Computation Method for Regulating Unsteady Flow in Open Channels

COMPUTATION M E T H O D FOR REGULATING UNSTEADY F L O W IN O P E N CHANNELS Downloaded from ascelibrary.org by Ku Leuven - Campusbibliotheek on 04/16/15. Copyright ASCE. For personal use only; all rights reserved.

By Fubo Liu, 1 Jan Feyen, 2 and Jean Berlamont 3 ABSTRACT: Unsteady flow problems in open channels can be classified as simulation- and operation-type problems, according to the objectives of the study. The simulation problem is for predicting the discharge and water level in the channel during the future time series under given conditions. The operation problem, on the other hand, is for defining inflow at the upstream end of the channel or the operation schedule of controlling structures in order to get the desired outflow at the downstream end of the channel. In this paper, a finite difference computation method is derived for solving operation-type problems in open channels. The method is explicit and numerically stable. It was applied to solve the St. Venant equations, discretized by the Preissmann scheme. The computation results were compared with the results obtained using the double-sweep method, and there was good agreement between the two methods. The complete momentum equation was used in both methods. This computation method is called a backward-operation method since it is for solving operation-type problems, and the computation is performed backward both in space and time.

INTRODUCTION

Computational hydraulics is very well developed for the simulation of one-dimensional unsteady flow, such as flood forecasting. Computer models also have been developed for operation-type problems, such as irrigation scheme control (Ahn 1990; Gichuki et al. 1990; Swain et al. 1991; Husain et al. 1991). Many of the models developed to solve operation-type problems were based on the simulation methods since they can describe the complete dynamics of unsteady flow. The simulation methods were developed to predict the changes of the flow when the upstream or downstream boundary conditions are changed. The methods become very tedious when they are applied to solve operation problems for complicated canal networks. Many different upstream inflow patterns and controlling structure schedules, as well as all kinds of combinations of them, need to be simulated to obtain the desired outflow. This is a very time-consuming process, and an exact solution cannot be easily obtained. Wylie (1969) proposed a transient control technique, known as gate stroking, to set gate movements and upstream inflow according to the prescribed downstream flow hydrograph. The technique used the method of characteristics. It was first demonstrated with a single canal pool (Wylie 1969), and further implemented for a series of pools separated by control gates (Bodley and Wylie 1978). Modification and application of this technique have been reported by Gientke (1974), Falvey et al. (1979), and Falvey (1987). However, the method of characteristics •Res. Asst, Ctr. for Irrig. Engrg., Katholieke Universiteit Leuven, Vital Decosterstraat 102, 3000 Leuven, Belgium. 2 Prof., Ctr. for Irrig. Engrg., Dept. of Agric. Sci., Katholieke Universiteit Leuven, Vital Decosterstraat 102, 3000 Leuven, Belgium. 3 Prof., Ctr. for Irrig. Engrg., Dept. of Civ. Engrg., Katholieke Universiteit Leuven, de Croylaan 2, 3001 Leuven (Heverlee), Belgium. Note. Discussion open until March 1,1993. To extend the closing date one month, a written request must be filed with the ASCE Manager of lournals. The manuscript for this paper was submitted for review and possible publication on June 20, 1991. This paper is part of the Journal of Irrigation and Drainage Engineering, Vol. 118, No. 10, September/October, 1992. ©ASCE, ISSN 0733-9437/92/0010-0674/$1.00 + $.15 per page. Paper No. 2120. 674

J. Irrig. Drain Eng. 1992.118:674-689.

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used in the gate-stroking algorithm is relatively more complex and unwieldy. The backward operation method presented in this paper, being an explicit finite difference method, can provide an easier solution. BASIC EQUATIONS

The St. Venant equations are the governing equations describing water flow in open channels. The differential form of the St. Venant equations may appear in many different ways when different independent variables are used (Cunge et al. 1980). Nevertheless, they essentially consist of the continuity equation and the equation of motion. Assuming no lateral inflows or losses, one may put the equations as dA dt dQ dt

dX

0

1 01

(1)

QA

dh_

(2)

dx

dx \ A

where A = wetted cross-sectional area; t = time; Q = discharge; x horizontal coordinate in the flow direction; g = acceleration due to gravity; h = water depth; S, = bottom slope of the channel; and Sf = friction slope. SIMULATION METHODS

Unsteady flow in open channels can be simulated by solving the St. Venant equations at a finite number of grid points in the x-t plane. The two types of schemes developed for the discretization of the St. Venant equations are explicit and implicit. Since the implicit schemes have the advantage of being unconditionally stable numerically, they are now more widely applied than the explicit schemes. The Preissmann scheme is the most widely used implicit method. It was introduced by Preissmann (1961), and has been adapted by a number of researchers. According to Abbott (1979), the general form of the Preissmann-type schemes (Fig. 1) can be expressed as

t

0AX

( 1 - 0 ) AX

o unknown grid x known grid

n+1 (1-fl)At

+

8 tit

M

j FIG. 1.

J+1

•

x

Preissmann Scheme Applied in Double-Sweep Method

675

J. Irrig. Drain Eng. 1992.118:674-689.


fx= h m'/++i1 ~f,/+1) + (1"e){f,/+i ~f/)] ft = xt [*(f?x - M + a - M/T 1 -m / = efo/y+i1 + (i - m+1]

+ (i - e)H>#n + (i - 4>)^']

(3)

(4) (5)

where/'/ = /(/Ax, «Af); 9 and c)> = weighting coefficients; Ax = space interval; and At = time interval. Introducing the Preissmann scheme into (1) and (2) yields the following linear algebraic equations for each two adjacent grid points (Liggett and Cunge 1975; Hromadka II et al. 1985): Axq, + Biqj+l

+ dyj + Diy,+1

+ £, = 0

(6)

A2qj + B2qj+1 + C2y,- + D2yj+1 + E2 = 0

(7)

where qf and yt = discharge and water-level increments from time level n to (n + 1) at grid point /; qJ+l and yj+1 are these at grid point (/ + 1); and A, B, C, D, and E are coefficients computed with known values at time level n. Each grid point has two unknowns, q and y. If there are p grid points, there will be 2p variables. There are (2p — 2) equations like (6) and (7). The equations are solved together with one upstream boundary condition and one downstream boundary condition in order to find the 2p variables at time level (n + 1). Any standard method of solving a set of linear algebraic equations can be used, and the double-sweep method is a very efficient one. A detailed explanation of the double-sweep method can be found in Cunge et al. (1980) or Hromadka II et al. (1985). BACKWARD-OPERATION METHOD

For simulation problems, one upstream boundary condition (e.g., discharge hydrograph) and one downstream boundary condition (e.g., discharge-water-level relationship) must be known to predict the flow condition in the channel as a function of time. A set of algebraic equations need to be solved simultaneously when an implicit scheme is applied, in order to define the values of all the unknowns. It is noticed that for operation problems, there are two downstream boundary conditions specified: the expected discharge and water level at the dowsnstream outlet. Knowing qp and yp between any two time levels at the last section of the channel, one can apply (6) and (7) for the last two sections p — 1 and p (Fig. 2), and solve qp_l and vp_x explicitly: „ - Q(^2g P + D2yP + E2) - C2(B,qp + Dxyp + EQ q l "- ~ AXC2-CXA2

(8)

„ - MB2qP ^ ^

(9)

+ D2yP + E2) - A2{Bxqp + D.y, + EQ CXA2-AXC2

With the calculated value of qp_1 and yp-u the value of qp^2 and yp-2 can be computed. This computation process is continued until the upstream boundary is reached, as shown in Fig. 2. The foregoing is a backward computation in space. This computing procedure cannot be applied easily because the changes of flow at the downstream end of the channel are caused by the changes 676

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o unknown grid x known grid n+2 n+1

FIG. 2. Sketch of Backward Computation in Space

o+n&t o+(n-1)at

////////////////////////////////////////////// FIG. 3. Changes of Flow Transferred from Upstream to Downstream

upstream (Fig. 3). If the time taken by the first propagation to travel from upstream to the downstream end of the channel is tc, the discharge and water level at the downstream-end section remain the same before the time level (f0 + fc), and the process between t0 and (t0 + tc) cannot be computed. Since the flow state at the time level (t0 + tc) is required for the computation after that time level, the problem cannot be solved. Fortunately, the aforementioned difficulty can be overcome by backward computation in time. Considering the time level (t0 + nAt) as the final condition, and knowing qp and yp between any two time levels at the downstream-end section, one may proceed backward in both space and time to compute the discharge and water-level profile at the time level [t0 + (« - l)At], as shown in Fig. 4. This computation process is continued until time level t0. The solution gives the discharge and water level in the channel at each section and moment, and defines the flow pattern at the upstream intake required to match the specification of the downstream flow. The physical meaning of the backward-computation method is clear. Knowing the expected outflow at the downstream outlet, one has to look backward, in both space and time, for the necessary upstream inflow. With the backward-computation method, the Preissmann scheme (Fig. 5) is written as dx

Ax

[(l - e)(/; - fu) + ©(/"

fU)\

677

J. Irrig. Drain Eng. 1992.118:674-689.

(10)


,

p-1

P


x

FIG. 4. Sketch of Backward Computation in Space and Time

t

(1-0)&X

h

•x

0ax

I )

$

•

4n-1

-)fAX

T 04


'"(1-0 *t

" ' ' .S»n

j-1

j

x

FIG. 5. The Preissmann Scheme Used in BO Method

f=if(l-W

/r 1 ) + w - i - /n1)]

/ = (i - 0)[(i - $)f] + w^]

• • (ii)

+ e[(i - 40/T1 + ¥U]

• • (12) With the Preissmann scheme in this form, the coefficients A, B, C, D, and E in (8) and (9) are computed from the later time level instead of the earlier time level. COMPARISON BETWEEN BACKWARD-OPERATION METHOD AND DOUBLE-SWEEP METHOD

The backward-operation (BO) method is compared with the double-sweep method to check its correctness. The test is for the unsteady state flow in a canal with the following physical characteristics: canal length = 2,500 m, side slope = 1.5, invert slope = 0.001, bottom width = 5 m, Manning's roughness coefficient = 0.025, and a fixed overflow weir with free flow condition as the downstream outlet. The discharge at the downstream outlet is expected to increase from 5 m3 s _ 1 to 10 m3 s~x in one hour, remain at 10 m3 s~x for two hours, then decrease to 5 m3 s~x in one hour (line BO downstream in Fig. 6). The water level at the downstream-end section (line BO downstream in Fig. 7) is deduced from the discharge using the discharge-water-level relationship of the fixed weir under free-flow condition. With the specified discharge and water level at the downstream-end section, 678

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o11 BO DS BO DS

0

3600

7200

10800

14400

downstream downstream upstream upstream

18000 21600

25200

time (sec) FIG. 6.

Comparison of Discharge between BO Method and Double-Sweep Method

BO downstream S downstream

BO upstream //DS upstream

I I I I I | 1 1 1 ! ) I l l I I I | || | | I

10800

14400

18000

21600 25200 time (sec)

FIG. 7. Comparison of Water-Level between BO Method and Double-Sweep Method

the discharge and water level at the upstream intake (line BO upstream in Figs. 6 and 7) are computed. Then, the upstream discharge hydrograph (line BO upstream in Fig. 6) obtained by the BO method is used as the upstream boundary condition, and the fixed overflow weir at the end of the canal is used as the downstream boundary condition, to simulate the flow in the canal with the double-sweep (DS) method. The simulation leads to the discharge hydrograph at the downstream-end section (line DS downstream in Fig. 6) and stage hydrographs at both the upstream- and downstreamend sections (lines DS upstream and DS downstream in Fig. 7). For both computations the complete momentum equation is used, with the weighting coefficient = 0.5, time interval At = 300 s, and space interval Ax = 500 m. 0 = 1.0 in the computation with the BO method, and 0 = 0.7 in the 679

J. Irrig. Drain Eng. 1992.118:674-689.


computation with the double-sweep method. Fig. 6 shows the comparison between the expected discharge used in the BO method and the discharge simulated by the double-sweep method at the downstream outlet. Since the upstream discharge hydrograph obtained with the BO method is used as the boundary condition in the simulation procedure, the discharge hydrograph at the upstream intake related to both methods is on the same line. Fig. 7 shows the comparison with respect to the water depth at both the upstream- and the downstream-end sections, and Fig. 8 shows the comparison of water surface profiles. As illustrated by these figures, the BO and DS methods yield very similar results. This proved that the upstream inflow defined by the BO method can generate the expected outflow. As seen in Figs. 6 and 7, during both the increasing and decreasing of the flow, there are spikes in the upstream discharge and water-level hydrographs before steady flow is reached. This is to satisfy the abrupt changes of the specified flow at the downstream end of the canal, and will be discussed in detail in the following. STABILITY OF BACKWARD-OPERATION METHOD

The stability of the Preissmann scheme applied to the simulation of unsteady flow has been discussed by a number of authors. The latest significant investigation was reported by Samuels et al. (1990), who concluded that the scheme must be forward weighted, with a time weighting coefficient 6 & 0.5, and that the flow must be physically stable. They used the standard technique of Fourier analysis in their investigation. From the work of Fread (1974), Lyn et al. (1987), and Samuels et al. (1990), it can be seen that the stability of a computation method depends on the formulation of the discretization scheme. Since the discretization form of the BO method expressed in (10)-(12) is essentially the same as the general form of the Preissmann scheme in (3)-(5), it can be inferred that the BO method has the same stability limits as the Preissmann scheme applied in the simulation

BO t=7200 ,DS t=7200

500

1000

1500

2000 2500 distance (m)

FIG. 8. Comparison of Water-Surface Profile between BO Method and DoubleSweep Method

680

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methods, although it is an explicit method. This is tested with numerical experiments by verifying the following parameters: the weighting coefficient 0 of the Preissmann scheme, the computation space interval Ax, and time interval At. The computation is to define the upstream inflow in order to meet the specification of downstream outflow. The canal on which the computations are performed has the same dimensions and characteristics as the canal used for the previous computation. The computation weighting coefficients 0 and $ and the space and time intervals are all the same as before, except for the parameter tested. The definition of the weighting coefficient 0 of the Preissmann scheme used in the BO method can be seen in Fig. 5. It has been found that the stability of the computation can be influenced considerably by the weighting factor 0 . Fig. 9 shows that the stability is improved when 0 increases from 0.6 to 1.0. The computation is not stable for 0 < 0.5, being the same as found for the implicit simulation methods. Fig. 10 shows very little change of the computed upstream discharge in the tested case when the space interval Ax is reduced from 500 m to 5 m. Numerical experiments under different situations reveal that the space interval has the same significance in the computation with the BO method as with the conventional simulation method. It is commonly believed that the explicit methods must obey the Courant condition (Courant et al. 1928) to ensure numerical stability, and the smaller the time interval the better the numerical stability. In apparent contrast to this, Fig. 11 shows that the BO method finds a smoother upstream hydrograph with a larger time interval. But, as a matter of fact, the oscillations of the upstream hydrograph do not indicate the instability of the BO method. For simulation problems, any upstream inflow hydrograph may be given, and a smooth downstream flow can always be found. As for operation problems, where the BO method is applied, strictly speaking, the specified downstream flow may never be exactly realized. In principle, the flow variation at the downstream end of the canal is a smooth process when changes are transferred from the upstream. However, the flow specification given

2

"|i T 111 n u n [ ii I I I i i i T t i | I I I I I I 11 ii i | I I I i H I i i i n m m II i i rfiTrri i ri n 11 n 1111 i r n r

0

3600

7200

10800

14400

18000 21600

25200

time (sec) FIG. 9,

Comparison of Computed Upstream Discharge with Different 6 Factors

681

J. Irrig. Drain Eng. 1992.118:674-689.

^

11

AX=5 AX=500


10

downstream

876-

3600

7200

10800

14400

18000

21600 25200 time (sec)

FIG. 10. Comparison of Computed Upstream Discharge with Different Distance Intervals

• n • • 11 • i • • 11 • i n • 111. • i n , , n i • • • • i m i • • i . i n 11 n i • i • i • • • n • i • • • • i

3600 FIG. 11. tervals

7200

10800

14400

18000

21600 25200 time (sec)

Comparison of Computed Upstream Discharge with Different Time In-

to the BO method always has abrupt changes and can never be ideally smooth. So with a smaller time interval, the upstream inflow can be defined more accurately, but this means more oscillations in order to meet more specifications of the downstream flow. That is shown in Fig. 12. The upstream flow is computed using the BO method with a time interval of 300 s and 60 s, respectively, and the obtained upstream discharge hydrographs are used as the upstream boundary condition to simulate the downstream flow using the double-sweep method with the same time interval of 60 s. As can be seen, the upstream flow hydrograph obtained with a smaller time interval has more oscillations and can provide a downstream flow closer to 682

J. Irrig. Drain Eng. 1992.118:674-689.


•^12 \ ?11 E

i .upstream at=60 ¥ upstream At=300 /downstream At=60 [1/ I / / d o w n s t r e a m at=30 300| expected discharge

rio u n ••o

8

765 4 3

iiiiiii|iilitiiiiiniiiiuiiiii|luliiiiiinnuiiiiiii|miiiinii|iiiiiiiiin

0

3600

7200

10800 14400 18000 21600 25200 time (sec)

FIG. 12. Comparison of Accuracy of BO Method with Different Time Intervals 5> 1200

downstream 2 km up 4 km up 6 km up 8 km up 10 km up

It 111 H [ I I I II IIII l l f l l I 111 It I I I j I t I I I U I I I

3600 7200 10800 14400 18000 21600 25200 28800 time (sec) FIG. 13. Discharge Computed by BO Method at Different Upstream Sections

what is expected. So, oscillations caused by smaller time interval do not indicate the instability of the BO method but its superior accuracy. However, the oscillations can cause the failure of the computation. Nevertheless, from the operational point of view, the larger time interval should be preferred. Besides the computational parameters, the length of the channel also contributes to the oscillations of the upstream flow obtained with the BO method. The results shown in Fig. 13 are for a canal length of 10,000 m and a bottom width of 20 m. The computation space interval is 1,000 m, and the time interval is 300 s. The discharge hydrographs at different upstream sections are computed with the BO method according to the desired discharge at the downstream outlet. It can be seen that the oscillations in 683

J. Irrig. Drain Eng. 1992.118:674-689.


the computed upstream discharge hydrograph are amplified with the distance from the downstream section. Again, the oscillations here have nothing to do with the stability of the computation method, although they may jeopardize the computation. Rather, they indicate that the required upstream flow is more strict for a longer channel in order to provide the same downstream flow pattern. It can be generally concluded that the BO method, though explicit, is numerically stable. As discussed, oscillations may occur in the upstream flow pattern defined by the BO method with a small computing time interval. Further, the oscillations can be amplified from downstream to the upstream of the channel, and finally may cause the instability of the flow. This is because, physically, the arbitrary specification of the downstream flow pattern is not always easy to achieve. A reasonably larger computation time interval can reduce the oscillations at a nonsignificant sacrifice of computational accuracy. In the example considered, discharge at the downstream outlet was arbitrarily specified to change linearly and abruptly. Methods to define the downstream outflow in an optimal way are being investigated. APPLICATION OF BACKWARD OPERATION METHOD TO CHANNEL NETWORKS

The BO method can be applied to both water-delivery networks (to find the amount of upstream inflow needed to ensure the downstream requirements), and to sewer systems (to define the limits of upstream release to minimize the downstream damage). Fig. 14 shows a water-delivery or a sewer system, in which water is distributed from one place to several downstream locations. If the outflow at each outlet is specified, with the BO method the operator will be able to find how the flow can be released from the intake, and how each controlling hydraulic structure in the system should be adjusted. The computation is performed backward from each outlet, and at the branch junction point the following set of equations can be applied:

(13)

G3 = fii + 2 2

FIG. 14. Flow Distributed from One Place to Several Downstream Locations

684

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H3l = MQx, HJ

(14)

#32 = f2(Q2,

(15)

H2)

H3 = max(H31, H32)

(16)

where Q = the discharge; and H = the water level. Qx and Hl are computed backward from branch 1, and H31, the controlling structure at position 1 needs to be adjusted to prevent the discharge released to branch 1 from exceeding the expected value Q1. The same principle applies whtn H31 > H32. Such a process can be continued until the most upstream intake is reached. The procedure explained herein is demonstrated for the sample canal network shown in Fig. 14. The physical characteristics of branches 2, 4, and 6 are identical with the first example. There is a rectangular sluice gate at sections 2, 5, and 8 to control the flow released from the main canal to branches 2, 4, and 6. The width of the gate equals 5 m and the discharge coefficient is 0.7. Branches 1, 3, 5, and 7 are also assumed to have the same physical characteristics as in the first example, except that the length of each branch is 1,000 m and the bottom width is 7 m. At the downstream side of branch 1 is a reservoir with constant water level. From the same moment, the discharge at outlets B, C, and D is expected to increase from 5 m3 s _ 1 to 10 m3 s _ 1 in 1 hr, remain at 10 m3 s _ 1 for 2 hr, and then decrease to 5 m3 s^ 1 in 1 hr. At the outlet A of the main canal, the discharge is expected to increase from 20 m3 s^ 1 to 40 m3 s^ 1 in 1 hr, remain constant for 2 hr, and then decrease to 20 m3 s _ 1 in 1 hr. With the BO method, the required discharge and water level at sections 3, 6, 9, and upstream intake section 10, as well as the gate opening at sections 2, 5, and 8 (Figs. 15,16, and 17), are computed. As seen in this example, the upstream inflow and the settings of each controlling structure can be found directly with the BO method, ^ 80 rO < JB. 70

section 10 section 9

n

e

0 J:

60

//

section 6. section 3.

« 50 .2 •a

outlet A

40 30 20

outlet B, C and Dv

10

0 " vynmiftiii n n n m i i i i ) i i n i i n n i | i n i i i i i u ifrrrrm 1111| 11111111111111111111

0 FIG. 15.

3600

7200

10800 14400 18000 21600 25200 time (sec)

Discharge Computed for Branched Canal Network

685

J. Irrig. Drain Eng. 1992.118:674-689.


section 10 v

1.2

I I I MIt I I I I I I I

3600

7200

10800 14400 18000 21600 25200 time (sec)

FIG. 16. Water Depth Computed for Branched Canal Network u.o -

c

A

0.70.6-

gate 2 / / \ gate 5 ^ / / /\

0.5-

gate S