..

- ... TA 416 .B87 HYD-325 1951

UNITED ST ATES DEPARTMENT OF THE INTERIOR

BUREAU OF RECLAMATION

STABLE CHANNEL PROFILES

Hydraulic Laboratory Report No. Hyd-325

ENGINEERING LABORATORIES BRANCH

COMMISSIONER'S OFFICE DENVER, COLORADO

September 27, 1951

..

_,

\

DATE DUE

t..o I Uol o 3

-GAYLORD

PRINTED IN U.S.A.

The information contained in this re -

port may not be used in any publica­

tion, advertising, or other prom otion

in such a manner as to constitute an

endorsement by the Government or the

Bureau of Reclamation, either e xpl ic i t

or implicit, of any material, p roduct,

device, or process that may be re ­

ferred to in the report.

-- -._ .. -

Introduction ••••• Basis ot Computations

CONTENTS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotation • • • . . . • . . • . . . • • • • • . • • . • Stable Channel Shape with Drag Proportional to Depth. Stable Channel with a Drag Force Proportional to

the Square ot the Velocity and no Lateral · Transter ot Drag • • • • • • • • • • • • • • • •

Stable Channel Shape with Lateral Drag Transter Accounted tor ••••••••••••••••••

Discussion ot Compilation of Data and Curves ••••• Example,· • • • • • • • • • • • • • • • • • • • • • • • Acknowledgments • • • • . • • • • • • • • • • • • • •

Computation tor S • 0~3 Computation tor S • 0.5 Computation tor S • o.6 Computation tor S • o.8 Computation tor S • 1.0

LIST OF TABLES

. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LIST OF FIGURES

Channel cross section • • • • • • • • • • • • • • • • Element of a wide channel ••••••••••••••

. 2 Comparison of values of (J-) for the drag relation

0

and the modified Chezy relation ••••••••• Properties of stable channels •••••••••••• Stable channel profiles found as a solution tor the

example • • • • • • • • • • • • • • • • tit • • • •

1 1 2 4

5

7 13 15 19

Table

1 and 2 3 and 4 5 and 6 7 and 8 9 and 10

Figure

1 2

3 through 7 8 through 12

13

UNITED STATES DEPARTMENT OF THE INTERIOR

BUREAU OF RECLAMATION

Design and Construction Division Engineering Laboratories Branch Denver, Colorado

Laboratory- Report No. Hyd-.325 Hydraulic Laboratory-Compiled by: R. E. Glover and

Q. L. Florey Reviewed by: E. W. Lane

September 27, 1951

Subject: Stable channel profiles

INTRODUCTION·

In order to deal intelligently with certain problems or maintenance of channels in erodible materials, it is important to know what shape and dimensions a channel should have in order to avoid changes of cross section due to scour or sediment deposition. A knowledge of stable shapes finds application to the problem or designing canals in earth and to the problems or river regimen. The computations of stable shape patterns described herein have been made at the request of Mr. E.W. Lane who has also laid down the basis for the computations.

BASIS OF COMPUTATIONS

The problem of finding a stable channel shape is here based on the following assumptions:

a. The particle is held against the bed by the canponent or the submerged weight or the particle acting normal to the bed.

b. At the edge of the stream the particles are at the angle of repose under the action or gravity.

c. At the· center or the stream the drag force of the flowing water holds the particles at the point or incipient instability.

d. At points between the edge and the center the particles are held at the point or incipient instability by the resultant of the gravity component or the particle's submerged.weight and the drag force or the fiowing water.·

In dealing with the drag force of th~ flowing water, two assumptions have been used as follows: ·· ·

e. The drag fore~ ·acting ··on ·an ar~a of the bed of the stream is produced by the component of the weight of the water directly above the.area, which acts in the di,r~ction of flow. This component is equf:1,1 to :the above weignt multiplied by the .. stream gradient. · · · ·

f. Tqe drag force acting· on a particle is proportional to the square of the mean velocity in the stream at the point where the particle is located.

Use of assumption (e) permits a simpler treatment than is possible with assumption (f) but does not permit the lateral transfer of drag due to velocity gradients in_the .horizontal direction.to be taken into account. This. factor can b,e included,. in the computat_ions when assumption Ji) fs used. ' · · ·

NOTATION ,·· .

' ' . '

A _ r,epresents the _ cross sectional area of the channel

a=; or ~alf the top width

c, Chezy's coefficient

D, depth of_ an in_finitely wide channel

f, longitudinal slope or friction slope

m, _ factor of, ~roportionality

n, coe_f_f'icient of .. roµghness

P, perimeter

Q; quanti,ty . of fl.ow ( cu:t,ic ft./ se_c)

q, drag force on a surface of a lamina of uriit len~h A , . . :

.r • P' tb,e hydr,ulic radi~s

s = tan g, the transver·~.e slope at the outer edge of the channel · · ·

2

T, top width

u. v2

v, mean velocity of the lamina at a distance x from the center of a channel

Ve, mean velocity of the flow in an infinitely wide channel of depth D = Yo

Vm, mean velocity of the flow in a channel of finite width and of depth Yo

V0 , mean velocity of the lamina at the center of a channel

W, submerged weight of a particle

x, distance measured horizontally from the center of a channel

y, depth of a channel at the distance x from the center; or, in the evaluation of m, y represents the distance from the top of the channel of depth D to a horizontal lamina of thickness dy

y0 , depth of a channel at the center

CZ, a variable

J3, a variable

n=L Yo

Q,. angle of repose for the material

l = ~ Yo

.p, density of water

-r.,, an average drag force

i, maximum allowable drag force for soil in lb/ft2

~, angle the tangent at x makes with the horizontal

3

STABLE CHANNEL SHAPE WITH DRAG PROPORTIONAL TO DEPTH

The type of stable channel studied herein has the property that all the particles of its bed are in a state of incipient instability under tpe action of the forces acting.on them. At the edge the drag force is zero; and incipient instability is due to the force of gravity acting on particles resting on the slope S, which is at the angle of repose. At this point the_~atioof the componen,t of force down the slope to the component normal to the slope is· s. At the middle of the channel, where the slope gz is zero, the particles are held in

. . dx the state of incipient instability by the drag force of the flowing water. In this case, also, the ratio of the force along the;bed to the force acting on the particle in the direction nonnal to the bed is s. At all other points the resultant of the drag force and the gravitational component along the bed act on the particle, and the ratio of this resultant to the gravitational force component normal to the bed must also bes. These r~lations may be expressed mathe­matically in the following way. If W represents the submerged weight of the particle, then, with the notation of Figure 1, the force component normal to the bed is W cos¢. The gravitational component along the bed is W sin ¢ and the drag force of the flowing water per unit of area of the bed is

•L- ¢ .WS Yo cos

Then the requirement that the condition of incipient "instability shall prevail everywhere is

Since

- gz • tan¢ dx

This relation can be put in the form 2 · 2 <f )_ . + s2 i"2 . s2

0

4

• • •

• • •

(1)

(2)

A solution of this differential equation which satisfies the condition that y • y0 when x • 0 is

• • •

Then it mq be concluded that a simple cosine curve represents the shape of a stable channel under the conditions specified in assumption (e).

STABLE CBADEL WITH A DRAG FQRCE PROPORTIORAL TO THE SQUARE OF THE TELOCITY ilD BO LATERAL TRANSFER or DRAG

In this case the drag force tel'lll is assumed t.o have the fol'lll:

wsL .y 2

0

(3)

where V represents the mean velocity in the stream at the distance x from the center1 and V represents the mean velocity at the center. Then the condition ~hat the particles everywhere be on the verge of motion is:

. . . (4)

But since tan ¢ =--! - (dy')

dx and sin¢= , cos¢ • 1 . . . (5)

V 1 + 2 V1 + (~)2 (~)

dx dx

5

\'he above expression ~an. be put in the form! . ,, .· .. . { .. . .

. . . (6)

In order to obtain an expression for the velocities,we may apply the Chezy f'ol'lll\lla

• • • . (7)

to compute the flow through an element such as the shaded element of Figure 1.. Since the as.sumption of .no ;i.ate;-al. tran~fer of drag is equivalent to an assumption of no drag fo~es·on the sides of the element, the hydraulic radius r·inay be computed by dividing the area of the element yd,x by the length of. bed _in contact _with its base. This length is

Then y

(...X.)' . r =v d 2 · 1 +. dx ·

.. ( 8)

and from (7)

(9)

Since

then, by division

(10)

6

If this expression is substituted into Equation (6), it becomes:

This is identical with Equation (2) and, of course, has the same solution, namely:

y • y cosL o Yo

It may be concluded therefore, that if the lateral transfer of drag is neglected, both assumptions (e) and (f) lead to the same shape for the stable channel. If this seems surprising, it should be recalled that the Chezy formula itself is derived by equating a frictional drag force proportional to the square of the velocity to the component of the weight of the water acting in the direction of flow. Therefore, if the Chezy relation is used with the above restrictions to determine the relation between velocity and drag, assumptions (e) and (f) become identical.

STABLE CHANNEL SHAPE WITH LATERAL DRAG TRANSFER ACCOUNTED FOR

Before the effect of lateral drag transfer can be introduced into the equations, it will be necessary to develop some means of computing it. In the present case it will be assumed that the shear force acting on any section through the stream is proportional to the gradient of the square of the velocity. The factor of pr~portionality, here called m, will be determined by computing the mean velocity in an infinitely wide channel according to these relations and then determining the factor m by comparison with the Chezy formula. The modifications needed to introduce the effect of lateral drag transfer can be made as soon as the factor m is evaluated. The detail of the evaluation of m is as follows:

In a channel of depth D and having a friction slope f, such as is shown in Figure 2, the shear on a unit area of the plane at a distance y below the top is

7

- . . (11)

),,

UBRr\RY.

SEP 2 O tUUI

-Bureau of Reclamallon Reclamation Service Cente,

where U = v2 and V repre~ents. the stream velocity at . y. The gravitational component acting on the volume included between unit areas of the planes y and y + dy is

ffdy

where -p represents the weight of a unit volume of water. Then if this gravitational component is to be held in balance by the difference between the drag forces on the top and bottom of the element:

d dU - - (m -) = .of dy dy '

(12)

In setting up,this expression;·rorces in the downstream direction have been considered positive •. By integration

and

dU - m - = .nfy + c1 dy r

where c1 and c2 are constants.

If .

then

Hence

or

dU - m dy = ,prn at y = D

U = 0 at y = D

mU == fr:2 -·4 U = ~ (D2 - y2) (13)

8

and

The mean velocity is D

Vmean .1: -Jvdy = 1_ ~[z . D . D la 2 .o

or

vmean = ~ 1'D T

• • •

D

Vn2 - I-+ B:sin-1 z] · . 2 D

0

-(14)

(15)

A rearrangement which will permit a comparison :wtth the Chezy formula is

For this case Chezy's formula has the form:

Vmean = C '{'Dr'

Then by comparison

C= ,p v~ and :J.t follows that:

~D7T2 m = 32c2

(16)

(17)

. . . {18)

{19)

It will now be possible to modify Equation (9) ,by adding to the original ·gravitation driving force the difference of the frictional drag forces on the two sides of the shaded element of Figure 1. The drag force acting on the side of the shaded element · nearest to the center of the channel is, for a unit length of channel,

q = y'( (20)

9

and acts in the downstream direction if the velocity gradient dV is dx .

negative. Under the same conditions the force on the side away from the origin is given by a similar expression, but here the·force acts upstream. The difference between the two forces is

dq = ~ (y7) dx dx ·

. . . (21)

Modification of' Eq~ation(9)will:be most easily made if this difference is expressed in tenns of an equivalent channel gradient. To do this ae.t

Then

f 1fydX = ~ (yT) dx dx

1 d -r. fl = -· - (yJ)

-pY dx • . • (22)

and the equation as modified to account for the lateral transfer of' drag becomes

Since

v2 -- y Vc2 Yo vl + {~)2

dx .

• • • (23)

(24)

. . . (25)

[ l + m d (Y.~).l pt- Y dX UA ~

10

or

• • (26)

To put this expression into dimensionless form let

Yod7) = dy • • • (27)

!= !_ Yo

and since

then

Yods = dx

2 m. fYo7f

32c2

• • • (28)

v22 - TJ ~ i. + ;~ ::: :1r [ 11:. c:/J} . . . <29J

Ye Vi+ c:J>2 i s s

The qomplicated nature of this expression would make it difficult to obtain a direct solution by integrating the differential equation which.might be obtained by eliminating the ratio of' L between this

Vo equation and Equation ( 6) • A trial method of solution will therefore be adopted inste,ui. By this process a trial profile wi.11 be chosen and the ratio(L(\iill be computed for it from both the Drag Relation

. Vo or· Equation (6) and the Modified Chezy Relation of' Equation (29). It will be considered that a satisfactory solution has been obtained when the two curves agree closely. An interesting feature of' Equation (29) is that the quantity m does not occur in it. This indicates that the stable shape will f'ix a value for the ratio ~} This shape will also

be independent of' size. ·

11

For these purposes, if' the relation between n and .$ can be assumed to be:

or

If

then

1'J = i + al 2 + /3 s4 q- = 2 a§+ 4 _/3 !3

2 4 L.1+a(~) +/3(.!...) Yo Yo Yo

! = 2 a(.!...) + 4/3 ,~,3 Yo Yo

L = O and ~ • - S when x • a Yo dx

· a 2 /3 a 4 0 = 1 +ct(-) + (-) Yo Yo

- s = 2 ac!....> + 4/3c!....>3 . Yo . Yo

from which

c§ !.... - 2> a = _2 _Yi_o --

c!.... ,2 Yo

( S a . ) - -- + 1 /3 = 2 Yo

. c;o>4

12

. . .

. . .

. . .

• • •

. . .

. . .

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

Discussio• or coMPILATI011 oF DATA m·cuana

The computations in tlle tables at the end ot this report are those used to obtain data tor the curves shown on Figures 3 through 7. They consist ot the development of the ratios of the •an velocity of the lamina at the distance x from the center of the channel to the mean velocity at the center tor various top widths and transverse slopes. The procedure followed in making these calcu­lations was to assume a value ot· . ; tor each value of s and to

0

determine the ratio of ~ from the Drag !{elation and from the 0

Modified Chezy Relation. When close agreement. between the two curves was obtained, a satisfactory solution of Equation (29) was assumed to have been attained. In computing the values at t:' from the Modified

.. 0

Chezy Relation, it was necessary to choose a value for the constant y 2m . o which would make the.curve agree with the one obtained from the n-

Drag Relation. rt agreement was not obtained, it was necessary to

choos~ a new value of and make a new set of calculations.

It was found that the curves could be made to agree quite

cloself although !_ was allowed to vary. a great deal. Consequently, Yo .

the greatest source of error was in choo.sing the value o't r· It is . . - 0

· I ·. a . believed that by careful adjustment of Y

O the curves., could be made

to coincide still more closely; however, the agreement shown in this set of curves is as close as is warranted by the approximations used in the development of these equations.

It may be noted that the solutions obtained by the modification for lateral transfer of drag agree very closely with tbe·cosine solution. Since this is t~e case, thE! use of the cosine solution may give equally ·1sattsfactory results· with considerably less work. ·

The range of transverse slope covered by this set of curves is from S • 0.3 to S • 1.0.

13

Ym Graphical integration was used to evaluate -

Vo' p -,

and Q as follows: Yo

a

. . .

0

a

r 2 p

2 l + (dr\.) ~! - . . . Yo df

0

a

A 2 f~ Y\ d_J y 2 - • • •

0

0

a -· 2 /Yo V.

Q - 2V y Y\. d ! . . . 0 0 Yo

0

14

(38)

(39)

(40)

(41)

It will be. not.ed that the curve fot p . is . 11ot . used for solution .· . -Yo .. · ··. ··<· .· ....

ot a problem by this method·. '1'b1.i is because the concept of the wetted perimeter and the h74i'aulicradiua is not in agreement w:l,tli the drag distribution a11umed in thia, solution. However·, the -c,ur:v:e is inserted to permit a comparison of results obtained by this method with similar results obtained from the ordinary formulas ot hydraulics.

EXAMPLE

The procedure involved in the design of a stable channel can best be illustrated by the solution of an example.

For the particular type of soil through which the channel will run, the engineer will know or will be able to determine certain physical characteristics on which to base his design. It will be necessary to know the angle of repose of the soil and the maximum allowable drag force. An appropriate coef'.ficient of roughness can be found in hydraulics tables, and the longitudinal slope on which the channel will run will be a predetermined factor. With these values specified, the quantity of wate.r. which the stable channel will carry will be a fixed value. Since. it .is· desirable to specify the amount of water to be carried by the channel, it is necessary to have some method of modifyingthe stab:te channel shape to increase or decrease the flow. If the desired flow is greater than the ,stable channel shape·would carry, it is proposed to insert a rectangular section at the center; and if the flow is to be les_s than the stable channel shape would carry, then a portion would be removed from the center. The portion to be removed in the latter case is easily determined by-use of Figure 12.

The conditions assumed for the design of this particular channel are:

a. The tangent of. the angle of r.epose of the soil is o.6 (S • 0.6).

b. The material through which the channel is to run will stand a drag force of 0.1 pound per square· foot ( ~ • 0.1) 0

c. The longitudinal slope of the channel will be 0.0004 (f • 0.0004).

d. A reasonable value for the coefficient of roughness would be n • 0.020.

15

In a wide channel of uniform depth the drag force per unit of area ot the bed would be 7' •.,Om and the velocity would be 'c· At the center of a stable channel, ot equal depth, the lateral trauater ot drag reduces the velocity from Ve to Y and the drag relation there becomes 0

An expression for the center depth can then be obtained from the above relation in the form

,,-y O - --.,--2-

.,0f( ~) V

C . V

For the prese1,1t case the upper curve of Figure 9 gives Vo• 0.983. Then c

Y • O.l • 4.146 feet 0 (62.4)(0.0004)(0.983)2

For an infinitely wide channel the following relation holds

I • t to,b l tA \~ (., ~- - \lo.>)

. ~

:, o-:.. D and for the conditions~assumed above the value of C • 94.4 is obtained from hydraulic tables.

Then

'c • 94.4 vc4.146)(o.ooo4)

• 94.4 (0.04072)

• 3.844 tt/sec

16

•

From Figure 9

Then

From Figure 8

Or

Vm • (0.869)(0.983)(3.844)

V • 3.28 ft/sec m

_ .... · '

A • (3.42) (~ •. 146) 2

A• 58.79 ft2

Then since Q • ·Av - ; , . m

Q • (58.79)(3.28)

• Q • i92.83 r.t3 /sec

From Figure 8 the value of !_ is found to be 5.38, so that the Yo

top width of the st_abl~ channel is 5.38 Yo or (5.38)(4 .• 11!,6) .• 22.30. Having now obtained the top width and the center depth of' the stable channel, it remains only to obtain the . x and y coordinates of· the channel bed. This is readily done by making use of Figure 11. For a transverse slope of S • Q.6 the following values of !...c

' . ' . ~ and ;

0 are obta,ined '711d the values of x and y are calculated.

17

X L. Yo X Yo y

0 0 1.000 4.146 0.5 a.073 0.958 3.972 1.0 4.146 0.839 3.478 1.5 6.219 o.649 2.691 2.0 8.292 o.4oo 1.658 2.5 10.365 0.117 o.485

11.15 0

If the channel is to carry leBS th~ the sta"t?le channel will carry, it is necessary to remove a portion of the channel. Suppose the channel is to carry 75 rt3/sec. Then the ratio of Q to Q0 is 75 or 0.389. From Figure 12 the value of !_

192.83 · a is found to be 0.369, that is, 36.9 percent of the top width is to be removed from the center of the channel. The modified section with a top width of 14.07 feet is shown on Figure 13.

If the channel is to carry more than the stable channel will carry, it is necessary to add a rectangular section at the center. Assume that it is desired for the channel to carry 300 tt3/sec, then the amount to be carried by the center section is 300 - 192.83 or 107.17 tt3/sec. The mean velocity of the flow in an infinitely wide channel of depth y0 is Ve, which was determined to be 3.844 ft3/sec in this case. For a center depth of 4.146 fe~t each foot of width would carry (4.146)(3.844) ft3/sec or 15.94 ftj/sec. The necessary width of the center section woµld then be l~t ~z or 6. 72 feet.

It should be noted that there is a discrepancy between the velocity of the center section and the maximum velocity of the stable section. This discrepancy is relatively unimportant, however, since the difference is less than 2 percent of the maximum velocity tor the center section; and, the lateral transfer of drag being small, the velocity at the edge of the center section would soon reach its maximum value at points nearer the center. The modified section with a top width of 29.02 feet is shown on Figure 13.

18

In conclusion, it may be noted that if water flowing in these channels carries sediment of the type coming from soil through which the channels run, it could be expected that the channels as designed would neither scour nor silt up. However, if clear water is to be carried, these channels would be stable to~ any slope less than 0.0004.

ACKNOWLEDGMENTS

The fundamental principles on which this work is based were laid ~own by Mr. E.W. Lane. This analysis has profited by the previous work of Mr. R. G. Conard who made many computations of the type shown in the tables and who succeeded in obtaining a close agreement between the Modified Chezy and Drag relations before he left the Bureau. The tables shown herein and the J:igures 3 to 7, inclusive, represent a further refinement of Mr. Conard's work by Mr. Florey •

19

Tab

le

l

Slo

pe=

0.3

a

= -

.0388

11-3

c;o

> =

5.

5 .f

l =

.0

0019

124

(12)

I+

{II

J

(1)

: (2

) :

(3)

: (4

) :

(5)

: (6

) .. :

(7

) :

(8)

(9)

: (1

0)

: ( 1

1)

: (1

2)

: (1

3)

. .

. .

. .

. .

. .

. .

(!..

..);

(!..

..)2

; (!

....)

3 ;

(.!...

.)4

Yo

: Y

o :

Yo

: Y

o !a

(.!.

...)2

!Jf

(.!....)q

.! (L

) !2

a(.!

....)

!4P

C.!...>

3! (d

y)

!cay/

!1 --

*-<d

y)2

~ cL

)4

• Y

o •

Yo

• Y

o •

Yo

• Y

o •

. d

x

• d

x

• s2

dx

•

V0

. . .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

0:

0.:

0

: 0

: 0

: 0

:1.0

0000

: 0

: .4

: .1

6:

.064

: .0

256:

-.

0062

1: .0

0000

5:

-993

79:

-.03

107:

.8

: .6

4:

.512

: .4

096:

-.

0248

6: .0

0008

:

.975

22:

-.06

215:

1.

2: 1

.44:

1.

728:

2.

0736

: -.

0559

3:.0

0040

: .9

4447

: -.

0932

2:

1.6:

2.5

6:

4.09

6:

6.55

36:

--09

944:

.001

25

: .9

0181

: ·-.

1242

9:

2.0:

4.0

0:

8.00

0:

16.0

000:

-.

1553

7:.0

0306

: .8

4769

: -.

1553

7:

2.4:

5.

76:

13.8

24:

33.1

776:

-.

2237

4:.0

0634

: .7

8260

: -.

1864

4:

2.8:

7.8

4:

21.9

52:

61.4

656:

-.

3045

3:.0

1175

: .7

0722

: -.

2175

1:

3.2:

10.2

4: 3

2.76

8:10

4.85

76:

--39

775:

.020

05:

.622

30:

-.24

858:

3.

6:12

.96:

46.

656:

167.

9616

: -.

5034

0:.0

3212

: .5

2872

: -.

2796

6:

4.0:

16.0

0: 6

4.00

0:25

6.00

00:

-.62

149:

.048

96:

.427

47:

-.31

073:

4.

4:19

.36:

85

.184

:374

.809

6:

-.75

200:

.071

68:

.319

68:

-.34

18o:

4.

8:23

.o4:

110.

592:

530.

8416

: -.

8949

4: .1

0152

:

.206

58:

-.37

288:

5.

2:27

.o4:

140.

608:

731.

1616

:-1.

0503

0:.l

3983

: .0

8953

: -.

4039

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