guess, a set of 300
test-runs was performed using as many equally spaced values of critical flow-depth. Out
of these critical depth values, half were meant for the condition Y c :S De and the
remaining half (i.e. I 50) were meant for the condition Y c > De. In all, 55 such test sets
were run and hence 16500 (i.e. 55x300) test runs were performed. Tables (1.4.1) and
(1.4.2) summarizes the results of these tests. Maximum absolute errors were found to be
9.9494x10'3, 9.9372xl04, 9.173lxl04
, 1.4066xl0'5 and 1.66893xl0-6 for the
convergence tolerances of 10·1, 10'2, 10'3, 104
, and 10'5 respectively (Table 1.4.2).
Summary & Conclusions
An iterative procedure was presented for quickly and accurately solving the
implicit problem of determining the critical depth in CCSs. Several thousand test runs
were performed to evaluate the effectiveness of the proposed iterative procedure. It was
verified during the test runs that the algorithm always converged for a convergence
tolerance of I 0-6 or more and the absolute errors were not affected by initial flow depth
guess. However, tolerances above 10-6 are not recommended for single precision values
because truncation errors may be such that the procedure never converges. Absolute
-54-
errors decreased with decreased tolerance. Also, method was not found to be much
sensitive to the starting point (Table I .4. I).
Tests similar to those devised for the iterative procedure were also devised for the
Newton-Raphson method. It was observed that the iterative procedure is computationally
efficient comparable to the Newton-Raphson method with suitable starting position
(number of iterations and computation times were generally a little larger for the Newton
Raphson method) and very insensitive to starting position.
In conclusion; the test results have shown that the iterative procedure presented
here meets the requirement of guaranteed convergence, highest computational efficiency
(speed and accuracy wise) and ability to handle both trapezoidal and compound canal
cross sections.
A computer program has been written to implement the procedure on a computer.
The source code was written in Fortran for efficiency and portability, especially among
personal computers.
* * *
-55-
Figure (1.4.1)
Typical Compound Canal Section (CCS) used for Performance Study
LB, 14 •I
(1.22)
Yc (orYn)
B(3 m) 14 •I Z (Average) ~ 1.20 Z1 (Average)~ 1.00
-56-
ZONEII(Yc>Dc)
+ ZONE I (Yc :SDc)
1
Figure (1.4.2). Flow-chart of the Proposed Algorithm for the Computation of Critical-Depth in Compound Canals
Compute C~, Cz
Yes
Compute Yc from Eq. (1.4.7)
Is Yes ABS(Yc-Y) > CT Y = Yc; I= I+ I
?
-57-
Initial Flow
Depth Guess
104
JO"' 10-2
JO"' I 0 10 50 102
103
1 o•
Table (1.4.1)
Number oflterations Required for
Various Convergence Tolerances & Initial Depth Guess
No. oflterations Required for the Convergence Tolerance of:
10"' 10-2 1 o·3 104 1 o·' i
' YcSDi Yc>D YcsD iYc>D YcsD Yc>D YcsD i Yc>D Ycsoi Yc>D
3 3 4 ; 4 5 5 6 6 7 \ 7 3 3 4 i 4 5 5 6 6 7
) 7 i
3 3 4 ' 4 5 5 6 6 7 7 l j
3 3 4 4 5 5 6 6 7 ' 7 2 3 3 4 4 5 6 6 7 7 3 3 4 4 5 5 6 6 7 ; 7 3 3 4 4 5 6 7 6 8 j 7 4 4 5 5 6 6 7 7 8 8 4 4 5 5 6 6 7 7 8 8 4 4 5 5 6 6 7 7 8 8 4 4 5 s 6 6 8 8 9 ' 9
- 58-
Table (1.4.2) Maximum Absolute Errors for Various Convergence Tolerances
and Initial Deptb-Guess
Initial Maximum Absolute Errors for the Convergence Tolerances of: Depth Guess w·' w-2 w-3 10-4 w·'
104 6.070lx10-3 7.76!4x104 8.3804xl o·' 8.8214x10-6 1.2516x10-6
1 o-3 6.0672xl o·3 7.7545x1 04 8.3804xW5 8.8214x!0-6 1.2516x10-6
w-2 6.0405xl0-3 7. 7545x1 04 8.3327x1 o·' 8.8214xl0-6 l.2516xl0-6
w·' 6.9257xl0-3 7.9929x104 8.4519x1 o·s 8.9406xl 0-6 l.l920x 1 0-6
1 9.9494xl o·3 9.9372xW4 1.0228x!O·' l.0490x1 o-5 1.6689x1 0-6
0 6.0701 xl o·' 7764lxl04 8.3804xlo·' 8.8214xl 0-6 1.2516x 10-6
10 8.3601xl0-3 7.7486x104 8.7738xro·' l.0252xl o·' 1.6689x1 0-6
50 9.8021 x1 o-3 1. 0268x 1 04 1.0442x!O·' 1.0728xl o-5 1.1920x10-6
102 8.3690x 1 o·3 9.1683x1 04 9.6797x1 o·' 1.00l3xl 0-6 1.1920x10-6
103 7.3902x1 o·3 8.6158x1 04 9.1731x104 1.4066xl o·' 1. 6689x 1 0-6
104 8. 9287x I o-3 9.4243x I 04 1.1754xl04 I .4066x1 o·' I. 6689x 1 0-6
-59-
(1.5) STANDARD LINED CANALS
This section analyses a rapidly convergmg & non-dimensionalised iterative
procedure* for quickly and accurately solving the implicit problem of determining the
critical flow depths in Standard Lined Canals (SLCs). The test results show that the
iterative procedure presented here meets the requirements of guaranteed convergence to a
unique solution and high computational efficiency (accuracy and speed-wise). The
findings will be helpful in contributing directly to the development of flexible
mathematical simulation model by providing an efficient algorithm with guaranteed
convergence for computing critical flow depths when necessary. Proposed algorithm can
also be used over a simple calculator.
*This procedure was presented by Rathore (l994b) in his paper entitled Critical Depth
Calculations in Standard
Association.
Lined Cannals in the (l) Indian Water Works
- 60-
Introduction
It is now an accepted practice to line all new 1J18.jor canals and the need for lining
of canals, whenever feasible, is well recognized (Subramanya & Sahu, 1991). For such
lined canals, the Bureau of Standards (IS: 4745-1968) has recommended a standard lined
canal sections which is a trapezoidal section with corners rounded off with radius equal to
full supply depth (Figure 1.5.1 ). The limiting case of this section with zero bottom width
is termed as the standard lined triangular section and is recomnrended by the Central
Water Commission (India) for discharge less than 55m3/second (Figure 1.5.2).
It is felt that a simple procedure for computing the critical depth in such canals is
warranted for use by hydraulic engineers; consequently it has drawn the attention of
many researchers and several graphical and tabular relationships correlating the critical
flow parameters have come out of efforts made in this direction. Although the concepts
behind these methods are still valid, there is an imperative need for replacing these
particular approaches by an efficient computational algorithm for the ease of
computation.
The purpose of this section is to accomplish the following:
1. Develop an unfaltering iterative procedures for quickly and accurately finding normal
flow depth in various channel cross sections;
2. Find conditions (for each case) guaranteeing both a solution that is unique and an
iterative procedure that will converge to the solution;
3. Perform test runs to verify convergence and computational efficiency (acc\lracy and
speed-wise) of the proposed algorithm.
- 61 -
4. Present flow-chart of the proposed algorithm.
Numerical analysis
Consider a SLC cross-section satisfying the following conditions:
Q, A and T are non negative, continuous, and strictly increasing.
2. Q(O) =A (0) = 0, and Q (oo) =A (oo) = oo
In such SLCs, eq. (1.2. I) is to be solved for the following two conditions:
(I) Yc > YLor Q > QL [T i" f(<j>, Drs)]
(II) Y c :0: Y1. or Q :0: QL [T = f( <j>, Drs)]
Where,
Y L = Dr •. tan(e /2).Sin(9)
AL = Dr •. tan(e /2).Sin((9)
QL = ..Jg.h3/TL = ;/[g.AL3/(BJ +Z YL)]
Condition I: [Y c > YL or Q > QL {T :;t f( <j>, Drs)})
From the geometry of the SLCS (Figure 1.5.1 ), one gets:
A(Y) = B1.Y + Z.Y2- K1.Z ... .. ........................................................ (1.5.1)
Where,
B1 = B + 2.Dr •. tan(9 /2), and
Ko = Dr,2/Z[2 tan(e /2)- 9]
Also,
T(Y) = 81 + 2.ZYc (1 52)
-62-
By introducing the values of "A" & "T" from equation (1.5.1) & (1.5.2)
respectively in ( 1.2.1) , one gets:
Q2 (---)u3(B1 + 2 Z.Y c)113 = B1.Y + Z.Yc2
- Ko.Z g
2.Q2 =~(--------) 113(K2 + Yc) 113
= 2Kz.Yc + Yc2- Ko
gZ2
Where,
Kz = Bt12Z = (B + 2.Drs.tan(6 /2))/2Z
Adding Kl on both sides,
2.Q2 =~(--------)''3(K2 + Yc)''J
gZ2
=~ I Y c = V (K, +{&(Kz + Y c)113}] - K2 ................................................. (1.5.3) I
Where,
K, = Ko + K/ , and
2.Q2 K3 =(--·-----)
gZ2
Non-dimensionalised version of equation (1.5.3) can also be presented while
preserving its rapidly converging behavior:
=~ I Yc = V[KJ. +{fu(Kz +Yc)113}] - fu .............................................. (1.5.3A) I
Where,
_fu = Btf2BZ = (B + 2.Drs.tan(6 /2))/2BZ
IS;,= Dr/IZ[2 tan(e /2)- f]/B2 + K22, and
- 63 -
2Q2
}(3 =-(---------) gZ2B5
For standard lined triangular sections (Figure 1.5.2), equations (1.5.3) &
(I 5.3A) can be made applicable by substituting:
B = 0 or 81 = 2.Drs.tan(e /2)
Condition ll: [Y c S Y L or Q :S QL {T = f( (jl, Drs)})
Equations (1.5.3) and (1.5.3A) are not valid for partial discharge conditions
t.e. Yc :0: Y1. which may occur in a channel (Subramanya & Sahu, 1991) as:
(a) A uniform flow under conditions of specific regulations, and
(b) A non-uniform flow in various gradually varied flow situations. Partial flow
parameters corresponding to uniform flow would be needed in many solution
procedures of such gradually varied flows.
For the condition Y c :0: Y Lor Q :0: QL [T = f( 1\l, Drs)]
A(Y) = B.Yc+Dr/<!J-[{Drs-Yc}Drs.Sin((jl)] ................................................. (l.5.4)
T(Y) = B + 2Dr5.Sin(<!J) ..................................................................................... (1.5.5)
By introducing the values of "A" & "T" from equation (1.5.5) & (1.5.6)
respectively in (I .2.1 ), a simplified iterative scheme can be expressed in the following
form:
I Yc =- KfC((jl) .............................................................................. (1.5 6)
Where,
-.1(2Drs- Y c - Y c2)
4> =- tan -I [ -----·---------)
D-Ye
-64-
[B + 2Drs Sin(l\>)t' K(l\>) ---------------------------------------------------
B + Drs21\>/Yc- [{Dr,JYc- I )Drs.Sin(!j>}]
Q2
]1/3 K = [ ----
g
and
Non-dimensionalised version of equation (1.5.6) can also be presented while
preserving its converging behaviour:
Xc = K-K(4>) ................................................................................................. (1.5.6A~ Where,
Yc = Yc/ Drs
[B/Drs + 2.Sin(4>)] 113
K( 4>) = ----------------------------------------------- and B1Drs+4>/Yc- [{1/Yc-l}Sin(!j>)]
Q2
K = [ --------t' gDrs5
Performance & Evaluation
Test runs of the proposed procedure were performed so as to determine its speed as
a function of initial flow-depth guess and convergence tolerance. For various values of Q,
B, Z, and D as shown in Table (1.5.1.); critical depths for 1320 such combinations and
also for selected convergence tolerances (I o·1, 10·2, 10·3, 10-4, and 10'5) and flow-depth
guess were computed. As such, 55 sets comprising of 1119 test runs in each of them were
performed for the condition Q > Q~.. As the test results do not converge for the value of
- 65-
initial flow-depth guess greater than YL when Q sOL> five sets of 201 test runs were
performed for various convergences tolerances (I o·1 to 10"5) using an initial depth guess
equal to YL for the condition Q ~ QL. In all, 62,550 (i.e. 55 x 1119 + 5 x 201) test runs
were performed on an IBM compatible microcomputer to observe the unfaltering
convergence and computational efficiency of the proposed iterative procedure. Test
results are shown in Table (1.5.2). Flow-chart of the proposed algorithm is depicted as
Figure (1.5.3).
Summary & Conclusions
An iterative procedure was presented for quickly and accurately solving the
implicit problem of determining the critical depth in SLCs. Several thousand test runs
were performed to evaluate the effectiveness of the proposed iterative procedure. It was
verified during the test runs that the algorithm always converged for a convergence
tolerance of 10-6 or more and the absolute errors were not affected by initial flow depth
guess However, tolerances above I 0-6 are not recommended for single precision values
because truncation errors may be such that the procedure never converges. Absolute
errors decreased with decreased tolerance. Also, method was not found to be much
sensitive to the starting point (Table 1.5.2).
Tests similar to those devised for the iterative procedure were also devised for the
Newton-Raphson method. It was observed that the iterative procedure is computationally
efficient comparable to the Newton-Raphson method with suitable starting position
(number of iterations and computation times were generally a little larger for the Newton
Raphson method) and very insensitive to starting position
- 66-
In conclusion; the test results have shown that the iterative procedure presented
here meets the requirement of guaranteed convergence, high computational efficiency
(speed and accuracy wise) and ability to handle both trapezoidal and triangular standard
lined canal cross-sections.
A computer program has been written to implement the procedure on a computer.
The source code was written in Fortran for efficiency and portability, especially among
personal computers.
* * *
- 67-
Figure (1.5.1)
Standard Lined Trapezoidal Channel Section for Q >55m3/sec.
Full Supply Depth Line
1
B
1+-------- BJ
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Figure ( 1.5.2)
Standard Lined Triangular Channel Section for Q :5 55 m3/sec.
'.
' . ZONE II ~I ~t
: 24> 'Dts I . I ' Drs
I I \
I ',
j4 ~\
i Drstan(8/2)
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Figure (1.5.3) Flow Chart of the Proposed Algorithm
YES
COMPUTE K~, K2, K,
Yctd=Yc: COMPUTEYc from Eq. (1.5.3)
IS ABS(Yc-Yctd)
>CT ?
NO
READ Dis, Q, g, CT, Z, B, Y,,"
YES
COMPUTE QL, e
IS Q>QL
?
WRITE Yc
YES
NO
COMPUTE K, Y1,: Yctd=Y1
COMPUTE <jl, K(<jl)
Yctd=Yc: COMPUTEYc from Eq. (1.5.6)
IS ABS (YcYctd) >CT
?
NO
Table (1.5.1) Range of Flow-Parameters Used In Test Runs
Variable Lower Value Upper Value Step
Q 1.00 501 50.00
z 0.5 1.50 0.20
Dfs 0.50 6.50 2.00
B l.OOm ll.OOm 2.00m
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Table (1.5.2)
Maximum Number of Iterations Required for
Various Convergence Tolerances & Initial Depth Guess
Initial Max. No. oflterations Required for Convergence Tolerance of Flow Depth 10"' 10-2 w-3 10-4 10-5
Guess Condition I (Q > QL)
10·4 4 6 7 8 10 10"3 4 6 7 8 10 1.0 4 6 7 8 9 2.0 4 5 7 8 9 3 4 5 7 8 9 6 4 5 6 7 9
20 4 5 6 8 9 50 4 6 7 8 9 102 4 6 7 8 10 103 5 6 7 9 10 104 5 6 8 9 10
Condition II (Q :5 Qd
YL 4 5 7 9 10
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(1.6) LOCATION OF CRITICAL FLOW SECTION
(THE SVF MODEL)
The present analysis deals with the case of a collector channel having zero inflow
at the discharge upstream and the discharge increasing downstream resulting in a change
of flow regime from subcritical to supercritical in an SVF. An efficient mathematical model
as such. has been developed for the location of critical flow section in a collector channel.
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Introduction
A steady spatially-vmied flow represents a gradually-varied flow with nonuniform
discharge. The discharge in the channelva1ies along the length of the channel due to lateral
addition or withdrawal. Thus Spatially-Varied Flow (SVF) can be classified into two
categmies:
i) SVF with increasing discharge, and
ii) SVFwith decreasing discharge.
SVF with increasing discharge finds considerable practical applications. Flows
in side-channel spillway. wash-water troughs in filler plants, roof gutters, highway
gutters are some of the typical instances. If the flow is subcritical eve1ywhere in the
channel, the control of the profile will be located at the downstream end of cham1el.
However, for all other flow situations, the determination of the control point is a
necessity to sta1t the computation. In an SVF with increasing discharges, the critical
depth line is not a straight line parallel to the bed as in GVF but is a curved line.
Depending upon the combination of the bottom slope, channel roughness and channd
geometly, the CJitical depth of spatially-va1ied flow can occur at a location somewhere
between the ends of the channel, giving rise to a profile which may he subcritical
during the first pa1t and ~upei"Clitical in the subsequent pa1t ofthe channel (Subramanya.
1990). A method of calculation of the critical depth and its location based on the
concept of' equivalent CJitical depth channel" has been proposed by Hind ( 1926 ). An
altemative method based on transitional profiles suggested by Smith ( l<l67) has the
advantages like simplicity and less tedious ~:olculations compared to Hind's method.
As ,;~vera] hydraulic engineering acti1·iti,., in open channel flow involve· the
computation of Spati8ll~ Varied Flo1~ (S\T). a .:omputatinn model ha,; been
de1·eloped for SVF in this chapter ·n1e discharge in a collector channel. having 7CJO
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inflow at the slatting point upstream, is the product of the constant lateral inflow
discharge per unit length and the length of the channel fi·om the slatting point upstream
to the section of interest. Thus one can write:
Q=~············································· .............................................. ~.6~
Where,
Q =discharge at distance x measured fi-om the slatting point upstream and,
q =lateral inflow discharge per unit length.
According to Chow( 1959), in case of the lateral flow beingnomllll to the collector
channel (assuming hydrostatic pressure and uniform velocity distribution), the dynamic
equation of at1 SVF is written in tenns of the vaiiation offree smface profile :
. d)'
dx
\Vhere,
S,-S-2(Dix)P
1-P
So= challllel bed slope, and
S = fiiction slope.
l11e Fronde number(F) is given by:
Q!A
F= =
/Qr
..................................... (1.6.2)
Solution of equation (I 6.2) can be obtain.:d using a tail Wdter condition or conn·ol
section. As the present analysis deal,; \lith tlu! ca;e where the discharge increases
d(mmtt eam resulting in a change of flow re.J!int-· li·ont su!K-ritical to supercritical in an
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SVF, the critical flow section serves as a .:ontrol section. Thus, equation (1.6:2) is
applied to the critical flow section, where Fronde number F is unity. Thus, one gets
S.,- S,- (D/x) = 0 .................................... ............. (1.6.4)
In the above, subscript 'c' refers to the parameters associated with the critical flow
section. At the critical flow section equation ( 1.6.6) becomes:
.................. ........................... 06~
The location of the critical flow section x,. can be obtained using equations( 1.6. 3) and
( 1.6.5) as:
.......................................................... (1.6.6)
As no physical solution can be obtained iiom (I. 6. 4) for cettain design parameters.
designers find it difficult to confum the existence of a ctitical section without any
mathematical guidance. To establish such guidance for obtaining feasible solution of equation
( 1.6.4) substituting equation ( 1.6.6) into equation ( 1.6.4) and nonnalizing, one gets:
s 2g
-+ -=) s, ;a, ....................................................... (/.6. 7)
\\here g = q 1(sjf"i15! l =Ali, t! = M 2• and I= l~ng.th parameter of channel. Also
equation (I 6 6) can be expressed in a nonHali7<:d fonn as·
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.. ........... . .......... ~6~
where~;= x.s jl. In the present study, the tenn S, is determined using the flow resistance
equations of Von I<annan (for rough regime) and Jain (I 976) (for transitional & smooth
regimes).
Rough Regime
For a complete rough regime, the Von Katman equation of flow resistance can be
used for the determination offiiction slopeS. The Von Karman eqation is: '
I kP ' '
[;: = I.I4- 0.86/n .................. " . ····· ......................... {1. 6 9)
4A '
The fiiction factor can be extracted from the equation ofbed shear as (Dey, 2000 ):
1:= p(Q/ At= (A/ P) S, ................................... (1. 6. 10) 8
Using equations(l.6.9) and( 1.6. 10), one can WJite:
P,Q',
s = .......................... (1.6.11) ,.
Replacing Qc from equation ( 1.6.5) into equation ( 1.6.11) and nomtalizing. }ields
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p -,
s ~ .................... ········· (1.6. 1 2) '
8T [ 114-086h• ( ~~'- ~' -,
where P = P!l, and K =KIf - -.~ ,\
Transitional Regime
In transitional regime, the Colebrook-White equation is applicable (Dey, 2000).
While the fiictionallosEes in full-bore flow pipes have been fully investigated by Colebrook
& White, a similar complete investigation ofthe Chezy coefficient C has not been completed.
· Titis is not only due to the extra vaiiables involved in fi"ee smface flows but also to the
wide range of surface rougbnesses met in practice and the difficulty in achieving steady
unifonn conditions outside the laboratmy. Defects in the setting of chamlelor sewer slopes,
obstmctions due to fuulty jii.Ilctions and the inherently II.IlsteadynatiJ.I·e of much fi·ee SII.Iface
flow combine to make such a study difficult (Douglas et. a!., 1999). The Ameiican Society '
of Civil Engineers 1963 study concruded that the behaviour ofChezy C could be infeiTed
directly from the full-bore flow fiiction factor as :
c ~ v (2g!A.},
where the approptiate value of the fiiction factor A was to be detemlined from the
Colebrook-White eqation with the charactetistic length being taken as the hydraulic mean
radius. R =AlP Nonnally the Colebrook-Wltite equation is recognized in its full-bore
flo\\ ti mn \dtere the appropriate value of R isd -1. Hence the coefficients will change in
the: equation as·
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l ......................... (1.6.13A)
Where Re is defi1~ed as pVR!p , V being the local cross-section mean flow velocity and
K, being the local roughness (Douglas et.aL, 1999).
Experimental investigations have shown that for partially filled pipe flow, where
the pipe diameter is less than a metre, the Colebrook-White-based resistance is more
accurate than the predictions of the Manning expression. At constant channel slope
Manning's n was found to vary with flow depth in pa1tially filled pipe flows where the
maximum depth was restiicted to less than a metre (Douglas et.al., 1999).
The advantage of using the Colebrook-White equation is that it covers the regime
of flow Jiomhydraulic smooth regime to rough regime (Dey, 2000). lbe rough or smooth
regime can be verified fl-om the values of relative roughness E ( = 0. 25k/A'1) and flow
Reynolds number Re (= 4QP1 V 1) using the well-known Moody diagram However, as
the Colebrook-White equation is an implicit equation oO .. , the use of flow resistance
equation proposed by Jain ( 1976)- which is the explicit from of the Colebrook-White
equation-makes the solution simpler. The equation of Jain ( 1976) is:
I ( KP 6./ro'>p f) f) l ' ' ' .................... (1.6.13) --r: =114-0.86/n + 4A Q,/1')
'
I 'sing equations ( I. 6. 5) and ( I 6. I 0 ). equ<i! ion ( I. tl 13) is e'\1nessed innonnalized fonn
as
- 79-
p -,
s ~ "' .. (1.6. 14) ,.
6. 1 !3", E,', I,'·'' ] 2
4A 135 -,.
where,
f3 ~ vl(g"'flS)
Smooth Regime
The complete smooth regime is seldom obtained in practice. The channel is con-
sidered to be smooth when the roughness elements are submerged by the laminar-sub layer.
l11e Nikuradse equation is applicable in smooth regime. As this equation is an implicit
solution ofA, Jain (1976) proposed an explicit fotm of the Nikuradse equation. The
equation is :
-~ - /.5146 .................................... (1.6.15)
Using equations( 1.6.5) & ( 1.6. 10), equation (I. 6. 15) is expressed innonnalized fotm as:
Ec s = ,.
1.5146 ]'
...................... ( 1. 6. I 6)
- 80-
Computational Steps
The equations developed in the preceding pages are used to design a collector
channel. As input data, the values of q, So, K, and other channel characteiistic parameters
are needed, such as for rectangular channel B; for trapezoidal challllel B, Z1
and Z2, and
for circular channel d The steps involved for the computation are given below :
I. Compute fL K, and !3.
2. Consider COlllplete rough regime. SubstituteS, from equation ( 1. 6.12) to equation
( 1.6. 7).
3. Compute .I>umerically, using the Muller method (Conte and de Boor,
1987)
4. Computed_,, f.,. and I,.·
5. Using(I.6.8)computeK,
6 Compute x, ( = K,!So./).
· 7. Compute~: and R.
8. Using the Moody diagram, check the hydraulic regime considered in
step 2.
9. lfagreeable, stop computation. Othe1wise. in step 2, use equation (1.6.14) or
( 1.6.16) in place of equation ( 1.6.12) and repeat steps 2 to 8.
Summary & Conclusion
In this section a numerical procedure has been presented for locating the ciitical
section of a collector channel having zero in flo\\ at the stalling point upstream and the
discharge in erasing dm\11;.1rcam resulting in a d1angc oflhm 1cgimc fiom subciitical to
supclcJitical in an SVF As the Colebrook- White equation is an implicit equation ofi ..
th c tcnn -". has hecn dc:t c:nnincd using the llo\\ -1 c:<;i>'l a nee equal ions nf\'on Kannan ti11
- 81 -