flow resistance in simulated irrigation borders and furrows · 2010-11-20 · flow resistance in...

Flow Resistance in Simulated

Irrigation Borders and Furrows

Conservation Research Report No. 3

Agricultural Research Service

U.S. DEPARTMENT OF AGRICULTURE

In Cooperation With

Colorado Agricultural Experiment Station

CONTENTS

Introduction Review of literature

Flow resistance formulas Laminar flow_ Transition from laminar to turbulent flow Turbulent flow Bed elevation Channel geometry

Requirements if flow resistance studies are to be applicabledesign

Equipment and procedures Channel construction Experimental procedures

Results Laminar flow Transition from laminar to turbulent flow Turbulent flow Prediction errors

Applications to field conditions Summary and conclusions Literature cited Symbols.. Appendix

Figures 25 to 42 Tables 8 to 11

to surface irrigation

Page

12223344

5557

101019193538404142434347

Washington, D.C. Issued November 1965

FLOW RESISTANCE IN SIMULATED IRRIGATIONBORDERS AND FURROWS

By E. G. KRUSE, Soil and Water Conservation Research Division, Agricultural Research Service; C. W. HUNTLEY, AgriculturalEngineering Department, Colorado Agricultural Experiment Station; and A. R. RoBINsoN, Soil and Water ConservationResearch Division, Agricultural Research Service

INTRODUCTIONThe efficient application and distribution of

water by irrigation furrows or borders is highlydependent on the rate of advance of water in thesechannels. The rate of advance is governed by theintake rate of the soil, the resistance offered bythe channels to the flow of water, and the dischargerate into the channels. A knowledge of thesefactors is essential for the design of efficient irri-gation systems. Intake rate has been the objectof much study and methods are available for itsmeasurement before the construction of irrigationsystems. Previous flow resistance studies havedealt with either artificially roughened boundaries,conduits intended for uses other than irrigation,or discharges much greater than are likely tooccur in small irrigation channels. Results ofstudies of flow resistance are not available for thetypes of roughness, sizes of channels, and dis-charges that are likely to be encountered in surfaceirrigation systems. Discharge can be regulatedto correspond to other design conditions.

Channel boundaries and flow conditions inirrigation furrows and borders differ from condi-tions in most other open channels in several ways.Discharges carried by furrows and borders aresmall, possibly in the laminar flow range in somecases. Boundary roughness is relatively great;at the lowest rates of flow, height of roughnessmay be of the same order of magnitude as flowdepth. Size, shape, and spacing of the roughnesselements are not uniform. Under field conditions,irrigation flows are further complicated by changesin boundary roughness and channel cross sectionwith time and distance because of erosion.

Resistance to flow in irrigation channels maybe caused by several factors. In earth channels,the boundary roughness is the primary cause offlow resistance. In vegetated channels, plantstems and leaves may have a greater effect onflow retardance than the soil roughness. Thecross-sectional shape and alinement of channelsmay also affect resistance.

Many previous flow resistance studies haveinvolved measurements of energy losses in con-duits intended for a specific use. Others havebeen attempts to determine the fundamentalrelationships between boundary roughness, chan-nel shape, channel alinement, and resistance to

flow. Results of the specific-use type of testscannot be generalized to include conduits of othersizes or other boundary roughnesses. The fun-damental studies have been conducted, for themost part, on artificially devised roughness, withroughness elements of uniform size, shape, andspacing. The effect of the roughness on resistanceis often expressed in terms of the equivalent sand-grain size, by using the rough boundary equationof Nikuradse (12). 1 However, it has not yet beenpossible to find a general relation between thedimensions of the roughness (even uniform rough-ness) and the equivalent sand grain size.

It is currently necessary to make trial resistancemeasurements on every type of conduit before theresistance of that conduit to flow can be accuratelyknown. In the installation of irrigation systemsit is often impractical to base design on trial re-sistance measurements. A procedure is neededwhereby resistance in irrigation channels can beestimated while the system is being designed inorder that changes will not be necessary after thesystem is constructed.

The study reported in this bulletin was con-ducted to determine the resistance to flow inchannels similar to irrigation furrows and borders.Channels having soil boundaries with differentdegrees of roughness were constructed in the lab-oratory: The relationships discovered betweenflow resistance and boundary roughness param-eters, after field verification, will provide a methodof estimating flow resistance for design of surfaceirrigation systems.

The specific objectives of the study were:1. To determine if both laminar and turbulent

flow are likely to occur in irrigation systems and,if so, under what conditions of discharge, tempera-ture, slope, roughness, etc., each type of flowoccurs.

2. To investigate the effects of boundaryroughness on resistance to flow in channels withroughness elements formed of soil, and thus mitni-lar m roughness to irrigation furrows and borders

3. To determine the effect of channel shape onflow resistance, within the range of shapes char•acteristic of irrigation furrows and borders.

I Italic numbers in parentheses refer to LitorsturrCited, p. 41.

2

CONSERVATION RESEARCH REPORT 3, U.S. DEPT. OF AGRICULTURE

The results of this study will provide the ad-ditional knowledge necessary for predicting therate of advance of irrigation streams, which inturn will permit the design of systems with maxi-mum water application efficiency. The specificcontribution of the study is to develop the relationsnecessary for predicting resistance to flow inirrigation borders and furrows without trial re-sistance runs.

Methods are being developed concurrently (21)for determining the surface profile of an advancingwater stream when resistance to flow is known.The volume of water in surface storage in thestream can be calculated if the water surfaceprofile is known. Rate of advance is determinedby equating the volume of water delivered to thechannel to the volume infiltrated plus the volumein surface storage (4, 6, 19).

REVIEW OF LITERATURE

Flow Resistance Formulas

Numerous studies have been made of resistanceto open channel flow. Both laminar and turbulentflow have been investigated. Most laboratorystudies have considered only the effects of arti-ficial roughness elements, uniform in size, shape,and spacing. The work most pertinent to thepresent study will be reviewed in this section.

Resistance equations have a different form forlaminar and turbulent flow. In laminar flow oversmooth boundaries the velocity is proportional toslope to the first power and to depth to the secondpower. Definitive equations have not been de-veloped for laminar flows over rough boundaries.In turbulent flow, velocity is proportional todepth and slope to other powers.

A theoretical equation for uniform laminar flowin a wide, smooth, rectangular channel was de-rived by Cornish (3). It can be written: 2

V= n S (1)

where:

V is the mean flow velocity,-y is the unit weight of fluid,d is the depth of flow,S is the slope of the energy line, and/I is the absolute viscosity of the fluid.

This relationship has been verified experimentally(I, 3, 13).

Resistance to uniform turbulent flow in openchannels is often expressed in terms of the re-sistance coefficient from one of the followingequations:

Manning:

V= 1.486 R2As 1 /2

n .(2)

Chezy :

V= Clilay (3)

2 A list of all symbols used in this bulletin can be foundon p. 42.

or Darcy-Weisbach:8gRS

V'where:

R is the hydraulic radius,g is the gravitational acceleration, andn, C, and f are the resistance coefficients in the

different equations.

The resistance coefficients can be related to eachother and to the mean flow ratio as follows:

1-7T7 =C/A/g=:--1 486 WI'

(5)

where: V,1, is the shear velocity VgRS.

Laminar Flow

The relationship between the Darcy-Weisbachfriction factor and the Reynolds number forsmooth boundaries-1=24/Re—can be obtainedby combining equations 1 and 4. Several in-vestigators (15, 20, 23), have plotted f against Refor laminar flows over rough boundaries andfound that the relationship is different from thatover smooth boundaries. The f-versus-Re line forrough boundaries falls parallel to and above theline for flows over smooth boundaries, indicatinggreater flow resistance.

A criterion for pipe flow has been presented thatspecifies the height of roughness that will causeflow resistance greater than that caused by smoothboundaries (5). The criterion is based on theassumption that flow separation occurs when theReynolds number at the tip of the roughnesselement (tip Reynolds number) reaches somecritical value and that the eddies caused byseparation increase the flow resistance but do notspread throughout the flow to cause generalturbulence. The criterion also assumes that flowvelocities have the same distribution as for smoothboundaries and are not affected by the presence ofthe roughness elements. The height of roughness

(4)

FLOW RESISTANCE IN SIMULATED IRRIGATION BORDERS AND FURROWS

3

e

h

i)

necessary for separation to occur in pipes, based onthese assumptions, is:

r Reee-

At2 Re

where:

r is the pipe radius,Re is the Reynolds number of the flow, andRek, is a critical value of Reynolds number at

the tip of the roughness element.

Transition From Laminar to Turbulent Flow

Several investigators have studied the tran-sition from laminar to turbulent flow in openchannels with smooth boundaries (1, 7, 9, 14).The range of critical Reynolds numbers variesfrom 300 to 1,400. The critical Reynolds numberis apparently affected by the amount of initialdisturbance in the flowing streams.

For rough boundaries, the critical Reynoldsnumber may be defined as that for which theDarcy-Weisbach resistance coefficient ceases tobe inversely proportional to the first power of theReynolds number. This critical Reynolds num-ber is generally lower than that for smoothboundaries and has been found by differentinvestigators (15, 20, 23) to be a function ofroughness height, channel slope, shape, etc.Parsons (15) presents the following criterion forcritical Reynolds number for an earth channelwith random roughness:

7.5Re c — S2/3 (7)

At a slope of 0.001 the critical Reynolds numberhas a value of 750. Woo and Brater (23) foundcritical Reynolds numbers of 400 for roughboundary conditions.

Turbulent Flow

The studies of Nikuradse (12, 13) show thatthe theoretical logarithmic resistance formulasof Prandtl and von Karman are applicable toturbulent flow. The studies also show thatresistance to turbulent flow depends on boundaryroughness and fluid viscosity. For smooth bound-aries, flow resistance is a function of fluid viscosity.For rough boundaries, the relative roughness hasthe primary effect upon flow resistance. Inter-mediate cases exist where both roughness andviscosity affect resistance. The boundary con-dition is determined by the thickness of thelaminar sublayer, 6', relative to the roughnessheight. Nikuradse (13) found:

k,>66', for a rough boundary,

k,<-6' for a smooth boundary, and4

66'>k,>-6'

for a transitional boundary, (8)4

where k, is the diameter of the equivalent sandgrain roughness.

Keulegan (10) integrated the Prandtl-vonKarman universal velocity distribution law overthe cross sections of open channels of severalshapes. He considered the effects of channelshape and free surface, as well as the boundaryconditions, on flow resistance. The following areKeulegan's equations:

For rough boundaries—

', ••\ V 1 , 2. 03, R ,

k 11-E) — log -I-

V* K K

For wavy boundaries—

-, V 1 , 2.30 , RV* ,(1+e) —=aw --+— tog --r—K K K

And for smooth boundaries-

(1-Fe) -17.=a,-1+-2.30 log —RV*1- (11)

K K V K

where a represents the effect of channel shape ondistribution of boundary shear, 0 represents theeffect of channel shape on the flow-area-to-velocity-distribution relationship, V representskinematic viscosity, and K is the universal tur-bulence constant. The symbols a„ a,„, and a,,are hydraulic characteristics of the boundaries;a„ depends on the ratio of height to spacing of theboundary roughness elements; a, is a constant;a, is probably a function of roughness spacing.

Keulegan suggested that the boundary condi-V

tions can be determined by a plotting of 7

*

23 0K

Vlog R against log '`. For smooth and wavy

boundaries, such a plotting of data would describean inclined straight line having the equation:

V _2.30 lo RV*_FAV* K g

The theoretical value of A for very wide, smoothchannels is 3.0. Using the experimental data ofBazin (2), Keulegan found values of A equal to—1.3 and —3.0 for different degrees of waviness.

For rough boundaries, resistance is independent

h

d

!ed

rrle

g

6teh

e

as

y,t,

b

Is

(6)

(9)

(10)

(12)

4 CONSERVATION RESEARCH REPORT 3, U.S. DEPT. OF AGRICULTURE

of viscosity. An equation of the following formresults:

V 2.30, — — log Rd- Bv K

where, for Bazin's data, B is a function of theheight of the roughness.

Using Bazin's data, Keulegan was also able toderive the following criterion for distinguishingbetween wavy and rough boundaries:

k,V *>42.2 for a rough boundary, and

k'V*<42.2 for a wavy boundary, (14)

where k, is the diameter of the equivalent sandgrain roughness.

Sayre and Albertson (17) studied the effects ofspacing of baffle plate roughness on flow resistance.Their resistance measurements can be representedby the formula:

c =6.06 log –d319

where:d is the normal flow depth in a wide rec-

tangular channel,x is a resistance parameter with a different

value for each boundary, andC is the resistance coefficient from Chezy's

formula.The significance of x can be better understood

by comparing equation 15 with equation 9 of30.Keulegan. If a, is equal to 2

— log A, and 0-1is equal to 2.30 log B, and if ; is assumed negligible(as Keulegan found), equation 9 becomes :

V _2.30 R K log lc/AB

C 2.30 Rlog --x-

This equation is identical to equation 19 if K isgiven the value 0.38 and it is assumed that d canbe replaced by R for channels that are not infinitelywide. Therefore, x represents effects of roughnessheight and spacing and also the effects of channelshape.

Many other studies have resulted in similalogarithmic-type formulas. Investigators have n1)1

agreed on the proper value of K for these logaritlimic formulas. From Nikuradse's studies it wa -assumed that K is a universal constant with thevalue 0.40. However, later studies have indicatedvalues in the range of 0.36 and 0.38. The appareilivalue of K determined from experimental data callvary with the shape of the channel and accordingto the way that the depth of flow is measured.

Bed Elevation

Previous investigators have measured depth offlow to different datums relative to the mean bedelevation. In studies of artificial roughness, theroughness elements are usually fastened to a planewall that becomes the natural datum for depthmeasurements. However, for some studies (18,22), other datums have been chosen. A datumnearer the tops of the roughness elements is oftenused when the elements occupy a large volume(i.e., are densely spaced). Sayre and Albertson(17) found no depth correction necessary andattributed this to the small spacing density of theroughness elements used in their studies. Theyconcluded that the roughness elements did notgreatly inhibit flow turbulence near the bed.Schlichting (18) used a datum such that the volumeoccupied by roughness above the datum was equalto the volume open to flow below the datum.

Channel Geometry

Several studies (2, 10, 11, 16 , 20) have been madeto determine the effect of channel cross section onresistance. Most of these studies (2, 10, 11, 16)have found that the use of hydraulic radius torepresent the dimensions of irregularly shapedchannels is sufficient to allow for the effect ofshape on resistance.

The studies of Keulegan, already discussed,indicated that the effect of nonuniform boundaryshear on flow resistance was negligible and thatthe shape effect represented by B (equation 9)could be accounted for by use of a small, constantvalue for this term. Powell (16) found that theerror involved in neglecting channel shape effectdid not exceed 5 percent. Straub and coworkers(20) found that channel shape had some effect onresistance in rough channels, but a lesser effect insmooth channels.

(13)

(15)

or, say :(16)

FLOW RESISTANCE IN SIMULATED IRRIGATION BORDERS AND FURROWS 5

REQUIREMENTS IF FLOW RESISTANCE STUDIES ARE TO BE APPLICABLE TO SURFACEIRRIGATION DESIGN

The boundaries of irrigation channels are rougherthan those of most commercial conduits. Hence,the results of previous research on smooth bound-aries are not generally applicable to the studypresented in this bulletin. For the laminar flowrange, boundary roughness affects both the resist-ance to flow and the transition to turbulence.Both of these effects need to be studied over aboundary roughness similar to that of irrigationchannels.

Most fundamental studies of turbulent flow haveconsidered artificially roughened conduits whereall roughness elements on a given boundary haveidentical size and spacing. Even when consideringuniform boundary roughness, investigators havenot agreed on the roughness dimension, i.e.,height or spacing, that exerts the primary influenceon flow resistance, although most investigatorsconsider roughness height to have the greatereffect. Variations in spacing of roughness ele-ments, for roughness of a constant height, willcause variations in flow resistance. In studies offlow resistance, the roughness height and longi-tudinal spacing should both be measured as thevariables most likely to affect flow resistance.Transverse spacing and shape of the roughnesselements should have a lesser effect on resistance.

Previous studies of flow resistance have shownthat it is difficult to relate the effective dimensionsof natural or artificial roughness elements to aroughness standard without the use of trial resist-ance runs. The relationship between the dimen-sions of the roughness elements and the resistanceto flow can be applied directly to field conditionswith the least difficulty if the roughness con-structed in the laboratory is as similar as possibleto natural soil roughness.

It is apparent from past investigations that theresistance coefficients of uniform flow formulassuch as Manning's or Chezy's are not constant fora given channel, but vary with the depth of flow.Therefore, study of these coefficients shouldinvolve the full range of flow conditions likely tobe encountered in practical applications, especiallylow depths of flow such as occur in irrigationchannels.

The effects of channel shape on flow resistanceare less important than the effects of boundaryroughness. However, the magnitude of channelshape effects seems to vary with flow and boundaryconditions. Shape effects should be studied underconditions of low flow rates and very roughboundaries.

EQUIPMENT AND PROCEDURES

It would be difficult to control and measure thevariables necessary to evaluate flow resistance insmall irrigation channels with sufficient precisionin the field. For the experiments described andevaluated in this bulletin, flow resistance in smallchannels was studied in the hydraulics laboratory.Flow boundaries created in the laboratory weresimilar to those of irrigation borders and furrows;hence, the relation of resistance to roughnessdimensions found in this study should find directapplication in the field.

Channel Construction

A wooden flume, 4 feet wide, 60 feet long, andwith an adjustable slope, was used to support theflow channels used in this study. A supportingstructure constructed inside the flume containedthe soil-like material used for the experimentalchannels. This structure extended the full 60-footlength of the flume. The flume channels emptiedinto a weir box where discharge could be measuredby a calibrated, 90°, V-notch weir; to measuresmall flows, the discharge over a timed intervalwas caught and weighed. The headbox of theflume was supplied with water by pump from a

large sump. Rails on the flume walls provided adatum from which channels could be constructedand experimental measurements taken. Figure 1is a sketch of the flume and two types of channelused.

Three pumps-3 inches, 4 inches, and 8 inchesin size—were used to supply water to the soilchannels at steady rates, ranging from 0.01 to1.1 c.f.s. Higher rates of discharge sometimescaused erosion of the channel boundaries and, forthis reason, were not used.

For the first group of tests, channels were builtwith a rectangular cross section, to correspond tothe cross section of irrigation borders. The widthof the channels was limited to 1.88 feet. Parallelsheets of masonite formed the side walls of therectangular channels. The floor of the channel,hereafter called the bed, was composed of a soil-like material from which different roughnessescould be formed. A polyethylene sheet insertedbeneath this soil-like material prevented seepagefrom the channel. For most of the soil roughnessesformed, the side walls were much smoother thanthe channel bed. Figures 25 to 35 in the appendixshow the rectangular channels constructed; theidentifying letters in the legends for these figures

7FLOW RESISTANCE IN SIMULATED IRRIGATION BORDERS AND FURROWS

correspond to the identifying letters used in thetables?

For the second group of tests, channels wereconstructed with parabolic cross sections approxi-mating the range of common shapes of irrigationfurrows. The shape of each parabolic channel wasdescribed by the formula:

y=ax2 (17)

Channels were constructed having values of aequal to 0.40, 0.65, and 0.90. Figures 36 to 42 inthe appendix show the parabolic channels con-structed; the identifying letters in the legends forthese figures correspond to the identifying lettersused in the tables.

The material used in the channels was not anatural soil but a sandy material washed from thegravel at a local gravel pit. It originally containedlumps of clay. Before the sandy material was usedin the channel, the largest of these clay lumps wereremoved by sieving all the material through ah-inch screen. The material, after removal of theclay lumps, is shown by mechanical analysis to be avery fine sand (see fig. 2).

All channels were formed by a plywood screedfastened to a cart that ran on the flume rails.For rectangular channels (borders) the screed hada straight edge; for furrows the screed had thedesired parabolic shape. Channel elevation wasconstant relative to the rails, which had been presetat the desired slope.

After the channels were screeded, they weremodified in different ways to produce the range ofroughness necessary for the study. The range ofroughness exceeded that likely to be found inactual irrigation channels. The smoothest bound-aries were formed by hand troweling the screededchannel. The roughest boundaries were formedby using a tillage tool that provided an irregular,cloddy surface. Some boundaries were modifiedby ponding water on the surface for a short lengthof time. To form others, flows of water just largeenough to cause small amounts of soil movementwere run over the surface until the desired bound-ary was produced.

So that the change of resistance with dischargefor a given bed form could be investigated, thedifferent boundaries were stabilized with a chemi-cal treatment of the type developed by Vanoni andBrooks (22). Spray applications of sodium sili-cate, sodium aluminate, and calcium chloridesolutions prevented any alteration of the boundaryroughness for the slopes and discharges used inthis study.

3 Descriptive tabulations of many of the boundaryroughnesses used in this study are included in E. G.Kruse's Ph. D. dissertation—EFFECTS OF BOUNDARYROUGHNESS AND CHANNEL SHAPE ON RESISTANCE TOFLOW OF WATER IN VERY SMALL OPEN CHANNELS. On file,Library, Colo. State University, Fort Collins. 1962.

After the roughness was stabilized, elevations ofthe rectangular channels were determined withpoint gage readings at 0.05-foot intervals trans-verse to the direction of flow at each 5-foot station.The average of these readings was assumed to bethe mean bed elevation at the station where taken.

The cross-sectional profiles of each parabolicchannel were measured in a similar manner. Thecoefficient a of a least-squares parabola throughthe set of points was determined, the coordinatesof the vertex being fixed at the same depth for allstations for a given channel. By integrating theleast-squares parabolic equation, it was possible toobtain the area and wetted perimeter of the chan-nel as functions of the flow depth.

Experimental Procedures

Experimental runs were taken for two to fourchannel slopes for each boundary. The group ofruns for a given channel at a given slope constitutesa series. The first run of each series was made atthe lowest depth of discharge that would submergealmost all of the roughness on the channel bound-ary. Discharge was approximately doubled foreach subsequent run. Before the elevation of thewater surface was measured, weir readings werechecked every 5 minutes for 15 minutes, to makesure that all variation had ceased. During thistime, water temperature was measured at theheadbox of the channel.

When discharge became steady, the tailgate wasadjusted to give uniform flow in the channel. Thisadjustment was the one requiring the most judg-ment and the one most subject to error duringthe tests. As has been mentioned, screeding leftthe channel soil surface parallel to the flume rails.However, roughening the soil left its averageelevation slightly above the overall mean at somestations and slightly below it at others. Thiscondition was especially evident in the rippledbeds. For this reason, flow was judged to beuniform when the water surface elevation wasmost nearly constant relative to the flume rails.Both bed and water surface elevations were readwith a point gage mounted on the movable carriageused for screeding the boundaries. Thus, allelevations were measured from the same datumparallel to the channel slope and could be com-pared directly. The slope of the flume rails, setbefore each series of runs, was assumed equal tothe energy line slope for computational purposes.The measured depth of flow sometimes variedconsiderably (as much as 25 percent) from stationto station. When resistance coefficients werecalculated, the error due to this variability wasreduced by averaging results for at least fourstations between station 20 and station 50 (of the61 stations located at 1-foot intervals alongthe 60-foot flume). (Readings of water surface

771-814 0-05-2


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8

OO

0

C

O E

o o o o o 0co in ro

Jali!d wowed

0 0 1,0

9FLOW RESISTANCE IN SIMULATED I RRIGATION BORDERS AND FURROWS

elevation were taken every 5 feet between sta-tion 10 and station 55.)

Slight variations in the roughness of some sec-tions caused unavoidable nonuniformities in flowdepth for some channels. For instance, the waterdepths in one channel were slightly lower in themiddle of the test section than at stations furtherupstream or downstream. In such cases themean water surface was adjusted to the sameelevation at the two ends of the channel bychanging the tailgate setting.

The water surface elevation was measured tothe nearest 0.001 foot at every 5-foot stationalong the channel. When the water surface wasrough, three readings of water surface elevationwere taken across the channel and averaged foreach station.

After a series of runs at one slope, the flume wasset at a different slope by means of the adjustablescrew jacks. that supported it. For most channels,runs were made at slopes of 0.005, 0.001 and0.0005. Slopes of 0.0001 and 0.0003 were includedfor some channels. At slopes of 0.0005 or less,rail elevations were estimated to the nearest 0.0001foot in order to set the slope as precisely aspossible. At a slope of 0.0001 there was only a0.006-foot drop in elevation between the upstreamand downstream ends of the channel, making itextremely difficult to be sure when flow wasuniform.

After the last run for each boundary, six plastercasts, each 1 foot long and about 0.2 foot wide,were made in order to be able to make additionalroughness measurements, if necessary, after theboundary was changed. The location of eachcast was chosen at random along the test reachof the channel. The long dimensions of thecasts were aligned with the direction of flow sothat both height and longitudinal spacing of theroughness could be obtained from them.

Because of the random nature of the roughness,a statistical representation of roughness dimen-sions was necessary. The measure used forroughness height was the standard deviationof evenly spaced surface elevation measurementsabout the mean elevation. This was computedfor the rectangular channels by use of the fol-lowing formula:

=.1 2 (Yi-7)2n-1

where:a is the standard deviation,y i is the individual bed elevation measure-

ment,y is the mean bed elevation,n is the number of elevation measurements.

The values of a could be calculated directly frommeasurements of the casts for rectangularchannels because the cast measurements werereferenced to the true bed elevations.

A different procedure was used to calculatestandard deviation from casts of the parabolicchannels. Because readings were not referencedto bed elevations, a regression line was first com-puted through the elevations read from the casts.This regression line was assumed to be parallel tothe slope of the channel. The standard devia-tion of elevation measurements about this regres-sion line was then calculated by the formula:

VI(yi —a+bxf) 2n-2

where:Zxv—ZxZyin

b— •Lx2 (zx)2in

xi is the distance from some datum at whichthe elevation y i is measured.

Standard deviations were computed for casts fromboth the sides and bottom of the channels and theresults averaged. The significance of a is indicateddiagrammatically in figure 3.

Considerable variation existed in values of a fromdifferent casts for the same channel boundary.In extreme cases the value of a for a given castwas as much as twice as great as the mean valuefor the channel. No consistent variation wasevident in values of a measured on the sides orin the bottoms of parabolic channels.

Measurements of the longitudinal roughnessspacing were made only on the rectangular chan-nels. Spacing was defined and measured in twoways. Both measurements were taken along aline parallel to the direction of flow. To obtainthe first spacing measure, X„, the number ofroughness elements that projected more than laabove the mean bed elevation was counted andthe average spacing computed. The second spac-ing measure, X,„ was obtained by counting allthe roughness element crests along a length ofboundary and determining the average spacing.Results obtained from this procedure depend tosome extent on the judgment of the person makingthe measurement. Both spacing values wereobtained from measurements along 6 linear feetof each boundary. The significance of the twomeasurements of longitudinal spacing can beseen in figure 3.

No procedure was available for evaluating therugosity of the individual roughness elements.Rugosity did vary over a wide range, being repre-sented at one extreme by a cloddy-type roughnessthat was stabilized in the same condition in whichit was formed by a tillage tool. At the otherextreme were the roughnesses that had beensmoothed and rounded by low-velocity flows ofwater.

(18)

(19)

mean bed elevation


V ROUGHNESS HEIGHT: STANDARD DEVIATION OFEVENLY SPACED SURFACE ELEVATION MEASURE-MENTS ABOUT THE MEAN ELEVATION.

X., AVERAGE LONGITUDINAL SPACING OFROUGHNESS ELEMENTS PROJECTING MORE THAN la

ABOVE THE MEAN BED ELEVATION.

Xp AVERAGE LONGITUDINAL SPACING OF ALL ROUGHNESS-ELEMENT CRESTS.

FIGURE 3.—Diagrammatic representation of measured roughness dimensions.

RESULTS

The experimental data obtained are given intables 8, 9, 10, and 11 in the appendix. Forchannels where the mean bed elevations weredifferent at different stations, it was difficult todetermine when flow was uniform. To determinewhether any runs were nonuniform, mean depthof flow was plotted against discharge for all turbu-lent runs in the rectangular channels. For furrowsthe hydraulic radius was plotted against discharge.A well defined relationship existed between Q(total discharge) and d or R. The relationshipwas represented by a different line for each channelslope. The lines for all slopes were parallel.If it was assumed that the lines relating dischargeto depth represented data for uniform flows, devia-tions of individual points from the lines could beattributed to nonuniform flow or other errorsin taking the data. Data that varied by morethan 5 percent from the lines were considered tobe of questionable value and are not included inthe following analysis.

Laminar Flow

A plotting of the Darcy-Weisbach resistance co-efficient, f, against Reynolds number is often usedto illustrate resistance characteristics of the vari-ous regimes of pipe flow. The same type of datacan be plotted for open channel flows. If theReynolds number (Re) for open channels is definedas being equal to RV Iv (where v represents kine-matic viscosity) and d is assumed equal to R forwide channels, equations 1 and 4 can be combinedto yield :

24(20)

A line corresponding to this equation is plottedon each of figures 4 to 9 and represents the theoreti-cal resistance relation for laminar flows oversmooth boundaries.

In figures 4 to 9, coefficient of friction is plottedagainst Reynolds number, for six rectangularchannels with some runs at Reynolds numbers inthe 30 to 1,000 range (channels D, E, I, K, L,M). The curves for the data for five of thesechannels (D, E, I, K, M) lie above the theoreticalstraight line for laminar flows over smoothboundaries. Data for each channel slope is repre-sented by a different straight line for Reynoldsnumbers less than 500. For these five channelsthe resistance to flow was apparently increased bythe boundary roughness. The data for bed L(fig. 8) and for the lowest slope of bed K (fig. 7)indicate that resistance for these runs was the sameas for theoretically smooth channels. Althoughbed L functioned as a smooth bed, it was not atrue plane. The boundary had a very-fine-sand-grain roughness and a slight waviness.

The lines defined by the data for low Reynoldsnumbers have a slope of 45° and are parallel tothe theoretical line (solid line) for smooth bound-aries, indicating that the flows were similar totrue laminar flows.

Critical Roughness Height

On the basis of the assumptions used byGoldstein (5) a criterion can be developed for theheight of roughness necessary . for separation tooccur in laminar flow in wide, open channels.The velocity distribution for two-dimensionallaminar flow is given by :

v = (yd—y0 (21)

where:

v is the velocity at a distance y from theboundary, and

d is the depth of flow.


S

)

1-in r---

00 _ .4.— 0 0 0cS 0 0 0.

oO o6 .. i. ii

4- (r) (r) kr)ca)El, I ril 1

Q. .: IIll i i

4/

4

II

/ / /ii ll

•

,i 7 7

/

7 6(.)

.c 4.-.-.46 .3)

o tiE wv)

O— o...- .0,- co ms oE

/. •

/ /- 7

I,

/, •

14

/, ,/

/„

•

/

0-J *

q-

C \1

(1)

et• i•

,7

/

11

4..II

000

0

000...

00

00 I

0

00 0

o _ Lo

000 000

d cisti 11

(r) cf)

• • 4a)

Ea)G.X

000,0

I

1

000,0

I

00

O0

A•

000.

00

00



000It)

60 _ in0 070 0 o~E 0 ow 6 6

II II' (r)(r)1)9- ► •

LiJ

in5 oo o6 6,I II

(r) Q)n •

P/

I fI/

IfiLd607rAIL

1111,011.11INICKIIIII

BUNION/MI

rik . 300 t-,E00 w.c„..-IIM 0 ,,

4- Z1111 P snrLi

2MIMI.

.--JA

.LIIII c\i ....sor

00o"

00Lo"

00

00

0

0 0 0

50 100 500 1poo 5,000 iopooVR /I)

FIGURE 7.—Resistance coefficient f as a function of Reynolds number, VRly, for rectangular channel R.

Experimental Data♦ S = 0.0001

♦ 5 = 0.0003

• S = 0.0005

Laminar f low , smoothboundary theoretical

I.0

0.5

0.I

0z

1;1zz

1j

0

1-3

co

0Li

0

0

50,000

100 500 ip o o10 50

FIGURE 8.—Resistance coefficient f as a function of Reynolds number, VR/v, for rectangular channel L. Cn

4

VR/0

Experimental Data♦ S = 0.0001• 5 = 0.0003• S= 0.0005

S = 0.001•

•

f= 24 ••Re. 1

Laminar flow , smooth •boundary theoretical

•

I.0

0.5

0.1


• •OOOr

•8to

OCE0

0

C0

to

O OO

c3O

•O

4—

(1)

00)•cn

OO

O


17

The velocity at a level equal to the roughnessheight k is:

Vk=3V d(k 2d2

for laminar flow is:CI) 2.58d

irrfe(25)

or, if is small:

Vk =3V d-

Now, the tip Reynolds number is defined as:

Rek =V k

2=3V ci;

d k2=3V ; d-2

=3Re (D2

Let RekC be the value of tip Reynolds number atwhich separation occurs at the tops of the rough-ness elements (critical value). Then the height ofroughness necessary for separation to occur inopen channel flow is:

L d (Reker (23)15 Re

The data collected and the concept of a criticaltip Reynolds number were used to derive anexpression for the height of roughness necessary toaffect the flow (to cause head losses greater thanthe theoretical for laminar flow). Rectangularchannel L was not rough enough to cause headlosses greater than the theoretical. The highestvalue of tip Reynolds number observed for thischannel was 16. Likewise, bed K behaved as asmooth bed at a slope of 0.0001. Only one tipReynolds number at this slope was greater than20. With steeper slopes, resistances were greaterthan predicted by the theoretical equation forsmooth boundaries. All but one of the tipReynolds numbers for these slopes were greaterthan 20. Critical values, therefore, were assumedequal to 20 and were substituted into equation 23to find the critical roughness height, expressed interms of the standard deviation of bed elevationmeasurements:

V2 16

120 kRe)

2.58dare

Therefore, the height of roughness necessary tocause head losses greater than the theoretical

One set of data does not agree with criterion 25.For a slope of 0.0001, the data indicated bed Ito be rough. However, tip Reynolds numbersat this slope were all less than 20. Bed I wascomposed of sharp-edged angular roughness ele-ments. Apparently, separation occurred overthese elements, although the tip Reynolds numberwas low.

Resistance Equation

It has been shown that the resistance equationfor smooth boundaries is not applicable to laminarflows over most of the soil roughnesses studied.However, the data presented in figures 4 to 9 wereused to develop an empirical equation. First,the data for each channel were combined intoa single equation by expressing the intercept ofeach parallel straight line in terms of the channelslope. The resulting equation was:

f =AS° .5Re- 1

where A has a different value for each channel.The channels differed only in degree of channel

roughness. Therefore, A was apparently a func-tion of the roughness dimensions. By repeatedtrials, A was related to the measured roughnessheight, longitudinal spacing, roughness density,and various combinations of these terms. Therelation that best describes the experimental

data is A=6.0X 104 (. Therefore, the valueX,of the resistance coefficient can be predicted bythe following equation:

j_ 6.0X104S") .5kx

(26)Re ,)

where ty represents the roughness height, measuredas shown on page 9, and X, is the average longi-tudinal spacing between roughness crests. Acomparison between measured values of f andvalues computed from equation 26 is shown infigure 10.

Equation 26 is a purely empirical relationshipexpressing the resistance coefficient in terms ofthe slope of the energy line, a roughness parameter,and the Reynolds number for flows in the laminarrange. The equation indicates that as the heightof roughness is increased, both the disturbance toflow and the resistance coefficient increase. Anincrease in roughness spacing decreases both thenumber of disturbances and the resistance coef-ficient. The effect, of slope can be illustrated by-considering a constant discharge. At small slopes

(22)

(24)

CC 0

5 I0

0 80 o

o

1.0E

0

U

0.5

0

0

00 •C6 0

•000

U

0.5 10_ 6 X 10

4 S 0.5 (crA) f, predictedRe

0.10.1

0

0


10

5

0

FIGURE 10.—Comparison between predicted and measured values of f for low Reynolds number flows in wide,rectangular channels.

the flow will be deep and low in velocity; the eddiesformed at the roughness crests will be weak andwill cause turbulence only throughout a smallportion of the flow depth. At higher slopes flowdepth will be smaller and velocities greater. Theturbulent wakes will occupy a relatively largeportion of flow depth and resistance will be ac-cordingly greater.

Equation 26 is a desirable one for the computationof resistance coefficient because all variables canbe determined. The slope is generally pre-determined or can be measured, the roughnessparameter can be computed from measurementstaken of the boundary, and the Reynolds numberis merely the unit discharge divided by the vis-cosity of the water. The computed resistance


a1

SSr

coefficient can then be combined with the Darcy-Weisbach equation, equation 4, and the continuityequation (q=Vd) to find the mean velocity offlow or depth of flow. The following equation isobtained for depth of flow:

.V7.5X108qv a\

gars kxpl(27)

type are applicable to turbulent flows. In thisstudy, all flows with Reynolds numbers greaterthan 500 were analyzed by assuming that logarith-mic resistance formulas were appropriate. Fig-ures 4 to 9 show coefficient of friction plottedagainst Reynolds number for six of these channels.

Turbulence Constant

Different investigators have found differentvalues for the universal turbulence constant, K.

Values of 0.40 and 0.38 are most commonlyproposed. These values correspond to valuesfor the coefficient of the logarithmic term ofequation 20 equal to 5.75 and 6.06, respectively.

Values of K can be determined experimentallyfrom either velocity distribution data or meanflow data. In this study, since no valid velocityprofiles could be obtained, it was necessary toestimate K values from the mean flow data.Values of K determined in this way varied fromone channel to the next. All but one value weresmaller than 0.40, the constant found by Nik-uradse. Apparent values of K as determinedfrom mean flow data can differ for two reasons—nonuniform boundary shear or inconsistent meas-urement of flow depth. If boundary shear is notuniformly distributed, the value of e in equation9 will differ from zero and the apparent value ofK will differ from the true value by the factor1+i. Keulegan (10), from examinations ofBazin's data, found the effects represented by; to be negligible. Powell (16), however, found

values varying from —0.137 to —0.221, fordifferent channel shapes. If the assumption madefor rectangular channels in this study is valid,i.e., flow is unaffected by the side walls, the valueof a should be zero.

Values of K were determined in this study byplotting Chra, resistance coefficient, against u/R,a measure of the relative roughness (figs. 11 to 20).The data for nine of the rectangular channels andall of the parabolic channels plot as straight lineson the semilog graph paper. They can thereforebe represented by an equation of the form:

2.30 logl At

This equation has the same form as equation 9,the theoretical equation for rough-boundariedopen channels.

The value of 2.30/K, the coefficient of the loga-rithmic term in equation 28, ranges from 5.13 to11.95 for the different channels. Values for allbut one boundary are larger than the most com-monly accepted value of 5.75. The boundarywith the small value of 2.30/K (channel H) is onewith a relatively small value of a.

Transition From Laminar to Turbulent Flow

The transition from laminar to turbulent flow• was studied by observing streams of dye injected

into the flow. At the very lowest Reynoldsnumbers, no turbulence could be detected. At

• higher Reynolds numbers, wakes and generalinstability of laminar flow over the rough bound-aries began in a region close to the roughnesselements. Dye streams were straight near thesurface but undulating and beOnning to break upnear the bottom. At still higher Reynolds num-bers, turbulence was observed throughout theentire flow depth. The critical Reynolds numberfor laminar flow over rough boundaries was be-tween 400 and 500 for the range of slopes androughnesses studied. The critical Reynolds num-ber for laminar flow over a smooth boundary wasgreater than 500 and as high as 766.

The value of the critical. Reynolds number wasalso determined from the point at which the plotof the experimental data diverged from the theo-retical 45° line in figures 4 to 9. For channels withrough boundaries (channels D, E, I, K, and M)the transition from laminar to turbulent flow asindicated by these plots occurred at Reynoldsnumbers of 400 to 500. For a smoother boundary(channel L, the troweled soil surface), the transi-tion occurred at a Reynolds number of 700. Theseresults are in agreement with the dye streamobservations.

Observations of critical Reynolds number inthis study are in general agreement with the valuesreported for rough boundaries by previous in-

* vestigators: Jeffreys (9), 310; Hopf (7), 300 to330; and Allen (1), 300. For the smooth boundaryused in this study, the critical Reynolds number

4 was above 700. Horton and coworkers (8)observed laminar flow conditions up to a Reynoldsnumber of 548 and Owen (14) observed the sameup to a Reynolds number of 1,000.

Turbulent FlowThe analysis of critical Reynolds number has

shown that flows in the type of boundaries likelyto be encountered in irrigation channels are in atransitional or turbulent state at Reynolds num-bers greater than 500. Previous studies haveshown that resistance formulas of the logarithmic

(28)

.20 .60 .80.06 .08 .10

cf/ RFIGURE 11.—Relation of resistance coefficient (C/J) to relative roughness (cr/R) for rectangular channels N and B.

.40.01 .02 .04

O

oo Rectangular0

0.94 —7.40

Channel

l og

N

Gr

_

o o T--,c

=

Oo

o

_

ooco

O

_

_Rectangular Channel

r -0.97 — 7.80 log

B

cf-R

0c

00

_ _

12

10

8

c 1U.,z .h 6

4

2

0

0

L.4

0

tm

0

1-3

L.1

0

0"21

L.1

.006 .008 .01.004 .02 .04 .06 .08 .10

07RFIGURE 12.—Relation of resistance coefficient (C/.4) to relative roughness (olli) for rectangular channel C.

_0

Rectangular

Channel C

-

0

0 0 0 0 _

-

00

00 -

-0

0

0

-

r

18

16

14

12

10

0


II

10

9

8

c)1 1'

7

6

5

0.0I 0.02 0.04 0.06 0.08 0.1

Cr/ R

FIGURE 13.—Relation of resistance coefficient (C/ifi) to relative roughness (a1R) for rectangular channel D.

0.2

o

RectangularChannel D

d—vi=0.53 — 6.68 log --R-

9)0

l


0

o

O0

LLI

73)ccv

.cU,..ciScnCaten

CC

O 0

\ )1 eta'0

wc

—N-

•'tNIii

ql-%'

II- NO I etO

O -41/C

C0=C.)

z-e-- 0N

/---.,O

CO

r`:i

o111/

.-0 0s N

o•

a' 1

C 11

o c)ILZ'w

0

' X

CO cD r.

0

0

0

coO

CDO

vrO

O

O

9

771-814 0 - 65 - 4


(!

Tvcco.c0

..2alccs

45a)Cr

b I ct

cho

a)0)(Xi

0II

u I ist:/

• / y

0

• 00— 101 k

71; gO

c —c co0 o.co 6

I

°o

0► dc ic)

0 C) ILO!a)Cr

0

0

•0

i

/°

0

I I I 1

O

0N

0

coO

CD

O.

O

N0

Ct

N 0 N 0

0

0

coO

vrO

O

b

5


•

AI

bl etu. cno i o

as — r0c su0 c4- c in0 0cp .= Icr o cn

in

ct:II

vl

O

O

OO

0

O

O

0

1 I 1 N___

co

O

0

I I

0-5

II

bl ec

0

(0Y —

0)Tv 5

c co= oa

-= d0 111C)11_%1

II I 1

c nc0ta)CC

0/0

0/

tOP C1.1 O co

Ogt.

O

2 6 CONSERVATION RESEARCH REPORT 3, U.S. DEPT. OF AGRICULTURE

0


.06 .08 .10

0/R

FIGURE 18.—Relation of resistance coefficient (CAC!) to relative roughness (aIR) for parabolic channels (:, , and A.

.01 .02 .04 .20

•• • Parabolic_•

•

•1

el

Channel

•

G

_

o Parabolic Channel EON -

CM 0 c C

.on - =-5.95—& II g

11.95 log—R0

0 .1410 . _

Parabolic Channel A 0h11...g

= -2.80 — 8.30 log n-8

• _n 1

n

14

12

I0

8

6

4

2

0

co0


•

•

\01 etcri

0 0— ct

•a) toCC (6c3 •

0

.c0 =

-0 11'1 •7) —

•0 0

D

_a01.-.

0

CL

AL.: ,

•

•O

CO b I er0)

-4) 0.-cc .4.0 (0.c (C;0

I

0 cm

0 (D..0 N0 ii

6 cd.''a_

•

o

O

0

4,

0

/2c

(0 N O O C

CD0

Oct

0

cD09

4

0

0coOO

OO

srOO

O"(1) N

0

0 CO

•

L.L.

-cr)cQ

_c

0.O.000a.

)0 I Cc

c:nO

cr(D

6I

NIn

IIvlic.:.,

O

SO

o 0

o _4)

E0.c0

(2-0-.00

0CL

—

0

/00

o

b I Ct

cn2

0

0.

• ••

••0

o

•• •co

ocsIt

q IsZt

•

•

•

i i I I 1 si`---

co0

O

srO

NO


29

30


If it is assumed that the value of K is constantand that the corresponding proper value of 2.30

— issc

either 5.75 or 6.06, a depth correction can befound for each channel such that the plot of

'VTagainst ir will have the chosen slope. Values of the

correction for —2.30 equal to 5.75 are given in table 1both in absolute amounts and as fractions of theroughness height. There is no simple relation-ship between the depth correction and the heightof roughness.

Viscosity and Roughness Effects

Figures 11 to 20 indicate that since the resistanceof most channels is represented adequately as afunction of relative roughness, the boundaries arehydraulically rough. However, for any onechannel, depth increases and relative roughnessdecreases as discharge increases. As dischargeincreases, Reynolds number also increases. A re-lationship between resistance coefficient andReynolds number could therefore be confused witha relationship between resistance coefficient andrelative roughness. A more positive method ofdetermining the type of flow resistance occurringis necessary.

2.30

Keulegan (10) plotted V log R against

* K

log For channels with rough boundaries,flow resistance was independent of viscosity andV 2.30-- log R was a constant. Viscosity effectsV* K

were evident for hydraulically smooth channelsand the line formed by plotting data from suchchannels was inclined. This line can be repre-sented by the equation:

TV_ 2t30 log R.A+ 2.30 lo V*K g V

orV 2.30 RV*-1-7-*=A+ log v (29)

as would be expected from theory. Keuleganalso found some boundaries which he termed"wavy." Data for wavy boundaries describedlines parallel to and below the smooth boundarylines, indicating a higher resistance for a given

value of RV*V

The data for the channels studied were plottedin the form suggested by Keulegan (figs. 21, 22, and

TABLE 1.—Depth corrections

Channel type andidentifying letter

Roughnessheight, a

Depth cor-rection, c I

Relativedepth cor-rection c/a

Feet FeetRectangular:

B 0. 0187 0. 0083 0. 445D . 0118 . 00923 . 782E . 0090 . 0024 . 267F . 0141 . 0120 . 852G . 0117 . 0096 . 820H 0024 (2)I 0064 . 0010 . 156K . 0054 . 0050 . 926N . 0198 . 0080 . 404

Parabolic:A . 0080 . 0130 1. 62B . 0013 . 0007 . 539C . 0029 . 0045 1.55D . 0030 . 0015 . 500E . 0092 . 0236 2. 57F . 0015 . 0008 . 533

c is the depth correction necessary to make CAC pro-portional to 5.75 log crIR, where R is the hydraulic radiusbased on the corrected depth.

2 Negative value.

2.3023). A value of — equal to 6.06 was used becausethis value approached more nearly the valuesfound for the experimental channels in figures 11to 20 than the more commonly accepted 5.75.The boundaries are rough for all but one of thechannels, as indicated by the independence of

V-6.06

V—6.06 log R and the log —* term. Much of*

the scatter of the experimental points about themean value of the ordinate for each of thesechannels is due to the difference between 6.06and the best-fit value of 2.30

— for the channel.K

The data for rectangular channel C show a

functional relationship between 119— —6.06 log R

Vand —I% This relationship is represented in figureV

21 by a straight line with slope of 6.06. Theintercept of the straight line with the y axis islower than would occur for hydraulically smoothboundaries. The relationship is therefore typicalof that for wavy boundaries as presented byKeulegan. Data were taken for only one channelwith wavy boundaries; therefore, an expressionrelating the constant to the degree of wavinesscould not be derived. However, the boundaryfor channel C was smoother than any likely to beencountered in natural irrigation channels and,in this respect, has no practical importance.


0 0N0 CON

6oi 90'9-

NN

I '

_ cr<t_J _1

D LLI • • 00 Z II II iiZ Z co 0 w

_ a aH I0 0Wcr

1 '

• 4al II— Z

1 I 1 1

0 ra

<

•to

a

••

•

_O

• 1 43

•

I

d

_•\

•

010 1

1

0

I

•- I

•1 a f

•II •

II •

•

•

8, 0

0

••I••

I

A

<J<3 .

<

d1/43

I

1,3

•

•.c11 •

••

• 0

i , , , , , ,

Nro

sr

cr.

52720

664

9

0

3

I-s

1I.

eIf

ee6

a

e

e

h

n

e,

Osr

0

(0.ro

4.0 4.2 4.43.0

3.2

3.4

3.6 3.8

I og v'FIGURE 22.—C/ 471-6.06 log R as a function of log V* 11, for rectangular channels. Horizontal lines represent hydraulically rough boundaries.

2

1 1

RECTANGULAR

CHANNEL

I ,

= 0

= •

= L

= •

I

_

H

K

G

D

CD0 0

F=_

n

----•

•• •

•00

A A A AA

A_

47-4\••

•A z,

Ma

Ar7n,• „AC

a

•••

, . , ,

an

, , , ,

2

2

2

a,0

0

co

0zt.1

0

517:1

51

0

1-3

rn

!"30

0

0-3

51

3.2 3.3 3.4 3.5 4.23.6 3.7v *

log u

3.8 3.9 4.0 4.1

24

22

20

cr,

2 18

0

16

14

12

0

FIGURE 23.—CNT1-6.06 log R as a function of log 174,/p for parabolic channels. Horizontal lines represent hydraulically rough boundaries. Ch,GJ

0

.••

o.,• •

0---4,-- o

•

• o CX)•A

•

•

0

on W

A

• •n (

LI

•C

0

*0

•A

n

A

o 0

AA

•

00

♦

0

• •• ••n 0

00 PARABOLIC

CHANNEL

0 F=o. •= A

•= 0= •= X

C=n

n III X nn

•D

GAEX

X1

LX

X

X

1-3

z

02

rb0

Ltk:)02azt:1

34


On the basis of figures 21 to 23, a criterion canbe established for distinguishing between roughand wavy boundaries, using standard deviationof bed elevation measurements as an estimate ofroughness height. It is assumed that the thick-ness of the laminar sublayer, S', can be defined bythe same formula as is used for pipe flows:

E=11.6 V (30)

In table 2, values of the ratio of o to 5' are shownfor the one wavy boundary and for the two roughboundaries having the smallest value of a. Only

atwo values of exceed 0.520 for the wavy bound-

ary, channel C. There are two values of Fa smallerthan 0.520 for the rough boundaries, channels Hand K, which would be expected to most nearlyapproach the wavy condition. For these data,the distinguishing criterion for the transition fromwavy to rough boundaries would then be:

cr-0.5205'

or, combining this equation with equation 30:

V*a=6.03 (31)

Resistance ParameterThe use of an equivalent resistance term (x) to

take account of all factors affecting resistance wasdiscussed in the Review of Literature section.Such a term can be calculated most easily fromequation 16. The equivalent resistance term, x,in this equation was shown to represent effects ofboth roughness and channel shape on flow resist-ance. By using equation 16, values of x can becalculated for the channels with rough boundariesfrom data contained in figures 21 to 23. Theaverage value of the ordinate for each channel,-6.06 to R, is equal to -6.06 to X. Valuesof x were calculated in this way for each roughchannel. A tabulation of x values is shown intable 3.

The values of x may now be related to thephysical characteristics of the channel that affectresistance. Results of previous research haveindicated that channel roughness will exert theprimary effect on flow resistance. The methodsused to measure channel roughness have beendescribed in the section on Experimental Pro-cedures. Values of the following roughness meas-ures are listed in table 3: a, the standard deviation

TABLE 2.-Ratio of roughness height (a) to laminarsublayer thickness (5') for three rectangular chan-nels with small roughness height'

Channel C Channel H Channel K

Run eir Run alai Run alai

2-C 0. 422 2-H 1.29 4-K 0. 5303-C . 500 3-H 1.55 6-K 5004-C 545 6-H 565 7-K 6019-C_ _ _ .279 7-H 660 8-K . 67410-C_ _ _ . 336 8-H 820 9-K . 805

11-C.. _ . 403 9-H 983 10-K.. _ _ _ 1.0112-C_ _ _ _ . 441 11-H_ _ _ _ 1. 53 16-K_ _ _ _ 1.8513-C_ _ _ _ . 553 14-H _ _ _ _ . 468 31-K_ _ _ _ . 69018-C.. _ .178 15-H.. _ _ _ . 570 33-K_ ___ .85519-C_ _ - _ .197 16-H _ _ _ _ . 688

20-C.. _ _ _ . 221 17-H.. _ _ _ . 85623-C_ _ _ _ . 429 18-H_ _ 1.0524-C_ _ _ _ . 517 19-H_ _ _ _ 1.32

20-H _ _ 1.51

1 The boundary of channel C was wavy. Channels Hand K had hydraulically rough boundaries.

TABLE 3.-Calculated values of resistance param-eter and measured roughness dimensions forrough channels

Channel type andidentifying letter

xl a2 A X I,'

Rectangular: Feet Feet Feet FeetB 0. 0155 0. 0187 0. 197 0. 141D . 00740 . 0118 .375 . 099E .00166 .0090 .860 .257F .0107 .0141 .625 .358G . 00561 . 0117 . 667 .275H .00089 .0024 .215 I . 00691 . 0064 . 240 . 054K .00115 .0054 .400 .139N .0100 .0198 .286 .133

Parabolic : 5A . 00525 0084 B . 000323 .0013 C . 00100 0029 D . 000512 .0030 E . 00933 0092 F . 000316 0015 G . 00204 0044

1 Resistance parameter.2 Roughness height: standard deviation of evenly

spaced surface elevation measurements about the meanelevation.

3 Average longitudinal spacing of roughness elementsprojecting more than 1 a above the mean bed elevation.

4 Average longitudinal spacing of all roughness elementcrests.

5 No measurements of longitudinal roughness spacingmade.


35

TABLE 4.-Differences between measured and predicted values of the resistance parameter, and relatedmeasured roughness-spacing indexes

5

Rectangularchannel

x,measured

x,predicted

Absolutedifference

Relativedifference

X, X,

Feet Feet Feet PercentB 0. 0155 0. 0174 + 0.0019 +12. 3 0. 197 0. 141D .00740 .00813 +.00073 +9. 9 .375 .099E .00166 .00516 +.00350 +210.0 .860 .257F .0107 .0112 +.0005 +4.7 .625 .358G .00561 .00812 +.00253 +45.7 .667 .275H 00089 00058 - . 00031 -34. 8 .215 I . 00691 .00297 -.00394 -57. 0 .240 .054K . 00115 .00219 +. 00104 +99. 0 .400 .139N_ 0100 .0194 +. 0094 +94. 0 .286 . 133

5

1

r

V

L.t

of equally spaced measurements of bed elevations;X„, the average longitudinal spacing of roughnessesthat projected more than one standard deviation(1a) above the mean bed elevation; and X„, theaverage longitudinal spacing of all roughnesscrests.

Attempts were made to correlate values of x withmeasured dimensions of roughness. A correlationexisted only between x and o; the calculated valueof the correlation coefficient is 0.93. The rela-tionship of the resistance parameter, x, to rough-ness height, a, is plotted in figure 24. Theregression line representing the relationship of xto a was determined by least-squares methods.The equation of this line is:

x=12.9cr' . " (32)

The spacing of the roughness elements cannot berelated directly to the flow resistance parameter,x. However, previous studies (17, 18) have shownthat the flow resistance of conduits with the sameroughness height but different spacing differs.Therefore, it would be expected that the scatter ofthe data about the line relating x and a, figure 24,might be correlated with the roughness spacing.Table 4 includes both absolute and percentagedeviations of measured x values from those pre-dicted by equation 27, and values of the twomeasured roughness-spacing indexes. The rela-tions are not well enough defined so that thesemeasures of spacing can be used to improve theestimate of x.

Channel Shape EffectsThe value of x theoretically expresses the effects

of both roughness dimensions and channel shape onflow resistance (see equation 16). Values of x forparabolic channels with three different shapes areplotted against o in figure 24. In this plot, theshape of the channels is represented by use of thehydraulic radius as a length parameter. There is

no relationship between the deviation of theindividual points from the best-fit line in this plotand the shape of the channels.

Supporting Data

Parsons (15) measured the resistance to turbulentflow over a short section of rough channel. Thechannel roughness was formed by directing a waterspray on a soil-cement mixture, which formedsmall, closely spaced craters. The resulting sur-face was similar to that of soil roughened byfalling raindrops.

Data are available for the computation of x anda for two series of Parsons' runs. For series IVthe computed value of x is 0.00138 foot. Thevalue of u, computed from the bed roughnessmeasurements, is 0.00490 foot. For series V runsthe corresponding values are: x=0.00160 foot,a=0.00590 foot. These values of x and a areplotted on figure 24. Parsons' data are repre-sented by equation 32 equally as well as the datafrom the present study.

Prediction Errors

Equations 16, 26, and 32 may now be used topredict flow conditions for natural channels ifmeasurements of a and X for these channels canbe made.

When an irrigation system is being designed,slope is determined by topographic and land form-ing limitations, channel shape depends on the cropto be grown, and discharge rate may be assumed.The problem is then to predict the depth of flow.The magnitude of error likely to be encounteredin predicting depth of laminar flow over roughboundaries can be determined by reference tofigure 10. Examination of this figure indicatesthat for 66 of the 73 values, the error in predictingf (the Darcy-Weisbach resistance coefficient) byequation 26 was less than 30 percent.

19785

493

0.

0.00

0.00

0.000

)1

5

1

5

•

• 0

•V• 0

•

•

x

xo

0

0

•

•• I X . 12.901.66o Rectangular Channels

• Parabolic Channels

x Rectangular Channels,Parsons' Data

0.010.000

0.001 0.005cr ft.


FIGURE 24.—Relationship of resistance parameter, x, to roughness height, a.


37

The defining equation for f can be written:

8gdS 8gd3Sq2

Taking logarithms of both sides and differentiating:

df-3dd dS 2dqd S

Each of the terms in this equation represents therelative error in one of the variables. If q and Sare accurately known, dS/S and dq/q are negligibleand:

dd 1 df

Thus, a 30-percent error in f results in only a10-percent error in d.

The best method found for estimating x was torelate it to a measure of the roughness height, a,by means of equation 32. The correlation coeffi-cient determined for x and a was 0.93, indicatinga highly significant relationship. Although otherdimensions of roughness affect resistance, theycould not be measured accurately enough to be ofpractical value. The error in using equation 32to estimate x is indicated in figure 24, where theplotted points represent measured values and thestraight line represents estimated values. Theerror is quite large for some channels—as much as210 percent for rectangular channel E. However,because x appears only in a logarithmic term inthe flow resistance formula, equation 16, the errorsinvolved in practical applications of the formulaare smaller.

Normal flow depth for very wide channels canbe calculated from equation 16 written in theform:

q 6.06 log d= -6.06 log x (16a)„ad3/2

Some trial calculations of d were made, using bothmeasured and estimated values of x, so that theerror in d could be determined. An intermediatevalue of slope, 0.001, was assumed for all the cal-culations. Two values of q were assumed—onethe lowest for which turbulent flow would occur,and the other near the upper limit of the range ofdischarges used in this study.

Table 5 lists the errors found in predicted normaldepth of turbulent flow for two rates of discharge,when using the equations developed from thisstudy. The maximum error for any channel is27.5 percent at the lower turbulent unit discharge.At the higher rate of discharge, errors in determin-ing depth are about half those at the lower rate.

The error in using equations 16 and 32 for cal-culating the hydraulic radii of parabolic channelswas also investigated. The computation for

TABLE 5.—Error in calculating normal flow depthin rectangular channels

Assuming q= 0.0060 c.f.s./ft. and S=0.001

Channel d, meas-ured

d, pre-dicted

Error

Feet Feet PercentB 0. 0508 0. 0533 +4. 9D . 0402 . 0413 +2. 7

. 0283 . 0361 +27. 5F . 0452 . 0457 +1. 1G . 0369 . 0412 +11. 6H . 0254 . 0238 —6. 3I . 0392 . 0318 —18. 9K . 0266 . 0299 +12. 4N .0443 . 0556 +25. 2

Assuming q= 0.500 c.f.s./f . and 8=0.001

B . 462 . 470 +1.7D . 412 . 417 +1. 2E . 341 . 391 +14.7F . 434 . 436 +O. 5G . 396 . 416 +5. 5H . 318 . 306 —3. 8I . 407 . 365 —10. 3K . 328 . 352 +7. 3N . 429 . 478 +11. 4

parabolic channels is more difficult than for rec-tangular channels. The relation between areaand hydraulic radius must be known so that thevelocity can be calculated from the Q/A relation-ship and the equation can be solved by trial anderror. Values of R were determined by this pro-cedure for both small and large discharges.Values of hydraulic radius based on both predictedand measured X values are shown in table 6.The maximum error for these channels was 12.4percent, at the low rate of discharge. At thehigher rate of discharge, the error was reduced,just as it was in calculating normal flow depth forrectangular channels.

The common method of determining the re-sistance of an irrigation channel in the field is toestimate Manning's resistance coefficient, n, fromvisual observation of the roughness, alinement,vegetation, etc.; the estimate is based on thedesigner's experience. The resistance coefficientis usually given a single value for each channel andits variation with depth of flow is not considered.The range of values of Manning's n measured forturbulent flows in each of the rectangular channelsstudied is shown in table 7. A large range ofvalues of n existed for the channels with the largervalues of a. Thus, any one value of n assumed forthese channels would be considerably in error atdepths of flow other than that to which it applies.These data illustrate the inadequacy of a single nvalue to characterize the resistance in a small,rough channel.

38


TABLE 6.—Error in calculAting hydraulic radius ofparabolic channels from predicted x values

ASSUMING Q=0.0030 c.f.s. AND S=0.001

Channel R,measured

R,predicted

Error

Feet Feet PercentA 0. 0349 0. 0343 —1. 7B .0225 .0220 —2.2C . 0261 . 0254 —2. 7D . 0209 .0220 +5. 3E . 0348 . 0305 —12.4F . 0216 . 0214 —0. 9G . 0276 . 0266 —3. 6

ASSUMING Q=0.350 c.f.s. AND S=0.001

A . 239 .236 —1.3B . 188 .185 —1. 6C . 204 . 201 —1.5D . 172 . 178 +3.5E .230 . 215 —6. 5F .180 . 179 —0. 6

.208 . 204 —1.9

The error involved in assuming a constant valueof n for a channel can be compared with the errorof the prediction equations. Assuming q and Sknown, taking logarithms of both sides of theManning equation (equation 2), and differentiating:

dq_5 dd dS _dnq 3 d 2 S n

If q and S are constant:

dd_3 dod 5 n

TABLE 7.—Range of measured Manning's n forturbulent flows

Channel type and identifying letter n

Rectangular:0. 022-0. 049

C . 010- . 013D . 016- . 029E . 014- . 021F . 019- . 031

. 016- . 039H . 012- . 017I . 016- . 032K . 012- . 017N . 18- . 045

Parabolic:A . 19- . 030B . 011- . 014C .013- .018D . 012- . 014E . 011- . 052F . 011- . 013G . 015- . 017

The maximum relative error in estimating flowdepth for the rectangular channels was 27.5percent, using equations 16 and 32. This isequivalent to an error in n of 46 percent. Nowconsider the range of observed n values. For someof the rough channels one would have errors as highas 70 percent in n at low flow depths even if,through extreme good fortune, the average valuefor the range of discharges encountered had beenoriginally estimated correctly. Therefore, use ofmeasured roughness heights in conjunction withequations 16 and 32 should produce resistanceestimates having less error than the procedurecurrently in common use.

APPLICATIONS TO FIELD CONDITIONS

The equations for uniform flow over small earthchannels in both the laminar and turbulentregimes that were developed from the laboratorystudies reported in this bulletin might be appliedto the calculation of flow parameters in irrigationborders and furrows. It should be remembered,however, that the equations have not yet beenverified under field conditions.

Where channels have no vegetation, boundarieswill be hydraulically rough for almost all laminarand turbulent flows. Therefore, the first step inanalyzing such flow is to measure roughness heightand spacing. The following procedure is suggested.

For the measurement of roughness height, six toten 1-foot sections of the bed parallel to the direc-tion of flow may be selected at random. Bedelevations should be measured, with a precision of0.001 foot, at 0.05-foot intervals relative to somearbitrary datum. The value of CI can be calculatedby using equation 19 for each of the 1-foot strips

and these values can be averaged to obtain therepresentative value for the channel.

The same sections of the channel can be used tomeasure the average longitudinal spacing, X,.The number of roughness crests divided into thetotal length of the sections sampled will givethe value of X,. Some judgment must be used incounting the roughness crests: crests too small ortoo low to contribute materially to the flow re-sistance should not be counted.

Next, the regime of flow should be determined.For two-dimensional flows, the Reynolds numbercan be calculated from known values of unit dis-charge and viscosity. Thus:

q= Vd=P

For Reynolds numbers greater than 500, the tur-bulent flow equations (equations 16 and 32) should

d_V7.5X108qv ( a)g AITS \Xpl

To check the roughness assumption use criterion25.

and

(27)=0.0032

are -11:17-

2.58d2.58(0.027)


be used. For Reynolds numbers less than 500,the laminar flow equations (equations 26 or 27)are applicable.

If flows are in the laminar regime and it is as-sumed that boundary roughness is great enough toaffect flow resistance (a> 2.58d/VR e), normaldepth can be calculated from equations 26 or 27.Then :

6.0X 104S° •5 X, f- Re

Since Re<500, the flow is laminar. As-suming that the boundary is rough enough toaffect flow, calculate depth using equation 27:

d=__r7.5 X 108 qvL g1/3

/

_17.5 X 10 8 (0.005) (1.06X 10 -5) k( 0.20.0101"32.2(0.0316)

=0.027 ft.(26)

With this estimate of d, the criterion for boundaryroughness, defined by equation 25, can be checked :

a> 2.58d for a rough boundary;- cre

a<2.85d for a smooth boundary.AIR e

(25)

If the criterion does indicate the channel to berough, the assumption is vindicated and the esti-mate of d may be assumed to be correct. In theevent that the boundary is not rough, the flowdepth may be calculated from equation 1, thetheoretical equation for laminar flow in a wide,smooth, rectangular channel, rewritten in thefollowing form:

d= 301!gS

The following sample calculation illustrates indetail an analysis of flow in the laminar regime,where it is assumed that the irrigation border ishydraulically rough, that is, where o, the roughnessheight, is great enough to cause head losses greaterthan the theoretical for laminar flow(a >2.58d/ i/Re).

Given :

q= 0.005 c.f .s./f t.S=0.001a=0.010 ft.

X,=0.20 ft.T=70° F.

Solution:Compute the Reynolds number:

Vd 0.005Re =--- q-=-1.06X -472V V 10 -5

cr>2.58d

Al Re

Therefore, the boundary is hydraulically rough, asassumed, and the calculated value of d is correct.

If the Reynolds number is greater than 500,turbulent flow can be assumed to exist. Forirrigation borders or very wide channels, equation16 can be written in the form:

q =6.06 log d-6.06 log x

where:X=12.9cr i." (32)

From equations 16 and 32, d can be found bytrial and error, if q, S and o are known. To assurethat equation 16 is valid for the case being con-sidered, the computed value of d can be inserted incriterion 31, where VoRrS is substituted for V.:

417Sa>6.0, for a rough boundary;V

Vird a<6.0, for a smooth boundary.

The following sample calculation illustrates indetail an analysis of flow in the turbulent regime,where it is assumed that the irrigation borderboundary is hydraulically rough.

Given:

q= 0.040 c.f.s./ft.S=0.001o=0.010 ft.T=70° F.

(1)

Irg7Sid3/2


Solution:

Compute the Reynolds number:

R e

— q — 0.040 ==3 780v - 1.06X10 -5 '

Since Re> 500, turbulent flow exists. Assum-ing that the boundary is hydraulically rough,compute x using equation 32:

x-=-12.90.1 •66

=12.9(0.010)"6=0.00618

Compute depth by trial and error, using equa-tion 16, which can be written as follows:

rd3 ,2 6.06 log d=-6.06 log xor:

Algs2=d"[6.06 log d-6.06 log x]

For the left-hand side of the equation:

q 0.040 _0.224g 32.2 0.001

For the right-hand side:

Assume d= 0.10 ft.d3/2 [6.06 log d- 6.06 log x]

(0.10)" [6.06 log (0.10) —6.06log (0.00618)1=0.234

Assume d=0.090 ft.

d3'2 [6.06 log d-6.06 log x]=0.190

Assume d= 0.098 ft.

& I' [6.06 log d— 6.06 log x ] =0.224

Therefore, d= 0.098 ft.

Check criterion 31, the criterion for distin-guishing between rough and wavy boundaries.

V*o- -4E/So- --=-02.2(0.098)(0.001)(0.010) -52.91.06X 10-5

Since V* >6, the boundary is hydraulicallyrough, as assumed.

Calculation of the flow variables for irrigationfurrows is slightly more difficult. Equation 16can be written:

-4TIS

The relationship between flow area and hydraulicradius must be known either in graphical orequation form, for the channel being considered.Then for a given Q, S, and x, a value of R isassumed, V is calculated, and the value of Adetermined. If these values satisfy Q= AV ,the assumption for R was correct. During thelaboratory testing of channel resistance, no runswere made on parabolic channels with wavyor transitional boundaries. However, it is as-sumed that criterion 31 is valid for these channelsas well as for the rectangular channels.

V =6.06 log R-6.06 log x

SUMMARY AND CONCLUSIONSThe factors affecting flow resistance in irri-

gation borders and furrows are difficult to evalu-ate, owing largely to the nonuniformity of theroughness characteristics and to the wideranges in rate of water discharge and size ofroughness elements that occur. A review ofliterature revealed that past research has notprovided sufficient information on hydraulicresistance of these channels to allow for rationaldesign of surface irrigation systems. Therefore,a study of the hydraulics of simulated irrigationborders and furrows was conducted in the labora-tory.

Small channels having boundaries formed ofsoil and chemically stabilized to prevent altera-tion of the roughness, were formed in a laboratoryflume. Rectangular channels with smooth side-walls and parabolic channels represented irri-gation borders and furrows, respectively. Obser-

vations were made of dimensions of the boundaryroughness elements, channel shapes, flow regimeover a wide range of flow rates, and the flowparameters necessary for calculating flow re-sistance coefficients. Flow resistance coefficientsthen were empirically related to dimensions of theboundary roughness elements. Analysis of theexperimental results leads to the following corielusions.

1. Both laminar and turbulent flows can occurwhen flow rates and flow boundaries are similar tothose of surface irrigation systems.

2. Critical Reynolds numbers for the transitionfrom laminar to turbulent flow are 500 for roughboundaries and 700 for smoother boundaries. Acritical Reynolds number of 500 will probably beapplicable to all wide irrigation channels.

3. The following criterion can be used to deter-mine the height of roughness such that flow


resistance will be greater than the theoreticalsmooth boundary equation for laminar flow:

cr> 2. 58d (25)ArRe

4. The resistance coefficient for rough, laminarflow can be ptedicted by the following equation:

6.0X 104S" (X„,

where Cr and X, are measures of the roughnessheight and longitudinal spacing, respectively.

5. The transition from wavy to rough boundaryconditions for turbulent flows occurs when

All surface irrigation channels would be expectedto be hydraulically rough when flow is turbulent.

6. The resistance coefficient for turbulent,hydraulically rough flows can be predicted forboth wide rectangular and parabolic channels bythe following equations:

(16)--0=--6.06 log -x

andx= 12 . 9a1.66

(32)

7. The variation in flow resistance caused by arange of parabolic cross sections is negligible com-pared to the effects of boundary roughness.

8. The magnitude of error involved in predictingresistance to flow using equations 16, 26, and 32 issmall enough to allow practical application of theequations. Larger errors would result from thecurrently used procedure of estimating a singlevalue of Manning's n to represent a channel for alldischarges.

f- Re(26)

LITERATURE CITED(13)ALLEN, J.

1934. STREAMLINE AND TURBULENT FLOW INOPEN CHANNELS. Phil. Mag. and Jour.Sci. 116: 1081-1112.

BAZIN, M.1865. RECHERCHES EXPERIMENTALES SUR

L'ECOULEMENT DE L'EAU DANS LES CANAUXDECOUVERTS. [Paris] Acad. des Sci. Math.et Phys. 19.

CORNISH, R. J.1928. FLOW IN A PIPE OF RECTANGULAR CROSS

SECTION. Roy. Soc. London, Proc. seriesA, 120: 691-700.

DAVIS, JOHN R.1961. ESTIMATING RATE OF ADVANCE FOR IRRI-

GATION FURROWS. Amer. Soc. Agr. Engin.Trans. 4: 52-54, 57.

GOLDSTEIN, S.1938. MODERN DEVELOPMENTS IN FLUID DYNAM-

ICS. v. 1, 330 pp. Oxford.HALL, W. A.

1956. ESTIMATING IRRIGATION BORDER FLOW. Agr.Engin. 37: 263-265.

HOPF, L.1925. TURBULENZ BEI EINEM FLUME [TURBU-

LENCE IN A FLOW]. Ann. der Phys. 32:777-808.

HORTON, R. E., LEACH, H. R., and VAN VLIET, R.1934. THEORY OF LAMINAR SHEET FLOW. Amer.

Geophys. Union Trans. 15: 393-404.JEFFREYS, H.

1925. FLOW OF WATER IN AN INCLINED CHANNELOF RECTANGULAR SECTION. Phil. Mag.and Jour. Sci. 49: 793-807.

KEULEGAN, G. H.1938. LAWS OF TURBULENT FLOW IN OPEN CHAN-

NELS. [U.S.] Natl. Bur. Standards Jour.Res. 21: 707-741.

NIKURADSE, J.1930. TURBULENTE STROMUNG IN NICHT KREIS-

FORMIGEN ROHREN. Ingen.-Archiv V. 1,306 pp.

1932. GESETZMASSIGKEITEN DER TURBULENTENSTROMUNG IN GLATTEN ROHREN. Ver.Deut. Ingen., Forshungsheft 356: 301-315.

1933. STROMUNGSGESETZE IN RAUHEN ROHREN.Verein Deut. Ingen. Forshungsheft 361,22 pp.

(14) OWEN, W. M.1954. LAMINAR TO TURBULENT FLOW IN A WIDE

OPEN CHANNEL. Amer. SOC. Civ. Engin.Trans. 119: 1157-1164.

(15) PARSONS, D. A.1949. DEPTHS OF OVERLAND FLOW. U.S. Soil

Conserv. Serv. Tech. Pub. 82,33 pp.(16) POWELL, R. W.

1949. RESISTANCE TO FLOW IN SMOOTH CHANNELS.Amer. Geophys. Union. Trans. 30: 875-878.

(17) SAYRE, W. W., and ALBERTSON, M. L.1961. ROUGHNESS SPACING IN RIGID OPEN CHAN-

NELS. Amer. Soc. Civ. Engin. Proc. 87(HY 3): 121-150.

(18) SCHLICHTING, H.1936. EXPERIMENTELLE UNTERSUCHUNGEN ZUM

RAUHIGKEITSPROBLEM. Ingen.-Archiv 3: 1.(19) Siluiz, HOLLIS.

1960. FURROW HYDRAULICS STUDY AT THE SOUTH-WESTERN IRRIGATION FIELD STATION. InProceedings of the ARS-SCS Workshop_onHydraulics of Surface Irrigation. U.S.Dept. Agr. ARS 41-43: 56-62.

(20) STRAUB, L. G., SILBERMAN, E., and NELSON, H. C.1956. SOME OBSERVATIONS ON OPEN CHANNEL FLOW

AT SMALL REYNOLDS NUMBERS. Amer. Soc.Civ. Engin. Proc. 82 (EM3): 1-23.

(21) TINNEY, E. R., and BASSErF, D. L.1961. TERMINAL SHAPE OF A SHALLOW LIQUID

FRONT. Amer. Soc. Civ. Engin. Proc. 85(HY5): 117-133.

(22) VANONI, V. A., and BROOKS, N. 11.1957. LABORATORY STUDIES OF THE ROUGHNESS

AND SUSPENDED LOAD OF ALLUVIA'. STRF,A MS.Calif. Inst. Technol. Sedimentation LabRpt. E-68, 121 pp.

(23) Woo, I)., and BRATER, E. F.1961. LAMINAR FLOW IN ROOM! RECI'ANOULAR

CHANNELS. Jour. Geophys. Iles. 66. 42074217.

DescriptionShear velocity, 11 gRS.Velocity at a level corresponding

to the tops of the roughnesselements.

Horizontal coordinate for measur-ing cross section of parabolicchannels.

Distance from some datum atwhich elevation y, is measured.

Depth of flow at center of para-bolic channels.

Distance from boundary.Channel elevation at distance x

from center line of parabolicchannels.

Individual bed elevation measure-ment.

Channel shape factor.Unit weight of water.Thickness of laminar sublayer.Height of roughness necessary

for separation to occur in flowthrough pipes.

Channel shape factor.von Karman universal turbulence

constant.Average longitudinal spacing of

roughness elements projectingmore than to above mean bedelevation.

Average longitudinal spacing ofall roughness-element crests.

Absolute viscosity.Kinematic viscosity.Standard deviation of boundary

elevation measurements aboutthe mean elevation; a descrip-tion of roughness height.

Resistance parameter from Sayre-Albertson equation; a parameterrepresenting effects of bothroughness and channel shape.

DimensionsLT-'LT-'

L

L

L

LL

L

FL-2LL

L

L

FTL-2LIT- 1L

L


SYMBOLSSymbolV.V k

Dimensions DescriptionL-' Coefficient in equation for

parabolas.L Constant in boundary regression

equation. Hydraulic characteristic for wavy

boundaries. Hydraulic characteristic for rough

boundaries. Hydraulic characteristic for smooth

boundaries. Constant, defined where used.LICross-sectional area.

Constant.Constant.Depth correction.Resistance coefficient from Chezy

formula.Normal flow depth.Resistance coefficient from Darcy-

Weisbach formula.Gravitational acceleration.Roughness height.Diameter of equivalent sand grain

roughness.Resistance coefficient from Man-

ning's equation.Number of measurements or obser-

vation s.Unit discharge.Total discharge.Radius of circular pipe.Hydraulic radius.Reynolds number, VRly.Tip Reynolds number.Value . of tip Reynolds number at

which separation occurs at thetops of the roughness elements(the critical value).

Slope.Velocity of flow at distance y from

the boundary.Mean flow velocity.

Symbola

a

a,

a,

a,

AAbB

C

df

k,

n

n

QrRReRekRe k ,

S

L

L

LT-2LL

LIT-'LL

LT-1

V LT-'

x

xi

y

yy

yi

fi

a'

X,

X,

x

APPENDIX

Figures 25 to 35 show the rectangular channels(simulated borders) constructed for this study.(The roughness preparation for channel B, notshown, was similar to that of channel D.) Figures

36 to 42 show the parabolic channels (simulatedfurrows) constructed.

Tables 8 to 11 summarize the experimental dataobtained.

FIGURE 25.—Rectangular channel C. The boundary wastroweled to produce the smoothest surface possible.

FIGURE 27.—Rectangular channel E. The boundary wasroughened. A stream of water then formed small,widely spaced ripples.

FIGURE 26.—Rectangular channel D. The boundary wasroughened, then flooded with a nonerosive stream ofwater before stabilization.

FIGURE 28.—Rectangular channel F. A large, erosivestream of water was used to form relatively largeripples.


FIGURE 29.—Rectangular channel G. The ripples aresmaller than those in channel F.

FIGURE 31.—Rectangular channel I. Small particles ofsoil were sieved onto the screeded boundary surface toform small, loosely spaced, very irregular roughnesselements.

FIGURE 30.—Rectangular channel H. This boundary wasstabilized in the condition existing after it was screeded.Small longitudinal grooves were formed by particlescatching on the edge of the screed.

FIGURE 32.—Rectangular channel K. After being rough-ened, this boundary was modified by a very low flow ofwater.


45

FIGURE 33.—Rectangular channel L. This boundary wastroweled. It is very similar to rectangular channel C.

FIGURE 35.—Rectangular channel N. After being rough-ened, this boundary was sprayed lightly and uniformlywith a garden hose.

FIGURE 34.—Rectangular channel M. This boundarywas roughened in a regular pattern by making indenta-tions with the end of a trowel. A low flow of water wasused to smooth the indentations.

FIGURE 36.—Parabolic channel A. This channel wasformed with a furrowing shovel. It should be verysimilar to a newly formed furrow in the field.

46


FIGURE 37.—Parabolic channel B. This channel wasscreeded to the desired shape and left with a smalldegree of roughness.

FIGURE 39.—Parabolic channel D. Preparation of thischannel was similar to that of channel B.

FIGURE 38.—Parabolic channel C. After being screededthis channel was roughened and then modified with alow flow of water. Channels B and C were the narrowestand deepest of the channels studied.

FIGURE 40.—Parabolic channel E. Preparation of thischannel was similar to that of channel C. ChannelsD and E were the widest and shallowest of the channelsstudied.

,Db...,......t cr?7??9°Tnnonnonn

.. ....o 1

' I I

I

°AO°

–,.....r.,Tri"? 0

tc.....,,-T90-.40(!)(1)01'?ww

n

g WW

0• •

.-

T cr tww–wto by to dt14

n

CD

• •

SO (")00. Co

g0 d5

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0 DIANI--, 00000AWNOMWI--,AWWWW=1.....000M 0c0..-4

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WW..t-L 0000MWW.-I M&WW

1..1...+000c00/1-, WW 000010/W

N.....0©OCOW00Ww1-.1-,00 .43WWWWw-4r M.

MWIA=WWW= 00MWOWN WoAW 01.-. W .7 -, 41.011W0/00000=WOM 000WWW WW- 000000 000001W,..,. 1-,1-, i-n ,,... 0

02 '41W0-4WWWWW W00 n4WW 00c0M 000.4010 00=0/WW1-, ft..1-. Wc0=c0W0c00001&=0.-, 14.

0.14, n-. W•AIA00010".-n 010)0)00(=MIA WW ...11-. W

oo0cowWW., M&MP nnA05 ..400.-,

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48


TABLE 8.-Summary of measured data-rectangular TABLE 8.-Summary of measured data-rectangularchannels-Continuedchannels-Continued

Channeland run

q d V T vx105

Ft a/ Ft./ Ft.=/Channel E: Sec.1Ft. Ft. Sec. °F. Sec.

4-E 005 . 0506 . 0540 943 55 1. 315-E 1 005 . 102 . 0780 1. 31 55 1. 316-E 005 . 202 . 118 1. 71 56 1. 297-E 1 005 . 380 . 179 2. 13 57 1. 278-E ' 005 . 596 . 235 2. 54 56 1. 29

9-E 001 . 00236 . 020 118 58 1. 2510-E 001 . 00530 026 204 58 1. 2511-E 1 001 . 0114 . 0388 . 296 57 1. 2712-E 001 . 0299 . 0664 . 452 56 L 2913-E .001 .0650 . 101 . 644 56 1. 2914-E I . 001 . 140 . 172 . 809 57 1. 2715-E 1 . 001 . 292 254 1. 15 57 1. 2716-E . 001 . 538 . 387 1. 39 57 1. 27

17-E . 0005 00294 025 118 58 1. 2518-E . 0005 0053 0 031 171 59 1. 2319-E . 0005 . 0110 . 0478 . 231 58 1. 2520-E . 0005 .0233 0710 328 58 1. 2521-E . 0005 0482 106 456 58 1. 2522-E . 0005 0640 126 508 57 1. 2723-E . 0005 127 190 671 57 1. 2724-E . 0005 256 290 884 58 1. 2525-E I . 0005 505 458 1. 10 58 1. 2526-E 0005 00162 020 081 63 1 15

Channel F:1-F 1 . 0005 0179 0949 191 59 1. 232-F 1 . 0005 0105 0765 . 139 59 1. 233-F . 0005 0430 140 . 310 57 1. 274-F_ _ ...... _ _ . 0005 . 146 . 260 . 561 57 1. 275-F __ -___ _ .0005 . 0835 . 196 . 428 57 1. 276-F . 0005 . 294 . 391 . 753 58 1. 25

7-F 1 . 001 0165 0780 . 214 59 1. 238-F 001 0320 102 . 315 59 1. 239-F 001 129 204 . 633 57 1. 2710-F . 001 254 290 . 877 57 1. 2711-F 001 . 0675 . 146 . 466 57 1. 27

Dhannel G:1-G . 0005 0. 00750 0. 0519 0. 146 62 1. 172-G . 0005 0158 .0760 .208 62 1. 173-G 1 _ ___ _ . 0005 . 0340 . 102 . 334 60 1. 214-G . 0005 . 0686 . 149 .462 60 1.215-G 1 _ ___ _ . 0005 . 146 .237 .618 60 1.216-G 1 0005 .292 .349 .839 60 1.217-G'------- 0005 .463 .468 .992 60 1.21

8-G 001 .00777 0441 . 178 60 1.219-G 1 _ .001 .0167 .0605 .277 60 1.2110-G ".. . 001 . 0331 .0829 . 402 60 1. 2111-G 001 .0686 . 123 .560 60 1.2112-G 001 . 142 . 186 .768 59 1.2313-G 1 . 001 .270 .270 1.00 59 1.2314-G 001 .565 .427 1.32 59 1.23

15-G .005 .0101 .0355 .288 60 1.2116-G 005 0224 0495 . 458 60 1. 2117- G 005 .0435 .0664 .659 60 1.2118-G 005 .0930 .0960 .973 60 1.2119-G 005 . 187 . 137 1. 37 60 1. 2120-G 005 .368 . 198 1.87 60 1.2121-G'------ 005 .570 .262 2.18 60 1.22

:hannel H:1-H 1 005 . 0107 .0245 .438 61 1. 192-TI A05 0214 11:142 R20 R(1 1.21

See footnote at end of table, page 49.

Channeland run

S q d V T vx103

Ft s/ Ft./ Ft'/Channel H: Sec./Ft. Ft. Sec. °F. Sec.

3-H . 005 .0430 .0499 .862 60 1.214-H 1 . 005 . 122 .0856 1.43 60 1.21

6-11 . 001 .0105 .0312 .339 62 1. 177-H . 001 . 0181 . 0444 . 409 61 1. 198-H . 001 .0390 .0680 .573 61 1. 199-H . 001 .0776 . 101 .769 60 1.2111-H .001 .304 .236 1.30 61 1.1912-H 1 _ . 001 . 520 . 349 1. 50 61 1. 19

14-H __ .0005 .0112 .0408 .276 64 1.1515-H------ .0005 .0221 .0622 .356 63 1.1616-H .0005 .0439 .0905 .485 63 1.1617-H .0005 .0878 .192 .622 62 1.1718-H . 0005 . 172 . 214 . 807 62 1. 1719-H . 0005 . 360 . 332 1. 09 63 1. 1620-H . 0005 .543 .436 1.25 63 1. 16

Channel I:1-I .0005 . 00146 . 023 . 0634 68 1.092-I 0005 . 00269 . 032 . 0842 68 1. 093-I 0005 . 00735 . 0496 . 148 68 1. 094-I .0005 .0151 .0701 .215 68 1.095-I 0005 . 0276 . 0970 .285 68 1. 096-I .0005 .0585 .153 .383 68 1.097-I 0005 . 121 . 225 . 537 67 1. 108-I .0005 .277 .354 .783 67 1.109-I .0005 .451 .465 .972 67 1.1010-I .0005 .00092 .022 .0418 70 1.0611-I .0005 .00301 .031 .0971 70 1.0612-I 0005 . 00767 . 0494 . 155 70 1. 0613-I .0005 .145 .245 .595 68 1.09

21-I 1 ___ _ _ _ . 0001 .0158 . 150 . 105 69 1.0722-I 0001 . 00124 .037 .0334 70 1.0623-I 0001 . 00043 . 022 . 0194 68 1. 0924-I .0001 .00314 .048 .0654 70 1.0625-I I 0001 .00596 .0610 . 0979 70 1.0626-I'------- 0001 .00702 .0760 .0927 69 1.0727-I 0001 .0308 . 181 . 170 68 1.0928-I 0001 . 0554 . 249 . 223 68 1. 0929-I 0001 . 104 .342 . 305 68 1.0930-I .0001 . 187 .466 .401 68 1.09

' 31-I .001 .00301 .028 . 1074 69 1.0732-I 001 . 00618 . 0379 . 164 69 1. 0733-I 001 . 0147 . 0571 . 258 69 1. 0734-I 001 . 0336 . 0864 . 389 69 1. 0735-I 001 . 0640 . 127 . 504 68 1. 0936-I 001 . 128 . 187 . 685 68 1. 0937-I 001 . 260 . 277 . 939 68 1. 0938-I I _ _.... _ . 001 . 520 . 421 1. 24 68 1. 09

39-I I _ _ - _ . 005 . 00606 . 0269 . 227 69 1. 0740-I 1 005 . 0120 . 0363 . 333 69 1. 0741-I I _ . 005 . 0236 . 0493 . 480 69 1. 0742-I 005 . 0500 . 0706 . 708 69 1. 0743-I 005 . 0970 . 0980 . 990 68 1. 0944-I 005 • 187 . 141 1. 33 68 1. 09

Jhannel K:1-K 0001 .0006E . 019 . 0356 60 1. 212-K . 0001 . 00197 . 031 . 0636 60 1. 213-K . 0001 . 00311 . 032 . 0972 59 1. 234-K . 0001 . 0082C . 062 . 132 59 1. 235-K .0001 .00451 "045 .1003 59 1.23R-IC OW11 flf1R7R nAR 19R AO 1 92


49

TABLE 8.-Summary of measured date-rectangular TABLE 8.--Summary of measured date-rectangularchannels-Continuedchannels-Continued

Channeland run

S q d V T Fx103

Ft'/ Ft./ Ft?/Channel K: Sec./Ft. Ft. Sec. °F. Sec.

7-K 0001 0143 083 . 172 58 1. 258-K 0001 0246 110 . 223 56 1. 299-K 0001 0470 154 . 305 57 1. 2710-K_

- __ .0001 . 100 . 241 . 416 57 1. 27

11-K 1 _ _ _ _ . 0001 . 191 . 365 . 523 56 1. 2912-K 1 _ _ _ _ . 0001 . 315 . 481 . 656 56 1. 29

13-K 1 _ _ _ _ . 0005 . 0178 . 054 . 328 57 1. 2714-K 1 _ _ _ _ . 0005 . 0289 . 072 . 402 56 1. 2915-K 1 _ _ _ _ . 0005 . 0500 . 099 . 505 57 1. 2716-K 0005 101 . 155 . 649 58 1. 2517-K 1 _ _ _ _ . 0005 . 174 . 216 . 806 58 1. 2518-K 1 _ _ _ _ . 0005 . 437 . 396 1. 10 58 1. 2519-K 0005 00145 . 019 . 0765 58 1. 2520-K 0005 . 00083 . 016 . 052 58 1. 2521-K 0005 00252 . 023 . 1097 58 1. 2522-K 0005 00345 . 025 . 1379 58 1. 2523-K 0005 00449 . 028 . 1605 58 1. 2524-K 0005 00616 . 032 . 1930 58 1. 25

25-K 0003 00064 . 017 . 0376 57 1. 2726-K 0003 . 00196 . 023 . 0852 57 1. 2727-K 0003 00219 . 024 . 0911 58 1. 2528-K 0003 . 00154 . 022 . 0702 58 1. 2529-K 0003 00114 . 020 . 0572 58 1. 2530-K 0003 00580 . 036 . 1612 58 1. 2531-K 0003 00693 . 039 . 1781 56 1. 2932-K 0003 00372 . 029 1281 56 1. 29

Channel L:1-L 0001 00178 . 0360 0496 56 1. 292-L 0001 00229 . 0315 0727 56 1. 293-L 0001 00637 0400 1591 55 1. 314-L 0001 . 00036 . 0185 . 0199 56 1. 295-L 0001 00707 . 0225 0312 56 1. 296-L 0001 00463 . 0430 1075 56 1. 297-L 0001 00173 . 0280 0620 54 1. 338-L 0001 . 00924 . 0545 . 1697 54 1. 339-L 0001 . 00688 . 0465 1482 54 1. 3310-L 0001 00556 . 0400 1388 54 1. 33

11-L 0003 00131 . 0185 0709 54 1. 3312-L 0003 00087 . 0145 0599 54 1. 3313-L 0003 . 00276 . 0225 1225 52 1. 3714-L 0003 00436 . 0275 1589 52 1. 3715-L 0003 . 00617 . 0310 1993 51 1. 3916-L 0003 00720 . 0320 2253 51 1. 3917-L 0003 . 00330 . 0255 1296 56 1. 2918-L 0003 . 00746 . 0335 2230 51 1. 3919-L 0003 . 01066 . 0390 . 2730 51 1. 3920-L 0003 . 00978 . 0365 2685 52 1. 37

21-L 0005 . 00091 . 0120 0764 52 1. 3722-L 0005 . 00063 . 0105 0599 54 1. 3323-L 0005 . 00479 . 0240 2000 52 1. 3724-L 0005 . 00325 . 0205 159 52 1. 3725-L 0005 . 00522 . 0240 . 2173 52 1. 3726-L 0005 . 00680 . 0270 2523 52 1. 3727-L 0005 . 00735 . 0280 2626 52 1. 37

28-L 001 . 00062 . 0085 . 0725 54 1. 3329-L 001 . 00208 . 0130 1607 54 1. 3330-L 001 . 00096 . 0100 0963 54 1. 3331-L 001 . 00394 0170 2323 54 1. 3332-L 001 . 00293 . 0150 . 1961 54 1. 3333-L 001 00508 0185 2723 Azt 1 RR

Channeland run

S q d V T px105

Ft.'/ Ft./ Ft. 21Channel L: Sec./Ft. Ft. Sec. °F. Sec.

34-L 001 . 00595 . 0200 . 2976 53 1. 3535-L 001 . 00608 . 0195 . 3126 54 1. 3336-L 001 . 00681 . 0210 . 3251 53 1. 3537-L 001 . 00818 . 0220 . 3729 52 1. 37

Channel M:3-M 001 . 00702 . 0315 . 2230 53 1. 354-M 001 . 00474 . 0260 . 1826 54 1. 335-M_ _ . 001 . 00576 . 0290 . 1985 54 1. 336-M 001 . 00106 . 0160 . 0663 56 1. 297-M 001 . 00095 . 0160 . 0593 56 1. 298-M 001 . 00083 . 0150 . 0553 56 1. 299-M 001 . 00122 . 0165 . 0742 56 1. 2910-M 001 . 00347 . 0220 . 1579 56 1. 2911-M 001 . 00243 . 0200 . 1215 56 1. 29

12-M 0005 . 00198 . 0235 . 0844 50 1. 4113-M 0005 . 00124 . 0200 . 0622 50 1. 4114-M . 0005 . 00267 . 0255 . 1048 50 1. 4115-M . 0005 . 00347 . 0275 . 1266 50 1. 4116-M _ 0005 . 00590 . 0330 . 1789 50 1. 4117-M 0005 . 00428 . 0285 . 1578 52 1. 3718-M_ _ . 0005 . 00659 . 0345 . 1914 51 1. 3919-M . 0005 . 00816 . 0395 . 2069 51 1. 39

20-M . 0003 . 00045 . 0155 . 0292 53 1. 3521-M . 0003 . 00265 . 0270 . 0984 53 1. 3522-M . 0003 . 00104 . 0215 . 0483 53 1. 3523-M . 0003 . 00151 . 0230 . 0655 53 1. 3524-M . 0003 . 00441 . 0320 . 1378 45 1. 5425-M . 0003 . 00582 . 0360 . 1618 50 1. 4126-M . 0003 . 00735 . 0400 . 1839 50 1. 4127-M . 0003 . 01020 . 0485 . 2105 50 1. 41

Channel N:3-N 1 . 0003 . 00715 . 0663 . 107 54 1. 334-N . 0003 . 0108 . 07 58 . 142 54 1. 335-N . 0003 . 0199 . 104 . 190 51 1. 396-N 1 . 0003 . 111 . 258 . 432 50 1. 417-N . 0003 . 146 . 280 . 520 49 1. 448-N 1 . 0003 . 280 , . 425 . 657 51 1. 39

9-N 1 005 . 0114 . 0443 . 256 53 1. 3510-N . 005 . 0244 . 0593 . 410 53 1. 3511-N . 005 . 0576 . 0882 . 655 52 1. 3712-N . 005 . 0779 . 102 . 766 51 1. 3913-N . 005 . 109 . 120 . 908 51 1. 3914-N . 005 . 205 . 166 1. 23 51 1. 3915-N 005 . 353 . 223 1. 58 48 1. 4616-N 005 . 00950 . 0372 0. 255 50 1. 41

20-N 001 . 00842 . 0510 . 165 52 1. 3721-N . 001 . 0189 . 0733 . 257 52 1. 3722-N . 001 . 0305 . 0925 . 330 50 1. 4123-N . 001 . 0671 . 143 . 469 49 1. 4424-N . 001 . 115 . 190 . 606 49 1. 4425-N I . 001 . 219 . 266 . 820 50 1. 41

27-N . 0005 . 00810 . 0588 . 137 52 1. 3728-N . 0005 . 0162 . 0798 . 203 62 1. 3729-N . 0005 . 0292 . 111 . 262 50 I. 4130-N_ ___ _ _ . 0005 . 0661 . 161 . 4 11 50 1. 4131-N 1 . 0005 . 135 . 258 . 525 50 I. 4132-N 2 . 0005 . 267 . 373 . 714 50 1. 4133-N . 0005 . 00720 . 0538 . 133 52 1. 37

1 Data of questionable value, see page 10.

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FLOW RESISTANCE IN SIMULATED IRRIGATION BORDERS AND FURROWS 51TABLE 9.-Resistance coefficients and calculated

parameters-rectangular channels-Continued

Channel andrun

n CI SID f Re old

Channel H:4-H 1 . 0143 12. 2 0546 10, 100 . 0284

6-H . 0139 10.8 .0706 905 .07797-H . 0145 10.9 .0685 1, 520 .05488-H . 0136 12. 3 0535 3, 270 . 03589-H . 0133 13.5 0440 6, 410 . 024111-H . 0138 14.9 .0364 25, 800 . 010312-H 1 . 0155 14. 2 0403 44, 000 . 00696

14-H .0143 10.7 .0691 980 .059615-H . 0146 11.3 .0635 1, 910 . 039116-H . 0138 12.7 .0495 3, 780 .026917-H . 0145 13. 0 . 0473 7, 550 . 017118-H . 0147 13.8 0423 14, 800 . 011419-H . 0147 14.9 0362 31, 200 0073220-H .0153 14.9 .0362 47, 000 .00557

Channel I:1-I .737 134 2-I 581 247 3-I . 0302 5. 75 240 674 . 1294-I . 0262 6. 38 196 1,380 . 09105-I . 0245 7. 22 153 2, 530 . 06586-I .0248 7. 70 . 134 5, 370 . 04177-I . 0229 8.92 100 11, 000 . 02848-I .0213 10.7 .0694 25, 200 .01809-I .0205 11.2 .0638 41, 100 .013710-I 1.62 87 11-I .423 283 12-I . 0288 5. 50 . 264 721 . 12913-I . 0219 9. 46 0890 13, 400 . 0260

21-I 1 0400 4. 79 . 346 1, 470 042622-I .851 116 23-I 1. 51 39 24-I . 289 296 25-I 1 0234 6. 99 . 163 563 10526-I 1 0288 5. 94 . 227 659 084027-I 0278 7. 05 . 160 2, 820 035328-I .0264 7.88 . 128 5, 100 025629-I .0239 9. 19 .0942 9, 570 .018730-I 0222 10.4 .0737 17, 200 0137

31-I .625 281 32-I 0326 4. 72 . 359 580 16833-I 0270 6. 03 . 218 1, 380 . 11234-I 0236 7. 39 . 146 3, 140 074035-I 0235 7. 90 . 128 5, 870 050236-I .0224 8.87 .101 11,700 .034137-I .0212 9.96 .0800 23, 800 023038-I 1 0214 10. 7 . 0692 48, 000 0152

39-I 1 . 0419 3. 45 . 671 571 . 23740-I 1 0347 4. 37 . 416 1, 130 . 17641-I 1 0294 5.40 .271 2, 210 13042-I .0255 6.61 . 183 4, 670 .090543-I .0225 7.86 . 129 8, 900 .065244-I . 0214 8. 80 . 103 17, 200 0453

Dhannel K:1-K 385 562-K 198 1633-K 087 2535-K 115 3679-K . 0138 13. 7 0428 3, 700 034810-K 0424 15. 0 . 0370 7, 900 . 022211-K 1 . 0145 15. 2 0344 14, 800 014712-K 1 . 0140 le, 7 0200 24 Ann ni l 1

TABLE 9.-Resistance coefficients and calculatedparameters-rectangular channels-Continued

Channel andrun

n C14 f Re old

Channel K:13-K 1 0143 11. 1 . 0610 1,390 . 099014-K 1 0144 1L 8 . 0534 2, 240 . 074315-K 1 0141 12. 6 . 0454 3, 940 . 054016-K 0147 13. 0 . 0406 8, 050 . 034617-K 1 0153 13. 6 . 0347 13, 900 . 024818-K 1 0163 13. 7 . 0298 34, 800 . 013519-K . 418 116 20-K . 763 64 21-K . 247 201 22-K . 168 276 23-K . 140 360 24-K . 111 494

25-K 929 50 26-K 245 154 27-K 223 175 28-K 345 124 29-K 473 92 30-K 107 464 31-K 095 538 32-K 137 288 33-K 0143 11. 3 . 0626 1, 190 . 0940

Channel L:1-L 377 138 2-L 153 178 3-L 041 487 4-L 1. 20 28 5-L .598 55 6-L 096 359 7-L 188 131 8-L .049 695 9-L 054 518 10-L 054 417 _____ _

11-L .284 99 12-L 312 65 13-L 116 201 14-L 084 319 15-L 060 444 16-L 049 519 17-L 117 256 18-L 052 537 19-L 041 766 20-L 039 715

21-L 265 67 _ . ....22-L - 37723-L 077 350 24-L 104 238 25-L 065 381 26-L 055 497 -___,27-L . 052 537 _ ,--_,

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TABLE 10.-Summary of measured data-parabolic channels-Continued

Channel and run S Q y R A V T vx105

Channel B: Ft.3ISec. Ft. Ft. Ft.' Ft./Sec. °F. Ft.ilSec.1-B 0003 . 00585 . 061 . 0402 . 0220 . 266 60 . 212-B 0003 . 02037 . 116 . 0730 . 0565 . 361 56 . 293-B 0003 . 0403 . 154 . 0960 . 0870 .464 56 . 294-B 0003 . 0647 . 196 . 118 . 124 . 522 56 . 295-B 0003 . 0899 . 234 .139 .161 . 558 56 . 296-B . 0003 . 134 . 279 . 164 . 210 . 644 55 . 317-B 0003 . 176 . 314 . 181 . 251 . 700 55 . 31

8-B 0005 . 00727 . 062 . 0400 . 0220 . 330 60 . 219•-B . 0005 .0240 . 108 . 0676 .0500 . 480 58 . 2510-B 0005 . 0563 . 161 . 0969 . 0900 . 626 57 . 2711-B 0005 . 0916 . 202 . 121 .129 . 710 56 . 2912-B 0005 . 149 . 258 . 151 . 185 . 803 56 . 2913-B 0005 . 226 . 318 . 182 . 254 . 891 56 . 29

14-B 001 . 00673 . 053 . 0343 . 0174 . 387 60 . 2115-B 001 . 01697 . 078 . 0499 .0310 . 548 58 . 2516-B 001 . 0301 . 102 . 0640 . 0460 . 654 56 . 2917-B 001 . 0728 . 154 . 0947 . 0860 . 847 56 . 2918-B 001 . 132 . 206 . 123 . 132 . 996 56 . 2919-B 001 . 228 . 265 . 156 . 194 1. 178 56 . 2920-B 001 . 276 . 295 . 171 . 228 1. 211 55 . 31

21-B .005 .00721 .039 .0256 .0111 .653 59 .2322-B 005 . 0197 . 060 . 0389 . 0210 . 940 59 . 2323-B 005 . 0658 . 103 . 0659 . 0475 1. 386 57 . 2724-B 005 . 1259 . 139 .0850 1. 22 1. 718 56 . 2925-B - 005 . 231 . 184 .112 1. 28 2. 063 56 . 2926-B 005 . 354 . 224 . 126 1. 34 2. 353 56 . 29

Channel C:1-C 0003 . 0076 . 079 . 0492 . 0314 . 248 62 . 172-C 0003 . 0174 . 118 . 0739 . 0574 . 302 61 . 193-C 0003 . 0331 . 160 . 0979 . 0910 . 364 60 . 214-C 0003 . 0590 . 213 . 128 .140 . 423 59 .235-C 0003 . 0816 . 250 . 147 . 177 . 460 58 .256-C 0003 . 1388 . 299 . 172 . 232 .600 58 . 257-C 0003 . 229 . 396 . 221 .355 . 646 58 . 25

8-C 0005 . 00692 . 070 .0447 . 0260 . 266 60 . 219-C 0005 . 0296 . 133 . 0829 . 0688 . 429 60 2110-C 0005 . 0583 . 181 . 110 . 109 . 534 59 2111-C 0005 . 0924 . 225 . 134 . 151 . 612 59 2112-C 0005 . 152 . 277 . 161 . 207 . 734 58 2513-C 0005 . 234 . . 338 . 192 . 279 . 839 58 . 25

14-C 001 . 0204 . 096 . 0608 . 0422 . 483 59 . 2115-C 001 . 0517 . 149 . 0917 . 0816 . 632 59 .2116-C 001 . 0928 . 197 . 119 . 124 . 748 58 .2517-C 001 . 158 . 249 . 148 . 176 . 897 58 .2518-C 001 . 290 . 327 . 187 . 266 1. 091 58 . 2519-C 001 . 409 . 378 . 211 . 329 1. 243 58 .25

20-C 005 . 0189 . 066 . 0423 . 0240 .790 60 .2121-C 005 . 0571 . 115 .0720 .0552 1. 034 51) .2:122-C 005 . 100 . 142 . 0863 . 0755 1. 428 59 .2123-C 005 . 235 . 207 . 124 . 134 1.761 58 . 2524-C 005 . 391 . 262 . 153 . 190 2. 061 58 . 25

Channel D:1-D .001 .00359 .037 .0245 .0151 .238 62 .172-D .001 .00744 .047 .0312 .0219 .340 62 .173-D .001 .0135 .061 .0402 .0320 .422 60 .214-D .001 .0374 .097 .0630 .0638 .586 60 . 215-D . 001 . 0775 .137 . 0884 . 108 . 721 68 . 256-D . 001 . 161 . 188 . 120 . 172 .936 58 .257-D . 001 . 226 . 226 . 143 . 227 . 995 59 .2:14_1-1 11111I • • 3275 9R9 1 fi4 2)44 I. 153 58 . 25


54


TABLE 10.-Summary of measured data-parabolic channels-Continued

Channel and run S Q Y R A V T 2'x105

Channel D: Ft.3I Sec. Ft. Ft. Ft. 2 Ft. I Sec. °F. Ft.abSec.9-D .0005 .00650 .052 .0344 .0253 .257 64 1.1510-D .0005 .0146 .078 .0540 .0462 .316 60 1.2111-D .0005 .0328 .106 .0690 .0734 .446 60 1.2112-D . 0005 . 0700 . 148 . 0953 . 121 . 578 60 1. 2113-D . .0005 .1415 .222 .140 .222 .639 60 1.2114-D ' .0005 .234 .259 .162 .280 .836 60 1.2115-D .0005 .383 .337 .208 .415 .926 80 1.21

Channel E:1-E .0005 .00353 .076 .0497 .0432 .0818 68 1.092-E .0005 .00566 .092 .0601 .0577 .0980 68 1.093-E .0005 .0164 .126 .0813 .0926 .177 68 1.094-E .0005 .0312 .154 .0984 .125 .250 62 1.175-E .0005 .0746 .208 .132 .197 .380 61 1.196-E .0005 .150 .282 .176 .310 .484 61 1.197-E 1 . 0005 .412 .338 .207 .406 1.014 81 1. 19

9-E .001 .00431 .071 .0475 .0394 .110 66 1.1210-E .001 .0127 .096 .0619 .0610 .209 66 1.1211-E .001 .0264 .124 .0800 .0900 .292 66 1.1212-E .001 .0692 .178 .113 .155 .447 62 1.1713-E 001 . 1462 . 239 . 150 . 241 . 607 62 1. 1714-E .001 .238 .294 .181 .330 .721 62 1.17

Channel F:1-F . 001 00598 (2) (2)2-F .001 .0168 (2) (2)3-F . 001 . 0448 . 118 . 0751 .0675 . 660 65 1. 134-F .001 .1089 .171 .106 .116 .937 64 1.155-F . 001 . 197 . 225 . 138 . 177 1. 116 64 1. 156-F . 001 .394 . 319 . 189 . 298 1. 321 65 1. 137-F .001 .580 .398 .231 .417 1.544 65 1.13

8-F .00043 .00765 .085 .0421 .0273 .280 68 1.099-F_ .00043 .01955 .092 .0624 .0463 .422 66 1.1210-F .00043 .0420 .136 .0859 .0830 .506 65 1.1311-F .00043 .0926 .195 .121 .143 .648 64 1.1512-F .00043 .1852 .273 .164 .236 .784 64 1.1513-F .00043 .593 .479 .271 .550 1.077 65 1.13

Channel G:1-G . 00043 00428 (2) (2)2-0 . 00043 00855 (2) (2)3-G . 00043 021 (2) (2)4-G .00043 .0431 .158 .0991 .106 .408 67 1.105-G .00043 .0856 .214 .131 .166 .516 65 1.136-G . 00043 . 167 . 303 . 180 . 278 . 600 65 1. 137-G .00043 .234 .350 .206 .346 .676 65 1.138-G . 00043 . 374 . 449 . 256 . 503 . 744 66 1. 12

9-G . 001 . 00503 (2) (2)10-G . 001 . 0127 (2) (2)11-G . 001 (2) (2)12-G .001 .0694 .182 .113 .130 .534 66 1.1213-G .001 .1426 .233 .142 .188 .760 66 1.1214-G .001 .238 .302 .180 .277 .860 66 1.1215-G .001 .480 .413 .238 .444 1.082 66 1.12

Data of questionable value; see page 10.2 Nonuniform flow.

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56


TABLE 11.—Resistance coefficients and calculatedparameters—parabolic channels--Continued

Channel andrun

n C/ A/17 f Re (rIR

Channel G:1-G (2) (I)2-G (2) (2)3-G__ (2) (2)4-G . 0161 11.0 . 0612 3, 680 . 04405-0 . 0155 12.1 . 0511 6, 000 . 03336-G 0164 12. 0 . 0517 9, 550 . 02427-0 0159 12. 7 . 0460 12, 330 . 02128-G 0167 12. 5 . 0479 17, 000 . 0170

9-G (2) 10-G 11-G ('

2))(2) (2)

(2)12-G 0206 8. 9 . 102 5, 390 . 038613-G 0168 11. 2 . 0632 9, 640 . 030714-G 0174 11. 3 . 0626 13, 830 . 024215-G 0167 12. 4 . 0524 23, 000 . 0183

Data of questionable value; see page 10.2 Nonuniform flow.

U.S. GOVERNMENT PRINTING OFFICE:1965 0-771-614

flow resistance in simulated irrigation borders and furrows · 2010-11-20 · flow resistance in...

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