ELSEVIER

Advances in Engineering Software Vol. 29. NO. 1, pp. 77-86, 1998

8 1998 Elwier Science Ltd

PU:SO965-9978(97)00027-6

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Technical Note

Recursive algorithm for steady flow in a canal network

Rajeev Misra Department of Civil Engineering, Indian Institute of Technology, Bombay 400 076, India

(Received 20 July 1994; revised 26 March 1997; accepted 22 May 1997)

A recursive-iterative algorithm to solve the equation for one-dimensional gradually varied steady flow in a canal network is developed. The recursion is used to sweep up and down the network, whereas iterations are used to obtain convergence for the discharge distribution at canal junctions. The convergence characteristics of different discharge updating techniques are discussed. The application of the algorithm is demonstrated through eight test problems on two networks, by starting the computations from different initial guesses of discharge distribution in the network. 0 1998 Elsevier Science Ltd.

NOTATION . A

bc Cd

C”

kfab

Ki

n

P Q R

SO Sf

T X

Y AQ ff

area of flow width of the control structure coefficient of discharge coefficient of approach velocity acceleration due to gravity head loss at a simple canal junction discharge distribution coefficient Manning’s coefficient height of control structure crest discharge hydraulic radius bed slope friction slope given by Manning’s equation top width distance along the canal depth of flow imbalance in junction continuity relaxation coefficient

1 INTRODUCTION

The numerical modeling of steady flow in a canal network can play an important role in irrigation scheduling of exist- ing canal systems and in design and planning of proposed canal projects. By and large, canals are designed based on the uniform flow concept, but uniform flow at all locations

in the network, even under design discharges, is not feasible. Analysis of steady flow in a single canal reach for a given discharge is simple and straight forward. Many numerical methods are available to solve gradually varied steady flow equations in a single canal reach.277 However, these methods cannot be applied directly to a network problem owing to the unknown discharge distribution in the network and the added complexities at canal junctions. Chow2 suggested a graphical procedure to compute a gradually varied flow profile across a river island assuming the continuity of the water surface at the channel junction. Wylie’ suggested an iterative procedure for the river island problem. Misra4 pre- sented an algorithm to compute steady flow, based on the assumption of uniform flow occurrence at upstream reaches of all the canals. This procedure leads to anomalies at canal junctions due to the presence of non-uniform flow. By and large, steady flow solutions in canal networks are obtained using a transient approach. In this approach unsteady flow computations are carried out for a long time, keeping all the control structures unmoved.4*6 Furthermore, it requires the specification of initial conditions, to start the unsteady flow computations. By and large, an over simplified solution of steady flow equations is used as the initial condition.’ The transient approach requires a considerably large computa- tional time. Many unsteady flow models have been devel- oped in the past.3 Interestingly, no attempt has been made to solve steady flow equations directly for canal/channel

77

78 R. Misra

a Canal number

@ Junction

[* Canal junction

b Down stream network boundary

*Head

Fig. 1. Layout of the study canal network.

networks. Such an approach may be computationally economical over the transient approach which is conven- tionally used.

This paper presents a recursive iterative algorithm for the analysis of steady flow in canal networks (Fig. 1). The recursion is used to sweep up and down the network. Itera- tions are performed at canal junctions to obtain a converged solution by redistributing discharges using different tech- niques. The model solves the dynamic equation of gradually varied steady flow using an explicit four point Runge-Kutta method. The model can handle any realistic combinations of locations and types of control structures at canal junctions. A large variety of control structures such as cross regulators, notches, weirs, gates, flumes, falls etc. are used. These struc- tures may discharge under submerged, unsubmerged and no flow conditions. The results obtained on eight test problems are presented. The convergence characteristics of different discharge distribution techniques are discussed.

2 GOVERNING EQUATIONS

The dynamic equation of gradually varied steady flow is written as ‘,

(1)

where y is the depth of flow, x is the distance along the canal, S, is the longitudinal bed slope, Q is the discharge in the canal, T is the top width, A is the area of flow, g is the acceleration due to gravity and Sr is the friction slope given by Manning’s equation written as,

(2)

where n is Manning’s roughness, R is the hydraulic radius.

-0- Computational node

- -

a b

Main canal

Fig. 2. Schematic diagram of a typical canal junction.

at the downstream (upstream) end for subcritical (super- critical) flow. However, in a network, the discharge in any canal is unknown and hence the boundary conditions at both the ends of all the canals in the network are to be specified. A large variety of control structures such as weirs, notches, gates, cross-regulators, flumes etc. are used to regulate/distribute water in different parts of the network. These structures may discharge under submerged, unsub- merged and no flow conditions. The water surface may fall below the gate. The gates may be kept at fully opened, partially opened or closed. The steady flow discharge for these controls are used as boundary conditions. The sub- mergence limit defined for a particular type of control struc- ture’ decides whether the structure is submerged or unsubmerged and the discharge equation to be used. These boundaries may occur in different combinations of types and location of control structures at canal junctions. At all canal junctions, the boundary conditions of the con- necting canal ends are to be satisfied simultaneously along with the junction continuiq equation. At junction locations, where there is no control structure, the energy equation is taken as the boundary condition. The set of boundary condition equations to be satisfied at a typical canal junction (Fig. 2) are written as follows.

Junction continuity equation

Q,-Qb-Qc=O Junction energy equation

(3)

Weir discharge equation

(4)

where subscripts a, b, c stand for nodes a, b and c (Fig. 2), AHrab is the loss of head between nodes a and b, Cd is the coefficient of discharge, p is the height of the weir, b, is width of the control structure, and g, is the gate opening.

3 BOUNDARY CONDITIONS 4 RECURSIVE METHOD

For a gradually varied flow in a single channel, the dis- charge Q is known and one boundary condition is specified In this method the recursion is used to sweep up and down

Recursive algorithm for steady jaw in a canal network 79

in the network whereas iterations are performed at canal junctions to obtain a converged solution by redistributing discharges using different techniques. The model solves the steady flow eqn (l), using an explicit fourth order Runge-Kutta method. The algorithmic structure is such that the local convergence at different canal junctions is obtained before obtaining the converged solution in the entire network.

4.1 Description of the method

For the current or assumed discharge distribution in the net- work, the flow of algorithm starts at the headworks (A in Fig. 1). The algorithmic control is shifted to the immediate/ next downstream junction. At the junction, the algorithmic control is further shifted to the next downstream junc- tions, until a downstream network boundary is reached (for example F in Fig. 1). At the network boundary, the flow depth for the current canal discharge is computed from the network boundary condition using Newton’s method. For this computed end depth, the gradually varied flow profile is evaluated to obtain the flow depth at the upstream end of the canal. This procedure is repeated for all the downstream canals of the current junction (D in Fig. 1). The computed depths downstream of the junction, when used with the junction boundary conditions, may not yield a unique upstream depth as there may be an error in the current discharge distribution. The discharges in all the canals downstream of the junction are corrected in a way to satisfy the junction continuity eqn (3) and boundary condi- tions eqns (4) and (5). Having obtained the new discharge distribution at the junction, the algorithmic procedure of computing end depths followed by gradually varied flow analysis is repeated until the convergence is reached at this junction (D in Fig. 1), i.e. the junction continuity and boundary conditions are satisfied.

Once, for a given junction inflow, the converged solution is obtained, the gradually varied flow computations are performed for the canal upstream of the junction. The con- trol is shifted to the upstream canal junction and gradually varied flow computations are performed for all canals downstream of the junction. A new distribution is obtained at this junction and the algorithmic procedure of com- puting flow is repeated for all the junctions and network boundaries downstream of this junction. This procedure is repeated until the convergence is reached at the head- works.

It is obvious from the algorithm outlined above that fewer iterations are performed at the junctions upstream in the network compared with those downstream, i.e. the least number of iterations is performed at headworks whereas the number of iterations performed is maximum at downstream network boundaries. This algorithmic struc- ture of achieving localized convergence before achieving the global convergence is to be adopted to ensure conver- gence even if the initial assumption for discharge distribu- tion is poor.

4.2 Implementation of the method

The algorithmic structure suggests the use of recursion for its implementation. Recursion is supported by most of the modem programming languages such as Pascal, Algol, C etc. However this feature is not supported by FORTRAN. The recursion is the technique of defining/addressing a procedure (SUBROUTINE) to itself. It may be possible to code the present algorithm using iterations. However, the use of recursion in coding the present algorithm results in an elegant and efficient program. The shifting of the algo- rithmic control up and down in the network is performed by recursion in the code without any need for extra data or tracking routines.

To give an insight to the algorithmic structure, a high level pseudo-code of the recursive procedure, and flow of algorithmic control in analyzing the network given in Fig. 1, are presented in Figs 3 and 4. The following are the nota- tions used in the pseudo-code: junc is the variable used to denote both junctions and network boundary; canahjunci) is the global canal number of the ith downstream canal at junc.

The pseudo-code in Fig. 3 shows that the recursive calls of the procedure STEADY-FLOW are made only at the headworks and canal junctions. At any junc, the computa- tional procedure is repeated until the convergence is reached. At network boundaries no recursive calls are made and once the depth is calculated at this location, the gradually varied flow computations are performed for the canal. At junctions and headworks the procedure of computing the upstream depth and discharge distribution is repeated until junction continuity and boundary condi- tions are satisfied. The gradually varied flow computations are performed in the canal upstream of junc, only when the converged solution at junc is obtained. Figure 4 shows that the end depth and gradually varied flow computations in canals 4 and 5 are repeated until convergence is reached at junc = 3. A new distribution of discharges is obtained at junc = 2 and the computations in canals 2,3,4 and 5 are repeated. Figure 4 also shows that for a new distribution at junc = 1 the computations are repeated in the entire network.

4.3 Discharge distribution at junctions

At a junction (Fig. 2) the depths at all nodes downstream of it are available from gradually varied flow profile compu- tations as discussed earlier. For these downstream depths and discharges, the corresponding upstream depths (yai) are calculated from the boundary conditions. The boundary conditions are usually non-linear and are solved using Newton’s method. The average of these depths is taken as the depth at the node upstream of the junction as seen in the pseudo-code given in Fig. 3. At a typical canal junction given in Fig. 2, this procedure reduces to the following.

l Compute upstream depth yah’ for known Qb and yb using eqn (4).

R. Misra

Procedure STEADY-FLOW (junc :integer);

Begin convergence='NOT REACHED'; Repeat

If junc='Head works' Then Call STEADY-FLOW(junc = D/S junction to canal[junc,'l]); correct discharge at the headworks set convergence flag Print "Discharge at Head works=",junc Print 'Convergence =n,convergence

Else If junc='Canal Junction' Then For i:=l to no-d/s-canals[junc] do

Call STEADY-FLOW(junc = D/S boundary to canal[junc,i]); End For compute average u/s depth distribute discharges at canal junction set convergence flag Print "Depth and Discharge at junction =",junc Print "Convergence =n,convergence If convergence -'REACHED' Then

compute C.V. flow profiles in canal U/S of junc Print "C.V. Flow in canal =",U/S of junc

End If Else If junc='d/s Network Boundary' Then

solve for end depth using Boundary Conditions compute C.V. flow profiles in canal U/S of junc Print "Depth at N.B. =",junc Print "G.V. Flow in canal =", canal U/S of junc

End If Until (convergence='REACIIED') end { Recursive Algorithm ]

Fig.3. Pseudo-code of the recursive algorithm.

l Compute upstream depth yac’ for known Q, and yc using eqn (5).

l The depth at node a is given by

ya = Lb’ +y*,'1/2.0 (6)

Using the average value of upstream depth y,, and depths yab’ and yac’ calculated from boundary conditions, the dis- charges are redistributed at the junction (Fig. 2) as

af’=Ql=a[AQ’+AQF*] i=b,c (7)

where n is the iteration number and (Y the relaxation coefficient. AQi* is the discharge correction due to the imbalance in junction continuity equation given by

(8)

where AQ is the net imbalance in junction continuity equation given by

AQ=Qi-Qi-@ (9)

In eqn (7), AQi** is the discharge correction due to the imbalance in depths y& and yoc at the upstream node and

is given by

AQT* = Ki(ya - y,i) i = b, c

where Ki is a positive number.

(10)

For a relaxation coefficient CY = 1 .O and the same Ki value for all the canals at the junction (i = b,c), the junction continuity equation will not be violated with iterations. That is, AQi* = AQ = 0, if the initial discharge distribution satisfies the junction continuity equations. This can be veri- fied from eqns (6)-(g). However, when different Ki values are used for different canals at the junction and/or the relaxation coefficient is other than unity, the junction con- tinuity equation is violated. As the iterations progress, both the discharge corrections AQi* and AQi** become smaller and smaller, and after certain iterations convergence is achieved to the desired accuracy. The choice of a! and Ki

is discussed in the subsequent sections.

4.4 Choice of Ki

The discharge correction coefficient Ki can be chosen in many ways to arrive at a convergent discharge distribution at junctions and a family of numerical schemes can be developed based on the method outlined above. In the pre- sent analysis three different schemes are tried. In scheme 1

Recursive algorithm for steady pow in a canal network

{ Starting Steady flow Analysis Using Recursive Iterative Algorithm 1

- Depth at N.B.= E; C.V.Flow in canal = 4; - Depth at N.B.= F; C.V.Flow in canal = 5; - Depth and Discharge at junction = D; convergence = NOT Reached - Depth at N.B.= E; G.V.Flow in canal = 4; - Depth at N.B.= F; C.V.Flow in canal = 5; - Depth and Discharge at junction = D; convergence = NOT Reached

- Depth at N.B.= E; G.V.Flow in canal = 4; - Depth at N.B.= F; C.V.Flow in canal = 5; - Depth and Discharge at junction = D; convergence = REACHED - G.V.Flow in canal = 3; - Depth at N.B.= C; C.V.Flow in canal = 2; - Depth and Discharge at junction = B; convergence = NOT Reached - Depth at N.B.= E; G.V.Flow in canal = 4; - Depth at N.B.= F; C.V.Flow in canal = 5; - Depth and Discharge at junction = D; convergence = NOT Reached

- Depth and Discharge at junction = D; convergence p REACHED - C.V.Flow in canal = 3; - Depth at N.B.= C; G.V.Flow in canal = 2; - Depth and Discharge at junction = B; convergence = REACHED - G.V.Flow in canal = 1; - Discharge at Head woiks = A; convergence = NOT Reached - Depth at N.B.= E; G.V.Flow in canal = 4;

- Depth at N.B.= F; G.V.Flow in canal = 5; - Depth and Discharge at junction = D; convergence = REACHED - G.V.Flow in canal = 3; - Depth at N.B.= C; G.V.Flow in canal = 2; - Depth and Discharge at junction = B; convergence = REACHED - G.V.Flow in canal = 1; - Discharge at Head works = A; convergence = REACHED

{ Terminate the execution converged solution achieved )

Fig. 4. Flow of algorithm control.

at a typical canal junction (Fig. 2), Ki is obtained as,

Ki= ff i=b,c (11) (1

In scheme 2, Ki is obtained as,

K.=!hi=b ( I 9 (12) Yai

It should be noted here that in scheme 1, Ki is only a function of flow variables in canals upstream of the junc- tion and hence is constant for all the canals downstream of the junction. Scheme 2 gives different values of Ki for different canals at the junction. In scheme 3, the expressions for Ki are derived by differentiating the sim- plified form of boundary conditions with respect to the

81

upstream depth. For an over-shot type of control structure (e.g. weir), the boundary condition can be expressed as,

Qi = C(Yoi - Pui)3’2 (13)

= constant (assumed). Differentiat- to yai, we obtain the expression

hi

Using eqn (10) and eqn (14), Ki can be written as,

(14)

(15)

Similarly, for free flow through an under-shot type of

R. Msra

Table 1. Geometric elements of canals in network I

Bed width Side slope Bed slope Manning’s n Length (m)

Canal 1

10.0 I:2

1 in 5000 0.030

4000

Canal 2

5.0

1:2 1 in 2000 0.030 3000

Canal 3

5.0 1:2

1 in 2000 0.030

2000

Canal 4

2.0 1:1.5

1 in 500 0.030

1000

Canal 5

2.0

1:1.5 1 in 500 0.030 700

control structure (sluice gate), the discharge equation is written as,

Q= J2gCdbWJOt,i -PUN) - &V

and Ki is obtained from,

(16)

Ki=: Qi

2 Olai -Ihi) - 6w > (17)

For a submerged gate, the discharge equation is written as,

Q = q’%Cdy/~ (18)

and for this case, Ki is obtained from,

(19)

In eqn (16) and eqn (18), w is the gate opening, 6 is a contraction coefficient, pU is the sill height above the upstream canal bed.

For no control structure boundary Ki is taken as,

(20)

5 APPLICATION 5.2 Initial guess

The model is verified through applications on symmetrically bifurcating canal networks and other idealized systems where the model results can be verified through direct checks. The results of other cases are verified using a tran- sient approach.5 A very good agreement of results between the two approaches is obtained.

5.1 Test problems

To demonstrate the applicability of the algorithm and to discuss the convergence characteristics, the results of

analysis on eight different problems on two networks are presented. The layout for both the networks is the same as given in Fig. 1. The geometric elements of different canals of networks I and II are given in Table 1 and Table 2 respectively. All the canals in both networks are trapezoidal in shape. The details of all eight problems studied on two networks are given in Table 3. In problems 1 and 5 there is no control structure at both the canal junctions B and D (Fig. 1). Problems 2 and 6 simulate the effect of submerged gates at all the downstream canals at junctions B and D. In problem 3 and 7 weirs (free or submerged) are taken as boundary conditions at both the canal junctions B and D (Fig. 1). Problems 4 and 8 are designed to study the con- vergence characteristics of a mixed junction boundary problem with different types of control structures. An unsubmerged gate with a constant reservoir head of 3.5 m is taken as the boundary condition at headworks for all eight problems studied. The uniform flow equations are used as boundary conditons at network tail ends C and E. A broad crusted weir discharging free is taken as the bound- ary condition at F for all the problems studied. For control structures such as weirs and gates, suitable values of para- meters such as widths, crest heights and crest lengths and gate openings are assumed.

To start the recursion/iteration procedure the start values of discharge distribution and water surface are to be speci- fied. The major problem in solving directly the steady flow equations is to obtain the unreasonable initial guess of unknowns and in view of this, in many cases a transient approach is used to obtain the steady flow solution.5’6 The convergence characteristics of the algorithm pre- sented are tested for a wide range of initial guess values of discharge distribution on several networks. The robust- ness of the algorithm is demonstrated through depth and discharge convergence plots obtained for problem 1 at

Table 2. Geometric elements of canals in network II

Bed width Side slope Bed slope Manning’s n Length (m)

Canal 1

10.0 1:2 1 in 7500 0.015 4000

canal2 Canal 3 Canal 4 Canal 5

8.0 5.0 3.0 2.0 1:2 1:2 1:1.5 1:1.5 1 in5000 1 in4000 1 in2000 1 in2000 0.020 0.022 0.025 0.030 3000 2000 1000 700

Recursive algorithm for steady jlow in a canal network 83

Table 3. Detsils of the problems studied

Problem Junction and network boundaries Network

A B C D E F

Canal I Canal 1 Canal 2 Canal 3 Canal 2 Canal 3 Canal 4 Canal 5 Canal 4 Canal 5

1 UG - NC NC UF - NC NC UF W I 2 UC - G G UF - G G UF W I 3 UG - W W UF - W W UF W I 4 UG - NC G UF - W G UF W I 5 UG - NC NC UF - NC NC UF W II 6 UG - G G UF - G G UF W II 7 UG - W W UF - W W UF W II 8 UG - NC G UF - W G UF W II

Note: UG unsubmerged gate; G submerged gate; UF uniform flow; NC no control structure; W weir.

junction D for the following initial guesses for discharge distributions.

l Initial guess on discharge distribution is an over- estimation.

l Initial guess on discharge distribution is an under- estimation.

l Initial guess is obtained from the approximate steady flow method, which is based on the assump- tion of uniform flow at all the nodes downstream of a junction4.

The variation in average upstream depth at junction D, as obtained from eqn (5), with iterations is plotted in Fig. 5 for all three initial guesses. Figure 6 presents the variation in discharge in canal 4 for the same problem. Scheme 2 is used in the computations. Steep jumps in depth and discharge values in Figs 5 and 6 indicate the effects of redistributions made at junction B or headworks (Fig. 1). Convergence characteristics similar to those in Figs 5 and 6 are observed

--------

- -- Over estimation ----.- Under estimation

- Approximate analysis

40 60 80 100 120 140

ItoratIons

Fig. 5. Convergence path of depth upstream otjunction D for different initial guesses of discharge distribution (problem 1).

for schemes 1 and 3 and for all the problems. It is generally observed that the approximate steady flow method provides a reasonable guess of initial discharge distribution. Hence, this algorithm is recommended to give the initial distribu- tion of discharges.

5.3 Comparison of different discharge distribution schemes

The performance of the present algorithm largely depends on the initial guess, discharge correction coefficient Ki

and relaxation coefficient CL It is observed that the per- formance of a scheme (selection of Ki) largely depends on the combination of control boundaries present at canal junctions. Control structures occur in different combina- tions of types and locations at canal junctions. These struc- tures may discharge under submerged, unsubmerged and no flow conditions. However, algorithmically canal junc- tions can be broadly classified as the following.

0 20 40 60 80 100 120 140 lteratlons

Fig. 6. Convergence path of discharge in canal 4 for different initial guesses of discharge distribution (problem 1).

84 R. Msra

2 60- .0 z r =

40-

20- A Scheme 1 o Scheme 2 * Scheme 3

01 I I I I I 0.4 0.6 0.8 1.0 1.2 1.4

a 1.t

Fig. 7. Convergence characteristics of different schemes for a canal network with type I canal junctions (problem 5).

Type I: no control structure at canal junction (pro- blems 1 and 5). Type II: under-shot type of control structures (e.g. gates) in all the canals downstream of a junction (problems 2 and 6). Type III: over-shot type of control structures (e.g. weirs) in all the canals downstream of a junction (problmes 3 and 7). Type IV: a combination of different types (problems 4 and 8).

The performance characteristics of all three schemes for problems 5, 6 and 7 are presented in Figs 7-9. Similar

02 0.4 0.6 0.8 1.0 1.2 1.4 a

Fig. 8. Convergence characteristics of different schemes for a canal network with type II canal junctions (problem 6).

0 scheme 2 n Scheme 3

ot I I I I I 0.4 0.6 0.8 1.0 1.2 1.4

02 1.6

Fig. 9. Convergence characteristics of different schemes for a canal network with type III canal junctions (problem 7).

characteristics are observed for other problems studied. The performance index for the present comparison is taken as the total number of discharge redistribution itera- tions performed at junctions B and D. It should be noted here that every discharge redistribution iteration performed at junctions B or D is followed by gradually varied flow computations in the two downstream canals. Hence, the total number of redistribution iterations is directly a mea- sure of the total work load involved to arrive at the final solution. For the sake of comparison in the present analysis, the work load involved in computing varied flow profiles in different canals is assumed to be the same, though it depends on the number of computational nodes considered. Scheme 1 performs best for problems with type I and type III junctions as in Figs 7 and 9. However, this scheme is very sensitive to the value of relaxation coefficient (a) for pro- blems with type II junctions (Fig. 8). Scheme 2 exhibits average convergence characteristics for problems with type I and type II canal junctions (Figs 7 and 8). However, its performance for problems with type III canal junctions is not satisfactory (Fig. 9). Scheme 3, which is based on a simplified form of boundary conditions, performed well for all the problems studied. Moreover, for this scheme, the optimal relaxation coefficient is found to be fairly constant for all problems with similar types of canal junctions. For scheme 3, the optimal relaxation coefficient for problmes with type I junction is found to be between 1 .O- 1.1 (Fig. 7). For problems with type II canal junctions, the optimal value of the relaxation coefficient is between 0.7 and 0.8 (Fig. 8), and for problems with type III canal junctions, the optimal value of the relaxation coefficient is close to 1.0 (Fig. 9). For problems with mixed control boundaries at junctions (type II), a single relaxation coefficient as the average of optimal values for different types of boundaries is recommended.

Recursive algorithm for steady jlow in a canal network 85

2.85 0 10 20 30 40 50 60 70 a0 90

lterotlon

Fig. 10. Convergence path of depth for different relaxation coefficients (problem 3).

Figures 10 and I 1 present the typical convergence paths followed by the average depth upstream at junction D, for optimal, under and over relaxations. These plots are obtained for problem 3 using scheme 3. The convergence of the over relaxation case is slowed down owing to oscilla- tions as seen in Figs 10 and 11 as may be expected. There may be situations when using over relaxation, where these oscillations amplify and the algorithm may fail to converge. With under relaxation, the convergence is guaranteed though it may be slow. The flatter portions in convergence paths (Figs 10 and 11) show the redistribution iterations performed at junction D, whereas the steep changes indicate the effect of redistribution made at junction B.

‘O *Ii-------l --- Over relaxation X=1.4 ------ Under relaxahon orz0.6 - Ophum relaxation a:l-0

aa1 I I I I I I

0 10 20 30 40 50 60 70 a0 90 ltefations

Fig. 11. Convergence path of discharge in canal 5 for different relaxation coefficients (problem 3).

Table 4. Discharge distribution in the networks

Problem Discharge in cumecs

Canal 1 Canal 2 Canal 3 Canal 4 Canal 5

1 36.205 17.828 18.376 11.666 6.710 2 36.205 18.643 17.562 9.678 7.885 3 36.205 17.825 18.380 9.094 9.286 4 36.205 23.877 12.329 8.322 4.007 5 36.205 20.460 15.745 9.269 6.477 6 36.205 24.473 11.732 7.378 4.354 7 36.205 22.652 13.553 8.255 5.298 8 36.205 25.665 10.541 5.607 4.934

5.4 Discharge distribution

The results of the analysis made for the eight problems (Table 3) on two canal networks are presented in Table 4. The discharge at a canal junction is distributed depending on the type of control structure and its discharging condition.

5.5 Acceleration of convergence

All the results presented so far are obtained with a fixed criterion of lop4 for bcth depth and discharge changes at all the junctions. The computational efficiency can be con- siderably improved by using a course convergence criterion for initial iterations at downstream junctions (e.g. D in Fig. 1) and improving it by a fixed factor each time dis- charge corrections are made at the headworks until the desired accuracy limit (e.g. 10w4) is reached. Table 5 presents a comparison of efficiency, in terms of total number of discharge distribution iterations performed at junctions B and D. Scheme 3 is used. The results suggest that the efficiency of the algorithm can be considerably improved.

6 CONCLUSION

A recursive iterative algorithm to solve directly gradually varied steady flow equations in a canal network is devel- oped. The results obtained for eight test problems on two

Table 5. Comparison of algorithmic efficiency with gradual change in convergence limit

Problem

1 2 3 4 5 6 7 8

Number of iterations

Recursive method Modified method

35 20 41 23 26 22 27 20 34 23 34 22 32 24 58 24

86 R. Misra

canal networks are presented. The structure of the proposed algorithm is such that convergence is achieved in a phased manner avoiding numerical oscillations. At canal junctions the discharges are corrected in each iteration using three alternate schemes. The convergence characteristics of the three schemes are discussed in terms of the variation of the number of iterations with relaxation coefficient. Scheme 3 derived from a simplified form of boundary conditions is recommended, as it is found to be efficient for different types of canal junctions. The robustness of the algorithm is demonstrated by obtaining the solution starting from different initial guess values.

2. Chow, V. T., Open Channel Hydraulics. McGraw-Hill, 1959. 3. Contractor, D.N. and Schuurmans, W. Informed use and

potential pitfalls of canal model. J. Zrrig. Drain. Eng., AXE, 1993, 119(4), 663-673.

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5. Misra, R., Sridharan, K. and Mohan Kumar, M. S. Transients in canal networks. J. brig. Drain. Eng. ASCE,, 1992, 118(S), 690-707.

6. Navarro, P.G. and Saviron, J.M. McCormack method for numerical simulation of one dimensional discontinuous unsteady open channel flow. J. Hydrol. Res., 1992, 30(l), 95-105.

7. Paine, J.N. Open channel flow algorithm in Newton Raphson form. J. Irrig. Drain. Eng., ASCE, 1991, 118(2), 306-3 19.

8. The ASCE Task Committee on Irrigation Canal System Hydraulic Modeling, Unsteady flow modeling of irrigation canals, J. Irrig. Drain. Eng., ASCE, 193, 119(4), 615-631.

9. Wylie, E. Water surface profiles in divided open channels. J. Hydrol. Res., 1972, 10(3), 325-341

1. Bos, M. G., Discharge Measuring Structures. Oxford and IBH Publishing Co., New Delhi, 1976.