NOTE ON FLOW IN CANALS*

B y W . H . J U R N E Y (Bureau of Reclamation, Denver, Colorado)

Introduction. The design of irrigation systems to service a certain area sometimes necessitates the installation of pumping stations between each of which is a canal section with water being pumped in at one end and out at the other at constant rates.

Starting or stopping the pumps produces long waves which are sometimes referred to as “surge waves” or “bore waves,” with associated changes in height of the water surface. Such waves have been considered by M asse1 and Deymie.2 It is the prediction of these changes in height of the water surface, under constant inflow, constant out­

flow, or any combination of the two, which is the problem solved in this note. This is done by first considering an infinite canal with one source of constant inflow and using a method of “image” sources to produce the effect of reflections a t the ends of a finite canal. Simplifying assumptions3 are as follows:

1). The canal is of constant cross section throughout its length.

2). The effect of fluid velocity in the canal on the wave velocity is negligible.

3). Vertical accelerations of water are negligible.

4). The frictional resistance to flow is proportional to the velocity.

5). The height of water surface above normal depth, due to wave action, is small compared to normal depth.

Fundamental wave equations. The usual equations of motion and continuity a re:4

dr] dll B dll dq

= ■—— 1---m, (1); = -— > (2)

d x gd l g d x hdl

respectively, where x is the horizontal distance along the canal (ft.), t is the time (sec.), h is the normal depth of the water in the canal (ft.), q is the height of the water surface above normal (ft.), u is the horizontal velocity of the fluid (ft./sec.), g is the accelera­

tion of gravity (ft./sec.2) and A is a constant to be determined. In the computations on a special case for a finite canal, B was taken as igA/Q where A is the area of a vertical cross section of the canal (sq. ft.) and i is the slope of the canal computed from Chezy’s formula to give a flow of Q cu. ft. per sec. One alternative procedure is to find the frictional force resisting flow in the form4 X = cu2 and approximate this

* R e c e iv e d A p r il 18, 1944. T h i s in v e stig a tio n w a s c a rrie d o u t b y th e w rit e r in the T e c h n ic a l E n g in e e r ­ in g S e c tio n of the B u re a u of R e c la m a tio n w h ich is u n d e r th e d ire c tio n of S e n io r E n g in e e rs R . E . G lo v e r a n d I v a n E . H o u k . A ll d esigns a n d in v e s tig a tio n s of the B u re a u a re co n d u cted u n d e r th e d ire c tio n of J . L . S a v a g e , C h ie f D e sig n in g E n g in e e r. A l l en g in eerin g a n d co n stru c tio n w o rk is u n d e r the d ire c tio n of S . O . H a rp e r , C h ie f E n g in e e r, w ith h e a d q u a rte rs in D e n v e r, C o lo ra d o ; a n d a ll a c t iv it ie s of th e B u r e a u a re u n d e r th e d ire c tio n of H . W . B a sh o re , C o m m issio n e r, w ith h e a d q u a rte rs in W a sh in g to n , D . C .

1 P . M a s sé , Hydrodynamique fluviale, régimes variables, H e rm a n n , P a r is , 1935.

2 P h . D e y m ié , Proc. 5lh Int. Congress A ppl. MecJt., Cambridge, Mass., 1938, W ile y , 1939, p p . 5 3 7 -5 4 3 .

5 L a m b , Hydrodynamics (6th e d .), C a m b rid g e U n iv e r s it y P re ss, 1932, p. 254, e t seq.

4 L a m b , lo c . c it .

force by 0.75 cuQ/A. This gives the “least square fit” to the force X in the range of . velocities Q /A .

It is easily seen that, for the infinite canal of uniform cross section with constant inflow at a point, the boundary condition is w(0, t)= Q /A =constant, where Q is the constant inflow.

In order to reduce the problem to dimensionless form, it is convenient to make the substitutions arbitrary velocity at the origin, th at is, w(0, t). In particular, to complete the solution for the infinite canal problem as proposed, it is merely necessary to choose

Hence, after integrating and adding the initial effect, we have

£7(0,

= f Y < “-{>/o(a - Q tfft)# +

= 1. (12)

The solution of this integral equation is easily shown to be6

F'(a) = e-“{lo(a) + /i(«)}. (13)

The effect of applying increments of //(0 , a) may be represented approximately by A£708, a) k ¿ - < - » 7 0{ \ / ( a - 0 2 ~ 0 2} / v (£)<*£, (14)

0= { ^ ( a - { ) + / , ( « - { ) }

A £ 7 ( 0, a ) = - 0 { /o(a - £) + 7i(« - Q } +

a — £

1 - 3 0 5 { / 2( a - £ ) + 7 3( a - £ ) }

5 ! ( a - £ ) 2 ^{+ • ■ •}

] r

Thus, integration and the addition of the initial effect yields the solution of the prob­

lem as

r* cc—p _ _ _ _ _ _ _ _ _ _ _ _ _

e“ £7(0, a ) = I 7 0{V(< * - £)2 ~ 0 2} {7o(£) + 7i(£)}<*£ + 70{ V a 2 - 0 2} , (16)

j 0

r “H* r f , /33 { /i(a - £) + 72(a - £)}

e°/7(0, a) = I e“ f — 0 { 7 o(a — £) + I i ( a — £)} + — 7

-J o L 3 ! a : — £

1 - 3 0 5 { 7 2( a - £ ) + 7 3( a - £ ) }

5 ! ( a - £ ) 2 + ; 7 o ( £ ) + 71( £ ) } < f £

, . »1, / s i r / \ ) r ? { M + /.(«)} l-3 05{ 72( a ) + 73( a ) } _ -f ea — 0{7o(o;) + 7i(a) j + — —--- (17)

3! a 5! or

Equation (17), on which attention is now focused, may be transformed into slightly better form for computation, as follows. We write

r <*-» { / „ ( a - £) + 7n+1(a - £)}

A ; - - - -7T { 7 o ( £ ) + 7 x ( £ ) } 7 £

Jo (a - £)n

= e~a

f

J ³ S n

° { I n(s) + I n+i(s)}

' 0 Therefore,

7 7 (0 , a ) = 1 +

f

J 0

fi3 1-3/35

+ — { G i0 9 , a ) + U ( a ) } - — — {G 2( 0 , a ) + K , ( a ) } + • • • , ( 1 9 )

3! 5!

! H . T . D a v is , A survey of methods for the inversion of integrals of Volterra type, In d ia n a U n iv e r s it y S tu d ie s, N o s. 76 a n d 77, 1927, p . 51.

where

K n(a) = e-° {/»(«) + W « ) }

(20)

H(ß, a) as given by Eq. (19) is tabulated in Table 1 for the range of values three terms of each series involved in Eq. (19) giving the results to three decimal places. Tables of the modified Bessel function,7 in conjunction with a Simp­

son’s rule for five intervals, were used in making the computations. The results were checked by graphical integration. I t is to be noted that a horizontal row in Table 1 gives the history of the height above normal a t a fixed time, while a vertical column gives the height history at a fixed point.

T a b l e 1. H(ß, a)

0.0 0 .1 0.2 0 .3 0 .4 0 .5 0.6 0 .7 0.8 0 .9 1.0

0.0 1.000

0.1 1 .0 9 8 0 .9 0 5

0.2 1.1 9 1 0 .9 9 8 0 .8 1 9

0 .3 1 .2 8 0 1 .0 8 6 0 .9 0 7 0 .7 4 1 0 .4 1 .3 6 5 1 .1 7 1 0 .9 9 1 0 .8 2 4 0 .6 7 0 0 .5 1 .4 4 6 1 .2 5 3 1 .0 7 2 0 .9 0 4 0 .7 4 9 0 .6 0 6

0.6 1 .5 2 5 1 .3 3 1 1 .1 5 0 0 .9 8 1 0 .8 2 5 0 .6 8 1 0 .5 4 9 0 .7 1 .6 0 1 1 .4 0 7 1 .2 2 5 1 .0 5 6 0 .8 9 8 0 .7 5 3 0 .6 1 9 0 .4 9 6

0.8 1 .6 7 4 1 .4 8 0 1 .2 9 8 1 .1 2 7 0 .9 6 9 0 .8 2 2 0 .6 8 7 0 .5 6 2 0 .4 4 9 0 .9 1 .7 4 5 1 .5 5 0 1 .3 6 8 1 .1 9 7 1 .0 3 7 0 .8 8 9 0 .7 5 2 0 .6 2 6 0 .5 1 1 0 .4 0 6

1.0 1 .8 1 3 1 .6 1 9 1 .4 3 6 1 .2 6 4 1 .1 0 4 0 .9 5 4 0 .8 1 6 0.688 ^{0 .5 7 1} 0 .4 6 4 0 .3 6 6 N o t e : T h e ta b le w a s co m p u ted to m o re figures an d in th e rang e 0 b u t sin ce it is u sed h e re fo r illu s tr a t iv e purp oses o n ly , th e a b b re v ia te d form is g iven .

Finite canal section. The flow condition in a finite canal section of length L with constant flow Q a t one end (origin) is simulated by considering an infinite canal with sources of constant inflow Q located a t points 0, +2L, ±4Z , • • • . In the type of in­

vestigation for which this problem was solved, the maximum height in the canal was the prime consideration, and only a few reflections are needed to determine this maxi­

mum. Table 1 suffices to carry out the necessary computations. For the outflow case it is merely necessary to reverse results for inflow.

Remarks. Strictly speaking, the results of course apply only to a canal initially a t rest, and if in the case of a finite canal with a sloping bottom it is desired to com­

pute heights subsequent to a shut down, these should be referred to the surface in running position which, in the case of a properly designed canal, will be parallel to the bottom.

Heaviside solved the analogous electromagnetic problem for the infinite telegraph cable by use of operational calculus. But so far as the writer knows, the solution as applied to canals is unavailable elsewhere. It is seen th a t by the elimination of U or H from Eqs. (7) and (8), the differential equation which either one satisfies is

7 G r a y a n d M a th e w s, Treatise on Bessel functions, M a c m illa n an d C o ., L o n d o n , 1922.

d 2<t> 2 dp d 2p

— + — = — • (21)

da- da

dfl-Thus the problem for the infinite canal, of which Eq. (19) is the solution, is equivalent to th at of solving Eq. (21) (a, /3^0) with the boundary conditions

0(0, 0) = 0, 0(0, a ) = * - « [ * «

{3Jo(a)